In systems that have a circular aperture (such as an optical microscope), the modulation and/or contrast transfer function is often computed or measured with star and bar targets similar to the one illustrated in Figure 6(b). Targets of this type have both radial and tangential patterns that are orthogonal to each other and are also useful for detecting focus errors and aberrations such as astigmatism. Variations of the basic star target design contain paired lines and dots that allow determination of objective diffraction patterns both in and out of focus and are useful for measurements conducted in brightfield, reflection contrast, or epifluorescence illumination modes. The wedge and bar spacing period ranges from 0.1 micrometer to tens of microns with spatial frequencies between 0.2 and 25 line pairs per millimeter. Radial modulation transfer targets are ideal for high-resolution measurements using photographic film or analog sensors, but the horizontal and vertical pixelated nature of CCD detectors benefits from analysis utilizing targets that are geometrically consistent with the pixel rows and columns of the imaging device.

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This phenomenon is illustrated in Figure 8(a) for the periodic knobs imaged from the curved surface of a diatom frustule. As the microscope focus is changed, the knobs undergo inversion of contrast, producing a ripple effect in the relative modulation (compare knobs (1) through (5) in Figure 8(a)). Increasing the degree of defocus will produce a corresponding increase in the oscillations observed with a modulation transfer function plot, with contrast reversals affecting increasingly larger features in the image. As the specimen plane is defocused, contrast drops rapidly for microscopic feature having high spatial frequencies and more slowly for those with low frequencies. It is often useful to measure contrast at a particular spatial frequency and then follow contrast as a function of distance on either side of the image plane. This analysis is sometimes termed the through-focus transfer function and is a measure of the depth of focus for a particular objective.

The highest spatial frequencies that can be imaged by a microscope objective are proportional to the numerical aperture and are based on the distribution size of the point spread function. Objectives with low numerical apertures produce point spread functions that have a wider intensity distribution at the image plane than those formed by objectives with higher numerical apertures. At the limit of resolution, adjacent Airy disks or point spread functions start to overlap, obscuring the ability to distinguish between individual intensities. Narrower intensity distributions (at higher numerical apertures) can approach each other more closely and still be resolved by the microscope. This implies that a narrow point spread function corresponds to a high spatial frequency. In fact, the optical transfer function, a measure of spatial frequency response for an optical system, is the mathematical Fourier transform of the point spread function.

Sometimes the best way to be sure of something is just to go and look at it. That’s the basis of the drift-testing method of determining TFoV. Drift testing is simple in concept. You place a star on one edge of the field of view of an eyepiece and then use a stopwatch to measure how long it takes to drift all the way to the opposite edge of the field of view. Because stellar motion (actually, Earth’s motion) is known very precisely, it’s easy to convert the elapsed time to an accurate TFoV.

These are the usual instructions for drift testing. Place the chosen star just outside of the field of view of the eyepiece. When the star appears at the edge of the field of view, start your stopwatch. Just as the star disappears on the other side of the field of view, stop the stop watch and record the elapsed time. Calculate the TFoV in arcminutes using the formula:

The modulation transfer function of an optical system that contains a cascading series of components (microscope, digital video camera, video capture board, computer monitor, etc.) can be calculated by multiplying the individual MTF's of each component. By conducting a careful analysis of the combined system modulation transfer functions, a prediction about performance of the system can be obtained. In the same manner, the system phase response can be obtained by adding the phase transfer functions of individual components (Note: phase transfer functions are summed while modulation transfer functions are multiplied). Together, the modulation and phase transfer functions define the optical transfer function of the system. It is important to point out that the contrast transfer function does not have the same mathematical properties as the modulation transfer function and cannot be obtained simply by multiplying the CTF of individual components.

When we test a new finder, we simply point it at a bright star when we first set up the scope and then go about our business. If the star is anywhere near the celestial equator, we know about how long the half-diameter drift test should take, based on the published FOV of the finder. For example, if the finder supposedly has a 6° field, we know that our half-diameter drift test should take about 12 minutes (the 3° half-diameter at about four minutes per degree). We center the star in the crosshairs and start our stopwatch. About 10 minutes later, we wander back to the scope and watch as the star drifts out of the field of view. We jot down the elapsed time and which star we used, and do the calculations later at our convenience.

