Abstract

We use coupled mode theory, adequately incorporating optical losses, to model ultra-broadband terahertz (THz) waveguide emitters (0.1-20 THz) based on difference frequency generation of femtosecond infrared (IR) optical pulses. We apply the model to a generic, symmetric, five-layer, metal/cladding/core waveguide structure using transfer matrix theory. We provide a design strategy for an efficient ultra-broadband THz emitter and apply it to polymer waveguides with a nonlinear core composed of a poled guest-host electro-optic polymer composite and pumped by a pulsed fiber laser system operating at 1567 nm. The predicted bandwidths are greater than 15 THz and we find a high conversion efficiency of 1.2 × 10−4 W−1 by balancing both the modal phase-matching and effective mode attenuation.

© 2013 OSA

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  1. X. Zheng, C. V. McLaughlin, P. D. Cunningham, and L. M. Hayden, “Organic broadband terahertz sources and sensors,” J. Nanoelectron. Optoelectron.2(1), 58–76 (2007).
    [CrossRef]
  2. K. Liu, J. Xu, and X.-C. Zhang, “GaSe crystals for broadband terahertz wave detection,” Appl. Phys. Lett.85(6), 863–865 (2004).
    [CrossRef]
  3. U. Peschel, K. Bubke, D. C. Hutchings, J. S. Aitchison, and J. M. Arnold, “Optical rectification in a traveling-wave geometry,” Phys. Rev. A60(6), 4918–4926 (1999).
    [CrossRef]
  4. Y. J. Ding, “Terahertz parametric converters by use of novel metallic-dielectric hybrid waveguides,” J. Opt. Soc. Am. B23(7), 1354–1359 (2006).
    [CrossRef]
  5. A. Marandi, T. E. Darcie, and P. P. M. So, “Design of a continuous-wave tunable terahertz source using waveguide-phase-matched GaAs,” Opt. Express16(14), 10427–10433 (2008).
    [CrossRef] [PubMed]
  6. M. Cherchi, A. Taormina, A. C. Busacca, R. L. Oliveri, S. Bivona, A. C. Cino, S. Stivala, S. R. Sanseverino, and C. Leone, “Exploiting the optical quadratic nonlinearity of zinc-blende semiconductors for guided-wave terahertz generation: a material comparison,” IEEE J. Quantum Electron.46(3), 368–376 (2010).
    [CrossRef]
  7. C. Staus, T. Kuech, and L. McCaughan, “Continuously phase-matched terahertz difference frequency generation in an embedded-waveguide structure supporting only fundamental modes,” Opt. Express16(17), 13296–13303 (2008).
    [CrossRef] [PubMed]
  8. Z. Ruan, G. Veronis, K. L. Vodopyanov, M. M. Fejer, and S. Fan, “Enhancement of optics-to-THz conversion efficiency by metallic slot waveguides,” Opt. Express17(16), 13502–13515 (2009).
    [CrossRef] [PubMed]
  9. T. Chen, J. Sun, L. Li, and J. Tang, “Proposal for efficient terahertz-wave difference frequency generation in an AlGaAs photonic crystal waveguide,” J. Lightwave Technol.30(13), 2156–2162 (2012).
    [CrossRef]
  10. Y. Li, X. Hu, F. Liu, J. Li, Q. Xing, M. Hu, C. Lu, and C. Wang, “Terahertz waveguide emitters in photonic crystal fiber form,” J. Opt. Soc. Am. B29(11), 3114–3118 (2012).
    [CrossRef]
  11. Z. Wang, H. Liu, N. Huang, Q. Sun, and J. Wen, “Efficient terahertz-wave generation via four-wave mixing in silicon membrane waveguides,” Opt. Express20(8), 8920–8928 (2012).
    [CrossRef] [PubMed]
  12. F. F. Lu, T. Li, J. Xu, Z. D. Xie, L. Li, S. N. Zhu, and Y. Y. Zhu, “Surface plasmon polariton enhanced by optical parametric amplification in nonlinear hybrid waveguide,” Opt. Express19(4), 2858–2865 (2011).
    [CrossRef] [PubMed]
  13. S. B. Hasan, C. Rockstuhl, T. Pertsch, and F. Lederer, “Second-order nonlinear frequency conversion processes in plasmonic slot waveguides,” J. Opt. Soc. Am. B29(7), 1606–1611 (2012).
    [CrossRef]
  14. F. M. Pigozzo, D. Modotto, and S. Wabnitz, “Second harmonic generation by modal phase matching involving optical and plasmonic modes,” Opt. Lett.37(12), 2244–2246 (2012).
    [CrossRef] [PubMed]
  15. S. B. Bodrov, I. E. Ilyakov, B. V. Shishkin, and A. N. Stepanov, “Efficient terahertz generation by optical rectification in Si-LiNbO3-air-metal sandwich structure with variable air gap,” Appl. Phys. Lett.100(20), 201114 (2012).
    [CrossRef]
  16. M. I. Bakunov, S. B. Bodrov, A. V. Maslov, and M. Hangyo, “Theory of terahertz generation in a slab of electro-optic material using an ultrashort laser pulse focused to a line,” Phys. Rev. B76(8), 085346 (2007).
    [CrossRef]
  17. Y.-C. Huang, T.-D. Wang, Y.-H. Lin, C.-H. Lee, M.-Y. Chuang, Y.-Y. Lin, and F.-Y. Lin, “Forward and backward THz-wave difference frequency generations from a rectangular nonlinear waveguide,” Opt. Express19(24), 24577–24582 (2011).
    [CrossRef] [PubMed]
  18. V. Berger and C. Sirtori, “Nonlinear phase matching in THz semiconductor waveguides,” Semicond. Sci. Technol.19(8), 964–970 (2004).
    [CrossRef]
  19. H. Cao, R. A. Linke, and A. Nahata, “Broadband generation of terahertz radiation in a waveguide,” Opt. Lett.29(15), 1751–1753 (2004).
    [CrossRef] [PubMed]
  20. 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. Express15(25), 16308–16315 (2007).
    [CrossRef] [PubMed]
  21. F. Peter, S. Winnerl, H. Schneider, and M. Helm, “Excitation wavelength dependence of phase matched terahertz emission from a GaAs slab,” Opt. Express18(19), 19574–19580 (2010).
    [CrossRef] [PubMed]
  22. K. Saito, T. Tanabe, Y. Oyama, K. Suto, and J.-i. Nishizawa, “Terahertz-wave generation by GaP rib waveguides via collinear phase-matched difference-frequency mixing of near-infrared lasers,” J. Appl. Phys.105(6), 063102 (2009).
    [CrossRef]
  23. V. A. Kukushkin, “Generation of terahertz pulses from tightly focused single near-infrared pulses in double-plasmon waveguides,” J. Opt. Soc. Am. B25(5), 818–824 (2008).
    [CrossRef]
  24. K. Saito, T. Tanabe, and Y. Oyama, “Elliptically polarized THz-wave generation from GaP-THz planar waveguide via collinear phase-matched difference frequency mixing,” Opt. Express20(23), 26082–26088 (2012).
    [CrossRef] [PubMed]
  25. P. Bienstman, “Rigorous and efficient modeling of wavelength scale photonic components,” (Universiteit Gent 2000–2001).
  26. Y.-F. Li and J. W. Y. Lit, “General formulas for the guiding properties of a multilayer slab waveguide,” J. Opt. Soc. Am. A4(4), 671–677 (1987).
    [CrossRef]
  27. X. Ying and I. Katz, “A simple reliable solver for all the roots of a nonlinear function in a given domain,” Computing41(4), 317–333 (1989).
    [CrossRef]
  28. R. E. Smith, G. W. Forbes, and S. N. Houde-Walter, “Unfolding the multivalued planar waveguide dispersion relation,” IEEE J. Quantum Electron.29(4), 1031–1034 (1993).
    [CrossRef]
  29. K. D. Singer, M. G. Kuzyk, and J. E. Sohn, “Second-order nonlinear-optical processes in orientationally ordered materials: relationship between molecular and macroscopic properties,” J. Opt. Soc. Am. B4(6), 968–976 (1987).
    [CrossRef]
  30. A. Nahata, J. Shan, J. T. Yardley, and C. Wu, “Electro-optic determination of the nonlinear-optical properties of a covalently functionalized Disperse Red 1 copolymer,” J. Opt. Soc. Am. B10(9), 1553–1564 (1993).
    [CrossRef]
  31. Ş. E. Kocabaş, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B79(3), 035120 (2009).
    [CrossRef]
  32. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  33. P. D. Cunningham, N. N. Valdes, F. A. Vallejo, L. M. Hayden, B. Polishak, X.-H. Zhou, J. Luo, A. K.-Y. Jen, J. C. Williams, and R. J. Twieg, “Broadband terahertz characterization of the refractive index and absorption of some important polymeric and organic electro-optic materials,” J. Appl. Phys.109(4), 043505 (2011).
    [CrossRef]
  34. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, 1998), Vol. I - III.
  35. M. A. Ordal, R. J. Bell, R. W. Alexander, L. A. Newquist, and M. R. Querry, “Optical properties of Al, Fe, Ti, Ta, W, and Mo at submillimeter wavelengths,” Appl. Opt.27(6), 1203–1209 (1988).
    [CrossRef] [PubMed]
  36. S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater.29(11), 1481–1490 (2007).
    [CrossRef]
  37. M. Dellnitz, O. Schütze, and Q. Zheng, “Locating all the zeros of an analytic function in one complex variable,” J. Comput. Appl. Math.138(2), 325–333 (2002).
    [CrossRef]
  38. P. N. Robson and P. C. Kendall, eds. Rib Waveguide Theory by the Spectral Index Method (Electronic and Electrical Engineering Research Studies: Optoelectronics Series) (Wiley, 1990).

