Genetic Optimization of Photonic Bandgap Structures
Genetic Optimization of Photonic
Bandgap Structures
Joel Goh, Ilya Fushman, Dirk Englund, Jelena Vučković
Ginzton Laboratory, Stanford University, Stanford, CA 94305, USA
joelgoh@stanfordalumni.org; ifushman@stanford.edu; englund@stanford.edu; jela@stanford.edu
Abstract: We investigate the use of a Genetic Algorithm (GA) to
design a set of photonic crystals (PCs) in one and two dimensions. Our
flexible design methodology allows us to optimize PC structures which
are optimized for specific objectives. In this paper, we report the results of
several such GA-based PC optimizations. We show that the GA performs
well even in very complex design spaces, and therefore has great potential
for use as a robust design tool in present and future applications.
© 2018 Optical Society of America
OCIS codes: (130) Integrated optics; (130.2790) Guided waves; (130.3210) Integrated optics
devices; (140) Lasers and laser optics; (140.3410) Laser resonators; (140.5960) Semiconductor
lasers; (230) Optical devices; (230.5750) Resonators; (230.6080) Sources; (250) Optoelectron-
ics; (250.5300) Photonic integrated circuits; (260) Physical optics; (260.5740) Resonance;
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1. Introduction
Photonic crystals (PCs) describe a class of semiconductor structures which exhibit a periodic
variation of refractive index in 1, 2, or 3 dimensions. As a result of this periodic variation,
PCs possess a photonic band gap – a range of frequencies in which the propagation of light
is forbidden [1, 2]. This is the analog of the electronic bandgap in traditional semiconductors.
This unique characteristic of PCs enables them to be used to effectively manipulate light. PCs
have already been used for applications such as modifying the spontaneous emission rate of
emitters [3, 4], slowing down the group velocity of light [5, 6], and designing highly efficient
nanoscale lasers [7].
Given that Photonic Crystals find applications in a myriad of areas, we proceed to investigate
the question: What is the best possible PC design for a given application? Traditionally, the
design of optimal PC structures has been largely done by either trial-and-error, iterative searches
through a design space, by physical intuition, or some combination of the above methods [8, 9].
However, such methods of design have their limitations, and recent developments in PC design
optimization have instead taken on a more systematic and algorithmic nature [10, 11, 12, 13].
In this work, we report the results of a Genetic Algorithm to optimize the design of a set of
one and two-dimensional PC structures. We show that the Genetic Algorithm can effectively
optimize PC structures for any given design objective, and is thus a highly robust and useful
design tool.
2. Genetic Algorithms
Genetic Algorithms (also known as Evolutionary Algorithms) are a class of optimization algo-
rithms that apply principles of natural evolution to optimize a given objective [14, 15, 16]. In
the genetic optimization of a problem, different solutions to the problem are picked (usually
randomly), and a measure of fitness is assigned to each solution. On a given generation of the
design, a set of operations, analogous to mutation and reproduction in natural selection, are per-
formed on these solutions to create a new generation of solutions, which should theoretically
be “fitter” than their parents. This process is repeated until the algorithm terminates, typically
after a pre-defined number of generations, or after a particularly “fit” solution is found, or more
generally, when a generation of solutions meets some pre-defined convergence criterion.
3. Implementation
Genetic Algorithms have already been used in PC design - to find non-intuitive large-bandgap
designs [12, 17] and for designing PC fibers [18]. In our work, we performed the genetic opti-
mization by varying the sizes of circular holes in a triangular lattice. This approach was chosen
because the search space is conveniently constrained in this paradigm, and the optimized struc-
tures can be easily fabricated, if desired. A freely available software package [19] was used to
simulate the designed structures.
