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# Hermite–Gaussian Modes

Author: the photonics expert

Definition: propagation modes or resonator modes which are described with Hermite–Gaussian functions

More general term: modes

When light propagates in free space or in a homogeneous optical medium, its intensity profile will generally change during propagation. For certain electric field amplitude distributions, however, which are called modes, this is not the case: the shape of the amplitude profile remains constant, even though there may be a re-scaling of the profile, an overall change in optical phase, and possibly also a change in the total optical power.

For each combination of an optical frequency, a beam axis, a focus position, and some beam radius of a Gaussian beam in the focus, there is a whole family of Hermite–Gaussian modes (TEMnm modes, Gauss–Hermite modes). These are approximate solutions of the wave equation, valid for weak focusing (→ paraxial approximation). Their electric field distributions are essentially given by the product of a Gaussian function and a Hermite polynomial, apart from the phase term:

$$\begin{array}{l} {E_{nm}}(x,y,z) = {E_0}\frac{{{w_0}}}{{w(z)}}\;\\ \quad \cdot {H_n}\left( {\sqrt 2 \frac{x}{{w(z)}}} \right)\exp \left( { - \frac{{{x^2}}}{{w{{(z)}^2}}}} \right) \cdot {H_m}\left( {\sqrt 2 \frac{y}{{w(z)}}} \right)\exp \left( { - \frac{{{y^2}}}{{w{{(z)}^2}}}} \right)\\ \quad \cdot \exp \left( { - i\left[ {kz - \left( {1 + n + m} \right)\arctan \frac{z}{{{z_{\rm{R}}}}} + \frac{{k\left( {{x^2} + {y^2}} \right)}}{{2R(z)}}} \right]} \right) \end{array}$$

where <$H_n(x)$> is the Hermite polynomial with the non-negative integer index <$n$>. The indices <$n$> and <$m$> determine the shape of the profile in the <$x$> and <$y$> direction, respectively. The quantities <$w$> and <$R$> evolve in the <$z$> direction as described in the article on Gaussian beams.

The intensity distribution of such a mode (Figure 1) has <$n$> nodes in the horizontal direction and <$m$> nodes in the vertical direction. For <$n = m$> = 0, a Gaussian beam is obtained. This mode is called the fundamental mode or axial mode, and it has the highest beam quality with an M2 factor of 1. Other Hermite–Gaussian modes with indices <$n$> and <$m$> have an <$M^2$> factor of <$(1 + 2 n)$> in the <$x$> direction, and <$(1 + 2 m)$> in the <$y$> direction.

### Mode Calculations

The software RP Resonator is a particularly flexible tool for calculating all kinds of resonator mode properties. Both standing-wave and ring resonators are supported. You can take into account alignment sensitivity and thermal lensing.

A further generalization of the equation above would allow for different mode sizes and focus positions (astigmatism) for the <$x$> and <$y$> directions. The direction of the electric field, not specified in the equation above, determines the polarization.

The electric field distributions of the Hermite–Gaussian modes are a system of functions which are mutually orthogonal. Arbitrary field distributions can be decomposed into Hermite–Gaussian functions, where the amplitude content of each one is determine by an overlap integral.

Hermite–Gaussian modes can often be used to represent the modes of an optical resonator, if the optical elements in the resonator only do simple changes to the phase and intensity profiles (e.g., approximately preserving parabolic phase profiles) and the paraxial approximation is satisfied. As these conditions are very often fulfilled in laser resonators, laser resonator modes are often of Hermite–Gaussian kind. If such a laser operates on a single mode, a characteristic intensity profile of the output beam as in Figure 1 can be observed.

A single Hermite–Gaussian resonator mode may be excited by incident quasi-monochromatic laser light if (a) the incident light has a non-zero field overlap with the mode and a frequency matching a resonance, and (b) if all other resonator modes either have a zero field overlap or resonance frequencies not fitting to the laser light. This combination of conditions may be fulfilled even for an incident beam shape not fitting well to the Hermite–Gaussian mode shape. For example, one could excite a TEM01 mode with a suitably aligned Gaussian input beam of suitable frequency, if that is not resonant with other modes.

Another frequently used mode family is that of Laguerre–Gaussian modes, which fit better to situations with a rotational symmetry because they are based on polar coordinates. Note, however, that lasers often contain components placed at a slight angle against the beam, which breaks rotational symmetry. Therefore, Hermite–Gaussian modes are often better suited for laser beams.

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## Questions and Comments from Users

2022-01-12

Many laser providers give a <$M^2$> to be less or equal to a non-integer value, e.g. <$M^2 < 1.2$>. Does it mean there is a “non-integer” mode?

The beam quality factor is generally not an integer, but can be any value above for equal unity. For Hermite–Gaussian modes, you get integer values, but you can have any superpositions of those modes and then generally get non-integer values.

2024-04-16

Why do TEM modes have zero intensity nodes?

That just follows from the differential equation which determines them.

2024-04-24

Is it possible – for a given real light beam – to find the corresponding mode superposition (or an approximation to it in a least squares sense)? Assuming it consists only of HG modes and e.g. no rotationally symmetric modes.

Experimentally, this is difficult. You may (for a monochromatic beam) think that you could measure the power content of the HG modes by checking transmitted powers when scanning the resonator length, but mode frequency degeneracies are difficult to handle.

If you have a complex amplitude profile (which you may measure, e.g. with a Shack–Hartmann sensor), you can do the mode decomposition numerically – by numerically calculating overlap integrals with the mode functions.

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