Abstract

Formulas for the calculation of the direction cosines of refracted and internally reflected rays in anisotropic uniaxial crystals are presented. The method is based on a transformation to a nonorthonormal coordinate system in which the normal surface associated with the extraordinary ray is of spherical shape. A numerical example for the case of refraction and internal reflection in calcite is given.

© 1998 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  4. J. D. Trollinger, R. A. Chipman, D. K. Wilson, “Polarization ray tracing in birefringent media,” Optical Engineering 30, 461–466 (1991).
    [CrossRef]
  5. W. Q. Zhang, “General ray-tracing formulas for crystal,” Appl. Opt. 31, 7328–7331 (1992).
    [CrossRef] [PubMed]
  6. W. Q. Zhang, “Photon tunneling in a uniaxial crystal film,” Appl. Opt. 37, 79–83 (1998).
    [CrossRef]
  7. S. C. McClain, L. W. Hillman, R. A. Chipman, “Polarization ray tracing in anisotropic optically active media: I. Algorithms,” J. Opt. Soc. Am. A 10, 2371–2382 (1993).
    [CrossRef]
  8. S. C. McClain, L. W. Hillman, R. A. Chipman, “Polarization ray tracing in anisotropic optically active media: II. Theory and physics,” J. Opt. Soc. Am. A 10, 2383–2393 (1993).
    [CrossRef]
  9. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).
  10. R. D. Guenther, Modern Optics (Wiley, New York, 1990).

1998 (1)

W. Q. Zhang, “Photon tunneling in a uniaxial crystal film,” Appl. Opt. 37, 79–83 (1998).
[CrossRef]

1993 (2)

1992 (1)

W. Q. Zhang, “General ray-tracing formulas for crystal,” Appl. Opt. 31, 7328–7331 (1992).
[CrossRef] [PubMed]

1991 (1)

J. D. Trollinger, R. A. Chipman, D. K. Wilson, “Polarization ray tracing in birefringent media,” Optical Engineering 30, 461–466 (1991).
[CrossRef]

1990 (1)

1986 (1)

1983 (1)

Chipman, R. A.

Echarri, R. M.

Guenther, R. D.

R. D. Guenther, Modern Optics (Wiley, New York, 1990).

Hillman, L. W.

Liang, Q.-T.

McClain, S. C.

Simon, M. C.

Trollinger, J. D.

J. D. Trollinger, R. A. Chipman, D. K. Wilson, “Polarization ray tracing in birefringent media,” Optical Engineering 30, 461–466 (1991).
[CrossRef]

Wilson, D. K.

J. D. Trollinger, R. A. Chipman, D. K. Wilson, “Polarization ray tracing in birefringent media,” Optical Engineering 30, 461–466 (1991).
[CrossRef]

Yariv, A.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

Yeh, P.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

Zhang, W. Q.

W. Q. Zhang, “Photon tunneling in a uniaxial crystal film,” Appl. Opt. 37, 79–83 (1998).
[CrossRef]

W. Q. Zhang, “General ray-tracing formulas for crystal,” Appl. Opt. 31, 7328–7331 (1992).
[CrossRef] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

Refraction in an uniaxial crystal. An incident ray is refracted into an uniaxial crystal with optic axis ê 3. For simplifying the drawing ê 3 is assumed to lie within the plane of incidence. The propagation and ray vectors, and ŝ, of the e ray are shown. is the normal vector to the surface. The refractive indices are m (1) = 1, m o 2 = 1.5, and m e 2 = 2; the angle of incidence is 50°. The curve shows the cross section of the normal surface with the drawing plane. All vectors are unit vectors; for the graphical representation the dimension of the normal surface was normalized to unity along the optic axis.

Fig. 2
Fig. 2

Same as Fig. 1 but in a coordinate system K ′. Note that the vector ′ is no longer normal to the crystal surface.

Fig. 3
Fig. 3

Internal reflection in an uniaxial crystal. A ray is reflected within an uniaxial crystal with optic axis ê 3. For simplifying the drawing ê 3 is assumed to lie within the plane of incidence. and ŝ are the propagation and ray vectors of the e ray. The refractive indices are m o 2 = 1.5 and m e 2 = 2; the angle of incidence is 40°. The curve shows the cross section of the normal surface with the drawing plane. All vectors are unit vectors; for the graphical representation the dimension of the normal surface was normalized to unity along the optic axis.

