Abstract

The absorption and scattering losses at 6328 Å in MBBA thin-film waveguides were measured. The absorption loss was studied with a thermal-lens technique and was found to be highly dichroic. Thermal self-focusing was observed for the ordinary ray. The measured scattering losses were between 15 and 40 dB cm−1, depending on the polarization and the liquid-crystal orientation. A modified bulk-scattering loss caused by the thermal fluctuation of the long-range-crystal ordering is calculated and compared with the measurements.

© 1974 Optical Society of America

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  1. J. M. Schmur and T. G. Giallorenzi, (U. S. Government Printing Office, Washington, D. C., 1972).
  2. C. Hu and J. R. Whinnery, IEEE J. Quantum Electron. 10, 556 (1974).
    [Crossref]
  3. T. P. Sosnowski, Opt. Commun. 4, 408 (1972).
    [Crossref]
  4. D. J. Channin, Appl. Phys. Lett. 21, 365 (1973).
    [Crossref]
  5. J. P. Sheridan, J. M. Schmur, and T. G. Giallorenzi, Appl. Phys. Lett. 22, 560 (1973).
    [Crossref]
  6. C. Hu, J. R. Whinnery, and N. M. Amer, IEEE J. Quantum Electron. 10, 218 (1974).
    [Crossref]
  7. C. Hu and J. R. Whinnery, Appl. Opt. 12, 72 (1973).
    [Crossref] [PubMed]
  8. I. L. Fabelinskii, Molecular Scattering of Light (Plenum, New York, 1968), Table V.
    [Crossref]
  9. P. G. De Gennes, Mol. Cryst. Liq. Cryst. 7, 325 (1969).
    [Crossref]
  10. K. O. Hill, A. Watanabe, and J. G. Chambers, Appl. Phys. Lett. 11, 1952 (1972).
  11. P. Pieranski, F. Brochard, and E. Guyon, J. Phys. (Paris) 33, 681 (1972).
    [Crossref]
  12. P. G. De Gennes, Mol. Cryst. Liq. Cryst. 12, 193 (1971).
    [Crossref]
  13. I. Haller, H. A. Huggins, and M. J. Freiser, Mol. Cryst. Liq. Cryst. 16, 53 (1972).
    [Crossref]
  14. C. W. Oseen, Trans. Faraday Soc. 29, 883 (1933).
    [Crossref]
  15. P. Chatelain, Acta Crystallogr. 1, 315 (1948).
    [Crossref]
  16. Orsay Liquid Crystal Group, Mol. Cryst. Liq. Cryst. 13, 187 (1971); I. Haller and J. D. Lister, Phys. Rev. Lett. 24, 1550 (1970).
    [Crossref]
  17. T. W. Stinson and J. D. Lister, J. Phys. (Paris) 33,(1972).
    [Crossref]
  18. I. Haller, J. Chem. Phys. 57, 1400 (1972).
    [Crossref]
  19. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Addison–Wesley, Reading, Mass., 1961), Sec. 91.
  20. T. G. Giallorenzi, J. Appl. Phys. 44, 242 (1973).
    [Crossref]
  21. D. Marcuse, Bell Syst. Tech. J. 48, 3187 (1969).
    [Crossref]
  22. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), p. 449.
  23. P. K. Tien and R. Ulrich, J. Opt. Soc. Am. 60, 1325 (1970).
    [Crossref]
  24. D. F. Nelson and J. McKenna, J. Appl. Phys. 38, 4057 (1967).
    [Crossref]
  25. Model 500 piezo-optical kit. Vari-Light Corp., 9770 Conklin Rd., Cincinnati, Ohio 45242.
  26. H. Gasparoux, B. Regaya, and J. Prost, C. R. Acad. Sci., Ser. B 272, 1168 (1971).
  27. H. Gruler and G. Meier, Mol. Cryst. Liq. Cryst. 16, 299 (1972).
    [Crossref]
  28. L. T. Creagh, Proc. IEEE 61, 814 (1973).
    [Crossref]
  29. J. P. Sheridan, Naval Research Laboratory, private communication.
  30. D. Marcuse, Bell Syst. Tech. J. 51, 429 (1972).
    [Crossref]
  31. S. Jen, Div. of Engineering and Applied Physics, Harvard University, private communication.
  32. F. J. Kahn, G. N. Taylor, and H. Schonhorn, Proc. IEEE 61, 823 (1973).
    [Crossref]

1974 (2)

