journal of modern optics, 2002, vol. 49, no. 9, 1437–1452
Rigorous derivation of coupled mode equations for short,
high-intensity grating-coupled, co-propagating pulses
J. E. SIPE{, C. MARTIJN DE STERKE{} and
B. J. EGGLETON}
{ Department of Physics, University of Toronto, Toronto, Canada,
M5S1A7
{ School of Physics, University of Sydney 2006, Australia
} Australian Photonics Cooperative Research Centre, Australian
Technology Park, Eveleigh, 1430, Australia
} Bell Laboratories, Lucent Technologies, 600 Mountain Ave., Murray
Hill, NJ 07974, USA
(Received 31 December 2000; revision received 15 August 2001)
Abstract.
We present a rigorous derivation of coupled mode equations for
the grating-assisted coupling of short, co-propagating pulses. Since this is a copropagating geometry, the required grating period is much longer than the
wavelength of the light and is typically 0:1–10 mm. This can exceed the spatial
extent of the pulses substantially, and the validity of the usual assumptions that
lead to coupled mode theory is therefore not guaranteed. Here we show
rigorously that the standard nonlinear coupled mode equations can be used to
describe these experiments and we give expressions for the parameters.
1.
Introduction
In 1988, Trillo et al. [1] first demonstrated all-optical polarization switching in
a birefringent fibre with a rocking filter. In these experiments, the rocking filter
acts as coupling element between the two orthogonal polarization modes of the
fibre. The typical operation of such a device is that at low intensities the rocking
filter resonantly couples the two modes, whereas at high intensities the propagation
constants of the modes are affected by the fibre nonlinearity, shifting the resonance, and thus effectively decoupling the modes. This leads to a power dependence of the coupling ratio into the two modes. A rocking filter is an example of a
more general class of devices in which modes propagating in the same direction are
coupled by a periodic structure. The two modes can be the two polarization modes
of a birefringent fibre coupled by a rocking filter [1–3], or can be totally different
modes of a circular fibre coupled by a long-period grating [4, 5], or can be the
modes of different, unequal cores.
Since the work by Trillo et al. [1], nonlinear optical switching between two copropagating modes has also been reported by Krautchik et al. [2], Psaila et al. [3],
and Eggleton et al. [4, 5]. The more recent work was partly driven by the discovery
of the photosensitivity of optical fibres, which allows one to write conveniently
gratings of essentially arbitrary period [6–8]. A characteristic feature of many of
these experiments is that the length of the optical pulses is much shorter than the
Journal of Modern Optics ISSN 0950–0340 print/ISSN 1362–3044 online # 2002 Taylor & Francis Ltd
http://www.tandf.co.uk/journals
DOI: 10.1080/09500340110103878
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Figure 1.
Schematic of the type of experiments considered. The solid line indicates the
electric field of a short pulse as a function of position at a fixed time. The dashed
line indicates the periodicity, for example the refractive index, or the orientation of
the principal axes of the fibre. The pulse envelope contains many optical cycles, yet
is shorter than the period.
grating period. In the experiments of [3] for example, 4 ps pulses, corresponding to
a spatial extent of less than 1 mm, are coupled by a grating with a period exceeding
10 mm. This is illustrated in figure 1, in which the solid line shows the electric field
associated with a short pulse versus position, at a fixed time. The dashed line
indicates the periodicity; this line could indicate the orientation of the birefringence axes [1–3], or the refractive index [4, 5].
The analysis of experiments of the type discussed above is usually based on
coupled mode theory [9–14]. Two basic types of derivations of the coupled mode
equations are known, both of which require that the pulse is long enough to
contain many optical cycles. The first type consists of CW calculations for the
response of a linear grating [9, 10]. Although, using Fourier analysis, such results
can be extended to include time-dependent calculations in linear problems, such
an approach of course does not apply to cases where the optical nonlinearity plays a
role. The second type of derivation of coupled mode equations [12–14] explicitly
addresses pulses propagating in a nonlinear medium, but also has its limitations.
The first is that the medium is often taken to be one-dimensional from the outset,
so that the light propagates as a plane wave [12–14]. Although this may be
satisfactory for core modes, when dealing with cladding modes one requires
explicit expressions for the coupling coefficient in terms of the refractive index
perturbation and the overlap of the relevant modes. The other limitation of the
second type of derivation is that the optical pulses are implicitly taken to be longer
than the period of the grating. One method, for example, expresses the electric
field as the sum of the products of Bloch functions, the magnitude of which has the
period of the grating, and a slowly varying envelope [13]. This is justified when
this is applied to the coupling between counter-propagating modes, since the
grating period is then smaller than the wavelength; therefore if the pulse contains
many optical periods, it also extends over many grating periods. However, such an
approach is clearly suspicious in the cases we are considering here, where the pulse
length, which still contains many optical periods, can be much smaller than the
grating period. The other methods have similar problems. Therefore, these
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derivations are suspicious when applied to short pulses since the phase accumulation that leads to the reflection differs in the two cases. When the pulses are much
longer than a period then this phase accumulation can be thought of as occurring
gradually over the length of the pulse. In contrast, for short pulses the phase
accumulation is generally uniform over the length of the pulse and it must be
thought of as occuring period-by-period in an abrupt manner. The envelope
functions, the evolution of which is given by the coupled mode equations that are
to be derived here, thus describe this phase accumulation as smoothed out over
many periods, rather than the usual interpretation as a gradual process that occurs
along the length of a pulse. There is thus no really satisfactory justification of the
use of coupled mode theory for the type of experiments discussed above.
