Imaging through scattering media is one of the most challenging problems in optics. The extensive scattering of light by scattering media is a significant obstacle for imaging and focusing. Suppression of the scattering effect can be realized by modulating the shape of the wavefront by transmission matrix measurement^{[13" target="_self" style="display: inline;">–3]}, iterative wavefront optimization^[4,5], and optical phase conjugation (OPC)^{[6–9" target="_self" style="display: inline;">–9]}. Among them, OPC^[10] with its time-reversal nature determines the optimum wavefront without time-consuming iterations. Generally, OPC can be generated by adaptive optics, named digital OPC (DOPC)^[11]. In addition, OPC is also achieved by using the analog OPC technique^{[1214" target="_self" style="display: inline;">–14]} based on volume grating in the photorefractive crystals^[15,16], which can inherently compensate the distorted wavefront. In contrast to DOPC, analog OPC is more convenient for achieving conjugating phase without inversion and the fine adjustments of relative phase differences.

In principle, the analog OPC based on volume grating of the photorefractive crystals can be the proper way to compensate the distortion effectively. However, the modulations with high spatial frequency in the wavefront are always introduced by scattering media. Limited by the recording structure of the photorefractive volume grating, the diffraction efficiency of the high-frequency modulations in the distorted wavefront is weak, and the compensation effect of wavefront distortion is greatly reduced. In photorefractive crystals, the effective wave vector of the photorefractive grating is determined by the angle between the vector direction of the volume grating and the optical axis direction of the photorefractive crystal^{[1719" target="_self" style="display: inline;">–19]}. Taking into account the above problems in analog OPC, optimization of the recording structure to obtain a high-quality compensation effect with high signal to noise ratio (SNR) is worthy of study.

In this Letter, we optimize the volume grating structure in analog OPC generated by the photorefractive LiNbO3 crystal. The compensation effect for the high spatial frequency in the distorted wavefront is theoretically analyzed and experimentally verified. Based on the couple-wave equation of volume grating, we theoretically analyze the effective wave vector ratio of high spatial frequency in the distorted wavefront and the diffraction efficiency of the signal volume grating with the recording angle. In the experiment, the analog OPC system based on the photorefractive grating is constructed to verify the compensation effects of the distorted wavefront with the recording angle θ at 7.5°, 30°, and 45°, respectively. Combining the theoretical analysis and the experimental results, the SNR of the compensated images by the OPC with different recording angles is further analyzed, and the optimal recording structure of the photorefractive volume grating is concluded.

Analyses for optimizing the structure of the photorefractive volume grating are focused on two steps: the recording and reconstructing of the distorted wavefront by the photorefractive volume grating. In the recording process, the geometry structure of the photorefractive volume grating is depicted in Fig. 1. The electro-optic axis c∧ is oriented along the z direction. The bisector of the angle between the two recording beams is parallel to the x direction. The reference beam Er intersects with signal beam Eo, and the complex field amplitudes can be written as $$\begin{gathered} Er(x,z)=A(x,z)exp(−ikrr), \\ Eo(x,z)=a(x,z)exp(−ikor), \end{gathered}$$(1)where A, a and kr, ko represent the complex amplitude and the wave vector of Er and Eo, respectively. The wave vector k of the signal volume grating formed by the two waves parallel to the z direction in the crystal can be expressed as k=kr−ko,(2)as the distortion generated by the scattering medium is introduced in the signal wave. For the recording information of the distorted wavefront, the original signal information can be regarded as the low spatial frequency component. Correspondingly, the scattered noise information can be considered as the high spatial frequency component. A scattered noise beam Eos is supposed, with the angle α relative to the x axis, and the complex field amplitude can be written as Eos(x,z)=B(x,z)exp(−ikosr),(3)where B and kos are the complex amplitude and the wave vector of Eos, respectively. The wave vector ks of the photorefractive volume grating with high spatial frequency formed by Eos and Er is ks=kr−kos. ks′ is defined as the component of wave vector ks parallel to the c axis in the crystal; it can be written as ks′=kscos[(α+θ)/2].(4)

Fig. 1. Recording geometry of scattered beams by photorefractive volume grating. Volume gratings 1 and 2 represent the signal volume grating and the photorefractive volume grating with high spatial frequency. Er, Eo, and Eos represent the reference beam, signal beam, and scattered noise beam, respectively. θ is the recording angle, WR and WS are the widths of Er and Eo, respectively. α is the angle between Eos and the x axis.

