  We present a high-precision temporal-spatial phase-demodulation algorithm for
phase-shifting interferometry (PSI) affected by random/systematic
phase-stepping errors. Laser interferometers in standard optical-shops suffer
from several error sources including random phase-shift deviations. Even
calibrated phase-shifters do not achieve floating-point linear accuracy, as
routinely obtained in multimedia video-projectors for fringe-projection
profilometry. In standard optical-shops, calibrated phase-shifting
interferometers suffer from nonlinearities due to vibrations, turbulence, and
environmental fluctuations (temperature, pressure, humidity, air composition)
still under controlled laboratory conditions. These random phase-step errors
(even if they are small), increases the uncertainty of the phase measurement.
This is particularly significant if the wavefront tolerance is tightened to
high precision optics. We show that these phase-step errors precludes
high-precision wavefront measurements because its uncertainty increases to
around lambda/10. We develop an analytical expression based on
optical-wavefront formalism showing that these phase-step nonlinearities appear
as a spurious conjugate signal degrading the desired wavefront. Removing this
spurious conjugate constitutes the central objective of the proposed nonlinear
phase-shifting algorithm (nPSA). Using this nPSI algorithm we demodulate
experimental interferograms subject to small vibrations and phase-shifter
nonlinearities, obtaining a high-precision spurious-free, demodulated
wavefront. We show that our artifact-free, temporal-spatial quadrature
filtering, accomplishes an equivalent wavefront precision as the one obtained
from floating-point linear phase-shifting interferometry.
