## the Gaussian beam (interactive)

The Gaussian beam “is a convenient, widespread model in laser optics.” [1] It is indeed convenient that such a mathematically relatively simple model can describe the evolution of laser light in free space so well. Given the wavelength of the laser, it can be described by a single parameter. This can be either the beam waist $w_0$, its far-field divergence angle $\theta$ or numerical aperture $\mathrm{NA}$, the Rayleigh range $z_R$ or the confocal parameter $b$. Given one, the others are easily determined. The graphic below will aid this process. Start by entering a wavelength or frequency, and then enter any of the various parameters and the rest will be calculated.

Below the graphic, various plots will give the beam cross-section (amplitude, intensity and phase) and the evolution of beam width and wavefront curvature near the waist position $z=0$. At the bottom of the page, the various formulas used for the calculations here are given.

Also note that the Gaussian beam is a solution of the paraxial Helmholtz (wave) equation, i.e. it is only a good approximation as long as the divergence angle is not too large. This is the case when approximately $w_0 k > 4$ or $w_0 > 2\lambda / \pi$.

Check out the entry on Paraxiality for more information.

Show cross-sections for distance $z=$ from beam waist.

Plots of the normalized transverse amplitude $|E / E_0|$ (red) and intensity $I / I_0$ (blue) evolution at longitudinal position $z$ = 0μm from beam waist. The horizontal line indicates the $1/e$ amplitude, respective $1/e^2$ intensity level.

Plot of the transverse phase evolution at longitudinal position $z$ = 0μm from beam waist.

Plots of the longitudinal evolution of beam width $w(z)$ and radius of wavefront curvature $R(z)$ for the Gaussian beam defined above. The vertical line denotes the Rayleigh range $z_R$.

#### Formulas

The complex electric field amplitude is given by [1]

E(\mathbf{r}, z) = E_0 \frac{w_0}{w(z)} \exp \left( -\frac{\mathbf{r} \cdot \mathbf{r}}{w^2(z)} -ikz -ik \frac{\mathbf{r} \cdot \mathbf{r}}{2R(z)} +i \zeta(z) \right)

with transverse coordinate $\mathbf{r} = (x,y)^T$ and longitudinal coordinate $z$, defined such that the location of minimal beam width is at $z=0$. The quantity $k = 2\pi / \lambda$ is called the wavenumber.

The (time-averaged) intensity of the beam is then given by

I(\mathbf{r},z) = \frac{ E(\mathbf{r},z) E^*(\mathbf{r},z)}{2 \eta} = I_0 \left( \frac{w_0}{w(z)} \right)^2 \exp \left( -\frac{2 \mathbf{r} \cdot \mathbf{r}}{w^2(z)} \right)

where $*$ denotes the complex conjugate, and $\eta$ is the characteristic impedance of the medium. The initial amplitude and intensity are

\begin{equation*}
E_0 = E(x=0,y=0,z=0) \quad \text{and} \quad I_0 = I(x=0,y=0,z=0)
\end{equation*}

respectively. The $1/e$ beam width is given by

\label{beam-width}
w(z) = w_0 \, \sqrt{ 1+ {\biggl( \frac{z}{z_R} \biggr)}^2 } = w_0 \, \sqrt{ 1+ {\biggl( \frac{2 z}{k w_0^2} \biggr)}^2 }

where

\label{rayleigh-range}
z_R = \frac{\pi w_0^2}{\lambda} = \frac{k w_0^2}{2}

is called the Rayleigh range and $w_0 = w(0)$ is the minimum beam width, or beam waist. At the Rayleigh range, $w(\pm z_R) = \sqrt{2} w_0$.

The wavefront curvature $R(z)$ is given by

\label{curvature}
R(z) = z \left[{ 1+ {\biggl( \frac{z_R}{z} \biggr)}^2 } \right] = z \left[{ 1+ {\biggl( \frac{k w_0^2}{2 z} \biggr)}^2 } \right]

At the waist, the wavefronts are plane, $R(0) = \infty$; the curvature radius decreases to a minimum at the Rayleigh range, and far from $z_R$ increases again as $R(z) \approx z$.

The confocal parameter, or depth of focus, $b$ is the distance between the Rayleigh ranges on both sides of the waist,

b = 2 z_R = k w_0^2

The longitudinal phase delay $\zeta(z)$, or Guoy phase, is

\label{guoy}
\zeta(z) = \mathrm{arctan} \left( \frac{z}{z_R} \right) = \mathrm{arctan} \left( \frac{2 z}{k w_0^2} \right)

which imposes an additional phase shift of π on the beam as it passes through the focus.

The far-field divergence angle $\theta$ (in radians) of the beam is given by

\label{divergence-angle}
\theta \simeq \frac{\lambda}{\pi w_0} = \frac{2}{k w_0} = \frac{w_0}{z_R}

and the numerical aperture is

\mathrm{NA} = \sin \left(\theta \right)

At the waist, the Gaussian beam is a plane wave with a Gaussian apodization, while in the far-field it is a spherical wave, also with Gaussian apodization, making it cone-shaped.

#### Canonical Forms

Interestingly, all spatial quantities in the above formulas scale with wavelength, even if it’s not immediately apparent. Hence, we can normalize the equations describing the properties of the Gaussian beam to the wavelength of the radiation to obtain the canonical, or standard, or normalized forms. Starting with the normalized quantities

\begin{equation*}
z’ = 2\pi \frac{z}{\lambda} = k \cdot z \quad \mathbf{r}’ = k \cdot \mathbf{r} \quad w’_0 = k \cdot w_0
\end{equation*}

we obtain from $\eqref{rayleigh-range}$

\begin{equation*}
z’_R = k \cdot z_R = \frac{k^2 w_0^2}{2} = \frac{{w^\prime_0}^2}{2}
\end{equation*}

and from $\eqref{beam-width}$ and $\eqref{curvature}$

\begin{gather*}
w'(z’) = k \cdot w\biggl(z = \frac{z’}{k}\biggr) = w’_0 \, \sqrt{ 1+ {\biggl( \frac{2 z’}{{w’_0}^2} \biggr)}^2 } \\
R'(z’) = k \cdot R\biggl(z = \frac{z’}{k}\biggr) = z’ \left[{ 1+ {\biggl( \frac{{w’_0}^2}{2 z’} \biggr)}^2 } \right]\\
\end{gather*}

Also, from $\eqref{divergence-angle}$ and $\eqref{guoy}$

\begin{gather*}
\theta \simeq \frac{2}{w’_0} = \frac{w’_0}{z’_R}\\
\zeta'(z’) = \zeta\biggl(z = \frac{z’}{k}\biggr) = \mathrm{arctan} \left( \frac{2 z’}{{w’_0}^2} \right)
\end{gather*}

Note that since $\zeta’$ is not a coordinate or length, it does not scale with $k$ – it is simply the Guoy phase written in terms of $z’$.
Then finally, the canonical form of the complex field amplitude is

\begin{equation*}
E(\mathbf{r’}, z’) = E_0 \frac{w’_0}{w'(z’)} \exp \left( -\frac{\mathbf{r’} \cdot \mathbf{r’}}{{w’}^2(z’)} -iz’ -i \frac{\mathbf{r’} \cdot \mathbf{r’}}{2R'(z’)} + i \zeta'(z’) \right)
\end{equation*}

and analogously the beam intensity.

#### Notes

Also check out post #145 which takes a closer look at the difference between the Gaussian beam model and the real beam resulting from diffraction.

#### References

[1] Gaussian beam [Wikipedia]

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