Abstract

The paraxial theory of spherical refracting surfaces, spherical lenses, and slabs with one birefringent medium is investigated analytically: using walk-off effects in the paraxial domain, a number of relations between objects and images are deduced, along with cardinal elements, in the case where the optic axis is parallel to the optical axis. This method naturally shows that in some cases first-order astigmatism appears. An argument based on the wavefront (and phase) transformation shows that any spherical birefringent thin lens is stigmatic in the paraxial domain, because the first-order astigmatisms due to the two surfaces of such a lens compensate each other. This is a priori not the case with thick birefringent lenses--but two such cases are detailed.

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  1. M. C. Simon, "Ray tracing formulas for monoaxial optical components," Appl. Opt. 22, 354-360 (1983).
  2. M. C. Simon and R. M. Echarri, "Ray tracing formulas for monoaxial optical components: vectorial formulation," Appl. Opt. 25, 1935-1939 (1986).
  3. H. Shimomura, H. Kikuta, and K. Iwata, "First-order aberration of a double focus lens made of a uniaxial crystal," J. Opt. Soc. Am. A 9, 814-819 (1992).
  4. J. P. Lesso, A. J. Duncan, W. Sibbett, and M. J. Padgett, "Aberrations introduced by a lens made from a birefringent material," Appl. Opt. 39, 592-598 (2000).
  5. M. Avendaño-Alejo and M. Rosete-Aguilar, "Paraxial theory for birefringent lenses," J. Opt. Soc. Am. A 22, 881-891 (2005).
  6. M. J. Downs, W. H. McGivern, and H. J. Ferguson, "Optical system for measuring the profiles of super smooth surfaces," Precis. Eng. 7, 211-215 (1985).
  7. C. Chou, J. Shyu, Y. Huang, and C. Yuan, "Common-path optical heterodyne profilometer: a configuration," Appl. Opt. 37, 4137-4142 (1998).
  8. K. Kinnstatter, M. Ojima, and S. Yonezawa, "Amplitude detection for focus error in optical disks using a birefringent lens," Appl. Opt. 29, 4408-4413 (1990).
  9. J. A. Ghosh and A. K. Chakraborty, "High frequency enhancement using a birefringent lens," Opt. Commun. 40, 329-331 (1982).
  10. W. Fiala, "Multifocal intraocular lenses fabricated from media exhibiting tuned birefringence," Optom. Vision Sci. 69, 329-332 (1992).
  11. O. N. Stavroudis, "Ray tracing formulas for uniaxial crystals," J. Opt. Soc. Am. 52, 187-191 (1962).
  12. Q.-T. Liang, "Simple ray tracing formulas for uniaxial optical crystals," Appl. Opt. 29, 1008-1010 (1990).
  13. J. Lekner, "Reflection and refraction by uniaxial crystals," J. Phys. Condens. Matter 3, 6122-6133 (1991).
  14. W. Q. Zhang, "General ray-tracing formulas for crystals," Appl. Opt. 31, 7328-7331 (1992).
  15. Z. Shao and C. Yi, "Behavior of extraordinary rays in uniaxial crystals," Appl. Opt. 33, 1209-1212 (1994).
  16. Z. Shao, "Refractive indices for extraordinary waves in uniaxial crystals," Phys. Rev. E 52, 1043-1048 (1995).
  17. E. Cojocaru, "Direction cosines and vectorial relations for extraordinary-wave propagation in uniaxial media," Appl. Opt. 36, 302-306 (1997).
  18. E. Cojocaru, "Explicit relations for the extraordinary-ray trajectory at the back of a rotating uniaxial birefringent plate," Appl. Opt. 36, 8886-8888 (1997).
  19. G. Beyerle and I. S. McDermid, "Ray-tracing formulas for refraction and internal reflection in uniaxial crystal," Appl. Opt. 37, 7947-7953 (1998).
  20. E. Cojocaru, "Characteristics of ray traces at the back of biaxial crystals at normal incidence," Appl. Opt. 38, 4004-4010 (1999).
  21. M. Avendaño-Alejo, O. Stavroudis, and A. R. Boyain, "Huygens' principle and rays in uniaxial anisotropic media I. Crystal axis normal to refracting surface," J. Opt. Soc. Am. A 19, 1668-1673 (2002).
  22. M. Avendaño-Alejo and O. Stavroudis, "Huygens' principle and rays in uniaxial anisotropic media II. Crystal axis with arbitrary orientation," J. Opt. Soc. Am. A 19, 1674-1679 (2002).
  23. M. C. Simon and K. V. Gottschalk, "Optical path in birefringent media and Fermat's principle," Pure Appl. Opt. 7, 1403-1410 (1998).
  24. M. Avendaño-Alejo and M. Rosete-Aguilar, "Optical path difference in a plane-parallel uniaxial plate," J. Opt. Soc. Am. A 23, 926-932 (2006).
  25. G. Chartier, Manuel d'Optique (Hermès, 1997), pp. 240-244.
  26. S. Huard, Polarisation de la Lumière (Masson, 1994), pp. 76-84.
  27. G. Bruhat, Cours de Physique Générale--Optique (Masson, 1992).
  28. L. Dettwiller, "Absence of internal conical refraction with the spatially dispersive index surface of fluorine; discussion of the orthogonality of the Poynting vector to the index surface," Opt. Express 14, 3339-3344 (2006).
  29. But any refracting surface (D) intersecting (Δg) at an umbilic S--called the "vertex" of (D)--and having at S an osculating sphere of center C on (Δg) also, has the same paraxial optical properties as a spherical refracting surface of center C and vertex S.
  30. In parametric form and with an x axis parallel to (Δo), the equation of the ellipse (centered at the origin O) that is the section of (Σe) by the incidence plane (Fig. ) is x=no cos ,i>t, y=ne sin t, hence the radius of curvature is R=(x˙2 + y˙2)3/2x˙y¨ − x¨y˙=(no2 sin2 t + ne2 cos2 t)3/2none, see, for example, E. W. Weisstein, CRC Concise Encyclopedia of Mathematics, 2nd ed. (Chapman & Hall/CRC, 2003), p. 869. Because U is an umbilic, nA is this radius R at the point U; it corresponds to t=0, whence nA=ne2/no.
  31. M. Avendaño-Alejo, "Analysis of the refraction of the extraordinary ray in a plane-parallel uniaxial plate with an arbitrary orientation of the optical axis," Opt. Express 13, 2549-2555 (2005).
  32. If M is a regular point of (Σ′) of class C2, the matrix (C) is symmetric. Moreover, if M is in a symmetry plane (πS) of (Σ′), and if (C) is expressed in a basis (ux,uy) where ux is parallel to (πS) and uy normal to it, then (C) is diagonal.
  33. See, for example, M. V. Klein, Optics (Wiley, 1970), pp. 84-105.
  34. L. Dettwiller, "Walk-off near optic axes of spatially dispersive media of fluorite symmetry: the case of conical points," J. Mod. Opt. 54, 1173-1185 (2007).
  35. J. W. Goodman, Introduction à l'Optique de Fourier et à l'Holographie (Masson, 1972), pp. 73-78.
  36. See, for example, L. Dettwiller, Les Instruments d'Optique--Étude Théorique, Expérimentale et Pratique, 2nd ed. (Ellipses, 2002), p. 42.
  37. J. H. Burnett, Z. H. Levine, and E. L. Shirley, "Intrinsic birefringence in calcium fluoride and barium fluoride," Phys. Rev. B 64, 241102(R) (2001).
  38. L. Dettwiller, "Observation récente d'une forme de biréfringence dans certains cristaux à symétrie cubique--théorie et conséquences pratiques," Bull. Un. Prof. Phys. Chim. 99(2), 77-103 (2005).
  39. M. C. Simon, "Image formation through monoaxial plane-parallel plates," Appl. Opt. 27, 4176-4182 (1988).
  40. A. A. Lebedeff, "L'interféromètre à polarisation et ses applications," Rev. d'Opt. 9, 385-413 (1930).

