Abstract:
An improved apparatus and method are described for accurately &#34;reconstructing&#34; steep surface profiles, such as for aspheric surfaces, using improved two-wavelength phase-shifting interferometry, wherein single-wavelength precision is obtained over surfaces having departures of hundreds of visible wavelengths from a reference surface. The disclosed technique avoids cumulative summing of detector errors over a large detector array by computing the &#34;equivalent&#34; phase for each detector independently of the intensities of other detectors. Inaccurate phase data points having an equivalent fringe contrast less than a predetermined threshold are eliminated from data from which contour maps of the aspheric surface are displayed or plotted.

Description:
BACKGROUND OF THE INVENTION 
     The invention relates to improvements in two-wavelength phase-shifting interferometry, and more particularly, to improvements which make possible the obtaining of single-wavelength precision in interferometric measurements with the dynamic range of two-wavelength interferometric measurements, and application thereof to testing aspheric surfaces. 
     Two-wavelength holography and phase-shifting interferometry are known techniques for nondestructively testing optical surfaces. Numerous phase-shifting interferometry techniques and apparatus are known, wherein the phase of a wavefront is determined using a single wavelength with very high measurement precision. One such system is described in copending application Ser. No. 6/781,261, filed Sept. 27, 1985, now U.S. Pat. No. 4,639,139, issued Jan. 27, 1987 by inventors Wyant &amp; Prettyjohns, entitled &#34;Optical Profiler Using Improved Phase-shifting Interferometry&#34;, and incorporated herein by reference. The single-wavelength phase-shifting interferometry techniques, wherein computers are utilized to record data, compute and subtract surface errors, and compute surface height variations in the measured surface from correct phase data, are fast, require no intermediate recording step, as is required in two-wavelength holography, and avoid the inconvenience of using photographic chemicals. 
     Two-wavelength phase-shifting interferometry is a recent technique that extends the measurement range of single-wavelength phase-shifting interferometry, allowing the measurement of the profiles of deeper surfaces than has been previously possible with single-wavelength phase-shifting interferometry. The two-wavelength phase-shifting interferometry techniques were derived by applying phase measurement techniques, in place of intermediate recording steps wherein interference patterns were recorded on photographic film, developed, and then illuminated from the same surface with a different wavelength source, producing interference patterns referred to as MOIRE patterns, the phases of which were computed by a computer. This technique represents the closest prior art to the present invention, and is described in detail in &#34;Two-Wavelength Phase-Shifting Interferometry&#34;, by Y. Cheng and co-inventor Wyant, &#34;Applied Optics&#34;, Volume 23, No. 24, page 4539, Dec. 15, 1984. 
     The overall state-of-the-art in interferometric optical testing is well presented in the article &#34;Recent Advances in Interferometric Optical Testing&#34;, Laser Focus/Electro-Optics, November 1985, page 118 to 132, by co-inventors Wyant and Creath, incorporated herein by reference. 
     As pointed out in the above-mentioned Cheng and Wyant paper that introduces the concept of two-wavelength phase-shifting interferometry, ordinary single-wavelength phase-shifting interferometry provides very high precision in the range from λ/100 λ/1000, peak-to-valley. In phase-shifting interferometry, the phase distribution across the interferogram is measured &#34;modulo 2π&#34;. In other words, the measured phase distribution will contain 2π discontinuities, which can only be eliminated as long as the slope of the wavefront being measured is small enough that the phase changes by less than π between adjacent detectors or pixels of the detector array. If the latter condition is met, the phase discontinuities can be removed by adding or subtracting 2π to the measured phase until the resulting phase difference between adjacent pixels is always less than π. 
     Unfortunately, it is frequently desirable to be able to test surfaces that are so steep that the measured phase change between adjacent pixels will be greater than π, and the 2π ambiguities cannot be eliminated by simple addition or subtraction. 
     In the Cheng and Wyant paper, a technique for solving the 2π ambiguity problem is introduced, wherein two sets of phase data, with 2π ambiguities present, are stored in a computer, which then calculates the phase difference between pixels for a longer &#34;equivalent&#34; wavelength λ eq . The paper describes an algorithm for computing the phase, wherein an elaborate mathematical summation is performed wherein the difference in phase computed at each of the two-wavelengths is computed for each pixel element, multiplied by a certain number, and the differences between the computed differences for successively adjacent pixels are summed. The problem with that technique is that errors in the computed differences for various pixels also are summed. As a practical matter, the technique described in the Cheng and Wyant paper is much less accurate than is desired, and requires far more computation time than is desirable, even though the technique represents an advance over the two-wavelength holographic techniques of the prior art, because the recording of an interferogram on film is not required, and the alignment problems associated with two-wavelength holography are avoided. In the technique of the Cheng and Wyant paper, the pixel errors increase with approximately the square root of the number of detector points in the detector array, and certain arithmetic round-off errors and electronic noise associated with the detector elements are cumulatively summed over the entire detector array. As a result, in applications wherein two-wavelength phase-shifting interferometry might be advantageous without the above-mentioned problems, such as in testing certain aspheric surfaces, it often will be necessary to instead rely on prior techniques for measuring of contouring aspheric surfaces. 
