Abstract

Measurement of the linear electro-optic effect in poled polymer thin films with two parallel electrodes is analyzed. A single-beam polarization interferometry is adopted in a reflection configuration. The dependence of the linear electro-optic modulated intensity on linear optical parameters is investigated experimentally and is compared with the theoretical analysis. Specifically, the electro-optic modulated intensity is measured as a function of optical bias, optical polarization direction, and incidence angle. In particular, the dependence of the electro-optic modulated signal on the incidence angle shows that the linear electro-optic effect in the polymer thin film induces the modulations of both the refractive angle and the linear refractive index. Also, the amount of Fabry–Perot effect influencing the modulation signal is analyzed.

© 1997 Optical Society of America

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References

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  1. K. D. Singer, M. G. Kuzyk, W. R. Holland, J. E. Sohn, and S. J. Lalama, Appl. Phys. Lett. 53, 1800 (1988).
    [CrossRef]
  2. K. S. Lee, J. W. Wu, M. H. Lee, H. K. Kim, and Y. H. Won, Opt. Quantum Electron. 27, 347 (1995).
    [CrossRef]
  3. C. C. Teng and H. T. Man, Appl. Phys. Lett. 56, 1734 (1990).
    [CrossRef]
  4. Y. Shuto and M. Amano, J. Appl. Phys. 77, 4632 (1995).
    [CrossRef]
  5. L. M. Hayden, G. F. Sauter, F. R. Ore, P. L. Pasillas, J. M. Hoover, G. A. Lindsay, and R. A. Henry, J. Appl. Phys. 68, 456 (1990).
    [CrossRef]
  6. J. S. Schildkraut, Appl. Opt. 29, 19 (1990).
    [CrossRef]
  7. M. A. Mortazavi, A. Knoesen, S. T. Kowel, R. A. Henry, J. M. Hoover, and G. A. Lindsay, Appl. Phys. B 53, 287 (1991).
    [CrossRef]
  8. R. N. DeMartino, D. E. Allen, R. Keosian, G. Khanarian, and D. R. Haas, Mater. Res. Soc. Symp. Proc. 228, 39 (1992).
    [CrossRef]
  9. Y. Levy, M. Dumont, E. Chastaing, P. Robin, P. A. Chollet, G. Gadret, and F. Kajzar, Mol. Cryst. Liq. Cryst. Sci. Technol. Sect. B 4, 1 (1993).
  10. P.-A. Chollet, G. Gadret, F. Kajzar, and P. Raimond, Thin Solid Films 242, 138 (1994).
    [CrossRef]
  11. D. Morichere, P.-A. Chollet, W. Fleming, M. Jurich, B. A. Smith, and J. D. Swalen, J. Opt. Soc. Am. B 10, 1984 (1993).
    [CrossRef]
  12. C. A. Eldering, A. Knoesen, and S. T. Kowel, J. Appl. Phys. 69, 3676 (1991).
    [CrossRef]
  13. A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1987).
  14. K. D. Singer, M. G. Kuzyk, and J. E. Sohn, J. Opt. Soc. Am. B 4, 968 (1987).
    [CrossRef]
  15. J. W. Wu, J. Opt. Soc. Am. B 8, 142 (1991).
    [CrossRef]
  16. See the caption of Fig. 8 and discussions at the top of pages 12 and 14 of Ref. 9. The discussion at the bottom of page 12, however, attributes the increasing oscillation of the modulation signal to the increasing absorption. See also discussions in Ref. 11.

1995

K. S. Lee, J. W. Wu, M. H. Lee, H. K. Kim, and Y. H. Won, Opt. Quantum Electron. 27, 347 (1995).
[CrossRef]

Y. Shuto and M. Amano, J. Appl. Phys. 77, 4632 (1995).
[CrossRef]

1994

P.-A. Chollet, G. Gadret, F. Kajzar, and P. Raimond, Thin Solid Films 242, 138 (1994).
[CrossRef]

1993

D. Morichere, P.-A. Chollet, W. Fleming, M. Jurich, B. A. Smith, and J. D. Swalen, J. Opt. Soc. Am. B 10, 1984 (1993).
[CrossRef]

Y. Levy, M. Dumont, E. Chastaing, P. Robin, P. A. Chollet, G. Gadret, and F. Kajzar, Mol. Cryst. Liq. Cryst. Sci. Technol. Sect. B 4, 1 (1993).