2007

L. Dettwiller, "Walk-off near optic axes of spatially dispersive media of fluorite symmetry: the case of conical points," J. Mod. Opt. 54, 1173-1185 (2007).

2006

2005

2002

2001

J. H. Burnett, Z. H. Levine, and E. L. Shirley, "Intrinsic birefringence in calcium fluoride and barium fluoride," Phys. Rev. B 64, 241102(R) (2001).

2000

1999

1998

1997

1995

Z. Shao, "Refractive indices for extraordinary waves in uniaxial crystals," Phys. Rev. E 52, 1043-1048 (1995).

1994

1992

1991

J. Lekner, "Reflection and refraction by uniaxial crystals," J. Phys. Condens. Matter 3, 6122-6133 (1991).

1990

1988

1986

1985

M. J. Downs, W. H. McGivern, and H. J. Ferguson, "Optical system for measuring the profiles of super smooth surfaces," Precis. Eng. 7, 211-215 (1985).

1983

1982

J. A. Ghosh and A. K. Chakraborty, "High frequency enhancement using a birefringent lens," Opt. Commun. 40, 329-331 (1982).

1962

1930

A. A. Lebedeff, "L'interféromètre à polarisation et ses applications," Rev. d'Opt. 9, 385-413 (1930).

Appl. Opt.

M. C. Simon, "Ray tracing formulas for monoaxial optical components," Appl. Opt. 22, 354-360 (1983).

M. C. Simon and R. M. Echarri, "Ray tracing formulas for monoaxial optical components: vectorial formulation," Appl. Opt. 25, 1935-1939 (1986).

J. P. Lesso, A. J. Duncan, W. Sibbett, and M. J. Padgett, "Aberrations introduced by a lens made from a birefringent material," Appl. Opt. 39, 592-598 (2000).

C. Chou, J. Shyu, Y. Huang, and C. Yuan, "Common-path optical heterodyne profilometer: a configuration," Appl. Opt. 37, 4137-4142 (1998).

K. Kinnstatter, M. Ojima, and S. Yonezawa, "Amplitude detection for focus error in optical disks using a birefringent lens," Appl. Opt. 29, 4408-4413 (1990).

