Holographic Encryption Applications Using Composite Orbital Angular Momentum Beams
Holographic Encryption Applications Using Composite Orbital Angular Momentum Beams
Zhang, Nian;Xiong, Baoxing;Zhang, Xiang;Yuan, Xiao
2022-08-26 00:00:00
hv photonics Communication Holographic Encryption Applications Using Composite Orbital Angular Momentum Beams 1 , 2 , 3 , 4 1 , 2 , 3 , 4 1 , 2 , 3 , 4 1 , 2 , 3 , 4 , Nian Zhang , Baoxing Xiong , Xiang Zhang and Xiao Yuan * School of Optoelectronic Science and Engineering, Soochow University, Suzhou 215006, China Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China Key Lab of Advanced Optical Manufacturing Technologies of Jiangsu Province, Suzhou 215006, China Key Lab of Modern Optical Technologies of Education Ministry of China, Suzhou 215006, China * Correspondence: xyuan@suda.edu.cn Abstract: Optical orbital angular momentum (OAM) holography has been developed and imple- mented as a vital method for optical encryption. However, OAM holography can only be encoded and decoded with an OAM beam, which limits the level of optical encryption. Here, composite OAM beams are introduced using a computer-generated hologram (CGH) for holographic encryption. The target image is encoded with composite helical mode indices, and the OAM holographic image can only be reconstructed under a specific illuminating composite OAM beam. The experimental results are consistent with the theoretical design and numerical simulations, verifying that composite OAM beams can provide a higher security level for optical holographic encryption. The proposed method can be used to enhance anti-counterfeiting applications, secure communication systems, and imaging systems. Keywords: orbital angular momentum holography; composite vortex beams; holographic encryption Citation: Zhang, N.; Xiong, B.; 1. Introduction Zhang, X.; Yuan, X. Holographic Optical holography technology provides a vital strategy for reconstructing both the in- Encryption Applications Using tensity and phase information of an object with computer-generated holograms (CGH), and Composite Orbital Angular has been widely used for optical tweezers [1,2], optical encryption [3], three-dimensional Momentum Beams. Photonics 2022, 9, 605. https://doi.org/10.3390/ displays [4,5], and data storage [6]. For conventional holographic systems, the different photonics9090605 physical properties of light, including wavelength, polarization, and incidence angle, have been explored and have been proven to carry independent information channels for op- Received: 13 July 2022 tical holographic encryption [7–10]. Moreover, orbital angular momentum (OAM), as an Accepted: 22 August 2022 essential physical dimension of light, has attracted considerable attention [11]. In general, Published: 26 August 2022 OAM is characterized by a helical phase factor, exp(ilq), where l represents the topological Publisher’s Note: MDPI stays neutral charge number and q indicates the azimuthal angle of a helical wave-front. OAM states with regard to jurisdictional claims in can enhance information capacity due to its physically unlimited orthogonal helical modes published maps and institutional affil- and can be widely applied in optical communications [2,12], meta-surfaces [13], and so on. iations. Recently, OAM beams have been experimentally implemented for holographic en- cryption, including the field of linear optics and nonlinear optics [14,15]. Ruffato et al. made the first attempt to encode and decode information by designing CGH with OAM beams and phase singularities [16]. After that, OAM multiplexing holography for optical Copyright: © 2022 by the authors. encryption was explored [17–20]. Unlike the work of Ruffato et al., OAM multiplexing Licensee MDPI, Basel, Switzerland. holography preserves the doughnut intensity distribution of the OAM mode throughout This article is an open access article the spatial-frequency transform by discrete sampling. However, current methods only en- distributed under the terms and code a single OAM phase mode into the information, which limits the level of holographic conditions of the Creative Commons encryption. Optical vortices with a superimposed helical phase have been explored in the Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ past few years. Composite OAM beams are expected to enhance encryption capabilities 4.0/). Photonics 2022, 9, 605. https://doi.org/10.3390/photonics9090605 https://www.mdpi.com/journal/photonics Photonics 2022, 9, x FOR PEER REVIEW 2 of 11 encode a single OAM phase mode into the information, which limits the level of holo- graphic encryption. Optical vortices with a superimposed helical phase have been ex- plored in the past few years. Composite OAM beams are expected to enhance encryp- Photonics 2022, 9, 605 2 of 11 tion capabilities by superimposing different OAM modes. To the best of our knowledge, holography with different superimposed OAM modes has not yet been explored or im- plemented. by superimposing different OAM modes. To the best of our knowledge, holography with In this study, we conducted an experiment utilizing