Some polarizers eliminate the non-passed polarization component (Ey in the above example) by absorbing it, while others reflect this component. Absorbing polarizers are convenient when it is desirable to completely eliminate one polarization component from the system. A disadvantage  of absorbing polarizers is that they are not very durable and may be damaged by high intensity light (as found in many laser applications).When a reflective polarizer is operated in such a way that the blocked (i.e., reflected) polarization component is deflected into a convenient direction, such as 90° relative to the transmitted polarization component, then the polarizer acts like a polarizing beamsplitter, as shown below.

What is polarizationofwaves in Physics

Suppose the two components have equal amplitudes again, but now consider the case where these two components are not in phase, such that the angles of the sine functions are different. In particular, suppose there is a constant phase difference of p/2 between them, which corresponds to a distance of l/4 in the “fixed time” picture. The x component is

The polarization of light reflected and transmitted at an interface between two media or at a thin-film multilayer coating can be altered dramatically. These two cases are considered below.

Following table gives design specifics for doublets achromats consisting of BK7 crown and F2 flint, with all measures in units of the focal length. The specs are based on 100mm f/10 objectives, but are scalable within the normal range of refractor apertures, up to about 50% faster or any slower objective (by applying the desired focal number ratio to 10 directly to the radii), with only minor raytrace adjustments. Substituting similar glasses should also require only minor adjustments; glass thickness is generally not a significant factor, the exception being the Gauss objective.

A long time standard for doublet achromats is the Fraunhofer doublet. It is relatively easy to make, free from coma and, as any other doublet achromat, about as well corrected for secondary spectrum as a doublet made of ordinary glasses can be. The doublet consists from the positive front crown element, and negative rear flint element. The radii of lens curvature vary somewhat with the particulars of a doublet; for the standard crown and flint combination (BK7/F2) they are approximately R1~0.61f, R2~-0.35f, with the focal length of about 0.44f, and R3~-0.36f and R4~-1.48f, with the focal length of about 0.78f, R1-4 being the lens surface radii from the front to the rear, and f being the final system focal length (alternatively, the curvatures can be expressed in the inverse form, f/Ri, as c1~1.64, c2~-2.86, c3~-2.78 and c4~-0.68). This also approximates the lens element focal lengths as f1~0.43f for the crown and f2~-0.76f for the flint.

As the examples above indicate, required change of the basic (front) doublet achromat in an optimized 2-doublet arrangement are relatively small. It mostly limits to bending the lenses to obtain either flat field or cancel astigmatism. Its chromatic correction can remain unchanged (infinity-corrected) if the rear achromat's is set for its object distance (which is, effectively, equal to the separation between rear doublet and virtual image formed by the front doublet). The former generally needs to be somewhat overcorrected in the blue (i.e. with blue and green having nearly a common focus, and the red focusing farther away), as well as somewhat overcorrected spherical-aberration-wise vs. doublet corrected for infinity (unless the front doublet induces offsetting aberrations). Use of special glasses for the rear corrector does not appreciably improves chromatic correction of such systems, because it is already being generated by the front doublet. For significantly improved chromatic correction, rear doublet has to be designed so that it offsets the chromatism induced by the front doublet, which requires complex lens systems. Alternately, both doublets have to be made with special glasses. A system with apochromatic correction can be made with common crown and flint, but such system would require three groups of lenses (two separated doublets and a positive lens closer to the focal plane for lateral color correction), and would be significantly longer than the effective focal length. It would also require strongly curved surfaces, generating significant higher order spherical aberration. Use of special low-dispersion glasses in combination with common glass types makes possible much higher level of chromatic correction in a doublet, triplet or Petzval refractor. The degree of improvement is determined by the respective properties of the two glasses combined, and can vary significantly. Somewhat informally, such refracting objectives are referred to as semi-apo and apochromatic (apo). Following page gives several examples, including some that were, or still are  marketed, but before that a quick look on the effects of central line error and stopping down an achromat. EFFECT OF THE CENTRAL LINE ERROR ON CHROMATIC CORRECTION Presence of spherical aberration in the central line means that it is inevitably carries over to all others. Since spherical aberration in the red and blue are usually of opposite sign at the minimized central line error, any induced sphrical aberration (from miscollimation, or radius error) will add to one, and subtract from the other one. In general, spherochromatism in achromats is low, dominated by chromatic defocus, hence a small to moderate central line error will not significantly affect their chromatic correction. As raytrace below shows, that is the case with as fast as a 100mm f/5 unit (top). 1/5 wave P-V of spherical aberration only slightly increased P-V error in the red, and have it slightly reduced in the blue. However, in a correction mode with colors tightly bound together - i.e. apochromatic correction - the effect can be significant. As little as 1/7 wave of overcorrection in a triplet shown (Ascar 185mm f/7) caused the error in the central line to exceede error in the red C line, which has the error reduced, while also appreciably increasing the blue and violet error. EFFECT OF STOPPING DOWN ACHROMAT Stoping down achromat generally reduces all its aberrations, but it will likely cause chromatic disbalance which may not be significant, but could be noticeable vs. achromat unit with the same aperture and focal ratio that has the colors optimally balanced. The longitudinal aberration plot shows that the color curves are being literally trimmed off at the mask radius level. Despite no change in the respective paraxial foci location, that effectively pulled the blue closer to the green focus, while pushing the red farther out. As the OPD plots show, the error is significantly reduced in both, F and C line, due to the larger Airy disc and decreased defocus sensitivity, but significantly more in the former. Spherical aberration is also significantly smaller. It is of no practical consequence in this case, with it being negligible at a full aperture, but can be significant if the error at full aperture is not negligible. For primary spherical aberration, the wavefront error changes in proportion to the 4th power of the aperture, but in fast achromats with a mix of lower and higher order SA, reduction factor will vary somewhat; in general, it is smaller.   ◄ 9. REFRACTING TELESCOPES   ▐    9.2. Refracting telescope objectives: Apo and semi-apo ► Home  |  Comments

It is a small 32mm f/24 lens with low residual spherical aberration. As most of the objectives from that time, it is somewhat overcorrected (vs. singlet, which has shorter wavelengths focusing closer to the objective than the longer ones). The likely reason is that it was compensating for chromatism of the 5-element terestrial eyepiece used at that time (An investigation of the eighteenth-century achromatic telescope, D.H. Jaecks 2009). Strictly talking, it is not an achromat, since no two widely separated wavelengths come to the same focus, but the level of correction is close to it. For full correction, it is necessary to make the negative element weaker (bottom box, left, R1=-3250mm). While spherical aberration is only slightly higher, significant portion of the F and C-line error comes from it. With corrected spherical aberration (R3=-211mm and, to keep F and C close together, R1=-1900mm), an optimized version of this objective (bottom right) can be compared to the modern standard achromat: 1/16 wave P-V in the F and C line, for given aperture and focal ratio, correspond to those of f/61 standard achromat, with the correction in the violet g-line (0.22 wave P-V) corresponds to that of f/89 achromat. In other words, Dollond's doublet has 2.5 times better correction in F/C, and 3.7 times better g-line correction. Some coma is present, but at this focal ratio and aperture size not noticeable (in the original configuration, 0.083 wave RMS at 1° off axis), hence Dollonds didn't need to know the calculation for minimizing it. They certainly had some knowledge about controlling spherical aberration, but judging on multiple factors - their keeping relative aperture very small, aspherizing, triplet arrangement - it was incomplete. Spherical aberration is easily controlled by balancing the two inner radii. After that, a number of doublet types have been developed. An achromat, by definition, uses two common glasses - crown and flint - to reduce primary chromatism (chromatism of a single lens). Since chromatic correction of a doublet depends mainly on the glass combination, and common glasses span relatively narrow range of properties (i.e. refractive index and dispersion), doublets achromats have similar level of chromatism and differ mostly in their correction of monochromatic aberrations.

The longitudinal aberration plot shows that the color curves are being literally trimmed off at the mask radius level. Despite no change in the respective paraxial foci location, that effectively pulled the blue closer to the green focus, while pushing the red farther out. As the OPD plots show, the error is significantly reduced in both, F and C line, due to the larger Airy disc and decreased defocus sensitivity, but significantly more in the former. Spherical aberration is also significantly smaller. It is of no practical consequence in this case, with it being negligible at a full aperture, but can be significant if the error at full aperture is not negligible. For primary spherical aberration, the wavefront error changes in proportion to the 4th power of the aperture, but in fast achromats with a mix of lower and higher order SA, reduction factor will vary somewhat; in general, it is smaller.   ◄ 9. REFRACTING TELESCOPES   ▐    9.2. Refracting telescope objectives: Apo and semi-apo ► Home  |  Comments

However, if the optical system is in any way sensitive to polarization, even when the incident light is unpolarized, it is important to recognize that the beamsplitter can transmit and reflect different amounts of the “s” and “p” polarization states, as shown below.

