Abstract

The inverse design of a three-dimensional nanophotonic resonator is presented. The design methodology is computationally fast (10 minutes on a standard desktop workstation) and utilizes a 2.5-dimensional approximation of the full three-dimensional structure. As an example, we employ the proposed method to design a resonator which exhibits a mode volume of 0.32(λ/n)3 and a quality factor of 7063.

© 2011 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. A. B. Miller, “Rationale and challenges for optical interconnects to electronic chips,” Proc. IEEE 88, 728–749 (2000).
    [CrossRef]
  2. J. Lu and J. Vuckovic, “Inverse design of nanophotonic structures using complementary convex optimization,” Opt. Express 18, 3793–3804 (2010).
    [CrossRef] [PubMed]
  3. U. Inan and A. Inan, Electromagnetic Waves (Prentice Hall, 2000), p. 296.
  4. K. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. Mag. 14, 302–307 (1966).
    [CrossRef]
  5. S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University Press, 2004).
  6. S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein are preparing a manuscript to be called, “Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers,” http://www.stanford.edu/~boyd/papers/distr_opt_stat_learning_admm.html .
  7. Y. Chen, T. A. Davis, W. W. Hager, and S. Rajamanickam, “Algorithm 887: CHOLMOD, supernodal sparse Cholesky factorization and update/downdate,” ACM Trans. Math. Software 35(3), 2 (2009).
  8. M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming , version 1.21. http://cvxr.com/cvx , January 2011.
  9. D. Englund, I. Fushman, and J. Vuckovic, “General recipe for designing photonic crystal cavities,” Opt. Express 13, 5961–5975 (2005).
    [CrossRef] [PubMed]
  10. J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Optimization of Q-factor in photonic crystal microcavities,” IEEE J. Quantum Electron. 38, 850–856 (2002).
    [CrossRef]

2010

2009

Y. Chen, T. A. Davis, W. W. Hager, and S. Rajamanickam, “Algorithm 887: CHOLMOD, supernodal sparse Cholesky factorization and update/downdate,” ACM Trans. Math. Software 35(3), 2 (2009).

2005

2004

S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University Press, 2004).

2002

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Optimization of Q-factor in photonic crystal microcavities,” IEEE J. Quantum Electron. 38, 850–856 (2002).
[CrossRef]

2000

D. A. B. Miller, “Rationale and challenges for optical interconnects to electronic chips,” Proc. IEEE 88, 728–749 (2000).
[CrossRef]

U. Inan and A. Inan, Electromagnetic Waves (Prentice Hall, 2000), p. 296.

1966

K. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. Mag. 14, 302–307 (1966).
[CrossRef]

Boyd, S.

S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University Press, 2004).

Chen, Y.

Y. Chen, T. A. Davis, W. W. Hager, and S. Rajamanickam, “Algorithm 887: CHOLMOD, supernodal sparse Cholesky factorization and update/downdate,” ACM Trans. Math. Software 35(3), 2 (2009).

Davis, T. A.

Y. Chen, T. A. Davis, W. W. Hager, and S. Rajamanickam, “Algorithm 887: CHOLMOD, supernodal sparse Cholesky factorization and update/downdate,” ACM Trans. Math. Software 35(3), 2 (2009).

Englund, D.

Fushman, I.

Hager, W. W.

Y. Chen, T. A. Davis, W. W. Hager, and S. Rajamanickam, “Algorithm 887: CHOLMOD, supernodal sparse Cholesky factorization and update/downdate,” ACM Trans. Math. Software 35(3), 2 (2009).

Inan, A.

U. Inan and A. Inan, Electromagnetic Waves (Prentice Hall, 2000), p. 296.

Inan, U.

U. Inan and A. Inan, Electromagnetic Waves (Prentice Hall, 2000), p. 296.

Loncar, M.

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Optimization of Q-factor in photonic crystal microcavities,” IEEE J. Quantum Electron. 38, 850–856 (2002).
[CrossRef]

Lu, J.

Mabuchi, H.

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Optimization of Q-factor in photonic crystal microcavities,” IEEE J. Quantum Electron. 38, 850–856 (2002).
[CrossRef]

Miller, D. A. B.

D. A. B. Miller, “Rationale and challenges for optical interconnects to electronic chips,” Proc. IEEE 88, 728–749 (2000).
[CrossRef]

Rajamanickam, S.

