The interferometers consist of two high-precision mirrors with a flatness better than λ/40 over 90% of the aperture, having broadband coatings and spacers of different thickness. The working aperture is either 22 mm or 48 mm, and the reflectivity finesse is > 120. The standard spectral range is 450 nm … 1100 nm, others on request. Combined with our CCD linear array, the interferometers form a versatile spectral analysis system.

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For an optical resonance, the amplitude distribution not only has to maintain its shape after one round trip, but also to experience a phase shift which is an integer multiple of <$2\pi$>. This is possible only for certain optical frequencies. Therefore, the modes are characterized by a set of three indices: the transverse mode indices <$n$> and <$m$>, plus an axial mode number <$q$> (the round-trip phase shift divided by <$2\pi$>). A notation such as TEM_{nmq} includes the axial mode number in cases where it is important. Modes with <$n = m$> = 0 are called axial modes (or fundamental modes, Gaussian modes), whereas all other are called higher-order modes or higher-order transverse modes. Note that due to the Gouy phase shift the optical frequencies depend not only on the axial mode number, but also on the transverse mode indices <$n$> and <$m$> (see Figure 4). For now ignoring chromatic dispersion, we have:

Resonator modes are very important e.g. in the context of laser resonators, Fabry–Pérot interferometers and resonant mode cleaners.

Motion pictures make limited use of aperture control; to produce a consistent image quality from shot to shot, cinematographers usually choose a single aperture setting for interiors (e.g., scenes inside a building) and another for exteriors (e.g., scenes in an area outside a building), and adjust exposure through the use of camera filters or light levels. Aperture settings are adjusted more frequently in still photography, where variations in depth of field are used to produce a variety of special effects.

The hyperfocal distance has a property called "consecutive depths of field", where a lens focused at an object whose distance from the lens is at the hyperfocal distance H will hold a depth of field from H/2 to infinity, if the lens is focused to H/2, the depth of field will be from H/3 to H; if the lens is then focused to H/3, the depth of field will be from H/4 to H/2, etc.

As distance or the size of the acceptable circle of confusion increases, the depth of field increases; however, increasing the size of the aperture (i.e., reducing f-number) or increasing the focal length reduces the depth of field. Depth of field changes linearly with f-number and circle of confusion, but changes in proportion to the square of the distance to the subject and inversely in proportion to the square of the focal length. As a result, photos taken at extremely close range (i.e., so small u) have a proportionally much smaller depth of field.

Depth of field vs depth of focusmicroscope

In optics and photography, hyperfocal distance is a distance from a lens beyond which all objects can be brought into an "acceptable" focus. As the hyperfocal distance is the focus distance giving the maximum depth of field, it is the most desirable distance to set the focus of a fixed-focus camera.[41] The hyperfocal distance is entirely dependent upon what level of sharpness is considered to be acceptable.

Image sensor size affects DOF in counterintuitive ways. Because the circle of confusion is directly tied to the sensor size, decreasing the size of the sensor while holding focal length and aperture constant will decrease the depth of field (by the crop factor). The resulting image however will have a different field of view. If the focal length is altered to maintain the field of view, while holding the f-number constant, the change in focal length will counter the decrease of DOF from the smaller sensor and increase the depth of field (also by the crop factor). However, if the focal length is altered to maintain the field of view, while holding the aperture diameter constant, the DOF will remain constant. [6][7][8][9]

For 35 mm motion pictures, the image area on the film is roughly 22 mm by 16 mm. The limit of tolerable error was traditionally set at 0.05 mm (0.0020 in) diameter, while for 16 mm film, where the size is about half as large, the tolerance is stricter, 0.025 mm (0.00098 in).[15] More modern practice for 35 mm productions set the circle of confusion limit at 0.025 mm (0.00098 in).[16]

Other authors such as Ansel Adams have taken the opposite position, maintaining that slight unsharpness in foreground objects is usually more disturbing than slight unsharpness in distant parts of a scene.[20]

Due to chromatic dispersion and diffraction effects, the mode spacings actually have a (weak) frequency dependence, which, however, is often not of interest.

