Vol. 11, No. 4/April 1994/J. Opt. Soc. Am. A
Sipe et al.
1307
Propagation through nonuniform grating structures
J. E. Sipe*
School of Physics, University of Sydney, NSW 2006, Australia
L. Poladian
Australian Photonics Cooperative Research Centre and School of Physics, University of Sydney, NSW 2006, Australia
C. Martijn de Sterke
School of Physics and Australian Photonics Cooperative Research Centre, University of Sydney, NSW 2006, Australia
Received April 26, 1993; revised manuscript received August 30, 1993; accepted September 2, 1993
We consider linear propagation through shallow, nonuniform gratings, such as those written in the core
of photosensitive optical fibers. Though, of course, the coupled-mode equations for such gratings are well
known, they are often derived heuristically. Here we present a rigorous derivation and include effects that
are second order in the grating parameters. While the resulting coupled-mode equations can easily be solved
numerically, such a calculation often does not give direct insight into the qualitative nature of the response.
Here we present a new way of looking at nonuniform gratings that immediately does yield such insight and, as
well, provides a convenient starting point for approximate treatments such as WKB analysis. Our approach,
which is completely within the context of coupled-mode theory, makes use of an effective-medium description,
in which one replaces the (in general, nonuniform) grating by a medium with a frequency-dependent refractive
index distribution but without a grating.
1.
INTRODUCTION
The properties of uniform gratings are well understood.
If the incident field has a wavelength A close to the
Bragg wavelength AB of the grating, it is strongly reflected
through constructive interference of the wavelets reflected
by each period of the grating. The Bragg wavelength for
a uniform grating with average index 71and grating period
d is given by AB 2d. Note that AB depends on the
optical path length within each period and thus depends
on both the period and the average refractive index. The
wavelength range AA over which the grating is highly
reflecting is given by 8A/AB = n/i, where Sn is the depth
of modulation of the refractive index. This reflection
band is associated with the opening of a photonic band
gap by the grating, i.e., a frequency interval in which no
running-wave solutions for the electromagnetic field can
be found.
For a mathematical description of the properties of
gratings one often uses coupled-mode theory.` 3 This
approximation, valid for small modulation depths, permits one to work with the amplitudes of forward- and
backward-propagating waves, often referred to as the
modes, rather than with the fields themselves. At frequencies within the photonic band gap these waves are
evanescent, leading to the strong reflection mentioned
above, whereas at frequencies outside the gap the waves
are oscillatory, and most of the light is transmitted. For
uniform gratings and for some other simple cases such as
a linear chirp the resulting coupled-mode equations can be
solved analytically, but for more general nonuniformities
they must be solved numerically. Though the coupledmode equations are well known, surprisingly they are
0740-3232/94/041307-14$06.00
often not derived rigorously but are usually obtained by
the use of a simple heuristic argument. The argument
makes use of the assumption that the mode amplitudes
vary slowly on the scale of a single grating period.
Interest in grating structures has dramatically increased in recent years with the possibility of writing
periodic variations in the index of refraction directly
into the core of optical fibers. 4 Of course, it has been
possible to write gratings on the surfaces of waveguide
structures for a long time, but such processes are often
complicated, requiring several fabrication steps. These
surface gratings have usually been assumed to be uniform
for the purposes of a fairly simple analysis. The gratings
that are currently being written in fiber cores can
be strongly nonuniform-they essentially follow the
intensity distribution of the writing beams and often
have a rectified Gaussian profile.5 They also differ in a
qualitative way from simpler grating structures in that
an increase in the spatially averaged index of refraction
is also induced, following the Gaussian profile associated
with the grating. In general, by a nonuniform grating
we shall mean a grating for which any one or more of the
period of the grating, the depth of modulation, and the
average index of refraction are slowly varying.
Increased interest in nonuniform gratings is also
being driven by possibilities for grating devices, such
as wavelength-selective mirrors, wavelength-selective
couplers, filters, frequency references, mode converters,
pulse compressors, and dispersion compensators. 1
3 6 11
-
Although some of the devices require only uniform
gratings, permitting nonuniformities often provides
extra degrees of freedom in the design process. Some
applications, however, such as the dispersion compen©1994 Optical Society of America
1308
J. Opt. Soc. Am. A/Vol. 11, No. 4/April 1994
sator, work by virtue of the nonuniformity-a uniform
grating cannot compensate for a large amount of
dispersion. Both techniques mentioned above permit
one to fabricate specified nonuniform gratings by making
use, for example, of specially designed phase masks to
deform the wave fronts' 2 or, in the case of fibers, by
suitably curving them.
Motivated by the current interest in novel grating
structures, in the present paper we consider the theoretical analysis of nonuniform gratings. To derive the
coupled-mode equations rigorously, we use the method of
multiple scales. The advantage of this method is that
it permits one to separate the different length scales in
a systematic way. It thus not only leads to a rigorous
derivation of the standard coupled-mode equations but
also leads to a nontrivial extension to include effects
that are proportional to the square of the refractive-index
modulation, contributions that are not always negligible
in the strongest gratings. Such a result is impossible to
obtain with the heuristic approach commonly used.
The coupled-mode equations cannot be solved in closed
form for most nonuniformities. Nonetheless, much
physical insight into the qualitative nature of the
solutions can be gained by the introduction of an
effective-medium picture to describe the properties of
a grating. We show in this paper that propagation
through a (nonuniform) grating is equivalent to that
through an effective medium without a grating but
with a nonvanishing relative dielectric constant Eeff and
a magnetic permeability Aeff and hence an effective
refractive index neff. Here e
and ,e4ff depend on
the local grating parameters-the effective medium is
thus inhomogeneous if the grating is nonuniform. With
this approach the properties of uniform gratings are
easily recovered. Uniform gratings map directly onto
Fabry-Perot filters, and all the results for such systems
can thus be transferred over immediately.
The effective-medium picture also naturally allows one
to identify a local photonic band gap for a grating by
considering the frequencies for which the effective refractive index neff is imaginary. For slowly varying gratings
one can thus define, at each point along the grating, a
local Bragg wavelength and a local photonic band gap.
When light of a given wavelength travels through the
grating, it encounters, then, two types of region: regions
where the wavelength is outside the local band gap, and
thus where it propagates freely, and regions where it is
inside the local band gap, where reflection occurs. This
insight leads one to the use of band diagrams, showing the
photonic band gap of the grating as a function of position.
Such diagrams provide a simple physical approach to
understanding the qualitative response of a nonuniform
grating.
The effective-medium approach thus leads to the insight that the properties of nonuniform gratings can be
understood in terms of regions of free propagation (where
neff is real) separated by regions that act as barriers
(where neff is imaginary). It is then perhaps not surprising that one can implement the WKB method to find
approximately the transmission and reflection coefficients
of nonuniform gratings.
The contents of this paper are as follows. For
completeness, and to establish our notation, we review in
Sipe et
al.
Section 2 the heuristic derivation of the coupled-mode
equations. In Section 3 we derive the coupled-mode
equations more rigorously, using the method of multiple
scales; the contributions of second-order grating effects
to the coupling constants are also identified. The reader
not interested in this more rigorous analysis may proceed
directly from Section 2 to Section 4, where we introduce
the effective-medium approach. Although we begin there
with the second-order coupled-mode equations derived in
Section 3 [Eqs. (45)], and the approach can, in principle,
include such effects, for simplicity we neglect them in
our examples. Thus the reader can just as well begin
Section 4 with the heuristically derived Eqs. (11), rather
than with the rigorously derived and more accurate
Eqs. (45), by simply omitting the overbars in au,-v, and
k in Eqs. (47), (49), and (50) and by replacing d by .
The application of the effective-medium approach to a
uniform grating is discussed in Section 4, and the analogy
of a uniform grating to a simple Fabry-Perot structure is
developed. In Section 5 we discuss nonuniform gratings;
we introduce band diagrams, and, using the particular
example of a Gaussian profile grating, we show the
usefulness of the WKB approximation to find expressions
for the reflection and transmission coefficients.
2.
COUPLED-MODE EQUATIONS:
HEURISTIC DERIVATION
Although many grating geometries of interest involve
planar waveguides or fibers, it is usual to idealize the
fields as depending on only a single spatial variable, say
z, reducing the problem to a one-dimensional analysis.
