Abstract

The angle-dependent reflectance at the boundary between an isotropic and a uniaxial material has been fully analyzed, and as a consequence a new technique for complete characterization of the optical tensor of a uniaxial crystal has been developed and demonstrated. For the general uniaxial case, for both positive and negative uniaxial crystals both analytic formulation and numerical modeling show that if the TE-to-TM conversion reflectivity is measured then two sharp cusps appear at critical angles that correspond to the limits of generation of the ordinary and the extraordinary modes in the uniaxial crystal. (For the more conventional polarization-conserving reflectivity there is only one sharp critical edge, the second critical edge being rounded owing to polarization conversion.) It is also found that, for a general orientation of the uniaxial axis with respect to both the plane of incidence and the crystal surface, rotating the plane of incidence by 180° results in a different polarization-conversion signal, thereby permitting a completely unambiguous determination of the Euler angles of the optical tensor. Based on the analytic and numerical modeling results, a novel and accurate optical technique, the polarization-conversion reflectivity technique, has been developed and used to characterize the optical tensor of a single crystal of calcite. The results show that, as well as an accurate optical-tensor characterization, a precise determination of the axis of symmetry of the crystal is simple without recourse to x-ray diffraction.

© 1994 Optical Society of America

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References

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  1. D. Riviere, Y. Levy, and C. Imbert, “Determination of liquid crystal refractive indices from critical angle measurements,” Opt. Commun. 25, 206–210 (1978).
    [Crossref]
  2. Y. Levy, D. Riviere, C. Imbert, and M. Boix, “Détermination des angles d’obliquitié d’un crystal liquide en phase nématique an voisinage d’une surface,” Opt. Commun. 26, 225–227 (1978).
    [Crossref]
  3. K. Eidner, G. Mayer, and R. Schuster, “Determination of liquid crystal parameters by the attenuated total reflection technique,” Phys. Lett. A 118, 149–151 (1986).
    [Crossref]
  4. F. Bloisi, L. Vicari, and F. Simoni, “Nonlinear liquid interfaces: determination of the local director orientation,” Nuovo Cimento 12D, 1273 (1990).
    [Crossref]
  5. F. Bloisi, L. Vicari, and F. Simoni, “Determination of the director orientation inside a hybrid nematic cell by total internal reflection,” Mol. Cryst. Liq. Cryst. 179, 45 (1990).
  6. M. Warenghem, M. Ismaili, and D. Hector, “Experimental determination of nematic director distribution in the vicinity of the interface by reflectivity measurements,” J. Phys. III (Paris) 2, 765–775 (1992).
  7. F. Yang and J. R. Sambles, “Critical angles for reflectivity at an isotropic–anisotropic boundary,” J. Mod. Opt. 40, 1131–1142 (1993).
    [Crossref]
  8. M. Born and E. Wolfe, Principles of Optics, 5th ed. (Pergamon, New York, 1975).
  9. F. Yang and J. R. Sambles, “Optical characterization of liquid crystals by means of half-leaky guided modes,” J. Opt. Soc. Am. B 10, 858–866 (1993).
    [Crossref]

1993 (2)

F. Yang and J. R. Sambles, “Critical angles for reflectivity at an isotropic–anisotropic boundary,” J. Mod. Opt. 40, 1131–1142 (1993).
[Crossref]

F. Yang and J. R. Sambles, “Optical characterization of liquid crystals by means of half-leaky guided modes,” J. Opt. Soc. Am. B 10, 858–866 (1993).
[Crossref]

1992 (1)

M. Warenghem, M. Ismaili, and D. Hector, “Experimental determination of nematic director distribution in the vicinity of the interface by reflectivity measurements,” J. Phys. III (Paris) 2, 765–775 (1992).

1990 (2)

F. Bloisi, L. Vicari, and F. Simoni, “Nonlinear liquid interfaces: determination of the local director orientation,” Nuovo Cimento 12D, 1273 (1990).
[Crossref]

F. Bloisi, L. Vicari, and F. Simoni, “Determination of the director orientation inside a hybrid nematic cell by total internal reflection,” Mol. Cryst. Liq. Cryst. 179, 45 (1990).

