This document outlines a presentation on photonic crystals and nonlinear optics. It discusses: - What photonic crystals are and how they inhibit light propagation through periodic refractive index patterns. Maxwell's equations are used to model light propagation in these structures. - Common photonic crystal topologies in 1D, 2D and 3D, including photonic bandgap properties. Applications like mirrors and waveguides are mentioned. - How nonlinear optical effects like the Pockels and Kerr effects modify a material's refractive index with an electric field. Nonlinear photonic crystals combine these effects. - The document concludes by proposing nonlinear photonic crystals can act as optical limiters that regulate light transmission intensity.

1. Supervised by: Prof. Dr. Hassan Elkamchouchi
Dr. Adel Elfahhar
Presented by: Eng. Islam Mohammed Salah Kotb
Alexandria University
Faculty of Engineering
Electrical Engineering Department

2. Outlines
 Introduction
 What are Photonic Crystals?
 Maxwell Equations and Bandgap
 Famous topologies
 Electro-optics and Nonlinear Optics
 Nonlinear Photonic Crystals
 Optical limiter

3. Outlines
 Introduction







4. Is it possible to have an All-Optical
Processor?

5. The answer is
Photonic Crystals

6. Outlines

 What are Photonic Crystals?






7.  Photonic crystals are regular arrays of materials
with different refractive indices arranged in such
a way to inhibit the propagation of light “Optical
Insulators”

8. 3µm
Photonic Crystals in Nature
wing scale:
Morpho rhetenor butterfly
[ P. Vukosic et al.,
Proc. Roy. Soc: Bio.
Sci. 266, 1403
(1999) ]
Peacock feather
[J. Zi et al, Proc. Nat. Acad. Sci. USA,
100, 12576 (2003) ]
[figs: Blau, Physics Today 57, 18 (2004)]
http://www.bugguy012002.com/MORPHIDAE.html
[ also: B. Gralak et al., Opt. Express 9, 567 (2001) ]

9. Electronic and Photonic Crystals
atoms in diamond structure
wavevector
electronenergy
Periodic
Medium
Blochwaves:
BandDiagram
dielectric spheres, diamond lattice
wavevector
photonfrequency

10. planewave
E,H ~ ei(k×x-wt)
k = w / c =
2p
l
k
scattering
It is known that light scatters when collides with
atoms.

11. here: scattering off three specks of silicon

12. What about many particles?

13. Bloch Theorem states waves in a periodic
medium can propagate without scattering
the light seems to form several coherent beams
that propagate without scattering
… and almost without diffraction (supercollimation)

14. • • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
•••
•••
•••
•••
•••
•••
•••
•••
•••
•••
•••
•••
•••
•••
for most λ, beam(s) propagate
through crystal without scattering
(scattering cancels coherently)
...but for some λ (~ 2a), no light can propagate: a photonic band gap
a
planewave
E,H ~ ei(k×x-wt)
k = w / c =
2p
l
k

15. Outlines


 Maxwell Equations and Bandgap





16. 0H.
E.















JE
t
H
H
t
E
Maxwell Equations
Maxwell equation in linear, isotropic, non-magnetic and homogeneous media
can be written as:

17. In optical materials there is no free charges or currents included so:
0H.
0E.










E
t
H
H
t
E
J








18. 00
2
2
2
2
2
2
0
2
2
0
1
where
1
)(
1
1
)(
1
)(
















c
H
tc
H
r
E
tc
E
r
r
E
t
E



Eigen Operator
(Hermitian):
•ω are real (lossless)
•eigen-states are
orthogonal
•eigen-states are complete
(give all solutions)

19. Bloch Theorem
The Bloch-Floquet theorem tells us that, for a Hermitian eigenproblem
whose operators are periodic functions of position, the solutions can always
be chosen of the form
)(),( ).(
rHetrH k
trki 




Corollary 1: k is conserved, i.e. no scattering of Bloch wave
Corollary 2: given by finite unit cell,
so ω are discrete ωn(k)
Hk
rki
e


20. k
k
H
c
Hki
r
ki
E
c
Ekiki
r


2
2
)(
)(
1
)(
)()(
)(
1


















•The field is periodic so solve over a finite domain a unit cell
(First Brilluin Zone)
•Solve for all frequencies which are : ωn(k) (for n=1,2,3,…. ).
•Plot as a function of the wavevector k, to form the band
structure of the crystal.
•Solution can be obtained by iterative numerical methods such
as Finite Difference Time Domain (FDTD).

