Coupled resonator slow-Coupled resonator slow-wave optical structures wave optical structures

Parma, 5/6/2007

Jiří Petráček, Jaroslav Čáp

petracek@fme.vutbr.cz

all-optical high-bit-rate communication systems

- optical delay lines

- memories

- switches

- logic gates

- ....

“slow” light

nonlinear effects increased efficiency

Outline

• Introduction: slow-wave optical structures (SWS)• Basic properties of SWS

– System model – Bloch modes– Dispersion characteristics– Phase shift enhancement– Nonlinear SWS

• Numerical methods for nonlinear SWS– NI-FD– FD-TD

• Results for nonlinear SWS

Outline

• Introduction: slow-wave optical structures (SWS)• Basic properties of SWS

– System model – Bloch modes– Dispersion characteristics– Phase shift enhancement– Nonlinear SWS

• Numerical methods for nonlinear SWS– NI-FD– FD-TD

• Results for nonlinear SWS

Slow light

• the light speed in vacuum c

• phase velocity v

• group velocity vg

How to reduce the group velocity of light?

Electromagnetically induced transparency - EIT

Stimulated Brillouin scattering

Slow-wave optical structures (SWS) –

– pure optical way

Miguel González Herráez, Kwang Yong Song, Luc Thévenaz: „Arbitrary bandwidth Brillouin slow light in optical fibers,“ Opt. Express 14 1395 (2006)

Ch. Liu, Z. Dutton, et al.: „Observation of coherent opticalinformation storage in an atomic medium using halted light pulses,“ Nature 409 (2001) 490-493

A. Melloni and F. Morichetti, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. And Quantum Electron. 35, 365 (2003).

Slow-wave optical structure (SWS)

- chain of directly coupled resonators (CROW - coupled resonator optical waveguide)

- light propagates due to the coupling between adjacent resonators

coupled Fabry-Pérot cavities

1D coupled PC defects

2D coupled PC defects

coupled microring resonators

Various implementations of SWSs

Outline

• Introduction: slow-wave optical structures (SWS)• Basic properties of SWS

– System model – Bloch modes– Dispersion characteristics– Phase shift enhancement– Nonlinear SWS

• Numerical methods for nonlinear SWS– NI-FD– FD-TD

• Results for nonlinear SWS

A. Melloni and F. Morichetti, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. And Quantum Electron. 35, 365 (2003).

J. K. S. Poon, J. Scheuer, Y. Xu and A. Yariv, “Designing coupled-resonator optical waveguide delay lines", J. Opt. Soc. Am. B 21, 1665-1673, 2004.

System model of SWS

nb

nf 1nf

1nbM

d

System model of SWS

nf

nb

1nb

1nf

rt

nb

nf

1nb

1nf

tr r

Relation between amplitudes

1 nnn tbrfb

11 nnn rbtff

122 tr

Transmission matrix

n

n

n

n

b

fM

b

f

1

1

1 nnn tbrfb

11 nnn rbtff

nb

nf 1nf

1nbM

1

1 22

r

rrt

tM

For lossless SWS it follows from symmetry:

nb

nf 1nf

1nbM

)exp(

)exp(1

1

1

ikdr

rikd

is

1

1 22

r

rrt

tM

d

ikdist exp

ikdrr exp1

1221 sr

real – (coupling ratio)

real

Propagation in periodic structure

nb

nf 1nf

1nbM

nfdi )exp(

nbdi )exp(

d

Bloch modes

b

fdi

b

fM )exp(

nb

nf 1nf

1nbM

nfdi )exp(

nbdi )exp(

d

s

kdd

)sin()cos(

eigenvalue eq. for the propagation constant of Bloch modes

A. Melloni and F. Morichetti, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. And Quantum Electron. 35, 365 (2003).

J. K. S. Poon, J. Scheuer, Y. Xu and A. Yariv, “Designing coupled-resonator optical waveguide delay lines", J. Opt. Soc. Am. B 21, 1665-1673, 2004.