We follow this practice and write
E(r, t) = 9E(z)exp(-iwt) + c.c.,
H(r, t) = 9H(z)exp(-iwt) + c.c.,
(1)
where c.c denotes complex conjugation. Then Maxwell's
equations become
dE = iploH(z),
dz
dz = icoe(z)E(z) = icocOn2(z)E(z),
(2)
where Ao and eO are the permeability and the permittivity
of free space, respectively. We neglect any magnetic
effects and have introduced a spatially varying index of
refraction n(z) = [e(z)/eol2, where e(z) is the spatially
varying permittivity. We take n(z) to be purely real here
for simplicity, though extinction that is due to absorption
or scattering can easily be taken into account phenomeno-
logically by a simple extension of our treatment. Taking
the second derivative of the first of Eqs. (2) and substituting into the second, we find that
d2 E +k2F
dz 2
n(z) ]2
[ nwj E
z)= 0,
(3)
where k = cono/c, c = (oeo)-" 2 , and no is a reference
refractive index. We introduce a gradual variation in the
background refractive index and a grating by taking
Sipe
Vol. 11, No. 4/April 1994/J. Opt. Soc. Am. A
et al.
n(z)/no = 1 + o-(koz) + 2K(koz)cos[2koz + 0(koz)],
(4)
where 5, a-, and K are slowly varying functions of their
variables. The spatially varying phase •0 and the backcan be positive or
ground refractive-index variation
negative; the grating amplitude K is taken to be positive,
and we assume that a-I, K << 1. The wave number
ko identifies the nominal wave number of light at the
Bragg scattering resonance, and the corresponding nominal resonance frequency is wo cko/no. Introducing the
detuning parameter A with respect to this frequency,
A
we have k2
=
k 2 [n(z)/no]2
( - wJo)/wJo,
k02 (1 + A)2
k
-
2(1
(5)
+ 2A) and
ko 2{1 + 21a-(6) + A] + 4K(6lcos[26 + 0(ej},
(6)
where we put 4 koz, and we have also assumed that
A << 1. Using relation (6) in Eq. (3), we find that
d 2E
d
+ {1 + 2[a-(4:) + A] + 2K(4:)exp(iy)
+2K(e)exp(-iy)}E = 0,
(7)
where for convenience we have defined y = 2f + 0(4:).
We then proceed with the usual heuristic derivation
by noting that, in the absence of the grating and the
background index variation (ar = 0), the solution to
Eq. (7) at vanishing detuning would be of the form
E(4:) = a+(e)exp(ie) + a_()exp(-i:),
(8)
where the functions a-- are uniform. Therefore one attempts to take into account the effects of small a, K, and
A by permitting a+(f) to vary slowly and approximates
the left-hand side of Eq. (7) by
d2 E
-:=
_ a+({) + 2i
d ca+
Iexp(+i4:)
+ -a-() - 2i dajexp(-i6),
(9)
neglecting the terms involving the second derivatives of
the a,. Using relation (9) in Eq. (7) and introducing new
functions u(4) and v(6) by
find that u(4:) and v(f) must satisfy the coupled-mode
equations
du:) = +i[&(49U(4) + K(f&V(e)],
dv(()
de:
= -i[&()v(e)
+
(11)
a4:
(12)
(6) = Ca(4:) + A - 2
Equation (12) demonstrates that variations in the background refractive index have the same effect as a detuning
and a grating chirp. This implies, for example, that a
Gaussian profile grating (see Section 5) behaves as if the
grating were chirped, even though the actual period of
the grating is constant. Note that Eqs. (11) satisfy the
relation
(13)
d [u(6)2 - Iv(e)12] = 0.
Identifying Iu(4:)1 2 as proportional to the power in the
forward-propagating mode [see Eqs. (8) and (10)] and
I V(4) l2 as proportional to that in the backward mode,
Eq. (13) expresses energy conservation through the grating, despite the scattering of the forward- and backwardpropagating modes off the grating, described by Eqs. (11).
Although it is simple and straightforward, this derivation of the coupled-mode equations is problematic. If
the exp(± 3/2iy) terms discussed directly below Eqs. (10)
were not neglected in an ad hoc way, they would lead
to rapidly varying contributions to the a(f) and thus
to such contributions to u(6) and v(4:). Yet the use of
approximation (9) requires that the a, (4) not contain such
rapidly varying contributions, and thus the derivation
given here is internally inconsistent. The idea of separating out the slowly varying and rapidly varying terms
and effects is obviously physically sound, but it requires a
more careful separation of these different scales. To this
we turn in Section 3.
3. COUPLED-MODE EQUATIONS:
MULTIPLE-SCALES DERIVATION
We now return to the original Eqs. (2). Before we even
specify the form of n(z), it is useful to rewrite those equations as coupled-local-mode amplitude equations. Recall
that, if n(z) were uniform, a wave traveling toward z = +x
would have magnetic and electric fields related by H =
nE/Zo, where Zo is the vacuum impedance, whereas a
wave traveling toward z = -
would have H
=-nE/Zo.
So we are led to introduce
FEW + o Hz
z
]
1 [n(z)
a -(>)= v(e) exp[- 40(f) ]'
respectively,
+ K(4)U(4)],
where
A+ (z) = 2a, () = u(6)exp
1309
2
)
H(z)]
(14)
(10)
we find terms involving exp[+ (1/2)iy],
exp[-(1/2)iy], exp[+(3/2)iy], and exp[-(3/2)iy]. Neglecting the last two exponentials as having rapid spatial
variations and thus presumably being unimportant, we
expecting A+(z) to relate to the component of a field
propagating in the forward direction and A_(z) to describe that associated with propagation in the backward
direction. The factor n(z)/no]V2 , where again no is a
reference refractive index, is introduced because the flux
1310
J. Opt. Soc. Am. A/Vol. 11, No. 4/April 1994
Sipe
in a plane wave in a uniform medium is proportional
to nIE 2 = ffE 2 . With the definitions in Eqs. (14) we
thus expect that the flux toward z = +- will be described
by A+ (z)1 2 , and that traveling toward z = - will be
described by IA_(z) 2 .
Using Eqs. (2), we can easily derive the pair of exact
equations that the A+(z) satisfy:
dA+
dA_
dz
W n(z)A (z) +
=+i
.
n1)~z
-W
=
c
(d{ln[n(z)]})A-(z),
±dfln[n(z)]}A+Z
2\dz)
(15)
From these equations it is clear that, for n(z) uniform,
the A+(z) are uncoupled and do describe the amplitudes
of waves propagating toward z = ±-. Further, for real
n(z) our current case of interest, Eqs. (15) lead, in general, to
d [IA+ (z)12 - IA_ 12] =
0
(16)
[cf. Eq. (13)], expressing energy conservation and confirming the identification of A. (z)J2 discussed in the
preceding paragraph.
We now turn to an expression for n(z). Instead of the
cosine function used in Eq. (4), we use a more general
periodic function G(y), such that Gy + 2) = G(y).
Expanding G in a Fourier series, we write
G(y) =
Y gm
exp(imy),
g-m =
In the same spirit we restrict ourselves to small detunings from the nominal Bragg resonance frequency by
putting
(W - Wo)/o=
ovn(4)/cko = 1 + 4q[a-(Qq)
+ A + K(-q)G(y)]
2
+ 77 A[-(_ql)
(-koz)].
(18)
Note that here we take - and K to be functions of order
unity-the smallness of the background index variation
and the grating amplitude are explicitly displayed by
the factor . Likewise, the functions - and K defined
here, as well as , are taken to vary significantly as
their parameters range over unity-again, their slow
variation is described by the inclusion of X in those
parameters. By this device we can carefully keep track
of the different length scales in the problem. At the end
of the calculation, but only then, we let - 1, and the
functions -, K, and
revert to the functions defined in
Section 2.
(20)
+ K4)G(y)],
where, as in Section 2, we have put f = koz, and here
y =2 +
(,76).