21. Bloch’s theorem: solutions
are periodic in k
kx
kyfirst Brillouin zone
= minimum |k|
“primitive cell”
2p
aG
M
X
irreducible Brillouin zone: reduced by symmetry
Frequency(2πc/a)=a/λ
G GX M
a
2d periodicity,
=12:1

22. Outlines



 Famous topologies




23. 1 D Photonic Crystals
ε1 ε2 ε1 ε2 ε1 ε2 ε1 ε2 ε1 ε2 ε1 ε2
ε(x) = ε(x+a)a
band gap
k
0 π/a–π/a
irreducible Brillouin zone

24. Famous Applications
 Dielectric mirrors
 Fiber Bragg grating
2
3
4
5
6
7
8
9
10
Number of
periods
n1/n2 = 1.44
Transmission(dB)
Frequency (c/a)
0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0
-5
-10
-15
-20
-25
Varying number of layers
Transmission(dB)
Varying index ratio
Frequency (c/a)
0
-5
-10
-15
-20
-25
-30
-35
-40
-45
0 0.1 0.2 0.3 0.4 0.5 0.6
1.3
1.5
2
3
4
Index ratio

25. 2 D Photonic Crystals
 There are 2 famous topologies that are mostly used:
 Square Lattice
 Triangular Lattice

26. Square Lattice
 The irreducible Brillouin Zone of this lattice is an
isosceles right angled triangle
 TM band gaps are found using isolated regions of high
dielectric constant material imbedded in a
background of lower dielectric constant.

27.  TE band gaps, significant bands are found for isolated
regions of lower dielectric constant immersed in a
higher dielectric background.

28. Triangular Lattice
 TM band gaps tend to occur in lattices formed by
isolated high-permittivity regions, and TE band gaps
in connected lattices.
 the connected lattice tends to combine these two
advantages in the case where the diameter of the holes
approaches the period of the lattice.
 Easy to fabricate using semiconductors fabrication
techniques.

30. Point defects
 Introducing a point defect in any of the above mentioned topologies
will increase the field localization at this point.

32. Line defects
 When a line defect is introduced into a photonic crystal waveguides
may be created. Such waveguides can guide light at optical wavelengths
with minimal propagation losses.

33. Applications
 Beam Splitters

34. resonantfilters
channel-dropfilters
high-transmission
sharpbends
waveguidesplitters

35. 3 D Photonic Crystals
Yablonovite:
 The first experimental observation of a 3D complete photonic bandgap
was made by Eli Yablonovitch in 1991 using a variant of the diamond
lattice structure, now known as the Yablonovite.
 This structure has a complete gap when the refractive index is n = 3.6

36. Woodpile

37. Rod-hole (MIT)
• The structure is an fcc lattice of air (or low index) cylinders in
dielectric, oriented along the 111 direction. Such a structure results in a
system of two types: triangular lattices of air holes in dielectric and
dielectric cylinders (“rods”) in air.

38.  This structure offers a bandgap of 21% and even
over 8% for Si:SiO2 contrast (ε = 12:2); the PBG
persists down to ε contrasts of 4:1 (2:1 index
contrast).

39. Outlines




 Electro-optics and Nonlinear Optics



40. Electro-optics
 The electro-optic effect is a change in the refractive index
that results from the application of a steady or low-
frequency electric field. An electric field applied to an
anisotropic optical material modifies its refractive indexes
and thereby the effect that it has on polarized light passing
through it.
 The dependence of the refractive index on the applied
electric field has one of the two following forms:
 The refractive index changes in proportion to the applied
electric field, which is known as the linear electro-optic
effect or Pockets effect.
 The refractive index changes in proportion to the square of
the applied electric field, known as the quadratic electro-
optic effect or Kerr effect.

