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Vibration of low amplitude imaged in amplitude and phase by sideband versus carrier correlation digital holography N. Verrier,1,2 L. Alloul,1 and M. Gross1,* 1
Laboratoire Charles Coulomb—UMR 5221, CNRS-UM2 Université Montpellier II Bat 11. Place Eugène Bataillon 34095, Montpellier Cedex 5, France 2 Laboratoire Hubert Curien, UMR 5516, CNRS-Université Jean Monnet, 18 Rue du Professeur Benoît Lauras, Saint-Etienne 42000, France *Corresponding author: [email protected] Received December 5, 2014; revised December 19, 2014; accepted December 20, 2014; posted December 23, 2014 (Doc. ID 229072); published January 30, 2015 Sideband holography can be used to get field images (E0 and E1 ) of a vibrating object for both the carrier (E0 ) and the sideband (E1 ) frequency with respect to vibration. Here we propose to record E0 and E 1 sequentially and to image the product E1 E 0 or the correlation hE1 E0 i. We show that these quantities are insensitive to the phase related to the object roughness and directly reflect the phase of the mechanical motion. The signal to noise can be improved by averaging E1 E0 over a neighbor pixel, yielding hE1 E0 i. Experimental validation is made with a vibrating cube of wood and a clarinet reed. At 2 kHz, vibrations of amplitude down to 0.01 nm are detected. © 2015 Optical Society of America OCIS codes: (120.7280) Vibration analysis; (090.1995) Digital holography; (040.2840) Heterodyne; (120.2880) Holographic interferometry. http://dx.doi.org/10.1364/OL.40.000411
There is a big demand for full-field vibration measurements, in particular in industry. Different holographic techniques are able to image and analyze such vibrations. Pulsed holography [1–3] records several holograms with time separation in the 1…1000 μs range, getting the instantaneous velocity from the phase difference. For small vibration frequencies, one can directly track the vibration of the object with fast CMOS cameras [4,5]. The analysis of the motion can be done by phase difference or by Fourier analysis in the time domain. For periodic vibration motions, measurements can be done with a slow camera. Indeed, an harmonically vibrating object illuminated by a laser yields alternate dark and bright fringes [6] that can be analyzed by time averaged holography [7]. It is also possible to use a phase modulated reference to shift these fringes [8], and to determine vibration amplitude by fringe shift algorithms [9]. With phase modulated reference, it is also possible to detect vibration harmonics of high range [10], and to select a given harmonic by adjusting the phase modulation amplitude [11]. Although the time averaged method provides a way to determine the amplitude of vibration [12], quantitative measurement remains quite difficult for low and high vibration amplitudes. We have developed a variant of phase shifting holography [13] called digital heterodyne holography (DHH, not to be confused with heterodyne holography with plates [14]), in which the phase shift is replaced by a frequency shift controlled by acousto-optic modulators (AOM) driven by numerical synthesizers [15,16]. This gives much better control of the phase [17]. DHH is extremely versatile, since the frequency, phase, and amplitude of both reference and signal beams can be controlled. By shifting the frequency of the local oscillator ωLO with respect to illumination ω0 , it is for example possible to detect the holographic signal at a frequency ω different from illumination ω0 . This ability is extremely useful to analyze vibration, since DHH can detect 0146-9592/15/030411-04$15.00/0
selectively the signal that is scattered by an object on a vibration sideband of frequency ωm ω0 mωA , where ωA is the vibration frequency and m and integer index. As was reported by Ueda et al. [18], the detection of the sideband m 1 is advantageous when the vibration amplitude is small. Nanometric vibration amplitude measurements were achieved with sideband digital holography on the m 1 sideband [19], and comparison with single point laser interferometry has been made [20]. Verrier and Atlan [21] have shown that one can simultaneously measure E 0 and E 1 by using a local oscillator with two frequency components. One can thus infer the mechanical phase of the vibration [22]. However, Bruno et al. [23] show that this simultaneous detection of E 0 and E1 can lead to cross talk, and to a loss of detection sensitivity, which becomes annoying when the vibration amplitude is small. In this Letter, we show that simultaneous detection of E 0 and E 1 is not necessary, and that equivalent or superior performances can be obtained by detecting E 0 and E 1 sequentially. Indeed, the cross talk effects seen by Bruno et al. [23] disappear in that case. We also show that the random phase variations caused by the roughness of the object can be eliminated by calculating the product E 1 E 0 . It is then possible to increase the signal-to-noise ratio (SNR) by averaging E 1 E 0 over neighboring pixels yielding hE 1 E0 i. The sequential measurement of E 0 and E 1 , and the calculation of E 1 E 0 and hE 1 E 0 i, make it possible to image the vibration “full field,” and to measure quantitatively its amplitude and phase. Maximum sensitivity is achieved by focusing the illumination in the studied point and by averaging the E 1 E 0 in that region. Finally, we prove that the sensitivity is limited by a sideband signal of one photo-electron per demodulated image sequence. The device and method were validated experimentally by studying a cube of wood vibrating at ≃20 kHz, and a clarinet reed at ≃2 kHz. Measurement sensitivity of 0.01 nm for a © 2015 Optical Society of America
