The nonlinear directional coupler: an analytic solution - CCM

Abstract
The nonlinear directional coupler: an analytic solution
R. Vilela Mendes *
Universidade Tecnica de Lisboa and Grupo de Fısica-Matematica, Complexo Interdisciplinar,
Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal
Received 13 November 2003; received in revised form 19 December 2003; accepted 21 December 2003
Linear and nonlinear directional couplers are currently used in fiber optics communications. They may also play a
role in multiphoton approaches to quantum information processing if accurate control is obtained over the phases and
polarizations of the signals at the output of the coupler. With this motivation, the constants of motion of the coupler
equation are used to obtain an explicit analytical solution for the nonlinear coupler.
Ó 2003 Elsevier B.V. All rights reserved.
PACS: 42.30.Qw; 42.80.Vc; 42.81.My
Keywords: Nonlinear directional coupler
1. Introduction
Directional couplers are useful devices currently
used in fiber optics communications. Because of
the interaction between the signals in the input fibers,
power fed into one fiber is transferred to the
other. The amount of power transfer can be controlled
by the coupling constant, the interaction
length or the phase mismatch between the inputs.
If, in addition, the material in the coupler region
has nonlinearity properties, the power transfer will
also depend on the intensities of the signals [1,2]. A
large number of interesting effects take place in
nonlinear directional couplers [3–7] with, in particular,
the possibility of performing all classical
* Tel.: +351-914739442; fax: +351-217954288.
E-mail address: vilela@cii.fc.ul.pt (R. Vilela Mendes).
Optics Communications 232 (2004) 425–427
0030-4018/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved.
doi:10.1016/j.optcom.2003.12.056
www.elsevier.com/locate/optcom
logic operations by purely optical means [8]. They
may also play a role in quantum information
processing.
The use of the intensity-dependent phase shifts
associated to the Kerr nonlinearity was, in the
past, proposed for the construction of quantum
gates [9–11]. However they rely on one-photon
processes and therefore would require very strong
nonlinearities, not available in the low loss optical
materials. On the other hand, the quantum computation
scheme based on linear optics of Knill,
Laflamme and Milburn [12] is probabilistic and
relies on a delicate sensitivity of one-photon detectors.
For this reason multiphoton approaches
have been explored based either on the quantumlike
behavior [14–16] of optical modes on a fiber
[13] (the role of time played by a space coordinate)
or on coherent states [17]. For light beams on a

426 R. Vilela Mendes / Optics Communications 232 (2004) 425–427
fiber, sizable nonlinear effects are easy to achieve
with available materials. In particular the directional
coupler might provide an already available
tool for the implementation of linear or nonlinear
gates. 1
For quantum information purposes one would
require accurate information on the phases and
polarizations of the signals at the output of the
coupler. Analytic solutions are ideal for this purpose
although, in general, difficult to obtain for
nonlinear systems. Here, by exploring the constants
of motion of the coupler equation, an explicit
analytical solution is obtained for the
nonlinear coupler.
Jensen [1] had already obtained an elliptic integral
solution for the power propagating in each
fiber. However, for information processing purposes,
detailed information on both amplitude and
phase is required. In addition, by parametrizing all
the nonlinear information in the constants of
motion, a simpler solution is obtained.
2. An analytic solution
Consider two linear optical fibers coming together
in a coupler of nonlinear material. The
equation for the electric field is
o
DE l0e0 2E ot2 ¼ l o
0
2PL ot2 þ l o
0
2PNL ot2 ; ð1Þ
PL and PNL being the linear and nonlinear components
of the medium polarization
PLðr; tÞ ¼e0v ð1Þ Eðr; tÞ: ð2Þ
For symmetric molecules (like SiO2) the leading
nonlinear term is
PNLðr; tÞ ¼e0v ð3Þ jEðr; tÞj 2 Eðr; tÞ; ð3Þ
where an instantaneous nonlinear response may be
assumed (except for extremely short pulses) because
in current fibers the electronic contribution
to v ð3Þ occurs on a time scale of 1–10 fs.
1 There have been some speculations [18] that nonlinear
quantum(like) effects might endow quantum computation with
yet additional power.
Separating fast and slow (time) variations
Eðr; tÞ ¼1 2fEðr; tÞe ix0t þ c:c:g;
PNLðr; tÞ ¼1 2fPNLðr; tÞe ix ð4Þ
0t
þ c:c:g;
and using Eqs. (3) and (4) one obtains for the e ix0t part of a transversal mode
PNL 1;2 ðr; tÞ ¼ 3e0
8 vð3Þ e ix 0t
jE1;2j 2 þ 2
3 jE2;1j 2 E1;2
þ 1
3 E2;1E2;1E 1;2 þ c:c: : ð5Þ
The labels 1 and 2 denote two orthogonal polarizations.
The dependence on transversal coordinates
ðx; yÞ is separated by considering
E ðiÞ
k ðr; tÞ ¼WðiÞ
k ðx; y; zÞeibiz ix0t
e ; ð6Þ
W ðiÞ
k ðx; y; zÞ being an eigenmode of the coupler with
slow variation along z
D2W ðiÞ
k þ x2 0
c2 1 þ vð1Þ b ðiÞ2 W ðiÞ
k ¼ 0; ð7Þ
ðiÞ denotes the mode number, k the polarization and
D2 ¼ o2 o2
þ
ox2 oy2 :
Neglecting 2 o2W ðiÞ =oz2 one obtains
ðiÞ
oWðiÞ
1;2
2ib
oz ¼ 3x2 0
4c
2 vð3Þ
W ðiÞ
2
1;2 þ 2
3 WðiÞ
2
2;1
W ðiÞ
1;2
þ 1
3 WðiÞ
2;1WðiÞ 2;1WðiÞ 1;2 : ð8Þ
In directional couplers the propagating beams are
made to overlap along one of the transversal coordinates
ðxÞ. Typically, in the nonlinear region of
the directional coupler, the eigenmodes are symmetric
(+) and antisymmetric ()) functions of x,
the amplitudes in each fiber at the input and output
of the coupler being recovered by
W ð1Þ
k
W ð2Þ
k
1
¼
2 WðþÞ
Þ
k þ Wð
k
¼ 1
2 WðþÞ
ð Þ
k Wk ;
:
ð9Þ
2 Justified for slow variations of the refractive index along the
beam axis over distances of the order of one wavelength.

