Abstract

A vector formalism is used to deduce the formulas for calculating the refractive indices as functions of the angle of incidence and the direction of the refracted rays when light is refracted by a biaxial, birefringent crystal.

© 1987 Optical Society of America

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References

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  1. M. C. Simon, “Ray tracing formulas for monoaxial optical components,” Appl. Opt. 22, 354 (1983).
    [Crossref] [PubMed]
  2. M. C. Simon, R. M. Echarri, “Ray tracing formulas for monoaxial optical components: vectorial formulation,” Appl. Opt. 25, 1935 (1986).
    [Crossref] [PubMed]
  3. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 681.

1986 (1)

1983 (1)

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Figures (4)

Fig. 1
Fig. 1

Vector diagram for a plane wave in a biaxial crystal. The vectors ′ and ″ are perpendicular to the ( D , R ^ ) and ( D , R ^) planes, respectively.

Fig. 2
Fig. 2

Coordinate systems. For x < 0 the medium is isotropic, and for x > 0 the medium is anisotropic.

Fig. 3
Fig. 3

Curves for the indices of refraction in mica as functions of the angle of incidence α. The angle between the plane of incidence and the plane containing the optical axes is γ.

Fig. 4
Fig. 4

Rays and normals in the case of light refraction in a biaxial crystal.

Equations (45)

