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High accuracy Fourier transform fringe pattern analysis
Optics and Lasers in Engineering 14 (1991) 325—339
______
High Accuracy Fourier Transform Fringe Pattern Analysis
M. Kujawinska & J. Wójciak Institute of Design of Precise and Optical Instruments, Warsaw University of Technology, 8 Chodkiewicza Str., 02-525 Warsaw, Poland
ABSTRACT The principle of the Fourier transform method offringe pattern analysis is described with emphasis on the refinements for high accuracy measurements. Characteristics of the main sources of errors of the retrieved phase in respect to the interferogram features and the modified software procedure are given. The quantitative results of the analysis of computer-generated and real interferograms are presented. The possibility of obtaining the overall accuracy of Al 100 is proved.
1 INTRODUCTION Two different approaches are taken to phase measuring methods for fringe pattern analysis. The first one deals with the temporal phase measurement methods (TPM), defined as methods which combine three or more intensity measurements from a time series of images or points to provide a direct measurement of the phase distribution of an interferogram. The TPM include the phase shifting method,”2 the heterodyning method3 and phase locking interferometry.4 The most popular method in the majority of recent applications and commercial instruments is the phase shifting method. In spatial phase measurement methods (SPM) all the information necessary to reduce the interferogram to a phase map is recorded simultaneously. The information can be retrieved from one fringe pattern with a spatial carrier (Fourier transform method ETM5’6) or 325
Optics and Lasers in Engineering 0143-8166/91/$03.50 © 1991 Elsevier Science Publishers Ltd, England. Printed in Northern Ireland
326
M. Kujawinska, J. Wójciak
from three phase-shifted interferograms separated in space (spatial phase-stepped method, SPSM7). Considering the application of the fringe pattern analysis methods in high accuracy measurements performed by commercially available systems, two methods are preferable: the phase shifting PSM and Fourier transform methods. Several advantages and disadvantages of each technique are based upon available computer power, analysis time, feedback to fringe generator (i.e. the possibility of modifying the experimental arrangement), costs, etc. Relative advantages and disadvantages of the spatial technique in comparison with the temporal one may be summarized as follows: Advantages: 1.
2.
Only one interferogram is needed, while the TPM requires multiple interferograms taken at different times (this enables the analysis of transient events). No special devices for carrier generation are needed whereas the temporal technique requires a phase shifting device.
Disadvantages: 1. 2. 3.
The detector array must have a higher spatial resolution than in the TPM. The spatial distribution of the detector sensitivity must be uniform over the array. SPM requires more sophisticated processing (longer computational time).
The features of FTM and TPM supplement each other, although until now, most commercial systems were based on the phase shifting method. The recent rapid development in the fabrication technology for high resolution image sensors and the implementation of pipe-line processing in personal computers seems to be changing the situation in favour of the Fourier transform technique. However, the FIM, as based on a global (not point-wise) operation and using FF1 carries several traps for the unwary user. Careless operation with the spatial frequency space, where all the signals are mixed up, may cause the loss of accuracy or even significant modification of information. Here we focus on some of the errors that are particular to the spatial carrier (FTM) technique and ways to avoid or decrease their influence on the accuracy of the retrieved phase. This paper reviews the Fourier fringe analysis technique for the reconstruction of surfaces underlying optically generated contour maps.
High accuracy Fourier transform fringe pattern analysis
327
2 A REVIEW OF THE TECHNIQUE
The prerequisite for using the Fourier transform method is the addition of a phase term to the interferogram which is the linear function of co-ordinates, e.g. by tilting one of the mirrors in the interferometer. The general form of the intensity pattern after its recording by a detector is given by: 1(r)
D(r)[A(r) + ~C~(r)] + N(r)
(1)
where r is the position vector of the point (x, y) in the interferogram, C~(r)= 0.5B~(r)exp [in(2~rf0r + 43(r))]
(2)
n is the integer value and f0 direction of a carrier frequency, 43(r) is the
unknown phase. The interferogram extends over a domain D(r) defined as: ~r1 Ji 1.0
inside the outside theinterferogram interferogram
—
3
A(r) is the background intensity distribution, B~(r)is the nth harmonic amplitude of the local contrast of the pattern, N(r) is the random noise term. If the fringe pattern is Fourier transformed it becomes: i(f)
=
d(f) ® [a(f) + ~ c~(f)]+ n(f)
(4)
where f is the position vector of spatial frequencies. It has to be assumed that: —the background and contrast function are slowly varying —the spatial spectrum of C1(r) does not overlap in the Fourier plane with spectra of A(r) and higher harmonics of C(r), so that only the c1(f) spectral component may be filtered and taken for the further analysis. The spectrum term c1(f) is selected and translated by f0 on the frequency axis toward the origin to take out the carrier heterodyning. Applying the Fourier transform to c1(f), C1(r) is obtained and the phase function may be calculated 43(r)
=
Im [C1(r)] arctan (Re [C,(r)})
where Re denotes the real, and Im the imaginary part of C1(r).
(5)
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M. Kujawinska, J. Wójciak
The phase is obtained in the form of phase fringes (43 modulo [2111) and has to be unwrapped in order to obtain a continuous phase function. This may be done by scanning through all the data and adding or subtracting 211 every time a discontinuity is detected. The FTM described above proposed by Takeda et a!. ~was extended by Macy5 to handle two-dimensional data. Recently Bone er al., Roddier and Roddier1°and Kujawinska’1 presented several refinements of FTM. The method can be modified, as proposed by Kreis,’ to obtain a version which does not use a high spatial carrier frequency in the fringe pattern being analysed. But, in order to avoid the sign ambiguity, the technique requires two phase shifted interferograms. Here we consider the case of a Fourier transform method which needs only one interferogram to determine the full phase distribution, and may be used to analyse fringe patterns obtained in static and dynamic processes, as well as those recorded on photographic films. The procedure proposed should enable the automatic analysis of a fringe pattern obtained in an interferometer for testing optical elements. The accuracy of the retrieved, slowly varying phase within the whole domain of an interferogram should be about A/lOt).
