Download Guided modes of integrated optical guides. A mathematical study

IMA Journal of Applied Mathematics (1998) 60,225-261
Guided modes of integrated optical guides.
A mathematical study
A.S. BONNET-BENDHIAt
Laboratoire de Simulation et de Mode’lisation desphknomsnesde Propagation,
ENSTA, Centre de 1‘Yvette, Cheminde la Hun&e, 9 1120Palaiseau,France
AND
G.CALOZ$AND
F. MAHI?§
IRMAR, Campusde Beaulieu, Universite’de RennesI, 35042 RennesCedex
[Received 10 August 1996and in revised form 7 August 19971
A waveguide in integrated optics is defined by its refractive index. The guide is assumed
to be invariant in the propagation direction while in the transverse direction it is supposed
to be a compact perturbation of an unbounded stratified medium. We are interested in
the modes guided by this device, which are waves with a transverse energy confined in a
neighbourhood of the perturbation.
Our goal is to analyse the existence of such guided modes. Under the assumptions of
weak guidance the problem reduces to a two-dimensional eigenvalue problem for a scalar
field. The associated operator is unbounded, selfadjoint, and bounded from below. Its spectrum consists of the discrete spectrum corresponding to the guided modes and of the essential spectrum corresponding to the radiation modes. We present existence results of guided
modes and an asymptotic study at high frequencies, which shows that contrarily to the case
of optical fibers, the number of guided modes can remain bounded. The major tools are the
min-max principle and comparison of results between different eigenvalue problems. The
originality of the present study lies in the stratified character of the unbounded reference
medium.
1. Introduction
During the last decadehigh speedtransmissionof information has undergoneimportant
changeswith the development of optical fibres and more generally of waveguides.These
have been the object of numerous studieseither from a theoretical or from a numerical
point of view; see,for instance,(Marcuse, 1982; Wilcox, 1984; Bonnet, 1988; Bamberger
& Bonnet, 1990; Weder, 1991) and the referencestherein.
In the field of integrated optics the sameproperties of light propagation are used to
improve the performanceswhich up to now were reservedto electronic devices.
A microguide in integrated optics is defined by its refractive index. In our paper we
shall considerwaveguidesinvariant in the propagationdirection, which are composedof a
stratified medium with a compact perturbation, in the transversesectioncalled the core of
the guide; seeFig. 1.
The stratified medium is the reference medium (sometimes called the cladding by anat ([email protected])
$ (caloz@univ-rennes
1.fr)
5 (fmahe@ univ-rennes 1 .fr)
@ Oxford University
Press 1998
226
A. S. BONNET-BENDHIA
FIG.
1. Stratified
optical
ETAL.
guide
logy with optical fibres) and is intended to guide electromagnetic waves in one layer. The
unboundedness of the reference medium will put obstacles in the theoretical and numerical
studies as well as induce particular phenomena at high frequency. The compact perturbation of the reference medium is designed to confine waves inside a layer in a neighbourhood of the perturbation.
Our goal is to study electromagnetic waves guided by such devices, which are waves
of the form #(xi, X2)ei(wf-@3), where (XI, x2) are the transverse coordinates and x3 is the
longitudinal coordinate. The guided modes correspond to waves of finite transverse energy
which propagate without attenuation, that is, u and /? are real. Our analysis is limited to
lossless isotropic dielectrics. We will determine existence conditions for guided modes and
bounds for the number of guided modes.
Although we do not emphasize the computational point of view here let us mention
that numerical studies of such problems have been considered in the same and even more
general configurations; see (Bastiaansen et al., 1992; Koshiba, 1992; Rahman & Davies,
1984). In a subsequent paper, we will present numerical results based on the localized finite
element method; see (Mahe, 1977).
To our knowledge there are no theoretical studies of guided modes for such perturbed
stratified media. For the case of a uniform exterior medium, mathematical analyses have
been done; see (Bamberger & Bonnet, 1990; Urbach, 1996) for instance. In contrast to this
case of optical fibres, there are no simple geometries for which analytical computations
can be done. Our study is carried out with spectral theory tools for selfadjoint operators.
Under the assumptions of weak guidance, the study of the Maxwell system reduces
through an approximation of order 0 to the study of a scalar equation in the plane; see
(Bonnet, 1988; Snyder & Love, 1983) or (Vassalo, 1985). For the dielectric microguides
used in practice, this scalar approximation is generally not valid, due to the great variation
between the refractive index of the air occupying the upper medium and the one of the
dielectric materials. However, the study of this scalar model has to be carried out before
starting with the vectorial model, since it allows to solve in a much simpler situation a lot of
mathematical difficulties due to the stratification of the unbounded medium. Moreover the
results established in this paper will be used as technical tools for the study of the vectorial
problem.
The paper is organized as follows. In Section 2 we show how the study of guided modes
under the weak guidance assumption is reduced to a two-dimensional scalar eigenvalue
problem, for a one-parameter dependent selfadjoint linear operator.
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227
The properties of the operator are presented in Section 3. Its spectrum consists of an essential spectrum corresponding to a continuum of radiation modes and of a point spectrum
corresponding to guided modes. We determine the lower bound of the essential spectrum,
which is given by the fundamental mode of the reference medium.
In Section 4 we recall the min-max principle which will be extensively used to characterize the point spectrum below the essential spectrum. Thanks to the min-max principle
we can compare the original problem with eigenproblems which are simpler to solve. Original comparison techniques are introduced.
Section 5 is devoted to the study of the first mode, also called the fundamental mode. We
discuss thoroughly conditions for its existence with particular attention at low frequencies.
An asymptotic study at high frequencies is developed in Section 6. We prove that the
number of modes can be bounded as the wave number tends to infinity. In some cases all
guided modes will disappear beyond a critical value, which is contrary to what happens for
optical fibres.
In the Appendix we report the results for the one-dimensional study of the reference
stratified medium.
Problems of guided waves are also related to research fields other than optics. In elastodynamics we can refer to (Bamberger, Dermenjian & Joly, 1990; Bamberger, Joly &
Kern, 1991). Hydrodynamics offers similarities with optics in stratified media. In (BonnetBenDhia & Joly, 1993), the authors study the waves guided by the sea coast. The depth
plays the role of the refractive index. Its variations confine the surface waves close to the
coast. As for electromagnetic waves in stratified media, the propagation of surface waves
presents a dispersive character. Some results are similar in both fields and the technique
used here is inspired by the one used in hydrodynamics. Nevertheless in optics the study is
much more involved because the wave confinement depends on the refractive index which
is a function of two variables while the guidance of the surface waves only depends on the
depth. The difference becomes evident at high frequencies. For instance the example presented at the end of Section 6, where the fundamental mode disappears at high frequencies,
has no equivalent in hydrodynamics.
Finally let us introduce the standard notation we use through the paper. We write Iw+
for the non-negative real numbers and R$ for the positive real numbers. For m E Z,
l<p<oo,andB~IW”(n~~),W
“qP(L?) is the Sobolev space of functions with derivatives up to order m in U’(Q) endowed with the norm 11.Ilm,p,a and seminorm I. Int,p,Q. For
P = 2, Wm*2(s2) is a Hilbert space also denoted by Hm(i2) endowed with the scalar product (., .)m,a and the norm II.Ilm,Q. We shall also use the standard differential operators div,
rot, V, A.
2. Preliminaries
The propagation of light waves is described by the Maxwell equations
rotlE = -aB/llt,
divD = 0,
rotW = D/at,
divIE5= 0,
where IE is the electric field, W is the magnetic field, D is the electric induction, and II3 is
the magnetic induction related by
IEB=pW
and
ID = &.
A.S.BONNET-BENDHIA
228
ETAL.
The magnetic permeability p and the dielectric permittivity 6 characterize the electromagnetic behaviour of the material. It is classicalin guided optics to assumethat p is a constant
equal to ~0, the permeability of the vacuum. The permittivity 6 is related to the refractive
index by the formula c = con*, 60being the permittivity of the vacuum.
Before presentingthe mathematicalmodel, we describethe guide. We assumethat the
guide is invariant in one direction (say 0x3) which will be the propagation direction, and
that the guide is a perturbation of a stratified medium. If the function y2denotesthe refractive index, then y2is a function of xi, x2 only, y1= yt(xl, x2). Outside the perturbation, the
function n dependsonly on x2. A guide is representedin Fig. 1.
We shall study electromagneticwavesharmonic in x3, solater on we will only represent
a transversesection of the guide.
Let ii E La be a function with positive values,
such that
ii(~) =
(
if
~:, if
6 > c,
~ ( -c,
inf ii(c) > 0,
Qa
for somepositive c. The values nt, nh play the samerole and without lossof generality
choose
nb
’
nt
consideringthat the upper material is the air. The refractive index n satisfiesthe following
assumptions:
n E Lm(R2),
inf n(x) > 0,
(24
XEIR?
and there exists a compact set K c IK* such that
n(x> = ii(~)
for all x = (xl, x2) 4 K.
cw
Let n+ denote the essentialsupremumof n,
n+ = Il4loo.W*(= supn)*
The index function ii defines a planar waveguide associatedto the guide; it represents
the stratified medium without perturbation.
Theoretical studiesof the one-dimensionalproblem of the planar waveguide are consideredin (Guillot, 1985; Schechter, 1981) or (Wilcox, 1984). In the Appendix we report
basic resultsof this problem, which are not given in the referencesabove.
