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Abstract
The orbital angular momentum (OAM) holography has been identified as a vital approach for achieving ultrahigh-capacity multiplexation without a theoretical helical phase index limit. However, the encoding and decoding of an OAM hologram require a complete helical phase mode, which does not take full utilization of the angular space. In this paper, the partial OAM holography is proposed by dividing an OAM mode into several partial orbital angular momentums and encode each partial mode with a different target image. An image can only be reconstructed using an appropriate partial OAM mode within a specific illuminating angular range, henceforth holographic multiplexation of images can be realized. This method can significantly increase the holographic information capacity and find widespread applications.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
As a freedom of light except for the well-known parameters such as amplitude, phase, wavelength, and polarization, the orbital angular momentum (OAM) has attracted intensive attentions. In general, OAM is used to characterize the “twisted” helical phase front of a light beam when its wavevector spirals around the beam axis. Such a helical phase front of an OAM-carrying beam is usually represented as exp(ilφ), where l denotes the helical phase index and φ is the azimuthal angle [1]. Due to the inherent orthogonality of different OAM modes, OAM can thus enhance the information capacity and be used for optical manipulation [2,3], optical tweezers [4,5], optical communications [1,6], and so on.
Recently, the OAM holography has been proposed and experimentally demonstrated in linear optics [7–9], and it continues to extend into the field of nonlinear optics [10,11]. The core of the OAM holography is to preserve and select appropriate OAM modes in the reconstruction of holographic images. For this purpose, the Fourier transform of the designed hologram, i.e., the target image, should be sampled by a two-dimensional Dirac comb array, where the sampling period is determined by the Fourier transform of the OAM mode [12]. Due to the discrete sampling, the reconstruction resolution and the information capacity are limited. Although some multiplexation methods such as the modulated OAM holography [13] have been proposed, the performance needs to be improved significantly further. In addition, a complete OAM mode is applied in both image encoding and decoding, which does not take full utilization of the angular space of a helical phase mode.
For this reason, we enhance the utilization of the angular space by dividing an OAM mode into several partial OAM modes, under the condition that the Fourier transform of a partial OAM mode is structurally stable and controllable. The rest of the paper is organized as follows. The principle of the partial OAM holography is introduced in Section 2. The experimental implementation is presented in Section 3. The reconstruction characterization is discussed in Section 4, and the paper is summarized in Section 5.
2. Principle of partial OAM-multiplexation
2.1 Spatial frequency distribution of partial helical phase modes
The reconstructed image can be expressed as a convolution between the sampled holographic image and the Fourier transform of the OAM mode. As a result, the spot associated with an OAM mode can be regarded as a kernel function in convolution. In order to avoid spatial overlapping between adjacent kernel functions, the Fourier transform of the designed hologram, i.e., the target image, should be sampled by a two-dimensional Dirac comb array, where the sampling period is determined by the kernel size [12]. The spatial frequency distribution of a complete OAM mode is doughnut-shaped, which can be expressed as [14]
A partial helical phase mode within a limited angular range -φ0/2<φ≤φ0/2 is defined, which can be expressed as the product of a helical phase function and an angular rectangle function Rect(φ/φ0)
Then Rect(φ/φ0) can be decomposed into a Fourier series. As a result, a partial helical phase mode can be expressed as [14,15]
As a result, each term in the Fourier series is rotated by ± π/2 with respect to the angular rectangle function, resulting in a rotated partial vortex pattern. The rotating direction depends on the sign of the helical phase index. A comparison of the spatial frequency distribution of a complete helical phase mode and that of a partial helical phase mode is shown in Fig. 1(a). It can be seen the latter has a partial doughnut intensity distribution that is counter-clockwisely rotated by π/2. The evolution of the partial OAM beam in free space is very complex. However, the far-field pattern of the partial OAM mode is structurally stable and controllable, which provides the feasibility of manipulating partial OAM modes.
