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| Averaged Results for Twenty Interferograms of a 15" F/4.5 Newtonian Primary Mirror |
Background
In response to several questions about precision and accuracy in interferometric testing I have analyzed twenty interferograms of the same mirror and averaged the results. Before I present the results I should say something about how the interferometer and fringe analysis software work. I will also comment on how systematic and random errors affect the test results.
The interferometers we use at OMI acquire a digital image (interferogram) of the interference pattern that is produced by combining an accurate reference wavefront with the wavefront generated be the mirror being testes (test wavefront). The fringe analysis software “traces” the fringes in the interferogram by picking points along each interference fringe. Then the software creates a model of the test wavefront surface by fitting an arbitrary surface to the points generated in the fringe tracing process. A set of equations called Zernike Polynomials is used to create the model surface to represent the wavefront. The Zernike polynomials are chosen for this calculation because they provide a convenient way of calculating spherical aberration, coma, astigmatism… and so on. (There are 36 terms in the Zernike polynomials used by the software, each one representing a specific type of aberration.)
In order to obtain the best fit between the model wavefront surface and the traced points, the software uses a linear regression technique to minimize the least square error between the calculated model surface and the points created in the fringe tracing process. This is a statistical calculation so more points produce a better fit and hence more accurate results. The accuracy of the results are limited however, by the precision of the test hardware and systematic and random errors.
Errors in the Interferometric Test
The accuracy of measuring of any natural phenomenon will be limited by systematic and random errors. In our case, systematic errors include things like the accuracy of the reference wavefront, the ability of the fringe tracing software to accurately follow the fringes, and the quality of the fit between the calculated model wavefront and the input points. Systematic errors should be measured or estimated, and subtracted if they are significant.
Random errors that affect the test include vibration, air currents and thermal gradients in the air in the test chamber (the refractive index of air is a function its density, which varies with temperature, pressure and even humidity). Random errors should be reduced by averaging an appropriate number of data sets.
Systematic errors
Here are some details about systematic errors in the interferometric tests made for this article.
Interferometer Reference Element: We purchased the interferometer reference element from Tucson Optical Research Corporation (TORC). It was supplied with a Zygo interferogram showing that it is about 1/20th wave P-V across its F/2 aperture. Using the central F/4.5 portion of the F/2 beam results in an error contribution of about 1/100th wave P-V.
Precision Optical Flat: The flat mirror used in this test is 1/10th wave P-V across its 22” aperture. The error is pure curvature (power). If the mirror were perfectly flat (which is impossible) it would project the image of a point source placed at the focus of a parabola in auto-collimation back to infinity. Since it is not perfectly flat the image of the point source is projected back to a finite conjugate very close to infinity. If you could figure the parabola to a perfect null against this flat (also impossible) the parabola would be very slightly under-corrected. This amounts to a very small fraction of a wave of spherical aberration. For the sake of argument let’s say its 1/20th wave P-V.
Automatic and Manual Fringe Tracing: If the contrast in the interferogram is low, or if there are spurious interference patterns due to internal reflections or dust, the automatic fringe tracing software may get confused and misplace the data points on the fringe. It may even wander off the fringe entirely along some spurious feature in the image. If this happens, the software allows the operator to edit the placement of the points. The precision of the placement of the points therefore is limited to the operators ability to center the point on the interference fringe. To estimate the precision/error in manual fringe placement lets consider a specific case.
The interferogram to the right is one of the twenty used in this analysis. As you can see, the contrast is relatively low and there are spurious diffraction patterns in the image. [Achieving a clean, evenly illuminate beam is one of the biggest challenges in interferometry. We have been through several iterations of design, prototype and test to get to where we are today. We still have a ways to go to achieve interferograms that will consistently auto-fringe. The next iteration will include some interesting tricks using ?-waveplates, ?-waveplates, attenuators and polarizers to bring up the contrast and eliminate background noise. This is really exiting stuff (to me anyway) that I will write about later…] As a result, some manual fringe editing was required. In this particular case, the image was scaled so that its diameter on the computer screen was 7.5”. The average fringe spacing on the screen was ~½”. By clicking points onto the fringes and measuring with a ruler on the computer screen we estimate that we can center a point on the fringe within +/- 1/32” with normal effort. In this example, which is fairly typical, the point placement error is about 6.25% of the fringe spacing. Given that the test was made in double-pass we know that the fringe spacing in the interferogram is ?-wave. With this we can estimate that the point placement accuracy is +/- (0.0625*1/4-wave) or +/- 1/64th wave P-V.
