Abstract

We develop a coupled mode theory (CMT) model of the behavior of a polarization source in a general photonic structure, and obtain an analytical expression for the resulting generated electric field; loss, gain and/or nonlinearities can also be modeled. Based on this treatment, we investigate the criteria needed to achieve an enhancement in various nonlinear effects, and to produce efficient sources of terahertz radiation, in particular. Our results agree well with exact finite-difference time-domain (FDTD) results. Therefore, this approach can also in certain circumstances be used as a potential substitute for the more numerically intensive FDTD method.

© 2008 Optical Society of America

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  1. P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, "Generation of Optical Harmonics," Phys. Rev. Lett. 6, 118-119 (1961).
    [CrossRef]
  2. M. Bass, P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, "Optical Mixing," Phys. Rev. Lett. 8, 18-18 (1962).
    [CrossRef]
  3. J. A. Giordmaine, "Mixing of Light Beams in Crystals," Phys. Rev. Lett. 8, 19-20 (1962).
    [CrossRef]
  4. P. D. Maker, R. W. Terhune, M. Nisenhoff, and C. M. Savage, "Effects of Dispersion and Focusing on the Production of Optical Harmonics," Phys. Rev. Lett. 8, 21-22 (1962).
    [CrossRef]
  5. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between Light Waves in a Nonlinear Dielectric," Phys. Rev. 127, 1918-1939 (1962).
    [CrossRef]
  6. N. Bloembergen and Y. R. Shen, "Quantum-Theoretical Comparison of Nonlinear Susceptibilities in Parametric Media, Lasers, and Raman Lasers," Phys. Rev. 133, A37-A49 (1964).
    [CrossRef]
  7. G. Eckardt, R. W. Hellwarth, F. J. McClung, S. E. Schwartz, D. Weiner, and E. J. Woodbury, "Stimulated Raman Scattering From Organic Liquids," Phys. Rev. Lett. 9, 455 (1962).
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  10. Yuri Kivshar, "Spatial solitons: Bending light at will," Nature Phys. 2, 729-730 (2006).
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    [CrossRef]
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    [CrossRef]
  13. B. Ferguson and X. -Cheng Zhang, "Materials for terahertz science and technology," Nature Mater. 1, 26-33 (2002).
    [CrossRef]
  14. E. Mueller, "Terahertz radiation sources for imaging and sensing applications," Photon. Spectra. 40, 60 (2006).
  15. E. Purcell, "Spontaneous emission probabilities at radio frequencies," Phys. Rev. 69, 681 (1946).
  16. M. Solja�?i�?, and J. D. Joannopoulos, "Enhancement of non-linear effects using photonic crystals," Nature Mater. 3, 211-219 (2004).
    [CrossRef] [PubMed]
  17. G. DAguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, "Photonic band edge effects in finite structures and applications to |(2) interactions," Phys. Rev. E. 64, 016609 (2001).
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  20. T. Laroche, F. I. Baida, and D. Van Labeke, "Three-dimensional finite-difference time-domain study of enhanced second-harmonic generation at the end of a apertureless scanning near-field optical microscope metal tip," J. Opt. Soc. Am. B 22, 1045-1051 (2005).
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    [CrossRef] [PubMed]
  26. D. H. Auston, K. P. Cheung, J. A. Valdmanis, and D. A. Kleinman, "Cherenkov Radiation from Femtosecond Optical Pulses in Electro-Optic Media," Phys. Rev. Lett. 53, 1555-1558 (1984).
    [CrossRef]
  27. M. Ibanescu, E. J. Reed, and J. D. Joannopoulos, "Enhanced Photonic Band-Gap Confinement via Van Hove Saddle Point Singularities," Phys. Rev. Lett. 96, 033904 (2006).
    [CrossRef] [PubMed]
  28. G. Chang, C. J. Divin, J. Yang, M. A. Musheinish, S. L. Williamson, A. Galvanauskas, and T. B. Norris, "GaP waveguide emitters for high power broadband THz generation pumped by Yb-doped fiber lasers," Opt. Express 15, 16308-16315 (2007).
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    [CrossRef]
  30. S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell�??s equations in a planewave basis, " Opt. Express 8, 173-190 (2001).
    [CrossRef] [PubMed]
  31. R. W. Boyd, Nonlinear Optics (Academic Press, 2002).
  32. A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. Burr, "Improving accuracy by subpixel smoothing in the finite-difference time domain," Opt. Lett. 31, 2972-2974 (2006).
    [CrossRef] [PubMed]