1986 (1)

K. Eidner, G. Mayer, and R. Schuster, “Determination of liquid crystal parameters by the attenuated total reflection technique,” Phys. Lett. A 118, 149–151 (1986).
[Crossref]

1978 (2)

D. Riviere, Y. Levy, and C. Imbert, “Determination of liquid crystal refractive indices from critical angle measurements,” Opt. Commun. 25, 206–210 (1978).
[Crossref]

Y. Levy, D. Riviere, C. Imbert, and M. Boix, “Détermination des angles d’obliquitié d’un crystal liquide en phase nématique an voisinage d’une surface,” Opt. Commun. 26, 225–227 (1978).
[Crossref]

Bloisi, F.

F. Bloisi, L. Vicari, and F. Simoni, “Nonlinear liquid interfaces: determination of the local director orientation,” Nuovo Cimento 12D, 1273 (1990).
[Crossref]

F. Bloisi, L. Vicari, and F. Simoni, “Determination of the director orientation inside a hybrid nematic cell by total internal reflection,” Mol. Cryst. Liq. Cryst. 179, 45 (1990).

Boix, M.

Y. Levy, D. Riviere, C. Imbert, and M. Boix, “Détermination des angles d’obliquitié d’un crystal liquide en phase nématique an voisinage d’une surface,” Opt. Commun. 26, 225–227 (1978).
[Crossref]

Born, M.

M. Born and E. Wolfe, Principles of Optics, 5th ed. (Pergamon, New York, 1975).

Eidner, K.

K. Eidner, G. Mayer, and R. Schuster, “Determination of liquid crystal parameters by the attenuated total reflection technique,” Phys. Lett. A 118, 149–151 (1986).
[Crossref]

Hector, D.

M. Warenghem, M. Ismaili, and D. Hector, “Experimental determination of nematic director distribution in the vicinity of the interface by reflectivity measurements,” J. Phys. III (Paris) 2, 765–775 (1992).

Imbert, C.

D. Riviere, Y. Levy, and C. Imbert, “Determination of liquid crystal refractive indices from critical angle measurements,” Opt. Commun. 25, 206–210 (1978).
[Crossref]

Y. Levy, D. Riviere, C. Imbert, and M. Boix, “Détermination des angles d’obliquitié d’un crystal liquide en phase nématique an voisinage d’une surface,” Opt. Commun. 26, 225–227 (1978).
[Crossref]

Ismaili, M.

M. Warenghem, M. Ismaili, and D. Hector, “Experimental determination of nematic director distribution in the vicinity of the interface by reflectivity measurements,” J. Phys. III (Paris) 2, 765–775 (1992).

Levy, Y.

Y. Levy, D. Riviere, C. Imbert, and M. Boix, “Détermination des angles d’obliquitié d’un crystal liquide en phase nématique an voisinage d’une surface,” Opt. Commun. 26, 225–227 (1978).
[Crossref]

D. Riviere, Y. Levy, and C. Imbert, “Determination of liquid crystal refractive indices from critical angle measurements,” Opt. Commun. 25, 206–210 (1978).
[Crossref]

Mayer, G.

K. Eidner, G. Mayer, and R. Schuster, “Determination of liquid crystal parameters by the attenuated total reflection technique,” Phys. Lett. A 118, 149–151 (1986).
[Crossref]

Riviere, D.

D. Riviere, Y. Levy, and C. Imbert, “Determination of liquid crystal refractive indices from critical angle measurements,” Opt. Commun. 25, 206–210 (1978).
[Crossref]

Y. Levy, D. Riviere, C. Imbert, and M. Boix, “Détermination des angles d’obliquitié d’un crystal liquide en phase nématique an voisinage d’une surface,” Opt. Commun. 26, 225–227 (1978).
[Crossref]

Sambles, J. R.

