Abstract

We report a theoretical analysis of nonlinear optical sum-frequency generation from the bulk of a chiral liquid in the dipole approximation. In our theoretical formulation the circular birefringence effect of a chiral medium was properly taken into account. The angular dependence of the reflected and transmitted sum-frequency signals on the incident angles of two input beams was calculated to yield the optimal geometry for probing bulk chirality. We also derived a microscopic expression for the totally antisymmetric part of a second-order nonlinear optical susceptibility to elaborate unique features in the studies of chirality-related properties with sum-frequency generation.

© 1998 Optical Society of America

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References

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  1. I. Gutman, V. Babovic, and S. Jokic, “The origin of biomolecular chirality: the generalized Frank model with arbitrary initial conditions,” Chem. Phys. Lett. 144, 187–190 (1988).
    [CrossRef]
  2. See, for example, A. W. Hall, J. Hollingshurst, and J. W. Goodby, “Chiral and achiral calamitic liquid crystals for display applications,” in Handbook of Liquid Crystal Research, P. J. Collings and J. S. Patel, eds. (Oxford U. Press, Oxford, 1997), Chap. 2.
  3. E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
    [CrossRef]
  4. J. D. Byers, H. I. Yee, and J. M. Hicks, “A second harmonic generation analog of optical rotatory dispersion for the study of chiral monolayers,” J. Chem. Phys. 101, 6233–6241 (1994).
    [CrossRef]
  5. T. Petralli-Mallow, T. M. Wong, J. D. Byers, H. I. Yee, and J. M. Hicks, “Circular dichroism spectroscopy at interfaces: a surface second harmonic generation study,” J. Phys. Chem. 97, 1383–1388 (1993).
    [CrossRef]
  6. J. D. Byers, H. I. Yee, T. Petralli-Mallow, and J. M. Hicks, “Second-harmonic generation circular-dichroism spectroscopy from chiral monolayers,” Phys. Rev. B 49, 14, 643–16, 647 (1994).
    [CrossRef]
  7. J. D. Byers and J. M. Hicks, “Electronic spectral effects on chiral surface second harmonic generation,” Chem. Phys. Lett. 231, 216–224 (1994).
    [CrossRef]
  8. J. M. Hicks, T. Petralli-Mallow, and J. D. Byers, “Consequences of chirality in second-order non-linear spectroscopy at surfaces,” Discuss. Faraday Soc. 99, 341–357 (1994).
    [CrossRef]
  9. T. Verbiest, M. Kauranen, J. J. Maki, M. N. Teerenstra, A. J. Schouten, R. J. M. Nolte, and A. Persoons, “Linearly polarized probes of surface chirality,” J. Chem. Phys. 103, 8296–8298 (1995).
    [CrossRef]
  10. J. J. Maki, T. Verbiest, M. Kaurenen, S. V. Elshocht, and A. Persoons, “Comparison of linearly and circularly polarized probes of second-order optical activity of chiral surfaces,” J. Chem. Phys. 105, 767–772 (1996).
    [CrossRef]
  11. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap. 6.
  12. K. B. Eisenthal, “Equilibrium and dynamic processes at in terfaces by second harmonic and sum frequency generation,” Annu. Rev. Phys. Chem. 43, 627–661 (1992).
    [CrossRef]
  13. J. Y. Huang and Y. R. Shen, “Sum-frequency generation as a surface probe,” in Laser Spectroscopy and Photochemistry on Metal Surfaces, H. L. Dai and W. Ho, eds. (World Scientific, Singapore, 1995), Vol. 1, pp. 5–53.