2007

2006

C. Luo, M. Ibanescu, E. J. Reed, S. G. Johnson, and J. D. Joannopoulos, "Doppler Radiation Emitted by an Oscillating Dipole Moving inside a Photonic Band-Gap Crystal," Phys. Rev. Lett. 96, 043903 (2006).
[CrossRef] [PubMed]

M. Ibanescu, E. J. Reed, and J. D. Joannopoulos, "Enhanced Photonic Band-Gap Confinement via Van Hove Saddle Point Singularities," Phys. Rev. Lett. 96, 033904 (2006).
[CrossRef] [PubMed]

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

Yuri Kivshar, "Spatial solitons: Bending light at will," Nature Phys. 2, 729-730 (2006).
[CrossRef]

Günter Steinmeyer, "Laser physics: Terahertz meets attoscience," Nature Phys. 2, 305-306 (2006).
[CrossRef]

E. Mueller, "Terahertz radiation sources for imaging and sensing applications," Photon. Spectra. 40, 60 (2006).

2005

2004

M. Solja�?i�?, and J. D. Joannopoulos, "Enhancement of non-linear effects using photonic crystals," Nature Mater. 3, 211-219 (2004).
[CrossRef] [PubMed]

2003

T. Tanabe, K. Suto, J.-ichi Nishizawa, K. Saito, and T. Kimura, "Frequency-tunable terahertz wave generation via excitation of phonon-polaritons in GaP," J. Phys. D: Applied Physics 36, 953-957 (2003).
[CrossRef]

2002

B. Ferguson and X. -Cheng Zhang, "Materials for terahertz science and technology," Nature Mater. 1, 26-33 (2002).
[CrossRef]

2001

G. DAguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, "Photonic band edge effects in finite structures and applications to |(2) interactions," Phys. Rev. E. 64, 016609 (2001).
[CrossRef]

J. Squier and M. Müller, "High resolution nonlinear microscopy: a review of sources and methods for achieving optimal imaging," Rev. Sci. Instrum. 72, 2855-2867 (2001).
[CrossRef]

S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell�??s equations in a planewave basis, " Opt. Express 8, 173-190 (2001).
[CrossRef] [PubMed]

1996

K. Sakoda and K. Ohtaka, "Optical response of three-dimensional photonic lattices: Solutions of inhomogeneous Maxwells equations and their applications," Phys. Rev. B 54, 5732-5741 (1996).
[CrossRef]

K. Sakoda and K. Ohtaka, "Sum-frequency generation in a two-dimensional photonic lattice," Phys. Rev. B 54, 5742-5749 (1996).
[CrossRef]

1984

D. H. Auston, K. P. Cheung, J. A. Valdmanis, and D. A. Kleinman, "Cherenkov Radiation from Femtosecond Optical Pulses in Electro-Optic Media," Phys. Rev. Lett. 53, 1555-1558 (1984).
[CrossRef]

1964

N. Bloembergen and Y. R. Shen, "Quantum-Theoretical Comparison of Nonlinear Susceptibilities in Parametric Media, Lasers, and Raman Lasers," Phys. Rev. 133, A37-A49 (1964).
[CrossRef]

1962

G. Eckardt, R. W. Hellwarth, F. J. McClung, S. E. Schwartz, D. Weiner, and E. J. Woodbury, "Stimulated Raman Scattering From Organic Liquids," Phys. Rev. Lett. 9, 455 (1962).
[CrossRef]

M. Bass, P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, "Optical Mixing," Phys. Rev. Lett. 8, 18-18 (1962).
[CrossRef]