Polarisationmeaning in Physics

A system with apochromatic correction can be made with common crown and flint, but such system would require three groups of lenses (two separated doublets and a positive lens closer to the focal plane for lateral color correction), and would be significantly longer than the effective focal length. It would also require strongly curved surfaces, generating significant higher order spherical aberration. Use of special low-dispersion glasses in combination with common glass types makes possible much higher level of chromatic correction in a doublet, triplet or Petzval refractor. The degree of improvement is determined by the respective properties of the two glasses combined, and can vary significantly. Somewhat informally, such refracting objectives are referred to as semi-apo and apochromatic (apo). Following page gives several examples, including some that were, or still are  marketed, but before that a quick look on the effects of central line error and stopping down an achromat. EFFECT OF THE CENTRAL LINE ERROR ON CHROMATIC CORRECTION Presence of spherical aberration in the central line means that it is inevitably carries over to all others. Since spherical aberration in the red and blue are usually of opposite sign at the minimized central line error, any induced sphrical aberration (from miscollimation, or radius error) will add to one, and subtract from the other one. In general, spherochromatism in achromats is low, dominated by chromatic defocus, hence a small to moderate central line error will not significantly affect their chromatic correction. As raytrace below shows, that is the case with as fast as a 100mm f/5 unit (top). 1/5 wave P-V of spherical aberration only slightly increased P-V error in the red, and have it slightly reduced in the blue. However, in a correction mode with colors tightly bound together - i.e. apochromatic correction - the effect can be significant. As little as 1/7 wave of overcorrection in a triplet shown (Ascar 185mm f/7) caused the error in the central line to exceede error in the red C line, which has the error reduced, while also appreciably increasing the blue and violet error. EFFECT OF STOPPING DOWN ACHROMAT Stoping down achromat generally reduces all its aberrations, but it will likely cause chromatic disbalance which may not be significant, but could be noticeable vs. achromat unit with the same aperture and focal ratio that has the colors optimally balanced. The longitudinal aberration plot shows that the color curves are being literally trimmed off at the mask radius level. Despite no change in the respective paraxial foci location, that effectively pulled the blue closer to the green focus, while pushing the red farther out. As the OPD plots show, the error is significantly reduced in both, F and C line, due to the larger Airy disc and decreased defocus sensitivity, but significantly more in the former. Spherical aberration is also significantly smaller. It is of no practical consequence in this case, with it being negligible at a full aperture, but can be significant if the error at full aperture is not negligible. For primary spherical aberration, the wavefront error changes in proportion to the 4th power of the aperture, but in fast achromats with a mix of lower and higher order SA, reduction factor will vary somewhat; in general, it is smaller.   ◄ 9. REFRACTING TELESCOPES   ▐    9.2. Refracting telescope objectives: Apo and semi-apo ► Home  |  Comments

Presence of spherical aberration in the central line means that it is inevitably carries over to all others. Since spherical aberration in the red and blue are usually of opposite sign at the minimized central line error, any induced sphrical aberration (from miscollimation, or radius error) will add to one, and subtract from the other one. In general, spherochromatism in achromats is low, dominated by chromatic defocus, hence a small to moderate central line error will not significantly affect their chromatic correction. As raytrace below shows, that is the case with as fast as a 100mm f/5 unit (top). 1/5 wave P-V of spherical aberration only slightly increased P-V error in the red, and have it slightly reduced in the blue. However, in a correction mode with colors tightly bound together - i.e. apochromatic correction - the effect can be significant. As little as 1/7 wave of overcorrection in a triplet shown (Ascar 185mm f/7) caused the error in the central line to exceede error in the red C line, which has the error reduced, while also appreciably increasing the blue and violet error. EFFECT OF STOPPING DOWN ACHROMAT Stoping down achromat generally reduces all its aberrations, but it will likely cause chromatic disbalance which may not be significant, but could be noticeable vs. achromat unit with the same aperture and focal ratio that has the colors optimally balanced. The longitudinal aberration plot shows that the color curves are being literally trimmed off at the mask radius level. Despite no change in the respective paraxial foci location, that effectively pulled the blue closer to the green focus, while pushing the red farther out. As the OPD plots show, the error is significantly reduced in both, F and C line, due to the larger Airy disc and decreased defocus sensitivity, but significantly more in the former. Spherical aberration is also significantly smaller. It is of no practical consequence in this case, with it being negligible at a full aperture, but can be significant if the error at full aperture is not negligible. For primary spherical aberration, the wavefront error changes in proportion to the 4th power of the aperture, but in fast achromats with a mix of lower and higher order SA, reduction factor will vary somewhat; in general, it is smaller.   ◄ 9. REFRACTING TELESCOPES   ▐    9.2. Refracting telescope objectives: Apo and semi-apo ► Home  |  Comments

Using this description of a single transverse orientation of a light wave, we can now consider multiple orientations to describe different states of polarization.

That is, E appears to oscillate along a line oriented at 45° with respect to the x axis. Hence this situation is called linear polarization.Notice that equivalently we could view the wave at a particular location (“fixed position”) and watch its amplitude evolve with time. Suppose we sit at the position z = 0. Then we see that

If the difference between the two phase values is p/2, then the wave emerging from the material (say into air) will be circularly polarized. This occurs when

As mentioned on the previous page, the idea of a doublet achromat probably comes from Chester Moore Hall in the 1730s, but it was John Dollond who had it patented in 1758 and succeeded in producing working doublet achromats. The original Dollond's doublet - possibly as the one claimed by Hall (order not specified) - had flint (negative element) in front, followed by crown, with all surfaces spherical. His son Peter modified it by putting crown in front, and correcting residual spherical aberration by aspherizing the 4th surface, polishing off its peripheral area (Dollond and Son's pursuit of achromaticity, R.Sorrenson 2001). On the other hand, there is also evidence that he was trying to minimize spherical aberration by making focal length of the central region of the crown front radius longer by removing glass from it (New Light on the Invention of the Achromatic Telescope Objective, R. Willach 1996). That would, however, be ineffective, since most of the aberration is generated on the outer lens area (in addition to the front radius generating only a small fraction of spherical aberration vs. inner radius). From what is avalable, it seems fair to conclude that it was John Dollond who made the crucial step from the idea of combining two different glasses to actually producing lens objective with chromatism reduced to the level of an achromat (in court proceedings from 1765, Hall's doublet was described as consisting of a plano-convex crown and plano-concave flint element, with the latter having focal length three times longer; such doublet would not just have only marginally lower chromatism than a singlet lens, but would also suffer from unacceptably high spherical aberration due to a large differential between two radii). Experimenting with prisms, John Dollond determined approximate value of the focal lenghts ratio for the two lenses as 3:2 (flint-to-crown), and minimized spherical aberration by making the flint element a negative meniscus, which allowed for the stronger concave radius, nearly equal to the crown element's inner radius. Below is raytrace of one of his achromats (data from Remarks to the article: New Light on the Invention of the Achromatic Telescope Objective, I. Nesterenko 2017). Note that only e, F and C-line index is measured, with the g and r values being determined by raytracing software (OSLO Edu). It is a small 32mm f/24 lens with low residual spherical aberration. As most of the objectives from that time, it is somewhat overcorrected (vs. singlet, which has shorter wavelengths focusing closer to the objective than the longer ones). The likely reason is that it was compensating for chromatism of the 5-element terestrial eyepiece used at that time (An investigation of the eighteenth-century achromatic telescope, D.H. Jaecks 2009). Strictly talking, it is not an achromat, since no two widely separated wavelengths come to the same focus, but the level of correction is close to it. For full correction, it is necessary to make the negative element weaker (bottom box, left, R1=-3250mm). While spherical aberration is only slightly higher, significant portion of the F and C-line error comes from it. With corrected spherical aberration (R3=-211mm and, to keep F and C close together, R1=-1900mm), an optimized version of this objective (bottom right) can be compared to the modern standard achromat: 1/16 wave P-V in the F and C line, for given aperture and focal ratio, correspond to those of f/61 standard achromat, with the correction in the violet g-line (0.22 wave P-V) corresponds to that of f/89 achromat. In other words, Dollond's doublet has 2.5 times better correction in F/C, and 3.7 times better g-line correction. Some coma is present, but at this focal ratio and aperture size not noticeable (in the original configuration, 0.083 wave RMS at 1° off axis), hence Dollonds didn't need to know the calculation for minimizing it. They certainly had some knowledge about controlling spherical aberration, but judging on multiple factors - their keeping relative aperture very small, aspherizing, triplet arrangement - it was incomplete. Spherical aberration is easily controlled by balancing the two inner radii. After that, a number of doublet types have been developed. An achromat, by definition, uses two common glasses - crown and flint - to reduce primary chromatism (chromatism of a single lens). Since chromatic correction of a doublet depends mainly on the glass combination, and common glasses span relatively narrow range of properties (i.e. refractive index and dispersion), doublets achromats have similar level of chromatism and differ mostly in their correction of monochromatic aberrations.

