Abstract

We describe a novel approach based on topology optimization that enables automatic discovery of wavelength-scale photonic structures for achieving high-efficiency second-harmonic generation (SHG). A key distinction from previous formulation and designs that seek to maximize Purcell factors at individual frequencies is that our method aims to not only achieve frequency matching (across an entire octave) and large radiative lifetimes, but also optimizes the equally important nonlinear-coupling figure of merit β¯, involving a complicated spatial overlap-integral between modes. We apply this method to the particular problem of optimizing micropost and grating-slab cavities (one-dimensional multilayered structures) and demonstrate that a variety of material platforms can support modes with the requisite frequencies, large lifetimes Q>104, small modal volumes (λ/n)3, and extremely large β¯102, leading to orders of magnitude enhancements in SHG efficiency compared to state-of-the-art photonic designs. Such giant β¯ alleviate the need for ultranarrow linewidths and thus pave the way for wavelength-scale SHG devices with faster operating timescales and higher tolerance to fabrication imperfections.

© 2016 Optical Society of America

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Corrections

7 April 2016: A correction was made to the funding section.


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References

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2015 (3)

Z. Lin, S. G. Johnson, A. W. Rodriguez, and M. Lončar, “Design of diamond microcavities for single photon frequency down-conversion,” Opt. Express 23, 25279–25294 (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, 374–377 (2015).
[Crossref]

B. Shen, P. Wang, and R. Menon, “An integrated-nanophotonics polarization beamsplitter with 2.4 × 2.4 μm2 footprint,” Nat. Photonics 9, 378–382 (2015).
[Crossref]

2014 (6)

A. Y. Piggott, J. Lu, T. M. Babinec, K. G. Lagoudakis, J. Petykiewicz, and J. Vuckovic, “Inverse design and implementation of a wavelength demultiplexing grating coupler,” Sci. Rep. 4, 7210 (2014).
[Crossref]

H. Men, K. Y. K. Lee, R. M. Freund, J. Peraire, and S. G. Johnson, “Robust topology optimization of three-dimensional photonic-crystal band-gap structures,” Opt. Express 22, 22632–22648 (2014).
[Crossref]

S. Buckley, M. Radulaski, J. L. Zhang, J. Petykiewicz, K. Biermann, and J. Vučković, “Multimode nanobeam cavities for nonlinear optics: high quality resonances separated by an octave,” Opt. Express 22, 26498–26509 (2014).
[Crossref]

C. Wang, M. J. Burek, Z. Lin, H. A. Atikian, V. Venkataraman, I.-C. Huang, P. Stark, and M. Lončar, “Integrated high quality factor lithium niobate microdisk resonators,” Opt. Express 22, 30924–30933 (2014).
[Crossref]

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

M. Minkov and V. Savona, “Automated optimization of photonic crystal slab cavities,” Sci. Rep. 4, 5124 (2014).
[Crossref]

2013 (3)

2012 (4)

S. Zaske, A. Lenhard, C. A. Keßler, J. Kettler, C. Hepp, C. Arend, R. Albrecht, W.-M. Schulz, M. Jetter, P. Michler, and C. Becher, “Visible-to-telecom quantum frequency conversion of light from a single quantum emitter,” Phys. Rev. Lett. 109, 147404 (2012).
[Crossref]

W. H. P. Pernice, C. Xiong, C. Schuck, and H. X. Tang, “Second harmonic generation in phase matched aluminum nitride waveguides and micro-ring resonators,” Appl. Phys. Lett. 100, 223501 (2012).
[Crossref]

Z.-F. Bi, A. W. Rodriguez, H. Hashemi, D. Duchesne, M. Lončar, K.-M. Wang, and S. G. Johnson, “High-efficiency second-harmonic generation in doubly-resonant X(2) microring resonators,” Opt. Express 20, 7526–7543 (2012).
[Crossref]

M. Lermer, N. Gregersen, F. Dunzer, S. Reitzenstein, S. Hofling, J. Mork, L. Worschech, M. Kamp, and A. Forchel, “Bloch-wave engineering of quantum dot micropillars for cavity quantum electrodynamics experiments,” Phys. Rev. Lett. 108, 057402 (2012).
[Crossref]

2011 (3)

F. Wang, B. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidiscip. Optim. 43, 767–784 (2011).
[Crossref]

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

J. S. Levy, M. A. Foster, A. L. Gaeta, and M. Lipson, “Harmonic generation in silicon nitride ring resonators,” Opt. Express 19, 11415–11421 (2011).
[Crossref]

2010 (4)

J. U. Fürst, 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, 153901 (2010).
[Crossref]

B. S. Darki and N. Granpayeh, “Improving the performance of a photonic crystal ring-resonator-based channel drop filter using particle swarm optimization method,” Opt. Commun. 283, 4099–4103 (2010).
[Crossref]