An explicit analytic solution for the nonlinear
coupler Eq. (8) is now obtained by noticing that it
has two constants of motion, namely
o
oz WðiÞ
2
1
o
oz WðiÞ
1 WðiÞ
n
2
þ W ðiÞ
2
Therefore, defining
W ðiÞ
1
2
W ðiÞ
1 WðiÞ
2
þ W ðiÞ
2
2
WðiÞ
1 WðiÞ
2
2
¼ 0;
W ðiÞ
1 WðiÞ
o
2 ¼ 0:
¼ a ðiÞ ;
¼ icðiÞ ;
ð10Þ
ð11Þ
one obtains for the electrical field of the eigenmodes
i oEðiÞ
1
oz
i oEðiÞ
2
oz
with
b ðiÞ
¼ bðiÞE
ðiÞ
1
ik ðiÞ E ðiÞ
2 ;
¼ bðiÞE
ðiÞ
2 þ ikðiÞE ðiÞ
1 ;
¼ b ðiÞ þ 3x2 0
8c 2
k ðiÞ ¼ x2 0
8c2 vð3Þ b ðiÞ cðiÞ :
v ð3Þ
b ðiÞ aðiÞ ;
Notice that, through a ðiÞ and c ðiÞ , b ðiÞ
E ðiÞ
2 z ¼ e ibðiÞ z E ðiÞ
1 0 sin k ðiÞ z
ð12Þ
ð13Þ
and jðiÞ depend
on the material properties, on the geometry
of the mode and also on its intensity. One may
now obtain, for each eigenmode, the input–output
relation of the nonlinear coupler
E ðiÞ
1 ðÞ¼e z
ibðiÞz ðiÞ
E1 0 ð Þcos kðiÞ n
z
E ðiÞ
2 0 sin k ðiÞ o
z ;
n
ð14Þ
þ E ðiÞ
2 0 cos k ðiÞ z
(ðiÞ ¼ ðþÞ or ())), the nonlinearity being embedded
into b ðiÞ
and k ðiÞ
b ðiÞ
¼ b ðiÞ þ 3x2 0
8c 2
k ðiÞ ¼ x2 0
4c 2
v ð3Þ
v ð3Þ
b ðiÞ
Im EðiÞ
ðiÞ 1
b
E ðiÞ
2
ð Þ þ E ðiÞ
2
ð Þ
1 0
ð0ÞE ðiÞ
ð Þ :
R. Vilela Mendes / Optics Communications 232 (2004) 425–427 427
2 0
2 0
;
ð15Þ
To obtain the corresponding input–output relations
in the two fibers (1) and (2) one uses Eq.
(9), namely
E ð1Þ
k ðzÞ ¼1
2 EðþÞ
ð Þ
k ðÞþ z Ek ðzÞ ;
E ð2Þ
k ðzÞ ¼1
2 EðþÞ
k ðÞ z
ð Þ
Ek ðzÞ :
ð16Þ
In conclusion, Eqs. (14)–(16) provide an analytic
solution for the nonlinear directional coupler, from
which phases and polarizations may be obtained
explicitly. Through the constants of motion a ðiÞ and
c ðiÞ , which depend on the material properties and
the geometry of the mode, the nonlinearity of the
system is completely parametrized. The solution
(14)–(16) may then be used as a convenient guide
for experimental implementation.
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