Topics in polarization ray tracing for image projectors

Many subtle effects arise when tracing polarization along rays that converge or diverge to form an image. This paper concentrates on a few examples that are notable for the challenges they pose in properly analyzing vector imaging problems. A striking example is the Federov-Imbert shift, in which coating phase-shifts cause a reflected beam to actually be deviated "sideways" out of the plane of incidence. A second example involving groups of coated surfaces is the correction of contrast loss from skew-angle depolarization in the optics of data projectors that use reflective polarization-modulating light valves. We show that phase-controlled coatings can collectively correct the contrast loss by exploiting a symmetry that arises when the coatings are operated in double-pass (due to use of reflective light valves). In lowest order, this symmetry causes any ellipticity that the coatings may introduce in the polarization of illuminating skew-rays to cancel in the return pass from the light valve back through the optics. Even beyond this first order reversibility result, we have shown elsewhere that, for NA less than about 0.2, the computation involved in calculating beam contrast can be reduced to the equivalent of tracing a single ray. We show here that the Federov-Imbert shift can be derived in a straightforward way using this formalism. Even a non-polarizing system will show vector effects when the numerical aperture is sufficiently high, as in photolithographic lenses. Wavefront quality in these deep-UV lenses is of order λ/100, and simulations to account for the complexities of the image transfer steps during IC manufacture must be accurate to better than a part in 1E2 or 1E3; hence small polarization distortions in the superposed image rays become very significant. An interesting source of such distortions is spatial dispersion in CaF2 lens elements, which gives rise to intrinsic birefringence at the ppm level. Polarization ray tracing must then contend with the phenomenon of double refraction, wherein a given ray splits into two rays each time it passes through an element, giving rise in principle to an exponentially extended family of rays in the exit pupil. However, we show that it is possible to merge each coherent family of rays into a single plane-wave component of the image. (This is joint work with colleagues at Carl Zeiss SMT.1) Generalizing beyond the analysis of birefringence, such a plane-wave component can be identified with the particular subset of rays that are converged through a common pupil point and transferred to the image after diffracting from the object points within an isoplanatic patch. Thin-film amplitude transfer coefficients implicitly take into account the prismatic change in beam-width that occurs when such a ray bundle refracts through a lens surface, but these coefficients do not include the focusing effect arising from power in the surfaces; hence polarization ray-tracing by sequential application of thin-film transfer coefficients does not by itself provide the correct amplitude distribution over the pupil.