J. A. Giordmaine, "Mixing of Light Beams in Crystals," Phys. Rev. Lett. 8, 19-20 (1962).
[CrossRef]

P. D. Maker, R. W. Terhune, M. Nisenhoff, and C. M. Savage, "Effects of Dispersion and Focusing on the Production of Optical Harmonics," Phys. Rev. Lett. 8, 21-22 (1962).
[CrossRef]

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between Light Waves in a Nonlinear Dielectric," Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

1961

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, "Generation of Optical Harmonics," Phys. Rev. Lett. 6, 118-119 (1961).
[CrossRef]

1946

E. Purcell, "Spontaneous emission probabilities at radio frequencies," Phys. Rev. 69, 681 (1946).

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between Light Waves in a Nonlinear Dielectric," Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

Auston, D. H.

D. H. Auston, K. P. Cheung, J. A. Valdmanis, and D. A. Kleinman, "Cherenkov Radiation from Femtosecond Optical Pulses in Electro-Optic Media," Phys. Rev. Lett. 53, 1555-1558 (1984).
[CrossRef]

Baida, F. I.

Bass, M.

M. Bass, P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, "Optical Mixing," Phys. Rev. Lett. 8, 18-18 (1962).
[CrossRef]

Bermel, P.

Bloembergen, N.

N. Bloembergen and Y. R. Shen, "Quantum-Theoretical Comparison of Nonlinear Susceptibilities in Parametric Media, Lasers, and Raman Lasers," Phys. Rev. 133, A37-A49 (1964).
[CrossRef]

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between Light Waves in a Nonlinear Dielectric," Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

Bravo-Abad, J.

Burr, G.

Chang, G.

Cheng Zhang, X.

B. Ferguson and X. -Cheng Zhang, "Materials for terahertz science and technology," Nature Mater. 1, 26-33 (2002).
[CrossRef]

Cheung, K. P.

D. H. Auston, K. P. Cheung, J. A. Valdmanis, and D. A. Kleinman, "Cherenkov Radiation from Femtosecond Optical Pulses in Electro-Optic Media," Phys. Rev. Lett. 53, 1555-1558 (1984).
[CrossRef]

Divin, C. J.

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between Light Waves in a Nonlinear Dielectric," Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

Dudley, J. M.

J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, "Self-similarity in ultrafast nonlinear optics," Nature Phys. 3, 597-603 (2007).
[CrossRef]

Eckardt, G.

G. Eckardt, R. W. Hellwarth, F. J. McClung, S. E. Schwartz, D. Weiner, and E. J. Woodbury, "Stimulated Raman Scattering From Organic Liquids," Phys. Rev. Lett. 9, 455 (1962).
[CrossRef]

Farjadpour, A.

Ferguson, B.

B. Ferguson and X. -Cheng Zhang, "Materials for terahertz science and technology," Nature Mater. 1, 26-33 (2002).
[CrossRef]

Finot, C.

J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, "Self-similarity in ultrafast nonlinear optics," Nature Phys. 3, 597-603 (2007).
[CrossRef]

Franken, P. A.

M. Bass, P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, "Optical Mixing," Phys. Rev. Lett. 8, 18-18 (1962).
[CrossRef]

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, "Generation of Optical Harmonics," Phys. Rev. Lett. 6, 118-119 (1961).
[CrossRef]

Galvanauskas, A.

Giordmaine, J. A.

J. A. Giordmaine, "Mixing of Light Beams in Crystals," Phys. Rev. Lett. 8, 19-20 (1962).
[CrossRef]

Hellwarth, R. W.

G. Eckardt, R. W. Hellwarth, F. J. McClung, S. E. Schwartz, D. Weiner, and E. J. Woodbury, "Stimulated Raman Scattering From Organic Liquids," Phys. Rev. Lett. 9, 455 (1962).
[CrossRef]

Hill, A. E.

M. Bass, P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, "Optical Mixing," Phys. Rev. Lett. 8, 18-18 (1962).
[CrossRef]

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, "Generation of Optical Harmonics," Phys. Rev. Lett. 6, 118-119 (1961).
[CrossRef]

Ibanescu, M.