Hansma and Peterson have discussed determining the combined effects of defocus and diffraction using a root-square combination of the individual blur spots.[30][31] Hansma's approach determines the f-number that will give the maximum possible sharpness; Peterson's approach determines the minimum f-number that will give the desired sharpness in the final image and yields a maximum depth of field for which the desired sharpness can be achieved.[d] In combination, the two methods can be regarded as giving a maximum and minimum f-number for a given situation, with the photographer free to choose any value within the range, as conditions (e.g., potential motion blur) permit. Gibson gives a similar discussion, additionally considering blurring effects of camera lens aberrations, enlarging lens diffraction and aberrations, the negative emulsion, and the printing paper.[27][e] Couzin gave a formula essentially the same as Hansma's for optimal f-number, but did not discuss its derivation.[32]

If a subject is at distance s and the foreground or background is at distance D, let the distance between the subject and the foreground or background be indicated by

Hopkins,[33] Stokseth,[34] and Williams and Becklund[35] have discussed the combined effects using the modulation transfer function.[36][37]

Such modes exist whether or not the resonator is geometrically stable, but the mode properties of unstable resonators are fairly complicated. In the following, only modes of stable resonators are considered.

Are you interested in photography? We have various encyclopedia articles on such topics, explaining physical foundations and technological aspects:

The depth of field (DOF) is the distance between the nearest and the farthest objects that are in acceptably sharp focus in an image captured with a camera. See also the closely related depth of focus.

The FPI 100 is a confocal, scanning Fabry–Perot interferometer with a built-in photodetector unit, designed for measuring and controlling the mode profiles of continuous wave (cw) lasers. The FPI is available with different mirror sets and photodetectors for wavelength ranges between 330 nm and 3000 nm.

Light Scanning Photomacrography (LSP) is another technique used to overcome depth of field limitations in macro and micro photography. This method allows for high-magnification imaging with exceptional depth of field. LSP involves scanning a thin light plane across the subject that is mounted on a moving stage perpendicular to the light plane. This ensures the entire subject remains in sharp focus from the nearest to the farthest details, providing comprehensive depth of field in a single image. Initially developed in the 1960s and further refined in the 1980s and 1990s, LSP was particularly valuable in scientific and biomedical photography before digital focus stacking became prevalent.[23][24]

Single-frequency operation of a laser means that only a single resonator mode (nearly always a Gaussian one) is excited; this leads to a much lower emission bandwidth than in cases where multiple resonator modes are excited.

If different modes of a laser resonator are simultaneously excited in continuous-wave operation, there is usually the phenomenon of mode competition.

Depth of field vs depth of focusnikon

Moreover, traditional depth-of-field formulas assume equal acceptable circles of confusion for near and far objects. Merklinger[c] suggested that distant objects often need to be much sharper to be clearly recognizable, whereas closer objects, being larger on the film, do not need to be so sharp.[19] The loss of detail in distant objects may be particularly noticeable with extreme enlargements. Achieving this additional sharpness in distant objects usually requires focusing beyond the hyperfocal distance, sometimes almost at infinity. For example, if photographing a cityscape with a traffic bollard in the foreground, this approach, termed the object field method by Merklinger, would recommend focusing very close to infinity, and stopping down to make the bollard sharp enough. With this approach, foreground objects cannot always be made perfectly sharp, but the loss of sharpness in near objects may be acceptable if recognizability of distant objects is paramount.

Photographers can use the lens scales to work backwards from the desired depth of field to find the necessary focus distance and aperture.[38] For the 35 mm lens shown, if it were desired for the DOF to extend from 1 m to 2 m, focus would be set so that index mark was centered between the marks for those distances, and the aperture would be set to f/11.[f]

LightMachinery manufactures the world's finest solid and air spaced etalons. Our fluid jet polishing systems allow us to routinely create surfaces that are better than λ/100 peak to valley.

where <$\varphi_\textrm{G}$> is the Gouy phase shift per round trip. The magnitude of that Gouy phase depends on the resonator design.

