An interesting phenomenon of Sub-Bragg diffreaction in 2D photonic crystals would be discussed here. I would like to acknowledge Dr. R.V Nair, IIT Ropar for providing knowledge about such an interesting topic.

1. Sub-Bragg Diffraction in Two-
dimensional Photonic Crystals
Aditya Bhardwaj

2. Introduction
Photonic crystals (PCs) are materials which have a periodic dielectric
constant in three-orthogonal spatial directions. Based on variation in
refractive index, PCs are classified as 1D, 2D and 3D.
1D-periodic in
one direction
2D-periodic in
two direction
3D-periodic in
three direction
High ε-region Low ε-region

3. Reciprocal Space
a
2π/a
REAL SPACE
RECIPROCAL SPACE
,
Real space
lattice vectors
Reciprocal space
lattice vectors
a
a
a



3
2
1
,
,
b
b
b



3
2
1
,
,
)
.(
2
=
3
2
1
3
2
1
a
a
a
a
a
b 








)
.(
2
=
1
3
2
1
3
2
a
a
a
a
a
b 








)
.(
2
=
2
1
3
2
1
3
a
a
a
a
a
b 









4. Brillouin Zone in 2D
Wigner -Seitz cell in reciprocal space is known as brillouin zone. Wigner cell
is a region in space that is closest to a given lattice point, than to any other
lattice point in reciprocal space.
Wigner -Seitz cell for a 2D crystal
1 st, 2 nd and 3 rd Brillouin zone

5. Diffraction
Bragg and Von Laue condition
θ
θ
d
Crystal planes
Incident X-ray Reflected X-ray
Let are the primitive vectors of crystal
lattice. The three Laue conditions for scattering vector,
Laue equations for integer values of a reflection’s
reciprocal lattice indices (h,k,l) are as follows. a.∆k =
2πh, b .∆k = 2πk and c.∆k = 2πl
c
b
a



,
,
g
k
k i





0
is the condition for Bragg diffraction
in reciprocal space
k

0
is the wave vector for outgoing diffracted beam
ki

is the wave vector for incident beam
g= k-k’
-k
θ θ
K
k'
)
cos(
2 
 d
m 

cos
d

6. Photonic Stop Gap
Due to Bragg diffraction, frequency windows are formed that are forbidden for
propagation of light in certain directions. These frequency windows are known as stop
gaps, which are associated with absence of light propagation along specific directions in
the photonic crystals.
Formation of stop gap
0 π/d 2π/d
ω
k

7. Photonic Band Gap
A photonic band gap (PBG) crystal is a structure that could manipulate beams of light
in the same way semiconductors control electric currents. It is a forbidden energy range
caused due to variation in refractive index, in which light of specific frequency can
transmit.
Formation of photonic band gap
R. V. Nair et a. Phys. Rev. B, 83, 205313, 2011

8. Sub-Bragg Diffraction
• Diffraction due to a single Bragg plane is known as simple Bragg
diffraction and diffraction due to more than one Bragg plane is called
multiple Bragg diffraction. The diffraction that occurs below the frequency
of simple Bragg diffraction is known as Sub-Bragg diffraction.
• To study about Sub-Bragg diffraction, we will investigate the diffraction
conditions of 2D photonic crystals using reflectance spectroscopy in high-
symmetry directions.

9. First order
simple
Bragg
Reflectivity peak between
4700 and 7300 cm-1 is for
simple Bragg diffraction
and between 9800 and
11100 cm-1 .
Bragg Diffraction

10. Sub-Bragg First order
simple Bragg Reflectivity peak between 5400
and 6900 cm-1 is for Sub-Bragg
diffraction and between 8100
and 10000 cm-1 is for Simple
Bragg diffraction
Sub-Bragg Diffraction
c
a
n a
c
eff
2
2
1
1
1
)
cos(
2
tan







R. V. Nair at al. Phys. Rev. Lett. 108, 083901 (2012)

11. Conclusions
THANK YOU
We studied about :
• Concept of brillouin zone, Bragg diffraction
• Photonic crystals, formation of photonic stop gap, photonic
band gap.
• Sub-Bragg diffraction in 2D photonic crystals.