![xvi Preface
velocity, dispersion, and diffraction. Eikonal methods are then introduced in order
to study waves in nonuniform plasma. The treatment here is very traditional, using
asymptotic series and brute force calculation.
In Chapter 2, we introduce two important tools that will be used throughout
the later chapters: variational methods for wave equations and the Weyl symbol
calculus. The use of a variational principle allows a unified treatment of later
topics, and provides an elegant derivation of the wave-action conservation law
using Noether’s theorem. (Further discussion of variational methods and Noether’s
theorem is provided in Appendix B.) The Weyl theory of operator symbols underlies
everything done later in the book.1
This theory was first developed in the context
of quantum mechanics, but the methods are completely general and they provide a
systematic means for deriving wave equations that are local in both x and k. These
methods are needed for dealing with caustics, tunneling, and mode conversion, all
of which involve a breakdown of the eikonal approximation that is local in ray
phase space.
In Chapter 3, we begin our discussion of eikonal theory for vector wave equations
in earnest. In this chapter, we transition by stages from the more familiar x-space
discussion of eikonal theory to a covariant phase space treatment. In many settings,
the covariant formulation is not needed, and we strive to keep things as concrete as
possible. Therefore, in this book we tend to use a preferred laboratory frame and
work with physical fields (e.g. E(x, t) and B(x, t)), rather than allowing arbitrary
frames and using four-vector notation and the vector potential Aμ(x). However,
covariant formulations are needed in some astrophysical and space plasma appli-
cations, so we include them even if they are not a major focus of the book. The
conservation of energy, momentum, and wave action is covered in this chapter.
Chapter 4 discusses visualization and wave-field construction. The modern the-
ory of ray tracing is a geometrical theory. Geometrical theories appeal to the visual
intuition. A visual picture can help guide us through a calculational thicket by
providing a map. Mixing metaphors, the goal of this chapter is to help bring visual
intuition onto the battlefield as a valued ally along with the more analytical methods
of calculation. This chapter also provides examples of the construction of wave
fields from ray-based data. This section connects us directly back to the original
motivation of eikonal theory: to find solutions of wave equations.
The next three chapters concern situations where the eikonal approximation
breaks down in various ways.2
1 The theory of operator symbols is based upon the representation theory for the Heisenberg–Weyl group. A
short, self-contained, introduction to this topic is provided in Appendix D.
2 By an eikonal solution, we mean a wave field that has well-defined phase, amplitude, and polarization:
ψ(x, t) = A(x, t) exp[iθ(x, t)]ê(x, t).](https://image.slidesharecdn.com/2339176-250523215407-0dd59a91/85/Ray-Tracing-And-Beyond-Phase-Space-Methods-In-Plasma-Wave-Theory-Tracy-Er-22-320.jpg)
![1
Introduction
The science of optics, like every other physical science, has two different
directions of progress, which have been called the ascending and the
descending scale, the inductive and the deductive method, the way of
analysis and of synthesis. In every physical science, we must ascend from
facts to laws, by the way of induction and analysis; and must descend
from laws to consequences, by the deductive and synthetic way. We must
gather and groupe appearances, until the scientific imagination discerns
their hidden law, and unity arises from variety: and then from unity must
re-deduce variety, and force the discovered law to utter its revelations of
the future.
William Rowan Hamilton (1805–1865)1
It is a fact of immediate importance to our everyday experience that light nearly
always travels in straight lines from the source to our eyes, perhaps scattering off
some object along the way.2
Without the ability to assume this as a fact about the
world around us, our extraordinary talent for instinctively comprehending spatial
relationships in everyday life would be severely compromised. Consider how much
computer power must be expended to disentangle the multiple images of distant
galaxies3
to map the dark matter distribution in the visible universe [MRE+
07].
Imagine what life would be like if we had to do similar mental computations just
to navigate around the furniture in our living room.4
1 From “On a general method of expressing the paths of light, and of the planets, by the coefficients of a
characteristic function,” by WR Hamilton (1833) [Ham33].
2 That rays travel in straight lines was fully appreciated by the ancient world. See, for example, Euclid’s Optics
(ca. 300 BCE) [Bur45] which begins with the statement: “Let it be assumed that lines drawn directly from the
eye pass through a space of great extent.” (emphasis added) From this simple insight, Euclid lays the groundwork
for geometrical optics and projective geometry, which form the basis for modern fields like computer animation.
See Darrigol [Dar12] for a recent survey of the history of optics from antiquity to the nineteenth century.
3 Using Einstein’s theory of gravitational lensing.
4 This leads to many other interesting questions, such as: What types of spatial imagery do animals possess
that evolved in the dark, or in murky environments such as muddy water? Some, like bats and dolphins, use
echolocation, which in many cases can provide a spatial image since ultrasound waves locally travel in straight
lines, too. But what type of spatial imagery do animals possess that live largely by sense of smell, such as ants?
1](https://image.slidesharecdn.com/2339176-250523215407-0dd59a91/85/Ray-Tracing-And-Beyond-Phase-Space-Methods-In-Plasma-Wave-Theory-Tracy-Er-29-320.jpg)
![2 Introduction
How do we build upon this insight that light nearly always travels in straight
lines in order to develop a theory with predictive power? More importantly, how
can we develop a theory that can encompass those cases where light does not
travel in straight lines? And, looking further, can we develop a theory that can be
extended to other types of waves in other settings? In the following sections, we
selectively discuss some foundational concepts that we will find useful for the rest
of the book. This is not a historical survey of the development of ray tracing. The
history of optics and wave theory is vast, and the story too complex, for us to do
more than touch upon the most relevant highlights to begin stocking our toolkit.
Some suggestions for further reading are given along the way for readers who want
more detail.
1.1 Fermat’s principle of stationary time
1.1.1 General comments
A major unifying theme of this book concerns the power of variational principles.
These have a venerable history. In the first century CE, the mathematician and
inventor Heron of Alexandria posed, and solved, the following problem in planar
geometry: Given a line and two points not on the line, what is the shortest path
between the two points that touches the line? If the points lie on opposite sides
of the line, then the path connecting them crosses the line and the shortest path
is simply a straight line. However, if they lie on the same side of the line, then
Heron proved (without the use of calculus!) that the shortest path obeys the law of
reflection. This basic principle of optics was therefore known in antiquity, and it
was known to satisfy a minimization principle.
Given that light travels in straight lines, and that rays obey the law of reflection,
why invoke a least-time principle? Because the path of least time and the path
of shortest length are only equivalent if the speed of light is constant along the
path. Feynman famously pointed out that lifeguards must solve for the least-time
path every time they rescue a swimmer: they must determine how far to run along
the beach before they dive into the water, where their speed of propagation drops
dramatically. If they are good at their job, their path obeys Snell’s Law, as we’ll
discuss.5
Let’s start with the original form of Fermat’s principle of least time, and improve
it as we pursue the implications. The least-time principle asserts that of all possible
paths light might take from the source to the point of observation, it will take the
path that requires the least time. We will adopt the convention that the actual paths
light follows are called rays, to distinguish them from all the possible paths we
5 More recently, ants have been found to follow the least-time path as well. See Oettler et al. [OSZ+13] .](https://image.slidesharecdn.com/2339176-250523215407-0dd59a91/85/Ray-Tracing-And-Beyond-Phase-Space-Methods-In-Plasma-Wave-Theory-Tracy-Er-30-320.jpg)
![1.1 Fermat’s principle of stationary time 3
Figure 1.1 The point source S and the point of observation O can be connected
by an infinite number of paths, γ . Fermat’s principle of least time states that the
light ray follows that particular path between S and O for which the time taken is
minimal.
might imagine. We should emphasize, of course, that the source will in general
emit light in many different directions. The principle relates to that particular ray
which travels from the source point to the observation point and does not concern
itself with the rest, although in principle the observation point is arbitrary. We will
return to this important issue in a moment.
Fermat used the least-time principle around 1657 to derive what we now call
Snell’s Law, which we will discuss in a moment. Here we note that, although
Galileo famously described an attempt to measure the speed of light by flashing
lanterns about a mile apart in the 1630s, the finiteness of the speed of light was
not firmly established until 1676. The astronomer Roemer had detected a regular
variation over some years in the timing of the observed eclipses of Io, a moon of
Jupiter, relative to the predicted times. The slight advance or retardation of the times
for Io to disappear and reappear behind Jupiter depended upon whether Earth and
Jupiter were on the same or opposite sides of the Sun. Roemer correctly attributed
this apparent change to the finite speed of light, which led to the estimate c ≈
140,000 miles per second (see Hockey et al. [Sch07]).
1.1.2 Uniform media
Start with the simplest case, where light has the constant speed c and no obstacles
are present. We assume the light is emitted by a point source at S, and the observer
is at the point O (see Figure 1.1). The travel time from S to O is
T [γ ] =
L[γ ]
c
, (1.1)
where L[γ ] is the length of the path γ from source to observer. It is blindingly
obvious6
to everyone but theoretical physicists and mathematicians that the shortest
path between two points is a straight line. But, of course, it is a worthwhile exercise
in variational principles to prove it (see Problem 1.1).
6 Pun intended.](https://image.slidesharecdn.com/2339176-250523215407-0dd59a91/85/Ray-Tracing-And-Beyond-Phase-Space-Methods-In-Plasma-Wave-Theory-Tracy-Er-31-320.jpg)
![4 Introduction
1.1.3 Snell’s Law
Thus, the least-time path and the shortest-length path are the same in the simplest
case where the light speed is constant. But the least-time principle leads to some-
thing new: Suppose S and O lie in two different regions. The source lies in region 1,
where the speed of light is c/n1. The observation point is in region 2, where the
speed of light is c/n2.7
We leave it as an exercise for the reader to prove that the
least-time principle in this case leads to what we now call Snell’s Law8
for the
bending of rays at the interface
n1 sin θ1 = n2 sin θ2. (1.2)
An approximate form of Snell’s Law had been established by Ptolemy (ca. CE
90–168), in terms of the ratios of the angles. This is only correct for rays that are
nearly perpendicular to the interface. It appears that the correct form involving the
sines of the angles was discovered by the Persian astronomer-mathematician Ibn
Sahl around the year 984 CE (see Hockey et al. [Ber07]), though this form was
not known in the West until the early seventeenth century, when the invention of
the telescope (1608) would have motivated the development of improved theories
and measurements of light refraction for lens design.9
Snell’s Law was established
empirically, of course, and there are multiple claims to primacy in the literature
of the early mid 1600s. What is certain is that Fermat was aware of the law of
refraction (1.2), and showed that his principle of least time could be used to derive it.
The combination of straight-line rays within uniform regions, and Snell’s Law
at the interface between regions, forms the basis for ray optics and lens theory.
Kepler was the first to provide a theoretical explanation for the telescope, using ray
theory for compound lens systems. This theory first appeared in Kepler’s Dioptrice
(1611).10
Observation shows that the refractive index depends upon the color of light.
When this effect is included, the theory of prisms emerges and – through the
possibility of a double internal reflection within raindrops – the theory of the
rainbow.11
Thus, many of the design principles for microscopes, telescopes,
7 The constants n1 and n2 are the refractive indices; θ1 and θ2 are the angles formed by rays in the two regions
and the local normal at the interface. See Figure 3.4, Section 3.5.
8 See Problem 1.2.
9 See Willach [Wil08] for an interesting account of the evolving crafts of glass and lens manufacture in the late
Middle Ages and how these contributed to the invention of the telescope. Going even further back, for those
interested in the history of lenses in the classical world, a survey of what is known about the use of lenses in
the Graeco-Roman world can be found in Plantzos [Pla97]. We thank our colleague Professor Lily Panoussi
for bringing this reference to our attention.
10 This book, written in Latin, has not yet been translated into English, although high-quality scans of the original
are available online. It is interesting to note that the first figure to appear in the text concerns a means to measure
the refractive properties of materials.
11 Descartes discussed what is now considered the correct geometrical explanation in his 1637 Discourse on
Method. Earlier scholars in China and the Middle East also realized the importance of internal reflection for
explaining the rainbow by using glass spheres as laboratory models of raindrops.](https://image.slidesharecdn.com/2339176-250523215407-0dd59a91/85/Ray-Tracing-And-Beyond-Phase-Space-Methods-In-Plasma-Wave-Theory-Tracy-Er-32-320.jpg)
![1.1 Fermat’s principle of stationary time 5
cameras, eyeglasses, and the explanation of some of the most beautiful of atmo-
spheric phenomena, follow from the principle of least time.
Isaac Newton (1643–1727) made important contributions to the theory of optics
and revolutionized the design of telescopes using methods based upon the assump-
tion that light was composed of particles, supplemented by the notion that the speed
depended upon their color [New10]. The ray theory of light fits nicely with this
hypothesis, and the success of the predictions using a ray theory seemed to confirm
the particle hypothesis. (However, Fermat and the least-time principle are not even
mentioned in Newton’s Optics.)
It was not until the work of Thomas Young (1773–1829) and Fresnel (1788–
1827) that light was shown convincingly to be a wave phenomenon, capable of
diffraction and interference like water waves.12
This built upon the much earlier
work of Huygens (1629–1695).13
This then leads to a puzzle: if light is a wave, why
does ray theory work so well? We will see in later sections that Hamilton provided
an answer to this question by showing how to construct wave fields, including
interference patterns, using an entirely new type of ray theory.
1.1.4 Distributed sources
We should point out that nonpoint sources are dealt with at this stage by simple
superposition. Each point on the extended source S
is treated as a point source
independent of the others. This is easy to understand if light is composed of
particles, but it is a more subtle issue if light is a wave. Use of the superposition
assumption leads to the theory of imaging optics.
This simple approach to the analysis of distributed sources is valid only if the
light emission is incoherent from one point on the source to the next. Speaking
imprecisely for the moment, by a coherent source we mean one whose rays have a
well-defined phase θ at almost all points along each ray, and that this phase function
is smoothly varying along the ray and between neighboring rays. Coherent wave
fields are the primary topic of this book, though we will return to incoherent fields
in Section 3.5.5, where we summarize a ray phase space theory for them. We note
here that the incoherence of visible light in everyday life is almost always a good
assumption.14
What is lacking in the theory so far, of course, is that we have not discussed
how to compute the light intensity. A ray could arrive at the observation point with
zero intensity, in which case the existence of the ray itself is largely meaningless.
12 Young was able to produce coherent light by using a pinhole smaller than the transverse coherence length of
sunlight. The coherence length of an extended incoherent source is the wavelength divided by the solid angle
of the source [Wol07].
13 See, for example, Fresnel’s essay in [Fre00].
14 The reader should verify that the previous statement is correct. When, outside of a physics laboratory, do you
encounter coherent visible light? What physical conditions are required to produce it?](https://image.slidesharecdn.com/2339176-250523215407-0dd59a91/85/Ray-Tracing-And-Beyond-Phase-Space-Methods-In-Plasma-Wave-Theory-Tracy-Er-33-320.jpg)
![6 Introduction
Figure 1.2 The point source S and the point of observation O can no longer be
connected by a straight-line path, but the paths must instead go around the obstacle
B. A mirror M is present and we consider only paths that reflect from the mirror.
The least-time path is the light ray that obeys the law of reflection.
It is an observational fact that the convergence of rays increases the intensity of
light. Think of the focusing of sunlight using a lens.15
Likewise, as rays diverge, the
intensity decreases. It is physically reasonable to conjecture that a conservation law
applies, for example one modeled on treating the energy density in the light field
as a conserved fluid. If light were composed of discrete and long-lived particles,
the conservation law would follow directly from particle number conservation.
But if light is a wave, the derivation is less obvious. This will be discussed at
length later in the book, where we discuss wave energy and wave momentum
(Section 3.3.2), and we derive action conservation laws for wave fields, using both
coherent (Section 3.1) and incoherent (Section 3.5.5) formulations.
1.1.5 Stationarity vs. minimization: the law of reflection
Return to a point source S and a fixed point of observation O, but now block the
direct straight-line ray path and add a plane mirror (see Figure 1.2). This will turn
out to be a situation where the naive formulation of the principle of least time will
fail us, but it will help guide us to a better formulation. If we consider only paths
that pass from S to O after reflection from M, the path of least time reflects from M
in such a way that the angle of incidence equals the angle of reflection (as measured
15 The first mention of the use of lenses for burning and cauterizing for medicinal purposes is believed to be
Aristophanes, The Clouds, first performed ca. 423 BCE. The reference occurs in lines 767–769, see p. 64
of [Ari12]. The use of mirrors to focus light for the same purpose was also understood, as evidenced by the
famous legend of Archimedes setting fire to the Roman fleet at Syracuse, ca. 214–212 BCE.](https://image.slidesharecdn.com/2339176-250523215407-0dd59a91/85/Ray-Tracing-And-Beyond-Phase-Space-Methods-In-Plasma-Wave-Theory-Tracy-Er-34-320.jpg)
![1.1 Fermat’s principle of stationary time 7
Figure 1.3 The point source S and the point of observation O separated by the
obstacle B, but now we consider all paths. The sequence of paths shown that do
not reflect from the mirror all have travel times less than the ray.
with respect to the local normal at the point of reflection).16
This is simply Heron’s
problem, mentioned earlier. With the law of reflection, a host of new applications
emerge: optical instruments such as reflecting telescopes (building upon Newton’s
original breakthrough design) can be analyzed.
But why should we restrict ourselves to paths that reflect from M? Why can’t
we proceed as before and consider all paths that go from S to O? In that case, it
should be clear that some of the paths shown in Figure 1.3 have travel times less
than the ray that reflects from M. In fact, of the sequence of paths shown, the one
with the shortest travel time is the one that travels in a straight line from S to the
edge of B, and then on to O. Why don’t we use that path as our least-time ray and
ignore the path that reflects from M?17
Suppose we now remove the obstacle. The straight-line path is once again the
least-time path, but we also know from experience that the ray that reflects off the
mirror will also reach the point O. A helpful way to view what is going on – and
one that is quite physical – is to think of the source S as emitting rays that travel in
straight lines in all possible directions. If they encounter the mirror M, they bounce
and satisfy the law of reflection. Almost all of the infinitude of rays emitted from
S will miss the point O. There are two that make it to O: the straight-line path and
the one that bounces off the mirror and satisfies the law of reflection along the way.
16 See Problem 1.3.
17 There is good reason to consider the path that bends around the edge of B as a ray, but it requires careful
treatment at the edge where it encounters the obstacle. In fact, the light can diffract around the edge if it
is sharp enough, so some light could reach O from S by this route. But this takes us beyond a simple ray
picture. However, we note that the Huygens–Fresnel theory of wave propagation and diffraction starts with
such considerations. See Fresnel [Fre00].](https://image.slidesharecdn.com/2339176-250523215407-0dd59a91/85/Ray-Tracing-And-Beyond-Phase-Space-Methods-In-Plasma-Wave-Theory-Tracy-Er-35-320.jpg)
![1.1 Fermat’s principle of stationary time 9
that the ray is the shortest path (among those that reflect from the mirror) for the
flat mirror, but the longest straight-line path that reflects once off the curved mirror
shown to the right in the figure. This can be seen by starting with the straight-line
path (the ray) from S to O, and then considering a nearby path that reflects once just
before it reaches O. This path must have longer flight time than the straight-line
path. Now move the reflection point away from O toward S. When the reflection
point is in the neighborhood of S, the straight-line path becomes minimal once
again. Therefore, the flight time must have reached a maximum in between. These
two examples once again emphasize that we should formulate Fermat’s principle
as a principle of stationary time.
Before leaving the topic of reflection, we should mention that the specular
reflection we have described here is typical of highly polished surfaces. Rough
surfaces (rough on the length scale of a wavelength of light) can lead to more
diffuse types of reflection, first characterized by Lambert (1728–1777). This type
of reflection is, in fact, more common for everyday surfaces, and diffusion models
for reflection by textured surfaces are commonly used in computer graphics.18
1.1.6 Smoothly varying media
Thus, Fermat’s principle, now properly understood as a stationarity principle rather
than a minimization principle, unifies many important results that had appeared to
be distinct and brings them under one theory. It is natural, then, to extend Fermat’s
theory to situations where the refractive index varies continuously. Let’s consider
an important special case of a two-dimensional layered medium.
Suppose we have a two-dimensional system that is uniform in the x-direction,
but the refractive index varies in y: n(y). Our source lies at x = 0, and could be
extended in y (for example, it could be a building or a mountain). The point of
observation is at x1 and y = y1. Draw a path, γ = y : x → y(x) from a point on
the source to the point of observation
r(x) = [x, y(x)], r(0) = (0, y0), r(x1) = (x1, y1). (1.3)
The time required for light to travel along the path is given by the integral
T [y] =
1
c
x1
0
n[y(x)]
1 + [y(x)]2 dx, y
(x) ≡
dy
dx
. (1.4)
18 Rayleigh scattering should also be mentioned because it also involves a breakdown in the simple law of
reflection. Rayleigh scattering involves the scattering of light by particles that are smaller than a wavelength.
This type of scattering shows a strong frequency dependence and explains why the sky is blue; see for example
Jackson [Jac98].](https://image.slidesharecdn.com/2339176-250523215407-0dd59a91/85/Ray-Tracing-And-Beyond-Phase-Space-Methods-In-Plasma-Wave-Theory-Tracy-Er-37-320.jpg)
![10 Introduction
Here, we use the notation T [y] to denote the fact that the elapsed time is a functional
of the path y(x).19
Requiring T [y] to be stationary with respect to small variations
in the path leads to the Euler–Fermat equation
1 + [y(x)]2
dn
dy
−
d
dx
n y
1 + [y(x)]2
= 0, (1.5)
which can be reorganized as
y
(x) =
1 + [y
(x)]2
d ln n(y)
dy
. (1.6)
Hence, in a uniform medium (n
= 0), we recover the previous result that the light
paths are straight lines: y
(x) = 0. In a nonuniform medium (n
= 0), however,
the light paths curve toward the region of higher refractive index: concave upward
[y
(x) 0] for n
(y) 0 and concave downward [y
(x) 0] for n
(y) 0. Fur-
ther aspects of the continuous case, including a derivation of Snell’s Law for
a continuous layered medium, the trapping of waves in channels, and mirages
are examined in Problems 1.5 through 1.8. In Problem 1.9 the general three-
dimensional case is examined.
