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Abstract
The use of the diffractive optical element (DOE) can often significantly reduce the size and enhance the performance of the optical system, but it is mostly prevented by low diffraction efficiency and serious speckle noise due to the quantization error. In this paper, an error tracking-control-reduction (ETCR) algorithm is proposed to suppress the quantization error, which adjusts the accumulative action, controls the current state and predicts the trend of the error. The simulation and experiment results verify that the ETCR algorithm has high diffraction efficiency which can be comparable with the Gerchberg-Saxton (GS) and Modified GS algorithms. Furthermore, the root-mean-square error (RMSE) of the proposed algorithm is significantly lower than that of the GS and MGS algorithms. Based on the proposed method, a 2-level DOE has been designed and fabricated to generate several grey images with only 0.05 RMSE.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The diffractive optical element (DOE) of micro-nano profiles can modulate the wavefront of incident light, thereby generating a desired light field. It has been widely applied in augmented reality (AR), virtual reality (VR) [1–4], light trapping [5], beam shaping [6–7], laser fabrication [8] and holographic display [9–12] because of its miniaturization and integration. It is a phase retrieval process [13–18] to design the DOE. Based on the phase retrieval algorithms, we can calculate the wavefront of incident light to obtain the corresponding phase-only hologram, i.e. the phase distribution of DOE. By using the traditional photolithography technique, the phase-only hologram can be processed on optical substrate to obtain the phase-only DOE. In general, the profile of the DOE is staircaselike. During the design process, the quantization operation is always performed to reduce the phase levels of the DOE from dispersed staircaselike levels to ${2^L}$ levels, where L is a positive integer. As a result, severe quantization error will be generated, thereby resulting in serious speckle noise. In addition, the diffraction efficiency of the DOE is also affected by the quantization error. Therefore, it is significant to optimize the quantization error for designing high-efficiency low-noise DOE.
In 1978, Hsueh et al. proposed a double-phase holograms (DPH) algorithm [19], which considered that an arbitrary complex value could be decomposed into two complex values with constant amplitude. Hence, each cell of a complex hologram was divided into two subcells which were encoded with phase-only quantities. Without iteration, the DPH algorithm can easily convert a complex hologram into a phase-only hologram, but it constraints the space-bandwidth product [20,21]. In 2016, based on the DPH algorithm, Qi et al. [22] proposed a modified scheme at the expense of increasing system complexity, which solved the problem that the original algorithm had constrained bandwidth. Same as the DPH algorithm, the bidirectional error diffusion (BERD) algorithm, proposed by Tsang et al. [23], could also convert a complex hologram into a phase-only hologram without the requirement of iteration. This algorithm scanned each pixel of the complex hologram in sequence and forced the magnitude of each visited pixel to a constant value. Meanwhile, the resulting error was diffused to the neighboring pixels that had not been visited before. In 2019, Yang et al. [24] analyzed the four error diffusion coefficients in the algorithm, and obtained a more suitable combination of coefficients. Both the DPH and BERD algorithms have extremely fast computation speed due to their non-iterative character. However, both of them are affected by severe quantization error, thereby generating serious speckle noise. In addition, they have extremely low diffraction efficiency. For example, the diffraction efficiency of the DOE designed by the BERD algorithm is about 22% [25].
In 1972, the iterative scheme for phase retrieval, termed Gerchberg-Saxton (GS) algorithm [26], was proposed to design the DOE. The GS algorithm has extremely high diffraction efficiency, which can design a 16-level DOE with over 90% diffraction efficiency. However, during the iteration, the GS algorithm is prone to converge to a local optimal solution resulting in calculation error. Liu et al. [27] proposed a modified GS (MGS) algorithm which further reduced the calculation error of the GS algorithm. The same as the GS algorithm, the MGS algorithm also has extremely high diffraction efficiency. In 2016, based on the MGS algorithm, Wang et al. [28] proposed a hybrid Gerchberg–Saxton-like (HGSL) algorithm which introduces the strategy of gradient descent and weighting technique to the GS algorithm. But the DOE designed by the three algorithms are also affected by the quantization error so that the quality of the reconstructed images will be deteriorated.
