- Email: [email protected]
Automated fringe analysis techniques in Japan
15 (1991) 79-91
Optics and Lasers in Engineering
Automated
Fringe Analysis Toyohiko
University of Tsukuba,
Techniques
in Japan
Yatagai
Institute of Applied Physics, Tsukuba,
Ibaraki 305, Japan
(Received 29 March 1989; accepted 17 October 1990)
ABSTRACT Research efforts for interferometric fringe analysis techniques and their applications in Japan are reviewed with an emphasis on recent developments. Digital fringe analysis techniques and theoretical approaches to phase measuring interferometry are described. Automatic fringe analysis of speckle interferometry are also presented.
1 INTRODUCTION Optical measurement techniques have been a key ingredient in greatly improving optical and precision engineering. Especially, stable and versatile interferometers have been developed and interferometric testing techniques have played very important roles. This paper reviews recent progress of interferometric testing and automated fringe analysis techniques in Japan. Much work in automatic phase detection and interferometric fringe analysis has been done, with relation to development of digital data processing equipments. Some engineering applications of interferometric testing have been investigated, including an automatic inspection system using holographic interferometer in an automobile industry, an aspheric surface testing system for micro-optics, a surface finishing testing system in high precision machining, a magnetic field measuring system using an electron holography, a flatness tester for VLSI wafers and optical discs, optical fiber sensors and so on. Fringe patterns generated by interferometric techniques have been used extensively in various fields of optical measurements. The fringe analyzing procedures, including fringe peak detection and fringe order determination, are tedious and time consuming. Research on automatic 79 Optics and Lasers in Engineering 0143-8166/91/$03.50
Ltd, England.
Printed in Northern
Ireland
0 1991 Elsevier Science Publishers
80
Toyohiko
Yatagai
fringe analysis and precision phase measuring techniques is very important in applying interferometric technique to practical uses. Various kinds of techniques have been investigated in Japan.
2 AUTOMATIC
FRINGE
ANALYSIS
2.1 Grey level fringe analysist’*‘i To estimate fringe peaks, the fringe density binarization technique is commonly used in many fringe analysis systems because of simple algorithms. On the other hand, we employ the grey level method in which local variation of the fringe density is mainly considered. This grey level method is sensitive to noise, but can detect peaks by processing only local areas smaller than those in the binary case. Software for the fringe analysis system based on the grey level analysis has been developed in modular form to make sequential execution of programs, of which typical procedure and program names are shown in Fig. 1. In order to extract fringe peaks in the grey level method, it is very important to diminish the influence of noise, including speckle noise. One of the most simplest and most effective methods is unweighted averaging: the grey level at each point (x, y) of the output pattern is the average of the grey levels in the input pattern over a neighbourhood of (x, y) as shown in Fig. 2(b) Usually, 3 x 3 pixels are a convenient size of the neighbourhood because of the noise size and the average width of the fringes. of extracting fringe central lines-i.e. The procedure fringe skeletons-consists of three steps: extracting fringe skeletons, thinning the skeletons, and eliminating false skeletons and connecting the disconnected skeletons. By using a 5 X 5 pixel matrix, we perform the peak detections with respect to the four directions: x, y, +45” and -45” directions. We adopt the skeletoning criterion that if the peak conditions for two of the four directions are satisfied, the central point of the 5 x 5 pixel matrix is the skeleton point, as shown in Fig. 2(c). In order to make fringe thinning, we use a modified Hilditch thinning algorithm for grey level skeleton patterns. Some of the isolated short skeleton lines are eliminated and disconnected fringe skeletons are connected by a man-machine interactive method. The result of the thinning procedure is shown in Fig. 2(d). Fringe order assignment is also performed by a man-machine interactive operation. At the first step of the fringe order determina-
Automated
fringe analysis techniques
in Japan
81
Fringe Data Input
lHashinn;;f+ngr
Data ]
Fr ingc Analysis SKLT THNNFR I Fringe Order Delcrminalion FRNOD, FRCOR FRPATC, FRTVPT Interpola\ion INTP, FRMSH L ’ Display of Analyzed
Data
GMFICE
Fig. 1.
Typical program flow for grey level fringe analysis.
tion, temporal fringe orders are automatically assigned, and then these temporal fringe orders are corrected using an errata list made by an operation. Fringe order is judged from the direction of fringe movement due to reference phase modifications by a piezo translator. Next, interpolation methods are employed to determine the fractional fringe orders of pixels between fringe skeletons. Figure 3 shows a perspective plot of fringe orders determined. The spline interpolation is effective for evaluating local surface irregularity, while the least-square interpolation is effective for global surface shape evaluation.
