Abstract

We derive general conditions for 100% frequency conversion in any doubly resonant nonlinear cavity, for both second- and third-harmonic generation via χ (2) and χ (3) nonlinearities. We find that conversion efficiency is optimized for a certain “critical” power depending on the cavity parameters, and assuming reasonable parameters we predict 100% conversion using milliwatts of power or less. These results follow from a semi-analytical coupled-mode theory framework which is generalized from previous work to include both χ (2) and χ (3) media as well as inhomogeneous (fully vectorial) cavities, analyzed in the high-efficiency limit where down-conversion processes lead to a maximum efficiency at the critical power, and which is verified by direct finite-difference time-domain (FDTD) simulations of the nonlinear Maxwell equations. Explicit formulas for the nonlinear coupling coefficients are derived in terms of the linear cavity eigenmodes, which can be used to design and evaluate cavities in arbitrary geometries.

© 2007 Optical Society of America

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2006 (5)

M. Liscidini and L. A. Andreani, "Second-harmonic generation in doubly resonant microcavities with periodic dielectric mirrors," Phys. Rev. E 73, 016,613 (2006).
[CrossRef]

L. Fan, H. Ta-Chen, M. Fallahi, J. T. Murray, R. Bedford, Y. Kaneda, J. Hader, A. R. XZakharian, J. Moloney, S. W. Koch, and W. Stolz, "Tunable watt-level blue-green vertical-external-cavity surface-emitting lasers by intracavity frequency doubling," Appl. Phys. Lett. 88, 2251,117 (2006).
[CrossRef]

L. Scaccabarozzi, M. M. Fejer, Y. Huo, S. Fan, X. Yu, and J. S. Harris, "Enchanced second-harmonic generation in AlGaAs/AlxOy tightly confining waveguides and resonant cavities," Opt. Lett. 31, 3626-3630 (2006).
[CrossRef]

A. Di Falco, C. Conti, and G. Assanto, "Impedance matching in photonic crystal microcavities for secondharmonic generation," Opt. Lett. 31, 250 (2006).
[CrossRef] [PubMed]

A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. Burr, J. D. Joannopoulos, and S. G. Johnson, "Improving accuracy by subpixel smoothing in the finite-difference time domain," Opt. Lett. 31, 2972-2974 (2006).
[CrossRef] [PubMed]

2005 (4)

2004 (3)

M. Liscidini and L. A. Andreani, "Highly efficient second-harmonic generation in doubly resonant planar microcavities," Appl. Phys. Lett. 85, 1883 (2004).
[CrossRef]

P. P. Markowicz, H. Tiryaki, H. Pudavar, P. N. Prasad, N. N. Lepeshkin, and R. W. Boyd, "Dramatic enhancement of third-harmonic generation in three-dimensional photonic crystals," Phys. Rev. Lett. 92, 083903 (2004).
[CrossRef] [PubMed]

W. Suh, Z. Wang, and S. Fan, "Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities," IEEE J. Quantum Electron. 40, 1511-1518 (2004).
[CrossRef]

2003 (2)

P. Scotto, P. Colet, and M. San Miguel, "All-optical image processing with cavity type II second-harmonic generation," Opt. Lett. 28, 1695 (2003).
[CrossRef] [PubMed]

A. M. Malvezzi, G. Vecchi, M. Patrini, G. Guizzeti, L. C. Andreani, F. Romanato, L. Businaro, E. D. Fabrizio, A. Passaseo, and M. D. Vittorio, "Resonant second-harmonic generation in a GaAs photonic crystal waveguide," Phys. Rev. B 68, 161,306 (2003).
[CrossRef]

2002 (5)

2001 (6)

X. Mu, Y. J. Ding, H. Yang, and G. J. Salamo, "Cavity-enhanced and quasiphase-matched mutli-order reflectionsecond-harmonic generation from GaAs/AlAs and GaAs/AlGaAs multilayers," Appl. Phys. Lett. 79, 569 (2001).
[CrossRef]

J. Hald, "Second harmonic generation in an external ring cavity with a Brewster-cut nonlinear cystal: theoretical considerations," Opt. Commun. 197, 169 (2001).
[CrossRef]

S. Fan, S. G. Johnson, J. D. Joannopoulos, C. Manolatou, and H. A. Haus, "Waveguide branches in photonic crystals," J. Opt. Soc. Am. B 18, 162-165 (2001).
[CrossRef]

G. McConnell, A. I. Ferguson, and N. Langford, "Cavity-augmented frequency tripling of a continuous wave mode-locked laser," J. Phys. D: Appl. Phys 34, 2408 (2001).
[CrossRef]

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, and A. P. Shkurinov, "Enhancement of sum frequency generation near the photonic band edge under the quasiphase matching condition," Phys. Rev. E 63, 046,609 (2001).
[CrossRef]

G. D. Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidavovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J.W. Haus, and M. Bertolotti, "Photonic band edge effects in finite structures and applications to χ(2) interactions," Phys. Rev. E 64, 016,609 (2001).

2000 (1)

I. I. Zootoverkh, K. N. V., and E. G. Lariontsev, "Enhancement of the efficiency of second-harmonic generation in microlaser," Quantum Electron. 30, 565 (2000).
[CrossRef]

1999 (2)

1997 (2)

1996 (1)

1995 (2)

G. T. Moore, K. Koch, and E. C. Cheung, "Optical parametric oscillation with intracavity second-harmonic generation," Opt. Commun. 113, 463 (1995).
[CrossRef]

N. Hashizume, M. Ohashi, T. Kondo, and R. Ito, "Optical harmonic generation in multilayered structures: a comprehensive analysis," J. Opt. Soc. Am. B 12, 1894-1904 (1995).
[CrossRef]

1994 (2)

R. Paschotta, K. Fiedler, P. Kurz, and J. Mlynek, "Nonlinear mode coupling in doubly resonant frequency doublers," Appl. Phys. Lett. 58, 117 (1994).

W. J. Kozlovsky, W. P. Risk,W. Lenth, B. G. Kim, G. L. Bona, H. Jaeckel, and D. J. Webb, "Blue light generation by resonator-enhanced frequency doubling of an extended-cavity diode laser," Appl. Phys. Lett. 65, 525-527 (1994).
[CrossRef]

1993 (3)

1990 (2)

M. J. Collet and R. B. Levien, "Two-photon loss model of intracavity second-harmonic generation," PRA 43(9), 5068-5073 (1990).

