- Email: [email protected]
Advances in real-time holographic interferometry for the measurement of vibrations and deformations
Optics and Lasers in Engineering 32 (2000) 515}527
Advances in real-time holographic interferometry for the measurement of vibrations and deformations J. Frejlich*, P.M. Garcia Laborato& rio de O! ptica, Universidade Estadual de Campinas, Instituto de Fisica, Caixa Postal 6165, 13083-970 Campinas-SP, Brazil
Abstract We describe the latest improvements in holographic interferometry that enable the real-time measurement of vibrational modes and static deformations in surfaces using low-power laser illumination and a photorefractive Bi TiO crystal as the recording medium. An e$cient setup has been developed where the most critical elements have been optimized: target illumination and backscattered light collection, distribution of light between the object and reference beams, and stabilization system operation. Experimental results for vibration and deformatiom measurement are reported. 2000 Elsevier Science Ltd. All rights reserved.
The use of photorefractive materials as real time, reversible holographic recording media has been shown to eliminate most of the handicaps of holography as a practical tool for vibration and deformation measurement. Low-frequency perturbations and changes in the setup are adaptively accounted for [1] because of the relatively fast response of these materials. Higher frequency perturbations instead can be compensated by the use of an active stabilization feedback opto-electronic loop [2] as described below. The e$cient illumination of the target's surface and collection of the backscattered light from it, is very important for maximizing the intensity of the holographically reconstructed object wave containing the required information about vibration and deformation. Such optimization needs the retro-re#ectivity of the target's surface to be taken into account. The negative feedback opto-electronic loop
* Corresponding author. Fax: #55-19-289-31-37. Academia da Forc7 a AeH rea, Pirassununga-SP, Brazil. E-mail address: frejlich@i".unicamp.br (J. Frejlich). 0143-8166/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 3 - 8 1 6 6 ( 9 9 ) 0 0 0 7 3 - 1
516
J. Frejlich, P.M. Garcia / Optics and Lasers in Engineering 32 (2000) 515}527
used for stabilizing the setup is essential for the performance of the setup and has been rearranged in order to decrease the level of parasitic signals in it. The setup is used to measure vibrational modes by time-average holographic interferometry [3] or alternatively to measure static deformations by the so-called double holographic exposure method [4].
1. Measurement of vibration and deformation Several techniques allow to measure vibrations and deformations using holography. We have chosen time-average holographic interferometry for vibrations [3] and double exposure for deformations [4]. These are general methods that have been adapted to the special features of photorefractive recording media. The good results reported in this communication are possible because of the particular features of the setup, including e$cient target illumination and light distribution, as well as the use of a negative opto-electronic feedback loop for stabilization. The use of a retrore#ective painting on the target's surface largely contributes to the performance of the setup. 1.1. Vibrations The measurement of vibration by time-average holographic interferometry is based on the fact that the di!raction e$ciency of the hologram recorded (during a time large compared to the period of the vibration under analysis) by the light backscattered from a surface vibrating with amplitude d and frequency ) can be written as 2b g(d)"g mJ (4pd/j), m" , b"I /I, 0 1 (1#b)
(1)
where g is the di!raction e$ciency (using equal intensity recording beams) of the surface at rest, m is the visibility of the interference fringes, I , I are the intensities of 0 1 the reference and object beams incident on the crystal, and J is the Bessel function of order zero. The holographically reconstructed target's surface image is therefore superimposed to a pattern of dark and bright fringes corresponding to the di!erent maxima and minima of the Bessel function as shown in Fig. 1. The position of these fringes allows computing the map of the local values of the amplitude vibration d over the target's surface with the help of a table of Bessel functions as shown in Table 1. Note that each point of local maximum amplitude of vibration in the membrane is at the center of a pattern of approximately concentric fringes. The measurement is particularly simple when the excitation frequency corresponds to one of the normal vibration modes as is the case illustrated in Figs. 1 and 4a}c. To evaluate the performance of this technique, the response of some points of local maximum amplitude of vibration in a loud-speaker membrane as a function of the applied voltage are shown in Figs. 2 and 3 for two di!erent frequencies. The vibration of a thin (0.2 mm thick) phosphorous}bronze metallic plate was also visualized using the above referred real-time holographic interferometry technique.
