MEASURING A NOVEL

OPTICAL SPRING EFFECT

by

Baylee Danz

A senior thesis submitted to the faculty of

Brigham Young University - Idaho

in partial fulfillment of the requirements for the degree of

Bachelor of Science

Department of Physics

Brigham Young University - Idaho

April 2018

Copyright c© 2018 Baylee Danz

All Rights Reserved

BRIGHAM YOUNG UNIVERSITY - IDAHO

DEPARTMENT APPROVAL

of a senior thesis submitted by

Baylee Danz

This thesis has been reviewed by the research advisor, research coordinator,and department chair and has been found to be satisfactory.

Date Richard Hatt, Advisor

Date Evan Hansen, Comittee Member

Date Jon Johnson, Comittee Member

Date Stephen McNeil, Department Chair

Date R. Todd Lines, Senior Thesis Coordinator

ABSTRACT

MEASURING A NOVEL

OPTICAL SPRING EFFECT

Baylee Danz

Department of Physics

Bachelor of Science

Current gravitational wave detectors employ extremely high laser power levels

in order to reduce the counting error of photons, a type of quantum noise

called shot noise, that arises from quantum mechanics. As a consequence of

this high power, radiation pressure creates a significant opto-mechanical cou-

pling between the laser field and the mechanical motion of the mirrors (also

known as test masses). This optical spring effect can in principle be used to

reduce the quantum noise below the Standard Quantum Limit (SQL). In this

experiment, we aim to learn more about the optical spring effect to broaden

our knowledge of its behavior and possible applications in future gravitational

wave detector advancements. This research focuses specifically on measuring

the optical spring effect in a configuration without an optical cavity. A de-

tailed description of the experimental process of assembling the configuration

is described.

ACKNOWLEDGMENTS

I would like to thank my family for supporting me through all of the chal-

lenges I have faced and for their unyielding faith in me even through my

doubts.

I would like to thank my mentor at Louisiana State University, Dr. Thomas

Corbitt, for accepting me as a research student for this project. I would also

like to thank Jonathan Cripe at Louisiana State University for all the effort

and time he put into helping me, as well as my research group from that REU. I

am incredibly grateful to the National Science Foundation for supporting this

work through the REU Site in Physics and Astronomy (NSF grant number

1560212) at Louisiana State University.

A special thanks to my professors at Brigham Young University Idaho who

have encouraged, guided, and shaped my undergraduate studies; I could not

have succeeded without your help. Thank you to my advisor Brother Hatt, my

other committee members, Brother Hansen, Brother Johnson, Brother McNeil,

and my senior thesis coordinator Brother Lines.

Thank you to all the people who made this possible. I couldn’t have done

it alone.

Contents

Table of Contents xi

List of Figures xiii

1 Introduction 11.1 Gravitational Waves and LIGO . . . . . . . . . . . . . . . . . . . . . 11.2 Noise and the Standard Quantum Limit . . . . . . . . . . . . . . . . 41.3 The Optical Spring Effect . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Design 92.1 Basic Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Experiment 153.1 Configuration Distances and Optics . . . . . . . . . . . . . . . . . . . 153.2 Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3 Locking the Interferometer . . . . . . . . . . . . . . . . . . . . . . . . 31

4 Results 33

5 Conclusion 35

Bibliography 36

A MatLab Code to Optimize Optics’ Positions 41

xi

List of Figures

1.1 Basic Michelson interferometer used in LIGO. . . . . . . . . . . . . . 31.2 Constructive and destructive interference. . . . . . . . . . . . . . . . 41.3 LIGO quantum noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1 Version of a Michelson-Sagnac Interferometer that was used as thebasic configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Photograph of a cantilever chip. . . . . . . . . . . . . . . . . . . . . . 112.3 Diagram and photo of the structure of one cantilever mirror. . . . . . 122.4 Representation of the optical fields incident on and resulting from the

cantilever. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1 Basic Mach-Zehnder configuration. . . . . . . . . . . . . . . . . . . . 163.2 Configuration schematic. . . . . . . . . . . . . . . . . . . . . . . . . . 183.3 Photograph of the configuration built as seen from above. . . . . . . . 213.4 Example of a camera image shown from a port at minimum power. . 243.5 Photodetector signals from Port A and the dark port at equivalent

time steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.1 Gain vs frequency and phase vs frequency graphs at 50 mW, 100 mW,200 mW, and 360 mW. . . . . . . . . . . . . . . . . . . . . . . . . . . 33

xiii

Chapter 1

Introduction

The detection of gravitational waves has sparked numerous research projects focused

on improving gravitational wave detection technology. This technology is currently

limited by the standard quantum limit, but efforts are being made to develop im-

provements to minimize the quantum noise. This chapter will focus on the reasons

behind investigating the optical spring effect.

1.1 Gravitational Waves and LIGO

The Laser Interferometer Gravitational-Wave Observatory (LIGO) made the first de-

tection in 2015 of a gravitational wave, predicted 100 years prior by Albert Einstein′s

theory of relativity. A gravitational wave is a perturbation in spacetime caused by

a large event involving massive astronomical objects [1]. LIGO has announced six

detected events as of January 2018. Five of the gravitational waves detected have

been the result of black hole mergers. These black holes have ranged from 7 to 36

solar masses. The merging released an enormous amount of energy: the first detected

merger produced waves that radiated 3M�c2. The first frequencies detected were

1

2 Chapter 1 Introduction

35 − 450 Hz. In addition to the five black hole mergers, a gravitational wave was

detected as a result of a binary neutron star system. The most recent announced

detection was a black hole merger in June 2017, announced November 2017 [2–7].

There are currently two LIGO facilities, one in Livingston, Louisiana and the

other in Hanford, Washington. LIGO works closely with the Virgo Collaboration

and European Gravitational Observatory (EGO) in Cascina, Italy, a similar inter-

ferometric facility aimed at detecting gravitational waves [8]. LIGO is based on a

Michelson-Sagnac configuration comprised of a 200-Watt laser split by a beamsplit-

ter along two perpendicular 4-kilometer arms. In order to increase the sensitivity of

the detector, Fabry-Perot cavities with signal recycling mirrors were constructed on

each interferometric arm. These cavities serve two purposes:

• Light is reflected about 280 times within these cavities, effectively increasing the

length of LIGO to magnify any perturbations that are detected. This improves

LIGO sensitivity to gravitational waves.

