Fast Surface Proler by White-Light Interferometry. Tokyo Institute of Technology, O-okayama, Meguro-ku, Tokyo , Japan ABSTRACT

Transcription

1 Fast Surface Proler by White-Light Interferometry Using a New Algorithm, the SEST Algorithm Hirabayashi Akira a, Ogawa Hidemitsu b, and Kitagawa Katsuichi c a Yamaguchi University, Tokiwadai, Ube , Japan b Tokyo Institute of Technology, -1-1 O-okayama, Meguro-ku, Tokyo , Japan c Toray Engineering Co.,Ltd., , Oe, Otsu-shi, Shiga , Japan ABSTRACT We devise a fast algorithm for surface proling by white-light interferometry. It is named the SEST algorithm after Square Envelope function estimation by Sampling Theory. Conventional methods for surface proling by white-light interferometry based their foundation on digital signal processing technique, which is used as an approximation of continuous signal processing. Hence, these methods require narrow sampling intervals to achieve good approximation accuracy. In this paper, we introduce a totally novel approach using sampling theory. That is, we provide a generalized sampling theorem that reconstructs a square envelope function of a white-light interference fringe from sampled values of the interference fringe. A sampling interval in the SEST algorithm is 6-14 times wider than those of conventional methods when an optical lter of the center wavelength 600nm and the bandwidth 60nm is used. The SEST algorithm has been installed in a commercial system which achieved the world's fastest scanning speed of 4:75m=s. The height resolution of the system lies in the order of 10nm for a measurement range of greater than 100m. Keywords: Surface proling, surface proler, white-light interferometry, interference fringe, bandpass signal, sampling theorem, the SEST algorithm 1. INTRODUCTION White-light interferometry is an elegant method for measuring surface proles of objects such as semiconductors, liquid crystal displays (LCDs), plastic lms, and precision machinery parts. 1{13 One of the most important advantages of white-light interferometry is that its measurement range is virtually unlimited, while that of phase-shifting interferometry is limited to not more than half a wavelength. In order to obtain surface proles, many conventional methods utilize an envelope function of an interference fringe or the square of the envelope function, which we call a square envelope function. For example, Kino et al. and Chim et al. proposed methods of determining the envelope function by using digital ltering technique such as the discrete Fourier transform or the discrete Hilbert transform. 1,3 Caber et al. developed a communication theory approach to determine a square envelope function. 4,5 Larkin proposed a nonlinear algorithm based on phase-shifting interferometry. 7 Other methods, which do not utilize these envelope functions, have been proposed as well. For example, a method by de Groot et al. 8,9 uses the phase gradient calculated from the discrete Fourier transform of sampled data. A method by means of a wavelet lter is presented by Recknagel et al. 11 All of these conventional methods use discrete signal processing technique, although white-light interference fringe and these envelope functions are continuous. Hence, discrete signal processing can be considered as an approximation of continuous signal processing. Therefore, these methods require narrow sampling intervals to achieve good approximation accuracy. That results in a large number of sampled data and high computational costs. In order to reduce both of them, we introduce a novel approach using sampling theory. That is, we provide a generalized sampling theorem that directly reconstructs a square envelope function from sampled values of the Further author information: (Send correspondence to H.A.) H.A.: a-hira@csse.yamaguchi-u.ac.jp O.H.: ogawa@og.cs.titech.ac.jp K.K.: katsuichi kitagawa@toray-eng.co.jp 1

