Transcript Cristaux photoniques non

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:
At high light densities:



P   0 E





(1)
( 2)
(3)
P(r, t )   0  E(r, t )   EE   EEE  ...
Types of non-linear response in PC






(1)
( 2)
(3)
P(r, t )   0  E(r, t )   EE   EEE  ...
With periodic modulation of the non-linear material properties
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
Non linear response due to optical Kerr effect
(2) modulated NPC
1. Second harmonic generation (SHG) and phase matching
2. Quasi phase matching (QPM)
3. Phenomenological approach
4. Analytical approach
5. Fabrication techniques
6. Some devices and applications
7. 2D QPM-NPC
8. Natural QPM-NPC
SHG
Non-linear polarization:
Second harmonic polarization:
Where 2deff = (2)
Second harmonic polarization (vectorial representation):
SHG
SHG gained over the traveled distance (l):
Dk  2k  k2
Coherence length:
Dk=0
QPM for SHG
Proposed by N. Bloembergen in 1962
QPM for SHG
Maximal efficiency for 50/50 duty cycle and:

( 2)
b
The effective efficiency is reduced by factor of p/2
 
( 2)
a
QPM for SHG
Second harmonic of the electric field:
(2) susceptibility in Fourier representation:
Where
QPM for SHG
After integration:
QPM when Dk’=0
0  2k1  k2  K N
0  2k1  k2  NDkG
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 2 →2: semiconductor materials: GaAs, GaN
•Periodic modulation of pump intensity (corrugated capillary waveguide for
High Harmonic Generation)
•Periodic-poling of ferroelectrics, switching 2 →-2: LiBaNO3, etc…
•Many more…
Fabrication of PPLN
~30 mm
References:
•Easy to fabricate
•The change could be either temporary or
permanent
Fabrication of PPLN
SEM top view of PPLN grating
100 mm
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
Experiment
Theory
Pioneering papers:
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
Reciprocal lattice (RL) representation
Phase matching condition
(momentum conservation law):
While deff ~
kG
2D QPM NPC
Nonlinear Ewald construction
Gmn
In the RL space:
1. Draw 2.k in 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.
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:
kG ~ p (  )
Where p() is the probability of existence of domain size =G/p
SHG in natural 2D QPM NPC
SHG in natural 2D QPM NPC
Interesting 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. Definition
2. Analytical considerations
3. Photonic crystals with Kerr type defects
4. Kerr effect super-prism
5. Kerr type PC - optical response
6. Non-linear modes, spatial optical solitons
7. 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
Periodic modulation of the linear part of the refractive index as standard PC
The optical response is based on that of a linear PC
Dynamical switching of the optical response based on AC Kerr effect:
Types:
Insertion of defects exhibiting Kerr type non-linearity
The material exhibits high Kerr non-linearity
Studied phenomena:
Switching of the properties of photonic crystal using high intensity control beam
Mode self generated changes of the optical properties: soliton waves
High 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
One of the materials is considered non-linear:
 NL (r , t )     (3) I (r , t )
nNL  n0  n2 I (r , t )
Kerr non-linearity is small:
Dnmax / n  0.001
Kerr non-linearity can be considered in perturbation theory
Diversity of Kerr type defects
A – Symmetric optical filter
B – Asymmetric optical filter
C – Optical bend
D – Channel drop filter
E – Waveguide branch
In absence of high power excitation – standard defect response
In presence of high power excitation – switched defect response due to changed
refractive index
Some literature
Theoretical proposals and descriptions:
S. F. Mingaleev and Yu.S.Kivshar
Effective equations for photonic-crystal waveguides and circuits
Opt. Lett. 27, 231 (2002)
M Soljacic, M Ibanescu, S G Johnson, Y Fink, and J. D. Joannopoulos
Optimal bistable switching in nonlinear photonic crystals
Phys. Rev. E 66, 055601R (2002)
M Soljacic, C Luo, S Fan, and J. D. Joannopoulos
Nonlinear photonic crystal microdevices for optical integration
Opt. Lett. 28, 637 (2003)
Experimental observations:
Somebody should do them …
Linear Drop-off filter
2 waveguides
2 high Q factor microcavities
High index rods
Filing factor - 0.2
Q  res 2
res  0.3697(2pc) / a
In – Out symmetric transmission given by:

T4 ()   /     res 
2
2
No power dependence
2

Bistable Drop-off filter
Rods from Non-linear Kerr material
Dnmax / n  0.001
For carrier frequency:
0  res
  (res  0 ) /   3
Expected bistability of the carrier
transmission due to « resonance
shift »
1-4 Transmission for high intensity signal
4-3 Transmission for the reflected weak signal
Bistable Drop-off filter
Non-linear transmission:
1
T4 
2
1  P4 / P0   
Where P0 is a characteristic power of the process
1
P0  f (Q 2 , res , Dnmax
)
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)
l0 = 1.55 mm
Required conditions:
P0 = 15 mW
Working power 25 mW
Kerr effect super-prism
GaAs-based PC slab:
Kerr coefficient n2 = 3x10-16 m2/W.
r/a 0.33
Dependence of the diffraction
angle on the signal power
Controllable diffraction angle via
pump pulse
“Optically tunable superprism effect in nonlinear photonic crystals”,
N. - C. Panoiu, M. Bahl, and R. M. Osgood, Jr., Opt. Lett. 28, 2503 (2003).
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 PC
Can design PC for their interaction
Can use them for loss-less bends
Analytical description
Description in coupled-mode theory
Solution of the corresponding non-linear Schrödinger equation:
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