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Phase hologram optimization with bandwidth constraint strategy for speckle-free optical reconstruction

Lizhi Chen, Songzhi Tian, Hao Zhang, Liangcai Cao, and Guofan Jin

Open Access Open Access

Abstract

An iterative method with bandwidth constraint strategy is proposed to design phase holograms for high-quality speckle-free optical reconstruction. The bandwidth properties of the reconstructed field are analyzed theoretically based on the sampling theory, which helps in properly allocating the sampling resources for efficiently describing the speckles and artifacts in the reconstructed field. Iterative calculation with bandwidth constraint strategy of the reconstructed field and quadratic initial phase can optimize the phase hologram without stagnation problem, which provides effective controls of the reconstructed intensity fluctuations and helps to suppress the speckles and artifacts. Numerical and optical experiments have been performed to validate the proposed method can achieve excellent image fidelity.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Computer-generated phase holograms can reconstruct the desired intensity distribution in the image plane with high optical efficiency by modulating the phase component of the incident light [1]. The development of phase-only spatial light modulators (SLMs) makes it possible to dynamically upload computer-generated phase holograms. Nowadays, phase holograms have been widely used in many applications due to their high optical efficiency and flexible configuration, such as beam shaping, optical tweezer, three-dimensional (3D) displays and near-eye displays [2–6]. Whereas, the phase hologram synthesis is an ill-posed inverse problem. Severe speckles or artifacts would occur in the optical reconstruction if they are not handled properly. In recent decades, it has been under intense investigation to optimize phase holograms for high-quality optical reconstruction.

To improve the reconstruction quality of phase holograms, various optimization methods have been proposed. These methods can be mainly categorized into two types of optimization scheme, namely non-iterative [7–13] and iterative [14–19] methods. Non-iterative methods possess the merit of low computation load; whereas, the robustness of these methods would easily be affected due to the lack of further optimization according to different target images. Iterative methods try to search and optimize the phase profile of phase holograms according to specific target image, which would be more effective for enhancing the reconstruction quality.

The error-reduction iterative algorithm was widely used to optimize phase holograms due to its simplicity and flexibility, which was introduced by Gerchberg and Saxton to solve phase retrieval problems [20]. The error-reduction algorithm is consequently also referred to as the Gerchberg-Saxton (GS) algorithm in the literature. The error-reduction algorithm was extensively studied by Fienup who introduced the input-output class of algorithms to speed up convergence [21]. To further improve the performance of these algorithms, some modified iterative algorithms have been proposed to optimize phase holograms by introducing amplitude freedom into the image plane [22–24]. In these modified algorithms, the reconstruction quality can be further improved in the signal region by relaxing the amplitude constraint of the non-signal region. However, due to the lack of overall and effective constraint of the whole reconstructed intensity distribution, the optical reconstruction would still be contaminated by the speckle noise. The time-average method [25] and the pixel separation method [26,27] can be combined with the iterative methods to effectively suppress the speckles in the optical reconstruction; whereas, these methods require a dynamic device with high frame rates. Recently, double-constraint iterative algorithms were introduced to suppress the speckle noise by simultaneously constraining the amplitude and phase distribution of the signal region [28,29]. However, the bandwidth properties of the reconstructed field were not considered in these methods, which would cause artifacts in the optical reconstruction.

In order to effectively suppress the speckles as well as artifacts in the optical reconstruction, the bandwidth property of the reconstructed intensity distribution should be considered during the phase hologram optimization. For example, the bandwidth of the reconstructed intensity distribution is larger than the bandwidth of the reconstructed complex amplitude distribution in the case of random phase. To effectively control the intensity fluctuations of the reconstructed field, smaller sampling interval should be chosen and more sampling points require to be constrained during the phase hologram optimization; otherwise, the intensity distribution of the non-sampling points might be far away from the ideal distribution, which would cause severe speckles or artifacts in the optical reconstruction. To solve such a problem, Wyrowski et al. masterly designed an iterative Fourier transform algorithm applied to computer holography for speckle-free reconstruction [30–32]. In this method, the speckles can be effectively eliminated by keeping the reconstructed complex amplitude field band-limited. Whereas, this method was not used to optimize phase holograms but to optimize the bandwidth of the reconstructed field, which needs a further coding process according to the modulation type of the holograms. Later, Wyrowski et al. introduced the theory of speckles in diffractive optics and applied it to the design of diffractive optical elements (DOEs) for beam shaping [33]. Whereas, the design of the initial signal phase used in the iterative process was not analyzed, which would affect the convergence of the iterative algorithm. In addition, optical experiments were not implemented for demonstrating the optical reconstruction quality. In recent years, some derived methods were also presented to design DOEs for speckle-suppressed beam shaping [34,35]. In these methods, the sampling intervals were reduced to constrain more sampling points in the image plane. Whereas, the bandwidth properties of the reconstructed field were not paid enough attention. Random phase was used as the initial phase of the iterative process, which would affect the band-limit constraint of the reconstructed field and cause the stagnation problem of the iterative algorithm. Phase singularities widely existing in the random initial phase cannot be eliminated during the iterative process, which would still cause severe speckles in the optical reconstruction [36].

