21

2.2 Coupled-Mode Equations 39

An important feature of coupled-mode equations is phase matching, which represents the degree to which the induced mode coupling terms in the po- larization have the same phase as the ?eld modes they a?ect. To the extent that the phases di?er, the mode coupling is reduced. Phase matching involves di?erences in wave vectors and amounts to conservation of momentum. This is distinct from the frequency di?erences of the last section, which amount to conservation of energy.
Indeed, the resonance denominators appearing in the nonlinear suscepti- bilities can be interpreted as a consequence of energy conservation. In con- trast, the spatial phase factors are a result of momentum conservation. This is particularly apparent when the electromagnetic ?eld is quantized, as will be the case in the second part of the book, because in that case, it is easy to show that the energy and momentum of a photon of frequency ? and mo- mentum K are k? and kK, respectively. In a vacuum, one has that ? = Kc for all frequencies, so that energy conservation automatically guarantees mo- mentum conservation. But this ceases to be true in dispersive media, where the factor of proportionality between ? and K is frequency-dependent.
Consider for instance the case of di?erence-frequency generation, where
two incident waves at frequencies ?1 and ?2 combine in the nonlinear medium to generate a wave at the di?erence frequency ?d. In the slowly-varying am- plitude approximation and in steady state (we neglect the ?/?t terms), (1.43) becomes

dEn = iKn

, (2.12)

dz 2?n Pn
where ?n is the permittivity at the frequency ?n of the host medium in which our oscillators are found. From (1.64, 1.65), we use the linear solutions for modes 1 and 2

dE n

N (z)e2 En

dz c iKn

2?nm


2 2
n ?

2i?n

= ??? En . (2.13)


Note that in this example we assume that the host medium is purely dis- persive; otherwise another absorption term would have to be included. Us- ing (2.10, 2.12), we ?nd for the di?erence-frequency term at the frequency
?d = ?1 ? ?2

. (1)


(1)

i(K1 ?K2 ?Kd )z .

dE d = iKd N (z)e

(e/m)Ed ? ax1 [x2 ]?e

. (2.14)

dz 2?d

?2 ? ?2 ? 2i?d?



The coupled-mode equations (2.13, 2.14) take a rather simple form, since we have neglected the back action of Ed on E1 and E2. Equation (2.13) sim- ply describes the linear absorption and dispersion of E1 and E2 due to the
nonlinear oscillators, as does the ?rst term on the right-hand side of (2.14) for Ed. In many cases these are small e?ects compared to those of the host medium accounted for by ?n, and we neglect them in the following. E1 and E2


then remain constant and (2.14) (without the leading term) can be readily integrated over the length L of the nonlinear medium to give
ei?KL ? 1

Ed(L) = GE1E ?

i?K , (2.15)

where


iKd aN e . x(1)[x(1)]?/E1E ? .

G = ? 2? ?2 ,
d ? ?2 ? 2i?d?
and the K-vector mismatch ?K = K1 ? K2 ? Kd. The resulting intensity
Id = |Ed|2 is

Id(L) = |G|2I1I2

sin2(?KL/2)
(?K/2)2


. (2.16)

If ?K = 0, Id(L) reduces to |G|2I1I2L2, but for ?K ƒ= 0, it oscillates periodically. To achieve e?cient frequency conversion, it is thus crucial that
(K1 ?K2)L be close to KdL. For ?K ƒ= 0 the maximum intensity Id is reached
for a medium of length L = ?/?K. For larger values of ?KL, the induced
polarization at the frequency ?d and the wave propagation at that frequency start to interfere destructively, attenuating the wave. For still larger values L, the interference once again becomes positive, and continues to oscillate in this fashion. Since nonlinear crystals are expensive, it is worth trying to achieve the best conversion with the smallest crystal, namely for ?KL = ?. In the plane-wave, collinear propagation model described here, perfect phase matching requires that the wave speeds un = vn/Kn all be equal, as would be the case in a dispersionless medium. More generally we have the di?erence
?K = K1 ? K2 ? Kd = (n1?1 ? n2?2 ? nd?d)/c ƒ= 0 since the n’s di?er. For
noncollinear operation the vectorial phase matching condition
?KL = |K1 ? K2 ? Kd|L c ? (2.17)

must be ful?lled for maximum Id. There are a number of ways to achieve this, including appropriate geometry, the use of birefringent media, and tem- perature index tuning.


2.3 Cubic Nonlinearity

We already mentioned that quadratic nonlinearities such as described in the preceding sections to not occur in isolated atoms, for which the lowest order nonlinear e?ects are cubic in the ?elds. These can be described in our classical model by keeping the bx3 term instead of ax2 in the nonlinear oscillator equation (2.1). In the presence of two strong pump ?elds at frequencies ?1 and ?2, the third-order polarization given by bx3 includes contributions at
the frequencies ?1 and ?2 and at the sideband frequencies ?0 = ?1 ? ? and
?3 = ?2 + ? as well, where

2.3 Cubic Nonlinearity 41
? = ?2 ? ?1 . (2.18)
The generation of these sidebands is an example of four-wave mixing. To describe the initial growth of the sidebands, we write the anharmonic term bx3 to third-order in x1 and x2, and ?rst-order in the small displacements x0 and x3, that is,

[x(1)]3 = 1 [x(1) ei(K1 z??1 t) + x(1)

i(K2 z??2 t)

8 1 2 e

+3x(1)

(1)

0 ei(K0 z??0 t) + 3x3 ei(K3 z??3 t) + c.c.]
,

× 2x(1)

(1)

1 x2 ei[(K1 +K2 )z?(?1 +?2 )t]

+2x(1)

(1)

1 [x2 ]? ei[(K1 ?K2 )z?(?1 +?2 )t]
2 .
+ .{[x(1)]2 e2i(Kn z??n t) + |x(1) 2

n n=1

n | } + c.c.

