# Appendix B

# The Scalar Theory of Diffraction

## B.1 Full Scalar Theory

In order to introduce the concept of the scalar theory of diffraction, let us consider for a start an arbitrary wavefront *U*(*x, y*; *z*) propagating from *z* = 0 along the positive *z*-axis of a Cartesian referential [1]. The Fourier transform of such a wavefront is given by

The same arbitrary wavefront *U*(*x, y*; *z*) is thus the inverse Fourier transform of *U*(*u, v*; *z*), defined as

Therefore, *U*(*x, y*;*z*) can be described as an infinite composition of the set of functions *e*^{−2πj(ux + vy)} weighted by the coefficients of *U*(*u, v*; *z*). Bearing in mind the complex representation of plane waves (Equation (A.7)), it is obvious that these functions can be understood as sets of plane waves propagating in the *z* direction with the cosine direction

This means that *U*(*x, y*; *z*) can be decomposed into an angular spectrum of plane waves *U*(*u, v*; *z*). This angular spectrum of plane waves is also the Fourier transform of *U*(*u, x*; *z*), or the far-field representation of that same function.

Now let us examine how the angular spectrum propagates from a plane at *z* = 0 to a plane at *z* = *z*_{0}. We have therefore to find a relation between *U*(*u, v*; 0) and *U*(

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