Interferometric diagnostics of ablation craters formed by femtosecond laser pulses

Transcription

1 Interferometric diagnostics of ablation craters formed by femtosecond laser pulses A. S. Zakharov, M. V. Volkov, and I. P. Gurov St. Petersburg State Institute of Precision Mechanics and Optics (Technical University), St. Petersburg V. V. Temnov Institute for Laser and Plasma Physics at the University of Essen, Essen, Germany; Institute of Applied Physics of the Russian Academy of Sciences, Nizhni Novgorod K. Sokolovski-Tinten and D. von der Linde Institute for Laser and Plasma Physics at the University of Essen, Essen, Germany Submitted January 24, 2002 Opticheski Zhurnal 69, July 2002 It is proposed to use methods of interferometry for quantitative analysis of the removal of material from the surface of a solid under the action of femtosecond laser pulses laser ablation craters. The characteristics of the two-dimensional discrete nonlinear Kalman filter for noise-resistant recovery of the deconvolved phase of the pattern of interference fringes characterizing the ablation craters. We present the results of a processing of the two-dimensional interferograms obtained in reflection of light from gallium arsenide and zinc surfaces before and after a high-power femtosecond laser pulse acted on them. The experiment results confirm the high noise resistance of the proposed method of recovering the phase of the two-dimensional interference fringes Optical Society of America INTRODUCTION The removal of material from a solid surface under the action of ultrashort laser pulses laser ablation is of particular interest from the standpoint of investigating little-studied fundamental mechanisms of interaction of high-power laser radiation with matter. 1,2 The technology of surface modification by femtosecond laser pulses is widely used to process various materials and for microstructuring of surfaces, and also in medicine, in the restoration of art objects, and other areas. The interferometric diagnostic technique presented in this paper, which uses the illumination of an interferometer by femtosecond laser pulses, permits a quantitative characterization of the relief of the ablation craters, which have a typical depth of several tens of nanometers. A special role in this is played by the computer algorithms needed for processing and reconstruction of the two-dimensional phase function of the interference fringes. The widely used methods of phase-shearing interferometry 3 are unsuitable for dynamic measurements, since they require recording, with a time separation, several fringe patterns with a specified phase shift. The methods based on the Fourier Hilbert transform 4,5 in many cases does not provide the necessary noise resistance in the processing and deconvolution of the two-dimensional phase function of the real distorted patterns of interference fringes. Noise resistance of the processing of the fringe patterns is made possible by the use of parametric models for the signals that determine the interference fringes, in particular, models of stochastic processes in the form stochastic difference equations. 6,7 On this basis it is possible to construct recursive algorithms with a high resolving power, ensuring noise-resistant recovery of the deconvolution total phase of the interference fringes. 7 9 An example of a recursive algorithm of this type is discrete linear Kalman filtering. 10 However, the linear filter does not permit optimal estimates of parameters on which the fringe signal depends nonlinearly, in particular, the value of the phase of the fringes. In this paper we consider the possibility of using the method of nonlinear discrete Kalman filtering for processing the pattern of interference fringes with recovery of the variations of the deconvolved two-dimensional phase function of the fringes characterizing the relief of the craters form in various materials during femtosecond laser ablation. PROCESSING OF TWO-DIMENSIONAL PATTERNS OF INTERFERENCE FRINGES BY THE METHOD OF NONLINEAR KALMAN FILTERING A nonlinear Kalman filter is intended for obtaining dynamic estimates of parameters that are nonlinearly related to the signal values. Such a filter is specified 9,10 by the observation equation s k h k n k 1 and the equation of the system k 1 k f k x w k. 2 In Eqs. 1 and 2 s(k) is the observed signal vector, (k) is the parameter vector, h( ) and f( ) are known nonlinear vector functions, x is the discretization step, n(k) is the noise of the observations, and w(k) is the forming noise. The filtering algorithm has a recursive character and is based on prediction of the values of the parameter vector pr (k) at the next discretization step from the observation data. If one is able to find an estimate of the parameter vector pr (k), then for determination of the error of this estimate cr (k) (k) 478 J. Opt. Technol. 69 (7), July /2002/ $ The Optical Society of America 478