J. Bravo-Abad, A. W. Rodriguez, J. D. Joannopoulos, P. T. Rakich, S. G. Johnson, and M. Soljacic, “Efficient low-power terahertz generation via on-chip triply-resonant nonlinear frequency mixing,” Appl. Phys. Lett. 96, 101110 (2010).
[Crossref]

R. Krischek, W. Wieczorek, A. Ozawa, N. Kiesel, P. Michelberger, T. Udem, and H. Weinfurter, “Ultraviolet enhancement cavity for ultrafast nonlinear optics and high-rate multiphoton entanglement experiments,” Nat. Photonics 4, 170–173 (2010).
[Crossref]

2009 (2)

K. Rivoire, Z. Lin, F. Hatami, W. T. Masselink, and J. Vučković, “Second harmonic generation in gallium phosphide photonic crystal nanocavities with ultralow continuous wave pump power,” Opt. Express 17, 22609–22615 (2009).
[Crossref]

P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Lončar, “High quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. 94, 121106 (2009).
[Crossref]

2007 (1)

2006 (3)

A. Gondarenko, S. Preble, J. Robinson, L. Chen, H. Lipson, and M. Lipson, “Spontaneous emergence of periodic patterns in a biologically inspired simulation of photonic structures,” Phys. Rev. Lett. 96, 143904 (2006).
[Crossref]

P. S. Kuo, K. L. Vodopyanov, M. M. Fejer, D. M. Simanovskii, X. Yu, J. S. Harris, D. Bliss, and D. Weyburne, “Optical parametric generation of a mid-infrared continuum in orientation-patterned GaAs,” Opt. Lett. 31, 71–73 (2006).
[Crossref]

K. L. Vodopyanov, M. M. Fejer, X. Yu, J. S. Harris, Y.-S. Lee, W. C. Hurlbut, V. G. Kozlov, D. Bliss, and C. Lynch, “Terahertz-wave generation in quasi-phase-matched GaAs,” Appl. Phys. Lett. 89, 141119 (2006).
[Crossref]

2005 (2)

S. Tanzilli, W. Tittel, M. Halder, O. Alibart, P. Baldi, N. Gisin, and H. Zbinden, “A photonic quantum information interface,” Nature 437, 116–120 (2005).
[Crossref]

Q. Xu and M. Lipson, “Carrier-induced optical bistability in silicon ring resonators,” Opt. Lett. 31, 341–343 (2005).
[Crossref]

2004 (3)

2003 (2)

M. Soljacic, C. Luo, J. D. Joannopoulos, and S. Fan, “Nonlinear photonic crystal microdevices for optical integration,” Opt. Lett. 28, 637–639 (2003).
[Crossref]

M. F. Yanik, S. Fan, and M. Soljacic, “High-contrast all-optical bistable switching in photonic crystal microcavities,” Appl. Phys. Lett. 83, 2739–2741 (2003).
[Crossref]

2002 (3)

M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in non-linear photonic crystals,” Phys. Rev. E 66, 055601(R) (2002).

A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89, 240401 (2002).
[Crossref]

K. Svanberg, “A class of globally convergent optimization methods based on conservative convex separable approximations,” SIAM J. Optim. 12, 555–573 (2002).
[Crossref]

1997 (5)

M. H. MacDougal, P. D. Dapkus, A. E. Bond, C.-K. Lin, and J. Geske, “Design and fabrication of VCSELs with AlxOy-GaAs DBRs,” IEEE J. Sel. Top. Quantum Electron. 3, 905–915 (1997).
[Crossref]

M. Wu, G. Li, W. Yuen, and C. Chang-Hasnain, “Widely tunable 1.5 μm micromechanical optical filter using AlOx/AlGaAs DBR,” Electron. Lett. 33, 1702–1704 (1997).
[Crossref]

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[Crossref]

G. D. Miller, R. G. Batchko, W. M. Tulloch, D. R. Weise, M. M. Fejer, and R. L. Byer, “42%-efficient single-pass cw second-harmonic generation in periodically poled lithium niobate,” Opt. Lett. 22, 1834–1836 (1997).
[Crossref]

M. A. Arbore, A. Galvanauskas, D. Harter, M. H. Chou, and M. M. Fejer, “Engineerable compression of ultrashort pulses by use of second-harmonic generation in chirped-period-poled lithium niobate,” Opt. Lett. 22, 1341–1343 (1997).
[Crossref]

1994 (1)

1982 (1)

T. F. Heinz, C. K. Chen, D. Ricard, and Y. R. Shen, “Spectroscopy of molecular monolayers by resonant second-harmonic generation,” Phys. Rev. Lett. 48, 478–481 (1982).
[Crossref]

Albrecht, R.

S. Zaske, A. Lenhard, C. A. Keßler, J. Kettler, C. Hepp, C. Arend, R. Albrecht, W.-M. Schulz, M. Jetter, P. Michler, and C. Becher, “Visible-to-telecom quantum frequency conversion of light from a single quantum emitter,” Phys. Rev. Lett. 109, 147404 (2012).
[Crossref]

Alibart, O.