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vibration at 2 kHz, comparable to the sensitivity obtained by Bruno et al. [23] at 40 kHz, is demonstrated. Consider an object illuminated by a laser at frequency ω0 that vibrates at frequency ωA with an out of plane vibration amplitude zmax . The out of plane coordinate is zt zmax sinωA t. The field scattered by the object is E Eteiω0 t c:c:, where c.c. is the complex conjugate and Et is the field complex amplitude. In reflection geometry, we have Et E wo ejΦj sinωA targ Φ , where E wo is the complex field without movement, and Φ is a complex quantity that describes the phase modulation. The phase of this modulation is arg Φ, while the amplitude is jΦj 4πzmax ∕λ. Because P of the Jacobi–Anger expansion, we have Et E wo m J m jΦjejmωA targ Φ , where J m is the mth-order Bessel function of the first kind. The scattered field E is then the sum of field components E m E m eiωm t c:c: of frequency ωm ω0 mωA , where m is the sideband index (m 0 for the carrier) with E m E wo J m jΦjejm arg Φ . For small vibration amplitudes jΦj, the energy within sidebands decreases very rapidly with the sideband indexes m, and one has only to consider the carrier m 0, and the first sideband m 1. We have thus E 0 Φ E wo J 0 jΦj;
E 1 Φ Ewo J 1 jΦjej arg Φ :
(1)
Note that time averaged holography [7] that detects the carrier field E0 is not efficient in detecting small amplitude vibration jΦj, because E 0 varies quadratically with jΦj. On the other hand, sideband holography that is able to detect selectively the sideband field E 1 is much more sensitive, because E 1 varies with linearly with jΦj. One must notice that the field scattered by the sample without vibration E wo depends strongly on the x, y position. In a typical experiment, the sample rugosity is such that this field is a fully developed speckle. The reconstructed fields E 0 and E 1 are thus random in phase, from one pixel to the next. One cannot thus simply extract the phase of the vibration from a measurement made on a single sideband. To remove the pixel-to-pixel random phase of E wo , we propose recording the hologram successively on the carrier m 0 and the sideband m 1, to reconstruct the corresponding field image of the object E 0 x; y and E 1 x; y, and to calculate and image the product E 1 E 0 , since this product does not involve Ewo , but jE wo j2 , which is real and has no phase. Indeed, we have E 1 E 0 jE wo j2 J 1 jΦjJ 0 jΦjej arg Φ :
(2)
For small vibration amplitudes (jΦj ≪ 1), E 1 E 0 simplifies to jE wo j2 Φ∕2. Product E 1 E 0 is a powerful tool since it gives directly the phase of mechanical motion arg Φ. Nevertheless, problems can be encountered when the signal jE wo j2 scattered without vibration vanishes. Figure 1 shows the holographic experimental setup used to measure successively E 0 and E 1 to get E 1 E 0 . The main laser L is a Sanyo DL-7140-201 diode laser (λ 785 nm, 50 mW), split into an illumination beam (frequency ωI , complex field E I ), and in an LO beam (ωLO , E LO ). The illumination light scattered by the object interferes with the reference beam on the camera
Fig. 1. DHH setup applied to analyze vibration. L, main laser; AOM1, AOM2, acousto-optic modulators; M, mirror; BS, beam splitter; CCD, camera.
(Lumenera 2-2: 1616 × 1216 pixels of 4.4 × 4.4 μm) whose frame rate is ωCCD 10 Hz. To simplify further the Fourier transform calculations, the 1616 × 1216 measured matrix is truncated to 1024 × 1024. The illumination and LO beam frequencies ωLO and ωI are tuned by using two Bragg cells AOM1 and AOM2, and we have ωLO ωL ωAOM1 and ωI ωL ωAOM2 , where ωAOM1∕2 ≃ 80 MHz are the frequencies of RF signals that drive the AOMs. The RF signals are tuned to have ωLO − ωI ωCCD ∕4 to get E 0 , and to have ωLO − ωI ωA ωCCD ∕4 to get E 1 . Successive sequences of nmax 128 camera frames (i.e., I 0 ; I 1 …I 127 ) are recorded by tuning the RF signals first on the carrier (E 0 ), then on the sideband (E 1 ). The carrier and sideband complex hologram H are obtained from these sequences by four-phase demodulation with nmax frames: Hx; y
nn max −1 X n0
j n I n x; y:
(3)
The field images of the object E 0 x; y and E 1 x; y are then reconstructed from H using a single FFT scheme [24]. The product E 1 E 0 is finally calculated. Figure 2(a) shows the reconstructed product E 1 E 0 images of a cube of wood (2 cm × 2 cm) vibrating at its resonance frequency ωA ∕2π 21.43 kHz. Brightness is jE 1 E 0 j) and color E 1 E 0 phase: arg E 1 E 0 . As seen, the neighbor points have the same phase (same color). To get better SNR for the phase, E 1 E 0 is averaged over neighbor x, y points by using a 2D Gaussian blur filter of radius 4 pixels. Figure 2(b) displays the phase of the averaged product E 1 E 0 in 3D. As seen, the opposite corners (upper left and bottom right for example) vibrate in phase, while the neighbor corners (upper left and upper right for example) vibrate in phase opposition. Note that the cube
Fig. 2. Reconstructed images of a cube of wood vibrating at ωA ∕2π 21.43 kHz. (a) E1 E0 product image: brightness is jE1 E0 j and color is arg E 1 E 0 . (b) 3D display of arg E 1 E 0 .