2 pr (k) one can use the well-known method of onedimensional discrete linear Kalman filtering. 6,10 Recovery of the phase of two-dimensional interference fringe patterns requires a two-dimensional nonlinear Kalman filter. Let us first discuss the procedure for nonlinear Kalman filtering of a one-dimensional discrete signal. According to the observation equation 1, the error of prediction of the signal values is given as s er k h pr k er k h pr k n k. 3 Here the equation of the system 2 is transformed to er k 1 er k f pr k er k f pr k x w k. 4 The functions h( ) and f( ) in 1 and 2 can be expanded in a Taylor series in the form h pr k er k h pr k h pr k er k, 5 f pr k er k f pr k f pr k er k. From Eqs. 3 6 we obtain s er k h pr k er k n k, er k 1 er k f pr k pr k x w k. 8 Taking into account that (k) pr (k) er (k), we obtain dynamic estimates of the unknown parameter vector (k) from Eq. 8. Expressions 7 and 8 can be used to synthesize a onedimensional quasi-optimal discrete nonlinear Kalman filter that can be used for row-by-row processing of twodimensional patterns of interference fringes. The features of the row-by-row dynamic filtering of fringe patterns are discussed in detail in Ref. 9. The recorded fringe pattern is a two-dimensional matrix of discrete readings. Independent filtering of the data sequences along individual rows can lead to distortion of the overall fringe pattern, since at two adjacent points of a column of data the errors of estimation of the phase can be significantly different from each other if the filtering of the individual rows is done independently, and significant errors in the determination of the spatial frequency in the vertical direction can arise. A quasi-two-dimensional processing alternately along rows and columns does not solve this problem, since then one cannot completely eliminate large phase jumps. Twodimensional processing of the interference pattern is necessary, with the predictions based not only on the parameter values at the previous point of the row but also on those at the previous point of the column of data containing the values of the intensity of the light in the pattern of interference fringes. For two-dimensional processes of the interference fringes by means of a discrete nonlinear Kalman filter it is necessary to estimate a parameter vector of the form (s 0,s m,,u H,u V ) T, where s 0 is the background component of the interference pattern, s m is the amplitude of the fringes, is the deconvolved phase of the fringes, u H is the spatial frequency in the horizontal direction, and u V is the spatial frequency in the vertical direction. In our version of 6 7 the two-dimensional nonlinear Kalman filter the signal parameters at a point with discrete coordinates (i, j) were predicted according to the simple rules s 0 i, j s 0 i 1,j s 0 i, j 1 /2, 9 s m i, j s m i 1,j s m i, j 1 /2, i, j i 1,j 2 u V i 1,j y i, j 1 2 u H i, j 1 x /2, u H i, j u H i 1,j u H i, j 1 /2, u V i, j u V i 1,j u V i, j 1 /2. 13 In Eq. 11 and below, x and y denote the discretization steps in the horizontal and vertical directions, respectively. As the observation vector at the point (i, j) we used a vector of the form s(i, j),s(i, j 1),s(i 1,j),s(i 1,j 1) T. For prediction of the vector at each point we have used the formulas s i, j s 0 i, j s m i, j cos i, j, 14 s i, j 1 s 0 i, j s m i, j cos i, j 2 u H i, j x, s i 1,j s 0 i, j s m i, j cos i, j 2 u V i, j y, s i 1,j 1 s 0 i, j s m i, j cos i, j u H i, j x 2 u V i, j y. 17 The initial conditions for the calculations using formulas 9 17 were determined by starting from preliminary rough estimates of the parameter vector. In particular, the estimate of the background component of the fringe pattern was determined by the average value of the intensity, the estimate of the fringe amplitude by the variance of the variable component, the spatial frequencies were calculated according to the average number of fringes in the pattern, and the value of the phase at the initial point was assumed equal to zero. Because of the locally adaptive character of the algorithm for two-dimensional recursive filtering, the accuracy of these preliminary estimates was sufficient for subsequent stable deconvolution of the two-dimensional phase function of the fringe pattern. EXPERIMENTAL APPARATUS FOR THE FORMATION AND INTERFEROMETRIC DIAGNOSTICS OF ABLATION CRATERS In the experimental apparatus Fig. 1, a detailed description of which is given in Ref. 2, a high-power femtosecond pump pulse is used for excitation of the sample. After a fixed delay time another weak femtosecond pulse probe pulse is fed into a Linnik microinterferometer, and a portion of the radiation that has passed through a microobjective is used to illuminate the excited part of the surface, which is a plane object. A CCD recording camera is placed in the image plane, and at the exit from the objective the reflected probe pulse interferes with the reference pulse, which has passed 479 J. Opt. Technol. 69 (7), July 2002 Zakharov et al. 479

