Ronchi grating

The early writers discussed here treated vision more as a geometrical than as a physical, physiological, or psychological problem. The first known author of a treatise on geometrical optics was the geometer Euclid (c. 325 BC–265 BC). Euclid began his study of optics as he began his study of geometry, with a set of self-evident axioms.

This theory of the active power of rays had an influence on later scholars such as Ibn al-Haytham, Robert Grosseteste and Roger Bacon.[11]

Setting aside the issues of epistemology and theology, Grosseteste's cosmogony of light describes the origin of the universe in what may loosely be described as a medieval "big bang" theory. Both his biblical commentary, the Hexaemeron (1230 x 35), and his scientific On Light (1235 x 40), took their inspiration from Genesis 1:3, "God said, let there be light", and described the subsequent process of creation as a natural physical process arising from the generative power of an expanding (and contracting) sphere of light.[25]

In obtaining the various frames of Fig. 7 we have fixed all the system parameters and only varied the tilt of the reference beam. Note that the characteristic fringes of coma in Fig. 7 are quite different from those of coma in the Ronchi test, shown in Figs. 4(d)-(f). Incidentally, the patterns of Fig. 7 show similarities with the Ronchigrams of spherical aberration displayed in Fig. 5. This is not a coincidence, and is rooted in the algebraic forms of the aberration function for third order coma (r3 cos f) and spherical aberration (r4 ), and also in the fact that a Ronchigram, being a kind of shearing interferogram (albeit with a large shear), is related to the derivative of the wavefront aberration function. Knife-edge and wire tests. A schematic diagram of the knife-edge method of testing optical systems is shown in Fig. 8. A geometric-optical interpretation of this test suffices for most practical purposes: the knife-edge blocks different groups of rays in its various positions along the optical axis, allowing the remaining rays to reach the observation plane.3 Another method of testing, known as the wire test, is quite similar to the knife-edge method, being obtained from it by the substitution of the knife-edge with a fine wire.3   Figure 8. In the knife-edge test a certain region in the vicinity of focus is blocked by a knife-edge; the nature and the magnitude of aberrations are then inferred from the resulting patterns of intensity distribution at the observation plane. (The knife-edge may be moved both along and perpendicular to the optical axis.) The wire test is similar to the knife-edge test except that a fine wire is used instead to block certain groups of rays.   Since the grating in the Ronchi test may be thought of as a series of parallel knife-edges or, more aptly, a series of parallel wires, it should not come as a surprise that similarities exist between Ronchigrams and the patterns observed in these other tests. In fact, early attempts at explaining the results of Ronchi?s method were based on geometrical optics, and considered the grating as a set of parallel wires whose shadows produced the observed patterns.5 We will not delve into these matters, but simply draw the reader?s attention to Figs. 9 and 10, where we show several computed patterns of intensity distribution for the knife-edge and wire tests, respectively.4     Figure 9. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the knife-edge test carried out with a laser beam. The lens under test (NA = 0.5, f = 6000l0) and the pupil relay lens (NA = 0.5) are assumed to be fixed in their respective positions, while the knife-edge moves along the optical axis. (The tip of the knife remains on the axis at all times.) The lens under test is assumed to have 3 waves of primary spherical aberration. In frames (a) to (d) the distance of the knife-edge from paraxial focus is Dz = -15l0, 0, +15l0, and +20l0, respectively. (Positive distances are in the direction of the marginal focus.)   The results of the simulated knife-edge test depicted in Fig. 9 assume a laser as the light source. Consequently, frames (a) and (b) of Fig. 9 exhibit several dark lines which, with a less coherent light source, would have been absent. The results of the simulated wire test shown in Fig. 10 assume an extended light source, since the small amount of spherical aberration present in the system under consideration would render the test useless with a wire that, fine as it may be, is still wider than the focused spot produced by a laser beam. Note the similarities between the patterns of Figs. 9 and 10 on the one hand, and those of Figs. 5(d)-(f) on the other.   Figure 10. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the wire test with an extended, quasi-monochromatic light source. The lens under test (NA = 0.5, f = 6000l0) has 3 waves of primary spherical aberration. The assumed wire diameter is 15l0, which is comparable to the size of the image of the extended light source, as measured in the vicinity of focus. In (a) the wire is centered on axis and is 25l0 away from paraxial focus (in the direction of the marginal focus). In (b) the wire is again centered on axis, but is 20l0 away from paraxial focus. In (c) the wire is shifted 0.5l0 off-axis while its distance from paraxial focus remains at 20l0.   Extensions of the Ronchi test. Several modifications and extensions of the Ronchi test have appeared over the years, and have helped to solve specific problems in testing of optical systems.3 As an example we mention the "double-frequency grating lateral shear interferometer" invented by James Wyant in the early 1970?s. The grating in this device has two slightly different frequencies, which give rise to two +1st order beams as well as two -1st order beams; the beams in each pair are slightly shifted relative to each other. Moreover, the (average) pitch of the grating is such that there is no overlap between the 0th, +1st, and -1st orders. Consequently, interference occurs between the two +1st order beams (and, likewise, between the two -1st order beams). One can thus obtain an arbitrarily small lateral shear of the wavefront under test, and use the results to achieve accurate quantitative measurements. A two-dimensional version of the double-frequency grating has also been employed to generate lateral wavefront shear simultaneously along the X and Y axes. (Remember that beam propagation is along Z and, therefore, X and Y are orthogonal axes in the plane of the grating.) In the absence of  a 2-D grating, one must rotate a 1-D grating by 90° to obtain  wavefront shear first along the X- and then along the Y-axis. Acknowledgment. I am grateful to Professor Roland Shack of the Optical Sciences Center for many illuminating discussions, and also for suggesting some of the examples presented in this article.   References V. Ronchi, "Le Frange di Combinazioni Nello Studio delle Superficie e dei Sistemi Ottici," Riv. Ottica Mecc. Precis. 2, 9 (1923). V. Ronchi, "Due Nuovi Metodi per lo Studio delle Superficie e dei Sistemi Ottici," Ann. Sc. Norm. Super. Pisa, 15 (1923). D. Malacara, ed., Optical Shop Testing, second edition, Wiley, New York, 1992. The computer simulations reported in this article were performed by DIFFRACT?, a product of MM Research, Inc., Tucson, Arizona. G. Toraldo di Francia, "Geometrical and interferential aspects of the Ronchi Test," in Optical Image Evaluation, National Bureau of Standards Circular 526, issued April 29, 1954. J. C. Wyant, "Double frequency grating lateral shear interferometer," Appl. Opt. 12, 2057 (1973). << Back to list of Articles Home | About MM Research, Inc. | Online Publications Diffract | SIM 3D_Max | Multilayer | Temprofile © Copyright 1987-2011, MM Research, Inc. 5748 N. Camino del Conde, Tucson, Arizona 85718

Figure 5. Computed plots of intensity distribution at the observation plane of Fig. 1, showing the patterns obtained by sliding the grating along the optical axis. The lens under test (NA = 0.5, f = 6000l0) is assumed to have 3 waves of primary spherical aberration, and its paraxial focus is coincident with the focal point of the relay lens. The grating is moved along the optical axis by an amount Dz relative to the (common) focal plane; positive distances are towards the marginal focus. (a)  Dz= -10l0, (b)  Dz = 0, (c)  Dz = 10l0, (d) Dz = 15l0, (e)  Dz= 20l0, (f)  Dz= 25l0.

The last three frames in Fig. 4 represent the effects of third order coma. A change in the orientation of this aberration causes the interference pattern to change drastically. Figures 4(d)-(f) correspond to 3 waves of coma oriented at 0°, 45°, and 90°, respectively. Sliding the grating along the optical axis. A change in the position of the grating relative to the focal plane influences the observed fringe pattern. We limit our discussion to the case of spherical aberration, although similar analyses could be performed for other aberrations as well. Assuming 3 waves of spherical aberration as before, we obtain the patterns displayed in Fig. 5 as we slide the grating along the optical axis in the system of Fig. 1.4 Once again, we have taken the lens under test to have NA = 0.5 and f = 6000l0. The paraxial focus of the lens under test coincides with the front focal point of the relay lens, and the grating is shifted by different amounts Dz relative to this common focus. Frames (a)-(f) in Fig. 5 correspond to different values of Dz, starting at Dz = -10l0 in (a) and moving forward to Dz = +25l0 in (f). In the process, as the grating moves through paraxial focus and towards marginal focus, we observe a rich variety of patterns that aid us in determining the nature and the magnitude of the aberration.     Figure 5. Computed plots of intensity distribution at the observation plane of Fig. 1, showing the patterns obtained by sliding the grating along the optical axis. The lens under test (NA = 0.5, f = 6000l0) is assumed to have 3 waves of primary spherical aberration, and its paraxial focus is coincident with the focal point of the relay lens. The grating is moved along the optical axis by an amount Dz relative to the (common) focal plane; positive distances are towards the marginal focus. (a)  Dz= -10l0, (b)  Dz = 0, (c)  Dz = 10l0, (d) Dz = 15l0, (e)  Dz= 20l0, (f)  Dz= 25l0.   To be sure, the Ronchi test is not the only scheme used during fabrication and evaluation of optical systems; several other tests exist and their relative merits have been expounded in the literature.3 It is useful here to examine some of these alternative methods and to compare the resulting patterns (interferograms or otherwise) with those obtained with the Ronchi test. Testing by interfering with a reference plane-wave. Figure 6 shows the schematic diagram of a Mach-Zehnder interferometer, which is one among many that can be used to evaluate the aberrated wavefronts directly. In this system a coherent, monochromatic beam of light is sent through the lens under test, is collected and recollimated by a well-corrected lens, and is made to interfere with a reference beam that has been split off from the original, incident wavefront.   Figure 6. Schematic diagram of a Mach-Zehnder interferometer that might be set up for a direct measurement of wavefront aberrations. The pupil relay lens (itself free from aberrations) forms at the observation plane an image of the exit pupil of the lens under test. A fraction of the incident beam is diverted from its original path and sent to the observation plane by means of the various mirrors and beam-splitters. The observed fringes are characteristic of the aberrations present at the exit pupil of the lens under test. A small tilt of the mirror shown at the lower left side of the figure would introduce a linear phase shift on the reference beam. This tilt is generally useful in producing signature fringe patterns at the observation plane.   The flat mirror shown in the lower left side of the interferometer is mounted on a tip-tilt stage that allows the introduction of a small amount of tilt in the reference beam. Figure 7 shows the computed patterns of intensity distribution at the observation plane of the Mach-Zehnder interferometer corresponding to 3 waves of primary coma.4         Figure 7. Computed plots of intensity distribution at the observation plane of Fig. 6. The lens under test (NA = 0.5, f = 6000l0) is assumed to have 3 waves of primary coma, and its nominal focus is coincident with the focal point of the relay lens. The tilt angle y of the reference beam increases progressively from (a) to (f). (a) y = -0.1°, (b) y = 0°, (c) y  = 0.05°, (d)y = 0.07°, (e) y = 0.1°, (f)y  = 0.18°.   In obtaining the various frames of Fig. 7 we have fixed all the system parameters and only varied the tilt of the reference beam. Note that the characteristic fringes of coma in Fig. 7 are quite different from those of coma in the Ronchi test, shown in Figs. 4(d)-(f). Incidentally, the patterns of Fig. 7 show similarities with the Ronchigrams of spherical aberration displayed in Fig. 5. This is not a coincidence, and is rooted in the algebraic forms of the aberration function for third order coma (r3 cos f) and spherical aberration (r4 ), and also in the fact that a Ronchigram, being a kind of shearing interferogram (albeit with a large shear), is related to the derivative of the wavefront aberration function. Knife-edge and wire tests. A schematic diagram of the knife-edge method of testing optical systems is shown in Fig. 8. A geometric-optical interpretation of this test suffices for most practical purposes: the knife-edge blocks different groups of rays in its various positions along the optical axis, allowing the remaining rays to reach the observation plane.3 Another method of testing, known as the wire test, is quite similar to the knife-edge method, being obtained from it by the substitution of the knife-edge with a fine wire.3   Figure 8. In the knife-edge test a certain region in the vicinity of focus is blocked by a knife-edge; the nature and the magnitude of aberrations are then inferred from the resulting patterns of intensity distribution at the observation plane. (The knife-edge may be moved both along and perpendicular to the optical axis.) The wire test is similar to the knife-edge test except that a fine wire is used instead to block certain groups of rays.   Since the grating in the Ronchi test may be thought of as a series of parallel knife-edges or, more aptly, a series of parallel wires, it should not come as a surprise that similarities exist between Ronchigrams and the patterns observed in these other tests. In fact, early attempts at explaining the results of Ronchi?s method were based on geometrical optics, and considered the grating as a set of parallel wires whose shadows produced the observed patterns.5 We will not delve into these matters, but simply draw the reader?s attention to Figs. 9 and 10, where we show several computed patterns of intensity distribution for the knife-edge and wire tests, respectively.4     Figure 9. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the knife-edge test carried out with a laser beam. The lens under test (NA = 0.5, f = 6000l0) and the pupil relay lens (NA = 0.5) are assumed to be fixed in their respective positions, while the knife-edge moves along the optical axis. (The tip of the knife remains on the axis at all times.) The lens under test is assumed to have 3 waves of primary spherical aberration. In frames (a) to (d) the distance of the knife-edge from paraxial focus is Dz = -15l0, 0, +15l0, and +20l0, respectively. (Positive distances are in the direction of the marginal focus.)   The results of the simulated knife-edge test depicted in Fig. 9 assume a laser as the light source. Consequently, frames (a) and (b) of Fig. 9 exhibit several dark lines which, with a less coherent light source, would have been absent. The results of the simulated wire test shown in Fig. 10 assume an extended light source, since the small amount of spherical aberration present in the system under consideration would render the test useless with a wire that, fine as it may be, is still wider than the focused spot produced by a laser beam. Note the similarities between the patterns of Figs. 9 and 10 on the one hand, and those of Figs. 5(d)-(f) on the other.   Figure 10. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the wire test with an extended, quasi-monochromatic light source. The lens under test (NA = 0.5, f = 6000l0) has 3 waves of primary spherical aberration. The assumed wire diameter is 15l0, which is comparable to the size of the image of the extended light source, as measured in the vicinity of focus. In (a) the wire is centered on axis and is 25l0 away from paraxial focus (in the direction of the marginal focus). In (b) the wire is again centered on axis, but is 20l0 away from paraxial focus. In (c) the wire is shifted 0.5l0 off-axis while its distance from paraxial focus remains at 20l0.   Extensions of the Ronchi test. Several modifications and extensions of the Ronchi test have appeared over the years, and have helped to solve specific problems in testing of optical systems.3 As an example we mention the "double-frequency grating lateral shear interferometer" invented by James Wyant in the early 1970?s. The grating in this device has two slightly different frequencies, which give rise to two +1st order beams as well as two -1st order beams; the beams in each pair are slightly shifted relative to each other. Moreover, the (average) pitch of the grating is such that there is no overlap between the 0th, +1st, and -1st orders. Consequently, interference occurs between the two +1st order beams (and, likewise, between the two -1st order beams). One can thus obtain an arbitrarily small lateral shear of the wavefront under test, and use the results to achieve accurate quantitative measurements. A two-dimensional version of the double-frequency grating has also been employed to generate lateral wavefront shear simultaneously along the X and Y axes. (Remember that beam propagation is along Z and, therefore, X and Y are orthogonal axes in the plane of the grating.) In the absence of  a 2-D grating, one must rotate a 1-D grating by 90° to obtain  wavefront shear first along the X- and then along the Y-axis. Acknowledgment. I am grateful to Professor Roland Shack of the Optical Sciences Center for many illuminating discussions, and also for suggesting some of the examples presented in this article.   References V. Ronchi, "Le Frange di Combinazioni Nello Studio delle Superficie e dei Sistemi Ottici," Riv. Ottica Mecc. Precis. 2, 9 (1923). V. Ronchi, "Due Nuovi Metodi per lo Studio delle Superficie e dei Sistemi Ottici," Ann. Sc. Norm. Super. Pisa, 15 (1923). D. Malacara, ed., Optical Shop Testing, second edition, Wiley, New York, 1992. The computer simulations reported in this article were performed by DIFFRACT?, a product of MM Research, Inc., Tucson, Arizona. G. Toraldo di Francia, "Geometrical and interferential aspects of the Ronchi Test," in Optical Image Evaluation, National Bureau of Standards Circular 526, issued April 29, 1954. J. C. Wyant, "Double frequency grating lateral shear interferometer," Appl. Opt. 12, 2057 (1973). << Back to list of Articles Home | About MM Research, Inc. | Online Publications Diffract | SIM 3D_Max | Multilayer | Temprofile © Copyright 1987-2011, MM Research, Inc. 5748 N. Camino del Conde, Tucson, Arizona 85718

Knife-edge and wire tests. A schematic diagram of the knife-edge method of testing optical systems is shown in Fig. 8. A geometric-optical interpretation of this test suffices for most practical purposes: the knife-edge blocks different groups of rays in its various positions along the optical axis, allowing the remaining rays to reach the observation plane.3 Another method of testing, known as the wire test, is quite similar to the knife-edge method, being obtained from it by the substitution of the knife-edge with a fine wire.3   Figure 8. In the knife-edge test a certain region in the vicinity of focus is blocked by a knife-edge; the nature and the magnitude of aberrations are then inferred from the resulting patterns of intensity distribution at the observation plane. (The knife-edge may be moved both along and perpendicular to the optical axis.) The wire test is similar to the knife-edge test except that a fine wire is used instead to block certain groups of rays.   Since the grating in the Ronchi test may be thought of as a series of parallel knife-edges or, more aptly, a series of parallel wires, it should not come as a surprise that similarities exist between Ronchigrams and the patterns observed in these other tests. In fact, early attempts at explaining the results of Ronchi?s method were based on geometrical optics, and considered the grating as a set of parallel wires whose shadows produced the observed patterns.5 We will not delve into these matters, but simply draw the reader?s attention to Figs. 9 and 10, where we show several computed patterns of intensity distribution for the knife-edge and wire tests, respectively.4     Figure 9. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the knife-edge test carried out with a laser beam. The lens under test (NA = 0.5, f = 6000l0) and the pupil relay lens (NA = 0.5) are assumed to be fixed in their respective positions, while the knife-edge moves along the optical axis. (The tip of the knife remains on the axis at all times.) The lens under test is assumed to have 3 waves of primary spherical aberration. In frames (a) to (d) the distance of the knife-edge from paraxial focus is Dz = -15l0, 0, +15l0, and +20l0, respectively. (Positive distances are in the direction of the marginal focus.)   The results of the simulated knife-edge test depicted in Fig. 9 assume a laser as the light source. Consequently, frames (a) and (b) of Fig. 9 exhibit several dark lines which, with a less coherent light source, would have been absent. The results of the simulated wire test shown in Fig. 10 assume an extended light source, since the small amount of spherical aberration present in the system under consideration would render the test useless with a wire that, fine as it may be, is still wider than the focused spot produced by a laser beam. Note the similarities between the patterns of Figs. 9 and 10 on the one hand, and those of Figs. 5(d)-(f) on the other.   Figure 10. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the wire test with an extended, quasi-monochromatic light source. The lens under test (NA = 0.5, f = 6000l0) has 3 waves of primary spherical aberration. The assumed wire diameter is 15l0, which is comparable to the size of the image of the extended light source, as measured in the vicinity of focus. In (a) the wire is centered on axis and is 25l0 away from paraxial focus (in the direction of the marginal focus). In (b) the wire is again centered on axis, but is 20l0 away from paraxial focus. In (c) the wire is shifted 0.5l0 off-axis while its distance from paraxial focus remains at 20l0.   Extensions of the Ronchi test. Several modifications and extensions of the Ronchi test have appeared over the years, and have helped to solve specific problems in testing of optical systems.3 As an example we mention the "double-frequency grating lateral shear interferometer" invented by James Wyant in the early 1970?s. The grating in this device has two slightly different frequencies, which give rise to two +1st order beams as well as two -1st order beams; the beams in each pair are slightly shifted relative to each other. Moreover, the (average) pitch of the grating is such that there is no overlap between the 0th, +1st, and -1st orders. Consequently, interference occurs between the two +1st order beams (and, likewise, between the two -1st order beams). One can thus obtain an arbitrarily small lateral shear of the wavefront under test, and use the results to achieve accurate quantitative measurements. A two-dimensional version of the double-frequency grating has also been employed to generate lateral wavefront shear simultaneously along the X and Y axes. (Remember that beam propagation is along Z and, therefore, X and Y are orthogonal axes in the plane of the grating.) In the absence of  a 2-D grating, one must rotate a 1-D grating by 90° to obtain  wavefront shear first along the X- and then along the Y-axis. Acknowledgment. I am grateful to Professor Roland Shack of the Optical Sciences Center for many illuminating discussions, and also for suggesting some of the examples presented in this article.   References V. Ronchi, "Le Frange di Combinazioni Nello Studio delle Superficie e dei Sistemi Ottici," Riv. Ottica Mecc. Precis. 2, 9 (1923). V. Ronchi, "Due Nuovi Metodi per lo Studio delle Superficie e dei Sistemi Ottici," Ann. Sc. Norm. Super. Pisa, 15 (1923). D. Malacara, ed., Optical Shop Testing, second edition, Wiley, New York, 1992. The computer simulations reported in this article were performed by DIFFRACT?, a product of MM Research, Inc., Tucson, Arizona. G. Toraldo di Francia, "Geometrical and interferential aspects of the Ronchi Test," in Optical Image Evaluation, National Bureau of Standards Circular 526, issued April 29, 1954. J. C. Wyant, "Double frequency grating lateral shear interferometer," Appl. Opt. 12, 2057 (1973). << Back to list of Articles Home | About MM Research, Inc. | Online Publications Diffract | SIM 3D_Max | Multilayer | Temprofile © Copyright 1987-2011, MM Research, Inc. 5748 N. Camino del Conde, Tucson, Arizona 85718

Equality of numerical apertures means that only the zero-order diffracted beam will be fully transmitted to the observation plane. Of the ±1st order beams only those portions that overlap the zero order will reach the observation plane. The period of the grating in this example has been a little less than l0/NA, leaving a small gap between +1st and -1st orders.4 Absence of aberrations means that the phase distribution over the cross-sections of the various diffracted orders is uniform and, therefore, no interference fringes are to be expected.

