Designing Nanoparticle Self-Propulsion With
ARCHIVES
Nonequilibrium Casimir Physics
MASSAC HLJSET
NS TUTE
OF FECHNIOLOLGY
by
AUG 10 2015
Eric D. Tomlinson
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree of
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Bachelor of Science
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2015
@ 2015 Massachusetts Institute of Technology. All rights reserved.
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Department of Physics
May 8, 2015
Certified by.
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(-Ateven G. Johnson
Professor of Applied Mathematics
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.
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Thesis Supervisor
..................
Robert L. Jaffe
Professor of Physics
Thesis Supervisor
Accepted by . .
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....
Professor Nergis Mavalvala
Senior Thesis Coordinator, Department of Physics
2
Designing Nanoparticle Self-Propulsion With Nonequilibrium
Casimir Physics
by
Eric D. Tomlinson
Submitted to the Department of Physics
on May 8, 2015, in partial fulfillment of the
requirements for the degree of
Bachelor of Science
Abstract
This work presents an analysis of thermal self-propulsion behavior in nanoparticles
using several recent advancements in the field of nonequilibrium Casimir physics.
We compute fundamental limits on the thermal power emission and thermal selfpropulsion force that is attainable for particles of a given size. The limits that we
obtain are valid for photon emission at a single frequency; however, they allow us to
estimate the maximum total power emission and self-propulsion force that we can expect to achieve for a wide range of materials that are commonly used in nanoparticle
manufacturing. We provide a detailed description of the role that particle temperature, material composition, and geometry play in generating thermal self-propulsion
forces and use this information to develop a general procedure for designing efficient
self-propulsion behavior using the SCUFF-EM software package [24]. Finally, we
present the results of our exploratory design study amongst silicon dioxide nanoparticles and identify three candidates that exhibit strong self-propulsion.
Thesis Supervisor: Steven G. Johnson
Title: Professor of Applied Mathematics
Thesis Supervisor: Robert L. Jaffe
Title: Professor of Physics
3
4
Acknowledgments
In February 2014, I was re-admitted into MIT after a much-needed leave of absence.
My first three years of college had been unexpectedly rocky and my time away had left
me feeling unsure of myself and confused about my academic aspirations. Fortunately,
my return to academia has been a wonderful, tremendously encouraging experience.
I am sincerely grateful for the friendship and guidance that I have received from
Homer Reid. Homer is an engaging lecturer, a gifted researcher, and a generous
donor of his time and energy. Working with Homer has truly been the highlight of
my undergraduate education. I would like to thank my academic advisor Bob Jaffe
both for his assistance in my re-admission and for inviting me to attend his weekly
Casimir theory research group. The information and feedback that I gained from
attending these meetings has been instrumental in the development of this thesis. I
would also like thank my thesis advisor Steven Johnson for helping to support my
research with Homer over the past year.
In addition to those that I have already mentioned above, I wish to acknowledge a
few individuals that have directly contributed to my research by providing useful insights and helpful discussions. In no particular order, they are: Owen Miller, Matthias
Kruger, Thorsten Emig, and Mehran Kardar. In the realm of personal and emotional
support, I would like to thank my parents and my wonderful group of friends whodespite being scattered around the country-have kept me going through spirited
phone calls and surprise visits.
This thesis is dedicated to Drew Ames. Your friendship and support over the past
few years has meant the world to me. Here's hoping that our paths continue on their
surprisingly parallel trajectories.
5
6
Contents
1
Overview
13
2
General Concepts
15
2.1
2.2
3
16
2.1.1
Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
Computing Power and Force in Fluctuational Electrodynamics . . . .
18
2.2.1
20
Discretization
. . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3
Computing Power and Force in the T-Matrix Formalism
. . . . . . .
21
2.4
The Boundary Element Method . . . . . . . . . . . . . . . . . . . . .
22
Fundamental Limits On Thermal Self-Propulsion
25
3.1
Upper Bound on Thermal Power Emission . . . . . . . . . . . . . . .
27
3.1.1
Accounting for Particle Size . . . . . . . . . . . . . . . . . . .
29
3.1.2
Implications for Particle Design . . . . . . . . . . . . . . . . .
31
3.2
3.3
4
Computing Power and Force in Deterministic Electrodynamics . . . .
. . . . . . . . . . . . . . .
32
3.2.1
Computing the Gradient . . . . . . . . . . . . . . . . . . . . .
33
3.2.2
Symmetries of the T-Matrix . . . . . . . . . . . . . . . . . . .
34
3.2.3
Conservation of Energy . . . . . . . . . . . . . . . . . . . . . .
36
3.2.4
Results and Discussion . . . . . . . . . . . . . . . . . . . . . .
38
Experimental Predictions . . . . . . . . . . . . . . . . . . . . . . . . .
41
Upper Bound on Thermal Self-Propulsion
Designing Thermal Self-Propulsion With SCUFF-EM
45
4.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
4.1.1 - Particle Temperature . . . . . . . . . . . . . . . . . . . . . . .
46
4.1.2
Material Composition
. . . . . . . . . . . . . . . . . . . . . .
47
4.1.3
G eom etry . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
4.2
Design Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
4.3
Initial Design Results . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
Design Parameters
7
5
4.3.1
Prototype 1: Death Star . . . . . . . . . . . . . . . . . . . . .
55
4.3.2
Prototype 2: Sorcerer's Hat
. . . . . . . . . . . . . . . . . . .
59
4.3.3
Prototype 3: Martini Glass . . . . . . . . . . . . . . . . . . . .
61
4.3.4
Final Comparison: Total Self-Propulsion . . . . . . . . . . . .
63
Conclusions and Future Work
65
A Dyadic Green's Functions
67
8
List of Figures
2-1
3-1
3-2
3-3
Typical triangular-element surface discretization scheme that is used
in conjunction with the boundary element method. . . . . . . . . . .
23
Arbitrarily-shaped particle of radius R along with a rough estimate of
the angle AO over which the shape of the particle varies. . . . . . . . 30
Plots displaying the convergence properties of our nonlinear self-propulsion
force optimization scheme for 1 < emax < 4 . . . . . . . . . . . . . . . 39
Best estimates obtained for the dimensionless analogues of maximum
thermal power emission, maximum thermal self-propulsion force, and
the thermal power emissio corresponding to the maximum thermal selfpropulsion force. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4-1 Energy spectrum of thermal photons at various temperatures. . . . .
4-2 Schematic diagram of the Death Star particle prototype. . . . . . . .
4-3 Parameter sweep over possible Death Star shapes at the smallest length
scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-4 Parameter sweep over possible Death Star length scales for the best
particle in Figure 4-3. . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-5 Schematic diagram of the Sorcerer's Hat particle prototype. . . . . .
4-6 Parameter sweep over possible shapes for the Sorcerer's Hat prototype.
4-7 Schematic diagram of the Martini Glass particle prototype. . . . . . .
4-8 Parameter sweep over possible angles for the Martini Glass prototype.
9
48
56
57
58
59
60
61
62
10
List of Tables
3.1
3.2
3.3
3.4
4.1
4.2
4.3
An (incomplete) collection of T-matrix symmetry properties that hold
in the spherical wave basis 119]. . . . . . . . . . . . . . . . . . . . . .
List of the number of independent elements in T-matrices that satisfy
various symmetry properties . . . . . . . . . . . . . . . . . . . . . . .
Ratio of the average magnitude of negative eigenvalues to positive
eigenvalues associated with the rightmost point (100,000 constraint
vectors) in each of the four plots from Figure 3-2. . . . . . . . . . . .
Collection of the data represented graphically in Figure 3-3. . . . . .
Parameters used in the oscillator model (4.10) for the permittivity of
SiO 2 [17]. All values are given in units of radians per second. . . . . .
Details of our search through the Death Star parameter space. The
parameter d has three start and stop values, each corresponding to a
different value of the parameter r. . . . . . . . . . . . . . . . . . . . .
Final statistics on the total thermal self-propulsion force and thermal
acceleration at room temperature for the best candidate particle from
each class. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
35
36
40
43
55
56
63
12
Chapter 1
Overview
A particle whose internal temperature differs from the temperature of the surrounding
medium will experience a self-propulsion force if it preferentially emits or absorbs
electromagnetic radiation in a single direction. At a conceptual level, this behavior
is easy to understand. In order to equilibrate with its surroundings, the particle
will exchange energy with the medium in a process known as radiative heat transfer.
The thermal radiation that mediates this exchange of energy is also endowed with
linear momentum. If there happens to be a directional imbalance in the flow of linear
momentum between the particle and the medium, then the particle will feel a recoil
force in accordance with Newton's third law of motion. This recoil force will persist
until the internal temperature of the particle matches the temperature of the medium.
This means that the particle will remain in a self-propelled state until it has reached
thermal equilibrium.
Despite its conceptual simplicity, thermal self-propulsion has never been explored
in detail. There are two primary reasons for the lack of research on this topic at
the present time. First of all, the effect is small and would be challenging for an
experimentalist to observe. Nanoparticle manufacturing is a rapidly developing field
of research, but the ability to precisely engineer optical properties at this length scale
is still outside of our grasp. Secondly, it turns out that this phenomenon is very
difficult to predict in realistic systems. To understand why, we must delve into the
mechanism that gives rise to radiative heat and momentum transfer.
Quantum and thermal fluctuations in otherwise neutral bodies create stochastic
electromagnetic fields that are present everywhere in space. In systems at thermal equilibrium, these fluctuating electromagnetic fields give rise to Casimir forces,
which are a generalized version of the van der Waals interaction that exist between
macroscopic bodies [5]. In nonequilibrium systems, the fields additionally mediate
13
the transfer of energy from hotter to colder bodies 129]. This flow of energy from
hot bodies to cold bodies modifies the Casimir force in a highly non-trivial fashion.
Fortunately, recent advances in the field have made it possible to compute nonequilibrium Casimir forces between arbitrary compact objects [16, 291. In this thesis, we
make use of these advances to study nanoparticle self-propulsion by computing the
nonequilibrium Casimir force between a particle and its surrounding medium.
Our study of thermal self-propulsion is divided into four chapters. In Chapter 2,
we develop general mathematical techniques for computing the emitted power and
self-propulsion force from an isolated, radiating material body. These techniques set
the stage for a brief description of the main computational tools that we will utilize in
our study: the T-matrix formalism and the boundary element method. In Chapter 3,
we use the T-matrix formalism to obtain fundamental limits on the thermal power
emission and thermal self-propulsion that is attainable for particles of a given size.
Strictly speaking, the limits that we obtain are only valid for photon emission at a single frequency. However, they allow us to estimate the maximum total power emission
and self-propulsion force that we can expect to achieve for a wide range of materials
that are commonly used in nanoparticle manufacturing. In Chapter 4, we perform
an exploratory design study of thermal self-propulsion in silicon dioxide nanoparticles
using the SCUFF-EM software package. Here we provide a detailed description of
the role that is played by particle temperature, material composition, and geometry
in generating thermal self-propulsion forces. After developing a general approach to
nanoparticle design, we use the software to locate three novel self-propulsion geometries and compare their relative performance. Finally, we conclude in Chapter 5 with
a summary of our findings and present our suggestions for future work.
14
Chapter 2
General Concepts
In this chapter we develop general techniques for computing the emitted power and
self-propulsion force from an isolated, radiating material body. We first consider
the case where the source of radiation is a deterministic volume density of electric
current. Using the dyadic Green's functions from classical electromagnetic theory, we
construct explicit formulas for the power and force from the electric current density
alone. Our goal in this section is to convince the reader that the emitted power
and self-propulsion force both depend quadraticallyon the volume density of electric
current. Once this has been accomplished, we consider the case where the radiation
is produced by thermal current fluctuations that are present in the body when it is
held at finite temperature.
For stochastic current sources, we lose the ability to make predictions regarding
the instantaneous values of any quantities of interest. Furthermore, any quantity
that depends linearly on the current density will average over time to zero. Using the
fluctuation-dissipation theorem of statistical physics, it is possible to derive a twopoint correlation function that relates fluctuations in the product of vector components
of the current density to the temperature and material properties of the radiating
body. We make use of this two-point correlation function to derive expressions for
the thermal power emission and thermal self-propulsion force for an isolated material
body. Our development of the aforementioned material will take inspiration from
references [16, 22, 25].
15
2.1
Computing Power and Force in Deterministic
Electrodynamics
The reader will recall that, in the presence of a known volume density of electric
current J(x), the components of the electric and magnetic fields are given by linear
integral relations of the form:
E (w, x) = JG (w; x, x')J(w, x') dx',
(2.1)
Hi(w, x) =
(2.2)
G (w; x, x')Jj (w,x') dx',
where G3 and G3 are the electric and magnetic dyadic Green's functions (DGFs)
and we have assumed that all currents and fields have a time dependence that is
proportional to e-t. For a brief review of the dyadic Green's functions, we refer the
reader to Appendix A.