Much relational work, in addition to that mentioned above, has been done to design DOE. For instance, Zhang et al. [29] and Wyrowski et al [30] proposed the modified Y-G algorithm and the three step method to design DOE which can be used to convert the Guassian profile into the uniform beam with high uniformity. Although many algorithms can be used to design multilevel-phase DOE, they have not further optimized the quantization error of DOE. In order to design a DOE with high diffraction efficiency and low speckle noise, an error tracking-control-reduction (ETCR) algorithm is proposed in this paper. The most ingenious part of the proposed algorithm lies in a special ETCR constraint that can be used to control the error of the “past”, “present” and “future”. Furthermore, iteration quantization is performed rather than the direct quantization which is performed only during the final iteration. The ETCR algorithm proposed in this paper can be regarded as an optimized scheme of the iterative Fourier transform algorithm. Therefore, comparing it with the classic iterative Fourier transform algorithm (GS and MGS algorithm) can show the superiority of our algorithm. Adopting the proposed ETCR algorithm, we designed and fabricated a 2-level DOE whose diffraction efficiency was close to the theoretical limiting diffraction efficiency [31] of DOEs, and its root-mean-square error (RMSE) was 5-6 times lower than the GS and MGS algorithms. This paper is organized with five sections. After this introduction, section 2 demonstrates the principle of the proposed ETCR algorithm. Section 3 shows the simulation and experiment results. In this section, we compare the proposed ETCR algorithm with the GS and MGS algorithms. Section 4 discusses the influence of zero-padding on the image reconstruction quality of the DOE designed by the proposed algorithm. Section 5 is the summary.
2. Principle and method
In this section, after describing the detailed calculation process of the algorithm, the error control principle of the ETCR constraint is introduced, followed by the method of determining the relevant parameters.
2.1 The calculation process of ETCR algorithm
The flow chart of the ETCR algorithm is shown in Fig. 1. When phase hologram or DOE is designed, the imaging plane is usually divided into signal domain and noise domain, wherein the signal domain is used to place the target image. Due to the separation of noise and signal domains, the quality of the reconstructed light field can be improved [32]. To avoid the influence of the twin image [33] and the center zero-order on the experimental results, the signal domain is placed at a position offset from the center of the imaging plane. In Fig. 1, $({x_0},{y_0})$ and $({x_1},{y_1})$ represent the coordinates on the phase hologram plane and the imaging plane, respectively. There are two key steps in the ETCR algorithm. One is to perform the ETCR constraint on the amplitude of the reconstructed light field on the imaging plane [see step 2 in Fig. 1], the other is to quantize the phase hologram during the iteration [see step 4 in Fig. 1]. The ETCR constraint can be used to correct the error of the reconstructed light field, containing three means: proportional control, integral control and derivative control. Among them, the proportional control is performed to control the current state of the error so that the calculation error can be decreased; the integral control is performed to eliminate the steady-state error (SSE, i.e. the error generated when the algorithm converges to the local optimal solution) that cannot be eliminated by the proportional control; the derivative control is performed to predict the trend of the error, which weakens the error oscillation caused by the proportional and integral controls. Through the operations of the ETCR constraint and the iterative quantization, the quantization error can be controlled, so that it can be reduced or eliminated. The specific steps of the proposed algorithm can be described as follows.
The initial phase distribution on the phase hologram plane is random, denoted as ${\varphi ^{(1)}}({x_0},{y_0})$. First, the complex light field U on the phase hologram plane, which has amplitude of 1 and phase of ${\varphi ^{(k)}}({x_0},{y_0})$, propagates to the imaging plane to obtain a complex light field ${E^{(k)}}({x_1},{y_1})$ where k represents the number of iterations. The fast Fourier transform (FFT) is selected as the propagating function. Then, the ETCR constraint is performed to correct the magnitude of ${E^{(k)}}({x_1},{y_1})$ but remain its phase, which can be described by Eqs. (1)–(3).
During the iteration, we need to determine whether the reconstructed light field meets the error requirement which can be expressed by the RMSE function, as shown in Eq. (4). If so, the iteration is terminated and the quantized pure phase distribution ${\varphi ^{(k)}}({x_0},{y_0})$ becomes the output of the iteration; otherwise, the iteration continues.