2.1 Actually the transversesection of the guide is bounded. We can assume
it is unbounded becausethe dimensionsof K are very small compared with the onesof
the stratified medium and becausewe are looking for waves confined in a neighbourhood
of K.
REMARK
A model case,to which we shall often refer, is the caseof the three layers guide; seeFig.
2. To that category belongsthe rib guide often usedin applications; seeFig. 3.
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ii
-w-w-
---w
--a--
---w
229
= n,
w-“-e
-w-M.
E(x2)= n,
--we-
---w.
X(x2)= nb
FIG.
2. Three layers guide and planar guide associated
FIG.
3. Three layers rib guide
REMARK 2.2 Notice that when y2is constant outside K, the cladding surrounding the
core is homogeneousand we have an optical fibre. For a mathematicalstudy of fibres see
(Bamberger& Bonnet, 1990; Poirier, 1994).
When the cladding is composedof two different homogeneousmaterials, then we have
a perturbed diopter; see,for instance,(Bonnet & Djellouli, 1988; Gmati, 1992).
The electromagnetic waves which propagateinside the guide would be a solution of the
Maxwell equations. The ones we are looking for are the modesand have the following
form:
E
@1,X2,&t)
=
s
(( >
H
(Xl
9 x2>e
i(kcot-j?q)
9
(2.4)
where co is the speedof light in the vacuum, k is the wave number,and B is the propagation
constant of the mode.
We say that a modeis guided if it propagateswithout attenuation andhasfinite transverse
energy, that is,
k, B E R
and E, H E (L2(IK2))3.
The guided mode with the largest propagationconstant p is called thefundamental mode.
If the transverseenergy is not finite or if p is complex, then we have a radiation mode,
which dispersesinto the cladding.
2.3 Given an incident lightwave (k is given), several guided modes(corresponding to different values of B) could propagateinside the guide at different speeds.
REMARK
A.S.BONNET-BENDHIA
230
ETAL.
The velocity of a mode is v = kco/p, which is the velocity of a planar wave in a material
of constant index q = j?/ k. The ratio /?/k is called the eflective index of the mode.
We want to look for solutions of the form (2.4) to the Maxwell system under the assumption of weak guidance (that is, large wave number and weak variations of the index).
Then approximations of such solutionsare obtained with the expressions
B
and H = kCoP0
E=
and H=--
where u E L2(R2) is a distribution solution of the scalarequation
-Au - k2n2u = -/12u
in R2;
(2.5)
for zero-order approximation of the Maxwell systemsee,for instance,(Bonnet, 1988; Snyder & Love, 1983) or (Vassalo, 1985).
Thus looking for guided modesunder the assumptionof weak guidanceconsistsin determining the real numbers/?, k, the functions u E L2(R2) such that (-p2, u) is an eigenpair
of the operator (-A - k2n2). A way to study thesemodesis to fix k and look for (-p2, u).
By varying k we then get the dispersionrelation k w B(k). This choice hasphysical motivation becausek representsthe wave number of a lightwave running through the guide.
2.4 Mathematicians generally prefer to consider p as a parameterand k as a
function of /J; see(Wilcox, 1984) or (Weder, 1991). In the scalarcase(2.5) both points of
view are similar but for the Maxwell systemthe study hasbeendone so far only when p is
a parameter;see(Bamberger& Bonnet, 1990).
REMARK
3. General properties
Let n be a refractive index, that is, n E Loo(R2) satisfies(2.1) and (2.2). For a given nonnegative value of the parameter k, the problem we shall study reads: find B E IEX+and
u E H’(lR2), u $0, such that
a(k; u, v) = -p2(u, v)ow2
for all v E H’(R2),
,
(34
where the bilinear form a(k; ., .) : H’ (R2) x H’ @X2)+ R is given by: for u, v E H’ (R2)
a(k; u, v) =
(VuVv - k2n2uv) dx.
sw2
(W
In fact (3.1) is the variational formulation of our guided mode problem (2.5).
Let the unboundedoperator Ak : D(Ak) c L2(R2) + L2(R2) be given by
D(Ak) = {v E H’(R”); Av E L2(R2)}
and Akv = -Av
- k2n2v for v E V(Ak);
PROPAGATION
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231
we consider the problem: find p E IR? and u E D(Ak), u $0, such that
Problem (3.3) is the spectral formulation of our guided mode problem. We can check that
(3.1) and (3.3) are equivalent, since for all u E D(Ak) and u E H’ (R2)
(ALU,
~)OJ@
= a(k; u, v).
To study the eigenproblem (3.3) or (3.1) we need a rather detailed study of the bilinear
form a(k; ., .) and of the operator A k. The notation related to spectral theory is the one
used in (Schechter, 1971). We will recall the major examples when needed.
3.1 The bilinear form a(k; ., .) defined in (3.2) is continuous symmetric
and bounded from below
PROPOSITION
for all 2) E H1(R2)
a(k; u, v) > ~~VV~~&~
7 - k2n$llull&2.,
WV
The proof of Proposition 3.1 is immediately deduced from the definition of the bilinear
form a(k; . , .) and the assumptions on n.
PROPOSITION
(1)
D(Ak)
=
3.2
The following propertiesfor the operator Ak hold:
H2(R2),
(2) Ak is bounded from below, selfadjoint, and its spectrumsatisfies
o(&)
C [-k2&
00).
Proof The property (1) is a direct consequenceof the definition of D(Ak). From the
estimate(3.4) we deducethat for u E D(Ak)
that is, Ak is bounded from below. Since a(k; ., .) is symmetric and satisfies(3.4), we
deducethat Ak is regularly accretive and that Ak is selfadjoint; see(Schechter, 1971, pp.
24, 33). The estimate for the spectrum is immediate since for h < -k2nt, the bilinear
0
form a(k; u, v) - A(u, U)O.~2 is coercive.
One difficulty in the study of the spectrumof Ak lies in the fact that Ak hasno compact
resolvent. Indeed let h < -k2n:; then h is in the resolvent set of & and (Ak - AZ)-*
is clearly not compact since the Sobolev embeddingsH2(R2) c H1 (lR2) c L2(R2) are
not compact. Consequently (Ak - AZ)-’ is not compact for all h in p(Ak).
We know since & is selfadjoint that the spectrum of Ak consistsof a continuum, the
essentialspectrum o&&),
and of a discrete set, the discrete spectrum ad(&), which
is the set of isolated eigenvaluesof finite multiplicity. We determine now the essential
spectrumof Ak while the discrete spectrumwill be studiedlater.
The essentialspectruma,,,(Ak) is characterized asfollows; see(Schechter, 1971,p. 34):
h belongsto De&&) if and only if there exists a sequence(up)p~i C 2>(&) suchthat
llupllo,~2= 1,
(Ak - hZ)u, --+ 0,
up - 0 weakly in L 2 (R 2 ).
Such a sequenceis called a singular sequenceassociatedto Ak - AI.
232
A. S. BONNET-BENDHIA
ETAL.
PROPOSITION
3.3 Let & : V(&)
c L2(R2) -+ L2(R2) be the operator associated to
the refractive index of the non-perturbed stratified medium. Then
oess (Ak)
Proof
=
*ess (xk
)
l
The operator Ak can be decomposed into
Ak
=&+B,
where for 21E H2(R2)
&
=
-Av-
k2n2v
-k2(n2
- n2)w.
and for UJ E L2(R2)
Bw=
To prove the result it is sufficient now to check that B is &-compact,
sequence (up}~~i C H2(lR2) satisfies
that is, whenever a
then {Bu&,~l
has a subsequence convergent in L2(R2); see (Schechter, 1971, p.16). Indeed let (~~}~>i be such a sequence. Then (~~}~>i is bounded in H*(lR”). Since n2 - fi2
has a compact support there exists a subsequence (upj}~~i such that { (n2 - fi2)~,j}i21
converges in L2 (IR2).
0
PROPOSITION
3.4
Let & be as defined in Proposition 3.3. Then
aess (Ak)
=
a,ss(~k)
=
[y(k),
m>,
W)
is a characteristic value of the associatedplanar waveguide (seethe Appendix).
Proof The first equality in (3.5) hasbeenproved in Proposition 3.3. Let us now check the
secondone.
We check first that (&ss(&) C [T(k), 00). Given q E H1 (R2), we have
and so by the definition of Y(k)
Consequently u-(&)
c
[y(k), 00).
PROPAGATION
OF ELECTROMAGNETIC
-k2n2,
-k2nz,
7(k)
,~t++*
--------
In the resolventset
233
WAVES
x Likely eigenvalues
4-----------)
R
Essentialspectrum
FIG. 4. The spectrum of Ak
To prove the secondinclusion, we will build a singular sequence{LQ,}~>1 for all h >
y(k). It is sufficient to take h > y(k) since a,,,(&) is closed. We set v =‘i+/m
E
IRZ. It is proved in the Appendix that y(k) < -k2ni; if y(k) < -k2n$ then y(k) is an
eigenvalue of -d2/dy2 - k2fi2 with an eigenfunction p E H2(R); if y(k) = -k2ni then
there exists a singular sequence(~p)p~ t c H2(R) associatedwith -d2/dy2 - k2ii2-y(k).
If y(k) < -k2n2b we set
1
up(x)
=
9l
Xl
iuxl -
~e’p’e
[
co(x2)
1
and if y(k) = -k2nE we set
up(x)
=
8
[
1 Xl
33(pk1vx’Pp(X2)
*
9
I
where 8 E C~(IR) is chosensuchthat
IP Ilo,wIl~llo,IIt= 1 and lleIl0,~ = 1 respectively.
