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Abstract
We propose a holographic display system for complex amplitude modulation (CAM) using a phase-only spatial light modulator (SLM) and two polarization gratings (PG). The two sub-holograms of the complex-amplitude computed generated hologram (CGH) are loaded in different regions of SLM. Two diffractive components couple in space after longitudinal migration from the double PGs, and finally interfered through the line polarizer. The influence of the system error on the reconstructed image quality is analyzed, which provides a theoretical assessment for adding pre-compensation to CGH to compensate the system error. Moreover, on the base of the proposed system, a large depth of field and enlarged display area display is realized and the real-time display can be achieved because of the analytical complex-amplitude computed generated hologram. The optical experimental results show that the proposed system has high energy efficiency, and can provide high-quality holographic display with a large depth of field and enlarged display area.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Using conventional spatial light modulators (SLMs) for complex amplitude modulation (CAM) is a great challenge and receives much attention in the last decades [1,2]. Because current SLMs cannot modulate the amplitude and the phase simultaneously, the complex- amplitude holograms must be converted into amplitude-only holograms or phase-only holograms before uploading on the SLM in holographic display [3–6]. To obtain precise wavefront modulation, researchers have made efforts to achieve CAM. The cascade of the SLMs [7–9] can be employed to achieve CAM, such as the interference method [10], the iteration method [11] and so on. However, this method always suffers from the pixel-to-pixel alignment problem because a slight misalignment will have a great influence on the reconstruction quality. In order to avoid the problem, several CAM technologies based on single SLM are proposed. A well-known method to generate the complex field is the bleaching method [12–14], which records an off-axis hologram on the phase-only SLM. The essence of the bleaching methods is the first order of the expansion which happens to be the desired complex field. In other words, this method will introduce extra orders, which is useless for holographic display and reduces the effective space bandwidth product. Another common method is the superpixel method [15–17], which combine a few pixels of the SLM to act as one superpixel and resamples the hologram by filtering system for CAM. However, the shifting noise cannot be eliminated completely and degrades the reconstruction quality. In recent years, an advanced superpixel has been developed [3], where a structured half-wave plate, a polarization sensitive component and a polarizer are used to combine two adjacent pixels of the SLM to realize CAM. This method avoids the shifting noise in traditional surperpiel method. However, the processing of the structure will be more difficult as the pixel size of the SLM becomes smaller. Moreover, the sinusoidal grating filter and binary phase grating [18–20] are also used as the auxiliary optical element to realize the superposition of two sub-holograms for CAM. Different from the superpixel method, the decomposed two holograms are in different regions of the SLM instead of staggered arrangement. However, the low diffraction efficiency limits the application in practical display system.
In recent years, polarization grating (PG) [21], which has polarization selectivity and can concentrate most of the diffracted energy in the ±1 order, has been introduced into holographic display systems to improve the performance. Yoo et al. [22] utilizes the combination of multiplexed HOEs, a PG and the polarization-dependent eyepiece lens to realize a pupil movement with an extended eyebox. Lin et al. [23] uses a polarization grating to realize the 2D pupil duplication for Maxwellian near-eye display (NED) systems. Shi et al. [24] uses a pair of parallel PGs to provide an extended eyebox and eliminate the multiple or blank image problems in Maxwellian NED.
In this paper, a novel CAM method is proposed for the first time by utilizing a pair of PGs. The proposed method can achieve CAM with high diffraction efficiency compared with sinusoidal grating. The relationship between alignment error and imaging quality is simulated and analyzed. According to this analysis, pixel offset is used for pre-compensation to improve alignment. Moreover, through the improvement of the system and the multiplex of holograms, we realized the holographic display of different information with multiple depth and expanded display area, which will be illustrated in section 4.
2. Principle and system
2.1 Polarization gratings
The PG, which is fabricated by an anisotropic medium, has very high diffraction efficiencies in ±1 orders and can selectively diffract the incident light in +1 or −1 order depending on the polarization state of the incident light. Figure 1 shows the polarization selectivity characteristics of the transmissive PG. As shown in Fig. 1 (a), the left-handed circular polarization (LCP) light will be diffracted with a tilted angle $\theta $ and converted to the RCP after passing through the PG. In contrast, the right-handed circular polarization (RCP) light will be diffracted with a tilted angle $- \theta $ and converted to the RCP when it is incident on the PG as shown in Fig. 1 (b). Therefore, the linear polarization (LP) light, which can be divided into the LCP light and the RCP light with equal intensity, will be converted into LCP beam and a RCP beam with the same intensity and opposite tilted angles after modulated by the PG as shown in Fig. 1(c).
Fig. 1. The working principle of the transmissive PG when the incident light is (a) the LCP light; (b) the RCP light and (c) the LP light.