MTFtest

Once you determine an accurate TFoV for a particular eyepiece in one scope, you can back-convert that TFoV to a field stop diameter for that eyepiece by multiplying the TFoV in decimal degrees by the focal length of the scope in mm and then dividing by (180/φ) or about 57.2958. For example, to back-convert the 64.1 arcminute field we determined using a scope with a focal length of 1,255mm, convert the 64.1 arcminute field to 1.07° and use the formula 1.07 * (1,255/57.2958) = 23.44mm.

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The modulation transfer function is useful for characterizing not only traditional optical systems, but also photonic systems such as analog and digital video cameras, image intensifiers, and film scanners. This concept is derived from standard conventions utilized in electrical engineering that relate the degree of modulation of an output signal to a function of the signal frequency. In optical microscopy, signal frequency can be equated to a periodicity observed in the specimen, ranging from a metal line grating evaporated onto a microscope slide or repeating structures in a diatom frustule to subcellular particles observed in living tissue culture cells.

where T is the elapsed time in seconds and Cos(Dec) is the cosine of the declination in decimal degrees of the star you used for testing. That’s fine as far as it goes, but it does require a bit more explanation. Let’s look at each of these terms:

Because they represent a spherical surface (the sky) as a plane (the view in the eyepiece), all eyepieces must introduce some distortion. That means that even if the AFoV is specified accurately, it cannot easily be translated to an accurate TFoV. For example, Tele Vue Panoptic eyepieces have a nominal AFoV of 68°, which is accurate insofar as it goes. But the distortion present in a Panoptic eyepiece means that if you measure the TFoV accurately you’ll find that it corresponds to an undistorted AFoV of less than 65°. (We’re not knocking Tele Vue or Panoptics here; Tele Vue is a first-rate optical maker, and Panoptics are superb, world-class eyepieces. Distortion is simply an optical fact of life.)

in which the imaginary term represents the phase transfer function (PTF), or the change in phase position as a function of spatial frequency. Therefore, the optical transfer function is a spatial frequency-dependent complex variable whose modulus is the modulation transfer function, and whose phase is described by the phase transfer function. If the phase transfer function is linear with frequency, it represents a simple lateral displacement of the image as would be observed with and aberration such as geometric distortion. However, if the phase transfer function is nonlinear, it can adversely affect image quality. In the most dramatic example, a phase shift of 180 degrees produces a reversal of image contrast, where light and dark patterns are inverted.

MTF

which is more commonly referred to as the Rayleigh criterion, or the resolution limit of the microscope. Because r is inversely proportional to numerical aperture and directly proportional to the illuminating wavelength, it follows that r and f(c) are also inversely proportional, a fundamental property of Fourier transforms (the width of a function is inversely proportional to the width of its transform).

When there are no significant aberrations present in an optical system, the modulation transfer function is related to the size of the diffraction pattern, which is a function of the system numerical aperture and wavelength of illumination. In quantitative terms, the modulation transfer function for an optical system with a uniformly illuminated circular aperture can be expressed as:

In situations where the specimen is a periodic line grating composed of alternating black and white lines of equal width (square waves), a graph relating the percentage of specimen contrast transferred to the image is known as the contrast transfer function (CTF). Most specimens display a composition of sinusoidally varying intensities having differing spatial frequencies instead of distinct sharp profiles in the form of square waves. In this case, a graph relating output as a fraction of input intensity versus signal (spatial) frequency is analogous to the modulation transfer function. As the spatial frequency approaches very large values, the square wave response resembles that of a sinusoid, yielding graphs of the contrast transfer function and the modulation transfer function that are virtually identical.

The modulation transfer function has not yet been established for several contrast enhancing modes commonly utilized in optical microscopy (such as polarized light), which await more highly perfected theories of image formation and appropriate test patterns (or specimens) to determine, by experiment, the MTF values.

