19

1 Classical Electromagnetic Fields








In this book we present the basic ideas needed to understand how laser light interacts with various forms of matter. Among the important consequences is an understanding of the laser itself. The present chapter summarizes clas- sical electromagnetic ?elds, which describe laser light remarkably well. The chapter also discusses the interaction of these ?elds with a medium con- sisting of classical simple harmonic oscillators. It is surprising how well this simple model describes linear absorption, a point discussed from a quantum mechanical point of view in Sect. 3.3. The rest of the book is concerned with nonlinear interactions of radiation with matter. Chapter 2 generalizes the classical oscillator to treat simple kinds of nonlinear mechanisms, and shows us a number of phenomena in a relatively simple context. Starting with Chap. 3, we treat the medium quantum mechanically. The combination of a classical description of light and a quantum mechanical description of matter is called the semiclassical approximation. This approximation is not always justi?ed (Chaps. 13–19), but there are remarkably few cases in quantum op- tics where we need to quantize the ?eld.
In the present chapter, we limit ourselves both to classical electromagnetic ?elds and to classical media. Section 1.1 brie?y reviews Maxwell’s equations in a vacuum. We derive the wave equation, and introduce the slowly-varying amplitude and phase approximation for the electromagnetic ?eld. Section 1.2 recalls Maxwell’s equations in a medium. We then show the roles of the in- phase and in-quadrature parts of the polarization of the medium through which the light propagates, and give a brief discussion of Beer’s law of light absorption. Section 1.3 discusses the classical dipole oscillator. We introduce the concept of the self-?eld and show how it leads to radiative damping. Then we consider the classical Rabi problem, which allows us to introduce the classical analog of the optical Bloch equations. The derivations in Sects. 1.1–1.3 are not necessarily the simplest ones, but they correspond as closely as possible to their quantum mechanical counterparts that appear later in the book.
Section 1.4 is concerned with the coherence of the electromagnetic ?eld.
We review the Young and Hanbury Brown-Twiss experiments. We intro- duce the notion of nth order coherence. We conclude this section by a brief


comment on antibunching, which provides us with a powerful test of the quantum nature of light.
With knowledge of Sects. 1.1–1.4, we have all the elements needed to un- derstand an elementary treatment of the Free-Electron Laser (FEL), which
is presented in Sect. 1.5. The FEL is in some way the simplest laser to un- derstand, since it can largely be described classically, i.e., there is no need to quantize the matter.


1.1 Maxwell’s Equations in a Vacuum

In the absence of charges and currents, Maxwell’s equations are given by
?•B = 0 , (1.1)
?•E = 0 , (1.2)
?B
?×E = ? ?t , (1.3)
?E
?×B = ?0?0 ?t , (1.4)
where E is the electric ?eld, B is the magnetic ?eld, ?0 is the permeability of the free space, and ?0 is the permittivity of free space (in this book we use MKS units throughout). Alternatively it is useful to write c2 for 1/?0?0, where c is the speed of light in the vacuum. Taking the curl of (1.3) and
substituting the rate of change of (1.4) we ?nd
1 ?2E
?×?×E = ? c2 ?t2 . (1.5)
2

This equation can be simpli?ed by noting that ?×? = ?(?•) ??
using (1.2). We ?nd the wave equation

and

2 1 ?2E
? E ? c2 ?t2 = 0 . (1.6)
This tells us how an electromagnetic wave propagates in a vacuum. By direct substitution, we can show that
E(r, t) = E0f (K•r ? ?t) (1.7) is a solution of (1.6) where f is an arbitrary function, E0 is a constant, ?
is an oscillation frequency in radians/second (2? × Hz), K is a constant
vector in the direction of propagation of the ?eld, and having the magnitude K ? |K| = ?/c. This solution represents a transverse plane wave propagating along the direction of K with speed c = ?/K.
A property of the wave equation (1.6) is that if E1(r, t) and E2(r, t) are solutions, then the superposition a1E1(r, t) + a2E2(r, t) is also a solution,

1.1 Maxwell’s Equations in a Vacuum 3

where a1 and a2 are any two constants. This is called the principle of super- position. It is a direct consequence of the fact that di?erentiation is a linear operation. In particular, the superposition
E(r, t) = . Ek f (Kk •r ? ?t) (1.8)
k

is also a solution. This shows us that nonplane waves are also solutions of the wave equation (1.6).
Quantum opticians like to decompose electric ?elds into “positive” and “negative” frequency parts
E(r, t) = E+(r, t) + E?(r, t) , (1.9)
where E+(r, t) has the form