     Up to now, phase-shifting interferometric techniques that utilize longer equivalent wavelengths have resulted in a substantial loss of accuracy and, as a practical matter, have not been applicable to measurement of many aspheric surfaces. 
     Those skilled in the art recognize that economical, accurate measuring of aspheric optical components has been an important objective in the optics art. Those skilled in the art know that most optical surfaces presently are spherical surfaces. However, if it were possible to make economical aspheric surfaces, better optical performance often could be obtained. Optical systems designed with aspheric optical components may be lighter in weight, have fewer elements, and therefore have the potential for being less expensive. Designing aspheric optical components is not a major problem, as computer software for so doing has been available for quite some time. Up to now, however, fabrication of aspheric optical components has been very expensive, because there has been no inexpensive, practical means of testing them. 
     One prior technique for testing aspheric surfaces has been to utilize &#34;null lenses&#34; wherein an aspheric optical component is fabricated that exactly cancels the asphericity in the optical element being tested. Unfortunately, the aspheric null lens has to be manufactured first, using testing techniques that are very expensive. For every new optical aspheric component that is to be constructed, a separate expensive null lens must be manufactured, usually for that purpose only. 
     Typically, the null lens is manufactured and tested by techniques that require many spherical components precisely assembled in a manner known to those skilled in the art. Providing such an assembly of spherical lenses to test aspheric elements is known to be a difficult undertaking. 
     For many years, co-inventor Wyant has worked on producing computer-generated holograms that each can be used for testing a certain aspheric element of an optical system. This technique has worked, but it has been expensive and difficult because a new hologram is needed for every asphere to be tested. In order to make a single hologram, an expensive, high precision plotting device is required, and many hours are required to plot a single hologram. The prior art for testing aspheric optical components frequently involves set up costs of tens of thousands of dollars for each new aspheric optical element to be tested. 
     There is a substantial unmet need for a technique of improving the accuracy of two-wavelength phase-shifting interferometry to increase accuracy and reduce speed of computation. 
     There also is a great unmet need for an inexpensive, improved technique of testing steep aspheric surfaces without the use of null lens and holograms. 
     There is also a great unmet need for an improved technique of testing steep aspheric surfaces with a single test system, and for reducing the time required to test each aspheric surface. 
     A problem that has existed in prior phase-shifting interferometers at the present state-of-the-art has been that of operating on data stored in a computer to correct data errors produced by various sources, such as stray reflections and scattered light. Prior techniques have included the technique of determining the intensity difference between data frames computed at a single pixel or detector as the phase is shifted, and if the computed difference was too small, eliminating the computed phase. This approach often resulted in discarding good data as well as bad because the phases may have been computed from intensity measurements at points that are near a peak or valley of a fringe where the intensity does not change much between consecutive measurements. 
     SUMMARY OF THE INVENTION 
     Accordingly, it is an object of the invention to provide an improved two-wavelength phase-shifting interferometry apparatus and method that avoids the inaccuracy and long computation times of the closest prior art. 
     It is another object of the invention to provide an improved apparatus and technique for measuring contours of aspheric surfaces that are steeply sloped relative to a spherical surface. 
     It is another object of the invention to provide a single system that can accurately test a large number of different aspheric surfaces without major modification for differing aspheric surfaces to be tested. 
     It is another object of the invention to provide an apparatus and technique for avoiding the effect of averaging of fringes across individual detectors of a detector array having large surface areas. 
     It is another object of the invention to exclude invalid data points in mathematical reconstruction of a measured surface such as a steeply aspheric surface while avoiding discarding of good data points. 
     It is another object of the invention to provide an apparatus and technique for extending the measurement range of an interferometer while maintaining the precision and accuracy characteristic of single-wavelength phase-shifting interferometry. 