J. Frejlich, P.M. Garcia / Optics and Lasers in Engineering 32 (2000) 515}527
517
Fig. 1. Loudspeaker membrane excited at a 3.0 kHz electric signal and analysed by the time-average holographic interferometry technique. The brighter areas are those at rest, the "rst dark fringe indicates a vibration amplitude of 0.12 lm, the second one 0.28 lm, the third one 0.44 lm, and so on according to data in Table 1. Table 1 Vibration amplitudes d (nm)
0 120.9 191.4 277.05 352.6 435.7 511.3 594.4 670.0 750.6 831.2
Zero (Rad) x
* 2.4 * 5.5 * 8.65 * 11.8 * 14.9 *
Max (rad) x
J (x)
0 * 3.8 * 7.0 * 10.15 * 13.3 * 16.5
1 * 0.16 * 0.09 * 0.062 * 0.048 * 0.038
The plate was tightly "xed to the external metallic ring of a commercial loud-speaker, using a plastic (PVC) double ring with a clear 79.5 mm internal diameter. The density of the plate was 6.24 g/cm. The plate was painted with a retro-re#ective ink to increase the amount of backscattered light collected by the optical setup and focused onto the photorefractive crystal. The loud-speaker was used to excite the plate. Figs. 4a}c show the interference patterns obtained for the frequencies of the electric
518
J. Frejlich, P.M. Garcia / Optics and Lasers in Engineering 32 (2000) 515}527
Fig. 2. Amplitude of vibration at a point of local maximum in the membrane of a loudspeaker as a function of the applied voltage for a signal of 4.2 kHz.
Fig. 3. Amplitude of vibration at two di!erent points of local maximum in the membrane of a loudspeaker as a function of the applied voltage for a signal of 1.4 kHz.
signals feeding the loudspeaker that lead to the "rst, third and fourth normal vibration modes respectively. The amplitudes of the local maxima can be approximately estimated from the number of fringes and Table 1. 1.2. Deformations Photorefractive crystals can be used as double exposure recording media because the recording takes a "nite time (inversely proportional to the total amount of light onto the crystal) so that it is possible to record the "rst image of the target under study and then the second image of the deformated target as is usually done in classical interferometry. In the case of photorefractive crystals, however, the latter one is being recorded while the former one begins to fade. At a certain moment in this process both images reach similar intensities and a maximum contrast of the fringes arising from the interference of both wavefronts ("rst and second object images) is obtained as shown in Fig. 5. It is convenient to interrupt the light onto the crystal between both
J. Frejlich, P.M. Garcia / Optics and Lasers in Engineering 32 (2000) 515}527
519
Fig. 4. Thin phosphorous}bronze metallic plate excited with the loudspeaker being operated at 255 Hz and 90 mV (A), 600 Hz and 30 mV (B) and 800 Hz and 60 mV (C).
exposures in order to avoid the continuous recording instead of the recording of two well-de"ned states of the surface. The BTO crystal used in these experiments is particularly well suited because it exhibits a rather low dark conductivity that grants no sensible changes in the already recorded image once the light is cut-o! between both exposures.
2. Experimental setup The actual experimental setup used in these experiments is described in Fig. 6. The input laser beam is divided into a reference and an object one, using a polarization
520
J. Frejlich, P.M. Garcia / Optics and Lasers in Engineering 32 (2000) 515}527
Fig. 5. Deformation of a thin glass plate painted with retro-re#ective ink. The 1 cm reference mark in the upper side of the plate is for size scaling leading to an approximately constant slope of 1.47 lm/cm perpendicular to the direction of fringes.
beam-splitter cube (PBS). The amplitude ratio between both beams is controlled using a half-wave retardation plate (HWP) at the PBS input. The polarization direction of the PBS-exiting beams are made parallel by the use of another HWP at the output. A low-power microscope objective lens is used to expand the object beam in order to illuminate the whole target surface. A device formed by a PBS, two HWP and one-quarter wave retardation plate (QWP) are used to direct all the light onto the target surface and then allow to get the whole backscattered collected light through the PBS directly onto the recording photorefractive crystal (BTO) with minimum losses. Two photographic good-quality objectives are used to produce a reduced target image onto the BTO and then an adequately sized image onto the ccd-camera for observation and/or image acquisition and processing. The reference beam is also directed onto the BTO to interfere with the object beam to produce the required hologram for recording. The hologram is produced in real time in the BTO crystal and at the same time is reconstructed by the same reference beam used for recording it: the di!racted reference beam is actually the reconstructed object beam carrying all the information needed about the target vibration or deformation. 2.1. Reading of dynamic holograms The reading of holograms written in real-time reversible recording media as photorefractive crystals, requires special techniques because the uniform reference beam erases the hologram during reading. Several possibilities do exist for reading these dynamic holograms as they are called. We have chosen an e$cient technique based on the anisotropic di!raction properties of some crystals among which are the
J. Frejlich, P.M. Garcia / Optics and Lasers in Engineering 32 (2000) 515}527
521
Fig. 6. PBS: polarizing beamsplitter cube; HWP and QWP: halfwave and quarterwave retardation plates, respectively; M: "rst surface mirrors; PZT: piezo-electric supported mirror; PLC: path length compensator; EOM: eletro-optical modulator; SF: spatial "lter; BTO: photorefractive Bi TiO crystal; D: photodetec tor; P1 e P2: polarizers; CCD: image detector; LA: lock-in ampli"er; INT: integrator; HV: high-voltage source for the PZT.
sillenites and in particular the Bi TiO used in our experiments. In fact, under certain experimental contitions, the transmitted and di!racted (holographically reconstructed) beams are orthogonally polarized [5]. In this case, the latter beam carrying the necessary information about vibration and deformation can be separated from the transmitted beam that carries no information, just using a simple polarizer (P1 in Fig. 6).