C. Luo, M. Ibanescu, E. J. Reed, S. G. Johnson, and J. D. Joannopoulos, "Doppler Radiation Emitted by an Oscillating Dipole Moving inside a Photonic Band-Gap Crystal," Phys. Rev. Lett. 96, 043903 (2006).
[CrossRef] [PubMed]

M. Ibanescu, E. J. Reed, and J. D. Joannopoulos, "Enhanced Photonic Band-Gap Confinement via Van Hove Saddle Point Singularities," Phys. Rev. Lett. 96, 033904 (2006).
[CrossRef] [PubMed]

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

Joannopoulos, J. D.

J. Bravo-Abad, A. Rodriguez, P. Bermel, S. G. Johnson, J. D. Joannopoulos, and M. Solja�?i�?, "Enhanced nonlinear optics in photonic-crystal microcavities," Opt. Express 15, 16161-16176 (2007).
[CrossRef] [PubMed]

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

M. Ibanescu, E. J. Reed, and J. D. Joannopoulos, "Enhanced Photonic Band-Gap Confinement via Van Hove Saddle Point Singularities," Phys. Rev. Lett. 96, 033904 (2006).
[CrossRef] [PubMed]

C. Luo, M. Ibanescu, E. J. Reed, S. G. Johnson, and J. D. Joannopoulos, "Doppler Radiation Emitted by an Oscillating Dipole Moving inside a Photonic Band-Gap Crystal," Phys. Rev. Lett. 96, 043903 (2006).
[CrossRef] [PubMed]

M. Solja�?i�?, and J. D. Joannopoulos, "Enhancement of non-linear effects using photonic crystals," Nature Mater. 3, 211-219 (2004).
[CrossRef] [PubMed]

S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell�??s equations in a planewave basis, " Opt. Express 8, 173-190 (2001).
[CrossRef] [PubMed]

Johnson, S. G.

Kleinman, D. A.

D. H. Auston, K. P. Cheung, J. A. Valdmanis, and D. A. Kleinman, "Cherenkov Radiation from Femtosecond Optical Pulses in Electro-Optic Media," Phys. Rev. Lett. 53, 1555-1558 (1984).
[CrossRef]

Laroche, T.

Luo, C.

C. Luo, M. Ibanescu, E. J. Reed, S. G. Johnson, and J. D. Joannopoulos, "Doppler Radiation Emitted by an Oscillating Dipole Moving inside a Photonic Band-Gap Crystal," Phys. Rev. Lett. 96, 043903 (2006).
[CrossRef] [PubMed]

Maker, P. D.

P. D. Maker, R. W. Terhune, M. Nisenhoff, and C. M. Savage, "Effects of Dispersion and Focusing on the Production of Optical Harmonics," Phys. Rev. Lett. 8, 21-22 (1962).
[CrossRef]

McClung, F. J.

G. Eckardt, R. W. Hellwarth, F. J. McClung, S. E. Schwartz, D. Weiner, and E. J. Woodbury, "Stimulated Raman Scattering From Organic Liquids," Phys. Rev. Lett. 9, 455 (1962).
[CrossRef]

Millot, G.

J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, "Self-similarity in ultrafast nonlinear optics," Nature Phys. 3, 597-603 (2007).
[CrossRef]

Mueller, E.

E. Mueller, "Terahertz radiation sources for imaging and sensing applications," Photon. Spectra. 40, 60 (2006).

Müller, M.

J. Squier and M. Müller, "High resolution nonlinear microscopy: a review of sources and methods for achieving optimal imaging," Rev. Sci. Instrum. 72, 2855-2867 (2001).
[CrossRef]

Musheinish, M. A.

Nisenhoff, M.

P. D. Maker, R. W. Terhune, M. Nisenhoff, and C. M. Savage, "Effects of Dispersion and Focusing on the Production of Optical Harmonics," Phys. Rev. Lett. 8, 21-22 (1962).
[CrossRef]

Norris, T. B.

Ohtaka, K.

