Download FDTD – Example (1)

Modelling techniques and
applications
Qing Tan
EPFL-STI-IMT-OPTLab
24.07.2009
Content
1. Introduction
2. Frequency domain method-- RCWA
3. Time domain method -- FDTD
4. Conclusion
Introduction (1)
1. Why Modelling ?
A complementary tool to design specific optical functions and
prediction of optical properties for nanostructures.
2. What?
A process of solving Maxwell equations by computer, combined
with boundary conditions.




 D
B
 E  
  D   ev   H 
J
t
t




D  E
B  H

B  0
Introduction (2)
3. How?
Two main category
a. Frequency domain method


E (r , t )  E (r )e  jwt


H (r , t )  H (r )e  jwt

1
w 2 
 [
  H (r )]  ( ) H (r )
 (r )
c
Example:
Plane wave expansion (PWE),
Rigorous coupled wave analysis (RCWA),
Finite element method (FEM), ...
Advantage:
Solution for each frequency, readable result
Disadvantage: Complicated eigenmodes solving process
Introduction (3)
3. How?
Two main category
b. Time domain method


B
 E  
t
 D
 H 
t



D D(t  t )  D(t )

t
t
Example:
Finite difference time domain method (FDTD)
Advantage:
Applicable for any arbitrary structure
Disadvantage: Result understanding for varying frequency,
Time consuming
RCWA – Introduction
1. Rigorous coupled wave
(RCWA) started in 1980s.
analysis
2. Suitable structure:
periodic grating structure
3. Calculation process:
a. Slicing structure into layers so that each
layer is homogeneous in propagation z
direction.
b. For each layer, permittivity and EM
components are represented by Fourier
expansion.
c. Boundary conditions are used for
neighbouring layers to form a matrix .
d. Calculation of coupling coefficient for
each Fourier component.
M.G. Moharam and T.K. Gaylord, J. Opt. Soc. Am. 72, 1385-1392 (1982)
RCWA – Discussion
1. Rigorous calculation: the accuracy is determined by the
truncation of the Fourier expansion. In other words, the
accuracy tends to be infinite close to reality by increasing the
Fourier expansion orders.
2. Structure approximation is made for non-binary grating.
3. Instability of matrix inversion will cause calculation error.
4. Efforts have been paid to improve the stability.
• N. Chateau and J.-P. Hugonin, “Algorithm for the rigorous coupled-wave analysis of
grating diffraction,” J. Opt. Soc. Am. A 11, 1321–1331 (1994).
• M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation
of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance
matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995).
• L. Li, ‘‘Use of Fourier series in the analysis of discontinuous periodic structures,’’ J. Opt.
Soc. Am. A 13, 1870 – 1876 (1996).
RCWA – Example (1)
A polarizing beam splitter that uses the anisotropic spectral reflectivity
(ASR) characteristics of a high spatial frequency multilayer binary grating.
R.C. Tyan, P. C. Sun, Y. Fainman - SPIE MILESTONE SERIES MS, 2001
RCWA – Example (2)
Numeric results of the reflectivity for TE and TM polarized waves vs.
wavelength of a 7-layer PBS designed for normally incident waves.
R.C. Tyan, P. C. Sun, Y. Fainman - SPIE MILESTONE SERIES MS, 2001
FDTD – Introduction
1. FDTD: Finite difference time domain method
time domain method, widely used, time consuming
2. Discretize the Maxwell
equation in Time and Space domain


B

 E  

D
t
 H 


t
 iwt
E  E ( r )e
Time:
Space:






B B(t  t )  B(t )
D D(t  t )  D(t )


t
t
t
t

E y Ex
Ex Ez
Ez E y
 E  (

) xˆ  (

) yˆ  (

) zˆ
y
z
z
x
x
y
E x z, t  E x z  z / 2, t   E x z  z / 2, t 

z
z
FDTD – Algorithm
Yee’s algorithm
1. Maxwell boundary condition
between adjacent cells is self
satisfied in this algorithm.
2.
Each
field
component
depends on the field of the
previous time step itself and the
surrounding component in Yee’s
algorithm.
K. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic
media,” IEEE Trans. Antennas and Propag. 14, 302–307 (1966).
FDTD – Accuracy
1. Approximation is made for the derivative conversion.
2. Accuracy is determined by the space and time step size. The
smaller the step size, the more accurate the result.
In practice:
Space step:
   / 10
Time step:
1
1
1 1/ 2
 * t  ( 2  2  2 )
x
y
z
3. This method is applicable for arbitrary structure.
4. Calculation is made within the finite domain for finite structure.
A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite Difference Time-Domain
Method, Third Edition (Artech House Publishers, 2005).
FDTD – ‘Finite’
1. To start the calculation, ‘specified’ incidence is required.
Incidence:
Space: Plane wave, waveguide mode, ...
Time: Pulse with finite time
2. Outside the finite structure?
Edge condition: Perfectly
matched layer (PML) is the
most commonly used layer.
Finite structure
PML
PEC
PML: Strongly absorbing
region for incident waves
while minimum light is
reflected back.
Attention: Evanescent wave
FDTD – Discussions
1. FDTD is a popular numerical method, because of relatively
easy implementation, arbitrary structure applicability.
2. FDTD is a time consuming method due to the iterative
calculation process.
3. FDTD is a broadband calculation process. The spectrum is
decided by the time pulse shape. The frequency band spectrum
is realized by one single simulation.
4. FDTD is also limited in the application for dispersive materials.
Because the dispersion model is in spectrum domain. Finite
element method (FEM) is a good frequency domain method to
solve this problem.
FDTD – Example (1)
Goal: Design tunable photonic crystal cavity by means of liquid
crystal, which is tunable by temperature.
Structure: Photonic crystal waveguide
which is filled with liquid crystal
W1 with coupling holes,
Methodology: Maximize the field concentration in the cavity holes
Tools: FDTD based commercial software Microwave studio
FDTD – Example (2)
Incident: Fundamental waveguide mode, Gaussian pulse (working
at wavelength of 1.5 µm)
Edge condition: Distance 1.5 µm to the PML layer to avoid
evanescent wave reaching PML region.
Step size:
   / 20
Refractive index: nSi = 3.48, nLC = 1.5 or 1.55
FDTD – Example (3)
Final design:
period [nm]
430
r0 [nm] r1 [nm] r2 [nm] p1 [nm] p2 [nm]
111
131
131
470
850
  14nm ext  100
FDTD – Example (4)
Extinction ratio improvement:
period [nm]
430
r0 [nm] r1 [nm] r2 [nm] r3 [nm] p1 [nm] p2 [nm] p3[nm]
111
151
126
101
485
865
1295
  14.74nm ext  4039
Conclusion
1. Modelling is necessary to understand optical properties of
nanostructures and optical device realization.
2. Each modelling method has its corresponding strengths and
weaknesses.
3. Choosing a proper modelling tool is required.
4. The modelling error must be understood to make correct
calculation.
Thank you for your attention!