Abstract

In this manuscript we propose an easily scalable true all-optical AND logic gate for pulsed signal operation based on band-gap transmission within nonlinear realistic air-hole type coupled photonic crystal waveguides (C-PCW). We call it “true” all-optical AND logic gate, because all AND gate topologies operate with temporal solitons that maintain a stable pulse envelope during the optical signal processing along the different C-PCW modules yielding ultrafast full-optical digital signal processing. We directly use the registered (output) signal pulse as new input signal between multiple concatenated nonlinear C-PCW modules (i.e. AND gates) to setup a multiple-input true all-optical AND logic gate. Extensive full-wave computational electromagnetic analysis proves the correctness of our theoretical studies and the proposed operation principle of the multiple-input AND logic gate is vividly demonstrated for realistic C-PCWs.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
Ultracompact all-optical XOR logic gate in a slow-light silicon photonic crystal waveguide

C. Husko, T. D. Vo, B. Corcoran, J. Li, T. F. Krauss, and B. J. Eggleton
Opt. Express 19(21) 20681-20690 (2011)

All-optical picoseconds logic gates based on a fiber optical parametric amplifier

David Ming Fai Lai, C. H. Kwok, and Kenneth Kin-Yip Wong
Opt. Express 16(22) 18362-18370 (2008)

All-optical logic gates based on two-dimensional low-refractive-index nonlinear photonic crystal slabs

Ye Liu, Fei Qin, Zi-Ming Meng, Fei Zhou, Qing-He Mao, and Zhi-Yuan Li
Opt. Express 19(3) 1945-1953 (2011)

References

  • View by:
  • |
  • |
  • |

  1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059 (1987).
    [Crossref] [PubMed]
  2. K. Yasumoto ed., Electromagnetic Theory and Applications for Photonic Crystals (CRC Press, 2005).
    [Crossref]
  3. J.-M. Brosi, C. Koos, L. C. Andreani, M. Waldow, J. Leuthold, and W. Freude, “High-speed low-voltage electro-optic modulator with a polymer-infiltrated silicon photonic crystal waveguide,” Opt. Express 16, 4177–4191 (2008).
    [Crossref] [PubMed]
  4. Y. Ishizaka, Y. Kawaguchi, K. Saitoh, and M. Koshiba, “Design of ultra compact all-optical XOR and ANG logic gates with low power consumption,” Optics Communication 284, 3528–3533 (2011).
    [Crossref]
  5. P. Rani, S. Fatima, Y. Kalra, and R. K. Sinha, “Realization of all optical logic gates using universal NAND gates on photonic crystal platform,” Superlattices and Microstructures 109, 619–625 (2017).
    [Crossref]
  6. P. Andalib and N. Granpayeh, “All-optical ultracompact photonic crystal AND gate based on nonlinear ring resonators,” J. Opt. Soc. Am. B 26, 10–16 (2009).
    [Crossref]
  7. P. Andalib and N. Granpayeh, “All-optical ultracompact photonic crystal NOR gate based on nonlinear ring resonators,” Journal of Opt. A: Pure Appl. Opt. 11, 085203 (2009).
    [Crossref]
  8. Y. Fu, X. Hu, and Q. Gong, “Silicon photonic crystal all-optical logic gates,” Phys. Lett. A 377, 329–333 (2013).
    [Crossref]
  9. C. Husko, T. D. Vo, B. Corcoran, J. Li, T. Krauss, and B. Eggleton, “Ultracompact all-optical XOR logic gate in a slow-light silicon photonic crystal waveguide,” Opt. Express 19, 20681–20690 (2011).
    [Crossref] [PubMed]
  10. Q. Liu, Z. Ouyang, C. Wu, C. Liu, and J. Wang, “All-optical half adder based on cross structures in two-dimensional photonic crystals,” Opt. Express 16, 18992–19000 (2008).
    [Crossref]
  11. F. Geniet and J. Leon, “Energy transmission in the forbidden band gap of a nonlinear chain”, Phys. Rev. Lett. 89, 134102 (2002).
    [Crossref] [PubMed]
  12. R. Khomeriki, “Nonlinear bandgap transmission in optical waveguide arrays,” Phys. Rev. Lett. 92, 063905 (2004).
    [Crossref]
  13. F. Wang, Z. Gong, X. Hu, X. Yang, H. Yang, and Q. Gong, “Nanoscale on-chip all-optical logic parity checker in integrated plasmonic circuits in optical communication range,” Sci. Rep. 6, 24433 (2016).
    [Crossref] [PubMed]
  14. M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, “Numerical studies of mode gaps and coupling efficiency for line-defect waveguides in two-dimensional photonic crystals,” Phys. Rev. B 64, 155113 (2001).
    [Crossref]
  15. C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. Pelusi, D. Moss, B. Eggleton, T. White, and T. Krauss, “Slow light enhanced nonlinear optics in silicon photonic crystal waveguides,” IEEE Journal of Selected Topics in Quantum Electronics,  16, 344–356 (2010).
    [Crossref]
  16. A. Blanco-Redondo, C. Husko, D. Eades, Y. Zhang, J. Li, T. Krauss, and B. Eggleton, “Observation of soliton compression in silicon photonic crystals,” Nature Communications 5, 3160 (2014).
    [Crossref] [PubMed]
  17. N. N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman and Hall, London, 1997).
  18. T. Krauss, “Slow light in photonic crystal waveguides”, J. Phys. D: Appl. Phys. 40, 2666 (2007).
    [Crossref]
  19. R. Khomeriki and J. Leon, “All-optical amplification in metallic subwavelength linear waveguides,” Phys. Rev. A 87, 053806 (2013).
    [Crossref]
  20. V. Jandieri and R. Khomeriki, “Linear amplification of optical signal in coupled photonic crystal waveguides,” IEEE Photonics Technology Letters 27, 639–641 (2015).
    [Crossref]
  21. V. Jandieri, R. Khomeriki, D. Erni, and W. C. Chew, “Realization of All-Optical Digital Amplification in Coupled Nonlinear Photonic Crystal Waveguides,” Progress in Electromagnetics Research,  158, 63–72 (2017).
    [Crossref]
  22. J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics,  4, 535–544 (2010).
    [Crossref]
  23. C. Lacava, M. A. Ettabib, and P. Petropoulos, “Nonlinear silicon photonic signal processing devices for future optical networks,” Apl.Sci. 7, 103–107 (2017).
  24. Y. Liu, F. Qin, Z. Meng, F. Zhou, Q. Mao, and Z. Li, “All-optial logic gates based on two-dimensional low-refractive-index nonlinear photonic crystal slabs,” Opt. Express 19, 1945–1953 (2011)
    [Crossref] [PubMed]
  25. Y. Tanaka, H. Nakamura, Y. Sugimoto, N. Ikeda, K. Asakawa, and K. Inoue, “Coupling properties in a 2-D photonic crystal slab directional coupler with a triangular lattice of air holes,” IEEE Journal of Quantum Electronics 41, 76–84 (2005).
    [Crossref]
  26. P. Strasser, R. Fluckiger, R. Wuest, F. Robin, and H. Jackel, “InP-based compact photonic crystal directional coupler with large operation range,” Opt. Express 15, 8472–8478 (2007).
    [Crossref] [PubMed]
  27. K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Transactions on Antennas and Propagation 52, 2603–2611 (2004).
    [Crossref]
  28. V. Jandieri, K. Yasumoto, and B. Gupta, “Directivity of radiation from a localized source coupled to electromagnetic crystals”, Journal of Infrared, Millimetre and Terahertz Waves 301102–1112 (2009).
    [Crossref]
  29. V. Jandieri and K. Yasumoto, “Electromagnetic Scattering by Layered Cylindrical Arrays of Circular Rods,” IEEE Transaction on Antennas and Propagation 59, 2437–2441 (2011).
    [Crossref]
  30. R. Khomeriki, L. Chotorlishvili, B. Malomed, and J. Berakdar, “Creation and amplification of electromagnon solitons by electric field in nanostructured multiferroics,”, Phys. Rev. B. 91, 041408(R) (2015).
    [Crossref]
  31. M. Malishava and R. Khomeriki, “All-Phononic digital transistor on the basis of gap-soliton dynamics in an anharmonic oscillator ladder”, Phys. Rev. Lett. 115, 104301 (2015).
    [Crossref] [PubMed]
  32. A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method, (Artech House, 1995).
  33. R. Kappeler, Reducing the propagation losses of slab photonic crystal waveguides for active photonic devices. Diss. ETH Zurich, No. 20485, Zurich, July13, 2012.