(21)
Now, following in the spirit of Section 2 [cf. Eqs. (8) and
(10)], we define new functions u(4) and v(g) by putting
A+ () = u(4:)exp(i4)exp + 2
A_(6) = v( 4 )exp(-i
gm* to ensure that G is real.
n(z)/nO = 1 + 7oa-(7koz) + 2JK(fkoz)G[2koz +
(19)
A
[cf. Eq. (5)], where, as in Section 2, we have wo ecko/no,
and now A is of order unity (until the end of the calculation, when we let j - 1 and A reverts to the actual small
detuning). Condition (19) restricts our discussion to fundamental Bragg scattering from the grating; since G in
general, contains higher harmonics (gm for Iml > 1), there
will also be scattering from the grating at frequencies
near those corresponding to wave numbers Imiko. They
do not satisfy Eq. (19); yet we can derive such harmonic
scattering by replacing Eq. (19) by the corresponding condition and proceeding along the lines that we develop.
But this we do not do here. Our present interest in the
high grating harmonics is not in the scattering that they
produce at new frequencies but in their modification of
the fundamental Bragg scattering [see Eqs. (43) below].
Using Eqs. (18) and (19), we may write
(17)
We take
go = 0 and choose the amplitude of the function such that
Ig-11 = Igil = 1. In particular, we take g 1 = g = 1,
which can always be achieved by an appropriate translation of a function G that does not satisfy that particular
condition. Then, in the special case gm = 0 for m + 1,
we have G(y) - 2 cos y.
We effect the separation of different length scales implicit in the discussion in Section 2 by here introducing a
positive parameter 7J<< 1.13 In place of Eq. (4) we now
write
where
et al.
4
)exp -
4()]
0
( ).
(22)
Then Eqs. (15), (20), and (22) yield the equations that u()
and v(4:) must satisfy:
du(e)
d4e
+i4q[&(el) + K(4l)G(y)]u()
+ iq2A[-(:j) + K(4l)G(y)]u()
+
x (d {In[l + na-(4:) +
dv(4:)
d4
=
exp(-iy)
2K(l)G(y)]})v(e),
-in[&(4:) + K(4:)G(y)]v()
2
- iq A7('A(4:)
x ({ln[1
+ K(4l)G(y)]v() +
+ va-(4:)
+
2
exp(+iy)
-qK(l)G(y)1} u(e),
(23)
where we have put 4:
[cf. Eq. (12)]
7
f(4l) = -( 1) + A -
[see Eq. (26) below] and
2 a 1
(24)
Up to this point we have made no approximations;
Eqs. (23) are equivalent to Eqs. (2) with expression (18)
used for n(z).
Vol. 11, No. 4/April 1994/J. Opt. Soc. Am. A
Sipe et al.
We now seek approximate solutions for u(4) and u(4) in
a form that identifies their behavior on different length
scales. For a typical function f () we do this by writing' 3
where F is assumed to vary significantly as each of its
parameters varies over a range of order unity. Then the
variation of f over different length scales is captured by
the variation of F on its different parameters. Putting
4
p=
side of Eq. (32) there are terms that involve 4: through
their dependence on y and terms that involve only e:1, 2,
etc. The first set must be equal to dul)(:o, 4,. - .)/04a,
2,.. .)/o4:
since au)(4,
f (6) = F(6, 277,cog,...),(25)
u(1)(,
*:*l) =
+ K (l)
f() = F(eo, l, 2.... )
(27)
and
mX l
0
V( )(el, 2 ,...)
2(m - 1)
(33a)
p[i(
-
while the second requires that
aF +
df((6)
F +
2
aF + . .
(28)
a u(°)(61,
Actually, we seek an approximate solution of Eqs. (23) by
assuming a form for u(4) and v(f) that is more general
than Eq. (25). We take
U(4)
0
U(l)(60,
= U( )(1,42 ... ) +
2
2
+ 77U( )(4o,61, , .... ) +
l,
..
2,...)
(29)
,
and likewise for v(4:). This identifies not only the different length scales over which u(6) and v(f) vary but
also the different scales of the contributions of their
components. From Eq. (29) we have
~+
a(°)
aedl
du(6)
F
de
(7') .
+
v()(:o,4:,,.*.)
=
)
+ K(4:l)G'(y)exp(-iyv(O)(4l,6,...).
(31)
Next we use the Fourier expansion (17) and the values
go = 0,g = gl = 1 to rewrite the right-hand side of
Eq. (31):
0.4(:eo,
4()(
,1t2 ...
+
ael
aeo
0
iK(4l)U( )(:,,42 ...) Y gm
m#o
+ iK(el)V(0)(64:,,
>
2 ...)
exp(imy)
mgm
exp[i(m
-
1)y]
ml+,1
+ i1I&(4:,) 0 (4:,
where G'(y)
:2,. ..)
dG(y)/dy.
0
- K(4l)V( )(4:, 42, .. )
+ K(4:,)v 0 (4
2
4:(x
**)] ,
E
gm exp(imy)
+ K(61)U(0)(61,
- mgm
2, ***)
exp[i(m + 1)y]
2(m + 1)exim
(34a)
(30)
au(')(1 6 2,...) + 0u(')(4o,4el,..*)
4:2, ...
(33b)
M*o 2
0v(O)(4,,
i[& (el) + K(4)G()]uu(o)4(1,
2,.... *)]-
K(61)0S)(e,
Applying the same approach to the second of Eqs. (23),
we find the corresponding two equations:
m#o,-I
+
i[&(e:)z(O)(I1, 42, **)
+
X
Note that, unlike the higher-order u(P), u(°} is chosen
not to depend on do. The reason is that the dominant
contributions to u(4) and v(f) are slowly varying, the
rapid variations having been taken out in Eqs. (22) (the
validity of this separation of slow and fast scales is
justified by our final results).
We now substitute Eq. (30) into the first of Eqs. (23)
and expand the right-hand side in powers of 77. Equating
the coefficients of 77on both sides of the equation, we find
-
=
+
++
O04
2,... )
l
I()
aeo
+
u
+ 70 d04:2+
-
g Žm exp(imy)
K(el)4(o)(4:1, 6:2,...)
rn*o 2
we have
a
is, by assumption, independent
of o; au(°/4: must then be equal to the second
set. Thus Eq. (32) yields two equations; a solution for
UM(6'No, l .i...)
to the first can easily be found:
(26)
e,
1311
(32)
We see that on the right-hand
4:2,**)
= i[fr
OU(:)uelX4,4:2...)
+ K(4:l)V() (el, 6,2 , *)] -
(34b)
We can now, at least in principle, solve for the dependence of u0') and v(l) on 4:0 and 4el and that of u(°) and v0 ) on
el: Eqs. (33b) and (34b) must first be solved consistently,
and the results then must be used in Eqs. (33a) and (34a).
If we choose, we can stop the development at this point,
keeping only the lowest-order term in Eq. (29). Then,
letting 77- 1, we find that Eqs. (33b) and (34b) agree
with the coupled-mode equations (11) derived heuristically in Section 2. But we can also go on and determine a better approximation for the fields. We return
to the first of Eqs. (23) and, using Eq. (30), write out
the equation that results from keeping all the terms
0
of order 772. On the left-hand side will be 0u(°)/042,
2)
ouM1)/804l, and au( /g0o [see Eq. (30)]. Now 0u~l/4l +
au(2 )/0a4o depends on 4:0[see Eqs. (29), (33a), and (34a)],
as does a subset of the terms on the right-hand sides of
the equations. That subset of terms then must equal
ou5l/o4:l + au(2)/4o, since, by assumption, 0u( 0 /a02 is
independent of 4:0. Since 0u~l)/0 4l can be determined
from Eqs. (33a) and (34a), we can now find u(2)(:o ,4, .. )
just as we found u'l)(4o, l, .. .) above [Eq. (33a)]. We can
next identify au(0)/042, which must equal the terms on the
right-hand side of our equation that are independent of
4:o. This yields
1312
J. Opt. Soc. Am. A/Vol. 11, No. 4/April 1994
0u(°)(4l, 2,
) = i[&()U(0(:(l,
.
Sipe et
42 ... )
u ()
04:2
+
0
(6:)v
al.