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The effect of increasing spatial frequency on image contrast in a diffraction-limited optical microscope is illustrated in Figure 1. A periodic line grating consisting of alternating white and black rectangular bars (representing 100 percent contrast) is presented at two spatial frequencies on the left-hand side of the figure. The resulting image produced in the microscope is shown on the right side of each objective, and appears as a sinusoidal intensity that has reduced contrast, which is plotted in the graph below the image in terms of a relative percentage of the object contrast. One hundred percent contrast represents regular white and black repeating bars, while zero percent contrast is manifested by gray bars that blend into a gray background of the same intensity. After the contrast value reaches zero, the image becomes a uniform shade of gray, and remains as such for all higher spatial frequencies.

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The limit of resolution with an optical microscope is reached when the spatial frequency approaches 5000 lines per millimeter (spatial period equal to 0.2 microns), using an illumination wavelength of 500 nanometers at high numerical aperture (1.4). At this point, contrast would be barely detectable and the image would appear a neutral shade of gray. In real specimens, the amount of contrast observed in a microscope depends upon the size, brightness, and color of the image, but the human eye ceases to detect periodicity at contrast levels below about three to five percent for closely spaced stripes and may not reach the 0.2-micron limit of resolution.

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If you have a Dobsonian or other alt-az mount, simply center the star in the eyepiece field and start your stopwatch. The human eye is very, very good at centering objects. If you start with the star centered in the eyepiece field, by definition it will cross a half-diameter of the eyepiece field as it drifts to the edge. You can use the same calculation described previously and simply double the result to determine the TFoV of the eyepiece.

Let’s assume that you’ve drift tested using Antares, which took exactly 4:43 or 283 seconds to drift across the diameter of the eyepiece field. What is the TFoV of that eyepiece in arcminutes?

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If you have an alt-az scope but prefer to use a full-diameter drift test, first do a half-diameter drift test described in the preceding bullet point. Watch where the star leaves the field of view, and then restart the test with the star entering the view at the opposite point on the clock face. (We prefer to use half-diameter drift tests for alt-az scopes; we can do twice as many tests in the same time and average the results.)

Any combination of telescope and eyepiece has a specific true field of view (TFoV), which is determined solely by the focal length of the telescope and the field stop diameter of the eyepiece. TFoV quantifies the amount of sky visible in a particular scope with a particular eyepiece. For example, if a particular telescope/eyepiece combination provides a 1° TFoV, two stars that are separated by exactly 1° will just fit into the eyepiece field, with each star on opposite edges of the field.

You must be certain that the star drifts across the actual full diameter of the field of view, rather than across a chord.

A perfect aberration-free optical system is termed diffraction limited, because the effects of light diffraction at the pupils limit the spatial frequency response and establish the limits of resolution. Presented in Figure 2 is a graph relating the modulation transfer function of a repeating specimen imaged with incoherent illumination by visible light with several different diffraction-limited microscope objectives having a circular pupil. In this case, objective quality affects the modulation response as a function of spatial frequency. Higher quality objectives (red line in Figure 2) exhibit greater performance than those of a lower quality (yellow line), and are able to transfer contrast more effectively at higher spatial frequencies. The objective represented by the yellow curve has the highest performance at low spatial frequencies, but falls short of the high numerical aperture objective at larger frequencies. Beneath the graph is a representation of relative feature size versus spatial frequency with respect to the Rayleigh criteria and Sparrow limit. Also presented is a series of sine waves representing a specimen (object) and the resulting image produced in a typical microscope as the frequency of the sinusoid increases.

