coupling light into photonic-crystal waveguides
TRANSCRIPT
Coupling Light into Photonic-Crystal Waveguides
by
Randall Ball
B.S. University of Colorado, 2018
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirements for the degree of
Bachelor of Science
College of Engineering
2018
This thesis entitled:Coupling Light into Photonic-Crystal Waveguides
written by Randall Ballhas been approved for the College of Engineering
Prof. Cindy Regal
Prof. John Cumalat
Prof. Erica Ellingson
Date
The final copy of this thesis has been examined by the signatories, and we find that both thecontent and the form meet acceptable presentation standards of scholarly work in the above
mentioned discipline.
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Ball, Randall (B.S., Engineering Physics)
Coupling Light into Photonic-Crystal Waveguides
Thesis directed by Prof. Cindy Regal
The development of systems with strong atom-photon interactions has many applications
including quantum simulation and quantum computation. The hallmark of these systems is the
coupling of single atoms to single photons. When conducting experiments with these systems, the
measured signal is on the single photon level, making the measuring process extremely important.
In this thesis, we discuss the development of a new scheme for coupling photons into and out of
our experiment, including the requirements for such a system to be practical. We successfully
demonstrate one way coupling of 50% over the range of wavelengths necessary for our experiment.
Finally, we discuss future improvements to the system necessary for implementing it in our current
experiment. My contributions to this project are helping design the optical system for coupling
light into the waveguides, and testing the various optical system designs.
Dedication
To my lab mates who will continue this project after I leave.
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Acknowledgements
First and foremost, I want to thank Professor Cindy Regal for her guidance and her patience
over the past three years. She has looked after me since my freshman year at CU, and I will be
forever grateful.
I also want to thank Professor Jeff Kimble at Caltech for being a great role model, whose
lessons went far beyond physics.
This project wasn’t possible without my outstanding lab-mates at JILA. Tobias Thiele has
been a driving force behind the project, and his passion for physics is inspiring. It’s amazing that
after a long day of working in the lab, he still has the energy invite everyone over to talk about
science. Xingsheng Luan was a huge help over the past year, with many great insights that helped
the project move forward. He also has many cool T-shirts. I’m especially grateful to Ting-Wei
Hsu, who made lab work exciting and fun. He might be the most practical grad student I’ve ever
met. The man can fix anything, plus he wears a tool belt. Finally, Greg Miller has been a great
help on the project, and I wish him luck as he takes it over when I leave.
I also want to thank my friends outside of the lab. Carlos Lopez-Abadia and Konnor
VonEschen have been close friends throughout college, and their support has made this possible.
I’m also grateful to Kamran Shahbaz, whom I’ve known forever, for his encouragement.
Finally, I want to thank my family for everything they’ve given to me. My parents have
always been there for me, and are the main reason for any and all of my success. From a young
age, my dad has given me a passion for math and science, and my mom has taught me hard work
and determination. Also, my brother’s pretty cool.
Contents
Chapter
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Project Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Waveguide Design 4
2.1 Performance Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Photonic Crystal Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 Waveguide Coupler Requirements . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.1 Photonic Crystal Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.2 Waveguide Coupler Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Fiber-Coupling Scheme 14
3.1 Fiber-Coupling Advantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Fiber-Coupling Disadvantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 Fiber-Coupling Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4 Free-Space-Coupling Scheme 19
4.1 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.2 Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
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4.3 Properties of Free-Space-Coupled Waveguides . . . . . . . . . . . . . . . . . . . . . . 20
4.3.1 Chromatic Focal Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.3.2 Waveguide ”Field of View” . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.3.3 Waveguide Spatial Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.4 Geometric Cnstraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.5 Choosing Lenses For Free-Space-Coupling . . . . . . . . . . . . . . . . . . . . . . . . 28
4.5.1 Aspherical Lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.5.2 Achromatic Lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.5.3 Meniscus-Achromat Lens Pair . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.6 Lens Mount . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.7 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5 Conclusion and Outlook 36
Bibliography 37
Figures
Figure
2.1 Diagram of the waveguide with fiber coupling. The photonic crystal is in the middle,
where the atoms couple to the light in the waveguide. The waveguide couplers on
either end couple light into the waveguide from an optical fiber. Image from Su-
Peng’s thesis[19] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Qualitative dispersion relation for a rectangular waveguide. The group velocity of
the light is given by the slope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Dispersion relation for our photonic crystal waveguides. kx is given in units of the
photonic crystal lattice constant a, and represents the absolute amount of phase
gained over one unit cell of the photonic crystal. . . . . . . . . . . . . . . . . . . . . 8
2.4 We can model the photonic-crystal as a stack of dielectric with different indexes of
refraction. Where there is more material, the index of refraction is higher, and where
there is less material, the index of refraction is lower. For any value of k, there are
two possible configurations of the electric field that satisfy both the periodicity and
the symmetry of the system. One solution has higher intensity in the areas of high
index of refraction, and the other solution has higher intensity in the areas of low
index of refraction. The different indexes of refraction cause these two solutions to
have different energies, and therefore different frequencies. . . . . . . . . . . . . . . . 10
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2.5 Dispersion relation for photonic-crystal waveguide on the left. The frequencies where
the group velocity approach zero are placed at two Cesium atomic transitions by
modifying the waveguide design. When light is propagating along the waveguide,
the reflection coefficient at the photonic-crystal is plotted on the the right for an
ideal infinite photonic-crystal and the real finite photonic-crystal. . . . . . . . . . . 11
2.6 Electric field norm for the waveguide mode. The electric field is strongest around
the edges of the waveguide, and falls off exponentially outside the waveguide. The
right plot is the electric field norm along a cut through the center of the waveguide.