S. Tanzilli, W. Tittel, M. Halder, O. Alibart, P. Baldi, N. Gisin, and H. Zbinden, “A photonic quantum information interface,” Nature 437, 116–120 (2005).
[Crossref]

Almeida, V. R.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431, 1081–1084 (2004).
[Crossref]

Andersen, U. L.

J. U. Fürst, 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, 153901 (2010).
[Crossref]

Arbore, M. A.

Arend, C.

S. Zaske, A. Lenhard, C. A. Keßler, J. Kettler, C. Hepp, C. Arend, R. Albrecht, W.-M. Schulz, M. Jetter, P. Michler, and C. Becher, “Visible-to-telecom quantum frequency conversion of light from a single quantum emitter,” Phys. Rev. Lett. 109, 147404 (2012).
[Crossref]

Atikian, H. A.

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, 374–377 (2015).
[Crossref]

A. Y. Piggott, J. Lu, T. M. Babinec, K. G. Lagoudakis, J. Petykiewicz, and J. Vuckovic, “Inverse design and implementation of a wavelength demultiplexing grating coupler,” Sci. Rep. 4, 7210 (2014).
[Crossref]

Baldi, P.

S. Tanzilli, W. Tittel, M. Halder, O. Alibart, P. Baldi, N. Gisin, and H. Zbinden, “A photonic quantum information interface,” Nature 437, 116–120 (2005).
[Crossref]

Barrios, C. A.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431, 1081–1084 (2004).
[Crossref]

Batchko, R. G.

Becher, C.

S. Zaske, A. Lenhard, C. A. Keßler, J. Kettler, C. Hepp, C. Arend, R. Albrecht, W.-M. Schulz, M. Jetter, P. Michler, and C. Becher, “Visible-to-telecom quantum frequency conversion of light from a single quantum emitter,” Phys. Rev. Lett. 109, 147404 (2012).
[Crossref]

Bendose, M. P.

M. P. Bendose and O. Sigmund, Topology Optimization (Springer, 2004).

Bi, Z.-F.

Biermann, K.

Bliss, D.

P. S. Kuo, K. L. Vodopyanov, M. M. Fejer, D. M. Simanovskii, X. Yu, J. S. Harris, D. Bliss, and D. Weyburne, “Optical parametric generation of a mid-infrared continuum in orientation-patterned GaAs,” Opt. Lett. 31, 71–73 (2006).
[Crossref]

K. L. Vodopyanov, M. M. Fejer, X. Yu, J. S. Harris, Y.-S. Lee, W. C. Hurlbut, V. G. Kozlov, D. Bliss, and C. Lynch, “Terahertz-wave generation in quasi-phase-matched GaAs,” Appl. Phys. Lett. 89, 141119 (2006).
[Crossref]

Bloemer, M. J.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
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A. Y. Piggott, J. Lu, T. M. Babinec, K. G. Lagoudakis, J. Petykiewicz, and J. Vuckovic, “Inverse design and implementation of a wavelength demultiplexing grating coupler,” Sci. Rep. 4, 7210 (2014).
[Crossref]

Piggott, A. Y.

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, 374–377 (2015).
[Crossref]

A. Y. Piggott, J. Lu, T. M. Babinec, K. G. Lagoudakis, J. Petykiewicz, and J. Vuckovic, “Inverse design and implementation of a wavelength demultiplexing grating coupler,” Sci. Rep. 4, 7210 (2014).
[Crossref]

Preble, S.

A. Gondarenko, S. Preble, J. Robinson, L. Chen, H. Lipson, and M. Lipson, “Spontaneous emergence of periodic patterns in a biologically inspired simulation of photonic structures,” Phys. Rev. Lett. 96, 143904 (2006).
[Crossref]

Radulaski, M.

Rakich, P. T.

J. Bravo-Abad, A. W. Rodriguez, J. D. Joannopoulos, P. T. Rakich, S. G. Johnson, and M. Soljacic, “Efficient low-power terahertz generation via on-chip triply-resonant nonlinear frequency mixing,” Appl. Phys. Lett. 96, 101110 (2010).
[Crossref]

Reitzenstein, S.

M. Lermer, N. Gregersen, F. Dunzer, S. Reitzenstein, S. Hofling, J. Mork, L. Worschech, M. Kamp, and A. Forchel, “Bloch-wave engineering of quantum dot micropillars for cavity quantum electrodynamics experiments,” Phys. Rev. Lett. 108, 057402 (2012).
[Crossref]

Ricard, D.

T. F. Heinz, C. K. Chen, D. Ricard, and Y. R. Shen, “Spectroscopy of molecular monolayers by resonant second-harmonic generation,” Phys. Rev. Lett. 48, 478–481 (1982).
[Crossref]

Rivoire, K.

Robinson, J.