The relationship between spatial frequency and the modulation transfer function for the diatom is illustrated in Figure 8(b). The graph represents a series of varying focus levels where the measured MTF is plotted against spatial frequency (number of sinusoidal features per unit distance). A drop in relative modulation values with defocus at fixed spatial frequencies is obvious in the figure, as well as the contrast reversal at focus levels 4 and 5 where the reduced spatial frequency drops into negative values of the MTF. Curve number 1 represents the diatom frustule in focus, and curves 2 through 5 present the results with successively increasing levels of defocus. The dotted line corresponds to the approximate spatial frequency of the knobs illustrated in Figure 8(a). Contrast is at a minimum where the dotted line crosses curve 4 and is reversed where curve 5 dips below zero on the y-axis.

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Divide the AFoV, 50°, by the magnification, 48X, to yield a TFoV of 50/48, or about 1.0417°. You can multiply the result in degrees by 60 to convert it to arcminutes, which are used more commonly to refer to an eyepiece field of view. A 1.0417° field is 62.5 arc-minutes, which is abbreviated 62.5’.

The actual focal lengths of both the telescope and eyepiece often differ 5% or more from nominal stated value because of normal manufacturing variances. For example, a telescope with a stated focal length of 1,200mm may actually be 1,173mm (or 1,234mm), and an eyepiece labeled 10mm may in fact be 9.7mm (or 10.4mm).

where I(max) is the maximum intensity displayed by a repeating structure and I(min) is the minimum intensity found in the same specimen. By convention, the modulation transfer function is normalized to unity at zero spatial frequency. Modulation is typically less in the image than in the specimen and there is often a slight phase displacement of the image relative to the specimen. By comparing several specimens having differing spatial frequencies, it can be determined that both image modulation and phase shifts will vary as a function of spatial frequency. By definition, the modulation transfer function (MTF) is described by the equation:

Divide the focal length of the scope, 1,200mm, by the focal length of the eyepiece, 25mm, to determine that that combination provides 48X magnification.

The Solar year is 365.2425 days long, which is 525,949.2 minutes or 31,556,952 seconds. A full circle is 360°, or 21,600 arcminutes. The correction for sidereal time therefore adds (21,600 arcminutes / 31,556,952 seconds) = 0.00068447675+ arcminutes/second to the apparent drift speed of a star located on the celestial equator. (This figure is only a very close approximation because Earth’s orbital speed around the Sun varies according to its orbital position; when Earth is near Sol, it moves faster than when it is more distant.) Adding the ~0.0006845 correction to the Solar time constant 0.25 and rounding the result gives us 0.2507.

In a cascaded series of devices that work together to produce an image, contrast is lost in certain frequency regions at each step, generally at the higher end of the spatial frequency range. In this regard, each detector or image processing function can also be used to cut off or boost the modulation transfer function at certain frequencies. At each stage, noise introduced by image transfer and processing is also a function of spatial frequency. Therefore, fine-tuning the response for optimum image contrast and system performance is dependent not only upon the type of image information desired, but also the frequency dependence of noise levels in the image. In addition, because the modulation transfer function of a detector is wavelength-dependent, it must be determined under carefully defined conditions of illumination.

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The resolution and performance of an optical microscope can be characterized by a quantity known as the modulation transfer function (MTF), which is a measurement of the microscope's ability to transfer contrast from the specimen to the intermediate image plane at a specific resolution. Computation of the modulation transfer function is a mechanism that is often utilized by optical manufacturers to incorporate resolution and contrast data into a single specification.

Utilization of contrast enhancing techniques such as phase contrast and differential interference contrast (DIC) results in unique modulation transfer functions that display curves markedly different from those observed in brightfield illumination using the objective's full numerical aperture (Figure 5). For example, the narrow illumination produced by phase rings in phase contrast microscopy produces a modulation transfer function curve that oscillates above and below the brightfield curve, while the curves generated by DIC objectives vary with the angle between the specimen period and the shear direction of the Wollaston or Nomarski prisms. Also illustrated in Figure 5 is the curve produced by a single-sideband edge enhancement microscope (developed by Dr. Gordon W. Ellis), which yields images of superior contrast at high spatial frequencies.