M. A. Persaud, J. M. Tolchard, and A. I. Ferguson, "Efficient generation of picosecond pulses at 243 nm," IEEE J. Quantum Electron. 26, 1253-1258 (1990).
[CrossRef]

1989 (2)

1988 (1)

G. S. Dutta and J. Jolly, "Third harmonic generation in layered media in presence of optical bistability of the fundamental," Pramana J. Phys. 50, 239 (1988).

1987 (1)

1982 (1)

J. C. Berquist, H. Hemmati, and W. M. Itano, "High power second harmonic generation of 257 nm radiation in an external ring cavity," Opt. Commun. 43, 437-442 (1982).
[CrossRef]

1981 (1)

M. Brieger, H. Busener, A. Hese, F. V. Moers, and A. Renn, "Enhancement of single frequency SHG in a passive ring resonator," Opt. Commun. 38, 423-426 (1981).
[CrossRef]

1980 (1)

P. D. Drummond, K. J. McNeil, and D. F. Walls, "Non-equilibrium transitions in sub/second harmonic generation I: Semiclassical theory," Opt. Acta. 27, 321-335 (1980).
[CrossRef]

1977 (1)

A. I. Gerguson and M. H. Dunn, "Intracavity second harmonic generation in continuous-wave dye lasers," IEEE J. Quantum Electron. 13, 751-756 (1977).
[CrossRef]

1970 (1)

R. G. Smith, "Theory of intracavity optical second-harmonic generation," IEEE J. Quantum Electron. 6, 215-223 (1970).
[CrossRef]

1966 (1)

A. Ashkin, G. D. Boyd, and J. M. Dziedzic, "Resonant optical second harmonic generation and mixing," IEEE J. Quantum Electron. 2, 109-124 (1966).
[CrossRef]

1962 (1)

J. A. Armstrong, N. loembergen, J. Ducuing, and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

Aguanno, G. D.

G. D. Aguanno, M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, "Generalized coupled-mode theory for |(2) interactions in finite multi-layered structures," J. Opt. Soc. Am. B 19, 2111-2122 (2002).
[CrossRef]

G. D. Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidavovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J.W. Haus, and M. Bertolotti, "Photonic band edge effects in finite structures and applications to χ(2) interactions," Phys. Rev. E 64, 016,609 (2001).

Aktsipetrov, O. A.

Andreani, L. A.

M. Liscidini and L. A. Andreani, "Second-harmonic generation in doubly resonant microcavities with periodic dielectric mirrors," Phys. Rev. E 73, 016,613 (2006).
[CrossRef]

M. Liscidini and L. A. Andreani, "Highly efficient second-harmonic generation in doubly resonant planar microcavities," Appl. Phys. Lett. 85, 1883 (2004).
[CrossRef]

Andreani, L. C.

A. M. Malvezzi, G. Vecchi, M. Patrini, G. Guizzeti, L. C. Andreani, F. Romanato, L. Businaro, E. D. Fabrizio, A. Passaseo, and M. D. Vittorio, "Resonant second-harmonic generation in a GaAs photonic crystal waveguide," Phys. Rev. B 68, 161,306 (2003).
[CrossRef]

Armstrong, J. A.

J. A. Armstrong, N. loembergen, J. Ducuing, and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

Ashkin, A.

A. Ashkin, G. D. Boyd, and J. M. Dziedzic, "Resonant optical second harmonic generation and mixing," IEEE J. Quantum Electron. 2, 109-124 (1966).
[CrossRef]

Assanto, G.

Baets, R.

Balakin, A. V.

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, and A. P. Shkurinov, "Enhancement of sum frequency generation near the photonic band edge under the quasiphase matching condition," Phys. Rev. E 63, 046,609 (2001).
[CrossRef]

Bedford, R.

L. Fan, H. Ta-Chen, M. Fallahi, J. T. Murray, R. Bedford, Y. Kaneda, J. Hader, A. R. XZakharian, J. Moloney, S. W. Koch, and W. Stolz, "Tunable watt-level blue-green vertical-external-cavity surface-emitting lasers by intracavity frequency doubling," Appl. Phys. Lett. 88, 2251,117 (2006).
[CrossRef]

Berger, V.

Bermel, P.

Berquist, J. C.

J. C. Berquist, H. Hemmati, and W. M. Itano, "High power second harmonic generation of 257 nm radiation in an external ring cavity," Opt. Commun. 43, 437-442 (1982).
[CrossRef]

Bertolotti, M.

G. D. Aguanno, M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, "Generalized coupled-mode theory for |(2) interactions in finite multi-layered structures," J. Opt. Soc. Am. B 19, 2111-2122 (2002).
[CrossRef]

G. D. Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidavovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J.W. Haus, and M. Bertolotti, "Photonic band edge effects in finite structures and applications to χ(2) interactions," Phys. Rev. E 64, 016,609 (2001).

Bethune, D. S.

Bienstman, P.

Bloemer, M. J.

G. D. Aguanno, M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, "Generalized coupled-mode theory for |(2) interactions in finite multi-layered structures," J. Opt. Soc. Am. B 19, 2111-2122 (2002).
[CrossRef]

G. D. Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidavovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J.W. Haus, and M. Bertolotti, "Photonic band edge effects in finite structures and applications to χ(2) interactions," Phys. Rev. E 64, 016,609 (2001).

Boley, T. K. W.

Bona, G. L.

W. J. Kozlovsky, W. P. Risk,W. Lenth, B. G. Kim, G. L. Bona, H. Jaeckel, and D. J. Webb, "Blue light generation by resonator-enhanced frequency doubling of an extended-cavity diode laser," Appl. Phys. Lett. 65, 525-527 (1994).
[CrossRef]

Bowden, C. M.

G. D. Aguanno, M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, "Generalized coupled-mode theory for |(2) interactions in finite multi-layered structures," J. Opt. Soc. Am. B 19, 2111-2122 (2002).
[CrossRef]

G. D. Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidavovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J.W. Haus, and M. Bertolotti, "Photonic band edge effects in finite structures and applications to χ(2) interactions," Phys. Rev. E 64, 016,609 (2001).

Boyd, G. D.

A. Ashkin, G. D. Boyd, and J. M. Dziedzic, "Resonant optical second harmonic generation and mixing," IEEE J. Quantum Electron. 2, 109-124 (1966).
[CrossRef]

Boyd, R. W.

