Abstract

In general, a caustic by refraction at an arbitrary surface is commonly known as a diacaustic. We study the formation of the diacaustic in a plane interface between an isotropic medium and a uniaxial crystal, for both ordinary and extraordinary rays, when the crystal axis is perpendicular to the plane of incidence and when it lies in the plane of incidence. For the latter case two special positions of the crystal axis with respect to the normal to the refracting surface for the extraordinary rays are treated.

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  2. A. Yariv and P. Yeh, Optical Waves in Crystals, Propagation and Control of Laser Radiation (Wiley, 1983), Chap. 4, pp. 69-120.
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    [CrossRef]
  12. M. C. Simon and K. V. Gottschalk, “Waves and rays in uniaxial birefringent crystals,” Optik (Stuttgart) 118, 457-470 (2007).
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  13. Y. Wang, L. Liang, H. Xin, and L. Wu, “Complex ray tracing in uniaxial absorbing media,” J. Opt. Soc. Am. A 25, 653-657 (2008).
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  14. O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics, The K-Function and Its Ramifications (Wiley-VCH Verlag GmbH & Co. KGaA, 2006), Chap. 12, pp. 179-186.
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  16. O. Stavroudis, “Refraction of wavefronts: a special case,” J. Opt. Soc. Am. 59, 114-115 (1969).
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  17. S. Cornbleet, Microwave and Optical Ray Geometry (Wiley, 1984), Chap. 2, pp. 11-35.
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  19. J. J. Stoker, Differential Geometry (Wiley-Interscience, 1969), Chap. 2, pp. 12-52.
  20. J. Morgan, Introduction to Geometrical and Physical Optics (Mc-Graw Hill, 1953), Chap. 1, pp. 1-24.
  21. J. W. Rutter, Geometry of Curves (CRC, 2000), Chaps. 13, 14, pp. 210-269.

2008

2007

M. C. Simon and K. V. Gottschalk, “Waves and rays in uniaxial birefringent crystals,” Optik (Stuttgart) 118, 457-470 (2007).
[CrossRef]

2006

O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics, The K-Function and Its Ramifications (Wiley-VCH Verlag GmbH & Co. KGaA, 2006), Chap. 12, pp. 179-186.
[CrossRef]

2005

H. Ren, L. Liu, D. Liu, and Z. Song, “Double refraction and reflection of sequential crystal interfaces with arbitrary orientation of the optic axis and application to optimum design,” J. Mod. Opt. 52, 529-539 (2005).
[CrossRef]

2002

2000

J. W. Rutter, Geometry of Curves (CRC, 2000), Chaps. 13, 14, pp. 210-269.

1999

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), Chap. 15, pp. 790-852.

1998

1997

1994

1991

J. D. Troliger, R. A. Chipman, and D. K. Wilson, “Polarization ray tracing in birefringent media,” Opt. Eng. (Bellingham) 30, 461-466 (1991).
[CrossRef]

1990

1984

S. Cornbleet, Microwave and Optical Ray Geometry (Wiley, 1984), Chap. 2, pp. 11-35.

1983

M. C. Simon, “Ray tracing formulas for monoaxial optical components,” Appl. Opt. 22, 354-360 (1983).
[CrossRef] [PubMed]

A. Yariv and P. Yeh, Optical Waves in Crystals, Propagation and Control of Laser Radiation (Wiley, 1983), Chap. 4, pp. 69-120.

1976

F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (Mc-Graw Hill, 1976), Chap. 2, pp. 36-40.

1975

1969

O. Stavroudis, “Refraction of wavefronts: a special case,” J. Opt. Soc. Am. 59, 114-115 (1969).
[CrossRef]

J. J. Stoker, Differential Geometry (Wiley-Interscience, 1969), Chap. 2, pp. 12-52.

1958

M. Herzberger, Modern Geometrical Optics (Interscience Publishers, 1958), Chap. 5, pp. 40-57.

1953

J. Morgan, Introduction to Geometrical and Physical Optics (Mc-Graw Hill, 1953), Chap. 1, pp. 1-24.

Avendaño-Alejo, M.

Beyerle, G.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), Chap. 15, pp. 790-852.

Chipman, R. A.