It is important to emphasize once more that this type of ray theory depends only
upon the refractive index n(x), and implicitly assumes the waves are incoherent
(due to the lack of any reference to a phase function). There is no dispersion relation
between the wave frequency and wavevector, because there is no wavevector in
Fermat’s theory.20
The “wave equation” was unknown to Fermat and contem-
poraries. All we need to know to apply Fermat’s theory is the wave speed,
v(x) = c/n(x), so we can compute the travel time along any path.
Ray tracing of this sort can also be applied to other types of waves. For example,
computer aided tomography (CAT) reconstructs “images” by measuring the atten-
uation of X-rays along ray paths, while positron emission tomography (PET) scans
map the spatial distribution of the intensity of gamma ray emission. In acoustics,
ray theory is important for the theory of reverberation. It is used in the design of
concert halls and recording studios, and it forms the basis for ultrasound imaging.
A striking example of these ideas is the design of whispering galleries, where
sound rays skim along the gently curved wall of a room.21
19 See Appendix B for a discussion of functionals.
20 The wavevector is the gradient of the phase θ(x), as we will discuss in the next section.
21 In the audible range of frequencies (approximately 20 Hz to 20 kHz), and in typical rooms (a few, to a few
tens of meters across) the lower frequencies are not well modeled using rays, but the higher frequencies often
can be treated as traveling in straight lines, satisfying the law of reflection. This is because the wavelength of
sound waves is λ = cs/f and, with a sound speed of ∼ 300 m/s, we have λ ranging from 15 m for the lowest
audible frequencies down to 1.5 cm for the highest audible frequencies. The information needed for human
speech recognition lies in the mid-range of frequencies.](https://image.slidesharecdn.com/2339176-250523215407-0dd59a91/85/Ray-Tracing-And-Beyond-Phase-Space-Methods-In-Plasma-Wave-Theory-Tracy-Er-38-320.jpg)
![1.2 Hamilton’s principle of stationary phase 11
Viewed through the modern lens of the statistical mechanics of light–matter
interactions, we now interpret the refractive index as a quantity that summarizes the
average macroscopic outcome of the complex microphysics when electromagnetic
fields interact with matter.22
For the multicomponent wave equations encountered
in plasma applications, a scalar refractive index is often too simple. Magnetized
plasmas are nonisotropic, and kinetic effects lead to wave equations that are integro-
differential (hence nonlocal) in space and time. Additionally, in many cases we need
to deal with coherent waves; hence we need to introduce the phase. But we will see
that the ray concept generalizes nicely to this more complex situation, and that the
geometrical picture the ray theory provides will prove useful as well. In particular,
we will retain the notion that rays are special paths that satisfy some stationarity
principle, but now instead of Fermat’s stationary time principle, we introduce a
stationary phase principle that was first used by Hamilton in his studies of optics.23
1.2 Hamilton’s principle of stationary phase
So that great ocean of ether which bathes the farthest stars is ever newly
stirred by waves that spread and grow, from every source of light, till they
move and agitate the whole with their minute vibrations: yet like sounds
through air, or waves on water, these multitudinous disturbances make
no confusion, but freely mix and cross, while each retains its identity,
and keeps the impress of its proper origin.
William Rowan Hamilton (1805–1865)24
Fermat’s principle is a geometric variational principle in the geodesic sense; that
is, it prescribes the path of light as it travels in a nonuniform medium by extremizing
the optical length L ≡ c T . The most important difference between the Fermat and
Hamilton approaches is the role of the phase function, θ(x, t). The phase, which
has been absent in our discussion until now, will come to play a central role in
Hamilton’s theory. We will, of course, exploit complex formulations since they
simplify life significantly, so by a “phase” we mean that the light wave field is
assumed to be of the form of a rapid variation, characterized by the oscillation
22 For example, for some exotic media the refractive index is negative [SSS01]. This does not imply faster-than-
light travel, or causal paradoxes! Instead, it shows that the concept of refractive index has been decoupled from
the notion of wave speed and is instead interpreted directly from the geometry of rays and Snell’s Law.
23 N.B. Hamilton’s principle of stationary phase should not be confused with the stationary phase method for
evaluating oscillating integrals. Although both arise frequently in discussions of eikonal theory, the former
is a principle for deriving dynamical equations in wave mechanics, while the latter is a general method for
computing asymptotic limits of certain types of oscillatory integrals. See Appendix C.1.1 for a discussion of
the stationary phase method.
24 From “On a general method of expressing the paths of light, and of the planets, by the coefficients of a
characteristic function,” by WR Hamilton (1833) [Ham33].](https://image.slidesharecdn.com/2339176-250523215407-0dd59a91/85/Ray-Tracing-And-Beyond-Phase-Space-Methods-In-Plasma-Wave-Theory-Tracy-Er-39-320.jpg)
![12 Introduction
Figure 1.6 [Left] A conceptual figure showing a region of three-space. The wave-
fronts, θ(x, t0) = const., are assumed to foliate the region, meaning that at each
spatial point there is a well-defined phase. Only the level set θ(x, t0) = 0 is shown,
for clarity. Also shown is an arbitrary path x(σ), which is assumed to intersect
the surfaces of constant-θ transversely. [Right] The level sets θ(x, t0) = 0 and
θ(x, t0 + dt) = 0 are shown. The path is held fixed here, but the wavefronts move
along the path at each spatial point.
exp[iθ(x, t)], multiplying a more slowly varying real amplitude function we will
denote as A(x, t).25
It was Hamilton who took the revolutionary step of asking how the phase
function, considered as a smooth function of space and time, could be written as
the integral of some function along one-dimensional paths (these will become the
rays in Hamilton’s theory), and how this function changes as the initial and final
points of the rays are varied. This leads to the concept of the phase integral, which
is central to the theory. This mathematical breakthrough demonstrated how a ray
theory could be used to compute wave patterns.26
In this section on the stationary phase principle, we emphasize a few of the con-
ceptual building blocks of Hamilton’s theory. The rest of the book is an elaboration
of these ideas, particularly with an eye toward applications in plasma wave theory.
1.2.1 Phase speed
We start by focusing attention on θ(x, t), a given smooth function of space and
time, and consider Figure 1.6. We wish to understand how the phase evolves in
time.
25 In this book, we almost always assume that the phase θ(x, t) is a real function. For a different approach using
a complex phase function to study evanescence and damping, see for example Choudhary and Felsen [CF73]
and Maj et al. [MMPF13].
26 Hamilton was well aware that the same mathematical theory could be used to study particle dynamics. In a
series of papers, he laid out his ray theory, dipping into optics or particle mechanics as examples of his general
theoretical approach [Ham28].](https://image.slidesharecdn.com/2339176-250523215407-0dd59a91/85/Ray-Tracing-And-Beyond-Phase-Space-Methods-In-Plasma-Wave-Theory-Tracy-Er-40-320.jpg)
![14 Introduction
Recall that the phase function and the path are assumed to be given, so this “phase
speed” is the rate at which the wavefront moves along a given path. If we change
the path, the “phase speed” changes.
Example 1.1 Consider a particle moving along the unperturbed trajectory x(t) in
the wave field A exp[iθ]. The condition (1.12) is the condition for the particle to
see a constant phase. Hence, this is a Landau resonance.
Example 1.2 Choose the path x(t) to lie along the local magnetic field. The
condition (1.12) becomes v ≡ ω/k. That is, vp ≡ v is the phase speed along the
magnetic field.
Later in the book we will encounter certain phenomena that arise in plasmas
when phase matching occurs due to a resonance. For example, in Chapter 6 we
discuss mode conversion, tunneling, and wave emission from coherent sources.
In addition, in Chapter 7 we discuss gyroresonance. In each of these cases, in
the multidimensional setting, the resonance requires a matching of phase fronts
throughout a multidimensional spatial region, not just a matching of the phase
speeds at a single point.
1.2.2 Phase integrals and rays
Suppose we are once again given the phase function, θ(x, t). Choose a fixed time
t0 and consider how the phase changes following the path x(σ). That is, given
θ[x(σ); t0] find dθ/dσ. The chain rule gives
d
dσ
θ[x(σ); t0] =
dx
dσ
· ∇θ[x(σ), t0], (1.13)
which we write as
dθ
dσ
≡
dx
dσ
· k[x(σ)], (1.14)
where we have introduced the wavevector k[x(σ)] by first taking the gradient of
the given phase field k(x) ≡ ∇θ(x, t0), and then evaluating it at x(σ).28
Notice that
by defining k as the gradient of the scalar θ(x, t), we have the identity
∇ × k(x, t) = ∇ × ∇θ = 0, (1.15)
for an eikonal wave.
28 It is important to note that ∇θ and dx/dσ have no intrinsic relation to one another, so k(σ) will, in general,
not be tangent to the path at x(σ). This is true even if dx/dσ is the group velocity (defined in Section 1.5). We
will give examples later.](https://image.slidesharecdn.com/2339176-250523215407-0dd59a91/85/Ray-Tracing-And-Beyond-Phase-Space-Methods-In-Plasma-Wave-Theory-Tracy-Er-42-320.jpg)
![1.2 Hamilton’s principle of stationary phase 15
Equivalently, given only a curl-free k(x), and the surface θ(x, t0) = 0,29
we can
choose a particular path and, starting at the point x0 where it punctures the surface
θ(x, t0) = 0, holding t fixed, we can integrate (1.14) along the path x(σ) to find
θ[x(σ), t0] =
σ
0
k[x(σ
)] ·
dx
dσ
(σ
) dσ
. (1.16)
It is important to emphasize that if we reparameterize the path σ = σ(s), the
numerical value of the integral does not change. Given that the path x(σ) is arbitrary,
and the choice of θ[x(σ); t0] = 0 as the base point for the integral was too, the
integral (1.16) is seen as another way of writing k(x) = ∇θ(x; t0), a way which
explicitly involves integration along spatial paths. (These are not yet rays, they are
still arbitrary smooth paths in x-space.) (See Problem 1.10.)
We now ask the question: how does the expression (1.16) have to be modified
so it will be correct at a slightly later time, t0 → t0 + dt? To simplify, let’s assume
that the path does not change in time. Only the phase evolves in time, and the
wavefronts move along the fixed path. The picture to have in mind is the right
panel of Figure 1.6, showing how the wavefront θ = 0 has shifted in the short time
dt.
If the phase oscillation exp[iθ(x, t)] locally looks like a plane wave exp[i(k · x −
ωt)] for some frequency ω, then we should use dθ ≡ ∇θ · dx + θt dt to identify
the local quantities
k(x, t) = ∇θ and ω(x, t) = −θt . (1.17)
This line of reasoning suggests that the general time-dependent form of the phase
integral is
θ[x(t), t] =
t
0
k[x(t
)] ·
dx
dt
(t
) − ω[x(t
), t
] dt
. (1.18)
Here we have used the fact that the k · dx/dσ integral is independent of param-
eterization to change from the arbitrary orbit parameter σ to the physical time t.
At this stage, (1.18) is simply an integral form of the phase function θ(x, t) that
satisfies the identities (1.17).
Up until now, we have concerned ourselves with arbitrary smooth paths in
physical space, and their relation to a given phase function θ(x, t). The next set of
moves involves an important shift in perspective, so we want to call attention to
their importance.
29 That is, we now assume that we do not know θ(x, t). We only know where the θ = 0 surface lies at time t0,
and the curl-free wavevector field k(x).](https://image.slidesharecdn.com/2339176-250523215407-0dd59a91/85/Ray-Tracing-And-Beyond-Phase-Space-Methods-In-Plasma-Wave-Theory-Tracy-Er-43-320.jpg)
![16 Introduction
1. Double the dimensionality of our system by treating k as independent of x. This
new space (x, k) is called ray phase space.
2. Introduce (x, k, t), a smooth function on ray phase space and time.
3. Introduce a smooth path in ray phase space, parameterized by the physical
time t: [x(t), k(t)]. (We choose to use the physical time here just to keep things
one bit less abstract. A more general parameterization will be used later in the
book.)
4. Define the phase integral , which is a functional of the phase space path
[x(t), k(t)] ≡
t
0
k(t
) · dx(t
) − [x(t
), k(t
), t
] dt
. (1.19)
5. Now invoke a stationarity principle30
for the phase integral (1.19), allowing
variation of x(t) and k(t) separately, holding the endpoints of the path fixed.
Stationarity with respect to x(t) → x(t) + εy(t), holding k(t) fixed, implies
(after integration by parts, and suppressing the arguments for clarity)
0 =
d [x + εy, k]
dε ε=0
= −
t
0
y · {dk + ∇x dt} , (1.20)
while stationarity with respect to k(t) → k(t) + εκ(t), holding x(t) fixed,
implies
0 =
d [x, k + εκ]
dε ε=0
=
t
0
κ · {dx − ∇k dt} . (1.21)
Together, these imply
dx
dt
= ∇k ,
dk
dt
= −∇x . (1.22)
Equations (1.22) are called Hamilton’s equations, and is the ray Hamiltonian.
Given the smooth function (x, k, t) and the initial conditions [x(t0), k(t0)], Hamil-
ton’s equations determine a unique path in ray phase space. Paths that satisfy
Hamilton’s equations we will call rays once more, in order to distinguish them
from arbitrary paths in ray phase space. But note that the dimensionality of the
space the ray inhabits has doubled from Fermat’s formulation.
Constructing the eikonal phase θ(x, t) using rays that satisfy (1.22) in the phase
integral (1.19) requires that we follow a family of rays. For that family of rays, if
at each point x there is only one ray, then k is well-defined as a function of x, and
30 See Appendix B for a brief discussion of variational methods.](https://image.slidesharecdn.com/2339176-250523215407-0dd59a91/85/Ray-Tracing-And-Beyond-Phase-Space-Methods-In-Plasma-Wave-Theory-Tracy-Er-44-320.jpg)
![1.2 Hamilton’s principle of stationary phase 17
we can write k(x, t) once more. The phase space integral (1.19) then reduces to
the previous form of the phase integral (1.18), thereby showing that k = ∇θ once
more. For this family of rays, we also have the identity
∂θ
∂t
≡ −ω(x, t) = − [x, k(x), t]. (1.23)
Therefore, from (1.23) and Hamilton’s equations (1.22), following a ray the local
frequency changes as
dω
dt
= ẋ · ∇x + k̇ · ∇k +
∂
∂t
=
∂
∂t
. (1.24)
A careful discussion of the construction of the eikonal phase function θ(x, t) from
a family of rays must be postponed until Section 3.2.1, where we discuss lifts and
projections.
There is much we have left out of this first encounter with Hamilton’s theory. For
example, we have not discussed how to compute the wave amplitude. In plasma
wave problems, we must also find the polarization of the electric and magnetic
fields at each point. Hamilton’s work concerned light rays, but we now know that
the ray theory he invented is the basis for all theories of short-wave asymptotics,
that is, it applies to all types of electromagnetic waves, sound waves, elastic waves
in solids, etc. We will have much more to say about these matters later in this
chapter, and in the rest of the book.
Before leaving this section, we should also mention the fundamental contri-
bution of Emmy Noether (1882–1935) concerning symmetries and conservation
laws. She was led to this topic during her time at Göttingen by Hilbert and Klein,
who were puzzling over some features of the then new general theory of relativ-
ity concerning the energy conservation law. Noether showed that the proper form
of the conservation law followed from a symmetry of the Hilbert action princi-
ple for the theory, but she went on to show that the method she had uncovered
was general.31
Specifically, what is now called Noether’s theorem shows that if a
variational principle, like Hamilton’s stationary phase principle, has a continuous
global symmetry, then making the variation local results in a conservation law. As
we shall see in Chapter 2, Noether’s theorem provides a simple and elegant means
to derive the wave-action conservation law, a result that is usually derived by brute
force calculation. This is also discussed in Appendix B.
31 For more of this interesting history, see Weyl’s memorial address for Noether, given at Bryn Mawr in 1935,
reprinted in [Wey70], and the recent article [Bye98].](https://image.slidesharecdn.com/2339176-250523215407-0dd59a91/85/Ray-Tracing-And-Beyond-Phase-Space-Methods-In-Plasma-Wave-Theory-Tracy-Er-45-320.jpg)
![18 Introduction
1.3 Modern developments
1.3.1 Quantum mechanics and symbol calculus
Hamilton developed his formulation of optics using systems of rays, but he was
also fully aware that it provided a new formulation of the Euler–Lagrange theory of
particle mechanics (see, for example, his expository article [Ham33]). Jumping for-
ward now to the 1920s, it is significant that when Erwin Schrödinger (1887–1961)
began his search to find the wave equation governing de Broglie’s matter waves,
he returned to Hamilton’s theory to guide his thinking. Schrödinger’s important
breakthrough is described in Moore [Moo94]. We take a moment to recount some of
these important developments of the 1920s and early 1930s, because Schrödinger’s
revolutionary ideas concerning wave mechanics, paired with the even more abstract
matrix mechanics of Heisenberg, provide a conceptual bridge between Hamilton
and Hermann Weyl (1885–1955), who is just about ready to enter our story. In
an attempt to understand the underlying mathematical nature of these two very
different theories of quantum mechanics, Weyl laid the foundations for what is
now called the symbol calculus, which plays a large role in this book.
The de Broglie relation (1924) is
p =
h
λ
≡ h̄k. (1.25)
Here λ is the matter wavelength, p the particle momentum, h is Planck’s constant,32
with h̄ ≡ h/2π. Recall, also, Einstein’s formula (1905) relating the energy and
frequency of a photon
E = hν ≡ h̄ω. (1.26)
These previously known relations were Schrödinger’s starting point.
Initially, in late 1925, in an attempt to understand the physical meaning of de
Broglie’s hypothesis, Schrödinger tried to construct the electron wave function for
the hydrogen atom, using only (1.25) and the classical Kepler orbit.33
This attempt
ran into problems because of caustics, a topic we will discuss in Chapter 5. These
technical difficulties led him to seek a wave equation that would govern the particle
dynamics, which he could then solve directly without recourse to the classical
orbits.
It is not possible to reconstruct the sequence of events that led to the final form
of what we now call the Schrödinger equation, and in any event he did not derive
the equation. Instead, over the course of only a few weeks, he used Hamilton’s
32 For completeness: h = 6.626 × 10−34 joule-seconds = 4.136 × 10−15 eV-seconds.
33 That is, he assumed the orbit for a classical point-charge electron in a 1/r potential.](https://image.slidesharecdn.com/2339176-250523215407-0dd59a91/85/Ray-Tracing-And-Beyond-Phase-Space-Methods-In-Plasma-Wave-Theory-Tracy-Er-46-320.jpg)
![1.3 Modern developments 19
ideas and well-known operator correspondences (see below) to reason his way to
two plausible candidates for a matter wave equation. The candidate equations were
then used to make physical predictions, which could be compared with experiment.
It was assumed that the wave equation should be linear, so that solutions could be
superposed.
It appears that the first equation Schrödinger derived was a relativistic equation
(now called the Klein–Gordon equation). Start with the Einstein mass–energy
relation for a free particle
E2
= p2
c2
+ m2
c4
, (1.27)
where m is the rest mass of the particle. In a 1/r potential, Schrödinger reasoned
that this should become
E +
e2
r
2
= p2
c2
+ m2
c4
. (1.28)
Now consider the following correspondences, which were well-known from
Fourier analysis
k ↔ −i∇ and ω ↔ i
∂
∂t
. (1.29)
Therefore, using the de Broglie and Einstein relations (1.25) and (1.26), the mass–
energy relation (1.28) suggested consideration of the following relativistic wave
equation
ih̄
∂
∂t
+
e2
r
2
ψ = −h̄2
c2
∇2
ψ + m2
c4
ψ. (1.30)
Inserting an eikonal ansatz ψ = A exp[iθ] (and neglecting the derivatives of A),
the mass–energy relation (1.27) is now seen to be the eikonal equation for the
phase θ.34
According to Moore [Moo94], Schrödinger thought this was too easy.
He rejected this equation because it also did not give the right energy levels for the
hydrogen atom in the nonrelativistic limit.35
Schrödinger then decided to look for the nonrelativistic theory, and he began by
considering the classical Hamiltonian for a particle in an external potential
H(x, p) =
p2
2m
+ V (x) = E. (1.31)
34 We will have much more to say about the eikonal equation in Chapter 2.
35 We now know that this is because the electron has spin, and its relativistic theory is governed by the Dirac
equation. The Klein–Gordon equation is adequate for spin-0 particles.](https://image.slidesharecdn.com/2339176-250523215407-0dd59a91/85/Ray-Tracing-And-Beyond-Phase-Space-Methods-In-Plasma-Wave-Theory-Tracy-Er-47-320.jpg)
![20 Introduction
Using similar reasoning as in the relativistic case, Schrödinger then conjectured
that the appropriate quantum wave equation should be
−
h̄2
2m
∇2
+ V (x)
ψ = ih̄
∂ψ
∂t
, (1.32)
which is the famous equation that now bears his name. From beginning to end,
this took only a few weeks (that is, starting with his attempts to construct the wave
function using only Kepler orbits and the de Broglie relation to the “derivation” of
the wave equation). However, Schrödinger was unable to compute the full hydrogen
spectrum from this wave equation because of technical difficulties in the continuum
region. For this part of the calculation, he turned to his close friend Hermann Weyl
for help, and together they quickly completed the calculation.
Heisenberg’s matrix mechanics had been developed in 1925, by Heisenberg,
Born, and Jordan, as an attempt to create a quantum theory involving only observ-
ables. Heisenberg’s breakthrough was to realize that these observables should not
have to commute as algebraic objects, reflecting the idea that the sequence in which
different observations were taken mattered. Hence, Born suggested the observables
should be represented by matrices, and the result of an observation would be one
of the eigenvalues. Because the results of a measurement are real quantities (in the
mathematical sense), the matrices representing observables must be self-adjoint.
