RP Photonics logo
VL logo part of the
Virtual
Library

Encyclopedia of Laser Physics and Technology

Hermite–Gaussian Modes

previous  |  next  |  feedback

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

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:

Hermite--Gaussian beam

where Hn(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 M2 factor of (2n + 1) in the x direction, and (2m + 1) in the y direction.

intensity distributions of Hermite–Gaussian modes

Figure 1: Intensity profiles of the lowest-order Hermite–Gaussian modes, starting with TEM00 (lower left-hand side) and going up to TEM33 (upper right-hand side).

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.

Another frequently used mode family is that of Laguerre–Gaussian modes.

See also: modes, higher-order modes, Gaussian beams


cover of print encyclopedia

Since October 2008, the Encyclopedia of Laser Physics and Technology is also available in the form of a two-volume book. Maybe you would enjoy reading it also in that form! The print version has a carefully designed layout and can be considered a must-have for any institute library, laser research group, or laser company.

You may order the print version via Wiley-VCH.

arrow
Home New articles Spotlight Feedback Advertising
Categories Search Quiz Links Page hits
H

This encyclopedia is provided by
RP Photonics Consulting GmbH.

You can get technical consulting from the author, Dr. Rüdiger Paschotta.

A.L.S. logo

A.L.S. GmbH

Picosecond laser diodes
<30 ps, 375 – 1600 nm, >1 Wp, single shot – 120 MHz

Onefive logo

Onefive

Low-noise
femtosecond,
picosecond,
and tunable single-frequency lasers for OEM and R&D applications.

High Q logo

A leading supplier of all-diode-pumped picosecond and femtosecond laser and amplifier systems for scientific, medical and industrial markets.

RP Fiber Power 2.0

RP Fiber Power

This software is a powerful tool for designing fiber amplifiers and fiber lasers.
See the comprehensive description!

Field Guide to Lasers

This compact book by Dr. Paschotta explains principles and types of lasers.

Your Advertisement at This Place

will be seen by many thousands of visitors per month. These banners receive far over 100'000 page hits per month. Check the details.