Sliding the grating along the optical axis. A change in the position of the grating relative to the focal plane influences the observed fringe pattern. We limit our discussion to the case of spherical aberration, although similar analyses could be performed for other aberrations as well. Assuming 3 waves of spherical aberration as before, we obtain the patterns displayed in Fig. 5 as we slide the grating along the optical axis in the system of Fig. 1.4 Once again, we have taken the lens under test to have NA = 0.5 and f = 6000l0. The paraxial focus of the lens under test coincides with the front focal point of the relay lens, and the grating is shifted by different amounts Dz relative to this common focus. Frames (a)-(f) in Fig. 5 correspond to different values of Dz, starting at Dz = -10l0 in (a) and moving forward to Dz = +25l0 in (f). In the process, as the grating moves through paraxial focus and towards marginal focus, we observe a rich variety of patterns that aid us in determining the nature and the magnitude of the aberration.     Figure 5. Computed plots of intensity distribution at the observation plane of Fig. 1, showing the patterns obtained by sliding the grating along the optical axis. The lens under test (NA = 0.5, f = 6000l0) is assumed to have 3 waves of primary spherical aberration, and its paraxial focus is coincident with the focal point of the relay lens. The grating is moved along the optical axis by an amount Dz relative to the (common) focal plane; positive distances are towards the marginal focus. (a)  Dz= -10l0, (b)  Dz = 0, (c)  Dz = 10l0, (d) Dz = 15l0, (e)  Dz= 20l0, (f)  Dz= 25l0.   To be sure, the Ronchi test is not the only scheme used during fabrication and evaluation of optical systems; several other tests exist and their relative merits have been expounded in the literature.3 It is useful here to examine some of these alternative methods and to compare the resulting patterns (interferograms or otherwise) with those obtained with the Ronchi test. Testing by interfering with a reference plane-wave. Figure 6 shows the schematic diagram of a Mach-Zehnder interferometer, which is one among many that can be used to evaluate the aberrated wavefronts directly. In this system a coherent, monochromatic beam of light is sent through the lens under test, is collected and recollimated by a well-corrected lens, and is made to interfere with a reference beam that has been split off from the original, incident wavefront.   Figure 6. Schematic diagram of a Mach-Zehnder interferometer that might be set up for a direct measurement of wavefront aberrations. The pupil relay lens (itself free from aberrations) forms at the observation plane an image of the exit pupil of the lens under test. A fraction of the incident beam is diverted from its original path and sent to the observation plane by means of the various mirrors and beam-splitters. The observed fringes are characteristic of the aberrations present at the exit pupil of the lens under test. A small tilt of the mirror shown at the lower left side of the figure would introduce a linear phase shift on the reference beam. This tilt is generally useful in producing signature fringe patterns at the observation plane.   The flat mirror shown in the lower left side of the interferometer is mounted on a tip-tilt stage that allows the introduction of a small amount of tilt in the reference beam. Figure 7 shows the computed patterns of intensity distribution at the observation plane of the Mach-Zehnder interferometer corresponding to 3 waves of primary coma.4         Figure 7. Computed plots of intensity distribution at the observation plane of Fig. 6. The lens under test (NA = 0.5, f = 6000l0) is assumed to have 3 waves of primary coma, and its nominal focus is coincident with the focal point of the relay lens. The tilt angle y of the reference beam increases progressively from (a) to (f). (a) y = -0.1°, (b) y = 0°, (c) y  = 0.05°, (d)y = 0.07°, (e) y = 0.1°, (f)y  = 0.18°.   In obtaining the various frames of Fig. 7 we have fixed all the system parameters and only varied the tilt of the reference beam. Note that the characteristic fringes of coma in Fig. 7 are quite different from those of coma in the Ronchi test, shown in Figs. 4(d)-(f). Incidentally, the patterns of Fig. 7 show similarities with the Ronchigrams of spherical aberration displayed in Fig. 5. This is not a coincidence, and is rooted in the algebraic forms of the aberration function for third order coma (r3 cos f) and spherical aberration (r4 ), and also in the fact that a Ronchigram, being a kind of shearing interferogram (albeit with a large shear), is related to the derivative of the wavefront aberration function. Knife-edge and wire tests. A schematic diagram of the knife-edge method of testing optical systems is shown in Fig. 8. A geometric-optical interpretation of this test suffices for most practical purposes: the knife-edge blocks different groups of rays in its various positions along the optical axis, allowing the remaining rays to reach the observation plane.3 Another method of testing, known as the wire test, is quite similar to the knife-edge method, being obtained from it by the substitution of the knife-edge with a fine wire.3   Figure 8. In the knife-edge test a certain region in the vicinity of focus is blocked by a knife-edge; the nature and the magnitude of aberrations are then inferred from the resulting patterns of intensity distribution at the observation plane. (The knife-edge may be moved both along and perpendicular to the optical axis.) The wire test is similar to the knife-edge test except that a fine wire is used instead to block certain groups of rays.   Since the grating in the Ronchi test may be thought of as a series of parallel knife-edges or, more aptly, a series of parallel wires, it should not come as a surprise that similarities exist between Ronchigrams and the patterns observed in these other tests. In fact, early attempts at explaining the results of Ronchi?s method were based on geometrical optics, and considered the grating as a set of parallel wires whose shadows produced the observed patterns.5 We will not delve into these matters, but simply draw the reader?s attention to Figs. 9 and 10, where we show several computed patterns of intensity distribution for the knife-edge and wire tests, respectively.4     Figure 9. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the knife-edge test carried out with a laser beam. The lens under test (NA = 0.5, f = 6000l0) and the pupil relay lens (NA = 0.5) are assumed to be fixed in their respective positions, while the knife-edge moves along the optical axis. (The tip of the knife remains on the axis at all times.) The lens under test is assumed to have 3 waves of primary spherical aberration. In frames (a) to (d) the distance of the knife-edge from paraxial focus is Dz = -15l0, 0, +15l0, and +20l0, respectively. (Positive distances are in the direction of the marginal focus.)   The results of the simulated knife-edge test depicted in Fig. 9 assume a laser as the light source. Consequently, frames (a) and (b) of Fig. 9 exhibit several dark lines which, with a less coherent light source, would have been absent. The results of the simulated wire test shown in Fig. 10 assume an extended light source, since the small amount of spherical aberration present in the system under consideration would render the test useless with a wire that, fine as it may be, is still wider than the focused spot produced by a laser beam. Note the similarities between the patterns of Figs. 9 and 10 on the one hand, and those of Figs. 5(d)-(f) on the other.   Figure 10. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the wire test with an extended, quasi-monochromatic light source. The lens under test (NA = 0.5, f = 6000l0) has 3 waves of primary spherical aberration. The assumed wire diameter is 15l0, which is comparable to the size of the image of the extended light source, as measured in the vicinity of focus. In (a) the wire is centered on axis and is 25l0 away from paraxial focus (in the direction of the marginal focus). In (b) the wire is again centered on axis, but is 20l0 away from paraxial focus. In (c) the wire is shifted 0.5l0 off-axis while its distance from paraxial focus remains at 20l0.   Extensions of the Ronchi test. Several modifications and extensions of the Ronchi test have appeared over the years, and have helped to solve specific problems in testing of optical systems.3 As an example we mention the "double-frequency grating lateral shear interferometer" invented by James Wyant in the early 1970?s. The grating in this device has two slightly different frequencies, which give rise to two +1st order beams as well as two -1st order beams; the beams in each pair are slightly shifted relative to each other. Moreover, the (average) pitch of the grating is such that there is no overlap between the 0th, +1st, and -1st orders. Consequently, interference occurs between the two +1st order beams (and, likewise, between the two -1st order beams). One can thus obtain an arbitrarily small lateral shear of the wavefront under test, and use the results to achieve accurate quantitative measurements. A two-dimensional version of the double-frequency grating has also been employed to generate lateral wavefront shear simultaneously along the X and Y axes. (Remember that beam propagation is along Z and, therefore, X and Y are orthogonal axes in the plane of the grating.) In the absence of  a 2-D grating, one must rotate a 1-D grating by 90° to obtain  wavefront shear first along the X- and then along the Y-axis. Acknowledgment. I am grateful to Professor Roland Shack of the Optical Sciences Center for many illuminating discussions, and also for suggesting some of the examples presented in this article.   References V. Ronchi, "Le Frange di Combinazioni Nello Studio delle Superficie e dei Sistemi Ottici," Riv. Ottica Mecc. Precis. 2, 9 (1923). V. Ronchi, "Due Nuovi Metodi per lo Studio delle Superficie e dei Sistemi Ottici," Ann. Sc. Norm. Super. Pisa, 15 (1923). D. Malacara, ed., Optical Shop Testing, second edition, Wiley, New York, 1992. The computer simulations reported in this article were performed by DIFFRACT?, a product of MM Research, Inc., Tucson, Arizona. G. Toraldo di Francia, "Geometrical and interferential aspects of the Ronchi Test," in Optical Image Evaluation, National Bureau of Standards Circular 526, issued April 29, 1954. J. C. Wyant, "Double frequency grating lateral shear interferometer," Appl. Opt. 12, 2057 (1973). << Back to list of Articles Home | About MM Research, Inc. | Online Publications Diffract | SIM 3D_Max | Multilayer | Temprofile © Copyright 1987-2011, MM Research, Inc. 5748 N. Camino del Conde, Tucson, Arizona 85718

Figure 3. Distribution of intensity at the observation plane of Fig. 1 in the absence of aberrations. The pupil relay lens is chosen to have the same numerical aperture as the object under test, thereby limiting the collected light to the zero-order beam and to those portions of the ±1st orders that overlap the 0th order.

Figure 10. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the wire test with an extended, quasi-monochromatic light source. The lens under test (NA = 0.5, f = 6000l0) has 3 waves of primary spherical aberration. The assumed wire diameter is 15l0, which is comparable to the size of the image of the extended light source, as measured in the vicinity of focus. In (a) the wire is centered on axis and is 25l0 away from paraxial focus (in the direction of the marginal focus). In (b) the wire is again centered on axis, but is 20l0 away from paraxial focus. In (c) the wire is shifted 0.5l0 off-axis while its distance from paraxial focus remains at 20l0.   Extensions of the Ronchi test. Several modifications and extensions of the Ronchi test have appeared over the years, and have helped to solve specific problems in testing of optical systems.3 As an example we mention the "double-frequency grating lateral shear interferometer" invented by James Wyant in the early 1970?s. The grating in this device has two slightly different frequencies, which give rise to two +1st order beams as well as two -1st order beams; the beams in each pair are slightly shifted relative to each other. Moreover, the (average) pitch of the grating is such that there is no overlap between the 0th, +1st, and -1st orders. Consequently, interference occurs between the two +1st order beams (and, likewise, between the two -1st order beams). One can thus obtain an arbitrarily small lateral shear of the wavefront under test, and use the results to achieve accurate quantitative measurements. A two-dimensional version of the double-frequency grating has also been employed to generate lateral wavefront shear simultaneously along the X and Y axes. (Remember that beam propagation is along Z and, therefore, X and Y are orthogonal axes in the plane of the grating.) In the absence of  a 2-D grating, one must rotate a 1-D grating by 90° to obtain  wavefront shear first along the X- and then along the Y-axis. Acknowledgment. I am grateful to Professor Roland Shack of the Optical Sciences Center for many illuminating discussions, and also for suggesting some of the examples presented in this article.   References V. Ronchi, "Le Frange di Combinazioni Nello Studio delle Superficie e dei Sistemi Ottici," Riv. Ottica Mecc. Precis. 2, 9 (1923). V. Ronchi, "Due Nuovi Metodi per lo Studio delle Superficie e dei Sistemi Ottici," Ann. Sc. Norm. Super. Pisa, 15 (1923). D. Malacara, ed., Optical Shop Testing, second edition, Wiley, New York, 1992. The computer simulations reported in this article were performed by DIFFRACT?, a product of MM Research, Inc., Tucson, Arizona. G. Toraldo di Francia, "Geometrical and interferential aspects of the Ronchi Test," in Optical Image Evaluation, National Bureau of Standards Circular 526, issued April 29, 1954. J. C. Wyant, "Double frequency grating lateral shear interferometer," Appl. Opt. 12, 2057 (1973). << Back to list of Articles Home | About MM Research, Inc. | Online Publications Diffract | SIM 3D_Max | Multilayer | Temprofile © Copyright 1987-2011, MM Research, Inc. 5748 N. Camino del Conde, Tucson, Arizona 85718

Light is made up of particles called photons and hence inherently is quantized. Quantum optics is the study of the nature and effects of light as quantized photons. The first indication that light might be quantized came from Max Planck in 1899 when he correctly modelled blackbody radiation by assuming that the exchange of energy between light and matter only occurred in discrete amounts he called quanta. It was unknown whether the source of this discreteness was the matter or the light.[63]: 231–236  In 1905, Albert Einstein published the theory of the photoelectric effect. It appeared that the only possible explanation for the effect was the quantization of light itself. Later, Niels Bohr showed that atoms could only emit discrete amounts of energy. The understanding of the interaction between light and matter following from these developments not only formed the basis of quantum optics but also were crucial for the development of quantum mechanics as a whole. However, the subfields of quantum mechanics dealing with matter-light interaction were principally regarded as research into matter rather than into light and hence, one rather spoke of atom physics and quantum electronics.

In his Catoptrica, Hero of Alexandria showed by a geometrical method that the actual path taken by a ray of light reflected from a plane mirror is shorter than any other reflected path that might be drawn between the source and point of observation.

Optics began with the development of lenses by the ancient Egyptians and Mesopotamians, followed by theories on light and vision developed by ancient Greek philosophers, and the development of geometrical optics in the Greco-Roman world. The word optics is derived from the Greek term τα ὀπτικά meaning 'appearance, look'.[1] Optics was significantly reformed by the developments in the medieval Islamic world, such as the beginnings of physical and physiological optics, and then significantly advanced in early modern Europe, where diffractive optics began. These earlier studies on optics are now known as "classical optics". The term "modern optics" refers to areas of optical research that largely developed in the 20th century, such as wave optics and quantum optics.

His more general consideration of light as a primary agent of physical causation appears in his On Lines, Angles, and Figures where he asserts that "a natural agent propagates its power from itself to the recipient" and in On the Nature of Places where he notes that "every natural action is varied in strength and weakness through variation of lines, angles and figures."[26]

Figure 9. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the knife-edge test carried out with a laser beam. The lens under test (NA = 0.5, f = 6000l0) and the pupil relay lens (NA = 0.5) are assumed to be fixed in their respective positions, while the knife-edge moves along the optical axis. (The tip of the knife remains on the axis at all times.) The lens under test is assumed to have 3 waves of primary spherical aberration. In frames (a) to (d) the distance of the knife-edge from paraxial focus is Dz = -15l0, 0, +15l0, and +20l0, respectively. (Positive distances are in the direction of the marginal focus.)   The results of the simulated knife-edge test depicted in Fig. 9 assume a laser as the light source. Consequently, frames (a) and (b) of Fig. 9 exhibit several dark lines which, with a less coherent light source, would have been absent. The results of the simulated wire test shown in Fig. 10 assume an extended light source, since the small amount of spherical aberration present in the system under consideration would render the test useless with a wire that, fine as it may be, is still wider than the focused spot produced by a laser beam. Note the similarities between the patterns of Figs. 9 and 10 on the one hand, and those of Figs. 5(d)-(f) on the other.   Figure 10. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the wire test with an extended, quasi-monochromatic light source. The lens under test (NA = 0.5, f = 6000l0) has 3 waves of primary spherical aberration. The assumed wire diameter is 15l0, which is comparable to the size of the image of the extended light source, as measured in the vicinity of focus. In (a) the wire is centered on axis and is 25l0 away from paraxial focus (in the direction of the marginal focus). In (b) the wire is again centered on axis, but is 20l0 away from paraxial focus. In (c) the wire is shifted 0.5l0 off-axis while its distance from paraxial focus remains at 20l0.   Extensions of the Ronchi test. Several modifications and extensions of the Ronchi test have appeared over the years, and have helped to solve specific problems in testing of optical systems.3 As an example we mention the "double-frequency grating lateral shear interferometer" invented by James Wyant in the early 1970?s. The grating in this device has two slightly different frequencies, which give rise to two +1st order beams as well as two -1st order beams; the beams in each pair are slightly shifted relative to each other. Moreover, the (average) pitch of the grating is such that there is no overlap between the 0th, +1st, and -1st orders. Consequently, interference occurs between the two +1st order beams (and, likewise, between the two -1st order beams). One can thus obtain an arbitrarily small lateral shear of the wavefront under test, and use the results to achieve accurate quantitative measurements. A two-dimensional version of the double-frequency grating has also been employed to generate lateral wavefront shear simultaneously along the X and Y axes. (Remember that beam propagation is along Z and, therefore, X and Y are orthogonal axes in the plane of the grating.) In the absence of  a 2-D grating, one must rotate a 1-D grating by 90° to obtain  wavefront shear first along the X- and then along the Y-axis. Acknowledgment. I am grateful to Professor Roland Shack of the Optical Sciences Center for many illuminating discussions, and also for suggesting some of the examples presented in this article.   References V. Ronchi, "Le Frange di Combinazioni Nello Studio delle Superficie e dei Sistemi Ottici," Riv. Ottica Mecc. Precis. 2, 9 (1923). V. Ronchi, "Due Nuovi Metodi per lo Studio delle Superficie e dei Sistemi Ottici," Ann. Sc. Norm. Super. Pisa, 15 (1923). D. Malacara, ed., Optical Shop Testing, second edition, Wiley, New York, 1992. The computer simulations reported in this article were performed by DIFFRACT?, a product of MM Research, Inc., Tucson, Arizona. G. Toraldo di Francia, "Geometrical and interferential aspects of the Ronchi Test," in Optical Image Evaluation, National Bureau of Standards Circular 526, issued April 29, 1954. J. C. Wyant, "Double frequency grating lateral shear interferometer," Appl. Opt. 12, 2057 (1973). << Back to list of Articles Home | About MM Research, Inc. | Online Publications Diffract | SIM 3D_Max | Multilayer | Temprofile © Copyright 1987-2011, MM Research, Inc. 5748 N. Camino del Conde, Tucson, Arizona 85718

To determine the appropriate grating period P, one needs to know the wavelength l0 of the beam used for testing, and the numerical aperture NA of the focused cone of light. (By definition, NA = sin q, where q is the half-angle subtended by the exit pupil of the lens at its focal point. If the lens under test is being used at full aperture, NA will also be equal to 0.5/f-number.) To avoid multiple overlaps among diffracted orders, the angle between adjacent orders must exceed the focused cone?s half-angle. Now, it is well-known in the theory of diffraction gratings that, at normal incidence, sin qn = nl0/P where n, an integer, is the order of diffraction, and qn is the corresponding deviation angle from the surface normal. Therefore, we arrive at the conclusion that P should be less than or equal to l0/NA. For example, assume that the lens under test has a numerical aperture NA = 0.5. Then, if the grating period is chosen to be 2l0, each diffracted order will deviate from the zero-order by 30°, making the +1st order just touch the -1st order in the far field. Figure 3 shows the computed intensity distribution at the observation plane of an aberration-free system in which the relay lens has the same numerical aperture as the lens under test (NA = 0.5).     Figure 3. Distribution of intensity at the observation plane of Fig. 1 in the absence of aberrations. The pupil relay lens is chosen to have the same numerical aperture as the object under test, thereby limiting the collected light to the zero-order beam and to those portions of the ±1st orders that overlap the 0th order.     Equality of numerical apertures means that only the zero-order diffracted beam will be fully transmitted to the observation plane. Of the ±1st order beams only those portions that overlap the zero order will reach the observation plane. The period of the grating in this example has been a little less than l0/NA, leaving a small gap between +1st and -1st orders.4 Absence of aberrations means that the phase distribution over the cross-sections of the various diffracted orders is uniform and, therefore, no interference fringes are to be expected. Ronchigrams for primary or Seidel aberrations. Figure 4 shows the computed patterns of intensity distribution at the observation plane of Fig. 1, corresponding to different types of primary (Seidel) aberrations of the lens.   Figure 4. Computed plots of intensity distribution at the observation plane of Fig. 1. The lens under test is assumed to have 3 waves of primary (Seidel) aberrations, and the grating is at the nominal focal plane of the lens. (a) defocus, (b) spherical, (c) astigmatism oriented at 45°,  (d) coma at 0°, (e) coma at 45°, (f) coma at 90°.     For these calculations we fixed the distance between the lens under test and the relay lens; we then placed the grating at the paraxial focus of the converging wavefront.4 The pattern in Fig. 4(a) was obtained when we assumed the presence of 3 waves of curvature (or defocus) at the exit pupil of the lens. Different amounts of defocus would create essentially the same pattern albeit with a different number of fringes. In Fig. 4(b) we observe the fringes arising from the presence of 3 waves of third order spherical aberration in the test system. The shapes of these fringes depend not only on the magnitude of the aberration, but also on the position of the grating relative to the focal plane. (We will have more to say about this point later.) Figure 4(c) shows the fringes that would arise when 3 waves of primary astigmatism are present. When the orientation of astigmatism changes, the fringes will remain straight lines, but their orientation within the observation plane will change accordingly. The last three frames in Fig. 4 represent the effects of third order coma. A change in the orientation of this aberration causes the interference pattern to change drastically. Figures 4(d)-(f) correspond to 3 waves of coma oriented at 0°, 45°, and 90°, respectively. Sliding the grating along the optical axis. A change in the position of the grating relative to the focal plane influences the observed fringe pattern. We limit our discussion to the case of spherical aberration, although similar analyses could be performed for other aberrations as well. Assuming 3 waves of spherical aberration as before, we obtain the patterns displayed in Fig. 5 as we slide the grating along the optical axis in the system of Fig. 1.4 Once again, we have taken the lens under test to have NA = 0.5 and f = 6000l0. The paraxial focus of the lens under test coincides with the front focal point of the relay lens, and the grating is shifted by different amounts Dz relative to this common focus. Frames (a)-(f) in Fig. 5 correspond to different values of Dz, starting at Dz = -10l0 in (a) and moving forward to Dz = +25l0 in (f). In the process, as the grating moves through paraxial focus and towards marginal focus, we observe a rich variety of patterns that aid us in determining the nature and the magnitude of the aberration.     Figure 5. Computed plots of intensity distribution at the observation plane of Fig. 1, showing the patterns obtained by sliding the grating along the optical axis. The lens under test (NA = 0.5, f = 6000l0) is assumed to have 3 waves of primary spherical aberration, and its paraxial focus is coincident with the focal point of the relay lens. The grating is moved along the optical axis by an amount Dz relative to the (common) focal plane; positive distances are towards the marginal focus. (a)  Dz= -10l0, (b)  Dz = 0, (c)  Dz = 10l0, (d) Dz = 15l0, (e)  Dz= 20l0, (f)  Dz= 25l0.   To be sure, the Ronchi test is not the only scheme used during fabrication and evaluation of optical systems; several other tests exist and their relative merits have been expounded in the literature.3 It is useful here to examine some of these alternative methods and to compare the resulting patterns (interferograms or otherwise) with those obtained with the Ronchi test. Testing by interfering with a reference plane-wave. Figure 6 shows the schematic diagram of a Mach-Zehnder interferometer, which is one among many that can be used to evaluate the aberrated wavefronts directly. In this system a coherent, monochromatic beam of light is sent through the lens under test, is collected and recollimated by a well-corrected lens, and is made to interfere with a reference beam that has been split off from the original, incident wavefront.   Figure 6. Schematic diagram of a Mach-Zehnder interferometer that might be set up for a direct measurement of wavefront aberrations. The pupil relay lens (itself free from aberrations) forms at the observation plane an image of the exit pupil of the lens under test. A fraction of the incident beam is diverted from its original path and sent to the observation plane by means of the various mirrors and beam-splitters. The observed fringes are characteristic of the aberrations present at the exit pupil of the lens under test. A small tilt of the mirror shown at the lower left side of the figure would introduce a linear phase shift on the reference beam. This tilt is generally useful in producing signature fringe patterns at the observation plane.   The flat mirror shown in the lower left side of the interferometer is mounted on a tip-tilt stage that allows the introduction of a small amount of tilt in the reference beam. Figure 7 shows the computed patterns of intensity distribution at the observation plane of the Mach-Zehnder interferometer corresponding to 3 waves of primary coma.4         Figure 7. Computed plots of intensity distribution at the observation plane of Fig. 6. The lens under test (NA = 0.5, f = 6000l0) is assumed to have 3 waves of primary coma, and its nominal focus is coincident with the focal point of the relay lens. The tilt angle y of the reference beam increases progressively from (a) to (f). (a) y = -0.1°, (b) y = 0°, (c) y  = 0.05°, (d)y = 0.07°, (e) y = 0.1°, (f)y  = 0.18°.   In obtaining the various frames of Fig. 7 we have fixed all the system parameters and only varied the tilt of the reference beam. Note that the characteristic fringes of coma in Fig. 7 are quite different from those of coma in the Ronchi test, shown in Figs. 4(d)-(f). Incidentally, the patterns of Fig. 7 show similarities with the Ronchigrams of spherical aberration displayed in Fig. 5. This is not a coincidence, and is rooted in the algebraic forms of the aberration function for third order coma (r3 cos f) and spherical aberration (r4 ), and also in the fact that a Ronchigram, being a kind of shearing interferogram (albeit with a large shear), is related to the derivative of the wavefront aberration function. Knife-edge and wire tests. A schematic diagram of the knife-edge method of testing optical systems is shown in Fig. 8. A geometric-optical interpretation of this test suffices for most practical purposes: the knife-edge blocks different groups of rays in its various positions along the optical axis, allowing the remaining rays to reach the observation plane.3 Another method of testing, known as the wire test, is quite similar to the knife-edge method, being obtained from it by the substitution of the knife-edge with a fine wire.3   Figure 8. In the knife-edge test a certain region in the vicinity of focus is blocked by a knife-edge; the nature and the magnitude of aberrations are then inferred from the resulting patterns of intensity distribution at the observation plane. (The knife-edge may be moved both along and perpendicular to the optical axis.) The wire test is similar to the knife-edge test except that a fine wire is used instead to block certain groups of rays.   Since the grating in the Ronchi test may be thought of as a series of parallel knife-edges or, more aptly, a series of parallel wires, it should not come as a surprise that similarities exist between Ronchigrams and the patterns observed in these other tests. In fact, early attempts at explaining the results of Ronchi?s method were based on geometrical optics, and considered the grating as a set of parallel wires whose shadows produced the observed patterns.5 We will not delve into these matters, but simply draw the reader?s attention to Figs. 9 and 10, where we show several computed patterns of intensity distribution for the knife-edge and wire tests, respectively.4     Figure 9. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the knife-edge test carried out with a laser beam. The lens under test (NA = 0.5, f = 6000l0) and the pupil relay lens (NA = 0.5) are assumed to be fixed in their respective positions, while the knife-edge moves along the optical axis. (The tip of the knife remains on the axis at all times.) The lens under test is assumed to have 3 waves of primary spherical aberration. In frames (a) to (d) the distance of the knife-edge from paraxial focus is Dz = -15l0, 0, +15l0, and +20l0, respectively. (Positive distances are in the direction of the marginal focus.)   The results of the simulated knife-edge test depicted in Fig. 9 assume a laser as the light source. Consequently, frames (a) and (b) of Fig. 9 exhibit several dark lines which, with a less coherent light source, would have been absent. The results of the simulated wire test shown in Fig. 10 assume an extended light source, since the small amount of spherical aberration present in the system under consideration would render the test useless with a wire that, fine as it may be, is still wider than the focused spot produced by a laser beam. Note the similarities between the patterns of Figs. 9 and 10 on the one hand, and those of Figs. 5(d)-(f) on the other.   Figure 10. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the wire test with an extended, quasi-monochromatic light source. The lens under test (NA = 0.5, f = 6000l0) has 3 waves of primary spherical aberration. The assumed wire diameter is 15l0, which is comparable to the size of the image of the extended light source, as measured in the vicinity of focus. In (a) the wire is centered on axis and is 25l0 away from paraxial focus (in the direction of the marginal focus). In (b) the wire is again centered on axis, but is 20l0 away from paraxial focus. In (c) the wire is shifted 0.5l0 off-axis while its distance from paraxial focus remains at 20l0.   Extensions of the Ronchi test. Several modifications and extensions of the Ronchi test have appeared over the years, and have helped to solve specific problems in testing of optical systems.3 As an example we mention the "double-frequency grating lateral shear interferometer" invented by James Wyant in the early 1970?s. The grating in this device has two slightly different frequencies, which give rise to two +1st order beams as well as two -1st order beams; the beams in each pair are slightly shifted relative to each other. Moreover, the (average) pitch of the grating is such that there is no overlap between the 0th, +1st, and -1st orders. Consequently, interference occurs between the two +1st order beams (and, likewise, between the two -1st order beams). One can thus obtain an arbitrarily small lateral shear of the wavefront under test, and use the results to achieve accurate quantitative measurements. A two-dimensional version of the double-frequency grating has also been employed to generate lateral wavefront shear simultaneously along the X and Y axes. (Remember that beam propagation is along Z and, therefore, X and Y are orthogonal axes in the plane of the grating.) In the absence of  a 2-D grating, one must rotate a 1-D grating by 90° to obtain  wavefront shear first along the X- and then along the Y-axis. Acknowledgment. I am grateful to Professor Roland Shack of the Optical Sciences Center for many illuminating discussions, and also for suggesting some of the examples presented in this article.   References V. Ronchi, "Le Frange di Combinazioni Nello Studio delle Superficie e dei Sistemi Ottici," Riv. Ottica Mecc. Precis. 2, 9 (1923). V. Ronchi, "Due Nuovi Metodi per lo Studio delle Superficie e dei Sistemi Ottici," Ann. Sc. Norm. Super. Pisa, 15 (1923). D. Malacara, ed., Optical Shop Testing, second edition, Wiley, New York, 1992. The computer simulations reported in this article were performed by DIFFRACT?, a product of MM Research, Inc., Tucson, Arizona. G. Toraldo di Francia, "Geometrical and interferential aspects of the Ronchi Test," in Optical Image Evaluation, National Bureau of Standards Circular 526, issued April 29, 1954. J. C. Wyant, "Double frequency grating lateral shear interferometer," Appl. Opt. 12, 2057 (1973). << Back to list of Articles Home | About MM Research, Inc. | Online Publications Diffract | SIM 3D_Max | Multilayer | Temprofile © Copyright 1987-2011, MM Research, Inc. 5748 N. Camino del Conde, Tucson, Arizona 85718

