Four types of backward iterative quantization (BIQ) methods were proposed to design multilevel diffractive optical elements (DOEs). In these methods, the phase values first quantized in the early quantization steps are those distant from the quantization levels, instead of the neighboring ones that the conventional iterative method began with. Compared with the conventional forward iterative quantization (FIQ), the Type 4 BIQ achieved higher efficiencies and signal-to-noise ratios for 4-level unequal-phase DOEs. For equal-phase DOEs, the Type 4 BIQ performed better when the range increment of each quantization step was large (>15°), while the FIQ performed better when the range increment was small (<15°).
© 2005 Optical Society of America
Multilevel diffractive optical elements (DOEs), taking advantage of binary optics technology to implement non-spherical and irregular profiles, are able to manipulate the wavefront of an incident light to produce a diffractive image required by various applications, such as beam shaping, image reconstruction, multi-wavelength operation, and optical interconnections [1,2]. Implementations of these applications generally need complex configurations when using conventional refractive optics. On the contrary, a single multilevel DOE can provide an equivalent function and hence reduce the system volume, complexity, and cost. The profiles of the multilevel DOEs contain a staircase-like relief of microstructured features that modifies the wavefront of the incident light, and the modified wavefront generates a diffractive image on the signal plane. The etching depth of each unit cell, or pixel, on the surface profile of a multilevel DOE is calculated by the wavelength of the incident light, the refractive indices of the surrounding and/or the DOE dielectric, and the corresponding phase value that is obtained by conducting a design procedure.
Many design methods, such as the iterative Fourier-transform algorithm (IFTA) [3–7], the simulated annealing (SA) method , the genetic algorithm (GA) , the direct search method , etc, have been developed to calculate the phase profile of a multilevel DOE. Among them, the IFTA offers a fast and simple learning procedure to obtain the phase profile. A classical IFTA to design a multilevel DOEs contains two steps : First, a continuous-phase profile achieving high diffraction efficiency is obtained by applying the amplitude constraints on both the object and the signal domains. Second, the resulted continuous-phase profile is quantized by assigning the phase values to the close quantization phase levels that are specified beforehand.
Although the multilevel DOEs with the quantization levels equally spaced in [0, 2π] have been commonly used, multilevel DOEs with the quantization levels unequally spaced in the same range are able to achieve even higher diffraction efficiencies [11–13] and to realize specific applications [14,15]. To obtain a multilevel DOE of optimal phase levels, an IFTA-based method was proposed in which the quantization levels are calculated at each quantization step by using a probability density function of phase and amplitude . The quantization levels are optimized to ensure a minimum wavefront difference when the quantization range is enlarged. It has been shown that the dynamic optimizations of phase in the iterative processes result in higher diffraction efficiencies and signal-to-noise ratios that are also robust against the initial conditions. In this paper, I employed the schemes of the classical IFTA  and the IFTA-based  procedures to illustrate the performance of different quantization approaches for the DOE profiles with equally- and unequally-spaced quantization levels.
When the IFTA is applied to calculate the phase profile of a multilevel DOE, the procedure stagnates to one of the local minima in a few iteration cycles . The performance of the resulting DOE varies greatly with the initial conditions. An iterative quantization method, widely used in IFTA, overcomes the stagnation by quantizing the phase values in a smaller range centered around the quantization levels and progressively enlarging the range of quantization in the following steps . Although the total number of iterations increases, the diffraction efficiency and other performances of the designed DOE improve. This method is termed, in this paper, the forward iterative quantization (FIQ), for comparison with the later proposed methods. When the FIQ method is applied to multilevel DOEs, the quantization levels are equally spaced and fixed in the entire quantization procedure. When the phase levels need to be calculated in the subsequent quantization steps, however, the diffraction efficiency of the designed DOEs varies not only with the initial conditions but also with the phase ranges of the iterative quantization. Two situations might result in variation of the diffraction efficiency. When the range increment is large and thus fewer quantization steps are conducted, a large number of pixels are quantized to the quantization levels in each step. The quantization levels, calculated after each quantization step, only change slightly from the previous step. The performance of the designed DOE, therefore, is similar to the DOE of fixed quantization levels. If, however, the range increment is small, only a small number of pixels are quantized in each step. The updated quantization levels may change dramatically from the previous levels, and the diffraction efficiency may therefore regress as well. In order to solve the two problems, I propose backward iterative quantization (BIQ) methods in which the phase values first quantized in the early steps are those distant from the quantization levels. Figure 1 illustrates the phase distribution, in the complex-valued plane, in an intermediate step of designing a 4-level DOE using the FIQ and BIQ methods.
