Birefringence at an Isotropic-Uniaxial Interface: Waves, Rays, and Fresnel Coefficients

Optically anisotropic dielectrics are birefringent if their index of refraction for a given wavelength depends on the polarization state and the propagation direction of the refracted wave. Ordered materials like crystals have three principal indices of refraction (specific to the crystal structure) in the directions known as the principal axes. Together these constitute the principal coordinate system. If all three indices are the same, the material is characterized as isotropic; if two are the same, as uniaxial; and if all three are different, as biaxial.

We focus on the second case (uniaxial), where the crystal has one axis of symmetry (as in the trigonal, tetragonal, and hexagonal crystal systems), known as the optical axis (, principal axis). In the index ellipsoid notation (optical indicatrix, electric impermeability surface), the index of refraction along the direction of the optical axis is known as extraordinary, , and the two equal indices in directions perpendicular to the are referred to as ordinary, . Birefringence in an isotropic-uniaxial interface for an incoming linearly polarized light wave is better seen as double refraction; the incoming wave splits into two refracted waves inside the uniaxial medium. Their propagation depends on their state of polarization.

In this Demonstration, the is allowed to rotate not only on the plane of incidence (the - plane in the fixed laboratory frame) but also perpendicular to that plane (elevation, around the axis). The coordinate system used for rotation is a fixed , , laboratory system where the interface is the fixed - plane, the normal to the interface is the lab axis, and the lab axis is normal to the plane of incidence. The incidence plane is important since it contains all the wave vectors and the normal to the interface. The crystal is held fixed in space, with the crystal structure rotated. A transformation of coordinate systems has been taken into account in order to translate the crystal geometry from the principal coordinate system that rotates with the to the fixed laboratory frame. The rotation transformation used always assumes that rotation in the plane of incidence () precedes the elevation angle regardless of the order each slider was moved.

A key concept in birefringence is the dispersion relation between energy and momentum ( being the wave vector) of a wave. Maxwell's equations lead to a dispersion relation that can be visualized as a surface in -space, formally known as the normal surface or the -surface. In uniaxial crystals, the -surface consists of two sheets: a sphere for the wave and an ellipsoid of revolution for the wave. The intersection of the -surfaces with the plane of incidence (the - plane in this case) collapses the 3D constructions to a circle and an ellipse. Poynting vectors are normal to their corresponding -surface. In this Demonstration, the ray Poynting vector are found through the geometric construction of the normal to the ellipse at its intersection with the wave vector.

The wave's direction of propagation is found analytically using Snell's law with as the corresponding index of refraction. Its polarization is perpendicular to the plane formed by the and the wave vector. The wave's direction of propagation coincides with the corresponding energy flow (the direction of the Poynting vector or ray direction). The refracted wave satisfies a modified version of Snell's law; the index of refraction the wave experiences depends not only on the values of and of the refracting medium but is also a function of the direction of propagation. Here, the extraordinary wave's direction of propagation is solved for numerically. Its polarization vector lies in the plane formed by , , and (Poynting vector) and the equation of this plane is provided in the "data" section. The waves have different directions for wave vectors and energy flow (walkoff angle, , and angle).

Assuming that the incoming wave's electric field has unit amplitude, then the reflected field will have amplitude in the -pol. direction ( axis) and amplitude in the -pol. direction (plane of incidence) while the transmitted field will have amplitude (in the direction defined by the cross product of the wave vector and the ) and the transmitted field will have amplitude (on the plane formed by , , and and normal to ). These four quantities are known as the Fresnel coefficients. We can then get the power ratios , , , and with the standard procedure (intensity ratios in a unit area). The value is also calculated (sum of the four power ratios) and should always be 1, since it represents conservation of energy.