     Briefly described, and in accordance with one embodiment thereof, the invention provides an improved phase-shifting interferometer and method of operating wherein a light beam having a first wavelength illuminates a reference mirror via a reference arm of the interferometer, and simultaneously illuminates a test surface via a test arm of the interferometer to produce an interference pattern on an array of photodetectors, measuring the intensities of the interference wavefront on a detector of the array at a plurality of different phases of the interference pattern that are produced by shifting the reference mirror, computing a phase of the interference for the pattern at that detector for that wavelength, repeating the same procedure for a second wavelength beam, and computing an equivalent phase for that detector by computing the difference between the phases computed for the first and second wavelengths, respectively. An equivalent optical path difference for the detector is computed by multiplying the equivalent phase by an equivalent wavelength corresponding to an imaginary interference pattern between the first and second wavelength beams, and repeating the entire procedure for each detector of the array. In one described embodiment of the invention, 2π ambiguities are corrected for either the equivalent phases or the equivalent optical path difference by adding or subtracting integral multiples of 2π to the equivalent phases as necessary to make the phase difference between adjacent detectors less than π, or by adding or subtracting multiples of the equivalent wavelength to the equivalent optical path difference as needed to make the difference in the equivalent optical path difference between adjacent detectors less than one-half of the equivalent wavelength. In one embodiment of the invention, the equivalent optical path difference for all of the detectors is used to plot or display a two-dimensional or three-dimensional display of the test surface, which may be a steeply aspheric surface. 
     In another embodiment of the invention, the equivalent optical path difference data is utilized to determine the amount of correction needed to resolve 2π ambiguities in the phase difference data or optical path difference data corresponding to the first wavelength which, after correction for such 2π ambiguities, is then utilized to plot or display a two-dimensional or three-dimensional display of the test surface. The foregoing technique provides much greater accuracy and, in effect, provides two-wavelength phase-shifting interferometry for the large range that can be achieved for the equivalent wavelength, and the accuracy that can be achieved for single-wavelength phase-shifting interferometry at the first wavelength, without cumulative summing errors associated with prior two-wavelength phase-shifting interferometry, which summing errors are due to electronic noise, detector noise, and/or round-off errors. 
     In one embodiment of the invention, a mask plate is aligned over the detector array, with an array of small apertures therein, aligned with each of the detectors to effectively reduce the area of each detector such that its diameter is much less than the equivalent fringe spacing. A light intensifier may be positioned between the mask plate and the photodetector array to compensate for loss of ambient energy due to the mask, and a fiberoptic bundle is disposed between the intensifier and the photodetector array to prevent further loss of light energy and ensure that the interference pattern is imaged onto the face of the photodetector array. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a schematic diagram of a two-wavelength phase-shifting interferometer of the present invention. 
     FIG. 2 shows a pair of graphs useful in describing the operation of the two-wavelength phase-shifting interferometer of FIG. 1. 
     FIGS. 3A and 3B show two graphs of three-dimensional countour plots of an aspheric surface, one measured by prior art single-wavelength phase-shifting interferometry and another measured by the apparatus of FIG. 1 in accordance with the present invention. 
     FIGS. 4A and 4B constitute a flow chart of a program executed by the computer of the two-wavelength phase-shifting interferometer of FIG. 1 to carry out the method of the present invention. 
     FIG. 5A is a diagram of a point mask used in conjunction with the photodetector array of FIG. 1 to reduce the effective size of each of the photodetectors of the array. 
     FIG. 5B is a schematic diagram illustrating the use of the point mask and an intensifier in conjunction with the photodetector array in the interferometer of FIG. 1. 
    
    
     DESCRIPTION OF THE INVENTION 
     Referring now to the drawings, two-wavelength interferometer 1 of FIG. 1 includes two lasers having different wavelengths λ a  and λ b , respectively. Laser 2 can be an argon laser producing a beam 4 of light having a wavelength λ a  impinging upon a mirror 5. Laser 2 can be switched on and off by means of a signal on bus 41 in response to control circuitry in block 29. Similarly, laser 3, which can be a helium-neon laser producing a beam 6 having a wavelength λ b , also is switched on and off in response to a difference signal or bus 41. Beam 6 passes through attenuator 7 to produce beam 9 when laser 3 is switched on. Beam 8 is reflected from mirror 5 and also from the surface of mirror 7 to produce beam 9 when laser 2 is on. Beam 9 is reflected from mirror 10 to produce beam 11, which enters a microscope objective 12. Microscope objective 12 focuses beam 11 onto a pinhole 12A and into a collimating lens 14, as indicated by reference numeral 13. The beam 15 emerging from collimating lens 14 impinges upon a beam splitter 16. 
