STRUCTURES
AND THE COMMITTEE ON GRADUATE STUDIES
OF STANFORD UNIVERSITY
FOR THE DEGREE OF
© 2016 by Linxiao Zhu. All Rights Reserved.
Re-distributed by Stanford University under license with the author.
This work is licensed under a Creative Commons Attribution- Noncommercial 3.0 United States License.
Shanhui Fan, Primary Adviser
I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.
James Harris
I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.
David Miller
Patricia J. Gumport, Vice Provost for Graduate Education
This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives.
iii
Abstract
Electromagnetic heat transfer is a fundamental and ubiquitous energy process. For example, sunlight
that constitutes most of the energy on Earth is the thermal radiation from the sun at around 5800
Kelvin, the light of an incandescent light bulb is the thermal radiation from the hot filament, and
what an infrared camera detects is the thermal radiation of an object. Controlling electromagnetic
heat transfer is essential for improving the energy efficiency and can point to novel applications.
However, electromagnetic heat transfer in naturally occurring materials is largely limited by the
properties of existing materials. Photonic structures provide possibilities to achieve electromagnetic
heat transfer properties that are unattainable using naturally occurring materials.
This thesis explores the opportunities for controlling electromagnetic heat transfer using non-
reciprocal photonic structures, tailoring thermal radiation for the applications of radiative cooling,
active control of near field heat transfer, and understanding thermal radiation from a single thermal
emitter. I will also discuss angle-selective perfection with 2D materials, in particular in the thermal
infrared.
In Chapter 1, I will introduce electromagnetic heat transfer, discuss different numerical methods
for calculating electromagnetic heat transfer in nano-structures, and introduce the general consid-
eration in radiative cooling. Chapter 1 serves as the background for the following chapters.
In the first part, including Chapter 2 and Chapter 3, I will discuss new opportunities in controlling
electromagnetic heat transfer by using non-reciprocal photonic structures. In particular, we will re-
examine some constraints that were thought to be intrinsic for electromagnetic heat transfer, and
introduce ways to eliminate them.
In Chapter 2, we discuss the possibility to maximally violate detailed balance of thermal radia-
tion. The general form of Kirchhoff’s law states that for a thermal emitter, at the same wavelength
and angle, the specular angular emissivity equals the specular angular absorptivity. This is usually
termed as detailed balance for thermal emission in some textbooks. However, this detailed balance
relation is not a consequence of the second law, and rather is only true for reciprocal structures that
consist of materials that satisfy Lorentz reciprocity. On the other hand, violating detailed balance
is of both practical and fundamental interest. Practically, violation of detailed balance points to
a pathway for fundamental improvement for energy conversion processes such as solar cells. We
iv
will introduce the general conditions to achieve maximal violation of detailed balance in thermal
radiation, and numerically demonstrate a magneto-optical photonic crystal structure that exhibits
near-complete violation of detailed balance by performing a direct calculation of thermal emission
based on fluctuational electrodynamics.
In Chapter 3, we discuss the theoretical discovery of thermal supercurrent, i.e. persistent direc-
tional heat current at thermal equilibrium, in non-reciprocal many-body near field electromagnetic
heat transfer. Transport processes including charge, mass etc. usually signify that the system is
away from equilibrium. Similarly, for radiative heat transport, typically a temperature gradient is
needed in order to have a non-zero net radiative heat transfer. On the other hand, the discoveries
of superconductivity, superfluidity and quantum Hall effects represent some of the most important
discoveries of physics, where there is supercurrent at zero bias. Thus, it is of particular fundamental
physics importance to examine whether there can exist a thermal supercurrent in heat transfer.
Apart from its fundamental physics importance, the demonstration of thermal supercurrent in near
field heat transfer also points to fundamentally new method for controlling heat flow in the nanoscale.
In the second part, including Chapter 4, Chapter 5 and Chapter 6, I will discuss controlling ther-
mal radiation for the application of radiative cooling. Earth’s atmosphere has a transparency window
for electromagnetic waves between 8 and 13 microns, which coincides with the peak wavelength of
the blackbody radiation from an object at typical terrestrial temperatures. By using the cold of
outer space as a cold heat sink, a terrestrial object can send out its heat through this transparency
window to achieve passive radiative cooling.
In Chapter 4, we discuss color-preserving daytime radiative cooling, in which one aims to lower
the temperature of a structure as much as possible, while the amount of sunlight absorption needs
to be maintained for functional or aesthetic considerations. This concept is attractive for radiative
cooling of clothes, cars and outdoor electronics.
In Chapter 5, we theoretically study radiative cooling for solar cells. A solar cell, under the sun,
heats up. The heating is undesirable, and leads to reduced efficiency and also reduces reliability. A
solar cell by necessity naturally has radiative access to the sky. Thus, it would be very attractive to
lower the temperature of a solar cell by sending its heat to the cold outer space, while simultaneously
maintaining its full amount of sunlight absorption.
In Chapter 6, we experimentally demonstrate radiative cooling of a solar absorber using a visibly
transparent thermal blackbody based on a photonic crystal, and show a temperature reduction as
large as 13 C using radiative cooling. We also show that radiative cooling can synergize with other
cooling mechanisms.
In the third part, we discuss active control of near field electromagnetic heat transfer, including
ultrahigh contrast and large bandwidth thermal rectification in near field heat transfer between
nanoparticles in Chapter 7, and negative differential thermal conductance in near field heat transfer
in Chapter 8.
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In Chapter 9, we introduce a simple temporal coupled mode theory to discuss thermal emission
from a single thermal emitter, with arbitrary geometry and composition complexity. We numerically
verify that for an arbitrarily shaped thermal emitter with arbitrary material complexity, its thermal
emission can be described by a simple formula.
In Chapter 10, we introduce a general approach to achieve angle-selective perfect absorption with
two-dimensional materials, and experimentally demonstrate record-high absorption of mid-infrared
light in single-layer graphene. Such an angle-selective perfect absorption can be important for energy,
photo-detection and sensing applications.
In Chapter 11, we summarize the studies in this thesis, and provide suggestions for future work.
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Acknowledgments
I would like to first thank my research advisor Professor Shanhui Fan. I thank him for leading me
to the exciting research field of thermo-photonics. I especially want to thank him for supporting
me for trying different things, from which I learned many new things. I also treasure a lot from the
discussion with him about many things, which will be helpful for my future life.
I would like to thank Professor David A. B. Miller for being my co-advisor, Professor Mark
Brongersma for serving the chair for my PhD defense, Professor James S. Harris and Professor Amir
Safavi-Naeini for serving in my defense committee. I would also like to thank Professor Shanhui
Fan, Professor James Harris, and Professor David Miller for careful reading of my thesis. I also
would like to thank Professor Evan Reed, Professor Steven Block, Professor Daniel Fisher, Professor
Benjamin Lev and Professor Jenela Vuckovic.
I would like to thank all of the labmates in the Shanhui Fan group. I would like to thank Clayton
for helping me a lot when I started working in near field electromagnetic heat transfer, and Sunil,
Kaifeng, Yu Guo and Parthi for collaboration on near field heat transfer studies. I would like to
thank Aaswath, Zhen and Eli for collaboration in radiative cooling projects. I would also like to
thank Zongfu for his valuable suggestions and collaboration. I would like to thank Wonseok and
Victor for helping on numerical codes. Without them, many of the works could not have been done.
I am also fortunate to collaborate with other groups. I would like to thank Professor Michal
Lipson and Raphael from Cornell University (now in Columbia University) for collaboration for
experimental demonstration of near field heat transfer between parallel structures. I would like
thank Professor Paul Braun and Kevin from UIUC for collaboration in experimental demonstration
of a high-temperature-stable tungsten inverse opal photonic crystal, which is important for high-
temperature thermo-photovoltaic application. I would also like to thank Professor Xinran Wang and
Fengyuan from Nanjing University, and Professor Juejun Hu and Hongtao from MIT for collaboration
on the work of angle-selective perfect absorption with 2D materials. I would like to thanks Michael
B. Sinclair and Ting Shan Luk from Sandia National Laboratories for collaboration on the work of
temporal coupled mode model for thermal emission from a single thermal emitter.
I am grateful to the staff in Stanford Nanofabrication Facility (SNF), and Stanford Nano Shared
Facility (SNSF), for great suggestions and generous help.
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Finally, I would like to thank my family and my wife Zaiyue, who supports and encourages me
always.
