## Abstract

A curved multiplexing method based on the curved computer-generated hologram (CCGH) is proposed theoretically and demonstrated experimentally to increase field of view (FOV) and spatial bandwidth. Point source method is used to calculate the CCGH. Curved multiplexing method can be used to reconstruct different 3D objects at the same position by bending a composed hologram which is synthesized of several CCGHs with different central angles. Numerical simulations and optical experiments demonstrate that the method is feasible. It could have a good prospect by combining with the curved display screen and flexible display materials.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Three-dimensional (3D) display has become an international research hotpot with the development of display technology and market demand. Holographic 3D display is regarded as the ultimate 3D display technology which is promising to achieve true 3D display without any wearable devices. Computer-generated hologram (CGH) is a key technique to realize holographic display by recording hologram digitally. Normally the CGH can be calculated for 3D virtual object and reconstructed object by loading on a spatial light modulator (SLM). However, the field of view (FOV) is limited by the shape and the pixel pitch of SLM according to Nyquist theory [1] and hence it can’t meet the display requirements of large viewing angle. The method of splicing multiple SLMs is used to increase FOV of holographic display. The system is very complicated, and seamless splicing is also a great challenge [2,3]. Curved hologram is an effective way to overcome the constraints of FOV without the need of splicing SLM. Most reports of curved hologram are about cylindrical hologram [4–7]. The cylindrical hologram has a 360° look-around property and can be observed from any direction. Some fast calculation methods [8,9] for a cylindrical CGH are proposed to reduce computing time. Curved hologram can be implemented by combining with high refractive index materials [10] and the microrelief of a DOE [11]. An improved method of complex amplitude modulation [12]is used for holographic display with a wide viewing angle. The other problem is the limited spatial bandwidth of the current SLM devices and information capacity needs to be improved at the same time. An ultra-thin holographic display based on the phase modulation of the topological insulating material [13] can be used to expand the spatial bandwidth effectively. The graphene-based material [14] and metasurfaces [15] combined with the CGH have enable write-once phase manipulation for 3D holographic image with the potential of wide viewing angles and spatial bandwidth. The multiplexing encoding method [16] can improve the spatial bandwidth and simplify the system by encoding the light waves at different wavelength into a pure-phase hologram. Some other multiplexing methods is used to simplify the system, like the time-divided method [17], depth-divided method [18], the space-divide method [19] and etc.

In this paper, curved multiplexing method based on curved computer-generated hologram (CCGH) is proposed to increase the FOV and spatial bandwidth. In order to combine with the curved display screen and flexible display materials, the shape of CCGH is a part of the cylindrical surface. It is calculated by point source method for simulation wave propagation in the cylindrical coordinate system. The curved hologram can be fabricated on the flexible material by the femto-second laser direct writing (DLW) [10]. The curved multiplexing method is that different objects can be reconstructed at the same position by bending a composed hologram synthetized by some CCGHs of different central angles. The reconstructed process of a CCGH is shown in the Fig. 1(a). In order to reconstruct the object on the original position, the CCGH should be illuminated by the conjugate beam of the reference beam. The reconstruction of a composed hologram based on the curved multiplexing method is shown in the Fig. 1(b). When composed hologram is placed in a plane, the reference is parallel light. When the composed hologram is bent into curved hologram with different central angle, the corresponding cylindrical reference beam is used to ensure the light illuminate each point of the hologram vertically. The different objects are reconstructed one by one at the same positon. We analyzed the principle that CCGH is used to expand the FOV of the reconstructed objects. Spatial bandwidth and information capacity can be multiplying by the curved multiplexing method. Numerical simulations and optical experiments are performed to demonstrate that the method is feasible.

