Abstract

In a conventional digital phase conjugation system, only the phase of an input light is time-reversed. This deteriorates phase conjugation fidelity and restricts application fields to specific cases only when the input light has uniformly-distributed scattered wavefront. To overcome these difficulties, we present a digital phase conjugate mirror based on parallel alignment of two phase-only spatial light modulators (SLMs), in which both amplitude and phase of the input light can be time-reversed. Experimental result showed that, in the phase conjugation through a holographic diffuser with diffusion angle of 0.5 degree, background noises decrease to 65% by our digital phase conjugation mirror.

© 2014 Optical Society of America

1. Introduction

Since the first observation in 1972 of phase compensation effect by a phase conjugate light [1], it has been applied in numerous fields including imaging through a distortion medium [2] and laser resonator [3]. A recently-fashionable technique for generating the phase conjugate light is to use the combination of wavefront detection with an image sensor and wavefront modulation with a spatial light modulator (SLM), which named digital phase conjugation [47]. As compared with traditional optical phase conjugation based on photorefractive effect and brillouin scattering [8,9], the digital phase conjugation has various advantages such that phase conjugation reflectivity can be freely controlled, any laser wavelength and intensity can be used, and a phase conjugate wavefront can be flexibly adjusted. In typical digital phase conjugation system, however, only the phase of an input light is time-reversed by a single phase-only SLM while the amplitude of the input light is assumed constant. The uniformalization of the amplitude considerably decreases phase conjugation fidelity [10] and restricts application fields to specific cases only when the importance of the phase is ensured [11,12], for example, the input light has uniformly-distributed scattered wavefront due to the transmission through the multi-mode fiber and random scattering medium [6,7]. In contrast, C. Bellanger et al. have reported a different digital phase conjugation system in which both amplitude and phase of the input light are time-reversed by displaying an off-axis hologram on the SLM [5]. However, so-called computer-generated holograms such as the off-axis hologram inherently entail the loss of the spatial resolution of the SLM [1315], making the significant decrease of the phase conjugation fidelity.

For realizing high phase conjugation fidelity, the wavefront modulation method should be capable of full complex modulation with the same resolution as that of the SLM, but this demand cannot be satisfied by the single SLM. A reasonable solution is therefore to combine two SLMs in some fashion. Some groups have proposed a method in which two SLMs with coupled amplitude and phase response connect in tandem so that arbitrary complex modulation is conducted as a product of each modulation [16,17]. However, this additionally requires polarizer, analyzer, and 4f relay optics, meaning that the digital phase conjugation system becomes large. Different group has also presented another method in which two SLMs with coupled amplitude and phase response are arranged in parallel so that the complex modulation is conducted as an addition of each modulation [18]. However, this method causes the loss of optical power due to the direct amplitude modulation by the SLM and also has difficulty in achieving the full complex modulation because amplitude and phase responses by the SLM are coupled.

In this work, we first propose a full complex modulation method by a parallel arrangement of two phase-only SLMs, named dual-phase modulation method (DPMM). We also develop a new digital phase conjugation system based on the DPMM that permits high phase conjugation fidelity. Unlike the conventional system, this system can generate time-reversed complex wavefront without the loss of the amplitude and therefore can be regarded as digital phase conjugate 'mirror' in nature. The DPMM implements arbitrary complex modulation as the addition of two phase modulation response, and the generated complex wavefront has the same resolution as that of the SLM. This method also relies on the simple setup consisting of two phase-only SLMs and a beam splitter. These advantages enable us to increase the phase conjugation fidelity with simplifying the system size.

In Sect. 2.1, the basic principle of the DPMM is described. In Sect. 2.2, we explain practical ways for accurately working the DPMM. In Sect. 2.3, we experimentally demonstrate that the DPMM can perform the full complex modulation with the same resolution as that of the SLM. In Sect. 3.1 and Sect. 3.2, the basic operation of our digital phase conjugation system and practical ways for its accurate operation are described. In Sect. 3.3, it is experimentally demonstrated that our system can improve the phase conjugation fidelity in the case of the phase conjugation through a holographic diffuser with the spread angle of 0.5 degree.

2. Dual-phase modulation method by parallel arrangement of two phase-only SLMs

2.1. Basic operation

Mathematically, a given complex amplitude can be expressed as the addition of two phasors.

Aexp(iφ)=exp(iθ1)+exp(iθ2).
The DPMM performs this idea optically and spatially with the compact setup consisted of two phase-only SLMs (PSLMs) and a beam splitter (BS), as shown in Fig. 1(a). Assuming each PSLM can modulate the phase in the range [0, 2π] without any change of the amplitude, as shown in Figs. 1(b) and 1(c), the addition of two modulation curves enables to access all complex values in the complex plane, as shown in Fig. 1(d). Two phase images θ1(x,y) and θ2(x,y) to be displayed on PSLM1 and PSLM2 are ideally calculated as follows;
θ1(x,y)=φ(x,y)+cos1[A(x,y)2],
θ2(x,y)=φ(x,y)cos1[A(x,y)2],
where A(x,y) and φ(x,y) denote a desired amplitude image and phase image. For the precise operation of the DPMM, however, we should also consider initial deformations of the PSLMs and the difference of optical path length between two PSLMs and the BS. Considering these practical issues, Eqs. (2) and (3) are transformed as follows;
θ1(x,y)=φ(x,y)+cos1[A(x,y)2]α(x,y)l(z),
θ2(x,y)=φ(x,y)cos1[A(x,y)2]β(x,y).
Here, α(x,y) and β(x,y) are the initial deformations of each PSLM, and l(z) is the difference of the optical path length. The ways for addressing the above problems are detailed in the following section.

 figure: Fig. 1

Fig. 1 Dual-phase modulation method. (a) Conceptual diagram, (b) Modulation curve of phase-only SLM1, (c) Modulation curve of phase-only SLM2, (d) Accessible complex values in complex plane.