Fig. 4
Fig. 4

Same as Fig. 3 but in a coordinate system K ′.

Tables (2)

Tables Icon

Table 1 Numerical Example of Vectors ŝ and for a Ray with Incidence Angle θ(1) on Calcite (mo = 1.54426, me = 1.55335)a

Tables Icon

Table 2 Numerical Example of Vectors ŝ and for a Ray Internally Reflected on a Calcite–Air Boundary (mo = 1.54426, me = 1.55335) with Incidence Angle θ(1) a

Equations (48)

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D = E .
k ˆ = D × H | D × H | ,     s ˆ = E × H | E × H | .
k ˆ x 2 m 2 - x / 0 + k ˆ y 2 m 2 - y / 0 + k ˆ z 2 m 2 - z / 0 = k ˆ 2 m 2 ,
k ˆ x 2 m e 2 + k ˆ y 2 m e 2 + k ˆ z 2 m o 2 - k ˆ 2 m 2 k ˆ 2 m o 2 - k ˆ 2 m 2 = 0 .
m 1 sin   θ 1 = m 2 sin   θ 2     m 1 m 2 2 = 1 - n ˆ · k ˆ 2 2 1 - n ˆ · k ˆ 1 2
k ˆ o 2 = s ˆ o 2 = m 1 m o 2 k ˆ 1 + n ˆ · k ˆ 1 2 - 1 + m o 2 m 1 2 1 / 2 - n ˆ · k ˆ 1 n ˆ .
s ˆ = k ˆ | k ˆ | .
γ = O 1 / m e 2 0 0 0 1 / m e 2 0 0 0 1 / m o 2 O T
O = e ˆ 1 · o ˆ 1 e ˆ 2 · o ˆ 1 e ˆ 3 · o ˆ 1 e ˆ 1 · o ˆ 2 e ˆ 2 · o ˆ 2 e ˆ 3 · o ˆ 2 e ˆ 1 · o ˆ 3 e ˆ 2 · o ˆ 3 e ˆ 3 · o ˆ 3 .
γ 2 = O 1 / m e 2 2 0 0 0 1 / m e 2 2 0 0 0 1 / m o 2 2 O T = 0 m o 2   m e 2 2 .
k ˆ 2 = m 1 | γ k ˆ 1 | k ˆ 1 + n ˆ · k ˆ 1 2 - 1 + 1 m 1 | γ k ˆ 1 | 2 1 / 2 - n ˆ · k ˆ 1 n ˆ .
k ˆ 1 = γ k ˆ 1 | γ k ˆ 1 | ,     n ˆ = γ n ˆ | γ n ˆ | .
k ˆ 2 = γ - 1 · k ˆ 2 | γ - 1 · k ˆ 2 | .
s ˆ 2 = · k ˆ 2 | · k ˆ 2 | = γ 2 · k ˆ 2 | γ 2 · k ˆ 2 | = γ · k ˆ 2 | γ · k ˆ 2 | .
k ˆ 2 = | γ n ˆ | 2 k ˆ 1 + - γ n ˆ · γ k ˆ 1 n ˆ | γ n ˆ | 2 k ˆ 1 + - γ n ˆ · γ k ˆ 1 n ˆ | ,
( γ n ˆ · γ k ˆ 1 2 - | γ n ˆ | 2 | γ k ˆ 1 | 2 + | γ n ˆ | 2 / m 1 2 ) 1 / 2 .
m 2 = 1 | γ k ˆ 2 | .
| γ k ˆ 2 | = k ˆ 2 x m e 2 2 + k ˆ 2 y m e 2 2 + k ˆ 2 z m o 2 2 1 / 2 = 1 m 2 2 1 / 2 = 1 m 2 ,
| γ k ˆ 2 k 2 | = | γ k 2 | = const ,
k ˆ 2 = k ˆ 1 - 2 n ˆ · k ˆ 1 n ˆ .
