20

With the cross-correlation function rewritten as
G(1)(r1t1, r2t2) = |G(1)(r1t1, r2t2)|ei?(r1 t1 ,r2 t2 ) , (1.84) (1.81) becomes
(|E+(r, t)|2) = G(1)(r1t1, r1t1) + G(1)(r2t2, r2t2)
2|G(1)(r1t1, r2t2)| cos ?. (1.85)

The third term in (1.83) is responsible for the appearance of interferences.
We say that the highest degree of coherence corresponds to a light ?eld that produces the maximum contrast on the screen, where contrast is de?ned as
V = Imax ? Imin . (1.86)
Imax + Imin
Substituting (1.83) with cos ? = 1, we readily obtain
2|G(1)(r1t1, r2t2)|

V =
G(1)(r1t1, r1t1) + G(1)(r2t2, r2t2)

. (1.87)


The denominator in (1.85) doesn’t play an important role; G(1)(riti, riti) is just the intensity on the detector due to the ith slit and the denominator acts as a normalization constant. To maximize the contrast for a given source and
geometry, we need to maximize the numerator 2|G(1)(r1t1, r2t2)|. To achieve
this goal we note that according to the Schwarz inequality
G(1)(r1t1, r1t1)G(1)(r2t2, r2t2) ? |G(1)(r1t1, r2t2)|2 . (1.88)

The coherence function is maximized when equality holds, that is when
|G(1)(r1t1, r2t2)| = [G(1)(r1t1, r1t1)G(1)(r2t2, r2t2)]1/2 , (1.89)

which is the coherence condition used by Born and Wolf. As pointed out by Glauber, it is convenient to replace this condition by the equivalent expression
G(1)(r1t1, r2t2) = E ?(r1t1)E (r2t2) , (1.90)
where the complex function E (r1t1) is some function, not necessarily the electric ?eld. If G(1)(r1t1, r2t2) may be expressed in the form (1.88), we say that G(1) factorizes. This factorization property de?nes ?rst-order coherence: when (1.88) holds, the fringe contrast V is maximum.
This de?nition of ?rst-order coherence can be readily generalized to higher orders. A ?eld is said to have nth-order coherence if its mth-order correlation functions
G(m)(x1 ... xm, ym ... y1) = (E?(x1) ••• E?(xm)E+(ym) ••• E+(y1)) (1.91) factorize as


G(m)(x1 ... xm, ym ... y1) = E ?(x1) •••E ?(xm)E (ym) •••E (y1) (1.92)
for all m ? n. Here we use the compact notation xj = (rj , tj ), yj = (rm+j , tm+j ), and G(m) is a direct generalization of (1.80).
Before giving an example where second-order correlation functions play
a crucial role, we point out that although a monochromatic ?eld is coherent to all orders, a ?rst-order coherent ?eld is not necessarily monochromatic. One might be led to think otherwise because we often deal with stationary light, such as that from stars and cw light sources. By de?nition, the two- time properties of a stationary ?eld depend only on the time di?erence. The corresponding ?rst-order correlation function thus has the form
G(1)(t1, t2) = G(1)(t1 ? t2) . (1.93)

If such a ?eld is ?rst-order coherent, then with (1.88), we ?nd
G(1)(t1 ? t2) = E ?(t1)E (t2) , (1.94)


which is true only if


E (t1) ? e?i?t1 , (1.95)

that is, stationary ?rst-order coherent ?elds are monochromatic!
Let us now turn to the famous Hanbury Brown-Twiss experiment Fig. 1.5, which probes higher-order coherence properties of a ?eld. In this experiment, a beam of light (from a star in the original experiment) is split into two beams, which are detected by detectors D1 and D2. The signals are multiplied and averaged in a correlator. This procedure di?ers from the Young two-slit experiment in that light intensities, rather than amplitudes, are compared. Two absorption measurements are performed on the same ?eld, one at time t and the other at t + ? . It can be shown [Cohen-Tannoudji et al. 1989] that
this measures |E+(r,t + ?, )E+(r, t)|2 . Dropping the useless variable r and
averaging, we see that this is precisely the second-order correlation function
G(2)(t, t + ?, t + ?, t) = (E?(t)E?(t + ? )E+(t + ? )E+(t)) , (1.96)













Fig. 1.5. Diagram of Hanbury Brown-Twiss experiment


or for a stationary process,
G(2)(? ) = (E?(0)E?(? )E+(? )E+(0)) . (1.97)

According to (1.89), the ?eld is second-order coherent if (1.92) holds and
G(2)(? ) = E ?(0)E ?(? )E (? )E (0) . (1.98)

It is convenient to normalize this second-order correlation function as


(2)
g(2)(? ) =


(? )


