Optics in non homogeneous waveguide arrays

R. MORANDOTTI1*, H.S. EISENBERG2, D. MANDELIK3 ,Y. SILBERBERG3, D. MODOTTO4,

M. SOREL5, C. R. STANLEY5, and J.S. AITCHISON6

1INRS-EMT, Varennes J3X 1S2, Canada2University of California, Department of Physics, Santa Barbara, CA 93106, USA

3Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel4Dipartimento di Elettronica per l’Automazione, Universita’ di Brescia, via Branze 38, 25123 Brescia, Italy

5Department of Electronics and Electrical Engineering, University of Glasgow, Glasgow, G12 8QQ, Scotland6Department of Electrical and Computer Engineering, University of Toronto, Toronto, Ontario M5S 3G4, Canada

Optical discrete solitons possess extremely interesting dynamical properties. Such properties lead to the realization of several

ultrafast, all-optical switching mechanisms, based on the use of uniform and nonuniform waveguides array.

Keywords: discrete solitons, waveguide arrays, integrated optics, all-optical switching, semiconductor devices.

Nonlinear discrete systems have recently attracted a lot of

interest, due to their generality and appearance in a wide

variety of areas. Many systems in nature and on a variety of

scales share the same dynamical nonlinear properties, and

can support stable energy localizations. These localiza-

tions, named discrete solitons, can interact with each other

or with the structure in which they propagate, leading to a

variety of useful applications. A particular class of optical

discrete solitons, propagating in arrays of nonlinear

waveguides, has proven to be a convenient system for the

experimental study of nonlinear discrete phenomena. In ad-

dition, their fast response and mutual interactions could be

exploited for all-optical switching applications. These

solitons are similar to their continuous counterparts, and

can be generated in media through the Kerr nonlinearity.

However, unlike continuous spatial solitons, they are not

invariant for translation and rotation, and therefore exhibit

a number of novel dynamical properties.

In particular, theoretical studies have predicted that

soliton reflection, transmission and trapping can be induced

by inserting suitable defects in the array. Several switching

configurations and applications can be designed by slightly

modifying the geometry of the array.

In this work we will introduce the dynamic properties

of discrete solitons, both teoretically and experimentally,

and then we will study the interaction of discrete solitons

with structural defects in arrays of AlGaAs waveguides.

We fabricated both attractive and repulsive defects in oth-

erwise uniform arrays. When a beam is injected in proxim-

ity of a weak defect, little difference is observed between

attractive and repulsive defects, as they reflect and transmit

the beam accordingly to its transverse velocity. However,

when the beam is injected inside the defect, a strong trap-

ping takes place in the attractive defect, while the repulsive

defect is very unstable and drives the soliton away. We will

present a series of different experimental situations, with

the correspondent numerical analysis.

From a more general point of view, the properties stated

above rise from the peculiar dynamics of nonlinear dynam-

ics in discrete systems. In particular, it was observed that

stable energy localization, or discrete solitons, can propa-

gate without broadening, overcoming the general spreading

due to discrete diffraction. Christodoulides and Joseph sug-

gested that discrete optical solitons are examples of such

localized states, and can be generated in weakly coupled

nonlinear waveguide arrays [1]. Since then, the characteris-

tics of such waveguide arrays have been extensively stud-

ied, both theoretically and experimentally.

For example, arrays of waveguides support both stable

and unstable soliton solutions. The stable states correspond

to solitons that are centred on a waveguide and propagate

along the waveguide direction, while solitons that are sym-

metrically centred between two neighbour waveguides

(which carry the same power) are unstable, and tend to

shift away from the waveguide direction.

In order to better explain these concepts, we will start

with a brief description of some basic properties of discrete

solitons. Consider an infinite array of single-mode

waveguides that are weakly coupled through their evanes-

cent field. We assume that the modes of the individual

waveguides remain unchanged and that only the mode am-

plitudes En evolve during propagation. In the ideal case,

Opto-Electron. Rev., 13, no. 2, 2005 R. Morandotti 103

7th International Workshop on Nonlinear Optics Applications

OPTO-ELECTRONICS REVIEW 13(2), 103–106

*e-mail: morandotti@inrs-emt.uquebec.ca

where no losses are present and the excitation has no time

dependence (continuous wave excitation), the field evolu-

tion is described by the discrete nonlinear Schrödinger

equation (DNLS)

idE

dzC E E E

nn n n n+ + + =+ -( )1 1

20g E , (1)

where C is the coupling coefficient and g is the nonlinear

parameter, accounting for the optical Kerr effect. In fact, a

model based on Eq. (1) reproduces qualitatively most of

the effects observed very well, even when it fails some-

times to produce good quantitative agreement with the ex-

perimental results. We restrict most of the discussion to

Eq. (1).