The angle of the reflected ray,θr, is always equal to the angle of the incident ray, θi, this result is called the “law of reflection.” The angle of the transmitted (or refracted) ray, θT, is related to the angle of incidence by the well-known “Snell’s Law” relationship: ni sin θinbsp;= nt sin θT. It turns out that s-polarized light is always more highly reflected than p-polarized light. In fact, at a special angle called “Brewster’s Angle,” denoted θB, the p-polarized component sees no reflection, or is completely transmitted. Brewster’s angle is given by θB = arctan(nt/ni). The power or intensity reflection coefficients for a light wave (i.e., the squares of the amplitude reflection coefficients) for air-to-glass (left) and glass-to-air (right) look like:

If Ax  Ay , the total wave E is linearly polarized, but it is no longer oriented at 45° with respect to the x axis. In fact we can see that it is oriented at an angle  where

To understand the polarization of light, we must first recognize that light can be described as a classical wave. The most basic parameters that describe any wave are the amplitude and the wavelength. For example, the amplitude of a wave represents the longitudinal displacement of air molecules for a sound wave traveling through the air, or the transverse displacement of a string or water molecules for a wave on a guitar string or on the surface of a pond, respectively. We will refer to the amplitude of a light wave with the letter “E.” The amplitude of a light wave represents the potential for a charged particle (such as an electron) to feel a force – formally it may represent the “electric field” of an electromagnetic wave. Because this potential vibrates along the directions transverse to the direction the wave is traveling, light is a “transverse wave,” just like the waves on a string or water surface.Because light is a transverse wave, and because there are two transverse dimensions, there are fundamentally two distinct directions in which the light wave may oscillate. Let’s call these the x and y directions for a light wave traveling along the z direction. We’ll call the two distinct waves Ex and Ey, where we denote these by vectors to remind us that they point in (or oscillate along) a certain direction (the x and y directions, respectively).The amplitude of the light wave describes how the wave propagates in position and time. Mathematically, we can write it as a “sine wave” where the angle of the sine function is a linear combination of both position and time terms:

Polarization examples

When light is incident on an interface between two different media with different indexes of refraction, some of the light is reflected and some is transmitted. When the angle of incidence is not normal, different polarizations are reflected (and transmitted) by different amounts. This dependence was first properly described by Fresnel, and hence it is often called “Fresnel Reflection.” It is simplest to describe the polarization of the incident, reflected, and transmitted (refracted) light in terms of a vector component perpendicular to the plane of incidence, called the “s” component, and a component parallel to the plane of incidence, called the “p” component. The “plane of incidence” is the plane which contains the incident ray and the transmitted and reflected rays (i.e., all of these rays lie on one plane). In the example in the diagram below, the plane of incidence is the plane containing the x and z axes. That is, Es || y, while Ep lies in the x-z plane.

Presence of spherical aberration in the central line means that it is inevitably carries over to all others. Since spherical aberration in the red and blue are usually of opposite sign at the minimized central line error, any induced sphrical aberration (from miscollimation, or radius error) will add to one, and subtract from the other one. In general, spherochromatism in achromats is low, dominated by chromatic defocus, hence a small to moderate central line error will not significantly affect their chromatic correction. As raytrace below shows, that is the case with as fast as a 100mm f/5 unit (top). 1/5 wave P-V of spherical aberration only slightly increased P-V error in the red, and have it slightly reduced in the blue. However, in a correction mode with colors tightly bound together - i.e. apochromatic correction - the effect can be significant. As little as 1/7 wave of overcorrection in a triplet shown (Ascar 185mm f/7) caused the error in the central line to exceede error in the red C line, which has the error reduced, while also appreciably increasing the blue and violet error. EFFECT OF STOPPING DOWN ACHROMAT Stoping down achromat generally reduces all its aberrations, but it will likely cause chromatic disbalance which may not be significant, but could be noticeable vs. achromat unit with the same aperture and focal ratio that has the colors optimally balanced. The longitudinal aberration plot shows that the color curves are being literally trimmed off at the mask radius level. Despite no change in the respective paraxial foci location, that effectively pulled the blue closer to the green focus, while pushing the red farther out. As the OPD plots show, the error is significantly reduced in both, F and C line, due to the larger Airy disc and decreased defocus sensitivity, but significantly more in the former. Spherical aberration is also significantly smaller. It is of no practical consequence in this case, with it being negligible at a full aperture, but can be significant if the error at full aperture is not negligible. For primary spherical aberration, the wavefront error changes in proportion to the 4th power of the aperture, but in fast achromats with a mix of lower and higher order SA, reduction factor will vary somewhat; in general, it is smaller.   ◄ 9. REFRACTING TELESCOPES   ▐    9.2. Refracting telescope objectives: Apo and semi-apo ► Home  |  Comments

where A is called the “amplitude factor,” the variable l (“lambda”) is the “wavelength” (units of nm), and the variable v (“nu”) is the “frequency” (units of Hz, or sec–1). If a snapshot of the wave could be taken at a fixed time, l would be the distance from one wave peak to the next. If one sits at a fixed point in space and counts the wave peaks as they pass by, v gives the frequency of these counts, or 1/v gives the time between peaks. The sign between the position and time terms determines the direction the wave travels: when the two terms have the opposite sign (i.e., the “–” sign is chosen), the wave travels in the positive z direction. For convenience we often use two new variables called the “wavenumber” k = 2p/l and the “angular frequency” 2pv (“omega”), which absorb the factor of 2p, so that the wave amplitude can now be written more compactly as

1/5 wave P-V of spherical aberration only slightly increased P-V error in the red, and have it slightly reduced in the blue. However, in a correction mode with colors tightly bound together - i.e. apochromatic correction - the effect can be significant. As little as 1/7 wave of overcorrection in a triplet shown (Ascar 185mm f/7) caused the error in the central line to exceede error in the red C line, which has the error reduced, while also appreciably increasing the blue and violet error. EFFECT OF STOPPING DOWN ACHROMAT Stoping down achromat generally reduces all its aberrations, but it will likely cause chromatic disbalance which may not be significant, but could be noticeable vs. achromat unit with the same aperture and focal ratio that has the colors optimally balanced. The longitudinal aberration plot shows that the color curves are being literally trimmed off at the mask radius level. Despite no change in the respective paraxial foci location, that effectively pulled the blue closer to the green focus, while pushing the red farther out. As the OPD plots show, the error is significantly reduced in both, F and C line, due to the larger Airy disc and decreased defocus sensitivity, but significantly more in the former. Spherical aberration is also significantly smaller. It is of no practical consequence in this case, with it being negligible at a full aperture, but can be significant if the error at full aperture is not negligible. For primary spherical aberration, the wavefront error changes in proportion to the 4th power of the aperture, but in fast achromats with a mix of lower and higher order SA, reduction factor will vary somewhat; in general, it is smaller.   ◄ 9. REFRACTING TELESCOPES   ▐    9.2. Refracting telescope objectives: Apo and semi-apo ► Home  |  Comments

Polarization is a fundamental property of light. While many optical applications are based on systems that are “blind” to polarization, a very large number are not. Some applications rely directly on polarization as a key measurement variable, such as those based on how much an object depolarizes or rotates a polarized probe beam. For other applications, variations due to polarization are a source of noise, and thus throughout the system light must maintain a fixed state of polarization – or remain completely depolarized – to eliminate these variations. And for applications based on interference of non-parallel light beams, polarization greatly impacts contrast. As a result, for a large number of applications control of polarization is just as critical as control of ray propagation, diffraction, or the spectrum of the light. Yet despite its importance, polarization is often considered a more esoteric property of light that is not so well understood. In this article our aim is to answer some basic questions about the polarization of light, including: what polarization is and how it is described, how it is controlled by optical components, and when it matters in optical systems.

It is also possible to take advantage of an appreciable difference in reflected or transmitted phase for p- and s-polarized light over a region of the spectrum where the reflected and transmitted intensities are essentially equal, thus forming a waveplate.

Triplet achromats require more glass and work, but offer no significantly better correction of aberrations - either monochromatic or chromatic - than doublet achromat. The only possibly beneficial use of the triplet would be for very fast, large achromats with significant level of higher-order spherical aberration.

