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Normally, to incorporate two binary conventional computer-generated holograms (CGHs) into a single polarization-selective computer-generated hologram (PSCGH), the respective pixels of the conventional CGHs will result in 4 different combinations of the phase values. Thus, the 4 phase combinations have to be realized by 4 types of pixel structures in a PSCGH. In this paper, we propose a method to reduce the PSCGH’ s 4 phase combinations to 3 using an optimization approach. The PSCGH’s first-order diffraction efficiency is 30% and the contrast ratio is 28 after the optimization.
©2003 Optical Society of America
1. Introduction
Conventional phase-only computer-generated holograms (CGHs), whose feature size is much greater than the incident wavelength, are normally independent of the light polarization state. They have been widely used in wavefront correction, beam shaping, beam coupling, optical interconnection and other applications. With the design feasibilities and the development of micro- and nano-fabrication technologies, polarization-selective computer-generated holograms (PSCGHs) have been a topic for some years [1–5]. The creation of the PSCGHs is mainly based on the birefringence of the materials or the form-birefringence by fabrication of subwavelength sized structures in an isotropic substrate [4, 6]. Based on the birefringence properties, the PSCGHs have demonstrated some polarization multiplexing functions, where the results confirmed that the phase delays of two conventional CGHs could be physically encoded into one PSCGH, and the reconstructions could be multiplexed for the two orthogonal polarizations respectively. Therefore, one PSCGH has replaced two conventional CGHs.
Fabrication of PSCGHs is a challenging work. This is because the subwavelength sized structures or birefringent materials are a must to achieve the polarization-selection. To implement the multiplexing function, we need 4 phase combinations for a PSCGH for TE and TM polarizations, i.e., (0, 0), (0, π), (π, 0) and (π, π). In parenthesis, the first entries are the phase delays relating to a TE incident light, while the second entries are relating to a TM incident light. Generally, we have to use 4 different types of structures to realize them. To realize the combinations of (0, π) and (π, 0) selectively, the subwavelength sized structures or birefringent materials are essential. Binary subwavelength sized horizontal/vertical gratings are the simplest form-birefringent structures. Hence, they have been widely adopted to realize the two combinations. The other two combinations of (0, 0) and (π, π) could be implemented by multilevel depths as Ref. [2] or by using another two subwavelength sized structures as Refs. [4, 5]. However, these implementations will increase the complexities in fabrication.
There is an alternative way to avoid the use of the multilevel structures or more than 2 subwavelength structures to tackle with the problem. Namely, if we reduce the phase combinations of (0, 0) and (π, π) to one, i.e. either (0, 0) or (π, π), the design and fabrication will be significantly simpler. The PSCGHs would only need 3 different types of structures to achieve the polarization multiplexing functions.
In this paper, we propose an optimization approach to reduce the phase combinations. The combinations of (0, π) and (π, 0) are remained while the other two combinations of (0, 0) and (π, π) are forced to (0, 0) through an optimization process.
As the fabrication of subwavelength sized structure is more difficult than that of the super wavelength structures, an advantage of the optimization method is that the total number of the required subwavelength structures in a form-birefringent PSCGH can be minimized. Similarly, a multilevel material-birefringent PSCGH can be simplified as a binary. Furthermore, the multiplexing contrast ratio can be taken into account in the optimization simultaneously.
2. Theoretical analysis
Let us consider two conventional CGHs as input functions in the polarization multiplexing. As an example, the CGHs consist of 128×128 binary phase-only pixels respectively. The phase value of each pixel is either 0 or π. Without losing generality, the desired reconstructions of the conventional CGHs are considered as a flat-top beam and a Bessel beam as shown in Fig. 1 respectively.
Fig. 1. The desired reconstructed patterns for the two conventional CGHs, (a) a flat-top beam, (b) a Bessel beam.
We designed the phase distributions of the two conventional CGHs using the Gerchberg-Saxton (G-S) algorithm [7]. As a result, the two CGHs will reconstruct a flat-top beam and a Bessel beam respectively. The phase distributions of the two CGHs are shown in Figs. 2(a) and (b). The black and white dots in Fig. 2 represent the phase values 0 and π respectively.
Fig. 2. CGHs’ phase distributions, (a) the CGH for the flat-top beam, (b) the CGH for the Bessel beam.
The reconstructed images of the two CGHs are shown in Fig. 3. Due to the binary phase-only structures, the CGHs reconstructed twin images in the ±1st orders.
To incorporate the two CGHs into a single PSCGH, the respective pixels of the CGHs will result in 4 different combinations of the phase values, as listed in Table 1.