. (1.99)

|G(1)(0)|2

Since a stationary ?rst-order coherent ?eld is monochromatic and satis?es (1.93), second-order coherence implies that

g(2)(? ) = 1 . (1.100)

that is, g(2)(? ) is independent to the delay ? .
The original experiment on Hanbury Brown-Twiss was used to measure the apparent diameter of stars by passing their light through a pinhole. A second-order correlation function like that in Fig. 1.6 was measured. Although the light was ?rst-order coherent, we see that it was not second-order coher- ent. The energy tended to arrive on the detector in bunches, with strong statistical correlations.
In contrast to the well-stabilized laser with a unity g(2) and the star-light with bunching, recent experiments in resonance ?uorescence show antibunch- ing, with the g(2) shown in Fig. 1.7. Chapter 16 discusses this phenomenon



g(2)(?)







1




0 ?

Fig. 1.6. Second-order correlation function (1.97) for starlight in original Hanbury Brown-Twiss experiment


g(2)(?)

1





0 ?

Fig. 1.7. Second-order correlation function showing antibunching found in reso- nance ?uorescence


in detail; here we point out that such behavior cannot be explained with classical ?elds. To see this, note that
(2)(0) ? |G(1)(0)|2

g(2)(0) ? 1 = G

In terms of intensities, this gives


|G(1)(0)|2

. (1.101)

g(2)(0) ? 1 = (I

2)? (I)2
(I)2


((I ? (I))2) (I)2


, (1.102)


where we do not label the times, since we consider a stationary system with ? = 0. Introducing the probability distribution P (I) to describe the average over ?uctuations, we ?nd for (1.100)


g(2)(0) ? 1 =

1 ¸
(I)2


dIP (I)(I ? (I))2 . (1.103)

Classically this must be positive, since (I ? (I))2 ? 0 and the probability
distribution P (I) must be positive. Hence g(2) cannot be less than unity, in
contradiction to the experimental result shown in Fig. 1.7. At the beginning of this chapter we say that the ?elds we use can usually be treated classically. Well we didn’t say always! To use a formula like (1.101) for the antibunched case, we need to use the concept of a quasi -probability function P (I) that permits negative values. Quantum mechanics allows just that (see Sect. 13.6).


1.5 Free-Electron Lasers

At this point we already have all the ingredients necessary to discuss the basic features of free-electron lasers (FEL). They are extensions of devices such as klystrons, undulators, and ubitrons, which were well-known in the millimeter regime many years ago, long before lasers existed. In principle, at least, nothing should have prevented their invention 30 or 40 years ago.


As shown in Chap. 7, conventional lasers rely on the inversion of an atomic or molecular transition. Thus the wavelength at which they oper- ate is determined by the active medium they use. The FEL eliminates the atomic “middle-man”, and does not rely on speci?c transitions. Potentially, FEL’s o?er three characteristics that are often hard to get with conventional lasers, namely, wide tunability, high power, and high e?ciency. They do this by using a relativistic beam of free electrons that interact with a periodic structure, typically in the form of a static magnetic ?eld. This structure exerts a Lorentz force on the moving electrons, forcing them to oscillate, sim- ilarly to the simple harmonic oscillators of Sect. 1.3. As discussed at the end of that section, oscillating electrons emit radiation with the ?eld shown in Fig. 1.3. In the laboratory frame, this radiation pattern is modi?ed according to Lorentz transformations. Note that in contrast to the case of radiative de- cay discussed in Sect. 1.3, the FEL electron velocity approaches that of light
and the v×B factor in the Lorentz force of (1.65) cannot be neglected.
The emitted radiation is mostly in the forward direction, within a cone
of solid angle ? = 1/? (see Fig. 1.8). Here ? is the relativistic factor
? = [1 ? v2/c2]?1/2 , (1.104)

where v is the electron velocity. For ? = 200, which corresponds to electrons with an energy on the order of 100 MeV, ? is about 5 milliradians, a very small angle.
In general for more that one electron, each dipole radiates with its own phase, and these phases are completely random with respect to one another.
The total emitted ?eld is ET = E+ + E?, where
T T
















Fig. 1.8. Highly directional laboratory pattern of the radiation emitted by a rela- tivistic electron in circular orbit in the x-y plane while moving along the z axis at the speed v = 0.9c. The x axis is de?ned to be that of the instantaneous accelera- tion. Equation (14.44) of Jackson (1999) is used for an observation direction n in the x-z plane (the azimuthal angle ? = 0). In the nonrelativistic limit (v c), this formula gives the butter?y pattern of Fig. 1.3



N
T =
k=1


E+ei?k


, (1.105)


and the sum is over all electrons in the system.
The total radiated intensity IT is proportional to |E+|2, which is
. N .2

IT = .. E+ei?k .