Use of special glasses for the rear corrector does not appreciably improves chromatic correction of such systems, because it is already being generated by the front doublet. For significantly improved chromatic correction, rear doublet has to be designed so that it offsets the chromatism induced by the front doublet, which requires complex lens systems. Alternately, both doublets have to be made with special glasses. A system with apochromatic correction can be made with common crown and flint, but such system would require three groups of lenses (two separated doublets and a positive lens closer to the focal plane for lateral color correction), and would be significantly longer than the effective focal length. It would also require strongly curved surfaces, generating significant higher order spherical aberration. Use of special low-dispersion glasses in combination with common glass types makes possible much higher level of chromatic correction in a doublet, triplet or Petzval refractor. The degree of improvement is determined by the respective properties of the two glasses combined, and can vary significantly. Somewhat informally, such refracting objectives are referred to as semi-apo and apochromatic (apo). Following page gives several examples, including some that were, or still are  marketed, but before that a quick look on the effects of central line error and stopping down an achromat. EFFECT OF THE CENTRAL LINE ERROR ON CHROMATIC CORRECTION Presence of spherical aberration in the central line means that it is inevitably carries over to all others. Since spherical aberration in the red and blue are usually of opposite sign at the minimized central line error, any induced sphrical aberration (from miscollimation, or radius error) will add to one, and subtract from the other one. In general, spherochromatism in achromats is low, dominated by chromatic defocus, hence a small to moderate central line error will not significantly affect their chromatic correction. As raytrace below shows, that is the case with as fast as a 100mm f/5 unit (top). 1/5 wave P-V of spherical aberration only slightly increased P-V error in the red, and have it slightly reduced in the blue. However, in a correction mode with colors tightly bound together - i.e. apochromatic correction - the effect can be significant. As little as 1/7 wave of overcorrection in a triplet shown (Ascar 185mm f/7) caused the error in the central line to exceede error in the red C line, which has the error reduced, while also appreciably increasing the blue and violet error. EFFECT OF STOPPING DOWN ACHROMAT Stoping down achromat generally reduces all its aberrations, but it will likely cause chromatic disbalance which may not be significant, but could be noticeable vs. achromat unit with the same aperture and focal ratio that has the colors optimally balanced. The longitudinal aberration plot shows that the color curves are being literally trimmed off at the mask radius level. Despite no change in the respective paraxial foci location, that effectively pulled the blue closer to the green focus, while pushing the red farther out. As the OPD plots show, the error is significantly reduced in both, F and C line, due to the larger Airy disc and decreased defocus sensitivity, but significantly more in the former. Spherical aberration is also significantly smaller. It is of no practical consequence in this case, with it being negligible at a full aperture, but can be significant if the error at full aperture is not negligible. For primary spherical aberration, the wavefront error changes in proportion to the 4th power of the aperture, but in fast achromats with a mix of lower and higher order SA, reduction factor will vary somewhat; in general, it is smaller.   ◄ 9. REFRACTING TELESCOPES   ▐    9.2. Refracting telescope objectives: Apo and semi-apo ► Home  |  Comments

Unpolarized light can be polarized using a “polarizer” or “polarizing beamsplitter,” and the state of already polarized light can be altered using a polarizer and/or optical components that are “birefringent.” In this section we explore some examples of these types of components.

Polarization is a critical property of light for many optical systems and applications. This brief tutorial summarizes some of the most basic aspects of polarization, including how it is described, the impact of polarizing and birefringent elements on light, and how optical interfaces and filters can change the polarization of light.

In other words, if we look down the propagation axis in the positive x direction, the vector E at various locations (and at t = 0) now looks like:

We can see that the tip of E traces out a circle as we follow the wave along the z axis at a fixed time. Similarly, if we sit at a fixed position, the tip of E appears to trace out a circle as time evolves. Hence this type of polarization is called circular polarization.

A polarizer transmits only a single orientation of linear polarization, and blocks the rest of the light. For example, a polarizer oriented along x passes x and blocks Ey.

We can see that in general the light emerges in a different state of elliptic polarization. In fact, for the example illustrated above, the particular choice of L for a given difference between nx and ny causes the linearly polarized light at the input end to be converted to circularly polarized light at the other end of the birefringent material. How did this happen? Let’s look at the math. Consider the phases accumulated by the two component waves as they travel through the birefringent material. The waves can be described by

1/5 wave P-V of spherical aberration only slightly increased P-V error in the red, and have it slightly reduced in the blue. However, in a correction mode with colors tightly bound together - i.e. apochromatic correction - the effect can be significant. As little as 1/7 wave of overcorrection in a triplet shown (Ascar 185mm f/7) caused the error in the central line to exceede error in the red C line, which has the error reduced, while also appreciably increasing the blue and violet error. EFFECT OF STOPPING DOWN ACHROMAT Stoping down achromat generally reduces all its aberrations, but it will likely cause chromatic disbalance which may not be significant, but could be noticeable vs. achromat unit with the same aperture and focal ratio that has the colors optimally balanced. The longitudinal aberration plot shows that the color curves are being literally trimmed off at the mask radius level. Despite no change in the respective paraxial foci location, that effectively pulled the blue closer to the green focus, while pushing the red farther out. As the OPD plots show, the error is significantly reduced in both, F and C line, due to the larger Airy disc and decreased defocus sensitivity, but significantly more in the former. Spherical aberration is also significantly smaller. It is of no practical consequence in this case, with it being negligible at a full aperture, but can be significant if the error at full aperture is not negligible. For primary spherical aberration, the wavefront error changes in proportion to the 4th power of the aperture, but in fast achromats with a mix of lower and higher order SA, reduction factor will vary somewhat; in general, it is smaller.   ◄ 9. REFRACTING TELESCOPES   ▐    9.2. Refracting telescope objectives: Apo and semi-apo ► Home  |  Comments

From what is avalable, it seems fair to conclude that it was John Dollond who made the crucial step from the idea of combining two different glasses to actually producing lens objective with chromatism reduced to the level of an achromat (in court proceedings from 1765, Hall's doublet was described as consisting of a plano-convex crown and plano-concave flint element, with the latter having focal length three times longer; such doublet would not just have only marginally lower chromatism than a singlet lens, but would also suffer from unacceptably high spherical aberration due to a large differential between two radii). Experimenting with prisms, John Dollond determined approximate value of the focal lenghts ratio for the two lenses as 3:2 (flint-to-crown), and minimized spherical aberration by making the flint element a negative meniscus, which allowed for the stronger concave radius, nearly equal to the crown element's inner radius. Below is raytrace of one of his achromats (data from Remarks to the article: New Light on the Invention of the Achromatic Telescope Objective, I. Nesterenko 2017). Note that only e, F and C-line index is measured, with the g and r values being determined by raytracing software (OSLO Edu). It is a small 32mm f/24 lens with low residual spherical aberration. As most of the objectives from that time, it is somewhat overcorrected (vs. singlet, which has shorter wavelengths focusing closer to the objective than the longer ones). The likely reason is that it was compensating for chromatism of the 5-element terestrial eyepiece used at that time (An investigation of the eighteenth-century achromatic telescope, D.H. Jaecks 2009). Strictly talking, it is not an achromat, since no two widely separated wavelengths come to the same focus, but the level of correction is close to it. For full correction, it is necessary to make the negative element weaker (bottom box, left, R1=-3250mm). While spherical aberration is only slightly higher, significant portion of the F and C-line error comes from it. With corrected spherical aberration (R3=-211mm and, to keep F and C close together, R1=-1900mm), an optimized version of this objective (bottom right) can be compared to the modern standard achromat: 1/16 wave P-V in the F and C line, for given aperture and focal ratio, correspond to those of f/61 standard achromat, with the correction in the violet g-line (0.22 wave P-V) corresponds to that of f/89 achromat. In other words, Dollond's doublet has 2.5 times better correction in F/C, and 3.7 times better g-line correction. Some coma is present, but at this focal ratio and aperture size not noticeable (in the original configuration, 0.083 wave RMS at 1° off axis), hence Dollonds didn't need to know the calculation for minimizing it. They certainly had some knowledge about controlling spherical aberration, but judging on multiple factors - their keeping relative aperture very small, aspherizing, triplet arrangement - it was incomplete. Spherical aberration is easily controlled by balancing the two inner radii. After that, a number of doublet types have been developed. An achromat, by definition, uses two common glasses - crown and flint - to reduce primary chromatism (chromatism of a single lens). Since chromatic correction of a doublet depends mainly on the glass combination, and common glasses span relatively narrow range of properties (i.e. refractive index and dispersion), doublets achromats have similar level of chromatism and differ mostly in their correction of monochromatic aberrations.

Some materials have a different index of refraction for light polarized along different directions. This phenomenon is called birefringence. For example, suppose light polarized along the x direction sees an index of nx, while light polarized along the y direction sees an index ny. Now suppose linearly polarized light passes through a piece of such a material of length L, where the linear polarization axis is oriented at 45° with respect to the x and y axes. The fixed time picture thus looks like:

Typical astronomical doublet is air-spaced. Air space makes it possible to optimally correct for monochromatic aberrations with any glass combination. Cemented doublets, on the other side, require specific glass combinations that allow optimal correction with equal inner radii (for instance, BK7 and SF5; such doublets perform just as well with a small air gap). Another obstacle for cemented doublets fabrication is that in larger sizes - generally around 100mm in diameter, and larger - thermal expantion and contraction of the glass becomes detrimental for the bond between lenses. A long time standard for doublet achromats is the Fraunhofer doublet. It is relatively easy to make, free from coma and, as any other doublet achromat, about as well corrected for secondary spectrum as a doublet made of ordinary glasses can be. The doublet consists from the positive front crown element, and negative rear flint element. The radii of lens curvature vary somewhat with the particulars of a doublet; for the standard crown and flint combination (BK7/F2) they are approximately R1~0.61f, R2~-0.35f, with the focal length of about 0.44f, and R3~-0.36f and R4~-1.48f, with the focal length of about 0.78f, R1-4 being the lens surface radii from the front to the rear, and f being the final system focal length (alternatively, the curvatures can be expressed in the inverse form, f/Ri, as c1~1.64, c2~-2.86, c3~-2.78 and c4~-0.68). This also approximates the lens element focal lengths as f1~0.43f for the crown and f2~-0.76f for the flint. The inner two radii of the Fraunhofer can be equalized, without significant change in the correction level, on or off axis (in order to minimize ensuing spherical aberration, R4 is slightly weakened, and lens spacing slightly widened). Such modification is known as Baker doublet. Another coma-free doublet with reversed order (flint in front) is the Steinheil, which requires significantly more strongly curved surfaces (R1~0.43f, R2~-0.224f, R3~-0.223f and R4~-f for F2/BK7 glasses). Two other doublet achromat types of mostly historical significance are the Littrow, requiring even more strongly curved surfaces than the Steinheil, with more coma than comparable paraboloid, and the Clark, with somewhat less coma than the Littrow, but more lateral chromatism. Another older doublet type is the Cooke, which consists of the biconvex front and biconcave rear element; it has more than double the coma of Littrow, while no advantage of easier fabrication. Diagrams below illustrate basic properties of the main achromatic doublet types: longitudinal aberration plot for five spectral lines spanning most of the visual spectrum (g-436nm, F-486nm, e-546nm, C-656nm and r-707nm), axial F-e-C ray spots, P-V wavefront error at 0.5° off-axis (e-line) and best image curvature radius. FIGURE 146: Doublet achromat objective types for the common crown/flint combination, 100mm f/10 examples. The standard choice is Fraunhofer doublet, an aplanat consisting from biconvex front lens followed by a negative meniscus, with the secondary spectrum, defined as the longitudinal separation between the red/blue (C/F) focus and green (e-line) focus, of about 0.00055f (about 0.0005f measured from d-line). The only remaining monochromatic aberration is negligible astigmatism; best image curvature is about -0.36f (concave toward the objective). Its inverse form, with the flint element in front, known as the Steinheil, is nearly identical in performance, but requires more strongly curved lenses for given focal length. Another variation, with the inner radii equalized, and somewhat wider spacing to compensate for the spherical aberration it induced, is known as the Baker doublet and does not differ in its performance level from the other two. Unlike them, it is not strictly a contact doublet, but the air gap is relatively small. The original Littrow objective consists from the equconvex front lens followed by the negative lens of the same inner radius and flat last surface. With three identical radii and a flat surface, it is the easiest to fabricate, but at a price of some residual coma and spherical aberration. Since the astigmatism is nearly identical in all doublets and near negligible, nearly all of the 0.133 wave RMS at 0.5° off axis is coma, which makes it more than twice greater than in a comparable paraboloid (since the size of linear quality field with respect to coma changes with the third power of mirror's F-number, it makes this Littrow's coma comparable to that in a paraboloid with 30% larger f-ratio (f/7.7 for f/10 lens). Not a concern in visual observing, and neither is the residual spherical, which is here little over 1/12 wave P-V. The air gap is reduced to zero, in order to keep this residual spherical at its minimum. With larger lenses, the latter may become significant, in which case one of the two inner radii may need to be slightly changed in order to have it minimized or cancelled. Note that the last surface in the common crown/flint combination cannot be flat; it is mildly convex, since the -f1/f2=V2/V1 achromatic condition (Eq. 42) for the Littrow implies V2=(n2-1)V1/2(n1-1), where n, V and 1, 2 refer to the refractive index, Abbe number, front and rear lens, respectively, hence such lens configuration requires specific glass match (the only nearly matching flint for the flat-rear Littrow using BK7 is F15). A Littrow modification by Clark&Sons, known as Clark doublet, has the two lenses more widely separated (~15% of the focal length, according to Sidgwick), in order to make the inner surfaces accessible for cleaning without taking lenses out. Since widening the separation effectively weakens the rear relative to front lens, with the former in the Littrow already slightly too weak for cancelling spherical aberration, the Clark requires somewhat stronger (relative to R1 and R2) third surface (like the Littrow, the last surface can be flat only with a specific glass match; for the common crown/flint combinations it is mildly convex). The Clark's coma is somewhat lower than Littrow's, but its lateral color is, due to the wider lens separation, significantly larger, although still acceptable (less than 1/4 of the Airy disc diameter between F and C lines at 0.5° off axis for the above system). Also, best image surface is somewhat more strongly curved, indicating a bit stronger astigmatism: the respective Zernike coefficients for primary coma and astigmatism at 0.5° of 0.252 and 0.107 indicate 0.252/√8=0.088 and 0.107/√6= 0.044 wave RMS error, thus astigmatism about half as strong as coma, which is about a third lower than in the Littrow. Clark configuration can be coma-free with the first radius made stronger than the second one, and also stronger fourth radius. Finally, the alternative aplanatic doublet solution, consisting of a positive and negative meniscus, is the Gauss doublet. Since a meniscus requires significantly more strongly curved surfaces to achieve given power, this objective type is, in addition to be more difficult for fabrication, unsuitable for all but long focal ratio instruments. As raytrace shows, higher order spherical originating at the strongly curved surfaces - particularly R4 - makes it unacceptable already at 100mm f/10, with 0.057 wave RMS design limit in the e-line, and more than a third larger chromatic error due to spherochromatism (residual coma noticeable at the off-axis wavefront map cannot be further minimized without strengthening the radii even more, which would double the spherical error). Vice versa, accepting significantly more coma would allow for the weaker radii and reducing higher-order spherical to insignificant, but it would be still inferior to the other doublets, both, for more difficult fabrication and lower field quality. The positive is somewhat lower astigmatism - about 1/3 of the Clark objective, and nearly half of the other doublets' - and correspondingly less strongly curved best image surface.   Following table gives design specifics for doublets achromats consisting of BK7 crown and F2 flint, with all measures in units of the focal length. The specs are based on 100mm f/10 objectives, but are scalable within the normal range of refractor apertures, up to about 50% faster or any slower objective (by applying the desired focal number ratio to 10 directly to the radii), with only minor raytrace adjustments. Substituting similar glasses should also require only minor adjustments; glass thickness is generally not a significant factor, the exception being the Gauss objective. TABLE 11:  ACHROMAT DOUBLETS: GENERAL SPECIFICATIONS FOR BK7/F2 GLASS (in units of focal length) Type aplanat R1 t1 M1 R2 AIR (M2) R3 t3 M3 R4 FRAUNHOFER YES 0.6 0.011 BK7 -0.36 0.001 -0.363 0.007 F2 -1.51 STEINHEIL YES 0.442 0.007 F2 0.229 0.001 0.2285 0.009 BK7 -15 BAKER YES 0.582 0.011 BK7 -0.363 0.003 -0.363 0.007 F2 -1.59 LITTROW NO 0.45 0.01 BK7 -0.45 0.00005 -0.45 0.006 F2 -7.7 CLARK NO 0.428 0.01 BK7 -0.428 0.015 -0.4 0.0065 F2 -8 GAUSS (~f/14 and slower) YES 0.129 0.00665 BK7 0.3345 0.0003 0.14206 0.0042 F2 0.10357 COOKE NO 0.373 0.009 BK7 -0.563 0.001 -0.527 0.006 F2 4.7 A doublet with air gap wide as Clark's can be made aplanatic, in which case it is jus another variant of the Baker, with the third radius somewhat more strongly curved. Relatively unusual achromat designs are triplet achromats, as well as those with more than one group of lenses. The latter include Petzval-type achromats and those with the second lens group closer to the focal plane. By their basic form, they belong to dialyte objectives, defined as those employing widely separated elements. Triplet achromats require more glass and work, but offer no significantly better correction of aberrations - either monochromatic or chromatic - than doublet achromat. The only possibly beneficial use of the triplet would be for very fast, large achromats with significant level of higher-order spherical aberration. Petzval-type achromat is a design that uses two groups of lenses, with the rear group at 1/3 to 2/3 of the focal length of the front group behind (approximately; in the original Petzval configuration rear doublet is at half the focal length of the front doublet apart, its focal length half that of the front doublet, and the combined f.l. also half that of the front doublet). If using common crown and flint glasses, such arrangement can reduce secondary spectrum by approximately 15%; apparently, generally not considered worth the extra expense. With the second lens group closer to the focus of the front group, which is not Petzval configuration, rather one with sub-aperture corrector, secondary spectrum can be reduced somewhat more, up to about 30%. The reason for this is that the blue and red exit the front lens group - which is assumed to be doublet achromat - only slightly separated, but at different angles: as FIG. 96 hints, the red just below, and blue just above the green ray, with the former two converging to a common focus (or nearly so) and emerging above the green ray at some distance toward the focus. Thus there is no appreciable effect on longitudinal chromatism by the second group of lens, until it is far back enough for the red/blue rays to raise above the green ray, and get refracted more strongly at the rear lens group, focusing slightly shorter relative to the green light. Additional advantage of the dialyte form in an achromat is that astigmatism can be manipulated, either cancelled for less curved image field, or added in the opposite sign, still very low, in order to flatten the field, as illustrated below (FIG. 147C). FIGURE 147: Standard Fraunhofer doublet achromat can be combined with an auxiliary widely separated second doublet for enhanced correction (A) 100mm f/10 objective alone, for comparison (B) Petzval-type achromat with a cemented rear doublet. The front lens is an f/10 doublet. The only significant change is that the front doublet is bent to generate coma cancelling the coma of rear doublet, which in turn cancels out astigmatism. As a result field curvature equals the focal length, and there is no other monochromatic aberrations to speak of. Since the astigmatism is cancelled, best image surface coincides with the Petzval surface. (C) Another f/10 doublet in front, with a doublet sub-aperture (field) air spaced corrector. Again, the main modification is bending the front doublet to induce some coma, offsetting that of the corrector which, in turn, flattens the field w/o introducing significant astigmatism. In fact, astigmatism is lower than in a comparable achromat (D). Chromatism is also lower in both double achromats (somewhat more in the one with field corrector), without appreciably larger lateral color (note that the Airy disc for e-line spot at 0.5° off-axis is enlarged for clarity). As the examples above indicate, required change of the basic (front) doublet achromat in an optimized 2-doublet arrangement are relatively small. It mostly limits to bending the lenses to obtain either flat field or cancel astigmatism. Its chromatic correction can remain unchanged (infinity-corrected) if the rear achromat's is set for its object distance (which is, effectively, equal to the separation between rear doublet and virtual image formed by the front doublet). The former generally needs to be somewhat overcorrected in the blue (i.e. with blue and green having nearly a common focus, and the red focusing farther away), as well as somewhat overcorrected spherical-aberration-wise vs. doublet corrected for infinity (unless the front doublet induces offsetting aberrations). Use of special glasses for the rear corrector does not appreciably improves chromatic correction of such systems, because it is already being generated by the front doublet. For significantly improved chromatic correction, rear doublet has to be designed so that it offsets the chromatism induced by the front doublet, which requires complex lens systems. Alternately, both doublets have to be made with special glasses. A system with apochromatic correction can be made with common crown and flint, but such system would require three groups of lenses (two separated doublets and a positive lens closer to the focal plane for lateral color correction), and would be significantly longer than the effective focal length. It would also require strongly curved surfaces, generating significant higher order spherical aberration. Use of special low-dispersion glasses in combination with common glass types makes possible much higher level of chromatic correction in a doublet, triplet or Petzval refractor. The degree of improvement is determined by the respective properties of the two glasses combined, and can vary significantly. Somewhat informally, such refracting objectives are referred to as semi-apo and apochromatic (apo). Following page gives several examples, including some that were, or still are  marketed, but before that a quick look on the effects of central line error and stopping down an achromat. EFFECT OF THE CENTRAL LINE ERROR ON CHROMATIC CORRECTION Presence of spherical aberration in the central line means that it is inevitably carries over to all others. Since spherical aberration in the red and blue are usually of opposite sign at the minimized central line error, any induced sphrical aberration (from miscollimation, or radius error) will add to one, and subtract from the other one. In general, spherochromatism in achromats is low, dominated by chromatic defocus, hence a small to moderate central line error will not significantly affect their chromatic correction. As raytrace below shows, that is the case with as fast as a 100mm f/5 unit (top). 1/5 wave P-V of spherical aberration only slightly increased P-V error in the red, and have it slightly reduced in the blue. However, in a correction mode with colors tightly bound together - i.e. apochromatic correction - the effect can be significant. As little as 1/7 wave of overcorrection in a triplet shown (Ascar 185mm f/7) caused the error in the central line to exceede error in the red C line, which has the error reduced, while also appreciably increasing the blue and violet error. EFFECT OF STOPPING DOWN ACHROMAT Stoping down achromat generally reduces all its aberrations, but it will likely cause chromatic disbalance which may not be significant, but could be noticeable vs. achromat unit with the same aperture and focal ratio that has the colors optimally balanced. The longitudinal aberration plot shows that the color curves are being literally trimmed off at the mask radius level. Despite no change in the respective paraxial foci location, that effectively pulled the blue closer to the green focus, while pushing the red farther out. As the OPD plots show, the error is significantly reduced in both, F and C line, due to the larger Airy disc and decreased defocus sensitivity, but significantly more in the former. Spherical aberration is also significantly smaller. It is of no practical consequence in this case, with it being negligible at a full aperture, but can be significant if the error at full aperture is not negligible. For primary spherical aberration, the wavefront error changes in proportion to the 4th power of the aperture, but in fast achromats with a mix of lower and higher order SA, reduction factor will vary somewhat; in general, it is smaller.   ◄ 9. REFRACTING TELESCOPES   ▐    9.2. Refracting telescope objectives: Apo and semi-apo ► Home  |  Comments