2 interference fringe. Based on the theorem, we propose a new surface proling algorithm, the SEST algorithm, named after Square Envelope function estimation by Sampling Theory. The algorithm requires a small number of sampled data and low computational costs. In fact, its sampling interval is extended to 1:45m when an optical lter of the center wavelength 600nm and the bandwidth 60nm is used, which is 6-14 times wider than those of conventional methods. Furthermore, the algorithm requires only arithmetical calculations, no transcendental calculations except for only one cosine function. The SEST algorithm has been installed in a commercial system which achieved the world's fastest vertical scanning speed of 4:75m=s. The height resolution of the system lies in the order of 10nm for a measurement range of greater than 100m.. SURFACE PROFILING BY WHITE-LIGHT INTERFEROMETRY In this section, surface proling by white-light interferometry is outlined. First, we shall show a basic set up of a white-light interferometer used for surface proling. Then, a mathematical model of the white-light interference fringe is reviewed. Finally, we shall give an explicit denition of a square envelope function of an interference fringe..1. White-Light Interferometer Fig.1 shows a schematic of a white-light interferometer. In this gure, the Michaelson interferometer is used. The Mirau interferometer can be used as well. A beam from the white-light source in Fig.1 crosses through the beam splitter at the point O, and divided into two portions. One of the beams, indicated by the dashed line, is transmitted to a surface of an object being observed. The other beam, indicated by the dotted line, is transmitted to the reference mirror whose distance from the point O is L 1. E is a virtual plane whose distance from the point O along the dashed line is L 1. z is the distance of the plane E from the stage. It is referred to as the height of the interferometer. Two beams are reected from the object and the reference mirror, respectively, and recombined at the beam splitter. The resultant beam intensity produced by the interference eect is observed by a charge-coupled device (CCD) video camera which has, for example, detectors. Each detector corresponds to a point on the surface of the object, which is denoted by (x; y). The height of the surface of the object at the point (x; y) from the stage of the interferometer is denoted by z p (x; y). As the interferometer is scanned along the vertical axis, the z-axis, the intensity observed by each detector is varied. The intensity along the z-axis is shown in Fig. by a dotted line. The graph is called a white-light interference CCD camera Reference mirror Beam splitter Filter O 100 L 1 (xed) L 1 White-light source Object Stage E z Figure. An example of a white-light interference fringe g(z). Its sampled values are also shown by `'. Figure 1. Schematic of a white-light interferometer

3 fringe or simply an interference fringe. It appears in the right side in Fig. if the height z p (x; y) is high, while it appears in the left side if z p (x; y) is low. Hence, the maximum position of an interference fringe provides the height of the point on the surface. The CCD camera outputs the intensity of an interference fringe, for example, every 1=30 second. Hence, we can utilize only discrete sampled values of the interference fringe as shown by `' in Fig.. From these sampled values, we have to estimate the maximum position of the interference fringe. Therefore, sampling theory naturally plays an important role... Mathematical model of an interference fringe We shall review a mathematical model of an interference fringe following Refs. 1,1. The center wavelength and the bandwidth of the optical lter in Fig.1 are denoted by c and b, respectively. Letb() be an amplitude distribution of the beam at the beam-splitter, where is the wavelength. b() is restricted into an interval [ c 0 b ; c + b ]by a narrowband optical lter. That is, it holds that b() =0 (0 << c 0 b ;> c + b ): (1) For example, for a typical lter c = 600nm and b =60nm. The parameter is commonly used among practical engineers. For mathematical analysis, however, the angular wavenumber, k, is suitable. It is dened by Let a(k) be an expression of b() in terms of k: Let k l and k u be respectively. Eqs.(1), (3), and (4) yield k l = k = : () a(k) =b : (3) k ; k u = ; (4) c + b c 0 b a(k) =0 (0 <k<k l ;k>k u ): (5) Averaged attenuation rates of two beams along the the dashed line and the dotted line in Fig.1 are denoted by q o (k) and q r (k), respectively. Let (k) be (k) = fa(k)g q o (k)q r (k) (k >0); 0 (k 0): (6) Eqs.(5) and (6) yield (k) =0 (k <k l ;k>k u ): (7) That is, (k) is restricted into an interval [k l ;k u ]. By using these notations, a model of an interference fringe is given as g(x; y; z) =f(x; y; z)+c; (8) where f(x; y; z) and C are dened by f(x; y; z) = C = Z ku Z ku k l (k) cos kfz 0 z p (x; y)g dk; (9) k l fa(k)g fq o (k)g + fq r (k)g 3 dk; (10) 3

