External Conical Refraction - MM Research, Inc.

External Conical Refraction

by Masud Mansuripur
Published in Optics & Photonics News, pp. 50-52, August 1997

In a previous column (June 1997) we described the phenomenon of internal conical refraction: A collimated beam of light, upon entering a biaxial birefringent crystal, spreads out into a hollow cone and exits the crystal slab in the form of two concentric cylinders of light. The case of external conical refraction, which, in a way, is the same phenomenon in reverse, is the subject of the present article. Specifically, we will describe the conditions under which a hollow cone of light, converging towards a point on the surface of a biaxial crystal slab, will become collimated along the optic ray axis of the crystal, and continue to propagate along that axis for as long as the beam remains within the crystal slab. When the beam reaches the opposite facet of the slab, it emerges as an expanding cone of light. In a sense, the focused cone remains in focus in its entire path through the crystal, and diverges only after it exits the slab.

There are certain subtle differences between internal and external conical refractions; for instance, the optic axis of wave normals along which the beam propagates in the former case is not the same as the optic ray axis in the latter. This and other differences will become clear in the course of the following discussions.

The phenomena of conical refraction have been known for over 160 years,^1,2 and a complete explanation of them based on Maxwell�s electromagnetic theory has emerged, which is accessible through the published literature.^3,4 The complexity of the physics involved, however, is such that it prevents us from attempting to give a simple explanation. We shall, therefore, confine our efforts to presenting a descriptive picture of external conical refraction by way of computer simulations based on Maxwell equations.⁵

Biaxial birefringent crystals and their optic ray axes. In general, a birefringent crystal has three different refractive indices, one along each of its three principal axes. Assuming that the principal axes are the X-, Y-, and Z-axes of a Cartesian coordinate system, the principal indices may be denoted as n_x, n_y, and n_z. The ray ellipsoid of this crystal, shown in Fig. 1, will have semi-axis lengths 1/n_x, 1/n_y, 1/n_z along the coordinate axes. (Note that the ray ellipsoid is different from the index ellipsoid whose semi-axis lengths are the refractive indices themselves. While the index ellipsoid is relevant to the discussion of internal conical refraction, it is the ray ellipsoid that plays a central role in the case of external conical refraction.)

Figure 1. The ray ellipsoid has, by definition, semi-axes of length 1/n_x 1/n_y 1/n_z along the principal axes X, Y, Z of the crystal. For a ray propagating in a given direction, a plane through the center of the ellipsoid and perpendicular to the ray will have an elliptical cross-section with the ellipsoid. A propagation direction for which the cross-sectional ellipse becomes a circle is known as an optic ray axis. In general, biaxial crystals have two such optic axes.

For a ray propagating along a given direction, the plane passing through the center of the ray ellipsoid and perpendicular to the ray will, in general, have an elliptical cross-section with the ellipsoid. If a ray happens to be in such a direction that its corresponding cross-sectional ellipse becomes a circle, then the direction of that ray will define an optic ray axis. (We must emphasize that the optic ray axis is different from the optic axis, which is obtained in a similar fashion from the index ellipsoid.) Assuming n_y < n_x < n_z it is not difficult to show that the optic ray axis is in the YZ-plane, making an angle q with the Z-axis, where tan q = Ö(n_x² - n_y²) / (n_z² - n_x²)

Crystals in which the three principal indices of refraction are different from each other exhibit two optic ray axes (as well as two optic axes); these crystals are therefore known as biaxial. Internal and external conical refractions occur only in biaxial birefringent materials.

External conical refraction. Consider the system of Fig. 2, which consists of a focusing lens, a slab of biaxial birefringent crystal, a pinhole, and a collimating lens. The incident beam is uniform, coherent, and monochromatic with a vacuum wavelength of l0. The crystal slab has refractive indices n_x= 1.533, n_y = 1.500, n_z = 1.565, and its thickness is 5000 l0. (For the red HeNe wavelength of 633 nm, for example, this slab would be 3.165 mm thick.) The optic ray axes of this crystal are located symmetrically in the YZ-plane at an angle q = 45.14° from Z. The slab is cut with one of its optic ray axes perpendicular to its polished flat surfaces. (Starting at this point and continuing through the remainder of the article, the coordinate system is redefined such that the incident beam will propagate along the Z-axis, and the polished surfaces of the crystal will be parallel to the XY-plane.)

Figure 2. Schematic diagram of an optical system that demonstrates the phenomenon of external conical refraction. A coherent, monochromatic beam of light (wavelength = l0) is brought to focus by a lens. The focused spot enters a biaxial birefringent crystal slab, which is cut such that its polished surfaces are perpendicular to one of its optic ray axes. The exit facet of the crystal is painted black except for a small aperture in the middle that is left open to allow rays that propagate near the optic ray axis to exit the crystal. The exiting rays propagate to a second lens where they are collected and recollimated. In the simulations reported in this article the incident beam is uniform over the entrance pupil of the focusing lens, both lenses have NA = 0.075 and f = 46,667 l0, the crystal slab has thickness = 5000 l0 and principal refractive indices n_x = 1.533, n_y = 1.500, n_z = 1.565, and the pinhole diameter is d = 100 l0.

