Abstract

We have developed a three-dimensional model to study the properties of terahertz emission from a one-dimensional, χ(2)-doped photonic crystal. We exploit difference-frequency generation in a collinear configuration and find an enhancement factor of up to 20 with respect to difference-frequency conversion from an equivalent bulk structure.

© 2006 Optical Society of America

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  1. N. M. Froberg, B. B. Hu, X.-C. Zhang, and D. H. Auston, "Terahertz radiation from a photoconducting antenna array," IEEE J. Quantum Electron. 28, 2291-2301 (1992).
    [CrossRef]
  2. Y. C. Shen, P. C. Upadhya, E. H. Linfield, and H. E. Beere, "Terahertz generation from coherent optical phonons in biased GaAs photoconductive emitter," Phys. Rev. E 69, 235325 (2004).
  3. L. Xu, X.-C. Zhang, and D. H. Auston, "Terahertz beam generation by femtosecond optical pulses in electro-optic materials," Appl. Phys. Lett. 61, 1784-1787 (1992).
    [CrossRef]
  4. K. H. Yang, P. L. Richards, and Y. R. Shen, "Generation of far-infrared radiation by picosecond light pulses in LiNbO3," Appl. Phys. Lett. 19, 320-323 (1971).
    [CrossRef]
  5. A. S. Nikoghosyan, E. M. Laziev, R. M. Martirosyan, and A. A. Hakhoumian, "Efficient ultrashort light pulse conversion in GHz-THz pulses in ZnTe, GaAs, and DAST crystals," iCONO 2001: Ultrafast Phenomena and Strong Laser Fields, V.M.Gordienko, A.A.Afanas'ev, and V.V.Shuvalov, eds., Proc. SPIE 4752, 40-48 (2002).
  6. A. Nahata, D. H. Auston, C. Wu, and J. T. Yardley, "Generation of terahertz radiation from poled polymer," Appl. Phys. Lett. 67, 1358-1361 (1995).
    [CrossRef]
  7. A. Tredicucci, F. Capasso, G. Gmachi, D. L. Sivco, A. L. Hutchinson, and A. Y. Chao, "High performance interminiband quantum cascade lasers with grades superlattices," Appl. Phys. Lett. 73, 2101-2103 (1998).
    [CrossRef]
  8. M. A. Piestrup and R. N. Fleming, "Continuosusly tunable submillimeter wave source," Appl. Phys. Lett. 26, 418-421 (1975).
    [CrossRef]
  9. A. M. Weiner and D. E. Leaird, "Generation of terahertz-rate trains of femtosecond pulses by phase-only filtering," Opt. Lett. 15, 51-53 (1990).
    [CrossRef] [PubMed]
  10. T. Taniuchi and H. Nakanishi, "Continuously tunable terahertz-wave generation in GaP crystal by collinear difference frequency mixing," Electron. Lett. 40, 327-328 (2004).
    [CrossRef]
  11. W. Shi and Y. J. Ding, "A monochromatic and high-power terahertz source tunable in the ranges of 2.7-38.4 and 58.2-3540 µm for variety of potential application," Appl. Phys. Lett. 84, 1635-1638 (2004).
    [CrossRef]
  12. T. Taniuchi, H. Adachi, S. Okda, T. Sasaki, and H. Nakanishi, "Continuously tunable THz and far-infrared wave generation fron DAST crystal," Electron. Lett. 40, 549-551 (2004).
    [CrossRef]
  13. P. E. Powers, R. A. Alkuwari, J. W. Haus, K. Suizu, and H. Ito, "Terahertz generation with tandem seeded optical parametric generators," Opt. Lett. 30, 640-642 (2005).
    [CrossRef] [PubMed]
  14. M. Bertolotti, C. M. Bowden, and C. Sibilia, "Nanoscale linear and nonlinear optics," AIP Conf. Proc. 560, 1-31 (2001).
    [CrossRef]
  15. Y. Lu, M. Xiao, and G. J. Salamo, "Coherent microwave generation in nonlinear photonic crystal," IEEE J. Quantum Electron. 38, 481-485 (2002).
    [CrossRef]
  16. N. Bloembergen and A. J. Sievers, "Nonlinear optical properties of periodic laminar structures," Appl. Phys. Lett. 17, 483-486 (1970).
    [CrossRef]
  17. G. D'Aguanno, M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, "Generalized coupled-mode theory for chi(2) interactions in finite multilayered structures," J. Opt. Soc. Am. B 19, 2111-2121 (2002).
    [CrossRef]
  18. M. Centini, G. D'Aguanno, C. Sibilia, L. Sciscione, M. Bertolotti, M. Scalora, and M. J, Bloemer, "Non-phase-matched enhancement of second-harmonic generation in multilayer structures with internal reflections," Opt. Lett. 29, 1924-1926 (2004).
    [CrossRef] [PubMed]
  19. J. W. Haus, P. Powers, M. Torres-Cisneros, M. Scalora, M. J. Bloemer, N. Akozbek, and M. A. Meneses-Nava, "Enhancement tunable terahertz generation in photonic band-gap structures," Laser Phys. 14, 635-642 (2004).
  20. G. D'Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, "Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures," Phys. Rev. E 63, 036610 (2001).
    [CrossRef]
  21. O. Di Stefano, S. Savasta, and R. Girlanda, "Mode expansion and photon operators in dispersive and absorbing dielectrics," J. Mod. Opt. 48, 67-84 (2001).
    [CrossRef]
  22. G. D'Aguanno, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, "Density of modes and tunneling times in finite, one-dimensional, photonic crystals: a comprehensive analysis," Phys. Rev. E 70, 016612 (2004).
    [CrossRef]
  23. J. Lekner, Theory of Reflection (Martin Nijhoff, 1987), Chap. 12.
  24. W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, 1990), pp. 375-378.
  25. Y. R. Shen, The Principle of Nonlinear Optics (Wiley, 1984), pp. 110-113.
  26. M. Centini, G. D'Aguanno, M. Scalora, M. J. Bloemer, C. M. Bowden, C. Sibilia, N. Mattiucci, and M. Bertolotti, "Dynamics of counterpropagating pulses in photonic crystals: enhancement and suppression of stimulated emission processes," Phys. Rev. E 67, 036617 (2003).
    [CrossRef]

2005 (1)

2004 (7)

M. Centini, G. D'Aguanno, C. Sibilia, L. Sciscione, M. Bertolotti, M. Scalora, and M. J, Bloemer, "Non-phase-matched enhancement of second-harmonic generation in multilayer structures with internal reflections," Opt. Lett. 29, 1924-1926 (2004).
[CrossRef] [PubMed]

Y. C. Shen, P. C. Upadhya, E. H. Linfield, and H. E. Beere, "Terahertz generation from coherent optical phonons in biased GaAs photoconductive emitter," Phys. Rev. E 69, 235325 (2004).

