2021 27 August BACHELOR OF SCIENCE ASTROPHYSICS

Bruce Schumm Lab of Adaptive Optics Director Philip Hinz Santa Cruz

# Analyzing Variation in Phase Delays Across Phase Plates With a Quadrature
Polarization Interferometer

Cesar Laguna

###### Abstract

In order for telescopes to obtain good and precise images they need to see
through atmospheric turbulence. To accomplish this and compensate for
atmospheric turbulence we use Adaptive Optics technologies. In this thesis we
analyze the variations in phase delays across phase plates which simulate
atmospheric turbulence in order to characterize them and determine how well
these phase plates reproduce the phase delay variation of the atmosphere. This
experiment was conducted using the Quadrature Polarization Interferometer
(QPI) testbed in the Lab of Adaptive Optics (LAO) at the University of
California Santa Cruz (UCSC). Using the QPI lab setup allowed us to be able to
develop and refine a final algorithm for determining the phase delay of the
phase plates. The characterization of the phase plates was accomplished by
using the interference patterns between a test and reference path and
measuring their pathlength variations. This was achieved by calculating the
intensity of the two paths of the Helium-Neon (HeNe) laser, modifying those
values in order to obtain phase values between the beams, and from the phase
values we are able to determine the path length variations, or phase delays,
of the phase plates. This allows us to determine Fried’s parameter, $r_{o}$,
by analyzing any given separation on the phase plate and determining the path
length variations between those separations.

###### Contents

  1. 0 Introduction
    1. 1 Background: Adaptive Optics in Astronomy
    2. 2 Testbed
  2. 1 Methodology
    1. 1 Quadrature Polarization Interferometer (QPI) Intro
    2. 2 Assembly
      1. 1 Laser and Non-Polarizing Beam Splitters
      2. 2 Wave Plates
      3. 3 Focusing Lens’ and the Final Alignment
  3. 2 Data
    1. 1 Image Registration and Calibration
    2. 2 Data Acquisition Process
      1. 1 Defining a Data Set
      2. 2 Taking the Images
  4. 3 Results
    1. 1 Image Analysis
      1. 1 Image Processing
      2. 2 Calculating $r_{o}$
  5. 4 Conclusion
    1. 1 Summary
    2. 2 Future Direction

###### List of Figures

  1. 1 Quadrature Polarization Interferometer Layout
  2. 2 Wave Plates Operation: The figure above demonstrates the manner in which the wave plates create the trig functions sin and cos and the phase difference between them.
  3. 1 1951 USAF Resolution Test Chart [Taken from Wikipedia ”1951 USAF resolution test chart”]
  4. 2 USAF Resolution Chart Comparison. These images show the two camera images added together. Left image the test charts are not aligned therefore the cameras have to be adjusted; the right images shows the test charts correctly aligned.
  5. 3 Saturation Cuts. Left images has peaks which are cut off because the pixels are heavily saturated whereas the right images the beam intensity has been adjusted and the pixels are no longer saturated.
  6. 4 Width of 1 line for USAF Chart [Taken from Wikipedia ”1951 USAF resolution test chart”]
  7. 5 Complete Data Set. A data set consists of 6 total images; 3 image sets. One row in the figure above is considered an image set. All images are required in order to be able to determine $r_{o}$.
  8. 1 Max Flux for Both Cameras. Normalization of the raw images for transmitted and reflected cameras.
  9. 2 Max Intensity for Both Cameras. We have to adjust the normalization in order to obtain values that will work for the trig functions Sine and Cosine.
  10. 3 Phase Estimate for Phase Plate
  11. 4 Unwrapped Phase Estimate for Phase Plate
  12. 5 Comparison of $r_{o}$ Calculations. The left method calculates $r_{o}$ for multiple random pixels and averages them out in order to obtain a good averaged $r_{o}$ for the phase plate which was $r_{o}$=1250microns. The right image calculates $r_{o}$ once by measuring the entire phase plate however the final $r_{o}$ tends to be inaccurate, as for the right image it gives $r_{o}$ = 1600microns
  13. 6 Data Set for Smaller $r_{o}$
  14. 7 Comparison of $r_{o}$ Calculations: The method shown in the left image calculates $r_{o}$ to be 650 microns while the method in the right image calculates $r_{o}$ to be 1380 microns.

###### List of Tables

  1. 1 Components Required for QPI

To my parents,

Cesar and Sarai Laguna

###### Acknowledgements.

Thank you to my thesis advisor, family and friends for all their support and
help throughout not only this thesis but also my academic career.