. (1.106)

. .
. k .
.k=1 .

Expanding the absolute value in (1.104), we obtain

N
IT = . |E+|2 + . E?E+e?i(?k ??j ) . (1.107)

k
k=1

k j
kƒ=j


Assuming that the amplitudes of the ?elds emitted by each electron are the same



we obtain

|Ek |2 = I , (1.108)

IT = NI + I . e?i(?k ??j ) . (1.109)
kƒ=j

For random phases, the second term in (1.107) averages to zero, leaving

IT = NI , (1.110)

that is, the total intensity is just the sum of the individual intensities. The contributions of the electrons add incoherently with random interferences, as is the case with synchrotron radiation.
However if we could somehow force all electrons to emit with roughly the same phase, ?k c ?j for all k and j, then (1.107) would become
IT = NI + N (N ? 1)I = N 2I . (1.111)

Here the ?elds emitted by all electrons would add coherently, i.e., with con- structive interference, giving an intensity N times larger than with random phases.
The basic principle of the FEL is to cause all electrons to have approxi-
mately the same phase, thereby producing constructive interferences (stim- ulated emission). A key feature of these lasers is that the wavelength of the emitted radiation is a function of the electron energy. To understand this, note that an observer moving along with the electrons would see a wiggler moving at a relativistic velocity with a period that is strongly Lorentz con- tracted. To this observer the ?eld appears to be time-dependent, rather than static, since it ties by. In fact, the wiggler magnetic ?eld appears almost as an electromagnetic ?eld whose wavelength is the Lorentz-contracted period


of the wiggler. It is well-known that an electron at rest can scatter electro- magnetic radiation. This is called Thomson scattering. Because the electron energy is much higher than that of the photons, at least in the visible range, we can neglect their recoil, and hence the wavelength of the scattered radia- tion equals that of the incident radiation

s = ?w . (1.112)
?r r

Here we use primes to mean that we are in the electron rest frame. Going back to the laboratory frame, we examine the radiation emitted in the for- ward direction. As Prob. 1.16 shows, this is also Lorentz contracted with the wavelength
?s = ?w /2?2 , (1.113)
where ?w is the period of the wiggler and
?z = (l ? v2/c2)?1/2 . (1.114)

Here we use ?z rather than ? because the relevant velocity for the Lorentz transformation is the component along the wiggler (z) axis. Since v is directed primarily along this axis, ?s is to a good approximation given by ?w /2?2.
An alternative way to obtain the scattered radiation wavelength ?s of (1.111) is to note that for constructive interference of scattered radiation, ?s + ?w must equal the distance ct the light travels in the transit time t = ?w /vz it takes for the electrons to move one wiggler wavelength. This gives ?s + ?w = c?w /vz , and (1.111) follows with the use of (1.112).
We see that two Lorentz transformations are needed to determine ?s. Since ?z c ? is essentially the energy of the electron divided by mc2, we can
change the wavelength ?s of the FEL simply by changing the energy of the electrons. The FEL is therefore a widely tunable system. In principle the FEL should be tunable continuously from the infrared to the vacuum ultraviolet.
We now return to the problem of determining how the electrons are forced of emit with approximately the same phase, so as to produce constructive
interferences. We can do this with Hamilton’s formalism in a straightforward way. For this we need the Hamiltonian for the relativistic electron interacting with electric and magnetic ?elds. We note that the energy of a relativistic electron is
E = ,m2c4 + p2c2 (1.115)

where p is the electron momentum. For an electron at rest, p = 0, giving Einstein’s famous formula E = mc2. For slow electrons (p mc), we expand
the square root in (1.113) ?nding E c mc2 + p2/2m, which is just the rest en-
ergy of the electron plus the nonrelativistic kinetic energy. For the relativistic
electrons in FEL, we need to use the exact formula (1.113).
To include the interaction with the magnetic and electric ?elds, we use the principle of minimum coupling, which replaces the kinetic momentum p by the canonical momentum


p ? P ? eA . (1.116)

Here A is the vector potential of the ?eld. Using the prescription (1.113), we ?nd the required Hamiltonian
H = c[(P ? eA)2 + m2c2]1/2 ? ?mc2 . (1.117)

Hamilton’s equations of motion are

P? ?H
i
q? = ? ?H
i


(1.118)

, (1.119)


where the three components of the canonical momentum, Pi, and the three electron coordinates, qi, completely describe the electron motion. To obtain their explicit form, we need to know A. This consists of two contributions, that of the static periodic magnetic ?eld, and that of the scattered laser ?eld.
If the transverse dimensions of the electron beam are su?ciently small compared to the transverse variations of both ?elds, we can treat the ?elds
simply as plane waves. A then has the form


1
A = 2 eˆ?[Aw e?


iKw z


+ Ase


?i(ws t?Ks z)


] + c.c. (1.120)

Here Aw and As are the amplitudes of the vector potential of the wiggler and the laser, respectively, and
?

eˆ? = eˆ?

