Design of Antenna-Coupled Lumped-Element

Design of Antenna-Coupled Lumped-Element Titanium Nitride KIDs for LongWavelength Multi-Band Continuum Imaging - CaltechTHESIS
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Design of Antenna-Coupled Lumped-Element Titanium Nitride KIDs for LongWavelength Multi-Band Continuum Imaging
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Ji, Chenguang
(2015)
Design of Antenna-Coupled Lumped-Element Titanium Nitride KIDs for LongWavelength Multi-Band Continuum Imaging.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/Z9W66HQ7.
Abstract
Many applications in cosmology and astrophysics at millimeter wavelengths including CMB polarization, studies of galaxy clusters using the Sunyaev-Zeldovich effect (SZE), and studies of star formation at high redshift and in our local universe and our galaxy, require large-format arrays of millimeter-wave detectors. Feedhorn and phased-array antenna architectures for receiving mm-wave light present numerous advantages for control of systematics, for simultaneous coverage of both polarizations and/or multiple spectral bands, and for preserving the coherent nature of the incoming light. This enables the application of many traditional "RF" structures such as hybrids, switches,
and lumped-element or microstrip band-defining filters.
Simultaneously, kinetic inductance detectors (KIDs) using high-resistivity materials like titanium nitride are an attractive sensor option for large-format arrays because they are highly multiplexable and because they can have sensitivities reaching the condition of background-limited detection. A
KID is a LC resonator. Its inductance includes the geometric inductance and kinetic inductance of the inductor in the superconducting phase. A photon absorbed by the superconductor breaks a Cooper pair into normal-state electrons and perturbs its kinetic inductance, rendering it a detector
of light. The responsivity of KID is given by the fractional frequency shift of the LC resonator per unit optical power.
However, coupling these types of optical reception elements to KIDs is a challenge because of the impedance mismatch between the microstrip transmission line exiting these architectures and the high resistivity of titanium nitride. Mitigating direct absorption of light through free space coupling to the inductor of KID is another challenge. We present a detailed titanium nitride KID
design that addresses these challenges. The KID inductor is capacitively coupled to the microstrip in such a way as to form a lossy termination without creating an impedance mismatch. A parallel plate capacitor design mitigates direct absorption, uses hydrogenated amorphous silicon, and yields acceptable noise. We show that the optimized design can yield expected sensitivities very close to
the fundamental limit for a long wavelength imager (LWCam) that covers six spectral bands from 90 to 400 GHz for SZE studies.
Excess phase (frequency) noise has been observed in KID and is very likely caused by two-level systems (TLS) in dielectric materials. The TLS hypothesis is supported by the measured dependence of the noise on resonator internal power and temperature. However, there is still a lack of a unified microscopic theory which can quantitatively model the properties of the TLS noise. In this thesis we
derive the noise power spectral density due to the coupling of TLS with phonon bath based on an existing model and compare the theoretical predictions about power and temperature dependences with experimental data. We discuss the limitation of such a model and propose the direction for future study.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
sensors, low-temperature detectors, bolometers, submillimeter-wave and millimeter-wave receivers and detectors, kinetic inductance detectors, radio telescopes and instrumentation
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Materials Science
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Golwala, Sunil
Group:
Astronomy Department
Thesis Committee:
Golwala, Sunil (chair)
Goddard, William A., III
Johnson, William L.
Rutledge, David B.
Zmuidzinas, Jonas
Defense Date:
18 May 2015
Non-Caltech Author Email:
jcg051987 (AT) gmail.com
Record Number:
CaltechTHESIS:05282015-121920930
Persistent URL:
DOI:
10.7907/Z9W66HQ7
Default Usage Policy:
No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:
8896
Collection:
CaltechTHESIS
Deposited By:
Chenguang Ji
Deposited On:
28 May 2015 23:57
Last Modified:
10 Mar 2020 19:19
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Design of antenna-coupled lumped-element titanium nitride
KIDs for long-wavelength multi-band continuum imaging

Thesis by

Chenguang Ji

In Partial Fulfillment of the Requirements
for the Degree of
Doctor of Philosophy

California Institute of Technology
Pasadena, California

2015
(Defended May 18th, 2015)

c 2015
Chenguang Ji

iii

Dedicated to everyone who accompanied me through the ups and downs along the way

Acknowledgments
First and foremost I would like to express the deepest gratitude to my advisor, Sunil Golwala. Thank
you for creating such a relaxed and positive atmosphere to discuss science and for granting me the
freedom to pursue my interest in research. I have great appreciation for your unfailing encouragement
during the many hardships and respect for your perseverance and tenacity for leading our project
through the tough times.
I am also extremely grateful to Jonas Zmuidzinas. Jonas introduced me the TLS noise project
in late 2012 and we had a series of very interesting talks since then despite his heavy obligation
at JPL. Also, as the founder of MKID, Jonas provided me with critical comments on the LWCam
design work at different stages.
I am really thankful to David Rutledge, Bill Johnson, and Bill Goddard for serving on my thesis
defense committee and the valuable comments from an interdisciplinary perspective.
I am very fortunate to have been working in a highly supportive environment in the submillimeter/millimeter observational astronomy group. Chris McKinney, back then, was one of Jonas’
postdocs working on another subproject MAKO for CCAT. We got paired up to experimentally determine the TLS noise properties in the parallel-plate structure MKID. Chris turned out to be an
amazing mentor and I benefited so much by working closely with such an enthusiastic and excellent
experimentalist. I was also later granted with the privilege to use his privately (and secretly) owned
drills. Erik Shirokoff, another postdoc working on Superspec, has always been very patient in listening to my problems with Sonnet simulation and good at suggesting smart solutions. Matt Hollister,
the mechanical engineer, successfully maintained the old blue dewar to work for the hot/cold measurement on the antenna. Mark Gonzales, the shop course instructor, taught me the “nice and easy”
machining skills hand by hand meticulously. Andrew Beyer, our collaborator at JPL, fabricated the
device for LWCam. Seth Siegel, my fellow graduate student, shared with me his powerful MCMC
code for detector efficiency characterization. Bade Uzgil, my officemate, said hello to me everyday
with a bright smile. Albert Lam, an undergraduate, renovated the dewar and took turns with me to

do the refills and never complained. Ran Duan, a senior graduate student, helped me to ease into
the role during my early stage of research. I am also indebted to Jiansong Gao, a previous graduate
student of the group. Though we only got the chance to chat over the phone, Jiansong gave me lots
of sincere advices. He is also the person who laid the groundbreaking foundation work about MKID
TLS noise theory, the basis of Chapter 2 of this thesis.
Among the many courses I took over the years at Caltech, some made a huge impact on me.
Thanks to these outstanding lecturers. You not only transfered knowledge and skills but also imparted wisdom, fostered the passion for science and changed my way of thinking. They are Brent
Fultz, Sossina Haile, Matthew Fisher, Sergei Gukov, and Alexei Kitaev.
During my PhD I’ve also been teaching undergraduate courses as an assistant and meet a few
fantastic persons. Thanks Jessica Hsu, Julia Su, and Pinting Chen for being my faithful (and
amazing) audiences in physics 2 a and b at Caltech. It was a very delighted experience to have you
all as my students. Thanks Frank Rice for being such an awesome leader in physics 6 and 7. As
somebody who is witty, remarkably fond of his job, and professional in detail, you left a very deep
impression on me. Thanks Rebecca Wernis, Daniel DeFelippis, and Nico Salzetta. I greatly enjoyed
your partnership.
I also want to mention some of my best friends who make my Caltech life fun and unforgettable.
They are Ding Ding, Yinglu Tang, Xiaowei Deng, Jing Zhang, Jonathan Hood, Andres Goza, and
the whole 2009 class of Caltech C members. I will cherish the friendship I have with you all.
Lastly I would like to thank my parents and members in my extended family. Thank you for
always being with me in such a long journey. Without your sacrifice and unconditional love, this
thesis would never be possible.

Abstract
Many applications in cosmology and astrophysics at millimeter wavelengths including CMB polarization, studies of galaxy clusters using the Sunyaev-Zeldovich effect (SZE), and studies of star
formation at high redshift and in our local universe and our galaxy, require large-format arrays of
millimeter-wave detectors. Feedhorn and phased-array antenna architectures for receiving mm-wave
light present numerous advantages for control of systematics, for simultaneous coverage of both polarizations and/or multiple spectral bands, and for preserving the coherent nature of the incoming
light. This enables the application of many traditional “RF” structures such as hybrids, switches,
and lumped-element or microstrip band-defining filters.
Simultaneously, kinetic inductance detectors (KIDs) using high-resistivity materials like titanium
nitride are an attractive sensor option for large-format arrays because they are highly multiplexable
and because they can have sensitivities reaching the condition of background-limited detection. A
KID is a LC resonator. Its inductance includes the geometric inductance and kinetic inductance
of the inductor in the superconducting phase. A photon absorbed by the superconductor breaks a
Cooper pair into normal-state electrons and perturbs its kinetic inductance, rendering it a detector
of light. The responsivity of KID is given by the fractional frequency shift of the LC resonator per
unit optical power.
However, coupling these types of optical reception elements to KIDs is a challenge because of
the impedance mismatch between the microstrip transmission line exiting these architectures and
the high resistivity of titanium nitride. Mitigating direct absorption of light through free space
coupling to the inductor of KID is another challenge. We present a detailed titanium nitride KID
design that addresses these challenges. The KID inductor is capacitively coupled to the microstrip
in such a way as to form a lossy termination without creating an impedance mismatch. A parallelplate capacitor design mitigates direct absorption, uses hydrogenated amorphous silicon, and yields
acceptable noise. We show that the optimized design can yield expected sensitivities very close to
the fundamental limit for a long wavelength imager (LWCam) that covers six spectral bands from

vii
90 to 400 GHz for SZE studies.
Excess phase (frequency) noise has been observed in KID and is very likely caused by two-level
systems (TLS) in dielectric materials. The TLS hypothesis is supported by the measured dependence
of the noise on resonator internal power and temperature. However, there is still a lack of a unified
microscopic theory which can quantitatively model the properties of the TLS noise. In this thesis we
derive the noise power spectral density due to the coupling of TLS with phonon bath based on an
existing model and compare the theoretical predictions about power and temperature dependences
with experimental data. We discuss the limitation of such a model and propose the direction for
future study.

viii

Contents
Acknowledgments

Abstract

List of Figures

List of Tables

xiii

1 Background

1.1

1.2

1.3

Scientific Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.1.1

Introduction to mm/submm astrophysics . . . . . . . . . . . . . . . . . . . .

1.1.2

Scientific motivation of long wavelength imager . . . . . . . . . . . . . . . . .

Review of light-coupling architectures . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2.1

Feedhorn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2.2

Phased array antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Introduction to kinetic inductance detector . . . . . . . . . . . . . . . . . . . . . . .

1.3.1

Principles of operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3.2

Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3.3

Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Study of two-level-system noise in dielectric materials
2.1

2.2

General properties of TLS noise in KID . . . . . . . . . . . . . . . . . . . . . . . . .

2.1.1

Noise measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1.2

Experimental results of TLS noise in KID . . . . . . . . . . . . . . . . . . . .

Standard model of two level systems . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2.1

Localized representation and energy representation . . . . . . . . . . . . . . .

2.2.2

Analysis of interaction with external fields . . . . . . . . . . . . . . . . . . . .

ix
2.2.3
2.3

2.4

Rigorous solution of electric susceptibiities . . . . . . . . . . . . . . . . . . . .

Model of TLS noise spectral density . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3.1

Model of the dielectric constant due to TLS . . . . . . . . . . . . . . . . . . .

2.3.2

Power spectral density of σ̂z for a single TLS . . . . . . . . . . . . . . . . . .

Theoretical results about TLS noise . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4.1

Power law dependence on noise frequency . . . . . . . . . . . . . . . . . . . .

2.4.2

Power law dependence on amplitude of the electric field . . . . . . . . . . . .

2.4.3

Power law dependence on temperature . . . . . . . . . . . . . . . . . . . . . .

2.4.4

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Architecture of the millimeter wave coupler and KID design of LWCam
3.1

3.2

3.3

3.4

Architecture of the millimeter wave coupler . . . . . . . . . . . . . . . . . . . . . . .

3.1.1

Structure of the coupler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1.2

Model of the coupling process . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1.3

Uniform power deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1.4

Efficiency of the coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Why parallel plate structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2.1

TLS noise reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2.2

Elimination of stray light absorption . . . . . . . . . . . . . . . . . . . . . . .

Design of the resonator and readout circuit . . . . . . . . . . . . . . . . . . . . . . .

3.3.1

Components of the resonator and readout circuit . . . . . . . . . . . . . . . .

3.3.2

Transmission coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Procedures of fabrication

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Mapping-speed based optimization of the LWCam design
4.1

4.2

55
58

Optimization of mapping speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1.1

Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1.2

Constant quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1.3

Independent parameters swept during the optimization

. . . . . . . . . . . .

4.1.4

Intermediate variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1.5

Determination of optimal parameters . . . . . . . . . . . . . . . . . . . . . . .

Fundamental tradeoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Conclusion and outlook

Bibliography

List of Figures
1.1

Typical spectra of galaxy emission from the radio to IR wave band. . . . . . . . . . .

1.2

Thermal SZ effect, relativistic, and kinetic corrections. . . . . . . . . . . . . . . . . . .

1.3

Emissivity history of the universe as a function of redshift. . . . . . . . . . . . . . . .

1.4

A design of feedhorn and the predicted band-averaged beam pattern.

. . . . . . . . .

1.5

Single scale and multi-scale phased array antenna. . . . . . . . . . . . . . . . . . . . .

1.6

Principle of KID operation and frequency domain multiplexing. . . . . . . . . . . . . .

2.1

A diagram of the homodyne readout system used for the noise measurement. . . . . .

2.2

Resonance circle and noise ellipse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3

Noise spectra of a KID resonator in the phase (frequency) and amplitude directions. .

2.4

Frequency noise versus internal power. . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.5

Frequency noise versus temperature under several readout power. . . . . . . . . . . . .

2.6

Double potential wells and wave functions of a two-level system. . . . . . . . . . . . .

2.7

Theoretical frequency noise spectral density at several temperatures. . . . . . . . . . .

2.8

Theoretical frequency noise versus the amplitude of electric field. . . . . . . . . . . . .

2.9

Theoretical spectral density of σz at several amplitudes of electric field. . . . . . . . .

2.10

Theoretical frequency noise versus temperture. . . . . . . . . . . . . . . . . . . . . . .

2.11

Theoretical spectral density of σz at several temperatures. . . . . . . . . . . . . . . . .

3.1

Cross-sectional view of the KID device. . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2

Plan view of the KID device’s left end in the top metal (Nb) layer. . . . . . . . . . . .

3.3

Plan view of the KID device’s right end in the top metal (Nb) layer. . . . . . . . . . .

3.4

Plan view of the KID device’s left end in the middle metal (TiN) layer. . . . . . . . .

3.5

Plan view of the KID device’s right end in the middle metal (TiN) layer. . . . . . . .

3.6

Distributed element model of the coupling scheme. . . . . . . . . . . . . . . . . . . . .

3.7

Lumped element model of the coupling scheme. . . . . . . . . . . . . . . . . . . . . . .

xii
3.8

Attenuation length of current in Nb transmission line. . . . . . . . . . . . . . . . . . .

3.9

Symmetric sources of the TiN transmission line. . . . . . . . . . . . . . . . . . . . . .

3.10

Longitudinal attenuation lengths for the 6 millimeter-wave bands. . . . . . . . . . . .

3.11

Adiabatic coupling scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.12

Current density in the Nb transmission lines for the six millimeter-wave bands. . . . .

3.13

Layout of bottom plates in the 16 coupling capacitors. . . . . . . . . . . . . . . . . . .

3.14

Current density in the TiN transmission line for 90 GHz band (unit: A/m). . . . . . .

3.15

Current density in the TiN transmission line for 150 GHz band (unit: A/m). . . . . .

3.16

Current density in the TiN transmission line for 230 GHz band (unit: A/m). . . . . .

