Hartmann Mask Analysis
Scott Hesser and Kate Perry
Experimental Physics
May 2000
Abstract:
This lab describes how to use simple masks to collimate reflecting telescopes and determine the quality of the mirror. An image is taken on either side of good focus and the masks allow the software to determine the wavefront error of the system. Two masks are needed, one that is strictly for collimating, and another that is used to analyze the aberrations of the mirror.
Introduction:
In the past amateur astronomers had to test their optics using the traditional “star test.” This was accomplished by using a high magnification eyepiece and examining the diffraction fringes of a star. Aberrations and collimation problems could easily be recognized. However, this procedure was time consuming, and gave no numerical data or printouts that could be shared. Problems were very much a matter of personal judgement. The CCD revolution has changed the way amateur astronomers operate. A Charged Coupling Device camera and the proper software allow amateurs to conduct the Hartmann Mask test. This test was developed around 1900, but it was only used on extremely large aperture telescopes because the test was expensive to conduct. The mathematics involved used to take a several days with a whole team working on them. Today the mathematics of this test can be handled quickly by a computer. The CCD camera allows a quick transfer of digital information to the software. Amateurs can now make accurate and quantitative measurements of their optical systems.
Theory:
A Hartmann mask is a cover with small holes arranged in a pattern and is placed over the aperture of the telescope. Each small hole is its own aperture and produces its own image. If the aperture of the telescope is covered with a Hartmann mask, its out of focus image will resemble the pattern of the mask. By taking images of a star on both sides of focus, the Hartmann Mask Analysis software can make ray traces from each individual spot and give a numerical measurement of the error in the optics. The following discussion is divided into two parts. The first relates to the possible errors that can be found in an optical system. The second part describes the Hartmann Mask Analysis and what the software is able to do.
Optical Errors:
Nearly all large aperture telescopes made to day are reflecting telescopes. Mirrors are easier to make and they are lighter so they are easier to mount. However, a drawback to the reflecting telescope is the need for a secondary mirror. The mirror focuses an image in front of itself. In order to view the image a person would have to stand in front of the mirror, blocking out the incoming light and defeating the purpose of the telescope. A solution to this is to mount a smaller secondary mirror to reflect the image out of the incoming light path. This lab examines a Newtonian design telescope. The main aperture mirror is parabolic in shape and the secondary is a flat diagonal mirror. See Figure 1.
The central obstruction degrades the image only marginally if it is not too large. The larger problem of having the secondary is aligning it with the primary mirror, a process known as collimation. Alignment consists of two parts: first, the centers of the two mirrors should be on an axis that is parallel with incoming light rays; secondly, the mirrors should be tilted so that their planes are perpendicular to this axis. This means that rotating either mirror would not alter the image. The first part is known as centering, and the second is squaring-on.
To collimate a reflecting telescope it is easiest to observe a bright star out of focus. This should produce a circular image with a dark circle inside of it, something that roughly looks like a donut. An uncollimated telescope will have a skewed donut. See Figure 2.
To align the telescope, one tries to make the out of focus image perfectly symmetrical. However, doing this by looking at the donut is accurate to a limited degree.
Refracting telescopes are prone to chromatic aberration. Because the index of refraction through the glass lens is different for every wavelength, each color produces a slightly different image. Reflecting telescopes do not have this problem. However, they are prone to geometrical aberrations caused by their shape. Geometrical aberrations are generally a problem of rays from different parts of the mirror being focused at slightly different points. The ideal telescope would focus all rays of light to one point, but this is impossible.
Spherically curved mirrors suffer from spherical aberration. Rays hitting the inside of the mirror are focused to a different point than rays hitting the outside of the mirror. This can be corrected with a specially designed lens placed over the aperture. Spherical telescopes are generally expensive because of the correcting lens. Astigmatism occurs when the curvature along the horizontal axis of the mirror is different from the curvature along the vertical. This was the problem the Hubble Space Telescope experienced.* It can be corrected by a lens with varying curvatures.
Parabolic mirrors suffer from coma. Stars at the edge of the field of view are skewed to look as if they had tails like comets, hence coma. See Figure 3. Coma is worse for fast scopes, or scopes with small f-ratios. The f-ratio is the focal length divided by the diameter of the mirror. A fast scope is considered anything below f/5. Coma can be lessened to some degree with a field flattener lens.
Optical quality is defined by the deviations the mirror forces on to the converging wavefront. A wavefront is the forward boundary of a wave. It is essentially a line traced along a waves crests or troughs. If light is examined as a collection of photons, at a given instant these photons can be represented by points, and an equal radius can be drawn around each to represent possible directions. The wavefront is the forward boundary of these radii. A flat ideal wavefront is perpendicular to the direction of the light. See Figure 4.
