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Abstract
Liquid crystal spatial light modulator (LC-SLM) holographic display is affected by its structure, which products multi-level diffracted image with zero-order spot, resulting in low light energy utilization and poor uniformity of the reproduced image. This paper presents a method to improve the uniformity of light energy distribution in the reproduced image by using phase compensation, and the uniformity of the image can be effectively improved by using digital blazed grating to deviate the image and performing phase compensation according to the light energy distribution. Analyzing the uniformity of light energy distribution, the phase distribution is compensated, and experiments verify the phase compensation. The experimental results show that the uniformity and light energy utilization of the reproduced image after compensation has been improved. The results show that the proposed phase compensation method can be applied to both Fresnel holography and Fourier holography; both can effectively improve the uniformity and efficiency of light energy. Therefore, this method has a specific application value to enhance the quality of holographic reproduction and light field modulation based on LC-SLM.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
With the development of holographic display technology [1–4], holographic display based on liquid crystal spatial light modulator (LC-SLM) comes into being. Scholars have widely recognized its advantages of convenience and easy control. However, due to the limitation of its own processing conditions, its internal structure will lead to zero-order spot and multi-level diffraction images in the reproduced result, leading to uniform energy distribution and reducing the quality of the holographic display results.
Most scholars [5–11] are to improve the quality of holographic reproduced results from the design algorithm of hologram and holographic reproduced optical path [12]. However, in metasurface holography, there are also studies [13] that use voltage to control the polarization state to modulate the intensity change of the reconstructed image. LC-SLM is influenced by its own structure, which can cause an uneven distribution of the reproduced image energy and reduce the holographic reproduction results, so there is less research on this type of study. Therefore, this paper is based on the above problem, in order to get the LC-SLM holographic display results without the influence of zero-order spot and multi-level diffraction image, and to get high quality holographic reproduced image, a phase compensation method based on the reproduced image light energy distribution is proposed. First, the blazed grating separates the zero-order spot from the reproduced image. Then, by setting the size of the reproduced region, it is found that the energy distribution is not uniform after the holographic reproduction, so the phase compensation can make the energy distribution of the image more uniform, and finally get high quality holographic display results. Experimental verification is performed by near-field Fresnel holography and far-field Fourier holographic optical paths. The results are helpful to the application of LC-SLM in high quality holographic display.
2. Basic principles
2.1 Design principle of CGH
According to the diffraction reconstruction distance as the object of study, there are two main methods in the design of holograms: Fresnel hologram design and Fourier hologram design. A schematic diagram of the holographic recording reproduction is shown in Fig. 1 below.
In Fig. 1, the object light is interference superimposed on the holographic plane with the reference light, and then the phase is retained on the holographic plane, which is a recording process. Then, the reference light reproduces the holographic plane, and the desired reproduction target can be obtained.
Let $O({{x_1},{y_1}} ) R({{x_1},{y_1}} )$ be the object plane complex amplitude and the reference light complex amplitude respectively, and D be the recording distance.
In the Fresnel diffraction field, the complex amplitude distribution of the holographic plane is [14]:
In the Fraunhofer diffraction field, the complex amplitude distribution of the holographic plane is:
During the calculation, the phase distribution of the final holographic plane can be obtained by discretizing and quantizing according to the specific calculation parameters.
2.2 Holographic reproduction principle of LC-SLM
When the holographic reproduction is performed by LC-SLM, the “Grid effect” of its internal structure will lead to the generation of zero-order spot and multi-level diffraction image, the quality of the reproduced image is seriously affected. For the internal structure of LC-SLM, it is essentially equivalent to a two-dimensional grating. We analyze the light field distribution of the output plane by grating diffraction theory.