= (xˆ ? iyˆ)/

2 , (1.121)


where xˆ and yˆ are the unit vectors along the transverse axes x and y, respec- tively. This form of the vector potential is appropriate for circularly polarized magnets. Also Kw = 2?/?w , where ?w is the wiggler period, and ?s and Ks are the frequency and wave number of the scattered light.
With this form of the vector potential, the Hamiltonian (1.115) doesn’t
depend explicitly on x and y. Hence from (1.116), we have

?
x = ? ?x = 0 ,
?
y = ? ?z = 0 , (1.122)
that is, the transverse canonical momentum is constant. Furthermore, this constant equals zero if the electrons have zero transverse canonical momen- tum upon entering the wiggler

PT = 0 . (1.123)

This gives the kinetic transverse momentum


pT = ?eA , (1.124)

which shows that the transverse motion of the electron is simply a circular orbit, as might be expected intuitively.
For the longitudinal motion, the Hamilton equations of motion reduce to

z? = pz /m? , (1.125)
e2 ? 2
p?z = ? 2m? ?z (A ) . (1.126)

Equation (1.123) just gives the usual formula pz = ?mvz for relativistic particles. Equation (1.124) is more informative and states that the time rate of change of the longitudinal electron momentum is given by the spatial derivative of the square of the vector potential. Potentials proportional to A2 are common in plasma physics where they are called ponderomotive potentials. Computing ?(A2)/?z explicitly, we ?nd

?(A2)



where

= 2iKA? Asei(Kz??s t) + c.c., (1.127)
?z


K = Kw + Ks . (1.128)

Thus in the longitudinal direction, the electron is subject to a longitudinal force moving with the high speed
v = ?s . (1.129)
s K
Since according to (1.111) Ks Kw , vs is almost the speed of light.
In the laboratory frame, both the electrons and the potential move at close to the speed of light. It is convenient to rewrite the equations of motion (1.123, 1.124) in a frame moving at velocity vs, that is, riding on the pondero- motive potential. For this we use
? = z ? vst + ?0 ? ? , (1.130) which is the position of the electron relative to the potential and K? is
the phase of the electron in the potential. ?0 is determined by A? As =
|Aw As| exp(iK?0) and K?0 is the phase of the electron relative to the pon-
deromotive potential at z = t = 0. This gives readily

?? = vz = vs , (1.131)

which is the electron velocity relative to the potential. To transform (1.124) we have to take into account that ? is not constant. First, taking Aw and As real, we readily ?nd


p?z = ?


2Ke2
m? |Aw As| sin K? . (1.132)


This is a nonlinear oscillator equation that includes all odd powers of the displacement K?. Noting further that p?z = m??2v?z (see Prob. 1.17) and

that v?z = ?¨, we obtain




where



2K e2
? = ? 2 4 |Aw As| sin K? , (1.133)
s


M = m[1 + (eA/mc)2]1/2 (1.134)


is the e?ective (or shifted) electron mass, and we have at the last stage of the derivation approximated ?z by ?s = [1 ? v2/c2]1/2. Equation (1.131)

is the famous pendulum equation. Thus in the frame moving at velocity

vs,

the dynamics of the electrons is the same as the motion of a particle in a
sinusoidal potential. Note that the shifted mass M is used rather than the electron mass m.
The pendulum equation describes the motion of particles on a corrugated rooftop. In the moving frame, the electrons are injected at some random
position (or phase) ?0 with some relative velocity ??(0). Intuitively, we might
expect that if this velocity is positive, the electron will decelerate, transferring
energy to the ?eld, while if the velocity is negative, the electron will accelerate, absorbing energy from the ?eld. However as we know from the standard pendulum problem, the relative phase ?0 with respect to the ?eld also plays a crucial role. From (1.130), we see that p?z is negative if and only if sin K?


??






?? ? ?









