## Abstract

A non-iterative algorithm is proposed to generate phase holograms with optimized phase modulation. A quadratic initial phase with continuous distributed spectrum is utilized to iteratively optimize the phase modulation in the reconstruction plane, which can be used as an optimized phase distribution for arbitrary target images. The phase hologram can be calculated directly according to the modulated wave field distribution in the reconstruction plane. Fast generation of the phase holograms can be achieved by this non-iterative implementation, and the avoidance of the random phase modulation helps to suppress the speckle noise. Numerical and optical experiments have demonstrated that the proposed method can efficiently generate phase holograms with quality reconstructions.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Holographic display is regarded as one of the most promising display techniques and has been under intense investigation in recent decades [1–4]. The development of spatial light modulators (SLMs) provides an efficient way to dynamically reconstruct computer-generated holograms (CGHs). Since the SLMs directly modulate the amplitude or phase of the incident wave, perfect complex modulation is difficult to realize in the current devices without sacrificing the space-bandwidth product of the system [5,6]. Compared to the amplitude modulations, phase holograms have higher optical efficiency and no conjugate image, which are more preferable in the applications where optical efficiency and reconstruction quality are demanded [7].

In the calculation of phase holograms, various iterative phase retrieval algorithms have been proposed to optimize the phase distribution on the hologram plane for improving the reconstruction quality [8–13]. In general cases, multiple iterations with forward and backward diffraction calculations are needed to achieve convergence during optimization, hence they are often time-consuming for synthesizing holograms with quality reconstructions. In addition, most of them still suffer from the speckle noise in optical reconstructions due to the random phase distribution of the reconstructed images.

Recently, interesting non-iterative methods have been proposed to calculate phase holograms. Double-phase method [14,15] and error diffusion method [16,17] were proposed to convert a complex hologram into a phase hologram. The double-phase method encodes a complex amplitude distribution into a pure phase distribution by decomposing a complex value into two phase quantities. This method would sacrifice the space-bandwidth product of the display devices due to the decomposition implementation of each complex amplitude value. The error diffusion method provides another way to convert a complex hologram into a phase hologram by compensating the modulation errors through adjacent phase values. Due to the limited compensation range of the adjacent sampling points, the light efficiency and the reconstructed image would be affected in optical reconstruction. In addition, the implementation of the error diffusion algorithm is sequential, hence the generation of the hologram is time-consuming.

Apart from generating phase holograms from the complex wave fields, some researchers have attempted to enhance the reconstruction quality by imposing the designed phase distribution to the target image. In the calculation of phase holograms, random phase (RAP) method was widely used to diffuse the object wave while yielding severe speckle noise. To improve the reconstruction quality, the optimized random phase (ORAP) method was presented to generate phase-only Fourier holograms by applying a pre-optimized random phase to the target image [18]. This method was effective for improving the reconstruction quality. Whereas, speckle noise in optical reconstruction is related to the phase mutations and vortices, which exist widely in random phase distribution [12,13]. Hence strong intensity fluctuations would occur in optical reconstruction due to the random phase distribution of the reconstructed images. Patterned-phase-only holograms (PPOH) with non-iterative calculation was also proposed to improve the reconstruction quality of the RAP method [19]. In this method, a phase mask with periodic random phase pattern was added to the target image for improving the visual quality of the reconstructed image. Recently, the optimized random phase tiles (ORPT) method was proposed to improve the PPOH by pre-optimization procedure [20]. The ORPT can be generated for arbitrary distance between the hologram plane and object, which is helpful for improving reconstruction quality in holographic 3D display. Whereas, due to the periodic replication of the phase mask, the reconstructed image would be contaminated by the pixelated pattern in these methods. The gradient-limited random phase addition method was also proposed to enhance the reconstruction quality of the RAP method [21]. The target image was segmented into low frequency region and high frequency region. Then, wide and narrow optimum ranges of a limited random phase are added to the low and high frequency regions, respectively. In the above methods, due to the random and erratic phase distribution of the reconstructed image, severe speckle noise could not be suppressed effectively in optical reconstruction. Recently, the random phase-free methods were presented to improve the reconstruction quality by multiplying the object wave with a virtual convergence light [22–26]. Owing to the avoidance of the random phase, these methods can effectively reduce the speckle in optical reconstruction. They were more preferable for generating the amplitude holograms due to the direct and convenient coding process [22,23]. To implement the random phase-free methods in phase hologram generation, they were combined with the error diffusion method [24–26], which would cause low light efficiency and sacrifice the computational efficiency. In addition, synthetic amplitude method was proposed recently for improving the reconstruction quality of non-iterative phase holograms by applying a synthetic amplitude to the phase hologram [27]. This method is innovative, but the reconstructed images are degraded by the artifacts, which requires to be improved further. Therefore, it is still a challenge to generate a phase hologram with high quality and low speckle reconstruction in a fast and robust way.