The internal structure of LC-SLM is shown in Fig. 2(a). Its transmittance function is:
${t_{pi}}({x,y} )$ is the transfer function of LC-SLM liquid crystal region (active region), which can be expressed as:
Therefore, $T({{f_x},{f_y}} )$ in the above equation represents the complex amplitude distribution of the output plane, and its structure leads to the complex amplitude intensity distribution showing $sinc$ function distribution, which also leads to the existence of zero-order spot and multi-level diffraction image in the reproduction results. The grating diffraction pattern is shown as the single slit diffraction envelope in Fig. 2(b), which is determined by the effective region size $d^{\prime}$ of the pixel. Taking $\sin \theta /\lambda$ as the horizontal coordinate and $\theta$ as the diffraction angle, the width of the central bright region of the single slit diffraction is $2/d^{\prime}$, and the width of the secondary bright lines is $1/d^{\prime}$. From the grating equation, it can be seen that the conditions for the interference maximization are satisfied by the grating equation [15]:
In Eq. (7), d is the period of the grating (pixel size), and still using $\sin \theta /\lambda$ as the horizontal coordinate, the position of the interference maximization can be obtained as shown in Fig. 2(b), and the coordinates of the interference maximization position at each level are $m/d,m = 0, \pm 1, \pm 2, \pm 3\ldots $. When holographic reproduction is performed with LC-SLM, the reproduced diffracted images at all levels lie between the adjacent maxima of each interference, as shown by the“Sun” icon in Fig. 2(b), with the coordinates of $m/2d,m ={\pm} 1, \pm 3, \pm 5\ldots $, the energy distribution of these reproduced images is affected by the single slit diffraction patterns.
2.3 Principle of phase compensation of reproduced image light energy distribution
2.3.1 Analysis of the effect of blazed grating on the reproduced image
The following is an analysis of the relationship between the position change of the zero-order spot and the multi-level diffraction image in the output plane when holographic reproduction is performed with LC-SLM, and the schematic diagram is shown in Fig. 3 below. As shown in the figure below, when holographic reproduction is performed, there will be zero-order spots and multi-level diffraction images in the output plane, but ideally, we need the best reproduced image. The theoretical analysis shows that four mutually symmetrical first-level reproduced images with relatively high energy will appear near the optical axis. Their positions are determined by the internal structure size of the LC-SLM, but their sizes are controllable in the threshold range.
For the position of the reproduced image, we can move it according to the blazed grating, which is superimposed on the designed hologram. After adding the digital blazed grating to the hologram, the relationship between the angle of reproduced image offset $\theta$ and the minimum characteristic size (LC-SLM pixels) of the hologram d and T pixels is as follows:
Therefore, we obtain the phase structure expression discretized two-dimensional digital blazed gratings as follows:
The total number of pixels in the holographic plane is $M\mathrm{\ \times }N$, where $m = 1,2,3\ldots ..,M$; $n = 1,2,3\ldots ..,N$; $T$ is the number of pixels corresponding to the digital blazed gratings, $a = 1,b = 0$ are the digital blazed grating in the vertical slot direction, and $a = 0,b = 1$ are the digital blazed grating in the horizontal slot direction.
We superpose the calculated phase structure $\varphi ({m,n} )$ with the phase structure of the digital blazed grating ${\varphi _1}({m,n} )$, and then take the remainder of the $2\pi$ to obtain the phase distribution of the holographic plane ${\varphi ^{\prime}}$ is:
2.3.2 Phase compensation analysis of reproduced image light energy distribution
Firstly, the reproduced image needs to be filled, in order to avoid the overlap of the reproduced image and the zero-order spot, . The idea of filling is to divide output surface calculation region into a signal region and a noise region. The signal region is the region where the reproduced image is located, and the noise region is the useless information of the image, which can be filled with 0. The specific model diagram is shown in Fig. 4 below.
Taking “Xi’an Technological University logo” as an example, the overall region is the size of the output plane, and the size of the output plane is set to $\frac{{\lambda D}}{d} \times \frac{{\lambda D}}{d}$ according to the maximum sampling theorem setting in the calculation. In order to avoid the interaction between the zero-order spot and the reproduced image, the output plane is divided into Signal region and Noise region, which avoids the interaction and provides a large range of movement when the blazed grating deviates from the reproduced image.