Fig. 1.9. Initial phase-space con?guration of the electrons relative to the pon- deromotive potential. The phases (plotted horizontally) are shown only between
?? ? ? ? ?. The vertical axis gives the electron energies. Initially, the electron
beam is assumed to have vanishing energy spread and random phase


is positive. Hence the electrons initially absorb energy for 0 ? ?0 ? ?/K and give up energy for ?/K ? ?0 ? 2?/K. This is illustrated in Fig. 1.9, which
shows the phase space of the pendulum. The abscissa is the phase ? of the
electron relative to the potential, while the ordinate is the relative velocity ??. Initially all electrons have the same velocity (or energy) ??. Since there is no way to control their initial phases, they are distributed uniformly between
?? and ?.
Electrons with phases between ?? and 0 accelerate, while others decel-
erate, so that after a small time, the phase-space distribution looks like that
in Fig. 1.10. Three important things have occurred. First, the electrons now have di?erent energies, more or less accelerated or decelerated, depending on their initial phases. Thus the initially monoenergetic electron beam now has

an energy spread. Second, the average relative velocity (??

of the electrons

has decreased, giving an average energy loss by the electrons. Conservation
of energy shows that the ?eld energy has increased by the same amount. The
recoil of the electrons leads to gain. Third, the electron distribution is no longer uniformly distributed between ? = ??/K and ?/K. The electrons are
now bunched in a smaller region. Instead of producing random interferences
with an emitted intensity proportional to N , they are redistributed by the ponderomotive potential to produce constructive interference as discussed for (1.109). These three e?ects, recoil, bunching, and spread, are key to under- standing FEL’s. They always occur together, and a correct FEL description must treat them all.

??






?? ? ?









Fig. 1.10. As in Fig. 1.9, but a small time after injection into the wiggler. We see a bunching in position (horizontial axis) and spread in energy (vertical axis)


??






?? ? ?









Fig. 1.11. As in Fig. 1.9, but at the instant of maximum energy extraction

??






?? ? ?









Fig. 1.12. As in Fig. 1.9, but for longer times such that the electron absorb energy from the laser ?eld


What happens for even longer times is shown in Fig. 1.11. The bunching, recoil, and spread have all increased. Note that the spread increases much faster than the recoil. This is a basic feature of the FEL that makes it hard to operate e?ciently. Since pendulum trajectories are periodic, still longer times cause electrons that ?rst decelerated to accelerate and vice versa as shown in Fig. 1.12. For such times the average electron energy increases, that


is, laser saturated. To maximize the energy transfer from the electrons to the laser ?eld, the length of the wiggler should be chosen just short enough to avoid this backward energy transfer. This kind of saturation is quite di?erent from that for two-level media discussed in Chap. 5. For the latter, the gain is bleached toward zero, but does not turn into absorption. Here the saturation results from the onset of destructive interference in a fashion analogous the phase matching discussed in Sect. 1.2. To maintain the constructive interfer- ence required for (1.111) as the electrons slow down, some FEL’s gradually decrease the wiggler wavelength along the propagation direction. This kind of wiggler is called a tapered wiggler.
In this qualitative discussion, we have assumed that the initial relative velocity of the electrons was positive, i.e., that the average electron moves
faster than the ponderomotive potential. If the initial velocity is negative, the average electron initially absorbs energy. These trends are depicted in Fig. 1.13, which plots the small-signal gain of the FEL versus the relative electron velocity.










Gain

0.6


0.3


0.0




?0.3

?0.6
?2? ?? 0
x






? 2?


Fig. 1.13. FEL Gain function versus initial electron velocity relative to the pon- deromotive potential


This elementary discussion of the FEL only consider the electrons, and uses conservation of energy to determine whether the ?eld is ampli?ed. A more complete FEL theory would be self-consistent, with the electrons and ?eld treated on the same footing. Such a theory of the FEL is beyond the scope of this book and the reader is referred to the references for further discussion. A self-consistent theory of conventional lasers is given in Chap. 7.


Problems


1.1 Derive the wave equation from microscopic Maxwell’s equations that include a charge density and current. For this, (1.2, 1.4) become

?

?•E = ,
0



1 ?E

?×B = ?0J + c2 ?t ,
respectively. Hint: ?rst show that the conservation of charge equation

?? + J = 0
?t

is solved by ? = ??•P + ?free, J = ?P/?t + ?×M + Jfree. For our purposes, assume M = Jfree = ?free = 0, and neglect a term proportional to ??.
Show using the divergence theorem that

¸
dV (?• P)r =

¸
(P • d?)r ?


P dV .


Given the relation ? = ??• P from Prob. 1.1, show that the polarization P
can be interpreted as the dipole moment per unit volume.

Derive the slowly-varying amplitude and phase equations of motion (1.31, 1.32) by substituting (1.26, 1.28) into the wave equation (1.25). Specify which terms you drop and why.

Derive the equations (1.56, 1.57) of motion for the classical Bloch vector components U and V by substituting (1.28) into (1.44) and using the slowly- varying approximation. Calculate the evolution and magnitude of the classical Bloch vector in the absence of decay.