• Power is magnified within the cavities. Although a 200 W laser is used as the

input, the signal recycling mirrors increase power up to 750 kW. This improves

the resolution of any signals that LIGO detects [9].

1.1 Gravitational Waves and LIGO 3

Figure 1.1 Basic Michelson interferometer used in LIGO. The input laser,seen left, is incident on a beamsplitter and split along two perpendicular 4km arms (indicated by green brackets, up and right). The signal recyclingcavitites are labeled as Fabry Perot cavities, evident by the signal recyclingmirrors facing one another along the arms. The resulting signal from beaminterference is gathered at the dark port, shown by a black dot (bottom) [9].

Through partially reflective mirrors, the high-power light is able to return to the

intersection between the 4 km arms. Without any perturbation, the two beams are

perfectly out of phase and interfere destructively at the dark port. As a gravitational

wave passes over the facility and bends spacetime, the arm lengths are distorted by

a fraction of the size of a proton. This length difference changes the phase of the

beams, and they can interfere constructively at the dark port. These changes in the

4 Chapter 1 Introduction

interference pattern is how LIGO sees gravitational waves.

=

+ +

=

Figure 1.2 Two beams that are in phase interfere constructively, creating apattern with an even taller amplitude (left). If the beams are not in phase andare perfectly out of phase, they interfere destructively, canceling amplitudesout and resulting in no visible signal (right). Adapted from [10].

1.2 Noise and the Standard Quantum Limit

LIGO is limited by various noise sources because of its high sensitivity. Scientists are

aware of these noise sources and the threat they pose by masking gravitational wave

signals, so all known noise sources are taken into account while analyzing signals.

These sources are targeted and removed from the final data using Wiener filtering

and specific noise subtraction analyses [2].

Some of the most prominent noise sources include:

• Seismic vibrations. These limit detection at low frequencies and can include vi-

brations in the Earths crust caused by tectonic movement, human movement,

storms, or ocean waves hitting the shore hundreds of miles away. Human move-

ments that cause noise may include anything from footsteps to nearby traffic.

• Thermal noise. This limits detection at midrange frequencies. Thermal noise

arises from the increased movement of mirrors particles when heated by the

1.2 Noise and the Standard Quantum Limit 5

laser as well as the distortions in materials that the temperature causes. Cooling

systems are in place to minimize this noise.

• Quantum noise. Two types of quantum noise are:

– Radiation pressure noise. This is caused by a quantum uncertainty in the

position of laser photons imparting a radiation pressure force on the mir-

rors. This limits detection at low frequencies.

– Shot noise. This noise is caused by a quantum uncertainty in the position of

photons at output. Shot noise limits detection at high frequencies [11, 12].

Due to the improvements of Advanced LIGO, the main limiting factor of current

and future detection is quantum noise, though steps are still being taken to minimize

the other types of noise.

Due to the Heisenberg Uncertainty Principle in quantum mechanics, the position

and momentum of a particle are impossible to measure simultaneously to high preci-

sion, creating an uncertainty in the prediction of where a laser photon will be. This

principle leads to uncertainty in predicting the behavior of photons interacting with

the test masses. Due to the fluctuations in photons, there is uncertainty in determin-

ing how much and where radiation pressure force will be exerted on the mirrors. The

inability to determine exactly where a mass will be on the quantum level creates the

standard quantum limit (SQL) [12]. An example of a LIGO quantum noise budget

and standard quantum limit is shown in Figure 1.3.

6 Chapter 1 Introduction

Figure 1.3 LIGO quantum noise. Measured noise from LIGO is shown inblue (LHO) and design sensitivity is shown in green (SRD). Calculated quan-tum noise at three different input power levels is also shown (QN). Standardquantum limit (SQL) is shown in red [11].

1.3 The Optical Spring Effect

Projects to improve gravitational wave detecting technology are focusing on various

methods of overcoming the SQL. Two such methods are squeezed light injection and

utilizing an optical spring effect, the latter of which is the focus of this thesis. The

optical spring effect is a result of optomechanical coupling between the test masses

(mirrors) and optical field [12]. The basis of this effect is a linear coupling between

the radiation pressure force that the optical field exerts on the test mass, and the

cavity length. This linear relationship exerts a restoring force on the mirrors with a

calculable spring constant. This effect can be seen when LIGO′s Fabry-Perot signal

1.3 The Optical Spring Effect 7

recycling cavities are detuned from resonance. The cavities are currently controlled

by feedback loops to stay in resonance and prevent the optical spring′s antidamping

force from overwhelming the mechanical damping force [13].

8 Chapter 1 Introduction

Chapter 2

Design

In this chapter we outline the basic schematic of the configuration used in this exper-

iment. A brief discussion of the mathematical theory behind this configuration [13]

will also be discussed.

2.1 Basic Configuration

Previous experiments investigating the optical spring effect have done so using a sig-

nal recycling cavity. However, the optical spring effect can also be created in any

configuration that creates a linear coupling between radiation pressure force and dis-

placement of the mirrors [13]. Our study investigates the optical spring effect in a

configuration without a cavity. We began by using a Mach-Zehnder configuration, a

type of interferometer which consists of two optical fields incident on a beamsplitter

at a 45-degree angle. However, when we proceeded with this configuration we encoun-

tered physical conflicts in the positioning of the beamsplitter mount and the spacing

of the lenses (see Chapter 3 for more details). Because of these conflicts, we decided

to use a variation of the Michelson-Sagnac interferometer, controlling the phase of

9

10 Chapter 2 Design

two laser arms incident on a beamsplitter mirror at 180 degrees. A schematic of the

basic configuration is shown in Figure 2.1.

Laser

Lens 1

Lens 2

Dark Port

Camera & Photodetector

Lens 3

Lens 4

P.T. Mirror B

Cantilever

Mirror A & Piezo

Mirror B

P.T. Mirror A

Port A

Camera & Photodetector

Port B

Camera & Photodetector

Figure 2.1 Version of a Michelson-Sagnac Interferometer that was used asthe basic configuration.

The cantilever is a beamsplitter made of layers of gallium arsenide. It is a partially

transmissive mirror. Similar cantilevers were used by Cole, et al. (2008), and the

gallium arsenide layers are candidates for future LIGO test mass coatings [14]. In

our configuration, a cantilever chip was placed at the intersection between the two

optical fields. This cantilever chip contains rows of cantilevers of varying mechanical

properties and sizes, as shown in the photographs below.

