# Gaussian Beams

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

More general term: light beams

German: Gauß-Strahlen

Author: Dr. Rüdiger Paschotta

Cite the article using its DOI: https://doi.org/10.61835/mla

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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. The definition of Gaussian beams concerns both the intensity and phase profile, as explained in the following:

### Intensity Profile

The transverse profile of the optical intensity of the beam with an optical power <$P$> can be described with a Gaussian function:

$$I(r,z) = \frac{P}{{\pi w{{(z)}^2}/2}}\;\exp \left( { - 2\frac{{{r^2}}}{{w{{(z)}^2}}}} \right)$$Here, the beam radius <$w(z)$> is the distance from the beam axis where the intensity drops to <$1/e^2$> (≈ 13.5%) of the maximum value. If the beam hits a hard aperture with radius <$w$>, ≈ 86.5% of the optical power can get through the aperture. For an aperture radius of 1.5 <$w$> or 2 <$w$>, this fraction is increased to 98.9% and 99.97%, respectively. (A common error in the integration leads to substantially different results – see section “Questions and Comments from Users” below.)

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

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.

### Phase Profile

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 (change of beam direction).
- A quadratic phase variation is associated with a curvature of the optical wavefronts. It is related to divergence or convergence of the beam. If there is no quadratic phase variation, one is at the focus position of the beam. The beam will nevertheless diverge, since the wavefronts get curved during further propagation.

Note that there are also multimode beams with Gaussian intensity profile but complicated phase patterns, and these are *not* called Gaussian beams.

### Polarization

There are no requirements concerning polarization of a Gaussian beam, i.e., it may e.g. be linear, circular, elliptical, radial, azimuthal, or not polarized at all. In many cases, a scalar description of a Gaussian beam, ignoring polarization, is used.

## Propagation of Gaussian Beams

Gaussian beams are usually (and also in this article) considered in situations where the beam divergence is relatively small (i.e., the beam waist radius sufficiently large), 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 the paraxial approximation, a Gaussian beam propagating in free space (or in a homogeneous medium) remains Gaussian, except that of course its parameters evolve. For a monochromatic beam, propagating in the <$z$> direction with the wavelength <$\lambda$>, the complex electric field amplitude (phasor) is

with the peak amplitude <$|E_0|$> and beam radius <$w_0$> at the beam waist, the wavenumber <$k = 2\pi / \lambda$>, the Rayleigh length <$z_\textrm{R}$> (see below) and the radius of curvature <$R(z)$> of the wavefronts. The oscillating real electric field is obtained by multiplying the phasor with <$\exp(-i \omega t) = \exp(-i 2\pi c t / \lambda)$> and taking the real part.

(Note: the equation above is based on the physicists' sign convention for wave phasors, rather than the one which is more common in engineering, although in the optics one often finds the latter in the scientific literature, where the last phase term is <$\exp(-j \: [...])$>. If that is used, the signs in some other equations need to be adapted. See the article on sign conventions in wave optics for more details.)

Due to the basic phenomenon of diffraction, the beam radius cannot simply remain constant – it varies along the propagation direction. This can be described mathematically as

$$w(z) = {w_0}\sqrt {1 + {{\left( {z/{z_{\rm{R}}}} \right)}^2}} $$with the **Rayleigh length**

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.

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

where a positive <$R$> means that the wavefronts are leaning back.

Note that for <$z \rightarrow 0$> one obtains <$R \rightarrow \infty$> rather than <$r \rightarrow 0$>, as we have <$r^2$> in the denominator.

For propagation in transparent media, <$\lambda$> 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.

The term <$-\arctan z/z_\textrm{R}$> in the expression for the phase of 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 <$z_\textrm{R}$>) is

$$\theta = \frac{\lambda }{{\pi {w_0}}}$$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 <$\lambda /\pi$>, i.e., it depends only the wavelength. For light beams with non-ideal beam quality (see below), that value is larger.

In terms of Gaussian beam parameters, the paraxial approximation requires that the beam radius at the focus is large compared with the wavelength. (However, it does not need to be far larger for reasonable accuracy of the equations.) 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 substantially more complex method is required for calculating the beam propagation. In that regime, one may also need to consider the vector character of the electromagnetic field, including a longitudinal polarization component.

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 *M*^{2} factor.

