SURFACE MONITORING

A technological program for processing interferograms by Fourier transformation

V. A. Gorshkov,a) A. S. Nevrov, and D. A. Novikov

Optika Scientific Manufacturing Organization, Moscow

A. G. Lomakinb)

All-Russia Scientific Research Institute of Optophysical Measurements, Moscow

(Submitted November 12, 2010)Opticheskiı̆ Zhurnal 78, 44–50 (April 2011)

This paper describes the production of topographical maps of the surfaces of optical items byFourier transformation. Software is presented that is used in fabricating high-accuracy asphericoptics. A comparative analysis is carried out of this method with the amplitude method usedearlier in the process of shaping and monitoring the surfaces of optical items. It is concludedthat this method is suitable for monitoring the surfaces of optical items. c© 2011 OpticalSociety of America.

INTRODUCTION

The energy losses of optical systems close to the focus,which characterize the quality of the optics, are proportional tothe rms wave-front deformation, which in turn is determinedby the accuracy with which the surfaces of the optical elementsof the system are fabricated. The shaping of the surface ofoptical items with high accuracy is an important technologicalproblem, which is solved by using interference monitoringmethods.1,2

The technological monitoring of the surfaces of opticalitems, especially aspheric ones, must cover a range of devia-tions from the specified parameters of tens and even hundredsof micrometers at the aspherization stage and to hundredthsof a micrometer at the lapping and certification stage. Themost informative and accurate method of monitoring is theinterference method. This includes interferometry with areference wave front, shearing interferometry, interferometryin the IR range, etc.3 At the same time, interference is anindirect method, in which information on the deviations of themonitored surface from a comparison surface is determinedby the shape and position of the interference fringes in theinterference field.

As a rule, to obtain a topographic map of the surfacedeviations, the interference pattern must be expanded; i.e., thecoordinates and orders of the interference fringes must bedetermined.

Automatic systems for computer processing of interfero-grams are known that make it possible to obtain informationon the surface topography. These systems usually operate incombination with interferometers that directly digitize a signalfrom a video camera. However, the interference pattern musthave a character close to straight fringes.

Information on the surface topography at the intermediatestages of the processing – aspherization or lapping of thesurface – is needed for technological purposes. The interfer-ence pattern of the surfaces at these stages can substantially

differ from straight fringes and can be complex. It is impos-sible to automatically process such data. The interferogramin this case is usually interpreted by the operator manually,with subsequent computer processing of the data, and thissubstantially increases the time to monitor the given surface,especially at the lapping stage.

The goal of this project was to create a program thatcan process interference patterns of increased complexity byprocessing interferograms in the automatic regime by Fouriertransformation. This would make it possible without digitizingthe orders of the fringes to increase the throughput of themeasurements at the intermediate stages of the processing andlapping of the surfaces of optical items.3,4

GENERAL DESCRIPTION OF THE PROGRAM

The program INTPROC thus developed is executedunder the Windows operating system. Its starting data arethe parameters of the item and a digitized image of theinterferogram in the form of a graphic file in bmp format.This means that the program need not be restricted to aspecific monitoring device but can use arbitrary data from avideo camera, scanner, or digital camera or data obtained viaInternet.

The result of the operation of the INTPROC program isto create a topographic map of the deviations of the testsurface from a comparison surface in ADK format, which issubsequently needed in the technological process of automaticshaping.

A number of sequential operations are executed duringthe operation of the program: bringing the interferograminto coincidence with the geometrical contour of the item,eliminating noise, calculating the surface topography, andreviewing and analyzing the results. Its operation is organizedaccording to the principle of a single-window interface. Theprogram allows different contours of the perimeter of the itemsto be specified.

257 J. Opt. Technol. 78 (4), April 2011 1070-9762/2011/040257-05/$15.00 c© 2011 Optical Society of America 257

( ( () ) )

FIG. 1. Specification of the shape of the perimeter of the working aperture: (a) circular, (b) oval, (c) polygonal.

The specification of the contour includes specifying theparameters of the item and a grid. The present version ofthe program provides four types of contours—circular, oval,rectangular, and polygonal (Fig. 1).

