Slow Light, Fast Light, and Optical Solitons inStructured Optical Waveguides

Robert W. Boyd and John E. Heebner

with

Nick Lepeshkin, Aaron Schweinsberg, Matt Bigelow, and Q-Han Park

University of Rochesterhttp://optics.rochester.edu

Suresh Periera, Philip Chak, and John SipeUniversity of Toronto

Presented at Nonlinear Guided Waves and Their Applications,Stresa Italy, September 1-4, 2002.

Interest in Slow Light

Fundamentals of optical physics

Intrigue: Can (group) refractive index really be 106?

Optical delay lines, optical storage, optical memories

Implications for quantum information

Challenge/Goal

Slow light in room-temperature solid-state material.

• Slow light in a structured waveguide

• Slow light in room temperature ruby

(facilitated by a novel quantum coherence effect)

Slow Light

group velocity ≠≠≠≠ phase velocity

Artificial Materials for Nonlinear Optics

0

Artifical materials can produce Large nonlinear optical response Large dispersive effects

ExamplesFiber/waveguide Bragg gratingsPBG materialsCROW devices (Yariv et al.)SCISSOR devices

Weak pulses spread because of dispersion

050

100150

0 20 40 60 80 100 120 t (ps)z (resonator #)

inte

nsity

050

100150

0 20 40 60 80 100 120 t (ps)z (resonator #)

inte

nsity

But intense pulses form solitons through balance of dispersion and nonlinearity.

Shows slow-light, tailored dispersion, and enhanced nonlinearity

NLO of SCISSOR Devices(Side-Coupled Integrated Spaced Sequence of Resonators)

Optical solitons described by nonlinear Schrodinger equation

Ultrafast All-Optical Switch Based On Arsenic Triselenide Chalcogenide Glass

We excite a whispering gallery mode of a chalcogenide glass disk.

The nonlinear phase shift scales as the square of the finesse F of the resonator. (F ≈ 10 2 in our design)

Goal is 1 pJ switching energy at 1 Tb/sec.

r5050

RingResonator

Input-2

Output-1

J. E. Heebner and R. W. Boyd, Opt. Lett. 24, 847, 1999. (implementation with Dick Slusher, Lucent)

Input-1

Output-2

FDTD simulation

Motivation

To exploit the ability of microresonators to enhance nonlinearities and induce strong dispersive effects for creating structured waveguides with exotic properties.

Currently, most of the work done in microresonators involves applications such as disk lasers, dispersion compensators and add-drop filters. There's not much nonlinear action!

A cascade of resonators side-coupled to an ordinary waveguide can exhibit:

slow light propagation induced dispersion enhanced nonlinearities

Intensity Enhancement ( |E3 / E1|2 )

Modified Dispersion Relation ( β vs. ω )

Properties of a Single Microresonator

frequency (ω)

effe

ctiv

e pr

opag

atio

nco

nsta

nt (

β)B

uild

-up

Fac

tor

|E3

/ E

1|2

ωR

r2 = 0.90

r2 = 0.75

r2 = 0.25

r2 = 0.00

r2 = 0.00 r2 = 0.25

r2 = 0.75 r2 = 0.90

ωR − 2πT

ωR + 2πT

E

R

3E

4

E1

E2

Assuming negligible attenuation, this resonator is, unlike a Fabry-Perot, of the "all-pass" device -

there is no reflected or drop port.

Definitions

Finesse

Transit Time

E4E2

= r it

rit

E3E1

F = π1−r

T = n2πRc

Nonlinear Schrödinger Equation (NLSE)

Fundamental Soliton Solution

Propagation Equation for a SCISSOR

L

∂∂z A = −i 12 β2 ∂2

∂t2A + iγ |A|2A

A(z,t) = A0 sech tTp

ei12

γ|A0 |2z

By arranging a spaced sequence of resonators, side-coupled to an ordinary waveguide, one can create an effective, structured waveguide that supports pulse propagation in the NLSE regime.

Propagation is unidirectional, and there is NO photonic bandgap to produce the enhancement. Feedback is intra-resonator and not inter-resonator.

ω R

γ eff

β2

Balancing Dispersion & Nonlinearity

A0 = |β2 |

γ Tp2= T2

3 γ2πRTp2

Resonator-induced dispersion can be 5-7 orders of magnitude greater than the material dispersion of silica!

Resonator enhancement of nonlinearity can be 3-4 orders of magnitude!

An enhanced nonlinearity may be balanced by an induced anomalous dispersion at some detuning from resonance to form solitons

A characteristic length, the soliton period may as small as the distance between resonator units!

soliton amplitude

adjustable by controlling ratio of transit time to pulse width

Fundamental Soliton

Weak Pulsepulse

disperses

pulsepreserved

Soliton Propagation

050

100150

200250

0 20 40 60 80 100

050

100150

200250

0 20 40 60 80 100

0

0.1

0.2

0.3

t (ps)

z (resonator #)

pow

er (

µW)

0

0.1

0.2

0.3

t (ps)

z (resonator #)

pow

er (

W)

simulation assumes a chalcogenide/GaAs-

like nonlinearity

5 µm diameter resonators with a

finesse of 30

SCISSOR may be constructed from 100 resonators spaced by

10 µm for a total length of 1 mm

soliton may be excited via a 10 ps, 125mW

pulse

Dark Solitons

0100300

20

60

100

0

20

t (ps)z (resonator #)

po

wer

(m

W)

SCISSOR system also supports the propagation of dark solitons.