2.2 CGH calculation
To generate CGH, we adopt an angular spectrum layer-based method. In the angular spectrum layer-based method, a 3D object is sliced into several parallel layers with depth information. For each layer, the complex amplitude distribution in the hologram plane is calculated by the angular spectrum method, which decomposes the optical wavefront into many plane waves with different spatial frequencies and superimposes them in the observation plane. By adding the diffraction distribution of all sliced layers, the complex amplitude distribution of CGH is obtained and can be described as
Then, the complex amplitude distribution of the desired object on the hologram can be expressed as:
The $A({x,y} )$ and $\varphi ({x,y} )$ represent the amplitude and phase distribution of the desired wavefront, and j is the imaginary unit. In DPH approach, after the amplitude $A({x,y} )$ is normalized to [0,1], the complex field can be decomposed into the sum of two phase-only holograms (POHs) as follows:
The decomposed two POHs can be solved analytically as
The CAM can be realized by loading two POHs onto different regions of the SLM and modulated through our system. The proposed holographic display system is analyzed in detail in section 2.3.
2.3 Principle of complex amplitude modulation system
The basic concept of our approach is the phase collinear interference. The modulated wavefronts with the same amplitude and different phases from different partitions of the same phase-only modulating SLM interfere to recover complex amplitude wavefronts. Figure 2 shows the principle of our proposed CAM system. The system consists of a phase-only SLM, a combined quarter-wave plate (CQWP), two identical PGs and a linear polarizer (LP). The CQWP consists of two quarter-wave plates (QWPs) whose optical axes are perpendicular to each other. Firstly, the decomposed two POHs $\textrm{exp}[{j{\theta_1}} ]$ and $\exp [{j{\theta_2}} ]\; $ are uploaded on the SLM at different positions with a separated distance. The $\exp [{j{\theta_1}} ]\; \; $ is uploaded on the upper part of the SLM. The LP light with phase distribution of $\textrm{exp}[{j{\theta_1}} ]$ will be converted to the LCP light after modulated by the upper part of CQWP, where two QWPs with the optical axis at 45° and -45° to the vibration direction). Then, the light fields of $\textrm{exp}[{j{\theta_1}} ]$ are converted to the RCP light with a titled angle -θ after passing through the first PG. Finally, the light fields of $\textrm{exp}[{j{\theta_1}} ]$ are converted to the LCP light parallel to the z axis after modulated by the second PG. Similarly, the light fields of $\textrm{exp}[{j{\theta_2}} ]$ are converted to the RCP light parallel to the z axis after the PG2. When the distance between the two PGs is appropriate, the lateral displacement of the light field of $\exp [{j{\theta_1}} ]$ and $\exp [{j{\theta_2}} ]$ is ensured to match the pixel of the two separated sub-holograms on the SLM. Under the condition of perfect alignment, the ±1 order images will be combined behind the second PG. Then, they are coherently superimposed to generate the desired complex field in the central region after passing through the LP.
The process can be further explained mathematically in detail. When the incident light is circularly polarized, we can define the transmission matrix of the PG as
3. Results
3.1 Numerical simulations
From the principle illustrated in section 2.3, we can conclude that the target object can be reconstructed accurately via the proposed CAM system when the two PGs are placed at exactly the right places and the separated distance satisfies the specific condition of Eq. (10). However, the assembling errors in the system will degrade the reconstruction quality. Specifically, a small position shift of the PG will cause an imperfect overlap between the decomposed two POHs. The misaligned output light field caused by the position shift can be defined as
Where Δy is the translation mismatch error of two POHs at the output plane caused by the position shift of the PG $\varDelta d$. And they have the following relationshipIn addition, the tilt of the PG will also significantly affect the reconstruction quality and the misaligned output light field caused by the tilt can be expressed as
In order to evaluate the influence of the assembling errors on the reconstruction quality in our proposed CAM system, we perform numerical simulations. The ‘CAM’ image is set as the target complex image as shown in Fig. 3(a). The computation parameters are listed below. The resolution of the POHs is 800 × 1000 and the pixel pitch of the SLM is 8µm × 8µm. The distance between the center of two POHs is set as 960 pixels as shown in Fig. 3(b).
In the simulation, as shown in Fig. 4(a), two POHs can perfectly reproduce the complex amplitude information of the target object without considering the errors. However, when there are large translation mismatch error and grating tilt error in the system, it can be seen from Fig. 4(b) and Fig. 4(c) that the reconstructed image ‘CAM’ will have obvious dislocation and moire fringe, which will affect the imaging quality of the reconstructed target.
Fig. 4. Simulation reproduction results of the target image with (a) no errors, (b) translation mismatch error of 15 pixels, (c) grating tilt error of 0.05 degree.