_ ,e
=)Uu(6exp 2
(35)
)(el,, 2 , *...)],
v(°)(f)-(e)exp - 2
'
(42)
where
Eqs. (39) are satisfied if we have
AaS(4:)
Em-1, 2(m
j~2(4)
-
(36)
+ 1)
di((6)
+ie(fl(4:)+ W(4:)v(4)],
d
=:
-
is real and
ul(4:) = [(ei) +
1
0 (el) 1
2
04:, j
2m 3 -2m+ 1
2m(m- 1)
2()
m•,A1
dli(4:)
.0aK(e)
2
a4l
_ 1
d:
(37)
042
=
0
i&(4:,vM (4:1, 4:2...)
-
+ *(41)(0)(:1, 4:2,...)].
(38)
Equations (35) and (38) now allow for the determination
of the variations of u) and u(°) over the longer length
scale described by 2. It is over this scale that the
higher (Iml > 1 Fourier components of the grating become
important, as described by Eqs. (36) and (37).
It is, of course, possible to move on to variations over
even longer length scales if one considers the 3 terms
in Eqs. (23). But we stop our analysis here. Combining
Eqs. (33b), (34b), (35), and (38), we find that
a__0)
-ilff(:)(4: + W7Wi(:].-
(43)
To determine the actual fields from the solution of the
coupled-mode equations (43), we note first from Eqs. (14)
that
is, in general, complex. We note that the summations
over m in Eqs. (36) and (37) include both positive and
negative integers. Proceeding in the same way in sorting
out the 772 terms in the second of Eqs. (23), we find that
aVCO)(:1, 2 .... )
-
E(z) = [no/n(z)h 2[A+(z) + A_(z)],
H(z) = [non(z)]"12 (1/Zo) [A+ (z) - A_(z)],
(44)
where the A+(z) are given in terms of u(4:) and v(4:)
by Eqs. (22). A moment's consideration confirms that,
if the effective coupling constants Th and 71 in Eqs. (43)
are obtained by a calculation to second order in 77,it is
appropriate to keep contributions to A, (z) up to first order
in 7, analogous to the correspondence in perturbation
theory of second-order terms in energy with first-order
terms in the wave function in a calculation in quantum
mechanics. Using Eq. (29) in Eqs. (22) and the results
for 5') and 51), we then find that
A+(z)
={U(4:)F1 + K()>j
g
exp(imy)J
+ V(4)exp[-iT(4)]K()
=
+i[77 fr_(4:) +
2
7 a&(4:)]iP'
• y
mg'1) exp[i(m -1)y]
m•#O,1 2(m
-1
+ i[77KN(l) + 772L(4l)]?(0)
x exp(+i4:)ex
-
i[qKc(4l) + 772 *(4:,)]U(o).
(39)
A_ (z) =
To write our results in their final forms, we take the limit
- 1 in the spirit of perturbation theory and find that
T) +
7)(a
= Uk ) +
=(\
Gs
+ Aa-(4)
-
EI
2 ()
mt-to
-1
2
K()
=
~m002mj
mgm 1)exp[i(m + 1)y]
grI
)< exp(-i4)exp-[
-
2(m + 1)
]
(45)
where, since we have let 77- 1, we have y = 24: + 0(4),
and
49(4:) a 1(s) + CO(4).
+K
(4) +
095(4:)
2m3 -2m+
where 71(4:) and
-
moo, -1 2(m +
i 0,K4:)
2(e
- 2 f¢f
K)
04a +K(:
mO1
gm ep(im)]
1 - K(e )
iU(4)exp[- i(4)]K(4:)
X y
a4
+ V (4)
K(){1
tL
+
We(4)
(40)
ic(4:)exp[i5(4:)]
U(6)
+ 2 (4)]
2m(m - 1)
() are real and 71(4) > 0; with
(41)
(46)
From Eqs. (44) and (45) we see that there indeed
are rapidly varying terms in the amplitudes a+ defined
by Eq. (8), as indicated in the discussion at the end of
Section 2. Yet these terms are small and, to lowest
order, do not affect the form of the coupled-mode equations: If &() and (4:) are neglected, Eqs. (43) reduce
to Eq. (11). Nonetheless, to higher order they do modify
those equations, leading to new coupling constants
Sipe
Vol. 11, No. 4/April 1994/J. Opt. Soc. Am. A
et al.
0
0.05
0.1
0.15
0.2
0.25
K
-0.002
-0.004
1313
K
0.25,
I
-0.006
I
-0.008
I
-0.01:
0.05
0.1
0.2
0.15
(a)
0.25 K
(b)
Fig. 1. Parameters (a) ff and (b) 7?as a function of the grating depth K for a cosinusoidal structure with oa= 0 and constant 0. The
dotted lines indicate lowest-order results, in which K = K and if = 0, while the dashed line in (a) indicates our second-order results.
The solid curves indicate exact results obtained by integration of the wave equation directly.
7(4:) and K1(f). There are contributions to the corrections if(4) - a(6) and 71(4)- K(4:) from the fundamental
Fourier components of the grating g±i themselves, from
the higher grating harmonics, from an interplay between
the detuning and the gradual variation in the background
refractive index, from an interplay between the grating
amplitude and the background index variation, and from
variations in the amplitude and the phase of the grating.
We now illustrate some of our results for the simple
case of a uniform, purely cosinusoidal grating with a
constant background refractive index; we therefore take
a = 0 and K and 0 to be constant. Figures 1(a) and 1(b)
show, respectively, the effective detuning 13and the effective grating strength
K
0 the summation is zero, and the second-order corrections
to both Wand 71vanish, so that K1= K and wu= oa.
4.
EFFECTIVE-MEDIUM PICTURE
For a specified ir(z) and
K(z), Eqs. (43) must, in general,
be solved numerically, subject to the appropriate boundary conditions. But physical insight can be brought to
bear for an understanding of the qualitative nature of
the solutions if we write Eqs. (43) in a slightly different
way. If we define effective electric and magnetic fields
according to
Eeff = 1(4 + -U(6)
Heff = U(4) -U(4:),
following from Eqs. (40) and (41),
respectively, as a function of K for this case. The dotted
lines in these figures show the results used in conventional coupled-mode theory, the dashed curve in (a) shows
the results of our second-order treatment, and the solid
curves show the exact results obtained by integration of
respectively, we find that Eqs. (43) may be written as
dEeff = ieff ( 4 )Heff (),
the wave equation directly without the use of an envelope
function approach.
(47)
Turning first to Fig. 1(a), we note
that, since we took a = 0, the first-order result (dotted
line) coincides with the horizontal axis. Our secondorder result clearly initially follows the exact result and
deviates significantly only when K
0.1.
Considering
next Fig. 1(b), we point out that 71= K to lowest order,
as indicated by the dotted line. It is easy to see from
Eq. (41) that the second-order contribution to 71vanishes
dHeff
- ieeff(Eeff(),
(48)
de:
where we have defined an effective permeability and an
effective permittivity by
Aeff
T(e) -
K(()
,eff =
(4:)
+7(4:),
(49)
for our case, the lowest-order correction to 71being of third
order.
As expected, for deep gratings an exact treatment
must be used; yet our approach is clearly an improvement
over standard coupled-mode theory.
In finishing this section, we note that for grating structures with a discontinuous refractive index distribution,
such as those consisting of alternating uniform layers of
different refractive indices, the summation in the last
term in Eq. (40) is conditionally convergent; the result
thus depends on the order of summation. One can easily
establish the proper order by considering the closely related case in which the refractive-index jump is somewhat
smoothed out, so that all the Fourier components such
that
Iml > m', where m' depends on the degree of smooth-
ing, vanish.
This implies that terms with ±m should be
paired before summation.
By finally letting m'
- cc, we
can then evaluate the summation in Eq. (40). For the
particular example of a uniform periodic structure consisting of alternating uniform layers, a = 0, and constant
respectively. Clearly, Eeff and Heff are not the electric
and magnetic fields; the actual fields E(z) and H(z) are
given by Eqs. (44) and (45). Likewise, Jeff and eeff are,
of course, not the actual permeability and permittivity,
respectively, of the medium; in fact, we took the medium
to be nonmagnetic. Yet, by comparing Eqs. (48) with
Eqs. (2), we see that, with respect to an effective spatial
variable 4:and with w formally set to unity, one may use
an effective medium with electric and magnetic properties
(49) to understand the behavior of the fields Eeff and Heff
and thus that of ii(6) and v5(f). As an aside we point out
that Eeff and Heff actually correspond to the amplitudes
of the local Bloch functions of the periodic structure. It is
then perhaps not surprising that, when they are written
in terms of these functions, the resulting equations take
on a simple form.
