Abstract

We report on the application, for the first time to our knowledge, of spectroscopic generalized ellipsometry to liquid crystal materials. We have measured the three normalized elements of the Jones transmission matrix t at various sample temperatures within the spectral range from 340 to 1700 nm (0.73 to 3.65 eV) on thin cells filled with twisted nematic mixtures 4-cyano-4′-pentylbiphenyl (5CB) and 4-cyano-4′-(2 methyl)-butylbiphenyl (CB15) (Ref. 4). The Berreman 4 × 4 matrix for electromagnetic plane waves in a biaxial medium homogeneously twisted along the sample normal is derived and presented. Analytic expressions in the case of light propagation along the helical axis permit the calculation of the transmission and reflection coefficients simultaneously without numerical approximations. This solution is valid for any biaxial configuration of chiral liquid crystals, including the case of absorption. We have fully analyzed the measured Jones transmission matrix elements and obtained the geometrical sample properties and, as a function of the photon energy and the temperature, the refractive indices no, and ne of the chiral liquid crystals. We found that within the experimental error the main refractive indices of the mixtures 5CB and CB15 are those of pure 5CB at the same reduced temperatures. The handedness of the optical activity of the samples can be obtained immediately from the phase information of the Jones transmission matrix coefficients.

© 1996 Optical Society of America

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References

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  1. R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. 31, 488–493 (1941).
    [CrossRef]
  2. R. M. A. Azzam, N. M. Bashara, “Generalized ellipsometry for surfaces with directional preference: application to diffraction gratings,” J. Opt. Soc. Am. 62, 1521–1523 (1972).
    [CrossRef]
  3. M. Schubert, B. Rheinländer, J. A. Woollam, B. Johs, C. Herzinger, “Extension of rotating-analyzer ellipsometry to generalized ellipsometry: determination of the dielectric function tensor from uniaxial TiO2,” J. Opt. Soc. Am. A 13, 875–883 (1996).
    [CrossRef]
  4. S. Chandrasekhar, Liquid Crystals, 2nd ed. (Cambridge U. Press, Cambridge, 1992).
    [CrossRef]
  5. C. Gu, P. Yeh, “Extended Jones matrix method II,” J. Opt. Soc. Am. A 10, 966–973 (1993) and references therein. See also D. W. Berreman, T. J. Scheffer, “Bragg reflection of light from single-domain cholesteric liquid-crystal films,” Phys. Rev. Lett. 25, 577–581 (1970); K. Eidner, G. Mayer, M. Schmidt, H. Schmiedel, “Optics in stratified media—the use of optical eigenmodes of uniaxial crystals in the 4 × 4-matrix formalism,” Mol. Cryst. Liq. Cryst. 172, 191–200 (1989); H. Wöhler, M. Fritsch, G. Haas, D. A. Mlynski, “Characteristic matrix method for stratified anisotropic media: optical properties of special configurations,” J. Opt. Soc. Am. A 8, 536–540 (1991).
    [CrossRef]
  6. D. L. Jaggard, X. Sun, “Theory of chiral multilayers,” J. Opt. Soc. Am. A 9, 804–813 (1992).
    [CrossRef]
  7. S. Bassani, C. H. Papas, 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]
  8. W. S. Weiglhofer, A. Lakhatakia, “Wave propagation in a continuously twisted biaxial dielectric medium parallel to the helical axis,” Optik 96, 179–183 (1994).
  9. D. W. Berreman, “Optics in stratified and anisotropic media,” J. Opt. Soc. Am. 62, 502–510 (1972).
    [CrossRef]
  10. M. Schubert, “Polarization dependent parameters of arbitrarily anisotropic homogeneous epitaxial systems,” Phys. Rev. B 53, 4265–4274 (1996).
    [CrossRef]
  11. P. Yeh, “Optics of anisotropic layered media: a new 4 × 4 matrix algebra,” Surf. Sci. 96, 41–53 (1980).
    [CrossRef]
  12. P. Drude, The Theory of Optics (Longmans, Green, New York, 1922).
  13. G. Pelzl, H. Sackmann, “Double refraction of the cholesterol and smectic modifications of cholesteryl esters of n-fatty acids,” Z. Phys. Chem. (Leipzig) 254, 354–372 (1973).
  14. Z. Yaniv, N. A. P. Vaz, G. Chidichimo, J. W. Doane, “Direct nuclear-magnetic-resonance measurements of biaxiality in the cholesteric liquid crystalline phase,” Phys. Rev. Lett. 47, 46–49 (1981).
    [CrossRef]
  15. H. Wöhler, M. Fritsch, G. Haas, D. A. Mlynski, “Faster 4 × 4 matrix method for uniaxial inhomogeneous media,” J. Opt. Soc. Am. A 5, 1554–1557 (1988).
    [CrossRef]
  16. G. Heppke, F. Oestreicher, “Determination of the cholesteric screw sense,” Mol. Cryst. Liq. Cryst. (Lett.) 41, 245–249 (1978).
    [CrossRef]
  17. R. Cano, “Optical rotary power of cholesteric liquid crystals,” Bull. Soc. Franc. Mineral. Cristallogr. 90, 333–351 (1967).
  18. C. Cramer, H. Binder, M. Schubert, B. Rheinländer, H. Schmiedel, “Optical properties of microconfined liquid crystals,” Mol. Cryst. Liq. Cryst. 282, 395–405 (1996).
    [CrossRef]
  19. I. Haller, “Thermodynamic and static properties of liquid crystals,” Prog. Sol. State Chem. 10, 103–112 (1975). See also St. Limmer, “Physical principles underlying the experimental methods for studying the orientational order of liquid crystals,” Fortschr. Phys. 37, 879–931 (1989).
    [CrossRef]
  20. S.-T. Wu, “A semiempirical model for liquid-crystal refractive index dispersions,” J. Appl. Phys. 69, 2080–2087 (1991).
    [CrossRef]

