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Gaussian Beams

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Definition: light beams where the electric field profile in a plane perpendicular to the beam axis can be described with a Gaussian function, possibly with an added parabolic phase profile

German: Gauß-Strahlen

Category: general optics

How to cite the article; suggest additional literature

In optics and particularly in laser physics, laser beams often occur in the form of Gaussian beams, which are named after the mathematician and physicist Johann Carl Friedrich Gauß. Here, the transverse profile of the optical intensity of the beam with a power P can be described with a Gaussian function:

I(r,z) for Gaussian beam

where the beam radius w(z) is the distance from the beam axis where the intensity drops to 1/e2 (≈ 13.5%) of the maximum value. A hard aperture with radius w can transmit ≈ 86.5% of the optical power. For an aperture radius of 1.5 w or 2 w, this fraction is increased to 98.9% and 99.97%, respectively.

Note that the factor 1 / 2 in the denominator in the equation is unfortunately often forgotten, so that the on-axis intensity of the beam is underestimated by a factor of 2. For example, quoted numbers for the measured damage threshold of optical components are often affected by that problem; the peak intensity at the damage threshold in terms of optical power may have be calculated with or without the mentioned factor, so that a substantial quantitative uncertainty remains for the reader.

The full width at half maximum (FWHM) of the intensity profile is ≈1.18 times the Gaussian beam radius w(z).

In addition to the Gaussian shape of the intensity profile, a Gaussian beam has a transverse phase profile which can be described with a polynomial of at most second order. A linear phase variation in one direction (not considered further here) describes a tilt, and a quadratic phase variation is associated with divergence or convergence of the beam.

Propagation of Gaussian Beams

Gaussian beams are usually considered in situations where the beam divergence is relatively small, so that the so-called paraxial approximation can be applied. This approximation allows the omission of the term with the second-order derivative in the propagation equation (as derived from Maxwell's equations), so that a first-order differential equation results. Within this approximation, a Gaussian beam propagating in free space remains Gaussian, except that of course its parameters evolve. For a monochromatic beam, propagating in the z direction with the wavelength λ, the complex electric field amplitude (phasor) is

E(r,z) for Gaussian beam

with the peak amplitude |E0| and beam radius w0 at the beam waist, the wavenumber k = 2π / λ, the Rayleigh length zR (see below) and the radius of curvature R(z) of the wavefronts. The oscillating real electric field is obtained by multiplying the phasor with exp(i 2πc t / λ) and taking the real part.

Gaussian beam

Figure 1: Snapshot of the electric field distribution around the focus of a Gaussian beam. In this example, the beam radius is only slightly larger than the wavelength, and the beam divergence is strong. According to the equation above, the field pattern is moving from left to right (i.e., toward larger z).

The beam radius varies along the propagation direction according to

w(z) for Gaussian beam

with the Rayleigh length

Rayleigh length of Gaussian beam

which determines the length over which the beam can propagate without diverging significantly. (The older literature often deals with the confocal length b, which is just twice the Rayleigh length.) A so-called collimated beam (with approximately constant beam radius) has to have a large Rayleigh length, compared with the envisaged propagation distance.

beam radius of Gaussian beam

Figure 2: Evolution of the beam radius of a Gaussian beam (blue curve). The two vertical lines indicate the Rayleigh length, and the dashed lines show the asymptotic behavior far from the beam waist.

The position z = 0 in the equation above corresponds to the beam waist or focus where the beam radius is at its minimum, and the phase profile is flat. The radius of curvature R of the wavefronts evolves according to

R(z) for Gaussian beam

For propagation in transparent media, λ is the wavelength in the medium (i.e., not the vacuum wavelength). Otherwise, the formalism explained above can be used without modification, assuming that the medium is homogeneous, isotropic and lossless.

Gaussian beam with curved wavefronts

Figure 3: Gaussian beam with curved wavefronts. The curvature is weak both very close to the focus and far from the focus.

The term with the arctan function in the expression for the electric field describes the Gouy phase shift, which is important e.g. for the resonance frequencies of optical resonators.

The beam divergence in the far field (i.e., for z values much larger than zR) is

divergence of Gaussian beam

which shows that the smaller the waist radius and the longer the wavelength, the stronger is the divergence of the beam far from the waist. The beam parameter product (product of waist radius and far-field divergence angle) of a Gaussian beam is λ/π, i.e., it depends only the wavelength. For laser beams with non-ideal beam quality (see below), that value is larger.