P. P. Markowicz, H. Tiryaki, H. Pudavar, P. N. Prasad, N. N. Lepeshkin, and R. W. Boyd, "Dramatic enhancement of third-harmonic generation in three-dimensional photonic crystals," Phys. Rev. Lett. 92, 083903 (2004).
[CrossRef] [PubMed]

D. D. Smith, G. Fischer, R. W. Boyd, and D. A. Gregory, "Cancellation of photoinduced absorption in metal nanoparticle composites through a counterintuitive consequence of local field effects," J. Opt. Soc. Am. B 14, 1625 (1997).
[CrossRef]

Brieger, M.

M. Brieger, H. Busener, A. Hese, F. V. Moers, and A. Renn, "Enhancement of single frequency SHG in a passive ring resonator," Opt. Commun. 38, 423-426 (1981).
[CrossRef]

Burr, J.

Busener, H.

M. Brieger, H. Busener, A. Hese, F. V. Moers, and A. Renn, "Enhancement of single frequency SHG in a passive ring resonator," Opt. Commun. 38, 423-426 (1981).
[CrossRef]

Bushuev, V. A.

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, and A. P. Shkurinov, "Enhancement of sum frequency generation near the photonic band edge under the quasiphase matching condition," Phys. Rev. E 63, 046,609 (2001).
[CrossRef]

Businaro, L.

A. M. Malvezzi, G. Vecchi, M. Patrini, G. Guizzeti, L. C. Andreani, F. Romanato, L. Businaro, E. D. Fabrizio, A. Passaseo, and M. D. Vittorio, "Resonant second-harmonic generation in a GaAs photonic crystal waveguide," Phys. Rev. B 68, 161,306 (2003).
[CrossRef]

Centini, M.

G. D. Aguanno, M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, "Generalized coupled-mode theory for |(2) interactions in finite multi-layered structures," J. Opt. Soc. Am. B 19, 2111-2122 (2002).
[CrossRef]

G. D. Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidavovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J.W. Haus, and M. Bertolotti, "Photonic band edge effects in finite structures and applications to χ(2) interactions," Phys. Rev. E 64, 016,609 (2001).

Cheung, E. C.

G. T. Moore, K. Koch, and E. C. Cheung, "Optical parametric oscillation with intracavity second-harmonic generation," Opt. Commun. 113, 463 (1995).
[CrossRef]

Clemen, A. E. M.

Colet, P.

Collet, M. J.

M. J. Collet and R. B. Levien, "Two-photon loss model of intracavity second-harmonic generation," PRA 43(9), 5068-5073 (1990).

Conti, C.

Cowan, A. R.

A. R. Cowan and J. F. Young, "Mode matching for second-harmonic generation in photonic crystal waveguides," Phys. Rev. E 65, 085,106 (2002).

de Sterke, C. M.

Di Falco, A.

Ding, Y. J.

X. Mu, Y. J. Ding, H. Yang, and G. J. Salamo, "Cavity-enhanced and quasiphase-matched mutli-order reflectionsecond-harmonic generation from GaAs/AlAs and GaAs/AlGaAs multilayers," Appl. Phys. Lett. 79, 569 (2001).
[CrossRef]

Dixon, G. J.

Dolgova, T. V.

Drummond, P. D.

P. D. Drummond, K. J. McNeil, and D. F. Walls, "Non-equilibrium transitions in sub/second harmonic generation I: Semiclassical theory," Opt. Acta. 27, 321-335 (1980).
[CrossRef]

Dumeige, Y.

G. D. Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidavovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J.W. Haus, and M. Bertolotti, "Photonic band edge effects in finite structures and applications to χ(2) interactions," Phys. Rev. E 64, 016,609 (2001).

Dunn, M. H.

A. I. Gerguson and M. H. Dunn, "Intracavity second harmonic generation in continuous-wave dye lasers," IEEE J. Quantum Electron. 13, 751-756 (1977).
[CrossRef]

Dutta, G. S.

G. S. Dutta and J. Jolly, "Third harmonic generation in layered media in presence of optical bistability of the fundamental," Pramana J. Phys. 50, 239 (1988).

Dziedzic, J. M.

A. Ashkin, G. D. Boyd, and J. M. Dziedzic, "Resonant optical second harmonic generation and mixing," IEEE J. Quantum Electron. 2, 109-124 (1966).
[CrossRef]

Fabrizio, E. D.

A. M. Malvezzi, G. Vecchi, M. Patrini, G. Guizzeti, L. C. Andreani, F. Romanato, L. Businaro, E. D. Fabrizio, A. Passaseo, and M. D. Vittorio, "Resonant second-harmonic generation in a GaAs photonic crystal waveguide," Phys. Rev. B 68, 161,306 (2003).
[CrossRef]

Fallahi, M.

L. Fan, H. Ta-Chen, M. Fallahi, J. T. Murray, R. Bedford, Y. Kaneda, J. Hader, A. R. XZakharian, J. Moloney, S. W. Koch, and W. Stolz, "Tunable watt-level blue-green vertical-external-cavity surface-emitting lasers by intracavity frequency doubling," Appl. Phys. Lett. 88, 2251,117 (2006).
[CrossRef]

Fan, L.

L. Fan, H. Ta-Chen, M. Fallahi, J. T. Murray, R. Bedford, Y. Kaneda, J. Hader, A. R. XZakharian, J. Moloney, S. W. Koch, and W. Stolz, "Tunable watt-level blue-green vertical-external-cavity surface-emitting lasers by intracavity frequency doubling," Appl. Phys. Lett. 88, 2251,117 (2006).
[CrossRef]

Fan, S.

Farjadpour, A.

Fedyanin, A. A.

Fejer, M. M.

Ferguson, A. I.

G. McConnell, A. I. Ferguson, and N. Langford, "Cavity-augmented frequency tripling of a continuous wave mode-locked laser," J. Phys. D: Appl. Phys 34, 2408 (2001).
[CrossRef]

M. A. Persaud, J. M. Tolchard, and A. I. Ferguson, "Efficient generation of picosecond pulses at 243 nm," IEEE J. Quantum Electron. 26, 1253-1258 (1990).
[CrossRef]

Fiedler, K.