Toyohiko Yatagai
82
(4
(4
(cl
Fig. 2. Skeletoning and thinning of interferometric fringes: (a) Original interferometric fringe pattern; (b) skeletons of (a) in the case of detecting peaks for one of the four directions; (c) skeleton of (a) in the case of detecting peaks for two of four directions and (d) thinned skeletons of (c) with the area outside the boundary removed.
Fig. 3.
Example
of 3-D plot of interpolated fringe order, pattern is shown in Fig. 2(a).
of which
original
fringe
Automated fringe analysis techniques in Japan
83
2.2 Binary fringe analysis3
In a binary version of fringe analysis, we can employ a special-purpose hardware for digital image processing. A flow diagram of the fringe analysis is shown in Fig. 4. Procedures shown in Fig. 4 are carried out by means of the hardware modules. At the first step, the original interferometric fringe pattern is masked with the standard mask pattern so that the outside area of the wafer is excluded. The masked fringe pattern is subjected to contrast enhancement. In order to reduce the pepper-and-salt noise in the outside area of the fringe pattern, a level sliced wafer pattern is superimposed on the masked fringe pattern before contrast enhancement. The masked fringe pattern is binarized after contrast enhancement. Since the outside analyzing area is indeterminate, the standard mask pattern is superimposed on the binarized fringe pattern. To make a thinning of the dark parts of interferometric fringes, the fringe contrast is inverted. Then the inverted fringe pattern is subjected to thinning. In the next step of the fringe analysis, the fringe order is assigned. Since the sufficient tilt introduced removes the fringe order ambiguity, the fringes are numbered in a certain order. The fringe ordering procedure is performed by the function of region labelling of the hardware module. Level slice
Mask pattern
Original image
Enhancement
Binarization ndetrmlnatr
Inverse
Thinning
Fig. 4.
Fringe
analysis
procedures
for binary fringe analysis.
Toyohiko Yatagai
84
3 PHASE
MEASURING
INTERFEROMETRY
In the phase measuring interferometry, a bias phase term is introduced. fringe pattern with the bias phase 6 Suppose the interference introduced can be written as; f(x,
y, 6) =
a(x, y) + b(x, Y) cm b#a Y> - 61
(1)
where @(x, y) is the phase to be evaluated and a(x, y) and b(x, y) are the average fringe intensity and the fringe contrast, respectively. From the intensity variation of Z(x, y, S) due to the change of the bias phase 6, the phase +(x, y) is evaluated without the influence of the average fringe intensity and the fringe contrast. The phase measuring method is divided into three classes; the heterodyne method, the fringe scanning method and the carrier fringe analysis method.4 3.1 Heterodyne
method
In the heterodyne method, the bias phase 6 is introduced by using the beat effect of two wavefronts with slightly different frequencies. That is, the bias phase is given by 6=2xAvt
(2)
where Av is the frequency difference. Equation (2) means that the bias phase is introduced in the time domain in the heterodyne interferometry. The phase 4 to be estimated is measured by using an electronic phase meter in this case. The electronic phase meter usually has a high accuracy in phase estimation. 3.2 Fringe scanning method’ In the fringe scanning method, one of the mirrors is stepwisely moved through half of the wavelength so that the relative phase S, of the interferogram is changed; 6, = 2~rnlN
(n = 1, 2, . . . ,N - 1)
(3)
where N denotes the number of mirror movement. The irradiance at each point in the interference pattern goes through one cycle of periodic variation. The computer determines a best-fit sine function for the irradiance versus the amount of phase shift at each point of the interference pattern. The phase of the best-fit function is a direct measure of the test wavefront.
85
Automated fringe analysis techniques in Japan
According to the fringe scanning phase detection tions with sinusoidal weights,
principle,
summa-
c = c Z(X, y, 6,) cos (2n7GnlN)
(4)
s = C Z(X, y, 6,) sin (2~rnlZV)
(5)
are calculated to extract sinusoidal parts of intensity phases of the interferogram are given by
variation.