For our purposes in this brief conceptual survey, the most important example of
noncommuting observables are the position
x and the momentum
p.36
These two
observables obey the commutation relation
x
p −
p
x ≡ [
x,
p] = ih̄
Id, (1.33)
where
Id is the identity. It is easy to show that there are no finite-dimensional real-
izations of this commutation relation.37
There is, however, the familiar realization
in terms of the operator associations
x ↔ x,
p ↔ −ih̄
∂
∂x
. (1.34)
The operators act on functions of x. These types of associations are familiar from
Fourier analysis (but without the h̄).
Therefore, by 1926 there were two successful formulations of quantum mechan-
ics, matrix mechanics and wave mechanics. Of the two, wave mechanics had the
more direct link with classical mechanics via Hamilton’s theory of ray systems.
36 We denote these objects with carats to distinguish them from ordinary numbers. We will have much more to
say about these matters in Chapter 2.
37 The proof is by contradiction. Assume that
x,
p, and
Id are N × N matrices with N finite. Take the trace
of (1.33). The commutator has zero trace, but the trace of the identity matrix is N. Therefore, N cannot be
finite.](https://image.slidesharecdn.com/2339176-250523215407-0dd59a91/85/Ray-Tracing-And-Beyond-Phase-Space-Methods-In-Plasma-Wave-Theory-Tracy-Er-48-320.jpg)
![1.3 Modern developments 21
Weyl was in the very center of this ferment, through his close friendship with
Schrödinger, and was already considered one of the leading mathematicians of his
generation.38
From his work on Lie groups, Weyl recognized (1.33) as a Lie algebra. Elements
of the Lie algebra generate the Lie group through exponentiation.39
An arbitrary
element of the Lie algebra is the linear combination a
x + b
p, where a and b are
real constants, and the associated group element is
T ≡ ei(a
x+b
p)
. (1.35)
Because
x and
p are self-adjoint, the operator
T is unitary for real a and b (see
Section 2.3, Section D.3.1, and Appendix D for more details).40
The operator
p generates shifts in x-space, while
x generates shifts in k-space; therefore the
combination is a phase space shift. This Lie group is now called the Heisenberg–
Weyl group, the group of noncommutative shifts on classical phase space. By 1926,
Weyl was able to show that any of Heisenberg’s matrix observables could be
written as a linear superposition of operators of the type (1.35).41
The expansion
coefficients can be found using what is now called the symbol of the operator.
We will discuss Weyl’s ideas in Chapter 2, but in the context of classical wave
equations, where they are equally powerful.
We should also mention here the third formulation of quantum mechanics using
path integrals. This approach is identified with Feynman, who introduced the idea
in his Ph.D. thesis [FB42]. Feynman was motivated by the problem of quantizing
systems that had no Hamiltonian.42
Without a Hamiltonian, neither the Heisenberg
nor Schrödinger approaches could be used to construct the time evolution operator.
In Feynman’s theory, there is assumed to be a rule for assigning a phase exp[iθ]
to each point on any path in x-space, including nonsmooth paths. The phases for
all possible paths – starting from fixed initial and ending points – are summed over,
and only those paths for which the phase is locally stationary will interfere con-
structively and survive in the asymptotic limit h̄↓0. When the rule for constructing
the phase at each point is based upon the classical action, θ ≡ h̄−1
Ldt with L
the Lagrangian, then the paths which survive to dominate the quantum transition
probabilities are the classical ones, that is, the paths for which the classical action
is stationary.
38 In his Weyl Centenary Lecture, Roger Penrose writes that Weyl was the greatest mathematician who worked
entirely in the twentieth century. This allowed Penrose to avoid arguments about Hilbert, Poincaré, or Cartan.
39 Strictly speaking, it is i
x and i
p that are the generators.
40 Because
x and
p are self-adjoint operators, their linear combination a
x + b
p is also self-adjoint, with real
eigenvalues.
41 Weyl’s theory first appeared in the 1927 article [Wey27] and was later expanded into the 1931 book [Wey31].
42 For example, in cases where the Legendre transformation from (x, ẋ) to the canonical coordinates (x, p) was
not well-defined.](https://image.slidesharecdn.com/2339176-250523215407-0dd59a91/85/Ray-Tracing-And-Beyond-Phase-Space-Methods-In-Plasma-Wave-Theory-Tracy-Er-49-320.jpg)
![22 Introduction
The path integral approach to the quantum theory of photons actually has an
earlier history, and can be traced to a little-known paper by Wentzel [Wen24], which
appeared in the very same year as de Broglie’s work (1924) and, therefore, preceded
even the work of Heisenberg and Schrödinger.43
This is also of note for our history,
because Wentzel (1898–1978) is the “W” in “WKB.”44
Although mathematical
approximation schemes similar to what are now called “WKB methods” were
used in the nineteenth century, the papers of Wentzel [Wen26a, Wen26b, Wen26c],
Kramers [Kra26], and Brillouin [Bri26] are noteworthy because they are the first
systematic attempts to use Schrödinger’s newly proposed wave equation to compute
quantum corrections to classical theories.45
Feynman’s original work concerned paths in x-space, as in Fermat’s original
least-time principle. For systems that have a Hamiltonian, it is possible to refor-
mulate the path-integral theory entirely in phase space, replacing L by pq̇ − H,
with H the Hamiltonian. This is not just a formal development. If we attempt
to use Weyl’s symbol theory to compute the symbol of the evolution operator
U ≡ exp[it
H] in terms of the symbol of the Hamiltonian
H, then the phase space
path integral arises directly, not the x-space path integral (see, for example, Berezin
and Shubin [BS91], DeWitt-Morette et al. [DMMN77], or Richardson [Ric08]).46
The fact that Weyl’s theory originated in the context of quantum physics can
obscure the fact that the mathematical ideas are completely general, applicable
to all manner of wave equations, including those in plasma theory. This early
identification of the symbol calculus with the mathematical foundations of quantum
mechanics may explain why Weyl’s ideas took over fifty years to find their way
into the plasma physics literature.47
1.3.2 Ray phase space and plasma wave theory
The following discussion is drawn from the 1991 review article by AN Kauf-
man [Kau91].48
The theoretical study of plasma dynamics utilizes several different spaces. There
is the three-dimensional physical space (denoted by x) and four-dimensional
space-time (denoted by x or xμ
= (x, t)). These are the natural base-spaces
for the Maxwell field, for various plasma densities (particle density, current
43 A summary of the paper, and a discussion of its historical significance, can be found in [AL97].
44 We also mention in passing, that Wentzel was a Ph.D. supervisor for one of the co-authors of this book (ANK),
who therefore has a “W-number” of one.
45 See [FGNO09].
46 We thank our colleague Nahum Zobin for his help in understanding the theory of path integrals.
47 The first paper that we are aware of that uses the Weyl symbol calculus in plasma theory is the 1988 paper by
McDonald [McD88].
48 See also the more recent reviews from the 2009 KaufmanFest [BT09, TB09, Bri09, Kau09].](https://image.slidesharecdn.com/2339176-250523215407-0dd59a91/85/Ray-Tracing-And-Beyond-Phase-Space-Methods-In-Plasma-Wave-Theory-Tracy-Er-50-320.jpg)
![1.3 Modern developments 23
density, momentum-flux density), and for wave fields. On the other hand, plasma
kinetic equations deal with distributions in six-dimensional single-particle position-
velocity space (r, v) or six-dimensional single-particle phase space (r, p), or with
their eight-dimensional extensions (r, p).
Plasma physicists are accustomed to dealing with these spaces for these entities.
But plasma waves, too, have a natural ray phase space, which we have already
encountered in our discussion of Hamilton’s ray optics. Although Hamilton’s phase
space approach to optics is ancient, it drew little attention in the plasma community
until the 1980s. Its renaissance was due to the impact of the spectacular development
of semi-classical dynamics. Classical physicists (which almost all plasma physicists
are) can now see how their understanding of classical particle orbits in phase space
relates to the quantum wave function in position space.
The Berkeley plasma theory group, led by AN Kaufman, was strongly influenced
by the comprehensive review articles of Berry [Ber77a, BU80, BT76, Ber77b] and
of Percival [Per77], and by the research of their groups. Those reviews pointed
the way back to the pioneering work of Keller [Kel58, Kel85], Keller and Rubi-
now [KR60], and of Maslov [MF02].
The Berkeley group’s entry to the field was sparked by the paper of Wersinger,
Finn, and Ott [WFO80], discussing how plasma waves can exhibit chaotic ray
orbits. This naturally led to the question of what the corresponding wave field
looked like. The stadium-billiard was selected as the simplest system to investi-
gate, and, in his Ph.D. thesis, McDonald obtained the first chaotic wave eigen-
function [MK79, MK85]. To relate this x-space wave field to phase space, a
coarse-grained Wigner function was used, whose properties for a chaotic system
had been recently predicted by Berry [Ber77a, Ber77b]. This led to a more detailed
study [MK88, McD88] of Wigner functions and of the Weyl symbol calculus
relating phase space functions and x-space operators.49
From this study, a simple
derivation of the wave kinetic equation for action-density in phase space was uncov-
ered [MK85], as well as Hamiltonian formulations for wave interactions [MK82].50
Up to this point, there had been no occasion to utilize variational principles.
In plasma physics, variational principles had appeared in two disparate forms.
For fields on x-space, they had been used by Crawford and co-workers Kim
and Galloway [KC77a, KC77b, GC77], by Dougherty [Dou70, Dou74], and
by Dewar [Dew72b, Dew70, Dew72a] as a very powerful approach to x-space
problems, such as nonlinear wave interactions. On the other hand, for single-
particle motion, Littlejohn had employed a phase space variational principle to
obtain a simpler derivation [Lit83] of his non-canonical Hamiltonian theory of
49 See Section 2.3 and Appendix D for a discussion of the Weyl symbol calculus.
50 See Section 3.5.5 for a discussion of the derivation of the wave kinetic equation.](https://image.slidesharecdn.com/2339176-250523215407-0dd59a91/85/Ray-Tracing-And-Beyond-Phase-Space-Methods-In-Plasma-Wave-Theory-Tracy-Er-51-320.jpg)
![24 Introduction
guiding-center motion [Lit79, Lit81]. The Hamiltonian and action principle for-
mulations of plasma physics have been reviewed by Morrison, who made significant
contributions to this area [Mor05].
Four other important theoretical developments led to the formulation of the
plasma phase space variational principle. The first was the Hamiltonian theory
of the oscillation center (OC), introduced by Dewar [Dew73] (see also John-
ston [Joh76]) as a systematic resolution of then current confusion about quasilinear
diffusion [Kau72]. The second was the concept of ponderomotive potential, for the
nonlinear low-frequency effects of high-frequency fields. These lines of research
led to the realization [JKJ78] that the ponderomotive potential was the zero-velocity
limit of the quadratic term in the oscillation-center Hamiltonian, called the pon-
deromotive Hamiltonian.
The third development was the (Hamiltonian) Lie transform, introduced into
plasma physics by Dragt and Finn [DF76, DF79] and by Dewar [Dew76]. The
Berkeley group adopted this technique wholeheartedly, applying it to a range of
problems. In particular, Cary converted Dewar’s OC transformation from a mixed-
variable generating function to the far simpler Lie transform [CK81]. The resulting
expression for the ponderomotive Hamiltonian was observed to be identical in
form (except for a sign) to the linear susceptibility [CK77, JK78]. This astounding
relation between a nonlinear expression and a linear one (now called the K-χ
relation) meant that something deep remained to be discovered; the minus sign
pointed in the Lagrangian direction.
The fourth development [Lit82] was Littlejohn’s Lagrangian Lie transform
(L3
T). Although the Hamiltonian Lie transform works quite well, its formalism
leads to expressions that are not manifestly gauge-invariant, and thus to potentially
unphysical interpretations. In contrast, L3
T makes full use of modern differential
geometry in physical (as opposed to canonical) phase space; it can embrace the
Hamiltonian Lie transform as a special case, and it generates the OC transformation
by a vector field in phase space, which is the physical oscillation perturbation.
The spark for further progress came from the paper of Dubin et al. [DKOL83]
who constructed self-consistent evolution equations for a distribution of guid-
ing centers and for the electric-potential wave field. They demonstrated this self-
consistency by finding an energy conservation law, by trial and error. It seemed
to ANK that self-consistent equations and their conservation laws should come
automatically from an appropriate variational principle; this had been shown, in
the x-space context, by Dewar [Dew77]. It had also been shown, by Dominguez
and Berk [DB84], that a variational principle was the natural vehicle for deriving
self-consistent equations subject to a systematic approximation scheme.
But the transformations to guiding centers (developed by Littlejohn) and to
oscillation centers (developed by Dewar) required phase space coordinate changes.](https://image.slidesharecdn.com/2339176-250523215407-0dd59a91/85/Ray-Tracing-And-Beyond-Phase-Space-Methods-In-Plasma-Wave-Theory-Tracy-Er-52-320.jpg)
![1.3 Modern developments 25
Littlejohn then suggested to ANK that the single-particle phase space action prin-
ciple, which was the starting point for L3
T, was also a natural starting point for a
theory of the self-consistent evolution of electromagnetic fields and particles. Now
everything fell into place. Starting with the total action for the system, and applying
a Lie transform to the action of each particle, led Kaufman and Boghosian to the
K-χ theorem [KB84, Kau87]), to the ponderomotive effects on oscillation centers,
and to linear wave propagation in the OC medium. The results were in complete
agreement with the previous Hamiltonian formulations of McDonald, Grebogi,
Kaufman, and Omohundro [GKL79, MGK85, Kau82, Omo86], and provided in
addition the evolution of the quasistatic background field, which had been missing
from the previous Hamiltonian approach.
Two further developments contributed importantly to the formulation. One was
the covariant formalism: treating the particle motion relativistically and covariantly,
rather than in 3 + 1 notation; this simplified the equations and led to additional
insights.51
Whereas the unmagnetized plasma (see Kaufman and Holm [KH84])
posed no problems in generalizing to covariance, the magnetized case was quite
challenging and rewarding. Building on Littlejohn’s covariant Poisson brackets,
a covariant single-particle action principle in the guiding-center representation
was formulated [Sim85]. The second development was Similon’s construction
of a Lagrangian density (in x-space) for this phase space action principle, and
adaptation of Noether’s methods to finding an algorithm for energy-momentum
conservation laws [Sim85] (see Appendix B, and Section 3.3.1).
Building on these foundations, the covariant OC transformation for gyrating
particles interacting nonresonantly with a single eikonal wave in a weakly nonuni-
form background field was derived, which led to a manifestly gauge-invariant
and Lorentz-covariant expression for the ponderomotive Hamiltonian and lin-
ear susceptibility (see Boghosian [Bog87]). Taking variations with respect to the
background four-potential led to the covariant equations for the background field,
including the effects of wave-induced magnetization. The latter effects were cru-
cial in analyzing the ponderomotive stabilization of low-frequency modes by high-
frequency waves [SK84, SKH86].
It remained to incorporate the resonant interaction of particles with an eikonal
wave. As a first approach, an approximate interaction Lagrangian was derived
by integrating a linearized version across the resonant region of particle phase
space [GK83]. Variation then led to the self-consistent change of the back-
ground field, resulting from wave–particle resonance, and to coarse-grained energy-
momentum conservation laws (see Ye and Kaufman [YK92]).
51 See Section 3.3.1 for a discussion of the Lorentz-covariant ray theory, and Section 3.4 for the fully covariant
theory on ray phase space.](https://image.slidesharecdn.com/2339176-250523215407-0dd59a91/85/Ray-Tracing-And-Beyond-Phase-Space-Methods-In-Plasma-Wave-Theory-Tracy-Er-53-320.jpg)
![26 Introduction
Application to ion gyroresonant heating led to the realization that the situa-
tion is considerably more complex. It is generally recognized that, when an elec-
tromagnetic plasma wave crosses a resonance layer transversely, two additional
waves are created, and a large fraction of the incident wave action is converted
to quasilinear diffusion (in phase space) of the resonant particles. Following Ye
and Kaufman [Ye90, YK88a, YK88b], we now interpret this phenomenon in terms
of Friedland’s formulation of mode conversion, which is discussed at length in
Chapters 6 and 7.
The basic idea underlying Friedland’s work is that a multicomponent wave field
supports modes with different polarizations and different dispersion functions.
Conversion occurs when two modes can exist at the same point in ray phase space.
Wave action can then be exchanged in its vicinity, due to their coupling. While
the fraction of conversion had been known for the case of one-dimensional spatial
variation of the medium, Friedland solved the coupled partial differential equations
in x-space, for a full four-dimensional space-time variation of the medium (see
Friedland et al. [FGK87]). In his result, a Poisson bracket appeared, showing
that phase space methods were waiting to be applied. Accordingly, ANK utilized
methods which Littlejohn had developed [Lit86] for semi-classical analysis of the
Schrödinger equation. In place of the standard (x, k) coordinates of ray phase
space, a locally linear canonical transformation to new coordinates (q, p) is used,
in which one set (q1, p1) was related to (k, x) by the two dispersion functions.
The wave field is then expressed as a function on q-space instead of x-space,
as a generalization of the Fourier transform. With this technique, the conversion
problem could be completely solved [KF87, TK90]. This theory has now been
generalized, and adapted for use in the numerical code RAYCON (see Chapter 6, as
well as the Appendices E through G).
Friedland prepares the equations for the multicomponent wave field by system-
atically eliminating components, in such a way as to preserve the desirable feature
of having slowly varying coefficients. His algorithm for accomplishing this, termed
congruent reduction [FK87], is based on a Hermitian-form variational principle,
and utilizes the Weyl symbol calculus. When this technique is applied to linearized
kinetic equations, as in gyroresonance, one is led to the dispersion functions of
ballistic modes (see Friedland and Goldner [FG86]), which represent perturbations
of the particle density (in phase space) in the absence of coupling to the electro-
magnetic perturbation. The resonance of particles with a collective mode is then
interpreted as a linear conversion to a continuum of these ballistic waves. These
waves can, in turn, transfer action to other collective modes (see Ye and Kauf-
man [Ye90, YK88a, YK88b]). The residue of the ballistic modes undergoes phase
mixing and is interpreted as collisionless absorption. Using these ideas, the compu-
tation of the conversion of an incoming fast magnetosonic wave to a minority-ion](https://image.slidesharecdn.com/2339176-250523215407-0dd59a91/85/Ray-Tracing-And-Beyond-Phase-Space-Methods-In-Plasma-Wave-Theory-Tracy-Er-54-320.jpg)
Ray Tracing And Beyond Phase Space Methods In Plasma Wave Theory Tracy Er
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- 7. RAY TRACING AND BEYOND This complete introduction to the use of modern ray-tracing techniques in plasma physics describes the powerful mathematical methods generally applicable to vec- tor wave equations in nonuniform media, and clearly demonstrates the application of these methods to simplify and solve important problems in plasma wave theory. Key analytical concepts are carefully introduced as needed, encouraging the development of a visual intuition for the underlying methodology, with more advanced mathematical concepts succinctly explained in the appendices, and sup- porting MATLAB code available online. Covering variational principles, covariant formulations, caustics, tunneling, mode conversion, weak dissipation, wave emis- sion from coherent sources, incoherent wave fields, and collective wave absorption and emission, all within an accessible framework using standard plasma physics notation, this is an invaluable resource for graduate students and researchers in plasma physics. e. r. tracy is the Chancellor Professor of Physics at the College of William and Mary, Virginia. a. j. brizard is a Professor of Physics at Saint Michael’s College, Vermont. a. s. richardson is a Research Scientist in the Plasma Physics Division of the US Naval Research Laboratory (NRL). a. n. kaufman is an Emeritus Professor of Physics at the University of California, Berkeley.
