Abstract

We report new closed-form expressions for Maker fringes of anisotropic and absorbing poled polymer thin films in multilayer structures that include back reflections of both fundamental and second-harmonic waves. The expressions, based on boundary conditions at each interface, can be applied to multilayer structures containing a buffer and a transparent conducting oxide layer, which might enhance multiple reflections of fundamental and second-harmonic waves inside a nonlinear thin film layer. This formulation facilitates Maker fringe analysis for a sample containing additional multilayer structures on either side of a poled polymer thin film. Experimental data and numerical simulations are given to indicate the importance of inclusion of such a reflective layer in analyses for reliable characterization of second-harmonic tensor elements.

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References

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  1. G. A. Lindsay and K. D. Singer, eds., Polymers for Second-Order Nonlinear Optics, Vol 601 of ACS Symposium Series (ACS, 1995).
  2. W. N. Herman, S. R. Flom, and S. H. Foulger, eds., Organic Thin Films for Photonic Applications, Vol. 1039 of ACS Symposium Series (ACS, 2010).
  3. D. M. Burland, R. D. Miller, and C. A. Walsh, “Second-order nonlinearity in poled-polymer systems,” Chem. Rev. 94(1), 31–75 (1994).
    [CrossRef]
  4. C. C. Teng and H. T. Man, “Simple reflection technique for measuring the electro-optic coefficient of poled polymers,” Appl. Phys. Lett. 56(18), 1734–1736 (1990).
    [CrossRef]
  5. J. S. Schildkraut, “Determination of the electro optic coefficient of a poled polymer film,” Appl. Opt. 29(19), 2839–2841 (1990).
    [CrossRef] [PubMed]
  6. D. H. Park, C. H. Lee, and W. N. Herman, “Analysis of multiple reflection effects in reflective measurements of electro-optic coefficients of poled polymers in multilayer structures,” Opt. Express 14(19), 8866–8884 (2006).
    [CrossRef] [PubMed]
  7. 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(1), 21–22 (1962).
    [CrossRef]
  8. J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41(4), 1667–1681 (1970).
    [CrossRef]
  9. N. Okamoto, Y. Hirano, and O. Sugihara, “Precise estimation of nonlinear-optical coefficients for anisotropic nonlinear films with C∞v symmetry,” J. Opt. Soc. Am. B 9(11), 2083–2087 (1970).
    [CrossRef]
  10. W. N. Herman and L. M. Hayden, “Maker fringes revisited: second-harmonic generation from birefringent or absorbing materials,” J. Opt. Soc. Am. B 12(3), 416–427 (1995).
    [CrossRef]
  11. T. K. Lim, M.-Y. Jeong, C. Song, and D. C. Kim, “Absorption effect in the calculation of a second-order nonlinear coefficient from the data of a maker fringe experiment,” Appl. Opt. 37(13), 2723–2728 (1998).
    [CrossRef] [PubMed]
  12. H. Hellwig and L. Bohaty, “Multiple reflections and Fabry-Perot interference corrections in Maker fringe experiments,” Opt. Commun. 161(1-3), 51–56 (1999).
    [CrossRef]
  13. N. A. Sanford and J. A. Aust, “Nonlinear optical characterization of LiNbO3. I. Theoretical analysis of Maker fringe patterns for x-cut wafers,” J. Opt. Soc. Am. B 15(12), 2885–2908 (1998).
    [CrossRef]
  14. M. Abe, I. Shoji, J. Suda, and T. Kondo, “Comprehensive analysis of multiple-reflection effects on rotational Maker-fringe experiments,” J. Opt. Soc. Am. B 25(10), 1616–1624 (2008).
    [CrossRef]
  15. M. Braun, F. Bauer, Th. Vogtmann, and M. Schwoerer, “Precise second-harmonic generation Maker fringe measurements in single crystals of the diacetylene NP/4-MPU and evaluation by a second-harmonic generation theory in 4×4 matrix formulation and ray tracing,” J. Opt. Soc. Am. B 14(7), 1699–1706 (1997).
    [CrossRef]
  16. M. Braun, F. Bauer, Th. Vogtmann, and M. Schwoerer, “Detailed analysis of second-harmonic-generation Maker fringes in biaxially birefringent materials by a 4×4 matrix formulation,” J. Opt. Soc. Am. B 15(12), 2877–2884 (1998).
    [CrossRef]
  17. V. Rodriguez and C. Sourisseau, “General Maker-fringe ellipsometric analyses in multilayer nonlinear and linear anisotropic optical media,” J. Opt. Soc. Am. B 19(11), 2650–2664 (2002).
    [CrossRef]
  18. S. Lee, B. Park, S.-D. Lee, G. Park, and Y. D. Kim, “Second-harmonic generation in poled films of nonlinear optical polymer composites,” Opt. Quantum Electron. 27(5), 411–420 (1995).
    [CrossRef]
  19. I. P. Kaminow, An Introduction to Electrooptic Devices (Academic, 1974).
  20. J. P. Drummond, S. J. Clarson, J. S. Zetts, F. K. Hopkins, and S. J. Caracci, “Enhanced electro-optic poling in guest–host systems using conductive polymer-based cladding layers,” Appl. Phys. Lett. 74(3), 368–370 (1999).
    [CrossRef]
  21. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).
  22. P. Yeh, Optical Waves in Layered Media (Wiley, 1988)
  23. D. Guo, R. Lin, and W. Wang, “Gaussian-optics-based optical modeling and characterization of a Fabry-Perot microcavity for sensing applications,” J. Opt. Soc. Am. A 22(8), 1577–1588 (2005).
    [CrossRef] [PubMed]
  24. K. D. Singer, M. D. Kuzyk, and J. E. Sohn, “Second-order nonlinear-optical processes in orientationally ordered materials: relationship between molecular and macroscopic properties,” J. Opt. Soc. Am. B 17, 566–571 (2000).
  25. R. C. Hoffman, A. G. Mott, M. J. Ferry, T. M. Pritchett, W. Shensky, J. A. Orlicki, G. R. Martin, J. Dougherty, J. L. Leadore, A. M. Rawlett, and D. H. Park, “Poling of visible chromophores in millimeter-thick PMMA host,” Opt. Mater. Express 1(1), 67–77 (2011).
    [CrossRef]
  26. J. Zhang, G. Wang, Z. Liu, L. Wang, G. Zhang, X. Zhang, Y. Wu, P. Fu, and Y. Wu, “Growth and optical properties of a new nonlinear Na3La9O3(BO3)8 crystal,” Opt. Express 18(1), 237–243 (2010).
    [CrossRef] [PubMed]
  27. C. Chen, Z. Shao, J. Jiang, J. Wei, J. Lin, J. Wang, N. Ye, J. Lv, B. Wu, M. Jiang, M. Yoshimura, Y. Mori, and T. Sasaki, “Determination of the nonlinear optical coefficients of YCa4O(BO3)3 crystal,” J. Opt. Soc. Am. B 17(4), 566–571 (2000).
    [CrossRef]
  28. S. Ramo, J. R. Whinnery, and T. V. Duzer, Fields and Waves in Communication Electronics (Wiley, New York, 1965), Chap. 6.

