1302
OPTICS LETTERS / Vol. 40, No. 7 / April 1, 2015
Self-adaptive vibrometry with CMOS-LCOS digital holography Umberto Bortolozzo,1,* Daniel Dolfi,2 Jean-Pierre Huignard,3 Stéphanie Molin,2 Arnaud Peigné,4 and Stefania Residori1 1
INLN, Université de Nice-Sophia Antipolis, CNRS, 1361 route des Lucioles, 06560 Sophia Antipolis, France 2
Thales Research Technology, Campus Polytechnique, 1 av Augustin Fresnel, 91767 Palaiseau, France 3 Jphopto-consultant, 20 rue Campo Formio, 75013 Paris, France 4
Thales Underwater System, 525 Route Dolines, 06901 Sophia Antipolis, France *Corresponding author: [email protected]
Received January 14, 2015; revised February 17, 2015; accepted February 21, 2015; posted February 23, 2015 (Doc. ID 232434); published March 20, 2015 A self-adaptive interferometer based on digital holography is here reported for applications involving measurements of very small amplitude vibrations. The two-beam coupling gain is optimized through an electronic feedback, while the dynamic character of the hologram allows reaching a high sensitivity of the interferometric measurements even in unstable environments and with strongly distorted wave-fronts. The frequency bandwidth of the adaptive interferometer and its spatial resolution are determined, respectively, by the maximum frame rate and the pixel size of the camera and of the spatial light modulator used to build the digital holographic setup. © 2015 Optical Society of America OCIS codes: (090.2880) Holographic interferometry; (090.1995) Digital holography; (190.4223) Nonlinear wave mixing; (230.3720) Liquid-crystal devices. http://dx.doi.org/10.1364/OL.40.001302
Since its early introduction by Goodman [1], digital holography is now an important and developed technology in industry, scientific instrumentation, and biophotonics, with a diversity of applications in metrology, non-destructive testing, microscopy, and 3D imaging [2]. Even in the conditions of very low signal intensities, a shot noise-limited image reconstruction has been demonstrated [3]. These recent developments are mainly due to the availability of very-high-resolution visible-nearinfrared coupled-charged device (CCD) and complementary metal-oxide semiconductor (CMOS) matrix detectors with micron pixel size, as well as to very performant computing facilities for full size image processing and reconstruction from a complex fringe interferogram. More recently, a novel scheme of digital holography was introduced for real-time phase conjugation of a segmented wavefront and demonstration of coherent combining of fiber lasers [4]. In this approach, the digital holographic system consists of a CCD detection matrix that drives a liquid crystal, LC, spatial light modulator, and SLM. Beside coherent fiber combining, this configuration was also used for demonstrating phase conjugation in a multimode fiber, in highly scattering media, and for focusing and scanning through a multimode fiber [5–9]. The purpose of this Letter is to show that a digital holographic structure based on a CMOS-LCOS (Liquid Crystal on Silicon) SLM can be employed as a nonlinear twobeam coupling medium, where self-diffraction and energy exchange take place between the incident interacting waves. The intensity fringe pattern is recorded by a CMOS matrix, while the corresponding phase hologram is displayed on a LCOS SLM, from which the two incident waves self-diffract in the Raman–Nath regime due to the small thickness of the LC layer. The setup exhibits interesting properties, like two-wave mixing (2WM) gain and nonlinear properties usually encountered in photorefractive crystals or in Kerr-like media [10]. The digital 0146-9592/15/071302-04$15.00/0