Here we present such a justification. Briefly, it relies on the physical process
that at each of the interfaces, or, more generally, at each nonuniformity in the
guided-wave structure, a small fraction of the pulse is coupled into other modes of
the structure. By considering the local coupled mode of the structure, this coupling
can be quantified. By superimposing these small contributions in a periodic
structure, with the correct phases, applying a phase matching condition, and
focusing on the field envelope functions, a set of coupled mode equations can be
derived rigorously. The result confirms the general form of the standard coupled
mode equations, and the expressions for the coefficients. Although few would
probably have doubted the general form of the coupled mode equations, rigorous
determination of the coefficients is essential if quantitative comparison between
theory and experiment is to be carried out.
Our derivation is somewhat similar to the local coupled mode theory presented
by Marcuse, [9] that of Snyder and Love [10] and the more recent work, done in
the context of second-harmonic generation, of Isoshima and Tada [15]. However,
compared to the work of Marcuse [9] and Snyder and Love [10], ours can deal with
pulses and is not limited to cw signals, it explicitly deals with Kerr nonlinearities,
it has the appealing physical basis outlined above, and clearly indicates the
approximations made.
The outline of this paper is as follows. We derive (exact) coupled modal
amplitude equations for a non-uniform waveguide in section 2, introduce equations for the electric field envelopes in section 3 and then in section 4 focus on
periodic nonuniformities and derive final equations.
2.
Derivation of the coupled-amplitude equations
Consider a guided-wave structure with a refractive index n ¼ nðx; y; zÞ, where z
is the propagation axis. At (angular) frequency !, the Maxwell equations read
r E ÿ i!0 H ¼ 0;
ð1Þ
r H þ i!"0 n2 ðx; y; zÞE ¼ ÿi!P;
where E, H and P are the electric and magnetic fields, and the nonlinear
polarization respectively, and "0 and 0 are the vacuum permeability and permittivity, respectively. In equation (1), P includes only the nonlinear polarization,
since the linear component enters via the refractive index n. We first consider the
modes of a guided-wave structure, and thus set P ¼ 0, initially.
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In the simple case in which n ¼ nðx; yÞ and P ¼ 0, the solutions of equations (1)
are of the form [9, 10]
Em ðx; yÞ exp ðiým zÞ;
Hm ðx; yÞ exp ðiým zÞ;
ð2Þ
where ým is the propagation constant of mode m, and refer to the propagation
direction. The ým include discrete values, associated with bound modes, and also a
continuous spectrum associated with radiation modes. Substituting (2) into
Maxwell’s equations (1), it is then found that Em and Hm satisfy the coupled
equations
iým z^ Em þ r? Em ÿ i!0 Hm ¼ 0;
ð3Þ
iým z^ Hm þ r? Hm þ i!"0 n2 ðx; yÞEm ¼ 0;
where x^, y^ and z^ are the unit vectors in the indicated directions, and
r? ¼ x^ @=@x þ y^ @=@y.
We now return to the general case where n ¼ nðx; y; zÞ, and define the local
modes Em , Hm and the local propagation constant ým ðzÞ as the solutions to
equations (3), but with n ¼ nðx; y; zÞ replacing n ¼ nðx; yÞ. Thus at some value of z,
z0 say, one finds the modes for a structure with nðx; y; zÞ ¼ nðx; y; z0 Þ. Then Em ,
Hm , and ým acquire a z-dependence by simply varying z0 . Of course these are not
solutions to Maxwell’s equations. This can be shown explicitly by substituting
am ðzÞEm ðx; y; zÞ exp ðiþm ðzÞÞ;
and a similar expression for H, where
þm ðzÞ ¼
ðz
0
ð4Þ
ým ðz 0 Þ dz 0 ;
ð5Þ
into Maxwell equations (1). From its definition, þm ðzÞ represents the lowest order
approximation to the phase. Making use of relations (3), we establish that
r ½am Em exp ðiþm ފ ÿ i!0 ½am Hm exp ðiþm ފ
dam
@Em
exp ðiþm Þ;
¼
z^ Em þ am z^ dz
@z
r ½am Hm exp ðiþm ފ þ i!"0 n2 ½am Em exp ðiþm ފ
dam
@Hm
¼
exp ðiþm Þ;
z^ Hm þ am z^ dz
@z
ð6Þ
where, for convenience, all spatial arguments have been dropped. Clearly these
equations are not satisfied unless the am depend on z.