With algebraic manipulationswe can check that {LQ.,}~~~is a singular sequence.
cl
REMARK
3.1 The values of h in o&&)
correspondto radiation modes,that is, to solutions of (2.5) which are not in L2(R2). Indeed considerfor positive v
u(x) = 8 [v(x2)eivx1]
and
- p2 = Y(k) + v2,
where q(x2) is the function introduced in the proof of Proposition 3.4. Then u satisfiesthe
equation (2.5) with ii.
In fact we have proved in Proposition 3.4 that o&&)
= o (&) = [v(k), 00). We can
already notice that the existence of a discrete spectrumcan occur if the reference medium
is perturbed.
We have representedin Fig. 4, the resultsalready known on the spectrumof Ak.
It is time now to turn our attention to the point spectrum of Ak, namely a,(Ak), which
gives the guided modes.It consistsof the discrete spectrumq(Ak) which are eigenvalues
below y(k) (the lower boundof the essentialspectrumof Ak) and of eigenvaluesembedded
in the essentialspectrum.
Since & f L(H2(R2); L2(R2)) is selfadjoint and boundedfrom below by -k2nt, we
deducethat
q,(&)
We have the following
result.
C [-k2n:,
00).
A.S.BONNET-BENDHIA
234
PROPOSITION 3.5
ITAL.
Let h be an eigenvalue of Ak, then necessarily
-k2n2 +’ < A ’< -k2n2 b’
W)
Proof We have already argued on the first inequality. To check the upper bound, we follow
cl
the method used in (Weder, 199 1).
In the next two remarks we make some comments and give physical interpretation of the
eigenvalues.
REMARK 3.2
If the eigenvalue -p2 is below the essential spectrum, that is,
-k2n:
G -b2 < y(k),
then the velocity of the associated mode v = kco/b has the bounds
co/n+ G v < C(k),
where T(k) is given by the relation -r(k)i7(k)2
= cik2. The lower bound is the velocity of
a planar wave in a material of index y2+ and the upper bound is the velocity of the slowest
wave propagating in the perfectly stratified medium.
When the cladding is a planar waveguide with guided modes, see the Appendix, the
slowest velocity is the one of the fundamental mode T(k), which depends on k nonlinearly.
This is a difficulty of our problem, which shows the dispersive character of the stratified
medium: two lightwaves with two different frequencies will propagate at different speeds.
This property will have a direct effect on the number of modes at high frequencies; see
Section 6.
REMARK 3.3 In the case of optical fibre and diopter we have F(k) = -k2nz and so from
Proposition 3.5 we cannot have eigenvalues in the essential spectrum. In the general case
of waveguide we can find eigenvalues in the essential spectrum, which means we can find
guided modes propagating faster than some mode in the stratified medium.
We will not present examples of guides with eigenvalues embedded in the essential spectrum in this paper; such eigenvalues lie in the interval [y(k), -k2ni]. An example is given
in (Bonnet-BenDhia & Mahe, 1997). The guided modes embedded in the radiation modes
are very sensitive to imperfections in the guide and are less interesting from a practical
point of view than the modes below.
The question of existence of eigenvalues embedded in the essential spectrum is a problem encountered in various fields of application. Non-existence results have been proved
in acoustics in a stratified medium; see (Weder, 1988) or for the Schrodinger operators see
(Reed & Simon, 1978). Examples of embedded eigenvalues are presented for perturbed
guides by Witsch (1990), for electromagnetic gratings by Bonnet & Starling (1994), in
elastodynamics by Joly & Weder (1992).
To go further in the study of the point spectrum of Ak below the essential spectrum
[y(k), oo), we shall introduce in the next section the techniques adequate for selfadjoint
operators bounded from below, namely the min-max principle and comparison principles.
PROPAGATION
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4. The min-max principle and comparison principles
A major result to characterize eigenvaluesof selfadjoint operatorsboundedfrom below is
the min-max principle; see(Reed & Simon, 1978; Dunford & Schwartz, 1963).
Corresponding to the problem (3.3), we define the min-max quantities h,(k), m > 1,
bY
=
Al(k)
a(k v, v)
inf
VEH1(W2)
(v, v)o,Iw*
u#O
and form > 1
b?lw =
a(k v, 20
inf
SUP
fUHm(H’(~*))HUH,,,
(v, v)o,R*’
u#O
(4.2)
where H, (H *(1w2))is the set of m-dimensionalsubspacesof H1 (lR2). Then
-k2n$
< Al(k)
< AZ(k) < . . . < k,,,(k) < . . . < y(k)
and if hj(k) = y(k) for somej > 1 then Ak has at most j - 1 eigenvaluesbelow T(k).
If hj(k) < y(k), then hi (k), . . . , hj (k) are the first j eigenvaluesof Ak. Moreover the
numbers3Lm
(k), m >+ 1, can be characterized by
Ll(k) =
211 v...,
u m -1~~*(~*)
a@;w, u-0
inf
SUP
w~H’(~*),w#O
WEbl
(WY
W)O,R*
’
,...JJm-*I*
where [VI, . . . , v,-r]l
is the orthogonal complement in L2(Iw2) to vi, . . . , ~~-1. In
particular when h,-i (k) < y(k) then
h,(k) =
inf
w~H'(W*),w#0
waCO1
a&; w w>
(w,
w)O,@
'
. . ..Jp.-Ill
where qq, . . . , pm- 1 are eigenvectorsassociatedwith the eigenvalueshi (k), . . . , h,-1 (k).
In other words, if N(k) is the number of eigenvaluesstrictly below y(k), then
N(k) = sup{m E Pi; A,(k)
< y(k)},
and N(k) is called the number of guided modes.
We presentnow a comparisonmethod useful when we try to get boundson the number
of guided modes.We comparethe solutions of problemsstated in somesubdomainsS2of
IK2to the one stated in Iw2.The method hasbeenalready usedto study wave phenomenain
(Bonnet-BenDhia & Joly, 1993) when G is bounded. In our applications the domains 52
are not necessarilybounded.
Let n E P(Iw2) be an index function satisfying to (2.1) and (2.2), and 52 c Iw2be
an open set containing K. We assumethat the boundary dQ of 52 is regular. The exterior
normal unit vector on aln is denotedby u and the complementof Q in Iw2by QC.
On the set Q we consider eigenproblemswith homogeneousDirichlet or Neumann
boundary conditions.
236
A. S. BONNET-BENDHIA
ETAL.
We define the problem: find h E R, u E Hd (n), u + 0, such that
a&k; u, U) = k(u, zl)o,~ for all v E Hi(D),
(44
where the bilinear form aa (k; u, v) : H1 (Q) x H1 ($2) -+ IEXis given by
aa(k; u, v) =
I sz
(VuVv - k2n2uv] dx.
(43
We denote by Af the operator given by the spectral formulation. Associated to the problem
(4.4) we define the quantities
ND(k) = sup (m E lV; h.;(k) < T(k)},
W)
where y(k) is the lowest bound of the essential spectrum of Ak.
We now define the problem: find h E IR, u E H’ (Q), u + 0, such that
a&k; u, v) = A(u, V)O,Q for all v E H’(D).
W)
We denote by A; the operator given by the spectral formulation. As before we associate
to the eigenproblem (3.8) the quantities
?$(k) =
a&; v, v>
inf
&EW~(H~WN v,zH,,,
sup (v9 V>O,Q
V#O
NN(k) = sup (m E RI; k:(k)
PROPOSITION
4.1
< y(k)}.
(4.10)
We assume that Q is a piecewise C* open set. For all m E lV*, k E IRT_
h,(k) < A:(k)
and ND(k) < N(k).
Proof The proof is immediate when we identify Hd (In) with a subspaceof H1 (lR2).
•I
In the case of the Neumann boundary conditions our results are not so general. The
extension by 0 of cpE H1 (X2) to @ E H1 (IR2) does not work anymore.
Let a, b E lEX+be such that K c (-a, a) x (-6, b). We define the three different sets
where d = max(b, c).
521= b E R2; lxal -c 4,
02 = b E R2; Id < a),
(4.12)
523= (- a, a> x C-d, 4,
(4.13)
(4.11)
PROPAGATION
OF ELECTROMAGNETIC
237
WAVES
PROPOSITION
4.2 Let 52 be one of the three sets defined above in (4.1 l), (4.12), or
43.13). Then for all m E FJ*, k E IEY&the following holds:
ND(k)
< N(k)
< NN(k).
(4.15)
Proof
Both estimates involving the Dirichlet problem in (4.14), (4.15) are proved in
Proposition 4.1 for any piecewise C’ open set. Since the estimate (4.15) is an immediate consequence of (4.14), it suffices to check
(4.16)
mw,N w9 WN < Lz (k)
for Q in (4.1 l), (4.12) and (4.13).
Case 1: i2 = i2l. Given u E H’ (R*) we have, since d 2 c,
s
- k*n*u*) dx 2 -k*n; /
s2c ((Vul*
i2c
u*dx.
