Spherical self-diffraction for speckle suppression of a spherical phase-only hologram

Bingyi Li; Jun Wang; Chun Chen; Yuejia Li; Ruoxue Yang; Ni Chen

doi:10.1364/OE.401679

1. Introduction

Holographic display is a promising technique to reconstruct three-dimensional (3D) scenes [1,2] since it can provide all the depth cues that human eyes can perceive. The information capacity and field of view are affected by the size and shape of the recording surface. Compared with the traditional planar holograms, curved holograms can provide larger information capacity and field of view, which is of great significance for the development of holographic 3D display. Soares and Fernandes first reported the wide-field holographic display using curved holograms [3], making it possible to reconstruct and observe images from a 360° horizontal direction. However, the optical recording system of curved holograms is generally complex. Computer-generated hologram (CGH) is a promising solution using numerical computing of the diffraction on non-planar observation surfaces [4].

The calculation of light field distribution on curved surface has wide applications. In addition to calculating a curved hologram, it can also be used in a wave-front recording surface (WRS) method to improve the calculation speed [5], or as a pre-calculation of the light field on a plane [6]. The calculation process of curved hologram is more complicated than a planar one. Therefore, there are many fast calculation methods proposed, which are designed based on curves. Some of them are proposed to obtain the computer-generated cylindrical holograms [7–10]. More recently, a curved multiplexing method based on cylindrical computer-generated holograms is also proposed to increase field of view and spatial bandwidth [11]. Meanwhile, sphere is also a special curved surface. Compared with cylinders, spheres have a more symmetrical structure and a larger field of view. Spherical holography can realize an omnidirectional reconstruction of light field and provide full view theoretically. Therefore, many researches have also been carried out based on spherical holography. M. Tachiki et al. proposed a fast convolution algorithm based on FFT for the fast calculation of spherical holograms, [12]. B. Jackin and T. Yatagai proposed a method of spherical wave spectrum [13], which is based on wave propagation defined in spectral domain and spherical coordinates. Besides, G. Li et al. proposed an acceleration method for computer generated spherical hologram calculation of real objects using graphics processing unit (GPU) [14]. Moreover, Y. Sando et al. proposed a calculation method based on spherical harmonic transform [15]. H. Cao and E. Kim introduced a method of faster generation of holographic videos of objects moving in space using a spherical hologram based on 3D rotational motion compensation scheme [16]. However, the speckle noise problem of the reconstructed images has never been considered in the conventional fast calculation methods of spherical holograms. In holographic display, speckle noise seriously degrades the quality of the reconstructed image especially for a phase-only CGH, and many efforts have been devoted to obtain a high-quality reconstruction [17–19]. The spherical computer-generated hologram is inevitably suffered from the speckle noise since it is necessary to add random phase to the object to ensure the scattering characteristic of reconstructed image.

In this paper, spherical self-diffraction iteration (SSDI) algorithm is proposed to suppress the speckle noise in the spherical phase-only hologram (SPOH). The algorithm is based on a spherical self-diffraction (SSD) model in which the object and the diffraction distribution are on the same sphere. Compared with the traditional spherical front-propagation (SFP) between two concentric spheres, using SSD as the basic diffraction model in iteration algorithm can achieve efficient speckle suppression. Though SSD cannot reconstruct the image on a whole sphere since part of the sphere is occupied by the original object, it can remain the large-viewing-area characteristic of curved hologram. SSD is a special case of spherical back-propagation (SBP) at limit condition. The fast calculation of SSD is achieved by using the spherical wave spectrum method. The correctness of SBP and SSD as well as the effectiveness of SSDI algorithm are verified by numerical simulations. The simulation results show that the SSDI algorithm can reconstruct an image with high quality after 5 iterations, which can greatly reduce the calculation time.

2. Theoretical analysis

2.1 Random phase and speckle noise

In the past simulation experiments of spherical holography, the reconstruction is mostly based on complex amplitude of diffraction field. However, the holograms need to be encoded in practical terms. To calculate the SPOH, it is necessary to add random phase to the object, which can smooth the image and simulate the process of scattering. As shown in Fig. 1(c), if the reconstructed image losses the scattering characteristic, human eyes can only capture the light emitted from a small certain range due to the directionality of reconstructed light. As shown in Figs. 1(a) and (e), if the random phase is added to the object, the reconstructed image on the sphere will have the scattering characteristic, therefore, the pattern on a larger range of the sphere can be observed. Figures 1(b), (d) and (f) show the reconstruction effect after speckle suppression. It can realize the observation of high-quality reconstructed image at different positions.