##  Chapter 0 Introduction

###  1 Background: Adaptive Optics in Astronomy

Atmospheric turbulence or wavefront distortion is produced by light passing
through Earth’s atmosphere which has turbulent air flow thus the name
atmospheric turbulence. This distortion is caused by a varying refractive
index which in turn is brought about by temperature gradient, the direction
and rate at which temperature changes in a given area. The difference in
temperature creates different delays in the light passing through it which
results in atmospheric turbulence. This atmospheric turbulence, “seeing”,
affects the focus or sharpness of an image; the “twinkling” of stars is a
byproduct of these distortions. Adaptive Optics (AO) is a technology that is
used to compensate for atmospheric turbulence. AO removes wavefront
distortions by using “counter wavefront distortions” in the form of a
deformable mirror which makes corrections to the wavefront distortions in real
time and is also able to “extends diffraction-limited imaging to much fainter
and complex objects” (Beckers,14).

The resolution of optical instruments is primarily limited by the diffraction
limit, the resolution power of an optical imaging system; even a “perfect”
lens will be limited by its diffraction limit. Assuming no disturbances
(turbulence, noise, etc) or imperfections in the mirror the diffraction limit
is given by:

$\theta=1.22*\frac{\lambda}{D}$ (1)

where $\theta$ is the resolvable angle, $\lambda$ is the wavelength of the
light, and D is the diameter of the lens. However, due to the effect that
atmospheric turbulence produces, D is now replaced with $r_{o}$ or Fried’s
parameter. Fried’s parameter, $r_{o}$, is the measurement of the strength of
optical transmission through turbulent media due to the differences in
temperature gradient in the atmosphere; inhomogeneities in the atmospheres
refractive index caused by temperature gradients. More precisely it is defined
as the size on the wavefront, usually the diameter of a circular
aperture/area, where the root-mean-square variation (phase) is 1 radian. By
replacing D with $r_{o}$, the angular resolution is now limited to
approximately:

$\theta=\frac{\lambda}{r_{o}}$ (2)

Since the diameter for most telescopes today is D$>>r_{o}$, AO must be
implemented in order to obtain image resolution results that approach the
diffraction limit. AO allows us to modify telescopes and fine tune them until
we are able to to reach the diffraction limit of a lens and thus be able to
obtain sharp images of the universe without the worry of turbulence.

###  2 Testbed

To begin our testing we want to replicate and simulate atmospheric turbulence,
or $r_{o}$, in the lab. By simulating an atmosphere in the lab we are in a
position to test and refine AO techniques with more ease. The precision of the
simulation relies on how large or small our replicated $r_{o}$ is. In the
Earth’s atmosphere the $r_{o}$ is relatively small because of the tiny
variations in temperature in small areas. When comparing the small
replicated/simulated $r_{o}$ it is not the same “small” $r_{o}$ that is seen
in the atmosphere. In the atmosphere $r_{o}$ is typically 100 - 200mm however
this is with our various large telescopes, such as the Keck telescope which is
10 meters in diameter. When simulating this in the lab environment we are
constructing and using an optical setup which is quite smaller. This means
that when we simulate atmospheric turbulence it will be far more convenient if
the “telescope”, or beam, is only 2mm in size which in turn means that the
$r_{o}$ we want to simulate should range between .5 - 1.5mm in size, keeping
the ratio consistent $r_{o}$ should be 1mm in size.

In order to simulate atmospheric turbulence in the lab we use phase plates.
Phase plates are made from a transparent material, such as glass or plastic,
and have a precise coating of clear acrylic spray paint on them; depending on
how the coating of acrylic spray paint is applied affects how large or small
$r_{o}$ will be [1]. In our case we want $r_{o}$ to be small, relative to our
optical setup, in order to properly and accurately simulate atmospheric
turbulence. Since the application of the acrylic paint effects $r_{o}$ we are
able to produce new phase plates of various $r_{o}$’s until we have the
technique to create proper atmospheric turbulence; an $r_{o}$ which follows a
5/3 power law. After the acrylic is applied to the phase plates we then
characterize them and determine their $r_{o}$. This is generally accomplished,
in simple terms, by first measuring the interference patterns, obtaining a
surface, and measure the root mean square (RMS) variation of the surface.

To acquire our data we use an interferometer, which after carrying out various
steps which will be explained later, we can measure the phase and determine
the $r_{o}$ of any phase plate of our choosing. An interferometer uses a
single light source that is split into two beams and when polarized creates an
interference pattern. By measuring an interference pattern we can recreate the
surface of phase plates or a mirror and determined on a technique that measure
the RMS variation in order to calculate a $r_{o}$. Using an interferometer
allows us to improve the process of determining phases’ and $r_{o}$s’ with
greater accuracy since we are able to do everything in a controlled
environment and are able to repeat the process with various phase plates
multiple times.

In order to confirm that our phase plates accurately simulate atmospheric
turbulence we verify $r_{o}$ by making sure it fits Kolmogorov’s power law.
Kolmogorov’s power law is a power law of 5/6, “a variance in temperature
between two points a distance x apart” [2]. When testing phase plates they may
either not follow this power law or they may have an $r_{o}$ that is either
far too high or far too low. Because of this we make new phase plates in order
to be able to obtain the desired $r_{o}$ that accurately simulates atmospheric
turbulence.