Simulation done by XingSheng Luan . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.1 Current fiber-coupled waveguide chip. The chip has 10 waveguides, and fibers are
glued to two of them. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Fiber sits 30 µm away from the waveguide coupler. The thin horizontal line ending
at a T is the waveguide, and the large white cylinder is the optical fiber. Light is
scattering off of the waveguide coupler. . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.1 Diagram of free-space-coupling. The light from an optical fiber is collimated by lens
1, then focused by lens 2 onto the end of the waveguide. We can change the size
of the Gaussian beam at the waveguide coupler by adjusting the ratio of the focal
lengths of lens 1 and lens 2. By using identical lenses, we project the optical fiber
mode onto the waveguide coupler with no magnification, and in theory achieve the
same coupling as fiber coupling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.2 Image of a free-space-coupled waveguide chip. These chips are much smaller than
the fiber-coupled chips so that the light exiting the waveguide doesn’t clip the end
of the chip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
x
4.3 Diagram of the test setup used to measure the reflection signal from the waveguides.
We use a SLED (super-luminescent diode) as a broadband light source. It emits
broadband light in the frequency range 800nm−900nm. The polarization is cleaned
up and set by a PBS and a broadband λ/2 waveplate. Then the light is sent through a
beam splitter and coupled into the waveguide. The light reflected from the waveguide
is measured on the output port of the BS on an optical spectrum analyzer, which
reads out the optical power at each wavelength. . . . . . . . . . . . . . . . . . . . . . 22
4.4 Reflection signal when coupling with f = 11.0mm asphere lenses. The vertical axis is
normalized reflected light, which is the amount of light reflected from the waveguide
divided by the input power. Because this light was coupled into the device, then
coupled out again, the peak coupling efficiency is the square root of 0.25, or 50%. . . 22
4.5 Reflection signal vs lens position for an f = 11 mm aspherical lens. Positive distance
means the lens is translated towards the waveguide. The longer wavelengths focus
slightly farther away from the lens than the shorter wavelengths, causing the skewing
of the reflection signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.6 Normalized reflected power vs input angle. At ±2 degrees away from normal, the
reflection efficiency only drops 4%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.7 Diagram of the setup for imaging the mode of the waveguide. Light is coupled into
the waveguide on the left (not shown), then the light coming out on the right is
imaged onto a camera. The ratio of the focal lengths of the two lenses (f2f1
) gives the
magnification of the imaging system. . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.8 Raw image of the waveguide mode. The mode shape is Gaussian, and the waist can
be extracted from the image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.9 Imaged beam waist vs camera position. We can fit the waist as a function of camera
position using equation 4.1 to find the beam waist at the focus. After accounting for
the magnification of the imaging system, we find the waveguide mode field diameter
to be 2w0 = 3.45 µm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
xi
4.10 Diagram of the vacuum chamber. The free-space-coupling lenses are below the vac-
uum chamber. The light travels from the lenses, off the green mirrors, and couples
into the waveguide in the middle. The minimum distance between the lenses and
the waveguide coupler is 27mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.11 Explanation of spherical aberration. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.12 Normalized reflection signal vs wavelength while free-space-coupling with a f =
50mm asphere (ThorLabs AL2550H). The peak coupling efficiency (√
0.3 = 55%) is
high, but the chromatic focal shift is too large to couple light at 852nm and 894nm. 30
4.13 Diagram of chromatic focal shift. Different wavelengths of light focus to different
points. To compensate for this, multiple lenses are compounded together to form an
achromatic lens. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.14 Reflection signal when free-space-coupling with f = 50mm achromatic lenses. The
coupling efficiency is√
0.18 = 42%, and is broadband, although it drops off below
860nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.15 Reflection signal when free-space-coupling with f = 60mm achromatic lens combined
with a meniscus lens. The peak coupling efficiency is√
0.30 = 55%. . . . . . . . . . . 32
4.16 Reflection signal when free-space-coupling with f = 60mm achromatic lens combined
with a meniscus lens into a bad waveguide. Polarization of the input light was
changed between the two measurements. . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.17 Drawings of the lens mount and translation stages. First drawing is the view from
the front, and the second drawing is the view from below. The stages are mounted
to the bottom of the vacuum chamber for stability. They hold the lens tube where
the free-space-coupling optics are held. . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Chapter 1
Introduction
In this thesis, I will be presenting the work I’ve done with the Regal group at JILA during
the summer and fall of 2017. I’ll start by explaining the motivation for our experiment, followed
by a summary of the experiment itself. Then I’ll focus in on my project within the experiment.
1.1 Motivation
A common goal in experimental quantum physics is to create quantum systems that we can
control, and to use those quantum systems to investigate phenomena that we don’t understand.
There is a huge range of applications for these controlled quantum systems, but the underlying
concept is the same: leverage the power of quantum mechanics to do something we couldn’t do
classically.
One powerful way to use these quantum systems is simulating complex quantum systems.
To motivate this, let’s start with a classical analogue. If you throw a tennis ball horizontally off
a cliff, where will it land? You could figure it out by writing down all the kinematic equations
you learned in your freshman year physics class and calculate the landing position, or you could go
stand on top of the cliff and throw the tennis ball. Better yet, you don’t even need the actual cliff,
you could simulate the problem by standing on a building of the same height. Even with a problem
this simple, it is much easier to simulate the problem than it is to calculate what will happen. Now
imagine the kinematic equations you write down are impossible to solve - now there is even more
reason to simulate the problem. This is the case when investigating quantum systems. We could
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write down equations that describe the system, but often times they are impossible to solve, so we
must resort to simulating the problem with our own quantum system.
The way different groups engineer their quantum systems differ in implementation and in-
tended application, but they are all structured in a similar way that reflects how quantum systems
appear in nature. Quantum systems in nature have two key ingredients: Quantum objects, and
interactions between them. The quantum objects can be atoms, fundamental particles, molecules,
or any other object that requires quantum mechanics to . These quantum objects interact via one
of the fundamental forces: Electromagnetic force, strong force, weak force, or gravitational force.