2012

2011

2010

M. Cherchi, A. Taormina, A. C. Busacca, R. L. Oliveri, S. Bivona, A. C. Cino, S. Stivala, S. R. Sanseverino, and C. Leone, “Exploiting the optical quadratic nonlinearity of zinc-blende semiconductors for guided-wave terahertz generation: a material comparison,” IEEE J. Quantum Electron.46(3), 368–376 (2010).
[CrossRef]

F. Peter, S. Winnerl, H. Schneider, and M. Helm, “Excitation wavelength dependence of phase matched terahertz emission from a GaAs slab,” Opt. Express18(19), 19574–19580 (2010).
[CrossRef] [PubMed]

2009

Z. Ruan, G. Veronis, K. L. Vodopyanov, M. M. Fejer, and S. Fan, “Enhancement of optics-to-THz conversion efficiency by metallic slot waveguides,” Opt. Express17(16), 13502–13515 (2009).
[CrossRef] [PubMed]

K. Saito, T. Tanabe, Y. Oyama, K. Suto, and J.-i. Nishizawa, “Terahertz-wave generation by GaP rib waveguides via collinear phase-matched difference-frequency mixing of near-infrared lasers,” J. Appl. Phys.105(6), 063102 (2009).
[CrossRef]

Ş. E. Kocabaş, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B79(3), 035120 (2009).
[CrossRef]

2008

2007

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. Express15(25), 16308–16315 (2007).
[CrossRef] [PubMed]

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater.29(11), 1481–1490 (2007).
[CrossRef]

M. I. Bakunov, S. B. Bodrov, A. V. Maslov, and M. Hangyo, “Theory of terahertz generation in a slab of electro-optic material using an ultrashort laser pulse focused to a line,” Phys. Rev. B76(8), 085346 (2007).
[CrossRef]

X. Zheng, C. V. McLaughlin, P. D. Cunningham, and L. M. Hayden, “Organic broadband terahertz sources and sensors,” J. Nanoelectron. Optoelectron.2(1), 58–76 (2007).
[CrossRef]

2006

2004

H. Cao, R. A. Linke, and A. Nahata, “Broadband generation of terahertz radiation in a waveguide,” Opt. Lett.29(15), 1751–1753 (2004).
[CrossRef] [PubMed]

