Abstract

We present new closed-form expressions for analysis of Teng-Man measurements of the electro-optic coefficients of poled polymer thin films. These expressions account for multiple reflection effects using a rigorous analysis of the multilayered structure for varying angles of incidence. The analysis based on plane waves is applicable to both transparent and absorptive films and takes into account the properties of the transparent conducting electrode layer. Methods for fitting data are presented and the error introduced by ignoring the transparent conducting layer and multiple reflections is discussed.

© 2006 Optical Society of America

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    [CrossRef]
  3. M. A. Mortazavi, A. Knoesen, S. T. Kowel, B. G. Higgins,and A. Dienes, "Second-harmonic generation and absorptionstudies of polymer-dye films oriented by corona-onset poling at elevated temperatures," J. Opt. Soc. Am. B 6, 733-741 (1989).
    [CrossRef]
  4. R. A. Norwood, M. G. Kuzyk, and R. A. Keosian, "Electro-optic tensor ratio determination of side-chain copolymers with electro-optic interferometry," J. Appl. Phys. 75, 1869-1874 (1994).
    [CrossRef]
  5. Y.-P. Wang, J.-P. Chen, X.-W. Li, J.-X. Hong, X.-H. Zhang, J.-H. Zhou, and A.-L. Ye, "Measuring eletro-optic coefficients of poled polymers using fiber-optic Mach-Zehnder interferometer," Appl. Phys. Lett. 85, 5102-5103 (2004).
    [CrossRef]
  6. M. J. Shin, H. R. Cho, S. H. Han, and J. W. Wu, "Analysis of Mach-Zehnder interferometry measurement of the Pockels coefficients in a poled polymer film with a reflection configuration," J. Appl. Phys. 83, 1848-1853 (1998).
    [CrossRef]
  7. H. Uchiki and T. Kobayashi, "New determination method of electro-optic constants and relevant nonlinear susceptibilities and its application to doped polymer," J. Appl. Phys. 64, 2625-2629 (1988).
    [CrossRef]
  8. D. Morichère, P.-A. Chollet, W. Fleming, M. Jurich, B. A. Smith, and J. D. Swalen, "Electro-optic effects in two tolane side-chain nonlinear-optical polymers: comparison between measured coefficients and second-harmonic generation," J. Opt. Soc. Am. B 10, 1894-1900 (1993).
    [CrossRef]
  9. L. M. Hayden, G. F. Sauter, F. R. Ore, P. L. Pasillas, J. M. Hoover, G. A. Lindsay, and R. A. Henry, "Second-order nonlinear optical measurements in guest-host and side-chain polymers," J. Appl. Phys. 68, 456-465 (1990).
    [CrossRef]
  10. S. Kalluri, S. Garner, M. Ziari, W. H. Steier, Y. Shi, and L. R. Dalton, "Simple two-slit interference electro-optic coefficients measurement technique and efficient coplanar electrode poling of polymer thin films," Appl. Phys. Lett. 69, 275-277 (1996).
    [CrossRef]
  11. C. C. Teng and H. T. Man, "Simple reflection technique for measuring the electro-optic coefficient of poled polymers," Appl. Phys. Lett. 56, 1734-1736 (1990).
    [CrossRef]
  12. J. S. Schildkraut, "Determination of the electro optic coefficient of a poled polymer film," Appl. Opt. 29, 2839-2841 (1990).
    [CrossRef] [PubMed]
  13. K. Clays and J. S. Schildkraut, "Dispersion of the complex electro-optic coefficient and electrochromic effects in poled polymer films," J. Opt. Soc. Am. B 9, 2274-2282 (1992).
    [CrossRef]
  14. P. M. Lundquist, M. Jurich, J.-F. Wang, H. Zhou, T. J. Marks, and G. K. Wong, "Electro-optic characterization of poled-polymer films in transmission," App. Phys. Lett. 69, 901-903 (1996).
    [CrossRef]
  15. F. Michelotti, A. Belardini, M. C. Larciprete, M. Bertolotti, A. Rousseau, A. Ratsimihety, G. Schoer, and J. Mueller, "Measurement of the electro-optic properties of poled polymers at λ = 1.55 μm by means of sandwich structures with zinc oxide transparent electrode," Appl. Phys. Lett. 83, 4477-4479 (2003).
    [CrossRef]
  16. Y. Levy, M. Dumont, E. Chastaing, P. Robin, P.-A. Chollet, G. Gadret, and F. Kajzar, "Reflection method for electro-optical coefficient determination in stratified thin film structures," Mol. Cryst. Liq. Cryst. Sci. Technol. Sect. B 4, 1-19 (1993).
  17. P.-A. Chollet, G. Gadret, F. Kajzar and P. Raimond, "Electro-optic coefficient determination in stratified organized molecular thin films: application to poled polymers," Thin Solid Films,  242, 132-138 (1994).
    [CrossRef]
  18. G. Khanarian, J. Sounik, D. Allen, S. F. Shu, C. Walton, H. Goldberg and J. B. Stamatoff, "Electro-optic characterization of nonlinear-optical guest-host films and polymers," J. Opt. Soc. Am. B 13, 1927-1934 (1996)
    [CrossRef]
  19. S. H. Han and J. W. Wu, "Single-beam polarization interferometry measurement of the linear electro-optic effect in poled polymer films with a reflection configuration," J. Opt. Soc. Am. B 14, 1131-1137 (1997).
    [CrossRef]
  20. F. Michelotti, G. Nicolao, F. Tesi, and M. Bertolotti, "On the measurement of the electro-optic properties of poled side-chain copolymer films with a modified Teng-Man technique," Chem. Phys. 245, 311-325 (1999).
    [CrossRef]
  21. Y. Shuto and M. Amano, "Reflection measurement technique of electro-optic coefficients in lithium niobate crystals and poled polymer films," J. Appl. Phys. 77, 4632-4638 (1995).
    [CrossRef]
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    [CrossRef]
  24. 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, 1577-1588 (2005).
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  26. For a discussion of wave impedances, see, for example, Ramo, Whinnery, and Van Duzer, Fields and Waves in Communication Electronics, (John Wiley and Sons, New York, 1965), Chap. 6.
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  28. R. A. Synowicki, "Spectroscopic ellipsometry characterization of indium tin oxide film microstructure and optical constants," Thin Solid Films,  313-314, 394-397 (1998).