     The beam 15 is split to produce beams 17 and 22. Beam 22 passes through a lens 23 that matches &#34;the best fit sphere&#34; of an aspheric test surface 25 so that the rays 24 impinge approximately perpendicularly onto aspheric test surface 25, and therefore are reflected back through lens 23 and beam splitter 16, contributing to interference beam 26 that passes through imaging lens 27, focusing interference beam 28, onto for example, a 256 by 256 detector array included in block 29. (In our experiments, however, a 100 by 100 detector array was used.) Reference numeral 41A designates a nonparallel window that reduces effects of interference between surfaces of the window and the detector array. 
     The portion of the interferometer including beam 22, lens 23, and aspheric test surface 25 is referred to as the test arm. 
     The other arm of the interferometer, referred to as the reference arm, includes beam 17 emerging from the lower surface of beam splitter 16 and reference mirror 18. 
     Beam 17 impinges upon reference mirror 18, which can be linearly moved in accordance with incorporated-by-reference patent application Ser. No. 781,261 (now U.S. Pat. No. 4,639,139) mentioned above. Beam 20 is reflected from reference mirror 18 onto the surface of a wedge 21 having approximately the same reflectivity as aspheric test surface 25. Beam 20 is reflected back to reference mirror 18, and then back to the lower surface of beam splitter 16, which then reflects the reference beam so that it interferes with the reflected test surface beam 22. Imaging lens 27 receives the interference beam 26 and focuses the test surface 25 onto the 256 by 256 detector array in block 29, as indicated by 28. 
     Bus 42 connects piezoelectric transducer 19 to control circuitry in block 29, synchronizing linear movement of reference mirror 18 with the scanning of the 256 by 256 photodiode detector array 29A in block 29 in accordance with the teachings of above-mentioned co-pending application Ser. No. 781,261. The intensity measured by each photodiode detector in the 256 by 256 array is converted to an analog signal as also described in application Ser. No. 781,261, which analog signal is operated upon by suitable programs stored in computer 40 to compute the optical path difference (OPD) between the surface of reference mirror 18 and aspheric test surface 25. 
     An equivalent wavelength λ eq  is computed for the present values of λ a  and λ b  being used in accordance with the equation: ##EQU1## 
     Four currents are measured for each photodiode at each of the two-wavelengths λ a  and λ b , and for each wavelength λ a  or λ b , the phase of the wavefront detected is computed from the following equation: ##EQU2## where A, B, C, and D represent the measured currents from the present photodetector at equally spaced time increments as the piezoelectric transducer 19 is linearly shifted. This equation is derived in &#34;Installation Et Utilisation Du Comparateur Photoelectrique Et Interferential Du Bureau International Des Poides Et Mesures&#34;, Carre&#39;, 1966, Metrologia, Vol. 2, pages 13-23. If this equation is used, the same ramping rate can be used for the piezoelectric transducer 19, regardless of the wavelengths of lasers 2 and 3. It should be noted that there are various other techniques for measuring the photodetector array output and shifting the piezoelectric transducer, with different equations for computing the phase at each photodetector. For example, see &#34;Contouring Aspheric Surfaces Using Two-Wavelength Phase-Shifting Interferometry&#34;, by Katherine Creath, Yeou-Yen Cheng, and James C. Wyant, Optica Acta, 1985, Vol. 32, No. 12, p. 1455-1464, incorporated herein by reference. Also see the above-mentioned co-pending patent application Ser. No. 781,261. If the intensity measurements and phase are obtained using the equations other than the Carre equations, the piezoelectric transducer linear ramping voltage slope may have to be adjusted for each wavelength to ensure that the phase is shifted by a precise amount. The computations will be less complex, but the accuracy will be slightly reduced. 
     It has been previously shown, in Wyant, 1971, Applied Optics, Vol. 10, page 2113 that an interferogram obtained with two wavelengths has an intensity distribution given by the equation: ##EQU3## 
     The argument of the cosine term is the equivalent phase φ eq  (X,Y) and can be written as: ##EQU4## wherein φ a  and φ b  are the phases measured for λ a  and λ b , respectively. 
     Thus, the difference between φ a  and φ b  measured at the two wavelengths λ a  and λ b  yields the phase φ eq  associated with the equivalent wavelength of the &#34;interference&#34; pattern that would be generated between λ a  and λ b  if two-wavelength holography techniques were utilized. 
