Abstract

A computationally-fast inverse design method for nanophotonic structures is presented. The method is based on two complementary convex optimization problems which modify the dielectric structure and resonant field respectively. The design of one- and two-dimensional nanophotonic resonators is demonstrated and is shown to require minimal computational resources.

© 2010 Optical Society of America

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References

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  1. K. Yee, "Numerical solution of initial boundary value problems involving Maxwells equations in isotropic media," IEEE Trans. Antennas Propag. Mag. 14, 302-307 (1966).
    [CrossRef]
  2. M. Albani and P. Bernardi, "A Numerical Method Based on the Discretization of Maxwell Equations in Integral Form," IEEE Trans. Microwave Theory Tech. 22, 446-450 (1974).
    [CrossRef]
  3. J. M. Gerardy and M. Ausloos, "Absorption spectrum of clusters of spheres from the general solution of Maxwell’s equations. The long-wavelength limit," Phys. Rev. B 22, 4950-4959 (1979).
    [CrossRef]
  4. P. Deotare, M. McCutcheon, I. Frank, M. Khan, and M. Loncar, "High quality factor photonic crystal nanobeam cavities," Appl. Phys. Lett. 94, 121106 (2009).
    [CrossRef]
  5. J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, "Design of photonic crystal microcavities for cavity QED," Phys. Rev. E 65, 1-11 (2002).
  6. Y. Akahane, T. Asano, B. Song, and S. Noda, "Fine-tuned high-Q photonic-crystal nanocavity," Opt. Express 13, 1202-1214 (2005).
    [CrossRef] [PubMed]
  7. A. Gondarenko and M. Lipson, "Low modal volume dipole-like dielectric slab resonator," Opt. Express 16, 17689-17694 (2008).
    [CrossRef] [PubMed]
  8. A. Hakansson and J. Sanchez-Dehesa, "Inverse designed photonic crystal de-multiplex waveguide coupler," Opt. Express 13, 5440-5449 (2005).
    [CrossRef] [PubMed]
  9. P. Borel, A. Harpth, L. Frandsen, M. Kristensen, P. Shi, J. Jensen, and O. Sigmund, "Topology optimization and fabrication of photonic crystal structures," Opt. Express 12, 1996-2001 (2004).
    [CrossRef] [PubMed]
  10. D. Englund, I. Fushman, and J. Vuckovic. "General Recipe for Designing Photonic Crystal Cavities," Opt. Express 12, 59615975 (2005).
    [CrossRef]
  11. CHOLMOD software package, accessed via MatLab.
  12. Intel Core 2 Quad 2.5GHz, 8Gb RAM.
  13. S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwells equations in a planewave basis," Opt. Express 8, 967-970 (1999).
  14. S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University Press, 2004).
  15. M. Grant and S. Boyd, CVX: MatLab software for disciplined convex programming, http://stanford.edu/~boyd/cvx, June 2009.
  16. K. Hennessy, C. Hogerle, E. Hu, A. Badolato, and A. Imamoglu, "Tuning photonic nanocavities by atomic force microscope nano-oxidation," Appl. Phys. Lett. 89, 041118 (2006).
    [CrossRef]
  17. B. -S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double-heterostructure nanocavity," Nat. Mater. 4, 207-210 (2005).
    [CrossRef]
  18. K. Rivoire, Z. Lin, F. Hatami, W. Ted Masselink, and J. Vuckovic, "Second harmonic generation in gallium phosphide photonic crystal nanocavities with ultralow continuous wave pump power," Opt. Express 17, 22609-22615 (2009).
    [CrossRef]

2009 (2)

2008 (1)

2006 (1)

K. Hennessy, C. Hogerle, E. Hu, A. Badolato, and A. Imamoglu, "Tuning photonic nanocavities by atomic force microscope nano-oxidation," Appl. Phys. Lett. 89, 041118 (2006).
[CrossRef]

2005 (4)