A. Gondarenko, S. Preble, J. Robinson, L. Chen, H. Lipson, and M. Lipson, “Spontaneous emergence of periodic patterns in a biologically inspired simulation of photonic structures,” Phys. Rev. Lett. 96, 143904 (2006).
[Crossref]

Rodriguez, A.

Rodriguez, A. W.

Savona, V.

M. Minkov and V. Savona, “Automated optimization of photonic crystal slab cavities,” Sci. Rep. 4, 5124 (2014).
[Crossref]

Scalora, M.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[Crossref]

Schrempel, F.

S. Diziain, R. Geiss, M. Zilk, F. Schrempel, E.-B. Kley, A. Tunnermann, and T. Pertsch, “Second harmonic generation in free-standing lithium niobate photonic crystal l3 cavity,” Appl. Phys. Lett. 103, 051117 (2013).
[Crossref]

Schuck, C.

W. H. P. Pernice, C. Xiong, C. Schuck, and H. X. Tang, “Second harmonic generation in phase matched aluminum nitride waveguides and micro-ring resonators,” Appl. Phys. Lett. 100, 223501 (2012).
[Crossref]

Schulz, W.-M.

S. Zaske, A. Lenhard, C. A. Keßler, J. Kettler, C. Hepp, C. Arend, R. Albrecht, W.-M. Schulz, M. Jetter, P. Michler, and C. Becher, “Visible-to-telecom quantum frequency conversion of light from a single quantum emitter,” Phys. Rev. Lett. 109, 147404 (2012).
[Crossref]

Shen, B.

B. Shen, P. Wang, and R. Menon, “An integrated-nanophotonics polarization beamsplitter with 2.4 × 2.4 μm2 footprint,” Nat. Photonics 9, 378–382 (2015).
[Crossref]

Shen, Y. R.

T. F. Heinz, C. K. Chen, D. Ricard, and Y. R. Shen, “Spectroscopy of molecular monolayers by resonant second-harmonic generation,” Phys. Rev. Lett. 48, 478–481 (1982).
[Crossref]

Sigmund, O.

F. Wang, B. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidiscip. Optim. 43, 767–784 (2011).
[Crossref]

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

M. P. Bendose and O. Sigmund, Topology Optimization (Springer, 2004).

Simanovskii, D. M.

Soljacic, M.

J. Bravo-Abad, A. W. Rodriguez, J. D. Joannopoulos, P. T. Rakich, S. G. Johnson, and M. Soljacic, “Efficient low-power terahertz generation via on-chip triply-resonant nonlinear frequency mixing,” Appl. Phys. Lett. 96, 101110 (2010).
[Crossref]

A. Rodriguez, M. Soljačić, J. D. Joannopoulos, and S. G. Johnson, “X(2) and X(3) harmonic generation at a critical power in inhomogeneous doubly resonant cavities,” Opt. Express 15, 7303–7318 (2007).
[Crossref]

M. F. Yanik, S. Fan, M. Soljačić, and J. D. Joannopoulos, “All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry,” Opt. Lett. 28, 2506 (2004).
[Crossref]

M. Soljacic, C. Luo, J. D. Joannopoulos, and S. Fan, “Nonlinear photonic crystal microdevices for optical integration,” Opt. Lett. 28, 637–639 (2003).
[Crossref]

M. F. Yanik, S. Fan, and M. Soljacic, “High-contrast all-optical bistable switching in photonic crystal microcavities,” Appl. Phys. Lett. 83, 2739–2741 (2003).
[Crossref]

M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in non-linear photonic crystals,” Phys. Rev. E 66, 055601(R) (2002).

Solomon, S. G.

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

Stark, P.

Strekalov, D. V.

J. U. Fürst, 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, 153901 (2010).
[Crossref]

Svanberg, K.

K. Svanberg, “A class of globally convergent optimization methods based on conservative convex separable approximations,” SIAM J. Optim. 12, 555–573 (2002).
[Crossref]

Tang, H. X.

W. H. P. Pernice, C. Xiong, C. Schuck, and H. X. Tang, “Second harmonic generation in phase matched aluminum nitride waveguides and micro-ring resonators,” Appl. Phys. Lett. 100, 223501 (2012).
[Crossref]

Tanzilli, S.

S. Tanzilli, W. Tittel, M. Halder, O. Alibart, P. Baldi, N. Gisin, and H. Zbinden, “A photonic quantum information interface,” Nature 437, 116–120 (2005).
[Crossref]

Tittel, W.

S. Tanzilli, W. Tittel, M. Halder, O. Alibart, P. Baldi, N. Gisin, and H. Zbinden, “A photonic quantum information interface,” Nature 437, 116–120 (2005).
[Crossref]

Trebino, R.

Tulloch, W. M.

Tunnermann, A.

S. Diziain, R. Geiss, M. Zilk, F. Schrempel, E.-B. Kley, A. Tunnermann, and T. Pertsch, “Second harmonic generation in free-standing lithium niobate photonic crystal l3 cavity,” Appl. Phys. Lett. 103, 051117 (2013).
[Crossref]

Udem, T.

