Inverse Design of Tapers by Bio-Inspired Algorithms

The efficiency of optical power transferred between two bidimensional waveguides, continuous waveguide and periodic subwavelength waveguide has been designed and optimized by using a taper composed by variable length segments (0.03 – 0.27 μm). Waveguides with variable width (270 – 330 nm) have been considered and the number of segments on the taper region has been varied from 10 to 15. The optimized taper was efficiently designed using bio-inspired algorithms based on the genetic algorithms and the artificial immune system. The power coupled has been computed using the frequency domain finite element method. The best solution is 20% shorter than previously existent tapers with coupling light efficiency above 90% over a broadband interval of frequencies and it exhibits high fabrication error tolerances.

I. INTRODUCTION The taper waveguide coupler is a passive device composed by a photonic waveguide whose shape or transversal cross section varying along the direction of propagation [1]. Waveguide couplers are generally used to improve or enhance the light coupling efficiency between two waveguides with different transversal sections propagation characteristics. For tapers with long length, the light can be coupled with negligible power loss into the output waveguide [2]- [4]. A drastic reduction of the taper's size will increase the power loss [1], [3].
A good alternative to increase the coupling efficiency between continuous waveguides and periodically segmented waveguides (PSW) is to use a segmented waveguide as a taper waveguide coupler by varying its geometric parameters (length size). One of the main advantages of working with PSW's is the possibility of manipulating the core effective refractive index along the optical circuit. Another desired property is the confinement of light in the PSW filters higher orders modes [5], [6]. In [4]- [8], it is proposed the use of periodically segmented waveguides to change the effective Inverse Design of Tapers by Bio-Inspired Algorithms guide index and therefore manipulation the field distribution within waveguides.
The aim of this work is to extend the analysis presented in [1], where the number of segments and the segment width were considered constant and the size of taper region was fixed to 4.5 µm, which is smallest than other previously published. Here, the number of segments in the taper region will be swept from 10 to 15 making the structures of this work ranging from 3.0 to 4.5 µm being smaller than those found in [1]. Another contribution of this work is the analysis of the coupling efficiency varying the segments width in the values of 270 nm, 300 nm and 330 nm. The influence of these parameters is analyzed in details aiming to obtain a robust optimized taper with a good tolerance of fabrication errors and a footprint smaller than previously designed tapers.
For the numerical modeling of these tapers, the FEM-2D formulation [9]- [12], is used to compute the field distribution and hence the coupled power can be obtained. On the other hand, the genetic algorithm (GA) and the artificial immune system (AIS) were used for the optimization of several photonic devices such as photonic crystals [11], [12], tapers in conventional waveguides [13], [14], and segmented tapers [1].
Some advantages of the models obtained with AIS and GA are the reduced time and computational effort, its simplicity and also its use in design and synthesis problems [1]. The GA and AIS parameters such as: Individual numbers, number of generations, mutation rates, and affinity have been adopted to be the optimized ones obtained in [1].
The structure to be optimized is displayed in Fig. 1  To demonstrate the validity and usefulness of artificial immune system and the genetic algorithm for this kind of problem, the evolution and the shape of the best tapers are presented.

II. BIO-INSPIRED ALGORITHMS
The bio-inspired algorithms based on the AIS and the GA, reported in [1], are used here and they are shown in Fig. 2. The fundamental Ex polarized mode of the CWG is considered as the incident field of the input taper. We considered three different widths for the CWG, 270nm, 300nm and 330nm. The resulting FEM meshes have about 55,000 elements and 110,000 nodes.
The fitness evolution of the best individuals by using AIS and GA, varying the number of segments (10 to 15) and several width w (270, 300 and 330 nm) as a function of the number of generations is shown in Fig. 3. In all cases, the algorithm based on artificial immune system and genetic algorithms presented good convergence.
It is observed that high values of coupling efficiency are obtained with the AIS and GA with a small number of segments n, almost independent of the width w. Table I    The best optimized tapers obtained by AIS with their respective number of segments (10 to 15) and width w (270, 300 and 330 nm), are displayed in Fig. 4, where we also show the spatial distributions of the field. In Fig. 4(a), it can be observed that high power efficiency is coupled into the taper when the waveguide width is 270 nm.    Fig. 7, we presented the curves of the coupling efficiency as a function of the wavelength for the best structures optimized by using AIS and GA varying the number of segments n (10 to 15) and the width w (270, 300 and 330 nm). A variation on the coupling efficiency of about 10% can be observed, but they are still high (above 83%).
In addition, the robustness of the designed and optimized tapers with w=270nm, 300nm and 330nm, is analyzed by varying (increasing and decreasing) the size of all segment lengths, of the best tapers founded, up to 10nm. This tolerance is about 10 times the step length used for the segments and it is also compatible with the existent fabrication technologies. The results are displayed in Fig. 9. A variation in coupling efficiency of less than 10% can be observed. A 0.5% improvement in coupling efficiency is noted when all segments increase 2nm. The AIS obtained the best results for w=270 nm while the GA obtained the best results for w=270 nm and w=330 nm.
A high coupling efficiency can be observed for the tapers designed using AIS and GA for various wavelengths in this study. Previous reported tapers presents lower coupling efficiency of 81.4% in [4] while tapers found in [1] and in this work using the AIS and GA presents coupling efficiency above 94%.
The coupler waveguides obtained by the bio-inspired algorithms can be used to design low loss and efficient tapers.  Consequently, the 2D approach can be used to obtain an insight of the behavior of 3D taper.
Moreover, the wavelength dependence of the GA optimized waveguide, under the effective index theory, with width w=270 nm is presented in Fig. 8. It can be seen that the coupling efficiency is above 96% in all the interval of wavelengths.  We can observe that both algorithms converge to values above 96% after running them ten times

IV. CONCLUSIONS
In conclusion, the optimization of planar waveguide couplers using PSW composed by silicon and silica, using the 2D-FEM in association with bio-inspired algorithms based on the AIS and the GA is presented. The results presented are better than the ones previously reported in literature considering sizes and materials with refractive indexes close to the ones used in this work.
It can be observed a pattern in the optimized structures: the closest segments of the input waveguide and the output waveguide have a larger filling fraction of silicon and the segments closer to the center of the taper have more silica.
A new set of tapers are presented, varying the number of the segments (10-15) and varying the width (270, 300 and 330 nm) and the optimized structures with w=270 nm show better results than the others presented in [1] and [4]. The footprint of the best solution is 20% smaller than previously ones reported in the literature.
Coupling efficiencies above 94% have been found for specific operating wavelengths and above 84% for the wavelength interval [1.5 μm -1.6 μm]. In addition, the designed models can be used to design optical waveguide couplers considering different materials, number of sections, period variation, different waveguides and 3D simulations. The proposed solutions exhibit high fabrication error tolerances.