Higher-order Hermite-Gauss modes

In general, any solution \(u(x,y,z)\) of the paraxial wave equation, can represent the transverse properties of a scalar electric field representing a beam-like electro-magnetic wave. Complete and countable sets of functions that are solutions to the paraxial wave equation are called transverse electromagnetic modes (TEM). One such set is the Hermite Gauss modes denoted by \(HG_{nm}\), that can be used to describe any solution of the paraxial wave equation \(u'\) by a linear superposition of Hermite–Gauss modes:

\[u'(x,y,z) = \sum_{n,m} a_{jnm} u_{nm}(x,y,z)\]

Which in turn allows us to describe any laserbeam using a sum of these modes:

\[E(t,x,y,z) = \sum_j \sum_{n,m} a_{jnm} u_{nm}(x,y,z) exp(i(\omega_jt - k_jz))\]

The “lowest-order” Hermite Gauss mode \(HG_{00}\) is usually called a Gaussian Beam. Such a beam profile (for a beam with a given wavelength \(\lambda\)) can be completely determined by two parameters: the size of the minimum spot size \(\omega_0\) (called the beam waist) and the position \(z_0\) of the beam waist along the z-axis.

The Hermite-Gauss modes are usually given in their orthonormal form as:

\[u_{n,m}(x,y,z) &= (2^{n+m-1} n! m! \pi)^{-1/2} \frac{1}{\omega(z)} \exp(i(n+m+1) \psi(z) \\ &\times H_n \left( \frac{\sqrt{2} x}{\omega(z)} \right) H_m \left(\frac{\sqrt{2} y}{\omega(z)}\right) \exp\left(-i \frac{k(x^2 + y^2)}{2R_C(z)} - \frac{x^2 + y^2}{\omega^2(z)}\right)\]

with n, m being the mode numbers (or mode indices). In this case n refers to the modes in the y-z plane (saggital) and m to the x-z plane (tangential). The following functions are used in the equation above:

  • \(H_n(x)\) : Hermite polynomial of the order n (unnormalised),

  • \(\omega(z)\) : beam radius or spot size,

  • \(R_C(z)\) : radius of curvature of the phase front,

  • \(\psi(z)\) : Gouy phase (see Accumulated Gouy Phase)

The Hermite-Gauss modes as given above are orthonormal and thus:

\[\int \int dx dy \; u_{nm} u^*_{n'm'} = \delta_{nn'} \delta_{mm'}\]

Therefore the power of a beam, being detected on a single-element photodetector (provided that the area of the detector is large with respect to the beam) can be computed as:

\[P = \sum_{n,m} a_{nm} a^*_{nm}\]

The x and y dependencies can be separated so that:

\[u_{nm}(x,y,z) = u_n(x,z) u_m(y,z)\]

For more information, see the [18].