Abstract

In this work, we derive general conditions to achieve high efficiency cascaded third harmonic generation and three photon parametric down conversion in Kerr nonlinear resonant cavities. We employ the general yet rapid temporal coupled-mode method, previously shown to accurately predict electromagnetic conversion processes in the time domain. In our study, we find that high-efficiency cascaded third harmonic generation can be achieved in a triply resonant cavity. In contrast, high-efficiency cascaded three-photon parametric down conversion cannot be achieved directly in a triply resonant cavity, although a combination of two doubly resonant cavities and three waveguides is an effective alternative. The stabilities of the calculated steady-state solutions for both processes are revealed by applying Jacobian matrices. Finally, we find that the inclusion of self- and cross- phase modulation introduces multi-stable solutions. Further study is required to find a simple way to reliably achieve stable conversion at the highest possible efficiency.

© 2015 Optical Society of America

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2015 (1)

I. Ricciardi, S. Mosca, M. Parisi, P. Maddaloni, L. Santamaria, P. De Natale, and M. De Rosa, “Frequency comb generation in quadratic nonlinear media,” Phys. Rev. A 91, 063839 (2015).
[Crossref]

2014 (2)

O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek, C. Lange, U. Huttner, D. Golde, T. Meier, M. Kira, S. Koch, and R. Huber, “Sub-cycle control of terahertz high-harmonic generation by dynamical bloch oscillations,” Nature Photon. 8, 119–123 (2014).
[Crossref]

R. Shugayev and P. Bermel, “Time-domain simulations of nonlinear interaction in microring resonators using finite-difference and coupled mode techniques,” Opt. Express 22, 19204–19218 (2014).
[Crossref] [PubMed]

2012 (1)

C. Weedbrook, B. Perrett, K. V. Kheruntsyan, P. D. Drummond, R. C. Pooser, and O. Pfister, “Resonant cascaded down-conversion,” Phys. Rev. A 85, 033821 (2012).
[Crossref]

2011 (3)

2010 (3)

K. Rivoire, Z. Lin, F. Hatami, and Vučković, “Sum-frequency generation in doubly resonant gap photonic crystal nanocavities,” Appl. Phys. Lett 97, 043103 (2010).
[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] [PubMed]

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

2009 (4)

H. Hashemi, A. W. Rodriguez, J. Joannopoulos, M. Soljačić, and S. G. Johnson, “Nonlinear harmonic generation and devices in doubly resonant kerr cavities,” Phys. Rev. A 79, 013812 (2009).
[Crossref]

B. Corcoran, C. Monat, C. Grillet, D. Moss, B. Eggleton, T. White, L. O’Faolain, and T. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photon. 3, 206–210 (2009).
[Crossref]

I. B. Burgess, A. W. Rodriguez, M. W. McCutcheon, J. Bravo-Abad, Y. Zhang, S. G. Johnson, and M. Lončar, “Difference-frequency generation with quantum-limited efficiency in triply-resonant nonlinear cavities,” Opt. Express 17, 9241–9251 (2009).
[Crossref] [PubMed]

I. B. Burgess, Y. Zhang, M. W. McCutcheon, A. W. Rodriguez, J. Bravo-Abad, S. G. Johnson, and M. Lončar, “Design of an efficient terahertz sourceusing triply resonant nonlinearphotonic crystal cavities,” Opt. Express 17, 20099–20108 (2009).
[Crossref] [PubMed]

2008 (3)

F. Gravier and B. Boulanger, “Triple-photon generation: comparison between theory and experiment,” J. Opt. Soc. Am. B 25, 98–102 (2008).
[Crossref]

A. Hayat and M. Orenstein, “Photon conversion processes in dispersive microcavities: Quantum-field model,” Phys. Rev. A 77, 013830 (2008).
[Crossref]

M. Bieler, “Thz generation from resonant excitation of semiconductor nanostructures: Investigation of second-order nonlinear optical effects,” IEEE J. Sel. Top. Quantum Electron. 14, 458–469 (2008).
[Crossref]

2007 (5)

P. Bermel, A. Rodriguez, J. D. Joannopoulos, and M. Soljačić, “Tailoring optical nonlinearities via the purcell effect,” Phys. Rev. Lett. 99, 053601 (2007).
[Crossref] [PubMed]

A. A. Kalachev and Y. Z. Fattakhova, “Generation of triphotons upon spontaneous parametric down-conversion in a resonator,” IEEE J. Quant. Electron 37, 1087–1090 (2007).
[Crossref]