E+(r, t) = 1 .
2
n


En(r)e?i?n t , (1.10)


where En(r) is a complex function of r, ?n is the corresponding frequency,
and in general
E?(r, t) = [E+(r, t)]? . (1.11)
In itself this decomposition is just that of the analytic signal used in classical coherence theory [see Born and Wolf (1970)], but as we see in Chap. 13, it has deep foundations in the quantum theory of light detection. For now we consider this to be a convenient mathematical trick that allows us to work with exponentials rather than with sines and cosines. It is easy to see
that since the wave equation (1.6) is real, if E+(r, t) is a solution, then so is E?(r, t), and the linearity of (1.6) guarantees that the sum (1.9) is also a
solution.
In this book, we are concerned mostly with the interaction of monochro- matic (or quasi-monochromatic) laser light with matter. In particular, con-
sider a linearly-polarized plane wave propagating in the z-direction. Its elec-
tric ?eld can be described by

E+(z, t) = 1 xˆE (z, t)ei[Kz??t??(z,t)] , (1.12)
2 0

where xˆ is the direction of polarization, E0(z, t) is a real amplitude, ? is the central frequency of the ?eld, and the wave number K = ?/c. If E(z, t) is truly monochromatic, E0 and ? are constants in time and space. More
generally, we suppose they vary su?ciently slowly in time and space that the
following inequalities are valid:

. ?E0 .
. .
. .
. .
. ?E0 .
. .
. ?z . KE0 , (1.14)
. .
. .
. .
. ?t . ? , (1.15)
. .
. ?? .
. ?z . K . (1.16)
. .

These equations de?ne the so-called slowly-varying amplitude and phase ap- proximation (SVAP), which plays a central role in laser physics and pulse propagation problems. Physically it means that we consider light waves whose amplitudes and phases vary little within an optical period and an optical wavelength. Sometimes this approximation is called the SVEA, for slowly- varying envelope approximation.
The SVAP leads to major mathematical simpli?cations as can be seen by substituting the ?eld (1.12) into the wave equation (1.6) and using (1.13–1.16)
to eliminate the small contributions E¨0, ?¨, Err, ?rr, and E? ??. We ?nd
?E0 + 1 ?E0 = 0 , (1.17)
?z c ?t
?? + 1 ?? = 0 , (1.18)
?z c ?t
where (1.17) results from equating the sum of the imaginary parts to zero and (1.18) from the real parts. Thus the SVAP allows us to transform the second-order wave equation (1.6) into ?rst-order equations. Although this does not seem like much of an achievement right now, since we can solve (1.6) exactly anyway, it is a tremendous help when we consider Maxwell’s equations in a medium. The SVAP is not always a good approximation. For example, plasma physicists who shine light on targets typically must use the second-order equations. In addition, the SVAP approximation also neglects the backward propagation of light.


1.2 Maxwell’s Equations in a Medium

Inside a macroscopic medium, Maxwell’s equations (1.1–1.4) become
?•B = 0 , (1.19)
?•D = ?free , (1.20)
?B
?×E = ? ?t , (1.21)
?D

?×H = J +

. (1.22)
?t


These equations are often called the macroscopic Maxwell’s equations, since they relate vectors that are averaged over volumes containing many atoms but which have linear dimensions small compared to signi?cant variations in the applied electric ?eld. General derivations of (1.19–1.22) can be very complicated, but the discussion by Jackson (1999) is quite readable. In (1.20, 1.22), the displacement electric ?eld D is given for our purpose by
D = ?E + P , (1.23)
where the permittivity ? includes the contributions of the host lattice and P is the induced polarization of the resonant or nearly resonant medium we wish to treat explicitly. For example, in ruby the Al2O3 lattice has an index
of refraction of 1.76, which is included in ?. The ruby color is given by Cr ions
which are responsible for laser action. We describe their interaction with light
by the polarization P. Indeed much of this book deals with the calculation of P for various situations. The free charge density ?free in (1.20) consists of
all charges other than the bound charges inside atoms and molecules, whose e?ects are provided for by P. We don’t need ?free in this book. In (1.22), the
magnetic ?eld H is given by