2005

2004

Y.-P. Wang, J.-P. Chen, X.-W. Li, J.-X. Hong, X.-H. Zhang, J.-H. Zhou, and A.-L. Ye, "Measuring eletro-optic coefficients of poled polymers using fiber-optic Mach-Zehnder interferometer," Appl. Phys. Lett. 85, 5102-5103 (2004).
[CrossRef]

2003

F. Michelotti, A. Belardini, M. C. Larciprete, M. Bertolotti, A. Rousseau, A. Ratsimihety, G. Schoer, and J. Mueller, "Measurement of the electro-optic properties of poled polymers at λ = 1.55 μm by means of sandwich structures with zinc oxide transparent electrode," Appl. Phys. Lett. 83, 4477-4479 (2003).
[CrossRef]

1999

F. Michelotti, G. Nicolao, F. Tesi, and M. Bertolotti, "On the measurement of the electro-optic properties of poled side-chain copolymer films with a modified Teng-Man technique," Chem. Phys. 245, 311-325 (1999).
[CrossRef]

1998

R. A. Synowicki, "Spectroscopic ellipsometry characterization of indium tin oxide film microstructure and optical constants," Thin Solid Films,  313-314, 394-397 (1998).

M. J. Shin, H. R. Cho, S. H. Han, and J. W. Wu, "Analysis of Mach-Zehnder interferometry measurement of the Pockels coefficients in a poled polymer film with a reflection configuration," J. Appl. Phys. 83, 1848-1853 (1998).
[CrossRef]

1997

1996

G. Khanarian, J. Sounik, D. Allen, S. F. Shu, C. Walton, H. Goldberg and J. B. Stamatoff, "Electro-optic characterization of nonlinear-optical guest-host films and polymers," J. Opt. Soc. Am. B 13, 1927-1934 (1996)
[CrossRef]

S. Kalluri, S. Garner, M. Ziari, W. H. Steier, Y. Shi, and L. R. Dalton, "Simple two-slit interference electro-optic coefficients measurement technique and efficient coplanar electrode poling of polymer thin films," Appl. Phys. Lett. 69, 275-277 (1996).
[CrossRef]

P. M. Lundquist, M. Jurich, J.-F. Wang, H. Zhou, T. J. Marks, and G. K. Wong, "Electro-optic characterization of poled-polymer films in transmission," App. Phys. Lett. 69, 901-903 (1996).
[CrossRef]

1995

Y. Shuto and M. Amano, "Reflection measurement technique of electro-optic coefficients in lithium niobate crystals and poled polymer films," J. Appl. Phys. 77, 4632-4638 (1995).
[CrossRef]

F. Wang, E. Furman, and G.H. Haertling, "Electro-optic measurements of thin-film materials by means of reflection differential ellipsometry," J. Appl. Phys. 78, 9-15 (1995).
[CrossRef]

1994

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

P.-A. Chollet, G. Gadret, F. Kajzar and P. Raimond, "Electro-optic coefficient determination in stratified organized molecular thin films: application to poled polymers," Thin Solid Films,  242, 132-138 (1994).
[CrossRef]

R. A. Norwood, M. G. Kuzyk, and R. A. Keosian, "Electro-optic tensor ratio determination of side-chain copolymers with electro-optic interferometry," J. Appl. Phys. 75, 1869-1874 (1994).
[CrossRef]

1993

Y. Levy, M. Dumont, E. Chastaing, P. Robin, P.-A. Chollet, G. Gadret, and F. Kajzar, "Reflection method for electro-optical coefficient determination in stratified thin film structures," Mol. Cryst. Liq. Cryst. Sci. Technol. Sect. B 4, 1-19 (1993).

D. Morichère, P.-A. Chollet, W. Fleming, M. Jurich, B. A. Smith, and J. D. Swalen, "Electro-optic effects in two tolane side-chain nonlinear-optical polymers: comparison between measured coefficients and second-harmonic generation," J. Opt. Soc. Am. B 10, 1894-1900 (1993).
[CrossRef]

1992

1990

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

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

L. M. Hayden, G. F. Sauter, F. R. Ore, P. L. Pasillas, J. M. Hoover, G. A. Lindsay, and R. A. Henry, "Second-order nonlinear optical measurements in guest-host and side-chain polymers," J. Appl. Phys. 68, 456-465 (1990).
[CrossRef]

1989

1988

H. Uchiki and T. Kobayashi, "New determination method of electro-optic constants and relevant nonlinear susceptibilities and its application to doped polymer," J. Appl. Phys. 64, 2625-2629 (1988).
[CrossRef]

Allen, D.

Amano, M.

Y. Shuto and M. Amano, "Reflection measurement technique of electro-optic coefficients in lithium niobate crystals and poled polymer films," J. Appl. Phys. 77, 4632-4638 (1995).
[CrossRef]

Belardini, A.

F. Michelotti, A. Belardini, M. C. Larciprete, M. Bertolotti, A. Rousseau, A. Ratsimihety, G. Schoer, and J. Mueller, "Measurement of the electro-optic properties of poled polymers at λ = 1.55 μm by means of sandwich structures with zinc oxide transparent electrode," Appl. Phys. Lett. 83, 4477-4479 (2003).
[CrossRef]

Bertolotti, M.

F. Michelotti, A. Belardini, M. C. Larciprete, M. Bertolotti, A. Rousseau, A. Ratsimihety, G. Schoer, and J. Mueller, "Measurement of the electro-optic properties of poled polymers at λ = 1.55 μm by means of sandwich structures with zinc oxide transparent electrode," Appl. Phys. Lett. 83, 4477-4479 (2003).
[CrossRef]

F. Michelotti, G. Nicolao, F. Tesi, and M. Bertolotti, "On the measurement of the electro-optic properties of poled side-chain copolymer films with a modified Teng-Man technique," Chem. Phys. 245, 311-325 (1999).
[CrossRef]

Burland, D. M.

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

Chastaing, E.

Y. Levy, M. Dumont, E. Chastaing, P. Robin, P.-A. Chollet, G. Gadret, and F. Kajzar, "Reflection method for electro-optical coefficient determination in stratified thin film structures," Mol. Cryst. Liq. Cryst. Sci. Technol. Sect. B 4, 1-19 (1993).

Chen, J.-P.

Y.-P. Wang, J.-P. Chen, X.-W. Li, J.-X. Hong, X.-H. Zhang, J.-H. Zhou, and A.-L. Ye, "Measuring eletro-optic coefficients of poled polymers using fiber-optic Mach-Zehnder interferometer," Appl. Phys. Lett. 85, 5102-5103 (2004).
[CrossRef]

Cho, H. R.