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Contents
1.2 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.3 Scattering approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.5 Finite-difference time-domain method . . . . . . . . . . . . . . . . . . . . . . 6
1.2.6 Other methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Radiative cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.1 Earth’s atmospheric transparency window . . . . . . . . . . . . . . . . . . . . 9
1.3.2 Radiative cooling while preserving the amount of sunlight absorption . . . . . 12
1.3.3 Subambient radiative cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Near-complete violation of detailed balance 15
2.1 Thermodynamic constraints on thermal emitter . . . . . . . . . . . . . . . . . . . . . 16
2.2 Conditions to maximally violate detailed balance . . . . . . . . . . . . . . . . . . . . 17
2.3 Numerical demonstration of a near-complete violation of detailed balance . . . . . . 19
3 Thermal supercurrent 24
3.1 Necessary conditions for achieving thermal supercurrent . . . . . . . . . . . . . . . . 25
3.2 A minimal analytic model based on coupled mode theory . . . . . . . . . . . . . . . 26
3.3 Physical system for achieving thermal supercurrent . . . . . . . . . . . . . . . . . . . 29
3.4 Scattering formalism for non-reciprocal many-body near field radiative heat transfer 30
3.5 Compare scattering approach and coupled mode model . . . . . . . . . . . . . . . . . 31
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3.6 Thermal supercurrent in physical structures from scattering formalism . . . . . . . . 32
3.7 Final remarks and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.8 Supplementary: proof for reciprocal heat transfer in many-body reciprocal systems . 35
3.8.1 Green’s function for many-body system . . . . . . . . . . . . . . . . . . . . . 35
3.8.2 Identity relation from many-body Green’s function . . . . . . . . . . . . . . . 36
3.8.3 Reciprocal heat transfer in many-body reciprocal system . . . . . . . . . . . . 38
4 Color-preserving daytime radiative cooling 41
4.1 Power balance equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2 Design principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.3 A specific design for color-preserving daytime radiative cooling . . . . . . . . . . . . 45
5 Radiative cooling of solar cells 49
5.1 Combined optical and thermal simulations . . . . . . . . . . . . . . . . . . . . . . . . 51
5.2 Design principle and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6 Demonstration of radiative cooling of a solar absorber 59
6.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
7 Ultrahigh-contrast & broadband thermal rectification 73
7.1 Brief review of photon-based thermal rectifiers . . . . . . . . . . . . . . . . . . . . . 74
7.2 Thermal rectification between two nanospheres . . . . . . . . . . . . . . . . . . . . . 74
7.2.1 Material properties and modal structures of SiC nanospheres . . . . . . . . . 75
7.2.2 Mechanism for thermal rectification in two-sphere system . . . . . . . . . . . 77
7.2.3 Numerical demonstration of rectification in two-sphere system . . . . . . . . 80
7.2.4 Dynamic study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
8.1 Mechanism for negative differential thermal conductance . . . . . . . . . . . . . . . . 86
8.2 Numerical demonstration of negative differential thermal conductance . . . . . . . . 87
8.3 Thermal bistability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
9 Thermal emission from a single thermal emitter 91
9.1 Temporal coupled mode theory for thermal emission from a single emitter . . . . . . 92
9.2 Validating the coupled mode theory formula . . . . . . . . . . . . . . . . . . . . . . . 93
9.2.1 Thermal emission from a dielectric sphere . . . . . . . . . . . . . . . . . . . . 94
9.2.2 Thermal emission from a single emitter with complex geometry . . . . . . . . 94
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10.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
10.4 Supplementary Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
10.4.2 Angle-selective perfect absorption in 2D material for s-polarization . . . . . . 106
10.5 Angle-selective perfect absorption in 2D material for p-polarization . . . . . . . . . . 109
10.6 Parasitic loss and angular spread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
11 Summary and suggestions for future work 114
11.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
11.2.1 Calculating non-reciprocal heat transfer with arbitrary geometries . . . . . . 116
11.2.2 Integration of radiative cooling with high-efficiency solar cells . . . . . . . . . 116
11.2.3 Theoretical approach for achieving angle-selective thermal emission . . . . . . 117
11.2.4 Experimental demonstration of active control of near field heat transfer . . . 117
Bibliography 118
List of Figures
1.1 The microscopic picture of electromagnetic heat transfer due to thermally excited
fluctuating current. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 A schematic for the geometry for calculating radiative heat transfer between patterned
periodic layers. Here, the system is periodic in the transverse directions. . . . . . . 5
1.3 A realistic atmospheric transmittance spectrum (magenta), normalized AM 1.5 solar
spectrum (yellow), and a normalized blackbody radiation spectrum at 300 K. . . . . 10
1.4 A schematic showing the energy balance between thermal radiation and absorption
of background blackbody radiation in equilibrium. . . . . . . . . . . . . . . . . . . . 12
2.1 Energy flow diagrams in the cases of (a) a reciprocal emitter, and (b) a non-reciprocal
emitter. The emitter undergoes radiative exchange with two separate blackbodies
labelled A and B, respectively. The emitter and the blackbodies are at the same
temperature T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 A schematic of a photonic crystal structure for maximal violation of detailed balance.
The structure consists of an n-InAs grating structure atop a uniform metal layer.
The structure is periodic in x-direction, and has the following geometry parameters:
p = 7.24 µm, w = 3.2 µm, t1 = 1.981 µm and t2 = 0.485 µm. External magnetic field
is applied in z-direction. TM polarization with electric field in x-y plane is considered. 18
2.3 Absorptivity for different parallel wave vectors and frequencies, for the structure in
Fig. 2.2 with PEC as mirror, at B = 3 T . The black and blue solid curves denote
the peaks of absorptivities at positive and negative parallel wave vectors, respectively.
The green solid curve is the mirror reflection of the blue solid curve. The grey solid
curves denote the folded band structure for a 1.361 µm-thick uniform n-InAs atop
PEC mirror without external B field. . . . . . . . . . . . . . . . . . . . . . . . . . . 20
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2.4 Reflectivity spectra of rθ→−θ and r−θ→θ, for the structure in Fig. 2.2 with PEC
mirror, at B = 3 T and θ = 61.28. The parameters for the coupled mode theory
(CMT) fitting using Eqs. 2.5 and 2.6 are: ω(kx) = 456.6 × 1011 rad/s, ω(−kx) =
463.7 × 1011 rad/s, γi,kx = 1.22 × 1011 rad/s, γe,kx = 9.78 × 1010 rad/s, γi,−kx =
1.33× 1011 rad/s and γe,−kx = 1.42× 1011 rad/s. . . . . . . . . . . . . . . . . . . . 21
2.5 Absorptivity (α) and emissivity (e) spectra, for the structure atop PEC mirror (Fig. 2.2),
at θ = 61.28, and (a) B = 0 T or (b) B = 3 T . (c) Absorptivity and emissivity
spectra, for the structure atop aluminum (Al) mirror (Fig. 2.2), at θ = 61.28 and
B = 3 T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.1 (a) Scheme of thermal supercurrent in a many-body system, in thermal equilibrium
of temperature T . (b) Heat transfer between bodies 1 and 2 at the same temperature
T , while the remaining part of the system body 3 is at temperature T3. . . . . . . . 25
3.2 (a) Schematic of a coupled mode theory model for heat exchange among three-bodies.
Each body supports two counter-rotating modes. (b) and (c) The heat transfer spectra
for S2→1 and S1→2, as calculated using the coupled mode model of Eq. 3.1. Vσ =
−5.37γ, Vπ = 3.78γ. The vertical lines denote the frequencies for the normal modes.