1992

R. N. DeMartino, D. E. Allen, R. Keosian, G. Khanarian, and D. R. Haas, Mater. Res. Soc. Symp. Proc. 228, 39 (1992).
[CrossRef]

1991

M. A. Mortazavi, A. Knoesen, S. T. Kowel, R. A. Henry, J. M. Hoover, and G. A. Lindsay, Appl. Phys. B 53, 287 (1991).
[CrossRef]

C. A. Eldering, A. Knoesen, and S. T. Kowel, J. Appl. Phys. 69, 3676 (1991).
[CrossRef]

J. W. Wu, J. Opt. Soc. Am. B 8, 142 (1991).
[CrossRef]

1990

L. M. Hayden, G. F. Sauter, F. R. Ore, P. L. Pasillas, J. M. Hoover, G. A. Lindsay, and R. A. Henry, J. Appl. Phys. 68, 456 (1990).
[CrossRef]

J. S. Schildkraut, Appl. Opt. 29, 19 (1990).
[CrossRef]

C. C. Teng and H. T. Man, Appl. Phys. Lett. 56, 1734 (1990).
[CrossRef]

1988

K. D. Singer, M. G. Kuzyk, W. R. Holland, J. E. Sohn, and S. J. Lalama, Appl. Phys. Lett. 53, 1800 (1988).
[CrossRef]

1987

Allen, D. E.

R. N. DeMartino, D. E. Allen, R. Keosian, G. Khanarian, and D. R. Haas, Mater. Res. Soc. Symp. Proc. 228, 39 (1992).
[CrossRef]

Amano, M.

Y. Shuto and M. Amano, J. Appl. Phys. 77, 4632 (1995).
[CrossRef]

Chastaing, E.

Y. Levy, M. Dumont, E. Chastaing, P. Robin, P. A. Chollet, G. Gadret, and F. Kajzar, Mol. Cryst. Liq. Cryst. Sci. Technol. Sect. B 4, 1 (1993).

Chollet, P. A.

Y. Levy, M. Dumont, E. Chastaing, P. Robin, P. A. Chollet, G. Gadret, and F. Kajzar, Mol. Cryst. Liq. Cryst. Sci. Technol. Sect. B 4, 1 (1993).

Chollet, P.-A.

P.-A. Chollet, G. Gadret, F. Kajzar, and P. Raimond, Thin Solid Films 242, 138 (1994).
[CrossRef]

D. Morichere, P.-A. Chollet, W. Fleming, M. Jurich, B. A. Smith, and J. D. Swalen, J. Opt. Soc. Am. B 10, 1984 (1993).
[CrossRef]

DeMartino, R. N.

R. N. DeMartino, D. E. Allen, R. Keosian, G. Khanarian, and D. R. Haas, Mater. Res. Soc. Symp. Proc. 228, 39 (1992).
[CrossRef]

Dumont, M.

Y. Levy, M. Dumont, E. Chastaing, P. Robin, P. A. Chollet, G. Gadret, and F. Kajzar, Mol. Cryst. Liq. Cryst. Sci. Technol. Sect. B 4, 1 (1993).

Eldering, C. A.

C. A. Eldering, A. Knoesen, and S. T. Kowel, J. Appl. Phys. 69, 3676 (1991).
[CrossRef]

Fleming, W.

D. Morichere, P.-A. Chollet, W. Fleming, M. Jurich, B. A. Smith, and J. D. Swalen, J. Opt. Soc. Am. B 10, 1984 (1993).
[CrossRef]

Gadret, G.

P.-A. Chollet, G. Gadret, F. Kajzar, and P. Raimond, Thin Solid Films 242, 138 (1994).
[CrossRef]

Y. Levy, M. Dumont, E. Chastaing, P. Robin, P. A. Chollet, G. Gadret, and F. Kajzar, Mol. Cryst. Liq. Cryst. Sci. Technol. Sect. B 4, 1 (1993).

