White-light interferometry, Hilbert transform, and noise Pavel Pavlíček *a, Václav Michálek a a Institute of Physics of Academy of Science of the Czech Republic, Joint Laboratory of Optics, 17. listopadu 50a, Olomouc, 772 07, Czech Republic; ABSTRACT White-light interferometry is an established and proved method for the measurement of the geometrical shape of objects. The advantage of white-light interferometry is that it is suitable for the measurement of the shape of objects with smooth as well as rough surface. The information about the longitudinal coordinate of the surface of the measured object is obtained from the white-light interferogram. The interferogram is the intensity at the detector expressed as the function of the position of the object. (The object is moved along the optical axis during the measurement process.) If the shape of an object with rough surface is measured, the phase of the interferogram is not evaluated because it is a random value. The information about the longitudinal coordinate is obtained from the center of the interferogram envelope. A classical method for the calculation of the envelope of white-light interferogram is the demodulation by means of Hilbert transform. However, the electric signal at the output of the camera is influenced by the noise. Therefore, as expected, the calculated envelope is also influenced by the noise. The result is that the measured longitudinal coordinate of the surface of the object is affected by an error. In our contribution, we look for the answer on following questions: How does the noise of the evaluated envelope differ from the noise of the interferogram? What is the minimal measurement uncertainty that can be achieved? Keywords: White-light interferometry, Hilbert transform, noise, measurement uncertainty 1. INTRODUCTION White-light interferometry is used to measure the geometrical shape of objects. The height variations of the measured object can be found in range from nanometers up to millimeters. White-light interferometry can be used for the measurement of the shape of objects with smooth as well as with rough surface. In this paper, we will deal with whitelight interferometry on rough surfaces [1]. The experimental setup of white-light interferometry is illustrated in Fig. 1. Usually a Michelson interferometer is used. The light source has a broadband spectrum. Suitable light sources for white-light interferometry are light-emitting diode, superluminescent diode or an incandescent lamp. The measured object is placed in one arm of the interferometer. The other arm is terminated by reference mirror. The light from the light source passes the beam-splitter and is reflected from the measured object as well as from the reference mirror. The reflected light from both interferometer arms is transmitted through an imaging system to a CCD camera. During the measurement, the object is moved along the optical axis as indicated by the arrow in Fig. 1. The intensity in each pixel is recorded as a function of the longitudinal position of the measured object. Such a record is called interferogram. An example of the interferogram is shown in Fig. 2(a). White-light interferometry on rough surfaces is based on interferometry in individual speckles. Therefore it is important that the speckle size corresponds roughly with the size of the pixel of the camera. The speckle size can be controlled by the stop in the imaging system. Thus, an interferogram is acquired for each individual speckle. The longitudinal coordinate of an object point is determined from the position of the center of the interferogram. The center of the interferogram is denoted by letter C in Fig. 2(a). As the measured surface is rough, the phase of the interferogram is a random value [1]. Therefore it has no significance to evaluate the phase of the interferogram. The longitudinal coordinate of the object surface is determined from the interferogram envelope only. The envelope of the * Pavel.Pavlicek@upol.cz; phone +420 585 631 680; fax +420 585 631 531; jointlab.upol.cz
interferogram from Fig. 2(a) is depicted in Fig. 2(b). The process of extracting the envelope from the interferogram is called demodulation. A classical method for the demodulation of the interferogram is the detection by means of Hilbert transform [2]. Figure 1. The schematic of the experimental setup of white-light interferometry. Figure 2. (a) Example of white-light interferogram - the original signal. (b) The envelope of white-light interferogram the signal after demodulation. (c) The noise extracted from the original signal. (d) The noise extracted from the signal after demodulation. In the real measurement, the interferogram is corrupted by the noise. Therefore, as one can expect, the interferogram envelope is also corrupted by the noise. Our goal is to determine how the noise of the interferogram affects the interferogram envelope. We investigate the influence of the noise on the measurement uncertainty of white-light interferometry on rough surfaces.
2. MEASUREMENT UNCERTAINTY The measurement uncertainty of white-light interferometry on rough surface is given by the ability of the evaluation algorithm to find the center of the interferogram envelope. The center of the interferogram envelope cannot be found absolutely accurately because the interferogram is affected by the noise. The theoretical limit of the evaluation algorithm can be calculated by means of Cramer-Rao inequality [3]. It shows that the theoretical limit of the search for the center of a modulated signal with a Gaussian envelope is given by 2 I δz = 2 4 l z, A where σ is the standard deviation of the noise, I A is the amplitude of the signal, l c is the coherence length of the used light and z is the sampling step. The meaning of the amplitude I A, the coherence length l c and the sampling step z is illustrated in Fig. 3. Coherence length is defined as the distance from the center of the interferogram to the value of longitudinal coordinate at which the envelope of the interferogram is equal to 1/e of its maximal value. c (1) Figure 3. The meaning of the amplitude I A, the coherence length l c, the sampling step Δz, and the wavelength λ. The marks indicate the sampled values. 3. CORRELATION OF THE NOISE The noise extracted from the interferogram is shown in Fig. 2(c). We assume that the noise of the interferogram is an uncorrelated noise with standard deviation equal to σ. The noise of the interferogram may be assumed to be uncorrelated because the individual intensities measured in one pixel of the camera for various positions of the measured object are independent. Figures 4(a) and (c) show the noise of the interferogram and its spectral density. Because the noise is uncorrelated, its spectral density is constant and equal to the variance σ 2 of the noise [4]. The envelope of the interferogram is calculated by means of Hilbert transform. The demodulation by Hilbert transform changes the form of the spectral density of the noise as shown in Fig. 4(d). The spatial frequency shift ν 0 that is observed in Fig. 4(d) is given by the wavelength λ of the used light source: ν 0 = 2/λ. Because of the Michelson arrangement of the interferometer, the period of the modulation of the interferogram is equal to λ/2 as shown in Fig. 3. The maximal spatial frequency ν max is determined by the sampling step: ν max = 2/z. The noise extracted from the envelope is shown in Fig. 4(b). The variance of the noise is equal to the mean value of the spectral density [4]. It is apparent from Fig. 4(d) that the variance of the noise of the envelope is equal to σ 2. This means that the standard deviation of the noise of the envelope is the same as that of the noise of the interferogram. Further, it is apparent from Fig. 4(d) that the noise of the envelope is no more uncorrelated because its spectral density is not constant. The process of demodulation by means of Hilbert transform conserves the variance of the noise but causes the correlation of the noise. The correlation of the noise is also apparent in Fig. 4(b).