, (2.19)

where the terms in {} are similar to those in (2.9). The factor of 3 results from the three ways of choosing the x0 and x3 from the triple product.
The curly braces in (2.19) contain two dc terms, a contribution oscillat-
ing at the di?erence frequency ?, and three rapidly oscillating contributions oscillating at the frequencies ?n + ?m. These time-dependent terms are some- times called (complex) index gratings, and the nonlinear polarization may
be interpreted as the scattering of a light ?eld En from the grating produced by two ?elds Em and Ek . In this picture, the dc terms are “degenerate” grat- ings produced by the ?elds Em and E ? . Equation (2.19) readily gives the third-order contributions to the components of the polarization Pn at the
frequencies of interest.
One can interpret (2.19) as the scattering of components in the [ ] of the ?rst lines o? the slowly varying terms in the {}. Speci?cally the |x(1)|2 terms in (2.19) contribute nonlinear changes at the respective frequencies of the
components in the [ ]. In contrast, the scattering o? the “Raman-like” term exp[i(K2 ? K1)z ? i?t] and its complex conjugate contribute corrections at frequencies shifted by ±?. Taking ?2 > ?1, we see that the ?2 term in the [ ] scatters producing components at both the lower frequency ?1 (called a Stokes shift) and the higher frequency ?3 = ?2 + ? (called an anti-Stokes shift).
Similarly the ?1 term in the [ ] leads to contributions at the frequencies ?0 = ?1 ? ? and at ?2. The induced polarization components at the frequencies ?0 and ?3 are called combination tones. They are generated in the nonlinear
medium from other frequencies. If the two pump beams at ?1 and ?2 are copropagating, the index grating represented by the K2 ?K1 term propagates
at approximately the velocity of light in the host medium, but if the beams are counterpropagating, the grating propagates at the relatively slow speed
v = ??/(K1 + K2). In particular, it becomes stationary for the degenerate
case ?1 = ?2. (Compare with the ponderomotive force acting on the electrons


in the free electron laser, (1.126)! –Can you draw an analogy between the two situations?)
We are often only interested in induced polarizations near or at the fun- damental frequencies ?n,n = 0, 1, 2, 3. Keeping only these in (2.19) and neglecting combination tones involving x0 and x1 in the pump-mode polar- izations (Prob. 2.7), we ?nd

3

|[x(1)]3|fund =

x(1)(|x(1)|2 + 2|x(1)|2) ei(K1 z??1 t)

8 1
+ x(1)

1

(1) 2

2

(1) 2


i(K2 z??2 t)

8 2 (|x2 |
6

+ 2|x1 | )e

+ [|x(1)|2 + |x(1)|2][x(1) ei(K0 z??0 t) + x(1) ei(K3 z??3 t)]

8 1
+ 6 x(1)


(1)

2

(1)

0 3

i[(K1 +K2 ?K3 )z??0 t]

8 1 x2 [x3 ]? e
6

+ x(1)

(1)

(1)

i[(K1 +K2 ?K0 )z??3 t]

8 1 x2 [x0 ]? e
3

+ [x(1) 2

(1)

i[(2K1 ?K2 )z??0 t]

8 1 ] [x2 ]? e
3

+ [x(1) 2

(1)

i[(2K2 ?K1 )z??3 t]

8 2 ] [x1 ]? e

+ c.c. (2.20)


Combining the various terms, we ?nd that the third-order polarization com- ponents are given by


(3) 6


(1) 2


(1) 2


(1)

P0 = 8 N eb [|x1 |
6

+ |x2 | ]x0

+ N eb x(1)x(1)

(1)

i(K1 +K2 ?K3 ?K0 )z

8 1 2 [x3 ]? e
3
8 1 2

(3) 3

(1)

(1) 2

(1) 2

P1 = 8 N eb x1 [|x1 |

+ 2|x2 | ] (2.21b)

(3) 3

(1)

(1) 2

(1) 2

P2 = 8 N eb x2 [2|x1 |

+ |x2 | ] (2.21c)

(3) 6

(1) 2

(1) 2

(1)

P3 = 8 N eb [|x1 |
6

+ |x2 | ]x3

+ N eb x(1)x(1)

(1)

i(K1 +K2 ?K0 ?K3 )z

8 1 2 [x0 ]? e
3
+ N eb [x(1)]2[x(1)]? ei(2K2 ?K1 ?K3 ) . (2.21d)
8 2 1
The polarization components P(3) and P(3) are solely due to the exis-
0 3
tence of index gratings, which are also responsible for the factors of 2 in the
cross coupling terms for P(3) and P(3). This asymmetry is sometimes called
1 2
nonlinear nonreciprocity and was discovered in quantum optics by Chiao


et al. (1966). It also appears in the work by van der Pol (1934) on coupled vacuum-tube tank circuits. In the absence of index gratings, the factors of 2
in (2.21b, c) are replaced by 1, and |x(1)|2 and |x(1)|2 play symmetrical roles

in P(3)

1 2
(3)

1 and P2 .
The polarizations Pn lead to coupled-mode equations for the ?eld en-
velopes. The procedure follows exactly the method of Sect. 2.2 and we obtain
(Prob. 2.2)


dE0 =


[? ?


