1. Non-linear photonic crystals
Resumed by: D. Simeonov
PO-014 Photonic crystals

2. 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:

3. Types of non-linear response in PC
With periodic modulation of the non-linear material properties
Modulated c(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

4. c(2) modulated NPC Second harmonic generation (SHG) and phase matching
Quasi phase matching (QPM)
Phenomenological approach
Analytical approach
Fabrication techniques
Some devices and applications
2D QPM-NPC
Natural QPM-NPC

5. SHG Non-linear polarization: Second harmonic polarization:
Where 2deff = c(2)
Second harmonic polarization (vectorial representation):

6. SHG
SHG gained over the traveled distance (l):
Dk=0
Coherence length:

7. QPM for SHG
Proposed by N. Bloembergen in 1962

8. QPM for SHG Maximal efficiency for 50/50 duty cycle and:
The effective efficiency is reduced by factor of p/2

9. QPM for SHG Second harmonic of the electric field:
c(2) susceptibility in Fourier representation:
Where

10. QPM for SHG QPM when Dk’=0 After integration:
The lattice reciprocal vectors can help for momentum conservation

11. QPM generalized
For any frequency conversion process in media with periodic c(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

12. Theory details

13. Some benefits of QPM

14. Methods and materials
Periodic E field (via segmented electrode) + field-induced c(2)
‘Frozen-in' field-induced c(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 c(2) →-c(2): semiconductor materials: GaAs, GaN
Periodic modulation of pump intensity (corrugated capillary waveguide for High Harmonic Generation)
Periodic-poling of ferroelectrics, switching c(2) →-c(2): LiBaNO3, etc…
Many more…

15. Fabrication of PPLN ~30 mm References: Easy to fabricate
The change could be either temporary or permanent

16. Fabrication of PPLN
SEM top view of PPLN grating
100 mm

17. PPLN tuning

18. Some results PPLN

19. Some results PPLN
Review for different techniques:

20. Some results PPLN

21. Some results PPLN

22. Some results PPLN

23. Some results PPLN

24. Some results PPLN

25. 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 , (1976) (2-5 plates, m = 3).
2nd proposition – growth inversion:
Ex: O. Levi et al Optics Lett. 27, 2091, (2002)

26. Fabrication of GaAs QPM NPC

27. Some results on GaAs QPM NPC

28. GaN QPM NPC
Very large transparency window
Low efficiency

29. 2D QPM NPC Interesting for :
Compensation of very large phase mismatches
Simultaneous phase matching of several parametric processes
Very broad band OPO
Pioneering papers:
Theory
Experiment

30. 2D QPM NPC Constant linear dielectric constant
Periodically modulated c(2) constant
Where r is an in-plane vector

31. 2D QPM NPC ~ Parametric process (SHG) in 2D:
The periodically modulated c(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 ~

32. 2D QPM NPC Reciprocal lattice (RL) representation
Phase matching condition (momentum conservation law):
While deff ~ kG

33. 2D QPM NPC Nonlinear Ewald construction In the RL space:
Draw 2.kw in the right direction finishing at an origin;
Draw a circle with center Ce.s.;
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.

34. Observation of SHG in 2D QPM NPC
Hexagonally Poled Lithium Niobate: A Two-Dimensional Nonlinear Photonic Crystal
k2w - 2kw - Gmn = 0

35. Natural 2D QPM NPC Existence of natural structures 2D QPM NPC
At a Currie temperature the SBN crystal exhibit a phase transition to form random size (given distribution) of needle like domains with opposite sign c(2)
Sr0.61Ba0.39Nb2O6 (SBN)
Such crystals are natural 2D QPM NPC and for:
Where p(L) is the probability of existence of domain size L=G/p

36. SHG in natural 2D QPM NPC

37. 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

38. Conical SHG

39. c(3) NPC Definition Analytical considerations
Photonic crystals with Kerr type defects
Kerr effect super-prism
Kerr type PC - optical response
Non-linear modes, spatial optical solitons
Analytical description

40. c(2) NPC conclusion
Used for assure the momentum conservation law for various non-linear parametric processes
Experimental techniques demonstrated it utility
Widely used and commercially available
A Fourier representation of c(2) gives both the available vectors in the reciprocal space and the efficiency coeficients

41. c(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

42. Some literature Photonic Crystals with Kerr nonlinear effects:
Existence of stable nonlinear localized modes in 2D & 3D PC
S.John et al., PRL, (1993)
Controlling transmission in 1D PC
M.Scalora et al., PRL, (1994), P.Tran , Opt. Lett, (1996)
Nonlinear guiding modes in 2D PC
A.R. McGurn, Phys. Lett. A, (1999)
Tunable microcavity for fast switching
P.R. Villeneuve, Opt. Lett., (1996)

43. Analytical considerations
One of the materials is considered non-linear:
Kerr non-linearity is small:
Kerr non-linearity can be considered in perturbation theory

44. 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

45. 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, R (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 …

46. Linear Drop-off filter
2 waveguides
2 high Q factor microcavities
High index rods
Filing factor - 0.2
In – Out symmetric transmission given by:
No power dependence

47. Bistable Drop-off filter
Rods from Non-linear Kerr material
For carrier frequency:
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

48. Bistable Drop-off filter
Non-linear transmission:
Where P0 is a characteristic power of the process

49. 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

50. 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).

51. 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

52. Kerr type PC - optical response

53. 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

54. Analytical description
Description in coupled-mode theory
Solution of the corresponding non-linear Schrödinger equation:

55. Some literature

56. Some more literature

57. 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

58. Thank you for
Your patience

59. 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.

60. Anomalous (negative) dispersion
Temporal solitons
Anomalous (negative) dispersion +
Kerr effect =
Temporal soliton
Can propagate without changing form
Does not change during collision
Can interact with other solitons