2011 (1)

2010 (1)

2008 (1)

2006 (1)

2005 (1)

2002 (1)

2000 (2)

1999 (2)

J. P. Drummond, S. J. Clarson, J. S. Zetts, F. K. Hopkins, and S. J. Caracci, “Enhanced electro-optic poling in guest–host systems using conductive polymer-based cladding layers,” Appl. Phys. Lett. 74(3), 368–370 (1999).
[CrossRef]

H. Hellwig and L. Bohaty, “Multiple reflections and Fabry-Perot interference corrections in Maker fringe experiments,” Opt. Commun. 161(1-3), 51–56 (1999).
[CrossRef]

1998 (3)

1997 (1)

1995 (2)

S. Lee, B. Park, S.-D. Lee, G. Park, and Y. D. Kim, “Second-harmonic generation in poled films of nonlinear optical polymer composites,” Opt. Quantum Electron. 27(5), 411–420 (1995).
[CrossRef]

W. N. Herman and L. M. Hayden, “Maker fringes revisited: second-harmonic generation from birefringent or absorbing materials,” J. Opt. Soc. Am. B 12(3), 416–427 (1995).
[CrossRef]

1994 (1)

D. M. Burland, R. D. Miller, and C. A. Walsh, “Second-order nonlinearity in poled-polymer systems,” Chem. Rev. 94(1), 31–75 (1994).
[CrossRef]

1990 (2)

C. C. Teng and H. T. Man, “Simple reflection technique for measuring the electro-optic coefficient of poled polymers,” Appl. Phys. Lett. 56(18), 1734–1736 (1990).
[CrossRef]

J. S. Schildkraut, “Determination of the electro optic coefficient of a poled polymer film,” Appl. Opt. 29(19), 2839–2841 (1990).
[CrossRef] [PubMed]

1970 (2)

J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41(4), 1667–1681 (1970).
[CrossRef]

N. Okamoto, Y. Hirano, and O. Sugihara, “Precise estimation of nonlinear-optical coefficients for anisotropic nonlinear films with C∞v symmetry,” J. Opt. Soc. Am. B 9(11), 2083–2087 (1970).
[CrossRef]

1962 (1)

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(1), 21–22 (1962).
[CrossRef]

Abe, M.