  14. See, for example, G. Arfken, Mathematical Methods for Physicists (Academic, Orlando, Fla., 1985), Chap. 3.
  15. J. A. Giordmaine, “Nonlinear optical properties of liquids,” Phys. Rev. A 138, 1599–1606 (1965).
    [CrossRef]
  16. P. M. Rentzepis, J. A. Giordmaine, and K. W. Wecht, “Coherent optical mixing in optically active liquids,” Phys. Rev. Lett. 16, 762–794 (1966).
    [CrossRef]
  17. D. A. Kleinman, “Nonlinear dielectric polarization in optical media,” Phys. Rev. 126, 1977–1979 (1962).
    [CrossRef]
  18. S. N. Volkov, N. I. Koroteev, and V. A. Makarov, “Sum-frequency generation by reflection of light from the surface of a nonabsorbing isotropic and gyrotropic medium,” Quantum Electron. 25, 1183–1187 (1995).
    [CrossRef]
  19. N. I. Koroteev, V. A. Makarov, and S. N. Volkov, “Sum fre-quency generation by reflection of light from the surface of a chiral medium,” Nonlinear Opt. 17, 247–269 (1997).
  20. P. Pelet, and N. Engheta, “The theory of chirowaveguides,” IEEE Trans. Antennas Propag. 38, 90–98 (1990).
    [CrossRef]
  21. S. Bassiri, C. H. Papas, and N. Engheta, “Electromagnetic wave propagation through a dielectric-chiral interface and through a chiral slab,” J. Opt. Soc. Am. A 5, 1450–1459 (1988).
    [CrossRef]
  22. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap. 2.
  23. J. F. Nicoud and R. J. Twieg, “Organic EFISH hyperpolarizability data,” in Nonlinear Optical Properties of Organic Molecules and Crystals, D. S. Chemla and J. Zyss, eds. (Academic, Orlando, Fla., 1987), Vol. 2, pp. 255–267.
  24. Q. Du, R. Superfine, E. Freysz, and Y. R. Shen, “Vibrational spectroscopy of water at the vapor/water interface,” Phys. Rev. Lett. 70, 2313–2316 (1993).
    [CrossRef] [PubMed]
  25. C. D. Stanners, Q. Du, R. P. Chin, P. Cremer, G. A. Somorjai, Y.-R. Shen, “Polar ordering at the liquid-vapor interface of n-alcohols (C1–C8),” Chem. Phys. Lett. 232, 407–413 (1995).
    [CrossRef]
  26. P. J. Stephens and M. A. Lowe, “Vibrational circular dichroism,” Annu. Rev. Phys. Chem. 36, 213–241 (1985).
    [CrossRef]
  27. T. B. Freedman and L. A. Nafie, “Stereochemical aspects of vibrational optical activity,” in Topics in Stereochemistry, E. Eliel and S. Wilen, eds. (Wiley, New York, 1987), pp. 113–206.
  28. D. Barron, “Methyl group as a probe of chirality in Raman optical activity,” Nature (London) 255, 458–460 (1975).
    [CrossRef]
  29. M. Diem, E. Photos, H. Khouri, and L. A. Nafie, “Vibrational circular dichroism in amino acids and peptides: 3. Solution- and solid-phase spectra of alanine and serine,” J. Am. Chem. Soc. 101, 6829–6837 (1979).
    [CrossRef]
  30. N. I. Koroteev, “New schemes for nonlinear optical spectroscopy of solutions of chiral biological macromolecules,” JETP 79, 681–690 (1994).
  31. J. L. Finney, “Hydration processes in biological and macromolucular systems,” Discuss. Faraday Soc. 103, 1–18 (1996).
    [CrossRef]

1997 (1)

N. I. Koroteev, V. A. Makarov, and S. N. Volkov, “Sum fre-quency generation by reflection of light from the surface of a chiral medium,” Nonlinear Opt. 17, 247–269 (1997).