Let us consider the case where the electric current distribution J(x) is entirely
contained within the volume of a single material body B. The radiation that is
produced by the electric current carries energy and momentum away from the body.
The time-averaged value of the emitted power P is obtained by integrating the mean
Poynting flux over any surface S that entirely surrounds the body 1 :
P =I Re
E*(x)
x
H(x)
-n(x) dx ,
(2.3)
where fi(x) is the outward-pointing unit normal to the surface S at the point x.
Similarly, we can compute the time-averaged value of the self-propulsion force F by
integrating the real part of the Maxwell stress tensor over the same surface:
=
I Re
{eoE (x)Ej (x) + poH (x)Hj (x)
-j
EoIE(x) 2 + polH(x))
}
ii
3(x) dx
.
F
By including the free-space permittivity co and permeability /Lo in (2.4), we have made
the assumption that the material body is embedded in vacuum.
We now wish to rewrite our expressions for the emitted power and self-propulsion
force in terms of the electric current density. We will demonstrate this procedure for
'Here and for the remainder of the chapter, any dependence on angular frequency w will be
dropped from our expressions. All currents and fields are assumed to vary harmonically with time.
16
the emitted power (2.3). Inserting our expressions for the electric (2.1) and magnetic
(2.2) fields into (2.3), we obtain:
Pr(
Eijk
(x)Pdx'
f G*(x, x')J*(x')
2 JslJ}
x
j
(2.5)
Gm(x,
x")Jm(x") dx"}
hi (x) dx
,
1
P = -Re
where we have used the Levi-Civita symbol Eijk to write the components of the cross
product. Although this expression (2.5) is rather complicated, it demonstrates that
the emitted power is a bilinear function of the vector components of the electric
current density J(x). Furthermore, if we change the order of integration and group
the terms in the following way:
P = Rejdx'jdx"J*(x')
G
*(x,X')G'm(x,
x")hi (x) dx
Jm(x")
(2.6)
we will see that the power can be thought of as the real part of a bilinear convolution
operation
P
=
Re J* QP *J
= Re
dx'
dx"J*(x')Q
m(x', X")Jm(X")
,
(2.7)
where the components of the tensor QP (x', x") are given by the quantity in curly
brackets (2.6). Applying this procedure to the self-propulsion force (2.4), one can
derive a similar expression:
F= Re J** QF* J = Re
dx'
dx"J*(x') m(x' x")J(x") ,
(2.8)
where the components of the tensor Q m(x', x") are given by
Gi'm
'
x
=
Gt*(x, x')Gjm(x, x")
- -6jGD(x, x')Gm(x, x")
(2.9)
+2 G*(xx')Gjm(xX")
t-
17
Gm* (x, x')Gm~x x") hj (x) dx.
2.1.1
Discretization
Equations (2.7) and (2.8) allow us to compute the emitted power and self-propulsion
force for any electric current distribution J(x). In practice, however, we will rarely
find ourselves in possession of an exact expression for the current distribution within
a material body. In deterministic electrodynamics, we will commonly be faced with
scenarios where we must solve for the electric current that is induced within the object
by an incident electromagnetic field. This first step in the calculation is typically
accomplished by discretizing the current density with a well-chosen set of vectorvalued basis functions ba(x):
(2.10)
J(x) = Zjaba(x),
a
and solving an integral equation for the expansion coefficients ja in terms of the
incident field. If we plug the discretized current density (2.10) into our expressions
for the power (2.7) and force (2.8), we obtain vector-matrix-vector products:
P= Re [jtQPj] ,
(2.11)
Fi = Re [jtQFj] ,
(2.12)
where the j is the vector of expansion coefficients and the matrix elements of QP and
are given by 2:
Q
QP
Q[
2.2
1
=
b* * QP * b, =
af1
= b*
* QF * bo =
dx'
JB
dx"
dx'
dx" b*(x')Qm(x', x")bm(x").
b*at(I
(x')Qim(x',fBx")bam(x"),
(2.13)
(2.14)
Computing Power and Force in Fluctuational
Electrodynamics
As we mentioned in the introduction to this chapter, if the current distribution J(x)
in a material body is fluctuating randomly with time-as is the case for a particle
held at finite temperature-then we lose the ability to make predictions regarding the
instantaneousvalue of any quantities of interest. Moreover, any measurable quantity
2
The notation here is unavoidably complex. When we write boldface ba with a single Greek
index, we are referring to the ath basis vector. When we write bae with Greek and Roman indices,
we are referring to the fth component of the ath basis vector.
18
that depends linearly on the current, such as the electric (2.1) or magnetic (2.2)
field, will average over time to zero. Fortunately, quantities that depend bilinearly
or quadraticallyon the current, like the emitted power (2.7) and self-propulsion force
(2.8), will typically have time-averaged values that are non-zero. This comports with
our intuition, as we know that finite-temperature material bodies emit energy-andmomentum-carrying photons off to infinity. How then, do we compute the emitted
power and self-propulsion force for thermally fluctuating current distributions?
The answer to this question was originally developed by Russian physicist S. M.
Rytov in the early 1950s [301. Rytov made use of the fluctuation-dissipation theorem
of statistical physics to derive a two-point correlation function relating fluctuations
in the product of vector components of the current density to the temperature T and
the material properties of the radiating body [31]:
CO
F
2(2kBT
_
KJi(x)J(x')
1
=
-6(x
-x')
w
hw
- coth
hw~
w Im eCi (w,x) .
(2.15)
We will have more to say about the Rytov correlation function (2.15) in Section 4.1,
but for now it will suffice the know that our ( ), notation implies an average over all
possible phases 3 , and the material properties of the radiating object are encoded in
the frequency-dependent permittivity tensor ey (w, x).
Taking the average of both sides of our equations for the power (2.7) and the force
(2.8), plugging in for the Rytov correlation function (2.15) and simplifying we obtain:
(P)
w 9(w,T)
KFi)
= w(w, T)
dx' Re Qm(x', x')
Im em(wx') ,
dx' Re Qm(x', x') [Im e m(w,
x')
(2.16)
,
(2.17)
where we have chosen to define a new function 8(w, T) that returns the quantity in
square brackets (2.15):
E(w, T)= -- coth
2
2kBT
.
(2.18)
Furthermore, by pulling the factor of O(w, T) outside of the volume integrals in (2.16)
3
This is simply the natural extension of time-averaging into the frequency domain. More technically, we can define the phase-average of any time-varying quantity f(t) by:
Kf)
e
=
t
f(t) dt),
where the angle brackets on the right-hand side indicate an average over all possible values of the
start time to [23].
19
and (2.17), we have made the assumption that the temperature distribution within
the radiating body is spatially constant. Equations (2.16) and (2.17) give us the
power spectral density and force spectral density, respectively. In other words, they
provide us with the contribution to thermal power emission and self-propulsion force
coming from fluctuations at a single angular frequency. If we wish to compute the
total thermal power emission and self-propulsion force, we must integrate the spectral
densities over all frequencies:
2.2.1
(Fi) =
(P), du ,
(P) =
(Fi) dw.
(2.19)
Discretization
To obtain the discrete forms of our expressions for the spectral densities (2.16) and
(2.17), we will have to first compute the discrete analogue of the Rytov correlation
function (2.15). Inverting our expression for the discretized current density (2.10),
we can write the expansion coefficients in the form:
ja
=
fb
(2.20)
(x) Jj (x) dx,
where the integral is again taken over the volume of the radiating body. If we take the
outer product of the vector of expansion coefficients j with itself, we obtain a matrix
that is populated with the elements:
[jj
= jdx
Xdx' b& (x) Ji (x) J (x
bm(x').
(2.21)
Taking the average of both sides of (2.21) and inserting the Rytov correlation function
(2.15), we obtain an object that we will refer to as the Rytov matrix:
[RI
--
Kjj
=
'wIT)
dx br(x) Im etm(w,
x)
b*m(x),
(2.22)
where we have temporarily dropped the w subscript from our angle brackets for notational clarity. The Rytov matrix (2.22) is the natural extension of the Rytov correlation function into the discrete domain.
Now we perform a subtle trick with our discrete formulas for the power (2.11)
and force (2.12). We will demonstrate the trick for the power, but it is applicable to
any physical quantity that can be expressed as a vector-matrix-vector product. First,
since the power (2.11) is a scalar quantity, we are free to express the right-hand-side
20
as the trace over what is effectively a one-dimensional matrix:
jf QPjI
P= Re Tr
(2.23)
Then, taking advantage of the invariance of the trace under cyclic permutations, we
re-express (2.23) as the trace of an alternative matrix:
P = ReTr QP (jjt) J,
(2.24)
where we have grouped the terms in a suggestive way. Equation (2.24) holds equally
well in deterministic electrodynamics as it does in fluctuational electrodynamics.
However, in the deterministic case, the object (jjt ) is a rank-one matrix, which
would make (2.24) an extremely inefficient method for computing the power.
Taking the average of both sides of equation (2.24), we are able to obtain an
expression for the power spectral density in terms of the Rytov matrix (2.22):
(P) =ReTr QPRJ,
Fi) =ReTr[QFR].
(2.25)
In (2.25), we have included the analogously obtained expression for the force spectral density. Finally, if we wish to compute the total thermal power emission and
thermal self-propulsion force, we must integrate the spectral densities (2.25) over all
frequencies. We include here the explicit formulas for later reference:
(P)
2.3
=
Ref
Tr [QPRdo,
(F) = Ref
Tr [Q R]d
.
(2.26)
Computing Power and Force in the T-Matrix
Formalism
There are a number of different ways in which one can formulate expressions for the
thermal power emission and self-propulsion force. As we will see in Section 2.4, our
trace formulas (2.26) are particularly well-suited for efficient numerical implementation. Here we provide a brief description of an alternative trace formulation that
arises out of the scattering theory of nonequilibrium Casimir forces and radiative
heat transfer 116]. The scattering matrix theory that is used in this approach will
highlight certain aspects of the underlying phenomenology that we will find useful in
our analysis of fundamental limits in Chapter 3.
21
The scattering theory is completely general, allowing one to compute fluctuationinduced energy and momentum transfer between any number of objects having arbitrary temperatures, shapes, material properties, and separations. Thermal power
emission and thermal self-propulsion arise as a special case in this formalism in which
only a single, finite-temperature object is present in the vacuum. We make no attempt to derive the trace formulas that are obtained in this context, as the process
is quite involved. Here will simply present the relevant formulas and provide a brief
qualitative discussion to familiarize the reader with their meaning and interpretation.
The trace formulas for thermal power emission and thermal self-propulsion force,
obtained in [16] and [21] respectively, are given by:
P)=2
7r
(F) =
7
1
00
f
e kBT
fo
-
(T +
- T
Im Tr PTT
T,
(2.28)
Tr
1
ekBT-I
2
})
(2.27)
where T is the transition matrix (T-matrix) of the radiating body, and p is referred
to as the infinitesimal translationmatrix. We will have much more to say about the
T-matrix formalism in Chapter 3, but for now it will suffice to know that the T-matrix
is a mathematical object that encodes the geometric and material properties of the
radiating body by telling us how it scatters incident light [19, 35]. The infinitesimal
translation matrix p is a purely off-diagonal matrix that is universal and entirely
independent of the properties of the radiating object 4 . Equations (2.27) and (2.28)
are known to be entirely equivalent to the formulas that we obtained above (2.26).
However, a direct correspondence between the two formulations has never been shown
in the literature.
2.4
The Boundary Element Method
The boundary element method (BEM) is a well-established technique in computational electromagnetism for efficiently solving integral equations [9]. In the context
of thermal power emission and thermal self-propulsion, the BEM provides us with
an efficient choice of basis functions for use in evaluating our trace formulas (2.26).
The matrices Q and R in these formulas are computed by integrating products of
basis functions over the volume of the radiating body. However, the BEM uses lo4
Explicit formulas for p are rather long and complicated, so we will not repeat them here. We
refer the interested reader to Appendix E of [16].
22
calized tangential-vector-valued basis functions that are restricted to the surface of
the object. It is common to use basis functions that are defined over small triangular
elements that approximate the surface of the object. Two examples of this type of
surface discretization can be seen in Figure 2-1.
The restriction of currents to the surface is made without loss of generality using
the well-known equivalence principle of electromagnetism, which states that the fields
within the object can be completely specified by a fictitious distribution of source
currents that are confined to the boundary [23]. In other words, the BEM involves
computing "effective" electric and magnetic surface currents that are mathematically
equivalent to the volume currents that would be realistically flowing through the
body. In Chapter 4, we will utilize a specialized piece of BEM software known as
SCUFF-NEQ to simulate thermal self-propulsion in a variety of different nanoparticle
configurations using precisely the trace formulas (2.26) that we have derived above.