下载图片 查看所有图片

The recorded information of the photorefractive volume grating with high spatial frequency is a component of the scattered noise information because of the recording intensity determined by the wave vector. Thus, in order to obtain a clear physical description about this grating, we define the effective wave vector ratio as ks′ks=cos[(α+θ)/2], which represents the recorded modulations with high spatial frequency parallel to the z direction.

Due to the photorefractive effect, the signal volume grating with the modulation n1 is formed by Er and Eo in the crystal. The wave number is k=πn1/λ, where λ is the wavelength of the recording waves. n1, the refractive index modulation in the photorefractive crystal, is assumed constant with n1=0.15×10−4. The recorded intensity and diffraction efficiency^[19] of the signal volume grating are determined by the recording structures. Similarly, the photorefractive volume grating with high spatial frequency is also recorded in the same way.

To analyze the diffraction behavior of the signal volume grating and photorefractive volume grating with high spatial frequency during the conjugate reconstruction, we take the signal volume grating as an example due to the similar physical mechanism. In the conjugate reconstruction process, the wave field in the signal volume grating can be written as E(x,z)=a*(x,z)eikor+Q(x,z)eikqr,(5)where a*(x,z) and −ko represent the complex amplitude and the wave vector of the reconstructed wave. Q and kq are the complex amplitude and the wave vector of the first-order diffraction wave, respectively. The two-dimensional coupled-wave equations^[19] can be given as cosθ∂a*∂x+sin∂a*∂y+ikAaQ=0,cosθ∂Q∂x−sin∂Q∂y+ikAaa*=0.(6)

Similarly, the above formulas are applicable for the analysis of the photorefractive volume grating with high spatial frequency. However, k obeys the vector triangle rule, and the coupling efficiency is related to the contribution of the effective wave vector ratio in the recording process. In addition, the reconstructed intensity is limited by the effective wave vector ratio.

Taking the traditional volume grating as example to analyze the diffraction efficiency of the signal grating and the photorefractive volume grating with high spatial frequency, the diffraction efficiency η is expressed in Eqs. (7) and (8): η(w0)=1−J02(w0)−J12(w0),(7)w0=πn1λWRWS2sin2θ,(8)where WR=2WS is the widths of the two intersected beams Er and Eo. J0 and J1 are the zero and first-order Bessel functions.

Based on above analysis, the effective wave vector ratio of the photorefractive volume grating with high spatial frequency in different recording angles is shown in Fig. 2, with α taken as 50° for convenience. It increases with the recording angle decreasing, which indicates that the recorded modulations with high spatial frequency can be significantly improved in the small recording angle. In addition, as shown in Fig. 2, the higher diffraction efficiency can be acquired under the condition of the smaller recording angle.

Fig. 2. Effective wave vector ratio of photorefractive volume grating with high spatial frequency and diffraction efficiency of the traditional volume grating in different recording angles.

下载图片 查看所有图片

In order to further verify the above analysis, three recording angles θ are chosen as 45°, 30°, and 7.5° to study the compensation effect of the modulations with high spatial frequency in the distorted wavefront in the experiment. Taking θ=45° as an example, the experimental setup of analog OPC is depicted in Fig. 3. A 45°-cut LiNbO3 crystal is used to generate the analog OPC, and the c axis is depicted by the arrow, as shown in Fig. 3. The illumination beam from a continuous laser (100 mW) with a wavelength of 532 nm is split into two by a beam splitter (BS1). The beam attenuators A1, A2 are used to adjust the two beams to the proper intensities. In the object beam “a” with the diameter of 0.75 mm, a negative United States Air Force (USAF) target is utilized for the purpose of generating a two-dimensional wide-field image pattern. A ground glass placed 5 cm behind the L2 acts as the scattering medium. The other beam of the interferometer is split further into the reference beam “A” and the conjugate reference beam “A*” by BS2 after expanding to a diameter of 15 mm. In recording, the object beam intersects with the reference beam in the crystal simultaneously for 5 min. A photorefractive volume grating with the information of the distorted wavefront is recorded in the crystal. In conjugate reconstruction, an OPC light field is generated with a conjugate reference beam to retrace the path of the original transmission. The image after OPC compensation is detected by a complementary metal–oxide–semiconductor (CMOS, Mako G-419B, pixel size 5.5μm×5.5μm). The focal lengths of L2, L4, L6, L7 are 100 mm, and L1, L3, L5 are 50, 75, 150 mm, respectively. Keeping the object beam unchanging, the recording angle θ of the reference beam is varied at 30° and 7.5°, respectively, to repeat the above experiment steps.

