Download SWS

Coupled resonator slow-wave
optical structures
Jiří Petráček, Jaroslav Čáp
[email protected]
Parma, 5/6/2007
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
Ch. Liu, Z. Dutton, et al.: „Observation of coherent optical
information storage in an atomic medium using halted light pulses,“
Nature 409 (2001) 490-493
Stimulated Brillouin scattering
Miguel González Herráez, Kwang Yong Song, Luc Thévenaz:
„Arbitrary bandwidth Brillouin slow light in optical fibers,“ Opt.
Express 14 1395 (2006)
Slow-wave optical structures (SWS) –
– pure optical way
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
Various implementations
of SWSs
coupled Fabry-Pérot cavities
1D coupled PC defects
2D coupled PC defects
coupled microring resonators
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
System model of 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
fn
bn
M
d
f n 1
bn 1
Relation between amplitudes
fn
f n 1
r
t
bn
r
bn  rf n  tbn1
bn 1
fn
bn 1
t
r
bn
f n1  tfn  rbn1
f n 1
r  t 1
2
2
Transmission matrix
fn
bn
M
 f n 1 
 fn 

  M  
 bn 1 
 bn 
1t2  r2 r 

M  
t  r
1 
f n 1
f n1  tfn  rbn1
bn 1
bn  rf n  tbn1
For lossless SWS it follows from symmetry:
fn
bn
M
d
f n 1
t  is exp  ikd 
bn 1
real – (coupling ratio)
r  r1 exp  ikd 
real
r  s 1
2
1
2
r1

1  t 2  r 2 r  1   exp( ikd )
  

M  

 r1
exp( ikd ) 
t  r
1  is 
Propagation in periodic structure
fn
bn
M
d
f n 1  exp( id ) f n
bn 1  exp( id )bn
Bloch modes
fn
bn
M
f n 1  exp( id ) f n
bn 1  exp( id )bn
d
eigenvalue eq. for the
propagation constant
of Bloch modes
f
f
M    exp( id ) 
b
b
sin( kd )
cos( d ) 
s
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)
1.6
s = 0.1
s = 0.3
s = 0.5
s = 0.7
s = 0.9
1.4
kd/2
1.2
FSR
f res
nd
c
f res
nd
c
f res
nd
c
nd
c
1
nd
FSR
c
0.8
0.6
0.4
-0.5
0
 d/2
0.5
Re( d)/2
Dispersion curves
0.4
s = 0.1
s = 0.3
s = 0.5
s = 0.7
s = 0.9
sin( kd )
cos( d ) 
s
0.2
0
0
0.1
0.2
0.3
0.4
0.5
kd/2
0.1
0.2
0.3
kd/2
0.4
0.5
Im( d)/2
-0.1
-0.2
-0.3
-0.4
-0.5
0
0.6
0.7
0.8
0.9
1
Re( d)/2
Bandwidth, B
0.4
s = 0.1
s = 0.3
s = 0.5
s = 0.7
s = 0.9
sin( kd )
cos( d ) 
s
0.2
0
0
0.1
0.2
0.3
at the edges of pass-band
0.4
0.5
kd/2
0.6
nd
B
c
s   sin( kd )
kd  2 arcsin s
B2
FSR

arcsin s
0.7
0.8
0.9
1
Re( d)/2
Group velocity
0.4
s = 0.1
s = 0.3
s = 0.5
s = 0.7
s = 0.9
sin( kd )
cos( d ) 
s
0.2
0
0
0.1
0.2
0.3
d
dk
vg 
v
d
d
s  sin (kd)

v
cos(kd)
vg
2
for resonance
frequency
2
vg
v
 s
0.4
0.5
kd/2
0.6
0.7
0.8
0.9
1
Re( d)/2
Group velocity
0.4
s = 0.1
s = 0.3
s = 0.5
s = 0.7
s = 0.9
sin( kd )
cos( d ) 
s
0.2
0
0
0.1
0.2
0.3
0.4
0.5
kd/2
0.6
0.7
0.4
0.5
kd/2
0.6
0.7
1
s 2  sin 2 (kd)

v
cos(kd)
0.8
0.6
g
v /v
vg
0.4
0.2
0
0.3
GVD: very strong
minimal
very strong
0.8
0.9
1
Infinite vs. finite structure
dispersion relation
f / f res
d / 
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-dimensional
Fabry-Perot cavity placed between two distributed Bragg reflectors
DBR
DBR
Finite structure consisting 1, 3 and 5 resonators
1
M=1
M=23
M=35
Transmittance
0.8
0.6
0.4
0.2
0
-150
-100
-50
0
Frequency f-f
res
50
[GHz]
100
150
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
number of
resonators
theory
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.
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
L Ndn Leff n
g  