In mode-locked operation, the optical spectrum is a frequency comb, consisting of exactly equidistant spectral lines (ignoring possible laser noise), where the line spacing is the inverse pulse repetition rate. Due to the not exactly equidistant mode frequencies, there is some amount of mismatch between emission frequencies and mode frequencies; the larger that mismatch is, the stronger needs to be the effect of the mode-locking device, for example a saturable absorber. Based on this insight, it is easy to understand why in cases with substantial chromatic dispersion it is hard to achieve mode locking with a broad emission bandwidth and a correspondingly short pulse duration.

Depth of field vs depth of focusreddit

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More so than in the case of the zero swivel camera, there are various methods to form criteria and set up calculations for DOF when swivel is non-zero. There is a gradual reduction of clarity in objects as they move away from the POF, and at some virtual flat or curved surface the reduced clarity becomes unacceptable. Some photographers do calculations or use tables, some use markings on their equipment, some judge by previewing the image.

Traditional depth-of-field formulas can be hard to use in practice. As an alternative, the same effective calculation can be done without regard to the focal length and f-number.[b] Moritz von Rohr and later Merklinger observe that the effective absolute aperture diameter can be used for similar formula in certain circumstances.[19]

Other technologies use a combination of lens design and post-processing: Wavefront coding is a method by which controlled aberrations are added to the optical system so that the focus and depth of field can be improved later in the process.[25]

In the simplest case of a resonator containing only parabolic mirrors and optically homogeneous media, the resonator modes (cavity modes) are Hermite–Gaussian modes. The simplest of those are the Gaussian modes, where the field distribution is defined by a Gaussian function (→ Gaussian beams). The evolution of the beam radius and the radius of curvature of the wavefronts is determined by the details of the resonator.

Another approach is focus sweep. The focal plane is swept across the entire relevant range during a single exposure. This creates a blurred image, but with a convolution kernel that is nearly independent of object depth, so that the blur is almost entirely removed after computational deconvolution. This has the added benefit of dramatically reducing motion blur.[22]

The Fabry–Pérot interferometers offered by ALPHALAS are high-resolution spectroscopic instruments having applications in spectral analysis of narrowband light sources like gas discharge lamps and lasers. These are especially useful for analyzing the spectral content of pulsed lasers, because they allow for single-shot measurements.

Depth of focusin photography

When a laser has a poor beam quality, this is usually (although not always) the result of the excitation of higher-order transverse cavity modes. When the output light is sent to a fast photodiode, one can detect beat notes involving higher-order modes, i.e., with frequencies which substantially deviate from integer multiples of the round-trip frequency.

This section covers some additional formula for evaluating depth of field; however they are all subject to significant simplifying assumptions: for example, they assume the paraxial approximation of Gaussian optics. They are suitable for practical photography, lens designers would use significantly more complex ones.

There is no well-defined number of transverse or longitudinal modes, so we cannot compare those numbers. Longitudinal refers to different optical frequencies while referring to a specific spatial shape, usually the Gaussian one for the fundamental modes. Transverse refers to different spatial shapes, as shown in Figure 3. According to that figure, you can say that for a given longitudinal mode, there are many transverse modes with similar optical frequency. However, you can also say that for any of those spatial shapes, there are many versions with different optical frequencies.

Note that M T = − f u − f {\textstyle M_{T}=-{\frac {f}{u-f}}} is the transverse magnification which is the ratio of the lateral image size to the lateral subject size.[5]

For certain values of the Gouy phase shift, mode frequency degeneracies can occur. In a laser, these can lead to a strong deterioration of beam quality by resonant coupling of the axial modes to higher-order modes. With proper resonator design, it is possible to avoid at least the particularly sensitive frequency degeneracies and thus to improve laser beam quality [3]. Such degeneracies also can have useful properties; e.g. when a Fabry–Pérot interferometer is used as an optical spectrum analyzer, precise adjustment of the mirrors (e.g. in a confocal configuration) allows the use without mode matching. Also, degenerate cavities can be used for Herriot-type multipass cells, which can be used e.g. for strongly increasing the round-trip path length in a laser resonator without changing the overall resonator design.