Aust, J. A.

Bauer, F.

Bohaty, L.

H. Hellwig and L. Bohaty, “Multiple reflections and Fabry-Perot interference corrections in Maker fringe experiments,” Opt. Commun. 161(1-3), 51–56 (1999).
[CrossRef]

Braun, M.

Burland, D. M.

D. M. Burland, R. D. Miller, and C. A. Walsh, “Second-order nonlinearity in poled-polymer systems,” Chem. Rev. 94(1), 31–75 (1994).
[CrossRef]

Caracci, S. J.

J. P. Drummond, S. J. Clarson, J. S. Zetts, F. K. Hopkins, and S. J. Caracci, “Enhanced electro-optic poling in guest–host systems using conductive polymer-based cladding layers,” Appl. Phys. Lett. 74(3), 368–370 (1999).
[CrossRef]

Chen, C.

Clarson, S. J.

J. P. Drummond, S. J. Clarson, J. S. Zetts, F. K. Hopkins, and S. J. Caracci, “Enhanced electro-optic poling in guest–host systems using conductive polymer-based cladding layers,” Appl. Phys. Lett. 74(3), 368–370 (1999).
[CrossRef]

Dougherty, J.

Drummond, J. P.

J. P. Drummond, S. J. Clarson, J. S. Zetts, F. K. Hopkins, and S. J. Caracci, “Enhanced electro-optic poling in guest–host systems using conductive polymer-based cladding layers,” Appl. Phys. Lett. 74(3), 368–370 (1999).
[CrossRef]

Ferry, M. J.

Fu, P.

Guo, D.

Hayden, L. M.

Hellwig, H.

H. Hellwig and L. Bohaty, “Multiple reflections and Fabry-Perot interference corrections in Maker fringe experiments,” Opt. Commun. 161(1-3), 51–56 (1999).
[CrossRef]

Herman, W. N.

Hirano, Y.

Hoffman, R. C.

Hopkins, F. K.

J. P. Drummond, S. J. Clarson, J. S. Zetts, F. K. Hopkins, and S. J. Caracci, “Enhanced electro-optic poling in guest–host systems using conductive polymer-based cladding layers,” Appl. Phys. Lett. 74(3), 368–370 (1999).
[CrossRef]

Jeong, M.-Y.

Jerphagnon, J.

J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41(4), 1667–1681 (1970).
[CrossRef]

Jiang, J.

Jiang, M.

Kim, D. C.

Kim, Y. D.

S. Lee, B. Park, S.-D. Lee, G. Park, and Y. D. Kim, “Second-harmonic generation in poled films of nonlinear optical polymer composites,” Opt. Quantum Electron. 27(5), 411–420 (1995).
[CrossRef]

Kondo, T.

Kurtz, S. K.

J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41(4), 1667–1681 (1970).
[CrossRef]

Kuzyk, M. D.

Leadore, J. L.

Lee, C. H.

Lee, S.

S. Lee, B. Park, S.-D. Lee, G. Park, and Y. D. Kim, “Second-harmonic generation in poled films of nonlinear optical polymer composites,” Opt. Quantum Electron. 27(5), 411–420 (1995).
[CrossRef]

Lee, S.-D.

S. Lee, B. Park, S.-D. Lee, G. Park, and Y. D. Kim, “Second-harmonic generation in poled films of nonlinear optical polymer composites,” Opt. Quantum Electron. 27(5), 411–420 (1995).
[CrossRef]

Lim, T. K.

Lin, J.

Lin, R.

Liu, Z.

Lv, J.

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(1), 21–22 (1962).
[CrossRef]

Man, H. T.

C. C. Teng and H. T. Man, “Simple reflection technique for measuring the electro-optic coefficient of poled polymers,” Appl. Phys. Lett. 56(18), 1734–1736 (1990).
[CrossRef]

Martin, G. R.

Miller, R. D.