4 respectively. In the following analysis, explicit (x; y) variation is ignored. Hence, g(x; y; z), f(x; y; z), and z p (x; y) are denoted by g(z), f(z), and z p, respectively, hereafter: g(z) =f(z)+c; (11) f(z) = Z ku k l (k)cosk(z 0 z p ) dk: (1) Since f(z) diers from g(z) only in the constant C, f(z) is also called an interference fringe. When it is necessary to distinguish them, we use terminologies `an interference fringe g(z)' and `an interference fringe f(z)'. It can be shown that it holds that for z 6= z p g(z) <g(z p ): (13) That is, an interference fringe g(z) has the maximum only at z = z p. In practice, however, it is hard to obtain the maximum position z p from g(z), because g(z) has high-frequency components..3. Square Envelope Function In order to overcome the problem mentioned above, an envelope function of a white-light interference fringe or the square of the envelope function are usually used, because they have the following two properties: (i) It has the maximum only at z = z p. (ii) It is smoother than the interference fringe. In order to dene the envelope function, let us introduce a function F (z) given by F (z) = Z ku k l (k)e ik(z0z p) dk: (14) It is a complex valued function. The enveloped function, m(z), is dened by the magnitude of F (z) 1 : m(z) =jf (z)j: (15) It can be recognized from Fig.3, where f(z) and m(z) are shown by dotted and solid lines, respectively. Eq.(15) implies that the envelope function is calculated by the square root of the sum of squares of the real and imaginary parts of F (z). Hence, from the practical point of view, the square of the envelope function is more convenient. We denote the function by r(z), and call a square envelope function: r(z) =jf (z)j : (16) Figure 3. The interference fringe f(z) shown in Fig. and its envelope function m(z). 4

5 Lemma.1. z 6= z p, The square envelope function r(z) has the maximum only at z = z p. That is, it holds that for any r(z) <r(z p ): (17) This lemma mathematically guarantees that the square envelope function has the property (i). We shall show that r(z) has the property (ii). Let ^f(!) be the Fourier transform of f(z): From Eq.(1), we have Let ^f(!) = Z 1 01 Eqs.(19) and (7) mean that f(z) is a bandpass signal such that f(z)e 0i!z dz: (18) ^f(!) = e0i!z p j!j : (19)! l =k l = 4 c + b ;! u =k u = 4 c 0 b : (0) ^f(!) =0 (j!j <! l ; j!j >! u ): (1) On the other hand, the Fourier transform of r(z) is given by ^r(!) = e0i!z p Z!u! l! 0 0! 0!0 Hence, the square envelope function r(z) is a lowpass signal such that d! 0 : () ^r(!) =0 (j!j >! u 0! l ): (3) Eq.(0) implies that! u 0! l <! l if and only if c > 3 b. In this case, r(z) is smoother than f(z) because of Eqs.(1) and (3). For example, when c = 600nm and b =30nm,! u 0! l =:10[1=m];! l =19:95[1=m]: Hence, r(z) is much smoother than f(z), and r(z) has the property (ii). 3. SAMPLING THEOREM FOR SQUARE ENVELOPE FUNCTIONS In order to detect the peak of the envelope function m(z) or the square envelope function r(z), conventional methods utilize digital signal processing techniques. 1{11 Since m(z) andr(z) are continuous signals, such techniques can be considered as an approximation of continuous signal processing. Hence, these methods require narrow sampling intervals to achieve good approximation accuracy. In this section, we introduce a new approach using sampling theory. That is, we shall provide a generalized sampling theorem that reconstructs the square envelope function r(z) directly from sampled values of the interference fringe f(z), not those of the square envelope function r(z) itself. First, a sampling theorem for the interference fringe f(z) is given by using a sampling theorem for bandpass signals. Based on the theorem, we shall derive the generalized sampling theorem for the square envelope function r(z). Finally, a sampling requirement will be discussed Sampling Theorem for Interference Fringes As mentioned in the previous section, the interference fringe f(z) is a bandpass signal such that its Fourier transform ^f(!) is supported in [0! u ; 0! l ] [ [! l ;! u ]. Hence, we can utilize sampling theorems for bandpass signals. 14{17 There are two types of such sampling theorems. The rst type has a restriction on frequency band, 14,15 while the other has no such restriction. 15{17 We utilize the former because it has a computationally ecient property. 5

