Comparison of Simulation to Actual Photos
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| Galilean Telescope Homepage | Site Map |
The experienced amateur astronomer learns to evaluate the quality of a telescope, in part, by comparing the images seen a little inside and outside focus. For a good quality, well-corrected telescope, the changes are quite symmetric. When operating in white light (as opposed to at a single wavelength), the Galilean telescope does not have this ideal symmetric behavior. This page uses JavaScript to create animations intended to give you an impression of what one sees while adjusting the focus by lengthening or shortening the drawtube of the Galilean telescope with both positive and negative eyepieces. It includes both simulations and actual photographs of the effects to be expected. If your browser is not set to permit JavaScripts, you will see only the middle image of the animations and the buttons below them will not work. In that case, please see the focus sequences presented in thumbnail format later on the page. A subset of the full size images can then be accessed by clicking on the thumbnails. The main reason for using JavaScript is to permit you to actively select the images you want to see, so that they do not all have to be downloaded each time you visit this page.
Assuming JavaScript is working, you should be able to click through the individual images you are interested in seeing using the buttons on the screen; or, alternatively, to animate the entire sequence in a slide show format, simulating what you would see through the telescope eyepiece as you pull the drawtube in and out. The first time through you may experience some delay since a temporary copy of each requested image has to be downloaded to your computer from the internet. If your browser is set to cache images, subsequent image transitions should be nearly instantaneous. For the animations to work properly, each image must load fully before the next is requested. If your internet connection is too slow to do this in real time, you may have to manually cache the images by stepping through them: for each sequence you are interested in animating, go to the first frame (by clicking "|<" under the picture), then go through the rest (by clicking ">" repeatedly) until you come to the last frame (indicated by the ">" button going gray). The animation should then work properly using the cached images regardless of your connection speed.
As much as possible, related images have been shown next to each other. However, you may well wish to compare images appearing on different parts of the page. The most efficient way to do this is to use your browser to open a second copy of the present page (e.g., CTRL-N in Internet Explorer), then adjust the size and position of the two windows to juxtapose the images however you desire.
This page is divided into the following major sections: observed focusing characteristics with a Galilean eyepiece, observed focusing characteristics with a Keplerian eyepiece, simulation of images by geometric ray tracing (a simple and easily-implemented technique for predicting the appearance of extended targets), and simulation of images by Fourier transformation (a much more rigorous technique for predicting the appearance of point sources).
The following images show the field visible when the Galilean telescope is focused on a power pole approximately 587 feet away. The images are reproduced at one-half their original pixel scale. That is, each pixel represents approximately 1.5 arc-seconds. The full width of the 550 pixel frames is about 13.4 arc-minutes. As explained elsewhere this encompasses the full field of the Galilean telescope, but only the central part of the visible image when the positive (Keplerian) eyepiece is used.
Use the buttons below the picture to step manually between images, or click "Animate" to automatically cycle back and forth through the focus sequence.
The numbers in the upper left indicate the sequence in distance of the eyepiece from the "best" focus. - = motion towards objective; + = motion away from objective. The steps are equal and the total range is about 10 mm.
The silhouetted power lines have little color of their own. Rather they assume the color that bleeds into them from the edge of the out-of-focus sky. The objective refracts blue light more strongly than red. Inside focus (eyepiece closer to objective) the blue light is better focused and the white sky is fringed by a reddish haze. Outside focus the situation is reversed and the shadow areas assume a bluish hue from the spill-over of unfocussed blue light from the white sky.
These pictures were taken on the same day and are cropped from the center of the field of view with the positive eyepiece. They have been inverted to match the upright orientation of the Galilean images. They are also reproduced at one-half the original pixel scale.
The numbers in the upper left indicate the sequence in distance of the eyepiece from the "best" focus. - = motion towards objective; + = motion away from objective. The steps are the same as for the negative eyepiece, but the average position is approximately 100 mm farther from the objective.
The color fringing is produced primarily by the chromatic aberration of the objective, and there is no detectable difference in its appearance at comparable distances from the best focus with the two eyepieces.