Then from the secondcharacterization of h,(k) we get
&n(k) 2
j’a (IVul* - k*n*u*) dx - k*n; jQc u* d.x
inf
sup
jQ u* dx + lQc u* dx
Vm-’Evm-‘(L*(W))uq;-‘nH* CR*)
(d. 17)
U#O
where V,-&,*(IEt*))
is the set of (m - I)-dimensional subspacesof L*(IR*). Notice that
for 2)E L*(R*) with u = 0 in a, the above quotient reduced to -k*n$ Therefore
&n(k) 2
inf
sup
v,-‘EV,-‘(L*(i2)) uq;-‘nmm
U#O
m>
with
a(u) = min
j’Q (IVul* - k*n*u*) dx
jQ u* dx
, -k*ni
.
Notice that here we have usedthe estimate,for al, a*, a3, a4 E Et, a3, a4 > 0,
a1
+a2
->min
a3 +
a4
a1
a2
-,--
( a3 a4 >
.
We then deduce that
h,(k) 2 min(At(k), -k*nt).
Since we already know that y(k) < -k*n$ it follows that
238
A.S.BONNET-BENDHIAET
AL.
Case2: D = Q2. Let u E H’(R*) be given. Then for almost all x1, IxiI > a, we get from
the definition (3.6) of y(k)
s, ( ~T3X~‘X2~~ -k2n2u2(xl,x2))
dX2 >T(k)l.*(x1,x2)dX2.
By integration in xi over (-00, -a) U (a, OQ),
*n*u*) dx > y(k)
s LF
u* dx.
We argueas in case 1 to check (4.16).
Case3: D = 523.We proceed in two steps.We compare first h:(k) with hi(k)(&)
one in case2) with the argumentof case 1. So
At(k)(&)
(the
> min (At(k), -k*nfj) .
Then from the analysisof case2, we can check (4.16).
cl
4.1 If we denote by h:(k), NN (k) the quantities (4.9), (4.10) associatedwith
522,and hgN (k), NNN(k) the onesassociatedwith 523then in the proof of Proposition 4.2
we have checked that
REMARK
N(k) < NN(k) < NNN(k).
(4.19)
In the next proposition we presentsomeadditional results for both eigenvalueproblems
(4.4) and (4.8) when Q is one of the sets(4.1 l), (4.12) or (4.13).
PROPOSITION
4.3
The following assertionshold.
(i) When D = 521= (X E R*; 1x21< d}, then
a,,(A:>= WD(k),
001, W) < YD(0
~-ess(A;)
= [5iN(k),
m), TNw < W),
(4.20)
(4.21)
where rD(k), TN(k) are defined in (A.26), (A.27) for m = 1.
(ii) When 52 = Q2 = (X E lR*; 1x1I < a}, then
a,,,(Af)
= [-k*nE + n2/4a2, m),
(4.22)
o&A:)
= [-k*n& 00).
(4.23)
(iii) When $2 = 523 = (- a,a) x (-d,d), then the spectrum o(AF) or a(AF) is an
increasing sequenceof eigenvaluestending to 00.
PROPAGATION
OF ELECTROMAGNETIC
239
WAVES
We will not present a proof of Proposition 4.3 here since all the arguments are classical
or have been used before.
As direct consequences of the comparison principles given above we get bounds on the
number of guided modes.
4.4 For a fixed value k of the wave number a guide has a finite number of
guided modes. Moreover there exists a k* > 0 such that
PROPOSITION
NW
for all 0 < k < k*.
< 1
proof Let k E IR> be given and J2 = (-a, a) x (-d, d), see(4.13). From Proposition 4.2
we deducethat N(k) < NN (k) and from Proposition 4.3(iii), that NN (k) c 00.
cl
We can prove the existence of k* by contradiction.
Let us apply the comparison principle in Proposition 4.2 to the three layers rib guide,
seeFig. 3, with the refractive index
nb
n(x)
=
n+
nt
if x < -c,
if x E (-a, a) x (-c,
otherwise,
H - c) or -c < x2 < c,
where H > 2c. We assumethat n+ > nb > nt. We can operatein two different ways: either
comparewith respectto the domain (-a, a) x (-c, H - c) or with respectto (-a, a) x R.
The secondway gives more accurate boundsbut needsmore computations.
Let Q be the set (-a, a) x (-c, H - c). It is not difficult to check that
2
h;Jk)
t
= -k2n:
+ $$
+ 5
$$(k)
9
= -k2nt
+ $
+ s
2
for p,q E N
2
2
for p, q E N*.
So we have the bounds
ND(k)
< N(k)
< NN(k)
(4.24)
with
2
ND(k)
= Card (p, q) E N* x N*; -k2n$
+ $$
2
+ 5
(4.25)
2
NN(k)
= Card (p, q) E N x IV; -k2nt
+ 5
2
2
+ 5
2
< y(k)
I
.
(4.26)
Notice that we can computey(k) with the formulae presentedin the Appendix. Then with
(4.25) and (4.26) it is simpleto compute ND(k) and NN (k).
Let Q be the set (-a, a) x R. We can compute the eigenvaluesnumerically when Q
is a strip. In the example A.1 of the Appendix we have presenteda way of studying the
three layers planar waveguide. If ~4(k, H) are the min-max associatedwith the three layers
planar waveguide of thicknessH, see(A.@ and (A.9), then
A.S.BONNET-BENDHIA
240
ETAL
So we have the bounds
ND(k)
< N(k)
(4.27)
< NN(k)
2
ND(k)
2
NN(k)
2
= Card (p, q) E IV* x W*; y&k, H) + $$-
< ?vG
I
(4.28)
9
2
= Card (p, q) E IV x Pi; y&k, H) + $$-
(4.29)
4.2 To illustrate both estimates(4.24) and (4.27), we present two computed
examples.In both of them we have chosen
REMARK
%- ~3,
n,=l.
nb=2,
Then we have modified the size of the bump.
Case 1. We choose a = 1, c = 1, and H = 3. The estimates(3.24) and (3.27) give
respectively
ifk E [Ol,O.8],
0 < N(k)
< 1,
if k E [0*9, lOO],
0 < N(k)
< 2,
ifk E [Ol, lOO],
0 < N(k)
< 1.
Case 2. We choosea = 3, c = 1, and H = 6. The estimates(4.24) give
ifk
ifk
ifk
ifk
ifk
ifk
E [O-l, 0.21
E [O-3,0.4]
E [0*5,0.8]
[O-9, 1.21
E
E [ 1*3,6.6]
E [6-7, lOO]
0<
0<
1<
1<
NW
NW
NO
NW
<
<
<
<
1,
3,
4,
6,
3 < NW < 8,
4 < NW < 9,
and
ifk
ifk
ifk
ifk
ifk
ifk
E [O-l, 0.321
E [0*33,0.44]
E [0*45,0.8]
E [O-9, 1.11
E [l-2,5.4]
E [5*5, 1001
0<
1<
1<
2<
3<
4<
NW
N(k)
NW
NW
N(k)
<
<
<
<
<
1,
2,
3,
4,
5,
N(k)
< 6.
The numberspresentedabove were obtained by solving the dispersionequation given
in the example A. 1. More details and further examplescan be found in the work of Mahe
(1993).
The results above illustrate Proposition 4.4. Moreover we notice that the number of
modesis not increasing indefinitely with k. This result will be proved in a more general
framework in Section 6.
PROPAGATIONOFELECTROMAGNETICWAVES
241
REMARK 4.3 We will look in the next sectionsfor solutionsof (3.1) (or (3.3)) for given
-+ . We could addressalsothe problem of the dependenceon k of the modes.
valuesof%-& -R,
Let h,(k) be defined in (4.1) and (4.2). When h,(k) -is an eigenvalue of Ak, the function
k I+ h,(k) is called the mth dispersioncurve. Extending the results of Bonnet (1988), we
can prove that the function k t+ Am(k) is continuous, almost everywhere differentiable,
and decreasing.
5. The fundamental mode
From the minmax principle, we can immediately discusssomeexistence conditions for
the fundamentalmode.
PROPOSITION
5.1
If the refractive index n satisfies
n(x) < ii(xz)
for all x E R2
6 1)
then there are no guided modes,N(k) = 0 for all k > 0.
Proof By the definition of Y(k) in (3.6) and the hypothesis,we obtain for all cpE Cr(lR2)
and x1 E R
We integrate the inequality over xi and usethe density of C,“(R2) in H1 (R2) to get
This completesthe proof.
PROPOSITION
5.2
cl
We assumethat the refractive index n satisfies
n(x) > fi(x2)
for all x E R2
measure(R = {x E R2; n(x) > 3(x2)}) # 0.
If the associatedplanar waveguide hasa guided wave
r(k) < -k2n2b,
then the guide hasat least one guided mode, that is,
b(k)
and then N(k) 2 1.
<
?w)
A.S.BONNET-BENDHIA
242
ITAL.
Proof We will prove the following inequality:
a&; v, VI
inf
< no
UdP(W2)(v, v)oJp
u#O
It is sufficient thus to prove that there exists a function v E H1 (R2) suchthat
a(k; v, v) =
s
w2(WA2 -k 2n2v2) dx < y(k)
v2dx.
s w2
The assumption(5.4) implies that T(k) is an eigenvalueof -d2/dy2 - k2fi2 with an eigenfunction ?&seethe Appendix. We take
I,
v(x) = (p(x2)e--LyJx1
where a is a positive number which will be fixed later. Then
a(k; v, v) =
((@I>2+ (a2 _ k2fiz)qz) e-2aIXlIdx _
k2(n2 - fi2)~2e-2alxlI &
sIit2
s w*
and
1
a(k; v, v) = -~ R(F(k) + a2)q2 dx2 k2(n2 - ii2)~2e-2aix1idx.
s
sk
wt.2have4Ml; . = llalg IR’ So to check that hi (k) < F(k) it suffices to notice that
R2
k2(n2
_ ~2)~2em2”IXlI
dx
sif
cl
for a > 0 small enoughsince (5.3) holds and p is almosteverywhere non-zero in R.
