In optics, a Gaussian beam is a beam of electromagnetic radiation whose transverse electric field and intensity (irradiance) distributions are described by Gaussian functions. Many lasers emit beams with a Gaussian profile, in which case the laser is said to be operating on the fundamental transverse mode, or "TEM₀₀ mode" of the laser's optical resonator. When refracted by a lens, a Gaussian beam is transformed into another Gaussian beam (characterized by a different set of parameters), which explains why it is a convenient, widespread model in laser optics.

The mathematical function that describes the Gaussian beam is a solution to the paraxial form of the Helmholtz equation. The solution, in the form of a Gaussian function, represents the complex amplitude of the electric field, which propagates along with the corresponding magnetic field as an electromagnetic wave in the beam.

Mathematical form

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For a Gaussian beam, the complex electric field amplitude, measured in volts per meter, at a distance r from its centre, and a distance z from its waist, is given by

$E(r,z)\; =\; E\_0\; \backslash frac\{w\_0\}\{w(z)\}\; \backslash exp\; \backslash left(\; \backslash frac\{-r^2\}\{w^2(z)\}\backslash right)\; \backslash exp\; \backslash left(\; -ikz\; -ik\; \backslash frac\{r^2\}\{2R(z)\}\; +i\; \backslash zeta(z)\; \backslash right)\backslash \; ,$

where

$i\; =\; \backslash sqrt\{-1\}\; \backslash ,$ is the imaginary unit, and

$k\; =\; \{\; 2\; \backslash pi\; \backslash over\; \backslash lambda\; \}$ is the wave number (in radians per meter).

The functions w(z), R(z), and ζ(z) are parameters of the beam, which we define below.

The corresponding time-averaged intensity (or irradiance) distribution, measured in watts per square meter, is

$I(r,z)\; =\; \{\; |E(r,z)|^2\; \backslash over\; 2\; \backslash eta\; \}\; =\; I\_0\; \backslash left(\; \backslash frac\{w\_0\}\{w(z)\}\; \backslash right)^2\; \backslash exp\; \backslash left(\; \backslash frac\{-2r^2\}\{w^2(z)\}\; \backslash right)\backslash \; ,$

where w(z) is the radius at which the field amplitude and intensity drop to 1/e and 1/e², respectively. This parameter is called the beam radius or spot size of the beam. E₀ and I₀ are, respectively, the electric field amplitude and intensity at the center of the beam at its waist, i.e. $E\_0\; \backslash equiv\; |E(0,0)|$ and $I\_0\; \backslash equiv\; I(0,0)$. The constant $\backslash eta\; \backslash ,$ is the characteristic impedance of the medium in which the beam is propagating. For free space, $\backslash eta\; =\; \backslash eta\_0\; \backslash approx\; 377\; \backslash \; \backslash mathrm\{ohms\}$.

Beam parameters

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The geometry and behavior of a Gaussian beam are governed by a set of beam parameters, which are defined in the following sections.

Beam width or "spot size"

For a Gaussian beam propagating in free space, the spot size w(z) will be at a minimum value w₀ at one place along the beam axis, known as the beam waist. For a beam of wavelength λ at a distance z along the beam from the beam waist, the variation of the spot size is given by

$w(z)\; =\; w\_0\; \backslash ,\; \backslash sqrt\{\; 1+\; \{\backslash left(\; \backslash frac\{z\}\{z\_0\}\; \backslash right)\}^2\; \}\; \backslash \; .$

where the origin of the z-axis is defined, without loss of generality, to coincide with the beam waist, and where

$z\_0\; =\; \backslash frac\{\backslash pi\; w\_0^2\}\{\backslash lambda\}$

is called the Rayleigh range or depth of focus.