Given that we have photographs taken through the modern Galilean telescope, it may seem pointless to try to simulate what one should see through it. The reason for being interested in a simulation is that the present telescope represents only one of countless possible configurations. One might wonder how the performance might change if we altered the aperture, focal length or glass characteristics (dispersion). If one is able to successfully simulate the image forming properties of present telescope, then it should be possible to explore these questions without actually having to build the alternative telescopes (some of which may be very difficult or impossible to realize in practice).
As a first attempt at simulation, we instructed a computer to trace rays through the two lenses of the Galilean telescope and tabulate the angles at which they emerged from the eyepiece. Each ray was assigned a random color (wavelength) between 386 and 696 nanometers from a solar-like distribution that simulates white light (i.e., the sum of a large number of randomly selected rays gives a neutral white or gray on the computer screen). The refraction angles were calculated using dispersion data appropriate for that wavelength. The lenses are assumed to have perfectly spherical surfaces. Each emerging ray impinged on a particular pixel in the simulated image, adding a certain amount of red, green and blue intensity as needed to render that color. To simulate the effects of diffraction, the geometrically-calculated position was randomly perturbed by an amount based on the known size and distribution of light in the nominal diffraction disk. As additional rays were traced, the computer kept track of total amounts of red, green and blue intensity falling on each pixel of the image. Finally, these were normalized to the 0..256 range required by the computer hardware, and displayed as an image.
Although this may seem fairly straightforward, a number of questions arise both as to how to properly render the colors, and as to whether the effects of diffraction are being properly represented at out-of-focus points. The accurate rendering of colors is especially troublesome because most computer displays have a highly non-linear response. The simulated images were converted a format (gamma = 0.5) that should display more-or-less properly on a conventional PC monitor with a gamma of about 2; but there are a number of ways in which this correction can be made (e.g., correcting all channels by the same factor or correcting the three channels separately), each giving a rather different appearance.
As to the target, for analyzing the performance of the present telescope we have found it convenient to imagine the telescope imaging a distant 2 arc-minute square white target on a black background. On the card is a pattern of three black bars whose centers are separated by 10 arc-seconds. When viewed at 20X, the white test card is magnified to an apparent size of 40 arc-minutes, or a little less than one degree. In the following simulations, we show the predicted image over a full one-degree square (as seen through the eyepiece), which accounts for the black border around the target.
Although on this page we show only the images expected looking down the axis at the particular target just described, the software developed for this purpose is equally capable of predicting the image expected with other target patterns, at other viewing angles, and with non-spherical lenses.
In the following pair of animations we compare the expected image for a 23 mm aperture Galilean telescope with BK-7 lenses (objective FL = +1000 mm; eyepiece FL = -50 mm), to that from a perfect telescope of the same aperture and focal length. The yellow numbers in the upper left are the distance in millimeters from the front surface of the objective to the front surface of the -50 mm FL negative eyepiece.
| Perfect Telescope | 1000 mm FL Galilean Telescope |
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The animation on the left represents a "perfect" Galilean telescope, and shows that in this case the simulation correctly renders the neutral tones of the test target. Each frame was produced by tracing 10 million rays. By "perfect" we mean a telescope made to the same specifications as the website telescope, but substituting spherical lenses made of an imaginary very high index dispersion-free glass. Such lenses would be very thin and would suffer from neither spherical nor chromatic aberration. The remaining fuzziness comes from a combination of defocus and the intrinsic diffraction of the 23 mm diameter objective. This image is essentially what you would expect to see through a modern well-corrected reflector or refractor stopped to a 23 mm entrance aperture. In order to resolve the three bars on the test target, the drawtube has to be set quite close to the best focus.
The animation on the right is our best effort at simulating the present real telescope by geometric ray tracing. Each frame was produced by tracing 100 million rays. The color fringing arises from unfocused light from the white card spilling into the dark areas of the scene. A complementary color is seen at the edge of the white area. Although the three bars are never as sharply resolved, their existence can be detected over a larger range of focus than in a "perfect" telescope. Virtually identical images are expected with a positive eyepiece if it is placed 100 mm farther from the objective.