According to the above analysis, the energy distribution of the reproduced image will be affected by the light energy distribution of the output plane, that is, the $sinc$ function, so in different positions, the energy distribution of the reproduced image is not uniform. The energy distribution of the reproduced images in two positions are shown in Fig. 5 below,Fig. 5(a) shows the reproduced image below the zero-order spot and Fig. 5(b) shows the reproduced image above the zero-order spot.
The relative position of the reproduced image and the zero-order spot directly affects the energy distribution, so it is necessary to compensate for it to improve the uniformity of the reproduced image and obtain high quality holographic reproduction results.
First, we need to discuss the energy distribution of the reproduced image after the blazed grating deviation, then according to the energy distribution, the phase compensation is performed to improve the uniformity of the reproduced image. The specific idea is to calculate the required compensation amount according to the light energy distribution in the reproduced image region (signal region) of the output plane, so as to compensate the phase of the original target image. The specific calculation steps are as follows:
First determine the initial holographic calculation parameters, including the reproduction of the original image, the reproduction distance and the size of the reproduced image.
Delineating a square region containing all the energy distributions of the reproduced image based on the size of the reproduced image, which called the reproduced image region (that is, the signal region analyzed above). As shown in Fig. 8, the white area is the region of the reproduced image.
Set all the values in the reproduced image region to 1, the rest to 0, carry on the holographic reproduction calculation, and load the calculated phase structure with the digital blazed grating in the vertical (or horizontal) slot direction, use the spatial light modulator to get the results.
The obtained reproduction result is the reproduced image region pattern with uneven energy distribution, and the obtained image result is fitted with normalization process to obtain a complete compensation amount, and the compensation amount is inverted in order to compensate for the unevenness of the reproduction result.
The result of the inversion is superimposed on the original image to obtain the new compensation result. Then the phase hologram is calculated again, and the hologram is loaded again with the same parameters of the digital blazed grating. The reproduced image not only has high light energy utilization, but also has uniform light intensity distribution and no zero-order spot.
In order to express the compensation process more intuitively, the traditional calculation process and the calculation steps of the proposed method are represented by flow charts. As shown in Fig. 6, Fig. 6(a) is the calculation flow chart of the traditional holographic design process, and Fig. 6(b) is the calculation flowchart of the phase compensation method for reproducing the light energy distribution. The red dotted line box is the phase compensation method of light energy distribution in the reproduction domain we mentioned.
3. Simulation experiments
3.1 Initial parameter setting
The two images in Fig. 7 are respectively used as the original image of the holographic reproduction, the wavelength $\lambda = 520nm$, the total number of pixels of the holographic plane and the output plane $M\mathrm{\ \times }N = 512 \times 512$, the reproduction distance in the near-field Fresnel diffraction region $d = 200mm$, the resolution of the holographic plane $\Delta \xi = \Delta \zeta = 8\mu m$, and the resolution of the object plane $\Delta x = \Delta y = 0.032mm$.
3.2 Reproduced image light energy distribution and phase compensation calculation
First, we determine the range of the reproduced image, which is the range of the signal area, according to the range of the reconstructed image. The focus of our study is on the uniformity of the light energy in the signal region, as shown below:
By calculating the region of the reproduced image, the light energy distribution is obtained, and the compensation amount is synthesized according to the region of the reproduced image. The synthesis process needs to make trade-off, interpolation, smoothing, and synthesis of the reproduced discrete data results, and the results obtained are shown in Fig. 9(a), and the compensation amount after inversion is shown in Fig. 9(b). According to the result of the light energy distribution, it is obvious that the output plane is compensated in the reverse direction.
