## Abstract

Herein, we propose a band-limited double-phase method to improve the quality of reconstructed images encoded by double-phase holograms (DPHs) derived from complex-amplitude light waves. Although the quality of images produced by DPHs was improved compared to that of conventional holographic images, it still suffered from degradation because of the spatial shifting noise generated during the conversion from complex-amplitude holograms to phase-only holograms. The proposed method overcomes this shortcoming by defining a band-limiting function according to the spatial distribution of DPHs in the frequency domain to remove the specific spatial frequency components severely affected by the spatial shifting of DPHs. The sharpness of images reconstructed from band-limited DPHs with appropriate optical filtering showed an improvement of 36.84% in simulations and 51.67% in experiments evaluated by 10-90% intensity variation.

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

## 1. Introduction

Holographic display is a form of three-dimensional displays that has shown promising prospects of development because of its advantages in wavefront reconstruction [1,2]. The advancement of spatial light modulators (SLMs) and computing technologies presents a new avenue for electro-holography, which allows the use of SLMs to present computer-generated holograms without a photosensitive medium and complicated interference recording procedure [3–5]. Because conventional SLMs cannot perform full complex modulation in a single panel, the complex holograms generated by computers are typically converted to amplitude-only or phase-only holograms. Using conventional SLMs for complex modulation of coherent light beams is a challenging task for core techniques in various fields such as holographic display [6], optical encryption [7], and optical tweezers [8].

In 1978, Hsueh and Sawchuk described a method of generating double-phase holograms (DPHs) on the basis of the decomposition of a complex value into two phases [9]. Thus, two phases could be encoded in a single phase-only hologram by combining sub-cells. Furthermore, the coherent superposition of adjacent sub-cells enabled the production of a good approximation of the desired complex field with spatial filtering of a 4-*f* system. DPHs have garnered significant attention because they can be implemented using phase-only SLMs [10]. In 2003, Arrizón modified double-phase holographic encoding with double-pixel cells for better implementation in liquid-crystal displays [11]. In 2014, Mendoza-Yero et al. utilized binary gratings to sample phase matrices and combine them into a single-pixel DPH. It afforded the initial idea of approximately encoding a complex field into a phase-only optic element without a cell-based structure [12]. Various improvements and modifications such as noise suppression [13,14], modulation for a 3-D scene [15], and lensless reconstruction for holographic projection [16] have since been carried out, based on the single-pixel on-axis configuration in DPHs. The reconstruction based on existing DPHs is still strongly dependent on the 4-*f* system and filtering. In 1997, Mendlovic et al. attributed the main cause of the reduction in performance of DPHs to the spatial shifting between the two different phases [17]. In 2002, Arrizón mathematically separated the spatial shifting noise from a DPH in the frequency domain and revealed it to have a higher intensity on the edge of the signal term [18]. This explained the necessity of a 4-*f* spatial filtering system in the reconstruction of a DPH, which blocked off the frequency components severely affected by the noise term. This smoothing filter reduced the image sharpness in reconstruction and, thus, strictly limited the spatial-bandwidth product (SBP). In 2019, a DPH with a weight factor was introduced to suppress the spatial shifting noise [19]. Despite these innovations, the reconstruction of DPHs still suffers from limited spatial resolution because of their inherent restriction in SBP.

The double-phase method has been introduced in holographic displays to realize complex wavefront modulation. Currently, most calculation methods for propagation between parallel planes are mainly based on fast Fourier transform, including the single Fourier-transform-based Fresnel method (SFT-FR), convolution-based Fresnel method (CV-FR), the angular spectrum (AS) method, and many other modified methods [20–23]. The AS method is derived directly from the Rayleigh–Sommerfeld diffraction theory, while the CV-FR and SFT-FR methods include a Fresnel approximation in the paraxial field. Considering the implementation of a near-field propagation-based holographic display, the AS method is more applicable for the generation of complex holograms.

Herein, we propose a band-limited double-phase method based on a single-pixel on-axis double-phase technique. The quantitative relationship between the signal term and the noise term of a DPH was theoretically analyzed on the basis of the spectrum distribution of double-phase codes. Furthermore, we defined a band-limiting function to efficiently separate the signal term from the spatial shifting noise in the frequency domain. Considering that band limitation was capable of previously removing the high-noise spatial frequency and enabling the further extension of expressible spectral components in reconstruction, the proposed method leads to an improved reconstruction with higher marginal sharpness and spatial resolution.