##  Chapter 1 Methodology

###  1 Quadrature Polarization Interferometer (QPI) Intro

An interferometer is an instrument that uses one single light source split
into two to create an interference pattern which can then be analyzed. The
interferometer used in this experiment is based on the Mach-Zehnder
Interferometer. The Mach-Zehnder Interferometer is an instrument used to
measure and determine the phase shift between two collimated beams of light
that are derived from one single light source [3]. The operation of a Mach-
Zehnder interferometer goes as follows: the single light source is properly
focused and collimated using a lens then using a beam splitter it is split
into the two beams, the test and reference beams. From here the two separate
beams are reflected using mirrors towards a second beam splitter where they
are then registered by a single detector. The Mach-Zehnder interferometer
allows for the light to travel through an equal optical length in both the
test and reference beams which then results in constructive interference.
Since the Mach-Zehnder interferometer is also a highly adaptable/configurable
instrument, it is a great interferometer to derive our interferometer from.

Building off of a Mach-Zehnder interferometer, a phase shifting interferometry
allows us to ”measure the phase variation across the beam and then convert
them into height variations across the sample” [4]. This phase difference is
determined by measuring the intensity of the interference patterns from both
cameras which we know to have a 90° phase change between them. This 90° phase
change between them typically results from moving the mirror in the reference
beam back and forth to change the path length which results in a phase change.
A phase shifting interferometer uses the intensities across the interference
frames, converts this to the phase variations, and converts this to the final
step of height variations, optical path length, in order to recreate surface
of the imaged object which in turn allows us to calculate the $r_{o}$ of said
object.

For this thesis work we use a Quadrature Polarization Interferometer (QPI). A
Quadrature Polarization Interferometer lets us measure the phase delay in
phase plates which we can then use to determine the $r_{o}$ of the phase
plate. QPI makes this possible because it is a type of phase shifting
interferometer, like described above, which allows us to get two interference
patterns simultaneously, which is phase shifted by 90°. The QPI setup used in
this thesis is shown in the following figure:

Figure 1: Quadrature Polarization Interferometer Layout

QPI Components
---
Number | Component
1 | HeNe Laser Source
2 | Optic Fiber Cable
3 | Cable to Laser Mount
4 | Thorlabs Lens f=400mm
5 | Linear Polarizer
6 | 2” Non-Polarized Beam Splitter
7 | Mirror
8 | 2” $\frac{\lambda}{2}$ Waveplate
9 | 2” $\frac{\lambda}{4}$ Waveplate
10 | Phase Plate
11 | 11” Polarized Beam Splitter
12 | Thorlabs Lens f=100mm
13 | HT-5000-S Emergent Camera
14 | Motorized Flip Mount

Table 1: Components Required for QPI

By using this method we can test more than just a phase plate; by removing the
mirror in part 7 of Figure 1 and replacing it with a small deformable mirror
we are able to test the deformable mirror after the phase plates have been
characterized. Although I will not be testing a small deformable mirror, once
QPI is complete and I have shown that it provides accurate/expected results
for $r_{o}$ this is a possibility for the future.

As described in the Mach-Zehnder interferometer description above, a single
light source is split into two beams, the test and reference beams. By
introducing a quarter-wave plate into the reference beam or by changing the
optical length of the reference beam, this allows for the measurement of phase
difference or phase delay. This is the basis for how we will obtain the phase
delays using QPI, by introducing a quarter-wave plate into the reference beam.
QPI is a bit different as it introduces two more non-polarized beam splitters,
a polarized beam splitter, two wave plates, and relocates both mirrors. The
introduction of all the pieces allows us to better create constructive
interference and to better detect phase delay.

###  2 Assembly

####  1 Laser and Non-Polarizing Beam Splitters

Using the same principles of the Mach-Zehnder interferometer but now adopted
to QPI, our goal is to be able to characterize phase plates. In order to
characterize phase plates we first need to derive the phase by using two
polarization’s that travel the same path length but it differring beams, a
test beam and a reference beam. In QPI we do this with a HeNe laser (633 nm
wavelength) however before introducing this HeNe laser we first need to do our
alignments, starting with the laser alignment. The laser alignment is done
with a handheld laser pointer; handheld laser pointers allow us to make better
alignments as the beam is thin and focused which lets us make finer
adjustments in our upcoming procedure. Using the handheld laser we first
“mate” the laser to the work bench meaning we make sure the laser is parallel
across the entire length of the work bench. Since the length of the work bench
is far greater than the length the laser has to travel in the setup, this
means that the laser is parallel and straight for the length of the QPI
system.