3.17

Current density in the TiN transmission line for 275 GHz band (unit: A/m). . . . . .

3.18

Current density in the TiN transmission line for 350 GHz band (unit: A/m). . . . . .

3.19

Current density in the TiN transmission line for 400 GHz band (unit: A/m). . . . . .

3.20

Histogram of the current density from 1 µm×1 µm squares in TiN microstrip. . . . .

3.21

Millimeter wave absorption efficiency of TiN inductor. . . . . . . . . . . . . . . . . . .

3.22

Lumped element model of KID’s readout circuit. . . . . . . . . . . . . . . . . . . . . .

3.23

Layout of the readout circuit (90 GHz band). . . . . . . . . . . . . . . . . . . . . . . .

3.24

Zoomed-in view of the readout circuit (90 GHz). . . . . . . . . . . . . . . . . . . . . .

3.25

Layout of the readout circuit (150 GHz band). . . . . . . . . . . . . . . . . . . . . . .

3.26

Layout of the readout circuit (230 GHz band). . . . . . . . . . . . . . . . . . . . . . .

3.27

Layout of the readout circuit (275 GHz band). . . . . . . . . . . . . . . . . . . . . . .

3.28

Layout of the readout circuit (350 GHz band). . . . . . . . . . . . . . . . . . . . . . .

3.29

Layout of the readout circuit (400 GHz band). . . . . . . . . . . . . . . . . . . . . . .

3.30

Transmission coefficients S11 and S21 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1

Intersections of the two Qi -related functions. . . . . . . . . . . . . . . . . . . . . . . .

4.2

Inductance L and side length lC of the capacitor. . . . . . . . . . . . . . . . . . . . . .

4.3

Filling fraction of live area ALive /AP ixel and number of pixels NP ixel . . . . . . . . . .

4.4

LS
Responsivity and TLS noise equivalent power NEPTphase
.. . . . . . . . . . . . . . . . .

4.5

ot
Fundamental NEP NEPf und and total NEP NEPTphase
. . . . . . . . . . . . . . . . . .

4.6

Critical current density Jc and actual current density J. . . . . . . . . . . . . . . . . .

4.7

Width of TiN inductor/absorber wabs and mapping speed. . . . . . . . . . . . . . . .

4.8

att
Dependence of mapping speed on λatt
T iN /0.5λSi and tabs . . . . . . . . . . . . . . . . . .

4.9

Tradeoff between filling fraction of live area and TLS noise. . . . . . . . . . . . . . . .

xiii

List of Tables
3.1

Impedances of several components evaluated at ν=90 GHz. . . . . . . . . . . . . . . .

3.2

Transverse attenuation lengths in a 1 µm×20 nm TiN microstrip transmission line. . .

3.3

Coupling capacitance Cc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1

Relevant physical constants in the optimization. . . . . . . . . . . . . . . . . . . . . .

4.2

Results of optimization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 1

Background
1.1

Scientific Motivation

1.1.1

Introduction to mm/submm astrophysics

Most of the submillimeter- and millimeter- wave photons in the universe are emitted by the dense
interstellar gases and dusts, which are the “cradles” of new stars[1]. Interestingly, these interstellar
materials are almost transparent to the sub/millimeter waves, but many orders more opaque for
the optical wave bands. Therefore the sub/millimeter wave astronomy is an important platform to
study star and galaxy formations.
Deep surveys at sub/millimeter wavelengths also make it possible to study the characteristics
of galaxies as a function of the red shift. Figure 1.1 shows a typical spectra of emission from the
galaxies. The discovery that the radiation intensity is a fast increasing function of frequency at
sub/millimeter wavelength (∼200-1000 µm) establishes the fact that the flux intensity of emission
from a galaxy measured in this wave band is almost independent of the redshift, which is considerably
different from other wavelengths and opens up the possibility of discovering and studying very distant
galaxies.

Figure 1.1: Typical (normalized) spectra of galaxy emission at frequencies from the radio to IR wave
bands[2].

Moreover, observations at sub/millimeter wavelengths provide important information about
CMB physics. The cosmic microwave background (CMB), which peaks at 2 mm, is the thermal
radiation left over from the Big Bang of the universe. The tiny anisotropy or temperature fluctuation corresponds to regions of slightly different densities, which evolved into the stars and galaxies
of today. Study of CMB physics provides a crucial test of the cosmological models.
Another important application of sub/millimeter wave astronomy is through the Sunyaev-Zeldovich
effect (SZE). SZE states that the CMB photons are inverse Compton scattered to high energy state
when interacting with hot electron gases, resulting in a decrease in the CMB intensity below 218
GHz and a corresponding increase at higher frequencies. This phenomenon is shown in Figure 1.2.
The SZE consists of the thermal component, kinetic correction, and relativistic correction. The
kinetic component of SZE is due to the relative motion of the cluster with respect to the rest frame
of the CMB, so it can provide measurements of cluster peculiar velocities. The thermal SZ spectrum
is temperature-independent in non-relativistic limit (only the amplitude depends on temperature),
but relativistic correction makes its shape temperature dependent. The relativistic SZE correction
therefore provides measurements of cluster temperature.

(arbitrary units)

Figure 1.2: Thermal SZE model spectra (dashed red line) (credit to Mike Zemcov) with y=1 × 10−4 .
The Kompaneets y parameter is defined as y = τ × (kB T /mc2 ), where τ is the optical depth or
the fraction of photons scattered and (kB T /mc2 ) is the electron temperature in unit of the rest
mass of the electron. Also shown is the distorted spectrum (solid black line) after the addition
of relativistic corrections with Te = 5 keV (dashed orange line), and kinetic SZE corrections with
vpec =+500 km/s (dashed blue line). The sub-mm source line (another solid black line) refers to
the typical spectrum of the dust-obscured galaxies. The radio source line (dashed black line) refers
to the galaxies that are bright at radio wavelengths, due usually to large amounts of synchrotron
radiation from electrons accelerated by energetic phenomena like shock waves and active galactic
nuclei (super massive black holes at the centers of galaxies). The color bars refer to typical spectral
bands based on the atmospheric transmission windows.

1.1.2

Scientific motivation of long wavelength imager

We propose a design study for the long-wavelength imager for Cerro Chajnantor Atacama Telescope
(CCAT): LWCam. CCAT is a 25 meter telescope at an excellent cite in Chile and would be one of
the highest permanent, ground-based telescopes in the world. LWCam will cover a 200 field-of-view
in six spectral bands (required for subtraction of radio and mm/submm galaxy foregrounds) from
0.75 to 3.3 mm. The six bands are expected to have 14080, 14080, 3520, 3520, 880, and 880 pixels
with per-pixel sensitivities of 5.9, 3.7, 1.6, 1.8, 1.7, 1.8 mJy s1/2 . The fine angular resolution (0.240
at 1.1 mm), wide field-of-view, broad spectral coverage, and large mapping speed of LWCam will
enable a variety of scientific studies, including the dusty star-forming galaxy (DSFG) population
and the intra-cluster medium (ICM) in galaxy clusters.
The dusty star-forming galaxy population plays a crucial role in galaxy evolution over cosmic

time. It is known that the flux ratio of wavelengths 350 µm to 850 µm S350 /S850 drops from 4-7 at
z (redshift)∼2 to 2-4 at z∼4 and 1-2 at z∼6. Therefore only with the 850 µm data from LWCam,
low-luminosity z∼2 DSFG, and ultra-luminous higher-z sources can be distinguished. Also, the
large pixel counts and high mapping speed of LWCam would yield thousands of z>4 DSFG and
enable the first measurement of high-z clustering, which is to be compared with strong clustering
of lower z-DSFG, and hundreds of z>5 DSFGs that can connect the epoch of dusty star formation
with the end of re-ionization. Finally, the multiple spectral bands of LWCam provide approximate
redshift information and probe the highest redshifts that can most incisively test models of galaxy
formation because of the wavelength-dependent nature of the emissivity history, as shown in Figure
1.3. Note that those sources are so optically obscured that one cannot obtain the redshift by optical
spectroscopy. Submm spectroscopy is quite difficult though, and it is only possible with the Atacama
Large Millimeter/submillimeter Array (ALMA) and on small samples of sources so far.

Figure 1.3: The emissivity history of the universe as a function of redshift at a range of
wavelengths[4].

Imaging in multiple spectral bands in the 80-420 GHz (0.715 to 3.75 mm) range will also enable
new studies of the ICM in galaxy clusters via SZ effects, specifically the mapping of thermal and nonthermal pressure using the thermal SZ effect, the detection and study of high temperature regions
using its relativistic corrections, and the study of unvirialized bulk velocities in the ICM and the
peculiar motions of entire galaxy clusters using the kinetic SZ effect.

1.2

Review of light-coupling architectures

In this section we consider the way in which the mm/submm radiation described in the previous
section can be coupled to the detectors. Among the many existing coupling architectures (direct
absorption, lens-coupled dual-slot-dipole antenna, sinuous antenna, etc.), we will briefly review feedhorn and phased-array antenna.

1.2.1

Feedhorn

A horn antenna is an antenna consisting of a flaring structure that directs the waves in a beam.
The smooth-walled horn can accommodate a wide spectrum of signals since it does not contain any
resonant elements. Horns have the advantages of moderate directivity, simple construction, and
adjustment. The different flare angles and expansion curves also make possible a variety of different
beam profiles. The common types of horns include pyramidal horns, sectoral horns, conical horns,
exponential horns, and corrugated horns. Figure 1.4 shows a picture of a corrugated horn. There
are two techniques for the horn to couple light to the detector: direct absorption and microstrip
coupling. In the direct absorption scheme the absorber/detector is placed in a cavity behind the
horn. In the microstrip coupling scheme, waveguide probes are placed at the output of the horn and
connect to the microstrip that terminates in a detector.

Figure 1.4: Left: A drawing of a preliminary horn design incorporating ring loaded slots[5]. The
zoom shows the geometry of the ring-loaded grooves more clearly. Three photographs show prototype
layers etched using a three layer mask and a deep reactive ion-etch (DRIE) machine. Right: The
predicted band-averaged beam pattern in both the 90 and 150 GHz bands. These patterns were
constructed by simulating the beam pattern at 5 GHz increments and averaging these results within
the predicted detector passband. These simulations show the input reflection to be below -20 dB
and the cross-polarization below -30 dB across both bands.

1.2.2

Phased array antenna

A phased array antenna is an array of antenna, the signal phases of which are varied spatially so
that the combined radiation pattern is reinforced in one direction and suppressed in other directions.
There are dual-polarization single band designs and single-polarization multi-band designs. In both
cases, the light is received through the silicon substrate. The advantages of phased array antenna
include great directivity and excellent steering ability. Figure 1.5 (Left) shows an array of single-scale
slot antennas.

Figure 1.5: Left: schematic layout of single-scale phased-array antenna array showing slot dipoles,
taps, summing tree, bandpass filters, and coplanar waveguide microwave kinetic inductance detectors. Right: conceptual design of multi-scale phased array.

A multi-scale phased array antenna as shown in Figure 1.5 (Right) is an extension of the above
mentioned single-scale phased array antenna. To match the pixel size to the wavelength, larger
pixels at longer wavelengths can be synthesized from the smaller pixels matched to the shortest
wavelengths. For LWCam, it would be optimal to have three scales of pixel size to ensure good Airy
function matching. The bandwidth of the antenna is set by the feed density (smallest wavelength)
and the slot length (largest wavelength). The width of the microstrip at the slots is about 1 µm to
match the slot impedance. The width expands in the summing tree (not trivially: every summing
junction doubles the width, but then it is tapered back down before the next summing junction so
that the tree does not get too wide). The microstrip transmission line at the output of the antenna
is 4 µm wide.
LWCam will use multi-scale phased array antenna for the following reasons.
• The phased array can be fabricated on the same substrate as the detectors so that a separate light
coupling structure is not required.

• One can cover many spectral bands with the same focal plane because the array can be designed
to have a very broad intrinsic bandwidth (up to 10:1).

1.3

Introduction to kinetic inductance detector

Light received at the antenna is transferred to the photon detector. The design of LWCam is based
on the concept of the kinetic inductance detector (KID) due to the excellent multiplexability, which
motivates their use in applications that require a large arrays of detectors. In this section we will
review KID’s principles of operation, limiting factors of sensitivity, and applications.

1.3.1

Principles of operation

Superconductors have zero dc resistance below the transition temperature Tc . At absolute zero
temperature electrons in the superconducting phase stay in the form of Cooper pairs with a bonding
energy 2∆ via phonon mediated attractive interaction. The Cooper pairs accelerate under an external electric field like free electrons and acquire a kinetic energy. Such an energy can be retrieved
by reversing the direction of the electric field. Therefore the exchange between electron kinetic energy and the electromagnetic energy induces a reactive impedance for an ac field, called the kinetic
inductance Lk . On the other hand, the excitations of the BCS ground state, the quasiparticles,
experience dissipation as normal-state electrons, resulting in a real part of the impedance.
The superconductor can be engineered to form the inductor of a LC resonator, whose resonant
frequency f0 and quality factor Qr are determined by the ac impedance Z. When a photon with
sufficient energy hν > 2∆ is absorbed by the superconductor, Cooper pairs will be broken and quasiparticles are created, altering both the imaginary and real parts of the impedance. The change δZ is
therefore translated into a shift δf0 and δQr , which can be read out by examining the transmission
of the probe signal (normally in the microwave band). Such a photon detector is called the kinetic
inductance detector (KID)[6].
The most attractive part of KID is its multiplexing ability. The traditional cryogenic detectors
like transition-edge sensor are generally used with individual preamplifiers and wiring for the output. Multiplexing schemes have been developed along the way but require complex, custom-designed
superconducting electronics, located close to the detector array. In contrast, KID allows a straightforward frequency domain approach to multiplexing. This results in a dramatic simplification of
the detector arrays and the associated cryogenic electronics, making it possible to produce a large

format array involving thousands of detectors.

Figure 1.6: Left: The basic operation of a KID[6]. (a) Photons with energy hν are absorbed in a
superconducting film, producing a number of excitations, called quasiparticles. (b) To sensitively
measure these quasiparticles, the film is placed in a high frequency planar resonant circuit. (c) The
increase in the kinetic inductance and surface resistance of the film following a photon absorption
event pushes the resonance to lower frequency and changes its amplitude. (d) If the detector (resonator) is excited with a constant on-resonance microwave signal, the energy of the absorbed photon
can be determined by measuring the degree of phase and amplitude shift. Right: An example of
frequency domain multiplexed (FDM) KIDs.