R. Krischek, W. Wieczorek, A. Ozawa, N. Kiesel, P. Michelberger, T. Udem, and H. Weinfurter, “Ultraviolet enhancement cavity for ultrafast nonlinear optics and high-rate multiphoton entanglement experiments,” Nat. Photonics 4, 170–173 (2010).
[Crossref]

Vahala, K.

K. Vahala, Optical Microcavities (World Scientific, 2004).

Vaziri, A.

A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89, 240401 (2002).
[Crossref]

Venkataraman, V.

Viswanathan, R.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[Crossref]

Vodopyanov, K. L.

K. L. Vodopyanov, M. M. Fejer, X. Yu, J. S. Harris, Y.-S. Lee, W. C. Hurlbut, V. G. Kozlov, D. Bliss, and C. Lynch, “Terahertz-wave generation in quasi-phase-matched GaAs,” Appl. Phys. Lett. 89, 141119 (2006).
[Crossref]

P. S. Kuo, K. L. Vodopyanov, M. M. Fejer, D. M. Simanovskii, X. Yu, J. S. Harris, D. Bliss, and D. Weyburne, “Optical parametric generation of a mid-infrared continuum in orientation-patterned GaAs,” Opt. Lett. 31, 71–73 (2006).
[Crossref]

Vuckovic, J.

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, 374–377 (2015).
[Crossref]

A. Y. Piggott, J. Lu, T. M. Babinec, K. G. Lagoudakis, J. Petykiewicz, and J. Vuckovic, “Inverse design and implementation of a wavelength demultiplexing grating coupler,” Sci. Rep. 4, 7210 (2014).
[Crossref]

S. Buckley, M. Radulaski, J. L. Zhang, J. Petykiewicz, K. Biermann, and J. Vučković, “Multimode nanobeam cavities for nonlinear optics: high quality resonances separated by an octave,” Opt. Express 22, 26498–26509 (2014).
[Crossref]

K. Rivoire, Z. Lin, F. Hatami, W. T. Masselink, and J. Vučković, “Second harmonic generation in gallium phosphide photonic crystal nanocavities with ultralow continuous wave pump power,” Opt. Express 17, 22609–22615 (2009).
[Crossref]

Wang, C.

Wang, F.

F. Wang, B. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidiscip. Optim. 43, 767–784 (2011).
[Crossref]

Wang, K.-M.

Wang, P.

B. Shen, P. Wang, and R. Menon, “An integrated-nanophotonics polarization beamsplitter with 2.4 × 2.4 μm2 footprint,” Nat. Photonics 9, 378–382 (2015).
[Crossref]

Weihs, G.

A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89, 240401 (2002).
[Crossref]

Weinfurter, H.

R. Krischek, W. Wieczorek, A. Ozawa, N. Kiesel, P. Michelberger, T. Udem, and H. Weinfurter, “Ultraviolet enhancement cavity for ultrafast nonlinear optics and high-rate multiphoton entanglement experiments,” Nat. Photonics 4, 170–173 (2010).
[Crossref]

Weise, D. R.

Weyburne, D.

White, W. E.

Wieczorek, W.

R. Krischek, W. Wieczorek, A. Ozawa, N. Kiesel, P. Michelberger, T. Udem, and H. Weinfurter, “Ultraviolet enhancement cavity for ultrafast nonlinear optics and high-rate multiphoton entanglement experiments,” Nat. Photonics 4, 170–173 (2010).
[Crossref]

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd Ed. (Princeton University, 2008).

Worschech, L.

M. Lermer, N. Gregersen, F. Dunzer, S. Reitzenstein, S. Hofling, J. Mork, L. Worschech, M. Kamp, and A. Forchel, “Bloch-wave engineering of quantum dot micropillars for cavity quantum electrodynamics experiments,” Phys. Rev. Lett. 108, 057402 (2012).
[Crossref]

Wu, M.

M. Wu, G. Li, W. Yuen, and C. Chang-Hasnain, “Widely tunable 1.5 μm micromechanical optical filter using AlOx/AlGaAs DBR,” Electron. Lett. 33, 1702–1704 (1997).
[Crossref]

Xiong, C.

W. H. P. Pernice, C. Xiong, C. Schuck, and H. X. Tang, “Second harmonic generation in phase matched aluminum nitride waveguides and micro-ring resonators,” Appl. Phys. Lett. 100, 223501 (2012).
[Crossref]

Xu, Q.

Yanik, M. F.

M. F. Yanik, S. Fan, M. Soljačić, and J. D. Joannopoulos, “All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry,” Opt. Lett. 28, 2506 (2004).
[Crossref]

M. F. Yanik, S. Fan, and M. Soljacic, “High-contrast all-optical bistable switching in photonic crystal microcavities,” Appl. Phys. Lett. 83, 2739–2741 (2003).
[Crossref]

Yu, X.