R. Paschotta, K. Fiedler, P. Kurz, and J. Mlynek, "Nonlinear mode coupling in doubly resonant frequency doublers," Appl. Phys. Lett. 58, 117 (1994).

Fischer, G.

Fröhlich, U.

Gerguson, A. I.

A. I. Gerguson and M. H. Dunn, "Intracavity second harmonic generation in continuous-wave dye lasers," IEEE J. Quantum Electron. 13, 751-756 (1977).
[CrossRef]

Gregory, D. A.

Guizzeti, G.

A. M. Malvezzi, G. Vecchi, M. Patrini, G. Guizzeti, L. C. Andreani, F. Romanato, L. Businaro, E. D. Fabrizio, A. Passaseo, and M. D. Vittorio, "Resonant second-harmonic generation in a GaAs photonic crystal waveguide," Phys. Rev. B 68, 161,306 (2003).
[CrossRef]

Hader, J.

L. Fan, H. Ta-Chen, M. Fallahi, J. T. Murray, R. Bedford, Y. Kaneda, J. Hader, A. R. XZakharian, J. Moloney, S. W. Koch, and W. Stolz, "Tunable watt-level blue-green vertical-external-cavity surface-emitting lasers by intracavity frequency doubling," Appl. Phys. Lett. 88, 2251,117 (2006).
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J. Hald, "Second harmonic generation in an external ring cavity with a Brewster-cut nonlinear cystal: theoretical considerations," Opt. Commun. 197, 169 (2001).
[CrossRef]

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Hashizume, N.

Haus, H. A.

Haus, J.W.

G. D. Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidavovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J.W. Haus, and M. Bertolotti, "Photonic band edge effects in finite structures and applications to χ(2) interactions," Phys. Rev. E 64, 016,609 (2001).

Hemmati, H.

J. C. Berquist, H. Hemmati, and W. M. Itano, "High power second harmonic generation of 257 nm radiation in an external ring cavity," Opt. Commun. 43, 437-442 (1982).
[CrossRef]

Hese, A.

M. Brieger, H. Busener, A. Hese, F. V. Moers, and A. Renn, "Enhancement of single frequency SHG in a passive ring resonator," Opt. Commun. 38, 423-426 (1981).
[CrossRef]

Huo, Y.

Ibanescu, M.

Itano, W. M.

J. C. Berquist, H. Hemmati, and W. M. Itano, "High power second harmonic generation of 257 nm radiation in an external ring cavity," Opt. Commun. 43, 437-442 (1982).
[CrossRef]

Ito, R.

Jaeckel, H.

W. J. Kozlovsky, W. P. Risk,W. Lenth, B. G. Kim, G. L. Bona, H. Jaeckel, and D. J. Webb, "Blue light generation by resonator-enhanced frequency doubling of an extended-cavity diode laser," Appl. Phys. Lett. 65, 525-527 (1994).
[CrossRef]

Joannopoulos, J. D.

Johnson, S. G.

Jolly, J.

G. S. Dutta and J. Jolly, "Third harmonic generation in layered media in presence of optical bistability of the fundamental," Pramana J. Phys. 50, 239 (1988).

Kaneda, Y.

L. Fan, H. Ta-Chen, M. Fallahi, J. T. Murray, R. Bedford, Y. Kaneda, J. Hader, A. R. XZakharian, J. Moloney, S. W. Koch, and W. Stolz, "Tunable watt-level blue-green vertical-external-cavity surface-emitting lasers by intracavity frequency doubling," Appl. Phys. Lett. 88, 2251,117 (2006).
[CrossRef]

Kim, B. G.

W. J. Kozlovsky, W. P. Risk,W. Lenth, B. G. Kim, G. L. Bona, H. Jaeckel, and D. J. Webb, "Blue light generation by resonator-enhanced frequency doubling of an extended-cavity diode laser," Appl. Phys. Lett. 65, 525-527 (1994).
[CrossRef]

Kimble, H. J.

Kira, G.

Koch, K.

K. Koch and G. T. Moore, "Singly resonant cavity-enhanced frequency tripling," J. Opt. Soc. Am. B 16, 448 (1999).
[CrossRef]

G. T. Moore, K. Koch, and E. C. Cheung, "Optical parametric oscillation with intracavity second-harmonic generation," Opt. Commun. 113, 463 (1995).
[CrossRef]

Kondo, T.

Kozlovsky, W. J.

W. J. Kozlovsky, W. P. Risk,W. Lenth, B. G. Kim, G. L. Bona, H. Jaeckel, and D. J. Webb, "Blue light generation by resonator-enhanced frequency doubling of an extended-cavity diode laser," Appl. Phys. Lett. 65, 525-527 (1994).
[CrossRef]

Kuramochi, E.

Kurz, P.

R. Paschotta, K. Fiedler, P. Kurz, and J. Mlynek, "Nonlinear mode coupling in doubly resonant frequency doublers," Appl. Phys. Lett. 58, 117 (1994).

Langford, N.

G. McConnell, A. I. Ferguson, and N. Langford, "Cavity-augmented frequency tripling of a continuous wave mode-locked laser," J. Phys. D: Appl. Phys 34, 2408 (2001).
[CrossRef]

Lenth, W.

W. J. Kozlovsky, W. P. Risk,W. Lenth, B. G. Kim, G. L. Bona, H. Jaeckel, and D. J. Webb, "Blue light generation by resonator-enhanced frequency doubling of an extended-cavity diode laser," Appl. Phys. Lett. 65, 525-527 (1994).
[CrossRef]

Lepeshkin, N. N.

P. P. Markowicz, H. Tiryaki, H. Pudavar, P. N. Prasad, N. N. Lepeshkin, and R. W. Boyd, "Dramatic enhancement of third-harmonic generation in three-dimensional photonic crystals," Phys. Rev. Lett. 92, 083903 (2004).
[CrossRef] [PubMed]

Levenson, J. A.

G. D. Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidavovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J.W. Haus, and M. Bertolotti, "Photonic band edge effects in finite structures and applications to χ(2) interactions," Phys. Rev. E 64, 016,609 (2001).

Levien, R. B.

M. J. Collet and R. B. Levien, "Two-photon loss model of intracavity second-harmonic generation," PRA 43(9), 5068-5073 (1990).

Lin, C.

A. Villeneuve, C. C. Yang, G. I. Stegeman, C. Lin, and H. Lin, "Nonlinear refractive-index and two-photon absorption near half the band gap in AlGaAs," Appl. Phys. Lett. 62, 2465-2467 (1993).
[CrossRef]

Lin, H.