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2.2. Experimental setup

Figure 2 shows an experimental setup for the dual-phase modulation method (DPMM). Amplitude and phase images generated by the DPMM are measured by a holographic diversity interferometry (HDI) [19], one of phase-shifting digital holography [20]. Experimental parameters are as follows; the laser wavelength is 532 nm, the phase-only SLM has the pixel size of 20 × 20 μm2 and the pixel number of 800 × 600, the CCD has the pixel size of 3.75 × 3.75 μm2 and the pixel number of 1280 × 960. Light utilization efficiency in the DPMM, defined as the power ratio of generated light to incident light, is nearly 45%. This value is determined mainly due to the power loss by the BS. The necessary steps for precisely operating the DPMM are; (i) pixel-to-pixel matching between PSLM1 and PSLM2, (ii) the compensation of deformations of PSLM1 and PSLM2, (iii) the elimination of the optical path difference between two PSLMs and the BS. For performing the pixel-to-pixel matching between two PSLMs, a relay optics using L1 and L2 is employed so that one pixel on the PSLM is imaged onto the region of 2 × 2 pixels on the CCDs, and then spatial coordinates of two images modulated by each PSLM are adjusted to match on the CCD1. For compensating the initial deformation of each PSLM, blank images are displayed on each PSLM, then the phase measurements of each blank image are separately performed. Since the measured phase distributions include the knowledge about the deformations of the two PSLM along with aberrations of L1 and L2, we use these distributions as α(x,y) and β(x,y) in Eqs. (4) and (5). The measured distributions are shown in Fig. 3 and also used in subsequent experiment. For addressing the optical path difference, blank images are displayed on each PSLM and then we control the intensity of the generated light after the BS to be maximum by monitoring the CCD1.

 figure: Fig. 2

Fig. 2 Experimental setup for dual-phase modulation method. Reference light is interfered with generated light for measurement of complex amplitude with holographic diversity interferometer (HDI). The HDI consists of two CCDs, PBS, and QWP. QWP makes phase difference of π/2 between transmitted light and reflected lights after PBS2, which is necessary for phase measurement with the HDI. Phase-only SLM provided by Hamamatsu can modulate the phase in the range [0, 2π] without any change of intensity. PSLM1 and PSLM2 offer phase modulation of 2π when gray levels are 85 and 157.

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 figure: Fig. 3

Fig. 3 Initial deformations of two PSLMs, each of which is measured by displaying blank image (no modulation) onto each PSLM.

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2.3. Experimental result

For evaluating quality of the complex modulation by the DPMM, we use signal to noise ratio (SNR) defined as follows;

SNR(dB)=10log10m=1Ndxn=1Ndyf(mΔx,nΔy)2m=1Ndxn=1Ndy{f(mΔx,nΔy)2g(mΔx,nΔy)2},
where Ndx and Ndy are respectively the number of data pixels in transversal plane, Δx and Δy are respectively the size of a data pixel in transversal plane, f(mΔx,nΔy) is the desired amplitude or phase image, and g(mΔx,nΔy) is the generated amplitude or phase image. First, we experimentally show that the DPMM can perform the complex modulation using a complex image. Figures 4(a) and 4(b) show desired amplitude image and phase image in which each data pixel is represented by 16 × 16 SLM pixels. Figures 4(c) and 4(d) show two phase images decomposed from the desired images based on Eqs. (4) and (5), which are respectively displayed on PSLM1 and PSLM2. Figures 4(e) and 4(f) show generated amplitude image and phase image. In Fig. 4(e), enough contrast was obtained to readily distinguish three amplitude values and the SNR value was 13.5 dB, but the amplitude value appears to be relatively lower in outer region than center region. We think that this is due to Gaussian beam profile or imperfect compensation of PSLM deformations. In Fig. 4(f), four phase values were clearly identifiable and the high SNR value of 19.2 dB was obtained.

 figure: Fig. 4

Fig. 4 Complex modulation using a complex image consisting of a multi-level amplitude image and a multi-level phase image. Each data pixel consists of 16 × 16 SLM pixels. (a) Desired amplitude image. Each data pixel in the amplitude image randomly takes either of three values: 1, 0.6, and 0.3. (b) Desired phase image. Each data pixel in the phase image has either of four values: π/2, π, 3π/2, and 2π. (c) Phase image displayed on PSLM1. (d) Phase image displayed on PSLM2. (e) Measured amplitude image. (f) Measured phase image.

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Next, we experimentally clarify through the independent modulation of the amplitude and phase images that the DPMM is capable of the full complex modulation and it has the same resolution as that of the SLM. Figures 5(a) and 5(b) show desired amplitude image and phase image in which one data pixel corresponds to one SLM pixel, that is, the resolution of the desired image is equivalent to that of the SLM. Here, it is noted that each image is separately generated, that is, no phase image is generated when the amplitude image is generated and vice versa. Figures 5(c) and 5(d) show generated amplitude image and phase distribution when only the modulation of the amplitude image is performed by the two PSLMs. In Fig. 5(c), although the generated amplitude image slightly blurs in lower region, the overall image was analogous to the desired one and high spatial frequency components were clearly generated. In Fig. 5(d), the measured phase distribution was mostly homogeneous, meaning that the independent amplitude modulation was accurately conducted. Figures 5(e) and 5(f) show generated phase image and amplitude distribution when only the modulation of the phase image is performed. In Fig. 5(e), the generated phase image seems to be a little noisy, but this is mainly due to the lack of phase unwrapping and is not inherent problem in the DPMM. High spatial frequency components were also generated clearly. In Fig. 5(f), although edges of the phase image was emphasized, the measured amplitude distribution was mostly uniform, meaning that the independent phase modulation was accurately conducted. These result demonstrates that the DPMM can achieve the same resolution as that of the SLM and can independently control the amplitude and phase. The experimental demonstration of the independent amplitude and phase control is also clear evidence that the full complex modulation is possible, as described below. The result of Figs. 5(c) and 5(d) indicates that the DPMM can modulate arbitrary amplitude in the range [0, 1] without any change of the phase, which is expressed in complex coordinate by a straight line in Fig. 6(a). The result of Figs. 5(e) and 5(f) also indicates that the DPMM can modulate arbitrary phase in the range [0, 2π] without any change of the amplitude, which is expressed in the complex coordinate by a circle in Fig. 6(a). It is thus evident that any complex value is accessible, that is, the full complex modulation can be achieved, as shown in Fig. 6(b).