1.3.2

Sensitivity

There are generally four types of noises in the kinetic inductance detectors[7, 8]. The first is the
photon noise. The incident millimeter wave photons from the background (dewar, telescope, and
sky) obey Bose-Einstein statistics and have an intrinsic fluctuation in each mode. The second is the
quasiparticle generation-recombination noise. In the steady state under optical loading the detector
maintains a dynamic equilibrium of the quasiparticles. The quasiparticles can recombine to emit
phonons, which is a point Poisson process and induces the fluctuation in quasiparticle density and
recombination noise. The reverse process, in which thermal phonons break Cooper pairs, gives
the generation noise. The first two types of noise are called fundamental noise. The third is the
amplifier noise. KID uses a cryogenic low noise amplifier, either a high-electron-mobility transistor
(HEMT) or a silicon-germanium bipolar-junction transistor (BJT), to amplify the transmitted signal.
The amplifier adds voltage fluctuations to the probe signal when amplifying it, which is usually
characterized by a noise temperature of a few Kelvin. The fourth is the two-level-system (TLS) noise.
There exist extensively the two-level states in amorphous materials, which respond to the external
field and contribute to the dielectric constant. A certain mechanism regarding the TLS, which is
conjectured to be TLS-TLS interaction in the most recent studies[11, 12, 13], causes the fluctuation
in the corresponding dielectric constant and leads to a jittering of KID’s resonant frequency[9]. Since

we are interested in the noise in the measurement of the incoming optical power, we convert all these
four types of noises to“noise-equivalent power” (NEP) or the effective noise on the incoming power
measurement due to each noise component, so that they can be compared with each other. Our goal
is to make the amplifier noise and TLS noise subdominant to the fundamental noise. Experiments
have shown that the amplifier noise can be well suppressed by using a strong probe signal. So
the difficult part is the TLS noise. Various methods have been developed along the way for its
reduction[10].
Initially, KIDs were fabricated from aluminium since it is a simple, well-behaved, and easyto-fabricate superconductor. The kinetic inductance fraction of Al was low (∼ 5%) and so was the
responsivity, which led to the TLS noise that is substantially larger than fundamental noises in noiseequivalent power unit. While it became clear one could reduce TLS noise using large capacitors, this
presented engineering challenges for detectors. So new materials were sought out that could offer
a larger kinetic inductance signal. Recently titanium nitride (TiN) arises as an excellent candidate
for KID materials since it promises greatly improved sensitivity relative to aluminum for a given
resonator geometry as discussed in [31]. Specifically:
• The kinetic inductance of a superconductor is proportional to its normal-state resistivity. The
highly resistive material TiN has a nearly unity kinetic inductance fraction α and leads to a lower
resonant frequency f0 for the resonator:

α=

Lk
Lk
∼1
Ltot
Lk + Lg

(1.1)

f0 = (Ltot C)−1/2 = ((Lk + Lg )C)−1/2
(Lg C)−1/2

(1.2)

where Lg is the geometric inductance which results from the magnetic energy stored in the structure. MKID’s responsivity benefits from both the larger α and lower f0 [7], rendering TLS noise
equivalent power much smaller for fixed resonator quality factor Qr .
• TiN’s high resistivity is a better match to the wave impedance of silicon

µ/Si than aluminum

or other low resistivity materials, making it much easier to build free space coupled KID[15, 16].
Because of these advantages, this thesis focuses on a design for LWCam that uses TiN KIDs1 .
1 It is worth pointing out that the metallic glass, for example NbSi[33], might be a better KID material than TiN.
NbSi has a normal-state resistivity several times larger than TiN. NbSi can make perfect thin film down to 10 nm free
of inhomogeneity (a common problem for TiN KID). The Tc is tunable: 15% silicon might reduce Tc down to 1 K.
We use TiN since its fabrication and testing techniques had been mature at JPL when this design work was carried
out.

1.3.3

Applications

The KID technology have been most intensively applied and tested in the area of sub/millimeter
imaging. The multi-wavelength sub/millimeter inductance camera (MUSIC) is designed to have
2304 detectors in 576 spatial pixels and four spectral bands at 0.87, 1.04, 1.33, and 1.98 mm[14].
The KIDs are made of Al and operate at several GHz. MUSIC is used to to study dusty starforming galaxies, galaxy clusters via the Sunyaev-Zeldovich effect and star formation in our own
and nearby galaxies. MUSIC has been deployed since 2012 at Caltech Submillimeter Observatory
(CSO). MAKO is a scalable 350 µm pathfinder instrument with a prototype of couple of hundreds
of pixels[15, 16]. MAKO uses lumped element kinetic inductance detectors (LEKID) patterned from
TiN films. The resonators are designed to operate at 100 MHz, which presents numerous advantages,
including an improved pixel noise equivalent power, a simplified analog readout circuit, and a higher
achievable multiplexing density. MAKO saw the first light in 2013 at CSO and reached the condition
of photon-noise-limited detection in 2014. Superspec is a ultra-compact spectrometer-on-a-chip for
high redshift observations[17, 18]. It applied the LEKID technology to R∼500 spectrometers covering
the 190-310 GHz band. Both MAKO and Superspec are originally proposed for CCAT too, aiming
ultimately towards 106 detector arrays. KID is also under development for other applications such
as optical/X-ray detection[19] and dark matter search[20, 21].

Chapter 2

Study of two-level-system noise in
dielectric materials
In Chapter 1 we discussed the fact that TLS noise is an important contributor to total NEP for KIDs.
While its behavior is by now well understood phenomenologically[8], providing engineering recipes
for its minimization, a fundamental understanding of TLS noise is desirable. In this chapter we
will first go through the experimentally established properties of TLS noise in KID and introduce
the standard tunneling model that serves historically as the theoretical basis for the many TLSrelevant phenomenons. We will then relate the dielectric constant to the state of each individual
TLS and derive an expression for its power spectral density (PSD). The evolution of individual TLS
that takes into account of the interaction with the phonon bath is subsequently elaborated. Finally
we carry out numerical analysis of the frequency dependence, power dependence, and temperature
dependence of TLS frequency noise and compare it with the data.

2.1

General properties of TLS noise in KID

2.1.1

Noise measurement

A diagram of the experimental set-up for noise measurement in KID is shown in Figure 2.1. A
synthesizer generates a microwave signal with frequency f as the probe. Part of the signal couples
with KID in the fridge, gets amplified (by a HEMT and a room temperature amplifier), and feeds
into the RF (radio frequency) port of an IQ mixer. The rest of the signal goes directly into the
LO (local oscillator) port of the mixer. The output I and Q (audio frequency ports) voltages are
proportional to the in-phase and in-quadrature amplitudes of the transmitted signal.

Figure 2.1: A diagram of the homodyne readout system used for the noise measurement.

When f is varied, the output ξ = [I, Q]T traces a circle in the 2D IQ plane called the resonance
circle as shown in figure 2.2. With f fixed, ξ is seen to fluctuate around its mean and the fluctuations
δξ(t) = [δI(t), δQ(t)]T are digitized for noise analysis. δξ(t) can be projected onto two directions:
the one that is tangent to the circle δξk (t) and the one in the orthogonal direction δξ⊥ (t). δξk (t) and
δξ⊥ (t) correspond to the fluctuations of the phase and amplitude of the resonator’s electric field E.
Their power spectral densities Sf f (ν) and Saa (ν) are therefore measures of the phase (frequency)
and amplitude noises in KID.

Figure 2.2: (a) Resonance circle of a 200 nm Nb on Si resonator at 120 mK (solid line)[8], quasiparticle trajectory calculated from the Mattis-Bardeen theory (dashed line). For this figure, the
readout point ξ = [I, Q]T is located at the resonance frequency fr . (b) Noise ellipse (magnified by
a factor of 30). Other parameters are fr = 4.35 GHz, Qr = 3.5 × 105 (coupling limited), readout
power Pr ≈-84 dBm, and internal power Pint ≈-30 dBm.

2.1.2

Experimental results of TLS noise in KID

Experiments have revealed several characteristic properties of TLS noise in KID[27, 28, 29]: the
frequency noise has a power law dependence on the noise frequency Sf f (ν) ∼ ν −1/2 ; the frequency
−1/2

noise has a power law dependence on the amplitude of the electric field Sf f (ν) ∼ Pint

~ −1 ; the
∼ |E|

frequency noise has a power law dependence on the temperature Sf f (ν) ∼ T −2 for T > 100 mK; the
frequency noise is several orders of magnitude stronger than the amplitude noise Sf f (ν)
Saa (ν).
The data are shown in Figure 2.3 to Figure 2.5.

Figure 2.3: Noise spectra[8] of a 200 nm Nb on Si resonator at 120 mK in the phase (frequency) (solid
line) and amplitude (dashed line) directions. Other parameters are fr = 4.35 GHz, Qr = 3.5 × 105 ,
readout power Pr ≈ −84 dBm, and internal power Pint ≈ −30 dBm.

Figure 2.4: Frequency noise[8] at 1kHz Sδfr (1kHz)/fr2 vs. internal power Pint falls on to straight lines
−1/2
of slope -1/2 in the log-log plot, indicating a power law dependence: Sδfr /fr ∝Pint . Data points
marked “+”, “” and “*” indicate the on-resonance (f = fr ) noise of three different resonators
(with different fr and Qr on the same chip) under four different readout power Pµω . Data points
marked with “◦” indicate the noise of resonator (marked with “*”) measured at half-bandwidth away
from the resonant frequency (f = fr ± fr /2Qr ) under the same four Pµω . The data is measured
from a 200 nm thick Al on sapphire device.

Figure 2.5: Frequency noise at 30 Hz as a function of temperature measured at Pint =-78, -86, -94,
and -102 dBm from Nb on sapphire with SiO2 dielectric microstrip device. At T > 100 mK, the
noise roughly scales as T −2 .

2.2

Standard model of two level systems

With the goal of explaining the TLS noise observed in KIDs, we will develop a model for the fluctuations in the dielectric properties of a medium containing TLS and their impact on the resonator
frequency and dissipation fluctuation spectra.

2.2.1

Localized representation and energy representation

Various experiments regarding the thermal properties and response to external electric field have
shown that the glass can behave in a substantially different manner compared with perfect crystal
at low temperature. The two-level-state tunneling model[22] was initially proposed in 1970s and
subsequently achieved huge success in explaining the experimental results. Such a model postulates
that a broad spectrum of two level systems extensively exists in glass, in which an atom or group
of atoms can occupy one of the two potential minima. The Hamiltonian of the system can be
conveniently expressed using the basis set φ1 and φ2 in the localized representation, where φ1 and
φ2 are the ground state wave functions of the two adjacent potential wells:
1 −∆
H0 = 
2 ∆

∆0 

(2.1)

where ∆ is the asymmetric energy which equals the energy difference between the right and left
potential wells and ∆0 is the tunneling energy.
In the standard TLS model, a uniform distribution in ∆ and a log uniform distribution in ∆0 is
usually assumed:

P (∆, ∆0 )d∆d∆0 =

P0
d∆d∆0
∆0

(2.2)

where P0 is the two level density of states found to be on the order of 1044 /J·m3 . The Hamiltonian
can be diagonalized to obtain the eigenstates. The wave functions of these eigenstates in the position
representation, ψ1 and ψ2 , are shown in Figure 2.6.

H0 =

σz

= (∆2 + ∆20 )−1/2

(2.3)
(2.4)

Figure 2.6: Double potential wells, localized wave functions φ1 and φ2 , and eigenstates ψ1 and ψ2
of a two-level system

2.2.2

Analysis of interaction with external fields

TLS can interact with the external electric field and strain field through the perturbation of the
asymmetric energy ∆.
In the electric problem, we can define the electric dipole moments of the two localized states φ1
~ though the atom
(lower energy state) and φ2 (higher energy state) as ∓d~0 where d~0 = − 21 ∇E~ ∆(E),
~ represents the gradient of the
or group of atoms under investigation has no net charge. Here ∇E~ ∆(E)
~ Note the consistency
asymmetric energy ∆ in the parametric space spanned by the electric field E.
~ is anti-parallel with d~0 , the asymmetric energy
between the definitions of φ1 , φ2 , and d~0 : when E
~ (by the definition of gradient). The interaction Hamiltonian has a simple
∆ is increased by d~0 · E
form in the localized representation:

Hint
= −σz d~0 · E

(2.5)

The same Hamiltonian has a slightly different form in the energy representation:

Hint
=−

∆0
~ ·E
~ = −( 1 σz d~0 + σx d)
σz +
σx d~0 · E

(2.6)

d~0 = 2d~0
is called the permanent electric dipole moment. It is “permanent” because it appears
in the thermodynamic equilibrium state of the TLS, which is a classical state, or in other words, a

17
~ and ∓ 1 d0 can
mixed state with absolutely no coherence. d~0 can be also expressed as d~0 = −∇E~ (E)
be regarded in the same fashion as the electric dipole moments of the two eigenstates ψ1 (ground
∆0
state) and ψ2 (excited state). d~ = d~0
is called the transition electric dipole moment. The
transition dipole moment only appears when the TLS is driven to be in a superposition of ψ1 and
~ is not vanishing). The factors ∆ and ∆0 in d~0 and d~ come from the change of
ψ2 (so that hσx di
basis from localized representation to energy representation.
In the acoustic problem, the interaction Hamiltonian can be similarly written as

Hint
=−

∆0
σz +
σx γe

(2.7)

where γ is the elastic dipole moment and e is the strain field. The interaction Hint
with phonon bath

can both relax the TLS back to its thermodynamic equilibrium state (the mixed state mentioned
above) and cause it to lose the coherence (mathematically hσx i) between ground state and excited
state (real dephasing effect). We characterize the strength of the relaxation by rate T1−1 . The real
dephasing rate is therefore 12 T1−1 according to the master equation. Then, in the absence of any
electric field, we have
dhσz (t)i
hσz (t)i − σzeq ()
=−
dt
T1
σzeq () = − tanh(
2kT
dhσ+ (t)i
hσ+ (t)i
=−
dt
2T1
hσ− (t)i
dhσ− (t)i
=−
dt
2T1

(2.8)
(2.9)
(2.10)
(2.11)

where the time-dependent operators are in the Heisenberg picture, σz represents the difference of
the probabilities in the excited state and ground state, σ+ and σ− represent the coherence. The
solutions are simply exponential decays.
Next we analyze how the state of the TLS would physically evolve under an external electric field
~ 0 (eiωt + e−iωt ). We consider two toy Hamiltonian H rel = − 1 σz d0 · E
~ and H res = −σx d · E,
E(t)
=E
int
int
the two components of the interaction Hamiltonian in Eq. 2.6 in the electric problem.
rel
rel
• Hint
: Hint
causes the eigenenergy of the TLS to oscillate with time sinusoidally with a small

~ 0 cos(ωt). The TLS would be simultaneously relaxed by the interaction
amplitude (t) = − 2d~0 · E
with the phonon bath Hint
toward the instantaneous equilibrium state, which results in a delayed

oscillation of the permanent dipole moment hd~0 σ̂z i also with angular frequency ω. In the low

18
frequency regime ω
T1−1 , the phase delay is approximately zero since the TLS relaxes relatively
fast. In the high frequency regime ω
T1−1 , in analogy to a classical damped harmonic oscillator,
the phase delay should approach π/2. The amplitude of the oscillation of hd~0 σ̂z i would, however,
become increasingly small as the frequency ω goes up since there is not enough time for the TLS
to relax to the instantaneous equilibrium state corresponding to (t). We call such a response the
relaxation response.
res
: This is the standard Rabi problem if we do not take into account of the acoustic relaxation
• Hint
induced by Hint
. The TLS is driven coherently by the external electric field to oscillate between
the ground state and excited state in a harmonic manner with frequency ΩR = Ω2 + ∆2d , where

~ is the Rabi frequency and ∆d = ωL − ωT LS = ω − /~ is the detuning of the electric
~ 0 · d/~
Ω = 2E
field with respect to the TLS. During the Rabi oscillation, the amplitude of the transition dipole
~ x i also swings harmonically between zero (in either ground state or excited
moment of the TLS hdσ̂
state) and the maximum (equal superposition between the ground state and excited state). The
Rabi oscillation will be modified but still continue if Hint
is incorporated. We call such a response

the resonance response.
Let’s move back to the full interaction Hamiltonian Hint
. Since the perturbation in provided
rel
by Hint
is so small that the Rabi oscillation is affected very little, the state of TLS would evolve
res
rel
basically as there is only Hint
. Although Hint
still induces the oscillation in the permanent dipole

moment hd~0 σ̂z i, it is actually minuscule because the microwave signal used by KID is in the high
frequency regime fL ∼ 1 GHz with respect to the dielectric materials with T1 > 1 µs.