According to the simulation, the real Galilean telescope has a kind of self-filtering property, such that at any given focal position some color will be in relatively sharp focus, while the other colors will be so out of focus as to be less bothersome than in a perfect system. That is, although the best image is not as good, the Galilean telescope is more tolerant to errors in focus. In the perfect telescope the bars can be distinguished over a range of less than 10 mm; whereas in the simulated real telescope they can be detected over a range of more than 20 mm.
For additional simulations of the in-focus central images expected with various combinations of commercially-available singlet lenses see How to Build a Galilean Telescope.
To test whether the preceding simulations accurately represent the performance of the real telescope, we printed on white photo quality paper a pattern of black bars identical to that used in the simulations, and to a scale such that when taped to a light pole 214 feet from the front of the telescope the bars would appear, to the unaided eye, to subtend an angle of 10 arc-seconds between centers. The target was cut to a size subtending a total angle of 2 arc-minutes and pasted to a black velvet background. We shot a focus sequence of this target using the website Galilean telescope and C-3000 camera at various exposure settings on January 12, 2005. As in the simulations, the drawtube was moved in steps of 5 millimeters, although the exact distance from objective to eyepiece was not accurately measured.
In the following, a thumbnail of the actual test photo is shown below a thumbnail of the corresponding simulation. Click on any of the thumbnails to see the full size image. The full-size photographs are cropped from the focus series, and shown at their original pixel scale. That is, their scale is twice as large as the (reduced) photographs the power pole insulators shown at the top of the page.
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The simulated colors seem similar to, but somewhat more intense (saturated), than the real ones, especially at the best focus. The resolution appears significantly better than that predicted by the simulation. The exposure sequence (not shown) reveals that the response of the camera to changing light levels is non-linear, especially at the high intensity end, where the pixel counts never quite saturates even for large increases in intensity. This is different from the photometric behavior assumed in the simulations, and can affect both the apparent sharpness of the images and the intensity (saturation) of the colors. The simulations can be made to more closely resemble the photos, or vice versa, by arbitrarily manipulating the gamma. Further analysis is required before we can reproduce the simulations and actual photos on the same photometric scale, however it seems unlikely they will ever match exactly.
The crisp bluish-white photograph shown at 945 mm in the lower row comes closest to duplicating the impression obtained looking through the telescope with the human eye, although the white appears much brighter to the eye.
Until the detailed photometric comparison can be completed, we have the following conclusions regarding the accuracy of the simulations by geometric ray tracing in comparison to the actual photos:
As our second attempt at simulation, we used a method, based on the fast Fourier transform, that should give a highly accurate prediction for the image produced by a point source at a single wavelength. The image produced by a white light point source (such as a star) can be estimated by summing (incoherently) the predicted images at a series of constituent wavelengths; and, at least in theory, the image from an extended (two-dimensional) source (such as a planet or test target) can be obtained by summing (again incoherently) the images from each constituent point.
In this technique, rays from the source, all initially in phase, are traced geometrically to the exit pupil of the telescope (which in the present case happens to be near the eyepiece, but inside the telescope). This produces a map of the expected distribution of the phase and intensity of light over that surface. The Fourier transform of this phase distribution is a prediction of the system's Fraunhofer diffraction pattern, and should be a close approximation to what is actually seen looking in the eyepiece. If the phase and intensity distribution is uniform, the pattern consists of a bright central patch surrounded by a series of increasingly faint rings, commonly referred to as the Airy disc, after 19th century British mathematician and Astronomer Royal G. B. Airy. The angular scale of this classic pattern is directly proportional to the wavelength and inversely proportional to the diameter of the exit pupil. If the distribution is non-uniform, the pattern will be distorted. The distortions can range from subtle changes in the intensity of the rings compared to the central spot, to major changes in shape. Prediction of this optical pattern by Fourier transformation of the phase map is far more rigorous than the results obtained by geometric ray tracing alone, and for light of a single color should give results essentially identical to the observations.