K. Sakoda and K. Ohtaka, "Optical response of three-dimensional photonic lattices: Solutions of inhomogeneous Maxwells equations and their applications," Phys. Rev. B 54, 5732-5741 (1996).
[CrossRef]

K. Sakoda and K. Ohtaka, "Sum-frequency generation in a two-dimensional photonic lattice," Phys. Rev. B 54, 5742-5749 (1996).
[CrossRef]

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between Light Waves in a Nonlinear Dielectric," Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

Peters, C. W.

M. Bass, P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, "Optical Mixing," Phys. Rev. Lett. 8, 18-18 (1962).
[CrossRef]

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, "Generation of Optical Harmonics," Phys. Rev. Lett. 6, 118-119 (1961).
[CrossRef]

Purcell, E.

E. Purcell, "Spontaneous emission probabilities at radio frequencies," Phys. Rev. 69, 681 (1946).

Reed, E. J.

M. Ibanescu, E. J. Reed, and J. D. Joannopoulos, "Enhanced Photonic Band-Gap Confinement via Van Hove Saddle Point Singularities," Phys. Rev. Lett. 96, 033904 (2006).
[CrossRef] [PubMed]

C. Luo, M. Ibanescu, E. J. Reed, S. G. Johnson, and J. D. Joannopoulos, "Doppler Radiation Emitted by an Oscillating Dipole Moving inside a Photonic Band-Gap Crystal," Phys. Rev. Lett. 96, 043903 (2006).
[CrossRef] [PubMed]

Richardson, D. J.

J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, "Self-similarity in ultrafast nonlinear optics," Nature Phys. 3, 597-603 (2007).
[CrossRef]

Rodriguez, A.

Roundy, D.

Sakoda, K.

K. Sakoda and K. Ohtaka, "Optical response of three-dimensional photonic lattices: Solutions of inhomogeneous Maxwells equations and their applications," Phys. Rev. B 54, 5732-5741 (1996).
[CrossRef]

K. Sakoda and K. Ohtaka, "Sum-frequency generation in a two-dimensional photonic lattice," Phys. Rev. B 54, 5742-5749 (1996).
[CrossRef]

Savage, C. M.

P. D. Maker, R. W. Terhune, M. Nisenhoff, and C. M. Savage, "Effects of Dispersion and Focusing on the Production of Optical Harmonics," Phys. Rev. Lett. 8, 21-22 (1962).
[CrossRef]

Schwartz, S. E.

G. Eckardt, R. W. Hellwarth, F. J. McClung, S. E. Schwartz, D. Weiner, and E. J. Woodbury, "Stimulated Raman Scattering From Organic Liquids," Phys. Rev. Lett. 9, 455 (1962).
[CrossRef]

Shen, Y. R.

N. Bloembergen and Y. R. Shen, "Quantum-Theoretical Comparison of Nonlinear Susceptibilities in Parametric Media, Lasers, and Raman Lasers," Phys. Rev. 133, A37-A49 (1964).
[CrossRef]

Solja??i??, M.

Squier, J.

J. Squier and M. Müller, "High resolution nonlinear microscopy: a review of sources and methods for achieving optimal imaging," Rev. Sci. Instrum. 72, 2855-2867 (2001).
[CrossRef]

Suto, K.

T. Tanabe, K. Suto, J.-ichi Nishizawa, K. Saito, and T. Kimura, "Frequency-tunable terahertz wave generation via excitation of phonon-polaritons in GaP," J. Phys. D: Applied Physics 36, 953-957 (2003).
[CrossRef]

Tanabe, T.

T. Tanabe, K. Suto, J.-ichi Nishizawa, K. Saito, and T. Kimura, "Frequency-tunable terahertz wave generation via excitation of phonon-polaritons in GaP," J. Phys. D: Applied Physics 36, 953-957 (2003).
[CrossRef]

Terhune, R. W.

P. D. Maker, R. W. Terhune, M. Nisenhoff, and C. M. Savage, "Effects of Dispersion and Focusing on the Production of Optical Harmonics," Phys. Rev. Lett. 8, 21-22 (1962).
[CrossRef]

Valdmanis, J. A.

D. H. Auston, K. P. Cheung, J. A. Valdmanis, and D. A. Kleinman, "Cherenkov Radiation from Femtosecond Optical Pulses in Electro-Optic Media," Phys. Rev. Lett. 53, 1555-1558 (1984).
[CrossRef]

Van Labeke, D.

