NationMaster - Encyclopedia: Gaussian beam

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. For the book by Sir Isaac Newton, see Opticks. ... Beam may refer to: Look up beam in Wiktionary, the free dictionary. ... Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. ... In physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field. ... In physics, intensity is a measure of the time-averaged energy flux. ... Irradiance, radiant emittance, and radiant exitance are radiometry terms for the power of electromagnetic radiation at a surface, per unit area. ... Gaussian curves parametrised by expected value and variance (see normal distribution) A Gaussian function (named after Carl Friedrich Gauss) is a function of the form: for some real constants a > 0, b, and c. ... For other uses, see Laser (disambiguation). ... A transverse mode of a beam of electromagnetic radiation is a particular intensity pattern of radiation measured in a plane perpendicular (i. ... This page meets Wikipedias criteria for speedy deletion. ... The straw seems to be broken, due to refraction of light as it emerges into the air. ... This article is about the optical device. ... Mathematical models are of great importance in physics. ...

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. In geometric optics, the paraxial approximation is an approximation used in ray tracing of light through an optical system (such as a lens). ... The Helmholtz equation, named for Hermann von Helmholtz, is the elliptic partial differential equation where is the Laplacian, is a constant, and the unknown function is defined on three-dimensional Euclidean space R3. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ... In physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field. ... Magnetic field lines shown by iron filings In physics, the space surrounding moving electric charges, changing electric fields and magnetic dipoles contains a magnetic field. ... Electromagnetic radiation is a propagating wave in space with electric and magnetic components. ...

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

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 Josephson junction array chip developed by NIST as a standard volt. ... The metre, or meter (symbol: m) is the SI base unit of length. ...

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

where

i

is the imaginary unit (for which

i 2 = - 1

), 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. In mathematics, the imaginary unit (or sometimes the Latin or the Greek iota, see below) allows the real number system to be extended to the complex number system . ... Wavenumber in most physical sciences is a wave property inversely related to wavelength, having units of inverse length. ... In mathematics and physics, the radian is a unit of angle measure. ...

The corresponding time-averaged intensity (or irradiance) distribution, measured in watts per square meter, is For other uses, see Watt (disambiguation). ... A square metre (US spelling: square meter) is by definition the area enclosed by a square with sides each 1 metre long. ...

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

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. $E 0$ and $I 0$ are, respectively, the electric field amplitude and intensity at the center of the beam at its waist, that is $E 0 = | E (0,0) |$ and $I 0 = I (0,0)$ . The constant $eta ,$ is the characteristic impedance of the medium in which the beam is propagating. For free space, $eta = eta_0 = 120pi mathrm{Omega} approx 377 mathrm{Omega}$ . The characteristic impedance of a uniform transmission line is the ratio of the amplitudes of a single pair of voltage and current waves propagating along the line in the absence of reflections. ...

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.

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 Diagram of Gaussian beam parameters: File links The following pages link to this file: User:DrBob/Figures Gaussian beam Categories: GFDL images ... For other uses, see Wavelength (disambiguation). ...

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

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.

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 or depth of focus of the beam:

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

Radius of curvature

R(z) is the radius of curvature of the wavefronts comprising the beam. Its value as a function of position is The distance from the center of a sphere or ellipsoid to its surface is its radius. ...

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

Beam divergence

The parameter $w (z)$ approaches a straight line for $z gg z_0$ . 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. In laser science, the beam parameter product (BPP) is the product of a laser beams divergence angle and the diameter of the beam at its narrowest point (the beam waist). The BPP quantifies the quality of a laser beam, and how well it can be focussed to a small...

Gouy phase

The longitudinal phase delay or Gouy phase of the beam is

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

Complex beam parameter

Main article: Complex beam parameter

The complex beam parameter is In optics, the complex beam parameter is a complex number that specifies of properties of a Gaussian beam at a particular point in space by its wavelength Î», radius of curvature R and beam radius w (defined at 1/e2 intensity). ...

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

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. A cavity resonator uses resonance to amplify a wave. ... Ray transfer matrix analysis (also known as ABCD matrix analysis) is a type of ray tracing technique used in the design of some optical systems, particularly lasers. ...

Power and intensity

Power through an aperture

The power P (in watts) passing through a circle of radius r in the transverse plane at position z is In physics, power (symbol: P) is the rate at which work is performed or energy is transferred. ...

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

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 $r = 1.224cdot w(z) ,$ .

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 $π r 2$ : In calculus, lHÃ´pitals rule (sometimes spelled as lHospitals rule) uses derivatives to help compute limits with indeterminate forms. ...

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

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

Notes

^ Siegman (1986) p. 630.

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.

Category: Lasers

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