Haas, D. R.

R. N. DeMartino, D. E. Allen, R. Keosian, G. Khanarian, and D. R. Haas, Mater. Res. Soc. Symp. Proc. 228, 39 (1992).
[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, J. Appl. Phys. 68, 456 (1990).
[CrossRef]

Henry, R. A.

M. A. Mortazavi, A. Knoesen, S. T. Kowel, R. A. Henry, J. M. Hoover, and G. A. Lindsay, Appl. Phys. B 53, 287 (1991).
[CrossRef]

L. M. Hayden, G. F. Sauter, F. R. Ore, P. L. Pasillas, J. M. Hoover, G. A. Lindsay, and R. A. Henry, J. Appl. Phys. 68, 456 (1990).
[CrossRef]

Holland, W. R.

K. D. Singer, M. G. Kuzyk, W. R. Holland, J. E. Sohn, and S. J. Lalama, Appl. Phys. Lett. 53, 1800 (1988).
[CrossRef]

Hoover, J. M.

M. A. Mortazavi, A. Knoesen, S. T. Kowel, R. A. Henry, J. M. Hoover, and G. A. Lindsay, Appl. Phys. B 53, 287 (1991).
[CrossRef]

L. M. Hayden, G. F. Sauter, F. R. Ore, P. L. Pasillas, J. M. Hoover, G. A. Lindsay, and R. A. Henry, J. Appl. Phys. 68, 456 (1990).
[CrossRef]

Jurich, M.

D. Morichere, P.-A. Chollet, W. Fleming, M. Jurich, B. A. Smith, and J. D. Swalen, J. Opt. Soc. Am. B 10, 1984 (1993).
[CrossRef]

Kajzar, F.

P.-A. Chollet, G. Gadret, F. Kajzar, and P. Raimond, Thin Solid Films 242, 138 (1994).
[CrossRef]

Y. Levy, M. Dumont, E. Chastaing, P. Robin, P. A. Chollet, G. Gadret, and F. Kajzar, Mol. Cryst. Liq. Cryst. Sci. Technol. Sect. B 4, 1 (1993).

Keosian, R.

R. N. DeMartino, D. E. Allen, R. Keosian, G. Khanarian, and D. R. Haas, Mater. Res. Soc. Symp. Proc. 228, 39 (1992).
[CrossRef]

Khanarian, G.

R. N. DeMartino, D. E. Allen, R. Keosian, G. Khanarian, and D. R. Haas, Mater. Res. Soc. Symp. Proc. 228, 39 (1992).
[CrossRef]

Kim, H. K.

K. S. Lee, J. W. Wu, M. H. Lee, H. K. Kim, and Y. H. Won, Opt. Quantum Electron. 27, 347 (1995).
[CrossRef]

Knoesen, A.

M. A. Mortazavi, A. Knoesen, S. T. Kowel, R. A. Henry, J. M. Hoover, and G. A. Lindsay, Appl. Phys. B 53, 287 (1991).
[CrossRef]

C. A. Eldering, A. Knoesen, and S. T. Kowel, J. Appl. Phys. 69, 3676 (1991).
[CrossRef]

Kowel, S. T.

C. A. Eldering, A. Knoesen, and S. T. Kowel, J. Appl. Phys. 69, 3676 (1991).
[CrossRef]

M. A. Mortazavi, A. Knoesen, S. T. Kowel, R. A. Henry, J. M. Hoover, and G. A. Lindsay, Appl. Phys. B 53, 287 (1991).
[CrossRef]

Kuzyk, M. G.

K. D. Singer, M. G. Kuzyk, W. R. Holland, J. E. Sohn, and S. J. Lalama, Appl. Phys. Lett. 53, 1800 (1988).
[CrossRef]

K. D. Singer, M. G. Kuzyk, and J. E. Sohn, J. Opt. Soc. Am. B 4, 968 (1987).
[CrossRef]

Lalama, S. J.

K. D. Singer, M. G. Kuzyk, W. R. Holland, J. E. Sohn, and S. J. Lalama, Appl. Phys. Lett. 53, 1800 (1988).
[CrossRef]

Lee, K. S.