1996 (3)

M. Schubert, B. Rheinländer, J. A. Woollam, B. Johs, C. Herzinger, “Extension of rotating-analyzer ellipsometry to generalized ellipsometry: determination of the dielectric function tensor from uniaxial TiO2,” J. Opt. Soc. Am. A 13, 875–883 (1996).
[CrossRef]

M. Schubert, “Polarization dependent parameters of arbitrarily anisotropic homogeneous epitaxial systems,” Phys. Rev. B 53, 4265–4274 (1996).
[CrossRef]

C. Cramer, H. Binder, M. Schubert, B. Rheinländer, H. Schmiedel, “Optical properties of microconfined liquid crystals,” Mol. Cryst. Liq. Cryst. 282, 395–405 (1996).
[CrossRef]

1994 (1)

W. S. Weiglhofer, A. Lakhatakia, “Wave propagation in a continuously twisted biaxial dielectric medium parallel to the helical axis,” Optik 96, 179–183 (1994).

1993 (1)

1992 (1)

1991 (1)

S.-T. Wu, “A semiempirical model for liquid-crystal refractive index dispersions,” J. Appl. Phys. 69, 2080–2087 (1991).
[CrossRef]

1988 (2)

1981 (1)

Z. Yaniv, N. A. P. Vaz, G. Chidichimo, J. W. Doane, “Direct nuclear-magnetic-resonance measurements of biaxiality in the cholesteric liquid crystalline phase,” Phys. Rev. Lett. 47, 46–49 (1981).
[CrossRef]

1980 (1)

P. Yeh, “Optics of anisotropic layered media: a new 4 × 4 matrix algebra,” Surf. Sci. 96, 41–53 (1980).
[CrossRef]

1978 (1)

G. Heppke, F. Oestreicher, “Determination of the cholesteric screw sense,” Mol. Cryst. Liq. Cryst. (Lett.) 41, 245–249 (1978).
[CrossRef]

