Modeling the free space propagation of laser beams using Fourier transforms

Question

I am trying to model the propagation of a laser beam in free space. I have an initial field $E_{in}(x,z=0)$ (a Gaussian beam) and need to find the fields at other points on the optical axis $E(x,z=d)$ for an arbitrary distance $d$.

By reading through a couple of texts, this is the approach that I have right now:

• Compute the Fourier transform of the initial field: $\hat{E}(k_x) = \mathscr{F}[E_{in}(x,z=0)]$
• Multiply $\hat{E}$ by the the free space transfer function $e^{i k_z z0}$ where $k_z = \sqrt{k^2 - k_x^2}$ to propagate it by a distance $z0$ along the optical axis.
• Inverse Fourier transform back to obtain $E(x,z=z0)$

This method makes sense to me. I think we are imagining the field as an infinite collection of plane waves and through the Fourier transform, we are essentially moving each of these plane waves by propagating through each of their respective wave numbers. I understand that the ABCD matrix method might be an easier technique, but I need a method that works for arbitrary beams and not just Gaussian beams.

I am implementing this on Mathematica at the moment and the resulting fields that I am getting do not match my expectations from Gaussian beam propagation (they do not follow the trends of spherical wave fronts). I would appreciate any help in figuring out if this is the right approach. I would also appreciate any help in finding other techniques that might be useful for this modeling.

Answer 1

The problem with using the actual free space Fourier propagator is aliasing.

I learned this through trial and error as well, after a few wavelengths the numerical model really begins to behave poorly, probably due to aliasing and roundoff error.

The Fresnel approximation actually does a better job if you are anywhere further than the very near field region, it is numerically more stable...I'm sure you could write software that corrects the errors, but just use Fresnel, it is very accurate...

$$\newcommand{\Four}{\mbox{$\mathcal{F}$}} u_z(\mathbf{r}) \approx \frac{e^{ikz}}{i\lambda z}e^{i\pi\frac{\mathbf{r}^2}{\lambda z}}\Four\left\{ u_0(\mathbf{r}_0) e^{i\pi\mathbf{r}_0^2/\lambda z}\right\}_{\mathbf{\rho} = \frac{\mathbf{r}}{\lambda z}}$$

where the following constraint must be met : $$z^3 \gg \|\mathbf{r}-\mathbf{r}_0\|^4/\lambda$$ and $\mathbf{r}:=(x,y)$; the coordinates in the plane at your particular $z$, perpendicular to the $z$-axis, $\mathbf{r}_0:=(x_0,y_0)$; the coordinates in the plane at your initial field position $z_0 = 0$, perpendicular to the $z$-axis.

Fraunhofer (far-field) is valid when $$z \gg \|\mathbf{r}-\mathbf{r}_0\|^2/\lambda$$

Answer 2

I already wrote elsewhere that Gaussian beams are just some approximations to solutions of the Maxwell equations. For this reason, I derived some exact solutions of the Maxwell equations that are approximated by Gaussian beams extremely well ("asymptotically accurately") when the beam waist is much larger than the wavelength. Please see https://arxiv.org/abs/physics/0405091, Eq. 22. The solution describes a circularly polarized beam, but it is not difficult to derive a solution describing a linearly polarized beam.

Answer 3

As mentioned above, aliasing is an issue with Fresnel propagation using Fourier transforms, but it is very manageable. SPIE has published some good books on writing code for Fresnel propagation and how to minimize the effects of aliasing. These two cover the propagation math and have lots of Matlab code: https://doi.org/10.1117/3.858456

https://doi.org/10.1117/3.866274

This one is less about propagation and more about doing Fourier transforms in Mathematica: https://doi.org/10.1117/3.2574956

A lot of universities have institutional subscriptions to the SPIE Digital Library, so you might be able to download these books for free.