When the incident rays enter the crystal slab they will, in general, propagate in various directions, but the rays that happen to be on a special cone, namely, the cone of external conical refraction, will propagate strictly along the optic ray axis and will emerge from a point opposite the point of entry into the crystal. A small pinhole (diameter = 100 l0 in the present example) on the exit facet of the crystal slab allows only these axial rays to emerge. The emergent rays diverge as they propagate towards a collecting lens, where they are recollimated and directed towards the observation plane.

Figure 3 shows computed plots of intensity distribution, polarization ellipticity, and polarization rotation angle at the observation plane, corresponding to a circularly polarized incident beam. Note that the emergent rings of light in Fig. 3(a) are in the bottom half of the pupil. (Had the crystal been cut with its other optic ray axis perpendicular to the polished surfaces, the rings would have appeared in the top half of the pupil instead.) The ellipticity plot in Fig. 3(b) is coded in gray-scale, with black corresponding to -45° (i.e., left circular polarization or LCP) and white corresponding to +45° (i.e., right circular polarization or RCP). The relevant part of the plot, which is the region in the bottom half of the pupil where the emergent beam�s intensity is non-vanishing, shows zero ellipticity. The emergent rings of light, therefore, are linearly polarized. The direction of this linear polarization varies over the rings, however, as the plot of polarization rotation angle in Fig. 3(c) indicates. (The gray-scale used here assigns black to -90° and white to +90°.)

Figure 3. Distributions of intensity (a), polarization ellipticity (b), and polarization rotation angle (c) at the observation plane. The incident beam at the entrance pupil of the focusing lens is assumed to be circularly polarized. The ellipticity plot in (b) is coded in gray-scale, with black corresponding to -45° (i.e., LCP) and white corresponding to +45° (i.e., RCP). Distribution of polarization rotation angle depicted in (c) is also coded in gray-scale, but the black pixels in this case represent -90° rotation from the X-axis, while the white pixels correspond to +90° rotation. The jaggedness of the transition from balck to white in the lower part of (c) is caused by small numerical errors; since the discontinuity represented by this transition is not a physical discontinuity, the jaggedness has no physical significance.

According to Fig. 3(c), over the circumference of the rings the polarization vector rotates from -90° at the bottom (i.e., E-field antiparallel to Y-axis), to 0° at the top (E-field parallel to X), and back to +90° at the bottom (E-field parallel to Y). The apparent discontinuity of polarization direction at the bottom of the rings does not signify a physical discontinuity, because the phase of the rings (not shown here) also exhibits a 180° change during one full cycle around the rings. The overall E-field distribution turns out to be continuous after all.

Character of the emergent beam at the pinhole. The beam emerging from the pinhole in the system of Fig. 2 possesses certain interesting features. Figure 4 shows computed plots of intensity distribution (a), polarization ellipticity (b), and polarization rotation angle (c) at the pinhole, corresponding to a circularly polarized incident beam. The intensity plot in Fig. 4(a) shows a bright spot at the center of the pinhole, surrounded by a diffuse, more or less uniform background distribution. The origin of the diffuse light may be traced back to those incident rays that were outside the cone of external refraction and, therefore, once inside the crystal, did not become aligned with the optic ray axis. The plot of polarization ellipticity in Fig. 4(b) shows that the state of polarization varies from RCP in the bright white rings to LCP in the dark rings, covering the full gamut of elliptical polarization in the intervening regions. The plot of polarization rotation angle in Fig. 4(c) indicates that the orientation of the ellipse of polarization is not uniform over the aperture, but rotates through 180° around certain circular bands. All in all, this is a complex and fascinating state of affairs, compared to the dull, uniform state of polarization of the focused spot that first entered the crystal.

Figure 4. Distributions of intensity (a), polarization ellipticity (b), and polarization rotation angle (c) within the pinhole at the exit facet of the crystal slab. The incident beam at the entrance pupil of the focusing lens is assumed to be circularly polarized. The ellipticity plot in (b) is coded in gray-scale, with black corresponding to -45° (i.e., LCP) and white corresponding to +45° (i.e., RCP). Distribution of polarization rotation angle depicted in (c) is also coded in gray-scale, but the black pixels in this case represent -90° rotation from the X-axis, while the white pixels correspond to +90° rotation.

Effect of incident polarization. As was the case with internal conical refraction, the full cone of external refraction appears only when the incident beam contains all possible polarization directions. This is the case with right- or left-circularly polarized light as well as with unpolarized light. When the incident beam happens to have linear polarization, however, certain parts of the emergent cone of light will be missing. This is shown in Fig. 5 for an incident beam that is linearly polarized along the X-axis. The distributions of intensity, polarization ellipticity, and polarization rotation angle at the observation plane shown in Fig. 5 are analogous to those in Fig. 3 corresponding to a circularly polarized incident beam. The lower part of the rings in Fig. 5(a), however, is missing simply because its required polarization, which is linear along Y, is not present in the incident beam. Aside from this missing segment, other features of the emergent beam shown in Fig. 5 are quite similar to those in Fig. 3.

Figure 5. Same as Fig. 3 except for the state of polarization of the incident beam, which is linear along X in the present case.

References

W.R. Hamilton, Trans. Roy. Irish Acad. 17, 1 (1833).
H. Lloyd, Trans. Roy. Irish Acad. 17, 145 (1833).
M. Born and E. Wolf, Principles of Optics, 6th edition, chapter 14, Pergamon Press, Oxford (1980).
M. V. Klein, Optics, Wiley, New York (1970).
The simulations reported in this article were performed by DIFFRACT�, a product of MM Research, Inc., Tucson, Arizona.

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