The autocovariance function of the noise is calculated as the inverse Fourier transform of the spectral density of the noise. The noise of the interferogram is uncorrelated, its spectral density is a constant function with the value σ 2. Accordingly, the autocovariance function of the noise of the interferogram has the value σ 2 at 0 and the value 0 elsewhere. The spectral density of the noise of the interferogram and the corresponding autocovariance function are shown in Fig. 5(a) and (c), respectively. The spectral density of the noise of the envelope is shown in Fig. 5(b). The autocovariance function of the noise of the envelope is shown in Fig. 5(d). The variance of the noise is equal to σ 2 in both cases: the noise of the interferogram and the noise of the envelope. (The variance of the noise is equal to the autocovariance function at 0.) For both cases, the noise of the interferogram and the noise of the envelope, the variance of the noise (which is equal to the autocovariance function at 0) is equal to σ 2. Figure 4. (a) The noise extracted from the original signal. (b) The noise extracted from the signal after demodulation. (c) Spectral density of the noise of the original signal. (d) Spectral density of the noise of the signal after demodulation. In this example λ/δz = 10 (10 samples within the distance of one wavelength). Figure 5. (a) Spectral density of the original signal noise. (b) Spectral density of the noise of the signal after demodulation. (c) Autocovariance function of the noise of the original signal. (d) Autocovariance function of the noise of the signal after demodulation.
4. INFLUENCE OF THE CORRELATION OF THE NOISE The noise of the envelope is correlated. Therefore, if the minimal achievable measurement uncertainty is calculated from the envelope by means of Cramer-Rao inequality, the inverse of the correlation matrix must be involved. The correlation matrix r k,l of the noise can be constructed from the autocovariance function j kl kl 0 r kl = (2) 2 shown in Figs. 5(c) and (d). The construction of the correlation matrix is illustrated in Fig. 6. Figure 6. The construction of the correlation matrix from the autocovariance function. For the uncorrelated noise, 0 = (0) = 2 and i = (iz) = 0 for all i > 0. Therefore, the correlation matrix of an uncorrelated noise is an identity matrix. According to Eq. (2), the correlation depends only on the distance between two points. Therefore the correlation matrix of the noise extracted from the interferogram envelope is a symmetric Toeplitz matrix. A Toeplitz matrix is a matrix in which each descending diagonal from left to right is constant as can be seen in Fig. 6 [5]. The inverse of a Toeplitz matrix is generally not a Toeplitz matrix. However, for large dimensions, the inverse correlation matrix has nearly Toeplitz form and can be easily expressed [4]. The inverse k,l of the correlation matrix is calculated from function j 2 kl = k l. (3) The autocovariance function is the inverse Fourier transform of the spectral density () = 1 FT (4) and the function is the inverse Fourier transform of the reciprocal of the spectral density = 1 1 FT (5). By inserting of the inverse correlation matrix into Cramer-Rao inequality, the measurement uncertainty is by a factor of square root of two (1.414) higher than that for the uncorrelated noise [3]. This is an important result. The measurement uncertainty of the interferogram is by factor square root of two higher than the measurement uncertainty of a signal with the same form as the interferogram envelope. The reason is the correlated noise of the envelope of the interferogram. The increase factor does not depend on the coherence length l c, wavelength and step size z.
5. CONCLUSIONS The envelope detection by means of Hilbert transform transforms the noise of the interferogram. If the interferogram is affected by a normally distributed uncorrelated noise, the envelope noise has the same standard deviation but it is correlated. The autocovariance function of the envelope noise depends on the ratio between the mean wavelength of the used light and the sampling step. The correlation of the noise causes that the measurement uncertainty is by factor of square root of two higher than it would be for a signal with the same form but without modulation. ACKNOWLEDGEMENTS This research was supported financially by the project TA 01010517. REFERENCES [1] Dresel, T., Häusler, G., and Venzke, H., "Three-dimensional sensing of rough surfaces by coherence radar," Appl. Opt. 31, 919 925 (1992). [2] Larkin, K. G., "Efficient nonlinear algorithm for envelope detection in white light interferometry," J. Opt. Soc. Am. A 13, 832 843 (1996). [3] Pavliček, P. and Michalek, V., "White-light interferometry Envelope detection by Hilbert transform and influence of noise," Opt. Lasers. Eng. 50, 1063 1068 (2012). [4] Helstrom, C. W., [Elements of signal detection & estimation], PTR Prentice Hall, Englewood Cliffs (1995). [5] Golub, G. H. and van Loan, C. F., [Matrix computation], The John Hopkins University Press, London (1996).