The last part of the calculation procedure, the transformation of the phase map to a predicted color image, is very similar in concept to the Aberrator Software developed some years ago by Dutch amateur and Registax author Cor Berrevoets, although the implementation is completely independent of his. In particular, the phase map of the Galilean telescope is freshly estimated at each wavelength.
As with the ray-tracing software, the program developed for this purpose can predict images at any entry angle. However, we show below only the predicted full color point response corresponding to the previous ray-tracing simulations of an on-axis target, over a range of +/-20 mm from the best focus. The predicted point response seems to confirm the self-filtering property of the Galilean telescope demonstrated in the previous simulations and test photos. Each simulated Airy disk is the sum of the diffraction predictions at 20 equally spaced colors (wavelengths). Because the brightness of the diffraction spot falls off as it increases in diameter, the out of focus images from the perfect telescope would be too dim to be easily seen on a computer screen. Therefore the gain of all images has been independently adjusted to bring the brightest point in each to saturation. Similarly, most of the simulated white light point images from the Galilean telescope show a strongly colored central spot. This spot is always surrounded by a unfocused haze of light in the complementary color (such that the sum of light from all the pixels adds to a fixed quantity of white). The haze is difficult to perceive on the computer screen, and, due to the normalization, all of the central spots appear of the same intensity even though they are not. For example, the violet colored central spot seen through the Galilean telescope with the drawtube at a short position (930 mm) is dimmer (and the haze brighter), than at the position (950 mm) where a yellowish spot is seen.
In the following, the simulation squares cover a width of exactly 40 arc-minutes as seen through the eyepiece. However, resolution is more commonly expressed in target space: correcting for the 20x magnification of the telescope, the width of the simulation in target space is 2 arc minutes (that is, the same as the width of the white card shown in the test photos and ray-tracing simulations). Since the ray-tracing simulations show both the card and the area around it, these images are slightly magnified compared to those.
To bring out detail in the outer rings, these simulations have not only been normalized to bring the central spot to saturation, but they are also displayed at a gamma of 0.5 (which exaggerates faint tones). As will be shown below, this makes the rings more prominent than they appear in our photographs of actual Airy disks. Whether they are also more prominent than they appear to the eye is hard to say: that appearance is very subjective.
The following three simulations all refer to a 23 mm aperture by 1000 mm focal length with the image formed by a -50 mm focal length singlet eyepiece. As before, the numbers in yellow are the distance in millimeters from the first surface of the objective to the first surface of the eyepiece.
| Perfect Telescope | Galilean Telescope | Galilean at 555 nm |
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The animation at the left represents "perfect" Galilean telescope, that is, one with the same aperture and focal length as the website telescope, but made from an imaginary very high index, dispersion-free glass (which, as explained above, means it suffers from neither chromatic nor spherical aberration). The telescope is focusing a white light point source. The colored diffraction ring around the central spot in the in-focus (950 mm) image is probably too faint to be seen on most computer screens. It is very similar in size to that seen in the center panel (Galilean telescope) at 950 mm, but shows a Newton's rings type color gradation. If you want to see the ring, so can save a copy of the photo to your local disc and manipulate it with any photo-processing software. The predicted images at equal distances on either side of the best focus are almost perfectly symmetric. That is, the image at 930 mm (-20 mm from focus) is nearly identical to that at 970 mm (+20 mm from focus). The central spot is itself seen to go through a distinct sequence of color changes as the outer rings are added. The sequence is: white - pale yellow - ruby red (not shown here because the focus step is too coarse) - nearly black - turquoise blue - then back to yellow. The intensity (saturation) of the colors fades with increasing distance, and far from focus the central spot reverts to white. As will be shown below, the predicted sequence is indeed seen in a well-corrected system.