Fig. 1. Principle of partial OAM holography. (a) The spatial frequency distributions of a complete helical phase mode and a partial helical phase mode. (b) The relationship between the helical phase index and the sampling period. (c) The design procedure of a partial OAM hologram. (d)-(e) Reconstruction results of partial OAM holograms with φ0=π/2 and φ0=π, respectively
2.2 Preservation of partial OAM mode
In digital holography, the complex amplitude on the image plane can be regarded as the superposition of an infinite number of plane-wave spatial frequency components according to the Fourier integral theorem, which can be expressed as [7,12]
Notably, the complete and the partial kernel functions have the same radius [Fig. 1(a)], but with different angular ranges. As a result, the partial OAM holography can have the same sampling period as the conventional OAM holography in the spatial frequency domain. Here, the sampling constant d is defined as the diameter of the main ring of the doughnut-shaped intensity distribution, which is calculated by Eq. (1) with the diameter of the OAM beam assigned as 8.64 µm. It is approximately proportional to the helical phase index l, d = ks|l|+d0 µm, with ks = 40/3 µm and d0 = 40 µm [Fig. 1(b)]. It should be noted that the diameter d will decrease gradually as the hologram size increases [9].
To design a partial OAM hologram, the target image is sampled first [Fig. 1(c)]. Via the Gerchberg-Saxton algorithm [16], a phase-only OAM-hologram with a discrete spatial-frequency distribution is obtained [Fig. 1(c)]. Therefore, the partial OAM hologram is generated by superimposing a part of the designed hologram onto the partial helical phase mode [right part in Fig. 1(c)]. Notably, a circular hologram is preferred, otherwise the resulting spots will have less regular intensity distributions. Then crosstalk between adjacent spots will occur, which severely affects the SNR of partial OAM multiplexing. Figures 1(d-e) show the numerically reconstructed image from the partial OAM hologram with φ0=π/2 and φ0=π respectively. It is seen that each spot of the reconstructed image shows a partial doughnut-shaped intensity distribution with a counter-clockwise rotation of π/2. Henceforth, the partial OAM can be preserved in each spot of the reconstructed images.
2.3 Selectivity of partial OAM mode
A hologram is generated by superimposing multiple partial OAM modes $H = \textrm{Rect}({{\varphi / {{\varphi_0}}}} )\sum\nolimits_{n = 1}^N {{H_n}{e^{i{l_n}\varphi }}}$, where N presents the number of image channels, ln is the n-th encoding helical index, and Hn is the n-th image channel. When the hologram is illuminated with a partial OAM beam $\textrm{Rect}({{\varphi / {{\varphi_0}}}} ){e^{i{l_{de}}\varphi }}$, the reconstructed image can be generated as ${E_{rec}} = F\left[ {\textrm{Rect}({{\varphi / {{\varphi_0}}}} )\sum\nolimits_{n = 1}^N {{H_n}{e^{i({{l_n} + {l_{de}}} )\varphi }}} } \right]$, which can be separated into two terms
Interestingly, only the mode lm=-lde is converted into the quasi-Gaussian spots F[Rect(φ/φ0)Hm], which corresponds to the desired spot-pattern image. While in the other channels in the latter term, each point is converted into an arc-shaped intensity distribution, which can equivalently be regarded as a partial OAM mode with a helical phase index ln+lde. As the peak intensity of these modes are much lower than that of the lm mode, these modes can be regarded as background noise and ignored. In another word, only the mode -lde is selected from multiple image channels by the decoding beam lde. As an example, a hologram composed of two spot-patterned target images is provided, where the two images are separately modulated by two partial helical phase modes l1 = 15 and l2=-20, respectively, as depicted in Fig. 2(a1). When the hologram is directly illuminated by a planar beam, the two spot-images are converted into a partial doughnut-like intensity distribution combining the two partial OAM patterns [Fig. 2(a2)]. As a contrary, when the hologram is decoded with a partial phase mode lde=-15, the reconstructed image can be converted into the solid intensity spot shown in [Figs. 2(a3)-(a4)]. As a result, the mode selectivity of l1 = 15 can be revealed. It is worth noting that the reconstructed spot is of an elongated quasi-Gaussian distribution, which is converted from the fan-shaped planar beam. The spot can be expressed as
Fig. 2. Partial OAM mode selectivity. (a1) Hologram of two spot-images encoded using two partial helical phase modes with l1 = 15 and l2=-20. (a2) Reconstructed image by a planar wave. (a3)-(a4) Image decoded by the partial helical phase mode lde=-15.(b1)-(b4) represent the case for l1 = 15 and l2 = 20, decoded with lde=-15. (c1)-(c4) are the case for the complete helical phase mode lde=-15. (d) Peak intensity of the converted fundamental mode as a function of angular range
As a result, the elongated solid spot is affected by the fan-shaped angle and fan-shaped azimuth of the partial OAM hologram, as shown in Fig. 2(d). Although the spot is no longer a fundamental Gaussian mode, the image resolution does not change.