 figure: Fig. 5

Fig. 5 Independent modulation of amplitude image and phase image with the same resolution as that of PSLM. Each image consists of 256 × 256 SLM pixels. (a) Desired amplitude image. This is represented by 256 gray levels and includes all gray levels. (b) Desired phase image. This is represented by 256 gray levels and includes all gray levels. (c) Measured amplitude image. SNR was 6.25 dB. (d) Measured phase distribution when only the amplitude image is modulated. (e) Measured phase image. SNR was 5.85 dB. This low value is mainly due to phase wrapping. (f) Measured amplitude distribution when only the phase image is modulated.

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 figure: Fig. 6

Fig. 6 Demonstration of full complex modulation using the result of Fig. 5.

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3. Digital phase conjugate mirror based on dual-phase modulation method

3.1. Basic operation

Figure 7 shows conceptual diagram of the digital phase conjugate mirror. Unlike conventional digital phase conjugation system, our system can time-reverse both the amplitude and phase of the input light. This ability increases the phase conjugation fidelity and also overcome the restriction that the input light should have the uniformly-distributed scattered wavefront, which diversifying application fields of the digital phase conjugation. As compared with the conventional system, although our system additionally requires one spatial light modulator (SLM) and one beam splitter, it still maintains simple setup and some advantages brought by our system are of great significance as mentioned above.

 figure: Fig. 7

Fig. 7 Conceptual diagram of digital phase conjugate mirror.

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A complex wavefront of the input light A(x,y)exp[(x,y)] is measured by a wavefront detector with a CCD via the interference with a plane wave. In a computer, the phase conjugation of the measured complex wavefront A(x,y)exp[-(x,y)] is calculated and then it is decomposed into two phase images exp[1(x,y)] and exp[2(x,y)] based on Eqs. (4) and (5). The two phase images are displayed onto the PSLM1 and the PSLM2, then those are irradiated by the plane wave. Two modulated lights exp[1(x,y)] and exp[2(x,y)] are synthesized after transmitting BS2 to be phase conjugate light A(x,y)exp[-(x,y)]. It should be noted that, in the conventional digital phase conjugation system, since only the phase of the input light is time-reversed, the phase conjugate light has only the phase distribution exp[-(x,y)].

3.2. Experimental setup

Figure 8 shows an experimental setup for the digital phase conjugate mirror. An input light as a plane wave transmits through a phase object with phase profile exp[(x,y)]. The distorted input light is propagated to wavefront detector via two relay optics using L1, L2, L3, and L4, then its complex wavefront A(x,y)exp[(x,y)] is measured by the holographic diversity interferometry (HDI) via the interference with a reference light 1. In a computer, the phase conjugation of the measured complex wavefront A(x,y)exp[-(x,y)] is calculated and then it is decomposed into two phase images exp[1(x,y)] and exp[2(x,y)] based on Eqs. (4) and (5). The two phase images displayed onto the PSLM1 and the PSLM2 are respectively modulated onto the reference light 1 as an irradiation light. Two modulated lights exp[1(x,y)] and exp[2(x,y)] are synthesized after transmitting BS2 to be phase conjugate light A(x,y)exp[-(x,y)]. The synthesized phase conjugate light transmits again through the phase object with the phase exp[(x,y)] via relay optics with L1 and L2. Then, the phase distribution exp[-(x,y)] of the phase conjugate light is cancelled, resulting in the recovery of the original plane wave. For verifying whether the phase compensation works correctly, we monitor the intensity peak of the phase conjugate light on a CCD4. For verifying whether the amplitude distribution of the input light is correctly reflected from the digital phase conjugate mirror, we also monitor it on a CCD3 placed on the focal plane of L1.

 figure: Fig. 8

Fig. 8 Experimental setup for digital phase conjugate mirror. A OHP sheet is overhead projector sheet used for the correct mapping between SLMs and CCDs. The amplitude object is USAF resolution target (Edmund) and it is arranged at the focal plane of L1. A phase object is a plane-convex lens with f = 500mm or a holographic diffuser with diffusion angle of 0.5 degree (Edmund) and it is arranged at a distance of 30 mm from an amplitude object.

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A key point for accurately working the digital phase conjugate mirror is precise mapping from wavefront detector (HDI) to wavefront modulator (DPMM). However, this is difficult because the light reflected from the modulator doesn’t propagate to the detector. For addressing this difficulty, we newly prepare a reference light 2 and take the following procedure in order; (i) collimating input light, reference light 1, and reference light 2 eachother, (ii) matching spatial positions on CCD1 of the image on the input light and the image on reference light 1 by a OHP sheet, (iii) matching spatial positions on the CCD3 of the image on the reference light 2 by the OHP sheet and images on the reference light 1 by the two SLMs. Several parameters used for the above procedure are illustrated in Fig. 9.

 figure: Fig. 9

Fig. 9 Several parameters used for precise mapping from wavefront sensor (HDI) to wavefront modulator (DPMM). (a) Image size printed on OHP sheet. (b) Image size displayed on SLM. (c) Spatial positions of OHP image on CCD1 and CCD2. (d) Spatial positions of OHP image and two SLM images on CCD3.