In addition, we used the following parameters for the implementation of our Genetic Algo-
rithm:
Chromosome Encoding. We used a direct-chromosome encoding, where the various opti-
mization parameters were stored in a vector. For the current simulations, for simplicity,
we only varied the radii of cylindrical holes in a triangular lattice. Our implementation
can be easily modified to include other optimization parameters as well, such as the po-
sitions of the various holes, or the refractive index of the dielectric material.
Selection. We used fitness-proportionate selection (also known as roulette-wheel selection),
to choose parent chromosomes for mating. In this selection scheme, a chromosome is
selected with a probability Pi that is proportional to its fitness fi, as shown in Eq. (1).
Mating. After a pair of parent chromosomes vparent,1 and vparent,2 were selected, they were
mated to produce a child chromosome vchild by taking a random convex combination of
the parent vectors, as in Eq. (2).
λ ∼ U(0,1)
~vchild = λ~vparent,1 +(1−λ )~vparent,2 (2)
Mutation. Mutation was used to introduce diversity in the population. We used two types of
mutation in our simulations, a random-point crossover and a gaussian mutation.
1) Random-point crossover: For an original chromosome vector ~vorig of length N, we
select a random index, k, from 0 to N as the crossover point, and swap the two halves of
~vorig to produce the mutated vector,~vmut , as represented in Eq. (3).
~vorig = (v1,v2, . . . ,vN)
k ∼ U{0,1,2, ....,N}
~vmut = (vk+1,vk+2, . . . ,vN ,v1,v2, . . . ,vk−1)
T (3)
2) Gaussian mutation: To mutate a chromosome vector by Gaussian mutation, we define
each element of ~vmut to be independent and identically distributed Gaussian Random
Variables with mean ~vorig and a standard deviation proportional to the corresponding
elements of ~vorig. This searches the space in the vicinity of the original chromosome
vector~vorig.
vmuti ∼ N
vorigi ,σ
, i ∈ {0,1,2, ....,N} (4)
σ2 is a algorithm-specific variance, and can be tuned to change the extent of parameter-
space exploration due to mutation.
Cloning. To ensure that the maximum fitness of the population was would never decrease, we
copied (cloned) the top few chromosomes with the highest fitness in each generation and
inserted them into the next generation.
4. Simulation Results
4.1. Optimizing Planar Photonic Cavity Cavities
4.1.1. Q-factor Maximization
One problem of interest in PC design is the inverse problem, where one tries to find a dielectric
structure to confine a given (target) electromagnetic mode. Here we consider the inverse design
problem of optimizing a linear-defect cavity in a planar photonic crystal cavity. The Q-factor is
a common figure of merit measuring how well a cavity can confine a given mode, and can be
approximated (assuming no material absorption) by the following expression:
Qtotal
where Q|| represents the Q-factor in the direction parallel to the slab, and Q⊥ represents
the Q-factor perpendicular to the slab. Q⊥ is usually the limiting factor for Qtotal. As was
shown previously [10, 20], the vertical mode confinement, which occurs through total internal
reflection (TIR), can be improved if the mode has minimal k-space components inside the light
cone.
In the subsequent sections, we report the results where we employed our GA to minimize
the light cone radiation of such cavities. We used one-dimensional photonic crystals as approx-
imations to these cavities [21], and simulated these cavities using the standard Transfer Matrix
method for the E-field [22].
4.1.2. Matching to a Target Function
In [10] it was noted that minimization of light cone radiation could be performed via mode-
matching to a target function which already possessed such a property. We therefore used a
fitness function that was equal (up to a normalizing factor) to the reciprocal of the mean-squared
difference between our simulated mode and a target mode (see Eq (6)). For this simulation, our
chromosome encoded the thicknesses of the dielectric slabs in the structure, and was a vector
of length 10. We used 100 chromosomes in each generation and allowed them to evolve for 80
generations.
f itness ∝
| fsim(x)− ftarget (x)|
We used target modes that were sinusoidal functions multiplied by sinc and sinc-squared
envelope respectively. Such target modes have theoretically no radiation at or near the Gamma
point and are therefore ideal candidates as target functions. The results, shown in Fig. 1, clearly
feature a suppression of k-vector components at low spatial-frequencies. Matching using the
the sinc-squared envelope target function appeared to produce a better match. From the k-
space plots, the GA evidently had difficulty matching the sharp edges for the sinc-envelope
target mode.