- 8. “Ray Tracing and Beyond is an encyclopedic and scholarly work on the linear theory of dispersive vector waves, summarizing the powerful general theory developed over the careers of four leading practitioners and teachers in theoretical plasma physics. It seems destined to become a ‘must-read’ classic for graduate students and researchers, not only specialists in plasma physics (a field which involves a myriad of wave problems in nonuniform media) but also the many other physicists and applied mathematicians working on problems involving waves.” Robert L. Dewar, Australian National University
- 9. RAY TRACING AND BEYOND Phase Space Methods in Plasma Wave Theory E. R. TRACY College of William and Mary, Virginia A. J. BRIZARD Saint Michael’s College, Vermont A. S. RICHARDSON US Naval Research Laboratory (NRL) A. N. KAUFMAN University of California, Berkeley
- 10. University Printing House, Cambridge CB2 8BS, United Kingdom Published in the United States of America by Cambridge University Press, New York Cambridge University Press is a part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9780521768061 © E. R. Tracy, A. J. Brizard, A. S. Richardson and A. N. Kaufman 2014 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2014 Printed in the United Kingdom by TJ International Ltd. Padstow, Cornwall A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Tracy, E. R. (Eugene Raymond), 1956– Ray tracing and beyond : phase space methods in plasma wave theory / E. R. Tracy, A. J. Brizard, A. S. Richardson, and A. N. Kaufman. pages cm Includes bibliographical references and indexes. ISBN 978-0-521-76806-1 (hardback) 1. Plasma waves. 2. Ray tracing algorithms. 3. Phase space (Statistical physics) I. Brizard, Alain Jean. II. Title. QC718.5.W3T72 2014 530.412 – dc23 2013028341 ISBN 978-0-521-76806-1 Hardback Additional resources for this publication at www.cambridge.org/raytracing Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
- 11. For Louise Kaufman She walked in beauty
- 13. Contents Preface page xiii Acknowledgements xix 1 Introduction 1 1.1 Fermat’s principle of stationary time 2 1.1.1 General comments 2 1.1.2 Uniform media 3 1.1.3 Snell’s Law 4 1.1.4 Distributed sources 5 1.1.5 Stationarity vs. minimization: the law of reflection 6 1.1.6 Smoothly varying media 9 1.2 Hamilton’s principle of stationary phase 11 1.2.1 Phase speed 12 1.2.2 Phase integrals and rays 14 1.3 Modern developments 18 1.3.1 Quantum mechanics and symbol calculus 18 1.3.2 Ray phase space and plasma wave theory 22 1.4 One-dimensional uniform plasma: Fourier methods 27 1.4.1 General linear wave equation: D(−i∂x, i∂t )ψ = 0 28 1.4.2 Dispersion function: D(k, ω) 29 1.4.3 Modulated wave trains: group velocity and dispersion 31 1.4.4 Weak dissipation 34 1.4.5 Far field of dispersive wave equations 35 1.5 Multidimensional uniform plasma 38 1.6 One-dimensional nonuniform plasma: ray tracing 40 1.6.1 Eikonal equation for an EM wave 40 1.6.2 Wave-action conservation 42 1.6.3 Eikonal phase θ(x) 43 vii
- 14. viii Contents 1.6.4 Amplitude A(x) 44 1.6.5 Hamilton’s equations for rays 45 1.6.6 Example: reflection of an EM wave near the plasma edge 46 1.7 Two-dimensional nonuniform plasma: multidimensional ray tracing 47 1.7.1 Eikonal equation for an EM wave 48 1.7.2 Wave-action conservation 48 1.7.3 Eikonal phase θ(x,y) and Lagrange manifolds 49 1.7.4 Hamilton’s equations for rays 49 Problems 51 References 55 2 Some preliminaries 62 2.1 Variational formulations of wave equations 62 2.2 Reduced variational principle for a scalar wave equation 63 2.2.1 Eikonal equation for the phase 64 2.2.2 Noether symmetry and wave-action conservation 64 2.3 Weyl symbol calculus 66 2.3.1 Symbols in one spatial dimension 66 2.3.2 Symbols in multiple dimensions 72 2.3.3 Symbols for multicomponent linear wave equations 74 2.3.4 Symbols for operator products: the Moyal series 74 Problems 76 References 78 3 Eikonal approximation 80 3.1 Eikonal approximation: x-space viewpoint 81 3.2 Eikonal approximation: phase space viewpoint 84 3.2.1 Lifts and projections 89 3.2.2 Matching to boundary conditions 92 3.2.3 Higher-order phase corrections 94 3.2.4 Action transport using the focusing tensor 95 3.2.5 Pulling it all together 97 3.2.6 Frequency-modulated waves 102 3.2.7 Eikonal waves in a time-dependent background plasma 104 3.2.8 Symmetries 105 3.2.9 Curvilinear coordinates 108 3.3 Covariant formulations 111 3.3.1 Lorentz-covariant eikonal theory 111 3.3.2 Energy-momentum conservation laws 119
- 15. Contents ix 3.4 Fully covariant ray theory in phase space 121 3.5 Special topics 128 3.5.1 Weak dissipation 129 3.5.2 Waveguides 132 3.5.3 Boundaries 134 3.5.4 Wave emission from a coherent source 139 3.5.5 Incoherent waves and the wave kinetic equation 142 Problems 146 References 151 4 Visualization and wave-field construction 154 4.1 Visualization in higher dimensions 155 4.1.1 Poincaré surface of section 155 4.1.2 Global visualization methods 157 4.2 Construction of wave fields using ray-tracing results 170 4.2.1 Example: electron dynamics in parallel electric and magnetic fields 173 4.2.2 Example: lower hybrid cutoff model 173 References 182 5 Phase space theory of caustics 183 5.1 Conceptual discussion 187 5.1.1 Caustics in one dimension: the fold 187 5.1.2 Caustics in multiple dimensions 191 5.2 Mathematical details 193 5.2.1 Fourier transform of an eikonal wave field 194 5.2.2 Eikonal theory in k-space 196 5.3 One-dimensional case 198 5.3.1 Summary of eikonal results in x and k 198 5.3.2 The caustic region in x: Airy’s equation 200 5.3.3 The normal form for a generic fold caustic 205 5.3.4 Caustics in vector wave equations 210 5.4 Caustics in n dimensions 212 Problems 218 References 226 6 Mode conversion and tunneling 228 6.1 Introduction 228 6.2 Tunneling 242 6.3 Mode conversion in one spatial dimension 247
- 16. x Contents 6.3.1 Derivation of the 2 × 2 local wave equation 247 6.3.2 Solution of the 2 × 2 local wave equation 252 6.4 Examples 258 6.4.1 Budden model as a double conversion 259 6.4.2 Modular conversion in magnetohelioseismology 261 6.4.3 Mode conversion in the Gulf of Guinea 263 6.4.4 Modular approach to iterated mode conversion 269 6.4.5 Higher-order effects in one-dimensional conversion models 273 6.5 Mode conversion in multiple dimensions 276 6.5.1 Derivation of the 2 × 2 local wave equation 276 6.5.2 The 2 × 2 normal form 279 6.6 Mode conversion in a numerical ray-tracing algorithm: RAYCON 283 6.7 Example: Ray splitting in rf heating of tokamak plasma 295 6.8 Iterated conversion in a cavity 301 6.9 Wave emission as a resonance crossing 303 6.9.1 Coherent sources 304 6.9.2 Incoherent sources 308 Problems 310 Suggested further reading 322 References 323 7 Gyroresonant wave conversion 327 7.1 Introduction 327 7.1.1 General comments 329 7.1.2 Example: Gyroballistic waves in one spatial dimension 331 7.1.3 Minority gyroresonance and mode conversion 333 7.2 Resonance crossing in one spatial dimension: cold-plasma model 335 7.3 Finite-temperature effects in minority gyroresonance 348 7.3.1 Local solutions near resonance crossing for finite temperature 359 7.3.2 Solving for the Bernstein wave 373 7.3.3 Bateman–Kruskal methods 379 Problems 385 References 392 Appendix A Cold-plasma models for the plasma dielectric tensor 394 A.1 Multifluid cold-plasma models 395 A.2 Unmagnetized plasma 397
- 17. Contents xi A.3 Magnetized plasma 399 A.3.1 k B0 400 A.3.2 k B0 401 A.4 Dissipation and the Kramers–Kronig relations 403 Problems 404 References 405 Appendix B Review of variational principles 406 B.1 Functional derivatives 406 B.2 Conservation laws of energy, momentum, and action for wave equations 408 B.2.1 Energy-momentum conservation laws 408 B.2.2 Wave-action conservation 409 References 411 Appendix C Potpourri of other useful mathematical ideas 412 C.1 Stationary phase methods 412 C.1.1 The one-dimensional case 412 C.1.2 Stationary phase methods in multidimensions 416 C.2 Some useful facts about operators and bilinear forms 421 Problem 424 References 424 Appendix D Heisenberg–Weyl group and the theory of operator symbols 426 D.1 Introductory comments 426 D.2 Groups, group algebras, and convolutions on groups 427 D.3 Linear representations of groups 430 D.3.1 Lie groups and Lie algebras 434 D.4 Finite representations of Heisenberg–Weyl 436 D.4.1 The translation group on n points 436 D.4.2 The finite Heisenberg–Weyl group 439 D.5 Continuous representations 442 D.6 The regular representation 445 D.7 The primary representation 445 D.8 Reduction to the Schrödinger representation 446 D.8.1 Reduction via a projection operator 446 D.8.2 Reduction via restriction to an invariant subspace 446 D.9 The Weyl symbol calculus 447 References 452
- 18. xii Contents Appendix E Canonical transformations and metaplectic transforms 453 E.1 Examples 453 E.2 Two-dimensional phase spaces 457 E.2.1 General canonical transformations 457 E.2.2 Metaplectic transforms 459 E.3 Multiple dimensions 466 E.3.1 Canonical transformations 466 E.3.2 Lagrange manifolds 467 E.3.3 Metaplectic transforms 469 E.4 Canonical coordinates for the 2 × 2 normal form 471 References 476 Appendix F Normal forms 479 F.1 The normal form concept 479 F.2 The normal form for quadratic ray Hamiltonians 482 F.3 The normal form for 2 × 2 vector wave equations 488 F.3.1 The Braam–Duistermaat normal forms 497 F.3.2 The general case 497 References 499 Appendix G General solutions for multidimensional conversion 500 G.1 Introductory comments 500 G.2 Summary of the basis functions used 500 G.3 General solutions 504 G.4 Matching to incoming and outgoing fields 506 Reference 510 Glossary of mathematical symbols 511 Author index 514 Subject index 517
- 19. Preface Waves exist in a great variety of media (in all phases of matter) as well as a vacuum (in the case of electromagnetic waves). All simple waves, on the one hand, share some basic properties such as amplitude, frequency and period, wavelength, and wave velocity (both phase and group velocity). Waves in a turbulent medium, or waves generated by random sources, on the other hand, are more appropriately described in terms of probability distributions of amplitude, and spectral densities in frequency and wavelength. In this book, we focus primarily upon coherent waves that are locally plane wave in character. That is, they have a well-defined amplitude, phase, and polarization at most (but not all) points. The regions where this local plane-wave approximation breaks down are important, and the development of appropriate local methods to deal with them is an important topic of the book. We include a very short discussion of phase space approaches for incoherent waves, for completeness. This is the first book to present modern ray-tracing theory and its application in plasma physics. The emphasis is on methods and concepts that are generally applicable, including methods for visualizing ray families in higher dimensions. A self-contained exposition is given of the mathematical foundations of ray-tracing theory for vector wave equations, based upon the Weyl theory of operator symbols. Variational principles are used throughout. These provide a means to derive a Lorentz-covariant ray theory, along with related conservation laws for energy, momentum, and wave action. Phase space variational principles are also used to provide a unified treatment of caustics, tunneling, mode conversion, and gyro- resonant wave–particle interactions. Many examples are presented to show the power of these ideas to simplify and solve problems in plasma wave theory. Each chapter ends with a set of problems that allows the reader to explore the topics in more depth. The major theme of the book concerns the use of phase space methods. Orig- inally developed by Hamilton for the study of optics, these methods became a xiii
- 20. xiv Preface familiar tool in the study of classical particle motion, and they are part of the standard toolkit for any physicist. The use of phase space methods in the study of plasma waves and the Weyl symbol calculus – and the underlying group the- oretical and geometrical ideas these methods are based upon – are more recent developments that are less well-known in the plasma physics community. The theory of short-wavelength asymptotics has advanced significantly since the 1960s. There is now a large literature on the topic in mathematics and certain subfields of physics, such as atomic, molecular, and optical (AMO) physics, and nuclear physics. This revolution in understanding has produced a relatively minor impact upon ray tracing as practiced by most plasma physicists. There are several reasons for this. First, the modern theory of short-wavelength asymptotics (which is called “semi- classical analysis” in the AMO literature) is couched in mathematical terms that are unfamiliar to scientists who are trained in traditional approaches to plasma wave theory. Traditional approaches tend to emphasize the particular, rather than the general. There is a large emphasis placed upon naming the multitudes of modes, and their classification. These ideas are important, but they can overwhelm the student and the researcher with details and obscure the underlying universal principles. Most students are introduced to plasma wave theory in the uniform setting where Fourier methods apply. They are presented with a survey of the various types of plasma waves. They then quickly skip over how Fourier methods must be modified in a nonuniform plasma. If they are lucky, they are given a superficial introduction to a form of ray-tracing theory that is one-dimensional and largely of nineteenth- century vintage. As a result, most plasma physicists are completely unaware of the revolution that has occurred in ray-tracing theory, and they are therefore poorly prepared to apply it to their own area of work. Second, if students explore ray tracing in the mathematics literature, they will find that there is a relatively limited range of examples studied, such as the single- particle Schrödinger equation, or the wave equation with a spatially dependent wave speed. Plasma wave equations have a much richer variety. Plasma wave equations include phenomena that the mathematical literature overlooks, such as wave–particle resonance, gyroresonance, and finite-temperature effects. Dissipa- tion is often ignored in the mathematical literature, as is wave emission, matching to boundary conditions, and Lorentz covariance. All of these topics are important in plasma applications, and all are covered in this book. Third, instead of introducing students to these important theoretical ideas in plasma wave theory, there has been a growing emphasis on teaching full-wave simulation methods. While computational tools for the study of waves are very important (and will continue to grow in importance), the lack of coverage of mod- ern eikonal theory leaves many students and researchers without a firm grasp of
- 21. Preface xv its use in wave problems. Ray-tracing codes, which do not include the modern improvements mentioned above, are commonly used for “quick-and-dirty” calcu- lations by experimentalists. This is done for the simple reason that following rays provides insight and promotes physical intuition in practical situations. But by following a collection of rays independently, as is commonly done in such calcu- lations, one is really treating the wave field as incoherent without regard to proper matching to boundary values. This also makes it impossible to correctly compute the wave amplitude. A coherent wave field, properly matched to given boundary values, must be synthesized using a family of rays that have a well-defined set of properties. With some effort (for example by properly dealing with mode conver- sion), the calculation can be kept “quick” but “cleaned up” so it can be used to accurately compute the phase, amplitude, and polarization along each ray, thereby providing the possibility for a full construction of the wave field. A major focus of the book is the investigation of the processes by which waves, or waves and particles, interact with each other, so that they may exchange energy, momentum, and wave action. These basic processes arise in various settings, and a unified treatment is possible. Examples include mode conversion, wave absorption, and emission by resonant particles in nonuniform plasma. In order for such conver- sion to take place, certain resonance conditions involving the participating waves, or waves and particles, must be met. These conditions are described in terms of the dynamics of Hamiltonian orbits in ray phase space, and particle orbits in particle phase space. While the focus will be on cold-plasma models for pedagogic pur- poses, we also include a discussion of finite-temperature effects for completeness (Chapter 7). The effects of finite temperature are studied using Case–van Kampen methods, with the theory adapted to the ray phase space setting. A reader already well-versed in plasma physics should find the examples very familiar, though the method of analysis pursued in later chapters is likely to be new, as well as our emphasis on the use of variational principles. A brief derivation of the cold-plasma fluid model is included in Appendix A for those readers who are not plasma physicists. More detailed discussions of the physical assumptions underlying the models can be found in the references cited. Chapter 1 begins with some introductory comments and examples. A brief historical survey is presented to provide context and to highlight some of the more recent developments in ray tracing. The historical survey here is highly selective, and the narrative provides a first introduction to topics such as the use of variational principles to derive wave equations and their conservation laws, Hamilton’s ray theory, Weyl’s theory of operator symbols, and the much more recent use of these foundational ideas in plasma theory. A brief introduction to eikonal theory for a scalar wave equation is then presented, starting with a quick review of waves in uniform plasma, Fourier methods, and the concepts of phase velocity, group
- 22. xvi Preface velocity, dispersion, and diffraction. Eikonal methods are then introduced in order to study waves in nonuniform plasma. The treatment here is very traditional, using asymptotic series and brute force calculation. In Chapter 2, we introduce two important tools that will be used throughout the later chapters: variational methods for wave equations and the Weyl symbol calculus. The use of a variational principle allows a unified treatment of later topics, and provides an elegant derivation of the wave-action conservation law using Noether’s theorem. (Further discussion of variational methods and Noether’s theorem is provided in Appendix B.) The Weyl theory of operator symbols underlies everything done later in the book.1 This theory was first developed in the context of quantum mechanics, but the methods are completely general and they provide a systematic means for deriving wave equations that are local in both x and k. These methods are needed for dealing with caustics, tunneling, and mode conversion, all of which involve a breakdown of the eikonal approximation that is local in ray phase space. In Chapter 3, we begin our discussion of eikonal theory for vector wave equations in earnest. In this chapter, we transition by stages from the more familiar x-space discussion of eikonal theory to a covariant phase space treatment. In many settings, the covariant formulation is not needed, and we strive to keep things as concrete as possible. Therefore, in this book we tend to use a preferred laboratory frame and work with physical fields (e.g. E(x, t) and B(x, t)), rather than allowing arbitrary frames and using four-vector notation and the vector potential Aμ(x). However, covariant formulations are needed in some astrophysical and space plasma appli- cations, so we include them even if they are not a major focus of the book. The conservation of energy, momentum, and wave action is covered in this chapter. Chapter 4 discusses visualization and wave-field construction. The modern the- ory of ray tracing is a geometrical theory. Geometrical theories appeal to the visual intuition. A visual picture can help guide us through a calculational thicket by providing a map. Mixing metaphors, the goal of this chapter is to help bring visual intuition onto the battlefield as a valued ally along with the more analytical methods of calculation. This chapter also provides examples of the construction of wave fields from ray-based data. This section connects us directly back to the original motivation of eikonal theory: to find solutions of wave equations. The next three chapters concern situations where the eikonal approximation breaks down in various ways.2 1 The theory of operator symbols is based upon the representation theory for the Heisenberg–Weyl group. A short, self-contained, introduction to this topic is provided in Appendix D. 2 By an eikonal solution, we mean a wave field that has well-defined phase, amplitude, and polarization: ψ(x, t) = A(x, t) exp[iθ(x, t)]ê(x, t).
- 23. Preface xvii Chapter 5 presents the phase space theory of caustics. These involve a breakdown of the eikonal approximation in x-space, but not in k-space. Hence, caustics are dealt with by moving between the x- and k-representations, solving for the local wave behavior near the caustic in the k-representation, then matching to the incoming and outgoing fields in the x-representation. This strategy works because of certain fundamental results from the theory of phase integrals; hence, we provide an extensive discussion of stationary phase methods in Appendix C. Chapter 5 involves the first encounter with what are called normal form methods, which also play an important role in the chapters that follow.3 Chapter 6 treats tunneling and mode conversion. These phenomena are due to a resonance in ray phase space. In the case of mode conversion, two distinct types of eikonal waves, associated with two polarizations, have dispersion functions that are nearly degenerate, meaning that – for a given wave frequency ω – at some point x these two wave types have nearly equal values of k. This causes a breakdown in the eikonal approximation, which is not valid in any representation near the mode conversion point. Weyl methods are used, in tandem with variational principles, to derive the appropriate local wave equation, which is a 2 × 2 vector wave equation involving the polarizations of the two wave types undergoing conversion.4 This local wave equation is then solved and matched to incoming and outgoing eikonal fields.5 Chapter 7 discusses the phase space theory of gyroresonant wave conversion. This phenomenon presents a significant challenge for full-wave simulation because of the wide range of spatial scales involved. The inclusion of finite-temperature effects makes the problem even more challenging numerically. A phase space theory, however, allows us to treat the problem in modular fashion, and to use matched asymptotics to construct a complete solution throughout the resonance region. The calculation is the most technical in the book, but self-contained, and it illustrates the power of phase space ideas. Examples are scattered throughout the book. They are drawn from a wide range of applications in plasma physics and beyond. The phase space theory of Budden- type resonances is covered in great detail at various places in the book, in one dimension and tokamak geometry, for cold and warm plasma. The ray-tracing algo- rithm RAYCON is described. This is the first ray-tracing algorithm that can deal with ray splitting due to mode conversion in tokamak geometry. In addition, mode con- version in magnetohelioseismology and equatorial ocean waves is briefly covered 3 In addition to the discussion of normal forms for phase integrals in Appendix C, a general introduction to those aspects of the theory of normal forms needed elsewhere in this book is provided in Appendix F. 4 Tunneling involves only one polarization, and hence can be reduced to a scalar problem. 5 The normal form theory for the local 2 × 2 wave equation is presented in Appendix F, and the general solution of the local wave equation is presented in Appendix G.
- 24. xviii Preface to illustrate the wide applicability of the methods. Many of the more technical details, and additional mathematical topics, have been relegated to an extended set of appendices. The power of the phase space viewpoint will become apparent as these various topics are developed. The discussions in the main part of the book are kept relatively brief, with an emphasis on the concepts. To avoid bogging down the discussion, many technical details are either presented in the appendices, or developed as exercises for the reader. An extensive list of citations is provided for readers who wish to learn even more. Those who are new to these ideas are strongly encouraged to attempt the problems, as the only way to learn is by doing. The many figures provided in the text are a key part of the discussion; they help to develop geometrical intuition. In particular, the MATLAB code RAYCON – which was used to generate the figures for ray tracing in tokamaks in Section 6.6 – is available online as a supplement to the text, and the reader is encouraged to use this code as well. Our goal is to make the material accessible and useful to a wide audience. We have written the material for graduate students in plasma physics and related fields, which should also make it accessible to researchers in these fields as well. We assume that the reader is mathematically sophisticated, but that the primary interest of the reader lies in understanding how to apply these methods to real physical problems. Comments and suggestions are welcome.