## 2. Band-limited double-phase method

The procedure for calculating a band-limited DPH is shown in Fig. 1. A two-dimensional object can be described as $I({x,y} )$. The AS of the two-dimensional image $I({x,y} )$ can be described as

where*F*represents the Fourier transform. According to the AS diffraction theory, transfer function ${H_F}({u,v} )$ can be expressed as where

*λ*is the wavelength and

*k*= 2

*π*

*/λ*is the wave number. Symbols

*u*and

*v*are the spatial frequencies in the

*x*and

*y*directions, respectively. Complex field after diffraction propagation ${U_d}({x,y} )$ can be derived from the AS of the initial field: Output complex hologram $U({x,y} )$ is the convolution of a diffracted object with the band-limiting function and is expressed as where $P({u,v} )$ is the band-limiting function, which is expounded in the subsequent section. The formula for a complex hologram is also defined as a simplification: This consists of amplitude $A({x,y} )$ and phase $\varphi ({x,y} )$. Because $A({x,y} )$ is normalized to [0, 1], it can be decomposed into the sum of two phases [8]: $\alpha ({x,y} )$ can be described by $A({x,y} )$ as Substituting the decomposed $A({x,y} )$ into Eq. (5), $U({x,y} )$ can be defined as

Considering the digital implementation of the complex hologram, we convert $U({x,y} )$ to a pixelated form:

The summation notation here represents the superposition of the modulated waves reflected by pixels on an SLM. ${A_{n,m}}$ and ${\varphi _{n,m}}$ are the amplitude and phase of the modulated plane wave reflected by the ${({n,m} )^{\textrm{th}}}$ pixel, respectively. According to the single-pixel on-axis double-phase method [8], the two pixelated decomposed waves—${h^{(1)}}({x,y} )$ and ${h^{(2)}}({x,y} )$—can be represented with their sampled values taken at the Nyquist limit: where*f*

_{S}is the sampling frequency and

*f*

_{N}is the highest frequency of the processed image (Nyquist frequency). $h_{n,m}^{(1)}$ and $h_{n,m}^{(2)}$ then can be synthesized into one DPH $h({x,y} )$, which is given by Phase quantities ${\theta _{n,m}}$ can be expressed by where phase shift ${\Delta _{n,m}}$ to encode sampled amplitude ${A_{n,m}}$ is Such a DPH consists of two constant-amplitude waves with phases:

## 3. Frequency domain analysis and definition of band-limiting function

We assume that the desired complex field is encoded using a phase-only SLM with a pixelated structure with a filling factor of 1. The dimensions of the rectangular pixel-active window are in line with inter-pixel distances *α* and *β*, along the horizontal and vertical axes, respectively. The complex field generated by the SLM is expressed by the function:

The product of envelope functions $E({u,v} )\cdot {P_s}({u,v} )$ restricts the DPH reconstructed signal to be within the on-axis centered band [shown in Fig. 2(a)], while $E({u,v} )\cdot {P_n}({u,v} )$ modulates the noise item [shown in Fig. 2(b)]. To block the high-noise area, a 4-*f* system with a spatial filter is usually introduced into the reconstruction.

Considering that Fourier frequencies *u* and *v* take values in the $({ - 1/2\alpha , 1/2\alpha } )$ and $({ - 1/2\beta , 1/2\beta } )$ domains, respectively, we define the boundary function of the signal envelope and noise envelope ${P_b}({u,v} )$ as

*K*is a parameter that determines the bandwidth of complex holograms. Boundary function ${P_b}({u,v} ) = K$ and band-limiting function $P({u,v} )$, taking various values of

*K*, are plotted in Fig. 3.

## 4. Results

#### 4.1 Numerical simulations

We conducted a numerical simulation to demonstrate the complex reconstruction of band-limited DPHs. The target two-dimensional patterns were two 1024×1024 grayscale images from the USC-SIPI Image Database [24]. We named these two images as ‘image 1’ and ‘image 2’, as is shown in Fig. 4. The two-dimensional complex field was a 1024×1024 complex array consisting of the amplitude component ranging [0, 1] and the phase component ranging [0, 2π]. In the first case, the pattern of amplitude component is constructed from ‘image 1’ while the pattern of phase component is constructed by ‘image 2’. We compared the reconstruction of a DPH encoded from the complex field after the band limitation and the reconstruction of a DPH encoded directly from the target complex field. The curves of the peak signal-to-noise ratios (PSNRs) changing with the diameter of the filter are shown in Fig. 5. The partial zoom-in view of the reconstructed amplitudes and phases come from the labeled points on the curves are also shown in Fig. 5. We observed that the reconstruction of a band-limited DPH was improved compared to that of a conventional DPH, especially for preserving the amplitude components from the spatial shifting noises.

