Abstract

A high-forward-transmission all-optical diode based on cascaded photonic crystal cavities is proposed. To obtain a high forward transmission and a large nonreciprocity, we generate a box-like spectrum by cascading two side-coupled photonic crystal cavities. We find that by appropriately adjusting the distance between two cavities, a high contrast of the transmission and a flat spectral lineshape with almost 100% transmission can be obtained. These characteristics are preferable for achieving a high forward transmission and a large nonreciprocity. Numerical results show that the designed all-optical diode performs perfect transmission with a large nonreciprocal transmission ratio of >50  dB.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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2016 (1)

2015 (1)

Y. Yu, Y. Chen, H. Hu, W. Xue, K. Yvind, and J. Mork, “Nonreciprocal transmission in a nonlinear photonic-crystal Fano structure with broken symmetry,” Laser Photon. Rev. 9, 241–247 (2015).
[Crossref]

2014 (2)

2013 (1)

2012 (2)

W. Ding, B. Luk’yanchuk, and C.-W. Qiu, “Ultrahigh-contrast-ratio silicon Fano diode,” Phys. Rev. A 85, 25806 (2012).
[Crossref]

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
[Crossref]

2011 (2)

L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photonics 5, 758–762 (2011).
[Crossref]

X. Huang and S. Fan, “Complete all-optical silica fiber isolator via stimulated Brillouin scattering,” J. Lightwave Technol. 29, 2267–2275 (2011).
[Crossref]

2008 (2)

2006 (2)

2004 (1)

2002 (1)

2001 (2)

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]

K. Gallo, G. Assanto, K. R. Parameswaran, and M. M. Fejer, “All-optical diode in a periodically poled lithium niobate waveguide,” Appl. Phys. Lett. 79, 314–316 (2001).
[Crossref]

2000 (2)

A. Chutinan and S. Noda, “Waveguides and waveguide bends in two-dimensional photonic crystal slabs,” Phys. Rev. B 62, 4488–4492 (2000).
[Crossref]

M. Koshiba, Y. Tsuji, and M. Hikari, “Time-domain beam propagation method and its application to photonic crystal circuits,” J. Lightwave Technol. 18, 102–110 (2000).
[Crossref]

1999 (1)

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
[Crossref]

1997 (1)

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[Crossref]

Asano, T.

Assanto, G.

K. Gallo, G. Assanto, K. R. Parameswaran, and M. M. Fejer, “All-optical diode in a periodically poled lithium niobate waveguide,” Appl. Phys. Lett. 79, 314–316 (2001).
[Crossref]

Bi, L.

L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photonics 5, 758–762 (2011).
[Crossref]

Chen, Y.

Y. Yu, Y. Chen, H. Hu, W. Xue, K. Yvind, and J. Mork, “Nonreciprocal transmission in a nonlinear photonic-crystal Fano structure with broken symmetry,” Laser Photon. Rev. 9, 241–247 (2015).
[Crossref]

Chu, S. T.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[Crossref]

Chutinan, A.

A. Chutinan and S. Noda, “Waveguides and waveguide bends in two-dimensional photonic crystal slabs,” Phys. Rev. B 62, 4488–4492 (2000).
[Crossref]

Ding, W.

W. Ding, B. Luk’yanchuk, and C.-W. Qiu, “Ultrahigh-contrast-ratio silicon Fano diode,” Phys. Rev. A 85, 25806 (2012).
[Crossref]

Dionne, G. F.

L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photonics 5, 758–762 (2011).
[Crossref]

Fan, L.

J. Wang, L. Fan, L. T. Varghese, H. Shen, Y. Xuan, B. Niu, and M. Qi, “A theoretical model for an optical diode built with nonlinear silicon microrings,” J. Lightwave Technol. 31, 313–321 (2013).
[Crossref]

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
[Crossref]

Fan, S.

Fejer, M. M.

K. Gallo, G. Assanto, K. R. Parameswaran, and M. M. Fejer, “All-optical diode in a periodically poled lithium niobate waveguide,” Appl. Phys. Lett. 79, 314–316 (2001).
[Crossref]

Foresi, J.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[Crossref]

Fujisawa, T.

Gaeta, A. L.

