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OPTICS LETTERS / Vol. 28, No. 20 / October 15, 2003
Time-averaged digital holography Pascal Picart, Julien Leval, Denis Mounier, and Samuel Gougeon École Nationale Supérieure d’Ingénieurs du Mans, Rue Aristote, 72085 Le Mans Cedex 9, France Received April 28, 2003 We demonstrate that it is possible to study the modal structures of a vibrating object with digitally recorded holograms by use of the time-averaging principle. We investigate the numerical reconstruction from a theoretical point of view, and we show that the numerically reconstructed object from a digital hologram is modulated by the zeroth-order Bessel function. Results of experiments in time-averaged digital holography are presented. © 2003 Optical Society of America OCIS codes: 090.0090, 090.2880, 100.2000, 120.3180, 120.0120, 120.4630.
Digital holography can be used advantageously to image objects, and many fascinating possibilities have been demonstrated. A single hologram can provide amplitude-contrast and phase-contrast microscopic imaging,1 and digital color holography is possible.2 Spatial multiplexing and demultiplexing for twin sensitivity measurements has also been demonstrated.3 In classical holography, analysis of the vibration of diffusely ref lecting surfaces began with the pioneering work of Powell and Stetson,4 who showed that the reconstructed image of a time-averaged hologram of a sinusoidally vibrating object is modulated by the J0 function under the assumption that the exposure time is much longer than the vibration period. Furthermore, the amplitude factor is related to the amplitude of vibration. However, to the best of our knowledge, the ability of digital holography for time averaging has not been demonstrated. In this Letter we demonstrate that a numerical vibrating object encoded by a time-averaged digital speckle f ield and reconstructed with the discrete Fresnel transform is also modulated by the J0 function. To do this, let us consider the setup described in Fig. 1. The object of interest is a loudspeaker that is 60 mm in diameter. Diverging lens L4 reduces the spatial frequency spectrum of the object, allowing a hologram of the full object to be recorded correctly. The virtual loudspeaker produced by lens L4 is placed at d0 苷 1030 mm in front of the detector, and it is reduced by a factor of 0.7 compared with the original. The introduction of the off-line holographic recording is accomplished with lens L2 . It is displaced out of the afocal axis by means of two micrometric transducers, allowing a translation dz of the lens in the z direction and a translation dy in the y direction. The reference wave also produced is written as R共x0 , y 0 兲 苷 ar exp关22ip共ur x0 1 vr y 0 兲兴. In our setup ur ⬵ 76.5 mm21 and vr ⬵ 269 mm21 . The detector is a 12-bit digital CCD with M 3 N 苷 1024 3 1360 pixels, each measuring Dx 苷 Dy 苷 4.65 mm. When the object is under sinusoidal excitation, the optical phase variation it produces is written as Dw共t兲 苷 Dw0 1 Dwm sin共v0 t 1 w0 兲, where Dw0 is an offset term, Dwm is the maximum amplitude of the vibration at pulsation v0 , and w0 is the phase of the vibration. Therefore the illuminated object produces an instantaneous object wave that can be written as A共t兲 苷 A0 exp共ic0 兲exp关iDw共t兲兴, with the 0146-9592/03/201900-03$15.00/0
spatial dependence 共x, y兲 of terms A, A0 , c0 , Dw0 , Dwm , and w0 voluntarily omitted. This object diffracts an instantaneous object f ield onto the detector, given by i exp共2i2pd0 兾l兲 O共x0 , y 0 , d0 , t兲 苷 ld0 ∑ ∏ ip 3 exp 2 共x0 2 1 y 0 2 兲 ld0 Z 1` Z 1` A共x, y, t兲 3 2`
2`
∏ ∑ ip 共x2 1 y 2 兲 3 exp 2 ld0 ∏ ∑ 2ip 0 0 共xx 1 yy 兲 dxdy . (1) 3 exp ld0 In the hologram plane the instantaneous power is given by Hi 共x0 , y 0 , d0 , t兲 苷 O0 共x0 , y 0 , d0 , t兲 1 R ⴱ 共x0 , y 0兲O共x0 , y 0 , d0 , t兲 1 R共x0 , y 0 兲O ⴱ 共x0 , y 0 , d , t兲 , 0
(2)
where O0 is the offset term contributing to the zeroth order. The energy recorded by the detector is the temporal integration of the instantaneous power during a lapse of time T , starting from time t1 . Assuming that
Fig. 1.
