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Phase-stepping technique with an electro-optic crystal in digital speckle pattern interferometry
15 April 1998
Optics Communications 149 Ž1998. 235–238
Phase-stepping technique with an electro-optic crystal in digital speckle pattern interferometry Luciano Angel 1, Myrian Tebaldi 2 , Rodrigo Henao, Alberto Tagliaferri 3, Marcelo Trivi 4 , Nestor Bolognini 2 , Roberto Torroba 4 ´ Centro de InÕestigaciones Opticas, P.O. Box 124, (1900) La Plata, Argentina Received 17 September 1997; revised 6 January 1998; accepted 7 January 1998
Abstract The use of an electro-optic crystal as phase-stepping device in a digital speckle interferometer is discussed. Phase stepping is introduced by varying the external voltage applied to the crystal. The crystal calibration procedure is outlined and errors are discussed. Experimental results compared to those obtained with a piezo-electrically driven mirror are presented. q 1998 Elsevier Science B.V. PACS: 42.25.Hz; 42.25.Lc; 42.30.Ms Keywords: BSO crystal; Phase stepping; Metrology; Digital speckle pattern interferometry
Interferometry can be used to produce a fringe pattern that represents the field surface displacement of an object in response to some change in mechanical loading w1x. Digital speckle pattern interferometry ŽDSPI. is one of the most modern techniques for depicting such fringe patterns. It combines real-time processing with the flexibility of software handling w2x. DSPI was developed by combining the well-known techniques of holographic and speckle interferometry by using an image hologram setup and following the methods of double-exposure holography. It utilizes a CCD camera interfaced to a computer to process the data. For comparison measurements, a reference frame stored in memory is continuously subtracted
1
Permanent address: Universidad EAFIT, Medellın, ´ Colombia. Also at Facultad de Ciencias Exactas, Universidad Nacional de La Plata, Argentina. 3 Permanent address: Instituto de Fisica, Universidade Federal Fluminense, Niteroi, Rio Janeiro, Brazil. 4 Also at Facultad de Ingenierıa, ´ Universidad Nacional de La Plata, Argentina. 2
from incoming data, and then the intensity difference is displayed. The fringes represent the correlation between the speckle patterns, with the same interpretation as for those fringes obtained by holographic interferometry. However, for quantitative purposes, interferograms must be analyzed so that the results can be presented in the required numerical form w3x. Progress has been made in fringe pattern analysis with a number of electronic aids. These devices allowed a substantial improvement in the accuracy of fringe location within an interferogram w4x. The wide availability of digital image processing equipment prompted a number of studies to investigate different alternatives of automatic interferogram analysis. Modern techniques have reached a point at which they can provide useful results, particularly with phase-stepping methods w5x. By introducing discrete shifts in the position of the fringes, we can calculate the phase map at pixel location Ž i, j . in a relatively simple way. The phase-stepping technique is a simplification of the intensity integration method and requires a minimum of three measurements to solve for three unknowns in the interference equation. The phase is usually changed when a mirror with a computer-
0030-4018r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 0 1 3 - 3
L. Angel et al.r Optics Communications 149 (1998) 235–238
236
controlled piezoelectric translator is moved. These phase shifts can be introduced either in the object or in the reference beam. Errors in phase measurements are introduced because the desired phase steps are not achieved either because of miscalibration or nonlinearity of the piezoelectric transducer w6x. There are many algorithms used to extract phase, and a certain combination of them can minimize phase-stepper miscalibrations and nonlinearity errors w7x. However, a reduction of these errors is desirable by using another method to introduce phase-steps without mechanical movements. To demonstrate the technique, we chose an electro-optic BSO crystal located in one beam of the DSPI. Its use in DSPI systems has been already implemented as a fringe visibility control device w8x. Our approach is based on the response of such crystal to the application of a constant electric voltage difference. According to the electro-optic effect, the application of such voltage induces a birefringence on the crystal w9x. We make use of this property to introduce the desired phase-steps without mechanical movements. In the following, we will discuss the characteristics and calibration procedure using a BSO crystal as phase-stepping device. Some experimental results will be shown, observing an adequate coincidence with those obtained by using a PZT under the same experimental conditions. The basic configuration that we use is shown in Fig. 1 w10x. The laser beam is divided in two by a beam splitter, and then each beam is expanded. Both beams are equivalent to obtaining two different speckle patterns from the same surface, one that serves as a reference wave and the other as an object wave. The BSO crystal is placed in one arm being uniformly illuminated with an expanded parallel beam, perpendicular to the incident face. The directions of the crystal, ²001:, ²110: and ²110: coincide with the XYZ axes respectively, being Z the propagation axis of the DSPI interferometer beam. The physical dimensions are L X s 10 mm, L Y s 4 mm and LZ s 11 mm. For a voltage V0 applied between the ²001: faces separated a distance L X , the crystal becomes birefringent. In this configuration, the expression of birefringence results:
d n s 12 n30 r41
V0 LX
,
Fig. 2. Ža. Interferometric fringes, Žb. phase map and Žc. 3D plot of a tilted plane surface obtained with the BSO crystal.