To verify the generality of the proposed band-limited DPH method, we exchanged the pattern of amplitude and phase for the complex field, where the pattern of amplitude component is constructed from ‘image 2’ while the pattern of phase component is constructed by ‘image 1’. We compared the reconstruction of conventional DPHs and band-limited DPHs once more. The curves of the PSNRs as well as the reconstructed amplitudes and phases on the labeled points are presented in Fig. 6. The results in Fig. 6 indicate a similar regularity with those in Fig. 5.

To illustrate the superiority of band limitation, we attempted numerical reconstructions of complex holograms encoded to conventional and band-limited DPHs. Because the numerical calculation of a complex hologram is based on the AS diffraction theory, the Nyquist theorem requires that the sampled transfer function is limited within the following region to avoid aliasing errors [20]. The propagation distance *z* must be limited to

The target object considered for testing was a binary star bar image [Fig. 7(a)], which had a continuous change in spatial resolution from the edge to the center. The conventional double-phase method and the proposed band-limited double-phase method generated a hologram with 1024×1024 pixels and ${\Delta _x} = {\Delta _y} = 3.74\;\mathrm{\mu }\textrm{m}$ pixel pitch. The parameter of the band-limiting function was artificially set as $K = {1 / 5}$, which comes from the subsequent comparison of band-limited DPHs with various *K* values. It can be intuitively understood as a filter passing the ${{|{{P_s}({u,v} )} |} / 5} \ge |{{P_n}({u,v} )} |$ spatial frequency components while blocking the ${{|{{P_s}({u,v} )} |} / 5} < |{{P_n}({u,v} )} |$ spatial frequency components.

To quantitatively describe the enhancement of the image sharpness in reconstruction, we considered the total variation of a 2D reconstructed hologram, which is given by [25]:

Figure 8 presents the numerical reconstructed complex holograms with and without band limitation. The reconstructed objects were obtained through a numerical 4-f filtering with various sizes of apertures and backpropagation. The reconstructed images by conventional and band-limited DPHs are presented in the 1^{st} and 2^{nd} rows, respectively. To verify the effect of the band-limiting function, we numerically reconstructed conventional and band-limited DPHs with the enlargement of the diameter of the filter. The images in the 1^{st}, 2^{nd}, and 3^{rd} columns were reconstructed with filters of 1.5, 2.5, and 3.5 mm in diameter (*D*), respectively. The total variation of the reconstructed images is labeled in Fig. 8. To evaluate the spatial resolution of the reconstructed images generated by the two methods, we plotted the intensity at the same radius on the reconstructed images, both by conventional and band-limited DPHs, which are shown as curves in the 3^{rd} row of Fig. 8. The radius of the plotted pixels are labeled in blue and red curves in the reconstructed images.

The images reconstructed by band-limited DPHs had a higher quality than those by conventional DPHs, including suppressed noises and improved spatial resolutions. With the enlargement of the filter, the reconstructed images by conventional DPHs had a severe reduction in the image definition, while band-limited DPHs were still able to generate faithful reconstructions. The trend of the total variation for band-limited and conventional DPHs showed that the image sharpness can be enhanced by increasing the size of the filter, as most of the spatial frequency components affected by the noise term were previously removed. Narrower lines can be distinguished in the zoom-in-views of reconstructed images by band-limited DPHs, compared to those by conventional DPHs, which indicates a higher spatial resolution in reconstruction as well as a greater capability of band-limited DPHs to preserve minute details.

For an overall comparison, we plotted the curves of the PSNRs for the reconstructed images changing with diameter *D* of a filter. Figure 9(a) shows the PSNRs for the reconstructed images by conventional DPHs and the proposed band-limited DPHs ($K = {1 / 5}$). The curves indicated much higher PSNRs for band-limited DPHs than those for conventional DPHs. The curve for conventional DPHs presented a more drastic reduction with the enlargement of the filter, while the curve for band-limited DPHs presented a gentler shape and maintained higher levels. This phenomenon ruled out the possibility that the improvement of the total variation for band-limited DPHs was caused by spurious details and excessive noise. This verifies the superiority of band-limited DPHs in removing noise and expanding the available SBP of DPHs. Figure 9(b) shows the PSNRs for the reconstructed images by band-limited DPHs with different values of band-limiting parameter *K*. The variation in *K* led to a variation in restricted image quality. We use this analysis as the basis for the modification of *K* while generating band-limited DPHs.

