Abstract

We present a simplified coupled mode theory (CMT), suited for high losses, to describe ultra-broadband THz generation through optical rectification (OR) of fs infrared pulses in waveguides. We derive a new expression that incorporates loss effects into the coherence length for OR. The simplified approach reproduces the results of a computationally rigorous integral CMT that must be used for broadband THz generation. With the new model we perform a parametric study to establish the optimal conditions for OR in symmetric, five-layer, metal/cladding/core structures with electro optic polymer cores. We find conversion efficiencies as high as 35 × 10−4 W−1 and bandwidths up to 20 THz when pumping at 1900 nm. We find that low-loss-cladding layers enhance the efficiency for phase-matched structures, increase the interaction length, and improve the stability of the efficiency with respect to variations in waveguide parameters.

© 2013 Optical Society of America

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References

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  1. F. A. Vallejo and L. M. Hayden, “Design of ultra-broadband terahertz polymer waveguide emitters for telecom wavelengths using coupled mode theory,” Opt. Express21(5), 5842–5858 (2013).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  4. 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]
  5. C. V. McLaughlin, L. M. Hayden, B. Polishak, S. Huang, J. Luo, T.-D. Kim, and A. K.-Y. Jen, “Wideband 15 THz response using organic electrooptic polymer emitter-sensor pairs at communications wavelengths,” Appl. Phys. Lett.92(15), 151107 (2008).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2013 (1)

2012 (1)

2011 (1)

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]

2009 (2)

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]

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 (1)

C. V. McLaughlin, L. M. Hayden, B. Polishak, S. Huang, J. Luo, T.-D. Kim, and A. K.-Y. Jen, “Wideband 15 THz response using organic electrooptic polymer emitter-sensor pairs at communications wavelengths,” Appl. Phys. Lett.92(15), 151107 (2008).
[CrossRef]

2006 (3)

2004 (1)

2002 (1)

H. Ma, A. K. Y. Jen, and L. R. Dalton, “Polymer-based optical waveguides: materials, processing, and devices,” Adv. Mater.14(19), 1339–1365 (2002).
[CrossRef]

1996 (1)

A. Nahata, A. S. Weling, and T. F. Heinz, “A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,” Appl. Phys. Lett.69(16), 2321–2323 (1996).
[CrossRef]

1993 (1)

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]

1989 (1)

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]

1987 (2)

Cao, H.

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]

Dalton, L. R.

H. Ma, A. K. Y. Jen, and L. R. Dalton, “Polymer-based optical waveguides: materials, processing, and devices,” Adv. Mater.14(19), 1339–1365 (2002).
[CrossRef]

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]

Günter, P.

Hayden, L. M.

F. A. Vallejo and L. M. Hayden, “Design of ultra-broadband terahertz polymer waveguide emitters for telecom wavelengths using coupled mode theory,” Opt. Express21(5), 5842–5858 (2013).
[CrossRef] [PubMed]

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]

C. V. McLaughlin, L. M. Hayden, B. Polishak, S. Huang, J. Luo, T.-D. Kim, and A. K.-Y. Jen, “Wideband 15 THz response using organic electrooptic polymer emitter-sensor pairs at communications wavelengths,” Appl. Phys. Lett.92(15), 151107 (2008).
[CrossRef]

Heinz, T. F.

A. Nahata, A. S. Weling, and T. F. Heinz, “A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,” Appl. Phys. Lett.69(16), 2321–2323 (1996).
[CrossRef]

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]

Huang, S.

C. V. McLaughlin, L. M. Hayden, B. Polishak, S. Huang, J. Luo, T.-D. Kim, and A. K.-Y. Jen, “Wideband 15 THz response using organic electrooptic polymer emitter-sensor pairs at communications wavelengths,” Appl. Phys. Lett.92(15), 151107 (2008).
[CrossRef]

Jen, A. K. Y.

H. Ma, A. K. Y. Jen, and L. R. Dalton, “Polymer-based optical waveguides: materials, processing, and devices,” Adv. Mater.14(19), 1339–1365 (2002).
[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]

C. V. McLaughlin, L. M. Hayden, B. Polishak, S. Huang, J. Luo, T.-D. Kim, and A. K.-Y. Jen, “Wideband 15 THz response using organic electrooptic polymer emitter-sensor pairs at communications wavelengths,” Appl. Phys. Lett.92(15), 151107 (2008).
[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]

Khan, R. U. A.

