Gaussian beam

In optics, a Gaussian beam is a beam of electromagnetic radiation with high monochromaticity whose amplitude envelope in the transverse plane is given by a Gaussian function; this also implies a Gaussian intensity (irradiance) profile. This fundamental (or TEM₀₀) transverse Gaussian mode describes the intended output of most (but not all) lasers, as such a beam can be focused into the most concentrated spot. When such a beam is refocused by a lens, the transverse ''phase'' dependence is altered; this results in a ''different'' Gaussian beam. The electric and magnetic field amplitude profiles along any such circular Gaussian beam (for a given wavelength and polarization) are determined by a single parameter: the so-called waist . At any position relative to the waist (focus) along a beam having a specified , the field amplitudes and phases are thereby determinedSvelto, pp. 153–5. as detailed below.

The equations below assume a beam with a circular cross-section at all values of ; this can be seen by noting that a single transverse dimension, , appears. Beams with elliptical cross-sections, or with waists at different positions in for the two transverse dimensions ( astigmatic beams) can also be described as Gaussian beams, but with distinct values of and of the location for the two transverse dimensions and .

Arbitrary solutions of the paraxial Helmholtz equation can be expressed as combinations of Hermite–Gaussian modes (whose amplitude profiles are separable in and using Cartesian coordinates) or similarly as combinations of Laguerre–Gaussian modes (whose amplitude profiles are separable in and using cylindrical coordinates).Siegman, p. 642.probably first considered by Goubau and Schwering (1961). At any point along the beam these modes include the same Gaussian factor as the fundamental Gaussian mode multiplying the additional geometrical factors for the specified mode. However different modes propagate with a different Gouy phase which is why the net transverse profile due to a superposition of modes evolves in , whereas the propagation of any ''single'' Hermite–Gaussian (or Laguerre–Gaussian) mode retains the same form along a beam.

Although there are other possible modal decompositions, these families of solutions are the most useful for problems involving compact beams, that is, where the optical power is rather closely confined along an axis. Even when a laser is ''not'' operating in the fundamental Gaussian mode, its power will generally be found among the lowest-order modes using these decompositions, as the spatial extent of higher order modes will tend to exceed the bounds of a laser's resonator (cavity). "Gaussian beam" normally implies radiation confined to the fundamental (TEM₀₀) Gaussian mode.

Mathematical form

The Gaussian beam is a transverse electromagnetic (TEM) mode.Svelto, p. 158. The mathematical expression for the electric field amplitude is a solution to the paraxial Helmholtz equation. Assuming polarization in the direction and propagation in the direction, the electric field in phasor (complex) notation is given by:

$= E_0 \, \hat \, \frac \exp \left( \frac\right ) \exp \left(\! -i \left(kz +k \frac - \psi(z) \right) \!\right)$

where

* is the radial distance from the center axis of the beam,
* is the axial distance from the beam's focus (or "waist"),
* is the imaginary unit,
* is the wave number (in radians per meter) for a free-space wavelength , and is the index of refraction of the medium in which the beam propagates,
*, the electric field amplitude (and phase) at the origin (, ),
* is the radius at which the field amplitudes fall to of their axial values (i.e., where the intensity values fall to of their axial values), at the plane along the beam,
* is the waist radius,
* is the radius of curvature of the beam's wavefronts at , and
* is the Gouy phase at , an extra phase term beyond that attributable to the phase velocity of light.

There is also an understood time dependence multiplying such phasor quantities; the actual field at a point in time and space is given by the real part of that complex quantity. This time factor involves an arbitrary sign convention, as discussed at .

Since this solution relies on the paraxial approximation, it is not accurate for very strongly diverging beams. The above form is valid in most practical cases, where .

The corresponding intensity (or irradiance) distribution is given by

$I(r,z) = = I_0 \left( \frac \right)^2 \exp \left( \frac\right),$

where the constant is the wave impedance of the medium in which the beam is propagating. For free space, ≈ 377 Ω. is the intensity at the center of the beam at its waist.

If is the total power of the beam,
$I_0 = .$

Evolving beam width

At a position along the beam (measured from the focus), the spot size parameter is given by a hyperbolic relation:
$w(z) = w_0 \, \sqrt,$
where
$z_\mathrm = \frac$
is called the Rayleigh range as further discussed below, and $n$ is the refractive index of the medium.