composite OAM beams for different superimposed OAM modes has not yet been explored or implemented. holographic encryption. In the experiment, we obtained good results without In this study, we conducted an experiment utilizing composite OAM beams for holo- post-processing of the reconstructed images. Holograms of target images with compo- graphic encryption. In the experiment, we obtained good results without post-processing site OAM beams were obtained using the adaptive weighted Gerchberg-Saxton (WGS) of the reconstructed images. Holograms of target images with composite OAM beams were algorithm. Our results show that OAM states with composite superimposed vortex obtained using the adaptive weighted Gerchberg-Saxton (WGS) algorithm. Our results beams provide higher security compared with those of the previous holography sys- show that OAM states with composite superimposed vortex beams provide higher security tems. For a specific composite OAM beam, the holographic image can be reconstructed compared with those of the previous holography systems. For a specific composite OAM clearly; otherwise, it can become a blurred image. We verify the feasibility of using such beam, the holographic image can be reconstructed clearly; otherwise, it can become a composite OAM beams using optical experiments and numerical simulations. blurred image. We verify the feasibility of using such composite OAM beams using optical experiments and numerical simulations. 2. Principle and Methods 2. Principle and Methods A schematic diagram of composite OAM phase holography is shown in Figure 1. A schematic diagram of composite OAM phase holography is shown in Figure 1. An An incident beam with the expected OAM beam illuminates the CGH phase pattern, incident beam with the expected OAM beam illuminates the CGH phase pattern, and the and the holographic image can be reconstructed clearly. Conversely, if the CGH phase holographic image can be reconstructed clearly. Conversely, if the CGH phase pattern is pattern is illuminated with the wrong beam, the noise becomes too high and blurs the illuminated with the wrong beam, the noise becomes too high and blurs the image. image. Figure 1. Schematic diagram of the composite OAM phase holography. Decoding of the CGH with Figure 1. Schematic diagram of the composite OAM phase holography. Decoding of the CGH with the correct the correct com composite posite OAM OAM be beam: am: th the e enco encoding ding image appears in the image appears in the far-field (the far-field (the logo logoof of Soocho Soochow w University). University). 2.1. Composite-State Laguerre-Gaussian Beams 2.1. Composite-State Laguerre-Gaussian Beams The complex amplitude of the Laguerre-Gaussian (LG) beam is written as [21]: The complex amplitude of the Laguerre-Gaussian (LG) beam is written as [21]: q h i h i h i jlj 2 2 2 2p! 22 jlj 2 w r 2 0 2r r ikr z 2! p w rr 22 r ikrz E(r, q, z) = L exp exp |l| 0 p 2 2 2 2 p p+jlj ! w(z) w(z) ( ) w (z) w (z) 2(z +z ) Er (,θ,zL ) = exp - exp - p R 22 2 2 (1) h i wz () wz () π pl+! wz () wz ( ) 2(z +z ) () R z exp i(2p +jlj + 1)tan exp( ilq) (1) -1 ×exp ip (2 + l + 1)ta q n exp(-ilθ) z 2 R where w(z) = w 1 + (z/z ) is waist radius at the distance z, w is the radius of the 0 R 0 LG beam at the distance z = 0, z = pw /l is the Rayleigh length, l is the wavelength, and k = 2p/l is the wavenumber. L () represents the generalized Laguerre polynomial, whereas p and l represent the radial and angular mode numbers, respectively. The LG beam distribution possesses (p + 1) concentric rings. If set p = 0, each mode is a single ring. Photonics 2022, 9, x FOR PEER REVIEW 3 of 11 where wz () = w 1+(z / z ) is waist radius at the distance z, w0 is the radius of the LG 0 R beam at the distance z = 0, zR = πw /λ is the Rayleigh length, λ is the wavelength, and k = 2π/λ is the wavenumber. L ⋅ represents the generalized Laguerre polynomial, where- () Photonics 2022, 9, 605 3 of 11 as p and l represent the radial and angular mode numbers, respectively. The LG beam distribution possesses (p + 1) concentric rings. If set p = 0, each mode is a single ring. Here, we discuss the situation where LG beams with a common waist radius at the Here, we discuss the situation where LG beams with a common waist radius at the source plane are coaxially superposed. The complex amplitude of the superposition of source plane are coaxially superposed. The complex amplitude of the superposition of two two LG beams of (p1, l1) and (p2, l2) can be written as: LG beams of (p , l ) and (p , l ) can be written as: 1 1 2 2 ll , l l 12 1 2 Er (,θ,zE ) = (r,θ,zE )+ (,r θ, z) (2) pp , p p 12 1 2 l ,l l l 1 2 1 2 E (r, q, z) = E (r, q, z) + E (r, q, z) (2) p ,p p p 1 2 1 2 Next, we focus on the generation of a high-quality composite-state LG beam through a Next, pha we s focus e-only spa on thetia generation l light modul of a a high-quality tor (SLM). The vi composite-state tal step is to genera LG beam te the de- through a sired phase-only hologra spatial m, which c light modulator an be obta(SLM). ined by [ The 22vital ]: step is to generate the desired