2 ? 2] + ?


2 ? ei(2K1 ?K2 ?K0 )z

dz ?E0

0 ? 01|E1| ?

02|E2|

0121E1 E2

3 E ei(K1 +K2 ?K3 ?K0 )z , (2.22a)

dE1 =
dz

?E [?1 ?

?1|E1|2

? ?12|E2|


2] , (2.22b)

dE2 =
dz

?E2

[?2 ?

?2|E2|2

? ?21|E1|

2] , (2.22c)

dE3 =


[? ?

2 ? 2 + ?

2 ? ei(2K2 ?K1 ?K3 )z

dz ?E3

3 ? 31|E1| ?

32|E2|

3212E2 E1

0 E ei(K1 +K2 ?K0 ?K3 )z . (2.22d)
Here En is the complex amplitude of the ?eld at frequency ?n, and the ??nEn
terms allow for linear dispersion and absorption.
Equations (2.22b, c) for the pump modes amplitudes are coupled by the cross-coupling (or cross-saturation) coe?cients ?nj . To this order of pertur-
bation, they are independent of the sidemode amplitudes E0 and E3. Because
E1 and E2 always conspire to create an index grating of the correct phase, the
evolution of these modes is not subject to a phase matching condition. Equa-
tions of this type are rather common in nonlinear optics and laser theory. In Sect. 7.4, we obtain an evolution of precisely this type for the counterpropa- gating modes in a ring laser. We show that the cross-coupling between modes can lead either to the suppression of one of the modes or to their coexistence, depending on the magnitude of the coupling parameter C = ?12?21/?1?2 and relative sizes of the ?n.
In contrast, the sidemodes E0 and E3 are coupled to the strong pump ?elds
E1 and E2 only, and not directly to each other. They have no back-action on
the pump modes dynamics, and their growth is subject to a phase-matching
condition.


Four-Wave Mixing with Degenerate Pump Frequencies

In many experimental situations, it is convenient to drive the nonlinear medium with two pump ?elds of the same frequency ?2, but with opposite propagation directions given by the wave vectors K2? and K2?. The pump


waves cannot by themselves generate polarization components at sideband frequencies. However one can still take advantage of the index gratings pro- duced by the pump beams with weak waves at frequencies symmetrically
detuned from ?2 by a small amount ±? (see Fig. 2.1). This procedure has
gained considerable popularity in connection with optical phase conjugation.
In optical phase conjugation, one of the sidebands is called the probe (at
?1 = ?2 ? ? ) and the other the signal (at ?3 = ?2 + ?), and we adopt this
notation here in anticipation of Chaps. 9, 10.










?1 ?2 ?3

Fig. 2.1. Mode spectrum in four-wave mixing for optical phase conjugation


We consider the wave confguration in Fig. 2.2 with two counterpropa- gating pump beams along one direction, and counterpropagating signal and “conjugate” waves along another direction, which we call the z axis. The electric ?eld for these four waves has the form

1
E(r, t) = 2 [E1 e


i(K1 z??1 t)


+ E2? e


i(K2? •r??2 t)


+ E2? e


i(K2? •r??2 t)

+E3 ei(?K3 z??3 t)] + c.c., (2.23)
where we take K2? = ?K2?. The ?eld fringe patterns resulting from interfer- ence between the various waves can induce index gratings. The corresponding linear displacement x(1)(t) contains components proportional to each of the ?eld amplitudes, and the third-order nonlinear displacement x(3) consists of the sum of all terms proportional to the products of three ?elds, each of which can be anyone of the four waves or their complex conjugates. This gives a
grand total of 8 • 8 • 8 = 512 terms. Fortunately, we’re only interested in a rel-
atively small subset of these terms, namely those with the positive frequency
?1 linear in E1. This gives a third-order signal polarization P(3) proportional
to




























Fig. 2.2. Diagram of interaction between standing-wave pump beam (?2) with probe (?1) and conjugate (?3) beams used in phase conjugation



E1E2?E ?

+ E1E ? E2? + E2?E1E ?

+ E ? E1E2?

2? 2?

2? 2?

+E2?E ? E1? + E ? E2?E1 + E1E2?E ?

+ E1E ? E2?

2? 2?

2? 2?

+E2?E1E ?

+ E ? E1E2? + E2?E ? E1 + E ? E2?E1

2? 2?

2? 2?

+[E ?E2?E2? + E ?E2?E2? + E2?E ?E2? + E2?E ?E2?
3 3 3 3
i(K3 ?K1 )z

+E2?E2?E ? + E2?E2?E ?]e

(2.24a)

3 3
2 2 ?

i(K3 ?K1 )z

= 6E1(|E2?|

+ |E2?| ) + 6E3 E2?E2? e

. (2.24b)


The various terms in (2.24a) have simple physical interpretations. For instance, the ?rst term results from the product in [x(1)]3


E1 ei(K1 z??1 t)E2

i(K2? • r??2 t)


? ?1(K2? • r??2 t) 2?