Weiner, D.

G. Eckardt, R. W. Hellwarth, F. J. McClung, S. E. Schwartz, D. Weiner, and E. J. Woodbury, "Stimulated Raman Scattering From Organic Liquids," Phys. Rev. Lett. 9, 455 (1962).
[CrossRef]

Weinreich, G.

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Figures (8)

Fig. 1.
Fig. 1.

(Color online) Color contour plot of the dielectric function ε(x,y) of the 2D PhC srtucture: (a) 1a×1a cell, and (b) 5a×5a cell, showing the optical beam through the central waveguide.

Fig. 2.
Fig. 2.

(Color online) (a) Projected band diagram of the first three bands. (b) Color contour plot of the second band, showing the saddle point where the band is narrowest.

Fig. 3.
Fig. 3.

(Color online) Color contour plot of w k y = 0 ( x , y ) .

Fig. 4.
Fig. 4.

(Color online) Periodically poled PhC structure. The red portions correspond to the original orientation of the nonlinear crystal when χ(2) is positive, while in the blue portions, the crystal’s orientation is flipped resulting in a negative χ(2).

Fig. 5.
Fig. 5.

(Color online) Overlap integral �� poled one period as a function of ky for modes at the narrowest portion (kx =0.1559(2π/a)) of the second band.

Fig. 6.
Fig. 6.

(Color online) CMT calculations: (a) Color contour plot of the unnormalized terahertz energy density ε(x,y)|E(x,y,t)|2 at t=1010(a/c), in a 2D box of size 1a×80a. (b) A zoom-in version of the plot in Fig. a, showing more details of the interval y∈±[8a,19a]. Note that the optical beam was originally sent through the waveguide at y=0

Fig. 7.
Fig. 7.

(Color online) FDTD calculation for the terahertz emitted energy in the PhC structure (not normalized to the bulk).

Fig. 8.
Fig. 8.

(Color online) FDTD calculations: (a) Color contour plot of the terahertz energy density ε(x,y)|E(x,y,t)|2 at t=1010(a/c), in a 2D box of size 1a×80a. (b) A zoom-in version of the plot in Fig. a, showing more details of the interval y∈±[8a,19a]. Note that the optical beam was originally sent through the waveguide at y=0.

Equations (35)