K. S. Lee, J. W. Wu, M. H. Lee, H. K. Kim, and Y. H. Won, Opt. Quantum Electron. 27, 347 (1995).
[CrossRef]

Lee, M. H.

K. S. Lee, J. W. Wu, M. H. Lee, H. K. Kim, and Y. H. Won, Opt. Quantum Electron. 27, 347 (1995).
[CrossRef]

Levy, Y.

Y. Levy, M. Dumont, E. Chastaing, P. Robin, P. A. Chollet, G. Gadret, and F. Kajzar, Mol. Cryst. Liq. Cryst. Sci. Technol. Sect. B 4, 1 (1993).

Lindsay, G. A.

M. A. Mortazavi, A. Knoesen, S. T. Kowel, R. A. Henry, J. M. Hoover, and G. A. Lindsay, Appl. Phys. B 53, 287 (1991).
[CrossRef]

L. M. Hayden, G. F. Sauter, F. R. Ore, P. L. Pasillas, J. M. Hoover, G. A. Lindsay, and R. A. Henry, J. Appl. Phys. 68, 456 (1990).
[CrossRef]

Man, H. T.

C. C. Teng and H. T. Man, Appl. Phys. Lett. 56, 1734 (1990).
[CrossRef]

Morichere, D.

D. Morichere, P.-A. Chollet, W. Fleming, M. Jurich, B. A. Smith, and J. D. Swalen, J. Opt. Soc. Am. B 10, 1984 (1993).
[CrossRef]

Mortazavi, M. A.

M. A. Mortazavi, A. Knoesen, S. T. Kowel, R. A. Henry, J. M. Hoover, and G. A. Lindsay, Appl. Phys. B 53, 287 (1991).
[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, J. Appl. Phys. 68, 456 (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, J. Appl. Phys. 68, 456 (1990).
[CrossRef]

Raimond, P.

P.-A. Chollet, G. Gadret, F. Kajzar, and P. Raimond, Thin Solid Films 242, 138 (1994).
[CrossRef]

Robin, P.

Y. Levy, M. Dumont, E. Chastaing, P. Robin, P. A. Chollet, G. Gadret, and F. Kajzar, Mol. Cryst. Liq. Cryst. Sci. Technol. Sect. B 4, 1 (1993).

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, J. Appl. Phys. 68, 456 (1990).
[CrossRef]

Schildkraut, J. S.

Shuto, Y.

Y. Shuto and M. Amano, J. Appl. Phys. 77, 4632 (1995).
[CrossRef]

Singer, K. D.

K. D. Singer, M. G. Kuzyk, W. R. Holland, J. E. Sohn, and S. J. Lalama, Appl. Phys. Lett. 53, 1800 (1988).
[CrossRef]

K. D. Singer, M. G. Kuzyk, and J. E. Sohn, J. Opt. Soc. Am. B 4, 968 (1987).
[CrossRef]

Smith, B. A.

D. Morichere, P.-A. Chollet, W. Fleming, M. Jurich, B. A. Smith, and J. D. Swalen, J. Opt. Soc. Am. B 10, 1984 (1993).
[CrossRef]

Sohn, J. E.

K. D. Singer, M. G. Kuzyk, W. R. Holland, J. E. Sohn, and S. J. Lalama, Appl. Phys. Lett. 53, 1800 (1988).
[CrossRef]

K. D. Singer, M. G. Kuzyk, and J. E. Sohn, J. Opt. Soc. Am. B 4, 968 (1987).
[CrossRef]

Swalen, J. D.

D. Morichere, P.-A. Chollet, W. Fleming, M. Jurich, B. A. Smith, and J. D. Swalen, J. Opt. Soc. Am. B 10, 1984 (1993).
[CrossRef]

Teng, C. C.

C. C. Teng and H. T. Man, Appl. Phys. Lett. 56, 1734 (1990).
[CrossRef]

Won, Y. H.

K. S. Lee, J. W. Wu, M. H. Lee, H. K. Kim, and Y. H. Won, Opt. Quantum Electron. 27, 347 (1995).
[CrossRef]

Wu, J. W.