1975 (1)

I. Haller, “Thermodynamic and static properties of liquid crystals,” Prog. Sol. State Chem. 10, 103–112 (1975). See also St. Limmer, “Physical principles underlying the experimental methods for studying the orientational order of liquid crystals,” Fortschr. Phys. 37, 879–931 (1989).
[CrossRef]

1973 (1)

G. Pelzl, H. Sackmann, “Double refraction of the cholesterol and smectic modifications of cholesteryl esters of n-fatty acids,” Z. Phys. Chem. (Leipzig) 254, 354–372 (1973).

1972 (2)

1967 (1)

R. Cano, “Optical rotary power of cholesteric liquid crystals,” Bull. Soc. Franc. Mineral. Cristallogr. 90, 333–351 (1967).

1941 (1)

Azzam, R. M. A.

Bashara, N. M.

Bassani, S.

Berreman, D. W.

Binder, H.

C. Cramer, H. Binder, M. Schubert, B. Rheinländer, H. Schmiedel, “Optical properties of microconfined liquid crystals,” Mol. Cryst. Liq. Cryst. 282, 395–405 (1996).
[CrossRef]

Cano, R.

R. Cano, “Optical rotary power of cholesteric liquid crystals,” Bull. Soc. Franc. Mineral. Cristallogr. 90, 333–351 (1967).

Chandrasekhar, S.

S. Chandrasekhar, Liquid Crystals, 2nd ed. (Cambridge U. Press, Cambridge, 1992).
[CrossRef]

Chidichimo, G.

Z. Yaniv, N. A. P. Vaz, G. Chidichimo, J. W. Doane, “Direct nuclear-magnetic-resonance measurements of biaxiality in the cholesteric liquid crystalline phase,” Phys. Rev. Lett. 47, 46–49 (1981).
[CrossRef]

Cramer, C.

C. Cramer, H. Binder, M. Schubert, B. Rheinländer, H. Schmiedel, “Optical properties of microconfined liquid crystals,” Mol. Cryst. Liq. Cryst. 282, 395–405 (1996).
[CrossRef]

Doane, J. W.

Z. Yaniv, N. A. P. Vaz, G. Chidichimo, J. W. Doane, “Direct nuclear-magnetic-resonance measurements of biaxiality in the cholesteric liquid crystalline phase,” Phys. Rev. Lett. 47, 46–49 (1981).
[CrossRef]

Drude, P.

P. Drude, The Theory of Optics (Longmans, Green, New York, 1922).

Engheta, N.

Fritsch, M.

Gu, C.

Haas, G.

Haller, I.

I. Haller, “Thermodynamic and static properties of liquid crystals,” Prog. Sol. State Chem. 10, 103–112 (1975). See also St. Limmer, “Physical principles underlying the experimental methods for studying the orientational order of liquid crystals,” Fortschr. Phys. 37, 879–931 (1989).
[CrossRef]

Heppke, G.

G. Heppke, F. Oestreicher, “Determination of the cholesteric screw sense,” Mol. Cryst. Liq. Cryst. (Lett.) 41, 245–249 (1978).
[CrossRef]

Herzinger, C.

Jaggard, D. L.

Johs, B.

Jones, R. C.

Lakhatakia, A.

W. S. Weiglhofer, A. Lakhatakia, “Wave propagation in a continuously twisted biaxial dielectric medium parallel to the helical axis,” Optik 96, 179–183 (1994).

Mlynski, D. A.

Oestreicher, F.

G. Heppke, F. Oestreicher, “Determination of the cholesteric screw sense,” Mol. Cryst. Liq. Cryst. (Lett.) 41, 245–249 (1978).
[CrossRef]

Papas, C. H.

Pelzl, G.