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N ^ × H = - u D ,
N ^ × E = μ 0 u H ,
N ^ · D = 0 ,
N ^ · H = 0 ,
D = ¯ E ,
R ˜ = E × H E × H .
N 1 2 ( u 2 - u 1 2 ) ( u 2 - u 2 2 ) + N 2 2 ( u 2 - u 3 2 ) ( u 2 - u 1 2 ) + N 3 2 ( u 2 - u 1 2 ) ( u 2 - u 2 2 ) = 0 ,
D · D = 0.
E = ¯ D ,             E = ¯ D ,
g 2 = 1 j = 1 3 N j 2 ( u 2 - u j 2 ) 2 ,
R j = [ u 2 + g 2 ( u 2 - u j 2 ) ] N j u 4 + g 2 ,             j = 1 , 2 , 3 ,
u j = c n j ,             u = c / n ,
R ^ = N ^ 1 + h 2 + h 2 1 + h 2 n 1 2 n 1 2 - n 2 ( N ^ · z ^ 1 ) z ^ 1 + h 2 1 + h 2 n 2 2 n 2 2 - n 2 ( N ^ · z ^ 2 ) z ^ 2 + h 2 1 + h 2 n 3 2 n 3 2 - n 2 ( N ^ · z ^ 3 ) z ^ 3 ,
h 2 = 1 1 3 n j 4 N j 2 ( n j 2 - n 2 ) 2 .
S ^ · y ^ = S ^ · y ^ ,             S ^ · z ^ = S ^ · z ^ ,
S ^ · y ^ u i = N ^ · y ^ u ,         S ^ · z ^ u i = N ^ · z ^ u ,
x ^ = n ^ ,
y ^ = n ^ × z ^ 1 n ^ × z ^ 1 ,
z ^ = n ^ × n ^ × z ^ 1 n ^ × z ^ 1 .
N ^ · y ^ = w ( S ^ · y ^ ) ,
N ^ · z ^ = w ( S ^ · z ^ ) ,
( N ^ · x ^ ) 2 + ( N ^ · y ^ ) 2 + ( N ^ · z ^ ) 2 = 1 ,
N 1 2 ( w 2 - w 2 2 ) ( w 2 - w 3 2 ) + N 2 2 ( w 2 - w 3 2 ) ( w 2 - w 1 2 ) + N 3 2 ( w 2 - w 1 2 ) ( w 2 - w 2 2 ) = 0 ,
w = u / u i ,
w 1 = u 1 / u i ,             w 2 = u 2 / u i ,             w 3 = u 3 / u i .
( N ^ · x ^ ) 2 = 1 - w 2 δ ,
δ = ( S ^ · y ^ ) 2 + ( S ^ · z ^ ) 2 .
N 1 = ( N ^ · x ^ ) ( z ^ 1 · x ^ ) + ( N ^ · y ^ ) ( z ^ 1 · y ^ ) + ( N ^ · z ^ ) ( z ^ 1 · z ^ ) , N 2 = ( N ^ · x ^ ) ( z ^ 2 · x ^ ) + ( N ^ · y ^ ) ( z ^ 2 · y ^ ) + ( N ^ · z ^ ) ( z ^ 2 · z ^ ) , N 3 = ( N ^ · x ^ ) ( z ^ 3 · x ^ ) + ( N ^ · y ^ ) ( z ^ 3 · y ^ ) + ( N ^ · z ^ ) ( z ^ 3 · z ^ ) .
( z ^ 1 · y ^ ) = 0 ,
N 1 2 = w 2 α 1 + 2 w ( N ^ · x ^ ) β 1 + γ 1 , N 2 2 = w 2 α 2 + 2 w ( N ^ · x ^ ) β 2 + γ 2 , N 3 2 = w 2 α 3 + 2 w ( N ^ · x ^ ) β 3 + γ 3 ,
α 1 = ( S ^ · z ^ ) 2 ( z ^ 1 · z ^ ) 2 - δ ( z ^ 1 · x ^ ) 2 , β 1 = ( S ^ · z ^ ) ( z ^ 1 · x ^ ) ( z ^ 1 · z ^ ) , γ 1 = ( z ^ 1 · x ^ ) 2 ,
α 2 = [ ( S ^ · y ^ ) ( z ^ 2 · y ^ ) + ( S ^ · z ^ ) ( z ^ 2 · z ^ ) ] 2 - δ ( z ^ 2 · x ^ ) 2 , β 2 = [ ( S ^ · y ^ ) ( z ^ 2 · y ^ ) + ( S ^ · z ^ ) ( z ^ 2 · z ^ ) ] 2 ( z ^ 2 · x ^ ) , γ 2 = ( z ^ 2 · x ^ ) 2 ,
α 3 = [ ( S ^ · y ^ ) ( z ^ 3 · y ^ ) + ( S ^ · z ^ ) ( z ^ 3 · z ^ ) ] 2 - δ ( z ^ 3 · x ^ ) 2 , β 3 = [ ( S ^ · y ^ ) ( z ^ 3 · y ^ ) + ( S ^ · z ^ ) ( z ^ 3 · z ^ ) ] 2 ( z ^ 3 · x ^ ) , γ 3 = ( z ^ 3 · x ^ ) 2 .
N 1 2 + N 2 2 + N 3 2 = 1 ,
α 1 + α 2 + α 3 = 0 , β 1 + β 2 + β 3 = 0 , γ 1 + γ 2 + γ 3 = 1.
w 4 ( 1 + a ) + w 2 ( p - d + c ) + r = 2 ( N ^ · x ^ ) ( - w 3 b - w q ) ,
- a = α 1 ( w 2 2 + w 3 2 ) + α 2 ( w 3 2 + w 1 2 ) + α 3 ( w 1 2 + w 2 2 ) , - b = β 1 ( w 2 2 + w 3 2 ) + β 2 ( w 3 2 + w 1 2 ) + β 3 ( w 1 2 + w 2 2 ) d - c = γ 1 ( w 2 2 + w 3 2 ) + γ 2 ( w 3 2 + w 1 2 ) + γ 3 ( w 1 2 + w 2 2 ) , p = α 1 w 2 2 w 3 2 + α 2 w 3 2 w 1 2 + α 3 w 1 2 w 2 2 , q = β 1 w 2 2 w 3 2 + β 2 w 3 2 w 1 2 + β 3 w 1 2 w 2 2 , r = γ 1 w 2 2 w 3 2 + γ 2 w 1 2 w 3 2 + γ 3 w 1 2 w 2 2 .
w 8 A + w 6 B + w 4 C + w 2 D + E = 0 ,
A = ( 1 + a ) 2 + 4 δ b 2 , B = 2 ( 1 + a ) ( p - d + c ) - 4 b 2 + 8 b q δ , C = ( p - d + c ) 2 + 2 ( 1 + a ) r - 8 b q + 4 δ q 2 , D = 2 ( p - d + c ) r - 4 q 2 , E = r 2 .
n = 1.5936 , β = 26.34° , n = 1.5900 , β = 26.40° .
ρ = 26.34° , ρ = 27.51° ,
n = 1.5868 , β = 26.46° , n = 1.5977 , β = 26.27° ,
ρ = 27.44° , ρ = 26.27° .
n = 1.5879 , β = 26.44° , n = 1.5961 , β = 26.30° .
ρ = 27.42° , δ = 44.50° , ρ = 26.37° , δ = 45.67° .

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