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is excited in one of its corners by a needle. This may explain why the opposite corners are not perfect in phase in Fig. 2(c). We can go further and use E 0 and E 1 to calculate the vibration complex amplitude Φ. To increase the SNR, let us average over neighbor reconstructed pixels E 1 E 0 and jE 0 j2 with jE 0 j2 jE wo j2 J 20 Φ ≃ jE wo j2 Φ∕2:
(4)
By averaging over pixels, we get the so called correlation hE 1 E 0 i, and auto correlation hjE 0 j2 i X hE 1 E 0 ix;y ≃ Φx; y∕2N pix jE wo x0 ; y0 j2 ; x0 ;y0
X hjE 0 j ix;y ≃ 1∕N pix jE wo x0 ; y0 j2 ; 2
(5)
Fig. 3. Ratio jhE 1 E 0 ij∕hjE0 j2 i calculated by Monte Carlo by decreasing the sideband signal jE1 j2 . Horizontal axis is the total sideband energy, nmax N pix hjE 1 j2 i in photo electron units. Simulation is made with nmax 400, N pix 502 , jE LO j2 104 , and jE0 j2 102 photo electrons.
x0 ;y0
P where x0 ;y0 is the summation over the N pix pixels of the averaging zone located around the point of coordinate x, y. The vibration amplitude Φ is then Φx; y 2hE 1 E 0 ix;y ∕hjE 0 j2 ix;y :
(6)
We get here Φ that gives both the amplitude zmax λjΦj∕4π and the phase arg Φ of the mechanical motion. Note that it is also possible to get Φ by calculating the ratio E 1 ∕E 0 ≃ Φ as done by Verrier and Atlan [21], but the ratio calculation is unstable for the points x, y of the object where E wo is close to zero. To evaluate the limits of sensibility of the correlation method, we have calculated by Monte Carlo the detection limit of jΦj, for an ideal holographic detection that is only limited by shot noise. The calculation is similar to one made by Lesaffre et al. [25]. For each pixel (x, y), each frame (n), and each sequence (m 0 or m 1), we have calculated the ideal camera signal I 0n in the absence of shot noise. We have I 0n jE LO jn E 00∕1 j2 , where the factor j n accounts for the phase shift of the local oscillator field E LO with respect to object field E 0∕1 for frame n. To account for the roughness of the sample, E 00 and E01 are taken proportional to a Gaussian speckle E 0wo x; y that is uncorrelated from one pixel to the next, but remains the same for all frames n and all sequences m 0 or 1. To 0 account for shot noise, we have padded to I n a Gaussian random noise s of variance of I 0n , where I 0n is expressed in electron photo electron units. The noise s is uncorrelated from one pixels x, y to another, from one frame n to another, and from one sequence m to another. In this way, we have obtained the Monte Carlo frame signals I n I 0n s with which we have performed the fourphase demodulation with nmax frames of Eq. (3). We have then calculated the reconstructed signal E 0 and E 1 , the products E 1 E0 and jE 0 j2 , and the correlation and auto correlation hE 1 E 0 i and hjE 0 j2 i. We have finally calculated the ratio hE 1 E 0 i∕hjE 0 j2 i that gives Φ using Eq. (6). Figure 3 gives the result of the Monte Carlo simulation for the ratio jhE1 E 0 ij∕hjE 0 j2 i. Each point corresponds to a simulation made with a double sequence m 0 and 1. The simulation is performed by decreasing the sideband averaged signal field intensity hjE 1 j2 i 1, 0.5, 0.25,…
photo electron per pixel and per frame. The other parameters of the simulation are nmax 400, N pix 502 , jE LO j2 104 , and hjE 0 j2 i 102 photo electrons. For each value of hjE 1 j2 i, 10 simulations are performed. As can be seen in Fig. 3, the ratio jhE 1 E 0 ij∕hjE 0 j2 i decreases with the sideband signal E 1 proportionally p with hjE 1 j2 i. When E1 becomes very small, the ratio reaches a noise floor related to shot noise. To simplify the present discussion, the results are displayed as a function of the total number of photo electrons on the sideband m 1. The x axis is thus nmax N pix hjE 1 j2 i. As seen in Fig. 3, the noise floor is reached for nmax N pix hjE 1 j2 i 1. We have verified by making simulation not displayed in Fig. 3 that this result does not depend on jE LO j2 and hjE 0 j2 i, provided that jELO j2 ≫ thus to a hjE 0 j2 i ≫ hjE1 j2 i. The noise floor corresponds q minimal
ratio
jhE 1 E 0 ij∕hjE 0 j2 i 1∕ nmax N pix hjE 0 j2 i
to a minimal detectable vibration (i.e., to 10−4 ), and q amplitude Φ 2∕ nmax N pix hjE 0 j2 i (i.e., to 210−4 ). We have tested the ability of the correlation method to measure low vibration amplitude in an experiment made with a vibrating clarinet reed excited at ωA 2 kHz with a loudspeaker. To increase the sensitivity, the vibration amplitude is measured on a single point x0 ; y0 [see Fig. 4(a)]. The illumination has been focused on that point, and the calculations have been made with an averaging region centered on that point, whose radius has been chosen to include the whole illumination zone. We have reconstructed the field image of the reed at
Fig. 4. (a) Clarinet reed with illumination beam focused in x0 , y0 . (b) Sideband m 1 reconstructed image of the vibrating reed. The display is made in arbitrary log scale for the field intensity jE1 x; yj2 .