In this study, we propose an iterative method with bandwidth constraint strategy to design phase holograms for high-quality speckle-free optical reconstruction. The bandwidth properties of the reconstructed field are analyzed both mathematically and physically based on the sampling theory, which helps in allocating sampling resources for efficiently describing the speckles and artifacts in the reconstructed field. During the phase hologram optimization, a proper sampling interval is applied to the reconstructed field based on the bandwidth property of the reconstructed intensity distribution. Then, iterative calculation with bandwidth constraint strategy of the reconstructed field is utilized to optimize phase holograms with speckle-free reconstruction. To avoid the stagnation problem of the iterative algorithm caused by the random initial phase, a band-limited quadratic phase is used as the initial phase distribution. The parameter setting of the quadratic initial phase and its influence on the convergence of the iterative algorithm are analyzed in this study. Our proposed method can provide overall and effective controls of the reconstructed intensity fluctuations and suppress the speckles and artifacts in the reconstructed field. Numerical and optical results confirm that our proposed method can be used to design phase holograms with excellent image fidelity.

2. Bandwidth analysis of the reconstructed field

In this study, we focus on the optimization of Fourier phase holograms. Iterative algorithms can be used to optimize Fourier phase holograms for improving the reconstruction quality. Whereas, the bandwidth properties of the reconstructed field were not paid attention in the majority of iterative algorithms, which would cause a large discrepancy between the numerical and optical reconstructions. In the following part, the bandwidth properties of the reconstructed field are analyzed in detail based on the sampling theory; and then a proper sampling strategy is used to efficiently describe the overall intensity distribution of the reconstructed field.

In the optical Fourier transform system, as shown in Fig. 1, the reconstructed complex amplitude field A in the image plane can be obtained by the Fourier transform of the complex amplitude field H in the hologram plane, where H = hexp(iФ), h is the amplitude distribution of the incident wave, and Ф is the phase distribution of the phase hologram. Due to the finite size of the hologram, the complex amplitude field in the hologram plane is expressed as

$$H(x,y) = h({x,y} )\textrm{exp}[{i\Phi ({x,y} )} ]\textrm{rect}\left( {\frac{x}{a}} \right)\textrm{rect}\left( {\frac{y}{b}} \right), $$
where x, y represent the coordinates in the hologram plane, respectively. a is the size of the hologram in x direction, and b is the size of the hologram in y direction. Due to the finite size of the hologram, the bandwidth of the reconstructed complex amplitude field A must be limited. For simplicity, one-dimensional analysis is adopted here. Under paraxial approximation, the spatial frequency range of the reconstructed complex amplitude field A in u direction can be described as
$$- \frac{a}{{2\lambda f}} \le {f_u}\{A \}\le \frac{a}{{2\lambda f}}, $$
where λ represents the light wavelength, and f represents the focal length of the Fourier lens in the optical Fourier transform system. The highest spatial frequency of the reconstructed complex amplitude field A in u direction is limited by the size of the hologram, which is easy to be understood from the point of the marginal light propagation in the hologram plane, as shown in Fig. 2. It shows that the highest spatial frequency of the reconstructed complex amplitude field A is decided by the maximum angle θ of the incident light in the image plane. While, the maximum incident angle θ is limited by the size of the hologram. Under the paraxial approximation, the maximum incident angle θ is calculated as
$$\theta \approx \frac{a}{{2f}}, $$
where a represents the size of the hologram in x direction, and f represents the focal length of the Fourier lens. Herein, the spatial frequency range of the reconstructed complex amplitude field A in u direction can also be deduced from Eq. (3), and the result is the same with Eq. (2).

 figure: Fig. 1.

Fig. 1. Schematic diagram for optical Fourier transform system: the hologram plane is located in the front focal plane of the Fourier lens, and the image plane is located in the back focal plane of the Fourier lens.

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 figure: Fig. 2.

Fig. 2. Schematic diagram for marginal light propagation model of the hologram plane in the optical Fourier transform system.

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In the most applications of computer holography, such as beam shaping and holographic display, the reconstructed intensity distribution is of the interest, which can be calculated as

$$I = A \cdot {A^\ast }, $$
where I represents the reconstructed intensity distribution, A represents the reconstructed complex amplitude field, and A* represents the complex conjugate of the reconstructed complex amplitude field. It deserves to be noted that the bandwidth of the reconstructed intensity distribution is different from that of the reconstructed complex amplitude field. According to the autocorrelation theorem [37], the spectra of the reconstructed intensity I and the spectra of the reconstructed complex amplitude A are related by an autocorrelation as
$$F\{I \}= F\{A \}{\bf \odot }F\{A \}, $$
where $\odot$ represents the autocorrelation operation, and F{} represents the Fourier spectra. Herein, the bandwidth of the reconstructed intensity I can be described as
$$B\{I \}\le 2B\{A \}, $$
where B{} represents the bandwidth. The possible bandwidth of the reconstructed intensity distribution would be larger than the bandwidth of the reconstructed complex amplitude field. Hence, the spatial frequency range of the reconstructed intensity I in u direction can be described as
$${f_u}\{I \}\le 2{f_{u(\rm max)}}\{A \}= \frac{a}{{\lambda f}}. $$
This can be explained from the point of light propagation model in the optical Fourier transform system as well. In Fig. 2, the highest spatial frequency of the reconstructed intensity distribution would occur when the separate incident light with maximum incident angle +θ and –θ interfere in the image plane. According to the principle of interference, the highest spatial frequency of the reconstructed intensity distribution in the image plane is calculated as
$${f_{u(\rm max)}}\{I \}= \frac{{2\sin \theta }}{\lambda } \approx \frac{2}{\lambda } \cdot \frac{a}{{2f}} = \frac{a}{{\lambda f}}, $$
which is the same with Eq. (7).