The specific cantilever we chose to utilize in our configuration was highly reflective

mirror about 100 microns wide with a mechanical frequency of about 800 Hz. The

2.1 Basic Configuration 11

Figure 2.2 (Top) Photograph of a cantilever chip. Five rows of individualmirrors are visible. (Bottom) Photograph of three different mirrors in oneof the rows of the cantilever chip. Each have different mechanical properties[15].

12 Chapter 2 Design

structure of a cantilever is shown in the diagram below:

Figure 2.3 Diagram and photo of the structure of one cantilever mirror [16].

Our optical configuration utilizes a laser of 1064 nm with maximum power of

500 mW. Power input is controlled by a waveplate not shown in the configuration

diagram, allowing us to record the optical spring effect at varying power levels. Lenses

were used along the laser beams path to focus the beam diameter to a target width

of 20 microns, or at least within the 100 µm-wide cantilever. The specific distances

and properties of the lenses and mirrors used in this configuration will be discussed

in the next chapter.

After being focused by Lenses 1 and 2, the input beam is incident on a beam-

splitter. This initial beamsplitter creates the two optical fields, beams A and B, that

are incident on opposite sides of the cantilever after being directed by mirrors and

focused through Lenses 3 and 4. The transmitted and reflected beams from the can-

tilever exit the configuration at three ports, as shown in the diagram: Port A, Port

B, and the dark port. Beamsplitters were used to analyze the signal from each of

these ports with a camera and a photodetector. Filters were used (not shown in the

2.2 Mathematical Model 13

diagram) when reducing the amount of power incident on a camera or photodetector

was necessary.

In order to control the phase difference of beams incident on the cantilever neces-

sary to create an optical spring effect, a piezo control device was attached to Mirror

A, which changes the length of beam A at a controlled frequency. A one-way filter

was also placed in front of the power input to ensure that reflected beams would not

damage the laser source (not shown in diagram). Specific alignment and procedures

will be discussed in greater detail in the next chapter.

2.2 Mathematical Model

We first define the individual optical fields interacting with the cantilever. To illus-

trate, consider Figure 2.4. Fields a and d are incident on the mirror from opposite

directions. Fields b and c are the combined transmitted beams through the cantilever

and reflected beams from the cantilever from fields d and a.

a c

b d

Figure 2.4 Representation of the optical fields incident on and resultingfrom the cantilever. Fields a and d are from incident beams a and b of theconfiguration. Fields b and c are a combination of the reflected and partiallytransmitted incident beams. Adapted from [13].

The normalized fields can then be mathematically defined as:

a =

√P0

2

b = ρa+ τd

14 Chapter 2 Design

c = τa− ρd

d =

√P0

2eiφ

φ =Lω0

c

with a − d representing the incident and resulting beams, L and φ representing the

length and phase difference between a and d, and ρ and τ representing the reflectivity

and transmissivity constants of the cantilever with the sum of their squares equal to

1 [13]. The net force on the cantilever is determined through the difference between

incident and resulting powers, or Pnet of the cantilever system:

Pnet = |b|2 − |c|2 = 2ρτP0 cosφ

Fnet = (Pa + Pb − Pc − Pd)/c

where P0 is the initial power input [13]. The radiation pressure force FRP can be

gathered from Fnet: if incident fields a and d are balanced and Pb – Pc is a nonzero

value, the radiation pressure force is Pnet/c. This force displaces the cantilever by

some small amount δL around an equilibrium position, creating a differential force

and spring constant KOS according to the equations:

δFRP =1

c

dPnetdL

δL

KOS = −1

c

dPnetdL

=2

c2ωP0ρτ sinφ

For a more detailed description of the optical spring mathematical model for this

configuration, see [13].

Chapter 3

Experiment

This chapter includes specific lengths and properties of the configuration used. We

will also explain the process followed to build the configuration and align the op-

tics. Various problems were encountered throughout the experimental process; these

problems were described along with their respective solutions.

3.1 Configuration Distances and Optics

As mentioned in Chapter 2, we initially began to build a Mach-Zehnder configuration

illustrated in Figure 3.1. However, we encountered complications due to the 45-degree

angle of incidence on the cantilever. At that angle, the mount holding the cantilever

chip would impede the beam’s access to the cantilever mirror. Because the mount

prevented a clear path, we were forced to reconsider what type of configuration to

build. Instead we used the version of a Michelson-Sagnac interferometer introduced

in Chapter 2 that allowed an unimpeded normal incidence onto the cantilever.

15

16 Chapter 3 Experiment

Figure 3.1 Basic Mach-Zehnder configuration [17]. This configuration wouldhave been used had the cantilever mount not interfered with the beam pathsincident onto the cantilever. The cantilever would have been placed at theintersection of red and blue beams closest to the output signals.

Before we began to physically set up the configuration, we needed a blueprint

to follow. After drawing a rough schematic to determine how many optics would

be necessary, we used a computer program to optimize lens positions. The program

took inputs of path length, number of lenses and a range of position that they could

be placed within, and the final desired beam width which was about 20 microns. It

output the positions along the path length of the lenses that would minimize the

beam to the desired width. This MatLab code is shown in Appendix A.

This program provided a viable basis for the configuration but failed to take into

account impossibilities in converting the computational model to a physical model.

For instance, sometimes the best theoretical placement of lenses was within one cen-

timeter of one another. This would prove impossible to recreate on an optics table,

as the thickness of the lenses and the lens mounts prevented such positions. The

program also didn′t take into account the properties of right triangles, such as the

right triangle in part of our configuration. Thus, various parameters had to be put

3.1 Configuration Distances and Optics 17

into the program to ensure the configuration was physically possible.

An optimized diagram of the configuration with optics’ distances in meters from

the origin is shown in Figure 3.2. The flip mirror marks the beginning of the con-

figuration path at 0.0 m, and the cantilever marks the end of the path at 0.636 m.

Also included in this diagram is the laser source. Although this experiment could be

carried out with the laser source directly input into the configuration, due to limited

table availability we diverted our laser from another configuration. The optics from

the other configuration are unnecessary and are not shown.

18 Chapter 3 Experiment

Figure 3.2 a) Configuration schematic. b) Configuration schematic labeledwith optics’ distances (in meters) from the origin at 0.0.