K. Liu, J. Xu, and X.-C. Zhang, “GaSe crystals for broadband terahertz wave detection,” Appl. Phys. Lett.85(6), 863–865 (2004).
[CrossRef]

V. Berger and C. Sirtori, “Nonlinear phase matching in THz semiconductor waveguides,” Semicond. Sci. Technol.19(8), 964–970 (2004).
[CrossRef]

2002

M. Dellnitz, O. Schütze, and Q. Zheng, “Locating all the zeros of an analytic function in one complex variable,” J. Comput. Appl. Math.138(2), 325–333 (2002).
[CrossRef]

1999

U. Peschel, K. Bubke, D. C. Hutchings, J. S. Aitchison, and J. M. Arnold, “Optical rectification in a traveling-wave geometry,” Phys. Rev. A60(6), 4918–4926 (1999).
[CrossRef]

1993

R. E. Smith, G. W. Forbes, and S. N. Houde-Walter, “Unfolding the multivalued planar waveguide dispersion relation,” IEEE J. Quantum Electron.29(4), 1031–1034 (1993).
[CrossRef]

A. Nahata, J. Shan, J. T. Yardley, and C. Wu, “Electro-optic determination of the nonlinear-optical properties of a covalently functionalized Disperse Red 1 copolymer,” J. Opt. Soc. Am. B10(9), 1553–1564 (1993).
[CrossRef]

1989

X. Ying and I. Katz, “A simple reliable solver for all the roots of a nonlinear function in a given domain,” Computing41(4), 317–333 (1989).
[CrossRef]

1988

1987

Aitchison, J. S.

U. Peschel, K. Bubke, D. C. Hutchings, J. S. Aitchison, and J. M. Arnold, “Optical rectification in a traveling-wave geometry,” Phys. Rev. A60(6), 4918–4926 (1999).
[CrossRef]

Alexander, R. W.

Arnold, J. M.

U. Peschel, K. Bubke, D. C. Hutchings, J. S. Aitchison, and J. M. Arnold, “Optical rectification in a traveling-wave geometry,” Phys. Rev. A60(6), 4918–4926 (1999).
[CrossRef]

Bakunov, M. I.

M. I. Bakunov, S. B. Bodrov, A. V. Maslov, and M. Hangyo, “Theory of terahertz generation in a slab of electro-optic material using an ultrashort laser pulse focused to a line,” Phys. Rev. B76(8), 085346 (2007).
[CrossRef]

Bell, R. J.

Berger, V.

V. Berger and C. Sirtori, “Nonlinear phase matching in THz semiconductor waveguides,” Semicond. Sci. Technol.19(8), 964–970 (2004).
[CrossRef]

Bivona, S.

M. Cherchi, A. Taormina, A. C. Busacca, R. L. Oliveri, S. Bivona, A. C. Cino, S. Stivala, S. R. Sanseverino, and C. Leone, “Exploiting the optical quadratic nonlinearity of zinc-blende semiconductors for guided-wave terahertz generation: a material comparison,” IEEE J. Quantum Electron.46(3), 368–376 (2010).
[CrossRef]

Bodrov, S. B.

S. B. Bodrov, I. E. Ilyakov, B. V. Shishkin, and A. N. Stepanov, “Efficient terahertz generation by optical rectification in Si-LiNbO3-air-metal sandwich structure with variable air gap,” Appl. Phys. Lett.100(20), 201114 (2012).
[CrossRef]

M. I. Bakunov, S. B. Bodrov, A. V. Maslov, and M. Hangyo, “Theory of terahertz generation in a slab of electro-optic material using an ultrashort laser pulse focused to a line,” Phys. Rev. B76(8), 085346 (2007).
[CrossRef]

Bubke, K.

U. Peschel, K. Bubke, D. C. Hutchings, J. S. Aitchison, and J. M. Arnold, “Optical rectification in a traveling-wave geometry,” Phys. Rev. A60(6), 4918–4926 (1999).
[CrossRef]

Busacca, A. C.

M. Cherchi, A. Taormina, A. C. Busacca, R. L. Oliveri, S. Bivona, A. C. Cino, S. Stivala, S. R. Sanseverino, and C. Leone, “Exploiting the optical quadratic nonlinearity of zinc-blende semiconductors for guided-wave terahertz generation: a material comparison,” IEEE J. Quantum Electron.46(3), 368–376 (2010).
[CrossRef]

Cao, H.

Chang, G.

Chen, T.

Cherchi, M.

M. Cherchi, A. Taormina, A. C. Busacca, R. L. Oliveri, S. Bivona, A. C. Cino, S. Stivala, S. R. Sanseverino, and C. Leone, “Exploiting the optical quadratic nonlinearity of zinc-blende semiconductors for guided-wave terahertz generation: a material comparison,” IEEE J. Quantum Electron.46(3), 368–376 (2010).
[CrossRef]

Chuang, M.-Y.

Cino, A. C.

M. Cherchi, A. Taormina, A. C. Busacca, R. L. Oliveri, S. Bivona, A. C. Cino, S. Stivala, S. R. Sanseverino, and C. Leone, “Exploiting the optical quadratic nonlinearity of zinc-blende semiconductors for guided-wave terahertz generation: a material comparison,” IEEE J. Quantum Electron.46(3), 368–376 (2010).
[CrossRef]

Cunningham, P. D.

P. D. Cunningham, N. N. Valdes, F. A. Vallejo, L. M. Hayden, B. Polishak, X.-H. Zhou, J. Luo, A. K.-Y. Jen, J. C. Williams, and R. J. Twieg, “Broadband terahertz characterization of the refractive index and absorption of some important polymeric and organic electro-optic materials,” J. Appl. Phys.109(4), 043505 (2011).
[CrossRef]

X. Zheng, C. V. McLaughlin, P. D. Cunningham, and L. M. Hayden, “Organic broadband terahertz sources and sensors,” J. Nanoelectron. Optoelectron.2(1), 58–76 (2007).
[CrossRef]

Darcie, T. E.

Dellnitz, M.