Because the actual TFoV is determined solely by the focal length of the telescope and the field-stop diameter of the eyepiece, it is possible to calculate very accurate TFoV figures if you have an accurate focal length for your scope and have the field-stop diameter for the eyepiece in question. Unfortunately, Tele Vue is the only eyepiece maker that routinely publishes field-stop diameters for its eyepieces. For some eyepieces, you can use calipers to measure the field-stop diameter yourself. However, the field stop of many eyepieces is inside the eyepiece rather than exposed, which means you have to disassemble the eyepiece to measure its field stop, which is not a recommended procedure.

The quickest way to determine TFoV is to divide the Apparent Field of View (AFoV) of the eyepiece by the magnification provided by that eyepiece in a given scope. For example, if your scope has a focal length of 1,200mm and you use a 25mm Plössl eyepiece with an AFoV of 50°, you can calculate the TFoV as follows:

It’s interesting to compare the results of this method with the AFoV method. Assuming a telescope focal length of 1,255mm and a 27mm Panoptic with a 68° AFoV, the AFoV method yields a true field of about 1.46°. Calculating based on the 27mm Panoptic field stop diameter of 30.5mm yields a true field of view of only 1.39°, which specifies accurately the amount of sky visible with that telescope/eyepiece combination. Back-converting the accurate TFoV to determine a “real” AFoV gives us about 64.72° rather than the nominal 68°. Again, this is not a knock on Tele Vue or Panoptic eyepieces; the AFoV in Panoptics simply “looks” wider than it really is.

The target presented in Figure 6(a) is designed specifically for testing the horizontal modulation transfer function of a macro imaging system such as a telescope, binoculars, video system, camera, or digital video recorder. It is composed of sinusoidal patterns having a spatial frequency range between 0.2 and 80 line pairs per millimeter with a grayscale optical density range varying between 0.2 and 1.2 and an 80 percent modulation of the sine waves. This type of target relays image quality information over a wide range of frequencies and contains on-target references for denoting the contrast levels of the sinusoidal frequencies. In video microscopy, microscopic test targets of sinusoidal targets are not readily available, so the contrast transfer function of a video detector coupled to the microscope is often determined rather than the modulation transfer function.

Individual objectives in a microscope display a specific modulation transfer function (or optical transfer function) that depends on numerical aperture, objective design, illumination wavelength, and the mode of contrast generation. When the numerical aperture of the condenser is equal to or greater than that of the objective, the spatial frequency cutoff value decreases with decreasing objective numerical aperture (Figure 4(a)). Holding the objective numerical aperture value constant and varying the condenser numerical aperture results in progressively lower cutoff values with decreasing condenser numerical aperture (Figure 4(b)).

In practice, the performance of a microscope objective or other lens system is often determined by tracing a large number of light rays emitted by a point source in a uniformly distributed array over the vignetted entrance pupil of the objective. After passing through the exit pupil and being distributed over the image plane, the ray intersections are used to plot a spot diagram of the light points at the image plane. In most cases, several hundred rays are utilized to construct a spot diagram, which may take into account optical aberrations if the spacings of light rays are so adjusted. The resulting spot diagram is then regarded as a point spread function and is converted into a graph of the modulation transfer function versus spatial frequency by means of a Fourier transform.

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Using a “slower” star without a correction factor would overstate the TFoV of an eyepiece. (The star takes longer to drift across the field, so the TFoV appears larger than it really is.) But you can use any star for drift testing if you calculate its declination correction factor. To do so, convert the star’s declination to a decimal value and determine the cosine of that value. For example, if you use Antares, convert its declination of -25°26’ to -25.433°. The cosine of -25.433° is about 0.9031.

For perfectly accurate results, the star you use must be located exactly on the celestial equator [Hack #17], at declination 0°0’0”. This is true because a star at other than exactly 0 declination has a different apparent drift rate. In practice, any star within ±5° or so of declination 0 yields sufficiently accurate results for all but the most critical work. If Orion is visible, use Mintaka (the belt star on the Bellatrix side), which is at declination about 0°18’. If Orion isn’t up, other good choices are Porrima (α-Virginis) at about 1°29', Sadalmelik (γ-Aquarii) at about -0°17', and σ-Serpens at about 1°00’. (You can use any star for precise drift testing if you apply a declination correction factor, described in the next section.)