F. Yang and J. R. Sambles, “Critical angles for reflectivity at an isotropic–anisotropic boundary,” J. Mod. Opt. 40, 1131–1142 (1993).
[Crossref]

F. Yang and J. R. Sambles, “Optical characterization of liquid crystals by means of half-leaky guided modes,” J. Opt. Soc. Am. B 10, 858–866 (1993).
[Crossref]

Schuster, R.

K. Eidner, G. Mayer, and R. Schuster, “Determination of liquid crystal parameters by the attenuated total reflection technique,” Phys. Lett. A 118, 149–151 (1986).
[Crossref]

Simoni, F.

F. Bloisi, L. Vicari, and F. Simoni, “Nonlinear liquid interfaces: determination of the local director orientation,” Nuovo Cimento 12D, 1273 (1990).
[Crossref]

F. Bloisi, L. Vicari, and F. Simoni, “Determination of the director orientation inside a hybrid nematic cell by total internal reflection,” Mol. Cryst. Liq. Cryst. 179, 45 (1990).

Vicari, L.

F. Bloisi, L. Vicari, and F. Simoni, “Determination of the director orientation inside a hybrid nematic cell by total internal reflection,” Mol. Cryst. Liq. Cryst. 179, 45 (1990).

F. Bloisi, L. Vicari, and F. Simoni, “Nonlinear liquid interfaces: determination of the local director orientation,” Nuovo Cimento 12D, 1273 (1990).
[Crossref]

Warenghem, M.

M. Warenghem, M. Ismaili, and D. Hector, “Experimental determination of nematic director distribution in the vicinity of the interface by reflectivity measurements,” J. Phys. III (Paris) 2, 765–775 (1992).

Wolfe, E.

M. Born and E. Wolfe, Principles of Optics, 5th ed. (Pergamon, New York, 1975).

Yang, F.

F. Yang and J. R. Sambles, “Optical characterization of liquid crystals by means of half-leaky guided modes,” J. Opt. Soc. Am. B 10, 858–866 (1993).
[Crossref]

F. Yang and J. R. Sambles, “Critical angles for reflectivity at an isotropic–anisotropic boundary,” J. Mod. Opt. 40, 1131–1142 (1993).
[Crossref]

J. Mod. Opt. (1)

F. Yang and J. R. Sambles, “Critical angles for reflectivity at an isotropic–anisotropic boundary,” J. Mod. Opt. 40, 1131–1142 (1993).
[Crossref]

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

J. Phys. III (Paris) (1)

M. Warenghem, M. Ismaili, and D. Hector, “Experimental determination of nematic director distribution in the vicinity of the interface by reflectivity measurements,” J. Phys. III (Paris) 2, 765–775 (1992).

Mol. Cryst. Liq. Cryst. (1)

F. Bloisi, L. Vicari, and F. Simoni, “Determination of the director orientation inside a hybrid nematic cell by total internal reflection,” Mol. Cryst. Liq. Cryst. 179, 45 (1990).

Nuovo Cimento (1)

F. Bloisi, L. Vicari, and F. Simoni, “Nonlinear liquid interfaces: determination of the local director orientation,” Nuovo Cimento 12D, 1273 (1990).
[Crossref]

Opt. Commun. (2)

D. Riviere, Y. Levy, and C. Imbert, “Determination of liquid crystal refractive indices from critical angle measurements,” Opt. Commun. 25, 206–210 (1978).
[Crossref]

Y. Levy, D. Riviere, C. Imbert, and M. Boix, “Détermination des angles d’obliquitié d’un crystal liquide en phase nématique an voisinage d’une surface,” Opt. Commun. 26, 225–227 (1978).
[Crossref]

Phys. Lett. A (1)

K. Eidner, G. Mayer, and R. Schuster, “Determination of liquid crystal parameters by the attenuated total reflection technique,” Phys. Lett. A 118, 149–151 (1986).
[Crossref]

Other (1)

M. Born and E. Wolfe, Principles of Optics, 5th ed. (Pergamon, New York, 1975).

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Figures (17)

Fig. 1
Fig. 1

Geometry for an isotropic–uniaxial system.