Note that the pump phase dependencies K2? • r cancel identically, as they do for all terms in (2.24). The ?rst and second terms represent contributions to
the nonlinear refraction of the ?eld E1 due the nonlinear index (Kerr e?ect)
induced by the pump ?eld intensity I2?. The third term can be understood as originating from the scattering of the ?eld E2? o? the grating produced by E1 and E2?, etc. Its e?ect on the polarization is precisely the same as that of the


?rst two terms, but we have intentionally written it separately in anticipation of the quantum mechanical discussion of Chap. 10, where the order in which the ?elds are applied to the medium matters. Indeed in our classical model, the terms in the ?rst three lines in (2.24a) are all proportional to the product
of a pump ?eld intensity and the probe ?eld E1, and can be globally described
as nonlinear absorption and refraction terms.
The last two lines in (2.24a) couple the sidemode E3 to E1 via the following scattering mechanism: the ?eld E ? interferes with the pump ?elds E2? and E2? to induce two complex index gratings that scatter E2? and E2?, respec- tively, into E1. This process, which is essentially the real-time realization of
holographic writing and reading, is called phase conjugation and is discussed
in detail in Chap. 10. The process retrore?ects a wavefront, sending it back along the path through which it came (see Fig. 10.1). It can be used to com- pensate for poor optics. Note that although the pump phase dependencies cancel one another as they do for the terms in the ?rst four lines of (2.24a),
the induced polarization has the phase exp[i(K3z ? ?1t)], while Maxwell’s equations require exp[i(K1z ? ?1t)]. This gives the phase mismatch factor exp[i(K3 ? K1)z], which is important except in the degenerate frequency case
?1 = ?2 = ?3, for which K3 = K1.
Neglecting the depletion of the pump beams E2? and E2?, we ?nd the coupled-mode equations for E1 and E ?

dE1 =
dz dE ?


??1E1


+ ?1E ? e2i?Kz , (2.25a)

3 ? ?

? ?2i?Kz



where

? dz = ??3 E3 + ?3 E1 e

, (2.25b)

?K = K3 ? K1 . (2.26)
2
Here ?dE ?/dz appears since E ? propagates along ?z, and we use ?n for the
3 3
coupling coe?cient to agree with later usage, although it is only a part of a
susceptibility.
To solve these equations, we proceed by ?rst transforming away the phase mismatch by the substitution

E1 = A1 e2i?Kz (2.27)

into (2.25). In particular, (2.25a) becomes


dA1 =
dz

?(?1

+ 2i?K)A1

+ ?1E ? . (2.28)


We seek solutions of (2.25b, 2.28) of the form e?z . Substituting A1 = e?z into (2.28), solving for E ?, and substituting the result into (2.25b), we ?nd the
eigenvalues


1 2 ? 1/2
?± = ? 2 (?1 ? ?? + 2i?K) ± [(?1 + ?? + 2i?K) /4 ? ?1? ]
3 3 3
= ?a ± [?2 ? ?1??]1/2 = ?a ± w . (2.29)

Hence the general solutions are
A1(z) = e?az [A ewz + B e?wz ] (2.30a)


and


E ? ?az wz


?wz

3 (z) = e

[C e

+ D e

] , (2.30b)

where the coe?cients A, B, C and D are determined by the boundary condi- tions of the problem.
We suppose here that a weak signal weak ?eld E1(0) is injected inside the
nonlinear medium at z = 0, and we study the growth of the counterpropa- gating conjugate wave E ?, which is taken to be zero at z = L. This means that A1(0) = E1(0) = constant, and E ?(L) = 0, in which case one has immediately B = E1(0)?A and D = ?C ewL. Matching the boundary conditions of (2.25b,
2.28) at z = L yields


1 wL
A = 2 A1(0) e?


(w ? ?)/(w cosh wL + ? sinh wL) , (2.31)

2wC ewL = ??(A sinh wL + A1(0) e?wL . (2.32)

Further manipulation yields ?nally


E1(z) = E1(0) e?(a+w+2i?K)z

. (w ?) ew(z?L) sinh wz .
1 +
w cos wL + ? sinh wL


, (2.33)


3 (z) = ?3E1(0)

e?az sinh w(z ? L)


. (2.34)

E ? ?

w cosh wL + ? sinh wL

In particular the amplitude re?ection coe?cient r = E ?(0)/E1(0) is given by

r = E3 (0) =


sinh wL ??


. (2.35)

E1(0)

? 3 w cosh wL + ? sinh wL

See Chap. 10 on phase conjugation for further discussion of these equations.


Coupled Modes and Squeezing

A popular topic is the “squeezing”, i.e., deamplifying, of noise in one quadra- ture of an electromagnetic wave at the expense of amplifying the noise in the orthogonal quadrature. One way to achieve such squeezing is through the use of mode coupling mechanisms such as described by (2.25a, b). To see under which conditions the ?n coupling factors can lead to this quadrature- dependent ampli?cation, let’s drop the ?n term in (2.25) and put the time dependencies back in. We ?nd for example the schematic equation



{?(3)E 2

?2i?t

}[E3 e

i?t]?