M. J. Shin, H. R. Cho, S. H. Han, and J. W. Wu, "Analysis of Mach-Zehnder interferometry measurement of the Pockels coefficients in a poled polymer film with a reflection configuration," J. Appl. Phys. 83, 1848-1853 (1998).
[CrossRef]

Chollet, P.-A.

P.-A. Chollet, G. Gadret, F. Kajzar and P. Raimond, "Electro-optic coefficient determination in stratified organized molecular thin films: application to poled polymers," Thin Solid Films,  242, 132-138 (1994).
[CrossRef]

D. Morichère, P.-A. Chollet, W. Fleming, M. Jurich, B. A. Smith, and J. D. Swalen, "Electro-optic effects in two tolane side-chain nonlinear-optical polymers: comparison between measured coefficients and second-harmonic generation," J. Opt. Soc. Am. B 10, 1894-1900 (1993).
[CrossRef]

Y. Levy, M. Dumont, E. Chastaing, P. Robin, P.-A. Chollet, G. Gadret, and F. Kajzar, "Reflection method for electro-optical coefficient determination in stratified thin film structures," Mol. Cryst. Liq. Cryst. Sci. Technol. Sect. B 4, 1-19 (1993).

Clays, K.

Dalton, L. R.

S. Kalluri, S. Garner, M. Ziari, W. H. Steier, Y. Shi, and L. R. Dalton, "Simple two-slit interference electro-optic coefficients measurement technique and efficient coplanar electrode poling of polymer thin films," Appl. Phys. Lett. 69, 275-277 (1996).
[CrossRef]

Dienes, A.

Dumont, M.

Y. Levy, M. Dumont, E. Chastaing, P. Robin, P.-A. Chollet, G. Gadret, and F. Kajzar, "Reflection method for electro-optical coefficient determination in stratified thin film structures," Mol. Cryst. Liq. Cryst. Sci. Technol. Sect. B 4, 1-19 (1993).

Fleming, W.

Furman, E.

F. Wang, E. Furman, and G.H. Haertling, "Electro-optic measurements of thin-film materials by means of reflection differential ellipsometry," J. Appl. Phys. 78, 9-15 (1995).
[CrossRef]

Gadret, G.

P.-A. Chollet, G. Gadret, F. Kajzar and P. Raimond, "Electro-optic coefficient determination in stratified organized molecular thin films: application to poled polymers," Thin Solid Films,  242, 132-138 (1994).
[CrossRef]

Y. Levy, M. Dumont, E. Chastaing, P. Robin, P.-A. Chollet, G. Gadret, and F. Kajzar, "Reflection method for electro-optical coefficient determination in stratified thin film structures," Mol. Cryst. Liq. Cryst. Sci. Technol. Sect. B 4, 1-19 (1993).

Garner, S.

S. Kalluri, S. Garner, M. Ziari, W. H. Steier, Y. Shi, and L. R. Dalton, "Simple two-slit interference electro-optic coefficients measurement technique and efficient coplanar electrode poling of polymer thin films," Appl. Phys. Lett. 69, 275-277 (1996).
[CrossRef]

Goldberg, H.

Guo, D.

Haertling, G.H.

F. Wang, E. Furman, and G.H. Haertling, "Electro-optic measurements of thin-film materials by means of reflection differential ellipsometry," J. Appl. Phys. 78, 9-15 (1995).
[CrossRef]

Han, S. H.

M. J. Shin, H. R. Cho, S. H. Han, and J. W. Wu, "Analysis of Mach-Zehnder interferometry measurement of the Pockels coefficients in a poled polymer film with a reflection configuration," J. Appl. Phys. 83, 1848-1853 (1998).
[CrossRef]

S. H. Han and J. W. Wu, "Single-beam polarization interferometry measurement of the linear electro-optic effect in poled polymer films with a reflection configuration," J. Opt. Soc. Am. B 14, 1131-1137 (1997).
[CrossRef]

Hayden, L. M.

L. M. Hayden, G. F. Sauter, F. R. Ore, P. L. Pasillas, J. M. Hoover, G. A. Lindsay, and R. A. Henry, "Second-order nonlinear optical measurements in guest-host and side-chain polymers," J. Appl. Phys. 68, 456-465 (1990).
[CrossRef]

Henry, R. A.

L. M. Hayden, G. F. Sauter, F. R. Ore, P. L. Pasillas, J. M. Hoover, G. A. Lindsay, and R. A. Henry, "Second-order nonlinear optical measurements in guest-host and side-chain polymers," J. Appl. Phys. 68, 456-465 (1990).
[CrossRef]

Higgins, B. G.

Hong, J.-X.

Y.-P. Wang, J.-P. Chen, X.-W. Li, J.-X. Hong, X.-H. Zhang, J.-H. Zhou, and A.-L. Ye, "Measuring eletro-optic coefficients of poled polymers using fiber-optic Mach-Zehnder interferometer," Appl. Phys. Lett. 85, 5102-5103 (2004).
[CrossRef]

Hoover, J. M.

L. M. Hayden, G. F. Sauter, F. R. Ore, P. L. Pasillas, J. M. Hoover, G. A. Lindsay, and R. A. Henry, "Second-order nonlinear optical measurements in guest-host and side-chain polymers," J. Appl. Phys. 68, 456-465 (1990).
[CrossRef]

Jurich, M.

Kajzar, F.

P.-A. Chollet, G. Gadret, F. Kajzar and P. Raimond, "Electro-optic coefficient determination in stratified organized molecular thin films: application to poled polymers," Thin Solid Films,  242, 132-138 (1994).
[CrossRef]

Y. Levy, M. Dumont, E. Chastaing, P. Robin, P.-A. Chollet, G. Gadret, and F. Kajzar, "Reflection method for electro-optical coefficient determination in stratified thin film structures," Mol. Cryst. Liq. Cryst. Sci. Technol. Sect. B 4, 1-19 (1993).

Kalluri, S.

S. Kalluri, S. Garner, M. Ziari, W. H. Steier, Y. Shi, and L. R. Dalton, "Simple two-slit interference electro-optic coefficients measurement technique and efficient coplanar electrode poling of polymer thin films," Appl. Phys. Lett. 69, 275-277 (1996).
[CrossRef]

Keosian, R. A.

R. A. Norwood, M. G. Kuzyk, and R. A. Keosian, "Electro-optic tensor ratio determination of side-chain copolymers with electro-optic interferometry," J. Appl. Phys. 75, 1869-1874 (1994).
[CrossRef]

Khanarian, G.

