ABSTRACT

The Seidel wavefront aberrations are defined as deviations from paraxial optics. Similarly, for the derivation of polarization aberrations, paraxial optics forms an excellent basis for deriving the low order forms of polarization aberration. For paraxial optics, three polarization aberrations occur, aberrations which are the diattenuation or retardance equivalents of defocus (quadratic), tilt (linear) and piston (constant). So the polarization aberration of most optical systems can often be described with just three terms, polarization defocus, polarization tilt, and polarization piston. In fact, the fraction of the étendue of an optical system which is well described by second order polarization aberrations is generally far larger than the fraction of the étendue described by the fourth order wavefront aberrations, i.e. the region where the contributions of spherical aberration, coma, astigmatism, and field curvature are much less than one wave of optical path length.

The paraxial ray trace is used to calculate the angle of incidence across the wavefront at each interface. The amplitude coefficients, such as Fresnel equations for uncoated interfaces, are fit to quadratic equations. The combination of the paraxial angle of incidence with these quadratic amplitude relations leads to surface contributions of the polarization aberrations forms, three for linear diattenuation and three for linear retardance, at each interface. These surface contributions can be summed over the interfaces to obtain a paraxial polarization aberration for the optical system. Several examples will demonstrate the utility of the paraxial polarization aberrations for approximating the Jones pupil of radially symmetric lens and mirror systems, and describing many off-axis systems, such as fold mirrors and off-axis telescopes.

Zernike polynomials are generalized into vector Zernike polynomials for the discussion of higher order polarization aberrations. Vectors repeat after a 360° rotation. Angle of incidence, linear retardance, and linear diattenuation are not vectors; their properties repeat after a 180° rotation. To account for this geometrical property of repetition after a 180° rotation, orientors are introduced which provide basis functions for the expansion of angle of incidence, linear retardance, and linear diattenuation. Orientors are defined using an “angle-doubled” approach. Given a Jones pupil, the pupil function is divided into a hermitian (diattenuating) matrix function and a unitary (retarding) matrix function, both of which have four degrees of freedom, using the polar decomposition. The linear retardance is expanded in orientors by doubling the orientation angles and expanding in the vector Zernike polynomials. The average phase contribution from coatings, a contribution to the “scalar” wavefront aberration is expanded in “scalar” or ordinary Zernike polynomials, as is usual, as is the circular retardance. A similar algorithm is applied to the diattenuation aberrations.

Optical system polarization aberrations can be measured by placing the system in the sample compartment of a Mueller matrix imaging polarimeter. Usually the exit pupil is imaged, measuring a Mueller matrix as a function of pupil coordinates. Maps are readily generated of linear diattenuation, linear retardance, and other metrics. Such a Mueller matrix pupil image is readily converted to a Jones pupil, but the absolute phase, the wavefront aberration is not measured by a non-interferometric Mueller matrix image set.