2017 (3)

P. Rani, S. Fatima, Y. Kalra, and R. K. Sinha, “Realization of all optical logic gates using universal NAND gates on photonic crystal platform,” Superlattices and Microstructures 109, 619–625 (2017).
[Crossref]

V. Jandieri, R. Khomeriki, D. Erni, and W. C. Chew, “Realization of All-Optical Digital Amplification in Coupled Nonlinear Photonic Crystal Waveguides,” Progress in Electromagnetics Research,  158, 63–72 (2017).
[Crossref]

C. Lacava, M. A. Ettabib, and P. Petropoulos, “Nonlinear silicon photonic signal processing devices for future optical networks,” Apl.Sci. 7, 103–107 (2017).

2016 (1)

F. Wang, Z. Gong, X. Hu, X. Yang, H. Yang, and Q. Gong, “Nanoscale on-chip all-optical logic parity checker in integrated plasmonic circuits in optical communication range,” Sci. Rep. 6, 24433 (2016).
[Crossref] [PubMed]

2015 (3)

V. Jandieri and R. Khomeriki, “Linear amplification of optical signal in coupled photonic crystal waveguides,” IEEE Photonics Technology Letters 27, 639–641 (2015).
[Crossref]

R. Khomeriki, L. Chotorlishvili, B. Malomed, and J. Berakdar, “Creation and amplification of electromagnon solitons by electric field in nanostructured multiferroics,”, Phys. Rev. B. 91, 041408(R) (2015).
[Crossref]

M. Malishava and R. Khomeriki, “All-Phononic digital transistor on the basis of gap-soliton dynamics in an anharmonic oscillator ladder”, Phys. Rev. Lett. 115, 104301 (2015).
[Crossref] [PubMed]

2014 (1)

A. Blanco-Redondo, C. Husko, D. Eades, Y. Zhang, J. Li, T. Krauss, and B. Eggleton, “Observation of soliton compression in silicon photonic crystals,” Nature Communications 5, 3160 (2014).
[Crossref] [PubMed]

2013 (2)

R. Khomeriki and J. Leon, “All-optical amplification in metallic subwavelength linear waveguides,” Phys. Rev. A 87, 053806 (2013).
[Crossref]

Y. Fu, X. Hu, and Q. Gong, “Silicon photonic crystal all-optical logic gates,” Phys. Lett. A 377, 329–333 (2013).
[Crossref]

2011 (4)

C. Husko, T. D. Vo, B. Corcoran, J. Li, T. Krauss, and B. Eggleton, “Ultracompact all-optical XOR logic gate in a slow-light silicon photonic crystal waveguide,” Opt. Express 19, 20681–20690 (2011).
[Crossref] [PubMed]

Y. Ishizaka, Y. Kawaguchi, K. Saitoh, and M. Koshiba, “Design of ultra compact all-optical XOR and ANG logic gates with low power consumption,” Optics Communication 284, 3528–3533 (2011).
[Crossref]

