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Designing photonic crystal cavities with high quality factors and low mode volumes is of great importance for maximizing interactions of light and matter in metamaterials. Previous work on photonic crystal cavities has revealed dramatic improvements in performance by fine-tuning the device design. In L3 cavities, slight shifts of the holes on the edge of the cavity have been found to greatly increase quality factors without significantly altering the mode volume. Here we demonstrate utilizing a nature inspired search algorithm to efficiently explore a large parameter space. The results converge upon a new cavity model with a high quality factor to mode volume ratio (Q/V = 798,000 (λ/n)−3).
© 2013 Optical Society of America
1. Introduction
Photonic crystals are novel metamaterials arising from periodic variations in material optical properties and allow researchers to create mirrors, waveguides and cavities on length scales comparable to the photon wavelength [1].
Optical nanocavities have been used extensively in studying light-matter interactions such as Purcell enhancement in quantum dots and second harmonic generation in GaAs [1–4]. By modifying the local density of optical states, they can enhance or suppress the spontaneous emission of quantum emitters [3,4]. This leads to a host of interesting physics with far-ranging applications from quantum information processing [5] to improved laser systems [4–7]. Ideally these systems have high quality factors in small mode volumes [2–8], but there is a large parameter space to explore to find the best possible cavity configuration.
Here we demonstrate a novel approach for cavity design and use, as a starting point, a cavity design that has been used extensively in recent years. Of particular interest are thin slabs of high permittivity material containing a hexagonal lattice of air holes. This planar photonic crystal design has advantages including compatibility with epitaxial growth and cleanroom fabrication techniques and the ease of designing in-plane waveguides, etc., for scalability. By filling in three linearly adjacent holes an optical cavity can be created, referred to as an L3 cavity [9].
Improvements to the quality factor to mode volume ratio (Q/V) have been shown by shifting the holes on the edge of the cavity outwards [8,10]. We expect the optical confinement of the cavity to also be influenced by the other holes bordering the top and bottom of the cavity. In this article we study if shifting the other holes adjacent to the cavity or changing their sizes might also increase Q/V.
Assuming that the ideal combination of hole placement and size will vary along the lines of symmetry of the cavity, we adjust the holes labeled A, B, and C in Fig. 1. Each of the holes can vary in radius and position along the x-direction. Holes labeled B and C are also allowed to move in the y-direction. This leads to a total of eight parameters to study. Realistic estimates of the necessary step sizes to fully explore this problem yield a parameter space on the order of 108-1010 entries. The computational time needed to evaluate each of the entries in this parameter space is beyond practical limits with current technology. As with other computationally complex problems, we use a more elegant approach to explore the parameter space.
Fig. 1 Photonic crystal cavity design parameters. The basic L3 design is modified by shifting the edge holes labeled “A” symmetrically along the x direction. Holes B and C are allowed to shift in both the x and y directions. All labeled holes are allowed to vary in radius.
We consider the parameters of our L3 cavity as inputs to some unknown deterministic fitting function, which in this case is the Q/V ratio. We determine the Q/V ratio by performing finite difference time domain (FDTD) simulations and cannot derive a useful heuristic for exploring the parameter space. This maps our problem to an uninformed search, which is in general computationally complex [11].
A number of new approaches to studying complex problems have been inspired by natural systems. Evolutionary and genetic algorithms have been used to design radio telescopes [12]. Studies of ant foraging have improved network router performance [13]. In nanophotonics, evolutionary algorithms have found bowtie shaped resonators with ultra-small mode volumes [14] and improved the band gap of photonic crystals [15]. Random walk has improved absorption in photovoltaics [16]. Stochastic optimization has improved robust, slow-light tapers [17]. In our work we use a new approach inspired by gravity and referred to as the gravitational search algorithm (GSA) [18].
2. Gravitational search algorithm
In the GSA we randomly place “agents” in our eight dimensional parameter space and evaluate each of them for their quality factor and mode volume. According to each Q/V ratio we assign a “mass” to each agent such that larger mass is linearly associated with higher Q/V. Every agent is then allowed to accelerate through the parameter space due to the masses and positions of the other agents for one time step. We then reevaluate each agent for its new Q/V ratio, assign a new mass, and repeat the process. The system evolves until it converges upon an optimized solution.
We begin by defining our parameter space based on the radii and shift in position of the holes in Fig. 1. This gives us a total of 8 parameters: rA, rB, rC, ΔxA, ΔxB, ΔxC, ΔyB, and ΔyC. For our current application, our parameters have consistent units, so calculating distances in parameter space from one agent to another can be straightforward. However we wish to fully disassociate our search algorithm from our fitness function and be able to use it for inputs of arbitrary types and scales. For each agent, i, we shift to the dimensionless parameters
where rmin and rmax are 0.1 a and 0.5 a, respectively, and a is the photonic crystal lattice parameter. All other parameters are scaled accordingly. For Δx (similarly for Δy) we have Δxmax = 0.5 a and Δxmin = −0.5 a. It is important to note that the choice of the minimum and maximum values will affect the scaling of our dimensionless parameters. We want the extrema large enough to encompass all possible optimized solutions yet as small as possible to improve convergence time and not miss small features in parameter space. If care is not taken in their choice and the global optimal solution is outside the initial placement, the system may converge on a local rather than global optimization. There is no limitation in the algorithm to stay within the regions 0≤si,j≤1 during evolution. Agents can and often do travel outside these bounds.We begin our simulation by giving 48 agents random initial position and velocity vectors. For each agent we perform FDTD simulations (described below) to determine the quality factor and mode volume for each point in parameter space. We define our fitness function, , as the final Q/V ratio as determined by the FDTD simulation.