C. Hu and J. R. Whinnery, IEEE J. Quantum Electron. 10, 556 (1974).
[Crossref]

C. Hu, J. R. Whinnery, and N. M. Amer, IEEE J. Quantum Electron. 10, 218 (1974).
[Crossref]

1973 (6)

C. Hu and J. R. Whinnery, Appl. Opt. 12, 72 (1973).
[Crossref] [PubMed]

D. J. Channin, Appl. Phys. Lett. 21, 365 (1973).
[Crossref]

J. P. Sheridan, J. M. Schmur, and T. G. Giallorenzi, Appl. Phys. Lett. 22, 560 (1973).
[Crossref]

T. G. Giallorenzi, J. Appl. Phys. 44, 242 (1973).
[Crossref]

L. T. Creagh, Proc. IEEE 61, 814 (1973).
[Crossref]

F. J. Kahn, G. N. Taylor, and H. Schonhorn, Proc. IEEE 61, 823 (1973).
[Crossref]

1972 (8)

D. Marcuse, Bell Syst. Tech. J. 51, 429 (1972).
[Crossref]

H. Gruler and G. Meier, Mol. Cryst. Liq. Cryst. 16, 299 (1972).
[Crossref]

T. W. Stinson and J. D. Lister, J. Phys. (Paris) 33,(1972).
[Crossref]

I. Haller, J. Chem. Phys. 57, 1400 (1972).
[Crossref]

I. Haller, H. A. Huggins, and M. J. Freiser, Mol. Cryst. Liq. Cryst. 16, 53 (1972).
[Crossref]

T. P. Sosnowski, Opt. Commun. 4, 408 (1972).
[Crossref]

K. O. Hill, A. Watanabe, and J. G. Chambers, Appl. Phys. Lett. 11, 1952 (1972).

P. Pieranski, F. Brochard, and E. Guyon, J. Phys. (Paris) 33, 681 (1972).
[Crossref]

1971 (3)

P. G. De Gennes, Mol. Cryst. Liq. Cryst. 12, 193 (1971).
[Crossref]

Orsay Liquid Crystal Group, Mol. Cryst. Liq. Cryst. 13, 187 (1971); I. Haller and J. D. Lister, Phys. Rev. Lett. 24, 1550 (1970).
[Crossref]

H. Gasparoux, B. Regaya, and J. Prost, C. R. Acad. Sci., Ser. B 272, 1168 (1971).

1970 (1)

1969 (2)

D. Marcuse, Bell Syst. Tech. J. 48, 3187 (1969).
[Crossref]

P. G. De Gennes, Mol. Cryst. Liq. Cryst. 7, 325 (1969).
[Crossref]

1967 (1)

D. F. Nelson and J. McKenna, J. Appl. Phys. 38, 4057 (1967).
[Crossref]

1948 (1)

P. Chatelain, Acta Crystallogr. 1, 315 (1948).
[Crossref]

1933 (1)

C. W. Oseen, Trans. Faraday Soc. 29, 883 (1933).
[Crossref]

Amer, N. M.

C. Hu, J. R. Whinnery, and N. M. Amer, IEEE J. Quantum Electron. 10, 218 (1974).
[Crossref]

Brochard, F.

P. Pieranski, F. Brochard, and E. Guyon, J. Phys. (Paris) 33, 681 (1972).
[Crossref]

Chambers, J. G.

K. O. Hill, A. Watanabe, and J. G. Chambers, Appl. Phys. Lett. 11, 1952 (1972).

Channin, D. J.

D. J. Channin, Appl. Phys. Lett. 21, 365 (1973).
[Crossref]

Chatelain, P.

P. Chatelain, Acta Crystallogr. 1, 315 (1948).
[Crossref]

Creagh, L. T.

L. T. Creagh, Proc. IEEE 61, 814 (1973).
[Crossref]

De Gennes, P. G.

P. G. De Gennes, Mol. Cryst. Liq. Cryst. 12, 193 (1971).
[Crossref]

P. G. De Gennes, Mol. Cryst. Liq. Cryst. 7, 325 (1969).
[Crossref]

Fabelinskii, I. L.

I. L. Fabelinskii, Molecular Scattering of Light (Plenum, New York, 1968), Table V.
[Crossref]

Freiser, M. J.

I. Haller, H. A. Huggins, and M. J. Freiser, Mol. Cryst. Liq. Cryst. 16, 53 (1972).
[Crossref]

Gasparoux, H.