G. Pelzl, H. Sackmann, “Double refraction of the cholesterol and smectic modifications of cholesteryl esters of n-fatty acids,” Z. Phys. Chem. (Leipzig) 254, 354–372 (1973).

Rheinländer, B.

Sackmann, H.

G. Pelzl, H. Sackmann, “Double refraction of the cholesterol and smectic modifications of cholesteryl esters of n-fatty acids,” Z. Phys. Chem. (Leipzig) 254, 354–372 (1973).

Schmiedel, H.

C. Cramer, H. Binder, M. Schubert, B. Rheinländer, H. Schmiedel, “Optical properties of microconfined liquid crystals,” Mol. Cryst. Liq. Cryst. 282, 395–405 (1996).
[CrossRef]

Schubert, M.

M. Schubert, “Polarization dependent parameters of arbitrarily anisotropic homogeneous epitaxial systems,” Phys. Rev. B 53, 4265–4274 (1996).
[CrossRef]

C. Cramer, H. Binder, M. Schubert, B. Rheinländer, H. Schmiedel, “Optical properties of microconfined liquid crystals,” Mol. Cryst. Liq. Cryst. 282, 395–405 (1996).
[CrossRef]

M. Schubert, B. Rheinländer, J. A. Woollam, B. Johs, C. Herzinger, “Extension of rotating-analyzer ellipsometry to generalized ellipsometry: determination of the dielectric function tensor from uniaxial TiO2,” J. Opt. Soc. Am. A 13, 875–883 (1996).
[CrossRef]

Sun, X.

Vaz, N. A. P.

Z. Yaniv, N. A. P. Vaz, G. Chidichimo, J. W. Doane, “Direct nuclear-magnetic-resonance measurements of biaxiality in the cholesteric liquid crystalline phase,” Phys. Rev. Lett. 47, 46–49 (1981).
[CrossRef]

Weiglhofer, W. S.

W. S. Weiglhofer, A. Lakhatakia, “Wave propagation in a continuously twisted biaxial dielectric medium parallel to the helical axis,” Optik 96, 179–183 (1994).

Wöhler, H.

Woollam, J. A.

Wu, S.-T.

S.-T. Wu, “A semiempirical model for liquid-crystal refractive index dispersions,” J. Appl. Phys. 69, 2080–2087 (1991).
[CrossRef]

Yaniv, Z.

Z. Yaniv, N. A. P. Vaz, G. Chidichimo, J. W. Doane, “Direct nuclear-magnetic-resonance measurements of biaxiality in the cholesteric liquid crystalline phase,” Phys. Rev. Lett. 47, 46–49 (1981).
[CrossRef]

Yeh, P.

Bull. Soc. Franc. Mineral. Cristallogr. (1)

R. Cano, “Optical rotary power of cholesteric liquid crystals,” Bull. Soc. Franc. Mineral. Cristallogr. 90, 333–351 (1967).

J. Appl. Phys. (1)

S.-T. Wu, “A semiempirical model for liquid-crystal refractive index dispersions,” J. Appl. Phys. 69, 2080–2087 (1991).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (5)

Mol. Cryst. Liq. Cryst. (1)

C. Cramer, H. Binder, M. Schubert, B. Rheinländer, H. Schmiedel, “Optical properties of microconfined liquid crystals,” Mol. Cryst. Liq. Cryst. 282, 395–405 (1996).
[CrossRef]

Mol. Cryst. Liq. Cryst. (Lett.) (1)

G. Heppke, F. Oestreicher, “Determination of the cholesteric screw sense,” Mol. Cryst. Liq. Cryst. (Lett.) 41, 245–249 (1978).
[CrossRef]

Optik (1)

W. S. Weiglhofer, A. Lakhatakia, “Wave propagation in a continuously twisted biaxial dielectric medium parallel to the helical axis,” Optik 96, 179–183 (1994).