After determining the fitness function for each agent, we assign a mass to each agent. It is important to note that the mass of an agent will change at each iteration of the GSA and it does not take into account any previous values (non-memory search). We assign the masses according to
where fmin is the minimum value of Q/V at the current iteration and δm is a small constant (10−6). Based on each agent’s position and assigned mass we can now calculate the acceleration through parameter space of each agent due to the other agents. We update the agent’s kinematics and calculate their new Q/V. This process is iterated until results converge.There are several features of the GSA that differ from calculation of true gravitational acceleration. The magnitude of the acceleration of each agent depends only on the masses of the other agents and not the distances between them. The positions in s space determine only the direction of the acceleration. In nature’s gravity, the strength of the interaction decreases as the square of the distance [19]. While this is a very significant difference between the two, previous studies have found the distance independent form to have faster convergence [18].
Additionally, the gravitational constant G is not held constant for our simulations. For large G agents are strongly attracted to one another and they quickly traverse large distances, especially those with low fitness function values (low mass). Early in the simulation this is preferable since we do not want to spend much time looking through regions of low interest. For small G, agents are less affected by one another and allowed to roam farther along their current velocities before being pulled towards the bulk of the other agents. By allowing agents more freedom to move away from the bulk, it allows local exploitation of the parameter space to fine tune the optimized solution. We model our constant G as decaying exponentially over the lifetime of the simulation.
Figure 2(a) shows the values of one parameter, rA, in our simulation for all 48 agents over 400 iterations. Each agent is shown in a different color. Early in the evolution of the system, the agents move large distances in rA. After 300 iterations they begin to converge on our optimized solution. Similar graphs of the other parameters all show the same trends. With 48 agents we have run 1.5x104 FDTD simulations, a significant improvement over the estimated 108 calculations required to fully explore the parameter space. However, there is no guarantee that the solution we have found is the global optimal solution.
Fig. 2 Evolution of the hole radius, rA, and Q/V ratio through the simulation. After approximately 200 iterations results begin to converge upon optimized solution. Horizontal line in Q/V represents benchmark for comparison.
3. Photonic crystal cavity results
The GSA is a fascinating approach to solving complex problems, but we must have some metric to determine if its results are actually any improvement over existing techniques. To compare our results we consider a common configuration of the L3 cavity in which the two holes on the edge of the cavity are shifted out by 0.15 a [8]. We evaluate our designs by calculating the quality factor of the cavity and the following definition of the mode volume [20].
FDTD simulations were performed using the MEEP c + + libraries [21]. The lattice surrounding the PC cavity was a triangular lattice of air holes in a GaAs slab. The radii of the holes and slab thicknesses were 0.3 a and 0.6 a, respectively. For the GSA iterations, simulations were performed at 10 pixels per lattice parameter with subpixel averaging. The PC cavity is surrounded by 8 holes, including those labeled in Fig. 1. Perfectly matched layers of thickness 1 a surround the entire computational grid. To check the accuracy of the final results, the calculations were performed using a resolution of 40 pixels per lattice parameter and 12 holes around the cavity.
Using our material parameters, we find this configuration has an Ey anti-node at the center of the cavity with a quality factor of approximately 68,000 and mode volume of 0.74 (λ/n)3. This Q/V of approximately 92,000 (λ/n)−3 is an extraordinary accomplishment for photonics and it is the benchmark to which we compare our results. Table 1 shows the final results from our parameter search. The mode volume of this cavity is 0.71 (λ/n)3, which is not a significant improvement over the benchmark. However, the quality factor of this cavity is 567,000, leading to Q/V of approximately 798,000 (λ/n)−3. This is a significant improvement and an excellent demonstration of how these nature inspired algorithms can find useful solutions.
Table 1. Optimized design parameters for cavity with Q/V = 798,000 (λ/n)−3
Figure 2(b) shows the evolution of the average Q/V values at each iteration and Fig. 3 shows the electric field (|E|2) mode profile of our final cavity design and the benchmark L3 with 0.15a shift [8]. The fundamental mode frequency for both cavities is 0.26 c/a. The B and C holes stay in roughly the same x location but shift toward the cavity center in the y direction. The shift is greater for the holes nearer the center. However holes A move slightly outward. All holes decrease in size.
Fig. 3 a) Electric field (Ey) mode profile for our final design parameters and b) the L3 cavity with 0.15a shift, which is our benchmark.
We believe there are two competing effects here that when properly balanced give our optimized result. First, by moving holes toward the cavity center the regions of low permittivity force the mode inward toward the center of the cavity, not unlike how H1 cavities have much smaller mode volumes than L3 [22]. However bringing these holes closer to the center will tend to decrease the quality factor since we are making a more abrupt change in the transition from the cavity to the lattice [8]. Second, by decreasing the radii of the holes we are creating a more gradual change from the cavity to the photonic crystal lattice, increasing the quality factor. This allows the mode to penetrate further into the lattice increasing the mode volume. Ultimately we have these two competing effects, so finding some combination of the two should yield an optimized cavity design. This is the solution our search discovers.
4. Conclusion
In conclusion, we have designed a new photonic crystal cavity structure with Q/V = 798,000 (λ/n)3. By treating the Q/V ratio of a cavity as the fitness function of an uninformed search, we allow a nature-inspired algorithm to explore the parameter space defining the cavity.
Acknowledgments
This material is based upon work supported by the National Science Foundation under Grant No. ECCS-0844908, and the Materials Research Science and Engineering Center program DMR-1120923.
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