3 SOME ERROR SOURCES In the Fourier transform method of fringe pattern analysis several sources of errors appear.~’~”1’ ~ These include: —the errors associated with the use of the FF1, namely: 1. 2. 3.
aliasing, if the sampling frequency is too low the picket fence effect, if the analysed wave front includes a frequency which is not one of the discrete frequencies leakage of energy from one frequency into adjacent ones, owing to fringe discontinuity (inappropriate truncation of the data)
—the errors due to incorrect filtering in the Fourier space, especially if nonlinear recording of a fringe pattern has to be considered —the influence of random noise and spurious fringes in the interferogram —the errors due to incorrect determination of the domain causing errors in the unwrapping and fringe extrapolation (if used) procedures.
High accuracy Fourier transform fringe pattern analysis
329
The possibility of eliminating or decreasing the above mentioned sources of errors depends on: —the features of the interferogram recorded and analysed (the carrier frequency, the spatial bandwidths of the background, the information content, ~ c~(f) and the gradients of the phase function 43(r), the signal-to-noise ratio (SIN), the complexity of the interferogram domain, as well as the digitization of the image) —the modifications of the basic procedure of FTM.5 The recognition of the features of an interferogram may be based on the assessment of the spatial spectrum of the fringe pattern.16 They can also be determined one-by-one by the methods described in several works.8’9~1”7”8 If the basic assumptions for FTM are fulfilled, the detector has sufficient resolution and the domain of an interferogram is correctly determined, the error in the retrieved phase may be estimated. Frankowski et al.’3 showed that the influence of the quantization errors is negligible (for 6 bits the phase error X/1000). The expectation value of the rms phase error due to random noise in an interferogram is given by6’9 E(ô43rms)
=
(O~l2)”2ô~/(Jrm)
(6)
where a~(= n/N) is the ratio of the number of spectral sample points, n, within the filter passband in frequency domain to the number of sample points, N; ô~is the rms value of the noise; m is the mean modulation amplitude. Bone et al.9 showed that, with an optimum filter window, the errors due to the random noise included in the filter window are approximately equal to the errors from information components lost from the filter window. These two sources of error account for most of the errors in the retrieved phase (we do not consider now the errors due to inappropriate truncation of data). This then provides an estimate of the error in the retrieved phase under optimum filtering conditions: (‘5ørms)est
=
(2cv)”2ô~/(~rm)
(7)
The accuracy of the retrieved phase may be increased by implementing several refinements of the fringe pattern analysis technique of Takeda et a!. specifically: ,~
—using a 2-D Fourier transform which permits better discrimination between wanted and unwanted components —using a 2-D analog of the Hamming window and the extension of data over the whole sampling area, which reduce significantly the
330
M. Kujawinska, J. Wójciak
errors due to inappropriate truncation of data~”and enable one to obtain accurate values of phase over the full data window —removing the heterodyning by subtracting the term 2jrf0r after the 9ls (if f,, is known and may be accurately whole phasefrom retrieval determined a region of unperturbed fringes) or by removing the total linear term” (if it is determined by the least-square method) —introducing the automatic procedures for the estimation of an interferogram domain including:’0” —a circular domain whose size is determined on the basis of a chosen threshold level and a binary filter (used mostly for optical element testing) —an arbitrary domain by thresholding the fringe contrast function (additional forward and inverse FT is required) —an arbitrary domain determined on the basis of two interferograms with different arbitrary phase shift; by thresholding the modulation —subtracting the errors caused by the experimental arrangement (field distortions, residual errors in the alignment, etc). These errors can be removed from the retrieved phase data provided this is done before the heterodyning is removed.
4 COMPUTER SIMULATIONS Several works consider, in various ways, the problem of errors in the retrieved phase. However, there is still not enough knowledge about the distribution of the error in the analysed domain under various modifications of the FTM applied and owing to various features of an interferogram. In high accuracy optical testing and other applications it is not enough to estimate an overall error in the retrieved phase but it is also necessary to know where the high accuracy results can be achieved. As the criterion for the assessment of the accuracy of the retrieved phase, two values are usually taken: 1.
Peak-to-valley value, P/V, i.e. the difference between the maximum and minimum error values. This parameter, in the presence of big boundary, errors, is not a good estimate of the accuracy achieved. After cutting off the boundary region, it may give the real estimate of the local phase accuracy
High accuracy Fourier transform fringe pattern analysis 2.
331
Root mean square deviation of the calculated phase from the theoretical one. This parameter is less sensitive for the boundary errors and gives the global information about the accuracy of the retrieved phase.
The detailed accuracy considerations are connected with the optical flatness measurement and aim at determination of the procedure for this measurement with accuracy better than A/50. The tests were performed under the following assumptions: —the analysed phase 43(x, y) <3jr and is slowly varying —FFT is performed for resolution 128 x 128 —the optimum carrier frequency equals 16—17 fringes in the circular domain with the radius R0 = 60 pixels —a random noise with the standard deviation of 12% of the mean value is present in the fringe pattern —the spectrum filtering window is square with the size equal to ~ of the carrier frequency. The error for the retrieved phase was determined for various variants of the algorithm applied: —Using the data without modification, multiplied by the Hamming window or both extrapolated over the whole sampling area and multiplied by the Hamming window. The extrapolation is performed by copying the boundary of the domain in the direction of the fringes (according to the direction of vector f~)and through adding linear fringes with the previously chosen11carrier The frequency algorithm in the direction perpendicular to the fringes. maintains the continuity of fringes and the level of the contrast and background in the considered region. In the case of low signal-tonoise ratio the data are smoothed at the boundary of the fringe pattern in order to avoid serious errors in the fringe extrapolation procedure. —Using various filtration windows: circular, square, and thresholdbased windows, of comparable size. Further variations refer to modifications of the interferogram features and are implemented by: —Introducing a spherical phase function 43(x, y) = w(x2 + y2). The function is modified by varying the coefficient w. The tests were performed for a spherical function with P/V = 0, 100, 300 and 400 nm.