A lens (or more generally, an optical system consisting of a number of lenses and mirrors) is placed in the position of the "object under test." The lens is then illuminated with a beam of light, which, for the purposes of the present article, will be assumed to be coherent and quasi-monochromatic. (These restrictions on the beam may be substantially relaxed in practice.3)

Optics documents Ptolemy's studies of reflection and refraction.[7] He measured the angles of refraction between air, water, and glass, but his published results indicate that he adjusted his measurements to fit his (incorrect) assumption that the angle of refraction is proportional to the angle of incidence.[8][9]

Ibn Sahl, a mathematician active in Baghdad during the 980s, is the first Islamic scholar known to have compiled a commentary on Ptolemy's Optics. His treatise Fī al-'āla al-muḥriqa "On the burning instruments" was reconstructed from fragmentary manuscripts by Rashed (1993).[12] The work is concerned with how curved mirrors and lenses bend and focus light. Ibn Sahl also describes a law of refraction mathematically equivalent to Snell's law.[13] He used his law of refraction to compute the shapes of lenses and mirrors that focus light at a single point on the axis.

Avicenna (980–1037) agreed with Alhazen that the speed of light is finite, as he "observed that if the perception of light is due to the emission of some sort of particles by a luminous source, the speed of light must be finite."[20] Abū Rayhān al-Bīrūnī (973-1048) also agreed that light has a finite speed, and stated that the speed of light is much faster than the speed of sound.[21]

Today's fields of interest among quantum optics researchers include parametric down-conversion, parametric oscillation, even shorter (attosecond) light pulses, use of quantum optics for quantum information, manipulation of single atoms and Bose–Einstein condensates, their application, and how to manipulate them (a sub-field often called atom optics).

Although disputed, archeological evidence has been suggested of the use of lenses in ancient times over a period of several millennia.[38] It has been proposed that glass eye covers in hieroglyphs from the Old Kingdom of Egypt (c. 2686–2181 BCE) were functional simple glass meniscus lenses.[39] The so-called Nimrud lens, a rock crystal artifact dated to the 7th century BCE, might have been used as a magnifying glass, although it could have simply been a decoration.[40][41][42][43][44]

Where Euclid had limited his analysis to simple direct vision, Hero of Alexandria (c. AD 10–70) extended the principles of geometrical optics to consider problems of reflection (catoptrics). Unlike Euclid, Hero occasionally commented on the physical nature of visual rays, indicating that they proceeded at great speed from the eye to the object seen and were reflected from smooth surfaces but could become trapped in the porosities of unpolished surfaces.[5] This has come to be known as emission theory.[6]

Ronchigrams for primary or Seidel aberrations. Figure 4 shows the computed patterns of intensity distribution at the observation plane of Fig. 1, corresponding to different types of primary (Seidel) aberrations of the lens.   Figure 4. Computed plots of intensity distribution at the observation plane of Fig. 1. The lens under test is assumed to have 3 waves of primary (Seidel) aberrations, and the grating is at the nominal focal plane of the lens. (a) defocus, (b) spherical, (c) astigmatism oriented at 45°,  (d) coma at 0°, (e) coma at 45°, (f) coma at 90°.     For these calculations we fixed the distance between the lens under test and the relay lens; we then placed the grating at the paraxial focus of the converging wavefront.4 The pattern in Fig. 4(a) was obtained when we assumed the presence of 3 waves of curvature (or defocus) at the exit pupil of the lens. Different amounts of defocus would create essentially the same pattern albeit with a different number of fringes. In Fig. 4(b) we observe the fringes arising from the presence of 3 waves of third order spherical aberration in the test system. The shapes of these fringes depend not only on the magnitude of the aberration, but also on the position of the grating relative to the focal plane. (We will have more to say about this point later.) Figure 4(c) shows the fringes that would arise when 3 waves of primary astigmatism are present. When the orientation of astigmatism changes, the fringes will remain straight lines, but their orientation within the observation plane will change accordingly. The last three frames in Fig. 4 represent the effects of third order coma. A change in the orientation of this aberration causes the interference pattern to change drastically. Figures 4(d)-(f) correspond to 3 waves of coma oriented at 0°, 45°, and 90°, respectively. Sliding the grating along the optical axis. A change in the position of the grating relative to the focal plane influences the observed fringe pattern. We limit our discussion to the case of spherical aberration, although similar analyses could be performed for other aberrations as well. Assuming 3 waves of spherical aberration as before, we obtain the patterns displayed in Fig. 5 as we slide the grating along the optical axis in the system of Fig. 1.4 Once again, we have taken the lens under test to have NA = 0.5 and f = 6000l0. The paraxial focus of the lens under test coincides with the front focal point of the relay lens, and the grating is shifted by different amounts Dz relative to this common focus. Frames (a)-(f) in Fig. 5 correspond to different values of Dz, starting at Dz = -10l0 in (a) and moving forward to Dz = +25l0 in (f). In the process, as the grating moves through paraxial focus and towards marginal focus, we observe a rich variety of patterns that aid us in determining the nature and the magnitude of the aberration.     Figure 5. Computed plots of intensity distribution at the observation plane of Fig. 1, showing the patterns obtained by sliding the grating along the optical axis. The lens under test (NA = 0.5, f = 6000l0) is assumed to have 3 waves of primary spherical aberration, and its paraxial focus is coincident with the focal point of the relay lens. The grating is moved along the optical axis by an amount Dz relative to the (common) focal plane; positive distances are towards the marginal focus. (a)  Dz= -10l0, (b)  Dz = 0, (c)  Dz = 10l0, (d) Dz = 15l0, (e)  Dz= 20l0, (f)  Dz= 25l0.   To be sure, the Ronchi test is not the only scheme used during fabrication and evaluation of optical systems; several other tests exist and their relative merits have been expounded in the literature.3 It is useful here to examine some of these alternative methods and to compare the resulting patterns (interferograms or otherwise) with those obtained with the Ronchi test. Testing by interfering with a reference plane-wave. Figure 6 shows the schematic diagram of a Mach-Zehnder interferometer, which is one among many that can be used to evaluate the aberrated wavefronts directly. In this system a coherent, monochromatic beam of light is sent through the lens under test, is collected and recollimated by a well-corrected lens, and is made to interfere with a reference beam that has been split off from the original, incident wavefront.   Figure 6. Schematic diagram of a Mach-Zehnder interferometer that might be set up for a direct measurement of wavefront aberrations. The pupil relay lens (itself free from aberrations) forms at the observation plane an image of the exit pupil of the lens under test. A fraction of the incident beam is diverted from its original path and sent to the observation plane by means of the various mirrors and beam-splitters. The observed fringes are characteristic of the aberrations present at the exit pupil of the lens under test. A small tilt of the mirror shown at the lower left side of the figure would introduce a linear phase shift on the reference beam. This tilt is generally useful in producing signature fringe patterns at the observation plane.   The flat mirror shown in the lower left side of the interferometer is mounted on a tip-tilt stage that allows the introduction of a small amount of tilt in the reference beam. Figure 7 shows the computed patterns of intensity distribution at the observation plane of the Mach-Zehnder interferometer corresponding to 3 waves of primary coma.4         Figure 7. Computed plots of intensity distribution at the observation plane of Fig. 6. The lens under test (NA = 0.5, f = 6000l0) is assumed to have 3 waves of primary coma, and its nominal focus is coincident with the focal point of the relay lens. The tilt angle y of the reference beam increases progressively from (a) to (f). (a) y = -0.1°, (b) y = 0°, (c) y  = 0.05°, (d)y = 0.07°, (e) y = 0.1°, (f)y  = 0.18°.   In obtaining the various frames of Fig. 7 we have fixed all the system parameters and only varied the tilt of the reference beam. Note that the characteristic fringes of coma in Fig. 7 are quite different from those of coma in the Ronchi test, shown in Figs. 4(d)-(f). Incidentally, the patterns of Fig. 7 show similarities with the Ronchigrams of spherical aberration displayed in Fig. 5. This is not a coincidence, and is rooted in the algebraic forms of the aberration function for third order coma (r3 cos f) and spherical aberration (r4 ), and also in the fact that a Ronchigram, being a kind of shearing interferogram (albeit with a large shear), is related to the derivative of the wavefront aberration function. Knife-edge and wire tests. A schematic diagram of the knife-edge method of testing optical systems is shown in Fig. 8. A geometric-optical interpretation of this test suffices for most practical purposes: the knife-edge blocks different groups of rays in its various positions along the optical axis, allowing the remaining rays to reach the observation plane.3 Another method of testing, known as the wire test, is quite similar to the knife-edge method, being obtained from it by the substitution of the knife-edge with a fine wire.3   Figure 8. In the knife-edge test a certain region in the vicinity of focus is blocked by a knife-edge; the nature and the magnitude of aberrations are then inferred from the resulting patterns of intensity distribution at the observation plane. (The knife-edge may be moved both along and perpendicular to the optical axis.) The wire test is similar to the knife-edge test except that a fine wire is used instead to block certain groups of rays.   Since the grating in the Ronchi test may be thought of as a series of parallel knife-edges or, more aptly, a series of parallel wires, it should not come as a surprise that similarities exist between Ronchigrams and the patterns observed in these other tests. In fact, early attempts at explaining the results of Ronchi?s method were based on geometrical optics, and considered the grating as a set of parallel wires whose shadows produced the observed patterns.5 We will not delve into these matters, but simply draw the reader?s attention to Figs. 9 and 10, where we show several computed patterns of intensity distribution for the knife-edge and wire tests, respectively.4     Figure 9. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the knife-edge test carried out with a laser beam. The lens under test (NA = 0.5, f = 6000l0) and the pupil relay lens (NA = 0.5) are assumed to be fixed in their respective positions, while the knife-edge moves along the optical axis. (The tip of the knife remains on the axis at all times.) The lens under test is assumed to have 3 waves of primary spherical aberration. In frames (a) to (d) the distance of the knife-edge from paraxial focus is Dz = -15l0, 0, +15l0, and +20l0, respectively. (Positive distances are in the direction of the marginal focus.)   The results of the simulated knife-edge test depicted in Fig. 9 assume a laser as the light source. Consequently, frames (a) and (b) of Fig. 9 exhibit several dark lines which, with a less coherent light source, would have been absent. The results of the simulated wire test shown in Fig. 10 assume an extended light source, since the small amount of spherical aberration present in the system under consideration would render the test useless with a wire that, fine as it may be, is still wider than the focused spot produced by a laser beam. Note the similarities between the patterns of Figs. 9 and 10 on the one hand, and those of Figs. 5(d)-(f) on the other.   Figure 10. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the wire test with an extended, quasi-monochromatic light source. The lens under test (NA = 0.5, f = 6000l0) has 3 waves of primary spherical aberration. The assumed wire diameter is 15l0, which is comparable to the size of the image of the extended light source, as measured in the vicinity of focus. In (a) the wire is centered on axis and is 25l0 away from paraxial focus (in the direction of the marginal focus). In (b) the wire is again centered on axis, but is 20l0 away from paraxial focus. In (c) the wire is shifted 0.5l0 off-axis while its distance from paraxial focus remains at 20l0.   Extensions of the Ronchi test. Several modifications and extensions of the Ronchi test have appeared over the years, and have helped to solve specific problems in testing of optical systems.3 As an example we mention the "double-frequency grating lateral shear interferometer" invented by James Wyant in the early 1970?s. The grating in this device has two slightly different frequencies, which give rise to two +1st order beams as well as two -1st order beams; the beams in each pair are slightly shifted relative to each other. Moreover, the (average) pitch of the grating is such that there is no overlap between the 0th, +1st, and -1st orders. Consequently, interference occurs between the two +1st order beams (and, likewise, between the two -1st order beams). One can thus obtain an arbitrarily small lateral shear of the wavefront under test, and use the results to achieve accurate quantitative measurements. A two-dimensional version of the double-frequency grating has also been employed to generate lateral wavefront shear simultaneously along the X and Y axes. (Remember that beam propagation is along Z and, therefore, X and Y are orthogonal axes in the plane of the grating.) In the absence of  a 2-D grating, one must rotate a 1-D grating by 90° to obtain  wavefront shear first along the X- and then along the Y-axis. Acknowledgment. I am grateful to Professor Roland Shack of the Optical Sciences Center for many illuminating discussions, and also for suggesting some of the examples presented in this article.   References V. Ronchi, "Le Frange di Combinazioni Nello Studio delle Superficie e dei Sistemi Ottici," Riv. Ottica Mecc. Precis. 2, 9 (1923). V. Ronchi, "Due Nuovi Metodi per lo Studio delle Superficie e dei Sistemi Ottici," Ann. Sc. Norm. Super. Pisa, 15 (1923). D. Malacara, ed., Optical Shop Testing, second edition, Wiley, New York, 1992. The computer simulations reported in this article were performed by DIFFRACT?, a product of MM Research, Inc., Tucson, Arizona. G. Toraldo di Francia, "Geometrical and interferential aspects of the Ronchi Test," in Optical Image Evaluation, National Bureau of Standards Circular 526, issued April 29, 1954. J. C. Wyant, "Double frequency grating lateral shear interferometer," Appl. Opt. 12, 2057 (1973). << Back to list of Articles Home | About MM Research, Inc. | Online Publications Diffract | SIM 3D_Max | Multilayer | Temprofile © Copyright 1987-2011, MM Research, Inc. 5748 N. Camino del Conde, Tucson, Arizona 85718