Modulation transfer function

If you have or can get accurate values for your scope’s focal length and your eyepieces’ field-stop diameters, you can calculate an accurate TFoV. To do so, divide the eyepiece field-stop diameter in mm by the focal length of the scope and multiply the result by (360/2φ) or about 57.2958. (There are 2φ radians in a full circle of 360°. Dividing 360° by 2φ yields about 57.2958° per radian.) For example, if you use a 27mm Tele Vue Panoptic eyepiece with a 30.5mm field stop in a scope of 1,255mm focal length, the TFoV is (30.5/1,255)*(360/2φ), or about 1.39°. Multiplying that result by 60 yields the TFoV in arcminutes, about 83.5’.

Up to this point, we’ve considered only Earth’s rotation on its axis, which is the basis of Solar time. But to calculate stellar motion properly, we have to use sidereal time (star time), which is slightly different from Solar time. The difference exists because as Earth rotates on its axis, it also orbits Sol. Over the course of one year, Earth makes one full additional rotation in its orbit, and this must be taken into account.

Direct measurements of the modulation transfer function are conducted by utilizing specific test pattern targets consisting of high-contrast periodic line gratings having a series of spacings that usually range from one or several millimeters down to 0.1 micrometer, as illustrated in Figure 8. These targets allow evaluation of microscope objective diffraction patterns, both in and out of focus, in a variety of contrast enhancing modes. Detector arrays are utilized to measure the distribution of light in the image plane by summation of the point spread functions, and a Fourier transform algorithm applied to the data to determine the modulation transfer function.

The factor 0.2507 is the constant that converts seconds of time to the TFoV in arcseconds. The earth rotates on its axis 360° in 24 hours, which translates to 15° per hour, 1° every four minutes, 1/4° per minute, and 1/240° per second. There are 60 arcminutes per degree, so Earth rotates at 60/240, or 0.25 arcminutes per second. (This is known as the Solar Time Constant.) So why the extra decimal places?

It is important to know the TFoV of your eyepieces in your scope and the TFoV of your finder because you use the true field to match the stars that are visible in the eyepiece to those on your charts and to plan and execute star hops [Hack #21]. There are several methods, of varying accuracy, to determine TFoV.

The relationship between the modulation transfer function and the point spread function for a diffraction-limited optical microscope is illustrated in Figure 3. As discussed above, the limiting cutoff frequency (f(c)) of the modulation transfer function is directly proportional to the objective numerical aperture and inversely proportional to the illumination wavelength. The radius of the first dark concentric ring surrounding the central intensity peak of a point spread function (or Airy disk) is expressed by the equation:

A typical intensity scan made from a star target measured with a high numerical aperture apochromatic objective operating in transmitted light mode is presented in Figure 7(a). Intensity values were averaged over the dimension parallel to the target grating lines. When these types of data are collected for a variety of objectives at varying numerical aperture and plotted as percent contrast versus spatial frequency, a graph similar to that illustrated in Figure 7(b) is obtained. Contrast transfer approaches 100 percent at very low spatial frequencies (wide spacing periods) and gradually drops with increasing spatial frequency. As spatial frequencies reach the Abbe limit (the imaging wavelength divided by twice the objective numerical aperture), contrast values are generally too low to detect individual spacings in the line grating.

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Most astronomers use drift testing only for eyepieces, but it’s just as accurate a means of determining the true field of view of optical finder scopes. Of course, finders have much wider fields of view than eyepieces, so it may take half an hour or more to do a full-diameter drift test on a finder, which you really don’t need to do. Finders have nicely centered crosshairs, so it’s trivially easy to do an accurate half-diameter drift test. If you center the star in the crosshairs, there’s no question that the star crossed the exact center of the field.