After the flat wavefront has reflected off the mirror it develops a spherical shape as it converges to a central point. Any aberrations or stigmatism on the lens surface will cause the wavefront to deviate from its ideal shape. The measure of this deviation is often regarded as the wavefront error. There are several different ways of expressing the wavefront error. Two of the most common, and the two used in the Hartmann Mask Analysis software, are the peak to valley error and the RMS error (root mean square error).
The wavefront is rarely flat, and as it is concentrated to the focal point it is further disturbed by the interference patterns that are set up. Lord Rayleigh once stated that difference between peak to valley was greater than º of the wavelength one could notice a drop in image quality. Figure 5 shows the wavefront as it approaches the focal point. The sinusoidal pattern is the result of interference from light’s wave like properties. This interference pattern puts an unavoidable limit on the size of the point which light can be focused to. This limit, or the diffraction limit as it is often referred to, was considered by Rayleigh to be ºl. Notice in Figure 5 the good wavefront fits within this boundary. The bad wavefront has a peak that extends beyond the ºl limit. This causes part of the wavefront to focus beyond the diffraction limit focal point. Similarly, the RMS error is an average of the overall deviation from the ideal wavefront. The RMS is used so negative and positive values will not cancel each other out, as they would if a mean function were used. The accepted limit for the RMS error is 1/14l.
As stated above a good set of diffraction limited optics does not focus light to a perfect point. The interference forms a central bright disk with dimmer fringes surrounding it. See Figure 6. The fringes quickly die off in intensity, so what is really noticed is the central disk, or airey disk or blur spot, there are numerous names for it. In this lab the image will be referred to as a blur spot. The size of the blur spot is related to the f/# of the mirror by the equation:
spot size = 2.44 ´ l ´ f/#.
The faster the telescope the smaller the spot size. The theoretical blur spot represents the perfect focus. In reality it is difficult to achieve a minimum blur spot this small. In the discussion about coma it was stated that faster telescopes tended to experience more coma. This is largely because the spot size becomes too small to hide the aberration.
Hartmann Mask Analysis:
*It will be helpful to refer to the software interface printouts at the back of the lab through out this discussion for an easier interpretation of the functions.
The Hartmann Mask Analysis works by taking two CCD pictures on either side of focus. This means one image is taken before the converging wavefront has reached focus, and another after the wavefront has reached focus and begins to diverge. In this lab the two sides of focus are designated down focus and up focus. When the image is out of focus the pattern of the mask is easily recognizable. The f/# is specified, and these images are loaded in to the software which uses a sophisticated process, which is beyond the scope of this lab, to trace rays from the spots on one side of focus to the corresponding spots on the other side. See Figure 7.
Theoretically these rays should pass through the point of focus, but as said before no optical system is perfect, so the rays do not all intersect at one point. The software then calculates the point of best focus; this is where the rays merge closest to one another and the blur spot would be the smallest. In the “Ray Intercepts” box the program shows a cross section of the focal plane, the plane perpendicular to the optical axis. The rays are given different colors in order to tell which spot they came from. Since the rays are pretty close together at best focus, there is an option to change the size of the box in order to zoom in for a better look. The program gives the RMS distance between these intercepts as well as the width along the x and y axis.
Each of these intercepts represents the center of a blur spot. The overall width can be compared to the theoretical blur spot size. Obviously the true spot size will always be larger than the theoretical one, but good optics will have only slightly larger ones. An error under a factor of two would be sufficient. In Figure 8 the combined spot size is greater than the theoretical by more than a factor of two, this points to poor alignment.
Collimation is the most easily controlled part of a good optical system. As stated before, collimating from investigation of the donut is accurate to a person’s ability to determine symmetry in the shape. More accurate measurements could be made if symmetry were more easily noticeable and if some quantitative values could be looked at. A Hartmann mask with holes at 90° to each other, along the x and y axis, can help accomplish that. See Figure 9.
For a properly aligned telescope, the image will be symmetrical and each spot will have a similar brightness and shape. Minimizing the ray intercept width is good quantitative indication of proper alignment. This mask makes determining the width easier because the program measures along the x and y axis and there is less to clutter the box.
In order to utilize the programs other measuring capabilities a mask with more holes than the collimation mask will be needed. To get a proper model of the optics the analyzing mask, as it will be called, should have as many holes as possible with out cluttering the mask. Also, the program cannot recognize over 32 holes, but only larger apertures can support a mask with that many. The holes should be roughly 1/15 the size of the aperture size and spaced far enough apart that their images will not blend into each other. Holes should be made on the inner radius of the mask as well as the outer. The software also does most of its measuring at angular intervals of 22.5°. It regards the 0° line as the horizon and takes measurements at 0°, 45°, 90°, and 135°. It is helpful if the holes on the mask are positioned at these angular intervals. Figure 10 is a sample analyzing mask. Remember that because of the central obstruction there is no point in putting holes at the center.
