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Abstract
Optical holography has been investigated for enhancing information capacity and encryption. Here, the multi-vortex geometric orbital angular momentum (MVG-OAM) multiplexing holography is proposed and experimentally implemented, which encodes information into MVG beams with different central OAM, sub-beam OAM, and coherent-state phase. The orthogonality of the above three parameters are analyzed, respectively, which point out the feasibility of using them for holographic multiplexing. A three-dimensional multiplexing holography is realized by combining these three parameters, which offers potential applications for information storage, optical encryption, and display.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Holography provides a vital solution for optical information encryption [1] or storage [2], three-dimensional displays [3,4], etc. The various physical dimensions of light, such as amplitude [5], polarization [6,7], and wavelength [8], have been studied to improve the optical information capacity. However, the bandwidth of these dimensions is still limited. The orbital angular momentum (OAM), as a new degree of freedom (DoF) of light, has attracted considerable attention due to its theoretically infinite orthogonal modes [9]. Recently, OAM holography with discrete sampling in the spatial frequency domain has been proposed and experimentally investigated, which provides a method to improve the bandwidth of holographic multiplexing [10,11]. However, since the beam divergence angle and beam size of the OAM beam increase with the increase of the OAM order, the maximum multiplexing capacity of the OAM mode set is limited by the space-bandwidth product (SBP) of the optical system [12]. Meeting the growing demand for data capacity while fully utilizing the SBP is challenging.
In OAM holography, exploring more DoFs for increasing information capacity has been widely studied. For instance, multiple images can be reconstructed on different OAM and polarization channels [13]. The multi-dimensional encrypted OAM multiplexing holography is proposed based on the modulated chiro-optical beams (MCOBs) [14]. By introducing the multiple parameters of MCOB, four-dimensional spatial multiplexing is achieved. Combining the radial and angular dimensions of perfect OAM beam, the ultra-dense perfect OAM multiplexing holography with integer and fractional topological charges (TCs) is investigated, and the resolution of TC in holographic multiplexing can reach 0.01 [15]. In ellipticity-encrypted OAM multiplexing holography, an elliptic vortex beams with inverse TC under a customized ellipticity are needed to reconstruct the target images [16]. In partial OAM holography, an integer OAM mode is divided into several partial phase modes to improve the multiplexing capacity [17]. The Laguerre Gaussian (LG) mode holography is implemented in the azimuthal mode index and radial index, which further broadens the capability of holographic multiplexing [18]. Although introducing more parameters in OAM multiplexing holography can improve the information capacity, the improvement is also limited. This is due to the fact that the introduction of these parameters does not significantly enhance the divergence degeneracy of beams, resulting in no further improvement of the number of multiplexed channels in optical systems with the limited SBP.
Recently, a class of ray-wave geometric beams (RWGBs) with wave-like and ray-like properties have attracted attention, which have distinct OAM characteristics and provide new controllable DoFs to enhance the divergence degeneracy [19–21]. In this paper, the multi-vortex geometric (MVG) beams with three independent encoded parameters [22], including central OAM, sub-beam OAM, and coherent-state phase, one type of RWGBs, are introduced into the OAM multiplexing holography, namely, the multi-vortex geometric orbital angular momentum (MVG-OAM) multiplexing holography is proposed. The orthogonality of the three parameters are theoretically analyzed, respectively, and the feasibility of these parameters as independent channels in holographic multiplexing are pointed out. The multi-dimensional multiplexing holography is experimentally implemented. The method proposed here can improve the capacity of the optical holographic systems. Section 2 is devoted to the principle of MVG-OAM multiplexing holography. In section 3, the experimental implementation is presented. Finally, we give the discussion and conclusion in Section 4.