Figure 3 shows the computed intensity distribution at the observation plane of an aberration-free system in which the relay lens has the same numerical aperture as the lens under test (NA = 0.5).     Figure 3. Distribution of intensity at the observation plane of Fig. 1 in the absence of aberrations. The pupil relay lens is chosen to have the same numerical aperture as the object under test, thereby limiting the collected light to the zero-order beam and to those portions of the ±1st orders that overlap the 0th order.     Equality of numerical apertures means that only the zero-order diffracted beam will be fully transmitted to the observation plane. Of the ±1st order beams only those portions that overlap the zero order will reach the observation plane. The period of the grating in this example has been a little less than l0/NA, leaving a small gap between +1st and -1st orders.4 Absence of aberrations means that the phase distribution over the cross-sections of the various diffracted orders is uniform and, therefore, no interference fringes are to be expected. Ronchigrams for primary or Seidel aberrations. Figure 4 shows the computed patterns of intensity distribution at the observation plane of Fig. 1, corresponding to different types of primary (Seidel) aberrations of the lens.   Figure 4. Computed plots of intensity distribution at the observation plane of Fig. 1. The lens under test is assumed to have 3 waves of primary (Seidel) aberrations, and the grating is at the nominal focal plane of the lens. (a) defocus, (b) spherical, (c) astigmatism oriented at 45°,  (d) coma at 0°, (e) coma at 45°, (f) coma at 90°.     For these calculations we fixed the distance between the lens under test and the relay lens; we then placed the grating at the paraxial focus of the converging wavefront.4 The pattern in Fig. 4(a) was obtained when we assumed the presence of 3 waves of curvature (or defocus) at the exit pupil of the lens. Different amounts of defocus would create essentially the same pattern albeit with a different number of fringes. In Fig. 4(b) we observe the fringes arising from the presence of 3 waves of third order spherical aberration in the test system. The shapes of these fringes depend not only on the magnitude of the aberration, but also on the position of the grating relative to the focal plane. (We will have more to say about this point later.) Figure 4(c) shows the fringes that would arise when 3 waves of primary astigmatism are present. When the orientation of astigmatism changes, the fringes will remain straight lines, but their orientation within the observation plane will change accordingly. The last three frames in Fig. 4 represent the effects of third order coma. A change in the orientation of this aberration causes the interference pattern to change drastically. Figures 4(d)-(f) correspond to 3 waves of coma oriented at 0°, 45°, and 90°, respectively. Sliding the grating along the optical axis. A change in the position of the grating relative to the focal plane influences the observed fringe pattern. We limit our discussion to the case of spherical aberration, although similar analyses could be performed for other aberrations as well. Assuming 3 waves of spherical aberration as before, we obtain the patterns displayed in Fig. 5 as we slide the grating along the optical axis in the system of Fig. 1.4 Once again, we have taken the lens under test to have NA = 0.5 and f = 6000l0. The paraxial focus of the lens under test coincides with the front focal point of the relay lens, and the grating is shifted by different amounts Dz relative to this common focus. Frames (a)-(f) in Fig. 5 correspond to different values of Dz, starting at Dz = -10l0 in (a) and moving forward to Dz = +25l0 in (f). In the process, as the grating moves through paraxial focus and towards marginal focus, we observe a rich variety of patterns that aid us in determining the nature and the magnitude of the aberration.     Figure 5. Computed plots of intensity distribution at the observation plane of Fig. 1, showing the patterns obtained by sliding the grating along the optical axis. The lens under test (NA = 0.5, f = 6000l0) is assumed to have 3 waves of primary spherical aberration, and its paraxial focus is coincident with the focal point of the relay lens. The grating is moved along the optical axis by an amount Dz relative to the (common) focal plane; positive distances are towards the marginal focus. (a)  Dz= -10l0, (b)  Dz = 0, (c)  Dz = 10l0, (d) Dz = 15l0, (e)  Dz= 20l0, (f)  Dz= 25l0.   To be sure, the Ronchi test is not the only scheme used during fabrication and evaluation of optical systems; several other tests exist and their relative merits have been expounded in the literature.3 It is useful here to examine some of these alternative methods and to compare the resulting patterns (interferograms or otherwise) with those obtained with the Ronchi test. Testing by interfering with a reference plane-wave. Figure 6 shows the schematic diagram of a Mach-Zehnder interferometer, which is one among many that can be used to evaluate the aberrated wavefronts directly. In this system a coherent, monochromatic beam of light is sent through the lens under test, is collected and recollimated by a well-corrected lens, and is made to interfere with a reference beam that has been split off from the original, incident wavefront.   Figure 6. Schematic diagram of a Mach-Zehnder interferometer that might be set up for a direct measurement of wavefront aberrations. The pupil relay lens (itself free from aberrations) forms at the observation plane an image of the exit pupil of the lens under test. A fraction of the incident beam is diverted from its original path and sent to the observation plane by means of the various mirrors and beam-splitters. The observed fringes are characteristic of the aberrations present at the exit pupil of the lens under test. A small tilt of the mirror shown at the lower left side of the figure would introduce a linear phase shift on the reference beam. This tilt is generally useful in producing signature fringe patterns at the observation plane.   The flat mirror shown in the lower left side of the interferometer is mounted on a tip-tilt stage that allows the introduction of a small amount of tilt in the reference beam. Figure 7 shows the computed patterns of intensity distribution at the observation plane of the Mach-Zehnder interferometer corresponding to 3 waves of primary coma.4         Figure 7. Computed plots of intensity distribution at the observation plane of Fig. 6. The lens under test (NA = 0.5, f = 6000l0) is assumed to have 3 waves of primary coma, and its nominal focus is coincident with the focal point of the relay lens. The tilt angle y of the reference beam increases progressively from (a) to (f). (a) y = -0.1°, (b) y = 0°, (c) y  = 0.05°, (d)y = 0.07°, (e) y = 0.1°, (f)y  = 0.18°.   In obtaining the various frames of Fig. 7 we have fixed all the system parameters and only varied the tilt of the reference beam. Note that the characteristic fringes of coma in Fig. 7 are quite different from those of coma in the Ronchi test, shown in Figs. 4(d)-(f). Incidentally, the patterns of Fig. 7 show similarities with the Ronchigrams of spherical aberration displayed in Fig. 5. This is not a coincidence, and is rooted in the algebraic forms of the aberration function for third order coma (r3 cos f) and spherical aberration (r4 ), and also in the fact that a Ronchigram, being a kind of shearing interferogram (albeit with a large shear), is related to the derivative of the wavefront aberration function. Knife-edge and wire tests. A schematic diagram of the knife-edge method of testing optical systems is shown in Fig. 8. A geometric-optical interpretation of this test suffices for most practical purposes: the knife-edge blocks different groups of rays in its various positions along the optical axis, allowing the remaining rays to reach the observation plane.3 Another method of testing, known as the wire test, is quite similar to the knife-edge method, being obtained from it by the substitution of the knife-edge with a fine wire.3   Figure 8. In the knife-edge test a certain region in the vicinity of focus is blocked by a knife-edge; the nature and the magnitude of aberrations are then inferred from the resulting patterns of intensity distribution at the observation plane. (The knife-edge may be moved both along and perpendicular to the optical axis.) The wire test is similar to the knife-edge test except that a fine wire is used instead to block certain groups of rays.   Since the grating in the Ronchi test may be thought of as a series of parallel knife-edges or, more aptly, a series of parallel wires, it should not come as a surprise that similarities exist between Ronchigrams and the patterns observed in these other tests. In fact, early attempts at explaining the results of Ronchi?s method were based on geometrical optics, and considered the grating as a set of parallel wires whose shadows produced the observed patterns.5 We will not delve into these matters, but simply draw the reader?s attention to Figs. 9 and 10, where we show several computed patterns of intensity distribution for the knife-edge and wire tests, respectively.4     Figure 9. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the knife-edge test carried out with a laser beam. The lens under test (NA = 0.5, f = 6000l0) and the pupil relay lens (NA = 0.5) are assumed to be fixed in their respective positions, while the knife-edge moves along the optical axis. (The tip of the knife remains on the axis at all times.) The lens under test is assumed to have 3 waves of primary spherical aberration. In frames (a) to (d) the distance of the knife-edge from paraxial focus is Dz = -15l0, 0, +15l0, and +20l0, respectively. (Positive distances are in the direction of the marginal focus.)   The results of the simulated knife-edge test depicted in Fig. 9 assume a laser as the light source. Consequently, frames (a) and (b) of Fig. 9 exhibit several dark lines which, with a less coherent light source, would have been absent. The results of the simulated wire test shown in Fig. 10 assume an extended light source, since the small amount of spherical aberration present in the system under consideration would render the test useless with a wire that, fine as it may be, is still wider than the focused spot produced by a laser beam. Note the similarities between the patterns of Figs. 9 and 10 on the one hand, and those of Figs. 5(d)-(f) on the other.   Figure 10. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the wire test with an extended, quasi-monochromatic light source. The lens under test (NA = 0.5, f = 6000l0) has 3 waves of primary spherical aberration. The assumed wire diameter is 15l0, which is comparable to the size of the image of the extended light source, as measured in the vicinity of focus. In (a) the wire is centered on axis and is 25l0 away from paraxial focus (in the direction of the marginal focus). In (b) the wire is again centered on axis, but is 20l0 away from paraxial focus. In (c) the wire is shifted 0.5l0 off-axis while its distance from paraxial focus remains at 20l0.   Extensions of the Ronchi test. Several modifications and extensions of the Ronchi test have appeared over the years, and have helped to solve specific problems in testing of optical systems.3 As an example we mention the "double-frequency grating lateral shear interferometer" invented by James Wyant in the early 1970?s. The grating in this device has two slightly different frequencies, which give rise to two +1st order beams as well as two -1st order beams; the beams in each pair are slightly shifted relative to each other. Moreover, the (average) pitch of the grating is such that there is no overlap between the 0th, +1st, and -1st orders. Consequently, interference occurs between the two +1st order beams (and, likewise, between the two -1st order beams). One can thus obtain an arbitrarily small lateral shear of the wavefront under test, and use the results to achieve accurate quantitative measurements. A two-dimensional version of the double-frequency grating has also been employed to generate lateral wavefront shear simultaneously along the X and Y axes. (Remember that beam propagation is along Z and, therefore, X and Y are orthogonal axes in the plane of the grating.) In the absence of  a 2-D grating, one must rotate a 1-D grating by 90° to obtain  wavefront shear first along the X- and then along the Y-axis. Acknowledgment. I am grateful to Professor Roland Shack of the Optical Sciences Center for many illuminating discussions, and also for suggesting some of the examples presented in this article.   References V. Ronchi, "Le Frange di Combinazioni Nello Studio delle Superficie e dei Sistemi Ottici," Riv. Ottica Mecc. Precis. 2, 9 (1923). V. Ronchi, "Due Nuovi Metodi per lo Studio delle Superficie e dei Sistemi Ottici," Ann. Sc. Norm. Super. Pisa, 15 (1923). D. Malacara, ed., Optical Shop Testing, second edition, Wiley, New York, 1992. The computer simulations reported in this article were performed by DIFFRACT?, a product of MM Research, Inc., Tucson, Arizona. G. Toraldo di Francia, "Geometrical and interferential aspects of the Ronchi Test," in Optical Image Evaluation, National Bureau of Standards Circular 526, issued April 29, 1954. J. C. Wyant, "Double frequency grating lateral shear interferometer," Appl. Opt. 12, 2057 (1973). << Back to list of Articles Home | About MM Research, Inc. | Online Publications Diffract | SIM 3D_Max | Multilayer | Temprofile © Copyright 1987-2011, MM Research, Inc. 5748 N. Camino del Conde, Tucson, Arizona 85718

In this centennial of Vasco Ronchi's birth it seemed appropriate to devote one of these columns to the well-known method of testing optical systems that he developed in the 1920's.1,2 The essential features of the Ronchi test may be described by reference to Fig. 1.

Like Hero, Claudius Ptolemy in his second-century Optics considered the visual rays as proceeding from the eye to the object seen, but, unlike Hero, considered that the visual rays were not discrete lines, but formed a continuous cone.

In the late 13th and early 14th centuries, Qutb al-Din al-Shirazi (1236–1311) and his student Kamāl al-Dīn al-Fārisī (1260–1320) continued the work of Ibn al-Haytham, and they were among the first to give the correct explanations for the rainbow phenomenon. Al-Fārisī published his findings in his Kitab Tanqih al-Manazir (The Revision of [Ibn al-Haytham's] Optics).[23]

For these calculations we fixed the distance between the lens under test and the relay lens; we then placed the grating at the paraxial focus of the converging wavefront.4 The pattern in Fig. 4(a) was obtained when we assumed the presence of 3 waves of curvature (or defocus) at the exit pupil of the lens. Different amounts of defocus would create essentially the same pattern albeit with a different number of fringes. In Fig. 4(b) we observe the fringes arising from the presence of 3 waves of third order spherical aberration in the test system. The shapes of these fringes depend not only on the magnitude of the aberration, but also on the position of the grating relative to the focal plane. (We will have more to say about this point later.) Figure 4(c) shows the fringes that would arise when 3 waves of primary astigmatism are present. When the orientation of astigmatism changes, the fringes will remain straight lines, but their orientation within the observation plane will change accordingly.

Figure 10. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the wire test with an extended, quasi-monochromatic light source. The lens under test (NA = 0.5, f = 6000l0) has 3 waves of primary spherical aberration. The assumed wire diameter is 15l0, which is comparable to the size of the image of the extended light source, as measured in the vicinity of focus. In (a) the wire is centered on axis and is 25l0 away from paraxial focus (in the direction of the marginal focus). In (b) the wire is again centered on axis, but is 20l0 away from paraxial focus. In (c) the wire is shifted 0.5l0 off-axis while its distance from paraxial focus remains at 20l0.

Another English Franciscan, John Pecham (died 1292) built on the work of Bacon, Grosseteste, and a diverse range of earlier writers to produce what became the most widely used textbook on optics of the Middle Ages, the Perspectiva communis. His book centered on the question of vision, on how we see, rather than on the nature of light and color. Pecham followed the model set forth by Alhacen, but interpreted Alhacen's ideas in the manner of Roger Bacon.[29]

Figure 8. In the knife-edge test a certain region in the vicinity of focus is blocked by a knife-edge; the nature and the magnitude of aberrations are then inferred from the resulting patterns of intensity distribution at the observation plane. (The knife-edge may be moved both along and perpendicular to the optical axis.) The wire test is similar to the knife-edge test except that a fine wire is used instead to block certain groups of rays.   Since the grating in the Ronchi test may be thought of as a series of parallel knife-edges or, more aptly, a series of parallel wires, it should not come as a surprise that similarities exist between Ronchigrams and the patterns observed in these other tests. In fact, early attempts at explaining the results of Ronchi?s method were based on geometrical optics, and considered the grating as a set of parallel wires whose shadows produced the observed patterns.5 We will not delve into these matters, but simply draw the reader?s attention to Figs. 9 and 10, where we show several computed patterns of intensity distribution for the knife-edge and wire tests, respectively.4     Figure 9. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the knife-edge test carried out with a laser beam. The lens under test (NA = 0.5, f = 6000l0) and the pupil relay lens (NA = 0.5) are assumed to be fixed in their respective positions, while the knife-edge moves along the optical axis. (The tip of the knife remains on the axis at all times.) The lens under test is assumed to have 3 waves of primary spherical aberration. In frames (a) to (d) the distance of the knife-edge from paraxial focus is Dz = -15l0, 0, +15l0, and +20l0, respectively. (Positive distances are in the direction of the marginal focus.)   The results of the simulated knife-edge test depicted in Fig. 9 assume a laser as the light source. Consequently, frames (a) and (b) of Fig. 9 exhibit several dark lines which, with a less coherent light source, would have been absent. The results of the simulated wire test shown in Fig. 10 assume an extended light source, since the small amount of spherical aberration present in the system under consideration would render the test useless with a wire that, fine as it may be, is still wider than the focused spot produced by a laser beam. Note the similarities between the patterns of Figs. 9 and 10 on the one hand, and those of Figs. 5(d)-(f) on the other.   Figure 10. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the wire test with an extended, quasi-monochromatic light source. The lens under test (NA = 0.5, f = 6000l0) has 3 waves of primary spherical aberration. The assumed wire diameter is 15l0, which is comparable to the size of the image of the extended light source, as measured in the vicinity of focus. In (a) the wire is centered on axis and is 25l0 away from paraxial focus (in the direction of the marginal focus). In (b) the wire is again centered on axis, but is 20l0 away from paraxial focus. In (c) the wire is shifted 0.5l0 off-axis while its distance from paraxial focus remains at 20l0.   Extensions of the Ronchi test. Several modifications and extensions of the Ronchi test have appeared over the years, and have helped to solve specific problems in testing of optical systems.3 As an example we mention the "double-frequency grating lateral shear interferometer" invented by James Wyant in the early 1970?s. The grating in this device has two slightly different frequencies, which give rise to two +1st order beams as well as two -1st order beams; the beams in each pair are slightly shifted relative to each other. Moreover, the (average) pitch of the grating is such that there is no overlap between the 0th, +1st, and -1st orders. Consequently, interference occurs between the two +1st order beams (and, likewise, between the two -1st order beams). One can thus obtain an arbitrarily small lateral shear of the wavefront under test, and use the results to achieve accurate quantitative measurements. A two-dimensional version of the double-frequency grating has also been employed to generate lateral wavefront shear simultaneously along the X and Y axes. (Remember that beam propagation is along Z and, therefore, X and Y are orthogonal axes in the plane of the grating.) In the absence of  a 2-D grating, one must rotate a 1-D grating by 90° to obtain  wavefront shear first along the X- and then along the Y-axis. Acknowledgment. I am grateful to Professor Roland Shack of the Optical Sciences Center for many illuminating discussions, and also for suggesting some of the examples presented in this article.   References V. Ronchi, "Le Frange di Combinazioni Nello Studio delle Superficie e dei Sistemi Ottici," Riv. Ottica Mecc. Precis. 2, 9 (1923). V. Ronchi, "Due Nuovi Metodi per lo Studio delle Superficie e dei Sistemi Ottici," Ann. Sc. Norm. Super. Pisa, 15 (1923). D. Malacara, ed., Optical Shop Testing, second edition, Wiley, New York, 1992. The computer simulations reported in this article were performed by DIFFRACT?, a product of MM Research, Inc., Tucson, Arizona. G. Toraldo di Francia, "Geometrical and interferential aspects of the Ronchi Test," in Optical Image Evaluation, National Bureau of Standards Circular 526, issued April 29, 1954. J. C. Wyant, "Double frequency grating lateral shear interferometer," Appl. Opt. 12, 2057 (1973). << Back to list of Articles Home | About MM Research, Inc. | Online Publications Diffract | SIM 3D_Max | Multilayer | Temprofile © Copyright 1987-2011, MM Research, Inc. 5748 N. Camino del Conde, Tucson, Arizona 85718

Johannes Kepler (1571–1630) picked up the investigation of the laws of optics from his lunar essay of 1600.[6] Both lunar and solar eclipses presented unexplained phenomena, such as unexpected shadow sizes, the red color of a total lunar eclipse, and the reportedly unusual light surrounding a total solar eclipse. Related issues of atmospheric refraction applied to all astronomical observations. Through most of 1603, Kepler paused his other work to focus on optical theory; the resulting manuscript, presented to the emperor on January 1, 1604, was published as Astronomiae Pars Optica (The Optical Part of Astronomy). In it, Kepler described the inverse-square law governing the intensity of light, reflection by flat and curved mirrors, and principles of pinhole cameras, as well as the astronomical implications of optics such as parallax and the apparent sizes of heavenly bodies. Astronomiae Pars Optica is generally recognized as the foundation of modern optics (though the law of refraction is conspicuously absent).[32]

Figure 4. Computed plots of intensity distribution at the observation plane of Fig. 1. The lens under test is assumed to have 3 waves of primary (Seidel) aberrations, and the grating is at the nominal focal plane of the lens. (a) defocus, (b) spherical, (c) astigmatism oriented at 45°,  (d) coma at 0°, (e) coma at 45°, (f) coma at 90°.

The earliest written record of magnification dates back to the 1st century CE, when Seneca the Younger, a tutor of Emperor Nero, wrote: "Letters, however small and indistinct, are seen enlarged and more clearly through a globe or glass filled with water."[45] Emperor Nero is also said to have watched the gladiatorial games using an emerald as a corrective lens.[46]

Figure 9. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the knife-edge test carried out with a laser beam. The lens under test (NA = 0.5, f = 6000l0) and the pupil relay lens (NA = 0.5) are assumed to be fixed in their respective positions, while the knife-edge moves along the optical axis. (The tip of the knife remains on the axis at all times.) The lens under test is assumed to have 3 waves of primary spherical aberration. In frames (a) to (d) the distance of the knife-edge from paraxial focus is Dz = -15l0, 0, +15l0, and +20l0, respectively. (Positive distances are in the direction of the marginal focus.)

Willebrord Snellius (1580–1626) found the mathematical law of refraction, now known as Snell's law, in 1621. Subsequently, René Descartes (1596–1650) showed, by using geometric construction and the law of refraction (also known as Descartes' law), that the angular radius of a rainbow is 42° (i.e. the angle subtended at the eye by the edge of the rainbow and the rainbow's centre is 42°).[33] He also independently discovered the law of reflection, and his essay on optics was the first published mention of this law.[34]

Choosing an appropriate grating. For best results the pitch of the grating should be chosen such that, as shown in Fig. 2, no more than two diffraction orders will overlap at any given point.     Figure 2. Diagram showing several diffracted orders in the far field of the grating of Fig. 1. When the grating?s period is chosen properly, each diffracted order (i.e., emergent cone of light) will overlap only with its nearest neighbors. Except for a lateral shift in position, the various orders are identical, carrying the amplitude and phase distribution of the beam as it appears at the exit pupil of the object under test.   To determine the appropriate grating period P, one needs to know the wavelength l0 of the beam used for testing, and the numerical aperture NA of the focused cone of light. (By definition, NA = sin q, where q is the half-angle subtended by the exit pupil of the lens at its focal point. If the lens under test is being used at full aperture, NA will also be equal to 0.5/f-number.) To avoid multiple overlaps among diffracted orders, the angle between adjacent orders must exceed the focused cone?s half-angle. Now, it is well-known in the theory of diffraction gratings that, at normal incidence, sin qn = nl0/P where n, an integer, is the order of diffraction, and qn is the corresponding deviation angle from the surface normal. Therefore, we arrive at the conclusion that P should be less than or equal to l0/NA. For example, assume that the lens under test has a numerical aperture NA = 0.5. Then, if the grating period is chosen to be 2l0, each diffracted order will deviate from the zero-order by 30°, making the +1st order just touch the -1st order in the far field. Figure 3 shows the computed intensity distribution at the observation plane of an aberration-free system in which the relay lens has the same numerical aperture as the lens under test (NA = 0.5).     Figure 3. Distribution of intensity at the observation plane of Fig. 1 in the absence of aberrations. The pupil relay lens is chosen to have the same numerical aperture as the object under test, thereby limiting the collected light to the zero-order beam and to those portions of the ±1st orders that overlap the 0th order.     Equality of numerical apertures means that only the zero-order diffracted beam will be fully transmitted to the observation plane. Of the ±1st order beams only those portions that overlap the zero order will reach the observation plane. The period of the grating in this example has been a little less than l0/NA, leaving a small gap between +1st and -1st orders.4 Absence of aberrations means that the phase distribution over the cross-sections of the various diffracted orders is uniform and, therefore, no interference fringes are to be expected. Ronchigrams for primary or Seidel aberrations. Figure 4 shows the computed patterns of intensity distribution at the observation plane of Fig. 1, corresponding to different types of primary (Seidel) aberrations of the lens.   Figure 4. Computed plots of intensity distribution at the observation plane of Fig. 1. The lens under test is assumed to have 3 waves of primary (Seidel) aberrations, and the grating is at the nominal focal plane of the lens. (a) defocus, (b) spherical, (c) astigmatism oriented at 45°,  (d) coma at 0°, (e) coma at 45°, (f) coma at 90°.     For these calculations we fixed the distance between the lens under test and the relay lens; we then placed the grating at the paraxial focus of the converging wavefront.4 The pattern in Fig. 4(a) was obtained when we assumed the presence of 3 waves of curvature (or defocus) at the exit pupil of the lens. Different amounts of defocus would create essentially the same pattern albeit with a different number of fringes. In Fig. 4(b) we observe the fringes arising from the presence of 3 waves of third order spherical aberration in the test system. The shapes of these fringes depend not only on the magnitude of the aberration, but also on the position of the grating relative to the focal plane. (We will have more to say about this point later.) Figure 4(c) shows the fringes that would arise when 3 waves of primary astigmatism are present. When the orientation of astigmatism changes, the fringes will remain straight lines, but their orientation within the observation plane will change accordingly. The last three frames in Fig. 4 represent the effects of third order coma. A change in the orientation of this aberration causes the interference pattern to change drastically. Figures 4(d)-(f) correspond to 3 waves of coma oriented at 0°, 45°, and 90°, respectively. Sliding the grating along the optical axis. A change in the position of the grating relative to the focal plane influences the observed fringe pattern. We limit our discussion to the case of spherical aberration, although similar analyses could be performed for other aberrations as well. Assuming 3 waves of spherical aberration as before, we obtain the patterns displayed in Fig. 5 as we slide the grating along the optical axis in the system of Fig. 1.4 Once again, we have taken the lens under test to have NA = 0.5 and f = 6000l0. The paraxial focus of the lens under test coincides with the front focal point of the relay lens, and the grating is shifted by different amounts Dz relative to this common focus. Frames (a)-(f) in Fig. 5 correspond to different values of Dz, starting at Dz = -10l0 in (a) and moving forward to Dz = +25l0 in (f). In the process, as the grating moves through paraxial focus and towards marginal focus, we observe a rich variety of patterns that aid us in determining the nature and the magnitude of the aberration.     Figure 5. Computed plots of intensity distribution at the observation plane of Fig. 1, showing the patterns obtained by sliding the grating along the optical axis. The lens under test (NA = 0.5, f = 6000l0) is assumed to have 3 waves of primary spherical aberration, and its paraxial focus is coincident with the focal point of the relay lens. The grating is moved along the optical axis by an amount Dz relative to the (common) focal plane; positive distances are towards the marginal focus. (a)  Dz= -10l0, (b)  Dz = 0, (c)  Dz = 10l0, (d) Dz = 15l0, (e)  Dz= 20l0, (f)  Dz= 25l0.   To be sure, the Ronchi test is not the only scheme used during fabrication and evaluation of optical systems; several other tests exist and their relative merits have been expounded in the literature.3 It is useful here to examine some of these alternative methods and to compare the resulting patterns (interferograms or otherwise) with those obtained with the Ronchi test. Testing by interfering with a reference plane-wave. Figure 6 shows the schematic diagram of a Mach-Zehnder interferometer, which is one among many that can be used to evaluate the aberrated wavefronts directly. In this system a coherent, monochromatic beam of light is sent through the lens under test, is collected and recollimated by a well-corrected lens, and is made to interfere with a reference beam that has been split off from the original, incident wavefront.   Figure 6. Schematic diagram of a Mach-Zehnder interferometer that might be set up for a direct measurement of wavefront aberrations. The pupil relay lens (itself free from aberrations) forms at the observation plane an image of the exit pupil of the lens under test. A fraction of the incident beam is diverted from its original path and sent to the observation plane by means of the various mirrors and beam-splitters. The observed fringes are characteristic of the aberrations present at the exit pupil of the lens under test. A small tilt of the mirror shown at the lower left side of the figure would introduce a linear phase shift on the reference beam. This tilt is generally useful in producing signature fringe patterns at the observation plane.   The flat mirror shown in the lower left side of the interferometer is mounted on a tip-tilt stage that allows the introduction of a small amount of tilt in the reference beam. Figure 7 shows the computed patterns of intensity distribution at the observation plane of the Mach-Zehnder interferometer corresponding to 3 waves of primary coma.4         Figure 7. Computed plots of intensity distribution at the observation plane of Fig. 6. The lens under test (NA = 0.5, f = 6000l0) is assumed to have 3 waves of primary coma, and its nominal focus is coincident with the focal point of the relay lens. The tilt angle y of the reference beam increases progressively from (a) to (f). (a) y = -0.1°, (b) y = 0°, (c) y  = 0.05°, (d)y = 0.07°, (e) y = 0.1°, (f)y  = 0.18°.   In obtaining the various frames of Fig. 7 we have fixed all the system parameters and only varied the tilt of the reference beam. Note that the characteristic fringes of coma in Fig. 7 are quite different from those of coma in the Ronchi test, shown in Figs. 4(d)-(f). Incidentally, the patterns of Fig. 7 show similarities with the Ronchigrams of spherical aberration displayed in Fig. 5. This is not a coincidence, and is rooted in the algebraic forms of the aberration function for third order coma (r3 cos f) and spherical aberration (r4 ), and also in the fact that a Ronchigram, being a kind of shearing interferogram (albeit with a large shear), is related to the derivative of the wavefront aberration function. Knife-edge and wire tests. A schematic diagram of the knife-edge method of testing optical systems is shown in Fig. 8. A geometric-optical interpretation of this test suffices for most practical purposes: the knife-edge blocks different groups of rays in its various positions along the optical axis, allowing the remaining rays to reach the observation plane.3 Another method of testing, known as the wire test, is quite similar to the knife-edge method, being obtained from it by the substitution of the knife-edge with a fine wire.3   Figure 8. In the knife-edge test a certain region in the vicinity of focus is blocked by a knife-edge; the nature and the magnitude of aberrations are then inferred from the resulting patterns of intensity distribution at the observation plane. (The knife-edge may be moved both along and perpendicular to the optical axis.) The wire test is similar to the knife-edge test except that a fine wire is used instead to block certain groups of rays.   Since the grating in the Ronchi test may be thought of as a series of parallel knife-edges or, more aptly, a series of parallel wires, it should not come as a surprise that similarities exist between Ronchigrams and the patterns observed in these other tests. In fact, early attempts at explaining the results of Ronchi?s method were based on geometrical optics, and considered the grating as a set of parallel wires whose shadows produced the observed patterns.5 We will not delve into these matters, but simply draw the reader?s attention to Figs. 9 and 10, where we show several computed patterns of intensity distribution for the knife-edge and wire tests, respectively.4     Figure 9. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the knife-edge test carried out with a laser beam. The lens under test (NA = 0.5, f = 6000l0) and the pupil relay lens (NA = 0.5) are assumed to be fixed in their respective positions, while the knife-edge moves along the optical axis. (The tip of the knife remains on the axis at all times.) The lens under test is assumed to have 3 waves of primary spherical aberration. In frames (a) to (d) the distance of the knife-edge from paraxial focus is Dz = -15l0, 0, +15l0, and +20l0, respectively. (Positive distances are in the direction of the marginal focus.)   The results of the simulated knife-edge test depicted in Fig. 9 assume a laser as the light source. Consequently, frames (a) and (b) of Fig. 9 exhibit several dark lines which, with a less coherent light source, would have been absent. The results of the simulated wire test shown in Fig. 10 assume an extended light source, since the small amount of spherical aberration present in the system under consideration would render the test useless with a wire that, fine as it may be, is still wider than the focused spot produced by a laser beam. Note the similarities between the patterns of Figs. 9 and 10 on the one hand, and those of Figs. 5(d)-(f) on the other.   Figure 10. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the wire test with an extended, quasi-monochromatic light source. The lens under test (NA = 0.5, f = 6000l0) has 3 waves of primary spherical aberration. The assumed wire diameter is 15l0, which is comparable to the size of the image of the extended light source, as measured in the vicinity of focus. In (a) the wire is centered on axis and is 25l0 away from paraxial focus (in the direction of the marginal focus). In (b) the wire is again centered on axis, but is 20l0 away from paraxial focus. In (c) the wire is shifted 0.5l0 off-axis while its distance from paraxial focus remains at 20l0.   Extensions of the Ronchi test. Several modifications and extensions of the Ronchi test have appeared over the years, and have helped to solve specific problems in testing of optical systems.3 As an example we mention the "double-frequency grating lateral shear interferometer" invented by James Wyant in the early 1970?s. The grating in this device has two slightly different frequencies, which give rise to two +1st order beams as well as two -1st order beams; the beams in each pair are slightly shifted relative to each other. Moreover, the (average) pitch of the grating is such that there is no overlap between the 0th, +1st, and -1st orders. Consequently, interference occurs between the two +1st order beams (and, likewise, between the two -1st order beams). One can thus obtain an arbitrarily small lateral shear of the wavefront under test, and use the results to achieve accurate quantitative measurements. A two-dimensional version of the double-frequency grating has also been employed to generate lateral wavefront shear simultaneously along the X and Y axes. (Remember that beam propagation is along Z and, therefore, X and Y are orthogonal axes in the plane of the grating.) In the absence of  a 2-D grating, one must rotate a 1-D grating by 90° to obtain  wavefront shear first along the X- and then along the Y-axis. Acknowledgment. I am grateful to Professor Roland Shack of the Optical Sciences Center for many illuminating discussions, and also for suggesting some of the examples presented in this article.   References V. Ronchi, "Le Frange di Combinazioni Nello Studio delle Superficie e dei Sistemi Ottici," Riv. Ottica Mecc. Precis. 2, 9 (1923). V. Ronchi, "Due Nuovi Metodi per lo Studio delle Superficie e dei Sistemi Ottici," Ann. Sc. Norm. Super. Pisa, 15 (1923). D. Malacara, ed., Optical Shop Testing, second edition, Wiley, New York, 1992. The computer simulations reported in this article were performed by DIFFRACT?, a product of MM Research, Inc., Tucson, Arizona. G. Toraldo di Francia, "Geometrical and interferential aspects of the Ronchi Test," in Optical Image Evaluation, National Bureau of Standards Circular 526, issued April 29, 1954. J. C. Wyant, "Double frequency grating lateral shear interferometer," Appl. Opt. 12, 2057 (1973). << Back to list of Articles Home | About MM Research, Inc. | Online Publications Diffract | SIM 3D_Max | Multilayer | Temprofile © Copyright 1987-2011, MM Research, Inc. 5748 N. Camino del Conde, Tucson, Arizona 85718

