Gaussian Beam

This part continues from the Propagation in an Anisotropic Medium tutorial.

Because the wave equation governs optical propagation, the transverse field distribution pattern and its variation along the longitudinal propagation direction have to satisfy this equation in order for the wave to exist and to propagate. A well-defined field pattern that can remain unchanged as the wave propagates is called a mode of wave propagation. Such a transverse field pattern is known as a transverse mode. The optical modes that exist in a given medium are determined by the optical properties of the medium together with any boundary conditions imposed on the wave equation by the optical structures in the medium. Here we consider the optical modes in a homogeneous medium. Modes in waveguides and optical fibers are discussed in the next two tutorials.

A monochromatic optical wave propagating in an isotropic, homogeneous medium is governed by the wave equation given in (82) [refer to the propagation in an isotropic medium tutorial]. Clearly, the monochromatic plane wave expressed in (58) [refer to the polarization of light tutorial] is a solution of this wave equation. Therefore, plane waves are normal modes in an isotropic, homogeneous medium.

They are not the only normal modes, however, as the wave equation governing wave propagation in such a medium has other normal-mode solutions. One such important set of modes is the Gaussian modes. Like plane waves, Gaussian modes are normal modes of wave propagation in an isotropic, homogeneous medium. Different from a plane wave, however, a Gaussian mode has a finite cross-sectional field distribution defined by its spot size. Being an unguided field with a finite spot size, a Gaussian mode differs from a waveguide mode, discussed in later tutorials, in that its spot size varies along its longitudinal axis, taken to be the z axis, of propagation though its pattern remains unchanged. Therefore, its transverse field distribution also changes with z though the field pattern does not change.

A Gaussian mode field at a frequency ω can thus be expressed as

\[\tag{132}\mathbf{E}_{mn}(\mathbf{r},t)=\pmb{\mathcal{E}}_{mn}(x,y,z)\exp(\text{i}\mathbf{k}\cdot\mathbf{r}-\text{i}\omega{t})=\hat{e}\mathcal{E}_{mn}(x,y,z)\exp(\text{i}\mathbf{k}\cdot\mathbf{r}-\text{i}\omega{t})\]

with a corresponding field distribution for its magnetic field component, where m and n are mode indices associated with the two transverse dimensions x and y, respectively. A Gaussian mode field has neither longitudinal electric nor longitudinal magnetic field components. It is a TEM mode that has only transverse electric and magnetic field components. Normal modes are orthonormal to each other and can be normalized, as is discussed in later tutorials. Gaussian modes are normalized by the following condition:

\[\tag{133}\frac{2k}{\omega\mu_0}\displaystyle\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}|\mathcal{E}_{mn}(x,y,z)|^2\text{d}x\text{d}y=1\]

The location, taken to be z = 0 for a beam propagating along the z axis, where the smallest spot size of the beam occurs, is known as the waist of a Gaussian beam. The minimum Gaussian beam spot size, w₀, is defined as the e^-2 radius of the Gaussian beam intensity profile at the beam waist. The diameter of the beam waist is d₀ = 2w₀.

As illustrated in figure 14 above, a Gaussian beam has a plane wavefront at its beam waist. The beam remains well collimated within a distance of

\[\tag{134}z_R=\frac{kw_0^2}{2}=\frac{\pi nw_0^2}{\lambda}\]

known as the Rayleigh range, on either side of the beam waist. In (134), k = 2πn/λ is the propagation constant of the optical beam in a medium of refractive index n. The parameter b = 2z_R is called the confocal parameter of the Gaussian beam. Because of diffraction, a Gaussian beam diverges away from its waist and acquires a spherical wavefront. As a result, both its spot size, w(z), and the radius of curvature, R(z), of its wavefront are functions of distance z from its beam waist:

\[\tag{135}w(z)=w_0\left(1+\frac{z^2}{z_R^2}\right)^{1/2}=w_0\left[1+\left(\frac{2z}{kw_0^2}\right)^2\right]^{1/2}\]

and

\[\tag{136}\mathcal{R}(z)=z\left(1+\frac{z_R^2}{z^2}\right)=z\left[1+\left(\frac{kw_0^2}{2z}\right)^2\right]\]

We see from (135) that at z = ±z_R, \(w=\sqrt{2}w_0\). At |z| ≫ z_R, far away from the beam waist, R(z) ≈ z and w(z) ≈ 2|z|/kw₀. Therefore, the far-field beam divergence angle is

\[\tag{137}\Delta\theta=2\frac{w(z)}{|z|}=\frac{4}{kw_0}=\frac{2\lambda}{\pi nw_0}\]

For the far field at |z| ≫ z_R, we find that the beam spot size w(z) is inversely proportional to the beam waist spot size w₀ but is linearly proportional to the distance |z| from the beam waist. This characteristic does not exist for the near field at |z| ≤ z_R.