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The authors wish to acknowledge ANR-ICLM (ANR2011-BS04-017-04) for financial support and M. Lesaffre for fruitful discussion and helpful comments.
Fig. 5. Ratio jhE 1 E 0 ij∕hjE0 j2 i as a function of the reed excitation voltage converted in vibration amplitude zmax in nm units. Measurements, dark gray points; and theory J 1 jΦj∕ J 0 jΦj ≃ jΦj∕2, light gray curve.
the carrier frequency (i.e., E 0 x; y) [see Fig. 4(b)], and verified on the holographic data that most of the energy jE 0 j2 is within the averaging zone x0 , y0 . Sequences of nmax 128 frames I n have been recorded for both carrier (E 0 ) and sideband (E 1 ), while the peak-to-peak voltage V pp of the loudspeaker sinusoidal signal has been decreased. We have then calculated H by Eq. (3), reconstructed the field images of the reed E 0 and E 1 , and calculated E 1 E 0 , jE 0 j2 , and hE1 E 0 i and hjE 0 j2 i. We have then calculated the ratio jhE 1 E0 ij∕hjE 0 j2 i. The latter is plotted in Fig. 5 as a function of the loudspeaker voltage V pp that is proportional to vibration amplitude Φ. The measured points follow jhE 1 E 0 ij∕hjE 0 j2 i ≃ jΦj∕2 ∝ V pp down to a vibration amplitude noise floor of jΦj ≃ 710−5 that corresponds to zmax ≃ 10−2 nm, i.e., λ∕78; 000. Note that the noise floor measure here is about ×20 lower than in previous holographic experiments [19,21] and lower than the limit λ∕3500 predicted by Ueda et al. [18]. A similar noise floor has been detected by holography by Bruno et al. [23], but at much higher vibration frequency (40 kHz). In future work, it would be interesting to explore the limits of sensitivity of the correlation technique for low vibration amplitude. This can be done by using a laser with lower noise, by increasing the vibration frequency ωA (and in all case by measuring the laser noise at ωA ). One could also increase the illumination power, and the number of frames of the coherent detection (nmax > 128). A better control of the vibration ωA versus camera ωCCD frequencies, and or a proper choice of the demodulation equation also could be important to avoid leak detection of E 0 , when detection is tuned on E 1 .