holographic structure presents a number of additional features, such as high control of the 2-WM gain, control of the phase shift Φ of the incident grating pattern with respect to the index grating displayed on the SLM, diversity of operating wavelengths and frame rates, high spectral sensitivity, and dynamic range of the detector matrix. Several of these characteristics are outlined in the hereafter-presented model and experiments, as well as in the interferometric demonstration of vibrometry. Adaptive holography was first demonstrated by using photoconductive crystals driven by vibrating interference light patterns [11]. Successively implemented in photorefractive materials, it had been proved as a useful tool for measuring very small vibration amplitudes of 3D objects with a scattering surface [12–15]. More recently, it has been employed to realize acousto-optic imaging through highly scattering biological media [16] and to achieve the detection of picometer displacements by using light valves [17]. Hereafter, we will give a principle demonstration of CMOS-LCOS digital adaptive holography applied to vibrometry. The experimental setup is shown in Fig. 1. The two components, the CMOS camera and the SLM, are symmetrically placed at 45 deg from a 50% beam splitter and precisely imaged one onto each other. The SLM (Holoeye PLUTO) is a LCOS-type phase-only spatial light modulator, 8 bit and 8 μm pixel size and operates in the reflection mode. In order to optimize the image matching, the CMOS (Photonfocus AG, MV1 D1312-80-G2) and the LCOS must have the same pixel size. The size of the image is 1312 × 1080 pixels. Two coherent light beams, a reference R and a speckled signal S waves, come from a solid-state laser (OXXIUS, single longitudinal mode DPSS laser) of wavelength λ 532 nm, and are collimated and enlarged with a beam size of ∼1 cm. They are separated by the beam splitter and, then sent to interfere onto the CMOS camera. The resulting interference pattern is displayed in real time © 2015 Optical Society of America
April 1, 2015 / Vol. 40, No. 7 / OPTICS LETTERS
Δφt φ~ δφ sinΩt;
1303
(2)
where φ~ accounts for the slowly varying phase perturbations, and δφ sinΩt is the phase modulation, for example a vibration, that we want to detect. Because the frequency Ω∕2π of the modulation is largely greater than the bandwidth of the adaptive hologram, the pattern acquired by the camera is jRj2 jSj2 2π ic jRSjJ 0 δφ cos x φ~ ; 2 2 Λ
Fig. 1. Digital adaptive holographic interferometer setup for vibrometry. The two beams R and S originate from the same laser at 532 nm. Low-amplitude phase modulation at frequency Ω of the speckle on the signal beam is due to propagation in a multimode (MM) fiber. Photodetectors are placed on I 0 and I −1 .
(video frame rate of 50 Hz in the experiment) on the LCOS to generate a complex phase hologram, which is the image of the incident holographic pattern on the CMOS. The SLM phase hologram diffracts in real time the incident interfering reference and signal waves, and they are reflected back by the rear mirror of the LCOS. Self-diffraction occurs in the Raman–Nath regime due to the small thickness of the LCOS liquid crystal layer (few μm) and a rather large hologram grating period. The output signal and reference intensities I 0 and I −1 are recorded by two photodiodes (Thorlabs, PDA36A) placed on the 0 and −1 order of diffraction. The SLM and the camera must be precisely aligned in order to match their respective pixels in a oneto-one correspondence. The procedure used in our experiment is as follows. At first, the interferometer is aligned with a white light source in such a way to fix the same distance between the beam splitter and the camera and between the beam splitter and the SLM. When the white light fringes are observed, the two arms of the equivalent Michelson interferometer are equal. Then, each pixel of the camera has to be aligned with the respective one of the SLM. This is achieved by imaging the CMOS sensor and the LCOS rear mirror on a second camera placed at the exit of the interferometer. Both, the CMOS and the SLM are mounted on three axis-translation stages and rotations stages. Once the device is aligned, the reference and signal beam are sent at the entrance of the interferometer with an angle of 2θ ≃ 11 mrad, the fringes spacing being Λ ≃ λ∕2θ 48 μm. The reference R and signal S jSjeiΔφt waves interfere producing an interference pattern on the CMOS camera