1996 (2)

J. J. Maki, T. Verbiest, M. Kaurenen, S. V. Elshocht, and A. Persoons, “Comparison of linearly and circularly polarized probes of second-order optical activity of chiral surfaces,” J. Chem. Phys. 105, 767–772 (1996).
[CrossRef]

J. L. Finney, “Hydration processes in biological and macromolucular systems,” Discuss. Faraday Soc. 103, 1–18 (1996).
[CrossRef]

1995 (3)

C. D. Stanners, Q. Du, R. P. Chin, P. Cremer, G. A. Somorjai, Y.-R. Shen, “Polar ordering at the liquid-vapor interface of n-alcohols (C1–C8),” Chem. Phys. Lett. 232, 407–413 (1995).
[CrossRef]

T. Verbiest, M. Kauranen, J. J. Maki, M. N. Teerenstra, A. J. Schouten, R. J. M. Nolte, and A. Persoons, “Linearly polarized probes of surface chirality,” J. Chem. Phys. 103, 8296–8298 (1995).
[CrossRef]

S. N. Volkov, N. I. Koroteev, and V. A. Makarov, “Sum-frequency generation by reflection of light from the surface of a nonabsorbing isotropic and gyrotropic medium,” Quantum Electron. 25, 1183–1187 (1995).
[CrossRef]

1994 (5)

J. D. Byers, H. I. Yee, T. Petralli-Mallow, and J. M. Hicks, “Second-harmonic generation circular-dichroism spectroscopy from chiral monolayers,” Phys. Rev. B 49, 14, 643–16, 647 (1994).
[CrossRef]

J. D. Byers and J. M. Hicks, “Electronic spectral effects on chiral surface second harmonic generation,” Chem. Phys. Lett. 231, 216–224 (1994).
[CrossRef]

J. M. Hicks, T. Petralli-Mallow, and J. D. Byers, “Consequences of chirality in second-order non-linear spectroscopy at surfaces,” Discuss. Faraday Soc. 99, 341–357 (1994).
[CrossRef]

J. D. Byers, H. I. Yee, and J. M. Hicks, “A second harmonic generation analog of optical rotatory dispersion for the study of chiral monolayers,” J. Chem. Phys. 101, 6233–6241 (1994).
[CrossRef]

N. I. Koroteev, “New schemes for nonlinear optical spectroscopy of solutions of chiral biological macromolecules,” JETP 79, 681–690 (1994).

1993 (2)

Q. Du, R. Superfine, E. Freysz, and Y. R. Shen, “Vibrational spectroscopy of water at the vapor/water interface,” Phys. Rev. Lett. 70, 2313–2316 (1993).
[CrossRef] [PubMed]

T. Petralli-Mallow, T. M. Wong, J. D. Byers, H. I. Yee, and J. M. Hicks, “Circular dichroism spectroscopy at interfaces: a surface second harmonic generation study,” J. Phys. Chem. 97, 1383–1388 (1993).
[CrossRef]

1992 (1)

K. B. Eisenthal, “Equilibrium and dynamic processes at in terfaces by second harmonic and sum frequency generation,” Annu. Rev. Phys. Chem. 43, 627–661 (1992).
[CrossRef]

1990 (1)

P. Pelet, and N. Engheta, “The theory of chirowaveguides,” IEEE Trans. Antennas Propag. 38, 90–98 (1990).
[CrossRef]

1988 (2)

I. Gutman, V. Babovic, and S. Jokic, “The origin of biomolecular chirality: the generalized Frank model with arbitrary initial conditions,” Chem. Phys. Lett. 144, 187–190 (1988).
[CrossRef]

S. Bassiri, C. H. Papas, and N. Engheta, “Electromagnetic wave propagation through a dielectric-chiral interface and through a chiral slab,” J. Opt. Soc. Am. A 5, 1450–1459 (1988).
[CrossRef]

1985 (1)

P. J. Stephens and M. A. Lowe, “Vibrational circular dichroism,” Annu. Rev. Phys. Chem. 36, 213–241 (1985).
[CrossRef]

1979 (1)

M. Diem, E. Photos, H. Khouri, and L. A. Nafie, “Vibrational circular dichroism in amino acids and peptides: 3. Solution- and solid-phase spectra of alanine and serine,” J. Am. Chem. Soc. 101, 6829–6837 (1979).
[CrossRef]