41. 3n
2
a
-
3n
1
a
2
23
2
13
2
1
)0()(
2)0()(
)2
22
1
1
(
3
2
2
10
asdefinedis)(lityimpermeabielectricThe
0
2
2
2
a,
0
1
a),0(Where
....,2
2
a
2
1
1
an(E)













































EnEnnEn
EEE
EaEa
n
n
dn
d
n
E
dE
nd
EdE
dn
nn
EEn
is called Packels Coefficient
is called Kerr Coefficient

42. Pockels Effect Kerr Effect

43. Nonlinear Optical Materials
 The refractive index, and consequently the speed of light in a nonlinear
optical medium, depends on light intensity.
 The principle of superposition is violated in a nonlinear optical
medium.
 The frequency of light is altered as it passes through a nonlinear optical
medium
 Photons interact within the confines of a nonlinear optical medium so
that light can indeed be used to control light.
 The properties of a dielectric medium through which an optical
electromagnetic wave propagates are described by the relation between
the polarization-density vector: P(r, t) and the electric-field vector E(r,
t). The mathematical relation between the vector functions P(r, t) and
E(r, t) which is governed by the characteristics of the medium, defines
the system. The medium is said to be nonlinear if this relation is
nonlinear.

45. where and are coefficients describing the strength of the second and third-order nonlinear effects,
Second-Order Nonlinear Optical Materials
•Second harmonic generation

46. Third-Order Nonlinear Optical Materials (kerr Medium)
•Third harmonic generation

47. Optical Kerr effect
n2=1+χ
is the optical intensity of the initial wave
η is the impedance of the medium
Δn=Δχ/2n
which is called Optical Kerr Coefficient

48. Outlines





 Nonlinear Photonic Crystals


49. Nonlinear optical Photonic Crystals
 Photonic Crystal (periodic dielectric) Photonic Bandgap
 Nonlinear Material (Kerr Medium) “n” changes with light intensity
New Outstanding Applications

50.  Nonlinear Photonic crystals can either be made of fully
nonlinear materials or by just adding nonlinear elements as
defects in a linear Photonic Crystal
=0
=0
Maxwell Equations
Nonlinear media

51. Maxwell Equations for nonlinear periodic media

52. This problem is solved using perturbation theory
ε(r)=ε0(r)+Δε

53. Outlines






 Optical limiter

54. Optical Limiter

55. 0.00E+00
2.00E-03
4.00E-03
6.00E-03
8.00E-03
1.00E-02
0.1 0.5 1 1.5 2 2.5 3 3.5 4 4.5
OutputPower
Input Power
1.16
1.23
1- Rectangular Lattice with line nonlinear defect
Lattice constant a = 0.57μm; rod radius r = 0.075 μm, RI of rods n = 3.5; the three nonlinear rods rnl = 0.12 μm,
n2 = 2.7x10-9 m2/W.

56. Rods are all nonlinear with nrod=1.87, nmedium=1.25 and χ(3)=0.0015.
2- Rectangular Lattice with two nonlinear defects

57. 3- Square Coupled Cavity waveguide (rods in low index dielectric):
•Dielectric rods with nr = 3.5, nb = 1.5. The
periodicity of the lattice is given by a, whereas
the radius of the holes is r = a/4.
• The resonant cavity has been created in this
system by increasing the radius of the central
hole to rd = 5a/3.
•The two line defects have been introduced by
reducing the radius of the corresponding holes to
r/3.

58. 4- Triangular Coupled Cavity waveguide (holes in dielectric substrate):
“a” is the lattice constant of the structure, the radius of the holes =0.3a, ns= 3.4 (GaAs),
nNL =2.6, n2 = 2.7×10-9 m2/W and the defect radius is rd = 0.25a, Defect period is 5a

59. All Optical AND gate

60. Future Work
 Controllable Threshold Optical Hard Limiter
 More Devices (XOR, OR,….)