6 The restriction in the theorem is that the lower cuto frequency! l must be an integer multiple of its bandwidth! u 0! l. However,! l and! u do not satisfy this relation in general. Then, we introduce two positive real numbers! c and! b with a nonnegative integer I such that! c =(I +1)! b ; (4) and consider a new interval [0! c 0! b ; 0! c +! b ] [ [! c 0! b ;! c +! b ]. Eq.(4) means that the lower cuto frequency! c 0! b of the new interval is given by an integer multiple of its bandwidth! b.if! c and! b satisfy! c 0! b! l ;! u! c +! b ; (5) f(z) is supported in the new interval [0! c 0! b ; 0! c +! b ] [ [! c 0! b ;! c +! b ], and we can apply a sampling theorem of the rst type to a white-light interference fringe. The following lemma gives a necessary and sucient condition for such! c,! b, and I. Lemma 3.1. Let! c and! b bepositive real numbers satisfying Eq.(4) with a nonnegative integer I. Eq.(5) holds if and only if! l 0 I ; (6)! u 0! l and, for any xed I in Eq.(6)! b satises 8 >< >:! u! b (I =0);! u (I +1)!! l b (I 6= 0): I (7) A proof of Lemma 3.1 is reserved for Appendix A. Eq.(6) implies! u =f(i +1)g! l =(I) wheni 6= 0. Hence, there certainly exists a real number! b which satises the second equation of Eq.(7). I =0 6 0! b! b 0! u 0! l! l! u -! I =1 6 04! b 0! b! b 4! b 0! u 0! l! l! u -! I = 6 06! b 04! b 4! b 6! b 0! u 0! l 0! b! b! l! u -! Figure 4. Examples of the interval [0! c 0! b ; 0! c +! b ] [ [! c 0! b ;! c +! b ]. If I =0; 1, and, then the interval becomes [0! b ;! b ], [04! b ; 0! b ] [ [! b ; 4! b ], and [06! b ; 04! b ] [ [4! b ; 6! b ], respectively. 6

7 Fig.4 shows examples of the new interval [0! c 0! b ; 0! c +! b ] [ [! c 0! b ;! c +! b ]. We can see from the gure that the width of the interval decreases as the integer I increases. Furthermore, even for a xed I there exist an innite number of intevals [0! c 0! b ; 0! c +! b ] [ [! c 0! b ;! c +! b ] such that Eq.(5) holds, because there exist an innite number of! b which satisfy Eq.(7) in general. By using the interval [0! c 0! b ; 0! c +! b ] [ [! c 0! b ;! c +! b ] with! c,! b, and I such that Eqs.(4), (6), and (7) hold, we can apply the sampling theorem for bandpass signals 14,15 to the interference fringe f(z). Lemma 3.. (Sampling theorem for interference fringes) Let I be aninteger such that Eq.(6) holds and! b beany real number which satises Eq.(7). Let! c beareal number dened by Eq.(4). Let 1 be a sampling interval such that 1=! b ; (8) and fz n g 1 be sample points dened by z n = n1: (9) Let sinc(z) be a function dened by and f n (z)g 1 be functions dened by sinc(z) = 8 < : sin z (z 6= 0); z 1 (z =0); (30) n (z) = sinc! b(z 0 z n ) Then, f(z) can be reconstructed from the sampled values ff(z n )g 1 by f(z) = 1X cos! c (z 0 z n ): (31) f(z n ) n (z): (3) 3.. Sampling Theorem for Square Envelope Functions Based on Lemma 3., we shall provide a generalized sampling theorem for square envelope functions. Let f s (z) bea function dened by f s (z) = Z ku k l (k) sin k(z 0 z p ) dk: (33) It is the imaginary part of F (z) in Eq.(14), while f(z) is its real part. Hence, Eq.(16) yields Let us dene functions f' n (z)g 1 by r(z) =ff(z)g + ff s (z)g : (34) ' n (z) = sinc! b(z 0 z n ) sin! c (z 0 z n ): (35) The functions f s (z) and ' n (z) are expressed by the Hilbert transforms of f(z) and n (z), respectively: f s (z) = 1 ' n (z) = 1 Z 1 01 Z 1 Hence, Lemma 3. yields the following sampling theorem for f s (z). 01 f(z 0 ) z 0 z 0 dz0 ; (36) n (z 0 ) z 0 z 0 dz0 : (37) 7