T. Taniuchi and H. Nakanishi, "Continuously tunable terahertz-wave generation in GaP crystal by collinear difference frequency mixing," Electron. Lett. 40, 327-328 (2004).
[CrossRef]

W. Shi and Y. J. Ding, "A monochromatic and high-power terahertz source tunable in the ranges of 2.7-38.4 and 58.2-3540 µm for variety of potential application," Appl. Phys. Lett. 84, 1635-1638 (2004).
[CrossRef]

T. Taniuchi, H. Adachi, S. Okda, T. Sasaki, and H. Nakanishi, "Continuously tunable THz and far-infrared wave generation fron DAST crystal," Electron. Lett. 40, 549-551 (2004).
[CrossRef]

J. W. Haus, P. Powers, M. Torres-Cisneros, M. Scalora, M. J. Bloemer, N. Akozbek, and M. A. Meneses-Nava, "Enhancement tunable terahertz generation in photonic band-gap structures," Laser Phys. 14, 635-642 (2004).

G. D'Aguanno, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, "Density of modes and tunneling times in finite, one-dimensional, photonic crystals: a comprehensive analysis," Phys. Rev. E 70, 016612 (2004).
[CrossRef]

2003 (1)

M. Centini, G. D'Aguanno, M. Scalora, M. J. Bloemer, C. M. Bowden, C. Sibilia, N. Mattiucci, and M. Bertolotti, "Dynamics of counterpropagating pulses in photonic crystals: enhancement and suppression of stimulated emission processes," Phys. Rev. E 67, 036617 (2003).
[CrossRef]

2002 (2)

2001 (3)

G. D'Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, "Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures," Phys. Rev. E 63, 036610 (2001).
[CrossRef]

O. Di Stefano, S. Savasta, and R. Girlanda, "Mode expansion and photon operators in dispersive and absorbing dielectrics," J. Mod. Opt. 48, 67-84 (2001).
[CrossRef]

M. Bertolotti, C. M. Bowden, and C. Sibilia, "Nanoscale linear and nonlinear optics," AIP Conf. Proc. 560, 1-31 (2001).
[CrossRef]

1998 (1)

A. Tredicucci, F. Capasso, G. Gmachi, D. L. Sivco, A. L. Hutchinson, and A. Y. Chao, "High performance interminiband quantum cascade lasers with grades superlattices," Appl. Phys. Lett. 73, 2101-2103 (1998).
[CrossRef]

1995 (1)

A. Nahata, D. H. Auston, C. Wu, and J. T. Yardley, "Generation of terahertz radiation from poled polymer," Appl. Phys. Lett. 67, 1358-1361 (1995).
[CrossRef]

1992 (2)

N. M. Froberg, B. B. Hu, X.-C. Zhang, and D. H. Auston, "Terahertz radiation from a photoconducting antenna array," IEEE J. Quantum Electron. 28, 2291-2301 (1992).
[CrossRef]

L. Xu, X.-C. Zhang, and D. H. Auston, "Terahertz beam generation by femtosecond optical pulses in electro-optic materials," Appl. Phys. Lett. 61, 1784-1787 (1992).
[CrossRef]

1990 (1)

1975 (1)

M. A. Piestrup and R. N. Fleming, "Continuosusly tunable submillimeter wave source," Appl. Phys. Lett. 26, 418-421 (1975).
[CrossRef]

1971 (1)

K. H. Yang, P. L. Richards, and Y. R. Shen, "Generation of far-infrared radiation by picosecond light pulses in LiNbO3," Appl. Phys. Lett. 19, 320-323 (1971).
[CrossRef]

1970 (1)

N. Bloembergen and A. J. Sievers, "Nonlinear optical properties of periodic laminar structures," Appl. Phys. Lett. 17, 483-486 (1970).
[CrossRef]

Adachi, H.

T. Taniuchi, H. Adachi, S. Okda, T. Sasaki, and H. Nakanishi, "Continuously tunable THz and far-infrared wave generation fron DAST crystal," Electron. Lett. 40, 549-551 (2004).
[CrossRef]

Akozbek, N.

J. W. Haus, P. Powers, M. Torres-Cisneros, M. Scalora, M. J. Bloemer, N. Akozbek, and M. A. Meneses-Nava, "Enhancement tunable terahertz generation in photonic band-gap structures," Laser Phys. 14, 635-642 (2004).

Alkuwari, R. A.

Auston, D. H.

A. Nahata, D. H. Auston, C. Wu, and J. T. Yardley, "Generation of terahertz radiation from poled polymer," Appl. Phys. Lett. 67, 1358-1361 (1995).
[CrossRef]

N. M. Froberg, B. B. Hu, X.-C. Zhang, and D. H. Auston, "Terahertz radiation from a photoconducting antenna array," IEEE J. Quantum Electron. 28, 2291-2301 (1992).
[CrossRef]

L. Xu, X.-C. Zhang, and D. H. Auston, "Terahertz beam generation by femtosecond optical pulses in electro-optic materials," Appl. Phys. Lett. 61, 1784-1787 (1992).
[CrossRef]

Beere, H. E.

Y. C. Shen, P. C. Upadhya, E. H. Linfield, and H. E. Beere, "Terahertz generation from coherent optical phonons in biased GaAs photoconductive emitter," Phys. Rev. E 69, 235325 (2004).

Bertolotti, M.

M. Centini, G. D'Aguanno, C. Sibilia, L. Sciscione, M. Bertolotti, M. Scalora, and M. J, Bloemer, "Non-phase-matched enhancement of second-harmonic generation in multilayer structures with internal reflections," Opt. Lett. 29, 1924-1926 (2004).
[CrossRef] [PubMed]

M. Centini, G. D'Aguanno, M. Scalora, M. J. Bloemer, C. M. Bowden, C. Sibilia, N. Mattiucci, and M. Bertolotti, "Dynamics of counterpropagating pulses in photonic crystals: enhancement and suppression of stimulated emission processes," Phys. Rev. E 67, 036617 (2003).
[CrossRef]

G. D'Aguanno, M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, "Generalized coupled-mode theory for chi(2) interactions in finite multilayered structures," J. Opt. Soc. Am. B 19, 2111-2121 (2002).
[CrossRef]

M. Bertolotti, C. M. Bowden, and C. Sibilia, "Nanoscale linear and nonlinear optics," AIP Conf. Proc. 560, 1-31 (2001).
[CrossRef]

G. D'Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, "Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures," Phys. Rev. E 63, 036610 (2001).
[CrossRef]

Bloembergen, N.