3.1 Configuration Distances and Optics 19

The following is a list of the optics and their properties that were used, beginning

from the laser and following the optical path to the cantilever:

• The laser has a maximum power of 500 mW. The input power to our configu-

ration was controlled by a half waveplate paired with a polarizing beamsplitter.

We adjusted the waveplate for varying power levels. 50 mW were initially used

while setting up our configuration, then varying power levels were recorded once

the configuration was operational and the optical spring effect was noticeable.

• The isolator was used to ensure that light travels only in one direction. If any

beams reflected off of optics and traveled back along the beam path, the laser

source would be damaged.

• The waveplate, paired with a polarizing beamsplitter, was used to change the

input power level by diverting certain polarization components of the laser.

• The following mirror and flip mirror were only in place to divert the laser from

an adjacent project into our configuration. A flip mirror was used to allow easy

laser access to the other configuration if needed.

• The polarizing beamsplitter (PBS) was used to decrease the power input. Spe-

cific polarization was unnecessary to this project, but the PBS diverted certain

polarizations from the beam path, thus diminishing the power input moving

forward. Beam blocks were placed on either side of the PBS to ensure that the

diverted polarized beams didnt interfere with other optics.

• Lens 1 has a focal length of 0.05 m.

• Lens 2 has a focal length of 0.0254 m.

20 Chapter 3 Experiment

• The following optic is a beamsplitter. All of the following beamsplitters are

50% reflective and 50% transmissive (50/50) according to their labels. However,

they were measured to be closer to 42% reflective and 45% transmissive. This

beamsplitter creates beams A and B which converge at the cantilever but also

allows a signal to exit the configuration at the dark port.

• The beamsplitter at the dark port is a 50/50 beamsplitter. It divides the signal

so that the dark port camera and photodetector can read it.

• Mirror B is highly reflective. It has a reflectivity value R > 99%.

• Mirror A also has a reflectivity value R > 99%. This mirror is smaller and

contains the piezoelectric device. The device is glued to the back of the mirror.

When a voltage is sent through the piezo at a certain frequency, it expands

and contracts, allowing control over the phase of beam A through minute path

length changes.

• Port (P.T.) mirror A is 94% reflective, allowing a signal to reach Port A camera

and photodetector.

• Port mirror B is 94% reflective, allowing a signal to reach Port B camera and

photodetector.

• The beamsplitter at Port A is a 50/50 beamsplitter. It divides the signal so

that the Port A camera and photodetector can read it.

• The beamsplitter at Port B is a 50/50 beamsplitter. It divides the signal so

that the Port B camera and photodetector can read it.

• Lens 3 has a focal length of 0.05 m.

• Lens 4 has a focal length of 0.05 m.

3.2 Alignment 21

• The cantilever is 65% reflective. It is made of gallium arsenide and has a

mechanical resonance of about 800 Hz.

Figure 3.3 Photograph of the configuration built as seen from above. Laserbeam paths are indicated by red lines. Ports are indicated by yellow arrows.

3.2 Alignment

The alignment of the configuration required the most attention of any step in this

research project. It consisted of five parts:

1. Optimizing the positions of optics during the initial setup, excluding the can-

tilever.

2. Aligning the beams as much as possible without the cantilever.

3. Setting up the cantilever.

4. Aligning beams individually onto cantilever.

5. Aligning both beams at the same time onto the cantilever.

Throughout each of these steps, measurements of optics positions and signals from

22 Chapter 3 Experiment

the ports were recorded for reference. Recording the configuration positions at each

step was vital; if a mistake was made in subsequent positioning, we could refer to the

previous positioning and retrace our steps to start anew.

Step 1. Initially placing the optics in positions corresponding to the theoretical

model obtained from the computer program proved more difficult than expected.

Once we decided to redo our configuration, as mentioned in Chapter 2, to the version

of the Michelson-Sagnac interferometer shown in the diagram, we had to take down

our previous work and start anew. Confined by limited table space, the computer

model positions, and human error, we had to retrace our steps a few times to ensure

the optics would be placed correctly.

We used a methodical approach of setting up one optic at a time, beginning closest

to the laser and working towards where the cantilever will be placed. While securing

the optics on the table, the laser was blocked for safety purposes. It was only used

to make sure the optics were placed properly: when placing a lens, we adjusted it

so that the beam passed through its center as much as we could determine with a

detector card. We then used a ruler to make sure that the resulting beam wasn′t

tilted horizontally or vertically by measuring its position at a close and far distance

and adjusting the lens accordingly. The goal was to make sure the beam was parallel

to a plane 3 inches above the table throughout the configuration, and perpendicular

to all of the optics. Placing beamsplitters was similar except we had to check that

the paths of both resulting beams were straight. When we optimized the positions of

the mirrors, we centered the beam on the mirror and used gridlines on the table to

ensure that the reflected beam was straight along the desired path.

Lenses 3 and 4 were initially on stationary lens mounts that provided minute

adjustments. These lenses′ positions are vital in focusing the beam width down to

20 microns before interacting with the cantilever. We were incapable of placing the

3.2 Alignment 23

optics in the exact position that the mathematical model suggested due to imperfect

estimations of distance. We were able to place the optics within about a centimeter

of the mathematically perfect position, but a range of error existed. Because of this,

we needed to be able to adjust the lenses positions along the path, and their current

mounts did not allow for this adjustment. We needed more dynamic lens mounts after

optimizing beam and optics alignment, but we knew that taking down parts of an

already good configuration to rebuild them in better ways would be more beneficial

to the experiment in the long run. The stationary lens mounts were replaced with

mounts that could move the lens along the path length towards or away from the

cantilever. These adjustable mounts proved vital to our eventual success.

Step 2. Cameras and photodetectors were placed at each of the ports. Initially,

due to limited resources, we placed cameras at two ports and a photodetector at Port

B. However, having both a camera and photodetector at each port proved necessary

to compare ports simultaneously, so we acquired both devices for each port. Cameras

proved most useful in the initial setup: two beams were visible at the ports if beams

A and B were not aligned properly. Optimizing various lens and mirror positions

showed the beams converging on the camera′s image.

The initial configuration did not contain the cantilever due to the delicacy of its

structure and placement. Because the cantilever is small and the beam width was not

yet minimized to 20 microns, we didnt want to risk placing the cantilever in the path

of the beam and breaking one of its mirrors with an uncontrolled radiation pressure

force. All correct beam alignments and correct optics placements were necessary

before placing the cantilever into the configuration.