A. Villeneuve, C. C. Yang, G. I. Stegeman, C. Lin, and H. Lin, "Nonlinear refractive-index and two-photon absorption near half the band gap in AlGaAs," Appl. Phys. Lett. 62, 2465-2467 (1993).
[CrossRef]

Lipson, M.

Liscidini, M.

M. Liscidini and L. A. Andreani, "Second-harmonic generation in doubly resonant microcavities with periodic dielectric mirrors," Phys. Rev. E 73, 016,613 (2006).
[CrossRef]

M. Liscidini and L. A. Andreani, "Highly efficient second-harmonic generation in doubly resonant planar microcavities," Appl. Phys. Lett. 85, 1883 (2004).
[CrossRef]

Liu, T.-M.

T.-M. Liu, C.-T. Yu, and C.-K. Sun, "2 Ghz repetition-rate femtosecond blue sources by second-harmonic generation in a resonantly enhanced cavity," Appl. Phys. Lett. 86, 061,112 (2005).
[CrossRef]

Lotem, H.

Maes, B.

Maidykovski, A. I.

Malvezzi, A. M.

A. M. Malvezzi, G. Vecchi, M. Patrini, G. Guizzeti, L. C. Andreani, F. Romanato, L. Businaro, E. D. Fabrizio, A. Passaseo, and M. D. Vittorio, "Resonant second-harmonic generation in a GaAs photonic crystal waveguide," Phys. Rev. B 68, 161,306 (2003).
[CrossRef]

Manolatou, C.

Mantsyzov, B. I.

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, and A. P. Shkurinov, "Enhancement of sum frequency generation near the photonic band edge under the quasiphase matching condition," Phys. Rev. E 63, 046,609 (2001).
[CrossRef]

Markowicz, P. P.

P. P. Markowicz, H. Tiryaki, H. Pudavar, P. N. Prasad, N. N. Lepeshkin, and R. W. Boyd, "Dramatic enhancement of third-harmonic generation in three-dimensional photonic crystals," Phys. Rev. Lett. 92, 083903 (2004).
[CrossRef] [PubMed]

Marowsky, G.

Martemyanov, M. G.

Mattei, G.

McConnell, G.

G. McConnell, A. I. Ferguson, and N. Langford, "Cavity-augmented frequency tripling of a continuous wave mode-locked laser," J. Phys. D: Appl. Phys 34, 2408 (2001).
[CrossRef]

McNeil, K. J.

P. D. Drummond, K. J. McNeil, and D. F. Walls, "Non-equilibrium transitions in sub/second harmonic generation I: Semiclassical theory," Opt. Acta. 27, 321-335 (1980).
[CrossRef]

Mitsugi, S.

Mlynek, J.

Moers, F. V.

M. Brieger, H. Busener, A. Hese, F. V. Moers, and A. Renn, "Enhancement of single frequency SHG in a passive ring resonator," Opt. Commun. 38, 423-426 (1981).
[CrossRef]

Moore, G. T.

K. Koch and G. T. Moore, "Singly resonant cavity-enhanced frequency tripling," J. Opt. Soc. Am. B 16, 448 (1999).
[CrossRef]

G. T. Moore, K. Koch, and E. C. Cheung, "Optical parametric oscillation with intracavity second-harmonic generation," Opt. Commun. 113, 463 (1995).
[CrossRef]

Mu, X.

X. Mu, Y. J. Ding, H. Yang, and G. J. Salamo, "Cavity-enhanced and quasiphase-matched mutli-order reflectionsecond-harmonic generation from GaAs/AlAs and GaAs/AlGaAs multilayers," Appl. Phys. Lett. 79, 569 (2001).
[CrossRef]

Murray, J. T.

L. Fan, H. Ta-Chen, M. Fallahi, J. T. Murray, R. Bedford, Y. Kaneda, J. Hader, A. R. XZakharian, J. Moloney, S. W. Koch, and W. Stolz, "Tunable watt-level blue-green vertical-external-cavity surface-emitting lasers by intracavity frequency doubling," Appl. Phys. Lett. 88, 2251,117 (2006).
[CrossRef]

Nakabayashi, S.

Norton, A. H.

Notomi, M.

Ohashi, M.

Ohta, N.

Ou, Z. Y.

Ozheredov, I. A.

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, and A. P. Shkurinov, "Enhancement of sum frequency generation near the photonic band edge under the quasiphase matching condition," Phys. Rev. E 63, 046,609 (2001).
[CrossRef]

Paschotta, R.

R. Paschotta, K. Fiedler, P. Kurz, and J. Mlynek, "Nonlinear mode coupling in doubly resonant frequency doublers," Appl. Phys. Lett. 58, 117 (1994).

Passaseo, A.

A. M. Malvezzi, G. Vecchi, M. Patrini, G. Guizzeti, L. C. Andreani, F. Romanato, L. Businaro, E. D. Fabrizio, A. Passaseo, and M. D. Vittorio, "Resonant second-harmonic generation in a GaAs photonic crystal waveguide," Phys. Rev. B 68, 161,306 (2003).
[CrossRef]

Patrini, M.

A. M. Malvezzi, G. Vecchi, M. Patrini, G. Guizzeti, L. C. Andreani, F. Romanato, L. Businaro, E. D. Fabrizio, A. Passaseo, and M. D. Vittorio, "Resonant second-harmonic generation in a GaAs photonic crystal waveguide," Phys. Rev. B 68, 161,306 (2003).
[CrossRef]

Pearl, S.

Persaud, M. A.

M. A. Persaud, J. M. Tolchard, and A. I. Ferguson, "Efficient generation of picosecond pulses at 243 nm," IEEE J. Quantum Electron. 26, 1253-1258 (1990).
[CrossRef]

Peters, A.

Petrov, E. V.

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, and A. P. Shkurinov, "Enhancement of sum frequency generation near the photonic band edge under the quasiphase matching condition," Phys. Rev. E 63, 046,609 (2001).
[CrossRef]

Prasad, P. N.

P. P. Markowicz, H. Tiryaki, H. Pudavar, P. N. Prasad, N. N. Lepeshkin, and R. W. Boyd, "Dramatic enhancement of third-harmonic generation in three-dimensional photonic crystals," Phys. Rev. Lett. 92, 083903 (2004).
[CrossRef] [PubMed]

Pudavar, H.