hologram, which can be obtained by [22]: ll, Φ=+ FA cos rgE 2πfx () (3) SLM amp pp, x 12 h i l ,l 1 2 F = F cos Arg E + 2p f x (3) SLM amp p ,p x ll, 1 where Famp = Abs( E ), and “Abs” and “Arg” are the amplitude and phase operations of p , p l ,l the electric field of compo 1 2 site-state LG beams in Equation (2), respectively. A blazed where F = Abs(E ), and “Abs” and “Arg” are the amplitude and phase operations of amp p ,p 1 2 grating with phase shift 2πfxx (fx is the grating frequency) is used to control the diffrac- the electric field of composite-state LG beams in Equation (2), respectively. A blazed grating tion angle of the light reflected from the phase-only SLM in the x direction. Here, the with phase shift 2pf x (f is the grating frequency) is used to control the diffraction angle of x x relevant holograms of the composite OAM beams are plotted in Figure 2(a1–a5). Evi- the light reflected from the phase-only SLM in the x direction. Here, the relevant holograms dently, the holograms contain both amplitude and phase modulation, and the spatial of the composite OAM beams are plotted in Figure 2(a1–a5). Evidently, the holograms phase-delay and grating depth determine the wave-front and intensity distribution of contain both amplitude and phase modulation, and the spatial phase-delay and grating the incident light, respectively. Although most SLMs are phase-only devices, there are depth determine the wave-front and intensity distribution of the incident light, respectively. still several Although most techniques SLMs are t phase o convert -only a devices, phase-only there ardevice e still several response into a f techniques to ull convert amplia - tude-phase r phase-only device esponse response [23,24] into . There a fullfor amplitude-phase e, in this experiment, the response [23 phase ,24]. Ther -only efor SLM w e, in this as cal experiment, ibrated intthe o a line phase-only ar 2π phase SLM respon was calibrated se over all into 256a gr linear ay leve 2 l, phase and inr t esponse his case,over the rall e- 256 gray level, and in this case, the residual amplitude modulation was negligible [25,26]. sidual amplitude modulation was negligible [25,26]. Figure 2. (a1–a5) The corresponding CGH phase patterns of composite OAM beams. (b1–b5) Sim- Figure 2. (a1–a5) The corresponding CGH phase patterns of composite OAM beams. (b1–b5) ulated and (c1–c5) experimental intensity distributions of composite OAM beams. Simulated and (c1–c5) experimental intensity distributions of composite OAM beams. The simulated and experimental intensity distributions of the composite OAM beams The simulated and experimental intensity distributions of the composite OAM b ar ee am shown s are sh inown in Figure Fi 2gur (b1–b e 2 5,c1–c5), (b1–b5),(r cespectively 1–c5), respect . Note ively.that Notin e th the at in case the c of apse of = p p1 and = p2 1 2 l = l , we can observe the composite vortex beam composed of an optical bright-ring and l1 = −l2, we can observe the composite vortex beam composed of an optical 1 2 lattice containing |l l | bright petals on each ring, and the helical phases disappear 1 2 (OAM = 0), owing to the destructive interference of two LG beams with contrary topological charges. However, there are shear phase singularities, where the phase abruptly switches by p [27,28]. In Figure 2(b3,b5,c3,c5), optical dark-ring lattices are observed where the phase singularities exist. The composite OAM beam has a dark vortex core with a topological Photonics 2022, 9, 605 4 of 11 charge of 1 at the beam center, accompanied by three |l -l | vortex singularities in the 1 2 periphery. The experimental results (Figure 2(c1–c5)) appear to be consistent with the numerical simulations (Figure 2(b1–b5)), and the slight differences are likely due to beam rotation while propagating. The beam rotation phenomenon can be explained from the perspective of the Gouy phase F = (2p + |l| + 1)arctg(z/z ). As p = p and l = l , there is no rotation R 1 2 1 2 because both components of the beam have the same Gouy phase, which is also the case when the composite OAM beam exhibits no vortices [27,28]. As generation of the CGH phase pattern is influenced by the incident light field, we consider the propagation effect of the beam in the following section. 2.2. Design of the Composite-State LG Beam Hologram An iterative Fourier transform algorithm is the best proper approach when designing an optimized CGH based on the forward and backward propagation of the light between the hologram plane and image plane [29,30]. We used a typical iterative Fourier transform algorithm (i.e., adaptive WGS algorithm) to suppress the speckle noise, reduce the appear- ance of artifacts, and improve the diffraction efficiency of the holographic image, based on Ref. [31]. Three different processed images with different levels of complexity were chosen, as shown in Figure 3a–c. The first image shows the letter “A”, characterized by a grayscale channel (Figure 3a). The second image is the logo of Soochow University with pure black and white pixels (Figure 3b), and the third is a grayscale image of a