References 1. G. Pedrini, Y. Zou, and H. Tiziani, J. Mod. Opt. 42, 367 (1995). 2. G. Pedrini, H. J. Tiziani, and Y. Zou, Opt. Laser Eng. 26, 199 (1997). 3. G. Pedrini, P. Froening, H. Fessler, and H. Tiziani, Opt. Laser Technol. 29, 505 (1998). 4. G. Pedrini, W. Osten, and M. E. Gusev, Appl. Opt. 45, 3456 (2006). 5. Y. Fu, G. Pedrini, and W. Osten, Appl. Opt. 46, 5719 (2007). 6. R. L. Powell and K. A. Stetson, J. Opt. Soc. Am. 55, 1593 (1965). 7. P. Picart, J. Leval, D. Mounier, and S. Gougeon, Opt. Lett. 28, 1900 (2003). 8. K. A. Stetson and W. R. Brohinsky, J. Opt. Soc. Am. A 5, 1472 (1988). 9. R. J. Pryputniewicz and K. A. Stetson, Proc. SPIE 1162, 456 (1990). 10. C. Aleksoff, Appl. Phys. Lett. 14, 23 (1969). 11. O. J. Løkberg, H. M. Pedersen, H. Valø, and G. Wang, Appl. Opt. 33, 4997 (1994). 12. P. Picart, J. Leval, D. Mounier, and S. Gougeon, Appl. Opt. 44, 337 (2005). 13. I. Yamaguchi and T. Zhang, Opt. Lett. 22, 1268 (1997). 14. R. Dändliker, B. Ineichen, and F. Mottier, Opt. Commun. 9, 412 (1973). 15. F. Le Clerc, M. Gross, and L. Collot, Opt. Lett. 26, 1550 (2001). 16. F. Le Clerc, L. Collot, and M. Gross, Opt. Lett. 25, 716 (2000). 17. M. Atlan, M. Gross, and E. Absil, Opt. Lett. 32, 1456 (2007). 18. M. Ueda, S. Miida, and T. Sato, Appl. Opt. 15, 2690 (1976). 19. P. Psota, V. Ledl, R. Dolecek, J. Erhart, and V. Kopecky, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 59, 1962 (2012). 20. P. Psota, V. Lédl, R. Doleček, J. Václavk, and M. Šulc, in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2012), paper DSu5B-3. 21. N. Verrier and M. Atlan, Opt. Lett. 38, 739 (2013). 22. F. Bruno, J.-B. Laudereau, M. Lesaffre, N. Verrier, and M. Atlan, Appl. Opt. 53, 1252 (2014). 23. F. Bruno, J. Laurent, D. Royer, and M. Atlan, Appl. Phys. Lett. 104, 083504 (2014). 24. U. Schnars and W. Juptner, Appl. Opt. 33, 179 (1994). 25. M. Lesaffre, N. Verrier, and M. Gross, Appl. Opt. 52, A81 (2013).
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Vibration of low amplitude imaged in amplitude and phase by sideband versus carrier correlation digital holography N. Verrier,1,2 L. Alloul,1 and M. Gross1,* 1
Laboratoire Charles Coulomb—UMR 5221, CNRS-UM2 Université Montpellier II Bat 11. Place Eugène Bataillon 34095, Montpellier Cedex 5, France 2 Laboratoire Hubert Curien, UMR 5516, CNRS-Université Jean Monnet, 18 Rue du Professeur Benoît Lauras, Saint-Etienne 42000, France *Corresponding author: [email protected] Received December 5, 2014; revised December 19, 2014; accepted December 20, 2014; posted December 23, 2014 (Doc. ID 229072); published January 30, 2015 Sideband holography can be used to get field images (E0 and E1 ) of a vibrating object for both the carrier (E0 ) and the sideband (E1 ) frequency with respect to vibration. Here we propose to record E0 and E 1 sequentially and to image the product E1 E 0 or the correlation hE1 E0 i. We show that these quantities are insensitive to the phase related to the object roughness and directly reflect the phase of the mechanical motion. The signal to noise can be improved by averaging E1 E0 over a neighbor pixel, yielding hE1 E0 i. Experimental validation is made with a vibrating cube of wood and a clarinet reed. At 2 kHz, vibrations of amplitude down to 0.01 nm are detected. © 2015 Optical Society of America OCIS codes: (120.7280) Vibration analysis; (090.1995) Digital holography; (040.2840) Heterodyne; (120.2880) Holographic interferometry. http://dx.doi.org/10.1364/OL.40.000411
There is a big demand for full-field vibration measurements, in particular in industry. Different holographic techniques are able to image and analyze such vibrations. Pulsed holography [1–3] records several holograms with time separation in the 1…1000 μs range, getting the instantaneous velocity from the phase difference. For small vibration frequencies, one can directly track the vibration of the object with fast CMOS cameras [4,5]. The analysis of the motion can be done by phase difference or by Fourier analysis in the time domain. For periodic vibration motions, measurements can be done with a slow camera. Indeed, an harmonically vibrating object illuminated by a laser yields alternate dark and bright fringes [6] that can be analyzed by time averaged holography [7]. It is also possible to use a phase modulated reference to shift these fringes [8], and to determine vibration amplitude by fringe shift algorithms [9]. With phase modulated reference, it is also possible to detect vibration harmonics of high range [10], and to select a given harmonic by adjusting the phase modulation amplitude [11]. Although the time averaged method provides a way to determine the amplitude of vibration [12], quantitative measurement remains quite difficult for low and high vibration amplitudes. We have developed a variant of phase shifting holography [13] called digital heterodyne holography (DHH, not to be confused with heterodyne holography with plates [14]), in which the phase shift is replaced by a frequency shift controlled by acousto-optic modulators (AOM) driven by numerical synthesizers [15,16]. This gives much better control of the phase [17]. DHH is extremely versatile, since the frequency, phase, and amplitude of both reference and signal beams can be controlled. By shifting the frequency of the local oscillator ωLO with respect to illumination ω0 , it is for example possible to detect the holographic signal at a frequency ω different from illumination ω0 . This ability is extremely useful to analyze vibration, since DHH can detect 0146-9592/15/030411-04$15.00/0