-7
w
Figure 2-1: Typical triangular-element surface discretization scheme that is used in
conjunction with the boundary element method.
23
24
Chapter 3
Fundamental Limits On Thermal
Self-Propulsion
Before we dive into the realm of computer simulation, there is a great deal of numerical and phenomenological insight to be gained from the scattering matrix theory
that we described in Section 2.3. In this chapter, we will make use of the T-matrix
trace formulas (2.27) and (2.28) to obtain fundamental limits on the thermal power
emission and self-propulsion force that is attainable for particles of a given size. The
most important result of this chapter is a set of explicit predictions-reported in
Table 3.4 on page 43-for the maximum self-propulsion force that is attainable for
any homogeneous dielectric nanoparticle of arbitrarily complex shape. In addition,
we will explore details of the T-matrix formalism which, in this context, will allow
us to extract useful design principles that will guide our numerical search for efficient
self-propulsion geometries in Chapter 4.
We will restrict our attention to contributions to the power and self-propulsion
force coming from photons emitted by the body at a single frequency. This approach
has obvious limitations, but will not significantly reduce the generality of our results.
We will demonstrate in Chapter 4 that the power and self-propulsion force are dominated by contributions coming from a small set of frequencies, particularly those
associated with dielectric resonances. As long as we tailor our results to the resonant
frequencies that are unique to each object, we will reap the benefits of reduced mathematical complexity and computation time with only a minimal loss of predictive
power.
For the purposes of our discussion, we will rewrite equations (2.27) and (2.28) in
25
the following simplified form:
P=_
-
_
(3.1)
<D,(W)
d,
oj{
F
/0
fo
ekBTl}
2
hw
7r e kBT
4D,
_
(3.2)
.
C
In these expressions, we have introduced the dimensionless quantities <D, and <PF
representing the temperature-independent power flux spectral density
<D,~(W)
= Tr
1(T + TV) - TV
,(3.3)
and force flux spectral density
=D
Tr I5T7'}
wJm
(3.4)
where j = (w/c)- 1 p is a non-dimensionalized version of the infinitesimal translation
matrix and the dependence on angular frequency w enters implicitly through the Tmatrix. For the remainder of this chapter, we will work directly with (3.3) and (3.4),
recognizing their unique dependence on particle shape and composition. For ease of
discussion, we also will refer to (3.3) and (3.4) as the "power" and "force", respectively.
It is our hope that any discomfort caused by our loose terminology will be alleviated
by the notable absence of the phrase "flux spectral density" in every sentence for the
remainder of the chapter.
Turning our attention to the form of these equations, we see that the force and
the power both depend quadratically on the T-matrix. This quadratic dependence
follows from the fact that the microscopic current densities in the radiating body
enter into the formalism twice: once as the source of the incident field and again as
the currents induced by the incident field. Due to the presence of an overall minus
sign, the power (3.3) is a concave function of the elements of the T-matrix. This
has the interesting consequence of ensuring the existence of a T-matrix that globally
maximizes the emitted power. We will explore this feature analytically and discuss
its physical significance in Section 3.1. The expression for the self-propulsion force
(3.4) is complicated by the presence of 5, which mixes the T-matrix elements in a
non-trivial way. It is unclear from the form of equation (3.4) alone whether the force
magnitude has a finite upper bound. We will demonstrate in Section 3.2, however,
that a global maximum force emerges when we impose constraints on the T-matrix
26
to ensure that it is physically realizable.
3.1
Upper Bound on Thermal Power Emission
Computing the T-matrix that globally maximizes the emitted power turns out to be a
straightforward exercise in multivariable calculus, with the only complication arising
from the fact that the elements of the matrix are complex numbers. We proceed by
defining the elements of the T-matrix T3 in terms of their real and imaginary parts:
T
To = XQ + iya,
= T*3=a
Xa - iyo,
Xa,Yaa e R .
(3.5)
Substituting (3.5) back into our expression for the emitted power (3.3) and simplifying, we arrive at the following equivalent expressioni:
= - Tr{X + XXT + yyT}.
(X)
(3.6)
In anticipation of computing the gradient, we re-express (3.6) in the form:
'Ip(X, Y)
= -Xaa - xaoxao - yaoyao)
(3.7)
.
where we have made use of the convention that each repeated index is summed over
all possible values2
Before we proceed, we should clarify what is meant when we speak of computing
the "gradient" of (3.3). We have written the emitted power in the form (3.7) to
emphasize the fact that the trace formula (3.3) is merely a multivariable function that
maps complex "vectors" of length (dim T) 2 [or two real "vectors" of length (dim T)2 ]
into real numbers. In this language, the meaning of the gradient becomes clear: we
are computing the sensitivity of the emitted power (3.3) to changes in the individual
elements of the T-matrix, treating the real and imaginary parts of each element as
distinct real variables. Now that the meaning is clear, the actual computation of the
'Moving forward, we will omit the implicit dependence on w from our expressions for power and
force. For our purposes, it will suffice to think of these quantities as functions of the T-matrix
elements alone.
2
To make the meaning of (3.7) absolutely clear, we repeat it here with the implied summations
re-inserted:
4p (,
)
X.
27
(.2
+y.,
gradient of (3.7) is quite simple:
aip = - ,ca va
&Dy _-
2 6 lia6
v
-
2
15a
6
v
,Y aO=
- o
,3Xa ,3=
2Y
-
MX n ,
qv .
(3.8)
(3.9)
Setting the gradient equal to zero and solving for X, and Yu, we obtain:
-13
8x
VX21V
a
=
0
2
Y 7 V= 0,
which tells us that the T-matrix which maximizes the emitted power is given by:
216OV .
-v =
(3.10)
Plugging this expression for the T-matrix (3.10) back in to the emitted power formula
(3.3) and recognizing that aa = dim T, we arrive at the following analytic expression:
max
P
=
4
dim T.
(3.11)
This formula (3.11) highlights one detail of the T-matrix that we have, as of yet,
glossed over. As an abstract mathematical object, the T-matrix is infinite-dimensional,
so our formula (3.11) predicts that the maximum power should be infinite. We should
not despair, however, as there is a subtle but important physical assumption lying
beneath this conclusion.
To understand the significance of this subtlety, it will help if we fix in our minds
a particular basis for the T-matrix. For the purposes of this discussion, the spherical
basis will be the most intuitive, so moving forward we will imagine that all fields
under consideration have been projected onto a basis of vector spherical wave functions3 . In this case each entry of the T-matrix, written Tpim,p'e'm', represents the
complex amplitude of an outgoing vector spherical wave Eo"' with polarization P
and multipole order (f, m) that is produced when the object under consideration is
illuminated by a regular vector spherical wave ETm, with unit amplitude. In this
basis, we can formally write down an expression for the dimension of the T-matrix,
3
For a review of the definitions and properties of vector spherical wave functions, we refer the
reader to any one of the following wonderful references on the topic [1, 11, 20].
28
as it is simply the total number of vector spherical multipoles:
dim T = 2
(2e+ 1).
(3.12)
f=1
Thinking now in terms of spherical wave functions, it is easier to see the physical
assumption being made in our expression for the maximum-power T-matrix (3.10):
it describes an object that couples equally to spherical multipoles fields of arbitrarily high order. We now present a heuristic argument that demonstrates why this
idealization can never be realized in practice.
3.1.1
Accounting for Particle Size
In Figure 3-1, we depict a two-dimensional slice of an arbitrarily-shaped particle of
radius R that is in the presence of an incident field {Einc, Hinc} that is traveling with
wave vector k. As we can see, the particle is not assumed to be spherically symmetric,
so the "radius" of the particle is meant to indicate the radius of the smallest sphere
that can completely contain the object within its interior. Vector spherical harmonics
of order f exhibit oscillations in the azimuthal coordinate 0 with period T~ 2i/fe.
For the incident field to excite spherical modes of order f within the particle, the
wavelength A = 27r/k = 2wc/w must be less than the distance RAO over which the
material and geometric properties of the particle vary, which in turn must be smaller
than the distance scale TR set by the period of the spherical mode. This length scale
comparison allows us to make a rough estimate fmax of the largest spherical mode
that can be supported in our particle's interior:
A < RAO < 2rR
max
= [wR .
(3.13)
In equation (3.13), we have made use of the floor function Lx], which returns the
largest integer that is less than or equal to x. In other words, any spherical modes of
order f > emax that are excited in the particle will rapidly decay, meaning that their
contribution to the scattered field will be negligibly small. We can now understand
why our prediction for the maximum emitted power was unphysical: an object can
only emit an infinite amount of power if it is infinitely large. For compact objects of
radius R, however, there is a well-defined upper limit to the amount of power that
can be emitted by photons with angular frequency w.
To compute the maximum power for compact objects, we must convert our upper
29
R
IEinc
H
n
k
Figure 3-1: Arbitrarily-shaped particle of radius R along with a rough estimate of
the angle AO over which the shape of the particle varies.
multipole bound (3.13) into the language of T-matrices. By saying that the incident
field in Figure 3-1 does not excite spherical modes of order f > fmax within the particle,
we are claiming that spherical vector waves of order f >
emax
pass right through the
particle without even "seeing" it. Recalling the physical interpretation of Tim,p'e'm',
we can see that this conclusion is equivalent to the requirement that all elements of
the T-matrix with f and f' greater than
Tpjm,ptym, ~ 0
emax
for
be equal to zero:
(3.14)
f, ' > fmax .
With this condition (3.14), our expression for the maximum-power T-matrix (3.10)
is modified to read (in the spherical basis):
I
-4'Af6mim
Tpfm,p'e'm, =
e, ' <
emax
.
2
(3.15)
otherwise
0
Plugging this modified T-matrix (3.15) back into the trace formula for emitted power
(3.3), we obtain the same result that we had before (3.11), except that dim T is reinterpreted as the dimension of the non-zero portion of the T-matrix. Fortunately, this
is precisely what we get when we truncate the sum (3.12) at f = fmax.
Computing
the partial sum and plugging (3.13) in for emax, we arrive at the following simple
expression for the maximum power that can be emitted at angular frequency w by an
object of radius R:
R)
<) max(w
P2
=
[wR+ 1
_c _
30
-1
(3.16)
3.1.2
Implications for Particle Design
We have obtained an upper bound on the thermal power emission for compact objects
(3.16), but can we design a particle that will achieve this maximum? In other words,
can we design an object whose T-matrix is given by (3.15)? This type of problem,
known in the scattering literature as inverse design, is normally very difficult to
solve. Luckily, however, the form of the maximum-power T-matrix is simple enough
to permit a solution without much work. Here we will find it helpful to describe the
particle by its S-matrix 135], which is related to the T-matrix by
S = I+ 2T,
(3.17)
where I is the identity matrix. The S-matrix provides a subtly different viewpoint by
telling us how incoming vector spherical waves Eietm, transform into outgoing vector
spherical waves E"t when they scatter off of the particle. According to (3.17), the
maximum-power S-matrix is given by:
0
f, f' <
max
(3.18)
Spem,pijm' =
1 Pppfvfomn
otherwise
The interpretation of the S-matrix is more straightforward: the particle described
by (3.18) absorbs all incoming spherical waves of order f < max but is otherwise
transparent. In the limit as max -+ oo, this S-matrix describes an idealized perfect
absorber, which is commonly referred to as a blackbody. A well-established corollary
of Kirchoff's law of thermal radiation states that an object in thermal equilibrium
with its surroundings will emit and absorb electromagnetic radiation at the same
rate [36]. In our case, this means that our perfect absorber will also be a perfect
emitter 4 . In essence, we have rediscovered the fact, known to Kirchoff over 150 years
ago, that-at a given temperature and frequency-a blackbody will emit more radiant
energy than any other object held at the same temperature.
Another interesting facet of the blackbody emitter is that it radiates photons
isotropically, so it will not exhibit thermal self-propulsion. After all, for an object
to experience self-propulsion, there must be a directional imbalance in the linear
4 In
Chapter 4, we will study particles that are held at room temperature (~ 300 K) in vacuum. In
this case, the warm particles are most definitely not in thermal equilibrium with their surroundings.
This interpretation will still be valid, however, as we will only compute quantities in the instant of
time after the particle is placed in vacuum and will assume that it was in thermal equilibrium with
a heat source immediately before.