Fig. 1. Scattering characteristic of reconstructed image to enlarge visible area and the necessity to suppress the speckle noise. (a)(e) Large range of the image can be observed while the speckle noise is obvious. (c) The viewing area is limited. (b)(d)(f) The observation effect after speckle suppression at different positions.

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In the next sections, SSDI algorithm is proposed to suppress speckle noise in SPOH, before which the calculation of SSD, which is the basic diffraction model of the algorithm, is analyzed in detail.

2.2 Spherical front and back propagation models

The propagation models between two concentric spheres are established based on the spherical wave spectrum method. Firstly, the basic solution of Helmholtz equation in the spherical coordinate system is considered. Ignoring the time factor of the solution, Helmholtz equation can be derived with only spatial variables as:

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

where ▽² is the Laplace operator, k is the wave number, and u is the amplitude distribution of the light field. Replacing x, y and z with r, θ and φ in the spherical coordinate system, the solution of Eq. (1) can be defined by the separation of variables as:

(2)$$u(r,\theta ,\varphi ) = R(r)\Theta (\theta )\Phi (\varphi ),$$

where the variation ranges of r, θ and φ are $[0, + \infty )$, $[0,\pi )$ and $[0,2\pi )$, respectively. From [20], the distribution of spatial light field on the sphere can be expressed as:

(3)$$u(r,\theta ,\phi ) = \sum\limits_{l = 0}^\infty {\sum\limits_{m ={-} l}^l {{C_{lm}}} } R(r)Y_l^m(\theta ,\phi ),$$

where C_lm is a constant for definite order and R(r) is the radial function. Y(θ,φ) is called spherical harmonic function, which is the angular eigenfunction of Helmholtz equation under the natural periodic condition and the finite boundary value condition. To solve the propagation problem between two concentric spheres, the spherical harmonic transformation (SHT) is defined by:

(4)$$U_l^m(r) = \int\limits_\Omega {u(r,\theta ,\phi )Y{{_l^m}^\ast }(\theta ,\phi )d\Omega } .$$

And the inverse spherical harmonic transformation (ISHT) can be given by replacing C_lmR(r) with spherical harmonic spectrum $U_l^m(r)$ in Eq. (3). By using SHT, the propagation of light field on the sphere can be equivalent to the propagation of spherical wave spectrum. And the expression of transfer function is determined by R(r). From [21], we can obtain the solution of R(r):

(5)$$R(r) = {R_1}h_l^{(1)}(kr) + {R_2}h_l^{(2)}(kr),$$

where $h_l^{(1)}(kr)$ and $h_l^{(2)}(kr)$ are the first and second kinds of spherical Hankel functions. To obtain the exact expression of Eq. (5), it is essential to consider the boundary conditions and the physical significance of propagation model. There are two different propagation directions for the inner sphere as shown in Figs. 2(a) and (b). One is along the exterior normal and corresponds to SFP, while the other is along the inner normal and corresponds to SBP. They have different physical meanings. Therefore, the transfer functions are variant for different propagation models.

Fig. 2. Spherical front-propagation and back-propagation models between concentric spheres. (a) SFP. (b) SBP. (c) Stereoscopic view.

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Assume that the radii of the inner sphere and the outer sphere are r₁ and r₂, respectively. When the light field distribution u(r₁) is known, the spherical wave spectrum U(r₁) can be calculated by Eq. (4). From Eq. (3), the propagation equations can be deduced as:

(6)$$u({r_2},\theta ,\phi ) = \sum\limits_{l = 0}^\infty {\sum\limits_{m ={-} l}^l {U({r_2})Y_l^m} } (\theta ,\phi ),\; U({r_2}) = \frac{{R({r_2})}}{{R({r_1})}}U({r_1}).$$

From Eq. (6), the transfer function can be defined as:

(7)$$TF = \frac{{R({r_2})}}{{R({r_1})}},$$

where R(r₁) and R(r₂) are determined by Eq. (5). Therefore, to determine the expression of TF, the function R(r) should be considered first. Hankel functions of different kinds have their own significance when describing the propagation process of waves. The first kind of Hankel function is used to describe the divergent waves while the second kind is used to describe the converging waves. The transfer functions of SFP and SBP are separately defined by:

(8)$$T{F_{SFP}} = \frac{{h_l^{(1)}(k{r_2})}}{{h_l^{(1)}(k{r_1})}},$$

(9)$$T{F_{SBP}} = \frac{{h_l^{(1)}(k{r_2})}}{{h_l^{(2)}(k{r_1})}}.$$

2.3 Spherical self-diffraction

SSD is proposed based on SBP. If the radius of the sphere is r, to obtain the calculation method of SSD, the only thing to do is to replace r₁ and r₂ with r in Eq. (9). However, when we deduce the propagation models between two concentric spheres, the radius constraint is r₁<r₂. Thus, the transfer function of the SSD is defined by:

(10)$$T{F_{SSD}} = \frac{{h_l^{(1)}(kr)}}{{h_l^{(2)}(kr)}}.$$

The calculation result of SSD can be obtained by preforming SHT and ISHT, and the transfer function TF_SSD describes the propagation process. If the initial complex amplitude distribution of the object is C₁, SSD calculation of C₁ is defined by:

(11)$${C_1}^\prime = {\textrm{ISHT}} [{{\textrm{SHT}} ({{C_1}} )\times T{F_{SSD}}} ].$$

As shown in Fig. 3, SSD is a special case of SBP at limit condition, that is, when r₂ approaches r₁ infinitely, the object diffracted to the back will fall on its own sphere, and the SSD process is completed. In this model, the object and diffraction distributions are on the same sphere. The object is illuminated by the convergent light. Though SSD cannot reconstruct the image on a whole sphere, since part of the sphere is occupied by the original object, it can remain the large-viewing-area characteristic of curved hologram. Compared with the traditional SFP model, SSD has a great advantage in suppressing speckle noise particularly. The correctness of both SBP and SSD is demonstrated in Section 3.1 through numerical simulations.

Fig. 3. Spherical self-diffraction model.

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2.4 Spherical self-diffraction iteration algorithm

The SSDI algorithm is proposed based on SSD, which can effectively suppress the speckle noise of reconstructed image of SPOH. The flowchart of SSDI algorithm is shown in Fig. 4. The subscript k represents the kth iteration process. If the number of iterations is n, k = 1, 2, 3, …, n. Firstly, the amplitude distribution A₀ of the object is added with the random phase θ₁ that ranges from 0 to 2π as the initial estimation phase. For the kth iteration, the complex amplitude distribution C_k can be obtained by:

(12)$${C_k} = {A_0} \times \exp (j{\theta _k}).$$

The SSD result of C_k is calculated by Eq. (11), which is expressed as C_k’. Then the phase φ_k of C_k’ is reserved. The function phase(·) is used to obtain the phase of the variable. Forcing the amplitude to be A₀ for amplitude constraint, the complex amplitude distribution D_k is expressed as:

(13)$${D_k} = {A_0} \times \exp (j{\varphi _k}).$$

Fig. 4. Flowchart of SSDI algorithm.

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Since SSD and ISSD are conducted for the same sphere, they essentially describe the same propagation process except the propagation direction. ISSD calculation of D_k is defined by:

(14)$${D_k}^\prime = {\textrm{ISHT}} \{{{\textrm{SHT}} [{{\textrm{conj}} ({D_k}) \times T{F_{SSD}}} ]} \},$$

where function conj(·) is used to obtain the conjugate of the variable. The phase θ_k₊₁ of D_k’ is reserved, and it is noted that the subscript k has changed to k+1 to distinguish from the last iteration:

(15)$${\theta _{k + 1}} = {\textrm{phase}} ({D_k}^\prime ).$$

Using Eq. (11) to Eq. (14), the amplitude distribution on the sphere in each iteration can be calculated orderly. The phase-only hologram H can be obtained after the iterations are finished:

(16)$$H = \exp (j{\varphi _n}).$$

The effectiveness of SSDI algorithm on speckle suppression can be proved from its characteristics of amplitude squared error (ASE) reduction and convergence. ASE reduction assures that the performance of the suppression is improved with the increase of iteration times, and fast convergence can greatly reduce the calculation time. The ASE of SSDI algorithm is defined by:

(17)$$ASE = \sum\limits_{\textrm{all}\;\textrm{points}} {{{|{\Delta {A_k}} |}^2}} \textrm{ = }\sum\limits_{\textrm{all}\;\textrm{points}} {{{|{{A_k} - {A_0}} |}^2}} ,$$

where △A_k is the difference between amplitudes of object and the result of SSD, ∑ represents the summation of the amplitude differences of all points. The main idea of SSDI algorithm refers to Gerchberg-Saxton iteration whose ASE reduction and convergence have been proved in [22]. It is pointed out that the ASE must decrease or remain constant after each FFT in Gerchberg-Saxton algorithm according to Parseval’s theorem. In our proposed algorithm, FFT or the traditional plane diffraction calculation is replaced by SSD to calculate the spherical hologram. ASE reduction with the increase of iteration times is greatly related to the conservation of energy. The structure of sphere which is the carrier of self-diffraction can assure that the energy remains unchangeable before and after SSD. Therefore, the iteration process shown in Fig. 4 can make the ASE decrease and ensure its convergence.