8 Lemma 3.3. The function f s (z) can be reconstructed from the sampled values ff(z n )g 1 of the interference fringe f(z) by f s (z) = 1X f(z n )' n (z): (38) Lemma 3., Lemma 3.3, and Eq.(34) yield the following generalized sampling theorem for square envelope functions: Theorem 3.4. (Sampling theorem for square envelope functions) Let I be aninteger such that Eq.(6) holds and! b be any real number which satises Eq.(7). Let! c beareal number dened by Eq.(4). Let 1 be a sampling interval given by Eq.(8), and fz n g 1 be sample points dened by Eq.(9). Then, it holds that 1. When z is a sample point z j, r(z j )=ff(z j )g + 4 ( 1 X ) f(z j+n+1 ) : (39) n +1. When z is not any sample point, r(z) = cos z ( 1X 1 f(z n ) z 0 z n ) cos z ( 1X 1 ) 3 f(z n+1 ) 5 : (40) z 0 z n+1 (Proof) It follows from Eqs.(3), (31), (30), and (4) that f(z) = 8 >< >: f(z j ) (z = z j ); " # " # sin!b z 1X f(z n ) cos cos! c z 0 (01) I!b z 1X f(z n+1 ) sin! c z (otherwise):! b z 0 z n! b z 0 z n+1 Similarly, it follows from Eqs.(38), (35), (30), and (4) that f s (z) = 8 >< >: (01) I sin! c z " 1X sin! b z! b f(z j+n+1 ) n +1 1X # f(z n ) +(01) I cos! c z z 0 z n " cos! b z! b 1X # f(z n+1 ) z 0 z n+1 (z = z j ); (otherwise): (41) (4) Eqs.(34), (41), (4), and (8) yield Eqs.(39) and (40). Theorem 3.4 is a generalized sampling theorem in the sense that it reconstructs r(z) directly from the sampled values of the interference fringe f(z), not those of the square envelope function r(z) itself. Eq.(41) is a computationally ecient form of Eq.(3). In fact, the second equation of Eq.(41) requires evaluation of only four trigonometric functions while Eq.(3) requires evaluation of an innite number of such functions. The second type of sampling theorems for bandpass signals mentioned at the beginning of Subsection 3.1 does not yield such an ecient form. This is the reason why we used the sampling theorem for bandpass signals of the rst type. This property resulted in a simple expression of the sampling theorem for square envelope functions. That is, Eq.(39) requires arithmetical calculations only. Eq.(40) requires evaluation of a single trigonometric function and arithmetical calculations Sampling Intervals In Theorem 3.4, the sampling interval 1 is dened by Eq.(8). Since I and! b are chosen so that Eqs.(6) and (7) hold, respectively, there are an innite number of the sampling intervals 1 as follows: Corollary 3.5. For any xed integer I such that 0 I c 0 b b ; (43) 8

