Theory
The derivations given here are treated in more detail in the relevant literature, e.g. [ST91] and [Goo05].
Optical transfer function
Let us consider a wave field \(u(\mathbf{r_0})\) whose values we know at an initial plane \(\mathbf{r_0}=(x_0,y_0,z_0)\) (\(z_0\) fixed). The field has a certain vacuum wavelength \(\lambda\) and is traveling through a homogeneous medium with refractive index \(n_\mathrm{m}\). From the knowledge of the wave field at the plane \(\mathbf{r_0}\) and its wavelength \(\lambda/n_\mathrm{m}\), we can infer the direction of propagation of the wave field for every point in \(\mathbf{r_0}\). We rewrite the field at \(\mathbf{r_0}\) as an angular spectrum, a sum over all possible directions \(\mathbf{s}=(p,q,M)\), assuming that the field is only traveling from left to right
The equation above describes the Huygens-Fresnel principle: the value of the field \(u\) at a certain position \(\mathbf{r_0}\) at the initial plane (point source) is defined as an integral over all possible plane waves with wavenumber \(k_\mathrm{m}=\frac{2\pi n_\mathrm{m}}{\lambda}\), weighted with the amplitude \(A(p,q)\).
Let us now consider the 2D Fourier transform of \(u(\mathbf{r_0})\).
Here we made use of the identity of the delta distribution
If we now perform the same procedure for a different position \(\mathbf{r_\mathrm{d}}=(x_0,y_0,z_\mathrm{d})\), we will see that the Fourier transform of the field becomes
Thus, the propagation of the field \(u(\mathbf{r_0})\) by a distance \(d=z_\mathrm{d}-z_0\) is described by a multiplication with the transfer function
in Fourier space. This is the basis of the convolution-based numerical propagation algorithms implemented in nrefocus. The process of numerical propagation with the angular spectrum method can be written as
with the Fourier transform \(\mathcal{F}\) and its inverse \(\mathcal{F}^{-1}\). With the convolution operator \(\ast\), we may rewrite this equation to
Fresnel approximation
The Fresnel approximation (or paraxial approximation) uses a Taylor expansion to simplify the exponent of the transfer function \(e^{ik_\mathrm{m}Md}\). The exponent can be rewritten as
If the angles of propagation \(\theta_\mathrm{x}\) and \(\theta_\mathrm{y}\) for each plane wave of the angular spectrum is small, then we can make the paraxial approximation:
We now Taylor-expand the exponent around small values of \(\theta\)
The Fresnel approximation discards the third term (\(\sim \theta^4\)) and the transfer function then reads:
Thus, the propagation by a distance distance \(d=z_\mathrm{d}-d\) in the Fresnel approximation can be written in the form of the convolution
Note that the Fresnel approximation results in paraboloidal waves \((p^2+q^2)\) whereas spherical waves are used with the Helmholtz equation.
Transfer functions in nrefocus
The numerical focusing algorithms in this package require the input data \(u_\text{in}\) to be normalized by the incident plane wave \(u_0(\mathbf{r_0})\) according to
As a result, the transfer functions change to