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3.3. Experimental result

First, we experimentally clarify that both the amplitude and phase of the input light can be correctly time-reversed by our digital phase conjugate mirror. In optical path of the input light, as shown in Fig. 8, USAF target is arranged as an amplitude object in front focal plane of L1 and also plane-convex lens with f = 500 mm is arranged as an phase object at a distance of 30 mm from the amplitude object. Figures 10(a) and 10(b) show intensity and phase distributions of the input light measured by the HDI. In Fig. 10(a), the measured intensity has a clear pattern of USAF target. In Fig. 10(b), the measured phase has a pattern in a concentric fashion and phase slope gradually increases from center to outer area like the surface of the plane-convex lens. Figure 10(c) shows intensity distribution on the CCD4 when no digital phase conjugate mirror works, that is, blank images are displayed on the PSLM1 and PSLM2. No focusing effect was observed and the distribution was widespread. Figures 10(d) and 10(e) respectively show intensity distributions of the phase conjugate light on the CCD3 and CCD4 when the digital phase conjugate mirror works. In Fig. 10(d), although the measured phase was slightly highlighted within the intensity distribution, the center pattern of the USAF target was exactly recovered by the conjugate mirror, which cannot be realized by the conventional digital phase conjugation system. In Fig. 10(e), extremely sharp focus spot was observed in center area, which means that the phase distortion was compensated by the retransmission through the plane-convex lens. The contrast of the focus, defined as the ratio between peak value and the average of background noise, was evaluated to be nearly 900. These results clarified that both amplitude and phase of the input light can be correctly time-reversed by our system. Figure 10(f) represents the profile of phase conjugated focus spot. The full-width at half-maximum (FWHM) was 41 µm and it was in agreement with that of plane wave, meaning that the compensation of the phase distortion was completely accomplished.

 figure: Fig. 10

Fig. 10 Focusing through plane-convex lens with f = 500 mm using digital phase conjugate mirror. (a) Measured intensity distribution with USAF target. (b) Measured phase distribution in a concentric fashion. (c) Widened intensity distribution on CCD4 when the blank image is displayed on the PSLM1. (d) Phase conjugated intensity distribution on CCD3 when the digital phase conjugate mirror works. (e) Phase conjugated focus spot on CCD4 when the digital phase conjugate mirror works. (f) Profile and FWHM of phase conjugated focus spots.

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Next, we demonstrate that the amplitude and phase control can improve the phase conjugation fidelity in comparison with the phase-only control. In the optical path of the input light, as shown in Fig. 8, no amplitude object is set and only a holographic diffuser with diffusion angle of 0.5 degree is arranged as the phase object at a distance of 170 mm from L1. This diffuser largely distorts amplitude and phase distributions of the input light due to free-space propagation. Thus, both the amplitude and phase in the input light should be time-reversed for the phase conjugation with high-fidelity. Figures 11(a) and 11(b) show measured amplitude and phase distributions of the input light distorted by the transmission through the diffuser. In Fig. 11(a), the measured intensity was not homogeneous but distributed like honeycomb. In Fig. 11(b), the measured phase distribution seems to be concavo-convex surface of the diffuser. Figure 11(c) shows intensity distribution on the CCD4 when blank image is displayed on the PSLM1. No focusing effect was observed and speckle patterns were observed over an entire area. Figures 11(d) and 11(e) show phase conjugated focus spots when the conventional digital phase conjugation system and the digital phase conjugate mirror are respectively performed. Both the focus spots were apparently sharp, which indicates that the compensation of the phase distortion by the diffuser was partially achieved at least. In Figs. 11(f) and 11(g), background noises in Figs. 11(d) and 11(e) are highlighted by increasing light sensitivity of the CCD4. Clearly, the background noise in Fig. 11(f) was larger than that in Fig. 11(g). In addition, the ratio of the average background was 1.57. This result revealed that controlling not only the phase but also the amplitude improves the phase conjugation fidelity. Figure 11(h) represents the FWHMs of the focus spots in both systems. We found that both the FWHMs were mostly in agreement with that of plane wave.

 figure: Fig. 11

Fig. 11 Focusing through random diffuser with the diffusion angle of 0.5 degree using digital phase conjugate mirror. (a) Measured intensity distribution. (b) Measured phase distribution. (c) Scattered intensity distribution on CCD4 when the blank image is displayed on the PSLM1. (d) Phase conjugated focus spot on CCD4 when conventional digital phase conjugation works, that is, conjugated phase distribution is displayed on PSLM1. (e) Phase conjugated focus spot on CCD4 when our digital phase conjugate mirror works. (f) Highlighted background noise on CCD4 when conventional digital phase conjugation works. (g) Highlighted background noise on CCD4 when our digital phase conjugate mirror works. (h) Profile and FWHM of phase conjugated focus spots.

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4. Conclusion

For the application to the digital phase conjugation, we first presented the DPMM where two PSLMs are parallely arranged. In the experiment, we succeeded in the full complex modulation with the same resolution as that of the SLM. We also developed the digital phase conjugate system based on the DPMM for time-reversing both the amplitude and phase of the input light. This ability can diversify the application fields unlike the conventional system where only the phase is controlled. For example, this system can be applied in the imaging through weekly-scattering medium and the compensation of modes distorted during the propagation through the multi-mode fiber. In the experiment on the phase conjugation using the plane-convex lens, the focus contrast of nearly 900 was obtained. In the phase conjugation using the diffuser with the spread angle of 0.5 degree, our phase conjugation system decreased the background noise to 65% in comparison with the conventional system.

Acknowledgments

We would like to thank Mr. Y. Kan and Mr. S. Shimizu for helpful discussions. This work was supported by JSPS KAKENHI Grant Number 25289110 and Grant-in-Aid for JSPS Fellows 25-1808.