2. Principle of MVG-OAM multiplexing holography
The electric field of the MVG beam can be expressed as [22]:
Fig. 1. (a) Schematic diagram of three parameters of MVG beam $|{E_{5,2}^{(\pi /2,\pi /2,\phi )}} \rangle _{5,0}^5$. (b) The numerical simulation of the inner product between $|{E_{15,0}^{(\pi /2,\pi /2,0)}} \rangle _{5,0}^5$ and $|{E_{{n_0},0}^{(\pi /2,\pi /2,0)}} \rangle _{5,0}^5$. (c) The numerical simulation of the inner product between $|{E_{5,15}^{(\pi /2,\pi /2,0)}} \rangle _{5,0}^5$ and $|{E_{5,{m_0}}^{(\pi /2,\pi /2,0)}} \rangle _{5,0}^5$. (d) The numerical simulation of the inner product between $|{E_{5,10}^{(\pi /2,\pi /2,\pi )}} \rangle _{5,0}^5$ and $|{E_{5,10}^{(\pi /2,\pi /2,\phi )}} \rangle _{5,0}^5$. (e) The numerical simulation of the inner product between $|{E_{5,10}^{(\pi /2,\pi /2,0)}} \rangle _{5,0}^5$ and $|{E_{5,10}^{(\pi /2,\pi /2,\pi /2)}} \rangle _{5,0}^N$.
The premise for using MVG beams in optical communication applications is that the three parameters are orthogonal. The degree of correlation (DOC) is used as an evaluation criterion and varies in the range of [0, 1]. 0 and 1 indicate the complete orthogonality and complete non-orthogonality of two MVG beams, respectively. The inner product can represent the DOC of two MVG beams, which can be written as:
In hologram, when an MVG beam is used for holographic reconstruction, the spatial frequency components on the hologram plane can be expressed as:
where (x, y) indicates the Cartesian coordinates in the hologram plane. Eh(·) and EMVG(·) are the complex amplitudes of the hologram and the MVG beam, respectively. Based on the Fourier integral theorem, a Fourier pair is formed between the electric field of the image plane and hologram plane, so the electric field of the image plane is: where (X, Y) indicates the Cartesian coordinates in the image plane. The operators $\Im $ and * are the Fourier transform and convolution, respectively. It can be seen that the Fourier transform of an incident MVG beam, which acts as a kernel function of the convolution, is simply generated into each pixel of the reconstructed image. Therefore, in order to avoid spatial overlap of adjacent pixels and to preserve the beam property in the reconstructed images, it is necessary to sample the target image with a 2D Dirac comb sampling array. Here, the sampling interval d is determined by the intensity distribution of the MVG beam in the Fourier plane, which can be obtained by the second moment of intensity [23], namely, d = 2 $\sqrt {{{2\int\limits_0^{2\pi } {\int\limits_0^\infty {{r^2}I(r,\phi )drd\phi } } } / {\int\limits_0^{2\pi } {\int\limits_0^\infty {I(r,\phi )drd\phi } } }}} $, and I(r, ϕ) is the intensity distribution of the MVG beam in the cross-section. As a result, the sampling interval d as a function of sub-beam OAM m0 is shown in Fig. 2(a). Note that the constant d increases with the increase of m0.Fig. 2. The design principle of MVG-OAM selective holography. (a) The relationship between the sub-beam OAM m0 and the sampling constant d. (b) The principle of the MVG-OAM selective holography. (c) The flow chart of MVG-OAM-preserved hologram based on AWGS algorithm. (d)-(e) Results of numerical reconstruction of MVG-OAM-preserved holograms with coherent-state phase 0 and π, respectively. Inset: intensity of selected pixels.