The program uses a tabular specification of the deviationsof the topography as a function of the surface coordinates ofthe item. The deviations are determined at selected regularsurface points called nodal points of the topographic grid. Thetopographic grid is determined by the characteristic spacingbetween the points, called the grid step. The specification ofthe grid is introduced for compatibility with the ADK processprogram for automatic shaping. The grid uses the ADK rulesto form a set of nodes on the surface of the item at which thedeviations of the actual surface from the comparison surfaceare determined.

The interferogram of the wave front to be analyzedis recorded from the interferometer by a digital camerain graphic format *.bmp or *.jpg. The interferogram isreduced to standardized form by using functions that rotatethe interferogram by an arbitrary angle or by a multiple of 90◦,that filter the noise on the image of the interferogram, and thatincrease the image contrast.

The configuration of the outer perimeter of the workingaperture is then determined in order to distinguish on theinterferogram image the region to be processed and inorder to prompt the program to differentiate the image intothose regions that belong only to the item and only to thesurrounding background (Fig. 1).

In the dialog regime, the size of the contour of the item isspecified on the image of the interferogram. The slope of thecontour may be specified for items with square and rectangularcontours, since in general the orientation of the boundaries ofthe item may not coincide with the direction of the rows andcolumns of the image.

The filtering of the image of the interferogram includesspecifying the parameters of the filter and suppressing thenoise on the interferogram image. The noise filtering is carriedout only within the region indicated by the item’s contour.

A nonlinear spatial filter determined by the size and formof a nonlinear bell is used to suppress noise on the image(for filtering). The filter parameters are selected in the dialog

FIG. 2. Phase pattern in 2D format from a processed interferogram with thefollowing topographic parameters: rms dev. = 0.013 µm, P-V = 0.100 µm.

regime, depending on the specific form of the image. Using thecorrectly selected filter makes it possible to eliminate noise.

The INTPROC program makes it possible to create adata file for analyzing the item’s topography using theADK program with subtraction of the slope and defocusingof the monitored wave front (Fig. 2). To visualize thesurface, INTPROC makes it possible to construct a 3D mapin pseudocolors, reconstructing the phase map from theprocessed interferogram (Fig. 3).

The following quantities are introduced in the dialogregime: the wavelength of the source radiation at which theinterferogram to be analyzed was obtained, the factor of themonitoring system that shows how much the deviations of therecorded wave front differ from the deviations of the surfacebeing monitored, and the value of the image distortion.

The program operates in the automatic regime, inwhich all the operations are executed sequentially, usingcriteria defined in the manual regime, and forms a 3D mapin pseudocolors. In this case, when the calculations arecompleted, it is possible to turn to an arbitrary intermediate

258 J. Opt. Technol. 78 (4), April 2011 Gorshkov et al. 258

FIG. 3. Phase pattern in 3D format with the following topographicparameters: rms dev. = 0.013 µm, P-V = 0.100 µm.

object and repeat the calculation with other settings of thecriteria.

A BRIEF DESCRIPTION OF A METHOD OFRECONSTRUCTING A TOPOLOGICAL MAP OF A SURFACE

Two interference patterns are needed to implement thegiven method—one of the object to be monitored and one of areference comparison surface (a plane). The reference plane isused to eliminate the aberrations of the optical system, as wellas to determine the carrier frequency.

An interference pattern can be represented in the formof a diffraction grating. The spectrum of interferograms infringes of finite width has three clearly expressed peaks,which in optics are called the orders of diffraction. Usingthis terminology, it can be said that useful information iscontained in the+1 and−1 orders of diffraction. These ordersdiffer from the zeroth order by an amount equal to the carrierfrequency of the fringes in the interferograms (the number offringes in the interference field).

Figures 4 and 5 show the spectra of actual interferencepatterns. Because of the nonlinear nature of the recorder,the spectrum of these gratings contains higher orders ofdiffraction, which are easily seen on the figures presented here.

The phase-reconstruction algorithm by Fourier transfor-mation includes the following operations:3

1. Preprocessing of the interferograms, apodization.2. Direct Fourier transformation of the interferograms of the

object and the reference plane.3. Band-pass frequency filtering with discrimination of the+1 order of the Fourier transform of the object and thereference plane.

4. Inverse Fourier transformation of the spectra of the objectand the reference plane.

5. Multiplication of the complex components of the spectra ofthe object and the reference plane, obtained as a result offiltering.