In order to engineer our own quantum system, we must chose a quantum object, and a way
for our quantum objects to interact. There are too many quantum objects used for me to list them
all, but atoms [10], ions [4], and electric circuits [3] are a few of the most popular categories. For
all of these quantum objects, we are limited to electromagnetic interactions, but the characteristics
of each interaction are different. Atoms interact by exchanging infrared or visible photons, ions
interact via coulomb repulsion, and electric circuits interact by exchanging microwave photons.
All of these interactions are essentially the coherent transfer of energy. When an atom
interacts with another atom, energy is transferred via a photon. However, this energy can also be
lost to the environment. For these quantum systems to be useful, the quantum objects need to
interact enough for us to observe interesting dynamics before the energy is lost to the environment.
Researchers have come up with lots of tricks to keep the energy in the system for longer, and
different quantum systems have different challenges. In the case of atoms interacting via photons,
the challenge is keeping the photons from flying away. Many groups place mirrors around the atoms
so that any photons emitted away from the quantum system are reflected back.
In our experiment, we couple infrared photons to cesium atoms to create our quantum system.
Atoms are convenient to use because unlike quantum circuits, every Cs atom is identical[5], and
unlike ions, atoms are neutral and aren’t as sensitive to noise[7]. We also choose this system because
infrared photons are stable at room temperature[14]. Microwave photons used in quantum circuit
experiments are not stable at room temperature, which in combination with the requirement that
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the circuits be superconducting forces those experiments to be operated below 4 K in a dilution
refrigerator.
We use patterned waveguides to enhance the interactions between atoms, and decrease the
energy lost to the environment. The waveguide is designed to slow down the speed of light so that
when an atom emits a photon, the photon doesn’t leave the system. Because we confine the emitted
photon near the atoms, atoms within the system interact very strongly by exchanging photons. We
can even enter a regime where the light doesn’t propagate along the waveguide, so that any light
emitted by an atom will only travel a finite distance[2, 8]. This allows us to tune the interaction
length between atoms, because the atoms interact via photons. This platform is powerful because
we can tune many parameters of our quantum systems by varying the design of the patterned
waveguide.
1.2 Project Goal
In order to use these waveguides as a platform for our atomic system, we need to get light to
efficiently propagate along the waveguides. In the current experiment, we glue optical fibers to the
end of the waveguides to couple light into the waveguide. This has many disadvantages which we
will investigate in the Waveguide Design section. The goal of this project is to upgrade the method
of coupling light into the waveguides.
Chapter 2
Waveguide Design
In this section, we will discuss the design of the waveguides used in our experiment. We’ll
start by looking at the performance requirements and constraints of the design, then look at the
design itself. The design (shown in Fig 2.1) is broken up into two components: the photonic-crystal
section in the middle where the light interacts with the atoms, and the coupler at the end of the
waveguide where light is coupled into the waveguide.
Figure 2.1: Diagram of the waveguide with fiber coupling. The photonic crystal is in the middle,where the atoms couple to the light in the waveguide. The waveguide couplers on either end couplelight into the waveguide from an optical fiber. Image from Su-Peng’s thesis[19]
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2.1 Performance Requirements
2.1.1 Photonic Crystal Requirements
The purpose of these waveguides is to enhance interactions between pairs of atoms and
between atoms and photons, and to decrease energy loss from the system. We increase interactions
between photons and atoms, so that nearby atoms strongly interact by exchanging photons. There
is a two step approach to increasing atom-light interactions. First, we confine the light around
the atoms[17]. Second, we slow down the light so that the atom and photon have more time to
interact[13].
Because the light is ”trapped” inside the waveguide, we can confine the light around the
atoms by decreasing the area of the waveguide cross-section. Reducing the speed of the photons
propagating in the waveguide requires cleverly patterning the waveguide.
Next, we want the rate at which energy is lost from the system to be much smaller than
the rate at which atoms interact with each other. It turns out that by reducing the speed of the
photons in the waveguide, we also decrease the rate at which energy is lost from the system[11].
To summarize, the goals for the photonic-crystal waveguide region are:
• Confine the light around the atoms
• Reduce the speed of the photons in the waveguide
• Reduce energy loss from the system
2.1.2 Waveguide Coupler Requirements
The waveguide coupler is the end of the waveguide where light either enters or exits the
waveguide. The performance goal for the coupler is to maximize the coupling efficiency between
the waveguide and the exterior optical-system. The ”exterior optical-system” is dependent on the
coupling scheme we are using, and will be fully explained in the Fiber Coupling Scheme and Free
Space Coupling Scheme sections. Basically, send light through some polarization optics and then
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couple that light into the waveguide either using an optical fiber or a lens. We want to maximize
the amount of light that couples into the waveguide. This is important because the signals we
receive from the atoms are single photons. We receive these signals through these couplers, and we
need good coupling efficiency to measure these signals.
The waveguide coupler is subject to an additional constraint if we use fiber-coupling. Because
the fiber sits 30 µm away from the coupler, the structural tethers attached to the coupler can’t
extend past the coupler. With free-space-coupling, the tethers can extend past the coupler and
balance the tension in the waveguide better.
To summarize, the design goal of the waveguide coupler is:
• High coupling efficiency into and out of the waveguide.
2.2 Design
2.2.1 Photonic Crystal Design
A detailed explanation of the photonic-crystal waveguide design is beyond the scope of this
thesis. Here I’ll present an intuitive explanation, and direct readers here [6, 19, 12] for a more
detailed analysis.
Let’s remind ourselves of the design goals for the photonic-crystal waveguide:
• Confine the light around the atoms
• Reduce the speed of the photons in the waveguide
• Reduce energy loss from the system
First, we need to confine the light around the atoms. It turns out that the index of refraction
in electromagnetism plays a similar role as the potential in quantum mechanics (see ch 2 of Photonic
Crystals [9]). Light is concentrated in areas of high index of refraction. Since the index of refraction
for our Silicon Nitride waveguides (n = 2.016) is higher than the vacuum surrounding it (n = 1),
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the light is ”trapped” within the waveguide. By changing the waveguide cross-section, we can
increase the intensity of the light propagating along it.