In fact the simplecaseof a uniform cladding is not included in Proposition 5.2. With the
assumption(5.2) and the modified assumption
measure(K =
ix E 1w2;M) ’ Ilao.R~> # 07
(5.3’)
we can use comparison techniques to get the existence of guided modes; see (Bonnet,
1988) in the caseof optical fibres with uniform cladding. The assumption(5.3) is in fact
weaker and we need to add (5.4) to ensurethe existenceof the fundamental mode.
The next two corollaries are consequencesof Proposition 5.2 for particular guides.
COROLLARY
5.1
Let y2be a refractive index such that (5.2) and (5.3) hold. We assume
further that
nb
= nt
IIn-II oo,IR’nb,
(53
C
(ii2
-ni)dy
> 0.
s -C
Then the guide hasat leastone guided mode, N(k) 3 1, for all k > 0.
w9
243
PROPAGATIONOFELECTROMAGNETICWAVES
Proof In the Appendix we prove that under the assumptions(5.6), (5.7) and (5.8), the fundamental mode of the planar waveguide associatedexists. We can conclude using Proposcl
ition 5.2.
COROLLARY
5.2
Let y2be a refractive index suchthat
nb
>
nt,
6%
then there exists a positive constant& suchthat
N(k) = 0
for all k -c k-*
If furthermore we assumethat (5.2) and (5.3) hold and that
(5.10)
II?‘I
- II0o.W > f’b
then there exists a positive constantz > & suchthat
N(k) > 1 for all k > k.
Proof In the Appendix we prove that under the assumption(5.9), there exists kl such that
Y(k) = -k*nE
for k < kl.
Let us prove now that N(k) = 0 for k sufficiently small. We define the refractive index z
bY
if x2 > d,
iflxzl
<4
ifxz c -d,
where the real numbersa, 6 aresuch that (-a, a) x (-6, b) 1 K and d = max(b, c). Then
clearly G(X) > n(x) for all x E R*. In fact i is the index of a planar guide with nt # nb.
So there exists -k > 0 suchthat
=
Y(k) = -k*nt
for k c & and p(k) < -k*nE
for k > k,
with obvious notation for p(k). So for v E H1 (R*) we have
s
Iw2(IVv1* - k*n*v*) dx > /
Iv
(lVv1* - k*ii*v*) dx
and for k < k and x1 E R
h [(E-Ik*E*v*]
d7c2> -k*nbLv*dx2.
By integrating we deduce that
s
~2(lVvl* - k*n*v*) dx > -k*ni /
lit*
v* dx
244
A. S. BONNET-BENDHIA
ET AL.
and then
Al(k)
> -k2nz
= y(k)
for k < -*k
This implies that N(k) = 0 for k < -k.
Furthermore if we assume (5.10), then from Proposition A.7, we deduce the existence
0
of k such that for k > k, jT(k) < -k2ni. Finally we conclude using Proposition 5.2.
In the next two results we give some kind of characterization for guides having a fundamental mode at low frequencies.
PROPOSITION
5.5
We assume the existence of a sequence {k&21
converging to 0 such
that
N(k,)
Then necessarily the following
(5.11)
> 1 for all p E N.
conditions hold
nb
IlR*(n2
-
=
(5.12)
nt,
n-2 ) dx > 0.
(5.13)
Proof The equality (5.12) is an immediate consequence of Corollary
now that (5.13) holds. From the comparison principle we deduce that
5.2. Let us check
where N N (k) is the number defined by (4.10) for the Neumann problem in 522 defined by
(4.12). From the assumption (5.11) we deduce that
NN(kp)
> 1 for all p E IV,
which gives the existence of a function tpp E H1 (Qz), 11~~/o,D3 = 1, such that
{IVvp12
-k;n2rp;}
dx
<Y(kp)
/ p;dx.
s~2
s
s
a2
Then from Y(kp)
< -king
IVq+J2 dx - k;
522
and
s 02
n2p; dx - Y(kp)
(pi dx < ki
s 522
we deducethat
J
IVp,12dx --+ 0 when p -+ 00
522
and then
1
cpp+ Ifl 3;I
in L2(&),
(n2 - ii2)t$ dx
523
(5.14)
PROPAGATION
OF ELECTROMAGNETIC
245
WAVES
where 1fi3 1denotes the measure of 523.
Finally, the definition of y(kp) implies that
I
(n2 - fi2)qi dx > 0
for all p,
523
cl
which implies (5.13) on passingto the limit.
We can prove somesort of reciprocal of Proposition 5.5.
5.6
ing hypotheses:
PROPOSITION
We assumethat the associatedplanar waveguide satisfiesthe follow= nt
nb
and
b%o,W
>
nb,
and that its fundamental mode exists at each frequency, that is,
C
-2
- ni) dy > 0.
(5.15)
(n2 - n2> dx > 0.
(5.16)
(n
s -C
Moreover we assumethat
s
IIt2
Then there exists a k- > 0 suchthat N(k) > 1 for 0 < k -c -*
k
We argue asin the proof of Proposition 5.2, by proving the existence of a function
v E H1 (R2) such that
Proof
a(k; v, v) =
s
Iw2(WI2 - k2n2v2) dx < y(k)
v2dx.
s IP
In Proposition A.4 we prove that the assumption(5.15) is necessaryand sufficient for the
existenceof the fundamental mode of the associatedplanar waveguide, at each frequency.
Therefore there exists a function (pk E Hi (R), with JiC co,dy = 2c, such that
The development then follows the proof of Proposition 5.2. Then to prove the existence
of a guided mode in the guide of index n for k small enough, it is sufficient to check that
s(n2
lFP
(5.17)
n2)i&x2) dx > 0.
We have
s lw
(n2 - ti2)&x2)
dx =
s Iv
(n2 - ii2) dx +
(n2 - fi2)(@(x2)
s IIt2
- 1) dx.
246
A. S. BONNET-BENDHIA
ETAL.
-------
----w-nb
FIG. 5. Particular
waveguide
The proof is complete if we prove that
2- ii2)(&2)
I w* (n
Since y(k)
< -k2ni
- 1) dx -+ 0
as k + 0.
we have
sw
Then d&/dy + 0 in L2(lEQ . From the normalization
in L2(-c, c) as k + 0.
of (pk we deduce that ok tends to 1
Cl
5.1 To illustrate the previous proposition we consider a three layers guide
whose shape is represented in Fig. 5, with nb, n+ E Ik, n+ > nb > 0.
We define
REMARK
K+ = (x E R2; n2(x) - fi2(x2) > 0)
and
K- = {x E R2; n2(x) - fi2&)
c 0).
If 1K+ 1, 1K- 1denote the measures of the sets K +, K-, then from Propositions 5.5 and 5.6,
we get the following. If IK+I > IK-1 then N(k) > 1 for low frequencies. If N(k) > 1 for
low frequencies then I K+ I > I K - I.
Similar results have been encountered for optical fibres (Bonnet, 1988) and in hydrodynamics (Bonnet-BenDhia & Joly, 1994).
REMARK
5.2
The assumption (5.16) in Proposition 5.6 can be relaxed to the following:
s(n2
Iv
see Proposition A.4.
ti2) dx > 0
and
sup(n2 - ii2) > 0;
(5.18)
PROPAGATION
6. Asymptotical
OF ELECTROMAGNETIC
WAVES
247
study at high frequencies
Results at high frequencies, that is, for large k, are developed now. When a guide has an
index yt whose maximum is reached only in a bounded domain, that is, yt+ > llii IIoo,R2,then
the perturbation K of the stratified medium traps the wave and the stratified medium does
not play any role for large k. The results presentedfor optical fibres or for guides with a
diopter cladding, see(Bamberger& Bonnet, 1990), can be generalized to stratified media.
In fact the number of guided modesis increasingto 00 when k is increasingand the modes
are more and more confined in the bounded set where the index is maximal.
Here our original results concern the casewhen the maximum of the index inside the
core is equal to the maximum of the index in the stratified medium.
First, upper bounds for the number of guided modesare derived. In particular we prove
that the number of guided modes with the rib waveguide remains bounded for large k.
Finally we presentthe example of a guide with no mode at high frequencies.
61
.
Upper boundsfor the number of modes
The main tool to get boundsfor the number of modesis the comparisonmethod. Here we
will useit in the following way.
6.1 Let n be an index, that is, n E Lo3(N2) satisfies(2.1) and (2.2). There
exists C suchthat for k E EQ+the following bound for the number of modesN(k) holds:
PROPOSITION
N(k) < C (y(k) + k’nt) ;
(64
we recall that here y(k), defined in (3.6), is the lower bound of the essentialspectrum of
Ak*
Proof We assumethat the perturbation K is included in the rectangle (-a, a) x (-6, b).
By Proposition 4.2, we know that
min@,N(k), Y(k)) < A, (k),
where A:(k) is the mth eigenvalue for the Neumann problem (4.8) with $2 = (-a, a) x
(-d, d), see(4.13). Moreover by the min-max formulae, it is obvious that At(k) < h:(k),
where AT(k) is the mth eigenvaluefor the Neumannproblem (4.8) with n replacedby n+
and Q = (- a, a) x (-d, d). Therefore
N(k) < sup(m E N; h;(k)
< y(k)}.