The acceptable circle of confusion depends on how the final image will be used. The circle of confusion as 0.25 mm for an image viewed from 25 cm away is generally accepted.[14]

Shallowdepth of field

The depth of field can be determined by focal length, distance to subject (object to be imaged), the acceptable circle of confusion size, and aperture.[2] Limitations of depth of field can sometimes be overcome with various techniques and equipment. The approximate depth of field can be given by:

b = f m s N x d s ± x d = d m s x d D . {\displaystyle b={\frac {fm_{\mathrm {s} }}{N}}{\frac {x_{\mathrm {d} }}{s\pm x_{\mathrm {d} }}}=dm_{\mathrm {s} }{\frac {x_{\mathrm {d} }}{D}}.}

Many lenses include scales that indicate the DOF for a given focus distance and f-number; the 35 mm lens in the image is typical. That lens includes distance scales in feet and meters; when a marked distance is set opposite the large white index mark, the focus is set to that distance. The DOF scale below the distance scales includes markings on either side of the index that correspond to f-numbers. When the lens is set to a given f-number, the DOF extends between the distances that align with the f-number markings.

Some methods and equipment allow altering the apparent DOF, and some even allow the DOF to be determined after the image is made. These are based or supported by computational imaging processes. For example, focus stacking combines multiple images focused on different planes, resulting in an image with a greater (or less, if so desired) apparent depth of field than any of the individual source images. Similarly, in order to reconstruct the 3-dimensional shape of an object, a depth map can be generated from multiple photographs with different depths of field. Xiong and Shafer concluded, in part, "... the improvements on precisions of focus ranging and defocus ranging can lead to efficient shape recovery methods."[21]

Are transverse modes lesser in number than longitudinal modes in a laser resonator? Why is it usually said that for a given transverse mode there can be a number of longitudinal modes, but not vice versa?

At the extreme, a plenoptic camera captures 4D light field information about a scene, so the focus and depth of field can be altered after the photo is taken.

Depth of fieldanddepth of focusPDF

For a given subject framing and camera position, the DOF is controlled by the lens aperture diameter, which is usually specified as the f-number (the ratio of lens focal length to aperture diameter). Reducing the aperture diameter (increasing the f-number) increases the DOF because only the light travelling at shallower angles passes through the aperture so only cones of rays with shallower angles reach the image plane. In other words, the circles of confusion are reduced or increasing the DOF.[10]

For cameras that can only focus on one object distance at a time, depth of field is the distance between the nearest and the farthest objects that are in acceptably sharp focus in the image.[1] "Acceptably sharp focus" is defined using a property called the "circle of confusion".

The DOF beyond the subject is always greater than the DOF in front of the subject. When the subject is at the hyperfocal distance or beyond, the far DOF is infinite, so the ratio is 1:∞; as the subject distance decreases, near:far DOF ratio increases, approaching unity at high magnification. For large apertures at typical portrait distances, the ratio is still close to 1:1.

When the POF is rotated, the near and far limits of DOF may be thought of as wedge-shaped, with the apex of the wedge nearest the camera; or they may be thought of as parallel to the POF.[17][18]

Precise focus is only possible at an exact distance from a lens;[a] at that distance, a point object will produce a small spot image. Otherwise, a point object will produce a larger or blur spot image that is typically and approximately a circle. When this circular spot is sufficiently small, it is visually indistinguishable from a point, and appears to be in focus. The diameter of the largest circle that is indistinguishable from a point is known as the acceptable circle of confusion, or informally, simply as the circle of confusion.

Depth of field vs depth of focusphotography

That's correct: we don't have Gaussian modes in such a case. We are just at the edge of a stability range. By the way, there is not even a clearly defined beam center position in such a case.

Ask RP Photonics for advice and calculations concerning the properties of cavity modes. For example, the powerful numerical software RP Resonator is ideal for laser resonator design.