M. Dellnitz, O. Schütze, and Q. Zheng, “Locating all the zeros of an analytic function in one complex variable,” J. Comput. Appl. Math.138(2), 325–333 (2002).
[CrossRef]

Ding, Y. J.

Divin, C. J.

Fan, S.

Ş. E. Kocabaş, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B79(3), 035120 (2009).
[CrossRef]

Z. Ruan, G. Veronis, K. L. Vodopyanov, M. M. Fejer, and S. Fan, “Enhancement of optics-to-THz conversion efficiency by metallic slot waveguides,” Opt. Express17(16), 13502–13515 (2009).
[CrossRef] [PubMed]

Fejer, M. M.

Forbes, G. W.

R. E. Smith, G. W. Forbes, and S. N. Houde-Walter, “Unfolding the multivalued planar waveguide dispersion relation,” IEEE J. Quantum Electron.29(4), 1031–1034 (1993).
[CrossRef]

Galvanauskas, A.

Hangyo, M.

M. I. Bakunov, S. B. Bodrov, A. V. Maslov, and M. Hangyo, “Theory of terahertz generation in a slab of electro-optic material using an ultrashort laser pulse focused to a line,” Phys. Rev. B76(8), 085346 (2007).
[CrossRef]

Hasan, S. B.

Hayden, L. M.

P. D. Cunningham, N. N. Valdes, F. A. Vallejo, L. M. Hayden, B. Polishak, X.-H. Zhou, J. Luo, A. K.-Y. Jen, J. C. Williams, and R. J. Twieg, “Broadband terahertz characterization of the refractive index and absorption of some important polymeric and organic electro-optic materials,” J. Appl. Phys.109(4), 043505 (2011).
[CrossRef]

X. Zheng, C. V. McLaughlin, P. D. Cunningham, and L. M. Hayden, “Organic broadband terahertz sources and sensors,” J. Nanoelectron. Optoelectron.2(1), 58–76 (2007).
[CrossRef]

Helm, M.

Houde-Walter, S. N.

R. E. Smith, G. W. Forbes, and S. N. Houde-Walter, “Unfolding the multivalued planar waveguide dispersion relation,” IEEE J. Quantum Electron.29(4), 1031–1034 (1993).
[CrossRef]

Hu, M.

Hu, X.

Huang, N.

Huang, Y.-C.

Hutchings, D. C.

U. Peschel, K. Bubke, D. C. Hutchings, J. S. Aitchison, and J. M. Arnold, “Optical rectification in a traveling-wave geometry,” Phys. Rev. A60(6), 4918–4926 (1999).
[CrossRef]

Ilyakov, I. E.

S. B. Bodrov, I. E. Ilyakov, B. V. Shishkin, and A. N. Stepanov, “Efficient terahertz generation by optical rectification in Si-LiNbO3-air-metal sandwich structure with variable air gap,” Appl. Phys. Lett.100(20), 201114 (2012).
[CrossRef]

Ivanov, C. D.

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater.29(11), 1481–1490 (2007).
[CrossRef]

Jen, A. K.-Y.

P. D. Cunningham, N. N. Valdes, F. A. Vallejo, L. M. Hayden, B. Polishak, X.-H. Zhou, J. Luo, A. K.-Y. Jen, J. C. Williams, and R. J. Twieg, “Broadband terahertz characterization of the refractive index and absorption of some important polymeric and organic electro-optic materials,” J. Appl. Phys.109(4), 043505 (2011).
[CrossRef]

Kasarova, S. N.

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater.29(11), 1481–1490 (2007).
[CrossRef]

Katz, I.

X. Ying and I. Katz, “A simple reliable solver for all the roots of a nonlinear function in a given domain,” Computing41(4), 317–333 (1989).
[CrossRef]

Kocabas, S. E.

Ş. E. Kocabaş, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B79(3), 035120 (2009).
[CrossRef]

Kuech, T.

Kukushkin, V. A.

Kuzyk, M. G.

Lederer, F.

Lee, C.-H.

Leone, C.

M. Cherchi, A. Taormina, A. C. Busacca, R. L. Oliveri, S. Bivona, A. C. Cino, S. Stivala, S. R. Sanseverino, and C. Leone, “Exploiting the optical quadratic nonlinearity of zinc-blende semiconductors for guided-wave terahertz generation: a material comparison,” IEEE J. Quantum Electron.46(3), 368–376 (2010).
[CrossRef]

Li, J.

Li, L.

Li, T.

Li, Y.

Li, Y.-F.

Lin, F.-Y.

Lin, Y.-H.

Lin, Y.-Y.

Linke, R. A.

Lit, J. W. Y.

Liu, F.

Liu, H.

Liu, K.

K. Liu, J. Xu, and X.-C. Zhang, “GaSe crystals for broadband terahertz wave detection,” Appl. Phys. Lett.85(6), 863–865 (2004).
[CrossRef]

Lu, C.

Lu, F. F.

Luo, J.

P. D. Cunningham, N. N. Valdes, F. A. Vallejo, L. M. Hayden, B. Polishak, X.-H. Zhou, J. Luo, A. K.-Y. Jen, J. C. Williams, and R. J. Twieg, “Broadband terahertz characterization of the refractive index and absorption of some important polymeric and organic electro-optic materials,” J. Appl. Phys.109(4), 043505 (2011).
[CrossRef]

Marandi, A.

Maslov, A. V.

M. I. Bakunov, S. B. Bodrov, A. V. Maslov, and M. Hangyo, “Theory of terahertz generation in a slab of electro-optic material using an ultrashort laser pulse focused to a line,” Phys. Rev. B76(8), 085346 (2007).
[CrossRef]

McCaughan, L.

McLaughlin, C. V.

X. Zheng, C. V. McLaughlin, P. D. Cunningham, and L. M. Hayden, “Organic broadband terahertz sources and sensors,” J. Nanoelectron. Optoelectron.2(1), 58–76 (2007).
[CrossRef]

Miller, D. A. B.

Ş. E. Kocabaş, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B79(3), 035120 (2009).
[CrossRef]

Modotto, D.