The design principle of an MVG-OAM selective hologram is depicted in Fig. 2(b). First, the image is sampled by a 2D Dirac comb array, leading to the generation of a sparse target image. An MVG mode preserved hologram is obtained by the adaptive weighted Gerchberg-Saxton (AWGS) algorithm [24]. The flow chart of MVG-OAM-preserved hologram based on AWGS algorithm is shown in Fig. 2(c). The initial input phase distribution is provided by the random phase and MVG phase. In the iteration process, the amplitude constraint on the hologram plane is unit amplitude, while a weighting factor w is embedded into each phase retrieval iteration to realize the amplitude constraint on the image plane. When the iteration is accomplished, the MVG-OAM-preserved hologram can be obtained. As an example, in Figs. 2(d) and 2(e), when incident MVG beams with different coherent-state phase 0 or π illuminate the corresponding MVG-OAM preserved hologram, the wavefront of the incident MVG beams can be reconstructed in each pixel of the reconstructed image.
In OAM holography, in addition to achieving the preservation of OAM characteristics, OAM mode selectivity is also key. Usually, OAM mode selective holograms can only convert the encoded target information into a quasi-Gaussian spot with relatively high intensity distribution under the illumination of an incident OAM beam with an inverse mode index. We know that the Hermite polynomials Hn(·) has no negative index. Do the three parameters of MVG beam have the same mode selectivity as the OAM mode selectivity? Fortunately, the mode selectivity of MVG beams can be achieved using the conjugate modulation demultiplexing method, thus a given MVG mode can be converted into a quasi-Gaussian intensity distribution (see Supplement 1 sec. 3), which is the basis of image multiplexing. Based on the strong mode selectivity, the MVG-OAM selective hologram can be designed by superimposing the MVG phase function and the mode-preserved hologram. As a result, the MVG-OAM multiplexing hologram can be obtained from the superposition of multiple MVG-OAM selective holograms.
To confirm that MVG-OAM multiplexing holography can achieve reconstruction of the target images with a higher signal-to-noise ratio (SNR), a 4-bit multiplexing hologram is designed and used to compare the demultiplexing performance of OAM-multiplexing holography and MVG-OAM multiplexing holography. The SNR is defined as 10log10(Isignal/Inoise), where Isignal and Inoise represent the intensity of holographic image reconstructed from the desired channel and other multiplexing channels, respectively. Table 1 and Table 2 list the SNRs on each multiplexing channel. Note that using MVG mode in holographic multiplexing can effectively improve SNR. It can be explained by the fact that when the decoded MVG mode does not match the encoded MVG mode, the complex phase structure of the MVG mode causes the intensities in the non-desired imaging channels to exhibit weak speckle patterns instead of the concentrated speckle patterns in the OAM modes. The reconstructed intensity distributions in the target holographic images are analyzed when different incident MVG beams and incident OAM beams illuminate the corresponding designed 4-bit MVG-OAM and OAM multiplexing holograms (see Supplement 1 sec. 4). In the desired imaging channel, the quasi-Gaussian spots with a stronger intensity distribution can be obtained. Conversely, the peak intensities in the other channel are lower than that of the desired channel, which can be regarded as background noise.
Table 1. Numerical simulation of reconstructed SNR for each channel in MVG-OAM multiplexing holography
Table 2. Numerical simulation of reconstructed SNR for each channel in OAM multiplexing holography
We numerically compared the pixel size of 100 lowest-order OAM beams and MVG beams on the image plane, as illustrated in Fig. 3(a). Here, the second moment of intensity of OAM beams and MVG beams is used to calculate the pixel size. Note that although the pixel size of the low-order MVG beams is larger than that of the low-order OAM beams, the pixel size of the high-order MVG beams changes slowly, which is beneficial to reduce inter-mode crosstalk and improve SNR. In conventional OAM multiplexing holography, the mode interval Δl is usually set very large to avoid multiplexing crosstalk, which sacrifices the multiplexing capability as well as limits the resolution of the reconstructed image [15–17,25]. However, the resolution of the reconstructed images can be improved by using high-order MVG modes, as shown in Fig. 3(b). The resolution of the image is defined as 25.4 mm/d, where d is the size of the incident MVG beam and OAM beam.