## 2. Methods

#### 2.1 CCGH generation

The diffraction propagation between CCGH and object is calculated by the point source method. The schematic diagram of the CCGH generation is shown in the Fig. 2. The geometric relationship between object and hologram in the Cartesian coordinates system is shown in the Fig. 2(a) on lateral view. The cylindrical coordinate system on the top view is shown in the Fig. 2(b). The complex amplitude distribution of the object plane and hologram is *O(x, y, z _{1})* and

*H(x, y, z)*in the Cartesian coordinates system. According to the spherical wave diffraction theory [20,21], the complex amplitude of a point

*h(x*of diffracted wavefront on the hologram can be given by:

_{p}, y_{q}, z_{s})*M*and

*N*are the resolution of the object,

*k = 2π/λ*is the wave number,

*λ*is the wavelength of the light wave,

*γ*is the distance between the point

_{mn}*O(x*on the object and the point

_{m}, y_{n}, z_{1})*h(x*on the hologram. It can be given by:

_{p}, y_{q}, z_{s})In the cylindrical coordinate system *H(θ, y)*, the point *h(x _{p}, y_{q}, z_{s})* can be given by

*R*is the curvature radius of the curve hologram. Taking the Eqs. (2) and (3) into Eq. (1), complex amplitude of a point

*h(θ, y*of hologram is given by

_{q})*i*is the number of the CCGH.

#### 2.2 The curved multiplexing of hologram

The curved multiplexing method based on the CCGH is proposed to improve the spatial bandwidth and information capacity. The flow chat of the curved multiplexing method is shown in Fig. 3. In the generation process of a composed hologram, three original objects are calculated to generate the CCGHs with different central angles respectively. The flat CGH can be regarded as a CCGH with the central angle 0°. The pixel sizes and pixel numbers of the three CCGH are same. The complex amplitude distribution of three CCGHs are added and synthetized into a composed hologram finally. The complex amplitude of the composed hologram is given by

In the reconstruction process, the composed hologram is bent into curved hologram with the different central angles. When the corresponding cylindrical wave illuminate the curved hologram as the reference beam, the corresponding objects are reconstructed one after one at the same position. In this way, the spatial bandwidth and information capacity can be increased several times over. The pixel number of the curved hologram can be increased to achieve a better performance in the future work.#### 2.3 The analyzation of FOV

In order to illustrate that the FOV of reconstructed object can be increased by CCGH, the plane CGH which is usually loaded on the flat SLM is compared with the CCGH. Ma X et al. studied the parameters of flat SLM and analysis the relationship among the resolution, the pixel pitch, the reconstructed distance and FOV of the 3D reconstructed object [22]. The FOVs of the plane CGH and the CCGH are shown in the Fig. 4.

Where the reconstructed process of the plane CGH is represented by the blue lines. *β _{max}* is the maximum diffraction angle of a single pixel of plane CGH loaded on the SLM. According to the diffraction theory, when the pixel shape of the hologram is square, the diffraction angle

*β*is calculated as [23]

_{max}*p*is the pixel pitch,

*λ*is the wavelength of the reference beam in free space. As we can see from Eq. (7), the bigger maximum diffraction angle can be obtained with the smaller the pixel pitch and the longer wavelength. According the triangular geometry relationship, the FOV of the plane CGH loaded on the SLM can be calculated as:Where

*H*is the width of the plane hologram.

*D*is the maximum size of the reconstructed object that can be observed totally. The reconstructed distance between the reconstructed object and the plane CGH can be calculated asWhere

*H > D. L*is the distance between the reconstructed image and the plane CGH to satisfy the maximum FOV.

The reconstructed process of the CCGH is represented by red line shown in Fig. 4. It is obvious that the FOV of CCGH (red line) is increased compared with plane CGH (blue line). According to the triangular geometry relationship, the FOV of the reconstructed object of CCGH can be calculated as:

*D/2<Rsin(α/2)*,

*α/2 + β*.

_{max}<π/2*α*is the central angle of the CCGH.

*R*is the curvature radius of the CCGH marked by red dotted line.

*D*is the maximum size of the reconstructed object. The width of the CCGH is

*H = Rα*. The reconstructed distance

*L′*between the reconstructed object and the CCGH can be calculated as

*α*is smaller and smaller, FOV and reconstructed distance

*L′*of the CCGH are tended to be consistent with that of plane CGH. The FOV of the reconstructed object and the reconstructed distance are related to the central angle α and the curvature radius R of the CCGH when maximum diffraction angle

*β*and the object size D is certain. Assuming the width

_{max}*H*of the hologram is 1.5 cm, pixel pitch is 3.74μm, the reconstructed image size is 1 cm, and the wavelength of reference beam is 532 nm for green. The relationship between the central angle

*α*and the FOV is shown in the Fig. 5(a). It is seen that the FOV of the reconstructed image augments with the increased the central angle. Therefore, CCGH can be used to enlarge the FOV of reconstructed object in the curved horizontal direction effectively.