According to the Shannon’s sampling theorem, when a discrete calculation is performed, the reconstructed intensity I requires to be sampled with sampling intervals

$$\Delta u \le \frac{{\lambda f}}{{2a}}\textrm{ },\textrm{ }\Delta v \le \frac{{\lambda f}}{{2b}}. $$

In the majority of iterative algorithms applied to the design of Fourier holograms, the fast Fourier transform (FFT) algorithm is implemented to speed up the calculation of the discrete Fourier transform. Hence, the sampling interval of the image plane in u direction is calculated as $\Delta u^{\prime}=\lambda f / a$, where a represents the size of the hologram in x direction. According to Eq. (9), the sampling interval $\Delta u^{\prime}$ is insufficient to describe the reconstructed intensity distribution in the image plane. Using the sampling interval $\Delta u^{\prime}$ to implement the iterative algorithm, the speckles and artifacts in the reconstructed field cannot be described and eliminated effectively. In this way, although seemingly “perfect” reconstruction can be obtained in numerical simulations, the optical reconstructions would be contaminated by the speckles and artifacts. Based on our theoretical analysis, a correct sampling interval $\Delta u^{\prime}=\lambda f / 2a$ in u direction should be chosen to effectively describe the reconstructed intensity distribution. To reduce the computation load, the sampling interval $\Delta u^{\prime}=\lambda f / 2a$ can be set and sufficient to sample the reconstructed intensity distribution in u direction. Similarly, the sampling interval in v direction can be set as $\Delta v^{\prime}=\lambda f / 2b$.

In the FFT based iterative algorithm, it can be realized to reduce the sampling interval of the reconstructed field by applying zero-padding operation to the hologram plane, as shown in Fig. 3. An optimized phase hologram is generated by the GS algorithm, to modulate a uniform incident plane wave for producing a desired intensity distribution in the image plane. The zero-padding operation is applied to the hologram plane, where the pixel pitch of the hologram plane is not changed, but the size of the hologram plane is zero-padded to 2a×2b. According to the FFT algorithm, the sampling interval of the reconstructed field is changed as

$$\Delta u^{\prime} = \frac{{\lambda f}}{{2a}}\textrm{ },\textrm{ }\Delta v^{\prime} = \frac{{\lambda f}}{{2b}}, $$
which is sufficient to describe the reconstructed intensity distribution. In Fig. 3, it is worth to mention that severe speckle noise in the reconstructed image can be seen only by reducing the sampling interval of the reconstructed field. Hence, to effectively suppress the speckles and artifacts in the reconstructed field, the correct sampling should be applied to the reconstructed field in the iterative design of holograms.

 figure: Fig. 3.

Fig. 3. Schematic diagram of reducing the sampling interval of the reconstructed field by applying zero-padding operation to the hologram plane.

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3. Method

Based on the bandwidth analysis of the reconstructed field, a proper sampling should be applied to the reconstructed field for efficiently describing the reconstructed intensity distribution. In the FFT based iterative algorithm, it can be realized to reduce the sampling interval of the reconstructed field by applying the zero-padding operation to the hologram plane. According to the Shannon’s sampling theorem as well as the bandwidth property of the reconstructed intensity distribution, the size of the hologram plane requires only to be zero-padded to twice the size of the original ones, which can largely reduce the computation load. In what follows, a simple and effective iterative algorithm with bandwidth constraint strategy of the reconstructed field is presented to design phase holograms with speckle-free reconstruction in the case of uniform incident plane wave.

3.1 Bandwidth constraint optimization algorithm

The working principle of our proposed iterative algorithm is shown in Fig. 4. Generally, parameters of the phase hologram are fixed before the phase hologram optimization. Here, assuming that the resolution of the phase hologram is M×N, the pixel pitch is p, and the size of the phase hologram is a×b, where a = Mp, b = Np. During the phase hologram optimization, the pixel pitch of the hologram plane is not changed, while the size of the hologram plane is zero-padded to 2a×2b for reducing the sampling interval of the reconstructed field. Then, the number of the sampling points in the hologram plane is changed as 2M×2N, which is the same with the number of the sampling points in the image plane. Therefore, a target image with the resolution of 2M×2N requires to be obtained by the interpolation method before the phase hologram optimization, such as a sinc interpolation [38]. It should be noted that the physical size of the phase hologram is still a×b, which is located in the central area of the zero-padded hologram plane. Due to the finite area of the phase hologram, the bandwidth of the reconstructed complex amplitude field A with the resolution of 2M×2N must be limited. Herein, a bandwidth constraint optimization algorithm is introduced here to design the phase hologram.

 figure: Fig. 4.

Fig. 4. Schematic diagram of bandwidth constraint optimization algorithm.

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In the iterative algorithm, the initial phase ${\varphi _0}$ is applied to the target image as ${A_0} = \sqrt {{I_0}} \textrm{exp}({i{\varphi_0}} )$, where ${I_0}$ is the intensity distribution of the target image, and the resulting complex amplitude A0 is set as the initial input of the iterative process. In kth iteration, the complex amplitude field ${H_k} = |{{H_k}} |\textrm{exp}({i{\mathrm{\Phi }_k}} )$ in the hologram plane is calculated by the inverse Fourier transform of the complex amplitude field Ak in the image plane. The phase hologram can be generated from the complex amplitude field Hk after a phase extraction and cropping operation. On the other hand, to further enhance the reconstruction quality, more iterations require to be implemented. Specifically, the phase component of the complex amplitude field Hk is reserved, while the amplitude component outside of the phase hologram area is cleared to ensure band-limit constraint of the reconstructed field, and the amplitude component inside of the phase hologram area is normalized. Then, a modified complex amplitude field $H^{\prime}_{k}$ is generated in the hologram plane. The above process is also equivalent to applying the zero-padding operation to the phase hologram. Subsequently, the reconstructed complex amplitude distribution $A^{\prime}_{k}$ in the image plane is calculated by the Fourier transform of the complex amplitude field $H^{\prime}_{k}$. The input complex amplitude distribution Ak+1 for the next iteration is generated from $A^{\prime}_{k}$ after the amplitude constraint in the image plane.

