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 Hermite-gaussian modes
5 Laguerre-Gaussian modes
6 See also
7 Notes
8 References

[edit] Mathematical form

For a Gaussian beam, the complex electric field amplitude 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

r

is the radial distance from the center axis of the beam,

z

is the axial distance from the beam's narrowest point (the "waist"),

i

is the imaginary unit (for which

i 2 = - 1

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

E 0 = | E (0,0) |

w (z)

is the radius at which the field amplitude and intensity drop to 1/e and 1/e² of their axial values, respectively, and

w 0 = w (0)

is the waist size (described in more detail below).

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 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 $I 0 = I (0,0)$ is the intensity at the center of the beam at its waist. The constant $\eta \,$ is the characteristic impedance of the medium in which the beam is propagating. For free space, $\eta = \eta_0 \approx 377 \ \mathrm{\Omega}$ .

[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

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

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

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

is called the Rayleigh range.

[edit] Rayleigh range and confocal parameter

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

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

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

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

[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

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

[edit] Beam divergence

The parameter $w (z)$ approaches a straight line for $z \gg z_R$ . The angle between this straight line and the central axis of the beam 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

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

[edit] Complex beam parameter

Main article: Complex beam parameter

The complex beam parameter is

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

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

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

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.

In terms of the complex beam parameter $q$ , a gaussian field with one transverse dimension is proportional to

${u}(x,z) = \frac{1}{\sqrt{{q}_x(z)}} \exp\left(-i k \frac{x^2}{2 {q}_x(z)}\right)$ .

In two dimensions one can write the potentially elliptical or astigmatic beam as the product

${u}(x,y,z) = {u}(x,z)\, {u}(y,z)$ ,

which for the common case of circular symmetry where $q x = q y = q$ and $x 2 + y 2 = r 2$ yields^[2]

${u}(r,z) = \frac{1}{{q}(z)}\exp\left( -i k\frac{r^2}{2 {q}(z)}\right)$ .

[edit] Power and intensity

[edit] Power through an aperture

The power P 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] Hermite-gaussian modes

See also: Transverse mode

In one transverse dimension higher order Hermite-gaussian modes exist. These are natural extensions of the fundamental lowest-order Gaussian solution. In terms of the previously defined complex $q$ parameter these modes have intensity distributions proportional to

${u}_n(x,z) = \left(\frac{2}{\pi}\right)^{1/4} \left(\frac{1}{2^n n! w_0}\right)^{1/2} \left( \frac{{q}_0}{{q}(z)}\right)^{1/2} \left[\frac{{q}_0}{{q}_0^\ast} \frac{{q}^\ast(z)}{{q}(z)}\right]^{n/2} H_n\left(\frac{\sqrt{2}x}{w(z)}\right) \exp\left[-i \frac{k x^2}{2 {q}(z)}\right]$

where the function $H n (x)$ is the Hermite polynomial of order $n$ , and the asterisk indicates complex conjugation. For the case $n = 0$ the equation yields a Gaussian transverse distribution.

For two dimensional rectangular coordinates one constructs a function $u m, n (x, y, z) = u m (x, z) u n (y, z)$ from a product of two one dimensional functions as given above. Mathematically this property is due to the separation of variables applied to the paraxial Helmholtz equation for Cartesian coordinates.^[3]

Hermite-Gaussian modes are typically designated "TEM_m,n", where m and n are the polynomial indices in the x and y directions. A Gaussian beam is thus TEM_0,0.

[edit] Laguerre-Gaussian modes

If we consider the problem in cylindrical coordinates we can write higher order modes using Laguerre- instead of Hermite-polynomials. In this manner two dimensional Laguerre-Gaussian modes may be written as

${u}(r,\theta,z)=\sqrt{\frac{2 p!}{(1+\delta_{0m})\pi(m+p)!}} \frac{\exp\left(i (2 p + m + 1)(\psi(z) - \psi_0)\right)}{w(z)} \times \left(\frac{\sqrt{2}r}{w(z)}\right)^m L_p^m\left(\frac{2 r^2}{w(z)^2}\right) \exp\left[ -i k \frac{r^2}{2{q}(z)}+i m \theta\right]$

where $L_p^m(r)$ are the generalised Laguerre polynomials, the radial index $p\ge 0$ , the azimuthal index is $m$ and $δ 0 m$ represents the Kronecker delta, $δ 0 m = 0$ if $m\neq0$ but 1 otherwise.^[4]

[edit] See also

[edit] Notes

^ Siegman (1986) p. 630.
^ See Siegman (1986) p. 639. Eq. 29
^ Siegman (1986), p645, eq. 54
^ Siegman (1986), p647, eq. 64

[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.

Mandel, Leonard and Wolf, Emil (1995). Optical Coherence and Quantum Optics. Cambridge: Cambridge University Press. ISBN 0-521-41711-2. Chapter 5, "Optical Beams," pp. 267.

Siegman, Anthony E. (1986). Lasers. University Science Books. ISBN 0-935702-11-3. Chapter 16.

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

F. Pampaloni and J. Enderlein (2004). "Gaussian, Hermite-Gaussian, and Laguerre-Gaussian beams: A primer". arXiv:physics/0410021.

Gaussian Beam Propagation, Melles-Griot

Gaussian Beam Optics Tutorial, Newport

Categories: Lasers

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