Abstract

We develop and implement a new inverse computational framework for designing photonic elements with one or more high-Q scattering resonances. The approach relies on solving for the poles of the scattering matrix, which mathematically amounts to minimizing the determinant of the matrix representing the Fredholm integral operator of the electric field with respect to the permittivity profile of the scattering element. We apply the method to design subwavelength gradient-permittivity structures with multiple scattering resonances and quality factors as high as 800. We also find the spectral scattering cross sections are consistent with Fano lineshapes. The compact form and computational efficiency of our formalism suggest it can be an effective tool for designing Fano-resonant structures with multiple high-Q resonances for applications such as frequency mixing and conversion.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (2)

M. Minkov, D. Gerace, and S. Fan, “Doubly resonant χ(2) nonlinear photonic crystal cavity based on a bound state in the continuum,” Optica 6(8), 1039–1045 (2019).
[Crossref]

F. Liu and L. Ying, “Sparsifying preconditioner for the time-harmonic Maxwell’s equations,” J. Comput. Phys. 376, 913–923 (2019).
[Crossref]

2018 (8)

S. Molesky, Z. Lin, A. Y. Piggott, W. Jin, J. Vuckovic, and A. W. Rodriguez, “Inverse design in nanophotonics,” Nat. Photonics 12(11), 659–670 (2018).
[Crossref]

T. W. Hughes, M. Minkov, I. A. Williamson, and S. Fan, “Adjoint method and inverse design for nonlinear nanophotonic devices,” ACS Photonics 5(12), 4781–4787 (2018).
[Crossref]

D. Liu, Y. Tan, E. Khoram, and Z. Yu, “Training deep neural networks for the inverse design of nanophotonic structures,” ACS Photonics 5(4), 1365–1369 (2018).
[Crossref]

M. Ma, F. Cheng, and Y. Liu, “Deep-learning-enabled on-demand design of chiral metamaterials,” ACS Nano 12(6), 6326–6334 (2018).
[Crossref]

Z. Liu, D. Zhu, S. P. Rodrigues, K. T. Lee, and W. Cai, “Generative model for the inverse design of metasurfaces,” Nano Lett. 18(10), 6570–6576 (2018).
[Crossref]

R. Pestourie, C. Perez-Arancibia, Z. Lin, W. Shin, F. Capasso, and S. G. Johnson, “Inverse design of large-area metasurfaces,” Opt. Express 26(26), 33732–33747 (2018).
[Crossref]

L. Pilozzi, F. A. Farrelly, G. Marcucci, and C. Conti, “Machine learning inverse problem for topological photonics,” Commun. Phys. 1(1), 57 (2018).
[Crossref]

C. Sitawarin, W. Jin, Z. Lin, and A. W. Rodriguez, “Inverse-designed photonic fibers and metasurfaces for nonlinear frequency conversion,” Photonics Res. 6(5), B82–B89 (2018).
[Crossref]

2017 (4)

Z. Lin, M. Loncar, and A. W. Rodriguez, “Topology optimization of multi-track ring resonators and 2D microcavities for nonlinear frequency conversion,” Opt. Lett. 42(14), 2818–2821 (2017).
[Crossref]

A. Y. Piggott, J. Petykiewicz, L. Su, and J. Vuckovic, “Fabrication-constrained nanophotonic inverse design,” Sci. Rep. 7(1), 1786 (2017).
[Crossref]

M. F. Limonov, M. V. Rybin, A. N. Poddubny, and Y. S. Kivshar, “Fano resonances in photonics,” Nat. Photonics 11(9), 543–554 (2017).
[Crossref]

F. Alpeggiani, N. Parappurath, E. Verhagen, and L. Kuipers, “Quasinormal-mode expansion of the scattering matrix,” Phys. Rev. X 7(2), 021035 (2017).
[Crossref]

2016 (4)

Z. Lin, X. Liang, M. Loncar, S. G. Johnson, and A. W. Rodriguez, “Cavity-enhanced second-harmonic generation via nonlinear-overlap optimization,” Optica 3(3), 233–238 (2016).
[Crossref]

D. P. Lake, M. Mitchell, H. Jayakumar, L. F. dos Santos, D. Curic, and P. E. Barclay, “Efficient telecom to visible wavelength conversion in doubly resonant gallium phosphide microdisks,” Appl. Phys. Lett. 108(3), 031109 (2016).
[Crossref]

Z. Lin, A. Pick, M. Loncar, and A. W. Rodriguez, “Enhanced spontaneous emission at third-order Dirac exceptional points in inverse-designed photonic crystals,” Phys. Rev. Lett. 117(10), 107402 (2016).
[Crossref]