? E1e

?i?t

, (2.36)


where (?3) is a third-order susceptibility. Suppose that at a time t, ei?t = 1 and that {} = 1. According to (2.36), this tends to amplify E1. Now wait until the orthogonal quadrature phasor exp(i?t ? i?/2) = 1. At this time, the second-harmonic (two-photon) phasor exp(?2i?t) has precessed through two times ?/2, that is, {} = –1. Hence a two-photon coupling {} ?ips the sign
of the coupling between orthogonal quadratures. This is the signature of a
coupling process that can lead to squeezing. It is equally possible that a ?(2) process with a single pump photon having the value 2? can cause squeezing. This ?(2) process is known as parametric ampli?cation. Chapter 17 discusses the squeezing of quantum noise by four-wave mixing.


Nonlinear Susceptibilities

So far we have used an anharmonic oscillator to introduce some aspects of nonlinear optics that are useful in the remainder of this book. Such a simple model is surprisingly powerful and permits us to understand numerous non- linear optics e?ects intuitively. In general, however, ?rst principle quantum mechanical calculations are needed to determine the response of a medium to a strong electromagnetic ?eld. A substantial fraction of this book ad- dresses this problem under resonant or near-resonant conditions, i.e., under conditions such that the frequency (ies) of the ?eld(s) are near an atomic transition. Perturbative analyzes such as sketched in this chapter are usually not su?cient to describe these situations.
In many cases, however, the incident radiation is far from resonance with any transition of interest, and/or the material relaxation rate is exceedingly fast. In such cases, perturbation theory based on the concept on nonlinear susceptibility may be of great advantage. This is the realm of conventional nonlinear optics, and the reader should consult the recent treatises by Shen (1984), by Hopf and Stegeman (1986), and by Boyd (1992), as well as the classic book by Bloembergen (1965), for detailed descriptions of these ?elds. Here we limit ourselves to a brief introduction to the formalism of nonlinear susceptibility.
In linear problems, the polarization of the medium is (by de?nition) a linear function of the applied electric ?elds. The most general form that it can take is given by the space-time convolution of a linear susceptibility tensor ?(1) with the electric ?eld:

¸
P(r, t) = ?0

¸ t
d3r
??


dtr?(1)(r ? rr,t ? tr) : E(rr, tr) . (2.37)


Taking the four-dimensional Fourier transform of this expression for a monochro- matic wave E(r, t) = E (K, ?)ei(K•r??t), we ?id

2.5 Nonlinear Susceptibilities 49
P(K, ?) = ?0?(1)(K, ?) : E(K, ?) , (2.38)

with

¸
?(1)(K, ?) =


¸ t
d3r
??



dtr?(1)(r, tr)ei(K • r??tr ) . (2.39)

The linear dielectric constant is related to ?(1)(K, ?) via

?(K, ?) = ?0[1 + ?(1)(K, ?)] . (2.40)

In the nonlinear case, and for electric ?elds su?ciently weak that perturbation theory is valid, one gets instead

¸
P (r, t) = ?0
¸

¸ t
d3r
??


dtr?(1)(r ? rr,t ? tr) • E(rr, tr)

+?0
¸
+?0

.

dr1dt1dr2dt2?(2)(r ? r1,t ? t1; r ? r2,t ? t2) : E(r1, t1)E(r2, t2)

dr1dt1dr2dt2dr3dt2?(3)(r ? r1,t ? t1; r ? r2,t ? t2; r ? r3,t ? t3)

. E(r1, t1)E(r2, t2)E(r3, t3) + ... , (2.41)

where ?(n) the nth-order susceptibility. If E(r, t) can be expressed as a sum of plane waves,
E(r, t) = . E(Kn, ?n)ei(Kn • r??n t) , (2.42)
n
then as in the linear case, the Fourier transform of (2.41) gives

P(K, ?) = P(1)(K, ?) + P(2)(K, ?) + P(3)(K, ?) + ... (2.43)

with P(1)(K, ?) given by (2.38) and

P(2)(K, ?) = ?(2)(K = Kn + Km,? = ?n + ?m) : E(Kn, ?n)E(Km, ?m)
P(3)(K, ?) = ?(3)(K = Kn + Km + KA,? = ?n + ?m + ?A)

.E(Kn, ?n)E(Km, ?m)E(KA, ?A)


and



?(n)(K = K1 + K2 + ... + Kn,? = ?1 + ?2 + ... + ?n)
¸

= d3r1dt1dr dt2 ... d3rndtn?(n)
×(r ? r1,t ? t1; r ? r2,t ? t2; ... r ? rn,t ? tn)
× exp[+iKi•(r ? r1) ? i?1(t ? t1) + ... + iKn•(r ? rn) ? i?n(t ? tn)] .


Problems

Solve the couplem. ode equations

dE1 =
dz


??1E1


+ ?1E ? , (2.44)

dE ? =

?? ?

+ ??


, (2.45)

dz ?

3 E3

3E1


valid for phase-matched forward three-wave mixing. Ans:
E1(z) = e?az [E1(0) cosh wz + (??E1(0) + ?1E ?(0)) sinh wz/w] (2.46)
E ? ?az ? ? ?

3 (z) = e

[E3 (0) cosh wz + (aE3 (0) + ?3E1(0)) sinh wz/w] , (2.47)

where a = (?1 + ??)/2,? = (?1 ? ??)/2, and w = ,?2 + ?1??.
3 3 3
Derive the coe?cients in the coupled-mode equations (2.22).

Calculate all wavelengths generated in a ?(3) nonlinear medium by a combination of 632.8 nm and 388 nm laser light.