The above is the core of the ETCR algorithm where Eqs. (1)–(3) are the main theoretical results of our paper. Next, the error control principle of the ETCR constraint will be described in detail.
2.2 The principle of the ETCR constraint
The schematic diagram of the ETCR constraint is shown in Fig. 2.
In the schematic diagram, ${E_d}$ and ${A^{(k)}}$ are the amplitudes of the desired light field and the reconstructed light field, respectively. ${e^{(k)}} = {E_d} - {A^{(k)}}$ represents the error of the ${k^{th}}$ iteration, i.e., the current error. “⊗” represents arithmetic symbol, indicating that there is computational relationship between the connected objects. On behalf of the error control operator after the proportional (P), integral (I) and derivative (D) controls, $A_s^{(k)}$ is used to correct the error.
If there is only the proportional control, the error control operator $A_s^{(k)}$ can be expressed by Eq. (5).
where y0 represents the initial error control operator. In this paper, $y0 = {E_d}$. ${K_p}$ is the proportional coefficient which can be used to adjust the effect of the proportional control. Obviously, the error control operator $A_s^{(k)}$ after the proportional control is linearly proportional to the current error ${e^{(k)}}$. If ${E_d} > {A^{(k)}}$, i.e., ${e^{(k)}} > 0$, then ${A^{(k)}}$ is positively controlled by $A_\textrm{s}^{(k)}$ and thus it is increased; if ${E_d} < {A^{(k)}}$, i.e., ${e^{(k)}} < 0$, then ${A^{(k)}}$ is reversely controlled by $A_\textrm{s}^{(k)}$ and thus it is reduced. In other words, the role of the proportional control is to optimize the reconstructed ${A^{(k)}}$ to approximate to the desired ${E_d}$, so that the calculation error can be rapidly decreased. However, the proportional control depends on the current error. If there is only proportional control, the algorithm will stagnate when the error is reduced to a certain extent, leading to the SSE. Therefore, in order to remove the SSE, the integral control is added on the basis of the proportional control, which can be expressed by Eq. (6).The integral control corresponds to ${K_i}\ast \sum\limits_{t = 1}^k {{e^{(\textrm{t})}}}$ in Eq. (6), which controls the accumulative errors, where ${K_i}$ represents the integral coefficient. Because of the accumulative effect of the errors, as long as an error occurred, the integral control will persist until the SSE is eliminated. However, error oscillation will occur which is caused by the proportional control and the integral control, resulting in that the algorithm cannot converge. In order to eliminate the oscillation, it is necessary to introduce derivative control into the proportional-integral control, which can be used to predict the trend of the error. Therefore, when the error of the next iteration has not occurred, the error control operator $A_s^{(k)}$ is modified in advance to avoid oscillation. The final proportional-integral-derivative control can be expressed as Eq. (7).
Continue to rewrite Eq. (8) to Eq. (9).
Let ${\omega _1} = ({K_p} + {K_i} + {K_d})$, ${\omega _2} ={-} ({K_p} + 2{K_d})$, ${\omega _3} = {K_d}$ and $\Delta A_S^{(k)} = A_S^{(k)} - A_S^{(k - 1)}$, then Eq. (9) can be rewritten to Eqs. (1) and (2). On the basis of the above analysis, we can conclude that the roles of the proportional control, integral control and derivative control are to reduce the calculation error, eliminate the SSE and suppress the oscillation, respectively. The values of ${K_p}$, ${K_i}$ and ${K_d}$ play significant roles in determining the effect of the proportional-integral-derivative control. It is necessary to determine the optimal combination of the coefficients.
2.3 Parameter design
In this paper, the trial-and-error testing method is adopted to design the optimal combination of the three coefficients.
