IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 15, NO. 7, JULY 2003
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Genetic Algorithm and Finite-Element Design of Short Single-Section Passive Polarization Converter D. Correia, Student Member, IEEE, J. P. da Silva, and H. E. Hernández-Figueroa, Senior Member, IEEE
Abstract—A very short passive polarization rotator is proposed, consisting of a single-section asymmetric waveguide. The use of genetic algorithm together with the finite-element method was crucial to achieve a structure with 99% polarization conversion and a length of 90 m. This optimum structure exhibits a low total insertion loss: 1.16 dB. Index Terms—Finite-element method, genetic algorithm (GA), polarization rotator.
I. INTRODUCTION
T
HE DESIGN of passive polarization converters or rotators has attracted considerable attention in the last decade or so, motivated by the increasing need of miniaturization for integrated optics devices. Based on this, several authors have reported efficient structures, each time smaller, capable to achieve almost full polarization conversion. Rotators several millimeters long and few decibel losses were presented in [1] and [2]. The fabrication process of such passive components was not simple, though it was less complex than the active ones. Van der Tol et al. [3] proposed a relatively simpler converter, based on a slant angle geometry. It had lower losses and was shorter and easier to fabricate than ones previously proposed [1], [2]. Tzolov and Fontaine [4] called attention to the possibility of efficient conversion being attained using a single-section polarization converter. Huang et al. [5] fabricated such single-section structure, obtaining 96% polarization conversion over a length of 720 m. Finally, Rahman et al. [6], by means of finite-element simulations, showed that by varying the transverse geometry dimensions of the structure realized in [5], the efficiency could be improved to 99% with a drastic length reduction to 320 m and 0.5-dB insertion losses. However, none of the authors cited above used a systematic optimization technique to achieve their results. The present work shows how the search for optimal structures can be facilitated by using a simple optimization scheme based on the widely recognized genetic algorithm (GA) strategy. GA has been intensively used in electromagnetics, antennas in particular, in the last ten years [7]. Designers of waveguide-based photonics devices have only recently [8] realized the benefits of the GA powerful features. Using the combined action of GA and a highly efficient Manuscript received August 26, 2002; revised March 31, 2003. This work was supported in part by the Brazilian Agencies FAPESP and CNPq under FAPESP Process 00/04371-3 and CNPq Process 301209/94-4. The authors are with the Department of Microwaves and Optics, School of Electrical and Computer Engineering, University of Campinas, 13083-970 Campinas, Brazil (e-mail:
[email protected];
[email protected];
[email protected]). Digital Object Identifier 10.1109/LPT.2003.813407
Fig. 1.
Geometry of passive polarization converter.
H-field full-vectorial finite-element eigenvalue mode-solver [9], we show here that even more significant improvements than those reported in [6] can be attained. In fact, polarization converters with lengths of about 90 m and polarization conversion efficiency as high as 99%, while keeping the total insertion loss as low as 1.16 dB, are possible to be obtained. II. PASSIVE POLARIZATION CONVERTER Following the polarization rotator’s design evolution described in the introduction, the slanted single section proposed in [5] and [6] summarizes previous improvement attempts reported in [1]–[4]; therefore, such geometry, illustrated in Fig. 1, was adopted in this work. Due to the asymmetry introduced by the inclined cut, the first two modes of this structure are hybrid; however, the maxima amplitude of each component practically coincide at the same position. Let us call the normalized maxima of these two modes and , with propagation constants and , , where , . The more hybrid respectively, and are. the modes, the nearer to unity the components Considering the total magnetic field as a linear combination of these two modes, its maximum amplitude writes as follows:
(1) . Expression (1) was obtained in such where, , coincides with a pure a way that at the input transverse electric excitation, where we have and . When , we have and . given by This situation defines the coupling distance
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IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 15, NO. 7, JULY 2003
TABLE I INTERVAL LIMITS
FOR VARIABLE PARAMETERS PROFILE SHOWN IN FIG. 1
RELATED
NEW INTERVAL LIMITS
TO THE
It is clear that the more hybrid the modes are, the coupling to the transverse magnetic polarization gets closer to the ideal sitand . The normalized uation, i.e., coupled to the component at the coupling distance power is given by (3) m and of the adopted proThe parameters file (see Fig. 1), were kept fixed, while the other five parameters were allowed to vary in the intervals shown in Table I. The chosen initial intervals were based on [6]. III. GENETIC ALGORITHM The first step to apply GA to our problem was to decode our variable parameters into a binary code. This process was done in a way similar to [8] and is shown in (4). Thus, if is one of the variable parameters of Table I, then (4) and are the minimum and maximum where , value of the variable parameter , respectively, and are the bits describing . We used a total of four bits to each variable parameter what resulted in a total of 20 bits to each possible solution. The second step is to choose a fitness or cost function to compare the searched profiles. This is one of the most important criteria of a GA-based algorithm. In our problem, two different, and usually conflicting, goals have to be satisfied. First, we desire a profile that has high hybrid nature, which means to make , given in (3), as close to 1 as possible. Second, we wish the polarization converter to be as short as possible, to integrate the component easily. So, to reduce the length, given in (2), we have . With these two goals, we tried the following fitto increase ness function: (5) is a sensitivity parameter to be adjusted. We first tried . During the GA process, particular care must be taken when the two first modes of the trial profiles are nearly cut off. These situations were handled allocating very low values to the fitness function and replacing the population exactly over those structures, for the first ten generations. The next ten generations had a random replacement of 40%. We used 25 individuals at each generation.