9 I = [m] Figure 5. Sampling intervals 1 belonging to these 10 blocks can be used for reconstruction of r(z) from sampled values of f(z) when the optical lter of c = 600nm and b =30nm is used. In this case, the Nyquist interval is 1.5m. any real number 1 which satises can be used as the sampling interval 1 in Theorem 3.4. The following is a direct consequence of Corollary I( 4 c + b ) 1 1(I +1)( 4 c 0 b ) (44) Corollary 3.6. The maximum, 1 max, of the sampling interval 1 is given by 1 max = 1 c + b ( c 0 b ); (45) 4 b where bxc is the maximum integer not exceeding a real number x. c and b are determined by the characteristics of the optical lter used in an interferometer as is shown in Eq.(1). Hence, Corollary 3.5 means that the characteristics of the optical lter completely determine the sampling interval 1. Since the square envelope function r(z) is a lowpass signal such that Eq.(3) holds, it can be reconstructed by so-called Someya-Shannon's sampling theorem 18,19 if sampled values of r(z) itself are available with the sampling interval less than or equal to the Nyquist interval =(! u 0! l ). However, since only sampled values of f(z) are available, not every sampling interval belonging to (0;=(! u 0! l )] can be used. It follows from Eq.(0) that Nyquist interval =! u 0! l = 1 4 c + b b ( c 0 b ): (46) Eqs.(45) and (46) mean that 1 max is less than or equal to the Nyquist interval of square envelope functions. Hence, Corollary 3.5 says that only a subset of sampling intervals less than the Nyquist interval can be used as the sampling interval 1 in Theorem 3.4. For example, when the optical lter of c = 600nm and b =30nm is used, only sampling intervals 1 belonging to 10 blocks in Fig.5 can be used. Note that in this case, the Nyquist interval is 1.50m. If an optical lter of c = 600nm and b =30nm is used, 1 max is 1:45m. It is much wider than sampling intervals used in conventional systems. For example, in the systems produced by Veeco Instruments Inc. and Zygo Corporation, sampling intervals are 0:4m and0:10m, respectively. 6,10 1 max is about 6 and 14 times wider than intervals of these systems. 4. NEW SURFACE PROFILING ALGORITHM AND SURFACE PROFILER Based on Theorem 3.4, we propose a new surface proling algorithm. In the theorem, it is assumed that (i) an innite number of sampled values can be used, and (ii) sampled values f(z n ) of the interference fringe f(z) are available. In practical applications, however, only a nite number of sampled values can be used, and only sampled values g(z n ) of the interference fringe g(z) = f(z) + C in Eq.(11) are available. Hence, we rst truncate the innite series in Theorem 3.4 from n =0toN 0 1. Then, we approximate the sampled values f(z n )by f n = g(z n ) 0 ^C; (47) where ^C is an estimate of C. For example, the average of fg(z n )g N01 can be used as ^C: 9

10 ^C = 1 N N01 X g(z n ): (48) Finally, the square envelope function r(z) is approximated by the following function r N (z): 1. When z is a sample point z j (j =0; 1;:::;N 0 1),. When z is not any sample point, r N (z) = cos z r N (z j )=(f j ) < : 1 8 < : X b N01 c f n z 0 z n 9 = ; b N0j0 c X b j+1 c + f j+n+1 n +1 1+cos z 1 9 = ; : (49) 8 < b N X c01 : f n+1 z 0 z n+1 9 = ; : (50) Based on these equations, we propose a new surface proling algorithm, the SEST algorithm named after Square Envelope function estimation by Sampling Theory. The SEST Algorithm 1. Acquire fg(z n )g N01 by a white-light interferometer at the sample points fz ng N01 in Eq.(9).. Repeat the following steps for fg(z n )g N01 3. Calculate ^C by Eq.(48). 4. Calculate ff n g N01 by Eq.(47). acquired by all CCD detectors. 5. Search for the maximum position of r N (z) in Eqs.(49) and (50), and use the position as an estimate of z p. This algorithm has been installed in a commercial system, 13 which is shown in Fig.6. Specications of the system is shown in Table 1. Fig.7 shows a three-dimensional image of IC bumps obtained by the surface proler. In order to evaluate the accuracy of the proler, we measured the surface prole of a step height standard of 9:947m. Its three dimensional image is shown in Fig.8. The dierence between the averages of estimated values of z p for the top and the bottom of the step is 9.933m. The relative error is 0:13%, which shows a good performance of the surface proler. Figure 6. Photograph of a surface proler in which the SEST algorithm has been installed. 10

11 Measurement technique Algorithm Range of measurement Vertical scanning speed Objectives Table 1. Specications of the system shown in Fig.6. Narrow-band white-light interferometry The SEST algorithm 0-100m (Optional: 0-350m) Dependent on the characteristics of an optical lter; [Example] 4.75m/sec when c = 600nm and b =30nm Selectable;.5X, 5X, 10X, 0X, 50X Image zoom Selectable; 0.35X, 0.45X, 0.6X Measurement speed Repeatability Measurement array Options Dependent on the various parameters; [Example] In case of 5m range & 1810 pixel array, Total : 0.8 sec (Scan 0.6sec + Calculation 0.sec) Dependent on the various parameters; [Example] Sigma(Average)= 0nm for standard step height Selectable; 51x480, 56x40, 18x10 Continuous zoom lens Vibration isolation table Phase-shift measurement software Bump measurement software Stitching software Figure 7. Three-dimensional image of IC bumps obtained by the surface proler in Fig Figure 8. Three-dimensional image of a step height standard of 9:947m obtained by the surface proler in Fig CONCLUSION We introduced a new approach to surface proling by white-light interferometry. That is, we provided a generalized sampling theorem that reconstructs a square envelope function of a white-light interference fringe directly from sampled values of the interference fringe. Based on the theorem, we proposed a new surface proling algorithm, the SEST algorithm, named after Square Envelope function estimation by Sampling Theory. The sampling interval of the algorithm is 1:45m when an optical lter of the center wavelength 600nm and the bandwidth 60nm is used, 11

