2.710 Optics
Light Propagation in Sub-wavelength
Modulated Media
6th May 2009
Chong Shau Poh, Naveen Kumar Balla & Kalpesh Mehta
Outline
•Photonic Crystals and Electromagnetism
•Understanding FDTD
•Simulation results
Photonic Crystals
• Photonic crystals are artificial media with
periodic index contrast.
• The periodicity or spacing determines the
relevant light frequencies.
Maxwell’s Equations
∇•B = 0
∇ • D = 4πρ
1 ∂B
∇× E +
=0
c ∂t
1 ∂D 4π
∇× H −
=
J
c ∂t
c
In the absence of free charges and
currents, we can set ρ = J = 0.
Electromagnetism as an
Eigenvalue Problem
E(r,t) = E(r)e iωt
H (r,t) = H (r)e iωt
1
ω 2
∇×(
∇ × H (r)) = ( ) H (r)
ε (r)
c
ΘH n = λn H
n
General Properties of the
Harmonic Modes
• ω2 is real and positive
• Two modes H1(r) and H2(r) at different
frequencies ω1 and ω2 are orthogonal
• Scale invariance – the solution at one scale
determines the solution at all other length
scale
Bloch Theorem for
electromagnetism
In a periodic dielectric medium,
i.e. ε(r+a)= ε(r),
then the solution to the Master’s Equation has to
satisfy:
H(r) = ei(k·r)uk(r)
where uk(r) = uk(r+a) is a periodic function.
FDTD
• Finite-difference Time-domain methods are
widely used in computational electromagnetics
to analyze interactions between
electromagnetic waves and complex dielectric
or metallic structures.
• Able to compute the respond of a linear system
at many frequencies with a single computation
by just taking the Fourier transform of the
response to a short pulse.
FDTD
• Approximating Maxwell’s equation in real
space using finite differences
• Imposing appropriate boundary conditions
• Explicitly time-marching the fields to obtain
the direct time-domain response
Yee’s Lattice
Difference Equations
f (x, t2 ) − f (x, t1 ) f (x, t2 ) − f (x, t1 )
∂f (x, t )
= lim
≈
Δt → 0
∂t
Δt
Δt
Image removed due to copyright restrictions.Please see
http://commons.wikimedia.org/wiki/File:Yee-cube.svg
∂f (x, t )
f (x2 , t ) − f (x1 , t1 ) f (x2 , t ) − f (x1 , t )
= lim
≈
Δx→0
∂x
Δx
Δx
∂Ex 1  ∂H z ∂H y
= 
−
∂t
ε  ∂y
∂z
∂E y 1  ∂H x ∂H z
= 
−
ε  ∂z
∂t
∂x
∂Ex 1  ∂H z ∂H y 
= 
−