T. Carmon and K. J. Vahala, “Visible continuous emission from a silica microphotonic device by third-harmonic generation,” Nat. Phys. 3, 430–435 (2007).
[Crossref]

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

A. Parini, G. Bellanca, S. Trillo, M. Conforti, A. Locatelli, and C. D. Angelis, “Self-pulsing and bistability in nonlinear bragg gratings,” J. Opt. Soc. Am. B 24, 2229–2237 (2007).
[Crossref]

2006 (2)

B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, and J. W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature 440, 1166–1169 (2006).
[Crossref] [PubMed]

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

2005 (3)

Y. Morozov, I. Nefedov, V. Aleshkin, and I. Krasnikova, “Terahertz oscillator based on nonlinear frequency conversion in a double vertical cavity,” Semiconductors 39, 113–118 (2005).
[Crossref]

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett. 95, 133901 (2005).
[Crossref] [PubMed]

B. Maes, P. Bienstman, and R. Baets, “Modeling second-harmonic generation by use of mode expansion,” J. Opt. Soc. Am. B 22, 1378–1383 (2005).
[Crossref]

2004 (4)

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

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-q toroid microcavity,” Phys. Rev. Lett. 93, 083904 (2004).
[Crossref] [PubMed]

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Nonlinear optics and crystalline whispering gallery mode cavities,” Phys. Rev. Lett. 92, 043903 (2004).
[Crossref] [PubMed]

M. Soljačić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3, 211–219 (2004).
[Crossref]

2003 (4)

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

A. R. Cowan and J. F. Young, “Optical bistability involving photonic crystal microcavities and fano line shapes,” Phys. Rev. E 68, 046606 (2003).
[Crossref]

V. S. Ilchenko, A. B. Matsko, A. A. Savchenkov, and L. Maleki, “Low-threshold parametric nonlinear optics with quasi-phase-matched whispering-gallery modes,” JOSA B 20, 1304–1308 (2003).
[Crossref]

M. Soljačić, M. Ibanescu, S. G. Johnson, J. D. Joannopoulos, and Y. Fink, “Optical bistability in axially modulated omniguide fibers,” Opt. Lett. 28, 516–518 (2003).
[Crossref]

2001 (1)

J. Hald, “Second harmonic generation in an external ring cavity with a brewster-cut nonlinear crystal: theoretical considerations,” Opt. Commun 197, 169–173 (2001).
[Crossref]

1999 (1)

Y. H. Avetisyan, “Cavity-enhanced terahertz region difference frequency generation in surface-emitting geometry,” Proc. SPIE 3795, 501–506 (1999).
[Crossref]

1998 (1)

T. Felbinger, S. Schiller, and J. Mlynek, “Oscillation and generation of nonclassical states in three-photon down-conversion,” Phys. Rev. Lett. 80, 492 (1998).
[Crossref]

1997 (1)

1995 (1)

G. T. Moore, K. Koch, and E. Cheung, “Optical parametric oscillation with intracavity second-harmonic generation,” Opt. Commun. 113, 463–470 (1995).
[Crossref]

1994 (1)

M. M. Fejer, “Nonlinear optical frequency conversion,” Phys Today 47, 25–33 (1994).
[Crossref]

1993 (1)

1991 (1)

H. Haus and W. Huang, “Coupled-mode theory,” Proc. IEEE 79, 1505–1518 (1991).
[Crossref]

1987 (1)

1979 (1)

R. A. Baumgartner and R. Byer, “Optical parametric amplification,” IEEE J. Quant. Electron 15, 432–444 (1979).
[Crossref]

1970 (1)

R. Smith, “Theory of intracavity optical second-harmonic generation,” IEEE J. Sel. Top. Quantum Electron. 6, 215–223 (1970).
[Crossref]

1966 (1)

A. Ashkin, G. Boyd, and J. Dziedzic, “Resonant optical second harmonic generation and mixing,” IEEE J. Sel. Top. Quantum Electron. 2, 109–124 (1966).
[Crossref]

1964 (1)

A. Hurwitz, “On the conditions under which an equation has only roots with negative real parts,” Selected papers on mathematical trends in control theory 65, 273–284 (1964).

1961 (1)

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

Aleshkin, V.

Y. Morozov, I. Nefedov, V. Aleshkin, and I. Krasnikova, “Terahertz oscillator based on nonlinear frequency conversion in a double vertical cavity,” Semiconductors 39, 113–118 (2005).
[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] [PubMed]

Angelis, C. D.