## Complex <$q$> Parameter

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

so that the complex electric field can be written as

$$E(r,z) \propto \exp \left[i k{r^2} / 2 q(z) \right].$$(With electrical engineer's sign convention, the sign of the term with <$\lambda / \pi w^2$> in the definition of <$q(z)$> would be opposite. Effectively, <$q$> is turned into its complex conjugate.)

Propagation over some distance then simply increases the <$q$> parameter by that distance. 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

$$q' = \frac{{Aq + B}}{{Cq + D}}$$where <$A$>, <$B$>, <$C$> and <$D$> are the components of the ABCD matrix. In textbooks, ABCD matrices for many kinds of optical elements are available. Note that with physicists' sign convention, all matrix components need to be turned into their complex conjugates; however, in many cases of interest, the matrix components are purely real.

There are some optical elements which can not be described with ABCD matrices, and which convert a Gaussian beam into a non-Gaussian beam; an example are axicons.

## 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 TEM_{00} 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 *M*^{2} factor. A Gaussian beam has the highest possible beam quality, which is related to the lowest possible beam parameter product, and corresponds to <$M^2$> = 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. One requires a focus at the fiber end with a beam radius which fits to the size of the fiber mode.

## Importance of Gaussian Beams

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

- Gaussian beams have a Gaussian intensity profile at any location along the beam axis (at least within the paraxial approximation); only the beam radius varies.
- A Gaussian beam remains Gaussian also after passing simple kinds of optical elements (e.g. lenses without optical aberrations).
- Gaussian beams are the lowest-order self-consistent field distribution in optical resonators (→
*resonator modes*) provided that there are no intracavity elements causing beam distortions. For that reason, the output beams of many lasers are Gaussian. - Single-mode fibers have beam profiles which are usually close to Gaussian. Even in cases with a less than perfect fit, the Gaussian approximation is popular because of the relatively simple rules for calculating the beam propagation.
- There are so-called
*higher-order modes*e.g. of Hermite–Gaussian type. These have more complicated field patterns and exhibit a larger beam parameter product. - For beams with poor beam quality, the Gaussian mode analysis can be generalized, using the so-called
*M*^{2}factor.

### Bibliography

[1] | H. Kogelnik and T. Li, “Laser beams and resonators”, Appl. Opt. 5 (10), 1550 (1966); https://doi.org/10.1364/AO.5.001550 |

[2] | P. A. Bélanger, “Beam propagation and the ABCD ray matrices”, Opt. Lett. 16 (4), 196 (1991); https://doi.org/10.1364/OL.16.000196 |

[3] | U. Levy, Y. Silberberg and N. Davidson, “Mathematics of vectorial Gaussian beams”, Advances in Optics and Photonics 11 (4), 828 (2019); https://doi.org/10.1364/AOP.11.000828 |

[4] | A. E. Siegman, Lasers, University Science Books, Mill Valley, CA (1986) |

[5] | J. Alda, “Laser and Gaussian beam propagation and transformation”, https://www.researchgate.net/publication/255041663_Laser_and_Gaussian_Beam_Propagation_and_Transformation |

See also: laser beams, Gouy phase shift, sign conventions in wave optics, modes, multimode beams, Hermite–Gaussian modes, beam quality, diffraction-limited beams, *M*^{2} factor, beam parameter product, beam radius, beam waist, collimated beams, ABCD matrix

This encyclopedia is authored by Dr. Rüdiger Paschotta, the founder and executive of RP Photonics AG. How about a tailored training course from this distinguished expert at your location? Contact RP Photonics to find out how his technical consulting services (e.g. product designs, problem solving, independent evaluations, training) and software could become very valuable for your business!

## Questions and Comments from Users

2020-05-28

Its is possible to flatten a Gaussian beam profile by defocusing the beam?

The author's answer:

No, it isn't – away from its focus, the beam will still have a Gaussian beam profile, just with larger diameter.

2020-05-31

What happens when a Gaussian beam goes into another medium, passing by a planar interface? Is it just a matter of correcting parameters like beam divergence, Rayleigh range and Gouy phase by considering the new refractive index and correcting the beam direction using Snell's law?

The author's answer:

Yes, you are right: the beam will again be a Gaussian beam in the medium, just with modified parameters.

2020-10-10

Is the cross-section of a TEM_{00} Gaussian beam always circular?