Table 1. Multiplexing two binary CGHs into one PSCGH.
Consequently, in theory, a PSCGH should have 4 basic pixels to realize the phase combinations. As discussed above, we will replace one combination by one of the other 3 combinations. In this case, we select the combination 4, i.e. (π, π) to be replaced.
3. Optimization approach
The optimization approach involves two steps: the replacement and random optimization. The first step of the optimization approach is the replacement of the 4th combination state (π, π) by one of the other 3 combination states (0, 0), (0, π) or (π, 0), as illustrated in Table 1. In this step, we search the 4th combination pixel by pixel from the first pixel to the last. Therefore, after the replacement, we can ensure that all the pixels of the 4th combination have been replaced by the other 3 combination states.
An immediate question is to determine which one of the 3 phase combinations will replace the 4th combination state, thus we need to compute the 3 combinations’ merit costs of the pixel respectively. The merit function for the replacement is described by Eq. (1) as follows,
where the IntTErec,TMrec represents the reconstructed image intensity of the PSCGH by the TE or TM incident light, and the IntTEdes,TMdes represents the desired reconstructed image intensity of the corresponding CGH. The replacement rules should be governed by the following criterion - (1) the PSCGH has the minimum merit cost with either TE or TM polarization, or (2) the PSCGH has the minimum sum of merit costs for both TE and TM polarizations. In addition to the natural differences of the original images in TE and TM polarizations, the CGH image reconstructions might introduce further discrepancies in terms of total intensities and image sizes. With such embedded variations in the cost values of TE and TM polarizations, a weighting factor can be appended onto the initial polarization results. In our case, the weighting factor of 2.0107 and 1.9894 were used for TE and TM polarizations respectively, and the results are shown in Fig. 4. Alternatively, the 4th combination state can be replaced randomly by any of the other three states during the replacement process, however, the subsequent computations would take a longer time.
The reconstructed patterns of the computed PSCGH, using only the first three combination states as mentioned earlier, are shown in Figs. 4(a) and (b) for TE and TM polarizations respectively. The desired reconstructed patterns (Flat-tops in Fig. 4(a) and Bessels in Fig. 4(b)) are discernibly clear. However, the undesired patterns (Bessels in Fig. 4(a) and Flat-tops in Fig. 4(b)) are also discernable. Hence, we need to further optimize the PSCGH to suppress the undesired patterns.
Fig. 4. Reconstructed images of the PSCGH after the combination 4s replaced, (a) reconstructed flat-top beams, (b) reconstructed Bessel beams.
In the second step, we optimize the PSCGH using a random search for TE and TM polarizations separately. This step is similar to the simulated annealing algorithm [8], but the criterion for acceptance or rejection is different. There is no probability control with the temperature variations in this step. We just accept all the changes if there is no merit value increasing. We implement the optimization for the TE polarization first. When the TE optimization is terminated, then we optimize the TM polarization. In this step, to ensure that the 4th combination will not appear again after the pixel change, we choose two of three combinations for each polarization state respectively, namely the (0,0) and (π,0) for TE and the (0,0) and (0, π) for TM. There is no crosstalk between the optimization processes for the TE and TM polarization states. Therefore, the two polarizations are optimized separately, and the 4th combination will not appear after the random optimizations. To increase the contrast ratio and diffraction efficiency, the second step can be repeated many times. The detailed optimization procedures for TE and TM polarizations are shown in Fig. 5.
The criterion for a pixel change in the above procedures is based on the merit functions, which are similar to that of Eq. (1). During the computations, if the cost value did not increase, the program would accept the phase change. Otherwise, the pixel’s phase will keep its original value.
We optimized the TE part and the TM part only once respectively. The non-uniformities of TE’s and TM’s compared to their respective desired patterns are plotted in Figs. 6(a) and (b).
Fig. 6. Non-uniformities vs. iterations (a) for TE reconstructed flat-top beams and (b) for TM reconstructed Bessel beams respectively.
For both TE and TM optimizations in Fig. 6, the computations took 1000 times within the outer loops and 2000 inner loops with each pixel being selected randomly. From Fig. 6 we can see that, after about 400 outer loops, the non-uniformities are nearly unchanged. To decrease the non-uniformities further, we can run the TE and TM optimization process repeatedly as shown in Fig. 5. This is because any phase change in TE or TM optimization process of Fig. 5 will affect the combination states. The final reconstructed images of the PSCGH after the second optimization step are shown in Figs. 7(a) and (b). It is seen in the figure that, not only the desired pattern is clearly discernible, but also the noise level has been reduced substantially. In addition, the reconstructed images after the above random optimization process have been improved significantly.
