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
Since the first observation in 1972 of phase compensation effect by a phase conjugate light , it has been applied in numerous fields including imaging through a distortion medium  and laser resonator . 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 [4–7]. 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  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 . However, so-called computer-generated holograms such as the off-axis hologram inherently entail the loss of the spatial resolution of the SLM [13–15], 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 . 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.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;Eqs. (2) and (3) are transformed as follows;
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) , one of phase-shifting digital holography . 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.
2.3. Experimental result
For evaluating quality of the complex modulation by the DPMM, we use signal to noise ratio (SNR) defined as follows;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.
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).
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.
A complex wavefront of the input light A(x,y)exp[iφ(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[-iφ(x,y)] is calculated and then it is decomposed into two phase images exp[iθ1(x,y)] and exp[iθ2(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[iθ1(x,y)] and exp[iθ2(x,y)] are synthesized after transmitting BS2 to be phase conjugate light A(x,y)exp[-iφ(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[-iφ(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[iξ(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[iφ(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[-iφ(x,y)] is calculated and then it is decomposed into two phase images exp[iθ1(x,y)] and exp[iθ2(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[iθ1(x,y)] and exp[iθ2(x,y)] are synthesized after transmitting BS2 to be phase conjugate light A(x,y)exp[-iφ(x,y)]. The synthesized phase conjugate light transmits again through the phase object with the phase exp[iξ(x,y)] via relay optics with L1 and L2. Then, the phase distribution exp[-iξ(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.
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.
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.
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.
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.
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.
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