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Description of Pulse Propagation in Optical Strip Waveguides by the Generalized Nonlinear Schrdinger Equation.

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Annalen der Physik. 7. Polge, Band 44,Heft ?, 1987, S. 120-1'26
J. A. Barth, k i p z i g
Description of Pulse Propagation in Optical Strip
Waveguides by the Generalized Nonlinear Schrodinger
Equation
I3y H. EICHHORN
Sektion Physik der Friedrich-Schiller-UniversitatJena, GDR
A b s t r a c t . Taking into consideration the transverse confinement, the dispersion of the linear
as well as of the nonlinear part of the refractive index and the nonlinear interaction of the electrom.ignetic field with the guiding material, the yielding equation for the envelope function of the fundamental mode propagating in a strip wawguide is the generalized nonlinear Schrodinger equation.
In contrast to the nonlinear Schrodinger equation, solitons of the generalized nonlinear Schrodinger
equcttion exist in t h e regimes of negative and positive group dispersion and are asymmetric.
Beschreibung der Impulsausbreitung in optischen Streifenleitern
mittels der verallgemeinertennichtlinearen Schrodingergleichung
I n h a l t s u b e r s i c h t . Unter Beriicksichtigung der transversalen Inhomogenitat, der Dispersion
sowohl des linearen als auch des nichtlinearen Anteilcs des Brechungsindexes und der nichtlinearen
Wechselwirkung des elektromagnetischm Feldes mit den1 leitenden Material e r h d t man die vernllgeineinerte nichtlineare Schrodingergleiehung nls Gleichung fur die Einhullende der Grundmode, die
sich in einem Streifenleiter ausbreitet . Im Gegensatz zur nichtlinearen Schrodingergleichung findet
man Solitonen der verallgemeinerten nichtlinearen Schrodingergleichung in den Bereichrn mit negat,iver und positiver Gruppendispersion, sie sind da.riiber hinilus asymmetrisch.
I . Introduction
To compensate the influence of group dispersion on the wave propagation in optical
fibers, Hasegawa and Tappert [ 11 proposed to use a field dependent refractive index,
72. ==
n("
+ )&'VL) p j 2 .
Seglecting the tlispersion of 7t,('NL), they found solitons as solutions of the iiorilinear
Schrodinger eqiiation ( N L S )in the regime of negative group dispersion, i.e. t?k/6W2 < 0,
k = C W ( ~ ) / C . This prediction has been verified by Mollenauer et al. [2]. Hasegawa/
Tappert took no detailed account of transverse inhomogeneity in their analysis Subsecjuently, ,Jaia/Tzoar [31 and C'rosigtiani/YalJas/Di~orto[4] have introduced approaches
which accounted for this effect in average fashions. In this paper, we use a perturbation
procedure which makes plain the close connection betn een the physical effects arising
from transverse confinement, t h e tionliiiear interaction of the electromagnetic field with
the guiding inaterial ant1 the dispersion not only of dL)but of d X Ltoo.
) The equation
for the envelope frinctiori of the funrlaniental mode propagating in a strip wrtveguide
results to be the generalized NLS ( G N L S ) ,
H. EWHHORS.
Pulse Propagation in Optical Strip Waveguides
121
The N-fold Eacldr~ndtransformation and, :ts a special case, the N-soliton solution
of eq. (L) are olitained in a subsequent paper [ 5 ] . We find that -- in constrast to the
NLS - solitons of (1)exisb not only in the regime of negativo group dispersioii but i n
the regiine of positive dispersion too and are asymmetrical. By the may, the initial
value problem fov the GNLS has been solved [(;I and the damping of the 1-soliton solution of (1)has heell calculated too [7].
2. General Formulation
The waveguides used in integrated optics are usually rectangular strips o i dielectric
material that are embedded in other dielectrics. Following [81, we analyse a structure that
is more general. The rectangular core consisting of a week dispersive weak nonlinear
material is surroiindetl by weak dispersive weak nonlinear materials different on all
four sides of it.
Fig. 1. Schematic of the five dielectric regiom of a rectangular dielectric wiveguide [S]
The theoretical description starts from the Maxwell equations and the following constitutive equations providing isotrope materials
B = lUoH,D = D(" f D(ATh)
(2)
-00
DiL)ir,t ) = E,, J
at d L ) ( r tt)
; En(r,t - t),
-m
-w
DpvL)(r,t ) =
-w
-52
dt, ... J dz, { q ( r t ;tl,tz,t,)Ek(r,t - t l ) E l ( r ,t
--m
x E ? ( t~, t,)
-T
+ z 2 ( r ltl,
,
t3)E3(r,t - q ) E k ( r t, - re)
+ x3(rt,r1,r2,t3d3)El(r,t t2,
X EJ(r9
t - t3)
~ )
(:3)
tl)
x E3(rj1 - r2)-%(r,t - .,I>.