There’s a caveat here. You might assume you could keep decreasing the waveguide cross-
sectional area and the intensity of the guided light would keep increasing, but there is a limit. The
light leaks out of the Silicon Nitride into vacuum, where the intensity drops exponentially. When
the waveguide cross-section is larger than the wavelength of light, shrinking the waveguide will
increase the intensity of the light. However, when the waveguide cross-section is much smaller than
the wavelength of the light, shrinking the waveguide allows the light to penetrate farther out into
vacuum, decreasing the intensity of the light. The waveguides in our experiment are on the order
of 200 nm wide, which is much smaller than the wavelength of light we use: λ ≈ 850 nm, so we
are in the regime where the majority of the light is outside the waveguide.
During the experiment, the atoms are trapped next to the waveguide where the intensity of
the light is high[18]. Simply by using a small waveguide we have confined the light around the
atoms, thus increasing the atom-photon interactions.
The next step is to slow the light down to increase the amount of time the atoms and the
photons interact. To achieve this, we pattern the waveguide to decrease the group velocity of the
light. First, we need to understand where the group velocity comes from.
When we solve Maxwell’s equations for a waveguide, we find wave solutions to the equations.
There are two important parameters for a wave: the frequency ω, and the wave-vector k. In this
case, we are interested in the projection of k along the waveguide, which we will label k without bold
font. The value k tells us how fast the phase of the wave evolves along the waveguide. Every wave
solution of a waveguide has a range of allowed k where it can propagate along the waveguide, and
corresponding ω. This relationship between k and ω for a system is called the dispersion relation.
Fig 2.2 shows qualitatively the dispersion relation for a rectangular waveguide. The dispersion
relation is a straight line, with a slope given by the index of refraction of the material. By using a
material with a high index of refraction, we can reduce the phase velocity of the light - which is the
same as the group velocity for a linear dispersion relation. However, for materials we can currently
8
fabricate the index of refraction is about 2, and even materials with high index of refraction such
as Germanium have an index of about 4. We want to slow down light by a larger factor so that
our atom-photon interaction is much stronger.
Figure 2.2: Qualitative dispersion relation for a rectangular waveguide. The group velocity of thelight is given by the slope.
Figure 2.3: Dispersion relation for our photonic crystal waveguides. kx is given in units of thephotonic crystal lattice constant a, and represents the absolute amount of phase gained over oneunit cell of the photonic crystal.
9
By adding a periodic structure to the waveguide, we can engineer the dispersion relation
to be flat for certain frequencies[20]. In Fig 2.3, the dispersion relation for our photonic crystal
waveguides is shown. The group velocity approaches 0 at νA and νD, which is much better than
slowing down the light by a factor of 2. By patterning the waveguide, we’ve slowed the light to a
stop, and enhanced interactions between the atoms and the photons in the waveguide. We design
the photonic-crystal so that the band edges occur at the cesium D1 and D2 lines - the atomic
transitions we want to use for our experiment.
The band gap - range of frequencies at which light can’t propagate - can be understood by
looking at the symmetry of the system. First, we model the photonic-crystal as a dielectric stack
with different indexes of refraction, as shown in Fig 2.4. Because the photonic-crystal is periodic
and symmetric, the electric field must be too. For a given k, there are two configurations of the
electric field that satisfy the periodicity and symmetry of the system. One has electric field nodes
in the areas of low n, and the other has electric field nodes in the areas of high n. Because the
n is different, the energies of the two configurations will be different, therefore they have different
frequencies. This causes the band gap in the dispersion relation.
It is important to note that the dispersion relation shown in Fig 2.2 is for light polarized in
the plane of the waveguide, with no transverse component - the TE mode of the waveguide. The
other polarization - the TM mode - has a different dispersion relation that is not shown.
When light is sent into the photonic-crystal at a frequency within the range of the dispersion
relation, the light will propagate along the waveguide at a speed given by the group velocity. As the
frequency of the light approaches νA or νD, the group velocity approaches 0 and the light comes to
a standstill. If we continue into the ”bandgap” - the range of frequencies with no corresponding real
k - the value of k becomes imaginary. This corresponds to a mode where the electric field strength
falls off exponentially as you move along the photonic-crystal. Because the light can’t propagate
along the photonic-crystal, it is reflected back along the waveguide. Fig 2.5 shows the reflected
signal from the waveguide. In the ideal case, the infinite photonic-crystal has a sharp cutoff at the
edges of the band gap. In reality, the photonic-crystal is finite, and the ends of the photonic-crystal
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region act as mirrors which form a cavity. The reflection signal in this case has cavity resonances
next to the band gap.
Figure 2.4: We can model the photonic-crystal as a stack of dielectric with different indexes ofrefraction. Where there is more material, the index of refraction is higher, and where there is lessmaterial, the index of refraction is lower. For any value of k, there are two possible configurationsof the electric field that satisfy both the periodicity and the symmetry of the system. One solutionhas higher intensity in the areas of high index of refraction, and the other solution has higherintensity in the areas of low index of refraction. The different indexes of refraction cause these twosolutions to have different energies, and therefore different frequencies.
The final performance goal for our waveguide is to decrease energy loss from the system.
Energy is lost from the system when the photons emitted from the atoms aren’t captured in the
waveguide. When an atom emits a photon, the photon couples into different modes in proportion
11
Figure 2.5: Dispersion relation for photonic-crystal waveguide on the left. The frequencies wherethe group velocity approach zero are placed at two Cesium atomic transitions by modifying thewaveguide design. When light is propagating along the waveguide, the reflection coefficient at thephotonic-crystal is plotted on the the right for an ideal infinite photonic-crystal and the real finitephotonic-crystal.
to the number of states available to the photon in that mode. If there are more states available
to the photon in a given mode, the photon is more likely to couple to that mode. The number of
states is given by the density of states, which tells you the range of k available for a range of ω, or
∂k
∂ω=
1
vG. Because the group velocity for the photonic crystal waveguide mode approaches 0 for
certain frequencies, the density of states approaches ∞, and the atom will only emit photons into
the photonic-crystal waveguide mode. Therefore, by decreasing the group velocity of the photons to
increase atom-photon interactions, we have also decreased the rate of energy loss from the system.