In Section 2, four types of the BIQ methods are described. The simulation results of a 4-level DOE designed using the FIQ and BIQ methods are shown in Section 3. The results include the diffraction efficiency and the signal-to-noise ratio of the DOEs by assigning 20 initial conditions (composed of 20 random phase arrays and the target intensity image) and 11 quantization ranges. In Section 4, discussions of these methods are presented. Instead of stagnation, a chaos-like behavior was observed in some of the designed DOEs in this study.
2. Backward iterative quantization (BIQ) methods
In a Fourier-transform optical system, the complex-valued field G(u,v) of the diffraction image on the signal plane is the Fourier transform of the field g(x,y) on the object plane, which is given by
where FT indicates the Fourier transform . The field function g(x,y) modulated by a multilevel DOE is a complex-valued function of a constant amplitude for the illumination of a plane wave, i.e., g(x,y)=exp(jϕ(x,y)). The goal of the IFTA is to calculate a discrete form of the phase function ϕ(x,y) by minimizing the error of the resulted signal G(u,v) and a target pattern F(u,v). It is achieved by applying the constraints of phase and amplitude to the object domain and the constraint of amplitude to the signal domain in the iterations of the IFTA. To calculate the phase profile of an N-level DOE, the quantization procedure is partitioned into Q steps with a maximal number K of iteration cycles in each step. The signal error can be effectively reduced by replacing the signal amplitude |Gk| of the kth iteration by the target amplitude |F| ,
Here, S denotes the signal window, containing non-zero elements in the signal plane. The field G’k(u,v) is inverse Fourier transformed and results in a complex-valued function g’k(x,y) that is generally complex-valued, i.e., g’k(x,y)=|g’k(x,y)|exp(jϕ’k(x,y)). In the signal domain, the new complex amplitude gk+1(x,y) is subject to the amplitude constraint of
In general, Eq. (3a) is used; however, using Eq. (3b) for un-quantized pixels in the BIQ methods can achieve higher diffraction efficiency. The phase constraint of using the FIQ is expressed by a projection of phase given by [5,6]
where Δ(n,q)=(ϕ(n,q) - ϕ(n-1,q))/2 and the quantization ratio is given by 0<ε (1)<…<ε (q)<…<ε (Q)=1. Since ϕ is cyclically periodic, Δ(N+1,q)=Δ(1,q). The parameters of the FIQ used for designing 4-level DOEs are illustrated in Fig. 2(a). Note that for designing equal-phase DOEs the quantization levels ϕ(n,q) are fixed for all q’s. For unequal-phase DOEs, quantization levels ϕ(n,q) are calculated by minimizing the mean-squared error of the object field at step q, which is defined as 
where the clipping boundaries C(n,q) are given by
and C(N+1,q)=C(1,q)+2π. The function paw(ϕ) is the product of the probability density function p(ϕ) and the mean of the amplitude |gk(x,y)| with respect to ϕ.
In order to solve the problems caused by the use of FIQ, previously described, the phase values distant from the quantization levels are projected onto the quantization ranges of the proposed BIQ methods. The effect of the BIQ methods is illustrated in Fig. 2(b) in which the upper and the lower bounds of ϕ(n,q) are given by
[ , ] defines a free window, , of ϕ(n,q) in which phase values will not be quantized.
