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| In [[optics]], the '''complex beam parameter''' is a [[complex number]] that specifies the properties of a [[Gaussian beam]] at a particular point z along the axis of the beam. It is usually denoted by ''q''. It can be calculated from the beam's vacuum [[wavelength]] λ<sub>0</sub>, the [[radius of curvature]] ''R'' of the [[phase front]], the [[index of refraction]] ''n'' (''n''=1 for air), and the beam radius ''w'' (defined at 1/''e''<sup>2</sup> intensity), according to:<ref name=Yariv>{{cite book |first=Amnon |last=Yariv |year=1989 |title=Quantum Electronics |edition=3rd |publisher=Wiley |isbn=0-471-60997-8}}</ref>
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| :<math> \frac{1}{q(z)} = \frac{1}{R(z)} - \frac{i\lambda_0}{\pi n w(z)^2} </math>.
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| Alternatively, ''q'' can be calculated according to
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| :<math> q(z) = z + z_0 i\ ,</math><ref name=Yariv/>
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| where ''z'' is the location at which ''q'' is calculated, relative to the location of the [[beam waist]], ''z<sub>0</sub>'' is the [[Rayleigh range]], and ''i'' is the [[imaginary unit]].
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| ==Uses of the complex beam parameter==
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| The complex beam parameter is usually used in [[ray transfer matrix analysis]], which allows the calculation of the beam properties at any given point as it propagates through an optical system, if the ray matrix and the initial complex beam parameter is known. This same method can also be used to find the fundamental mode size of a stable [[optical resonator]].
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| Given the initial beam parameter, ''q''<sub>i</sub>, one can use the ray transfer matrix of an optical system, <math>\begin{pmatrix} A & B \\ C & D \end{pmatrix}</math>, to find the resulting beam parameter, ''q''<sub>f</sub>, after the beam has traversed the system:<ref name=Yariv/>
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| :<math> q_f = \frac{Aq_i+B}{Cq_i+D}</math>.
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| It is often convenient to express this equation in terms of the reciprocals of ''q'':<ref name=Yariv/>
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| :<math> { 1 \over q_f } = \frac{C+D/q_i}{A+B/q_i}</math>.
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| To find the complex beam parameter of a stable [[optical resonator]], one needs to find the ray matrix of the cavity. This is done by tracing the path of beam in the cavity. Assuming a starting point, find the matrix that goes through the cavity and return until the beam is in the same position and direction as the starting point. With this matrix and by making ''q''<sub>i</sub> = ''q''<sub>f</sub>, a quadratic is formed as:
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| :<math>C{q_f}^2 + (D-A)q_f - B = 0</math>.
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| Solving this equation gives the beam parameter for the chosen starting position in the cavity, and by propagating, the beam parameter for any other location in the cavity can be found.
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| ==References==
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| <references/>
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| {{DEFAULTSORT:Complex Beam Parameter}}
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| [[Category:Optics]]
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