Current data networks rely on a mix of optical carriers and electronic circuitry. Signals are transported in the optical domain by fibers that support high bandwidths and low losses. Meanwhile, electronic devices distributed along the backbone provide the rest of the essential functions needed for a modern packet-switching network, such as routing, storing, and queuing the data traffic. The performance of such networks, however, is limited by the electronics, which have a maximum speed of only a few gigabits per second. To satisfy the increasing bandwidth requirements of emerging applications and to eliminate this bottleneck, we wish to replace the electronic circuitry with analogous optical systems that provide transparency and high bandwidth while consuming significantly less energy. Furthermore, we imagine a broad range of applications for such optical circuits beyond their use in networks, such as for optical computing or lightweight, highly integrated automotive and aeronautical monitoring.

Conventional optical circuits, however, must be large to reduce losses when guiding light around curves. Photonic crystals offer a handy alternative technology that allows us to design nanoscale circuits that can control the flow of light with innovative silicon patterns (see Figure figure 1). A photonic crystal is an engineered inhomogeneous periodic structure made of two or more materials with very different dielectric constants.^{1} It supports a complete photonic band gap: a frequency range within which no photons can propagate through the material. Irregularities (‘defects’) introduced in the structure form localized defect states, which allow light to be guided through the structure on a subwavelength scale with minimal losses.

**Figure 1. **Schematic view of basic functions that an optical circuit must be able to perform to manage the flow of light in a photonic crystal structure.

By introducing different forms of disorder in a photonic crystal, one can achieve more complex functions. Tailoring this disorder for a particular application, however, is extremely difficult. The traditional design approach starts with a proposed structure, then allows only a few degrees of freedom to vary and explores the whole landscape of values of these parameters. From this landscape, the most adequate structures are extracted. This approach is computationally costly and often impractical. On the other hand, one can elucidate the functions that a structure must fulfill, then use inverse design methods to find potential solutions. This is the approach that we are exploring.

We design photonic crystal circuits first by defining their optical functions. Then we use a combination of heuristic methods^{2, 3} and evolutionary algorithms,^{4} plus the well-known finite element method,^{5} to discover structures that best achieve these functions. Specifically, first we articulate the circuit's target properties, then we encode it as an objective function that measures the fitness of each candidate solution throughout the iterative process. Next, starting from a custom initial geometry, we let heuristic algorithms postulate some candidate solutions. These solutions are solved in the stationary linear regime using the finite element method, and they are sorted according to their fitness.

Our method handles all the variables involved in the design process at the same time using the advantages of heuristics, enhances the speed of the search process by exploiting the power of the finite element method, and can propose new geometries.

We use several heuristic methods (one at a time) as well as an evolutionary algorithm. These heuristic algorithms are based on dissimilar mechanisms. On the one hand, we use the improved harmony search algorithm^{2} inspired by the process of musical improvisation, in which musicians adjust the pitch or change a note to form an aesthetically pleasing melody. This algorithm treats the parameters involved in the optimization process as the notes of a musical score, and the set of degrees of freedom handled by the algorithm play the role of harmonies. Then the algorithm forces some changes in the pitch of these notes, i.e., it changes the values of the parameters according to stochastic operators that control the flow of the optimization process. In addition, it introduces randomization in the same way a musician improvises a new note in a jazz concert. We have also used the better-known fast simulated annealing (FSA) algorithm. FSA comes from the annealing process in metals, during which one heats and then slowly cools down the atoms, which then try to find the locations of minimum potential energy, in other words, a crystal structure. However, in the cooling process this method is allowed to perform uphill movements (in other words, it accepts more energetic configurations) according to the ‘Metropolis’ criterion. This may seem illogical, but Metropolis demonstrated long ago that these uphill movements enhance the search for the global optimum because they significantly reduce the probability of getting stuck in a local minimum.^{6} Moreover, the possible states an atom can access are controlled by a modified probability density function that speeds up the global optimization search process.

The design problem we address here constitutes a non-polynomial hard problem, that is, there is no way of solving it in finite polynomial time. Therefore, the computational time required for solving it using a direct brute-force search of all feasible solutions grows exponentially with the size of the photonic crystal device under consideration. Our method is faster because, unlike the direct approach, it does not explore all the possible solutions, but rather a subset of them dictated by the constraints we initially pose on the problem. Consequently, it finds the structures in a reasonable time using just a standard computer. To sum up, our method minimizes the computational effort and at the same time steers the search toward configurations that yield results as close as possible to our objectives.

