Solution to the issue of high-order diffraction images for cylindrical computer-generated holograms

Jie Zhou; Lei Jiang; Guangwei Yu; Jiabao Wang; Yang Wu; Jun Wang

doi:10.1364/OE.518935

1. Introduction

Holography is widely acknowledged as a leading three-dimensional (3D) technology, capable of reconstructing the entire light field of a 3D scene and providing the human eye with complete parallax and depth information [1–6]. With the evolution of computer technology, researchers have begun utilizing computers to simulate optical diffraction to generate the hologram of a given object, namely the computer-generated hologram (CGH). CGH has rapidly advanced in recent years, with significant improvements in reconstruction quality [7–11] and computational speed [12–16]. However, most of the research on CGH is still only in the plane, which has a highly restricted viewing zone.

Numerous endeavors have been focused on expanding the viewing zone of CGH [17–20]. Among the reported methods, the cylindrical computer-generated hologram (CCGH) is recognized as a particularly effective method [21–31]. Sando et al. first proposed a fast calculation method for CCGH by using a convolution algorithm in a cylindrical coordinate system [21]. Yamaguchi et al. realized a CCGH that is viewable at 360° by printing its segmental fringes with a prototype fringe printer and proposed a fast calculation method by segmentation and table for the horizontal direction [22]. Jackin et al. proposed another fast calculation method for CCGH based on wave propagation in the spectral domain in a cylindrical coordinate system [23]. Sando et al. proposed a method for calculating CCGH based on the spectral relation between a 3D object and its diffracted wavefront and used a Bessel function expansion to save computing time and memory usage [24–26]. Zhao et al. proposed a fast calculation method for CCGH by using a wave-front recording plane (WRP) [27]. Wang et al. proposed a fast calculation method of CCGH by convolution algorithm between two concentric cylindrical surfaces with analysis of non-constant obliquity factor and unified inside-out propagation (IOP) and outside-in propagation (OIP) models by the unified expression of their obliquity factors [28]. Li et al. proposed an occlusion culling method by using a horizontal optical path limitation function to improve the reconstruction quality for CCGH [29]. Zhou et al. proposed a conical hologram to expand the effective viewing zone in the vertical direction for CCGH [30]. Tan et al. proposed a scaled CCGH based on scaled convolution to overcome constraints on height [31]. However, the issue of high-order diffraction images due to pixelated structure in CCGH has not been previously reported and solved. High-order diffraction images issue in CCGH is more worthy of being solved than in planar CGH. There are two main reasons: One is that for a cylindrical model offering a 360° viewing zone in the horizontal direction, the high-order diffraction images always overlap with the reconstruction image, leading to quality degradation. The other is that the 4f system is commonly used to eliminate high-order diffraction images in planar CGH, but its implementation is predictably complex for a cylindrical model. Therefore, how to solve the issue of degradation of reconstruction image quality caused by high-order diffraction in CCGH is worth studying.

In this paper, we propose a solution to the issue of high-order diffraction images for CCGH. We derive the cylindrical diffraction formula from the outer hologram surface to the inner object surface in the spectral domain, and based on this, we subsequently analyze the effects brought by the pixel structure and propose the high-order diffraction model. Based on the proposed high-order diffraction model, we use the gradient descent method to optimize CCGH accounting for all diffraction orders simultaneously. Furthermore, we analyze the issue of circular convolution due to the periodicity of the Fast Fourier Transform (FFT) in cylindrical diffraction. The correctness of the proposed high-order diffraction model and the effectiveness of the proposed optimization method are demonstrated by numerical simulation. To our knowledge, this is the first time that the issue of high-order diffraction in CCGH has been proposed, and we believe our work can offer valuable guidance to practitioners in the field.

2. Principle

2.1. Proposed high-order diffraction model for CCGH

The Helmholtz equation is used to describe the propagation characteristics of electromagnetic fields, which is expressed by:

(1)$$({{\nabla^\textrm{2}}\textrm{ + }{k^2}} )u = 0, $$

where ${\nabla ^\textrm{2}}$ is the Laplace operator, k is the wave number, and u is the complex amplitude of the light field in space to be solved. The solution to the Helmholtz equation in a cylindrical coordinate system, as shown in Fig. 1(a), can be expressed by:

(2)$$u(r,\varphi ,y) = R(r)\Phi (\varphi )Y(y),$$

where R(r), Φ(φ), and Y(y) represent the radial, angle, and vertical components, respectively. From [23], the most general solution of Eq. (1) in the spectral domain can be expressed by:

(3)$$u(r,\varphi ,y) = \sum\limits_{n ={-} \infty }^\infty {{e^{in\varphi }}\frac{1}{{2\pi }}} \int\limits_{ - \infty }^\infty {{A_n}({k_y})} {e^{i{k_y}y}}H_n^{(1)}({k_r}r) + {B_n}({k_y}){e^{i{k_y}y}}H_n^{(2)}({k_r}r)d{k_y},$$

where A_n(k_y) and B_n(k_y) are both arbitrary constants, and H(1) n and H(2) n are respectively the first and second kind Hankel Functions. When a source point u_s(r_s, φ_s, y_s) is located on the hologram surface of radius r_s, and a destination point u_d(r_d, φ_d, y_d) is located on the object surface of radius r_d, which is less than r_s, we only need to consider the case of inward propagation. Therefore, Eq. (3) can be reduced to:

(4)$$u(r,\varphi ,y) = \sum\limits_{n ={-} \infty }^\infty {{e^{in\varphi }}\frac{1}{{2\pi }}} \int\limits_{ - \infty }^\infty {{B_n}({k_y})} {e^{i{k_y}y}}H_n^{(2)}({k_r}r)d{k_y}.$$

Fig. 1. (a) Diffraction model in cylindrical coordinate system. (b) Schematic diagram of pixelated structure.

Download Full Size | PDF

The Fourier transform pair in cylindrical coordinates is expressed by U_n(r, k_y) and u(r, φ, y):

(5)$${U_n}(r,{k_y}) = \frac{1}{{2\pi }}\int\limits_0^{2\pi } {d\varphi } \int\limits_{ - \infty }^\infty {u(r,\varphi ,y){e^{ - in\varphi }}{e^{ - i{k_y}y}}dy} ,$$

(6)$$u(r,\varphi ,y) = \sum\limits_{n ={-} \infty }^\infty {{e^{in\varphi }}} \frac{1}{{2\pi }}\int\limits_{ - \infty }^\infty {{U_n}{e^{i{k_y}y}}d{k_y}.}$$

Comparing Eq. (6) at r = r_s with Eq. (4), we get:

(7)$${U_n}({r_s},{k_y}) = {B_n}({k_y})H_n^{(2)}({k_r}{r_s}).$$

Combining Eq. (7) and Eq. (4), we get:

(8)$${u_d}({r_d},{\varphi _d},{y_d}) = \sum\limits_{\textrm{n ={-} }\infty }^\infty {{e^{in{\varphi _d}}}} \frac{1}{{2\pi }}\int\limits_{ - \infty }^\infty {{U_n}({r_s},{k_y}){e^{i{k_y}{y_d}}}\frac{{H_n^{(2)}({k_r}{r_d})}}{{H_n^{(2)}({k_r}{r_s})}}d{k_y}} .$$

Hence the propagation from the outside hologram surface to the inside object surface in a cylindrical coordinate system in the spectral domain can be calculated by Eq. (8), and the transfer function of it can be expressed by:

(9)$$TF = \frac{{H_n^{(2)}({k_r}{r_d})}}{{H_n^{(2)}({k_r}{r_s})}}.$$

This propagation process can be performed by the Fast Fourier Transform (FFT) and Inverse Fast Fourier Transform (IFFT) in numerical calculations:

(10)$${u_d}({\varphi _d},{y_d}) = IFFT[FFT[{u_s}({\varphi _s},{y_s})] \times TF],$$

However, when loading CCGH onto certain medium, such as the spatial light modulator (SLM) used in planar CGH, the issue posed by pixel structures must be considered. As shown in Fig. 1(b), the dot structure of the pixels creates gaps between the pixels that cannot be modulated, which brings about high-order diffraction. The issue of high-order diffraction images in CCGH will be analyzed below.

As shown in Fig. 2(a), a complex amplitude hologram with a random phase obtained by the original method [23] can reconstruct a high-quality target image. However, it is an ideal reconstruction without considering pixel structure. In the actual reconstruction, as shown in Fig. 2(b), after loading CCGH onto the medium, the optical field of the hologram surface is expressed by:

(11)$${u^{\prime}_s}({\varphi _s},{y_s}) = [{u_s}({\varphi _s},{y_s}) \times \textrm{comb} (\frac{{{\varphi _s}}}{{\Delta \varphi }},\frac{{{y_s}}}{{\Delta y}})] \ast rect (\frac{{{\varphi _s}}}{{\alpha \Delta \varphi }},\frac{{{y_s}}}{{\beta \Delta y}}) \times \textrm{rect} (\frac{{{\varphi _s}}}{{M\Delta \varphi }},\frac{{{y_s}}}{{N\Delta y}}),$$

where comb(·) and rect(·) are respectively two-dimensional Comb Function and Rectangular Window Function, Δφ and Δy represent respectively the pixel pitch in the azimuthal and vertical direction, α and β represent respectively the fill rate in the azimuthal and vertical direction, and M and N represent respectively pixels in the azimuthal and vertical direction. In its reconstruction image, obvious high-order diffraction images can be seen, which seriously degrade the reconstruction quality. Especially for a cylindrical model with a 360° viewing zone in the horizontal direction, no matter how far the diffraction distance is, the high-order diffraction images always overlap with the reconstruction image.