Phys. Rev. B (1)

M. Schubert, “Polarization dependent parameters of arbitrarily anisotropic homogeneous epitaxial systems,” Phys. Rev. B 53, 4265–4274 (1996).
[CrossRef]

Phys. Rev. Lett. (1)

Z. Yaniv, N. A. P. Vaz, G. Chidichimo, J. W. Doane, “Direct nuclear-magnetic-resonance measurements of biaxiality in the cholesteric liquid crystalline phase,” Phys. Rev. Lett. 47, 46–49 (1981).
[CrossRef]

Prog. Sol. State Chem. (1)

I. Haller, “Thermodynamic and static properties of liquid crystals,” Prog. Sol. State Chem. 10, 103–112 (1975). See also St. Limmer, “Physical principles underlying the experimental methods for studying the orientational order of liquid crystals,” Fortschr. Phys. 37, 879–931 (1989).
[CrossRef]

Surf. Sci. (1)

P. Yeh, “Optics of anisotropic layered media: a new 4 × 4 matrix algebra,” Surf. Sci. 96, 41–53 (1980).
[CrossRef]

Z. Phys. Chem. (Leipzig) (1)

G. Pelzl, H. Sackmann, “Double refraction of the cholesterol and smectic modifications of cholesteryl esters of n-fatty acids,” Z. Phys. Chem. (Leipzig) 254, 354–372 (1973).

Other (2)

S. Chandrasekhar, Liquid Crystals, 2nd ed. (Cambridge U. Press, Cambridge, 1992).
[CrossRef]

P. Drude, The Theory of Optics (Longmans, Green, New York, 1922).

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Figures (9)

Fig. 1
Fig. 1

Schematic representation of a cholesteric liquid crystal. P is the distance between one full turn of the director n. The sign of P refers to a right- (positive) or left- (negative) handed helical arrangement of the molecules.

Fig. 2
Fig. 2

Definition of the orientation of a biaxial phase at the lowest boundary of the twisted material. The first plane of molecules next to the interface is treated as an infinitesimally thin slab of an arbitrarily oriented biaxial material. Hence three main dielectric-function values (main refractive indices) and three Euler angles are necessary for obtaining the dielectric function tensor in laboratory coordinates as defined in Ref. 10.

Fig. 3
Fig. 3

Simulated three-dimensional surface plot of Tpp = tan Ψpp exp(−iΔpp) of a glass cell filled with the twisted nematic mixture 5CB and 10% CB15 as a function of the incident photon energy and the orientation angle ϕ at the reduced sample temperature tr = 0.985. The director n is not tilted toward the z axis. The optical constants determined in this work from the mixture 5CB and 10% CB15 are used for the calculation. The glass optical constants determined at the same cavity material are used to model the incident and exit media. Note our result that a sign change in P causes the transformation Δps(+P) = Δsp(−P). (0 ≤ ϕπ, ψ = θ = 0°, P = +1.48 μm, d = 5.8 μm, Tc = 301.4 K).

Fig. 4
Fig. 4

Same as Fig. 3, for Tps = tan Ψps exp(iΔps).

Fig. 5
Fig. 5

Simulated three-dimensional surface plot of Tps = tan Ψps exp(iΔps) of a glass cell filled with the twisted nematic mixture 5CB and 10% CB15 as a function of the incident photon energy and the reduced sample temperature tr at the orientation angle ϕ = 45°. The same optical constants as those for Fig. 3 are used for the calculation. Note the shift of the optical activity that increases with decreasing temperature, owing to the increasing difference between the main refractive indices. 0.964 ≤ tr ≤ 0.999; ψ = θ = 0°, P = +1.48 μm, d = 5.8 μm, Tc = 301.4 K).

Fig. 6
Fig. 6

Experimental and generated data for the cross-polarization coefficients Tps, as a function of the sample temperature obtained from our measurements and the best fit for no and ne, the respective director position at the lowest interface ϕ, the cavity thickness d, and the pitch P, at pure 5CB. P = +∞, d = 3.29 μm, Tc = 308.65 K. (a) Ψps, (b) Δps.