To broaden the optimization space of the iterative algorithm and further enhance the reconstruction quality, weighted constraint strategy is introduced into the iterative process [24]. The image plane is partitioned into two regions according to the intensity distribution of the target image. The signal region is the area where the signal pattern is located, and the non-signal region is the area where there is no designed signal. During the iterative process, the enforced amplitude constraint for the next iteration in the image plane is

$$|{{A_{k\textrm{ + }1}}} |= \left\{ {\begin{array}{c} {\textrm{ }\sqrt {{I_0}} \textrm{ , (}u,v\textrm{)} \in S}\\ {\textrm{ }|{A{^{\prime}_k}} |\textrm{ , (}u,v\textrm{)} \notin S} \end{array}} \right., $$
where (u,v) are the coordinates in the image plane, and S denotes the signal region. The amplitude constraint strategy in the image plane improves the reconstruction quality in the signal region by relaxing the amplitude constraint in the non-signal region.

In order to effectively suppress the noise in the non-signal region, total energy in the hologram plane is controlled based on the energy conservation principle. In the Fourier transform system, the energy in the reconstructed image plane is equal to the energy in the hologram plane:

$$\int\!\!\!\int {{{|{A{{^\prime}_k}(u,v)} |}^2}dudv = } \int\!\!\!\int {{{|{H{{^\prime}_k}(x,y)} |}^2}dxdy}. $$

By controlling the total energy in the hologram plane, the total energy in the reconstructed image plane is constrained. During the iterative process, the total energy in the hologram plane is set as

$$\int\!\!\!\int {{{|{H{{^\prime}_k}(x,y)} |}^2}dxdy} = \int\!\!\!\int {{{|{{A_0}(u,v)} |}^2}dudv}, $$
where |A0(u,v)| is the amplitude distribution of the target image. The amplitude of the incident plane wave in the hologram plane is easy to be calculated from the total energy in the hologram plane. According to Eqs. (12) and (13), the following formula can be deduced:
$$\int\!\!\!\int {{{|{A{{{^\prime}}_k}(u,v)} |}^2}dudv = } \int\!\!\!\int {{{|{{A_0}(u,v)} |}^2}dudv}. $$
In this way, the total energy in the reconstructed image plane is equal to the energy of the target image. Through the iterations, more and more energy is gathered in the signal region by the direct amplitude constraint of the signal region, so the noise in the non-signal region is suppressed effectively. The above operation provides a larger optimization space of the iterative algorithm while maintaining overall and effective constraint in the image plane.

3.2 Choice of initial phase

In the majority of iterative algorithms applied to the design of the holograms, the usual choice of the initial complex amplitude in the image plane is ${A_0} = \sqrt {{I_0}}\cdot \textrm{exp}({i{\varphi_0}} )$ with a random phase ${\varphi _0}$. However, a random initial phase introduces many phase singularities, which would violate the band-limit constraint of the reconstructed field and cause the stagnation problem of the bandwidth constraint optimization algorithm. Numerical simulations of our proposed iterative algorithm with a random initial phase are performed to illustrate this problem, as shown in Fig. 5. It shows that even with a large number of iterations, the speckles in the reconstructed image cannot be eliminated due to the stagnation problem.

 figure: Fig. 5.

Fig. 5. Numerical simulations of our proposed bandwidth constraint optimization algorithm with a random initial phase: (a) target image, sampled in 600×600 pixels; reconstructed image after (b) 5 iterations, (c) 20 iterations, and (d) 100 iterations.

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To avoid the stagnation problem caused by the random initial phase, a possible means is to use a band-limited initial phase. Such a band-limited initial phase ought to ensure the band-limit constraint of the reconstructed field and distribute the entire signal energy as uniformly as possible into the physical area of the holograms. Here, a band-limited quadratic phase is used as the initial phase to approximately satisfy these conditions, which is expressed as ${\varphi _0}({m,n} )= k{m^2} + l{n^2}$, where k and l are constant parameters, m and n are pixel coordinates of the image plane. The parameters of the quadratic initial phase are important to optimize phase holograms. For different values of parameters k and l, the bandwidth of the initial complex amplitude A0 is different, as shown in Fig. 6. The parameters k and l should be chosen carefully so that the size of the Fourier spectrum of the initial complex amplitude A0 is as close as possible to the physical size of the phase hologram. Assuming that the resolution of the signal region in the image plane is P×Q, and the sampling interval of the image plane is denoted by Δu and Δv, respectively. Then, the quadratic initial phase in the physical coordinates can be expressed as ${\varphi _0}({m \Delta u,n \Delta v} )= k{m^2} + l{n^2}$. The local spatial frequency of the quadratic initial phase ${\varphi _0}$ in u direction is calculated as

$${f_u}\{{{\exp} ({i{\varphi_0}} )} \}{ = }\frac{1}{{2\pi }} \cdot \frac{{\partial {\varphi _0}}}{{\partial u}} = \frac{{km}}{{\pi \Delta u}}. $$
The physical size of the quadratic initial phase is the same with the size of the signal region in the image plane. Hence, the maximum spatial frequency of the quadratic initial phase ${\varphi _0}$ in u direction is calculated as
$${f_{u({\max})}}\{{{\exp} ({i{\varphi_0}} )} \}{ = }\frac{{|k |P}}{{2\pi \Delta u}}. $$

 figure: Fig. 6.