Fig. 2. (a) Ideal reconstruction. (b) Actual reconstruction.

Download Full Size | PDF

To solve the above issue, we need to establish a model to describe the high-order diffraction in CCGH. The original diffraction model is modified to correctly calculate the actual diffraction field considering the pixel structure, i.e., Eq. (8) is changed to:

(12)$${u_d}({r_d},{\varphi _d},{y_d}) = \sum\limits_{\textrm{n = } - \infty }^\infty {{e^{in{\varphi _d}}}} \frac{1}{{2\pi }}\int\limits_{ - \infty }^\infty {{{U^{\prime}}_n}({r_s},{k_y}){e^{i{k_y}{y_d}}}\frac{{H_n^{(2)}({k_r}{r_d})}}{{H_n^{(2)}({k_r}{r_s})}}d{k_y}} .$$

where ${U^{\prime}_n}$ is the frequency spectrum of ${u^{\prime}_s}$, which is expressed by:

(13)$$\begin{aligned} {{U^{\prime}}_n} &= \frac{1}{{2\pi }}\int\limits_0^{2\pi } {d\varphi } \int\limits_{ - \infty }^\infty {{{u^{\prime}}_s}({r_s},{\varphi _s},{y_s}){e^{ - in\varphi }}{e^{ - i{k_y}y}}dy} \\ &= {U_n}\ast \textrm{comb} (\frac{{\Delta \varphi }}{{\textrm{2}\pi }}n,\frac{{\Delta y}}{{\textrm{2}\pi }}{k_y}) \times sinc (\frac{{\alpha \Delta \varphi }}{{\textrm{2}\pi }}n,\frac{{\beta \Delta y}}{{\textrm{2}\pi }}{k_y})\\ &\textrm{ = }\textrm{sinc} (\frac{{\alpha \Delta \varphi }}{{\textrm{2}\pi }}n,\frac{{\beta \Delta y}}{{\textrm{2}\pi }}{k_y})\sum\limits_{i,j ={-} (\gamma - 1)/2}^{(\gamma - 1)/2} {{U_n}} (n + \frac{{2\pi }}{{\Delta \varphi }}i,{k_y} + \frac{{2\pi }}{{\Delta y}}j), \end{aligned}$$

where sinc(·) is the Sinc Function and γ is the number of diffraction orders. With Eq. (13), the spectrum containing frequency components of all diffraction orders can be obtained, followed by Eq. (12), the diffraction field containing all diffraction orders can be obtained. At this point, we establish the high-order diffraction model for CCGH.

2.2. Proposed optimization method for CCGH

In the previous section, we introduced the issue of high-order diffraction images in CCGH and established its model. In this section, to solve this issue, we use the gradient descent method to optimize CCGH accounting for all diffraction orders simultaneously based on the proposed model. The specific implementation of the proposed model and the proposed optimization method are illustrated visually through the pipeline in Fig. 3. The process is as follows:

Fig. 3. Specific implementation of proposed high-order diffraction model and proposed optimization method.

Download Full Size | PDF

Step 1: obtain the spectrum U_n of the hologram u_s by FFT.

Step 2: extend the spectrum period to integrate high-order diffraction into the propagation pipeline.

Step 3: multiply this extended spectrum by the Sinc Function to modulate the frequency components of different orders to account for the fill rate.

The above three steps are the specific implementation of Eq. (13), through which we can obtain the spectrum ${U^{\prime}_n}$ containing frequency components of all diffraction orders can be obtained.

Step 4: multiply ${U^{\prime}_n}$ by the transfer function TF to propagate the spectrum.

Step 5 and Step 6: obtain reconstruction image by IFFT and downsample operation.

All the above steps are the specific implementation of the proposed model (Eq. (12)), through which we can obtain the reconstruction image containing high-order diffraction images.

Step 7: optimize all orders together to produce high reconstruction quality, a process performed by minimizing Loss in Eq. (14) with gradient descent to update the hologram,

(14)$$Loss = MSELoss(s \cdot Am{p_r},Am{p_t}),$$

where Amp_r and Amp_t are the reconstruction amplitude and target amplitude, respectively, and s is a global scale factor.

2.3. Analysis of circular convolution for CCGH

The issue of circular convolution due to the periodicity of FFT in cylindrical diffraction is worth discussing here. The circular convolution in the vertical and azimuthal directions are analyzed separately.