Calculate the coupling coe?cient ?n for four-wave mixing based on an anharmonic oscillator.

Write the propagation equations for second-harmonic generation. Com- ment on phase matching.

Calculate the phase mismatch for a conjugate wave of frequency ?3 =
?2 + (?2 ? ?1) generated by signal and pump waves with frequencies ?1 and
?2, respectively, and propagating in the same direction. Include the fact that
the indices of refraction for the three waves are in general di?erent, that is,
?(?1) = ?(?2) + ??1 and ?(?3) = ?(?2) + ??3.

Show that (2.20) contains all the fundamental contributions from the third-order expression (2.19).

Evaluate the re?ection coe?cient r of (2.35) in the limit of large L. Answer: r = ???/(w ± ?) for Re(w) ? 0.

3 Quantum Mechanical Background











Chapters 1, 2 describe the interaction of radiation with matter in terms of a phenomenological classical polarization P. The question remains as to when this approach is justi?ed and what to do when it isn’t. Unexcited systems interacting with radiation far from the system resonances can often be treated purely classically. The response of the system near and at resonance often deviates substantially from the classical descriptions. Since the laser itself and many applications involve systems near atomic (or molecular) resonances, we need to study them with the aid of quantum mechanics.
In preparation for this study, this chapter reviews some of the highlights of quantum mechanics paying particular attention to topics relevant to the interaction of radiation with matter. Section 3.1 introduces the wave function for an abstract quantum system, discusses the wave function’s probabilistic interpretation, its role in the calculation of expectation values, and its equa- tion of motion (the Schro¨dinger equation). Expansions of the wave function in various bases, most notably in terms of energy eigenstates, are presented and used to convert the Schro¨dinger partial di?erential equation into a set of ordinary di?erential equations. The Dirac notation is reviewed and used to discuss the state vector and how the state vector is related to the wave func- tion. System time evolution is revisited with a short review of the Schro¨dinger, Heisenberg and interaction pictures.
In Chaps. 4–12, we are concerned with the interaction of classical electro- magnetic ?elds with simple atomic systems. Section 3.2 lays the foundations
for these chapters by discussing wave functions for atomic systems and study- ing their evolution under the in?uence of applied perturbations. Time depen- dent perturbation theory and the rotating wave approximation are used to predict this evolution in limits for which transitions are unlikely. The Fermi Golden Rule is derived. Section 3.3 deals with a particularly simple atomic model, the two-level atom subject to a resonant or nearly resonant classi- cal ?eld. We ?rst discuss the nature of the electric-dipole interaction and then use the Fermi Golden Rule to derive Einstein’s A and B coe?cients for spontaneous and stimulated emission. We then relax the assumption that the interaction is weak and derive the famous Rabi solution.
In Chaps. 13–19, we discuss interactions for which the electromagnetic
?eld as well as the atoms must be quantized. In particular, Chap. 13 shows


that electromagnetic ?eld modes are described mathematically by simple har- monic oscillators. In addition, these oscillators can model the polarization of certain kinds of media, such as simple molecular systems. In preparation for such problems, Sect. 3.4 quantizes the simple harmonic oscillator. The section writes the appropriate Hamiltonian in terms of the annihilation and creation operators, and derives the corresponding energy eigenstates.
This chapter is concerned with the quantum mechanics of single systems in pure states. Discussions of mixtures of systems including the decay phe- nomena and excitation mechanisms that occur in lasers and their applications are postponed to Chap. 4 on the density matrix.


3.1 Review of Quantum Mechanics

According to the postulates of quantum mechanics, the best possible knowl- edge about a quantum mechanical system is given by its wave function ?(r, t). Although ?(r, t) itself has no direct physical meaning, it allows us to calcu-
late the expectation values of all observables of interest. This is due to the
fact that the quantity
?(r, t)??(r, t) d3r

is the probability of ?nding the system in the volume element d3r. Since the system described by ?(r, t) is assumed to exist, its probability of being somewhere has to equal 1. This gives the normalization condition
¸
?(r, t)??(r, t) d3r = 1 , (3.1)

where the integration is taken over all space.
An observable is represented by a Hermitian operator O(r) and its ex-
pectation value is given in terms of ?(r, t) by
¸

(O)(t) =

d3r??(r, t) O(r)?(r, t) . (3.2)



Experimentally this expectation value is given by the average value of the results of many measurements of the observable O acting on identically pre- pared systems. The accuracy of the experimental value for (O) typically de-
pends on the number of measurements performed. Hence enough measure- ments should be made so that the value obtained for (O) doesn’t change
signi?cantly when still more measurements are performed. It is crucial to
note that the expectation value (3.2) predicts the average from many mea- surements; in general it is unable to predict the outcome of a single event with absolute certainty. This does not mean that quantum mechanics in other ways is unable to make some predictions about single events.


The reason observables, such as position, momentum, energy, and dipole moment, are represented by Hermitian operators is that the expectation val- ues (3.2) must be real. Denoting by (?, ?) the inner or scalar product of two
vectors ? and ?, we say that a linear operator O is Hermitian if the equality
(?, O?) = (O?, ?) . (3.3)
holds for all ? and ?. In this notation, (3.2) reads (O) = (?, O?).
An important observable in the interaction of radiation with bound elec-
trons is the electric dipole er. This operator provides the bridge between the quantum mechanical description of a system and the polarization of the medium P used as a source in Maxwell’s equations for the electromagnetic ?eld. According to (3.2) the expectation value of er is
¸

(er)(t) =

d3 r r e| ? (r, t) |2 , (3.4)


where we can move er to the left of ?(r, t)? since the two commute (an opera- tor like ?cannot be moved). Here we see that the dipole-moment expectation value has the same form as the classical value if we identify ? = e|?(r, t)|2 as
the charge density.
In nonrelativistic quantum mechanics, the evolution of ?(r, t) is governed by the Schro¨dinger equation.