tiger with finer details (Figure 3c). Figure 3d exhibits the flow chart of the adaptive WGS algorithm, with ' and F representing phase distributions of the Fourier plane and the hologram plane in the ith iteration, respectively. The initial input field distribution U is provided by the OAM composite OAM beams. The initial phase value to start the iteration is [31]: j(x, y) = exp[i2p(pDX f + qDY f )] (4) x y where (p, q), (DX, DY) and (f , f ) are the pixel coordinates, sampling intervals of the spatial x y domain, and frequency distribution in the x and y directions, respectively. During the iteration process, the amplitude constraint in the image plane is [31]: A W signal domain A = (5) con A noise domain where W = exp(A A ) is the weighting factor, and A and A are the target image t r t r amplitude and reconstructed image amplitude, respectively. With the progression of iterations, the normalized root mean squared error (RMSE) and peak signal-to-noise ratio (PSNR) are used to check the convergence of the algorithm. The RMSE and PSNR are defined as [31]: M N u 2 u å å A A p=1 q=1 RMSE = (6) M N å å A p=1 q=1 8 9 > > > > > > < = (2 1) MN PSNR = 10log (7) M N > > > 2> > > å å (A A ) : t r ; p=1 q=1 where M and N are the pixel numbers of the target image in x and y directions, respectively, and b is the bit depth of images. Photonics 2022, 9, x FOR PEER REVIEW 5 of 11 b 2 () 21 − MN PSNR = 10log (7) 10 MN AA − () tr pq == 11 Photonics 2022, 9, 605 5 of 11 where M and N are the pixel numbers of the target image in x and y directions, respec- tively, and b is the bit depth of images. Figure 3. (a) Grayscale image of the letter “A”; (b) logo of Soochow University in pure white and Figure 3. (a) Grayscale image of the letter “A”; (b) logo of Soochow University in pure white and black black pixels, which includes the name and motto of the University in English and Chinese; (c) pixels, which includes the name and motto of the University in English and Chinese; (c) grayscale grayscale image of a tiger with finer details; (d) flow chart of WGS algorithm. image of a tiger with finer details; (d) flow chart of WGS algorithm. The simulation was performed to verify the feasibility of composite OAM phase The simulation was performed to verify the feasibility of composite OAM phase holog- holography, as shown in Figure 4. The intensity and phase distribution of the input raphy, as shown in Figure 4. The intensity and phase distribution of the input composite composite OAM beam are shown in Figure 4a,b, respectively. The logo of Soochow OAM beam are shown in Figure 4a,b, respectively. The logo of Soochow University was University was used as a test image, partitioned into a signal and noise domain, and its used as a test image, partitioned into a signal and noise domain, and its size was 400 400, size was 400 × 400, as shown in Figure 4c. The CGH of the logo is shown in Figure 4d. as shown in Figure 4c. The CGH of the logo is shown in Figure 4d. The wavelength of The wa the laser vel and ength of the samplingla intervals ser and sa ofmp thelihologram ng interval wer s ofe the hol 632.8 nm ogra and m were 12.5 63 m2.8 nm 12.5 a nm, d 12.5 μm × 12.5 μm, respectively. The focal length of the Fourier lens was 100 mm. As respectively. The focal length of the Fourier lens was 100 mm. As shown in Figure 4e,f, shown the simulation in Figure results 4e,f, the showed simulation the evolution results of showed the evolution of the RMSE the RMSE and PSNR during WGS algo- and PSNR during WGS rithm convergence algorit to an optimized hm convergence to design ofan the op CGH timized using desaigcomposite n of the CG OAM H usin beam g a Photonics 2022, 9, x FOR PEER REVIEW 6 of 11 with (p , l ) = (1, 1) and (p , l ) = (1, 3), indicating that encoded information in CGH with composite OAM beam with (p1, l1) = (1, 1) and (p2, l2) = (1, 3), indicating that encoded in- 1 1 2 2 composite OAM beams can be an effective method for optical encryption. formation in CGH with composite OAM beams can be an effective method for optical encryption. Figure 4. Figure 4. ((a a) Th ) The e intensity and intensity and((b b) phase ) phase di distribution stribution of the n of the numerically umerically sim simulated ulated com composite posite OA OAM M beam; (c) target image containing signal and noise domain; (d) CGH of the target image (c); (e) beam; (c) target image containing signal and noise domain; (d) CGH of the target image (c); (e) RMSE; RMSE; and (f) PSNR. and (f) PSNR. 3. Experimental Results The experimental setup of the composite OAM phase holography is shown in Fig- ure 5. A He-Ne laser (Lumentum-1107P-632.8 nm-0.8 mW) was used to generate a Gaussian beam. The beam was expanded and collimated by a beam expander. An aper- ture was used to adjust the size of the beam irradiating on SLM1 (Holo- eye-PLUTO-2-VIS-096, 1920 × 1080 pixels, pixel pitch of 8 μm). A polarizer was used to generate a P-polarized light. High quality composite OAM beams were generated by loading the corresponding CGH into SLM1. Then, the composite OAM beam was dif- fracted by SLM1 and irradiated onto SLM2 (Hamamatsu-X13138 series-07, 1272 × 1024 pixels, pixel pitch of 12.5 μm). Composite OAM phase holograms were generated by the WGS algorithm and loaded into SLM2. A CMOS (pco. edge-4.2 bi, 2048 × 2048 pixels, pixel pitch of 6.5 μm) was used to capture the images. To separate the zero point from the reconstructed image, a programmable zoom lens (PZL) (fPZL = 150 mm) was used, whose phase distribution is expressed as [32]: φ (, xy) =− (x + y ) (8) PZL λf PZL Photonics 2022, 9, x FOR PEER REVIEW 6 of 11 Figure 4. (a) The intensity and (b) phase distribution of the numerically simulated composite OAM beam; (c) target image containing signal and noise domain; (d) CGH of the target image (c); (e) Photonics 2022, 9, 605 6 of 11 RMSE; and (f) PSNR. 3. Experimental Results 3. Experimental Results The experimental setup of the composite OAM phase holography is shown in Fig- ure 5. A He-Ne laser (Lumentum-1107P-632.8 nm-0.8 mW) was used to generate a The experimental setup of the composite OAM phase holography is shown in Figure 5. Gaussian beam. The beam was expanded and collimated by a beam expander. An aper- A He-Ne laser (Lumentum-1107P-632.8 nm-0.8 mW) was used to generate a Gaussian beam. ture was used to adjust the size of the beam irradiating on SLM1 (Holo- The beam was expanded and collimated by a beam expander. An aperture was used to eye-PLUTO-2-VIS-096, 1920 × 1080 pixels, pixel pitch of 8 μm). A polarizer was used to adjust the size of the beam irradiating on SLM1 (Holoeye-PLUTO-2-VIS-096, 1920 1080 pixels, generatpixel e a P- pitch polar of ize 8d l m). ightA . H polarizer igh qualit was y composit used to e OA generate M bea a P-polarized ms were gen light. erateHigh d by quality loadingcomposite the correspondin OAM beams g CGH int wereogenerated SLM1. Then, by loading the composit the corr e OAM be esponding am CGH was into dif- SLM1. fracted by SLM1 a Then, the composite nd irradi OAM ated onto beamSLM2 (Ha was diffracted mamby atsu- SLM1 X131 and 38 seri irradiated es-07, 12 onto 72 × SLM2 1024 (Hamamatsu-X13138 pixels, pixel pitch of 12 series-07, .5 μm). Composite OAM phase hologr 1272 1024 pixels, pixel pitch am of s were 12.5 genera m). Composite ted by the OAM phase holograms were generated by the WGS algorithm and loaded into SLM2. A WGS algorithm and loaded into SLM2. A CMOS (pco. edge-4.2 bi, 2048 × 2048 pixels, CMOS (pco. edge-4.2 bi, 2048 2048 pixels, pixel pitch of 6.5 m) was used to capture the pixel pitch of 6.5 μm) was used to capture the images. To separate the zero point from images. To separate the zero point from the reconstructed image, a programmable zoom the reconstructed image, a programmable zoom lens (PZL) (fPZL = 150 mm) was used, lens (PZL) (f = 150 mm) was used, whose phase distribution is expressed as [32]: whose phase PZL distribution is expressed as [32]: 2 2 j (x, y) = (x + y ) (8) φ (, xy) =− (x + y ) PZL (8) PZL l f λPfZL PZL Figure 5. Schematic diagram of the experimental setup of composite OAM phase holography. BE: beam expander; Pin: pinhole aperture; P: linear polarizer; PBS: polarizing beam splitter; SLM: spatial light modulator. Four different composite OAM phase hologram patterns were computed: (p , l ) = (0, 3) 1 1 and (p , l ) = (0, 3) (Figure 6(a1–c1)), (p , l ) = (1, 1) and (p , l ) = (1, 3) (Figure 6(a2–c2)), 2 2 1 1 2 2 (p , l ) = (1, 1) and (p , l ) = (1, 3) (Figure 6(a3–c3)), and (p , l ) = (1, 3) and (p , l ) = (1, 3) 1 1 2 2 1 1 2 2 (Figure 6(a4–c4)). The simulated results for correct composite OAM beam illumination are shown in Figure 6a. Evidently, the reconstructed images will clearly appear for the correct beam illumination. The corresponding experimental results are shown in Figure 6b,c. For the correct beam illuminations, the experimental results (Figure 6b) were consistent with the numerical simulations (Figure 6a). However, when the holograms were irradiated by the wrong beam, the reconstructed images became blurred (Figure 6c). Photonics 2022, 9, x FOR PEER REVIEW 7 of 11 Figure 5. Schematic diagram of the experimental setup of composite OAM phase holography. BE: beam expander; Pin: pinhole aperture; P: linear polarizer; PBS: polarizing beam splitter; SLM: spa- tial light modulator. Four different composite OAM phase hologram patterns were computed: (p1, l1) = (0, 3) and (p2, l2) = (0, −3) (Figure 6(a1–c1)), (p1, l1) = (1, 1) and (p2, l2) = (1, 3) (Figure 6(a2– c2)), (p1, l1) = (1, 1) and (p2, l2) = (1, −3) (Figure 6(a3–c3)), and (p1, l1) = (1, 3) and (p2, l2) = (1, −3) (Figure 6(a4–c4)). The simulated results for correct composite OAM beam illumina- tion are shown in Figure 6a. Evidently, the reconstructed images will clearly appear for the correct beam illumination. The corresponding experimental results are shown in Figure 6b,c. For the correct beam illuminations, the experimental results (Figure 6b) were consistent with the numerical simulations (Figure 6a). However, when the holo- Photonics 2022, 9, 605 7 of 11 grams were irradiated by the wrong beam, the reconstructed images became blurred (Figure 6c). Figure 6. (a) Simulated results for correct illumination with (a1) (p1, l1) = (0, 3) and (p2, l2) = (0, −3), Figure 6. (a) Simulated results for correct illumination with (a1) (p , l ) = (0, 3) and (p , l ) = (0, 3), 1 1 2 2 (a2) (p1, l1) = (1, 1) and (p2, l2) = (1, 3), (a3) (p1, l1) = (1, 1) and (p2, l2) = (1, −3), (a4) (p1, l1) = (1, 3) and (a2) (p , l ) = (1, 1) and (p , l ) = (1, 3), (a3) (p , l ) = (1, 1) and (p , l ) = (1, 3), (a4) (p , l ) = (1, 3) 1 1 2 2 1 1 2 2 1 1 (p2, l2) = (1, −3). (b) Experimental results for the composite OAM beam with (b1) (p1, l1) = (0, 3) and and (p , l ) = (1, 3). (b) Experimental results for the composite OAM beam with (b1) (p , l ) = (0, 3) 2 2 1 1 (p2, l2) = (0, −3), (b2) (p1, l1) = (1, 1) and (p2, l2) = (1, 3), (b3) (p1, l1) = (1, 1) and (p2, l2) = (1, −3), (b4) (p1, and (p , l ) = (0, 3), (b2) (p , l ) = (1, 1) and (p , l ) = (1, 3), (b3) (p , l ) = (1, 1) and (p , l ) = (1, 3), 2 2 1 1 2 2 1 1 2 2 l1) = (1, 3) and (p2, l2) = (1, −3). (c1–c4) Experimental results with wrong Gaussian beam illumination (b4) (p , l ) = (1, 3) and (p , l ) = (1, 3). (c1–c4) Experimental results with wrong Gaussian beam 1 1 2 2 of (p, l) = (0, 0). illumination of (p, l) = (0, 0). Then, the tiger image with finer details was encoded into the CGH phase pattern Then, the tiger image with finer details was encoded into the CGH phase pattern and computed for illumination with a composite OAM beam with (p1, l1) = (1, 1) and (p2, and computed for illumination with a composite OAM beam with (p , l ) = (1, 1) and 1 1 l2) = (1, 3). Several different composite OAM beams were used to test the optical re- (p , l ) = (1, 3). Several different composite OAM beams were used to test the optical 2 2 sponse of the input composite OAM beams different from the optimal one, as shown in response of the input composite OAM beams different from the optimal one, as shown Figure 7. It is evident that for the correct composite OAM beam illumination, the finer in Figure 7. It is evident that for the correct composite OAM beam illumination, the finer details of the tiger are clearly displayed (Figure 7a). Conversely, when the wrong com- details of the tiger are clearly displayed (Figure 7a). Conversely, when the wrong composite Photonics 2022, 9, x FOR PEER REVIEW 8 of 11 posite OAM beam illuminations are used, noise increases and the details of the image OAM beam illuminations are used, noise increases and the details of the image are no are no longer clearly visible (Figure 7b–d). longer clearly visible (Figure 7b–d). Figure 7. Experimental results of the CGH encoding grayscale image of a tiger. (a) The correct il- Figure 7. Experimental results of the CGH encoding grayscale image of a tiger. (a) The correct lumination of design was (p1, l1) = (1, 1) and (p2, l2) = (1, 3). The wrong illuminations used were (b) illumination of design was (p , l ) = (1, 1) and (p , l ) = (1, 3). The wrong illuminations used were 1 1 2 2 (p1, l1) = (0, 1) and (p2, l2) = (0, −1), (c) (p1, l1) = (0, 3) and (p2, l2) = (0, −3), and (d) (p1, l1) = (1, 1) and (p2, (b) (p , l ) = (0, 1) and (p , l ) = (0, 1), (c) (p , l ) = (0, 3) and (p , l ) = (0, 3), and (d) (p , l ) = (1, 1) 1 1 2 2 1 1 2 2 1 1 l2) = (1, −3). and (p , l ) = (1, 3). 2 2 As the calculated CGH phase pattern is illuminated by a specific intensity and phase distribution, the misalignment of the decoded beam with respect to the hologram position seems to affect the quality of the reconstructed image. To analyze the effects of displacement on the quality of the reconstructed image, the Soochow University logo was encoded into the CGH phase pattern and decoded using a composite OAM beam with (p1, l1) = (1, 1) and (p2, l2) = (1, 3). The CGH was illuminated with the correct compo- site OAM beam mode and size and was displaced the longitudinal and lateral directions of the beam. Moreover, because the composite OAM beam is axially symmetric, we moved the CGH along the positive x-axis direction. The phase distributions of the com- posite OAM beam with different lateral misplacements are shown in Figure 8(a1–a5). The change in the positions of the white circle indicates that the beam is being moved. As Figure 8(b1–b5) show, the reconstructed image gradually deteriorates and the details are no longer clear as the displacement increases in the lateral direction. The relatively high RMSs (~0.5) could have originated from the noise of the CCD or could from the uniformity of the incident beam. Figure 8. Experimental results increasing lateral misplacement of the Soochow University logo CGH with the composite OAM beam (p1, l1) = (1, 1) and (p2, l2) = (1, 3). (a1–a5) The phase distribu- tions of different lateral misplacements, the white circle indicates the position of one of the phase Photonics 2022, 9, x FOR PEER REVIEW 8 of 11 Figure 7. Experimental results of the CGH encoding grayscale image of a tiger. (a) The correct il- lumination of design was (p1, l1) = (1, 1) and (p2, l2) = (1, 3). The wrong illuminations used were (b) (p1, l1) = (0, 1) and (p2, l2) = (0, −1), (c) (p1, l1) = (0, 3) and (p2, l2) = (0, −3), and (d) (p1, l1) = (1, 1) and (p2, Photonics 2022, 9, 605 8 of 11 