Fig. 2
Fig. 2

Relation between the refracted waves and the refractive-index ellipsoid.

Fig. 3
Fig. 3

Relations among the incident, reflected, and transmitted fields at an interface between an isotropic and a uniaxial medium.

Fig. 4
Fig. 4

Reflectivities plotted against the incidence angle β for an isotropic–positive uniaxial system for a tilt angle of θ = 45° with a twist angle of (a) ϕ = 0° and (b) ϕ = 90°.

Fig. 5
Fig. 5

Reflectivity Rsp plotted against the incidence angle β for five different combinations of θ and ϕ for an isotropic–positive uniaxial system. For every curve the tilt angle is ϕ = 45°, and the twist angles θ for curves 1, 2, 3, 4, and 5 are 0°, 20°, 40°, 60,° and 80°, respectively.

Fig. 6
Fig. 6

Reflectivity Rsp plotted against the incidence angle β for four different combinations of θ and ϕ for an isotropic–positive uniaxial system. For every curve the twist angle is ϕ = −45°, and the tilt angles θ for curves 1, 2, 3, and 4 are 90°, 85°, 80°, and 75°, respectively.

Fig. 7
Fig. 7

Reflectivity Rsp plotted against the incidence angle β for an isotropic–positive uniaxial system. The tilt angle is ϕ = 45°, and the twist angles are ϕ = 45° for the solid line and ϕ = −135° for the dashed line.

Fig. 8
Fig. 8

Reflectivity Rsp plotted against the incidence angle β for five different combinations of θ and ϕ for an isotropic–negative uniaxial system. For every curve the tilt angle is θ = 45°, and the twist angles ϕ for curves 1, 2, 3, 4, and 5 are 0°, 20°, 40°, 60,° and 80°, respectively.

Fig. 9
Fig. 9

Reflectivity Rsp plotted against the incidence angle β for four different combinations of θ and ϕ for an isotropic–negative uniaxial system. For every curve the twist angle is ϕ = −45° and the tilt angles θ for curves 1, 2, 3, and 4 are 90°, 85°, 80°, and 75°, respectively.

Fig. 10
Fig. 10

Reflectivity Rsp plotted against the incidence angle β for an isotropic–negative uniaxial system. The tilt angle is θ = 45°, and the twist angles are ϕ = 45° for the solid curve and ϕ = −135° for the dashed curve.

Fig. 11
Fig. 11

Sample geometry for the PCTR technique.

Fig. 12
Fig. 12

Angle-dependent reflectivity Rpp data for the calcite crystal when the twist angle ϕ = ±90°.

Fig. 13
Fig. 13

Angle-dependent reflectivity Rsp data for the calcite crystal when the twist angle ϕ = 0° (or ±180°).

Fig. 14
Fig. 14

Angle-dependent reflectivity Rsp data for the calcite crystal at various twist angles. For curve 1 the twist angle is ϕ = 0° (or ±180°), and the differences of the twist angles from curves 1 are 20°, 35°, 45°, and 75° for curves 2–5.

Fig. 15
Fig. 15

For the angle-dependent reflectivity Rsp, experimental data (small crosses that blend to form the heavy curve) shown in Fig. 14, curve 3, compared with numerically modeled curves 1 (ϕ = −145°) and 2 (ϕ = 35°).

Fig. 16
Fig. 16

Numerically modeled results of Rsp with a 3.0-μm-thick matching fluid of permittivity 1′ = 2.9756. For the rest of the geometry the parameters are 1 = 2.9950, θ = 44.76°, ϕ = 0°, || = 2.2043, and = 2.7414.