N. Bloembergen and A. J. Sievers, "Nonlinear optical properties of periodic laminar structures," Appl. Phys. Lett. 17, 483-486 (1970).
[CrossRef]

Bloemer, M. J

Bloemer, M. J.

G. D'Aguanno, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, "Density of modes and tunneling times in finite, one-dimensional, photonic crystals: a comprehensive analysis," Phys. Rev. E 70, 016612 (2004).
[CrossRef]

J. W. Haus, P. Powers, M. Torres-Cisneros, M. Scalora, M. J. Bloemer, N. Akozbek, and M. A. Meneses-Nava, "Enhancement tunable terahertz generation in photonic band-gap structures," Laser Phys. 14, 635-642 (2004).

M. Centini, G. D'Aguanno, M. Scalora, M. J. Bloemer, C. M. Bowden, C. Sibilia, N. Mattiucci, and M. Bertolotti, "Dynamics of counterpropagating pulses in photonic crystals: enhancement and suppression of stimulated emission processes," Phys. Rev. E 67, 036617 (2003).
[CrossRef]

G. D'Aguanno, M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, "Generalized coupled-mode theory for chi(2) interactions in finite multilayered structures," J. Opt. Soc. Am. B 19, 2111-2121 (2002).
[CrossRef]

G. D'Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, "Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures," Phys. Rev. E 63, 036610 (2001).
[CrossRef]

Bowden, C. M.

M. Centini, G. D'Aguanno, M. Scalora, M. J. Bloemer, C. M. Bowden, C. Sibilia, N. Mattiucci, and M. Bertolotti, "Dynamics of counterpropagating pulses in photonic crystals: enhancement and suppression of stimulated emission processes," Phys. Rev. E 67, 036617 (2003).
[CrossRef]

G. D'Aguanno, M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, "Generalized coupled-mode theory for chi(2) interactions in finite multilayered structures," J. Opt. Soc. Am. B 19, 2111-2121 (2002).
[CrossRef]

M. Bertolotti, C. M. Bowden, and C. Sibilia, "Nanoscale linear and nonlinear optics," AIP Conf. Proc. 560, 1-31 (2001).
[CrossRef]

G. D'Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, "Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures," Phys. Rev. E 63, 036610 (2001).
[CrossRef]

Capasso, F.

A. Tredicucci, F. Capasso, G. Gmachi, D. L. Sivco, A. L. Hutchinson, and A. Y. Chao, "High performance interminiband quantum cascade lasers with grades superlattices," Appl. Phys. Lett. 73, 2101-2103 (1998).
[CrossRef]

Centini, M.

M. Centini, G. D'Aguanno, C. Sibilia, L. Sciscione, M. Bertolotti, M. Scalora, and M. J, Bloemer, "Non-phase-matched enhancement of second-harmonic generation in multilayer structures with internal reflections," Opt. Lett. 29, 1924-1926 (2004).
[CrossRef] [PubMed]

M. Centini, G. D'Aguanno, M. Scalora, M. J. Bloemer, C. M. Bowden, C. Sibilia, N. Mattiucci, and M. Bertolotti, "Dynamics of counterpropagating pulses in photonic crystals: enhancement and suppression of stimulated emission processes," Phys. Rev. E 67, 036617 (2003).
[CrossRef]

G. D'Aguanno, M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, "Generalized coupled-mode theory for chi(2) interactions in finite multilayered structures," J. Opt. Soc. Am. B 19, 2111-2121 (2002).
[CrossRef]

G. D'Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, "Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures," Phys. Rev. E 63, 036610 (2001).
[CrossRef]

Chao, A. Y.

A. Tredicucci, F. Capasso, G. Gmachi, D. L. Sivco, A. L. Hutchinson, and A. Y. Chao, "High performance interminiband quantum cascade lasers with grades superlattices," Appl. Phys. Lett. 73, 2101-2103 (1998).
[CrossRef]

Chew, W. C.

W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, 1990), pp. 375-378.

D'Aguanno, G.

G. D'Aguanno, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, "Density of modes and tunneling times in finite, one-dimensional, photonic crystals: a comprehensive analysis," Phys. Rev. E 70, 016612 (2004).
[CrossRef]

M. Centini, G. D'Aguanno, C. Sibilia, L. Sciscione, M. Bertolotti, M. Scalora, and M. J, Bloemer, "Non-phase-matched enhancement of second-harmonic generation in multilayer structures with internal reflections," Opt. Lett. 29, 1924-1926 (2004).
[CrossRef] [PubMed]

M. Centini, G. D'Aguanno, M. Scalora, M. J. Bloemer, C. M. Bowden, C. Sibilia, N. Mattiucci, and M. Bertolotti, "Dynamics of counterpropagating pulses in photonic crystals: enhancement and suppression of stimulated emission processes," Phys. Rev. E 67, 036617 (2003).
[CrossRef]

G. D'Aguanno, M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, "Generalized coupled-mode theory for chi(2) interactions in finite multilayered structures," J. Opt. Soc. Am. B 19, 2111-2121 (2002).
[CrossRef]

G. D'Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, "Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures," Phys. Rev. E 63, 036610 (2001).
[CrossRef]

Di Stefano, O.

O. Di Stefano, S. Savasta, and R. Girlanda, "Mode expansion and photon operators in dispersive and absorbing dielectrics," J. Mod. Opt. 48, 67-84 (2001).
[CrossRef]

Ding, Y. J.

W. Shi and Y. J. Ding, "A monochromatic and high-power terahertz source tunable in the ranges of 2.7-38.4 and 58.2-3540 µm for variety of potential application," Appl. Phys. Lett. 84, 1635-1638 (2004).
[CrossRef]

Fleming, R. N.

M. A. Piestrup and R. N. Fleming, "Continuosusly tunable submillimeter wave source," Appl. Phys. Lett. 26, 418-421 (1975).
[CrossRef]

Froberg, N. M.

N. M. Froberg, B. B. Hu, X.-C. Zhang, and D. H. Auston, "Terahertz radiation from a photoconducting antenna array," IEEE J. Quantum Electron. 28, 2291-2301 (1992).
[CrossRef]

Girlanda, R.

O. Di Stefano, S. Savasta, and R. Girlanda, "Mode expansion and photon operators in dispersive and absorbing dielectrics," J. Mod. Opt. 48, 67-84 (2001).
[CrossRef]

Gmachi, G.