$(x, y) = arctan (s/c)
The (6)
The calculated values of arctangent are wrapped between ~JC radians. The unwrapped phase value gives the correct shape corresponding to the phase profile. 3.3 Carrier fringe analysis in real domain6 In the carrier fringe analysis the bias phase is introduced by tilt in an interferometric fringe pattern. The bias phase term is given by 6 = 2JGlx.X
(7)
where CYis the spatial frequency of the tilted carrier fringes. This means that the bias phase spatially changed is used in the case of the carrier fringe method. In real domain analysis of the carrier fringe method, the interferometric fringe is sampled with the sampling period of the average spacing of the carrier fringes. If the sampling phase is changed, different sampled fringes are obtained. Figure 5 shows examples of sampled fringes with different phases. These sampled fringes are subjected to be analyzed by the same procedure in the fringe scanning method to obtain the phase @(x, y). Some variations of real domain analysis of the carrier fringe method are described in Refs 7 and 8. 3.4 Carrier fringe analysis in Fourier domain9 Suppose the carrier fringe equation of eqn (1) is rewritten as Z(x, y) = a(~, y) + k(x, y) exp (2~c7Gi LXX) + k*(x, y) exp (--2k m)
(8)
where k(x, Y) = (1/2)&x, Y) exp W(x,
~91
(9) In the Fourier domain analysis, the carrier fringe pattern of eqn (8) is
Toyohiko Yutagui
86
,
deformed grating
phase
profi 1 e
Fig. 5.
Fourier
transformed.
Wx, where
Carrier
fringe analysis
in real domain.
So we have
vy> =Ah,
Z(Y~ Y,),
(a)
A(vx, v,)
v,) + K(v* - a, v,) + K*(vx + a, v,)
(10)
and
K(vX, Y,) are Fourier transforms of If the tilt is sufficiently large, the terms of a, K and K* are separated in the Fourier domain. Because the term K(vX - LY,Y,) has the information of the phase @(x, y), this term is shifted to the origin of the Fourier domain so that we have the term K(vz, Y,). Then this term is Fourier transformed again to obtain the term k(x, y) and the imaginary part of its logarithm gives the phase @, as follows;
i(x, y), a@, y) and k(x, y), respectively.
log W,
y) = log a(~, y)/2 - i$(x, Y>
4 SPECKLE
FRINGE
(11)
ANALYSIS
In speckle interferometry, low-resolution devices, for example, a TV camera can be used because of the large size of the speckles in the image plane which is adjusted with the aperture of an imaging lens. A TV detection and filtering technique for video signals makes it possible to perform the measurements in real time. This method is called
87
Automated fringe analysis techniques in Japan
He-Ne
I
*
’ Digital Frame M Memory
Y TV- ’ Moni - ’ lor Fig. 6.
MiniComputer PDP-11134 ~ 4
Laser
Disk
J
Schematic diagram of computer-aided
,
Teletype
Lighl Pen
speckle pattern interferometry.
electronic speckle pattern interferometry (ESPI) and has been developed using analog and digital signal processing techniques. 4.1 Semi-automatic fringe analysis in ESPPoY” A schematic diagram of speckle interferometer equipped with a computer system is shown in Fig. 6. The light from a He-Ne laser is expanded by an objective lens and split into object and reference beams by a beam splitter. Diffusely scattering light from object and reference surfaces is collected by an imaging lens and focused on a TV camera. The light intensity of the speckle image is converted to an electronic video signal and sampled to yield a digital picture. The digital picture is stored in a digital frame memory transferred to a minicomputer by direct memory access. A speckle interferogram is generated arithmetically between two digitized speckle patterns before and after the deformation of the object to measure object deformation. Numerical data about the deformation are extracted by analysis of the speckle interferogram by digital image processing. Figure 7(a) shows an example of a fringe pattern generated by taking absolute values of differences of grey level speckle patterns, and the effect of low pass filtering is shown in Fig. 7(b). In a high quality interferogram, local grey level information can be used for fringe peak detection. But this approach cannot be used in a speckle interferogram as shown in Fig. 7(b), because many false skeletons were extracted due to speckle noise. The speckle interferogram averaged over 15 x 15 area is
Toyohiko
88
(b)
(4 Fig. 7.
Yutugai
Speckle fringe pattern generated by (a) taking absolute between grey level speckle pattern, and (b) low-pass
values of differences filtering.
averaged again over 51 x 51 sample points to yield a threshold level pattern. Fringe binarization is performed so that levels greater than or equal to corresponding threshold levels are mapped into 1 and mapped into 0 otherwise. With a few iterations of expanding and shrinking the bright area with 8-neighbour points, disconnected areas are fused and notched boundaries of bright fringes are smoothed. The resultant pattern is shown in Fig. 8(a). Disconnected regions of the same fringe order numbers and fused area of two different fringe order numbers still remain. To achieve linkage of disconnected areas and removal of any extra area, man-machine interactive processing is used. Because the boundaries of the binary fringes corrected by this processing are notchy, a few iterations of expanding and shrinking the bright fringes with 8-neighbour points are performed to smooth the boundaries. The processed fringe pattern is shown in Fig. 8(b) and subsequently skeletonized to yield Fig. 8(c). Fringe order numbers are assigned interactively. 4.2 Fringe scanning in ESPIU The fringe scanning method of phase measuring can be applied to ESPI fringe analysis. Figure 9 shows a schematic diagram of the fringe
(4 Fig. 8.
