= Squared Rayleigh radius
{title2=$R^2/(2\sigma^2)\sim\operatorname{Exp}(1)$}
Squaring a <Rayleigh distribution> variable produces an <exponential distribution>. For $w\geq0$,
$$
\mathbb P(R^2>w)=e^{-w/(2\sigma^2)},\qquad
R^2\sim\operatorname{Exp}\left(\frac1{2\sigma^2}\right),
$$
where the <exponential distribution> parameter is a rate. Equivalently, $R^2/\sigma^2$ has <chi-squared distribution> with two <degrees of freedom (statistics)>. An amplitude and its square therefore have different <probability density functions>, even though one determines the other.
This makes the <Rayleigh-normal scale mixture> transparent: at unit scale, $R^2$ is exponential, allowing a random normal scale to produce a <Laplace distribution>. Another equivalent parametrization uses a <Weibull distribution> with shape two and scale $\sqrt2\sigma$. Conventions that call this latter scale the Rayleigh scale differ by a factor of $\sqrt2$; check the <probability density function> before comparing numerical parameter values.<NIST Dataplot Reference Manual/Rayleigh Probability Density Function>[NIST Dataplot Reference Manual; Description: chi distribution with two degrees of freedom; Weibull shape two and scale sqrt(2) times sigma; warning about Rayleigh scale conventions]{ref}
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