Diffraction causes images to lose sharpness at high f-numbers (i.e., narrow aperture stop opening sizes), and hence limits the potential depth of field.[27] (This effect is not considered in the above formula giving approximate DOF values.) In general photography this is rarely an issue; because large f-numbers typically require long exposure times to acquire acceptable image brightness, motion blur may cause greater loss of sharpness than the loss from diffraction. However, diffraction is a greater issue in close-up photography, and the overall image sharpness can be degraded as photographers are trying to maximize depth of field with very small apertures.[28][29]

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In addition to the Gaussian modes, a stable resonator also has higher-order modes with more complicated intensity distributions. At a beam waist, the electric field distribution can be written as a product of two Hermite polynomials with orders <$n$> and <$m$> (non-negative integers, corresponding to <$x$> and <$y$> directions) and two Gaussian functions. (We still assume a simple resonator with only parabolic mirrors and optically homogeneous media.) These modes are also called TEMnm modes; the article on Hermite–Gaussian modes describes the exact mathematical form. The optical intensity distribution of such a mode (Figure 1) has <$n$> nodes in the horizontal direction and <$m$> nodes in the vertical direction (Figure 3). If the Gaussian beam radius of the fundamental mode is known, it is easy to calculate the mode profiles of all higher-order modes.

The lens design can be changed even more: in colour apodization the lens is modified such that each colour channel has a different lens aperture. For example, the red channel may be f/2.4, green may be f/2.4, whilst the blue channel may be f/5.6. Therefore, the blue channel will have a greater depth of field than the other colours. The image processing identifies blurred regions in the red and green channels and in these regions copies the sharper edge data from the blue channel. The result is an image that combines the best features from the different f-numbers.[26]

where <$\Delta \nu$> is the free spectral range (axial mode spacing) and <$\delta \nu$> the transverse mode spacing. The latter can be calculated as

Depth of field vs depth of focuscamera

If the right end mirror were curved in the opposite direction (i.e., having a convex surface), no Gaussian mode exists. In such situations, we have an unstable resonator. Such resonators also have modes, but of far more complicated shape. Most lasers work with a resonator in the stable regime, where Gaussian modes exist.

Thomas Sutton and George Dawson first wrote about hyperfocal distance (or "focal range") in 1867.[42] Louis Derr in 1906 may have been the first to derive a formula for hyperfocal distance. Rudolf Kingslake wrote in 1951 about the two methods of measuring hyperfocal distance.

DOF ≈ 2 N c ( u f ) 2 = 2 N c ( 1 − 1 M T ) 2 {\displaystyle {\text{DOF}}\approx 2Nc\left({\frac {u}{f}}\right)^{2}=2Nc\left(1-{\frac {1}{M_{T}}}\right)^{2}}

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In some respects, such an OPO is similar to a laser. It may be that you misaligned the resonator such that the TEM21 mode is a better overlap with the pump beam than the fundamental resonator mode. So you probably need to optimize the alignment. There might also be a problem with the exact resonator length, which might be slightly different for higher-order modes. It is a special challenge in the initial alignment of a synchronously pumped OPO to get the transverse alignment and the resonator length correct at the same time.

For simple resonators as in the example above, one can use relatively simple equations for calculating the mode parameters. For more complicated resonator setups, e.g. involving multiple curved mirrors and possibly additional optical components such as a laser crystals with some thermal lensing, one usually requires numerical software to calculate the mode properties.

s = 2 D N D F D N + D F , {\displaystyle s={\frac {2D_{\mathrm {N} }D_{\mathrm {F} }}{D_{\mathrm {N} }+D_{\mathrm {F} }}},}

Solid etalons, air spaced etalons, piezo tunable etalons, Gires–Tournois etalons – LightMachinery has extensive expertise in the manufacturing and testing of all kinds of Fabry–Pérot etalons from 1 mm square to 100 mm in diameter. These devices require high quality, very flat optical surfaces and extreme parallelism to achieve high performance, making them a good match for the polishing and metrology at LightMachinery.

On a view camera, the focus and f-number can be obtained by measuring the depth of field and performing simple calculations. Some view cameras include DOF calculators that indicate focus and f-number without the need for any calculations by the photographer.[39][40]

Resonant enhancement, e.g. of an incident light wave, hitting a partially transmissive mirror of a resonator from outside, is possible within a range of optical frequencies. The width of that range is called the resonator bandwidth, and this quantity is determined by the rate of optical power losses.

Laser oscillation in continuous-wave operation usually occurs with one or several frequencies which correspond fairly precisely to certain mode frequencies. However, frequency-dependent gain can cause some frequency pulling (slightly nonresonant oscillation), and the mode frequencies themselves can be influenced e.g. by thermal lensing in the gain medium.