Musheinish, M. A.

Nahata, A.

Newquist, L. A.

Nikolov, I. D.

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater.29(11), 1481–1490 (2007).
[CrossRef]

Nishizawa, J.-i.

K. Saito, T. Tanabe, Y. Oyama, K. Suto, and J.-i. Nishizawa, “Terahertz-wave generation by GaP rib waveguides via collinear phase-matched difference-frequency mixing of near-infrared lasers,” J. Appl. Phys.105(6), 063102 (2009).
[CrossRef]

Norris, T. B.

Oliveri, R. L.

M. Cherchi, A. Taormina, A. C. Busacca, R. L. Oliveri, S. Bivona, A. C. Cino, S. Stivala, S. R. Sanseverino, and C. Leone, “Exploiting the optical quadratic nonlinearity of zinc-blende semiconductors for guided-wave terahertz generation: a material comparison,” IEEE J. Quantum Electron.46(3), 368–376 (2010).
[CrossRef]

Ordal, M. A.

Oyama, Y.

K. Saito, T. Tanabe, and Y. Oyama, “Elliptically polarized THz-wave generation from GaP-THz planar waveguide via collinear phase-matched difference frequency mixing,” Opt. Express20(23), 26082–26088 (2012).
[CrossRef] [PubMed]

K. Saito, T. Tanabe, Y. Oyama, K. Suto, and J.-i. Nishizawa, “Terahertz-wave generation by GaP rib waveguides via collinear phase-matched difference-frequency mixing of near-infrared lasers,” J. Appl. Phys.105(6), 063102 (2009).
[CrossRef]

Pertsch, T.

Peschel, U.

U. Peschel, K. Bubke, D. C. Hutchings, J. S. Aitchison, and J. M. Arnold, “Optical rectification in a traveling-wave geometry,” Phys. Rev. A60(6), 4918–4926 (1999).
[CrossRef]

Peter, F.

Pigozzo, F. M.

Polishak, B.

P. D. Cunningham, N. N. Valdes, F. A. Vallejo, L. M. Hayden, B. Polishak, X.-H. Zhou, J. Luo, A. K.-Y. Jen, J. C. Williams, and R. J. Twieg, “Broadband terahertz characterization of the refractive index and absorption of some important polymeric and organic electro-optic materials,” J. Appl. Phys.109(4), 043505 (2011).
[CrossRef]

Querry, M. R.

Rockstuhl, C.

Ruan, Z.

Saito, K.

K. Saito, T. Tanabe, and Y. Oyama, “Elliptically polarized THz-wave generation from GaP-THz planar waveguide via collinear phase-matched difference frequency mixing,” Opt. Express20(23), 26082–26088 (2012).
[CrossRef] [PubMed]

K. Saito, T. Tanabe, Y. Oyama, K. Suto, and J.-i. Nishizawa, “Terahertz-wave generation by GaP rib waveguides via collinear phase-matched difference-frequency mixing of near-infrared lasers,” J. Appl. Phys.105(6), 063102 (2009).
[CrossRef]

Sanseverino, S. R.

M. Cherchi, A. Taormina, A. C. Busacca, R. L. Oliveri, S. Bivona, A. C. Cino, S. Stivala, S. R. Sanseverino, and C. Leone, “Exploiting the optical quadratic nonlinearity of zinc-blende semiconductors for guided-wave terahertz generation: a material comparison,” IEEE J. Quantum Electron.46(3), 368–376 (2010).
[CrossRef]

Schneider, H.

Schütze, O.

M. Dellnitz, O. Schütze, and Q. Zheng, “Locating all the zeros of an analytic function in one complex variable,” J. Comput. Appl. Math.138(2), 325–333 (2002).
[CrossRef]

Shan, J.

Shishkin, B. V.

S. B. Bodrov, I. E. Ilyakov, B. V. Shishkin, and A. N. Stepanov, “Efficient terahertz generation by optical rectification in Si-LiNbO3-air-metal sandwich structure with variable air gap,” Appl. Phys. Lett.100(20), 201114 (2012).
[CrossRef]

Singer, K. D.

Sirtori, C.

V. Berger and C. Sirtori, “Nonlinear phase matching in THz semiconductor waveguides,” Semicond. Sci. Technol.19(8), 964–970 (2004).
[CrossRef]

Smith, R. E.

R. E. Smith, G. W. Forbes, and S. N. Houde-Walter, “Unfolding the multivalued planar waveguide dispersion relation,” IEEE J. Quantum Electron.29(4), 1031–1034 (1993).
[CrossRef]

So, P. P. M.

Sohn, J. E.

Staus, C.

Stepanov, A. N.

S. B. Bodrov, I. E. Ilyakov, B. V. Shishkin, and A. N. Stepanov, “Efficient terahertz generation by optical rectification in Si-LiNbO3-air-metal sandwich structure with variable air gap,” Appl. Phys. Lett.100(20), 201114 (2012).
[CrossRef]

Stivala, S.

M. Cherchi, A. Taormina, A. C. Busacca, R. L. Oliveri, S. Bivona, A. C. Cino, S. Stivala, S. R. Sanseverino, and C. Leone, “Exploiting the optical quadratic nonlinearity of zinc-blende semiconductors for guided-wave terahertz generation: a material comparison,” IEEE J. Quantum Electron.46(3), 368–376 (2010).
[CrossRef]

Sultanova, N. G.

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater.29(11), 1481–1490 (2007).
[CrossRef]

Sun, J.

Sun, Q.

Suto, K.

K. Saito, T. Tanabe, Y. Oyama, K. Suto, and J.-i. Nishizawa, “Terahertz-wave generation by GaP rib waveguides via collinear phase-matched difference-frequency mixing of near-infrared lasers,” J. Appl. Phys.105(6), 063102 (2009).
[CrossRef]

Tanabe, T.