S. Liu, M. B. Sinclair, S. Saravi, G. A. Keeler, Y. Yang, J. Reno, G. M. Peake, F. Setzpfandt, I. Staude, T. Pertsch, and I. Brener, “Resonantly enhanced second-harmonic generation using III–V semiconductor all-dielectric metasurfaces,” Nano Lett. 16(9), 5426–5432 (2016).
[Crossref]

2015 (2)

M. Celebrano, X. Wu, M. Baselli, S. Grobmann, P. Biagioni, A. Locatelli, C. D. Angelis, G. Cerullo, R. Osellame, B. Hecht, and L. Duo, “Mode matching in multiresonant plasmonic nanoantennas for enhanced second harmonic generation,” Nat. Nanotechnol. 10(5), 412–417 (2015).
[Crossref]

A. Y. Piggott, J. Lu, K. G. Lagoudakis, J. Petykiewicz, T. M. Babinec, and J. Vuckovic, “Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer,” Nat. Photonics 9(6), 374–377 (2015).
[Crossref]

2014 (3)

Z. Lin, T. Alcorn, M. Loncar, S. G. Johnson, and A. W. Rodriguez, “High-efficiency degenerate four-wave mixing in triply resonant nanobeam cavities,” Phys. Rev. A 89(5), 053839 (2014).
[Crossref]

P. S. Kuo, J. Bravo-Abad, and G. S. Solomon, “Second-harmonic generation using-quasi-phase matching in a GaAs whispering-gallery-mode microcavity,” Nat. Commun. 5(1), 3109 (2014).
[Crossref]

V. Ganapati, O. D. Miller, and E. Yablonovitch, “Light trapping textures designed by electromagnetic optimization for subwavelength thick solar cells,” IEEE J. Photovolt. 4(1), 175–182 (2014).
[Crossref]

2013 (2)

2012 (4)

2011 (3)

J. S. Jense and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photonics Rev. 5(2), 308–321 (2011).
[Crossref]

M. G. Banaee and K. B. Crozier, “Mixed dimer double-resonance substrates for surface-enhanced Raman spectroscopy,” ACS Nano 5(1), 307–314 (2011).
[Crossref]

Y. Chu, D. Wang, W. Zhu, and K. B. Crozier, “Double resonance surface enhanced Raman scattering substrates: an intuitive coupled oscillator model,” Opt. Express 19(16), 14919–14928 (2011).
[Crossref]

2010 (3)

Y. Chu, M. G. Banaee, and K. B. Crozier, “Double-resonance plasmon substrates for surface-enhanced Raman scattering with enhancement at excitation and stokes frequencies,” ACS Nano 4(5), 2804–2810 (2010).
[Crossref]

H. Men, N. C. Nguyen, R. M. Freund, P. A. Parrilo, and J. Peraire, “Bandgap optimization of two-dimensional photonic crystals using semidefinite programming and subspace methods,” J. Comput. Phys. 229(10), 3706–3725 (2010).
[Crossref]

J. U. Furst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[Crossref]

2009 (1)

M. Florescu, S. Torquato, and P. J. Steinhardt, “Designer disordered materials with large, complete photonic band gaps,” Proc. Natl. Acad. Sci. 106(49), 20658–20663 (2009).
[Crossref]

2007 (1)

2006 (1)

M. Liscidini and L. C. Andreani, “Second-harmonic generation in doubly resonant microcavities with periodic dielectric mirrors,” Phys. Rev. E 73(1), 016613 (2006).
[Crossref]

2004 (3)

2000 (1)

1998 (1)

M. T. Chu, “Inverse eigenvalue problems,” SIAM Rev. 40(1), 1–39 (1998).
[Crossref]

1995 (1)

M. Neviere, R. Reinisch, and E. Popov, “Electromagnetic resonances in linear and nonlinear optics: Phenomenological study of grating behavior through the poles and zeros of the scattering operator,” J. Opt. Soc. Am. 12(3), 513–523 (1995).
[Crossref]

1993 (2)

1944 (1)

K. Levenberg, “A method for the solution of certain non-linear problems in least squares,” Q. Appl. Math. 2(2), 164–168 (1944).
[Crossref]

Alaeian, H.

H. Alaeian, A. C. Atre, and J. A. Dionne, “Optimized light absorption in Si wire array solar cells,” J. Opt. 14(2), 024006 (2012).
[Crossref]

Alcorn, T.