B. -S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double-heterostructure nanocavity," Nat. Mater. 4, 207-210 (2005).
[CrossRef]

D. Englund, I. Fushman, and J. Vuckovic. "General Recipe for Designing Photonic Crystal Cavities," Opt. Express 12, 59615975 (2005).
[CrossRef]

A. Hakansson and J. Sanchez-Dehesa, "Inverse designed photonic crystal de-multiplex waveguide coupler," Opt. Express 13, 5440-5449 (2005).
[CrossRef] [PubMed]

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

2004 (1)

2002 (1)

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, "Design of photonic crystal microcavities for cavity QED," Phys. Rev. E 65, 1-11 (2002).

1999 (1)

S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwells equations in a planewave basis," Opt. Express 8, 967-970 (1999).

1979 (1)

J. M. Gerardy and M. Ausloos, "Absorption spectrum of clusters of spheres from the general solution of Maxwell’s equations. The long-wavelength limit," Phys. Rev. B 22, 4950-4959 (1979).
[CrossRef]

1974 (1)

M. Albani and P. Bernardi, "A Numerical Method Based on the Discretization of Maxwell Equations in Integral Form," IEEE Trans. Microwave Theory Tech. 22, 446-450 (1974).
[CrossRef]

1966 (1)

K. Yee, "Numerical solution of initial boundary value problems involving Maxwells equations in isotropic media," IEEE Trans. Antennas Propag. Mag. 14, 302-307 (1966).
[CrossRef]

Akahane, Y.

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

B. -S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double-heterostructure nanocavity," Nat. Mater. 4, 207-210 (2005).
[CrossRef]

Albani, M.

M. Albani and P. Bernardi, "A Numerical Method Based on the Discretization of Maxwell Equations in Integral Form," IEEE Trans. Microwave Theory Tech. 22, 446-450 (1974).
[CrossRef]

Asano, T.

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

B. -S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double-heterostructure nanocavity," Nat. Mater. 4, 207-210 (2005).
[CrossRef]

Ausloos, M.

J. M. Gerardy and M. Ausloos, "Absorption spectrum of clusters of spheres from the general solution of Maxwell’s equations. The long-wavelength limit," Phys. Rev. B 22, 4950-4959 (1979).
[CrossRef]

Badolato, A.

K. Hennessy, C. Hogerle, E. Hu, A. Badolato, and A. Imamoglu, "Tuning photonic nanocavities by atomic force microscope nano-oxidation," Appl. Phys. Lett. 89, 041118 (2006).
[CrossRef]

Bernardi, P.

M. Albani and P. Bernardi, "A Numerical Method Based on the Discretization of Maxwell Equations in Integral Form," IEEE Trans. Microwave Theory Tech. 22, 446-450 (1974).
[CrossRef]

Borel, P.

Deotare, P.

P. Deotare, M. McCutcheon, I. Frank, M. Khan, and M. Loncar, "High quality factor photonic crystal nanobeam cavities," Appl. Phys. Lett. 94, 121106 (2009).
[CrossRef]

Englund, D.

D. Englund, I. Fushman, and J. Vuckovic. "General Recipe for Designing Photonic Crystal Cavities," Opt. Express 12, 59615975 (2005).
[CrossRef]

Frandsen, L.

Frank, I.

P. Deotare, M. McCutcheon, I. Frank, M. Khan, and M. Loncar, "High quality factor photonic crystal nanobeam cavities," Appl. Phys. Lett. 94, 121106 (2009).
[CrossRef]

Fushman, I.

D. Englund, I. Fushman, and J. Vuckovic. "General Recipe for Designing Photonic Crystal Cavities," Opt. Express 12, 59615975 (2005).
[CrossRef]

Gerardy, J. M.

J. M. Gerardy and M. Ausloos, "Absorption spectrum of clusters of spheres from the general solution of Maxwell’s equations. The long-wavelength limit," Phys. Rev. B 22, 4950-4959 (1979).
[CrossRef]

Gondarenko, A.