3. Simulation results

3.1 Correctness of SBP and SSD

Firstly, to demonstrate the correctness of SBP, the diffraction and reconstruction process of SFP and SBP were simulated together. In this experiment, the reconstruction of original object is conducted using complex amplitude reconstruction and the speckle noise is not considered. We chose an object composed of 512×1024 pixels to assure that the sampling intervals are the same in the latitude and longitude directions. The radii of the inner and outer spheres were set to be 10 mm and 100 mm, respectively, and the wavelength was chosen to be 280 µm. The software for the simulation experiments is Python 3.7.0, and the package pyshtools of Python software is used to conduct a complex SHT operation.

The diffraction results of the object and its reconstruction results using different propagation models are shown in Fig. 5(a). SFP and SBP models are both used to reconstruct the object for each diffraction result, from which we can find out the influence caused by different propagation models in the reconstruction process. Only the propagation models are consistent in the diffraction and reconstruction process can the object be reconstructed correctly. Otherwise we can clearly observe the blurry reconstructed images. It is noted that a “shift” phenomenon is observed in the diffraction result of SBP and the two blurry images. The shift is caused by the same spherical expansion approach of different propagation models, which is shown in Fig. 5(b). Therefore, the calculation result of SFP is normal while the calculation result of SBP has a shift. The three images with a shift shown in Fig. 5(a) are all obtained after one calculation of SBP.

Fig. 5. Simulations of diffraction and reconstruction using SFP and SBP. (a) Simulation results. (b) Spherical expansion approach.

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Then the correctness of SSD was proved. The objects were all composed of 512×1024 pixels in this experiment. From Section 2.3, we know that the SSD is a special case of SBP. Therefore, the reversal will occur in the expanded picture of its diffraction result, which can be observed in Fig. 5(a). Though it is just caused by a certain spherical expansion approach, we can change the pattern of the object to assure the diffraction pattern located in the center of the expansion picture, which can be seen from the pictures of the first line of Fig. 6. Besides, it is meaningless to conduct SSD calculation for the object on a whole sphere since the object and the diffraction result of SSD will have partial overlap, which makes it hard to distinguish the diffraction result from the object. Therefore, the patterns only occupy hemi-sphere in the simulation.

Fig. 6. Diffraction results of SSD with different radii.

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The diffraction results of different objects with several discrete radii of the sphere are shown in Fig. 6. From the results, it shows that with the increase of radius, the diffraction result becomes clear, which is seemingly contrary to the traditional diffraction theory. However, it can be explained from the structure of sphere. When the radius increases, the influence of diffraction from other points on the diffraction sphere will be reduced for each sampling point. Therefore, the diffraction result presented as the spherical expansion picture seems closer to the object. Since the number of sampling points remained unchanged in the experiment, the area of a single pixel on the diffraction sphere becomes larger with the increase of radius. The stereoscopic simulation diagram of SSD is shown in Fig. 7, from which we can understand the process of SSD easily.

Fig. 7. Stereoscopic simulation diagram of SSD.

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3.2 Speckle suppression test

To show the effectiveness of SSDI algorithm on speckle suppression, the calculation time and Peak Signal to Noise Ratio (PSNR) value of the reconstructed images of SPOHs with different iterations are compared. The calculation of PSNR is defined by [23]:

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

where f and g are the amplitude of original and reconstructed images, respectively. M and N are separately the sampling numbers in the horizontal and vertical directions. PSNR can be used to evaluate the quality of reconstructed image compared to the original. When PSNR is higher, the reconstruction quality is better.

The object is composed of 512×1024 pixels in this experiment. The radii of the sphere are set to be 10 mm and 20 mm, and the results are shown in Fig. 8 and Fig. 9, respectively. It is obvious that with the increase of iterations, the reconstruction quality is improved. Different from the traditional iteration algorithm, SSDI needs less iteration times to achieve the improvement of reconstruction quality, which can greatly reduce the calculation time. The change of PSNR under different radii and calculation time with the increase of iterations are shown in Fig. 10. It shows that PSNR will quickly reach a gradual value, and the calculation time increases approximately linearly. Therefore, through only 5∼10 iterations we can obtain the phase-only hologram that can reconstruct a high-quality image.