The eigenvaluesAZ(k) are given in the numbering p, 4 by the expressions
2
h.‘,N,(k)
= -k2n: + 5
9
2
2
+ $$-
2
for p, q in PJ.
W)
Then the number of eigenvaluesA’,N,(k) inside the interval [-k2n:, y(k)] is bounded by
•I
its length y(k) + k2n: times a consiantdependingon a and 6. And (6.1) is proved.
248
A.S.BONNET-BENDHIA
REMARK
6.1
ITAL.
Notice that in the proof above we have usedpart of the following result. If
N(A) =
(p, q) E w*; ($+$)~*<A}1
then
ab
N(A) - AIt
as A + 00;
see(Courant & Hilbert, 1953, p. 430).
REMARK 6.2 We can state Proposition 6.1 in other words. In fact the quantity L =
y(k) + k*nt is the length of the interval [A*&
F(k)], where we can find guided modes;
seeFig. 4. The numberof guidedmodesinsidethis interval is lessthan the numberof eigenvalues for the Neumannproblem in (-a, a) x (-b, b) with an index rt+. Theseeigenvalues
are known and their number inside the interval is proportional to L, as L tendsto 00.
COROLLARY
6.2
Let y2be an index with K c (-a, a) x (-b, b).
(i) If the index satisfies
then there exist C, ko > 0 such that for all k > ko
N(k) < Ck*.
64)
(ii) If the index satisfies
fi(Y> = n+
fi(Y> < n+
for y E (-7, rl),
otherwise,
I
(6*9
then the number of guided modesremainsbounded as k tends to 00. Moreover we
get the following estimatefor the number of guided modes,for large k
N(k) < M+
66)
(m, p) E N* x TV;
where d = max(b, q).
(iii) If the index satisfies
n+2 -
NY)*
- IYI”, a > 0, in a neighbourhoodof y = 0,
iz(y) < n* c n+ otherwise,
I
W)
then there exist C, ko > 0 suchthat for all k > ko
N(k) < Ck4/(*+@.
Proof (i) We notice that
Y(k) + k*n: < Y(k) + k27T: + k*(n: - E:),
(6.8)
PROPAGATION
OF ELECTROMAGNETIC
249
WAVES
where E+ = Ilfilloo,~. Then from Proposition A.6 we deduce that (y(k) + k2n:)/k2
is
bounded.
(ii) If we use the index comparison principle, then to prove the estimate (6.6) it is sufficient
to check that the number of guided modes for the guide of index
ifx E (-a,a)
otherwise
x (-d,d),
(63
is bounded for large k by M+. In the set 52 = ( -a, a) x R, we consider the Neumann
eigenvalueproblem
-Au - k2E2u = pu
auav = 0 on an,
inn,
(6.10)
where v is the unit outward normal vector on an. Since 5 in fact dependsonly on x2, it is
not difficult using separationof variables to check that the eigenvaluesof (6.10) are
Pmp
where pm(k) is the mth eigenvalue, when it exists, of the planar waveguide with index
ti(0, .); see(A.9). Using the comparisonprinciple with the Neumannproblem, seeProposition 4.2, to check the bound given in (6.9), it is sufficient to prove that
NN(k) < M+,
(6.11)
where NN (k) is the number of eigenvaluesof (6.10) below the essentialspectrum of &
(and also of Ak), that is,
RN(k)
=
I{<m,
P> E I+?* X N;
Pm,,
<
v(k)}1
l
From Proposition A.8 we get
y(k)
Yr2
ask tends to oo
+ k2Fit + 4v2
m2n2
p2rt2
PmO+k2E~+~
+
p2YT2
ask tendsto 00.
-Q-+3
Now if l/q2 < m2/d2 + p2/a2, then fork large enough
2
T(k)
<
Pm(k)
+
$$
2
--
Pn1,p
and so (6.11) holds.
(iii) We can adapt the proof of Proposition A.6 by setting y, = 0 and pE = k-2/((r+2),
prove that under the assumptions(6.7) the term y(k) + k2n: is estimatedby
y(k)
+ k2n:
to
< Ck4/(2+(r);
seealso (Bonnet-BenDhia & Djellouli, 1994). Therefore (6.8) holds.
cl
250
A.S.BONNET-BENDHIA
ITAL.
REMARK 6.3 From a physical point of view we can understand why the number of
guided modes N(k) remains bounded for some stratified guides while it tends to infinity
for an optical fibre.
We have seen in Remark 3.2 that the velocity of a guided mode for a guide of index rt
satisfies
co/n+ < 1) < w
while the velocity of a guided mode for an optical fibre with an homogeneous cladding of
index ~26satisfies
In the case of the fibre the interval where u lies is independent of the frequency. When k
tends to infinity the size of the core becomes larger compared to the wavelength 2n/k and
the number of guided modes tends to 00.
In contrast under the assumptions in (5.5) the length of the interval
u(k) - 2 +O
n+
fork+oo;
see Proposition A.6. The difference between the velocity in the core and the one in the
cladding tends to 0; this balances the fact that the size of the core is getting larger and
larger compared to the wavelength.
6.2
No mode at high frequencies
Here we describe a guide with no mode at high frequencies. The idea is to choose a
guide for which the set where n achieves its maximum is strictly included inside the strip
IR x C-79 rl)PROPOSITION 6.3
We assume that
n(x) < n+
for almost all x2 4 (-q,
q)
(6.12)
and that
n(y) = n+
4x>
if
= yc(x2)
n(x) = n+
n(x) < n* -= n+
IYI < %
if 1x11> a,
if 1x1I < a and 1x21< t -Cq,
if 1x1I c a and 1x21> t.
(6.13)
Then there exists k, > 0 suchthat
fork > k,
N(k) = 0.
(6.14)
Proof We use the sameargumentsasin the proof of Corollary 6.2. We have
N(k) < AN(k) = {(m, p) E TV*x IV; j&,(k) + p2n2/4a2 < y(k)},
where j&(k) + p2n2/4a2 are the eigenvaluesof a Neumann eigenproblem identical to
(6.10).
PROPAGATION
OF ELECTROMAGNETIC
WAVES
251
n*
nb
-------
-e---e---.
-77
a
-a
3
-----w-w-.
FIG. 6. Guide with no mode at high frequencies
From Proposition A.& we get the following behaviour of j&(k) andy(k), for k + 00,
j&(k) + k2n$ + 7r2/4t2
and y(k) + k2nt + sr2/4q2.
We conclude that for k large enough N(k) = 0.
cl
EXAMPLE 6.1 Let n be the index defined in Fig. 6.
We assumethat n+ > n* > ?‘&and for simplicity that b - Q = 2a. The first eigenvalue
,z for -A in the square (-a, a) x (q, q + 2a) with homogeneousDirichlet boundary
conditions is p = n2/2a2. If p + 0 is a correspondingeigenvector, we extend it by 0 and
get @E H’ (8X2)satisfying
JR2(lW2
- k2n2q2) dx
x2 _ k2n2
JR&&
= 2a2
*i(’
For all k, -k2n$ < y(k) < -k2&
to have
For a given k, we could choosen* and a big enough
-k2nz + n2/2a2 -c y(k),
with n+ > n* > nb. From the comparisonprinciple we deduce hi (k) -Cy(k). So for that
particular value of k we have a guided mode at least. For large k, we will have no guided
modes.
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A., DERMENJIAN,
Y. & JOLY, P. 1990Mathematicalanalysisof the propagationof
elasticguidedwavesin heterogeneous
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BAMBERGER,
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H. J. M., BAKEN, N. H. G. & BLOK, H. 1992Domain-integral
analysisof channel
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in anisotropicmultilayeredmedia.IEEE Trans. Microwave Theory Techn. 40.
BONNET, A. S. 1988Analyse mathematique
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ETAL.
A. S. & DJELLOULI,
R. 1988 Etude mathematique des modes guides d’une fibre optique,
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A. S. & DJELLOULI,
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A. S. & MAHI?, F. 1994 A guided mode in the range of the radiation modes
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D. 1982 Light Transmission Optics, 2nd edition. Electrical/Computer
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B. M. A. & DAVIES, J. B. 1984 Finite-element analysis of optical and microwave waveguide problems. IEEE Trans. Microwave Theory Techn. 32.
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H. P. 1996 Analysis of the domain integral operator for anisotropic dielectric waveguides.
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C. 1985 Theorie des Guides d’Ondes Electromagnetiques, Tomes 1 et 2. Editions Eyrolles.
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R. 1988 Absence of eigenvalues of the acoustic propagator in deformed wave guides. Rockj
iMountain J. Math. l&495-503.
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R. 1991 Spectral and Scattering Theory for Wave Propagation in Perturbed Stratified
A4edia. Applied Mathematical Sciences 50. Berlin: Springer.
WILCOX,
C. H. 1984 Sound Propagation in Strutijied Fluids, Applied Mathematical Sciences 50.
Berlin: Springer.
WITSCH,
K. J. 1990 Examples of embedded eigenvalues for the Dirichlet Laplacian in perturbed
waveguides. A4ath. Meth. Appl. Sci. 12,91-93.
BONNET,
A. Appendix A: The planar waveguide
Several times within the paper we needed results for the stratified medium associated with the guide.