Hakansson, A.

Harpth, A.

Hatami, F.

Hennessy, K.

K. Hennessy, C. Hogerle, E. Hu, A. Badolato, and A. Imamoglu, "Tuning photonic nanocavities by atomic force microscope nano-oxidation," Appl. Phys. Lett. 89, 041118 (2006).
[CrossRef]

Hogerle, C.

K. Hennessy, C. Hogerle, E. Hu, A. Badolato, and A. Imamoglu, "Tuning photonic nanocavities by atomic force microscope nano-oxidation," Appl. Phys. Lett. 89, 041118 (2006).
[CrossRef]

Hu, E.

K. Hennessy, C. Hogerle, E. Hu, A. Badolato, and A. Imamoglu, "Tuning photonic nanocavities by atomic force microscope nano-oxidation," Appl. Phys. Lett. 89, 041118 (2006).
[CrossRef]

Imamoglu, A.

K. Hennessy, C. Hogerle, E. Hu, A. Badolato, and A. Imamoglu, "Tuning photonic nanocavities by atomic force microscope nano-oxidation," Appl. Phys. Lett. 89, 041118 (2006).
[CrossRef]

Jensen, J.

Joannopoulos, J. D.

S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwells equations in a planewave basis," Opt. Express 8, 967-970 (1999).

Johnson, S. G.

S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwells equations in a planewave basis," Opt. Express 8, 967-970 (1999).

Khan, M.

P. Deotare, M. McCutcheon, I. Frank, M. Khan, and M. Loncar, "High quality factor photonic crystal nanobeam cavities," Appl. Phys. Lett. 94, 121106 (2009).
[CrossRef]

Kristensen, M.

Lin, Z.

Lipson, M.

Loncar, M.

P. Deotare, M. McCutcheon, I. Frank, M. Khan, and M. Loncar, "High quality factor photonic crystal nanobeam cavities," Appl. Phys. Lett. 94, 121106 (2009).
[CrossRef]

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, "Design of photonic crystal microcavities for cavity QED," Phys. Rev. E 65, 1-11 (2002).

Mabuchi, H.

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, "Design of photonic crystal microcavities for cavity QED," Phys. Rev. E 65, 1-11 (2002).

McCutcheon, M.

P. Deotare, M. McCutcheon, I. Frank, M. Khan, and M. Loncar, "High quality factor photonic crystal nanobeam cavities," Appl. Phys. Lett. 94, 121106 (2009).
[CrossRef]

Noda, S.

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

B. -S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double-heterostructure nanocavity," Nat. Mater. 4, 207-210 (2005).
[CrossRef]

Rivoire, K.

Sanchez-Dehesa, J.

Scherer, A.

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, "Design of photonic crystal microcavities for cavity QED," Phys. Rev. E 65, 1-11 (2002).

Shi, P.

Sigmund, O.

Song, B.

Song, B. -S.

B. -S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double-heterostructure nanocavity," Nat. Mater. 4, 207-210 (2005).
[CrossRef]

Ted Masselink, W.

Vuckovic, J.

K. Rivoire, Z. Lin, F. Hatami, W. Ted Masselink, and J. Vuckovic, "Second harmonic generation in gallium phosphide photonic crystal nanocavities with ultralow continuous wave pump power," Opt. Express 17, 22609-22615 (2009).
[CrossRef]

D. Englund, I. Fushman, and J. Vuckovic. "General Recipe for Designing Photonic Crystal Cavities," Opt. Express 12, 59615975 (2005).
[CrossRef]

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, "Design of photonic crystal microcavities for cavity QED," Phys. Rev. E 65, 1-11 (2002).

Yee, K.