To analyze quantitatively, signal-to-noise ratio (SNR) is utilized in our numerical simulations. SNR is defined as
The degradation of the reconstruction brought by the $\varDelta y$ and $\varDelta \theta $ is evaluated as shown in Fig. 5. From the results, it can be observed that a small $\varDelta y$ or $\varDelta \theta $ will greatly degrade the reconstruction quality image, which indicates that the alignment is vital for our proposed CAM system.
Fig. 5. SNR variations for the misalignment factors of (a) Translation mismatch error, (b) Grating tilt error
3.2 Experiments
Optical experiments as shown in Fig. 6 are carried out to verify the proposed system. The illumination beam at a wavelength of 532 nm is emitted from a solid-state laser. The phase-only SLM used is Holoeye Pluto with 1920 × 1080 resolution, 8µm pixel pitch, and [0, 2π] phase modulation range. We use a 4-f system to filter out the background noise of the SLM. The focal lengths of lens1 and lens 2 are 60 mm and 75 mm respectively. Two identical PGs utilized have a first-order diffraction angle of ${\theta _{ {\pm} 1}} = 10^\circ $. The pixel resolution of the original complex object tested here is set to be 800 × 1000. POHs are calculated based on the DPH approach and then loaded into different regions on SLM. The function of the two PGs is synthesize the POHs into a complex hologram at the center of the back focal plane of the second Fourier lens in the 4-f system. And then the complex hologram needs to propagate a distance to reconstruct the original object. The results are displayed by the CCD with a camera lens (Nikon AF-S 105 mm).
Before carrying out the experiment, Pre-compensation needs to be performed to obtain the better complex amplitude display effect. We set the y-direction interval between the two POHs as ${h_{DPH}} = 900\textrm{pixels}$. According to Eq. (10) and the system parameters, the distance between the two PGs can be calculated that ${d_{PG}} = 20.42\textrm{mm}$. In an ideal situation, the holograms obtained by our calculation can be overlapped to fulfill CAM at the center of the optical axis only when the distance between the polarization gratings is correctly adjusted to the theoretical value and no additional phase differences is introduced because of the tilt. However, in actual experiment, due to the systematic and experimental errors, the two POHs can hardly coincide on the output surface perfectly. Therefore, pre-compensation is critical. In pre-compensation, first of all, we need to ensure that the incident surfaces of the two polarization gratings in the system are not tilted and relatively parallel, so as to minimize the influence of the grating tilt error. The specific operation is as follows. Before adding the polarization grating to the system, we need to mark the center position of the incident light in advance. After adding the first polarization grating, we correct the tilt of the first polarization grating so that the center of the plane where the positive and negative first-order diffracted light overlap is consistent with the center of the previous incident light. The same steps are used for the tilt correction of the second polarization grating. Second, the translation mismatch error of two POHs needs to be calibrated. The longitudinal position of the reconstructed image for two sub holograms is determined by the interval of sub holograms and the interval of two polarization grating. In actual operation, it is difficult to realize the accurate pixel alignment of the reconstructed image of two sub holograms by adjusting the distance between two gratings. In order to make the reconstructed images coincide better, after the grating position is fixed, we calibrate precisely by adjusting the interval of the POHs by pixel in the algorithm program, and observe the alignment through the image taken by CCD. In this experiment, the actual ${h_{DPH}}$ after our final calibration is 893pixels.
Figure 7 shows effect of pre-compensation for figures ‘CAM’. Figure 7 (a) shows the result without pre-compensation. Because of the mismatch of the reconstructed images, Blur and Mohr stripes are obvious. After the pre-compensation of the hologram, the reconstructed image becomes clear as shown in Fig. 7(b). The interval here between the two POHs is ${h_{DPH}} = 887\textrm{pixels}$.
3D reconstruction is also performed. The two sets of patterns are with the reproduction distance of 100 mm and 150 mm respectively. The holographic 3D display effect is shown in Fig. 8. It can be clearly observed that when the CCD is focused near, the left pattern becomes clear and right one becomes blurred, and vice versa. The experimental results show that the proposed system can show a high-quality 3D holographic display. At the same time, we measure the transmission efficiency of the system. The energy transmission efficiency is 45.6% in the process of synthesizing the target light field through complex amplitude modulation after the 4-f system.