P. P. Markowicz, H. Tiryaki, H. Pudavar, P. N. Prasad, N. N. Lepeshkin, and R. W. Boyd, "Dramatic enhancement of third-harmonic generation in three-dimensional photonic crystals," Phys. Rev. Lett. 92, 083903 (2004).
[CrossRef] [PubMed]

Renn, A.

M. Brieger, H. Busener, A. Hese, F. V. Moers, and A. Renn, "Enhancement of single frequency SHG in a passive ring resonator," Opt. Commun. 38, 423-426 (1981).
[CrossRef]

Risk, W. P.

W. J. Kozlovsky, W. P. Risk,W. Lenth, B. G. Kim, G. L. Bona, H. Jaeckel, and D. J. Webb, "Blue light generation by resonator-enhanced frequency doubling of an extended-cavity diode laser," Appl. Phys. Lett. 65, 525-527 (1994).
[CrossRef]

Rodriguez, A.

Romanato, F.

A. M. Malvezzi, G. Vecchi, M. Patrini, G. Guizzeti, L. C. Andreani, F. Romanato, L. Businaro, E. D. Fabrizio, A. Passaseo, and M. D. Vittorio, "Resonant second-harmonic generation in a GaAs photonic crystal waveguide," Phys. Rev. B 68, 161,306 (2003).
[CrossRef]

Roundy, D.

Salamo, G. J.

X. Mu, Y. J. Ding, H. Yang, and G. J. Salamo, "Cavity-enhanced and quasiphase-matched mutli-order reflectionsecond-harmonic generation from GaAs/AlAs and GaAs/AlGaAs multilayers," Appl. Phys. Lett. 79, 569 (2001).
[CrossRef]

San Miguel, M.

Scaccabarozzi, L.

Scalora, M.

G. D. Aguanno, M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, "Generalized coupled-mode theory for |(2) interactions in finite multi-layered structures," J. Opt. Soc. Am. B 19, 2111-2122 (2002).
[CrossRef]

G. D. Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidavovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J.W. Haus, and M. Bertolotti, "Photonic band edge effects in finite structures and applications to χ(2) interactions," Phys. Rev. E 64, 016,609 (2001).

Schiller, S.

Schneider, K.

Schnitzler, H.

Scotto, P.

Sheik-Bahae, M.

Shimony, Y.

Shinya, A.

Shkurinov, A. P.

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, and A. P. Shkurinov, "Enhancement of sum frequency generation near the photonic band edge under the quasiphase matching condition," Phys. Rev. E 63, 046,609 (2001).
[CrossRef]

Sibilia, C.

G. D. Aguanno, M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, "Generalized coupled-mode theory for |(2) interactions in finite multi-layered structures," J. Opt. Soc. Am. B 19, 2111-2122 (2002).
[CrossRef]

G. D. Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidavovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J.W. Haus, and M. Bertolotti, "Photonic band edge effects in finite structures and applications to χ(2) interactions," Phys. Rev. E 64, 016,609 (2001).

Smith, D. D.

Smith, R. G.

R. G. Smith, "Theory of intracavity optical second-harmonic generation," IEEE J. Quantum Electron. 6, 215-223 (1970).
[CrossRef]

Stegeman, G. I.

G. I. Stegeman, M. Sheik-Bahae, E. Van Stryland, and G. Assanto, "Large nonlinear phase shifts in second-order nonlinear-optical processes," Opt. Lett. 18, 13-15 (1993).
[CrossRef] [PubMed]

A. Villeneuve, C. C. Yang, G. I. Stegeman, C. Lin, and H. Lin, "Nonlinear refractive-index and two-photon absorption near half the band gap in AlGaAs," Appl. Phys. Lett. 62, 2465-2467 (1993).
[CrossRef]

Suh, W.

W. Suh, Z. Wang, and S. Fan, "Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities," IEEE J. Quantum Electron. 40, 1511-1518 (2004).
[CrossRef]

Sun, C.-K.

T.-M. Liu, C.-T. Yu, and C.-K. Sun, "2 Ghz repetition-rate femtosecond blue sources by second-harmonic generation in a resonantly enhanced cavity," Appl. Phys. Lett. 86, 061,112 (2005).
[CrossRef]

Ta-Chen, H.

L. Fan, H. Ta-Chen, M. Fallahi, J. T. Murray, R. Bedford, Y. Kaneda, J. Hader, A. R. XZakharian, J. Moloney, S. W. Koch, and W. Stolz, "Tunable watt-level blue-green vertical-external-cavity surface-emitting lasers by intracavity frequency doubling," Appl. Phys. Lett. 88, 2251,117 (2006).
[CrossRef]

Tanabe, T.

Tanner, C. E.

Tiryaki, H.

P. P. Markowicz, H. Tiryaki, H. Pudavar, P. N. Prasad, N. N. Lepeshkin, and R. W. Boyd, "Dramatic enhancement of third-harmonic generation in three-dimensional photonic crystals," Phys. Rev. Lett. 92, 083903 (2004).
[CrossRef] [PubMed]

Tolchard, J. M.

M. A. Persaud, J. M. Tolchard, and A. I. Ferguson, "Efficient generation of picosecond pulses at 243 nm," IEEE J. Quantum Electron. 26, 1253-1258 (1990).
[CrossRef]

Van Stryland, E.

Vecchi, G.

A. M. Malvezzi, G. Vecchi, M. Patrini, G. Guizzeti, L. C. Andreani, F. Romanato, L. Businaro, E. D. Fabrizio, A. Passaseo, and M. D. Vittorio, "Resonant second-harmonic generation in a GaAs photonic crystal waveguide," Phys. Rev. B 68, 161,306 (2003).
[CrossRef]

Vidavovic, P.

G. D. Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidavovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J.W. Haus, and M. Bertolotti, "Photonic band edge effects in finite structures and applications to χ(2) interactions," Phys. Rev. E 64, 016,609 (2001).

Villeneuve, A.

A. Villeneuve, C. C. Yang, G. I. Stegeman, C. Lin, and H. Lin, "Nonlinear refractive-index and two-photon absorption near half the band gap in AlGaAs," Appl. Phys. Lett. 62, 2465-2467 (1993).
[CrossRef]

Vittorio, M. D.