As shown in Fig. 4(a), physically, the diffraction range expands with the propagation distance in the vertical direction. However, in Eq. (13), the diffraction field has the same size as the input field in the vertical direction. Therefore, it becomes necessary to address the exceeding part of the diffraction field. Numerically, due to the circular property of FFT, the exceeding part enters the diffraction field from the other side, introducing a computational error. The size of the error portion increases with the propagation distance. This effect is often referred to as circular convolution error. To eliminate this error, we use zero padding to expand the computational window as shown in Fig. 4(b). Since the excess portion leads to error, we can control the number of zero padding N_P to prevent the excess portion from entering the computation window. N_P in the vertical direction can be calculated by:

(15)$${N_p} = \frac{{({r_s} - {r_d})}}{{\Delta y}} \cdot \tan (\arcsin (\frac{{\gamma \lambda }}{{2\Delta y}})).$$

However, in the azimuthal direction, the opposite is true. There is no need for zero padding to avoid the effect of circular convolution. As shown in Fig. 5(a), the diffraction range is a closed circulation circle in the azimuthal direction. Light from the part of the input field near +π enters the part of the diffraction field near -π, while light from the part of the input field near -π enters the part of the diffraction field near +π. Linear convolution does not take this into account. On the contrary, as shown in Fig. 5(b), the circular convolution introduced when using FFT to calculate linear convolution exactly matches it.

Fig. 4. Schematic diagram to show (a) effect of circular convolution error in vertical direction and (b) adaptive zero padding method to avoid it.

Download Full Size | PDF

Fig. 5. Schematic diagram to show (a) closed circulation circle in azimuthal direction and (b) circular convolution that can correctly calculate it.

Download Full Size | PDF

3. Simulation results and analysis

In this section, we will demonstrate the correctness of our proposed high-order diffraction model for CCGH in Sec. 3.1 and the effectiveness of our proposed optimization method to the issue of the high-order diffraction images for CCGH in Sec. 3.2, and we will discuss the issue of circular convolution for CCGH in Sec. 3.3. In this section, All CCGH algorithms are implemented in Python 3.7.1 and Pytorch 1.10.2. The central processing unit (CPU) and graphics processing unit (GPU) are 13th Gen Intel Core i7-13700KF and NVIDIA GeForce RTX 4070 with CUDA version 11.6, respectively. The specific parameter settings are as follows: The radius of the outer hologram surface r_s and inner object surface r_d is set to be 10 cm and 1 cm, respectively, and their heights are both 10 cm. The fill rate in the azimuthal direction α and in the vertical direction β are both 80%. The wavelength λ is chosen to be 25 µm. The resolution is chosen to be 1024 in each direction.

3.1. Correctness of proposed high-order diffraction model

To prove the correctness of our proposed model of high-order diffraction, we perform the simplest single-slit experiment and Young’s interference experiment in each direction. The diffraction field calculated by the point source method (PSM) is chosen to be the standard, and the correlation coefficient index (CC) is chosen to be the evaluation metric, which is expressed as:

(16)$$CC(f,g) = \frac{{\sum\limits_{i = 1}^M {\sum\limits_{j = 1}^N {({f_{ij}} - \overline f )({g_{ij}} - \overline g )} } }}{{\sqrt {\sum\limits_{i = 1}^M {\sum\limits_{j = 1}^N {{{({f_{ij}} - \bar{f})}^2} \cdot \sum\limits_{i = 1}^M {\sum\limits_{j = 1}^N {{{({g_{ij}} - \bar{g})}^2}} } } } } }},$$

where M and N are pixels in the azimuthal and vertical directions, respectively, and (i, j) is the pixel coordinate. f_ij and g_ij represent the amplitudes of the diffraction field with our proposed model and PSM, respectively. $\bar{f}$ and $\bar{g}$ are the averages of the amplitudes. In general, the closer CC is to 1 indicates a better correlation between these two diffraction field amplitudes.

Firstly, to demonstrate that our proposed model can accurately calculate high-order diffraction, the single-slit diffraction experiment is chosen for the first experiment, in which it is easy to distinguish different diffraction orders in its diffraction field. As shown in Fig. 6, in the first row, a single slit of 20 pixels is located at the center of the input field. The diffraction field is calculated by PSM and our proposed model with the different number of orders γ, respectively. In the second row, the diffraction field obtained by PSM is taken as the standard, in which different diffraction orders can be clearly distinguished. In the third row, there is only 0th-order in the diffraction field obtained by the proposed model with γ = 1, i.e., without considering high-order diffraction. In the fourth row, the 0th-order and ±1st-orders can be accurately calculated by the proposed model with γ = 3. In the fifth row, 0th-order, ± 1st-orders, and ±2nd-orders can be accurately calculated by the proposed model with γ = 5. For γ = 1, the CC is 0.9190 in the azimuthal direction and 0.9615 in the vertical direction. Compared to γ = 1, the CC with γ = 3 and γ = 5 is higher. For γ = 3, the CC is 0.9554 in the azimuthal direction and 0.9980 in the vertical direction. For γ = 5, the CC is 0.9989 in the azimuthal direction and 0.9990 in the vertical direction.