2.2.2 Waveguide Coupler Design
The goal for the waveguide coupler design is:
• High coupling efficiency into and out of the waveguide
• Coupling at at both λ = 852nm (Cs D2 line) and λ = 894nm (Cs D1 line)
Before we optimize the coupling efficiency, we need to know what the light looks like when
it is propagating along the waveguide. The waveguide at the coupler region has a rectangular
12
cross-section, 200nm deep and varying width (about 200nm wide). Fig 2.6 shows the mode profile
of the light propagating along the waveguide. The mode profile was calculated in a Comsol FDFD
simulation.
Figure 2.6: Electric field norm for the waveguide mode. The electric field is strongest around theedges of the waveguide, and falls off exponentially outside the waveguide. The right plot is theelectric field norm along a cut through the center of the waveguide. Simulation done by XingShengLuan
To maximize coupling, we want to send in light that matches the mode profile of the
waveguide[16]. For both fiber and free-space coupling schemes, the mode that we can send in
is limited to a Gaussian mode. Optical fibers operating at λ = 850 nm have a Gaussian mode
profile with a mode field diameter of 2w = 5 µm (w being the Gaussian beam waist or half the
1/e2 diameter). Now we need to adjust the waveguide coupler dimensions so that the mode of the
waveguide has a large overlap with the mode of the fiber. For free-space coupling, where we can
adjust the size of the Gaussian beam at the focus by choosing different lenses, we have more freedom
on our waveguide fabrication. The goal is still to make the waveguide mode a reasonable size so
that we can couple well with a lens. Because we currently use fiber-coupling in our experiment, all
of the waveguides we fabricate are designed to maximize mode overlap with the mode of an optical
fiber.
13
Achieving coupling at both λ = 852 nm and λ = 894 nm is easy in a fiber-coupling setup.
As long as the optical fiber guides both wavelengths, the coupling will be the same. In free-
space-coupling, we have to take special care that our optical system performs the same at both
wavelengths. Regardless of the coupling scheme, achieving coupling at both wavelengths is not
done with the waveguide coupler design, but with the external coupling scheme design.
Chapter 3
Fiber-Coupling Scheme
In the current atomic experiment, we use optical fibers to couple light into the waveguides.
Fig 3.1 shows the current generation chip with 10 waveguides. The waveguides are suspended above
grooves in the chip, which extend past the end of the waveguide. Fibers are glued into these grooves
to couple light into the waveguide. Fig 3.2 is an image of the fiber 30µm away from the end of the
waveguide.
3.1 Fiber-Coupling Advantages
Fiber-coupling has three main advantages:
• Once the fibers are glued, the coupling is stable
• No need for optical access to the waveguide coupler
• Coupling at λ = 852nm and λ = 894nm
Because the fibers are glued to the waveguides, once the glue dries, the fibers won’t move, so
the coupling is constant over time. Additionally, the fibers guide the light from the waveguide to the
outside of the experimental vacuum chamber, so there is no need for optical access to the waveguide
coupler. This saves space on the outside of the vacuum chamber for other optics. Finally, we don’t
have to worry about wavelength dependent performance, as long as the optical fiber guides light at
λ = 852nm and λ = 894nm.
15
Figure 3.1: Current fiber-coupled waveguide chip. The chip has 10 waveguides, and fibers are gluedto two of them.
3.2 Fiber-Coupling Disadvantages
While fiber-coupling is robust, it has many disadvantages:
• Fiber-coupling is limited to a few waveguides on a chip
• Can’t bake the vacuum chamber with fiber-coupling
• Easy to break the waveguides while installing optical fibers
• Adds constraints to waveguide fabrication
When we couple light into a waveguide, we need to coupler light into both ends of the
waveguide. This means that every waveguide we couple to has two optical fibers glued to it. In
16
Figure 3.2: Fiber sits 30 µm away from the waveguide coupler. The thin horizontal line ending ata T is the waveguide, and the large white cylinder is the optical fiber. Light is scattering off of thewaveguide coupler.
order to get the light to and from the waveguide, we need to feed the optical fibers into the vacuum
chamber. This increases the limit on the background pressure in the vacuum chamber. We also
have a limit on the number of feed-throughs we can install in the vacuum chamber. In the current
experiment, we are limited to 8 feed-throughs, or 4 waveguides. During the experiment, the Cs
atoms we trap next to the waveguides stick to the waveguide and degrade the optical performance.
Over time, the waveguides aren’t usable anymore and we need to switch to a new waveguide. With
only 4 usable waveguides in the vacuum chamber, we need to open the vacuum chamber and install
a new set of waveguides often, which adds overhead.
The second disadvantage is that we can’t bake the vacuum chamber with optical fibers inside
it. Optical fibers have a max operating temperature of 85 In order to get low pressure in
a vacuum chamber, we heat up the vacuum chamber so that any residual ”gunk” on the walls
17
evaporates off and we can pump it out of the chamber. It is common to heat up the vacuum
chamber up to 300 hile baking. Unfortunately we can’t bake our vacuum chamber because the
heat might damage the optical fibers. All the ”gunk” that might be baked out is left on the walls
of the vacuum chamber, and prevents us from reaching really low vacuum pressures.
The third disadvantage is that there’s a high risk of damaging the waveguides while gluing
the fiber to the chip. During the gluing process, we fit the fiber into the groove on the chip, then
put a drop of heat cured epoxy in the groove. The epoxy spreads out across the groove, then
dries. If too much epoxy is used, the epoxy will spread out into the waveguide, and the waveguide
will be ruined. Additionally, when the epoxy dries, it can apply stress to the chip that can break
the waveguides. It can also push the optical fiber into the waveguide and break it. There are
many ways to break the waveguides while gluing optical fibers to them, and unfortunately we have
experienced many of them.
