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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 18 — Sep. 9, 2013
  • pp: 20831–20836
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Optimizing nanophotonic cavity designs with the gravitational search algorithm

Timothy W. Saucer and Vanessa Sih  »View Author Affiliations


Optics Express, Vol. 21, Issue 18, pp. 20831-20836 (2013)
http://dx.doi.org/10.1364/OE.21.020831


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Abstract

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 OSA

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

1. J. D. Joannopoulos, R. D. Maede, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton Univ. Press, Princeton, 1995).

].

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

1. J. D. Joannopoulos, R. D. Maede, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton Univ. Press, Princeton, 1995).

4

4. K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003). [CrossRef] [PubMed]

]. By modifying the local density of optical states, they can enhance or suppress the spontaneous emission of quantum emitters [3

3. J. Lee, T. W. Saucer, A. J. Martin, J. M. Millunchick, and V. Sih, “Time-Resolved Two-Pulse Excitation of Quantum Dots Coupled to a Photonic Crystal Cavity in the Purcell Regime,” Phys. Rev. Lett. 110(1), 013602 (2013). [CrossRef] [PubMed]

,4

4. K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003). [CrossRef] [PubMed]

]. This leads to a host of interesting physics with far-ranging applications from quantum information processing [5

5. B. Jiang, Y. J. Zhang, W. J. Zhou, W. Chen, A. J. Liu, and W. H. Zheng, “Spontaneous-emission control by local density of states of photonic crystal cavity,” Chin. Phys. B 20(2), 024208 (2011). [CrossRef]

] to improved laser systems [4

4. K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003). [CrossRef] [PubMed]

7

7. O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-Dimensional Photonic Band-Gap Defect Mode Laser,” Science 284(5421), 1819–1821 (1999). [CrossRef] [PubMed]

]. Ideally these systems have high quality factors in small mode volumes [2

2. T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432(7014), 200–203 (2004). [CrossRef] [PubMed]

8

8. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003). [CrossRef] [PubMed]

], 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

9. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “Investigation of high-Q channel drop filters using donor-type defects in two-dimensional photonic crystal slabs,” Appl. Phys. Lett. 83(8), 1512 (2003). [CrossRef]

].

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

8. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003). [CrossRef] [PubMed]

,10

10. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “Fine-tuned high-Q photonic-crystal nanocavity,” Opt. Express 13(4), 1202–1214 (2005). [CrossRef] [PubMed]

]. 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
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.
. 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.

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

11. S. Russell and P. Norvig, Artificial Intelligence: A Modern Approach, 3rd ed. (Prentice Hall, 2009).

].

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 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, f(si), as the final Q/V ratio as determined by the FDTD simulation.

s¨i=jiGmj|sjsi|(sjsi)
(3)

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)
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.
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.

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

8. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003). [CrossRef] [PubMed]

]. We evaluate our designs by calculating the quality factor of the cavity and the following definition of the mode volume [20

20. R. Coccioli, M. Boroditsky, K. W. Kim, Y. Rahmat-Samii, and E. Yablonovitch, “Smallest possible electromagnetic mode volume in a diectric cavity,” IEEE Proc. Optoelectron. 145, 391 (1998).

].

Vmode=ϵ(r)|E(r)|2drmax(ϵ(r)|E(r)|2)
(4)

FDTD simulations were performed using the MEEP c + + libraries [21

21. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181(3), 687–702 (2010). [CrossRef]

]. 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

Table 1. Optimized design parameters for cavity with Q/V = 798,000 (λ/n)−3

table-icon
View This Table
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.

Figure 2(b) shows the evolution of the average Q/V values at each iteration and Fig. 3
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.
shows the electric field (|E|2) mode profile of our final cavity design and the benchmark L3 with 0.15a shift [8

8. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003). [CrossRef] [PubMed]

]. 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.

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.

References and links

1.

J. D. Joannopoulos, R. D. Maede, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton Univ. Press, Princeton, 1995).

2.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432(7014), 200–203 (2004). [CrossRef] [PubMed]

3.

J. Lee, T. W. Saucer, A. J. Martin, J. M. Millunchick, and V. Sih, “Time-Resolved Two-Pulse Excitation of Quantum Dots Coupled to a Photonic Crystal Cavity in the Purcell Regime,” Phys. Rev. Lett. 110(1), 013602 (2013). [CrossRef] [PubMed]

4.

