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In relativistic laser-plasma physics the '''relativistic similarity parameter''' '''''S''''' is a [[dimensionless parameter]] defined as

:<math> S=\frac{n_e}{a_0n_{cr}}</math>,

where <math>{n_e}</math> is the electron plasma density, <math>{n_{cr}=m_e\omega_0^2/4\pi e^2}</math> is the critical plasma density and <math>{a_0=eA/m_e c^2}</math> is the normalized vector potential. Here <math>{m_e}</math> is the [[Electron rest mass|electron mass]], <math>{e}</math> is the electron charge, <math>{c}</math> is the speed of light and <math>{\omega_0}</math> is the laser frequency. Note that [[CGS units]] were used above.

The concept of similarity and the similarity parameter <math>{S}</math> were first introduced in plasma physics by Sergey Gordienko.<ref>{{cite journal|author=S.Gordienko and A.Pukhov|title=Scaling for ultra-relativistic laser plasmas and quasi-monoenergetic electrons|journal=Physics of Plasmas|volume=12|pages=043109|year=2005|url= http://link.aip.org/link/PHPAEN/v12/i4/p043109/s1}}</ref> It allows distinguishing between relativistically overdense <math>{(S\gg 1)}</math> and underdense plasmas <math>{(S\ll 1)}</math>.

The similarity parameter is connected to basic symmetry properties of the collisionless [[Vlasov equation]] and is thus the [[relativistic plasma]] analog of the [[Reynolds number]] in [[fluid mechanics]]. Gordienko showed that in the relativistic limit (<math>{a_0\gg 1}</math>) the laser-plasma dynamics depends on three dimensionless parameters: <math>\omega_0\tau</math>, <math>{R\omega_0/c}</math> and <math>{S}</math>, where <math>{\tau}</math> is the duration of the laser pulse and <math>{R}</math> is the typical radius of the [[beam waist|laser waist]]. The main result of the relativistic similarity theory can be summarized as follows: if the parameters of the interaction (plasma density and laser amplitude) change simultaneously so that the <math>{S}</math> parameter remains constant, the dynamics of the electrons remains the same.

The similarity theory allows deriving non-trivial power-law scalings for the energy of fast electrons in underdense and overdense plasmas.<ref>{{cite journal|author=T.Baeva, S.Gordienko and A.Pukhov|title=Theory of high-order harmonic generation in relativistic laser interaction with overdense plasma|journal=Physical Review E|volume=74|pages=046404|year=2006|url= http://pre.aps.org/abstract/PRE/v74/i4/e046404|bibcode = 2006PhRvE..74a6404F |doi = 10.1103/PhysRevE.74.016404 }}</ref>

== References ==
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{{Reflist}}

== External links ==

[[Category:Plasma physics]]


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