The final disadvantage, and perhaps the most important, is that fiber-coupling adds con-
straints to the waveguide design. The first constraint affects the structural properties of the waveg-
uide. The waveguide is supported by a series of tethers, as shown in Fig 2.1. Because the fiber
sits so close to the end of the waveguide, the tethers on the end can’t extend past the end of the
waveguide. The waveguide has high tension, so ideally the end tethers would extend out at an
angle to properly balance the tension in the waveguide. Because we can’t do that, the waveguides
aren’t as structurally robust. Fiber-coupling also limits the number of waveguides per chip. Each
waveguide sits in a groove that has to be as wide as an optical fiber. We use optical fibers with a
diameter of 125µm, and the grooves are 150µm across. This size limits the number of waveguides
we can fit on a chip. We have fabricated next-generation free-space chips that hold 30 waveguides,
compared to the 10-15 waveguides on the fiber-coupled chips.
3.3 Fiber-Coupling Performance
There are a few performance benchmarks set by fiber-coupling that we will aim to meet with
free-space coupling. The first is coupling efficiency. In the current experiment, we measure 45%
18
coupling efficiency. This means that if we send 1 mW through the optical fiber, 0.45 mW will
propagate along the waveguide. The next benchmark is broadband performance. Fiber-coupling
works at both λ = 852nm and λ = 894 nm, spanning nearly 50 nm.
Chapter 4
Free-Space-Coupling Scheme
In this section, we will present the free-space-coupling scheme, and analyze its advantages
and disadvantages. Next, we will define goals for an implementation of free-space-coupling. Then
we will investigate the properties of free-space-coupled chips, and the design an optical system to
achieve our goals. Finally, we will look at an implementation of free space coupling and compare
our results to our defined goals.
4.1 Design
Free-space-coupling works by using a lens to focus a laser beam onto the end of the waveguide
coupler[15]. A diagram of free-space-coupling is shown in Fig 4.1. The free-space-coupling setup
consists of an optical fiber and two lenses. We designed the system this way to make it stable and
compact for use in the experiment. Other implementations[19] have added mirrors between the
two lenses to optimize alignment, but the setup is bulky.
4.2 Goals
The purpose of this project is to develop free-space-coupling so that future generations of
the experiment can upgrade to it from fiber-coupling. This defines a natural set of goals, where
we aim to meet the performance of fiber-coupling in terms of coupling efficiency, but still keep the
advantages of free-space-coupling such as coupling to every waveguide on a chip. Here are our goals
for free-space-coupling:
20
Figure 4.1: Diagram of free-space-coupling. The light from an optical fiber is collimated by lens1, then focused by lens 2 onto the end of the waveguide. We can change the size of the Gaussianbeam at the waveguide coupler by adjusting the ratio of the focal lengths of lens 1 and lens 2.By using identical lenses, we project the optical fiber mode onto the waveguide coupler with nomagnification, and in theory achieve the same coupling as fiber coupling.
• 45% coupling efficiency
• Coupling at λ = 852nm and λ = 894nm
• Easy to couple light into every waveguide on a chip
• Satisfies geometry requirements of the vacuum chamber
The first two goals are simply meeting the performance benchmarks of fiber-coupling. The
second two goals have to do with the advantages of free-space coupling. Coupling to every device
on a chip is one of the reasons we are upgrading to free-space-coupling, and satisfying the vacuum
chamber geometry requirements means keeping the entire free-space-coupling setup outside the
vacuum chamber. If we can meet these four goals, then we can eventually upgrade the experiment
to free-space-coupling without a drop in performance.
4.3 Properties of Free-Space-Coupled Waveguides
Before implementing free-space-coupling in the experiment, we need to understand the prop-
erties of the free-space-coupled waveguides. This will allow us to design an optimal coupling setup.
Free-space-coupled waveguides are different from the old generation fiber-coupled waveguides.
21
Because there is no fiber placing constraints on the waveguide design, they are more compact, and
the tethers are designed differently. Fig 4.2 is an image of a free-space-coupled chip. This chip is
much smaller than the fiber-coupled chips so that the light that exits the waveguide doesn’t clip
the end of the chip.
Figure 4.2: Image of a free-space-coupled waveguide chip. These chips are much smaller than thefiber-coupled chips so that the light exiting the waveguide doesn’t clip the end of the chip.
In order to test the free-space-coupled waveguides, we built a test setup to couple light into
the waveguides. The test setup used a pair of f = 11.0mm asphere lenses (Thorlabs A397TM-B)
in the configuration shown in Fig 4.1 to couple light from an optical fiber to the waveguides.
Fig 4.3 shows a diagram of the test setup for measuring the reflection signal from the waveg-
uide. The reflection signal measured is shown in Fig 4.4.
22
Figure 4.3: Diagram of the test setup used to measure the reflection signal from the waveguides.We use a SLED (super-luminescent diode) as a broadband light source. It emits broadband lightin the frequency range 800nm − 900nm. The polarization is cleaned up and set by a PBS and abroadband λ/2 waveplate. Then the light is sent through a beam splitter and coupled into thewaveguide. The light reflected from the waveguide is measured on the output port of the BS on anoptical spectrum analyzer, which reads out the optical power at each wavelength.
Figure 4.4: Reflection signal when coupling with f = 11.0mm asphere lenses. The vertical axis isnormalized reflected light, which is the amount of light reflected from the waveguide divided bythe input power. Because this light was coupled into the device, then coupled out again, the peakcoupling efficiency is the square root of 0.25, or 50%.
