qspec.gaussian_beam_3d  (  r ,  k ,  w0 ,  p0 ,  r0 = None ,  axis = -1  )[source]

The Gaussian beam intensity at the position $\vec{r} - \vec{r}_0$ $$\begin{aligned} I(\vec{r}) &= \frac{2P_0}{\pi w_z^2}\,\exp\!\left[-2\left(\!\frac{\rho}{w_z}\!\right)^{\! 2}\right]\\[2ex] w_z &= w_0\sqrt{1 + \left(\!\frac{z}{z_0}\!\right)^{\!2}},\qquad z_0 = \frac{1}{2}|\vec{k}|w_0^2\\[2ex] z &= (\vec{r} - \vec{r}_0)\cdot\hat{k},\qquad \rho = \sqrt{\left[(\vec{r} - \vec{r}_0)\cdot\hat{x}\right]^2 + \left[(\vec{r} - \vec{r}_0)\cdot\hat{y}\right]^2}, \end{aligned}$$ where $\hat{k}$ is the unit vector in $\vec{k}$ direction and $\hat{x}$, $\hat{y}$ are unit vectors orthogonal to $\hat{k}$.

Parameters:

rarray_like

The position 3-vector $\vec{r}$ where to calculate the beam intensity (m).

karray_like

The 3-vector $\vec{k}$ of light, where $|\vec{k}| = \omega / c$ (rad / m).

w0array_like

The beam waist $w_0$ (m).

p0array_like

The total power $P_0$ propagated by the gaussian beam (W).

r0array_like

The position 3-vector $\vec{r}_0$ of the beam waist. If r0 is None, it is [0., 0., 0.] (m).

axisint

The axis along which the vector components are aligned.

Returns:

I_rndarray

The intensity $I(\vec{r})$ (W/m² = μW/mm²).

Raises:

ValueError

r, k and r0 must have 3 components along the specified axis. The shapes of r, k, w0, r0 and p0 must be compatible.