(3)
where J 0 is the Bessel function of the first kind and of order zero. For the digital holography, the bandwidth of the adaptive hologram is determined by the frame rate of the CMOS and of the SLM, in our case it is 50 Hz. As noted before, the high-frequency phase modulation is not acquired. On the contrary, the adaptive interferom~ eter follows the slowly varying phase fluctuations φ. At this point, the sequence acquired by the camera is appropriately transformed and sent to the LCOS. The transformation used is a simple re-normalization of the sequence ic r; t:
ΔφSLM
x0 ; y0 ; t
2π 0 x − Φ φ~ ; φ0 φ1 cos Λ
(4)
where the renormalization parameters φ0 and φ1 are 8 bit numbers, φ1 is the amplitude of the grating, and Φ is a phase term introduced by adding a translation Λ x → x0 − 2π Φ. This term may be controlled through the signal processing unit or by translating the position of the LCOS with respect to the incident holographic fringes (see Fig. 1). The signal and reference beams are then diffracted by the phase grating ΔφSLM . The diffracted optical field can be written as ([17]) jSj jRj −iΦ e γ (5) E m J m φ1 eiδφ sinΩt iJ m1 φ1 2 2 ~ where γ im eiφ0 −mΦm1φ is a phase factor, m is the order of diffraction, and J m is the Bessel function of order m. We note that the parameter Φ can be used to change the relative phase between the diffracted signal and the diffracted reference beam. The output m-order intensity is
(1)
jSj2 jRj2 J 2m1 φ1 4 4 jSRj sinδφ sinΩt Φ: (6) J m φ1 J m1 φ1 2
where Δφt is the phase shift of the signal with respect to the reference beam. In order to take into account slowly varying environmental phase perturbations, the phase shift can be decomposed in two parts,
As observed experimentally, and also from the Eqs. 56, we see that the signal beam can be amplified because of the diffraction of the reference beam on to the LCOS. In order to calculate the 2WM gain, we neglect the high frequency phase modulations and we obtain
ix; y; t
jRj2 jSj2 2π jRSj cos x Δφt ; 2 2 Λ
I m J 2m φ1
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OPTICS LETTERS / Vol. 40, No. 7 / April 1, 2015
J 2 φ βJ 2 φ p J φ J φ I0 0 1 1 1 β 0 1 1 1 sinΦ; (7) 2 4 4 2 jSj where β jRj2 ∕jSj2 . The 2-WM gain as a function of β is shown in Fig. 2. The maximum value of β is determined by the dynamic range of the CMOS camera, in our case is of the order of ∼2500. By using a constant value for φ1 and if we choose the phase shift Φ 0, the diffracted signal and reference are in quadrature, and the gain increases linearly with β [Fig. 2(a)]. In the case that φ1 is proportional to jSRj, i.e., proportional to the fringe contrast, like is the case for Kerr-like media [18], the gain is maximized when the phase Φ π∕2 as shown in Fig. 2(b). As a characteristic of the adaptive holography, we see from the Eqs. 56 that the low frequency phase perturba~ tions φt are filtered out by the diffraction of the two input beams. On the contrary, the high-frequency modulation appears and, if we set Φ 0, its amplitude δφ is automatically kept linear with respect to the measured intensity. Complex wavefront signal beams can be used. As shown in Fig. 1, the signal beam is a phase-modulated speckle field due to propagation in a 5-m-long step index multimode fiber (core diameter 200 μm, numerical aperture 0.22). In Fig. 3 is shown the output intensity when the CMOS-LCOS are off (a) and then switched on (b). Because diffraction occurs in the Raman–Nath regime, several orders of diffraction are observed. By considering the expressions for the output intensities in the direction of the signal (m 0) and of the reference (m −1), we have in the conditions of linear detection jRj2 jSj2 J 20 φ1 4 4 jRjjSjJ 0 φ1 J 1 φ1 δφ sinΩt
I 0 J 21 φ1
(8)
Fig. 3. Diffraction of the reference and signal beams on the hologram. (a) The CMOS-LCOS are off and (b) on (φ1 2.2 rad, jRj2 5.1 mW∕cm2 , jSj2 0.4 mW∕cm2 ).