K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003). [CrossRef] [PubMed]

5.

B. Jiang, Y. J. Zhang, W. J. Zhou, W. Chen, A. J. Liu, and W. H. Zheng, “Spontaneous-emission control by local density of states of photonic crystal cavity,” Chin. Phys. B 20(2), 024208 (2011). [CrossRef]

6.

Z. Lin and J. Vučković, “Enhanced two-photon processes in single quantum dots inside photonic crystal nanocavities,” Phys. Rev. B 81(3), 035301 (2010). [CrossRef]

7.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-Dimensional Photonic Band-Gap Defect Mode Laser,” Science 284(5421), 1819–1821 (1999). [CrossRef] [PubMed]

8.

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003). [CrossRef] [PubMed]

9.

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “Investigation of high-Q channel drop filters using donor-type defects in two-dimensional photonic crystal slabs,” Appl. Phys. Lett. 83(8), 1512 (2003). [CrossRef]

10.

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “Fine-tuned high-Q photonic-crystal nanocavity,” Opt. Express 13(4), 1202–1214 (2005). [CrossRef] [PubMed]

11.

S. Russell and P. Norvig, Artificial Intelligence: A Modern Approach, 3rd ed. (Prentice Hall, 2009).

12.

G. S. Hornby, J. D. Lohn, and D. S. Linden, “Computer-Automated Evolution of an X-Band Antenna for NASA’s Space Technology 5 Mission,” Evol. Comput. 19(1), 1–23 (2011). [CrossRef] [PubMed]

13.

E. Bonabeau, G. Theraulaz, and M. Dorigo, Swarm Intelligence: From Natural to Artificial Systems (Oxford University Press, Oxford, 1999).

14.

A. Gondarenko and M. Lipson, “Low modal volume dipole-like dielectric slab resonator,” Opt. Express 16(22), 17689–17694 (2008). [CrossRef] [PubMed]

15.

S. Preble, M. Lipson, and H. Lipson, “Two-dimensional photonic crystals designed by evolutionary algorithms,” Appl. Phys. Lett. 86(6), 061111 (2005). [CrossRef]

16.

C. Lin and M. L. Povinelli, “Optimal design of aperiodic, vertical silicon nanowire structures for photovoltaics,” Opt. Express 19(S5Suppl 5), A1148–A1154 (2011). [CrossRef] [PubMed]

17.

A. Mutapcic, S. Boyd, A. Farjadpour, S. G. Johnson, and Y. Avniel, “Robust design of slow-light tapers in periodic waveguides,” Eng. Optim. 41(4), 365–384 (2009). [CrossRef]

18.

E. Rashedi, H. Nezamabadi-pour, and S. Saryazdi, “GSA: A Gravitational Search Algorithm,” Inf. Sci. 179(13), 2232–2248 (2009). [CrossRef]

19.

S. T. Thornton and J. B. Marion, Classical Dynamics of Particles and Systems, 5th ed. (Brooks Cole, 2003).

20.

R. Coccioli, M. Boroditsky, K. W. Kim, Y. Rahmat-Samii, and E. Yablonovitch, “Smallest possible electromagnetic mode volume in a diectric cavity,” IEEE Proc. Optoelectron. 145, 391 (1998).

21.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181(3), 687–702 (2010). [CrossRef]

22.

F. Römer, B. Witzigmann, O. Chinellato, and P. Arbenz, “Investigation of the Purcell effect in photonic crystal cavities with a 3D Finite Element Maxwell Solver,” Opt. Quantum Electron. 39(4-6), 341–352 (2007). [CrossRef]

OCIS Codes
(350.4238) Other areas of optics : Nanophotonics and photonic crystals
(230.5298) Optical devices : Photonic crystals

ToC Category:
Photonic Crystals

History
Original Manuscript: May 14, 2013
Revised Manuscript: August 17, 2013
Manuscript Accepted: August 17, 2013
Published: August 29, 2013

Citation
Timothy W. Saucer and Vanessa Sih, "Optimizing nanophotonic cavity designs with the gravitational search algorithm," Opt. Express 21, 20831-20836 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-18-20831