Compared to the fundamental Gaussian mode converted from the complete OAM hologram [Fig. 2(c)], the peak intensity of the converted partial OAM mode decreases slightly. Meanwhile, the spot size increases [Figs. 2(a4), 2(b4) and 2(c4)]. This phenomenon is explained as follows. The intensity distribution of each spot is converted from the illuminated fan-shaped part of the hologram, then a greater angular range has a larger fan-shaped area. Therefore, as the angular range φ0 increases, the converted partial OAM becomes more concentrated and the peak intensity can increase gradually [Fig. 2(d)]. In fact, the peak intensity can be numerically fitted with a parabola Imax= 4367φ02.
2.4 Multiplexation in the angular space
As we know, a complete circular helical phase mode has an angular range of 2π, which can be divided into K angular fans, and the k-th division can be defined as $\textrm{Rec}{\textrm{t}_k} = \textrm{Rect}[{\varphi /{\varphi_0} - k} ]$, with 0 ≤ k ≤ K-1. Every division can be encoded separately to generate a partial OAM hologram $H_k^{mul}\textrm{ = Rec}{\textrm{t}_k}\sum\nolimits_{n = 1}^N {{H_{k,n}}{e^{i{l_n}\varphi }}}$. These K partial OAM holograms can be combined into an OAM hologram, which can be expressed as
Therefore, if an image channel of the k-th partial OAM hologram needs to be decoded, the k-th fan-shaped division should be illuminated by the inverse helical phase mode. Only when the illuminated fan-shaped zone and inverse partial modes match, the desired image can be obtained. Consequently, the superimposed images can be reconstructed individually.
3. Experimental demonstration of partial OAM-multiplexation
To demonstrate the partial OAM multiplexation, an experimental system is built as shown in Fig. 3(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 a half-wave plate to control its power and polarization. After passing through a beam splitter and a partial block, the beam 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. To reconstruct the image on a given plane, a virtual Fourier transform lens with a focal length of f = 224.5 mm was added to the phase hologram [Fig. 3(b)]. The transmittance function of a FT lens can be expressed as
where f is the focal length, which determines the reconstruct length, kx and ky are transversal coordinates on the hologram plane. In addition, we equivalently superimpose two decoding partial phase modes on the hologram and illuminate an angular region of it with a planar beam [Fig. 3(a)]. The partial OAM can present a half doughnut-shaped intensity distribution on each spot of the reconstructed image.Fig. 3. The experimental demonstration. (a) Optical setup. P: linear polarizer; L: lens; BS: beam splitter; HWP: half-wave plate. (b) Encoding an FT lens to a hologram. (c) A hologram with φ0=π. (d) Result of partial OAM preservation.
As an example, three fan-shaped holograms with a limited angular range φ0 = 2π/3 are designed [Figs. 4(a-b)], the three fan-shaped zones are Rect1, Rect2, and Rect3, which can form a complete circular hologram [Fig. 4(c)]. Each partial OAM hologram has two image channels associated with two target images, which are separately encoded with the helical phase modes l1 = 15 and l2=-15. Six target images, namely letters “A-F” with 1080×1080 pixels are adopted. The partial OAM-multiplexation hologram in Rect1 is encoded with letters “A” and “B”, the partial OAM-multiplexation hologram in Rect2 is encoded with letters “C” and “D”, and the partial OAM-multiplexation hologram in Rect3 is encoded with letters “E” and “F”. The sampling period of the target image is 240 µm, corresponding to 240/p = 30 pixels, and p =8 µm is the pixel size of the SLM.
