ECE 695 Numerical Simulations Lecture 19: Transfer Matricespbermel/ece695/... · 3/26/2015 ECE 695,...
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ECE 695Numerical Simulations
Lecture 20: Transfer Matrices and S4
Prof. Peter Bermel
March 1, 2017
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Outline
• Periodic solution strategy
– Stacked periodicity
– Transverse periodicity
• S-Matrix Simulations
• S4 RCWA solver
– Capabilities
– Live demonstration
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S-Matrices
• For S-matrix, connect incoming to outgoing fields from boundaries of region
• Mathematically,
𝑢(𝑝+1)
𝑑(0)=
𝑇 𝑢𝑢(𝑝)
𝑅 𝑢𝑑(𝑝)
𝑅 𝑑𝑢(𝑝)
𝑇 𝑑𝑑(𝑝)
𝑢(0)
𝑑(𝑝+1)
• For input from below:
– transmission from bottom to top given by 𝑇 𝑢𝑢(𝑝)
– reflection at bottom given by 𝑅 𝑑𝑢(𝑝)
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S-Matrices: Periodicity
• What happens if one or more layers are periodic?
• Then there are two types of coupling:
– Layer-to-layer (refractive)
– Mode-to-mode (diffractive)
• Can both be treated in a single framework?
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S-Matrices: Stacked Periodicity
• Recall that:
𝑠(𝑝) =𝑡 11(𝑝)
− 𝑡 12(𝑝)
𝑡 22(𝑝) −1 𝑡 21
(𝑝)𝑡 12(𝑝)
𝑡 22(𝑝) −1
−𝑡 22(𝑝) −1 𝑡 21
(𝑝)𝑡 22(𝑝) −1
• For quarter wave stack, express in terms of t-matrix elements:
𝑠(𝑝) =1 0
0 exp −𝑗𝛽𝑚,−𝑝
Δ𝑦𝑝𝑠(𝑝) exp 𝑗𝛽𝑚,+
𝑝Δ𝑦𝑝 0
0 1
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S-Matrices: Transverse Periodic Solution
• Divide into layers, uniform in z-direction
• Find Bloch states in each layer
• Calculate transfer function for field amplitudes
• Iteratively develop S-matrix
• Choose inputs from both sides
• Calculate resulting outputs (transmission and reflection) and losses (absorption A=1-T-R)
2/27/2017
Whittaker & Culshaw, Phys. Rev. B 60, 2610 (1999)Tikhodeev et al., Phys. Rev. B 66, 045102 (2002)
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S-Matrices: Periodic Solution Strategy
• Can use a Fourier series expansion in real space of the H-fields:
• In momentum space, can represent as:
• And then the electric field as:
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S-Matrices: Periodic Solution Strategy
• Eigenvalue equation becomes
• More compactly represented as:
• Where the eigenvectors f have a unique orthonormalitycondition:
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S-Matrices: Periodic Solution Strategy
• Can rephrase:
• Where H-fields are written as:
• And where E-fields are given by:
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S-Matrices: Periodic Solution Strategy
• Interface matrix in WC notation:
• Where:
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S-Matrix Construction: Recap
• We can relate the s-interface matrix to the t-interface matrix from before:
𝑠(𝑝) =𝑡 11(𝑝)
− 𝑡 12(𝑝)
𝑡 22(𝑝) −1 𝑡 21
(𝑝)𝑡 12(𝑝)
𝑡 22(𝑝) −1
−𝑡 22(𝑝) −1 𝑡 21
(𝑝)𝑡 22(𝑝) −1
• Then iteratively construct next S-matrix via:
𝑇 𝑢𝑢(𝑝)
𝑅 𝑢𝑑(𝑝)
𝑅 𝑑𝑢(𝑝)
𝑇 𝑑𝑑(𝑝)
= 𝑡 𝑢𝑢(𝑝)
1 − 𝑅 𝑢𝑑(𝑝−1)
𝑟 𝑑𝑢(𝑝) −1
𝑇 𝑢𝑢(𝑝−1)
𝑅 𝑢𝑑(𝑝−1)
+ 𝑇 𝑑𝑑(𝑝−1)
𝑟 𝑑𝑢(𝑝)
1 − 𝑅 𝑢𝑑(𝑝−1)
𝑟 𝑑𝑢(𝑝) −1
𝑇 𝑢𝑢(𝑝−1)
𝑟 𝑢𝑑(𝑝)
+ 𝑡 𝑢𝑢(𝑝)
𝑅 𝑢𝑑(𝑝−1)
1 − 𝑟 𝑑𝑢(𝑝)
𝑅 𝑢𝑑(𝑝−1) −1
𝑡 𝑑𝑑(𝑝)
𝑇 𝑑𝑑(𝑝−1)
1 − 𝑟 𝑑𝑢(𝑝)
𝑅 𝑢𝑑(𝑝−1) −1
𝑡 𝑑𝑑(𝑝)
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S-Matrices: Periodic Solution Strategy
• In WC’s notation:
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S-Matrix Simulations
• Transmission through triangular lattice converges as number of plane waves NG increases
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Whittaker & Culshaw, Phys. Rev. B 60, 2610 (1999)
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Photonic Simulations with S4
Full-wave photonic simulations of arbitrary layered media, including thin-film and crystalline
PV cells
V. Liu, S. Fan, Comp. Phys. Comm. 183, 2233 (2012)
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Accuracy improves systematically with computing power
V. Liu, S. Fan, Comp. Phys. Comm. 183, 2233 (2012)
Photonic Simulations with S4
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S4: Lua Control Files
• Obtain a new, blank simulation object with no solutions:
S = S4.NewSimulation()
• Define all materials:
S:AddMaterial('name', {eps_real, eps_imag})
• Add all layers:
S:AddLayer('name', thickness, 'material_name')
• Add patterning to layers:
S:SetLayerPatternCircle('layer_name', 'inside_material', {center_x, center_y}, radius)
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S4: FMM Formulations
• Specify the excitation mechanism:
S:SetExcitationPlanewave(
{angle_phi, angle_theta}, -- phi in [0,180), theta in [0,360)
{s_pol_amp, s_pol_phase}, -- phase in degrees
{p_pol_amp, p_pol_phase})
• Specify the operating frequency:
S:SetFrequency(0.4)
• Obtain desired output:
forward_power, backward_power = S:GetPoyntingFlux('layer_name', z_offset)
print(forward_power, backward_power)
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S4: Input
Can choose several examples drawn from the literature
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S4: Output
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Transmission through multilayer stack matches analytical expression
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S4: Output
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Transmission through 1D square grating of silicon and air
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S4: Output
• Transmission from Fig. 4 of Tikhodeev et al., Phys. Rev. B 66, 045102 (2002).
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S-Matrix Method: Advantages
• No ad hoc assumptions regarding structures
• Applicable to wide variety of problems
• Suitable for eigenmodes or high-Q resonant modes at single frequency
• Can treat layers with large difference in length scales
• Computationally tractable enough on single core machines
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S-Matrix Method: Disadvantages
• Accurate solutions obtained more slowly as the following increase:
– Number of layers
– Absolute magnitude of Fourier components (especially for metals)
– Number of plane-wave components (~N3)
• Relatively slow for broad-band problems (time-domain is a good alternative)
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Next Class
• Is Friday, Mar. 3
• Next time, we will continue with transfer matrix models, focusing on CAMFR
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