Khurgin, J. B.

Kim, T.-D.

C. V. McLaughlin, L. M. Hayden, B. Polishak, S. Huang, J. Luo, T.-D. Kim, and A. K.-Y. Jen, “Wideband 15 THz response using organic electrooptic polymer emitter-sensor pairs at communications wavelengths,” Appl. Phys. Lett.92(15), 151107 (2008).
[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]

Kukushkin, V. A.

Kuzyk, M. G.

Li, Y.-F.

Linke, R. A.

Lit, J. W. Y.

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]

C. V. McLaughlin, L. M. Hayden, B. Polishak, S. Huang, J. Luo, T.-D. Kim, and A. K.-Y. Jen, “Wideband 15 THz response using organic electrooptic polymer emitter-sensor pairs at communications wavelengths,” Appl. Phys. Lett.92(15), 151107 (2008).
[CrossRef]

Ma, H.

H. Ma, A. K. Y. Jen, and L. R. Dalton, “Polymer-based optical waveguides: materials, processing, and devices,” Adv. Mater.14(19), 1339–1365 (2002).
[CrossRef]

McLaughlin, C. V.

C. V. McLaughlin, L. M. Hayden, B. Polishak, S. Huang, J. Luo, T.-D. Kim, and A. K.-Y. Jen, “Wideband 15 THz response using organic electrooptic polymer emitter-sensor pairs at communications wavelengths,” Appl. Phys. Lett.92(15), 151107 (2008).
[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]

Nahata, A.

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]

A. Nahata, A. S. Weling, and T. F. Heinz, “A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,” Appl. Phys. Lett.69(16), 2321–2323 (1996).
[CrossRef]

Neis, 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]

C. V. McLaughlin, L. M. Hayden, B. Polishak, S. Huang, J. Luo, T.-D. Kim, and A. K.-Y. Jen, “Wideband 15 THz response using organic electrooptic polymer emitter-sensor pairs at communications wavelengths,” Appl. Phys. Lett.92(15), 151107 (2008).
[CrossRef]

Ruan, Z.

Ruiz, B.

Schneider, A.

Singer, K. D.

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]

Sohn, J. E.

Stillhart, M.

Sun, G.

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.

F. A. Vallejo and L. M. Hayden, “Design of ultra-broadband terahertz polymer waveguide emitters for telecom wavelengths using coupled mode theory,” Opt. Express21(5), 5842–5858 (2013).
[CrossRef] [PubMed]

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.

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]

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]

Vodopyanov, K. L.

Weling, A. S.

A. Nahata, A. S. Weling, and T. F. Heinz, “A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,” Appl. Phys. Lett.69(16), 2321–2323 (1996).
[CrossRef]

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]

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]

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]

Adv. Mater. (1)

H. Ma, A. K. Y. Jen, and L. R. Dalton, “Polymer-based optical waveguides: materials, processing, and devices,” Adv. Mater.14(19), 1339–1365 (2002).
[CrossRef]

Appl. Phys. Lett. (2)

C. V. McLaughlin, L. M. Hayden, B. Polishak, S. Huang, J. Luo, T.-D. Kim, and A. K.-Y. Jen, “Wideband 15 THz response using organic electrooptic polymer emitter-sensor pairs at communications wavelengths,” Appl. Phys. Lett.92(15), 151107 (2008).
[CrossRef]

A. Nahata, A. S. Weling, and T. F. Heinz, “A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,” Appl. Phys. Lett.69(16), 2321–2323 (1996).
[CrossRef]

Computing (1)

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. (1)

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]

J. Appl. Phys. (1)

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. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (3)

Opt. Express (4)

Opt. Lett. (1)

Phys. Rev. B (1)

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]

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Figures (8)

Fig. 1
Fig. 1

(a) THz power density spectra, dPTHz, vs. THz frequencies, Ω, computed with the three-wave-approximation for the devices shown in Fig. 5(a) of [1]; the emitter length, L, is indicated with arrows. (b) Ratio of dPTHz obtained through both theories, dPFull/ dPApprox. vs Ω,; the results correspond to the same devices in (a). (c) Nonlinear THz conversion efficiencies, ηTHz vs. L, obtained for the results in (a) and in Fig. 5(a) of [1].