A. M. Malvezzi, G. Vecchi, M. Patrini, G. Guizzeti, L. C. Andreani, F. Romanato, L. Businaro, E. D. Fabrizio, A. Passaseo, and M. D. Vittorio, "Resonant second-harmonic generation in a GaAs photonic crystal waveguide," Phys. Rev. B 68, 161,306 (2003).
[CrossRef]

Walls, D. F.

P. D. Drummond, K. J. McNeil, and D. F. Walls, "Non-equilibrium transitions in sub/second harmonic generation I: Semiclassical theory," Opt. Acta. 27, 321-335 (1980).
[CrossRef]

Wang, Z.

W. Suh, Z. Wang, and S. Fan, "Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities," IEEE J. Quantum Electron. 40, 1511-1518 (2004).
[CrossRef]

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W. J. Kozlovsky, W. P. Risk,W. Lenth, B. G. Kim, G. L. Bona, H. Jaeckel, and D. J. Webb, "Blue light generation by resonator-enhanced frequency doubling of an extended-cavity diode laser," Appl. Phys. Lett. 65, 525-527 (1994).
[CrossRef]

Wieman, C. E.

Wu, L.-A.

Xiao, M.

Xu, Q.

Yakovlev, V. A.

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A. Villeneuve, C. C. Yang, G. I. Stegeman, C. Lin, and H. Lin, "Nonlinear refractive-index and two-photon absorption near half the band gap in AlGaAs," Appl. Phys. Lett. 62, 2465-2467 (1993).
[CrossRef]

Yang, H.

X. Mu, Y. J. Ding, H. Yang, and G. J. Salamo, "Cavity-enhanced and quasiphase-matched mutli-order reflectionsecond-harmonic generation from GaAs/AlAs and GaAs/AlGaAs multilayers," Appl. Phys. Lett. 79, 569 (2001).
[CrossRef]

Young, J. F.

A. R. Cowan and J. F. Young, "Mode matching for second-harmonic generation in photonic crystal waveguides," Phys. Rev. E 65, 085,106 (2002).

Yu, C.-T.

T.-M. Liu, C.-T. Yu, and C.-K. Sun, "2 Ghz repetition-rate femtosecond blue sources by second-harmonic generation in a resonantly enhanced cavity," Appl. Phys. Lett. 86, 061,112 (2005).
[CrossRef]

Yu, X.

Zootoverkh, I. I.

I. I. Zootoverkh, K. N. V., and E. G. Lariontsev, "Enhancement of the efficiency of second-harmonic generation in microlaser," Quantum Electron. 30, 565 (2000).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (7)

A. Villeneuve, C. C. Yang, G. I. Stegeman, C. Lin, and H. Lin, "Nonlinear refractive-index and two-photon absorption near half the band gap in AlGaAs," Appl. Phys. Lett. 62, 2465-2467 (1993).
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L. Fan, H. Ta-Chen, M. Fallahi, J. T. Murray, R. Bedford, Y. Kaneda, J. Hader, A. R. XZakharian, J. Moloney, S. W. Koch, and W. Stolz, "Tunable watt-level blue-green vertical-external-cavity surface-emitting lasers by intracavity frequency doubling," Appl. Phys. Lett. 88, 2251,117 (2006).
[CrossRef]

W. J. Kozlovsky, W. P. Risk,W. Lenth, B. G. Kim, G. L. Bona, H. Jaeckel, and D. J. Webb, "Blue light generation by resonator-enhanced frequency doubling of an extended-cavity diode laser," Appl. Phys. Lett. 65, 525-527 (1994).
[CrossRef]

X. Mu, Y. J. Ding, H. Yang, and G. J. Salamo, "Cavity-enhanced and quasiphase-matched mutli-order reflectionsecond-harmonic generation from GaAs/AlAs and GaAs/AlGaAs multilayers," Appl. Phys. Lett. 79, 569 (2001).
[CrossRef]

T.-M. Liu, C.-T. Yu, and C.-K. Sun, "2 Ghz repetition-rate femtosecond blue sources by second-harmonic generation in a resonantly enhanced cavity," Appl. Phys. Lett. 86, 061,112 (2005).
[CrossRef]

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[CrossRef]

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[CrossRef]

G. D. Aguanno, M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, "Generalized coupled-mode theory for |(2) interactions in finite multi-layered structures," J. Opt. Soc. Am. B 19, 2111-2122 (2002).
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S. Pearl, H. Lotem, and Y. Shimony, "Optimization of laser intracavity second-harmonic generation by a linear dispersion element," J. Opt. Soc. Am. B 16, 1705 (1999).
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G. McConnell, A. I. Ferguson, and N. Langford, "Cavity-augmented frequency tripling of a continuous wave mode-locked laser," J. Phys. D: Appl. Phys 34, 2408 (2001).
[CrossRef]

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P. D. Drummond, K. J. McNeil, and D. F. Walls, "Non-equilibrium transitions in sub/second harmonic generation I: Semiclassical theory," Opt. Acta. 27, 321-335 (1980).
[CrossRef]

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[CrossRef]

Opt. Express (1)

Opt. Lett. (10)

A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. Burr, J. D. Joannopoulos, and S. G. Johnson, "Improving accuracy by subpixel smoothing in the finite-difference time domain," Opt. Lett. 31, 2972-2974 (2006).
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[CrossRef]

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G. J. Dixon, C. E. Tanner, and C. E. Wieman, "432-nm source based on efficient second-harmonic generation of GaAlAs diode-laser radiation in a self-locking external resonant cavity," Opt. Lett. 14, 731-733 (1989).
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A. R. Cowan and J. F. Young, "Mode matching for second-harmonic generation in photonic crystal waveguides," Phys. Rev. E 65, 085,106 (2002).

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Figures (4)

Fig. 1.
Fig. 1.

Top: Schematic diagram of waveguide-cavity system. Input light from a waveguide (left) at one frequency (amplitude s 1+) is coupled to a cavity mode (amplitude a 1), converted to a cavity mode at another frequency (amplitude a 2) by a nonlinear process, and radiated back into the waveguide (amplitude s 2-). Reflections at the first frequency (s 1-) may also occur. Bottom: 1d example, formed by quarter-wave defect in a quarter-wave dielectric stack. Dielectric material is yellow, and electric field Ez of third-harmonic mode is shown as blue/white/red for positive/zero/negative amplitude.

Fig. 2.
Fig. 2.