The use of an effective-medium approach in grating
problems is not new. For the case of light propagating
1314
Sipe et
J. Opt. Soc. Am. A/Vol. 11, No. 4/April 1994
parallel to the grating, effective-medium approaches have
been discussed by MacLeod' 4 and de Sterke.' 5
However,
our present treatment is much more general and is easier
to use. It should further be noted that effective-medium
approaches have also been developed for different grating
problems in which the incident light is obliquely incident
onto the grating'
To develop the correspondence discussed above, we
introduce an effective refractive index neff(6) in the usual
way:
neff()
2
(eeff ,eff)1/
=
=
[- 2(f)
(50)
- 712(4:)]2
From Eqs. (24), (40), and (41),
f
[
(2)
-K2(
}
(51)
where, in Eq. (51) and henceforth in this section, we
restrict ourselves for simplicity to keeping only the lowestorder terms in if(54) and 71(4:) [cf. Eqs. (11)]. In the simple
case in which a and K are uniform and either is uniform
or has a constant derivative, we have a uniform effective
index
neff = (2
- K 2 )1/2,
where
(52)
1 a
(53)
Aecr-+ A - 2 04:
is a shifted detuning. The solutions to Eqs. (48) then
take the simple form
surrounded by media in which no grating is present (K = 0
and, say,
= 0). The effective-medium description of
this geometry is that of a uniform medium with refractive
index given by Eq. (50) surrounded by regions with an
effective index neff = AI. That is, the uniform grating
problem maps onto the simple Fabry-Perot filter.
Because the reflectivity from a uniform grating" 2
is qualitatively so different from that of the usual
Fabry-Perot filter,'7 it is interesting to follow through
the analogy in some detail. Consider first the usual
Fabry-Perot filter, where we take a medium of refractive
index n in the interval 0 < z c L surrounded by media of
refractive index unity. The reflection coefficient is given
by the well-known expression
r= r12[1- exp(2ikL)]
1 - r 2 l2 exp(2ikL)
where r1 2 is the Fresnel reflection coefficient from region 1
to region 2, r 2 = (1 - n)/(1 + n), and r12 = -r 21 . Furk = co/C,
ther,
Z-
l e)
C
(54)
,
(58)
where c is the frequency, C = c/n, and c is the speed of
light in vacuum. The reflectivity of the filter, R = Ir 2 ,
is shown in Fig. 2. The analytic structure of r is easily
specified. Its zeros are at frequencies such that kL = pr,
where p is an integer, or
c/C = pir/L,
(59)
and occur when all the contributions to the back-reflected
light add destructively. The poles of r are off the real
frequency axis (as they must be, as R ' 1 for real frequencies) at exp(2ikL) = r2l 2 , or
Eeff(e) = E exp(±inff ),
Heff(6)=
al.
= p-
L
-
I
2L
ln(r2j- 2 ) .
(60)
The poles and the zeros for a uniform grating are indi-
cated schematically in Fig. 3.
where E is a constant and we have introduced an effective
impedance
(. :)v2
A --
_
2
(55)
the second equality holding in the special case of a uniform grating.
A crucial point is that, although the actual refractive
index n(z) of the medium has been assumed to be positive
and real, the effective index neff(4 ) can be either real or
imaginary [see Eq. (50); we take the square root such that
Re(neff) Ž 0, Im(neff) Ž 0]. In fact, for a uniform grating,
Eq. (52) shows that neff is imaginary whenever
-K
< A
< K.
R
1
(56)
This precisely identifies the photonic band gap of the
uniform grating, which is the frequency range over which
propagating fields cannot exist in the structure. Thus it
is not surprising that neff is imaginary in the frequency
interval [see Eqs. (54)].
Despite this difference the formal similarity between
Eqs. (54) and the description of light propagating in a
uniform medium suggests that the latter treatment can be
carried over to treat a number of grating problems. The
simplest example would be a length of uniform grating
-10
.
U
0
10
5
Fig. 2. Reflectivity as a function of frequency for a Fabry-Perot
filter. The refractive-index contrast has been taken to be 5.
Im(coUC)
0.5r
0.25
0
0
90 Re(UC)
0
0
0
-0.25[
0.
I
Fig. 3. Location in the complex frequency plane of the poles
(filled circles) and the zeros (open circles) of the amplitude reflection coefficient for a Fabry-Perot filter with refractive-index
contrast equal to 5.
Sipe
Vol. 11, No. 4/April 1994/J. Opt. Soc. Am. A
et al.
assuming that the poles are close to the zeros leads to an
approximate expression for their positions:
R
AK
0.6.
0.4
~~~~~~~A/K
3
1
2
0
-2
-1
-3
Fig. 4. Reflectivity as a function of frequency of a uniform
grating with KL = 5
Next we consider a uniform grating in the region 0 '
c 1, surrounded by regions with no grating. Using the
effective-medium picture, we can immediately write down
the reflection coefficient of the structure,
- exp(2ineff )]1
(61)
1 - r2l2 exp(2ineffl)
where neff is given by Eq. (50) and
r12 = (Z - 1)/(Z + 1),
(62)
with Z given by the second equality in Eq. (55). We have
also used the fact that the impedance of region 1 is unity,
=
eeff
there. Also, note that r12
=
-r 2 l.
1
a =
il
p7p 1K
/[(P/Kl)
1
1
_]
) 2 + ]V2
(65)
In
]12 + p7/K1
[(pV/K 1)2 +
+ 111/2 -
[(p/KJ)2
2K1
(66)
pV/KI
For K large and p not too large, this assumption is
obviously justified. Unlike for the usual Fabry-Perot
filter, for which the poles in r are all the same distance
from the real axis, for the uniform grating the poles
move farther from the real axis as p increases. This
is associated with the decrease in the value of the peak
reflectivities as the detuning increases. The poles and
the zeros are schematically shown in Fig. 5. So we can
conclude that, while the occurrence of a photonic band
gap is typical for grating structures, the zeros in the
reflectivity are similar to those of Fabry-Perot filters.
Thus the difference between the reflectivity of a
Fabry-Perot filter and that of a uniform grating can
be attributed to the strong dependence of the effective
index of the latter on the detuning. It is interesting to
compare explicitly the dispersion relations of the uniform
medium [Eq. (59)] and of the uniform grating [Eq. (50)]:
It is
w = kC
easy to confirm that Eq. (61) agrees with the well-known
(uniform medium),
2
2
,A = (neff)2 + K
reflection coefficient of a uniform grating. In particular,
for
+112-i[(/V
where
<
since .eff
2 + 1]K
+[p/l
A
0.2
r
1315
(uniform grating).
(67)
= 0, the frequency for which the reflection coefficient
is largest, we find that
r(a=0) = i tanh(KI).
(63)
The reflectivity R = I 2 is shown in Fig. 4.
The differences between the behaviors of the usual
Fabry-Perot filter and of the uniform grating are highlighted by a comparison of the analytic structures of
the reflection coefficients. The zeros of the reflection
coefficient for a uniform grating are given by neff 1 = P7r
where here p must be a positive integer, since Re(neff) 2 0.
For each positive p there are two zeros:
a/K
±[(PI/K1)
2
+ 1]1
(64)
[cf. relation (56)]. Thus the detuning must be large
enough in magnitude to be outside the photonic band gap
before the fields can propagate in the effective-medium
picture (neff real) and to allow for the complete destructive
interference of backreflected light. The photonic band
gap provides a central region where the effective index is
purely imaginary, and thus the structure there is highly
reflective.