The author's answer:

It is often defined like that, but one may use a generalized definition where you require Gaussians in <$x$> and <$y$> direction, but not necessarily with the same width.

2021-02-15

For me a 86.5% level of the normal distribution would correspond to 1.5 standard deviations. Increasing the beam waist by 2 would mean 3 standard deviations and rather 99.7% than 99.97% intensity.

The author's answer:

Maybe you forgot about the two-dimensional characteristics of the problem.

2021-03-19

Where does the factor of 0.94 come from exactly?

The author's answer:

You find that value if you calculate the integral over the Gaussian function, e.g. with a peak value of 1 and a given FWHM, in order to normalize it to a certain energy.

2021-09-24

If I want to calculate the radius of a Gaussian beam using a knife edge between thresholds, say 10% - 90%, or 20% - 80%, or 25% - 75%, how do I do it? I have seen in scientific literature that for a threshold of 10% - 90%, the radius, r, of the beam r = 0.7803 · (t2 - t1), where t2, and t1 are distances at 90% and 10% thresholds. They did not show how it is done.

The author's answer:

You just need to integrate the intensity of the Gaussian beam over an area which is cut off to some extent on one side according to the area covered by the knife edge. That can be done with an analytical calculation or with numerical tools, as is in both cases not particularly difficult.

2021-10-21

You mentioned that adding phase 'in one direction' describes a tilt. Could you elaborate on this?

The author's answer:

Consider a plane wave propagating with an angle <$\varphi$> against the z direction. The wave vector has an x component, for example, with the magnitude <$k \: \sin\varphi$>. That component describes a linear phase variation in x direction.

2021-11-26

In the literature, some people define the intensity as you have, I(r,z) = Io * exp(...) where Io = 2P/pi*w^{2}.

Now, others define it with some other scaling, like; I(r,z) = 2Io * exp(...). Why would that be?

The author's answer:

They obviously use a two times smaller value of Io, which is then only half the peak intensity. I am not sure why one would like to do that.

2022-06-24

How are the transverse modes of a resonator related to the focusability of the laser beam emitted?

The author's answer:

Higher-order transverse modes have a poorer beam quality, i.e., they are harder to focus.

2022-11-15

Please help me identify the approach to finding the beam waist radius after it has undergone focusing by a lens of a given focal length, given the initial radius (before focusing) and its wavelength.

The author's answer:

You find that in the article on lenses, see the paragraph “Focusing a Collimated Beam”.

2023-01-04

If I have a pulsed laser with 10 μJ pulse energy and the beam is Gaussian, is that energy the average energy or the peak energy? Would the energy at tip of the distribution (the center of the beam) be equal to 10 μJ or less by a factor?

The author's answer:

Pulse energy is always the energy over the full beam area.
A local quantity would be the pulse fluence = energy per area in units of J/cm^{2}, for example. That thing integrated over the full beam area is the pulse energy.

2023-08-14

I want to calculate the total power encircled by a certain radius – how to do this without using a simulation software?

The author's answer:

Simply integrate the Gaussian beam intensity over the area of a circle with the given radius, and divided that by the full area integral. Use polar coordinates to obtain:

$$p_{\rm inside} = \frac{\int_{r = 0}^{R}{\exp(-2 (r / w)^2) \: r \: {\rm d}r}}{\int_{r = 0}^{\infty}{\exp(-2 (r / w)^2) \: r \: {\rm d}r}}$$Note the factor <$r$> in the integrand, which comes from the integration in azimuthal direction.

2023-11-27

There seems to be a unit mismatch: how can wavelength/pi result in mm mrad?

The author's answer:

Radians (or mrad) are not a real unit, as the radian measure is essentially a ratio (of circumference length to radius), thus a dimensionless ratio. We use the radians only to indicate that we use that radian concept to specify an angle. So the basic units on each side of the equation are meters.

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2020-04-07

I would expect the power transmission of a Gaussian beam through a 1-w radius aperture to be the integral of your expression for the intensity, which I'm calculating as 95.4%, not 86.5%.

The author's answer:

It seems that you got the integration wrong: you apparently simply integrated the Gaussian function in the radial direction from 0 to <$w$>. However, you need to integrate over the area; therefore, the integrand must be <$\exp(-2 (r / w)^2) \: 2\pi \: r \: \textrm{d}r$> (apart from the normalization factor). With that, you get the mentioned 86.5%.