For the reconstructed object of the CCGH, the visible area is cut off due to the limitation of the maximum diffraction angle of a pixel, as shown in the dash area of Fig. 6(a). We define a cut-off angle *γ* to calculate the relationship between FOV and the cut-off area shown in the Fig. 6(b). The angle *γ* is expressed as

*β*is the maximum diffraction angle.

_{max}*α*is the central angle. It can be seen from the Fig. 5(b) that the visible area of the reconstructed object reduces with central angle

*α*increasing. The visible area will not be cut off when the

*γ*is bigger than the FOV/2. When the

*γ*is smaller than the FOV/2, the visible area starts to be cut off. Assume the parameters are all as same as the calculation of FOV before. The relationship between the FOV/2 and

*γ*is shown in the Fig. 5(b). The point marked by red means that the FOV/2 and

*γ*are equal. That is to say parallel of boundary beams of the two angles is a critical state that determines if the visible area is cut off. Therefore the central angle of the hologram can’t be blindly increased even if large FOV need to be obtained. We should make a trade-off between the increasing of FOV and the viewable area.

## 3. Numerical simulations and optical experiments

#### 3.1 Optical experiment setup

The schematic of the optical experimental setup for reconstruction is shown in the Fig. 7. The green laser with wavelength 532nm are collimated by the collimator which consists of spatial filter and collimating lens. The CCGH is loaded on a phase-only SLM (JD8714.pixel pitch 3.74um, resolution: 3840 × 1920, 256 phase modulation levels). The 4f system that consisted of two lens and a filter is used to eliminate the impact of the zero order beam introduced by SLM on the reconstructed image [24]. The focal lengths of L1 and L2 are 350mm. The reconstructed image is recorded by the CCD after L2.

#### 3.2 Precompensation

The flat reflection-type phase-only SLM and plane reference wave is used in the optical experiment. The phase distribution Φ* _{CCGH}* of the CCGH need to be pre-compensated in two steps before loading on the SLM. Firstly, add the phase distribution Φ

*of cylindrical wave to the CCGH. That is equivalent to change the plane reference wave to cylindrical reference wave. Secondly, the wave-front difference that is caused by loading the CCGH on a flat SLM should be corrected. Therefore the phase variation Φ*

_{C}*of each pixel of hologram are compensated according to different distance between the CCGH and flat SLM. The phase distribution Φ*

_{P}**after compensation can be expressed as**

_{SLM}*of convergent cylindrical wave and the phase variation Φ*

_{C}*can be described as Where*

_{P}*k = 2π/λ*is the wavenumber in free space,

*λ*is wavelength,

*x*is the horizontal coordinate,

*R*is the curvature radius of the CCGH,

*b*is the distance distribution of point to point between the CCGH and flat SLM. Different point of CCGH has different distance to the flat SLM. Here the optical path difference is double due to the reflection-type SLM. Then the compensated CCGH can be load on the flat SLM to reconstruct objects.

#### 3.3 Numerical and optical experimental results

We carry out three numerical simulations to demonstrate the methods. All the simulations are running on the platform of Matlab R2016b. The simulation is according to the physical propagation process of the optical experiment. The point source method is used to calculated the CCGH. The CCGH is pre-compensated, therefore Angular spectrum theory [20] is used to reconstruct the objects in the reconstructed process.

The first experiment demonstrates that the CCGHs with different central angles can generate and reconstruct in the cylindrical coordinate system successfully. Three gray images of 1024 × 1024 pixels shown in Figs. 8(a)-8(c) are used to calculate three CCGHs of 3840 × 2160 pixels with the central angles of 0°, 8°, 15° respectively. The parameters used in the numerical simulation and the optical experiment is given as following: the sampling interval of the object and the hologram is 20 × 20μm and 3.74 × 3.74μm respectively. The wavelength of the reference light is 532nm. The distance between the object plane and the hologram is 200mm. The phase-only CCGHs are generated by using point source method. The numerical reconstructed images are shown in Figs. 8(d)-8(f). The quality of reconstructed image can be calculated by the peak signal-to-noise ratio (PSNR) and speckle contract (SC). The PSNR and SC of the three reconstructed image are 12.9 dB, 14.1 dB, 9.4 dB and 0.366, 0.305, 0.344, respectively. It can demonstrated that the calculation by point source method and reconstruction of compensated CCGH by angular spectrum theory work well. The compensated CCGH is loaded on the SLM. The optical experimental results are shown in Figs. 8(g)-8(i). The optical experimental results and numerical simulations match well.