The nonlinear term in Eq. (1) is only significant at the

high powers and can be ignored in the low-intensity re-

gime, where the equation can be solved analytically. This

low power linear solution describes discrete diffraction in

an array of waveguides. When the input beam is injected

into a single guide, a pronounced diffraction cone is ob-

tained, where most of the power is propagating in two

wings. Spectral analysis of Eq. (1) shows that these wings

are associated with propagation at zero spatial dispersion.

Hence, for this particular angle of propagation the beam

does not diffract up to second order. Even when the power

is increased the field never escapes from this cone. If we

restrict to consider fields that vary slowly across the array,

the DNLS appears as a natural discretisation of the

well-known non-linear Schrödinger equation. However,

while the latter is integrable, the behaviour of the DNLSE

must be described in terms of approximate analytical solu-

tions or through numerical methods. Two limiting cases,

which allow for an analytical description, can be defined.

The first, also known as the long wavelength limit, can be

described by the continuum approximation which leads to

the nonlinear Schrödinger equation and generates similar

solutions. These solitons are able to move in the array al-

most undisturbed. In the second case, the excitation is lo-

calized mostly in a single waveguide, which is effectively

decoupled from the rest of the array. In this limit the mu-

tual interaction and any energy transport between the

guides are blocked by the nonlinearity.

The difference in energy between the stable and unsta-

ble states, named the “Peierls-Nabarro potential” (PNP) [2],

explains the particular behaviours [3] recently observed in

uniform arrays of waveguides. These dynamics can be fur-

ther enhanced, or cancelled, in non-homogenous arrays.

Non-uniform arrays have been used in previous studies;

in particular, arrays with an (effective) transversal gradient

of the refractive index across the array have been used to

generate optical Bloch oscillations [4]. In an earlier experi-

ment, the behaviour of a single defect, formed by a narrow

waveguide embedded in a uniform array, was investigated.

In the linear regime, the defect traps the field, which be-

comes free to escape at the higher input when phase match-

ing with the neighbouring waveguides is achieved [5].

Various theoretical studies have predicted that soliton

reflection, transmission and trapping can be controlled by

inserting suitable defects in an array. We study ar-

ray-defects induced by a local change in the coupling

strength between a pair of adjacent waveguides. A local in-

crease in the coupling coefficient corresponds to an attrac-

tive defect, where the soliton tends to get trapped, while a

decrease in the coupling coefficient corresponds to a repul-

sive defect. There is a strong analogy between the behav-

iour of a soliton interacting with a structural defect, and a

classical particle in the presence of a potential. The case of

Optics in non homogeneous waveguide arrays

104 Opto-Electron. Rev., 13, no. 2, 2005 © 2005 COSiW SEP, Warsaw

Fig. 1. Experimental set-up and top view of the sample (N.B.: in the experiment we centred the beam on the defect; in the picture above we

intentionally positioned the beam 3 waveguides aside from the defect, visible in micrograph).

a repulsive defect, corresponding to a local decrease in the

effective index, is equivalent to the interaction of the

soliton with a potential barrier. If this is the case, the

soliton may be reflected from the defect site (for small in-

put angles) or transmitted through it (for larger angles). For

an intermediate situation, the soliton exhibits an inelastic

collision with the defect and it is partially transmitted and

partially reflected. The case of an attractive defect, corre-

sponding to a locally higher effective index, is equivalent

to an interaction of the soliton with a potential well. Here,

the soliton may be trapped by the defect, which induces an

attractive potential for the soliton, when the proper condi-

tions are met.

In this paper we study the interaction of discrete

solitons with structural defects in arrays of AlGaAs

waveguides. We fabricated both attractive and repulsive

centres by changing the distance between a single pair of

adjacent waveguides, embedded in a uniform array. A local

decrease in the distance corresponds to an attractive defect,

while a repulsive centre is created instead by increasing

such a distance. The measurements described in this paper

were performed by using the same set-up employed in pre-

vious experiments [3] (see Fig. 1). The pulses of 100-fsec

duration and with a wavelength of 1530 nm were injected

in several different arrays. The input power, the number of

waveguides excited, and the angle of propagation could be

all controlled.

The arrays employed in the experiment contained 41

waveguides etched into the top cladding of a 6-mm long

slab waveguide. The coupling coefficient of adjacent

waveguides was constant, with the only exception of the

defect pair, where it was increased for an attractive defect,

and decreased for a repulsive defect.

In our first experiment, we positioned the beam at the

centre of the two waveguides forming the defect and per-

formed the measurements at low and at high powers (70 W

and 800 W, as shown in Figs. 2 and 3, respectively). The

input angle was varied between –0.85 and +0.85 degrees.