332
M. Kujawinska, J. Wójciak
—Introducing the background Gaussian function A(x, y)
where A,
=
A0 exp {—[(x + x0)2 + (y + y0)2]/w~}
(8)
1 is the maximum of the background function, w,, is a parameter to modify the background function and x0, y,, are the shift of the co-ordinates x, y. By varying the described parameters, different reductions of intensity across the domain are achieved. The errors were tested for the cases of decreasing the intensity at the boundary of the domain to O~2A0,0~5A,,and O•8A~. —Introducing spurious fringes in the form of additive noise (Airy diffraction patterns).
An example of a computer-generated interferogram with spurious fringes and its extrapolated version are shown in Fig. 1(a) and (b), respectively. The intensity in an interferogram is calculated according to the equation: I(x,y)=D(x,y)A(x,y)[1
+cos(2~rf~y+43(x,y)]+N(x,y)
(9)
This form of intensity (using the background function of the form given by eqn (8)) describes in the best way the intensity distribution which is obtained in a Fizeau interferometer used for flatness testing. The error distribution is calculated as the difference: A43(x, y) = 43(x, y)
—
43,(x, y)
(10)
where 43~(x,y) is the theoretical phase value. The P/V and rms values of the error distribution are calculated for the whole circular domain, D(x, y), and for the domain reduced by cutting off 5, 10, 15, 20 and 25% of its radius, R0. This procedure enables one to track the changes in the estimates of the phase accuracy. It will also show the area within which the influence of the edge errors can be negligible. The series of diagrams show the changes of P/V
(a) (h) Fig. 1. An example of a computer-generated interferogram with spurious fringes (a) and its extrapolated version (h).
High accuracy Fourier transform fringe pattern analysis
333
and rms values of the error distribution as a function of phase domain reduction, specifically: —the comparison between the analyses of the unmodified and the modified data (Fig. 2(a)) —the influence of the background variation on the error distribution for phase function with P/V = 0 (Fig. 2(b)) —the comparison between the error distributions for various phase functions (Fig. 2(c)). P-v [nrnl RMS
~m] 70
60
6
50 5
(a) Fig. 2. The diagrams of P/V and rms values of the phase error distribution calculated for sequential reduction of the phase domain. (a) Comparison between the results of the analyses of unmodified and modified data: X, no data modification;•, multiplication by HW; 0, extrapolation of data and multiplication by HW.
334
M. Kujawinska, J. Wójciak
P-v
{nml
RMS
[nm] 70
7
30
(b) Fig. 2—contd. (b) The influence of the background variation for the phase functions with P/V = 0 nm. The diagrams are given for decreases of the intensity at the boundary of the domain to x—0~2A,,,S—tJ~5A and 0—O’8A.
The last two series of investigations were performed using data modified by multiplication with the Hamming window, but without data extension. Additionally, the influence of the shape of the filtering window and asymmetry in both the background and contrast function distributions are checked. Figure 3 shows the influence of diffraction noise in the form of spurious fringes. The Airy diffraction pattern introduced (as shown in Fig. 1(a)) causes significant local change of P/V error function (P/V 15 nm), indicated by the arrow in Fig. 3. The analysis of the results obtained gives several conclusions which may be helpful for quantitative, high accuracy measurements: —Applying both extension of the data and a 2-D Hamming window, HW, leads to the most accurate results.
High accuracy Fourier transform fringe pattern analysis p.’~,
RMS
1r~m~
(nm]
70
7
60
6
335
400nm
400nm 50 300nm 40
4
lOOnm
Fig. 2—contd. (c) The comparison between (c) the error distribution for various phase functions (P/V 100 (0), 300 (•), 400nm (x)).
—The edge error propagation is usually significant (10% or 20% of the maximum fringe shift for HW + data extension and HW only, respectively). If accuracy better than A/50 is required, the phase within the domain with radius 0~85only should be analyzed and the results for both types of data modification are similar. —The influence of the background variation is relatively significant in the case of small phase values, while it has secondary influence for bigger values (43 > ~r/4). In certain cases it may cause a decrease in the edge errors due to better local continuation of the fringe pattern. —The contrast variations in a fringe pattern up to 50% usually do not increase the phase errors. However, a more significant reduction of -—-
336
M. Kujawinska, J. Wójciak
l5iim
-,
~‘
~
~
~
:~
~
~‘
—~
_CC..•
~
II
,.. -. ..‘.
...—~I .-~
~
~ -
I
~
r~
,.-.
.‘
~~ *
.~ ,.
V
I1~.. .,,--,-
.‘
I I
I
~r
:-~~
r”
~.
I
__. ~..—
I
I
—~ —
-:~- -~~‘-
,~— ~.. ..‘-‘.-
~
Fig. 3. The 3-D phase error map obtained with the domain reduced by 10 pixels for
the phase retrieved from an interferogram with spurious fringes (see Fig. 1(a)). The arrow indicates the area of increased local P/V error due to the spurious fringes.
contrast may cause (especially in the presence of high random noise) a dramatic increase of the edge errors. —While the optimum size of the filter is chosen, its shape has a secondary influence on the accuracy of the phase retrieval. —The influence of the spurious fringes depends on their spatial frequency, as well as all the beat frequencies which are formed by interaction between the fringe pattern and the additional sets of fringes. Of course, the values of the phase errors will vary for different S/N ratios, phase and background functions, and domain complexity. However, the main tendencies in their distribution will remain similar.