V. Jandieri and K. Yasumoto, “Electromagnetic Scattering by Layered Cylindrical Arrays of Circular Rods,” IEEE Transaction on Antennas and Propagation 59, 2437–2441 (2011).
[Crossref]

Y. Liu, F. Qin, Z. Meng, F. Zhou, Q. Mao, and Z. Li, “All-optial logic gates based on two-dimensional low-refractive-index nonlinear photonic crystal slabs,” Opt. Express 19, 1945–1953 (2011)
[Crossref] [PubMed]

2010 (2)

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics,  4, 535–544 (2010).
[Crossref]

C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. Pelusi, D. Moss, B. Eggleton, T. White, and T. Krauss, “Slow light enhanced nonlinear optics in silicon photonic crystal waveguides,” IEEE Journal of Selected Topics in Quantum Electronics,  16, 344–356 (2010).
[Crossref]

2009 (3)

P. Andalib and N. Granpayeh, “All-optical ultracompact photonic crystal AND gate based on nonlinear ring resonators,” J. Opt. Soc. Am. B 26, 10–16 (2009).
[Crossref]

P. Andalib and N. Granpayeh, “All-optical ultracompact photonic crystal NOR gate based on nonlinear ring resonators,” Journal of Opt. A: Pure Appl. Opt. 11, 085203 (2009).
[Crossref]

V. Jandieri, K. Yasumoto, and B. Gupta, “Directivity of radiation from a localized source coupled to electromagnetic crystals”, Journal of Infrared, Millimetre and Terahertz Waves 301102–1112 (2009).
[Crossref]

2008 (2)

2007 (2)

2005 (1)

Y. Tanaka, H. Nakamura, Y. Sugimoto, N. Ikeda, K. Asakawa, and K. Inoue, “Coupling properties in a 2-D photonic crystal slab directional coupler with a triangular lattice of air holes,” IEEE Journal of Quantum Electronics 41, 76–84 (2005).
[Crossref]

2004 (2)

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Transactions on Antennas and Propagation 52, 2603–2611 (2004).
[Crossref]

R. Khomeriki, “Nonlinear bandgap transmission in optical waveguide arrays,” Phys. Rev. Lett. 92, 063905 (2004).
[Crossref]

2002 (1)

F. Geniet and J. Leon, “Energy transmission in the forbidden band gap of a nonlinear chain”, Phys. Rev. Lett. 89, 134102 (2002).
[Crossref] [PubMed]

2001 (1)

M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, “Numerical studies of mode gaps and coupling efficiency for line-defect waveguides in two-dimensional photonic crystals,” Phys. Rev. B 64, 155113 (2001).
[Crossref]

1987 (1)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059 (1987).
[Crossref] [PubMed]

Akhmediev, N. N.

N. N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman and Hall, London, 1997).

Andalib, P.

P. Andalib and N. Granpayeh, “All-optical ultracompact photonic crystal NOR gate based on nonlinear ring resonators,” Journal of Opt. A: Pure Appl. Opt. 11, 085203 (2009).
[Crossref]

P. Andalib and N. Granpayeh, “All-optical ultracompact photonic crystal AND gate based on nonlinear ring resonators,” J. Opt. Soc. Am. B 26, 10–16 (2009).
[Crossref]

Andreani, L. C.

Ankiewicz, A.

N. N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman and Hall, London, 1997).

Asakawa, K.

Y. Tanaka, H. Nakamura, Y. Sugimoto, N. Ikeda, K. Asakawa, and K. Inoue, “Coupling properties in a 2-D photonic crystal slab directional coupler with a triangular lattice of air holes,” IEEE Journal of Quantum Electronics 41, 76–84 (2005).
[Crossref]

Azizi, K.

M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, “Numerical studies of mode gaps and coupling efficiency for line-defect waveguides in two-dimensional photonic crystals,” Phys. Rev. B 64, 155113 (2001).
[Crossref]

Berakdar, J.

R. Khomeriki, L. Chotorlishvili, B. Malomed, and J. Berakdar, “Creation and amplification of electromagnon solitons by electric field in nanostructured multiferroics,”, Phys. Rev. B. 91, 041408(R) (2015).
[Crossref]

Blanco-Redondo, A.

A. Blanco-Redondo, C. Husko, D. Eades, Y. Zhang, J. Li, T. Krauss, and B. Eggleton, “Observation of soliton compression in silicon photonic crystals,” Nature Communications 5, 3160 (2014).
[Crossref] [PubMed]

Brosi, J.-M.

Chew, W. C.

V. Jandieri, R. Khomeriki, D. Erni, and W. C. Chew, “Realization of All-Optical Digital Amplification in Coupled Nonlinear Photonic Crystal Waveguides,” Progress in Electromagnetics Research,  158, 63–72 (2017).
[Crossref]

Chotorlishvili, L.

R. Khomeriki, L. Chotorlishvili, B. Malomed, and J. Berakdar, “Creation and amplification of electromagnon solitons by electric field in nanostructured multiferroics,”, Phys. Rev. B. 91, 041408(R) (2015).
[Crossref]

Corcoran, B.

C. Husko, T. D. Vo, B. Corcoran, J. Li, T. Krauss, and B. Eggleton, “Ultracompact all-optical XOR logic gate in a slow-light silicon photonic crystal waveguide,” Opt. Express 19, 20681–20690 (2011).
[Crossref] [PubMed]

C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. Pelusi, D. Moss, B. Eggleton, T. White, and T. Krauss, “Slow light enhanced nonlinear optics in silicon photonic crystal waveguides,” IEEE Journal of Selected Topics in Quantum Electronics,  16, 344–356 (2010).
[Crossref]

Eades, D.