To verify the applicability of band-limited DPHs in a two-dimensional grayscale object and a two-dimensional binary object, we carried out numerical reconstructions for the 1024×1024 grayscale image named ‘boat’ and the 1024×1024 binary image named ‘THU’. The curves of the PSNRs for the reconstructed images changing with diameter *D* of a filter are plotted in Fig. 10. As we had anticipated, the PSNRs of the reconstructed images by conventional DPHs decreased dramatically with the enlargement of the filter with both of the original objects, while the PSNRs of the reconstructed images by band-limited DPHs still maintained higher levels [Figs. 10(a) and 10(b)]. Figures 10(c) and 10(d) show the PSNRs of the reconstructed images by band-limited DPHs with different values of band-limiting parameter *K*, providing the basis for determining *K* in the calculation. It can be observed from Figs. 10(a) and 10(b) that the PSNR has a significant difference. This difference is caused by the intensity of the noise term. For a binary image with a large area of black background like the image ‘THU’, more pixels with low intensity are included even after the near-field propagation. The low-intensity components result in a relatively higher-intensity noise term, which can be conclude from Eq. (22). It explains why the reconstructions for ‘THU’ have lower PSNRs.

To illustrate the significance of extending expressible spectral components and expanding available SBP, we compared the qualities of the reconstructed images by conventional and band-limited DPHs at maximum points on the curves of PSNRs. As is shown in Figs. 10(a) and 10(b), the maximum points on the curves for conventional DPHs are labeled as ‘a’ (*D* = 2.66 mm and *D* = 2.09 mm), while the corresponding points on the curves for band-limited DPHs with the same filters are labeled as ‘A’. The maximum points on the curves for band-limited DPHs are labeled as ‘B’ (*D* = 3.35 mm and *D* = 2.81 mm), while the corresponding points on the curves for band-limited DPHs with the same filters are labeled as ‘b’. The points of ‘B’ (‘b’) exhibited PSNRs similar to those of the points of ‘A’ (‘a’) and larger diameters of filters, which indicates an expanded SBP in reconstruction.

We numerically reconstructed conventional DPHs at ‘a’ and ‘b’, and band-limited DPHs at ‘A’ and ‘B’, respectively. The total variations for reconstructed images are labeled in Fig. 11. The images on the 1^{st} column are reconstructed at points ‘A’ (‘a’), while the images on the 2^{nd} column are reconstructed at points ‘B’ (‘b’). The images on the 2^{nd} row and 4^{th} row of Fig. 11 are the reconstructed images by conventional DPHs. The zoom-in views of the reconstructed images by conventional DPHs at points ‘a’ include blurred outlines and details, resulting from the loss of high-spatial-frequency components and the ringing effect of a filter. The reconstructed images by conventional DPHs at points ‘b’ contain excessive spatial shifting noises which were included in the images with the enlargement of the filter. The images on the 3^{rd} row and 5^{th} row of Fig. 11 are the reconstructed images by band-limited DPHs. The zoom-in views of the reconstructed images by band-limited DPHs at point ‘A’ perform similarly with those by conventional DPHs. However, the reconstructed images by band-limited DPHs at points ‘B’ include better outlines with fewer blurs and noises. The reconstructed images by band-limited DPHs at points ‘B’ and conventional DPHs at points ‘b’ have higher total variations than those for DPHs at point ‘A’ (‘a’). However, the reconstructed images by conventional DPHs at points ‘b’ show an additional increasing of total variation caused by noises, while the reconstructed images by band-limited DPHs at points ‘B’ have closer values of total variation to the original images. As mentioned above, reconstruction for images with a large area of black background typically suffers from extremely severe noises, the reconstructions for the image ‘THU’ thus present relatively higher values of total variation. Considering the curves for the PSNRs in Fig. 10 and the improvement of total variation simultaneously, the band-limited DPHs can be numerically reconstructed with enhanced image sharpness while avoiding excessive noise.

The intensities of the pixels on the same line in different reconstructions are plotted in the 6^{th} row of Fig. 11. From Fig. 11(l), it is shown that the image sharpness, as defined by the 10–90% intensity variations at the edge of the flag post in the images reconstructed by simulation, increases from 3.8 pixels by the conventional DPH to 2.4 pixels by the band-limited DPH, respectively. This represents an increase in relative sharpness by 36.84% and the corresponding increase in spatial frequency (hence, spatial resolution). This expansion of available SBP exceeds the inherent limit of conventional DPHs and enables considerable promotion of DPHs.