Fig. 6. Different complex amplitude distribution in the hologram plane calculated by the inverse Fourier transform of the initial complex amplitude field with different quadratic phase in the image plane.

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In our method, Δu is calculated as

$$\Delta u = \frac{{\lambda f}}{{2a}}, $$
where a represents the size of the phase hologram in x direction, λ represents the light wavelength, and f represents the focal length of the Fourier lens.

To obtain a Fourier spectrum of the initial complex amplitude A0 as close as possible to the physical size of the phase hologram, we can derive the following equation from Eq. (2):

$${f_{u(\textrm {max})}}\{{{\exp} ({i{\varphi_0}} )} \}= \frac{a}{{2\lambda f}}. $$
Combining Eqs. (16)–(18), we can obtain:
$$k ={\pm} \frac{\pi }{{2P}}. $$
In a similar way, we can obtain:
$$l ={\pm} \frac{\pi }{{2Q}}. $$

4. Reconstruction results

To validate the feasibility of our proposed method, numerical simulations are performed for quantitative evaluation. The performance of our proposed method is compared with the GS algorithm [20]. To compare the simulation results, some evaluation functions are used here. The first one is root-mean-square error (RMSE) between the target image and the reconstructed image. The RMSE is calculated as:

$$\textrm{RMSE} = \sqrt {\sum\limits_{m = 1}^M {\sum\limits_{n = 1}^N {{{[{{I_r}({m,n} )- {I_t}({m,n} )} ]}^2}/\sum\limits_{m = 1}^M {\sum\limits_{n = 1}^N {{{[{{I_t}({m,n} )} ]}^2}} } } } }, $$
where M and N are the horizontal and vertical resolution of images, Ir(m,n) is the intensity distribution of the reconstructed image, and It(m,n) is the intensity distribution of the target image. The RMSE indicates how well the reconstructed image agrees with the target one. The lower the value of RMSE is, the better the reconstruction quality will be. The formula of RMSE used here is slightly different from that used in our previous research paper [13], but both these two formulas of RMSE can be used for quantitative evaluation, and the only difference lies in the normalization operation. Another well-known evaluation function is structural similarity index measure (SSIM) and it is given by
$$\textrm{SSIM} = \frac{{({2{\mu_t}{\mu_r} + {c_1}} )\cdot ({2{\sigma_{t,r}} + {c_2}} )}}{{({\mu_t^2 + \mu_r^2 + {c_1}} )\cdot ({\sigma_t^2 + \sigma_r^2 + {c_2}} )}}, $$
where µt and µr are the mean values of the target and reconstructed images, respectively. σt and σr are the standard deviations of the target and reconstructed images, respectively. σt,r is the covariance between the target image and the reconstructed image, c1 and c2 are positive constants used to avoid the null denominator. The value of SSIM varies from 0 to 1, and larger SSIM indicates better structural similarity.

In the comparisons of the GS algorithm and our proposed method, the resolution of the phase hologram is set as 600×600, which is the same with the resolution of original target images. In our proposed method, the modified sampling method is used, then the reconstructed image is sampled in 1200×1200 pixels; hence, the initial target image should be up-sampled to 1200×1200 pixels by the interpolation method. Here, the sinc interpolation is used to up-sample the target image [38]. Specifically, the FFT algorithm is applied to the original target image with the resolution of 600×600 to obtain its Fourier spectrum. Then, 300 zeros are padded on each side of the Fourier spectrum. The up-sampled target image with the resolution of 1200×1200 is generated by applying an inverse FFT algorithm to the zero-padded Fourier spectrum. While, in the GS algorithm, the reconstructed image is still sampled in 600×600 pixels, which is different from our proposed method. To compare the reconstruction results of the GS algorithm and our proposed method objectively, the reconstructed images by the GS algorithm should be sampled densely in 1200×1200 pixels, which can be realized by applying the zero-padding operation to the optimized phase hologram with the resolution of 600×600.

A ‘baboon’ image (from USC-SIPI image database) is set as the test pattern in the numerical comparisons. The target image plane is sampled in 1200×1200 pixels, which contains the signal region and non-signal region. The target image is located in the signal region. The resolution of the signal region is set as 800×800, and the surrounding area is set as non-signal region. According to Eqs. (19) and (20), the parameters k and l of the quadratic initial phase are calculated as 0.002. The number of iterations is set as 100. The reconstruction results are shown in Figs. 7(b)-(d). The RMSEs and the SSIMs are plotted in Figs. 7(e) and (f), respectively. Figure 7(b) shows that the reconstructed image by the GS algorithm is degraded by severe speckle noise. Figure 7(c) shows that the reconstruction quality is improved effectively in our proposed method; whereas, the speckles cannot be eliminated totally in the case of the random initial phase due to the stagnation problem. Figure 7(d) shows that the reconstructed image is almost exactly the same with the target image in our proposed method with quadratic initial phase, and the speckles are eliminated effectively. Figures 7(e) and (f) show that the convergence of our method is fast and stable, and the performance of our method exceeds the GS algorithm. In the GS algorithm, due to the lack of overall and effective constraints of the reconstructed field, the reconstruction quality cannot be improved effectively.

 figure: Fig. 7.

Fig. 7. Numerical simulations: (a) target image plane is sampled in 1200×1200 pixels; reconstructed images by (b) GS algorithm, (c) our proposed method with random initial phase and (d) our proposed method with quadratic initial phase; (e) comparison of the root-mean-square errors (RMSEs); (f) comparison of the structural similarity index measures (SSIMs).