“Mating” the laser to the bench is important to the setup, especially when we
introduce the non-polarized beam splitters. The non-polarized beam splitters,
as their name suggests, split the beam into two, with one going straight
through and another splitting perpendicular from the first. The laser “mating”
makes the introduction of the non-polarized beam splitters easier as we no
longer have to worry about the laser alignment and can primarily focus on the
placement of the beam splitters; placement meaning positioning, tip and tilt.
Once all four of our non-polarized beam splitters have been properly placed,
along with both mirrors which are placed in the same manner, we can then
replace our handheld laser pointer with a HeNe laser in order to obtain a
proper source rather than a point source.

We place our HeNe laser source at the same height at which we had our handheld
laser pointer in such a manner that the center of the HeNe laser is also the
center of the non-polarized beam splitters, as it was for the handheld laser.
In order to obtain a proper “beam” of light we first need to place a lens into
the path of our HeNe laser light source; the lens we chose has a focal length
of 400 millimeters (mm). Placing the lens at 400 mm away from the HeNe laser
source, we then used a shear plate to determine where exactly the beam is
properly collimated since this provides us with the exact/proper placement for
the lens.

Once we had our main beam aligned and properly collimated we added a polarizer
behind the focus lens that produced a linearly polarized beam which is
parallel to the testbed/bench. Since we have already accurately aligned all
four non-polarized beam splitters we can see how the beam travels through the
setup as of this step. The linearly polarized beam is split into two at the
first beam splitter creating what will be referred to as the test and
reference beams. From there two other beam splitters, along with both mirrors,
redirect the beams and allow for them to rejoin and merge together at the
final and fourth beam splitter. After the two beams have been recombined, a
smaller polarizing beam splitter is added to the system, following the same
procedure as for the non-polarized beam splitters, so that the beam can split
once more and separate into the two phases; these separated beams should be at
an angle difference of 90°s providing us with the trig functions cos and sin.
The polarized beam splitter should be aligned during the laser alignment
process with the handheld laser pointer in order to obtain the best alignment
results; if necessary it is possible to introduce it into the system and align
it once the HeNe laser is in place although this is slightly harder to get the
correct alignment but completely possible. I mentioned a phase difference
between the two beams however as of this step in the procedure there is no
phase difference. The phase difference between two beams is achieved by adding
wave plates into the system.

####  2 Wave Plates

A retarder or wave plate is an optic which changes the polarization of the
light that travels through it. In order to obtain the phase difference between
the two beams we introduce two wave plates into the system, a half-wave plate
and a quarter-wave plate into the test beam and reference beam respectively.
By using a half-wave plate and a quarter-wave plate we are able to make one
beam rotate by 45° and make the other beam circularly polarized respectively.
When we input the polarizer at the start of QPI it polarizes the light so that
it is parallel to the table/bench. The half-wave plate is placed in the test
beam and takes this linearly polarized light and rotates it by 45° which gives
us a polarized light that gives us sin and cos. However if we were to look at
our cameras in this step we would see the same interference patterns on both
cameras with no phase difference between the two, this is where the quarter-
wave plate comes into play. Placing the quarter-wave plate in the reference
beam, it takes the linearly polarized light and converts it to circularly
polarized light. Circularly polarized light means that as one trig function is
at its maximum the other is at its minimum, meaning that there is a 90° phase
difference between the trig functions, this is how we get the 90° phase shift
between cos and sin that we see in the cameras. The final step lies in the
polarized beam splitter which separates both trig functions by allowing only
one polarized direction to transmit through it to one camera while the other
polarized direction is reflected towards the other camera.

Figure 2: Wave Plates Operation: The figure above demonstrates the manner in
which the wave plates create the trig functions sin and cos and the phase
difference between them.

To make sure we set up the wave plates correctly we need to use a power meter
as well as a second polarizer. A power meter allows us to determine the
intensity of the beam, which allows us to set the wave plates accurately.

The wave plates are introduced into the system one at a time. We begin by
inserting the half-wave plate, however before the half-wave plate is inserted
into the system, we first want to place a second polarizer on the opposite
side of the beam splitter; opposite with respect to where the half-wave plate
will be placed. Once the second polarizer is set we then place the power meter
directly behind it. Using the power meter, we first find the angle of maximum
flux for the second polarizer. Once this angle is found we then insert the
half-wave plate into the test beam path between in place number 8 of Figure 1,
by doing so the angle of maximum flux has now changed. In order for the half-
wave plate to be aptly adjusted we want there to be a 45° angle change in the
polarizers angle of maximum flux when the half-wave plate is inserted and
absent; meaning if the angle of maximum flux of the polarizer is 25 degrees
when the half-wave plate is missing, it should then be 70 degrees with the
half-wave plate inserted. Since we know the angle of maximum flux for the
polarizer when the half-wave plate is absent, we can do the following: by
changing the angle of the polarizer by 45 degrees we can then focus only on
finding the angle of maximum flux for the half-wave plate. Once this is
complete, the half-wave plate has been adequately placed and we can move on to
the quarter-wave plate.