The number of spacings per unit interval in a specimen is referred to as the spatial frequency, which is usually expressed in quantitative terms of the periodic spacings (spatial period) found in the specimen. A common reference unit for spatial frequency is the number of line pairs per millimeter. As an example, a continuous series of black and white line pairs with a spatial period measuring 1 micrometer per pair would repeat 1000 times every millimeter and therefore have a corresponding spatial frequency of 1000 lines per millimeter.

If you know the focal length of your other scopes, you can determine their true fields of view without drift testing simply by using the field-stop diameter calculation described in the preceding section.

If the star you use for testing is exactly on the celestial equator (declination 0), this factor drops out because the cosine of 0 is 1. If the star is not on the celestial equator, its apparent motion is slower. The larger the absolute value of the declination, the slower the apparent motion. A star located at exactly +90° or -90° declination has no apparent motion at all because it is exactly on the pole of the axis around which Earth rotates. Polaris, for example, is located at declination +89°15', and it requires 24 hours to trace a 1.5° circle around the pole.

The phase response from an ideal imaging system demonstrates a linear dependence on spatial frequency, with a position shift that is independent of the frequency and normalized to zero at zero spatial frequency. In the ideal system, all sinusoidal image components are displaced by the same amount, resulting in a net position shift for the image without degradation of image quality. When the phase response deviates from ideal linear behavior, then some components will be shifted to a greater degree than others resulting in image degradation. This is especially critical in electronic video systems, which often possess less than ideal phase characteristics that can lead to noticeable loss of image quality. Fortunately, an ideal aberration-free optical system having a circular aperture and a centered optical axis (such as a high-performance microscope) will produce a phase transfer function that has a value of zero for all spatial frequencies in all directions. In this case, phase shifts occur exclusively for off-axis rays and only the modulation transfer function need be considered.

This quantity, as discussed above, is an expression of the contrast alteration observed in the image of a sinusoidal object as a function of spatial frequency. In addition, there is a position or phase shift of the sinusoid that is dependent upon spatial frequency in both the horizontal and vertical coordinates. A good example occurs in video microscopy where the raster scanning process produces slightly different responses resulting in a variation between the horizontal and vertical modulation transfer functions.

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Modulation of the output signal, the intensity of light waves forming an image of the specimen, corresponds to the formation of image contrast in microscopy. Therefore, a measurement of the MTF for a particular optical microscope can be obtained from the contrast generated by periodic lines or spacings present in a specimen that result from sinusoidal intensities in the image that vary as a function of spatial frequency. If a specimen having a spatial period of 1 micron (the distance between alternating absorbing and transparent line pairs) is imaged at high numerical aperture (1.40) with a matched objective/condenser pair using immersion oil, the individual line pairs would be clearly resolved in the microscope. The image would not be a faithful reproduction of the line pair pattern, but would instead have a moderate degree of contrast between the dark and light bars (Figure 1). Decreasing the distance between the line pairs to a spatial period of 0.5 microns (spatial frequency equal to 2000 lines per millimeter) would further reduce contrast in the final image, but increasing the spatial period to 2 microns (spatial frequency equal to 500 lines per millimeter) would produce a corresponding increase in image contrast.

A perfect optical system would have a modulation transfer function of unity at all spatial frequencies, while simultaneously having a phase transfer factor of zero. In cases where the image produced by the microscope (or other optical system) is sinusoidal and there is no significant phase shift, the modulus of the optical transfer function reverts to the modulation transfer function.

It’s surprising how much the actual FOV of a finder may differ from published specifications. We saw one finder with a nominal FOV of 6.5° that turned out to be more like 5.8°. Conversely, we remember another finder with a supposed 5° FOV that actually had a 5.5° FOV. That amount of difference can be significant, particularly if you are a dedicated star hopper, so it’s worth testing your own finders to determine their actual fields of view.

It is interesting to note that this equation expresses (in terms of spatial frequency) the fact that resolution increases with both numerical aperture and shorter wavelengths.

