\chapter{Abandon the random phase approximation} \label{cha:rpa}

\clearpage

Historically, we've assumed that OPAs have random phase on every shot.  %
This makes interference processes quickly average to zero over many shots---we rarely take fewer
than 100 shots per pixel.  %
Here I demonstrate that this assumption is very poor, at least for the femtosecond OPAs.  %

In these experiments, I simply send OPA1 and OPA2 simultaneously into the array detector.  %
The crucial detail is that the beams are exactly collinear---overlaped in a beamsplitter.  %
I then scan delay between them while collecting single shot spectra using the array detector.  %

\autoref{rpa:fig:delay} shows the results of these experiments for OPA2 vs itself
(``auto-interference'')  and vs OPA1 (``cross interference'').  %
At zero delay all colors arise simultaniously, so there are no modulations along the array axis
(vertical).  %
As I scan further from zero modulations set in as each wavelength within the pulse has a different
period in delay space.  %
It is crucial to remember that the monochromator acts like a stretcher, so we see interference
between the two pulses even when separated by 400 fs.  %

The fringe pattern is expected in the case of auto-interference, but it is also quite stable in
cross-interference. In the next experiment, I explore just how stable the cross-phase is.

\autoref{rpa:fig:time} shows the same single-shot spectrum taken 1000 times at a fixed delay of 500
fs.  %
The phase does drift, but it is certainly not quickly randomized.  %
In fact, the period shifts by 180 degrees in roughly one minute---much much longer than any single
pixel that we have taken.  %

This result forces us to reconsider our assumptions when identifying potential sources of artifact
in our measurements.  %

\begin{figure}
  \includegraphics[width=\textwidth]{"opa_phase/auto_cross_interference"}
  \caption[Auto-interference vs cross-interference.]{
    Interference between OPA outputs as function of relative arrival time.
    In the left hand plot, OPA2 interferes with itself.
    In the right hand plot, OPA1 interferes with OPA2.
    Signal is intensity level.
  }
  \label{rpa:fig:delay}
\end{figure}

\begin{figure}
  \includegraphics[width=\textwidth]{"opa_phase/time_interference"}
  \caption[Cross interference over 100 seconds.]{
    Cross interference at fixed delay of 500 fs.
    1000 single-shot acquisitions over a period of 100 seconds in lab time.
  }
  \label{rpa:fig:time}
\end{figure}