J. D. Troliger, R. A. Chipman, and D. K. Wilson, “Polarization ray tracing in birefringent media,” Opt. Eng. (Bellingham) 30, 461-466 (1991).
[CrossRef]

Cojocaru, E.

Cornbleet, S.

S. Cornbleet, Microwave and Optical Ray Geometry (Wiley, 1984), Chap. 2, pp. 11-35.

Gottschalk, K. V.

M. C. Simon and K. V. Gottschalk, “Waves and rays in uniaxial birefringent crystals,” Optik (Stuttgart) 118, 457-470 (2007).
[CrossRef]

Herzberger, M.

M. Herzberger, Modern Geometrical Optics (Interscience Publishers, 1958), Chap. 5, pp. 40-57.

Jenkins, F. A.

F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (Mc-Graw Hill, 1976), Chap. 2, pp. 36-40.

Liang, L.

Liang, Q. T.

Liu, D.

H. Ren, L. Liu, D. Liu, and Z. Song, “Double refraction and reflection of sequential crystal interfaces with arbitrary orientation of the optic axis and application to optimum design,” J. Mod. Opt. 52, 529-539 (2005).
[CrossRef]

Liu, L.

H. Ren, L. Liu, D. Liu, and Z. Song, “Double refraction and reflection of sequential crystal interfaces with arbitrary orientation of the optic axis and application to optimum design,” J. Mod. Opt. 52, 529-539 (2005).
[CrossRef]

McDermid, I. S.

Morgan, J.

J. Morgan, Introduction to Geometrical and Physical Optics (Mc-Graw Hill, 1953), Chap. 1, pp. 1-24.

Ren, H.

H. Ren, L. Liu, D. Liu, and Z. Song, “Double refraction and reflection of sequential crystal interfaces with arbitrary orientation of the optic axis and application to optimum design,” J. Mod. Opt. 52, 529-539 (2005).
[CrossRef]

Rutter, J. W.

J. W. Rutter, Geometry of Curves (CRC, 2000), Chaps. 13, 14, pp. 210-269.

Shao, Z.

Simon, M. C.

M. C. Simon and K. V. Gottschalk, “Waves and rays in uniaxial birefringent crystals,” Optik (Stuttgart) 118, 457-470 (2007).
[CrossRef]

M. C. Simon, “Ray tracing formulas for monoaxial optical components,” Appl. Opt. 22, 354-360 (1983).
[CrossRef] [PubMed]

Song, Z.

H. Ren, L. Liu, D. Liu, and Z. Song, “Double refraction and reflection of sequential crystal interfaces with arbitrary orientation of the optic axis and application to optimum design,” J. Mod. Opt. 52, 529-539 (2005).
[CrossRef]

Stavroudis, O.

Stavroudis, O. N.

O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics, The K-Function and Its Ramifications (Wiley-VCH Verlag GmbH & Co. KGaA, 2006), Chap. 12, pp. 179-186.
[CrossRef]

Stoker, J. J.

J. J. Stoker, Differential Geometry (Wiley-Interscience, 1969), Chap. 2, pp. 12-52.

Swindell, W.

Troliger, J. D.

J. D. Troliger, R. A. Chipman, and D. K. Wilson, “Polarization ray tracing in birefringent media,” Opt. Eng. (Bellingham) 30, 461-466 (1991).
[CrossRef]

Wang, Y.

White, H. E.

F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (Mc-Graw Hill, 1976), Chap. 2, pp. 36-40.

Wilson, D. K.

J. D. Troliger, R. A. Chipman, and D. K. Wilson, “Polarization ray tracing in birefringent media,” Opt. Eng. (Bellingham) 30, 461-466 (1991).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), Chap. 15, pp. 790-852.

Wu, L.

Xin, H.

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals, Propagation and Control of Laser Radiation (Wiley, 1983), Chap. 4, pp. 69-120.

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals, Propagation and Control of Laser Radiation (Wiley, 1983), Chap. 4, pp. 69-120.

Yi, C.

Appl. Opt.