31
momentum that is leaving the body. It turns out that this attribute, which follows
directly from the definition of a blackbody, is already encoded in the diagonal form
of the maximum-power T-matrix (3.10). We will discuss the symmetry properties of
the T-matrix in greater detail in Section 3.2.2, but for now it will suffice to know
that an object whose T-matrix is diagonal in the spherical basis will exhibit spherical
symmetry. In Section 2.3, we mentioned briefly that the infinitesimal translation
matrix fp is purely off-diagonal. By simple calculation, one can show that the trace
of the product of a diagonal matrix with a purely off-diagonal matrix is always zero,
which is sufficient to conclude that the self-propulsion force (3.4) for a spherically
symmetric particle will vanish.
This conclusion has interesting consequences with regard to the design of optimal
self-propulsion geometries. For example, one might naively expect that the best selfpropulsion design is a particle that emits all of its photons in a single direction. It is
true that, if we fix the total power emitted by the body, then unidirectional photon
emission, being maximally asymmetric, will result in the largest self-propulsion force.
This is not, however, the design problem we face in reality. To understand why, let us
consider first what would happen if we tried to work toward this goal with an actual
particle that is composed of a fixed amount of dielectric (or conducting) material.
We might start by molding our particle into the shape of a sphere, since have we just
learned that this shape will give us the greatest power emission that is available to
us. We will then proceed to deform the particle in some asymmetric fashion, with
the intent of concentrating photon emission in a certain direction. What we will find,
however, is that the further our particle deviates from spherical symmetry, the lower
its total power output will be5 . Therefore, there is a significant trade-off that occurs
between our ability to increase the concentration of photons in a particular direction
and the total number of photons available to us. In other words, even if we were to
somehow design a particle that emitted all of its photons in a single direction, the
total power that would be emitted by the object would be quite low, suggesting that
there is some less geometrically extreme particle that would perform better.
3.2
Upper Bound on Thermal Self-Propulsion
We now wish to use the techniques we have developed and the insight we have gained
in the previous section to compute the maximum thermal self-propulsion force that
5
This must be the case, since any deformation that moves the particle toward spherical symmetry
will increase the emitted power.
32
can be achieved by a particle of radius R that is emitting photons with angular frequency w. As we anticipated in the chapter introduction, this problem is significantly
more difficult than the one we encountered in Section 3.1, for a number of reasons.
First of all, we must now contend with the infinitesimal translation matrix ], or,
p-matrix for short. The complexity of the p-matrix is reduced considerably when
we work in the spherical basis and restrict ourselves to self-propulsion forces along
the z-axis; however, it is still formidable and will make calculations involving the
force and its gradient more laborious. Secondly, we will discover that if we naively
attempt to compute the maximum force by setting the gradient of (3.4) equal to
zero and solving for the elements of the T-matrix, we will obtain a T-matrix that
does not correspond to a physically realizable particle. As it turns out, there are
additional symmetries and properties of the T-matrix that must be satisfied for our
upper bound to be meaningful. In our maximum-power calculation, these conditions
were automatically satisfied due to the diagonal, spherically symmetric form of the
solution. In the realm of force calculations, we no longer have the luxury of a simple
T-matrix, and must therefore devise a strategy for imposing the extra constraints.
Finally, there is no reason a priori to believe that a closed-form, analytic expression
for the maximum force T-matrix even exists. With all of this in mind, we will focus
our efforts on molding this problem into a form that can be efficiently be handled by
NLopt, an open source library for nonlinear optimization [14].
3.2.1
Computing the Gradient
Although it is not strictly required, many of the nonlinear optimization routines that
are available to us can be sped up dramatically by supplying an analytic formula
for the gradient. This eases the computational burden by reducing the total number
of times that the objective function (3.4) must be evaluated while the search for a
maximum is conducted. We will again define T and Tt according to (3.5), and we
will decompose the p-matrix in a similar way:
Paf =QPaa + iQa ,
Pcp , Q,, cE R .
(3.19)
Before moving on, it is important to reiterate that despite their complicated origin,
the matrices P and Q are strictly constant, and will simply be carried along in our
computation of the gradient. With this mind, we can plug (3.19) and (3.5) into the
33
original expression for the force (3.4) to obtain:
DF(X, Y) = T- (3.20)
Following the steps we took in Section 3.1, we write (3.20) in component form:
(DF(X, Y) = PaYO-yXa-j + QaOX,3yXay - Po~X)3yY
+ Qay,Y6YcY.-
(3.21)
At this point, computing the gradient of (3.21) is simply a matter of applying the
product rule and keeping close track of indices. The result is given by the following
two expressions:
3.2.2
-
(Q77 + Q Q?l)XQv + (P77
-YJ
-(P~l
-
-
-Pa(3.22)
Pca)Xav + (Q77a + Qa??)Ycw.
(3.23)
Symmetries of the T-Matrix
Few techniques in the scattering theorist's arsenal match the versatility, accuracy,
and elegance of the T-matrix. We have seen evidence of its symbolic power: equations (3.1) and (3.2) hold true for any object that one could imagine. However, the
simplicity and compactness of these formulas can be quite misleading. It is true that
all of the mathematical difficulties that arise when dealing with realistic materials
and geometries have been abstracted away, but they certainly have not disappeared
from the calculation. They have merely been reassigned to the poor soul in charge of
computing the elements of the T-matrix. For those of us who must, for one reason
or another; "look under the hood" and grapple with the elements themselves, the
T-matrix can seem rather intimidating. Working in the spherical basis and retaining
only elements corresponding to the lowest-order mode, the T-matrix is composed of
36 complex numbers that collectively describe the geometric and material features of
the object that can be resolved by e = 1 spherical waves. If we add in the fact that
the number of elements in the T-matrix grows like (fMax + 1)4, it may sound enticing
to abandon our efforts before they've even begun. Fortunately, however, the elements
of the T-matrix are not all independent.
We have collected in Table 3.1 a short list of symmetry properties of the T-matrix
in the spherical wave basis 119]. The first property in our list must be satisfied in
6
In the engineering literature, this property is said to follow from electromagnetic reciprocity [19].
Reciprocity, however, is a direct consequence of the time-reversal invariance of Maxwell's equations.
34
T-Matrix Property
Symmetry
Time-reversal 6
TPem,P'em'
=
( -1)mm'7p,,,p,_m
Azimuthal
Tpem,p',tm' = 6 m'mTfm,p'e'm
Reflection: x - y plane
I
TPm,Pem' = 0 unless (_ J+f =
Tpipme/m, = 0 unless (Spherical
Tpmpe',m'
1 )'+"
= -1
= 6 p'p 6e' 6 mmTpim,pem
Table 3.1: An (incomplete) collection of T-matrix symmetry properties that hold in
the spherical wave basis 1191.
any system that is symmetric under a reversal of the direction of time. The theory
of electrodynamics, as described by Maxwell's equations, is completely unchanged by
the transformation t -+ -t, so we must take care to ensure that this property holds
in any T-matrix that we claim maximizes self-propulsion force. Knowing this ahead
of time, we can pre-program this symmetry property into the T-matrix by restricting
the space of possible matrices that our nonlinear optimization software can explore.
We are also free to restrict this space further by requiring that our particle exhibit
spatial symmetries. Looking at the second property in our list, we see that one can
ensure azimuthal symmetry by setting all T-matrix elements with m -, m' equal to
zero, and we will find it advantageous to do so. As we discussed in Section 3.1.2,
thermal self-propulsion forces are a result of asymmetric photon emission. Imagining
a coordinate system whose origin lies at our particle's center of mass, it stands to
reason that self-propulsion forces directed along the z-axis arise when our particle
is asymmetric under reflections about the x - y plane. It is clear, then, that the
requirement of azimuthal symmetry neatly confines our particle to motion along the
z-axis. It is also clear from this line of reasoning that our space of possible T-matrices
.
should not include those that satisfy properties three and four in our list7
A quick glance at Table 3.2 will reveal that by requiring that our particle be
azimuthally symmetric and time-reversal invariant, we have dramatically reduced the
number of independent T-matrix elements. The addition of azimuthal symmetry
comes with the auxiliary benefit of setting a large number of independent elements
equal to zero. We can directly exploit the sparsity of our T-matrix in costly linear
7
Property four in Table 3.1, which is simply a combination of properties two and three, was
discussed in detail in Section 3.1.2.
35
Number of independent T-matrix elements
emax
No symmetry
Time-reversal
Time-reversal + Azimuthal
1
36
21
7
2
256
136
30
3
900
465
77
4
2304
1176
156
Table 3.2: List of the number of independent elements in T-matrices that satisfy
various symmetry properties.
algebra operations to achieve a substantial reduction in computation time. At this
stage, we appear to be in pretty good shape. We have worked hard to ensure that our
particle is not engaging in egregious violations of physical law, and have reduced the
computation time for our optimization problem from weeks to hours. However, if we
were to attempt the computation now, we would find that the optimizer would return
infinity. There is one more condition that the T-matrix must satisfy to ensure that
it is physically realizable, and it turns out that imposing this condition efficiently is
rather difficult.
3.2.3
Conservation of Energy
As it stands, there is nothing stopping our nonlinear optimization software from
sending all of the independent elements of the T-matrix to infinity. Let us use the
S-matrix to consider the physical implications of this process. As we have mentioned
before, the elements of the S-matrix represent the complex amplitudes of outgoing
spherical waves that result from incoming spherical waves with unit strength. These
complex amplitudes provide us with the expansion coefficients of the scattered field.
We can use the magnitude of the scattered field coefficients to compute the Poynting
vector, which, when integrated over any surface that completely encloses the object,
tells us the rate at which energy is being carried off to infinity. It is clear, then, that
if we multiply the T-matrix by a number that is greater than one, we will increase the
total amount of energy that is leaving the body per unit time. Since the amplitude
of incoming waves is fixed at unity, there is surely a point at which the rate of energy
leaving the body will exceed the rate of energy that is being supplied. An object
with this property is said to be an active or gain medium, and generally relies on an
external power source to achieve field amplification.
36
In contrast, the particles that we are interested in consist of passive or lossy
media. Each particle constitutes an isolated physical system whose only source of
energy comes from thermal current fluctuations. In this case, it is obvious that there
must be an upper bound to the rate at which energy (and momentum) can leave the
body. How, then, is this bound encoded in the properties of the S-matrix? It turns
out that the statement of conservation of energy is equivalent to the requirement that
the matrix I - SSI be positive semi-definite 119]:
I - SSt > 0.
(3.24)
Although it's not immediately obvious, this condition (3.24) ensures that the S-matrix
maps all non-zero vectors into vectors with a smaller Euclidean norm8 . This statement
is precisely what we had in mind when we discussed using the S-matrix to compute the
Poynting vector. We can now use equation (3.17) to derive an alternative expression
of energy conservation in terms of the T-matrix:
2 (7T + V) - TVf > 0 .(3.25)
We can see immediately that (3.25) places a condition on precisely the same matrix
whose trace gives us the emitted power (3.3).
At long last, our final task has emerged. In order to compute a physically meaningful upper bound on the thermal self-propulsion force that can be attained by a
compact object emitting photons with angular frequency w, we must perform a numerical optimization of the force (3.4), subject to the constraint that the T-matrix is
azimuthally symmetric, time-reversal invariant, and that a particular nonlinear function of the T-matrix (3.25) is positive semi-definite. As we discussed in Section 3.2.2,
symmetries of the T-matrix are easily enforced by restricting the space of possible
matrices that our optimization software can explore. Efficiently imposing the positive
semi-definiteness constraint (3.25), however, will bring us to the forefront of applied
mathematics.
Semidefinite programming (SDP) is a rich and exciting field of applied mathematics that has primarily developed in the past twenty years [7]. The objective of
linear SDP is to minimize a linear function F(M) -+ R subject to the constraint
that the matrix M > 0. It was demonstrated in the 1990's that linear SDP is ef8
To understand why this is true, consider the following alternative statement of (3.24):
(I - StS) x > 0 for all complex vectors x. By simple linear algebra manipulations, one can
show that this expression is equivalent to: x1 2 > ISx1 2
.
xt
37
ficiently solvable using a class of techniques known as interior point methods [7].
As a consequence, a number of different software packages have been developed for
solving problems in linear SDP, and a good review of the techniques they implement
can be found in [34]. In contrast, however, the problem posed by nonlinear SDP is
still an area of active research. A few algorithms have been proposed for efficiently
solving problems in nonlinear SDP 12, 4], but there is no consensus in the field, and
open-source implementations of these algorithms are hard to find.