Fig. 2
Fig. 2

(a) Symmetric five-layer slab waveguide, the core is made out of the guest-host EO polymer composite AJTB203/APC, the arrows represent the dipole moment of the chromophores. (b) Second order susceptibility vs. wavelength for AJTB203 in the telecom range. Inset shows the chemical structure of the chromophore.

Fig. 3
Fig. 3

For structures in Fig. 2(a) with t = 10 μm, and distinct cores sizes (d = 5-10 μm): (a) Effective index Neff,0 (inset respective mode attenuation α0) vs THz frequencies (Ω), for the fundamental, m = 0, TEM-like THz mode. The dispersion for AJTB203 (attenuation in inset) is indicated with arrows. PS index is ~1.59 remaining flat for the THz frequencies. (b) Effective group indices, ng,eff vs core half-thickness (d), for the fundamental, l = 0, and first excited, l = 1, even IR modes for the pumping wavelength λ0, indicated with arrows.

Fig. 4
Fig. 4

For structures in Fig. 2(a) with t = 10 µm: (a) Coherence length, Lc vs THz frequencies (Ω), computed neglecting losses (solid black and red), and with our numerical method including losses (dotted black and red) for structures pumped at 1567 nm (d = 9.5 µm) and 1900 nm (d = 9.125 µm). (b) and (c) coherence length contour plots, Lc vs (Ω, d) with our numerical method pumping at 1567 nm (b) and 1900 nm (c). The plot color range in Lc at 1567 nm is almost twice that at 1900 nm and, the blank regions near the origin are out of range, Lc−1.

Fig. 5
Fig. 5

(a), (b) and (c), THz power density spectrum, dPTHz vs. THz frequencies, Ω, for OR between the l = 0 IR mode and the TEM-like (m = 0) THz mode for structures in Fig. 2(a) with t = 10 μm, d (indicated in figure), different emitter lengths, L (indicated with arrows), and pumped at λ0 (indicated in figure). (d) Nonlinear THz conversion efficiency, ηTHz, vs emitter length, L, for the structures in (a), (b) and (c). (e) and (f) efficiency contour plots, ηTHz vs. (L, d), for OR mediated respectively by the l = 0 and l = 1 even IR modes pumped at 1567 nm.

Fig. 6
Fig. 6

For structures in Fig. 2(a) with t = 10 μm: (a) Mode profiles for the l = 0 and l = 1 even IR modes for a structure with d = t (i.e., no cladding) at 1567 nm. (b) OR effective THz attenuation, αeff = αTHz + Δβi, and its different contributions for OR between the l = 0 IR mode and the TEM-like THz mode for a structure d = 10 µm, pumped at 1900 nm. ΔβIR, is defined by Δβi = ΔβIR - αTHz /2. (c) IR contribution to the imaginary part of the phase mismatch, ΔβIR vs core half-size, d, for the (l = 1) even IR mode evaluated at 7.5 THz for different pumping wavelenghts. Inset: Δβi,IR vs. d, for the l = 0 IR mode, the units are the same as in (c).

Fig. 7
Fig. 7

For OR between the l = 0 and l = 1 IR mode (indicated with arrows) and the (m = 0) THz mode for structures with t = 10 μm: (a) Nonlinear THz conversion efficiency, ηmax vs core half-thickness, d, for structures with optimal emitter length, Lopt, pumped at 1567 nm (solid lines) and 1900 nm (dotted lines); ηmax = ηTHz(Lopt,d) is maximum for any L value. (b) Maximum nonlinear THz conversion efficiency, ηMAX vs pumping wavelength, λ. ηMAX is the maximum of ηmax for any d, in (a) ηMAX is indicated with asterisks (l = 0) and red dots (l = 1). (c) Overall optimal emitter length, LOPT, and L80%, vs λ0, ηTHz decreases to 80% of ηMAX at L80%.