4.1.3. Direct Minimization of Light Cone Radiation
In the preceding subsection, we observed that when we formulated our objective as a matching
problem, in the case of the sinc-envelope, the GA sacrificed the desired low spatial-frequency
suppression in an effort to match the overall shape of the function. The preceding formulation
therefore poses an implicit constraint on our optimization. By reformulating the optimization
problem, we were able to effectively remove this constraint, and obtain a better result.
0 100 200 300 400 500
−3 −2 −1 0 1 2 3
k (a/λ)
0 100 200 300 400 500
−3 −2 −1 0 1 2 3
k (a/λ)
Fig. 1: Top-left: Real-space mode profile after optimizing for closest-match to a sinc-envelope
target mode. Top-right: k-space mode profile of optimized simulated mode and a sinc-envelope
target mode. Bottom: Real-space and k-space mode profiles for matching against a sinc2-
envelope target mode.
Our reformulation directly minimized the k-vector components in the light cone, by min-
imizing the integrated square-magnitude of the simulated E-field mode in k-space inside the
light cone. The fitness function that we used is given as in Eq (7), where V represents the set of
k-vectors within the light cone.
f itness =
|F(k)|2dk
The final, evolved structure, together with the corresponding real-space and k-space mode
profiles are shown in Fig 2. The k-space mode profile features a strong suppression of radiation
at low frequencies, to a greater extent as compared to the optimized fields from the preced-
ing simulations. By relaxing our constraint and performing a direct optimization, our GA has
designed a structure that achieves better light cone suppression than before. Our direct opti-
mization paradigm has exploited the extreme generality of the GA, which simply requires that
a fitness function be defined, with little further constraint thereafter.
4.2. Maximal Gap at any k-vector Point
Moving on to the more generic case of 2D photonic crystals, we will proceed to show the results
of simulations for maximizing the TE bandgap at any point in k-space for a 2-Dimensional PC
structure with a triangular lattice of air holes. This could be useful for PC design applications
where the target mode to be confined is centered around a particular point in k-space [10]. By
maximizing the bandgap at that k-space point, we would effectively design a better mirror for
a mode resonating along this k-space direction.
0 50 100 150 200 250 300 350 400 450
Optimized Mode − real space
−3 −2 −1 0 1 2 3
k (a/λ)
Optimized Mode − k space
Fig. 2: Top: Real-space mode profile of optimized resonant E-field mode. Bottom: Correspond-
ing k-space mode profile of optimized mode
We used a supercell which was three periods wide in each dimension and varied the radii of
the nine holes in total, and we encoded the chromosome as a vector of these nine holes. We
used a population size of 60 chromosomes for each generation, and allowed the optimization to
run for a total of 100 generations.
To evaluate the fitness of each chromosome, we used the eigensolver in Ref [19] to calculate
the gap-to-midgap ratio at the K-point of the band diagram. We then scaled the calculated ratio
exponentially to tune the selection pressure of the optimization. Figure 3 shows the variation of
the gap-to-midgap ratio of our structures as the algorithm progressed.
Our Genetic Algorithm performs as expected, and we get a general increase of fitness as
the algorithm progresses. All the four runs do not show any significant increase in fitness af-
ter Generation 80, at which point they have maximum fitnesses (i.e. ratio of their bandgap to
midgap value) of around 72%. All the optimized structures after the run have similar dielectric
structures and band diagrams. The dielectric structures and a sample band diagram is shown in
Figure 4.