l2) = (1, −3). As the calculated CGH phase pattern is illuminated by a specific intensity and As the calculated CGH phase pattern is illuminated by a specific intensity and phase phase distribution, the misalignment of the decoded beam with respect to the hologram distribution, the misalignment of the decoded beam with respect to the hologram position position seems to affect the quality of the reconstructed image. To analyze the effects of seems to affect the quality of the reconstructed image. To analyze the effects of displacement displacement on the quality of the reconstructed image, the Soochow University logo on the quality of the reconstructed image, the Soochow University logo was encoded into was encoded into the CGH phase pattern and decoded using a composite OAM beam the CGH phase pattern and decoded using a composite OAM beam with (p , l ) = (1, 1) 1 1 with (p1, l1) = (1, 1) and (p2, l2) = (1, 3). The CGH was illuminated with the correct compo- and (p , l ) = (1, 3). The CGH was illuminated with the correct composite OAM beam 2 2 site OAM beam mode and size and was displaced the longitudinal and lateral directions mode and size and was displaced the longitudinal and lateral directions of the beam. of the beam. Moreover, because the composite OAM beam is axially symmetric, we Moreover, because the composite OAM beam is axially symmetric, we moved the CGH moved the CGH along the positive x-axis direction. The phase distributions of the com- along the positive x-axis direction. The phase distributions of the composite OAM beam posite OAM beam with different lateral misplacements are shown in Figure 8(a1–a5). with different lateral misplacements are shown in Figure 8(a1–a5). The change in the The change in the positions of the white circle indicates that the beam is being moved. positions of the white circle indicates that the beam is being moved. As Figure 8(b1–b5) As Figure 8(b1–b5) show, the reconstructed image gradually deteriorates and the details show, the reconstructed image gradually deteriorates and the details are no longer clear are no longer clear as the displacement increases in the lateral direction. The relatively as the displacement increases in the lateral direction. The relatively high RMSs (~0.5) high RMSs (~0.5) could have originated from the noise of the CCD or could from the could have originated from the noise of the CCD or could from the uniformity of the uniformity of the incident beam. incident beam. Figure 8. Experimental results increasing lateral misplacement of the Soochow University logo Figure 8. Experimental results increasing lateral misplacement of the Soochow University logo CGH CGH with the composite OAM beam (p1, l1) = (1, 1) and (p2, l2) = (1, 3). (a1–a5) The phase distribu- with the composite OAM beam (p , l ) = (1, 1) and (p , l ) = (1, 3). (a1–a5) The phase distributions of 1 1 2 2 tions of different lateral misplacements, the white circle indicates the position of one of the phase different lateral misplacements, the white circle indicates the position of one of the phase singularities. (b1–b5) Experimental far-field images for different lateral misplacements. A pixel is 12.5 m. Furthermore, a parameter in the range of (0, 1] is represented by the different weights of the modes constituting the input beam. As shown in Figure 9(a1–a5,b1–b5), the dif- ferent weights of the input beam result in different intensity distributions, but the phase distribution profiles are similar. From Figure 9(c1–c5), we can see that noise increases and the quality of the images decrease at smaller weights. The effect of the intensity dis- tribution on the reconstructed image appears to be unimportant because the CGH is a phase-only holographic pattern, so the phase distribution of the beam plays a key role in the reconstructed image. Photonics 2022, 9, x FOR PEER REVIEW 9 of 11 singularities. (b1–b5) Experimental far-field images for different lateral misplacements. A pixel is 12.5 μm. Furthermore, a parameter in the range of (0, 1] is represented by the different weights of the modes constituting the input beam. As shown in Figure 9(a1–a5),(b1–b5), the different weights of the input beam result in different intensity distributions, but the phase distribution profiles are similar. From Figure 9(c1–c5), we can see that noise in- creases and the quality of the images decrease at smaller weights. The effect of the inten- sity distribution on the reconstructed image appears to be unimportant because the Photonics 2022, 9, 605 9 of 11 CGH is a phase-only holographic pattern, so the phase distribution of the beam plays a key role in the reconstructed image. Figure 9. Analysis of the Soochow University logo CGH for different composite OAM beam energy Figure 9. Analysis of the Soochow University logo CGH for different composite OAM beam energy ratios. (a1–a5) The intensity and (b1–b5) phase distributions of the composite OAM beams; (c1–c5) ratios. (a1–a5) The intensity and (b1–b5) phase distributions of the composite OAM beams; (c1–c5) experimental far-field images for different beam energy ratios. experimental far-field images for different beam energy ratios. 