The center panel shows the predicted performance of the website telescope operating under identical conditions. Instead of showing a rapid increase in diameter and a colorful pattern of rings as one moves off focus, the simulated Galilean telescope shows primarily a change in the color of the central spot, plus a slight wavelength-related increase in size (going from blue to red). Intuitively, this behavior occurs because, due to the chromatic aberration of the objective, different colors (wavelengths) come to focus at different positions of the drawtube. At any given position, one color is in focus and forms a (relatively) intense Airy spot which strongly dominates over the haze of highly defocused light from the other (complementary) colors. Hence, if one does not mind the color change, the actual Galilean telescope is much more tolerant of focus errors than a perfect one, but at the same time, the contrast is poorer because of the haze of unfocussed light. Somewhat surprisingly, as also shown here, over most of the range the haze is quite dim compared to the focused point of light, and not nearly so bothersome as one might have thought. The fact that the focus deteriorates much sooner on the red side than on the blue side seems to be related to the changing slope of the dispersion curve for BK7.
To clarify how the white light image is a composite of the individual wavelengths, the left hand panel represents the predicted performance of the website telescope operating at a single wavelength, in this case 555 nanometers (5550 Å), the color best focused at the nominal 950 mm drawtube position. As shown, at a single wavelength the tolerance of the telescope to errors in focus is essentially as small as for a perfect telescope. Unlike the perfect telescope, there is an slight asymmetry in the patterns seen on either side of the best focus, and that asymmetry increases with distance. This is an indication of the very slight spherical aberration inherent to a singlet objective (even of long focal length) with a spherical first surface. It should be kept in mind that these images have been normalized to constant apparent intensity. In reality, as the diameter of the diffraction pattern increases, the intensity falls off in such a way as to keep the total amount of light in the image constant. If the images were not normalized, the monochromatic diffraction disk would rapidly fade into invisibility as its diameter increases.
It may seem peculiar that the only slightly off-white test card predicted by the ray tracing simulation (and seen by the camera looking through the real telescope) can be produced by the superposition of such strongly colored Airy spots as those shown in the center panel above. The explanation, again, comes from the faint background haze of the complementary color, which is not really visible on the computer screen. For example, in predicting the color of the white test card at a focal position of 945 mm, the light from any particular point, by itself, would produce a strong green spot in the image, as shown above. But there is added to this the contribution of the complementary color from all the surrounding points. The contribution from any one of these is slight, but there are so many surrounding points that the total effect is the only very slightly off-white color shown in the ray tracing simulation.
In the ray-tracing section, the resolution of the website telescope was discussed in terms of its ability to detect the presence of bars on a test target. Amateur astronomers are more accustomed to considering resolution in terms of the telescope's ability to split double stars (two closely placed point sources). This can be easily simulated with the present software by superimposing the predicted Airy patterns for independent sources at two different entry angles. In the examples shown below, the sources (of equal brightness) were separated by the 2, 3, 5 and 7 arc seconds and the resulting image predicted over a width of 10 arc minutes (corresponding to a total width of 30 arc seconds in target space, after correcting for the 20X magnification of the hypothetical telescope with a -50 mm focal length eyepiece). The Airy spots are shown twice as large as in the preceding animations.
The predicted results are shown for two different apertures and rendered with two different gammas (to better show the rings). All images are full-color simulations calculated assuming the eyepiece was 945 mm from the objective.
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| Gamma | 3 arc sec | 5 arc sec | 7 arc sec | 3 arc sec | 5 arc sec | 7 arc sec |
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The examples on the left, with the 23 mm diameter aperture, correspond to the current website telescope. The simulations agree very well with the result obtained by photographing a distant NBS bar target: bars with centers separated by 5.6 arc seconds were easily resolved, while bars separated by 4.0 arc seconds were seen as a continuous blur. While resolving bars is not precisely the same as resolving point sources, the present simulations also suggest that separate objects merge into an unresolvable mass somewhere between 3 and 5 arc seconds. Since the resolution limit is evidently determined very much by the diffraction properties of the objective, the resolution estimates made by the Fourier transform technique are presumably far more reliable than those obtained by geometric ray tracing (where the diffractional blurring was handled quite arbitrarily). The present focal position has been chosen because the green images show nicely on the computer screen. The observed resolution at 945 mm is a little better than the Rayleigh limit given on the Galilean Optics Page, because the predominant wavelength at 945 mm is a little shorter than the one assumed in the calculation of the Rayleigh limit. The diameter of the Airy spot is slightly smaller still in the blue (see previous section), so slightly higher angular resolution even than that shown here could be obtained by placing the eyepiece a little closer to the objective.
