Abstract

A formalism which allows finite ray tracing through monoaxial crystals was developed starting from Maxwell’s equations. The derived formulas were applied to a Wollaston prism in convergent light and spot diagrams were obtained.

© 1983 Optical Society of America

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References

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  1. A. Sommerfeld, in Optik Vorlesungen uber Theoretische Physik. Band IV (Akademische Verlagsgesellschaft. Geest & Portig K-G, Leipzig, 1959.)
  2. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975).
  3. J. M. Simon, M. C. Simon, Appl. Opt. 17, 3352 (1978).
    [Crossref] [PubMed]
  4. W. Bartholomeyczyk, Z. Instrumentenkd. 68, 208 (1960).

1978 (1)

1960 (1)

W. Bartholomeyczyk, Z. Instrumentenkd. 68, 208 (1960).

Bartholomeyczyk, W.

W. Bartholomeyczyk, Z. Instrumentenkd. 68, 208 (1960).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975).

Simon, J. M.

Simon, M. C.

Sommerfeld, A.

A. Sommerfeld, in Optik Vorlesungen uber Theoretische Physik. Band IV (Akademische Verlagsgesellschaft. Geest & Portig K-G, Leipzig, 1959.)

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975).

Appl. Opt. (1)

Z. Instrumentenkd. (1)

W. Bartholomeyczyk, Z. Instrumentenkd. 68, 208 (1960).

Other (2)

A. Sommerfeld, in Optik Vorlesungen uber Theoretische Physik. Band IV (Akademische Verlagsgesellschaft. Geest & Portig K-G, Leipzig, 1959.)

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975).

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Figures (9)

Fig. 1
Fig. 1

Diagram of the vectors that describe the plane wave (normal plane to the vector H).

Fig. 2
Fig. 2

Diagram of the vectors that describe the two plane waves in birefringent uniaxial medium. z3 is the optical axis. The vector Ř′ is parallel to Ň.

Fig. 3
Fig. 3

Reflection and refractive of a ray of light that falls on a birefringent uniaxial medium. The x = 0 plane, perpendicular to the plane of the paper, is the plane of discontinuity. z3, the optical axis, lies in the plane (y = 0) that is parallel to the paper.

Fig. 4
Fig. 4

Huygens’ construction for a uniaxial positive crystal (uo > ue).

Fig. 5
Fig. 5

Huygens’ construction for a uniaxial negative crystal (ue > uo).

Fig. 6
Fig. 6

Qualitative diagram of a passage of rays through a Wollaston prism.

Fig. 7
Fig. 7

Spot diagram for images ① and ②.

Fig. 8
Fig. 8

Spot diagram for images ③ and ④.

Fig. 9
Fig. 9

Straight lines that represent the distortion.

Equations (63)