1975 (1)

D. Barron, “Methyl group as a probe of chirality in Raman optical activity,” Nature (London) 255, 458–460 (1975).
[CrossRef]

1966 (1)

P. M. Rentzepis, J. A. Giordmaine, and K. W. Wecht, “Coherent optical mixing in optically active liquids,” Phys. Rev. Lett. 16, 762–794 (1966).
[CrossRef]

1965 (1)

J. A. Giordmaine, “Nonlinear optical properties of liquids,” Phys. Rev. A 138, 1599–1606 (1965).
[CrossRef]

1962 (1)

D. A. Kleinman, “Nonlinear dielectric polarization in optical media,” Phys. Rev. 126, 1977–1979 (1962).
[CrossRef]

1937 (1)

E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
[CrossRef]

Annu. Rev. Phys. Chem. (2)

K. B. Eisenthal, “Equilibrium and dynamic processes at in terfaces by second harmonic and sum frequency generation,” Annu. Rev. Phys. Chem. 43, 627–661 (1992).
[CrossRef]

P. J. Stephens and M. A. Lowe, “Vibrational circular dichroism,” Annu. Rev. Phys. Chem. 36, 213–241 (1985).
[CrossRef]

Chem. Phys. Lett. (3)

C. D. Stanners, Q. Du, R. P. Chin, P. Cremer, G. A. Somorjai, Y.-R. Shen, “Polar ordering at the liquid-vapor interface of n-alcohols (C1–C8),” Chem. Phys. Lett. 232, 407–413 (1995).
[CrossRef]

I. Gutman, V. Babovic, and S. Jokic, “The origin of biomolecular chirality: the generalized Frank model with arbitrary initial conditions,” Chem. Phys. Lett. 144, 187–190 (1988).
[CrossRef]

J. D. Byers and J. M. Hicks, “Electronic spectral effects on chiral surface second harmonic generation,” Chem. Phys. Lett. 231, 216–224 (1994).
[CrossRef]

Discuss. Faraday Soc. (2)

J. M. Hicks, T. Petralli-Mallow, and J. D. Byers, “Consequences of chirality in second-order non-linear spectroscopy at surfaces,” Discuss. Faraday Soc. 99, 341–357 (1994).
[CrossRef]

J. L. Finney, “Hydration processes in biological and macromolucular systems,” Discuss. Faraday Soc. 103, 1–18 (1996).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

P. Pelet, and N. Engheta, “The theory of chirowaveguides,” IEEE Trans. Antennas Propag. 38, 90–98 (1990).
[CrossRef]

J. Am. Chem. Soc. (1)

M. Diem, E. Photos, H. Khouri, and L. A. Nafie, “Vibrational circular dichroism in amino acids and peptides: 3. Solution- and solid-phase spectra of alanine and serine,” J. Am. Chem. Soc. 101, 6829–6837 (1979).
[CrossRef]

J. Chem. Phys. (3)

T. Verbiest, M. Kauranen, J. J. Maki, M. N. Teerenstra, A. J. Schouten, R. J. M. Nolte, and A. Persoons, “Linearly polarized probes of surface chirality,” J. Chem. Phys. 103, 8296–8298 (1995).
[CrossRef]

J. J. Maki, T. Verbiest, M. Kaurenen, S. V. Elshocht, and A. Persoons, “Comparison of linearly and circularly polarized probes of second-order optical activity of chiral surfaces,” J. Chem. Phys. 105, 767–772 (1996).
[CrossRef]

J. D. Byers, H. I. Yee, and J. M. Hicks, “A second harmonic generation analog of optical rotatory dispersion for the study of chiral monolayers,” J. Chem. Phys. 101, 6233–6241 (1994).
[CrossRef]

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

J. Phys. Chem. (1)

T. Petralli-Mallow, T. M. Wong, J. D. Byers, H. I. Yee, and J. M. Hicks, “Circular dichroism spectroscopy at interfaces: a surface second harmonic generation study,” J. Phys. Chem. 97, 1383–1388 (1993).
[CrossRef]

JETP (1)

N. I. Koroteev, “New schemes for nonlinear optical spectroscopy of solutions of chiral biological macromolecules,” JETP 79, 681–690 (1994).