A. Tredicucci, F. Capasso, G. Gmachi, D. L. Sivco, A. L. Hutchinson, and A. Y. Chao, "High performance interminiband quantum cascade lasers with grades superlattices," Appl. Phys. Lett. 73, 2101-2103 (1998).
[CrossRef]

Hakhoumian, A. A.

A. S. Nikoghosyan, E. M. Laziev, R. M. Martirosyan, and A. A. Hakhoumian, "Efficient ultrashort light pulse conversion in GHz-THz pulses in ZnTe, GaAs, and DAST crystals," iCONO 2001: Ultrafast Phenomena and Strong Laser Fields, V.M.Gordienko, A.A.Afanas'ev, and V.V.Shuvalov, eds., Proc. SPIE 4752, 40-48 (2002).

Haus, J. W.

P. E. Powers, R. A. Alkuwari, J. W. Haus, K. Suizu, and H. Ito, "Terahertz generation with tandem seeded optical parametric generators," Opt. Lett. 30, 640-642 (2005).
[CrossRef] [PubMed]

J. W. Haus, P. Powers, M. Torres-Cisneros, M. Scalora, M. J. Bloemer, N. Akozbek, and M. A. Meneses-Nava, "Enhancement tunable terahertz generation in photonic band-gap structures," Laser Phys. 14, 635-642 (2004).

G. D'Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, "Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures," Phys. Rev. E 63, 036610 (2001).
[CrossRef]

Hu, B. B.

N. M. Froberg, B. B. Hu, X.-C. Zhang, and D. H. Auston, "Terahertz radiation from a photoconducting antenna array," IEEE J. Quantum Electron. 28, 2291-2301 (1992).
[CrossRef]

Hutchinson, A. L.

A. Tredicucci, F. Capasso, G. Gmachi, D. L. Sivco, A. L. Hutchinson, and A. Y. Chao, "High performance interminiband quantum cascade lasers with grades superlattices," Appl. Phys. Lett. 73, 2101-2103 (1998).
[CrossRef]

Ito, H.

Laziev, E. M.

A. S. Nikoghosyan, E. M. Laziev, R. M. Martirosyan, and A. A. Hakhoumian, "Efficient ultrashort light pulse conversion in GHz-THz pulses in ZnTe, GaAs, and DAST crystals," iCONO 2001: Ultrafast Phenomena and Strong Laser Fields, V.M.Gordienko, A.A.Afanas'ev, and V.V.Shuvalov, eds., Proc. SPIE 4752, 40-48 (2002).

Leaird, D. E.

Lekner, J.

J. Lekner, Theory of Reflection (Martin Nijhoff, 1987), Chap. 12.

Linfield, E. H.

Y. C. Shen, P. C. Upadhya, E. H. Linfield, and H. E. Beere, "Terahertz generation from coherent optical phonons in biased GaAs photoconductive emitter," Phys. Rev. E 69, 235325 (2004).

Lu, Y.

Y. Lu, M. Xiao, and G. J. Salamo, "Coherent microwave generation in nonlinear photonic crystal," IEEE J. Quantum Electron. 38, 481-485 (2002).
[CrossRef]

Martirosyan, R. M.

A. S. Nikoghosyan, E. M. Laziev, R. M. Martirosyan, and A. A. Hakhoumian, "Efficient ultrashort light pulse conversion in GHz-THz pulses in ZnTe, GaAs, and DAST crystals," iCONO 2001: Ultrafast Phenomena and Strong Laser Fields, V.M.Gordienko, A.A.Afanas'ev, and V.V.Shuvalov, eds., Proc. SPIE 4752, 40-48 (2002).

Mattiucci, N.

G. D'Aguanno, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, "Density of modes and tunneling times in finite, one-dimensional, photonic crystals: a comprehensive analysis," Phys. Rev. E 70, 016612 (2004).
[CrossRef]

M. Centini, G. D'Aguanno, M. Scalora, M. J. Bloemer, C. M. Bowden, C. Sibilia, N. Mattiucci, and M. Bertolotti, "Dynamics of counterpropagating pulses in photonic crystals: enhancement and suppression of stimulated emission processes," Phys. Rev. E 67, 036617 (2003).
[CrossRef]

Meneses-Nava, M. A.

J. W. Haus, P. Powers, M. Torres-Cisneros, M. Scalora, M. J. Bloemer, N. Akozbek, and M. A. Meneses-Nava, "Enhancement tunable terahertz generation in photonic band-gap structures," Laser Phys. 14, 635-642 (2004).

Nahata, A.

A. Nahata, D. H. Auston, C. Wu, and J. T. Yardley, "Generation of terahertz radiation from poled polymer," Appl. Phys. Lett. 67, 1358-1361 (1995).
[CrossRef]

Nakanishi, H.

T. Taniuchi and H. Nakanishi, "Continuously tunable terahertz-wave generation in GaP crystal by collinear difference frequency mixing," Electron. Lett. 40, 327-328 (2004).
[CrossRef]

T. Taniuchi, H. Adachi, S. Okda, T. Sasaki, and H. Nakanishi, "Continuously tunable THz and far-infrared wave generation fron DAST crystal," Electron. Lett. 40, 549-551 (2004).
[CrossRef]

Nikoghosyan, A. S.

A. S. Nikoghosyan, E. M. Laziev, R. M. Martirosyan, and A. A. Hakhoumian, "Efficient ultrashort light pulse conversion in GHz-THz pulses in ZnTe, GaAs, and DAST crystals," iCONO 2001: Ultrafast Phenomena and Strong Laser Fields, V.M.Gordienko, A.A.Afanas'ev, and V.V.Shuvalov, eds., Proc. SPIE 4752, 40-48 (2002).

Okda, S.

T. Taniuchi, H. Adachi, S. Okda, T. Sasaki, and H. Nakanishi, "Continuously tunable THz and far-infrared wave generation fron DAST crystal," Electron. Lett. 40, 549-551 (2004).
[CrossRef]

Piestrup, M. A.

M. A. Piestrup and R. N. Fleming, "Continuosusly tunable submillimeter wave source," Appl. Phys. Lett. 26, 418-421 (1975).
[CrossRef]

Powers, P.

J. W. Haus, P. Powers, M. Torres-Cisneros, M. Scalora, M. J. Bloemer, N. Akozbek, and M. A. Meneses-Nava, "Enhancement tunable terahertz generation in photonic band-gap structures," Laser Phys. 14, 635-642 (2004).

Powers, P. E.

Richards, P. L.

K. H. Yang, P. L. Richards, and Y. R. Shen, "Generation of far-infrared radiation by picosecond light pulses in LiNbO3," Appl. Phys. Lett. 19, 320-323 (1971).
[CrossRef]

Salamo, G. J.