The number of iterations is selected as 50. First, let ${K_p} = [a:L:b]$, ${K_i}\textrm{ = }0$ and ${K_d}\textrm{ = }0$, where a and b represent the minimum and maximum values of ${K_p}$, respectively, and L represents the pitch of value. For instance, at the beginning, let ${K_p} = [\textrm{ - }10:1:10]$. The 21 values of ${K_p}$ are substituted into the iteration in turn and thus 21 RMSEs can be calculated. Assuming that when the value of ${K_p}$ is between 0 and 3, the corresponding RMSE is relatively small. Then, update the value range of ${K_p}$ and let L = 0.1, that is, ${K_p} = [0:0.1:3]$. Among these 31 values, the corresponding value range of ${K_p}$ can be found when the RMSE is relatively small. Continue to update the value range of ${K_p}$, and let L = 0.01 until the final ${K_p}$ (termed ${K^{\prime}_p}$) is found corresponding to the minimum RMSE. Next, let ${K_p} = {K^{\prime}_p}$, ${K_i}\textrm{ = }[{a:L:b} ]$, ${K_d}\textrm{ = }0$, in the same manner, an optimal ${K_i}$ can be found, denoted as ${K^{\prime}_i}$. Final, adopting the above method, an optimal ${K_d}$ can also be found, denoted as ${K^{\prime}_d}$.
Multigroup optimal values of ${K_p}$, ${K_i}$ and ${K_d}$ are shown in Table 1, corresponding to the DOEs with different phase levels.
Table 1. The optimal values of ${K_p}$, ${K_i}$ and ${K_d}$
3. Simulation and experiment results
3.1 Simulation analysis
In order to verify the validity of our algorithm in controlling the error, we performed simulation analysis by using the parameters obtained above. The error control effect of the three components (proportional control, integral control and derivative control) of the ETCR constraint is shown in Fig. 3.
Fig. 3. The error control effect of the ETCR constraint. The curve “None” corresponds to ${k_p} = 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {k_i} = 0{\kern 1pt} {\kern 1pt} {\kern 1pt} and{\kern 1pt} {\kern 1pt} {k_d} = 0$; the curve “P” corresponds to ${k_p} = {k^{\prime}_p},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {k_i} = 0{\kern 1pt} {\kern 1pt} {\kern 1pt} and{\kern 1pt} {\kern 1pt} {k_d} = 0$; the curve “PI” corresponds to ${k_p} = {k^{\prime}_p},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {k_i} = {k^{\prime}_{i{\kern 1pt} }}{\kern 1pt} {\kern 1pt} and{\kern 1pt} {\kern 1pt} {k_d} = 0$; the curve “PID” corresponds to ${k_p} = {k^{\prime}_p},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {k_i} = {k^{\prime}_{i{\kern 1pt} }}{\kern 1pt} {\kern 1pt} and{\kern 1pt} {\kern 1pt} {k_d} = {k^{\prime}_d}$.
In Fig. 3., ${k^{\prime}_p},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {k^{\prime}_{i{\kern 1pt} }}{\kern 1pt} {\kern 1pt} and{\kern 1pt} {\kern 1pt} {k^{\prime}_d}$ represent the optimal values of ${k_p},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {k_{i{\kern 1pt} }}{\kern 1pt} {\kern 1pt} and{\kern 1pt} {\kern 1pt} {k_d}$, respectively. When ${k_p} = 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {k_i} = 0{\kern 1pt} {\kern 1pt} {\kern 1pt} and{\kern 1pt} {\kern 1pt} {k_d} = 0$, the ETCR constraint does not work. It can be seen from the black curve, the RMSE is extreme high due to the lack of ETCR constraint. When ${k_p} = {k^{\prime}_p},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {k_i} = 0{\kern 1pt} {\kern 1pt} {\kern 1pt} and{\kern 1pt} {\kern 1pt} {k_d} = 0$, the proportional control (P) starts to work and the RMSE is rapidly decreased, which shows that the role of the P control is to suppress the error. When ${k_p} = {k^{\prime}_p},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {k_i} = {k^{\prime}_i}{\kern 1pt} {\kern 1pt} {\kern 1pt} and{\kern 1pt} {\kern 1pt} {k_d} = 0$, the RMSE is further reduced to a extremely low extent because of action of the I control. However, affected by PI control, the error oscillation will occur. In order to eliminate the oscillation, the D control is introduced on the basis of the PI control. Obviously, becuase of action of the ETCR constraint, the error oscillation disappears and the error converges to a very low extent. Based on the above analysis, we can draw a conclusion that both the P and I control can be used to suppress the errors, and the D control can be used to prevent error oscillation. Therefore, accoding to the change law of the error curve, we can reasonably adjust the weight of each component of the ETCR constraint to stably reduce the error. Such a process of observing the variation law of the error curve is called “tracking”; The process of adjusting the weight of each component of the ETCR constraint to control the change law of error is called “cotrol”; Controlled by the ETCR constraint, the error can be effectively suppressed, which is called “reduction”.