A. Blanco-Redondo, C. Husko, D. Eades, Y. Zhang, J. Li, T. Krauss, and B. Eggleton, “Observation of soliton compression in silicon photonic crystals,” Nature Communications 5, 3160 (2014).
[Crossref] [PubMed]

Ebnali-Heidari, M.

C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. Pelusi, D. Moss, B. Eggleton, T. White, and T. Krauss, “Slow light enhanced nonlinear optics in silicon photonic crystal waveguides,” IEEE Journal of Selected Topics in Quantum Electronics,  16, 344–356 (2010).
[Crossref]

Eggleton, B.

A. Blanco-Redondo, C. Husko, D. Eades, Y. Zhang, J. Li, T. Krauss, and B. Eggleton, “Observation of soliton compression in silicon photonic crystals,” Nature Communications 5, 3160 (2014).
[Crossref] [PubMed]

C. Husko, T. D. Vo, B. Corcoran, J. Li, T. Krauss, and B. Eggleton, “Ultracompact all-optical XOR logic gate in a slow-light silicon photonic crystal waveguide,” Opt. Express 19, 20681–20690 (2011).
[Crossref] [PubMed]

C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. Pelusi, D. Moss, B. Eggleton, T. White, and T. Krauss, “Slow light enhanced nonlinear optics in silicon photonic crystal waveguides,” IEEE Journal of Selected Topics in Quantum Electronics,  16, 344–356 (2010).
[Crossref]

Erni, D.

V. Jandieri, R. Khomeriki, D. Erni, and W. C. Chew, “Realization of All-Optical Digital Amplification in Coupled Nonlinear Photonic Crystal Waveguides,” Progress in Electromagnetics Research,  158, 63–72 (2017).
[Crossref]

Ettabib, M. A.

C. Lacava, M. A. Ettabib, and P. Petropoulos, “Nonlinear silicon photonic signal processing devices for future optical networks,” Apl.Sci. 7, 103–107 (2017).

Fatima, S.

P. Rani, S. Fatima, Y. Kalra, and R. K. Sinha, “Realization of all optical logic gates using universal NAND gates on photonic crystal platform,” Superlattices and Microstructures 109, 619–625 (2017).
[Crossref]

Fluckiger, R.

Freude, W.

Fu, Y.

Y. Fu, X. Hu, and Q. Gong, “Silicon photonic crystal all-optical logic gates,” Phys. Lett. A 377, 329–333 (2013).
[Crossref]

Geniet, F.

F. Geniet and J. Leon, “Energy transmission in the forbidden band gap of a nonlinear chain”, Phys. Rev. Lett. 89, 134102 (2002).
[Crossref] [PubMed]

Gong, Q.

F. Wang, Z. Gong, X. Hu, X. Yang, H. Yang, and Q. Gong, “Nanoscale on-chip all-optical logic parity checker in integrated plasmonic circuits in optical communication range,” Sci. Rep. 6, 24433 (2016).
[Crossref] [PubMed]

Y. Fu, X. Hu, and Q. Gong, “Silicon photonic crystal all-optical logic gates,” Phys. Lett. A 377, 329–333 (2013).
[Crossref]

Gong, Z.

F. Wang, Z. Gong, X. Hu, X. Yang, H. Yang, and Q. Gong, “Nanoscale on-chip all-optical logic parity checker in integrated plasmonic circuits in optical communication range,” Sci. Rep. 6, 24433 (2016).
[Crossref] [PubMed]

Granpayeh, N.

P. Andalib and N. Granpayeh, “All-optical ultracompact photonic crystal NOR gate based on nonlinear ring resonators,” Journal of Opt. A: Pure Appl. Opt. 11, 085203 (2009).
[Crossref]

P. Andalib and N. Granpayeh, “All-optical ultracompact photonic crystal AND gate based on nonlinear ring resonators,” J. Opt. Soc. Am. B 26, 10–16 (2009).
[Crossref]

Grillet, C.

C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. Pelusi, D. Moss, B. Eggleton, T. White, and T. Krauss, “Slow light enhanced nonlinear optics in silicon photonic crystal waveguides,” IEEE Journal of Selected Topics in Quantum Electronics,  16, 344–356 (2010).
[Crossref]

Gupta, B.

V. Jandieri, K. Yasumoto, and B. Gupta, “Directivity of radiation from a localized source coupled to electromagnetic crystals”, Journal of Infrared, Millimetre and Terahertz Waves 301102–1112 (2009).
[Crossref]

Hu, X.

F. Wang, Z. Gong, X. Hu, X. Yang, H. Yang, and Q. Gong, “Nanoscale on-chip all-optical logic parity checker in integrated plasmonic circuits in optical communication range,” Sci. Rep. 6, 24433 (2016).
[Crossref] [PubMed]

Y. Fu, X. Hu, and Q. Gong, “Silicon photonic crystal all-optical logic gates,” Phys. Lett. A 377, 329–333 (2013).
[Crossref]

Husko, C.

A. Blanco-Redondo, C. Husko, D. Eades, Y. Zhang, J. Li, T. Krauss, and B. Eggleton, “Observation of soliton compression in silicon photonic crystals,” Nature Communications 5, 3160 (2014).
[Crossref] [PubMed]

C. Husko, T. D. Vo, B. Corcoran, J. Li, T. Krauss, and B. Eggleton, “Ultracompact all-optical XOR logic gate in a slow-light silicon photonic crystal waveguide,” Opt. Express 19, 20681–20690 (2011).
[Crossref] [PubMed]

Ikeda, N.

Y. Tanaka, H. Nakamura, Y. Sugimoto, N. Ikeda, K. Asakawa, and K. Inoue, “Coupling properties in a 2-D photonic crystal slab directional coupler with a triangular lattice of air holes,” IEEE Journal of Quantum Electronics 41, 76–84 (2005).
[Crossref]

Inoue, K.