The adjustment of the quarter-wave plate is done using the same equipment used
for the half-wave plate, a power meter and a polarizer. We start by moving the
power meter and the polarizer to opposite side of the non-polarized beam
splitter from where the quarter-wave plate will be placed. We then rotate the
polarizer to see the expected flux variation and confirm that the light is
linearly polarized. By rotating the polarizer we are able to find two angles,
the minimum flux and maximum flux angle. Once we have these two angles we
insert the quarter-wave plate in the reference beam path in part number 9 of
Figure 1. We then want to find the angle for the quarter-wave plate that
provides no change in flux for the minimum and maximum flux angles we had
previously found for the polarizer. Meaning we want to find an angle for the
quarter-wave plate that when we switch between the max and min angles on the
polarizer there is no variation/change in their flux. After this is complete,
the setup of the wave plates is complete and we remove the second polarizer as
well as the power meter from the setup as they are no longer needed.

####  3 Focusing Lens’ and the Final Alignment

Once the beam has traveled through the system and through the polarized beam
splitter we need to focus the light to image it onto the cameras. In order to
do this we will use lens’ that have a focal length of $\sim$100mm. Placing the
lens’ in positions 12 of Figure 1 we adjust the height of them so that their
center is at the same height as the center of the polarized beam splitter and
place them at an equal distance from the polarized beam splitter. From here we
begin constructing the setup for the cameras. In order to make adjustments
easier, we mount the camera on 3 different movable bases. The 3 bases allow
for us to move the camera small amounts in any direction, up and down,
forwards and backwards, left and right. After mounting the bases to one
another and finally the camera to them, we place them $\sim$100mm away from
the lens’.

From here we need to see the live feed of the cameras to be able to adjust
their positioning properly. Following their setup instructions we connected
them to the computer and were able to see their live feed. Next we place an
iris in part number 10 of Figure 1, where the phase plates will eventually go,
and use a motorized mount to block the beam from traveling in the reference
path. The iris aperture can be any size you wish but we chose a medium size
aperture as it makes the next step slightly easier. Looking at the live feed
of the cameras now we see that they are not focused or properly aligned; they
have different magnifications, different positioning (imagining of light onto
camera), and are unfocused. By using the iris as the reference we use the
bases to move the camera until the iris aperture edge is seen as a sharp edge
instead of a blurry edge on the live feed; this is when the focus is best. We
do this for both cameras and once both cameras have the best focus we can
manually adjust for, we switch over to adjusting for the magnification of the
cameras.

In order to adjust the magnification we need to change how close/far the lens
is to the polarizing beam splitter; however by doing so it messes up the focus
of the cameras. This means that as we correct the magnification we also need
to adjust the positioning of the cameras in order to regain the focus. One of
the best ways to do this is to try and place both lenses equally far away from
the polarized beam splitter, which we did. From here we make slight
adjustments until we are able to get very similar magnification and great
focus on both cameras. After this process is complete, QPI’s setup has
officially been completed and we can now begin testing.

##  Chapter 2 Data

In this section we describe the processes of data acquisition with QPI. This
includes calibrating the system, describing a data set, and a walk-through of
how to take the images for said data set.

###  1 Image Registration and Calibration

In order to begin taking images, or data, we first need to register and check
the pixel scale of each camera. To check calibration we make sure both cameras
are centered the same and their images perfectly overlap with one another.
This is accomplished by using a 1951 USAF resolution test chart:

Figure 1: 1951 USAF Resolution Test Chart [Taken from Wikipedia ”1951 USAF
resolution test chart”]

By placing the test chart where the phase plate should go in part number 10 of
Figure 1 and blocking the reference beam, to avoid creating interference, we
are able to adjust the positioning of the lenses (up/down or left/right) until
both test chart images are aligned over one another. This process is easier to
complete if there is a program that displays the live feed of both cameras on
top of each other so that we can make lens adjustments in real time.

(a)

(b)

Figure 2: USAF Resolution Chart Comparison. These images show the two camera
images added together. Left image the test charts are not aligned therefore
the cameras have to be adjusted; the right images shows the test charts
correctly aligned.

Once the resolution test chart shows us that the images overlap over one
another well enough we move on to our next and final calibration step. We take
an image of the non-plate interference pattern, using either camera, and plot
the x-axis of the image. By doing this we are able to see which, if any, of
the pixels are saturated. Saturation occurs when too much light is being
registered by a pixel and if any pixels are saturated, or if pixels are close
to being saturated, we lower the intensity of the HeNe laser.

(a)

(b)

Figure 3: Saturation Cuts. Left images has peaks which are cut off because the
pixels are heavily saturated whereas the right images the beam intensity has
been adjusted and the pixels are no longer saturated.