2.2.3

Rigorous solution of electric susceptibiities

We can solve the equations of motion of the TLS with the full interaction Hamiltonian Hint
rigor-

ously. Note that the Hamiltonian of TLS in an external electric field

H = H0 + Hint

(2.12)

has a formal analogy to that of a spin 1/2 system in a magnetic field

~ ·S
~ = −~γ(B
~ 0 · S)
~ − ~γ(B
~ 0 · S)
H = −~γ B

(2.13)

19
~ 0 is the static magnetic field, B
~ 0 is the oscillating perturbation field, and S
~ = ~σ /2. We can
where B
identify the following correspondence:

~ 0 = (0, 0, )
−~γ B

(2.14)

~ 0 = (−2d~ · E,
~ 0, −d~0 · E)
−~γ B

(2.15)

Without any relaxation or dephasing process, the dynamic equation for a free spin in a magnetic
field is
d~
~ ×B
S(t) = γ S
dt

(2.16)

where S(t)
can be interpreted as the spin operator in Heisenberg picture.
In order to precisely describe the evolution of the state of TLS, the relaxation process, real
dephasing process, and ensemble dephasing process must all be incorporated. The relaxation process
and real dephasing process have been discussed above. We will focus on the ensemble dephasing
process here. It is known that TLS can interact with each other via strain field. The interaction leads
to a random fluctuation of the eigenenergy δ(t). The coherence operator in the Heisenberg picture
of a free TLS evolves as σ̂+ (t) = σ̂+ (0) exp(−j ~ t). When the TLS-TLS interaction is incorporated,
the independent fluctuations δ(t) will cause destructive interferences among the TLS if we consider
the ensemble average, which can be regarded as an effective dephasing process. We characterize the
strength of such a process by rate (T2ens )−1 ∼ ∆/~, where ∆ is the amplitude of the fluctuation.
We use T2−1 = (2T1 )−1 + (T2ens )−1 to denote the total dephasing rate.
Having all the three processes included, the evolution of the ensemble average of spin operator
hSi i can be described by the following Bloch equations:
hSx i
hSx (t)i = γ(hSy iBz − hSz iBy ) −
=0
dt
T2
hSy i
hSy (t)i = γ(hSz iBx − hSx iBz ) −
=0
dt
T2
hSz i − Szeq [Bz (t)]
hSz (t)i = γ(hSx iBy − hSy iBx ) −
=0
dt
T1

(2.17)

where

Szeq [Bz (t)] =

~γBz (t)
tanh(
2kT

(2.18)

20
is the instantaneous equilibrium value of Sz .
The electric susceptibility tensors for the relaxation response χrel (ω) and resonance response
χres (ω) can be calculated by solving the Bloch equations[8] and substitute the B field components
by their counterparts in the electric problem:

~0
hd~0 i = χrel (ω) · E

(2.19)

~ = χ (ω) · E
~0
hdi
res

(2.20)

dσzeq () 1 − jωT1 ~0 ~0
dd
d 1 + ω 2 T12
σ0
χres (ω) = − z
d~d~
~ ω − ω + jT2−1
ω + ω − jT2−1
χrel (ω) = −

(2.21)
(2.22)

~ is the induced
where 12 hd~0 i is the induced oscillation of the permanent dipole moment and hdi
~ have the same angular frequency ω
oscillation of the transition dipole moment. Both 21 hd~0 i and hdi
as the driving field. σzeq () and σz0 are the expectation values of σ̂z in the steady state without and
with the driving field.
2kT
1 + (ω − ω)2 T22
σz0 =
σ eq ()
1 + Ω2 T1 T2 + (ω − ω)2 T22 z

σzeq () = − tanh(

(2.23)
(2.24)

~ 0 /~ is the Rabi frequency. When the drive amplitude goes to zero,
where ω = /~ and Ω = 2d~ · E
the Rabi frequency vanishes and σz0 → σzeq . When the drive amplitude gets large, σz0 → 0 indicating
equal probability in the excited and ground states.
The susceptibility χres (ω) ∼ d~d~ is a tensor. One of d~ determines the strength of the coupling
~ 0 /~(ω − ω + jT −1 ) and the other d~ is a measure of the transition
with external electric field ∼ d~ · E
dipole moment of the particular TLS. For the same TLS, d~ is fixed and its resonance response is
always along this direction, and the magnitude of the response depends on the alignment between
~ 0 via d~ · E
~ 0 . Similar physical meanings hold for χrel (ω) and d~0 .
d~ and E
The forms of χrel (ω) in the low and high frequency limits are consistent with the analysis in
section 2.2.2:
dσzeq () ~0 ~0
dd
d
dσzeq () 1 ~0 ~0
χrel (ω
T1−1 ) = −j
dd →0
d
ωT1
χrel (ω → 0) →

(2.25)
(2.26)

21
where in the low frequency limit, the phase delay is zero, in the high frequency limit the phase delay
is π/2 and the amplitude of response is vanishing.
The first term ∼ (ω − ω + jT2−1 )−1 and second term ∼ (ω + ω − jT2−1 )−1 of χres (ω) correspond
to the contributions from rotating wave and counter rotating wave components in the precession of
the Bloch vector ~r = (rx , ry , rz ), where rx = 2Re(ρ12 ), ry = 2Im(ρ12 ), rz = ρ22 − ρ11 . ρij with
i = 1, 2 and j = 1, 2 are the elements of the density matrix of the TLS. Geometrically, rx (t) and
ry (t) are the projection of ~r(t) in the x − y plane whose amplitude |~r| is mainly determined by its z
component rz . That is why the susceptibility χres (ω) is proportional to the expectation value of σ̂z .
We are exploring the hypothesis that TLS noise in resonators comes from the fluctuation of the
dielectric constant due to TLS of the KID’s capacitor. We therefore want to consider the electric
susceptibility due to TLS and its fluctuation.

2.3

Model of TLS noise spectral density

In Gao’s thesis[8], the power and temperature dependences of MKID’s resonant frequency f0 and
internal quality factor Qi have been successfully explained with the standard TLS model we reviewed
in the last section. In that work, it is also speculated that the observed TLS noise should be
attributed to the contribution from each individual independently fluctuating TLS and the TLSTLS interaction is weak enough that it does not produce noticeable correlation between different
TLS’s responses to the external electric field. The random state switching of each TLS comes from
either the coupling with phonon bath or some other mechanisms. A specific form for the operator of
the dielectric constant due to TLS ε̂T LS was suggested. Instead of deriving the noise spectral density
from ε̂T LS , Gao proposed an empirical ansatz as the solution, which well matched the experimental
data in some aspects. In this thesis we will examine the validity of ε̂T LS and explore its theoretical
implications for the case that the fluctuation of individual TLS is entirely due to the coupling with
phonon bath.

2.3.1

Model of the dielectric constant due to TLS

We use εT LS to represent the contribution to the overall dielectric constant from the TLS, that is

εtot = εother + εT LS

(2.27)

22
where εother include the contribution from vaccum (1) and from the non-TLS dielectrics present. In
Gao’s thesis the dielectric constant operator ε̂T LS (ω, ~r, t) due to TLS is defined by extending Eq.
2.22 as:

ε̂T LS (ω, ~r, t) = −

rα ∈Vh

d~α d~α δ(~r − ~rα )

σ̂z,α (ω, t)
α − ~ω + jΓα
α + ~ω − jΓα

(2.28)

ε̂T LS (ω, ~r, t) is the dielectric constant operator due to TLS for an external electric field with angular
frequency ω at position ~r and time t. α is the label of the particular TLS. ~rα indicates the position
of the TLS. Vh is the TLS-host volume. α = (∆2α + ∆20,α )1/2 is the energy level separation.
d~α = dˆα d0 ∆0,α /α is the transition dipole moment where dˆα is the dipole orientation unit vector
which is assumed to be random and isotropically distributed. Γα is the real dephasing rate of the
−1
TLS: Γα = 21 T1,α
. Tα = Γ−1
α instead is used in the following derivation. We assume Tα is the same

for all the TLS.
There is, however, an obvious problem with such a definition. Note that Eq. 2.22 only applies
for an ensemble average of TLS in the steady state. Therefore, the abstraction from the steady state
solution for the ensemble average of TLS as given by Eq. 2.22 to the instantaneous operator relation
for a set of individual TLS shown in Eq. 2.28 is mathematically insuffucient and physically incorrect.
Mathematically, when we have an equality between the expectation values for two operators hÂi =
hB̂i, one may not conclude that the two operators are equal. To demonstrate this point from a
physics point of view, let’s consider the evolution of a single TLS starting in its ground state for a
time scale that is much longer than the period of the electric field but much shorter than the period
of Rabi oscillation. First, we know that the dielectric constant due to the single TLS is physically
well-defined and measurable. Secondly, the TLS can be simply treated as being in the ground state
all the time by the assumption (no time allowed to go through the full Rabi oscillation). Eq. 2.28
gives a finite value for hε̂(ω, ~r, t)i since hσ̂z (t)i = −1. But physically the induced transition dipole
moment is vanishing since the TLS is not in any superposition of the ground state and excited state.
Such an obvious discrepancy is just an example of why the definition in Eq. 2.28 is wrong and
should not be used to compute ε(ω, ~r, t)’s time correlation (TLS noise). The right way to define the
instantaneous dielectric constant operator should be through the transition dipole moment operator
of TLS σ̂x rather than σ̂z , which, however, we won’t be able to further study in this thesis.
While there are physical deficiencies of the above choice of operator, we pursue a derivation of
TLS noise expectations for it for two reasons: 1) these deficiencies were not recognized initially and
2) it is valuable to see what this model predicts (and how it fails) so that we may obtain some

23
guidance as to how a correct and complete model of TLS noise can be constructed. So we ignore
the above problem and stick with the definition in Eq. 2.28, from which we can derive the power
spectral density of TLS noise with some straightforward algebra.
Define the average dielectric constant operator due to TLS over the region Vh as
ε̂ave
T LS (ω, t) =

=−

ˆT LS (ω, ~r, t)d~r
dα dα
σ̂z,α (ω, t)
α − ~ω + jΓα
α + ~ω − jΓα

(2.29)

(2.30)

rα ∈Vh

1 X ~ ~
dα dα χα (ω , ω)σ̂z,α (ω, t)
Vh

(2.31)

~(ω − ω + jTα−1 ) ~(ω + ω − jTα−1 )

(2.32)

=−

rα ∈Vh

χα (ω , ω) =

The power spectral density of ε̂ave
T LS (ω, t) is defined as
Sεave
(ν) = lim
T LS
T →∞ 2T

Z ∞

Z T
dt
−T

−∞

ave
dτ e−iντ hε̂ave†
T LS (ω, t + τ )ε̂T LS (ω, t)iss

(2.33)

where the symbol hiss means expectation value in the steady state.
ave
The time correlation ε̂ave†
T LS (ω, t + τ )ε̂T LS (ω, t) can be simplified by averaging dα isotropically and

replacing the sum of TLS with an integration over the density of states. Since only the component
of the induced transition dipole moment along the external electric field matters, we treat ε̂ave
T LS (ω, t)
and its time correlation as a scalar in the following discussion:

ave
ε̂ave†
(2.34)
T LS (ω, t + τ )ε̂T LS (ω, t)
= 2
d~α d~α d~α d~α |χα (ω , ω)|2 σ̂z,α (ω, t + τ )σ̂z,α (ω, t)
(2.35)
Vh
rα ∈Vh
Z Z Z
(~e · d~α )(~e · d~α )(d~α · ~e)(d~α · ~e)|χα (ω , ω)|2 σ̂z,α (ω, t + τ )σ̂z,α (ω, t)ddˆα
d∆0 d∆
Vh
∆0

(2.36)

Z ∆max

Z ∆0,max
d∆

∆0,min

d∆0
∆0

Z π2

sin θdθ cos4 θ

∆0

d40 |χα (ω , ω)|2 σ̂z,α (ω, t + τ )σ̂z,α (ω, t)
(2.37)

where ê is a unit vector indicating the direction of the external electric field, ∆max is the cutoff of the
asymmetric energy ∆, ∆0,max , and ∆0,min are the upper and lower limits of the tunneling energy
∆0 , θ is the angle between the direction of the electric field and the transition dipole moment.

24
Let u = ∆0 / and apply the following changes of variables.
Z ∆max Z ∆0,max

d∆0 d∆ =
∆0

∆0,min

Z max Z 1

√ 0
dud
− u2
umin

(2.38)

The time correlation can be reduced to

ave
ε̂ave†
(2.39)
T LS (ω, t + τ )ε̂T LS (ω, t)
Z max Z 1
Z π2
√ 0
d
u4 du
cos4 θ sin θdθd40 |χα (ω , ω)|2 σ̂z,α (ω, t + τ )σ̂z,α (ω, t) (2.40)
Vh 0
umin u 1 − u
Z max
2 P0 d40
d|χ(ω , ω)|2 σ̂z (ω, t + τ )σ̂z (ω, t)
(2.41)
15 Vh 0

The subscript α in Eq. 2.40 indicates the dependence on , u, and θ of the evolution of the TLS. Note
that the major contribution to the integration over u and θ comes from the region u ∼ 1 and θ ∼ 0,
~ ∼ |d~0 | and d~ k E
~ 0 . Since the dependence of the evolution of
which corresponds to the case that |d|
~ 0 /~ = 2|d~0 ||E
~ 0 |u cos θ, we can safely
the TLS on u and θ is through the Rabi frequency Ω = 2d~ · E
~ 0 | to simplify the calculation. This is
assume all the TLS have the same Rabi frequency Ω = 2|d~0 ||E
a good approximation for our problem. The subscript α is hence dropped from Eq. 2.41.
The power spectral density of ε̂ave
T LS (ω, t) is therefore
2 P0 d40
Sεave
(ν)
T LS
15 Vh

Z max

d|χ(ω , ω)|2 Sσz (ν)

(2.42)

The dielectric constant can be decomposed into the real part and imaginary part:

00
ε̂ave
T LS = ε̂T LS − j ε̂T LS

(2.43)

Their power spectral densities are
2 P0 d40
15 Vh

Z max

2 P0 d40
Sε00T LS (ν) =
15 Vh

Z max

Sε0T LS (ν) =

2.3.2

d|Reχ(ω , ω)|2 Sσz (ν)

(2.44)

d|Imχ(ω , ω)|2 Sσz (ν)

(2.45)

Power spectral density of σ̂z for a single TLS

To calculate the power spectral density Sσz (ν), we need to know the evolution of the density operator
of the TLS, as it contains the information of the state populations and their time correlation. The

25
evolution of the density operator of the TLS can be described by the Heisenberg equation
ρ̂˙ = [Ĥ, ρ̂] + L̂ρ̂

(2.46)

Ĥ is the Hamiltonian of a TLS driven by an external electric field. Ĥ can be written in the interaction
picture as

Ĥ = −∆d σ̂z +

(σ̂+ + σ̂− )

(2.47)

~ 0 /~ is the Rabi frequency.
where ∆d = ωL − ωT LS = ω − ω is the detuning and Ω = 2d~ · E
L̂ is the Lindblad operator that describes the effect on the TLS of its coupling with the phonon
bath[23, 24, 25]. L̂ = L̂(ĉem ) + L̂(ĉabs ) + L̂(ĉd ) can be decomposed into three components:
L̂(ĉem )ρ̂ = ĉem ρ̂ĉ†em − (ĉ†em ĉem ρ̂ + ρ̂ĉ†em ĉem )
L̂(ĉabs )ρ̂ = ĉabs ρ̂ĉabs − (ĉ†abs ĉabs ρ̂ + ρ̂ĉ†abs ĉabs )
1 †
L̂(ĉd )ρ̂ = ĉd ρ̂ĉd − (ĉd ĉd ρ̂ + ρ̂ĉ†d ĉd )

(2.48)
(2.49)
(2.50)

where ĉem , ĉabs , and ĉd correspond, respectively, to the phonon emission induced projection, phonon
absorption induced projection, and ensemble dephasing induced projection. We have

ĉem = [(1 + n)Γ] 2 σ̂−

ĉabs = (nΓ) 2 σ̂+

ĉd = (2Γd ) 2 σ̂z

(2.51)
(2.52)
(2.53)

where ωL is the angular frequency of the electric field or the probe signal (the subscript L indicates
“laser” for general purpose), n = (e~ωL /kB T − 1)−1 is the number of phonon quanta at angular
frequency ωL in the thermal bath at temperature T , Γ = 2n+1
T1−1 is the spontaneous decay rate

of the TLS due to its coupling with the phonon bath, and Γd = (T2ens )−1 = T2−1 − 12 T1−1 is the
ensemble dephasing rate due to TLS-TLS interaction. As we discussed before, such a dephasing
process is only meaningful for the ensemble average of many TLS. Here we consider the evolution of
a single TLS, so Γd should not be included. The following discussion keeps the ensemble dephasing
term as an extra degree of freedom in order to best fit the experimental data.
The terms in Lindblad operator have very clear physical interpretations. Take L̂(ĉem ) as the

26
example. ĉem ρ̂ĉ†em represents the increase rate of the ground state probability due to phonon emission
induced decay from the excited state. − 21 (ĉ†em ĉem ρ̂ + ρ̂ĉ†em ĉem ) represents the decrease rate of the
excited state and superposition probability in the same process.
Having the equations of motion for the evolution of the density operator of the TLS, we can
compute Sσz (ν) very efficiently with the aid of quantum optics toolbox[26].