Y. Lu, M. Xiao, and G. J. Salamo, "Coherent microwave generation in nonlinear photonic crystal," IEEE J. Quantum Electron. 38, 481-485 (2002).
[CrossRef]

Sasaki, T.

T. Taniuchi, H. Adachi, S. Okda, T. Sasaki, and H. Nakanishi, "Continuously tunable THz and far-infrared wave generation fron DAST crystal," Electron. Lett. 40, 549-551 (2004).
[CrossRef]

Savasta, S.

O. Di Stefano, S. Savasta, and R. Girlanda, "Mode expansion and photon operators in dispersive and absorbing dielectrics," J. Mod. Opt. 48, 67-84 (2001).
[CrossRef]

Scalora, M.

G. D'Aguanno, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, "Density of modes and tunneling times in finite, one-dimensional, photonic crystals: a comprehensive analysis," Phys. Rev. E 70, 016612 (2004).
[CrossRef]

J. W. Haus, P. Powers, M. Torres-Cisneros, M. Scalora, M. J. Bloemer, N. Akozbek, and M. A. Meneses-Nava, "Enhancement tunable terahertz generation in photonic band-gap structures," Laser Phys. 14, 635-642 (2004).

M. Centini, G. D'Aguanno, C. Sibilia, L. Sciscione, M. Bertolotti, M. Scalora, and M. J, Bloemer, "Non-phase-matched enhancement of second-harmonic generation in multilayer structures with internal reflections," Opt. Lett. 29, 1924-1926 (2004).
[CrossRef] [PubMed]

M. Centini, G. D'Aguanno, M. Scalora, M. J. Bloemer, C. M. Bowden, C. Sibilia, N. Mattiucci, and M. Bertolotti, "Dynamics of counterpropagating pulses in photonic crystals: enhancement and suppression of stimulated emission processes," Phys. Rev. E 67, 036617 (2003).
[CrossRef]

G. D'Aguanno, M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, "Generalized coupled-mode theory for chi(2) interactions in finite multilayered structures," J. Opt. Soc. Am. B 19, 2111-2121 (2002).
[CrossRef]

G. D'Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, "Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures," Phys. Rev. E 63, 036610 (2001).
[CrossRef]

Sciscione, L.

Shen, Y. C.

Y. C. Shen, P. C. Upadhya, E. H. Linfield, and H. E. Beere, "Terahertz generation from coherent optical phonons in biased GaAs photoconductive emitter," Phys. Rev. E 69, 235325 (2004).

Shen, Y. R.

K. H. Yang, P. L. Richards, and Y. R. Shen, "Generation of far-infrared radiation by picosecond light pulses in LiNbO3," Appl. Phys. Lett. 19, 320-323 (1971).
[CrossRef]

Y. R. Shen, The Principle of Nonlinear Optics (Wiley, 1984), pp. 110-113.

Shi, W.

W. Shi and Y. J. Ding, "A monochromatic and high-power terahertz source tunable in the ranges of 2.7-38.4 and 58.2-3540 µm for variety of potential application," Appl. Phys. Lett. 84, 1635-1638 (2004).
[CrossRef]

Sibilia, C.

M. Centini, G. D'Aguanno, C. Sibilia, L. Sciscione, M. Bertolotti, M. Scalora, and M. J, Bloemer, "Non-phase-matched enhancement of second-harmonic generation in multilayer structures with internal reflections," Opt. Lett. 29, 1924-1926 (2004).
[CrossRef] [PubMed]

M. Centini, G. D'Aguanno, M. Scalora, M. J. Bloemer, C. M. Bowden, C. Sibilia, N. Mattiucci, and M. Bertolotti, "Dynamics of counterpropagating pulses in photonic crystals: enhancement and suppression of stimulated emission processes," Phys. Rev. E 67, 036617 (2003).
[CrossRef]

G. D'Aguanno, M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, "Generalized coupled-mode theory for chi(2) interactions in finite multilayered structures," J. Opt. Soc. Am. B 19, 2111-2121 (2002).
[CrossRef]

M. Bertolotti, C. M. Bowden, and C. Sibilia, "Nanoscale linear and nonlinear optics," AIP Conf. Proc. 560, 1-31 (2001).
[CrossRef]

G. D'Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, "Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures," Phys. Rev. E 63, 036610 (2001).
[CrossRef]

Sievers, A. J.

N. Bloembergen and A. J. Sievers, "Nonlinear optical properties of periodic laminar structures," Appl. Phys. Lett. 17, 483-486 (1970).
[CrossRef]

Sivco, D. L.

A. Tredicucci, F. Capasso, G. Gmachi, D. L. Sivco, A. L. Hutchinson, and A. Y. Chao, "High performance interminiband quantum cascade lasers with grades superlattices," Appl. Phys. Lett. 73, 2101-2103 (1998).
[CrossRef]

Suizu, K.

Taniuchi, T.

T. Taniuchi and H. Nakanishi, "Continuously tunable terahertz-wave generation in GaP crystal by collinear difference frequency mixing," Electron. Lett. 40, 327-328 (2004).
[CrossRef]

T. Taniuchi, H. Adachi, S. Okda, T. Sasaki, and H. Nakanishi, "Continuously tunable THz and far-infrared wave generation fron DAST crystal," Electron. Lett. 40, 549-551 (2004).
[CrossRef]

Torres-Cisneros, M.

J. W. Haus, P. Powers, M. Torres-Cisneros, M. Scalora, M. J. Bloemer, N. Akozbek, and M. A. Meneses-Nava, "Enhancement tunable terahertz generation in photonic band-gap structures," Laser Phys. 14, 635-642 (2004).

Tredicucci, A.

A. Tredicucci, F. Capasso, G. Gmachi, D. L. Sivco, A. L. Hutchinson, and A. Y. Chao, "High performance interminiband quantum cascade lasers with grades superlattices," Appl. Phys. Lett. 73, 2101-2103 (1998).
[CrossRef]

Upadhya, P. C.

Y. C. Shen, P. C. Upadhya, E. H. Linfield, and H. E. Beere, "Terahertz generation from coherent optical phonons in biased GaAs photoconductive emitter," Phys. Rev. E 69, 235325 (2004).

Weiner, A. M.

Wu, C.

A. Nahata, D. H. Auston, C. Wu, and J. T. Yardley, "Generation of terahertz radiation from poled polymer," Appl. Phys. Lett. 67, 1358-1361 (1995).
[CrossRef]

Xiao, M.

Y. Lu, M. Xiao, and G. J. Salamo, "Coherent microwave generation in nonlinear photonic crystal," IEEE J. Quantum Electron. 38, 481-485 (2002).
[CrossRef]

Xu, L.

