Gaussian Beam (Ch. 3) Oct. 16, 2017

• Envelop equation: TEM wave$\to$ no field in longitudinal dir (z); transverse field$\text{ }{E}_{t}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}A\left(x,y,z\right){e}^{-jkz}$
Nonparaxial wave equation$\text{ }{\nabla }_{t}^{2}A\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\partial }^{2}A/\partial {z}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}2jk\partial A/\partial z\text{\hspace{0.17em}}=\text{\hspace{0.17em}}0$
• Paraxial wave equation:
Envelop varies slowly along z
+$\text{ }{\nabla }_{t}^{2}A\text{\hspace{0.17em}}-\text{\hspace{0.17em}}j2k\partial A/\partial z\text{\hspace{0.17em}}=\text{\hspace{0.17em}}0$
+ Normalized with$\text{ }{r}_{N}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}r/{w}_{o}$ ,$\text{ }Z\text{\hspace{0.17em}}=\text{\hspace{0.17em}}z/\left(2{z}_{o}\right)$ and$\text{ }\psi \text{\hspace{0.17em}}=\text{\hspace{0.17em}}A/{A}_{o}$ ,$\text{ }\frac{1}{{r}_{N}}\frac{\partial }{{r}_{N}}\left({r}_{N}\frac{\partial \psi }{\partial {r}_{N}}\right)\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{1}{{r}_{N}^{2}}\frac{{\partial }^{2}\psi }{\partial {\phi }^{2}}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}2j\frac{\partial \psi }{\partial Z}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}0$
where$\text{ }{w}_{o}$ is the initial (minimum) beam waist (radius),$\text{ }{z}_{o}$ is the diffraction or Rayleigh length and$\text{ }2{z}_{o}$ is the depth of focus.
• Particular soln: fundamental Gaussian beam$TE{M}_{0,0}$ mode$\text{ }A\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{A}_{o}{e}^{-{r}^{2}/w\left(z{\right)}^{2}}{e}^{-j\left[k{r}^{2}/2R\left(z\right)\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{\text{tan}}^{-1}\left(z/{z}_{o}\right)\right]}{w}_{o}/w\left(z\right)$
spot size (radius)$\text{ }w\left(z\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{w}_{o}\sqrt{1\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\left(z/{z}_{o}{\right)}^{2}}$ ;
radius of curvature$\text{ }R\left(z\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}z\left(1\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\left({z}_{o}/z{\right)}^{2}\right)$
Rayleigh range or diffraction length$\text{ }{z}_{o}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\pi {w}_{o}^{2}/\lambda$
where$\text{ }{w}_{o}$ is the min. spot size (radius) at 1/e relative amplitude of the field and$\text{ }\lambda \text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\lambda }_{o}/n$ the wavelength in the medium
• Important parameters of Gaussian beam:
q parameter:$\text{ }q\text{\hspace{0.17em}}=\text{\hspace{0.17em}}z\text{\hspace{0.17em}}+\text{\hspace{0.17em}}j{z}_{o}$$\text{ }\to \text{\hspace{0.17em}}1/q\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1/R\left(z\right)\text{\hspace{0.17em}}-\text{\hspace{0.17em}}j\lambda /\left(\pi {w}^{2}\left(z\right)\right)$
Divergence angle (full)$\text{ }2{\theta }_{o}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}2\lambda /\left(\pi {w}_{o}\right)$
Depth of focus$\text{ }2{z}_{o}$ within which beam size expands by$\text{ }\sqrt{2}$ times of$\text{ }{w}_{o}$
Transverse phase:$\text{ }-k{r}^{2}/2R\left(z\right)$ , min$\text{ }R\text{\hspace{0.17em}}=\text{\hspace{0.17em}}2{z}_{o}$ , max$\text{ }R\to \infty$ (plane wave) when z=0 and$\text{ }z->\infty$
Longitudinal phase:$\text{ }{\text{tan}}^{-1}\left(z/{z}_{o}\right)$ ; ranging from$\text{ }-\pi /2$ to$\text{ }\pi /2$ (Gouy effect).
Power: total$\text{ }{P}_{o}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{I}_{o}\pi {w}_{o}^{2}/2$ (indep of z) where$\text{ }{I}_{o}$ is the peak intensity.
Fraction power within an aperture$\text{ }1\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{e}^{-2{r}^{2}/{w}^{2}}$
Focusing Gaussian beam (approx. equation) by thin lens:
Consider far field, i.e.$\text{ }f\text{\hspace{0.17em}}\gg \text{\hspace{0.17em}}z{\prime }_{o}$