B
H = ? ? M , (1.24)
where ? is the permeability of the host medium and M is the magnetization of the medium. For the media we consider, M = 0 and ? = ?0. The current
density J is often related to the applied electric ?eld E by the constitutive relation J = ?E, where ? is the conductivity of the medium.
The macroscopic wave equation corresponding to (1.6) is given by com- bining the curl of (1.21) with (1.23, 1.24). In the process we ?nd ?×?×E =
2 2
?(?•E) ?? E c ?? E. In optics ?•E c 0, since most light ?eld vectors
vary little along the directions in which they point. For example, a plane- wave ?eld is constant along the direction it points, causing its ? • E to vanish
identically. We ?nd

2 ?J

1 ?2E

?2P

?? E + ? ?t + c2 ?t2 = ?? ?t2 , (1.25)
where c = 1/??? is now the speed of light in the host medium. In Chap. 7 we use the ?J/?t term to simulate losses in a Fabry–Perot resonator.
We drop this term in our present discussion.
For a quasi-monochromatic ?eld, the polarization induced in the medium is also quasi-monochromatic, but generally has a di?erent phase from the ?eld. Thus as for the ?eld (1.9) we decompose the polarization into positive and negative frequency parts
P(z, t) = P+(z, t) + P?(z, t) ,
but we include the complex amplitude P(z, t) = N| X(z, t), that is,


P+(z, t) = 1 xˆ
2
1


(z, t)ei[Kz??t??(z,t)]

= xˆN (z) X(z, t)ei[Kz??t??(z,t)] . (1.26) 2

Here N (z) is the number of systems per unit volume, | is the dipole mo- ment constant of a single oscillator, and X(z, t) is a complex dimensionless
amplitude that varies little in an optical period or wavelength. In quantum mechanics, | is given by the electric dipole matrix element ?. Since the po-
larization is real, we have

P?(z, t) = [P+(z, t)]? . (1.27)
It is sometimes convenient to write X(z, t) in terms of its real and imaginary parts in the form
X ? U ? iV . (1.28)

The classical real variables U and V have quantum mechanical counterparts that are components of the Bloch vector U eˆ1 + V eˆ2 + W eˆ3, as discussed
in Sect. 4.3. The slowly-varying amplitude and phase approximation for the
polarization is given by
. ?U .
. ?t . ?|U | , (1.29)
. .
. ?V .
. ?t . ?|V | . (1.30)
. .


or equivalently by

. ?X .
. ?t . ?|X| .

. .

We generalize the slowly-varying Maxwell equations (1.17, 1.18) to include the polarization by treating the left-hand side of the wave equation (1.25) as before and substituting (1.26) into the right-hand side of (1.25). Using
(1.29, 1.30) to eliminate the time derivatives of U and V and equating real
imaginary parts separately, we ?nd
?E0 + 1 ?E0 = K K

?z c ?t

? 2? Im(P) = 2? N (z)| V (1.31)

. ??

1 ?? . K K

E0 +
?z c ?t

= ? 2? ReP = ? 2? N (z)| U (1.32)


These two equations play a central role in optical physics and quantum optics. They tell us how light propagates through a medium and speci?cally how the real and imaginary parts of the polarization act. Equation (1.31) shows that the ?eld amplitude is driven by the imaginary part of the polarization. This in-quadrature component gives rise to absorption and emission.


Equation (1.32) allows us to compute the phase velocity with which the electromagnetic wave propagates in the medium. It is the real part of the polarization, i.e, the part in-phase with the ?eld, that determines the phase velocity. The e?ects described by this equation are those associated with the index of refraction of the medium, such as dispersion and self focusing.
Equations (1.31, 1.32) alone are not su?cient to describe physical prob-
lems completely, since they only tell us how a plane electromagnetic wave responds to a given polarization of the medium. That polarization must still be determined. Of course, we know that the polarization of a medium is in?uenced by the ?eld to which it is subjected. In particular, for atoms or molecules without permanent polarization, it is the electromagnetic ?eld it- self that induces their polarization! Thus the polarization of the medium drives the ?eld, while the ?eld drives the polarization of the medium. In gen- eral this leads to a description of the interaction between the electromagnetic ?eld and matter expressed in terms of coupled, nonlinear, partial di?eren- tial equations that have to be solved self-consistently. The polarization of a medium consisting of classical simple harmonic oscillators is discussed in Sect. 1.3 and Chap. 2 discusses similar media with anharmonic (nonlinear) oscillators. Two-level atoms are discussed in Chaps. 3–7.
There is no known general solution to the problem, and the art of quantum optics is to make reasonable approximations in the description of the ?eld
and/or medium valid for cases of interest. Two general classes of problems reduce the partial di?erential equations to ordinary di?erential equations:
1) problems for which the amplitude and phase vary only in time, e.g., in
a cavity, and 2) problems for which they vary only in space, i.e., a steady state exists. The second of these leads to Beer’s law of absorption,1 which we consider here brie?y. We take the steady-state limit given by
?E0 = 0
?t
in (1.31). We further shine a continuous beam of light into a medium that responds linearly to the electric ?eld as described by the slowly-varying com- plex polarization
P = N (z)| (U ? iV ) ? N (z)| X = ?(?r + i?rr)E0(z) , (1.33) where ?r and ?rr are the real and imaginary parts of the linear susceptibility
?. This susceptibility is another useful way of expressing the polarization.
Substituting the in-quadrature part of P into (1.31), we obtain