Fig. 6. Single-slit experiment with 80% fill rate and r_d = 1 cm.

Download Full Size | PDF

Secondly, to better demonstrate the correctness of the proposed model, Young’s interference experiment is performed, which has a more complex diffraction field compared to single-slit. As shown in Fig. 7, in the first row, there are only two points in the input field located at -π/64 and +π/64 in the azimuthal direction, and at −1.56 mm and +1.56 mm in the vertical direction. In the second row, the diffraction field obtained by PSM is taken as the standard. As can be seen in the third, fourth, and fifth rows, compared to γ = 1, the diffraction field obtained by the proposed model with γ = 3 and γ = 5 is more correct. In the azimuthal direction, CC can reach 0.9633 and 0.9960 with γ = 3 and γ = 5, respectively, while it is only 0.8773 with γ = 1. In the vertical direction, CC can reach 0.9822 and 0.9992 with γ = 3 and γ = 5, respectively, while it is only 0.8741 with γ = 1. The above experiment results are obtained only with a fill rate of 80% and the inner object surface radius r_d of 1 cm. CC with different fill rates is shown in Fig. 8(a) and (b). For different fill rates, both in azimuthal and vertical direction, CC is always the highest with γ = 5, followed by γ = 3, and lowest with γ = 1. As the fill rate increases, CC with γ = 1 and γ = 3 will increase. This is because as the fill rate increases, in the diffraction field, the proportion of amplitude of high orders will decrease, and the proportion of the amplitude of the 0th-order will increase. CC with different r_d is shown in Fig. 8(c) and (d). With different r_d, both in the azimuthal and vertical direction, CC is always the highest with γ = 5, followed by γ = 3, and lowest with γ = 1.

Fig. 7. Young’s interference experiment with 80% fill rate and r_d = 1 cm.

Download Full Size | PDF

Fig. 8. CC of Young’s interference experiment (a)(b) with different fill rates and (c)(d) with different r_d.

Download Full Size | PDF

Finally, as shown in Fig. 9, we conduct the two-dimensional rectangular hole experiment and Young’s interference experiment. The first row is the rectangular hole experiment. A rectangular hole of 20 × 20 pixels is located at the center of the input field, and its magnification is in the red box. Different diffraction orders can be clearly distinguished in the diffraction field obtained by PSM. Only the 0th-order can be calculated by the proposed model with γ = 1. The 0th-order and ±1st-orders can be accurately calculated by the proposed model with γ = 3. The 0th-order, ± 1st-orders, and ±2nd-orders can be accurately calculated by the proposed model with γ = 5. CC with γ = 1 is 0.8907, with γ = 3 is 0.9800, and with γ = 5 is 0.9937. The second row is Young’s interference experiment. There are two points located at (-π/64,0) and (+π/64,0) in the input field. The diffraction field obtained by the proposed model with γ = 5 fits best with the diffraction field obtained by PSM, second best with γ = 3, and worst with γ = 1, and CC are 0.9951, 0.9764, and 0.8143, respectively. At this point, the series of experiments presented above is sufficient to prove the correctness of our proposed model of high-order diffraction.

Fig. 9. Two-dimensional rectangular hole experiment and Young’s interference experiment with 80% fill rate and r_d = 1 cm.

Download Full Size | PDF

3.2. Effectiveness of proposed optimization method

After demonstrating the correctness of the proposed high-order diffraction model, we will demonstrate that the issue of higher-order diffracted images does exist in CCGH and leads to serious degradation of the reconstruction quality, while our proposed optimization method can effectively solve this issue. For a better illustration, the Peak Signal-to-Noise Ratio (PSNR) and Structural Similarity Index (SSIM) are chosen to evaluate the quality of the reconstruction. The evaluation calculation formulas are as follows:

(17)$$PSNR(f,g) = 10\lg \left\{ {{{255}^2}/\left[ {\frac{1}{{MN}}\sum\limits_{i = 1}^M {\sum\limits_{j = 1}^N {{{({f_{ij}} - {g_{ij}})}^2}} } } \right]} \right\}.$$

(18) $$SSIM(f,g) = l(f,g)c(f,g)s(f,g),$$

(19)$$l(f,g) = \frac{{2{\mu _f}{\mu _g} + {C_1}}}{{\mu _f^2 + \mu _g^2 + {C_1}}},c(f,g) = \frac{{2{\sigma _f}{\sigma _g} + {C_2}}}{{\sigma _f^2 + \sigma _g^2 + {C_2}}},s(f,g) = \frac{{{\sigma _f}_g + {C_3}}}{{{\sigma _f} + {\sigma _g} + {C_3}}},$$

where M and N are pixels in the azimuthal and vertical directions, respectively, and (i, j) is the pixel coordinate. f_ij and g_ij represent the amplitude of the target image and the reconstruction image, respectively. µ_f, µ_g and σ_f, σ_g are the mean luminance and the standard deviation of the target image and reconstruction image, respectively. And C₁, C₂ and C₃ are constants to avoid a null denominator. It can be inferred that reconstruction quality is better when the PSNR is higher and the SSIM is closer to 1.

