11. Reflection/Transmission spectra Contents 1.Normal incidence on a simple dielectric slab 2. Normal incidence on a photonic crystal slab 3. Normal incidence.

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

0 2000 4000 6000 8000 10000

-8

-6

-4

-2

0

2

4

6

8

Hx

fiel

dFDTD time step

0 2000 4000 6000 8000 10000-6

-4

-2

0

2

4

6

FDTD time step

Hx

fiel

d

Without slab

With slab

Planewave source

Detect

Slab (T=1a)

1. Simple dielectric slab

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.

We will not use DS in this example

(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

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)

Now let’s perform the following calculations.

FT[ R(t)-R0(t) ] / FT[ R0(t) ] = Refelctance spectrum in FT[ T(t) ] / FT[ T0(t) ] = Transmission spectrum in

,where FT denotes Fourier Transformation.

** How to obtain R & T spectra

(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)

0.1 0.2 0.3 0.4 0.5 0.60.0

0.2

0.4

0.6

0.8

1.0

Tra

nsm

itta

nce

Normalized Frequency0.1 0.2 0.3 0.4 0.5 0.6

0.0

0.2

0.4

0.6

0.8

1.0

Ref

lect

ance

Normalized Frequency

Result

2. Square lattice photonic crystal slab

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)

We will not use DS in this example

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

0.2 0.3 0.4 0.50.0

0.2

0.4

0.6

0.8

1.0

Ref

lect

ance

Normalized Frequency0.2 0.3 0.4 0.5

0.0

0.2

0.4

0.6

0.8

1.0

Tra

nsm

itta

nce

Normalized Frequency

Result

3. Distributed Bragg Reflector

20 pairs of AlAs/GaAs

Target wavelength = 950 nm

GaAs AlAs

Grid resolution ∆z = 2.5 nm

Periodic boundary condition for x-y directions

z

Planewave source

PML PML

In this example, DD should be carefully chosen. The time required to reach the steady-state could be

longer than you initially thought.

We will not use DS in this example

You can change this as 0.1

In this example, we will only get reflectance spectrum

750 800 850 900 950 1000 1050 1100 11500.0

0.2

0.4

0.6

0.8

1.0R

efle

ctan

ce

Wavelength (nm)

- 10 pairs- 14 pairs- 20 pairs

Result

900 920 940 960 980 1000

0.85

0.9

0.95

1

Ref

lect

ance

(L

og

)

Wavelength (nm)

- 10 pairs- 14 pairs- 20 pairs 2

1 2tanh ln( / )R m n n

Accuracy

Analytic expression

(at the Bragg condition)

Exact FDTD

0.832424 0.83231

0.947996 0.94795

0.991636 0.99162

10

14

20

m

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

800 850 900 950 1000 1050 11000.0

0.2

0.4

0.6

0.8

1.0R

efle

ctan

ce

Wavelength (nm)

Single excitation (950nm)

Ey field

3 QWs