6. Computation of the phase distribution as an argument of thecomplex image.

FIG. 4. Fourier spectrum of the interference image of a reference plane. A is+1 order, B is zeroth order, and C is −1 order.

FIG. 5. Fourier spectrum of the interference image of an object. A is+1 order,B is zeroth order, and C is −1 order.

7. Transformation of the phase into deviations of themonitored wave front from the reference comparison wavefront (topography), postprocessing.

MATHEMATICAL MODELLING OF THEPHASE-RECONSTRUCTION ACCURACY

To evaluate the accuracy of the phase reconstruction,an ideal interferogram was created (for wavelength λ =

0.6328 µm) in the form of an image of dimension 600 ×600 points, containing fifty absolutely straight horizontalinterference fringes. The image of the fringes was apodized bya 7th-order Blackman window to minimize the influence of theedge effect. The same shape was assumed for the window ofthe frequency bandpass transmission filter. In order to estimatehow random error affects the phase reconstruction, additivenoise was added to the image. Gaussian white noise whosespectral density amplitude was varied during the modellingwas selected as a model of the noise.

259 J. Opt. Technol. 78 (4), April 2011 Gorshkov et al. 259

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FIG. 6. Dependence of (a) the rms deviation and (b) the P-V amplitude of thereconstructed phase (k fractions of wavelength) on rms noise (in percentage ofthe signal amplitude α) in reconstruction regions of various widths. 1—Regionat the center of the interferogram, 2—region with departure from the edge of≈ 15 readings, 3—region without departure.

Phase reconstruction was carried out over three regionsof the interferogram image. The first region was at the centerof the image far from the edge, the second was taken witha spacing from the edge of 15 readings, and the third wasthe interferogram as a whole. The radius of the Fourier filterfor separating out the +1 order equalled 25 readings. Foreach reconstructed phase, values were calculated for the rmsdeviations and the P-V (peak-value) amplitude.

Figure 6 shows graphs of the dependences of the rmsdeviations and P-V amplitudes of the reconstructed phaseon the rms noise in the initial interferogram. It can be seenfrom the graphs that, the wider the region of reconstruction,the larger the resulting values of the rms deviation and P-Vamplitude. This obvious conclusion is associated with the factthat the edge effect substantially decreases from the edge to thecenter of the interferogram. On both graphs, curve 1 containsno values for the rms noise equal to zero. This is because, forthe given value, both the rms deviation and the P-V amplitudeof the reconstructed phase were close to zero at the centerof the interferogram. When the rms noise is equal to 10%of the amplitude of the interference fringes, the influence ofthe noise becomes dominant over the edge effect. Therefore,all three graphs merge into one, with rms dev. ≈ λ/2000,P-V ≈ λ/300 (for curves 1 and 2), and λ/100 (for curve 3). It

can be concluded from the results of the modelling that curve2 in Figs. 6(a) and 6(b) demonstrates the most suitable choiceof the reconstruction parameters.

The half-width of the band-pass transmission filter(i.e., the filtering window) in this case was selected inaccordance with the conclusions explained in Ref. 5—namely,the half-width of the filter’s transmission band must not exceedhalf the value of the carrier frequency. If the filter window istaken to be wider, a larger number of frequency components ofthe noise extends into the resulting phase, while the frequencytail of the adjacent orders begins to have an appreciableinfluence.

For a Fourier window with a radius of 60 counts, whenthe rms noise equals 10%, we get rms dev. ≈ λ/800 andP-V ≈ λ/50 for curve 2. These results illustrate the negativeconsequences of increasing the half-width of the band-passtransmission filter—i.e., of increasing the filtering window.

When the filter has a narrow transmission window, astrong limitation of the spectrum begins to have an effect. Inthis case, the edge effect stops being local and spreads towardthe center of the interference pattern, significantly increasingthe values of the rms deviation and the P-V amplitude in thiscase. For example, if the radius of the filter is taken to be 15readings, the values for curve 2 are rms dev. ≈ λ/1500 andP-V ≈ λ/50 for rms noise equal to 10%.