We now search for solutions to the full Maxwell equations, i.e. with P 6¼ 0 in
equations (1), of the form
X
~ ðx; y; zÞ;
Eðx; y; zÞ ¼
am ðzÞ Em ðx; y; zÞ exp ðiþm ðzÞÞ þ E
m
Hðx; y; zÞ ¼
X
m
ð7Þ
am ðzÞ Hm ðx; y; zÞ exp ðiþm ðzÞÞ;
~ ðx; y; zÞ needs to be introduced here because of the
where ¼ . The field E
inclusion of the nonlinear polarization P in equations (1). As will be seen below, it
~ ¼ E~z~ and r E
~ ¼ ÿ^
^.
can be taken to point in the z-direction, so E
z r? E
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Substituting (7) into Maxwell equations (1) leads to two equations, only the second
of which has a z-component. For the z-component of the equation to be satisfied
~ ðx; y; zÞ ¼ ÿ Pz ðx; y; zÞ :
E
"0 n2 ðx; y; zÞ
ð8Þ
Using this relation and equations (6), it can then be shown that the remaining,
tranverse components of the Maxwell equations attain the form
Xdam
z^ Em exp ðiþm ðzÞÞ ¼ f E ðx; y; zÞ;
dz
m
Xdam
m
dz
ð9Þ
z^ Hm exp ðiþm ðzÞÞ ¼ f H ðx; y; zÞ;
where the driving terms f E and f H are given by
X
Pz
@Em
f E ¼ ÿ^
z r?
ÿ
am z^ exp ðiþm ðzÞÞ;
2
"0 n
@z
m
f H ¼ ÿi!P? ÿ
X
m
@Hm
exp ðiþm ðzÞÞ;
am z^ @z
ð10Þ
in obvious notation.
Following standard practice, [10] we set Em ¼ em? z^emz , and Hm ¼
hm? þ z^ hmz in equations (9), again, in obvious notation, and take the dot
product of the first of equations (9) with h?n? , and of the second with e?n? . We
then integrate transversely, and use mode orthogonality [9, 10],
ð
Pn
nm ;
ð11Þ
ds z^ ðem? h?n? Þ ¼
2
where P is the power carried in mode n, and ds is short for dx dy. Although
restricted to bound modes, this could easily be relaxed. Adding and subtracting the
resulting equation then finally leads to
ð
dan
Pn
exp ðiþn ðzÞÞ ¼ ds ½H?n f E ÿ E?n f H Š:
ð12Þ
dz
Here En , Hn , replace en , hn on the right-hand side because driving terms f E
and f h have no z-components.
The next task is to work out the right-hand sides of equations (12). This is done
in Appendix A, where it is shown that equations (12) can be written as
ð
X
dan
exp ðiþn ðzÞÞ ¼ i! ds E?n P Pn
am exp ðiþm ðzÞÞ
dz
m
@Hm
@Em
?
?
En ÿ
Hn :
z^ ds
@z
@z
ð
ð13Þ
Note that, unlike in one-dimensional treatments, these equations contain the zcomponents of all modal fields and of the induced nonlinear polarization.
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To cast equation (13) in its final form we define the coupling constants
ð i
@Hm 0
@Em 0
?
?
cn;m 0 ðzÞ ¼ ÿ pffiffiffiffiffiffiffiffiffiffiffiffi z^ ds
En ÿ
Hn :
@z
@z
PnPm
ð14Þ
Thus the coupling constants between the various modes depends on the spatial
variations of the electric and magnetic fields. However, at this stage it does not
explicitly depend on variationsÐin the propagation constants. Using the fact that in
a lossless medium the integral ds z^ ðEm 0 H?n Þ does not depend on z, it can be
shown that cn;m 0 ¼ c?m 0 ;n . We further define the normalized modal amplitudes
pffiffiffiffiffiffi
gn ðzÞ ¼ P n an ðzÞ;
ð15Þ
so that g2 has units of power, and we put
ð
i!
fn ðzÞ ¼ pffiffiffiffiffiffi exp ðÿiþn ðzÞÞ ds E?n P
Pn
ð16Þ
corresponding, in general, to nonlinear source terms. These equations have been
derived at a fixed frequency !. In the upcoming sections it is desirable to express
this explicitly in the notation. Hence, we write, ýðz; !Þ for ýðzÞ, and similarly for
the other parameters. In this notation, then, the final equations, which are exact,
read
@
gn ðz; !Þ
@z
¼ fn ðz; !Þ þ i
X
exp ðÿiþn ðz; !Þ þ i 0 þm ðz; !ÞÞ cn;m 0 ðz; !Þgm 0 ðz; !Þ: ð17Þ
m 0
Their interpretation is that changes in the modal amplitudes gn are associated
with nonlinear effects (first term on the right-hand side), or with waveguide
variations upon propagation (second term on right-hand side), leading to mode
coupling. The exponential factor expresses the required phase matching for the
efficient coupling between modes. Of course the nonlinear contributions also
require phase matching, but this is not expressed explicitly in equation (17).
Note that rapidly varying components are still included at this stage, and that we
have also not yet specified the relation between P and E.
3.