Dispersion curves (band diagram)

-0.5 0 0.5

0.4

0.6

0.8

1

1.2

1.4

1.6

d/2

kd/2

s = 0.1s = 0.3s = 0.5s = 0.7s = 0.9

c

ndfres

c

ndfres

c

ndfres

c

ndFSR

c

ndFSR

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

Re( d

)/2

kd/2

s = 0.1s = 0.3s = 0.5s = 0.7s = 0.9

Dispersion curves

0 0.1 0.2 0.3 0.4 0.5-0.5

-0.4

-0.3

-0.2

-0.1

Im(

d)/2

kd/2

s

kdd

)sin()cos(

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

Re( d

)/2

kd/2

s = 0.1s = 0.3s = 0.5s = 0.7s = 0.9

Bandwidth, B

c

ndB

)sin(kds at the edges of pass-band

s

kdd

)sin()cos(

sFSR

B

skd

arcsin2

arcsin2

Group velocity

)cos(

)(sin 22

kd

kds

v

vg

d

dkv

d

dvg

sv

vg for resonance frequency

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

Re( d

)/2

kd/2

s = 0.1s = 0.3s = 0.5s = 0.7s = 0.9

s

kdd

)sin()cos(

GVD: very strong very strongminimal

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

Re( d

)/2

kd/2

s = 0.1s = 0.3s = 0.5s = 0.7s = 0.9

Group velocity

0.3 0.4 0.5 0.6 0.70

0.2

0.4

0.6

0.8

1

v g/v

kd/2

)cos(

)(sin 22

kd

kds

v

vg

s

kdd

)sin()cos(

Infinite vs. finite structure

/d

resff /

dispersion relation

Jacob Scheuer, Joyce K. S. Poonb, George T. Paloczic and Amnon Yariv, “Coupled Resonator Optical Waveguides (CROWs),” www.its.caltech.edu/~koby/

COST P11 task on slow-wave structures

One period of the slow-wave structure consists of one-dimensionalFabry-Perot cavity placed between two distributed Bragg reflectors

DBR DBR

-150 -100 -50 0 50 100 1500

0.2

0.4

0.6

0.8

1

Frequency f-fres

[GHz]

Tra

nsm

itta

nce

M=1M=2M=3

Finite structure consisting 1, 3 and 5 resonators

35

Fengnian Xia,a Lidija Sekaric, Martin O’Boyle, and Yurii Vlasov: “Coupled resonator optical waveguides based on silicon-on-insulator photonic wires,” Applied Physics Letters 89, 041122 2006.

experiment

theory

number of resonators

Fengnian Xia,a Lidija Sekaric, Martin O’Boyle, and Yurii Vlasov: “Coupled resonator optical waveguides based on silicon-on-insulator photonic wires,” Applied Physics Letters 89, 041122 2006.

1550 nm

Fengnian Xia,a Lidija Sekaric, Martin O’Boyle, and Yurii Vlasov: “Coupled resonator optical waveguides based on silicon-on-insulator photonic wires,” Applied Physics Letters 89, 041122 2006.

Delay, losses and bandwidth

c

nL

cs

Ndn

v

L eff

gg

sFSRB

Buse

2(usable bandwidth, small coupling)

s

NdLeff

11

loss per unit length

Jacob Scheuer, Joyce K. S. Poon, George T. Paloczi and Amnon Yariv, “Coupled Resonator Optical Waveguides (CROWs),” www.its.caltech.edu/~koby/

c

ng

1

loss

Tradeoffs among delay, losses and bandwidth

Jacob Scheuer, Joyce K. S. Poon, George T. Paloczi and Amnon Yariv, “Coupled Resonator Optical Waveguides (CROWs),” www.its.caltech.edu/~koby/

10 resonators

FSR = 310 GHz

propagation loss = 4 dB/cm

s

Phase shift ...

... is enhanced by the slowing factor

s

kdd

)sin()cos(

deff effective phase shift experienced by the optical field propagating in SWS over a distance d

kd

g

eff

v

v

Nonlinear phase shift

Total enhancement:

2

g

eff

v

v

J.E. Heebner and R. W. Boyd, JOSA B 4, 722-731, 2002

intensity dependent phase shift is induced through SPM and XPM intensities of forward and backward propagating waves inside cavities of SWS are increased (compared to the uniform structure) and this causes additional enhancement of nonlinear phase shift

Advantage of non-linear SWS:

S. Blair, “Nonlinear sensitivity enhancement with one-dimensional photonic bandgap structures,” Opt. Lett. 27 (2002) 613-615.