Fig. 4. The experimental implementation of the partial OAM multiplexation. (a)-(b) Design of partial OAM holograms with an angular range φ0 = 2/3π. (c) The superimposed hologram. (d) Reconstruction image by a planar wave. (e) Decoded results.
We first illuminate the whole hologram using a planar beam [Fig. 4(c)]. Six image channels will interfere with each other, leading to a complex image pattern in the CCD [Fig. 4(d)]. When the hologram is decoded separately with different inverse partial OAM modes and illuminated at proper regions, six distinct target images can be individually reconstructed in high quality, as shown in Fig. 4(e). Notably, due to the complex evolution of partial OAM beams in free space, a partial OAM beam is not directly utilized to decode the partial OAM-multiplexation hologram. Instead, we directly impose the decoding partial helical phase on the hologram and then illuminate the hologram with fan-shaped planar beams. Compared to the traditional reconstructed OAM hologram, each spot in the reconstructed image will not show the fundamental Gauss intensity distribution [Fig. 3(a)]. Instead, an elliptical intensity distribution is caused by the fan-shaped hologram. The specific illuminated region is controlled by rotating the fan-shaped partial block with 2π/3 rad [Fig. 3(a)]. For example, when decoding the letter “A”, we need to first rotate the partial block to the zone of the partial OAM-multiplexation hologram, i.e., the fan-shaped zone Rect1 and then select the helical phase mode lde=-15 to superimpose onto this hologram for decoding.
4. Reconstruction characterization discussion
According to the Eq. (5), the SNR of the reconstructed images from partial OAM-multiplexation is influenced by the limited angular range Rectk and the encoded helical phase index difference △l = ln - ln-1. The difference △l is not dependent on the index n, which is identical for all the helical phase indices. Because a larger angular range can enhance the use of the hologram, the peak intensity of each spot in the reconstructed image can change significantly. Meantime, the noise caused by other channels does not change. As a result, the reconstruction quality can be improved by increasing the angle range of the fan-shaped partial OAM hologram [ Fig. 5(b)]. In addition, by increasing the helical phase index difference △l, the crosstalk caused by other channels can be attenuated. Then the SNR can be improved [Fig. 5(a)].
Fig. 5. Analysis of partial OAM-multiplexation. (a) SNR as a function of helical phase index difference. (b) SNR as a function of angular range. (c) Relationship between multiplexation channel number and pixel size of the hologram. (d) Image resolution as a function of the pixel size of the hologram.
In terms of the number of image channels, more angular divisions are preferred, which means that a complete hologram can be divided into any number of partial holograms. However, in practice, the multiplexation number is limited by the resolution of the hologram. In our experiment, the pixel size p is a limiting factor. The maximal diffraction angle of a hologram can be expressed as βmax = tan-1(λ/2p) [9]. The diameter of the reconstructed image can be calculated as 2f tan(βmax). For example, if sampling a = 40 points along each row in the target image, the multiplexation number can be calculated as 2×[f tan(βmax) /a-d0]/(△l×ks). The relationship between the multiplexation number and the hologram resolution is shown in Fig. 5(c). Therefore, the multiplexation number of a partial OAM hologram becomes greater by increasing the hologram resolution. In addition, the relationship of the reconstructed resolution and the helical phase index is provided, which is approximately 1/d [Fig. 5(d)]. As a result, the reconstruction resolution can be improved by decreasing the encoded helical phase index.
5. Summary
In summary, by utilizing a new degree of freedom in the angular space, the partial orbital angular momentum holography is proposed and experimentally demonstrated. By dividing a helical OAM phase mode into multiple partial phase modes along the azimuthal direction, multiplexation of partial OAM modes can be realized. As a result, the partial OAM holography will enhance the information capacity and provide diverse methods for high-security optical encryption. The partial OAM holography will find widespread applications in 3D displays, data storage [17], data encryption [18], optical manipulation and so on.