Y. Chen, T. A. Davis, W. W. Hager, and S. Rajamanickam, “Algorithm 887: CHOLMOD, supernodal sparse Cholesky factorization and update/downdate,” ACM Trans. Math. Software 35(3), 2 (2009).

Scherer, A.

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Optimization of Q-factor in photonic crystal microcavities,” IEEE J. Quantum Electron. 38, 850–856 (2002).
[CrossRef]

Vandenberghe, L.

S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University Press, 2004).

Vuckovic, J.

Yee, K.

K. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. Mag. 14, 302–307 (1966).
[CrossRef]

ACM Trans. Math. Software

Y. Chen, T. A. Davis, W. W. Hager, and S. Rajamanickam, “Algorithm 887: CHOLMOD, supernodal sparse Cholesky factorization and update/downdate,” ACM Trans. Math. Software 35(3), 2 (2009).

IEEE J. Quantum Electron.

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Optimization of Q-factor in photonic crystal microcavities,” IEEE J. Quantum Electron. 38, 850–856 (2002).
[CrossRef]

IEEE Trans. Antennas Propag. Mag.

K. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. Mag. 14, 302–307 (1966).
[CrossRef]

Opt. Express

Proc. IEEE

D. A. B. Miller, “Rationale and challenges for optical interconnects to electronic chips,” Proc. IEEE 88, 728–749 (2000).
[CrossRef]

Other

M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming , version 1.21. http://cvxr.com/cvx , January 2011.

U. Inan and A. Inan, Electromagnetic Waves (Prentice Hall, 2000), p. 296.

S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University Press, 2004).

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein are preparing a manuscript to be called, “Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers,” http://www.stanford.edu/~boyd/papers/distr_opt_stat_learning_admm.html .

Supplementary Material (1)

» Media 1: MOV (388 KB)     

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

(a) For computational feasibility, the resonant fields of a planar nanophotonic device are approximated as those of a truncated holey fiber with identical dielectric structure. (b) An example of our approximation using an L3 photonic crystal resonator. Most of the characteristics of the full three-dimensional field at the center of the slab appear in the approximate solution.

Fig. 2
Fig. 2

The (a) dielectric structure, ε, and (b) target field, E (Ey shown only), produced after 75 iterations (10 minutes) on a 160 × 160 grid. The resulting ε is almost completely binary, and relatively smooth (Media 1).

Fig. 3
Fig. 3

Value of the physics residual (blue) and design objective, or mode volume, (red) at each iteration. The physics residual seems to exhibit linear convergence, while the mode volume quickly saturates after roughly 25 iterations.

Fig. 4
Fig. 4

Comparison (Ey ) of (a) the target field from the inverse design method (from Fig. 2), (b) the actual 2.5-dimensional fiber mode, and (c) the field from the full three-dimensional FDTD simulation. The target field matches well with the full three-dimensional field.

Fig. 5
Fig. 5

Comparison of the cross sections of the Ey field along the x-axis from the target field (blue), the actual 2.5-dimensional fiber mode (green), and the field from the full three-dimensional FDTD simulation (red). There is some discrepancy between the target field and the full three-dimensional field, but even that is confined to the edges and is only on the order of ∼ 1% of the maximum field amplitude.

Fig. 6
Fig. 6

Comparison of the Fourier transforms of the Ey fields of the (a) the target field from the inverse design method, (b) the actual 2.5-dimensional fiber mode, and (c) the field from the full three-dimensional FDTD simulation. The error in our approximation introduces some small additional Fourier components into the light cone.

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

× × E μ ω 2 ɛ E = 0 , where
E = E ( x , y , z ) .
E = E ( x , y ) e i β z ,
t = 2 β tan 1 α β , where
α = k x 2 ( ω c ) 2 , and
k x = ( n eff ω c ) 2 β 2 .
n eff = | | ɛ E 2 | | | | E 2 | | .
minimize | | ɛ E 2 | | max { ɛ E 2 }
subject to × × E μ ω 2 ɛ E = 0
ɛ E = 0
FT lightcone { E } = 0 ,
ɛ = { ɛ air , ɛ silicon }
minimize E | | × × E μ ω 2 ɛ E | | + η | | ɛ E 2 | |
subject to E center = 1
ɛ E = 0
FT lightcone { E } = 0 ,
minimize ɛ | | × × E μ ω 2 ɛ E | |
subject to ɛ air ɛ ɛ silicon ,

Metrics