Behavior of extraordinary rays in uniaxial crystals

Non-refracted extraordinary rays in a uniaxial crystal

We theoretically investigate non-refracted extraordinary rays where the ray does not change its direction when transmitted from a vacuum into a uniaxial crystal, even at oblique incidence. This is possible only when the optic axis is in the plane of incidence. We find that within the usual birefringence range of n_o≤n_e≤no2, for a given angle of cut, there exits only one angle of incidence that produces a non-refracted extraordinary ray. Beyond this range, the number of solutions may be up to three. Numerical examples are given. All cases can be solved algebraically from a quartic equation.

Caustics caused by refraction in the interface between an isotropic medium and a uniaxial crystal

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.

Paraxial theory for birefringent lenses

The generalized ray tracing for the extraordinary ray through uniaxial crystals developed by Avendaño-Alejo and Stavroudis [J. Opt. Soc. Am. A 19, 1674 (2002)] has been applied to derive paraxial refracting equations. Paraxial equations are derived for three cases where the incident, ordinary, and extraordinary rays lie in the incident plane: (a) the crystal axis is parallel to the optical axis, (b) the crystal axis is orthogonal to the optical axis and lies in the plane of incidence, and (c) the crystal axis is orthogonal to both the optical axis and the incident plane. The paraxial ray-tracing equations for the extraordinary ray are represented by matrix operators. The elements of the matrix system give all the information of the focal points and of the principal points. Gaussian formulas are derived, and some examples are presented.

Huygens's principle and rays in uniaxial anisotropic media. I. Crystal axis normal to refracting surface

Huygens's principle is used to derive equations for tracing the extraordinary ray in a uniaxial crystal when the crystal axis is normal to the refracting surface. Snell's law is used to trace ordinary rays that lead to an array of ordinary spherical wavelets centered on the refracting surface. Each spherical wavelet is then replaced by an extraordinary wavelet in the shape of a rotationally symmetric ellipsoid whose major axis is in the direction of the crystal axis. The envelope of these is the extraordinary wave front; the extraordinary ray is a vector from the center of the wavelet to its point of contact with the extraordinary wave front. The inverse problem is also solved, yielding expressions for the ordinary ray in terms of the extraordinary ray, making it possible to trace the extraordinary ray out of the system by using a fictive ordinary ray. We found the geometrical invariant for the uniaxial crystals.

Characteristics of ray traces at the back of biaxial crystals at normal incidence

Ray vector traces that can be obtained at the back of a biaxial crystal through crystal rotation are calculated numerically by variation of one of the three Euler angles. Negative and positive lossless biaxial crystals are considered at normal incidence. Interesting features of these traces are evidenced by reference to the circular ring that forms at internal conical refraction.

Explicit relations for the extraordinary-ray trajectory at the back of a rotating uniaxial birefringent plate

Theoretical formulas are presented for calculating the trajectory of the extraordinary ray at the back of a rotating uniaxial birefringent plate.

Precise and versatile formula for birefringent filters

In an investigation of extraordinary- (E-) ray behavior and the index of refraction for E waves in a uniaxial crystal, a precise and versatile formula for birefringent filters, based on the exact construction of the optical path difference, is set up with neither the approximation Δn = n(o) - n(e) ≤ (or n(e)), nor the ambiguity sin(θ)sin(r(ω)) = n(e). The exact construction gives the correct variation of the position and the dimension in each path, yielding the path difference while the filter is tuning. The formula is applicable not only to a filter with its optical axis parallel to the entrance surface (PAPS) but also to a filter with its axis inclined to the surface (PAIS). Also, the formula indicates that a FAIS allows laser wavelengths to be tuned over a wider range than does a FAPS. The origin of the wider range is interpreted to be the greater variation in the index for the FAIS while the filter is tuning. With the help of the formula we design a FAIS for tuning a cw Ti:sapphire laser.