jRj2 jSj2 J 21 φ1 4 4 −jRjjSjJ 0 φ1 J 1 φ1 δφ sinΩt:
I −1 J 20 φ1
(9)
The two above equations are obtained in the case of small amplitude δφ ≪ 1. The signal beam issued from the multimode fiber is phase modulated at a frequency Ω 10 KHz. Correspondingly, an example of the spectrum measured for the diffracted reference beam I −1 is shown in Fig. 4. The modulation is clearly detected directly by the photodiode in a linear regime, corresponding to Φ 0. Finally, we would like to emphasize that CMOS-LCOS digital holography provides another substantial advantage with respect to conventional nonlinear media, that is, the amplitude of the phase grating on the SLM can be optimized through the signal processing transformation. As an example, φ1 can be chosen by controlling the electronic gain of the CMOS and, respectively, the LCOS. For example, by choosing φ1 1.435 rad such that J 0 φ1 J 1 φ1 , and by using a balanced detection as proposed in [19], we obtain the following expression for the output differential intensity ΔI I 0 − I −1 ∝ jRjjSjδφ sinΩt:
(10)
In this case, the signal-to-noise ratio is increased because the zero frequency is suppressed by the differential measurement. Another advantage is offered by the possibility of using a spatial transformation in order to optimize the diffraction efficiency, for example, by introducing an asymmetric grating profile.
Fig. 2. (a) CMOS-LCOS two-beam coupling gain and (b)– (d) comparison with a Kerr-like media in Raman–Nath regime of diffraction (as for example liquid crystal light valve [17] or photorefractive crystals [13]). Black squares: experimental points; continuous line: theoretical prediction. The parameters are: (a) φ1 2.2 rad, Φ 0; (b) φ1 αjSRj, Φ π∕2; (c) φ1 αjSRj, Φ 0; (d) φ1 αjSRj, Φ −π∕2; where α 4πλ n2 d, jRj2 5.1 mW∕cm2 , n2 1 cm2 ∕W, and d 6 μm.
Fig. 4. Spectrum of the detected intensity I −1 . Same parameters as before, Ω 10 KHz, δφ ≃ 0.3 mrad.
April 1, 2015 / Vol. 40, No. 7 / OPTICS LETTERS
In conclusion, digital CMOS-LCOS adaptive holography has been demonstrated in a relatively simple and compact setup. The performances in terms of two-beam coupling show an improvement of the 2-WM gain. We highlight the great flexibility of this approach: the setup is well suited to a large range of visible or IR wavelengths, due to progress of highly sensitive large size matrix sensors operating at video or much higher frame rates, and SLM technology is also available with improved performances in terms of number of pixels and speed, either with LC or MEMS arrays. Moreover, the beam coupling of the two interfering waves and phase shift between fringes and index grating is easily controlled and can be optimized. CMOS-LCOS digital holography seems, therefore, a viable and performant solution to realize interferometric measurements of very-small-phase amplitude modulations, in particular, on optical beams characterized by complex wave-fronts. D. Martina and S. Gigan (Laboratoire Kastler-Brossler de l’ENS, Paris) are gratefully acknowledged for helpful discussions. This work is supported by the DGA under the contract ANR-11-ASTR-0012, MEDUSE. References 1. J. W. Goodman and R. W. Laurence, Appl. Phys. Lett. 11, 77 (1967). 2. U. Shnars and W. Juptner, Digital Holography (Springer, 2005). 3. M. Gross and M. Atlan, Opt. Lett. 32, 909 (2007).