Fig. 17
Fig. 17

Angle-dependent reflectivity Rsp plotted against the incidence angle β for an isotropic–negative uniaxial system. The parameters used in the modeling are 1 = 3.2400, || = 2.2430, = 2.8500, ϕ = 0°, and θ = 89.5°, 89.25°, 89°, and 88.75° for curves 1, 2, 3, and 4, respectively.

Equations (136)

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cos ψ = cos θ cos α - sin θ sin ϕ sin α .
n e = ( cos 2 ψ e + sin 2 ψ e ) 1 / 2 .
n 1 sin β = n e sin α e
cot α e = Δ sin θ cos θ sin ϕ + Δ cos 2 θ + ( / 1 ) 1 / 2 ( sin β ) ( + Δ cos 2 θ ) [ ( + Δ cos 2 θ ) - 1 ( sin 2 β ) ( - Δ sin 2 θ cos 2 ϕ ) ] 1 / 2 ,
β c e = sin - 1 [ ( + Δ cos 2 θ ) 1 ( - Δ sin 2 θ cos 2 ϕ ) ] 1 / 2 ,
α e max = cot - 1 ( Δ sin θ cos θ sin ϕ + Δ cos 2 θ )
β c o = sin - 1 ( / 1 ) 1 / 2 .
cos γ o x = sin α o cos θ + sin θ sin ϕ cos α o sin ψ o ,
cos γ o y = - sin θ cos ϕ cos α o sin ψ o ,
cos γ o z = - sin θ cos ϕ sin α o sin ψ o ,
cos ψ o = cos θ cos α o - sin θ sin ϕ sin α o ,
sin α o = n 1 n o sin β .
cos γ e x = sin θ cos ϕ sin ψ e ,
cos γ e y = ( cos θ sin α e + sin θ sin ϕ cos α e ) cos α e sin ψ e ,
cos γ e z = ( cos θ sin α e + sin θ sin ϕ cos α e ) sin α e sin ψ e ,
cos ψ e = cos θ cos α e - sin θ sin ϕ sin α e
tan δ = tan ψ e ,
η = ψ - δ = ψ e - tan - 1 ( tan ψ e ) .
sin η = ( - ) sin ψ e cos ψ e ( 2 cos 2 ψ e + 2 sin 2 ψ e ) 1 / 2 ,
cos η = cos 2 ψ e + sin 2 ψ e ( 2 cos 2 ψ e + 2 sin 2 ψ e ) 1 / 2 .
( r - A ) cos β = t o cos γ o y + t e ( cos η cos γ e y + sin η sin α e ) ,
r + A = t o cos γ o x + t e cos η cos γ e x ,
n 1 ( A - r ) cos β = n o t o cos γ o x cos α o + n e t e cos η cos γ e x cos α e ,
- n 1 ( r + A ) = n o t o ( cos γ o y cos α o + cos γ o z sin α o ) + n e t e cos η ( cos γ e y cos α e + cos γ e z sin α e ) .
r p p = r A = n 1 ( d 1 - d 2 ) - ( cos β ) ( d 3 - d 4 ) D ,
r s s = r A = n 1 ( d 1 - d 2 ) + ( cos β ) ( d 3 - d 4 ) D ,
r s p = r A = 2 n 1 ( sin θ cos ϕ cos β ) [ ( cos θ ) q - ( sin θ sin ϕ ) q ] Δ D ,
