ABSTRACT
Chapter 12, Fresnel Aberrations, explores the polarization aberrations of several example optical systems which arise strictly from the Fresnel equations, for example systems of metal coated mirrors and uncoated lenses. Polarization change occurs at each reflecting and refracting surface due to differences between the s and p-components of the light’s reflectance and transmission coefficients. Multilayer coated surfaces produce polarization aberrations with similar functional forms and have scalable magnitudes.
Fresnel aberrations are visualized in terms of the diattenuation and retardance variations associated with ray paths. Across a set of rays, the angles of incidence and planes of incidence vary, and thus the polarization change varies, so that a uniformly polarized input beam has polarization variations when exiting. For an uncoated lens with an on-axis beam, the diattenuation map is all oriented radially in the plane of incidence. The magnitude of the diattenuation increases quadratically from zero in the middle of the pupil. If the lens is placed in between crossed linear polarizers (oriented at 0° and 90°), the 0° component of the light is removed and only the polarization-coupled 90° component is transmitted. The pupil is now dark along the x and y-axes and the transmitted light is the brightest at the edge of the pupil along the ±45° directions. The corresponding flux distribution (amplitude squared) is referred to the Maltese cross. For a Cassegrain telescope with bare aluminum coatings, the phase changes due to the mirrors are toroidal, curving upwards (phase advancing) quadratically along one axis across the exit pupil and downward (delayed) along the other. This toroidal form indicates that a small amount of astigmatism is introduced by the primary and secondary mirrors into the on-axis beam, contrary to conventional aberration theory where wavefront aberration theory for radially symmetric systems, there is no astigmatism on-axis! The telescope’s point spread function between crossed polarizers is dark in the center with four islands of light in a square.
Fold mirrors coated with metals such as aluminum or silver are a significant source of polarization aberration, diattenuation and retardance, which can be compensated by using individual components with aberrations of the opposite sign, minimizing the resultant aberration. For a high numerical aperture beam reflecting from a fold mirror viewed between two 0° polarizers, the point spread function deviates from an Airy disk by being slightly elliptical with incomplete diffraction rings. The fold mirror polarization aberration can be compensated by using mirror orientations with aberrations of the opposite sign. Sequences of one, two, and four gold-coated fold mirrors are considered to understand configurations which reduce the polarization aberrations. One clever application of the Fresnel equations is the Fresnel rhomb, a total internal reflection-based quarter wave retarder.
