Integrals are transformed when moving from $S$ to $S\u2032$ with the following equation:^{17}Display Formula
$\u222cSf(x,y)dxdy=\u222cS\u2032f(x\u2032,y\u2032)|\u2202(x,y)\u2202(x\u2032,y\u2032)|dx\u2032dy\u2032,$(40)
where $|\u2202(x,y)\u2202(x\u2032,y\u2032)|$ is the Jacobian determinant given by Display Formula$|\u2202(x,y)\u2202(x\u2032,y\u2032)|=|\u2202x\u2202x\u2032\u2202y\u2202x\u2032\u2202x\u2202y\u2032\u2202y\u2202y\u2032|=|\alpha 001|=\alpha ,$(41)
and $\alpha $ is defined as Display Formula$\alpha =wx(z)wy(z).$(42)
Equation (40) can be rewritten by changing to polar coordinates and substituting Eq. (41) for the Jacobian determinant Display Formula$\alpha \u222cS\u2032f(r\u2032,\theta \u2032)rdr\u2032d\theta \u2032.$(43)
Therefore, the transformation from region $S$ to region $S\u2032$ requires the integral to be multiplied by the scalar $\alpha $. However, the integrals in the scale factor derivation are always found as a ratio [Eqs. (2) and (9)], so the factor $\alpha $ cancels out when the scale factor derivation is performed in region $S\u2032$ with the transformed elliptical Gaussian. Using the results of Eqs. (38) and (43), Eq. (18) becomes Display Formula$\sigma r\u2032(z)=1\tau \psi 2(\nu ),$(44)
and using Eq. (1) Display Formula$w\sigma \u2032(z)=2\sigma r\u2032=1\psi (\nu )2\tau .$(45)
Substituting Eq. (39) for $\tau $ and solving for $wy(z)$ gives Display Formula$wy(z)=w\sigma \u2032(z)\psi (\nu ).$(46)
Using Eqs. (42), (46) becomes Display Formula$wx(z)=\alpha w\sigma \u2032(z)\psi (\nu ).$(47)
Transforming $w\sigma \u2032(z)$ from region $S\u2032$ back to region $S$ along the $x$ and $y$ axes yields Display Formula$w\sigma \u2032(z)=1\alpha w\sigma x(z),$(48)
Display Formula$w\sigma \u2032(z)=w\sigma y(z),$(49)
where $w\sigma x$ and $w\sigma y$ are the measured second moment radii along the $x$ and $y$ axes, respectively. Substituting Eqs. (48) into (47) and Eqs. (49) into (46) gives Display Formula$wx(z)=w\sigma x(z)\psi (\nu ),$(50)
Display Formula$wy(z)=w\sigma y(z)\psi (\nu ).$(51)
Thus, the scale factor is invariant to the ellipticity of a $TEM00$ Gaussian beam.