non-linear photonic crystals
DESCRIPTION
Non-linear photonic crystals. Resumed by: D. Simeonov PO-014 Photonic crystals. Definition. Nonlinear photonic crystals (NPC) are periodic structures whose optical response depends on the intensity of the optical field that propagates into the crystal. At low light densities:. - PowerPoint PPT PresentationTRANSCRIPT
Non-linear photonic crystals
Resumed by: D. Simeonov
PO-014 Photonic crystals
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...),(),( )3()2()1(0 EEEEEtrEtrP
Definition
Nonlinear photonic crystals (NPC) are periodic structures whose optical response depends on the intensity of the optical field that propagates into the crystal.
At low light densities:
At high light densities:
Types of non-linear response in PC
...),(),( )3()2()1(0 EEEEEtrEtrP
With periodic modulation of the non-linear material properties
Non linear response due to optical Kerr effect
Modulated (2) for quasi-phase matching (QPM)Applications: harmonic generation, wave mixing, optical parametric amplifiers etc
Without periodic modulation of the non-linear material properties
(2) modulated NPC
1. Second harmonic generation (SHG) and phase matching2. Quasi phase matching (QPM)3. Phenomenological approach4. Analytical approach5. Fabrication techniques6. Some devices and applications7. 2D QPM-NPC8. Natural QPM-NPC
Where 2deff = (2)
SHGNon-linear polarization:
Second harmonic polarization:
Second harmonic polarization (vectorial representation):
SHG
Coherence length:
k=0
SHG gained over the traveled distance (l):
22 kkk
QPM for SHG
Proposed by N. Bloembergen in 1962
QPM for SHG
Maximal efficiency for 50/50 duty cycle and: )2()2(ab
The effective efficiency is reduced by factor of /2
QPM for SHG
Where
Second harmonic of the electric field:
(2) susceptibility in Fourier representation:
QPM for SHG
NKkk 2120
QPM when k’=0
GkNkk 2120
After integration:
The lattice reciprocal vectors can help for momentum conservation
QPM generalized
For any frequency conversion process in media with periodic (2) it can be generalized:
Energy conservation law:
Momentum conservation law:
Such formalism can be derived for both 1D, 2D or 3D QPM-NPC crystals
Theory details
Some benefits of QPM
Methods and materials
•Periodic E field (via segmented electrode) + field-induced (2)
•‘Frozen-in' field-induced (2), in optical fibers
•Periodic destruction/reduction of nonlinearity via ion-implantation through a
mask
•Overgrowth on a template having periodic modulation of substrate
orientation →: semiconductor materials: GaAs, GaN
•Periodic modulation of pump intensity (corrugated capillary waveguide for
High Harmonic Generation)
•Periodic-poling of ferroelectrics, switching →-: LiBaNO3, etc…
•Many more…
Fabrication of PPLN
~30 m
•Easy to fabricate•The change could be either temporary or permanent
References:
Fabrication of PPLN
100 m
SEM top view of PPLN grating
PPLN tuning
Some results PPLN
Some results PPLN
Review for different techniques:
Some results PPLN
Some results PPLN
Some results PPLN
Some results PPLN
Some results PPLN
Fabrication of GaAs QPM NPC
Why GaAs?●Large nonlinearity, d14~ 100pm /V●Extensive transparency, 0.9 μm -17 μm●Mature technology
1st proposition – stacking thin plates (wafers):
A. Szilagyi, A. Hordvik, and H. Schlossberg, “A quasi-phase matching technique for efficient optical mixing and frequency doubling,” J. Appl. Phys., vol. 47, pp. 2025-2032, (1976) (2-5 plates, m = 3).
2nd proposition – growth inversion:
Ex: O. Levi et al Optics Lett. 27, 2091, (2002)
Fabrication of GaAs QPM NPC
Some results on GaAs QPM NPC
GaN QPM NPC
•Very large transparency window•Low efficiency
2D QPM NPC
Interesting for :•Compensation of very large phase mismatches•Simultaneous phase matching of several parametric processes•Very broad band OPO
Pioneering papers:
The
ory
Exp
erim
ent
2D QPM NPC
•Constant linear dielectric constant•Periodically modulated (2) constant
)()2()2( r
Where r is an in-plane vector
2D QPM NPC
Parametric process (SHG) in 2D:
The periodically modulated (2) constant can be represented as a Fourier series:
Where G are the available vectors from the reciprocal lattice (RL), and kG is its corresponding Fourier coefficient
~
2D QPM NPC
Phase matching condition (momentum conservation law):
While deff ~ kG
Reciprocal lattice (RL) representation
2D QPM NPCNonlinear Ewald construction
In the RL space:1. Draw 2.kin the right direction
finishing at an origin;2. Draw a circle with center Ce.s.;3. Where the circle passes trough
an origin – successful phase matching is possible.Gmn
In 2D basis:Gmn = m Gx + n Gy
Can be generalized for of plane incident light.