Experimental setup.
© 2003 Optical Society of America
October 15, 2003 / Vol. 28, No. 20 / OPTICS LETTERS
v0 T ¿ 1, we get, for the time averaging of the term of interest in Eq. (2), Z t1 1T i exp共2i2pd0 兾l兲 R ⴱ 共x0 , y 0 兲O共x0 , y 0, d0, t兲dt 苷 ld0 t1 ∑ ∏ ip 3 exp 2 共x0 2 1 y 0 2 兲 R ⴱ 共x0 , y 0 兲 ld0 ∏ ∑ Z 1` Z 1` 2ip 0 0 ˜ 3 共xx 1 yy 兲 dxdy , (3) F 共x, y兲exp ld0 2` 2` with F˜ 共x, y兲 苷 TA0 共x, y兲J0 关Dwm 共x, y兲兴exp关ic0 共x, y兲兴 ∏ ∑ ip 2 2 3 exp关iDw0 共x, y兲兴exp 2 共x 1 y 兲 . ld0
(4)
The term J0 共 兲 is the zeroth-order Bessel function. The numerical restitution of the object is based on the diffraction integral, considering the hologram as a transmittance. Therefore, in the first-order diffraction of interest and considering that the detector records a discrete version of the hologram, i.e., x0 苷 kDx and y 0 苷 lDy , we have AR 共X, Y , dR 兲 苷
i exp共2i2pdR 兾l兲 ldR ∑ ∏ ip 3 exp 2 共X 2 1 Y 2 兲 ldR " l苷L21 k苷K21 X X Z t1 1T R ⴱ 共kDx , lDy 兲 3 k苷0
l苷0
t1
#
3 O共kDx , lDy , d0, t兲dt ∏ ∑ ip 2 2 2 2 共k Dx 1 l Dy 兲 3 exp 2 ldR ∑ ∏ 2ip 3 exp 共kXDx 1 lY Dy 兲 , ldR
(5)
where 共X, Y 兲 is a set of coordinates associated with the reconstruction plane at distance dR and 共K, L兲 $ 共N, M兲. If we want to retrieve the initial object plane, we have to put dR 苷 2d0 in Eq. (5) to yield 共X, Y 兲 苷 共x, y兲. In this case Eq. (5) reduces to ∏ ∑ ip AR 共x, y, 2d0兲 苷 ar exp 共x2 1 y 2 兲 ld0 ˜ 3 F 共x 2 lur d0 , y 2 lvr d0 兲 fNM 共x, y兲 , ⴱW
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Equation (6) indicates that the reconstructed object is localized at the coordinates x 苷 lur d0 and y 苷 lvr d0 . It is clear that the reconstructed object is not rigorously similar to the initial object or to that obtained in classical holography with a photographic plate, because it is convoluted by the f iltering function of the discrete Fourier transform and thus spatially enlarged. HowfNM 共x, y兲 ever, because M and N are relatively large, W decreases rapidly to zero with increasing 共x, y兲 and is thus localized near coordinates 共0, 0兲. Therefore we fNM 共x, y兲 艐 NM d共x, y兲. When the can consider that W image is numerically reconstructed with a fast Fourier transform, it is sampled with spatial sampling steps equal to Dj 苷 ld0 兾KDx and Dh 苷 ld0 兾LDy. From Eq. (6), with x 苷 nDj and y 苷 mDh, we get the numerical version of the reconstructed object: AR 共n, m, 2d0 兲 ⬵ NMTR ⴱ 共n, m兲exp关2ipld0 共ur 2 1 vr 2 兲兴 3 Amod 共n 2 Kur Dx , m 2 Lvr Dy 兲 ,
(8)
where Amod 苷 A0 J0 共Dwm 兲exp共ic0 兲exp共iDw0 兲. This indicates that the numerically reconstructed object is localized in the image plane at the pixel coordinates 共Kur Dx, Lvr Dy兲 and that the complex amplitude of the initial object is modulated by the J0 function, which relates the amplitude modulation to the amplitude of vibration Dwm . Moreover, the offset phase Dw0 of the vibration is conserved in the numerical reconstruction process. To verify the theoretical investigation, we recorded a digital hologram of the loudspeaker in a vibrating state. The loudspeaker was sinusoidally excited at a frequency of 3900 Hz with an