H. Gasparoux, B. Regaya, and J. Prost, C. R. Acad. Sci., Ser. B 272, 1168 (1971).

Giallorenzi, T. G.

T. G. Giallorenzi, J. Appl. Phys. 44, 242 (1973).
[Crossref]

J. P. Sheridan, J. M. Schmur, and T. G. Giallorenzi, Appl. Phys. Lett. 22, 560 (1973).
[Crossref]

J. M. Schmur and T. G. Giallorenzi, (U. S. Government Printing Office, Washington, D. C., 1972).

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), p. 449.

Gruler, H.

H. Gruler and G. Meier, Mol. Cryst. Liq. Cryst. 16, 299 (1972).
[Crossref]

Guyon, E.

P. Pieranski, F. Brochard, and E. Guyon, J. Phys. (Paris) 33, 681 (1972).
[Crossref]

Haller, I.

I. Haller, H. A. Huggins, and M. J. Freiser, Mol. Cryst. Liq. Cryst. 16, 53 (1972).
[Crossref]

I. Haller, J. Chem. Phys. 57, 1400 (1972).
[Crossref]

Hill, K. O.

K. O. Hill, A. Watanabe, and J. G. Chambers, Appl. Phys. Lett. 11, 1952 (1972).

Hu, C.

C. Hu and J. R. Whinnery, IEEE J. Quantum Electron. 10, 556 (1974).
[Crossref]

C. Hu, J. R. Whinnery, and N. M. Amer, IEEE J. Quantum Electron. 10, 218 (1974).
[Crossref]

C. Hu and J. R. Whinnery, Appl. Opt. 12, 72 (1973).
[Crossref] [PubMed]

Huggins, H. A.

I. Haller, H. A. Huggins, and M. J. Freiser, Mol. Cryst. Liq. Cryst. 16, 53 (1972).
[Crossref]

Jen, S.

S. Jen, Div. of Engineering and Applied Physics, Harvard University, private communication.

Kahn, F. J.

F. J. Kahn, G. N. Taylor, and H. Schonhorn, Proc. IEEE 61, 823 (1973).
[Crossref]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Addison–Wesley, Reading, Mass., 1961), Sec. 91.

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Addison–Wesley, Reading, Mass., 1961), Sec. 91.

Lister, J. D.

T. W. Stinson and J. D. Lister, J. Phys. (Paris) 33,(1972).
[Crossref]

Marcuse, D.

D. Marcuse, Bell Syst. Tech. J. 51, 429 (1972).
[Crossref]

D. Marcuse, Bell Syst. Tech. J. 48, 3187 (1969).
[Crossref]

McKenna, J.

D. F. Nelson and J. McKenna, J. Appl. Phys. 38, 4057 (1967).
[Crossref]

Meier, G.

H. Gruler and G. Meier, Mol. Cryst. Liq. Cryst. 16, 299 (1972).
[Crossref]

Nelson, D. F.

D. F. Nelson and J. McKenna, J. Appl. Phys. 38, 4057 (1967).
[Crossref]

Oseen, C. W.

C. W. Oseen, Trans. Faraday Soc. 29, 883 (1933).
[Crossref]

Pieranski, P.

P. Pieranski, F. Brochard, and E. Guyon, J. Phys. (Paris) 33, 681 (1972).
[Crossref]

Prost, J.

H. Gasparoux, B. Regaya, and J. Prost, C. R. Acad. Sci., Ser. B 272, 1168 (1971).

Regaya, B.

H. Gasparoux, B. Regaya, and J. Prost, C. R. Acad. Sci., Ser. B 272, 1168 (1971).

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), p. 449.

Schmur, J. M.

J. P. Sheridan, J. M. Schmur, and T. G. Giallorenzi, Appl. Phys. Lett. 22, 560 (1973).
[Crossref]

J. M. Schmur and T. G. Giallorenzi, (U. S. Government Printing Office, Washington, D. C., 1972).

Schonhorn, H.

F. J. Kahn, G. N. Taylor, and H. Schonhorn, Proc. IEEE 61, 823 (1973).
[Crossref]

Sheridan, J. P.

J. P. Sheridan, J. M. Schmur, and T. G. Giallorenzi, Appl. Phys. Lett. 22, 560 (1973).
[Crossref]

J. P. Sheridan, Naval Research Laboratory, private communication.

Sosnowski, T. P.

T. P. Sosnowski, Opt. Commun. 4, 408 (1972).
[Crossref]

Stinson, T. W.