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

E ( r , t ) = v a v ( t ) E v ( r ) d 3 r ε ( r ) 2 E v ( r ) 2
d a v d t = i ω v a v ( Γ rad v + Γ abs v Γ g v ) a v + κ v s + v
κ v s + v = d 3 r J ( r , t ) · E v * ( r ) d 3 r ε ( r ) 2 E v ( r ) 2
d d t ( a v ( t ) Ξ ( t ) ) = Ξ ( t ) κ v s + v
a v ( t ) = a v ( t o ) e i ω v ( t t o ) e ( Γ rad v + Γ abs v Γ g v ) ( t t o )
+ t o t d t all space d 3 r J ( r , t ) · E v * ( r ) d 3 r ε ( r ) 2 E v ( r ) 2 e i ω v ( t t ) e ( Γ rad v + Γ abs v Γ g v ) ( t t )
E ( r , t ) = v E v ( r ) d 3 r ε ( r ) 2 E v ( r ) 2 a v ( t o ) e i ω v ( t t o ) e ( Γ ras v + Γ abs v Γ g v ) ( t t o ) +
v E v ( r ) d 3 r ε ( r ) 2 E v ( r ) 2 t o t d t all space d 3 r J ( r , t ) · E v * ( r ) e i ω v ( t t ) e ( Γ rad v + Γ abs v Γ g v ) ( t t )
A S · d a = A { Real [ E ( r , t ) ] × Real [ H ( r , t ) ] } · d a
a v = i ω s p · E v * ( r o ) i ( ω v ω s ) + ( Γ rad v + Γ abs v ) e i ω s t d 3 r ε ( r ) 2 E v ( r ) 2
𝒫 = v Γ rad v ( ω v ω s ) 2 + ( Γ v ) 2 2 ( ω s ) 2 p · E v * ( r o ) 2 d 3 r ε ( r ) 2 E v ( r ) 2
𝒫 = 4 π ( ω s ) 2 p 2 ε ( r o ) v ε ( r o ) p ̂ · E v * ( r o ) 2 δ ( ω v ω s ) d 3 r ε ( r ) E v ( r ) 2
𝒫 = 4 π ( ω s ) 2 p 2 ε ( r o ) g ( ω s , r o )
g ( ω s , r o ) = v ε ( r o ) E v ( r o ) 2 δ ( ω v ω s ) d 3 r ε ( r ) E v ( r ) 2
J ( r , t ) = j o e i ω s t δ ( r v t )
E ( r , t ) = v E v ( r ) d 3 r ε ( r ) 2 E v ( r ) 2 t d t e i ω s t j o · E v * ( v t ) e i ω v ( t t )
E ( r , t ) = n k σ E n k σ ( r ) d 3 r ε ( r ) 2 E n k σ ( r ) 2 G j o · e n k , G * t d t e i ω s t e i ( k + G ) · v t e i ω n k σ ( t t )
E ( r , t ) = n k σ E n k σ ( r ) d 3 r ε ( r ) 2 E n k σ ( r ) 2 G i [ j o · ( e n k , G ) * ] ω n k σ [ ω s + ( k + G ) · v ] e i [ ω s + ( k + G ) · v ] t
J T H z ( x , y ; t ) = z ̂ e i k x s x y 2 2 ζ 2 e i ω s t ( t 500 a c ) 2 2 τ 2
J T H z ( x , y ; t ) = J space T H z ( x , y ) · 𝓕 ( t )
J space T H z ( x , y ) = z ̂ e i k x s x y 2 2 ζ 2
𝓕 ( t ) = e i ω s t ( t 500 a c ) 2 2 τ 2
w k y ( x , y ) ε ( x , y ) E n = 2 ; ( k x = 0.1559 ( 2 π a ) , k y ; σ = T M ) ( x , y ) 2
𝒪 n k all space d 3 r J space T H z ( r ) · E n k * ( r )
𝒪 all space poled ( k y ) all space d 3 r q ( x ) J space T H z ( r ) · E n = 2 ; ( k x s , k y ) * ( r )
= all space d 3 r q ( x ) e i k x s x y 2 2 ζ 2 z ̂ . E n = 2 ; ( k x s , k y ) * ( r )
𝒥 t ( k y ) e i ω ( n = 2 ; k x s , k y ) t 0 t d t 𝓕 ( t ) e i ω ( n = 2 , k x s , k y ) t
= e i ω ( n = 2 ; k x s , k y ) t 0 t d t e i ( ω s ω ( n = 2 ; k x s , k y ) ) t ( t 500 a c ) 2 2 τ 2
E n k ( r ) = e i k · r u n k ( r )
𝒪 all space poled ( k y ) = all space d 3 r q ( x ) e i k y y y 2 2 ζ 2 z ̂ · u n = 2 ; ( k x s , k y ) * ( r )
𝒪 all space poled ( k y ) = 𝒩 x · 𝒪 one period poled ( k y )
𝒪 one period poled ( k y ) = a 2 a 2 d x a 2 a 2 d y q ( x ) e i k y y y 2 2 ζ 2 z ̂ · u n = 2 ; ( k x s , k y ) * ( r )
E ( r , t ) = k y E n = 2 ; ( k x s , k y ) ( r ) 𝒩 x 𝒩 y 2 𝒩 x · 𝒪 one period poled ( k y ) 𝓘 t ( k y )
E ( r , t ) = a 𝒩 y 2 π 0.5 ( 2 π a ) 0.5 ( 2 π a ) d k y E n = 2 ; ( k x s , k y ) ( r ) 𝒩 x 𝒩 y 2 𝒩 x · 𝒪 one period poled ( k y ) 𝓘 t ( k y )
= a π · π a π a d k y E n = 2 ; ( k x s , k y ) ( r ) 𝒪 one period poled ( k y ) 𝓘 t ( k y )

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