After aligning the optics and beam path, we placed a beam splitter plate in place

of where the cantilever would be to mimic the cantilever′s partially reflective and par-

tially transmissive behavior. This allowed us to use the ports as tools for optimizing

24 Chapter 3 Experiment

alignment; given that the configuration now matched our mathematical model, the

dark port should be dark if the beam paths were equal. We chose a couple different

optics, namely the beamsplitter that creates beams A and B and arm mirror B, to

optimize. Aligning the beams by moving the optics by hand would be impossible due

to the extreme sensitivity of placement. Instead, we used the small knobs on the optic

mounts to adjust minute placements. At this point the camera at the dark port did

not provide the accuracy we needed in aligning beams, so we used the photodetector.

By slowly adjusting the optics knobs, the power signal shown from the photodetector

increased or decreased. Adjusting two or more optics showed greater minimization of

the signal at the dark port. Our mathematical model predicted that Ports A and B

should be in phase and that the dark port should be shifted in phase by π from each

of these ports. If the dark port was at minimum power, then Ports A and B should

be at maximum power. On this assumption we used the photodetectors to estimate

positions that created a maximum signal at Ports A and B and a minimum signal at

the dark port.

Figure 3.4 Example of a camera image shown from a port at minimumpower.

3.2 Alignment 25

Before placing the cantilever in the configuration, we also tested the piezo control

device to make sure that we could see a change in frequency when the device was

turned on. Using a PID controller, we sent a voltage through the device with a certain

frequency and were able to see that frequency through the photodetector signals at

the ports. This confirmed to us that our configuration was aligned and that we were

able to control the length of beam A.

Figure 3.5 Photodetector signals from Port A and the dark port at equiva-lent time steps. The yellow wave is from the dark port and the red wave isfrom Port A. The two ports are π out of phase: when the yellow signal is ata trough, the red signal is at a peak, and vice versa. The port signals showdifferent powers due to filters placed in front of the dark port detector.

Step 3. Placing the cantilever into the setup proved to be a difficult thought

process because the cantilever chip was small and of unique shape. The chip had

to be able to move in all directions to adjust the very small beam to the very small

mirror, and the mirrors within the chip could not be impeded by any type of mount.

In addition to these limitations, the small space that was left to us between lens 3 and

4 mounts would make a tight squeeze for any cantilever mount we chose. One original

idea to mount the chip was to glue it onto a long focal-length lens. This would provide

26 Chapter 3 Experiment

a nearly flat clear surface that would not impede the beams path to the mirrors,

and the lens mounts were small and dynamic. However, due to a limited number

of cantilever chips, we were reluctant to use such a permanent method as gluing.

Instead, we used an optics mount that could be adjusted in all directions. It was a

large mount, however, and more adjustments had to be made to the configuration in

order to fit the mount in the given space. Once these adjustments were made and

the beams were realigned, the mount proved vital in the final adjustments.

Because the chip contained several rows of mirrors, we chose to focus on a larger

highly reflective mirror so that the reflected beam would be easier to see from the port

cameras. A bright reflection alerted us to when the beam was focused on a mirror

or not. This was not the mirror from which we collected optical spring data but it

helped us refine our configuration before focusing on the final mirror.

One of the initial tasks after mounting the cantilever was to determine if the mirror

we selected was at all bent or tilted. If the mirror′s surface was not perpendicular

to the beam path, the reflected beam would veer to the side or vertically after some

distance. Each cantilever mirror had the potential to be tilted a different way and

affect the reflected beam differently, so we had to choose one mirror to focus on while

adjusting the mount. If a reflected beam was not aligned with the transmitted beam,

we adjusted the pitch and yaw of the cantilever chip until the beams aligned. We

would use the port cameras and photodetectors to make sure the beams were visibly

aligned and that we were receiving the reflected beams′ signals in all ports. To ensure

that the signals from the ports were both transmitted and reflected beams, we would

block one beam at a time and observe the power drop shown by the photodetectors.

Adjusting the pitch and yaw of the cantilever chip was a process that would need to

be repeated once the beam width was minimized and focused onto the final mirror.

Step 4. To align the beams onto the cantilever, we focused on one beam at a

3.2 Alignment 27

time, blocking beam A to adjust beam B′s position and blocking beam B to adjust

beam A′s position. When a beam is blocked, transmitted light is seen from one port

and reflected light on the other. When beam A is blocked, transmitted light is seen

at Port A and reflected light is seen at Port B and the dark port. This knowledge

of what should be seen at each port was vital while adjusting the beam widths and

position on the cantilever.

The most important step at this point was to minimize the beam width to less

than the dimensions of the cantilever mirror. We had to interpret what was seen from

the cameras and photodetectors to determine the width of the beam.

If the beam was significantly bigger than the mirror:

• A shadow could be seen from the port cameras that showed the outline of each

mirror or the rows between mirrors.

• Transmitted light would be visible at the port that is supposed to detect only

reflected light. Reflected light would be minimal.

• Depending on how much wider the beam was than the mirror, a circular in-

terference pattern could be seen from transmitted light interfering with itself

around the circular mirror. This type of pattern was seen in the beginning of

our cantilever adjustments.

If the beam was smaller than the mirror:

• Reflected light would be obvious at one port and at the dark port. At first

the reflected beam was not bright enough to detect from port cameras but

was significant enough to detect on photodetectors. Eventually we were able

to clearly see the reflected beam at the respective port once the camera and

cantilever were adjusted.

28 Chapter 3 Experiment

• Transmitted light would not be visible at the respective port when covering one

beam.

Minimizing beam size was the first step. Determining onto which cantilever mirror

a beam was focused was the next step. The only way to determine onto which mirror a

beam was focused was to move the cantilever chip and observe the behavior of reflected

and transmitted beams when scanning across the chip (refer to cantilever image in

Chapter 2). Relying heavily on the images of reflected and transmitted beams that

the port cameras provided us, we used the vertical and horizontal adjustment knobs

on the cantilever mount to determine our placement on the chip.

Turning the horizontal adjustment knob scanned across the chip row. If the beam

was originally focused on a mirror, we would see the reflected beam disappear and

transmitted beam appear when focusing in the space left or right of the mirror.