2.4

Theoretical results about TLS noise

In this section we examine the theoretical dependences of the normalized TLS frequency noise
−1
2 P0 d0
spectral density Sεn0 = Sε0T LS (ν) 15
on noise frequency ν[27], amplitude of the electric field
Vh
|E|[28],
and temperature T [29]. We adjust the two independent free parameters Γ and Γd in the
above model to match the experimental data.

2.4.1

Power law dependence on noise frequency

Figure 2.7 shows the theoretical normalized TLS frequency noise spectral density for fL = 1 GHz,
Γ = 100 Hz, Γd = 100 Hz, Ω = 0 Hz (zero electric field) as a function of noise frequency ν at several
temperatures from T = 1~ωL /kB (50mK) to T = 10~ωL /kB (500mK). For the curve corresponding
to the lowest temperature, we observe power law dependence Sεn0 (ν) ∼ ν −1/2 for ν = 10 ∼ 1000Hz,
which is qualitatively consistent with the experimental data as shown in Figure 2.3.

Figure 2.7: Theoretical frequency noise spectral density for fL = 1 GHz, Γ = 100 Hz, Γd = 100 Hz,
Ω = 0 Hz at several temperatures.

2.4.2

Power law dependence on amplitude of the electric field

Figure 2.8 shows the theoretical normalized TLS frequency noise spectral density for fL = 1 GHz,
Γ = 100 Hz, Γd = 100 Hz, T = 1~ωL /kB at noise frequency ν = 50 Hz as a function of Rabi
~ We observe power law dependence S n0 (ν) ∼ Ω−1 for Ω = Γ ∼ 10Γ = 100 ∼ 1000
frequency Ω ∼ |E|.
Hz, which is qualitatively consistent with the experimental data as shown in Figure 2.4.
The underlying physics of the monotonic dependence of TLS frequency noise on amplitude of
the electric field is illustrated by Figure 2.9: the electric field shifts the peak of the spectrum of
σz by roughly the Rabi frequency Ω. Therefore the frequency range of interest for astronomical
observation (below 1000 Hz) moves further toward the tails of the spectrum, resulting in the ∼ Ω−1
dependence.

Figure 2.8: Theoretical frequency noise versus Rabi frequency for fL = 1 GHz, Γ = 100 Hz, Γd = 100
Hz, T = 1ωL =50 mK, ν = 50 Hz.

Figure 2.9: Theoretical spectral density of σz for fL = 1 GHz, Γ = 100 Hz, Γd = 100 Hz, T = 1ωL =50
mK at several Rabi frequencies.

2.4.3

Power law dependence on temperature

Figure 2.10 shows the theoretical normalized TLS frequency noise spectral density for fL = 1 GHz,
Γ = 100 Hz, Γd = 100 Hz, Ω = 0 Hz (zero electric field) at noise frequency ν = 50 Hz as a function of
temperature. We observe power law dependence Sεn0 (ν) ∼ T −2.5 from T = 1~ωL /kB to 10~ωL /kB ,
which is qualitatively consistent with the experimental data as show in Figure 2.5.
The underlying physics of the non-monotonic dependence on temperature is illustrated by Figure
R∞
2.11. From the definition of power spectral density, we know that −∞ Sσz (ν)dν = 2πhσ̂z2 iss . In
the low temperature regime T ≤ ~ωL /kB when the TLS stays close to the ground state most of
the time, the variance hσ̂z2 iss increases as temperature goes higher and so does Sσz (ν). In the high
temperature regime T ≥ ~ωL /kB when the TLS stays in the ground state and excited state with
equal probability, the variance hσ̂z2 iss ≈ 1, but the bandwidth of the spectrum grows proportionally
to the temperature. So Sσz (ν) for a fixed ν would decrease as temperature goes higher.

Figure 2.10: Theoretical frequency noise versus temperature for fL = 1 GHz, Γ = 100 Hz, Γd = 100
Hz, Ω = 0 Hz, ν = 50 Hz.

Figure 2.11: Theoretical spectral density of σz for fL = 1 GHz, Γ = 100 Hz, Γd = 100 Hz, Ω = 0
Hz, ν = 50 Hz at several temperatures.

2.4.4

Discussion

We have successfully chosen the values for Γ1 and Γd to match the theoretical predictions of TLS
frequency noise with experimental data. However these “optimal” values ∼100 Hz are orders of
magnitude off from the actual values ∼10 to 100 kHz for Γ estimated with Fermi’s golden rule[30]
and 10 MHz for Γd , implying that the demonstrated consistency might just be pure coincidence.
We can also derive the relative magnitude between the TLS frequency noise and amplitude noise
by comparing the integrals in Eq. 2.44 and Eq. 2.45.
Z max
Sε0T LS (ν) ∼

d|Reχ(ω , ω)|2 Sσz (ν) ∼ Tα ∼

Z max

d|Imχ(ω , ω)|2 Sσz (ν) ∼ Sε00T LS (ν) (2.54)

Therefore, the model predicts that the frequency noise is comparable with amplitude noise. However,
the experimental data shows that they are generally off by several orders of magnitude as illustrated
by Figure 2.3, revealing another discrepancy.
Such inefficacy of the model can be attributed to the bad “abstraction” discussed in section
2.3.1. To understand the intrinsic noise of a single TLS free of interaction with other TLS, we
might have to numerically simulate a long time series of the TLS’s response to the external electric
field. Recent studies suggest that to fully explain TLS noise the TLS-TLS interaction must also

31
be incorporated[11, 12]. Further work is still needed to justify this theory, such as whether it can
reasonably relate material property to the absolute strength of the TLS noise.

Chapter 3

Architecture of the millimeter wave
coupler and KID design of LWCam
In this chapter we describe 1) the architecture of the millimeter wave coupling from the microstrip
transmission line exiting the antenna to the KID inductor and 2) the design of the KID resonator
(inductor and capacitor) and its coupling with the microwave readout line.

3.1

Architecture of the millimeter wave coupler

In Chapter 1 we briefly explained the way that the millimeter light is received by the phased array
slot antenna. A signal is created upon the absorption of a photon in the microstrip transmission
line that exits the antenna. The primary prior implementation of coupling of millimeter wave
from microstrip transmission line to a KID was in MUSIC[14]. MUSIC uses single-scale, singlepolarization phased-array antennas, covering four spectral bands (150, 220, 290, and 350 GHz).
It uses KIDs that combine a coplanar waveguide (CPW) inductor with an interdigitated capacitor
(IDC). The majority of the KID is niobium, but there is a 350 µm length of aluminum at the shorted
end of the CPW. To couple from microstrip transmission line to the Al KID, the Nb microstrip runs
over this Al section. Since Al with Tc ∼ 1 K is absorptive for photons with hν > 2∆, the millimeter
wave power can be dissipated in Al to break Cooper pairs, modifying its kinetic inductance as
described in earlier sections.
For LWCam, we seek to use TiN KIDs because of their higher responsivity and thus greater
prospects for fundamental-noise-limited sensitivity. However, the above mentioned coupling scheme
does not work for the highly resistive material TiN as opposed to Al. Normally the impedance of
Nb microstrip transmission line with dimensions of a few microns is on the order of 50 Ω. Switching
to TiN ground plane dramatically increases this value by a factor of ten[31] as given in Table 3.1.

33
The resultant impedance mismatch would cause reflection of a major fraction of the millimeter wave
power from the antenna. The topic of this section is therefore about how to efficiently couple the
power from the Nb microstrip transmission line to the TiN inductor/absorber[34].
We will first illustrate the general structure of the coupling circuit, including the constituent
materials and geometries. The coupling process is subsequently modeled both analytically and with
finite element simulation. Also in order to maximize the KID responsivity, the millimeter wave
power must be guaranteed to deposit uniformly over the entire absorber. Special features involving
a power splitter and half-wavelength phase shifter are proposed to address this issue. At last we
present a study about the coupling efficiency over wide (several tens of GHz) bands to validate the
overall performance of the design. Throughout this section, the “coupling capacitor” refers to the
millimeter wave coupling capacitor unless specifically emphasized by “readout coupling capacitor”.

3.1.1

Structure of the coupler

We will first describe the structure following the cross-sectional view as given by Figure 3.1. The
device/coupler is composed of three dielectric layers and three patterned metal layers. Several functional components are formed within this structure: KID inductor/absorber, KID (parallel-plate)
capacitor, microstrip transmission line from the antenna, and (parallel-plate) coupling capacitor (its
role will be elaborated on in the latter part of this section). The thicknesses and materials of these
layers (from top to bottom) are listed together with descriptions of their roles in the corresponding
functional components.
• 150nm Nb: microstrip from the antenna, top plates of coupling capacitor
• 270nm amorphous Si1 : dielectric material between the plates of coupling capacitor, dielectric
material between the microstrip from the antenna and its ground plane
• 150nm Nb/20nm TiN: top plate of KID capacitor/KID inductor, bottom plates of coupling capacitor
• 800nm amorphous Si: dielectric material between the top plates and floating virtual ground of
KID capacitor, dielectric material between microstrip from the antenna and its ground plane
• 200nm Nb: ground plane of the microstrip transmission from the antenna, ground plane of TiN
microstrip transmission line, floating virtual ground of KID capacitor
1 We use hydrogenated amorphous silicon α:Si-H since hydrogen can significantly reduce the dangling bonds and
thus two level systems in the amorphous silicon, improving KID’s sensitivity.

34
• crystalline Si: substrate

Figure 3.1: Cross-sectional view of the KID device.

Next we will describe the structure layer by layer following the plan view. In the top metal
layer the 4 µm wide microstrip that carries the millimeter wave signal generated from the antenna
splits into two branches with equal width. The upper (in the plan view) branch is attached to a
row of 3 µm × 6 µm rectangles separated by 1 µm wide gap. Those are the top plates of the
coupling capacitors. The lower branch (in the plan view), instead of being directly connected to the
rectangles, first goes through a meandered region with length of half of a wavelength, which works
as a phase shifter. Figure 3.2 is the plan view of the left end in the top metal layer. Both of the two
branches are left unterminated at the right ends, as shown in Figure 3.3.

Figure 3.2: Plan view of the KID device’s left end in the top metal (Nb) layer. Solid red is Nb in
the top layer and dashed green is the projection of TiN from the middle layer.

Figure 3.3: Plan view of the KID device’s right end in the top metal (Nb) layer. Solid red is Nb in
the top layer and dashed green is the projection of TiN from the middle layer.

In the middle metal layer, the 1 µm wide meandered TiN microstrip forms the KID inductor/absorber and a lossy transmission line with the Nb ground for the millimeter wave. A row of
polygons with varying sizes are attached to the microstrip, which are the bottom plates of the
coupling capacitors. The plan views of the middle TiN layer are shown in Figure 3.4 and 3.5.

Figure 3.4: Plan view of the KID device’s left end in the middle metal (TiN) layer. Dashed red is
the projection of Nb from the top layer and solid green is TiN in the middle layer.

Figure 3.5: Plan view of the KID device’s right end in the middle metal (TiN) layer. Dashed red is
the projection of Nb from the top layer and solid green is TiN in the middle layer.

3.1.2

Model of the coupling process

The basic idea of the coupling scheme is to establish a quasi-continuous capacitive coupling Cc
between the Nb microstrip transmission line from the antenna and the TiN microstrip transmission
line. We make Cc by creating a series of parallel-plate coupling capacitors, with the top plates (3
µm×6 µm rectangles as mentioned above) attached to Nb microstrip in the top metal layer and
bottom plates (polygons as mentioned above) attached to TiN microstrip in the middle metal layer.
A distributed element model is shown in Figure 3.6.

36
ZN b

ZN b

Cc1

Cc2

Cc3

Cc4

ZT iN ZT iN ZT iN ZT iN ZT iN ZT iN ZT iN ZT iN ZT iN
Cc1

Cc2

Cc3

Cc4

ZN b

Figure 3.6: Distributed element model of the coupling scheme. ZN b denotes the impedance of Nb
transmission line and ZT iN denotes the impedance of TiN transmission line.

We will analyze the coupling mechanism qualitatively next based on the lumped element model
as shown in Figure 3.7 and sort out which physical factor dominates the coupling process and
determines the efficiency. The several relevant parameters are listed below.
• At =18 µm2 : area of coupling capacitor’s top plate; the subscript t means “in the top metal layer”
• Am : area of coupling capacitor’s bottom plate; the subscript m means “in the middle metal layer”
• tt : thickness of the top dielectric layer
• tm : thickness of the middle dielectric layer
• CN
b : capacitance between Nb microstrip and ground plane induced by the rectangle

• CTg iN : capacitance between TiN microstrip and ground plane induced by the polygon

37
ZN b
ZN b

ZT iN

CTg iN

ZT iN

Figure 3.7: Lumped element model of the coupling scheme. ZN b denotes the impedance of Nb
transmission line and ZT iN denotes the impedance of TiN transmission line.

The capacitances can be expressed in terms of the geometric parameters:
Am
Si At Am
tt
tt At
Si At
CN
b = Si
tt + tm
tt 1 + tm /tt
Si t Am
CTg iN = Si
tm
tt At tm /tt
Cc = Si

(3.1)
(3.2)
(3.3)

Table 3.1 shows the impedances of relevant components in Figure 3.7.
Component
Impedance (Ω)

ZN b
50

CN
0.1 × 104

Cc
0.4 × 104

CTg iN
1.3 × 104

ZT iN
800

Table 3.1: Impedances of relevant components evaluated at ν=90 GHz, tt = 270 nm, tm = 800 nm
and Am =1 µm2 .

Let’s consider the millimeter-wave signal generated at the antenna traveling along Nb transmission line with amplitude V0 . At the junction (point A) where Nb microstrip is capacitively coupled
to TiN microstrip via Cc , a fraction of the power is transferred into TiN transmission line. Since

38
the impedance between A and B, mainly contributed by Z(Cc ), is much larger than ZN b , such a
fraction is very small, ensuring that no impedance mismatch is created and the millimeter wave can
continue to propagate down the Nb microstrip transmission line.
Given that the coupling is weak, we know VAC ≈ V0 . The power dissipation in the TiN transmission line is therefore

P = I 2 ZT iN = (V0 /Z(Cc ))2 ZT iN ∝ t−2
t tm

(3.4)

We neglect the Z(CTg iN ) term since Z(CTg iN )
ZT iN . So the ratio of dielectric thicknesses tt /tm
also affects the coupling strength. We simulated the attenuation length of current in Nb transmission
line for tt /tm = 3 and tt /tm = 1/3 with several sizes of coupling capacitor. tt + tm = 1080 nm is
fixed. The results are shown in Figure 3.8. For size of 1 µm2 , the ratio between power attenuation
lengths is (22 mm/6.3 mm)2 ∼ 12, qualitatively consistent with the above analysis as in Eq. 3.4:
((3−2 × 1)/(1−2 × 3))−1 = 27. The actual design takes tt /tm =270 nm/800 nm=1/3, as it provides
the right coupling strength.

Figure 3.8: Attenuation length of current in Nb transmission line for tt /tm = 3 (left) and tt /tm = 1/3
(right).