Polarizationof lightnotes PDF

It is a small 32mm f/24 lens with low residual spherical aberration. As most of the objectives from that time, it is somewhat overcorrected (vs. singlet, which has shorter wavelengths focusing closer to the objective than the longer ones). The likely reason is that it was compensating for chromatism of the 5-element terestrial eyepiece used at that time (An investigation of the eighteenth-century achromatic telescope, D.H. Jaecks 2009). Strictly talking, it is not an achromat, since no two widely separated wavelengths come to the same focus, but the level of correction is close to it. For full correction, it is necessary to make the negative element weaker (bottom box, left, R1=-3250mm). While spherical aberration is only slightly higher, significant portion of the F and C-line error comes from it. With corrected spherical aberration (R3=-211mm and, to keep F and C close together, R1=-1900mm), an optimized version of this objective (bottom right) can be compared to the modern standard achromat: 1/16 wave P-V in the F and C line, for given aperture and focal ratio, correspond to those of f/61 standard achromat, with the correction in the violet g-line (0.22 wave P-V) corresponds to that of f/89 achromat. In other words, Dollond's doublet has 2.5 times better correction in F/C, and 3.7 times better g-line correction. Some coma is present, but at this focal ratio and aperture size not noticeable (in the original configuration, 0.083 wave RMS at 1° off axis), hence Dollonds didn't need to know the calculation for minimizing it. They certainly had some knowledge about controlling spherical aberration, but judging on multiple factors - their keeping relative aperture very small, aspherizing, triplet arrangement - it was incomplete. Spherical aberration is easily controlled by balancing the two inner radii. After that, a number of doublet types have been developed. An achromat, by definition, uses two common glasses - crown and flint - to reduce primary chromatism (chromatism of a single lens). Since chromatic correction of a doublet depends mainly on the glass combination, and common glasses span relatively narrow range of properties (i.e. refractive index and dispersion), doublets achromats have similar level of chromatism and differ mostly in their correction of monochromatic aberrations.

FIGURE 147: Standard Fraunhofer doublet achromat can be combined with an auxiliary widely separated second doublet for enhanced correction (A) 100mm f/10 objective alone, for comparison (B) Petzval-type achromat with a cemented rear doublet. The front lens is an f/10 doublet. The only significant change is that the front doublet is bent to generate coma cancelling the coma of rear doublet, which in turn cancels out astigmatism. As a result field curvature equals the focal length, and there is no other monochromatic aberrations to speak of. Since the astigmatism is cancelled, best image surface coincides with the Petzval surface. (C) Another f/10 doublet in front, with a doublet sub-aperture (field) air spaced corrector. Again, the main modification is bending the front doublet to induce some coma, offsetting that of the corrector which, in turn, flattens the field w/o introducing significant astigmatism. In fact, astigmatism is lower than in a comparable achromat (D). Chromatism is also lower in both double achromats (somewhat more in the one with field corrector), without appreciably larger lateral color (note that the Airy disc for e-line spot at 0.5° off-axis is enlarged for clarity).

After that, a number of doublet types have been developed. An achromat, by definition, uses two common glasses - crown and flint - to reduce primary chromatism (chromatism of a single lens). Since chromatic correction of a doublet depends mainly on the glass combination, and common glasses span relatively narrow range of properties (i.e. refractive index and dispersion), doublets achromats have similar level of chromatism and differ mostly in their correction of monochromatic aberrations.

Polarized and unpolarizedlight

Petzval-type achromat is a design that uses two groups of lenses, with the rear group at 1/3 to 2/3 of the focal length of the front group behind (approximately; in the original Petzval configuration rear doublet is at half the focal length of the front doublet apart, its focal length half that of the front doublet, and the combined f.l. also half that of the front doublet). If using common crown and flint glasses, such arrangement can reduce secondary spectrum by approximately 15%; apparently, generally not considered worth the extra expense. With the second lens group closer to the focus of the front group, which is not Petzval configuration, rather one with sub-aperture corrector, secondary spectrum can be reduced somewhat more, up to about 30%.

Another coma-free doublet with reversed order (flint in front) is the Steinheil, which requires significantly more strongly curved surfaces (R1~0.43f, R2~-0.224f, R3~-0.223f and R4~-f for F2/BK7 glasses).

Because of this relationship, a material with birefringence Dn of the appropriate thickness L to convert linear polarization to circular polarization is called a quarter-wave plate.What causes materials to be birefringent? Some materials, especially crystals, are naturally anisotropic at microscopic (sub-wavelength) size scales. For example, Calcite (CaCO3) is shown in the drawing below. The structure, and hence the response to polarized light, along the c direction is markedly different than that along the a and b directions, thus leading to a different index of refraction for light polarized along this direction.

Other materials are nominally isotropic, but when they are bent or deformed in some way, they become anisotropic and therefore exhibit birefringence. This effect is widely used to study the mechanical properties of materials with optics.