It appears, then, that we are left to our own devices.
implementation of one of these nonlinear SDP algorithms is
have instead decided to attack the problem from a different
based on the observation that the positive semi-definiteness
equivalently written:
xt
(T + Tt) - TT
x >
0
While writing our own
certainly an option, we
angle. Our approach is
condition (3.25) can be
(3.26)
for all complex vectors x. By recasting the requirement of energy conservation in this
form, we have traded a single, difficult-to-implement constraint (3.25) for an infinite
number of constraints 9 that are trivial to enforce (3.26). We certainly cannot ensure
that (3.26) be satisfied for every possible vector x, but we can require that it hold for
a very large number of them. By repeatedly performing the nonlinear optimization
with increasing numbers of randomly generated input vectors x, we can monitor the
evolution of the eigenvalues of the matrix in square brackets (3.26) and declare that
the maximum force has been reached when they are all positive or zero.
3.2.4
Results and Discussion
We are now ready to solve the full constrained, nonlinear optimization problem that
will provide us with an upper bound on the maximum thermal self-propulsion force
that is achievable by finite-sized particles. For this purpose, we have chosen to employ a local, gradient-based algorithm known as SLSQP (Sequential Least-Squares
Quadratic Programming), which is included in the NLopt library [14, 15]. In order to
understand the convergence properties of our nonlinear optimization scheme, we have
monitored both the maximum force obtained by the optimizer and the eigenvalues of
9
Why is the number of constraints infinite and not, say, limited by the dimension of the T-matrix?
If we were to construct N independent basis vectors x1 ,.. . , xN where N = dim T and ensure that
(3.26) be satisfied for each one, we would only scratch the surface of the full positive-definiteness
requirement. To see why, imagine constructing a new vector y =
cx1 from coefficients ci E C
and plugging it into (3.26). All of the xi[. ..]xj terms with i = j would surely be greater than zero,
but cross-terms with i # j are under no such obligation.
38
the matrix in square brackets (3.26) as we increased the number of independent vectors satisfying (3.26) from 100 to 100,000. The results of this procedure for em. = 1
through ma. = 4 can be seen in Figure 3-2.
I.==1, dimT=6
fl.3E.
0.33
0.32
-...
.
::::
0.30
---...
.......
-....
- --
0.28
r
0
0.27
:
-0.2
A
1.4
0
..
0.15
.
........
.-.....
-----..
-.-...-..... -..
-..
-.
................
......
..
.....
... ~
....
- .---...
....--.
---......
.....
..
.-...
-
1.3
.
.-.-.- .
-.
..
-..-----..
-----..-.-----
........
M
0.10
.-........1.2
0 .2
0.05
-0.00
-0.05
0.26
102
f.
'-4
4.0
0A
s
10'
C
p4
1.5
.
0.29
0.25
.
rz4
1.0
0.20
0.31
1,..=2, dimT=16
0.30
=3,dimT=30
a)
-0
0
1or
i0
7.5-
0.2
7.01
0.
6.5
11 5
10
103
0.5
0.0
-..
..
--6---0 .5
-
.2
.... ....
6.0
-0 .4
-1
2. 5 -
10'
.
.- --.
----.
-----..-- -1
...
....
...
...
.. 1 0
..
......
10
10
.8
5.0
.0
4.5
1.2
40
1
Number of Independent Vectors Satisfying xt
-- -
-- -- ------- ..
...
. ..
-..
.-.-1.5
.
-0 .6
.
- -- --.
" . -..
-.
-.-..-
-2.0
-
5.5
-
.6
4 dimT=48
1.
0.4
3.
S-
1.0
103
le
105
(T+ V') - TV] x > 0
Figure 3-2: Plots displaying the convergence properties of our nonlinear optimization
scheme for 1 < fm. 5 4. The blue circles, corresponding to the left axis, give the
maximum force obtained by the optimizer. The colored dotted lines, corresponding
to the right axis, display the eigenvalues of the matrix in square brackets (3.26). The
T-matrix conservation of energy criterion dictates that all eigenvalues be positive,
corresponding to the requirement that all dotted curves in the figure lie above the
horizontal black line.
We can see a clear picture of the optimization process in the top-left plot of
Figure 3-2. As we increase the number of constraint vectors from 100 to 100,000,
the maximum force, represented in the figure by blue circles, steeply decreases before
leveling off and converging to its final value. On the same plot, we have used colored,
dotted lines to display the eigenvalues of the matrix in (3.26). We can see a clear
division in the eigenvalues: four of them are positive and the other two (overlapping)
are negative. As we discussed in Section 3.2.3, a negative eigenvalue would allow the
particle to output more power than it is receiving from thermal fluctuations, resulting
39
.......
...
......
..
in an artificially inflated maximum force. This is precisely what we see on the left
end of the plot: the large negative eigenvalues are driving the "maximum" to a higher
value than would be physically realizable in an isolated system. From this information
we conclude that 100, or even 1000, constraint vectors are not sufficient to ensure that
the energy in our system is being conserved. Once we make our way to the right end
of the plot, however, we see that the maximum force has converged and the negative
eigenvalues have been pushed to zero. In other words, imposing 100,000 additional
inequality constraints (3.26) with randomly generated complex constraint vectors is
enough to ensure that our particle is physically realizable and that the rightmost
point on the plot is a realistic upper bound.
The keen-eyed observer will note that negative eigenvalues in the top-left plot
never actually reach zero. This is to be expected, since our method of choice for
enforcing conservation of energy (3.26) is an approximation that only becomes exact
when we use an infinite number of constraint vectors. However, this should not be
cause for concern, as our approximation scheme errs on the side of overestimation.
We are confident that the value we obtained through our optimization procedure is
a fundamental upper bound to the achievable thermal self-propulsion force, but in
order to find the least upper bound, we would need to extend the x-axis on these plots
to infinity. The important question to ask, then, is how close we are getting to the
least upper bound. It is difficult to provide a satisfying, quantitative answer to this
question without knowing the exact value of the least upper bound. However, we have
found that a useful relative measure of closeness can be obtained by computing the
average magnitude of the negative eigenvalues (A-) and dividing it by the average
magnitude of the positive eigenvalues (A+). The value of this ratio for our best
estimate (rightmost point) in each plot is given in Table 3.3.
emax
1
2
3
4
(A~)/(A+)
0.05
0.24
0.72
1.03
Table 3.3: Ratio of the average magnitude of negative eigenvalues to positive eigenvalues associated with the rightmost point (100,000 constraint vectors) in each of the
four plots from Figure 3-2.
In our case, the least upper bound on the force corresponds to (A-)/(A+) = 0, so
the magnitude of this ratio for our best estimate provides us with a rough measure of
the tightness of our bound. From Table 3.3, we can see that 100,000 constraint vectors
provides us with a tight upper bound for emax = 1, but this number of constraints
40
becomes less and less sufficient as we go to higher values of em'. This comports with
our intuition, as higher values of m correspond to larger T-matrices which have
larger numbers of eigenvalues that must be individually sent to zero. We conclude
from this procedure that our brute force method of enforcing conservation of energy
works well for fm. = 1, but increasingly overestimates the least upper bound as
we go to higher values of fm.. In order to improve our estimates, we need either
to increase our computing power to allow for the imposition of larger numbers of
constraint vectors, or we need to take a more advanced approach by implementing
one of the nonlinear SDP algorithms we mentioned above [2, 41.
Finally, we present for visual comparison our best estimates of the maximum thermal self-propulsion force and power emission in Figure 3-3. We have also included
in the figure the power emission that is attained when the particle is achieving maximum self-propulsion. Two features of this plot immediately stand out. First of all,
we see that the maximum power is increasing with max at a much greater rate than
the maximum force. We can understand this feature in the following way: let us first
remind ourselves that thermal power emission comes from photons carrying energy
away from the body and thermal self-propulsion force comes from those very same
photons carrying momentum away from the body. Energy is a scalar quantity which
grows directly with the number of photons escaping to infinity. Momentum, however,
is a vector-valued quantity, which means that the net force on the object can be
raised or lowered depending on the direction that each emitted photon is traveling.
Therefore, the only way that the maximum force could increase at the same rate as
the maximum power would be if all of the photons were being emitted in a single
direction. As we discussed in Section 3.1.2, this is not physically possible for reasons
that lead us to the next interesting feature of this plot: the power emission associated
with maximal self-propulsion is significantly lower than the maximum power. We explained above that maximal power emission is necessarily isotropic, resulting in zero
self-propulsion force. To achieve thermal self-propulsion, one must asymmetrically
deform the particle, which automatically lowers the total emitted power.
3.3
Experimental Predictions
We conclude this chapter with a much-needed return to reality. Non-dimensionalized
versions of the temperature-independent power flux spectral density <D, and force flux
spectral density <DF are wonderful for simplifying theory, but how do we relate our
results from Figure 3-3 to measurable quantities? What predictions can we make
41
10 -
--
-
-
-a-
- -.
.
-
-- --
-
12
P
- - - - - - - - - - ... ..
. . . . ..
. . . ..
.....
..
.
6
,nax
F
. .. .. . . .. . .. .. .. .. ..
... ..
- -- - ---- --2 --- ---
imax
Figure 3-3: Best estimates obtained for the dimensionless analogues of maximum
thermal power emission (red), maximum thermal self-propulsion force (green), and
the thermal power emission corresponding to the maximum thermal self-propulsion
force (blue).
to guide future experimentalists in their search for efficient, self-propelling nanoparticles? We must now return our attention to the starting point of this discussion:
the trace formulas for the total power emission (3.1) and total self-propulsion force
(3.2). Unfortunately, computing experimentally measurable quantities is not as simple as plugging our expressions for <Dm' and <bm' into these formulas. We went to
great lengths in Section 3.2 to ensure that the T-matrix corresponding to maximal
self-propulsion was physically realizable, but our efforts only hold true for photon
emission at a single frequency. In order to realistically compute the maximum total
power and force, one would have to perform a functional optimization of the integrals
(3.1) and (3.2) for the frequency-dependent T-matrix, which introduces a host of new
complexities' 0 . Luckily, it turns out that we are still able to use our results to make
reasonably accurate predictions for the maximum total force and power, for reasons
we briefly mentioned at the beginning of the chapter.
We will present evidence in Chapter 4 that for dielectric particles, the force and
"0For example, in addition to satisfying all of the properties we discussed in Section 3.2, a
frequency-dependent T-matrix must also obey causality. For a discussion of causality in the context
dielectric media, consult [11].
42
power integrand in equations (3.1) and (3.2) is sharply peaked at individual frequencies corresponding to material resonances within the body. Moreover, we will see in
the case of silicon dioxide (SiO 2 ), a material that is commonly used for manufacturing nanoparticles, that a single material resonance can dominate over all others,
resulting in a contribution to the self-propulsion force that is at least one order of
magnitude larger than the contributions coming from all other frequencies combined.
This empirical fact makes clear the utility of our single-frequency approach. If we
know the frequency wo and bandwidth Awo of the largest material resonance in a
dielectric particle, then we can approximate the maximum total power emission (3.1)
and self-propulsion force (3.2) that is possible with a particle of radius R held at
temperature T by:
Hm a(R, T, wo, Awo) ~
-
O
7e kB
Fmax (R, T, wo, Awo) ~
D
(wo, R)
--
4a
-
7ekBT
Awo
(3.27)
Awo,
(3.28)
I
(wo, R)
-1
where the values of 4m'7 and #bm up to fmax = 4 are given in Table 3.4. Equation
(3.28) is the most important result of this chapter. It provides us with an absolute
metric by which we can compare the performance of different nanoparticle designs. In
the following chapter, we will use numerical tools to investigate the extent to which
specific nanoparticle designs achieve these theoretical limits.
emax
=
[
max(w,
R)
#Da(,
1
1.5
0.26
2
4.0
0.94
3
7.5
2.19
4
12.0
4.12
R)
Table 3.4: Collection of the data represented graphically in Figure 3-3.
43
44
Chapter 4
Designing Thermal Self-Propulsion
With SCUFF-EM
In Chapter 3, we used the T-matrix formalism to obtain several guidelines and limitations on the design of efficient thermal self-propulsion geometries. In this chapter, we put these principles to work by performing an exploratory study of thermal
self-propulsion in nanoparticles using the SCUFF-EM (Surface Current/Field Formulation of Electromagnetism) computational physics suite [24]. The SCUFF-EM
software package uses the boundary element method (BEM) to solve a wide range of
problems in deterministic and fluctuational electrodynamics. Originally developed to
simulate Casimir interactions between arbitrary compact objects [23, 27, 28], SCUFFEM has been extended in recent years to include software tools that are capable of
predicting near field radiative heat transfer [29] and nonequilibrium Casimir forces.
We will use the nonequilibrium tool SCUFF-NEQ to accurately predict the thermal
self-propulsion forces that would be felt by asymmetric nanoparticles held at room
temperature in vacuum.
The typical work flow for running nanoparticle simulations with SCUFF-NEQ
takes on the following structure:
1. Create . geo file that describes the geometry of the nanoparticle in terms of
points, lines, and surfaces.