k ˆ 2 = | γ n ˆ | 2 k ˆ 1 - 2 γ n ˆ · γ k ˆ 1 n ˆ | γ n ˆ | 2 k ˆ 1 - 2 γ n ˆ · γ k ˆ 1 n ˆ | .
k ˆ 2 = w s ˆ 1 + n ˆ · s ˆ 1 2 - 1 + 1 w 2 1 / 2 - n ˆ · s ˆ 1 n ˆ .
Aw 4 - Bw 2 + C = 0 .
A = ( 1 + b 1 - s ˆ 1 · n ˆ 2 - s ˆ 1 · n ˆ × e ˆ 3 2 ) 2 - 4 b ( 1 - s ˆ 1 · n ˆ 2 1 - e ˆ 3 · n ˆ 2 - s ˆ 1 · n ˆ × e ˆ 3 2 ) , B = 2 ( 1 + b 1 - s ˆ 1 · n ˆ 2 - s ˆ 1 · n ˆ × e ˆ 3 2 ) × b n ˆ · e ˆ 3 2 + m 1 m e 2 2 - 4 b m 1 m e 2 2 × ( 1 - s ˆ 1 · n ˆ 2 1 - e ˆ 3 · n ˆ 2 - s ˆ 1 · n ˆ × e ˆ 3 2 ) , C = b n ˆ · e ˆ 3 2 + m 1 m e 2 2 2 ,
b = m 1 2 m e 2 2 - m o 2 2 / m o 2 2 m e 2 2 .
w 2 = B 2 A ± B 2 A 2 - 4 AC 1 / 2 .
s ˆ 2 = k ˆ 2 / m e 2 2 + 1 / m o 2 2 - 1 / m e 2 2 k ˆ 2 · e ˆ 3 e ˆ 3 ( 1 - k ˆ 2 · e ˆ 3 2 / m e 2 4 + k ˆ 2 · e ˆ 3 2 / m o 2 4 ) 1 / 2 .
s ˆ x = cos   α   cos   θ e + sin   θ e   sin   α e ˆ 3 x   sin   θ e - e ˆ 3 y   cos   θ e e ˆ 3 z 2 + e ˆ 3 x   sin   θ e - e ˆ 3 y   cos   θ e 2 1 / 2 , s ˆ y = cos   α   sin   θ e - cos   θ e   sin   α e ˆ 3 x   sin   θ e - e ˆ 3 y   cos   θ e e ˆ 3 z 2 + e ˆ 3 x   sin   θ e - e ˆ 3 y   cos   θ e 2 1 / 2 , s ˆ z = e ˆ 3 z   sin   α e ˆ 3 z 2 + e ˆ 3 x   sin   θ e - e ˆ 3 y   cos   θ e 2 1 / 2 ,
cot   θ e = 2 e ˆ 3 x e ˆ 3 y m o 2 - m e 2 2 m o 2 + e ˆ 3 x 2 m e 2 - m o 2 ± 2 m o ( m o 2 m e 2 + m e 2 e ˆ 3 x 2 m e 2 - m o 2 / sin 2   θ 1 - m o 2 + m e 2 - m o 2 e ˆ 3 x 2 + e ˆ 3 y 2 ) 1 / 2 2 m o 2 + e ˆ 3 x 2 m e 2 - m o 2 , tan   α = m e 2 - m o 2 tan   θ m e 2 + m o 2   tan 2   θ , cos   θ = e ˆ 3 x cos   θ e + e ˆ 3 y sin   θ e .
k ˆ 2 = cos   θ e ,   sin   θ e ,   0 .
sin   θ 2 sin   θ 1 2 = m 1 2 ( 1 / m e 2 2 + 1 / m o 2 2 - 1 / m e 2 2 × k ˆ 2 z 2 ) .
k ˆ 2 = k ˆ 2 x ,   k ˆ 2 y ,   k ˆ 2 z
s ˆ 2 = V e 2 k ˆ 2 x V r V pe ,   V e 2 k ˆ 2 y V r V pe ,   V o 2 k ˆ 2 z V r V pe ,
