qspec.gaussian_doppler_3d  (  r ,  k ,  w0 ,  v ,  r0 = None ,  axis = -1  )[source]

The length |$\vec{k}^\prime$| of the Doppler-shifted 3-vector k in the rest frame of the atom $$\begin{aligned} |\vec{k}^\prime| &= |\vec{k}|\gamma\left[1 - \beta\cos(\alpha)\left(1 - \frac{w_0^2}{2z_+^2} - \frac{\rho^2z_-^2}{2z_+^4}\right) - \beta\sin(\alpha)\frac{\rho z}{z_+^2}\right]\\[2ex] z_\pm^2 &= z^2 \pm z_0^2\\[2ex] z_0 &= \frac{1}{2}|\vec{k}|w_0^2\\[2ex] z &= (\vec{r} - \vec{r}_0)\cdot\hat{k}\\[2ex] \rho &= \sqrt{\left[(\vec{r} - \vec{r}_0)\cdot\hat{x}\right]^2 + \left[(\vec{r} - \vec{r}_0)\cdot\hat{y}\right]^2}, \end{aligned}$$ where $\beta = |\vec{v}| / c$ is the relativistic velocity, $\gamma$ the time-dilation factor and $\alpha$ the angle between $\vec{k}$ and $\vec{v}$.

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).

varray_like

The velocity 3-vector $\vec{v}$ (m/s).

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:

k_absndarray

The length |$\vec{k}^\prime$| (rad / m).

Raises:

ValueError

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