Self-weight Deflection of the Mirror During Test: The largest systematic error contribution in the test comes from self-weight deflection of the mirror while it is hanging vertically in the test rig. Two-inch thick Pyrex mirrors in particular tend to show a certain amount of third order astigmatism, depending on the aperture of the mirror. (A 15” mirror typically shows 0.2 waves of astigmatism in this orientation, a 20” will typically exhibit 0.4 waves of astigmatism.) If the astigmatism in the mirror is due to self-weight deflection, the astigmatism angle indicated in the interferometer output will be in the vertical axis (90 or 270 degrees in this case). The fact that the astigmatism is due to self-weight deflection can be verified by rotating the mirror 90 degrees. If the astigmatism does not rotate with the mirror (stays in the vertical axis) then it is confirmed that it is due to self-weight deflection. So, under the right conditions it is justifiable to subtract the astigmatism term out of the interferometric results.
That said, there are some practical caveats to the above statement: If the astigmatism angle differs significantly from vertical you may have a problem that requires further investigation. If the astigmatism value is larger than normal you may again have a problem that requires further investigation. In many cases, the result of this further investigation leads to a complete re-polish with a full size lap to remove the astigmatism.
To complicate matters further, astigmatism can also be induced in the test by misalignment of the mirror to the optical flat. This necessitates some method of determining the quality of the optical alignment. Fortunately, coma is also an alignment induced error. If the coma term is low you know that the alignment is good and the astigmatism value is believable. Even though the coma term is subtracted from the results, we have found that it must be lower than ~0.2 wave. Otherwise, the additional astigmatism and higher order aberrations due to misalignment will skew the results.
To summarize, in waves P-V, we have the following values for systematic error contribution:
- Interferometer Reference Element: 0.01 wave
- Optical Flat: .05
- Fringe Tracing: 0.016 wave
- Self-weight Deflection: Subtracted when reasonable to do so.
These errors add in quadrature (root sum square: square the errors, add them, then take the square root of the sum). The result is 0.0534 or approximately 1/20th wave P-V. This is a typical limit of precision (or test resolution) for an interferometer. Other optical tests will have similar or worse limits of precision when their systematic error is taken into account.
Random Errors
As you will see from the test data, the largest errors in the interferometric tests are random errors. The main contributions come from air currents and vibration.
Air Currents: As observers you are well aware of the detrimental effects of air currents (atmospheric seeing) on image quality. We suffer the same effects, but to a lesser degree in the optics shop. Even under the seemingly steady conditions inside the test tunnel we see air current defects as large as ? wave P-V when testing a 15” mirror. All optical tests are subject to these air current errors.
Vibration: Vibrations contribute to the total error by decreasing fringe contrast. Vibrations cause the interference fringes to move rapidly (typically in the up/down direction because the vibrations are transmitted into the system by oscillations in the shop floor). The fringes are in motion during the exposure of the interferogram, causing them to be blurred.
Vibration can also affect the fringe spacing across the mirror. The mirror hangs in a sling of chain. The top of the mirror is unsupported. When the mirror and support system vibrate the top of the mirror may make larger excursions than the bottom. This causes the fringes at the top of the mirror to spread, which looks like astigmatism in the interferogram.
The affects of random errors must be reduced by averaging data sets. How many data sets do we have to average to obtain an accurate representation of the shape of the wavefront? Obviously the more the better but, as you will see from the test data, you reach a point of diminishing returns relatively quickly. Specifically, if you take steps to minimize vibration and air currents, an average of five interferograms produces acceptable results.
Averaging Interferometric Data Sets
When the fringes are analyzed the software produces two sets of numbers. The first is a set of coefficients for the Zernike polynomials. There are 36 of these coefficients and when they are plugged into the Zernike polynomials the result is a model of the shape of the wavefront. The second set of numbers is the output representing P-V, rms, Strehl ratio, spherical aberration, coma and astigmatism (3rd order Seidel aberrations). These values are calculated from the Zernike coefficients. [There are higher order terms such as trefoil and clover leaf. These are 5th order aberrations. In fact, each of the 36 Zernike coefficients represents a specific type of 3rd, 5th or 7th order aberration.]