There are four BIQ alternatives that are based on different ways of projection of the phase values. In the Type 1 BIQ method, the phase values outside de are projected onto the close available phase values, i.e., and . The projection of ϕ’k(x,y) is given by
As the quantization proceeds, i.e., as q increases, the phase values concentrate progressively toward the quantization levels, and finally ϕ(x,y)=ϕ(n,q) at q=Q. The Type 2 BIQ method linearly projects the phase values onto new phase values in . The linear projection of the phase values is given by
where β (n,q)=(-)/(C (n+1,q)-C(n-q) is a compression ratio of the quantization step q. At q=Q, = and β (n,q)=0 means the free windows close and ϕ(x,y)=ϕ(n,Q). In the Type 3 BIQ, the phase values outside are directly projected onto the quantization levels and the others remain unchanged. This projection is given by
In the Type 4 BIQ method, moving the phase values resembles scrolling back a reel with a roller equivalent to the quantization level. All the phase values in the clipping range of ϕ(n,q) sequentially shift toward ϕ(n,q), the quantization level. The scroll-back projection of the phase values is given by
Figure 3 illustrates the phase projections of these methods between two adjacent levels ϕ(1,q) and ϕ(2,q).
3. Simulation results
These methods were applied to design a 4-level phase-only DOE which produces a binary target pattern of 64×64 pixels. The initial condition was a complex-valued array, composed of the amplitude distribution of the target image and one of 20 phase distributions generated randomly by the computer. By first using the classical IFTA to generate 20 continuous-phase DOEs, the iterative quantization begins. The optimal design and the stable state were determined by evaluating a cost function of the signal field Gk(u,v), given by
where w 1 and w 2 are real constants. Here, ηk and SNRk are the relative diffraction efficiency and the signal-to-noise ratio of the DOE in iteration cycle k, respectively, defined as
When stagnation occurs, f k becomes constant. The solution of the multilevel DOE is selected when the largest f k results in the final quantization step.
3.1. Four-level equal-phase DOEs
For the equal-phase designs, the 4 quantization phase levels are equally spaced in [0, 2π], i.e., [0, π/2, π, and 3π/2]. Eleven quantization steps, Q=[1, 2, 3, 4, 5, 6, 7, 9, 12, 15, 30], are chosen in the simulations. When Q=6, for example, the quantization ratio ε (q)=q/6 and the quantization range increases by a value of π/12, or 15°, at each step. Note that the direct iterative quantization is conducted when Q=1. In order to achieve a higher diffraction efficiency, the amplitude constraint Eq. (3a) was used for all designs of equal-phase DOEs and the designs of the unequal-phase DOEs using the FIQ method. Table 1 lists the overall descriptive statistics of the diffraction efficiencies and the SNRs of 220 DOEs (20 initial arrays and 11 Q’s) with 4 discrete phase levels designed by the above-described methods. Among the BIQ methods, only the Type-4 BIQ performs competitively with the FIQ method (ηavg=0.8165 and SNRavg=0.7774 vs. ηavg=0.8162 and SNRavg=0.7882). Figure 4 shows the details of η‘s and SNRs of the 4-level DOEs with respect to Q and the individual DOE. As shown in Fig. 4(a), the FIQ method achieved a higher η for large Q’s, Q≥7, indicating small increments of quantization range. The Type-4 BIQ method, however, performed better when
Q’s were less than 6 (large increments of quantization range). In fact, all BIQ methods achieved a stable η for all Q’s. The SNRs of DOEs, as shown in Fig. 4(b), designed using the FIQ and Type-4 BIQ were in general larger than those using the other three BIQs, and all of them did not vary greatly with the value Q. Figures 4 (c) and (d) show the efficiencies and the SNRs of 10 values of Q’s, excluding the Q=1 case (direct quantization), with respect to the individual DOE. The standard deviations of the efficiency of the DOEs by the FIQ were large because of their great variations at small and large Q’s.