4. 4. Disc Discussion ussion W We veri e verified fied t the he fea feasibility sibility of us of using ing com composite positOAM e OAM phase phase holo holograms grams t to encode o encod infor e in- - mation through simulation and experimental results. In our experiment, we generated a formation through simulation and experimental results. In our experiment, we generat- high-quality ed a high-qua composite lity composite OAM beam by l OAM beam by loading o the adCGH ing the CGH i into SLM1, nto SLM1 and the, angle and the a between ngle the incident light and the surface of SLM1 was less than 10 [33]. A circular aperture between the incident light and the surface of SLM1 was less than 10° [33]. A circular ap- was used to choose the positive first-order diffracted light beam from SLM1. Using the erture was used to choose the positive first-order diffracted light beam from SLM1. Us- described method, we could generate high-quality composite OAM beams with any radial ing the described method, we could generate high-quality composite OAM beams with and angular indices. Notaly, the waist size of the beam was a crucial parameter. In Ref. [34], any radial and angular indices. Notaly, the waist size of the beam was a crucial parame- for given beam indices, the image of the CGH encoding information could only be formed ter. In Ref. [34], for given beam indices, the image of the CGH encoding information clearly in the neighborhood of the optimal beam waist. In contrast to beam indices, which could only be formed clearly in the neighborhood of the optimal beam waist. In contrast can be represented as discrete integer values, the beam waist is a continuous parameter to beam indices, which can be represented as discrete integer values, the beam waist is a that can also be used as a dimension of the decoding key, further increasing security of the continuous parameter that can also be used as a dimension of the decoding key, further encoded information. increasing security of the encoded information. In this paper, we focused on the Fourier CGH, and the image was reconstructed by a Fourier lens. The misalignment of the lens introduced aberration distortion to the reconstructed image. However, further improvements may be considered. One method to achieve a lens-less reconstruction of a holographic image is by using the Fresnel CGH encoding. IAnother effective method is by adding the phase function of a Fourier transform lens on the Fourier CGH. In experiments, misalignment and beam energy ratio analyses can provide a good tolerance between the hologram and the phase distribution of the beam, and better images can be obtained will even micron-level precision. Hence, by designing composite OAM phase holographic experiments, information security can be improved and have wide application in the field of security and anti-counterfeiting. Photonics 2022, 9, 605 10 of 11 5. Conclusions In this paper, we present the detailed design and procedure of CGH information encod- ing for illumination with composite OAM beams. A high-quality composite OAM beam can be generated by loading the CGH into a phase-only SLM. The weighted Gerchberg-Saxton algorithm was implemented for the computation of the optimized CGH phase pattern. The calculation of the CGH for a given composite OAM beam resulted in a one-to-one correspondence between the holographic phase pattern and the beam, and the encoded information of CGH could be decoded with the correct illumination. The experimental results showed that the reconstructed images only clearly appeared; otherwise, the images appeared blurred. This proposed method has potential applications in anti-counterfeiting applications, secure communication systems, and imaging systems. Author Contributions: Conceptualization, X.Y.; methodology, N.Z. and X.Y.; software, N.Z.; vali- dation, N.Z. and B.X.; formal analysis, N.Z. and B.X.; investigation, N.Z.; resources, N.Z. and B.X.; data curation, N.Z.; writing—original draft preparation, N.Z.; writing—review and editing, X.Z. and X.Y.; visualization, N.Z.; supervision, X.Z. and X.Y.; project administration, X.Z. and X.Y.; funding acquisition, X.Z. and X.Y. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by National Natural Science Foundation of China (NSFC) (61775153, 61705153), NSAF Joint Fund (U1930106), Natural Science Research of Jiangsu Higher Education Institutions of China (19KJA210001) and Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The available data has been stated in the article. Conflicts of Interest: The authors declare no conflict of interest. References 1. Dholakia, K.; Cižmár, T. Shaping the future of manipulation. Nat. Photonics 2011, 5, 335–342. [CrossRef] 2. Padgett, M.; Di Leonardo, R. Holographic optical tweezers and their relevance to lab on chip devices. Lab Chip 2011, 11, 1196–1205. [CrossRef] [PubMed] 3. Li, J.; Kamin, S.; Zheng, G.; Neubrech, F.; Zhang, S.; Liu, N. Addressable metasurfaces for dynamic holography and optical information encryption. Sci. 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