References and links

1. B. Y. Zeldovich, V. I. Popovichev, V. V. Ragulskii, and F. S. Faizullov, “Connection between the wave fronts of the reflected and exciting light in stimulated Mandel'-shtam-Brillouin scattering,” Sov. Phys. JETP Lett. 15, 109–113 (1972).

2. H. Kogelnik and K. S. Pennington, “Holographic imaging through a random medium,” J. Opt. Soc. Am. 58(2), 273–274 (1968). [CrossRef]  

3. S. MacCormack and J. Feinberg, “High-brightness output from a laser-diode array coupled to a phase-conjugating mirror,” Opt. Lett. 18(3), 211–213 (1993). [CrossRef]   [PubMed]  

4. M. Cui and C. Yang, “Implementation of a digital optical phase conjugation system and its application to study the robustness of turbidity suppression by phase conjugation,” Opt. Express 18(4), 3444–3455 (2010). [CrossRef]   [PubMed]  

5. C. Bellanger, A. Brignon, J. Colineau, and J. P. Huignard, “Coherent fiber combining by digital holography,” Opt. Lett. 33(24), 2937–2939 (2008). [CrossRef]   [PubMed]  

6. C. L. Hsieh, Y. Pu, R. Grange, and D. Psaltis, “Digital phase conjugation of second harmonic radiation emitted by nanoparticles in turbid media,” Opt. Express 18(12), 12283–12290 (2010). [CrossRef]   [PubMed]  

7. I. N. Papadopoulos, S. Farahi, C. Moser, and D. Psaltis, “Focusing and scanning light through a multimode optical fiber using digital phase conjugation,” Opt. Express 20(10), 10583–10590 (2012). [CrossRef]   [PubMed]  

8. T. Omatsu, A. Katoh, K. Okada, S. Hatano, A. Hasegawa, M. Tateda, and I. Ogura, “Investigation of photorefractive phase conjugate feedback on the lasing spectrum of a broad-stripe laser diode,” Opt. Commun. 146(1-6), 167–172 (1998). [CrossRef]  

9. V. Wang and C. R. Giuliano, “Correction of phase aberrations via stimulated Brillouin scattering,” Opt. Lett. 2(1), 4–6 (1978). [CrossRef]   [PubMed]  

10. I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett. 101, 120601 (2008).

11. A. V. Oppenheim and J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69(5), 529–541 (1981). [CrossRef]  

12. L. B. Lesem, P. M. Hirch, and J. A. Jordan Jr., “The kinoform: A new wavefront reconstruction device,” IBM J. Res. Develop. 13(2), 150–155 (1969). [CrossRef]  

13. A. J. MacGovern and J. C. Wyant, “Computer generated holograms for testing optical elements,” Appl. Opt. 10(3), 619–624 (1971). [CrossRef]   [PubMed]  

14. J. M. Florence and R. D. Juday, “Full complex spatial filtering with a phase mostly DMD,” Proc. SPIE 1558, 487–498 (1991). [CrossRef]  

15. A. Shibukawa, A. Okamoto, M. Takabayashi, and A. Tomita, “Spatial cross modulation method using a random diffuser and phase-only spatial light modulator for constructing arbitrary complex fields,” Opt. Express 22(4), 3968–3982 (2014). [CrossRef]   [PubMed]  

16. L. G. Neto, D. Roberge, and Y. Sheng, “Full-range, continuous, complex modulation by the use of two coupled-mode liquid-crystal televisions,” Appl. Opt. 35(23), 4567–4576 (1996). [CrossRef]   [PubMed]  

17. D. A. Gregory, J. C. Kirsch, and E. C. Tam, “Full complex modulation using liquid-crystal televisions,” Appl. Opt. 31(2), 163–165 (1992). [CrossRef]   [PubMed]  

18. R. D. Juday and J. M. Florence, “Full complex modulation with two one-parameter SLMs,” Proc. SPIE 1558, 499–504 (1991). [CrossRef]  

19. A. Okamoto, K. Kunori, M. Takabayashi, A. Tomita, and K. Sato, “Holographic diversity interferometry for optical storage,” Opt. Express 19(14), 13436–13444 (2011). [CrossRef]   [PubMed]  

20. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22(16), 1268–1270 (1997). [CrossRef]   [PubMed]  