When the software averages data sets, it is averaging the 36 Zernike coefficients, not the P-V, rms, Strehl and 3rd order output values.
It is important to note this well. The software is modeling the wavefront for each interferogram with all of its random errors. It is these modeled wavefronts that are being averaged to reduce the affect of random error. Averaging only the P-V, rms, Strehl and 3rd order aberrations does nothing to reduce random error because these numbers do not contain direct information about these errors.
Data Analysis
I am attaching a spreadsheet containing output data (P-V, rms, Strehl, sphere, coma and astigmatism) for twenty interferograms labeled A through T. There are three sets of rows in the spreadsheet: In the first set of rows the output includes all aberrations. In the second set of rows the coma coefficient is removed. In the third set of rows coma and astigmatism are removed. In each set of rows, the output is listed left to right for the first five data sets, followed by the output for the average of the first five sets of Zernike coefficients. This is followed by the next five data sets, then the output for the average of the first ten sets,… and so on until we have added all twenty interferograms to the average.
The following table summarizes the averaged data in sets of five, ten, fifteen and twenty interferograms.
| Averaged Sets of Five, Ten, Fifteen and Twenty Interferograms |
| Data Sets |
5 Data sets
Ave A-E |
10 Data sets
Ave A-J |
15 Data sets
Ave A-O |
20 Data sets
Ave A-T |
| RMS |
0.064 |
0.067 |
0.063 |
0.06 |
| Peak-Valley |
0.374 |
0.375 |
0.37 |
0.349 |
| Strehl Ratio |
0.852 |
0.838 |
0.854 |
0.869 |
| Spherical Aberration |
-0.138 |
-0.154 |
-0.166 |
-0.176 |
| Astigmatism, Angle |
0.272 -80.8 deg |
0.282 -82.7 deg |
0.266 -83.7 deg |
0.244 -82.7 deg |
| Coma, Angle |
0.177 -83.9 deg |
0.197 -84.3 deg |
0.17 -84.9 deg |
0.169 -85.6 deg |
Coma Removed
|
| Data Sets |
Ave A-E |
Ave A-J |
Ave A-O |
Ave A-T |
| RMS |
0.06 |
0.063 |
0.06 |
0.056 |
| Peak-Valley |
0.352 |
0.346 |
0.339 |
0.331 |
| Strehl Ratio |
0.866 |
0.854 |
0.867 |
0.882 |
| Spherical Aberration |
-0.138 |
-0.154 |
-0.166 |
-0.176 |
| Astigmatism, Angle |
0.272 -80.8 deg |
0.282 -82.7 deg |
0.266 -83.7 deg |
0.244 -82.7 deg |
| Coma, Angle |
Removed |
Removed |
Removed |
Removed |
Coma / Astigmatism Removed
|
| Data Sets |
Ave A-E |
Ave A-J |
Ave A-O |
Ave A-T |
| RMS |
0.024 |
0.025 |
0.026 |
0.026 |
| Peak-Valley |
0.153 |
0.153 |
0.156 |
0.161 |
| Strehl Ratio |
0.977 |
0.975 |
0.973 |
0.973 |
| Spherical Aberration |
-0.138 |
-0.154 |
-0.166 |
-0.176 |
| Astigmatism, Angle |
Removed |
Removed |
Removed |
Removed |
| Coma, Angle |
Removed |
Removed |
Removed |
Removed |
This most important thing to note from the above table is that the difference between the average of 5 , 10, 15 and 20 sets is relatively small. Under the test conditions in our shop, averaging five data sets gets you to the point of diminishing returns.
