GaussianBeam¶

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Specifies a time-harmonic vectorial Gaussian beam.

# define a Gaussian beam with a propagating Fourier spectrum
GaussianBeam {
  Waist = 6.2831e-5
  Focus = [0 0 0]

  Incidence = FromBelow
  Lambda0 = 6.2831e-6
  SP = [1 0]
  ThetaPhi = [25 0]

  OpticalSystem {
    NumericalAperture = 1.0
  }
}

This section allows to specify the parameters of a vectorial Gaussian beam defined by its Fourier spectrum. The following figure indicates the modeled setup of an incident beam passing through an optical system prior to reaching the computational domain.

A macroscopic beam passes an optical system that focusses it into the computational domain. The beam is determined by its spatial distribution in the cross section plane in front of the optical system. The optical axis can tilted by the angles $\theta$ and $\phi$ .¶

The beam profile is determined by its spatial cross-section in a plane perpendicular to the optical axis and the optical system. The corresponding spectrum in k-space and the out of plane vector components are determined automatically.

Definition of the spatial cross section

We use the 2-vector parameter SP, $[\VField{\VField{E}}_s, \VField{\VField{E}}_p]$ to specify the polarization of the beam:

$\begin{eqnarray*} \VField{A}_\perp = \VField{\VField{E}}_s \cdot \hat{s}+\VField{\VField{E}}_p \cdot \hat{p}, \end{eqnarray*}$

where $\hat{s},\hat{p}$ are defined as for the plane wave case. This amplitude is modulated by a Gaussian profile determined by the Waist (optionally a 2-vector parameter $[W_s, W_p]$ along $\hat{s},\hat{p}$ directions):

$\begin{eqnarray*} A(\pvec{x}_\perp)= \exp\left( -\frac{(\pvec{x}_\perp \cdot \hat{s})^2}{W_s^2} - \frac{(\pvec{x}_\perp \cdot \hat{p})^2}{W_p^2}\right) \end{eqnarray*}$

In the standard case with the optical axis aligned with the z-axis and a symmetric waist the resulting beam shape is the familiar:

$\begin{eqnarray*} \VField{A}_\perp =\VField{A}_\perp A(\pvec{x}_\perp) = (\VField{E}_s \hat{y} + \VField{E}_p \hat{x}) \exp\left(-\frac{x^2}{W^2} - \frac{y^2}{W^2}\right) \end{eqnarray*}$

where $[\VField{\VField{E}}_s, \VField{\VField{E}}_p]$ are the entries of SP and the Waist $W$ .

Definition of the optical system

The OpticalSystem is used to model the transfer of the beam in the sampling cross section to the focal plane. The focal plane is described by the Focus and a rotation of the default optical axis $\hat{z}$ by the angles ThetaPhi. With the rotation matrices $M_{\varphi}$ and $M_{\theta}$ and the unit vector $\hat{z}$ (the direction is determined by the parameter Incidence $z$ (FromBelow) or $-z$ (FromAbove) ) the optical axis $\hat{k}$ is calculated as follows:

$\begin{eqnarray*} \hat{k} = \frac{2 \pi n}{\lambda_0} M_{\varphi} \cdot M_{\theta} \cdot \hat{z} \end{eqnarray*} \begin{eqnarray*} M_{\varphi} = \begin{pmatrix} \cos(\varphi) & -\sin(\varphi) & 0 \\ \sin(\varphi) & \cos(\varphi) & 0 \\ 0 & 0 & 1 \end{pmatrix} \qquad M_{\theta} = \begin{pmatrix} \cos(\theta) & 0 & \sin(\theta) \\ 0 & 1 & 0 \\ -\sin(\theta) & 0 & \cos(\theta) \end{pmatrix} \end{eqnarray*}$

.

The OpticalSystem has several parameters to describe aberrations as well as parameters defining the SpotMagnification and NumericalAperture to describe focusing system or restrict the Fourier spectrum.

Note

As the image is magnified by the value of SpotMagnification you must use a value $<1.0$ to describe a focusing system.

By default, the full Fourier spectrum is passed through a perfect system without abberations and an infinite numerical aperture.

Numerical parameters

The sampling rate is automatically adapted to suit the beam parameters. The parameter DeltaK and NRefinements can be used for a user-defined sampling.