Note: this box searches only for keywords in the titles of articles, and for acronyms. For full-text searches on the whole website, use our search page.

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

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 <$M^2$> = 1.

Beamquality

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

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.

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.

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.

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.

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.

Beamdivergence

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.)

(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.)

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

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.

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:

Laser M2

Please do not enter personal data here. (See also our privacy declaration.) If you wish to receive personal feedback or consultancy from the author, please contact him, e.g. via e-mail.

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.

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

Laguerre Gaussbeam

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:

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.

Here you can submit questions and comments. As far as they get accepted by the author, they will appear above this paragraph together with the author’s answer. The author will decide on acceptance based on certain criteria. Essentially, the issue must be of sufficiently broad interest.

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.

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.

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.

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:

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!

Laser cutting

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.

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?

By submitting the information, you give your consent to the potential publication of your inputs on our website according to our rules. (If you later retract your consent, we will delete those inputs.) As your inputs are first reviewed by the author, they may be published with some delay.

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/cm2, for example. That thing integrated over the full beam area is the pulse energy.

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.

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%.

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.

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

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

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.

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.

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%.

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.

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.

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

Gaussianbeam

Note: the article keyword search field and some other of the site's functionality would require Javascript, which however is turned off in your browser.

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.

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

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.

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.

(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.)

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.

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

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?

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.