Funding
National Natural Science Foundation of China (51875107); Shanghai Academy of Spaceflight Technology (2019-086).
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.
References
1. A. E. Willner, K. Pang, H. Song, K. Zou, and H. Zhou, “Orbital angular momentum of light for communications,” Appl. Phys. Rev. 8(4), 041312 (2021). [CrossRef]
2. D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003). [CrossRef]
3. K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics 5(6), 335–342 (2011). [CrossRef]
4. K. T. Gahagan and G. A. Swartzlander, “Optical vortex trapping of particles,” Opt. Lett. 21(11), 827–829 (1996). [CrossRef]
5. M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011). [CrossRef]
6. J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012). [CrossRef]
7. H. Ren, G. Briere, X. Fang, P. Ni, R. Sawant, S. Héron, S. Chenot, S. Vézian, B. Damilano, V. Brändli, S. A. Maier, and P. Genevet, “Metasurface orbital angular momentum holography,” Nat. Commun. 10(1), 1–8 (2019). [CrossRef]
8. H. Zhou, B. Sain, Y. Wang, C. Schlickriede, R. Zhao, X. Zhang, Q. Wei, X. Li, L. Huang, and T. Zentgraf, “Polarization-Encrypted Orbital Angular Momentum Multiplexed Metasurface Holography,” ACS Nano 14(5), 5553–5559 (2020). [CrossRef]
9. H. Ren, X. Fang, J. Jang, J. Bürger, J. Rho, and S. A. Maier, “Complex-amplitude metasurface-based orbital angular momentum holography in momentum space,” Nat. Nanotechnol. 15(11), 948–955 (2020). [CrossRef]
10. X. Fang, H. Yang, W. Yao, T. Wang, Y. Zhang, M. Gu, and M. Xiao, “High-dimensional orbital angular momentum multiplexing nonlinear holography,” Adv. Photonics 3(1), 1–7 (2021). [CrossRef]
11. X. Fang, H. Wang, H. Yang, Z. Ye, Y. Wang, Y. Zhang, X. Hu, S. Zhu, and M. Xiao, “Multichannel nonlinear holography in a two-dimensional nonlinear photonic crystal,” Phys. Rev. A 102(4), 043506 (2020). [CrossRef]
12. X. Fang, H. Ren, and M. Gu, “Orbital angular momentum holography for high-security encryption,” Nat. Photonics 14(2), 102–108 (2020). [CrossRef]
13. F. Wang, X. Zhang, H. Yuan, R. Xiong, and X. Jiang, “Enhancing the information capacity with modulated orbital angular momentum holography,” IEEE Photonics J. 14(1), 5212905 (2022). [CrossRef]
14. D. M. Cottrell, J. A. Davis, and T. J. Hernandez, “Fraunhofer diffraction of a partially blocked spiral phase plate,” Opt. Express 19(14), 12873–12878 (2011). [CrossRef]
15. S. Qiu, Y. Ren, T. Liu, Z. Liu, C. Wang, Y. Ding, Q. Sha, and H. Wu, “Directly observing the skew angle of a Poynting vector in an OAM carrying beam via angular diffraction,” Opt. Lett. 46(14), 3484–3487 (2021). [CrossRef]
16. R. W. Gerchberg and W. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg). 35, 237–250 (1971).
17. Q. Song, A. Baroni, P. C. Wu, S. Chenot, V. Brandli, S. Vézian, B. Damilano, P. de Mierry, S. Khadir, P. Ferrand, and P. Genevet, “Broadband decoupling of intensity and polarization with vectorial Fourier metasurfaces,” Nat. Commun. 12(1), 1–9 (2021). [CrossRef]
18. Q. Song, M. Odeh, J. Zúñiga-Pérez, B. Kanté, and P. Genevet, “Plasmonic topological metasurface by encircling an exceptional point,” Science 373(6559), 1133–1137 (2021). [CrossRef]