selectively the signal that is scattered by an object on a vibration sideband of frequency ωm ω0 mωA , where ωA is the vibration frequency and m and integer index. As was reported by Ueda et al. [18], the detection of the sideband m 1 is advantageous when the vibration amplitude is small. Nanometric vibration amplitude measurements were achieved with sideband digital holography on the m 1 sideband [19], and comparison with single point laser interferometry has been made [20]. Verrier and Atlan [21] have shown that one can simultaneously measure E 0 and E 1 by using a local oscillator with two frequency components. One can thus infer the mechanical phase of the vibration [22]. However, Bruno et al. [23] show that this simultaneous detection of E 0 and E1 can lead to cross talk, and to a loss of detection sensitivity, which becomes annoying when the vibration amplitude is small. In this Letter, we show that simultaneous detection of E 0 and E 1 is not necessary, and that equivalent or superior performances can be obtained by detecting E 0 and E 1 sequentially. Indeed, the cross talk effects seen by Bruno et al. [23] disappear in that case. We also show that the random phase variations caused by the roughness of the object can be eliminated by calculating the product E 1 E 0 . It is then possible to increase the signal-to-noise ratio (SNR) by averaging E 1 E 0 over neighboring pixels yielding hE 1 E0 i. The sequential measurement of E 0 and E 1 , and the calculation of E 1 E 0 and hE 1 E 0 i, make it possible to image the vibration “full field,” and to measure quantitatively its amplitude and phase. Maximum sensitivity is achieved by focusing the illumination in the studied point and by averaging the E 1 E 0 in that region. Finally, we prove that the sensitivity is limited by a sideband signal of one photo-electron per demodulated image sequence. The device and method were validated experimentally by studying a cube of wood vibrating at ≃20 kHz, and a clarinet reed at ≃2 kHz. Measurement sensitivity of 0.01 nm for a © 2015 Optical Society of America
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vibration at 2 kHz, comparable to the sensitivity obtained by Bruno et al. [23] at 40 kHz, is demonstrated. Consider an object illuminated by a laser at frequency ω0 that vibrates at frequency ωA with an out of plane vibration amplitude zmax . The out of plane coordinate is zt zmax sinωA t. The field scattered by the object is E Eteiω0 t c:c:, where c.c. is the complex conjugate and Et is the field complex amplitude. In reflection geometry, we have Et E wo ejΦj sinωA targ Φ , where E wo is the complex field without movement, and Φ is a complex quantity that describes the phase modulation. The phase of this modulation is arg Φ, while the amplitude is jΦj 4πzmax ∕λ. Because P of the Jacobi–Anger expansion, we have Et E wo m J m jΦjejmωA targ Φ , where J m is the mth-order Bessel function of the first kind. The scattered field E is then the sum of field components E m E m eiωm t c:c: of frequency ωm ω0 mωA , where m is the sideband index (m 0 for the carrier) with E m E wo J m jΦjejm arg Φ . For small vibration amplitudes jΦj, the energy within sidebands decreases very rapidly with the sideband indexes m, and one has only to consider the carrier m 0, and the first sideband m 1. We have thus E 0 Φ E wo J 0 jΦj;
E 1 Φ Ewo J 1 jΦjej arg Φ :
(1)
Note that time averaged holography [7] that detects the carrier field E0 is not efficient in detecting small amplitude vibration jΦj, because E 0 varies quadratically with jΦj. On the other hand, sideband holography that is able to detect selectively the sideband field E 1 is much more sensitive, because E 1 varies with linearly with jΦj. One must notice that the field scattered by the sample without vibration E wo depends strongly on the x, y position. In a typical experiment, the sample rugosity is such that this field is a fully developed speckle. The reconstructed fields E 0 and E 1 are thus random in phase, from one pixel to the next. One cannot thus simply extract the phase of the vibration from a measurement made on a single sideband. To remove the pixel-to-pixel random phase of E wo , we propose recording the hologram successively on the carrier m 0 and the sideband m 1, to reconstruct the corresponding field image of the object E 0 x; y and E 1 x; y, and to calculate and image the product E 1 E 0 , since this product does not involve Ewo , but jE wo j2 , which is real and has no phase. Indeed, we have E 1 E 0 jE wo j2 J 1 jΦjJ 0 jΦjej arg Φ :
(2)
For small vibration amplitudes (jΦj ≪ 1), E 1 E 0 simplifies to jE wo j2 Φ∕2. Product E 1 E 0 is a powerful tool since it gives directly the phase of mechanical motion arg Φ. Nevertheless, problems can be encountered when the signal jE wo j2 scattered without vibration vanishes. Figure 1 shows the holographic experimental setup used to measure successively E 0 and E 1 to get E 1 E 0 . The main laser L is a Sanyo DL-7140-201 diode laser (λ 785 nm, 50 mW), split into an illumination beam (frequency ωI , complex field E I ), and in an LO beam (ωLO , E LO ). The illumination light scattered by the object interferes with the reference beam on the camera
Fig. 1. DHH setup applied to analyze vibration. L, main laser; AOM1, AOM2, acousto-optic modulators; M, mirror; BS, beam splitter; CCD, camera.