Y. Tanaka, H. Nakamura, Y. Sugimoto, N. Ikeda, K. Asakawa, and K. Inoue, “Coupling properties in a 2-D photonic crystal slab directional coupler with a triangular lattice of air holes,” IEEE Journal of Quantum Electronics 41, 76–84 (2005).
[Crossref]

Ishizaka, Y.

Y. Ishizaka, Y. Kawaguchi, K. Saitoh, and M. Koshiba, “Design of ultra compact all-optical XOR and ANG logic gates with low power consumption,” Optics Communication 284, 3528–3533 (2011).
[Crossref]

Jackel, H.

Jandieri, V.

V. Jandieri, R. Khomeriki, D. Erni, and W. C. Chew, “Realization of All-Optical Digital Amplification in Coupled Nonlinear Photonic Crystal Waveguides,” Progress in Electromagnetics Research,  158, 63–72 (2017).
[Crossref]

V. Jandieri and R. Khomeriki, “Linear amplification of optical signal in coupled photonic crystal waveguides,” IEEE Photonics Technology Letters 27, 639–641 (2015).
[Crossref]

V. Jandieri and K. Yasumoto, “Electromagnetic Scattering by Layered Cylindrical Arrays of Circular Rods,” IEEE Transaction on Antennas and Propagation 59, 2437–2441 (2011).
[Crossref]

V. Jandieri, K. Yasumoto, and B. Gupta, “Directivity of radiation from a localized source coupled to electromagnetic crystals”, Journal of Infrared, Millimetre and Terahertz Waves 301102–1112 (2009).
[Crossref]

Jaskorzynska, B.

M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, “Numerical studies of mode gaps and coupling efficiency for line-defect waveguides in two-dimensional photonic crystals,” Phys. Rev. B 64, 155113 (2001).
[Crossref]

Kalra, Y.

P. Rani, S. Fatima, Y. Kalra, and R. K. Sinha, “Realization of all optical logic gates using universal NAND gates on photonic crystal platform,” Superlattices and Microstructures 109, 619–625 (2017).
[Crossref]

Kappeler, R.

R. Kappeler, Reducing the propagation losses of slab photonic crystal waveguides for active photonic devices. Diss. ETH Zurich, No. 20485, Zurich, July13, 2012.

Karlsson, A.

M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, “Numerical studies of mode gaps and coupling efficiency for line-defect waveguides in two-dimensional photonic crystals,” Phys. Rev. B 64, 155113 (2001).
[Crossref]

Kawaguchi, Y.

Y. Ishizaka, Y. Kawaguchi, K. Saitoh, and M. Koshiba, “Design of ultra compact all-optical XOR and ANG logic gates with low power consumption,” Optics Communication 284, 3528–3533 (2011).
[Crossref]

Khomeriki, R.

V. Jandieri, R. Khomeriki, D. Erni, and W. C. Chew, “Realization of All-Optical Digital Amplification in Coupled Nonlinear Photonic Crystal Waveguides,” Progress in Electromagnetics Research,  158, 63–72 (2017).
[Crossref]

V. Jandieri and R. Khomeriki, “Linear amplification of optical signal in coupled photonic crystal waveguides,” IEEE Photonics Technology Letters 27, 639–641 (2015).
[Crossref]

M. Malishava and R. Khomeriki, “All-Phononic digital transistor on the basis of gap-soliton dynamics in an anharmonic oscillator ladder”, Phys. Rev. Lett. 115, 104301 (2015).
[Crossref] [PubMed]

R. Khomeriki, L. Chotorlishvili, B. Malomed, and J. Berakdar, “Creation and amplification of electromagnon solitons by electric field in nanostructured multiferroics,”, Phys. Rev. B. 91, 041408(R) (2015).
[Crossref]

R. Khomeriki and J. Leon, “All-optical amplification in metallic subwavelength linear waveguides,” Phys. Rev. A 87, 053806 (2013).
[Crossref]

R. Khomeriki, “Nonlinear bandgap transmission in optical waveguide arrays,” Phys. Rev. Lett. 92, 063905 (2004).
[Crossref]

Koos, C.

Koshiba, M.

Y. Ishizaka, Y. Kawaguchi, K. Saitoh, and M. Koshiba, “Design of ultra compact all-optical XOR and ANG logic gates with low power consumption,” Optics Communication 284, 3528–3533 (2011).
[Crossref]

Krauss, T.

A. Blanco-Redondo, C. Husko, D. Eades, Y. Zhang, J. Li, T. Krauss, and B. Eggleton, “Observation of soliton compression in silicon photonic crystals,” Nature Communications 5, 3160 (2014).
[Crossref] [PubMed]

C. Husko, T. D. Vo, B. Corcoran, J. Li, T. Krauss, and B. Eggleton, “Ultracompact all-optical XOR logic gate in a slow-light silicon photonic crystal waveguide,” Opt. Express 19, 20681–20690 (2011).
[Crossref] [PubMed]

C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. Pelusi, D. Moss, B. Eggleton, T. White, and T. Krauss, “Slow light enhanced nonlinear optics in silicon photonic crystal waveguides,” IEEE Journal of Selected Topics in Quantum Electronics,  16, 344–356 (2010).
[Crossref]

T. Krauss, “Slow light in photonic crystal waveguides”, J. Phys. D: Appl. Phys. 40, 2666 (2007).
[Crossref]

Kushta, T.

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Transactions on Antennas and Propagation 52, 2603–2611 (2004).
[Crossref]

Lacava, C.

C. Lacava, M. A. Ettabib, and P. Petropoulos, “Nonlinear silicon photonic signal processing devices for future optical networks,” Apl.Sci. 7, 103–107 (2017).

Leon, J.