522
J. Frejlich, P.M. Garcia / Optics and Lasers in Engineering 32 (2000) 515}527
2.2. Optimization of illumination The amount of light available for illuminating the target, record the hologram in the crystal and read it, is limited by the power of the laser source being used. A powerful source is interesting because: E it speeds up the holographic recording because recording time is roughly inversely proportional to the average light onto the crystal, E the speed up of recording allows to adaptively cope with perturbations of higher frequency, E allows illuminating a larger target's surface. In order to optimize the available amount of light we must e$ciently illuminate and collect the light from the target, and adequately divide the input beam between the object and reference beams in the setup. 2.2.1. Target illumination The illumination and light collection from the target's surface is described in Fig. 6: a polarization beamsplitter cube (PBS), a halfwave retardation plate (HWP) and a quarterwave plate (QWP) are used. The incident light (TE-polarized) is completely re#ected towards the target by the PBS and on its way forth and back from it crosses twice the QWP thus rotating its polarization direction by 903 and therefore being transmitted through the PBS to the crystal. In this way, the limited available light from the laser source is e$ciently used. To further improve light collection from the target it is painted with a special retro-re#ective ink. 2.2.2. Distribution of light among reference and object beams Fig. 7 shows a simpli"ed schema of the light distribution between the reference and the object beams in the setup that allows to calculate the di!racted reference beam
Fig. 7. Simpli"ed schema showing the distribution of light between reference and object beams: BS, beamsplitter; M mirror; I e I reference and object beams at the BS output; I e I, reference and object 0 1 0 1 beams e!ectively incident on the crystal.
J. Frejlich, P.M. Garcia / Optics and Lasers in Engineering 32 (2000) 515}527
523
Fig. 8. Optimization of the target illumination: I", di!racted reference beam measured (arbitrary units) as 0 a function of R"I/I (circles), and the best "tting to theory (continuous line). From "tting we get 1 0 f/c"1.15 for our presently retro-re#ective painted loudspeaker membrane.
intensity I" that is to be maximized. 0 I""gI , g"g m, 0 0
2b m" , (1#b)
b"I /I, R"I/I , 0 1 1 0
(2)
where g is the maximum di!raction e$ciency that can be obtained for a hologram in the crystal. I "I#I , I "fI , I"cI, 1 0 0 0 1 1 Rf/c I""4g I f , 0 (R#f/c)(1#R)
(3) (4)
I" is maximum for 0 f/c"R(1#(1#2/R#1/R).
(5)
Fig. 8 shows the values measured for I" (circles) as a function of R for a loud-speaker 0 membrane painted with retro-re#ected ink. The continuous curve is the best theoretical "tting with Eq. (4) and from this "tting the value f/c"1.15 is obtained. From the independently measured f"0.14 value the e!ectively collected retro-re#ected light can be estimated to be c"0.12 for that target in our setup. As seen from data in Fig. 8 the maximum value for I" is obtained for R"0.61, in good agreement with what 0 can be deduced from Eq. (5). 2.3. Stabilization feedback loop The light intensity propagating along the object beam direction, behind the crystal, can be written as I "Ig#I (1!g)$2g(g(1!g)(II cos u, 1 1 1 0 0
(6)
524
J. Frejlich, P.M. Garcia / Optics and Lasers in Engineering 32 (2000) 515}527
where u is the phase shift between the reference and object beams behind the crystal, g is the di!raction e$ciency of the hologram, I are the object or 1 0 reference beams, respectively, incident onto the crystal, and g is a parameter depending on mutual polarization and coherence relations between reference and object beams. All measurements are carried out behind the crystal, so that bulk absorption need not be considered throughout. If no external electric "eld is applied to the crystal (the present case) we can show that u"0. For actively stabilizing the setup, it is necessary to modulate the phase of one of the interfering beams (the reference one in our case) in order to produce a homodyne signal that is selectively detected and ampli"ed for use as error signal in the feedback loop. In fact, a phase modulation of amplitude t and frequency ) (2p10 kHz here) in the phase u of Eq. (6) will result in harmonic components in X where the amplitude of the "rst two ones are IX"4gJ (t )(g(1!g)(II sin u, 1 1 0
(7)
IX"4gJ (t )(g(1!g)(II cos u, 1 1 0
(8)
where J is the Bessel function of order 1 or 2. In nonperturbated conditions and in the absence of an externally applied "eld on the crystal, u"0 and consequently IX"0. As soon as a perturbation on the setup appears that is faster 1 than the crystal's response time uO0 and also IXO0. Therefore, the latter signal can 1 be used as error signal in the negative feedback stabilization loop. The operation of the stabilization feedback loop is schematically represented by the block-diagram in Fig. 9. The optical signal is transformed into an electrical signal in a photodetector and the "rst harmonic
(9)
Fig. 9. Block diagram showing the operation of the negative feedback stabilization loop: D, photodetector with conversion factor k ; LA, lock-in ampli"er with gain G; INT, signal integrator with integration time q ; " G PM, phase modulator with conversion factor K; u, phase shift;