J. Mod. Opt.

H. Ren, L. Liu, D. Liu, and Z. Song, “Double refraction and reflection of sequential crystal interfaces with arbitrary orientation of the optic axis and application to optimum design,” J. Mod. Opt. 52, 529-539 (2005).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Eng. (Bellingham)

J. D. Troliger, R. A. Chipman, and D. K. Wilson, “Polarization ray tracing in birefringent media,” Opt. Eng. (Bellingham) 30, 461-466 (1991).
[CrossRef]

Optik (Stuttgart)

M. C. Simon and K. V. Gottschalk, “Waves and rays in uniaxial birefringent crystals,” Optik (Stuttgart) 118, 457-470 (2007).
[CrossRef]

Other

J. J. Stoker, Differential Geometry (Wiley-Interscience, 1969), Chap. 2, pp. 12-52.

J. Morgan, Introduction to Geometrical and Physical Optics (Mc-Graw Hill, 1953), Chap. 1, pp. 1-24.

J. W. Rutter, Geometry of Curves (CRC, 2000), Chaps. 13, 14, pp. 210-269.

O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics, The K-Function and Its Ramifications (Wiley-VCH Verlag GmbH & Co. KGaA, 2006), Chap. 12, pp. 179-186.
[CrossRef]

M. Herzberger, Modern Geometrical Optics (Interscience Publishers, 1958), Chap. 5, pp. 40-57.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), Chap. 15, pp. 790-852.

A. Yariv and P. Yeh, Optical Waves in Crystals, Propagation and Control of Laser Radiation (Wiley, 1983), Chap. 4, pp. 69-120.

F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (Mc-Graw Hill, 1976), Chap. 2, pp. 36-40.

S. Cornbleet, Microwave and Optical Ray Geometry (Wiley, 1984), Chap. 2, pp. 11-35.

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Figures (8)

Fig. 1
Fig. 1

Point source located at P = ( 0 , d ) immersed in a denser medium and the process of refraction to form an apparent image of the point source.

Fig. 2
Fig. 2

Family of refracted rays and the positions of the apparent images for n i = 1 and n o = 1.54421 . The dashed lines form the diacaustic.

Fig. 3
Fig. 3

Ellipse whose evolute is related to the envelope of the family of refracted ordinary rays. The focus of this ellipse coincides with the position of the point source. We consider that the source is placed at ( 0 , 1 ) coordinates with arbitrary units, for n i = 1 and n o = 1.658 .

Fig. 4
Fig. 4

Ellipses whose evolute is related to the envelope of the family of refracted rays, for extraordinary rays when the crystal axis is parallel and perpendicular to the normal to the refracting surface for positive and negative uniaxial crystals quartz and calcite, respectively. The continuous curves correspond to the case when the crystal axis is perpendicular to and the dashed curves to the case when the crystal axis is parallel to the normal to the refracting surface.

Fig. 5
Fig. 5

Point source placed at P = ( 0 , d ) immersed in a rarer medium, and the process of refraction to form an apparent image of the point source.

Fig. 6
Fig. 6

Family of refracted rays and the positions of the apparent images for n i = 1 and n o = 1.54421 . The dashed lines form the diacaustic when the point source is located in a rarer medium.

Fig. 7
Fig. 7

Hyperbola whose evolute is related to the envelope of the family of refracted rays for n i = 1 and n o = 1.658 for calcite. A focus of this hyperbola coincides with the location of the point source, which is placed at ( 0 , 1 ) with arbitrary units.

Fig. 8
Fig. 8

Hyperbola whose evolute is related to the envelope of the family of refracted rays, or diacaustic, for both ordinary and extraordinary rays when the crystal axis is perpendicular to the plane of incidence, for positive and negative uniaxial crystals. The solid curves correspond to the extraordinary rays, and the dashed curves to the ordinary rays.