An analyzing mask and the “Ray Intercept” box can be used to detect aberrations. The colors that indicate which spot the ray came from are important for this. If the outer spots’ rays focus at different points from the inner spots, it is evidence of spherical aberration. When the outer spots’ rays have a group center to the side of the inner spots’ center, there is coma. If the vertical spots’ rays focus at a different point than the horizontals, there is astigmatism. Also, if the rays seem lopsided to one size it may be an indication of poor alignment in a given direction. The program has an option to change the focus position that the “Ray Intercept” box displays. The percentage of distance from one image to the other is given in terms of percentage. As the distance is changed the rays will spread out. Pulling away from best focus slightly may help recognize aberrations more easily.
The program can also measure the RMS wavefront error as well as the peak to valley error. These errors are at a minimum when they are calculated at the best focus point. As you pull away from best focus they increase dramatically. The software measures the wavefront as a line across the aperture. The angle of this line, or slice, as it is referred to, can be changed to 0°, 45°, 90°, and 135°, as mentioned before. Looking at these different angles can help to localize where a defect might be.
The software also gives a visual picture of the wavefront. See an interface printout at the back of the lab. The picture is centered on the optical axis and extends two wavelengths to either side. At best focus the ideal wavefront is flat. (Although it was stated before that the wavefront is converging and should be spherical, a cross section this small will not show that curvature.) As the focus position is pulled away from best focus the wavefront bends upward or downward, depending on which direction the focus is pulled. The bending wave front should be symmetrical. If it is not this may also be an indication of poor alignment or an aberration.
Before concluding this section, we should mention a word about gathering the images. Obviously a point source at infinity or near infinity should be used. This provides an easily identifiable image and the telescope is designed to focus paraxial rays of light. An artificial point source taken indoors is ideal because problems with tracking and air currents can be avoided. If the image is taken using a star, the exposure should be fairly lengthy, at least 10 seconds. The atmosphere distorts the incoming wavefront from its original flat shape. A long exposure averages out this distortion. However, if one does want to measure the distortion cause by the atmosphere, a short exposure, around 1/10 of a second, could theoretically measure it.
Procedure:
Because the weather has been against us the entire semester, we used an LED with a size distance ratio of 0.00003 (that is roughly 3arcsec. which is a good approximation of a star) and turned off all the lights in the hallway to simulate a star. As you can see from our results on the following pages, this is a very accurate representation. The first mask was used to collimate the telescope. The holes made should be just slightly larger than 1/15th of the aperture size. Smaller holes will diffract the light and larger holes will make the images run together. The Hartmann software indicated that 1/15th the aperture would work well but that was just a little small. For the six inch telescope we used, we found that ½” holes worked very well.
Each hole creates an image of our ‘star’ and if the image is skewed to one side, we then know which way to turn the main mirror in the telescope to center the images and hence collimate the telescope. The second mask is used to determine what aberrations your telescope has as discussed above.
The holes in this mask need not be in any particular pattern although it may be easier to see the distinction between Up focus and Down focus if a pattern is used. An image is taken on both sides of actual focus (see Figure __). We termed ours Up and Down focus because of the way the camera was situated. Those images are then loaded into the Hartmann software and the data is analyzed.
Results and Discussion:
The ray intercept box on the Hartmann printouts shows the Minimum Spot Size location or the point of best focus. As can be seen, the points do not all converge showing that this mirror does not have perfect focus. As can be seen on Printout 2, the wavefront error, ideally zero, is rather large at 45°. This is an indication of poor collimation on the 45° axis. Collimation is also out slightly on the 90° axis as seen on Printout 3. The Spot Size is also larger than predicted by about 2.5 times. The theoretical or blur spot size is, 2.44(l)(f/#) = 2.44(.633mm)(5) @ 8mm. The actual spot size is indicated on the Printouts below the ray intercept box. This is another indication of a collimation problem.
When the outer spots focus in a group off to the side of the inner spots, it is evidence of coma. The visual indication of coma is tales on the stars nearest the edges of the field of view. Printout 5 shows the inner ring focusing in a long, tight group to the lower left of the ray intercept box where the outer ring sits off to the upper right. This is a very good indication of coma. The wavefront error on this printout also shows just how skewed to the left the 45° axis is.
Conclusion:
Even though the collimation is out on the 45° axis, the overall optical quality of the telescope is Diffraction Limited, meaning good. The wavefront error was within the parameters for a mirror of that size and quality.
References:
Holmes, Alan. SBIG Memorandum Using the SBIG Hartmann Mask Software.
Published by Santa Barbara Instrument Group. 3/1/99.
http://orion.wheatonma.edu/sky_survey/optics/focallength.html. Optics Primer. Maintained
Spencer Zawasky. 5/5/00.
http://sbig.com.sbwhtmls/hartmann.html. New Product Announcement. Maintained by
Santa Barbara Instrument Group. 4/21/00.
Schroeder, Daniel J. Astronomical Optics. New York: Academic Press Inc.
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Suiter, Harold R. Star Testing Astronomical Telescopes. Richmond: William-
Bell, Inc. 1994.