2. Convert the .geo file into a .msh file using a finite-element surface mesh generator. We use the open-source software package Gmsh for this purpose 18].
3. Create a . scuf f geo file that specifies the material properties of the nanoparticle
and the nature of the embedding medium (vacuum in our case).
45
4. Run SCUFF-NEQ with the . scuf fgeo file and temperature of the nanoparticle
as inputs and request that it compute all quantities of interest, such as power,
force, and torque (PFT).
Furthermore, SCUFF-NEQ can compute contributions to PFT coming from individual frequencies, corresponding in the case of power and force to the quantities in curly
brackets in equations (3.1) and (3.2), or it can adaptively integrate over all frequencies
and return to the total PFT. Full frequency-integrated calculations of the total PFT
are computationally expensive, so we will search for efficient self-propulsion geometries at single frequencies and save total PFT results for a final comparison between
nanoparticle candidates at the end of the chapter.
4.1
Design Parameters
Our search for efficient self-propulsion geometries is, to our knowledge, the first of
its kind. As a consequence, our work will be largely exploratory. Due to our limited
computational resources, we are unable to conduct an exhaustive search through all
particle designs. To compensate for this limitation, we must restrict our exploration
to small regions of the design space that we predict will contain useful information. If
we focus our search on warm particles in vacuum, we are left with three experimentally tunable parameters: particle temperature, material composition, and geometry.
Each one of these parameters has a direct effect on the strength of the thermal selfpropulsion force, and the particular effect that they have is, in most cases, difficult
to predict ahead of time. For example, we will find that efficient self-propulsion is
typically traceable to a complex interplay between a particle's geometry and its material properties. In order to maximize the utility of our computational resources,
we must carefully consider the role that each of these parameters play in thermal
self-propulsion and then use this information to narrow the boundaries of our search.
4.1.1
Particle Temperature
Out of the three design parameters, the effect of particle temperature on thermal selfpropulsion is the easiest to understand. By changing the temperature of the particle,
we alter its thermal emission spectrum. This information is beautifully encoded in
46
the Rytov correlation function (2.15), where the quantity in square brackets
I +1
h hw
hw
E)(w, T) = -- coth
(4.1)
2
/hw
2kBT
____
=
e
_B
2
gives the average energy contained in an electromagnetic mode of frequency w as
a function of T, the temperature of the host medium [26]. This expression (4.1)
remains valid at all temperatures and all frequencies, smoothly interpolating between
the prediction of classical equipartition in the high-temperature limit:
lim
T-*oo
(w, T) = kBT,
(4.2)
and the presence of quantum-mechanical, zero-point energy modes in the low-temperature
limit:
lim E(w, T)=(4.3)
2
T-O
As we discussed in Chapter 3, the formulas for thermal power emission (3.1) and
thermal self-propulsion (3.2) can be neatly factored at each frequency into a term
4bP/F that encodes the contribution of particle geometry and material composition,
and a term
(4.4)
&
= E(w, T) ,
ekBT -
12
that weights each contribution as a function of particle temperature. We have written
the weighting factor in the form (4.4) to emphasize that it is merely the energy spectral
density (4.1) minus the vacuum energy, which contributes nothing to the energy and
momentum content of the far-field. In Figure 4-1, we have plotted the weighting factor
(4.4) at several different temperatures for visual comparison. The takeaway message
from Figure 4-1 is that the role of temperature in thermal self-propulsion is, in a way,
maximally uninteresting. If we are given an existing self-propulsion geometry, we can
tune the rate of acceleration by changing the particle's temperature, but temperature
alone has no effect on the underlying self-propulsion mechanism. For this reason, we
are free to conduct our search at room temperature with no loss of generality.
4.1.2
Material Composition
At the present moment, SCUFF-NEQ is only capable of handling particles with homogeneous material properties. We will touch briefly in Chapter 5 on the topic of
thermal self-propulsion in layered media, but for now we are limited to the study
47
i
.; N
20
-1
0
15
0
0
900 K
10
700 K
14-
Ce
500 K'
5
300 K
1c 0 K
b.0
0.5
w [3
1.0
14
x 10 rad
1.5
2.0
-s]
Figure 4-1: Energy spectrum of thermal photons at various temperatures.
of homogeneous particles. Assuming that our particle is non-magnetic, its optical
behavior will be largely determined by its complex permittivity E(w). The functional
form of the permittivity will differ for dielectric and conducting media, so we must
consider each of these cases separately.
Dielectrics
The important optical properties of dielectrics can be understood on the basis of
a simple classical model, whereby the bound electrons are treated as a collection
of damped harmonic oscillators that are forced into oscillation by incoming electromagnetic waves. In general, based on their orbital placement, the electrons in each
molecule will experience different natural frequencies and damping coefficients. If we
assume that there are N molecules per unit volume in the material and each molecule
contains fj electrons with natural frequency wj and damping constant -Yj, the model
predicts a relative permittivity of the form:
E(w)
60
_
Ne 2 E
COM
f.
Lj
48
-
W
(4.5)
-
zWyj
where e and m are the electron charge and mass, respectively [11]. Assuming that
the damping constants yj are small, the relative permittivity (4.5) is predominately
real across the majority of the spectrum. Near the resonant frequencies wj, however,
the real part of the denominator in the sum vanishes and (4.5) becomes large and
imaginary. It can be shown that the absorption coefficient a(w), whose magnitude
serves as a measure of the material's preference for absorbing light at frequency W, is
related to the imaginary part of the relative permittivity by:
b) ~ -
Re
,(4.6)
W
a (W) = -c Im
e(w)
EO
~
WIme(w)
c Re e(w)
[e(w)~
R eO _
where the approximation assumes that Im e(w) > Re e(w), making it valid near resonant frequencies. This model, therefore, predicts that dielectric media will experience
enhanced, or, resonant absorption near select frequencies whose precise values are
characteristic of the material's underlying molecular structure. Despite the model's
conceptual simplicity, this prediction gives an accurate representation of the optical
properties of dielectrics and, provided that appropriate quantum-mechanical definitions are supplied for fj, wj, and 7j, the form of the relative permittivity (4.5) is
astonishingly accurate [10, 11].
The presence of fixed-frequency resonant absorption peaks in dielectric media will
greatly simplify the search for efficient self-propulsion geometries. As we mentioned
in Section 3.1.2, Kirchoff's law of thermal radiation implies that an object that is
classified as a resonant absorber can be equally classified as a resonant emitter. This
means that the thermal emission spectrum for dielectric particles will be dominated
by a discrete set of frequencies. Since the thermal self-propulsion force is largely
dependent on the form of the emission spectrumi, this provides ripe justification
for conducting our search at individual frequencies. Secondly, we will find that the
emission spectrum of our particles is, in general, more complex and nuanced than
our simplified model (4.5) would seem to suggestion. Surprisingly, this deviation is
not explained by a failing of our model to capture the essential physics of dielectrics.
Instead, it points to the existence of an additional resonance phenomenon that arises
when the interacting medium is confined to a compact region of space with welldefined boundaries.
Geometric, or, morphology-dependent resonances (MDR) occur when precise
wavelength-matching conditions are met in the particle's interior, allowing it to act
'If this is not obvious from the form of equation (3.2) alone, remember that the emission spectrum
is encoded in the frequency-dependent elements of the T-matrix, which are present in the flux < (W)-
49
like an optical cavity for a small range of frequencies [13]. Unlike the material resonances of (4.5), MDRs allow for the possibility of directional sensitivity in absorption
and thermal emission. The existence of thermal self-propulsion, then, is entirely
dependent on the excitation of anisotropic, or, directionally sensitive MDRs. The
functional form and frequency of MDRs is notoriously difficult to predict from the
appearance of particle geometry alone. Moreover, any change that is made to the
particle's shape and size will adjust, for better or for worse, every internal length
scale that could potentially function as an optical cavity. With this complicated picture in mind, the utility of geometrically insensitive material resonances in dielectrics
becomes clear. Each material resonance provides us with a fixed length scale that we
can attempt to accommodate within the particle as we mold it into an asymmetric
shape. We believe that the key to efficient self-propulsion lies in a particle that can
support directionally sensitive MDRs at frequencies that are already enhanced by
material absorption.
Conductors
The optical properties of conductors differ from the properties of dielectrics in two
significant ways. The first way in which they differ is that the imaginary part of the
permittivity decreases monotonically with frequency, so there is no resonant material
absorption. The second is due to the influence of electron density oscillations known
as plasmons. Assuming that there are no bound electrons in the bulk medium, and
that some fraction fo of the N molecules per unit volume are "free" to move around,
one can derive the following expression for the relative permittivity:
2
1 Ne
fo
-2yI
e(w)
Eo
.
comw(-yo - iW)
where 'yo/fo is an effective damping constant that introduces loss into the system [11].
Defining the plasma frequency of the metal by w2 = Ne 2 fo/Eom, we can write the
real and imaginary parts of (4.7) as:
[-
(4.8)
Re
Eo .
=
+
+j73'_e(W2 +702)
E2
It can be shown from the form of the permittivity given in (4.8) that the optical
properties of conducting media naturally split into two regimes depending on whether
in incident field frequency w is above or below the plasma frequency wp [3]. At
frequencies w < wp, the conductor is highly reflective, with light penetrating only a
50
.
very short distance into the medium. At frequencies w > wp, the reflectivity becomes
negligibly small and the conductor becomes almost entirely transparent 2
As was the case with dielectrics, interesting optical behaviors arise when the conducting medium is restricted to a finite spatial region with well-defined boundaries.
In this situation, the same mechanism that keeps externally incident radiation out
of the particle also keeps internally generated thermal radiation in. In other words,
the interior of a conducting particle that is held at a finite temperature acts like
a cavity for thermal radiation. This cavity has the ability to support MDRs, and
the subsequent field enhancement will create a signature on the fluctuating surface
currents, leading to a well-defined peak in the thermal emission spectrum. Again, if
the particle geometry is anisotropic, these MDRs allow for the possibility of directed
thermal emission.
In addition to cavity modes, there are modes that are associated with the collective, oscillatory motion of electrons along the particle's surface. These modes, known
as surface plasmons, are non-propagating, localized resonances whose frequency and
spatial distribution are strongly dependent on the particle's geometric features. A
full description of surface plasmons in compact metallic particles would take us far
outside the scope of this thesis3 . For the purposes of this discussion we can simply regard them as an additional class of MDRs that contribute non-trivially to the thermal
emission spectrum.
By mentioning the complex role of surface plasmons, we intend to convey the idea
that conducting particles have the potential to support a much richer, directionally
sensitive thermal emission spectrum than their dielectric counterparts. However, this
potential for richness comes with a steep cost. Since all of the optical resonances in
conducting particles are morphology-dependent, it is not sufficient to conduct a search
for efficient self-propulsion geometries at a single frequency. An appropriate scan over
potential particle shapes would require the evaluation of the force contribution at all
frequencies, since the location of thermal emission peaks would change with every
iteration. Based on this discussion, we feel that restricting our search to dielectric
particles will allow us to get the most out of our limited computational resources.
2
3
For most metals, the transition to transparency occurs in the ultraviolet part of the spectrum [3].
For an excellent introduction to the subject of surface plasmons in the context of small spherical
particles, please consult [6].
51
4.1.3
Geometry
In Section 4.1.2, we discussed in detail the significant role that particle geometry
plays in the generation of thermal self-propulsion. The geometry of the particle
determines the spectrum of MDRs, whose spatial distribution, if anisotropic, allow
for the possibility of directed thermal emission. At this point, our tasks seems crystal
clear: we select for our particle a commonly-used dielectric material, identify the
most prominent material resonance, and search for strong thermal self-propulsion as
we vary the particle's shape and size. However, before we proceed, we must consider
the effect that particle volume will have on thermal emission.
In general, the scaling of thermal emission with particle size is quite complicated.
However, for particles that are small compared to the wavelength of emission, it can
be shown that emission scales roughly with the volume of the object [3]. This is
simple to understand on the basis of classical intuition: as we increase the volume of
the particle, we are increasing the total number of microscopic emitters in the object
that are capable of radiating thermal energy. If we naively monitor the magnitude of
the self-propulsion force as we vary the particle's shape and size, it may be difficult
to distinguish between an increase associated with MDR excitation and one that is
merely the result of an increase in volume.
This facet of geometric optimization highlights the importance of proper normalization in identifying novel self-propulsion behavior. Without a consistent normalization scheme, it is difficult to meaningfully compare the performance of two different
particles. If we could somehow manage to fix the volume of the particles as we distort
their shape, then we could gauge their relative performance on the basis of force magnitude alone. Since this is rather difficult to achieve in practice, we will instead choose
to normalize the self-propulsion force by the mass of the particle. This is an ideal
choice of normalization for two reasons. First of all, it will allow us to completely
factor out any increase in force that is solely the result of an increase in volume.