12 which is 6-14 times wider than those of conventional methods. The algorithm requires only arithmetical calculations, no transcendental calculations except for only one cosine function. The SEST algorithm has been installed in a commercial system which achieved the world's fastest vertical scanning speed of 4:75m=s. The height resolution of the system lies in the order of 10nm for a measurement range of greater than 100m. APPENDIX A. PROOF OF LEMMA 3.1 Assume that Eq.(5) holds. It follows from Eq.(4) and the rst equation in Eq.(5) that I! l! b : (51) Eq.(5) also implies that! b! u 0! l : Hence, Eq.(51) yields Eq.(6). When I = 0, Eq.(4) means! c =! b. Hence, the second equation of Eq.(5) yields the rst equation of Eq.(7). When I 6= 0, Eqs.(4) and (5) yield the second equation in Eq.(7). We shall prove the converse. Assume that I and! b satisfy Eqs.(6) and (7), respectively. WhenI = 0, Eq.(4) yields! c =! b. Hence, the rst equation of Eq.(5) holds because! l 0. Furthermore, the rst equation of Eq.(7) yields the second equation of Eq.(5). When I 6= 0, Eq.(6) implies Hence, the second equation of Eq.(7) is well dened, and it yields Eq.(5).! u (I +1)! l I : (5) REFERENCES 1. G. S. Kino and S. S. C. Chim, \Mirau correlation microscope," Applied Optics 9(6), pp. 3775{3783, S. S. C. Chim and G. S. Kino, \Phase measurements using the mirau correlation microscope," Applied Optics 30(16), pp. 197{01, S. S. C. Chim and G. S. Kino, \Three-dimensional image realization in interference microscopy," Applied Optics 31(14), pp. 550{553, P. J. Caber, \Interferometric proler for rough surfaces," Applied Optics 3(19), pp. 3438{3441, P. J. Caber, S. J. Martinek, and R. J. Niemann, \A new interferometric proler for smooth and rough surfaces," Proceedings of SPIE 088, pp. 195{03, Veeco Instruments Inc. Catalog, No. NTS , Wyko NT Series Ultrafast Optical Prolers, K. G. Larkin, \Ecient nonlinear algorithm for envelope detection in white light interferometry," Journal of Optical Society of America 13(4), pp. 83{843, P. de Groot and L. Deck, \Three-dimensional imaging by sub-nyquist sampling of white-light interferograms," Optics letters 18(17), pp. 146{1464, P. de Groot and L. Deck, \Surface proling by analysis of white-light interferograms in the spatial frequency domain," Journal of Modern Optics 4(), pp. 389{401, \ 11. R.-J. Recknagel and G. Notni, \Analysis of white light interferograms using wavelet methods," Optics Communicatins 148, pp. 1{18, A. Hirabayashi, H. Ogawa, T. Mizutani, K. Nagai, and K. Kitagawa, \Fast surface proling by white-light interferometry using a sampling theorem for band-pass signals," Trans. Society of Instrument and Control Engineers 36(1), pp. 16{5, 000. (in Japanese). 13. \ 14. A. Kohlenberg, \Exact interpolation of band-limited functions," Journal of Applied Physics 4, pp. 143{1436, R. Marks II, Introduction to Shannon Sampling and Interpolation Theory, Springer-Verlag, New York, A. J. Jerri, \The Shannon sampling theorem - its various extensions and applications: A tutorial review," Proc. IEEE 65(11), pp. 1565{1596, A. I. Zayed, Advances in Shannon's Sampling Theory, CRC Press, New York, I. Someya, Waveform Transmission, Shukyosha, Tokyo, C. E. Shannon, \Communications in the presence of noise," Proc. IRE 37, pp. 10{1,