Knoesen, A.

Kobayashi, T.

H. Uchiki and T. Kobayashi, "New determination method of electro-optic constants and relevant nonlinear susceptibilities and its application to doped polymer," J. Appl. Phys. 64, 2625-2629 (1988).
[CrossRef]

Kowel, S. T.

Kuzyk, M. G.

R. A. Norwood, M. G. Kuzyk, and R. A. Keosian, "Electro-optic tensor ratio determination of side-chain copolymers with electro-optic interferometry," J. Appl. Phys. 75, 1869-1874 (1994).
[CrossRef]

Larciprete, M. C.

F. Michelotti, A. Belardini, M. C. Larciprete, M. Bertolotti, A. Rousseau, A. Ratsimihety, G. Schoer, and J. Mueller, "Measurement of the electro-optic properties of poled polymers at λ = 1.55 μm by means of sandwich structures with zinc oxide transparent electrode," Appl. Phys. Lett. 83, 4477-4479 (2003).
[CrossRef]

Levy, Y.

Y. Levy, M. Dumont, E. Chastaing, P. Robin, P.-A. Chollet, G. Gadret, and F. Kajzar, "Reflection method for electro-optical coefficient determination in stratified thin film structures," Mol. Cryst. Liq. Cryst. Sci. Technol. Sect. B 4, 1-19 (1993).

Li, X.-W.

Y.-P. Wang, J.-P. Chen, X.-W. Li, J.-X. Hong, X.-H. Zhang, J.-H. Zhou, and A.-L. Ye, "Measuring eletro-optic coefficients of poled polymers using fiber-optic Mach-Zehnder interferometer," Appl. Phys. Lett. 85, 5102-5103 (2004).
[CrossRef]

Lin, R.

Lindsay, G. A.

L. M. Hayden, G. F. Sauter, F. R. Ore, P. L. Pasillas, J. M. Hoover, G. A. Lindsay, and R. A. Henry, "Second-order nonlinear optical measurements in guest-host and side-chain polymers," J. Appl. Phys. 68, 456-465 (1990).
[CrossRef]

Lundquist, P. M.

P. M. Lundquist, M. Jurich, J.-F. Wang, H. Zhou, T. J. Marks, and G. K. Wong, "Electro-optic characterization of poled-polymer films in transmission," App. Phys. Lett. 69, 901-903 (1996).
[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, 1734-1736 (1990).
[CrossRef]

Marks, T. J.

P. M. Lundquist, M. Jurich, J.-F. Wang, H. Zhou, T. J. Marks, and G. K. Wong, "Electro-optic characterization of poled-polymer films in transmission," App. Phys. Lett. 69, 901-903 (1996).
[CrossRef]

Michelotti, F.

F. Michelotti, A. Belardini, M. C. Larciprete, M. Bertolotti, A. Rousseau, A. Ratsimihety, G. Schoer, and J. Mueller, "Measurement of the electro-optic properties of poled polymers at λ = 1.55 μm by means of sandwich structures with zinc oxide transparent electrode," Appl. Phys. Lett. 83, 4477-4479 (2003).
[CrossRef]

F. Michelotti, G. Nicolao, F. Tesi, and M. Bertolotti, "On the measurement of the electro-optic properties of poled side-chain copolymer films with a modified Teng-Man technique," Chem. Phys. 245, 311-325 (1999).
[CrossRef]

Miller, R. D.

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

Morichère, D.

Mortazavi, M. A.

Mueller, J.

F. Michelotti, A. Belardini, M. C. Larciprete, M. Bertolotti, A. Rousseau, A. Ratsimihety, G. Schoer, and J. Mueller, "Measurement of the electro-optic properties of poled polymers at λ = 1.55 μm by means of sandwich structures with zinc oxide transparent electrode," Appl. Phys. Lett. 83, 4477-4479 (2003).
[CrossRef]

Nicolao, G.

F. Michelotti, G. Nicolao, F. Tesi, and M. Bertolotti, "On the measurement of the electro-optic properties of poled side-chain copolymer films with a modified Teng-Man technique," Chem. Phys. 245, 311-325 (1999).
[CrossRef]

Norwood, R. A.

R. A. Norwood, M. G. Kuzyk, and R. A. Keosian, "Electro-optic tensor ratio determination of side-chain copolymers with electro-optic interferometry," J. Appl. Phys. 75, 1869-1874 (1994).
[CrossRef]

Ore, F. R.

L. M. Hayden, G. F. Sauter, F. R. Ore, P. L. Pasillas, J. M. Hoover, G. A. Lindsay, and R. A. Henry, "Second-order nonlinear optical measurements in guest-host and side-chain polymers," J. Appl. Phys. 68, 456-465 (1990).
[CrossRef]

Pasillas, P. L.

L. M. Hayden, G. F. Sauter, F. R. Ore, P. L. Pasillas, J. M. Hoover, G. A. Lindsay, and R. A. Henry, "Second-order nonlinear optical measurements in guest-host and side-chain polymers," J. Appl. Phys. 68, 456-465 (1990).
[CrossRef]

Raimond, P.

P.-A. Chollet, G. Gadret, F. Kajzar and P. Raimond, "Electro-optic coefficient determination in stratified organized molecular thin films: application to poled polymers," Thin Solid Films,  242, 132-138 (1994).
[CrossRef]

Ratsimihety, A.

F. Michelotti, A. Belardini, M. C. Larciprete, M. Bertolotti, A. Rousseau, A. Ratsimihety, G. Schoer, and J. Mueller, "Measurement of the electro-optic properties of poled polymers at λ = 1.55 μm by means of sandwich structures with zinc oxide transparent electrode," Appl. Phys. Lett. 83, 4477-4479 (2003).
[CrossRef]

Robin, P.

Y. Levy, M. Dumont, E. Chastaing, P. Robin, P.-A. Chollet, G. Gadret, and F. Kajzar, "Reflection method for electro-optical coefficient determination in stratified thin film structures," Mol. Cryst. Liq. Cryst. Sci. Technol. Sect. B 4, 1-19 (1993).

Rousseau, A.

F. Michelotti, A. Belardini, M. C. Larciprete, M. Bertolotti, A. Rousseau, A. Ratsimihety, G. Schoer, and J. Mueller, "Measurement of the electro-optic properties of poled polymers at λ = 1.55 μm by means of sandwich structures with zinc oxide transparent electrode," Appl. Phys. Lett. 83, 4477-4479 (2003).
[CrossRef]

Sauter, G. F.