J. Frejlich, P.M. Garcia / Optics and Lasers in Engineering 32 (2000) 515}527
525
where G is the lock-in ampli"er gain and k is the conversion factor of the photodetec" tor. The error signal < is integrated to produce the correction signal # 1 R < " < dt. (10) ! q # ' The correction voltage is applied to a phase modulator device PM (piezo-electric, electro-optic, etc.) produzing a feedback phase correction
u "K< , (11) ! where K is a characteristic of the device. The equilibrium of the output phase shift u can be formulated as follows: u"u !u , (12) , R KGk " 2J (t )g(I I(g(1!g), (13) u "A sin u dt, A" 0 1 q G where u is the e!ect of perturbations upon the setup output phase shift. In the , absence of noise (u "0) the equilibrium is u"0 as expected. An e$cient negative , feedback should lead to
du +0. dt
(14)
Substituting Eq. (13) into Eq. (12) and time-derivating u as indicated in Eq. (14), we obtain a di!erential equation the solution of which is 1 du , +0 assuming sin u+u. u"u e\ R# , A dt
(15)
Eq. (15) is veri"ed for the condition A<1 that represents a large negative feedback. In this case, the integrated error signal is a fundamental feature that largely improves the stabilization performance because it allows to keep the phase-shift condition u"0 (equilibrium) and still cope with steadily growing perturbations. An element contributing to the good performance of stabilization is the choice of an error signal that is obtained from the beams propagating along the object beam direction behind the crystal. In this way, we avoid detecting the transmitted reference beam that is phase-modulated and residually amplitude-modulated to some extent due to unavoidable misalignment of the EOM. Such an amplitude-modulated (at the same frequency )) signal in the feedback loop would seriously interfere in the stabilization process.
3. Discussion The experimental setup used in these experiments is relatively complicated but its operation is very simple and can be carried out by non-specialized technicians, once
526
J. Frejlich, P.M. Garcia / Optics and Lasers in Engineering 32 (2000) 515}527
the optical components are adjusted and "xed. The only adjustment left for the operator is to place the target in the correct position to have it adequately focused on the TV screen and eventually to correct the illumination of the target by gently acting on the screw of a mirror in the setup. The analysis of the pattern of fringes can be carried out on its photographic image or alternatively, the pattern can be transferred to a personal computer for analysis with an adequate standard commercial software. The use of a (nearly) real-time reversible photorefractive crystal as the holographic recording medium is essential in this experiment and allows overcoming most of the handicaps of classical holography: the operator can forget that a hologram is being recorded somewhere, and all changes in the target can be observed almost in real time. The setup allows to measure alternatively vibrations or deformations, with a very simple modi"cation in the operation procedure, without any change in the setup. The photorefractive crystal used in this instrument has been chosen among other possible materials, because of its suitability concerning spectral sensitivity, recording speed, di!raction e$ciency, optical quality, availability on the international market, etc., plus other speci"c properties (anisotropic di!raction e$ciency) that makes it particularly interesting for these purposes. The actively stabilized opto-electronic circuit described in this paper is also essential to enable the operation of this instrument in moderately perturbated environment. Without stabilization it would be quite hard to observe well-de"ned interference patterns except for occasional moments during the experiment. The operation of the stabilization loop is rather simple and involves simple electronics only.
4. Conclusions We have reported a new instrument based on holographic interferometry and allowing to measure both vibrations and deformations in nearly real time. The handling of the setup is very simple and can be carried out by non-specialized operators. Several examples of vibration analysis and static deformation measurement were discussed.
Acknowledgements We acknowledge "nancial support from FINEP/PADCT (Proc. 56.96.0297.00) as well as partial support from FAPESP and CNPq. We are thankful to the LaboratoH rio de Crescimento de Cristais/IFSC-USP that grew the crystals which enabled this work.
References [1] Stepanov SI. Applications of photorefractive crystals. Rep Prog Phys 1994;57:39}116.
J. Frejlich, P.M. Garcia / Optics and Lasers in Engineering 32 (2000) 515}527
527
[2] Freschi AA, Garcia PM, Frejlich J. Phase controlled photorefractive running holograms. Opt Commun 1997;143:257}60. [3] Huignard JP, Herriau JP, Valentin T. Time average holographic interferometry with photoconductive electrooptic Bi SiO crystals. Appl Opt 1977;16:2796}8. [4] Collier RJ, Burckhardt CB, Lin LH. Optical holography. New York: Academic Press, 1971. [5] Kamshilin AA, Petrov MP. Continuous reconstruction of holographic interferograms through anisotropic di!raction in photorefractive crystals. Opt Commun 1985;53:23}6.