K. S. Lee, J. W. Wu, M. H. Lee, H. K. Kim, and Y. H. Won, Opt. Quantum Electron. 27, 347 (1995).
[CrossRef]

J. W. Wu, J. Opt. Soc. Am. B 8, 142 (1991).
[CrossRef]

Appl. Opt.

Appl. Phys. B

M. A. Mortazavi, A. Knoesen, S. T. Kowel, R. A. Henry, J. M. Hoover, and G. A. Lindsay, Appl. Phys. B 53, 287 (1991).
[CrossRef]

Appl. Phys. Lett.

K. D. Singer, M. G. Kuzyk, W. R. Holland, J. E. Sohn, and S. J. Lalama, Appl. Phys. Lett. 53, 1800 (1988).
[CrossRef]

C. C. Teng and H. T. Man, Appl. Phys. Lett. 56, 1734 (1990).
[CrossRef]

J. Appl. Phys.

Y. Shuto and M. Amano, J. Appl. Phys. 77, 4632 (1995).
[CrossRef]

L. M. Hayden, G. F. Sauter, F. R. Ore, P. L. Pasillas, J. M. Hoover, G. A. Lindsay, and R. A. Henry, J. Appl. Phys. 68, 456 (1990).
[CrossRef]

C. A. Eldering, A. Knoesen, and S. T. Kowel, J. Appl. Phys. 69, 3676 (1991).
[CrossRef]

J. Opt. Soc. Am. B

K. D. Singer, M. G. Kuzyk, and J. E. Sohn, J. Opt. Soc. Am. B 4, 968 (1987).
[CrossRef]

J. W. Wu, J. Opt. Soc. Am. B 8, 142 (1991).
[CrossRef]

D. Morichere, P.-A. Chollet, W. Fleming, M. Jurich, B. A. Smith, and J. D. Swalen, J. Opt. Soc. Am. B 10, 1984 (1993).
[CrossRef]

Mater. Res. Soc. Symp. Proc.

R. N. DeMartino, D. E. Allen, R. Keosian, G. Khanarian, and D. R. Haas, Mater. Res. Soc. Symp. Proc. 228, 39 (1992).
[CrossRef]

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

Y. Levy, M. Dumont, E. Chastaing, P. Robin, P. A. Chollet, G. Gadret, and F. Kajzar, Mol. Cryst. Liq. Cryst. Sci. Technol. Sect. B 4, 1 (1993).

Opt. Quantum Electron.

K. S. Lee, J. W. Wu, M. H. Lee, H. K. Kim, and Y. H. Won, Opt. Quantum Electron. 27, 347 (1995).
[CrossRef]

Thin Solid Films

P.-A. Chollet, G. Gadret, F. Kajzar, and P. Raimond, Thin Solid Films 242, 138 (1994).
[CrossRef]

Other

See the caption of Fig. 8 and discussions at the top of pages 12 and 14 of Ref. 9. The discussion at the bottom of page 12, however, attributes the increasing oscillation of the modulation signal to the increasing absorption. See also discussions in Ref. 11.

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1987).

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Figures (8)

Fig. 1
Fig. 1

Single-beam polarization interferometry. The analyzer is set cross polarized to the optical polarization determined by the half-wave plate. The quarter-wave plate provides the optical bias. ITO, indium thin oxide; W. P., wave plate; pol, polarization.

Fig. 2
Fig. 2

Reflection geometry of the polymer thin film, in detail. np is larger than ns in the current geometry. The p wave traces BCCDDG, and the s wave traces BCCD.

Fig. 3
Fig. 3

Linear absorption spectrum of DCM/PIQ-2200 thin film after curing. O.D., optical density.

Fig. 4
Fig. 4

Schematic diagram of the experimental setup is shown.

Fig. 5
Fig. 5

Modulated intensity plotted as a function of the modulation voltage. The straight line is the least-squares fit, confirming that the measured signal is from the linear EO effect.

Fig. 6
Fig. 6

Modulated light intensity with the signal phase as a function of the phase retardation (bias) introduced by the quarter-wave plate. The dashed and dotted curves correspond to the base functions sin Γsp and sin(Γsp/2) respectively. The solid curve is the theoretical fit, I(Γsp)=sin Γsp+ sin(Γsp/2). is found to be 0.3 and 0.05 at (a) λ=633 nm and (b) λ =830 nm, respectively.