Nature (London) (1)

D. Barron, “Methyl group as a probe of chirality in Raman optical activity,” Nature (London) 255, 458–460 (1975).
[CrossRef]

Nonlinear Opt. (1)

N. I. Koroteev, V. A. Makarov, and S. N. Volkov, “Sum fre-quency generation by reflection of light from the surface of a chiral medium,” Nonlinear Opt. 17, 247–269 (1997).

Phys. Rev. (1)

D. A. Kleinman, “Nonlinear dielectric polarization in optical media,” Phys. Rev. 126, 1977–1979 (1962).
[CrossRef]

Phys. Rev. A (1)

J. A. Giordmaine, “Nonlinear optical properties of liquids,” Phys. Rev. A 138, 1599–1606 (1965).
[CrossRef]

Phys. Rev. B (1)

J. D. Byers, H. I. Yee, T. Petralli-Mallow, and J. M. Hicks, “Second-harmonic generation circular-dichroism spectroscopy from chiral monolayers,” Phys. Rev. B 49, 14, 643–16, 647 (1994).
[CrossRef]

Phys. Rev. Lett. (2)

P. M. Rentzepis, J. A. Giordmaine, and K. W. Wecht, “Coherent optical mixing in optically active liquids,” Phys. Rev. Lett. 16, 762–794 (1966).
[CrossRef]

Q. Du, R. Superfine, E. Freysz, and Y. R. Shen, “Vibrational spectroscopy of water at the vapor/water interface,” Phys. Rev. Lett. 70, 2313–2316 (1993).
[CrossRef] [PubMed]

Quantum Electron. (1)

S. N. Volkov, N. I. Koroteev, and V. A. Makarov, “Sum-frequency generation by reflection of light from the surface of a nonabsorbing isotropic and gyrotropic medium,” Quantum Electron. 25, 1183–1187 (1995).
[CrossRef]

Rev. Mod. Phys. (1)

E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
[CrossRef]

Other (7)

J. Y. Huang and Y. R. Shen, “Sum-frequency generation as a surface probe,” in Laser Spectroscopy and Photochemistry on Metal Surfaces, H. L. Dai and W. Ho, eds. (World Scientific, Singapore, 1995), Vol. 1, pp. 5–53.

See, for example, G. Arfken, Mathematical Methods for Physicists (Academic, Orlando, Fla., 1985), Chap. 3.

See, for example, A. W. Hall, J. Hollingshurst, and J. W. Goodby, “Chiral and achiral calamitic liquid crystals for display applications,” in Handbook of Liquid Crystal Research, P. J. Collings and J. S. Patel, eds. (Oxford U. Press, Oxford, 1997), Chap. 2.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap. 6.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap. 2.

J. F. Nicoud and R. J. Twieg, “Organic EFISH hyperpolarizability data,” in Nonlinear Optical Properties of Organic Molecules and Crystals, D. S. Chemla and J. Zyss, eds. (Academic, Orlando, Fla., 1987), Vol. 2, pp. 255–267.

T. B. Freedman and L. A. Nafie, “Stereochemical aspects of vibrational optical activity,” in Topics in Stereochemistry, E. Eliel and S. Wilen, eds. (Wiley, New York, 1987), pp. 113–206.

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Figures (5)

Fig. 1
Fig. 1

Schematic showing the s- and p-polarized directions and the propagation directions of the incident, refracted, and reflected waves.