D. M. Burland, R. D. Miller, and C. A. Walsh, “Second-order nonlinearity in poled-polymer systems,” Chem. Rev. 94(1), 31–75 (1994).
[CrossRef]

Mori, Y.

Mott, A. G.

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(1), 21–22 (1962).
[CrossRef]

Okamoto, N.

Orlicki, J. A.

Park, B.

S. Lee, B. Park, S.-D. Lee, G. Park, and Y. D. Kim, “Second-harmonic generation in poled films of nonlinear optical polymer composites,” Opt. Quantum Electron. 27(5), 411–420 (1995).
[CrossRef]

Park, D. H.

Park, G.

S. Lee, B. Park, S.-D. Lee, G. Park, and Y. D. Kim, “Second-harmonic generation in poled films of nonlinear optical polymer composites,” Opt. Quantum Electron. 27(5), 411–420 (1995).
[CrossRef]

Pritchett, T. M.

Rawlett, A. M.

Rodriguez, V.

Sanford, N. A.

Sasaki, T.

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(1), 21–22 (1962).
[CrossRef]

Schildkraut, J. S.

Schwoerer, M.

Shao, Z.

Shensky, W.

Shoji, I.

Singer, K. D.

Sohn, J. E.

Song, C.

Sourisseau, C.

Suda, J.

Sugihara, O.

Teng, C. C.

C. C. Teng and H. T. Man, “Simple reflection technique for measuring the electro-optic coefficient of poled polymers,” Appl. Phys. Lett. 56(18), 1734–1736 (1990).
[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(1), 21–22 (1962).
[CrossRef]

Vogtmann, Th.

Walsh, C. A.

D. M. Burland, R. D. Miller, and C. A. Walsh, “Second-order nonlinearity in poled-polymer systems,” Chem. Rev. 94(1), 31–75 (1994).
[CrossRef]

Wang, G.

Wang, J.

Wang, L.

Wang, W.

Wei, J.

Wu, B.

Wu, Y.

Ye, N.

Yoshimura, M.

Zetts, J. S.

J. P. Drummond, S. J. Clarson, J. S. Zetts, F. K. Hopkins, and S. J. Caracci, “Enhanced electro-optic poling in guest–host systems using conductive polymer-based cladding layers,” Appl. Phys. Lett. 74(3), 368–370 (1999).
[CrossRef]

Zhang, G.

Zhang, J.

Zhang, X.

Appl. Opt. (2)

Appl. Phys. Lett. (2)

J. P. Drummond, S. J. Clarson, J. S. Zetts, F. K. Hopkins, and S. J. Caracci, “Enhanced electro-optic poling in guest–host systems using conductive polymer-based cladding layers,” Appl. Phys. Lett. 74(3), 368–370 (1999).
[CrossRef]

C. C. Teng and H. T. Man, “Simple reflection technique for measuring the electro-optic coefficient of poled polymers,” Appl. Phys. Lett. 56(18), 1734–1736 (1990).
[CrossRef]

Chem. Rev. (1)

D. M. Burland, R. D. Miller, and C. A. Walsh, “Second-order nonlinearity in poled-polymer systems,” Chem. Rev. 94(1), 31–75 (1994).
[CrossRef]

J. Appl. Phys. (1)

J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41(4), 1667–1681 (1970).
[CrossRef]

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

J. Opt. Soc. Am. B (9)

K. D. Singer, M. D. Kuzyk, and J. E. Sohn, “Second-order nonlinear-optical processes in orientationally ordered materials: relationship between molecular and macroscopic properties,” J. Opt. Soc. Am. B 17, 566–571 (2000).

C. Chen, Z. Shao, J. Jiang, J. Wei, J. Lin, J. Wang, N. Ye, J. Lv, B. Wu, M. Jiang, M. Yoshimura, Y. Mori, and T. Sasaki, “Determination of the nonlinear optical coefficients of YCa4O(BO3)3 crystal,” J. Opt. Soc. Am. B 17(4), 566–571 (2000).
[CrossRef]

N. Okamoto, Y. Hirano, and O. Sugihara, “Precise estimation of nonlinear-optical coefficients for anisotropic nonlinear films with C∞v symmetry,” J. Opt. Soc. Am. B 9(11), 2083–2087 (1970).
[CrossRef]

W. N. Herman and L. M. Hayden, “Maker fringes revisited: second-harmonic generation from birefringent or absorbing materials,” J. Opt. Soc. Am. B 12(3), 416–427 (1995).
[CrossRef]