K. L. Vodopyanov, M. M. Fejer, X. Yu, J. S. Harris, Y.-S. Lee, W. C. Hurlbut, V. G. Kozlov, D. Bliss, and C. Lynch, “Terahertz-wave generation in quasi-phase-matched GaAs,” Appl. Phys. Lett. 89, 141119 (2006).
[Crossref]

P. S. Kuo, K. L. Vodopyanov, M. M. Fejer, D. M. Simanovskii, X. Yu, J. S. Harris, D. Bliss, and D. Weyburne, “Optical parametric generation of a mid-infrared continuum in orientation-patterned GaAs,” Opt. Lett. 31, 71–73 (2006).
[Crossref]

Yuen, W.

M. Wu, G. Li, W. Yuen, and C. Chang-Hasnain, “Widely tunable 1.5 μm micromechanical optical filter using AlOx/AlGaAs DBR,” Electron. Lett. 33, 1702–1704 (1997).
[Crossref]

Zaske, S.

S. Zaske, A. Lenhard, C. A. Keßler, J. Kettler, C. Hepp, C. Arend, R. Albrecht, W.-M. Schulz, M. Jetter, P. Michler, and C. Becher, “Visible-to-telecom quantum frequency conversion of light from a single quantum emitter,” Phys. Rev. Lett. 109, 147404 (2012).
[Crossref]

Zbinden, H.

S. Tanzilli, W. Tittel, M. Halder, O. Alibart, P. Baldi, N. Gisin, and H. Zbinden, “A photonic quantum information interface,” Nature 437, 116–120 (2005).
[Crossref]

Zeilinger, A.

A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89, 240401 (2002).
[Crossref]

Zhang, J. L.

Zhang, Y.

Y. Zhang and M. Lončar, “Sub-micron diameter micropillar cavities with high quality factors and ultra-small mode volumes,” arXiv:0902.2553 (2009).

Zilk, M.

S. Diziain, R. Geiss, M. Zilk, F. Schrempel, E.-B. Kley, A. Tunnermann, and T. Pertsch, “Second harmonic generation in free-standing lithium niobate photonic crystal l3 cavity,” Appl. Phys. Lett. 103, 051117 (2013).
[Crossref]

Appl. Phys. Lett. (6)

K. L. Vodopyanov, M. M. Fejer, X. Yu, J. S. Harris, Y.-S. Lee, W. C. Hurlbut, V. G. Kozlov, D. Bliss, and C. Lynch, “Terahertz-wave generation in quasi-phase-matched GaAs,” Appl. Phys. Lett. 89, 141119 (2006).
[Crossref]

M. F. Yanik, S. Fan, and M. Soljacic, “High-contrast all-optical bistable switching in photonic crystal microcavities,” Appl. Phys. Lett. 83, 2739–2741 (2003).
[Crossref]

J. Bravo-Abad, A. W. Rodriguez, J. D. Joannopoulos, P. T. Rakich, S. G. Johnson, and M. Soljacic, “Efficient low-power terahertz generation via on-chip triply-resonant nonlinear frequency mixing,” Appl. Phys. Lett. 96, 101110 (2010).
[Crossref]

W. H. P. Pernice, C. Xiong, C. Schuck, and H. X. Tang, “Second harmonic generation in phase matched aluminum nitride waveguides and micro-ring resonators,” Appl. Phys. Lett. 100, 223501 (2012).
[Crossref]

P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Lončar, “High quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. 94, 121106 (2009).
[Crossref]

S. Diziain, R. Geiss, M. Zilk, F. Schrempel, E.-B. Kley, A. Tunnermann, and T. Pertsch, “Second harmonic generation in free-standing lithium niobate photonic crystal l3 cavity,” Appl. Phys. Lett. 103, 051117 (2013).
[Crossref]

Electron. Lett. (1)

M. Wu, G. Li, W. Yuen, and C. Chang-Hasnain, “Widely tunable 1.5 μm micromechanical optical filter using AlOx/AlGaAs DBR,” Electron. Lett. 33, 1702–1704 (1997).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

M. H. MacDougal, P. D. Dapkus, A. E. Bond, C.-K. Lin, and J. Geske, “Design and fabrication of VCSELs with AlxOy-GaAs DBRs,” IEEE J. Sel. Top. Quantum Electron. 3, 905–915 (1997).
[Crossref]

J. Opt. Soc. Am. B (2)

Laser Photon. Rev. (1)

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

Nat. Commun. (1)

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

Nat. Photonics (3)

R. Krischek, W. Wieczorek, A. Ozawa, N. Kiesel, P. Michelberger, T. Udem, and H. Weinfurter, “Ultraviolet enhancement cavity for ultrafast nonlinear optics and high-rate multiphoton entanglement experiments,” Nat. Photonics 4, 170–173 (2010).
[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, 374–377 (2015).
[Crossref]