     It was not recognized by co-inventor Wyant, in spite of considerable reflection by him on the matter, that φ eq  could, as a practical matter, be obtained by equation (4), because both φ a  and φ b  are computed &#34;modulo 2π&#34; and, in accordance with the nature of the arctangent function, have discontinuities every 360 degrees. It was thought that the discontinuities would be indeterminate for φ a  and φ b  since the intensity measurements were taken at different wavelengths λ a  and λ b , respectively. One would ordinarily not consider it a practical thing to add or subtract terms with indeterminate or undefined discontinuities in order to obtain a term such as φ eq  that needs to be extremely precise. 
     Once the value of φ eq  is obtained in accordance with equation (4), with φ a  and φ b  being obtained in accordance with equation (2), the next step (if desired) is to compute the equivalent optical path difference in accordance with the equation: ##EQU5## for each detector element. 
     To operate the two-wavelength phase-shifting interferometer of FIG. 1, lasers 2 and 3 are aligned so that the beams 9 and 11, are colinear up to collimating lens 14. Using an uncoated aspheric test surface 25, and using the above-mentioned RETICON 100 by 100 photodiode array, the number of interference fringes appearing on the detector array 29A was controlled by positioning the detector array 29A and the imaging lens 27. The wavelength λ a  was either 0.4880 microns or 0.5145 microns. λ b  was 0.6328 microns. At each detector, argon laser 2 is turned on, thereby illuminating the array 29A, and the piezoelectric transducer 19 is shifted by linearly advancing it to allow three or four intensity measurements (depending on the method of phase computation used) from that photodiode to be measured and digitized by an analog-to-digital converter and then read by computer 40. The argon laser 2 then is switched off, helium-neon laser 3 is switched on, the piezoelectric transducer returned to its initial point, and the above procedure is repeated for the same photodiode. 
     Using an appropriate display or plotting routine that can be executed by computer 40, a three-dimensional contour map of the aspheric surface 25 can be displayed on a screen of computer 40 or plotted on a plotter. FIG. 3A shows the displayed result obtained using the above-described method, the individual curves representing the equivalent optical path difference OPD eq  (X,Y), with tilt being subtracted by the software in a conventional manner. It can be seen that the contour map of FIG. 3A has no discontinuities. 
     FIG. 3B shows a contour map of the same aspheric test surface 25 generated using single-wavelength phase-shifting interferometry for the λ a  laser wavelength only. It can be readily seen that there are many discontinuities and inaccuracies in the contour map of FIG. 3B. 
     It should be noted that φ eq  as computed by equation (4) must be adjusted by eliminating the above-mentioned 2π ambiguities by adding or subtracting integer multiples of 2π until the difference in φ eq  between successive photodiodes is less than π. Alternately, 2π ambiguities can be adjusted in OPD eq  by adding or subtracting integer multiples of λ eq  until the difference in OPD eq  is less than one-half of λ eq . 
     It also should be noted that the above technique is only valid if the condition that λ eq  is sufficiently large that there is no more one-half of an &#34;equivalent&#34; fringe (i.e., λ eq  wavelengths) between adjacent photodetectors in array 29A. (&#34;Equivalent&#34; fringes are fringes that would be produced by an interference pattern produced by λ a  and λ b  using two-wavelength holography, as described above.) 
     It should be appreciated that the accuracy of OPD eq  (X,Y) is less than the accuracy of OPD a  or OPD b  because inaccuracies due to photodetector noise and electronic noise are scaled up by the ratio of OPD eq  to the single-wavelength optical path difference. If greater accuracy is needed, the technique of the invention can be expanded by using the equivalent OPD eq  data to determine precisely where to correct the 2π amibiguities in the single-wavelength OPD data. How this is accomplished can be understood with reference to FIG. 2. 
     In FIG. 2, curves 30, 31, and 32 designate the optical path difference computed for the single-wavelength &#34;interferogram&#34; 2π discontinuities by measuring aspheric test surface 25. Reference numeral 31A designates the first 2π ambiguity, the amplitude of which is indeterminate. Reference numeral 30A designates the values of OPD a  computed until the next indeterminate 2π ambiguity occurs, as indicated by reference numeral 31A. Curve 32 designates the values of OPD a  continuing until the next indeterminate 2π ambiguity 32A. 
     It should be appreciated that the actual optical path difference for the aspheric surface 25 might change a great deal, i.e., much more than λ a , between adjacent pixels or photodetector elements, if the surface being measured is &#34;steep&#34;, i.e., varies greatly from a spheric surface. Therefore, there is no way of knowing how many multiples of λ a  need to be added to (or subtracted from) curve 31 to move its left most point up to match point 47, and thereby eliminate the 2π ambiguity 30A. However, the information in curve 33, although much less accurate than that in curves 30, 31, and 32 (because of the above mentioned scaling), can be utilized to determine how many multiples of λ a  need to be added to curve 31 to eliminate the 2π discontinuity 30A. 