Envelope function equations
Here we rewrite the exact equation (17) as a set of approximate equations for the
mode envelopes. The time-dependent envelope function gn ðz; tÞ is introduced
through
ð1
ÞŠÞ exp ðÿið! ÿ !
ÞtÞ d!;
gn ðz; tÞ gn ðz; !Þ exp ði½þn ðz; !Þ ÿ þðz; !
ð18Þ
0
is a reference frequency, chosen to be close to the centre of the pulse
where !
in section 4.2). We define fn ðz; tÞ in the
spectrum (we return to the choice of !
one may write
same way. Now for frequencies ! close to !
@þn ðz; !Þ
Þ;
Þ þ ð! ÿ !
Þýn0 ðz; !
ýn ðz; !
@z
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Þ is the spatially varying
where definition (5) for the þm was used. Here ýn0 ðz; !
. Higher order terms in expansion (19), correinverse group velocity @ý=@! at !
sponding to quadratic and higher-order dispersion of the homogeneous structure,
are neglected. This is justified for the short distances in the relevant experiments
[1–5]. Differentiating (18) with respect to z, using equation (17) for @gn =@z
and (19) for @þn =@z, we then find
@
@
Þ gn ðz; tÞ
gn ðz; tÞ ¼ fn ðz; tÞ ÿ ýn0 ðz; !
@z
@t
X
Þ ÿ 0 þm ðz; !
ފÞ
þ i
exp ðÿi½þn ðz; !
m 0
ð1
0
ފÞ
exp ðþi½ 0 þm ðz; !Þ ÿ 0 þm ðz; !
ÞtÞ d!:
cn;m 0 ðz; !Þgm 0 ðz; !Þ exp ðÿið! ÿ !
ð20Þ
Making the approximation that the frequency dependence of the coupling constants is small allows us to make the Taylor expansion
0
Þ þ cn;m
Þ;
cn;m ðz; !Þ cn;m ðz; !
ðz; !Þð! ÿ !
ð21Þ
0
. This is justified if the
where cn;m
ðz; !Þ ¼ dcn;m ðz; !Þ=d!, evaluated at !
; note
spectrum of the electric field is sufficiently narrow, and centred close to !
that the latter of these can always be satisfied for the type of problems considered;
this is discussed in more detail below equation (46). Approximation (21) is
assumed to have been made in all of the following.
Next consider the nonlinear terms fn in equation (20), and relate P and E.
Using definition equation (16), and in analogy with equation (18), we find that
t ÿ þn ðz; !
ފÞ
fn ðz; tÞ ¼ exp ði½!
ð
ð1
i!
?
d! pffiffiffiffiffiffi exp ðÿi!tÞ ds En ðr; !Þ Pðr; !Þ :
Pn
0
ð22Þ
We now make a futher approximation, also justified for fields that are spectrally
, that the modal profiles Em do not depend on
narrow and centred around !
Þ in equation (22), and ! by !
for the
frequency. En ðr; !Þ is replaced by En ðr; !
same reason. Defining now
ð1
Pþ ðr; tÞ ¼
Pðr; !Þ exp ðÿi!tÞ d!;
ð23Þ
we can write (22) as
0
!
t ÿ þn ðz; !
ÞŠÞ pffiffiffiffiffiffi
fn ðz; tÞ i exp ði½!
Pn
ð
Þ Pþ ðr; tÞ:
ds E?n ðr; !
ð24Þ
Henceforth we consider the particular case where P is due to the Kerr effect
[16, 17], the nonlinear switching mechanism in the pertinent experiments [1–5].
However, our method also applies to other switching mechanisms, such as
cascaded quadratic nonlinearities [18–20]. Taking now a Kerr nonlinearity due
to a third-order susceptibility ð3Þ ðrÞ, P is of the form [16, 17]
Pðr; tÞ ¼ "0 ð3Þ ðrÞ Eðr; tÞEðr; tÞEðr; tÞ;
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again, in obvious notation. Now according to the first of equations (7), Eðr; !Þ
~ ), by equation (8), is due to
consists of two contributions. The second of these (E
the nonlinear polarization, and, for the case considered, is cubic in E. Since we are
interested in the dominant contribution of the nonlinearity, which is cubic in the
field, we neglect it, as, in effect, it leads to phenomena of quintic and higher order
in the electric field. However, when studying mode coupling in media exhibiting
nonlinear polarization that can be described by a combination of ð3Þ and ð5Þ , to
model, for example, a saturating nonlinearity, the leading contribution associated
~ must be included. Neglecting these contributions now in evaluating the
with E
right-hand side of equation (25), we write
X ð1
Eþ ðr; tÞ am ðz; !Þ Em ðr; !Þ exp ði½þm ðz; !Þ ÿ !tŠÞ d!
m
X
m
0
Þ
Em ðr; !
ð1
0
am ðz; !Þ exp ði½þm ðz; !Þ ÿ !tŠÞ d!;
ð26Þ
where, again, the frequency dependence of the modal profiles is neglected.
Defining now
Þ gm ðz; tÞ
Em ðr; !
pffiffiffiffiffiffi
ÞÞ;
exp ðiþm ðz; !
