Coupled resonator slow-wave optical structures

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Coupled resonator slow-wave optical structures
Jirí Petrácek, Jaroslav Cáp petracek_at_fme.vutbr.cz
Parma, 5/6/2007
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• all-optical high-bit-rate communication systems
• optical delay lines
• memories
• switches
• logic gates
• ....

slow light
increased efficiency
nonlinear effects
3
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

4
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

5
Slow light

• the light speed in vacuum c
• phase velocity v
• group velocity vg

6
How to reduce the group velocity of light?
Electromagnetically induced transparency - EIT
Ch. Liu, Z. Dutton, et al. Observation of
coherent opticalinformation 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).
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Slow-wave optical structure (SWS)

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

- light propagates due to the coupling between
adjacent resonators
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Various implementations of SWSs
coupled Fabry-Pérot cavities
1D coupled PC defects
2D coupled PC defects
coupled microring resonators
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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

10
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.
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System model of SWS
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Relation between amplitudes
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Transmission matrix
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For lossless SWS it follows from symmetry
real (coupling ratio)
real
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Propagation in periodic structure
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Bloch modes
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.
17
Dispersion curves (band diagram)
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Dispersion curves
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Bandwidth, B
at the edges of pass-band
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Group velocity
for resonance frequency
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Group velocity
GVD very strong
very strong
minimal
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Infinite vs. finite structure
dispersion relation
Jacob Scheuer, Joyce K. S. Poonb, George T.
Paloczic and Amnon Yariv, Coupled Resonator
Optical Waveguides (CROWs), www.its.caltech.edu/
koby/
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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
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Finite structure consisting 1, 3 and 5 resonators
3
5
25
Fengnian Xia,a Lidija Sekaric, Martin OBoyle,
and Yurii Vlasov Coupled resonator optical
waveguides based on silicon-on-insulator photonic
wires, Applied Physics Letters 89, 041122 2006.
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experiment
number of resonators
theory
1550 nm
Fengnian Xia,a Lidija Sekaric, Martin OBoyle,
and Yurii Vlasov Coupled resonator optical
waveguides based on silicon-on-insulator photonic
wires, Applied Physics Letters 89, 041122 2006.
27
Fengnian Xia,a Lidija Sekaric, Martin OBoyle,
and Yurii Vlasov Coupled resonator optical
waveguides based on silicon-on-insulator photonic
wires, Applied Physics Letters 89, 041122 2006.
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Delay, losses and bandwidth
loss per unit length
loss
(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/
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Tradeoffs among delay, losses and bandwidth
10 resonators FSR 310 GHz propagation loss 4
dB/cm
Jacob Scheuer, Joyce K. S. Poon, George T.
Paloczi and Amnon Yariv, Coupled Resonator
Optical Waveguides (CROWs), www.its.caltech.edu/
koby/
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Phase shift ...
effective phase shift experienced by the optical
field propagating in SWS over a distance d

• ... is enhanced by the slowing factor

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

34
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
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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.
Petrácek Modelling of one-dimensional nonlinear
periodic structures by direct integration of
Maxwells equations in frequency domain. In
Frontiers in Planar Lightwave Circuit Technology
(Eds S. Janz, J. Ctyroký, S. Tanev) Springer,
2005.
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Maxwell Eqs.
Now it is necessary to formulate boundary
conditions.
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Analytic solution in linear outer layers
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Boundary conditions
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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. Petrácek Frequency-domain
simulation of electromagnetic wave propagation in
one-dimensional nonlinear structures, Optics
Communications 265 (2006) 331-335.
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new ODE systems for
and
and
The equations can be decoupled in case of
lossless structures (real n)
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Lossless structures (real n)
is conserved
decoupled
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Technique
known
?
?
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• Advantage
• Speed - for lossless structures only 1
  equation
• Disadvantage
• Switching between p and q formulation during the
  numerical integration

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FD-TD
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FD-TD phase velocity corrected algorithm
A. Christ, J. Fröhlich, and N. Kuster, IEICE
Trans. Commun., Vol. E85-B (12), 2904-2915 (2002).
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FD-TD convergence
common formulation
corrected algorithm
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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

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Results for COST P11 SWS structure
is the same in both layers
nonlinearity level
F. Morichetti, A. Melloni, J. Cáp, J. Petrácek,
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).
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Transmission spectra
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1 period
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2 periods
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3 periods
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? 1.5505 µm
Transmittance
normalized incident intensity
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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.
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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)
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Maximum normalized intensity inside the structure
normalized incident intensity
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2 periods
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3 periods
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Selfpulsing
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Selfpulsing
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Conclusion
SWS could play an important role in the
development of nonlinear optical components
suitable for all-optical high-bit-rate
communication systems.