Fig. 3. Experimental setup of analog OPC. M1–M5 are reflecting mirrors, SPF represents the spatial filter, and A1, A2 are the beam attenuators. BS1–BS3 are beam splitters, and L1–L7 are the lenses. A, a, and A* represent the reference beam, the object beam, and the conjugate reference beam of A.

下载图片 查看所有图片

The experimental images after OPC compensation in three recording angles and the normalized one-dimensional intensity distributions are shown in Fig. 4. Figures 4(a) and 4(b) are the images of the USAF target and the conjugate reconstructed image without the scattering medium, respectively. When the scattering medium is introduced, the random scattering inside the ground glass severely distorts the spatial phase profile and the spatial intensity distribution. The measured intensity pattern appears blurry and is filled with speckles, as shown in Fig. 4(c). The intensity distribution in the right column of Fig. 4(c) turns up as clutter and splits into multiple peaks in the rectangle area. Moreover, the maximum noise value of the background area is up to 0.95. The SNR, defined as the ratio between the maximum intensity of the rectangle area and the maximum intensity of the background area, is evaluated at approximately 1.05. Figures 4(d)–4(f) show the images after OPC compensation in the recording angles of 45°, 30°, and 7.5°, respectively. Compared with Fig. 4(b), Figs. 4(d)–4(f) show that the speckles are all reduced, and the maximum intensity of the background area decreases to 0.7, 0.48, and 0.22, respectively. Especially in Fig. 4(f), the image resolution is enhanced, and the number “5” is easy to distinguished, which indicates that the compensation effect of high spatial frequency is improved significantly in the small recording angle. The SNR corresponding to Figs. 4(d)–4(f) increases 1.4, 2, and 4.4 times that of Fig. 4(b), respectively.

Fig. 4. Experimental results of the USAF target (left column) and normalized one-dimensional intensity distribution (right column). The horizontal axis in the right column represents the coordinate value corresponding to the length of the red line, and the middle position of the red line is 0 μm. (a) Image without ground grass. (b) Conjugate reconstructed image without ground grass. (c) Image with the ground grass. (d)–(f) Images after OPC compensation in recording angles of 45°, 30°, and 7.5°, respectively.

下载图片 查看所有图片

Based on the three measured values of SNR, 1.43, 2.1, and 4.6, the SNR curve is fitted with the different recording angles, as shown in Fig. 5. Combining the curve of SNR and the diffraction efficiency, it is demonstrated that the high-efficiency compensation effect can be acquired by analog OPC with the smaller recording angle, which is in high consistence with the theoretical analysis.

Fig. 5. Numerical fitting SNR and diffraction efficiency in different recording angles. The error bar shows the standard deviation of three experimental measurements.

下载图片 查看所有图片

In conclusion, we have demonstrated that the compensation effect of modulations with high spatial frequency in the distorted wavefront can be improved by optimizing the recording angle. Based on the theoretical analysis, it is concluded that the effective wave vector ratio and the diffraction efficiency can be improved under the condition of the small recording angle. In the experiment, we choose the recording angles of 45°, 30°, and 7.5° to verify the compensation effect of high spatial frequency and to obtain the corresponding SNR. The fitted curve of the SNR is in good agreement with the theoretical diffraction efficiency. Consequently, under the condition of small recording angle, analog OPC has more efficient compensation effect on the distorted wavefront with high spatial frequency. We hope this work could be helpful for improving the compensation effect of analog OPC and provide a meaningful reference for optical imaging.