Fig. 8. Reconstructed images of the phase-only holograms with different iteration times under radius of 10 mm. (a) No iterations. (b) 1 iteration. (c) 3 iterations. (d) 5 iterations. (e) 10 iterations. (f) 20 iterations.

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Fig. 9. Reconstructed images of the phase-only holograms with different iteration times under radius of 20 mm. (a) No iterations. (b) 1 iteration. (c) 3 iterations. (d) 5 iterations. (e) 10 iterations. (f) 20 iterations.

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Fig. 10. The change of PSNR and calculation time with the increase of iterations. (a) The change of PSNR under different radii. (b) The change of calculation time.

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However, from the results we can see that if the number of sampling points of the object remains unchanged, with the increase of radius, the improvement of reconstruction quality using SSDI will decrease, which seemingly the SSDI algorithm losses the effectiveness to improve the reconstruction quality. Actually, with the increase of radius, the number of sampling points should also be increased. As shown in Fig. 11, the speckle suppression effect of SSDI algorithm is various with the number of sampling points and the radius of the sphere. From the experiment results we can deduce that it is necessary to increase the number of sampling points for the sphere with large radius.

Fig. 11. Comparison of reconstructed images of sunflowers before and after using SSDI algorithm. (a) 512×1024 sampling points and r=10 mm. (b) 512×1024 sampling points and r=20 mm. (c) 1024×2048 sampling points and r=20 mm.

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To further demonstrate the effectiveness of SSDI algorithm, variant pictures are used to compare the reconstruction quality before and after the iteration as shown in Fig. 12. The results show that through 5 iterations we can obtain a high-quality reconstructed image.

In the end, a simulation experiment is conducted to verify that the iteration algorithm based on SSD is much more effective to suppress the speckle noise than the iteration method based on SFP. That is the reason why we propose SSD and SSDI iteration algorithm. The objects are both composed of 512×1024 pixels and the radii r and R are set to be 10 mm and 100 mm, respectively. From the results shown in Fig. 13, the iteration method based on SFP can hardly improve the quality of reconstructed image through few iterations. Its maximum PSNR value that can be increased is close to the PSNR value with no iterations. In contrast, SSDI has a significant advantage in improving the quality of reconstructed image.

Fig. 12. Quality comparison between the reconstructed images using SSDI or without iterations. (a)(b)(c) No iterations. (d)(e)(f) SSDI through 5 iterations.

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Fig. 13. The speckle-suppression ability comparison between SFP iteration and SSD iteration algorithm. (a)(b)(c) The reconstruction effect using SFP iteration algorithm under different iteration times. (d)(e)(f) The reconstruction effect using SSD iteration algorithm under different iteration times.

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4. Conclusion

In this paper, the speckle noise in spherical holography is considered firstly, which greatly degrades the quality of reconstructed image. It can be effectively suppressed by the proposed SSDI algorithm based on the SSD model. Compared with using the iteration algorithm based the SFP model, which is a traditional spherical propagation model between concentric spheres, significant improvement is achieved by using the SSDI algorithm based on SSD in suppressing speckle noise. SSD is a special case of SBP when the radii of the concentric spheres are the same and the object and diffraction result are on the same sphere. The correctness of SBP and SSD as well as the effectiveness of SSDI algorithm are verified by numerical simulations. The simulation results show that the SSDI algorithm can reconstruct an image with high quality after 5 iterations, which can greatly reduce the calculation time. In addition, SSD model can be applied to other optimization iterative algorithms [24] to improve the quality of reconstructed image in the future. Although we don’t have the optical devices of spherical holography at the moment to provide the optical experimental evidence, proposed propagation models and speckle suppression method in this paper have greatly enriched the theory of spherical holography. With the development of meta-surface holography [25] and flexible materials [26], spherical holography could develop gradually from theory to practice, and could realize holographic 3D display with a large view in the future.

Funding

National Natural Science Foundation of China (U1933132); Chengdu Science and Technology Program (2019-GH02-00070-HZ).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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Spherical self-diffraction for speckle suppression of a spherical phase-only hologram

Abstract

1. Introduction

2. Theoretical analysis

2.1 Random phase and speckle noise

2.2 Spherical front and back propagation models

2.3 Spherical self-diffraction

2.4 Spherical self-diffraction iteration algorithm

3. Simulation results

3.1 Correctness of SBP and SSD

3.2 Speckle suppression test

4. Conclusion

Funding

Disclosures

References

Cited By

Figures (13)

Equations (18)

Optics Express