(b) Non-reciprocal case with Vp = −25γ. (c) Reciprocal case with Vp = 0. (d)
Map of coupling in the subsystem for achieving thermal supercurrent in different
directions. The red and blue regions denote thermal supercurrent in counter-clockwise
and clockwise directions for a subsystem, respectively. Here, we assume V < 0, which
is valid for dipoles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3 (a) Geometry of three magneto-optical spheres, forming an equilateral triangle. Ex-
ternal magnetic field is applied in the vertical direction. The spheres consists of
n-doped InSb, with the same doping level. Each sphere has a radius of 200 nm, and
the distance between the centers of two spheres is 500 nm. (b) and (c) The heat trans-
fer spectra of S1→2 and S2→1, from fluctuational electrodynamics. The system is at
thermal equilibrium of 300 K. (b) Non-reciprocal case with B = 3 T . (c) Reciprocal
case with B = 0 T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4 The heat transfer spectra of S1→2 and S2→1, from the coupled mode model (lines),
and from scattering approach (circles) taking into account only the dipole modes
that lie in the plane that contains the sphere centers. The system consists of three
magneto-optical spheres made of n-doped InSb, forming an equilateral triangle. Each
sphere has a radius of 200 nm, and external magnetic field is applied in the vertical
direction. (a) The case where there is a 100 nm gap between each pair of spheres and
B = 3 T . (b) The case where there is a 200 nm gap between each pair of spheres and
B = 1 T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
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3.5 Magnitude of Poynting flux, including all thermal noise sources in the spheres and
environment, at 300 K, λ = 10.49 µm and B = 3 T . The geometry is the same as in
Fig. 3.3. The blue arrows denote the energy streamlines. . . . . . . . . . . . . . . . 33
3.6 (a) The heat transfer spectra for S1→2 and S2→1, with B = 0.7T . (b) The heat
transfer spectra at B = 0T . For both (a) and (b), the radius of each sphere is
500 nm, and the distance between the centers of the spheres is 2200 nm. . . . . . . 34
4.1 (a) The black curve is the emissivity/absorptivity spectrum the original structure
which, for simplicity, absorbs sunlight at a flat value, and emits no thermal radi-
ation. The normalized solar radiation spectrum of AM1.5 spectrum is shown in
yellow. The normalized atmosphere transmission spectrum is shown in magenta. (b)
The black curve is the ideal emissivity/absorptivity spectrum of a modified struc-
ture, which has unity emissivity at thermal wavelengths. (c) The difference be-
tween the equilibrium temperatures for the original and the modified structures
(T = Teq,original−Teq,modified), at Tamb = 300 K, as a function of non-radiative heat
exchange coefficient hc, for different amounts of solar absorption Psun. (d) Teq−Tamb as a function of hc, for Psun = 100 W/m2, at Tamb = 300 K. . . . . . . . . . . . . . 44
4.2 (a) Schematic of the original structure, with silicon nanowire array on top of an
aluminum substrate. A disproportionate schematic is also shown to clarify the geom-
etry. The color for reflection at normal incidence under sunlight is shown. (b) The
black curve is the emissivity/absorptivity spectrum of the original structure. The
normalized solar radiation spectrum of the AM1.5 spectrum is shown in yellow. The
normalized atmosphere transmission spectrum is shown in magenta. (c) Schematic
of the modified structure, with quartz bar array on top of the original structure of
silicon nanowires. (d) The black curve is the emissivity/absorptivity spectrum of the
modified structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3 (a) Equilibrium temperature (Teq) subtracted by ambient temperature Tamb = 300 K,
for the modified structure with quartz bar array (blue curve) and the original struc-
ture (green curve), as a function of non-radiative heat exchange coefficient hc. (b)
The difference (T ) between the equilibrium temperatures for the original and the
modified structures, as a function of hc. The modified structure has a meaningfully
lower equilibrium temperature even for large values of hc. . . . . . . . . . . . . . . . 47
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5.1 Three-dimensional crystalline silicon solar cell structures. (a) Bare solar cell with
200 µm-thick uniform silicon layer, on top of an aluminum back reflector. (b) A thin
visibly-transparent ideal thermal emitter on top of the bare solar cell. (c) A 5mm-
thick uniform silica layer on top of the bare solar cell. (d) A two-dimensional square
lattice of silica pyramids and a 100 µm-thick uniform silica layer, on top of the bare
solar cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.2 The schematic of thermal simulation. h1 and h2 is the non-radiative heat exchange
coefficients at the upper and lower surfaces, respectively. Ambient temperature is
Tamb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.3 Operating temperature of solar cell with thermal emitter designs in Fig. 5.1, for
different solar heating power. The non-radiative heat exchange coefficients are h1 =
12 W/m2/K (corresponding to 3 m/s), and h2 = 6 W/m2/K (corresponding to
1 m/s). The ambient temperatures at the top and the bottom are both 300 K. . . 54
5.4 The emissivity and absorptivity spectra of solar cells with different thermal emitter
designs in Fig. 5.1, for normal direction and after averaging over polarizations. The
temperature of solar cells is 300 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.5 (a) The operating temperature of the solar cell under different emitter designs, for
different h1, and fixed h2 = 6 W/m2/K. (b) The operating temperature of the solar
cell under different emitter designs, for different h2, and fixed h1 = 12 W/m2/K. The
ambient temperature at both sides of solar cell is 300 K. The solar heating power is
800 W/m2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.6 Solar cell operating temperature, with 5 mm-thick uniform silica layer (blue curve),
and with silica pyramid structure (green curve), where the silica has been artificially
added a constant absorbance for solar wavelengths. h1 = 12 m/s, h2 = 6 m/s. The
ambient temperatures at both sides of solar cell is 300 K. The solar heating power is
800 W/m2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
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6.1 Rooftop setup and silica photonic crystal. (a), Photo of the apparatus and solar
absorbers during a test on a rooftop in Stanford, California. The solar absorbers
from left to right are the absorber structure with the planar silica layer, the absorber
structure with the silica photonic crystal, and two bare solar absorbers, respectively.
(b), Cut-out schematic of the apparatus through the middle. Mylar is polyethylene
terephthalate. The 12.5-µm-thick polyethylene film used to cover the opening of
the chambers is removed for the exposed test. (c), Normal-view scanning electron
microscope (SEM) image of the two-dimensional silica photonic crystal structure that
is fabricated and tested in our experiments. It consists of a square-lattice photonic
crystal structure with a periodicity of 6 µm made by etching 10-µm-deep air holes into
a 500-µm-thick double-side-polished fused silica wafer. (d), A side-view SEM image
of the photonic crystal structure along the cut denoted by the white dashed line in
(c). (e), Photo of the photonic crystal, showing the Stanford logo clearly visible and
lying underneath. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.2 Emissivity/absorptivity of solar absorbers from the ultraviolet to mid-infrared. (a),
Measured emissivity/absorptivity at 8 angle of incidence of solar absorbers over
optical and near-infrared wavelengths using an unpolarized light source, with the
normalized AM1.5 solar spectrum plotted for reference. (b), Measured emissiv-
ity/absorptivity of solar absorbers at 10 angle of incidence over mid-infrared wave-
lengths, averaged over both polarizations, with a realistic atmospheric transmittance
model for winter in California plotted for reference. In both (a) and (b), the black,
blue and red curves show the measured emissivity/absorptivity for the bare absorber
structure, the absorber structure with the planar silica layer, and the absorber struc-
ture with the silica photonic crystal, respectively. . . . . . . . . . . . . . . . . . . . . 62
6.3 Angular emissivity of solar absorbers. (a), Measured emissivity of the absorber struc-
ture with the silica photonic crystal at variable angles of incidence, over mid-infrared
wavelengths, averaged over both polarizations, with a realistic atmospheric transmit-
tance model for winter in California plotted for reference. (b), Average measured
emissivity ε of the solar absorbers between 8 µm and 13 µm (the atmospheric trans-
parency window) plotted as a function of polar angle of incidence. The yellow, blue
and red circles show the average emissivity for the bare absorber structure, the ab-
sorber structure with the planar silica layer, and the absorber structure with the silica
photonic crystal. The emissivity for the absorber structure with the silica photonic
crystal remains near-unity between 10 and 50 (96.2% at 10, and 94.1% at 50),
and remains high even at larger angles of incidence. At all angles of incidence, the
absorber structure with silica photonic crystal shows a substantially higher emissivity
than the bare structure and the absorber structure with planar silica layer. . . . . . 63
xvii
6.4 Steady-state temperature of solar absorbers with a wind shield. Rooftop measurement
of the performance of the bare absorber structure (black curve), the absorber structure
with the planar silica layer (blue curve), and the absorber structure with the silica
photonic crystal (red curve) against ambient air temperature (yellow curve) on a clear
winter day with a polyethylene cover in Stanford, California. The absorber structure
with the silica photonic crystal is on average 13 C cooler than the bare absorber
structure, and over 1 C cooler than the absorber structure with the planar silica
layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.5 Steady-state temperature of solar absorbers without a wind shield. Rooftop measure-
ment of the performance of the bare absorber structure (black curve), the absorber
structure with the planar silica layer (blue curve), and the absorber structure with
the silica photonic crystal (red curve) against ambient air temperature (yellow curve)
on a clear winter day without a polyethylene cover in Stanford, California. The ab-
sorber structure with the silica photonic crystal is on average 5.2 C cooler than the
bare absorber structure, and over 1.3 C cooler than the absorber structure with the
