The light source has a polygon mirror scanner and cannot be synchronized to an external source, however, the broadband EOM used is driven by an arbitrary waveform function generator (33120A, Agilent) and an RF amplifier (BTM00250, Tomco). The use of an arbitrary waveform generator allows the EOM to be synchronized to the light source k-sweep using an externally triggered sinusoidal tone burst, which then completely removes J_{offset2} in Eqs. 1,3 from each A-scan. The number of generated sinusoidal cycles should cover the duration of the A-scan sweep. The averaging of the measured surface Jones matrix
$J^{\prime \prime} _{\rm{surface}} $
$J surface \u2032\u2032$ across A-scans is then possible provided that we remove the global phase offset term
e^{i2αzε(n)} from each A-scan first. Under the reasonable assumption that all the Jones matrices in Eq.
3 are unit-determinant unitary (i.e., there is negligible polarization-dependent loss in the fibers and couplers)
$J^{\prime \prime} _{\rm{surface}} $
$J surface \u2032\u2032$ is normalized by using the matrix determinant which is expressed as
Display Formula5 \documentclass[12pt]{minimal}\begin{document}\begin{equation}
J'''_{\rm{surface}} = \frac{{J''_{\rm{surface}} }}{{\sqrt {\det ( {J''_{\rm{surface}} })} }},
\end{equation}\end{document}
$J surface \u2032\u2032\u2032=J surface \u2032\u2032det(J surface \u2032\u2032),$ where
$J^{\prime \prime} _{\rm{surface}} $
$J surface \u2032\u2032$ can be directly normalized to have the unit determinant since
J_{in} and
J_{out} can be treated as unitary matrices if the optical system represented by them is nondiattenuating.
^{3} The global phases of
$J^{\prime \prime \prime} _{\rm{surface}} $
$J surface \u2032\u2032\u2032$ are then normalized by the phases of their first column and first row elements to remove the sign ambiguity introduced in Eq.
5, because the root of determinant would decrease a range of the phase to be ±π/2.
$J^{\prime \prime \prime} _{\rm{surface}} $
$J surface \u2032\u2032\u2032$ can then be directly averaged across A-scans to provide a high accuracy estimate of
J_{out}J_{in}.