(b)
Interferograms obtained by improving the fringe quality 2) Fig. 7(b). (a) Result of 8-neighbour expansion and reshrinking after binarization of Fig. 7(b); (b) result of expansion and reshrinking after man-machine interactive processing of (a); (c) result of skeletoning of (b).
Automated
fringe analysis techniques -1
TVMoniIOf
Fig.
9.
Schematic
diagram
in Japan
He-Ne
Laser
1
TeleIwe
Lighl PW
of fringe scanning measurement.
89
ESPI
for
in-plane
deformation
scanning ESPI for measurement of in-plane deformation. The object measured is illuminated by two symmetrical beams with respect to the normal direction of the object surface. The mirror attached to a PZT is inserted in the path of one illuminating beam to change the path difference between two illumination beams. Speckle interferograms in each state of the object, with respect to the undeformed object, are shown in Fig. 10(a). Phase image proportional to deformation between adjacent states of the object are shown in Fig. 10(b).
5 CONCLUDING
REMARKS
Research efforts on interferometry and related fringe analysis techniques in Japan were reviewed, although not all could be included. As was shown, one can recognize many researchers active in optical measurements related to interferometry.
ACKNOWLEDGMENTS The author would like to thank his colleagues and the following people in Japan, who helped him prepare for this paper; Masanori Idesawa (Institute of Physical and Chemical Research), Mitsuo Takeda (University of Electra-Communications), Suezou Nakadate (Institute of Physical and Chemical Research), Kenji Nunome (NTN Toyo Bearing Co.),
90
Toyohiko
Yutagai
(4
(b) Fig. 10. (a) Low-pass filtered speckle interferograms in nine states of in-plane deformation of a brass plate with a hole. (b) Phase distribution proportional to the in-plane deformation between adjacent states of the deformation of (a).
Masane Suzuki and Kenji Yasuda (Fuji Photo-Optical Co.), Toshio Kanoh (Ricoh Co.), Nobuyuki Odagiri (Tokyo Seimitsu Co.), Akira Tonomura (Hitachi Co.).
REFERENCES 1. Yatagai, T., Nakadate, S., Idesawa, M. & Saito, H., Automatic fringe analysis using digital image processing techniques. Opt. Eng., 21 (1982) 432-5. 2. Yatagai, T., Idesawa, M., Yamaashi, Y. & Suzuki, M., Interactive fringe analysis system: applications to moire contourogram and interferogram. Opt. Eng., 21 (1982) 901-6.
91
Automated fringe analysis techniques in Japan
3. Yatagai, T., Inaba, S., Nakano, S. & Suzuki, M., Automatic flatness tester for very large scale integrated circuit wafers. Opt. Eng., 23 (1984) 401-5. 4. Yatagai, T., Recent development of optical interferometry. J. Jpn. Sot. Prec. Erg., 51 (1985) 695-702. (in Japanese). 5. Bruning, J. H., Herriott, D. R., Gallager, J. E., Rosenfeld, D. P., White, D. D. & Brangaccio, D. J., Digital wavefront interferometer for testing optical surfaces and lenses. Appl. Opt., 13 (1974) 2693-703. 6. Yatagai, T., Idesawa, M. & Saito, M., Automatic topography using high precision digital moire methods, Proc. SPZE, 361 (1983) 81-90. 7. Toyooka, S. & Tominage. M., Spatial fringe scanning for optical phase measurement. Opt. Commun., 51 (1984) 68-70. 8. Yatagai, T., Interferometric testing technology developments and applications in Japan, Pruc. SPIE, Vol816 (1987) 58-78. 9. Takeda, M., Ina, H. & Kobayashi, S., Fourier transform method of fringe pattern analysis for computer-based topography and interferometry. J. Opt. Sot. Amer., 72 (1982) 156-60. 10. Nakadate, S., Yatagai, T. & Saito,
H., Electronic speckle pattern using digital image processing techniques. Appl. Opt., 19
interferometry (1980) 1879-85.