R. Khomeriki and J. Leon, “All-optical amplification in metallic subwavelength linear waveguides,” Phys. Rev. A 87, 053806 (2013).
[Crossref]

F. Geniet and J. Leon, “Energy transmission in the forbidden band gap of a nonlinear chain”, Phys. Rev. Lett. 89, 134102 (2002).
[Crossref] [PubMed]

Leuthold, J.

Li, J.

A. Blanco-Redondo, C. Husko, D. Eades, Y. Zhang, J. Li, T. Krauss, and B. Eggleton, “Observation of soliton compression in silicon photonic crystals,” Nature Communications 5, 3160 (2014).
[Crossref] [PubMed]

C. Husko, T. D. Vo, B. Corcoran, J. Li, T. Krauss, and B. Eggleton, “Ultracompact all-optical XOR logic gate in a slow-light silicon photonic crystal waveguide,” Opt. Express 19, 20681–20690 (2011).
[Crossref] [PubMed]

Li, Z.

Liu, C.

Liu, Q.

Liu, Y.

Malishava, M.

M. Malishava and R. Khomeriki, “All-Phononic digital transistor on the basis of gap-soliton dynamics in an anharmonic oscillator ladder”, Phys. Rev. Lett. 115, 104301 (2015).
[Crossref] [PubMed]

Malomed, B.

R. Khomeriki, L. Chotorlishvili, B. Malomed, and J. Berakdar, “Creation and amplification of electromagnon solitons by electric field in nanostructured multiferroics,”, Phys. Rev. B. 91, 041408(R) (2015).
[Crossref]

Mao, Q.

Meng, Z.

Monat, C.

C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. Pelusi, D. Moss, B. Eggleton, T. White, and T. Krauss, “Slow light enhanced nonlinear optics in silicon photonic crystal waveguides,” IEEE Journal of Selected Topics in Quantum Electronics,  16, 344–356 (2010).
[Crossref]

Moss, D.

C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. Pelusi, D. Moss, B. Eggleton, T. White, and T. Krauss, “Slow light enhanced nonlinear optics in silicon photonic crystal waveguides,” IEEE Journal of Selected Topics in Quantum Electronics,  16, 344–356 (2010).
[Crossref]

Nakamura, H.

Y. Tanaka, H. Nakamura, Y. Sugimoto, N. Ikeda, K. Asakawa, and K. Inoue, “Coupling properties in a 2-D photonic crystal slab directional coupler with a triangular lattice of air holes,” IEEE Journal of Quantum Electronics 41, 76–84 (2005).
[Crossref]

Ouyang, Z.

Pelusi, M.

C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. Pelusi, D. Moss, B. Eggleton, T. White, and T. Krauss, “Slow light enhanced nonlinear optics in silicon photonic crystal waveguides,” IEEE Journal of Selected Topics in Quantum Electronics,  16, 344–356 (2010).
[Crossref]

Petropoulos, P.

C. Lacava, M. A. Ettabib, and P. Petropoulos, “Nonlinear silicon photonic signal processing devices for future optical networks,” Apl.Sci. 7, 103–107 (2017).

Pudo, D.

C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. Pelusi, D. Moss, B. Eggleton, T. White, and T. Krauss, “Slow light enhanced nonlinear optics in silicon photonic crystal waveguides,” IEEE Journal of Selected Topics in Quantum Electronics,  16, 344–356 (2010).
[Crossref]

Qin, F.

Qiu, M.

M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, “Numerical studies of mode gaps and coupling efficiency for line-defect waveguides in two-dimensional photonic crystals,” Phys. Rev. B 64, 155113 (2001).
[Crossref]

Rani, P.

P. Rani, S. Fatima, Y. Kalra, and R. K. Sinha, “Realization of all optical logic gates using universal NAND gates on photonic crystal platform,” Superlattices and Microstructures 109, 619–625 (2017).
[Crossref]

Robin, F.

Saitoh, K.

Y. Ishizaka, Y. Kawaguchi, K. Saitoh, and M. Koshiba, “Design of ultra compact all-optical XOR and ANG logic gates with low power consumption,” Optics Communication 284, 3528–3533 (2011).
[Crossref]

Sinha, R. K.

P. Rani, S. Fatima, Y. Kalra, and R. K. Sinha, “Realization of all optical logic gates using universal NAND gates on photonic crystal platform,” Superlattices and Microstructures 109, 619–625 (2017).
[Crossref]

Strasser, P.

Sugimoto, Y.

Y. Tanaka, H. Nakamura, Y. Sugimoto, N. Ikeda, K. Asakawa, and K. Inoue, “Coupling properties in a 2-D photonic crystal slab directional coupler with a triangular lattice of air holes,” IEEE Journal of Quantum Electronics 41, 76–84 (2005).
[Crossref]

Swillo, M.

M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, “Numerical studies of mode gaps and coupling efficiency for line-defect waveguides in two-dimensional photonic crystals,” Phys. Rev. B 64, 155113 (2001).
[Crossref]

Taflove, A.

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method, (Artech House, 1995).

Tanaka, Y.

Y. Tanaka, H. Nakamura, Y. Sugimoto, N. Ikeda, K. Asakawa, and K. Inoue, “Coupling properties in a 2-D photonic crystal slab directional coupler with a triangular lattice of air holes,” IEEE Journal of Quantum Electronics 41, 76–84 (2005).
[Crossref]

Toyama, H.

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Transactions on Antennas and Propagation 52, 2603–2611 (2004).
[Crossref]

Vo, T. D.

Waldow, M.

Wang, F.

F. Wang, Z. Gong, X. Hu, X. Yang, H. Yang, and Q. Gong, “Nanoscale on-chip all-optical logic parity checker in integrated plasmonic circuits in optical communication range,” Sci. Rep. 6, 24433 (2016).
[Crossref] [PubMed]

Wang, J.

White, T.