Continuing to scan left or right would bring the beam into contact with other mirrors

on the same row or the edge of the chip. Other mirrors were evidenced by alternating

reflected and transmitted beams, and the edge of the chip was seen as a large distorted

reflected image and no transmitted signal.

Turning the vertical adjustment knob scanned across multiple rows and the chip

edge between rows. If the beam was originally focused on the mirror, the reflected

beam disappeared and the transmitted beam appeared when focusing above or below

the mirror. The transmitted beam would then disappear and the reflected beam would

appear distorted when the beam focused on the chip edge between mirror rows. A

transmitted beam would reappear when the beam focused on the row above or below

our original mirror. If another cantilever mirror was on the row directly above or

below our original mirror, scanning vertically would show alternating transmitted

and reflected beams depending on whether the beam came into contact with the chip

edge or a mirror.

3.2 Alignment 29

Carefully observing the reflected and transmitted images while moving the can-

tilever chip determined onto which area of the chip the beam was focused. We indi-

vidually determined onto which mirror a beam was focused by repeating this process

while blocking one beam at a time.

The signal from the ports also confirmed that we were focused on a mirror when

we could see the image or the photodetector signal oscillating when the table was

bumped or a loud noise was made. This did not confirm onto which mirror the beam

was focused, but it did confirm that the beam was focused on a mirror. The sensitivity

of the cantilever mirrors to any type of vibration was evidenced by the photodetector

signal oscillating or the camera image vibrating slightly.

Step 5. Once we were fairly certain that the beams were focused on the same

mirror, we unblocked both beams. If both were focused on the same mirror, an

interference pattern would be visible at all the ports. The interference pattern was

a result of the transmitted beam through the cantilever and the reflected beam from

the cantilever, from both beams A and B.

When we initially unblocked both beams, a distinct interference pattern was not

visible. We also saw that only part of the camera image oscillated when a loud noise

was introduced to the cantilever. We eventually determined that despite our previous

efforts, beams A and B were not focused on the same cantilever mirror, and we had

to repeat the previous alignment process.

In time both beams focused on the same mirror and an interference pattern was

visible from all three ports. Despite both beams being focused on the same mirror,

they were not perfectly overlapping. More minor adjustments were made to various

optics in order to minimize the dark port and investigate the interference patterns

shown from the cameras. The final adjustments to minimize the dark port signal and

maximize Port A and B signals consisted of realigning various optics. Optimizing two

30 Chapter 3 Experiment

or more optics at once, lenses and mirrors and beamsplitter included, was necessary

to perfect the signals.

Miscellaneous problems encountered:

• Throughout this process we happened upon various stray beams whose source

we could not initially determine. Due to safety precautions, we set up beam

blocks whenever we found one of these stray beams veering from the configura-

tion. We determined that many of these beams were from reflections of optics

coatings.

• We also detected a strange interference pattern for transmitted light from the

cantilever, but only when one of the beams was blocked. We could not determine

the source but believe it was from an impurity or fingerprint on one of the optics

surfaces.

• Transmitted and reflected light should be inversely correlated, but they weren′t

initially. This was due to improper alignment of beams on one mirror. With

more readjustments, an inverse relationship could be seen between the dark

port and Ports A and B.

• Theoretically, Port A and B signals should be identical. However, they were

not. Some signals were more powerful than others, and some beams seen from

the port cameras had different shapes than the other ports. This could be due

to a number of factors, namely, human errors in the optics placements. This

could also be due the fact that the distances between P.T. Mirror 2 and the

Port A and B detection devices were not equal. The beams were distorted or

faded over some distance, and due to the confinement of our table, we did not

place the cameras and photodetectors at equal distances from their respective

port mirrors.

3.3 Locking the Interferometer 31

• Sometimes signals reaching the ports were stronger or weaker than expected. If

stronger, filters were placed in front of the cameras or photodetectors to prevent

damage to the devices. If weaker, settings on the cameras or photodetectors

needed to be adjusted to detect the low power.

• Optics would misalign easily due to the sensitivity of placement. An accidental

nudge to a lens mount or turning a knob a little bit too far would completely

misalign the beams. Each of the steps in the alignment process needed many

readjustments when beams would misalign.

• The pitch and yaw of the cantilever had to be readjusted when we chose the

final mirror onto which we would focus the beams. The initial mirror onto which

we focused was large and highly reflective; the final mirror was slightly smaller

and also highly reflective. We chose this mirror because we believed that we

would be able to see the optical spring effect more clearly due to its mechanical

properties.

3.3 Locking the Interferometer

The cantilever mirrors were extremely sensitive to noise produced by physical distur-

bances to the optics table or loud noises. When the configuration was aligned onto

the desired mirror, oscillations were visible from the photodetectors when the table

was bumped, when people spoke near the configuration, or when loud devices were

turned on elsewhere in the same room. Putting the configuration in a vacuum would

reduce noise and make the optical spring effect more visible. Due to limited time,

space, and resources, we did not use a vacuum to gather initial optical spring data.

However, future improvements of this project may include using a vacuum chamber

to minimize noise.

32 Chapter 3 Experiment

To begin taking data and recording the dark port signal, the configuration had

to be locked. This meant using the piezo device to counteract oscillations caused

by noise and locking the configuration at a specific frequency. We connected the

piezo and the dark port signal through a feedback loop using a PID controller and

an integrator. When noise produced an oscillation at the dark port photodetector

different than the set frequency, the feedback loop would calculate the difference and

signal to the piezo to counteract the noise. The phase difference between beams A

and B is controlled by setting the lock point on the PID controller. The feedback

loop is also how we measure the optical spring effect. By introducing a signal to the

loop, we measure the response of the system to the controlled signal, which will show

the spring effect.

Chapter 4

Results

The response of the dark port photodetector to an excitation on the piezo was mea-

sured in the frequency domain at varying input power levels controlled by the wave-

plate: 50 mW, 100 mW, 200 mW, and 360 mW. The results are shown in Figure

4.1.

Figure 4.1 Gain vs frequency and phase vs frequency graphs at 50 mW, 100mW, 200 mW, and 360 mW. Mathematical predictions are shown by dashedlines and measured values are shown by solid lines.

33

34 Chapter 4 Results

The amplitude graphs dip at the mechanical resonance around 800 Hz. The optical

spring frequency can be seen at 4.2 kHz. The theoretical model was created by other

members of this research group. The experimental data and theoretical model show

similar patterns; the mathematical prediction is shown on this graph as a dotted line.