Observation of SHG in 2D QPM NPC
Hexagonally Poled Lithium Niobate: A Two-Dimensional Nonlinear Photonic Crystal
k2 - 2k - Gmn = 0
Natural 2D QPM NPC
Existence of natural structures 2D QPM NPC
Sr0.61Ba0.39Nb2O6 (SBN) At a Currie temperature the SBN crystal exhibit a phase transition to form random size (given distribution) of needle like domains with opposite sign (2)
Such crystals are natural 2D QPM NPC and for:
)(~ pkG Where p() is the probability of existence of domain size =G/
SHG in natural 2D QPM NPC
SHG in natural 2D QPM NPCInteresting but complicated analytically:
Out of plane incident light
Central symmetry due to the random size distribution:•The G (kG) vector magnitudes are given by the domain size distribution•All possible G vectors exist in all directions perpendicular to the domains
Conical SHG
(3) NPC
1. Definition2. Analytical considerations3. Photonic crystals with Kerr type defects4. Kerr effect super-prism5. Kerr type PC - optical response
6. Non-linear modes, spatial optical solitons7. Analytical description
(2) NPC conclusion
1. Used for assure the momentum conservation law for various non-linear parametric processes
2. Experimental techniques demonstrated it utility
3. Widely used and commercially available
4. A Fourier representation of (2) gives both the available vectors in the reciprocal space and the efficiency coeficients
(3) NPC
Dynamical switching of the optical response based on AC Kerr effect:
Periodic modulation of the linear part of the refractive index as standard PCThe optical response is based on that of a linear PC
Types:Insertion of defects exhibiting Kerr type non-linearityThe material exhibits high Kerr non-linearity
Studied phenomena:
Switching of the properties of photonic crystal using high intensity control beamMode self generated changes of the optical properties: soliton wavesHigh order harmonic generation
Some literature
Photonic Crystals with Kerr nonlinear effects:
Existence of stable nonlinear localized modes in 2D & 3D PC S.John et al., PRL, 71 1168 (1993)
Controlling transmission in 1D PC M.Scalora et al., PRL, 73 1368 (1994), P.Tran , Opt. Lett, 21 1138
(1996)
Nonlinear guiding modes in 2D PC A.R. McGurn, Phys. Lett. A, 251 322 (1999)
Tunable microcavity for fast switching P.R. Villeneuve, Opt. Lett., 21 2017 (1996)
Analytical considerations
001.0/max nn
),(),( )3( trItrNL
Kerr non-linearity is small:
One of the materials is considered non-linear:
),(20 trInnnNL
Kerr non-linearity can be considered in perturbation theory
Diversity of Kerr type defects
A – Symmetric optical filterB – Asymmetric optical filterC – Optical bendD – Channel drop filterE – Waveguide branch
In absence of high power excitation – standard defect responseIn presence of high power excitation – switched defect response due to changed refractive index
Some literature
S. F. Mingaleev and Yu.S.KivsharEffective equations for photonic-crystal waveguides and circuitsOpt. Lett. 27, 231 (2002)
M Soljacic, C Luo, S Fan, and J. D. JoannopoulosNonlinear photonic crystal microdevices for optical integrationOpt. Lett. 28, 637 (2003)
M Soljacic, M Ibanescu, S G Johnson, Y Fink, and J. D. JoannopoulosOptimal bistable switching in nonlinear photonic crystalsPhys. Rev. E 66, 055601R (2002)
Theoretical proposals and descriptions:
Experimental observations:
Somebody should do them …
Linear Drop-off filter
2224 /)( resT
2 waveguides2 high Q factor microcavitiesHigh index rodsFiling factor - 0.2
2resQ
acres /)2(3697.0
In – Out symmetric transmission given by:
No power dependence
Bistable Drop-off filter
1-4 Transmission for high intensity signal4-3 Transmission for the reflected weak signal
Rods from Non-linear Kerr material
001.0/max nn
For carrier frequency:
res 0
3/)( 0 res
Expected bistability of the carrier transmission due to « resonance shift »
Bistable Drop-off filter
204
4/1
1
PPT
Non-linear transmission:
Where P0 is a characteristic power of the process ),,( 1max
20
nQfP res
Feasibility of Bistable Drop-off filter
Design parameters:
n2 = 1.5x10-17 m2/W (for GaAs n2 = 3x10-16 m2/W)
Q = 4000 (compatible with 10 Gbit/s)
0 = 1.55 m
Required conditions:
P0 = 15 mW
Working power 25 mW
Kerr effect super-prism
“Optically tunable superprism effect in nonlinear photonic crystals”,
N. - C. Panoiu, M. Bahl, and R. M. Osgood, Jr., Opt. Lett. 28, 2503 (2003).
GaAs-based PC slab:Kerr coefficient n2 = 3x10-16 m2/W. r/a 0.33
Dependence of the diffraction angle on the signal powerControllable diffraction angle via pump pulse
Kerr type PC - optical response
Calculated band structure of 1D GaAs – air PC (air gap DBR)
Solid curves – without switch beam
Dashed curves – with intense switch beam
Kerr type PC - optical response
Solitons in NPC
Temporal solitons:Kerr type PC (PC waveguide)Negative dispersion mode
Spatial solitons:Can exist in almost any Kerr type PCCan design PC for their interactionCan use them for loss-less bends
Analytical description
Solution of the corresponding non-linear Schrödinger equation:
Description in coupled-mode theory
Some literature
Some more literature
Conclusion
NPC structures offer VERY wide range of possibilities:
• Harmonic generations• All optically tunable PC optical response• Solitons and localized states• Very nice theoretical approaches
Thank you for Your patience
Introduction to solitons
In optics, the term soliton is used to refer to any optical field that does not change during propagation because of a delicate balance between nonlinear and linear effects in the medium. There are two main kinds of solitons:
Spatial solitons: the nonlinear effect can balance the diffraction. The electromagnetic field can change the refractive index of the medium while propagating, thus creating a structure similar to a graded-index fiber. If the field is also a propagating mode of the guide it has created, then it will remain confined and it will propagate without changing its shape
Temporal solitons: if the electromagnetic field is already spatially confined, it is possible to send pulses that will not change their shape because the nonlinear effects will balance the dispersion. Those solitons were discovered first and they are often simply referred as "solitons" in optics.
Temporal solitons
Anomalous (negative) dispersion+
Kerr effect=
Temporal soliton
Can propagate without changing form
Does not change during collision
Can interact with other solitons