The poles of the reflection coefficient of the uniform
grating are at frequencies such that exp(2ineff 1) = r2l ,
and, as with the usual Fabry-Perot filter, there is a
pole associated with each zero. But because both neff
and r21 depend on the detuning, an analytic expression
for the positions of the poles cannot be found. However,
Recalling that co and A are both frequency variables
and k and neff are both wave-number-like variables, we
can make an appealing correspondence between Eqs. (67)
and the dispersion relations for massless particles and
massive particles:
E = pc,
E2= p2 c2
2
4
(68)
+ m0 C ,
where E is the particle energy, p is the momentum, and
mo is the rest mass. Quantum mechanically, of course,
we identify E = hco and p = hk. From Eqs. (68) we see
that the effect of a uniform grating, characterized by the
amplitude parameter K, might be described as endowing
the effective-medium photon with an effective rest mass.
IM(A/K)
0.5.
0.25.
*
Re(A/c)
0
*
0
32
1
1
-2
-3
-0.25
* 0
-0.5
Fig. 5. Location in the complex frequency plane of the pole
(filled circles) and the zeros (open circles) of the amplitude
reflection for a uniform grating with KL = 5.
1316
J. Opt. Soc. Am. A/Vol. 11, No. 4/April 1994
Sipe et al.
In spite of the qualitative differences between the dispersion behavior of _teff and eeff from that of the permeability and the permittivity of usual materials, respectively, the form of Eqs. (48) guarantees that all the
standard formulas from the optics of multilayer thin films
can be immediately generalized to treat multigratings,
i.e., structures in which one region of uniform grating follows another. The generalization of Eq. (57) to Eq. (60)
is only the simplest of them. In Section 5 we turn to
the class of yet more complicated nonuniform grating
structures, where fteff and eeff are not piecewise uniform.
5.
order contributions to
and K and thus have omitted
the overbars on u and . To find the properties of these
gratings, one can, of course, solve these equations numerically. Instead, we here use the effective-medium picture
developed in Section 4 to gain insight into the general
nature of the solution as a function of A.
The effective-medium approach starts with Eqs. (48)
and (49). According to these equations, the propagation
of the modes through the grating is equivalent to that of a
wave through a grating-free medium but with parameters
ereff = 3Ko exp[-_ 2 /(kow) 2 ] + A,
NONUNIFORM GRATINGS
Areff
In this section we apply the effective-medium picture to
obtain qualitative and approximate results for nonuniform gratings. The approach developed here can be applied to grating structures with any nonuniformities, and
the essence of the analysis is summarized in Section 6.
But, for concreteness, we here consider in detail, as a
particular example, rectified Gaussian profile gratings,
such as those that would be written into a fiber core by
illumination in the ultraviolet with two interfering ideal
Gaussian beams. The writing intensity in the fiber core
is then given by
I(z) = Io exp(-z 2 /w 2 )[1 + cos(2koz)],
(69)
where the origin is taken to be at the intensity peak and
w is the width of the exposed region. We now assume
that this intensity distribution leads to a change in the
refractive-index distribution that is proportional to the
local intensity:
n(z) = no + An exp(-z 2 /w2 )[1 + cos(2koz)].
(70)
We refer to this as a rectified Gaussian grating, as the
change in the refractive index follows a Gaussian profile
and is nonnegative everywhere. It should be noted that
similar results are obtained if we take the index change to
be proportional to 12, though we do not consider this case
here. Comparing with Eq. (18), we find the parameter
functions associated with this refractive-index distribution to be
(f) = 2o exp[-:
K(4)
=
Ko
2
/(kow) 2],
exp[-4:2 /(kow) 2],
=
Ko
exp[-4: 2 /(kow) 2] +
so that we find the following for the effective refractive
index [Eq. (50)]:
neff
=
({2Ko
2
2
2
] + A}
2
(75)
- {Ko exp[- 6 /(RW) ]}2)1/2,
which, clearly, is imaginary for a certain range of detunings. We now construct the band diagram for this
grating structure: For every z we find the detunings for
which the argument of the square root is negative, so that
neff is imaginary. Clearly, at a given value for z this
range is
-3Ko
exp(-z 2 /W 2 ) < A < -Ko
exp(-z 2 /w 2 ),
(76)
where we return to using z as a spatial coordinate rather
than . This interval is shown schematically in the
band diagram in Fig. 6. As a function of position the
unshaded regions show the frequencies for which free
propagation occurs, while in the shaded regions the waves
are evanescent; more simply, the unshaded regions are
transparent, while the shaded regions act as distributed
mirrors.
From Fig. 6 we can immediately deduce some of the
properties of these gratings. For detunings A such that
A < -3KO,
A > 0,
(77)
light sees only real effective refractive indices, and the
reflection coefficient at these detunings is thus expected
to be small. On the other hand, for detunings such that
,Ko,
(78)
the light sees a single mirror surrounded by transparent
regions, and at these detunings the reflectivity is thus
0() = .
(71)
A/K
where
1
Ko =
Sn/2no.
(72)
We can now immediately use Eqs. (40), (41), and (43) to
find the coupled-mode equations
Z/W
= i{2Ko exp[-: 2 /(kow) 2 ]u()
+ Au(f) +
dv()
2
exp[-4: /ROw)
-3Ko < A <
G(y) = cos(y),
d()
d4:
(74)
,
= i{2Ko exp[-:
ko exp[-4:2 /(kow)2 ]V(4:)},
2
/(kow) 2]v()
Fig. 6. Band diagram for Gaussian profile grating showing as a
+ Av(4) + ko exp[-4:2 /(kow) 2 ]u(4:)},
(73)
where we have restricted ourselves again to the first-
function of position the frequencies for which light can propagate
and the grating is transparent (clear region) and the frequencies
for which light is evanescent and the grating acts as a distributed
mirror (shaded region).
Sipe
Vol. 11, No. 4/April 1994/J. Opt. Soc. Am. A
et al.
expected to be large, except where the reflecting region
becomes too narrow. The most interesting frequency interval is
-K 0
<
< .
(79)
For these frequencies the light sees two mirrors, with
a transparent region between and with a transparent
region on either side. The properties of such structures
are well known-for the range of detunings in interval
(79) the rectified Gaussian grating is like a Fabry-Perot
cavity! For most such frequencies, therefore, the reflectivity is high, but if the incident light has a frequency corresponding to one of the fringes of the Fabry-Perot cavity,
the reflectivity vanishes, and all the light is transmitted.
The number of resonances inside the Fabry-Perot cavity
increases with the modulation depth of the grating and
with w. The width of each of the resonances depends,
of course, on the finesse of the cavity at that frequency
and thus decreases with increasing reflectivity of each
of the mirrors. Rectified Gaussian gratings are thus
expected to exhibit a central high-reflectivity region, containing some narrow resonances for which the reflectivity
vanishes, surrounded by low-reflectivity regions without
much structure. Indeed, this behavior of Gaussian profile gratings has recently been observed experimentally. 5
It is worth distinguishing the Fabry-Perot effect
mentioned here from the Fabry-Perot-like behavior of
a uniform grating discussed in Section 4. The effect
discussed here occurs because there are, in the effectivemedium picture, two regions where the effective fields
are evanescent, bounding a region where they are
propagating. The mirrors are strongly reflecting, so
a high-finesse Fabry-Perot filter ensues. The weaker
Fabry-Perot-like behavior discussed in Section 4 occurred at frequencies at which the (uniform) grating
was sufficiently detuned from its Bragg wavelength that,
in the effective-medium picture, the effective fields were
(everywhere) propagating. The behavior discussed here
is thus akin to that of a Fabry-Perot filter made from
two mirrors, whereas that in Section 4 is akin to a
Fabry-Perot filter made of a dielectric slab. We refer
to these as Fabry-Perot effects of the first and second
kinds, respectively. Note that the presence of the weaker
Fabry-Perot effects of the second kind are not indicated
by band diagrams as shown in Fig. 6, which distinguish
only between evanescent and propagating regions (see
the discussion of Fig. 8 below).
Apart from solving Eqs. (73) numerically, once can
obtain more quantitative insight into the properties of
Gaussian profile gratings by adapting the well-known
WKB approximation'8 2 0 to the present problem. The
WKB method is an approximation that describes the
propagation of waves through slowly varying media, i.e.,
when
|n(6)
<< n()I.