In this study, we present a fast and efficient method to generate phase holograms with quality reconstructions. Optimized phase distribution is iteratively generated from a quadratic initial phase with continuous distributed spectrum according to the size of the reconstructed image. For a given optical system, the optimized phase distribution requires to be pre-generated only once. In hologram generation, the optimized phase is used to modulate the wave field in the reconstruction plane, and the phase hologram is calculated directly according to the modulated wave field distribution. Finally, numerical and experimental results confirm that our proposed method can improve the reconstruction quality effectively, and the avoidance of the random phase modulation helps to eliminate the speckle noise induced by the diffuse phase.

## 2. Method

Generally, most images contain a large amount of low frequency components. The low frequency parts of the object cannot spread light uniformly over the hologram plane without random phase modulation of the object. Hence, when the amplitude component of the complex amplitude distribution in the hologram plane is discarded for generating the phase hologram, only the high frequency components of the image can be reconstructed, as shown in Fig. 1(b). The RAP method provides an intuitive way to spread the spectrum of the image by diffusing the object wave with a random phase mask. However, the reconstruction quality of the image would be degraded significantly by the absence of the iteration optimization and the speckle noise induced by the diffuse property of the random phase, as shown in Fig. 1(c).

In order to suppress the speckle noise induced by the random phase and improve the reconstruction quality without iteration process, optimized phase modulation need to be generated to diffuse the object wave with smooth spectrum for arbitrary target images. Here, a modified iterative algorithm is used to obtain an optimized quadratic phase distribution from a rectangular aperture before phase hologram generation. The rectangular aperture is the area where the target images may distribute, which can be considered as the signal region. The process of the optimized quadratic phase generation is shown in Fig. 2. The rectangular aperture is multiplied with an initial quadratic phase distribution *φ _{t}*, and the resulting complex amplitude is set as the input of the iterative process. The initial quadratic phase is given by:

*a*and

*b*are positive constants,

*m*and

*n*are pixel coordinates of the target image plane. At ending of this section, it will be discussed how to determine the initial quadratic phase distribution. In the iteration process, the enforced amplitude constraint for the next iteration on the target image plane is

*T*=

*T*(

*m*,

*n*) is a two-dimensional function, which is unity at the signal region and zero at the non-signal region. |

*A*| is the amplitude distribution of the rectangular aperture, and |

_{t}*A*| is the amplitude distribution of the reconstructed image in

^{’}_{k}*k*th iteration. To exclude the division by zero, a small number ∼10

^{−10}should be added to |

*A*|. The weighted parameter

^{’}_{k}*α*is introduced here to enhance the convergence of the iteration process and given by The initial value

_{k}*α*

_{0}is a small number, ∼10

^{−8}. The value of

*α*would increase and approach to 1 with the increase of the iteration number.

_{k}In order to suppress the noise in the non-signal region of the reconstructed image, energy conservation between hologram plane and image plane is considered. Specifically, in the iteration process, the amplitude of incident wave on the hologram plane is calculated as

*M*and

*N*are the horizontal and vertical resolution of the target image. In the iteration process, the amplitude of incident plane wave on the hologram plane is replaced by |

*A*

_{holo}|, rather than replaced by unity. This operation leads a larger optimization space of the algorithm while maintaining the overall constraint on the intensity distribution of the reconstructed plane.