From here we need to determine pixel size to mm and we do this by looking at
the cameras we are using in QPI, which are HT-5000-SM Emergent cameras. We
take an image of the resolution test chart, with either camera as long as the
reference beam is blocked, and analyze the saved image. By looking up the
standard pattern and matching our selected group and element we are able to
look that up on a table which displays the width of 1 line:

Using these references and using group 0 element 2 we see that the width of
one of those lines is 445.45 microns; using our image we determine that the
width of that same line is $\sim$30 pixels. With these numbers we calculate
that 1 pixel is 14.8 microns; this is the conversion that will be used when
calculating $r_{o}$.

Figure 4: Width of 1 line for USAF Chart [Taken from Wikipedia ”1951 USAF
resolution test chart”]

###  2 Data Acquisition Process

####  1 Defining a Data Set

With the calibration and pixel conversion now done we can proceed to take the
images needed to determine the $r_{o}$, this set of images we refer to as a
data set. A data set consists of 6 total images, 3 image sets; an image set
being two images for: interference images, test beam images (block the
reference beam), and reference beam images (block the test beam). These images
are the actual measurements we will take where the interference images are the
ones we want and the test beam and reference beam images are needed in order
to scale the flux properly. In total this makes 6 images, 3 images taken with
the transmitted camera and 3 images taken with the reflected camera; this is a
data set.

When going to take a data set it is important to have both cameras synced with
one another so that when an image is saved both cameras capture the image
simultaneously, this is most important when capturing the interference images.
If the cameras are not synced and do not capture the images simultaneously
then because of vibration, the interference lines on the images will not be
100% aligned which may cause a small issue later on when analyzing the data
set. Camera synchronization ensures that we obtain very accurately aligned
images and rules out the effects of vibration and the slight fluctuations in
beam intensity.

####  2 Taking the Images

In order to begin taking a data set we first need to make sure that the
alignment is the same and nothing has been moved, either purposefully or
accidentally. If QPI has been slightly adjusted or accidentally moved, we use
the USAF resolution test chart in order to verify the camera images are
aligned correctly over one another. If using the Graphical User Interface
(GUI) software that the cameras came with it is also good practice to verify
that the pixel format for the cameras are in mono 12 instead of mono 8; if
using a GUI that was built for easier use like ours, this is something that
should be predefined but can be checked and modified if necessary.

After verifying the camera alignment we then proceed to take the data set
images. We place the phase plate that we wish to measure in part number 10 of
Figure 1 and make sure that both the test beam and reference beam blockers are
both up; up meaning the path(s) are not being blocked and down meaning the
path(s) are being blocked. Using the cameras’ GUI which allows us to save the
image from both cameras simultaneously we save the first image set, the
interference pattern images. Now by moving the test beam blocker down we save
and have our next set of images, the reference beam images (test beam blocked
images). Finally we move the test beam blocker up and the reference beam
blocker down and proceed with saving this set of images, the test beam images
(reference beam blocked images). After this set of images is taken we now have
our full data set for the phase plate such as in the following figure:

(a)

(b)

(c)

(d)

(e)

(f)

Figure 5: Complete Data Set. A data set consists of 6 total images; 3 image
sets. One row in the figure above is considered an image set. All images are
required in order to be able to determine $r_{o}$.

##  Chapter 3 Results

###  1 Image Analysis

In this section we walk-through the process of analyzing the images. This
section includes describing how we start from the acquired raw data images to
deriving the optical path delay across a phase plate, and our final method for
calculating $r_{o}$ for said phase plate.

All the image analysis as well as the $r_{o}$ calculations explained in the
following sections were accomplished using Python 3.0.

####  1 Image Processing

Once we have a full data set for a phase plate we can then proceed to analyze
it and determine its $r_{o}$. Since we have already made sure that the images
are aligned using our own custom GUI and the 1951 USAF resolution test chart
we have already confirmed that the two cameras images are aligned. Because of
this there is no image correction that we have to apply through the coding
process which means that we can begin the analysis.

In order to be able to determine $r_{o}$ there are a number of processes we
must first complete. The process begins by taking the raw data set images and
normalizing them in order to calculate the maximum flux for each pixel, a
value that should range between 0-2. Normalizing usually results in values
between 0-1 however we expect a range between 0-2 because 1 here defines the
approximate light expected from one beam path and since we have 2 beam paths
we therefore expect a range from 0-2. We normalize the image by using the
equation:

$I_{Norm}=\frac{I_{Int}}{(I_{ref}+I_{Test})}$ (1)

where $I_{Norm}$ is the normalized image, $I_{Int}$ in the interference image,
$I_{Ref}$ is the reference image, and $I_{Test}$ is the test image.

Figure 1: Max Flux for Both Cameras. Normalization of the raw images for
transmitted and reflected cameras.