(Lumenera 2-2: 1616 × 1216 pixels of 4.4 × 4.4 μm) whose frame rate is ωCCD 10 Hz. To simplify further the Fourier transform calculations, the 1616 × 1216 measured matrix is truncated to 1024 × 1024. The illumination and LO beam frequencies ωLO and ωI are tuned by using two Bragg cells AOM1 and AOM2, and we have ωLO ωL ωAOM1 and ωI ωL ωAOM2 , where ωAOM1∕2 ≃ 80 MHz are the frequencies of RF signals that drive the AOMs. The RF signals are tuned to have ωLO − ωI ωCCD ∕4 to get E 0 , and to have ωLO − ωI ωA ωCCD ∕4 to get E 1 . Successive sequences of nmax 128 camera frames (i.e., I 0 ; I 1 …I 127 ) are recorded by tuning the RF signals first on the carrier (E 0 ), then on the sideband (E 1 ). The carrier and sideband complex hologram H are obtained from these sequences by four-phase demodulation with nmax frames: Hx; y
nn max −1 X n0
j n I n x; y:
(3)
The field images of the object E 0 x; y and E 1 x; y are then reconstructed from H using a single FFT scheme [24]. The product E 1 E 0 is finally calculated. Figure 2(a) shows the reconstructed product E 1 E 0 images of a cube of wood (2 cm × 2 cm) vibrating at its resonance frequency ωA ∕2π 21.43 kHz. Brightness is jE 1 E 0 j) and color E 1 E 0 phase: arg E 1 E 0 . As seen, the neighbor points have the same phase (same color). To get better SNR for the phase, E 1 E 0 is averaged over neighbor x, y points by using a 2D Gaussian blur filter of radius 4 pixels. Figure 2(b) displays the phase of the averaged product E 1 E 0 in 3D. As seen, the opposite corners (upper left and bottom right for example) vibrate in phase, while the neighbor corners (upper left and upper right for example) vibrate in phase opposition. Note that the cube
Fig. 2. Reconstructed images of a cube of wood vibrating at ωA ∕2π 21.43 kHz. (a) E1 E0 product image: brightness is jE1 E0 j and color is arg E 1 E 0 . (b) 3D display of arg E 1 E 0 .
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is excited in one of its corners by a needle. This may explain why the opposite corners are not perfect in phase in Fig. 2(c). We can go further and use E 0 and E 1 to calculate the vibration complex amplitude Φ. To increase the SNR, let us average over neighbor reconstructed pixels E 1 E 0 and jE 0 j2 with jE 0 j2 jE wo j2 J 20 Φ ≃ jE wo j2 Φ∕2:
(4)
By averaging over pixels, we get the so called correlation hE 1 E 0 i, and auto correlation hjE 0 j2 i X hE 1 E 0 ix;y ≃ Φx; y∕2N pix jE wo x0 ; y0 j2 ; x0 ;y0
X hjE 0 j ix;y ≃ 1∕N pix jE wo x0 ; y0 j2 ; 2
(5)
Fig. 3. Ratio jhE 1 E 0 ij∕hjE0 j2 i calculated by Monte Carlo by decreasing the sideband signal jE1 j2 . Horizontal axis is the total sideband energy, nmax N pix hjE 1 j2 i in photo electron units. Simulation is made with nmax 400, N pix 502 , jE LO j2 104 , and jE0 j2 102 photo electrons.
x0 ;y0
P where x0 ;y0 is the summation over the N pix pixels of the averaging zone located around the point of coordinate x, y. The vibration amplitude Φ is then Φx; y 2hE 1 E 0 ix;y ∕hjE 0 j2 ix;y :
(6)
We get here Φ that gives both the amplitude zmax λjΦj∕4π and the phase arg Φ of the mechanical motion. Note that it is also possible to get Φ by calculating the ratio E 1 ∕E 0 ≃ Φ as done by Verrier and Atlan [21], but the ratio calculation is unstable for the points x, y of the object where E wo is close to zero. To evaluate the limits of sensibility of the correlation method, we have calculated by Monte Carlo the detection limit of jΦj, for an ideal holographic detection that is only limited by shot noise. The calculation is similar to one made by Lesaffre et al. [25]. For each pixel (x, y), each frame (n), and each sequence (m 0 or m 1), we have calculated the ideal camera signal I 0n in the absence of shot noise. We have I 0n jE LO jn E 00∕1 j2 , where the factor j n accounts for the phase shift of the local oscillator field E LO with respect to object field E 0∕1 for frame n. To account for the roughness of the sample, E 00 and E01 are taken proportional to a Gaussian speckle E 0wo x; y that is uncorrelated from one pixel to the next, but remains the same for all frames n and all sequences m 0 or 1. To 0 account for shot noise, we have padded to I n a Gaussian random noise s of variance of I 0n , where I 0n is expressed in electron photo electron units. The noise s is uncorrelated from one pixels x, y to another, from one frame n to another, and from one sequence m to another. In this way, we have obtained the Monte Carlo frame signals I n I 0n s with which we have performed the fourphase demodulation with nmax frames of Eq. (3). We have then calculated the reconstructed signal E 0 and E 1 , the products E 1 E0 and jE 0 j2 , and the correlation and auto correlation hE 1 E 0 i and hjE 0 j2 i. We have finally calculated the ratio hE 1 E 0 i∕hjE 0 j2 i that gives Φ using Eq. (6). Figure 3 gives the result of the Monte Carlo simulation for the ratio jhE1 E 0 ij∕hjE 0 j2 i. Each point corresponds to a simulation made with a double sequence m 0 and 1. The simulation is performed by decreasing the sideband averaged signal field intensity hjE 1 j2 i 1, 0.5, 0.25,…
