From b75d9d8f2ba798fbbadc975a789cf2615b743328 Mon Sep 17 00:00:00 2001
From: Blaise Thompson <blaise@untzag.com>
Date: Sun, 15 Apr 2018 00:16:26 -0500
Subject: 2018-04-15 00:16

---
 opa_phase/chapter.tex | 64 +++++++++++++++++++++++++++++++++------------------
 1 file changed, 41 insertions(+), 23 deletions(-)

(limited to 'opa_phase/chapter.tex')

diff --git a/opa_phase/chapter.tex b/opa_phase/chapter.tex
index 774ab45..ae1a9ea 100644
--- a/opa_phase/chapter.tex
+++ b/opa_phase/chapter.tex
@@ -1,35 +1,53 @@
 \chapter{Abandon the random phase approximation} \label{cha:rpa}
 
-Recently we've made some measurements that seem to imply phase stability between the fs OPAs.  %
-We've typically assumed that the OPAs have random phase on every shot, making coherent heterodyne
-processes average to zero.  %
-These measurements show that this is a very bad assumption.  %
-
-I've taken the interferogram of OPA1 vs OPA2, Clearly the OPAs remain phase locked for many shots.  %
-In 'over time' I show the spectral phase pattern (D = 500 fs) for 1000 single shot acquisitions
-over 430 seconds in lab time.  %
-The phase does drift, but it is certainly not quickly randomized.  %
+\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 simultaniously 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.  %
 
-I have more data showing:
-How the spectral phase changes over the course of hours.
-How the phase evolves as we scan the OPAs against each-other in color.
-The reproducibility of phase as the OPA motors move away and then return to a given color.
+\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.  %
 
-I'll work this data up and send out another email with many more details and thoughts once I have
-time.  %
-This quick note is just to let the group know that we must abandon the 'random phase' assumption
-when thinking about what heterodyne processes can happen as coherent artifacts.  %
+This result forces us to reconsider our assumptions when identifying potential sources of artifact
+in our measurements.  %
 
 \begin{figure}
-  \includegraphics[width=\textwidth]{"opa_phase/cross interference"}
-  \caption[CAPTION TODO]{
-    CAPTION TODO
+  \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/430 seconds"}
-  \caption[CAPTION TODO]{
-    CAPTION TODO
+  \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}
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