K. Yee, "Numerical solution of initial boundary value problems involving Maxwells equations in isotropic media," IEEE Trans. Antennas Propag. Mag. 14, 302-307 (1966).
[CrossRef]

Appl. Phys. Lett. (2)

P. Deotare, M. McCutcheon, I. Frank, M. Khan, and M. Loncar, "High quality factor photonic crystal nanobeam cavities," Appl. Phys. Lett. 94, 121106 (2009).
[CrossRef]

K. Hennessy, C. Hogerle, E. Hu, A. Badolato, and A. Imamoglu, "Tuning photonic nanocavities by atomic force microscope nano-oxidation," Appl. Phys. Lett. 89, 041118 (2006).
[CrossRef]

IEEE Trans. Antennas Propag. Mag. (1)

K. Yee, "Numerical solution of initial boundary value problems involving Maxwells equations in isotropic media," IEEE Trans. Antennas Propag. Mag. 14, 302-307 (1966).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

M. Albani and P. Bernardi, "A Numerical Method Based on the Discretization of Maxwell Equations in Integral Form," IEEE Trans. Microwave Theory Tech. 22, 446-450 (1974).
[CrossRef]

Nat. Mater. (1)

B. -S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double-heterostructure nanocavity," Nat. Mater. 4, 207-210 (2005).
[CrossRef]

Opt. Express (7)

Phys. Rev. B (1)

J. M. Gerardy and M. Ausloos, "Absorption spectrum of clusters of spheres from the general solution of Maxwell’s equations. The long-wavelength limit," Phys. Rev. B 22, 4950-4959 (1979).
[CrossRef]

Phys. Rev. E (1)

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, "Design of photonic crystal microcavities for cavity QED," Phys. Rev. E 65, 1-11 (2002).

Other (4)

CHOLMOD software package, accessed via MatLab.

Intel Core 2 Quad 2.5GHz, 8Gb RAM.

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

M. Grant and S. Boyd, CVX: MatLab software for disciplined convex programming, http://stanford.edu/~boyd/cvx, June 2009.

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Figures (9)

Fig. 1.
Fig. 1.

Inverse design of a one-dimensional structure using the unmodified least-squares method. The target field is a sinusoid within a Gaussian envelope. The computed dielectric structure (green area) supports a field (red circles) that exactly matches the target field (blue line). The entire design process is also extremely fast and takes less than 1 second to complete on a generic desktop computer [12]. The periodic singularities in the dielectric structure are non-physical and will be addressed later in the article.

Fig. 2.
Fig. 2.

Inverse design of one-dimensional structures using the regularized least-squares method. The same target field is used as in Fig. 1, and the computation time remains below 1 second. As the regularization parameter, η, is increased, ε is increasingly constrained to a chosen constant value of 10. At the same time, the mismatch between target and actual fields increases markedly. This illustrates the apparent trade-off between producing reasonable structures and accurately reproducing a fixed target field.

Fig. 3.
Fig. 3.

Inverse design of a one-dimensional structure using the complementary optimization method. The target field in Figs. 1 and 2 is used as the initial target field. The rates of change for both ε and H are controlled by regularization parameters η 1 = 10-4 and η 2 = 10-3 respectively. The 400 iterations used to achieve this result took 60 seconds to compute. This method results in a well-behaved ε that actually produces a field very similar to the original target field. Interestingly, the formation of a “steady-state” periodic structure toward the sides of the structure has emerged.

Fig. 4.
Fig. 4.

Inverse design of a one-dimensional structure using the complementary optimization method with bounded ε. The parameters are identical to those used to produce Fig. 3 with the exception that only one regularization term is now needed (η 2 = 10-3). The algorithm was run for 100 iterations, which took 100 seconds. The structure turns out to be almost completely binary-valued and looks like a periodic structure with tapered duty cycle. It produces an actual field which very closely matches the final target field.

Fig. 5.
Fig. 5.

Inverse design of an “S” resonator using the complementary optimization method without bounds on ε. The design was initialized by specifying an initial dielectric structure (ε= 1 everywhere) and a resonant field in the shape of an “S”. The final dielectric structure was produced after 50 iterations which took 90 seconds to complete in total. The grid size was 80×120. The final dielectric structure is quite unintuive, and yet reproduces the target field surprisingly well. This example demonstrates the versatility of the complementary optimization method in producing designs, from very simple specifications, which otherwise could be attained only with considerable difficulty.