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The parameters k and l of the quadratic initial phase are set as 0.002 in the numerical simulations, which can ensure the band-limit constraint of the reconstructed field and distribute the entire signal energy uniformly into the physical area of the holograms. Here, numerical simulations are also performed to illustrate the influence of the quadratic initial phase with different bandwidth property on the convergence of our proposed iterative algorithm. Figures 8(a)-(c) show the quadratic initial phase with different bandwidth property. The reconstruction results are shown in Figs. 8(d)-(f). The RMSEs are plotted in Fig. 8(g), and the SSIMs are plotted in Fig. 8(h). It can be seen that the bandwidth property of the quadratic initial phase would influence the convergence of the iterative algorithm with bandwidth constraint strategy. When the parameters k and l of the quadratic initial phase are calculated according to Eqs. (19) and (20), a higher reconstruction quality can be obtained. Otherwise, the band-limit constraint of the reconstructed field would be affected, which would degrade the convergence of the iterative algorithm and induce some speckles in the reconstructed image.

 figure: Fig. 8.

Fig. 8. Numerical simulations: quadratic initial phase with the parameters k and l calculated as (a) 0.0015, (b) 0.002, and (c) 0.0025; reconstructed images by our proposed method using different quadratic initial phase with (d) k, l = 0.0015, (e) k, l = 0.002, and (f) k, l = 0.0025; (g) comparison of the root-mean-square errors (RMSEs); (h) comparison of the structural similarity index measures (SSIMs).

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Some other images (from USC-SIPI image database) are also used as the target patterns to illustrate the robustness of our proposed method, shown in Fig. 9. The target image plane is still sampled in 1200×1200 pixels, which contains the signal region and non-signal region. The resolution of the signal region is set as 800×800, and the surrounding area is set as non-signal region. It can be seen that our proposed method can effectively enhance the reconstruction quality and eliminate the speckles for different target patterns.

 figure: Fig. 9.

Fig. 9. Numerical reconstructions with the GS algorithm (left), our proposed method using random initial phase (middle) and quadratic initial phase (right).

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To further validate the feasibility of our proposed method, optical experiments are also performed. The optical experimental setup is shown in Fig. 10. A laser beam at the wavelength of 532 nm is expanded, collimated and polarized for illumination. In the experiment, we use Meadowlark Optics 256 gray-scale level reflective phase-only SLM. The pixel pitch of the SLM is 9.2 µm, and the pixel resolution is 1920×1152. The Fourier lens with the focal length of 200 mm is used to form the optical Fourier transform system. The reconstructed images are located in the back focal plane of the Fourier lens and are captured directly by a camera sensor. To eliminate the zero-order noise of the SLM, a linear phase is added on the calculated hologram to separate the reconstructed image away from the zero-order interruption [39].

 figure: Fig. 10.

Fig. 10. Schematic of optical experimental setup.

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In the optical experiments, the phase hologram with the resolution of 1000×1000 is generated by the iterative algorithm and uploaded on the middle zone of the SLM. The optical reconstruction results of the GS algorithm and the proposed method are shown in Fig. 11. It shows that the reconstruction quality of our proposed method is better than the GS algorithm. The reconstructed image by the GS algorithm is degraded by severe speckles, while the reconstructed image by our proposed method is of higher visual quality and contain more optical details. It can be seen that the speckles can be nearly eliminated by our proposed method with a quadratic initial phase.

 figure: Fig. 11.

Fig. 11. Optical reconstructions of (a) GS algorithm, (b) our proposed method with random initial phase, (c) our proposed method with quadratic initial phase.

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We also chose some other patterns as test images in optical experiments to further illustrate the effectiveness of our proposed method, as shown in Fig. 12. It shows that better image quality and more details can be optically reconstructed by our proposed method, and the speckles can be effectively eliminated by our proposed method with a quadratic initial phase.

 figure: Fig. 12.

Fig. 12. Optical reconstructions of the GS algorithm (left), our proposed method with random initial phase (middle) and quadratic initial phase (right).

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A binary rectangle pattern is also set as the test pattern to evaluate the optical reconstruction performance of our proposed method. The measure of speckle contrast (SC) is used here to evaluate the speckle noise in the reconstructed images, which is defined as the ratio of standard deviation to mean value of the signal intensity. The lower SC indicates the less speckle noise occurs in the reconstructed image. Figure 13 shows the optical reconstruction results. It can be seen that the speckles can be eliminated effectively in our proposed method with a quadratic initial phase. Whereas, it should be noted that the speckles cannot be eliminated in our method with a random initial phase, which is caused by the remaining phase vortices during the iterative process [36].

 figure: Fig. 13.

Fig. 13. Optical reconstructions of a binary rectangle pattern: (a) GS algorithm, (b) our proposed method with random initial phase, (c) our proposed method with quadratic initial phase.

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5. Discussion

In some iterative algorithms, the quadratic phase was set as the initial phase of the iterative process [40,41], which helps to suppress the speckles in the reconstructed image. In our proposed method, the quadratic phase is also taken as the initial phase to iterate. The difference lies in that the bandwidth properties of the reconstructed field are analyzed in our method; then, a proper sampling method is applied to the reconstructed field, and iterative calculation with bandwidth constraint strategy of the reconstructed field is performed to optimize phase holograms with suppressed speckles and artifacts. The quadratic initial phase is used here to avoid the stagnation problem of the iterative process caused by the random initial phase. It should be noted that the quadratic initial phase is not the only choice to avoid the stagnation problem. While, in the previous iterative algorithms using a quadratic initial phase, the bandwidth of the reconstructed field was not considered; hence, the speckles and artifacts in the reconstructed field cannot be effectively described, which would cause ringing artifacts in the optical reconstruction. In order to illustrate this problem, numerical simulations and optical experiments are performed, and the reconstruction results are shown in Fig. 14. In the numerical reconstruction of the GS algorithm with a quadratic initial phase, the final optimized phase hologram should be zero-padded for densely sampling the reconstructed image, and the corresponding reconstructions are shown as Figs. 14(a) and (c). It can be seen that severe ringing artifacts occur in the reconstructed image by the GS algorithm with a quadratic initial phase, which is due to the lack of overall and effective constraints of the reconstructed intensity fluctuations during the iterations. While, in our proposed method, the speckles and artifacts in the reconstructed field can be effectively described, which helps to suppress the speckles and ringing artifacts in the reconstructions, shown as Fig. 14(b) and (d).

 figure: Fig. 14.

