INVERSE DESIGN FOR
NANO-PHOTONICS
June 4th 2013
Yablonovitch Group, UC Berkeley
Seminar outline

This morning: Theory
 General
presentation on inverse design
 Our approach: the adjoint method
 Step by step example

This afternoon:
 Software
presentation
 Code structure outline
 Hands on experience
Shape Optimization for photonic devices

Problem statement:
Given FOM(E,H), find eps(x) such that FOM is
maximum and eps(x) respects some constraints.
?
Typical figures of merit and constraints
Figures of Merit
- Transmission
- Mode matching
- Absorption
- Field intensity
- Scattering cross
section
Constraints
- Materials available
- Minimum dimensions
- Radius of curvature
- Periodicity
Typical figures of merit
Transmission
Mode-Match
Field intensity
Absorption
Typical constraints
Radius of curvature
Minimum dimension
Materials
Periodicity
Parameterization: describing shapes
Index maps
Level sets
?
Splines
Custom parameters
Optimizations: Heuristic methods
Genetic algorithms
Particle swarm optimization
Ant colony
Parameter sweeps
Opsis optimization: particle swarm
1500 2D simulations: -0.28 dB insertion loss
The problem:

Simulations required: ~C^(variables) !!!
The solution: deterministic optimization
Gradient descent
Calculating the gradient
With this information
we can calculate the
gradient for any
parameterisation
But how do we calculate the gradient?
But how do we calculate that?
Our Approach
Duality in Linear Algebra
Problem:
Compute gTb such that Ab=c
gT
such that
B
B is unknown
A
B
=
must solve for B first
c
Duality in Linear Algebra
Problem:
Compute gTb such that Ab=c
gT
such that
B
B
A
=
Substitution of Variables:
I could solve this dual problem instead
sT
such that
c
sTc = sTAB = (ATs)TB = gTB
AT
s =
g
c
Iterative Gradient Descent
Iterative Gradient Descent
Adjoint Method for Electromagnetics
Goal = Efficiently solve for the gradient of Merit(E,H)
Optimization Problem
light input
x0
Where should I add material?
light input
light input
x'
x0
Need to know dF/dx’ for all x’
light input
x'
x0
x'
x0
…
What happens when I add material?
Original
Perturbation = small sphere of material
Eperturbed
E0
ε1
ε1
P
ε2
How can I approximate this perturbation?
Eperturbed
Escattered
E0
≈
+
Gradient
light input
x'
x0
Island Perturbation Approximation
Boundary Perturbation Approximation
Key Trick 1: (approximate every perturbation as a dipole scatterer)
+
x'
x0
x0
+
x0
x'
x0
x'
x0
…
x'
…
x0
x'
x0
x'
x0
Reciprocity (Rayleigh-Carson)
Drive a dipole at x’
Find E at x0
Drive a dipole at x0
Find E at x’
equivalent
x'
x0
x'
x0
Reciprocity (Lorentz)
More generally:
equivalent
J1
E2
E1
J2
J1  E2 = J2  E1
Reciprocity (Lorentz)
Problem: Solve for E1 and E2
Dual Problem: Solve for E3
E3(x’’)
J1(x’)
E3(x’)
E1
equivalent
J3
J1  E3 = J3  E1
J2(x’’)
E2
J2  E3 = J3  E2
Gradient
light input
x'
Treat electric field at x0 as a current:
x0
Key Trick 2: (Lorentz reciprocity)
+
x'
x0
x0
+
x0
x'
x' x' x'
x'
x'
x'
x0
x0
x'
x'
x'
x'
x0
…
What does this achieve?
+
x0
x' x' x'
x'
x'
x'
x'
x'
x'
x0
2 Simulations
 Gradient of Merit Function with respect to changes in geometry and/or permittivity
is calculated everywhere in the simulation volume
 Did not have to vary any geometric parameters
Optimization Process
Every iteration = Calculate the gradient and step closer to a local optimum
Step by step example: direct simulation
Phase extracted=-137°
Adjoint simulation
Phase of
dipole:13
7°
The derivative field!
Add some material here!
Red = adding material will improve Merit Function
New geometry
Inclusion
of
index=2
Result
Incident field
Scattered field
Constructive
interference!
E2
Iteration
Repeat!
Software Implementation
Code Structure
Maxwell
Solver
Optimizer
Geometry
Merit Function
FreeForm
FieldEnergy
LevelSet
Transmission
ModeMatch
Gradient
Lumerical FDTD
Supports HPC cluster
with MPI
Optimization Region
Merit Function = ModeMatch
abs(Ez)
Source
(frequency = 830nm)
Geometry
(eps = Ta2O5, epsOut = SiO2)
(minimum dimension = 300nm)
(radius of curvature = 150nm)
How to run an optimization:
1. Create a setup file
setup.m
%% Lumerical Simulation
%% Frequency
%% Optimization Region
%% Geometry Properties
%% Merit Function
2. Create a Lumerical base file
baseFile.fsp
3. Run runOpt in Matlab
runOpt.m
Source
Optimization Region
Initial Geometry
Field and Index monitors
Merit Function monitors
FreeForm/LevelSet Geometries
Binary 2D Geometry:
1 = eps, 0 = epsOut
Constant thickness
Materials:
eps/epsOut = ‘material name’ or custom permittivity
Optional Geometric constraints:
minimum dimension, radius of curvature, minimum padding, symmetries
Optional Optimization constraints:
boundary changes only, allow new shapes to emerge
Optional Non-Designable Region:
1 = Designable, 0 = Non-Designable
Merit Functions
Transmission through a plane
transmission
Field Energy in a volume
E intensity
H intensity
absorption
Mode Match at a plane
E mode
ExH Propagating Mode
H mode
Complex Merit Function
(j,k,l) = user defined (1), frequency (2), monitor (3)
f = (transmission, field energy, mode match)
Example:
Waveguide Spectral Splitter: maximize the minimum transmission
through branch 1 at frequency 1, branch 2 at frequency 2, etc. for N branches
Custom Geometry Type
Install the software:
1. Download all Inverse Design files
2. Install Lumerical
3. Edit runOpt_params.m
Example
Waveguide Couplers for Silicon Photonics
Level set geometry class
Inside