The radius of the beam , at any position along the beam, is related to the full width at half maximum (FWHM) of the intensity distribution at that position according to:
$w(z)=.$

Wavefront curvature

The curvature of the wavefronts is largest at the Rayleigh distance, , on either side of the waist, crossing zero at the waist itself. Beyond the Rayleigh distance, , it again decreases in magnitude, approaching zero as . The curvature is often expressed in terms of its reciprocal, , the '' radius of curvature''; for a fundamental Gaussian beam the curvature at position is given by:

$\frac = \frac ,$

so the radius of curvature is
R(z) = z \left \right
Being the reciprocal of the curvature, the radius of curvature reverses sign and is infinite at the beam waist where the curvature goes through zero.

Gouy phase

The '' Gouy phase'' is a phase advance gradually acquired by a beam around the focal region. At position the Gouy phase of a fundamental Gaussian beam is given by
$\psi(z) = \arctan \left( \frac \right).$

The Gouy phase results in an increase in the apparent wavelength near the waist (). Thus the phase velocity in that region formally exceeds the speed of light. That paradoxical behavior must be understood as a near-field phenomenon where the departure from the phase velocity of light (as would apply exactly to a plane wave) is very small except in the case of a beam with large numerical aperture, in which case the wavefronts' curvature (see previous section) changes substantially over the distance of a single wavelength. In all cases the wave equation is satisfied at every position.

The sign of the Gouy phase depends on the sign convention chosen for the electric field phasor. With dependence, the Gouy phase changes from to , while with dependence it changes from to along the axis.

For a fundamental Gaussian beam, the Gouy phase results in a net phase discrepancy with respect to the speed of light amounting to radians (thus a phase reversal) as one moves from the far field on one side of the waist to the far field on the other side. This phase variation is not observable in most experiments. It is, however, of theoretical importance and takes on a greater range for higher-order Gaussian modes.

Elliptical and astigmatic beams

Many laser beams have an elliptical cross-section. Also common are beams with waist positions which are different for the two transverse dimensions, called astigmatic beams. These beams can be dealt with using the above two evolution equations, but with distinct values of each parameter for and and distinct definitions of the point. The Gouy phase is a single value calculated correctly by summing the contribution from each dimension, with a Gouy phase within the range contributed by each dimension.

An elliptical beam will invert its ellipticity ratio as it propagates from the far field to the waist. The dimension which was the larger far from the waist, will be the smaller near the waist.

Beam parameters

The geometric dependence of the fields of a Gaussian beam are governed by the light's wavelength (''in'' the dielectric medium, if not free space) and the following beam parameters, all of which are connected as detailed in the following sections.

Beam waist

The shape of a Gaussian beam of a given wavelength is governed solely by one parameter, the ''beam waist'' . This is a measure of the beam size at the point of its focus ( in the above equations) where the beam width (as defined above) is the smallest (and likewise where the intensity on-axis () is the largest). From this parameter the other parameters describing the beam geometry are determined. This includes the Rayleigh range and asymptotic beam divergence , as detailed below.

Rayleigh range and confocal parameter

The ''Rayleigh distance'' or ''Rayleigh range'' is determined given a Gaussian beam's waist size:

$z_\mathrm = \frac.$

Here is the wavelength of the light, is the index of refraction. At a distance from the waist equal to the Rayleigh range , the width of the beam is larger than it is at the focus where , the beam waist. That also implies that the on-axis () intensity there is one half of the peak intensity (at ). That point along the beam also happens to be where the wavefront curvature () is greatest.

The distance between the two points is called the ''confocal parameter'' or ''depth of focus'' of the beam.

Beam divergence

Although the tails of a Gaussian function never actually reach zero, for the purposes of the following discussion the "edge" of a beam is considered to be the radius where . That is where the intensity has dropped to of its on-axis value. Now, for the parameter increases linearly with . This means that far from the waist, the beam "edge" (in the above sense) is cone-shaped. The angle between that cone (whose ) and the beam axis () defines the ''divergence'' of the beam:
$\theta = \lim_ \arctan\left(\frac\right).$

In the paraxial case, as we have been considering, (in radians) is then approximately
$\theta = \frac$

where is the refractive index of the medium the beam propagates through, and is the free-space wavelength. The total angular spread of the diverging beam, or ''apex angle'' of the above-described cone, is then given by
$\Theta = 2 \theta\, .$

That cone then contains 86% of the Gaussian beam's total power.

Because the divergence is inversely proportional to the spot size, for a given wavelength , a Gaussian beam that is focused to a small spot diverges rapidly as it propagates away from the focus. Conversely, to ''minimize'' the divergence of a laser beam in the far field (and increase its peak intensity at large distances) it must have a large cross-section () at the waist (and thus a large diameter where it is launched, since is never less than ). This relationship between beam width and divergence is a fundamental characteristic of diffraction, and of the Fourier transform which describes