K. Saito, T. Tanabe, and Y. Oyama, “Elliptically polarized THz-wave generation from GaP-THz planar waveguide via collinear phase-matched difference frequency mixing,” Opt. Express20(23), 26082–26088 (2012).
[CrossRef] [PubMed]

K. Saito, T. Tanabe, Y. Oyama, K. Suto, and J.-i. Nishizawa, “Terahertz-wave generation by GaP rib waveguides via collinear phase-matched difference-frequency mixing of near-infrared lasers,” J. Appl. Phys.105(6), 063102 (2009).
[CrossRef]

Tang, J.

Taormina, A.

M. Cherchi, A. Taormina, A. C. Busacca, R. L. Oliveri, S. Bivona, A. C. Cino, S. Stivala, S. R. Sanseverino, and C. Leone, “Exploiting the optical quadratic nonlinearity of zinc-blende semiconductors for guided-wave terahertz generation: a material comparison,” IEEE J. Quantum Electron.46(3), 368–376 (2010).
[CrossRef]

Twieg, R. J.

P. D. Cunningham, N. N. Valdes, F. A. Vallejo, L. M. Hayden, B. Polishak, X.-H. Zhou, J. Luo, A. K.-Y. Jen, J. C. Williams, and R. J. Twieg, “Broadband terahertz characterization of the refractive index and absorption of some important polymeric and organic electro-optic materials,” J. Appl. Phys.109(4), 043505 (2011).
[CrossRef]

Valdes, N. N.

P. D. Cunningham, N. N. Valdes, F. A. Vallejo, L. M. Hayden, B. Polishak, X.-H. Zhou, J. Luo, A. K.-Y. Jen, J. C. Williams, and R. J. Twieg, “Broadband terahertz characterization of the refractive index and absorption of some important polymeric and organic electro-optic materials,” J. Appl. Phys.109(4), 043505 (2011).
[CrossRef]

Vallejo, F. A.

P. D. Cunningham, N. N. Valdes, F. A. Vallejo, L. M. Hayden, B. Polishak, X.-H. Zhou, J. Luo, A. K.-Y. Jen, J. C. Williams, and R. J. Twieg, “Broadband terahertz characterization of the refractive index and absorption of some important polymeric and organic electro-optic materials,” J. Appl. Phys.109(4), 043505 (2011).
[CrossRef]

Veronis, G.

Ş. E. Kocabaş, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B79(3), 035120 (2009).
[CrossRef]

Z. Ruan, G. Veronis, K. L. Vodopyanov, M. M. Fejer, and S. Fan, “Enhancement of optics-to-THz conversion efficiency by metallic slot waveguides,” Opt. Express17(16), 13502–13515 (2009).
[CrossRef] [PubMed]

Vodopyanov, K. L.

Wabnitz, S.

Wang, C.

Wang, T.-D.

Wang, Z.

Wen, J.

Williams, J. C.

P. D. Cunningham, N. N. Valdes, F. A. Vallejo, L. M. Hayden, B. Polishak, X.-H. Zhou, J. Luo, A. K.-Y. Jen, J. C. Williams, and R. J. Twieg, “Broadband terahertz characterization of the refractive index and absorption of some important polymeric and organic electro-optic materials,” J. Appl. Phys.109(4), 043505 (2011).
[CrossRef]

Williamson, S. L.

Winnerl, S.

Wu, C.

Xie, Z. D.

Xing, Q.

Xu, J.

Yang, J.

Yardley, J. T.

Ying, X.

X. Ying and I. Katz, “A simple reliable solver for all the roots of a nonlinear function in a given domain,” Computing41(4), 317–333 (1989).
[CrossRef]

Zhang, X.-C.

K. Liu, J. Xu, and X.-C. Zhang, “GaSe crystals for broadband terahertz wave detection,” Appl. Phys. Lett.85(6), 863–865 (2004).
[CrossRef]

Zheng, Q.

M. Dellnitz, O. Schütze, and Q. Zheng, “Locating all the zeros of an analytic function in one complex variable,” J. Comput. Appl. Math.138(2), 325–333 (2002).
[CrossRef]

Zheng, X.

X. Zheng, C. V. McLaughlin, P. D. Cunningham, and L. M. Hayden, “Organic broadband terahertz sources and sensors,” J. Nanoelectron. Optoelectron.2(1), 58–76 (2007).
[CrossRef]

Zhou, X.-H.

P. D. Cunningham, N. N. Valdes, F. A. Vallejo, L. M. Hayden, B. Polishak, X.-H. Zhou, J. Luo, A. K.-Y. Jen, J. C. Williams, and R. J. Twieg, “Broadband terahertz characterization of the refractive index and absorption of some important polymeric and organic electro-optic materials,” J. Appl. Phys.109(4), 043505 (2011).
[CrossRef]

Zhu, S. N.

Zhu, Y. Y.

Appl. Opt.

Appl. Phys. Lett.

K. Liu, J. Xu, and X.-C. Zhang, “GaSe crystals for broadband terahertz wave detection,” Appl. Phys. Lett.85(6), 863–865 (2004).
[CrossRef]

S. B. Bodrov, I. E. Ilyakov, B. V. Shishkin, and A. N. Stepanov, “Efficient terahertz generation by optical rectification in Si-LiNbO3-air-metal sandwich structure with variable air gap,” Appl. Phys. Lett.100(20), 201114 (2012).
[CrossRef]

Computing

X. Ying and I. Katz, “A simple reliable solver for all the roots of a nonlinear function in a given domain,” Computing41(4), 317–333 (1989).
[CrossRef]

IEEE J. Quantum Electron.