When the iteration is completed, the optimized phase *φ _{o}* is generated, which is the phase distribution of the signal region in the last iteration. Through iterations, the obtained quadratic phase can be viewed an equivalent of a physical diffuser, so that low frequency components of the image at the signal region can be uniformly diffused over the hologram plane. Due to dominance of the low frequency components in most common patterns, it is reasonable that the optimized quadratic phase can be applied to the target images for uniformly diffusing the object wave. It should be noted that for a given optical system, the optimized quadratic phase requires to be off-line generated only once according to the sampling parameters of the SLM and the window size of the target patterns, which would largely reduce the computation load.

If the optimized quadratic phase *φ _{o}* is directly applied to the target pattern for generating a phase hologram, the dynamic range of the reconstructed image would be degraded. To enhance the contrast of the reconstructed image, a feedback parameter

*β*is introduced here, as shown in Fig. 3(a). The parameter

*β*is given by

*A*| is the amplitude distribution of the target image, |

_{t}*A*| is the amplitude distribution of the output image, and $\varepsilon $ is a small number, ∼10

_{r}^{−10}, excluding the division by zero. Then, the optimized quadratic phase

*φ*and parameter

_{o}*β*are utilized to modulate the wave field in the reconstructed plane, as shown in Fig. 3(b). The modulated wave field distribution

*A*is calculated as follows:

_{m}Finally, the phase hologram is obtained via the Fourier transform and a phase extraction. The process in Fig. 3(a) can be viewed as a nonlinear “input-output” system with the input *A _{t}*, output

*A*, and nonlinear transform operator

_{r}*FP*

_{ac}F^{−1}, where

*P*denotes the amplitude constraint process. The feedback parameter

_{ac}*β*is calculated to collect the error between the output image and the target image, and then used to compensate the input error. In this way, the final reconstructed image is of higher visual quality. Importantly, the phase distribution of the reconstructed image is smoothly distributed without diffuse properties of the random phase, so that the speckle noise can be effectively suppressed in optical reconstruction [12,13].

The parameters of the initial quadratic phase distribution *φ _{t}* is important for generating the optimized phase

*φ*. In the following, how to determine the initial quadratic phase distribution is discussed in detail. For the initial quadratic phase ${\varphi _t} = a{m^2} + b{n^2}$, the parameters

_{o}*a*and

*b*cannot be too large, or the destructive interference would occur at some points. At the same time, the parameters

*a*and

*b*cannot be too small, since the spectrum should be smooth enough to avoid the stagnation [25]. For different values of

*a*and

*b*, the bandwidth of the spectrum is different, as shown in Fig. 4. Therefore, the parameters

*a*and

*b*should be chosen carefully so that the bandwidth of the spectrum is as close as possible to the hologram size. According to Eq. (1), the spatial frequency of the quadratic phase distribution

*φ*in

_{t}*x*direction can be expressed as:

*dx*is the sampling interval of the quadratic phase. Thereafter, the maximum spatial frequency of the quadratic phase

*φ*in

_{t}*x*direction is: where

*M*is the pixel number of the quadratic phase in

*x*direction. To obtain a spectrum as close as possible to the hologram size, we can derive the following equation: where

*S*represents the size of the hologram,

_{h}*λ*represents the light wavelength, and

*f*represents the focal length of the Fourier lens. To implement the fast Fourier transform (FFT) in our method, the sampling interval of the quadratic phase is deduced as:

Combining Eqs. (8)–(10), we can obtain:

In a similar way, we can obtain: where*N*is the pixel number of the quadratic phase in

*y*direction.

## 3. Reconstruction results

To demonstrate the effectiveness of the proposed method for generating phase holograms with quality reconstructions, numerical simulations are performed for quantitative evaluation. The performance of our proposed method is compared with the RAP method and the GS algorithm. To compare the simulation results, some evaluation functions are used here. The first one is peak signal-to-noise ratio (PSNR) between the target image and the reconstructed image. The PSNR for 8-bit gray-level image is defined as:

*M*and

*N*are the horizontal and vertical resolution of images,

*I*(

_{t}*m*,

*n*) is the intensity distribution of the target image, and

*I*(

_{r}*m*,

*n*) is the intensity distribution of the reconstructed image. The higher the value of PSNR is, the better the reconstruction quality will be. Another function is structural similarity index measure (SSIM) and it is given by

*μ*and

_{t}*μ*are the mean values of the target and reconstructed images, respectively.