Here,
TABLE II VARIABLE PARAMETERS BASED FIRST RESULT
FOR
ON THE
IV. NUMERICAL RESULTS AND DISCUSSION Our first results were m, , , m, and for a wavelength m. For this profile, we achieved a polarization conversion of 97% for a conversion length of 170 m. Due to the short length of this device, material loss can be neglected; therefore, only insertion loss is important. By means of a highly efficient full-vectorial finite-element beam propagation method (BPM) [10], the conversion length was confirmed and the computed total insertion loss was 1.18 dB. However, a careful analysis of this result showed that it could be improved. Comparing these values with the allowed values from Table I, we see that the only variable that is not at one extreme of its interval is . This strongly suggested that our intervals, based on [6], do not cover the optimum region of this problem. The above conclusion would hardly be possible without an optimization technique. If we just vary one variable, fixing the rest as done in [6], we will be restricted to local maxima. Such an approach would only work in linear problems, and obviously this is not the case here. The next step was to shift the intervals extremes to more appropriate values, for such variables whose computed values were at one extreme of its corresponding intervals. The intervals shown in Table I were then replaced by the ones given in Table II. With these new intervals, however, the solution became poor reached right from the bebecause of the high values of ginning, as a consequence, the best results achieved very high but low coupled normalized power . The reason was that , became unthe fitness function defined in (5), with , suitable. We then tried a new sensitivity parameter, . With the new interin order to decrease the influence of vals and the new fitness function we achieved a device length of m, 90 m for a polarization conversion of 99%, with , , m, and . The performance of this optimum geometry was then confirmed using the BPM simulator reported in [10], whose results are shown in Figs. 2 and 3. Additionally, the total insertion loss was found to be 1.16 dB. We observed that varying the slant angle by 2 , the solution remains practically the same; this variation is within the fabrication precision presently available for such a structure [5]. A careful analysis of sensitivity to fabrication errors was also performed for the other four variables. The results confirmed the robustness of our solutions. It is worth noting that this optimum , result was achieved at the expense of increasing but as a compensation the core’s size was reduced, preventing, therefore, the excitation of more modes than the two first hybrid ones. Insertion loss was reduced by 0.02 dB with respect to our first solution, but it is higher by 0.66 dB when compared with the
CORREIA et al.: GA AND FINITE-ELEMENT DESIGN OF SHORT SINGLE-SECTION POLARIZATION CONVERTER
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a drastic limitation of our optimization approach, since almost complete polarization conversion and very short coupling length are achieved at the expense of a tolerable small increment of insertion loss. Each optimization process was performed using 4000 linear elements, taking approximately 4 h in a Pentium II 800-MHz 128 MB RAM processor. To confirm the best structures’ characteristics, careful analyses were carried out using more than 8000 linear elements. All simulations were performed using free software libraries available in the LINUX system. V. CONCLUSION
Fig. 2. Normalized power curves for H (solid line) and H (dotted line) obtained via BPM simulation [10], along the optimum polarization converter. L = 90 m, with L = L = 30 m (see Fig. 1).
We have presented a powerful GA and finite-element methodbased approach for the efficient design of single-section passive polarization rotators. The use of GA proved to be crucial to achieve a structure with a length as short as 90 m and 99% of polarization conversion. However, close monitoring of the results along the optimization process is mandatory. In addition, verification of the solution’s robustness with respect to the available technology capability and also tolerable insertion loss values, must complete the whole procedure. We believe that the present approach may also be useful to improve the performance of other key photonics structures. REFERENCES
Fig. 3. Modulus of H (left side) and H (right side) related to power curves shown in Fig. 2, at (a) z = 15 m, (b) z = 75, and (c) z = 135 m.
best result given in [5]. It is interesting to observe that for these three structures, the insertion loss factor varies within a range of about 0.7 dB, while the coupling length varies about 230 m. This strongly suggests that ignoring losses does not constitute
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