K. Saha, Y. Okawachi, O. Kuzucu, M. Menard, M. Lipson, and A. L. Gaeta, “Chip-scale broadband optical isolation via Bragg scattering four-wave mixing,” in Conference on Lasers and Electro-Optics (CLEO), OSA Technical Digest (online) (Optical Society of America, 2013), paper QF1D.2.

Gallo, K.

K. Gallo, G. Assanto, K. R. Parameswaran, and M. M. Fejer, “All-optical diode in a periodically poled lithium niobate waveguide,” Appl. Phys. Lett. 79, 314–316 (2001).
[Crossref]

Haus, H. A.

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]

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
[Crossref]

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[Crossref]

Hikari, M.

Hu, H.

Y. Yu, Y. Chen, H. Hu, W. Xue, K. Yvind, and J. Mork, “Nonreciprocal transmission in a nonlinear photonic-crystal Fano structure with broken symmetry,” Laser Photon. Rev. 9, 241–247 (2015).
[Crossref]

Hu, J.

L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photonics 5, 758–762 (2011).
[Crossref]

Huang, Q.

Huang, X.

Huang, Z.

Jiang, P.

L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photonics 5, 758–762 (2011).
[Crossref]

Joannopoulos, J. D.

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]

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
[Crossref]

Johnson, S. G.

Khan, M. J.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
[Crossref]

Kim, D. H.

L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photonics 5, 758–762 (2011).
[Crossref]

Kimerling, L. C.

L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photonics 5, 758–762 (2011).
[Crossref]

Kivshar, Y. S.

Koshiba, M.

Kuzucu, O.

K. Saha, Y. Okawachi, O. Kuzucu, M. Menard, M. Lipson, and A. L. Gaeta, “Chip-scale broadband optical isolation via Bragg scattering four-wave mixing,” in Conference on Lasers and Electro-Optics (CLEO), OSA Technical Digest (online) (Optical Society of America, 2013), paper QF1D.2.

Laine, J.-P.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[Crossref]

Lan, S.

Li, D.

Lin, X.-S.

Lipson, M.

K. Saha, Y. Okawachi, O. Kuzucu, M. Menard, M. Lipson, and A. L. Gaeta, “Chip-scale broadband optical isolation via Bragg scattering four-wave mixing,” in Conference on Lasers and Electro-Optics (CLEO), OSA Technical Digest (online) (Optical Society of America, 2013), paper QF1D.2.

Little, B. E.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[Crossref]

Liu, Y.

H. K. Tsang and Y. Liu, “Nonlinear optical properties of silicon waveguides,” Semicond. Sci. Technol. 23, 064007 (2008).
[Crossref]

Luk’yanchuk, B.

W. Ding, B. Luk’yanchuk, and C.-W. Qiu, “Ultrahigh-contrast-ratio silicon Fano diode,” Phys. Rev. A 85, 25806 (2012).
[Crossref]

Makino, S.

Manolatou, C.

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]

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
[Crossref]

Menard, M.

K. Saha, Y. Okawachi, O. Kuzucu, M. Menard, M. Lipson, and A. L. Gaeta, “Chip-scale broadband optical isolation via Bragg scattering four-wave mixing,” in Conference on Lasers and Electro-Optics (CLEO), OSA Technical Digest (online) (Optical Society of America, 2013), paper QF1D.2.

Mingaleev, S. F.

Mizumoto, T.

B. J. H. Stadler and T. Mizumoto, “Integrated magneto-optical materials and isolators: a review,” IEEE Photon. J. 6, 1–15 (2014).
[Crossref]

Mork, J.

Y. Yu, Y. Chen, H. Hu, W. Xue, K. Yvind, and J. Mork, “Nonreciprocal transmission in a nonlinear photonic-crystal Fano structure with broken symmetry,” Laser Photon. Rev. 9, 241–247 (2015).
[Crossref]

Niu, B.

J. Wang, L. Fan, L. T. Varghese, H. Shen, Y. Xuan, B. Niu, and M. Qi, “A theoretical model for an optical diode built with nonlinear silicon microrings,” J. Lightwave Technol. 31, 313–321 (2013).
[Crossref]

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
[Crossref]

Noda, S.