ð27Þ
Eþ
m ðr; tÞ ¼
Pm
and using equations (15) and (18), equation (26) can simply be written as
X
tÞ
Eþ ðr; tÞ exp ðÿi!
Eþ
m :
ð28Þ
m
We are now ready to evaluate the nonlinear contribution. First set
Eðr; tÞ ¼ Eþ ðr; tÞ þ Eÿ ðr; tÞ in obvious notation [cf equation (23)], and substitute
into equation (25). We then collect all positive frequency parts, using equation
(28), and neglect terms oscillating at 3! because they are not phase-matched.
Finally, we assume full permutation symmetry of ð3Þ [16]. This then leads to the
expression
X
ÿ
þ
Pþ ðr; tÞ ¼ 3"0 ð3Þ ðrÞ
ð3Þ ðrÞ Eþ
ð29Þ
m1 1 ðr; tÞEm2 2 ðr; tÞEm3 3 ðr; tÞ;
m
~~
where m
~~
is short for m1 1 m2 2 m3 3 . Using equations (24) and (27), and defining
the nonlinear coupling constant
ð
3"0 !
p
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ds E?n ðrÞ ð3Þ ðrÞ Em1 1 ðrÞE?m2 2 ðrÞEm3 3 ðrÞ;
ð30Þ
Kn;~
ðzÞ
m~
P n P m1 P m2 P m3
the nonlinear source term fn can finally be written as
X
?
fn ðz; tÞ ¼ i
exp ði½ÿþn þ 1 þm1 ÿ 2 þm2 þ 3 þm3 ŠÞ Kn;~
m~
ðzÞgm1 1 gm2 2 gm3 3 ;
m
~~
ð31Þ
where, for brevity, the arguments z of the þm , and the arguments ðz; tÞ of the gm
have been dropped. Further, putting
X
dn ðz; tÞ i
exp ðÿi½þn ÿ 0 þm ŠÞ cn;m 0 ðzÞgm 0 ðz; tÞ
ð32Þ
m 0
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where, again, some of the arguments have been dropped, then the mode envelopes
gn satisfy the coupled equations
@
@
Þ gn ðz; tÞ ¼ dn ðz; tÞþfn :
gn ðz; tÞ þ ýn0 ðz; !
@z
@t
ð33Þ
Equations (33) are the final result of this section. They express that in the absence
of nonuniformities (dn ¼ 0) and nonlinearities (fn ¼ 0) the waves propagate at
their group velocity and do not couple. Nonuniformities, through variations in the
group velocity and scattering, and nonlinear effects lead to coupling between the
modes.
4.
Application to rocking filters and long-period gratings
Until now we have considered general longitudinal variations in the guided
wave structure. We now focus on rocking filters and long-period gratings used in
the pertinent experiments [1–5]. We do so in three steps. We first limit ourselves to
periodic variations (section 4.1), then apply the further restriction to consider only
the coupling of two modes propagating in the forward direction (section 4.2), and,
finally, remove rapidly varying terms to obtain the final coupled mode equations
(section 4.3).
4.1. Periodic structures
We first limit ourselves to periodic variations. If the properties of the structure
Þ and ýn0 ðz; !
Þ are also periodic. Denote their
vary periodically, then ýn ðz; !
0
averages by ýn and ýn , and put
Þ ¼ ýn þ ý~n ðzÞ;
ýn ðz; !
ð34Þ
Þ ¼ ýn0 þ ý~n0 ðzÞ;
ýn0 ðz; !
where ý~n ðzÞ and ý~n0 ðzÞ are periodic functions with zero average. With these
definitions and equation (5)
ðz
þn ðzÞ ¼ zýn þ ý~n0 ðz 0 Þ dz 0 zýn þ þ~ðzÞ;
ð35Þ
0
where þ~ðzÞ is periodic. We take cn;m 0 and Kn;~
m~
ðzÞ [equations (14) and (30)] also
periodic, with the same period.
We now rewrite equations (33) expressing this periodicity explictly. To do this
we first put
cn;m 0 ðzÞ ¼ exp ðÿi½~
þn ðzÞ ÿ 0 þ~m ðzފÞcn;m 0 ðzÞ;
n;~
K
þn ðzÞ þ 1 þ~m1 ðzÞ ÿ 2 þ~m2 ðzÞ þ 3 þ~m3 ðzފÞKn;~
m~
ðzÞ:
m~
ðzÞ ¼ exp ði½ÿ~
ð36Þ
Note that since cm 0 ;n ðzÞ ¼ c?n;m 0 ðzÞ, the same is true for cm 0 ;n ðzÞ. Then we can
write the final result (33) from section 3 as
@gn
@gn
@gn
þ ýn0
¼ ÿý~n0
þ dn þ fn ;
@z
@t
@t
ð37Þ
where we have suppressed the arguments, and where fn and dn are defined as in
equations (31) and (32) but with the functions þn ðzÞ in the exponentials replaced by
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n;~
the constants ýn , and K
n;m 0 replacing Kn;~
m~
and c
m~
and cn;m 0 . Equations (37)
are the most general coupled mode equations for a periodic structure.
