coupled resonator slow-wave optical structures
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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.
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