A. Melloni, F. Morichetti, M. Martinelli, „Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,“ Opt. Quantum Electron. 35 (2003) 365.

nonlinear processes are enhanced without affecting bandwidth

Outline

• Introduction: slow-wave optical structures (SWS)• Basic properties of SWS

– System model – Bloch modes– Dispersion characteristics– Phase shift enhancement– Nonlinear SWS

• Numerical methods for nonlinear SWS– NI-FD– FD-TD

• Results for nonlinear SWS

COST P11 task on slow-wave structures

One period of the slow-wave structure consists of one-dimensionalFabry-Perot cavity placed between two distributed Bragg reflectors

DBR DBR

Kerr non-linear layers

Integration of Maxwell Eqs. in frequency domain

One-dimensional structure:- Maxwell equations turn into a system of two coupled ordinary differential equations - that can be solved with standard numerical routines (Runge-Kutta).

H. V. Baghdasaryan and T. M. Knyazyan, “Problem of plane EM wave self-action in multilayer structure: an exact solution,“ Opt. Quantum Electron. 31 (1999), 1059-1072.

M. Midrio, “Shooting technique for the computation of plane-wave reflection and transmission through one-dimensional nonlinear inhomogenous dielectric structures,” J. Opt. Soc. Am. B 18 (2001), 1866-1981.

P. K. Kwan, Y. Y. Lu, “Computing optical bistability in one-dimensional nonlinear structures“ Opt. Commun. 238 (2004) 169-174.

J. Petráček: „Modelling of one-dimensional nonlinear periodic structures by direct integration of Maxwell’s equations in frequency domain.“ In: Frontiers in Planar Lightwave Circuit Technology (Eds: S. Janz, J. Čtyroký, S. Tanev) Springer, 2005.

inA

refAtrA

xLx 0x

Maxwell Eqs.

inA

refAtrA

xLx 0x

)()( xikcBxEx yz

)(,)(22 xExExiknxcB

x zzy

2

20

2, xExnxnxExn zz

Now it is necessary to formulate boundary conditions.

Analytic solution in linear outer layers

inA

refAtrA

xLx 0x

kxinAkxinAxEz )0(exp)0(exp)( refin

LxkLinAxEz )(exp)( tr

kxinAkxinAnxcBy )0(exp)0(exp)0()( refin

LxkLinALnxcBy )(exp)()( tr

Boundary conditions

inA

refAtrA

xLx 0x

refin)0( AAE z

inref)0()0( AAncBy

tr)( ALE z

tr)()( ALnLcBy

Admittance/Impedance concept

E. F. Kuester, D. C. Chang, “Propagation, Attenuation, and Dispersion Characteristics of Inhomogenous Dielectric Slab Waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23 (1975), 98-106.

J. Petráček: „Frequency-domain simulation of electromagnetic wave propagation in one-dimensional nonlinear structures,“ Optics Communications 265 (2006) 331-335.

z

y

E

icBpq 1

new ODE systems for

)()()( xExkqxEx zz

222 ,)()( xExnxqkxqx z

)()(,)()(22 xcBxpcBxnxkpxcB

x yyy

222 )(,)(1)( xpcBxnxpkxpx y

pycB

and qzE

and

The equations can be decoupled in case of lossless

structures (real n)

Lossless structures (real n)

LnAxqxE z

2

tr

2Im

qEeHES zx ImRe2

1 2

)()()( xExkqxEx zz

222 ,)()( xExnxqkxqx z

inA

refAtrA

xLx 0x

is conserved

decoupled

??

