Modelling techniques and applications Qing Tan
EPFL-STI-IMT-OPTLab 24.07.2009
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Content 1. Introduction 2. Frequency domain method-- RCWA 3.
Time domain method -- FDTD 4. Conclusion
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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.
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Introduction (2) 3. How? Two main category a. Frequency domain
method 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
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Introduction (3) 3. How? Two main category b. Time domain
method Example: Finite difference time domain method (FDTD)
Advantage: Applicable for any arbitrary structure Disadvantage:
Result understanding for varying frequency, Time consuming
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RCWA Introduction M.G. Moharam and T.K. Gaylord, J. Opt. Soc.
Am. 72, 1385-1392 (1982) 1.Rigorous coupled wave analysis (RCWA)
started in 1980s. 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.
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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, 13211331 (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, 10771086
(1995). L. Li, Use of Fourier series in the analysis of
discontinuous periodic structures, J. Opt. Soc. Am. A 13, 1870 1876
(1996).
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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
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Numeric results of the reflectivity for TE and TM polarized
waves vs. wavelength of a 7-layer PBS designed for normally
incident waves. RCWA Example (2) R.C. Tyan, P. C. Sun, Y. Fainman -
SPIE MILESTONE SERIES MS, 2001
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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 Time: Space:
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FDTD Algorithm Yees algorithm K. Yee, Numerical solution of
initial boundary value problems involving Maxwell's equations in
isotropic media, IEEE Trans. Antennas and Propag. 14, 302307
(1966). 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 Yees algorithm.
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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: Time step: 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).
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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? Finite
structure PML PEC Edge condition: Perfectly matched layer (PML) is
the most commonly used layer. PML: Strongly absorbing region for
incident waves while minimum light is reflected back. Attention:
Evanescent wave
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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.
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FDTD Example (1) Goal: Design tunable photonic crystal cavity
by means of liquid crystal, which is tunable by temperature.
Structure: Photonic crystal waveguide W1 with coupling holes, which
is filled with liquid crystal Methodology: Maximize the field
concentration in the cavity holes Tools: FDTD based commercial
software Microwave studio
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FDTD Example (2) Incident: Fundamental waveguide mode, Gaussian
pulse (working at wavelength of 1.5 m) Edge condition: Distance 1.5
m to the PML layer to avoid evanescent wave reaching PML region.
Step size: Refractive index: n Si = 3.48, n LC = 1.5 or 1.55
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FDTD Example (3) Final design: period [nm]r0 [nm] r1 [nm] r2
[nm] p1 [nm] p2 [nm] 430 111 131 131 470 850
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FDTD Example (4) period [nm]r0 [nm] r1 [nm] r2 [nm] r3 [nm] p1
[nm] p2 [nm]p3[nm] 430 111 151 126 101 4858651295 Extinction ratio
improvement:
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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.