11. Nakadate, S., Yatagai, T. & Saito, H., Computer-aided interferometry. Appl. Opt., 22 (1983) 237-43. 12. Nakadate, S. & Saito, H., Fringe scanning speckle-pattern Appi.
Opt.,
24 (1985) 2172-80.
speckle pattern interferometry.
Optics and Lasers in Engineering
Automated
Fringe Analysis Toyohiko
University of Tsukuba,
Techniques
in Japan
Yatagai
Institute of Applied Physics, Tsukuba,
Ibaraki 305, Japan
(Received 29 March 1989; accepted 17 October 1990)
ABSTRACT Research efforts for interferometric fringe analysis techniques and their applications in Japan are reviewed with an emphasis on recent developments. Digital fringe analysis techniques and theoretical approaches to phase measuring interferometry are described. Automatic fringe analysis of speckle interferometry are also presented.
1 INTRODUCTION Optical measurement techniques have been a key ingredient in greatly improving optical and precision engineering. Especially, stable and versatile interferometers have been developed and interferometric testing techniques have played very important roles. This paper reviews recent progress of interferometric testing and automated fringe analysis techniques in Japan. Much work in automatic phase detection and interferometric fringe analysis has been done, with relation to development of digital data processing equipments. Some engineering applications of interferometric testing have been investigated, including an automatic inspection system using holographic interferometer in an automobile industry, an aspheric surface testing system for micro-optics, a surface finishing testing system in high precision machining, a magnetic field measuring system using an electron holography, a flatness tester for VLSI wafers and optical discs, optical fiber sensors and so on. Fringe patterns generated by interferometric techniques have been used extensively in various fields of optical measurements. The fringe analyzing procedures, including fringe peak detection and fringe order determination, are tedious and time consuming. Research on automatic 79 Optics and Lasers in Engineering 0143-8166/91/$03.50
Ltd, England.
Printed in Northern
Ireland
0 1991 Elsevier Science Publishers
80
Toyohiko
Yatagai
fringe analysis and precision phase measuring techniques is very important in applying interferometric technique to practical uses. Various kinds of techniques have been investigated in Japan.
2 AUTOMATIC
FRINGE
ANALYSIS
2.1 Grey level fringe analysist’*‘i To estimate fringe peaks, the fringe density binarization technique is commonly used in many fringe analysis systems because of simple algorithms. On the other hand, we employ the grey level method in which local variation of the fringe density is mainly considered. This grey level method is sensitive to noise, but can detect peaks by processing only local areas smaller than those in the binary case. Software for the fringe analysis system based on the grey level analysis has been developed in modular form to make sequential execution of programs, of which typical procedure and program names are shown in Fig. 1. In order to extract fringe peaks in the grey level method, it is very important to diminish the influence of noise, including speckle noise. One of the most simplest and most effective methods is unweighted averaging: the grey level at each point (x, y) of the output pattern is the average of the grey levels in the input pattern over a neighbourhood of (x, y) as shown in Fig. 2(b) Usually, 3 x 3 pixels are a convenient size of the neighbourhood because of the noise size and the average width of the fringes. of extracting fringe central lines-i.e. The procedure fringe skeletons-consists of three steps: extracting fringe skeletons, thinning the skeletons, and eliminating false skeletons and connecting the disconnected skeletons. By using a 5 X 5 pixel matrix, we perform the peak detections with respect to the four directions: x, y, +45” and -45” directions. We adopt the skeletoning criterion that if the peak conditions for two of the four directions are satisfied, the central point of the 5 x 5 pixel matrix is the skeleton point, as shown in Fig. 2(c). In order to make fringe thinning, we use a modified Hilditch thinning algorithm for grey level skeleton patterns. Some of the isolated short skeleton lines are eliminated and disconnected fringe skeletons are connected by a man-machine interactive method. The result of the thinning procedure is shown in Fig. 2(d). Fringe order assignment is also performed by a man-machine interactive operation. At the first step of the fringe order determina-
Automated
fringe analysis techniques
in Japan
81
Fringe Data Input
lHashinn;;f+ngr
Data ]
Fr ingc Analysis SKLT THNNFR I Fringe Order Delcrminalion FRNOD, FRCOR FRPATC, FRTVPT Interpola\ion INTP, FRMSH L ’ Display of Analyzed
Data
GMFICE
Fig. 1.