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D j = ε j E j ; j = 1,2,3.
u j = 1 μ 0 ε j ; j = 1,2,3.
rot H = D ˙ ,
rot E = μ 0 H ˙ ,
div D = 0 ,
div H = 0.
E ( r , t ) = E exp [ i 2 π λ ( N · r u t ) ] ,
D ( r , t ) = D exp [ i 2 π λ ( N · r u t ) ] ,
H ( r , t ) = H exp [ i 2 π λ ( N · r u t ) ] ,
N H = u D ,
N E = μ 0 u H ,
N · D = 0 ,
N · H = 0.
S = E H ,
R = S / | S | .
D H E H R H N H N D R E .
υ = u / cos δ .
E = μ 0 u 2 D + ( N · E ) N .
μ 0 u 2 D = cos 2 δ · E + μ 0 u 2 ( R · D ) R .
μ 0 u 2 ( R · D ) = ( N · E ) cos δ .
μ 0 u 2 D = u 2 υ 2 E u υ ( N · E ) R .
E j = μ 0 u j 2 D j , j = 1,2,3.
μ 0 u j 2 D j = μ 0 u 2 D j + ( N · E ) N j ,
μ 0 u 2 D j = u 2 υ 2 μ 0 u j 2 D j u υ ( N · E ) R j .
R j = u υ { 1 + υ 2 u 2 u 2 u j 2 } N j , j = 1,2,3.
υ 2 u 2 = υ 2 1 3 R j 2 u 2 1 3 N j 2 .
υ 2 u 2 = u 2 ( υ 2 u 2 ) 2 { 1 3 N j 2 ( u 2 u j 2 ) 2 + 2 ( υ 2 u 2 ) 1 3 N j 2 ( u 2 u j 2 ) } .
1 3 N j ( u 2 u j 2 ) = 0.
υ 2 u 2 = 1 u 2 · 1 1 3 N j 2 ( u 2 u j 2 ) 2 .
g 2 = u 2 ( υ 2 u 2 ) .
υ 2 = g 2 / u 2 + u 2 .
g 2 = 1 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 2 u j 2 ) D j + ( i = 1 3 u i 2 N i D i ) N j = 0 , j = 1,2,3.
N 1 2 ( u 2 u 2 2 ) ( u 2 u 3 2 ) + N 2 2 ( u 2 u 1 2 ) ( u 2 u 3 2 ) + N 3 2 ( u 2 u 1 2 ) ( u 2 u 2 2 ) = 0.
D 1 D 2 = N 1 N 2 ( u 2 u 2 2 ) ( u 2 u 1 2 ) , D 3 D 2 = N 3 N 2 ( u 2 u 2 2 ) ( u 2 u 3 2 ) ,
D 1 D 2 = N 1 N 2 ( u 2 u 2 2 ) ( u 2 u 1 2 ) , D 3 D 2 = N 3 N 2 ( u 2 u 2 2 ) ( u 2 u 3 2 ) ,
u o = 1 μ 0 ε 1 = 1 μ 0 ε 2 , u e = 1 μ 0 ε 3 .
( u 2 u o 2 ) { ( 1 N 3 2 ) ( u 2 u e 2 ) + N 3 2 ( u 2 u o 2 ) } = 0 ,
u 2 = u o 2 ,
u 2 = ( 1 N 3 2 ) u e 2 + N 3 2 u o 2 .
D 3 = 0 ; D 1 / D 2 = N 2 / N 1 ,
E 3 = 0 ; E 1 / E 2 = N 2 / N 1 .
R j = N j , υ = u .
u 2 = u e 2 sin 2 γ + u o 2 cos 2 γ,
D 1 / D 2 = N 1 / N 2 , D 3 / D 2 = N 3 2 1 N 2 N 3 .
R 1 = N 1 u e 2 ( 1 N 3 2 ) u e 4 + N 3 2 u o 4 , R 2 = N 2 u e 2 ( 1 N 3 2 ) u e 4 + N 3 2 u o 4 , R 3 = N 3 u o 2 ( 1 N 3 2 ) u e 4 + N 3 2 u o 4 ,
υ 2 = u e 4 ( 1 N 3 2 ) + u o 4 N 3 2 u e 2 ( 1 N 3 2 ) + u o 2 N 3 2 .
S y = S y , S z = S z ,
N y / u = S y / u , N z / u = S z / u ,
N y / u = S y / u , N z / u = S z / u ,
N 2 = u / u S y ,
N 3 cos ϑ + N 1 sin ϑ = u / u S z .
W = u / u , b = u o 2 u e 2 u 2 ,
W 2 = b N 3 2 + u e 2 / u 2 , W S y = N 2 , W S z = N 3 cos ϑ + N 1 sin ϑ , N 1 2 + N 2 2 + N 3 2 = 1.
W 4 A W 2 B + C = 0 , A = ( b S z 2 + b S y 2 sin 2 ϑ + 1 ) 2 4 b S z 2 cos 2 ϑ , B = 2 ( b S z 2 + b S y 2 sin 2 ϑ + 1 ) ( b sin 2 ϑ + u e 2 / u 2 ) 4 b S z 2 u e 2 / u 2 cos 2 ϑ , C = ( b sin 2 ϑ + u e 2 / u 2 ) 2 ,
W = B B 2 4 A C 2 A ,
W + = B + B 2 4 A C 2 A .
N y = W S y , N z = W S z .
N x = 1 W 2 ( S y 2 + S z 2 ) .
N 1 = N x cos ϑ + N z sin ϑ , N 2 = N y , N 3 = N z cos ϑ N x sin ϑ .
R 1 = N 1 u e 2 ( 1 N 3 2 ) u e 4 + N 3 2 u o 4 , R 2 = N 2 u e 2 ( 1 N 3 2 ) u e 4 + N 3 2 u o 4 , R 3 = N 3 u o 2 ( 1 N 3 2 ) u e 4 + N 3 2 u o 4 .
R x = R 1 cos ϑ R 3 sin ϑ , R y = R 2 , R z = R 3 cos ϑ + R 1 sin ϑ .

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