Fig. 7
Fig. 7

Modulated light intensity plotted as a function of the optical polarization angle. The solid curve is the theoretical fit.

Fig. 8
Fig. 8

Modulated light intensity plotted as a function of the incidence angle. The solid curve is the function f(θ), and the dashed curve is the function f(θ). (a) λ=633 nm and (b) λ=830 nm, respectively.

Equations (37)

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

1no2(x2+y2)+1ne2z2=1,
Δnx=Δny=Δno=-no32r13Ez,
Δnz=Δne=-ne32r33Ez.
ns=no,
Δns=Δno;
1np2=cos2 αpno2+sin2 αpne2,
ΔnpΔno cos2 αp+Δne sin2 αp,
sin θ=ns sin αs=np sin αp.
Δnp sin αp+np cos αpΔαp
=Δns sin αs+ns cos αsΔαs,
(Δnp-Δns)sin α=-n (Δαp-Δαs)cos α.
Δαp-Δαs=-tan αn(Δnp-Δns).
Δψ=2πλΔ(sn)=2πλ(sΔn+nΔs),
ss=BC¯+CD¯,sp=BC¯+CD¯+DG¯,
ss=2d/cos αs,
sp=2d/cos αp+2d sin θ(tan αs-tan αp).
Γ0Γ(ϕ=π/4)=Δψp-Δψs.
Esin=E cos ϕ,Epin=E sin ϕ.
Esout=E cos ϕ exp(-iΓ0),Epout=E sin ϕ.
EAout=E sin ϕ cos ϕ[exp(-iΓ0)-1].
Iout=EAoutEAout*=E2 sin2 2ϕ sin2 Γ0/2.
Iout=E2 sin2 Γ(ϕ)2.
sin2 Γ(ϕ)2=sin2 2ϕ sin2 Γ02.
Γ(ϕ)=Γ0 sin 2ϕ.
Γ0=Γ0m sin ωmt=Δψp-Δψs.
Γ0m=2kdΔ(np/cos αp-ns/cos αs-tan αp sin θ+tan αs sin θ)=2kdΔ(np cos αp-ns cos αs)=2kd(Δnp cos αp-np sin αpΔαp-Δns cos αs+ns sin αsΔαs)2kd[(Δnp-Δns)cos α-n(Δαp-Δαs)sin α].
Γ0m=2kd1cos α(Δnp-Δns)=2kdcos α(Δno cos2 αp+Δne sin2 αp-Δno)=2kdcos α(Δne-Δno)sin2 αp2kdcos α(Δne-Δno)sin2 α.
Γ0m2kd(Δne-Δno) sin2 θn(n2-sin2 θ)1/2.
Γm(ϕ)=Γ0m sin 2ϕ=2kd(Δne-Δno) sin2 θn(n2-sin2 θ)1/2sin 2ϕ.
Γ=Γsp+Γ(ϕ).
IA=2Ic sin2Γsp+Γ(ϕ)2=2Ic sin2 Γsp2+Im.
Im=IcΓ(ϕ)sin Γsp=IcΓ0 sin 2ϕ sin Γsp.
Imo=IcΓ0m sin 2ϕ sin Γsp=Ic2kd(Δne-Δno) sin2 θn(n2-sin2 θ)1/2×sin 2ϕ sin Γsp=-Ic 2πλ(r33-r13)Ezd n sin2 θ(n2-sin2 θ)1/2×sin 2ϕ sin Γsp.
Imo=-Ic 4π3r33Vmλn sin2 θ(n2-sin2 θ)1/2sin 2ϕ sin Γsp=-Ic 4π3r33Vmλf(θ)sin 2ϕ sin Γsp,
f(θ)=n2 sin2 θ(n2-sin2 θ)1/2.
f(θ)=(n2 sin2 θ)(n2-2 sin2 θ)(n2-sin2 θ)3/2.
I(Γsp)=sin Γsp+ sin(Γsp/2),

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