When the input is a high contrast square wave, such as the periodic grating target illustrated in Figure 1, transfer of contrast is determined by the contrast transfer function. A majority of specimens observed in the microscope, however, do not display such a regular periodicity and consist of "square waves" that are sinusoidal to varying degrees at the sub-micron level. In this case, the modulation transfer function is utilized to calculate transfer of contrast from the specimen to the image produced by the microscope.

If you have a polar-aligned equatorial mount, center the chosen star in the eyepiece field, turn off your drive motor(s), and use your RA slowmotion control to put the star just outside the eastern edge of the eyepiece field. As the star appears in the field, start your stopwatch. Stop it just as the star exits the eyepiece field.

The specified AFoV is often an approximation. An eyepiece with a nominal 55° AFoV may actually provide as little as 50° or as much as 60°. The nominal AFoV of premium eyepieces tends to be reasonably accurate; that of cheap Chinese eyepieces tends to be inaccurate and very optimistic. Also, the true AFoV of different focal length eyepieces in a series with nominally identical AFoVs may differ significantly from one focal length to another.

The AFoV method suffices for a quick-and-dirty calculation of TFoV and is good enough for most purposes, but it is not accurate enough for critical work.

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In some instances, the modulation transfer function of an optical microscope can actually be less than zero. This occurs in an otherwise functional system when performance is degraded due to defocus, aberrations, and/or manufacturing errors. Often, the modulation transfer function will oscillate above and below zero as the microscope is racked through the point of best focus on a specimen having features with high spatial frequency. When the transfer function dips below zero, the image undergoes a phase reversal in which dark features become bright and vice versa.

Another important concept is the optical transfer function (OTF), which represents the ratio of image contrast to specimen contrast when plotted as a function of spatial frequency, taking into account the phase shift between positions occupied by the actual and ideal image. In general terms, the optical transfer function can be described as:

But the timing and calculations are the easy part. The first time you actually drift test an eyepiece, you’ll learn that the hard part is getting the star to drift across the full diameter of the eyepiece field instead of a chord. If you try to do this by trial and error, you’ll waste a lot of time and become quite frustrated. We know. We’ve watched people do it. However, there are some easy solutions:

In these equations, ν is the frequency in cycles per millimeter, λ is the wavelength of illumination, and NA is the numerical aperture. At low spatial frequencies, image contrast is the highest, but falls to zero as the spatial frequency is increased beyond a certain point (drawn in Figure 2 as a reduction in amplitude produced in the image). The cutoff (f(c)) is the spatial frequency at which contrast reaches zero and can be expressed by the equation:

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The modulation transfer function is also related to the point spread function, which is the image of a point source of light (commonly referred to as the Airy disk) from the specimen projected by the microscope objective onto the intermediate image plane. Optical aberrations and numerical aperture variations affect the distribution of light intensity observed at the image plane, and thus influence the shape of the point spread function. Also note that the sum of the point spread functions produced by a specimen in a diffraction-limited microscope comprises the diffraction pattern produced at the image plane.

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When a specimen is observed in an optical microscope, the resulting image will be somewhat degraded due to aberrations and diffraction phenomena, in addition to minute assembly and alignment errors in the optics. In the image, bright highlights will not appear as bright as they do in the specimen, and dark or shadowed areas will not be as black as those observed in the original patterns. The specimen contrast or modulation can be defined as:

All optical systems and supporting components including microscopes, digital and analog video systems, video capture boards, cables, computer monitors, photographic film emulsions, and the human eye each have a characteristic modulation transfer function. In the case of analog and digital electronic imaging detectors, the reciprocal relationship discussed above between spatial resolution and frequency response is valid. In this case, however, the point spread function is replaced by the time response to a very short electrical impulse, and the optical transfer function is replaced by the imaging system's response to the sinusoidal electrical signal with respect to amplitude and phase. Electronic systems lack the symmetry of optical systems, which introduces non-linear phase effects into the function. Regardless of these differences, the underlying concepts are similar between electronic and optical systems, and this allows optical microscopes coupled to digital (or analog) imaging equipment to be analyzed within a common framework.

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