R. E. Smith, G. W. Forbes, and S. N. Houde-Walter, “Unfolding the multivalued planar waveguide dispersion relation,” IEEE J. Quantum Electron.29(4), 1031–1034 (1993).
[CrossRef]

M. Cherchi, A. Taormina, A. C. Busacca, R. L. Oliveri, S. Bivona, A. C. Cino, S. Stivala, S. R. Sanseverino, and C. Leone, “Exploiting the optical quadratic nonlinearity of zinc-blende semiconductors for guided-wave terahertz generation: a material comparison,” IEEE J. Quantum Electron.46(3), 368–376 (2010).
[CrossRef]

J. Appl. Phys.

K. Saito, T. Tanabe, Y. Oyama, K. Suto, and J.-i. Nishizawa, “Terahertz-wave generation by GaP rib waveguides via collinear phase-matched difference-frequency mixing of near-infrared lasers,” J. Appl. Phys.105(6), 063102 (2009).
[CrossRef]

P. D. Cunningham, N. N. Valdes, F. A. Vallejo, L. M. Hayden, B. Polishak, X.-H. Zhou, J. Luo, A. K.-Y. Jen, J. C. Williams, and R. J. Twieg, “Broadband terahertz characterization of the refractive index and absorption of some important polymeric and organic electro-optic materials,” J. Appl. Phys.109(4), 043505 (2011).
[CrossRef]

J. Comput. Appl. Math.

M. Dellnitz, O. Schütze, and Q. Zheng, “Locating all the zeros of an analytic function in one complex variable,” J. Comput. Appl. Math.138(2), 325–333 (2002).
[CrossRef]

J. Lightwave Technol.

J. Nanoelectron. Optoelectron.

X. Zheng, C. V. McLaughlin, P. D. Cunningham, and L. M. Hayden, “Organic broadband terahertz sources and sensors,” J. Nanoelectron. Optoelectron.2(1), 58–76 (2007).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Express

K. Saito, T. Tanabe, and Y. Oyama, “Elliptically polarized THz-wave generation from GaP-THz planar waveguide via collinear phase-matched difference frequency mixing,” Opt. Express20(23), 26082–26088 (2012).
[CrossRef] [PubMed]

A. Marandi, T. E. Darcie, and P. P. M. So, “Design of a continuous-wave tunable terahertz source using waveguide-phase-matched GaAs,” Opt. Express16(14), 10427–10433 (2008).
[CrossRef] [PubMed]

C. Staus, T. Kuech, and L. McCaughan, “Continuously phase-matched terahertz difference frequency generation in an embedded-waveguide structure supporting only fundamental modes,” Opt. Express16(17), 13296–13303 (2008).
[CrossRef] [PubMed]

Z. Ruan, G. Veronis, K. L. Vodopyanov, M. M. Fejer, and S. Fan, “Enhancement of optics-to-THz conversion efficiency by metallic slot waveguides,” Opt. Express17(16), 13502–13515 (2009).
[CrossRef] [PubMed]

F. Peter, S. Winnerl, H. Schneider, and M. Helm, “Excitation wavelength dependence of phase matched terahertz emission from a GaAs slab,” Opt. Express18(19), 19574–19580 (2010).
[CrossRef] [PubMed]

F. F. Lu, T. Li, J. Xu, Z. D. Xie, L. Li, S. N. Zhu, and Y. Y. Zhu, “Surface plasmon polariton enhanced by optical parametric amplification in nonlinear hybrid waveguide,” Opt. Express19(4), 2858–2865 (2011).
[CrossRef] [PubMed]

Y.-C. Huang, T.-D. Wang, Y.-H. Lin, C.-H. Lee, M.-Y. Chuang, Y.-Y. Lin, and F.-Y. Lin, “Forward and backward THz-wave difference frequency generations from a rectangular nonlinear waveguide,” Opt. Express19(24), 24577–24582 (2011).
[CrossRef] [PubMed]

Z. Wang, H. Liu, N. Huang, Q. Sun, and J. Wen, “Efficient terahertz-wave generation via four-wave mixing in silicon membrane waveguides,” Opt. Express20(8), 8920–8928 (2012).
[CrossRef] [PubMed]

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. Express15(25), 16308–16315 (2007).
[CrossRef] [PubMed]

Opt. Lett.

Opt. Mater.

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater.29(11), 1481–1490 (2007).
[CrossRef]

Phys. Rev. A

U. Peschel, K. Bubke, D. C. Hutchings, J. S. Aitchison, and J. M. Arnold, “Optical rectification in a traveling-wave geometry,” Phys. Rev. A60(6), 4918–4926 (1999).
[CrossRef]

Phys. Rev. B

M. I. Bakunov, S. B. Bodrov, A. V. Maslov, and M. Hangyo, “Theory of terahertz generation in a slab of electro-optic material using an ultrashort laser pulse focused to a line,” Phys. Rev. B76(8), 085346 (2007).
[CrossRef]

Ş. E. Kocabaş, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B79(3), 035120 (2009).
[CrossRef]

Semicond. Sci. Technol.

V. Berger and C. Sirtori, “Nonlinear phase matching in THz semiconductor waveguides,” Semicond. Sci. Technol.19(8), 964–970 (2004).
[CrossRef]

Other

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

P. N. Robson and P. C. Kendall, eds. Rib Waveguide Theory by the Spectral Index Method (Electronic and Electrical Engineering Research Studies: Optoelectronics Series) (Wiley, 1990).

E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, 1998), Vol. I - III.

P. Bienstman, “Rigorous and efficient modeling of wavelength scale photonic components,” (Universiteit Gent 2000–2001).

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Figures (7)

Fig. 1
Fig. 1

(a) Symmetric five-layer slab waveguide. The arrows represent the dipole moment of the chromophores. (b) Second order susceptibility, χ xxx (2) , vs. wavelength for DAPC in the telecom range. Inset shows the chemical structure of the chromophore used in DAPC.

Fig. 2
Fig. 2

(a) Refractive index and extinction coefficient for Al at THz frequencies (Ω). (b) Extinction coefficient and (c) refractive index for Al at IR wavelengths (λ). The dots represent experimental data and the lines correspond to the fits in the model. A global Drude fit is used in the THz range whereas interpolation fits are used for the IR range.

Fig. 3
Fig. 3

(a) Effective index Neff,m and (b) respective mode attenuation αm, for even modes in the IR wavelength range (λ) for a DAPC structure with d = 1.579 μm, t = 20 μm and PS cladding. The horizontal dashed lines enclose the bandwidth of the IR pump used in our calculations. For structures with DAPC cores (d = 1.579 μm for PS cladding and d = 1.274 μm for TOPAS claddings), we show (c) the effective index Neff,m and (d) the respective mode attenuation αm, for the TEM-like mode for THz frequencies (Ω) for a range of PS cladding thicknesses (t = 2.3-2.7 µm) and for a TOPAS cladding where t = 1.9 µm. The index dispersion for the core and cladding are shown for comparison. Core and cladding losses are neglected for the IR modes and αm for m = 1 in (b) is scaled for comparison, the scaling factor is indicated next to the curve.