T. W. Stinson and J. D. Lister, J. Phys. (Paris) 33,(1972).
[Crossref]

Taylor, G. N.

F. J. Kahn, G. N. Taylor, and H. Schonhorn, Proc. IEEE 61, 823 (1973).
[Crossref]

Tien, P. K.

Ulrich, R.

Watanabe, A.

K. O. Hill, A. Watanabe, and J. G. Chambers, Appl. Phys. Lett. 11, 1952 (1972).

Whinnery, J. R.

C. Hu and J. R. Whinnery, IEEE J. Quantum Electron. 10, 556 (1974).
[Crossref]

C. Hu, J. R. Whinnery, and N. M. Amer, IEEE J. Quantum Electron. 10, 218 (1974).
[Crossref]

C. Hu and J. R. Whinnery, Appl. Opt. 12, 72 (1973).
[Crossref] [PubMed]

Acta Crystallogr. (1)

P. Chatelain, Acta Crystallogr. 1, 315 (1948).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (3)

D. J. Channin, Appl. Phys. Lett. 21, 365 (1973).
[Crossref]

J. P. Sheridan, J. M. Schmur, and T. G. Giallorenzi, Appl. Phys. Lett. 22, 560 (1973).
[Crossref]

K. O. Hill, A. Watanabe, and J. G. Chambers, Appl. Phys. Lett. 11, 1952 (1972).

Bell Syst. Tech. J. (2)

D. Marcuse, Bell Syst. Tech. J. 48, 3187 (1969).
[Crossref]

D. Marcuse, Bell Syst. Tech. J. 51, 429 (1972).
[Crossref]

C. R. Acad. Sci., Ser. B (1)

H. Gasparoux, B. Regaya, and J. Prost, C. R. Acad. Sci., Ser. B 272, 1168 (1971).

IEEE J. Quantum Electron. (2)

C. Hu, J. R. Whinnery, and N. M. Amer, IEEE J. Quantum Electron. 10, 218 (1974).
[Crossref]

C. Hu and J. R. Whinnery, IEEE J. Quantum Electron. 10, 556 (1974).
[Crossref]

J. Appl. Phys. (2)

T. G. Giallorenzi, J. Appl. Phys. 44, 242 (1973).
[Crossref]

D. F. Nelson and J. McKenna, J. Appl. Phys. 38, 4057 (1967).
[Crossref]

J. Chem. Phys. (1)

I. Haller, J. Chem. Phys. 57, 1400 (1972).
[Crossref]

J. Opt. Soc. Am. (1)

J. Phys. (Paris) (2)

T. W. Stinson and J. D. Lister, J. Phys. (Paris) 33,(1972).
[Crossref]

P. Pieranski, F. Brochard, and E. Guyon, J. Phys. (Paris) 33, 681 (1972).
[Crossref]

Mol. Cryst. Liq. Cryst. (5)

P. G. De Gennes, Mol. Cryst. Liq. Cryst. 12, 193 (1971).
[Crossref]

I. Haller, H. A. Huggins, and M. J. Freiser, Mol. Cryst. Liq. Cryst. 16, 53 (1972).
[Crossref]

Orsay Liquid Crystal Group, Mol. Cryst. Liq. Cryst. 13, 187 (1971); I. Haller and J. D. Lister, Phys. Rev. Lett. 24, 1550 (1970).
[Crossref]

P. G. De Gennes, Mol. Cryst. Liq. Cryst. 7, 325 (1969).
[Crossref]

H. Gruler and G. Meier, Mol. Cryst. Liq. Cryst. 16, 299 (1972).
[Crossref]

Opt. Commun. (1)

T. P. Sosnowski, Opt. Commun. 4, 408 (1972).
[Crossref]

Proc. IEEE (2)

L. T. Creagh, Proc. IEEE 61, 814 (1973).
[Crossref]

F. J. Kahn, G. N. Taylor, and H. Schonhorn, Proc. IEEE 61, 823 (1973).
[Crossref]

Trans. Faraday Soc. (1)

C. W. Oseen, Trans. Faraday Soc. 29, 883 (1933).
[Crossref]

Other (7)

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Addison–Wesley, Reading, Mass., 1961), Sec. 91.

J. M. Schmur and T. G. Giallorenzi, (U. S. Government Printing Office, Washington, D. C., 1972).

I. L. Fabelinskii, Molecular Scattering of Light (Plenum, New York, 1968), Table V.
[Crossref]

Model 500 piezo-optical kit. Vari-Light Corp., 9770 Conklin Rd., Cincinnati, Ohio 45242.