- 25. Acknowledgements Gene Tracy would like to start by noting David Hume’s belief that two of the purest pleasures in life are study and society. This book reflects my desire to acknowledge my gratitude to Allan Kaufman for his friendship in both physics research and scientific conversation over two decades, along with my other co-authors, Alain Brizard and Steve Richardson. The mention here is kept short, because the book is one long acknowledgement of the debt I owe to Allan. Andre Jaun also deserves mention as valued member of our collaboration, and the godfather of RAYCON. I would also very much like to thank Louise Kaufman for her wonderful dinners, and for being her wonderful self, but she is no longer with us. I had hoped to finish this book in time for her to see it, but that was not to be so. She will be greatly missed. Also, thanks to Robert Littlejohn, Wulf Kunkel, and the many others at UC Berkeley and LBL who made me feel welcome on my many visits, shared their offices and thoughts – and the desk of EO Lawrence – along with great coffee and spectacular views. I would like to thank Phil Morrison, Jim Hanson, and John Finn, for their support and encouragement, for sharing my curiosity about classical physics in its many forms, both linear and nonlinear, and for coming to all those poster presentations over the years, along with my friend Parvez Guzdar. I would like to thank Hsing Hen Chen for introducing me to the pleasures of mathematical physics, and Alex Dragt for inspiring my fascination with phase space and geometry. Thanks to Nahum Zobin for sharing his love of mathematics and for helping to bring clarity to things that had been murky (to me at least). The William and Mary mathematical physics discussion group, led by Nahum for over a decade, helped to stimulate new thinking for me on many of the topics in this book. I would also like to thank my colleague John Delos for his crystal-clear lectures, and for sharing his enthusiasm about semi-classical methods. The College of William and Mary has been generous with its support through research leaves during tight budget times, and the physics department always xix
- 26. xx Acknowledgements provided a collegial environment in which to work. I would also like to acknowl- edge the support of the American taxpayer through the NSF–DOE Partnership in Basic Plasma Physics, and the US-DOE Office of Fusion Energy Sciences, which provided much-needed funding during these many years of effort. This book is a result of modest – but sustained – funding over many years that allowed me to spend time in depth, and provided support for a series of Ph.D. students who shared the particular journey summarized in this book, or a journey down another scientific path (Tim Williams, Jay Larson, Alastair Neil, Yuri Krasniak, George Andrews, Chris Kulp, Haijian Chen, Steve Richardson, Karl Kuschner, Yanli Xiao, and Dave Johnston). I would like to thank Joan and Doug Workman for their generosity in lending the house in Flat Rock, North Carolina, where several key chapters were written while I was snowed in one January. Also, to my good friends Bill Cooke and Dennis Manos: thanks for tolerating my absence over too many months on too many fronts, and for their constant encouragement to finish this project. I want to also thank my good friends Kelly Joyce, Teresa Longo, Leisa Meyer, Steve Otto, and Silvia Tandeciarz, for their support and for showing me that when one door closes another can open. And thanks to Suzanne Raitt, for the breakfast club, and for her caution about the danger of writing an acknowledgement that doesn’t say enough. I would also like to thank my book club companions for their contribution to my sanity, and moral support when I needed it most: Bill Cooke and Robin Cantor-Cooke, Deborah Denenholz Morse and Charlie Morse, Arthur Knight and Martha Howard, Henry and Sarah Krakauer. And finally, I would like to thank my daughter Kathryn, and my wife, Maureen. They put up with my tendency toward abstracted behavior while composing text in my head – or grumpiness while stuck on a minus sign somewhere – for too long. I know they are just as glad as I am that this project is complete. I want to especially thank Maureen for her masterful help with many of the figures, and for giving up so much of her time to help us finish this project. And I would like to thank her for her love and support over many years, but words are not up to the task. This whole adventure would never have happened without her. Alain Brizard would like to thank Allan Kaufman, for more than twenty-five years of inspiration and friendship and Gene Tracy, Phil Morrison, and Robert Littlejohn, for sharing my love of theoretical plasma physics. And Carl Oberman, John Krommes, and Ralph Lewis, for supporting my earliest efforts as a graduate student in developing and applying Lagrangian and Hamiltonian methods in plasma physics. Lastly, I would like to acknowledge Tom Stix for his important role in my professional life (for allowing me to become a graduate student at the Princeton
- 27. Acknowledgements xxi Plasma Physics Laboratory and for teaching me about plasma waves) and my personal life (for allowing me to meet the love of my life, Dinah Larsen). Steve Richardson would first like to thank his teacher, mentor, and friend Gene Tracy. The patience and encouragement shown in answering my many – and often repeated – questions has not only taught me much about physics but is also an inspiration to me in my own research. I would also like to thank Nahum Zobin for his enthusiasm in helping us physi- cists understand the mathematics of group representation theory and its relation to the Weyl symbol calculus. I would also like to acknowledge the DoE Fusion Energy Postdoctoral Research Program for its support, and Paul Bonoli, who mentored me during my time at the Plasma Science and Fusion Center. I’d also like to thank John Finn for the hours spent discussing plasma physics, and for his hospitality and friendship. I learned so much from him in such a short time while in Los Alamos. I would also like to acknowledge some of the many great software tools that were used in this project. These include MATLAB, TeXShop, BibDesk, the PGF/TikZ and PGF- PLOTS packages, Inkscape, and Veusz. These programs represent hours of hard work put in by often unpaid volunteers, and the tools they created make our jobs easier. Thank you Katie for your help while I worked. And Ann, thank you for your love and support, and for your good humor about my nerdiness. I love you both. Allan Kaufman There are very many wonderful people for me to thank, who interacted with me over the years. I had the great pleasure of acknowledging their contributions, in a memoir which Gene and Alain invited me to write several years ago, in connection with a symposium (Kaufmanfest 2007), which they had very kindly organized to celebrate my eightieth birthday. The interested reader can find this memoir, entitled “A half-century in plasma physics,” published in IOP Journal of Physics: Conference Series 169 (2009) 012002. The editorial staff at Cambridge have been superb. They have been patient, supportive, and exceedingly professional in their work. Over the four years it took to write this book, the text grew to nearly double the original planned length, and the number of figures went from a few dozen to over a hundred. We would especially like to thank those members of the editorial team who have worked these last few months to bring the book to its final form: Anne Rix, Jessica Murphy, Elizabeth Horne, and Simon Capelin. Along the way we also had the pleasure to work with Zoe Pruce, Fiona Saunders, and Antoaneta Ouzounova. Finally, we would like to express our gratitude to John Fowler for his early encouragement to write this book. Without his invitation to submit a proposal, we would not have started down the path that led us to this place.
- 29. 1 Introduction The science of optics, like every other physical science, has two different directions of progress, which have been called the ascending and the descending scale, the inductive and the deductive method, the way of analysis and of synthesis. In every physical science, we must ascend from facts to laws, by the way of induction and analysis; and must descend from laws to consequences, by the deductive and synthetic way. We must gather and groupe appearances, until the scientific imagination discerns their hidden law, and unity arises from variety: and then from unity must re-deduce variety, and force the discovered law to utter its revelations of the future. William Rowan Hamilton (1805–1865)1 It is a fact of immediate importance to our everyday experience that light nearly always travels in straight lines from the source to our eyes, perhaps scattering off some object along the way.2 Without the ability to assume this as a fact about the world around us, our extraordinary talent for instinctively comprehending spatial relationships in everyday life would be severely compromised. Consider how much computer power must be expended to disentangle the multiple images of distant galaxies3 to map the dark matter distribution in the visible universe [MRE+ 07]. Imagine what life would be like if we had to do similar mental computations just to navigate around the furniture in our living room.4 1 From “On a general method of expressing the paths of light, and of the planets, by the coefficients of a characteristic function,” by WR Hamilton (1833) [Ham33]. 2 That rays travel in straight lines was fully appreciated by the ancient world. See, for example, Euclid’s Optics (ca. 300 BCE) [Bur45] which begins with the statement: “Let it be assumed that lines drawn directly from the eye pass through a space of great extent.” (emphasis added) From this simple insight, Euclid lays the groundwork for geometrical optics and projective geometry, which form the basis for modern fields like computer animation. See Darrigol [Dar12] for a recent survey of the history of optics from antiquity to the nineteenth century. 3 Using Einstein’s theory of gravitational lensing. 4 This leads to many other interesting questions, such as: What types of spatial imagery do animals possess that evolved in the dark, or in murky environments such as muddy water? Some, like bats and dolphins, use echolocation, which in many cases can provide a spatial image since ultrasound waves locally travel in straight lines, too. But what type of spatial imagery do animals possess that live largely by sense of smell, such as ants? 1
- 30. 2 Introduction How do we build upon this insight that light nearly always travels in straight lines in order to develop a theory with predictive power? More importantly, how can we develop a theory that can encompass those cases where light does not travel in straight lines? And, looking further, can we develop a theory that can be extended to other types of waves in other settings? In the following sections, we selectively discuss some foundational concepts that we will find useful for the rest of the book. This is not a historical survey of the development of ray tracing. The history of optics and wave theory is vast, and the story too complex, for us to do more than touch upon the most relevant highlights to begin stocking our toolkit. Some suggestions for further reading are given along the way for readers who want more detail. 1.1 Fermat’s principle of stationary time 1.1.1 General comments A major unifying theme of this book concerns the power of variational principles. These have a venerable history. In the first century CE, the mathematician and inventor Heron of Alexandria posed, and solved, the following problem in planar geometry: Given a line and two points not on the line, what is the shortest path between the two points that touches the line? If the points lie on opposite sides of the line, then the path connecting them crosses the line and the shortest path is simply a straight line. However, if they lie on the same side of the line, then Heron proved (without the use of calculus!) that the shortest path obeys the law of reflection. This basic principle of optics was therefore known in antiquity, and it was known to satisfy a minimization principle. Given that light travels in straight lines, and that rays obey the law of reflection, why invoke a least-time principle? Because the path of least time and the path of shortest length are only equivalent if the speed of light is constant along the path. Feynman famously pointed out that lifeguards must solve for the least-time path every time they rescue a swimmer: they must determine how far to run along the beach before they dive into the water, where their speed of propagation drops dramatically. If they are good at their job, their path obeys Snell’s Law, as we’ll discuss.5 Let’s start with the original form of Fermat’s principle of least time, and improve it as we pursue the implications. The least-time principle asserts that of all possible paths light might take from the source to the point of observation, it will take the path that requires the least time. We will adopt the convention that the actual paths light follows are called rays, to distinguish them from all the possible paths we 5 More recently, ants have been found to follow the least-time path as well. See Oettler et al. [OSZ+13] .
- 31. 1.1 Fermat’s principle of stationary time 3 Figure 1.1 The point source S and the point of observation O can be connected by an infinite number of paths, γ . Fermat’s principle of least time states that the light ray follows that particular path between S and O for which the time taken is minimal. might imagine. We should emphasize, of course, that the source will in general emit light in many different directions. The principle relates to that particular ray which travels from the source point to the observation point and does not concern itself with the rest, although in principle the observation point is arbitrary. We will return to this important issue in a moment. Fermat used the least-time principle around 1657 to derive what we now call Snell’s Law, which we will discuss in a moment. Here we note that, although Galileo famously described an attempt to measure the speed of light by flashing lanterns about a mile apart in the 1630s, the finiteness of the speed of light was not firmly established until 1676. The astronomer Roemer had detected a regular variation over some years in the timing of the observed eclipses of Io, a moon of Jupiter, relative to the predicted times. The slight advance or retardation of the times for Io to disappear and reappear behind Jupiter depended upon whether Earth and Jupiter were on the same or opposite sides of the Sun. Roemer correctly attributed this apparent change to the finite speed of light, which led to the estimate c ≈ 140,000 miles per second (see Hockey et al. [Sch07]). 1.1.2 Uniform media Start with the simplest case, where light has the constant speed c and no obstacles are present. We assume the light is emitted by a point source at S, and the observer is at the point O (see Figure 1.1). The travel time from S to O is T [γ ] = L[γ ] c , (1.1) where L[γ ] is the length of the path γ from source to observer. It is blindingly obvious6 to everyone but theoretical physicists and mathematicians that the shortest path between two points is a straight line. But, of course, it is a worthwhile exercise in variational principles to prove it (see Problem 1.1). 6 Pun intended.
- 32. 4 Introduction 1.1.3 Snell’s Law Thus, the least-time path and the shortest-length path are the same in the simplest case where the light speed is constant. But the least-time principle leads to some- thing new: Suppose S and O lie in two different regions. The source lies in region 1, where the speed of light is c/n1. The observation point is in region 2, where the speed of light is c/n2.7 We leave it as an exercise for the reader to prove that the least-time principle in this case leads to what we now call Snell’s Law8 for the bending of rays at the interface n1 sin θ1 = n2 sin θ2. (1.2) An approximate form of Snell’s Law had been established by Ptolemy (ca. CE 90–168), in terms of the ratios of the angles. This is only correct for rays that are nearly perpendicular to the interface. It appears that the correct form involving the sines of the angles was discovered by the Persian astronomer-mathematician Ibn Sahl around the year 984 CE (see Hockey et al. [Ber07]), though this form was not known in the West until the early seventeenth century, when the invention of the telescope (1608) would have motivated the development of improved theories and measurements of light refraction for lens design.9 Snell’s Law was established empirically, of course, and there are multiple claims to primacy in the literature of the early mid 1600s. What is certain is that Fermat was aware of the law of refraction (1.2), and showed that his principle of least time could be used to derive it. The combination of straight-line rays within uniform regions, and Snell’s Law at the interface between regions, forms the basis for ray optics and lens theory. Kepler was the first to provide a theoretical explanation for the telescope, using ray theory for compound lens systems. This theory first appeared in Kepler’s Dioptrice (1611).10 Observation shows that the refractive index depends upon the color of light. When this effect is included, the theory of prisms emerges and – through the possibility of a double internal reflection within raindrops – the theory of the rainbow.11 Thus, many of the design principles for microscopes, telescopes, 7 The constants n1 and n2 are the refractive indices; θ1 and θ2 are the angles formed by rays in the two regions and the local normal at the interface. See Figure 3.4, Section 3.5. 8 See Problem 1.2. 9 See Willach [Wil08] for an interesting account of the evolving crafts of glass and lens manufacture in the late Middle Ages and how these contributed to the invention of the telescope. Going even further back, for those interested in the history of lenses in the classical world, a survey of what is known about the use of lenses in the Graeco-Roman world can be found in Plantzos [Pla97]. We thank our colleague Professor Lily Panoussi for bringing this reference to our attention. 10 This book, written in Latin, has not yet been translated into English, although high-quality scans of the original are available online. It is interesting to note that the first figure to appear in the text concerns a means to measure the refractive properties of materials. 11 Descartes discussed what is now considered the correct geometrical explanation in his 1637 Discourse on Method. Earlier scholars in China and the Middle East also realized the importance of internal reflection for explaining the rainbow by using glass spheres as laboratory models of raindrops.
- 33. 1.1 Fermat’s principle of stationary time 5 cameras, eyeglasses, and the explanation of some of the most beautiful of atmo- spheric phenomena, follow from the principle of least time. Isaac Newton (1643–1727) made important contributions to the theory of optics and revolutionized the design of telescopes using methods based upon the assump- tion that light was composed of particles, supplemented by the notion that the speed depended upon their color [New10]. The ray theory of light fits nicely with this hypothesis, and the success of the predictions using a ray theory seemed to confirm the particle hypothesis. (However, Fermat and the least-time principle are not even mentioned in Newton’s Optics.) It was not until the work of Thomas Young (1773–1829) and Fresnel (1788– 1827) that light was shown convincingly to be a wave phenomenon, capable of diffraction and interference like water waves.12 This built upon the much earlier work of Huygens (1629–1695).13 This then leads to a puzzle: if light is a wave, why does ray theory work so well? We will see in later sections that Hamilton provided an answer to this question by showing how to construct wave fields, including interference patterns, using an entirely new type of ray theory. 1.1.4 Distributed sources We should point out that nonpoint sources are dealt with at this stage by simple superposition. Each point on the extended source S is treated as a point source independent of the others. This is easy to understand if light is composed of particles, but it is a more subtle issue if light is a wave. Use of the superposition assumption leads to the theory of imaging optics. This simple approach to the analysis of distributed sources is valid only if the light emission is incoherent from one point on the source to the next. Speaking imprecisely for the moment, by a coherent source we mean one whose rays have a well-defined phase θ at almost all points along each ray, and that this phase function is smoothly varying along the ray and between neighboring rays. Coherent wave fields are the primary topic of this book, though we will return to incoherent fields in Section 3.5.5, where we summarize a ray phase space theory for them. We note here that the incoherence of visible light in everyday life is almost always a good assumption.14 What is lacking in the theory so far, of course, is that we have not discussed how to compute the light intensity. A ray could arrive at the observation point with zero intensity, in which case the existence of the ray itself is largely meaningless. 12 Young was able to produce coherent light by using a pinhole smaller than the transverse coherence length of sunlight. The coherence length of an extended incoherent source is the wavelength divided by the solid angle of the source [Wol07]. 13 See, for example, Fresnel’s essay in [Fre00]. 14 The reader should verify that the previous statement is correct. When, outside of a physics laboratory, do you encounter coherent visible light? What physical conditions are required to produce it?
- 34. 6 Introduction Figure 1.2 The point source S and the point of observation O can no longer be connected by a straight-line path, but the paths must instead go around the obstacle B. A mirror M is present and we consider only paths that reflect from the mirror. The least-time path is the light ray that obeys the law of reflection. It is an observational fact that the convergence of rays increases the intensity of light. Think of the focusing of sunlight using a lens.15 Likewise, as rays diverge, the intensity decreases. It is physically reasonable to conjecture that a conservation law applies, for example one modeled on treating the energy density in the light field as a conserved fluid. If light were composed of discrete and long-lived particles, the conservation law would follow directly from particle number conservation. But if light is a wave, the derivation is less obvious. This will be discussed at length later in the book, where we discuss wave energy and wave momentum (Section 3.3.2), and we derive action conservation laws for wave fields, using both coherent (Section 3.1) and incoherent (Section 3.5.5) formulations. 1.1.5 Stationarity vs. minimization: the law of reflection Return to a point source S and a fixed point of observation O, but now block the direct straight-line ray path and add a plane mirror (see Figure 1.2). This will turn out to be a situation where the naive formulation of the principle of least time will fail us, but it will help guide us to a better formulation. If we consider only paths that pass from S to O after reflection from M, the path of least time reflects from M in such a way that the angle of incidence equals the angle of reflection (as measured 15 The first mention of the use of lenses for burning and cauterizing for medicinal purposes is believed to be Aristophanes, The Clouds, first performed ca. 423 BCE. The reference occurs in lines 767–769, see p. 64 of [Ari12]. The use of mirrors to focus light for the same purpose was also understood, as evidenced by the famous legend of Archimedes setting fire to the Roman fleet at Syracuse, ca. 214–212 BCE.
- 35. 1.1 Fermat’s principle of stationary time 7 Figure 1.3 The point source S and the point of observation O separated by the obstacle B, but now we consider all paths. The sequence of paths shown that do not reflect from the mirror all have travel times less than the ray. with respect to the local normal at the point of reflection).16 This is simply Heron’s problem, mentioned earlier. With the law of reflection, a host of new applications emerge: optical instruments such as reflecting telescopes (building upon Newton’s original breakthrough design) can be analyzed. But why should we restrict ourselves to paths that reflect from M? Why can’t we proceed as before and consider all paths that go from S to O? In that case, it should be clear that some of the paths shown in Figure 1.3 have travel times less than the ray that reflects from M. In fact, of the sequence of paths shown, the one with the shortest travel time is the one that travels in a straight line from S to the edge of B, and then on to O. Why don’t we use that path as our least-time ray and ignore the path that reflects from M?17 Suppose we now remove the obstacle. The straight-line path is once again the least-time path, but we also know from experience that the ray that reflects off the mirror will also reach the point O. A helpful way to view what is going on – and one that is quite physical – is to think of the source S as emitting rays that travel in straight lines in all possible directions. If they encounter the mirror M, they bounce and satisfy the law of reflection. Almost all of the infinitude of rays emitted from S will miss the point O. There are two that make it to O: the straight-line path and the one that bounces off the mirror and satisfies the law of reflection along the way. 16 See Problem 1.3. 17 There is good reason to consider the path that bends around the edge of B as a ray, but it requires careful treatment at the edge where it encounters the obstacle. In fact, the light can diffract around the edge if it is sharp enough, so some light could reach O from S by this route. But this takes us beyond a simple ray picture. However, we note that the Huygens–Fresnel theory of wave propagation and diffraction starts with such considerations. See Fresnel [Fre00].
- 36. 8 Introduction Figure 1.4 The point source S is now shown emitting rays in all directions, and no obstacle lies between S and O. In this case, it is clear that two ray paths make it from S to O, and that we should understand Fermat’s principle not as a global minimization principle, but as a local stationarity principle with respect to neighboring paths. Figure 1.5 The law of reflection from a mirror surface arises by selecting the shortest path for the flat mirror (left), but the longest for the curved one (right). The straight-line path is the global winner in the race, but the second path is locally minimal when compared to neighboring paths that also reflect off the mirror. With these results in mind, a better formulation of Fermat’s principle is to define rays as any path for which the travel time is stationary with respect to small variations, and accept that there will sometimes be more than one ray that travels from S to O, rather than insisting upon a unique global minimum. For rays encountering mirrors, the variation is carried out only among neighboring paths that also reflect; hence it is an example of a constrained variation in the family of paths, not a general variation. Only by using a constrained variation at a mirror do we recover the expected physical result. As another example which demonstrates that the least-time principle is inade- quate, consider the flat and curved mirrors in Figure 1.5. In Problem 1.4 it is shown
- 37. 1.1 Fermat’s principle of stationary time 9 that the ray is the shortest path (among those that reflect from the mirror) for the flat mirror, but the longest straight-line path that reflects once off the curved mirror shown to the right in the figure. This can be seen by starting with the straight-line path (the ray) from S to O, and then considering a nearby path that reflects once just before it reaches O. This path must have longer flight time than the straight-line path. Now move the reflection point away from O toward S. When the reflection point is in the neighborhood of S, the straight-line path becomes minimal once again. Therefore, the flight time must have reached a maximum in between. These two examples once again emphasize that we should formulate Fermat’s principle as a principle of stationary time. Before leaving the topic of reflection, we should mention that the specular reflection we have described here is typical of highly polished surfaces. Rough surfaces (rough on the length scale of a wavelength of light) can lead to more diffuse types of reflection, first characterized by Lambert (1728–1777). This type of reflection is, in fact, more common for everyday surfaces, and diffusion models for reflection by textured surfaces are commonly used in computer graphics.18 1.1.6 Smoothly varying media Thus, Fermat’s principle, now properly understood as a stationarity principle rather than a minimization principle, unifies many important results that had appeared to be distinct and brings them under one theory. It is natural, then, to extend Fermat’s theory to situations where the refractive index varies continuously. Let’s consider an important special case of a two-dimensional layered medium. Suppose we have a two-dimensional system that is uniform in the x-direction, but the refractive index varies in y: n(y). Our source lies at x = 0, and could be extended in y (for example, it could be a building or a mountain). The point of observation is at x1 and y = y1. Draw a path, γ = y : x → y(x) from a point on the source to the point of observation r(x) = [x, y(x)], r(0) = (0, y0), r(x1) = (x1, y1). (1.3) The time required for light to travel along the path is given by the integral T [y] = 1 c x1 0 n[y(x)] 1 + [y(x)]2 dx, y (x) ≡ dy dx . (1.4) 18 Rayleigh scattering should also be mentioned because it also involves a breakdown in the simple law of reflection. Rayleigh scattering involves the scattering of light by particles that are smaller than a wavelength. This type of scattering shows a strong frequency dependence and explains why the sky is blue; see for example Jackson [Jac98].