Fig. 7
Fig. 7

Same as Fig. 6 for Tps and Tsp and 10% CB15. P = + 1.45 μm, d = 5.80 μm, Tc = 301.45 K. (a) Ψps and Ψsp, (b) Δps (upper plots) and Δsp (lower plots).

Fig. 8
Fig. 8

Same as Fig. 7 at 5CB and 25% CB15. P = +0.53 μm, d = 3.19 μm, Tc = 293.15 K. (a) Ψps and Ψsp, (b) Δps (upper plots) and Δsp (lower plots). Excellent agreement between theoretical and experimental data is obtained. Owing to the decreasing length of the pitch P, a strongly varying optical behavior is seen through Figs. 6–8 with increased CB15 concentration. This figure shows the observed spectral region of selective reflection (~1.3–1.7 eV). (See comments in the text concerning the deviations between the measured and the calculated phases, Δps and Δsp, below 1.3 eV.)

Fig. 9
Fig. 9

Ordinary and extraordinary refractive indices of 5CB and 10% CB15 at particular reduced sample temperatures. The optical constants were obtained from a regression that modeled their spectral and temperature dependencies through a known parametrization and application of the 4 × 4 matrix algebra to predict the measured data as described in the text. The refractive indices of 5CB and of 5CB and 25% CB15 were found to be the same within our error limit of 1% as those from 5CB and 10% CB15 shown here if considered at the same reduced sample temperature.

Tables (2)

Tables Icon

Table 1 Sample Configurations and Results from Best Fit

Tables Icon

Table 2 Optical Constants of the Nematic Mixtures 5CB and CB15 As a Function of Reduced Sample Temperature tr = T (K)/Tc (K) at λ = 436 nm

Equations (33)