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4. C. Bellanger, A. Brignon, J. Colineau, and J. P. Huignard, Opt. Lett. 33, 2937 (2008). 5. M. Paurisse, M. Hanna, F. Druon, P. Georges, C. Bellanger, A. Brignon, and J. P. Huignard, Opt. Express 17, 13000 (2009). 6. M. Cui and C. Yang, Opt. Express 18, 3444 (2010). 7. I. Papadopoulos, S. Farahi, D. Psaltis, and C. Moser, Opt. Express 20, 10583 (2012). 8. I. Papadopoulos, O. Simandoux, S. Farahi, J. P. Huignard, E. Bossy, D. Psaltis, and C. Moser, Appl. Phys. Lett. 102, 211106 (2013). 9. I. Papadopoulos, S. Farahi, D. Psaltis, and C. Moser, Biomed. Opt. Express 4, 260 (2013). 10. U. Bortolozzo, S. Residori, and J. P. Huignard, J. Phys. D 41, 224007 (2008). 11. S. I. Stepanov, I. A. Sokolov, G. S. Trofimov, V. I. Vlad, D. Popa, and I. Apostol, Opt. Lett. 15, 1239 (1990). 12. J. P. Huignard and A. Marrakchi, Opt. Lett. 6, 622 (1981). 13. V. Petrov, C. Denz, J. Petter, and T. Tschudi, Opt. Lett. 22, 1902 (1997). 14. S. Di Girolamo, A. A. Kamshilin, R. V. Romashko, Y. N. Kulchin, and J. C. Launay, Opt. Express 15, 545 (2006). 15. K. Shcherbin and M. B. Klein, Opt. Commun. 282, 2580 (2009). 16. E. Benoit a la Guillaume, S. Farahi, E. Bossy, M. Gross, and F. Ramaz, Opt. Lett. 37, 3216 (2012). 17. U. Bortolozzo, S. Residori, and J. P. Huignard, Opt. Lett. 34, 2006 (2009). 18. A. Brignon, I. Bongrand, B. Loiseaux, and J.-P. Huignard, Opt. Lett. 22, 1855 (1997). 19. U. Bortolozzo, S. Residori, and J. C. Howell, Opt. Lett. 38, 3107 (2013).
OPTICS LETTERS / Vol. 40, No. 7 / April 1, 2015
Self-adaptive vibrometry with CMOS-LCOS digital holography Umberto Bortolozzo,1,* Daniel Dolfi,2 Jean-Pierre Huignard,3 Stéphanie Molin,2 Arnaud Peigné,4 and Stefania Residori1 1
INLN, Université de Nice-Sophia Antipolis, CNRS, 1361 route des Lucioles, 06560 Sophia Antipolis, France 2
Thales Research Technology, Campus Polytechnique, 1 av Augustin Fresnel, 91767 Palaiseau, France 3 Jphopto-consultant, 20 rue Campo Formio, 75013 Paris, France 4
Thales Underwater System, 525 Route Dolines, 06901 Sophia Antipolis, France *Corresponding author: [email protected]
Received January 14, 2015; revised February 17, 2015; accepted February 21, 2015; posted February 23, 2015 (Doc. ID 232434); published March 20, 2015 A self-adaptive interferometer based on digital holography is here reported for applications involving measurements of very small amplitude vibrations. The two-beam coupling gain is optimized through an electronic feedback, while the dynamic character of the hologram allows reaching a high sensitivity of the interferometric measurements even in unstable environments and with strongly distorted wave-fronts. The frequency bandwidth of the adaptive interferometer and its spatial resolution are determined, respectively, by the maximum frame rate and the pixel size of the camera and of the spatial light modulator used to build the digital holographic setup. © 2015 Optical Society of America OCIS codes: (090.2880) Holographic interferometry; (090.1995) Digital holography; (190.4223) Nonlinear wave mixing; (230.3720) Liquid-crystal devices. http://dx.doi.org/10.1364/OL.40.001302
Since its early introduction by Goodman [1], digital holography is now an important and developed technology in industry, scientific instrumentation, and biophotonics, with a diversity of applications in metrology, non-destructive testing, microscopy, and 3D imaging [2]. Even in the conditions of very low signal intensities, a shot noise-limited image reconstruction has been demonstrated [3]. These recent developments are mainly due to the availability of very-high-resolution visible-nearinfrared coupled-charged device (CCD) and complementary metal-oxide semiconductor (CMOS) matrix detectors with micron pixel size, as well as to very performant computing facilities for full size image processing and reconstruction from a complex fringe interferogram. More recently, a novel scheme of digital holography was introduced for