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Interferometrically Testing Two Celestron C14 Edge Telescopes


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Interferometrically Testing Two Celestron C14 Edge Telescopes

 

John Hayes, Ph.D., Adjunct Research Professor

College of Optical Sciences

University of Arizona

 

Introduction

I’ve wanted to interferometrically test the two C14s that I have in my shop to get some good data on their optical quality for some time but gathering all of the equipment needed for such a test isn’t easy and I was hesitant to take my scope out of action when the skies were clear.  Since it would probably be cloudy for months on end in winter, December seemed like a good time to try to get it done.  Fortunately I have a lot of friends who were willing to help with this project and I was able borrow all the equipment needed to do the test. 

 

Collecting the Equipment

For the interferometer, I arranged to use a 4D Technology PhaseCam 6000 dynamic interferometer with a 1Mpx phase sensor.  This interferometer is very similar to those used to test virtually all of the 8.4 m mirrors made at the Steward Observatory Mirror Lab, the Hi-Rise telescope now orbiting Mars and most of the major components for the James Webb Space Telescope (the primary segments and the carbon fiber backplane structure.)  The PhaseCam is a state of the art interferometer system for long path length optical testing.

 

 

Figure 1. The PhaseCam is a polarizing Twyman Green interferometer.  This allows variable signal to reference beam intensity to give a high contrast signal independent of the part reflectivity.  It also allows the use of a pixelated polarization sensor that generates the four 90-degree phase-shifted signals needed to compute optical phase using a single frame. (Click image to launch larger version in a new tab)

 

The PhaseCam is a polarizing Twyman-Green interferometer so it is very light efficient—no matter how much light is returned by the optics under test.  Figure 1 shows the layout.   Its key advantage is that it can measure optical phase using just a single frame so the exposure time can be made as short as the available light will allow.  These characteristics make the PhaseCam very insensitive to vibration.  The fringe patterns may dance all around when things vibrate; but each frame contains all of the information to compute the optical phase—so it mostly doesn’t matter.  Since data can be taken very quickly, it is possible to average out time-varying noise sources such as air turbulence.  It also does away with the need for complex and expensive vibration isolation tables and fixtures.  The references listed at the end of this article describe the phase sensing technology used in the PhaseCam in greater detail.

 

Since the PhaseCam solves the problem of mechanical vibration, the next challenge is the requirement for a full aperture, precision flat mirror needed to do a double pass test.  Locating a good quality flat that’s a bit larger than 14” isn’t always an easy task.  (Yes, this test could probably be done with a liquid flat but that presents another set of challenges that I won’t go into here.)  Fortunately, the folks in the large optics shop at the College of Optical Sciences at the University of Arizona were willing to loan me a 24” coated precision flat.  They weren’t sure exactly how accurate it is, but they guessed that it should be “pretty good” with less than 1/10 wave PV of irregularity (i.e. +/- 1/20 wv  ~ 0.010 wv rms.)   I was so happy to find a good flat that I didn’t think much about the size until it arrived on the back of a truck in a crate that must have weighed around 250 lbs.  (Oh, oh!) The flat itself was a 4” thick and it looked like it was made of Zerodur.  If that’s true, the weight computes to about 165 lbs—and that’s about what it felt like.  Without a crane, it took three of us to move it.  I had access to a 22” Unertl tip-tilt mount that would hold the flat without too much trouble.  The mount is made of steel and cast iron and it weighs about 200 lbs.  We didn’t have a crane and the flat was so heavy that even with three of us, we couldn’t safely lift it up onto the mount on the optical table!  I quietly estimated the value of that piece of glass to be around $150,000 and I sure didn’t want to even slightly risk damaging it!  So we moved the mount off the table and set up everything on the floor.  Figure 2 shows a drawing of the setup and figure 3 shows a photo. 

 

Figure 2.  A schematic diagram of the double pass test.  Outgoing light from the diverger is carefully positioned at the on-axis field point in the focal plane of the telescope. The back working distance (BWD) was set to the 146.05 mm (5.75”) specification for the Edge system.  The F/9 test beam was selected to slightly overfill the F/10.8 pupil.  The telescope expands the beam and it is reflected back through the system and into the interferometer by the return flat.

 

At one point, I looked at all the equipment I had assembled on the floor to do the test and it likely adds up to over $300k worth of stuff!  This probably isn’t a test that most amateurs could pull off very easily.  If you put aside RC Optical Systems (now out of the amateur telescope business,) I don’t think that anyone in the “small telescope business” has anything even close to this capability.  (Yes, RCOS had a PhaseCam!)