Fresnel coefficients are derived by explicit application of electromagnetic boundary conditions at the interface (continuity of the tangential components of the electric and magnetic fields). Both the Fresnel coefficients graphs and the reflectance-transmittance graphs are calculated by a looping construct that steps the incidence angle in increments specified by the user (the highest accuracy comes at a premium; a 0.01 step takes about 20 seconds to evaluate). The Brewster angle is then deduced by finding the minimum value of the -component of the reflected ray, . The incoming wave is taken as linearly polarized and its polarization direction can be decided by the user between completely -polarized ( axis, normal to the plane of incidence, 0° in the corresponding slider) and completely -polarized (the - plane of incidence, 90° in the slider) in steps of one degree.

The EM boundary conditions at the interface require all wave vectors (, , , and ) to be in the plane of incidence.

The first medium has a user-defined index of refraction () also. When properly chosen (with respect to and ), total internal reflection (TIR) phenomena are explored and the corresponding critical angles ( and ) are identified from the data (when the corresponding refraction angles for the rays, not the wave vectors, become complex).

The Poynting vector is computed by finding the intersection point of with the -surface and evaluating the normal to the surface there.

A brief description of the selection menu:

1. Raw data on wave, ray, and field directions; Fresnel coefficients; reflectance and transmittance.

2. The -surfaces of the ordinary and extraordinary waves inside the uniaxial medium (their intersection with the plane of incidence, - plane).

3. The corresponding directions of propagation and polarizations for the two wave vectors and and for the ray/Poynting vector . The ray/Poynting vector coincides with the ordinary wave vector . The polarizations refer to an in the plane of incidence.

4. Fresnel coefficients and (absolute values). Changing the orientation affects the shape of the observed curves, while changing the incoming wave's polarization has a profound effect on the magnitude of the coefficients.

5. Phase of the reflectance coefficients and defined as the argument of the coefficient. In the case of internal reflection, the phase can also take on values between 0 and radians when the incoming wave's angle with the normal to the interface exceeds the critical angle and the coefficients become complex numbers.

6. Reflectance and transmittance , . Except for the strictly -polarized incident wave, the Brewster angle is found (changing the angular step has a direct impact on the accuracy). In the case of internal reflection, the critical angles (TIR angles) are also reported. Total reflectivity at normal incidence is also noted. The angular step provides a way to increase the accuracy of the reported values at the "expense" of longer evaluation times.

Snapshot 1: projection of the and surfaces (sphere and ellipsoid of revolution) in the plane of incidence. The two curves are tangent along the for all rotation angles if there is no elevation.

Snapshots 2 and 3: air-glass and glass-air (isotropic-isotropic) interfaces. Total internal reflection conditions are met in the second case for angles of incidence larger than about 41.5°.

Snapshot 4: air-calcite interface (, ). Calcite exhibits large negative birefringence (). The power curves look very similar to the ones in the air-glass interface (Snapshot 2).

Snapshot 5: artificial isotropic-calcite interface (, ). Check the sharp rise of the component in the reflected ray in the angular interval between the two TIR angles although the incoming polarization is polarized at 45°. There are two TIR angles, one for each index of refraction.

Snapshot 6: again air-calcite interface (as in Snapshot 5) but this time the is not rotated but elevated at 45° from the plane of incidence. Notice that in the region between the two TIR angles, the reflected ray rotates its polarization to almost 45° despite the incoming ray being only -polarized.

References

[1] B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed., Hoboken, NJ: John Wiley & Sons Inc., 2007.

[2] M. Sluijter, Ray-Optics Analysis of Homogeneous Uniaxially Anisotropic Media, Technical note TN-2007-00025, Philips Research Eindhoven, 2007.

[3] M. Sluijter, Ray-Optics Analysis of Inhomogeneous Uniaxially Anisotropic Media, Technical note TN-2007-00892, Philips Research Eindhoven, 2008.

[4] Z. Zhang and H. J. Caulfield, "Reflection and Refraction by Interfaces of Uniaxial Crystals," Optics & Laser Technology, 28(7), 1996, pp. 549–553. doi:10.1016/S0030-3992(96)00022-9.

[5] E. Hecht, Optics, 4th ed., San Francisco, CA: Pearson Addison Wesley, 2002.