As shown in Fig. 10, we simulate the reconstruction of holograms generated by the original [23] and proposed methods, respectively. The first column is the target images, which are “usaf1951”, “moto”, and “parrot”, respectively. The second, third, and fourth columns are all reconstructions of holograms generated by the original method, which are the reconstruction of the complex amplitude hologram without random phase (complex w/o RP), the complex amplitude hologram with random phase (complex w/ RP), and the phase-only hologram with random phase (phase-only w/ RP), respectively. In the reconstruction of complex w/o RP, while the 0th-order image can be well-reconstructed, the high-order diffraction images overlap with it, seriously degrading the quality of the reconstruction, with PSNR of 11.39 dB for “usaf1951”, 17.89 dB for “moto”, and 19.50 dB for “parrot”. Compared to complex w/o RP, the reconstruction quality of complex w/ RP is lower, with PSNR of 9.81 dB for “usaf1951”, 16.77 dB for “moto”, and 15.91 dB for “parrot”. However, it is difficult to achieve complex amplitude modulation, thus the simulated reconstruction of phase-only holograms is more informative. Due to the loss of amplitude, the reconstruction quality of phase-only w/ RP is the lowest, with PSNR of 8.26 dB for “usaf1951”, 13.50 dB for “moto”, and 13.46 dB for “parrot”. Therefore, the issue of higher-order diffraction images does exist in CCGH and leads to serious degradation of the reconstruction quality. The fifth column is the reconstruction of the phase-only hologram generated by the proposed method. Based on the proposed high-order diffraction model, we use the gradient descent method to optimize CCGH accounting for all diffraction orders simultaneously. The learning rate is set to 0.04 for all phase variables, and the number of optimization iterations is set to 500. By our proposed method, high-quality reconstruction can be obtained, with PSNR of 19.66 dB for “usaf1951”, 26.80 dB for “moto”, and 32.56 dB for “parrot”. In addition, 100 images are randomly selected from the DIV2 K [32] dataset as samples. The average PSNR and SSIM curves of their reconstruction with different fill rates are shown in Fig. 11(a) and (b), respectively. With the fill rate below 40%, the amplitude of the high-order diffraction image is almost as high as that of the 0th-order, making it difficult to optimize. However, with the fill rate above 40%, the proposed method produces the highest quality. The average PSNR and SSIM curves of their reconstruction with different r_d are shown in Fig. 11(c) and (d), respectively. The proposed method always produces the highest quality with different r_d. Therefore, we demonstrate that the proposed method can effectively solve the issue of high-order diffraction images, improving the reconstruction quality.

Fig. 10. Reconstruction images of holograms generated by original and proposed methods with 80% fill rate and r_d = 1 cm.

Download Full Size | PDF

Fig. 11. Average PSNR and SSIM curves of reconstruction images of holograms generated by original and proposed methods (a)(b) with different fill rates (c)(d) with different r_d.

Download Full Size | PDF

3.3. Discussion of circular convolution

The issue of circular convolution in cylindrical diffraction algorithm will be discussed here. As shown in Fig. 12, in the first row, the input field is a single slit from pixel 1000 to pixel 1019 and the diffraction field is calculated by PSM and the proposed model with different numbers of diffraction order γ with and without zero padding, respectively. In the second row, the diffraction field calculated by PSM is taken as the standard, which does not introduce the convolution algorithm. In the third, fourth, and fifth rows, the diffraction field is calculated by our proposed model without zero padding. Since our proposed model is performed by FFT, the circular convolution is introduced. In the azimuthal direction, since the diffraction range is a closed circulation circle, the light from the +π side will enter the diffraction from the -π side, as evidenced by the result obtained by PSM. The diffraction field calculated by the proposed model without zero padding exactly matches this. Therefore, there is no need for zero padding in the azimuthal direction. However, in the vertical direction, the part that exceeds the computation window enters the diffraction field from the other side, introducing a computational error, as shown in the red box. Therefore, the zero-padding is needed in the vertical direction. In the sixth, seventh, and eighth rows, after zero padding, with the number of zero padding N_P of 120 with γ = 384 with γ = 3, and 768 with γ = 5 according to Eq. (15), this computational error can be avoided. In addition, N_P at different diffraction distances is shown in Fig. 13.