To demonstrate the process, we designed a photonic device with low velocity dispersion suited for telecommunications applications. Our final design is a relatively complex circuit consisting of a photonic crystal structure that commutes signals between 1470 and 1560nm while operating almost single mode and with a near-zero velocity dispersion.^{7–9} The photonic circuit is composed of devices that were each designed separately using the inverse design method. Then we merged them into a more complex circuit that tackles light paths as depicted in Figure figure 2.

**Figure 2. **An optical circuit formed by several passive components designed in a photonic crystal cluster framework. At the cross-connect, possible paths split, labeled a or b. PCWG: Photonic crystal waveguide.

The initial photonic crystal was a triangular lattice of holes drilled on a silicon substrate with a spacing of 420nm. At the input port of the photonic circuit there is a tapering section that guides the light entering from a typical silicon waveguide into a splitter. One branch sends the light through a fully bidirectional wavelength division multiplexer and optical cross-connect that is compatible with chip-scale silicon photonics. The latter optical path consists of two parallel photonic crystal waveguide sections joined by an inverse-designed interference stage that matches the light modes in both waveguides. This device routes light, depending on wavelength, through either the exit port to the right, or the resonator to the left, or the sharp adjacent bends. In this latter case, light then goes through other elements that perform wavelength multiplexing and affect the velocity of the light beam. Interestingly, the area required for the whole device is only 24×12μm^{2}, much smaller than a standard microchip.

This circuit illustrates the use of inverse design methods to develop photonic crystal devices that implement the essential functions required in an optical integrated circuit. This design method is promising for a number of reasons. It provides a fast tool to identify a suitable photonic structure for a particular application, and it usually outperforms previous design methods based on trial and error. Also, it can produce a structure that fulfills even conflicting objectives. Finally, the calculated structures can be constrained so that they satisfy the limitations imposed by lithographic manufacturing techniques, which renders them buildable. The next step in our research will use this inverse design method to develop optical logic gates and to implement nonlinear effects in the photonic circuits.

*The authors thank the Basque Government for financial support under the SAIOTEK 2012 (ref. S-PE12UN043) program. I.A. thanks the Vicerrectorado de Euskara y Plurilinguismo de la UPV/EHU for financial support under the PhD Fellowships 2011 program.*

Imanol Andonegui, Angel J. Garcia-Adeva

Department of Applied Physics I

University of the Basque Country (UPV/EHU)

Bilbao, Spain

Imanol Andonegui received his degree in telecommunications engineering in 2008. He is currently a PhD candidate. His research focuses on photonic crystals, slow light, and electronic band gap materials.

Angel J. Garcia-Adeva is a professor of applied physics. He received his PhD in science in 1998. He was the the recipient of the 2007 Sturge Prize for his contribution to our understanding of the structure and dynamical properties of conventional and photonic crystals. His research is focused on photonic crystals and geometrically frustrated antiferromagnets.

References:

1. J. D. Joannopoulos, P. R. Villeneuve, S. Fan, Photonic crystals: putting a new twist on light, *Nature* 386, p. 143-149, 1997.

2. M. Mahdavi, M. Fesanghary, E. Damangir, An improved harmony search algorithm for solving optimization problems, *Appl. Math. Comput.* 188, p. 1567-1579, 2007.

3. H. Szu, R. Hartley, Fast simulated annealing, *Phys. Lett. A* 122, p. 157-162, 1987.

4. K. Deb, A. Pratap, S. Agarwal, T. Meyarivan, A fast and elitist multiobjective genetic algorithm: NSGA-II, *IEEE Trans. Evol. Comput.* 6, p. 181-197, 2002.

5. I. Andonegui, A. J. Garcia-Adeva, The finite element method applied to the study of two-dimensional photonic crystals and resonant cavities, *Opt. Express* 20, p. 4072-4092, 2013.

6. N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, Equations of state calculations by fast computing machines, *J. Chem. Phys.* 21, p. 1087-1092, 1953.

7. I. Andonegui, A. J. Garcia-Adeva, Inverse design and topology optimization of novel photonic crystal broadband passive devices for photonic integrated circuits, *4th Int'l Conf. Metamater. Photon. Cryst. Plasmon. (META '13)*, 2013.

8. I. Andonegui, I. Calvo, A. Blanco, A. J. Garcia-Adeva, Inverse design of novel nanophotonic structures, *15th Int'l Conf. Transpar. Opt. Netw.*, 2013. (Invited paper.)

9. I. Andonegui, I. Calvo, A. J. Garcia-Adeva, Advanced photonic crystal structures for nano-scale photonic integrated circuity, *Prog. Electromagn. Res. Symp. (PIERS 2013)*, 2013.