Figure 10 below shows the reproduction results of the reproduced image region and the reproduction results after compensation, as shown in Fig. 10(a), can be seen that the light energy distribution before compensation is not uniform, and is affected by the energy distribution of the zero-order spot, after compensation, however, Fig. 10(b) shows that the energy distribution in the reproduced image region is very uniform.
In order to reproduce a holographic target image, the compensation amount and the original image are synthesized to obtain the compensated target reproduced image. The above two images are shown in Fig. 11. Fig. (a) and Fig. (c) are the calculated gray images and Fig. (b) and Fig. (d) are the normalized 3D intensity maps.
The synthesized Fig. 11 is the reproduced image after compensation, and then the hologram is calculated again. The desired hologram results can be obtained by superimposing the results on the above-mentioned digital blazed gratings.
4. Experimental verification
The following is an experimental verification of the above analysis, using the holographic reproduced optical path shown in Fig. 12 and Fig. 14 below. The light source is emitted from a semiconductor laser (wavelength 520 nm), and after passing through a beam expander (BE), and then through a polarizer (POL) at a small angle obliquely incident to a LC-SLM (LC-SLM, model HDSLM80R is a product of UPOLabs, resolution 1920 × 1200 pixels, pixel size 8 $\mu m$, fill factor >95%), which is reflected to the output plane and finally observed by CCD. The Fresnel hologram and Fourier hologram were verified during the experiment. The difference between the two holograms is whether there is a lens between the output plane and LC-SLM. Fresnel holography does not require a lens, while Fourier holography requires a lens.
When observing with CCD, its exposure needs to be controlled reasonably because the uniformity of the results is not reflected when the exposure reaches saturation. Figure 13 and Fig. 15 above show the experimental results before and after compensation of Fresnel and Fourier holographic reproduction results, respectively. By comparing the uncompensated and compensated results, it can be found that the consistency of the brightness of each stroke of these characters has greatly improved after the compensation of the holographic reproduced images, whether located in the central region or at the corners. The brightness of the central region before compensation is higher than that of the corner region. In contrast, the light energy distribution of the reconstructed image after compensation is more uniform, and the light use efficiency is also increased.
In order to measure the quality of the reconstructed image [16,17], the holographic reproduction experiment is carried out by using LC-SLM, and the reproduced image is recorded with a camera, and then performing gray processing and calculations on images. In order to make the expression more understandable, it is defined as the uniformity of light energy distribution ($LDU$) and light energy utilization ($eta$) in the “effective graphic region”. The “effective graphic region” is the region where the energy should be concentrated in the output plane.
The specific formulae for the uniformity of energy distribution ($LDU$) and light energy utilization ($eta$) are given in Eq. (11) below:
In Eq. (11), Q is the effective graphic region of the output plane reproduced image region (the region where the theoretical light energy is not zero), ${|{E({x^{\prime},y^{\prime}} )} |^2}$ is the light energy value of each sampling point in the effective graphic region of the output plane reproduced image region, ${|{\overline E } |^2}$ is the average light energy value in the effective graphic region of the output plane reproduced image region, ${|{{E_0}} |^2}$ is the input light energy value, ${N_Q}$ is the sum of sampling points in the effective graphic region of the output plane reproduced image region.
According to the above Eq. (11), the uniformity of light energy distribution and light energy utilization corresponding to Fig. 13 and Fig. 15 are calculated, and the results are obtained as shown in Table 1. As can be seen from the compensated calculations, the reproduced image is much more homogeneous and has increased light energy utilization. The effect of Fresnel holography is better. Compared with the images before and after the improvement, it can be found that the uniformity is improved greatly, and the utilization of light energy is also improved. For Fourier holography, although the uniformity of the reproduced image is improved to a certain extent, the light energy utilization is decreased. By controlling the incident laser power, we can vary the intensity of the reproduced result for holographic reproduction. In conclusion, compensation has improved the quality of the reproduced results.