Once we have the maximum flux we need to convert this to intensity.
Determining the intensity requires the modification of the normalization
through the following equation:

$I_{Sin}=I_{Norm}-1$ (2)

which should provide an intensity that ranges between -1 and 1. By doing so we
are converting to the expected ranges of the tragicomic functions sin and cos.

Figure 2: Max Intensity for Both Cameras. We have to adjust the normalization
in order to obtain values that will work for the trig functions Sine and
Cosine.

It is important to note that there are two cameras meaning that both these
equations will have to be calculated twice per data set; once for the
transmitted camera and once for the reflected camera.

For example if the transmitted images are used when defining $I_{Sin}$ then
the reflected images should be put through the exact same equations except
should be named $I_{Cos}$ instead of $I_{Sin}$ because of a phase shift. This
phase shift between the two is ideally 90°, thus their names $I_{Sin}$
compared to $I_{Cos}$. This ideal, perfect 90° phase shift is only possible if
the the images in an image set where taken simultaneously, if there was no
vibration in the optics, and if there was no fluctuation in beam intensity; if
these variables cause the phase shift to vary form 90° significantly then this
will lead to errors in the next step. Because of this it is very important to
align QPI to the best of our ability and try to make the cameras take their
images simultaneously.

Once we have the intensities, $I_{Cos}$ and $I_{Sin}$, for both cameras we
then proceed to calculate the phase values. Plugging in both intensities into
the equation:

$Phase=2*\arctan{(\frac{I_{Sin}}{I_{Cos}})}$ (3)

we are able to calculate the phase values which should range between -pi and
pi.

Figure 3: Phase Estimate for Phase Plate

If we had the ideal 90° phase shift between both cameras then if we were to
look at a cut of the plot we would have an asymmetrical sawtooth shaped graph.
The phase shift between the cameras is not a perfect 90° which may be a result
of vibrations in the optics and/or because we do not have perfect sync between
our two cameras; they, at the moment, cannot take images simultaneously.
Although the frame sync difference is already small in order to obtain better
synchronization, and hopefully the ideal 90° phase shift, we want the frame
sync difference to be .5-1ms.

After obtaining the phase values we take those values and unwrap them.
Unwrapping releases the constraint that the phase values had before, from - pi
to pi, and allows for the resulting values to be larger than a difference of
2*pi. Phase unwrapping takes the phase values and unwraps it into a surface;
this unwrapped surface should be the surface of what we imaged, in this case
the surface of the phase plate. The unwrapping algorithm works by looking for
discontinuities and either adding or subtracting 2pi from all values that
follow in order to correct for this discontinuity. In simpler terms the phase
unwrapping is the variation of path length from one area to another hence any
light that travels through more paint [of the phase plate] than air will
travel a longer distance which we will see in the unwrapped image. This is
what the phase unwrapping algorithm determines and based on this it recreates
the surface of the object which the light passed through, in this case the
phase plate. After attempting various python methods of unwrapping we settled
on:

$Unwr\\_Phase=unwrap\\_phase(Phase)$ (4)

where Unwrap_Phase is a module imported from a python package named skimage
[5]. Although this unwrapping algorithm is not perfect, it works remarkably
well and provides far less errors than any other unwrapping method we used.

Figure 4: Unwrapped Phase Estimate for Phase Plate

Once we unwrap the phase and have recreated the surface of the phase plate we
have finished the image processing process and can proceed with determining
the $r_{o}$.

####  2 Calculating $r_{o}$

Calculating an accurate $r_{o}$ via the unwrapped phase is another procedure
in itself. The basis for how we calculate the $r_{o}$ for these phase plates
was shown by [2]:

$\phi_{rms}=0.162(D/r_{o})^{5/6}\qquad{[waves]}$ (5)

where $\phi_{rms}$ is the root mean square for phase variation, D is the
diameter of the aperture, and$r_{o}$ is atmospheric turbulence. However the
equation above has the rms phase value in units of waves whereas we have it in
units of radians thus we multiply the equation above by 2pi radians and get:

$\phi_{rms}=0.162(D/r_{o})^{5/6}(2\pi)\qquad{[radians]}$ (6)

From here we take the equation above and rearranging to solve for $r_{o}$ we
obtain:

$r_{o}=\frac{D}{(\phi_{rms}/(0.162*2\pi))^{6/5}}\qquad{[cm]}$ (7)

Since our $\phi_{rms}$ values are in radians and so is 2 pi, the radians
cancel leaving us with units of length which is expected. An important note
about the equation above is that it assumes that all the phase plates we
test/measure follow Kolmogorov’s 5/6 power law. Once we have our equation we
can begin calculating for $r_{o}$. In order to calculate $r_{o}$ we need to
determine both the unknowns in the equation above. As mentioned previously
$\phi_{rms}$ is the root mean square for phase variation meaning that it is
the RMS for an aperture of [unwrapped] phase values; the diameter of this
aperture is D.