Advances in real-time holographic interferometry for the measurement of vibrations and deformations J. Frejlich*, P.M. Garcia Laborato& rio de O! ptica, Universidade Estadual de Campinas, Instituto de Fisica, Caixa Postal 6165, 13083-970 Campinas-SP, Brazil
Abstract We describe the latest improvements in holographic interferometry that enable the real-time measurement of vibrational modes and static deformations in surfaces using low-power laser illumination and a photorefractive Bi TiO crystal as the recording medium. An e$cient setup has been developed where the most critical elements have been optimized: target illumination and backscattered light collection, distribution of light between the object and reference beams, and stabilization system operation. Experimental results for vibration and deformatiom measurement are reported. 2000 Elsevier Science Ltd. All rights reserved.
The use of photorefractive materials as real time, reversible holographic recording media has been shown to eliminate most of the handicaps of holography as a practical tool for vibration and deformation measurement. Low-frequency perturbations and changes in the setup are adaptively accounted for [1] because of the relatively fast response of these materials. Higher frequency perturbations instead can be compensated by the use of an active stabilization feedback opto-electronic loop [2] as described below. The e$cient illumination of the target's surface and collection of the backscattered light from it, is very important for maximizing the intensity of the holographically reconstructed object wave containing the required information about vibration and deformation. Such optimization needs the retro-re#ectivity of the target's surface to be taken into account. The negative feedback opto-electronic loop
* Corresponding author. Fax: #55-19-289-31-37. Academia da Forc7 a AeH rea, Pirassununga-SP, Brazil. E-mail address: frejlich@i".unicamp.br (J. Frejlich). 0143-8166/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 3 - 8 1 6 6 ( 9 9 ) 0 0 0 7 3 - 1
516
J. Frejlich, P.M. Garcia / Optics and Lasers in Engineering 32 (2000) 515}527
used for stabilizing the setup is essential for the performance of the setup and has been rearranged in order to decrease the level of parasitic signals in it. The setup is used to measure vibrational modes by time-average holographic interferometry [3] or alternatively to measure static deformations by the so-called double holographic exposure method [4].
1. Measurement of vibration and deformation Several techniques allow to measure vibrations and deformations using holography. We have chosen time-average holographic interferometry for vibrations [3] and double exposure for deformations [4]. These are general methods that have been adapted to the special features of photorefractive recording media. The good results reported in this communication are possible because of the particular features of the setup, including e$cient target illumination and light distribution, as well as the use of a negative opto-electronic feedback loop for stabilization. The use of a retrore#ective painting on the target's surface largely contributes to the performance of the setup. 1.1. Vibrations The measurement of vibration by time-average holographic interferometry is based on the fact that the di!raction e$ciency of the hologram recorded (during a time large compared to the period of the vibration under analysis) by the light backscattered from a surface vibrating with amplitude d and frequency ) can be written as 2b g(d)"g mJ (4pd/j), m" , b"I /I, 0 1 (1#b)
(1)
where g is the di!raction e$ciency (using equal intensity recording beams) of the surface at rest, m is the visibility of the interference fringes, I , I are the intensities of 0 1 the reference and object beams incident on the crystal, and J is the Bessel function of order zero. The holographically reconstructed target's surface image is therefore superimposed to a pattern of dark and bright fringes corresponding to the di!erent maxima and minima of the Bessel function as shown in Fig. 1. The position of these fringes allows computing the map of the local values of the amplitude vibration d over the target's surface with the help of a table of Bessel functions as shown in Table 1. Note that each point of local maximum amplitude of vibration in the membrane is at the center of a pattern of approximately concentric fringes. The measurement is particularly simple when the excitation frequency corresponds to one of the normal vibration modes as is the case illustrated in Figs. 1 and 4a}c. To evaluate the performance of this technique, the response of some points of local maximum amplitude of vibration in a loud-speaker membrane as a function of the applied voltage are shown in Figs. 2 and 3 for two di!erent frequencies. The vibration of a thin (0.2 mm thick) phosphorous}bronze metallic plate was also visualized using the above referred real-time holographic interferometry technique.