- 38. 10 Introduction Here, we use the notation T [y] to denote the fact that the elapsed time is a functional of the path y(x).19 Requiring T [y] to be stationary with respect to small variations in the path leads to the Euler–Fermat equation 1 + [y(x)]2 dn dy − d dx n y 1 + [y(x)]2 = 0, (1.5) which can be reorganized as y (x) = 1 + [y (x)]2 d ln n(y) dy . (1.6) Hence, in a uniform medium (n = 0), we recover the previous result that the light paths are straight lines: y (x) = 0. In a nonuniform medium (n = 0), however, the light paths curve toward the region of higher refractive index: concave upward [y (x) 0] for n (y) 0 and concave downward [y (x) 0] for n (y) 0. Fur- ther aspects of the continuous case, including a derivation of Snell’s Law for a continuous layered medium, the trapping of waves in channels, and mirages are examined in Problems 1.5 through 1.8. In Problem 1.9 the general three- dimensional case is examined. It is important to emphasize once more that this type of ray theory depends only upon the refractive index n(x), and implicitly assumes the waves are incoherent (due to the lack of any reference to a phase function). There is no dispersion relation between the wave frequency and wavevector, because there is no wavevector in Fermat’s theory.20 The “wave equation” was unknown to Fermat and contem- poraries. All we need to know to apply Fermat’s theory is the wave speed, v(x) = c/n(x), so we can compute the travel time along any path. Ray tracing of this sort can also be applied to other types of waves. For example, computer aided tomography (CAT) reconstructs “images” by measuring the atten- uation of X-rays along ray paths, while positron emission tomography (PET) scans map the spatial distribution of the intensity of gamma ray emission. In acoustics, ray theory is important for the theory of reverberation. It is used in the design of concert halls and recording studios, and it forms the basis for ultrasound imaging. A striking example of these ideas is the design of whispering galleries, where sound rays skim along the gently curved wall of a room.21 19 See Appendix B for a discussion of functionals. 20 The wavevector is the gradient of the phase θ(x), as we will discuss in the next section. 21 In the audible range of frequencies (approximately 20 Hz to 20 kHz), and in typical rooms (a few, to a few tens of meters across) the lower frequencies are not well modeled using rays, but the higher frequencies often can be treated as traveling in straight lines, satisfying the law of reflection. This is because the wavelength of sound waves is λ = cs/f and, with a sound speed of ∼ 300 m/s, we have λ ranging from 15 m for the lowest audible frequencies down to 1.5 cm for the highest audible frequencies. The information needed for human speech recognition lies in the mid-range of frequencies.
- 39. 1.2 Hamilton’s principle of stationary phase 11 Viewed through the modern lens of the statistical mechanics of light–matter interactions, we now interpret the refractive index as a quantity that summarizes the average macroscopic outcome of the complex microphysics when electromagnetic fields interact with matter.22 For the multicomponent wave equations encountered in plasma applications, a scalar refractive index is often too simple. Magnetized plasmas are nonisotropic, and kinetic effects lead to wave equations that are integro- differential (hence nonlocal) in space and time. Additionally, in many cases we need to deal with coherent waves; hence we need to introduce the phase. But we will see that the ray concept generalizes nicely to this more complex situation, and that the geometrical picture the ray theory provides will prove useful as well. In particular, we will retain the notion that rays are special paths that satisfy some stationarity principle, but now instead of Fermat’s stationary time principle, we introduce a stationary phase principle that was first used by Hamilton in his studies of optics.23 1.2 Hamilton’s principle of stationary phase So that great ocean of ether which bathes the farthest stars is ever newly stirred by waves that spread and grow, from every source of light, till they move and agitate the whole with their minute vibrations: yet like sounds through air, or waves on water, these multitudinous disturbances make no confusion, but freely mix and cross, while each retains its identity, and keeps the impress of its proper origin. William Rowan Hamilton (1805–1865)24 Fermat’s principle is a geometric variational principle in the geodesic sense; that is, it prescribes the path of light as it travels in a nonuniform medium by extremizing the optical length L ≡ c T . The most important difference between the Fermat and Hamilton approaches is the role of the phase function, θ(x, t). The phase, which has been absent in our discussion until now, will come to play a central role in Hamilton’s theory. We will, of course, exploit complex formulations since they simplify life significantly, so by a “phase” we mean that the light wave field is assumed to be of the form of a rapid variation, characterized by the oscillation 22 For example, for some exotic media the refractive index is negative [SSS01]. This does not imply faster-than- light travel, or causal paradoxes! Instead, it shows that the concept of refractive index has been decoupled from the notion of wave speed and is instead interpreted directly from the geometry of rays and Snell’s Law. 23 N.B. Hamilton’s principle of stationary phase should not be confused with the stationary phase method for evaluating oscillating integrals. Although both arise frequently in discussions of eikonal theory, the former is a principle for deriving dynamical equations in wave mechanics, while the latter is a general method for computing asymptotic limits of certain types of oscillatory integrals. See Appendix C.1.1 for a discussion of the stationary phase method. 24 From “On a general method of expressing the paths of light, and of the planets, by the coefficients of a characteristic function,” by WR Hamilton (1833) [Ham33].
- 40. 12 Introduction Figure 1.6 [Left] A conceptual figure showing a region of three-space. The wave- fronts, θ(x, t0) = const., are assumed to foliate the region, meaning that at each spatial point there is a well-defined phase. Only the level set θ(x, t0) = 0 is shown, for clarity. Also shown is an arbitrary path x(σ), which is assumed to intersect the surfaces of constant-θ transversely. [Right] The level sets θ(x, t0) = 0 and θ(x, t0 + dt) = 0 are shown. The path is held fixed here, but the wavefronts move along the path at each spatial point. exp[iθ(x, t)], multiplying a more slowly varying real amplitude function we will denote as A(x, t).25 It was Hamilton who took the revolutionary step of asking how the phase function, considered as a smooth function of space and time, could be written as the integral of some function along one-dimensional paths (these will become the rays in Hamilton’s theory), and how this function changes as the initial and final points of the rays are varied. This leads to the concept of the phase integral, which is central to the theory. This mathematical breakthrough demonstrated how a ray theory could be used to compute wave patterns.26 In this section on the stationary phase principle, we emphasize a few of the con- ceptual building blocks of Hamilton’s theory. The rest of the book is an elaboration of these ideas, particularly with an eye toward applications in plasma wave theory. 1.2.1 Phase speed We start by focusing attention on θ(x, t), a given smooth function of space and time, and consider Figure 1.6. We wish to understand how the phase evolves in time. 25 In this book, we almost always assume that the phase θ(x, t) is a real function. For a different approach using a complex phase function to study evanescence and damping, see for example Choudhary and Felsen [CF73] and Maj et al. [MMPF13]. 26 Hamilton was well aware that the same mathematical theory could be used to study particle dynamics. In a series of papers, he laid out his ray theory, dipping into optics or particle mechanics as examples of his general theoretical approach [Ham28].
- 41. 1.2 Hamilton’s principle of stationary phase 13 In one-dimensional problems, the phase speed is the rate dx/dt = vp at which constant-phase points progress in x. Consider a wave function ψ(x, t) ≡ A(x, t)eiθ(x,t) , (1.7) where A and θ are smooth real functions. In a small time interval dt, we must shift any given point x0 → x0 + dx to keep the phase constant.27 Taking the differential dθ = θx dx + θt dt ≡ k(x, t) dx − ω(x, t) dt, (1.8) and setting this to zero, we arrive at the condition dx dt = ω k ≡ vp. (1.9) This is the phase speed in one spatial dimension. There is an analogous quantity in multidimensions, which is sometimes called the phase velocity, but this quantity does not have the properties of a vector. Let’s see why this is so. Returning to Figure 1.6, choose an arbitrary – but fixed – time t0, and assume the phase θ(x, t0) is smooth throughout some three-dimensional region. This means that the wavefronts foliate the region of interest, forming a stack of surfaces like the two-dimensional leaves of a book (distorted, of course, since the phase is in general not a linear function of x). At a slightly later time t0 + dt, these wavefronts will have shifted slightly. Choose an arbitrary path γ (not necessarily a ray) that punctures the wavefronts transversely. At each point on the path, there is a well- defined phase for each t. The wavefronts move along the path at a characteristic speed. In what follows, we hold the path fixed. Restrict attention to one particular wavefront surface, say θ = 0. Now, define x0 to be the puncture point of the surface θ = 0 of the path γ at t = t0. At a slightly later time t0 + dt, the θ = 0 surface moves, and the puncture point of the path moves to x0 + dx (see Figure 1.6). Then 0 = dθ = ∇θ · dx + ∂θ ∂t dt. (1.10) This is zero because we are following the motion of a constant-θ surface as it moves along the path. The vector dx is tangent to the path at x0, so write it as dx = v dt ≡ vp n̂ dt, (1.11) where n̂ is a unit vector. This leads to (using k ≡ ∇θ, ω ≡ −θt ) 0 = dθ dt ⇒ vp = ω n̂ · k . (1.12) 27 We will see later in this chapter that the amplitude translates at the group velocity.
- 42. 14 Introduction Recall that the phase function and the path are assumed to be given, so this “phase speed” is the rate at which the wavefront moves along a given path. If we change the path, the “phase speed” changes. Example 1.1 Consider a particle moving along the unperturbed trajectory x(t) in the wave field A exp[iθ]. The condition (1.12) is the condition for the particle to see a constant phase. Hence, this is a Landau resonance. Example 1.2 Choose the path x(t) to lie along the local magnetic field. The condition (1.12) becomes v ≡ ω/k. That is, vp ≡ v is the phase speed along the magnetic field. Later in the book we will encounter certain phenomena that arise in plasmas when phase matching occurs due to a resonance. For example, in Chapter 6 we discuss mode conversion, tunneling, and wave emission from coherent sources. In addition, in Chapter 7 we discuss gyroresonance. In each of these cases, in the multidimensional setting, the resonance requires a matching of phase fronts throughout a multidimensional spatial region, not just a matching of the phase speeds at a single point. 1.2.2 Phase integrals and rays Suppose we are once again given the phase function, θ(x, t). Choose a fixed time t0 and consider how the phase changes following the path x(σ). That is, given θ[x(σ); t0] find dθ/dσ. The chain rule gives d dσ θ[x(σ); t0] = dx dσ · ∇θ[x(σ), t0], (1.13) which we write as dθ dσ ≡ dx dσ · k[x(σ)], (1.14) where we have introduced the wavevector k[x(σ)] by first taking the gradient of the given phase field k(x) ≡ ∇θ(x, t0), and then evaluating it at x(σ).28 Notice that by defining k as the gradient of the scalar θ(x, t), we have the identity ∇ × k(x, t) = ∇ × ∇θ = 0, (1.15) for an eikonal wave. 28 It is important to note that ∇θ and dx/dσ have no intrinsic relation to one another, so k(σ) will, in general, not be tangent to the path at x(σ). This is true even if dx/dσ is the group velocity (defined in Section 1.5). We will give examples later.
- 43. 1.2 Hamilton’s principle of stationary phase 15 Equivalently, given only a curl-free k(x), and the surface θ(x, t0) = 0,29 we can choose a particular path and, starting at the point x0 where it punctures the surface θ(x, t0) = 0, holding t fixed, we can integrate (1.14) along the path x(σ) to find θ[x(σ), t0] = σ 0 k[x(σ )] · dx dσ (σ ) dσ . (1.16) It is important to emphasize that if we reparameterize the path σ = σ(s), the numerical value of the integral does not change. Given that the path x(σ) is arbitrary, and the choice of θ[x(σ); t0] = 0 as the base point for the integral was too, the integral (1.16) is seen as another way of writing k(x) = ∇θ(x; t0), a way which explicitly involves integration along spatial paths. (These are not yet rays, they are still arbitrary smooth paths in x-space.) (See Problem 1.10.) We now ask the question: how does the expression (1.16) have to be modified so it will be correct at a slightly later time, t0 → t0 + dt? To simplify, let’s assume that the path does not change in time. Only the phase evolves in time, and the wavefronts move along the fixed path. The picture to have in mind is the right panel of Figure 1.6, showing how the wavefront θ = 0 has shifted in the short time dt. If the phase oscillation exp[iθ(x, t)] locally looks like a plane wave exp[i(k · x − ωt)] for some frequency ω, then we should use dθ ≡ ∇θ · dx + θt dt to identify the local quantities k(x, t) = ∇θ and ω(x, t) = −θt . (1.17) This line of reasoning suggests that the general time-dependent form of the phase integral is θ[x(t), t] = t 0 k[x(t )] · dx dt (t ) − ω[x(t ), t ] dt . (1.18) Here we have used the fact that the k · dx/dσ integral is independent of param- eterization to change from the arbitrary orbit parameter σ to the physical time t. At this stage, (1.18) is simply an integral form of the phase function θ(x, t) that satisfies the identities (1.17). Up until now, we have concerned ourselves with arbitrary smooth paths in physical space, and their relation to a given phase function θ(x, t). The next set of moves involves an important shift in perspective, so we want to call attention to their importance. 29 That is, we now assume that we do not know θ(x, t). We only know where the θ = 0 surface lies at time t0, and the curl-free wavevector field k(x).
- 44. 16 Introduction 1. Double the dimensionality of our system by treating k as independent of x. This new space (x, k) is called ray phase space. 2. Introduce (x, k, t), a smooth function on ray phase space and time. 3. Introduce a smooth path in ray phase space, parameterized by the physical time t: [x(t), k(t)]. (We choose to use the physical time here just to keep things one bit less abstract. A more general parameterization will be used later in the book.) 4. Define the phase integral , which is a functional of the phase space path [x(t), k(t)] ≡ t 0 k(t ) · dx(t ) − [x(t ), k(t ), t ] dt . (1.19) 5. Now invoke a stationarity principle30 for the phase integral (1.19), allowing variation of x(t) and k(t) separately, holding the endpoints of the path fixed. Stationarity with respect to x(t) → x(t) + εy(t), holding k(t) fixed, implies (after integration by parts, and suppressing the arguments for clarity) 0 = d [x + εy, k] dε ε=0 = − t 0 y · {dk + ∇x dt} , (1.20) while stationarity with respect to k(t) → k(t) + εκ(t), holding x(t) fixed, implies 0 = d [x, k + εκ] dε ε=0 = t 0 κ · {dx − ∇k dt} . (1.21) Together, these imply dx dt = ∇k , dk dt = −∇x . (1.22) Equations (1.22) are called Hamilton’s equations, and is the ray Hamiltonian. Given the smooth function (x, k, t) and the initial conditions [x(t0), k(t0)], Hamil- ton’s equations determine a unique path in ray phase space. Paths that satisfy Hamilton’s equations we will call rays once more, in order to distinguish them from arbitrary paths in ray phase space. But note that the dimensionality of the space the ray inhabits has doubled from Fermat’s formulation. Constructing the eikonal phase θ(x, t) using rays that satisfy (1.22) in the phase integral (1.19) requires that we follow a family of rays. For that family of rays, if at each point x there is only one ray, then k is well-defined as a function of x, and 30 See Appendix B for a brief discussion of variational methods.
- 45. 1.2 Hamilton’s principle of stationary phase 17 we can write k(x, t) once more. The phase space integral (1.19) then reduces to the previous form of the phase integral (1.18), thereby showing that k = ∇θ once more. For this family of rays, we also have the identity ∂θ ∂t ≡ −ω(x, t) = − [x, k(x), t]. (1.23) Therefore, from (1.23) and Hamilton’s equations (1.22), following a ray the local frequency changes as dω dt = ẋ · ∇x + k̇ · ∇k + ∂ ∂t = ∂ ∂t . (1.24) A careful discussion of the construction of the eikonal phase function θ(x, t) from a family of rays must be postponed until Section 3.2.1, where we discuss lifts and projections. There is much we have left out of this first encounter with Hamilton’s theory. For example, we have not discussed how to compute the wave amplitude. In plasma wave problems, we must also find the polarization of the electric and magnetic fields at each point. Hamilton’s work concerned light rays, but we now know that the ray theory he invented is the basis for all theories of short-wave asymptotics, that is, it applies to all types of electromagnetic waves, sound waves, elastic waves in solids, etc. We will have much more to say about these matters later in this chapter, and in the rest of the book. Before leaving this section, we should also mention the fundamental contri- bution of Emmy Noether (1882–1935) concerning symmetries and conservation laws. She was led to this topic during her time at Göttingen by Hilbert and Klein, who were puzzling over some features of the then new general theory of relativ- ity concerning the energy conservation law. Noether showed that the proper form of the conservation law followed from a symmetry of the Hilbert action princi- ple for the theory, but she went on to show that the method she had uncovered was general.31 Specifically, what is now called Noether’s theorem shows that if a variational principle, like Hamilton’s stationary phase principle, has a continuous global symmetry, then making the variation local results in a conservation law. As we shall see in Chapter 2, Noether’s theorem provides a simple and elegant means to derive the wave-action conservation law, a result that is usually derived by brute force calculation. This is also discussed in Appendix B. 31 For more of this interesting history, see Weyl’s memorial address for Noether, given at Bryn Mawr in 1935, reprinted in [Wey70], and the recent article [Bye98].
- 46. 18 Introduction 1.3 Modern developments 1.3.1 Quantum mechanics and symbol calculus Hamilton developed his formulation of optics using systems of rays, but he was also fully aware that it provided a new formulation of the Euler–Lagrange theory of particle mechanics (see, for example, his expository article [Ham33]). Jumping for- ward now to the 1920s, it is significant that when Erwin Schrödinger (1887–1961) began his search to find the wave equation governing de Broglie’s matter waves, he returned to Hamilton’s theory to guide his thinking. Schrödinger’s important breakthrough is described in Moore [Moo94]. We take a moment to recount some of these important developments of the 1920s and early 1930s, because Schrödinger’s revolutionary ideas concerning wave mechanics, paired with the even more abstract matrix mechanics of Heisenberg, provide a conceptual bridge between Hamilton and Hermann Weyl (1885–1955), who is just about ready to enter our story. In an attempt to understand the underlying mathematical nature of these two very different theories of quantum mechanics, Weyl laid the foundations for what is now called the symbol calculus, which plays a large role in this book. The de Broglie relation (1924) is p = h λ ≡ h̄k. (1.25) Here λ is the matter wavelength, p the particle momentum, h is Planck’s constant,32 with h̄ ≡ h/2π. Recall, also, Einstein’s formula (1905) relating the energy and frequency of a photon E = hν ≡ h̄ω. (1.26) These previously known relations were Schrödinger’s starting point. Initially, in late 1925, in an attempt to understand the physical meaning of de Broglie’s hypothesis, Schrödinger tried to construct the electron wave function for the hydrogen atom, using only (1.25) and the classical Kepler orbit.33 This attempt ran into problems because of caustics, a topic we will discuss in Chapter 5. These technical difficulties led him to seek a wave equation that would govern the particle dynamics, which he could then solve directly without recourse to the classical orbits. It is not possible to reconstruct the sequence of events that led to the final form of what we now call the Schrödinger equation, and in any event he did not derive the equation. Instead, over the course of only a few weeks, he used Hamilton’s 32 For completeness: h = 6.626 × 10−34 joule-seconds = 4.136 × 10−15 eV-seconds. 33 That is, he assumed the orbit for a classical point-charge electron in a 1/r potential.
- 47. 1.3 Modern developments 19 ideas and well-known operator correspondences (see below) to reason his way to two plausible candidates for a matter wave equation. The candidate equations were then used to make physical predictions, which could be compared with experiment. It was assumed that the wave equation should be linear, so that solutions could be superposed. It appears that the first equation Schrödinger derived was a relativistic equation (now called the Klein–Gordon equation). Start with the Einstein mass–energy relation for a free particle E2 = p2 c2 + m2 c4 , (1.27) where m is the rest mass of the particle. In a 1/r potential, Schrödinger reasoned that this should become E + e2 r 2 = p2 c2 + m2 c4 . (1.28) Now consider the following correspondences, which were well-known from Fourier analysis k ↔ −i∇ and ω ↔ i ∂ ∂t . (1.29) Therefore, using the de Broglie and Einstein relations (1.25) and (1.26), the mass– energy relation (1.28) suggested consideration of the following relativistic wave equation ih̄ ∂ ∂t + e2 r 2 ψ = −h̄2 c2 ∇2 ψ + m2 c4 ψ. (1.30) Inserting an eikonal ansatz ψ = A exp[iθ] (and neglecting the derivatives of A), the mass–energy relation (1.27) is now seen to be the eikonal equation for the phase θ.34 According to Moore [Moo94], Schrödinger thought this was too easy. He rejected this equation because it also did not give the right energy levels for the hydrogen atom in the nonrelativistic limit.35 Schrödinger then decided to look for the nonrelativistic theory, and he began by considering the classical Hamiltonian for a particle in an external potential H(x, p) = p2 2m + V (x) = E. (1.31) 34 We will have much more to say about the eikonal equation in Chapter 2. 35 We now know that this is because the electron has spin, and its relativistic theory is governed by the Dirac equation. The Klein–Gordon equation is adequate for spin-0 particles.