 

 

 

Figure 3.  The actual test setup showing the PhaseCam on the left and the return flat on the right.  The box under the telescope simply acts as a spacer to get everything to about the right height.  One advantage of the large flat is that the telescope axis height does not have to be set very carefully.

 

Alignment and Testing

I set up the telescope to measure the on-axis field point.  First, I set the tilt of the flat mirror to null out the fringe pattern from the collimated output beam from the interferometer.  Then I inserted the telescope into the test cavity.  I put in a spacer (a 1 ¼” PVC plumbing fixture) on the back of the scope with an edge at the required back working distance of 146.05 mm (5.75”.)  With a piece of clear tape over the rear edge of the spacer tube, I could easily position the scope so that the focused spot was within less than 1 mm of the field center.  By tipping and tilting the scope, I could get the return beam to reflect back along the input path.  Sliding the telescope back and forth along the test axis brought the beam into focus.  The focal point was positioned within about +/- 2 mm from the correct back working distance spec.  The design tolerance for the on-axis focal position is quite low so the effect of such minor lateral positioning errors will be insignificant.  Once it was aligned, I removed the tape for final (minor) adjustments of the return mirror tilt to null the fringes.

 

I used the minimum exposure of 34 milliseconds allowed by the camera and it worked quite well in spite of a little vibration due to traffic on a nearby road.   In order to get the best results, I let both scopes thermally stabilize for a day (at about 20 C) before taking data.  I also used signal averaging to reduce errors due to air turbulence in the test path.  Averaging 64 phase frames produced frame to frame repeatability of about 5 milli-waves rms (<~1/40 wave PV.)  The interferometer/diverger combination typically has an absolute accuracy of better than +/- 0.05 waves PV over the full beam and I didn’t go to any special effort to calibrate out any residual errors in the system.

 

 

Figure 4. Centering the test beam at the correct back working distance.  The spot could be centered to within about 1 mm and set to the proper 5.75” BWD to within +/- 2 mm.  The clear tape allowed the beam to be precisely focused on the tape and made it easy to see the return beam from the flat as the telescope was aligned.  Look carefully and you can see the two spots.  The tape was removed before taking data.

 

 

 

Figure 5.  The difference between two time averaged measurements with the return flat at 0-degrees and rotated by 90-degrees.  The test 14” test beam is roughly 2-3” off-center on the 24” flat surface.  This data gives an indication of the quality of the return flat over the 14” aperture used for testing.  The PV of the difference is about 1/20 wave and the rms is a bit less than 0.01 waves. This appears to be an excellent flat over the 14” diameter test pupil. (Click image to launch larger version in a new tab)

 

Estimated Accuracy

Without performing a calibration against an absolute standard and lacking firm specs on the flat, it’s hard to give a firm statement of the absolute accuracy for this test set up.  In order to estimate the quality of the flat, I subtracted two wavefront measurements using an off-center region on the flat with a 90-degree rotation and found a little less than 1/20 wave PV and 0.01 waves rms difference between the two measurements—as shown in Figure 5.  So the flat is certainly quite good over the 14” pupil.  The PhaseCam/diverger combination is generally supplied with an accuracy of ~0.013 waves rms.  Since a certified calibration sphere was not available to calibrate the diverger optics, I measured the same system twice using two different diverger lenses as shown in Figure 6 to determine that the diverger lenses should be good to about 0.011 waves rms over the pupil.  Using these numbers, the estimated accuracy of these measurement is somewhere in the range of 0.015 - 0.020 waves rms, which is certainly good enough for this type of application. 

 

 

Figure 6.  Data taken of the first telescope with two different diverger lenses (and two different setups) to show the effect of the diverger optics on the measurement.  Wavefront A is the first data set taken with a long efl diverger.  Wavefront B is the same telescope measured with a shorter efl diverger after the telescope was repositioned in the test beam.  C shows the difference between the two measurements after resizing and aligning the data.  Edge mismatch phase errors have been masked (shown in gray.)  In this case, the field point was adjusted a little, which explains the tiny bit of coma (0.023 waves) that is evident in the C data.  In this case, coma has been introduced only because of an alignment difference so it can be subtracted as shown in D to get an estimate of the rms contribution due only to the difference in diverger optics.   This measurement shows good agreement between the two configurations with an upper bound of 0.011 waves rms agreement (0.035 waves PVq(99%). (Click image to launch larger version in a new tab)