Typical program flow for grey level fringe analysis.
tion, temporal fringe orders are automatically assigned, and then these temporal fringe orders are corrected using an errata list made by an operation. Fringe order is judged from the direction of fringe movement due to reference phase modifications by a piezo translator. Next, interpolation methods are employed to determine the fractional fringe orders of pixels between fringe skeletons. Figure 3 shows a perspective plot of fringe orders determined. The spline interpolation is effective for evaluating local surface irregularity, while the least-square interpolation is effective for global surface shape evaluation.
Toyohiko Yatagai
82
(4
(4
(cl
Fig. 2. Skeletoning and thinning of interferometric fringes: (a) Original interferometric fringe pattern; (b) skeletons of (a) in the case of detecting peaks for one of the four directions; (c) skeleton of (a) in the case of detecting peaks for two of four directions and (d) thinned skeletons of (c) with the area outside the boundary removed.
Fig. 3.
Example
of 3-D plot of interpolated fringe order, pattern is shown in Fig. 2(a).
of which
original
fringe
Automated fringe analysis techniques in Japan
83
2.2 Binary fringe analysis3
In a binary version of fringe analysis, we can employ a special-purpose hardware for digital image processing. A flow diagram of the fringe analysis is shown in Fig. 4. Procedures shown in Fig. 4 are carried out by means of the hardware modules. At the first step, the original interferometric fringe pattern is masked with the standard mask pattern so that the outside area of the wafer is excluded. The masked fringe pattern is subjected to contrast enhancement. In order to reduce the pepper-and-salt noise in the outside area of the fringe pattern, a level sliced wafer pattern is superimposed on the masked fringe pattern before contrast enhancement. The masked fringe pattern is binarized after contrast enhancement. Since the outside analyzing area is indeterminate, the standard mask pattern is superimposed on the binarized fringe pattern. To make a thinning of the dark parts of interferometric fringes, the fringe contrast is inverted. Then the inverted fringe pattern is subjected to thinning. In the next step of the fringe analysis, the fringe order is assigned. Since the sufficient tilt introduced removes the fringe order ambiguity, the fringes are numbered in a certain order. The fringe ordering procedure is performed by the function of region labelling of the hardware module. Level slice
Mask pattern
Original image
Enhancement
Binarization ndetrmlnatr
Inverse
Thinning
Fig. 4.
Fringe
analysis
procedures
for binary fringe analysis.
Toyohiko Yatagai
84
3 PHASE
MEASURING
INTERFEROMETRY
In the phase measuring interferometry, a bias phase term is introduced. fringe pattern with the bias phase 6 Suppose the interference introduced can be written as; f(x,
y, 6) =
a(x, y) + b(x, Y) cm b#a Y> - 61
(1)
where @(x, y) is the phase to be evaluated and a(x, y) and b(x, y) are the average fringe intensity and the fringe contrast, respectively. From the intensity variation of Z(x, y, S) due to the change of the bias phase 6, the phase +(x, y) is evaluated without the influence of the average fringe intensity and the fringe contrast. The phase measuring method is divided into three classes; the heterodyne method, the fringe scanning method and the carrier fringe analysis method.4 3.1 Heterodyne
method
In the heterodyne method, the bias phase 6 is introduced by using the beat effect of two wavefronts with slightly different frequencies. That is, the bias phase is given by 6=2xAvt
(2)
where Av is the frequency difference. Equation (2) means that the bias phase is introduced in the time domain in the heterodyne interferometry. The phase 4 to be estimated is measured by using an electronic phase meter in this case. The electronic phase meter usually has a high accuracy in phase estimation. 3.2 Fringe scanning method’ In the fringe scanning method, one of the mirrors is stepwisely moved through half of the wavelength so that the relative phase S, of the interferogram is changed; 6, = 2~rnlN
(n = 1, 2, . . . ,N - 1)
(3)
where N denotes the number of mirror movement. The irradiance at each point in the interference pattern goes through one cycle of periodic variation. The computer determines a best-fit sine function for the irradiance versus the amount of phase shift at each point of the interference pattern. The phase of the best-fit function is a direct measure of the test wavefront.
85
Automated fringe analysis techniques in Japan
According to the fringe scanning phase detection tions with sinusoidal weights,
principle,
summa-
c = c Z(X, y, 6,) cos (2n7GnlN)
(4)
s = C Z(X, y, 6,) sin (2~rnlZV)
(5)
are calculated to extract sinusoidal parts of intensity phases of the interferogram are given by
variation.
