By O'Shea D.C., et al.

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**Extra resources for Diffractive Optics: Design, Fabrication, and Test**

**Sample text**

Because a diffractive optical element will be located at the aperture plane, we will use a special notation for the ﬁeld at that point: P(x, y, 0) = P0 (x, y). This integral [Eq. 8)] is a useful tool for analyzing the scalar propagation of light. The eight assumptions listed earlier are valid for a wide range of optics problems. Assumptions 1 to 4 are true for vacuum, air, and most glasses and transparent media; while assumptions 5 to 8 are valid for most cases where a beam of light is propagating from an aperture, lens, or diffractive element to some target plane.

However, it is not useful to make it too speciﬁc. To do so, we would lose the ability to apply it to a large number of situations. In solving the Helmholtz equation, P is expressed as an integral over the aperture, known as the Kirchhoff integral. The derivation of this solution is somewhat involved and would divert this discussion. Instead we will describe the solution for a geometry that is appropriate for diffractive optics. 45. 1 A variety of optical phenomena that can be accurately analyzed using scalar diffraction theory.

14) There are two further simpliﬁcations that will make connections between this integral expression and the Fourier transform. 15) where α and β are the direction cosines of the propagation vector k. The third component, kz , has a direction cosine γ = z/R, but our integration is over only the aperture. 16) permitting Eq. 14) to be written in a compact and useful form: P(X, Y, Z) = u(x, y, 0) exp[−ik(xX/R)] exp[−ik(yY/R)]dxdy. 17) aperture It is usually the case for transmitting diffractive optical elements that the aperture includes phase variations, so phase should be included in the function u(x, y, 0) as u0 (x, y) exp[iφ0 (x, y)].