Scalar diffraction theory

The fundamental problem in scalar diffraction theory is to determine the value of a scalar wave xp at a point P given the value of xp and Vxp on a closed surface S surrounding P. A concise and lucid treatment of this theory is given by Wangsness (1963), who shows that... 

An amount of energy I a2 is removed from a beam with irradiance /, as a result of reflection, refraction, and absorption of the rays that are incident on the sphere that is, every ray is either absorbed or changes its direction and is therefore counted as having been removed from the incident beam. An opaque disk of radius a also removes an amount of energy I a2, and to the extent that scalar diffraction theory is valid, a sphere and an opaque disk have the same diffraction pattern. Therefore, for purposes of this analysis, we may replace the sphere by an opaque disk. 

Figure 7.4 The crosses are computed from Mie theory. Scalar diffraction theory and geometrical optics predict the limiting value 1.067. From Bohren and Herman, 1979.

According to scalar diffraction theory (Section 4.4) the scattering amplitude in the forward direction is proportional to the cross-sectional area of the particle, regardless of its shape, and is independent of refractive index. To the extent that diffraction theory is a good approximation, therefore, the radius corresponding to the response of an instrument that collects light scattered near the forward direction by a nonspherical particle is that of a sphere with equal cross-sectional area. The larger the particle, however, the more the... 

We have used scalar diffraction theory in this calculation, which is an approximation in two parts. The first part consists of approximating the electromagnetic field as a transverse field. We have derived the conditions under which it is permissible to do so. In the Appendix, we discuss the conditions under which it is possible to replace the vector Helmholtz equation by the scalar Helmholtz equation for transverse fields. In a sense, we have reduced the problem to a solution of the scalar Helmholtz equation. The second part of the approximation consists of exploiting the reduction of the vector Helmholtz equation to a scalar Helmholtz equation. Scalar diffraction theory is based on the scalar Helmholtz equation. Hence, when it is permissible to neglect the longitudinal and cross-polarized components of the Gaussian beam, we may use solutions of the scalar Helmholtz equation for transverse fields and may take over the results of scalar diffraction theory with confidence for this special case. 

It is possible to include phase transformers in scalar diffraction theory. The calculations are lengthy, however, and we refer the reader to Anan ev (1992) and Martin and Bowen (1993) for details. An alternative approach exists that is equivalent to the transfer matrix method of geometrical optics, although the results are justifiable in terms of diffraction theory (Anan ev, 1992 Martin and Bowen, 1993). The formalism is discussed, for example, in Hecht and Zajac (1979, pp. 171-175) and we will briefly outline the necessary results. 

We have now successfully reduced the vector Helmholtz equation to the scalar Helmholtz equation for transverse fields. Under the conditions derived in Section III, transverse fields are often an accurate description of a Gaussian beam. In order to study the effects of diffraction on transverse fields, we note that scalar diffraction theory is based on the scalar Helmholtz theory (Born and Wolf, 1980, pp. 370-386). Thus, we may use scalar diffraction theory with the function u to elucidate the effects of diffraction on Gaussian beams that are well approximated by transverse fields. 

This is the intensity of the exposure radiation in the plane of the wafer. The extended source method, or Hopkins method,is often used to predict the aerial image of a partially coherent, diffraction-limited, low-numerical-aperture-aherrated projection system based on scalar diffraction theory. For very high NA, vector calculations involving the complete solution of Maxwell s equation are used. The illumination may be of a single wavelength or it may be broadband. The illumination source may be a conventional disk shape or other more complicated shapes as in off-axis illumination. ... 

J. Durnin, "Exact solutions for diffraction free beams 1 the scalar theory" J. Opt. Soc. 

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