 

Results for the First Telescope

The first C14 Edge HD system is the telescope that I most often use for imaging, which was produced early in 2015.  The optics are stock but the system has undergone numerous modifications to address alignment and mechanical issues.  The test data shown in figure 7 was taken after the secondary was carefully aligned.  With tilt and power removed (the alignment induced errors,) the single pass wavefront accuracy is 0.065 wv rms and 0.322 wv PVq(99%).  PVq(99%) is a PV estimator that gives the range of values containing 99% of the data.  The Strehl ratio measures at 0.845, which exceeds the minimum Maréchal criterion of 0.8 for a diffraction-limited system at the test wavelength of 632.8 nm.

 

 

 

Figure 7.  The single pass wavefront for the first C14 Edge showing Strehl performance at 0.845.   This data shows that the system exceeds the requirements for diffraction-limited performance.  Astigmatic error is largest contributor at 0.125 wv (not shown on the plot.)  The fringes (in the lower plot) also show that the secondary may be very slightly out of alignment; though the residual coma term is only 0.048 waves (Click image to launch larger version in a new tab)

 

In case anyone is bothered by how the results relate to the Rayleigh ¼ wave PV criteria, it is important to understand a couple of things.  First, the Rayleigh criterion was originally derived from 3rd order spherical aberration and it has been extrapolated as a rough approximation for balanced 3rd order terms.  Second, when we look at real-world irregularity, it is entirely possible for the OPD (optical path difference) between two regions to exceed the Rayleigh criteria and still have a diffraction-limited system.  As long as the area is small, slightly high or low regions will simply redistribute the light a bit without seriously impacting the overall imaging performance.  In this case, only about 3.5% of the data lies outside of the 0.25 wave PV limit, which is minor in the face of normal atmospheric turbulence.  While interferometric data is quite precise, it can sometimes be susceptible to coherent noise, which may corrupt the PV value.  That’s why estimators such as PVq(99%) are used to reduce the likelihood of  noise variations producing a wildly incorrect PV value.  There are a number of other PV estimators that have been developed besides PVq, (such as robust PV which is called PVr) but that’s beyond the scope of this article.

 

For a “pretty good” system the Strehl value can be approximated from the variance of the wavefront (s2,), which is computed over all the data points (not just two) and that makes it a much better way to assess imaging performance than the Rayleigh criteria.  For Strehl values as low as about 0.3 (Mahajan, 1982), the geometric relationship between rms (s) wavefront irregularity and the Strehl ratio for an unobstructed circular pupil can be approximated by:

                                                   [1]

This relationship is what sets an upper limit of about 1/14 wave rms on wavefront irregularity for any diffraction-limited system.

 

 

Figure 8.  Fizeau testing a C14 secondary mirror on a WYKO 6000 interferometer requires a large transmission sphere, a beam attenuator, and a precision positioning mount.

 

In the Edge design, secondary misalignment (in tilt) introduces field independent coma, but coma can come from both secondary misalignment and surface irregularity.  I stopped adjusting the secondary when the 3rd order coma value fell below 0.05 wv (0.048 wv.)  After taking a second look at this data, it’s clear from the fringe pattern that adjusting the secondary a bit more may have reduced the coma a little further.  If that’s the case, the Strehl in this system might have been slightly improved to 0.855, which isn’t much.

 

I found that the wavefront is quite sensitive to secondary tilt, which needs to be aligned to within about 1/50 of a turn of the alignment screws to minimize the on-axis rms wavefront errors.  That represents a tilt of about +/- 15 arc-minutes.  This is a very tight tolerance that is difficult to achieve without a lot of care and a good alignment signal.  Achieving this level of accuracy with a star test requires very steady seeing, a very sharp eye, and a bit of practice.

 

One of the main reasons that I wanted to test this telescope is that I’ve disassembled it many times and I’ve been slightly concerned that the main mirror may have been stressed a little the last time it was reassembled.  Indeed, the dominant wavefront aberration is astigmatism at 0.125 wave, which is an amount that could easily be due to mechanical deformation.  Keep in mind that it would only take 1/16 wave of astigmatic error on the mirror surface to produce this result and that’s not very much!  Errors of this magnitude could also be introduced during the fabrication of the primary mirror, the secondary mirror, or the corrector plate.  I’ve previously tested the surface figure of the secondary surface on a WYKO 6000 phase shifting interferometer.  Figure 8 shows the test set up and figure 9 shows the results.  This secondary has a very high quality spherical surface so it can be ruled out as a source for any of the astigmatic error.