3.1.3

Uniform power deposition

KID’s responsivity (fractional shift in resonant frequency per unit optical power) is maximized only
with the quasi-particles generated uniformly within the entire inductor, because in the recombinationlimited regime uniform distribution of quasiparticles yields the longest effective lifetime for a given
input optical power, inductor volume, and choice of material. We must therefore make sure that
the power dissipation in the TiN meandered microstrip is uniform both vertically and horizontally

39
(from the plan view).
ZN b

ZN b

Cc1

ZN b

Cc2

ZN b

Cc3

ZN b
Cc4

ZT iN ZT iN ZT iN ZT iN ZT iN ZT iN ZT iN ZT iN ZT iN
Cc1
ZN b

Cc2
−V1

ZN b

Cc3
−V2

ZN b

Cc4
−V3

ZN b

−V4

ZN b

Figure 3.9: Symmetric sources of the TiN transmission line.

In the vertical direction the 50/50 power splitter and half-wavelength shifter introduced earlier in
this chapter are necessary to reach this goal. As illustrated in Figure 3.4 and Figure 3.9, the two ends
of each vertical section of TiN meandered microstrip are capacitively coupled to the upper and lower
Nb microstrips, respectively. The splitter and shifter make the voltages at the two Nb microstrips
equal in magnitude but opposite in sign. Therefore the power dissipation in TiN is guaranteed to be
symmetric around the central line. Moreover, in order for the power to be absorbed uniformly in the
vertical direction, the length of each vertical TiN section must also be shorter than or comparable to
the power attenuation length. According to transmission line theory, we have that the propagation
constant and field attenuation length are

γ = α + jβ =
λatt = 1/α

(R + jωL)(G + jωC) ≈

jRωC

(3.5)
(3.6)

where R, L, G, and C are the resistance, inductance, conductance, and capacitance per unit length
of the transmission line, and λatt is the attenuation length. We refer to this attenuation length as
transverse attenuation length in the following discussion. The transverse attenuation lengths of a 1
µm-wide 20 nm-thick TiN microstrip on top of 800 nm amorphous Si are listed for six millimeter
wave bands in Table 3.2.
In the horizontal direction the coupling capacitance Cc is adiabatically adjusted along the Nb
transmission line by changing the areas of the bottom plates (a series of polygons as shown in Figure
3.13). The coupling strength can be characterized by its equivalent attenuation length of power in

40
Parameter
ν (GHz)
R (Ω/m)
ωL (Ω/m)
G (/Ωm)
ωC (/Ωm)
λatt (µm)

band 1
90
5 × 107
9.0 × 104
74
23

band 2
150
5 × 107
1.5 × 105
126
18

band 3
230
5 × 107
2.3 × 105
195
15

band 4
275
5 × 107
2.8 × 105
232
13

band 5
350
5 × 107
3.5 × 105
289
12

band 6
400
5 × 107
4.0 × 105
333
11

Table 3.2: Transverse attenuation lengths in a 1 µm×20 nm TiN microstrip transmission line.

the Nb transmission line. We refer to this attenuation length as the longitudinal attenuation length.
We can straightforwardly derive the relation between coupling strength and position needed to
obtain uniform power absorption: the power P in Nb transmission line should decrease linearly as
the millimeter wave travels:

P (L) = P0 (1 −

dP (L)
P (L)
=−
dL
λ(L)

(3.7)
(3.8)

where P (L) and λ(L) are the power and longitudinal attenuation length in Nb transmission line at
position L, P0 = P (0) is the power at the starting point, and L0 is the length of Nb section that is
coupled with TiN. So we have
1 dP
=−
λ(L)
P dL
1 P0
L0 P
L0 1 − L/L0

(3.9)

We simulated the electric current density profiles in Nb microstrip with coupling capacitors
from 1 µm2 to 16 µm2 , as shown in Figure 3.13. We then fit this profile to a lossy open-ended
transmission line to extract the longitudinal attenuation length. The results are summarized in
Figure 3.10. Different lengths of vertical section of meandered TiN microstrip are used for different
millimeter wave bands to facilitate the “coupling strength engineering”, as will be discussed next.

Figure 3.10: Longitudinal attenuation lengths for the 6 millimeter-wave bands.

To obtain the desired linear power absorption profile, we used the results of the above simulation
to pick the coupling capacitor area that best matches the desired attenuation length (Eq. 3.9) as
a function of position along the Nb transmission line for each band individually. The results are
shown in Figure 3.11. The actual profile of coupling strength (staircase blue line) aligns well with
the ideal profile (green line).

Figure 3.11: Adiabatic coupling scheme. The staircase blue line indicates the actual profile of
coupling strength, the green line indicates the ideal profile, and the red line indicates 1/λ(0) in Eq.
3.9.

The resultant electric current density profiles in the two (upper and lower) Nb microstrips are
illustrated in Figure 3.12 for the six bands. Such a profile is in sharp contrast to the exponential
profile that one would expect for a constant coupling strength over the entire Nb transmission line.
Here the power decays approximately uniformly, validating the above engineering idea of “quasicontinuous coupling”. There are several other noticeable features in the profile too. First, the

43
current vanishes at the right end of Nb microstrip, as required by the open termination condition.
Secondly, the oscillation on the scale of a wavelength comes from the interference between right
traveling wave and its reflection at the open end: because of the discretization of the capacitor size
in units of 1 µm2 , the attenuation length profile does not perfectly match the desired one, and the
power profile is not perfectly linear, and thus some unabsorbed power remains at the end and is
reflected, creating the this long length scale interference pattern. Thirdly, the oscillation on the scale
of a few microns is due to the quasi-continuity of the coupling: the coupling consists of discrete steps,
creating small reflections at the points where such steps occur. Fourthly, there are some “smooth”
regions. It is found that these regions are coupled only with capacitors marked in red in Figure 3.13.
The actual mechanism needs further investigation but is unimportant to our goal here.

Figure 3.12: Current density in the Nb transmission lines for the six millimeter-wave bands.

Figure 3.13: Layout of bottom plates in the 16 coupling capacitors.

The electric current density profiles in TiN microstrip for the six bands are shown in Figure 3.143.19. The density in each vertical section is very uniform because the length is set to be comparable

45
with the transverse attenuation length. In the horizontal direction the wavelength-scale oscillations
shown in Figure 3.12 are also clearly visible.

Figure 3.14: Current density in the TiN transmission line for 90 GHz band (unit: A/m).

Figure 3.15: Current density in the TiN transmission line for 150 GHz band (unit: A/m).

Figure 3.16: Current density in the TiN transmission line for 230 GHz band (unit: A/m).

Figure 3.17: Current density in the TiN transmission line for 275 GHz band (unit: A/m).

Figure 3.18: Current density in the TiN transmission line for 350 GHz band (unit: A/m).

Figure 3.19: Current density in the TiN transmission line for 400 GHz band (unit: A/m).

We can examine the distribution of electric current density in TiN microstrip by creating a
histogram, as given by Figure 3.20. Since the quasiparticle density goes like P in the recombination
limited regime, the spread in electric current density gives the spread in quasiparticle density. The
distribution has a sharp peak and thus small standard deviation, indicating a uniform deposition of
the millimeter wave power.

Figure 3.20: Histogram of the current density from 1 µm×1 µm squares in TiN microstrip.

3.1.4

Efficiency of the coupling

The power absorption efficiency of TiN inductor for the six bands is shown in Figure 3.21. The
efficiency is generally more than 80% over the entire wide continuous band, attesting to the efficacy
of the coupling scheme. The less than 20% reflection is due to slight impedance mismatch at the
power splitter. If the microstrip from the antenna is 6 µm instead of 4 µm wide and still splits
into two 2 µm wide branches, then the efficiency will approach 100%. It is straightforward to

47
adiabatically widen the microstrip between the antenna and the power splitter to 6 µm, so this
near-100% efficiency can be easily obtained.

Figure 3.21: Millimeter wave absorption efficiency of TiN inductor.

3.2

Why parallel plate structure

One can make the KID capacitor using an interdigitated structure or a parallel-plate structure. The
interdigitated capacitor (IDC) has well controlled TLS noise[15, 16], and it is clear how to design such

48
a capacitor to render TLS noise subdominant. However, an IDC acts as an antenna, directly receiving
mm-wave radiation and routing it to the inductor in a way that bypasses the antenna. Though there
have not been lots of tests on TLS noise properties of parallel-plate capacitor, we believe, based on
TLS noise data[36] of the microstrip half wavelength resonator/KID using hydrogenated amorphous
silicon, that the TLS noise of a parallel-plate capacitor with the same dielectric material can be
made acceptably low too, therefore making it a promising substitute for IDC without any risk of
stray light absorption. We thus choose parallel plate capacitor for the design of LWCam. We will
discuss in detail its potential advantages in terms of TLS noise reduction and elimination of stray
light absorption next.

3.2.1

TLS noise reduction

As we reviewed in Chapter 2, experiments have established the fact that KID’s phase/frequency
noise due to TLS is inversely proportional to the amplitude of electric field and number of TLS
fluctuators[8]:
Sδfr (ν)
~ T LS
~ host
fr
|E|N
|E|V

(3.10)

~ is the microwave electric field, NT LS is the number of independent TLS fluctuators, and
where E
Vhost is the volume of dielectric material that hosts the many TLS. The parallel-plate structure have
significant advantages in terms of TLS noise reduction in both aspects: the electric field within the
capacitor can be made arbitrarily strong in principle with a sufficiently thin dielectric layer (though
practically there is a lower limit on the dielectric thickness due to the impedance constraint on the
Nb microstrip transmission line); the structure can easily extend to large area in fabrication.

3.2.2

Elimination of stray light absorption

Stray light absorption in KID’s inductor has three major impacts on its performance: increasing
the optical loading and therefore impairing the responsivity (as will be discussed in Chapter 4)[35],
adding extra photon noise, and circumventing the spectral and spatial filters. The first two factors
would further make it harder for KID to reach the condition of background limited detection.
KID in traditional designs[14] with an IDC suffers from stray light in two ways. First, photons
from the environment can be directly picked up by IDC due to its similar structure as the slot
antenna and arrive at the inductor. Secondly, any stray light that makes it way around to the

49
backside of the wafer can bounce around and be absorbed directly by the inductor. The parallel
plate capacitor adopted in our design can nicely prevent these from happening. It is obvious that
such a structure won’t collect light as an antenna and no light can leak through either. In addition,
~ = 0) condition,
the ground plane only 800nm below the inductor establishes a short circuit (E
rendering itself a good reflector of free-space electromagnetic field and protecting the inductor.

3.3

Design of the resonator and readout circuit

3.3.1

Components of the resonator and readout circuit

The KID (normally microwave) resonator consists of an inductor L and a capacitor C. The readout
line is a standard CPW with 50Ω impedance. The resonator is capacitively coupled with the readout
line via a coupling capacitor Cc as shown in Figure 3.22. Note that “coupling capacitor” and “Cc ”
will refer to the readout coupling capacitor throughout this section and is distinct from the millimeter
wave coupling capacitor discussed in section 3.1. Both C and Cc are created by forming a parallel
plate structure in the middle and bottom metal layers. The information about millimeter wave
intensity is contained in the shift of the resonant frequency, which can be accessed and acquired
by sending a (normally microwave) probe signal near the resonance through the readout line. Qi
denotes the internal quality factor of the resonator and Qc denotes the external quality factor. Qi
mainly comes from the internal dissipation due to the finite quasi-particle density in the inductor,
but could also be limited by metal quality or radiation. It is nonetheless not a problem for the
resonator geometry discussed here[31]. Qc comes from the coupling with the readout line via Cc .
The values of L, C, Qi , and Qc for six millimeter wave bands are determined by an optimization
aiming at maximizing the mapping speed, as will be discussed in the next chapter. Cc can be derived
based on the following relation[17]. The numbers are listed in Table 3.3.

Qc =

8C
ω0 Ce2 Z0

Ce = (Cc Cg )/(Cc + Cg )

(3.11)
(3.12)

where Cg is the capacitance between the inductor (TiN microstrip in our case) and ground plane,
ω0 is the angular frequency of the probe signal, and Z0 is the impedance of the readout CPW.
The dimensions of the resonator and readout circuit for six millimeter wave bands are shown in
Figure 3.23 to 3.29. Probe signal travels from port 1 to port 2, coupling with the resonator via Cc .

50
The ratio of center strip width to the width of gap in CPW is fixed as 3:2 to obtain 50 Ω impedance
for matching with the external readout cabling.
Parameter
ν (GHz)
C (pF)
Cc (pF)
Ce (pF)
Qi

band 1
90
27
1.16
0.619
3.9 × 104

band 2
150
17
1.37
0.518
3.1 × 104

band 3
230
18
1.66
0.575
2.2 × 104

band 4
275
2.51
0.375
2.3 × 104

band 5
350
4.89
0.404
1.8 × 104

band 6
400
11
15.3
0.520
1.4 × 104

Table 3.3: Coupling capacitance Cc . ν refers to the millimeter wave center frequency.

Cc
Z0

Figure 3.22: Lumped element model of KID’s readout circuit.

Figure 3.23: Layout of the readout circuit (90 GHz band) in the middle (left) and bottom (right)
metal layers (unit: µm). Green part is TiN and red part is Nb.

Figure 3.24: Zoomed-in view of the readout circuit (90 GHz) in the middle metal layer (unit: µm).
Green part is TiN and red part is Nb.

Figure 3.25: Layout of the readout circuit (150 GHz band) in the middle (left) and bottom (right)
metal layers (unit: µm). Green part is TiN and red part is Nb.

Figure 3.26: Layout of the readout circuit (230 GHz band) in the middle (left) and bottom (right)
metal layers (unit: µm). Green part is TiN and red part is Nb.

Figure 3.27: Layout of the readout circuit (275 GHz band) in the middle (left) and bottom (right)
metal layers (unit: µm). Green part is TiN and red part is Nb.

Figure 3.28: Layout of the readout circuit (350 GHz band) in the middle (left) and bottom (right)
metal layers (unit: µm). Green part is TiN and red part is Nb.

Figure 3.29: Layout of the readout circuit (400 GHz band) in the middle (left) and bottom (right)
metal layers (unit: µm). Green part is TiN and red part is Nb.

3.3.2

Transmission coefficients

The transmission coefficient S21 of a standard resonator coupled to a readout line can be expressed
as follows. We say the resonator is critically coupled when Qr = Qc /2 and at critical coupling we
have |S21 | = |S11 | = 1/2.

S21 (f ) = a(1 −

Qr /Qc ejφ0
1 + 2jQr ((f − f0 )/f0 )

(3.13)

54
where a is a complex constant accounting for the gain and phase shift through the system, Qr is the
total quality factor, Qc is the external quality factor, φ0 is the rotation factor, f0 is the resonant
frequency, and f is the frequency of the probe signal.
KID’s responsivity dS21 /dPopt and internal power (and thus amplitude of electric field in the
capacitor) are both maximized when the resonator is critically coupled[8] and our circuits are therefore designed to meet this criteria. We simulated the transmission coefficients from port 1 to port
2 and the results are shown in Figure 3.30. We see that such a requirement is satisfied. The small
deviations can be attributed to the non-ideality of the components with a finite size.

Figure 3.30: Transmission coefficients S11 and S21 , voltage ratios of reflected and transmitted signals
to input signal.