Okawachi, Y.

K. Saha, Y. Okawachi, O. Kuzucu, M. Menard, M. Lipson, and A. L. Gaeta, “Chip-scale broadband optical isolation via Bragg scattering four-wave mixing,” in Conference on Lasers and Electro-Optics (CLEO), OSA Technical Digest (online) (Optical Society of America, 2013), paper QF1D.2.

Parameswaran, K. R.

K. Gallo, G. Assanto, K. R. Parameswaran, and M. M. Fejer, “All-optical diode in a periodically poled lithium niobate waveguide,” Appl. Phys. Lett. 79, 314–316 (2001).
[Crossref]

Qi, M.

J. Wang, L. Fan, L. T. Varghese, H. Shen, Y. Xuan, B. Niu, and M. Qi, “A theoretical model for an optical diode built with nonlinear silicon microrings,” J. Lightwave Technol. 31, 313–321 (2013).
[Crossref]

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
[Crossref]

Qiu, C.-W.

W. Ding, B. Luk’yanchuk, and C.-W. Qiu, “Ultrahigh-contrast-ratio silicon Fano diode,” Phys. Rev. A 85, 25806 (2012).
[Crossref]

Ross, C. A.

L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photonics 5, 758–762 (2011).
[Crossref]

Saha, K.

K. Saha, Y. Okawachi, O. Kuzucu, M. Menard, M. Lipson, and A. L. Gaeta, “Chip-scale broadband optical isolation via Bragg scattering four-wave mixing,” in Conference on Lasers and Electro-Optics (CLEO), OSA Technical Digest (online) (Optical Society of America, 2013), paper QF1D.2.

Saitoh, K.

T. Sato, S. Makino, T. Fujisawa, and K. Saitoh, “Design of a reflection-suppressed all-optical diode based on asymmetric L-shaped nonlinear photonic crystal cavity,” J. Opt. Soc. Am. B 33, 54–61 (2016).
[Crossref]

T. Sato, T. Fujisawa, and K. Saitoh, “High-forward-transmission all-optical diode based on cascaded L3 photonic crystal cavities,” in The 24th Congress of International Commission for Optics (ICO-24) (2017), paper Th3E-03.

Sato, T.

T. Sato, S. Makino, T. Fujisawa, and K. Saitoh, “Design of a reflection-suppressed all-optical diode based on asymmetric L-shaped nonlinear photonic crystal cavity,” J. Opt. Soc. Am. B 33, 54–61 (2016).
[Crossref]

T. Sato, T. Fujisawa, and K. Saitoh, “High-forward-transmission all-optical diode based on cascaded L3 photonic crystal cavities,” in The 24th Congress of International Commission for Optics (ICO-24) (2017), paper Th3E-03.

Shen, H.

J. Wang, L. Fan, L. T. Varghese, H. Shen, Y. Xuan, B. Niu, and M. Qi, “A theoretical model for an optical diode built with nonlinear silicon microrings,” J. Lightwave Technol. 31, 313–321 (2013).
[Crossref]

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
[Crossref]

Song, B.-S.

Stadler, B. J. H.

B. J. H. Stadler and T. Mizumoto, “Integrated magneto-optical materials and isolators: a review,” IEEE Photon. J. 6, 1–15 (2014).
[Crossref]

Tsang, H. K.

H. K. Tsang and Y. Liu, “Nonlinear optical properties of silicon waveguides,” Semicond. Sci. Technol. 23, 064007 (2008).
[Crossref]

Tsuji, Y.

Uesugi, T.

Varghese, L. T.

J. Wang, L. Fan, L. T. Varghese, H. Shen, Y. Xuan, B. Niu, and M. Qi, “A theoretical model for an optical diode built with nonlinear silicon microrings,” J. Lightwave Technol. 31, 313–321 (2013).
[Crossref]

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
[Crossref]

Villeneuve, P. R.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
[Crossref]

Wang, J.

J. Wang, L. Fan, L. T. Varghese, H. Shen, Y. Xuan, B. Niu, and M. Qi, “A theoretical model for an optical diode built with nonlinear silicon microrings,” J. Lightwave Technol. 31, 313–321 (2013).
[Crossref]

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
[Crossref]

Wang, Y.