For from whatsoever distances fires can throw us their light and breathe their warm heat upon our limbs, they lose nothing of the body of their flames because of the interspaces, their fire is no whit shrunken to the sight.[4]

Abu 'Abd Allah Muhammad ibn Ma'udh, who lived in Al-Andalus during the second half of the 11th century, wrote a work on optics later translated into Latin as Liber de crepisculis, which was mistakenly attributed to Alhazen. This was a "short work containing an estimation of the angle of depression of the sun at the beginning of the morning twilight and at the end of the evening twilight, and an attempt to calculate on the basis of this and other data the height of the atmospheric moisture responsible for the refraction of the sun's rays." Through his experiments, he obtained the value of 18°, which comes close to the modern value.[22]

A two-dimensional version of the double-frequency grating has also been employed to generate lateral wavefront shear simultaneously along the X and Y axes. (Remember that beam propagation is along Z and, therefore, X and Y are orthogonal axes in the plane of the grating.) In the absence of  a 2-D grating, one must rotate a 1-D grating by 90° to obtain  wavefront shear first along the X- and then along the Y-axis. Acknowledgment. I am grateful to Professor Roland Shack of the Optical Sciences Center for many illuminating discussions, and also for suggesting some of the examples presented in this article.   References V. Ronchi, "Le Frange di Combinazioni Nello Studio delle Superficie e dei Sistemi Ottici," Riv. Ottica Mecc. Precis. 2, 9 (1923). V. Ronchi, "Due Nuovi Metodi per lo Studio delle Superficie e dei Sistemi Ottici," Ann. Sc. Norm. Super. Pisa, 15 (1923). D. Malacara, ed., Optical Shop Testing, second edition, Wiley, New York, 1992. The computer simulations reported in this article were performed by DIFFRACT?, a product of MM Research, Inc., Tucson, Arizona. G. Toraldo di Francia, "Geometrical and interferential aspects of the Ronchi Test," in Optical Image Evaluation, National Bureau of Standards Circular 526, issued April 29, 1954. J. C. Wyant, "Double frequency grating lateral shear interferometer," Appl. Opt. 12, 2057 (1973). << Back to list of Articles Home | About MM Research, Inc. | Online Publications Diffract | SIM 3D_Max | Multilayer | Temprofile © Copyright 1987-2011, MM Research, Inc. 5748 N. Camino del Conde, Tucson, Arizona 85718

The flat mirror shown in the lower left side of the interferometer is mounted on a tip-tilt stage that allows the introduction of a small amount of tilt in the reference beam. Figure 7 shows the computed patterns of intensity distribution at the observation plane of the Mach-Zehnder interferometer corresponding to 3 waves of primary coma.4         Figure 7. Computed plots of intensity distribution at the observation plane of Fig. 6. The lens under test (NA = 0.5, f = 6000l0) is assumed to have 3 waves of primary coma, and its nominal focus is coincident with the focal point of the relay lens. The tilt angle y of the reference beam increases progressively from (a) to (f). (a) y = -0.1°, (b) y = 0°, (c) y  = 0.05°, (d)y = 0.07°, (e) y = 0.1°, (f)y  = 0.18°.   In obtaining the various frames of Fig. 7 we have fixed all the system parameters and only varied the tilt of the reference beam. Note that the characteristic fringes of coma in Fig. 7 are quite different from those of coma in the Ronchi test, shown in Figs. 4(d)-(f). Incidentally, the patterns of Fig. 7 show similarities with the Ronchigrams of spherical aberration displayed in Fig. 5. This is not a coincidence, and is rooted in the algebraic forms of the aberration function for third order coma (r3 cos f) and spherical aberration (r4 ), and also in the fact that a Ronchigram, being a kind of shearing interferogram (albeit with a large shear), is related to the derivative of the wavefront aberration function. Knife-edge and wire tests. A schematic diagram of the knife-edge method of testing optical systems is shown in Fig. 8. A geometric-optical interpretation of this test suffices for most practical purposes: the knife-edge blocks different groups of rays in its various positions along the optical axis, allowing the remaining rays to reach the observation plane.3 Another method of testing, known as the wire test, is quite similar to the knife-edge method, being obtained from it by the substitution of the knife-edge with a fine wire.3   Figure 8. In the knife-edge test a certain region in the vicinity of focus is blocked by a knife-edge; the nature and the magnitude of aberrations are then inferred from the resulting patterns of intensity distribution at the observation plane. (The knife-edge may be moved both along and perpendicular to the optical axis.) The wire test is similar to the knife-edge test except that a fine wire is used instead to block certain groups of rays.   Since the grating in the Ronchi test may be thought of as a series of parallel knife-edges or, more aptly, a series of parallel wires, it should not come as a surprise that similarities exist between Ronchigrams and the patterns observed in these other tests. In fact, early attempts at explaining the results of Ronchi?s method were based on geometrical optics, and considered the grating as a set of parallel wires whose shadows produced the observed patterns.5 We will not delve into these matters, but simply draw the reader?s attention to Figs. 9 and 10, where we show several computed patterns of intensity distribution for the knife-edge and wire tests, respectively.4     Figure 9. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the knife-edge test carried out with a laser beam. The lens under test (NA = 0.5, f = 6000l0) and the pupil relay lens (NA = 0.5) are assumed to be fixed in their respective positions, while the knife-edge moves along the optical axis. (The tip of the knife remains on the axis at all times.) The lens under test is assumed to have 3 waves of primary spherical aberration. In frames (a) to (d) the distance of the knife-edge from paraxial focus is Dz = -15l0, 0, +15l0, and +20l0, respectively. (Positive distances are in the direction of the marginal focus.)   The results of the simulated knife-edge test depicted in Fig. 9 assume a laser as the light source. Consequently, frames (a) and (b) of Fig. 9 exhibit several dark lines which, with a less coherent light source, would have been absent. The results of the simulated wire test shown in Fig. 10 assume an extended light source, since the small amount of spherical aberration present in the system under consideration would render the test useless with a wire that, fine as it may be, is still wider than the focused spot produced by a laser beam. Note the similarities between the patterns of Figs. 9 and 10 on the one hand, and those of Figs. 5(d)-(f) on the other.   Figure 10. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the wire test with an extended, quasi-monochromatic light source. The lens under test (NA = 0.5, f = 6000l0) has 3 waves of primary spherical aberration. The assumed wire diameter is 15l0, which is comparable to the size of the image of the extended light source, as measured in the vicinity of focus. In (a) the wire is centered on axis and is 25l0 away from paraxial focus (in the direction of the marginal focus). In (b) the wire is again centered on axis, but is 20l0 away from paraxial focus. In (c) the wire is shifted 0.5l0 off-axis while its distance from paraxial focus remains at 20l0.   Extensions of the Ronchi test. Several modifications and extensions of the Ronchi test have appeared over the years, and have helped to solve specific problems in testing of optical systems.3 As an example we mention the "double-frequency grating lateral shear interferometer" invented by James Wyant in the early 1970?s. The grating in this device has two slightly different frequencies, which give rise to two +1st order beams as well as two -1st order beams; the beams in each pair are slightly shifted relative to each other. Moreover, the (average) pitch of the grating is such that there is no overlap between the 0th, +1st, and -1st orders. Consequently, interference occurs between the two +1st order beams (and, likewise, between the two -1st order beams). One can thus obtain an arbitrarily small lateral shear of the wavefront under test, and use the results to achieve accurate quantitative measurements. A two-dimensional version of the double-frequency grating has also been employed to generate lateral wavefront shear simultaneously along the X and Y axes. (Remember that beam propagation is along Z and, therefore, X and Y are orthogonal axes in the plane of the grating.) In the absence of  a 2-D grating, one must rotate a 1-D grating by 90° to obtain  wavefront shear first along the X- and then along the Y-axis. Acknowledgment. I am grateful to Professor Roland Shack of the Optical Sciences Center for many illuminating discussions, and also for suggesting some of the examples presented in this article.   References V. Ronchi, "Le Frange di Combinazioni Nello Studio delle Superficie e dei Sistemi Ottici," Riv. Ottica Mecc. Precis. 2, 9 (1923). V. Ronchi, "Due Nuovi Metodi per lo Studio delle Superficie e dei Sistemi Ottici," Ann. Sc. Norm. Super. Pisa, 15 (1923). D. Malacara, ed., Optical Shop Testing, second edition, Wiley, New York, 1992. The computer simulations reported in this article were performed by DIFFRACT?, a product of MM Research, Inc., Tucson, Arizona. G. Toraldo di Francia, "Geometrical and interferential aspects of the Ronchi Test," in Optical Image Evaluation, National Bureau of Standards Circular 526, issued April 29, 1954. J. C. Wyant, "Double frequency grating lateral shear interferometer," Appl. Opt. 12, 2057 (1973). << Back to list of Articles Home | About MM Research, Inc. | Online Publications Diffract | SIM 3D_Max | Multilayer | Temprofile © Copyright 1987-2011, MM Research, Inc. 5748 N. Camino del Conde, Tucson, Arizona 85718

Acknowledgment. I am grateful to Professor Roland Shack of the Optical Sciences Center for many illuminating discussions, and also for suggesting some of the examples presented in this article.   References V. Ronchi, "Le Frange di Combinazioni Nello Studio delle Superficie e dei Sistemi Ottici," Riv. Ottica Mecc. Precis. 2, 9 (1923). V. Ronchi, "Due Nuovi Metodi per lo Studio delle Superficie e dei Sistemi Ottici," Ann. Sc. Norm. Super. Pisa, 15 (1923). D. Malacara, ed., Optical Shop Testing, second edition, Wiley, New York, 1992. The computer simulations reported in this article were performed by DIFFRACT?, a product of MM Research, Inc., Tucson, Arizona. G. Toraldo di Francia, "Geometrical and interferential aspects of the Ronchi Test," in Optical Image Evaluation, National Bureau of Standards Circular 526, issued April 29, 1954. J. C. Wyant, "Double frequency grating lateral shear interferometer," Appl. Opt. 12, 2057 (1973). << Back to list of Articles Home | About MM Research, Inc. | Online Publications Diffract | SIM 3D_Max | Multilayer | Temprofile © Copyright 1987-2011, MM Research, Inc. 5748 N. Camino del Conde, Tucson, Arizona 85718

The earliest known working telescopes were the refracting telescopes that appeared in the Netherlands in 1608. Their inventor is unknown: Hans Lippershey applied for the first patent that year followed by a patent application by Jacob Metius of Alkmaar two weeks later (neither was granted since examples of the device seemed to be numerous at the time). Galileo greatly improved upon these designs the following year. Isaac Newton is credited with constructing the first functional reflecting telescope in 1668, his Newtonian reflector.[51]

Figure 6. Schematic diagram of a Mach-Zehnder interferometer that might be set up for a direct measurement of wavefront aberrations. The pupil relay lens (itself free from aberrations) forms at the observation plane an image of the exit pupil of the lens under test. A fraction of the incident beam is diverted from its original path and sent to the observation plane by means of the various mirrors and beam-splitters. The observed fringes are characteristic of the aberrations present at the exit pupil of the lens under test. A small tilt of the mirror shown at the lower left side of the figure would introduce a linear phase shift on the reference beam. This tilt is generally useful in producing signature fringe patterns at the observation plane.

Ronchitest

The results of the simulated knife-edge test depicted in Fig. 9 assume a laser as the light source. Consequently, frames (a) and (b) of Fig. 9 exhibit several dark lines which, with a less coherent light source, would have been absent. The results of the simulated wire test shown in Fig. 10 assume an extended light source, since the small amount of spherical aberration present in the system under consideration would render the test useless with a wire that, fine as it may be, is still wider than the focused spot produced by a laser beam. Note the similarities between the patterns of Figs. 9 and 10 on the one hand, and those of Figs. 5(d)-(f) on the other.   Figure 10. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the wire test with an extended, quasi-monochromatic light source. The lens under test (NA = 0.5, f = 6000l0) has 3 waves of primary spherical aberration. The assumed wire diameter is 15l0, which is comparable to the size of the image of the extended light source, as measured in the vicinity of focus. In (a) the wire is centered on axis and is 25l0 away from paraxial focus (in the direction of the marginal focus). In (b) the wire is again centered on axis, but is 20l0 away from paraxial focus. In (c) the wire is shifted 0.5l0 off-axis while its distance from paraxial focus remains at 20l0.   Extensions of the Ronchi test. Several modifications and extensions of the Ronchi test have appeared over the years, and have helped to solve specific problems in testing of optical systems.3 As an example we mention the "double-frequency grating lateral shear interferometer" invented by James Wyant in the early 1970?s. The grating in this device has two slightly different frequencies, which give rise to two +1st order beams as well as two -1st order beams; the beams in each pair are slightly shifted relative to each other. Moreover, the (average) pitch of the grating is such that there is no overlap between the 0th, +1st, and -1st orders. Consequently, interference occurs between the two +1st order beams (and, likewise, between the two -1st order beams). One can thus obtain an arbitrarily small lateral shear of the wavefront under test, and use the results to achieve accurate quantitative measurements. A two-dimensional version of the double-frequency grating has also been employed to generate lateral wavefront shear simultaneously along the X and Y axes. (Remember that beam propagation is along Z and, therefore, X and Y are orthogonal axes in the plane of the grating.) In the absence of  a 2-D grating, one must rotate a 1-D grating by 90° to obtain  wavefront shear first along the X- and then along the Y-axis. Acknowledgment. I am grateful to Professor Roland Shack of the Optical Sciences Center for many illuminating discussions, and also for suggesting some of the examples presented in this article.   References V. Ronchi, "Le Frange di Combinazioni Nello Studio delle Superficie e dei Sistemi Ottici," Riv. Ottica Mecc. Precis. 2, 9 (1923). V. Ronchi, "Due Nuovi Metodi per lo Studio delle Superficie e dei Sistemi Ottici," Ann. Sc. Norm. Super. Pisa, 15 (1923). D. Malacara, ed., Optical Shop Testing, second edition, Wiley, New York, 1992. The computer simulations reported in this article were performed by DIFFRACT?, a product of MM Research, Inc., Tucson, Arizona. G. Toraldo di Francia, "Geometrical and interferential aspects of the Ronchi Test," in Optical Image Evaluation, National Bureau of Standards Circular 526, issued April 29, 1954. J. C. Wyant, "Double frequency grating lateral shear interferometer," Appl. Opt. 12, 2057 (1973). << Back to list of Articles Home | About MM Research, Inc. | Online Publications Diffract | SIM 3D_Max | Multilayer | Temprofile © Copyright 1987-2011, MM Research, Inc. 5748 N. Camino del Conde, Tucson, Arizona 85718

In the fifth century BCE, Empedocles postulated that everything was composed of four elements; fire, air, earth and water. He believed that Aphrodite made the human eye out of the four elements and that she lit the fire in the eye which shone out from the eye making sight possible. If this were true, then one could see during the night just as well as during the day, so Empedocles postulated an interaction between rays from the eyes and rays from a source such as the sun. He stated that light has a finite speed.[2]

Galileo Galilei (also sometimes cited as a compound microscope inventor) seems to have found after 1609 that he could close focus his telescope to view small objects and, after seeing a compound microscope built by Drebbel exhibited in Rome in 1624, built his own improved version.[59][60][61] The name microscope was coined by Giovanni Faber, who gave that name to Galileo Galilei's compound microscope in 1625.[62]

The effects of diffraction of light were carefully observed and characterized by Francesco Maria Grimaldi, who also coined the term diffraction, from the Latin diffringere, 'to break into pieces', referring to light breaking up into different directions. The results of Grimaldi's observations were published posthumously in 1665.[36][37] Isaac Newton studied these effects and attributed them to inflexion of light rays. James Gregory (1638–1675) observed the diffraction patterns caused by a bird feather, which was effectively the first diffraction grating. In 1803 Thomas Young did his famous experiment observing interference from two closely spaced slits in his double slit interferometer. Explaining his results by interference of the waves emanating from the two different slits, he deduced that light must propagate as waves. Augustin-Jean Fresnel did more definitive studies and calculations of diffraction, published in 1815 and 1818, and thereby gave great support to the wave theory of light that had been advanced by Christiaan Huygens and reinvigorated by Young, against Newton's particle theory.

Figure 1. A beam of coherent, quasi-monochromatic light is brought to focus by an optical system that is undergoing tests to determine its aberrations. A diffraction grating, placed perpendicular to the optical axis in the vicinity of focus, breaks up the incident beam into several diffraction orders. The diffracted orders propagate, independently of each other, and are collected by a pupil relay lens, which forms an image of the exit pupil of the object under test at the observation plane.

Ronchiscreen

Hero demonstrated the equality of the angle of incidence and reflection on the grounds that this is the shortest path from the object to the observer. On this basis, he was able to define the fixed relation between an object and its image in a plane mirror. Specifically, the image appears to be as far behind the mirror as the object really is in front of the mirror.

Christiaan Huygens (1629–1695) wrote several works in the area of optics. These included the Opera reliqua (also known as Christiani Hugenii Zuilichemii, dum viveret Zelhemii toparchae, opuscula posthuma) and the Traité de la lumière.

As laser science needed good theoretical foundations, and also because research into these soon proved very fruitful, interest in quantum optics rose. Following the work of Dirac in quantum field theory, George Sudarshan, Roy J. Glauber, and Leonard Mandel applied quantum theory to the electromagnetic field in the 1950s and 1960s to gain a more detailed understanding of photodetection and the statistics of light (see degree of coherence). This led to the introduction of the coherent state as a quantum description of laser light and the realization that some states of light could not be described with classical waves. In 1977, Kimble et al. demonstrated the first source of light which required a quantum description: a single atom that emitted one photon at a time. Another quantum state of light with certain advantages over any classical state, squeezed light, was soon proposed. At the same time, development of short and ultrashort laser pulses—created by Q-switching and mode-locking techniques—opened the way to the study of unimaginably fast ("ultrafast") processes. Applications for solid state research (e.g. Raman spectroscopy) were found, and mechanical forces of light on matter were studied. The latter led to levitating and positioning clouds of atoms or even small biological samples in an optical trap or optical tweezers by laser beam. This, along with Doppler cooling was the crucial technology needed to achieve the celebrated Bose–Einstein condensation.