Fn

Slow Light and SCISSOR Structures

frequency, ω

FnR

c c

FnR

cnR

grou

p in

dex

buil

d-u

pk

eff −k

0

Fncslope =

F

n

2π

F2π n

0

FnR

c

2πL

FnR

c

cslope =

E3

E4

E1

E2 L

R

Frequency Dependence of GVD and SPM Coefficients

0

-4π 0

-0.5

0.0

0.5

1.0

F -2πF

4πF

2πF

normalized detuning, φ0

γeffk''eff

(2FT/π)2/L γ (2F/π)2 (2πR/L)

π3F

soliton condition

(GVD) (SPM)

Pulse Distortion on Propagation through SCISSOR Structure

0

strong pulse, at GVD extremum

weak pulse, on resonance

pow

er (

µW)

weak pulse, at GVD extremum

6

4

2

0

0 1 2 3 4time (ns)

8

input output

input

input

output

output (soliton)

pow

er (

µW)

6

4

2

0

8

pow

er (

µW)

6

4

2

0

Maximum delay, but pulse distorted by higher-orderdispersion.

Slightly reduced delay, no distortion, but spreading due to GVD.

Slightly reduced delay, no distortion, SPM com-pensates for spreading

"Fast" (Superluminal) Light in SCISSOR Structures

0

pow

er (

arb.

uni

ts) 1.0

overcoupledcoupledcritically

undercoupled

overcoupled

0.5

0

k eff

−k 0

frequency detuning, ω-ωR

2πL

400delay (ps)

propagationthroughvacuum

(time-advanced)propagation

throughundercoupled

SCISSOR

-40-80-120

0 8 δω- 8 δω

Requires loss in resonator structure

L 2L2L

SCISSOR Dispersion Relations

20 0

1.498

1.522

1.546

1.571

1.597π π- π

0

Bloch vector (keff)intensity build-up

Single-Guide SCISSOR

wav

elen

gth

(µm

)

Bragg gap

resonator gap

Bragg + resonator gap

No bandgap

Lπ

Large intensity buildup

-

Double-Guide SCISSORBandgaps occur Reduced intensity buildup

L 2L2L20 0 π π- π0

Bloch vector (keff)intensity build-up

Lπ-

separation = 1.5 π R

GaAs

AlxGa1-xAs(x = 0.4)

Microdisk Resonator Design

(Not drawn to scale)All dimensions in microns

0.5

2.0

10 10

0.450.25

s1 = 0.08s2 = 0.10s3 = 0.12s4 = 0.20

25

2.0

trench

gap

lineartaper

20

trench

d = 10.5

25lineartaper

J. E. Heebner and R. W. Boyd

Photonic Device Fabrication Procedure

RWB - 10/4/01

PMMA

AlGaAs-GaAs

structure

Oxide (SiO2)

AlGaAs-GaAs

structure

Oxide (SiO2)

PMMA

AlGaAs-GaAs

structure

PMMA

Oxide (SiO2)

(2) Deposit oxide

AlGaAs-GaAs

structure

(1) MBE growth

(3) Spin-coat e-beam resist

(4) Pattern inverse with

e-beam & develop(6) Remove PMMA

(7) CAIBE etch AlGaAs-GaAs

(8) Strip oxide

AlGaAs-GaAs

structure

Oxide (SiO2)

AlGaAs-GaAs

structure

AlGaAs-GaAs

structure

Oxide (SiO2)

(5) RIE etch oxide

AlGaAs-GaAs

structure

Oxide (SiO2)

Photonic Devices Writteninto PMMA Resist

Pattern Etched Into Silica Mask

AFM

Slow Light in Ruby

Need a large dn/dω. (How?)

Kramers-Kronig relations: Want a very narrow absorption line.

Well-known (to the few people how know itwell) how to do so:

Make use of “spectral holes” due topopulation oscillations.

Hole-burning in a homogeneouslybroadened line; requires T2 << T1.

1/T2 1/T1

inhomogeneously broadened medium

homogeneously broadened medium(or inhomogeneously broadened)

Γba

=1

T1

2γba

=2

T2

a

b b

a

cω

Γca

Γbc

atomicmedium

ω

ω + δω + δ

E1,

E3,

measure absorption

Spectral Holes Due to Population Oscillations

ρ ρbb aa w−( ) = δ δ δ δi t i tw t w w e w e≈ + +− −( ) ( ) ( ) ( )0

Population inversion:

δ T≤ / 11population oscillation terms important only for

ρ ω δ µω ω

δba

ba

ba i TE w E w+ =

− ++[ ]( )

/( ) ( )

23

01

1

Probe-beam response:

α ω δδ β

β

wT

T+ ∝ −

+

=

( )

( /

( )02

2

12 2

1

1

Ω

TT T T12

1 21) ( )+ Ω

Probe-beam absorption:

linewidth

h

Argon Ion LaserRuby

f = 40 cm

Function Generator

EO modulator

Digital

Oscilloscope

f = 7.5 cm

Diffuser

Reference Detector

Experimental Setup Used to Observe SLow-Light in Ruby

7.25 cm ruby laser rod (pink ruby)

512 µs

Time (ms)

0

0.2

0.4

0.6

0.8

1.0

1.2

No

rma

lize

d I

nte

nsity

Input Pulse

Output Pulse

0 10 20-10-20 30

Gaussian Pulse Propagation Through Ruby

No pulse distortion!

v = 140 m/s

ng = 2 x 106

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Dela

y (

ms)

2000 400 600 800 1000

Modulation Frequency (Hz)

P = 0.25 W

P = 0.1 W

Measurement of Delay Time for Harmonic Modulation

solid lines are theoretical predictions

For 1.2 ms delay, v = 60 m/s and ng = 5 x 106

Summary

Artificial materials hold great promise for applications in photonics because of

• large controllable nonlinear response

• large dispersion controllable in magnitude and sign

Demonstration of slow light propagation in ruby