Relativistic similarity parameter

In relativistic laser-plasma physics the relativistic similarity parameter S is a dimensionless parameter defined as
where formula_2 is the electron plasma density, formula_3 is the critical plasma density and formula_4 is the normalized vector potential. Here formula_5 is the electron mass, formula_6 is the electron charge, formula_7 is the speed of light and formula_8 is the laser frequency. Note that CGS units were used above.
The concept of similarity and the similarity parameter formula_9 were first introduced in plasma physics by Sergey Gordienko. It allows distinguishing between relativistically overdense formula_10 and underdense plasmas formula_11.
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 (formula_12) the laser-plasma dynamics depends on three dimensionless parameters: formula_13, formula_14 and formula_9, where formula_16 is the duration of the laser pulse and formula_17 is the typical radius of the 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 formula_9 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.