Abstract

A method of ray-tracing calculations for uniaxial optical components with curved surfaces is presented. A set of simple ray-tracing formulas is derived. With the spatial ray-tracing method in geometrical optics, by using a computer we plot spot diagrams of extraordinary-ray images formed by some crystal systems.

© 1991 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, 354–360 (1983).
    [Crossref] [PubMed]
  2. M. C. Simon, R. M. Echarri, “Ray tracing formulas for monoaxial optical components; vectorial formulation,” Appl. Opt. 25, 1935–1939 (1986).
    [Crossref] [PubMed]
  3. Z. X. Zhang, “The trajectory of the extraordinary ray as the crystal rotates,” Acta Phys. Sin. 29, 1483–1486 (1980).
  4. Q. T. Liang, “Simple ray tracing formulas for uniaxial optical crystals,” Appl. Opt. 29, 1008–1010 (1990).
    [Crossref] [PubMed]
  5. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1980), Chap. 14, p. 668.
  6. J. P. Mathieu, Optics, 1st ed. (Pergamon, London, 1975), Chap. 3, p. 86.
  7. A. Cox, A System of Optical Design, 1st ed. (Focal Press, London, 1964), Chap. 3, p. 133.

1990 (1)

1986 (1)

1983 (1)

1980 (1)

Z. X. Zhang, “The trajectory of the extraordinary ray as the crystal rotates,” Acta Phys. Sin. 29, 1483–1486 (1980).

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1980), Chap. 14, p. 668.

Cox, A.

A. Cox, A System of Optical Design, 1st ed. (Focal Press, London, 1964), Chap. 3, p. 133.

Echarri, R. M.

Liang, Q. T.

Mathieu, J. P.

J. P. Mathieu, Optics, 1st ed. (Pergamon, London, 1975), Chap. 3, p. 86.

Simon, M. C.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1980), Chap. 14, p. 668.

Zhang, Z. X.

Z. X. Zhang, “The trajectory of the extraordinary ray as the crystal rotates,” Acta Phys. Sin. 29, 1483–1486 (1980).

Acta Phys. Sin. (1)

Z. X. Zhang, “The trajectory of the extraordinary ray as the crystal rotates,” Acta Phys. Sin. 29, 1483–1486 (1980).

Appl. Opt. (3)

Other (3)

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1980), Chap. 14, p. 668.

J. P. Mathieu, Optics, 1st ed. (Pergamon, London, 1975), Chap. 3, p. 86.

A. Cox, A System of Optical Design, 1st ed. (Focal Press, London, 1964), Chap. 3, p. 133.

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Figures (7)

Fig. 1
Fig. 1

Refraction by the crystal surface.

Fig. 2
Fig. 2

Directions of the vectors k ̂ 1, k ̂ e, ŝ, ŵ, and N ̂.

Fig. 3
Fig. 3

Refractions by the two surfaces of the crystal component.

Fig. 4
Fig. 4

Two typical lenses of a uniaxial crystal (calcite).

Fig. 5
Fig. 5

Incident points of traced rays on the first surface of a lens.

Fig. 6
Fig. 6

Spot diagrams of e-rays.

Fig. 7
Fig. 7

Spot diagrams for isotropic lenses.

Tables (1)

Tables Icon

Table I Structural Parameters of the Two Lenses Shown In Fig. 4a

Equations (43)

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K 1 r = K r ,
k ̂ 1 ( α , β , γ )
k ̂ e ( α , β , γ )
N ̂ ( t , u , υ )
n 1 sin θ 1 = n sin θ e
n = n 0 n e ( n 0 2 sin 2 θ + n e 2 cos 2 θ ) 1 / 2 ,
cos 2 θ e = B + T cos 2 θ ,
B = n 1 2 n e 2 sin 2 θ 1 ,
T = n 1 2 ( n 0 2 n e 2 ) n 0 2 n e 2 sin 2 θ 1 .
cos θ = α x 0 + β y 0 + γ z 0 ,
cos θ e = t α u β + υ γ .
B + T ( α x 0 + β y 0 + γ z 0 ) 2 = ( t α + u β + υ γ ) 2 .
| α β γ α β γ t u υ | = 0 ,
α ( u γ υ β ) + β ( α υ t γ ) + γ ( β t u α ) = 0 .
α 2 + β 2 + γ 2 = 1 .
α i = g 1 β i + g 2 γ i , i = 1 , 2 , 3 , 4 ,
β 1 , 2 = H 1 γ 1 , 2 , β 3 , 4 = H 2 γ 3 , 4 ,
γ 1 , 2 = ± ( 1 g 6 H 1 2 + g 7 + g 8 H 1 ) 1 / 2 , γ 3 , 4 = ± ( 1 g 6 H 2 2 + g 7 + g 8 H 2 ) 1 / 2 ,
g 1 = ( α υ t γ ) / ( β υ u γ ) ,
g 2 = ( β t u α ) / ( β υ u γ ) ,
g 3 = T ( g 1 x 0 + y 0 ) 2 ( t g 1 + u ) 2 ,
g 4 = T ( g 2 x 0 + z 0 ) 2 ( t g 2 + υ ) 2 ,
g 5 = 2 [ T ( g 1 x 0 + y 0 ) ( g 2 x 0 + z 0 ) ( t g 1 + u ) ( t g 2 + υ ) ] ,
g 6 = g 1 2 + 1 ,
g 7 = g 2 2 + 1 ,
g 8 = 2 g 1 g 2 ,
H 1 , 2 = ( g 5 + B g 8 ) ± [ ( g 5 + B g 8 ) 2 4 ( g 3 + B g 6 ) ( g 4 + B g 7 ) ] 1 / 2 2 ( g 3 + B g 6 ) .
α > 0 , ( β u ) ( β u ) > 0 , ( γ υ ) ( γ υ ) > 0 . }
cos ϕ = α s x + β s y + γ s z .
| s x x y s z x 0 y 0 z 0 α β γ | = 0 .
s x 2 + x y 2 + s z 2 = 1 .
s x 1 , 2 = ( b 2 b 3 + c 1 c 2 ) ± [ ( b 2 b 3 + c 1 c 2 ) 2 ( b 1 2 + b 2 + c 1 ) 2 ( b 3 2 + c 2 2 b 1 2 ) ] 1 / 2 b 1 2 + b 2 2 + c 1 2 ,
s y 1 , 2 = ( b 3 b 2 s xi ) / b 1 , i = 1 , 2 ,
s z 1 , 2 = ( c 1 s x i c 2 ) / b 1 , i = 1 , 2 ,
b 1 = ( x 0 β y 0 α ) β ( z 0 α x 0 γ ) γ ,
b 2 = ( x 0 β y 0 α ) α ( y 0 γ z 0 β ) γ ,
b 3 = ( x 0 β y 0 α ) cos ϕ ,
c 1 = ( z 0 α x 0 γ ) α ( y 0 γ z 0 β ) β ,
c 2 = ( z 0 α x 0 γ ) cos ϕ .
ϕ = θ θ ,
tan ϕ = ( n 2 2 n 0 2 ) tan θ n 2 2 + n 0 2 tan 2 θ .
cos θ < cos θ , when ϕ > 0 , cos θ > cos θ , when ϕ < 0 . }
s ̂ = k ̂ e .

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