J. Frejlich, P.M. Garcia / Optics and Lasers in Engineering 32 (2000) 515}527
517
Fig. 1. Loudspeaker membrane excited at a 3.0 kHz electric signal and analysed by the time-average holographic interferometry technique. The brighter areas are those at rest, the "rst dark fringe indicates a vibration amplitude of 0.12 lm, the second one 0.28 lm, the third one 0.44 lm, and so on according to data in Table 1. Table 1 Vibration amplitudes d (nm)
0 120.9 191.4 277.05 352.6 435.7 511.3 594.4 670.0 750.6 831.2
Zero (Rad) x
* 2.4 * 5.5 * 8.65 * 11.8 * 14.9 *
Max (rad) x
J (x)
0 * 3.8 * 7.0 * 10.15 * 13.3 * 16.5
1 * 0.16 * 0.09 * 0.062 * 0.048 * 0.038
The plate was tightly "xed to the external metallic ring of a commercial loud-speaker, using a plastic (PVC) double ring with a clear 79.5 mm internal diameter. The density of the plate was 6.24 g/cm. The plate was painted with a retro-re#ective ink to increase the amount of backscattered light collected by the optical setup and focused onto the photorefractive crystal. The loud-speaker was used to excite the plate. Figs. 4a}c show the interference patterns obtained for the frequencies of the electric
518
J. Frejlich, P.M. Garcia / Optics and Lasers in Engineering 32 (2000) 515}527
Fig. 2. Amplitude of vibration at a point of local maximum in the membrane of a loudspeaker as a function of the applied voltage for a signal of 4.2 kHz.
Fig. 3. Amplitude of vibration at two di!erent points of local maximum in the membrane of a loudspeaker as a function of the applied voltage for a signal of 1.4 kHz.
signals feeding the loudspeaker that lead to the "rst, third and fourth normal vibration modes respectively. The amplitudes of the local maxima can be approximately estimated from the number of fringes and Table 1. 1.2. Deformations Photorefractive crystals can be used as double exposure recording media because the recording takes a "nite time (inversely proportional to the total amount of light onto the crystal) so that it is possible to record the "rst image of the target under study and then the second image of the deformated target as is usually done in classical interferometry. In the case of photorefractive crystals, however, the latter one is being recorded while the former one begins to fade. At a certain moment in this process both images reach similar intensities and a maximum contrast of the fringes arising from the interference of both wavefronts ("rst and second object images) is obtained as shown in Fig. 5. It is convenient to interrupt the light onto the crystal between both
J. Frejlich, P.M. Garcia / Optics and Lasers in Engineering 32 (2000) 515}527
519
Fig. 4. Thin phosphorous}bronze metallic plate excited with the loudspeaker being operated at 255 Hz and 90 mV (A), 600 Hz and 30 mV (B) and 800 Hz and 60 mV (C).
exposures in order to avoid the continuous recording instead of the recording of two well-de"ned states of the surface. The BTO crystal used in these experiments is particularly well suited because it exhibits a rather low dark conductivity that grants no sensible changes in the already recorded image once the light is cut-o! between both exposures.
2. Experimental setup The actual experimental setup used in these experiments is described in Fig. 6. The input laser beam is divided into a reference and an object one, using a polarization
520
J. Frejlich, P.M. Garcia / Optics and Lasers in Engineering 32 (2000) 515}527
Fig. 5. Deformation of a thin glass plate painted with retro-re#ective ink. The 1 cm reference mark in the upper side of the plate is for size scaling leading to an approximately constant slope of 1.47 lm/cm perpendicular to the direction of fringes.
beam-splitter cube (PBS). The amplitude ratio between both beams is controlled using a half-wave retardation plate (HWP) at the PBS input. The polarization direction of the PBS-exiting beams are made parallel by the use of another HWP at the output. A low-power microscope objective lens is used to expand the object beam in order to illuminate the whole target surface. A device formed by a PBS, two HWP and one-quarter wave retardation plate (QWP) are used to direct all the light onto the target surface and then allow to get the whole backscattered collected light through the PBS directly onto the recording photorefractive crystal (BTO) with minimum losses. Two photographic good-quality objectives are used to produce a reduced target image onto the BTO and then an adequately sized image onto the ccd-camera for observation and/or image acquisition and processing. The reference beam is also directed onto the BTO to interfere with the object beam to produce the required hologram for recording. The hologram is produced in real time in the BTO crystal and at the same time is reconstructed by the same reference beam used for recording it: the di!racted reference beam is actually the reconstructed object beam carrying all the information needed about the target vibration or deformation. 2.1. Reading of dynamic holograms The reading of holograms written in real-time reversible recording media as photorefractive crystals, requires special techniques because the uniform reference beam erases the hologram during reading. Several possibilities do exist for reading these dynamic holograms as they are called. We have chosen an e$cient technique based on the anisotropic di!raction properties of some crystals among which are the
J. Frejlich, P.M. Garcia / Optics and Lasers in Engineering 32 (2000) 515}527
521
Fig. 6. PBS: polarizing beamsplitter cube; HWP and QWP: halfwave and quarterwave retardation plates, respectively; M: "rst surface mirrors; PZT: piezo-electric supported mirror; PLC: path length compensator; EOM: eletro-optical modulator; SF: spatial "lter; BTO: photorefractive Bi TiO crystal; D: photodetec tor; P1 e P2: polarizers; CCD: image detector; LA: lock-in ampli"er; INT: integrator; HV: high-voltage source for the PZT.
sillenites and in particular the Bi TiO used in our experiments. In fact, under certain experimental contitions, the transmitted and di!racted (holographically reconstructed) beams are orthogonally polarized [5]. In this case, the latter beam carrying the necessary information about vibration and deformation can be separated from the transmitted beam that carries no information, just using a simple polarizer (P1 in Fig. 6).