r p s = r A = 2 n 1 ( sin θ cos ϕ cos β ) [ ( cos θ ) q + ( sin θ sin ϕ ) q ] Δ D ,
D = n 1 ( d 1 + d 2 ) + ( cos β ) ( d 3 + d 4 ) ,
d 1 = ( cos 2 β ) ( 2 + m q ) ,
d 2 = q ( l q - q 2 + p ) ,
d 3 = 1 ( l q - q 2 + 33 ) ,
d 4 = ( l q - m q 2 + ) ,
l = [ ( 33 - m 1 sin 2 β ) ] 1 / 2 ,
33 = + Δ cos 2 θ ,
m = - Δ sin 2 θ cos 2 ϕ ,
p = - Δ sin 2 θ sin 2 ϕ ,
q = ( - 1 sin 2 β ) 1 / 2 ,
q = n 1 sin β .
r s p = U + i V S + i T ,
R s p = r s p r s p * = U 2 + V 2 S 2 + T 2 .
R s p β = 2 ( S 2 + T 2 ) ( U U β + V V β ) - 2 ( U 2 + V 2 ) ( S S β + T T β ) ( S 2 + T 2 ) 2 .
U = 2 n 1 2 ( sin θ cos θ cos ϕ cos β sin β ) Δ ,
V = - 2 n 1 ( sin 2 θ cos ϕ sin ϕ cos β ) q Δ ,
S = n 1 l cos 2 β + n 1 l q 2 + n 1 2 ( cos β ) ( 33 - q 2 ) + ( cos β ) ( - m q 2 ) ,
T = q [ n 1 m cos 2 β + n 1 ( p - q 2 ) + n 1 2 l cos β + l cos β ] ,
q = ( n 1 2 sin 2 β - ) 1 / 2 .
l β | l 0 = - m 1 sin β cos β l - .
L 1 = n 1 ( cos β ) { [ 1 33 + - ( 1 + m ) 1 sin 2 β ] × [ 2 - ( 1 + ) sin 2 β ] + ( 1 + ) × [ p + m - ( 1 + m ) sin 2 β ] ( 1 sin 2 β - ) } ,
S = n 1 2 ( cos β ) [ ( 33 - q 2 ) - l q ] + ( cos β ) ( - m q 2 - l q ) ,
T = n 1 ( cos 2 β ) ( l + m q ) + n 1 q [ ( p - q 2 ) - l q ] ,
l = n o n e ( m e 1 sin 2 β - 33 ) 1 / 2 .
l β | l 0 = m 1 sin β cos β l + .
L 2 = q { 1 [ p + m - ( 1 + m ) sin 2 β ] × [ 2 - ( 1 + ) sin 2 β ] - ( cos 2 β ) [ 1 33 + - ( 1 + m ) 1 sin 2 β ] × ( 1 + ) } ,
L 1 = n 1 ( cos β ) Δ ( 1 - ) 1 ( - Δ sin 2 θ cos 2 ϕ ) × [ Δ ( sin 2 θ cos 2 ϕ ) ( cos 2 θ + 1 sin 2 θ ) + ( 1 cos 2 θ + 2 sin 2 θ cos 2 ϕ ) ] ,
L 2 = q ( 1 - ) 1 ( - Δ sin 2 θ cos 2 ϕ ) × { Δ [ ( 1 - ) ( 1 - ) cos 2 ϕ + sin 2 ϕ ] × sin 2 θ cos 2 θ + Δ × [ 2 sin 2 ϕ + ( 1 - ) 2 cos 2 ϕ ] sin 4 θ + 2 ( 1 - ) } ,
R s p β | β β c e - + ,
R s p β | β β c e + - .
U = 2 n 1 ( sin θ cos ϕ cos β ) [ ( cos θ ) q - ( sin θ sin ϕ ) q ] × Δ ,
V = 0 ,
S = n 1 [ ( cos 2 β ) ( l + m q ) + q ( l q - q 2 + p ) ] + ( cos β ) [ n 1 2 ( I q - q 2 + 33 ) + ( l q - m q 2 - ) ] ,
T = 0.
q β | q 0 = - 1 sin β cos β q - .