Z. Lin, T. Alcorn, M. Loncar, S. G. Johnson, and A. W. Rodriguez, “High-efficiency degenerate four-wave mixing in triply resonant nanobeam cavities,” Phys. Rev. A 89(5), 053839 (2014).
[Crossref]

Alpeggiani, F.

F. Alpeggiani, N. Parappurath, E. Verhagen, and L. Kuipers, “Quasinormal-mode expansion of the scattering matrix,” Phys. Rev. X 7(2), 021035 (2017).
[Crossref]

Andersen, U. L.

J. U. Furst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[Crossref]

Andreani, L. C.

M. Liscidini and L. C. Andreani, “Second-harmonic generation in doubly resonant microcavities with periodic dielectric mirrors,” Phys. Rev. E 73(1), 016613 (2006).
[Crossref]

Angelis, C. D.

M. Celebrano, X. Wu, M. Baselli, S. Grobmann, P. Biagioni, A. Locatelli, C. D. Angelis, G. Cerullo, R. Osellame, B. Hecht, and L. Duo, “Mode matching in multiresonant plasmonic nanoantennas for enhanced second harmonic generation,” Nat. Nanotechnol. 10(5), 412–417 (2015).
[Crossref]

Atre, A. C.

H. Alaeian, A. C. Atre, and J. A. Dionne, “Optimized light absorption in Si wire array solar cells,” J. Opt. 14(2), 024006 (2012).
[Crossref]

Babinec, T. M.

A. Y. Piggott, J. Lu, K. G. Lagoudakis, J. Petykiewicz, T. M. Babinec, and J. Vuckovic, “Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer,” Nat. Photonics 9(6), 374–377 (2015).
[Crossref]

Banaee, M. G.

M. G. Banaee and K. B. Crozier, “Mixed dimer double-resonance substrates for surface-enhanced Raman spectroscopy,” ACS Nano 5(1), 307–314 (2011).
[Crossref]

Y. Chu, M. G. Banaee, and K. B. Crozier, “Double-resonance plasmon substrates for surface-enhanced Raman scattering with enhancement at excitation and stokes frequencies,” ACS Nano 4(5), 2804–2810 (2010).
[Crossref]

Barclay, P. E.

D. P. Lake, M. Mitchell, H. Jayakumar, L. F. dos Santos, D. Curic, and P. E. Barclay, “Efficient telecom to visible wavelength conversion in doubly resonant gallium phosphide microdisks,” Appl. Phys. Lett. 108(3), 031109 (2016).
[Crossref]

Baselli, M.

M. Celebrano, X. Wu, M. Baselli, S. Grobmann, P. Biagioni, A. Locatelli, C. D. Angelis, G. Cerullo, R. Osellame, B. Hecht, and L. Duo, “Mode matching in multiresonant plasmonic nanoantennas for enhanced second harmonic generation,” Nat. Nanotechnol. 10(5), 412–417 (2015).
[Crossref]

Bhargava, S.

Bi, Z. F.

Biagioni, P.

M. Celebrano, X. Wu, M. Baselli, S. Grobmann, P. Biagioni, A. Locatelli, C. D. Angelis, G. Cerullo, R. Osellame, B. Hecht, and L. Duo, “Mode matching in multiresonant plasmonic nanoantennas for enhanced second harmonic generation,” Nat. Nanotechnol. 10(5), 412–417 (2015).
[Crossref]

Borel, P. I.

Bravo-Abad, J.

P. S. Kuo, J. Bravo-Abad, and G. S. Solomon, “Second-harmonic generation using-quasi-phase matching in a GaAs whispering-gallery-mode microcavity,” Nat. Commun. 5(1), 3109 (2014).
[Crossref]

Brener, I.

S. Liu, M. B. Sinclair, S. Saravi, G. A. Keeler, Y. Yang, J. Reno, G. M. Peake, F. Setzpfandt, I. Staude, T. Pertsch, and I. Brener, “Resonantly enhanced second-harmonic generation using III–V semiconductor all-dielectric metasurfaces,” Nano Lett. 16(9), 5426–5432 (2016).
[Crossref]

Cai, W.

Z. Liu, D. Zhu, S. P. Rodrigues, K. T. Lee, and W. Cai, “Generative model for the inverse design of metasurfaces,” Nano Lett. 18(10), 6570–6576 (2018).
[Crossref]

Capasso, F.

Celebrano, M.

M. Celebrano, X. Wu, M. Baselli, S. Grobmann, P. Biagioni, A. Locatelli, C. D. Angelis, G. Cerullo, R. Osellame, B. Hecht, and L. Duo, “Mode matching in multiresonant plasmonic nanoantennas for enhanced second harmonic generation,” Nat. Nanotechnol. 10(5), 412–417 (2015).
[Crossref]

Cerullo, G.