vg
sc
c
  1Leff 
1 Nd
n
g 
1c
s
loss per unit length
loss
B FSR
Buse  
s
2

(usable bandwidth, small coupling)
Jacob Scheuer, Joyce K. S. Poon, George T. Paloczi and Amnon Yariv, “Coupled Resonator
Optical Waveguides (CROWs),” www.its.caltech.edu/~koby/
Tradeoffs among delay, losses and bandwidth
10 resonators
FSR = 310 GHz
propagation loss = 4 dB/cm
s
Jacob Scheuer, Joyce K. S. Poon, George T. Paloczi and Amnon Yariv, “Coupled Resonator
Optical Waveguides (CROWs),” www.its.caltech.edu/~koby/
Phase shift ...
effective phase shift experienced by the optical field
propagating in SWS over a distance d
eff  d
  kd
sin( kd )
cos( d ) 
s
eff
v


vg
... is enhanced by the slowing factor
Nonlinear phase shift
 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
Total enhancement:
eff  v 
 
v 

 g
J.E. Heebner and R. W. Boyd, JOSA B 4, 722-731, 2002
2
Advantage of non-linear SWS:
nonlinear processes are enhanced
without affecting bandwidth
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.
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-dimensional
Fabry-Perot cavity placed between two distributed Bragg reflectors
DBR
DBR
Kerr non-linear layers
Integration of Maxwell Eqs. in frequency domain
Ain
Atr
Aref
x0
xL
x
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.
Maxwell Eqs.
Ain
Atr
Aref
x0
xL
x

E z ( x)  ikcB y ( x)
x





2
2
cB y ( x)  ikn x, E z x  E z ( x)
x

n x, E z  x 
2
  n x   n x   E x 
0
2
2
z
Now it is necessary to formulate boundary conditions.
Analytic solution in linear outer layers
Ain
Atr
Aref
x0
xL
x
Ez ( x)  Ainexp  in(0)kx  Aref exp in(0)kx
cB y ( x)  n(0) Ain exp  in (0)kx  Aref exp in (0)kx
Ez ( x)  Atrexp in( L)k x  L 
cBy ( x)  n( L ) Atr exp  in ( L )k x  L 
Boundary conditions
Ain
Atr
Aref
x0
E z (0)  Ain  Aref
cB y (0)  n(0) Aref  Ain 
xL
x
E z ( L)  Atr
cB y ( L)  n( L ) Atr
Admittance/Impedance concept
q p
1

icB y
Ez
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.
new ODE systems for
q and E z

E z ( x)  kq( x) E z ( x)
x



2
2
2
q ( x )   k q ( x )  n x, E z  x 
x
p and cB y





2

2
cBy ( x)  kp( x)n x, pcBy ( x) cBy ( x)
x


2

2
2
p( x)  k 1  p ( x)n x, pcBy ( x)
x

The equations can be decoupled in case of lossless
structures (real n)
Lossless structures (real n)
Ain
Atr
Aref
x0
xL


 1
 

2
S  Re E  H  e x E z Imq 
2
x
is conserved
E z  x  Imqx   Atr nL  
2
2

E z ( x)  kq( x) E z ( x)
x



2
2
2
q ( x )   k q ( x )  n x, E z  x 
x

decoupled
Technique
known
Ain
Atr ?
Aref ?
x0
xL
Aref in (0)  q(0)
r

Ain in (0)  q(0)
x
q( L)  in ( L )


q ( x)  k q 2 ( x)  n 2
x
T  RT   1  0

Advantage
Speed - for lossless structures – only 1
equation
Disadvantage
Switching between p and q formulation
during the numerical integration
FD-TD
1
D=24
n_eff, D=24
D=48
n_eff, D=48
D=72
n_eff, D=72
analyticky
Transmission
0,8
0,6
0,4
0,2
0
1,549
1,55
1,551
1,552
wavelength [ m]
1,553
1,554
1,555
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
100,00%
10000
common formulation
chyba Christ
10,00%
1000
chyba C
čas C
1,00%
100
0,10%
10
time [min]
relative error
čas Christ
corrected algorithm
0,01%
0
20
40
60
80
100
relative step [ x/ ]
120
140
1
160
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
  Ain n2
2
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).
Transmission spectra
 0
1 period
2 periods
3 periods
Transmittance
λ =1.5505 μm
normalized incident intensity

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.
Here incident intensity is about 10-6
However usually 10-4 - 10-3
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.
Are the high intensity effects important?
(e.g. multiphoton absorption)
normalized incident intensity
Maximum normalized intensity inside the structure
2 periods
3 periods
Selfpulsing
3
75%
2,5
50%
2
1,55015
25%
1,5
1,55017
1,55019
1,55021
0%
1
1,55023
1,55025
-25%
0,5
-50%
0
-75%
-0,5
-100%
0
50
100
150
Čas [ps]
200
250
-1
300
Fáze [p]
Propustnost
100%
Selfpulsing
3
75%
2,5
50%
2
25%
1,5
1,55030
0%
1
1,55035
1,55040
1,55045
-25%
0,5
1,55050
-50%
0
-75%
-0,5
-100%
0
50
100
150
Čas [ps]
200
250
-1
300
Fáze []
Propustnost
100%
Conclusion
SWS could play an important role in the
development of nonlinear optical
components suitable for all-optical high-bitrate communication systems.