real-time phase conjugation of a segmented wavefront and demonstration of coherent combining of fiber lasers [4]. In this approach, the digital holographic system consists of a CCD detection matrix that drives a liquid crystal, LC, spatial light modulator, and SLM. Beside coherent fiber combining, this configuration was also used for demonstrating phase conjugation in a multimode fiber, in highly scattering media, and for focusing and scanning through a multimode fiber [5–9]. The purpose of this Letter is to show that a digital holographic structure based on a CMOS-LCOS (Liquid Crystal on Silicon) SLM can be employed as a nonlinear twobeam coupling medium, where self-diffraction and energy exchange take place between the incident interacting waves. The intensity fringe pattern is recorded by a CMOS matrix, while the corresponding phase hologram is displayed on a LCOS SLM, from which the two incident waves self-diffract in the Raman–Nath regime due to the small thickness of the LC layer. The setup exhibits interesting properties, like two-wave mixing (2WM) gain and nonlinear properties usually encountered in photorefractive crystals or in Kerr-like media [10]. The digital 0146-9592/15/071302-04$15.00/0
holographic structure presents a number of additional features, such as high control of the 2-WM gain, control of the phase shift Φ of the incident grating pattern with respect to the index grating displayed on the SLM, diversity of operating wavelengths and frame rates, high spectral sensitivity, and dynamic range of the detector matrix. Several of these characteristics are outlined in the hereafter-presented model and experiments, as well as in the interferometric demonstration of vibrometry. Adaptive holography was first demonstrated by using photoconductive crystals driven by vibrating interference light patterns [11]. Successively implemented in photorefractive materials, it had been proved as a useful tool for measuring very small vibration amplitudes of 3D objects with a scattering surface [12–15]. More recently, it has been employed to realize acousto-optic imaging through highly scattering biological media [16] and to achieve the detection of picometer displacements by using light valves [17]. Hereafter, we will give a principle demonstration of CMOS-LCOS digital adaptive holography applied to vibrometry. The experimental setup is shown in Fig. 1. The two components, the CMOS camera and the SLM, are symmetrically placed at 45 deg from a 50% beam splitter and precisely imaged one onto each other. The SLM (Holoeye PLUTO) is a LCOS-type phase-only spatial light modulator, 8 bit and 8 μm pixel size and operates in the reflection mode. In order to optimize the image matching, the CMOS (Photonfocus AG, MV1 D1312-80-G2) and the LCOS must have the same pixel size. The size of the image is 1312 × 1080 pixels. Two coherent light beams, a reference R and a speckled signal S waves, come from a solid-state laser (OXXIUS, single longitudinal mode DPSS laser) of wavelength λ 532 nm, and are collimated and enlarged with a beam size of ∼1 cm. They are separated by the beam splitter and, then sent to interfere onto the CMOS camera. The resulting interference pattern is displayed in real time © 2015 Optical Society of America
April 1, 2015 / Vol. 40, No. 7 / OPTICS LETTERS
Δφt φ~ δφ sinΩt;
1303
(2)
where φ~ accounts for the slowly varying phase perturbations, and δφ sinΩt is the phase modulation, for example a vibration, that we want to detect. Because the frequency Ω∕2π of the modulation is largely greater than the bandwidth of the adaptive hologram, the pattern acquired by the camera is jRj2 jSj2 2π ic jRSjJ 0 δφ cos x φ~ ; 2 2 Λ
Fig. 1. Digital adaptive holographic interferometer setup for vibrometry. The two beams R and S originate from the same laser at 532 nm. Low-amplitude phase modulation at frequency Ω of the speckle on the signal beam is due to propagation in a multimode (MM) fiber. Photodetectors are placed on I 0 and I −1 .