Fig. 3. Numerical simulations of pixel size (a) and image resolution (b) of OAM beams and MVG beams.
3. Experimental results and discussions of MVG-OAM multiplexed holography
To verify the feasibility of the proposed MVG-OAM holography, an experimental setup diagram is built, as shown in Fig. 4. A He-Ne laser (Research Electro-Optics, R-31007-633 nm-0.8 mW) is used as the light source. The Lens L1 (20 × objective) and lens L2 (f = 400 mm) are used as beam expanders. An aperture A and polarizer P are used to adjust the beam size and polarization to ensure that the incident beam matches the phase-only spatial light modulator (SLM, Hamamatsu-X13138 series-07, 1272 × 1024 pixels, pixel pitch of 12.5 µm), respectively. The beam is modulated by the SLM and passes through a lens L3 (f = 100 mm) to a CMOS (pco. edge-4.2 bi, 2048 × 2048 pixels, pixel pitch of 6.5 µm) used to capture the reconstructed holographic images. In this paper, the experimental system is simplified since we only have one SLM. The hologram pattern is not directly illuminated by a MVG beam. Alternatively, the decoded MVG phase distribution is superimposed into the hologram, which is illuminated by a planar beam, as shown in Fig. 4(a). In the decrypted process, the hologram used is represented as the superposition of the decoded MVG phase and the MVG-OAM hologram, so the mathematical phase-only hologram can be described as:
where Φi, ψi-de and N represent the phase information of each image channel and the decoded MVG phase distribution, the number of multiplexing channels, respectively. The design principle of the hologram loaded into the SLM is shown in Fig. 4(b). The images can be reconstructed when the ϕ-, n0- and m0-MVG-OAM preserved holograms are illuminated by different incident MVG beams with phases φ(ϕ, n0, m0), namely, φ(0, 7, 2), φ(0, 1, 2), and φ(π, 7, 1) (see Supplement 1 sec. 5).Fig. 4. (a) Schematic diagram of the experimental setup of MVG-OAM holography. L1, L2, L3: lens; A: aperture; P: polarizer; BS: beam splitter; SLM: spatial light modulator. (b) The hologram loaded into the SLM contains the decoded MVG phase and MVG-OAM hologram.
The ϕ-MVG-OAM selective hologram is designed, as shown in Fig. 5(a). First, the target image is sparsely sampled using a 2D Dirac comb, and a designed ϕ-MVG-OAM preserved hologram is obtained by the AWGS algorithm. Then the MVG phase function φ(0, 7, 2) is superimposed into the ϕ-MVG-OAM-preserved hologram, resulting in a generation of ϕ-dependent MVG-OAM selective hologram. Only a given input conjugate MVG phase mode with ϕ = 0, the quasi-Gaussian spot with stronger intensity distribution can be obtained, leading to strong mode selectivity with respect to ϕ, as shown in Fig. 5(b). The experimental results of ϕ-dependent MVG-OAM selective holographic image are shown in Fig. 5(c). In the reconstructed holographic image, the complex intensity distribution of the correctly incident MVG mode is converted into a quasi-Gaussian mode with a solid-spot intensity distribution. In addition, the experimental results of the n0- and m0-dependent MVG-OAM selective holographic image can be found in Supplement 1 sec. 6.
Fig. 5. ϕ-dependent MVG-OAM selective holography. (a) The design process of a ϕ-MVG-OAM selective hologram. (b) The conversion of coherent-state phase ϕ of MVG modes. (c) Experimental reconstruction results of a ϕ-MVG-OAM selective holographic image based on incident MVG beams with ϕ = 0, 0.5π, π, and 1.5π, respectively.