planar silica layer. The inset shows a running average of the temperature data over
an averaging period of 8 minutes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.6 Modeling of steady-state temperature of solar absorbers with a wind shield. The
modeled steady-state temperatures for the bare solar absorber, the absorber struc-
ture with planar silica layer, and the absorber structure with silica photonic crystal,
are shown as the grey, light blue, and the light red bands, respectively, for an hc
value range from 6.5 Wm−2K−1 to 9.1 Wm−2K−1. The experimentally observed
values for the temperatures of the bare solar absorber, the absorber structure with
planar silica layer, and the absorber structure with silica photonic crystal, are shown
by the black, blue and red curves respectively, with the ambient air temperature as
the yellow curve. The solar irradiance measured in the same course of time is shown
as the green curve. The experimental observations match quite well with the model-
ing. The inset shows a zoomed-in view of the experimentally measured steady-state
temperatures of the absorber structures with planar silica layer and silica photonic
crystal, as compared with modeling using a combined non-radiative heat exchange
coefficient hc as 7.3 Wm−2K−1, showing excellent agreement between modeling and
experiments. In this case, we use a thin polyethylene film to cover the opening of the
chamber, to reduce effects from winds. . . . . . . . . . . . . . . . . . . . . . . . . . 70
xviii
6.7 Modeling of steady-state temperature of solar absorbers without a wind shield. The
modeled steady-state temperatures for the bare solar absorber, the absorber structure
with planar silica layer, and the absorber structure with silica photonic crystal, are
shown as the grey, light blue, and the light red bands, respectively, for an hc value
range from 11.6 Wm−2K−1 to 16 Wm−2K−1. The experimentally observed values
for the temperatures of the bare solar absorber, the absorber structure with planar
silica layer, and the absorber structure with silica photonic crystal, are shown by the
black, blue and red curves respectively, with the ambient air temperature as the yel-
low curve. The solar irradiance measured in the same course of time is shown as the
green curve. The experimental observations match quite well with the modeling. The
inset shows a zoomed-in view of the experimentally measured steady-state tempera-
tures of the absorber structures with planar silica layer and silica photonic crystal,
as compared with modeling using a combined non-radiative heat exchange coefficient
hc as 13.6 Wm−2K−1, showing reasonably good agreement between modeling and
experiments. The measured temperatures in the inset are with running average over
an averaging period of 8 minutes. In this case, the absorbers are subject to effects of
winds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
7.1 (a) Conventional device geometry for thermal rectification between two bodies con-
sisting of two parallel plates. (b) The red solid and blue dashed lines denote the tem-
perature dependence of a resonance of each body, respectively. The vertical dashed
lines mark the operating temperatures Th and Tl. The red circles and the blue rings
mark the resonance frequencies of two bodies at the operating temperatures, respec-
tively. (c) The (overlapping) resonance frequencies of two bodies in the forward bias
scenario in the conventional mechanism. (d) The (nonoverlapping) resonance fre-
quencies of two bodies in the reverse bias scenario in the conventional mechanism. (e)
Our device geometry for thermal rectification. For our representative implementation
in our study, two spheres have the radii of 96 nm and 10 nm respectively, and the
sphere-sphere distance is 26 nm. (f)-(g) Illustration of our mechanism for thermal
rectification. The symbols have the same meaning as (b)-(d). Notice that the two
bodies are on resonance in both the forward and reverse bias scenarios. . . . . . . . 75
7.2 (a) The solid black lines denote the temperature dependence of resonance frequencies
of m = 0 modes with different L for a deep-subwavelength 3C-SiC sphere. At a given
temperature, the resonance frequencies increase with L. We show here modes from
L = 1 to L = 9. The vertical dashed lines mark the operating temperatures Th and
Tl. (b)-(c) The electric field Ez distribution of L = 1 and L = 2 modes, with m = 0,
in x-z plane with y = 0. The z axis is the horizontal direction. The radii of the
spheres are 96 nm in (b) and 10 nm in (c). . . . . . . . . . . . . . . . . . . . . . . . 77
xix
7.3 (a) The field distribution Ez of |2large and |1small. (b) The field distribution (Ez)
of |1large and |2small. In (a) and (b), for the mode of the small sphere we only
show the field inside the small sphere. (c) Coupling constants between the four lowest
order modes of the two spheres in the m = 0 channel. L1 and L2 are the total angular
momenta of the modes of the large and the small spheres, respectively. (a)-(c) use
the same parameters of the representative device implementation [Fig. 7.1(e)]. . . . 78
7.4 (a) Net heat transfer magnitude as a function of the larger sphere temperature T1 and
the small sphere temperature T2. (b) Rectification contrast ratio for different sets of Th
and Tl. (a)-(b) use the same parameters of the representative device implementation
[see Fig. 7.1(e)]. (c) Maximum rectification contrast ratio as a function of the contrast
between the radii of the spheres (rlarge/rsmall), as the sphere-sphere distance is varied,
for a temperature bias Th = 700 K and Tl = 200 K. The small sphere has a radius
of 10 nm. (d) The net heat transfer between two spheres as a function of the large
sphere temperature T1, where the small sphere is maintained at T2 = 500 K, using
the same parameters of the representative device implementation [see Fig. 7.1(e)]. . 80
7.5 (a) Net heat transfer spectra in forward bias scenario (blue line), and reverse bias
scenario (magenta line), for a bias condition Th = 700 K, Tl = 200 K. (b) The net
heat transfer spectrum (blue line), far field radiation spectra of the large sphere (green
dashed line) and the small sphere (red dash-dotted line), in the forward bias scenarios.
(c) The net heat transfer spectrum (magenta line), far field radiation spectra of the
large sphere (green dashed line) and the small sphere (red dash-dotted line), in the
reverse bias scenarios. (a)-(c) use the same parameters of the representative device
implementation [see Fig. 7.1(e)]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
7.6 (a) Wave forms for the temperatures of the contacts τ1 and τ2, the temperatures of
two spheres T1 and T2, and net heat transfer, in three frequencies, i.e., 10, 100 and
500 MHz, as denoted by the circles in (b). The first row shows wave forms of the
temperatures of the contacts τ1 (blue line) and τ2 (green dashed line). The second
row shows wave forms of the temperatures of the two spheres T1 (blue line) and T2
(green dashed line). The third row shows wave forms of net heat transfer Q(T1, T2)
(red line). The contact conductances are h1=4512 nW , h2=5.1 nW . (b) Frequency
dependence of rectification contrast ratio for A = 250 K. (a) and (b) use the same
parameters of the representative device implementation [see Fig. 7.1(e)]. . . . . . . 84
xx
8.1 (a) The schematic of a system made of body 1 and body 2. (b), (c) Two bodies are
made of the same material and the black lines denote the temperature dependence of
electromagnetic resonance of such material. (b) illustrates the case of T1 = T2 and
(c) illustrates the case of T1 6= T2. For concreteness, the curve shown here is the
surface phonon-polariton frequency of silicon carbide of the 6H polytype, with the
extraordinary axis normal to the surface. . . . . . . . . . . . . . . . . . . . . . . . . 86
8.2 (a) Spectral heat flux between body 1 and body 2 at different T ≡ T1 − T2, for the
device made of two SiC-6H slabs as shown in the inset of (b), with T1 = 700K and
separation d = 100nm. In each subplot, the red dashed arrow and green solid arrow
denote the surface phonon-polariton frequencies for body 1 and body 2 respectively,
as shown in Figs. 8.1b and 8.1c. (b) The blue solid line, pertaining to the left y axis, is
the net heat flux between body 1 and body 2 as a function of T for the device shown
in the inset, with T1 = 700K and d = 100nm. The black dashed line, pertaining to
the right y axis, shows [Θ(ω, T1) − Θ(ω, T2)] as a function of T , with T1 = 700K
and ω corresponding to the surface phonon-polariton frequency of 6H silicon carbide
at 700K (Figs. 8.1b and 8.1c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
8.3 (a) The blue solid line is the net heat flux between body 1 and body 2 as a function
of T1, for the device made of two SiC-6H slabs as shown in the inset of Fig. 8.2b, with
T2 = 700K and d = 10nm. The black dashed line shows [Θ(ω, T1) − Θ(ω, T2)] as a
function of T1, with T2 = 700K and ω corresponding to the surface phonon-polariton
frequency of 6H silicon carbide at 700K (Figs. 8.1b and 8.1c). (b) Operating region
of a bi-stable thermal switch, corresponding to the rectangle in (a). Q0 is a constant
external heat flux into body 1. Here we assume that Q0 = 1.1185× 107W/m2. . . . 89
9.1 Thermal emission power spectrum comparison among coupled mode theory prediction
(red line), exact analytic equation result (green squares), and finite-difference time-
domain simulation result averaged over 60 runs (blue crosses), for a spherical single
emitter of radius 500 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
9.2 (a) Geometry for a single emitter. The top cross-shaped layer has dimensions l =
1.7 µm, w = 0.4 µm, with 0.1 µm in thickness. The sizes of the central and bottom
layers are 2 µm×2 µm×0.19 µm and 2 µm×2 µm×0.1 µm, respectively. (b) Electric
field intensity distribution (|E|2) for the mode at ω = 0.195(2πc/a). (c) Thermal
emission power spectrum comparison between coupled mode theory prediction (red
line) and finite-difference time-domain simulation result averaged over 100 runs (blue
crosses). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
xxi
9.3 (a) Modified geometry for an emitter. The top cross-shaped layer has dimensions
l1 = 1.86 µm, l2 = 1.62 µm, and w = 0.4 µm, with 0.1 µm in thickness. The
sizes of the central and bottom layers are 2 µm × 2 µm × 0.19 µm and 2 µm × 2 µm × 0.1 µm, respectively. (b) Thermal emission power spectrum comparison
between coupled mode theory prediction (red line) and finite-difference time-domain
simulation result averaged over 100 runs (blue crosses). . . . . . . . . . . . . . . . . 96
10.1 Decay rates and critical coupling. (a), Schematic of a structure consisting of a two-
dimensional material, separated from a mirror (light gray) by a dielectric spacer layer
(dark gray). As an example, the two-dimensional material is a single-layer graphene.