Equations (30)

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θ o = arcsin ( n i sin θ i n o ) = arctan ( n i sin θ i n o 2 n i 2 sin 2 θ i ) .
θ i = arcsin ( n o sin θ o n i ) = arctan ( n o sin θ o n i 2 n o 2 sin 2 θ o ) ,
tan θ e ia = n e n i n o sin θ i + ( n e 2 n o 2 ) sin ϕ cos ϕ n e 2 cos 2 ϕ + n 0 2 sin 2 ϕ n i 2 sin 2 θ i ( n e 2 cos 2 ϕ + n o 2 sin 2 ϕ ) n e 2 cos 2 ϕ + n o 2 sin 2 ϕ n i 2 sin 2 θ i ,
tan θ e ai = n e 2 cos ϕ sin ( θ e ϕ ) + n o 2 sin ϕ cos ( θ e ϕ ) n i 2 ( n o 2 cos 2 ( θ e ϕ ) + n e 2 sin 2 ( θ e ϕ ) ) ( n e 2 cos ϕ sin ( θ e ϕ ) + n o 2 sin ϕ cos ( θ e ϕ ) ) 2 ,
z = 1 tan θ i ( y d tan θ o ) ,
z n o y n i 2 n o 2 sin 2 θ o sin θ o + d n i 2 n o 2 sin 2 θ o cos θ o = 0 .
n i 2 y cos 3 θ o + d ( n i 2 n o 2 ) sin 3 θ o cos 2 θ o sin 2 θ o n i 2 n o 2 sin 2 θ o = 0 .
n i 2 y cos 3 θ o + d ( n i 2 n o 2 ) sin 3 θ o = 0 ,
θ o = arctan [ n i 2 y d ( n o 2 n i 2 ) ] 1 3 .
z n o y n i 2 n o 2 sin 2 { arctan [ n i 2 y d ( n o 2 n i 2 ) ] 1 3 } sin { arctan [ n i 2 y d ( n o 2 n i 2 ) ] 1 3 } + d n i 2 n o 2 sin 2 { arctan [ n i 2 y d ( n o 2 n i 2 ) ] 1 3 } cos { arctan [ n i 2 y d ( n o 2 n i 2 ) ] 1 3 } = 0 ,
1 = [ y n o 2 n i 2 d n i ] 2 3 + [ n o z d n i ] 2 3 ,
y o ( θ ) = d n i sin 3 θ n o 2 n i 2 , z o ( θ ) = d n i cos 3 θ n o ,
y ( t ) = ( a 2 b 2 ) sin 3 t a , z ( t ) = ( b 2 a 2 ) cos 3 t b .
y e ( θ ) = d n i sin 3 θ n e 2 n i 2 , z e ( θ ) = d n i cos 3 θ n e ,
z = 1 tan θ e ai [ y d tan θ e ] .
z n e 2 = [ y sin θ e d cos θ e ] n i 2 n o 2 [ n e 4 + n i 2 ( n o 2 n e 2 ) ] sin 2 θ e .
y e = d n i n o sin 3 θ n e n e 2 n i 2 , z e = d n i n o cos 3 θ n e 2 ,
z n o 2 = [ y sin θ e d cos θ e ] n i 2 n e 2 [ n o 4 + n i 2 ( n e 2 n o 2 ) ] sin 2 θ e ,
y e = d n i n e sin 3 θ n o n o 2 n i 2 , z e = d n i n e cos 3 θ n o 2 ,
z n i y n o 2 n i 2 sin 2 θ i sin θ i + d n o 2 n i 2 sin 2 θ i cos θ i = 0 .
θ i = arctan [ n o 2 y d ( n i 2 n o 2 ) ] 1 3 .
1 = [ y n i 2 n o 2 d n o ] 2 3 + [ n i z d n o ] 2 3 .
y o ( θ ) = d n o sinh 3 θ n o 2 n i 2 , z o ( θ ) = d n o cosh 3 θ n i .
y ( t ) = ( a 2 + b 2 ) sinh 3 t a , z ( t ) = ( a 2 + b 2 ) cosh 3 t b ,
y e ( θ ) = d n e sinh 3 θ n e 2 n i 2 , z e ( θ ) = d n e cosh 3 θ n i .
z = 1 tan θ e ia [ y d tan θ i ] .
z n i n o n e = ( y sin θ i d cos θ i ) n e 2 n i 2 sin 2 θ i .
y e ( θ ) = d n e sinh 3 θ n e 2 n i 2 , z e ( θ ) = d n e 2 cosh 3 θ n i n o ,
z n e n i n o = ( y sin θ i d cos θ i ) n o 2 n i 2 sin 2 θ i .
y e ( θ ) = d n o sinh 3 θ n o 2 n i 2 , z e ( θ ) = d n o 2 cosh 3 θ n e n i ,

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