 

 

 

Figure 9.  Test data measured by a WYKO 6000 interferometer showing the surface accuracy for the secondary mirror from the first telescope.  This is an excellent spherical surface with only very minor zonal circular zones. (Click image to launch larger version in a new tab)

 

Since both telescopes ultimately displayed similar astigmatic error oriented in the same direction, rotating the flat by 90 degrees made certain that the error was not in the flat.  There was less than 1/20 wave PV and 0.01 wave rms difference after rotation (as shown in figure 5).  Without removing the primary mirror for measurement, it’s hard to say if the error is in the primary or in the corrector plate.   Some folks have been successful at reducing the total astigmatic error in the wavefront by rotating the corrector relative to the primary to minimize the aberration but I didn’t try to make that adjustment.

 

Remember that the imaging performance of any system with a Strehl ratio of 0.8 is generally limited more by the diffraction of light than by optical aberration.  If the all of the astigmatism could be removed, the Strehl could theoretically climb to 0.946 or 0.957 (with the secondary perfectly aligned.)   As it is, the wavefront is fairly smooth and well corrected so the imaging performance of this system should be quite good.    This conclusion agrees with the excellent imaging performance that I have seen with this system both before and after this test.  Figure 10 shows the PSF performance of the system.

 

 

Figure 10.   The diffraction PSF performance for the first system computed from the measured wavefront data. A) The Airy disk displayed as I=Log(1+PSF) along with image scale circles. Keep in mind that even with good seeing, the long exposure FWHM blur size is only rarely smaller than about 1.5 arc-seconds.  B) The PSF displayed as I=log(1+k*PSF), where k =1000 to amplify the low intensity outer ring structure.  C) The same data as middle figure with the aberration errors scaled to zero to demonstrate a “perfect” pupil similarly sampled.  The bright second ring and low intensity of the third diffraction ring is due to the size of the central obscuration.  The slight irregularity in the rings is simply due to a sampling artifact.  80% of the total energy is contained within a 0.75 arc-sec aperture and 90% of the total energy fits within a 1.45 arc-sec aperture.

 

Results for the Second Telescope

The second system was likely produced in the first half of 2014 and acquired in August of that year.  The test results shown in figure 11 were taken after carefully aligning the secondary tilt.  This time, I simply looked at the 3rd order coma term and dialed it down to a minimum in each axis.  The values were very repeatable and it was pretty easy to minimize the value to within 0.010 waves with very tiny adjustments.  It was clear that this strategy quickly produced the straightest fringes in each direction to minimize the total wavefront error.  Once aligned, this telescope did not perform quite as well as the other scope with a PVq(99%) of 0.406 wv, an rms of 0.087 wv, and a Strehl of 0.741 so it falls short of the Maréchal criteria for a diffraction-limited system.  At 0.151 wv, astigmatic error is the largest contributor with spherical aberration a close second at 0.117 wv.  A previous test of the secondary mirror in this system using a WYKO 6000 (shown in Figure 12) revealed zones with a relatively large amount of correction so it appears that spherical aberration is not as well corrected as in the first system.  Still, simply correcting the astigmatism would bring the Strehl ratio up to 0.874.  Given most common seeing conditions, this system will certainly work just fine for long exposure imaging; however, under ideal conditions a sharp-eyed observer might notice a small difference in crispness in the finest image detail.  Figure 13 shows the PSF performance for the second system.

 

I experimented a bit with the focusing system, which introduces so much tilt that reversing focus direction introduces so many fringes that they can’t be resolved by the system.  Keep in mind that it’s a double pass test so the beam reflects from the primary twice, which amplifies any tilt by a factor of two.  Still, it appeared that the Strehl could be changed by as much as 0.1 by simply tilting the primary via the focus knob.  This simply confirms the common knowledge that these systems should always be aligned and focused by turning the focus knob in the same direction (CCW.)