References

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  1. B. Y. Zeldovich, V. I. Popovichev, V. V. Ragulskii, and F. S. Faizullov, “Connection between the wave fronts of the reflected and exciting light in stimulated Mandel'-shtam-Brillouin scattering,” Sov. Phys. JETP Lett. 15, 109–113 (1972).
  2. H. Kogelnik and K. S. Pennington, “Holographic imaging through a random medium,” J. Opt. Soc. Am. 58(2), 273–274 (1968).
    [Crossref]
  3. S. MacCormack and J. Feinberg, “High-brightness output from a laser-diode array coupled to a phase-conjugating mirror,” Opt. Lett. 18(3), 211–213 (1993).
    [Crossref] [PubMed]
  4. M. Cui and C. Yang, “Implementation of a digital optical phase conjugation system and its application to study the robustness of turbidity suppression by phase conjugation,” Opt. Express 18(4), 3444–3455 (2010).
    [Crossref] [PubMed]
  5. C. Bellanger, A. Brignon, J. Colineau, and J. P. Huignard, “Coherent fiber combining by digital holography,” Opt. Lett. 33(24), 2937–2939 (2008).
    [Crossref] [PubMed]
  6. C. L. Hsieh, Y. Pu, R. Grange, and D. Psaltis, “Digital phase conjugation of second harmonic radiation emitted by nanoparticles in turbid media,” Opt. Express 18(12), 12283–12290 (2010).
    [Crossref] [PubMed]
  7. I. N. Papadopoulos, S. Farahi, C. Moser, and D. Psaltis, “Focusing and scanning light through a multimode optical fiber using digital phase conjugation,” Opt. Express 20(10), 10583–10590 (2012).
    [Crossref] [PubMed]
  8. T. Omatsu, A. Katoh, K. Okada, S. Hatano, A. Hasegawa, M. Tateda, and I. Ogura, “Investigation of photorefractive phase conjugate feedback on the lasing spectrum of a broad-stripe laser diode,” Opt. Commun. 146(1-6), 167–172 (1998).
    [Crossref]
  9. V. Wang and C. R. Giuliano, “Correction of phase aberrations via stimulated Brillouin scattering,” Opt. Lett. 2(1), 4–6 (1978).
    [Crossref] [PubMed]
  10. I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett. 101, 120601 (2008).
  11. A. V. Oppenheim and J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69(5), 529–541 (1981).
    [Crossref]
  12. L. B. Lesem, P. M. Hirch, and J. A. Jordan., “The kinoform: A new wavefront reconstruction device,” IBM J. Res. Develop. 13(2), 150–155 (1969).
    [Crossref]
  13. A. J. MacGovern and J. C. Wyant, “Computer generated holograms for testing optical elements,” Appl. Opt. 10(3), 619–624 (1971).
    [Crossref] [PubMed]
  14. J. M. Florence and R. D. Juday, “Full complex spatial filtering with a phase mostly DMD,” Proc. SPIE 1558, 487–498 (1991).
    [Crossref]
  15. A. Shibukawa, A. Okamoto, M. Takabayashi, and A. Tomita, “Spatial cross modulation method using a random diffuser and phase-only spatial light modulator for constructing arbitrary complex fields,” Opt. Express 22(4), 3968–3982 (2014).
    [Crossref] [PubMed]
  16. L. G. Neto, D. Roberge, and Y. Sheng, “Full-range, continuous, complex modulation by the use of two coupled-mode liquid-crystal televisions,” Appl. Opt. 35(23), 4567–4576 (1996).
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  17. D. A. Gregory, J. C. Kirsch, and E. C. Tam, “Full complex modulation using liquid-crystal televisions,” Appl. Opt. 31(2), 163–165 (1992).
    [Crossref] [PubMed]
  18. R. D. Juday and J. M. Florence, “Full complex modulation with two one-parameter SLMs,” Proc. SPIE 1558, 499–504 (1991).
    [Crossref]
  19. A. Okamoto, K. Kunori, M. Takabayashi, A. Tomita, and K. Sato, “Holographic diversity interferometry for optical storage,” Opt. Express 19(14), 13436–13444 (2011).
    [Crossref] [PubMed]
  20. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22(16), 1268–1270 (1997).
    [Crossref] [PubMed]

2014 (1)

2012 (1)

2011 (1)

2010 (2)

2008 (2)

C. Bellanger, A. Brignon, J. Colineau, and J. P. Huignard, “Coherent fiber combining by digital holography,” Opt. Lett. 33(24), 2937–2939 (2008).
[Crossref] [PubMed]

I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett. 101, 120601 (2008).

1998 (1)

T. Omatsu, A. Katoh, K. Okada, S. Hatano, A. Hasegawa, M. Tateda, and I. Ogura, “Investigation of photorefractive phase conjugate feedback on the lasing spectrum of a broad-stripe laser diode,” Opt. Commun. 146(1-6), 167–172 (1998).
[Crossref]

1997 (1)

1996 (1)

1993 (1)

1992 (1)

1991 (2)

R. D. Juday and J. M. Florence, “Full complex modulation with two one-parameter SLMs,” Proc. SPIE 1558, 499–504 (1991).
[Crossref]

J. M. Florence and R. D. Juday, “Full complex spatial filtering with a phase mostly DMD,” Proc. SPIE 1558, 487–498 (1991).
[Crossref]

1981 (1)

A. V. Oppenheim and J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69(5), 529–541 (1981).
[Crossref]

1978 (1)

1972 (1)

B. Y. Zeldovich, V. I. Popovichev, V. V. Ragulskii, and F. S. Faizullov, “Connection between the wave fronts of the reflected and exciting light in stimulated Mandel'-shtam-Brillouin scattering,” Sov. Phys. JETP Lett. 15, 109–113 (1972).

1971 (1)

1969 (1)

L. B. Lesem, P. M. Hirch, and J. A. Jordan., “The kinoform: A new wavefront reconstruction device,” IBM J. Res. Develop. 13(2), 150–155 (1969).
[Crossref]

1968 (1)

Bellanger, C.

Brignon, A.

Colineau, J.

Cui, M.

Faizullov, F. S.

B. Y. Zeldovich, V. I. Popovichev, V. V. Ragulskii, and F. S. Faizullov, “Connection between the wave fronts of the reflected and exciting light in stimulated Mandel'-shtam-Brillouin scattering,” Sov. Phys. JETP Lett. 15, 109–113 (1972).

Farahi, S.

Feinberg, J.

Florence, J. M.

J. M. Florence and R. D. Juday, “Full complex spatial filtering with a phase mostly DMD,” Proc. SPIE 1558, 487–498 (1991).
[Crossref]

R. D. Juday and J. M. Florence, “Full complex modulation with two one-parameter SLMs,” Proc. SPIE 1558, 499–504 (1991).
[Crossref]

Giuliano, C. R.

Grange, R.

Gregory, D. A.

Hasegawa, A.

T. Omatsu, A. Katoh, K. Okada, S. Hatano, A. Hasegawa, M. Tateda, and I. Ogura, “Investigation of photorefractive phase conjugate feedback on the lasing spectrum of a broad-stripe laser diode,” Opt. Commun. 146(1-6), 167–172 (1998).
[Crossref]

Hatano, S.