The above description of the Ronchi test relied on its modern interpretation based on our current understanding of physical optics and the theory of diffraction gratings. Historically, however, the gratings used in the early days were quite coarse, and the results obtained with them required no more than a simple geometric optical theory for their interpretation. Typically, one would place the eye at the focus of the lens and hold a grating (e.g., a wire grid) in front of the eye, moving the grating in and out until a clear pattern became visible. At this point the beam would be illuminating several of the wires simultaneously. By looking through the grating and observing the shadows that the wires cast on the exit pupil, one could determine the type of aberration present in the system. The coarseness of the grating, of course, caused several of the diffracted orders (as we understand them today) to overlap each other, thus resulting in reduced contrast and smearing of the patterns near the boundaries. These problems were eventually overcome when finer gratings became available and the diffraction theory of the Ronchi test was better understood. Choosing an appropriate grating. For best results the pitch of the grating should be chosen such that, as shown in Fig. 2, no more than two diffraction orders will overlap at any given point.     Figure 2. Diagram showing several diffracted orders in the far field of the grating of Fig. 1. When the grating?s period is chosen properly, each diffracted order (i.e., emergent cone of light) will overlap only with its nearest neighbors. Except for a lateral shift in position, the various orders are identical, carrying the amplitude and phase distribution of the beam as it appears at the exit pupil of the object under test.   To determine the appropriate grating period P, one needs to know the wavelength l0 of the beam used for testing, and the numerical aperture NA of the focused cone of light. (By definition, NA = sin q, where q is the half-angle subtended by the exit pupil of the lens at its focal point. If the lens under test is being used at full aperture, NA will also be equal to 0.5/f-number.) To avoid multiple overlaps among diffracted orders, the angle between adjacent orders must exceed the focused cone?s half-angle. Now, it is well-known in the theory of diffraction gratings that, at normal incidence, sin qn = nl0/P where n, an integer, is the order of diffraction, and qn is the corresponding deviation angle from the surface normal. Therefore, we arrive at the conclusion that P should be less than or equal to l0/NA. For example, assume that the lens under test has a numerical aperture NA = 0.5. Then, if the grating period is chosen to be 2l0, each diffracted order will deviate from the zero-order by 30°, making the +1st order just touch the -1st order in the far field. Figure 3 shows the computed intensity distribution at the observation plane of an aberration-free system in which the relay lens has the same numerical aperture as the lens under test (NA = 0.5).     Figure 3. Distribution of intensity at the observation plane of Fig. 1 in the absence of aberrations. The pupil relay lens is chosen to have the same numerical aperture as the object under test, thereby limiting the collected light to the zero-order beam and to those portions of the ±1st orders that overlap the 0th order.     Equality of numerical apertures means that only the zero-order diffracted beam will be fully transmitted to the observation plane. Of the ±1st order beams only those portions that overlap the zero order will reach the observation plane. The period of the grating in this example has been a little less than l0/NA, leaving a small gap between +1st and -1st orders.4 Absence of aberrations means that the phase distribution over the cross-sections of the various diffracted orders is uniform and, therefore, no interference fringes are to be expected. Ronchigrams for primary or Seidel aberrations. Figure 4 shows the computed patterns of intensity distribution at the observation plane of Fig. 1, corresponding to different types of primary (Seidel) aberrations of the lens.   Figure 4. Computed plots of intensity distribution at the observation plane of Fig. 1. The lens under test is assumed to have 3 waves of primary (Seidel) aberrations, and the grating is at the nominal focal plane of the lens. (a) defocus, (b) spherical, (c) astigmatism oriented at 45°,  (d) coma at 0°, (e) coma at 45°, (f) coma at 90°.     For these calculations we fixed the distance between the lens under test and the relay lens; we then placed the grating at the paraxial focus of the converging wavefront.4 The pattern in Fig. 4(a) was obtained when we assumed the presence of 3 waves of curvature (or defocus) at the exit pupil of the lens. Different amounts of defocus would create essentially the same pattern albeit with a different number of fringes. In Fig. 4(b) we observe the fringes arising from the presence of 3 waves of third order spherical aberration in the test system. The shapes of these fringes depend not only on the magnitude of the aberration, but also on the position of the grating relative to the focal plane. (We will have more to say about this point later.) Figure 4(c) shows the fringes that would arise when 3 waves of primary astigmatism are present. When the orientation of astigmatism changes, the fringes will remain straight lines, but their orientation within the observation plane will change accordingly. The last three frames in Fig. 4 represent the effects of third order coma. A change in the orientation of this aberration causes the interference pattern to change drastically. Figures 4(d)-(f) correspond to 3 waves of coma oriented at 0°, 45°, and 90°, respectively. Sliding the grating along the optical axis. A change in the position of the grating relative to the focal plane influences the observed fringe pattern. We limit our discussion to the case of spherical aberration, although similar analyses could be performed for other aberrations as well. Assuming 3 waves of spherical aberration as before, we obtain the patterns displayed in Fig. 5 as we slide the grating along the optical axis in the system of Fig. 1.4 Once again, we have taken the lens under test to have NA = 0.5 and f = 6000l0. The paraxial focus of the lens under test coincides with the front focal point of the relay lens, and the grating is shifted by different amounts Dz relative to this common focus. Frames (a)-(f) in Fig. 5 correspond to different values of Dz, starting at Dz = -10l0 in (a) and moving forward to Dz = +25l0 in (f). In the process, as the grating moves through paraxial focus and towards marginal focus, we observe a rich variety of patterns that aid us in determining the nature and the magnitude of the aberration.     Figure 5. Computed plots of intensity distribution at the observation plane of Fig. 1, showing the patterns obtained by sliding the grating along the optical axis. The lens under test (NA = 0.5, f = 6000l0) is assumed to have 3 waves of primary spherical aberration, and its paraxial focus is coincident with the focal point of the relay lens. The grating is moved along the optical axis by an amount Dz relative to the (common) focal plane; positive distances are towards the marginal focus. (a)  Dz= -10l0, (b)  Dz = 0, (c)  Dz = 10l0, (d) Dz = 15l0, (e)  Dz= 20l0, (f)  Dz= 25l0.   To be sure, the Ronchi test is not the only scheme used during fabrication and evaluation of optical systems; several other tests exist and their relative merits have been expounded in the literature.3 It is useful here to examine some of these alternative methods and to compare the resulting patterns (interferograms or otherwise) with those obtained with the Ronchi test. Testing by interfering with a reference plane-wave. Figure 6 shows the schematic diagram of a Mach-Zehnder interferometer, which is one among many that can be used to evaluate the aberrated wavefronts directly. In this system a coherent, monochromatic beam of light is sent through the lens under test, is collected and recollimated by a well-corrected lens, and is made to interfere with a reference beam that has been split off from the original, incident wavefront.   Figure 6. Schematic diagram of a Mach-Zehnder interferometer that might be set up for a direct measurement of wavefront aberrations. The pupil relay lens (itself free from aberrations) forms at the observation plane an image of the exit pupil of the lens under test. A fraction of the incident beam is diverted from its original path and sent to the observation plane by means of the various mirrors and beam-splitters. The observed fringes are characteristic of the aberrations present at the exit pupil of the lens under test. A small tilt of the mirror shown at the lower left side of the figure would introduce a linear phase shift on the reference beam. This tilt is generally useful in producing signature fringe patterns at the observation plane.   The flat mirror shown in the lower left side of the interferometer is mounted on a tip-tilt stage that allows the introduction of a small amount of tilt in the reference beam. Figure 7 shows the computed patterns of intensity distribution at the observation plane of the Mach-Zehnder interferometer corresponding to 3 waves of primary coma.4         Figure 7. Computed plots of intensity distribution at the observation plane of Fig. 6. The lens under test (NA = 0.5, f = 6000l0) is assumed to have 3 waves of primary coma, and its nominal focus is coincident with the focal point of the relay lens. The tilt angle y of the reference beam increases progressively from (a) to (f). (a) y = -0.1°, (b) y = 0°, (c) y  = 0.05°, (d)y = 0.07°, (e) y = 0.1°, (f)y  = 0.18°.   In obtaining the various frames of Fig. 7 we have fixed all the system parameters and only varied the tilt of the reference beam. Note that the characteristic fringes of coma in Fig. 7 are quite different from those of coma in the Ronchi test, shown in Figs. 4(d)-(f). Incidentally, the patterns of Fig. 7 show similarities with the Ronchigrams of spherical aberration displayed in Fig. 5. This is not a coincidence, and is rooted in the algebraic forms of the aberration function for third order coma (r3 cos f) and spherical aberration (r4 ), and also in the fact that a Ronchigram, being a kind of shearing interferogram (albeit with a large shear), is related to the derivative of the wavefront aberration function. Knife-edge and wire tests. A schematic diagram of the knife-edge method of testing optical systems is shown in Fig. 8. A geometric-optical interpretation of this test suffices for most practical purposes: the knife-edge blocks different groups of rays in its various positions along the optical axis, allowing the remaining rays to reach the observation plane.3 Another method of testing, known as the wire test, is quite similar to the knife-edge method, being obtained from it by the substitution of the knife-edge with a fine wire.3   Figure 8. In the knife-edge test a certain region in the vicinity of focus is blocked by a knife-edge; the nature and the magnitude of aberrations are then inferred from the resulting patterns of intensity distribution at the observation plane. (The knife-edge may be moved both along and perpendicular to the optical axis.) The wire test is similar to the knife-edge test except that a fine wire is used instead to block certain groups of rays.   Since the grating in the Ronchi test may be thought of as a series of parallel knife-edges or, more aptly, a series of parallel wires, it should not come as a surprise that similarities exist between Ronchigrams and the patterns observed in these other tests. In fact, early attempts at explaining the results of Ronchi?s method were based on geometrical optics, and considered the grating as a set of parallel wires whose shadows produced the observed patterns.5 We will not delve into these matters, but simply draw the reader?s attention to Figs. 9 and 10, where we show several computed patterns of intensity distribution for the knife-edge and wire tests, respectively.4     Figure 9. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the knife-edge test carried out with a laser beam. The lens under test (NA = 0.5, f = 6000l0) and the pupil relay lens (NA = 0.5) are assumed to be fixed in their respective positions, while the knife-edge moves along the optical axis. (The tip of the knife remains on the axis at all times.) The lens under test is assumed to have 3 waves of primary spherical aberration. In frames (a) to (d) the distance of the knife-edge from paraxial focus is Dz = -15l0, 0, +15l0, and +20l0, respectively. (Positive distances are in the direction of the marginal focus.)   The results of the simulated knife-edge test depicted in Fig. 9 assume a laser as the light source. Consequently, frames (a) and (b) of Fig. 9 exhibit several dark lines which, with a less coherent light source, would have been absent. The results of the simulated wire test shown in Fig. 10 assume an extended light source, since the small amount of spherical aberration present in the system under consideration would render the test useless with a wire that, fine as it may be, is still wider than the focused spot produced by a laser beam. Note the similarities between the patterns of Figs. 9 and 10 on the one hand, and those of Figs. 5(d)-(f) on the other.   Figure 10. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the wire test with an extended, quasi-monochromatic light source. The lens under test (NA = 0.5, f = 6000l0) has 3 waves of primary spherical aberration. The assumed wire diameter is 15l0, which is comparable to the size of the image of the extended light source, as measured in the vicinity of focus. In (a) the wire is centered on axis and is 25l0 away from paraxial focus (in the direction of the marginal focus). In (b) the wire is again centered on axis, but is 20l0 away from paraxial focus. In (c) the wire is shifted 0.5l0 off-axis while its distance from paraxial focus remains at 20l0.   Extensions of the Ronchi test. Several modifications and extensions of the Ronchi test have appeared over the years, and have helped to solve specific problems in testing of optical systems.3 As an example we mention the "double-frequency grating lateral shear interferometer" invented by James Wyant in the early 1970?s. The grating in this device has two slightly different frequencies, which give rise to two +1st order beams as well as two -1st order beams; the beams in each pair are slightly shifted relative to each other. Moreover, the (average) pitch of the grating is such that there is no overlap between the 0th, +1st, and -1st orders. Consequently, interference occurs between the two +1st order beams (and, likewise, between the two -1st order beams). One can thus obtain an arbitrarily small lateral shear of the wavefront under test, and use the results to achieve accurate quantitative measurements. A two-dimensional version of the double-frequency grating has also been employed to generate lateral wavefront shear simultaneously along the X and Y axes. (Remember that beam propagation is along Z and, therefore, X and Y are orthogonal axes in the plane of the grating.) In the absence of  a 2-D grating, one must rotate a 1-D grating by 90° to obtain  wavefront shear first along the X- and then along the Y-axis. Acknowledgment. I am grateful to Professor Roland Shack of the Optical Sciences Center for many illuminating discussions, and also for suggesting some of the examples presented in this article.   References V. Ronchi, "Le Frange di Combinazioni Nello Studio delle Superficie e dei Sistemi Ottici," Riv. Ottica Mecc. Precis. 2, 9 (1923). V. Ronchi, "Due Nuovi Metodi per lo Studio delle Superficie e dei Sistemi Ottici," Ann. Sc. Norm. Super. Pisa, 15 (1923). D. Malacara, ed., Optical Shop Testing, second edition, Wiley, New York, 1992. The computer simulations reported in this article were performed by DIFFRACT?, a product of MM Research, Inc., Tucson, Arizona. G. Toraldo di Francia, "Geometrical and interferential aspects of the Ronchi Test," in Optical Image Evaluation, National Bureau of Standards Circular 526, issued April 29, 1954. J. C. Wyant, "Double frequency grating lateral shear interferometer," Appl. Opt. 12, 2057 (1973). << Back to list of Articles Home | About MM Research, Inc. | Online Publications Diffract | SIM 3D_Max | Multilayer | Temprofile © Copyright 1987-2011, MM Research, Inc. 5748 N. Camino del Conde, Tucson, Arizona 85718

Testing by interfering with a reference plane-wave. Figure 6 shows the schematic diagram of a Mach-Zehnder interferometer, which is one among many that can be used to evaluate the aberrated wavefronts directly. In this system a coherent, monochromatic beam of light is sent through the lens under test, is collected and recollimated by a well-corrected lens, and is made to interfere with a reference beam that has been split off from the original, incident wavefront.   Figure 6. Schematic diagram of a Mach-Zehnder interferometer that might be set up for a direct measurement of wavefront aberrations. The pupil relay lens (itself free from aberrations) forms at the observation plane an image of the exit pupil of the lens under test. A fraction of the incident beam is diverted from its original path and sent to the observation plane by means of the various mirrors and beam-splitters. The observed fringes are characteristic of the aberrations present at the exit pupil of the lens under test. A small tilt of the mirror shown at the lower left side of the figure would introduce a linear phase shift on the reference beam. This tilt is generally useful in producing signature fringe patterns at the observation plane.   The flat mirror shown in the lower left side of the interferometer is mounted on a tip-tilt stage that allows the introduction of a small amount of tilt in the reference beam. Figure 7 shows the computed patterns of intensity distribution at the observation plane of the Mach-Zehnder interferometer corresponding to 3 waves of primary coma.4         Figure 7. Computed plots of intensity distribution at the observation plane of Fig. 6. The lens under test (NA = 0.5, f = 6000l0) is assumed to have 3 waves of primary coma, and its nominal focus is coincident with the focal point of the relay lens. The tilt angle y of the reference beam increases progressively from (a) to (f). (a) y = -0.1°, (b) y = 0°, (c) y  = 0.05°, (d)y = 0.07°, (e) y = 0.1°, (f)y  = 0.18°.   In obtaining the various frames of Fig. 7 we have fixed all the system parameters and only varied the tilt of the reference beam. Note that the characteristic fringes of coma in Fig. 7 are quite different from those of coma in the Ronchi test, shown in Figs. 4(d)-(f). Incidentally, the patterns of Fig. 7 show similarities with the Ronchigrams of spherical aberration displayed in Fig. 5. This is not a coincidence, and is rooted in the algebraic forms of the aberration function for third order coma (r3 cos f) and spherical aberration (r4 ), and also in the fact that a Ronchigram, being a kind of shearing interferogram (albeit with a large shear), is related to the derivative of the wavefront aberration function. Knife-edge and wire tests. A schematic diagram of the knife-edge method of testing optical systems is shown in Fig. 8. A geometric-optical interpretation of this test suffices for most practical purposes: the knife-edge blocks different groups of rays in its various positions along the optical axis, allowing the remaining rays to reach the observation plane.3 Another method of testing, known as the wire test, is quite similar to the knife-edge method, being obtained from it by the substitution of the knife-edge with a fine wire.3   Figure 8. In the knife-edge test a certain region in the vicinity of focus is blocked by a knife-edge; the nature and the magnitude of aberrations are then inferred from the resulting patterns of intensity distribution at the observation plane. (The knife-edge may be moved both along and perpendicular to the optical axis.) The wire test is similar to the knife-edge test except that a fine wire is used instead to block certain groups of rays.   Since the grating in the Ronchi test may be thought of as a series of parallel knife-edges or, more aptly, a series of parallel wires, it should not come as a surprise that similarities exist between Ronchigrams and the patterns observed in these other tests. In fact, early attempts at explaining the results of Ronchi?s method were based on geometrical optics, and considered the grating as a set of parallel wires whose shadows produced the observed patterns.5 We will not delve into these matters, but simply draw the reader?s attention to Figs. 9 and 10, where we show several computed patterns of intensity distribution for the knife-edge and wire tests, respectively.4     Figure 9. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the knife-edge test carried out with a laser beam. The lens under test (NA = 0.5, f = 6000l0) and the pupil relay lens (NA = 0.5) are assumed to be fixed in their respective positions, while the knife-edge moves along the optical axis. (The tip of the knife remains on the axis at all times.) The lens under test is assumed to have 3 waves of primary spherical aberration. In frames (a) to (d) the distance of the knife-edge from paraxial focus is Dz = -15l0, 0, +15l0, and +20l0, respectively. (Positive distances are in the direction of the marginal focus.)   The results of the simulated knife-edge test depicted in Fig. 9 assume a laser as the light source. Consequently, frames (a) and (b) of Fig. 9 exhibit several dark lines which, with a less coherent light source, would have been absent. The results of the simulated wire test shown in Fig. 10 assume an extended light source, since the small amount of spherical aberration present in the system under consideration would render the test useless with a wire that, fine as it may be, is still wider than the focused spot produced by a laser beam. Note the similarities between the patterns of Figs. 9 and 10 on the one hand, and those of Figs. 5(d)-(f) on the other.   Figure 10. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the wire test with an extended, quasi-monochromatic light source. The lens under test (NA = 0.5, f = 6000l0) has 3 waves of primary spherical aberration. The assumed wire diameter is 15l0, which is comparable to the size of the image of the extended light source, as measured in the vicinity of focus. In (a) the wire is centered on axis and is 25l0 away from paraxial focus (in the direction of the marginal focus). In (b) the wire is again centered on axis, but is 20l0 away from paraxial focus. In (c) the wire is shifted 0.5l0 off-axis while its distance from paraxial focus remains at 20l0.   Extensions of the Ronchi test. Several modifications and extensions of the Ronchi test have appeared over the years, and have helped to solve specific problems in testing of optical systems.3 As an example we mention the "double-frequency grating lateral shear interferometer" invented by James Wyant in the early 1970?s. The grating in this device has two slightly different frequencies, which give rise to two +1st order beams as well as two -1st order beams; the beams in each pair are slightly shifted relative to each other. Moreover, the (average) pitch of the grating is such that there is no overlap between the 0th, +1st, and -1st orders. Consequently, interference occurs between the two +1st order beams (and, likewise, between the two -1st order beams). One can thus obtain an arbitrarily small lateral shear of the wavefront under test, and use the results to achieve accurate quantitative measurements. A two-dimensional version of the double-frequency grating has also been employed to generate lateral wavefront shear simultaneously along the X and Y axes. (Remember that beam propagation is along Z and, therefore, X and Y are orthogonal axes in the plane of the grating.) In the absence of  a 2-D grating, one must rotate a 1-D grating by 90° to obtain  wavefront shear first along the X- and then along the Y-axis. Acknowledgment. I am grateful to Professor Roland Shack of the Optical Sciences Center for many illuminating discussions, and also for suggesting some of the examples presented in this article.   References V. Ronchi, "Le Frange di Combinazioni Nello Studio delle Superficie e dei Sistemi Ottici," Riv. Ottica Mecc. Precis. 2, 9 (1923). V. Ronchi, "Due Nuovi Metodi per lo Studio delle Superficie e dei Sistemi Ottici," Ann. Sc. Norm. Super. Pisa, 15 (1923). D. Malacara, ed., Optical Shop Testing, second edition, Wiley, New York, 1992. The computer simulations reported in this article were performed by DIFFRACT?, a product of MM Research, Inc., Tucson, Arizona. G. Toraldo di Francia, "Geometrical and interferential aspects of the Ronchi Test," in Optical Image Evaluation, National Bureau of Standards Circular 526, issued April 29, 1954. J. C. Wyant, "Double frequency grating lateral shear interferometer," Appl. Opt. 12, 2057 (1973). << Back to list of Articles Home | About MM Research, Inc. | Online Publications Diffract | SIM 3D_Max | Multilayer | Temprofile © Copyright 1987-2011, MM Research, Inc. 5748 N. Camino del Conde, Tucson, Arizona 85718

Other remarkable results are the demonstration of quantum entanglement, quantum teleportation, and (recently, in 1995) quantum logic gates. The latter are of much interest in quantum information theory, a subject which partly emerged from quantum optics, partly from theoretical computer science.

Al-Kindi (c. 801–873) was one of the earliest important optical writers in the Islamic world. In a work known in the west as De radiis stellarum, al-Kindi developed a theory "that everything in the world ... emits rays in every direction, which fill the whole world."[10]

Ibn al-Haytham (known in as Alhacen or Alhazen in Western Europe), writing in the 1010s, received both Ibn Sahl's treatise and a partial Arabic translation of Ptolemy's Optics. He produced a comprehensive and systematic analysis of Greek optical theories.[15] Ibn al-Haytham's key achievement was twofold: first, to insist, against the opinion of Ptolemy, that vision occurred because of rays entering the eye; the second was to define the physical nature of the rays discussed by earlier geometrical optical writers, considering them as the forms of light and color.[16] He then analyzed these physical rays according to the principles of geometrical optics. He wrote many books on optics, most significantly the Book of Optics (Kitab al Manazir in Arabic), translated into Latin as the De aspectibus or Perspectiva, which disseminated his ideas to Western Europe and had great influence on the later developments of optics.[17][6] Ibn al-Haytham was called "the father of modern optics".[18][19]

Figure 7. Computed plots of intensity distribution at the observation plane of Fig. 6. The lens under test (NA = 0.5, f = 6000l0) is assumed to have 3 waves of primary coma, and its nominal focus is coincident with the focal point of the relay lens. The tilt angle y of the reference beam increases progressively from (a) to (f). (a) y = -0.1°, (b) y = 0°, (c) y  = 0.05°, (d)y = 0.07°, (e) y = 0.1°, (f)y  = 0.18°.

The Indian Buddhists, such as Dignāga in the 5th century and Dharmakirti in the 7th century, developed a type of atomism which defined the atoms which make up the world as momentary flashes of light or energy. They viewed light as being an atomic entity equivalent to energy, though they also viewed all matter as being composed of these light/energy particles.

The earliest known examples of compound microscopes, which combine an objective lens near the specimen with an eyepiece to view a real image, appeared in Europe around 1620.[52] The design is very similar to the telescope and, like that device, its inventor is unknown. Again claims revolve around the spectacle making centers in the Netherlands including claims it was invented in 1590 by Zacharias Janssen and/or his father, Hans Martens,[53][54][55] claims it was invented by rival spectacle maker, Hans Lippershey,[56] and claims it was invented by expatriate Cornelis Drebbel who was noted to have a version in London in 1619.[57][58]

Euclid did not define the physical nature of these visual rays but, using the principles of geometry, he discussed the effects of perspective and the rounding of things seen at a distance.

Ronchi gratingfor sale

Isaac Newton (1643–1727) investigated the refraction of light, demonstrating that a prism could decompose white light into a spectrum of colours, and that a lens and a second prism could recompose the multicoloured spectrum into white light. He also showed that the coloured light does not change its properties by separating out a coloured beam and shining it on various objects. Newton noted that regardless of whether it was reflected or scattered or transmitted, it stayed the same colour. Thus, he observed that colour is the result of objects interacting with already-coloured light rather than objects generating the colour themselves. This is known as Newton's theory of colour. From this work he concluded that any refracting telescope would suffer from the dispersion of light into colours. He went on to invent a reflecting telescope (today known as a Newtonian telescope), which showed that using a mirror to form an image bypassed the problem. In 1671 the Royal Society asked for a demonstration of his reflecting telescope. Their interest encouraged him to publish his notes On Colour, which he later expanded into his Opticks. Newton argued that light is composed of particles or corpuscles and were refracted by accelerating toward the denser medium, but he had to associate them with waves to explain the diffraction of light (Opticks Bk. II, Props. XII-L). Later physicists instead favoured a purely wavelike explanation of light to account for diffraction. Today's quantum mechanics, photons and the idea of wave-particle duality bear only a minor resemblance to Newton's understanding of light.