A planar waveguide associated to a guide with refractive index n E L”“(R*) satisfying (2.1) and
PROPAGA-TION
OF ELECTROMAGNETIC
253
WAVES
(2.2), is characterized by the refractive index fi E Loo(Iw2) such that
ii(y) =
Itt
i nb
if y > c,
ify<-c,
inf Isi(y) > 0
.vEIw
(A. 1)
for some positive c. Let F+, vt- denote the supremum of n, the infimum of n respectively
z+ = Iliill,,~,
vt- = Ilfi-‘l~;ol~.
(A2
In a planar waveguide the guided modes can be either transverse electric (TE, the electric field is in a
plane transverse to the propagation direction) or transverse magnetic (TM). The problem to find TE
guided modes of a planar guide is the following: determine the real numbers p, k, and the functions
u E H2(Iw), u $0, such that
d2u
-dy2 - k2A4 = _B2u
in Iw.
(A3
The associated mode propagates with the velocity u = kc&Y. We refer to (Marcuse, 1974) for a
presentation of the model and do not emphasize on a similar study of TM modes.
Remark that the scalar equation (A.3) is derived directly from the Maxwell system without any
assumption on the refractive index, in contrast to the two-dimensional case where the scalar model
is an approximation.
Like in the general case we consider k as a parameter. Then for a given non-negative value of k,
the variational problem consists in finding B E IF and u E H’ (R) such that
b(k; u, v) = -p2(u,
zQo,~ for all v E H’(R),
(A4
where the bilinear form b(k; ., .) : H’ (R) x H’ (W) + R is given for u, v E H’ (Iit) by
1
b(k; u, v) =
dy.
Problem (A.4) can be written in the equivalent spectral form: find /3 E IK+ and u E H2(Iw) such that
where the unbounded operator Bk : D( Bk) = H2(W) + L2(rW) is defined for u E H2(IK) by
Bku = --
d2u
- k2n2u.
dy2
REMARK
A. 1 We insist on the fact that a mode of a planar waveguide is not a guided mode in the
sense presented in Section 2 because here a mode u is only depending on one variable, u = u(y) E
H2(R).
Actually the study of the associated planar waveguide shows how the stratified medium confines
the lightwave in the vertical direction.
Although the study follows the same steps as for the general problem, it is much easier here.
The analogue of Propositions 3.1 and 3.2 holds for the bilinear form b(k; ., .) and its associated
unbounded operator Bk. The bilinear form b(k; ., .) is continuous, symmetric and bounded from
below
b(k; u, u) > -k2Fi$ll~R t for all u E H’(W).
The corresponding
operator Bk is bounded from below, selfadjoint, and its spectrum CY(Bk) satisfies
a(Bk) = [-k2E;,
00).
(A@
To study the spectrum a(Bk) we use a method similar to the one to study the spectrum of Ak in
Section 3. It consists of the essential spectrum Us,,(Bk) and of the point spectrum a,(Bkh
PROPOSITION
A. 1
The essential spectrum of Bk is the interval [-k2n&
%s(&)
= [--k2&
@.
oo),
254
A. S. BONNET-BENDHIA
ETAL.
We first check that a,,,(&)
> [-k2n& co) by building
Bk - hl. For h > -k2ni we define the sequence (vJ~~~ by
Proof
a singular sequence associated to
witha = J h + k2ni and p E Coo (IQ with compact support in (--00, -c). Let us check now that the
sequence of up = vP/llvP 110,~is a singular sequence associated to Bk - hl. With some calculation
we can show that
0 < B1 < llvpIlO,~
II mp
-~~,I~o,IR-+
< i32 forall
0
asp+
p,
00,
and that for all v E L2(R)
(vp,
--+ 0
V)O,IR
as p -+ 00.
This means that (uJP~i is a singular sequence associated to Bk - AZ.
Let us now check the inclusion oess(Bk) c [-k2& 00). To any h E oess(Bk) we can associate a
singular sequence (u&>i
C H2(IR). In particular
from which we deduce that
h > ~~~[-k2nt-S_:k2(n2-n:)ui,dy].
lim
Since the sequence (u~)~>~ is bounded in H’ (R), up converges to 0 in L2(-c,
h >/ -k2n2
c) and consequently
bw
We turn our attention now to the point spectrum of Bk, namely c+,( Bk).
PROPOSITION A.2
The following
assertions hold:
(1) ifE+ = n,$ then Bk has no eigenvalue;
(2) if 7i + > nb then 0 P (Bk) C (-k2ii2 +, -k2ni).
Proof
From the relation (A.6), we know that a,(Bk) is in [-k2Et, 00). Let us check that -k2iTt
cannot be an eigenvalue. If -k”Tz’t is an eigenvalue, this would imply the existence of a function
~0 E H2(R) not identically 0 such that
We deduce then that p is identically 0. So
up(BI;)
Y'
Let h be an eigenvalue of Bk, h > -k2n&
-c, p satisfies
-v
(-k2Fi;,
c
oo).
and q E H2(R) be a corresponding
d2v =
eigenvector. For
(A + k2n&,
so q has the form in (-co, -c
P(Y)
with a! = J h + k2ni.
= fS1 cos(ay)
Since p s in L2(lR),
+
62sin(ay)
it follows that (o = 0 in (-co,
-c). We then deduce from
PROPAGATION
OF ELECTROMAGNETIC
255
WAVES
the uniqueness of initial value problems that ye = 0 in Iw. We argue in a same way when a! = 0. This
means that
q,(Bk) c (-k*ii&
-k*nE).
cl
REMARK
A.2
If /? E W+ and u E H*(R)
are a solution of (A.3), then
-k*Tii
< -#I* < -k*ni.
So the velocity of the guided wave (for the planar waveguide) satisfies
that is, the guided wave propagates faster than in a medium of index Ti-+ and slower than in a medium
of index nb.
Since our goal is to prove the existence of eigenvalues for Bk, from now on we shall assume that
z+ > nb.
(A*V
In the case of a planar waveguide all the eigenvalues are below the essential spectrum and are
characterized by the min-max principle. This was not the case in Section 3, see Remark 3.3.
We define the numbers ‘ym(k), m 2 1, by
and
YnlW =
(Bkv,
v)O,w.
inf
&I ~HrnW2(@) ~EH,
sup (v, v)o,Iw ’
u#O
(A*%
here H,(H*@))
is the set of all m-dimensional subspaces of ZY*(IEX).From Proposition A.1 and
the min-max principle given in Propostion 4.1, we know that ym(k) is the mth eigenvalue of Bk if
y&k) c -k*n$
PROPOSITION
A.3
The number of eigenvalues of Bk is non-decreasing in k.
Proof It is sufficient to check that the function k E Iit+ t+ ym(k) + k*ni is decreasing. Given a
v E H*(IIZ), the function
k E Iw+ t+ min
0,
(Bkv,
vj0.R
+ k*n;(v,
(v9 V)o,R
is decreasing. With the min-max characterization
v)O,&t
>
of ym(k) the result follows.
When the planar waveguide satisfies
nb = nt, then we have a simple necessary
condition for the existence of at least one guided mode for any value of k.
PROPOSITION
A.4
sufficient
We assume that (A.7) holds and that
nb
Then y1 (k) < -k*n&
only if
cl
= nt.
(A.10)
that is, the fundamental mode for the planar guide exists, for any k > 0 if and
C
(ii2 - ni) dy > 0.
s -C
Proof
We first assume that
C
s -C
(ii2 - ni) dy > 0.
(A.1 1)
256
A. S. BONNET-BENDHIA
ETA,!,.
For a given k > 0,
YlW
-k2ni
=
+
(BRv,
inf
UEH2(lR)
+ k2n&
h v)o,w
V)O,R
V)O,IR
.
We define the function vR by
1
if Iyl < c,
IYI
vR(y)
=
0”
- R
-R
if c < Iyl -C R,
if lyl 2 R.
Then
b(k;
VR, VR) + k2n;(VR,
c
2
v&R
= R -
- k2
s -c
(ii2 - n;) dy < 0
for R sufficiently large. Since H2(R) is dense in H’ (IQ, we conclude that yi (k) < -k2ni. When
(A.1 1) holds with the equality we choose a test function ?.uR= vR + aw, where w f H’ (W) satisfies
C
C
(ii2 - ni)w dy = 0
and
s -C
Then R and a can be chosen such that
W;
WR,
WI?)
+
s -C
(E2 - nz)w2dy
k2Q2(WR,
wZ?)O,~
<
> 0.
0.
So the condition (A.1 1) is sufficient.
Let us check that (A.1 1) is necessary. If the fundamental mode exists for any k, then we can
consider a sequence (kp},+
converging to 0 and another sequence (v~}~>~ c H1 (IQ such that
(A.12)
and
C
v;dy = 1.
s -C
Thus the sequence (du,/dy),~
converges to 0 in L2(R). By compactness we deduce that the sequence (v&,~i converges to a non-zero constant in L2(-c, c). Then from (A. 12) we conclude that
0
(A. 11) holds.
PROPOSITION
A.5
We assume that (A.7) holds and that
nb > nt.
(A. 13)
Then there exists kl > 0 such that
Yl(k)=-k2ni
forO<k<ki,
(A. 14)
which means that Bk has no eigenvalues for 0 < k < kl.