Log-log plot of |s 3-|2/|s 1+|2 vs. n 2|s 1+|2 for the coupled-mode theory (grey) and FDTD (black squares), where n 2 is being varied. Also shown are the corresponding Δn/n (dashed blue) and Δω 1/ω 1 (solid red) curves.

Fig. 3.
Fig. 3.

Plot of first and second harmonic efficiency, |s 1-|2 /|s 1+|2 (black) and |s 2-|2 /|s 1+|2 (red), vs. χ (2)|s 1+|. 100% power transfer from ω 1 to ω 2 = 2ω 1 is achieved at χ (2)|s 1+| = 1.8 × 10-3.

Fig. 4.
Fig. 4.

Plot of first and third harmonic efficiency, |s 1-|2 /|s 1+|2 (black) and |s 3-|2 /|s 1+|2 (red), vs. n 2|s 1+|2. 100% power transfer from ω 1 to ω 3 = 3ω 1 is achieved at n 2|s 1+|2 = 2.8 × 10-4.

Equations (29)

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

d a k dt = ( i ω k 1 τ k ) a k + 2 τ s , k s +
s = s + + 2 τ s , k a k
d a 1 dt = ( i ω 1 1 τ 1 ) a 1 i ω 1 β 1 a 1 * a 2 + 2 τ s , 1 s +
d a 2 dt = ( i ω 2 1 τ 2 ) a 2 i ω 2 β 2 a 1 2 + 2 τ s , 2 s +
d a 1 dt = ( i ω 1 ( 1 α 11 a 1 2 α 13 a 3 2 ) 1 τ 1 ) a 1 i ω 1 β 1 ( a 1 * ) 2 a 3 + 2 τ s , 1 s +
d a 3 dt = ( i ω 3 ( 1 α 33 a 3 2 α 31 a 1 2 ) 1 τ 3 ) a 3 i ω 3 β 3 a 1 3 + 2 τ s , 3 s +
δω ω = 1 2 d 3 x δε E 2 d 3 x ε E 2 = 1 2 d 3 x E * δ P d 3 x ε E 2
δ ω 1 ω 1 = 1 4 d 3 x ijk ε χ ijk ( 2 ) [ E 1 i * ( E 2 j E 1 k * + E 1 j * E 2 k ) ] d 3 x ε E 1 2
δ ω 2 ω 2 = 1 4 d 3 x ijk ε χ ijk ( 2 ) E 2 i * E 1 j E 1 k d 3 x ε E 2 2
δ ω 1 ω 1 = 1 8 [ d 3 x ε χ ( 3 ) ( E 1 E 1 2 + 2 E 1 E 1 * 2 + 2 ( E 1 E 1 * ) ( E 3 E 3 * ) + 2 E 1 E 3 2 + 2 E 1 E 3 * 2 + 3 ( E 1 * E 1 * ) ( E 1 * E 3 ) ) d 3 x ε E 1 2 ]
δ ω 3 ω 3 = 1 8 [ d 3 x ε χ ( 3 ) ( E 3 E 3 2 + 2 E 3 E 3 * 2 + 2 ( E 1 E 1 * ) ( E 3 E 3 * ) + 2 E 1 E 3 2 + 2 E 1 E 3 * 2 + ( E 1 E 1 ) ( E 1 E 3 * ) ) d 3 x ε E 3 2 ]
α ij = 0
β 1 = 1 4 d 3 x ijk ε χ ijk ( 2 ) [ E 1 i * ( E 2 j E 1 k * + E 1 j * E 2 k ) ] [ d 3 x ε E 1 2 ] [ d 3 x ε E 2 2 ] 1 2
β 2 = 1 4 d 3 x ijk ε χ ijk ( 2 ) E 2 i * E 1 j E 1 k [ d 3 x ε E 1 2 ] [ d 3 x ε E 2 2 ] 1 2
α ii = 1 8 d 3 x ε χ ( 3 ) E i E i 2 + E i E i * 2 [ d 3 x ε E i 2 ] 2
α 31 = 1 4 d 3 x ε χ ( 3 ) E 1 2 E 3 2 + E 1 E 3 2 + E 1 E 3 * 2 [ d 3 x ε E 1 2 ] [ d 3 x ε E 3 2 ]
α 13 = α 31
β 1 = 3 8 d 3 x ε χ ( 3 ) ( E 1 * E 1 * ) ( E 1 * E 3 ) [ d 3 x ε E 1 2 ] 3 2 [ d 3 x ε E 3 2 ] 1 2
β 3 = 1 8 d 3 x ε χ ( 3 ) ( E 1 E 1 ) ( E 1 E 3 * ) [ d 3 x ε E 1 2 ] 3 2 [ d 3 x ε E 3 2 ] 1 2
s 1 + 2 = 2 ω 1 2 β 1 2 τ s , 2 τ s , 1 2 = ω 1 2 β 1 2 Q 2 Q 1 2
s 1 + 2 = [ 4 ω 1 2 β 1 2 τ s , 1 3 τ s , 3 ] 1 2 = [ ω 1 ω 3 4 β 1 2 Q 1 3 Q 3 ] 1 2
s 1 + 2 = 2 τ s , 1 ω 1 2 β 1 2 τ 1 3 τ 2
s 2 2 s 1 + 2 = τ 1 τ 2 τ s , 1 τ s , 2
s 2 2 s 1 + 2 1 ( τ s , 2 τ e , 2 + τ s , 1 τ e , 1 ) .
s 1 + 2 = [ 4 τ s , 1 2 ω 1 2 β 1 2 τ 1 5 τ 3 ] 1 2
s 3 2 s 1 + 2 = τ 1 τ 3 τ s , 1 τ s , 3
1 τ e , 1 NL ω 1 Im [ α 11 τ s , 1 2 s 1 + 2 + α 13 τ s , 3 2 τ s , 1 3 8 ω 3 2 β 3 2 s 1 + 6 ]
1 τ e , 3 NL ω 3 Im [ α 31 τ s , 1 2 s 1 + 2 + α 33 τ s , 3 2 τ s , 1 3 8 ω 3 2 β 3 2 s 1 + 6 ] .
s 3 2 s 1 + 2 1 τ s , 3 β 1 τ s , 3 τ s , 1 Im [ α 11 + 3 α 13 τ s , 3 + α 13 + 3 α 33 τ s , 1 ]

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