N. A. Sanford and J. A. Aust, “Nonlinear optical characterization of LiNbO3. I. Theoretical analysis of Maker fringe patterns for x-cut wafers,” J. Opt. Soc. Am. B 15(12), 2885–2908 (1998).
[CrossRef]

M. Abe, I. Shoji, J. Suda, and T. Kondo, “Comprehensive analysis of multiple-reflection effects on rotational Maker-fringe experiments,” J. Opt. Soc. Am. B 25(10), 1616–1624 (2008).
[CrossRef]

M. Braun, F. Bauer, Th. Vogtmann, and M. Schwoerer, “Precise second-harmonic generation Maker fringe measurements in single crystals of the diacetylene NP/4-MPU and evaluation by a second-harmonic generation theory in 4×4 matrix formulation and ray tracing,” J. Opt. Soc. Am. B 14(7), 1699–1706 (1997).
[CrossRef]

M. Braun, F. Bauer, Th. Vogtmann, and M. Schwoerer, “Detailed analysis of second-harmonic-generation Maker fringes in biaxially birefringent materials by a 4×4 matrix formulation,” J. Opt. Soc. Am. B 15(12), 2877–2884 (1998).
[CrossRef]

V. Rodriguez and C. Sourisseau, “General Maker-fringe ellipsometric analyses in multilayer nonlinear and linear anisotropic optical media,” J. Opt. Soc. Am. B 19(11), 2650–2664 (2002).
[CrossRef]

Opt. Commun. (1)

H. Hellwig and L. Bohaty, “Multiple reflections and Fabry-Perot interference corrections in Maker fringe experiments,” Opt. Commun. 161(1-3), 51–56 (1999).
[CrossRef]

Opt. Express (2)

Opt. Mater. Express (1)

Opt. Quantum Electron. (1)

S. Lee, B. Park, S.-D. Lee, G. Park, and Y. D. Kim, “Second-harmonic generation in poled films of nonlinear optical polymer composites,” Opt. Quantum Electron. 27(5), 411–420 (1995).
[CrossRef]

Phys. Rev. Lett. (1)

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(1), 21–22 (1962).
[CrossRef]

Other (6)

G. A. Lindsay and K. D. Singer, eds., Polymers for Second-Order Nonlinear Optics, Vol 601 of ACS Symposium Series (ACS, 1995).

W. N. Herman, S. R. Flom, and S. H. Foulger, eds., Organic Thin Films for Photonic Applications, Vol. 1039 of ACS Symposium Series (ACS, 2010).

I. P. Kaminow, An Introduction to Electrooptic Devices (Academic, 1974).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).

P. Yeh, Optical Waves in Layered Media (Wiley, 1988)

S. Ramo, J. R. Whinnery, and T. V. Duzer, Fields and Waves in Communication Electronics (Wiley, New York, 1965), Chap. 6.

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Figures (3)

Fig. 1
Fig. 1

(a) Multilayer structure geometry containing buffer and TCO layers. Black solid arrows represent fundamental waves and red arrows SH waves. First and second subscripts on n and θ are used to represent dependence of wavelength and layer symbols, respectively. For NLO film, o and e represent ordinary and extraordinary refractive indices, respectively. (b) Equivalent three layer structure using virtual layers (dashed lines). The numeral subscript/superscript i represents a wavelength (i = 1: fundamental and i = 2: second-harmonic). For example, t aS (1p) means a p-polarization transmission coefficient from air to substrate at the fundamental (i = 1) wavelength.

Fig. 2
Fig. 2

Second harmonic intensities of X-cut quartz at varying angles of incidence in three cases (Rotation is about the z axis): (a) Quartz only, (b) Quartz/Au, and (c) Au/Quartz/Au. Experimental results (red dots) are plotted with two different simulated Maker fringe patterns (blue: multiple reflections, cyan: no multiple reflections).

Fig. 3
Fig. 3

(a) Complex index of refraction of ITO, (b) Complex anisotropic indices of refraction of NLOP measured using ellipsometer (inset figure: absorption profile of the multilayered sample by UV-VIS spectrophotometer), (c) pp and (d) sp Maker fringe profiles. Experimental results (red circles) were plotted with simulation results by various types of configurations. For line colors in (c) and (d), Cyan: multilayer structure with multiple reflections of both fundamental and SH. Black: multilayer structure without any multiple reflections. Green dash: multilayer structure with multiple reflections of SH only. Blue: three layer model (air/NLOP/substrate) without any multiple reflections.

