Ligand cone angle: Difference between revisions
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[[File:GaussianBeamWaist.svg|thumb|350px|right|Gaussian beam width <math>w(z)</math> as a function of the axial distance <math>z</math>. <math>w_0</math>: beam waist; <math>b</math>: confocal parameter; <math>z_\mathrm{R}</math>: Rayleigh length; <math>\Theta</math>: total angular spread]] | |||
In [[optics]] and especially [[laser science]], the '''Rayleigh length''' or '''Rayleigh range''' is the distance along the propagation direction of a [[light beam|beam]] from the [[beam waist|waist]] to the place where the area of the [[Cross_section_(geometry)|cross section]] is doubled.<ref name="Siegman1986">{{cite book | last = Siegman | first = A. E. | title = Lasers | publisher = University Science Books | date = 1986 | pages = 664–669 | isbn = 0-935702-11-3 }}</ref> A related parameter is the '''confocal parameter''', ''b'', which is twice the Rayleigh length.<ref name="Damask" /> The Rayleigh length is particularly important when beams are modeled as [[Gaussian beam]]s. | |||
==Explanation== | |||
{{more|Gaussian beam}} | |||
For a Gaussian beam propagating in free space along the <math>\hat{z}</math> axis, the Rayleigh length is given by <ref name="Damask">{{cite book | last = Damask | first = Jay N. | title = Polarization Optics in Telecommunications | publisher = [[Springer Science+Business Media|Springer]] | date = 2004 | pages = 221–223 | isbn = 0-387-22493-9 }}</ref> | |||
:<math>z_\mathrm{R} = \frac{\pi w_0^2}{\lambda} ,</math> | |||
where <math>\lambda</math> is the [[wavelength]] and <math>w_0</math> is the [[beam waist]], the radial size of the beam at its narrowest point. This equation and those that follow assume that the waist is not extraordinarily small; <math>w_0 \ge 2\lambda/\pi</math>.<ref>Siegman (1986) p. 630.</ref> | |||
The radius of the beam at a distance <math>z</math> from the waist is <ref>{{cite book | last = Meschede | first = Dieter | title = Optics, Light and Lasers: The Practical Approach to Modern Aspects of Photonics and Laser Physics | publisher = Wiley-VCH | date = 2007 | pages = 46–48 | isbn = 3-527-40628-X }}</ref> | |||
:<math>w(z) = w_0 \, \sqrt{ 1+ {\left( \frac{z}{z_\mathrm{R}} \right)}^2 } . </math> | |||
The minimum value of <math>w(z)</math> occurs at <math>w(0) = w_0</math>, by definition. At distance <math>z_\mathrm{R}</math> from the beam waist, the beam radius is increased by a factor <math>\sqrt{2}</math> and the cross sectional area by 2. | |||
==Related quantities== | |||
The total angular spread of a Gaussian beam in [[radian]]s is related to the Rayleigh length by<ref name="Siegman1986"/> | |||
:<math>\Theta_{\mathrm{div}} \simeq 2\frac{w_0}{z_R}.</math> | |||
The [[Beam diameter|diameter]] of the beam at its waist (focus spot size) is given by | |||
:<math>D = 2\,w_0 \simeq \frac{4\lambda}{\pi\, \Theta_{\mathrm{div}}}</math>. | |||
These equations are valid within the limits of the [[paraxial approximation]]. For beams with much larger divergence the Gaussian beam model is no longer accurate and a [[physical optics]] analysis is required. | |||
==See also== | |||
* [[Beam divergence]] | |||
* [[Beam parameter product]] | |||
* [[Gaussian function]] | |||
* [[Electromagnetic wave equation]] | |||
* [[John Strutt, 3rd Baron Rayleigh]] | |||
* [[Robert Strutt, 4th Baron Rayleigh]] | |||
* [[Depth of field]] | |||
==References== | |||
{{reflist}} | |||
*[http://www.rp-photonics.com/rayleigh_length.html Rayleigh length] ''RP Photonics Encyclopedia of Optics'' | |||
[[Category:Optics]] | |||
Revision as of 16:12, 24 May 2013
In optics and especially laser science, the Rayleigh length or Rayleigh range is the distance along the propagation direction of a beam from the waist to the place where the area of the cross section is doubled.[1] A related parameter is the confocal parameter, b, which is twice the Rayleigh length.[2] The Rayleigh length is particularly important when beams are modeled as Gaussian beams.
Explanation
Template:More For a Gaussian beam propagating in free space along the axis, the Rayleigh length is given by [2]
where is the wavelength and is the beam waist, the radial size of the beam at its narrowest point. This equation and those that follow assume that the waist is not extraordinarily small; .[3]
The radius of the beam at a distance from the waist is [4]
The minimum value of occurs at , by definition. At distance from the beam waist, the beam radius is increased by a factor and the cross sectional area by 2.
Related quantities
The total angular spread of a Gaussian beam in radians is related to the Rayleigh length by[1]
The diameter of the beam at its waist (focus spot size) is given by
- .
These equations are valid within the limits of the paraxial approximation. For beams with much larger divergence the Gaussian beam model is no longer accurate and a physical optics analysis is required.
See also
- Beam divergence
- Beam parameter product
- Gaussian function
- Electromagnetic wave equation
- John Strutt, 3rd Baron Rayleigh
- Robert Strutt, 4th Baron Rayleigh
- Depth of field
References
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- Rayleigh length RP Photonics Encyclopedia of Optics
- ↑ 1.0 1.1 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 2.0 2.1 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ Siegman (1986) p. 630.
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534