     Note that the &#34;roughness&#34; of curve 33 represents the relative inaccuracy of OPD eq  compared to OPD a . Due to the fact that λ eq  is much longer than λ a , there are no 2π discontinuities in curve 33. Therefore, point 47A of curve 33B can be compared with point 47 of curve 30. Note that point 47 corresponds to the first photodetector diode or pixel before the 2π discontinuity 30A occurs in FIG. 2, and point 47A corresponds to the first detector after the 2π ambiguity 30A. An integral number of λ a  wavelengths determined by equation (6) below are added to or subtracted from OPD a  for curve 31 to cause point 47A to closely match point 47, within one half of λ a . 
     It should be noted that the above explanation with reference to FIG. 2 of resolving 2π ambiguities can be performed on φ eq  and φ a  which differ from the optical path differences only by the scaling factor 2π/λ eq  or 2π/λ a . 
     The following equation expresses this computation of the number N of λ a  wavelengths added or subtracted to &#34;resolve&#34; the above-described 2π ambiguity 30A: ##EQU6## where N is an integer. 
     Stated differently, what expression (6) means is that an integral number of λ a  &#39;s are added to and/or subtracted from a OPD a  until the difference between that quantity and the equivalent optical path difference OPD eq  is less than half of λ a . (Note that this technique yields proper results if there is no chromatic aberration present in the optics of the interferometer. If such errors are present, then a more complex technique is required). 
     Thus, the present invention can result in nearly the same accuracy in measuring a surface as is obtained using short (visible), single-wavelength phase-shifting interferometry, while also obtaining the much larger range of optical path differences that can be obtained with two-wavelength holography. 
     Comparing this technique with the two-wavelength phase-shifting interferometry techniques described in the above-mentioned Cheng and Wyant article, much greater accuracy is achieved, and much shorter computation times are achieved. This is because of the simplicity of the computation of equation (4) above, wherein the equivalent phase φ eq  is computed by simply subtracting the computed quantity φ b  from φ a  and computing OPD eq  from equation (5) for each photodiode. 
     If a particular photodiode produces an erroneous signal, due to externally induced noise or an intrinsic junction defect, that error will affect only one value of OPD eq . All other values of OPD eq  obtained will be independent of that error. In contrast, in the Cheng and Wyant reference, the differences in the OPDs between each adjacent pair of photodiodes is obtained by subtraction, and these differences are summed, as indicated in equations (9) in the Cheng and Wyant reference, wherein any error occurring as a result of nise or other inaccuracy, including computational round-off errors, occurring in any first pixel or photodiode will be added to the optical phase difference computed for every subsequently scanned photodetector. 
     Thus, while the technique described in the Cheng and Wyant article represents a significant advance in the art, the present above-described invention results in a significant improvement that finally makes two-wavelength phase-shifting interferometry a highly practical means of reducing the costs of testing aspheric optical components, making the total manufacturing costs thereof much less than has been the case up to now. 
     A computer algorithm, subsequently described with reference to the flow chart of FIG. 4A, was written to make the computations of equations (2), (4), and (7). During the phase calculations, saturated data points, wherein overly large photodiode currents were generated, and points wherein the modulation of the currents or intensity from one pixel or photodiode to the next were too small, were neglected. 
     Referring now to FIG. 4A, which shows the above-mentioned flow chart, the algorithm is entered at label 50. In block 51, laser 2 is switched on. The program then goes to block 52 and causes the control circuitry 29B to cause appropriate ramping of the reference mirror 18 by the piezoelectric transducer 19 and scanning of the detector array 29A to obtain the four measurement A, B, C, and D. The program stores A, B, C, and D and then goes to block 53, switches off laser 2, and switches on laser 3. In block 54, the program initializes the position of the piezoelectric transducer 19. In block 55, the program again causes ramping of the piezoelectric transducer and scanning of the detector array 29A, exactly as in block 52, and stores the four resulting measurements of E, F, G, and H. 
     Next, the program goes to block 56 and computes the phases φ a  and φ b , wherein φ a  is a function A, B, C, and D and φ b  is a function of E, F, G, and H. These calculations are made in accordance with equation (2) above; the details of the computations are shown in the flow chart of FIG. 4B. 
     Next, the program goes to block 57 and uses equation (4) to compute φ eq . 
     The program then goes to block 58, and computes λ eq  in accordance with equation 1 and computes OPD eq  in accordance with equation (5). Then the program goes to block 59 and removes the 2π ambiguities from λ eq  and from λ a  in accordance with the above-described techniques. Finally, the program goes to block 60, displays the corrected optical path differences, and exits at label 61. 