Theodoric of Freiberg (ca. 1250–ca. 1310) was among the first in Europe to provide the correct scientific explanation for the rainbow phenomenon,[31] as well as Qutb al-Din al-Shirazi (1236–1311) and his student Kamāl al-Dīn al-Fārisī (1260–1320) mentioned above.

Between the 11th and 13th centuries, so-called "reading stones" were invented. Often used by monks to assist in illuminating manuscripts, these were primitive plano-convex lenses, initially made by cutting a glass sphere in half. As the stones were experimented with, it was slowly understood that shallower lenses magnified more effectively. Around 1286, possibly in Pisa, Italy, the first pair of eyeglasses was made, although it is unclear who the inventor was.[50]

To be sure, the Ronchi test is not the only scheme used during fabrication and evaluation of optical systems; several other tests exist and their relative merits have been expounded in the literature.3 It is useful here to examine some of these alternative methods and to compare the resulting patterns (interferograms or otherwise) with those obtained with the Ronchi test. Testing by interfering with a reference plane-wave. Figure 6 shows the schematic diagram of a Mach-Zehnder interferometer, which is one among many that can be used to evaluate the aberrated wavefronts directly. In this system a coherent, monochromatic beam of light is sent through the lens under test, is collected and recollimated by a well-corrected lens, and is made to interfere with a reference beam that has been split off from the original, incident wavefront.   Figure 6. Schematic diagram of a Mach-Zehnder interferometer that might be set up for a direct measurement of wavefront aberrations. The pupil relay lens (itself free from aberrations) forms at the observation plane an image of the exit pupil of the lens under test. A fraction of the incident beam is diverted from its original path and sent to the observation plane by means of the various mirrors and beam-splitters. The observed fringes are characteristic of the aberrations present at the exit pupil of the lens under test. A small tilt of the mirror shown at the lower left side of the figure would introduce a linear phase shift on the reference beam. This tilt is generally useful in producing signature fringe patterns at the observation plane.   The flat mirror shown in the lower left side of the interferometer is mounted on a tip-tilt stage that allows the introduction of a small amount of tilt in the reference beam. Figure 7 shows the computed patterns of intensity distribution at the observation plane of the Mach-Zehnder interferometer corresponding to 3 waves of primary coma.4         Figure 7. Computed plots of intensity distribution at the observation plane of Fig. 6. The lens under test (NA = 0.5, f = 6000l0) is assumed to have 3 waves of primary coma, and its nominal focus is coincident with the focal point of the relay lens. The tilt angle y of the reference beam increases progressively from (a) to (f). (a) y = -0.1°, (b) y = 0°, (c) y  = 0.05°, (d)y = 0.07°, (e) y = 0.1°, (f)y  = 0.18°.   In obtaining the various frames of Fig. 7 we have fixed all the system parameters and only varied the tilt of the reference beam. Note that the characteristic fringes of coma in Fig. 7 are quite different from those of coma in the Ronchi test, shown in Figs. 4(d)-(f). Incidentally, the patterns of Fig. 7 show similarities with the Ronchigrams of spherical aberration displayed in Fig. 5. This is not a coincidence, and is rooted in the algebraic forms of the aberration function for third order coma (r3 cos f) and spherical aberration (r4 ), and also in the fact that a Ronchigram, being a kind of shearing interferogram (albeit with a large shear), is related to the derivative of the wavefront aberration function. Knife-edge and wire tests. A schematic diagram of the knife-edge method of testing optical systems is shown in Fig. 8. A geometric-optical interpretation of this test suffices for most practical purposes: the knife-edge blocks different groups of rays in its various positions along the optical axis, allowing the remaining rays to reach the observation plane.3 Another method of testing, known as the wire test, is quite similar to the knife-edge method, being obtained from it by the substitution of the knife-edge with a fine wire.3   Figure 8. In the knife-edge test a certain region in the vicinity of focus is blocked by a knife-edge; the nature and the magnitude of aberrations are then inferred from the resulting patterns of intensity distribution at the observation plane. (The knife-edge may be moved both along and perpendicular to the optical axis.) The wire test is similar to the knife-edge test except that a fine wire is used instead to block certain groups of rays.   Since the grating in the Ronchi test may be thought of as a series of parallel knife-edges or, more aptly, a series of parallel wires, it should not come as a surprise that similarities exist between Ronchigrams and the patterns observed in these other tests. In fact, early attempts at explaining the results of Ronchi?s method were based on geometrical optics, and considered the grating as a set of parallel wires whose shadows produced the observed patterns.5 We will not delve into these matters, but simply draw the reader?s attention to Figs. 9 and 10, where we show several computed patterns of intensity distribution for the knife-edge and wire tests, respectively.4     Figure 9. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the knife-edge test carried out with a laser beam. The lens under test (NA = 0.5, f = 6000l0) and the pupil relay lens (NA = 0.5) are assumed to be fixed in their respective positions, while the knife-edge moves along the optical axis. (The tip of the knife remains on the axis at all times.) The lens under test is assumed to have 3 waves of primary spherical aberration. In frames (a) to (d) the distance of the knife-edge from paraxial focus is Dz = -15l0, 0, +15l0, and +20l0, respectively. (Positive distances are in the direction of the marginal focus.)   The results of the simulated knife-edge test depicted in Fig. 9 assume a laser as the light source. Consequently, frames (a) and (b) of Fig. 9 exhibit several dark lines which, with a less coherent light source, would have been absent. The results of the simulated wire test shown in Fig. 10 assume an extended light source, since the small amount of spherical aberration present in the system under consideration would render the test useless with a wire that, fine as it may be, is still wider than the focused spot produced by a laser beam. Note the similarities between the patterns of Figs. 9 and 10 on the one hand, and those of Figs. 5(d)-(f) on the other.   Figure 10. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the wire test with an extended, quasi-monochromatic light source. The lens under test (NA = 0.5, f = 6000l0) has 3 waves of primary spherical aberration. The assumed wire diameter is 15l0, which is comparable to the size of the image of the extended light source, as measured in the vicinity of focus. In (a) the wire is centered on axis and is 25l0 away from paraxial focus (in the direction of the marginal focus). In (b) the wire is again centered on axis, but is 20l0 away from paraxial focus. In (c) the wire is shifted 0.5l0 off-axis while its distance from paraxial focus remains at 20l0.   Extensions of the Ronchi test. Several modifications and extensions of the Ronchi test have appeared over the years, and have helped to solve specific problems in testing of optical systems.3 As an example we mention the "double-frequency grating lateral shear interferometer" invented by James Wyant in the early 1970?s. The grating in this device has two slightly different frequencies, which give rise to two +1st order beams as well as two -1st order beams; the beams in each pair are slightly shifted relative to each other. Moreover, the (average) pitch of the grating is such that there is no overlap between the 0th, +1st, and -1st orders. Consequently, interference occurs between the two +1st order beams (and, likewise, between the two -1st order beams). One can thus obtain an arbitrarily small lateral shear of the wavefront under test, and use the results to achieve accurate quantitative measurements. A two-dimensional version of the double-frequency grating has also been employed to generate lateral wavefront shear simultaneously along the X and Y axes. (Remember that beam propagation is along Z and, therefore, X and Y are orthogonal axes in the plane of the grating.) In the absence of  a 2-D grating, one must rotate a 1-D grating by 90° to obtain  wavefront shear first along the X- and then along the Y-axis. Acknowledgment. I am grateful to Professor Roland Shack of the Optical Sciences Center for many illuminating discussions, and also for suggesting some of the examples presented in this article.   References V. Ronchi, "Le Frange di Combinazioni Nello Studio delle Superficie e dei Sistemi Ottici," Riv. Ottica Mecc. Precis. 2, 9 (1923). V. Ronchi, "Due Nuovi Metodi per lo Studio delle Superficie e dei Sistemi Ottici," Ann. Sc. Norm. Super. Pisa, 15 (1923). D. Malacara, ed., Optical Shop Testing, second edition, Wiley, New York, 1992. The computer simulations reported in this article were performed by DIFFRACT?, a product of MM Research, Inc., Tucson, Arizona. G. Toraldo di Francia, "Geometrical and interferential aspects of the Ronchi Test," in Optical Image Evaluation, National Bureau of Standards Circular 526, issued April 29, 1954. J. C. Wyant, "Double frequency grating lateral shear interferometer," Appl. Opt. 12, 2057 (1973). << Back to list of Articles Home | About MM Research, Inc. | Online Publications Diffract | SIM 3D_Max | Multilayer | Temprofile © Copyright 1987-2011, MM Research, Inc. 5748 N. Camino del Conde, Tucson, Arizona 85718

In the 4th century BC Chinese text, credited to the philosopher Mozi, it is described how light passing through a pinhole creates an inverted image in a "collecting-point" or "treasure house".[3]

This changed with the invention of the maser in 1953 and the laser in 1960. Laser science—research into principles, design and application of these devices—became an important field, and the quantum mechanics underlying the laser's principles was studied now with more emphasis on the properties of light, and the name quantum optics became customary.

In his Optics Greek mathematician Euclid observed that "things seen under a greater angle appear greater, and those under a lesser angle less, while those under equal angles appear equal". In the 36 propositions that follow, Euclid relates the apparent size of an object to its distance from the eye and investigates the apparent shapes of cylinders and cones when viewed from different angles. Pappus believed these results to be important in astronomy and included Euclid's Optics, along with his Phaenomena, in the Little Astronomy, a compendium of smaller works to be studied before the Syntaxis (Almagest) of Ptolemy.

In his Hypothesis of Light of 1675, Newton posited the existence of the ether to transmit forces between particles. In 1704, Newton published Opticks, in which he expounded his corpuscular theory of light. He considered light to be made up of extremely subtle corpuscles, that ordinary matter was made of grosser corpuscles and speculated that through a kind of alchemical transmutation "Are not gross Bodies and Light convertible into one another, ...and may not Bodies receive much of their Activity from the Particles of Light which enter their Composition?"[35]

Figure 2. Diagram showing several diffracted orders in the far field of the grating of Fig. 1. When the grating?s period is chosen properly, each diffracted order (i.e., emergent cone of light) will overlap only with its nearest neighbors. Except for a lateral shift in position, the various orders are identical, carrying the amplitude and phase distribution of the beam as it appears at the exit pupil of the object under test.   To determine the appropriate grating period P, one needs to know the wavelength l0 of the beam used for testing, and the numerical aperture NA of the focused cone of light. (By definition, NA = sin q, where q is the half-angle subtended by the exit pupil of the lens at its focal point. If the lens under test is being used at full aperture, NA will also be equal to 0.5/f-number.) To avoid multiple overlaps among diffracted orders, the angle between adjacent orders must exceed the focused cone?s half-angle. Now, it is well-known in the theory of diffraction gratings that, at normal incidence, sin qn = nl0/P where n, an integer, is the order of diffraction, and qn is the corresponding deviation angle from the surface normal. Therefore, we arrive at the conclusion that P should be less than or equal to l0/NA. For example, assume that the lens under test has a numerical aperture NA = 0.5. Then, if the grating period is chosen to be 2l0, each diffracted order will deviate from the zero-order by 30°, making the +1st order just touch the -1st order in the far field. Figure 3 shows the computed intensity distribution at the observation plane of an aberration-free system in which the relay lens has the same numerical aperture as the lens under test (NA = 0.5).     Figure 3. Distribution of intensity at the observation plane of Fig. 1 in the absence of aberrations. The pupil relay lens is chosen to have the same numerical aperture as the object under test, thereby limiting the collected light to the zero-order beam and to those portions of the ±1st orders that overlap the 0th order.     Equality of numerical apertures means that only the zero-order diffracted beam will be fully transmitted to the observation plane. Of the ±1st order beams only those portions that overlap the zero order will reach the observation plane. The period of the grating in this example has been a little less than l0/NA, leaving a small gap between +1st and -1st orders.4 Absence of aberrations means that the phase distribution over the cross-sections of the various diffracted orders is uniform and, therefore, no interference fringes are to be expected. Ronchigrams for primary or Seidel aberrations. Figure 4 shows the computed patterns of intensity distribution at the observation plane of Fig. 1, corresponding to different types of primary (Seidel) aberrations of the lens.   Figure 4. Computed plots of intensity distribution at the observation plane of Fig. 1. The lens under test is assumed to have 3 waves of primary (Seidel) aberrations, and the grating is at the nominal focal plane of the lens. (a) defocus, (b) spherical, (c) astigmatism oriented at 45°,  (d) coma at 0°, (e) coma at 45°, (f) coma at 90°.     For these calculations we fixed the distance between the lens under test and the relay lens; we then placed the grating at the paraxial focus of the converging wavefront.4 The pattern in Fig. 4(a) was obtained when we assumed the presence of 3 waves of curvature (or defocus) at the exit pupil of the lens. Different amounts of defocus would create essentially the same pattern albeit with a different number of fringes. In Fig. 4(b) we observe the fringes arising from the presence of 3 waves of third order spherical aberration in the test system. The shapes of these fringes depend not only on the magnitude of the aberration, but also on the position of the grating relative to the focal plane. (We will have more to say about this point later.) Figure 4(c) shows the fringes that would arise when 3 waves of primary astigmatism are present. When the orientation of astigmatism changes, the fringes will remain straight lines, but their orientation within the observation plane will change accordingly. The last three frames in Fig. 4 represent the effects of third order coma. A change in the orientation of this aberration causes the interference pattern to change drastically. Figures 4(d)-(f) correspond to 3 waves of coma oriented at 0°, 45°, and 90°, respectively. Sliding the grating along the optical axis. A change in the position of the grating relative to the focal plane influences the observed fringe pattern. We limit our discussion to the case of spherical aberration, although similar analyses could be performed for other aberrations as well. Assuming 3 waves of spherical aberration as before, we obtain the patterns displayed in Fig. 5 as we slide the grating along the optical axis in the system of Fig. 1.4 Once again, we have taken the lens under test to have NA = 0.5 and f = 6000l0. The paraxial focus of the lens under test coincides with the front focal point of the relay lens, and the grating is shifted by different amounts Dz relative to this common focus. Frames (a)-(f) in Fig. 5 correspond to different values of Dz, starting at Dz = -10l0 in (a) and moving forward to Dz = +25l0 in (f). In the process, as the grating moves through paraxial focus and towards marginal focus, we observe a rich variety of patterns that aid us in determining the nature and the magnitude of the aberration.     Figure 5. Computed plots of intensity distribution at the observation plane of Fig. 1, showing the patterns obtained by sliding the grating along the optical axis. The lens under test (NA = 0.5, f = 6000l0) is assumed to have 3 waves of primary spherical aberration, and its paraxial focus is coincident with the focal point of the relay lens. The grating is moved along the optical axis by an amount Dz relative to the (common) focal plane; positive distances are towards the marginal focus. (a)  Dz= -10l0, (b)  Dz = 0, (c)  Dz = 10l0, (d) Dz = 15l0, (e)  Dz= 20l0, (f)  Dz= 25l0.   To be sure, the Ronchi test is not the only scheme used during fabrication and evaluation of optical systems; several other tests exist and their relative merits have been expounded in the literature.3 It is useful here to examine some of these alternative methods and to compare the resulting patterns (interferograms or otherwise) with those obtained with the Ronchi test. Testing by interfering with a reference plane-wave. Figure 6 shows the schematic diagram of a Mach-Zehnder interferometer, which is one among many that can be used to evaluate the aberrated wavefronts directly. In this system a coherent, monochromatic beam of light is sent through the lens under test, is collected and recollimated by a well-corrected lens, and is made to interfere with a reference beam that has been split off from the original, incident wavefront.   Figure 6. Schematic diagram of a Mach-Zehnder interferometer that might be set up for a direct measurement of wavefront aberrations. The pupil relay lens (itself free from aberrations) forms at the observation plane an image of the exit pupil of the lens under test. A fraction of the incident beam is diverted from its original path and sent to the observation plane by means of the various mirrors and beam-splitters. The observed fringes are characteristic of the aberrations present at the exit pupil of the lens under test. A small tilt of the mirror shown at the lower left side of the figure would introduce a linear phase shift on the reference beam. This tilt is generally useful in producing signature fringe patterns at the observation plane.   The flat mirror shown in the lower left side of the interferometer is mounted on a tip-tilt stage that allows the introduction of a small amount of tilt in the reference beam. Figure 7 shows the computed patterns of intensity distribution at the observation plane of the Mach-Zehnder interferometer corresponding to 3 waves of primary coma.4         Figure 7. Computed plots of intensity distribution at the observation plane of Fig. 6. The lens under test (NA = 0.5, f = 6000l0) is assumed to have 3 waves of primary coma, and its nominal focus is coincident with the focal point of the relay lens. The tilt angle y of the reference beam increases progressively from (a) to (f). (a) y = -0.1°, (b) y = 0°, (c) y  = 0.05°, (d)y = 0.07°, (e) y = 0.1°, (f)y  = 0.18°.   In obtaining the various frames of Fig. 7 we have fixed all the system parameters and only varied the tilt of the reference beam. Note that the characteristic fringes of coma in Fig. 7 are quite different from those of coma in the Ronchi test, shown in Figs. 4(d)-(f). Incidentally, the patterns of Fig. 7 show similarities with the Ronchigrams of spherical aberration displayed in Fig. 5. This is not a coincidence, and is rooted in the algebraic forms of the aberration function for third order coma (r3 cos f) and spherical aberration (r4 ), and also in the fact that a Ronchigram, being a kind of shearing interferogram (albeit with a large shear), is related to the derivative of the wavefront aberration function. Knife-edge and wire tests. A schematic diagram of the knife-edge method of testing optical systems is shown in Fig. 8. A geometric-optical interpretation of this test suffices for most practical purposes: the knife-edge blocks different groups of rays in its various positions along the optical axis, allowing the remaining rays to reach the observation plane.3 Another method of testing, known as the wire test, is quite similar to the knife-edge method, being obtained from it by the substitution of the knife-edge with a fine wire.3   Figure 8. In the knife-edge test a certain region in the vicinity of focus is blocked by a knife-edge; the nature and the magnitude of aberrations are then inferred from the resulting patterns of intensity distribution at the observation plane. (The knife-edge may be moved both along and perpendicular to the optical axis.) The wire test is similar to the knife-edge test except that a fine wire is used instead to block certain groups of rays.   Since the grating in the Ronchi test may be thought of as a series of parallel knife-edges or, more aptly, a series of parallel wires, it should not come as a surprise that similarities exist between Ronchigrams and the patterns observed in these other tests. In fact, early attempts at explaining the results of Ronchi?s method were based on geometrical optics, and considered the grating as a set of parallel wires whose shadows produced the observed patterns.5 We will not delve into these matters, but simply draw the reader?s attention to Figs. 9 and 10, where we show several computed patterns of intensity distribution for the knife-edge and wire tests, respectively.4     Figure 9. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the knife-edge test carried out with a laser beam. The lens under test (NA = 0.5, f = 6000l0) and the pupil relay lens (NA = 0.5) are assumed to be fixed in their respective positions, while the knife-edge moves along the optical axis. (The tip of the knife remains on the axis at all times.) The lens under test is assumed to have 3 waves of primary spherical aberration. In frames (a) to (d) the distance of the knife-edge from paraxial focus is Dz = -15l0, 0, +15l0, and +20l0, respectively. (Positive distances are in the direction of the marginal focus.)   The results of the simulated knife-edge test depicted in Fig. 9 assume a laser as the light source. Consequently, frames (a) and (b) of Fig. 9 exhibit several dark lines which, with a less coherent light source, would have been absent. The results of the simulated wire test shown in Fig. 10 assume an extended light source, since the small amount of spherical aberration present in the system under consideration would render the test useless with a wire that, fine as it may be, is still wider than the focused spot produced by a laser beam. Note the similarities between the patterns of Figs. 9 and 10 on the one hand, and those of Figs. 5(d)-(f) on the other.   Figure 10. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the wire test with an extended, quasi-monochromatic light source. The lens under test (NA = 0.5, f = 6000l0) has 3 waves of primary spherical aberration. The assumed wire diameter is 15l0, which is comparable to the size of the image of the extended light source, as measured in the vicinity of focus. In (a) the wire is centered on axis and is 25l0 away from paraxial focus (in the direction of the marginal focus). In (b) the wire is again centered on axis, but is 20l0 away from paraxial focus. In (c) the wire is shifted 0.5l0 off-axis while its distance from paraxial focus remains at 20l0.   Extensions of the Ronchi test. Several modifications and extensions of the Ronchi test have appeared over the years, and have helped to solve specific problems in testing of optical systems.3 As an example we mention the "double-frequency grating lateral shear interferometer" invented by James Wyant in the early 1970?s. The grating in this device has two slightly different frequencies, which give rise to two +1st order beams as well as two -1st order beams; the beams in each pair are slightly shifted relative to each other. Moreover, the (average) pitch of the grating is such that there is no overlap between the 0th, +1st, and -1st orders. Consequently, interference occurs between the two +1st order beams (and, likewise, between the two -1st order beams). One can thus obtain an arbitrarily small lateral shear of the wavefront under test, and use the results to achieve accurate quantitative measurements. A two-dimensional version of the double-frequency grating has also been employed to generate lateral wavefront shear simultaneously along the X and Y axes. (Remember that beam propagation is along Z and, therefore, X and Y are orthogonal axes in the plane of the grating.) In the absence of  a 2-D grating, one must rotate a 1-D grating by 90° to obtain  wavefront shear first along the X- and then along the Y-axis. Acknowledgment. I am grateful to Professor Roland Shack of the Optical Sciences Center for many illuminating discussions, and also for suggesting some of the examples presented in this article.   References V. Ronchi, "Le Frange di Combinazioni Nello Studio delle Superficie e dei Sistemi Ottici," Riv. Ottica Mecc. Precis. 2, 9 (1923). V. Ronchi, "Due Nuovi Metodi per lo Studio delle Superficie e dei Sistemi Ottici," Ann. Sc. Norm. Super. Pisa, 15 (1923). D. Malacara, ed., Optical Shop Testing, second edition, Wiley, New York, 1992. The computer simulations reported in this article were performed by DIFFRACT?, a product of MM Research, Inc., Tucson, Arizona. G. Toraldo di Francia, "Geometrical and interferential aspects of the Ronchi Test," in Optical Image Evaluation, National Bureau of Standards Circular 526, issued April 29, 1954. J. C. Wyant, "Double frequency grating lateral shear interferometer," Appl. Opt. 12, 2057 (1973). << Back to list of Articles Home | About MM Research, Inc. | Online Publications Diffract | SIM 3D_Max | Multilayer | Temprofile © Copyright 1987-2011, MM Research, Inc. 5748 N. Camino del Conde, Tucson, Arizona 85718