4.2. Forward propagating waves
Here we limit ourselves further to the case directly applicable to the nonlinear
switching experiments discussed in section 1 [1–5]. In these experiments the
interacting modes all propagate in the same direction, which we take to be the
forward direction. Moreover, for each frequency there are only two relevant
modes. Note that in the experiments of Eggleton et al. [4, 5] one of the modes is
the fundamental fibre mode, whereas the other is a cladding mode. Since these are
closely spaced, the cladding mode that is involved depends on frequency.
Thus we concentrate on the interaction of two forward propagating modes,
labelled 1 and 2. Since the modes are forward propagating, we have i ¼ þ
everywhere. We first need to find expressions for the linear and nonlinear coupling
coefficients (dn and fn , respectively) in coupled mode equations (37). Starting
with the former, we find from equation (32), with the replacements discussed in
the last paragraph of section 4.1
d1 ðz; tÞ ¼ i
c11 g1 ðz; tÞ þ i
c12 exp ðÿiðý1 ÿ ý2 ÞÞg2 ðz; tÞ;
d2 ðz; tÞ ¼ i
c21 exp ðÿiðý2 ÿ ý1 ÞÞg1 ðz; tÞ þ i
c22 g2 ðz; tÞ;
ð38Þ
with c21 ðzÞ ¼ c12? ðzÞ, corresponding to the linear terms in equations (37).
We now evaluate the nonlinear coupling coefficients fn in equations (37). To
do so we make the additional assumptions that the nonlinear effect is small, and
that the z-dependence of the modal profiles En ðrÞ is small as well. With these
assumptions the only nonlinear contribution in coupled mode equations (37) arises
from terms that are slowly varying (phase matched [16]). From equation (31), with
the replacements discussed in the last paragraph of section 4.1, for such terms we
require that
ÿýn þ ým1 ÿ ým2 þ ým3 ¼ 0:
ð39Þ
Note that the only residual variation to the fn is now due to the gm terms in
equation (31). Now first taking n ¼ 1 in equation (39), we have either m1 ¼
m2 ¼ m3 ¼ 1, or m1 ¼ m2 ¼ 2 and m3 ¼ 1, or m3 ¼ m2 ¼ 2 and m1 ¼ 1. The last
two contribute the same, and so
1111 jg1 j2 g1 þ 2iK
1221 jg2 j2 g1 ;
s1 ðz; tÞ ¼ iK
ð40Þ
2222 jg2 j2 g2 þ 2iK
2112 jg1 j2 g2 :
s2 ðz; tÞ ¼ iK
For convenient comparison with the literature we set
ð
1111 ¼ 30 !
ds E?1 ð3Þ E1 E?1 E1 ;
ÿ1s K
P 21
ð
1221 ¼ 30 !
ÿ K
ds E?1 ð3Þ E2 E?2 E1 :
P1P2
ð41Þ
We define ÿ2s similarly to ÿ1s , and note that because of the permutation symmetry
2112 ¼ ÿ .
[16] of ð3Þ , K
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1447
With equations (38) for the linear coupling terms, and equations (39) and (41)
for the nonlinear coupling terms, coupled mode equations (37) for two coupled
forward propagating waves can now be written as
@g1
@g1
@g1
þ ý10
¼ ÿý~10
þ i
c11 g1 þ ic12 exp ðÿiðý1 ÿ ý2 ÞzÞg2
@z
@t
@t
þ iÿ1s jg1 j2 g1 þ 2iÿ jg2 j2 g1 ¼ 0;
@g2
@g2
@g2
þ ý20
¼ ÿý~20
þ i
c11 g2 þ ic12 exp ðþiðý1 ÿ ý2 ÞzÞg1
@z
@t
@t
ð42Þ
þ iÿ2s jg2 j2 g2 þ 2iÿ jg1 j2 g2 ¼ 0:
Although equations (42) may look familiar, they contain rapidly varying
components. To show these explicitly, we rewrite equations (42) to express the
periodicity of the linear coupling coefficients cnm and the periodic component of the
inverse group velocity ý~n0 (see equations (34)), both having period . In the
experiments we consider [1–5], ¼ 0:1–10 mm. Because of the periodicity we
may expand
X
c^nm ðlÞ exp ðilKzÞ;
ð43Þ
cnm ðzÞ ¼
l
ý~n0 ðzÞ
¼
X
l6¼0
ý^n0 ðlÞ exp ðilKzÞ;
ð44Þ
where l ¼ 0; 1; 2; . . . ; and
K¼
2
:
ð45Þ
such that
Further, we choose our reference frequency !
ý1 ÿ ý2 ¼ K:
ð46Þ
so that
As an aside, note that for the problems considered, we can always choose !
equation (46) is satisfied and so that it is also close to the centre of the spectrum of
the pulse. This latter requirement ensures some coupling between the modes. If it
were not satisfied the resulting situation is not of interest here.