C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. Pelusi, D. Moss, B. Eggleton, T. White, and T. Krauss, “Slow light enhanced nonlinear optics in silicon photonic crystal waveguides,” IEEE Journal of Selected Topics in Quantum Electronics,  16, 344–356 (2010).
[Crossref]

Wu, C.

Wuest, R.

Yablonovitch, E.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059 (1987).
[Crossref] [PubMed]

Yang, H.

F. Wang, Z. Gong, X. Hu, X. Yang, H. Yang, and Q. Gong, “Nanoscale on-chip all-optical logic parity checker in integrated plasmonic circuits in optical communication range,” Sci. Rep. 6, 24433 (2016).
[Crossref] [PubMed]

Yang, X.

F. Wang, Z. Gong, X. Hu, X. Yang, H. Yang, and Q. Gong, “Nanoscale on-chip all-optical logic parity checker in integrated plasmonic circuits in optical communication range,” Sci. Rep. 6, 24433 (2016).
[Crossref] [PubMed]

Yasumoto, K.

V. Jandieri and K. Yasumoto, “Electromagnetic Scattering by Layered Cylindrical Arrays of Circular Rods,” IEEE Transaction on Antennas and Propagation 59, 2437–2441 (2011).
[Crossref]

V. Jandieri, K. Yasumoto, and B. Gupta, “Directivity of radiation from a localized source coupled to electromagnetic crystals”, Journal of Infrared, Millimetre and Terahertz Waves 301102–1112 (2009).
[Crossref]

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Transactions on Antennas and Propagation 52, 2603–2611 (2004).
[Crossref]

Zhang, Y.

A. Blanco-Redondo, C. Husko, D. Eades, Y. Zhang, J. Li, T. Krauss, and B. Eggleton, “Observation of soliton compression in silicon photonic crystals,” Nature Communications 5, 3160 (2014).
[Crossref] [PubMed]

Zhou, F.

Apl.Sci. (1)

C. Lacava, M. A. Ettabib, and P. Petropoulos, “Nonlinear silicon photonic signal processing devices for future optical networks,” Apl.Sci. 7, 103–107 (2017).

IEEE Journal of Quantum Electronics (1)

Y. Tanaka, H. Nakamura, Y. Sugimoto, N. Ikeda, K. Asakawa, and K. Inoue, “Coupling properties in a 2-D photonic crystal slab directional coupler with a triangular lattice of air holes,” IEEE Journal of Quantum Electronics 41, 76–84 (2005).
[Crossref]

IEEE Journal of Selected Topics in Quantum Electronics (1)

C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. Pelusi, D. Moss, B. Eggleton, T. White, and T. Krauss, “Slow light enhanced nonlinear optics in silicon photonic crystal waveguides,” IEEE Journal of Selected Topics in Quantum Electronics,  16, 344–356 (2010).
[Crossref]

IEEE Photonics Technology Letters (1)

V. Jandieri and R. Khomeriki, “Linear amplification of optical signal in coupled photonic crystal waveguides,” IEEE Photonics Technology Letters 27, 639–641 (2015).
[Crossref]

IEEE Transaction on Antennas and Propagation (1)

V. Jandieri and K. Yasumoto, “Electromagnetic Scattering by Layered Cylindrical Arrays of Circular Rods,” IEEE Transaction on Antennas and Propagation 59, 2437–2441 (2011).
[Crossref]

IEEE Transactions on Antennas and Propagation (1)

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Transactions on Antennas and Propagation 52, 2603–2611 (2004).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Phys. D: Appl. Phys. (1)

T. Krauss, “Slow light in photonic crystal waveguides”, J. Phys. D: Appl. Phys. 40, 2666 (2007).
[Crossref]

Journal of Infrared, Millimetre and Terahertz Waves (1)

V. Jandieri, K. Yasumoto, and B. Gupta, “Directivity of radiation from a localized source coupled to electromagnetic crystals”, Journal of Infrared, Millimetre and Terahertz Waves 301102–1112 (2009).
[Crossref]

Journal of Opt. A: Pure Appl. Opt. (1)

P. Andalib and N. Granpayeh, “All-optical ultracompact photonic crystal NOR gate based on nonlinear ring resonators,” Journal of Opt. A: Pure Appl. Opt. 11, 085203 (2009).
[Crossref]

Nat. Photonics (1)

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics,  4, 535–544 (2010).
[Crossref]

Nature Communications (1)

A. Blanco-Redondo, C. Husko, D. Eades, Y. Zhang, J. Li, T. Krauss, and B. Eggleton, “Observation of soliton compression in silicon photonic crystals,” Nature Communications 5, 3160 (2014).
[Crossref] [PubMed]

Opt. Express (5)

Optics Communication (1)

Y. Ishizaka, Y. Kawaguchi, K. Saitoh, and M. Koshiba, “Design of ultra compact all-optical XOR and ANG logic gates with low power consumption,” Optics Communication 284, 3528–3533 (2011).
[Crossref]

Phys. Lett. A (1)

Y. Fu, X. Hu, and Q. Gong, “Silicon photonic crystal all-optical logic gates,” Phys. Lett. A 377, 329–333 (2013).
[Crossref]

Phys. Rev. A (1)

R. Khomeriki and J. Leon, “All-optical amplification in metallic subwavelength linear waveguides,” Phys. Rev. A 87, 053806 (2013).
[Crossref]

Phys. Rev. B (1)

M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, “Numerical studies of mode gaps and coupling efficiency for line-defect waveguides in two-dimensional photonic crystals,” Phys. Rev. B 64, 155113 (2001).
[Crossref]

Phys. Rev. B. (1)

R. Khomeriki, L. Chotorlishvili, B. Malomed, and J. Berakdar, “Creation and amplification of electromagnon solitons by electric field in nanostructured multiferroics,”, Phys. Rev. B. 91, 041408(R) (2015).
[Crossref]