(video frame rate of 50 Hz in the experiment) on the LCOS to generate a complex phase hologram, which is the image of the incident holographic pattern on the CMOS. The SLM phase hologram diffracts in real time the incident interfering reference and signal waves, and they are reflected back by the rear mirror of the LCOS. Self-diffraction occurs in the Raman–Nath regime due to the small thickness of the LCOS liquid crystal layer (few μm) and a rather large hologram grating period. The output signal and reference intensities I 0 and I −1 are recorded by two photodiodes (Thorlabs, PDA36A) placed on the 0 and −1 order of diffraction. The SLM and the camera must be precisely aligned in order to match their respective pixels in a oneto-one correspondence. The procedure used in our experiment is as follows. At first, the interferometer is aligned with a white light source in such a way to fix the same distance between the beam splitter and the camera and between the beam splitter and the SLM. When the white light fringes are observed, the two arms of the equivalent Michelson interferometer are equal. Then, each pixel of the camera has to be aligned with the respective one of the SLM. This is achieved by imaging the CMOS sensor and the LCOS rear mirror on a second camera placed at the exit of the interferometer. Both, the CMOS and the SLM are mounted on three axis-translation stages and rotations stages. Once the device is aligned, the reference and signal beam are sent at the entrance of the interferometer with an angle of 2θ ≃ 11 mrad, the fringes spacing being Λ ≃ λ∕2θ 48 μm. The reference R and signal S jSjeiΔφt waves interfere producing an interference pattern on the CMOS camera
(3)
where J 0 is the Bessel function of the first kind and of order zero. For the digital holography, the bandwidth of the adaptive hologram is determined by the frame rate of the CMOS and of the SLM, in our case it is 50 Hz. As noted before, the high-frequency phase modulation is not acquired. On the contrary, the adaptive interferom~ eter follows the slowly varying phase fluctuations φ. At this point, the sequence acquired by the camera is appropriately transformed and sent to the LCOS. The transformation used is a simple re-normalization of the sequence ic r; t:
ΔφSLM
x0 ; y0 ; t
2π 0 x − Φ φ~ ; φ0 φ1 cos Λ
(4)
where the renormalization parameters φ0 and φ1 are 8 bit numbers, φ1 is the amplitude of the grating, and Φ is a phase term introduced by adding a translation Λ x → x0 − 2π Φ. This term may be controlled through the signal processing unit or by translating the position of the LCOS with respect to the incident holographic fringes (see Fig. 1). The signal and reference beams are then diffracted by the phase grating ΔφSLM . The diffracted optical field can be written as ([17]) jSj jRj −iΦ e γ (5) E m J m φ1 eiδφ sinΩt iJ m1 φ1 2 2 ~ where γ im eiφ0 −mΦm1φ is a phase factor, m is the order of diffraction, and J m is the Bessel function of order m. We note that the parameter Φ can be used to change the relative phase between the diffracted signal and the diffracted reference beam. The output m-order intensity is
(1)
jSj2 jRj2 J 2m1 φ1 4 4 jSRj sinδφ sinΩt Φ: (6) J m φ1 J m1 φ1 2
where Δφt is the phase shift of the signal with respect to the reference beam. In order to take into account slowly varying environmental phase perturbations, the phase shift can be decomposed in two parts,
As observed experimentally, and also from the Eqs. 56, we see that the signal beam can be amplified because of the diffraction of the reference beam on to the LCOS. In order to calculate the 2WM gain, we neglect the high frequency phase modulations and we obtain
ix; y; t
jRj2 jSj2 2π jRSj cos x Δφt ; 2 2 Λ
I m J 2m φ1
1304
OPTICS LETTERS / Vol. 40, No. 7 / April 1, 2015
J 2 φ βJ 2 φ p J φ J φ I0 0 1 1 1 β 0 1 1 1 sinΦ; (7) 2 4 4 2 jSj where β jRj2 ∕jSj2 . The 2-WM gain as a function of β is shown in Fig. 2. The maximum value of β is determined by the dynamic range of the CMOS camera, in our case is of the order of ∼2500. By using a constant value for φ1 and if we choose the phase shift Φ 0, the diffracted signal and reference are in quadrature, and the gain increases linearly with β [Fig. 2(a)]. In the case that φ1 is proportional to jSRj, i.e., proportional to the fringe contrast, like is the case for Kerr-like media [18], the gain is maximized when the phase Φ π∕2 as shown in Fig. 2(b). As a characteristic of the adaptive holography, we see from the Eqs. 56 that the low frequency phase perturba~ tions φt are filtered out by the diffraction of the two input beams. On the contrary, the high-frequency modulation appears and, if we set Φ 0, its amplitude δφ is automatically kept linear with respect to the measured intensity. Complex wavefront signal beams can be used. As shown in Fig. 1, the signal beam is a phase-modulated speckle field due to propagation in a 5-m-long step index multimode fiber (core diameter 200 μm, numerical aperture 0.22). In Fig. 3 is shown the output intensity when the CMOS-LCOS are off (a) and then switched on (b). Because diffraction occurs in the Raman–Nath regime, several orders of diffraction are observed. By considering the expressions for the output intensities in the direction of the signal (m 0) and of the reference (m −1), we have in the conditions of linear detection jRj2 jSj2 J 20 φ1 4 4 jRjjSjJ 0 φ1 J 1 φ1 δφ sinΩt
I 0 J 21 φ1
(8)
Fig. 3. Diffraction of the reference and signal beams on the hologram. (a) The CMOS-LCOS are off and (b) on (φ1 2.2 rad, jRj2 5.1 mW∕cm2 , jSj2 0.4 mW∕cm2 ).