R s p β = 2 U ( S U β - U S β ) S 3 .
L 3 = - L 32 L 31 [ ( sin θ sin ϕ ) L 33 + ( cos θ ) L 34 ] ,
L 31 = ( n 1 L 33 ) 3 ,
L 32 = 8 ( sin 2 θ cos θ cos 2 ϕ ) × 2 1 ( Δ ) 2 ( 1 - ) ( 1 ) 1 / 2 > 0 ,
L 33 = 1 n 1 ( 1 - 1 ) × [ n 1 1 l + Δ 2 ( 1 cos 2 θ + sin 2 θ sin 2 ϕ ) ] > 0 ,
L 34 = ( 1 ) 1 / 2 × { n 1 [ m ( 1 - 1 ) + Δ ( 1 - sin 2 θ cos 2 ϕ ) ] + ( 1 + ) ( 1 - 1 ) 1 / 2 l } > 0.
R s p β | β β c o - + .
R s p β | β β c o - + .
R s p β | β β c o - - ,
R s p β | β β c o -
U = 2 n 1 2 ( sin θ cos θ cos ϕ cos β sin β ) Δ ,
V = - 2 n 1 ( sin 2 θ sin ϕ cos ϕ cos β ) q Δ ,
S = n 1 ( cos 2 β ) l + n 1 2 ( cos β ) ( 33 - q 2 ) + ( cos β ) ( - m q 2 ) - n 1 l ( q ) 2 ,
T = [ n 1 m cos 2 β + n 1 ( p - q 2 ) + n 1 2 l cos β + l cos β ] q .
q β | q 0 = 1 sin β cos β q + .
S q 0 = [ n 1 ( cos 2 β ) l + ( cos β ) × Δ ( 1 cos 2 θ + sin 2 θ cos 2 ϕ ) ] > 0.
R s p β | β β c o +
( cos θ ) q = ( sin θ sin ϕ ) q ,
β * = sin - 1 [ sin 2 θ sin 2 ϕ 1 ( cos 2 θ + sin 2 θ sin 2 ϕ ) ] 1 / 2 β c o .
U = 2 n 1 ( sin θ cos ϕ cos β ) × [ n 1 cos θ sin β - ( sin θ sin ϕ ) q ] Δ ,
V = 0 ,
S = n 1 [ ( cos 2 β ) ( l + m q ) + q ( l q - q 2 + p ) ] + ( cos β ) [ n 1 2 ( l q - q 2 + 33 ) + ( l q - m q 2 + ) ] ,
T = 0.
l β | l 0 = - 1 m sin β cos β l - .
L 4 = L 41 [ n 1 q L 42 + ( cos β ) L 43 ] ,
L 41 = n 1 cos 2 β + n 1 q 2 + n 1 2 q cos β + q cos β > 0 ,
L 42 = - Δ ( 1 33 sin 2 θ cos 2 ϕ - Δ sin 2 θ cos 2 ϕ + cos 2 θ ) > 0 ,
L 43 = 1 1 m { ( 1 - ) ( sin 2 θ cos 2 ϕ ) [ ( Δ ) 2 sin 2 θ - 2 Δ ] + 2 [ ( 1 - ) - ( - ) sin 2 θ sin 2 ϕ ] } > 0 ,
R s p β | β β c e - + .
S = n 1 q [ m + p - ( 1 + m ) sin 2 β ] + ( cos β ) [ 1 33 + - ( 1 + m ) 1 sin 2 β ] ,
T = [ n 1 cos 2 β + n 1 q 2 + ( n 1 2 + ) ( cos β ) q ] l ,
l = n o n e ( m 1 sin 2 β - 33 ) 1 / 2 .
l β | l 0 = m 1 sin β cos β l + .
T T β | l 0 = [ n 1 ( cos 2 β + q 2 ) + ( 1 + ) ( cos β ) q ] 2 × m 1 sin β cos β ,
S l 0 = n 1 q L 42 + ( cos β ) L 43 > 0 ;
R s p β | β β c e + .