Fig. 14. Numerical reconstructions of (a) GS algorithm with quadratic initial phase, and (b) our proposed method with quadratic initial phase; optical reconstructions of (c) GS algorithm with quadratic initial phase, and (d) our proposed method with quadratic initial phase.

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In our proposed method, the size of the phase hologram is zero-padded to twice of the original ones, then the sampling interval of the reconstructed field is reduced to half of the original ones. According to our theoretical analysis, the modified sampling interval is sufficient to sample the reconstructed intensity distribution due to the finite bandwidth of the reconstructed field. To verify our view, another numerical simulation is also performed, as shown in Fig. 15. Through our proposed method, the optimized phase hologram is generated. Then, the phase hologram is zero-padded to quadruple the size of the original ones, so the sampling interval of the reconstructed field is reduced to quarter of the original ones. From Fig. 15, it can be seen that even with a smaller sampling interval set into the reconstructed field, the reconstruction result is not changed, which is due to the finite bandwidth of the reconstructed intensity distribution. Hence, in the optimization process, the size of the phase hologram requires only to be zero-padded to twice of the original ones for reducing the sampling interval of the reconstructed field to half of the original ones, which can largely reduce the computation load.

 figure: Fig. 15.

Fig. 15. Numerical reconstructions with different sampling interval set into the reconstructed field by our proposed method.

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In our previous work [13], a non-iterative method was proposed to design phase holograms with optimized phase modulation. The optimized phase distribution is iteratively pre-generated from a quadratic initial phase with continuous distributed spectrum. In the hologram generation, the optimized phase is used to modulate the wave field in the reconstructed plane, and the phase hologram is calculated directly according to the modulated wave field. Fast generation of the phase holograms can be achieved by this non-iterative implementation, and the avoidance of the the random phase modulation helps to eliminate the speckle noise induced by the diffuse phase. Whereas, the bandwidth properties of the reconstructed field were not paid attention, which would still cause the ringing artifacts in the optical reconstruction. In this study, we propose an iterative method with bandwidth constraint strategy to design the phase holograms, which can effectively suppress the speckles and ringing artifacts in the optical reconstruction. Furthermore, our proposed method in this study can be combined with the non-iterative method proposed in our previous work to achieve a non-iterative phase hologram generation with suppressed speckles and artifacts, which is called as non-iterative algorithm with bandwidth constraint strategy. Here, some numerical results are given to provide a preliminary verification of this idea, as shown in Fig. 16. It can be seen that the non-iterative algorithm with bandwidth constraint strategy can be used to design phase holograms with suppressed speckles and ringing artifacts. While, owing to the lack of further optimization according to the specific target images in the non-iterative algorithm, the contrast of the reconstructed image is slightly degraded.

 figure: Fig. 16.

Fig. 16. Numerical reconstructions of (a) GS algorithm with quadratic initial phase, (b) non-iterative algorithm with bandwidth constraint strategy, and (c) our proposed iterative algorithm with bandwidth constraint strategy.

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Although our proposed method is based on the Fourier transform system, it can also be easily extended to the case of single FFT-based Fresnel transform. When the intensity distribution is of the interest in the reconstructed image plane, the single FFT-based Fresnel transform differs from the Fourier transform only by a quadratic phase factor which would not affect the band-limit constraint of the reconstructed field. Furthermore, some advanced 3D optimization method can be combined with the bandwidth constraint strategy to design phase holograms for high-quality speckle-free holographic 3D displays [42].

6. Conclusion

In conclusion, we propose an iterative method with bandwidth constraint strategy to design phase holograms with high-quality speckle-free optical reconstruction. The bandwidth properties of the reconstructed field are analyzed theoretically based on the sampling theory, which helps in allocating sampling resources for efficiently describing the speckles and artifacts in the reconstructed field. During the phase hologram optimization, a proper sampling method is used to reduce the sampling interval of the reconstructed field for effectively controlling the reconstructed intensity fluctuations, which is easy to realize in the FFT based algorithm by applying the zero-padding operation to the hologram plane. Iterative calculation with bandwidth constraint strategy of the reconstructed field is utilized to optimize the phase hologram with speckle-free reconstruction. To solve the stagnation problem caused by random initial phase, a band-limited quadratic phase is used as the initial phase distribution. Numerical and optical experiments are performed to validate the feasibility of our proposed method for providing effective controls of the reconstructed intensity fluctuations and suppressing the speckles and artifacts in the optical reconstruction. In our proposed method with a quadratic initial phase, the speckles and the ringing artifacts can be both suppressed effectively. Although this method is based on the Fourier transform system, it can be further extended to Fresnel region as well as phase hologram synthesis of 3D scenes.

Funding

National Natural Science Foundation of China (62035003, 61875105); National Key Research and Development Program of China (2017YFF0106400).

Disclosures

The authors declare no conflicts of interest.