The following table shows averages of the interferograms in separate sets of five.
| Averaged Sets of Five Interferograms |
| Data Sets |
5 Data sets
Ave A-E |
5 Data sets
Ave F-J |
5 Data sets
Ave K-O |
5 Data sets
Ave P-T |
| RMS |
0.064 |
0.071 |
0.057 |
0.05 |
| Peak-Valley |
0.374 |
0.38 |
0.362 |
0.335 |
| Strehl Ratio |
0.853 |
0.818 |
0.881 |
0.905 |
| Spherical Aberration |
-0.138 |
-0.17 |
-0.191 |
-0.204 |
| Astigmatism, Angle |
0.272 |
0.293 -84.4 deg |
0.234 -86.2 deg |
0.181 -78.2 deg |
| Coma, Angle |
0.177 |
0.216 -84.6 deg |
0.116 -87.2 deg |
0.166 -87.6 deg |
Coma Removed
|
| Data Sets |
Ave A-E |
Ave F-J |
Ave K-O |
Ave P-T |
| RMS |
0.06 |
0.067 |
0.055 |
0.047 |
| Peak-Valley |
0.352 |
0.351 |
0.343 |
0.324 |
| Strehl Ratio |
0.866 |
0.838 |
0.887 |
0.917 |
| Spherical Aberration |
-0.138 |
-0.17 |
-0.191 |
-0.204 |
| Astigmatism, Angle |
0.272 -80.8 deg |
0.293 -84.4 deg |
0.234 -86.2 deg |
0.181 -78.2 deg |
| Coma, Angle |
Removed |
Removed |
Removed |
Removed |
Coma / Astigmatism Removed
|
| Data Sets |
Ave A-E |
Ave F-J |
Ave K-O |
Ave P-T |
| RMS |
0.024 |
0.028 |
0.028 |
0.028 |
| Peak-Valley |
0.152 |
0.16 |
0.18 |
0.196 |
| Strehl Ratio |
0.977 |
0.969 |
0.969 |
0.969 |
| Spherical Aberration |
-0.138 |
-0.17 |
-0.191 |
-0.204 |
| Astigmatism, Angle |
Removed |
Removed |
Removed |
Removed |
| Coma, Angle |
Removed |
Removed |
Removed |
Removed |
You can see that there is some fluctuation in the results when comparing one average of five to the next.
It is very interesting to note how much fluctuation there is from one individual data set to the next. The following table shows the Mean, Standard Deviation and Maximum/Minimum of the output values for the set of twenty interferograms. [I should make it clear that the values in this table show the Mean, etc., of the P-V, rms, Strehl and 3rd order output values, not the average of the 36 Zernike coefficients as in the averages shown above.]
| Statistical Fluctuations in Individual Interferometric Tests |
|
Mean |
Std Dev |
Max |
Min |
| RMS |
0.06565 |
0.01647734 |
0.104 |
0.047 |
| Peak-Valley |
0.39745 |
0.09162939 |
0.678 |
0.273 |
| Strehl Ratio |
0.83815 |
0.07457619 |
0.915 |
0.65 |
| Spherical Aberration |
-0.17565 |
0.11183648 |
0.052 |
-0.359 |
| Astigmatism, Angle |
0.2517 |
0.09161481 |
0.462 |
0.112 |
| Coma, Angle |
0.1783 |
0.07388547 |
0.371 |
0.062 |
| Data Points |
151.4 |
42.3971201 |
225 |
82 |
Coma Removed
|
|
Mean |
Std Dev |
Max |
Min |
| RMS |
0.0619 |
0.01671085 |
0.101 |
0.042 |
| Peak-Valley |
0.3785 |
0.08659251 |
0.64 |
0.265 |
| Strehl Ratio |
0.854 |
0.07372495 |
0.934 |
0.667 |
| Spherical Aberration |
-0.17565 |
0.11183648 |
0.052 |
-0.359 |
| Astigmatism, Angle |
0.2517 |
0.09161481 |
0.462 |
0.112 |
Coma / Astigmatism Removed
|
|
Mean |
Std Dev |
Max |
Min |
| RMS |
0.0329 |
0.00676601 |
0.046 |
0.024 |
| Peak-Valley |
0.21685 |
0.06860511 |
0.437 |
0.127 |
| Strehl Ratio |
0.95675 |
0.01760943 |
0.978 |
0.919 |
| Spherical Aberration |
-0.17565 |
0.11183648 |
0.052 |
-0.359 |
As you can see from the attached spreadsheet and the above table the P-V, rms, Strehl and third order aberration output for the individual interferograms fluctuates wildly. However, the average of five or more data sets is quite stable. This demonstrates the importance of the averaging process in obtaining results that represent the actual wavefront to an acceptable degree of accuracy.
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