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z Ψ ( z ) = i k 0 Δ ( z ) Ψ ( z ) , Ψ ( z ) = ( E x , E y , H x , H y ) T ( z ) ,             k 0 ω c ,
Δ = [ - k x z x z z - k x z y z z 0 1 - k x 2 z z 0 0 - 1 0 y z z x z z - y x k x 2 - y y + y z z y z z 0 k x y z z z x x - x z z x z z x y - x z z y z z 0 - k x x z z z ] .
Ψ ( z + d ) = exp ( i k 0 Δ d ) Ψ ( z ) = T p Ψ ( z ) , T p exp ( i k 0 Δ d ) .
A = [ cos ζ - sin ζ 0 sin ζ cos ζ 0 0 0 1 ] ,             ζ = 2 π P z ,
( ζ ) A ( ζ ) [ x x x y x z y x y y , y z z x z y z z ] A ( ζ ) - 1 ,
Ψ ˜ ( ζ ) Γ Ψ ( ζ ) ,             Γ = [ 1 i 0 0 1 - i 0 0 0 0 1 i 0 0 1 - i ] ,
Ψ ˜ ( ζ ) = ( E + , E - , H + , H - ) T ( ζ ) , H ± = H x ± i H y , E ± = E x ± i E y .
F ( ζ ) diag [ exp ( i ζ ) ,             exp ( - i ζ ) ,             exp ( i ζ ) ,             exp ( - i ζ ) ] ,
Ψ ˜ ( ζ ) = F ( ζ ) Φ ˜ ( ζ ) .
z Φ ˜ ( ζ ) = i k 0 Δ ˜ ( ζ ) Φ ˜ ( ζ ) , Δ ˜ ( ζ ) = U + k x V ( ζ ) + k x 2 2 W ( ζ ) .
Δ ˜ ( k x = 0 ) U = [ - n 0 - i 0 0 n 0 i i p s - i r - n 0 s + i r - i p 0 n ] ,
n = 2 π k 0 P , p = 1 2 z z [ z z ( x x + y y ) - x z 2 - y z 2 ] , r = 1 2 z z [ z z ( y y - x x ) + x z 2 - y z 2 ] , s = 1 z z ( x z y z - x y z z ) .
q ± n 2 + p ± χ ,             χ = s 2 + r 2 + 4 n 2 p ,
Φ ˜ ( d ) = exp ( i k 0 U d ) Φ ˜ ( 0 ) = ( β 0 E + β 1 U + β 2 U 2 + β 3 U 3 ) Φ ˜ ( 0 ) ,
exp ( i k 0 q k d ) = β j q k j , j , k = 1 , , 4 , q 1 / 2 = ± q + , q 3 / 4 = ± q - .
β 0 = ( q + 2 cos κ - - q - 2 cos κ + ) / ( 2 χ ) , β 1 = i [ ( q + 2 / q - ) sin κ - - ( q - 2 / q + ) sin κ + ] / ( 2 χ ) , β 2 = ( cos κ + - cos κ - ) / ( 2 χ ) , β 3 = i [ ( 1 / q + ) sin κ + - ( 1 / q - ) sin κ - ] / ( 2 χ ) ,
κ ± k 0 d q ± .
T p ( d ) = Γ - 1 F ( ζ = 2 π P d ) ( β 0 E + β 1 U + β 2 U 2 + β 3 U 3 ) Γ .
T p - 1 ( d ) = Γ - 1 ( β 0 E - β 1 U + β 2 U 2 - β 3 U 3 ) × F ( ζ = - 2 π P d ) Γ .
T = L a - 1 T p L f .
[ A s B s A p B p ] T = T [ C s 0 C p 0 ) T ,
L a - 1 = 1 2 [ 0 1 - 1 / n a 0 0 1 1 / n a 0 1 0 0 1 / n a - 1 0 0 1 / n a ] , L f = [ 0 0 1 0 1 0 0 0 - n f 0 0 0 0 0 n f 0 ] ,
[ C p C s ] = [ t p p t s p t p s t s s ] [ A p A s ] .
t p s t p p T p s = tan Ψ p s exp ( i Δ p s ) , t s p t s s T s p = tan Ψ s p exp ( i Δ s p ) .
t p p t s s T p p = tan Ψ p p exp ( i Δ p p ) .
T p p = T 11 T 33 ,             T p s = - T 31 T 11 ,             T s p = - T 13 T 33 .
I ( t ) = I 0 ( 1 + α cos 2 Ω t + β sin 2 Ω t ) ,
α = T p p + T s p tan P exp ( i Δ 0 ) 2 - T p p T p s + tan P exp ( i Δ 0 ) 2 T p p + T s p tan P exp ( i Δ 0 ) 2 + T p p T p s + tan P exp ( i Δ 0 ) 2 ,
β = 2 Re { [ T p p + T s p tan P exp ( i Δ 0 ) ] [ T p p T p s + tan P exp ( i Δ 0 ) ] } T p p + T s p tan P exp ( i Δ 0 ) 2 + T p p T p s + tan P exp ( i Δ 0 ) 2 .
MSE = 1 N i = 1 N { [ α i m - α i calc ( T p p , T p s , T s p ; P , Δ 0 ) δ α i m ] 2 + [ β i m - β i calc ( T p p , T p s , T s p ; P , Δ 0 ) δ β i m ] 2 } ,
n i 2 [ λ ( n m ) , T ( K ) T ct ( K ) ] = ( c 0 + c 1 λ 2 ) 2 + κ i ( e 0 + e 1 λ 2 + e 2 λ 4 ) 2 × ( 1 - T T ct ) α ,             i = e , o ,
κ e = 2 3 ,             κ o = - 1 3 .
MSE = 1 N i = 1 N j = 1 M [ ( T p p - T p p calc δ T p p ) 2 + ( T p s - T p s calc δ T p s ) 2 + ( T s p - T s p calc δ T s p ) 2 ] ( ω i , T j ) ,

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