The examples on the right, with the 70 mm aperture, correspond to the Galilean telescope on the l'Ecole Supérieure d'Optique website. This example is shown to counter the often-repeated claim that increasing the aperture of a Galilean telescope much beyond an inch will destroy the image through a combination of spherical and chromatic aberration. According to the simulation, a well made 70 mm singlet of 1000 mm focal length should be able to resolve the 5 arc second point sources quite comparably to the 23 mm aperture telescope. In addition, although it does so with imperfect contrast (i.e., it does not cleanly separate the sources), it can detect the double nature of the source to below 2 arc seconds. While it is difficult to be sure the photos were taken under comparable seeing conditions, this accords well with the higher resolution Galilean photos shown on l'Ecole Supérieure d'Optique's website. Indeed, the simulation software suggests that, although the contrast will suffer, the resolution (star-splitting ability) will continue to increase even beyond 70 mm if the objective has good surfaces.
As a final example we compare our photograph of the double star Mizar to the predictions for 1000 mm focal length Galilean telescopes with various apertures. The two components of Mizar are separated by 14.4 arc seconds and have a magnitude difference of 1.7 (corresponding to an intensity ratio of 4.8::1). To show as much detail as possible, the photograph taken through the website telescope (23 mm aperture) has been magnified to twice its original size, giving a scale of 0.36 arc seconds per pixel. The simulations of two white light stars with this separation and intensity difference are shown for the website telescope, and for that telescope with the aperture increased to 38 and 70 mm, at the same scale as the photo. The simulations were stored at gamma = 0.5, which should make the intensity pattern seen on a typical computer display with gamma = 2 approximate what would be seen by the eye looking through the telescope. The simulated telescopes were set at a focal position of 945 mm to give a greenish image matching the color seen in the photo. Please note that the photo is a little overexposed compared to the simulations.
| Actual Photo | 23 mm Aperture | 38 mm Aperture | 70 mm Aperture |
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The 38 mm aperture corresponds to that of Galileo's broken "Discover" lens (although the focal length of that lens is actually a little longer than the 1000 mm assumed here), while the 70 mm aperture corresponds to the Ecole Supérieure d'Optique telescope described above. The diameter of the diffraction ring in the photo, taken under conditions of imperfect atmospheric seeing, is not completely clear. Judging from the few fragments visible, it appears to be slightly larger than that predicted in the 23 mm simulation. However, poor seeing has a tendency to scatter light into the outer rings, which might increase its size. Galileo described this system as consisting of two stars with apparent diameters of 4 and 6 arc-seconds separated by a gap of 10 arc seconds. Such a description seems to fall somewhere between the 23 and 38 mm simulations, probably closer to the 38 mm one. The contrast problems of the 70 mm telescope are perhaps more apparent here than in the previous simulations with equal intensity stars. Its contrast could, of course, be greatly improved by extending the focal length. At a 1000 mm focal length, the 38 mm aperture is predicted to provide a cleaner image than either the 23 or the 70 mm aperture.
As with the previous purely geometric ray tracing, the subtleties of color rendition on a computer screen create problems in comparing the simulations to either photographs or visual observations. We were initially surprised to find that for a perfect system the software developed for this purpose predicted a rather colorful Newton's rings like pattern around a white central spot. Our recollection, from looking through telescopes, had been that the rings around the central Airy disc look white (colorless), but we weren't completely sure; and searching the internet were unable to find any clear color photographs of an ideal Airy disk. Photographing this pattern afocally is difficult because any dust on the eyepiece or camera lens will generate its own rings, disrupting the pattern. To test the accuracy of the prediction, we instead used a Meade Lunar Planetary Imager to photograph a pinhole (<0.5 mm diameter) in a sheet of aluminum foil illuminated by a slide projector and viewed through a distant (50 feet away) well corrected 75 mm diameter by 1000 mm focal length achromatic lens stopped to a small aperture (7.1 mm). Since the image is formed directly on the CCD surface, with only one optical element, most of the dust problems are eliminated. Making the aperture small increases the image scale of the rings to an easily resolvable size without (in theory) changing their color or pattern. A sampling of the results is shown below (click on the images for a larger and clearer view).
