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COMSOL Conference 2016 Boston
Center for Metamaterials and Integrated Plasmonics
Transformation Optics Simulation Method for Stimulated Brillouin Scattering
Roberto Zecca, Patrick T. Bowen,David R. Smith, and Stéphane Larouche
October 6th 2016
R. Zecca et al., Phys. Rev. A (2016), in review
COMSOL Conference 2016 Boston
Center for Metamaterials and Integrated Plasmonics
Stimulated Brillouin scattering
2
• Nonlinear coupling of light and elastic waves (optical forces, scattering)
• Applications: optical lasers, amplifiers, strain sensors, slow light
• Potential for high gains and SNR in (nano-)structured materials/devices
• Numerical modeling is key to device design
COMSOL Conference 2016 Boston
Center for Metamaterials and Integrated Plasmonics
Finite-element SBS modeling
3
Stokes process:
MECHANICS OPTICS
Displacement vectorial field
Mass density variation scalar field
Bi-chromatic electromagnetic field
COMSOL Conference 2016 Boston
Center for Metamaterials and Integrated Plasmonics
optical forces (external)internal force (stress)
Finite-element SBS modeling
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mass x acceleration
OPTICS
MECHANICS
COMSOL Conference 2016 Boston
Center for Metamaterials and Integrated Plasmonics
Finite-element SBS modeling
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• Frequency-domain EM solver cannot take moving geometry into account
• Significant problem in the case of SBS
• Time-domain too onerous: (10 ps vs. fs)
• Different strategy needed
COMSOL Conference 2016 Boston
Center for Metamaterials and Integrated Plasmonics
Transformation optics: designing an invisibility cloak
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-2 -1 1 2
-2
-1
1
2
-2 -1 1 2
-2
-1
1
2
Topological Material
Given a coordinate transform
Using form-invariance of Maxwell’s equations
COMSOL Conference 2016 Boston
Center for Metamaterials and Integrated Plasmonics
Time-dependent TO metric
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• Represent movement by equivalent, time-dependent material properties
• Apply time-dependent metric, using deformation gradient as Jacobian
COMSOL Conference 2016 Boston
Center for Metamaterials and Integrated Plasmonics 9
Optical forces and potentialOptical mode
COMSOL Conference 2016 Boston
Center for Metamaterials and Integrated Plasmonics 10
Mass density/permittivity variation Permeability variation
Optical forces and potentialOptical mode
COMSOL Conference 2016 Boston
Center for Metamaterials and Integrated Plasmonics 11
Longitudinalcomponents
Transverse components
COMSOL Conference 2016 Boston
Center for Metamaterials and Integrated Plasmonics
Transformed Maxwell’s equations
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Maxwell’s equations
Apply metric
Transformed wave-like equation
Combine equations
Transformed Maxwell’s equations
• COMSOL master equation
• Apply TO metric to Maxwell’s equations and obtain similar expression
COMSOL Conference 2016 Boston
Center for Metamaterials and Integrated Plasmonics
Transformed Maxwell’s equations
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• COMSOL master equation
• Apply TO metric to Maxwell’s equations and obtain similar expression
COMSOL Conference 2016 Boston
Center for Metamaterials and Integrated Plasmonics
Test: 1D bulk amplifier
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Modified wave-like equationsUndepleted pump approximation
Theory TO method Simple coupling
Standard wave-like equations
R. W. Boyd, Nonlinear Optics.
COMSOL Conference 2016 Boston
Center for Metamaterials and Integrated Plasmonics
Test: 1D bulk amplifier
15
Modified wave-like equationsUndepleted pump approximation
Theory TO method Simple coupling
Standard wave-like equations
R. W. Boyd, Nonlinear Optics.
COMSOL Conference 2016 Boston
Center for Metamaterials and Integrated Plasmonics
Test: dielectric slab waveguide
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Theory calculations based on C. Wolff et al., Phys. Rev. A 92 (2015).
COMSOL Conference 2016 Boston
Center for Metamaterials and Integrated Plasmonics
Conclusions and acknowledgments
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• SBS in structures can be simulated using special techniques
• TO-based method can handle loss, anisotropy, complicated optical forces, solid-liquid interactions
• Future directions: application to metamaterial and plasmonic systems
Further acknowledgments: Robert L. Bryant, Duke University
COMSOL Conference 2016 Boston
Center for Metamaterials and Integrated Plasmonics
Proposed topics of discussion
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• Details of COMSOL simulations
• Transformation optics
• Brillouin scattering and optical forces
• Details of method derivation
• Theoretical descriptions of SBS
• Applications
Poster session at 6:00 pm Contacts:Roberto [email protected]
R. Zecca et al., Phys. Rev. A (2016), in review
To take a closer look…
COMSOL Conference 2016 Boston
Center for Metamaterials and Integrated Plasmonics
Metric coefficients
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COMSOL Conference 2016 Boston
Center for Metamaterials and Integrated Plasmonics
Details of COMSOL simulation
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• 2 Electromagnetic Waves (RF) modules
• 1 Solid Mechanics module
• 1 Math (DODE) module (to solve for H-fields)
• Frequency “load-ramping” to ease convergence when necessary
COMSOL Conference 2016 Boston
Center for Metamaterials and Integrated Plasmonics
Details of COMSOL simulation: TO metric
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COMSOL Conference 2016 Boston
Center for Metamaterials and Integrated Plasmonics
Details of COMSOL simulation: recasting PDEs
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COMSOL Conference 2016 Boston
Center for Metamaterials and Integrated Plasmonics
Details of COMSOL simulation: coupling
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COMSOL Conference 2016 Boston
Center for Metamaterials and Integrated Plasmonics
Details of COMSOL simulation: slab model
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Silicon
COMSOL Conference 2016 Boston
Center for Metamaterials and Integrated Plasmonics
Details of COMSOL simulation: slab model
25
Pump
Pu
mp
PM
L
Emp
ty space
ContinuityPML
COMSOL Conference 2016 Boston
Center for Metamaterials and Integrated Plasmonics
Details of COMSOL simulation: slab model
26
Signal
Sign
al P
ML
Emp
ty s
pac
eContinuity PML
COMSOL Conference 2016 Boston
Center for Metamaterials and Integrated Plasmonics
Details of COMSOL simulation: slab model
27
MechanicsPML PML
COMSOL Conference 2016 Boston
Center for Metamaterials and Integrated Plasmonics
Test: dielectric slab waveguide (details)
28
Theory calculations based on C. Wolff et al., Phys. Rev. A 92 (2015).
COMSOL Conference 2016 Boston
Center for Metamaterials and Integrated Plasmonics
Signal fit and gain extraction
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Theory calculations based on R. W. Boyd, Nonlinear Optics.
COMSOL Conference 2016 Boston
Center for Metamaterials and Integrated Plasmonics
Stress tensors
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• Engineering stress tensor
• Cauchy stress tensor
• First Piola-Kirchhoff stress tensor
• Second Piola-Kirchhoff stress tensor