(80)
The fields are expressed as a product of a slowly varying
envelope and a rapidly oscillating function that is locally
the exact solution for a uniform medium having the same
local values of the refractive index. Though the WKB
approximation is valid only when the refractive index is
1317
slowly varying, it often gives surprisingly good results
even when inequality (80) is not strictly satisfied.
Clearly, condition (80) fails near n(4) = 0. In a corresponding quantum-mechanical problem this corresponds
to the classical turning points, whereas here it corresponds to the edges of the local photonic band gap. The
solutions on either side of such a point must be related
through connection formulas.' 9 We show below how the
equation for a nonuniform grating can be recast into the
form of the Schrddinger equation-this then permits us
to use well-known WKB results from quantum mechanics.
The WKB approach, which we here apply within the
framework of coupled-mode theory, starts with effectivemedium equations (48). Differentiating each equation
once and cross substituting yield two independent
equations for the effective electric and magnetic fields:
d2 Eeff(4)
dfln[eff()3} dEeff(6)
2
d4
d4:
d
+Leff(4:)Eeff(4:)Eeff()
d2 Heff()
d62
_
0,
=
d{ln[eeff(:)]} dHeff(4:)
d4:
d4
+ eff()eeff(e)Heff(g)
=
0, (81)
respectively. We can eliminate the first derivatives of
the fields by defining new fields
Beff () = [eff(:)]H1/ 2 Eeff
(:)
Reff (4:) = [eff (4)]Il/2Heff()
(82)
giving
2
d Eeff()
2
+
___
e 1 ()
_
4[aeff (4)]2
2
d feff (e) +
d4
[n eff(4)]2 +
2
Fd
eff(4:)
ILd
1
2 Eeff (6)
3
4[,eeff(4:)]2
2
d4Aeff(e)
[deelff(4)
d
2'
Bf
6)=0
]j }1 eff ()
=
0
d2eeff (4)
de(2
i2
j J f e f g) , ( 3
respectively, where Eq. (50) has been used for neff(4).
The solution can be determined from either of Eqs. (83) on
its own, where we use Eqs. (48) and (82) to obtain both
of the original fields.
We now assume that eeff(4:), ,ueff(6), and thus neff(4)
are all slowly varying functions of 4eand consider a WKB
analysis of Eqs. (81). Without repeating the details of
the multiple-scales analysis, we note that the terms involving derivatives of eeff(4) and 1jeff() in Eqs. (83) are
of order 772 (where q is the small parameter related to the
slow variation) compared with the term [neff (4)]2 and do
not enter into the first-order WKB approximation. For
a given grating either of Eqs. (83) may be used. Note,
however, that when ,Ieff(4:) = 0 the first of Eqs. (83) has
an apparent singularity, and it is thus more convenient,
though not necessary, to use the second of Eqs. (83).
Likewise, when eeff(4:) = 0 the solution is obtained with
1318
Sipe
J. Opt. Soc. Am. A/Vol. 11, No. 4/April 1994
the first of Eqs. (83) rather than the second. Both equations lead to the same solution at all the other points,
the singularities are only apparent, and thus these terms
can always be neglected. Equations (83) take on the
standard Schrddinger form
2
2
d:E
+ [neff (4)] 2eff (4) =
of the right-hand part of condition (76). The total number of transmission fringes lying in the range given in
inequality (79) is given by
N
=[
.
_., koneff(z)dz + -2
=(3 1) an o+
XT
-
d2fef}(e) + [neff (4)]2 fleff (6) = 0
(84)
if one is interested only in extracting from them the
first-order WKB solutions. These solutions then take the
simple form
0neff
Heff (4:)
t[Z(4)]lEeff(4:),
(4)] E exp
i f neff(6)de
(85)
where E is a constant, the impedance Z(4:) is given by
Eq. (55), and we used Eqs. (82) to obtain the original
fields. Note the similarity to the solutions of the uniform
grating in Eqs. (54). When neff (4) is real, Eqs. (85) represent forward- and backward-traveling waves, and when
neff (4) is imaginary, they represent exponentially growing
and decaying solutions, in agreement with the discussion
above.
The WKB solutions in Eqs. (85) lose their validity
when neff(4) = 0, which occurs when either eeff(4 : ) = 0
or
(eff4(:)
= 0. This occurs precisely at the edges of the
local photonic band gap-for each value of A we refer to
the values of z at the edges of the band gap as turning
points, in direct analogy to the nomenclature in quantum
mechanics. The WKB solutions on either side of these
turning points must be connected with the use of the
WKB connection formulas. With the equations in this
standard Schrddinger form the solutions to a vast number
of problems are available directly from most standard
textbooks of quantum mechanics.18
For the range of detunings in inequality (78) there are
two turning points with an evanescent region between;
this is the well-known single-barrier tunneling problem,
and the WKB result for the reflectivity isi9
[
RWKB =
Eo
r
1 + exp[-2 Ej- kon'ff(z)dz
] }-1
(86)
where the position of the turning points is given by
zo = w[-ln(-A/3Kof]S2 with the use of the left-hand part
of condition (76) (recall that, for the frequency interval
under consideration, A is negative).
For the more interesting range of detunings in inequality (79) there are four turning points; this corresponds
to a double-barrier tunneling problem. The location of
the transmission fringes [see the discussion following
inequality (79)] is given by the cavity condition
kone (z)dz = m + I )7,
(87)
kow +
12-
(88)
where the integral has been evaluated at A = 0. It is
convenient to define for a nonuniform grating a quantity analogous to the grating strength KL for a uniform
grating:
(KL)eff =
Eeff~g~
et al.
T:K(:)d
f . ~
2
kow,
2 no
=
(89)
where the second equality holds for the Gaussian grating given in Eq. (70). Then the number of transmission
fringes [Eq. (88)] is simply
N =(
/3v) (KL)eff +
2
(90)
Figure 7 compares the location of the transmission
fringes as given by the WKB result and by direct numerical solution of the coupled-mode equations. The normalized detuning for the transmission fringes is shown as
a function of the effective grating strength. For large
grating strengths the results coincide; for small grating
strengths the mirrors at the ends of the Fabry-Perot
cavity are weak, and there is a phase change on reflection (analogous to the Goos-Hanchen shift) that cannot
be determined within the WKB approximation, leading
to the discrepancy seen in Fig. 7. The integral for the
cavity condition in Eq. (87) cannot be evaluated exactly
in simple closed form, but the following simple ad hoc
power-law form gives excellent agreement over the entire
range in inequality (79):
V (KL)eff(l + A/Ko) 51 4
(
+ 1/ 2 )7 .
(91)
Ahco
10.8
0.6
0.4
0.2
0
.
I
2
4
6
8
....10
'
12
(KL)eff
Fig. 7. Location of the Fabry-Perot-like fringes in a Gaussian
profile grating as a function of the effective grating strength
(KL)eff. The solid curves indicate the results according to the
WKB approximation, while the circles give exact results.
with m a whole number, where in this case the turning
points are given by z = w[-ln(-A/Ko)]i" 2 with the use
As
discussed in the text, the deviations, which are largest for small
negative detunings, are due to Goos-Hanschen shifts, which
cannot be correctly determined within the WEB approximation.
Sipe
Vol. 11, No. 4/April 1994/J. Opt. Soc. Am. A
et al.
R
F1
-4
-3
-2
-1
0
A/ o
Fig. 8. Reflectivity as a function of frequency for a Gaussian
profile grating with (KL)eff = 5.
The two sharp resonances for
small negative detunings are due to resonance transmission.
Energy density
15,
I
I
I
I
I
I
\1
I
Z/W
Fig. 9. Energy density as a function of position for the strongest
transmission resonance of a Gaussian profile grating with
(KL)eff = 5 at a detuning A/K -0.7 (top curve). This detuning
is indicated by the dotted line in the band diagram. The dashed
lines indicate the borders of the regions where the grating is
reflective at this detuning. The units are chosen such that the
incoming beam and thus also the transmitted beam have unit
energy densities.