_{r}*σ*and

_{t}*σ*are the standard deviations of the target and reconstructed images, respectively.

_{r}*σ*is the covariance between the target image and the reconstructed image,

_{t,r}*c*

_{1}and

*c*

_{2}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 RAP method, GS algorithm and our proposed method, some images with the dimensions of 1000×1000 were set as the test patterns. The resolution of the signal region was set as 800×800. According to Eqs. (11) and (12), the parameters of initial quadratic phase were set $a = 0.004$, $b = 0.004$. The distributions of the initial quadratic phase and the optimized quadratic phase are shown as Fig. 5. It shows that all the optimized quadratic phase is similar to the initial quadratic phase, which is smoothly distributed. Moreover, it shows that the intensity of the rectangular aperture is precisely reconstructed after 50 iterations. The reconstruction quality would be further improved by more iterations, but the improvement is limited due to the stagnation of the iterative algorithm. Considering both of the computational efficiency and the reconstruction quality, we choose 50 iterations for the optimized phase generation in our work.

A ‘parrot’ image was set as the test pattern firstly in the numerical comparisons. The iteration number of the GS algorithm was set as 50. The numerical reconstructions of different algorithms are shown as Fig. 6. It shows that the reconstructed image with the RAP method is severely degraded by the speckle noise. The reconstruction quality of our proposed method is greatly superior to the RAP method. Figure 6(c) shows that the reconstruction with unoptimized quadratic phase is degraded by the ringing artifacts. In our method, by additionally applying some small amount of randomness to the initial quadratic phase, the ringing artifacts could be suppressed more effectively, shown as Fig. 6(d). The reconstructed phase distributions of the signal region with these methods are shown as Figs. 6(e)–6(h). Although the numerical reconstruction of our proposed method is slightly lower than the GS algorithm, the diffuse property of the random initial phase in the GS algorithm would induce obvious speckle noise in optical reconstruction [12,13]. On the contrary, the optimized phase modulation of the reconstructed image with our method is smoothly distributed and contains less phase mutations without diffuse properties of the random phase, which is helpful for suppressing the speckle noise in optical reconstruction. In addition, we compare the efficiency of these methods. The efficiency, which is the ratio of energy in the signal region, is 0.927, 0.994, 0.996 and 0.996 for RAP method, GS algorithm, unoptimized quadratic phase, and our method, respectively.

Some other images were also used as the target patterns to illustrate the effectiveness of our proposed method, shown as Fig. 7. It can be seen clearly that our proposed method can effectively improve the reconstruction quality.

To further demonstrate the optical effects of our proposed algorithm, experiments are also performed. The schematic of the experimental setup is shown in Fig. 8. A laser beam at the wavelength of 532 nm is expanded and collimated for illumination. The SLM used here is Holoeye GAEA-2 with pixel resolution of 3840×2160 and pixel pitch of 3.74 µm. The focal length of the Fourier lens is 200 mm, which is the same as the focal length *f* in Eq. (9). The reconstructed images are captured directly by a sensor located in the back focal plane of the Fourier lens. To capture the entire image on the sensor, the resolution of the signal region is set as 1280×720. The iteration number of the GS algorithm was set as 50, both for the optimized quadratic phase generation. In order to eliminate the zero-order noise of SLM, a linear phase is added on the calculated hologram to separate the reconstructed image away from the interruption [28].

The optical reconstruction results of RAP method, GS algorithm and our proposed method are shown in Fig. 9. It can be seen that the reconstruction quality of our proposed method is better than RAP method and the GS algorithm. Owing to the optimized phase distribution of the reconstructed image with our proposed method is smoothly distributed and contains less phase mutations, the reconstructed image is of higher visual quality, as shown in Fig. 9(c). On the contrast, the random initial phase in the GS algorithm changes drastically, inducing obvious speckle noise in optical reconstruction.

We also chose some other patterns as test images in experiments to further illustrate the effectiveness of our proposed method, as shown in Fig. 10. It shows that the reconstructed images of our proposed method are of higher visual quality and contain less speckle noise.

