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 $w_0 > 2\lambda / \pi$.

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*}
\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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