Gaussian beam

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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.

1 Mathematical form
2 Beam parameters
3 Power and intensity
- 3.1 Power through an aperture
- 3.2 Peak and average intensity
4 See also
5 Notes
6 References

[edit] Mathematical form

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

$E(r,z) = E_0 \frac{w_0}{w(z)} \exp \left( \frac{-r^2}{w^2(z)}\right) \exp \left( -ikz -ik \frac{r^2}{2R(z)} +i \zeta(z) \right)\ ,$

where

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

$k = { 2 \pi \over \lambda }$ 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

$I(r,z) = { |E(r,z)|^2 \over 2 \eta } = I_0 \left( \frac{w_0}{w(z)} \right)^2 \exp \left( \frac{-2r^2}{w^2(z)} \right)\ ,$

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. $E_0 \ \stackrel{\mathrm{def}}{=}\ |E(0,0)|$ and $I_0 \ \stackrel{\mathrm{def}}{=}\ I(0,0)$ . The constant $\eta \,$ is the characteristic impedance of the medium in which the beam is propagating. For free space, $\eta = \eta_0 \approx 377 \ \mathrm{ohms}$ .

[edit] Beam parameters

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

[edit] 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

$w(z) = w_0 \, \sqrt{ 1+ {\left( \frac{z}{z_0} \right)}^2 } \ .$

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

$z_0 = \frac{\pi w_0^2}{\lambda}$

is called the Rayleigh range or depth of focus.

[edit] Rayleigh range and confocal parameter

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

$w(\pm z_0) = w_0 \sqrt{2} \,$

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

$b = 2 z_0 = \frac{2 \pi w_0^2}{\lambda}\ .$

[edit] Radius of curvature

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

$R(z) = z \left[{ 1+ {\left( \frac{z_0}{z} \right)}^2 } \right] \ .$

[edit] Beam divergence

The parameter $w (z)$ approaches a straight line for $z \gg z_0$ . The angle between this straight line and the beam's central axis is called the divergence of the beam. It is given by

$\theta \simeq \frac{\lambda}{\pi w_0} \qquad (\theta \mathrm{\ in\ radians.})$

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

$\Theta = 2 \theta\ .$

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 propagation^[1]. 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.

[edit] Gouy phase

The longitudinal phase delay or Gouy phase of the beam is

$\zeta(z) = \arctan \left( \frac{z}{z_0} \right) \ .$

[edit] Complex beam parameter

Main article: Complex beam parameter

The complex beam parameter is

$q(z) = z + q_0 = z + iz_0 \ .$

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

${ 1 \over q(z) } = { 1 \over z + iz_0 } = { z \over z^2 + z_0^2 } - i { z_0 \over z^2 + z_0^2 } = {1 \over R(z) } - i { \lambda \over \pi w^2(z) }$

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.

[edit] Power and intensity

[edit] Power through an aperture

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

$P(r,z) = P_0 \left[ 1 - e^{-2r^2 / w^2(z)} \right]\ ,$

where

$P_0 = { 1 \over 2 } \pi I_0 w_0^2$

is the total power transmitted by the beam.

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

${ P(z) \over P_0 } = 1 - e^{-2} \approx 0.865\ .$

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

[edit] Peak and average intensity

The peak intensity at an axial distance $z$ from the beam waist is calculated using L'Hôpital's rule as the limit of the enclosed power within a circle of radius $r$ , divided by the area of the circle $π r 2$ :

$I(0,z) =\lim_{r\to 0} \frac {P_0 \left[ 1 - e^{-2r^2 / w^2(z)} \right]} {\pi r^2} = \frac{P_0}{\pi} \lim_{r\to 0} \frac { \left[ -(-2)(2r) e^{-2r^2 / w^2(z)} \right]} {w^2(z)(2r)} = {2P_0 \over \pi w^2(z)}.$

The peak intensity is thus exactly twice the average intensity, obtained by dividing the total power by the area within the radius $w (z)$ .

[edit] See also

[edit] Notes

^ Siegman (1986) p. 630.

[edit] References

Saleh, Bahaa E. A. and Teich, Malvin Carl (1991). Fundamentals of Photonics. New York: John Wiley & Sons. ISBN 0-471-83965-5. Chapter 3, "Beam Optics," pp. 80–107.

Siegman, Anthony E. (1986). Lasers. University Science Books. ISBN 0-935702-11-3. Chapter 6.

Yariv, Amnon (1989). Quantum Electronics, 3rd Edition, Wiley. ISBN 0-471-60997-8.

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Category: Optics