S = n 1 ( m cos 2 β + p - q 2 ) q + ( cos β ) × [ 1 33 + - ( 1 + m ) q 2 ] ,
T = [ n 1 ( cos 2 β + q 2 ) + ( cos β ) ( 1 + ) q ] l .
q β | q 0 = - 1 sin β cos β q - .
R s p β = 2 U [ ( S 2 + T 2 ) U β - U ( S S β + T T β ) ] ( S 2 + T 2 ) 2 .
L 5 = - L 51 L 52 [ ( sin θ sin ϕ ) L 53 + ( cos θ ) L 54 ] ,
L 51 = 8 ( sin 2 θ cos 2 ϕ cos θ ) 1 2 ( Δ ) 2 ( 1 - ) × ( 1 ) 1 / 2 > 0 ,
L 52 = 1 L 53 2 ,
L 53 = ( 1 - ) n 1 × [ 1 ( Δ ) 2 ( 1 cos 2 θ + sin 2 θ cos 2 ϕ ) 2 + l 2 ] > 0 ,
L 54 = ( 1 ) 1 / 2 ( L 55 + L 56 ) ,
L 55 = ( - ) ( 1 - sin 2 θ sin 2 ϕ ) × ( 1 cos 2 θ + sin 2 θ cos 2 ϕ ) > 0 ,
L 56 = [ ( 1 - ) + 1 cos 2 θ ] sin 2 θ cos 2 ϕ + cos 2 θ + ( sin 4 θ cos 2 ϕ ) × ( 1 - cos 2 ϕ + cos 2 ϕ ) > 0.
R s p β | β β c o - + .
R s p β | β β c o - + ,
R s p β | β β c o - - ,
R s p β | β β c o -
S = n 1 2 ( cos β ) [ ( 33 - q 2 ) - l q ] + ( cos β ) ( - m q 2 - l q ) ,
T = n 1 ( cos 2 β ) ( l + m q ) + n 1 q [ ( p - q 2 ) - l q ] .
q β | q 0 = 1 sin β cos β q + .
q β | q 0
L 6 = l 1 ( cos 2 β ) [ ( 1 - ) + ( - ) × 1 ( 1 cos 2 θ + sin 2 θ cos 2 ϕ ) ] > 0 ,
R s p β | β β c o + - ,
R s p β | q 0 = 2 V ( V / β ) S 2 + T 2 > 0
V V β | q 0 = [ sin 2 θ sin ( 2 ϕ ) Δ ] 2 ( 1 - ) × [ ( 1 - ) ] 1 / 2 > 0.
1 sin 2 β 1 = cos 2 θ + sin 2 θ .
= 1 sin 2 β 2 ,
= 1 sin 2 β 3 ,
θ = sin - 1 ( sin 2 β 3 - sin 2 β 1 sin 2 β 3 - sin 2 β 2 ) 1 / 2 .
= 1 sin 2 β 2 ,
= 1 sin 2 β 3 ,
θ = sin - 1 ( sin 2 β 1 - sin 2 β 2 sin 2 β 3 - sin 2 β 2 ) 1 / 2 .
1 sin 2 β 1 = ( + Δ cos 2 θ ) - Δ sin 2 θ cos 2 ( 90° + Δ ϕ ) ,
1 sin 2 β 2 = ( + Δ cos 2 θ ) - Δ sin 2 θ cos 2 ( Δ ϕ ) ,
Δ θ 1 2 ( tan θ ) ( sin 2 β 1 sin 2 β 2 + sin 2 β 2 sin 2 β 1 ) ( Δ ϕ ) 2 ,
Δ sin 2 β 2 sin 2 β 1 ( - ) sin 2 θ ( Δ ϕ ) 2 .
Δ = 1 sin ( 2 β 2 ) Δ β 2 ,
Δ = 1 sin ( 2 β 3 ) Δ β 3 ,
Δ θ = [ ( sin 2 β 3 - sin 2 β 2 ) sin ( 2 β 1 ) Δ β 1 + ( sin 2 β 1 - sin 2 β 3 ) sin ( 2 β 2 ) Δ β 2 + ( sin 2 β 2 - sin 2 β 1 ) sin ( 2 β 3 ) Δ β 3 ] / [ ( sin 2 β 3 - sin 2 β 2 ) 2 sin ( 2 θ ) ] .

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