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References

  1. J. D. Joannopoulos, R. D. Maede, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton Univ. Press, Princeton, 1995).
  2. T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature432(7014), 200–203 (2004). [CrossRef] [PubMed]
  3. J. Lee, T. W. Saucer, A. J. Martin, J. M. Millunchick, and V. Sih, “Time-Resolved Two-Pulse Excitation of Quantum Dots Coupled to a Photonic Crystal Cavity in the Purcell Regime,” Phys. Rev. Lett.110(1), 013602 (2013). [CrossRef] [PubMed]
  4. K. J. Vahala, “Optical microcavities,” Nature424(6950), 839–846 (2003). [CrossRef] [PubMed]
  5. B. Jiang, Y. J. Zhang, W. J. Zhou, W. Chen, A. J. Liu, and W. H. Zheng, “Spontaneous-emission control by local density of states of photonic crystal cavity,” Chin. Phys. B20(2), 024208 (2011). [CrossRef]
  6. Z. Lin and J. Vučković, “Enhanced two-photon processes in single quantum dots inside photonic crystal nanocavities,” Phys. Rev. B81(3), 035301 (2010). [CrossRef]
  7. O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-Dimensional Photonic Band-Gap Defect Mode Laser,” Science284(5421), 1819–1821 (1999). [CrossRef] [PubMed]
  8. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature425(6961), 944–947 (2003). [CrossRef] [PubMed]
  9. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “Investigation of high-Q channel drop filters using donor-type defects in two-dimensional photonic crystal slabs,” Appl. Phys. Lett.83(8), 1512 (2003). [CrossRef]
  10. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “Fine-tuned high-Q photonic-crystal nanocavity,” Opt. Express13(4), 1202–1214 (2005). [CrossRef] [PubMed]
  11. S. Russell and P. Norvig, Artificial Intelligence: A Modern Approach, 3rd ed. (Prentice Hall, 2009).
  12. G. S. Hornby, J. D. Lohn, and D. S. Linden, “Computer-Automated Evolution of an X-Band Antenna for NASA’s Space Technology 5 Mission,” Evol. Comput.19(1), 1–23 (2011). [CrossRef] [PubMed]
  13. E. Bonabeau, G. Theraulaz, and M. Dorigo, Swarm Intelligence: From Natural to Artificial Systems (Oxford University Press, Oxford, 1999).
  14. A. Gondarenko and M. Lipson, “Low modal volume dipole-like dielectric slab resonator,” Opt. Express16(22), 17689–17694 (2008). [CrossRef] [PubMed]
  15. S. Preble, M. Lipson, and H. Lipson, “Two-dimensional photonic crystals designed by evolutionary algorithms,” Appl. Phys. Lett.86(6), 061111 (2005). [CrossRef]
  16. C. Lin and M. L. Povinelli, “Optimal design of aperiodic, vertical silicon nanowire structures for photovoltaics,” Opt. Express19(S5Suppl 5), A1148–A1154 (2011). [CrossRef] [PubMed]
  17. A. Mutapcic, S. Boyd, A. Farjadpour, S. G. Johnson, and Y. Avniel, “Robust design of slow-light tapers in periodic waveguides,” Eng. Optim.41(4), 365–384 (2009). [CrossRef]
  18. E. Rashedi, H. Nezamabadi-pour, and S. Saryazdi, “GSA: A Gravitational Search Algorithm,” Inf. Sci.179(13), 2232–2248 (2009). [CrossRef]
  19. S. T. Thornton and J. B. Marion, Classical Dynamics of Particles and Systems, 5th ed. (Brooks Cole, 2003).
  20. R. Coccioli, M. Boroditsky, K. W. Kim, Y. Rahmat-Samii, and E. Yablonovitch, “Smallest possible electromagnetic mode volume in a diectric cavity,” IEEE Proc. Optoelectron. 145, 391 (1998).
  21. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun.181(3), 687–702 (2010). [CrossRef]
  22. F. Römer, B. Witzigmann, O. Chinellato, and P. Arbenz, “Investigation of the Purcell effect in photonic crystal cavities with a 3D Finite Element Maxwell Solver,” Opt. Quantum Electron.39(4-6), 341–352 (2007). [CrossRef]

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