Fig. 2
Fig. 2

Vector diagram showing the relative orientations of the wave vectors for the incident, reflected, and refracted waves and the nonlinear polarization source. The splitting of the refracted wave vector by circular birefringence is neglected.  

Fig. 3
Fig. 3

Calculated sum-frequency intensities [(a), (b) in reflection; (c), (d) in transmission] plotted as a function of the incident angle of input beam 1. In the calculations, the incident angle of beam 2 was fixed at 40°. Left, the input polarizations for two input beams are p polarized. Right, the curves are generated with ps (p for beam 1 and s for beam 2) or sp input polarization combinations. The circular birefringence of the material is neglected.

Fig. 4
Fig. 4

(a) Reflected sum-frequency signal with sp input polarization combination plotted as a function of the incident angle of beam 1. The incident angle of beam 2 is increased as shown. (b) Similar results for the transmitted sum-frequency signal.

Fig. 5
Fig. 5

Calculated SFG intensity corrections for the circular birefringence of material. The dotted curves shown in Fig. 4 with θ2i=60° and a vanishing circular birefringence were chosen for the intensity references. The corrections for n+>n- are shown at the left; at the right the corrections for n+<n- are presented.

Equations (52)

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(A)χijk(2)=χa(2)εijk,
ks,1=k1t++k2t+,
ks,2=k1t++k2t-,
ks,3=k1t-+k2t+,
ks,4=k1t-+k2t-.
P(2)(ω3=ω1+ω2)
=j=14Pj(2)=j=14pj(2) exp[i(ks,j  r-ω3t)].
k1i sin θ1i+k2i sin θ2i=k3r sin θ3r=k3t+ sin θ3t+=k3t- sin θ3t-=ks,j sin θs,j=k1t± sin θ1t±+k2t± sin θ2t±=k1t± sin θ1t±+k2t sin θ2t.
D=εE+iξcB,
H=B/μ+iξcE,
  [E(ω3)+4πP(1)+4πPj,(2)]=0
××E(ω3)-2ω3μξc(ω3)×E(ω3)-ω32μεE(ω3)
=4πω32c2 Pj,±(2),
Eh=E3t+e^3t+ exp[i(k3t+  r-ω3t)]+E3t-e^3t- exp[i(k3t-  r-ω3t)].
Pj(2)=(pj,+e^j,++pj,-e^j,-+pj,k^s,j)×exp[i(ks,j  r-ω3t)],
Epar=4πω32pj,+c2(ks,j2-2ξ¯cks,jk3t-k3t2) e^j,++4πω32pj,-c2(ks,j2+2ξ¯cks,jk3t-k3t2) e^j,--4πpj,ε(ω3) k^s,j×exp[i(ks,j  r-ω3t)].
E3r=E3rse^3rs exp[i(k3r  r-ω3t)]+E3rpe^3rp exp[i(k3r  r-ω3t)]