amplitude voltage equal to 0.5 V in order to excite a vibration mode of its membrane. Digital holography has a lower spatial resolution than classical holography. Therefore we chose a small amplitude voltage because the excitation of the object must be performed with small amplitudes of vibration in order to achieve a correct resolution of the zeros of the J0 function. We recorded the timeaveraged digital hologram with an exposure time of 1.5 s, yielding v0 T 苷 36756.6, which is much greater than 1, thus justifying Eq. (4). The image amplitude obtained after the numerical reconstruction of the object with K 苷 L 苷 2048 is shown in Fig. 2. We have
(6)
fNM 共x, y兲 is the f ilwhere ⴱ represents convolution and W tering function of the two-dimensional discrete Fourier transform due to the finite extension of the recording. fNM 共x, y兲 is given by W ∏ ∑ fNM 共x, y兲 苷 exp 2ip共N 2 1兲 xDx 2 ip共M 2 1兲 yDy W ld0 ld0 3
sin共pNxDx 兾ld0 兲 sin共pMyDy 兾ld0 兲 . sin共pxDx 兾ld0 兲 sin共pyDy 兾ld0 兲
(7)
Fig. 2. Image amplitude of the reconstructed object.
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OPTICS LETTERS / Vol. 28, No. 20 / October 15, 2003
Fig. 3. Offset phase of the loudspeaker (modulo 2p).
共Kur Dx, Lvr Dy兲 ⬵ 共730, 2659兲. It can be seen that the localization and the modulated amplitude of the object match the theoretical prediction given by Eq. (8). In the first-order diffraction the modal structure of the membrane of the loudspeaker can be seen. The Bessel function is fully resolved and nodal lines appear. We computed the difference of the optical phase before and after excitation of the loudspeaker. The reference phase c0 was obtained with numerical recording and reconstruction of the loudspeaker in a static state. Referring to Eq. (8), the resulting phase difference is equal to Dw0 , which is the offset phase of the vibration. Figure 3 shows the modulo 2p offset phase, restricted
to its useful part, that we obtained with the data in Fig. 2 and the static image of the loudspeaker. It can be seen that the offset phase is not well resolved because of its high fringe density in some zones of the loudspeaker. In conclusion, we have demonstrated that it is possible to study vibrations with digitally recorded holograms by use of the time-averaging principle. The theoretical investigation showed that the numerical reconstruction from a time-averaged digital hologram is modulated by the zeroth-order Bessel function and that the object localization depends on the reconstructing distance, the number of sampling points, and the spatial frequencies of the carrier wave. The experimental results constitute, to the best of our knowledge, the f irst experiments in time-averaged digital holography. P. Picart’s e-mail address is
[email protected]. References 1. E. Cuche, F. Bevilacqua, and C. Depeursinge, Opt. Lett. 24, 291 (1999). 2. I. Yamaguchi, T. Matsumura, and J. Kato, Opt. Lett. 27, 1108 (2002). 3. P. Picart, E. Moisson, and D. Mounier, Appl. Opt. 42, 1947 (2003). 4. R. L. Powell and K. A. Stetson, J. Opt. Soc. Am. A 55, 1593 (1965).
