Classical Electromagnetic Fields

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.