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Figures (16)
Fig. 1.
Fig. 1. Schematic diagram for optical Fourier transform system: the hologram plane is located in the front focal plane of the Fourier lens, and the image plane is located in the back focal plane of the Fourier lens.

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Fig. 2.
Fig. 2. Schematic diagram for marginal light propagation model of the hologram plane in the optical Fourier transform system.

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Fig. 3.
Fig. 3. Schematic diagram of reducing the sampling interval of the reconstructed field by applying zero-padding operation to the hologram plane.

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Fig. 4.
Fig. 4. Schematic diagram of bandwidth constraint optimization algorithm.

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Fig. 5.
Fig. 5. Numerical simulations of our proposed bandwidth constraint optimization algorithm with a random initial phase: (a) target image, sampled in 600×600 pixels; reconstructed image after (b) 5 iterations, (c) 20 iterations, and (d) 100 iterations.

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Fig. 6.
Fig. 6. Different complex amplitude distribution in the hologram plane calculated by the inverse Fourier transform of the initial complex amplitude field with different quadratic phase in the image plane.

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Fig. 7.
Fig. 7. Numerical simulations: (a) target image plane is sampled in 1200×1200 pixels; reconstructed images by (b) GS algorithm, (c) our proposed method with random initial phase and (d) our proposed method with quadratic initial phase; (e) comparison of the root-mean-square errors (RMSEs); (f) comparison of the structural similarity index measures (SSIMs).

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Fig. 8.
Fig. 8. Numerical simulations: quadratic initial phase with the parameters k and l calculated as (a) 0.0015, (b) 0.002, and (c) 0.0025; reconstructed images by our proposed method using different quadratic initial phase with (d) k, l = 0.0015, (e) k, l = 0.002, and (f) k, l = 0.0025; (g) comparison of the root-mean-square errors (RMSEs); (h) comparison of the structural similarity index measures (SSIMs).

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Fig. 9.
Fig. 9. Numerical reconstructions with the GS algorithm (left), our proposed method using random initial phase (middle) and quadratic initial phase (right).

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Fig. 10.
Fig. 10. Schematic of optical experimental setup.

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Fig. 11.
Fig. 11. Optical reconstructions of (a) GS algorithm, (b) our proposed method with random initial phase, (c) our proposed method with quadratic initial phase.

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Fig. 12.
Fig. 12. Optical reconstructions of the GS algorithm (left), our proposed method with random initial phase (middle) and quadratic initial phase (right).

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Fig. 13.
Fig. 13. Optical reconstructions of a binary rectangle pattern: (a) GS algorithm, (b) our proposed method with random initial phase, (c) our proposed method with quadratic initial phase.

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Fig. 14.
Fig. 14. Numerical reconstructions of (a) GS algorithm with quadratic initial phase, and (b) our proposed method with quadratic initial phase; optical reconstructions of (c) GS algorithm with quadratic initial phase, and (d) our proposed method with quadratic initial phase.

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Fig. 15.
Fig. 15. Numerical reconstructions with different sampling interval set into the reconstructed field by our proposed method.

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Fig. 16.
Fig. 16. Numerical reconstructions of (a) GS algorithm with quadratic initial phase, (b) non-iterative algorithm with bandwidth constraint strategy, and (c) our proposed iterative algorithm with bandwidth constraint strategy.

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Equations (22)

Equations on this page are rendered with MathJax. Learn more.

H ( x , y ) = h ( x , y ) exp [ i Φ ( x , y ) ] rect ( x a ) rect ( y b ) ,
− a 2 λ f ≤ f u { A } ≤ a 2 λ f ,
θ ≈ a 2 f ,
I = A ⋅ A ∗ ,
F { I } = F { A } ⊙ F { A } ,
B { I } ≤ 2 B { A } ,
f u { I } ≤ 2 f u ( m a x ) { A } = a λ f .
f u ( m a x ) { I } = 2 sin ⁡ θ λ ≈ 2 λ ⋅ a 2 f = a λ f ,
Δ u ≤ λ f 2 a   ,   Δ v ≤ λ f 2 b .
Δ u ′ = λ f 2 a   ,   Δ v ′ = λ f 2 b ,
| A k  +  1 | = {   I 0  , ( u , v ) ∈ S   | A k ′ |  , ( u , v ) ∉ S ,
∫ ∫ | A ′ k ( u , v ) | 2 d u d v = ∫ ∫ | H ′ k ( x , y ) | 2 d x d y .
∫ ∫ | H ′ k ( x , y ) | 2 d x d y = ∫ ∫ | A 0 ( u , v ) | 2 d u d v ,
∫ ∫ | A ′ k ( u , v ) | 2 d u d v = ∫ ∫ | A 0 ( u , v ) | 2 d u d v .
f u { exp ( i φ 0 ) } = 1 2 π ⋅ ∂ φ 0 ∂ u = k m π Δ u .
f u ( max ) { exp ( i φ 0 ) } = | k | P 2 π Δ u .
Δ u = λ f 2 a ,
f u ( max ) { exp ( i φ 0 ) } = a 2 λ f .
k = ± π 2 P .
l = ± π 2 Q .
RMSE = ∑ m = 1 M ∑ n = 1 N [ I r ( m , n ) − I t ( m , n ) ] 2 / ∑ m = 1 M ∑ n = 1 N [ I t ( m , n ) ] 2 ,
SSIM = ( 2 μ t μ r + c 1 ) ⋅ ( 2 σ t , r + c 2 ) ( μ t 2 + μ r 2 + c 1 ) ⋅ ( σ t 2 + σ r 2 + c 2 ) ,
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