photo electron per pixel and per frame. The other parameters of the simulation are nmax 400, N pix 502 , jE LO j2 104 , and hjE 0 j2 i 102 photo electrons. For each value of hjE 1 j2 i, 10 simulations are performed. As can be seen in Fig. 3, the ratio jhE 1 E 0 ij∕hjE 0 j2 i decreases with the sideband signal E 1 proportionally p with hjE 1 j2 i. When E1 becomes very small, the ratio reaches a noise floor related to shot noise. To simplify the present discussion, the results are displayed as a function of the total number of photo electrons on the sideband m 1. The x axis is thus nmax N pix hjE 1 j2 i. As seen in Fig. 3, the noise floor is reached for nmax N pix hjE 1 j2 i 1. We have verified by making simulation not displayed in Fig. 3 that this result does not depend on jE LO j2 and hjE 0 j2 i, provided that jELO j2 ≫ thus to a hjE 0 j2 i ≫ hjE1 j2 i. The noise floor corresponds q minimal
ratio
jhE 1 E 0 ij∕hjE 0 j2 i 1∕ nmax N pix hjE 0 j2 i
to a minimal detectable vibration (i.e., to 10−4 ), and q amplitude Φ 2∕ nmax N pix hjE 0 j2 i (i.e., to 210−4 ). We have tested the ability of the correlation method to measure low vibration amplitude in an experiment made with a vibrating clarinet reed excited at ωA 2 kHz with a loudspeaker. To increase the sensitivity, the vibration amplitude is measured on a single point x0 ; y0 [see Fig. 4(a)]. The illumination has been focused on that point, and the calculations have been made with an averaging region centered on that point, whose radius has been chosen to include the whole illumination zone. We have reconstructed the field image of the reed at
Fig. 4. (a) Clarinet reed with illumination beam focused in x0 , y0 . (b) Sideband m 1 reconstructed image of the vibrating reed. The display is made in arbitrary log scale for the field intensity jE1 x; yj2 .
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The authors wish to acknowledge ANR-ICLM (ANR2011-BS04-017-04) for financial support and M. Lesaffre for fruitful discussion and helpful comments.
Fig. 5. Ratio jhE 1 E 0 ij∕hjE0 j2 i as a function of the reed excitation voltage converted in vibration amplitude zmax in nm units. Measurements, dark gray points; and theory J 1 jΦj∕ J 0 jΦj ≃ jΦj∕2, light gray curve.
the carrier frequency (i.e., E 0 x; y) [see Fig. 4(b)], and verified on the holographic data that most of the energy jE 0 j2 is within the averaging zone x0 , y0 . Sequences of nmax 128 frames I n have been recorded for both carrier (E 0 ) and sideband (E 1 ), while the peak-to-peak voltage V pp of the loudspeaker sinusoidal signal has been decreased. We have then calculated H by Eq. (3), reconstructed the field images of the reed E 0 and E 1 , and calculated E 1 E 0 , jE 0 j2 , and hE1 E 0 i and hjE 0 j2 i. We have then calculated the ratio jhE 1 E0 ij∕hjE 0 j2 i. The latter is plotted in Fig. 5 as a function of the loudspeaker voltage V pp that is proportional to vibration amplitude Φ. The measured points follow jhE 1 E 0 ij∕hjE 0 j2 i ≃ jΦj∕2 ∝ V pp down to a vibration amplitude noise floor of jΦj ≃ 710−5 that corresponds to zmax ≃ 10−2 nm, i.e., λ∕78; 000. Note that the noise floor measure here is about ×20 lower than in previous holographic experiments [19,21] and lower than the limit λ∕3500 predicted by Ueda et al. [18]. A similar noise floor has been detected by holography by Bruno et al. [23], but at much higher vibration frequency (40 kHz). In future work, it would be interesting to explore the limits of sensitivity of the correlation technique for low vibration amplitude. This can be done by using a laser with lower noise, by increasing the vibration frequency ωA (and in all case by measuring the laser noise at ωA ). One could also increase the illumination power, and the number of frames of the coherent detection (nmax > 128). A better control of the vibration ωA versus camera ωCCD frequencies, and or a proper choice of the demodulation equation also could be important to avoid leak detection of E 0 , when detection is tuned on E 1 .
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