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

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

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

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

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

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

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

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).

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

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).

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

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

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).

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).

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).

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).

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).

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).

M. Avendaño-Alejo and M. Rosete-Aguilar, "Paraxial theory for birefringent lenses," J. Opt. Soc. Am. A 22, 881-891 (2005).

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).

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

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

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

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).

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

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).

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

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

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
C^{2}, the matrix (C) is symmetric. Moreover, if M is in a symmetry plane
(πS) of
(Σ′), and if (C) is expressed in a basis
(u_{x},u_{y}) 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
(Δ_{o}), the equation of the ellipse (centered at the origin *O*) that is the section of
(Σ_{e}) by the incidence plane (Fig. ) is
*x*=*n*_{o} cos ,i>t, *y*=*n*_{e} sin *t*,
hence the radius of curvature is
R=(x˙^{2} + y˙^{2})^{3/2}x˙y¨ − x¨y˙=(n_{o}^{2} sin^{2} t + n_{e}^{2} cos^{2} t)^{3/2}n_{o}n_{e},
see, for example, E. W. Weisstein, *CRC Concise Encyclopedia of Mathematics*, 2nd ed. (Chapman & Hall/CRC, 2003), p. 869. Because *U* is an umbilic,
*n*_{A} is this radius *R* at the point *U*; it corresponds to
*t*=0, whence
*n*_{A}=n_{e}^{2}/n_{o}.

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).