L. M. Hayden, G. F. Sauter, F. R. Ore, P. L. Pasillas, J. M. Hoover, G. A. Lindsay, and R. A. Henry, "Second-order nonlinear optical measurements in guest-host and side-chain polymers," J. Appl. Phys. 68, 456-465 (1990).
[CrossRef]

Schildkraut, J. S.

Schoer, G.

F. Michelotti, A. Belardini, M. C. Larciprete, M. Bertolotti, A. Rousseau, A. Ratsimihety, G. Schoer, and J. Mueller, "Measurement of the electro-optic properties of poled polymers at λ = 1.55 μm by means of sandwich structures with zinc oxide transparent electrode," Appl. Phys. Lett. 83, 4477-4479 (2003).
[CrossRef]

Shi, Y.

S. Kalluri, S. Garner, M. Ziari, W. H. Steier, Y. Shi, and L. R. Dalton, "Simple two-slit interference electro-optic coefficients measurement technique and efficient coplanar electrode poling of polymer thin films," Appl. Phys. Lett. 69, 275-277 (1996).
[CrossRef]

Shin, M. J.

M. J. Shin, H. R. Cho, S. H. Han, and J. W. Wu, "Analysis of Mach-Zehnder interferometry measurement of the Pockels coefficients in a poled polymer film with a reflection configuration," J. Appl. Phys. 83, 1848-1853 (1998).
[CrossRef]

Shu, S. F.

Shuto, Y.

Y. Shuto and M. Amano, "Reflection measurement technique of electro-optic coefficients in lithium niobate crystals and poled polymer films," J. Appl. Phys. 77, 4632-4638 (1995).
[CrossRef]

Smith, B. A.

Sounik, J.

Stamatoff, J. B.

Steier, W. H.

S. Kalluri, S. Garner, M. Ziari, W. H. Steier, Y. Shi, and L. R. Dalton, "Simple two-slit interference electro-optic coefficients measurement technique and efficient coplanar electrode poling of polymer thin films," Appl. Phys. Lett. 69, 275-277 (1996).
[CrossRef]

Swalen, J. D.

Synowicki, R. A.

R. A. Synowicki, "Spectroscopic ellipsometry characterization of indium tin oxide film microstructure and optical constants," Thin Solid Films,  313-314, 394-397 (1998).

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, 1734-1736 (1990).
[CrossRef]

Tesi, F.

F. Michelotti, G. Nicolao, F. Tesi, and M. Bertolotti, "On the measurement of the electro-optic properties of poled side-chain copolymer films with a modified Teng-Man technique," Chem. Phys. 245, 311-325 (1999).
[CrossRef]

Uchiki, H.

H. Uchiki and T. Kobayashi, "New determination method of electro-optic constants and relevant nonlinear susceptibilities and its application to doped polymer," J. Appl. Phys. 64, 2625-2629 (1988).
[CrossRef]

Walsh, C. A.

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

Walton, C.

Wang, F.

F. Wang, E. Furman, and G.H. Haertling, "Electro-optic measurements of thin-film materials by means of reflection differential ellipsometry," J. Appl. Phys. 78, 9-15 (1995).
[CrossRef]

Wang, J.-F.

P. M. Lundquist, M. Jurich, J.-F. Wang, H. Zhou, T. J. Marks, and G. K. Wong, "Electro-optic characterization of poled-polymer films in transmission," App. Phys. Lett. 69, 901-903 (1996).
[CrossRef]

Wang, W.

Wang, Y.-P.

Y.-P. Wang, J.-P. Chen, X.-W. Li, J.-X. Hong, X.-H. Zhang, J.-H. Zhou, and A.-L. Ye, "Measuring eletro-optic coefficients of poled polymers using fiber-optic Mach-Zehnder interferometer," Appl. Phys. Lett. 85, 5102-5103 (2004).
[CrossRef]

Wong, G. K.

P. M. Lundquist, M. Jurich, J.-F. Wang, H. Zhou, T. J. Marks, and G. K. Wong, "Electro-optic characterization of poled-polymer films in transmission," App. Phys. Lett. 69, 901-903 (1996).
[CrossRef]

Wu, J. W.

M. J. Shin, H. R. Cho, S. H. Han, and J. W. Wu, "Analysis of Mach-Zehnder interferometry measurement of the Pockels coefficients in a poled polymer film with a reflection configuration," J. Appl. Phys. 83, 1848-1853 (1998).
[CrossRef]

S. H. Han and J. W. Wu, "Single-beam polarization interferometry measurement of the linear electro-optic effect in poled polymer films with a reflection configuration," J. Opt. Soc. Am. B 14, 1131-1137 (1997).
[CrossRef]

Ye, A.-L.

Y.-P. Wang, J.-P. Chen, X.-W. Li, J.-X. Hong, X.-H. Zhang, J.-H. Zhou, and A.-L. Ye, "Measuring eletro-optic coefficients of poled polymers using fiber-optic Mach-Zehnder interferometer," Appl. Phys. Lett. 85, 5102-5103 (2004).
[CrossRef]

Zhang, X.-H.

Y.-P. Wang, J.-P. Chen, X.-W. Li, J.-X. Hong, X.-H. Zhang, J.-H. Zhou, and A.-L. Ye, "Measuring eletro-optic coefficients of poled polymers using fiber-optic Mach-Zehnder interferometer," Appl. Phys. Lett. 85, 5102-5103 (2004).
[CrossRef]

Zhou, H.

P. M. Lundquist, M. Jurich, J.-F. Wang, H. Zhou, T. J. Marks, and G. K. Wong, "Electro-optic characterization of poled-polymer films in transmission," App. Phys. Lett. 69, 901-903 (1996).
[CrossRef]

Zhou, J.-H.

Y.-P. Wang, J.-P. Chen, X.-W. Li, J.-X. Hong, X.-H. Zhang, J.-H. Zhou, and A.-L. Ye, "Measuring eletro-optic coefficients of poled polymers using fiber-optic Mach-Zehnder interferometer," Appl. Phys. Lett. 85, 5102-5103 (2004).
[CrossRef]

Ziari, M.

S. Kalluri, S. Garner, M. Ziari, W. H. Steier, Y. Shi, and L. R. Dalton, "Simple two-slit interference electro-optic coefficients measurement technique and efficient coplanar electrode poling of polymer thin films," Appl. Phys. Lett. 69, 275-277 (1996).
[CrossRef]

App. Phys. Lett.