10-12-120cos θ3ri2 cos θ3t+-i2 cos θ3t-0-k3ri2 k3t+-i2 k3t-k3r cos θ3r012 k3t+ cos θ3t+12 k3t- cos θ3t- E3rsE3rpE3t+E3t-=j=14-S1,jS2,j cos θs,j+S3,j sin θs,jks,jS2,jks,jS1,j cos θs,j+iξc(ω3)ω3cS4,j,
S1,j=-12 4πω32pj,+c2(ks,j2-2ξ¯cks,jk3t-k3t2)+4πω32pj,-c2(ks,j2+2ξ¯cks,jk3t-k3t2),
S2,j=-i2 4πω32pj,+c2(ks,j2-2ξ¯cks,jk3t-k3t2)-4πω32pj,-c2(ks,j2+2ξ¯cks,jk3t-k3t2),
S3,j=4πpj,ε(ω3),
S4,j=S2,j cos θs,j+S3,j sin θs,j.
χa(2)=χxyz(2)=χyzx(2)=χzxy(2)=-χxzy(2)=-χyxz(2)=-χzyx(2).
E1t=E1tse^1ts+E1tpe^1tp,
E2t=E2tse^2ts+E2tpe^2tp.
px(2)=χa(2)(E1tsE2tp sin θ2t-E1tpE2ts sin θ1t),
py(2)=χa(2)E1tpE2tp sin(θ2t-θ1t),
pz(2)=χa(2)(-E1tpE2ts cos θ1t+E1tsE2tp cos θ2t).
αξηζ(2)(ω=ω1+ω2)
=-e32 g,n,n(rξ)gn(rη)nn(rζ)ng(ω-ωng+iΓng)(ω2-ωng+iΓng)+ (rξ)gn(rη)nn(rζ)ng(ω-ωng+iΓng)(ω1-ωng+iΓng)+ (rξ)gn(rη)nn(rζ)ng(ω+ωng+iΓng)(ω2+ωng+iΓng)+ (rξ)gn(rη)nn(rζ)ng(ω+ωng+iΓng)(ω1+ωng+iΓng)- (rξ)ng(rη)nn(rζ)gn(ω-ωnn+iΓnn) 1ω2+ωng+iΓng+ 1ω1-ωng+iΓng-(rξ)ng(rη)nn(rζ)gn(ω-ωnn+iΓnn)×1ω2-ωng+iΓng+1ω1+ωng+iΓngρg(0).
χa(2)=16 εijkχijk(2)=N16 εξηζαξηζ(2)=N 16 εξηζαξηζ(2)=Nαa(2),
χa(2)(ω=ω1+ω2)=-N 162 g,n,nμgn(μnn×μng)(ω-ωng+iΓng)(ω2-ωng+iΓng)(ω1-ωng+iΓng)-μgn(μnn×μng)(ω+ωng+iΓng)(ω2+ωng+iΓng)(ω1+ωng+iΓng)+μng(μnn×μgn)(ω-ωnn+iΓnn)(ω2+ωng+iΓng)(ω1+ωng+iΓng)-μng(μnn×μgn)(ω-ωnn+iΓnn)(ω1-ωng+iΓng)(ω2-ωng+iΓng)(ω1-ω2)ρg(0).
Eis+Ers=12 (Et++Et-),
-Eip cos θi+Erp cos θr=i2 (-Et+ cos θt++Et- cos θt-).
His+Hrs=12 (Ht++Ht-),
-Hip cos θi+Hrp cos θr=i2 (-Ht+ cos θt++Ht- cos θt-).
His=-cμ0ω kiEip,Hip=cμ0ω kiEis,
Hrs=-cμ0ω krErp,Hrp=cμ0ω krErs,
Ht+=-cμω ikt+Et+,Ht-=cμω ikt-Et-.
10-12-120cos θri2 cos θt+-i2 cos θt-0-kri2 kt+-i2 kt-kr cos θr012 kt+ cos θt+12 kt- cos θt-
×ErsErpEt+Et-=-EisEip cos θikiEipkiEis cos θi.
ErsErp=r11r12r21r22 EisEip,
Et+Et-=t11t12t21t22 EisEip.
t11=2D (n- cos θi+cos θt-)2 cos θi,
t12=-i2D (n- cos θt-+cos θi)2 cos θi,
t21=2D (n+ cos θi+cos θt+)2 cos θi,
t22=i2D (n+ cos θt++cos θi)2 cos θi;
r11=-1D [(n+n--1)(cos θt++cos θt-)cos θi-(n++n-)(cos2 θi-cos θt+ cos θt-)],
r12=2iD (n+ cos θt+-n- cos θt-)cos θi,
r21=-2iD (n- cos θt+-n+ cos θt-)cos θi,
r22=1D [(n+n--1)(cos θt++cos θt-)cos θi+(n++n-)(cos2 θi-cos θt+ cos θt-)],
D(n+n-+1)(cos θt++cos θt-)cos θi+(n++n-) (cos2 θi+cos θt+ cos θt-).

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