5 REAL FRINGE PATTERN ANALYSIS
The advantage of FTM is the possibility of using only one fringe pattern for full analysis of the interferogram. However, in the case of varying of environmental conditions (vibrations, temperature or illumination
High accuracy Fourier transform fringe pattern analysis
337
TABLE 1 The Results of the Phase Retrieval from the Series of Interferograms
Domain reduction (%)
0
10
20
Interferogram
P/V”
rm?
P/V
rms
P/V
rms
Ii
97
1F7
52
7~8
35
6~6
12
86
11~3
50
7~8
36
6~8
13 14 15
88 89 91
11~1 11•0 10~7
45 49 53
7.9 84 8~3
38 41 41
6~8 7~1 7~1
IAV
76
10~6
44
7.7
34
6~3
P/V and rms values in nm.
changes, etc.) the retrieved phase from a sequence of interferograms will vary. The overall accuracy of the measurement may be improved by averaging the results of n interferograms. The interferograms obtained from a Fizeau interferometer (43 180 mm diameter) used for optical flatness testing have features similar to those simulated by computer. Table 1 gives the results of P/V and rms of retrieved phase obtained for the series of five interferograms and their average (for the domain described by radius R0, 0~9R0,0.8R0). The main causes of the differences between the results for 11—15 are the environmental instabilities (additional phase). The other source of differences is different phase error distributions due to the fringe shift (vibrations) or random noise distribution. It is obvious that there are two ways to achieve, in real measurements, accuracy similar to that obtained for computer generated fringes, specifically: —increasing the stability of the arrangement —averaging several results in order to remove the influence of the random environmental instabilities. Figure 4 shows the 3-D plot of the averaged phase. The results compare well (for the domain within radius 0~8R0)with those obtained 9 by the traditional fringe tracking method used for flatness testing.’ 6 CONCLUSIONS This paper has reviewed the main sources of errors in the Fourier transform method of fringe pattern analysis and has shown the possible
338
M. Kujawinska. J. Wójciak ~_—r
r
—•~.
~
~
--, —.—— — -
-~~~-‘ --‘~-~
I
.
.~.
~
I
.~
•‘
-
.•~
Fig. 4. The 3-D plot of the retrieved shape of an optical flat (the average of 5 interferograms).
refinements of this technique in order to obtain high accuracy of the retrieved phase. The influence of the features of an interferogram is strongly emphasized. For data with random noise with a standard deviation of 20% of the mean value and contrast variations less than 50%, we have shown that it is possible to retrieve the phase with an rms error better than 1% of the maximum phase shift within the domain of radius 0~9.For small phases (43 <.~r)the overall accuracy of A/100 may be achieved within the same area. With the development of high resolution image sensors and the implementation of pipe-line processing in personal computers, FTM seems to be fully recommended for commercial systems of fringe pattern analysis.
REFERENCES
1. Creath, K., Phase-measurement interferometry techniques. In Progress in Optics, ed. E. Wolf. Elsevier Science Publishers. Amsterdam, (1988) 351—93.
2. Schwider, J., Elssner, K.-E., Burow, R. & Spolaczyk, R., Real-time
methods in optical testing interferometry. ZOS report 86/6, 1988, pp. 1—91.
High accuracy Fourier transform fringe pattern analysis
339
3. See for example: Massie, N. A., Nelson, R. D. & Holly, S., High performance real-time heterodyne interferometry. AppI. Opt., 18 (1979) 1797—803. 4. Johnson, G. W., Leiner, D. C. & Moore, D. T., Phase-locked interferometry. Proc. SPIE, 126 (1977) 152—60. 5. Takeda, M., Ina, H. & Kobayashi, S., Fourier-transform method of fringe pattern analysis for computer-based topography and interferometry. J. Opt. Soc. Am., 72 (1981) 156—60.
6. Takeda, M., Spatial-carrier fringe pattern analysis and its application in precision interferometry and profilometry: An Overview, industrial Metrology, 1 (1990) 79—99. 7. See for example: Kwon, 0. Y. & Sough, D. M., Multichannel grating phase-shift interferometers. Proc. SPIE, 599 (1985) 273—9. 8. Macy, W. W., Jr, Two-dimensional fringe pattern analysis. AppI. Opt., 22
(1983) 3898—902. 9. Bone, D. J., Bachor, H. A. & Sandeman, R. J., Fringe-pattern analysis using a 2-D Fourier transform. AppI. Opt., 25 (1986) 1653—60.
10. Roddier, C. & Roddier, F., Interferogram analysis using Fourier transform techniques. AppI. Opt., 26 (1987) 1668—73. 11. Kujawiñska, M., Spik, A. & Wojciak, J., Fringe pattern analysis using Fourier transform techniques. Proc. SPIE, 1121 (1989) 130—6. 12. Kreis, T., Digital holographic interference-phase measurement using the Fourier transform method. J. Opt. Soc. Am., 3 (1986) 847—55. 13. Frankowski, G., Stobbe, I., Tischer, W. & Schillke, F., Investigation of surface shapes using carrier frequency based analysing systems. Proc. SPIE, 1122 (1989) 89—100. 14. Malcom, A. A., Burton, D. R. & Lalor, M. J., A study of the effects of
windowing on the accuracy of surface measurements obtained from the Fourier analysis of fringe pattern. Proc. of FASIG Conf. Fringe Analysis ‘89, University of Technology, Loughborough, 4—5 April 1989. 15. Green, R. J., Walker, J. G. & Robinson, D. W., Investigation of the
Fourier transform method of fringe pattern analysis. Opt. and Lasers in Eng., 8 (1988) 29—44.
16. Choudry, A. & Kujawiñska, M., Fourier transform method for the automated analysis of fringe pattern. Proc. SPIE, 1135 (1989) 113—18. 17. Takeda, M. & Mutoh, K., Fourier transform profilometry for the automatic measurement of 3-D object shapes. App!. Opt., 22 (1983) 3977—82. 18. Nuget, K. A., Interferogram analysis using an accurate fully automatic algorithm. App!. Opt., 24 (1985) 3101—5. 19. Dew, G. D., The measurement of optical flatness. J. Sci. instrum., 34 (1966) 4094—6.