Finishing this section, we show in Fig. 8 the reflectiv-
ity as a function of wavelength for a Gaussian profile
grating with (KL)eff = 5. It clearly exhibits the expected
behavior in the various frequency intervals given in inequalities (77)-(79) discussed above. Note further the
small oscillation in reflectivity for A > 0, which is due
to Fabry-Perot effects of the second kind. Of course,
Eq. (90) gives the number of fringes in only the frequency
interval (79). Finally, in Fig. 9 the solid curves show
the energy density, which is proportional to 2 + Jv12,
for the mode occurring at the transmission resonance at
detuning A/K -0.7 in Fig. 8. The dotted line in Fig. 9
indicates the position of this resonance with respect to the
band diagram. The dashed lines give the positions of the
mU1
turning points, where the character of the waves changes
between evanescent and propagating. The figure thus
clearly shows that a strong resonance is set up inside the
grating, just as in Fabry-Perot filters. The resonances
become much stronger with increasing (KL)eff
6.
DISCUSSION AND CONCLUSIONS
Our rigorous derivation in Section 3 confirms the use of
the coupled-mode equations, even though these are often
derived in a more heuristic way, and identifies the second-
1319
order terms in the coupling constants. It also shows
that the basic idea behind such heuristic derivations is
essentially correct-if the grating parameters are slowly
varying in space on the scale of a single grating period,
then the largest component of the envelope functions of
the forward- and backward-propagating modes also vary
slowly on these scales. As indicated by the kinds of
power-series expression that the multiple-scales treatment generates, for gratings that are not slowly varying coupled-mode theory fails, and the problem must be
tackled exactly. Of course, if the grating consists of
piecewise slowly varying sections, then coupled-mode theory can be applied to each individual segment, while the
solutions in each segment are joined by use of the appropriate interface conditions. Coupled-mode theory also
fails for deep gratings, for which the modulation depth is a
sizable fraction of the background index, and globally for
gratings with parameters that vary significantly from one
end to the other, however slowly. A practical example
of the latter would be a linearly chirped grating with a
period on one end that differs significantly from that on
the other end. Such a grating may, for example, be used
to compensate dispersion over a wide band width. Then,
for a given incident wavelength, a fraction of the grating
would always be significantly detuned. But, although
the relative error in the reflectivity made in this way
may be substantial, the actual reflectivity that is due to
such segments is sufficiently small that the error may be
negligible.
We remind the reader that our analysis has been based
on one-dimensional equations. In fibers and waveguide
geometries of interest, effects that are higher dimen-
sional in nature may obviate the use of the coupled-mode
equations long before the limitations mentioned above
become important. In general, this happens when the
modal field is modified significantly by the presence of the
grating, for example when the refractive-index differences
associated with the grating are comparable with those
associated with the confinement of the radiation. Extreme examples include a model of a fiber with a cladding
that is more photosensitive than the core, and, though
apparently physically unrealistic, a model of a photosensitive effect that reduces the refractive index in the fiber
core. In either case, on sufficient ultraviolet illumination
the core-cladding index difference would vanish, and
the guided mode itself would disappear. Although the
change in background index and modulation depth would
be much less than that in the reference index, coupledmode analysis would clearly be inappropriate, since there
would no longer be any modes!
Our results in Sections 4 and 5 show that the
effective-medium approach can give good insight into the
properties of nonuniform gratings. At the level of the
coupled-mode equations the grating is formally identical
to a dielectric and magnetic medium but without
a grating. Though we considered only uniform and
rectified Gaussian gratings, the treatment is general
and, in principle, can be applied to any grating with
slowly varying parameters, such as gratings with a
linear chirp, or to Moir6 gratings. We briefly review
the approach here, based on the lowest-order coupledmode equations (11). An effective index of refraction
is found from Eq. (50). Where it is imaginary, the
1320
J. Opt. Soc. Am. A/Vol. 11, No. 4/April 1994
grating corresponds to an effective medium in which
the fields are evanescent; where it is real, the grating
corresponds to an effective medium in which the fields are
propagating. It is useful to construct a band diagram,
such as that shown in Fig. 6, with axes labeling z and
the detuning A, indicating by shading the parameter
space where neff is imaginary. For a given A, following
a line from z = to z = + then indicates the
presence of (distributed) effective mirrors for the light
as shaded regions are encountered. The boundaries
of the shaded regions are given by solution of Eq. (50)
for neff = 0, in general a trivial task compared with
numerically solving the coupled-mode equations. Yet
the qualitative nature of the response of a grating (or
class of gratings) can be easily ascertained simply by
examination of the band diagram. A more quantitative
analysis, yet still simpler than solution of the full
coupled-mode equations, can be extracted from the
WKB equations (85) and the corresponding connection
formulas. Fabry-Perot effects of the first kind appear
when mirrorlike regions bound transparentlike regions.
Regimes where they are present are clearly indicated
by the band diagram (shaded regions bounding clear
regions at fixed A); such effects can be described at the
WKB level of approximation. We note that the effectivemedium equations [Eqs. (48)] are fully equivalent to
the coupled-mode equations and thus describe as well
Fabry-Perot effects of the second kind [see the discussion
after relation (79)] such as those exhibited by a uniform
grating (Section 4), although regimes where these
generally weaker Fabry-Perot effects are present are
not indicated by the band diagrams as given here
and not described by the simple WKB approximation
used here.
ACKNOWLEDGMENTS
We appreciate the help of one of the referees in improving the presentation of our results. This study was
supported by the Australian Research Council, the Natural Sciences and Engineering Research Council, and
the Ontario Laser and Lightwave Research Centre. L.
Poladian thanks the Australian Research Council for a
Queen Elizabeth II Fellowship.
*Permanent address, Department of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada.
Sipe et al.
REFERENCES
1. H. Kogelnik, in Integrated Optics, T. Tamir, ed. (SpringerVerlag, Berlin, 1975), pp. 15-81.
2. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).
3. H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909-2947 (1969).
4. G. Meltz, W. W. Morey, and W. H. Glenn, "Formation of
Bragg gratings in optical fibers by a transverse holographic
method," Opt. Lett. 14, 823-825 (1989).
5. V. Mizrahi and J. E. Sipe, "Optical properties of photosensitive fiber phase gratings," J. Lightwave Technol. (to be
published).
6. C. Elachi, "Waves in active and passive periodic structures,"
Proc. IEEE 64, 1666-1698 (1976).
7. C. M. Ragdale, D. Reid, and I. Bennion, "Fiber grating
devices," in Fiber Laser Sources and Amplifiers, M. J.
Digonnet, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1171,
148-156 (1990).
8. K. Hill, "Aperiodic distributed-parameter waveguide for integrated optics," Appl. Opt. 13, 1853-1856 (1974).
9. G. I. Stegeman and D. G. Hall, "Modulated index structures,"
J. Opt. Soc. Am. A 7, 1387-1398 (1990).
10. F. Ouelette, "Dispersion cancellation using linearly chirped
Bragg gratings in optical waveguides," Opt. Lett. 12,
847-849 (1987).
11. D. Reid, C. M. Ragdale, I. Bennion, D. J. Robbins, J. Buus,
and W. J. Stewart, 'Phase-shifted Moir6 grating fibre resonators," Electron. Lett. 26, 10-12 (1990).
12. See, e.g., D. Z. Anderson, V. Mizrahi, T. Erdogan, and A.
E. White, "Production of in-fibre gratings using a diffractive
optical element," Electron Lett. 29, 566-568 (1993).
13. See, e.g., R. K Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C.
Morris, Solitons and Nonlinear Wave Equations (Academic,
London, 1982), Chap. 8.
14. H. A. Macleod, Thin-Film Optical Filters (Hilger, London,
1969).
15. C. M. de Sterke, "Simulations of gap soliton generation,"
Phys. Rev. A 45, 2012-2018 (1992).
16. See, e.g., T. K. Gaylord, W. E. Beard, and M. G. Moharan,
"Zero-reflectivity high spatial-frequency rectangular groove
dielectric surface-relief gratings," Appl. Opt. 25, 4562-4567
(1986).
17. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).
18. L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon, Oxford, 1977), Chap. 7.
19. N. Frbman and P. 0. Frbman, JWKB Approximation (NorthHolland, Amsterdam, 1976).
20. L. Poladian, "Graphical and WKB analysis of nonuniform
Bragg gratings," Phys. Rev. E 48, 4758-4767 (1993).