difference introduced by the crystal on the interferometer beam can be expressed as:
Ž1. Dws
where r41 is the electro-optic coefficient and n 0 is the refractive index without induced field. In our case n 0 s 2.54 and r41 s 4.4 pmrV. According to this, the phase
Fig. 1. Experimental setup.
2p
l
d n LZ ,
Ž2.
where l s 633 nm is the laser wavelength. In our scheme we adopt the three equations method requiring pr2 steps. To introduce every phase difference of pr2 we need to apply a voltage of 4 kV each time to the particular crystal we employed. Thus, to achieve a phase difference of p, we apply 8 kV, and so on. High contrast reference correlation fringes are obtained by introducing a slight tilt in the object. An external voltage is then applied to the crystal, which is increased until the reference fringe system drifts in pr2 at the TV monitor. The voltage value for this drift corresponds to the theoreti-
L. Angel et al.r Optics Communications 149 (1998) 235–238
cally predicted. The accuracy of the method depends on the accuracy of determining the voltage signal. To achieve this accuracy, the high voltage applied was driven by a ramp generator and measurements were made by comparing the input drive trace and the corresponding output from the high voltage unit. In our experiment, phase shifts of pr2 were measured with an accuracy of 0.28 and repeatability of 0.0002. We used this technique in many experiments employing the BSO crystal. We present here results for a tilted object and a deformation on a thin membrane loaded from the rear. The same experiments were conducted with a piezo-electrically driven mirror, as phase stepping device, observing discrepancies in the results below 2%. Fig. 2 shows Ža. the fringes, Žb. the phase map and Žc. the 3D plot of a tilted surface, obtained with the BSO crystal. Fig. 3 displays the case of the deformed membrane also showing Ža. the fringes, Žb. the phase map and Žc. the 3D plot. In conventional systems a calibration step is required according to the set-up geometry. In our case such calibration procedure has been carried out only in the crystal, because the phase change introduced by the crystal on the transmitted wave only depends on its own characteristics Želectro-optic coefficient, size, refraction index, etc... Once the calibration has been fulfilled, the phase step is independent of the particular set-up geometry involved. Besides, no extra drifts are introduced by external influences Žtemperature, humidity, etc.. under normal working laboratory conditions. Also, there is no need for a feedback mechanism to active compensation as that reported in other methods. When the incident light is linearly polarized, the polarization state of the light transmitted through the crystal depends on the thickness as well as on the applied field and the rotatory power. As was demonstrated in Ref. w11x, this should cause a decrease in fringe visibility. There, the visibility decay is analyzed in terms of the polarization state rotation, on the basis of a parameter related to the phase change value responsible for the speckle correlation fringes generation. In our experiment, this phase change corresponds to that introduced by object rotation or irregular deformation of the loaded plate. As in Ref. w11x, we maintained this phase change in the range of few degrees. Furthermore, according to the crystal used and our experimental conditions, the polarization state rotation of the beam emerging from the crystal, experimentally measured, was one and two degrees for an applied voltage of 4 kV and 8 kV, respectively. Then, the resulting polarization state lies within a tolerance range such that the visibility of the fringes is barely affected, less than 2%. In conclusion, the experiments conducted have shown that the use of a BSO crystal as a phase-stepping device in a digital speckle pattern interferometer is not only possible in practice, but is also simpler than using piezo-electrically driven mirrors. It was also demonstrated that it provides an
237
Fig. 3. Ža. Interferometric fringes, Žb. phase map and Žc. 3D plot of the clamped metal plate loaded with a point pressure applied on it.
alternative useful and convenient method of analyzing general deformations both quantitatively and qualitatively. Further research has been conducted on evaluating other type of crystals and working conditions to optimize the proposed technique.
