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.
The rapidly escalating complexity of resolution enhancement techniques (RET), now commonplace in leading edge lithography, requires accurate verification to avoid yield and performance problems on the patterned wafers. Model-based verification techniques that have been derived from optical proximity correction (OPC) obtain the required checking speed from sparse sampling of the layout at discrete evaluation points along the edges of layout patterns. This sparse sampling allows accurately calibrated models to be used for full chip checking applications. However, there is a demonstrated risk of missing significant patterning errors due to the sparse and edge-centric sampling of the layout. Grid-based simulation approaches which calculate the image on a fine grid over the entire layout space accurately detect patterning problems anywhere in the layout, but can be executed at reasonable runtimes for aerial image models only. The challenge for full-chip model-based verification of RET-enhanced layouts is, therefore, a trade-off between sparse, edge-centric simulation using accurate models versus simulations using approximate models over the entire layout space. This paper presents an approach, termed contourIFV, that has been demonstrated to overcome the aforementioned problems and has been shown to provide significant value in the verification of the RET and OPC prescription.
In conventional Optical and Process Correction (OPC), models are calibrated with the CD measurement from the “good” printable patterns. Predictions of process window loss are based on extrapolation from the “good” region into the failure region. The extrapolation is always a less accurate process than interpolation. In this paper, we utilize the experimental pass/fail data to build models that accurately identify and predict printing failures. We developed a methodology and a formal apparatus for failure modeling. It is found that two or more aerial image shape parameters are required to describe all failure mechanisms for a sub-100nm process. This empirical failure model is currently applied to Optical Rule Checking (ORC) of the post-OPC layout. It also can be used to constrain layout corrections in the future.
New degrees of freedom can be optimized in mask shapes
when the source is also adjustable, because required image symmetries can be provided by the source rather than the collected wave front. The optimized mask will often consist of novel sets of shapes that are quite different in layout from the target integrated circuit patterns. This implies that the optimization algorithm should have good global convergence
properties, since the target patterns may not be a suitable starting solution. We have developed an algorithm that can optimize mask and source without using a starting design. Examples are shown where the
process window obtained is between two and six times larger than that achieved with standard reticle enhancement techniques (RET). The optimized
masks require phase shift, but no trim mask is used. Thus far we can only optimize two-dimensional patterns over small fields (periodicities
of ;1 mm or less), though patterns in two separate fields can be jointly optimized for maximum common window under a single source. We also discuss mask optimization with fixed source, source optimization
with fixed mask, and the retargeting of designs in different mask regions to provide a common exposure level.
New degrees of freedom can be optimized in mask shapes when the source is also adjustable, because required image symmetries can be provided by the source rather than the collected wavefront. The optimized mask will often consist of novel sets of shapes that are quite different in layout from the target IC patterns. This implies that the optimization algorithm should have good global convergence properties, since the target patterns may not be a suitable starting solution. We have developed an algorithm that can optimize mask and source without using a starting design. Examples are shown where the process window obtained is between 2 and 6 times larger than that achieved with standard RET methods. The optimized masks require phase shift, but no trim mask is used. Thus far we have only optimized 2D patterns over small fields (periodicities of approximately 1 micrometer or less). We also discuss mask optimization with fixed source, source optimization with fixed mask, and the re-targeting of designs in different mask regions to provide a common exposure level.
Projectors that use LCOS lightvalves face special contrast requirements. Most configurations for reflective light valves employ tilted beam-dividing coatings that see both bright and dark polarization states. The optics must then be designed to eliminate polarization mixing at these coatings, which ordinarily arises when the S and P planes for different rays are non-parallel. We show how phase- controlled coatings can exploit the double-pass symmetry of the Plumbicon tri-prism geometry to correct this effect, reducing cross-polarized reflectivity to approximately 1E-3 when the light valve is mirror-like in black-state. Though contrast in different rays varies as a function of both ray skew component and coating angle of incidence, we show that for NA <EQ 0.2 the computation involved in calculating beam contrast is essentially equivalent to tracing a single ray. Light valves that use a normally-black TN mode exhibit a non-mirror-like phase dispersion in their black-state, complicating contrast control in the optics. Scatter depolarization at the edges of pixel electrodes is enhanced in these light valves, because the inherent twist causes the backplane polarization to be rotated out of alignment with pixel edges. We show that all of these contrast degradation mechanisms can be addressed by incorporating into the light valve a compensating layer having opposite birefringence to the black-state TN active layer. Moreover, when the compensating layer and driven layer are blue-shifted to a shorter LC thickness than would ordinarily be appropriate for the wavelength band of interest, a highly achromatic response is obtained at all gray levels.
Reflective x-Si backplanes allow projection displays to evolve toward higher pixel count and greater miniaturization, extending the range of competitive application. As light valve area A is reduced, projector output into solid angle S equalsV (pi) NA2 can in many cases be considered to decrease roughly as approximately (A*S)0.5, with the 0.5 exponent representing typical microdisplay operating in a regime that is neither purely brightness-limited nor purely power-limited. Polarization modulation entails a modified scaling approximately (A*S/2)0.5; color sequential operation, approximately (1/3)*(A*S)0.5; spatially divided single-light-valve RGB projection, approximately (A*S/3)0.5. Projection lenses for three-light-valve system must provide an increased working distance to accommodate a color recombiner. Zoom lens are often required in front projectors, and rear projection usually entails a short lens-to-screen distance. It has become cost-effective to use plastic aspherical elements to meet these requirements. Periodic strip-PBS arrays have been widely adopted for polarization recycling, but aperiodic homogenizers are sometimes used to correct the uneven magnification and symmetry limitations of conic reflectors. Bright-state and dark-state beams must occupy distinct etendues in the half space above a reflective light valve, creating a vulnerability to crosstalk. Crosstalk from a polarizing beamsplitter gives rise to a residual background intensity approximately 0.3*NA2, unless a quarterwave corrector is used. Crosstalk can also arise from stress birefringence in prism substrates. Stray light makes an indirect contribution to background, but can sometimes be corrected by filtering.
This course is directed towards presenting a methodology to include layout effects on circuit analysis. DFM imposes embellishments on the layout to ensure manufacturability with acceptable yields. Traditional circuit analysis and effects of process variability are performed at the schematics level using models for process corners and may lead to excessive guardbanding. Circuit Analysis that is able to predict impact and sensitivity of layout modifications uses circuit simulators together with information derived from litho simulations and helps the designer to ascertain that the layout accompanying the design meets the manufacturability criteria.