jRj2 jSj2 J 21 φ1 4 4 −jRjjSjJ 0 φ1 J 1 φ1 δφ sinΩt:
I −1 J 20 φ1
(9)
The two above equations are obtained in the case of small amplitude δφ ≪ 1. The signal beam issued from the multimode fiber is phase modulated at a frequency Ω 10 KHz. Correspondingly, an example of the spectrum measured for the diffracted reference beam I −1 is shown in Fig. 4. The modulation is clearly detected directly by the photodiode in a linear regime, corresponding to Φ 0. Finally, we would like to emphasize that CMOS-LCOS digital holography provides another substantial advantage with respect to conventional nonlinear media, that is, the amplitude of the phase grating on the SLM can be optimized through the signal processing transformation. As an example, φ1 can be chosen by controlling the electronic gain of the CMOS and, respectively, the LCOS. For example, by choosing φ1 1.435 rad such that J 0 φ1 J 1 φ1 , and by using a balanced detection as proposed in [19], we obtain the following expression for the output differential intensity ΔI I 0 − I −1 ∝ jRjjSjδφ sinΩt:
(10)
In this case, the signal-to-noise ratio is increased because the zero frequency is suppressed by the differential measurement. Another advantage is offered by the possibility of using a spatial transformation in order to optimize the diffraction efficiency, for example, by introducing an asymmetric grating profile.
Fig. 2. (a) CMOS-LCOS two-beam coupling gain and (b)– (d) comparison with a Kerr-like media in Raman–Nath regime of diffraction (as for example liquid crystal light valve [17] or photorefractive crystals [13]). Black squares: experimental points; continuous line: theoretical prediction. The parameters are: (a) φ1 2.2 rad, Φ 0; (b) φ1 αjSRj, Φ π∕2; (c) φ1 αjSRj, Φ 0; (d) φ1 αjSRj, Φ −π∕2; where α 4πλ n2 d, jRj2 5.1 mW∕cm2 , n2 1 cm2 ∕W, and d 6 μm.
Fig. 4. Spectrum of the detected intensity I −1 . Same parameters as before, Ω 10 KHz, δφ ≃ 0.3 mrad.
April 1, 2015 / Vol. 40, No. 7 / OPTICS LETTERS
In conclusion, digital CMOS-LCOS adaptive holography has been demonstrated in a relatively simple and compact setup. The performances in terms of two-beam coupling show an improvement of the 2-WM gain. We highlight the great flexibility of this approach: the setup is well suited to a large range of visible or IR wavelengths, due to progress of highly sensitive large size matrix sensors operating at video or much higher frame rates, and SLM technology is also available with improved performances in terms of number of pixels and speed, either with LC or MEMS arrays. Moreover, the beam coupling of the two interfering waves and phase shift between fringes and index grating is easily controlled and can be optimized. CMOS-LCOS digital holography seems, therefore, a viable and performant solution to realize interferometric measurements of very-small-phase amplitude modulations, in particular, on optical beams characterized by complex wave-fronts. D. Martina and S. Gigan (Laboratoire Kastler-Brossler de l’ENS, Paris) are gratefully acknowledged for helpful discussions. This work is supported by the DGA under the contract ANR-11-ASTR-0012, MEDUSE. References 1. J. W. Goodman and R. W. Laurence, Appl. Phys. Lett. 11, 77 (1967). 2. U. Shnars and W. Juptner, Digital Holography (Springer, 2005). 3. M. Gross and M. Atlan, Opt. Lett. 32, 909 (2007).
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