We now include equations (43)–(46) in equations (42). However, since in
section 4.3 we apply a multiple scales analysis to the results, we introduce a
notation that helps us identify those scales. We therefore set
¼ Kz;
¼ cKt;
ð47Þ
where c is the speed of light in vacuum, so that
@
@
¼K ;
@z
@
@
@
¼ cK :
@t
@
ð48Þ
Now applying the discussion in the previous paragraph to equations (42)
we find
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@g1
@g1
c^11 ð0Þ
c^12 ðÿ1Þ
þ cý10
¼i
g1 þ i
g2
@
@
K
K
X
ÿ1s
ÿ
jg1 j2 g1 þ 2i
jg2 j2 g1 þ i
c^11 ðsÞ exp ðisKzÞg1
K
K
s6¼0
X
þi
c^12 ðsÞ exp ðiðý1 ÿ ý2 þ sKÞzÞg2
þi
s6¼ÿ1
ÿ
X
s6¼0
@g1
;
ý^10 exp ðisKzÞ
@t
ð49Þ
@g2
@g2
c^22 ð0Þ
c^21 ðþ1Þ
þ cý20
¼i
g2 þ i
g1
@
@
K
K
X
ÿ2s
ÿ
jg2 j2 g2 þ 2i
jg1 j2 g2 þ i
c^22 ðsÞ exp ðisKzÞg2
K
K
s6¼0
X
þi
c^21 ðsÞ exp ðiðý2 ÿ ý1 þ sKÞzÞg2
þi
s6¼þ1
ÿ
X
s6¼0
@g2
:
ý^20 exp ðisKzÞ
@t
4.3. Multiple scales analysis
The final step in the derivation of the coupled mode equations is the application
of a multiple scales analysis [21] to equations (49), which allows one to separate
systematically terms varying on different length and time scales. In the present
context this is required since phase accumulation due to the presence of the
periodic structure occurs abruptly at the interfaces, rather than gradually over the
length over the pulse (see section 1). However, at the level of the slow time and
space coordinates z1 and t1 in terms of which the coupled mode equations are
written, and which are introduced below, the phase accumulation is gradual. The
‘non-secular’ contributions to the phase that average out over long times and
distances, expressed by the last three terms in equations (49), are then dropped at
the level at which we work. The formal justification of this is the multiple scales
analysis given below. Note that such a procedure is not required in the analysis of
Isoshima and Tada [15] since they deal with a problem that does not involve a
grating.
To start we assume the terms
c^nm ðlÞ
;
K
cý^n0 ðlÞ;
ÿ 2
jgj ;
K
ð50Þ
which occur on the right-hand side of equations (49), to be all of the same order
1, and look for solutions of the type
ð1Þ
gn ¼ gð0Þ
n ð0 ; 0 ; 1 ; 1 ; . . .Þ þ gn ð0 ; 0 ; 1 ; 1 ; . . .Þ þ ;
ð51Þ
where n ¼ 1; 2. Here 0 and 0 represent variations on the short length and time
scales, whereas 1 and 1 represent variations on longer scales. Therefore
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1449
ð0Þ
@gn
@gn
@gn
@gð0Þ
þ cýn0 n
þ cýn0
¼
@0
@0
@
@
ð0Þ
ð1Þ
ð0Þ
ð1Þ
@gn
@gn
0 @gn
0 @gn
þ cýn
þ
þ cýn
þ :
þ
@1
@1
@0
@0
ð52Þ
Now collecting terms of order 0 we find that
@gð0Þ
@gð0Þ
n
þ cýn0 n ¼ 0:
@0
@0
ð53Þ
Thus, to lowest order, pulses in each of the modes do not interact and propagate
with their (average) group velocity. We now collect the two terms in equation (52)
that are of order 1 . For the first of equations (49) we then find
"
#
ð0Þ
ð0Þ
@g1
@g
c^22 ð0Þ ð0Þ
c^21 ðþ1Þ ð0Þ
g þi
g1
þ cý10 1 ¼ i
@1
@1
K 2
K
þi
and
ð1Þ
ð1Þ
@g1
@g
þ cý10 1
@0
@0
¼i
ÿ2s ð0Þ 2 ð0Þ
ÿ ð0Þ 2 ð0Þ
jg j g2 þ 2i
jg j g2
K 2
K 1
X c^11 ðlÞ
l6¼0
K
ð0Þ
exp ðil0 Þg1 þ i
ð54Þ
X c^12 ðlÞ
l6¼ÿ1
K
ð0Þ
exp ði½ý1 ÿ ý2 þ lKŠ0 =KÞg2
ÿc
X
l6¼0
ð0Þ
@g
ý^10 ðlÞ exp ðil0 Þ 1 ;
@0
ð55Þ
and similarly for the second of equations (49). Now all phase factors on the
right-hand side of (55) are non-vanishing, and it thus contains no secular
terms [21]. In contrast, all terms on the right-hand side of equation (54) vary
slowly.