According to the previous analysis results, the three DoFs (ϕ, n0, m0) can be independently modulated and have an enormous potential in holographic multiplexing, which improves the information capacity. Figure 6(a) illustrates the encoding process of the ϕ-MVG-OAM multiplexing hologram. Four target images are multiplied by corresponding 2D Dirac comb sampling arrays in the image plane, and then the four-preserved holograms obtained are encoded by four MVG phase functions with φ(0, 7, 2), φ(0.5π, 7, 2), φ(π, 7, 2), and φ(1.5π, 7, 2), respectively, resulting in four MVG-OAM selective holograms. The superposition of all selective holograms can generate one MVG-OAM multiplexing hologram. A complex pattern can be obtained when a planar wave illuminates the multiplexing hologram, and no distinguishable image is obtained here, as depicted in Fig. 6(b). When the ϕ-MVG-OAM multiplexing hologram is sequentially illuminated by the incident MVG beams with conjugate phase distributions, four holographic images can be reconstructed, as shown in Fig. 6(c). The experimental results point out the feasibility of parameter ϕ as an independent information channel in the holographic multiplexing system.
Fig. 6. Schematic diagram of ϕ-MVG-OAM multiplexing holography. (a) The encoding process of the ϕ-MVG-OAM multiplexing hologram. (b) Reconstruction holographic image by a planar wave. (c) Experimental reconstruction results by different incident MVG beams.
In addition, the MVG-OAM multiplexing holography with the central OAM n0 key is designed, as shown in Fig. 7(a). Four target images are discrete sampled to obtain the corresponding preserved holograms, respectively, and then the MVG phase functions with φ(0, 3, 10), φ(0, 5, 10), φ(0, 7, 10), and φ(0, 9, 10) are superimposed into the corresponding holograms. Finally, one n0-dependent MVG-OAM multiplexing hologram can be obtained by combining all holograms. In the experiments, different reconstructed target images are captured by CMOS when the correctly incident MVG beams with central OAM n0 = 3, 5, 7, and 9 are used, respectively, as shown in Fig. 7(b). Further, Fig. 8 shows that the feasibility of the MVG-OAM multiplexing holography with the sub-beam OAM m0 key is verified. When the corresponding incident MVG beams are used to illuminate the m0-dependent MVG-OAM multiplexing hologram, the target images can be decoded. Therefore, it is confirmed that the central OAM n0 and the sub-beam OAM m0 can be used for holographic multiplexing.
Fig. 7. Schematic diagram of n0-MVG-OAM multiplexing holography. (a) The design process of the n0-MVG-OAM multiplexing hologram. (b) Experimental reconstruction results by different incident MVG beams with n0 = 3, 5, 7, and 9, respectively.
Fig. 8. Schematic diagram of m0-MVG-OAM multiplexing holography. (a) The design process of the m0-MVG-OAM multiplexing hologram. (b) Experimental reconstruction results by different incident MVG beams with m0 = 2, 4, 6, and 8, respectively.
The MVG-OAM multiplexing holography with ϕ, n0, and m0 keys have been implemented independently based on the above analysis. If the three DoFs are combined as the carriers of information, a multi-dimensional multiplexing holography is achieved, and the information capacity of the optical system is further improved. First, the two-parameter encrypted MVG-OAM multiplexing holography is implemented by combining any two of the three DoFs. Figure 9 shows the n0-ϕ-MVG-OAM multiplexing holography with the central OAM n0 and coherent-state phase ϕ key. The multiplexing hologram is generated by superimposing the mode-preserved holograms and the encoded MVG phases [Fig. 9(a)]. When the incident conjugate MVG modes with φ(0, 3, 10), φ(0.5π, 5, 10), φ(π, 7, 10), and φ(1.5π, 9, 10) illuminate the multiplexing hologram, four target images can be reconstructed. It can be seen that the image can be encrypted on the dual channel of n0 and ϕ.
Fig. 9. Schematic diagram of n0-ϕ-MVG-OAM multiplexing holography. (a) The design process. (b) Experimental reconstruction results by different incident MVG beams.