L. Xu, X.-C. Zhang, and D. H. Auston, "Terahertz beam generation by femtosecond optical pulses in electro-optic materials," Appl. Phys. Lett. 61, 1784-1787 (1992).
[CrossRef]

Yang, K. H.

K. H. Yang, P. L. Richards, and Y. R. Shen, "Generation of far-infrared radiation by picosecond light pulses in LiNbO3," Appl. Phys. Lett. 19, 320-323 (1971).
[CrossRef]

Yardley, J. T.

A. Nahata, D. H. Auston, C. Wu, and J. T. Yardley, "Generation of terahertz radiation from poled polymer," Appl. Phys. Lett. 67, 1358-1361 (1995).
[CrossRef]

Zhang, X.-C.

L. Xu, X.-C. Zhang, and D. H. Auston, "Terahertz beam generation by femtosecond optical pulses in electro-optic materials," Appl. Phys. Lett. 61, 1784-1787 (1992).
[CrossRef]

N. M. Froberg, B. B. Hu, X.-C. Zhang, and D. H. Auston, "Terahertz radiation from a photoconducting antenna array," IEEE J. Quantum Electron. 28, 2291-2301 (1992).
[CrossRef]

Zheltikov, A. M.

G. D'Aguanno, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, "Density of modes and tunneling times in finite, one-dimensional, photonic crystals: a comprehensive analysis," Phys. Rev. E 70, 016612 (2004).
[CrossRef]

AIP Conf. Proc. (1)

M. Bertolotti, C. M. Bowden, and C. Sibilia, "Nanoscale linear and nonlinear optics," AIP Conf. Proc. 560, 1-31 (2001).
[CrossRef]

Appl. Phys. Lett. (7)

W. Shi and Y. J. Ding, "A monochromatic and high-power terahertz source tunable in the ranges of 2.7-38.4 and 58.2-3540 µm for variety of potential application," Appl. Phys. Lett. 84, 1635-1638 (2004).
[CrossRef]

L. Xu, X.-C. Zhang, and D. H. Auston, "Terahertz beam generation by femtosecond optical pulses in electro-optic materials," Appl. Phys. Lett. 61, 1784-1787 (1992).
[CrossRef]

K. H. Yang, P. L. Richards, and Y. R. Shen, "Generation of far-infrared radiation by picosecond light pulses in LiNbO3," Appl. Phys. Lett. 19, 320-323 (1971).
[CrossRef]

A. Nahata, D. H. Auston, C. Wu, and J. T. Yardley, "Generation of terahertz radiation from poled polymer," Appl. Phys. Lett. 67, 1358-1361 (1995).
[CrossRef]

A. Tredicucci, F. Capasso, G. Gmachi, D. L. Sivco, A. L. Hutchinson, and A. Y. Chao, "High performance interminiband quantum cascade lasers with grades superlattices," Appl. Phys. Lett. 73, 2101-2103 (1998).
[CrossRef]

M. A. Piestrup and R. N. Fleming, "Continuosusly tunable submillimeter wave source," Appl. Phys. Lett. 26, 418-421 (1975).
[CrossRef]

N. Bloembergen and A. J. Sievers, "Nonlinear optical properties of periodic laminar structures," Appl. Phys. Lett. 17, 483-486 (1970).
[CrossRef]

Electron. Lett. (2)

T. Taniuchi and H. Nakanishi, "Continuously tunable terahertz-wave generation in GaP crystal by collinear difference frequency mixing," Electron. Lett. 40, 327-328 (2004).
[CrossRef]

T. Taniuchi, H. Adachi, S. Okda, T. Sasaki, and H. Nakanishi, "Continuously tunable THz and far-infrared wave generation fron DAST crystal," Electron. Lett. 40, 549-551 (2004).
[CrossRef]

IEEE J. Quantum Electron. (2)

Y. Lu, M. Xiao, and G. J. Salamo, "Coherent microwave generation in nonlinear photonic crystal," IEEE J. Quantum Electron. 38, 481-485 (2002).
[CrossRef]

N. M. Froberg, B. B. Hu, X.-C. Zhang, and D. H. Auston, "Terahertz radiation from a photoconducting antenna array," IEEE J. Quantum Electron. 28, 2291-2301 (1992).
[CrossRef]

J. Mod. Opt. (1)

O. Di Stefano, S. Savasta, and R. Girlanda, "Mode expansion and photon operators in dispersive and absorbing dielectrics," J. Mod. Opt. 48, 67-84 (2001).
[CrossRef]

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

Laser Phys. (1)

J. W. Haus, P. Powers, M. Torres-Cisneros, M. Scalora, M. J. Bloemer, N. Akozbek, and M. A. Meneses-Nava, "Enhancement tunable terahertz generation in photonic band-gap structures," Laser Phys. 14, 635-642 (2004).

Opt. Lett. (3)

Phys. Rev. E (4)

M. Centini, G. D'Aguanno, M. Scalora, M. J. Bloemer, C. M. Bowden, C. Sibilia, N. Mattiucci, and M. Bertolotti, "Dynamics of counterpropagating pulses in photonic crystals: enhancement and suppression of stimulated emission processes," Phys. Rev. E 67, 036617 (2003).
[CrossRef]

G. D'Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, "Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures," Phys. Rev. E 63, 036610 (2001).
[CrossRef]

G. D'Aguanno, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, "Density of modes and tunneling times in finite, one-dimensional, photonic crystals: a comprehensive analysis," Phys. Rev. E 70, 016612 (2004).
[CrossRef]

Y. C. Shen, P. C. Upadhya, E. H. Linfield, and H. E. Beere, "Terahertz generation from coherent optical phonons in biased GaAs photoconductive emitter," Phys. Rev. E 69, 235325 (2004).

Other (4)

A. S. Nikoghosyan, E. M. Laziev, R. M. Martirosyan, and A. A. Hakhoumian, "Efficient ultrashort light pulse conversion in GHz-THz pulses in ZnTe, GaAs, and DAST crystals," iCONO 2001: Ultrafast Phenomena and Strong Laser Fields, V.M.Gordienko, A.A.Afanas'ev, and V.V.Shuvalov, eds., Proc. SPIE 4752, 40-48 (2002).

J. Lekner, Theory of Reflection (Martin Nijhoff, 1987), Chap. 12.

W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, 1990), pp. 375-378.

