1. Inverse design for nano-photonics
Yablonovitch Group, UC Berkeley
June 4th 2013

2. Seminar outline This morning: Theory This afternoon:
General presentation on inverse design
Our approach: the adjoint method
Step by step example
This afternoon:
Software presentation
Code structure outline
Hands on experience

3. 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. ?

4. Typical figures of merit and constraints
- Transmission
- Mode matching
- Absorption
- Field intensity
- Scattering cross section
Constraints
- Materials available
- Minimum dimensions
- Radius of curvature
- Periodicity

5. Typical figures of merit
Transmission
Mode-Match
Field intensity
Absorption

6. Typical constraints Radius of curvature Minimum dimension Periodicity
Materials
Periodicity

7. Parameterization: describing shapes
Index maps
Level sets ?
Splines
Custom parameters

8. Optimizations: Heuristic methods
Genetic algorithms
Particle swarm optimization
Ant colony
Parameter sweeps

9. Opsis optimization: particle swarm
1500 2D simulations: dB insertion loss

10. The problem:
Simulations required: ~C^(variables) !!!

11. The solution: deterministic optimization
Gradient descent

12. Calculating the gradient
With this information we can calculate the gradient for any parameterisation

13. But how do we calculate the gradient?
But how do we calculate that?

14. Our Approach

15. Duality in Linear Algebra
Problem:
Compute gTb such that Ab=c gT B A B c
such that =
B is unknown
must solve for B first

16. Duality in Linear Algebra
Problem:
Compute gTb such that Ab=c gT B A B c
such that =
Substitution of Variables:
I could solve this dual problem instead sT c AT s g
such that =
sTc = sTAB = (ATs)TB = gTB

17. Iterative Gradient Descent

18. Iterative Gradient Descent

19. Adjoint Method for Electromagnetics
Goal = Efficiently solve for the gradient of Merit(E,H)

20. Optimization Problem
light input

21. … Where should I add material? light input light input light input
Need to know dF/dx’ for all x’

22. Perturbation = small sphere of material
What happens when I add material?
Original
Perturbation = small sphere of material
Eperturbed E0 ε1 ε1 ε2 P

23. How can I approximate this perturbation?
Eperturbed E0
Escattered ≈ +

24. Gradient
light input

25. Island Perturbation Approximation

26. Boundary Perturbation Approximation

27. Key Trick 1: (approximate every perturbation as a dipole scatterer)… + … +

28. Reciprocity (Rayleigh-Carson)
Drive a dipole at x’
Find E at x0
Drive a dipole at x0
Find E at x’
equivalent

29. Reciprocity (Lorentz)
More generally:
equivalent J2 E2 J1 E1
J1  E2 = J2  E1

30. Reciprocity (Lorentz)
Problem: Solve for E1 and E2
Dual Problem: Solve for E3
J1(x’) E1 J3
E3(x’)
E3(x’’)
equivalent
J1  E3 = J3  E1
J2(x’’) E2
J2  E3 = J3  E2

31. Gradient
light input
Treat electric field at x0 as a current:

32. Key Trick 2: (Lorentz reciprocity)… + +

33. + What does this achieve? 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

35. Step by step example: direct simulation
Phase extracted=-137°

36. Adjoint simulation
Phase of dipole:137°

37. The derivative field! Add some material here!
Red = adding material will improve Merit Function

38. New geometry
Inclusion of index=2

39. Constructive interference!
Result
Constructive interference!
Incident field
Scattered field E2
Iteration

40. Repeat!

41. Software Implementation

42. Code Structure Optimizer Maxwell Solver Geometry Merit Function
Gradient
Lumerical FDTD
FreeForm
FieldEnergy
Supports HPC cluster
with MPI
LevelSet
Transmission
ModeMatch

43. Optimization Region
Merit Function = ModeMatch
abs(Ez)
Geometry
(eps = Ta2O5, epsOut = SiO2)
(minimum dimension = 300nm)
(radius of curvature = 150nm)
Source
(frequency = 830nm)

44. 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
Source
Optimization Region
Initial Geometry
Field and Index monitors
Merit Function monitors 3.
Run runOpt in Matlab
runOpt.m

45. 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

46. Merit Functions Transmission through a plane Field Energy in a volume
E intensity
H intensity
absorption
Mode Match at a plane
E mode
H mode
ExH Propagating Mode

47. 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

48. Custom Geometry Type

49. Install the software: 1. Download all Inverse Design files 2.
Install Lumerical 3.
Edit runOpt_params.m

50. Example
Waveguide Couplers for Silicon Photonics

51. Level set geometry class
Inside  phi<0
Outside  phi>0

52. Example

53. 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)

54. 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

55. Example 1: 50%/50% splitter

57. Optimization video

58. Example
Optical Antenna for Heat-Assisted Magnetic Recording

59. 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)

60. Media Coupling Efficiency (%)
Optical Antenna Example 1
TE mode
Antenna Geometry
Media Coupling Efficiency (%)

61. Optical Antenna Example 1
storage layer
cross-section
800 nm
Media Coupling 2%
Wasted
Antenna Absorption
26%
Media Coupling 7%
Wasted
Antenna Absorption
27%
storage layer
cross-section
800 nm

62. 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)

63. Optical Antenna Example 2
TM mode
Antenna Geometry

64. Optical Antenna Example 2
media
cross-section
800 nm
Media Coupling 4%
Wasted
Antenna Absorption
31%
Media Coupling
10%
Wasted
Antenna Absorption
32%
media
cross-section
800 nm

65. Optical Antenna Example 3
TM mode
Merit Function =
Absorption in Storage Layer (FWHM)
Absorption in Antenna Peg

66. Example
Custom Wrapper for Silicon Photonics

67. 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

68. 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…

69. Matlab setup file “setup_EZSiPh_params”
Then type runOpt in matlab and enjoy!