Multiple DOEs with different phase levels have been designed by the proposed ETCR algorithm. Meanwhile, the design results have been compared with the GS [26] and MGS algorithms [27]. The diffraction efficiency (abbreviated as Eff in Table 2) and the RMSE of the DOEs designed by the three algorithms are given in Table 2.
Table 2. The Eff and the RMSE of the DOEs designed by the three algorithms, respectively
As can be seen from Table 2, the diffraction efficiency of the DOEs designed by the GS, MGS and ECTR algorithms is almost close to the theoretical limiting diffraction efficiency η [31] which can be expressed by Eq. (10). In Eq. (10), L is the number of phase levels of the DOE, where $\eta (2) = 40.5\%$, $\eta (4) = 81.6\%$, $\eta (8) = 95.0\%$ and $\eta (\infty ) = 100\%$.
The RMSE listed in Table 2 shows that although the MGS algorithm has a lower RMSE than the GS algorithm, the RMSE caused by the quantization error is still serious. In contrast, the proposed ETCR algorithm effectively suppresses the quantization error. The RMSE of the designed 2-level DOE can be reduced to 0.05, which is 5-6 times lower than that designed by the GS and MGS algorithms. Therefore, it is theoretically feasible to design high-efficiency low-noise DOE by taking advantage of the proposed ETCR algorithm.
In order to further explain the convergence of the ETCR algorithm, we have plotted the curves of RMSE and diffraction efficiency, as shown in Fig. 4.
As shown in Fig. 4., the curve of RMSE shows that the ETCR algorithm have good convergence and the RMSE will not increase with the increase of iteration. Meanwhile, as the number of phase levels of DOE increases, the corresponding RMSE decreases slightly. Comparing the efficiency curve with the RMSE curve, it can be seen that the number of phase levels mainly affects the diffraction efficiency of the DOE designed by ETCR algorithm, while the RMSE is less affected by the number of the phase levels. The main reason is that the proposed ETCR algorithm is extremely excellent and is able to reduce the RMSE of the 2-level DOE to a very low extent.
Two grey images “cameraman” and “lena” as shown in Fig. 5(a) and (b), which were all comprising of 400×400 pixels, were employed as the intensity ${I_d}({x_1},{y_1})$ of the desired light field to evaluate the proposed algorithm. Obviously, the target amplitude ${E_d}({x_1},{y_1})$ was equal to $\sqrt {{I_d}({x_1},{y_1})} $. Taking the design of the 2-level DOE as an example, we carried out the simulation analysis. The imaging plane was divided into two parts which were signal domain and noise domain as shown in Fig. 1. The resolutions of the noise domain and the signal domain were M×M and m×m, respectively, where M/m = 4.4 (The influence of M/m on the image reconstruction quality of the DOEs designed by the ETCR algorithm will be discussed in section 4). As mentioned above, the reconstructed light field of the 2-level DOE would be affected by the twin image, so the signal domain was placed at a position offset from the center of the imaging plane. MATLAB was selected as the encoding software. The simulation results are shown in Fig. 5.
Fig. 5. The desired images and the numerical reconstructed results of the 2-level DOEs which are calculated by the GS, MGS and proposed ETCR algorithms. (a) and (b) are intensities of the desired light fields which are represented by “cameraman” and “lena”, respectively. (c), (d), (g), (h), (k) and (l) represent the numerical reconstructed results, i.e., ${I_r}({x_1},{y_1})$. (e), (f), (i), (j), (m) and (n) represent the imaging error of the reconstructed light field, denoted as $|{{I_d}({x_1},{y_1}) - {I_r}({x_1},{y_1})} |$.