 

 

Figure 11.  The single-pass wavefront for the second C14 Edge showing Strehl performance at 0.741.   This data shows that the system does not meet the requirements for diffraction-limited performance.  Astigmatic error is largest contributor at 0.151 wvs (not shown on the plot.)  The other issue is spherical aberration that fits to 0.117 wv, which likely indicates a bit less than perfect wavefront correction by the corrector plate. (Click image to launch larger version in a new tab)

 

It’s relevant to understand the required manufacturing tolerance on optical surface accuracy in a SCT to achieve diffraction-limited performance.  In order to achieve a Strehl ratio of 0.8, the rms wavefront error must be at least 1/14 wave.   Surface errors on the mirrors must be doubled to get the effect on the wavefront.  Since the telescope is a two-mirror system, the rms surface errors add in quadrature to get the total error (equal to half of the wavefront error.)  If we assume that the surface errors are split evenly between the primary and secondary mirrors, we get the total wavefront rms error as a function of the rms surface errors:

 

                                                      [2]

 

Solving for the rms limit on each surface:

                                                                                                                                                          [3]

With 1/14 wave rms as the target wavefront error, the irregularity on each surface must be no more than 0.025 waves rms.  If we assume a ~4x relationship between PV and rms, each surface must have a PV surface accuracy of at least roughly 1/10 wave PV or better, which isn’t trivial at this diameter. 

 

 

Figure 12.  Surface accuracy for the secondary mirror from the second telescope measured by a WYKO 6000 interferometer.  Large zonal corrections are evident. (Click image to launch larger version in a new tab)

 

 

Figure 13.   Plots showing the calculated diffraction PSF performance for the second C14.  A) The Airy disk displayed as I=Log(1+PSF).  B) Shows the PSF displayed as I=log(1+k*PSF), where k =1000 to greatly amplify the low intensity outer diffraction ring structure.  C) Shows the same data as middle figure with the aberration errors scaled to zero to demonstrate a “perfect” pupil similarly sampled.  80% of the total energy is contained within a 0.75 arc-sec aperture and 90% of the total energy fits within a 1.44 arc-sec aperture, which is virtually identical to the first system in spite of the slightly reduced Strehl performance.

 

 

Additional Discussion

1)     Neither of these scopes had been very carefully aligned under the stars before testing.  However, the first system had been aligned under the sky using the imaging camera.  I had simply centered the secondary shadow at high magnification and confirmed good centration on either side of focus.  The seeing was so poor that it was difficult to see what was going on really close to focus, which turned out to be a problem.  The system initially measured at a Strehl of only 0.61 and it was obviously out of alignment on the interferometer.  I did some imaging with the system in this state with pretty good success; though it was difficult to tell if the image quality was limited more by poor seeing, vibration from the wind, or the mediocre secondary alignment.  Still, it demonstrates the inherent difficulty of aligning the secondary under the stars with poor seeing.

2)     Secondary alignment is traditionally accomplished by adjusting secondary tilt to center the secondary shadow through focus and to make the Airy disk look as “good” as possible (small, round and uniform.)  That approach works because the Airy disk will always be optimized when errors in the wavefront due to misalignment are minimized.  Using the interferometer to align the secondary mirror accomplishes that goal by directly measuring the shape of the wavefront in the exit pupil.  Both methods accomplish the same thing but the digital interferometer greatly reduces the role of judgment in the process.  Numbers show unambiguously when the wavefront errors are minimized.  Unfortunately, the expense of this technique makes it a one-time affair-for me and probably a “never-going-to-happen” for most everyone else.  The only consolation is that even though seeing may limit how well a system can be aligned visually under the sky, seeing effects may also limit any additional benefit gained from “perfect” alignment.

3)     I’ve centered the corrector plate on both of these telescopes and I see no evidence of significant shear errors due to the factory surface correction applied to the secondary mirrors.  That is almost certainly because the corrections applied by the factory on the secondary (ref: Celestron Edge HD White Paper) address low amplitude zonal errors with low spatial frequency variations.   I have contended for a long time that small shifts in the position of the corrector/secondary assembly are unimportant and these measurements bear that out.

4)     Overall, I’m impressed that Celestron can ship such high quality 14” systems.  Achieving the required surface accuracy isn’t trivial at this diameter and Celestron generally does a good job of shipping high quality large aperture systems, but, it’s hard to tell how many exceed the diffraction limit.  This sample of only two telescopes suggests that most are pretty close.  The newer scope is a little better than the older scope, which may indicate that Celestron is improving their process but it would take more data to say for sure.  It would certainly be interesting to measure a new system straight off the current assembly line.