Q.-T. Liang, "Simple ray tracing formulas for uniaxial optical crystals," Appl. Opt. 29, 1008-1010 (1990).

W. Q. Zhang, "General ray-tracing formulas for crystals," Appl. Opt. 31, 7328-7331 (1992).

Z. Shao and C. Yi, "Behavior of extraordinary rays in uniaxial crystals," Appl. Opt. 33, 1209-1212 (1994).

E. Cojocaru, "Direction cosines and vectorial relations for extraordinary-wave propagation in uniaxial media," Appl. Opt. 36, 302-306 (1997).

E. Cojocaru, "Explicit relations for the extraordinary-ray trajectory at the back of a rotating uniaxial birefringent plate," Appl. Opt. 36, 8886-8888 (1997).

G. Beyerle and I. S. McDermid, "Ray-tracing formulas for refraction and internal reflection in uniaxial crystal," Appl. Opt. 37, 7947-7953 (1998).

E. Cojocaru, "Characteristics of ray traces at the back of biaxial crystals at normal incidence," Appl. Opt. 38, 4004-4010 (1999).

M. C. Simon, "Image formation through monoaxial plane-parallel plates," Appl. Opt. 27, 4176-4182 (1988).

Bull. Un. Prof. Phys. Chim.

L. Dettwiller, "Observation récente d'une forme de biréfringence dans certains cristaux à symétrie cubique--théorie et conséquences pratiques," Bull. Un. Prof. Phys. Chim. 99(2), 77-103 (2005).

J. Mod. Opt.

L. Dettwiller, "Walk-off near optic axes of spatially dispersive media of fluorite symmetry: the case of conical points," J. Mod. Opt. 54, 1173-1185 (2007).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. Condens. Matter

J. Lekner, "Reflection and refraction by uniaxial crystals," J. Phys. Condens. Matter 3, 6122-6133 (1991).

Opt. Commun.

J. A. Ghosh and A. K. Chakraborty, "High frequency enhancement using a birefringent lens," Opt. Commun. 40, 329-331 (1982).

Opt. Express

Optom. Vision Sci.

W. Fiala, "Multifocal intraocular lenses fabricated from media exhibiting tuned birefringence," Optom. Vision Sci. 69, 329-332 (1992).

Phys. Rev. B

J. H. Burnett, Z. H. Levine, and E. L. Shirley, "Intrinsic birefringence in calcium fluoride and barium fluoride," Phys. Rev. B 64, 241102(R) (2001).

Phys. Rev. E

Z. Shao, "Refractive indices for extraordinary waves in uniaxial crystals," Phys. Rev. E 52, 1043-1048 (1995).

Precis. Eng.

M. J. Downs, W. H. McGivern, and H. J. Ferguson, "Optical system for measuring the profiles of super smooth surfaces," Precis. Eng. 7, 211-215 (1985).

Pure Appl. Opt.

M. C. Simon and K. V. Gottschalk, "Optical path in birefringent media and Fermat's principle," Pure Appl. Opt. 7, 1403-1410 (1998).

Rev. d'Opt.

A. A. Lebedeff, "L'interféromètre à polarisation et ses applications," Rev. d'Opt. 9, 385-413 (1930).

Other

J. W. Goodman, Introduction à l'Optique de Fourier et à l'Holographie (Masson, 1972), pp. 73-78.

See, for example, L. Dettwiller, Les Instruments d'Optique--Étude Théorique, Expérimentale et Pratique, 2nd ed. (Ellipses, 2002), p. 42.

If M is a regular point of (Σ′) of class C2, the matrix (C) is symmetric. Moreover, if M is in a symmetry plane (πS) of (Σ′), and if (C) is expressed in a basis (ux,uy) where ux is parallel to (πS) and uy normal to it, then (C) is diagonal.

See, for example, M. V. Klein, Optics (Wiley, 1970), pp. 84-105.

But any refracting surface (D) intersecting (Δg) at an umbilic S--called the "vertex" of (D)--and having at S an osculating sphere of center C on (Δg) also, has the same paraxial optical properties as a spherical refracting surface of center C and vertex S.

In parametric form and with an x axis parallel to (Δo), the equation of the ellipse (centered at the origin O) that is the section of (Σe) by the incidence plane (Fig. ) is x=no cos ,i>t, y=ne sin t, hence the radius of curvature is R=(x˙2 + y˙2)3/2x˙y¨ − x¨y˙=(no2 sin2 t + ne2 cos2 t)3/2none, see, for example, E. W. Weisstein, CRC Concise Encyclopedia of Mathematics, 2nd ed. (Chapman & Hall/CRC, 2003), p. 869. Because U is an umbilic, nA is this radius R at the point U; it corresponds to t=0, whence nA=ne2/no.

G. Chartier, Manuel d'Optique (Hermès, 1997), pp. 240-244.

S. Huard, Polarisation de la Lumière (Masson, 1994), pp. 76-84.

G. Bruhat, Cours de Physique Générale--Optique (Masson, 1992).

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