4.3. Optimal dual PC structures
As a more complex example, let us consider two similar PC designs, (1) a triangular lattice of
air holes in a dielectric slab, and (2) a triangular lattice of dielectric rods in air. Structure (1)
possesses a bandgap for TE light, but no bandgap for TM light, while structure (2) possesses a
bandgap for TM light, but not for TE light.
Our objective is to use the Genetic Algorithm to find a PC design in which the TE eigenmode
for structure (1) and the TM eigenmode for structure (2) are most similar. Maxwell’s equations
can be cast as eigenproblems for the Electric or Magnetic fields, and our approach could be po-
tentially useful in future PC design, because solving the inverse problem is analytically simpler
(at least intuitively) for the eigenproblem involving the E-field.
We used a 3x3 supercell for the optimization, and we minimize the mean-square difference
of the z-components of the electric and magnetic fields of the dual structures at the K-point
of the band diagram. We recognize a priori that a trivial solution, which we wish to avoid, is
0 10 20 30 40 50 60 70 80 90 100
Generation number
Fig. 3: Fitness (gap-to-midgap ratio at K-point of the band diagram) of maximally-fit struc-
ture of each generation for 100 generations. The maximum fitness is a monotonically non-
decreasing function due to cloning. A general increase in fitness arises as a result of various
genetic operations (selection, mating, mutation).
a structure that has a uniform refractive index (either dielectric or air) throughout, and so we
prevent the genetic algorithm from obtaining this by restricting our mutation to only a Gaussian
mutation (see Eq. 4). This preferentially searches the locality of points, and is a necessary trade-
off for obtaining a reasonable solution. This illustrates the versatility of the Genetic approach
- the extent of the search can be easily modified by a simple change of algorithm parameters.
Fig. 5 shows the optimal dual structures with the corresponding simulated fields.
5. Conclusion
From the results above, we have shown that our Genetic Algorithm is able to effectively opti-
mize PC designs to meet specific design criteria. Furthermore, by our choice of encoding, we
could easily impose constraints upon the design space to ensure that every design searched by
the algorithm could be realistically fabricated. Between different optimizations, all that needed
to be changed was the measure of how well a given structure complied with our design crite-
rion - the ”fitness function” in Genetic Algorithm parlance. Our Genetic Algorithm is therefore
highly robust and can be easily modified to optimize any user-defined objective function.
(a) Run 1 (b) Run 2
(c) Run 3 (d) Run 4
(e) Band Diagram - optimized
(f) Band Diagram - uniform holes, r/a = 0.3
Fig. 4: Dielectric structures (a-d), showing the optimal PC structures predicted by 4 runs our
Genetic Algorithm. The unit cell for each structure is depicted by the yellow bounding box. A
sample band diagram (for Run 3) is shown in (e). The optimized TE-bandgap, calculated as the
ratio of the size of the gap to the midgap value, was found to be ≃ 72%. The TE-bandgap for a
triangular lattice with uniform air holes (r/a = 0.3) is shown in (f ) for reference.
(a) Band 1, E-field (b) Band 1, H-field
(c) Band 2, E-field (d) Band 2, H-field
(e) Band 3, E-field (f) Band 3, H-field
(g) Band 4, E-field (h) Band 4, H-field
Fig. 5: Genetic Algorithm prediction of PC structures that have optimally matched E and H
fields, for the lowest 4 bands, at the K point. The E-fields are shown for structure with dielectric
rods, and have a TM bandgap, while the H-fields are shown for structures with air holes, and
have a TE bandgap. The shown fields are in the direction aligned with the rods. The fields for
the lowest 3 bands are very well matched, but begin to deviate significantly from each other at
band 4.
	Introduction
	Genetic Algorithms
	Implementation
	Simulation Results
	Optimizing Planar Photonic Cavity Cavities
	Q-factor Maximization
	Matching to a Target Function
	Direct Minimization of Light Cone Radiation
	Maximal Gap at any k-vector Point
	Optimal dual PC structures
	Conclusion