Fig. 8
Fig. 8

For structures in Fig. 2(a) with t = 10 μm, d = 9.85 μm (thin cladding layers ~150 nm) pumped at 1567 nm: (a) Mode profiles for the l = 0 and l = 1 even IR modes, Gaussian input profile with a 6.4 µm beam waist and corresponding mode expansion with the l = 0 and l = 1 modes. (b) Nonlinear THz conversion efficiency, ηTHz, vs emitter length, L, computed with integral DFG CMT [1] using the l = 0 and l = 1 IR modes and, with three-wave-approximation for OR mediated by the l = 0 and l = 1 IR modes. (c) THz power density spectrum, dPTHz vs. THz frequencies, Ω, for different emitter lengths, L computed with integral DFG CMT [1]. The THz power density spectrum computed with the three wave approximation have a similar behavior as observed (c) but reduced by a factor of 1/3 and 1/10 for the l = 0 and l = 1 modes respectively.

Tables (1)

Tables Icon

Table 1 Fit parameters for the six-oscillator-Lorentz model for AJTB203

Equations (12)

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

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+ Ω 2 ) A l IR (z,ω Ω 2 ) * Κ l,l' m (Ω,ω) .
Δβ= Ω c [ n g,eff ( ω 0 ) N eff,m (Ω)]+i[ α l ( ω 0 )(1 1 4 Ω 2 ω 0 2 )+ 1 4 Ω 2 ω 0 α l ( ω 0 ) α m (Ω) 2 ].
d P THz (z,Ω)=C(Ω)exp[( α m (Ω)+Δ β i )z] [cosh(Δ β i z)cos(Δ β R z)] Δ β R 2 +Δ β i 2 .
0=(( α m +Δ β i )(cosh(Δ β i L c )cos(Δ β R L c ))+Δ β i sinh(Δ β i L c )+Δ β R sin(Δ β R L c ))).
d P THz (z,Ω)=4π f rep m,m' [ ( d r z ^ ( e ^ m × h ^ m' )) A m (z,Ω) A m' (z,Ω) * e i( β m β m' * )z +c.c.].
A m THz (z,Ω)= π i ϵ 0 Ω τ 0 P m Κ l,l m ( ω 0 ) e iΔβ( ω 0 ,Ω)z/2 sin(Δβ( ω 0 ,Ω)z/2) Δβ( ω 0 ,Ω)/2i A l IR (z, ω 0 + Ω 2 ) A l IR (z, ω 0 Ω 2 ) * .
d P THz (z,Ω)=C(Ω) e ( α m (Ω)+Δ β i )z (cosh(Δ β i z)cos(Δ β R z)) Δ β R 2 +Δ β i 2 .
cΔβ=[( ω 0 +Ω/2) N eff,l ( ω 0 +Ω/2)( ω 0 Ω/2) N eff,l ( ω 0 Ω/2)Ω N eff,m (Ω)] +i[(ω+Ω/2) κ eff,l ( ω 0 +Ω/2)+( ω 0 Ω/2) κ eff,l ( ω 0 Ω/2) κ eff,m Ω(Ω)].
( ω 0 +Ω/2)f( ω 0 +Ω/2)±( ω 0 Ω/2)f( ω 0 Ω/2))={ 2 ω 0 f( ω 0 )+ Ω 2 2 f ( ω 0 )+O( Ω 2 ) Ω(f( ω 0 )+ ω 0 f ( ω 0 ))+O( Ω 3 ).
Δβ= Ω c [ n g,IR ( ω 0 ) N eff,m (Ω)]+i[ α l ( ω 0 )(1 1 4 Ω 2 ω 0 2 )+ 1 4 Ω 2 ω 0 α l ( ω 0 ) α m (Ω) 2 ].
ϵ(Ω)= ϵ 0 ( n b 2 + i=1 6 a i Ω i 2 ( Ω i 2 Ω) 2 2i Γ i Ω ).
(n(Ω)+ik(Ω)) 2 = ε b + i ε 0 Ω ( δ e 2 τ/ m e 1iΩτ )

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