$(x, y) = arctan (s/c)
The (6)
The calculated values of arctangent are wrapped between ~JC radians. The unwrapped phase value gives the correct shape corresponding to the phase profile. 3.3 Carrier fringe analysis in real domain6 In the carrier fringe analysis the bias phase is introduced by tilt in an interferometric fringe pattern. The bias phase term is given by 6 = 2JGlx.X
(7)
where CYis the spatial frequency of the tilted carrier fringes. This means that the bias phase spatially changed is used in the case of the carrier fringe method. In real domain analysis of the carrier fringe method, the interferometric fringe is sampled with the sampling period of the average spacing of the carrier fringes. If the sampling phase is changed, different sampled fringes are obtained. Figure 5 shows examples of sampled fringes with different phases. These sampled fringes are subjected to be analyzed by the same procedure in the fringe scanning method to obtain the phase @(x, y). Some variations of real domain analysis of the carrier fringe method are described in Refs 7 and 8. 3.4 Carrier fringe analysis in Fourier domain9 Suppose the carrier fringe equation of eqn (1) is rewritten as Z(x, y) = a(~, y) + k(x, y) exp (2~c7Gi LXX) + k*(x, y) exp (--2k m)
(8)
where k(x, Y) = (1/2)&x, Y) exp W(x,
~91
(9) In the Fourier domain analysis, the carrier fringe pattern of eqn (8) is
Toyohiko Yutagui
86
,
deformed grating
phase
profi 1 e
Fig. 5.
Fourier
transformed.
Wx, where
Carrier
fringe analysis
in real domain.
So we have
vy> =Ah,
Z(Y~ Y,),
(a)
A(vx, v,)
v,) + K(v* - a, v,) + K*(vx + a, v,)
(10)
and
K(vX, Y,) are Fourier transforms of If the tilt is sufficiently large, the terms of a, K and K* are separated in the Fourier domain. Because the term K(vX - LY,Y,) has the information of the phase @(x, y), this term is shifted to the origin of the Fourier domain so that we have the term K(vz, Y,). Then this term is Fourier transformed again to obtain the term k(x, y) and the imaginary part of its logarithm gives the phase @, as follows;
i(x, y), a@, y) and k(x, y), respectively.
log W,
y) = log a(~, y)/2 - i$(x, Y>
4 SPECKLE
FRINGE
(11)
ANALYSIS
In speckle interferometry, low-resolution devices, for example, a TV camera can be used because of the large size of the speckles in the image plane which is adjusted with the aperture of an imaging lens. A TV detection and filtering technique for video signals makes it possible to perform the measurements in real time. This method is called
87
Automated fringe analysis techniques in Japan
He-Ne
I
*
’ Digital Frame M Memory
Y TV- ’ Moni - ’ lor Fig. 6.
MiniComputer PDP-11134 ~ 4
Laser
Disk
J
Schematic diagram of computer-aided
,
Teletype
Lighl Pen
speckle pattern interferometry.
electronic speckle pattern interferometry (ESPI) and has been developed using analog and digital signal processing techniques. 4.1 Semi-automatic fringe analysis in ESPPoY” A schematic diagram of speckle interferometer equipped with a computer system is shown in Fig. 6. The light from a He-Ne laser is expanded by an objective lens and split into object and reference beams by a beam splitter. Diffusely scattering light from object and reference surfaces is collected by an imaging lens and focused on a TV camera. The light intensity of the speckle image is converted to an electronic video signal and sampled to yield a digital picture. The digital picture is stored in a digital frame memory transferred to a minicomputer by direct memory access. A speckle interferogram is generated arithmetically between two digitized speckle patterns before and after the deformation of the object to measure object deformation. Numerical data about the deformation are extracted by analysis of the speckle interferogram by digital image processing. Figure 7(a) shows an example of a fringe pattern generated by taking absolute values of differences of grey level speckle patterns, and the effect of low pass filtering is shown in Fig. 7(b). In a high quality interferogram, local grey level information can be used for fringe peak detection. But this approach cannot be used in a speckle interferogram as shown in Fig. 7(b), because many false skeletons were extracted due to speckle noise. The speckle interferogram averaged over 15 x 15 area is
Toyohiko
88
(b)
(4 Fig. 7.
