3. 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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4. Introduction (2)
3. How? Two main category
a. Frequency domain method
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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
5. Introduction (3)
3. How? Two main category
b. Time domain method
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Example: Finite difference time domain method (FDTD)
Advantage: Applicable for any arbitrary structure
Disadvantage: Result understanding for varying frequency,
Time consuming
6. 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.
7. 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).
8. 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
9. 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
10. 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
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11. FDTD – Algorithm
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).
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.
12. 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.
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A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite Difference Time-Domain
Method, Third Edition (Artech House Publishers, 2005).
13. 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
14. 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.
15. 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
16. 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:
Refractive index: nSi = 3.48, nLC = 1.5 or 1.55
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19. 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.