Fig. 12. Diffraction field of single slit from pixel 1000 to pixel 1019 with 80% fill rate and r_d = 1 cm with and without zero padding.

Download Full Size | PDF

Fig. 13. N_P at different diffraction distances.

Download Full Size | PDF

4. Conclusion

In this paper, we propose a solution to the issue of high-order diffraction images for CCGH. The proposed model can accurately calculate the high-order diffraction in cylindrical diffraction. And the proposed optimization method can effectively solve the issue of high-order diffraction images and improve reconstruction quality, with PSNR over 30 dB. Furthermore, the issue of circular convolution in cylindrical diffraction is analyzed and it is concluded that zero-padding is needed to avoid the circular convolution in the vertical direction, but circular convolution precisely matches the diffraction in the azimuthal direction. With advances in materials technology, such as metasurfaces [33–35], achieving light field modulation of cylindrical surfaces seems to be no longer difficult, and CCGH is expected to move towards practicality. We believe our work can offer valuable guidance to practitioners in the field.

Funding

National Natural Science Foundation of China (62275178); Chengdu Municipal Science and Technology Program (2022-GH02-00016-HZ); Open Fund of Jiangsu Engineering Research Center of Novel Optical Fiber Technology and Communication Network (SDGC2233).

Acknowledgments

The authors wish to thank the anonymous reviewers for their valuable suggestions.

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. P. Yu, Y. Liu, Z. Wang, et al., “Ultrahigh-density 3d holographic projection by scattering-assisted dynamic holography,” Optica 10(4), 481–490 (2023). [CrossRef]

2. S. So, J. Kim, T. Badloe, et al., “Multicolor and 3d holography generated by inverse-designed single-cell metasurfaces,” Adv. Mater. 35(17), 2208520 (2023). [CrossRef]

3. A. H. Dorrah, P. Bordoloi, V. S. de Angelis, et al., “Light sheets for continuous-depth holography and three-dimensional volumetric displays,” Nat. Photonics 17(5), 427–434 (2023). [CrossRef]

4. D. Pi, J. Liu, and Y. Wang, “Review of computer-generated hologram algorithms for color dynamic holographic three-dimensional display,” Light: Sci. Appl. 11(1), 231 (2022). [CrossRef]

5. J. Kim, D. Jeon, J. Seong, et al., “Photonic encryption platform via dual-band vectorial metaholograms in the ultraviolet and visible,” ACS Nano 16(3), 3546–3553 (2022). [CrossRef]

6. P. Georgi, Q. Wei, B. Sain, et al., “Optical secret sharing with cascaded metasurface holography,” Sci. Adv. 7(16), eabf9718 (2021). [CrossRef]

7. J. Zhang, N. Pégard, J. Zhong, et al., “3d computer-generated holography by non-convex optimization,” Optica 4(10), 1306–1313 (2017). [CrossRef]

8. P. Chakravarthula, Y. Peng, J. Kollin, et al., “Wirtinger holography for near-eye displays,” ACM Trans. Graph. 38(6), 1–13 (2019). [CrossRef]

9. Y. Peng, S. Choi, N. Padmanaban, et al., “Neural holography with camera-in-the-loop training,” ACM Trans. Graph. 39(6), 1–14 (2020). [CrossRef]

10. Z. Wang, T. Chen, Q. Chen, et al., “Reducing crosstalk of a multi-plane holographic display by the time-multiplexing stochastic gradient descent,” Opt. Express 31(5), 7413–7424 (2023). [CrossRef]

11. X. Xia, F. Yang, W. Wang, et al., “Investigating learning-empowered hologram generation for holographic displays with ill-tuned hardware,” Opt. Lett. 48(6), 1478–1481 (2023). [CrossRef]

12. L. Shi, B. Li, C. Kim, et al., “Towards real-time photorealistic 3d holography with deep neural networks,” Nature 591(7849), 234–239 (2021). [CrossRef]

13. K.X. Liu, J.C. Wu, Z.H. He, et al., “4K-DMDNet: diffraction model-driven network for 4 K computer-generated holography,” Opto-Electron. Adv. 6(5), 220135 (2023). [CrossRef]

14. R. Zhu, L. Chen, and H. Zhang, “Computer holography using deep neural network with Fourier basis,” Opt. Lett. 48(9), 2333–2336 (2023). [CrossRef]

15. Z. Dong, C. Xu, Y. Ling, et al., “Fourier-inspired neural module for real-time and high-fidelity computer-generated holography,” Opt. Lett. 48(3), 759–762 (2023). [CrossRef]