3.4

Procedures of fabrication

We anticipate fabricating the structure on high-resistivity silicon substrates as follows:
1. A 150-nm thick layer of Nb will be deposited by DC-magnetron sputtering and patterned by
plasma etching for the ground plane. The patterning will define the KID and readout coupling

56
capacitor bottom electrode islands. It will also define the edges of the ground plane of the
CPW readout feedline. This layer also serves as the ground plane for phased-array antenna,
and so slots will be cut in the ground plane for the antenna slots. A border around three
edges of the device will also be removed to allow a gold heat-sinking layer that makes direct
contact with the silicon substrate later. This heat-sinking layer is necessary to prevent the
silicon wafer from heating up due to the absorption of infrared power.
2. A 800-nm thick layer of α-Si:H will be deposited by chemical vapor deposition. No patterning
is necessary except to expose a border around three edges of the device for the gold heat-sink
layer and at the fourth edge to allow direct electrical contact to the ground plane via wire
bonds to the device holder.
3. A 20-nm thick layer of TiNx will be deposited and patterned by plasma etching for the KID
inductor.
4. The 150-nm thick layer of Nb for the KID and readout coupling capacitor top electrodes and
the CPW feedline center conductor will then be deposited and patterned. It must make direct
contact with the TiNx to form the KID. It would be simplest to use liftoff for this step, but
there are concerns that the Nb will become contaminated by photoresist: either during the ion
mill step that is necessary to remove any oxide on the TiNx so the TiNx -Nb contact is fully
conducting, or during the deposition of the Nb. Such contamination is known to be possible,
affecting the microstrip properties. An alternate process would be to deposit SiO2 and pattern
it by plasma etching to create a stencil identical to the liftoff stencil. Ion milling and Nb
deposition can then proceed as before. The Nb would be patterned using plasma etching, and
then the SiO2 removed by plasma etching.
5. A 270-nm thick layer of α-Si:H will be deposited by chemical vapor deposition to complete the
microstrip dielectric. Again, a border must be removed around the edge of the device.
6. A 450-nm thick layer of Nb will be deposited by DC-magnetron sputtering and patterned by
plasma etching to define the microstrip top layer of the microstrip-to-KID mm-wave coupler
and for the phased-array antenna. Again, a border around three edges of the device must be
removed, and also the fourth edge must be patterned to preserve the CPW feedline gap.
7. A 350-nm thick layer of Au will be deposited by DC-magnetron and patterned using liftoff
process. This film provides a 1-mm border around three edges of the device that is used to

57
heat sink the device using Au wire bonds to the device holder. The Au film overlaps the Nb
ground plane so that the wire bonds act to provide RF continuity between the ground plane
and the device holder.

Chapter 4

Mapping-speed based optimization
of the LWCam design
4.1

Optimization of mapping speed

In Chapter 3 we described the overall architecture of the millimeter wave coupling scheme, design
of KID resonator, and readout coupling scheme for LWCam. There we have used many physical
parameters such as the dimensions of TiN inductors, capacitances, and quality factors of KID resonator for six millimeter wave bands. In this chapter we will explain how we determined the values
of these parameters by performing an optimization to maximize the mapping speed. We start with
an introduction to the several realistic criteria that the optimization must satisfy. We then enumerate the constant quantities concerned in the optimization, identify the four parameters to be swept,
and derive the several intermediate variables indispensable for the calculation of mapping speed.
We walk through the determination of the optimal parameters, present the optimization result, and
finally discuss about the fundamental trade-off in our design.

4.1.1

Criteria

As discussed in Chapter 1, the next generation of SZE studies of galaxy clusters will require six
spectral bands from 90 GHz to 420 GHz. For this application, we are in parallel developing multiscale phased array antennas, which use a single antenna structure to cover this wide frequency range
in a single detector pixel. The pixel size is varied in a binary fashion with frequency so that all
spectral bands use pixels that approximately match the corresponding frequency-dependent Airy
function spot size at the focal plane. It is assumed that the antenna structure is 6.66 mm on a
side and that this corresponds to a single pixel at 90 GHz and 150 GHz, 4 pixels at 220 GHz and

59
290 GHz, and 16 pixels at 350 GHz and 420 GHz. Based on the field-of-view constraint and the
use of the 6.66 mm multi-scale antenna, the number of detectors that can be accommodated in the
focal plane in the absence of any dead area is 880, 880, 3500, 3500, 14000, and 14000. Our goal
is to maximize the mapping speed we obtain from this fixed available focal plane area, where the
mapping speed in a given band is Ndet Ωbeam /NEP2 . We also impose the following requirements:
• The mm wave power received at the antenna should be fully absorbed into the TiN absorber
rather than being dissipated in the lossy dielectrics. This requires that the equivalent loss tangent
of the TiN absorber be much larger than that of the dielectrics.
• The focal plane should be mainly filled with the antenna (live area) rather than KIDs (dead area).
• The detector should reach the fundamental-noise-limited sensitivity.
• The resonator bandwidth ∆f must provide enough signal bandwidth given the desired sampling
frequency fsample : ∆f ≥ fsample .

4.1.2

Constant quantities

The important constant quantities associated with each millimeter wave band are listed in Table 4.1
and we provide the corresponding detailed explanations below.
Band
Tsky (K)
Tdewar (K)
Ttel (K)
Tload (K)
λ (µm)
ν (GHz)
∆ν (GHz)
FWHMBeam (0 )
fsample (kHz)
ηopt
Popt (pW)
ηph
lantenna (mm)
Nantenna

14
13
32
3300
90
35
0.53
0.68
0.26
4.0
0.81
6.66
880

14
13
33
2000
150
47
0.37
0.97
0.40
8.6
0.57
6.66
880

10
17
13
40
1330
230
45
0.22
1.6
0.34
8.5
0.57
3.33
3500

14
20
13
47
1050
275
40
0.20
1.8
0.38
9.9
0.57
3.33
3500

30
41
27
98
850
350
34
0.14
2.6
0.26
12
0.57
1.66
14000

57
61
27
145
750
400
30
0.12
3.0
0.29
17
0.57
1.66
14000

Table 4.1: Relevant physical constants in the optimization.

• Loading temperatures Tsky , Tdewar and Ttel
Tsky , Tdewar , and Ttel are the equivalent Rayleigh-Jeans temperatures of the sky, the dewar, and

60
the telescope. Tsky is set by the measured atmospheric precipitable water vapor column at the
site (we assumed a 0.55 mm best case based on the data) and an atmospheric model that maps
this to opacities and thus optical loading in each band. Ttel is set by an estimate of how emissive
the telescope will be (5% at λ ≥1 mm, 10% at λ ≤1 mm) due to cracks, panel emissivity, and
the secondary feed leg structures (which scatter light to 300 K). Tdewar is chosen in a somewhat
ad hoc way such that (Ttel + Tsky + Tdewar ) = 3(Ttel + Tsky ). It’s hard to predict Tdewar ahead
of time, but this gives reasonable values based on experience. They together give the loading
temperature Tload at the detector.
• Millimeter wave center frequency ν and bandwidth ∆ν
ν and ∆ν are set by the atmospheric windows.
• Full width at half maximum of the Airy disk FWHMBeam
• Sampling frequency fsample
CCAT’s scanning velocity is v=1◦ /s.

fsample = (FWHMBeam /v/3)−1 × 2

(4.1)

• Optical efficiency ηopt
ηopt is the overall optical efficiency in the optics from the dewar window to the KID.
• Optical power Popt
Popt is the optical power received by the KID. Since the antenna is sensitive to only a single
polarization, we use the single-polarization form:

Popt = ηopt (kB Tload ∆ν)

(4.2)

• Photon conversion efficiency ηph
ηph is the fraction of photon energy that can be converted to excite quasiparticles in the superconductor.
• Side length of the antenna lantenna
• Number of antennas Nantenna
Nantenna is the number of antennas that can be placed in the focal plane for each millimeter wave
band assuming no dead area.

61
Next we list the constant quantities universal for all six millimeter wave bands.
• Superconducting critical temperature of TiN Tc and energy gap ∆
Tc is tunable pending on the detailed procedure of fabrication[31]. We use Tc = 1.1K such that
the 90GHz photon is energetically sufficient to break the Cooper pairs. The energy gap is related
to critical temperature in BCS theory by the following relation.

∆ = 1.76kB Tc

(4.3)

• Kinetic inductance fraction α
In KID resonator there are both kinetic inductance and geometric inductance. The ratio of kinetic
inductance to total inductance is defined as the kinetic inductance fraction, represented by α. TiN
film thinner than 100nm is demonstrated in experiments to exhibit large kinetic inductance with
α generally greater than 90%. We thus assume α = 0.9 throughout the optimization.
• The γ factor
γ is defined as the ratio of fractional surface impedance change to fractional conductivity change.
In the thin film limit and local limit[8], we know
δZs
δσ
=γ
Zs

(4.4)

γ = −1

(4.5)

• Quasiparticle recombination rate R
R is measured experimentally to be around 100 µm3 /s[31].
• Intrinsic quasiparticle lifetime τ0
τ0 is measured experimentally to be around 0.1 ms[31].
• Operating temperature of the dewar T
We use T = 100 mK
Tc = 1.1 K so that the density of thermally generated quasi-particles are
negligible.
• Single spin electron density of states at the Fermi level N0
N0 is experimentally measured to be around 3.5×1010 eVµm−3 [32].
• Thermal quasi-particle density nth
qp

62
According to BCS theory we know

nth
qp = 2N0 ∆

2πkB T

1/2
exp (−

) = 0.025 µm−3
kB T

(4.6)

We will see that this is negligible compared to the optically generated quasiparticle density.
• The readout power Pread and power dissipated in the resonator Pl
Pread is the power of the probe (normally microwave) signal in the readout line. Pl is the microwave
power dissipated in the inductor due to finite density of quasi-particles. Though high power can
reduce both TLS and amplifier noises, it does cause problems too. It is known that the kinetic
inductance of a superconductor is a nonlinear function of the current. Due to such a nonlinearity,
as the readout power is increased, the transmission curve S21 becomes distorted and asymmetric,
and eventually bifurcates. In the bifurcation regime, response exhibits discontinuous jumps[7].
Therefore, Pread and Pl should be kept low enough so that bifurcation does not occur. We choose
1 pW (-90 dBm) based on experience, but we will see later that it could be increased considerably
before the bifurcation starts for this design work.

Pread = 1 pW
Pl = Pread /2

(4.7)
(4.8)

• Amplifier noise temperature Tn
Tn is measured experimentally to be around 4 K for SiGe BJT amplifiers. We use the BJT because
it has good noise temperature in the 100 MHz regime, which is the driving requirement. HEMTs’
noise temperatures rise below 1 GHz. In addition, BJTs have good 1/f noise.
• Attenuation length of millimeter wave in amorphous Si λatt
Si
The loss tangent δ of a material is given by

tan δ =

ω00 + σ
ω0

(4.9)

where ω is the angular frequency of the wave, = 0 − j00 is the permittivity, and σ is the

63
conductivity. In the material power decays exponentially as

P (z) = P0 e−δkz

(4.10)

k = 2π/λ

(4.11)

where z is the distance that the wave travels in the material and k is the wave number. We can
derive the attenuation length for amorphous Si.

δSi = 7.1 × 10−4
λatt
Si =

1 λf ree
∼ 75λf ree
δSi 2π Si

(4.12)
(4.13)

where λf ree is the wavelength in free space. δSi is measured for JPL-deposited amorphous silicon
using microstrip resonator structure (private communication, Peter Day). It turns out that in our
design the dielectric loss is much smaller than the power dissipation in TiN inductor.

4.1.3

Independent parameters swept during the optimization

In this section we define the four independent parameters swept during the optimization. The
optimization will be carried out for six millimeter wave bands separately.
• Attenuation length of millimeter wave in TiN inductor/absorber λatt
T iN
att
For convenience, we scale λatt
T iN by 0.5λSi and consider a discrete set of values of the ratio
att
λatt
T iN /0.5λSi : 1/1, 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, 1/128, 1/256.

• Length of TiN inductor/absorber labs
att−1
att
att
−1
We sweep labs over 0.2λatt
+ λTatt−1
refers to the total attenuatot to 1.8λtot , where λtot = (λSi
iN )

tion length. Such a range corresponds to an absorption efficiency ηabs from 20% to 80% assuming
the exponential attenuation profile as indicated by Eq. 4.10. We do not go up to ηabs =100%
because that corresponds to an infinite labs , which is not doable for an optimization.
• Density of quasiparticles in TiN inductor/absorber nqp
We sweep nqp over 100 µm−3 to 5000 µm−3 in step of 100 µm−3 .
• Thickness of TiN inductor/absorber tabs
We sweep tabs over 20 nm to 100 nm in step of 10 nm. 20 nm is the minimum thickness of TiN
film that can be reliably fabricated so far[31].

64
While the parameters being swept are not the parameters we directly control physically, they are
the parameters over which it is most convenient to do the optimization. Then, from the optimized
values, we can extract the physical parameters of the design.

4.1.4

Intermediate variables

In this section we derive the intermediate variables for the calculation of mapping speed.
• Static quasiparticle lifetime τqp and dynamic quasiparticle lifetime τef f
τqp = (Rnqp + 1/τ0 )−1

(4.14)

τef f = (2Rnqp + 1/τ0 )−1

(4.15)

• Millimeter wave absorption efficiency in TiN inductor/absorber ηabs
λatt−1
labs
iN
ηabs = (1 − exp −
) × Tatt−1
λtot
λtot

(4.16)

• Volume of TiN inductor V
nqp V ∆
= ηph ηabs Popt
τqp
ηph ηabs Popt τqp
V =
nqp ∆

(4.17)
(4.18)

Note that these equations assume uniform distribution of quasiparticles, which is only approximately true.
• The area of TiN inductor/absorber AL
tabs

(4.19)

tabs labs

(4.20)

AL =

• The width of TiN inductor/absorber wabs

wabs =

• The resonant frequency f0 of KID resonator

65
Mattis-Bardeen theory gives the relation between f0 and Qi of a KID resonator.
α|γ|κ1 (f0 )nqp
2∆ 1/2
hf0
hf0
κ1 (f0 ) =
) sinh (
)K0 (
πN0 ∆ πkB T
2kB T
2kB T
Qi =

(4.21)
(4.22)

There is also the requirement that relates sampling frequency fsample with resonator bandwidth
∆f , as discussed at the beginning of this chapter.

Qi = 2Qr =

2f0
2f0
∆f
fsample

(4.23)

By coupling Eq. 4.21 and Eq. 4.23 we can find out a lower bound for f0 numerically. Figure 4.1
shows the intersection between the two functions of f0 .

Figure 4.1: Intersections of the two Qi -related functions. Blue line is the maximum Qi allowed
given the desired sampling frequency and red line is the Qi implied by the quasiparticle density.
The resonant frequency must therefore be in the region where the blue line is higher than the red
line. Responsivity increases as f0 decreases, so we always choose the f0 at which the two curves
intersect except for cases specifically emphasized.

66
• The internal quality factor Qi

Qi =

2f0
fsampling

(4.24)

• The ratio of the imaginary part to the real part of the complex conductivity rκ

κ=

δσ/|σ|
= κ1 + jκ2
δnqp

κ2
π 2πkB T 1/2
rκ =
= (
κ1

(4.25)
1+(

hf0
hf0
2∆ −1/2
exp (−
)I0 (
πkB T
2kB T
2kB T
hf0
hf0
sinh (
)K0 (
2kB T
2kB T

(4.26)

where σ is the complex conductivity of the superconductor.
• Total inductance of TiN inductor L

L = µ0 λef f (tabs )

labs 1
wabs α

(4.27)

where λef f (tabs ) is the effective penetration depth of TiN.
• Capacitance of KID capacitor C

(2πf0 )2 L

(4.28)

• Area of the ideal (parallel plate structure without virtual floating ground) KID capacitor AC

AC =

CtSi
Si
r 0

(4.29)

where tSi = 0.8 µm is the thickness of middle dielectric layer.
• Area of the actual (parallel plate structure with a virtual floating ground) KID capacitor ATCot
ATCot = 4AC

(4.30)

This factor of 4 appears because the true area of the capacitor is split into two capacitors and
they are put in series.

67
• Side length of the coupling capacitor lCc

Qc = Qi

(4.31)

Z0 = 50Ω
1/2
8C
Cc =
2πf0 Qc Z0
1/2
Cc
AC
lCc =

(4.32)
(4.33)
(4.34)

• Photon noise equivalent power NEPph
2Popt
2Popt hν +
∆ν

NEPph =

!1/2
(4.35)

The incident millimeter wave photons from the background (dewar, telescope, and sky) obey
Bose-Einstein statistics and have an intrinsic fluctuation in each mode: n̄ph (1 + n̄ph ), where
n̄ph = (ehν/kB T − 1)−1 . This is the photon noise.
• Recombination noise equivalent power NEPr
NEPr =

dPopt
dnqp

!1/2
τef
f Sr

V∆
ηabs ηph τef f

τef

Rn2qp

!1/2
(4.36)

where Sr is the power spectral density of quasiparticle recombination rate r(t). In the steady
state under optical loading the detector maintains a dynamic equilibrium of the quasiparticles.
R(t) = 12 r(t)V is the rate at which the recombination events occur in volume V . Such a process is
a point Poisson process hR(t)R(t0 )i = hR(t)iδ(t−t0 ), which induces the fluctuation in quasiparticle
density and recombination noise.
• Generation noise equivalent power NEPg
NEPr =

dPopt
dnqp

!1/2
τef
f Sg

V∆
ηabs ηph τef f

τef

2
R nth
qp

!1/2
(4.37)

The reverse process, in which thermal phonons break Cooper pairs, gives the generation noise.
The generation noise is often negligible since the operating temperature is usually much lower
than the transition temperature T
Tc and nth
qp
nqp .