dE0 =
dz

K
? 2 ?rrE0

= ?Re{?}E0 , (1.34)

1 Beer’s law is perhaps more accurately called Bouguier-Lambert-Beer’s law. We call it Beer’s law due to popular usage.



where


iK P


iKN (z)| X


N (z)|

? = 2? E = 2? E

= ?K 2?E

(V + iU ) (1.35)

is called the complex amplitude absorption coe?cient. We use an amplitude absorption coe?cient instead of an intensity coe?cient to be consistent with coupled-mode equations important for phase conjugation and other nonlinear
mode interactions. If ?rr is independent of E0, (1.34) can be readily integrated
to give
E0(z) = E0(0)e?Re{?}z . (1.36)
Taking the absolute square of (1.36) gives Beer’s law for the intensity
I(z) = I(0)e?2Re{?}z . (1.37)

We emphasize that this important result can only be obtained if ? is in- dependent of I, that is, if the polarization (1.33) of the medium responds linearly to the ?eld amplitude E0. Chapter 2 shows how to extend (1.33) to
treat larger ?elds, leading to the usual discussion of nonlinear optics. Time
dependent ?elds also lead to results such as (12.27) that di?er from Beer’s law. For these, (1.33) doesn’t hold any more (even in the weak-?eld limit) if the medium cannot respond fast enough to the ?eld changes. This can lead to e?ects such as laser lethargy, for which the ?eld is absorbed or ampli?ed according to the law


where b is some constant.

I(z) ? exp(?b?z) , (1.38)

The phase equation (1.32) allows us to relate the in-phase component of the susceptibility to the index of refraction n. As for the amplitude (1.34), we consider the continuous wave limit, for which ??/?t = 0. This gives
d?/dz = ?K?r/2 . (1.39) Expanding the slowly varying phase ?(z) c ?0 + zd?/dz, we ?nd the total
phase factor

Kz ? ?t ? ? c ?[(K ? d?/dz)z/? ? t] ? ?0
= ?[(1 + ?r/2)z/c ? t] ? ?0
= ?(z/v ? t) ? ?0 .
Noting that the velocity component2 v is also given by c/n, we ?nd the index of refraction (relative to the host medium)
n = 1 + ?r/2 . (1.40)

2 Note that the character v, which represents a speed, is di?erent from the char- acter ?, which represents a circular frequency (radians per second).


In coupled-mode problems (see Sects. 2.2, 11.2) and pulse propagation, in- stead of (1.12) it is more convenient to decompose the electric ?eld in terms
of a complex amplitude E (z, t) ? E0(z, t) exp(?i?), that is,


1
E(z, t) = 2 E (z, t)e


i(Kz??t)


+ c.c.. (1.41)

The polarization is then also de?ned without the explicit exp(i?) as


1
P (z, t) = 2 P(z, t)e


i(Kz??t)


+ c.c.. (1.42)


Substituting these forms into the wave equation (1.25) and neglecting small terms like ?2E /?t2, ?2P/?t2, and ?P/?t, and equating the coe?cients of ei(Kz??t) on both sides of the equation, we ?nd the slowly-varying Maxwell’s
equation

?E + 1 ?E = i

. (1.43)

?z c ?t

2? P

Note that in equating the coe?cients of ei(Kz??t), we make use of our assump- tion that P(z, t)varies little in a wavelength. Should it vary appreciably in a wavelength due, for example, to a grating induced by an interference fringe,
we would have to evaluate a projection integral as discussed for standing wave interactions in Sect. 5.3.
In a signi?cant number of laser phenomena, the plane-wave approximation
used in this chapter is inadequate. For these problems, Gaussian beams may provide a reasonable description. A simple derivation of the Gaussian beam
as a limiting case of a spherical wave exp(iKr)/r is given in Sect. 7.7.