Y. R. Shen, The Principle of Nonlinear Optics (Wiley, 1984), pp. 110-113.

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Figures (12)

Fig. 1
Fig. 1

Transmission spectrum of a 1-D periodic structure. The elementary cell is made by two layers of refractive index n 1 = 1.8 and n 2 = 1.4 at the frequency λ 0 = 1 μ m . The higher-index layer exhibits a normal dispersion. The thicknesses of the two layers are, respectively, d 1 = λ 0 2 n 1 , d 2 = λ 0 4 n 2 . The structure is surrounded by air ( n 0 = 1 ) . The transmission is a function of the normalized frequency ω ω 0 , where ω 0 = 2 π c λ 0 . In the inset is reported the spectrum of THz frequencies as a function of THz. The dark region in the inset is the part of the THz spectrum for which all the undesired second-order processes fall in the gap.

Fig. 2
Fig. 2

Conversion efficiency versus generated THz frequency. The solid curve is the total conversion efficiency, defined as η tot = η 2 ω 1 + η 2 ω 2 + η ω 1 + ω 2 + η ω 1 ω 2 . The long-dashed curve is the forward THz conversion efficiency η + THz multiplied by 100. The short-dashed curve is the backward THz conversion efficiency η THz multiplied by 100. The two pumps are arranged as shown in Fig. 3. The intensity is assumed to be 10 GW cm 2 for each pump.

Fig. 3
Fig. 3

Transmission spectrum as a function of the normalized frequency near the band edge. ω 1 is fixed on the band-edge resonance, whereas ω 2 is moved back in frequency.

Fig. 4
Fig. 4

Transmission and square modulus of the overlap integrals as a function of the second pump’s normalized frequency (lower scale) versus the THz-generated frequency (upper scale). Dark curve, the transmission (T); dark dashed curve, the sum of all the THz overlap integrals ( I THz tot); gray curve, the THz forward overlap integral ( I THz for); gray dashed curve, the backward overlap integral ( I THz back); and gray short-dashed curve, the sum of the overlap integrals of all the other generated frequencies, the optical ones ( I opt ) .

Fig. 5
Fig. 5

Overlap integrals as functions of the normalized first pump frequency ω 1 ω 0 , for four generated frequencies: (a) ω 1 ω 2 = 1 THz , (b) ω 1 ω 2 = 2.4 THz , (c) ω 1 ω 2 = 5 THz , (d) ω 1 ω 2 = 7.5 THz . Black curve, the transmission; black dashed curve, the sum of the THz overlap integrals (Overlap Tot); gray curve, the forward THz overlap integral (Overlap F); gray dashed curve, the backward overlap integral (Overlap B).

Fig. 6
Fig. 6

Configuration of the 3-D simulations. The pumps enter the structure at normal incidence; the interactive area is a cylinder of radius R equal to the spot size and length L equal to the photonic crystal length.—PBG, ∎.

Fig. 7
Fig. 7

Square modulus of the electric field, at a plane parallel to the y z plane, with distance of 4 cm from the source, in the forward direction as a function of the z coordinate. Solid curves, the photonic crystal emission; dashed curves, the equivalent dipole emission; short-dashed curves, the equivalent bulk emission. (a) ω 1 ω 2 = 1 THz , (b) ω 1 ω 2 = 2.5 THz , (c) ω 1 ω 2 = 5 THz , and (d) ω 1 ω 2 = 7.5 THz .

Fig. 8
Fig. 8

Overlap integrals (curves) and their equivalents for the 3-D model (points) as functions of the generated THz frequency. Solid curve and circles, the total (sum of the forward and backward) overlaps (Overlap Tot and EqOverlap Tot); dashed curve and squares, the forward overlaps (Overlap Forward and EqOverlap Forward); short-dashed curve and triangles, the backward overlaps (Overlap Backward and EqOverlap Backward).

Fig. 9
Fig. 9

Field localization as a function of the propagation coordinate x, for several incidence frequencies ν, in the plane-wave approximation. Solid curve, ν = 100 THz ; dashed curve, ν = 10 THz ; short-dashed curve, ν = 1 THz ; dashed–dotted curve, ν = 0.1 THz . Gray curve, the refractive index grating of the structure.

Fig. 10
Fig. 10

Field localization as a function of the propagation coordinate x, in the case of LTR incidence (solid curves) and RTL incidence (dashed curves). The frequency ν is (a) 1 THz, (b) 2.5 THz, (c) 5 THz, (d) 7.5 THz.

Fig. 11
Fig. 11

Overlap integrals (curves) and their equivalents for the 3-D model (points) in the case of both sides’ incidence. For both pumps, the phase difference between the LTR and the RTL beams is chosen in order to maximize the total conversion efficiency. Solid curve and circles, the total overlaps; the dashed curve and squares, the forward overlaps; the short-dashed curve and the triangles, the backward overlaps.

Fig. 12
Fig. 12

Overlap integrals (curves) and their equivalents for the 3-D model (points) in the case of both sides’ incidence. For both pumps, the phase difference between the LTR and the RTL beams is chosen in order to maximize the forward conversion efficiency. Solid curve and circles, the total overlaps; the dashed curve and squares, the forward overlaps; the short-dashed curve and the triangles, the backward overlaps.

Equations (35)