Arie, A.

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett. 95, 133901 (2005).
[Crossref] [PubMed]

Ashkin, A.

A. Ashkin, G. Boyd, and J. Dziedzic, “Resonant optical second harmonic generation and mixing,” IEEE J. Sel. Top. Quantum Electron. 2, 109–124 (1966).
[Crossref]

Avetisyan, Y. H.

Y. H. Avetisyan, “Cavity-enhanced terahertz region difference frequency generation in surface-emitting geometry,” Proc. SPIE 3795, 501–506 (1999).
[Crossref]

Baets, R.

Bahabad, A.

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett. 95, 133901 (2005).
[Crossref] [PubMed]

Bartal, G.

B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, and J. W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature 440, 1166–1169 (2006).
[Crossref] [PubMed]

Baumgartner, R. A.

R. A. Baumgartner and R. Byer, “Optical parametric amplification,” IEEE J. Quant. Electron 15, 432–444 (1979).
[Crossref]

Bellanca, G.

Bencheikh, K.

Berger, V.

Bermel, P.

R. Shugayev and P. Bermel, “Time-domain simulations of nonlinear interaction in microring resonators using finite-difference and coupled mode techniques,” Opt. Express 22, 19204–19218 (2014).
[Crossref] [PubMed]

P. Bermel, A. Rodriguez, J. D. Joannopoulos, and M. Soljačić, “Tailoring optical nonlinearities via the purcell effect,” Phys. Rev. Lett. 99, 053601 (2007).
[Crossref] [PubMed]

Bieler, M.

M. Bieler, “Thz generation from resonant excitation of semiconductor nanostructures: Investigation of second-order nonlinear optical effects,” IEEE J. Sel. Top. Quantum Electron. 14, 458–469 (2008).
[Crossref]

Bienstman, P.

Boulanger, B.

Boyd, G.

A. Ashkin, G. Boyd, and J. Dziedzic, “Resonant optical second harmonic generation and mixing,” IEEE J. Sel. Top. Quantum Electron. 2, 109–124 (1966).
[Crossref]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic Press, 2003).

Bravo-Abad, J.

Burgess, I. B.

Byer, R.

R. A. Baumgartner and R. Byer, “Optical parametric amplification,” IEEE J. Quant. Electron 15, 432–444 (1979).
[Crossref]

Carmon, T.

T. Carmon and K. J. Vahala, “Visible continuous emission from a silica microphotonic device by third-harmonic generation,” Nat. Phys. 3, 430–435 (2007).
[Crossref]

Chang, D. E.

M. W. McCutcheon, D. E. Chang, Y. Zhang, M. D. Lukin, and M. Lončar, “Broad-band spectral control of single photon sources using a nonlinear photonic crystal cavity,” arXiv:0903.4706 (2009).

Cheung, E.

G. T. Moore, K. Koch, and E. Cheung, “Optical parametric oscillation with intracavity second-harmonic generation,” Opt. Commun. 113, 463–470 (1995).
[Crossref]

Christodoulides, D. N.

B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, and J. W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature 440, 1166–1169 (2006).
[Crossref] [PubMed]

Claudio Andreani, L.

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T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-q toroid microcavity,” Phys. Rev. Lett. 93, 083904 (2004).
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M. Soljačić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3, 211–219 (2004).
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T. Carmon and K. J. Vahala, “Visible continuous emission from a silica microphotonic device by third-harmonic generation,” Nat. Phys. 3, 430–435 (2007).
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[Crossref]

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Figures (10)