The term "camera movements" refers to swivel (swing and tilt, in modern terminology) and shift adjustments of the lens holder and the film holder. These features have been in use since the 1800s and are still in use today on view cameras, technical cameras, cameras with tilt/shift or perspective control lenses, etc. Swiveling the lens or sensor causes the plane of focus (POF) to swivel, and also causes the field of acceptable focus to swivel with the POF; and depending on the DOF criteria, to also change the shape of the field of acceptable focus. While calculations for DOF of cameras with swivel set to zero have been discussed, formulated, and documented since before the 1940s, documenting calculations for cameras with non-zero swivel seem to have begun in 1990.

The blur disk diameter b of a detail at distance xd from the subject can be expressed as a function of the subject magnification ms, focal length f, f-number N, or alternatively the aperture d, according to

As an example, Figures 1 and 2 show the Gaussian resonator modes for two versions of a simple resonator with a plane mirror, a laser crystal, and a curved end mirror. For a more strongly curved end mirror (Figure 2), the mode radius on the left mirror becomes smaller. In any case, one can see that a beam starting e.g. on the left side with flat wavefronts (fitting to the flat mirror there) will somewhat expand on its path to the right side, where the wavefronts again match the shape of the other mirror, and it is easy to see that the field configuration will reproduce itself, including the shape of the wavefronts, after a full round trip. In addition to what can be seen in the drawing, there is the condition that the round-trip phase shift is an integer multiple of <$2\pi$>.

The modes of a passive resonator have a certain frequency bandwidth as a result of the damping of the intracavity field by power losses. If the optical power after one resonator round-trip is <$\rho$> times the original power (i.e., the fractional losses per round trip are <$1 - \rho$>) and the round-trip time is <$T_\textrm{rt}$>, the full width at half maximum bandwidth if the resonances is

the harmonic mean of the near and far distances. In practice, this is equivalent to the arithmetic mean for shallow depths of field.[44] Sometimes, view camera users refer to the difference vN − vF as the focus spread.[45]

Modes of laser resonator can differ significantly from those of an empty resonator because they are subject to transversely varying gain and loss. This not only results in some deformation of the spatial shape; it is also that the resonator modes are no longer mutually orthogonal. Instead, there is a set of adjoint modes, related to the actual resonator modes by some biorthogonality relations. This biorthogonal (non-normal) nature has a number of peculiar implications. For example, the total power circulating in the laser is no longer simply the sum of the powers propagating in the different modes. There are also effects on the laser noise.

I use an OPO pumped with a femtosecond Ti:sapphire laser. The beam profile I obtain is TEM21, even though it is supposed to yield a Gaussian mode. How to fix this?

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The blur increases with the distance from the subject; when b is less than the circle of confusion, the detail is within the depth of field.

Resonator modes are the modes of an optical resonator (cavity), i.e. electromagnetic field distributions which reproduce themselves after a full resonator round trip. More precisely, that means that the full amplitude profile (including the optical phase) must be unchanged after one round trip, apart from a possible loss of optical power.

You can see them on a screen, but usually not on a mirror, since the mirror should fully reflect that light rather than scattering it into your eyes.

For a given size of the subject's image in the focal plane, the same f-number on any focal length lens will give the same depth of field.[11] This is evident from the above DOF equation by noting that the ratio u/f is constant for constant image size. For example, if the focal length is doubled, the subject distance is also doubled to keep the subject image size the same. This observation contrasts with the common notion that "focal length is twice as important to defocus as f/stop",[12] which applies to a constant subject distance, as opposed to constant image size.

The software RP Resonator is a particularly flexible tool for calculating all kinds of mode properties, even including misalignment effects, and allowing sophisticated design optimizations. This is vital for laser development, for example. You can take into account alignment sensitivity and thermal lensing, and even design a resonator for operation in a specific stability zone.

When both mirrors are curved or only one is flat I see how a Gaussian mode 'fits' into the resonator. What happens when the mirrors are both flat? The Gaussian modes cannot have flat phase curvature in two places.