Weiner, A. M.

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
[Crossref]

Wu, L.-J.

Wu, W.-Q.

Wu, Y.

Xia, J.

Xuan, Y.

J. Wang, L. Fan, L. T. Varghese, H. Shen, Y. Xuan, B. Niu, and M. Qi, “A theoretical model for an optical diode built with nonlinear silicon microrings,” J. Lightwave Technol. 31, 313–321 (2013).
[Crossref]

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
[Crossref]

Xue, W.

Y. Yu, Y. Chen, H. Hu, W. Xue, K. Yvind, and J. Mork, “Nonreciprocal transmission in a nonlinear photonic-crystal Fano structure with broken symmetry,” Laser Photon. Rev. 9, 241–247 (2015).
[Crossref]

Yan, J.-H.

Yu, J.

Yu, Y.

Y. Yu, Y. Chen, H. Hu, W. Xue, K. Yvind, and J. Mork, “Nonreciprocal transmission in a nonlinear photonic-crystal Fano structure with broken symmetry,” Laser Photon. Rev. 9, 241–247 (2015).
[Crossref]

Yvind, K.

Y. Yu, Y. Chen, H. Hu, W. Xue, K. Yvind, and J. Mork, “Nonreciprocal transmission in a nonlinear photonic-crystal Fano structure with broken symmetry,” Laser Photon. Rev. 9, 241–247 (2015).
[Crossref]

Zeng, C.

Zhang, Y.

Zhou, H.

Zhou, K.-F.

Appl. Phys. Lett. (1)

K. Gallo, G. Assanto, K. R. Parameswaran, and M. M. Fejer, “All-optical diode in a periodically poled lithium niobate waveguide,” Appl. Phys. Lett. 79, 314–316 (2001).
[Crossref]

IEEE J. Quantum Electron. (1)

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Figures (11)

Fig. 1.
Fig. 1.

Schematic of n -cascaded side-coupled cavities.

Fig. 2.
Fig. 2.

Nonlinear transmission spectra of two symmetrically cascaded cavities for various input power and ϕ 12 when Q 1 , 2 = 10 4 , N 1 , 2 = 10 4    μm / pJ , and n 1 , 2 = 2.76 , where (a)  ϕ 12 = 0.5 π , (b)  ϕ 12 = 0.375 π , and (c)  ϕ 12 = 0.25 π .

Fig. 3.
Fig. 3.

Nonlinear transmission characteristics of two asymmetrically cascaded cavities when Δ ω 12 = 0.7 , Δ Q 12 = 2.0 , ( ω 1 = 2 π c / 1550    nm , Q 1 = 10,000 , where c is the speed of light in vacuum), ϕ 12 = 0.375 π , N 1 , 2 = 8.7 × 10 5    μm / pJ , and n 1 , 2 = 2.76 . (a) Transmission spectra for input power of 0 (linear) and 65 mW/μm (nonlinear), and (b) NTR corresponding to (a).

Fig. 4.
Fig. 4.

(a) Forward transmission, (b) backward transmission, and (c) NTR of two asymmetrically cascaded cavities for various input parameters (input power and detuning parameters), where structural parameters are the same as in Figs. 3(a) and 3(b).

Fig. 5.
Fig. 5.

FOM in the two asymmetrically cascaded cavities as a function of Δ ω 12 and Δ Q 12 for (a)  ϕ 12 = 0.375 π and (b)  ϕ 12 = 0.5 π .

Fig. 6.
Fig. 6.

Nonlinear transmission characteristics of two asymmetrically cascaded cavities when ϕ 12 = 0.375 π , N 1 , 2 = 8.7 × 10 5    μm / pJ , and n 1 , 2 = 2.76 , where the parameters ( Δ ω 12 , Δ Q 12 , P in ) are (a) (0.1, 1.0, 0.266 W/μm), (b) (0.6, 1.6, 0.108 W/μm), (c) (0.9, 2.0, 0.069 W/μm), and (d) (1.5, 4.0, 0.017 W/μm), respectively.

Fig. 7.
Fig. 7.

Schematics of the two asymmetrically cascaded L3 cavities.

Fig. 8.
Fig. 8.