S. Jen, Div. of Engineering and Applied Physics, Harvard University, private communication.

J. P. Sheridan, Naval Research Laboratory, private communication.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), p. 449.

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Figures (7)

Fig. 1
Fig. 1

Arrangement of the thermal-lens measurement method.

Fig. 2
Fig. 2

Irradiance at beam center vs time after opening shutter. Upper trace: laser polarized along n, thermal defocusing; lower trace: laser polarized normal to n, thermal focusing.

Fig. 3
Fig. 3

Coordinate system.

Fig. 4
Fig. 4

Calculated angular dependence of scattering intensity for n||x, TE wave incident, ϕ = 0, θm = π/2, n2 = 1.725, and n0 = 1.521.

Fig. 5
Fig. 5

Arrangement for measuring waveguide loss.

Fig. 6
Fig. 6

Flux vs propagation distance for three different substrate materials.

Fig. 7
Fig. 7

Loss for different configurations and substrate materials. Bars: calculated losses; dots: measured losses on fused-quartz substrates; crosses: Plexiglass substrates; circles: crown-glass substrates.

Equations (49)

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I bc ( t = 0 ) / I bc ( t = ) = 1 - θ + θ 2 / 2 ,
θ = P b l ( d n d T ) / λ k ;
I bc ( t = 0 ) / I bc ( t = ) = [ ( 1 - θ + θ 2 / 2 ) ( 1 - θ + θ 2 / 2 ) ] 1 2 ;
I bc ( t = 0 ) / I bc ( t = ) = 1 - ( θ + θ ) / 2 + ( θ + θ ) 2 / 8.
2 k = 1 k + 1 k .
b = 1.9 × 10 - 2 cm - 1 , b = 1.0 × 10 - 1 cm - 1 .
I bc ( t ) = I bc ( 0 ) [ 1 - θ / ( 1 + t c / 2 t ) + θ 2 / 2 ( 1 + t c / 2 t ) 2 ] - 1 ,
t c = w 2 ρ c / 4 k ,
Δ T = P b D 2 w k s ,
n y ( r ) = L - 3 q n y ( q ) e - q · r
n z ( r ) = L - 3 q n z ( q ) e - i q · r
q = x 2 l π L + y 2 m π L + z 2 n π L ;             l , m , n = 0 , ± 1 , .
n y ( q ) 2 = n z ( q ) 2 = k B T L 3 K q 2 ,
i j = δ i j + Δ ( n i n j - δ i j / 3 ) ,
= ( n e 2 + 2 n 0 2 ) 0 / 2 , Δ = ( n e 2 - n 0 2 ) 0 .
δ ɛ = Δ ( 0 n y n z n y 0 0 n z 0 0 ) .
d E = π e - i k R d v 0 λ 2 R k 2 ( δ E × E ) × k × k ,
d E = e e - i k R π d v 0 λ 2 R E y cos b m ( x - W ) e - i β m z Δ n y ( r ) ,
e = x × k × k / k 2 ,
E = e e - i k · r π Δ E y 0 λ 2 R L 3 d r e - i ( β m z - k · r ) cos b m ( x - W ) · q n y ( q ) e - i q · r
e π Δ E y e - i k · r 0 λ 2 R L 3 q n y ( q ) e - i ( q y - k y ) w - 1 q y - k y · e - i ( q z + β m - k z ) l - 1 q z + β m - k z · 2 b m b m 2 - ( q x - k x ) 2 · cos [ ( q z - k x ) d / 2 + a π ] ,