Group velocity

The preceding discussion introduced the velocity v = c/n, which is the veloc- ity at which the phase of a monochromatic wave of frequency ? propagates in a medium with index of refraction n(?), or phase velocity. Consider now the situation of two plane monochromatic waves of same amplitude E that
di?er slightly in frequency and wave number,
E(z, t) = E ei[(k0 +?k)z?(?0 +??)t] + E ei[(k0 ??k)z?(?0 ??)t]

.
= 2E ei(k0 z??0 t) cos ??

. ?k ..
t ? ?? z .


When adding a group of waves with a small spread of wave numbers and frequencies about k0 and ?0, we ?nd similarly that the total ?eld consists of a carrier wave with phase velocity v = c/n and group velocity

d?
vg = dk . (1.44)


In case the absorption of light at the frequency ?0 is su?ciently weak to be negligible, vg can be taken to be real and with k = ?n(?)/c we ?nd readily

d? c

vg = dk = (n + ?dn/d?)

. (1.45)

We observe that in regions of “normal dispersion”, dn/d? > 0, the group velocity is less than the phase velocity. However, the situation is reversed in
regions of “anomalous dispersion”, dn/d? < 0. Indeed vg can even exceed
c in this region. This has been the origin of much confusion in the past, in
particular it has been mentioned that this could be in con?ict with special
relativity. This, however, is not the case. This is incorrect, because the group velocity is not in general a signal velocity. This, as many other aspects of“fast light“ and “slow light,” is discussed very clearly in Milonni (2005).
Chapter 12 discusses how quantum interference e?ects such as electromag-
netically induced transparency can be exploited to dramatically manipulate the group velocity of light, resulting in particular in the generation of “slow light.”


Linear Dipole Oscillator

As a simple and important example of the interaction between electromag- netic waves and matter, let us consider the case of a medium consisting of classical damped linear dipole oscillators. As discussed in Chap. 3, this model describes the absorption by quantum mechanical atoms remarkably well. Speci?cally we consider a charge (electron) cloud bound to a heavy pos- itive nucleus and allowed to oscillate about its equilibrium position as shown
in Fig. 1.1. We use the coordinate x to label the deviation from the equilib- rium position with the center of charge at the nucleus. For small x it is a
good approximation to describe the motion of the charged cloud as that of a
damped simple harmonic oscillator subject to a sinusoidal electric ?eld. Such a system obeys the Abraham-Lorentz equation of motion
x¨(t) + 2?x? (t) + ?2x(t) = e E(t) , (1.46)
m
where ? is the natural oscillation frequency of the oscillator, and the dots stand for derivatives with respect to time. Note that since oscillating charges radiate, they lose energy. The end of this section shows how this process
leads naturally to a damping constant ?. Quantum mechanically this decay
is determined by spontaneous emission and collisions.
The solution of (1.44) is probably known to the reader. We give a deriva- tion below that ties in carefully with the corresponding quantum mechani- cal treatments given in Chaps. 4, 5. Chapter 2 generalizes (1.44) by adding
nonlinear forces proportional to x2 and x3 [see (2.1)]. These forces lead to




Fig. 1.1. Negative charge cloud bound to a heavy positive nucleus by Coulomb attraction. We suppose that some mysterious forces prevents the charge cloud from collapsing into the nucleus


coupling between ?eld modes producing important e?ects such as sum and di?erence frequency generation and phase conjugation. As such (1.44) and its nonlinear extensions allow us to see many “atom”-?eld interactions in a simple classical context before we consider them in their more realistic, but complex, quantum form.
We suppose the electric ?eld has the form


1
E(t) = 2 E0e?


i?t


+ c.c., (1.47)


where E0 is a constant real amplitude. In general the phase of x(t) di?ers from that of E(t). This can be described by a complex amplitude for x, that
is,

1
x(t) = 2 x0X(t)e?

i?t

+ c.c., (1.48)

where X(t) is the dimensionless complex amplitude of (1.26). In the following we suppose that it varies little in the damping time 1/?, which is a much more
severe approximation than the SVAP. Our problem is to ?nd the steady-state solution for X(t).
As in the discussion of (1.33, 1.34), we substitute (1.45, 1.46) into (1.44), neglect the small quantities X¨ and ?X? , and equate positive frequency com-
ponents. This gives


X? = ?[? + i(?2 ? ?2)/2?] X +


ieE0 2?mx0


. (1.49)

In steady state (X?