522
J. Frejlich, P.M. Garcia / Optics and Lasers in Engineering 32 (2000) 515}527
2.2. Optimization of illumination The amount of light available for illuminating the target, record the hologram in the crystal and read it, is limited by the power of the laser source being used. A powerful source is interesting because: E it speeds up the holographic recording because recording time is roughly inversely proportional to the average light onto the crystal, E the speed up of recording allows to adaptively cope with perturbations of higher frequency, E allows illuminating a larger target's surface. In order to optimize the available amount of light we must e$ciently illuminate and collect the light from the target, and adequately divide the input beam between the object and reference beams in the setup. 2.2.1. Target illumination The illumination and light collection from the target's surface is described in Fig. 6: a polarization beamsplitter cube (PBS), a halfwave retardation plate (HWP) and a quarterwave plate (QWP) are used. The incident light (TE-polarized) is completely re#ected towards the target by the PBS and on its way forth and back from it crosses twice the QWP thus rotating its polarization direction by 903 and therefore being transmitted through the PBS to the crystal. In this way, the limited available light from the laser source is e$ciently used. To further improve light collection from the target it is painted with a special retro-re#ective ink. 2.2.2. Distribution of light among reference and object beams Fig. 7 shows a simpli"ed schema of the light distribution between the reference and the object beams in the setup that allows to calculate the di!racted reference beam
Fig. 7. Simpli"ed schema showing the distribution of light between reference and object beams: BS, beamsplitter; M mirror; I e I reference and object beams at the BS output; I e I, reference and object 0 1 0 1 beams e!ectively incident on the crystal.
J. Frejlich, P.M. Garcia / Optics and Lasers in Engineering 32 (2000) 515}527
523
Fig. 8. Optimization of the target illumination: I", di!racted reference beam measured (arbitrary units) as 0 a function of R"I/I (circles), and the best "tting to theory (continuous line). From "tting we get 1 0 f/c"1.15 for our presently retro-re#ective painted loudspeaker membrane.
intensity I" that is to be maximized. 0 I""gI , g"g m, 0 0
2b m" , (1#b)
b"I /I, R"I/I , 0 1 1 0
(2)
where g is the maximum di!raction e$ciency that can be obtained for a hologram in the crystal. I "I#I , I "fI , I"cI, 1 0 0 0 1 1 Rf/c I""4g I f , 0 (R#f/c)(1#R)
(3) (4)
I" is maximum for 0 f/c"R(1#(1#2/R#1/R).
(5)
Fig. 8 shows the values measured for I" (circles) as a function of R for a loud-speaker 0 membrane painted with retro-re#ected ink. The continuous curve is the best theoretical "tting with Eq. (4) and from this "tting the value f/c"1.15 is obtained. From the independently measured f"0.14 value the e!ectively collected retro-re#ected light can be estimated to be c"0.12 for that target in our setup. As seen from data in Fig. 8 the maximum value for I" is obtained for R"0.61, in good agreement with what 0 can be deduced from Eq. (5). 2.3. Stabilization feedback loop The light intensity propagating along the object beam direction, behind the crystal, can be written as I "Ig#I (1!g)$2g(g(1!g)(II cos u, 1 1 1 0 0
(6)
524
J. Frejlich, P.M. Garcia / Optics and Lasers in Engineering 32 (2000) 515}527
where u is the phase shift between the reference and object beams behind the crystal, g is the di!raction e$ciency of the hologram, I are the object or 1 0 reference beams, respectively, incident onto the crystal, and g is a parameter depending on mutual polarization and coherence relations between reference and object beams. All measurements are carried out behind the crystal, so that bulk absorption need not be considered throughout. If no external electric "eld is applied to the crystal (the present case) we can show that u"0. For actively stabilizing the setup, it is necessary to modulate the phase of one of the interfering beams (the reference one in our case) in order to produce a homodyne signal that is selectively detected and ampli"ed for use as error signal in the feedback loop. In fact, a phase modulation of amplitude t and frequency ) (2p10 kHz here) in the phase u of Eq. (6) will result in harmonic components in X where the amplitude of the "rst two ones are IX"4gJ (t )(g(1!g)(II sin u, 1 1 0
(7)
IX"4gJ (t )(g(1!g)(II cos u, 1 1 0
(8)
where J is the Bessel function of order 1 or 2. In nonperturbated conditions and in the absence of an externally applied "eld on the crystal, u"0 and consequently IX"0. As soon as a perturbation on the setup appears that is faster 1 than the crystal's response time uO0 and also IXO0. Therefore, the latter signal can 1 be used as error signal in the negative feedback stabilization loop. The operation of the stabilization feedback loop is schematically represented by the block-diagram in Fig. 9. The optical signal is transformed into an electrical signal in a photodetector and the "rst harmonic
(9)
Fig. 9. Block diagram showing the operation of the negative feedback stabilization loop: D, photodetector with conversion factor k ; LA, lock-in ampli"er with gain G; INT, signal integrator with integration time q ; " G PM, phase modulator with conversion factor K; u, phase shift;