The intensity of the reconstructed light field is shown in Fig. 5(c), (d), (g), (h), (k) and (l), denoted as ${I_r}({x_1},{y_1})$. Obviously, the image reconstruction quality of the 2-level DOEs designed by the GS algorithm is particularly poor. Meanwhile the reconstructed light fields are nearly unrecognizable as shown in Fig. 5(c) and (d). Compared with the GS algorithm, the MGS algorithm has greater image reconstruction quality. But it is still affected by the speckle noise as shown in Fig. 5(g) and (h). The image reconstruction quality of the proposed algorithm is much greater than that of the GS and MGS algorithms [see Fig. 5(k) and (l)].
In order to intuitively display the imaging effect of the three algorithms, $|{{I_d}({x_1},{y_1}) - {I_r}({x_1},{y_1})} |$ in Fig. 5(e), (f), (i), (j), (m) and (n) is adopted to characterize the imaging error. The light fields recovered by the GS and MGS algorithms are affected by serious error, while the light fields recovered by the ETCR algorithm proposed in this paper have negligible error. Therefore, the numerical reconstructed results are consistent with the data in Table 2.
3.2 Experimental results
In order to verify the validity of the above simulation results, a 2-level DOE has been fabricated and experimental evaluation has been carried out.
The fabrication processes and the experiment results are as follows. The relief depth distribution $h({x_0},{y_0})$ of the DOE can be calculated by using the equation $h({x_0},{y_0}) = {{\lambda \cdot \varphi ({x_0},{y_0})} \mathord{\left/ {\vphantom {{\lambda \cdot \varphi ({x_0},{y_0})} {[2\pi (n - 1)]}}} \right.} {[2\pi (n - 1)]}}$ where $\varphi ({x_0},{y_0})$ represents the phase hologram, and n is refractive index. In this paper, the wavelength $\lambda$ was 650 nm, fused silica (JGS1) was chosen as the substrate material and the corresponding refractive index was 1.4565. Taking advantage of the traditional photolithography technique, several 2-level relief depth distributions were fabricated on the same substrate to fabricate a single DOE, which were designed by the GS, MGS and proposed ETCR algorithms. The feature size was 2 µm and the theoretical etching depth was 712 nm. Figure 6(a) shows the photo of the fabricated 2-level DOE. Figure 6(b) shows the microstructure of the DOE measured with an OLYMPUS BX51 microscope (×500). The step profiler (BRUKER DektakXT) was selected to test the etching depth of the fabricated 2-level DOE. The etching depth of the fabricated 2-level DOE was 723 nm [see Fig. 6(c)] which was close to the theoretical depth.
Fig. 6. Experiment measurement. (a) The 2-level DOE was fabricated by the photolithography technique. (b) The microstructure of the fabricated 2-level DOE measured with the microscope. (c) The etching depth of the fabricated 2-level DOE detected by the step profiler. (d) The experimental setup.
We constructed an optical setup to test the image reconstruction quality of the fabricated 2-level DOE, as shown in Fig. 6(d). A 650 nm laser was emitted by a collimator and irradiated on the fabricated DOE vertically, and then passed through a lens with a focal length of 300 mm. Meanwhile, the HR1600CTLGEC camera was used with resolution of 4896 × 3248 and pixel size of 7.4 × 7.4 µm. The calculated sampling interval of the reconstructed images was determined by $\lambda f/(Md) \approx 55$µm, which was larger than the pixel size of the digital camera. Accordingly, the reconstructed images were recorded correctly, as shown in Fig. 7.
Fig. 7. Experimental results. Optical reconstructed images of the 2-level DOE (a) and (b) designed by the GS algorithm; (c) and (d) designed by the MGS algorithm; (e) and (f) designed by the proposed ETCR algorithm.
In Fig. 7, the first row, the second row and the third row are the experimental results of the GS, MGS and proposed ETCR algorithms, respectively. The experimental results show that the reconstructed images of the 2-level DOE designed by the GS algorithm are difficult to be identified. Although the quality of the images reconstructed by the MGS algorithm is higher than that reconstructed by the GS algorithm, it is still affected by the speckle noise. In contrast, the images reconstructed by the proposed ETCR algorithm are extremely clear with little speckle noise. During the fabrication, the processes of coating, exposure, development and etching might lead to fabrication error. Due to the speckle noise caused by the fabrication error [34], the quality of the reconstructed image will be slightly lower than that of numerical simulation. Note that it is necessary to reasonably adjust the intensity of the incident light to avoid the received intensity of the CCD exceeding the threshold of the CCD, resulting that the reconstructed images are distorted.