(b), External and internal decay rates of the structure, as a function of angle of
incidence. The external decay rate (γe, blue line) and internal decay rate (γi, red
line) are evaluated from equations 10.1 and 10.2, respectively. (c), Calculated mid-
infrared absorptivity of the structure, for varying angles of incidence. (d), Calculated
peak absorptivity as a function of angle of incidence. In (b), (c) and (d), the structure
consists of a single-layer graphene, separated from a perfect electric conductor (PEC)
layer by a 1.9-µm-thick dielectric layer with a refractive index as 2.1. The graphene
is assumed to have a Fermi energy of −500 meV with a mobility as 750 cm2/(V · s). In (b), (c) and (d), s-polarization is considered. . . . . . . . . . . . . . . . . . . . . 99
10.2 Device and experimentally measured absorptivity. (a), Photo of the structure. It
consists of a doped single-layer graphene, separated from a gold reflector by a 1.9-
µm-thick Ge23Sb7S70 chalcogenide glass layer. Here the red rectangle denotes the
region that is coated with graphene. (b), Raman spectra for two structures with
different graphene layers with different doping levels. The experimental spectra are
shown in dots, and Lorentzian fittings are shown in black lines. The Raman spectra
are vertically displaced for clarity. The graphene with 5-minute (red dots) and 40-
second (blue dots) doping durations are estimated to have Fermi energy EF as −500
meV and −300 meV, respectively (see Section 10.4). (c), (d) and (e), Measured angle-
resolved absorptivity for structures. (c), The case with graphene at EF = −500 meV.
(d), The case with graphene at EF = −300 meV. (e), The case of a bare structure
without graphene. (f), Peak absorptivities for the whole structure (diamonds), the
absorption inside graphene (squares), and parasitic absorption (triangles). Red and
blue lines denote graphene with EF = −500 meV and EF = −300 meV, respectively.
(c), (d), (e) and (f) are for s-polarization. . . . . . . . . . . . . . . . . . . . . . . . . 101
xxii
10.3 Calculated critical angle for different graphene properties and wavelengths. (a), Cal-
culated critical angle as a function of graphene Fermi level and mobility. The structure
consists of a graphene layer on the top, a 1.9-µm-thick dielectric layer with refractive
index as 2.1 in the middle, and a PEC layer on the back. (b), Calculated critical
angle as a function of graphene Fermi level and resonance wavelengths. The struc-
ture consists of a graphene layer on the top, a dielectric layer with refractive index
as 2.1 in the middle, and a PEC layer on the back. The graphene has a mobility as
750 cm2/(V · s). In (a) and (b), s-polarization is considered. . . . . . . . . . . . . . 104
10.4 Fermi energy determination from Raman spectra. (a), G peak positions of graphene
Raman spectra. The red and blue bands denote the positions of the G peak on the
5-min chemically doped graphene and 40-second chemically doped graphene, respec-
tively. The black dots show the calibrated relation between G peak position and Fermi
level, replotted from literature. (b), 2D peak positions of graphene Raman spectra.
The red and blue bands denote the positions of the 2D peaks on the 5-min chemically
doped graphene and 40-second chemically doped graphene, respectively. The black
dots show the calibrated relation between 2D peak position and Fermi level, replot-
ted from literature. In both (a) and (b), the Raman spectra have been taken at five
randomly selected locations, as denoted by the bands. . . . . . . . . . . . . . . . . . 107
10.5 Geometry for a bare structure with light incident in s-polarization. The structure
consists of a lossless spacer on top of a perfect electric conductor (PEC) reflector.
The spacer layer has thickness d. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
10.6 Geometry for a bare structure with light incident in p-polarization. The structure
consists of a lossless spacer on top of a perfect magnetic conductor (PMC) reflector.
The spacer layer has thickness d. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
10.7 Influence of parasitic loss and angular spread. (a), Calculated absorptivity for a struc-
ture without angular spread. The pink line shows the absorptivity for the whole struc-
ture, and the cyan, grey and yellow lines show the absorptions inside graphene, the
metal layer and the chalcogenide glass (ChG), respectively. (b), Calculated absorptiv-
ities in the presence of angular spread. Taking into account of a Gaussian distribution
of angular spread with standard deviation of 7.5, the pink and cyan lines denote the
calculated absorptivities for the whole structure, and the part absorbed by graphene.
The green circles denote the measured absorptivity for the structure with graphene
at EF = −500 meV. In the calculations of (a) and (b), the structure consists of a
doped graphene layer with EF = −500 meV and 750 cm2/(V · s) mobility, on top
of a 1.9 µm-thick Ge23Sb7S70 layer and Au reflector. The light is incident at 88, in
s-polarization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Electromagnetic heat transfer is fundamentally and technologically important. The thermal radi-
ation from very hot object has a short peak wavelength, and becomes visible light. For example,
sunlight is the thermal radiation from the sun at around 5800 K, and an incandescent light bulb
emits light from from the hot filament at around 3000 K. Thermal radiation powers an array of
energy applications, including solar cells, and thermophotovoltaic system etc. For terrestrial ob-
jects, the typical temperature is around 300 K, with a peak wavelength in the mid infrared. In
this temperature range, thermal radiation is an important mechanism for heat dissipation. Also,
thermal emission through these mid infrared wavelengths provides information about the material
properties, and therefore is useful for thermal imaging, chemical and biological sensing applications.
Microscopically, electromagnetic heat transfer is sourced by thermal fluctuation. If a system has a
dissipation mechanism, fluctuation-dissipation relation requires that at thermal equilibrium, there is
fluctuation associated with the dissipation process. The dissipation can result from different physical
processes depending on the frequency. Loss in ultraviolet, visible and near-infrared wavelength
range, typically results from interband transitions, free electron absorption, exciton etc. Loss in
mid-infrared wavelength range can result from intraband transition, lattice vibration, molecular
rotation, interband transition etc. In the Lifshitz framework of the fluctuation-dissipation theorem,
as shown in Fig. 1.1, one can consider thermally excited fluctuating current J(r, ω) at position r and
frequency ω. While the ensemble average of the fluctuating current is zero, the ensemble average
of the correlation function of the fluctuating current is generally non-zero. We assume that the
materials are passive, linear, non-magnetic (relative magnetic permeability µ = 1), and local ( in
which case the permittivity at a spatial point only depends on ω). For isotropic material in which the
dielectric property is described by a scalar permittivity, the correlation function for the fluctuating
2
J
Figure 1.1: The microscopic picture of electromagnetic heat transfer due to thermally excited fluctuating current.
current is:
Jj(r, ω)J∗k (r′, ω′) = 4
π ωΘ(ω, T )Im [ε(r, ω)] δ(r− r′)δ(ω − ω′)ε0δjk, (1.1)
where Θ(ω, T ) = ~ω/ ( e
~ω kBT − 1
) is the ensemble average energy of a photon gas of frequency ω at
temperature T , and ε(r, ω) is the relative permittivity. Here, Jj(r, ω) denotes the j’th component
of the fluctuating current.
current is:
π ωΘ(ω, T )
2i δ(r− r′)δ(ω − ω′)ε0δjk, (1.2)
where ε(r, ω) is the relative permittivity tensor. This form can be used for calculating radiative heat
transfer in materials that violate Lorentz reciprocity.
With information about the fluctuating current source, from remaining side, what remains is to
solve the Maxwell equations, in which the equation of Ampere’s law includes the fluctuating current
source:
dt D(r, t) + J. (1.3)
The electromagnetic heat flux into an object can be obtained by integrating the Poynting flux over
the whole surface of the object, or by integrating the energy dissipation density over the whole
volume of the object.
CHAPTER 1. INTRODUCTION 4
1.2 Numerical methods
Due to the fact that calculation for electromagnetic heat transfer involves solving the Maxwell
equations, in the presence of fluctuating current at each position, the simulation is generally quite
complicated. There only exists semi-analytical formula for radiative heat transfer for a limited range
of geometries, including half space - half space, multilayer-multilayer, sphere-sphere etc. It is gen-
erally important to take advantage of the symmetry of the geometry in simulations. For instance,
for calculating the radiative heat transfer between periodically patterned layers, it is desirable to
express all the quantities including the permittivity profile and electromagnetic fields, in the Fourier
space. For calculating the radiative heat transfer between spherical objects, it is natural to ex-
press the electromagnetic fields in terms of vector spherical wave functions. On the other hand,
for arbitrary geometry without symmetry, general methods are needed. Methods that can handle
arbitrary geometry for radiative heat transfer includes boundary element method, finite-difference
time-domain method by directly updating the fluctuating current, and volume integral equation etc.
These methods derive from the mature counterparts in simulating usual electromagnetic wave equa-
tions without the fluctuating terms. Generally, the numerical methods that respect the symmetry
feature of the system greatly outperform the general methods for arbitrary geometries, while the
general methods are indispensable for treating system with general geometry.
In the following, we discuss various methods that are used in this thesis for calculating electro-
magnetic heat transfer.
1.2.1 Thermal radiation from reciprocal thermal emitter
We first consider thermal emission to the far field. If the emitter consists of materials that satisfy
the Lorentz reciprocity, the general form for Kirchhoff’s law of thermal radiation is valid, which
states that at the same angle and frequency, the angular spectral emissivity equals the absorptivity:
e(ω, θ, φ) = α(ω, θ, φ), (1.4)
where e is the directional spectral emissivity, α is the directional spectral absorptivity, ω is the
frequency, and θ and φ specify a direction. Thus, for reciprocal structure, one can calculate instead
the absorptivity, and infer the thermal emission properties from the absorption properties [1, 2, 3, 4].