     Referring now to FIG. 4B, the subroutine for computing the phases, as explained above with reference to block 56, is entered at label 65. First, the program sets a variable called PIXEL to 1. The program then goes to decision block 67 and determines if the present value of PIXEL is equal to 256 2  (assuming that a 256 by 256 photodetector array is being used). 
     If the determination of block 67 is affirmative, the algorithm is exited at label 68. Otherwise, the program goes to block 69 and computes the temporary variables D 1 , D 2 , and D 3  as illustrated. The program then goes to block 70 and computes the temporary variables D 4 , NUM (numerator), and DEN (denominator). 
     The algorithm then goes to block 71 and computes the quantity MOD (modulation), which is equivalent to equation (7) above. 
     Next, the program goes to decision block 72 and determines if any of A, B, C, or D has a value that exceeds a predetermined &#34;saturation&#34; level that indicates too much incident light on the corresponding photodetector, or if the modulation variable MOD is less than a predetermined minimum threshold. If the determination of block 72 is affirmative, the program goes to block 73 and determines that the data from the present pixel or photodetector is invalid by setting a flag corresponding to that pixel of the photodetector array, so that it can be later recognized as a bad point, increments PIXEL, and returns to decision block 67. If the determination of block 72 is negative, the program goes to decision block 74 and determines if DEN is equal to zero. If the determination of decision block 74 is affirmative, the program goes to decision block 75 and determines if the variable D 2  is greater than zero. If DEN is 0, then the phase has certain values meaning that the arctangent has become infinite, so depending on the sign of the sine term, the phase is given a value of π/2 or 3π/2, as indicated in blocks 76 and 77. 
     Referring to block 78, since an arctangent calculation will only give a phase modulo π because of the signs in the numerator and denominator of the equation, an ambiguity can occur there. Absolute values of NUM and DEN are computed to avoid this. 
     In blocks 79 through 85 it is determined in which cartesian coordinate quadrant 2, 3, or 4 to place the phase. Then π added to the phase or the phase is subtracted from 2π or π, as indicated. The final part of this calculation is to check if the numerator is equal to 0 in block 85. If the numerator is equal to 0, then the phase is set to either 0 or π depending on whether the denominator is less than or greater than 0. At that point, the program goes to block 75A and increments PIXEL. 
     Note that Appendix A, affixed hereto, includes a printout of an operating program executed by computer 40 and including the calculations corresponding to the flow charts of FIGS. 4A and 4B. 
     The three dimensional countour map shown in FIG. 3A was obtained for the two-wavelength technique described above, while the corresponding contour map of FIG. 3B was obtained using only the single-wavelength λ a . The technique described above with reference to equation 6 could be used to correct all of the obvious errors in FIG. 3B to provide a contour map resembling, but more accurate than, the contour map of FIG. 3A. 
     The fundamental limit to the above-described technique is the ratio of detector size to the fringe spacing. If many fringes are incident upon a single detector element, the intensity measured will, in effect, be an &#34;average&#34; over the detector area. Then, when the phase is shifted by the piezoelectric transducer 19, such points, or more particularly, the measured intensities or photocurrents, will not be &#34;modulated&#34; (i.e., change from one intensity measurement to the next as the phase is shifted by the motion of the piezoelectric transducer) sufficiently to allow an accurate phase computation to be made. 
     To avoid this problem, the size of the detector should be small compared to the λ a  and λ b  fringe spacing to ensure sampling of one fringe at a time as the phase is shifted by motion of the piezoelectric transducer. We are experimenting with holes 2, 5, and 10 microns in diameter in order to obtain a suitable trade off between accuracy of computation and loss of light received by the individual photodiodes. 
     The sensitivity of two-wavelength phase-shifting interferometry is limited by the equivalent wavelength used. Since single-wavelength phase-shifting interferometry is generally precise to λ a  /100 to λ a  /1000, the above described technique should measure surface defects of the order of 100 to 1000 angstroms for an equivalent wavelength λ eq  equal to 10 microns or 10 to 100 angstroms after correcting ambiguities in the single wavelength data. 