Fig. 1
Fig. 1 (a) schematic for cascaded third harmonic generation (b) schematic for three-photon parametric down conversion involving a coupled waveguide-cavity system. The nonlinear system contains a waveguide used for input (output) photon flow coupled to a resonant cavity with multiple resonant modes. |ai |2 represents the number of photons in ith mode, and |Si +/−|2 represents the number of input/output photons per second. In both figures, ω 1 = ω, ω 3 = 3ω, ω 9 = 9ω.
Fig. 2
Fig. 2 Conversion efficiency in the function of input power. The blue solid line shows the stable conversion efficiency, and red dash line shows the unstable calculated efficiency in steady-state. The maximum stable efficiency (37%) is much lower than the maximum efficiency without considering stability (80%). In unstable region, the system goes into limit cycle and the efficiency fluctuates with time which is shown in inset. The dark green dash dot line shows the bounds of limit cycle in the unstable region from time domain simulation, where the solid purple line shows the average efficiency.
Fig. 3
Fig. 3 Conversion efficiency versus field overlap ratio M 2/M 1 in two cases: the purple line ignores stability, while the blue line requires it. In the former, the maximum conversion efficiency increase monotonically with M 2/M 1. However, in the latter case, the maximum conversion efficiency peaks when M 2/M 1 near 1.
Fig. 4
Fig. 4 The procedure for exploring all types of ratio between three resonant modes.
Fig. 5
Fig. 5 The maximum conversion efficiency as a function of γi 0. Following the procedure of Fig. 4, when γi is varied, the other two decay rates are fixed. In figure (a) and (b), γ 1/γ 0 is varied. In figure (c) and (d), γ 3/γ 0 is varied. In figure (e) and (f), γ 9/γ 0 is varied.
Fig. 6
Fig. 6 Contour plot of maximum stable conversion efficiency as a function of M 2/M 1 and γ 3/γ 0. γ 1 = 0.1γ 0 and γ 9 = 10γ 0 are fixed. The right top inset figure shows the change of efficiency along the horizontal dash line. The right below inset figure shows the change of efficiency along the vertical dash line (M 2/M 1 is varied). C point in the figure represents the case where γ 1 : γ 3 : γ 9 = 1 : 1 : 100, M 2/M 1 = 100, and maximum stable conversion efficiency is 97%.
Fig. 7
Fig. 7 Plot of the conversion efficiency from 3ω to 1ω as a function of P/P 0. Here, P is the input source power, and P 0 is the power of critical point as shown in the figure. The critical power is the power for 100% efficiency. The threshold power in the figure means the lowest power for PDC to start. In other words, if P < PT , the conversion efficiency will always be zero. The inset shows time-dependent power ratios of the 1ω and 3ω modes at the critical power input for maximum conversion efficiency. The 3ω input source S 3 + ( t ) = S 3 + c r i t ( 1 e t / τ 1 ) e i 3 ω t gradually increases to a stable value, while the 1ω input S 1 + = S 1 + e ( t τ 2 ) 2 / Δ t 2 e i ω t is a Gaussian pulse that seeds the conversion process.
Fig. 8
Fig. 8 Schematic for cascaded 3PDC involving a coupled waveguide-cavity system. The nonlinear system contains three waveguides. The left waveguide is used for input (output) source at 9ω. The middle waveguide used to transfer 3ω photons between cavities. And the right waveguide is used to output 1ω photons. The decay rate in cavity one are set as γ 3 c a v ( 1 ) and γ 9 c a v ( 1 ) , and those in cavity two are set as γ 1 c a v 2 and γ 3 c a v 2 . Similarly, the nonlinear integral overlap in cavity one is Mcav (1), and in cavity two is Mcav (2).
Fig. 9
Fig. 9 (a) Contour plot of number of stable solutions ns as a function of ξ and ζ, and the system parameters are based on point C in Fig. 6(b). Contour plot of number of stable solutions ns (zero efficiency solution is not included) as a function of γ 3/γ 1 and m/M 1 for input power P = PT .
Fig. 10
Fig. 10 (a) Maximum stable efficiency as a function of α ≡ M 1/M 0 = M 2/M 0, as defined in Fig. 2. The inset shows the required input power for obtaining maximum stable efficiency. (b) Maximum stable efficiency as a function of β ≡ γ 1/γ 0 = γ 3/γ 0 = γ 9/γ 0. The inset shows the required input power for obtaining maximum stable efficiency.

Equations (14)