1.2.2 Thermal emission from non-reciprocal thermal emitter
The general form for Kirchhoff’s law of thermal radiation cannot be proven using the second law,
and rather is only true for reciprocal structure that satisfies Lorentz reciprocity. For non-reciprocal
CHAPTER 1. INTRODUCTION 5
emitter, there can be violation of detailed balance in thermal radiation:
e(θ, φ) 6= α(θ, φ). (1.5)
Thus, for thermal radiation from a non-reciprocal emitter, one can no longer infer the thermal emis-
sion properties from the absorption properties. For this non-reciprocal scenario, one need calculate
the thermal emission properties directly using fluctuational electrodynamics [5].
1.2.3 Scattering approach
A general formalism to calculate the heat transfer is the scattering approach, which takes into full
account all of the multiple reflections between the bodies, and embeds the information for each body
in T-matrix. As long as the T-matrix can be obtained, the scattering formalism applies to system
with arbitrary material composition and shape complexity. For example, scattering formalism is
used in calculating near field heat transfer between two nanospheres [6, 7]. In Chapter 3, we use the
scattering approach to demonstrate thermal supercurrent at thermal equilibrium in non-reciprocal
many-body near field radiative heat transfer. We note that the derived scattering approach applies
for both reciprocal and non-reciprocal near field radiative heat transfer.
1.2.4 Fourier-modal method for layered periodic structures
For calculating radiative heat transfer in layered periodic structures, a natural method is Fourier
modal method, or rigorous-coupled wave analysis (RCWA). This method has been used in Refs. [5,
8, 9] which are related to this thesis, and are used in Refs. [10, 11, 12, 13, 14, 15, 16] by other
researchers. A schematic geometry is shown in Fig. 1.2.
Gap
x
y
z
T1
T2
Figure 1.2: A schematic for the geometry for calculating radiative heat transfer between patterned periodic layers. Here, the system is periodic in the transverse directions.
CHAPTER 1. INTRODUCTION 6
In this method, all the quantities including the electromagnetic fields and the permittivity profile,
are described in the Fourier space. The first step is to construct the eigenmodes in each patterning
layer. The basic idea behind the Fourier modal method is to expand the fields within a patterning
layer into eigenmodes which have a simple exponential dependence in the vertical direction. One
then can construct the scattering matrix using the obtained eigenmodes as basis. Here, it is usually
necessary to use the numerically stable scattering matrix, rather than the transfer matrix. This
necessity for using numerically stable scattering matrix results from the fact that near field heat
transfer is typically dominated by the coupling of evanescent modes. Divergence of the field is
then quite common when applying the transfer matrix technique. The process for constructing
eigenmodes and scattering matrices are standard and can be found in Refs. [17, 18]. One then can
solve the electromagnetic fields in the presence of the fluctuating current, by matching the boundary
conditions are the spatial point of the fluctuating current, and calculate the resulting heat transfer
due to the presence of that fluctuating current.
As the thermal fluctuation here is local, thus, the heat transfer due to fluctuating current at
different spatial locations are completely independent from one another. Finally, one need take into
account of contributions of all the fluctuating currents, by performing a volumetric integration over
each patterning layer, in particular in the vertical direction. In the Fourier modal method, as the
eigenmodes have a simple exponential dependence in the vertical direction, this integration over the
source region can be analytically performed in a straightforward way.
To avoid such analytical integration over the vertical direction, alternatively, one can instead
start with an equivalent form for the fluctuation dissipation theorem [10, 11], which involves the
correlation function between fluctuating fields instead of currents.
In this Fourier modal method, the material can generally be anisotropic, for example magneto-
optical materials [5].
1.2.5 Finite-difference time-domain method
For arbitrary geometry without symmetry, general formalism can be useful, for example finite-
difference time-domain method.
Here we describe the finite-difference time-domain (FDTD) method [19] incorporating Langevin
approach to Brownian motion [20, 21, 22] which can be extended to calculate the thermal transfer
in a non-planar geometry.
The Langevin approach to Brownian motion models the polarization response P(t) of a system
to a local electric field E(t) and a random force term K(t) using the following equation of motion:
d2P
0P = σE + K (1.6)
where γ is the frictional coefficient of the polarization system, ω0 is its resonance frequency and σ
CHAPTER 1. INTRODUCTION 7
is its oscillation strength. The random force term K(t) models thermal fluctuations to the electric
field E(t) resulting in thermal emission.
After discretizing Eq. 1.6 and specifying K(t), we can perform time-stepping updates of P(t) and
E(t) using the conventional FDTD algorithm [20]. And from this numerical simulation of thermal
emission, we can calculate the heat flux spectrum in an arbitrarily shaped geometry. In the following
paragraphs, we give a brief description on how heat flux spectrum is calculated for a non-planar
geometry using FDTD.
We first describe how the random force term K(t) can be specified in order to model the thermal
fluctuations in the electromagnetic field. The polarization component of the displacement field
D = ε0E + P consist of a random component with spectral amplitude [21]:
Q(r, ω) = K(r, ω)
ω2 0 − ω2 − iγω
(1.7)
where K(r, ω) is the spectral amplitude of the random force term. The correlation function for
the components of Q(r, ω) satisfies the fluctuation-dissipation theorem given in Eq. 1.1 where the
fluctuating current jα(r, ω) = −iωQα(r, ω) and ε′′(ω) = Im [ ε0 + |P|
|E|
] = σγω
. After
combining Eq. 1.1 and 1.7, we get the following correlation function for K(r, ω) [21]:
Kα(r, ω)K∗β(r′, ω′) = 4
π σγΘ(ω, T )δαβδ(ω − ω′)δ(r− r′). (1.8)
This expression for the K correlation function gives us information about the distribution of the
random force term. For FDTD simulations, we generally need to Fourier transform Eq. 1.8 in
order to get the time-domain variance |Kα(r, t)|2 of the random force term distribution. However,
implementing this distribution of K in FDTD leads to a very high computational cost and is not
practical. In particular, the frequency dependent term Θ(ω, T ) in Eq. 1.8 leads to a complicated time-
dependence in the variance |Kα(r, t)|2. We next derive an alternative random force distribution
with a white-noise spectrum that is more practical to implement in FDTD while still allowing us to
calculate the heat flux spectrum of our non-planar geometry.
We first discuss the motivation behind using such an alternative random force term that has
a white-noise spectrum. This motivation arises from observing the form of the ensemble-averaged
heat flux spectrum S(ω) of a fluctuating electromagnetic field with correlation function given in
CHAPTER 1. INTRODUCTION 8
= 1
2
] =
] K ′j(r′, ω)K ′∗j (r′′, ω′) (1.9)
where εlmn is the Levi-Civita symbol, and K′(r, ω) = K(r, ω)/ √
Θ(ω, T ) has a white-noise spectrum.
In Eq. 1.9 we have used the following relations between the electromagnetic fields and the random
force term K:
H(r, ω) =
where ↔ GE and
↔ GH are the Green function dyad of the electric & magnetic fields, respectively.
The final form of Eq. 1.9 shows us that using a random force term with correlation function
K ′j(r′, ω)K ′∗j (r′′, ω′) in FDTD will allow us to simulate a normalized thermal emission in our
geometry, and consequently allow us to calculate a normalized heat flux spectrum S(ω)/Θ(ω, T ).
The actual heat flux spectrum S(ω) is then calculated by simply multiplying this normalized heat
flux spectrum by the known function Θ(ω, T ).
We next derive the time-domain form of this white-noise spectrum K′(r, ω). The discretized
correlation function of K′(r, ω) is:
K ′α(r, ω)K ′∗β (r′, ω′) = 4σγ
πV δαβδωω′δrr′ . (1.11)
Eq. 1.11 can be trivially Fourier transformed into the time-domain:
K ′α(r, t)K ′∗β (r′, t′) = 4σγ
πNV δαβδtt′δrr′ (1.12)
where N is the number of time steps used in the Fourier transform. Hence, by performing at each
time step a random drawing of K′ from a distribution with variance |K ′α(r, t)|2, we can simulate
a normalized thermal emission with normalized heat flux spectrum S(ω)/Θ(ω, T ) [Eq. 1.9]. We
emphasize that this variance |K ′α(r, t)|2 from Eq. 1.12 is time-independent. This is in contrast to
the complicated time-dependence of the variance |Kα(r, t)|2 derived from Eq. 1.8.
Since the specification of the variance |K ′α(r, t)|2 is the only constraint on the distribution of
CHAPTER 1. INTRODUCTION 9
K′, we can choose a uniform distribution with a range:[ − √
12σγ
] . (1.13)
Finally, we note that although here we used a permittivity that is modeled by a single-pole
Lorentz function, the method discussed can be implemented with material permittivity that has
multiple poles.
This method allows to obtain the heat transfer spectra over a broadband. Its tradeoff is that the
method is statistical in nature and needs averaging over around typically 40 runs. This method is
used in Chapter 9 to calculate the thermal emission from a single thermal emitter.