What are the units of Ke2/2m??? in (1.54), where ? is given by (1.74). What is the value of this quantity for the 632.8 nm line of the He–Ne laser (take ? = ?)? Calculate the absorption length (1/?) for a 1.06 ?m Nd:YAG laser beam propagating through a resonant linear medium with 1016 dipoles/m3.
A ?eld of the form E(z, t) cos(Kz??t) interacts with a medium. Using the “classical Bloch equations”, derive an expression for the index of refraction of the medium. Assume the oscillator frequency ? is su?ciently close to the ?eld frequency ? so that the rotating-wave approximation of (1.50) may be made.

In both laser physics and nonlinear optics, the polarization of a medium frequently results from the interaction of several separate ?elds. If P(r, t) is given by



P(r, t) = . Pn(r, t)ei(Kn •r) ,
n
solve for the polarization amplitude component Pn(r, t).

Problems 33

Find the magnetic ?eld B corresponding to the electric ?eld

1
E(z, t) = 2 xˆE U (z)e?

i?t

+ c.c. ,

for running-wave [U (z) = e?iKz ] and standing-wave [U (z) = sin Kz] ?elds. Draw a “3-D” picture showing how the ?elds look in space at one instant of time.
The change of variables z ? zr and t ? tr = tr ? z/c transforms the slowly-varying Maxwell’s equations from the laboratory frame to the so-called retarded frame. Write the slowly-varying Maxwell’s equations in this frame. Discuss Beer’s law in this frame.

In an optical cavity, the resonant wavelengths are determined by the constructive-interference condition that an integral number of wavelengths must occur in a round trip. The corresponding frequencies are determined by these wavelengths and the speed of light in the cavity. Given a cavity with a medium having anomalous dispersion, would it be possible to have more than one frequency resonant for a single wavelength? How?

Using Cartesian coordinates and using spherical coordinates show that the spherical wave exp(iKr ? i?t)/r satis?es the wave equation for free space.
Calculate the magnitudes of the electric and magnetic ?elds for a 3 mW
632.8 nm laser focussed down to a spot with a 2 ?m radius. Assume constant intensity across the spot. How does this result scale with wavelength?
Derive the index of refraction (1.40) for the case that ? = ?(t), i.e., not ?(z) as assumed in (1.39). The ?(t) case is generally more appropriate for lasers.
Solve (1.47) for X(t), i.e., as function of time.
Calculate the ?rst and second-order coherence functions for the ?eld
E+(r, t) = E0 e?(?+i?)(t?r/c)?(t r/c) ,
r
where ? is the Heaviside (step) function. This would be the ?eld emitted by an atom located at r = 0 and decaying spontaneously from time t = 0, if such a ?eld could be described totally classically.
Derive the FEL equation (1.111) using the Lorentz transformation zr =
?(z ? ?ct) and tr = ?(t ? ?z/c), where ? is given by (1.102) and ? = v/c.
Show that p?z = m??2v?z and proceed to derive the pendulum equa- tion (1.130). Use a personal computer to draw electron trajectories shown in Figs. 1.10–1.13 and discuss the trajectories.


The Kramers-Kronig relations allow one to calculate the real and imag- inary parts of a linear susceptibility ?(?) as integrals over one another as follows:

1 ¸ ?

d?r?r(?r)

?rr(?) = ? ? P.V.
??

?r ?

(1.135)
?

1
?r(?) =


P.V.

¸ ? d?r?rr(?r)


, (1.136)

? ? ?r ? ?

where P.V. means the principal value, i.e., the integral along the real axis excluding an arbitrarily small counterclockwise semicircle around the pole at
?r = ?. Equations (1.133, 1.134) are based on the assumption that ? has no
poles in the upper half plane; an equivalent set with a change in sign results for a ? that has no poles in the lower half plane. From (1.33, 1.51) we have the linear susceptibility


?(?) = ?r(?) + i?rr(?) = N ex0X(?) =

N e2 1

N e2

? ? ? ? i?

?E0

? 2??m ? ? ? + i?

= ? 2??m (?

?)2 + ?2 . (1.137)


Show that this ?(?) satis?es (1.133, 1.134). Hint: for (1.133) use the residue theorem as follows: the desired principal part = the residue for the pole at
?r = ? + i? minus the half residue for the pole at ?r = ?. It is interesting
to note that the power-broadened version of (1.135), namely (5.29), does not satisfy the Kramers-Kronig relations, since unlike (1.135), (5.29) does not reduce to a single pole in the lower half plane.