Extensions of the Ronchi test. Several modifications and extensions of the Ronchi test have appeared over the years, and have helped to solve specific problems in testing of optical systems.3 As an example we mention the "double-frequency grating lateral shear interferometer" invented by James Wyant in the early 1970?s. The grating in this device has two slightly different frequencies, which give rise to two +1st order beams as well as two -1st order beams; the beams in each pair are slightly shifted relative to each other. Moreover, the (average) pitch of the grating is such that there is no overlap between the 0th, +1st, and -1st orders. Consequently, interference occurs between the two +1st order beams (and, likewise, between the two -1st order beams). One can thus obtain an arbitrarily small lateral shear of the wavefront under test, and use the results to achieve accurate quantitative measurements. A two-dimensional version of the double-frequency grating has also been employed to generate lateral wavefront shear simultaneously along the X and Y axes. (Remember that beam propagation is along Z and, therefore, X and Y are orthogonal axes in the plane of the grating.) In the absence of  a 2-D grating, one must rotate a 1-D grating by 90° to obtain  wavefront shear first along the X- and then along the Y-axis. Acknowledgment. I am grateful to Professor Roland Shack of the Optical Sciences Center for many illuminating discussions, and also for suggesting some of the examples presented in this article.   References V. Ronchi, "Le Frange di Combinazioni Nello Studio delle Superficie e dei Sistemi Ottici," Riv. Ottica Mecc. Precis. 2, 9 (1923). V. Ronchi, "Due Nuovi Metodi per lo Studio delle Superficie e dei Sistemi Ottici," Ann. Sc. Norm. Super. Pisa, 15 (1923). D. Malacara, ed., Optical Shop Testing, second edition, Wiley, New York, 1992. The computer simulations reported in this article were performed by DIFFRACT?, a product of MM Research, Inc., Tucson, Arizona. G. Toraldo di Francia, "Geometrical and interferential aspects of the Ronchi Test," in Optical Image Evaluation, National Bureau of Standards Circular 526, issued April 29, 1954. J. C. Wyant, "Double frequency grating lateral shear interferometer," Appl. Opt. 12, 2057 (1973). << Back to list of Articles Home | About MM Research, Inc. | Online Publications Diffract | SIM 3D_Max | Multilayer | Temprofile © Copyright 1987-2011, MM Research, Inc. 5748 N. Camino del Conde, Tucson, Arizona 85718

Since the grating in the Ronchi test may be thought of as a series of parallel knife-edges or, more aptly, a series of parallel wires, it should not come as a surprise that similarities exist between Ronchigrams and the patterns observed in these other tests. In fact, early attempts at explaining the results of Ronchi?s method were based on geometrical optics, and considered the grating as a set of parallel wires whose shadows produced the observed patterns.5 We will not delve into these matters, but simply draw the reader?s attention to Figs. 9 and 10, where we show several computed patterns of intensity distribution for the knife-edge and wire tests, respectively.4     Figure 9. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the knife-edge test carried out with a laser beam. The lens under test (NA = 0.5, f = 6000l0) and the pupil relay lens (NA = 0.5) are assumed to be fixed in their respective positions, while the knife-edge moves along the optical axis. (The tip of the knife remains on the axis at all times.) The lens under test is assumed to have 3 waves of primary spherical aberration. In frames (a) to (d) the distance of the knife-edge from paraxial focus is Dz = -15l0, 0, +15l0, and +20l0, respectively. (Positive distances are in the direction of the marginal focus.)   The results of the simulated knife-edge test depicted in Fig. 9 assume a laser as the light source. Consequently, frames (a) and (b) of Fig. 9 exhibit several dark lines which, with a less coherent light source, would have been absent. The results of the simulated wire test shown in Fig. 10 assume an extended light source, since the small amount of spherical aberration present in the system under consideration would render the test useless with a wire that, fine as it may be, is still wider than the focused spot produced by a laser beam. Note the similarities between the patterns of Figs. 9 and 10 on the one hand, and those of Figs. 5(d)-(f) on the other.   Figure 10. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the wire test with an extended, quasi-monochromatic light source. The lens under test (NA = 0.5, f = 6000l0) has 3 waves of primary spherical aberration. The assumed wire diameter is 15l0, which is comparable to the size of the image of the extended light source, as measured in the vicinity of focus. In (a) the wire is centered on axis and is 25l0 away from paraxial focus (in the direction of the marginal focus). In (b) the wire is again centered on axis, but is 20l0 away from paraxial focus. In (c) the wire is shifted 0.5l0 off-axis while its distance from paraxial focus remains at 20l0.   Extensions of the Ronchi test. Several modifications and extensions of the Ronchi test have appeared over the years, and have helped to solve specific problems in testing of optical systems.3 As an example we mention the "double-frequency grating lateral shear interferometer" invented by James Wyant in the early 1970?s. The grating in this device has two slightly different frequencies, which give rise to two +1st order beams as well as two -1st order beams; the beams in each pair are slightly shifted relative to each other. Moreover, the (average) pitch of the grating is such that there is no overlap between the 0th, +1st, and -1st orders. Consequently, interference occurs between the two +1st order beams (and, likewise, between the two -1st order beams). One can thus obtain an arbitrarily small lateral shear of the wavefront under test, and use the results to achieve accurate quantitative measurements. A two-dimensional version of the double-frequency grating has also been employed to generate lateral wavefront shear simultaneously along the X and Y axes. (Remember that beam propagation is along Z and, therefore, X and Y are orthogonal axes in the plane of the grating.) In the absence of  a 2-D grating, one must rotate a 1-D grating by 90° to obtain  wavefront shear first along the X- and then along the Y-axis. Acknowledgment. I am grateful to Professor Roland Shack of the Optical Sciences Center for many illuminating discussions, and also for suggesting some of the examples presented in this article.   References V. Ronchi, "Le Frange di Combinazioni Nello Studio delle Superficie e dei Sistemi Ottici," Riv. Ottica Mecc. Precis. 2, 9 (1923). V. Ronchi, "Due Nuovi Metodi per lo Studio delle Superficie e dei Sistemi Ottici," Ann. Sc. Norm. Super. Pisa, 15 (1923). D. Malacara, ed., Optical Shop Testing, second edition, Wiley, New York, 1992. The computer simulations reported in this article were performed by DIFFRACT?, a product of MM Research, Inc., Tucson, Arizona. G. Toraldo di Francia, "Geometrical and interferential aspects of the Ronchi Test," in Optical Image Evaluation, National Bureau of Standards Circular 526, issued April 29, 1954. J. C. Wyant, "Double frequency grating lateral shear interferometer," Appl. Opt. 12, 2057 (1973). << Back to list of Articles Home | About MM Research, Inc. | Online Publications Diffract | SIM 3D_Max | Multilayer | Temprofile © Copyright 1987-2011, MM Research, Inc. 5748 N. Camino del Conde, Tucson, Arizona 85718

Ibn al-Haytham (Alhacen) wrote about the effects of pinhole, concave lenses, and magnifying glasses in his 11th century Book of Optics (1021 CE).[45][47][48] The English friar Roger Bacon, during the 1260s or 1270s, wrote works on optics, partly based on the works of Arab writers, that described the function of corrective lenses for vision and burning glasses. These volumes were outlines for a larger publication that was never produced, so his ideas never saw mass dissemination.[49]

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Several later works, including the influential A Moral Treatise on the Eye (Latin: Tractatus Moralis de Oculo) by Peter of Limoges (1240–1306), helped popularize and spread the ideas found in Bacon's writings.[28]

Figure 2. Diagram showing several diffracted orders in the far field of the grating of Fig. 1. When the grating?s period is chosen properly, each diffracted order (i.e., emergent cone of light) will overlap only with its nearest neighbors. Except for a lateral shift in position, the various orders are identical, carrying the amplitude and phase distribution of the beam as it appears at the exit pupil of the object under test.

The English Franciscan, Roger Bacon (c. 1214–1294) was strongly influenced by Grosseteste's writings on the importance of light. In his optical writings (the Perspectiva, the De multiplicatione specierum, and the De speculis comburentibus) he cited a wide range of recently translated optical and philosophical works, including those of Alhacen, Aristotle, Avicenna, Averroes, Euclid, al-Kindi, Ptolemy, Tideus, and Constantine the African. Although he was not a slavish imitator, he drew his mathematical analysis of light and vision from the writings of the Arabic writer, Alhacen. But he added to this the Neoplatonic concept, perhaps drawn from Grosseteste, that every object radiates a power (species) by which it acts upon nearby objects suited to receive those species.[27] Note that Bacon's optical use of the term species differs significantly from the genus/species categories found in Aristotelian philosophy.

The English bishop Robert Grosseteste (c. 1175–1253) wrote on a wide range of scientific topics at the time of the origin of the medieval university and the recovery of the works of Aristotle. Grosseteste reflected a period of transition between the Platonism of early medieval learning and the new Aristotelianism, hence he tended to apply mathematics and the Platonic metaphor of light in many of his writings. He has been credited with discussing light from four different perspectives: an epistemology of light, a metaphysics or cosmogony of light, an etiology or physics of light, and a theology of light.[24]

Figure 8. In the knife-edge test a certain region in the vicinity of focus is blocked by a knife-edge; the nature and the magnitude of aberrations are then inferred from the resulting patterns of intensity distribution at the observation plane. (The knife-edge may be moved both along and perpendicular to the optical axis.) The wire test is similar to the knife-edge test except that a fine wire is used instead to block certain groups of rays.

The lens brings the incident beam to a focus, in the vicinity of which a diffraction grating is placed perpendicular to the optical axis. (The optical axis will be denoted the Z-axis throughout this article.) The grating, also referred to as a Ronchi ruling, may be as simple as a low-frequency wire-grid, or as sophisticated as a modern short-pitched, phase/amplitude grating. The position of the grating should be adjustable in the vicinity of focus, so that it may be shifted back and forth along the optical axis. The grating breaks up the incident beam into multiple diffracted orders, which will subsequently propagate along Z and reach the lens labeled "pupil relay" in Fig. 1. (The pupil relay may simply be the lens of the eye, which projects the exit pupil of the object under test onto the retina of the observer. Alternatively, it may be a conventional lens that creates a real image of the exit pupil on a screen or on a CCD camera.) The diffracted orders from the grating will be collected by the relay lens and, within their overlap areas, will create interference fringes characteristic of the aberrations of the optical system under consideration. By analyzing these fringes, one can determine the type and, with some effort, the magnitude of the aberrations present at the exit pupil of the system. The above description of the Ronchi test relied on its modern interpretation based on our current understanding of physical optics and the theory of diffraction gratings. Historically, however, the gratings used in the early days were quite coarse, and the results obtained with them required no more than a simple geometric optical theory for their interpretation. Typically, one would place the eye at the focus of the lens and hold a grating (e.g., a wire grid) in front of the eye, moving the grating in and out until a clear pattern became visible. At this point the beam would be illuminating several of the wires simultaneously. By looking through the grating and observing the shadows that the wires cast on the exit pupil, one could determine the type of aberration present in the system. The coarseness of the grating, of course, caused several of the diffracted orders (as we understand them today) to overlap each other, thus resulting in reduced contrast and smearing of the patterns near the boundaries. These problems were eventually overcome when finer gratings became available and the diffraction theory of the Ronchi test was better understood. Choosing an appropriate grating. For best results the pitch of the grating should be chosen such that, as shown in Fig. 2, no more than two diffraction orders will overlap at any given point.     Figure 2. Diagram showing several diffracted orders in the far field of the grating of Fig. 1. When the grating?s period is chosen properly, each diffracted order (i.e., emergent cone of light) will overlap only with its nearest neighbors. Except for a lateral shift in position, the various orders are identical, carrying the amplitude and phase distribution of the beam as it appears at the exit pupil of the object under test.   To determine the appropriate grating period P, one needs to know the wavelength l0 of the beam used for testing, and the numerical aperture NA of the focused cone of light. (By definition, NA = sin q, where q is the half-angle subtended by the exit pupil of the lens at its focal point. If the lens under test is being used at full aperture, NA will also be equal to 0.5/f-number.) To avoid multiple overlaps among diffracted orders, the angle between adjacent orders must exceed the focused cone?s half-angle. Now, it is well-known in the theory of diffraction gratings that, at normal incidence, sin qn = nl0/P where n, an integer, is the order of diffraction, and qn is the corresponding deviation angle from the surface normal. Therefore, we arrive at the conclusion that P should be less than or equal to l0/NA. For example, assume that the lens under test has a numerical aperture NA = 0.5. Then, if the grating period is chosen to be 2l0, each diffracted order will deviate from the zero-order by 30°, making the +1st order just touch the -1st order in the far field. Figure 3 shows the computed intensity distribution at the observation plane of an aberration-free system in which the relay lens has the same numerical aperture as the lens under test (NA = 0.5).     Figure 3. Distribution of intensity at the observation plane of Fig. 1 in the absence of aberrations. The pupil relay lens is chosen to have the same numerical aperture as the object under test, thereby limiting the collected light to the zero-order beam and to those portions of the ±1st orders that overlap the 0th order.     Equality of numerical apertures means that only the zero-order diffracted beam will be fully transmitted to the observation plane. Of the ±1st order beams only those portions that overlap the zero order will reach the observation plane. The period of the grating in this example has been a little less than l0/NA, leaving a small gap between +1st and -1st orders.4 Absence of aberrations means that the phase distribution over the cross-sections of the various diffracted orders is uniform and, therefore, no interference fringes are to be expected. Ronchigrams for primary or Seidel aberrations. Figure 4 shows the computed patterns of intensity distribution at the observation plane of Fig. 1, corresponding to different types of primary (Seidel) aberrations of the lens.   Figure 4. Computed plots of intensity distribution at the observation plane of Fig. 1. The lens under test is assumed to have 3 waves of primary (Seidel) aberrations, and the grating is at the nominal focal plane of the lens. (a) defocus, (b) spherical, (c) astigmatism oriented at 45°,  (d) coma at 0°, (e) coma at 45°, (f) coma at 90°.     For these calculations we fixed the distance between the lens under test and the relay lens; we then placed the grating at the paraxial focus of the converging wavefront.4 The pattern in Fig. 4(a) was obtained when we assumed the presence of 3 waves of curvature (or defocus) at the exit pupil of the lens. Different amounts of defocus would create essentially the same pattern albeit with a different number of fringes. In Fig. 4(b) we observe the fringes arising from the presence of 3 waves of third order spherical aberration in the test system. The shapes of these fringes depend not only on the magnitude of the aberration, but also on the position of the grating relative to the focal plane. (We will have more to say about this point later.) Figure 4(c) shows the fringes that would arise when 3 waves of primary astigmatism are present. When the orientation of astigmatism changes, the fringes will remain straight lines, but their orientation within the observation plane will change accordingly. The last three frames in Fig. 4 represent the effects of third order coma. A change in the orientation of this aberration causes the interference pattern to change drastically. Figures 4(d)-(f) correspond to 3 waves of coma oriented at 0°, 45°, and 90°, respectively. Sliding the grating along the optical axis. A change in the position of the grating relative to the focal plane influences the observed fringe pattern. We limit our discussion to the case of spherical aberration, although similar analyses could be performed for other aberrations as well. Assuming 3 waves of spherical aberration as before, we obtain the patterns displayed in Fig. 5 as we slide the grating along the optical axis in the system of Fig. 1.4 Once again, we have taken the lens under test to have NA = 0.5 and f = 6000l0. The paraxial focus of the lens under test coincides with the front focal point of the relay lens, and the grating is shifted by different amounts Dz relative to this common focus. Frames (a)-(f) in Fig. 5 correspond to different values of Dz, starting at Dz = -10l0 in (a) and moving forward to Dz = +25l0 in (f). In the process, as the grating moves through paraxial focus and towards marginal focus, we observe a rich variety of patterns that aid us in determining the nature and the magnitude of the aberration.     Figure 5. Computed plots of intensity distribution at the observation plane of Fig. 1, showing the patterns obtained by sliding the grating along the optical axis. The lens under test (NA = 0.5, f = 6000l0) is assumed to have 3 waves of primary spherical aberration, and its paraxial focus is coincident with the focal point of the relay lens. The grating is moved along the optical axis by an amount Dz relative to the (common) focal plane; positive distances are towards the marginal focus. (a)  Dz= -10l0, (b)  Dz = 0, (c)  Dz = 10l0, (d) Dz = 15l0, (e)  Dz= 20l0, (f)  Dz= 25l0.   To be sure, the Ronchi test is not the only scheme used during fabrication and evaluation of optical systems; several other tests exist and their relative merits have been expounded in the literature.3 It is useful here to examine some of these alternative methods and to compare the resulting patterns (interferograms or otherwise) with those obtained with the Ronchi test. Testing by interfering with a reference plane-wave. Figure 6 shows the schematic diagram of a Mach-Zehnder interferometer, which is one among many that can be used to evaluate the aberrated wavefronts directly. In this system a coherent, monochromatic beam of light is sent through the lens under test, is collected and recollimated by a well-corrected lens, and is made to interfere with a reference beam that has been split off from the original, incident wavefront.   Figure 6. Schematic diagram of a Mach-Zehnder interferometer that might be set up for a direct measurement of wavefront aberrations. The pupil relay lens (itself free from aberrations) forms at the observation plane an image of the exit pupil of the lens under test. A fraction of the incident beam is diverted from its original path and sent to the observation plane by means of the various mirrors and beam-splitters. The observed fringes are characteristic of the aberrations present at the exit pupil of the lens under test. A small tilt of the mirror shown at the lower left side of the figure would introduce a linear phase shift on the reference beam. This tilt is generally useful in producing signature fringe patterns at the observation plane.   The flat mirror shown in the lower left side of the interferometer is mounted on a tip-tilt stage that allows the introduction of a small amount of tilt in the reference beam. Figure 7 shows the computed patterns of intensity distribution at the observation plane of the Mach-Zehnder interferometer corresponding to 3 waves of primary coma.4         Figure 7. Computed plots of intensity distribution at the observation plane of Fig. 6. The lens under test (NA = 0.5, f = 6000l0) is assumed to have 3 waves of primary coma, and its nominal focus is coincident with the focal point of the relay lens. The tilt angle y of the reference beam increases progressively from (a) to (f). (a) y = -0.1°, (b) y = 0°, (c) y  = 0.05°, (d)y = 0.07°, (e) y = 0.1°, (f)y  = 0.18°.   In obtaining the various frames of Fig. 7 we have fixed all the system parameters and only varied the tilt of the reference beam. Note that the characteristic fringes of coma in Fig. 7 are quite different from those of coma in the Ronchi test, shown in Figs. 4(d)-(f). Incidentally, the patterns of Fig. 7 show similarities with the Ronchigrams of spherical aberration displayed in Fig. 5. This is not a coincidence, and is rooted in the algebraic forms of the aberration function for third order coma (r3 cos f) and spherical aberration (r4 ), and also in the fact that a Ronchigram, being a kind of shearing interferogram (albeit with a large shear), is related to the derivative of the wavefront aberration function. Knife-edge and wire tests. A schematic diagram of the knife-edge method of testing optical systems is shown in Fig. 8. A geometric-optical interpretation of this test suffices for most practical purposes: the knife-edge blocks different groups of rays in its various positions along the optical axis, allowing the remaining rays to reach the observation plane.3 Another method of testing, known as the wire test, is quite similar to the knife-edge method, being obtained from it by the substitution of the knife-edge with a fine wire.3   Figure 8. In the knife-edge test a certain region in the vicinity of focus is blocked by a knife-edge; the nature and the magnitude of aberrations are then inferred from the resulting patterns of intensity distribution at the observation plane. (The knife-edge may be moved both along and perpendicular to the optical axis.) The wire test is similar to the knife-edge test except that a fine wire is used instead to block certain groups of rays.   Since the grating in the Ronchi test may be thought of as a series of parallel knife-edges or, more aptly, a series of parallel wires, it should not come as a surprise that similarities exist between Ronchigrams and the patterns observed in these other tests. In fact, early attempts at explaining the results of Ronchi?s method were based on geometrical optics, and considered the grating as a set of parallel wires whose shadows produced the observed patterns.5 We will not delve into these matters, but simply draw the reader?s attention to Figs. 9 and 10, where we show several computed patterns of intensity distribution for the knife-edge and wire tests, respectively.4     Figure 9. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the knife-edge test carried out with a laser beam. The lens under test (NA = 0.5, f = 6000l0) and the pupil relay lens (NA = 0.5) are assumed to be fixed in their respective positions, while the knife-edge moves along the optical axis. (The tip of the knife remains on the axis at all times.) The lens under test is assumed to have 3 waves of primary spherical aberration. In frames (a) to (d) the distance of the knife-edge from paraxial focus is Dz = -15l0, 0, +15l0, and +20l0, respectively. (Positive distances are in the direction of the marginal focus.)   The results of the simulated knife-edge test depicted in Fig. 9 assume a laser as the light source. Consequently, frames (a) and (b) of Fig. 9 exhibit several dark lines which, with a less coherent light source, would have been absent. The results of the simulated wire test shown in Fig. 10 assume an extended light source, since the small amount of spherical aberration present in the system under consideration would render the test useless with a wire that, fine as it may be, is still wider than the focused spot produced by a laser beam. Note the similarities between the patterns of Figs. 9 and 10 on the one hand, and those of Figs. 5(d)-(f) on the other.   Figure 10. Computed plots of intensity distribution at the observation plane of Fig. 8 corresponding to the wire test with an extended, quasi-monochromatic light source. The lens under test (NA = 0.5, f = 6000l0) has 3 waves of primary spherical aberration. The assumed wire diameter is 15l0, which is comparable to the size of the image of the extended light source, as measured in the vicinity of focus. In (a) the wire is centered on axis and is 25l0 away from paraxial focus (in the direction of the marginal focus). In (b) the wire is again centered on axis, but is 20l0 away from paraxial focus. In (c) the wire is shifted 0.5l0 off-axis while its distance from paraxial focus remains at 20l0.   Extensions of the Ronchi test. Several modifications and extensions of the Ronchi test have appeared over the years, and have helped to solve specific problems in testing of optical systems.3 As an example we mention the "double-frequency grating lateral shear interferometer" invented by James Wyant in the early 1970?s. The grating in this device has two slightly different frequencies, which give rise to two +1st order beams as well as two -1st order beams; the beams in each pair are slightly shifted relative to each other. Moreover, the (average) pitch of the grating is such that there is no overlap between the 0th, +1st, and -1st orders. Consequently, interference occurs between the two +1st order beams (and, likewise, between the two -1st order beams). One can thus obtain an arbitrarily small lateral shear of the wavefront under test, and use the results to achieve accurate quantitative measurements. A two-dimensional version of the double-frequency grating has also been employed to generate lateral wavefront shear simultaneously along the X and Y axes. (Remember that beam propagation is along Z and, therefore, X and Y are orthogonal axes in the plane of the grating.) In the absence of  a 2-D grating, one must rotate a 1-D grating by 90° to obtain  wavefront shear first along the X- and then along the Y-axis. Acknowledgment. I am grateful to Professor Roland Shack of the Optical Sciences Center for many illuminating discussions, and also for suggesting some of the examples presented in this article.   References V. Ronchi, "Le Frange di Combinazioni Nello Studio delle Superficie e dei Sistemi Ottici," Riv. Ottica Mecc. Precis. 2, 9 (1923). V. Ronchi, "Due Nuovi Metodi per lo Studio delle Superficie e dei Sistemi Ottici," Ann. Sc. Norm. Super. Pisa, 15 (1923). D. Malacara, ed., Optical Shop Testing, second edition, Wiley, New York, 1992. The computer simulations reported in this article were performed by DIFFRACT?, a product of MM Research, Inc., Tucson, Arizona. G. Toraldo di Francia, "Geometrical and interferential aspects of the Ronchi Test," in Optical Image Evaluation, National Bureau of Standards Circular 526, issued April 29, 1954. J. C. Wyant, "Double frequency grating lateral shear interferometer," Appl. Opt. 12, 2057 (1973). << Back to list of Articles Home | About MM Research, Inc. | Online Publications Diffract | SIM 3D_Max | Multilayer | Temprofile © Copyright 1987-2011, MM Research, Inc. 5748 N. Camino del Conde, Tucson, Arizona 85718

Like his predecessors, Witelo (born circa 1230, died between 1280 and 1314) drew on the extensive body of optical works recently translated from Greek and Arabic to produce a massive presentation of the subject entitled the Perspectiva. His theory of vision follows Alhacen and he does not consider Bacon's concept of species, although passages in his work demonstrate that he was influenced by Bacon's ideas. Judging from the number of surviving manuscripts, his work was not as influential as those of Pecham and Bacon, yet his importance, and that of Pecham, grew with the invention of printing.[30]