1.2.6 Other methods
There are other methods for calculating electromagnetic heat transfer in arbitrary geometries. One is
the boundary element method [23, 24, 7], where due to the equivalence principle one uses a fluctuating
surface current as the source. This method effectively reduces a system with unknowns in 3D space
to only unknowns on a 2D surface, which in many cases greatly accelerates the simulation and also
allows for a non-uniform grid naturally. The boundary element method is suitable for calculating
radiative heat transfer between compact homogeneous objects.
There are also methods using volumetric discretization in the frequency domain. One method
is volume-current formulation [25], which uses the volume-integral equation (VIE) method. This
method can be useful for treating the radiative heat transfer between objects with non-homogeneous
material composition. Another method is the discrete-dipole approximation method [26].
1.3 Radiative cooling
Here, we generally discuss several aspects of radiative cooling, including the atmospheric trans-
parency window and the thermal emitter.
1.3.1 Earth’s atmospheric transparency window
Earth’s atmosphere has a transparency window for electromagnetic waves between 8 and 13 microns.
A realistic atmospheric transmittance is shown as the magenta curve in Fig. 1.3. This atmospheric
transparency window also coincides with the peak wavelength of blackbody radiation spectrum of
terrestrial objects at typical temperatures, as can be seen in Fig. 1.3. By sending heat out in the
form of electromagnetic waves through this atmospheric transparency window, one can effectively
use the coldness of outer space as a heat sink, to passively cool an object on Earth. This technique
has been known as radiative cooling, and actually has been observed or taken advantage of centuries
CHAPTER 1. INTRODUCTION 10
0
0.5
1
Normalized
Figure 1.3: A realistic atmospheric transmittance spectrum (magenta), normalized AM 1.5 solar spectrum (yellow), and a normalized blackbody radiation spectrum at 300 K.
ago. For example, people have found that on nights with clear sky and still atmosphere, plants can
freeze even during the nights even when the ambient temperature is above the freezing point. This
is due to the fact that the plants are usually good thermal emitters, and can be cooled to below
ambient by radiative cooling on clear and windless nights. As another notable example, the Iranians
are believed to have used radiative cooling to make ice as early as 900 A. D. [27] on cloudless winter
nights, by filling a shallow pool with centimeter-deep water, protecting it from wind by building
adobe walls on the two sides, and allowing the pool to have a radiative access to the clear sky at
night.
These old arts are all for nighttime radiative cooling, where there is no influence from the sun.
However, there are more cooling demands in the daytime. The spectrum of solar irradiance on Earth
is mostly at wavelengths shorter than 4 µm, and there is nearly no spectral overlap between the
thermal radiation at 300 K and solar irradiance spectrum. Thus, theoretically one can control the
sunlight absorption and thermal emission properties of a structure independently.
We here consider a general thermal emitter structure, with a mirror on the back side. Here, the
thermal emitter is passive, linear, but generally can be non-reciprocal. The thermal emitter then
has a spectral directional emissivity as ε(λ,), and a spectral directional absorptivity as α(λ,),
where denotes direction.
We consider a structure at temperature T . The structure is exposed to a clear sky, and is subject
CHAPTER 1. INTRODUCTION 11
to solar irradiance, and atmospheric irradiance corresponding to an ambient temperature Tamb. The
net cooling power per unit area of a structure, Pnet(T ), is given by
Pnet(T ) = Prad(T )− Patm(Tamb)− Psun (1.14)
where
Prad(T ) =
∫ dcosθ
∫ ∞ 0
is the power radiated by the structure per unit area,
Patm(Tatm) =
∫ dcosθ
∫ ∞ 0
dλIBB(Tamb, λ)α(λ,)εatm(λ,) (1.16)
is the absorbed power per unit area emanating from the atmosphere, and
Psun =
∫ ∞ 0
dλα(λ, 0)IAM1.5(λ) (1.17)
is the incident solar power absorbed by the structure per unit area. Here ∫ d =
∫ π/2 0
0 dφ
is the angular integral over a hemisphere. IBB(T, λ) = (2hc2/λ5)/[ehc/(λkBT ) − 1] is the spectral
radiance of a blackbody at temperature T , where h, c, kB and λ, are the Planck constant, the velocity
of light, the Boltzmann constant, and wavelength, respectively. In Eq. 1.17, the solar illumination
is represented by AM1.5 Global Tilt spectrum. We assume the structure is facing the sun. Hence,
the term Psun is devoid of an angular integral, and the structure’s emissivity is represented by its
value in the zenith direction, θ = 0.
Here, Eq. 1.15 involves ε(λ,), while Eq. 1.16 involves α(λ,). Different from all previous studies
in radiative cooling, we here do not enforce the detailed balance relation for thermal radiation which
states that for the same angle and frequency the directional spectral emissivity of a thermal emitter
equals its absorptivity. This is because the detailed balance in thermal radiation is only a property
for reciprocal thermal emitter, and actually can be maximally violated by judicious design using
non-reciprocal structures as shown in Chapter 2.
Here what we do instead is to use only the constraint from the second law of thermodynamics.
For a thermal emitter, if the bottom side has no radiative exchange with the environment or the
thickness of the thermal emitter is sufficiently thick so that there is no radiative coupling between
the top surface and the bottom surface, at thermal equilibrium, the thermal radiation emitted from
the top after angular integration must equal the absorption of the background blackbody radiation
at the same temperature. A schematic is shown in Fig. 1.4.
This energy balance can be expressed as:∫ dcosθIBB(Tamb, λ)α(λ,) =
∫ dcosθIBB(Tamb, λ)ε(λ,). (1.18)
∫ dcosθε(λ,). (1.19)
Combine Eq. 1.19, with Eqs. 1.14, 1.15, 1.16 and 1.17, we then have
Pnet(T ) =
∫ dcosθ
− ∫
dλα(λ, 0)IAM1.5(λ), (1.20)
where we observe that for the thermal emitter the net cooling power expression can be solely ex-
pressed using the information of the directional spectral absorptivity.
1.3.2 Radiative cooling while preserving the amount of sunlight absorp-
tion
A solar absorber, under the sun is heated up. For a range of applications, for example solar cells
and outdoor structures, the amount of sunlight absorption is critical for functional or aesthetic
considerations, while the heating is undesirable. As a solar absorber by necessity must face the
sky, it naturally has radiative access to the coldness of the outer space through the atmospheric
transparency window. For these applications, it would be very attractive to use the cold outer space
as a cold heat sink to lower the temperature of the solar absorption, while maintaining the full
amount of sunlight absorption.
In a practical outdoor location the cooling power from the infrared radiation part when the ther-
mal emitter is at ambient temperature is generally around or smaller than 100 W/m2. As these cases
typically involve large amount of sunlight absorption, even when there is efficient radiative cooling,
sunlight absorption will dominate over the radiative cooling, heating up the absorber structure above
ambient temperature.
To optimally cool down these structures, one can maximize the net cooling power in Eq. 1.20 by
Figure 1.4: A schematic showing the energy balance between thermal radiation and absorption of background blackbody radiation in equilibrium.
CHAPTER 1. INTRODUCTION 13
optimizing the spectral angular absorptivity α(T, λ), at T > Tamb. As IBB(T, λ)−IBB(Tamb, λ)εatm(λ,) ≥ 0 at T > Tamb, to have optimal cooling the structure needs to be a perfect absorber for all angles and
frequencies in the thermal infrared. From Eq. 1.19, this also requires the structure to be a perfect
emitter, thus the ideal strategy is to be a thermal blackbody. On the other hand, the absorptivity
across the solar wavelengths needs to be maintained for functional and aesthetic considerations.
This scenario corresponds to the cases discussed in Chapter 4, Chapter 5, and Chapter 6.
1.3.3 Subambient radiative cooling
In other applications, one is directly targeting cooling a structure to below ambient under the sun.
The radiative cooling of a structure to sub-ambient temperature in the daytime has been recently
experimentally demonstrated in Ref. [4, 28]. It has also been proposed to harvest the energy from
the dark universe by using radiative cooling [29]. For this case, one should simultaneously engineer
the electromagnetic properties of the structure both at the thermal wavelengths and at the solar
wavelengths. On one hand, it requires reflection of over 95% of sunlight, and ideally to achieve perfect
reflection of sunlight. On the other hand, the optimal electromagnetic response of the structure in
the infrared is has subtle dependence on the temperature of the radiative cooler, the temperature
of the ambient, and also the transparency of the atmosphere.
One can consider the problem as maximizing the net cooling power by controlling the EM re-
sponse of the radiative cooler in the thermal wavelengths, for the given structure temperature,
ambient temperature, and atmospheric transparency. By examining Eq. 1.20, we observe that the
optimal value for α(λ,) depends on the sign of [IBB(T, λ)− IBB(Tamb, λ)εatm(λ,)]. The term
IBB(Tamb, λ)εatm(λ,) characterizes the thermal radiation emitted by Earth’s atmosphere towards
Earth’s surface, i.e. the strength of the downward radiation, when the ambient temperature is at
temperature Tamb. For a given set of T , Tamb and εatm(λ,), if the blackbody radiation intensity
at T exceeds the atmosphere downward radiation, it is optimal to have α(λ,) = 1 to maximize the
cooling power for the specific wavelength and angle. On the other hand, if the blackbody radiation
intensity at T is inferior to the atmosphere downward radiation, it is optimal to have α(λ,) = 0
to avoid any contribution for heating.
