Radiometric Consistency in Source Specifications for Lithography Alan E. Rosenbluth* a , Jaione Tirapu Azpiroz b , Kafai Lai b , Kehan Tian b , David O.S. Melville a , Michael Totzeck c , Vladan Blahnik c , Armand Koolen d , Donis Flagello e a IBM T.J. Watson Research Center, Yorktown Heights, NY; b IBM Semiconductor Research and Development Center, Hopewell Junction, NY c Carl Zeiss SMT, Oberkochen, Germany; d ASML Netherlands B.V., Veldhoven, The Netherlands; e ASML TDC, Tempe, Arizona ABSTRACT There is a surprising lack of clarity about the exact quantity that a lithographic source map should specify. Under the plausible interpretation that input source maps should tabulate radiance, one will find with standard imaging codes that simulated wafer plane source intensities appear to violate the brightness theorem. The apparent deviation (a cosine factor in the illumination pupil) represents one of many obliquity/inclination factors involved in propagation through the imaging system whose interpretation in the literature is often somewhat obscure, but which have become numerically significant in today's hyper-NA OPC applications. We show that the seeming brightness distortion in the illumination pupil arises because the customary direction-cosine gridding of this aperture yields non-uniform solid-angle subtense in the source pixels. Once the appropriate solid angle factor is included, each entry in the source map becomes proportional to the total |E|^2 that the associated pixel produces on the mask. This quantitative definition of lithographic source distributions is consistent with the plane-wave spectrum approach adopted by litho simulators, in that these simulators essentially propagate |E|^2 along the interfering diffraction orders from the mask input to the resist film. It can be shown using either the rigorous Franz formulation of vector diffraction theory, or an angular spectrum approach, that such an |E|^2 plane-wave weighting will provide the standard inclination factor if the source elements are incoherent and the mask model is accurate. This inclination factor is usually derived from a classical Rayleigh-Sommerfeld diffraction integral, and we show that the nominally discrepant inclination factors used by the various diffraction integrals of this class can all be made to yield the same result as the Franz formula when rigorous mask simulation is employed, and further that these cosine factors have a simple geometrical interpretation. On this basis one can then obtain for the lens as a whole the standard mask-to-wafer obliquity factor used by litho simulators. This obliquity factor is shown to express the brightness invariance theorem, making the simulator's output consistent with the brightness theorem if the source map tabulates the product of radiance and pixel solid angle, as our source definition specifies. We show by experiment that dose-to-clear data can be modeled more accurately when the correct obliquity factor is used. Keywords: Radiometry, obliquity factor, inclination factor, direction-cosine space, source radiance, mask diffraction. INTRODUCTION Overview To the authors' knowledge, all major litho simulators are based on fundamentally equivalent imaging equations. However, the radiometric interpretation of these equations has received little if any coverage in the lithographic literature. And though all litho simulators can be expected to handle input source patterns in a consistent way, there appears to be no clearcut common understanding of the exact physical quantity that a lithographic source should represent; at any rate we have not found any previous discussion of lithographic source radiometry in the literature. In our opinion the question has been somewhat clouded by the presence of a number of cosine factors in the standard imaging equations (referred to as obliquity or inclination factors) whose interpretation can be somewhat confusing. Such factors arise in describing diffraction of the E-field from the mask into the projection lens, and are not easy to relate conceptually to measurements of open frame irradiance patterns obtained with wafer-plane detectors. Such radiometric measurements involve cosine factors of their own, and we know of no previous work to reconcile the two sets of geometrical correction factors. (It should be noted that until recently the source-side obliquity factors in 4X reduction systems have involved fairly small angles.) *aerosen@us.ibm.com † Color copies of the figures are available from the author upon request. Optical Microlithography XXI, edited by Harry J. Levinson, Mircea V. Dusa, Proc. of SPIE Vol. 6924, 69240V, (2008) · 0277-786X/08/$18 · doi: 10.1117/12.775121 Proc. of SPIE Vol. 6924 69240V-1 2008 SPIE Digital Library -- Subscriber Archive Copy