The process of encryption by using the sub-beam OAM m0 and coherent-state phase ϕ is shown in Fig. 10. Four MVG phase modes with φ(0, 7, 2), φ(0.5π, 7, 4), φ(π, 7, 6), and φ(1.5π, 7, 8) are used to encrypt four target images. The four m0-ϕ MVG-OAM selective holograms can be obtained, and the superposition of these holograms can generate one m0-ϕ-MVG-OAM multiplexing hologram, as shown in Fig. 10(a). When the correctly incident MVG beams illuminate the multiplexing hologram, the reconstructed images can be captured by CMOS, as shown in Fig. 10(b). The m0-n0-MVG-OAM multiplexing hologram is designed in four different central OAMs (n0 = 3, 5, 7, 9) and four different sub-beam OAMs (m0 = 2, 4, 6, 8), as illustrated in Fig. 11. The results show that each target image is reconstructed from the suitable incident MVG beam.
Fig. 10. Schematic diagram of m0-ϕ-MVG-OAM multiplexing holography. (a) The design process. (b) Experimental reconstruction results by different incident MVG beams.
Fig. 11. Schematic diagram of m0-n0-MVG-OAM multiplexing holography. (a) The design process. (b) Experimental reconstruction results by different incident MVG beams.
Further, an optical holographic multiplexing system with high security can be realized by using three DoFs. Two different coherent-state phases ϕ (ϕ = 0 and π), five different central OAMs n0 (n0 = 5, 8, 11, 14, and 17), and five different sub-beam OAMs m0 (m0 = 14, 11, 8, 5, and 2) are used to encode ten target images in the experiment. The ten MVG mode phases with (ϕ, n0, m0) are used to encode the ten target images, respectively, as shown in Fig. 12(a). After that, a 10-bit MVG-OAM multiplexing hologram is obtained. In the experiment, the ten target images can be reconstructed when the correctly incident MVG beam illuminates the multiplexing hologram, as illustrated in Fig. 12(b). Here, it is noted that a holographic system can be achieved using any one DoF or combinations of the DoFs of MVG beam.
Fig. 12. Design and experimental results of a 10-bit ϕ-n0-m0-MVG-OAM multiplexing holography. (a) The design of 10bit MVG-OAM multiplexing hologram. (b) The experimental reconstruction results.
4. Discussion and conclusion
The MVG-OAM multiplexing holography is proposed and implemented by using the three parameters of the MVG beam, including central OAM, sub-beam OAM, and coherent-state phase. The conjugate modulation method is used to implement the demultiplexing task of the holographic system. The three parameters can be used independently as information carriers, which improve the capacity of holographic multiplexation. In the experiments, any one, two, or three parameters are chosen to encode and decode target information, which can provide a multi-dimensional multiplexing holography with high security. As an example, ten different MVG phase modes are used to encode the corresponding target images, respectively, realizing a 10-bit MVG-OAM multiplexing holography. As a result, the holography proposed here can have potential applications in optical information encryption/storage, 3D display, etc.
So far, different types of OAM holography have been achieved by increasing the DoFs of the laser beam (e.g., polarization [13]) or using a novel laser beam (e.g., multiramp helical-conical beam [26]). The OAM set is theoretically infinite orthogonal, which can improve the capacity of information and enhance the level of the information security. However, the increase of the mode order leads to a rapid increase in the divergence of OAM beam, which severely limits the number of channels for holographic multiplexing. Compared to OAM modes, MVG modes have a slower variation in divergence and the number of multiplexing channels is increased by two orders of magnitude (see Supplement 1 sec. 7). In our experiments, the 10 independent channels are implemented as an example for information multiplexing, which points to the possibility that MVG modes can be used to achieve a higher information capacity. In addition, the capacity of information can be further improved by combining other DoFs of light, such as amplitude and wavelength.
Funding
National Natural Science Foundation of China (61775153); Priority Academic Program Development of Jiangsu Higher Education Institutions.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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