Equations (34)

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

d =( 0 0 0 0 d 15 0 0 0 0 d 15 0 0 d 31 d 31 d 33 0 0 0 ).
Region I: { E 2a = e ^ 2a r r aS (pp) e i k 2a r r H 2a = y ^ n 2a r aS (pp) e i k 2a r r Region II: { E 2B = e ^ 2B A e i k 2B r + e ^ 2B r B e i k 2B r r H 2B = y ^ n 2B A e i k 2B r + y ^ n 2B B e i k 2B r r Region III: { E 2f = e ^ 2f C e i k 2f r + e ^ 2f r D e i k 2f r r + e b e 2i k 1f r + e b r e 2i k 1f r r H 2f = y ^ n 2f ( θ 2f )C e i k 2f r + y ^ n 2f ( θ 2f )D e i k 2f r r + h b e 2i k 1f r + h b r e 2i k 1f r r Region IV: { E 2T = e ^ 2T E e i k 2T r + e ^ 2T r F e i k 2T r r H 2T = y ^ n 2T E e i k 2T r + y ^ n 2T F e i k 2T r r Region V: { E 2S = e ^ 2S t aS (pp) e i k 2S r H 2S = y ^ n 2S t aS (pp) e i k 2S r ,
n if ( θ if )= ( cos 2 θ if n io 2 + sin 2 θ if n ie 2 ) 1/2 ,
t aS (ηp) = e i( ϕ 2SL + ϕ 2ST ) 4πi k 0 L n 2f c γ 2 c 2γ [ t aS (1η) E 1a t fTS (1η) ] 2 t aS (2p) t aBf (2p) × { d eff ( ηp ) [ e i Φ (ηp) + r fBa (2p) ( r fTS (1η) ) 2 e i Φ (ηp) ]sinc( Ψ (ηp) ) + d eff r( ηp ) [ ( r fTS (1η) ) 2 e i Ψ (ηp) + r fBa (2p) e i Ψ (ηp) ]sinc( Φ (ηp) ) },
Ψ (ημ) = ϕ 1 (η) ϕ 2 (μ) 2 , Φ (ημ) = ϕ 1 (η) + ϕ 2 (μ) 2 .
P 2ω = c 8π ( t Sa (2p) ) 2 | t aS (ηp) | 2 A cross ,
t aS (ηp) = e 2i k 0 n 2S c 2S ( L+ d T + l AD ) 4πi k 0 L n 2f c γ 2 c 2γ [ t aS (1η) E 1a t fS (1η) ] 2 t aS (2p) t af (2p) ×{ d eff ( ηp ) [ e i Φ (ηp) + r fa (2p) ( r fS (1η) ) 2 e i Φ (ηp) ]sinc( Ψ (ηp) ) + d eff r( ηp ) [ ( r fS (1η) ) 2 e i Ψ (ηp) + r fa (2p) e i Ψ (ηp) ]sinc( Φ (ηp) ) },
r aS (ηp) = e i ϕ 2aB ( t aS (1η) E 1a t fTS (1η) ) 2 4πi k 0 L n 2f c γ 2 c 2γ t aBf (2p) ×{ r aS (2p) [ d eff (ηp) ( e i Φ (ηp) + r fBa (2p) ( r fTS (1η) ) 2 e i Φ (ηp) )sinc( Ψ (ηp) ) + d eff r(ηp) ( r fBa (2p) e i Ψ (ηp) + ( r fTS (1η) ) 2 e i Ψ (ηp) )sinc( Φ (ηp) ) ] [ d eff (ηp) ( r aBf (2p) e i Φ (ηp) +( r aBf (2p) r fBa (2p) t aBf (2p) t fBa (2p) ) ( r fTS (1η) ) 2 e i Φ (ηp) )sinc( Ψ (ηp) ) + d eff r(ηp) ( ( r aBf (2p) r fBa (2p) t aBf (2p) t fBa (2p) ) e i Ψ (ηp) + r aBf (2p) ( r fTS (1η) ) 2 e i Ψ (ηp) )sinc( Φ (ηp) ) ] },
×× E 2 ( 2 k 0 ) 2 ( n 2o 2 0 0 0 n 2o 2 0 0 0 n 2e 2 ) E 2 = ( 2 k 0 ) 2 4π P NL ,
P NL = d : E 1 E 1 .