Rayleigh range and confocal parameter

At a distance from the waist equal to the Rayleigh range z₀, the width w of the beam is

$w(\backslash pm\; z\_0)\; =\; w\_0\; \backslash sqrt\{2\}\; \backslash ,$

The distance between these two points is called the confocal parameter of the beam:

$b\; =\; 2\; z\_0\; =\; \backslash frac\{2\; \backslash pi\; w\_0^2\}\{\backslash lambda\}\backslash \; .$

Radius of curvature

R(z) is the radius of curvature of the wavefronts comprising the beam. Its value as a function of position is

$R(z)\; =\; z\; \backslash left1+\; \{\backslash left(\; \backslash frac\{z\_0\}\{z\}\; \backslash right)\}^2\; \}\; \backslash right\backslash \; .$

Beam divergence

The parameter $w(z)$ approaches a straight line for $z\; \backslash gg\; z\_0$. The angle between this straight line and the beam's central axis is called the divergence of the beam. It is given by

$\backslash theta\; \backslash simeq\; \backslash frac\{\backslash lambda\}\{\backslash pi\; w\_0\}\; \backslash qquad\; (\backslash theta\; \backslash mathrm\{\backslash \; in\backslash \; radians.\})$

The total angular spread of the beam far from the waist is then given by

$\backslash Theta\; =\; 2\; \backslash theta\backslash \; .$

Because of this property, a Gaussian laser beam that is focused to a small spot spreads out rapidly as it propagates away from that spot. To keep a laser beam very well collimated, it must have a large diameter.

Since the gaussian beam model uses the paraxial approximation, it fails when wavefronts are tilted by more than about 30° from the direction of propagationSiegman (1986) p. 630.. From the above expression for divergence, this means the Gaussian beam model is valid only for beams with waists larger than about 2λ/π.

Laser beam quality is quantified by the beam parameter product (BPP). For a Gaussian beam, the BPP is the product of the beam's divergence and waist size $w\_0$. The BPP of a real beam is obtained by measuring the beam's minimum diameter and far-field divergence, and taking their product. The ratio of the BPP of the real beam to that of an ideal Gaussian beam at the same wavelength is known as M² ("M squared"). The M² for a Gaussian beam is one. All real laser beams have M² values greater than one, although very high quality beams can have values very close to one.

Gouy phase

The longitudinal phase delay or Gouy phase of the beam is

$\backslash zeta(z)\; =\; \backslash arctan\; \backslash left(\; \backslash frac\{z\}\{z\_0\}\; \backslash right)\; \backslash \; .$

Complex beam parameter

The complex beam parameter is

$q(z)\; =\; z\; +\; q\_0\; =\; z\; +\; iz\_0\; \backslash \; .$

It is often convenient to calculate this quantity in terms of its reciprocal:

$\{\; 1\; \backslash over\; q(z)\; \}\; =\; \{\; 1\; \backslash over\; z\; +\; iz\_0\; \}\; =\; \{\; z\; \backslash over\; z^2\; +\; z\_0^2\; \}\; -\; i\; \{\; z\_0\; \backslash over\; z^2\; +\; z\_0^2\; \}\; =\; \{1\; \backslash over\; R(z)\; \}\; -\; i\; \{\; \backslash lambda\; \backslash over\; \backslash pi\; w^2(z)\; \}$

The complex beam parameter plays a key role in the analysis of gaussian beam propagation, and especially in the analysis of optical resonator cavities using ray transfer matrices.

Power through an aperture

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The power P (in watts) passing through a circle of radius r in the transverse plane at position z is

$P(z)\; =\; P\_0\; \backslash left1\; -\; e^\{-2r^2\; /\; w^2(z)\}\; \backslash right\backslash \; ,$

where

$P\_0\; =\; \{\; 1\; \backslash over\; 2\; \}\; \backslash pi\; I\_0\; w\_0^2$

is the total power transmitted by the beam.

For a circle of radius $r\; =\; w(z)\; \backslash ,$, the fraction of power transmitted through the circle is

$\{\; P(z)\; \backslash over\; P\_0\; \}\; =\; 1\; -\; e^\{-2\}\; \backslash approx\; 0.865\backslash \; .$

Similarly, about 95 percent of the beam's power will flow through a circle of radius $r\; =\; 1.224\backslash cdot\; w(z)\; \backslash ,$.

References

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• Chapter 3, "Beam Optics," pp. 80–107.

• Chapter 6.

Notes

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See also

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• Gaussian function
• Electromagnetic wave equation
• Hermite polynomials

Optics

Gaußstrahl | Faisceau gaussien