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d 2 d z 2 E ω 1 + ω 1 2 ϵ ω 1 E ω 1 c 2 = 2 ω 1 2 c 2 ( d ( 2 ) E ω 1 * E 2 ω 1 + d ( 2 ) E ω 2 * E ω 1 + ω 2 + d ( 2 ) E ω 2 E ω 1 ω 2 ) ,
d 2 d z 2 E ω 2 + ω 2 2 ϵ ω 2 E ω 2 c 2 = 2 ω 2 2 c 2 ( d ( 2 ) E ω 2 * E 2 ω 2 + d ( 2 ) E ω 1 * E ω 1 + ω 2 + d ( 2 ) E ω 1 E ω 1 ω 2 * ) ,
d 2 d z 2 E 2 ω 1 + 4 ω 1 2 ϵ 2 ω 1 E 2 ω 1 c 2 = 4 ω 1 2 c 2 ( d ( 2 ) E ω 1 2 + d ( 2 ) E ω 1 + ω 2 E ω 1 ω 2 ) ,
d 2 d z 2 E 2 ω 2 + 4 ω 2 2 ϵ 2 ω 2 E 2 ω 2 c 2 = 4 ω 2 2 c 2 ( d ( 2 ) E ω 2 2 + d ( 2 ) E ω 1 + ω 2 E ω 1 ω 2 * ) ,
d 2 d z 2 E ω 1 + ω 2 + ( ω 1 + ω 2 ) 2 ϵ ω 1 + ω 2 E ω 1 + ω 2 c 2 = 2 ( ω 1 + ω 2 ) 2 c 2 ( d ( 2 ) E ω 1 E ω 2 + d ( 2 ) E ω 1 ω 2 E 2 ω 2 + d ( 2 ) E ω 1 ω 2 * E 2 ω 1 ) ,
d 2 d z 2 E ω 1 ω 2 + ( ω 1 + ω 2 ) 2 ϵ ω 1 ω 2 E ω 1 ω 2 c 2 = 2 ( ω 1 ω 2 ) 2 c 2 ( d ( 2 ) E ω 1 E ω 2 * + d ( 2 ) E ω 1 + ω 2 E 2 ω 2 + d ( 2 ) E ω 1 + ω 2 * E 2 ω 1 ) .
E ω 1 = A 1 Φ ω 1 + + B 1 Φ ω 1 ,
E ω 2 = A 2 Φ ω 2 + + B 2 Φ ω 2 ,
E 2 ω 1 = 4 ω 1 2 c 2 G 2 ω 1 ( ξ , z ) d ( 2 ) ( ξ ) E ω 1 2 ( ξ ) d ξ ,
E 2 ω 2 = 4 ω 2 2 c 2 G 2 ω 2 ( ξ , z ) d ( 2 ) ( ξ ) E ω 2 2 ( ξ ) d ξ ,
E ω 1 + ω 2 = 2 ( ω 1 + ω 2 ) 2 c 2 G ω 1 + ω 2 ( ξ , z ) d ( 2 ) ( ξ ) E ω 1 ( ξ ) E ω 2 ( ξ ) d ξ ,
E ω 1 + ω 2 = 2 ( ω 1 ω 2 ) 2 c 2 G ω 1 ω 2 ( ξ , z ) d ( 2 ) ( ξ ) E ω 1 ( ξ ) E ω 2 * ( ξ ) d ξ ,
1 + r LTR = ϕ LTR ( 0 ) , i ( ω c ) ( 1 r LTR ) = d d z ϕ LTR ( z ) z = 0 ,
t LTR = ϕ LTR ( L ) , i ( ω c ) t LTR = d d z ϕ LTR ( z ) z = L ,
t RTL = ϕ RTL ( 0 ) , i ( ω c ) t RTL = d d z ϕ RTL ( z ) z = 0 ,
1 + r RTL = ϕ RTL ( L ) , i ( ω c ) ( 1 + r RTL ) = d d z ϕ RTL ( z ) z = L ,
i ( ω c ) t LTR = d d z ϕ LTR ( z ) z = L ,
G ω ( z , ξ ) = 1 2 i k 0 t ( ω ) { Φ ω ( z ) Φ ω + ( ξ ) 0 z < ξ Φ ω + ( z ) Φ ω ( ξ ) ξ < z L } ,
η 2 ω 1 ± = π 2 ϵ 0 c d ( 2 ) 2 ( Q 1 + Q ) 2 I tot ( L λ 2 ω 1 ) 2 1 L 0 L grating ( z ) [ Φ ω 1 + ( z ) ] 2 Φ 2 ω 1 ( z ) d z 2 ,
η 2 ω 2 ± = π 2 ϵ 0 c d ( 2 ) 2 ( 1 1 + Q ) 2 I tot ( L λ 2 ω 2 ) 2 1 L 0 L grating ( z ) [ Φ ω 2 + ( z ) ] 2 Φ 2 ω 2 ( z ) d z 2 ,
η ω 1 + ω 2 ± = 4 π 2 ϵ 0 c d ( 2 ) 2 Q ( 1 + Q ) 2 I tot ( L λ ω 1 + ω 2 ) 2 1 L 0 L grating ( z ) Φ ω 1 + ( z ) Φ ω 2 + ( z ) Φ ω 1 + ω 2 ( z ) d z 2 ,
η ω 1 ω 2 ± = 4 π 2 ϵ 0 c d ( 2 ) 2 Q ( 1 + Q ) 2 I tot ( L λ ω 1 ω 2 ) 2 1 L 0 L grating ( z ) Φ ω 1 + ( z ) [ Φ ω 2 + ( z ) ] * Φ ω 1 ω 2 ( z ) d z 2 ,
I ± = 1 L 0 L grating ( z ) Φ ω 1 + ( z ) [ Φ ω 2 + ( z ) ] * Φ ω 1 ω 2 ( z ) d z 2 .
× × E ω 2 μ 0 ϵ ̂ E = 2 ϵ 0 d ̿ ( 2 ) : E 1 E 2 * .
E = μ 0 ω 2 V d V G ̂ ( r , r ) 2 ϵ 0 d ̿ ( 2 ) : E 1 E 2 * ,
G i j ( r , r ) = ( δ i j x i x j ) 1 4 π r 3 exp ( i k 0 r ) exp ( i k 0 r r r ) .
E = μ 0 ϵ 0 ω 2 d ( 2 ) A 1 A 2 * exp ( i k 0 r ) exp ( i ω t ) R k 0 ρ J 1 ( k 0 ρ R r ) I x ( x y r 2 1 y 2 r 2 y z r 2 ) ,
η = 0 a d ρ ρ 0 2 π d φ S x ( d , ρ , φ ) 0 a d ρ ρ 0 2 π d φ S x ( d , ρ , φ ) + a d d d x 0 2 π d φ [ S y ( x , a , φ ) cos φ + S z ( x , a , φ ) sin φ ] π R 2 ϵ 0 c ( A 1 2 + A 2 2 ) .
η + = 0 a d ρ ρ 0 2 π d φ S x ( d , ρ , φ ) π R 2 ϵ 0 c ( A 1 2 + A 2 2 ) ,
η = 0 a d ρ ρ 0 2 π d φ S x ( d , ρ , φ ) π R 2 ϵ 0 c ( A 1 2 , A 2 2 ) ,
η lat = a d d d x 0 2 π d φ [ S y ( x , a , φ ) cos φ + S z ( x , a , φ ) sin φ ] π R 2 ϵ 0 c ( A 1 2 + A 2 2 ) .
η + = 4 π 2 ϵ 0 c d ( 2 ) 2 Q ( 1 + Q ) 2 I tot ( L λ ) 2 I eq + ,
η = 4 π 2 ϵ 0 c d ( 2 ) 2 Q ( 1 + Q ) 2 I tot ( L λ ) 2 I eq ,
I eq + = 1 π L 2 s 1 y 2 + z 2 1 r 4 ( J 1 ) 2 [ r a ( a 2 + z 2 ) I x 2 + z 2 r Re ( I x L x * ) ] d s ,
I eq = 1 π L 2 s 1 y 2 + z 2 1 r 4 ( J 1 ) 2 [ r a ( a 2 + z 2 ) I x 2 + z 2 r Re ( I x L x * ) ] d s .

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