V pe 2 = V e 2 + V o 2 - V e 2 k ˆ 2 z 2 , V r 2 = V e 4 + V o 4 - V e 4 k ˆ 2 z 2 V pe 2 .
H     D × E = 0 m o 2 E y E z - m e 2 E y E z m e 2 E x E z - m o 2 E x E z m o 2 E x E y - m o 2 E x E y .
γ 2 k ˆ · E = 1 m o 2 m e 2 m o 2 k ˆ x E x + m o 2 k ˆ y E y + m e 2 k ˆ z E z = 1 0 m o 2 m e 2   k ˆ · D = 0 ,
γ 2 k ˆ · H = 1 m o 2 m e 2 m o 2 k ˆ x H x + m o 2 k ˆ y H y + m e 2 k ˆ z H z = 1 m e 2 k ˆ x H x + k ˆ y H y = 1 m e 2   k ˆ · H = 0 ,
s ˆ = γ 2 k ˆ | γ 2 k ˆ | .
1 - n ˆ · k ˆ 2 2 M 1 2 = 1 - n ˆ · k ˆ 1 2 | γ k ˆ 2 | 2 .
1 - n ˆ · k ˆ 2 2 = 1 - n ˆ · k ˆ 1 2 × | γ n ˆ | 4 γ n ˆ | 2 k ˆ 1 + - γ n ˆ · γ k ˆ 1 n ˆ | 2 ,
( γ n ˆ · γ k ˆ 1 2 - | γ n ˆ | 2 | γ k ˆ 1 | 2 + | γ n ˆ | 2 M 1 2 ) 1 / 2 ,
1 - n ˆ · k ˆ 1 2 M 1 2 | γ n ˆ | 4 γ n ˆ | 2 k ˆ 1 + - γ n ˆ · γ k ˆ 1 n ˆ | 2 - 1 - n ˆ · k ˆ 1 2 ( | γ n ˆ | 2 γ k ˆ 1 + - γ n ˆ · γ k ˆ 1 γ n ˆ ) 2 γ n ˆ | 2 k ˆ 1 + - γ n ˆ · γ k ˆ 1 n ˆ | 2 = ? 0 .
M 1 2 | γ n ˆ | 4 - ( | γ n ˆ | 2 γ k ˆ 1 + - γ n ˆ · γ k ˆ 1 γ n ˆ ) 2 = ? 0 .
M 1 2 | γ n ˆ | 4 - ( | γ n ˆ | 4 γ k ˆ 1 2 + γ n ˆ · γ k ˆ 1 2 γ n ˆ 2 - 2 γ n ˆ · γ k ˆ 1 2 γ n ˆ 2 + γ n ˆ · γ k ˆ 1 2 γ n ˆ 2 - | γ n ˆ | 4 γ k ˆ 1 2 + | γ n ˆ | 4 M 1 2 + 2 γ n ˆ · γ k ˆ 1 γ n ˆ 2 - γ n ˆ · γ k ˆ 1 γ n ˆ 2 ) = ? 0 .
1 - n ˆ · k ˆ 1 2 | γ k ˆ 2 | 2 - 1 - n ˆ · k ˆ 2 2 | γ k ˆ 1 | 2 = 0 .
1 - n ˆ · k ˆ 1 2 ( | γ n ˆ | 2 γ k ˆ 1 - 2 ( γ n ˆ ) · γ k ˆ 1 γ n ˆ ) 2 - [ γ n ˆ | 2 k ˆ 1 - 2 γ n ˆ · γ k ˆ 1 n ˆ | 2 - ( | γ n ˆ | 2 n ˆ · k ˆ 1 - 2 γ n ˆ · γ k ˆ 1 ) 2 ] | γ k ˆ 1 | 2 = ? 0 .
1 - n ˆ · k ˆ 1 2 ( | γ n ˆ | 2 γ k ˆ 1 - 2 γ n ˆ · γ k ˆ 1 γ n ˆ ) 2 - | γ n ˆ | 4 1 - n ˆ · k ˆ 1 2 | γ k ˆ 1 | 2 = ? 0 .
| γ n ˆ | 4 γ k ˆ 1 · γ k ˆ 1 + 4 γ n ˆ · γ k ˆ 1 2 γ n ˆ · γ n ˆ - 4 | γ n ˆ | 2 γ n ˆ · γ k ˆ 1 2 - | γ n ˆ | 4 | γ k ˆ 1 | 2 = ! 0 .

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