1. Introduction

The OAM holography proposed by Gu. et. al is identified as a vital approach for achieving ultrahigh-capacity multiplexing [1,2]. Owing to the theoretically unlimited helical mode index of an OAM beam, a large number of OAM-dependent information channels could be multiplexed in a single hologram, which holds great promise for ultrahigh-capacity holographic devices and systems. After that, complex-amplitude metasurface-based OAM holography [3], polarization-encrypted OAM holography [4], modulated OAM holography [5,6], perfect OAM holography [7], ellipticity-encrypted OAM holography [8], and high-dimensional nonlinear OAM holography [9,10] have been proposed in recent years. In addition, combining the deep learning or optical diffractive neural network with OAM holography has been investigated as well [11,12], which paves a way for optical communication, ultrahigh-security optical encryption and optical storage.

Depth is another possible multiplexing dimension for further increasing the information capacity. A relevant method is the ultra-dense perfect optical OAM multiplexed holography [7], in which the OAM modes are discriminated both radially and angularly by utilizing the “cone phase” degree of freedom. Another is the complex-amplitude metasurface-based OAM holography in the momentum space [3]. It is capable of multiplexing up to 200 independent channels on two depth planes, allowing holographic videos to be displayed. However, depth has not been widely applied as a multiplexing dimension mainly because of the serious coherence crosstalk between different image layers.

By utilizing the depth degree of freedom, we investigate the multi-layered depth multiplexing OAM holography (DM-OAMH). The rest of the paper is organized as follows. The DM-OAMH is introduced in Section 2. The method of suppressing the coherence crosstalk is presented in Section 3. Then simulation and experimental implementation are presented in Section 4, and the paper is summarized in Section 5.

2. Depth multiplexing OAM holography

In the OAM holography, the electric field on the hologram plane H(ξ, η) and that on the image plane E(x, y) form a Fourier pair, where x and y represent the abscissa coordinates on the image plane, and ξ and η denote those on the hologram plane. Given a target image E(x,y), the relationship can be expressed as

Here, a depth multiplexing OAM hologram can be generated by combining multiple OAM holograms and a Fresnel zone plate [1], where the Fresnel zone plate is used to control the reconstruction depth. A hologram encoded by multiple image channels can be expressed as

In order to decode the m-th channel image channel H_m, an OAM beam with a helical mode index l=-l_m should be used to illuminate the hologram. The resulting electric-field distribution after the hologram can be expressed as

The reconstructed electric-field distribution at z = f_m can be calculated by the Fresnel diffraction [13]

Here ${\otimes}$ denotes convolution and $\cal {F}$ denotes the Fourier transform, $Q^{\prime}(b )= \frac{{\textrm{exp} [{ik({{f_m} - {f_n}} )} ]}}{{i({{f_m} - {f_n}} )\lambda }}$ $\textrm{exp} [{i\pi b({{{x^{\prime}}^2} + {{y^{\prime}}^2}} )/\lambda } ]$. $\cal {F}\{{\textrm{exp} [{i({{l_n} - {l_m}} )\varphi } ]} \}$ is the doughnut-shaped intensity structure. The terms $\frac{{\textrm{exp}({ik{f_m}} )}}{{i{f_m}\lambda }}$ and $\frac{{\textrm{exp}({ik{f_n}} )}}{{i{f_n}\lambda }}$ in Eq. (5) have a relatively insignificant effect. Meantime, the terms $\textrm{exp}\left[ {\frac{{ik}}{{2{f_n}}}({x{^{\prime}{^2}} + y{^{\prime}{^2}}} )} \right]$ and $\textrm{exp}\left[ {\frac{{ik}}{{2{f_m}}}({x{^{\prime}{^2}} + y{^{\prime}{^2}}} )} \right]$ play as a low-pass filter [14], which are ignored here. As a result, Eq. (5) can be simplified as

The reconstruction intensity can thus be expressed as

The first term is the desired image with a pattern of Gaussian-shaped spots, while the last two terms are small due to the doughnut-shaped intensity structure of F_n. Each Gaussian spot in E_m corresponds to a doughnut-shaped spot in F_n. The inner projects of their complementary intensity distributions are close to zero. As a result, the last two terms can be ignored. The second term is the main coherence crosstalk resulting from other image channels, which can cause dramatic intensity oscillation and reduce the reconstruction quality.