Outside 
phi<0
phi>0
Example
Our implementation

First order accurate
Can enforce radius of curvature constraints
Can anchor points of the initial geometry
Works great for Silicon photonics!

Does not use the boundary derivative D/E calculation





Doesn’t (yet) support new shapes
Can be computationally demanding if many mesh points
(but is ok for most problems)
Examples: 50%/50% splitter
Figure of merit:
Mode-matching to
the TE mode of two
waveguides
Optimized using
level set geometry
representation and
an effective index
method
Example 1: 50%/50% splitter
Optimization video
Example
Optical Antenna for Heat-Assisted Magnetic Recording
Optical Antenna Example 1
TE mode
Optimization Region = Planar Gold Film
Arm = 650 x 150 nm2
Peg = 50 x 50 nm2
Thickness = 40 nm
Media Coupling =
Absorption in Storage Layer (FWHM)
Power injected into Waveguide
Media Stack
10nm thick Storage Layer (FePt)
Optical Antenna Example 1
TE mode
Antenna Geometry
Media Coupling Efficiency (%)
Optical Antenna Example 1
storage layer
cross-section
800 nm
Media Coupling
Media Coupling
2%
7%
Wasted
Antenna Absorption
Wasted
Antenna Absorption
26%
27%
storage layer
cross-section
800 nm
Optical Antenna Example 2
TM mode
Optimization Region = Planar Gold Film
Arm = 650 x 150 nm2
Peg = 50 x 50 nm2
Thickness = 40 nm
Media Coupling =
Absorption in Storage Layer (FWHM)
Power injected into Waveguide
Media Stack
10nm thick Storage Layer (FePt)
Optical Antenna Example 2
TM mode
Antenna Geometry
Optical Antenna Example 2
media
cross-section
800 nm
Media Coupling
Media Coupling
4%
10%
Wasted
Antenna Absorption
Wasted
Antenna Absorption
31%
32%
media
cross-section
800 nm
Optical Antenna Example 3
TM mode
Merit Function =
Absorption in Storage Layer (FWHM)
Absorption in Antenna Peg
Example
Custom Wrapper for Silicon Photonics
Easy Silicon Photonics Optimization

Extremely easy setup of an optimization

Only figure of merit possible: Mode-matching


Just one Lumerical file to set up and matlab file to
modify
Covers many Silicon Photonics applications
Easy Silicon Photonics Optimization
-Create a simulation file as if you were just going to test your own design
-Name your source “Source”
-Name your mode matching monitor “merit1”
-Add a monitor around the region to optimize named “Velocity”
-Use sources to create the mode you would like to couple into and call
them mode1, mode2, etc…
Matlab setup file “setup_EZSiPh_params”
Then type runOpt in matlab and enjoy!