The process for calculating $\phi_{rms}$ goes as follows: choose a pixel and
select all the pixels around it within an aperture of diameter D. The process
continues by taking all those pixels and taking the standard deviation of all
their unwrapped phase values, this is $\phi_{rms}$, however this is not the
correct $\phi_{rms}$ yet. Our HeNe laser is at a wavelength of 633 nm however
$r_{o}$ is generally calculated at a wavelength of 550 nm meaning we must take
our $\phi_{rms}$ and multiply it by (633/550). From here we now have the
correct $\phi_{rms}$ value.

Once we have $\phi_{rms}$ we take it and the diameter size D we used to
calculate it and input both of these values into equation 7. In doing so we
obtain the $r_{o}$ for that specific spot, for that aperture size. Now that we
can calculate the $r_{o}$ for a specific aperture we need to decide how we
will calculate $r_{o}$ for the entirety of the plate. In order to get a good
sense of what $r_{o}$ is across the entirety of the plate we decided to to
select multiple random pixels across the phase plate, choose a small aperture
size, calculate their individual $r_{o}$’s and average them all in order to
get one final averaged $r_{o}$ which will represent the $r_{o}$ of the entire
phase plate. We chose this method because by averaging numerous $r_{o}$’s of
all the same aperture but different locations on the phase plate we calculate
a more refined and averaged $r_{o}$ than if we were to calculate one $r_{o}$
that has an aperture size of the entire, if not most, of the phase plate.

(a)

(b)

Figure 5: Comparison of $r_{o}$ Calculations. The left method calculates
$r_{o}$ for multiple random pixels and averages them out in order to obtain a
good averaged $r_{o}$ for the phase plate which was $r_{o}$=1250microns. The
right image calculates $r_{o}$ once by measuring the entire phase plate
however the final $r_{o}$ tends to be inaccurate, as for the right image it
gives $r_{o}$ = 1600microns

We have explained how to calculate the $r_{0}$ of any phase plate, all images
above were examples of large $r_{o}$ and here is an example of a smaller
$r_{o}$:

Figure 6: Data Set for Smaller $r_{o}$

(a)

(b)

Figure 7: Comparison of $r_{o}$ Calculations: The method shown in the left
image calculates $r_{o}$ to be 650 microns while the method in the right image
calculates $r_{o}$ to be 1380 microns.

##  Chapter 4 Conclusion

###  1 Summary

This thesis work required setting up a Quadrature Polarization Interferometer
which is located in the Lab of Adaptive Optics at the University of Santa
Cruz. This thesis was the product of wanting to analyze variations in phase
delays across phase plates that simulate atmospheric turbulence in order to
demonstrate that we can characterize phase plates accurately and consistently.
The characterization of phase plates was accomplished by using QPI to examine
the interference patterns between two separate paths and measure their
pathlength variations. We were able to refine and create an algorithm that
determines Fried’s parameter, $r_{o}$, of any phase plate we wish to measure
using QPI and python code we developed. We did this by taking the raw images
and converting them to determine their maximum flux, then their intensity,
then their phase value, and finally their unwrapped phase values. The
unwrapped phase values recreated the surface of the phase plate and we were
able to calculate $r_{o}$ by analyzing the path length variations using the
unwrapped phase values.

###  2 Future Direction

Now that QPI is set up, the LAO can now use it to measure phase plates or even
test a small deformable mirror as I mentioned before. Future work will most
likely entail attempting to get an even better alignment for QPI however the
main focus will be the creation of phase plates. A side goal during this
thesis was creating a process that provides a consistent $r_{o}$, i.e. if we
want an $r_{o}$ of .5mm we want a phase plate creation process that
consistently results in an $r_{o}$ of .5mm. Since we have now developed an
algorithm that quickly calculates $r_{o}$ for any phase plates, we now create
and test phase plates quickly, efficiently, and effectively.

## References

  * [1] Rampy, R., Gavel, D., Dillon, D., and Thomas, S., “Production of phase screens for simulation of atmospheric turbulence,” ao 51, 8769 (Dec. 2012).
  * [2] Beckers, J. M., “Adaptive Optics for Astronomy: Principles, Performance, and Applications,” araa 31, 13–62 (Jan. 1993).
  * [3] Zetie, K. P., Adams, S. F., and Tocknell, R. M., “TEACHING PHYSICS: How does a Mach-Zehnder interferometer work?,” Physics Education 35, 46–48 (Jan. 2000).
  * [4] Wyant, J. C., “Dynamic Interferometry,” Optics & Photonics News 14, 36 (Apr. 2003).
  * [5] Herraez, M. A., Burton, D. R., Lalor, M. J., and Gdeisat, M. A., “Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path,” ao 41, 7437–7444 (Dec. 2002).