Fig. 6.
Fig. 6.

Inverse design of a doubly-resonant, degenerate “X” resonator produced by the complementary optimization method with unbounded ε. As in Fig. 5 the initial value of ε was 1 everywhere. The two initial target fields used are only slightly perturbed and are very similar to the two final actual target fields. Computationally, this design took 5 minutes to complete on a 120 × 120 grid and required 40 iterations. This example shows that the complementary optimization strategy can be extended to produce dielectric structures with multiple resonances. Such an “X” resonator is useful for polarization-entangled single-photon sources for example [16].

Fig. 7.
Fig. 7.

Inverse design of a single-to-dual beam waveguide coupler using the complementary optimization method without bounds on ε. Two degenerate modes with opposite symmetry (sine and cosine) are used as target fields (only one is shown). Only 4 iterations (14 seconds) are needed to achieve this solution on a 240×55 grid. This is a simple demonstration showing that the complementary optimization method can also be extended to design waveguiding devices.

Fig. 8.
Fig. 8.

Inverse design of a two-dimensional resonator using the complementary optimization method with strict bounds on ε. The initial specification is very simple and consists of an initial dielectric structure (ε = 12.25 everywhere), the frequency and mode-volume of the resonance field as well as a weighting factor, η, to avoid leaky field Fourier components. Additionally, the values of ε are only allowed to be modified within a central circular region and must be kept between 1 and 12.25. After 40 iterations on a 160 ×160 grid, which took 7 minutes to complete, a discrete structure emerged with excellent match between the predicted (x 40) and actual fields. The structure resembles a circular grating with a bowtie-like central structure for focusing the resonant energy to a single point.

Fig. 9.
Fig. 9.

Inverse design of a beam resonator in two dimensions using the complementary optimization method with bounded ε. The initial conditions are identical to those for Fig. 8, except that the initial dielectric structure is an unbroken waveguide, and ε can only be modified within that waveguide. The structure emerged after 40 iterations on a 320 × 40 grid, which took 5 minutes of computation. The bowtie-like structure has reappeared in the center. Interestingly, the effect of the Fourier term in Eq. (11), is seen in the outward tapering of the holes.

Equations (15)

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× ε 1 × H = ( ω c ) 2 H
AY Ax = ξx
By = d
minimize y ∣∣ By d ∣∣ 2 + η ∣∣ y y 0 ∣∣ 2 .
minimize y i ∣∣ B i 1 y i d i 1 ∣∣ 2 + η 1 ∣∣ y i y i 1 ∣∣ 2
minimize x i ∣∣ AY i Ax i ξx i 1 ∣∣ 2 + η 2 ∣∣ x i x i 1 ∣∣ 2
minimize y i ∣∣ B i 1 y i d i 1 ∣∣ 2
subject to ε max 1 y i ε min 1
minimize x i ∣∣ A Y i A x i ξ x i 1 ∣∣ 2 + η 2 ∣∣ x i x i 1 ∣∣ 2 .
minimize y i η 1 ∣∣ y i y i 1 ∣∣ 2 + j ∣∣ B i 1 ( j ) y i d i 1 ( j ) ∣∣ 2
minimize x i ( j ) ∣∣ AY i A x i ( j ) ξ x i 1 ( j ) ∣∣ 2 + η 2 ∣∣ x i ( j ) x i 1 ( j ) ∣∣ 2 .
minimize x i ∣∣ A Y i 1 A x i ξ x i ∣∣ 2 + η ∣∣ F x i ∣∣ 2
subject to ( A x i ) T Y i 1 ( A x i ) A mode
minimize y i ∣∣ B i y i d i ∣∣ 2
subject to ε max 1 y i ε min 1 .

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