M. Celebrano, X. Wu, M. Baselli, S. Grobmann, P. Biagioni, A. Locatelli, C. D. Angelis, G. Cerullo, R. Osellame, B. Hecht, and L. Duo, “Mode matching in multiresonant plasmonic nanoantennas for enhanced second harmonic generation,” Nat. Nanotechnol. 10(5), 412–417 (2015).
[Crossref]

Cheng, F.

M. Ma, F. Cheng, and Y. Liu, “Deep-learning-enabled on-demand design of chiral metamaterials,” ACS Nano 12(6), 6326–6334 (2018).
[Crossref]

Chu, M. T.

M. T. Chu, “Inverse eigenvalue problems,” SIAM Rev. 40(1), 1–39 (1998).
[Crossref]

Chu, Y.

Y. Chu, D. Wang, W. Zhu, and K. B. Crozier, “Double resonance surface enhanced Raman scattering substrates: an intuitive coupled oscillator model,” Opt. Express 19(16), 14919–14928 (2011).
[Crossref]

Y. Chu, M. G. Banaee, and K. B. Crozier, “Double-resonance plasmon substrates for surface-enhanced Raman scattering with enhancement at excitation and stokes frequencies,” ACS Nano 4(5), 2804–2810 (2010).
[Crossref]

Colton, D.

D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory (Springer Science & Business Media, 2012).

Conti, C.

L. Pilozzi, F. A. Farrelly, G. Marcucci, and C. Conti, “Machine learning inverse problem for topological photonics,” Commun. Phys. 1(1), 57 (2018).
[Crossref]

Crozier, K. B.

M. G. Banaee and K. B. Crozier, “Mixed dimer double-resonance substrates for surface-enhanced Raman spectroscopy,” ACS Nano 5(1), 307–314 (2011).
[Crossref]

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Figures (3)

Fig. 1.
Fig. 1. Inverse-designed cylinder with $2a/\lambda _0=0.4$ and a resonance at $\lambda _0/\lambda =1$. (a) Permittivity, (b) cross section, and electric field at (c) $\lambda _0/\lambda =1$ and (d) $\lambda _0/\lambda =2$.
Fig. 2.
Fig. 2. Inverse-designed cylinder with $2a/\lambda _0=0.4$ and a resonance at $\lambda _0/\lambda =2$. (a) Permittivity, (b) cross section, and electric field at (c) $\lambda _0/\lambda =1$ and (d) $\lambda _0/\lambda =2$.
Fig. 3.
Fig. 3. Inverse-designed cylinder with $2a/\lambda _0=0.4$ and resonances at $\lambda _0/\lambda =1$ and 2. (a) Permittivity, (b) cross section, and electric field at (c) $\lambda _0/\lambda =1$ and (d) $\lambda _0/\lambda =2$.

Equations (14)

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× ( × E ( x ) ) k 2 ( 1 m ( x ) ) E ( x ) = 0 ,
m ( x ) 1 ϵ r ( x ) ,
× ( × E i ( x ) ) k 2 E i ( x ) = 0 ,
lim | x | ( × E s ( x ) ) × x i k | x | E s ( x ) = 0.
E ( x ) = E i ( x ) k 2 Ω G ( x y ) m ( y ) E ( y ) d y Ω 1 1 m ( y ) m ( y ) E ( y ) G ( x y ) d y ,
p d ( x ) 1 1 m ( x ) m x d ( x ) , G d ( x ) G x d ( x ) , d = 1 , 2 , 3.
G i , j = h 3 G ( i h j h ) , G i , j d = h 3 G d ( i h j h ) ,
[ E 1 E 2 E 3 ] = [ E i 1 E i 2 E i 3 ] [ k 2 G m + G 1 p 1 G 1 p 2 G 1 p 3 G 2 p 1 k 2 G m + G 2 p 2 G 2 p 3 G 3 p 1 G 3 p 2 k 2 G m + G 3 p 3 ] [ E 1 E 2 E 3 ]
T [ k 2 G m + G 1 p 1 G 1 p 2 G 1 p 3 G 2 p 1 k 2 G m + G 2 p 2 G 2 p 3 G 3 p 1 G 3 p 2 k 2 G m + G 3 p 3 ] ,
E = E i + T E .
E = ( I T ) 1 E i ,
| det ( I T ) | = 0 ,
| det ( I + k 2 G m ) | = 0.
| det ( I + k 1 2 G ( k 1 ) m ) | = 0 , | det ( I + k 2 2 G ( k 2 ) m ) | = 0 ,