______
High Accuracy Fourier Transform Fringe Pattern Analysis
M. Kujawinska & J. Wójciak Institute of Design of Precise and Optical Instruments, Warsaw University of Technology, 8 Chodkiewicza Str., 02-525 Warsaw, Poland
ABSTRACT The principle of the Fourier transform method offringe pattern analysis is described with emphasis on the refinements for high accuracy measurements. Characteristics of the main sources of errors of the retrieved phase in respect to the interferogram features and the modified software procedure are given. The quantitative results of the analysis of computer-generated and real interferograms are presented. The possibility of obtaining the overall accuracy of Al 100 is proved.
1 INTRODUCTION Two different approaches are taken to phase measuring methods for fringe pattern analysis. The first one deals with the temporal phase measurement methods (TPM), defined as methods which combine three or more intensity measurements from a time series of images or points to provide a direct measurement of the phase distribution of an interferogram. The TPM include the phase shifting method,”2 the heterodyning method3 and phase locking interferometry.4 The most popular method in the majority of recent applications and commercial instruments is the phase shifting method. In spatial phase measurement methods (SPM) all the information necessary to reduce the interferogram to a phase map is recorded simultaneously. The information can be retrieved from one fringe pattern with a spatial carrier (Fourier transform method ETM5’6) or 325
Optics and Lasers in Engineering 0143-8166/91/$03.50 © 1991 Elsevier Science Publishers Ltd, England. Printed in Northern Ireland
326
M. Kujawinska, J. Wójciak
from three phase-shifted interferograms separated in space (spatial phase-stepped method, SPSM7). Considering the application of the fringe pattern analysis methods in high accuracy measurements performed by commercially available systems, two methods are preferable: the phase shifting PSM and Fourier transform methods. Several advantages and disadvantages of each technique are based upon available computer power, analysis time, feedback to fringe generator (i.e. the possibility of modifying the experimental arrangement), costs, etc. Relative advantages and disadvantages of the spatial technique in comparison with the temporal one may be summarized as follows: Advantages: 1.
2.
Only one interferogram is needed, while the TPM requires multiple interferograms taken at different times (this enables the analysis of transient events). No special devices for carrier generation are needed whereas the temporal technique requires a phase shifting device.
Disadvantages: 1. 2. 3.
The detector array must have a higher spatial resolution than in the TPM. The spatial distribution of the detector sensitivity must be uniform over the array. SPM requires more sophisticated processing (longer computational time).
The features of FTM and TPM supplement each other, although until now, most commercial systems were based on the phase shifting method. The recent rapid development in the fabrication technology for high resolution image sensors and the implementation of pipe-line processing in personal computers seems to be changing the situation in favour of the Fourier transform technique. However, the FIM, as based on a global (not point-wise) operation and using FF1 carries several traps for the unwary user. Careless operation with the spatial frequency space, where all the signals are mixed up, may cause the loss of accuracy or even significant modification of information. Here we focus on some of the errors that are particular to the spatial carrier (FTM) technique and ways to avoid or decrease their influence on the accuracy of the retrieved phase. This paper reviews the Fourier fringe analysis technique for the reconstruction of surfaces underlying optically generated contour maps.
High accuracy Fourier transform fringe pattern analysis
327
2 A REVIEW OF THE TECHNIQUE
The prerequisite for using the Fourier transform method is the addition of a phase term to the interferogram which is the linear function of co-ordinates, e.g. by tilting one of the mirrors in the interferometer. The general form of the intensity pattern after its recording by a detector is given by: 1(r)
D(r)[A(r) + ~C~(r)] + N(r)
(1)
where r is the position vector of the point (x, y) in the interferogram, C~(r)= 0.5B~(r)exp [in(2~rf0r + 43(r))]
(2)
n is the integer value and f0 direction of a carrier frequency, 43(r) is the
unknown phase. The interferogram extends over a domain D(r) defined as: ~r1 Ji 1.0
inside the outside theinterferogram interferogram
—
3
A(r) is the background intensity distribution, B~(r)is the nth harmonic amplitude of the local contrast of the pattern, N(r) is the random noise term. If the fringe pattern is Fourier transformed it becomes: i(f)
=
d(f) ® [a(f) + ~ c~(f)]+ n(f)
(4)
where f is the position vector of spatial frequencies. It has to be assumed that: —the background and contrast function are slowly varying —the spatial spectrum of C1(r) does not overlap in the Fourier plane with spectra of A(r) and higher harmonics of C(r), so that only the c1(f) spectral component may be filtered and taken for the further analysis. The spectrum term c1(f) is selected and translated by f0 on the frequency axis toward the origin to take out the carrier heterodyning. Applying the Fourier transform to c1(f), C1(r) is obtained and the phase function may be calculated 43(r)
=
Im [C1(r)] arctan (Re [C,(r)})
where Re denotes the real, and Im the imaginary part of C1(r).
(5)
328
M. Kujawinska, J. Wójciak
The phase is obtained in the form of phase fringes (43 modulo [2111) and has to be unwrapped in order to obtain a continuous phase function. This may be done by scanning through all the data and adding or subtracting 211 every time a discontinuity is detected. The FTM described above proposed by Takeda et a!. ~was extended by Macy5 to handle two-dimensional data. Recently Bone er al., Roddier and Roddier1°and Kujawinska’1 presented several refinements of FTM. The method can be modified, as proposed by Kreis,’ to obtain a version which does not use a high spatial carrier frequency in the fringe pattern being analysed. But, in order to avoid the sign ambiguity, the technique requires two phase shifted interferograms. Here we consider the case of a Fourier transform method which needs only one interferogram to determine the full phase distribution, and may be used to analyse fringe patterns obtained in static and dynamic processes, as well as those recorded on photographic films. The procedure proposed should enable the automatic analysis of a fringe pattern obtained in an interferometer for testing optical elements. The accuracy of the retrieved, slowly varying phase within the whole domain of an interferogram should be about A/lOt).