The data also showed that measurements taken at higher powers produce a larger

and clearer optical spring effect.

Chapter 5

Conclusion

We effectively measured the optical spring effect at four different power levels by

measuring the response of the system to an introduced signal. Our results matched our

theoretical models. The optical spring effect increases with power input. The piezo

control and feedback loop ensured the stability of the configuration by controlling the

phase difference between interferometer arms and suppressing noise.

Future applications of this research may include:

• Using a signal recycling cavity to increase the laser power in order to see a

greater optical spring effect.

• Increasing power levels to increase noise suppression.

• Putting the experiment in a vacuum to minimize disturbances caused by air

molecules and increase noise suppression.

• Investigate thermal noise of the system.

• Investigate properties of cantilevers, which may be used in future interferometric

technology.

35

36 Chapter 5 Conclusion

• Investigate more thoroughly how the optical spring effect can be used to reduce

noise below the SQL in LIGO advancements.

Although we successfully detected the optical spring effect in this configuration,

this research has not seen its end. Our configuration may continue to be used in

future projects to more clearly understand how the optical spring effect can improve

interferometric gravitational wave detection below the SQL.

Bibliography

[1] LIGO, “What are gravitational waves?.” https://www.ligo.caltech.edu/page/

what-are-gw. Accessed: 2018.

[2] B. P. Abbott et al., “Observation of gravitational waves from a binary black hole

merger,” Phys. Rev. Lett., vol. 116, p. 061102, Feb 2016.

[3] B. P. Abbott et al., “Gw151226: Observation of gravitational waves from a 22-

solar-mass binary black hole coalescence,” Phys. Rev. Lett., vol. 116, p. 241103,

Jun 2016.

[4] B. P. Abbott et al., “Gw170104: Observation of a 50-solar-mass binary black

hole coalescence at redshift 0.2,” Phys. Rev. Lett., vol. 118, p. 221101, Jun 2017.

[5] B. P. Abbott et al., “Gw170608: Observation of a 19 solar-mass binary black hole

coalescence,” The Astrophysical Journal Letters, vol. 851, no. 2, p. L35, 2017.

[6] B. P. Abbott et al., “Gw170814: A three-detector observation of gravitational

waves from a binary black hole coalescence,” Phys. Rev. Lett., vol. 119, p. 141101,

Oct 2017.

[7] B. P. Abbott et al., “Gw170817: Observation of gravitational waves from a binary

neutron star inspiral,” Phys. Rev. Lett., vol. 119, p. 161101, Oct 2017.

37

38 BIBLIOGRAPHY

[8] G. Gonzalez et al., “Memorandum of understanding between VIRGO on one side

and the laser interferometer gravitational wave observatory (LIGO) on the other

side,” LIGO Document M060038-v2, 2014.

[9] LIGO, “Ligo’s interferometer.” https://www.ligo.caltech.edu/page/ligos-ifo. Ac-

cessed: 2018.

[10] LIGO, “What is an interferometer.” https://www.ligo.caltech.edu/page/what-

is-interferometer. Accessed: 2018.

[11] T. Corbitt, Quantum Noise and Radiation Pressure Effects in High Power Op-

tical Interferometers. PhD thesis, Massachusetts Institute of Technology, Cam-

bridge, Massachusetts, 2008.

[12] T. Corbitt and N. Mavalvala, “Quantum noise in gravitational-wave interferom-

eters: Overview and recent developments,” vol. 5111, 06 2003.

[13] J. Cripe, B. Danz, B. Lane, M. Catherine Lorio, J. Falcone, G. Cole, and T. Cor-

bitt, “Observation of an optical spring with a beamsplitter,” 12 2017.

[14] G. Cole, S. Groblacher, K. Gugler, S. Gigan, and M. Aspelmeyer, “Monocrys-

talline al x ga 1 x as heterostructures for high-reflectivity high-q micromechanical

resonators in the megahertz regime,” Applied Physics Letters, vol. 92, p. 261108,

2008.

[15] A. Libson and T. Corbitt, “Ponderomotive squeezing and optomechanics.” Grav-

itational Wave Advanced Detector Workshop, 2015.

[16] R. Singh, G. D. Cole, J. Cripe, and T. Corbitt, “Stable optical trap from a single

optical field utilizing birefringence,” Phys. Rev. Lett., vol. 117, p. 213604, Nov

2016.

BIBLIOGRAPHY 39

[17] QuantumMoxie, “A simple but definitive guide to Mach-Zehnder inter-

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definitive-guide-to-mach-zehnder-interferometers/, May 2013. Accessed: 2018.

40 BIBLIOGRAPHY

Appendix A

MatLab Code to Optimize Optics’

Positions

%\begin {verbatum}

% −−−−−−−−−− Scr ipy us ing a l a mode mode matching u t i l i t i e s −−−−−−−−−−−−

% Fi r s t attempt at mode matching f o r new cav i ty

%Al l va lue s in m

c l o s e a l l

c l e a r c l a s s e s

% c r ea t e a new beam path ob j e c t

CAVITYypath = beamPath ;

% add components to the beam path

CAVITYypath . addComponent ( component . l e n s ( 0 . 05 , . 1 4 , ’ Lens1 ’ ) )

CAVITYypath . addComponent ( component . l e n s (0 .0254 , . 4826 , ’ Lens2 ’ ) )

CAVITYypath . addComponent ( component . l e n s ( 0 . 05 , . 4826 , ’ Lens3 ’ ) )

% l en s syntax : ( f o c a l length , z po s i t i on , s t r i n g l a b e l )

%CAVITYypath . addComponent ( component . l e n s ( . 05 , − . 03 , ’ SL2 ’ ) ) ;

%CAVITYypath . addComponent ( component . l e n s ( . 05 , − . 3 , ’ SLnew ’ ) ) ;

%Ste e r i ng mi r ro r s a f t e r the f i b e r output

CAVITYypath . addComponent ( component . f l a tM i r r o r ( 0 , ’ f l i pM i r r o r ’ ) )

CAVITYypath . addComponent ( component . f l a tM i r r o r ( . 2 5 , ’BS ’ ) )

CAVITYypath . addComponent ( component . f l a tM i r r o r ( . 3 171 , ’ armMirror1 ’ ) )

41

42 Chapter A MatLab Code to Optimize Optics’ Positions

CAVITYypath . addComponent ( component . f l a tM i r r o r ( . 3 848 , ’ beamsCross ’ ) )

CAVITYypath . addComponent ( component . f l a tM i r r o r ( . 5 317 , ’ armMirror2 ’ ) )

CAVITYypath . addComponent ( component . f l a tM i r r o r ( . 6 3 6 , ’ c an t i l e v e r ’ ) )

% f l a t mirror : ( z po s i t i on , s t r i n g l a b e l )

% f l a t mi r ro r s don ’ t change modematching but they l e t you know i f you ’ re going to be putt ing

% s t u f f on top o f eachother .