known

Technique

inA

refAtrA

xLx 0x

22 )()( nxqkxqx

)()( LinLq)0()0(

)0()0(

in

ref

qin

qin

A

Ar

01 TRT

Advantage

Speed - for lossless structures – only 1 equation

Disadvantage

Switching between p and q formulation during the numerical integration

FD-TD

0

0,2

0,4

0,6

0,8

1

1,549 1,55 1,551 1,552 1,553 1,554 1,555

wavelength [ m]

Tra

nsm

issi

on

D=24

n_eff, D=24

D=48

n_eff, D=48

D=72

n_eff, D=72

analyticky

FD-TD: phase velocity corrected algorithm

A. Christ, J. Fröhlich, and N. Kuster, IEICE Trans. Commun., Vol. E85-B (12), 2904-2915 (2002).

FD-TD: convergence

0,01%

0,10%

1,00%

10,00%

100,00%

0 20 40 60 80 100 120 140 160

relative step [ x/ ]

rela

tive

err

or

1

10

100

1000

10000

tim

e [m

in]

chyba Christ

chyba C

čas Christ

čas C

corrected algorithm

common formulation

Outline

• Introduction: slow-wave optical structures (SWS)• Basic properties of SWS

– System model – Bloch modes– Dispersion characteristics– Phase shift enhancement– Nonlinear SWS

• Numerical methods for nonlinear SWS– NI-FD– FD-TD

• Results for nonlinear SWS

Results for COST P11 SWS structure

is the same in both layers

2

2

in nA nonlinearity level

F. Morichetti, A. Melloni, J. Čáp, J. Petráček, P. Bienstman, G. Priem, B. Maes, M. Lauritano, G. Bellanca, „Self-phase modulation in slow-wave structures: A comparative numerical analysis,“ Optical and Quantum Electronics 38, 761-780 (2006).

0

Transmission spectra

1 period

2 periods

3 periods

Tra

nsm

ittan

ce

normalized incident intensity

λ =1.5505 μm

Here incident intensity is about 10-6

However usually 10-4 - 10-3

P. K. Kwan, Y. Y. Lu, “Computing optical bistability in one-dimensional nonlinear structures“ Opt. Commun. 238 (2004) 169-174.W. Ding, “Broadband optical bistable switching in one-dimensional nonlinear cavity structure,” Opt. Commun. 246 (2005) 147-152.J. He and M. Cada ,”Optical Bistability in Semiconductor Periodic structures,” IEEE J. Quant. Electron. 27 (1991), 1182-1188. S. Blair, “Nonlinear sensitivity enhancement with one-dimensional photonic bandgap structures,” Opt. Lett. 27 (2002) 613-615.A. Suryanto et al., “A finite element scheme to study the nonlinear optical response of a finite grating without and with defect,” Opt. Quant. Electron. 35 (2003), 313-332.

10-2

L. Brzozowski and E.H. Sargent, “Nonlinear distributed-feedback structures as passive optical limiters,” JOSA B 17 (2000) 1360-1365.

Upper limit of the most transparent materials 10-4

S. Blair, “Nonlinear sensitivity enhancement with one-dimensional photonic bandgap structures,” Opt. Lett. 27 (2002) 613-615.

Here incident intensity is about 10-6

However usually 10-4 - 10-3

Are the high intensity effects important?

(e.g. multiphoton absorption)

Max

imum

nor

mal

ized

inte

nsity

insi

de t

he s

truc

ture

normalized incident intensity

2 periods

3 periods

Selfpulsing

-100%

-75%

-50%

-25%

0%

25%

50%

75%

100%

0 50 100 150 200 250 300

Čas [ps]

Pro

pu

stn

os

t

-1

-0,5

0

0,5

1

1,5

2

2,5

3

Fá

ze [

p]

1,55015

1,55017

1,55019

1,55021

1,55023

1,55025

Selfpulsing

-100%

-75%

-50%

-25%

0%

25%

50%

75%

100%

0 50 100 150 200 250 300

Čas [ps]

Pro

pu

stn

os

t

-1

-0,5

0

0,5

1

1,5

2

2,5

3

Fá

ze [ ]

1,55030

1,55035

1,55040

1,55045

1,55050

Conclusion

SWS could play an important role in the development of nonlinear optical components suitable for all-optical high-bit-rate communication systems.