- 48. 20 Introduction Using similar reasoning as in the relativistic case, Schrödinger then conjectured that the appropriate quantum wave equation should be − h̄2 2m ∇2 + V (x) ψ = ih̄ ∂ψ ∂t , (1.32) which is the famous equation that now bears his name. From beginning to end, this took only a few weeks (that is, starting with his attempts to construct the wave function using only Kepler orbits and the de Broglie relation to the “derivation” of the wave equation). However, Schrödinger was unable to compute the full hydrogen spectrum from this wave equation because of technical difficulties in the continuum region. For this part of the calculation, he turned to his close friend Hermann Weyl for help, and together they quickly completed the calculation. Heisenberg’s matrix mechanics had been developed in 1925, by Heisenberg, Born, and Jordan, as an attempt to create a quantum theory involving only observ- ables. Heisenberg’s breakthrough was to realize that these observables should not have to commute as algebraic objects, reflecting the idea that the sequence in which different observations were taken mattered. Hence, Born suggested the observables should be represented by matrices, and the result of an observation would be one of the eigenvalues. Because the results of a measurement are real quantities (in the mathematical sense), the matrices representing observables must be self-adjoint. For our purposes in this brief conceptual survey, the most important example of noncommuting observables are the position x and the momentum p.36 These two observables obey the commutation relation x p − p x ≡ [ x, p] = ih̄ Id, (1.33) where Id is the identity. It is easy to show that there are no finite-dimensional real- izations of this commutation relation.37 There is, however, the familiar realization in terms of the operator associations x ↔ x, p ↔ −ih̄ ∂ ∂x . (1.34) The operators act on functions of x. These types of associations are familiar from Fourier analysis (but without the h̄). Therefore, by 1926 there were two successful formulations of quantum mechan- ics, matrix mechanics and wave mechanics. Of the two, wave mechanics had the more direct link with classical mechanics via Hamilton’s theory of ray systems. 36 We denote these objects with carats to distinguish them from ordinary numbers. We will have much more to say about these matters in Chapter 2. 37 The proof is by contradiction. Assume that x, p, and Id are N × N matrices with N finite. Take the trace of (1.33). The commutator has zero trace, but the trace of the identity matrix is N. Therefore, N cannot be finite.
- 49. 1.3 Modern developments 21 Weyl was in the very center of this ferment, through his close friendship with Schrödinger, and was already considered one of the leading mathematicians of his generation.38 From his work on Lie groups, Weyl recognized (1.33) as a Lie algebra. Elements of the Lie algebra generate the Lie group through exponentiation.39 An arbitrary element of the Lie algebra is the linear combination a x + b p, where a and b are real constants, and the associated group element is T ≡ ei(a x+b p) . (1.35) Because x and p are self-adjoint, the operator T is unitary for real a and b (see Section 2.3, Section D.3.1, and Appendix D for more details).40 The operator p generates shifts in x-space, while x generates shifts in k-space; therefore the combination is a phase space shift. This Lie group is now called the Heisenberg– Weyl group, the group of noncommutative shifts on classical phase space. By 1926, Weyl was able to show that any of Heisenberg’s matrix observables could be written as a linear superposition of operators of the type (1.35).41 The expansion coefficients can be found using what is now called the symbol of the operator. We will discuss Weyl’s ideas in Chapter 2, but in the context of classical wave equations, where they are equally powerful. We should also mention here the third formulation of quantum mechanics using path integrals. This approach is identified with Feynman, who introduced the idea in his Ph.D. thesis [FB42]. Feynman was motivated by the problem of quantizing systems that had no Hamiltonian.42 Without a Hamiltonian, neither the Heisenberg nor Schrödinger approaches could be used to construct the time evolution operator. In Feynman’s theory, there is assumed to be a rule for assigning a phase exp[iθ] to each point on any path in x-space, including nonsmooth paths. The phases for all possible paths – starting from fixed initial and ending points – are summed over, and only those paths for which the phase is locally stationary will interfere con- structively and survive in the asymptotic limit h̄↓0. When the rule for constructing the phase at each point is based upon the classical action, θ ≡ h̄−1 Ldt with L the Lagrangian, then the paths which survive to dominate the quantum transition probabilities are the classical ones, that is, the paths for which the classical action is stationary. 38 In his Weyl Centenary Lecture, Roger Penrose writes that Weyl was the greatest mathematician who worked entirely in the twentieth century. This allowed Penrose to avoid arguments about Hilbert, Poincaré, or Cartan. 39 Strictly speaking, it is i x and i p that are the generators. 40 Because x and p are self-adjoint operators, their linear combination a x + b p is also self-adjoint, with real eigenvalues. 41 Weyl’s theory first appeared in the 1927 article [Wey27] and was later expanded into the 1931 book [Wey31]. 42 For example, in cases where the Legendre transformation from (x, ẋ) to the canonical coordinates (x, p) was not well-defined.
- 50. 22 Introduction The path integral approach to the quantum theory of photons actually has an earlier history, and can be traced to a little-known paper by Wentzel [Wen24], which appeared in the very same year as de Broglie’s work (1924) and, therefore, preceded even the work of Heisenberg and Schrödinger.43 This is also of note for our history, because Wentzel (1898–1978) is the “W” in “WKB.”44 Although mathematical approximation schemes similar to what are now called “WKB methods” were used in the nineteenth century, the papers of Wentzel [Wen26a, Wen26b, Wen26c], Kramers [Kra26], and Brillouin [Bri26] are noteworthy because they are the first systematic attempts to use Schrödinger’s newly proposed wave equation to compute quantum corrections to classical theories.45 Feynman’s original work concerned paths in x-space, as in Fermat’s original least-time principle. For systems that have a Hamiltonian, it is possible to refor- mulate the path-integral theory entirely in phase space, replacing L by pq̇ − H, with H the Hamiltonian. This is not just a formal development. If we attempt to use Weyl’s symbol theory to compute the symbol of the evolution operator U ≡ exp[it H] in terms of the symbol of the Hamiltonian H, then the phase space path integral arises directly, not the x-space path integral (see, for example, Berezin and Shubin [BS91], DeWitt-Morette et al. [DMMN77], or Richardson [Ric08]).46 The fact that Weyl’s theory originated in the context of quantum physics can obscure the fact that the mathematical ideas are completely general, applicable to all manner of wave equations, including those in plasma theory. This early identification of the symbol calculus with the mathematical foundations of quantum mechanics may explain why Weyl’s ideas took over fifty years to find their way into the plasma physics literature.47 1.3.2 Ray phase space and plasma wave theory The following discussion is drawn from the 1991 review article by AN Kauf- man [Kau91].48 The theoretical study of plasma dynamics utilizes several different spaces. There is the three-dimensional physical space (denoted by x) and four-dimensional space-time (denoted by x or xμ = (x, t)). These are the natural base-spaces for the Maxwell field, for various plasma densities (particle density, current 43 A summary of the paper, and a discussion of its historical significance, can be found in [AL97]. 44 We also mention in passing, that Wentzel was a Ph.D. supervisor for one of the co-authors of this book (ANK), who therefore has a “W-number” of one. 45 See [FGNO09]. 46 We thank our colleague Nahum Zobin for his help in understanding the theory of path integrals. 47 The first paper that we are aware of that uses the Weyl symbol calculus in plasma theory is the 1988 paper by McDonald [McD88]. 48 See also the more recent reviews from the 2009 KaufmanFest [BT09, TB09, Bri09, Kau09].
- 51. 1.3 Modern developments 23 density, momentum-flux density), and for wave fields. On the other hand, plasma kinetic equations deal with distributions in six-dimensional single-particle position- velocity space (r, v) or six-dimensional single-particle phase space (r, p), or with their eight-dimensional extensions (r, p). Plasma physicists are accustomed to dealing with these spaces for these entities. But plasma waves, too, have a natural ray phase space, which we have already encountered in our discussion of Hamilton’s ray optics. Although Hamilton’s phase space approach to optics is ancient, it drew little attention in the plasma community until the 1980s. Its renaissance was due to the impact of the spectacular development of semi-classical dynamics. Classical physicists (which almost all plasma physicists are) can now see how their understanding of classical particle orbits in phase space relates to the quantum wave function in position space. The Berkeley plasma theory group, led by AN Kaufman, was strongly influenced by the comprehensive review articles of Berry [Ber77a, BU80, BT76, Ber77b] and of Percival [Per77], and by the research of their groups. Those reviews pointed the way back to the pioneering work of Keller [Kel58, Kel85], Keller and Rubi- now [KR60], and of Maslov [MF02]. The Berkeley group’s entry to the field was sparked by the paper of Wersinger, Finn, and Ott [WFO80], discussing how plasma waves can exhibit chaotic ray orbits. This naturally led to the question of what the corresponding wave field looked like. The stadium-billiard was selected as the simplest system to investi- gate, and, in his Ph.D. thesis, McDonald obtained the first chaotic wave eigen- function [MK79, MK85]. To relate this x-space wave field to phase space, a coarse-grained Wigner function was used, whose properties for a chaotic system had been recently predicted by Berry [Ber77a, Ber77b]. This led to a more detailed study [MK88, McD88] of Wigner functions and of the Weyl symbol calculus relating phase space functions and x-space operators.49 From this study, a simple derivation of the wave kinetic equation for action-density in phase space was uncov- ered [MK85], as well as Hamiltonian formulations for wave interactions [MK82].50 Up to this point, there had been no occasion to utilize variational principles. In plasma physics, variational principles had appeared in two disparate forms. For fields on x-space, they had been used by Crawford and co-workers Kim and Galloway [KC77a, KC77b, GC77], by Dougherty [Dou70, Dou74], and by Dewar [Dew72b, Dew70, Dew72a] as a very powerful approach to x-space problems, such as nonlinear wave interactions. On the other hand, for single- particle motion, Littlejohn had employed a phase space variational principle to obtain a simpler derivation [Lit83] of his non-canonical Hamiltonian theory of 49 See Section 2.3 and Appendix D for a discussion of the Weyl symbol calculus. 50 See Section 3.5.5 for a discussion of the derivation of the wave kinetic equation.
- 52. 24 Introduction guiding-center motion [Lit79, Lit81]. The Hamiltonian and action principle for- mulations of plasma physics have been reviewed by Morrison, who made significant contributions to this area [Mor05]. Four other important theoretical developments led to the formulation of the plasma phase space variational principle. The first was the Hamiltonian theory of the oscillation center (OC), introduced by Dewar [Dew73] (see also John- ston [Joh76]) as a systematic resolution of then current confusion about quasilinear diffusion [Kau72]. The second was the concept of ponderomotive potential, for the nonlinear low-frequency effects of high-frequency fields. These lines of research led to the realization [JKJ78] that the ponderomotive potential was the zero-velocity limit of the quadratic term in the oscillation-center Hamiltonian, called the pon- deromotive Hamiltonian. The third development was the (Hamiltonian) Lie transform, introduced into plasma physics by Dragt and Finn [DF76, DF79] and by Dewar [Dew76]. The Berkeley group adopted this technique wholeheartedly, applying it to a range of problems. In particular, Cary converted Dewar’s OC transformation from a mixed- variable generating function to the far simpler Lie transform [CK81]. The resulting expression for the ponderomotive Hamiltonian was observed to be identical in form (except for a sign) to the linear susceptibility [CK77, JK78]. This astounding relation between a nonlinear expression and a linear one (now called the K-χ relation) meant that something deep remained to be discovered; the minus sign pointed in the Lagrangian direction. The fourth development [Lit82] was Littlejohn’s Lagrangian Lie transform (L3 T). Although the Hamiltonian Lie transform works quite well, its formalism leads to expressions that are not manifestly gauge-invariant, and thus to potentially unphysical interpretations. In contrast, L3 T makes full use of modern differential geometry in physical (as opposed to canonical) phase space; it can embrace the Hamiltonian Lie transform as a special case, and it generates the OC transformation by a vector field in phase space, which is the physical oscillation perturbation. The spark for further progress came from the paper of Dubin et al. [DKOL83] who constructed self-consistent evolution equations for a distribution of guid- ing centers and for the electric-potential wave field. They demonstrated this self- consistency by finding an energy conservation law, by trial and error. It seemed to ANK that self-consistent equations and their conservation laws should come automatically from an appropriate variational principle; this had been shown, in the x-space context, by Dewar [Dew77]. It had also been shown, by Dominguez and Berk [DB84], that a variational principle was the natural vehicle for deriving self-consistent equations subject to a systematic approximation scheme. But the transformations to guiding centers (developed by Littlejohn) and to oscillation centers (developed by Dewar) required phase space coordinate changes.
- 53. 1.3 Modern developments 25 Littlejohn then suggested to ANK that the single-particle phase space action prin- ciple, which was the starting point for L3 T, was also a natural starting point for a theory of the self-consistent evolution of electromagnetic fields and particles. Now everything fell into place. Starting with the total action for the system, and applying a Lie transform to the action of each particle, led Kaufman and Boghosian to the K-χ theorem [KB84, Kau87]), to the ponderomotive effects on oscillation centers, and to linear wave propagation in the OC medium. The results were in complete agreement with the previous Hamiltonian formulations of McDonald, Grebogi, Kaufman, and Omohundro [GKL79, MGK85, Kau82, Omo86], and provided in addition the evolution of the quasistatic background field, which had been missing from the previous Hamiltonian approach. Two further developments contributed importantly to the formulation. One was the covariant formalism: treating the particle motion relativistically and covariantly, rather than in 3 + 1 notation; this simplified the equations and led to additional insights.51 Whereas the unmagnetized plasma (see Kaufman and Holm [KH84]) posed no problems in generalizing to covariance, the magnetized case was quite challenging and rewarding. Building on Littlejohn’s covariant Poisson brackets, a covariant single-particle action principle in the guiding-center representation was formulated [Sim85]. The second development was Similon’s construction of a Lagrangian density (in x-space) for this phase space action principle, and adaptation of Noether’s methods to finding an algorithm for energy-momentum conservation laws [Sim85] (see Appendix B, and Section 3.3.1). Building on these foundations, the covariant OC transformation for gyrating particles interacting nonresonantly with a single eikonal wave in a weakly nonuni- form background field was derived, which led to a manifestly gauge-invariant and Lorentz-covariant expression for the ponderomotive Hamiltonian and lin- ear susceptibility (see Boghosian [Bog87]). Taking variations with respect to the background four-potential led to the covariant equations for the background field, including the effects of wave-induced magnetization. The latter effects were cru- cial in analyzing the ponderomotive stabilization of low-frequency modes by high- frequency waves [SK84, SKH86]. It remained to incorporate the resonant interaction of particles with an eikonal wave. As a first approach, an approximate interaction Lagrangian was derived by integrating a linearized version across the resonant region of particle phase space [GK83]. Variation then led to the self-consistent change of the back- ground field, resulting from wave–particle resonance, and to coarse-grained energy- momentum conservation laws (see Ye and Kaufman [YK92]). 51 See Section 3.3.1 for a discussion of the Lorentz-covariant ray theory, and Section 3.4 for the fully covariant theory on ray phase space.
- 54. 26 Introduction Application to ion gyroresonant heating led to the realization that the situa- tion is considerably more complex. It is generally recognized that, when an elec- tromagnetic plasma wave crosses a resonance layer transversely, two additional waves are created, and a large fraction of the incident wave action is converted to quasilinear diffusion (in phase space) of the resonant particles. Following Ye and Kaufman [Ye90, YK88a, YK88b], we now interpret this phenomenon in terms of Friedland’s formulation of mode conversion, which is discussed at length in Chapters 6 and 7. The basic idea underlying Friedland’s work is that a multicomponent wave field supports modes with different polarizations and different dispersion functions. Conversion occurs when two modes can exist at the same point in ray phase space. Wave action can then be exchanged in its vicinity, due to their coupling. While the fraction of conversion had been known for the case of one-dimensional spatial variation of the medium, Friedland solved the coupled partial differential equations in x-space, for a full four-dimensional space-time variation of the medium (see Friedland et al. [FGK87]). In his result, a Poisson bracket appeared, showing that phase space methods were waiting to be applied. Accordingly, ANK utilized methods which Littlejohn had developed [Lit86] for semi-classical analysis of the Schrödinger equation. In place of the standard (x, k) coordinates of ray phase space, a locally linear canonical transformation to new coordinates (q, p) is used, in which one set (q1, p1) was related to (k, x) by the two dispersion functions. The wave field is then expressed as a function on q-space instead of x-space, as a generalization of the Fourier transform. With this technique, the conversion problem could be completely solved [KF87, TK90]. This theory has now been generalized, and adapted for use in the numerical code RAYCON (see Chapter 6, as well as the Appendices E through G). Friedland prepares the equations for the multicomponent wave field by system- atically eliminating components, in such a way as to preserve the desirable feature of having slowly varying coefficients. His algorithm for accomplishing this, termed congruent reduction [FK87], is based on a Hermitian-form variational principle, and utilizes the Weyl symbol calculus. When this technique is applied to linearized kinetic equations, as in gyroresonance, one is led to the dispersion functions of ballistic modes (see Friedland and Goldner [FG86]), which represent perturbations of the particle density (in phase space) in the absence of coupling to the electro- magnetic perturbation. The resonance of particles with a collective mode is then interpreted as a linear conversion to a continuum of these ballistic waves. These waves can, in turn, transfer action to other collective modes (see Ye and Kauf- man [Ye90, YK88a, YK88b]). The residue of the ballistic modes undergoes phase mixing and is interpreted as collisionless absorption. Using these ideas, the compu- tation of the conversion of an incoming fast magnetosonic wave to a minority-ion
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- 56. vizt: One suit of good curtains and fallens, one Rugg, one Quilt, one pair Blankets. Item. I give and bequeath unto my said Daughter Mary Ball two Diaper Table clothes marked M. B. with inck, and one Dozen of Diaper napkins, two towels, six plates, two pewter dishes, two basins, one large iron pott, one Frying pan, one old trunk. Item. I give and bequeath unto my said Daughter Mary Ball, one good young Paceing horse together with a good silk plush side saddle to be purchased by my Executors out of my Estate. Item. I give and bequeath unto my Daughter Elizabeth Bonum one suit of white and black callico, being part of my own wearing apparel. Item. All the rest of my wearing apparel I give and bequeath unto my said Daughter Mary Ball, and I do hereby appoint her (to) be under Tutiledge and government of Capt. George Eskridge during her minority. Item. My will is I do hereby oblige my Executors to pay to the proprietor or his agent for the securing of my said Daughter Mary Ball her land Twelve pounds if so much (be) due. Item. All the rest of my Estate real and personal whatsoever and wheresoever I give and devise unto my son John Johnson, and to his heirs lawfully to be begotten of his body, and for default of such Issue I give and devise the said Estate unto my Daughter Elizabeth Bonum, her heirs and assigns forever. Item. I do hereby appoint my son John Johnson and my trusty and well beloved friend George Eskridge Executors of this my last will and Testament and also revoke and Disannul all other former wills or Testaments by me heretofore made or caused to be made either by word or writing, ratifying and confirming this to be my last Will and Testament and no other. In witness whereof I have hereunto sett my hand and seal the Day and Date at first above written.
- 57. The mark and seal of Mary III Hewes. Sig. (Seal) Signed, Sealed and Published and Declared by Mary Hewes to be her last Will and Testament in presence of us. The mark of Robert × Bradley. The mark of Ralph × Smithurst David Stranghan. The chief witness to this will was a teacher of no mean repute who lived near Mrs. Hewes, And, says Dr. Beale, others might be named who followed the same calling in Mary Ball's girlhood and near her home. The son, John Johnson, named as joint executor in his mother's will, died very soon after her. His will and hers were recorded on the same day. The first bequest reveals his affection for his little half- sister. Imprimis. I give and bequeath unto my sister Mary Ball all my land in Stafford which my father-in-law Richard Hewes gave me, to the said Mary Ball and her heirs lawfully to be begotten of her body forever. The will of Samuel Bonum, husband of the Elizabeth mentioned in Mrs. Hewes's will, was probated in Westmoreland, Feb. 22, 1726, and contains an item bequeathing to my sister-in-law Mary Ball, my young dapple gray riding horse. Mary Ball was then eighteen years old. So it appears that the mother of Washington, although not rich, according to the standard of that day or this, was fairly well endowed with Virginia real estate. Also that she owned three or more riding-horses, her own maid, a few jewels, and house plenishing sufficient for the station of a lady in her day and generation.
- 59. CHAPTER VII MARY BALL'S CHILDHOOD It is easy to imagine the childhood of Mary Ball. Children in her day escaped from the nursery at an early age. They were not hidden away in convents or sent to finishing schools. There were no ostentatious débuts, no coming-out teas. As soon as a girl was fairly in her teens she was marriageable. Little girls, from early babyhood, became the constant companions of their mothers, and were treated with respect. Washington writes gravely of Miss Custis, six years old. They worked samplers, learned to edge handkerchiefs with a wonderful imitation of needle- point, plaited lace-strings for stays, twisted the fine cords that drew into proper bounds the stiff bodices, knitted garters and long hose, took lessons on the harpsichord, danced the minuet, and lent their little hands to clap muslins on the great clearstarching days, when the lace steenkirk, and ruffled bosoms, and ample kerchiefs, were gotten up and crimped into prescribed shape. No lounging, idleness, or loss of time was permitted. The social customs of the day enforced habits of self-control. For long hours the little Mary was expected to sit upon high chairs, with no relenting pillows or cushions, making her manners as became a gentleman's daughter throughout the stated dining days, when guests arrived in the morning and remained until evening. Nor was her upright figure, clad in silk coat and mittens, capuchin and neckatees, ever absent from the front seat of the yellow chariot as it swung heavily through the sands to return these stately visits, or to take her mother and sister to old St. Stephen's church. Arriving at the latter, she might possibly have had a glimpse now and then of other little girls as she paced the gallery on her way to the high-backed family pew, with its
- 60. railing of brass rods with damask curtains to prevent the family from gazing around when sitting or kneeling. Swallowed up in the great square pew she could see nothing. An Old Doll. From the viewpoint of a twentieth-century child, her small feet were set in a hard, if not thorny, path. The limits of an early colonial house allowed no space for the nursery devoted exclusively to a child, and filled with every conceivable appliance for her instruction and amusement. There were no wonderful mechanical animals, lifelike in form and color, and capable of exercising many of their functions. One stiff-jointed, staring, wooden effigy was the only prophecy of the enchanting doll family,—the blue-eyed, brown-eyed, flaxen-curled, sleeping, talking, walking, and dimpled darlings of latter-day children,—and the wooden-handled board, faced with horn and bound with brass, the sole representative of the child's picture-book of to-day. No children's books were printed in England
- 61. until the middle of the eighteenth century; but one Thomas Flint, a Boston printer, appreciating the rhymes that his mother-in-law, Mrs. Goose, sang to his children, published them in book form and gave them a name than which none is more sure of immortality. This, however, was in 1719—too late for our little Mary Ball. She had only the horn-book as resource in the long, dark days when the fairest of all books lay hidden beneath the snows of winter—the horn-book, immortalized by Thomas Tickell as far back as 1636:— Thee will I sing, in comely wainscot bound, And golden verge enclosing thee around: The faithful horn before, from age to age Preserving thy invulnerable page; Behind, thy patron saint in armor shines With sword and lance to guard the sacred lines. The instructed handles at the bottom fixed Lest wrangling critics should pervert the text. The sword and lance were in allusion to the one illustration of the horn-book. When the blue eyes wearied over the alphabet, Lord's prayer, and nine digits, they might be refreshed with a picture of St. George and the Dragon, rudely carved on the wooden back. The instructed handle clasped the whole and kept it together.