The resulting equations then finally read
@g1
@g1
þ ý10
¼ i c^11 ð0Þg1 þ i c^12 ðÿ1Þg2 þ i ÿ1s jg1 j2 g1 þ 2i ÿ jg2 j2 g1 ;
@z
@t
@g2
@g2
þ ý20
¼ i c^22 ð0Þg2 þ i c^21 ðþ1Þg1 þ i ÿ2s jg2 j2 g2 þ 2i ÿ jg1 j2 g2 ;
@z
@t
ð56Þ
which is the standard form of the coupled equations for coupled co-propagating
modes [9, 10]. Thus, as foreshadowed in section 1, the equations derived here
rigorously are of the same form as the standard result in the literature [9, 10].
However, of particular interest are the expressions for the various coefficients
entering equations (56). In fact, the form we obtain for linear constants c^ij is
identical to equations (3.3-8)–(3.3-10) found by Marcuse [9]. Similarly, the
expressions for nonlinear coefficients ÿis and ÿ are consistent with the existing
literature [17].
5.
Discussion and conclusions
Using a rigorous, but we feel intuitively appealing method, we have derived the
nonlinear coupled mode equations for co-propagating modes coupled by a periodic
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structure. They are identical to results used previously in the literature even
though they were based upon assumptions that are not all justified for the
geometries considered here. Our work thus fully justifies the application of these
equations to the experiments of Trillo et al. [1], Krautchik et al. [2], Psaila et al.
[3], and Eggleton et al. [4, 5], in which the pulse lengths can be smaller than the
grating period. Note that similar features occur in ring lasers [22] and in fibre
systems with periodic amplification [23] or dispersion [24], where the pulse length
can be many orders of magnitude larger than the relevant period. Admittedly, such
systems can clearly be considered to be one-dimensional, and also do not involve a
backward propagating field.
Novel features of our derivation include the fact that we allow for general
time dependence of the field envelope functions, and that the system is not
considered to be one-dimensional from the start. Note the careful treatment
of E~ in equation (8), which is lacking in most analyses. Finally, we explicitly
allow for pulses being shorter than the grating period, so that the usual
approach, in which the field is written as the product of a function that varies
on the scale of the periodic structure and a slowly varying envelope, is clearly
suspicious.
From our derivation it is clear what approximations need to be made to
obtain the results. Although none of the approximations are novel, they are
presented systematically for the first time. A first class of approximations is
justified for fields with narrow spectra. This allows one to drop the frequency
dependence of the modal profiles, and lets one make expansion (19); it is
justified for many-cycle optical pulses [1, 3–5]. The next set of approximations
concern the nonlinearity: it is assumed to be small so that only the dominant
contributions needs to be included. Third-harmonic generation can be neglected
because of lack of phase matching [16]. The final set of approximations
involves dropping rapidly varying terms in the coupled mode equations. Such
terms lead to higher corrections to the coefficents in the coupled mode equations,
and although we have not evaluated them here, our derivation allows their
straightforward inclusion by an extension [14] to higher order terms in the
multiple scales expansion. Nonetheless, the coupled mode equations derived
herein apply to all experiments that have been reported to date and can be used
for their analysis.
Acknowledgments
We thank David Psaila for discussions related to this work. JES acknowledges
the support of the Natural Sciences and Engineering Research Council of Canada,
and of Photonics Research Ontario. CMdS acknowledges support from the
Australian Research Council.
Appendix: Evaluation of the right-hand side of equation (12)
Here we show explicitly that equations (12) and (13) are equivalent. We first
consider the first term in the integrand of equation (12), where f E is defined in (10).
The first term in f E can be written as
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1451
Pz
ÿ ds z^ r?
H?n
"0 n2
ð Pz ?
Pz
?
H
ÿ
r? Hn
:
ds r? ¼ ÿ^
z
"0 n2 n
"0 n2
ð
ðA 1Þ
Now the integral over the first term on the right-hand side of (A1) vanishes
because we take P to vanish at infinity. In the second term we substitute the second
Maxwell equation (1) (note that since we take the z-component only, we may
replace r? by r). Combining the result with that from the second term in f E we
find that
ð
ð
ð
X
@Em
ds H?n f E ¼ i! ds Pz z^ E?n ÿ
am exp ðiþm ðzÞÞ^
z ds
H?n : ðA 2Þ
@z
m
Similarly, the second term in the integrand of equations (12) can be written as
ð
ÿ ds E?n f H ¼
ð
i! ds P? E?n
þ
X
m
ð
am exp ðiþm ðzÞÞ^
z ds
@Hm
E?n : ðA 3Þ
@z
Combining results (A 2) and (A 3) immedately leads to equation (13), as claimed.
References
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[20] He, H., and Drummond, P. D., 1997, Phys. Rev. Lett., 78, 4311.
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[21] Dodd, R. K., Eilbeck, J. C., Gibbon, J. D., and Morris, H. C., 1982, Solitons and
Nonlinear Wave Equations (London: Academic).
[22] Kelly, S. M. J., Smith, K., Blow, K. J., and Doran, N. J., 1991, Opt. Lett., 16, 1337.
[23] Hasegawa, A., and Kodama, Y., 1990, Opt. Lett., 15, 1443.
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Technol., 15, 1808.
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