% The other u s e f u l component part i s curved mirror , to make a curved mirror :

% component . curvedMirror ( rad iu s o f curvature , z po s i t i on , l a b e l )

%CAVITYypath . addComponent ( component . l e n s ( .025 , − . 015 , ’ Glued lens2 ’ ) ) ;

% de f i n e ” input beam” but i t doesn ’ t have to be at the input , i t can be anywhere in the beam path

CAVITYypath . seedWaist (124 . 9 e−6 ,− .24);

% seedWaist syntax : ( wais t width , z p o s i t i o n )

% de f i n e the beam you are t ry ing to match into , the t a r g e t .

CAVITYypath . targetWaist (20 e−6, . 6 3 6 ) ;

% targetWaist syntax : ( wais t width , z p o s i t i o n )

% s l i d e components to opt imize mode over lap .

%CAVITYypath = CAVITYypath . optimizePath ( ’ Col l imator1 ’ , [−1 − . 9 2 ] , . . .

% ’ Col l imator2 ’ , [ − . 88 − . 72 ] , ’ SL1 ’ , [ − . 2 − . 07 ] , ’ SL2 ’ , [ − . 1 − . 03 ] ) ;

%CAVITYypath = CAVITYypath . optimizePath ( ’ Glued lens ’ , [− .07 − .03 ] , ’ Outside1 ’ , [− .95 − . 7 ] , . . .

% ’ Outside2 ’ , [− .85 − .65 ])

CAVITYypath = CAVITYypath . optimizePath ( ’ Lens1 ’ , [ . 1 4 . 2 4 ] , ’ Lens3 ’ , [ . 3 7 . 6 ] , ’ Lens2 ’ , [ . 1 5 . 2 3 ] )

% optimizePath syntax ( component name , [ ( lower bound ) ( upper bound ) ] , another component name , . . . )

% you can choose to opt imize as many components as you would wish . Result i s s e n s i t i v e to i n i t i a l

% cond i t i on s which are de f ined by the z po s i t i o n o f the components be f o r e running opt imize path .

% You can make i t unbounded on e i t h e r or both s i d e s by us ing i n f .

% i f a component i s not named , i t w i l l s tay put .

%% a f t e r y path opt imized

% dup l i c a t e the opt imized beampath in order to work with the components e x c l u s i v e to the x path

CAVITYxpath = CAVITYypath . dup l i c a t e ;

% I f you j u s t did PSLxpath = PSLypath ; you would j u s t have two names f o r the same object ,

43

% changing one would change the other . ( th ink po i n t e r s )

%the x path has a d i f f e r e n t s t a r t i n g wais t than the y path

CAVITYxpath . seedWaist (124 . 9 e−6 ,− .24);

% add a c y l i n d r i c a l l e n s

%CAVITYxpath . addComponent ( component . l e n s ( . 7 5 2 , . 3 , ’CL1 ’ ) ) ;

% opt imize the po s i t i o n o f the c y l i n d r i c a l l e n s

%CAVITYxpath = CAVITYxpath . optimizePath ( ’CL1 ’ , [ . 2 5 . 2 9 ] ) ;

% the targetOver lap method c a l c u l a t e s the mode over lap assuming i t ’ s doing an

% x and y i n t e g r a l , because these are a c t ua l l y the two dimensions o f the same

% beam we square root and mult ip ly them toge the r .

modematch = sq r t (CAVITYxpath . targetOver lap ∗CAVITYypath . targetOver lap ) ;

d i sp ( [ ’ modematching = ’ , num2str (modematch ) ] )

%% plo t

% de f i n e p l o t t i n g domain

zdomain = − . 3 : . 0 0 1 : . 7 ;

f i g u r e (1 )

subplot ( 2 , 1 , 1 )

hold on % r i gh t now a l l the p l o t commands act l i k e the matlab p l o t command and w i l l

% overwr i t e the e x i s t i n g f i g u r e un l e s s you turn hold on

% The p lo t commands a c t ua l l y p l o t s two t race s , the top and bottom of the beam .

% The output o f the p l o t commands r e tu rn s the p l o t handle o f the top so when we make

% the legend we don ’ t have to put a l a b e l on the top and bottom of the beam .

yp lot = CAVITYypath . plotBeamWidth ( zdomain , ’ b ’ ) ;

xp lo t = CAVITYxpath . plotBeamWidth ( zdomain , ’ r ’ ) ;

CAVITYxpath . plotComponents ( zdomain , 0 , ’ r ∗ ’ ) ;

a x i s t i g h t

legend ( [ yp lo t xp lot ] , ’Y’ , ’X’ ) % i f we didn ’ t use handles we would need to do

44 Chapter A MatLab Code to Optimize Optics’ Positions

% legend ( ’Y top ’ , ’Y bottom ’ , ’X top ’ , ’X bottom ’ ) or something .

y l ab e l ( ’Beam width (m) ’ )

g r i d on

hold o f f

subplot ( 2 , 1 , 2 )

hold on

CAVITYypath . plotGouyPhase ( zdomain , ’ wrap ’ , ’ b ’ ) ;

CAVITYxpath . plotGouyPhase ( zdomain , ’ wrap ’ , ’ r ’ ) ;

CAVITYxpath . plotComponents ( zdomain , 0 , ’ r ∗ ’ ) ;

a x i s t i g h t

g r id on

hold o f f

y l ab e l ( ’Gouy Phase ( degree s ) ’ )

x l ab e l ( ’ a x i a l d i s t ance from MOPA aperture (m) ’ )

s e t ( f i n d a l l ( gcf , ’ type ’ , ’ axes ’ ) , ’ f o n t s i z e ’ , 2 0 )

s e t ( f i n d a l l ( gcf , ’ type ’ , ’ text ’ ) , ’ f on tS i z e ’ , 2 0 )

}