B. Shen, P. Wang, and R. Menon, “An integrated-nanophotonics polarization beamsplitter with 2.4 × 2.4 μm2 footprint,” Nat. Photonics 9, 378–382 (2015).
[Crossref]

Nature (2)

S. Tanzilli, W. Tittel, M. Halder, O. Alibart, P. Baldi, N. Gisin, and H. Zbinden, “A photonic quantum information interface,” Nature 437, 116–120 (2005).
[Crossref]

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431, 1081–1084 (2004).
[Crossref]

Opt. Commun. (1)

B. S. Darki and N. Granpayeh, “Improving the performance of a photonic crystal ring-resonator-based channel drop filter using particle swarm optimization method,” Opt. Commun. 283, 4099–4103 (2010).
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Opt. Express (10)

X. Liang and S. G. Johnson, “Formulation for scalable optimization of microcavities via the frequency-averaged local density of states,” Opt. Express 21, 30812–30841 (2013).
[Crossref]

D. Liu, L. H. Gabrielli, M. Lipson, and S. G. Johnson, “Transformation inverse design,” Opt. Express 21, 14223–14243 (2013).
[Crossref]

J. S. Levy, M. A. Foster, A. L. Gaeta, and M. Lipson, “Harmonic generation in silicon nitride ring resonators,” Opt. Express 19, 11415–11421 (2011).
[Crossref]

C. Wang, M. J. Burek, Z. Lin, H. A. Atikian, V. Venkataraman, I.-C. Huang, P. Stark, and M. Lončar, “Integrated high quality factor lithium niobate microdisk resonators,” Opt. Express 22, 30924–30933 (2014).
[Crossref]

Z. Lin, S. G. Johnson, A. W. Rodriguez, and M. Lončar, “Design of diamond microcavities for single photon frequency down-conversion,” Opt. Express 23, 25279–25294 (2015).
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Z.-F. Bi, A. W. Rodriguez, H. Hashemi, D. Duchesne, M. Lončar, K.-M. Wang, and S. G. Johnson, “High-efficiency second-harmonic generation in doubly-resonant X(2) microring resonators,” Opt. Express 20, 7526–7543 (2012).
[Crossref]

S. Buckley, M. Radulaski, J. L. Zhang, J. Petykiewicz, K. Biermann, and J. Vučković, “Multimode nanobeam cavities for nonlinear optics: high quality resonances separated by an octave,” Opt. Express 22, 26498–26509 (2014).
[Crossref]

A. Rodriguez, M. Soljačić, J. D. Joannopoulos, and S. G. Johnson, “X(2) and X(3) harmonic generation at a critical power in inhomogeneous doubly resonant cavities,” Opt. Express 15, 7303–7318 (2007).
[Crossref]

K. Rivoire, Z. Lin, F. Hatami, W. T. Masselink, and J. Vučković, “Second harmonic generation in gallium phosphide photonic crystal nanocavities with ultralow continuous wave pump power,” Opt. Express 17, 22609–22615 (2009).
[Crossref]

H. Men, K. Y. K. Lee, R. M. Freund, J. Peraire, and S. G. Johnson, “Robust topology optimization of three-dimensional photonic-crystal band-gap structures,” Opt. Express 22, 22632–22648 (2014).
[Crossref]

Opt. Lett. (6)

Phys. Rev. A (1)

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[Crossref]

Phys. Rev. E (1)

M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in non-linear photonic crystals,” Phys. Rev. E 66, 055601(R) (2002).

Phys. Rev. Lett. (6)

S. Zaske, A. Lenhard, C. A. Keßler, J. Kettler, C. Hepp, C. Arend, R. Albrecht, W.-M. Schulz, M. Jetter, P. Michler, and C. Becher, “Visible-to-telecom quantum frequency conversion of light from a single quantum emitter,” Phys. Rev. Lett. 109, 147404 (2012).
[Crossref]

T. F. Heinz, C. K. Chen, D. Ricard, and Y. R. Shen, “Spectroscopy of molecular monolayers by resonant second-harmonic generation,” Phys. Rev. Lett. 48, 478–481 (1982).
[Crossref]

A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89, 240401 (2002).
[Crossref]

J. U. Fürst, 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, 153901 (2010).
[Crossref]

A. Gondarenko, S. Preble, J. Robinson, L. Chen, H. Lipson, and M. Lipson, “Spontaneous emergence of periodic patterns in a biologically inspired simulation of photonic structures,” Phys. Rev. Lett. 96, 143904 (2006).
[Crossref]

M. Lermer, N. Gregersen, F. Dunzer, S. Reitzenstein, S. Hofling, J. Mork, L. Worschech, M. Kamp, and A. Forchel, “Bloch-wave engineering of quantum dot micropillars for cavity quantum electrodynamics experiments,” Phys. Rev. Lett. 108, 057402 (2012).
[Crossref]

Sci. Rep. (2)