3 SOME ERROR SOURCES In the Fourier transform method of fringe pattern analysis several sources of errors appear.~’~”1’ ~ These include: —the errors associated with the use of the FF1, namely: 1. 2. 3.
aliasing, if the sampling frequency is too low the picket fence effect, if the analysed wave front includes a frequency which is not one of the discrete frequencies leakage of energy from one frequency into adjacent ones, owing to fringe discontinuity (inappropriate truncation of the data)
—the errors due to incorrect filtering in the Fourier space, especially if nonlinear recording of a fringe pattern has to be considered —the influence of random noise and spurious fringes in the interferogram —the errors due to incorrect determination of the domain causing errors in the unwrapping and fringe extrapolation (if used) procedures.
High accuracy Fourier transform fringe pattern analysis
329
The possibility of eliminating or decreasing the above mentioned sources of errors depends on: —the features of the interferogram recorded and analysed (the carrier frequency, the spatial bandwidths of the background, the information content, ~ c~(f) and the gradients of the phase function 43(r), the signal-to-noise ratio (SIN), the complexity of the interferogram domain, as well as the digitization of the image) —the modifications of the basic procedure of FTM.5 The recognition of the features of an interferogram may be based on the assessment of the spatial spectrum of the fringe pattern.16 They can also be determined one-by-one by the methods described in several works.8’9~1”7”8 If the basic assumptions for FTM are fulfilled, the detector has sufficient resolution and the domain of an interferogram is correctly determined, the error in the retrieved phase may be estimated. Frankowski et al.’3 showed that the influence of the quantization errors is negligible (for 6 bits the phase error X/1000). The expectation value of the rms phase error due to random noise in an interferogram is given by6’9 E(ô43rms)
=
(O~l2)”2ô~/(Jrm)
(6)
where a~(= n/N) is the ratio of the number of spectral sample points, n, within the filter passband in frequency domain to the number of sample points, N; ô~is the rms value of the noise; m is the mean modulation amplitude. Bone et al.9 showed that, with an optimum filter window, the errors due to the random noise included in the filter window are approximately equal to the errors from information components lost from the filter window. These two sources of error account for most of the errors in the retrieved phase (we do not consider now the errors due to inappropriate truncation of data). This then provides an estimate of the error in the retrieved phase under optimum filtering conditions: (‘5ørms)est
=
(2cv)”2ô~/(~rm)
(7)
The accuracy of the retrieved phase may be increased by implementing several refinements of the fringe pattern analysis technique of Takeda et a!. specifically: ,~
—using a 2-D Fourier transform which permits better discrimination between wanted and unwanted components —using a 2-D analog of the Hamming window and the extension of data over the whole sampling area, which reduce significantly the
330
M. Kujawinska, J. Wójciak
errors due to inappropriate truncation of data~”and enable one to obtain accurate values of phase over the full data window —removing the heterodyning by subtracting the term 2jrf0r after the 9ls (if f,, is known and may be accurately whole phasefrom retrieval determined a region of unperturbed fringes) or by removing the total linear term” (if it is determined by the least-square method) —introducing the automatic procedures for the estimation of an interferogram domain including:’0” —a circular domain whose size is determined on the basis of a chosen threshold level and a binary filter (used mostly for optical element testing) —an arbitrary domain by thresholding the fringe contrast function (additional forward and inverse FT is required) —an arbitrary domain determined on the basis of two interferograms with different arbitrary phase shift; by thresholding the modulation —subtracting the errors caused by the experimental arrangement (field distortions, residual errors in the alignment, etc). These errors can be removed from the retrieved phase data provided this is done before the heterodyning is removed.
4 COMPUTER SIMULATIONS Several works consider, in various ways, the problem of errors in the retrieved phase. However, there is still not enough knowledge about the distribution of the error in the analysed domain under various modifications of the FTM applied and owing to various features of an interferogram. In high accuracy optical testing and other applications it is not enough to estimate an overall error in the retrieved phase but it is also necessary to know where the high accuracy results can be achieved. As the criterion for the assessment of the accuracy of the retrieved phase, two values are usually taken: 1.
Peak-to-valley value, P/V, i.e. the difference between the maximum and minimum error values. This parameter, in the presence of big boundary, errors, is not a good estimate of the accuracy achieved. After cutting off the boundary region, it may give the real estimate of the local phase accuracy
High accuracy Fourier transform fringe pattern analysis 2.
331
Root mean square deviation of the calculated phase from the theoretical one. This parameter is less sensitive for the boundary errors and gives the global information about the accuracy of the retrieved phase.
The detailed accuracy considerations are connected with the optical flatness measurement and aim at determination of the procedure for this measurement with accuracy better than A/50. The tests were performed under the following assumptions: —the analysed phase 43(x, y) <3jr and is slowly varying —FFT is performed for resolution 128 x 128 —the optimum carrier frequency equals 16—17 fringes in the circular domain with the radius R0 = 60 pixels —a random noise with the standard deviation of 12% of the mean value is present in the fringe pattern —the spectrum filtering window is square with the size equal to ~ of the carrier frequency. The error for the retrieved phase was determined for various variants of the algorithm applied: —Using the data without modification, multiplied by the Hamming window or both extrapolated over the whole sampling area and multiplied by the Hamming window. The extrapolation is performed by copying the boundary of the domain in the direction of the fringes (according to the direction of vector f~)and through adding linear fringes with the previously chosen11carrier The frequency algorithm in the direction perpendicular to the fringes. maintains the continuity of fringes and the level of the contrast and background in the considered region. In the case of low signal-tonoise ratio the data are smoothed at the boundary of the fringe pattern in order to avoid serious errors in the fringe extrapolation procedure. —Using various filtration windows: circular, square, and thresholdbased windows, of comparable size. Further variations refer to modifications of the interferogram features and are implemented by: —Introducing a spherical phase function 43(x, y) = w(x2 + y2). The function is modified by varying the coefficient w. The tests were performed for a spherical function with P/V = 0, 100, 300 and 400 nm.