23
4.3.1 Chromatic Focal Shift
Once we have a reflection signal, we can start looking at the characteristics of free-space-
coupling. The first measurement is how the coupling depends on the distance between the lens and
the waveguide. To measure this, we start with the lens aligned so that the reflection signal looks
like Fig 4.4. We define this position as 0, then move the lens closer and farther from the waveguide.
From calculations, we expect the longer wavelengths to focus slightly farther away than the shorter
wavelengths. Fig 4.5 shows how the reflection signal changes when the lens is translated towards
or away from the waveguide. The longer wavelengths (lower frequency) have a higher coupling
efficiency when the lens is translated away from the waveguide.
When we design the long working distance free-space-coupling setup, we will need to take
this chromatic focal shift into account.
4.3.2 Waveguide ”Field of View”
Next, we investigate the ”field of view” of the waveguide. We adjust the angle of the input
light and measure the change in coupling efficiency. For each input angle, I adjusted the position
of the lens to maximize the reflection signal at λ = 880nm, which is roughly the center of the band
gap, and recorded the maximum reflection efficiency. Fig 4.6 shows the normalized reflection signal
vs input angle. The reflection efficiency decreases by 4% when the angle is adjusted ±2 degrees
away from normal. From this measurement, we know that input angle is not a sensitive degree
of freedom. This will make a final free-space-coupling setup easier to design. On the other hand,
translating the lens systems has a huge effect on the coupling. Since the waveguide mode size is
roughly 5 µm in diameter, translating the beam on the micron scale is enough to decrease the
coupling efficiency.
4.3.3 Waveguide Spatial Mode
The next step in designing a free-space-coupling system is determining the size and shape of
the waveguide mode. This is done by coupling light into the waveguide, and imaging the light that
24
Figure 4.5: Reflection signal vs lens position for an f = 11 mm aspherical lens. Positive distancemeans the lens is translated towards the waveguide. The longer wavelengths focus slightly fartheraway from the lens than the shorter wavelengths, causing the skewing of the reflection signal.
comes out the other side. Fig 4.7 shows a diagram of the waveguide-mode imaging setup.
We expect the waveguide-mode size to be roughly 5µm in diameter. The camera we used
(DataRay WinCamD-UCD15) has 4.4µm pixels, so if we had a magnification of 1 the beam would
only hit one pixel. In order to use a large number of pixels for the image, we used a f = 11mm
asphere lens and a f = 150mm achromat lens to image the waveguide mode with a magnification
25
-4 -2 0 2 4
0.35
0.40
0.45
0.50
0.55
Input Angle (Degrees)
MaxCouplingEfficiency
Coupling Efficiency vs Input Angle
Figure 4.6: Normalized reflected power vs input angle. At ±2 degrees away from normal, thereflection efficiency only drops 4%.
of 13.64. With this magnification, the image should be roughly 14 pixels across, which is enough
to get good resolution. A raw image of the waveguide mode is shown in Fig 4.8.
To get an accurate mode size measurement, we want the waveguide coupler to be at the focal
point of lens 1, and the camera to be at the focal point of lens 2. First, we send a collimated
beam through lens 1 and couple the light into the waveguide, so that the focus of lens 1 is at the
waveguide coupler. Next, we add lens 2, and place the camera approximately at the focus. Then we
sweep the position of the camera and measure the Gaussian beam waist at the different positions,
as shown in Fig 4.9. As a Gaussian beam propagates, its waist changes according to equation 1[1].
w(z) = w0
√1 + (
z
zR)2 (4.1)
Where w0 is the beam waist at the focus, w(z) is the beam waist a distance z from the focus,
26
Figure 4.7: Diagram of the setup for imaging the mode of the waveguide. Light is coupled into thewaveguide on the left (not shown), then the light coming out on the right is imaged onto a camera.
The ratio of the focal lengths of the two lenses (f2f1
) gives the magnification of the imaging system.
Figure 4.8: Raw image of the waveguide mode. The mode shape is Gaussian, and the waist can beextracted from the image.
27
and zR is the Rayleigh range of the beam, given by equation 2.
zR =πw2
0
λ(4.2)
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50.0
0.5
1.0
1.5
2.0
2.5
3.0
Camera Position (mm)
BeamWaist
(μm) Minor Axis Data
Major Axis Data
Minor Axis
Fit
Major Axis Fit
Figure 4.9: Imaged beam waist vs camera position. We can fit the waist as a function of camera po-sition using equation 4.1 to find the beam waist at the focus. After accounting for the magnificationof the imaging system, we find the waveguide mode field diameter to be 2w0 = 3.45 µm.
We fit the data using equation 4.1, and find that the waveguide mode diameter is 2w0 =
3.45 µm. Note that we are assuming the waveguide mode is approximately a Gaussian beam, so
that we can model its propagation using Gaussian optics.
We now have a list of properties about free-space-coupling to take into account when designing
our final free-space-coupling setup:
• Focal length is dependent on wavelength
• Coupling efficiency is not very sensitive to the angle of incidence of the light
• Waveguide mode is approximately Gaussian, with a waist of 3.45 µm
28
4.4 Geometric Cnstraints
The final constraint on designing our free-space-coupling system is the geometry of the vac-
uum chamber. Figure 4.10 shows a diagram of the vacuum chamber, with the free-space-coupling
lenses. The lenses have to be outside the vacuum chamber, which restricts us to lenses with a
working distance - the distance between the front face of the lens and the focal point - that is
greater than 27 mm.
Figure 4.10: Diagram of the vacuum chamber. The free-space-coupling lenses are below the vacuumchamber. The light travels from the lenses, off the green mirrors, and couples into the waveguidein the middle. The minimum distance between the lenses and the waveguide coupler is 27mm.
4.5 Choosing Lenses For Free-Space-Coupling
We need our free-space-coupling lenses to have three properties:
• Small aberrations
29
• Small chromatic focal shift
• Working distance longer than 27mm
Aberrations are imperfections in the lens that lead to a distorted focal spot. When choosing
lenses that satisfy these requirements, Ting-Wei calculated the lens properties using OSLO.