The inner two radii of the Fraunhofer can be equalized, without significant change in the correction level, on or off axis (in order to minimize ensuing spherical aberration, R4 is slightly weakened, and lens spacing slightly widened). Such modification is known as Baker doublet.

Stoping down achromat generally reduces all its aberrations, but it will likely cause chromatic disbalance which may not be significant, but could be noticeable vs. achromat unit with the same aperture and focal ratio that has the colors optimally balanced. The longitudinal aberration plot shows that the color curves are being literally trimmed off at the mask radius level. Despite no change in the respective paraxial foci location, that effectively pulled the blue closer to the green focus, while pushing the red farther out. As the OPD plots show, the error is significantly reduced in both, F and C line, due to the larger Airy disc and decreased defocus sensitivity, but significantly more in the former. Spherical aberration is also significantly smaller. It is of no practical consequence in this case, with it being negligible at a full aperture, but can be significant if the error at full aperture is not negligible. For primary spherical aberration, the wavefront error changes in proportion to the 4th power of the aperture, but in fast achromats with a mix of lower and higher order SA, reduction factor will vary somewhat; in general, it is smaller.   ◄ 9. REFRACTING TELESCOPES   ▐    9.2. Refracting telescope objectives: Apo and semi-apo ► Home  |  Comments

Polarisation of lightin physics

The longitudinal aberration plot shows that the color curves are being literally trimmed off at the mask radius level. Despite no change in the respective paraxial foci location, that effectively pulled the blue closer to the green focus, while pushing the red farther out. As the OPD plots show, the error is significantly reduced in both, F and C line, due to the larger Airy disc and decreased defocus sensitivity, but significantly more in the former. Spherical aberration is also significantly smaller. It is of no practical consequence in this case, with it being negligible at a full aperture, but can be significant if the error at full aperture is not negligible. For primary spherical aberration, the wavefront error changes in proportion to the 4th power of the aperture, but in fast achromats with a mix of lower and higher order SA, reduction factor will vary somewhat; in general, it is smaller.   ◄ 9. REFRACTING TELESCOPES   ▐    9.2. Refracting telescope objectives: Apo and semi-apo ► Home  |  Comments

Additional advantage of the dialyte form in an achromat is that astigmatism can be manipulated, either cancelled for less curved image field, or added in the opposite sign, still very low, in order to flatten the field, as illustrated below (FIG. 147C).

Polarizationofelectromagnetic waves pdf

When the electric field of a light wave encounters the sheet, the component parallel to the chains causes electrons to oscillate along the direction of that component (Ey in the above example), thus absorbing energy and inhibiting the component from passing through the sheet. Because electrons can not respond to the other component (Ex), it is readily transmitted.

What if the two components Ex and Ey have unequal amplitude factors? We can see that the light wave is still linearly polarized.

Notice from the graph above on the right that for the case of reflection from a higher-index region to a lower-index region (in this case glass-to-air, or ni = 1.5 and nt = 1.0), the reflectivity becomes 100% for all angles greater than the “critical angle” θc = arcsin(nt/ni) and for both polarizations. This phenomenon is known as “Total Internal Reflection” (TIR).For angles of incidence below the critical angle only the amplitudes of the different polarization components are affected by reflection or transmission at an interface. Except for discrete changes of p (or 180°), the phase of the light is unchanged. Thus, the state of polarization can change in only limited ways. For example, linearly polarized light remains linearly polarized, although its orientation (angle ) may rotate. However, for angles greater than θc, different polarizations experience different phase changes, and thus TIR can affect the state of polarization of a light wave in the same way birefringence does. Thus linearly polarized light may become elliptical, or vice versa, in addition to changes in the orientation.

When an optical filter is used at a non-normal angle of incidence, as is common with so-called “plate beamsplitters,” the filter can impact the polarization of the light. If the incident light is incoherent and unpolarized, and the optical system is “blind” to polarization, the standard intensity reflection and transmission functions R(l) and T(l) may be determined for the new angle of incidence, and they are sufficient to characterize the two emerging beams.

The amplitude E, or the potential for a charged particle to feel a force, is vibrating along both the x and y directions. An actual charged particle would feel both of these fields simultaneously, or it would feel

Because the polarization response of a tilted multilayer thin-film coating can be very strong, optical filters can make excellent polarizers. For example, a basic edge filter at a high angle of incidence exhibits “edge splitting” – the edge wavelength for light at normal incidence shifts to a different wavelength for p-polarized light than it does for s-polarized light. As a result, there is a range of wavelengths for which p-polarized light is highly transmitted while s-polarized light ishighly reflected, as shown below.

and where, as before, E = Ex< + Ey. The three special cases described in sections a, b, and c above thus correspond to: (a) Ax = Ay and  = 0 (linear polarization; equal amplitudes); (b) 

A doublet with air gap wide as Clark's can be made aplanatic, in which case it is jus another variant of the Baker, with the third radius somewhat more strongly curved.

The reason for this is that the blue and red exit the front lens group - which is assumed to be doublet achromat - only slightly separated, but at different angles: as FIG. 96 hints, the red just below, and blue just above the green ray, with the former two converging to a common focus (or nearly so) and emerging above the green ray at some distance toward the focus. Thus there is no appreciable effect on longitudinal chromatism by the second group of lens, until it is far back enough for the red/blue rays to raise above the green ray, and get refracted more strongly at the rear lens group, focusing slightly shorter relative to the green light.

Polarisation of lightequation

Most polarizing beamsplitters are very efficient polarizers for the transmitted light (i.e., the ratio of desired to undesired polarization is very high); however, the reflected light generally contains some of both polarization components.How does a polarizer work? There are different ways of making a polarizer, and they are not described in detail here (see [1] for more examples). However, as an example consider one of the most popular absorbing polarizers: the well-known Polaroid “H-Sheet.” This polarizer, invented by E. H. Land in 1938, is a plastic, Poly-Vinyl Alcohol (PVA) sheet that has been heated and then stretched in one direction, forming long, nearly parallel hydrocarbon molecule chains. After dipping the sheet into an iodine-rich ink, long iodine chains form along the hydrocarbon molecules. Electrons freely move along the iodine chains, but do not easily move perpendicular to the chains. This ability for electrons to move freely in one direction but not the perpendicular direction is the key principle upon which most absorbing polarizers are based.

The amount of light output in each polarization state can be determined by simply breaking up the incident light into its two polarization components (s and p), and then calculating how much of each intensity is transmitted and reflected. For systems based on incoherent light, this level of detail is usually sufficient to keep track of the impacts of components like optical filters on polarization.For some optical systems – particularly those based on coherent light and that utilize or are sensitive to interference effects, for example – the complete state of polarization should be tracked at every point through the system. In that case, it is important to understand that optical filters based on multilayer thin-film coatings not only reflect and transmit different amounts of intensity for the s and p polarization states, but also impart different phases to the two different states. And both the amplitude and phase contributions can depend strongly on the wavelength of light. Thus, in general, an optical filter can act like the combination of a partial polarizer and a birefringent waveplate, for both reflected and transmitted light.To determine the effect of an optical filter on the light in such a system, the incident light should first be broken up into the two fundamental components associated with the plane of incidence of the filter (s and p components). Then, the amplitude and phase responses of the filter for the s and p components should be applied separately to each of the incident light components to determine the amplitudes and phases of the reflected and transmitted light components. Finally, the reflected s and p components can be recombined to determine the total reflected light and its state of polarization, and likewise for the transmitted light. These steps are illustrated in the diagram below.

Use of special low-dispersion glasses in combination with common glass types makes possible much higher level of chromatic correction in a doublet, triplet or Petzval refractor. The degree of improvement is determined by the respective properties of the two glasses combined, and can vary significantly. Somewhat informally, such refracting objectives are referred to as semi-apo and apochromatic (apo). Following page gives several examples, including some that were, or still are  marketed, but before that a quick look on the effects of central line error and stopping down an achromat. EFFECT OF THE CENTRAL LINE ERROR ON CHROMATIC CORRECTION Presence of spherical aberration in the central line means that it is inevitably carries over to all others. Since spherical aberration in the red and blue are usually of opposite sign at the minimized central line error, any induced sphrical aberration (from miscollimation, or radius error) will add to one, and subtract from the other one. In general, spherochromatism in achromats is low, dominated by chromatic defocus, hence a small to moderate central line error will not significantly affect their chromatic correction. As raytrace below shows, that is the case with as fast as a 100mm f/5 unit (top). 1/5 wave P-V of spherical aberration only slightly increased P-V error in the red, and have it slightly reduced in the blue. However, in a correction mode with colors tightly bound together - i.e. apochromatic correction - the effect can be significant. As little as 1/7 wave of overcorrection in a triplet shown (Ascar 185mm f/7) caused the error in the central line to exceede error in the red C line, which has the error reduced, while also appreciably increasing the blue and violet error. EFFECT OF STOPPING DOWN ACHROMAT Stoping down achromat generally reduces all its aberrations, but it will likely cause chromatic disbalance which may not be significant, but could be noticeable vs. achromat unit with the same aperture and focal ratio that has the colors optimally balanced. The longitudinal aberration plot shows that the color curves are being literally trimmed off at the mask radius level. Despite no change in the respective paraxial foci location, that effectively pulled the blue closer to the green focus, while pushing the red farther out. As the OPD plots show, the error is significantly reduced in both, F and C line, due to the larger Airy disc and decreased defocus sensitivity, but significantly more in the former. Spherical aberration is also significantly smaller. It is of no practical consequence in this case, with it being negligible at a full aperture, but can be significant if the error at full aperture is not negligible. For primary spherical aberration, the wavefront error changes in proportion to the 4th power of the aperture, but in fast achromats with a mix of lower and higher order SA, reduction factor will vary somewhat; in general, it is smaller.   ◄ 9. REFRACTING TELESCOPES   ▐    9.2. Refracting telescope objectives: Apo and semi-apo ► Home  |  Comments