Acknowledgements This research was performed under the auspices of the National Research Council, Grant Nr. 6825r96, PICT 0041 and 0249 ŽArgentina., Fundacion ´ Antorchas ŽArgentina., CICPBA ŽArgentina., Facultad de Ingenierıa ´ Universidad Nacional de La Plata ŽArgentina., the Na-
238
L. Angel et al.r Optics Communications 149 (1998) 235–238
tional Research Council CNPq, FAPERJ and FINEP ŽBrazil.. M. Trivi is researcher of Comision ´ de Investigaciones Cientıficas de la Provincia de la Buenos Aires ´ ŽArgentina.. L. Angel also acknowledges Instituto Colombiano para el Desarrollo de la Ciencia y la Tecnologıa ´ ŽColombia.. References w1x R. Jones, C. Wykes, Holographic and Speckle Interferometry, Cambridge Univ. Press, Cambridge, UK, 1983, pp. 183– 188.
w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x
K. Creath, G. Slettemoen, J. Opt. Soc. Am. A 2 Ž1985. 1629. D.W. Robinson, Appl. Optics 22 Ž1983. 2169. D. Robinson, D. Williams, Optics Comm. 57 Ž1986. 26. R. Henao, H. Rabal, A. Tagliaferri, R. Torroba, Appl. Optics 36 Ž1997. 2066. J. Van Wingerden, H.J. Frankena, C. Smorenberg, Appl. Optics 30 Ž1991. 2718. C. Joenathan, Appl. Optics 33 Ž1994. 4147. J. Pomarico, R. Torroba, N. Bolognini, Optik 99 Ž1995. 89. A.M. Glass, Opt. Eng. 17 Ž1978. 470. C. Joenathan, R. Torroba, Optics Lett. 15 Ž1990. 1159. R. Henao, A. Tagliaferri, R. Torroba, Optics Comm. 127 Ž1996. 14.
Optics Communications 149 Ž1998. 235–238
Phase-stepping technique with an electro-optic crystal in digital speckle pattern interferometry Luciano Angel 1, Myrian Tebaldi 2 , Rodrigo Henao, Alberto Tagliaferri 3, Marcelo Trivi 4 , Nestor Bolognini 2 , Roberto Torroba 4 ´ Centro de InÕestigaciones Opticas, P.O. Box 124, (1900) La Plata, Argentina Received 17 September 1997; revised 6 January 1998; accepted 7 January 1998
Abstract The use of an electro-optic crystal as phase-stepping device in a digital speckle interferometer is discussed. Phase stepping is introduced by varying the external voltage applied to the crystal. The crystal calibration procedure is outlined and errors are discussed. Experimental results compared to those obtained with a piezo-electrically driven mirror are presented. q 1998 Elsevier Science B.V. PACS: 42.25.Hz; 42.25.Lc; 42.30.Ms Keywords: BSO crystal; Phase stepping; Metrology; Digital speckle pattern interferometry
Interferometry can be used to produce a fringe pattern that represents the field surface displacement of an object in response to some change in mechanical loading w1x. Digital speckle pattern interferometry ŽDSPI. is one of the most modern techniques for depicting such fringe patterns. It combines real-time processing with the flexibility of software handling w2x. DSPI was developed by combining the well-known techniques of holographic and speckle interferometry by using an image hologram setup and following the methods of double-exposure holography. It utilizes a CCD camera interfaced to a computer to process the data. For comparison measurements, a reference frame stored in memory is continuously subtracted
1
Permanent address: Universidad EAFIT, Medellın, ´ Colombia. Also at Facultad de Ciencias Exactas, Universidad Nacional de La Plata, Argentina. 3 Permanent address: Instituto de Fisica, Universidade Federal Fluminense, Niteroi, Rio Janeiro, Brazil. 4 Also at Facultad de Ingenierıa, ´ Universidad Nacional de La Plata, Argentina. 2