3.2. Four-level unequal-phase DOEs when ϕ(1)=0
To prevent the implementation difficulty of the bias phase ϕ(1), we set ϕ(1)=0 here. In the optimization of the discrete phase levels of the unequal-phase DOE, only two variables, i.e.,
the etching depths d1 and d2, are calculated to construct 4 phase levels. As in Section 3.1, 11 Q values were chosen, however, the quantization ratio ε (q) of a specified Q does not increase linearly. In the designs of unequal-phase DOEs, the amplitude constraint Eq. (3a) and the FIQ method were used, and Eq. (3b) was used with the four BIQ methods. The descriptive statistics of these simulation results are listed in Table 2. The efficiency and SNR of the 4-level DOEs using the Type 4 BIQ method increase significantly (ηavg=0.8242 and SNRavg=0.8152). Figure 5 shows these performances with respect to the number of quantization steps Q and the individual DOE. The Type 4 BIQ method was able to produce 4-level DOEs with the largest efficiencies in all cases, and higher SNRs in some cases than in others. The largest efficiencies obtained by the FIQ and Type 4 BIQ were 0.8247 (DOE#7, Q=30) and 0.8323 (DOE#14, Q=30), and the highest SNR by the FIQ and Type 4 BIQ were 1.318 (DOE#3, Q=2) and 1.234 (DOE#20, Q=12), respectively. Although the highest SNR of 1.318 was obtained using the FIQ method, the corresponding efficiency was 0.8108 which is lower than the efficiency 0.8272 of DOE#20 at Q=12 obtained using the Type 4 BIQ. On the contrary, the lowest SNRs obtained using the FIQ and Type 4 BIQ were 0.2078 (DOE#17, Q=4) and 0.4401 (DOE#11, Q=12), respectively. These results show that the performance of the DOEs designed by using the Type 4 BIQ method is not only higher but is also more stable than the performance obtained using the FIQ and the other BIQs.
3.3. Four-level unequal-phase DOEs when ϕ(1)≠0
To completely investigate the designs of the unequal-phase DOEs, we have conducted simulations for the cases when ϕ(1)≠0. For the unequal-phase DOEs of 4 discrete phase levels, three variables ϕ(1), d1, and d2 were calculated to optimize the 4-level DOE. Equations (3a) and (3b) were used for the amplitude constraint when applying the FIQ and BIQ methods, respectively, to obtain a higher diffraction efficiency. The overall descriptive statistics of the simulation results are listed in Table 3. The means of the efficiencies and SNRs were on the average slightly higher than those of the DOEs when ϕ(1)=0. The largest efficiency obtained by the FIQ and Type 4 BIQ were 0.8244 (DOE#2, Q=12) and 0.8307 (DOE#4, Q=9), and the highest SNR by the FIQ and Type 4 BIQ were 1.223 (DOE#14, Q=12) and 1.281 (DOE#12,
Q=5), respectively. Compared with the cases in which ϕ(1)=0, these results show that the influence of the bias phase ϕ1 is not significant when the etching depths (d1 and d2 in 4-level DOEs) are optimized to obtain multilevel DOEs. Figure 6 shows these performances with respect to the number of quantization steps Q and the individual DOE. Still, the DOEs designed using the Type 4 BIQ method outperformed the DOEs designed using the other methods.
Figure 7 shows the intensity distributions of the target image and the diffractive field of the 4-level DOE of the largest efficiency designed by using the Type 4 BIQ method (DOE#14, Q=30: η=0.8323, SNR=0.9816). The four quantization levels obtained were 0π, 0.493π, 1.013π, and 1.506π, corresponding to two etching depths 1.013π (d1) and 0.493π (d2).
There are two factors in the design procedure of the proposed methods that influence the performance of the designed DOE: the quantity of pixels changed and the changes of the phase value by the quantization operators. In the Type 1 BIQ method, the changes of phase values were fixed, but the number of pixels changed increased as the quantization proceeded. The number of pixels changed was maximized using the Type 2 BIQ method accompanied by small changes of the phase values, which resulted in higher efficiencies than those obtained when using the Type 1 method. Then the phase values outside the free windows are offset directly to the quantization levels as in the Type 3 BIQ method. The number of pixels changed did not increase as the quantization proceeded, but the phase changes were large, especially in the first few quantization steps. This modification increased the efficiency but lowered the SNR as shown in Tables 1~3. The efficiencies and SNRs of DOEs obtained using the three BIQ methods, however, were not higher than those using the FIQ method. The large changes of either the number of pixels or the phase values, or both, resulted in instability in the iterative procedure. Instead of stagnation occurring frequently, as it does in the conventional IFTA, an oscillation of several states, even a chaos-like phenomenon, was observed when using these BIQ methods as shown in Fig. 8. It resulted in a great variation of the efficiencies, as DOEs #8 and #15 in Fig. 5(c) and DOE#12 in Fig. 6(c). Although this problem occurred when the DOE was designed by using the Type 4 BIQ method, the performance did not regress. Both the number of pixels changed and the changes of phase values decreased as the quantization proceeded because, in this method, the phase values close to the quantization levels were quantized first and the rest of the phase values were progressively shifted toward the quantization levels. These are especially important at the final steps of the iterative quantization. The efficiencies and SNRs of the DOEs designed using this method were the highest among all unequal-phase cases.