In the second experiment, three binary images of 512 × 512 pixels shown in Figs. 9(a)-9(c) are calculated to demonstrate the curved multiplexing method. A single phase-only hologram of 2048 × 2048 pixels is composed of three CCGHs with central angles 0°, 15°, 30°. The composed hologram is compensated into three curved holograms according to the different central angles by the pre-compensation method. The numerical reconstructed images of curved hologram with the central angles 0°,15° and 30° are shown in Figs. 9(d)-9(f), respectively. The correspondent optical experimental results are shown in Figs. 9(g)-9(i). The PSNR and SC of the three reconstructed image are 7.5dB, 7.0 dB, 6.9 dB and 0.328, 0.332, 0.337, respectively. The optical experimental results are in good agreement with numerical simulations. It is obvious that curved multiplexing method is feasible. It is noted that there are some crosstalk and background noise on the reconstructed images of numerical and experimental results due to the influence among three CCGHs.

The third experiment show the reconstruction of the 3D scene which is divided into multiple 2D slices of 400 × 200 pixels at different distances. The 3D objects are calculated to generate a hologram of 3840 × 2160 pixels composed of three CCGHs with the central angles 0°, 15° and 30°. The schematic diagram of the composed hologram generation is shown in the Fig. 10. The first plane CGH is calculated by 3D object of two slices which are letter ‘B’ and Chinese character ‘北’ focused on the distance z_{1} = 200mm and z_{2} = 250mm. By that analogy, the other two CCGHs of central angle 15° and 30° are calculated in the same way. The composed hologram of the three CCGHs synthetized by the curved multiplexing method. The numerical and optical reconstructed 3D images are shown in the Fig. 11. The pre-compensated CCGH with central angle 15° is loaded on the SLM. Correspondingly the letter ’I’ becomes from in-focus to blurred, while the Chinese character ‘理’ become from blurred to in-focus. By that analogy, the reconstructed images of the other two CCGHs are displayed in the same way. When the CCD focus at 200mm, the reconstructed results of numerical and optical experiment are shown in the Figs. 11(a) and 11(b). The letters ’B’, ’I’ and ’T’ are displayed one by one at same position according to the curved holograms with central angles 0°, 15° and 30° . When the CCD focus at 250mm, the reconstructed results of numerical and optical experiment are shown in the Figs. 11(c) and 11(d). The Chinese character ‘北’, ‘理’ and ‘工’ are displayed one by one at same position according to the curved holograms with central angles 0°, 15° and 30°. It is obvious that the 3D scene with the depth information can be reconstructed from a composed hologram generated by the curved multiplexing method. There are still some crosstalk and background noise on the reconstructed images.

In this section, the first experimental results demonstrate that the objects can be successfully reconstructed by CCGHs with different central angles based on the point source method in cylindrical coordinate system. The second and third experimental results demonstrate that the 2D and 3D objects can be reconstructed by bending a composed hologram into different central angles based on the curved multiplexing method. The spatial bandwidth and information capacity can be multiplied by the proposed method. The quality of the reconstructed objects are slightly low because the crosstalk and background noise generated by the influence among the CCGHs with different central angles. The image quality might be influenced by the increasing central angle. The influence in this paper is not obvious because the range of central angle is small.

## 4. Conclusion

The curved multiplexing method based on CCGH is proposed to increase the FOV and the spatial bandwidth. The CCGH is generated by point source method in cylindrical coordinate system. The increase of the FOV of CCGH is theoretically analyzed compared with the plane CGH. The curved multiplexing method is that a composed hologram is synthesized by CCGHs with different central angles. The numerical and experimental results both indicate that the 2D and 3D objects can be reconstructed base on the method correctly. The CCGH can provide a promising prospect to achieve the large FOV by combining with the curved display screen and flexible display materials. Complex amplitude modulation could also be used to achieve better performance in the curved multiplexing system in the future.

## Funding

National Key R&D Program of China (2017YFB1002900); National Natural Science Foundation of China (NSFC) (61575024, 61420106014); and the United Kingdom Government’s Newton Fund.

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