## 4. Discussion

The parameter settings of the proposed method for generating phase holograms need to be discussed for guiding the system design. During calculation, fast Fourier transform (FFT) was implemented to perform the Fourier transform. According to the sampling properties of FFT, the numbers of sampling points in the hologram plane and the reconstruction plane are identical. For a given optical reconstruction system, the number of the sampling points in the reconstructed image is fixed. To adjust the sampling parameters according to the target image, regional processing in the reconstruction plane is introduced. The number of sampling points in the signal region is set as the number of pixels in the target image, where a rectangular aperture is addressed. And other part of the reconstruction plane is set as the non-signal region. Numerical reconstructions with different sampling numbers are used to analyze their influence on the image quality. For simplicity, the sampling number of the hologram is set as 1500×1500. The reconstruction results with different sampling numbers of the signal region are shown as Fig. 11. It shows that all the reconstructed images are of high quality, and all the reconstructed phase distributions are smoothly distributed in the signal region. From the results we can see that the reconstruction quality of the signal region can maintain well in different size of the signal region, which is one of the advantages of our proposed method.

To enhance the dynamic range of the reconstructed image in our method, feedback operation in the phase hologram generation is implemented, and the feedback parameter *β* is calculated as Eq. (5). If the optimized quadratic phase is directly applied to the target pattern for generating a phase hologram, the dynamic range of the reconstructed image would be reduced. Herein, we introduce the feedback parameter *β* to improve the dynamic range of the reconstructed image. The feedback parameter *β* is calculated to collect the error between the output image and the target image, and then used to compensate the input error and enhance the dynamic range. In this way, the final reconstructed image is of higher visual quality. Numerical reconstructions with and without the feedback operation are shown as Fig. 12. It can be seen that the dynamic range of the reconstructed image would be effectively enhanced after introducing a feedback operation.

## 5. Conclusion

In conclusion, we propose a non-iterative and efficient method for calculating phase holograms with quality reconstructions. The optimized phase of the reconstructed image is smoothly distributed without diffuse properties of the random phase, which effectively suppresses the speckle noise in optical reconstruction. On the other hand, the optimized phase distribution requires to be pre-generated only once, which largely reduces the computation load. Numerical and experimental results show that reconstructed images of our proposed method are of higher visual quality and contain less speckle. Although this method is based on the Fourier transform system, it can be further extended to Fresnel region. In the future, we will extend this work to the generation of phase holograms in three-dimensional occasions.

## Funding

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

## Disclosures

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

## References

**1. **S. A. Benton and V. M. Bove, * Holographic Imaging* (Wiley, 2007).

**2. **Y. Pan, Y. Wang, J. Liu, X. Li, and J. Jia, “Fast polygon-based method for calculating computer-generated holograms in three-dimensional display,” Appl. Opt. **52**(1), A290–A299 (2013). [CrossRef]

**3. **H. Zhang, Y. Zhao, L. Cao, and G. Jin, “Fully computed holographic stereogram based algorithm for computer generated holograms with accurate depth cues,” Opt. Express **23**(4), 3901–3913 (2015). [CrossRef]

**4. **W. Qu, H. Gu, and Q. Tan, “Holographic projection with higher image quality,” Opt. Express **24**(17), 19179–19184 (2016). [CrossRef]

**5. **A. Jesacher, C. Maurer, A. Schwaighofer, S. Bernet, and M. Ritsch-Marte, “Near-perfect hologram reconstruction with a spatial light modulator,” Opt. Express **16**(4), 2597–2603 (2008). [CrossRef]

**6. **M. Makowski, A. Siemion, I. Ducin, K. Kakarenko, M. Sypek, A. Siemion, J. Suszek, D. Wojnowski, Z. Jaroszewicz, and A. Kolodziejczyk, “Complex light modulation for lensless image projection,” Chin. Opt. Lett. **9**(12), 1–3 (2011). [CrossRef]

**7. **Z. Zhang, Z. You, and D. Chu, “Fundamentals of phase-only liquid crystal on silicon (LCOS) devices,” Light: Sci. Appl. **3**(1), e135 (2014). [CrossRef]