     One technique for reducing the ratio between the effective photodetector area (size of pixel) and the distance between &#34;equivalent&#34; fringes is to utilize a mask plate 43, shown in FIG. 5A. Mask plate 43 includes an array of minute holes 44 that are centered over and aligned with each of the photodetector array elements in array 29A. Unfortunately, this technique reduces the amount of light that reaches the photodetector cells, reducing the photocurrents produced by each, possibly resulting in an undesirable reduction in accuracy. To offset the reduction in photocurrents produced by the interference pattern focused by the optics onto the detector array 29A, either more laser power is needed, or else a light intensifier device 46 can be mounted between mask plate 43 and the detector array 29A, FIG. 5B, and a fiberoptic bundle 45 can be aligned with and positioned between the intensifier 46 in the detector array 29A, as also shown in FIG. 5B, preventing further loss of light rays. Various suitable image intensifiers are commercially available, for example from ITT or Litton. Such intensifiers have a photocathode which responds to incident light followed by a microchannel plate which intensifies the signal, whose output activates a phosphor, producing larger amounts of light. 
     It should be appreciated that in using phase-shifting interferometry techniques, if the light signal falling on a particular photodetector cell is low for any reason, or if for some reason one of the two interferometer beams is much brighter than the other, there can be a considerable inaccuracy in the computed phase and optical path difference for that photodiode. As far as we know, no one has published a really good technique for determining whether or not the phase computations are accurate or inaccurate as a result of too low light intensity or saturation of a particular photodetector cell. In the past, we have compared the intensity measurements from adjacent photocells, and eliminated data points corresponding to photocells from which the measured intensity varies less than a preselected amount from that of the previously scanned photocell. As mentioned previously, this technique results in accidental discarding of a considerable amount of good data, reducing the accuracy of the overall countour map of the aspheric surface 25. 
     In accordance with a further aspect of the present invention, the following equation is utilized to compute the fringe modulation, or contrast: ##EQU7## 
     This expression is easily derived from the above Carre equations (equations 10) of the Creath et al., assuming that the phase-shift is close to π/2. The quantity γ is the constant preceding the cosine terms in the above mentioned equations for A, B, C, and D of the above mentioned Carre&#39; equations. Thus, γ is the amplitude of the variation of the quantities A, B, C, and D, and can be interpreted as the contrast of the interference fringes. We have found that if the fringe contrast is less than an experimentally determined threshold value, which depends on the ambient level of electrical noise and the amount of intrinsic photodetector junction noise, the accuracy of the phase plots and/or contour plots obtained is increased considerably over that previously available using prior techniques for eliminating bad data points. 
     It should be noted that chromatic aberration of interferometers can be minimized by proper design thereof. First, beam splitter 16 can be a membrane beam splitter called a pellicle. The microscope objective collimating lens combination 12 and 14, lens 23, and lens 27 all should be designed so that they are achromatic at the wavelengths λ a  and λ b . This means that the design is such that the image of the aspheric test surface is the same size and location for both λ a  and λ b . 
     While the invention has been described with reference to several particular embodiments thereof, those skilled in the art will be able to make various modifications to the described embodiments of the invention without departing from the true spirit and scope thereof. For example, there may be various techniques for computing the λ eq  at each detector without computing the phase for that detector at each of the two wavelengths, and instead computing φ eq  from the intensity measurements of that detector at each wavelength. 
     Such techniques nevertheless would distinguish over the Cheng and Wyant reference by providing computation of φ eq  at each detector point independently of the values computed for the other detector points, and thereby avoid the accumulation of any errors due to electronic noise, PN junction noise, or round-off errors at the other detector points. 
     Although an array of photodetectors is disclosed, it would be possible to instead use a single detector and means for providing relative movement between a single detector and the interferogram to accomplish the intensity measurements. 
     It should be appreciated that those skilled in the art can derive a variety of expressions for φ eq  as a function of the intensities measured at each of the two wavelengths λ a  and λ b  that do not involve computing φ a  and φ b . Such expressions can be derived from the above-mentioned Carre&#39; equations or from various other equations representing intensities of the interferogram at different phase shifts that are known to those skilled in the art. It should be appreciated that equation (7) will be different for each different mathematical technique for computing phase. It also should be noted that various other techniques for obtaining phase shifts can be accomplished other than by shifting the reference mirror. Appendix 2 provides an example of such a derivation. ##SPC1## 
     γ and I o  are normalized to be the same for both λ a  and λ b . The particular intensity equations are well-known equations that require a 90° phase shift between measurements. 
     The equations (1) above correspond to four intensity measurements at each detector at a first wavelength λ a . Equations (2) represent four corresponding intensity measurements at λ b  for the same phase shifts. Equations (3) through (6) show a way of combining equations (1) and (2). Equations (7) and (8) represent a way of combining equations (3) and (6) from which equation (9), for the equivalent phase. Equation (10) shows that φ eq  can be computed without computing φ a  and φ b .