16. G. Yu, J. Wang, H. Yang, et al., “Asymmetrical neural network for real-time and high-quality computer-generated holography,” Opt. Lett. 48(20), 5351–5354 (2023). [CrossRef]

17. Y.Z. Liu, X.N. Pang, S. Jiang, et al., “Viewing-angle enlargement in holographic augmented reality using time division and spatial tiling,” Opt. Express 21(10), 12068–12076 (2013). [CrossRef]

18. Z. Zeng, H. Zheng, Y. Yu, et al., “Full-color holographic display with increased-viewing-angle [Invited],” Appl. Opt. 56(13), F112–F120 (2017). [CrossRef]

19. R. Kang, J. Liu, D. Pi, et al., “Fast method for calculating a curved hologram in a holographic display,” Opt. Express 28(8), 11290–11300 (2020). [CrossRef]

20. D. Wang, N. N. Li, Z. S. Li, et al., “Color curved hologram calculation method based on angle multiplexing,” Opt. Express 30(2), 3157–3171 (2022). [CrossRef]

21. Y. Sando, M. Itoh, and T. Yatagai, “Fast calculation method for cylindrical computer-generated holograms,” Opt. Express 13(5), 1418–1423 (2005). [CrossRef]

22. T. Yamaguchi, T. Fujii, and H. Yoshikawa, “Fast calculation method for computer-generated cylindrical holograms,” Appl. Opt. 47(19), D63–D70 (2008). [CrossRef]

23. B. J. Jackin and T. Yatagai, “Fast calculation method for computer-generated cylindrical hologram based on wave propagation in spectral domain,” Opt. Express 18(25), 25546–25555 (2010). [CrossRef]

24. Y. Sando, D. Barada, and T. Yatagai, “Fast calculation of computer-generated holograms based on 3-D Fourier spectrum for omnidirectional diffraction from a 3-D voxel-based object,” Opt. Express 20(19), 20962–20969 (2012). [CrossRef]

25. Y. Sando, D. Barada, B. J. Jackin, et al., “Fast calculation method for computer-generated cylindrical holograms based on the three-dimensional Fourier spectrum,” Opt. Lett. 38(23), 5172–5175 (2013). [CrossRef]

26. Y. Sando, D. Barada, B. J. Jackin, et al., “Bessel function expansion to reduce the calculation time and memory usage for cylindrical computer-generated holograms,” Appl. Opt. 56(20), 5775–5780 (2017). [CrossRef]

27. Y. Zhao, M. Piao, G. Li, et al., “Fast calculation method of computer-generated cylindrical hologram using wave-front recording surface,” Opt. Lett. 40(13), 3017–3020 (2015). [CrossRef]

28. J. Wang, Q.H. Wang, and Y. Hu, “Unified and accurate diffraction calculation between two concentric cylindrical surfaces,” J. Opt. Soc. Am. A 35(1), A45–A52 (2018). [CrossRef]

29. Y. Li, J. Wang, C. Chen, et al., “Occlusion culling for computer-generated cylindrical holograms based on a horizontal optical-path-limit function,” Opt. Express 28(12), 18516–18528 (2020). [CrossRef]

30. Z. Zhou, J. Wang, Y. Wu, et al., “Conical holographic display to expand the vertical field of view,” Opt. Express 29(15), 22931–22943 (2021). [CrossRef]

31. C. Tan, J. Wang, Y. Wu, et al., “Fast scaled cylindrical holography based on scaled convolution,” Displays 81, 102619 (2024). [CrossRef]

32. E. Agustsson and R. Timofte, “Ntire 2017 challenge on single image super-resolution: dataset and study,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) Workshops, 126–135 (2017).

33. J. Kim, J. Seong, W. Kim, et al., “Scalable manufacturing of high-index atomic layer–polymer hybrid metasurfaces for metaphotonics in the visible,” Nat. Mater. 22(4), 474–481 (2023). [CrossRef]

34. J. Kim, W. Kim, D. K. Oh, et al., “One-step printable platform for high-efficiency metasurfaces down to the deep-ultraviolet region,” Light: Sci. Appl. 12(1), 68 (2023). [CrossRef]

35. M. Q. Mehmood, J. Seong, M. A. Naveed, et al., “Single-cell-driven tri-channel encryption meta-displays,” Adv. Sci. 9(35), 2203962 (2022). [CrossRef]

Solution to the issue of high-order diffraction images for cylindrical computer-generated holograms

Abstract

1. Introduction

2. Principle

2.1. Proposed high-order diffraction model for CCGH

2.2. Proposed optimization method for CCGH

2.3. Analysis of circular convolution for CCGH

3. Simulation results and analysis

3.1. Correctness of proposed high-order diffraction model

3.2. Effectiveness of proposed optimization method

3.3. Discussion of circular convolution

4. Conclusion

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (13)

Equations (19)

Optics Express