2 Classical Nonlinear Optics











Many problems of interest in optics and virtually all of those in this book in- volve nonlinear interactions that occur when the electromagnetic interaction becomes too large for the medium to continue to respond linearly. We have already seen how a nonlinearity plays an essential role in the free electron laser pendulum equation (1.133). In another example that we discuss in detail in Chap. 7, the output intensity of a laser oscillator builds up until satura- tion reduces the laser gain to the point where it equals the cavity losses. In situations such as second harmonic generation, one uses the fact that non- linearities can couple electromagnetic ?eld modes, transferring energy from one to another. Such processes can be used both to measure properties of the nonlinear medium and to produce useful applications such as tunable light sources.
In this chapter we extend Sect. 1.3’s discussion of the simple harmonic oscillator to include quadratic and cubic nonlinearities, i.e., nonlinearities
proportional to x2 and x3, respectively. Such nonlinearities allow us to un- derstand phenomena such as sum and di?erence frequency generation, mode coupling, and even chaos in a simple classical context. Subsequent chapters treat these and related phenomena in a more realistic, but complex, quantized environment.


2.1 Nonlinear Dipole Oscillator

Section 1.3 discusses the response of a linear dipole oscillator to a monochro- matic electric ?eld. When strongly driven, most oscillators exhibit nonlinear- ities that can be described by equations of motion of the form [compare with (1.46)]

x¨(z, t) + 2?x? (z, t) + ?2x(z, t) + ax2(z, t) + bx3(z, t) + •••
= e E(z, t) . (2.1)
m

Here we include a z dependence, since the polarization modeled by x(z, t) is a function of z. Speci?cally, x describes the position of an electron relative to the nucleus (internal degree of freedom) while z is the location of the


dipole in the sample (external degree of freedom). Such oscillators are called “anharmonic”. In many cases such as for isolated atoms, the coe?cient a vanishes, leaving bx3 as the lowest order nonlinear term.
We can determine the e?ect of the nonlinear terms on the response by a
process of iteration that generates an increasingly accurate approximation to
x(t) in the form of a power series
x(t) c x(1)(t) + x(2)(t) + ••• + x(n)(t) + ••• , (2.2) The leading term in this series is just the linear solution (1.51) itself or a
linear superposition (1.63) of such solutions. To obtain the second-order con-
tributions, we substitute the linear solution into (2.1), assume an appropriate form for the second-order contributions, and solve the resulting equations in a fashion similar to that for the linear solution. In the process, we ?nd that new ?eld frequencies are introduced. In general, the nth order term is given by solving the equation assuming that the nonlinear terms can be evalu-
ated with the (n ? 1)th order terms. Many of the phenomena in this book
require solutions that go beyond such a perturbative approach, since the cor-
responding series solution may fail to converge. Nevertheless, the subjects usually considered under the heading “nonlinear optics” are very useful and are typically described by second- and third-order nonlinearities.
We consider ?rst the response of a medium with an ax2 nonlinearity subjected to a monochromatic ?eld of frequency ?n, that is, (1.62) with a
single amplitude En(z). Choosing x(1) to be given by the corresponding linear
solution (1.51), we ?nd for x2 the approximate value


x2 c [x(1)]2 =


1 1
|x(1) 2


(1) 2


2i(Kn z??n t)

4 n |

+ [ ] e
4 n

+ c.c., (2.3)


where the slowly varying amplitude x(1) = x0nXn


and Xn


is given by (1.50).

We see that this nonlinear term contains both a dc contribution and one at
twice the initial frequency. The dc term gives the intensity measured by a square-law detector and is the origin of the Kerr electro-optic e?ect in crys- tals, while the doubled frequency term leads to second harmonic generation. Observation of the latter in quartz subjected to ruby laser light kicked o? the ?eld known as nonlinear optics [see Franken et al. (1961)]. With an an- harmonic forcing term proportional to (2.3), the second-order contribution x(2)(t) has the form

x(2)(t) = 1 x(2) + 1 x(2) ei(2Kn z?2?n t) + c.c.. (2.4)

2 dc

2 2?n

According to our iteration method, we determine the second-order coe?-

cients x(2)

and x(2) by substituting (2.2) into (1.1) approximating x2 by the

?rst-order expression (2.3). By construction, the terms linear in x(1) cancel the driving force (e/m)E, and we are left with a simple harmonic oscillator equation for the x(2) coe?cient, namely,

2.1 Nonlinear Dipole Oscillator 37
x¨(2)(t) + 2?x? (2)(t) + ?2x(2)(t) = a[x(1)(t)]2 . (2.5)
Equating coe?cients of terms with like time dependence, we ?nd

x(2)

a (1) 2

dc = ? 2?2 |xn |
a . eE n (z)/m .2

.
. 2 2
n ?

.
2i?n .

, (2.6)

x(2)

a (1) 2 1

2?n = ? 2 (xn )

?2 ? (2?n)2

— 2i(2?n)?