**8. **R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik **35**, 237–246 (1972).

**9. **J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. **19**(3), 193297 (1980). [CrossRef]

**10. **A. V. Kuzmenko, “Weighting iterative Fourier transform algorithm of the kinoform synthesis,” Opt. Lett. **33**(10), 1147–1149 (2008). [CrossRef]

**11. **W. Hsu and S. Lin, “Iterative pixelwise approach applied to computer-generated holograms and diffractive optical element,” Appl. Opt. **57**(1), A189–A196 (2018). [CrossRef]

**12. **R. Bräuer, F. Wyrowski, and O. Bryngdahl, “Diffusers in digital holography,” J. Opt. Soc. Am. A **8**(3), 572–578 (1991). [CrossRef]

**13. **F. Wyrowski and O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A **5**(7), 1058–1065 (1988). [CrossRef]

**14. **C. K. Hsueh and A. A. Sawchuk, “Computer-generated double-phase holograms,” Appl. Opt. **17**(24), 3874–3883 (1978). [CrossRef]

**15. **O. Mendoza-Yero, G. Mínguez-Vega, and J. Lancis, “Encoding complex fields by using a phase-only optical element,” Opt. Lett. **39**(7), 1740–1743 (2014). [CrossRef]

**16. **R. Eschbach and Z. G. Fan, “Complex-valued error diffusion for off-axis computer-generated holograms,” Appl. Opt. **32**(17), 3130–3136 (1993). [CrossRef]

**17. **P. W. M. Tsang and T.-C. Poon, “Novel method for converting digital Fresnel hologram to phase-only hologram based on bidirectional error diffusion,” Opt. Express **21**(20), 23680–23686 (2013). [CrossRef]

**18. **A. Velez-Zea, J. F. Barrera-Ramirez, and R. Torroba, “Optimized random phase only holograms,” Opt. Lett. **43**(15), 3558 (2018). [CrossRef]

**19. **P. Tsang, Y. T. Chow, and T.-C. Poon, “Generation of patterned-phase only holograms (PPOHs),” Opt. Express **25**(8), 9088–9093 (2017). [CrossRef]

**20. **A. Velez-Zea and R. Torroba, “Optimized random phase tiles for non-iterative hologram generation,” Appl. Opt. **58**(32), 9013–9019 (2019). [CrossRef]

**21. **T. Zhao, J. Liu, J. Duan, X. Li, and Y. Wang, “Image quality enhancement via gradient-limited random phase addition in holographic display,” Opt. Commun. **442**, 84–89 (2019). [CrossRef]

**22. **T. Shimobaba and T. Ito, “Random phase-free computer-generated hologram,” Opt. Express **23**(7), 9549–9554 (2015). [CrossRef]

**23. **Y. Nagahama, T. Shimobaba, T. Kakue, N. Masuda, and T. Ito, “Speeding up image quality improvement in random phase-free holograms using ringing artifact characteristics,” Appl. Opt. **56**(13), F61–F66 (2017). [CrossRef]

**24. **T. Shimobaba, T. Kakue, Y. Endo, R. Hirayama, D. Hiyama, S. Hasegawa, Y. Nagahama, M. Sano, M. Oikawa, T. Sugie, and T. Ito, “Random phase-free kinoform for large objects,” Opt. Express **23**(13), 17269–17274 (2015). [CrossRef]

**25. **H. Pang, J. Wang, M. Zhang, A. Cao, L. Shi, and Q. Deng, “Non-iterative phase-only Fourier hologram generation with high image quality,” Opt. Express **25**(13), 14323–14333 (2017). [CrossRef]

**26. **D. Fischer and S. Sinzinger, “Evaluation of quadratic phase hologram calculation algorithms in the Fourier regime,” Appl. Opt. **59**(6), 1501–1506 (2020). [CrossRef]

**27. **A. Velez-Zea and R. Torroba, “Synthetic amplitude for improved reconstruction of noniterative phase holograms,” Appl. Opt. **58**(22), 6144–6151 (2019). [CrossRef]

**28. **H. Zhang, Q. Tan, and G. Jin, “Holographic display system of a three-dimensional image with distortion-free magnification and zero-order elimination,” Opt. Eng. **51**(7), 075801 (2012). [CrossRef]