332
M. Kujawinska, J. Wójciak
—Introducing the background Gaussian function A(x, y)
where A,
=
A0 exp {—[(x + x0)2 + (y + y0)2]/w~}
(8)
1 is the maximum of the background function, w,, is a parameter to modify the background function and x0, y,, are the shift of the co-ordinates x, y. By varying the described parameters, different reductions of intensity across the domain are achieved. The errors were tested for the cases of decreasing the intensity at the boundary of the domain to O~2A0,0~5A,,and O•8A~. —Introducing spurious fringes in the form of additive noise (Airy diffraction patterns).
An example of a computer-generated interferogram with spurious fringes and its extrapolated version are shown in Fig. 1(a) and (b), respectively. The intensity in an interferogram is calculated according to the equation: I(x,y)=D(x,y)A(x,y)[1
+cos(2~rf~y+43(x,y)]+N(x,y)
(9)
This form of intensity (using the background function of the form given by eqn (8)) describes in the best way the intensity distribution which is obtained in a Fizeau interferometer used for flatness testing. The error distribution is calculated as the difference: A43(x, y) = 43(x, y)
—
43,(x, y)
(10)
where 43~(x,y) is the theoretical phase value. The P/V and rms values of the error distribution are calculated for the whole circular domain, D(x, y), and for the domain reduced by cutting off 5, 10, 15, 20 and 25% of its radius, R0. This procedure enables one to track the changes in the estimates of the phase accuracy. It will also show the area within which the influence of the edge errors can be negligible. The series of diagrams show the changes of P/V
(a) (h) Fig. 1. An example of a computer-generated interferogram with spurious fringes (a) and its extrapolated version (h).
High accuracy Fourier transform fringe pattern analysis
333
and rms values of the error distribution as a function of phase domain reduction, specifically: —the comparison between the analyses of the unmodified and the modified data (Fig. 2(a)) —the influence of the background variation on the error distribution for phase function with P/V = 0 (Fig. 2(b)) —the comparison between the error distributions for various phase functions (Fig. 2(c)). P-v [nrnl RMS
~m] 70
60
6
50 5
(a) Fig. 2. The diagrams of P/V and rms values of the phase error distribution calculated for sequential reduction of the phase domain. (a) Comparison between the results of the analyses of unmodified and modified data: X, no data modification;•, multiplication by HW; 0, extrapolation of data and multiplication by HW.
334
M. Kujawinska, J. Wójciak
P-v
{nml
RMS
[nm] 70
7
30
(b) Fig. 2—contd. (b) The influence of the background variation for the phase functions with P/V = 0 nm. The diagrams are given for decreases of the intensity at the boundary of the domain to x—0~2A,,,S—tJ~5A and 0—O’8A.
The last two series of investigations were performed using data modified by multiplication with the Hamming window, but without data extension. Additionally, the influence of the shape of the filtering window and asymmetry in both the background and contrast function distributions are checked. Figure 3 shows the influence of diffraction noise in the form of spurious fringes. The Airy diffraction pattern introduced (as shown in Fig. 1(a)) causes significant local change of P/V error function (P/V 15 nm), indicated by the arrow in Fig. 3. The analysis of the results obtained gives several conclusions which may be helpful for quantitative, high accuracy measurements: —Applying both extension of the data and a 2-D Hamming window, HW, leads to the most accurate results.
High accuracy Fourier transform fringe pattern analysis p.’~,
RMS
1r~m~
(nm]
70
7
60
6
335
400nm
400nm 50 300nm 40
4
lOOnm
Fig. 2—contd. (c) The comparison between (c) the error distribution for various phase functions (P/V 100 (0), 300 (•), 400nm (x)).
—The edge error propagation is usually significant (10% or 20% of the maximum fringe shift for HW + data extension and HW only, respectively). If accuracy better than A/50 is required, the phase within the domain with radius 0~85only should be analyzed and the results for both types of data modification are similar. —The influence of the background variation is relatively significant in the case of small phase values, while it has secondary influence for bigger values (43 > ~r/4). In certain cases it may cause a decrease in the edge errors due to better local continuation of the fringe pattern. —The contrast variations in a fringe pattern up to 50% usually do not increase the phase errors. However, a more significant reduction of -—-
336
M. Kujawinska, J. Wójciak
l5iim
-,
~‘
~
~
~
:~
~
~‘
—~
_CC..•
~
II
,.. -. ..‘.
...—~I .-~
~
~ -
I
~
r~
,.-.
.‘
~~ *
.~ ,.
V
I1~.. .,,--,-
.‘
I I
I
~r
:-~~
r”
~.
I
__. ~..—
I
I
—~ —
-:~- -~~‘-
,~— ~.. ..‘-‘.-
~
Fig. 3. The 3-D phase error map obtained with the domain reduced by 10 pixels for
the phase retrieved from an interferogram with spurious fringes (see Fig. 1(a)). The arrow indicates the area of increased local P/V error due to the spurious fringes.
contrast may cause (especially in the presence of high random noise) a dramatic increase of the edge errors. —While the optimum size of the filter is chosen, its shape has a secondary influence on the accuracy of the phase retrieval. —The influence of the spurious fringes depends on their spatial frequency, as well as all the beat frequencies which are formed by interaction between the fringe pattern and the additional sets of fringes. Of course, the values of the phase errors will vary for different S/N ratios, phase and background functions, and domain complexity. However, the main tendencies in their distribution will remain similar.
5 REAL FRINGE PATTERN ANALYSIS
The advantage of FTM is the possibility of using only one fringe pattern for full analysis of the interferogram. However, in the case of varying of environmental conditions (vibrations, temperature or illumination
High accuracy Fourier transform fringe pattern analysis
337
TABLE 1 The Results of the Phase Retrieval from the Series of Interferograms
Domain reduction (%)
0
10
20
Interferogram
P/V”
rm?