Phys. Rev. Lett. (4)

M. Malishava and R. Khomeriki, “All-Phononic digital transistor on the basis of gap-soliton dynamics in an anharmonic oscillator ladder”, Phys. Rev. Lett. 115, 104301 (2015).
[Crossref] [PubMed]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059 (1987).
[Crossref] [PubMed]

F. Geniet and J. Leon, “Energy transmission in the forbidden band gap of a nonlinear chain”, Phys. Rev. Lett. 89, 134102 (2002).
[Crossref] [PubMed]

R. Khomeriki, “Nonlinear bandgap transmission in optical waveguide arrays,” Phys. Rev. Lett. 92, 063905 (2004).
[Crossref]

Progress in Electromagnetics Research (1)

V. Jandieri, R. Khomeriki, D. Erni, and W. C. Chew, “Realization of All-Optical Digital Amplification in Coupled Nonlinear Photonic Crystal Waveguides,” Progress in Electromagnetics Research,  158, 63–72 (2017).
[Crossref]

Sci. Rep. (1)

F. Wang, Z. Gong, X. Hu, X. Yang, H. Yang, and Q. Gong, “Nanoscale on-chip all-optical logic parity checker in integrated plasmonic circuits in optical communication range,” Sci. Rep. 6, 24433 (2016).
[Crossref] [PubMed]

Superlattices and Microstructures (1)

P. Rani, S. Fatima, Y. Kalra, and R. K. Sinha, “Realization of all optical logic gates using universal NAND gates on photonic crystal platform,” Superlattices and Microstructures 109, 619–625 (2017).
[Crossref]

Other (4)

K. Yasumoto ed., Electromagnetic Theory and Applications for Photonic Crystals (CRC Press, 2005).
[Crossref]

N. N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman and Hall, London, 1997).

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method, (Artech House, 1995).

R. Kappeler, Reducing the propagation losses of slab photonic crystal waveguides for active photonic devices. Diss. ETH Zurich, No. 20485, Zurich, July13, 2012.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1 Schematic view of the three symmetric nonlinear C-PCWs embedded into a planar PhC with a triangular lattice made of air holes in a background nonlinear Kerr-type medium with a linear refractive index n0 = 2.95, where h is the period of the PhC and w = 3 h is the width of each of the coupled W1 defect waveguides. The holes are parallel to the z-axis. Number of layers of the upper and lower PhCs is 8. The quantity χ(3) stands for the third-order nonlinear optical susceptibility of the nonlinear background medium. The length of the photonic crystals is 30h.
Fig. 2
Fig. 2 Dispersion curves of the symmetric (blue and green lines) and the antisymmetric (red line) (super-) modes for the H-polarized field (Hz, Ex, Ey) of the three C-PCWs as shown in Fig. 1. The normalized operation frequency is ωh/2πc = 0.232 marked by the red dot. The distributions of the transversal magnetic field Hz for two symmetric (blue and green lines) and one antisymmetric (red line) modes are shown in the corresponding insets.
Fig. 3
Fig. 3 Schematic distributions (Hz) of the signal pulses propagating (d) inside the nonlinear C-PCWs (Fig. 1), when a CW input signal with the amplitude δ = 0.78 is launched through Port 2 and no input signals δ = 0 are injected into Port 1 (a) and Port 3 (c). Magnetic field Hz of the received signal at a distance x = 30h inside the C-PCW associated to Port 2 (e).
Fig. 4
Fig. 4 Schematic distributions (Hz) of the signal pulses propagating (d) inside the nonlinear C-PCWs (Fig. 1), when a CW signal with the amplitude δ = 0.78 is launched through Port 2 (b), and a Gaussian pulse with amplitude δ = 0.72 is injected into Port 1(a), whereas no single is injected into Port 3 (c). Magnetic field Hz of the received signal at a distance x = 30h inside the C-PCW associated to Port 2 (e). Amplitude transfer curve, which represents the dependence of the maximum of the output pulse envelope versus the maximum of the input Gaussian pulse envelope injected in Port 1 (f).
Fig. 5
Fig. 5 Schematic distributions (Hz) of the signal pulses propagating (d) inside the nonlinear C-PCWs (Fig. 1), when a CW signal with the amplitude δ = 0.78 is launched through Port 2 and Gaussian pulses with an amplitude δ = 0.72 are injected into Port 1 and Port 3. Magnetic field Hz of the received signal at a distance x = 30h inside the C-PCW associated to Port 2 (e). Schematic distributions (Hz) of the signal pulses propagating (f) inside the nonlinear C-PCWs (Fig. 1), when a CW input signal with the amplitude δ = 0.78 is launched through Port 2 and the output signal (e) is injected as a new input signal into Port 1 and Port 3 of a subsequent C-PCW stage. Magnetic field Hz of the received signal at a distance x = 30h inside the C-PCW associated to Port 2 (i). Schematic distributions (Hz) of the signal pulses propagating (g) inside the nonlinear C-PCWs (Fig. 1), when a CW input signal with the amplitude δ = 0.78 is launched through Port 2 and the output signal (e) is injected as a new input signal into Port 3. No single is injected into Port 1 of a subsequent C-PCW stage. Magnetic field Hz of the received signal at a distance x = 30h inside the C-PCW associated to Port 2 (h). The peak level contrast between the “1” and “0” overall output signal amounts to 16 dB.

Equations (3)

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

ω = ω S 1 ( β = π / h ) γ A t h 2 / 4 ,
H z ( x , y , t ) = F Φ S 1 ( x , y ) exp ( i δ ω t ) cosh [ ( x v S 1 t ) / Λ ] e i ( β x ω t ) + c . c .
Λ = 2 ω S 1 γ F 2 , ω S 1 = 2 ω S 1 β 2 , v S 1 = ω S 1 β , δ ω = γ F 2 4 .