The contrast ratio η is 28 in Fig. 7, and the first-order diffraction efficiency R equals 30%. The theoretical maximum first-order efficiency for a binary phase grating is 40%. Since the PSCGH is a combination of two binary phase distributions, the maximum theoretical diffraction efficiency of the PSCGH with orthogonal polarization lights will be the same as a single binary CGH, i.e., 40%. The root mean square (RMS) errors in Figs. 3(a) and (b) reconstructed by the conventional CGHs are 0.0772 for the flat-tops and 0.0516 for the Bessels respectively. After the first optimization step of the PSCGH, although the RMS errors in Figs. 4(a) and (b) are increased to 0.0909 and 0.0571 respectively, the RMS errors of the final PSCGH are remarkably decreased after the second optimization step. The RMS errors in Figs. 7(a) and (b) are 0.0398 and 0.0229 respectively, which are about half of the errors in Figs. 3(a) and (b).
Fig. 7. Reconstructed images after the second optimization step, (a) reconstructed flat-top beams, (b) reconstructed Bessel beams.
The 128×128-pixel PSCGH, compromising only the first three combination states of Table 1, are as shown in Fig. 8(a).
Fig. 8. Phase pattern of the optimized PSCGH, (a) the whole PSCGH, (b) the first 20×20 pixels of the PSCGH.
A close-up image shows the first 20×20 pixels of the PSCGH in Fig. 8(b). The blank pixel stands for (0, 0) and the horizontal and vertical grooves stand for (0, π) and (π, 0) respectively. Therefore, the PSCGH reconstructed two different clear patterns with just 3 phase combinations.
4. Summary
In this paper, two conventional computer-generated holograms (CGHs) were combined into one polarization-selective CGH (PSCGH) with only 3 phase combinations. The 4th phase combination is replaced by one of the other 3 combinations through an optimization process. The optimization involved two main steps: firstly the replacement of the 4th combination with one of the other 3 phase combinations and secondly further optimization of the TE and TM phase distributions respectively. The resultant PSCGH can reconstruct two clear patterns with orthogonal polarization lights. Furthermore, we can increase the diffraction efficiency and contrast ratio of the PSCGH by repeating the step 2. Therefore, this PSCGH can replace two conventional CGHs with simpler phase structures. Similarly, for other binary multiplexing optical devices, this method can also be applied to omit the most complicated pixel in all the 4 combinations. Hence, the design and fabrication of the multiplexing device will be simplified.
References and links
1. F. Xu, R. Tyan, P. Sun, Y. Fainman, C. Cheng, and A. Scherer, “Form-birefringent computer-generated holograms,” Opt. Lett. 21, 1513–1515 (1996). [CrossRef] [PubMed]
2. Joseph E. Ford, Fang Xu, Kristopher Urquhart, and Yeshaiahu Fainman, “Polarization-selective computer-generated holograms,” Opt. Lett. , 18, 456–458 (1993). [CrossRef] [PubMed]
3. Uwe D. Zeitner, Bernd Schnabel, E.-Bernhard Kley, and Frank Wyrowski, “Polarization multiplexing of diffractive elements with metal-stripe grating pixels,” Appl. Opt., 38, 2177–2181. [PubMed]
4. Wanji Yu, Tsuyoshi Konishi, Tetsuya Hamamoto, Hiroshi Toyota, Tsutom Yotsuya, and Yoshiki Ichioka, “Polarization-multiplexed diffractive optical elements fabricated by subwavelength structures,” Appl. Opt. , 41, 96–100 (2002). [CrossRef] [PubMed]
5. G. P. Nordin, P. Deguzman, J. Jiang, and J. T. Meier, “Polarization sensitive diffractive optics for integration with infrared photodetector arrays,” in Diffractive Optics and Micro-Optics, OSA Technical Digest (Optical Society of America, Washington DC, 2000), pp. 88–90 (Invited Paper).
6. R. M. A. Azzam, “Polarization beam splitters for infrared and millimeter waves using single-layer coated dielectric slab or unbacked films,” Appl. Opt. 25, 4225–4227 (1986). [CrossRef] [PubMed]
7. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik , 35, 237–246 (1972).
8. S. Kirkpatrick, C.D. Gellatt, and M.P. Vecchi, “Optimization by simulated annealing,” Science , 220, 671–680 (1983). [CrossRef] [PubMed]