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S i = S i + + 2 γ i a i ,
d a 1 d t = [ γ 1 + i ω 1 c a v ( 1 ω 1 2 m 11 | a 1 | 2 ω 1 ω 3 m 13 | a 3 | 2 ) ] a 1 + 2 γ s , 1 S 1 + i 3 M 1 ω 1 3 ω 3 a 3 ( a 1 * ) 2 d a 3 d t = [ γ 3 + i ω 3 c a v ( 1 ω 1 ω 3 m 31 | a 1 | 2 ω 3 2 m 33 | a 3 | 2 ω 3 ω 9 m 39 | a 9 | 2 ) ] a 3 + i M 1 * ω 1 3 ω 3 a 1 3 i 3 M 2 ω 3 3 ω 9 a 9 ( a 3 * ) 2 d a 9 d t = [ γ 9 + i ω 9 c a v ( 1 ω 3 ω 9 m 93 | a 3 | 2 ω 9 2 m 99 | a 9 | 2 ) ] a 9 i M 2 * ω 3 3 ω 9 a 3 3
M i = 2 8 d r ε 0 χ ( 3 ) ( E 3 i 1 * ) 3 E 3 i [ d r ε | E 3 i 1 | 2 ] 3 / 2 [ d r ε | E 3 i | 2 ] 1 / 2 m i j = 3 2 4 * ( 1 + δ i j ) d r ε 0 χ ( 3 ) | E i E j | 2 [ d r ε | E i | 2 ] [ d r ε | E j | 2 ]
d a 3 d t = [ γ 3 + i ω 3 c a v ( 1 ω 1 ω 3 m 31 | a 1 | 2 ω 3 2 m 33 | a 3 | 2 ) ] a 3 + i M 1 * ω 1 3 ω 3 a 1 3 + 3 γ s , 3 S 3 + d a 1 d t = [ γ 1 + i ω 1 c a v ( 1 ω 1 2 m 11 | a 1 | 2 ω 1 ω 3 m 13 | a 3 | 2 ) ] a 1 i 3 M 1 ω 1 3 ω 3 a 3 ( a 1 * ) 2
J r = [ ( γ 1 + 6 γ 3 r 3 0 r 1 0 + 18 γ 9 r 9 0 r 1 0 ) ( 3 γ 3 + 9 γ 9 r 9 0 r 3 0 ) 0 3 γ 3 r 3 0 r 1 0 + 9 γ 9 r 9 0 r 1 0 γ 3 6 γ 9 r 9 0 r 3 0 3 γ 9 0 3 γ 9 r 9 0 r 3 0 γ 9 ] , J ϕ = [ ( γ 1 + 6 γ 3 r 3 0 r 1 0 + 18 γ 9 r 9 0 r 1 0 ) ( 3 γ 3 r 3 0 r 1 0 + 9 γ 9 r 9 0 r 1 0 ) 0 3 γ 3 + 9 γ 9 r 9 0 r 3 0 ( γ 3 + 2 γ 9 r 9 0 r 3 0 ) 3 γ 9 r 9 0 r 3 0 0 3 γ 9 γ 9 ]
P c r i t = ω 3 | S 3 + c r i t | 2 = ω 3 2 3 M γ 1 3 γ 3 3 ω 1 3 ω 3 .
P T = ω 3 | S 3 + min | 2 = ω 3 8 27 M γ 1 3 γ 3 ω 1 3 ω 3 0.77 P c r i t
ω 1 c a v = ω 1 1 ω 1 ω 3 m 31 r 1 c r i t ω 3 2 m 33 r 3 c r i t ω 3 c a v = ω 3 1 ω 1 ω 3 m 31 r 1 c r i t ω 3 2 m 33 r 3 c r i t ω 3 ω 9 m 93 r 9 c r i t ω 9 c a v = ω 9 1 ω 3 ω 9 m 93 r 3 c r i t ω 9 2 m 99 r 9 c r i t
H = d r ε 0 [ χ ( 1 ) ( r ) + χ ( 3 ) ( r ) | E ( r ) | 2 ] [ ε E ( r ) 2 + 1 ε μ B ( r ) 2 ]
H k = 2 C k 2 d r ε ( r ) | g k ( r ) | 2 ( a ^ k a ^ k + a ^ k a ^ k )
C k = 1 2 ω k d r ε ( r ) | g k ( r ) | 2
H = k ω k ( a ^ k a ^ k + 1 2 ) + 2 i , j , k l ϕ = i , j , k , l C ϕ × [ a ^ ϕ g ϕ ( r ) e i ω ϕ t + a ^ ϕ g ϕ * ( r ) e i ω ϕ t ]
d a ^ 1 d t = i ω 1 a ^ 1 2 i C 1 [ g 1 * ( r ) e i ω t ] j , k , l ϕ = j , k , l C ϕ × [ a ^ ϕ g ϕ ( r ) e i ω ϕ t + a ^ ϕ g ϕ * ( r ) e i ω ϕ t ]
d a 1 d t = i ω 1 ( 1 ω 1 2 m 11 | a 1 | 2 ω 1 ω 3 m 13 | a 3 | 2 ) a 1 i M 1 ω 1 3 ω 3 a 3 ( a 1 * ) 2 ,

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