∂t
ε  ∂y
∂z 
∂H x
1  ∂E ∂E y 
=−  z −

∂t
∂z 
µ  ∂y



∂H x
1  ∂E ∂E y 
=−  z −

∂t
∂z 
µ  ∂y
∂H y
1  ∂E ∂E 
=−  x − z 
µ  ∂z
∂t
∂x 
∂Ez 1  ∂H y ∂H x 
= 
−

ε  ∂x
∂t
∂y 
∂H z
1  ∂E y ∂E x 
=− 
−

µ  ∂x
∂t
∂y 
n−1/ 2
Exn − Exn−1 1  ΔH zn−1/2 ΔH y 
= 
−


Δt
ε  Δy
Δz 
n
H xn+1/2 − H xn−1/2 1  ΔE zn ΔH y
= 
−
Δt
µ  Δy
Δz






• Periodic arranged rods of aluminium
• Spacing between adjacent rods a = 0.07
micrometer
y
x
Behavior at different frequencies
• Start wavelength = a/0.3
• Stop wavelength = a/0.1
y
x
Behavior at different frequencies
Dispersion diagram
1.4
1.2
Frequency (c/a)
1
0.8
0.6
0.4
0.2
0
0
0.1
0.2
0.3
Kx (2*pi/a)
0.4
0.5
• H(r) = ei(k·r)uk(r)
• If we take k = ko+2pi/a
• We will get similar
profile
• Thus we just need to
consider the k values
from the Brillouin zone
y
x
Lumerical example file
Controlling the light
Focusing effect
• At center spacing
between two adjacent
rows=a
• We increased the
spacing at the rate
0.15a, as we go further
away from the central
line
The focusing effect of graded index photonic crystals
H Kurt et al. Applied physics letters 93, 171108 (2008)
Focusing effect
Focusing effect
FDTD 1D Example
2
∂ 2u
2 ∂ u
= c
2
∂t 2
∂x
∂ 2u
∂t 2
∂ 2u
∂x 2
t +dt
x
u
x,t
u xt +dt − 2u xt + u xt −dt
=
dt 2
x,t
u xt +dx − 2u xt + u xt −dx
=
dx 2
 u xt +dx − 2u xt + u xt −dx 
t
t −
dt
=
(cdt ) 
 +
2u
x − u x
2
dx


2
Yee’s Lattice
Difference Equations
f (x, t2 ) − f (x, t1 ) f (x, t2 ) − f (x, t1 )
∂f (x, t )
= lim
≈
Δt → 0
∂t
Δt
Δt
Image removed due to copyright restrictions.Please see
http://commons.wikimedia.org/wiki/File:Yee-cube.svg
∂f (x, t )
f (x2 , t ) − f (x1 , t1 ) f (x2 , t ) − f (x1 , t )
= lim
≈
Δx→0
∂x
Δx
Δx
∂Ex 1  ∂H z ∂H y
= 
−
∂t
ε  ∂y
∂z
∂E y 1  ∂H x ∂H z
= 
−
ε  ∂z
∂t
∂x
∂Ex 1  ∂H z ∂H y 
= 
−

∂t
ε  ∂y
∂z 
∂H x
1  ∂E ∂E y 
=−  z −

∂t
∂z 
µ  ∂y



∂H x
1  ∂E ∂E y 
=−  z −

∂t
∂z 
µ  ∂y
∂H y
1  ∂E ∂E 
=−  x − z 
µ  ∂z
∂t
∂x 
∂Ez 1  ∂H y ∂H x 
= 
−

ε  ∂x
∂t
∂y 
∂H z
1  ∂E y ∂E x 
=− 
−

µ  ∂x
∂t
∂y 
n−1/ 2
Exn − Exn−1 1  ΔH zn−1/2 ΔH y 
= 
−


Δt
ε  Δy
Δz 
n
H xn+1/ 2 − H xn−1/ 2 1  ΔE zn ΔH y
= 
−
Δt
µ  Δy
Δz






Summary
• Photonic crystals have various interesting properties
and can be used to control the behavior of the light at
micro-scale
• Characterization of it requires use of numerical
techniques
References
• John D Joannopoulos, Johnson SG, Winn JN & Meade
RD (2008). Photonic Crystals: Molding the Flow of Light
(2nd ed.). Princeton NJ: Princeton University Press.
• Shanhui Fan. Thesis for the degree of Doctor of
Philosophy at the MIT. Photonic Crystals: Theory and
Device Applications.
• A. Taflove and S. C. Hagness, Computational
Electrodynamics: The Finite-Difference TimeDomain Method(3rd ed.). Norwood, MA: Artech
House, 1995.
• H. Kurt, E. Colak, O.Cakmak, H. Caglayan and E. Ozbay.
“The Focusing Effect of Graded Index Photonic
Cyrstals” Appl. Phys. Lett. 93, 171108(2008)
Acknowledgements
• Satoshi Takahashi
MIT OpenCourseWare
http://ocw.mit.edu
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Spring 2009
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