According to the above analysis, we can draw a conclusion that the reconstructed images of the DOE designed by our proposed algorithm are clearer and less contaminated with the speckle noise, which is very consistent with the simulation results.
4. Discussion
In this paper, the imaging plane has been divided into the noise domain and the signal domain to improve the image reconstruction quality of the DOEs designed by the proposed ETCR algorithm. The noise and signal domains were composed of M × M and m × m pixels, respectively. Moreover, the value of M/m plays a significant role in diffraction efficiency and RMSE of the DOE, as shown in Fig. 8.
Fig. 8. The influence of M/m on the diffraction efficiency and the RMSE of the DOEs designed by the proposed ETCR algorithm. (a) The relationship between M/m and diffraction efficiency. (b) The relationship between M/m and RMSE.
The influence of M/m on the diffraction efficiency of the 2-level DOE designed by the proposed algorithm is shown in Fig. 8(a). When M/m < 2, with the increase of M/m, the diffraction efficiency decreases slowly; when M/m = 2, the diffraction efficiency suddenly decreases by half because the generated twin image takes half of the energy; when M/m > 2, the diffraction efficiency is about 36%. The effect of M/m on the RMSE of the 2-level DOE is shown in Fig. 8(b). With the increase of M/m, the RMSE decreases rapidly. When M/m ≥ 3.5, the RMSE is reduced to 0.05∼0.08. Therefore, in order to balance the diffraction efficiency and RMSE, M/m needs to be greater than 3.5.
The effect of M/m on the diffraction efficiency of the 4-level DOE is shown in Fig. 8(a). When M/m < 2.5, the diffraction efficiency decreases slowly; when M/m ≥ 2.5, the diffraction efficiency is about 73%. The effect of M/m on the RMSE of the 4-level DOE is shown in Fig. 8(b). The RMSE decreases drastically as M/m increases. When M/m ≥ 3.6, the RMSE is about 0.02∼0.03. Therefore, in order to balance the diffraction efficiency and RMSE, M/m needs to be greater than 3.6.
The effect of M/m on the diffraction efficiency and the RMSE of the 8-level DOE is basically the same as that of the 4-level DOE, so no more analysis is required. As can be seen from Fig. 8(a) and (b), when the proposed ETCR algorithm is used to design an 8-level DOE, it is better to set M/m ≥ 3. At this time, the designed 8-level DOE has a diffraction efficiency of 83% and a RMSE of 0.008.
When the phase of the designed DOE is not quantized, the influence of M/m on its image reconstruction quality is different from the above cases. It can be seen from Fig. 8 that with the increase of M/m, the diffraction efficiency of the DOE without quantization declines quickly. However, when 1.1 ≤ M/m ≤ 1.4, the RMSE of the DOE without quantization is about ${10^{ - 4}}\sim {10^{ - 8}}$ and its diffraction efficiency is between 89% and 97%.
5. Conclusion
In summary, we have proposed a novel method, namely ETCR algorithm which can be used to design high-efficiency low-noise DOEs. Taking advantage of the specially designed ETCR constraint, the proposed ETCR algorithm can realize the proportional-integral-derivative control of the error, that is, it can be used to control the accumulative action, the current state and the trend of the error so that the quantization error can be decreased by iteration quantization. The proposed algorithm is expected to be used in AR, VR and holographic 3D display. The next work is to analyze the fuzzy relationship among the three coefficients (i.e., the proportional, integral and derivative coefficients), the error and the rate of change of the error based on the principle of fuzzy control [35,36], which is expected to further optimize the proposed algorithm.
Funding
National Natural Science Foundation of China (61605211, 6190525); The Instrument Development of Chinese Academy of Sciences (YJKYYQ20180008); The National R&D Program of China (2017YFC0804900); Sichuan Province Science and Technology Support Program (2019YJ0014); CAS “Light of West China” Program.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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