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The Design and Optimization of an Ion-Exchanged Polarization Converter Using a Genetic Algorithm Ismail M. Nassar, Hatem El-Refaei, Diaa Khalil, Senior Member, IEEE, and Omar A. Omar
Abstract—In this work, a genetic algorithm is used for the optimization of the design parameters of an integrated optical polarization converter on a glass substrate. The algorithm is applied to find the minimum possible length for the converter. The required asymmetry for the structure is realized with the aid of a cleaving process following the ion-exchange process. Simulation results show the possibility of realizing such a low-cost integrated converter. The obtained results demonstrate the importance of using such an algorithm in integrated optics problems where several design parameters such as the geometrical dimensions, the temperature, and the time duration are to be optimized.
Fig. 1. Proposed structure for the converter.
Index Terms—Diffusion equations, genetic algorithms, polarization.
technique for the diffusion equation and the boundary conditions are described. Simulation results are shown in Section V. Finally, the conclusion is mentioned in Section VI.
I. INTRODUCTION
I
ON diffusion into glass substrates enables the fabrication of cost-effective waveguides. Towards realizing an entirely integrated optical system, fabrication of several other optical components onto the same glass substrate is required. In this letter, we investigate the possibility of integrating a polarization converter with the shortest length. Such a converter is strongly required for future optical communication systems based on the polarization control. The design of a slanted angle polarization converter requires the optimization of several parameters. Genetic algorithms have been used for this purpose [1]. In order to couple an input quasitransverse-electric (TE) mode into a quasi-transverse-magnetic (TM) mode, this requires the fast and slow axes of the guide to have an angle of 45 with the input TE or TM mode. Also, for efficient power coupling between the modes, the number of guided propagating modes in the waveguide should be two [2], [3]. For a small device footprint, it is required to achieve the minimum beat length between the propagating modes. The waveguide could be realized by diffusing Ag ions or K into a glass substrate. The asymmetry causing rotation is then achieved by cleaving the diffused waveguide as shown in Fig. 1. A genetic algorithm [4] is used to optimize the temperature, time duration of the diffusion process, the location, and the angle of the cleaving process. The algorithm is designed to find the minimum beat length for the waveguide while satisfying both the rotation angle condition and the number of mode conditions. The letter is organized as follows: In Section II, a brief explanation for the proposed structure is included. In Section III, the genetic algorithm is explained. In Section IV, the solution Manuscript received September 29, 2006; revised April 20, 2007. The authors are with the Faculty of Engineering, Ain Shams University, Abbassia 11517, Cairo, Egypt (e-mail:
[email protected]). Color versions of one or more of the figures in this letter are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LPT.2007.901745
II. PROPOSED STRUCTURE Diffusing Ag ions or K ions into a glass substrate causes a slight increase in the refractive index of the glass. Such an increase is responsible for guiding the input mode. The maximum depends on the diffused ions and on the glass index change substrate. For Ag and K ions diffused into soda lime glass, is very low in the order of to [5], [6], which is one of the main problems in designing the converter because of the low confinement of the guided mode. The asymmetry of the field pattern of a mode decreases as confinement decreases. Asymmetry in the structure of the guide causes the input mode to rotate. Simulation showed that a larger change in refractive index on one side of the structure might be necessary to achieve the required asymmetry. This necessitates the cleaving process or an etching process while having control on the angle of the glass–air interface. The proposed structure is based on cleaving a part of the diffused waveguide with a specific angle, as shown in Fig. 1. Four design parameters were scanned for the minimum beat length. These parameters are the position of the cleaving , measured from the center of the mask, the cleaving angle , the temperature of the diffusion process , and the time of the diffusion. III. GENETIC ALGORITHM The genetic algorithm used for this investigation is combined with a semivectorial mode solver and a diffusion equation solver. All possible values for each one of the four design parameters of the waveguide are listed in Table I. The width of the mask opening was fixed at 4 m. The algorithm works as follows: A first generation consisting of ten different entities was set. Each entity, which is a combination of parameter values, was found by randomly assigning any possible value to each of the design parameters. For each one of these entities, the diffusion process was simulated according to its diffusion
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NASSAR et al.: DESIGN AND OPTIMIZATION OF AN ION-EXCHANGED POLARIZATION CONVERTER
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TABLE I PARAMETER VALUES USED BY THE GENETIC ALGORITHM
parameters. The resulting index profile was then passed to the mode solver to determine the rotation angle, the number of propagating modes, and the beat length. According to the beat length, the entity is assigned a probability of being included in the next generation. An entity with a number of modes other than two modes or a rotation angle less than 40 or greater than 50 was given a probability of zero. After assigning each entity its probability of survival, the next generation is found. Typical genetic algorithm operations such as crossover between the surviving entities of the next generation and mutation were also performed. Crossover enables exchanging parameter values between surviving entities. Mutation is the operation of randomly changing the value of any parameter to any other value included in the value set of the parameter. Mutation avoids resulting in local minima. The same procedure is repeated for several generations until it is recognized that shorter beat lengths no longer appear. This procedure requires running the diffusion equation solver and mode solver once for each entity. In order to save processing time, the results are saved for every entity, so that it is not recalculated for the same surviving entity in the next generation. Saving the results causes the calculations to be performed once for each entity during the first generation, then once for every new entity during the proceeding generations, which occurs very few times depending on the crossover and the probability of mutation. For a probability of mutation of 0.05 and a number of entities in a generation equal to 10, the design with minimum length could normally be found with an accuracy of less than 1% requiring around 10–20 generations while processing around 15–30 different entities out of 294 different combinations of parameter values according to Table I, which is the entire space being searched. This corresponds to a simulation time of 20 to 30 min on a 1.73-MHz processor.