The min. spot at the focus$\text{ }w{\prime }_{o}\text{\hspace{0.17em}}\text{≈}\text{\hspace{0.17em}}\lambda f/\left(\pi w\left(f\right)\right)$ where$\text{ }w\left(f\right)$ is the spot size at the lens.
• Gaussian beam interacts with mirror: Gauss beam inherits R from the radius of curvature of a mirror.
• Focused Gaussian beam: at focus$1/R\left(z\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}0$ , i.e.$\text{ }q\text{\hspace{0.17em}}=\text{\hspace{0.17em}}j{z}_{o}$ ; with this condition we can obtain the position and beam size at focus.
• Beam quality$\text{ }MM$ :$\text{ }M{M}^{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{2{w}_{m}2{\theta }_{m}}{4\lambda /\pi }\text{\hspace{0.17em}}\ge \text{\hspace{0.17em}}1$ measures how close is the beam to Gaussian shape.
• Exact description of Gaussian beam traversing a thin lens: A thin lens modifies phase of Gaussian beam by$\text{ }{e}^{jk{r}^{2}/\left(2f\right)}$ but not amplitude and beam size.
Relate plane 2 to plane 4$\text{ }{R}^{-1}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{f}^{-1}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}R{\prime }^{-1}$ Beam waist at plane 4$\text{ }w{\prime }_{0}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{w}{\sqrt{1\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\left(\pi {w}^{2}/\left(\lambda R\prime \right){\right)}^{2}}}$ and$\text{ }z\prime \text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{R\prime }{1\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\left(\lambda R\prime /\left(\pi {w}^{2}{\right)}^{2}}$
Relate plane 1 to plane 4 by substitute$\text{ }w\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{w}_{o}\sqrt{1\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\left(z/{z}_{o}{\right)}^{2}}$ and$\text{ }R\text{\hspace{0.17em}}=\text{\hspace{0.17em}}z\left(1\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\left({z}_{o}/z{\right)}^{2}\right)$
Result in magnification from ray optics${M}_{r}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}|f/\left(z\text{\hspace{0.17em}}-\text{\hspace{0.17em}}f\right)|$ while the precise magnification for Gaussian beam$\text{ }M\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{{M}_{r}}{\sqrt{1\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{r}^{2}}}$ where$\text{ }r\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{z}_{o}/\left(z\text{\hspace{0.17em}}-\text{\hspace{0.17em}}f\right)$
More equations$\text{ }w{\prime }_{o}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}M{w}_{o}$ ,$\text{ }2z{\prime }_{o}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{M}^{2}2{z}_{o}$ ,$\text{ }2\theta {\prime }_{o}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}2{\theta }_{o}/M$
• Gaussian beam propagating through optical components: e.g.$\text{ }n\left(r\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{n}_{o}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}f\left(r\right)$ ;$\text{ }k\text{\hspace{0.17em}}=\text{\hspace{0.17em}}2\pi n\left(r\right)/\lambda \text{\hspace{0.17em}}=\text{\hspace{0.17em}}{k}_{o}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{k}_{2}\left(r\right)$ ;
i.e. solving$\text{ }{\nabla }_{t}^{2}E\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\partial }^{2}E/\partial {z}^{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{k}_{o}{n}^{2}\left(r\right)E\text{\hspace{0.17em}}=\text{\hspace{0.17em}}0$ with$\text{ }E\left(x,y,z\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{A}_{o}\psi \left(x,y,z\right){e}^{-j{k}_{0}z}$ ,
A systematic approach$\to$ combine ABCD matrix and Gaussian beam
• ABCD Law: given a Gaussian beam with q parameter,$\text{ }{q}_{in}$ and an ABCD matrix for a component$\text{ }{q}_{out}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\left(A{q}_{in}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}B\right)/\left(C{q}_{in}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}D\right)$
We can apply the ABCD law for an optical system by cascading all optical elements and find the equivalent ABCD matrix
• High order transverse modes:
Rectangular (Hermite-Gaussian)