68
• Decay noise equivalent power NEPd
NEPd =

dPopt
dnqp

!1/2
τef
f Sd

V∆
ηabs ηph τef f

2 nqp
τef
V τ0

!1/2
(4.38)

The quasiparticles can decay through an intrinsic process (as opposed to recombination process)
characterized by a lifetime τ0 . Similarly such a point Poisson process induces the decay noise.
Decay noise is negligible compared with the recombination noise in the high quasiparticle density
limit.
• Amplifier noise equivalent power NEPamp
1/2
−1
dnqp /nqp
8kB Tn
V∆
HEMT
NEPamp = Sδnqp /nqp
nqp
dp
Pl rκ2
ηabs ηph τef f

(4.39)

KID uses uses a cryogenic low noise amplifier, either a high-electron-mobility transistor (HEMT)
or a silicon-germanium bipolar-junction transistor (BJT), to amplify the transmitted signal. The
amplifier adds white noise with the above NEP. The amplifier can also have 1/f gain fluctuation
noise, which we do not consider since it can be subtracted using off-resonance monitor tones.
• Amplitude of the electric field in KID capacitor E

Qi Pl
2πf0
1/2
2Energy
U=
E=
tabs

Energy =

(4.40)
(4.41)
(4.42)

where Energy is the maximum energy stored in KID capacitor, U is the voltage in KID capacitor.
• Actual current density in KID inductor J, critical current density Jc for bifurcation
1/2
2Energy
J=
wabs tabs
π∆tabs
ρ = µ0 λef f
1/2
πN0 ∆3
Jc = 0.42
~ρ

(4.43)
(4.44)
(4.45)
(4.46)

69
where I is the magnitude of current in KID inductor.
• TLS noise power spectral density SfT LS and associated noise equivalent power NEPphase
T LS
LS
SfTexp
= 4.1 × 10−18 Hz−1

(4.47)

Eexp = 6000 V/m

(4.48)

fexp = 1000 Hz

(4.49)

fLW Cam = 10 Hz
LS
SfT LS = SfTexp

(4.50)

LS
NEPphase
SfTexp
T LS

−1.7

−1

−1/2

ATc ot
fLW Cam
0.016 mm2 /2
fexp
−1
−1
1/2
rκ
LS
= SfTexp
δp
2Qi 2Popt

100 mK
1/2 δf /f

Eexp

−1
(4.51)
(4.52)

LS
where SfTexp
, Eexp , fexp are the experimentally measured TLS noise power spectral density, inferred

amplitude of the electric field, and operating frequency[36]. In the reference, the MKID is a half
wavelength microstrip resonator made of Al with amorphous Si as the dielectric material. One
end of the microstrip resonator overlaps with the center strip of the CPW (readout line) to create
a capacitive coupling while the other end is left unterminated. The electric field Eexp we used here
is the inferred amplitude of the electric field at the open end. fLW Cam is the operating frequency
or the frequency of the astronomical signal of LWCam.
• Fundamental noise equivalent power NEPf und
NEPf und = (NEP2ph + NEP2g + NEP2r + NEP2d )1/2

(4.53)

AP ixel = AAntenna + 25 × (ATCot + AL )

(4.54)

• Area of a single pixel

The factor 25 is to account for the fact that a single antenna couples to detectors from multiple
spectral bands. It comes from the following estimation: 25 = [ATCot (ν = 90 GHz) × 1 + ATCot (ν =
150 GHz) × 1 + ATCot (ν = 230 GHz) × 4 + ATCot (ν = 275 GHz) × 4 + ATCot (ν = 350 GHz) × 16 +
ATCot (ν = 400 GHz) × 16]/[ATCot (ν = 90 GHz) + ATCot (ν = 150 GHz) + ATCot (ν = 230 GHz) +
ATCot (ν = 275 GHz) + ATCot (ν = 350 GHz) + ATCot (ν = 400 GHz)].

70
• Actual number of pixels

Npixel =

AF ocalP lane
AP ixel

(4.55)

• Mapping speed vMapping

vMapping ∼

NP ixel

2 =
ot
NEPTphase

NP ixel
T LS 2
NEPphase + NEP2amp + NEP2f und

(4.56)

The mapping speed normally includes a factor of Ωbeam but we have dropped it because it is a
constant irrelevant to the optimization.

4.1.5

Determination of optimal parameters

Among the four parameters swept, we will first examine and interpret the dependence of several
intermediate variables and finally mapping speed of the six millimeter wave bands on nqp and labs
att
att
att
att
for fixed λatt
T iN /0.5λSi and tabs . We take λT iN /0.5λSi = 1/32 (or λT iN = 3.6 mm) and tabs = 20

nm of 90 GHz band as an example. Similar conclusions hold for the rest of the parameter space.
• Inductance L and side length lC of the capacitor

Figure 4.2: Inductance L and side length lC of the capacitor (90 GHz band) for the case of
att
λatt
T iN /0.5λSi = 1/32 and tabs = 20 nm.

71
We can make the following qualitative analysis for L and C.
labs
l2
∝ abs ∝ labs
n2qp
tabs labs
nqp
−2 −1
∝ labs
nqp
C=
(2πf0 ) L
L∝

labs
wabs

(4.57)

(4.58)

Therefore with low quasiparticle density and short inductor the area of KID capacitor tends to
be large and reduces the filling fraction of live (antenna) area in the focal plane.
• Filling fraction of live area ALive /AP ixel and number of pixels NP ixel

Figure 4.3: Filling fraction of live area ALive /AP ixel and number of pixels NP ixel (90 GHz band)
att
for the case of λatt
T iN /0.5λSi = 1/32 and tabs = 20 nm.

Figure 4.3 shows that the filling fraction of live area in the focal plane ALive /AP ixel is approximately inversely proportional to ATCot , consistent with our expectation. Therefore, with higher
quasiparticle density and longer inductor the focal plane can accomodate more pixels.
LS
• Responsivity and TLS noise equivalent power NEPTphase

LS
Figure 4.4: Responsivity and TLS noise equivalent power NEPTphase
(90 GHz band) for the case of
att
att
λT iN /0.5λSi = 1/32 and tabs = 20 nm.

We can make the following qualitative analysis for responsivity and TLS noise equivalent power
LS
NEPTphase

δfr /fr δnqp /nqp
γκ 1
δfr /fr
=−
∝ Q−1
∝ nqp
δp
δnqp /nqp
δp
2Qi 2Popt
NEPTLS ∝

δfr /fr
δp

(4.59)

−1

∝ n−1
qp

(4.60)

Therefore, the TLS noise equivalent power is approximately inversely proportional to the quasiparticle density. We also see that TLS noise has nothing to do with the lifetime of quasi-particles
δfr /fr
τqp in the system according to the conversion factor
. The low quasiparticle density correδp
sponds to a big inductor.
ot
• Fundamental noise equivalent power NEPf und and total noise equivalent power NEPTphase

ot
Figure 4.5: Fundamental noise equivalent power NEPf und and total noise equivalent power NEPTphase
att
att
(90 GHz band) for the case of λT iN /0.5λSi = 1/32 and tabs = 20 nm.

73
Figure 4.5 shows that the fundamental noise equivalent power is almost constant and the total
noise equivalent power rises up quickly when the quasiparticle density drops below certain value.
Such a rise is due to the dominance of TLS noise over the fundamental noise at low quasiparticle
density.
• Critical current density Jc and actual current density J

Figure 4.6: Critical current density Jc and actual current density J (90 GHz band) for the case of
att
λatt
T iN /0.5λSi = 1/32 and tabs = 20 nm.

Figure 4.6 shows that for high density of quasiparticles and long inductor the actual current
density J is significantly lower than the critical value 0.88Jc , where the bifurcation starts.
• Width of TiN inductor/absorber wabs and mapping speed

Figure 4.7: Width of TiN inductor/absorber wabs and mapping speed (90 GHz band) for the case
att
of λatt
T iN /0.5λSi = 1/32 and tabs = 20 nm.

Figure 4.7 shows that the mapping speed grows with higher density of quasiparticles and longer
inductor. We also know that inductors with width less than 1µm can not be reliably fabricated.

74
att
Therefore we can conclude that for fixed λatt
T iN /0.5λSi and tabs , the largest mapping speed corre-

sponds to wabs = 1 µm and labs = 1.8λatt
T iN .
att
Next we check the dependence of mapping speed on λatt
T iN /0.5λSi and tabs with fixed wabs = 1

µm and labs = 1.8λatt
T iN . Figure 4.8 shows that the maximum mapping speed corresponds to the
att
thinnest inductor (20 nm) and different λatt
T iN /0.5λSi . Table 4.2 summarizes the resultant parameters

of the six millimeter wave bands. In all cases, the total sensitivities are within 10% to 25% of the
fundamental noise limit and the number of pixels in each band is over 75% of the expected number
of pixels assuming no dead area, as was our goal.
Since our parameter sweep only goes up to ηabs =80%, a natural question to ask is would 100%
efficiency be better in terms of mapping speed. We think the answer is positive based on the trend
revealed by our analysis (and by common sense). This can be readily checked by tweaking the
parameter space to include ηabs =100% in this optimization. Nonetheless, it is worth pointing out
that the conclusion that ηabs =80% is optimal does not dictate the actual millimeter wave absorption
efficiency in the design. This is just a “guidance”. The actual efficiency depends on how we engineer
the profile of the coupling capacitors as discussed in last chapter and it is essentially decoupled from
the result of the optimization. It has the freedom to be any value between 0% and 100%. Assuming
we adopt the physical dimensions and quality factors listed in Table 4.2 for the KID, if we make
ηabs =80% by properly engineering the coupling capacitors, we then get the great mapping speed
as yielded exactly from the optimization. We can certainly make ηabs =100% too and the resultant
mapping speed might be slightly better.

att
Figure 4.8: Dependence of mapping speed on λatt
T iN /0.5λSi and tabs with fixed wabs = 1 µm and
att
labs = 1.8λT iN for six millimeter wave bands.

76
Band

labs (mm)

6.5

7.9

5.3

8.2

6.7

5.9

wabs (µm)

1.0

tabs (nm)

AC (mm2 )

0.83

0.51

0.53

0.26

0.27

0.34

f0 (MHz)

L (nH)

441

553

368

573

464

404

C (pF)

0.75

2632

10528

4.5

0.85

0.65

0.61

0.47

0.44

0.43

6.4 × 107

3.4 × 107

4.2 × 107

3.1 × 107

2.7 × 107

E (V/m)

2770

2959

2238

3050

2514

2059

Sδf /f (Hz−1 )

8.6 × 10−19

1.3 × 10−18

1.7 × 10−18

2.5 × 10−18

2.9 × 10−18

2.8 × 10−18

nqp (µm−3 )

2700

3050

3700

3200

3900

5000

τef f (µs)

1.8

1.6

1.3

1.5

1.3

1.0

118

172

NEPg (aW Hz−1/2 )

2.4 × 10−4

3.7 × 10−4

3.0 × 10−4

3.8 × 10−4

3.4 × 10−4

3.2 × 10−4

NEPd (aW Hz−1/2 )

3.4

5.5

5.0

5.8

6.2

NEPamp (aW Hz−1/2 )

6.0

NEPT LS (aW Hz−1/2 )

NEPf und (aW Hz−1/2 )

103

129

183

NEPT ot (aW Hz−1/2 )

115

140

193

1.8

2.2

2.4

2.6

5.6

8.1

2.2 × 10

ALive /AP ixel

0.75

NP ixel

658

Jc (mA/µm2 )

4.5

J (mA/µm2 )
df /f
dP

(W−1 )

NEPph (aW Hz−1/2 )
NEPr (aW Hz

−1/2

1/2

NEFD (mJy s

2.3 × 10

0.75

3.1 × 10

1.4 × 104

3.9 × 10

1.8 × 10

Table 4.2: Results of optimization.

4.2

Fundamental tradeoff

We find that the fundamental tradeoff in our design lies between the filling fraction of live area
in the focal plane and TLS noise. Individual detector sensitivity is maximized (NEP minimized)

77
when fundamental noise dominates over TLS and amplifier noise. The TLS noise contribution to
the individual detector NEP goes down as the capacitor area increases, but this also increases the
dead fraction of the focal plane and thus reduces mapping speed. Figure 4.9 demonstrates such a
tradeoff.
It is worth pointing out that the readout power could be increased by a good amount (factor of
5 to 10 in J and so 25 to 100 in Pread ), which would then reduce TLS noise contribution by a large
amount and push toward larger fill factor.

Figure 4.9: Tradeoff between filling fraction of live area and TLS noise.

Chapter 5

Conclusion and outlook
Future observational astronomy in the sub-millimeter/millimeter regime will demand a large format
of array of pixels, generally with 105 ∼ 106 or more detectors. The kinetic inductance detector
(KID) being actively developed over the past decade provides a promising route to easy and cheap
multiplexing. This thesis explores 1) the physical mechanism of two level system (TLS) noise, the
dominant limiting factor of KID’s sensitivity and 2) designs the interface that efficiently couples the
sub/millimeter photons collected at the antenna to the KID made of highly resistive material TiN
for simultaneously six wide continuous bands, as required by the study of dusty star-forming galaxy
population and galaxy clusters.
Chapter 2 starts from the microscopic model[8] of dielectric constant based on the independent
TLS assumption, derives the frequency noise spectral density resulting from the TLS-phonon bath
coupling and checks its dependence on the noise frequency, amplitude of the electric field in the KID
capacitor, as well as the system temperature, all of which have already been extensively calibrated
experimentally. A certain level of consistency between the model and data is found. However,
in-depth examination uncovers the incorrectness of the model and the demonstrated consistency is
speculated to be merely coincidence. The study of noise spectral density from a single independent
TLS might require numerical simulation of long time series of its dipole moment. Furthermore,
recent literature indicates that TLS-TLS interaction via strain field might be the actual origin of
TLS noise, and quantitative characterization of the noise strength has yet to be developed[11, 12, 13].
Chapter 3 describes in detail the millimeter wave photon coupling architecture from the antenna
to KID. The output of the antenna, a Nb microstrip, is generally in a huge impedance mismatch with
TiN, the material that generates the best responsivity of KID to date in the traditional coupling
scheme as in MUSIC[14]. In order to reconcile such a discrepancy, an adiabatic, efficient, and flexibly
tunable (to accommodate six spectral bands simultaneously) coupling method must be invented.

79
We performed a thorough study of the absorption process of millimeter wave power and verified
the absorption efficiency in the novel coupling scheme both analytically and numerically with finite
element software Sonnet[34]. The shortcoming of such a design lies in the complexity of the geometry
consisting of five layers of metal and dielectric material, which might present significant problems
in terms of device yield. Further systematic experiments under optical loading need definitely to be
carried out as the ultimate justification of the design viability.
Chapter 4 derives the physical dimensions of the TiN absorber/inductor, KID capacitor, and
readout coupling capacitor, which were used in chapter 3, for the six spectral bands by performing
an optimization pivoting around the goal of maximizing the mapping speed. Important independent parameters investigated include the length, width, millimeter wave power attenuation length,
and quasi-particle density of the TiN absorber/inductor. The solution ensures an absorption efficiency of millimeter wave power over 80% and a noise equivalent power close to background-limited
performance without any risk of bifurcation.
In conclusion, we have successfully achieved all the initial design expectations of the long wavelength imager (LWCam) proposed for CCAT in 2012 by developing the KID technology. We hope
the design work can be validated by future experiments.

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