Table 1. Results of light energy utilization and light intensity distribution inhomogeneity before and after compensation.
5. Discussion and analysis
The simulation and experiment results show that the proposed method is correct. In the following, we discuss the setting of its parameters, which are mainly divided into the analysis of the blazed grating and its compensation amount.
When the holographic reproduction is performed without the phase compensation method, the reproduction results are shown in Fig. 16 below, and the image is affected by the zero-order spot and multi-level diffraction image. Only the four symmetrical level reproduced images near the optical axis are studied in the Figure, and it can be seen from Fig. 16 that the energy distribution is strong near the zero-order spot and weak far from the zero-order spot, which resulting in uniformity distribution of the reproduced image light energy.
Fig. 16. LC-SLM reproduction results without the phase compensation method of the reproduced image light energy distribution
When we use the reproduced image light energy distribution compensation, the results are shown in Fig. 17 below. We use the blazed grating to deviate it, analyze the energy distribution of the reproduced image region, and then compensate it, which is the first-order reproduced image directly below the zero-order spot in the center. The reproduced image above the zero-order spot is not compensated. From the results of compensation in Fig. 17, it can be seen that the uniformity after compensation is greatly improved.
Fig. 17. LC-SLM reproduction results using phase compensation method of reproduced image light energy distribution.
In this paper, Fresnel diffraction reconstruction image is taken as an example. As shown in Fig. 16(a), there are four reconstructed images around the zero-order spot, so we compensate it in four positions. The results are as follows. To get a better look, the reproduced image is locally magnified, and the results are shown in Fig. 18 below.
Fig. 18. Local enlargement of the reproduced image before and after compensation.(a, c, e, and g are uncompensated results, and b, d, f, and h are compensated results, which also correspond to four positions, respectively, where the reconstructed image is at the under, top, right, and left of the zero-order spot)
The light energy utilization and light intensity distribution inhomogeneity before and after compensation for the above four cases are calculated below, and the calculation results are shown in Fig. 19 below. From the calculation results of four different positions in Fig. 19, it can be seen that the uniformity of light energy distribution is greatly improved and the light energy utilization is also improved.
6. Conclusion
To address the problem that the quality of the holographic reproduced image of LC-SLM is affected by zero-order diffraction image and its black grid structure, based on the analysis of light intensity distribution of the holographic reproduced image, this paper proposed a phase compensation method of reproduced image light energy distribution. The technique is to load the digital blazed grating on the designed phase hologram, then calculate the phase compensation amount based on the light intensity distribution in the reproduced image region, superimposing it with the original object light wave, recalculating it to obtain a new phase hologram, and adjusting the holographic reproduction image light intensity distribution. Finally, the verification and testing were carried out by LC-SLM holographic reproduction experiments, which showed that the reproduction results not only eliminated the effect of zero-order diffraction images, but also effectively improved uniformity and light energy utilization. Therefore, the proposed phase compensation method of the reproduced image light energy distribution is useful for improving the quality of the output results when the spatial light modulator is used for holographic reproduction or light field modulation.
However, in similar metasurface holography, the vector diffraction theory must be used because the minimum feature size of the metasurface hologram is smaller than the wavelength of light, and the scalar diffraction theory is not applicable in the design process. The reconstruction of metasurface holography is mostly machined into metasurface devices, so there is no “Grid effect” like LC-SLM. However, the reconstruction of metasurface holography is affected by other factors on the uniformity of light energy distribution of the reconstructed image, so this method can provide a reference for the design of metasurface holography.
Funding
Key Laboratory Research Program of Education Department of Shaanxi Province (18JS053); National Key Laboratory of Electromagnetic Space Security Project Fund.
Acknowledgments
This work was supported by Key Laboratory Research Program of Shaanxi Provincial Department of Education (18JS053); National Key Laboratory of Electromagnetic Space Security Project Fund.
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 may be obtained from the authors upon reasonable request.
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