from incoming data, and then the intensity difference is displayed. The fringes represent the correlation between the speckle patterns, with the same interpretation as for those fringes obtained by holographic interferometry. However, for quantitative purposes, interferograms must be analyzed so that the results can be presented in the required numerical form w3x. Progress has been made in fringe pattern analysis with a number of electronic aids. These devices allowed a substantial improvement in the accuracy of fringe location within an interferogram w4x. The wide availability of digital image processing equipment prompted a number of studies to investigate different alternatives of automatic interferogram analysis. Modern techniques have reached a point at which they can provide useful results, particularly with phase-stepping methods w5x. By introducing discrete shifts in the position of the fringes, we can calculate the phase map at pixel location Ž i, j . in a relatively simple way. The phase-stepping technique is a simplification of the intensity integration method and requires a minimum of three measurements to solve for three unknowns in the interference equation. The phase is usually changed when a mirror with a computer-
0030-4018r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 0 1 3 - 3
L. Angel et al.r Optics Communications 149 (1998) 235–238
236
controlled piezoelectric translator is moved. These phase shifts can be introduced either in the object or in the reference beam. Errors in phase measurements are introduced because the desired phase steps are not achieved either because of miscalibration or nonlinearity of the piezoelectric transducer w6x. There are many algorithms used to extract phase, and a certain combination of them can minimize phase-stepper miscalibrations and nonlinearity errors w7x. However, a reduction of these errors is desirable by using another method to introduce phase-steps without mechanical movements. To demonstrate the technique, we chose an electro-optic BSO crystal located in one beam of the DSPI. Its use in DSPI systems has been already implemented as a fringe visibility control device w8x. Our approach is based on the response of such crystal to the application of a constant electric voltage difference. According to the electro-optic effect, the application of such voltage induces a birefringence on the crystal w9x. We make use of this property to introduce the desired phase-steps without mechanical movements. In the following, we will discuss the characteristics and calibration procedure using a BSO crystal as phase-stepping device. Some experimental results will be shown, observing an adequate coincidence with those obtained by using a PZT under the same experimental conditions. The basic configuration that we use is shown in Fig. 1 w10x. The laser beam is divided in two by a beam splitter, and then each beam is expanded. Both beams are equivalent to obtaining two different speckle patterns from the same surface, one that serves as a reference wave and the other as an object wave. The BSO crystal is placed in one arm being uniformly illuminated with an expanded parallel beam, perpendicular to the incident face. The directions of the crystal, ²001:, ²110: and ²110: coincide with the XYZ axes respectively, being Z the propagation axis of the DSPI interferometer beam. The physical dimensions are L X s 10 mm, L Y s 4 mm and LZ s 11 mm. For a voltage V0 applied between the ²001: faces separated a distance L X , the crystal becomes birefringent. In this configuration, the expression of birefringence results:
d n s 12 n30 r41
V0 LX
,
Fig. 2. Ža. Interferometric fringes, Žb. phase map and Žc. 3D plot of a tilted plane surface obtained with the BSO crystal.
difference introduced by the crystal on the interferometer beam can be expressed as:
Ž1. Dws
where r41 is the electro-optic coefficient and n 0 is the refractive index without induced field. In our case n 0 s 2.54 and r41 s 4.4 pmrV. According to this, the phase
Fig. 1. Experimental setup.
2p
l
d n LZ ,
Ž2.