EFFECT OF THE CENTRAL LINE ERROR ON CHROMATIC CORRECTION Presence of spherical aberration in the central line means that it is inevitably carries over to all others. Since spherical aberration in the red and blue are usually of opposite sign at the minimized central line error, any induced sphrical aberration (from miscollimation, or radius error) will add to one, and subtract from the other one. In general, spherochromatism in achromats is low, dominated by chromatic defocus, hence a small to moderate central line error will not significantly affect their chromatic correction. As raytrace below shows, that is the case with as fast as a 100mm f/5 unit (top). 1/5 wave P-V of spherical aberration only slightly increased P-V error in the red, and have it slightly reduced in the blue. However, in a correction mode with colors tightly bound together - i.e. apochromatic correction - the effect can be significant. As little as 1/7 wave of overcorrection in a triplet shown (Ascar 185mm f/7) caused the error in the central line to exceede error in the red C line, which has the error reduced, while also appreciably increasing the blue and violet error. EFFECT OF STOPPING DOWN ACHROMAT Stoping down achromat generally reduces all its aberrations, but it will likely cause chromatic disbalance which may not be significant, but could be noticeable vs. achromat unit with the same aperture and focal ratio that has the colors optimally balanced. The longitudinal aberration plot shows that the color curves are being literally trimmed off at the mask radius level. Despite no change in the respective paraxial foci location, that effectively pulled the blue closer to the green focus, while pushing the red farther out. As the OPD plots show, the error is significantly reduced in both, F and C line, due to the larger Airy disc and decreased defocus sensitivity, but significantly more in the former. Spherical aberration is also significantly smaller. It is of no practical consequence in this case, with it being negligible at a full aperture, but can be significant if the error at full aperture is not negligible. For primary spherical aberration, the wavefront error changes in proportion to the 4th power of the aperture, but in fast achromats with a mix of lower and higher order SA, reduction factor will vary somewhat; in general, it is smaller.   ◄ 9. REFRACTING TELESCOPES   ▐    9.2. Refracting telescope objectives: Apo and semi-apo ► Home  |  Comments

All of the states of polarization described above are actually special cases of the most general state of polarization, called elliptical polarization, in which the tip of the electric field vector E traces out an ellipse in the x-y plane. The two components might have unequal amplitudes Ax  Ay , and also might contain a different relative phase, often denoted  That is, we may write generally

Two other doublet achromat types of mostly historical significance are the Littrow, requiring even more strongly curved surfaces than the Steinheil, with more coma than comparable paraboloid, and the Clark, with somewhat less coma than the Littrow, but more lateral chromatism. Another older doublet type is the Cooke, which consists of the biconvex front and biconcave rear element; it has more than double the coma of Littrow, while no advantage of easier fabrication.

Multilayer thin-film coatings have a large number of interfaces, since they are generally comprised of alternating layers of a high- and low-index layer materials. The fraction of incident light intensity Iin that is reflected (IR) and transmitted (IT) through a thin-film coating can be calculated from the indexes of refraction and the precise thicknesses of each layer. These intensity reflection and transmission functions R(l) and T(l), respectively, generally depend strongly on the wavelength of the light, because the total amount of light reflected from and transmitted through the coating comes from the interference of many individual waves that arise from the partial reflection and transmission at each interface. That is why optical filters based on thin-film coatings are called “interference filters.”

Relatively unusual achromat designs are triplet achromats, as well as those with more than one group of lenses. The latter include Petzval-type achromats and those with the second lens group closer to the focal plane. By their basic form, they belong to dialyte objectives, defined as those employing widely separated elements.

FIGURE 146: Doublet achromat objective types for the common crown/flint combination, 100mm f/10 examples. The standard choice is Fraunhofer doublet, an aplanat consisting from biconvex front lens followed by a negative meniscus, with the secondary spectrum, defined as the longitudinal separation between the red/blue (C/F) focus and green (e-line) focus, of about 0.00055f (about 0.0005f measured from d-line). The only remaining monochromatic aberration is negligible astigmatism; best image curvature is about -0.36f (concave toward the objective). Its inverse form, with the flint element in front, known as the Steinheil, is nearly identical in performance, but requires more strongly curved lenses for given focal length. Another variation, with the inner radii equalized, and somewhat wider spacing to compensate for the spherical aberration it induced, is known as the Baker doublet and does not differ in its performance level from the other two. Unlike them, it is not strictly a contact doublet, but the air gap is relatively small. The original Littrow objective consists from the equconvex front lens followed by the negative lens of the same inner radius and flat last surface. With three identical radii and a flat surface, it is the easiest to fabricate, but at a price of some residual coma and spherical aberration. Since the astigmatism is nearly identical in all doublets and near negligible, nearly all of the 0.133 wave RMS at 0.5° off axis is coma, which makes it more than twice greater than in a comparable paraboloid (since the size of linear quality field with respect to coma changes with the third power of mirror's F-number, it makes this Littrow's coma comparable to that in a paraboloid with 30% larger f-ratio (f/7.7 for f/10 lens). Not a concern in visual observing, and neither is the residual spherical, which is here little over 1/12 wave P-V. The air gap is reduced to zero, in order to keep this residual spherical at its minimum. With larger lenses, the latter may become significant, in which case one of the two inner radii may need to be slightly changed in order to have it minimized or cancelled. Note that the last surface in the common crown/flint combination cannot be flat; it is mildly convex, since the -f1/f2=V2/V1 achromatic condition (Eq. 42) for the Littrow implies V2=(n2-1)V1/2(n1-1), where n, V and 1, 2 refer to the refractive index, Abbe number, front and rear lens, respectively, hence such lens configuration requires specific glass match (the only nearly matching flint for the flat-rear Littrow using BK7 is F15). A Littrow modification by Clark&Sons, known as Clark doublet, has the two lenses more widely separated (~15% of the focal length, according to Sidgwick), in order to make the inner surfaces accessible for cleaning without taking lenses out. Since widening the separation effectively weakens the rear relative to front lens, with the former in the Littrow already slightly too weak for cancelling spherical aberration, the Clark requires somewhat stronger (relative to R1 and R2) third surface (like the Littrow, the last surface can be flat only with a specific glass match; for the common crown/flint combinations it is mildly convex). The Clark's coma is somewhat lower than Littrow's, but its lateral color is, due to the wider lens separation, significantly larger, although still acceptable (less than 1/4 of the Airy disc diameter between F and C lines at 0.5° off axis for the above system). Also, best image surface is somewhat more strongly curved, indicating a bit stronger astigmatism: the respective Zernike coefficients for primary coma and astigmatism at 0.5° of 0.252 and 0.107 indicate 0.252/√8=0.088 and 0.107/√6= 0.044 wave RMS error, thus astigmatism about half as strong as coma, which is about a third lower than in the Littrow. Clark configuration can be coma-free with the first radius made stronger than the second one, and also stronger fourth radius. Finally, the alternative aplanatic doublet solution, consisting of a positive and negative meniscus, is the Gauss doublet. Since a meniscus requires significantly more strongly curved surfaces to achieve given power, this objective type is, in addition to be more difficult for fabrication, unsuitable for all but long focal ratio instruments. As raytrace shows, higher order spherical originating at the strongly curved surfaces - particularly R4 - makes it unacceptable already at 100mm f/10, with 0.057 wave RMS design limit in the e-line, and more than a third larger chromatic error due to spherochromatism (residual coma noticeable at the off-axis wavefront map cannot be further minimized without strengthening the radii even more, which would double the spherical error). Vice versa, accepting significantly more coma would allow for the weaker radii and reducing higher-order spherical to insignificant, but it would be still inferior to the other doublets, both, for more difficult fabrication and lower field quality. The positive is somewhat lower astigmatism - about 1/3 of the Clark objective, and nearly half of the other doublets' - and correspondingly less strongly curved best image surface.

PAGE HIGHLIGHTS • Doublet achromat types   • Petzval-type achromat   • Effect of central line error   • Effect of stopping down

Diagrams below illustrate basic properties of the main achromatic doublet types: longitudinal aberration plot for five spectral lines spanning most of the visual spectrum (g-436nm, F-486nm, e-546nm, C-656nm and r-707nm), axial F-e-C ray spots, P-V wavefront error at 0.5° off-axis (e-line) and best image curvature radius.