T. Omatsu, A. Katoh, K. Okada, S. Hatano, A. Hasegawa, M. Tateda, and I. Ogura, “Investigation of photorefractive phase conjugate feedback on the lasing spectrum of a broad-stripe laser diode,” Opt. Commun. 146(1-6), 167–172 (1998).
[Crossref]

Hirch, P. M.

L. B. Lesem, P. M. Hirch, and J. A. Jordan., “The kinoform: A new wavefront reconstruction device,” IBM J. Res. Develop. 13(2), 150–155 (1969).
[Crossref]

Hsieh, C. L.

Huignard, J. P.

Jordan, J. A.

L. B. Lesem, P. M. Hirch, and J. A. Jordan., “The kinoform: A new wavefront reconstruction device,” IBM J. Res. Develop. 13(2), 150–155 (1969).
[Crossref]

Juday, R. D.

J. M. Florence and R. D. Juday, “Full complex spatial filtering with a phase mostly DMD,” Proc. SPIE 1558, 487–498 (1991).
[Crossref]

R. D. Juday and J. M. Florence, “Full complex modulation with two one-parameter SLMs,” Proc. SPIE 1558, 499–504 (1991).
[Crossref]

Katoh, A.

T. Omatsu, A. Katoh, K. Okada, S. Hatano, A. Hasegawa, M. Tateda, and I. Ogura, “Investigation of photorefractive phase conjugate feedback on the lasing spectrum of a broad-stripe laser diode,” Opt. Commun. 146(1-6), 167–172 (1998).
[Crossref]

Kirsch, J. C.

Kogelnik, H.

Kunori, K.

Lesem, L. B.

L. B. Lesem, P. M. Hirch, and J. A. Jordan., “The kinoform: A new wavefront reconstruction device,” IBM J. Res. Develop. 13(2), 150–155 (1969).
[Crossref]

Lim, J. S.

A. V. Oppenheim and J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69(5), 529–541 (1981).
[Crossref]

MacCormack, S.

MacGovern, A. J.

Moser, C.

Mosk, A. P.

I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett. 101, 120601 (2008).

Neto, L. G.

Ogura, I.

T. Omatsu, A. Katoh, K. Okada, S. Hatano, A. Hasegawa, M. Tateda, and I. Ogura, “Investigation of photorefractive phase conjugate feedback on the lasing spectrum of a broad-stripe laser diode,” Opt. Commun. 146(1-6), 167–172 (1998).
[Crossref]

Okada, K.

T. Omatsu, A. Katoh, K. Okada, S. Hatano, A. Hasegawa, M. Tateda, and I. Ogura, “Investigation of photorefractive phase conjugate feedback on the lasing spectrum of a broad-stripe laser diode,” Opt. Commun. 146(1-6), 167–172 (1998).
[Crossref]

Okamoto, A.

Omatsu, T.

T. Omatsu, A. Katoh, K. Okada, S. Hatano, A. Hasegawa, M. Tateda, and I. Ogura, “Investigation of photorefractive phase conjugate feedback on the lasing spectrum of a broad-stripe laser diode,” Opt. Commun. 146(1-6), 167–172 (1998).
[Crossref]

Oppenheim, A. V.

A. V. Oppenheim and J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69(5), 529–541 (1981).
[Crossref]

Papadopoulos, I. N.

Pennington, K. S.

Popovichev, V. I.

B. Y. Zeldovich, V. I. Popovichev, V. V. Ragulskii, and F. S. Faizullov, “Connection between the wave fronts of the reflected and exciting light in stimulated Mandel'-shtam-Brillouin scattering,” Sov. Phys. JETP Lett. 15, 109–113 (1972).

Psaltis, D.

Pu, Y.

Ragulskii, V. V.

B. Y. Zeldovich, V. I. Popovichev, V. V. Ragulskii, and F. S. Faizullov, “Connection between the wave fronts of the reflected and exciting light in stimulated Mandel'-shtam-Brillouin scattering,” Sov. Phys. JETP Lett. 15, 109–113 (1972).

Roberge, D.

Sato, K.

Sheng, Y.

Shibukawa, A.

Takabayashi, M.

Tam, E. C.

Tateda, M.

T. Omatsu, A. Katoh, K. Okada, S. Hatano, A. Hasegawa, M. Tateda, and I. Ogura, “Investigation of photorefractive phase conjugate feedback on the lasing spectrum of a broad-stripe laser diode,” Opt. Commun. 146(1-6), 167–172 (1998).
[Crossref]

Tomita, A.

Vellekoop, I. M.

I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett. 101, 120601 (2008).

Wang, V.

Wyant, J. C.

Yamaguchi, I.

Yang, C.

Zeldovich, B. Y.

B. Y. Zeldovich, V. I. Popovichev, V. V. Ragulskii, and F. S. Faizullov, “Connection between the wave fronts of the reflected and exciting light in stimulated Mandel'-shtam-Brillouin scattering,” Sov. Phys. JETP Lett. 15, 109–113 (1972).

Zhang, T.

Appl. Opt. (3)

IBM J. Res. Develop. (1)

L. B. Lesem, P. M. Hirch, and J. A. Jordan., “The kinoform: A new wavefront reconstruction device,” IBM J. Res. Develop. 13(2), 150–155 (1969).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

T. Omatsu, A. Katoh, K. Okada, S. Hatano, A. Hasegawa, M. Tateda, and I. Ogura, “Investigation of photorefractive phase conjugate feedback on the lasing spectrum of a broad-stripe laser diode,” Opt. Commun. 146(1-6), 167–172 (1998).
[Crossref]

Opt. Express (5)

Opt. Lett. (4)

Phys. Rev. Lett. (1)

I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett. 101, 120601 (2008).