- 62. Horn-book. All orphans and poor children in colonial Virginia were provided with public schools under the care of the vestries of the parishes—litle houses, says Hugh Jones in 1722, built on purpose where are taught English, writing, etc. Parents were compelled to send their children to these schools, and masters to whom children were bound were required to give them schooling until ye years of twelfe or thereabout without distinction of race or sex. For instance, in the vestry book of Petsworth Parish, in Gloucester County, is an indenture dated Oct. 30, 1716, of Ralph Bevis to give George Petsworth, a molattoe boy of the age of 2 years, 3 years' schooling; and carefully to instruct him afterwards that he may read well any part of the Bible. Having mastered the Bible, all literary possibilities
- 63. were open to the said George. The gentry, however, employed private tutors in their own families,—Scotchmen or Englishmen fresh from the universities, or young curates from Princeton or Fagg's Manor in Pennsylvania. Others secured teachers by indenture. In Virginia, says the London Magazine, a clever servant is often indentured to some master as a schoolmaster. John Carter of Lancaster directed in his will that his son Robert should have a youth servant bought for him to teach him in his books in English or Latin. Early advertisements in the Virginia Gazette assured all single men capable of teaching children to Read English, write or Cypher or Greek Latin and Mathematicks—also all Dancing Masters, that they would meet with good encouragement in certain neighborhoods. But this was after Mary Ball's childhood. Days of silent listening to the talk of older people were probably her early school days. In Virginia there were books, true, but the large libraries of thirty years later had not yet been brought over. There was already a fine library at Stratford in Westmoreland. Colonel Byrd's library was considered vast when it attained to 3600 titles. Books were unfashionable at court in England. No power in heaven or earth has been yet found to keep the wise and witty from writing them, but in the first years of the eighteenth century it was very bad form to talk about them. Later, even, the first gentleman in England was always furious at the sight of books. Old ladies used to declare that Books were not fit articles for drawing-rooms. Books! said Sarah Marlborough; prithee, don't talk to me about books! The only books I know are men and cards. But there were earnest talkers in Virginia, and the liveliest interest in all kinds of affairs. It was a picturesque time in the life of the colony. Things of interest were always happening. We know this of the little Mary,—she was observant and wise, quiet and reflective. She had early opinions, doubtless, upon the powers of the vestries, the African slave-trade, the right of a Virginia assembly to the privileges of parliament, and other grave questions of her time. Nor was the time without its vivid romances. Although no witch was ever burnt in
- 64. Virginia, Grace Sherwood, who must have been young and comely, was arrested under suspetion of witchcraft, condemned by a jury of old women because of a birth-mark on her body, and sentenced to a seat in the famous ducking-stool, which had been, in the wisdom of the burgesses, provided to still the tongues of brabbling women,—a sentence never inflicted, for a few glances at her tearful eyes won from the relenting justice the order that this ducking was to be in no wise without her consent, or if the day should be rainy, or in any way to endanger her health! Ducking-stool. Stories were told around the fireside on winter nights, when the wooden shutters rattled—for rarely before 1720 were windows sasht with crystal glass. The express, bringing mails from the north, had been scalped by Indians. Four times in one year had homeward- bound ships been sunk by pirates. Men, returning to England to receive an inheritance, were waylaid on the high seas, robbed, and murdered. In Virginia waters the dreaded Blackbeard had it all his own way for a while. Finally, his grim head is brought home on the bowsprit of a Virginia ship, and a drinking-cup, rimmed with silver, made of the skull that held his wicked brains. Of course, it could not
- 65. be expected that he could rest in his grave under these circumstances, and so, until fifty years ago (when possibly the drinking-cup was reclaimed by his restless spirit), his phantom sloop might be seen spreading its ghostly sails in the moonlight on the York River and putting into Ware Creek to hide ill-gotten gains in the Old Stone House. Only a few years before had the dreadful Tuscaroras risen with fire and tomahawk in the neighbor colony of North Carolina. The Old Stone House. Nearer home, in her own neighborhood, in fact, were many suggestive localities which a child's fears might people with supernatural spirits. Although there were no haunted castles with dungeon, moat, and tower, there were deserted houses in lonely places, with open windows like hollow eyes, graveyards half hidden by tangled creepers and wept over by ancient willows. About these there sometimes hung a mysterious, fitful light which little Mary, when a belated traveller in the family coach, passed with bated breath, lest warlocks or witches should issue therefrom, to say nothing of the interminable stretches of dark forests, skirting ravines
- 66. fringed with poisonous vines, and haunted by the deadly rattlesnake. People talked of strange, unreal lights peeping through the tiny port- holes of the old Stone House on York River—that mysterious fortress believed to have been built by John Smith—while, flitting across the doorway, had been seen the dusky form of Pocahontas, clad in her buckskin robe, with a white plume in her hair: keeping tryst, doubtless, with Captain Smith, with none to hinder, now that the dull, puritanic John Rolfe was dead and buried; and, as we have said, Blackbeard's sloop would come glimmering down the river, and the bloody horror of a headless body would land and wend its way to the little fortress which held his stolen treasure. Moreover, Nathaniel Bacon had risen from his grave in York River, and been seen in the Stone House with his compatriots, Drummond, Bland, and Hansford. Doubtless such stories inspired many of little Mary's early dreams, and caused her to tremble as she lay in her trundle-bed,—kept all day beneath the great four poster, and drawn out at night,—unless, indeed, her loving mother allowed her to climb the four steps leading to the feather sanctuary behind the heavy curtains, and held her safe and warm in her own bosom.
- 67. CHAPTER VIII GOOD TIMES IN OLD VIRGINIA Despite the perils and perplexities of the time; the irreverence and profanity of the clergy; the solemn warning of the missionary Presbyterians; the death of good Queen Anne, the last of the Stuarts, so dear to the hearts of loyal Virginians; the forebodings on the accession to the throne of the untried Guelphs; the total lack of many of the comforts and conveniences of life, Virginians love to write of the early years of the century as the golden age of Virginia. These were the days known as the good old times in old Virginia, when men managed to live without telegraphs, railways, and electric lights. It was a happy era! says Esten Cooke. Care seemed to keep away and stand out of its sunshine. There was a great deal to enjoy. Social intercourse was on the most friendly footing. The plantation house was the scene of a round of enjoyments. The planter in his manor house, surrounded by his family and retainers, was a feudal patriarch ruling everybody; drank wholesome wine—sherry or canary—of his own importation; entertained every one; held great festivities at Christmas, with huge log fires in the great fireplaces, around which the family clan gathered. It was the life of the family, not of the world, and produced that intense attachment for the soil which has become proverbial. Everybody was happy! Life was not rapid, but it was satisfactory. The portraits of the time show us faces without those lines which care furrows in the faces of the men of to-day. That old society succeeded in working out the problem of living happily to an extent which we find few examples of to-day. The Virginians of 1720, according to Henry Randall, lived in baronial splendor; their spacious grounds were bravely ornamented;
- 68. their tables were loaded with plate and with the luxuries of the old and new world; they travelled in state, their coaches dragged by six horses driven by three postilions. When the Virginia gentleman went forth with his household his cavalcade consisted of the mounted white males of the family, the coach and six lumbering through the sands, and a retinue of mounted servants and led horses bringing up the rear. In their general tone of character the aristocracy of Virginia resembled the landed gentry of the mother country. Numbers of them were highly educated and accomplished by foreign study and travel. As a class they were intelligent, polished in manners, high toned, and hospitable, sturdy in their loyalty and in their adherence to the national church. Another historian, writing from Virginia in 1720, says: Several gentlemen have built themselves large brick houses of many rooms on a floor, but they don't covet to make them lofty, having extent enough of ground to build upon, and now and then they are visited by winds which incommode a towering fabric. Of late they have sasht their windows with crystal glass; adorning their apartments with rich furniture. They have their graziers, seedsmen, brewers, gardeners, bakers, butchers and cooks within themselves, and have a great plenty and variety of provisions for their table; and as for spicery and things the country don't produce, they have constant supplies of 'em from England. The gentry pretend to have their victuals served up as nicely as the best tables in England. A quaint old Englishman, Peter Collinson, writes in 1737 to his friend Bartram when he was about taking Virginia in his field of botanical explorations: One thing I must desire of thee, and do insist that thee oblige me therein: that thou make up that drugget clothes to go to Virginia in, and not appear to disgrace thyself and me; for these Virginians are a very gentle, well-dressed people, and look, perhaps, more at a man's outside than his inside. For these and other reasons pray go very clean, neat and handsomely dressed to Virginia. Never mind thy clothes: I will send more another year.
- 69. Those were not troublous days of ever changing fashion. Garments were, for many years, cut after the same patterns, varying mainly in accordance with the purses of their wearers. The petticoats of sarcenet, with black, broad lace printed on the bottom and before; the flowered satin and plain satin, laced with rich lace at the bottom, descended from mother to daughter with no change in the looping of the train or decoration of bodice and ruff. There were no mails to bring troublesome letters to be answered when writing was so difficult and spelling so uncertain. Not that there was the smallest disgrace in bad spelling! Trouble on that head was altogether unnecessary. There is not the least doubt that life, notwithstanding its dangers and limitations and political anxieties, passed happily to these early planters of Virginia. The lady of the manor had occupation enough and to spare in managing English servants and negroes, and in purveying for a table of large proportions. Nor was she without accomplishments. She could dance well, embroider, play upon the harpsichord or spinet, and wear with grace her clocked stockings, rosetted, high-heeled shoes and brave gown of taffeta and moyre looped over her satin quilt. There was no society column in newspapers to vex her simple soul by awakening unwholesome ambitions. There was no newspaper until 1736. She had small knowledge of any world better than her own, of bluer skies, kinder friends, or gayer society. She managed well her large household, loved her husband, and reared kindly but firmly her many sons and daughters. If homage could compensate for the cares of premature marriage, the girl-wife had her reward. She lived in the age and in the land of chivalry, and her amiable qualities of mind and heart received generous praise. As a matron she was adored by her husband and her friends. When she said, Until death do us part, she meant it. Divorce was unknown; its possibility undreamed of. However and wherever her lot was cast she endured to the end; fully assured that when she went to sleep behind the marble slab in the garden an enumeration of her virtues would adorn her tombstone.
- 70. In the light of the ambitions of the present day, the scornful indifference of the colonists to rank, even among those entitled to it, is curious. Very rare were the instances in which young knights and baronets elected to surrender the free life in Virginia and return to England to enjoy their titles and possible preferment. One such embryo nobleman is quoted as having answered to an invitation from the court, I prefer my land here with plentiful food for my family to becoming a starvling at court. Governor Page wrote of his father, Mason Page of Gloucester, born 1718, He was urged to pay court to Sir Gregory Page whose heir he was supposed to be but he despised title as much as I do; and would have nothing to say to the rich, silly knight, who finally died, leaving his estate to a sillier man than himself—one Turner, who, by act of parliament, took the name and title of Gregory Page. Everything was apparently settled upon a firm, permanent basis. Social lines were sharply drawn, understood, and recognized. The court at home across the seas influenced the mimic court at Williamsburg. Games that had been fashionable in the days of the cavaliers were popular in Virginia. Horse-racing, cock-fighting, cards, and feasting, with much excess in eating and drinking, marked the social life of the subjects of the Georges in Virginia as in the mother country. It was an English colony,—wearing English garments, with English manners, speech, customs, and fashions. They had changed their skies only. Cœlum, non animum, qui trans mare currunt. It is difficult to understand that, while custom and outward observance, friendship, lineage, and close commercial ties bound the colony to England, forces, of which neither was conscious, were silently at work to separate them forever. And this without the stimulus of discontent arising from poverty or want. It was a time of the most affluent abundance. The common people lived in the greatest comfort, as far as food was concerned. Fish and flesh, game, fruits, and flowers, were poured at their feet from a liberal horn-of-plenty. Deer, coming down from the mountains to feed upon
- 71. the mosses that grew on the rocks in the rivers, were shot for the sake of their skins only, until laws had to be enforced lest the decaying flesh pollute the air. Painful and hazardous as were the journeys, the traveller always encumbered himself with abundant provision for the inner man. When the Knights of the Golden Horseshoe accomplished the perilous feat of reaching the summit of the Blue Ridge Mountains, they had the honor of drinking King George's health in Virginia red wine, champagne, brandy, shrub, cider, canary, cherry punch, white wine, Irish usquebaugh, and two kinds of rum,—all of which they had managed to carry along, keeping a sharp lookout all day for Indians, and sleeping on their arms at night. A few years later we find Peter Jefferson ordering from Henry Wetherburn, innkeeper, the biggest bowl of arrack punch ever made, and trading the same with William Randolph for two hundred acres of land. We are not surprised to find that life was a brief enjoyment. Little Mary Ball, demurely reading from the tombstones in the old St. Stephen's church, had small occasion for arithmetic beyond the numbers of thirty or forty years—at which age, having Piously lived and comfortably died, leaving the sweet perfume of a good reputation, these light-hearted good livers went to sleep behind their monuments. Of course the guardians of the infant colony spent many an anxious hour evolving schemes for the control of excessive feasting and junketing. The clergy were forced to ignore excesses, not daring to reprove them for fear of losing a good living. Their brethren across the seas cast longing eyes upon Virginia. It was an age of intemperance. The brightest wits of England, her poets and statesmen, were hard drinkers. All my hopes terminate, said Dean Swift in 1709, in being made Bishop of Virginia. There the Dean, had he been so inclined, could hope for the high living and hard drinking which were in fashion. There, too, in the tolerant atmosphere of a new country, he might—who knows?—have felt free to avow his marriage with the unhappy Stella.
- 72. In Virginia the responsibility of curbing the fun-loving community devolved upon the good burgesses, travelling down in their sloops to hold session at Williamsburg. We find them making laws restraining the jolly planters. A man could be presented for gaming, swearing, drunkenness, selling crawfish on Sunday, becoming engaged to more than one woman at a time, and, as we have said, there was always the ducking-stool for brabbling women who go about from house to house slandering their neighbors:—a melancholy proof that even in those Arcadian days the tongue required control.
- 73. CHAPTER IX MARY BALL'S GUARDIAN AND HER GIRLHOOD Except for the bequest in her brother-in-law's will, nothing whatever is known of Mary Ball for nine years—indeed, until her marriage with Augustine Washington in 1730. The traditions of these years are all based upon the letters found by the Union soldier,—genuine letters, no doubt, but relating to some other Mary Ball who, in addition to the flaxen hair and May-blossom cheeks, has had the honor of masquerading, for nearly forty years, as the mother of Washington, and of having her story and her letters placed reverently beneath the corner-stone of the Mary Washington monument. Mary Ball, only thirteen years old when her mother died, would naturally be taken to the Westmoreland home of her sister Elizabeth, wife of Samuel Bonum and only survivor, besides herself, of her mother's children. Elizabeth was married and living in her own house seven years before Mrs. Hewes died. The Bonum residence was but a few miles distant from that of Mrs. Hewes, and a mile and a half from Sandy Point, where lived the well-beloved and trusty friend George Eskridge. Major Eskridge seated Sandy Point in Westmoreland about 1720. The old house was standing until eight years ago, when it was destroyed by fire. He had seven children; the fifth child, Sarah, a year older than Mary Ball and doubtless her friend and companion. Under the tutelage and government of a man of wealth, eminent in his profession of the law, the two little girls would naturally be well and faithfully instructed. We can safely assume, considering all these circumstances, that Mary Ball's girlhood was spent in the Northern Neck of Virginia, and at the homes of Major Eskridge and her only sister; and that these faithful guardians provided her with
- 74. as liberal an education as her station demanded and the times permitted there cannot be the least doubt. Her own affectionate regard for them is emphatically proven by the fact that she gave to her first-born son the name of George Eskridge, to another son that of Samuel Bonum, and to her only daughter that of her sister Elizabeth. Tradition tells us that in the latter part of the seventeenth century, George Eskridge, who was a young law student, while walking along the shore on the north coast of Wales, studying a law-book, was suddenly seized by the Press Gang, carried aboard ship and brought to the colony of Virginia. As the custom was, he was sold to a planter for a term of eight years. During that time, he was not allowed to communicate with his friends at home. He was treated very harshly, and made to lodge in the kitchen, where he slept, because of the cold, upon the hearth. On the day that his term of service expired he rose early, and with his mattock dislodged the stones of the hearth. Upon his master's remonstrance, he said, The bed of a departing guest must always be made over for his successor; and throwing down his mattock he strode out of the house, taking with him the law-book which had been his constant companion during his years of slavery. He returned to England, completed his law studies, was admitted to the bar, and, returning to Virginia, was granted many thousand acres of land, held several colonial positions, and became eminent among the distinguished citizens of the Northern Neck,—the long, narrow strip of land included between the Potomac and the Rappahannock rivers. His daughter, Sarah, married Willoughby Newton, and lived near Bonum Creek in Westmoreland. The family intermarried, also, with the Lee, Washington, and other distinguished families in the Northern Neck.
- 75. CHAPTER X YOUNG MEN AND MAIDENS OF THE OLD DOMINION The social setting for Mary Ball—now a young lady—is easily defined. It matters little whether she did or did not visit her brother in England. She certainly belonged to the society of Westmoreland, the finest, says Bishop Meade, for culture and sound patriotism in the Colony. Around her lived the families of Mason, Taliaferro, Mountjoy, Travers, Moncure, Mercer, Tayloe, Ludwell, Fitzhugh, Lee, Newton, Washington, and others well known as society leaders in 1730. If she was, as her descendants claim for her, The Toast of the Gallants of Her Day, these were the Gallants,—many of them the fathers of men who afterward shone like stars in the galaxy of revolutionary heroes. The gallants doubtless knew and visited their tide-water friends,— the Randolphs, Blands, Harrisons, Byrds, Nelsons, and Carters,—and, like them, followed the gay fashions of the day. They wrote sonnets and acrostics and valentines to their Belindas, Florellas, Fidelias, and Myrtyllas—the real names of Molly, Patsy, Ann, and Mary being reckoned too homespun for the court of Cupid. These gallants wore velvet and much silk; the long vests that Charles the Second had invented as a fashion for gentlemen of all time; curled, powdered wigs, silver and gold lace; silken hose and brilliant buckles. Many of them had been educated abroad, or at William and Mary College,— where they had been rather a refractory set, whose enormities must be winked at,—even going so far as to keep race-horses at ye college, and bet at ye billiard and other gaming tables. Whatever their sins or shortcomings, they were warm-hearted and honorable, and most chivalrous to women. It was fashionable to present locks
- 76. of hair tied in true-lovers' knots, to tame cardinal-birds and mocking- birds for the colonial damsels, to serenade them with songs and stringed instruments under their windows on moonlight nights, to manufacture valentines of thinnest cut paper in intricate foldings, with tender sentiments tucked shyly under a bird's wing or the petal of a flower. With the youthful dames themselves, in hoop, and stiff bodice, powder and craped tresses, who cut watch-papers and worked book-marks for the gallants, we are on terms of intimacy. We know all their tricks and manners, through the laughing Englishman, and their own letters. An unpublished manuscript still circulates from hand to hand in Virginia, under oath of secrecy, for it contains a tragic secret, which reveals the true character of the mothers of Revolutionary patriots. These letters express high sentiment in strong, vigorous English, burning with patriotism and ardent devotion to the interests of the united colonies—not alone to Virginia. The spelling, and absurdly plentiful capitals, were those of the period, and should provoke no criticism. Ruskin says, no beauty of execution can outweigh one grain or fragment of thought. Beauty of execution and good spelling, according to modern standards, do not appear in the letters of Mary Ball and her friends, but they are seasoned with many a grain of good sense and thought. Of course we cannot know the names of her best friends. Her social position entitled her to intimacy with the sisters of any or all of the gallants we have named. She might have known Jane Randolph, already giving her heart to plain Peter Jefferson, and destined to press to her bosom the baby fingers that grew to frame the Declaration of Independence; or Sarah Winston, whose brilliant talents flashed in such splendor from the lips of Patrick Henry; or ill- starred Evelyn Byrd, whose beauty had fired the sluggish veins of George II and inspired a kingly pun upon her name, Much have I heard, lady, of thy fair country, but of the beauty of its birds I know but now,—all these and more; to say nothing of the mother of Sally and Molly Cary, of Lucy Grymes, of Betsy Fauntleroy, and of Mary
- 77. Bland, each of whom has been claimed by Lossing and others to be the Lowland beauty, to whom her illustrious son wrote such wonderful sonnets, but quite impossible in the case of Mary Bland, seeing she was born in 1704, and was some years older than his mother. They were a light-hearted band of maidens in these pre- Revolutionary days in the Old Dominion! They had no dreams sadder than mystic dreams on bride's cake, no duties except those imposed by affection, no tasks too difficult, no burdens too heavy. They sang the old-time songs, and danced the old-time dances, and played the old-time English games around the Christmas fires, burning nuts, and naming apple seeds, and loving their loves with an A or a B, even although my Lady Castlemaine, of whom no one could approve, had so entertained her very doubtful friends a hundred years before. They had the Pyrrhic dances, but they had the Pyrrhic phalanx as well! The nobler and manlier lessons were not forgotten in all the light-hearted manners of the age.
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