Linear transmission spectrum in the two cascaded L3 cavities where a = 420    nm , r = 0.29 a , d 1 = 4 , d 2 = 5 , and p = 20 . The dashed line and circles denote results calculated by CMT and FETD-BPM, respectively.

Fig. 9.
Fig. 9.

Forward and backward transmissions in the two cascaded L3 cavities as functions of input power where δ 1 = 2.1 ( ω in = 2 π c / 1560.169    nm ). The solid lines and circles with dashed lines denote results calculated by NL-CMT and NL-FETD-BPM, respectively.

Fig. 10.
Fig. 10.

Magnetic field distribution calculated by NL-FETD-BPM during the diode operation in the two cascaded L3 cavities, where input power is 0.1 W/μm and the other parameters are the same as for Fig. 9. (a) Forward and (b) backward propagation.

Fig. 11.
Fig. 11.

Time dependence of forward and backward transmissions in the two cascaded L3 cavities where the parameters are the same as for Fig. 10. (a) Input power and (b) normalized output power calculated by NL-FETD-BPM.

Tables (1)

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Table 1. Parameter Conversions and Corresponding Solutions

Equations (30)

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Q i = ω i τ i 2 ,
d d t a i = ( j n i n i + N i | a i | 2 ω i 1 τ i ) a i + κ i ( s + i ± s ( i + 1 ) ) ,
ϕ ( i ) ( i + 1 ) = β l i + l i + 1 2 ,
δ i = ( ω i ω in ) τ i ,
T = | s + 3 | 2 | s + 1 | 2 = δ 1 4 δ 1 4 + 4 ( δ 1 cos ϕ 12 + sin ϕ 12 ) 2 .
Δ ω i j = ( ω i ω j ) τ i ,
Δ Q i j = Q j / Q i .
FOM = max ( T FW ( δ , P ) | | T FW ( δ , P ) | | T BW ( δ , P ) | > 10 3 ) ,
d d t a i = ( j ω i 1 τ i ) a i + κ i ( s + i ± s ( i + 1 ) ) ,
s i = exp ( j β l i ) ( s ( i + 1 ) κ i * a i ) ,
s + ( i + 1 ) = exp ( j β l i ) ( s + i κ i * a i ) ,
κ i = 1 τ i exp ( j θ i ) ,
a i = κ i ( s + i ± s ( i + 1 ) ) j ( ω in ω i ) + 1 τ i .
Δ n NL N i | a i | 2 .
ω i n i n i + N i | a i | 2 ω i .
d d t a i = ( j n i n i + N i | a i | 2 ω i 1 τ i ) a i + κ i ( s + i ± s ( i + 1 ) ) .
a i = κ i ( s + i ± s ( i + 1 ) ) j ( ω in n i n i + N i | a i | 2 ω i ) + 1 τ i
[ s + i s i ] = [ Q i ] [ P i ] [ Q i ] [ s + ( i + 1 ) s ( i + 1 ) ] ,
[ P i ] = [ 1 j X i j X i + j X i 1 + j X i ] ,
[ Q i ] = [ exp ( + j β 2 l i ) 0 0 ± exp ( j β 2 l i ) ] ,
X i = 1 δ i + N i | a i | 2 n i 2 Q i .
T = | s + n | 2 | s + 1 | 2 ,
R = | s 1 | 2 | s + 1 | 2 .
a i = 2 Q i ω i ( s + i ± s ( i + 1 ) ) j [ ( Δ ω 1 i δ 1 ) ω 1 Δ Q 1 i ω i + 2 Q i N i | a i | 2 n i ] + 1 exp ( j θ ) ,
s i = exp ( j β l i ) ( s ( i + 1 ) ω i 2 Q i exp ( j θ i ) a i ) ,
s + ( i + 1 ) = exp ( j β l i ) ( s + i ω i 2 Q i exp ( j θ i ) a i ) ,
N i | a i | 2 n i + N i | a i | 2 N i | a i | 2 n i .
Q 1 ξ 1 Q 1 ,
( a i , s ± i ) ( ξ 1 / 2 a i , ξ s ± i ) ,
1 Δ ω 12 ( 1 ξ ) 2 Q 1 Δ ω 12 .