P. M. Lundquist, M. Jurich, J.-F. Wang, H. Zhou, T. J. Marks, and G. K. Wong, "Electro-optic characterization of poled-polymer films in transmission," App. Phys. Lett. 69, 901-903 (1996).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

F. Michelotti, A. Belardini, M. C. Larciprete, M. Bertolotti, A. Rousseau, A. Ratsimihety, G. Schoer, and J. Mueller, "Measurement of the electro-optic properties of poled polymers at λ = 1.55 μm by means of sandwich structures with zinc oxide transparent electrode," Appl. Phys. Lett. 83, 4477-4479 (2003).
[CrossRef]

Y.-P. Wang, J.-P. Chen, X.-W. Li, J.-X. Hong, X.-H. Zhang, J.-H. Zhou, and A.-L. Ye, "Measuring eletro-optic coefficients of poled polymers using fiber-optic Mach-Zehnder interferometer," Appl. Phys. Lett. 85, 5102-5103 (2004).
[CrossRef]

S. Kalluri, S. Garner, M. Ziari, W. H. Steier, Y. Shi, and L. R. Dalton, "Simple two-slit interference electro-optic coefficients measurement technique and efficient coplanar electrode poling of polymer thin films," Appl. Phys. Lett. 69, 275-277 (1996).
[CrossRef]

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

Chem. Phys.

F. Michelotti, G. Nicolao, F. Tesi, and M. Bertolotti, "On the measurement of the electro-optic properties of poled side-chain copolymer films with a modified Teng-Man technique," Chem. Phys. 245, 311-325 (1999).
[CrossRef]

Chem. Rev.

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

J. Appl. Phys.

R. A. Norwood, M. G. Kuzyk, and R. A. Keosian, "Electro-optic tensor ratio determination of side-chain copolymers with electro-optic interferometry," J. Appl. Phys. 75, 1869-1874 (1994).
[CrossRef]

Y. Shuto and M. Amano, "Reflection measurement technique of electro-optic coefficients in lithium niobate crystals and poled polymer films," J. Appl. Phys. 77, 4632-4638 (1995).
[CrossRef]

F. Wang, E. Furman, and G.H. Haertling, "Electro-optic measurements of thin-film materials by means of reflection differential ellipsometry," J. Appl. Phys. 78, 9-15 (1995).
[CrossRef]

M. J. Shin, H. R. Cho, S. H. Han, and J. W. Wu, "Analysis of Mach-Zehnder interferometry measurement of the Pockels coefficients in a poled polymer film with a reflection configuration," J. Appl. Phys. 83, 1848-1853 (1998).
[CrossRef]

H. Uchiki and T. Kobayashi, "New determination method of electro-optic constants and relevant nonlinear susceptibilities and its application to doped polymer," J. Appl. Phys. 64, 2625-2629 (1988).
[CrossRef]

L. M. Hayden, G. F. Sauter, F. R. Ore, P. L. Pasillas, J. M. Hoover, G. A. Lindsay, and R. A. Henry, "Second-order nonlinear optical measurements in guest-host and side-chain polymers," J. Appl. Phys. 68, 456-465 (1990).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Mol. Cryst. Liq. Cryst. Sci. Technol. Sect. B

Y. Levy, M. Dumont, E. Chastaing, P. Robin, P.-A. Chollet, G. Gadret, and F. Kajzar, "Reflection method for electro-optical coefficient determination in stratified thin film structures," Mol. Cryst. Liq. Cryst. Sci. Technol. Sect. B 4, 1-19 (1993).

Thin Solid Films

P.-A. Chollet, G. Gadret, F. Kajzar and P. Raimond, "Electro-optic coefficient determination in stratified organized molecular thin films: application to poled polymers," Thin Solid Films,  242, 132-138 (1994).
[CrossRef]

R. A. Synowicki, "Spectroscopic ellipsometry characterization of indium tin oxide film microstructure and optical constants," Thin Solid Films,  313-314, 394-397 (1998).

Other

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

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

For a discussion of wave impedances, see, for example, Ramo, Whinnery, and Van Duzer, Fields and Waves in Communication Electronics, (John Wiley and Sons, New York, 1965), Chap. 6.

M. T. Heath, Scientific Computing, 2nd ed. (McGraw Hill, 2002).

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

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Figures (9)

Fig. 1.
Fig. 1.

Schematic of the experimental Teng-Man setup. L is laser, P polarizer, A aperture, S slit, SBC Soleil-Babinet Compensator, and PD photodetector.

Fig. 2.
Fig. 2.

The optical bias curve and modulated intensity obtained as a function of SBC retardation setting x for a representative set of experimental data on a film with r 33=42 pm/V and s 33=0.3 pm/V using a peak voltage of 4 V. Points 1, 2, 3, and 4 correspond to compensator settings such that Ψ sp +Ω=π/2, 3π/2, 0, and π.

Fig. 3.
Fig. 3.

Multilayered structures in a simple model (a) and a rigorous model (b). For simplicity, subscripts s and p in the reflection coefficients are omitted in (b).

Fig 4.
Fig 4.

Error percentage plot for varying film thickness when the refractive indices of film and TCO are matched with glass (n=1.5) at 1.3 µm wavelength. The positive and negative envelopes are proportional to ±1/d 4 with a negative offset.

Fig. 5.
Fig. 5.

Optical properties (n+) of (solid) a representative ITO (Abrisa Corporation) measured by ellipsometry and (dashed) a representative polymer film selected for the simulation. (real part : black, imaginary part : red).

Fig. 6.
Fig. 6.

Error plots by varying the thickness of the film for fixed indices of refraction of ITO and the film and two thicknesses of ITO, 100 nm (a–d) and 50 nm (e–h) at various wavelengths and 45° angle of incidence under assumption of γ=1/3. For plots (a) and (e), s 33/r 33=1 (black solid), and s 33/r 33=2 (red dashed). For the wavelengths other than 0.8 µm, values of s 33/r 33 between 0 and 0.1 produce curves that are indistinguishable on this scale. For (a) and (e), the errors approach -107% and -110%, respectively. For (b), (f), (c), (g), (d), and (h), the error extremes are -18% to 12%, -15% to 10%, -86% to 38%, -40% to 25%, -80% to 350%, and 37% to 63%, respectively.

Fig. 7.
Fig. 7.