?
ik ?t ?(r, t) = H?(r, t) , (3.5)
where H is the Hamiltonian for the system and k = 1.054 × 10?34 joule- seconds is Planck’s constant divided by 2?. The Hamiltonian of an unper- turbed system, for instance an atom not interacting with light, is the sum of its kinetic and potential energies


2 2
= ? ?
2m


+ V (r) , (3.6)

where m is its mass and V (r) the potential energy. As we shall see shortly, in the coordinate representation we are considering here the momentum op- erator pˆ is expressed in terms of the system’s position operator r as
pˆ = ?ik? , (3.7)

so that we recognize that the ?rst term of the Hamiltonian (3.6) is nothing but the kinetic energy of the system. Note also the important relationship

[xˆi, pˆj ] = ik?ij (3.8)


where xˆi and pˆj are cartesian coordinates of the operators xˆ


and pˆ, and

[aˆ, ˆb] ? aˆˆb ? aˆˆb is the commutator between the operators aˆ and ˆb. Observ- ables which satisfy this commutation relation are generally called conjugate variables.


In view of (3.7), we see that the Schro¨dinger equation (3.5) is a partial di?erential equation. The time and space dependencies in (3.5) separate for functions having the form
?n(r, t) = un(r)e?i?n t (3.9) for which the un(r) satisfy the energy eigenvalue equation
Hun(r) ? k?nun(r) . (3.10)

The eigenfunctions un(r) can be shown to be orthogonal, and we take them to be normalized according to (3.1), so that they are then orthonormal,

¸ u? 3

. 1 n = m

and complete

n(r)um(r)d r = ?n,m =

(3.11)
0 n ƒ= m

. u? r r
n(r)un(r ) = ?(r ? r ) , (3.12)
n
where ?n,m and ?(r ? rr) are the Kronecker and Dirac delta functions, re- spectively. The completeness relation (3.12) means that any function can be
written as a superposition of the un(r). Problem 3.1 shows that this de?nition is equivalent to saying that any wave function can be expanded in a complete set of states.
In particular the wave function ?(r, t) itself can be written as the super-
position of the ?n(r, t):
?(r, t) = . Cn(t)un(r)e?i?n t . (3.13)
n

The expansion coe?cients Cn(t) are actually independent of time for prob- lems described by a Hamiltonian satisfying the eigenvalue equation (3.10). We have nevertheless included a time dependence in anticipation of adding an interaction energy to the Hamiltonian. Such a modi?ed Hamiltonian wouldn’t quite satisfy (3.10), thereby causing the Cn(t) to change in time.
Substituting (3.13) into the normalization condition (3.1) and using the
orthonormality condition (10), we ?nd
.
|Cn|2 = 1 . (3.14)
n
The |Cn|2 can be interpreted as the probability that the system is in the nth energy state. The Cn are complex probability amplitudes and completely determine the wave function. To ?nd the expectation value (3.2) in terms of the Cn, we substitute (3.13) into (3.2). This gives
(O) = . CnC? Omne?i?nm t , (3.15)
n,m


where the operator matrix elements Omn are given by
¸

Omn =

d3r u?

(r)Oun(r) , (3.16)



and the frequency di?erences



?nm = ?n ? ?m . (3.17)


Typically we consider the interaction of atoms with electromagnetic ?elds. To treat such interactions, we add the appropriate interaction energy to the Hamiltonian, that is
H = H0 + V . (3.18)

If we expand the wave function in terms of the eigenfunctions of the “unper- turbed Hamiltonian” H0, rather than those of the total Hamiltonian H, the
probability amplitudes Cn(t) change in time. To ?nd out just how, we sub- stitute the wave function (3.13) and Hamiltonian (3.18) into Schro¨dinger’s
equation (3.5) to ?nd
.(k?n + V)Cnun(r)e?i?n t = .(k?nCn + ikC? n)un(r)e?i?n t . (3.19)
n n

Cancelling the k?n terms, changing the summation index n to m, multiplying through by u? (r) exp(i?nt), and using the orthonormality property (3.11),
we ?nd the equation of motion for the probability amplitude Cn(t)

i

C? n(t) =
k

where the matrix element

(n|V|m)ei?nm tCm(t) , (3.20)
m


¸

(n|V|m) =

d3ru? (r)Vum(r) . (3.21)


Note that instead of the form (3.13), we can also expand the wave function
?(r, t) as
?(r, t) = . cn(t)un(r) , (3.22)
n
for which the k?n time dependence in (3.19) doesn’t cancel out. The cn(t) then obey the equation of motion

c? (t) = ?i? c (t) ? i .
k m


(n|V|m)cm(t) . (3.23)


In terms of the cn, the expectation value (3.2) becomes
(O) = . cnc? Omn . (3.24)
n,m