. (2.7)

Note here that if the applied ?eld frequency ?n is approximately equal to the natural resonance frequency ?, both the dc and second-harmonic coe?cients are divided by squares of optical frequencies. Hence these terms are usually very small, and second-order theory is a good approximation, for example, in noncentrosymmetric crystals.
Now consider the response of this nonlinear oscillator to an electric ?eld given by (1.62) with two frequency components at the frequencies ?1 and ?2. To lowest order (?rst order in the ?elds), we neglect the nonlinearities and
?nd x(t) c x(1)(t), which is given by the linear superposition (1.63) with two
modes. The approximate second-order nonlinearity [x(1)]2 has the explicit
form
[x(1)]2 = [x(1)ei(K1 z??1 t) + x(1) 2 2
1 2 ei(K z?? t)]2
1 1

= x(1)

(1)

i[(K1 ?K2 )z?(?1 ??2 )t]

(1)

(1)

i[(K1 +K2 )z?(?1 +?2 )t]

2 1 [x2 ]? e

+ x x e 2 1 2

2
+ .(|x(1)|2 + [x(1)]2 e2i(Kn z??n t)) + c.c.. (2.8)
4 n n n=1
As one expects from (2.4), each mode contributes a dc component and a com- ponent oscillating at its doubled frequency, 2?n. In addition, (2.8) contains
components at the sum and di?erence frequencies ?1 ± ?2. Hence the non-
linear dipoles can generate ?elds at these frequencies. More generally, when
higher orders of perturbation are considered, the polarization of a medium consisting of such anharmonic oscillators can generate all frequency com-
ponents of the form m?1 ± n?2, with m and n being integers. When such
combinations lead to frequencies other than harmonics of ?1 or ?2, they are
called combination tones. Such tones are responsible for self mode locking
in lasers (Sect. 11.3) and three- and four-wave mixing (rest of the present chapter and Chap. 10).
Generalizing the second-order contribution (2.4) to include the sum and di?erence frequencies, we have

x(2)(t) = 1 x(2)

i[(K1 +K2 )z?(?1 +?2 )t]

1 (2)

i[(K1 ?K2 )z?(?1 ??2 )t]

2 s e

+ x e
2 d

+ 1 . x(2)

i(2Kn z?2?n t) 1


(2)

2
n=1

2?n e

+ x + c.c.. (2.9)
2 dc


Substituting (2.8, 2.9) into (2.5) and equating coe?cients of like time de- pendence, we ?nd that x(2) is given by the sum of two terms like (2.6), the doubled frequency terms are given by (2.7), and the di?erence and sum fre- quency terms are given by


(1)


(1)

d = ?a ?2

x1 [x2 ]?
2

, (2.10)

? (?2 ? ?2)
(1)

? 2i(?1 ? ?2)?
(1)

s = ?a ?2

x1 x2
2

. (2.11)

? (?1 + ?2)

? 2i(?1 + ?2)?

Except for a factor of 2 for the degenerate frequency case ?1 = ?2 the di?er- ence frequency term (2.10) reduces to the dc term (2.6) and the sum frequency term (2.11) reduces to the second harmonic term (2.7).
Frequency combinations like those in (2.9) also appear in quantum me- chanical descriptions of the medium, which typically involve more intricate nonlinearities. In particular, di?erence frequency generation induces pulsa- tions of the populations in a medium consisting of two-level atoms irradiated by two beams of di?erent frequencies. These pulsations play an important role in saturation spectroscopy, as discussed in Chap. 9.


2.2 Coupled-Mode Equations

So far we have obtained the steady-state response of the nonlinear dipole to second order in the ?eld and have seen how combination tones at the fre- quencies m?1 + n?2 can be generated by such systems. The way in which the corresponding waves evolve is readily obtained from the wave equation (1.25) giving the propagation of an electromagnetic ?eld E(z, t) inside a medium of polarization P (z, t). For a medium consisting of nonlinear os- cillators, P (z, t) = N (z)ex(z, t), where N (z) is the oscillator density and z labels the position inside the medium. To analyze the growth of a wave at a frequency ?3, we consider three modes in the ?eld (1.62) and in the po- larization (1.65). The slowly varying amplitude approximation allows us to derive coupled di?erential equations for the evolution of the ?eld envelopes
En. These are called coupled-mode equations, and play an important role in
multiwave phenomena such as phase conjugation (Sect. 2.4 and Chap. 10) and
the generation of squeezed states (Chap. 17), in which case a fully quantum mechanical version of these equations is required. In general, coupled-wave equations form an in?nite hierarchy of ordinary di?erential equations, and some kind of approximation scheme is needed to truncate them. For instance, if we are only interested in the small signal build-up of the wave at frequency ?3, we can neglect its back-action on the nonlinear polarization P (z, t) – ?rst-
order theory in E3. This is the procedure used in Sect. 2.1. Another common
approximation assumes that the waves at frequencies ?1 and ?2 are so intense
that their depletion via the nonlinear wave-mixing process can be neglected.