3. Suppressing crosstalk by encoding random phases

For suppressing the coherence crosstalk, random phases Φ_n(x,y) are augmented into the target image, i.e, the target image E_n(x,y) is encoded as E_n(x,y)exp(iΦ_n) with -π<Φ_n≤π. According to the derivation in Section 2, the crosstalk term in Eq. (6) becomes

Meanwhile, the reconstructed intensity in Eq. (8) can be converted into

In Eq. (10), the first term is the desired image. The second term is composed of other image channels. Usually, the random phase has no effect on the image intensity, but Q’[1/(f_m-f_n)] plays an important role. It can disturb the amplitude, resulting in random crosstalk noise in the image [15]. To future explain this, as an example, a checkerboard diagram with random phases is designed, as shown in Fig. 1(a). As the diffraction distance increases, the diffraction pattern will be blurred. In contrast, when a random phase is not applied, the diffraction pattern will remain clear [Fig. 1(a)]. In addition, the last two terms in Eq. (11) have little effects on the image, and they almost vanish due to the orthogonality of random vectors [15].

 

Fig. 1. Principle of depth multiplexing OAM holography with random phase. (a) The role of random phase; (b) The normalized inner product of two random vectors; (c) The normalized inner product of two complementary chequerboard images as a function of the number of elements.

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Notably, the random phase technique has no significant effect on a single image, and it works well for an image with a large number of pixels. The orthogonality of two random images is affected by the pixel number Num of the images. As an example, the normalized inner product of two complementary chequerboard images is numerically evaluated, as shown in Fig. 1(c). As the pixel number increases, the normalized inner product will decrease gradually according to the central limit theorem, which is approximately 1/sqrt(Num) [Fig. 1(b-c)]. When the number of pixels approaches infinity, the normalized inner product will converge to zero, leading to coherence cancellation between any two reconstructed electric fields. Therefore, the reason for coherence crosstalk suppression can be simply explained as: when the depth image is reconstructed, the total energy of the other layer image channels remains constant, but it will sprinkle dispersedly in the reconstructed image. The coherence crosstalk from other image channels is converted into white noise.

The implementation of the DM-OAMH with random phases is shown in Fig. 2. A target image with random phases is sampled by a discrete array, where the sampling period is determined by the doughnut-shaped intensity distribution [1,2]. Here, the sampling spacing is adopted according to the diameter d of the main ring of the doughnut-shaped spots. Usually, the diameter d and the helical mode index l have a linear relationship, which is approximated as d = k_s|l|+d₀ µm, with a linear coefficient k_s= 20/3 µm and d₀= 20 µm. Notably, the size of the doughnut-shaped feature is affected by the numerical aperture (NA) of the holographic system [6]. Different holographic systems lead to different doughnut-shaped spot sizes and different linear relationships. When an image layer is sampled, an OAM preserving hologram with a discrete spatial-frequency distribution is obtained via the iterative Fourier transform algorithm with about 500 times [16]. A helical phase mode is added to the OAM-preserving hologram to create an OAM-selective hologram. Then a Fresnel zone plate is encoded onto the hologram to obtain an OAM depth selective hologram [Fig. 2(a)]. Finally, the holograms of all the layers are superimposed to obtain a depth multiplexing OAM hologram [Fig. 2(b)].

 

Fig. 2. The design of a depth multiplexing OAM hologram with random phases. (a) The generation of OAM depth selective hologram. (b) Generation of the OAM depth multiplexing hologram.

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4. Simulation and experimental implementation

As an example, 11 layers of a depth image are designed for numerical validation, as shown in Fig. 3. The target images are all cars [Fig. 3(a)] with 1024×1024 pixels and a pixel size of 8 µm. The sampling constant is 10 pixels and the depth range is 0.2 m. The encoding helical phase indexes are −10, −8, −6, −4, −2, 0, 2, 4, 6, 8, and 10, respectively. The 11 reconstructed layers have deviations of −0.1 m, −0.08 m, −0.06 m, …, 0.08 m, and 0.1 m, respectively from the central one. Comparison of the DM-OAMH is numerically conducted, as shown in Table 1. RMS (Root-Mean-Square), PV (Peak-to-Valley), PSNR (Peak Signal-to-Noise Ratio), and CCOEFF (cross-correlation coefficient) are used to evaluate the reconstruction quality. The RMS is defined as $RMS = \sqrt {\frac{1}{N}\sum\limits_{n = 1}^N {{\textrm{C}_n}^2} }$, where C_n is the n-th component of the background noise in Fig. 3(d-e). CCOEFF is defined as