To illustrate the performance of the proposed approaches for the DOEs with higher number of phase levels, I used DOE#1 and 2 Q’s (2 and 15) as the initial conditions to generate the DOEs with 8 discrete phase levels. As many might expect, the efficiencies of the obtained 8-level DOEs using the 5 quantization methods did not vary significantly, with a range from 0.9034 to 0.9090 and an average of 0.9069. The largest efficiency of 0.9090 (SNR=3.041) was obtained by using the Type 3 BIQ method when Q=15. However, the largest SNR of 4.308 (η=0.9077) was obtained by using the Type 2 BIQ method when Q=2, which increased significantly compared with the largest SNR of 3.294 for the equal-phase DOEs and 3.811 for the unequal-phase DOEs by using the FIQ method. As shown in Fig. 9, a clear image with rather low noise and high uniformity was constructed by the unequal-phase DOE with eight quantization levels (0π, 0.248π, 0.489π, 0.737π, 1.027π, 1.275π, 1.515π, and 1.763π), corresponding to three etching depths 1.027π (d1), 0.489π (d2), and 0.248π (d3). Note that the first three BIQ methods performed competitively with the Type 4 BIQ and FIQ methods for higher phase levels because the quantization ranges are much smaller than those of the 4-level cases. Correspondingly, both the number of pixels changed and the phase changes in each iteration were small, and the performance improved.
For the designs of binary DOEs, the SNR was increased significantly by using the Type 4 BIQ method, although the diffraction efficiency increased slightly. A different target pattern with a symmetric feature of a cross sign was used as the initial condition for the design of the binary DOEs at Q=60, corresponding to a range increment of 3°. As shown in Fig. 10, the largest efficiency and SNR obtained were 0.8694 and 1.028, respectively, by using the Type 4 BIQ, compared with 0.8675 (efficiency) and 0.6038 (SNR) obtained by using the FIQ method. In fact, the SNRs of the binary unequal-phase DOEs designed using the Type 4 BIQ were much larger than those using the other methods. Nevertheless, the diffraction efficiencies of the binary DOEs obtained using all these iterative quantization methods were close. The insignificant improvement of the efficiency of the binary DOEs might result from few phase freedoms that can be employed. There are two phase levels when ϕ(1)≠0 and only one phase level when ϕ(1)=0, which are dynamically optimized in the iterative process.
In this paper, I proposed 4 types of BIQ methods to overcome the problems caused by using the widely-used FIQ method. These problems usually occur when the FIQ quantization method is applied to the iterative optimization of unequal discrete phase levels. Instability occurred when using the Types 1 to 3 BIQ methods, and it impacted the efficiency and SNR of the designed DOE. By shifting the phase values progressively toward the quantization levels, the Type 4 BIQ method succeeded in increasing the efficiency and the SNR of the designed DOE. In the simulations of 4-level DOEs starting with 20 initial arrays and 11 quantization steps, the Type 4 BIQ method achieved the highest efficiency of unequal-phase DOEs, and the SNRs were on the average higher. In the designs of equal-phase DOEs, the Type 4 BIQ method achieved the largest efficiency at small Q’s and the FIQ achieved the largest efficiency at large Q’s.
The support of the National Science Council, Republic of China, under grant NSC 92-2215-E-027-007 is gratefully acknowledged. The author would like to thank for Mr. Kuo-Yuan Ho for his effort in completing the computer simulations. The author is also grateful for the unknown reviewer’s insightful and constructive comments and suggestions.
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