Fig. 2. Beat length dependence on diffusion time.
Using the finite-difference method, this equation can be solved explicitly by finding the concentration one time step after the other. Stability is assured by taking a time step given by (2) Unless is very close to unity, the time step results in reasonable simulation time. Image theory and the symmetry of the resulting diffusion index profile before cleaving is used to save processing time by restricting the window to one half of the structure then mirror imaging the resultant profile, after which cleaving is performed by simply replacing the cleaved part with ones as a refractive index. The difference in the refractive index caused by diffusion is relatively very small. This results in a weakly guided mode which means that the field pattern occupies a region much larger than the diffused waveguide. This necessitates a window much larger than the waveguide. In this case, the boundary condiat the window borders intions can simply be taken as side the glass. At the glass–air interface, on the other hand, the . The validity of boundary conditions are taken as these boundary conditions was verified by repeating some simulations with twice the window size. While the beat length error did not exceed 0.4%, the simulation time was multiplied by a factor of 12 on average.
IV. NONLINEAR DIFFUSION EQUATION
V. SIMULATION RESULTS
Including the difference in the values of the diffusion conand the original ions , the stants of both the incoming diffusion equation becomes nonlinear and is given by [7]
For the parameter set in Table I, the algorithm is run as a standard genetic algorithm on the first three parameters, resulting in three-dimensional space to be searched. For each entity (point in the space) with a certain temperature, time, and cleaving position, a complete sweep for all values of the cleaving angle is performed until the minimum beat length satisfying the rotation angle condition and the number of mode conditions is found. The cleaving angle is separated from the algorithm because the problem has extra constraints which are not typical in standard genetic algorithms. Taking and for Ag ions [8], the mask width was fixed at 4 m. The algorithm found a minimum beat length of about 610 m, a conversion efficiency of 86%, and an insertion loss C, s, m, and of 1.9 dB for . Parametric sweeps at these values are shown in Figs. 2–4, where a filled circle shows a valid entity satisfying the rotation
(1) where ; is the normalized concentration of the new ions; is the applied electric field; is the absolute temperature; is Boltzmann’s constant.
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Fig. 6. Field patterns and refractive index profiles. Fig. 3. Beat length dependence on temperature.
In Figs. 2 and 3, the algorithm found the minimum beat length for certain time and temperature values, respectively. In Fig. 4, on the other hand, the algorithm was bound by the number of mode conditions which could not be satisfied for m, which corresponds to a larger waveguide cross section allowing for more modes to propagate. In Fig. 6, contour lines for the -component and -component for the electric fields of the two modes are shown. The patterns belong to the minimum length converter. Similarity of the field patterns is due to the 45 rotation angle. At the right of Fig. 6, the solution of the diffusion equation for the minimum length converter is shown. VI. CONCLUSION
Fig. 4. Beat length dependence on the position of cleaving.
Simulation results show that etching or cleaving a part of the diffused waveguide, with good control of the cleaving angle, can result in an asymmetry sufficient to obtain optical axis rotation of 45 . The obtained results demonstrate the strength of a genetic algorithm in optimizing the structure. Adding new parameters, such as the insertion loss, to the optimization merit function could also be used to improve the structure performance. This will be the objective of future work. REFERENCES
Fig. 5. Beat length dependence on the cleaving angle.
angle and number of mode conditions, while an empty square stands for an invalid entity. The sweeps verify the existence of a minimum. Also, the sweep in Fig. 5 shows the dependency of the beat length for different wavelengths of the light source. As shown, the beat length increases monotonic with the wavelength and also increases monotonic with the cleaving angle.
[1] D. Correia, J. P. da Silva, and H. E. Hernandez-Figueroa, “Genetic algorithm and finite-element design of short-section passive polarization converter,” IEEE Photon. Technol. Lett., vol. 15, no. 7, pp. 915–917, Jul. 2003. [2] H. El-Refaei, D. Yevick, and T. Jones, “Slanted-rib waveguide InGaAsP–InP polarization converters,” J. Lightw. Technol., vol. 22, no. 5, pp. 1352–1357, May 2004. [3] H. Deng, D. O. Yevick, C. Brooks, and P. E. Jessop, “Design rules for slanted-angle polarization rotators,” J. Lightw. Technol., vol. 23, no. 1, pp. 432–445, Jan. 2005. [4] D. E. Goldberg, Genetic Algorithm in Search, Optimization, and Machine Learning. Reading, MA: Addison-Welsey, 1989. [5] S. I. Nagafi, Introduction to Glass Integrated Optics. Norwood, MA: Artech House, 1992. [6] G. L. Yip and J. Albert, “Characterization of planar optical waveguides by K+-ion exchange in glass,” Opt. Lett., vol. 10, no. 3, pp. 151–153, Mar. 1985. [7] J. Albert and J. W. Y. Lit, “Full modeling of field assisted ion exchange for graded index buried channel optical waveguide,” Appl. Opt., vol. 29, no. 18, pp. 2798–2804, Jun. 1990. [8] G. Stewart, C. A. Millar, P. J. R. Laybourn, C. D. W. Wilkinson, and R. M. DeLaRue, “Planar optical waveguides formed by silver-ion migration in glass,” IEEE J. Quantum Electron., vol. QE-13, no. 4, pp. 192–200, Apr. 1977.