 

Fig. 3. Comparisons of DM-OAMH with and without random phases. (a) Schematic diagram of the sampled target image. (b) The total crosstalk of 10 background layers with random phases. (c) The total crosstalk of 10 background layers without random phases. (d) A profile in (b). (e) A profile in (c).

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Table 1. Comparisons of DM-OAMH with and without random phases

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Then, an experimental system is built as shown in Fig. 4(a). A He-Ne laser with a wavelength of 632.8 nm is applied as the light source. The laser passes through a polarizer and beam expander and arrives at a spatial light modulator (SLM) Holoeye PLUTO-NIR-011. The SLM has 1080×1920 pixels with a pixel size of 8 µm. Modulated by the SLM and passing through a polarizer, the beam arrives at a CCD with 2048×2048 pixels and a pixel size of 5.5 µm. The polarizer in the experiment can filter out unwanted polarized light except the modulating one.

 

Fig. 4. Reconstruction result of OAM hologram with 2 layers. (a) Optical setup. (b) Reconstruction results. (b1-b2) Reconstruction result at z=0.200 m and 0.213 m, when decoded by the OAM mode with l=-10. (c1-c2) Reconstruction result at z=0.200 m and 0.213 m, when decoded by the OAM mode with l=10. (d1-d2) Reconstruction result at z=0.200 m and 0.213 m, when simultaneously decoded by the OAM mode with l=10 and OAM mode with l=-10.

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First, a two-layered depth multiplexing OAM hologram is designed, where the target images are a car and a plane, respectively. The pixel number is 1080×1080 and the pixel size of 8 µm. The encoding helical phase modes are l₁=−10 and l₂= 10 and the focal lengths of the Fresnel zone plates are f₁= 0.200 m and f₂= 0.213 m, respectively. The sampling constant of the target is adopted as 20 pixels. PSNR =20log(255/RMSE) is selected to measure the reconstructed image. When the OAM mode l₁= 10 is used to decode the hologram, only the car image can be selectively reconstructed at a depth of 0.2 m with PSNR = 24.9 dB and RMSE = 14.5, while the plane cannot be reconstructed at 0.213 m depth [Fig. 4(b1-b2)]. On the contrary, when decoded with the OAM mode l₂=−10, only the plane image can be reconstructed at a depth of 0.213 m, [Fig. 4(c1-c2)]. Interestingly, both can be reconstructed when two decoding modes are simultaneously used [Fig. 4(d1-d2)]. Henceforth, the method has a better reconstruction quality. Notably, DM-OAMH provides an option to select a particular image layer by choosing an appropriate decoding pattern, which is almost infeasible in the conventional holography.

Then, 5 images are adopted, where the encoded helical phase indexes are −15, −10, −5, 5, and 10, respectively. The sampling constant is 20 pixels and the depth range is from 0.21 m to 0.28 m. When the depth multiplexing OAM hologram is separately decoded by each OAM mode, 5 number or letter images can be sequentially reconstructed with high quality at different depths, as shown in Fig. 5. It is found the coherence noise is not significant in the reconstructed image and reconstruction image show well quality [Fig. 5(a-b)]. Notably, the resulting image size will gradually increase as the reconstruction length increases. The image size D is determined by the wavelength λ, pixel size of the hologram p, hologram width H, and reconstruction length z, D≈λz/p-H.

 

Fig. 5. Reconstruction results of OAM holograms with 5 layers. (a) The reconstruction results of 5 image layers “A”, “B”, “C”, “D” and “E”. (b) The reconstruction results of 5 images “1”, “2”, “3”, “4” and “5”.

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5. Summary

In summary, the depth multiplexing OAM holography is proposed based on random phase encoding. The method has three advantages:

As a result, this work can promote the development of OAM holography and may find widespread applications in dynamic OAM holography, optical stage, optical encryption and so on.