Laser Beam Calculations

Wavelength:
M2 factor: calc (1 for Gaussian beam)
Beam parameter product: calc (from M2)
Beam waist radius: calc (using the BPP)
Divergence half-angle: calc (using the BPP)
Rayleigh length: calc
Distance from focus:
Beam radius: calc
Curvature radius of wavefronts: calc

Enter input values with units, where appropriate. After you have modified some values, click a "calc" button to recalculate the field left of it.

In terms of Gaussian beam parameters, the paraxial approximation requires that the beam radius at the focus is large compared with the wavelength. This implies that the beam divergence does not become too large, and that the Rayleigh length is substantially larger than the beam radius. For very tightly focused beams, the paraxial approximation is not well satisfied, and a more complex method is required for calculating the beam propagation.

The article on laser beams contains a paragraph titled “Limitations for the Focusing of Laser Beams”. The presented rules can be applied to Gaussian beams but also to generalize beams with some larger M2 factor.

Complex q Parameter

The state of a Gaussian beam at a certain z position can be specified with a complex q parameter

q parameter of Gaussian beam

so that the complex electric field can be written as

E field with q parameter

Propagation over some length then simply increases the q parameter by that length. When a Gaussian beam passes an optical element such as a curved mirror or a lens, this can be described by transforming its parameters with an ABCD matrix according to

evolution of Gaussian beam

Astigmatic Beams

Gaussian beams can have different radii and divergence values for two perpendicular transverse directions, denoted e.g. x and y. Equations similar to those above can be used for describing the essentially independent evolution of beam radii in both directions. If the focus positions for both directions are not equal, the beam is called astigmatic.

Gaussian Beams and Resonator Modes

The modes of an optical resonator with the lowest order in the transverse direction (called TEM00 or fundamental transverse modes) are Gaussian modes, if the resonator is stable, all optical media in the resonator are homogeneous, and all surfaces between media are either flat or have a parabolic shape. Therefore, lasers emitting only on the fundamental transverse mode often emit beams with close to Gaussian shape. Deviations from the mentioned conditions, e.g. by thermal lensing in a gain medium, can cause non-Gaussian beam shapes and/or the simultaneous excitation of different transverse modes. Modes of higher transverse order can be described e.g. by Hermite–Gaussian or Laguerre–Gaussian functions. In any case, the deviation from a Gaussian beam shape can be quantified with the M2 factor. A Gaussian beam has the highest possible beam quality, which is related to the lowest possible beam parameter product, and corresponds to M2 = 1.

The fundamental propagation modes of fibers are generally not exactly Gaussian, but also not too far from that shape. Therefore, a Gaussian beam can usually be launched into a single-mode fiber with high efficiency (80% or larger), provided that suitable optics are used.

Importance of Gaussian Beams

The importance of Gaussian beams results from a number of special properties:

Bibliography

[1]H. Kogelnik and T. Li, “Laser beams and resonators”, Appl. Opt. 5 (10), 1550 (1966)
[2]P. A. Bélanger, “Beam propagation and the ABCD ray matrices”, Opt. Lett. 16 (4), 196 (1991)
[3]A. E. Siegman, Lasers, University Science Books, Mill Valley, CA (1986)
[4]J. Alda, “Laser and Gaussian beam propagation and transformation”, http://www.ucm.es/info/euoptica/org/pagper/jalda/docs/libr/laserandgaussian_eoe_03.pdf

(Suggest additional literature!)

See also: laser beams, Gouy phase shift, modes, Hermite–Gaussian modes, beam quality, diffraction-limited beams, M2 factor, beam parameter product, beam radius, beam waist, collimated beams, ABCD matrix
and other articles in the category general optics

In the RP Photonics Buyer's Guide, 25 suppliers for equipment for the characterization of Gaussian beams are listed.


Dr. R. Paschotta

This encyclopedia is authored by Dr. Rüdiger Paschotta, the founder and executive of RP Photonics Consulting GmbH. Contact this distinguished expert in laser technology, nonlinear optics and fiber optics, and find out how his technical consulting services (e.g. product designs, problem solving, independent evaluations, or staff training) and software could become very valuable for your business!

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