Secondly, by dividing the force by particle mass, we are predicting the acceleration
that each particle would experience if held at room temperature and placed in vacuum. Particle acceleration is a quantity that is, at least theoretically, easier for an
experimentalist to observer and measure.
52
4.2
Design Procedure
Based on our analysis in Section 4.1, we will focus our numerical search for selfpropelling particles in a relatively small region of the design space. To start, we will
choose a material from a list of commonly-used dielectrics in nanoparticle manufacturing 4 and identify its strongest material resonance. Next, we will run simulations
on a wide range of particle geometries, computing the thermal self-propulsion force
at room temperature for each one and dividing this number by the particle's mass.
Finally, after the simulations are completed, we will compare their predicted accelerations at room temperature and identify the best candidate out of the lot. In principle,
we would repeat this numerical search for a large number of different materials.
Our selection of particle geometries will be drawn from a few distinct classes,
with each class being based on a specific particle prototype. A prototype of degree
P is a particular geometric design that we have chosen to parameterize by P real
numbers {x 1, . . . , xp}. A class is defined as the set of particles that live within this
P-dimensional parameter space. For example, one might imagine designing a "cone
prototype" of degree P = 2 with independent parameters r and h for the radius of the
base and the height, respectively. The common choice of parameterization ensures
that each particle in the class will be geometrically distinct, but structurally similar
and topologically indistinguishable. For this reason, we can think of the parameter
space as the set of all possible geometric variations of the prototype. Ideally, there will
be a single point in the parameter space of each prototype that globally maximizes
thermal self-propulsion. It is our goal to locate that point by any means necessary.
If the degree of the prototype is small, we can get away with a large, coarse-grained
sweep of the parameter space. This will generally be accomplished in two steps:
1. Sweep each parameter xj from Xjtart to xi
AZyX
=
(xtoP -
X tart)/(N
-
in Nj equal increments of size
1).
2. Sweep a global scaling factor a from astart to astoP in M equal increments of
size Aa = (astoP - astart)/(M - 1). At each increment, replace {Xtart, Xst1P
with
saXjtart, axstP} and repeat step (1).
The values Xftart and xstop must be chosen carefully to ensure that the particle geometry remains well-defined throughout the sweep. The number T of independent
4
http: //www. us-nano. com/nanopowders
53
geometries that are sampled by this process is given by
P
T = MfJNj,
(4.9)
j=1
where P is the degree of the prototype. Let us compute this number for a realistic
scenario. Suppose that we have a prototype of degree P = 3 and we want to sample
N = 6 different values for each parameter at a total of M = 4 different length scales.
According to our formula (4.9), this will give us T = 864 distinct variations on the
prototype. On our local supercomputer cluster, each single-frequency simulation will
take roughly ten minutes, on average. The total simulation time for this coarsegrained sweep, then, is estimated at six days. In order to locate the best geometry in
this class, we will have to use the information gained from the coarse-grained sweep to
conduct one or more fine-grained sweeps that are centered around potential maxima
in the parameter space. As one can see, this process has the potential to be very
time consuming. For prototypes of degree P > 3, we will be forced to hold some of
the parameters fixed and conduct our sweep over the remainder. In this case, we will
attempt to explore sub-regions of the design space that we predict to have the highest
likelihood of containing the optimal self-propulsion geometry.
4.3
Initial Design Results
We have decided to conduct our exploratory search for efficient self-propulsion geometries amongst particles that are made out of silicon dioxide (SiO 2 ). We have selected
for its strong, infrared material resonances, which will dominate the thermal
emission spectrum at room temperature. We will use the oscillator model (4.5) from
Section 4.1.2 for the relative permittivity of Si0 2 :
Si0 2
3
ESio2(W) = E"
+
C2
W
j=1
2
.
,
(4.10)
I
where ex, = 2.04 and the parameters Cj, wj, and -yj are given in Table 4.1. From
the table, we can identify w 3 = 2.03 x 1014 rad/s as the strongest material resonance.
This angular frequency corresponds to a wavelength of A 3 = 9.29 x 10-6 m, leading
us to predict that we will begin to see strong thermal self-propulsion in particles with
54
.
geometric features that are on the order of 9 microns in length5
1
8.27 x 1013
8.54 x 1013
8.46 x 1012
2
3.40 x 1013
1.51 x 1014
8.33 x 1012
3
1.58 x 1014
2.03 x 1014
1.06 x 1013
Table 4.1: Parameters used in the oscillator model (4.10) for the permittivity of
SiO 2 [17]. All values are given in units of radians per second.
In the remaining sections of this chapter, we will apply our single-frequency design
procedure to three different particle prototypes. For each prototype, we describe the
region of the parameter space that we have chosen to explore and present the most
interesting findings from that region. We then select the best candidate from each
particle class and compute the total frequency-integrated thermal self-propulsion for
each one. From these results, we identify the best particle design overall and conclude
with a brief summary of the information that we have gained through the design
process.
4.3.1
Prototype 1: Death Star
We have chosen a simple, third-degree prototype (P = 3) as the starting point of
our search. In Figure 4-2, we present a schematic diagram of this particle along with
an example of the surface discretization that we use in our simulations with SCUFFNEQ. This geometry permits several unique parameterizations, but we will describe
it via the particle's radius R, the radius of the smaller spherical indent r, and their
center-to-center distance d. Although the inspiration for this geometry comes from a
galaxy far, far away 6 , the Death Star particle serves as a good starting point for our
numerical exploration for two reasons. The first reason is that as the indent depth
parameter d goes to R + r, the particle becomes a perfect sphere, which will radiate
isotropically. This serves as a vital consistency check for our numerical simulation,
since we can explicitly show that the force smoothly tends to zero as the particle
becomes symmetric.
Secondly, the parameter space of the particle is only three-
dimensional, which will allow us to capture the essential physics of this geometry
5
1n
6
http://en.wikipedia. org/wiki/DeathStar
addition, particles with geometric features that are on the order of 4.5 microns in length have
the potential to support half-wavelength MDRs.
55
-
r
rRR
It
Figure 4-2: On the left is a wire-frame representation of the Death Star prototype.
The design parameters are the radius of the particle R, the radius of the spherical
indent r, and the center-to-center distance d between the two. On the right is a
typical example of how we discretize or 'mesh' the object.
with a relatively small number of simulations.
We have performed a coarse sweep through the parameter space of the Death
Star, the details of which are collected in Table 4.2. Modeled after our discussion in
Section 4.2, the left half of Table 4.2 describes our search through possible variations
on the shape of the Death Star and the right half of the table describes the search
through possible variations on the size. With our formula (4.9), it is clear that we
have simulated thirty different shapes at ten different length scales, giving us a total
of T = 300 distinct variations on the Death Star prototype. At every length scale,
our results indicated that self-propulsion was greatest when the inner radius r had
a value that was 75% of the outer radius R. There were 100 geometries with this
property in our coarse-grained sweep, and we have condensed the interesting features
of that subset into two figures.
j
start
stop
1
R
1.0 Am
1.0 Pm
1
a start
astop
M
2
r
0.5 ym
1 .0
3
1.0
10.0
10
3
d
{0.52, 0.37, 0.25} pm
Pm
{1.15, 1.25, 1.5} pm
10
Table 4.2: Details of our search through the Death Star parameter space. The parameter d has three start and stop values, each corresponding to a different value of
the parameter r.
56
In Figure 4-3, we present the results of our sweep through the space of possible
shapes at the smallest length scale. We have included a few representative particles
in the figure, and they have been oriented so that their thermal self-propulsion will
propel them toward the top of the page. The acceleration is greatest in the object with
the largest spherical cavity and, as we anticipated, it decays to zero as the depth of the
cavity decreases and the particle approaches spherical symmetry. On first viewing, it
may seem as if the increase in acceleration from right to left is merely the result of a
steady decrease in volume. While the decrease in volume may play an important role,
we must remind ourselves that a decrease in volume also implies a decrease in total
radiative capacity. It is clear that the difference in performance between the leftmost
and rightmost particles in the figure is a complex balance between total mass and
thermal emission asymmetry.
In
-------
--
[I...nput
............
Material:
817
Si0
T=300.0 K
-
-
-
16 ----
-
..
----------.. ------------------
., 1 8 ---..- - - -
2
Parameter Space:
R = 1.0 JAM
r = 0.75 pm
d . -1.25Am
-i
- ---
14
-
-
x
12
10
C9
-
4
3
0.4
0.5
0.6
0.8
0.7
d
0.9
1.0
1.1
1.2
1.3
[..m.
Figure 4-3: Parameter sweep over possible Death Star shapes at the smallest length
scale.
In Figure 4-4, we have taken the best (leftmost) particle from Figure 4-3 and
computed its thermal acceleration over many different length scales. This task was
accomplished in two steps: first we performed a coarse sweep over the dimensionless
scaling factor in the range of 1.0 < a < 10.0, then we performed a finer sweep in the
range 1.0 < a < 3.0 to better resolve the peak. We believe that the sharp peak in
57
70
0
SptParameter
,
60 --
= a(1.0p)
Space:
Mate..aL:...
2r- =a(0.75 pm)
d=a(0.37pm)
T 300.0 K
A = 9.29 ism
a =1.0 - 10.0
--
-
50
R
Inu-,lus
-
-.
1-44
x
Best Particle
R = 1.6 pm
0
W
20
.. . . .......
....
.5pm
3
1Am
d= 0.592
....
--
- -- -- -
J
Eficie cy=85%
-
30 .... .....
0
1
2
3
4
6
5
7
8
9
10
a
Figure 4-4:
Parameter sweep over possible Death Star length scales for the best
particle in Figure 4-3.
this sweep over length scales is a clear indication of a directionally-sensitive MDR.
All of the data points in Figure 4-4 are taken from the same particle shape, but at
one particular size the acceleration is much greater than all others. We have included
the details of the best Death Star particle in a yellow box in the center of the figure.
While the precise details of the MDR that is being excited are currently unknown, we
wish to point out that the equatorial circumference of this particle is equal to 10.1 Am,
a number which is quite close to the wavelength of emission A = 9.29 1um. This is
right in line with our prediction that we would see strong thermal self-propulsion
occuring in particles with geometric features on the order of 9 microns. As a final
measure of the best Death Star's performance, we have determined from equation
(3.13) that this particle will couple to spherical waves with emax = 1 and have used
7
this information to compute its single-frequency efficiency , placing it at 8.5% of the
maximum possible thermal self-propulsion force for a particle of its size.
To compute the particle's efficiency at a single frequency, we compute the quantity in curly
brackets (3.28) and compare it to the thermal self-propulsion force given by SCUFF-NEQ.
7
58
Figure 4-5: Hybrid schematic diagram of the Sorcerer's Hat prototype. On the left we
detail the basic geometry and choice of parameterization and on the right we present
a typical example of its surface discretization.
4.3.2
Prototype 2: Sorcerer's Hat
The second class of particles that we have chosen to investigate are based off of the
fourth-degree prototype (P = 4) that is presented in Figure 4-5. This prototype,
which resembles the cartoonish headgear worn by a sorcerer's apprentice8 , is formed
by connecting two concentric parabolas with a semi-circular cap and revolving the
resulting shape around their shared axis. The object is parameterized by the height
Hin and radius Rin of the inner paraboloid and the height Hout and radius Rut
of the outer paraboloid, with the radius of the cap given by (Rut - Rin)/2. The
Sorcerer's Hat shares structural similarities with the Death Star from Section 4.3.1,
but is distinct in two important ways. First of all, every variation of the Sorcerer's Hat
geometry will experience a non-zero thermal self-propulsion force. This is in contrast
to the Death Star, which became spherically symmetric for a particular choice of its
parameters. Secondly, the Sorcerer's Hat does not contain any sharp edges or corners.
Sharp edges and corners pose a computational challenge, as they typically require a
very fine surface discretization to properly resolve. For this reason, we have learned
that we can get the most out of our computational resources if the particles that we
choose to simulate are geometrically smooth.
8
http://disney.wikia. com/wiki/TheSorcerer*27s-Apprentice
59
80
Parameter Space:
R.'t = 3.0 pm
MR,. = 2.1 ism
lo
.8,02:
H..t = 3.0 - 10.0pm
Hi. = 1.0 -9.0 pm
3=
......
T = 300.0 K
A= 9.29pm
4
0
R
.
60 ---
_.
_ _....
_.
_.._.
... ..
..........
_.
... ..
.
70
......
.......
=
3.0....=
H.=2.OpmJ
$0
0.1
0.2
0.3
0.5
0.4
0.6
0.7
0.9
0.8
1.0
Figure 4-6: Parameter sweep over possible shapes for the Sorcerer's Hat prototype.