= 0), this gives the amplitude
X = ieE0/2?mx0
? + i(?2 ? ?2)/2?




, (1.50)

and hence the displacement

i eE0

e?i?t

x(t) = 2 2m? ? + i(?2

+ c.c. (1.51)
? ?2)/2?


We often deal with the near resonance, that is, the situation where |? ? ? ? +?. For this case we can make the classical analog of the rotating-wave approximation de?ned in Sect. 3.2. Speci?cally we approximate ?2 ? ?2 by
?2 ? ?2 c 2?(? ? ?) . (1.52) This reduces (1.48, 1.49) to

X = ieE0/2?mx0
? + i(? ? ?)


, (1.53)

i eE0

ei?t

x(t) = 2 2m? ? + i(?

+ c.c. (1.54)
? ?)


Equation (1.52) shows that in steady state the dipole oscillates with the same frequency as the driving ?eld, but with a di?erent phase. At resonance
(? = ?), (1.52) reduces to

x(t, ? = ?) = eE0
2m??


sin ?t , (1.55)


that is, the dipole lags by ?/2 behind the electric ?eld (1.45), which oscillates as cos ?t. The corresponding polarization of the medium is P = N ex(t), where N is the number of oscillators per unit volume. Substituting this along
with (1.52) into (1.35), we ?nd the complex amplitude Beer’s law absorption
coe?cient
? = K N e ?
2?? 2m? ? + i(? ? ?)
or
? = ?0?[?? i(?? ?)] , (1.56)
?2 + (? ? ?)2
where the resonant absorption coe?cient ?0 = KN e2/4??m?. The real
part of this expression shows the Lorentzian dependence observed in actual
absorption spectra (see Fig. 1.2). The corresponding quantum mechanical absorption coe?cient of (5.29) di?ers from (1.54) in three ways:
1. ?2 + (? ? ?)2 is replaced by ?2(1 + I) + (? ? ?)2 ?

2. N becomes negative for gain media
3. e2/2m? is replaced by ?/k

? (1.57)
?


For weak ?elds interacting with absorbing media, only the third of these di?erences needs to be considered and it just de?nes the strength of the dipole moment being used. Hence the classical model mirrors the quantum mechanical one well for linear absorption (for a physical interpretation of this result, see Sect. 3.2).




?rr


?r


? ? ?




Fig. 1.2. Absorption (Lorentzian bell shape) and index parts of the complex ab- sorption coe?cient of (1.54)


Identifying the real and imaginary parts of (1–47) and using (1.33), we obtain the equations of motion for the classical Bloch-vector components U and V
U? = ?(? ? ?)V ? ?U , (1.58)
V? = (? ? ?)U ? ?V ? eE0/2m?x0 . (1.59)

Comparing (1.57) with (4.49) (in which ? = 1/T2), we see that the E0 term is multiplied by ?W , which is the third component of the Bloch vector. This
component equals the probability that a two-level atom is in the upper level
minus the probability that it is in the lower level. Hence we see that the classical (1.57) is reasonable as long as W c ?1, i.e., so long as the atom is
in the lower level.
From the steady-state value of X given by (1.51), we have the steady-state
U and V values

U = eE0

? ? ?

(1.60)


and

2m?x0 ?2 + (? ? ?)2
eE0 ?

V = ? 2m?x

?2 + (?

?)2 . (1.61)

Since (1.44) is linear, once we know the solution for the single frequency ?eld (1.45), we can immediately generalize to a multifrequency ?eld simply by taking a corresponding superposition of single frequency solutions. The
various frequency components in x(t) oscillate independently of one another.
In contrast the nonlinear media in Chap. 2 and later chapters couple the
modes. Speci?cally, consider the multimode ?eld

E(z, t) = 1 .
2
n


En(z) ei(Kn z??n t)


+ c.c., (1.62)


where we allow the ?eld amplitudes to be slowly varying functions of z and to
be complex since they do not in general have the same phases. The solution for the oscillator displacement x(t) at the position z is a superposition of
solutions like (1.46), namely,

x(t) = 1 .
2
n


x0nXne


i(Kn z??n t)


+ c.c., (1.63)

where mode nrs oscillator strength is proportional to x0n and the coe?cients
eE n /mx0n

Xn = ?2

? ?n ? 2i?n?

