Coupled-Mode Equations

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.