We note that for negative angles, the beam propagates from

left to right in the array, while positive angles correspond

to a right to left propagation. In each of the four graphs, the

horizontal axis stands for the input angle, while in the verti-

cal axis a linear grey scale represents the output intensity

profile. Plots (a) and (b) refer to the case of an attractive

defect, where experimental results (a) and 2D-BPM contin-

uous wave (CW) simulations (b) are compared. Similarly,

the case of a repulsive defect is shown in plots (c) and (d).

We discuss the linear case first (see Fig. 2). At low

power, a broad linear mode propagates along the attractive

defect, as shown in the experiment (a) and reproduced in

the simulation (b). The centre of the localization, corre-

sponding to the bright white area in the figure, moves be-

tween the pair of waveguides generating the defect, which

suggests weak trapping of the linear mode by the defect

site. On the other hand, the behaviour of a repulsive defect

[Figs. 2(c) and 2(d)], is consistently different. The beam

smoothly crosses the defect, and it is not trapped. Only a

slight decrease in the field intensity is noted around the de-

fect.

The intrinsic nature of the attractive and repulsive de-

fects becomes evident when the power is increased (Fig. 3).

Opto-Electron. Rev., 13, no. 2, 2005 R. Morandotti 105

7th International Workshop on Nonlinear Optics Applications

Fig. 2. Interaction of a low power beam with defects. The light is

injected at the defect site (dashed line); (a), (b) attractive defect,

experimental result and simulations; (c), (d) repulsive defect,

experimental results and simulations.

Fig. 3. Interaction of a discrete soliton with defects. Alike in Fig. 2,

the soliton is injected exactly into the defect, represented by a

dashed line; (a), (b), attractive case, experimental results and

simulations; (c), (d) repulsive defect, experimental results and

simulations.

In the former case, a narrow soliton locks firmly around the

defect (the soliton is trapped by the defect for input angles

as high as 0.4 degrees). Locking caused by the PNP also

occurs in a uniform array; however, the presence of an at-

tractive centre adds to the PNP and increases its effect. In

the case of a repulsive defect, we clearly observe an oppo-

site behaviour. A very sharp instability appears when the

soliton crosses the pair of waveguides forming the defect,

and the soliton flips abruptly from one side of the defect to

the other. In this case, the trapping induced by the PNP is

completely overcome. We also note the excellent qualita-

tive agreement between experimental results and simula-

tions, in spite of the fact that a continuous wave (CW)

model was used to perform the numerical calculations.

In conclusion, we have shown experimentally that engi-

neered structural defects in arrays may affect the motion of

discrete solitons, inducing important changes in their dy-

namical behaviour. These novel effects are clearly attrac-

tive for various all-optical switching operations.

Acknowledgements

The authors wish to thank Ulf Peschel for several useful

discussions on the subject. Financial support by the

US-Israel BSF, and by FQRNT and NSERC in Canada, is

gratefully acknowledged.

References

1. D.N. Christodoulides and R.I. Joseph, “Discrete

self-focusing in nonlinear arrays of coupled wave-guides”,

Opt. Lett. 13, 794–796 (1988); H.S. Eisenberg, Y.

Silberberg, R. Morandotti, A.R. Boyd and J.S. Aitchison,

“Discrete spatial optical solitons in waveguide arrays”,

Phys. Rev. Lett. 81, 3383–3387 (1998); J.W. Fleischer, M.

Segev, N.K. Efremidis, and D.N. Christodoulides, “Obser-

vation of two-dimensional discrete solitons in opti-

cally-induced nonlinear photonic lattices”, Nature 422,

147–150 (2003).

2. Y.S. Kivshar and D.K. Campbell, “Peierls-Nabarro poten-

tial barrier for highly localized nonlinear modes”, Phys.

Rev. E48, 3077–3081 (1993).

3. R. Morandotti, U. Peschel, J.S. Aitchison, H.S. Eisenberg,

and Y. Silberberg, “Dynamics of discrete solitons in optical

waveguide arrays”, Phys. Rev. Lett. 83, 2726–2729 (1999).

4. R. Morandotti, U. Peschel, J.S. Aitchison, H.S. Eisenberg,

and Y. Silberberg, “Experimental observations of linear and

nonlinear optical bloch oscillations”, Phys. Rev. Lett. 83,

4756–4759 (1999); T. Pertsch, P. Dannberg, W. Elflein, A.

Brauer and F. Lederer, “Optical bloch oscillations in tem-

perature tuned “”, Phys. Rev. Lett. 83, 4752–4755 (1999).

5. U. Peschel, R. Morandotti, J.S. Aitchison, H.S. Eisenberg

and Y. Silberberg, “Nonlinearly induced escape from

“Nonlinearly induced escape from a defect state in wave-

guide arrays”, Appl. Phys. Lett. 75, 1348–1350 (1999).

Optics in non homogeneous waveguide arrays

106 Opto-Electron. Rev., 13, no. 2, 2005 © 2005 COSiW SEP, Warsaw