| Exposure Sequence | Focus Sequence |
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The left hand image, shows that the exposure has to be lengthened far beyond its normal value for the ring pattern, especially the outer rings, to become visible. At 0.032 sec, only the bright core of the Airy disk can be seen. Depending on the settings of your monitor, you will probably start to see the first ring in the 0.5 or 1 second exposures. The second ring should be seen in the 2 to 4 second exposures, while up to five rings may be seen in the final frame. Note also that until the first ring is fully visible, an increase in exposure gives an illusion of an increasing diameter to the central spot. The color patterns in the rings are very difficult to appreciate visually (looking at the disk with a strong eyepiece in real time), apparently because the eye is so dazzled by the much brighter central white spot. The camera (much more apparent in the enlarged view) clearly sees the Newton's rings type gradations from bluish at the inner edges of the rings, to reddish at their outer edges. The origin of this pattern is easily understandable: the red light (longer wavelength) is diffracted through a greater angle than the blue.
The right hand image, shows how the appearance of the pattern changes with focal position. The image actually gets quite dim as one moves far from focus. To compensate for this, the exposure time generally increases from left to right. However the image at best focus (0 mm) has been overexposed (0.50 sec vs. 0.25 sec at +/-25 mm) in an effort to illustrate the color and diameter of the in-focus first ring. This makes the central spot at 0 mm appear larger than at +/-25 mm, when in reality it is slightly smaller. In contrast to the color gradations of the very dim in-focus diffraction rings, which are hard to perceive visually, the color changes at +/-75 mm and beyond are easily seen, and look to the eye (observing the pattern through a high power eyepiece) more-or-less as they do to the camera (receiving the light directly). We were a bit surprised to find, as shown here, a striking asymmetry in the color patterns observed at equal distances inside and outside focus. Intuitively this occurs because with the lens stopped to so small a diameter, the focal movements are quite large: the geometry with the imager 100 mm inside focus is significantly different from the geometry with the imager 100 mm outside focus. The physical distance required to move through the sequence of colored diffraction patterns increases with the square of the f-number, so, ironically, the geometric asymmetries are larger in the optically more perfect small aperture system.
The real question is: does the Fourier transform method properly predict the observed amount of asymmetry? The answer seems to be yes. However, it must be noted that the source used for the test photos was not at infinity, the color correction of the Melles Griot 3 inch achromat, although good is not perfect, and finally that the light from the objective was collected directly on the CCD of the LPI imager, and not re-imaged by an eyepiece, as is assumed by the current software.
| Simulated Perfect 6.4 mm Aperture Telescope |
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Acknowledging these possible problems, we show here a simulation of the Airy disks expected from a "perfect" (in the sense described above) small aperture Galilean telescope of 1000 mm focal length viewing a white light point source at infinity, as imaged with a "perfect" -50 mm focal length eyepiece. The simulation squares are 100 arc-minutes on a side as seen through the eyepiece, corresponding to 5 arc-minutes in target space (a small aperture produces a large diffraction disc). Initially a 7.1 mm aperture was tried for the simulations, matching the measured diameter of the aperture on the test telescope. However, the predicted diffraction patterns went through their color cycles in a slightly too short length of drawtube extension (a little over 100 mm of simulation corresponding to the observed change in 125 mm). This may have to do with the test source not being at infinity. To compensate for this, a 6.4 mm diameter aperture was selected for the simulations, and the resulting predicted patterns (including their asymmetry) fairly closely match the observations. As before, each prediction is normalized to bring the brightest point to saturation. In reality, the diffraction pattern becomes very dim as one moves away from the best focus (although the human eye compensates for this to a remarkable degree). Unlike the previous diffraction disk predictions, which are shown at gamma = 0.5 to bring out the faint ring patterns, these are shown at gamma = 1.0, which seems to match more closely the images recorded by the Meade imager.