3 terferograms is to extract information about the dynamic excitation and the final modification of the surface by comparison of the two-dimensional deconvolved phase functions, the initial and dynamic, and also of the initial and final interferograms. The difference of the two-dimensional phase functions of the fringes, which is recovered in the processing of the interferograms, yields quantitative information about the ablation crates. The reconstruction of the final relief of the craters as a result of the action of a femtosecond laser pulse on a substance is an important problem. In the present study it was solved by using the nonlinear Kalman filtering method discussed above. Recovery of the deconvolved two-dimensional phase functions of the initial and final interferograms was achieved, with a reconstruction of the final relief of the craters formed on gallium arsenide and zinc surfaces. FIG. 1. Diagram of experimental layout: 1 pump pulse, 2 probe pulse, 3 sample under study, M mirror. through an identical reference arm of the Linnik microinterferometer. Recorded as a result of each measurement, according to the technique of Ref. 2, are an initial interferogram of the unexcited surface, a dynamic interferogram of the excited surface after a fixed time delay following the action of the pump pulse with a typical time delay of up to several nanoseconds, and an interferogram of the final modification of the surface, obtained several seconds after the laser excitation. The main problem of processing the in- RESULTS OF EXPERIMENTS ON THE INTERFEROMETRIC DIAGNOSTICS OF ABLATION CRATERS MADE BY FEMTOSECOND LASER PULSES Figure 2 shows interferograms of a gallium arsenide surface before Fig. 2a and after Fig. 2b it was acted on by a high-power femtosecond laser pulse with an energy of several times the ablation threshold. 1 Figure 2c shows a halftone picture of the calculated difference of the deconvolved two-dimensional phase functions obtained with nonlinear Kalman filtering of the fringe patterns of Figs. 2a and 2b. Figure 2d shows the profile of the phase difference in the median horizontal cross section of the two-dimensional func- FIG. 2. Experimental interferograms obtained for gallium arsenide a,b, the difference of the deconvolved phase functions c, and the change in the phase difference in the median horizontal cross section d, characterizing the relief of the crater. 480 J. Opt. Technol. 69 (7), July 2002 Zakharov et al. 480

4 FIG. 3. Experimental interferograms for a zinc sample a,b, the reconstructed fringe pattern c, and the deviations of the phases of the fringe patterns d. tion in Fig. 2c; this profile gives a quantitative characterization of the profile of the crater right-hand scale in Fig. 2d. The crater depth is directly proportional to the measured phase difference, with a coefficient of proportionality /4, where 400 nm is the wavelength of the probe pulse radiation. The obtained crater depth of around 50 nm agrees with the results of direct measurements of the crater profiles with the aid of an atomic force microscope. The experimental interferograms obtained for an irregular zinc surface Figs. 3a and 3b that had been subjected to etching is more complicated to process. The characteristics of the interferograms are determined by the properties of the material and the conditions of light reflection. Some of the fringes have breaks and other local defects. The visibility of the interference fringes varies significantly over the field of the interferogram, especially for the interferogram obtained after formation of the crater Fig. 3b. These features do not permit stable recovery of the phase, to say nothing of its deconvolution by conventional methods including that based on the Fourier transform. For processing of the interferograms of Figs. 3a and 3b we used the method of preliminary amelioration of the fringe patterns, which is based on a modification of the local halftone histograms and has been successfully implemented in the processing of fringe patterns in moiré and holographic interferometry. 