E 2 = e 2 ( 8 π Δ E y 0 λ 2 R L 3 ) 2 q { n y ( q ) 2 sin 2 ( q y - k y ) w / 2 ( q y - k y ) 2 · sin 2 ( q z + β m - k z ) l / 2 ( q z + β m - k z ) 2 · b m 2 [ b m 2 - ( q x - k y ) 2 ] 2 · cos 2 [ d ( q x - k x ) / 2 + a π ] } .
q = - - - ( 2 π / L ) - 3 d q x d q y d q z .
E 2 = e 2 ( 8 π Δ E y 0 λ 2 R L 3 ) 2 k B T L 3 K ( L 2 π ) 3 ( w 2 ) ( l 2 ) π 2 · - b m 2 cos 2 [ d ( q x - k x ) / 2 + a π ] d q x [ b m 2 - ( q x - k x ) 2 ] 2 [ q x 2 + ( k y ) 2 + ( k z - β m ) 2 ] ,
A b m 2 ( k x + b m ) 2 + ( k y ) 2 + ( k z - β m ) 2 × - k x cos 2 [ ( q x - k x ) d / 2 + a π ] d q x [ b m 2 - ( q x - k x ) 2 ] 2 + b m 2 ( k x - b m ) 2 + ( k y ) 2 + ( k z - β m ) 2 × k x cos 2 [ ( q x - k x ) d / 2 + a π ] d q x [ b m 2 - ( q x - k x ) 2 ] 2 = d π 8 ( 1 ( k x + b m ) 2 + ( k y ) 2 + ( k z - β m ) 2 + 1 ( k x - b m ) 2 + ( k y ) 2 + ( k z - β m ) 2 ) .
E 2 = e 2 ( π Δ E y 2 0 λ 2 R ) 2 8 k B T l w K π A ;
e = sin θ ;
k x = n e k cos θ , k y = n e k sin θ sin ϕ , k z = n e k sin θ cos ϕ ;
β m = n 0 k sin θ m ,
b m = n 0 k cos θ m ,             θ m = cot b m / β m ;
A = d π 16 n 0 n e k 2 [ ( 1 + 1 2 δ 2 - cos θ cos θ m + sin θ sin θ m cos ϕ ) - 1 + ( 1 + 1 2 δ 2 - cos θ cos θ m - sin θ sin θ m cos ϕ ) - 1 ] ,
δ = ( n e - n 0 ) / n 0 .
P in = w d 4 n 0 0 c sin θ m E y 2 .
α = n e 0 R 2 c 2 P in l 0 π 0 π E 2 sin θ d ϕ d θ ,
α k B T π 4 n 0 2 K ( Δ 0 λ ) 2 ( sin θ m ) - 1 [ ( sin 2 θ m + 1 2 δ 2 ) × ln | 4 - 2 δ 2 - 2 cos 2 θ m δ 2 | - 1 - δ 2 - 5 cos 2 θ m 2 ] ,
α k B T π 4 n 0 2 K ( a 0 λ ) 2 ( 2 ln 2 / δ - 1 ) .
E 2 = ( π Δ 2 λ 2 0 R ) 2 i k B T w K π × A [ 2 E x 2 + sin 2 θ ( E z 2 - E x 2 ) ] ,
α TM = k B T π 4 n e 2 K ( Δ λ 0 ) 2 ( 2 ln 2 / δ + 1 ) .
α TE = k B T π 4 n e 2 K ( Δ 0 λ ) 2 ( 2 ln 2 / δ + 1 ) ,
α TM = k B T π 4 n 0 2 K ( Δ λ 0 ) 2 ( 2 ln 2 / δ - 1 ) .
A = d π 16 n 0 2 k 2 ( 1 - cos ψ sin θ m - cos η sin ψ cos θ m ) - 1 + ( 1 - cos ψ sin θ m + cos η sin ψ cos θ m ) - 1 ,
α TE = k B T π 4 n 0 2 K ( Δ λ 0 ) 2 sin θ m [ 2 cos 2 θ m ln | 1 sin S | - sin 2 θ m sin 2 S - cos 2 θ m cos 2 S + sin θ m ( 2 - sin S cos θ m ) ] .
α TE = k B T π 2 n 0 2 K ( Δ λ 0 ) 2 .
α TM = k B T π 2 n 0 2 K ( Δ λ 0 ) 2 .
n y ( q ) 2 = n z ( q ) 2 = k B T L 3 K q 2 + χ a H 2 + a E 2 ,
α = k B T π 4 n 0 2 K ( Δ λ 0 ) 2 × ( ( 2 + Δ 2 ) ln 1 + ( 1 + Δ 2 ) 1 2 Δ - ( 1 + Δ 2 ) 1 2 )
α = k B T π 4 n e 2 K ( Δ λ 0 ) 2 × [ ( 2 - Δ 2 ) ln 1 + ( 1 + Δ 2 ) 1 2 Δ + ( 1 + Δ 2 ) 1 2 ]
α = k B T π 4 n 0 2 K ( Δ λ 0 ) 2 × [ ( 4 + Δ 2 ) 1 2 - 2 Δ + Δ 2 2 ln 2 + ( 4 + Δ 2 ) 1 2 Δ ]
Δ ( δ 2 + χ a H 2 + a E 2 n 0 n k 2 K ) 1 2 .