A. Y. Piggott, J. Lu, T. M. Babinec, K. G. Lagoudakis, J. Petykiewicz, and J. Vuckovic, “Inverse design and implementation of a wavelength demultiplexing grating coupler,” Sci. Rep. 4, 7210 (2014).
[Crossref]

M. Minkov and V. Savona, “Automated optimization of photonic crystal slab cavities,” Sci. Rep. 4, 5124 (2014).
[Crossref]

SIAM J. Optim. (1)

K. Svanberg, “A class of globally convergent optimization methods based on conservative convex separable approximations,” SIAM J. Optim. 12, 555–573 (2002).
[Crossref]

Struct. Multidiscip. Optim. (1)

F. Wang, B. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidiscip. Optim. 43, 767–784 (2011).
[Crossref]

Other (5)

M. P. Bendose and O. Sigmund, Topology Optimization (Springer, 2004).

K. Vahala, Optical Microcavities (World Scientific, 2004).

Y. Zhang and M. Lončar, “Sub-micron diameter micropillar cavities with high quality factors and ultra-small mode volumes,” arXiv:0902.2553 (2009).

R. W. Boyd, Nonlinear Optics (Academic, 1992).

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd Ed. (Princeton University, 2008).

Supplementary Material (1)

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» Supplement 1: PDF (1084 KB)      Supplemental Document

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Figures (3)

Fig. 1.
Fig. 1.

Work flow of the design process. The DOFs in our problem consist of all of the pixels along the x direction of a 2D computational domain. Starting from a vacuum or uniform-dielectric slab, the optimization seeks to develop an optimal pattern of material layers (with a fixed thickness in the z direction) that can tightly confine light at the desired frequencies while ensuring maximal spatial overlap between the confined modes. The developed 2D cross-sectional pattern is truncated at a finite width in the y direction to produce a fully 3D micropost or grating cavity that is then simulated by FDTD methods to extract the resonant frequencies, quality factors, eigenmodes, and corresponding modal overlaps. Here, it must be emphasized that we merely performed 1D optimization (within a 2D computational problem) because of limited computational resources; consequently, our design space is severely constrained.

Fig. 2.
Fig. 2.

(a) Schematic illustration of a topology-optimized micropost cavity with alternating AlGaAs / Al 2 O 3 layers and dimensions h x × h y × h z = 8.4 × 3.5 × 0.84 ( λ 1 3 ) . For structural details, please refer to Supplement 1. (b)  x z cross section of the E y components of two localized modes of frequencies λ 1 = 1.5    μm and λ 2 = λ 1 / 2 .

Fig. 3.
Fig. 3.

Scatter plot of ( Q 1 rad ) 2 Q 2 rad versus nonlinear overlap | β ¯ | 2 for representative geometries, including WGMRs [26], microring and nanoring resonators [17,18], photonic crystal slabs, and nanobeam cavities [16,19]. A trend toward decreasing lifetimes and increasing overlaps is observed as devices become increasingly smaller. Meanwhile, it remains an open problem to discover structures with high Q , small V , and large | β ¯ | (shaded region).

Tables (2)

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Table 1. SHG Figures of Merit, Including Frequencies λ , Overall and Radiative Quality Factors Q ; Q r a d , and Nonlinear Coupling β Corresponding to the Fundamental and Harmonic Modes of Topology-Optimized Micropost and Grating Cavities of Different Material Systems

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Table 2. SHG Figures of Merit (see Table 1) of Representative, Hand-designed Geometries based on Bang-gap or Index-guided Confinementa

Equations (7)

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η max = ( 1 Q 1 Q 1 rad ) ( 1 Q 2 Q 2 rad ) ,
P 1 crit = 2 ω 1 ϵ 0 λ 1 3 ( χ eff ( 2 ) ) 2 | β ¯ | 2 Q 1 2 Q 2 ( 1 Q 1 Q 1 rad ) 1 ,
β ¯ = d r ϵ ¯ ( r ) E 2 * E 1 2 ( d r ϵ 1 | E 1 | 2 ) ( d r ϵ 2 | E 2 | 2 ) λ 1 3 ,
FOM 1 = Q 1 2 Q 2 | β ¯ | 2 ( 1 Q 1 Q 1 rad ) 2 ( 1 Q 2 Q 2 rad ) ,
FOM 2 = ( Q 1 rad ) 2 Q 2 rad | β ¯ | 2 ,
max ϵ ¯ α f ( ϵ ¯ α ; ω 1 ) = Re [ J 2 * · E 2 d r ] , M 1 E 1 = i ω 1 J 1 , M 2 E 2 = i ω 2 J 2 , ω 2 = 2 ω 1
J 1 = δ ( r α r 0 ) e ^ j , j { x , y , z } J 2 = ϵ ¯ ( r α ) E 1 j 2 e ^ j , M l = × 1 μ × ϵ l ( r α ) ω l 2 , l = 1,2 ϵ l ( r α ) = ϵ m + ϵ ¯ α ( ϵ d l ϵ m ) , ϵ ¯ α [ 0,1 ] ,

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