11,12 The features of the method used for amelioration of the interferograms is discussed in detail in Ref. 12. The ameliorated interferograms were subjected to twodimensional nonlinear Kalman filtering according to the algorithm described above for obtaining two-dimensional deconvolved phase functions of the fringes. Figure 3c shows a fringe pattern reconstructed from the values of the total phase of the interferogram of Fig. 3b after filtering by a nonlinear two-dimensional Kalman filter. A comparison of Fig. 3b and 3c clearly demonstrates the high efficiency of the interferogram processing methods developed here. Figure 3d shows a half-tone picture of the differences of the deconvolved phases of the interferogram of Figs. 3a and 3b. The profile of the phase differences in the median horizontal cross section of Fig. 3d is shown in Fig. 4. This represents the profile of an ablation crater whose depth is approximately 60 nm. It must be emphasized that the depth of the crater is substantially smaller than the characteristic amplitude of the initial surface roughness, confirming the high noise resistance of the interferogram processing methods used. It is important to note that the processing of the fringe patterns by the method of Kalman filtering does not introduce appreciable distortions at the boundaries of the fringe pattern and ensures stable filtering of the deconvolved phase under conditions when noise and distortions are present in the fringe pattern. CONCLUSION As a result of the processing of the experimental interferograms by a discrete nonlinear two-dimensional Kalman filter we obtained quantitative information about the structure of ablation craters formed on the gallium arsenide and zinc surfaces by the action of high-power single-shot femtosecond laser pulses with an energy several time greater than the ablation threshold. 1 For the results obtained one can conclude that a nonlinear Kalman filter may be used for processing the interference fringes with locally varying parameters under conditions where noise is present. An important advantage of a filter of this kind is the possibility of analyzing the interference field to obtain the optimal dynamic estimates of the deconvolved phase of the fringes at local points. In the case of twodimensional nonlinear filtering one achieves significantly lower rms and maximum errors of estimation of the parameters than for a row-by-row processing of the image. FIG. 4. Variation of the phase difference of Fig. 3d in the median horizontal cross section, characterizing the relief of the crater. 481 J. Opt. Technol. 69 (7), July 2002 Zakharov et al. 481

5 An advantage of the Kalman filter is the recovery of the two-dimensional total phase of the fringes, whereas the Fourier transform can yield only the reduced phase in the interval, and requires solving an additional problem of deconvolution of the phase. The Kalman filter, unlike the fast Fourier transform, does not require a specified format of the initial data dimensions of the matrix of readings. This property makes it possible to rapidly process fringe patterns obtained with professional cameras for scientific studies with any image format. 1 K. Sokolowski-Tinten, J. Bialkowski, A. Cavalleri, D. von der Linde, A. Oparin, J. Mayerter-Vehn, and S. I. Anisimov, Transient states of matter during short pulse laser ablation, Phys. Rev. Lett. 81, V. V. Temnov, K. Sokolowski-Tinten, and D. von der Linde, Ultrafast time- and space-resolved Michelson interferometry, J. Opt. Soc. Am. B 2002 to be published. 3 K. Creath, Phase measurement interferometry techniques, Progr. Opt. 26, M. Takeda, H. Ina, and S. Kobayashi, Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry, J. Opt. Soc. Am. A 72, C. Roddier and F. Roddier, Interferogram analysis using Fourier transform techniques, Appl. Opt. 26, V. N. Vasil ev and I. P. Gurov, Computer Processing of Signals in Application to Interferometric Systems in Russian, BKhV-Sankt-Peterburg, St. Petersburg I. Gurov and D. Sheynihovich, Interferometric data analysis based on Markov nonlinear filtering methodology, J. Opt. Soc. Am. A 17, I. P. Gurov, Analysis of interference fringes based on the recurrence nonlinear filtering methodology, Opt. Zh. No. 7, J. Opt. Technol. 67, I. Gurov and A. Zakharov, Dynamic nonlinear analysis of stochastic interference fields, International Conference on Coherent and Nonlinear Optics ICONO 01, Minsk, 2001; to be publ. in Proc. SPIE E. Lloyd and U. M. Liderman eds., Handbook of Applied Statistics, Vol. 2 Finansy i Statistika, Moscow, S. De Nicola, P. Ferraro, I. Gurov, R. Koviazin, and M. Volkov, Fringe analysis for moiree interferometry by modification of the local intensity histogram and use of a two-dimensional Fourier transform, Meas. Sci. Technol. No. 9, I. Gurov and M. Volkov, Distorted noisy interferograms enhancement and evaluation by the nonlinear 2D data-dependent fringe processing, in Optical Measurement Systems for Industrial Inspection. II: Application in Industrial Design, edited by O. W. Jueptner W.P.O. and M. Kujawinska, Proc. SPIE 4398, J. Opt. Technol. 69 (7), July 2002 Zakharov et al. 482