J. Frejlich, P.M. Garcia / Optics and Lasers in Engineering 32 (2000) 515}527
525
where G is the lock-in ampli"er gain and k is the conversion factor of the photodetec" tor. The error signal < is integrated to produce the correction signal # 1 R < " < dt. (10) ! q # ' The correction voltage is applied to a phase modulator device PM (piezo-electric, electro-optic, etc.) produzing a feedback phase correction
u "K< , (11) ! where K is a characteristic of the device. The equilibrium of the output phase shift u can be formulated as follows: u"u !u , (12) , R KGk " 2J (t )g(I I(g(1!g), (13) u "A sin u dt, A" 0 1 q G where u is the e!ect of perturbations upon the setup output phase shift. In the , absence of noise (u "0) the equilibrium is u"0 as expected. An e$cient negative , feedback should lead to
du +0. dt
(14)
Substituting Eq. (13) into Eq. (12) and time-derivating u as indicated in Eq. (14), we obtain a di!erential equation the solution of which is 1 du , +0 assuming sin u+u. u"u e\ R# , A dt
(15)
Eq. (15) is veri"ed for the condition A<1 that represents a large negative feedback. In this case, the integrated error signal is a fundamental feature that largely improves the stabilization performance because it allows to keep the phase-shift condition u"0 (equilibrium) and still cope with steadily growing perturbations. An element contributing to the good performance of stabilization is the choice of an error signal that is obtained from the beams propagating along the object beam direction behind the crystal. In this way, we avoid detecting the transmitted reference beam that is phase-modulated and residually amplitude-modulated to some extent due to unavoidable misalignment of the EOM. Such an amplitude-modulated (at the same frequency )) signal in the feedback loop would seriously interfere in the stabilization process.
3. Discussion The experimental setup used in these experiments is relatively complicated but its operation is very simple and can be carried out by non-specialized technicians, once
526
J. Frejlich, P.M. Garcia / Optics and Lasers in Engineering 32 (2000) 515}527
the optical components are adjusted and "xed. The only adjustment left for the operator is to place the target in the correct position to have it adequately focused on the TV screen and eventually to correct the illumination of the target by gently acting on the screw of a mirror in the setup. The analysis of the pattern of fringes can be carried out on its photographic image or alternatively, the pattern can be transferred to a personal computer for analysis with an adequate standard commercial software. The use of a (nearly) real-time reversible photorefractive crystal as the holographic recording medium is essential in this experiment and allows overcoming most of the handicaps of classical holography: the operator can forget that a hologram is being recorded somewhere, and all changes in the target can be observed almost in real time. The setup allows to measure alternatively vibrations or deformations, with a very simple modi"cation in the operation procedure, without any change in the setup. The photorefractive crystal used in this instrument has been chosen among other possible materials, because of its suitability concerning spectral sensitivity, recording speed, di!raction e$ciency, optical quality, availability on the international market, etc., plus other speci"c properties (anisotropic di!raction e$ciency) that makes it particularly interesting for these purposes. The actively stabilized opto-electronic circuit described in this paper is also essential to enable the operation of this instrument in moderately perturbated environment. Without stabilization it would be quite hard to observe well-de"ned interference patterns except for occasional moments during the experiment. The operation of the stabilization loop is rather simple and involves simple electronics only.
4. Conclusions We have reported a new instrument based on holographic interferometry and allowing to measure both vibrations and deformations in nearly real time. The handling of the setup is very simple and can be carried out by non-specialized operators. Several examples of vibration analysis and static deformation measurement were discussed.
Acknowledgements We acknowledge "nancial support from FINEP/PADCT (Proc. 56.96.0297.00) as well as partial support from FAPESP and CNPq. We are thankful to the LaboratoH rio de Crescimento de Cristais/IFSC-USP that grew the crystals which enabled this work.
References [1] Stepanov SI. Applications of photorefractive crystals. Rep Prog Phys 1994;57:39}116.
J. Frejlich, P.M. Garcia / Optics and Lasers in Engineering 32 (2000) 515}527
527
[2] Freschi AA, Garcia PM, Frejlich J. Phase controlled photorefractive running holograms. Opt Commun 1997;143:257}60. [3] Huignard JP, Herriau JP, Valentin T. Time average holographic interferometry with photoconductive electrooptic Bi SiO crystals. Appl Opt 1977;16:2796}8. [4] Collier RJ, Burckhardt CB, Lin LH. Optical holography. New York: Academic Press, 1971. [5] Kamshilin AA, Petrov MP. Continuous reconstruction of holographic interferograms through anisotropic di!raction in photorefractive crystals. Opt Commun 1985;53:23}6.