where l s 633 nm is the laser wavelength. In our scheme we adopt the three equations method requiring pr2 steps. To introduce every phase difference of pr2 we need to apply a voltage of 4 kV each time to the particular crystal we employed. Thus, to achieve a phase difference of p, we apply 8 kV, and so on. High contrast reference correlation fringes are obtained by introducing a slight tilt in the object. An external voltage is then applied to the crystal, which is increased until the reference fringe system drifts in pr2 at the TV monitor. The voltage value for this drift corresponds to the theoreti-
L. Angel et al.r Optics Communications 149 (1998) 235–238
cally predicted. The accuracy of the method depends on the accuracy of determining the voltage signal. To achieve this accuracy, the high voltage applied was driven by a ramp generator and measurements were made by comparing the input drive trace and the corresponding output from the high voltage unit. In our experiment, phase shifts of pr2 were measured with an accuracy of 0.28 and repeatability of 0.0002. We used this technique in many experiments employing the BSO crystal. We present here results for a tilted object and a deformation on a thin membrane loaded from the rear. The same experiments were conducted with a piezo-electrically driven mirror, as phase stepping device, observing discrepancies in the results below 2%. Fig. 2 shows Ža. the fringes, Žb. the phase map and Žc. the 3D plot of a tilted surface, obtained with the BSO crystal. Fig. 3 displays the case of the deformed membrane also showing Ža. the fringes, Žb. the phase map and Žc. the 3D plot. In conventional systems a calibration step is required according to the set-up geometry. In our case such calibration procedure has been carried out only in the crystal, because the phase change introduced by the crystal on the transmitted wave only depends on its own characteristics Želectro-optic coefficient, size, refraction index, etc... Once the calibration has been fulfilled, the phase step is independent of the particular set-up geometry involved. Besides, no extra drifts are introduced by external influences Žtemperature, humidity, etc.. under normal working laboratory conditions. Also, there is no need for a feedback mechanism to active compensation as that reported in other methods. When the incident light is linearly polarized, the polarization state of the light transmitted through the crystal depends on the thickness as well as on the applied field and the rotatory power. As was demonstrated in Ref. w11x, this should cause a decrease in fringe visibility. There, the visibility decay is analyzed in terms of the polarization state rotation, on the basis of a parameter related to the phase change value responsible for the speckle correlation fringes generation. In our experiment, this phase change corresponds to that introduced by object rotation or irregular deformation of the loaded plate. As in Ref. w11x, we maintained this phase change in the range of few degrees. Furthermore, according to the crystal used and our experimental conditions, the polarization state rotation of the beam emerging from the crystal, experimentally measured, was one and two degrees for an applied voltage of 4 kV and 8 kV, respectively. Then, the resulting polarization state lies within a tolerance range such that the visibility of the fringes is barely affected, less than 2%. In conclusion, the experiments conducted have shown that the use of a BSO crystal as a phase-stepping device in a digital speckle pattern interferometer is not only possible in practice, but is also simpler than using piezo-electrically driven mirrors. It was also demonstrated that it provides an
237
Fig. 3. Ža. Interferometric fringes, Žb. phase map and Žc. 3D plot of the clamped metal plate loaded with a point pressure applied on it.
alternative useful and convenient method of analyzing general deformations both quantitatively and qualitatively. Further research has been conducted on evaluating other type of crystals and working conditions to optimize the proposed technique.
Acknowledgements This research was performed under the auspices of the National Research Council, Grant Nr. 6825r96, PICT 0041 and 0249 ŽArgentina., Fundacion ´ Antorchas ŽArgentina., CICPBA ŽArgentina., Facultad de Ingenierıa ´ Universidad Nacional de La Plata ŽArgentina., the Na-
238
L. Angel et al.r Optics Communications 149 (1998) 235–238
tional Research Council CNPq, FAPERJ and FINEP ŽBrazil.. M. Trivi is researcher of Comision ´ de Investigaciones Cientıficas de la Provincia de la Buenos Aires ´ ŽArgentina.. L. Angel also acknowledges Instituto Colombiano para el Desarrollo de la Ciencia y la Tecnologıa ´ ŽColombia.. References w1x R. Jones, C. Wykes, Holographic and Speckle Interferometry, Cambridge Univ. Press, Cambridge, UK, 1983, pp. 183– 188.
w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x
K. Creath, G. Slettemoen, J. Opt. Soc. Am. A 2 Ž1985. 1629. D.W. Robinson, Appl. Optics 22 Ž1983. 2169. D. Robinson, D. Williams, Optics Comm. 57 Ž1986. 26. R. Henao, H. Rabal, A. Tagliaferri, R. Torroba, Appl. Optics 36 Ž1997. 2066. J. Van Wingerden, H.J. Frankena, C. Smorenberg, Appl. Optics 30 Ž1991. 2718. C. Joenathan, Appl. Optics 33 Ž1994. 4147. J. Pomarico, R. Torroba, N. Bolognini, Optik 99 Ž1995. 89. A.M. Glass, Opt. Eng. 17 Ž1978. 470. C. Joenathan, R. Torroba, Optics Lett. 15 Ž1990. 1159. R. Henao, A. Tagliaferri, R. Torroba, Optics Comm. 127 Ž1996. 14.