11. Reflection/Transmission spectra
Contents
1.Normal incidence on a simple dielectric slab
2. Normal incidence on a photonic crystal slab
3. Normal incidence on a Distributed Bragg mirror
4. Normal incidence on a 1-D photonic crystal cavity
1. Simple dielectric slab
8
Without slab
6
Hx field
Planewave
source
4
2
0
-2
-4
-6
-8
0
2000
4000
6000
8000
10000
FDTD time step
Slab (T=1a)
6
With slab
Detect
Hx field
4
2
0
-2
-4
-6
0
2000
4000
6000
FDTD time step
8000
10000
 We will not use DS in this example
 DD shouldn’t be this much long because the simple Fabry-Perot slab
does not have a high-Q resonance. As we can see from the field data,
DD may be set 4000.
 (1,1) may be set (0.1, 0.1).
 Gamma point periodic boundary condition. The use of
the planewave source ensures that the result will be only
for the exact kx=ky=0 point.
 Two point detectors were set,
one after the slab and the other between
the planewave source and the slab
** How to obtain R & T spectra
1) We now have modeR.dat and modeT.dat
2) We must run another simulation in the absence of the
dielectric slab. Then, we get modeR0.dat and modeT0.dat,
which will be used as reference data.
3) Let’s denote
modeR.dat = R(t)
modeT.dat = T(t)
modeR0.dat = R0(t)
modeT0.dat = T0(t)
(Important Node )
One must remember that the resolution of discrete FT
will depend on the length of input data.
We must add null values at the end of all ***.dat file
before performing FT.
I typically make the entire length of the input data file to
be about 1 million
(should be 2n format for accurate result)
Now let’s perform the following calculations.
FT[ R(t)-R0(t) ] / FT[ R0(t) ] = Refelctance spectrum in w
FT[ T(t) ] / FT[ T0(t) ] = Transmission spectrum in w
,where FT denotes Fourier Transformation.
1.0
1.0
0.8
0.8
Reflectance
Transmittance
Result
0.6
0.4
0.6
0.4
0.2
0.2
0.0
0.1
0.0
0.1
0.2
0.3
0.4
0.5
Normalized Frequency
0.6
0.2
0.3
0.4
0.5
Normalized Frequency
0.6
2. Square lattice photonic crystal slab
 We will not use DS in this example
 In this example, DD should be carefully chosen. The 2-D photonic-crystal
slab may contain very high-Q resonances
(See the Fan’s paper)
 You cannot change this (1,1)
 Two point detectors were set,
one after the slab and the other between
the planewave source and the slab
1.0
1.0
0.8
0.8
Transmittance
Reflectance
Result
0.6
0.4
0.6
0.4
0.2
0.2
0.0
0.0
0.2
0.3
0.4
Normalized Frequency
0.5
0.2
0.3
0.4
Normalized Frequency
0.5
3. Distributed Bragg Reflector
PML
PML
GaAs AlAs
z
Planewave
source
20 pairs of AlAs/GaAs
Target wavelength = 950 nm
Grid resolution ∆z = 2.5 nm
Periodic boundary condition for x-y directions
 We will not use DS in this example
 You can change this as 0.1
 In this example, DD should be carefully chosen.
The time required to reach the steady-state could be
longer than you initially thought.
 In this example, we will only get
reflectance spectrum
Result
1.0
- 10 pairs
- 14 pairs
- 20 pairs
Reflectance
0.8
0.6
0.4
0.2
0.0
750
800
850
900
950 1000 1050 1100 1150
Wavelength (nm)
Accuracy
- 10 pairs
- 14 pairs
- 20 pairs
Reflectance (Log)
1
Analytic expression
R  tanh2  m ln(n1 / n2 )
(at the Bragg condition)
0.95
0.9
m
0.85
900
920
940
960
Wavelength (nm)
980
1000
Exact
FDTD
10
0.832424
0.83231
14
0.947996
0.94795
20
0.991636
0.99162
Errors less than 1 part per 10,000 !
4. 1-D photonic-crystal cavity
GaAs AlAs
A cavity is formed
by the two Bragg mirrors
Result
1.0
Reflectance
0.8
0.6
0.4
0.2
0.0
800
850
900
950
1000
Wavelength (nm)
1050
1100
Single excitation (950nm)
Ey field
3 QWs