Equation (3.20) and equivalently (3.23) shows how the probability am- plitudes for the wave function written as a superposition of energy eigen- functions changes in time. They are equivalent to the original Schro¨dinger equation (3.5), but are no longer concerned with the precise position depen- dence, which is already accounted for by the r-dependence of the eigenfunc- tions un(r). In particular if we’re only concerned about how a system such as an atom absorbs energy from a light ?eld, this development is completely described by the changes in the Cn or cn.
The choice of using the relatively slowly varying Cn versus using the rapidly varying cn is a matter of taste and convenience. The time dependence of the
Cn is due to the interaction energy V alone, while that of the cn is due to the total Hamiltonian H. To distinguish between the two, we say that the
Cn are in the interaction picture, while the cn are in the Schr¨odinger picture.
We discuss this more formally at the end of this section.
Armed with (3.20) or (3.23), you can skip directly to Sect. 3.2, which shows how systems evolve in time due to various interactions. Before going ahead, we review the Dirac notation and some other aspects of the wave
function and of its more abstract form, the state vector |?). This material is
needed for our discussions involving quantized ?elds in Chaps. 13–19, and is
useful in proving various properties of the density operator in Chap. 4.
Up to now we have used the so-called coordinate representation, where all operators, as well as the wave function, are expressed as functions of the system’s position r. Alternatively, one can work in a number of other representations, a rather common one being the momentum representation. Here, operators and wave functions are expressed as functions of the system’s momentum p. As we shall see shortly, one can transform the system’s wave function from the coordinate to the momentum representation by a simple Fourier transformation of ?(r, t),


?(p, t) =

1 ¸
(2?k)3/2


d3r ?(r, t)e?ip•r/k . (3.25)


Here ?(p, t) describes the same dynamical state as ?(r, t). It doesn’t make any di?erence in principle which representation we choose to use, and as we
see with the C? n, we sometimes don’t have to worry about the coordinate
representation at all.
We now turn to a discussion of a general formalism which does away with the explicit use of representations, and allows us to switch from one to another representation, when it is desirable.


Dirac Notation

The formalism that permits one to achieve this goal is the Dirac notation. Roughly speaking Dirac’s formulation is analogous to using vectors instead of coordinates. The notation has an additional advantage in that one can label











Fig. 3.1. A two dimensional vector written in ordinary vector notation and in Dirac notation


the basis vectors much more conveniently than with ordinary vector notation. We start our discussion with a comparison between ordinary notation for a vector in a two-dimensional space and Dirac’s version. As shown in Fig. 3.1, a vector v can be expanded as

v = vxxˆ + vy yˆ , (3.26)

where xˆ and yˆ are unit vectors along the x and y axes, respectively. In Dirac notation, this reads
|v) = vx|x) + vy |y) , (3.27)
the component vx given in ordinary vector notation by the dot product xˆ • v
is given in Dirac notation by

vx = (x|v) . (3.28)
The Dirac vector |v) is called a “ket” and the vector (v| a “bra”, which come from calling the inner product (3.28) a “bra c ket”. With the notation (3.28), (3.27) reads

|v) = |x)(x|v) + |y)(y|v) . (3.29) This immediately gives the identity diadic (outer product of two vectors)
|x)(x| + |y)(y| = I . (3.30)

Equations (3.27, 3.30) can be immediately generalized to many dimensions as in




n
where the {|n)} are a complete orthonormal set of vectors, i.e., a basis. The inner products (n|v) are the expansion coe?cients of the vector |v) in this basis. The bra (n| is the adjoint of the ket |n) and the expansion coe?cients
have the property


(k|v) = (v|k)? . (3.33)
Unlike the real spaces of usual geometry, quantum mechanics works in a complex vector space called a Hilbert space, where the expansion coe?cients are in general complex.
The basis {|n)} is discrete. Alternatively, we can expand vectors in terms of the coordinate basis {|r)} which like the {|n)} forms a complete basis,
albeit a continuous one. For such a situation we need to use continuous sum-
mations in (3.31, 3.32), that is, integrals. For example, the identity operator of (3.32) can be expanded as
¸
I = d3r |r)(r| . (3.34)


One major advantage of the bra and ket notation is that you can label the vectors with as many letters as desired. For example, you could write |r??) in place of |r).
The vector of primary interest in quantum mechanics is the state vector
|?(t)). The wave function is actually the expansion coe?cient of |?) in the
coordinate basis




where the wave function

¸
|?) =

d3r|r)(r|?) , (3.35)

?(r, t) = (r|?) . (3.36)
Hence the state vector |?) is equivalent to the wave function ?(r, t), but doesn’t explicitly display the coordinate dependence.
Instead of using the position expansion of (3.35), we can expand the state vector in the discrete basis {|n)} as
|?) = . cn|n) . (3.37)
n

The most common basis to use consists of the eigenstates of the unperturbed Hamiltonian operator H0. For this basis, the expansion coe?cients cn are just
those in (3.22), and the energy eigenfunctions are related to the eigenvectors
by
un(r) = (r|n) . (3.38)
A useful trick in transforming from one basis to another is to think of the vertical bar as an identity operator expanded either as in (3.32) or as in (3.34). Using the form of (3.34) in (3.37) on both sides of the equation along with (3.38), we recover (3.22).
The expectation value of the operator O is given in terms of the state
vector by
(O)(t) = (?(t)|O|?(t)) . (3.39)