As the atmospheric transmittance depends on the angle (the atmosphere is less transparent at
larger angle due to larger optical path in the atmosphere), the optimal EM properties of a radiative
cooler has angular dependence. One also notes that the the optimal EM properties of a radiative
cooler also depends on the temperature of radiative cooler. One can easily imagine that the radiative
cooler that can reach the lowest steady state temperature, is not necessarily the one that cools down
fastest. The ultimate radiative cooler would need have temperature dependence that can maintain
its EM properties to the optimal set for all the temperatures. Phase change material whose material
properties change abruptly with temperature may provide an interesting platform to study these
effects.
CHAPTER 1. INTRODUCTION 14
One interesting and somewhat counter-intuitive fact is that the truly optimal EM properties of
the radiative cooler from the analysis above, can be significantly different from the emissivity profile
with unity emissivity between 8 and 13 µm, while perfectly reflecting outside. Also, we emphasize
that the derived optimal EM properties of a radiative cooler at the thermal wavelength range here
is actually a second law limit.
Chapter 2
balance∗
For thermal radiation, the principle of detailed balance leads to the general form of the Kirchhoff’s
law [30, 31, 32, 33], which states that
e(ω, θ, φ) = α(ω, θ, φ) (2.1)
where e is the directional spectral emissivity, α is the directional spectral absorptivity, ω is the
frequency, and θ and φ specify a direction. Seeking to violate detailed balance is fundamentally
important, because the principle of detailed balance implies the existence of an intrinsic loss mech-
anism that limits the efficiency of many energy conversion processes. For example, a solar absorber
absorbs light from the sun. The detailed balance then dictates that the solar absorber must therefore
radiate back to the sun. Such radiation back to the sun is an intrinsic loss mechanism that can only
be eliminated by maximal violation of detailed balance. The ability to significantly violate detailed
balance therefore points to a previously unexplored pathway for fundamental improvement of a
wide variety of energy conversion processes, including solar energy harvesting and thermal radiation
energy conversion.
Microscopically, Eq. 2.1 can be proven using the fluctuation-dissipation theorem, but only for
emitters consisting of materials satisfying Lorentz reciprocity [34, 35]. It has been noted theoretically
that non-reciprocal materials, such as magneto-optical materials, may not obey detailed balance [36]
and hence may not satisfy Eq. 2.1, without violating the second law of thermodynamics [37]. How-
ever, there has not been any direct experimental measurement or theoretical design of actual physical
structures that violate detailed balance.
∗Reprinted with permission from “Near-complete violation of detailed balance in thermal radiation” by L. Zhu and S. Fan, 2014. Phys. Rev. B, vol. 90, pp. 220301, Copyright [2014] by the American Physical Society.
15
CHAPTER 2. NEAR-COMPLETE VIOLATION OF DETAILED BALANCE 16
In recent years, significant efforts have been devoted to the use of engineered photonic structures,
including photonic crystals [38, 39, 40, 41, 42, 20, 43, 44, 45, 46, 47, 21, 48, 49, 50, 51, 52, 53], optical
antennas [54, 55, 56] and meta-materials [57, 58, 59], for the control of thermal radiation properties.
Photonic structures can exhibit thermal radiation properties that are significantly different from
naturally occurring materials. Notable examples include the creation of thermal emitters with
narrow spectrum [41, 43, 52] or enhanced coherence [40, 45]. All previous works on the thermal
radiation properties of photonic structures, however, consider only reciprocal materials.
In this chapter, using the formalism of fluctuational electrodynamics [60, 61, 62, 63, 64], we
present a direct numerical calculation of thermal emission from non-reciprocal photonic structures,
and introduce the theoretical conditions for such structures to maximally violate detailed balance,
i.e. to achieve a unity difference between directional spectral emissivity and absorptivity.
Non-reciprocal photonic structures represent an important emerging direction for the control of
thermal radiation. From a fundamental point of view, significant numbers of theoretical approaches
for the calculations of far-field thermal radiation use the Kirchhoff’s law of Eq. 2.1 by computing
the absorption properties [43, 44, 45, 46, 47, 48, 49, 51]. Such an approach is no longer applicable
for non-reciprocal thermal emitters, and direct calculations using the formalism of fluctuational
electrodynamics become essential. From a practical point of view, creating non-reciprocal thermal
emitters can have important implications for the enhancement of the efficiency for solar cells [65, 66]
and thermophotovoltaic systems [67].
We start by reviewing the general thermodynamic constraints on non-reciprocal thermal emitters.
Consider an emitter undergoes radiative exchange through two radiation channels, A and B, with
two separate blackbodies also labelled A and B, respectively. Part of the emission from either
blackbody A or blackbody B towards the emitter is absorbed, as described by absorptivities αA and
αB , respectively. The emitter also emits towards the blackbodies as described by emissivities eA
and eB , respectively. We consider the equilibrium situation where the emitter, and the blackbodies,
are at the same temperature T . The second law of thermodynamics then requires that there is no
net energy flow in or out of the emitter, independent of whether the emitter is reciprocal or not. In
the reciprocal case (Fig. 2.1a), αA,B = eA,B , and as a result the second law of thermodynamics is
satisfied. In the non-reciprocal case (Fig. 2.1b), consider the emission from blackbody A, through
channel A, we assume that the part of the emission that is not absorbed by the emitter is reflected
through channel B to blackbody B, with a reflectivity rA→B . As a result, we have
αA + rA→B = 1. (2.2)
CHAPTER 2. NEAR-COMPLETE VIOLATION OF DETAILED BALANCE 17
A B
α A
e A
e B
B
(a)
Figure 2.1: Energy flow diagrams in the cases of (a) a reciprocal emitter, and (b) a non-reciprocal emitter. The emitter undergoes radiative exchange with two separate blackbodies labelled A and B, respectively. The emitter and the blackbodies are at the same temperature T .
On the other hand, blackbody A receives emission both from the emitter and the part of emission
from blackbody B that is not absorbed by the emitter, i.e.
eA + rB→A = 1. (2.3)
Combining Eqs. 2.2 and 2.3 and similarly consider the energy balance of the blackbody B, we have
eA − αA = rA→B − rB→A = αB − eB (2.4)
For non-reciprocal systems, rA→B 6= rB→A [68, 69, 70]. As a result, eA,B 6= αA,B , and the detailed
balance is violated. On the other hand, from Eq. 2.4 there is no net energy flow in and out of the
emitter as well as bodies A and B, as required by the second law. Thus, for the non-reciprocal
structure considered here the second law in fact dictates the violation of detailed balance. The
argument here is equivalent to Ref. [36] which uses bidirectional reflectance distribution functions,
but simplified and generalized so that the argument can be directly applied to the physical system
that we will consider here that has only specular reflection.
2.2 Conditions to maximally violate detailed balance
As a main contribution, we next introduce the general conditions in order to achieve maximum
violation of detailed balance in a physical structure. As the emitter, we consider a photonic crystal
emitter structure that is periodic in x-direction, emitting to free space on top of the structure, with a
mirror at the back side (Fig. 2.2). For simplicity we consider only a two-dimensional case where both
the fields and the structure are assumed uniform along the z-direction. The principle described here
however is generalizable to three dimensions. For such a structure, its electromagnetic properties are
characterized by a photonic band structure ω(kx), where ω is the frequency, and kx is the parallel
wave vector.
t1 t2
z w
Figure 2.2: A schematic of a photonic crystal structure for maximal violation of detailed balance. The structure consists of an n-InAs grating structure atop a uniform metal layer. The structure is periodic in x-direction, and has the following geometry parameters: p = 7.24 µm, w = 3.2 µm, t1 = 1.981 µm and t2 = 0.485 µm. External magnetic field is applied in z-direction. TM polarization with electric field in x-y plane is considered.
Corresponding to the scenario as described in Fig. 2.1, we study the directional spectral emis-
sivity and absorptivity e(ω,±θ) and α(ω,±θ), respectively, where ±θ are the angles of incidence for
the two channels. Consider light incident with an angle of incidence θ, having a parallel wavevector
kx = ω/c sinθ. With a proper choice of periodicity that is sufficiently small, by momentum conser-
vation, light can only be reflected into the −θ channel (Fig. 2.2). Moreover, if the ω and kx of the
incident light satisfy the photonic band structure ω(kx) of the emitter, a mode inside the emitter
will be resonantly excited, as a result there will typically be strong absorption, with part of the
resonant excitation contributing to the reflected wave in the −θ channel. Similarly, the reflection
and absorption properties for light incident with an angle of incidence of −θ will be controlled by
the photonic band structure at ω(−kx).
For an emitter constructed from a reciprocal material, its photonic band structure is symmetric
in the kx-space [71], i.e. ω(kx) = ω(−kx). The resonance frequencies for light incident with an
incidence angle of either θ or −θ are the same, and rθ&