. (1.64)


Here we don’t make the resonance approximation of (1.50), since some of the modes may be o? resonance. The steady-state polarization P (z, t) of a
medium consisting of such oscillators is then given by

P (z, t) = 1 .
2
n


Pn(z)e


i(Kn z??n t)


+ c.c., (1.65)

where Pn(z) is given by N (z)ex0nXn. In Sect. 2.1, we ?nd that higher-order terms occur when nonlinearities are included in the equation of motion (1.44). These terms couple the modes and lead to anharmonic response. Finally, we note that the multimode ?eld (1.60) and the polarization (1.63) have the
same form in the unidirectional ring laser of Chap. 7, except that in a high-Q cavity the mode amplitudes En and polarization components Pn are functions of t, rather than z.


Radiative Damping

We now give a simple approximate justi?cation for the inclusion of a damping coe?cient ? in (1.44). As a charge oscillates it radiates electromagnetic energy and consequently emits a “self-?eld” Es. We need to ?nd the in?uence of this
self-?eld back on the charge’s motion in a self-consistent fashion. We ?nd
that the main e?ect is the exponential damping of this motion as given by (1.44). Speci?cally, we consider the equation governing the charge’s motion
under the in?uence of the self-?eld Es:

x¨ + ?2x = e E
m s

, (1.66)

which is just Newton’s law with the Lorentz force
Fs = e(Es + v × Bs) (1.67)
in the limit of small charge velocities (v c), where the magnetic part of the Lorentz force may be neglected.


While we don’t know the explicit form of Es, we can calculate its e?ects
using the conservation of energy. We evaluate the force Frad of the radiating
charge by equating the work it expends on the charge (during a time interval long compared to the optical period 1/?) to minus the energy radiated by
the charge during that time

¸ t+?t

t


Frad • v dtr = ?

¸ t+?t

t


(radiated power) dtr . (1.68)

To calculate the radiated power, we note that the instantaneous electromag- netic energy ?ow is given by the Poynting vector
1
S = Es×Bs , (1.69)
0
where for simplicity we suppose that the “host medium” is the vacuum. We note that the electric ?eld radiated in the far ?eld of the dipole is
e . n×(n×v? ) .

Es(R, t) = 4?? c2

R t?R/c

(1.70)

as shown in Fig. 1.3. The corresponding magnetic ?eld is Bs(R, t) =

c?1n×Es(R, t). In both expressions the dipole acceleration v?

is evaluated

at the retarded time t ? R/c and n is the unit vector R/R. Inserting these
expressions into (1.66), we ?nd the Poynting vector [ Jackson (1999)]

1

S =
?0c

(Es • Es)n

e2 1
=

1
(n×v? )2n

16?2?2c4 ?0c R2
e2?? 2 sin2 ?
= 16?2? c3R2 n . (1.71)














Fig. 1.3. Butter?y pattern given by (1.69) and emitted by an oscillating dipole. The vector gives the direction and relative magnitude of the Poynting vector S as a function of ?.


The total power radiated is given by integration of S over a sphere surround- ing the charge. Noting that

¸ 2?

0

we ?nd

¸ ?
d? d? sin3 ? = ?2?
0

¸ ?1

1


d(cos ?)(1 ? cos2 ?) = 8?/3 , (1.72)

¸ 2 e2 2

S • da = 3 4?? c3 ??

, (1.73)

which is the Larmor power formula for an accelerated charge. We now sub- stitute (1.71) into (1.66) and integrate by parts. We encounter the integral

¸ t+?t
t

.t+?t
dt v? • v? = v? • v.
.t

¸ t+?t
?
t


dt v • v¨ .


Since v and its derivatives are periodic, the constant of integration on the right hand side has a maximum magnitude, while the integrals continue to
increase as ?t increases. Hence the constant can be dropped. Equating the
integrands, we ?nd the radiation force

2 e2
Frad = 3 4?? c3 v¨ . (1.74)
A more detailed analysis of this problem is given in Sect. 19.3 of Jackson (1999), where the in?nities associated with point-like charges are also dis- cussed.
Assuming that the radiative damping is su?ciently small that the motion of the dipole remains essentially harmonic, (1.72) yields
2 e2?2
Frad = mx¨ = ? 3 4?? c3 v , (1.75)
which indicates that radiation reaction acts as a friction on the motion of the charge. This implies a damping rate constant

1 1 e2?2

1 ?2r0

? = 4??