I n (3), j, k = .c, !J, ;, r = (.c, y, i ) , rt = (.r, y), x , ==
x, = d&,
x3 =
o\ er repeating indices is sumed up. Keoaiim dispersion arid nonlinearity are assumed to
he weak, the derivation from the plane wave solution is small. Tlicreforc. the ansatz
for the electroniagnetic field reads as
&Lu,
1 .
Ek(r,t ) = --Ek(r, t ) ei(@z-md) -b c.c., HL(r,t ) = kk.r, t)ci(@z-Wo')
2
with
a
+ c.c.,
-
,?&, HL slowly var j ing in the longitudinal coorclinate z arid the time t. Transforming
t o k'oiirier space,
,6iL)(r,t ) ,
DiL) =
1
fiiL)e s p { i ( p z - coot)) + c.c.,
2
can he ~ r i t t e n
Ann. Physik Leipzig 44 (1987)2
122
in t,he form
1- do
t) :
E0
ALL)(.,
J
dw x ( L ) ( r t ,coo
-cc
+ m)&.(r, co)e-iml.
If the carrier frequency oj0 and the pulse-width are choosen in such a way that t,he
of dL)(
lat, coo
a)) is weak in the range where ik(c0
is)different from
dependence 011 zero then n =
can be developed into a Taylor series,
-+
idL)
+ m ) = no + n; w + T1n y m z + ....
)L(CO~
The primes indicate derivatives with respect to m, and the subscript 0 indicates evaluation a t coo. Hence,
-hiG)
=:
co{rti 4-
- [(n;)'
+ n07i:]8F
f ...} z k .
(-1)
The expression for DLNL)can be simplified in a simular manner. Using the rotating
wave apprulimation and dei-eloping xi into a Taylor series up to the term linear in
(0,
1 we obtain for DiNL),DLKL)= -D(NL)esp { - h o t }
2
axi 0
+ c.i.
Heie, xi,o = xi (coo, --coo, (o0), - is the derivation of
'mi
coi = 0, j = 1, ...,
+
xi (wo
+ ml,-(mo + a)'),
wo
m3) taken in
3, with respect to its arguments. An asterisk
indicates complex conjugation As mentioned, we restrict our consideration to conditions
where tho nonlinearity and dispersion are weak, i.e.
I ip"'
I/pjp)I = O(&),
-?&a,
1
-
= O(e), 1 n;;$ = O(&).
(6)
n0
n0
The nonlinearity is choosen to be of the same order of magnitude as the dispersion in
order to compensate the broadening of the pulse during propagation.
The int,roduced smalness parameter E is given by the quotient. of the frequencies of
the slowly varying envelope and of the carrier wave,
0;'
at
=;;
O(E), jS-'
a,
=:
O(E).
(7)
H. EICHHORX,
Pulse Propaga,tion in Optical Strip Waveguides
If we restriet
as
o w coiisiduration
W
Ek
=
to the i-nonomode case, the ansata for E.H reads
A
2 Bin)k ,
W
€It =
123
iiow
A
2 H f ) E”,
/I: = :L.,?/,
2.
(8)
n=O
n=O
3. Perturbation Solution for t.he Pulse
M’e obt,ain from the Maswell equations, the constitutive equations (2, 9) arid the
perturbation ansatz (8) in the zeroth order of E the transverse field components in terms
of the longitudinal components,
and the wave equat,ion for f
{a:
+ 2; + (k;
--
=
Eio),H;”),
p z ) } f = 0.
We have introduced the abbrel-iation
-
ko = k(rt, wo) = l/pozocoon(rt, coo).