Error contour plots at (a) λ=0.8 µm and s 33/r 33=1 (b) λ=1.3 µm and s 33/r 33=0.1 with thickness of ITO=100 nm, 45° angle of incidence, and γ=1/3. Each contour plot (a) and (b) shows asymptotic and cyclic behaviors with thickness of film irrespective of index of refraction of film, respectively.

Fig. 8.
Fig. 8.

EO coefficients r33 calculated by the simple model at various wavelengths and angles of incidence using thickness of film and ITO, 1.4 µm and 100 nm, respectively. EO coefficient r 33=100 pm/V was used for the simulation. The ratios s 33/r 33 are 2 at 0.8 µm and 0.1 at the other wavelength. Insets in (a) and (b) show the optical properties of ITO selected for the simulation. The crossover points of n and κ of ITO are around 1.54 and 1.92 µm in (a) and (b), respectively.

Fig. 9.
Fig. 9.

(a). δΨ sp and (b) δB/B versus angle of incidence at wavelengths 1.3 and 1.55 µm. The ratio s33/r 33=0.1 and film thickness=1.4 µm were used at both wavelengths for the ITO properties shown in Fig. 5.

Tables (1)

Tables Icon

Table 1. Parameter ranges to use the simple model with error less than ±20% at 45° angle of incidence.

Equations (66)