Yutugai
Speckle fringe pattern generated by (a) taking absolute between grey level speckle pattern, and (b) low-pass
values of differences filtering.
averaged again over 51 x 51 sample points to yield a threshold level pattern. Fringe binarization is performed so that levels greater than or equal to corresponding threshold levels are mapped into 1 and mapped into 0 otherwise. With a few iterations of expanding and shrinking the bright area with 8-neighbour points, disconnected areas are fused and notched boundaries of bright fringes are smoothed. The resultant pattern is shown in Fig. 8(a). Disconnected regions of the same fringe order numbers and fused area of two different fringe order numbers still remain. To achieve linkage of disconnected areas and removal of any extra area, man-machine interactive processing is used. Because the boundaries of the binary fringes corrected by this processing are notchy, a few iterations of expanding and shrinking the bright fringes with 8-neighbour points are performed to smooth the boundaries. The processed fringe pattern is shown in Fig. 8(b) and subsequently skeletonized to yield Fig. 8(c). Fringe order numbers are assigned interactively. 4.2 Fringe scanning in ESPIU The fringe scanning method of phase measuring can be applied to ESPI fringe analysis. Figure 9 shows a schematic diagram of the fringe
(4 Fig. 8.
(b)
Interferograms obtained by improving the fringe quality 2) Fig. 7(b). (a) Result of 8-neighbour expansion and reshrinking after binarization of Fig. 7(b); (b) result of expansion and reshrinking after man-machine interactive processing of (a); (c) result of skeletoning of (b).
Automated
fringe analysis techniques -1
TVMoniIOf
Fig.
9.
Schematic
diagram
in Japan
He-Ne
Laser
1
TeleIwe
Lighl PW
of fringe scanning measurement.
89
ESPI
for
in-plane
deformation
scanning ESPI for measurement of in-plane deformation. The object measured is illuminated by two symmetrical beams with respect to the normal direction of the object surface. The mirror attached to a PZT is inserted in the path of one illuminating beam to change the path difference between two illumination beams. Speckle interferograms in each state of the object, with respect to the undeformed object, are shown in Fig. 10(a). Phase image proportional to deformation between adjacent states of the object are shown in Fig. 10(b).
5 CONCLUDING
REMARKS
Research efforts on interferometry and related fringe analysis techniques in Japan were reviewed, although not all could be included. As was shown, one can recognize many researchers active in optical measurements related to interferometry.
ACKNOWLEDGMENTS The author would like to thank his colleagues and the following people in Japan, who helped him prepare for this paper; Masanori Idesawa (Institute of Physical and Chemical Research), Mitsuo Takeda (University of Electra-Communications), Suezou Nakadate (Institute of Physical and Chemical Research), Kenji Nunome (NTN Toyo Bearing Co.),
90
Toyohiko
Yutagai
(4
(b) Fig. 10. (a) Low-pass filtered speckle interferograms in nine states of in-plane deformation of a brass plate with a hole. (b) Phase distribution proportional to the in-plane deformation between adjacent states of the deformation of (a).
Masane Suzuki and Kenji Yasuda (Fuji Photo-Optical Co.), Toshio Kanoh (Ricoh Co.), Nobuyuki Odagiri (Tokyo Seimitsu Co.), Akira Tonomura (Hitachi Co.).
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Automated fringe analysis techniques in Japan
3. Yatagai, T., Inaba, S., Nakano, S. & Suzuki, M., Automatic flatness tester for very large scale integrated circuit wafers. Opt. Eng., 23 (1984) 401-5. 4. Yatagai, T., Recent development of optical interferometry. J. Jpn. Sot. Prec. Erg., 51 (1985) 695-702. (in Japanese). 5. Bruning, J. H., Herriott, D. R., Gallager, J. E., Rosenfeld, D. P., White, D. D. & Brangaccio, D. J., Digital wavefront interferometer for testing optical surfaces and lenses. Appl. Opt., 13 (1974) 2693-703. 6. Yatagai, T., Idesawa, M. & Saito, M., Automatic topography using high precision digital moire methods, Proc. SPZE, 361 (1983) 81-90. 7. Toyooka, S. & Tominage. M., Spatial fringe scanning for optical phase measurement. Opt. Commun., 51 (1984) 68-70. 8. Yatagai, T., Interferometric testing technology developments and applications in Japan, Pruc. SPIE, Vol816 (1987) 58-78. 9. Takeda, M., Ina, H. & Kobayashi, S., Fourier transform method of fringe pattern analysis for computer-based topography and interferometry. J. Opt. Sot. Amer., 72 (1982) 156-60. 10. Nakadate, S., Yatagai, T. & Saito,
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11. Nakadate, S., Yatagai, T. & Saito, H., Computer-aided interferometry. Appl. Opt., 22 (1983) 237-43. 12. Nakadate, S. & Saito, H., Fringe scanning speckle-pattern Appi.
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