The largest aberration for most lenses is spherical aberration. Spherical aberration occurs
when the focal length of the lens depends on how far from the center of the lens you are. Fig 4.11
shows the difference between a lens with spherical aberration, and a lens without. Lenses with
spherical aberration create a larger focal spot, and the focal spot won’t be a Gaussian, which will
decrease the coupling efficiency. The waveguide mode has a diameter of 3.45µm, so we need the
focal spot to be small.
Figure 4.11: Explanation of spherical aberration.
4.5.1 Aspherical Lens
Aspherical lenses are shaped to correct for spherical aberration. These lenses create a focal
spot that is nearly perfect, and are only limited by the diffraction of light (the wave nature of light
30
prevents an infinitesimal focal spot). We tested an asphere lens with a focal length of f = 5cm, as
shown in Fig 4.12. The peak coupling efficiency is 55%, but the chromatic focal shift is too large
to couple both 852nm and 894nm. This lens is useful for testing, but can’t be used in the final
free-space-coupling setup for the experiment.
Figure 4.12: Normalized reflection signal vs wavelength while free-space-coupling with a f = 50mmasphere (ThorLabs AL2550H). The peak coupling efficiency (
√0.3 = 55%) is high, but the chro-
matic focal shift is too large to couple light at 852nm and 894nm.
4.5.2 Achromatic Lens
It is clear that we need consider chromatic focal shifts when choosing a free-space-coupling
lens. In order to compensate for chromatic focal shifts, optical companies make compound lenses
called achromats. The engineers adjust the multiple surfaces to compensate for chromatic focal
shifts, as shown in Fig 4.13.
We measured the coupling efficiency with a pair of f = 50mm achromats (ThorLabs ACA254-
31
Figure 4.13: Diagram of chromatic focal shift. Different wavelengths of light focus to differentpoints. To compensate for this, multiple lenses are compounded together to form an achromaticlens.
050). The reflection signal is shown in Fig 4.14.
Figure 4.14: Reflection signal when free-space-coupling with f = 50mm achromatic lenses. Thecoupling efficiency is
√0.18 = 42%, and is broadband, although it drops off below 860nm.
32
The coupling efficiency is lower for the achromat (42%) than for the asphere (55%), but
the light is coupled across the band gap. This shows that the achromat designed to have lower
chromatic focal shift made a difference in performance.
4.5.3 Meniscus-Achromat Lens Pair
The final pair of lenses we tested is a combination of an achromatic lens, and a meniscus lens.
We used a f = 60mm achromat (Thorlabs ACA254-060) to achieve the broadband performance,
then corrected the beam for spherical aberration with the meniscus lens (Thorlabs LE1156). The
working distance of this lens pair is WD = 35 mm, so it satisfies the vacuum chamber geometry
requirements. Fig 4.15 shows two reflection signals when free-space-coupling with the meniscus-
achromat pairs. The polarization was adjusted between measuring the first and second plot, which
indicates there is some wavelength dependent polarization effect.
Figure 4.15: Reflection signal when free-space-coupling with f = 60mm achromatic lens combinedwith a meniscus lens. The peak coupling efficiency is
√0.30 = 55%.
The meniscus-achromat pair achieves broadband coupling ranging from 50% to 55% one
33
way coupling efficiency. This combines the high coupling efficiency of the aspherical lens with the
broadband performance of an achromatic lens.
When we first tested the meniscus-achromat pair, we used a bad waveguide chip, which
caused the spectrum to have wavelength dependent effects that we didn’t expect. Figure 4.16
shows the reflection spectrum from the bad waveguide. The difference between the two plots is
the polarization of the input light. This implies there is a wavelength dependent polarization effect
occurring at the waveguide.
4.6 Lens Mount
Because we limited the free-space-coupling setup to two lenses and an optical fiber, mounting
the setup to the vacuum chamber is much easier. We will mount all three optical elements in
a standard lens tube, and mount the lens tube to a translation stage. The translation stages are
custom made to be compact while still providing enough travel to couple light into every waveguide.
The translation stages will be mounted directly to the vacuum chamber to keep the system stable.
Fig 4.17 shows a drawing of the vacuum chamber with the free-space-coupling setup mounted.
4.7 Results
The goals we set for free-space-coupling were:
• 45% coupling efficiency
• Coupling at λ = 852nm and λ = 894nm
• Easy to couple light into every waveguide on a chip
• Satisfies geometry requirements of the vacuum chamber
The final lens pair we tested (the meniscus-achromat pair) met all of the goals relevant for the
lenses. The coupling efficiency was higher than 45%, the coupling was broadband, and it satisfies
the geometry requirements of the vacuum chamber. Being able to couple light into every waveguide
34
Figure 4.16: Reflection signal when free-space-coupling with f = 60mm achromatic lens combinedwith a meniscus lens into a bad waveguide. Polarization of the input light was changed betweenthe two measurements.
is dependent on the translation stage used to align the lenses. The translation stage we designed
will allow us to couple light into every waveguide, but hasn’t been tested yet.
35
Figure 4.17: Drawings of the lens mount and translation stages. First drawing is the view fromthe front, and the second drawing is the view from below. The stages are mounted to the bottomof the vacuum chamber for stability. They hold the lens tube where the free-space-coupling opticsare held.
Chapter 5
Conclusion and Outlook
The upgrade to free-space-coupling will improve our vacuum pressure, decrease experimental
overhead due to replacing waveguides, and remove a constraint on the fabrication of our waveguides.
We have investigated the properties of free-space-coupled waveguides, and have demonstrated high
coupling efficiency across the entire range of relevant wavelengths with free-space-coupling in a
setup similar to the experiment.
The next step in this project is to test the lenses with a complete vacuum chamber. This will
include testing the lens mounts discussed in section 5.6. After successfully testing the lenses with
a vacuum chamber, we will install the new vacuum chamber into the current experiment.
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