n 2f ( θ 2f )cos γ 2f e bx ±cos( θ 2f γ 2f ) h by = 4π k 0 L ϕ 1 (η) 1 2 ϕ 2 (p) { e ^ 2f P NL e ^ 2f r P NL
n 2f ( θ 2f )cos γ 2f e bx r ±cos( θ 2f γ 2f ) h by r = 4π k 0 L ϕ 1 (η) 1 2 ϕ 2 (p) { e ^ 2f P NLr e ^ 2f r P NLr ,
e ^ 2f P NL = E 1 2 e ^ 2f d : e ^ 1f e ^ 1f = E 1 2 d eff ( pp ) e ^ 2f r P NL = E 1 2 e ^ 2f r d : e ^ 1f e ^ 1f = E 1 2 d eff r( pp ) e ^ 2f P NLr = E 1r 2 e ^ 2f d : e ^ 1f r e ^ 1f r = E 1r 2 d eff r( pp ) e ^ 2f r P NLr = E 1r 2 e ^ 2f r d : e ^ 1f r e ^ 1f r = E 1r 2 d eff ( pp ) ,
d eff ( pp ) =2 d 15 c 2γ s 1γ c 1γ + s 2γ ( d 31 c 1γ 2 + d 33 s 1γ 2 ) d eff r( pp ) =2 d 15 c 2γ s 1γ c 1γ + s 2γ ( d 31 c 1γ 2 + d 33 s 1γ 2 )
c iγ =cos( θ if γ if ), s iγ =sin( θ if γ if ).
d eff ( sp ) = d 31 s 2γ = d eff r( sp ) .
E 1f = E 1s y ^ e i k 1f s r + E 1s r y ^ e i k 1f s,r r + E 1p e ^ 1p e i k 1f p r + E 1p r e ^ 1p r e i k 1f p,r r .
y ^ d : E 1f E 1f =2 d 15 s 1γ ( E 1s E 1p e i( k 1f s + k 1f p )r + E 1s E 1p r e i( k 1f s + k 1f p,r )r + E 1s r E 1p e i( k 1f s,r + k 1f p )r + E 1s r E 1p r e i( k 1f s,r + k 1f p,r )r ),
I/ II : e i ϕ 2aB D a ( 0 r aS (pp) )= D B B ( A B ), II / III : D B ( A B )= D f ( C D )+( e bx h by )+( e bx r h by r ), III / IV : D f 2 1 ( C D )+ e 2i ϕ 1 ( e bx h by )+ e 2i ϕ 1 ( e bx r h by r )= D T 2TL 1 ( E F ), IV /V : D T T 1 2TL 1 ( E F )= e i( ϕ 2SL + ϕ 2ST ) D S ( t aS (pp) 0 ),
D l =( cos( θ 2l ) cos( θ 2l ) n 2l n 2l ), D f =( cos( θ 2f γ 2f ) cos( θ 2f γ 2f ) n 2 cos( γ 2 ) n 2 cos( γ 2 ) ), m =( e i ϕ 2m 0 0 e i ϕ 2m ), f =( e i ϕ 2f 0 0 e i ϕ 2f ), and 2TL =( e i ϕ 2TL 0 0 e i ϕ 2TL ),
e i( ϕ 2SL + ϕ 2ST ) D aS ( t aS (pp) 0 )= e i ϕ 2aB ( 0 r aS (pp) ) 1 2 n 2f c γ 2 c 2γ D aBf ×{ ( 1 e i(2 ϕ 1 (p) ϕ 2 (p) ) 0 0 1 e i(2 ϕ 1 (p) + ϕ 2 (p) ) )( n 2f c γ 2 e bx + c 2γ h by n 2f c γ 2 e bx + c 2γ h by ) +( 1 e i(2 ϕ 1 (p) + ϕ 2 (p) ) 0 0 1 e i(2 ϕ 1 (p) ϕ 2 (p) ) )( n 2f c γ 2 e bx r + c 2γ h by r n 2f c γ 2 e bx r + c 2γ h by r ) },
D aBf = 1 t aBf (2p) ( 1 r fBa (2p) r aBf (2p) t aBf (2p) t fBa (2p) r aBf (2p) r fBa (2p) ), D aS = 1 t aS (2p) ( 1 r Sa (2p) r aS (2p) t aS (2p) t Sa (2p) r aS (2p) r Sa (2p) ).
r aS (iη) = r aB (iη) + r BS (iη) e 2i β B (iη) d B 1+ r aB (iη) r BS (iη) e 2i β B (iη) d B r BS (iη) = r Bf (iη) + r fTS (iη) e 2i β f (iη) L 1+ r Bf (iη) r fTS (iη) e 2i β f (iη) L r iTS (iη) = r fT (iη) + r TS (iη) e 2i β T (iη) d T 1+ r fT (iη) r TS (iη) e 2i β T (iη) d T