A complete set of Gaussian modes includes the fundamental TEM₀₀ mode and high-order TEM_mn modes. The specific forms of the mode fields depend on the transverse coordinates of symmetry: the mode fields are described by a set of Hermite-Gaussian functions in rectangular coordinates, whereas they are described by the Laguerre-Gaussian functions in cylindrical coordinates. Because there is no structurally determined symmetry in free space, either set is equally valid. Usually the Hermite-Gaussian functions in the rectangular coordinates are used. In a transversely isotropic and homogeneous medium, a normalized TEM_mn Hermite-Gaussian mode field propagating along the z axis can be expressed as

\[\tag{138}\begin{align}\hat{\mathcal{E}}_{mn}(x,y,z)&=\frac{A_{mn}}{w(z)}H_m\left[\frac{\sqrt{2}x}{w(z)}\right]H_n\left[\frac{\sqrt{2}y}{w(z)}\right]\exp\left[\text{i}\frac{k}{2}\frac{x^2+y^2}{q(z)}\right]\exp[\text{i}\xi_{mn}(z)]\\&=\frac{A_{mn}}{w(z)}H_m\left[\frac{\sqrt{2}x}{w(z)}\right]H_n\left[\frac{\sqrt{2}y}{w(z)}\right]\exp\left[-\frac{x^2+y^2}{w^2(z)}\right]\exp\left[\text{i}\frac{k}{2}\frac{x^2+y^2}{\mathcal{R}(z)}\right]\\&\times\exp[\text{i}\xi_{mn}(z)]\end{align}\]

where

\[A_{mn}=\left(\frac{\omega\mu_0}{\pi k}\right)^{1/2}(2^{m+n}\,m!\,n!)^{-1/2}\]

is the normalization constant, H_m is the Hermite polynomial of order m, q(z) is the complex radius of curvature of the Gaussian wave,

\[\tag{139}q(z)=z-\text{i}z_R\qquad\text{or}\qquad\frac{1}{q(z)}=\frac{1}{\mathcal{R}(z)}+\text{i}\frac{2}{kw^2(z)}\]

and \(\xi_{mn}(z)\) is the mode-dependent on-axis phase variation along the z axis given by

\[\tag{140}\xi_{mn}(z)=-(m+n+1)\tan^{-1}\frac{z}{z_R}=-(m+n+1)\tan^{-1}\left(\frac{2z}{kw_0^2}\right)\]

The Hermite polynomials can be obtained using the following relation:

\[\tag{141}H_m(\xi)=(-1)^m e^{\xi^2}\frac{\text{d}^m e^{-\xi^2}}{\text{d}\xi^m}\]

Some low-order Hermite polynomials are

\[\tag{142}H_0(\xi)=1,\qquad H_1(\xi)=2\xi,\qquad H_2(\xi)=4\xi^2-2,\qquad H_3(\xi)=8\xi^3-121\xi\]

We see from (138) and (142) that the transverse field distribution \(|\hat{\mathcal{E}}_{00}(x,y)|\) of the fundamental Gaussian mode, TEM₀₀, at any fixed longitudinal location z is simply a Gaussian function of the transverse radial distance r = (x² + y²)^1/2 and that the spot size w(z) is the e^-1 radius of this Gaussian field distribution at z.

The transverse field distribution of a high-order mode, TEM_mn, is the same Gaussian distribution spatially modulated by the Hermite polynomials H_m in x and H_n in y. As a result, its field distribution is more spread out radially than that of the fundamental TEM₀₀ mode. In general, the higher the order of a mode, the farther its transverse field distribution spreads out. This intensity patterns of some Hermite-Gaussian modes are shown in figure 15 below.

Example

A fundamental Gaussian beam in free space at the He-Ne laser wavelength of 632.8 nm has a spot size of w₀ = 500 μm at its beam waist. This beam has a Rayleigh range z_R = πw₀²/λ = 1.24 m and a confocal parameter b = 2z_R = 2.48 m. Using (135) and (136), we find the following spot sizes and radii of curvature at a few different locations:

\[\begin{align}&w=502\,\mu\text{m},\qquad\mathcal{R}=\pm15.5\,\text{m}{\qquad}\text{at }z=\pm10\,\text{cm},\\&w=642\,\mu\text{m},\qquad\mathcal{R}=\pm2.54\,\text{m}{\qquad}\text{at }z=\pm1\,\text{m},\\&w\approx40\,\text{cm},\qquad\mathcal{R}\approx\pm1\,\text{km}{\qquad}\text{at }z=\pm1\,\text{km}.\\ \end{align}\]

From these numerical examples, we see that a Gaussian beam diverges very slowly, much like a plane wave, within the Rayleigh range on both sides of its beam waist. At the beam waist, a Gaussian beam has a plane wavefront with R = ∞. At a distance much larger than the Rayleigh range on either side of the beam waist, a Gaussian beam approaches the characteristics of a spherical wave with R ≈ z. The Gaussian beam in this example has a far-field divergence angle of Δθ = 2λ/πw₀ = 0.8 mrad.

The next part continues with the Reflection and Refraction tutorial.