P/V
rms
P/V
rms
Ii
97
1F7
52
7~8
35
6~6
12
86
11~3
50
7~8
36
6~8
13 14 15
88 89 91
11~1 11•0 10~7
45 49 53
7.9 84 8~3
38 41 41
6~8 7~1 7~1
IAV
76
10~6
44
7.7
34
6~3
P/V and rms values in nm.
changes, etc.) the retrieved phase from a sequence of interferograms will vary. The overall accuracy of the measurement may be improved by averaging the results of n interferograms. The interferograms obtained from a Fizeau interferometer (43 180 mm diameter) used for optical flatness testing have features similar to those simulated by computer. Table 1 gives the results of P/V and rms of retrieved phase obtained for the series of five interferograms and their average (for the domain described by radius R0, 0~9R0,0.8R0). The main causes of the differences between the results for 11—15 are the environmental instabilities (additional phase). The other source of differences is different phase error distributions due to the fringe shift (vibrations) or random noise distribution. It is obvious that there are two ways to achieve, in real measurements, accuracy similar to that obtained for computer generated fringes, specifically: —increasing the stability of the arrangement —averaging several results in order to remove the influence of the random environmental instabilities. Figure 4 shows the 3-D plot of the averaged phase. The results compare well (for the domain within radius 0~8R0)with those obtained 9 by the traditional fringe tracking method used for flatness testing.’ 6 CONCLUSIONS This paper has reviewed the main sources of errors in the Fourier transform method of fringe pattern analysis and has shown the possible
338
M. Kujawinska. J. Wójciak ~_—r
r
—•~.
~
~
--, —.—— — -
-~~~-‘ --‘~-~
I
.
.~.
~
I
.~
•‘
-
.•~
Fig. 4. The 3-D plot of the retrieved shape of an optical flat (the average of 5 interferograms).
refinements of this technique in order to obtain high accuracy of the retrieved phase. The influence of the features of an interferogram is strongly emphasized. For data with random noise with a standard deviation of 20% of the mean value and contrast variations less than 50%, we have shown that it is possible to retrieve the phase with an rms error better than 1% of the maximum phase shift within the domain of radius 0~9.For small phases (43 <.~r)the overall accuracy of A/100 may be achieved within the same area. With the development of high resolution image sensors and the implementation of pipe-line processing in personal computers, FTM seems to be fully recommended for commercial systems of fringe pattern analysis.
REFERENCES
1. Creath, K., Phase-measurement interferometry techniques. In Progress in Optics, ed. E. Wolf. Elsevier Science Publishers. Amsterdam, (1988) 351—93.
2. Schwider, J., Elssner, K.-E., Burow, R. & Spolaczyk, R., Real-time
methods in optical testing interferometry. ZOS report 86/6, 1988, pp. 1—91.
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3. See for example: Massie, N. A., Nelson, R. D. & Holly, S., High performance real-time heterodyne interferometry. AppI. Opt., 18 (1979) 1797—803. 4. Johnson, G. W., Leiner, D. C. & Moore, D. T., Phase-locked interferometry. Proc. SPIE, 126 (1977) 152—60. 5. Takeda, M., Ina, H. & Kobayashi, S., Fourier-transform method of fringe pattern analysis for computer-based topography and interferometry. J. Opt. Soc. Am., 72 (1981) 156—60.
6. Takeda, M., Spatial-carrier fringe pattern analysis and its application in precision interferometry and profilometry: An Overview, industrial Metrology, 1 (1990) 79—99. 7. See for example: Kwon, 0. Y. & Sough, D. M., Multichannel grating phase-shift interferometers. Proc. SPIE, 599 (1985) 273—9. 8. Macy, W. W., Jr, Two-dimensional fringe pattern analysis. AppI. Opt., 22
(1983) 3898—902. 9. Bone, D. J., Bachor, H. A. & Sandeman, R. J., Fringe-pattern analysis using a 2-D Fourier transform. AppI. Opt., 25 (1986) 1653—60.
10. Roddier, C. & Roddier, F., Interferogram analysis using Fourier transform techniques. AppI. Opt., 26 (1987) 1668—73. 11. Kujawiñska, M., Spik, A. & Wojciak, J., Fringe pattern analysis using Fourier transform techniques. Proc. SPIE, 1121 (1989) 130—6. 12. Kreis, T., Digital holographic interference-phase measurement using the Fourier transform method. J. Opt. Soc. Am., 3 (1986) 847—55. 13. Frankowski, G., Stobbe, I., Tischer, W. & Schillke, F., Investigation of surface shapes using carrier frequency based analysing systems. Proc. SPIE, 1122 (1989) 89—100. 14. Malcom, A. A., Burton, D. R. & Lalor, M. J., A study of the effects of
windowing on the accuracy of surface measurements obtained from the Fourier analysis of fringe pattern. Proc. of FASIG Conf. Fringe Analysis ‘89, University of Technology, Loughborough, 4—5 April 1989. 15. Green, R. J., Walker, J. G. & Robinson, D. W., Investigation of the
Fourier transform method of fringe pattern analysis. Opt. and Lasers in Eng., 8 (1988) 29—44.
16. Choudry, A. & Kujawiñska, M., Fourier transform method for the automated analysis of fringe pattern. Proc. SPIE, 1135 (1989) 113—18. 17. Takeda, M. & Mutoh, K., Fourier transform profilometry for the automatic measurement of 3-D object shapes. App!. Opt., 22 (1983) 3977—82. 18. Nuget, K. A., Interferogram analysis using an accurate fully automatic algorithm. App!. Opt., 24 (1985) 3101—5. 19. Dew, G. D., The measurement of optical flatness. J. Sci. instrum., 34 (1966) 4094—6.