t aS (iη) = t aB (iη) t BS (iη) e i β B (iη) d B 1+ r aB (iη) r BS (iη) e 2i β B (iη) d B t BS (iη) = t Bf (iη) t fTS (iη) e i β f (iη) L 1+ r Bf (iη) r fTS (iη) e 2i β f (iη) L t iTS (iη) = t fT (iη) t TS (iη) e i β T (iη) d T 1+ r fT (iη) r TS (iη) e 2i β T (iη) d T ,
r jk (is) = Z k (is) Z j (is) Z k (is) + Z j (is) , r jk (ip) = Z j (ip) Z k (ip) Z j (ip) + Z k (ip) , t jk (iη) = 2 Z j (iη) Z j (iη) + Z k (iη) n j ( θ j )cos γ ij n k ( θ k )cos γ ik ,
Z (is) = 1 n io 2 sin 2 θ , Z (ip) = 1 n io 1 ( sinθ n ie ) 2 ,
t aS ( αs ) = e 2i k 0 n 2S c 2S ( L+ d T + l AD ) 1 2 n 2o c 2f t aS ( 2s ) t af ( 2s ) ×[ ( 1 e i( ϕ 1 (s) + ϕ 1 (p) ϕ 2 (s) ) )( n 2o c 2f e by h bx ) r fa (2s) ( 1 e i( ϕ 1 (s) + ϕ 1 (p) + ϕ 2 (s) ) )( n 2o c 2f e by + h bx ) +( 1 e i( ϕ 1 (s) + ϕ 1 (p) + ϕ 2 (s) ) )( n 2o c 2f e by r h bx r ) r fa (2s) ( 1 e i( ϕ 1 (s) + ϕ 1 (p) ϕ 2 (s) ) )( n 2o c 2f e by r + h bx r ) ],
n 2o c 2f e by ± h bx = 8π k 0 L ϕ 1 (s) + ϕ 1 (p) ± ϕ 2 (s) y ^ P NL ,
n 2o c 2f by r ± h bx r =± 8π k 0 L ϕ 1 (s) + ϕ 1 (p) ϕ 2 (s) y ^ P NLr ,
y ^ P NL =2 E 1s E 1p d 15 s 1γ , y ^ P NLr =2 E 1s r E 1p r d 15 s 1γ .
E 1f = E 1a ( y ^ cosα t aS (1s) t fS (1s) e i ϕ 1 (s) + e ^ 1f sinα t aS (1p) t fS (1p) e i ϕ 1 (p) )= y ^ E 1s + e ^ 1f E 1p E 1f r = E 1a ( y ^ cosα t aS (1s) t fS (1s) r fTS (1s) e i ϕ 1 (s) + e ^ 1f r sinα t aS (1p) t fS (1p) r fTS (1p) e i ϕ 1 (p) )= y ^ E 1s r + e ^ 1f r E 1p r .
y ^ P NL = ( E 1a ) 2 sin( 2α ) t aS (1s) t fS (1s) t aS (1p) t fS (1p) e i ϕ 1 (s) e i ϕ 1 (p) d 15 s 1γ y ^ P NLr = ( E 1a ) 2 sin( 2α ) t aS (1s) t fS (1s) t aS (1p) t fS (1p) r fS (1s) r fS (1p) e i ϕ 1 (s) e i ϕ 1 (p) d 15 s 1γ .
t aS ( αs ) = e 2i k 0 n 2S c 2S ( L+ d T + l AD ) 4πi k 0 L n 2o c 2f [ t aS (1s) t fS (1s) t aS (1p) t fS (1p) E 1a 2 ] t aS ( 2s ) t af ( 2s ) sin( 2α ) × d 15 s 1γ { [ e i Φ ( αs ) + r fa (2s) ( r fS (1s) r fS (1p) ) e i Φ ( αs ) ]sinc( Ψ ( αs ) ) +[ e i Ψ ( αs ) ( r fS (1s) r fS (1p) )+ r fa (2s) e i Ψ ( αs ) ]sinc( Φ ( αs ) ) },
Ψ (αs) = ϕ 1 (s) + ϕ 1 (p) ϕ 2 (s) 2 , Φ (αs) = ϕ 1 (s) + ϕ 1 (p) + ϕ 2 (s) 2 .

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