It is not clear if the slight residual mismatch in color tones in the simulations versus the photographs should be attributed to inadequacies in the color rendition scheme used in simulation algorithms or to a failure of the test telescopes to exactly replicate the conditions assumed in the predictions. In any event, the overall agreement between prediction and observation for both the ray-tracing and diffraction disk simulations seems rather good. We therefore feel fairly confident that the software gives reasonable predictions for the performance of Galilean telescope configurations that have not been checked experimentally.
Although the pinhole photographed above was typical, we observed that not all pinholes showed
exactly the same colors; some having a bluer overall color cast than the one shown.
This composite gives some impression of the range of variation that was observed.
The central disk in all of these is highly overexposed to bring out the colors of
the first ring.
We are not entirely sure of the origin of this color variation, but
images very similar to any of these can be obtained by manipulating the gain and
gamma of the simulation for a perfect in-focus Airy disk.
In other words, if we showed the simulated in-focus Airy disk for a perfect system
with the gain and gamma suitably manipulated it would be impossible to distinguish
the prediction from these actual photographs.
We have demonstrated both theoretically and experimentally that for the glass (BK7) and at the f/number (1000/23 = 43.5 = ratio of focal length to diameter) of the website telescope, the effect of the chromatic aberration of the singlet objective is to produce a sort of self filtering property with different colors coming into sharp focus as the distance from objective to eyepiece (the drawtube length) is changed. The dispersion (change of focal length with wavelength) of the glass leads primarily to a faint haze of unfocused light from the unfocused colors, which reduces the contrast of the image. The penalty in resolution is surprisingly slight. Indeed, in some circumstances it is possible that the resolution obtained with an uncorrected singlet objective might be better than that obtained with a modern achromat, since with a white light source, with the Galilean eyepiece relatively close to the objective, the average color contributing to the focused Galilean image will be bluer than the average color contributing to the "perfect" white light image produced by the achromat (and bluer light has intrinsically high resolution).
One might ask then, if even better self-filtering could be achieved with a still more dispersive objective. That is, since the in-focus light would be still more monochromatic, might one achieve still higher resolution (at the expense of contrast) by using a higher dispersion objective? The answer seems to be no.
We show below simulated images for the website telescope with its BK7 (optical crown) objective replaced by one made from the common Schott flint glass SF11 (with the front surface radius of curvature increased appropriately). An SF11 objective spreads the colors along the axis by more than twice the amount for a BK7 objective (see our table of refractive ratios). In all other respects these images are identical to the simulations shown for the normal (BK7) website telescope, to which they may be compared. The simulations are shown for a drawtube position of 945 mm.
| Bar Target | Point Sources | Gamma | |
|---|---|---|---|
| 3 arc sec | 5 arc sec | ||
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1.0 |
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0.5 | |
Despite the large increase in dispersion, the predicted changes in performance are slight. In the bar target image, the white background of the card is a more vivid green with the SF11 objective. It is also evident that the contrast is somewhat lower, but there is no obvious improvement in resolution. Both telescopes would exhibit slightly higher resolution with the drawtube a little closer to the objective, where the light forming the in-focus image is bluer.
Within the limitations of a reasonable webpage size, it has been possible to illustrate here only a few of the myriad aspects of the imaging properties of the Galilean telescope. Those wishing to explore other configurations (different glasses, apertures, focal lengths, Keplerian eyepiece, etc.) or the properties of the present configuration at other focal positions, image angles, etc. are welcome to request a free copy of the ray-tracing and/or Fourier transform simulation software to generate their own simulated images. We must point out, however, that this software, written in the Delphi 6 programming language, works only on Windows computers, handles only two-element singlet refractors, and may not be particularly user friendly or understandable.
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Images © Tom Pope and Jim Mosher
Last modified: July 14, 2006