Fig. 8. Reconstructed 3D images at different depths, (a) focus at ‘school badge’ at 100 mm, (b) focus at ‘BIT’ at 150mm
4. Multiple depth and display region enlarged
As for the system discussed in section 2.3, a multiple depth extended FOV display can be realized through removing the CQWP to make full use of the first order of diffraction based on the holographic multiple recording. The principle is shown in Fig. 9. Take ‘F’, ‘O’, ‘V’ as an example for the three different depths of image, and the information represented by the three letters is in different spatial positions of the input image plane. Here, ‘O’ is the information that we focus on. Two POHs of ‘O’ is calculated according to DPH method, and ‘F’ and ‘V’ is relatively calculated through GS method. Then, the final holograms can be obtained by multiplexing the holograms of three letters together. We set the holograms of the information ‘F’ and ‘V’ whose reproduction distances are ${l_1}$ and ${l_3}$ calculated by GS algorithm as $\textrm{exp}[{j{\varphi_F}} ]$ and $\textrm{exp}[{j{\varphi_V}} ]$. Two sub-holograms of the information ‘O’ calculated by the DPH with a reproduction distance of ${l_2}$ are recorded as $\textrm{exp}[{j{\varphi_{{O_1}}}} ]$ and $\textrm{exp}[{j{\varphi_{{O_2}}}} ]$. The multiplex holograms $\textrm{exp}[{j{\varphi_1}} ]$ and $\textrm{exp}[{j{\varphi_2}} ]$ which will be loaded onto different regions of the SLM can be expressed as:
Fig. 9. Principle of multi-depth-of-field extension FOV algorithm: (a) the proposed algorithm, (b) the system
When we load the multiplex hologram on SLM, the linearly polarized diffraction light of the two holograms will produce positive and negative diffraction images after passing through the dual-polarized grating system directly. In the center of the output plane, the two sub-holograms of ‘O’ that we pay more attention to will interfere at its reproduction distance ${l_2}$ to obtain a high-quality reproduction. ‘F’ and ‘V’ will also be reconstructed in other diffraction orders at ${l_1}$ and ${l_3}$. Although the multiplex holograms $\textrm{exp}[{j{\varphi_1}} ]$ and $\textrm{exp}[{j{\varphi_2}} ]$ will also produce the interference image of the information ‘F’ and ‘V’ in the center order at the distance of ${l_2}$ and ${l_3}$, we initially set the three letters in different spatial positions of the input image plane and we can remove unnecessary interference items in each order after the second PG by filtering. In addition, since ‘O’ is the information we pay more attention to, the influence of the interference term in the center order can be attenuated by multiplying the PHOs of $\textrm{exp}[{j{\varphi_{{O_1}}}} ]$ and $\textrm{exp}[{j{\varphi_{{O_2}}}} ]$ in Eqs. (15) and Eqs. (16) by a higher weight factor in the calculation of $\textrm{exp}[{j{\varphi_1}} ]$ and $\textrm{exp}[{j{\varphi_2}} ]$. Thus, holographic display with multiple depth of field and expanded display area can be realized.
In order to verify the effectiveness of proposed method, we perform optical experiments. The reconstruction distance of ‘F’, ‘O’, and ‘V’ is set to be 200 mm, 100 mm and 150 mm. In experiments, we filter out unnecessary terms in each order to obtain better observation results. As shown in Fig. 10, we can see the images are focused and blurred the same way as the real objects. Because the reconstructed image of ‘O‘ is the superposition of two diffraction components in the output plane, the intensity of the reconstructed image of O is twice as strong as that of ‘F’ and ‘V’. In fact, the function of the line polarizer here is only to make the two POHs of ‘O’ interfere and does not need to cover other diffraction orders. In this way, we can enlarge the display area by three times.
Fig. 10. The experimental results of Multiple depth and enlarged displaying region (a) Focusing on ‘F’, ${l_1} = 200\textrm{mm}$;(b) Focusing on ‘O’, ${l_2} = 100\textrm{mm}$;(c) Focusing on ‘V’, ${l_3} = 150\textrm{mm}$
5. Conclusion
In summary, a novel holographic 3D display system with complex amplitude modulation is proposed in this paper. Two polarization gratings are used to align the reconstructed images of two dual-phase sub-holograms to fulfill complex amplitude modulation. Meanwhile, the effect of alignment error on the final complex amplitude modulation image quality is explored, and the pixel offset method is used to pre-compensate the system error. Furthermore, on the basis of the proposed system, we propose a large depth of field and enlarged display area display method based on holographic multiple recording, which is verified by optical experiments. According to the theoretical analysis and experimental results, the system has high energy utilization, and high-quality complex amplitude modulation can be achieved through error analysis and pre-compensation. The proposed method is expected to be applied to holographic 3D display, such as FOV enlargement and image quality enhancement in holographic near-eye display.
Funding
National Natural Science Foundation of China (61975014, 62035003).
Acknowledgment
This work is supported by the National Natural Science Founding of China (NSFC) under Grant Nos. 61975014 and 62035003, Beijing Municipal Science & Technology Commission, Administrative Commission of Zhongguancun Science Park under Grant No. Z211100004821012.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but maybe obtained from the authors upon reasonable request.
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