An esact analytic treatment is not possible. Following an approximate approach
given in [8] which works for moclcs far from cutoff w0 ignore the shaded regions of Fig. 1
Ann. Physik Leipzig 44 (1987) 2
124
complctelv. Using the freeclnin in the acljustinent of the amplitude coefficients
&O) SO that i1.is”)
:
0 v e obtnin (cf. [ S ] )
cos(x*.r
cos(xrr
cos(x,.r
nith ( r = 2, 3 , s
=
+ @)cos(xuy+ a)
region 1 ,
+ p) cos 6
+- @) cos(lt,d + 8 )
region -4,
region 5,
e--Y5(v--d)
4,3 ni~mbersthe regions, cf. Pig. 1)
/yo= x.; + 7$ + 8 2 ,
0
ey4g
kiU),
== -@?E2
i&
=
, x,k. - @FE -
-rf
@rE
+ x; +
= TZ, tg
p2,
?go= lt:
@rE ~,Ix,,
- ys”
+
132,
=
the niiinhers v, p are integer. I n the linear theory of naveguiding, the amplitude A is
real and constant and is connected with the power transported by the corresponding
mode. Including the first order systems of the perturbation approach, d is a complex
function which is sloivly varying in 2 , t . The determining equation can be found in CkJSe
analogy to the 0(1) analgsis. I n the monomode case, we obtain for the trans~erse
components the expressions
H. EICHHORK,
Pulse Propagation in Optical Strip Waveguides
1%
Within the far from cutoff approximation (cf. [ S ] ) the nonlinear term in (LO) can be
siniplified,
&NL)(O)
+P (ia D(NL)(O)
+ jaY$ ( YN L ) ( o )
I%; *
3 x4,0 W , Wi 1 A l 2
biNL)(0)4
A
- p@'h)(o))
P
k..'
0
3 -WnW;;at ( I A
+4 awl
3x40
S
iF,f$SL)((J),
1=.L-,?/
2
A),
*r,y.
+
Here, itl = x1 ix2 xg. The introduced functions Wk(x,y), k = .c, y, 3 , are related to
@
, )
= AWL. The amplitude coefficient
= U.
can be calculated form
To obtain the equation determining A(z,t ) we multiply (10) by W,*a n d iritegrate over
the cross sectiorr of' the waveguide. Because
kko)by
Loo
j
-w
+m
ax
j-
dy
w:
{a:
$-
a;
+ (k; - p),)k p
-w
we obtain
with the real constants
(In a more extensive description, &) can be represented as a superposition of w-aveguide
modes. T t results eq. (1)for A again, if we make use of the orthogonality of the modes.)
Eq. (11) can be written in the standard form of the G N L S ,
if we introduce the variables
The upper sign in (12) corresponds to 71;' < 0, the lower one to nh' > 0 . K e remark
that the appearance of the term ib ( l q /2q),xin eq. (12) is justified only if
i.e. for materials with a remarkable frequency dependence of x , and/or for sriitabie
choices of the frequencies of the slowly varying envelope and the carrier wave. The
Ann. Physik Leipzig 44 (1987) 2
126
GNLS admits soliton solutions. I n particular, the 1-soliton solution is [5, 61
yiX, T )=
+
+
+
9
I/[x(l
b t ) e-v
eYl2 (by e-y)8
where y = 2y(X - X , ) - 8fqT. It contains the free real parameters &connected with
the velocity of the soliton, q-determining its amplitude and X,, &corresponding to
its phase. For b = 0, we obt,ain the 1-soliton solution of the NLS,
+
q ( X , T ) = 2q sech y exp i(2EX - 4(E2 - y2)T S},
if we choose x = +l. We coiiclude that - in contrast to the N L S - solitons of the
GNLS exist not only in the regime of negative group dispersion (x = +1) but in the
regime of positive dispersion ( X = -1) too. Further, the solitons of theGNLS are asymmetrical, whereas that one of the NLS are symmetrical.
Let us finally remark that the effect of transverse confinement on a pulse propagating in a strip waveguide is contained implicitly in t.he coefficients of the equation
for the envelope function. Therefore, the corresponding equation for fibers is expected
to be the GNLS too with different coefficients only.
References
[l]HASEGAWA,
A.; TAPPERT,
F.: Appl. Phys. Lett. 23 (1973) 1@.
MOLLENAUER,
L. F.; STOLEN,
R. H.; GORDON,J. P.: Phys. Rev. Lett. 46 (1980) 109.5.
JAIN,M.; Tzoan, N.: J. Appl. Phys. 49 (1978) 4649.
CROSIGNANI,
B.; PAPAS,
C. H.; DIPORTO,
P.: Opt. Lett. 6 (1981) 61.
EICHHORN,
H.: Ann. Phys. (Leipzig) 44 (1987).
[Gj EICRRORN,
H. : Application of the inverse scattering method to the generalized nonlinear Schrodinger equation, Inverse Problems 1 (1985) 193.
“71 EICHHORN,
H. : Damping of solutions of the generalized nonlinear Schrodinger equation (1984),
will be published.
[8] MARCUSE,D. : Theory of Dielectric Optical Waveguides. New York/London : Academic Press
1974.
[2]
[3]
[4]
[5]
Bei der Redaktion eingegangen am 6. Juni 1985.
Anschr. d. Verf.: H. EICHHORN
Sektion Physik der
Friedrich-Schiller-UniversitatJena
;Max-Wien-Platz1
Jena
DDR-6900
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