It should be noted that, with the enlargement of the filter, the spatial shifting noises in the band-limited DPHs are accumulated, because the band-limiting function can only remove high-noise frequency component rather than the whole noise term. As the result of noise accumulation, an excessive expansion of the filter results in the declination of the image quality. Thus, we suggest to carefully choose the size of the diameter of filter to reach an expected reconstruction.

#### 4.2 Optical experiments

To demonstrate the feasibility of the proposed band-limited double-phase method, we implemented the optical setup shown in Fig. 12. A coherent beam at a wavelength of 532 nm was emitted from a solid-state laser acting as a light source. It was attenuated and expanded before passing through the polarizer. Then, it was placed incident to a reflective liquid crystal on silicon phase-only SLM. The phase-only SLM we utilized had a pixel number of 3840 × 2160, pixel pitch of 3.74 µm, and frame rate of 60 Hz.

The original objects were two-dimensional amplitude-only arrays with a size of 1024 × 1024 pixels. Conventional and band-limited DPHs were computed previously with the simulated parameters, which were then uploaded onto the SLM.

The beam splitter (BS) allowed both the plane wave illumination of the SLM and the reflection toward a 4-f system with a filter blocking of unwanted diffraction orders and spatial frequency components. The complex hologram was reconstructed at the back focal plane of the second Fourier lens in the 4-f optical system. It was necessary for the optically reconstructed complex hologram to propagate for a distance to reconstruct the original object. The diameter of the filter was adjusted to attain the highest quality of reconstruction while uploading different DPHs. While we captured the reconstructions of the conventional DPHs, the corresponding band-limited DPHs were also captured at the same diameter of filter as a comparison. The same procedure was done when the reconstructions of band-limited DPHs were captured. Thus, all the reconstructed images by conventional and band-limited DPHs were captured with the same relatively small aperture and the same relatively large aperture.

Figure 13 shows the optically reconstructed images captured by a complementary metal oxide semiconductor (CMOS) detector. Their zoom-in views and curves for the intensities from a line of pixels are provided. It shows that the images reconstructed by band-limited DPHs with proper filtering retained most of the original features and avoid excessive noises, as is shown in Figs. 13(f) and 13(j). They exhibited better outlines and more details with less blurring compared to the images reconstructed by conventional DPHs. The total variation values for band-limited DHPs of the two patterns showed 28.61% and 9.17% improvement compared to those for conventional DPHs, respectively. This indicates the capability of band-limited DPHs to ensure noise removal and sharpness enhancement. From Fig. 13(l), it can be seen that the image sharpness, as defined by the 10–90% intensity variations at the edge of the flag post in the experimentally reconstructed images, increases from 12.0 pixels by the conventional DPH to 5.8 pixels by the band-limited DPH, respectively. This represents an increase in relative sharpness by 51.67% and the corresponding increase in spatial frequency (hence spatial resolution). The higher image sharpness of the optical reconstructions using the band-limited double-phase method verified its ability to improve spatial resolution.

## 5. Conclusion

The proposed band-limited method can remove undesirable high-frequency noise from DPHs with appropriate optical filtering during holographic reconstruction and generate high-quality images with much improved sharpness and spatial resolution. The corresponding image sharpness was evaluated by 10–90% intensity variation and it showed an improvement of 36.84% in simulation and 51.67% in experiment with the best diameter of filter. The implementation of band limitation in the frequency domain made it possible to overcome the inherent restriction to the image quality, as introduced by the noise in DPHs due to spatial shift. The removal of high-frequency noise components enabled the extension of expressible spectral components. Finer details in an image were reconstructed without excessive noises as the corresponding high-spatial-frequency components were preserved, thereby resulting in enhanced image sharpness. The star bar testing image and the total variation were specifically introduced to quantitatively evaluate the improvement in image sharpness. The numerical and optical reconstructions for band-limited DPHs confirmed the enhancement of image sharpness and, hence, highlighted the capacity of this method to expand SBP. This allows DPHs to be applied to high-precision, dynamic, and complex field reconstruction for high-quality holographic displays, holographic tweezers, and laser beam shaping.

## Funding

Tsinghua University (20193080075); National Natural Science Foundation of China (61775117, 62035003); Tsinghua-Cambridge Joint Research Initiative.

## Disclosures

The authors declare no conflicts of interest.

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