Modeling light trapping in nonlinear photonic structures

Alejandro AcevesDepartment of Mathematics and StatisticsThe University of New Mexico

Work in collaboration with Tomas Dohnal, ETH Zurich Center for Nonlinear Mathematics and Applications, AIMS, Poitiers, June 2006

What is a nonlinear photonic structure It is an “engineered” optical medium with

periodic properties. Photonic structures are built to manipulate

light (slow light, light localization, trap light) I’ll show some examples, but this talk will

concentrate on 2-dim periodic waveguides

Photonic structures with periodic index of refraction in the direction of propagation

I. 1D periodic structures – photonic crystal fibers fiber gratings

• Gap solitons (Theory: A.A, S. Wabnitz, Exp: Eggleton, Slusher (Bell Labs))

• Fiber gratings with defects – stopping of light (Goodman, Slusher, Weinstein 2002)

II. 2D periodic structures – waveguide gratings (T. Dohnal, A.A)• x- uniform medium with a z- grating

• z- grating, guiding in x2D soliton-like bullets

• Interactions of 2 bullets

• waveguide gratings with defects

Motivation of studyWe take advantage of periodic structures and nonlinear effects to propose new

stable and robust systems relevant to optical systems.

- Periodic structure - material with a periodically varying index of refraction (“grating”)

- Nonlinearity - dependence of the refractive index on the intensity of the electric field (Kerr nonlinearity)

The effects we seek:• existence of stable localized solutions – solitary waves, solitons• short formation lengths of these stable pulses• possibility to control the pulses – speed, direction (2D, 3D)

Prospective applications:• rerouting of pulses• optical memory• low-loss bending of light

Nonlinear optics in waveguides

Starting point : Maxwell’s equations Separation of variables, transverse mode satisfies eigenvalue problem

and envelope satisfies nonlinear wave equation derived from asymptotic theory

0||

))(,(

.),(),(),,,(

22

22

222

)(

2

AAAAi

UUc

yxnU

cceyxUtzAtzyxE

TZ

tkzi

This is the nonlinear Schroedinger equation known to describe opticalpulses in fibers and light beams in waveguides

Solitons in homogeneous systems

Elastic soliton interaction in theNonlinear Schroedinger Eq.

Optical fiber gratings

envelope) moving (backward ),(

envelope) moving (forward ),(

TZE

TZE

b

f

1-dim Coupled Mode Equations

EEEiEiEcE

EEEiEiEcE

zgt

zgt

)||2|(|

)||2|(|22

22

Gap solitons exist. Velocity proportional to amplitude mismatchBetween forward and backward envelopes.

1D Gap Soliton

II. Waveguide gratings – 2Din collaboration with Tomas Dohnal (currently a postdoc at ETH)

z

x

yAssumptions:• dynamics in y arrested by a fixed n(y) profile• xy-normal incidence of pulses• characteristic length scales of coupling, nonlinearity and diffraction are in balance

BARE 2D WAVEGUIDE

- 2D NL Schrödinger equation - collapse phenomena: point blow-up

WAVEGUIDE GRATING

- 2D CME - no collapse - possibility of localization

Dispersion relation for coupled mode equations

Frequency gap region.We will study dynamics in thisregime.

zk

Close, but outside the gapWell approximated by the2D NLSE + higher ordercorrections.Collapse arrest shown byFibich, Ilan, A.A (2002).But dynamics is unstable

Governing equations : 2D Coupled Mode Equations

EEEiEiEidEcE

EEEiEiEidEcE

xzgt

xzgt

)||2|(|

)||2|(|22

22

2

2

advection diffraction coupling non-linearity

.),0[],[],[: ,0 ,,, CLLLLEdc zzxxg

Stationary solutions via Newton’s iteration

Stationary case I

Stationary case I

Nonexistence of minima of the Hamiltonian

Defects in 1d gratings

Trapping of a gap soliton bullet in a defect. (this concept has been theoretically demonstrated by Goodman, Weinstein, Slusher for the 1-dim (fiber Bragg grating with a defect) case).

Ref: R. Goodman et.al., JOSA B 19, 1635 (July 2002)

.),0[],[],[: ,0 , CLLLLE zzxx

EEEiEzxVEzxiEiEE

EEEiEzxVEzxiEiEE

xzt

xzt

)||2|(|),(),(

)||2|(|),(),(22

22

2

2

advection diffraction coupling non-linearitydefect

Governing equations : 2D CME

κ=const., V=0 (no defect): Moving gap soliton bullet

2-D version of resonant trapping

3 trapping cases (GS: ω(v=0) ≈ 0.96)1. Trapping into two defect modes

|),,0(| tzxE |)(| tak

992.0 ,963.0 :modesdefect linear 2

16.0 and 180with

))tanh()(tanh(2

1);( and

)1)((tanh1

)(sech1

2

1 ),(sech2 where

),1)((tanh1 ),7;()()9;()(

22

222

222

1

2221

L

.k

cycycyTkzk

kzkkVxV

kzkxTzVzTxVV

Trapping into 2 modes (movie)

3 trapping cases (GS: ω(v=0) ≈ 0.96)

|),,0(| tzxE

2. Trapping into one defect mode

995.0 :modedefect linear 1

3.0,25.0,1.01 ,3.0 )()( 2222

L

bzaxbzax baeeV

Trapping into 1 mode (movie)

3 trapping cases (GS: ω(v=0) ≈ 0.96)3. No trapping

|),,0(| tzxE |)(| tak

870and 680470260470 :modesdefect linear 5

1 and 850with

))tanh()(tanh(2

1);( and

)1)((tanh1

)(sech1

2

1 ),(sech2 where

),1)((tanh1 ),5;()()5;()(

22

222

222

1

2221

. ., ., ., .-

.k

cycycyTkzk

kzkkVxV

kzkxTzVzTxVV

L

← ALL FAR FROM RESONANCE

No trapping (movie)

Finite dimensional approximation of the trapped dynamics

Less energy trapped

=> better agreement

Reflection from a “repelling” defect potential

Reflection with a tilt, “bouncing” beam

Conclusions

2d nonlinear photonic structures show promising properties for quasi-stable propagation of “slow” light-bullets

With the addition of defects, we presented examples of resonant trapping

Research in line with other interesting schemes to slow down light for eventually having all optical logic devices (eg. buffers)

Work recently appeared in Studies in Applied Math (2005)