Since the Sorcerer's Hat prototype has a larger number of independent parameters
(P ;> 3), we have decided to fix the inner and outer radii and conduct a sweep over
possible values of the inner and outer height. The result of this simulation can be
seen in Figure 4-6. Once again, we have oriented the particles in the figure so that
their thermal self-propulsion will drive them toward the top of the page. We have
plotted the thermal acceleration felt by each object at room temperature as a function
.
of the ratio of their inner height to their outer height. As a general rule of thumb,
whenever there are multiple data points with the same height ratio, the scale of the
9
object increases as the data points go to lower values of the acceleration
The results for the Sorcerer's Hat are somewhat less visually striking than the
results for the Death Star. While there does not appear to be an obvious resonant
effect occuring in this region of the parameter space, there is still a clear winner in
terms of thermal self-propulsion. We have provided the details of the best particle in
this class in a yellow box in the bottom right corner of Figure 4-6. The best Sorcerer's
Hat particle achieves an efficiency of 8.4% relative to the maximum self-propulsion
that is attainable for a particle of its size. This efficiency is almost identical to the
For example, there are four particles in our sweep with Hin/Hut = 0.5. Their inner and outer
heights are given by: (Hi,H0 t ) = (2jm, 4 pm), (31m, 6pm), (4 m, 8 pm) and (5 m, 1pm).
9
60
8.5% achieved by the best Death Star. However, the Sorcerer's Hat is physically larger
and can couple to spherical waves with fmax = 2. The fact that both particles achieve
an acceleration of roughly 70 pm/s 2 is a good consistency check on the accuracy of
our numerical upper bounds.
4.3.3
Prototype 3: Martini Glass
The final class of particles that we have studied is shown in Figure 4-7. This eighthdegree prototype (P = 8), which looks like it should hold beverages that are "shaken,
not stirred"' 0 , is the most exotic particle shape that we have attempted to explore.
With eight free parameters, it would be enormously time consuming to perform even
the coarsest of searches through the design space of this particle. We will demonstrate,
however, that useful information about this particle's thermal self-propulsion behavior
can be extracted from very small regions of the design space.
L2
Figure 4-7: Hybrid schematic diagram of the Martini Glass prototype. On the left we
detail the basic geometry and choice of parameterization and on the right we present
a typical example of its surface discretization.
lohttp://en.wikipedia.org/wiki/Martini_/28cocktail%29
61
The parameter of interest to us is the opening angle 0 of the Martini Glass.
Holding all of the other parameters at fixed values, we have computed the thermal
acceleration at room temperature as we varied the opening angle from 15' to 72'.
The result of these simulations can be seen in Figure 4-8. In this instance, the
example particles that we have included in the figure are oriented so that thermal
self-propulsion will carry them to the bottom of the page. The dependence of thermal
self-propulsion on opening angle is quite clear. The Martini Glass particle achieves
the largest thermal acceleration right in the middle of the figure, which corresponds to
an opening angle of 0 = 47.60. It is. worth pointing out that the peak in acceleration
for the Martini Glass is very gradual, and does not resemble the sharp resonance that
we saw in Figure 4-4 with the Death Star. It may be the case that these particles
are exhibiting two different types of MDRs; however, this comment will remain in
the realm of speculation until we are able to realistically visualize their internal mode
structure. Despite its exotic shape, the best particle that we have found in this class
only achieves an efficiency of 5.0% with fm = 1. We believe that this particle has
the potential to achieve efficient thermal self-propulsion, but the full realization of
this fact will require a more thorough exploration of its parameter space.
65
Parameter Space:
.....--..----....
-..-
L1 = 1.5
-
Ri = 0.3 ixm
R2 = 0.3pm
*
R3 = 0.2 pm
Efficiency =5.0%
55-- R 4 =0.62pm
D
-
-R=0.1-1.pm
= 15* -72*
= 47.6*
e
45
T - SWO. K
99pm
......
--...-.-.
.------...
..
-..
....
-...
.
..
-----------.
.
--..
.. -..-..
-
30 ........
35
pm
.
---.-.-.--.--.--
-----......-...-...-
-........
.........-.........
---
----
----------
...
'777
-
30
10
20
-
-
- -
30
-
pm
.......
---..
.--------.-. L
2=2.0
Best Particle:
Li = 1.5 pm
60 -- L2 = 2.0 m
RI = 0.3 Am
R2 = 0.3 pm
R3 = 0.2 pm
40
50
60
70
80
0
Figure 4-8: Parameter sweep over possible angles for the Martini Glass prototype.
62
4.3.4
Final Comparison: Total Self-Propulsion
As a final means of comparison, we have select the best candidate from each of the
three particle classes and have used SCUFF-NEQ to compute the full, frequencyintegrated self-propulsion force at room temperature. The result of these computations have been included in Table 4.3. In each of the three cases, the total thermal
acceleration is considerably lower than the acceleration per unit frequency that we
achieved at the material resonance. This is to be expected, as the bandwidth of the
resonance is extremely narrow. To determine the efficiency, we compared the total
force computed by SCUFF-NEQ with our estimate of the maximum total force (3.28),
using the Si0 2 resonant frequency w 3 and its bandwidth 7 3 from Table 4.1.
We find it encouraging that the total thermal acceleration calculation preserves
the ordering of particle design performance that we found in the single frequency
case. However, it is peculiar that the Sorcerer's Hat design achieves more than double the acceleration of the Death Star design, when they both exhibited the same
single-frequency acceleration and efficiency. We believe that this implies that there is
significant resonant behavior occuring at frequencies other than the primary material
resonance in the Sorcerer's Hat particle. Until an accurate method is developed for
computing the upper bound on thermal self-propulsion at all relevant frequencies, we
feel that it is best to exercise caution when comparing the total efficiency between different particle classes. In this case, the thermal acceleration gives a clear indication of
the relative performance between particle classes and it does so without qualification.
Prototype
Force [N]
Mass [kg]
Acceleration [pm/s2I
Efficiency [%]
Death Star
3.64 x 10-20
2.55 x 1014
1.43
5.1
Sorcerer's Hat
3.23 x 10-19
8.75 x 10-14
3.69
12.4
Martini Glass
1.64 x 10-20
1.67 x 10-14
0.98
2.3
Table 4.3: Final statistics on the total thermal self-propulsion force and thermal
acceleration at room temperature for the best candidate particle from each class.
63
64
Chapter 5
Conclusions and Future Work
This thesis has presented a technique for designing thermal self-propulsion in nanoparticles using several recent advancements in the field of nonequilibrium Casimir physics.
Our approach to nanoparticle design combines elements of the scattering matrix theory of nonequilibrium electromagnetic fluctuations [16, 18, 211 and cutting-edge numerical methods based on the fluctuating surface-current formalism [28, 29]. In Chapter 3, we developed a method for computing fundamental limits on thermal power
emission and thermal self-propulsion. The limits that we obtained are valid for photon emission at a single frequency and depend critically on the size of the particle.
However, we have argued that these results can be extended to the case of total
power emission and self-propulsion force if appropriate assumptions are made about
the particle's thermal emission spectrum.
In Chapter 4, we performed an exploratory design study of thermal self-propulsion
in nanoparticles using the SCUFF-EM software package [24]. We provided a detailed
description of the role played by particle temperature, material composition, and
geometry in generating thermal self-propulsion forces and used this information to
develop a strategy for restricting our focus to small regions of the design space that
we predicted to contain novel self-propulsion behavior. We explained the details of
our procedure for simulating a large number of variations on a particular particle
geometry and presented the results of this procedure as applied to three unique particle prototypes. We made use of two different methods for comparing dissimilar
particle designs. First, we normalized the thermal self-propulsion force computed by
SCUFF-EM by the mass of each particle. This normalization scheme, which gave
us the thermal acceleration that would be felt by each particle if held at room temperature in vacuum, allowed us to eliminate any bias in the thermal self-propulsion
force caused by a difference in particle volume. Secondly, we used the fundamental
65
limits that we obtained in Chapter 3 to estimate the self-propulsion efficiency that
was achieved by each particle design.
To our knowledge, our numerical exploration of thermal self-propulsion phenomena in nanoparticles is the first of its kind. For this reason, our work is intended to
serve as an entry point to the field for future researchers. There are several different
approaches that could be taken to improve on the quality and efficiency of our initial
designs. Most importantly, our search results could be extended to include particles
with heterogeneous material composition. At the time of this writing, the nonequilibrium tool SCUFF-NEQ is only capable of simulating homogeneous particles. However,
the developer has expressed an interest in adding support for piecewise-continuous
media, making it reasonable to assume that this feature will be available in the near
future. We have reason to believe that layered nanoparticles will play a significant
role in achieving truly efficient thermal self-propulsion. This opinion is informed both
by promising new advances in the field of directional nanoantennas [33] and by the
high degree of control that can be exerted over the flow of light in layered photonic
crystals [12].
There is also strong motivation to extend our search to include metallic nanoparticles. As we discussed in Section 4.1.2, compact conducting objects have the ability to
support a wide variety of directionally sensitive, morphology-dependent optical resonances, including surface plasmons, bulk plasmons, and ordinary cavity modes. Each
of these contributors to thermal self-propulsion pose a unique challenge that would
require a modification of our current design procedure, but we feel that there is potential for a significant payoff in performance if they can be efficiently exploited. Lastly,
there is ample opportunity to improve upon the upper bounds that we obtained for
the thermal self-propulsion force in Chapter 3. Our brute-force method for imposing
energy conservation in the T-matrix worked reasonably well for fmax = 1, but began
to falter for values of emax > 2. Efficient computation of the least upper bound on
thermal self-propulsion for higher values of em, will require a different approach altogether, one that takes advantage of existing algorithms in nonlinear semi-definite
programming [2, 4]. Furthermore, the scope of applicability of our limits can be
broadened by generalizing them to include contributions to the self-propulsion force
coming from photons emitted at all relevant frequencies. While the extension of our
numerical technique to multiple frequencies would add considerable complexity, it is
worth pursuing, as it would lead to a much stronger result regarding the maximum
total thermal power emission and self-propulsion force.
66
Appendix A
Dyadic Green's Functions
In Chapter 2, we make use of the electric and magnetic dyadic Green's functions
(DGFs) from classical electromagnetic theory. For a complete description of these
powerful functions and their many applications, we refer the reader to [32]. Here we
provide a brief overview that is intended to serve as a companion to the main text.
For reference, we repeat here equations (2.1) and (2.2) from Chapter 2 for the
electric and magnetic fields that are produced by an electric current density J(x):
E (w, x)
=
G (w; x, x')Jj(w, x') dx',
(A.1)
Hi(w, x)
=
G (w; x, x')Jj (w, x') dx',
(A.2)
In an infinite homogeneous medium that is characterized by a spatially-constant,
linear permittivity E(w) and permeability p(w), both of the DGFs can be written in
terms of a single tensor g:
GE (w; x, x')
=
GM (w; x, x') = V x !(k; x, x') ,
ikZg(k; x, x'),
(A.3)
where we have substituted in for the wavenumber k = wfijii and the wave impedance
lp/c in the medium. The tensor g, sometimes referred to as the "free space
Z =
dyadic Green's function," is defined by the following equation:
V x V x g(k; x, x') - k 2 g(k; x, x') = 16(x - x') .
(A.4)
The solution to (A.4) can be expressed concisely as
g(k; x, x')
=
1+
V0V
67
Go(k; x, x') ,
(A.5)
where the components of the tensor product are given by (V 0 V), 3 = &3&
the familiar scalar Green's function
and Go is
ik~x-x'I
Go(k; x, x') =
iklx-
2
47rlx - x'l
,
(A.6)
which satisfies the Helmholtz equation with a delta function source
(V 2 + k 2 )Go(k; x, x') = -6(x - x').
(A.7)
For completeness, we present the following explicit formulas for the components of
GE and GM, omitting the functional dependence on w to align with the conventions
of the main text' :
1 a
G'(x, x') = ikZ (6i + -k2i5
G' (x, x') = esayja
( 47rlx
eikx-x'l
- x |
(A.8)
(A.9)
( 47rlx
eikx-x'l
- x'I
'Equation (A.8) follows directly from the definition of Q (A.5). Equation (A.9), however, requires
a bit of manipulation. The curl of a two-index tensor is defined by:
(V x Q)\
= ftaago.
From this definition, we can write the components of GM as:
G'
(x, x')
=
e
0,c
(6f3J +
1a65
Go(x, x').
Looking at the second term in this expression, we see that the indices a and 3 are anti-symmetric
in the Levi-Civita symbol cjap and symmetric in the product of partial derivatives 0,6. This
combination uniquely vanishes, giving us the result seen in (A.9).
68
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