5)     I often use these two telescopes on a bench to set up and align my cameras, set filter offsets, and to calibrate my auto-focusing system.  I put a 15-micron pinhole on one of the scopes to create a near diffraction-limited collimator for the second scope.  I’ve been seeing on-axis astigmatic error in the whole system and it drove me crazy—mainly because I couldn’t see any similar error when I had each scope out under the sky.  These results show that both scopes have a bit of residual astigmatic error that roughly line up and that explains why I see it so easily on the bench with both scopes and not in the sky with each scope individually.  Mystery solved. 

I’ve since made a cradle that allows the collimator scope to be rotated by 90 degrees to fix the problem on the bench and it works quite well.  Through a high-powered eyepiece, the focused spot looks almost exactly the same as the diffraction calculation prediction.  That shouldn’t be a surprise but it’s a satisfying result nevertheless.

6)     I want to point out that these results only show the on-axis wavefront quality; however, this is the best way to determine the underlying residual optical errors in the system.   Due to weather limitations, I’ve only been able to take a few test images with the first system after this alignment session and the image quality appears to be excellent with sharp, round stars all the way into the corners of a 36 x 36 mm sensor (52 mm circle at the corners.)  Figure 14 shows the imaging performance over the entire field.

 

 

Figure 14.  A calibrated, stretched, unprocessed image with a 900 second exposure showing the field performance over a 36 x 36 mm sensor (16803.)  This data was taken with the first telescope under relatively poor seeing conditions after interferometric alignment.  Each panel is 500 x 500 px sub-sample demonstrating round, sharp stars over the entire 52 mm image circle. (Click image to launch larger version in a new tab)

 

 

Conclusion

High quality digital interferometric testing is a precise, unambiguous way to measure optical quality, which is why it is so widely used to produce the components for large-scale optical systems and space optics.  While this data very precisely demonstrates the optical performance of the telescopes at 20C, at the test wavelength, in the horizontal position, it is important to understand that these results also serve to set an upper limit on optical performance.  If the telescope components slightly change position when the system is pointed up into the sky or when the temperature changes, it will not perform any better than this data shows.  When properly designed, telescope performance should not radically change with either temperature or pointing angle, but at some level nothing is perfect.  So this data may not show the exact performance of either of these telescopes under the sky depending on the thermal and mechanical stability of each system.  In my experience, the first telescope (which I use the most often) has demonstrated exceptionally mechanical stability.  The data shown in Figure 14 shows how well it performs under the sky when it is thermally stable and my expectation is that it has the capability of delivering diffraction limited performance under real world conditions.  This report has demonstrated the kind of results that high-end digital dynamic interferometric testing can produce along with just some of the performance analysis that can be extracted from the data.

 

 

 

Suggested References

Edge HD, A Flexible Imaging Platform at An Affordable Price, The Celestron Engineering Team, April, 2013.

Virendra N Mahajan, “Strehl Ratio for Primary Aberrations:  some analytical results for circular and annular pupils,” J. Opt. Soc. Am., Vol 72,  No. 9, Sept., 1982.

J.  Hayes, J. Millerd, " Dynamic Interferometry, Getting Rid of the Jitters", Photonics Handbook, H-34, 2006

N. Brock, J. Hayes, B. Kimbrough, J. Millerd, M. North-Morris, M. Novak and J. C. Wyant, "Dynamic Interferometry", Proceedings of SPIE Vol. 5875 (SPIE, Bellingham, WA), page 58750F-1, 2005

M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, "Analysis of a  micropolarizer array-based  simultaneous phase-shifting  interferometer", APPLIED OPTICS, Vol. 44, page 6861, 10 November 2005

J. Millerd, N. Brock, J. Hayes, B. Kimbrough, M. Novak, M. North-Morris and J. C. Wyant, "Modern Approaches in Phase Measuring Metrology", Proceedings of SPIE Vol. 5856 (SPIE, Bellingham, WA), page 14-22, 2005

James Millerd, Neal Brock, John Hayes, Michael North-Morris, Matt Novak and James Wyant, “Pixelated Phase-Mask Dynamic Interferometer”,  Proceedings of SPIE Vol. 5531 (SPIE, Bellingham, WA), page 304-314, 2004

 

(Most of the papers about the PhaseCam (and many more) can be found on the 4D Website at:    www.4dtechnology.com/applications/reflibrary/)


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