The Broadband Fixed-Angle Source Technique (BFAST) · 2015. 12. 9. · available in the next release of FDTD Solutions (2016A) Accurate broadband results can be obtained from a single

The Broadband Fixed-Angle Source Technique (BFAST)

LUMERICAL SOLUTIONS INC.

© LUMERICAL SOLUTIONS INC1

OutlineIntroduction• Lumerical’s simulation products• Simulation of periodic structures

The new Broadband Fixed-Angle Source Technique (BFAST)• Details on BFAST• Limitations• Basic example: transmission through a dielectric stack• Performance considerations

Application Examples• Lamellar plasmonic grating (2D)• Plasmonic solar cell (3D)

Summary


Introduction


Our Products

FDTD SolutionsNANOPHOTONIC SOLVER (2D/3D)

MODE SolutionsWAVEGUIDE DESIGN ENVIRONMENT

spatial distribution of

charge carriers optical generation rate of charge

carriers

INTERCONNECTPHOTONIC INTEGRATED CIRCUIT SIMULATOR

DEVICECHARGE TRANSPORT

SOLVER (2D/3D)

System/Circuit Level

Component Level: Optical Component Level: Electrical



Simulation of Periodic Structures with FDTD

Large class of systems in photonics is periodic

Gratings

Photonics crystals

Meta-materials

CMOS sensor arrays and solar cells

…

Example of a basic grating, excited by a plane-wave Problem can be reduced to simulation of a single unit cell


… …

Periodicboundaryconditions

Reduction to a unit cell

Tilting the plane-wave source breaks the symmetry

Bloch’s theorem: 𝐸 𝑥 = 𝐸 𝑥 + 𝑎 𝑒𝑖 𝑘∥ 𝑎


Blochboundaryconditions

a

𝑘

𝑘∥

Changing the angle of incidence

https://en.wikipedia.org/wiki/Bloch_wave

Works well for narrow-band simulations around a given center frequency 𝒇𝟎:

Then 𝑘∥ =2𝜋𝑓0

𝑐sin(𝜃0) is constant

Length of the 𝑘-vector is frequency dependent, i.e. 𝑘 = k f =2𝜋𝑓

𝑐

For a given frequency range 𝑓min, 𝑓max → 𝑘min, 𝑘max

As a result, 𝜃 𝑓 = sin−1𝑘∥

𝑘= sin−1

𝑓0

𝑓sin(𝜃0)

In broadband simulations with Bloch boundary conditions, differentfrequency components are injected at different angles!

Wavelength-dependence of the angle


𝑘∥

For one or a few fixed angle(s): Run separate narrow-band simulations for each wavelength/frequency

100 frequency points 100x the computational time!

To compute an angle-wavelength map Sweep the angle and re-interpolate the data

Current solutions to obtain broadband results


https://kb.lumerical.com/en/ref_sim_obj_bloch_broadband_sweep.html

https://kb.lumerical.com/en/ref_sim_obj_bloch_broadband_sweep.html

The Broadband Fixed-Angle Source Technique (BFAST)

AVAILABLE IN RELEASE 2016A


Details on BFASTBFAST allows to inject light at a fixed angle over a broad spectrum!

BFAST is not just a new type of boundary condition. The core algorithm is different from standard FDTD!

It is based on the split-field method, but was customized to ensure compatibility with most existing material models and monitors!


Limitations of BFASTTwo fundamental limitations:

1. Nonlinear and all flexible material plugin materials will not function using BFAST.

2. Injection above the critical angles for total internal reflection (TIR) is not stable.


Basic ExampleTransmission through a dielectric stack (4 layers)


𝑙1 = 2.5𝜇𝑚 𝑙2 = 2.5𝜇𝑚Broadband source(0.8𝜇𝑚 – 1.6𝜇𝑚)

n=1.5 n=2.5 n=1.5n=1.0

Frequency domainmonitor

Results


Results (20 deg)


Results (40 deg)


Results (60 deg)


Performance ConsiderationsBFAST simulations take more time than identical simulations with Bloch boundary conditions.

Two contributions:

1. Angle-independent overhead: 1.5x - 4x

2. Angle-dependent factor: Δ𝑡~(1 − sin𝜃)

Rule of thumb: For angles > 𝟔𝟎∘, it mightstill be faster to use Bloch BCs instead of BFAST.


Angle 𝜽[degrees]

Simulationtime

0 1.0x

10 1.2x

20 1.5x

30 2.0x

40 2.8x

50 4.3x

60 7.5x

70 16.6x

80 65.8x

Application Examples


Gold surface with narrow but deep trenches

Acts as a “perfect absorber” around 𝜆 ≈ 3.2𝜇𝑚

Study angular dependence of reflectance spectrum

Lamellar Plasmonic Grating


2µm

……Gold

80nm

50

0n

m

F. J. Garcia-Vidal et al., "Localized Surface Plasmons in Lamellar Metallic Gratings," J. Lightwave Technol. 17, 2191-2195 (1999)

Nicolas Bonod et al., "Total absorption of light by lamellar metallic gratings," Opt. Express 16, 15431-15438 (2008)

FDTD Setup for the Lamellar GratingNarrow spectral features require high

frequency resolution

Also: a longer simulation time (20ps) and lower auto-shutoff tolerance (10−7)

Strong field gradients in the slot requirelocal mesh refinement


Broadband source(2.2μm – 4.2μm)

Frequency domainmonitor (1001 pts)

Mesh refinementregion (10nm)

Lamellar Plasmonic GratingReflection spectrum under normal incidence


Lamellar Plasmonic GratingComparison of Bloch BCs and BFAST (10 deg)


Broadband simulation withBloch BCs fails to accuratelysimulate this resonance(even at only 10 degrees).

Lamellar Plasmonic GratingBFAST allows convenient and accurate sweeps


For BFAST, computational time increases with angle:

Comparison with Bloch-sweep difficult due to varying auto-shutoff.

A broad-band Bloch simulation takes about 15s (at 10∘)!

Performance

Angle BFAST

0∘ 3s

10∘ 20s

20∘ 25s

30∘ 31s

40∘ 56s


Computer: Intel® Core™ i5-4460 (4 cores @ 3.2GHz)

Plasmonic enhanced solar cell


kb.lumerical.com/en/index.html?solar_cells_plasmonic_at_normal_and_oblique_incidence.html

Broadband simulation (400nm – 1100nm)

Contains highly dispersive media (silver and silicon)

Symmetries can be exploited to accelerate the simulations

https://kb.lumerical.com/en/index.html?solar_cells_plasmonic_at_normal_and_oblique_incidence.html

Plasmonic enhanced solar cell


Symmetries 4 x reduction

Materials Silicon, Silver

Wavelength 400 – 1100 nm (351 points)

Simulation time 90 seconds

Symmetries 2 x reduction

Simulation time 1800s


Summary

The new Broadband Fixed-Angle Source Technique (BFAST) will be available in the next release of FDTD Solutions (2016A)

Accurate broadband results can be obtained from a single simulation

Faster simulations for a fixed angle of incidence More convenient and/or accurate for angle-wavelength sweeps

Significant performance gains for broad spectra and moderate angles (below 45 degrees)

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Documents

The Broadband Fixed-Angle Source Technique (BFAST) · 2015. 12. 9. · available in the next release of FDTD Solutions (2016A) Accurate broadband results can be obtained from a single