Fig. 4
Fig. 4

(a) Group index n g,IR for the m = 0 IR mode and the value of Neff,0 for the TEM-like mode at 5 THz for structures with different half thickness values t. For the TOPAS structure d = 1.274 μm and d = 1.579 μm for the PS structure. (b) Nonlinear efficiency ηTHz vs emitter length L for the different structures.

Fig. 5
Fig. 5

For a structure with d = 1.579 μm and t = 2.5 μm and PS cladding: (a) Spectral power densities dPTHz vs. THz frequencies for different emitter lengths, L. (b) Output electric field at the center of the waveguide produced by the most efficient structure we found.

Fig. 6
Fig. 6

(a) Gaussian input profile (w0 = 1.5 μm) for the IR pump beam at 820 nm and corresponding mode decompositions using 1 and 2 modes and (b) mode profile for the fundamental TEM-like mode at 10 THz for a structure with (t = 5 μm, d = 1.5 μm).

Fig. 7
Fig. 7

Output THz electric field for a device with (d = 1.5 μm, t = 5 μm) for different propagation lengths L. (a) THz waveforms after being convolved with the detector response function R(Ω). (b) The corresponding THz spectral fields before and after being convolved with R(Ω), shown for comparison.

Equations (21)

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E ˜ = l A l (z) e ^ l ( r ,ω) e i β l (ω)z = l l e i β l z .
A l (z,ω) z = iω e i β l z P l (ω) S ( e ^ l, e ^ l,z ) P ˜ NL (ω)d r .
E ( r ,t)= d ω E ˜ ( r ,ω) e iωt +cc.= l d ω l e i( β l (ω)zωt) +cc..
P NL (t)= ϵ 0 χ (2) : E ( r ,t) E ( r ,t)= d Ω P ˜ NL (Ω) e iΩt = ϵ 0 d ωd ω l, l χ (2) :[ l l e i(( β l + β l )z(ω+ ω )t) + l l * e i(( β l β l )z(ω ω )t) +cc.].
P DFG (2) (Ω)= ϵ 0 l, l 0 d ω χ (2) : l l * e i( β l β l )z .
A m THz (z,Ω) z = i ϵ 0 Ω P m (Ω) 0 dω l, l e iΔβ(ω,Ω)z A l IR (z,ω) A l IR (z,ωΩ) * Κ l,l' m (Ω,ω) .
Κ l,l' m (Ω,ω)= S NL d A( e ^ m, e ^ m,z ) χ (2) (Ω;ω,ωΩ): e ^ l e ^ l * .
χ (2) : l l * =( χ xxx (2) l,x l ,x * + χ xzz (2) l,z l ,z * ) x ^ +( χ zzx (2) l,z l ,x * + χ zxz (2) l,x l ,z * ) z ^ .
E in ( r ,t)=Re[ x ^ E 0 exp( x 2 w 0 2 t 2 2 τ 0 2 i ω 0 t)].
U pulse = P pump T rep = T rep /2 T rep /2 S E in (t)× H in (t)d A dt π| E 0 | 2 L y w 0 τ 0 2 Z 0 2 .
E in (ω,x,z=0)= x ^ E 0 τ 0 2 2π exp( x 2 w 0 2 ((ω ω 0 ) τ 0 ) 2 2 ).
1 2 S d A e ^ i × h ^ j = δ ij = L y 2 n eff i Z 0 ρ (x) E i (x) E j (x)dx.
A l IR (ω)= L y 2 n eff l Z 0 ρ (x) E l (x) E in (ω,x)dx.
P ¯ THz = S E (t)× H (t)d A = 1 T rep T rep 2 T rep 2 dt S d A d Ωd Ω ( E ˜ (Ω)× H ˜ ( Ω ) e i(Ω+ Ω )t + E ˜ (Ω)× H ˜ ( Ω ) * e i(Ω Ω )t +cc. ) 0 4π T rep Re[ S E ˜ ( Ω)× H ˜ (Ω) * d A ]dΩ 0 d P THz (z,Ω)dΩ.
d P THz (z,Ω)= 4π T rep L y Z 0 l, l R e [ A l (z,Ω) A l (z,Ω) * e i( β l β l * )z ( n eff l ) * (n (x) ) 2 E l (Ω,x) E l * (Ω,x)dx].
( 2 x 2 + n i 2 k 0 2 β 2 ) E ( i ) ( x )=0 x i1 <x< x i .
E(x)= E (i) (x)= A i e i k i (x x i ) + B i e i k i (x x i )              x i1 <x< x i .
E ( i ) ( x i )= η i+1 E ( i+1 ) ( x i )and d E ( i ) dx | x= x i = d E ( i+1 ) dx | x= x i .
( 0 B 0 )=( m 1,1 m 1,2 m 2,1 m 2,2 )( A 4 0 ).
0=f( β 2 )= n 1 2 n 2 2 α 2 { tanh( α 2 d ) coth( α 2 d) }(1+ α ˜ 0 α 1 tanh( α 1 (td)))+( α ˜ 0 + α 1 tanh( α 1 (td))).
E= B 0 { sech( α 1 (td)) e α 0 (| x |t)                                                                             t<| x | sech( α 1 (td)) η 1 (cosh( α 1 (| x |t))+ α ˜ 0 α 1 sinh( α 1 (| x |t)))                      d<| x |<t 1 η 1 η 2 [(cosh( α 2 (x+d))+ α ˜ 1 α 2 tanh( α 1 (td))sinh( α 2 (x+d))) + α ˜ 0 α 1 (tanh( α 1 (td))cosh( α 2 (x+d))+ α ˜ 1 α 2 sinh( α 2 (x+d)))]         d<x<d.

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