${A}_{m,p}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{A}_{G}{H}_{m}\left(\sqrt{2}x/w\right){H}_{p}\left(\sqrt{2}y/w\right){e}^{j\left(p+m\right){\text{tan}}^{-1}\left(z/{z}_{o}\right)}$

Fundamental Gaussian mode envelop$\text{ }{A}_{G}$
Hermite polynomial$\text{ }{H}_{m}$ obeys$\frac{{d}^{2}{H}_{m}\left(u\right)}{d{u}^{2}}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}2u\frac{d{H}_{m}\left(u\right)}{du}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}2m{H}_{m}\left(u\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}0$ and can be generated recursively.$\text{ }{H}_{m+1}\left(u\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}2u{H}_{m}\left(u\right)-2m{H}_{m-1}\left(u\right)$ with$\text{ }{H}_{0}\left(u\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1$ ,$\text{ }{H}_{1}\left(u\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}2u$ ,$\text{ }{H}_{2}\left(u\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}4{u}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}2$
m -- x mode index; p -- y mode index

3D surface plot of Hermite Gaussian modes.
Cylindrical symmetric (Laguerre-Gaussian)

${A}_{p,l}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{A}_{G}{L}_{p}^{l}\left(2{r}^{2}/{w}^{2}\right)\left(\sqrt{2}r/w{\right)}^{l}{e}^{±jl\phi }{e}^{j\left(2p+l\right){\text{tan}}^{-1}\left(z/{z}_{o}\right)}$

Fundamental Gaussian mode envelop$\text{ }{A}_{G}$
Laguerre associated polynomial$\text{ }{L}_{p}^{l}$ obey$\text{ }\nu \frac{{d}^{2}{L}_{p}^{l}}{d{\nu }^{2}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\left(l\text{\hspace{0.17em}}+\text{\hspace{0.17em}}1\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\nu \right)\frac{d{L}_{p}^{l}}{d\nu }\text{\hspace{0.17em}}+\text{\hspace{0.17em}}p{L}_{p}^{l}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}0$ and can be generated by$\text{ }\frac{1}{p!}{e}^{\nu }\frac{{d}^{p}{e}^{-v}{\nu }^{1}{\nu }^{p}}{d{\nu }^{p}}$ with$\text{ }{L}_{0}^{l}\left(x\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1$ ,${L}_{1}^{l}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}l\text{\hspace{0.17em}}+\text{\hspace{0.17em}}1-x$ ,$\text{ }{L}_{2}^{l}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}0.5\left(l+1\right)\left(l+2\right)\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\left(l+2\right)x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}0.5{x}^{2}$
p -- radial mode index; l -- angular mode index
Bessel beam: The transverse field profile assumes variation like Bessel function, i.e.$\text{ }{A}_{m}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{A}_{o}{J}_{m}\left({k}_{t}r\right){e}^{jm\phi }$ Bessel beam satisfies the nonparaxial wave equation and is nondiffracting. However, it has infinite energy owing to its long tail, i.e. its RMS width is infinite.

Solving Helmholtz equations with separation of variables. (Ramo, Whinnery and Van Duzer, Fields and Waves in Communication Electronics, 2nd Ed. Wiley (1984).)
Obtaining HG mode solutions by solving the paraxial wave equation with separation of variables. (Milonni and Eberlay, Lasers, Wiley (1991).) (Notice that the paraxial wave equation has opposite sign for j or i since they assume that forward wave has propagation factor of amath e^{j k r}. endmath