It was thus possible to formulate the requirements onthe parameters of the reconstruction algorithm to achievethe best accuracy. These are that the edges of the initialinterferograms must be apodized and that, in selecting thereconstruction region, one should withdraw 15–20 readingsfrom the edges of the interferogram. The half-width of theband-pass transmission filter must not exceed half the carrierfrequency of the interference fringes. The noise level in theimages of the initial interferograms should be reduced by allavailable means.

EXPERIMENTAL TESTING OF THE PROGRAM

Working tests of the program in order to check itsreliability and the convergence and repeatability of its resultswere carried out at the final stage of the processing of aworkpiece (rms dev. = 0.05–0.0125λ) in the automaticregime on actual interferograms obtained by means ofshop interferometers of Fizeau or Twyman–Green typeby comparing them with the results obtained using theINTODIGITAL program.5,6

The images of the interferograms were digitized andprocessed using the program. It should be pointed out that theactual interferograms had a significant noise level.

A flat �250-mm mirror made from silicon carbide wasused in the testing. The shaping process of the working surfaceof the mirror was carried out on an APD-250 machine withcomputerized control. The working surface of the mirror wasoptically monitored after each processing session by obtainingits topological map. Images with a size of 4272× 2848 pixelswere used for this work. The interferograms with vertical,horizontal, and diagonal fringes were investigated, the numberof fringes varying from 24 to 40.

The interferograms were interpreted by means of theINTPROC and INTODIGITAL programs. The results were a

260 J. Opt. Technol. 78 (4), April 2011 Gorshkov et al. 260

TABLE 1. Results of processing interferograms during the shaping of a�250-mm mirror made from silicon carbide using programs INTPROC andINTODIGITAL.

Parameters of the topographic map of the workingsurface of the mirror

INTPROC INTODIGITAL

Session No. rms dev., µm P-V, µm rms, µm P-V, µm

1 0.017 0.125 0.018 0.1742 0.015 0.136 0.016 0.1703 0.012 0.100 0.013 0.0994 0.011 0.088 0.011 0.1045 0.091 0.091 0.009 0.092

topographic map of the surface and the values of its errors:the P-V amplitude and the rms deviation. To compare theresults obtained when these two programs were used forinterpretation, the file of topography data obtained usingthe INTPROC program was loaded into the ADK program,where the regular errors were subtracted: tilt, astigmatism,defocusing, and coma. The character of the distribution ofthe processing errors of the workpiece surface coincided[according to the P-V-amplitude and rms-deviation criteriaat the finishing stages of the workpiece (rms dev. =0.05–0.0125λ)]. The scatter of the error distribution was10−3 µm. The results of the interpretation are collected inTable 1.

CONCLUSION

A program has been developed that effectively processesinterference patterns with fringes of arbitrary direction. Thisallows the program to be used for technological purposes atthe stages of aspherization and lapping of aspheric surfaceswhen analyzing interferograms, and also makes it possibleto average a series of interferograms in order to reducethe influence of temperature gradients and air flows in themeasurement channel.

The use of built-in filters makes it possible to use theprogram to process extremely noisy interference patterns (ad-ditional parasitic interference, an image of the autocollimationpoints of the optical system of an interferometer, interference

spectra in an interference field being analyzed). This makesit efficient to obtain interferometric information concerningthe processing quality of optical surfaces, which is especiallyimportant when they are being lapped.

Testing of the program and a comparison of thetopographic maps with those calculated from other programsshows high convergence of the results, and this confirms thatthe program operates dependably and reliably. The method-ological error of this method corresponds to 0.003–0.002λaccording to the rms-deviation criterion.

It would be suitable to further improve this technologicalprogram in order to increase the range of the rms deviationsthat can be investigated when the shape errors of the testsurfaces increase and to supplement the analysis of theparameters of the wave front being studied in order to use itas a certification process.

In conclusion, the authors express gratitude to V. A.Zhavoronkov for providing interference measurements whileprocessing interferograms using the INTODIGITAL programduring the shaping of a flat �250 mirror from silicon carbide.

a)Email: optikal@npooptica.rub)Email: aglomakin@gmail.ru

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6M. Takeda, Q. Gu, M. Kinoshita, H. Takai, and Y. Takahashi, “Frequency-multiplex Fourier-transform profilometry: A single-shot three-dimensionalshape measurement of objects with large height discontinuities and/orsurface isolations,” Appl. Opt. 36, 5347 (1997).

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