Proof Ab absurd0 suppose that there exists a positive sequence (k&+
tending to 0 such that
yi (k,) < -k2ni. Let (JJ~)~~~ be a sequence of eigenvectors, vP E H2(rW) the eigenvector associated
to y1 (k,) an 8 normalized by
C
v;dy = 1.
s -C
Then
PROPAGATION
OF ELECTROMAGNETIC
WAVES
257
and with (A.13)
Consequently (du,/dy)pa
tends to 0 in L2(IR) and (vp)p>r is bounded in H1 (-c, c@. This implies
that (“Jeer converges to 0 in L2(-c,
oo), which is a contradiction with the normalization equation.
cl
We turn our attention now on to the number of guided modes as a function of the frequency k.
A. 6
PROPOSITION
The following
assertions hold for any m 2 1:
the function k E I%+ + Ynt(k)
lim
k-+oo
Proof
YmCk)
--
k2
-
+ k2Ti: is increasing
-2
(A.16)
n+*
By the definition of Z+, for any non-zero v E H2(R),
(Bkv,
kER+t-+
V)O,R
(A. 15)
+
the function
k2z,(v,
V)O,R
(v9 V)o,R
is non-negative increasing in k. Going back to the definition of ym(k) in (A@, (A.9), we conclude
that (A. 15) holds.
Let us check now the second assertion, (A. 16). For all 6 > 0, there exists an interval B(y,, p,)
such that
1
(Et - n(y>2> dy < 6.
-1
PC NYC*PC>
Then for a given E, we have
ym (k)
+ k’iit
<
SUP
VEH,,,
v#O
s
(dvldY>2
dY + k211vl12,
B(y,,p,)
Jqv
B(y”pc)
s R(,.s)
-cr p c )
(q
-
ii(Y12)
dY
v2 dy
where Hm is the space generated by the m functions
for y in B(y,,
u,,(y) =sin((y-y,)E)
whenpiseven,
z~,(y)=cos((y-yr)E)
whenpisodd,
p,) and extended by 0 outside. Then
m2x2
1
4k2p,2
i’c s B(Y,,Pc)
-+-
(Ti: -
fi2(Y>) dY.
We can conclude with (A.17) by choosing k big enough.
From Proposition
PROPOSITION
A.7
(A. 17)
0
A.6 we immediately deduce the following.
We assume that (A.7) holds. For any m >/ 1 there exists km such that
Yrn(k)=-
k2n2 )j
for all k <1 k m
and
ym (k) < -k2ni
for all k > k,,,.
For m > 1, the number km in Proposition A.7 is called the mth cutoff value.
Under some further assumptions on fi the results in Proposition A.6 can be made precise in the
following way.
258
A. S. BONNET-BENDHIA
PROPOSITION
A.8
ETAL.
We assume that (A.7) holds and that there exists 0 < q c c with
if y E [--t), ~1,
(A. 18)
otherwise.
Then
y,,,(k)
v,(k)
Proof
+ k2iii
-I- k2$
+
(A.19)
< m2n2/4q2
as k + 00.
m2n2/4v2
(A.20)
Working on the ball B(0, 77) we obtain the analogue of (A. 17):
m2n2
y,n(k) + k2E, < -
k2
q
@: - fi2(y>> dy,
-I- r7 s -rj
G2
from which we deduce (A.19) since ut+ = n(y) for y E [-r), Q].
The function k E R I+ y&k) + k2ii: is increasing and bounded. Let cc, be its limit. For any
k > k, we can associate to y&k) an eigenfunction u,(k) such that
W; u,(k),
u> = ym(Wm(k),
U)O,R forall u E H’(W
(A.2 1)
=
(A.22)
and
IIUm(k)llO,W
1.
From the relation (A.21), we deduce that
durn(k)
IIdy
k2
s IR
(E$
Ili,OgG Pm
-fi(Y)2)U~Ck)(Y)dY
<
(A.23)
Pm-
a subsequence (um (k&, 21 converging
has
BY compactness the family Iurn(k)lk>km
L2( -c, c). Now from (A.23) we also get that um has a support in [--r), ~1 and
t0
21,
in
rl
um(y)2dy = 1.
-rl
Then from the relation (A.21), we obtain that um is a non-zero solution of
s
-- d2um --PmUm
dy2
um(wr))
in
=
U,(q)
C--q,
~0,
=09
which implies that
m2n2
.e2n2
and there exists a e with pm = -.
Q2
Q2
sop1 = n2/4q2. We have no difficulty in checking that e = m with the min-max principle.
Pm
G
-
cl
REMARK
A.3 It is not difficult to generalize Proposition A.8 when we replace the assumption
(A. 18) by the following
ii(y) = 2+ ify E b1, a21 U b3, a41,
otherwise,
NY> -= 6
with -c -C al < a2 -c a3 < a4 < c. Then in (A.19) and (A.20) q is replaced by
r7*
= max(a2 - al, a4 - a3).
PROPAGATION
OF ELECTROMAGNETIC
259
WAVES
We end the Appendix by getting upper and lower bounds for the eigenvalues of Bk in terms of
eigenvalues of operators with compact resolvent. Let ti E Loo be the refractive index of a planar
waveguide satisfying (A. 1) and let d be bigger than c. To ii and ‘& we associate the two operators Bf
and Bt defined by
D(BF>=
D(BT)
H2(-d,
= (u E H2(-d,
d) n H;(-d,
d); u’(-d)
B,Du = -d2U/dy2
d),
= u’(d) = 0),
- k2i’i2u
B;u = -d2u/dy2
(A.24)
- k2fi2u.
(A.25)
Then we define the numbers for m 2 1
(A.26)
Y:(k) =
(B,NV,U)O,(-d,d)
inf
f&n~HrnW* C--W) ve~m
sup (v9 V)O,(-d,d) ’
v#O
(A.27)
PROPOSITION
A.9 Each of the operators BF and BF has an increasing sequence tending to 00 of
eigenvalues characterized by the relations (A.26) and (A.27) respectively.
For m 2 1, the following bounds hold:
min(y,NW, -k2n;) < ym(k) < v,“(k),
(A.28)
where v,(k) are defined in (A.8) and (A.9). Moreover when the fundamental mode exists, that is,
YlW
< -k2n2 b, then we have
Yl” (k)
(A.29)
< Yl (0
Proof The first part of the result is deduced from the compactness of the Sobolev embedding
H2 (-d, d) c H1 (-d, d) and the min-max principle.
To check the bound (A.28), we argue as in Proposition 4.2.
Let us check the bound (A.29). The assumption y1 (k) < -k2ni means that the fundamental mode
exists. So let pk be an eigenfunction corresponding to the eigenvalue yl(k) of Bk and +Qk be an
eigenfunction corresponding to the eigenvalue yIN (k) of B/.
For y E (-00, -d] U [d, oo), we have a formula for pk(y), indeed
(ok(y) = (Pk(-d)eKb(y+d),
pk(y) = (Pk(d)e-Kf’y-d’,
Kr, = J -yl(k)
Kt = J -yl(k)
- k2n&
- k2nf,
for y E (-00, -d],
for y E [d, CQ).
Then we deduce that
$p
-
4 = K&+(-d)
and
&A.
-(d)
dY
= -K&(d)
and pk satisfies
on
i-4
4,
(A.30)
-44
= K&Q(d).
260
A. S. BONNET-BENDHIA
The following
ETAL.
expression holds for yr (k):
yl(k) =
fd-d ((9I2’ -
k2fi2&) dy + /c&(-d)
+ I;
.
s-“d spk2
dY
Using the min-max
principle to characterize y:(k)
Yl w
2 Ylw
+
we get
KbP+o + Kt&o
s -“d
(ok2d Y
l
Consequently to have y1 (k) = y:(k), it is necessary that qk (-d) = pk(d) = 0. Since the function
pk satisfies (A.30), that implies pk = 0, Therefore we can conclude that (A.29) holds.
Cl
EXAMPLEA. 1 The above results can be illustrated in the case of the three layers planar waveguide, see Fig. 2, with the refractive index
4
nc
rt(Y) =
if y > c,
if-c<y<c,
if y < -c,
i ‘nb
where n, > nb 2 nt > 0. It is a simple matter to check that
= -k2nz +
Y;(k)
(m - 1)2n2
d-c2
and
To compute ‘ym(k), we solve the equation
--
d2v
- k2n2v = hv
dY2
successively in the intervals (--00, -c), (-c, c), and (c, oo), and impose the continuity of v and
its derivative at y = -c, y = c. Then the eigenvalues v,(k) are the solutions h of the dispersion
equation
tan(2c
(A.3 1)
(A + k2n:) - ,/x4-
with -k2n:
< h < -k2ni.
’
The dispersion equation (A.31) can be written in the form
tan x = F(x)
withx
= 2c,/m
and
F(x) x(J?z+ J&q
x2- J~JqTs
here VI = 2kc,/s,
(A.32)
q2 = 2kc,/s
;
The function F has a pole at the point
PROPAGATION
OF ELECTROMAGNETIC
WAVES
261
and is decreasing on both intervals [0, xP) and (x,, ~11. So there is a solution to (A.32) as soon as
either
In each interval (mn, (m + 1)~) there is at most one solution. To the solution x E (mn, (m + 1)x)
corresponds an eigenvalue ~~+~(k) = (x/~c)~ - k2nz with
m27r2
(m + 1j2n2
< Yfn+lW
< -k2f(
+
4c2
’
4c2
For small values of k the relation (A.33) does not hold. The lowest value for which (A.33) holds is
the first cutoff value and is given by
-k2n2 C +
kl = arctan (44)
2c $3.