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

δ n ˜ μ = 1 2 n ˜ μ 3 ( r μ 3 + i s μ 3 ) E 3 ,
I dc = I o 4 r s e i Ω r p 2 = A + B sin 2 ( Ψ sp + Ω 2 ) ,
A = I o 4 ( r s r p ) 2 , B = I o r s r p ,
r s = r s e i Ψ s , r p = r p e i Ψ p ,
Ψ sp = Ψ s Ψ p .
I m = δ A + δ B sin 2 ( Ψ sp + Ω 2 ) + B 2 sin ( Ψ sp + Ω ) δ Ψ sp .
δ Ψ sp = I m ( π 2 ) I m ( 3 π 2 ) 2 I c ,
δ B B = I m ( π ) I m ( 0 ) 2 I c .
B ˜ I o r s r p * = B e i Ψ sp
δ B ˜ B ˜ = δ B B + i δ Ψ sp = δ r s r s + ( δ r p r p ) * .
δ B ˜ B ˜ = 1 r s r s n ˜ o δ n ˜ o + ( 1 r p r p n ˜ o δ n ˜ o + 1 r p r p n ˜ e δ n ˜ e ) * H r r 33 + i H s s 33 .
( δ Ψ sp δ B B ) = ( Im ( H r ) Re ( H s ) Re ( H r ) Im ( H s ) ) ( r 33 s 33 ) .
r s = e 2 i β s d , r p = e 2 i β p d ,
β s = k o n ˜ s cos ( θ ˜ s ) , β p = k o n ˜ p cos ( θ ˜ p ) .
ψ sp = 2 ( β sr β pr ) d ,
δ ψ sp = 2 ( δ β sr δ β pr ) d
δ b = 2 b ( δ β si + δ β pi ) d .
δ b ˜ b ˜ = δ b b + i δ ψ sp = 2 id [ β s n ˜ o δ n ˜ o ( β p n ˜ o δ n ˜ o + β p n ˜ e δ n ˜ e ) * ] h r r 33 + i h s s 33 ,
( δ ψ sp δ b b ) = ( Im ( h r ) Re ( h s ) Re ( h r ) Im ( h s ) ) ( r 33 s 33 ) .
h r = iV [ ( k o n ˜ o 3 n ˜ e n ˜ e 2 N 2 ) * γ k o n ˜ o 4 n ˜ o 2 N 2 γ + ( k o n ˜ o n ˜ e N 2 n ˜ e 2 N 2 ) * ]
h s = iV [ k o n ˜ o 4 n ˜ o 2 N 2 γ + ( k o n ˜ o 3 n ˜ e n ˜ e 2 N 2 ) * γ + ( k o n ˜ o n ˜ e N 2 n ˜ e 2 N 2 ) * ] ,
δ ψ sp = ( n o 4 n o 2 N 2 n o 3 n e n e 2 N 2 ) k o γ r 33 V + n o n e N 2 n e 2 N 2 k o r 33 V = Im ( h r ) · r 33
δ b b = ( n o 4 n o 2 N 2 + n o 3 n e n e 2 N 2 ) k o γ s 33 V + n o n e N 2 n e 2 N 2 k o s 33 V = Im ( h s ) · s 33 ,
δ ψ sp = n 2 N 2 n 2 N 2 ( 1 γ ) r 33 k o V = Im ( h r ) · r 33
δ b b = ( 2 n 4 n 2 N 2 n 2 N 2 γ s 33 + n 2 N 2 n 2 N 2 s 33 ) k 0 V = Im ( h s ) · s 33 .
r = r 23 + r ̂ 34 e 2 i β 3 d 3 1 + r 23 r ̂ 34 e 2 i β 3 d 3 r ̂ 34 = r 34 + r ̂ 45 e 2 i β 4 d 4 1 + r 34 r ̂ 45 e 2 i β 4 d 4 r ̂ 45 = r 45 + r 56 e 2 i β 5 d 5 1 + r 45 r 56 e 2 i β 5 d 5 ,
r jk = Z k Z j Z k + Z j
Z s = 1 n ˜ o 2 N 2 , Z p = 1 n ˜ o 1 ( N n ˜ e ) 2 .
( δ Ψ sp ( θ 1 ) δ Ψ sp ( θ n ) δ B B ( θ 1 ) δ B B ( θ n ) ) = ( Im [ H r ( θ 1 ) ] Re [ H s ( θ 1 ) ] Im [ H r ( θ n ) ] Re [ H s ( θ n ) ] Re [ H r ( θ 1 ) ] Im [ H s ( θ 1 ) ] Re [ H r ( θ n ) ] Im [ H s ( θ n ) ] ) ( r 33 s 33 ) P = M · x .
M · x = P QR · x = P R · x = Q T P P ̂ .
R · x = P ̂ ( R 2 × 2 R lower ) · x = ( P 2 × 1 P lower ) x = R 2 × 2 1 · P 2 × 1 ,
Error = r 33 SM r 33 r 33 .
Error = Im ( h s ) [ Im ( H r ) Im ( h r ) ] + Re ( h s ) [ Re ( H r ) Re ( h r ) ] Im ( h r ) Im ( h s ) + Re ( h r ) Re ( h s )
+ Im ( h s ) Re ( H s ) Re ( h s ) Im ( H s ) Im ( h r ) Im ( h s ) + Re ( h r ) Re ( h s ) s 33 r 33 .
r = r 45 e 2 i β 3 d 3 e 2 i β 4 d 4 ,
f 234 q = e 2 i β q 3 d 3 , f 345 q = e 2 i β q 4 d 4 , g 345 q = 1 ( r 45 q ) 2 e 2 i β q 4 d 4 , g 456 q = 1 .
H r V ( n 4 ) 3 2 d 4 [ K 1 cos ( 2 β 4 d 4 ) i K 2 sin ( 2 β 4 d 4 ) + K 3 ] iV ( n 4 ) 3 K 4 ,
K 1 = [ 1 ( r 45 s ) 2 ] γ r 45 s r 34 s n ˜ o + [ 1 ( r 45 p * ) 2 ] ( γ r 45 p * r 34 p n o + 1 r 45 p * r 34 p n ˜ e )
K 2 = [ 1 + ( r 45 s ) 2 ] γ r 45 s r 34 s n ˜ o [ 1 ( r 45 p * ) 2 ] ( γ r 45 p * r 34 p n o + 1 r 45 p * r 34 p n e )
K 3 = γ r 45 s r 45 s n ˜ o + γ r 45 p * r 45 p * n o + 1 r 45 p * r 45 p * n ˜ e
K 4 = γ β 4 s n ˜ o γ β 4 p n o β 4 p n ˜ e ,
Error = 1 2 d 4 K 4 [ Im ( K 1 ) cos ( 2 β 4 d 4 ) Re ( K 2 ) sin ( 2 β 4 d 4 ) + Im ( K 3 ) ] .
δ n μ = n μ 3 2 [ ( 1 3 κ μ 2 n μ 2 ) r μ 3 + ( κ μ 2 n μ 2 3 ) κ μ n μ s μ 3 ] E 3
δ κ μ = n μ 3 2 [ ( 1 3 κ μ 2 n μ 2 ) s μ 3 + ( κ μ 2 n μ 2 3 ) κ μ n μ r μ 3 ] E 3 .
δ n μ 1 2 n μ 3 ( r μ 3 3 κ μ n μ s μ 3 ) E 3
δ κ μ 1 2 n μ 3 ( s μ 3 + 3 κ μ n μ r μ 3 ) E 3 .
δ n μ 1 2 n μ 3 r μ 3 E 3 .
δ κ μ 1 2 n μ 3 s μ 3 E 3 .
n ˜ s sin θ ˜ s = n ˜ p sin θ ˜ p = sin θ N ,
1 n ˜ p 2 = cos 2 θ ˜ p n ˜ o 2 + sin 2 θ ˜ p n ˜ e 2 .
β s = k o n ˜ o 2 N 2
β p = k o n ˜ o n ˜ e n ˜ e 2 N 2 .
δ β s = k o n ˜ o n ˜ o 2 N 2 δ n ˜ o
δ β p = k o n ˜ e n ˜ e 2 N 2 δ n ˜ o + k o n ˜ o N 2 n ˜ e 2 n e 2 N 2 δ n ˜ e .
δ β s k o n o n o 2 N 2 ( δ n o + i δ κ o )
δ β p k o n e n o 2 N 2 ( δ n o + i δ κ o ) + k o N 2 n o n e 2 n e 2 N 2 ( δ n e + i δ κ e ) .
H r = V 2 d 4 [ γ r s r s n ˜ o n ˜ o 3 + γ ( 1 r p r p n ˜ o n ˜ o 3 ) * + ( 1 r p r p n ˜ e n ˜ e 3 ) * ] .
H s = V 2 d 4 [ γ r s r s n ˜ o n ˜ o 3 γ ( 1 r p r p n ˜ o n ˜ o 3 ) * ( 1 r p r p n ˜ e n ˜ e 3 ) * ] .
1 r r n ˜ o , e = 1 r r r ̂ 34 ( r ̂ 34 r 34 r 34 n ˜ o , e + r ̂ 34 r ̂ 45 r ̂ 45 r 45 r 45 n ˜ o , e + r ̂ 34 β 4 β 4 n ˜ o , e ) ,
1 r s r s n ˜ o = f 234 s r s ( g 345 s r 34 s n ˜ o + f 345 s g 456 s r 45 s n ˜ o + 2 i d 4 r ̂ 45 s f 345 s β s 4 n ˜ o ) ,
1 r p r p n ˜ o = f 234 p r p ( g 345 p r 34 p n ˜ o + f 345 p g 456 p r 45 p n ˜ o + 2 i d 4 r ̂ 45 p f 345 p β p 4 n ˜ o ) ,
1 r p r p n ˜ e = f 234 p r p ( g 345 p r 34 p n ˜ e + f 345 p g 456 p r 45 p n ˜ e + 2 i d 4 r ̂ 45 p f 345 p β p 4 n ˜ e ) ,
f ijk q = [ 1 ( r ij q ) 2 ] exp ( 2 i β qj d j ) [ 1 + r ij q r ̂ jk q exp ( 2 i β qj d j ) ] 2 , g ijk q = 1 ( r ̂ jk q ) 2 exp ( 4 i β qj d j ) [ 1 + r ij q r ̂ jk q exp ( 2 i β qj d j ) ] 2
r 34 s n ˜ o = 2 n ˜ o Z 3 s ( Z 4 s ) 3 ( Z 3 s + Z 4 z ) 2 , r 34 p n ˜ o = 2 Z 3 p Z 4 p n ˜ o ( Z 3 p + Z 4 p ) 2 , r 34 p n ˜ e = 2 Z 3 p N 2 n ˜ o 2 n ˜ e 3 Z 4 p ( Z 3 p + Z 4 p ) 2 ,
r 45 s n ˜ o = 2 n ˜ o ( Z 4 s ) 3 Z 5 s ( Z 4 s + Z 5 s ) 2 , r 45 p n ˜ o = 2 Z 4 p Z 5 p n ˜ o ( Z 4 p + Z 5 p ) 2 , r 45 p n ˜ e = 2 Z 5 p N 2 n ˜ o 2 n e 3 Z 4 p ( Z 4 p + Z 5 p ) 2
β s 4 n ˜ o = k o Z 4 s n ˜ o , β p 4 n ˜ o = k o Z 4 p n ˜ o , β p 4 n ˜ e = k o 1 Z 4 p ( N 2 n ˜ e 3 ) .

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