Proc. IEEE (1)

A. V. Oppenheim and J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69(5), 529–541 (1981).
[Crossref]

Proc. SPIE (2)

R. D. Juday and J. M. Florence, “Full complex modulation with two one-parameter SLMs,” Proc. SPIE 1558, 499–504 (1991).
[Crossref]

J. M. Florence and R. D. Juday, “Full complex spatial filtering with a phase mostly DMD,” Proc. SPIE 1558, 487–498 (1991).
[Crossref]

Sov. Phys. JETP Lett. (1)

B. Y. Zeldovich, V. I. Popovichev, V. V. Ragulskii, and F. S. Faizullov, “Connection between the wave fronts of the reflected and exciting light in stimulated Mandel'-shtam-Brillouin scattering,” Sov. Phys. JETP Lett. 15, 109–113 (1972).

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Figures (11)

Fig. 1
Fig. 1 Dual-phase modulation method. (a) Conceptual diagram, (b) Modulation curve of phase-only SLM1, (c) Modulation curve of phase-only SLM2, (d) Accessible complex values in complex plane.
Fig. 2
Fig. 2 Experimental setup for dual-phase modulation method. Reference light is interfered with generated light for measurement of complex amplitude with holographic diversity interferometer (HDI). The HDI consists of two CCDs, PBS, and QWP. QWP makes phase difference of π/2 between transmitted light and reflected lights after PBS2, which is necessary for phase measurement with the HDI. Phase-only SLM provided by Hamamatsu can modulate the phase in the range [0, 2π] without any change of intensity. PSLM1 and PSLM2 offer phase modulation of 2π when gray levels are 85 and 157.
Fig. 3
Fig. 3 Initial deformations of two PSLMs, each of which is measured by displaying blank image (no modulation) onto each PSLM.
Fig. 4
Fig. 4 Complex modulation using a complex image consisting of a multi-level amplitude image and a multi-level phase image. Each data pixel consists of 16 × 16 SLM pixels. (a) Desired amplitude image. Each data pixel in the amplitude image randomly takes either of three values: 1, 0.6, and 0.3. (b) Desired phase image. Each data pixel in the phase image has either of four values: π/2, π, 3π/2, and 2π. (c) Phase image displayed on PSLM1. (d) Phase image displayed on PSLM2. (e) Measured amplitude image. (f) Measured phase image.
Fig. 5
Fig. 5 Independent modulation of amplitude image and phase image with the same resolution as that of PSLM. Each image consists of 256 × 256 SLM pixels. (a) Desired amplitude image. This is represented by 256 gray levels and includes all gray levels. (b) Desired phase image. This is represented by 256 gray levels and includes all gray levels. (c) Measured amplitude image. SNR was 6.25 dB. (d) Measured phase distribution when only the amplitude image is modulated. (e) Measured phase image. SNR was 5.85 dB. This low value is mainly due to phase wrapping. (f) Measured amplitude distribution when only the phase image is modulated.
Fig. 6
Fig. 6 Demonstration of full complex modulation using the result of Fig. 5.
Fig. 7
Fig. 7 Conceptual diagram of digital phase conjugate mirror.
Fig. 8
Fig. 8 Experimental setup for digital phase conjugate mirror. A OHP sheet is overhead projector sheet used for the correct mapping between SLMs and CCDs. The amplitude object is USAF resolution target (Edmund) and it is arranged at the focal plane of L1. A phase object is a plane-convex lens with f = 500mm or a holographic diffuser with diffusion angle of 0.5 degree (Edmund) and it is arranged at a distance of 30 mm from an amplitude object.
Fig. 9
Fig. 9 Several parameters used for precise mapping from wavefront sensor (HDI) to wavefront modulator (DPMM). (a) Image size printed on OHP sheet. (b) Image size displayed on SLM. (c) Spatial positions of OHP image on CCD1 and CCD2. (d) Spatial positions of OHP image and two SLM images on CCD3.
Fig. 10
Fig. 10 Focusing through plane-convex lens with f = 500 mm using digital phase conjugate mirror. (a) Measured intensity distribution with USAF target. (b) Measured phase distribution in a concentric fashion. (c) Widened intensity distribution on CCD4 when the blank image is displayed on the PSLM1. (d) Phase conjugated intensity distribution on CCD3 when the digital phase conjugate mirror works. (e) Phase conjugated focus spot on CCD4 when the digital phase conjugate mirror works. (f) Profile and FWHM of phase conjugated focus spots.
Fig. 11
Fig. 11 Focusing through random diffuser with the diffusion angle of 0.5 degree using digital phase conjugate mirror. (a) Measured intensity distribution. (b) Measured phase distribution. (c) Scattered intensity distribution on CCD4 when the blank image is displayed on the PSLM1. (d) Phase conjugated focus spot on CCD4 when conventional digital phase conjugation works, that is, conjugated phase distribution is displayed on PSLM1. (e) Phase conjugated focus spot on CCD4 when our digital phase conjugate mirror works. (f) Highlighted background noise on CCD4 when conventional digital phase conjugation works. (g) Highlighted background noise on CCD4 when our digital phase conjugate mirror works. (h) Profile and FWHM of phase conjugated focus spots.

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

Aexp( iφ )=exp( i θ 1 )+exp( i θ 2 ).
θ 1 (x,y)=φ(x,y)+ cos 1 [ A(x,y) 2 ],
θ 2 (x,y)=φ(x,y) cos 1 [ A(x,y) 2 ],
θ 1 (x,y)=φ(x,y)+ cos 1 [ A(x,y) 2 ]α(x,y)l(z),
θ 2 (x,y)=φ(x,y) cos 1 [ A(x,y) 2 ]β(x,y).
SNR(dB)=10 log 1 0 m=1 N dx n=1 N dy f ( mΔx,nΔy ) 2 m=1 N dx n=1 N dy { f ( mΔx,nΔy ) 2 g ( mΔx,nΔy ) 2 } ,

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