=
3 c3m 3

, (1.76)
c

where the classical radius of the electron is
e2 15
r0 = 4?? mc2 c 2.8 × 10?

meters . (1.77)

For 1 ?m radiation, ? = 2? × 1.8 MHz, which is in the range of decay values found in atoms. In cgs units the 4??0 in (1.74, 1.75).
With the replacement of e2/2m? by ?2/k, see (1.55), the classical decay rate (1.74) gives half the quantum mechanical decay rate (14.60). Here ? is the
reduced dipole matrix element between the upper and lower level transition.

1.4 Coherence 17

The other half of the decay rate is contributed by the e?ects of vacuum ?uctuations missing in a classical description. Note that in both the classical
and quantum mechanical cases, an ?2 term appears. In the quantum case,
this term results from the density of states of free space (14.46), while for the
classical case it comes from the acceleration of the electron. In some sense the density of states for the ?eld re?ects the fact that the ?eld itself is radiated by accelerating, oscillating charges. In free space the charge responsible for this ?eld is the bound electron itself, radiating a ?eld that acts back on the charge and causes it to emit radiation until no more downward transitions are possible. For further discussion, see Milonni (1986, 1984, 1994).


Coherence

Coherence plays a central role in modern physics. It is very hard to ?nd a single domain of physics where this concept is not applied. In this book we use it a great deal, speaking of coherent light, coherent transients, coherent propagation, coherent states, coherent excitation, etc. Just what is coherent? The answer typically depends on whom you ask! In a very general sense, a process is coherent if it is characterized by the existence of some well- de?ned deterministic phase relationship, or in other words, if some phase is not subject to random noise. This is a very vague de?nition, but general enough to encompass all processes usually called “coherent”. In this section and Sect. 13.5 we consider the coherence of classical light. Chapters 4, 12 discuss coherence in atomic systems.
The classic book by Born and Wolf (1970) gives a discussion of coherent
light in pre-laser terms. With the advent of the laser, a number of new e?ects have been discovered that have caused us to rethink our ideas about coherent light. In addition, the Hanbury Brown-Twiss experiment, which had nothing to do with lasers, plays an important role in this rethinking. Our discussion is based on the theory of optical coherence as developed by R. Glauber and summarized in his Les Houches lectures (1965).
We start with the famous Young double-slit experiment which shows how coherent light passing through two slits interferes giving a characteristic in- tensity pattern on a screen (see Fig. 1.4). Before going into the details of this experiment, we need to know how the light intensity is measured, either on a screen or with a photodetector. Both devices work by absorbing light. The absorption sets up a chemical reaction in the case of ?lm, and ionizes atoms or lifts electrons into a conduction band in the cases of two kinds of photodetec-tors. Section 13.5 shows by a quantum-mechanical analysis of
the detection process that these methods measure |E+(r, t)|2, rather than
|E(r, t)|2. This is why we performed the decomposition in (1.9).
Returning to Young’s double-slit experiment, we wish to determine
E+(r, t), where r is the location of the detector. E+(r, t) is made up of two
components, each coming from its respective slit

















Fig. 1.4. Young double-slit experiment illustrating how coherent light can interfere with itself

E+(r, t) = E+(r1, t1) + E+(r2, t2) , (1.78)
where r1 and r2 are locations of the slits and t1 and t2 are the retarded times
t1,2 = t ? s1,2/c (1.79)
s1 and s2 being the distances between the slits and the detector. From (1.76), the intensity at the detector is given by
|E+(r, t)|2 = |E+(r1, t1)|2 + |E+(r2, t2)|2
+2 Re[E?(r1, t1)E+(r2, t2)] , (1.80)

where we have made use of (1.9).
In general the light source contains noise. To describe light with noise we use a statistical approach, repeating the measurement many times and averaging the results. Mathematically this looks like
(|E+(r, t)2|) = (|E+(r1, t1)|2) + (|E+(r2, t2)|2)
+ 2Re(E?(r1, t1)E+(r2, t2)) , (1.81)
where the brackets (•••) stands for the ensemble average. Introducing the ?rst-order correlation function
G(1)(r1t1, r2t2) ? (E?(r1, t1)E+(r2, t2)) , (1.82)

we rewrite (1.79) as
(|E+(r, t)|2) = G(1)(r1t1, r1t1) + G(1)(r2t2, r2t2)
+2ReG(1)(r1t1, r2t2) . (1.83)
G(1)(riti, riti) is clearly a real, positive quantity, while G(1)(riti, rj tj ) is in general complex.