Download Numerical simulation of diffraction grating alignment and phase noise

FDTD Simulation of Diffraction
Grating Displacement Noise
Daniel Brown
University of Birmingham
AEI, Hanover - 14/12/2010
1
Overview
 Motivation: Diffraction grating displacements
 Finite-Difference Time-Domain(FDTD)
 Why use FDTD?
 How it works
 What problems there are
 Measuring the Gaussian beam diffraction phase noise
 Modelling the transmission on waveguide coatings
 Conclusion
2
Project Aim
 Try and simulate phase noise[1] from moving diffraction
grating.
 Simulate using a finite grating size using Gaussian beams
 Implement and verify a 2D/3D FDTD numerical solver for
Maxwell’s equations, this should be able to…
 Measure reactions to impulsive inputs like moving
gratings/sources
 Simulate time-domain interaction of Gaussian laser beams
with gratings
[1] A.Freise et al. Phase and alignment noise in grating interferometers.
New Journal of Physics 2007
3
The phase noise
What are we looking for:

Additional phase periodic
with grating period

Increase’s with diffraction
order linearly

Linear relationship to the
grating displacement
A.Freise et al. Phase and alignment noise in grating interferometers.
New Journal of Physics 2007
4
Phase noise in a cavity
 Allows for an all reflective component
optical cavity[1]
 Of particular use in GW detectors
 Potential to reduce thermal
disturbances compared to
transmissive optical cavity
 But, we see additional phase noise
added on each reflection from the
grating due to any lateral
movements[2]
[1] K. Sun et al. Byer. All-reflective Michelson, Sagnac, and Fabry-Perot interferometers based on grating
beam splitters.
[2] J Hallam et al. Lateral input-optic displacement in a diffractive Fabry-Perot cavity 2010
5
Previous and ongoing work
 Previous and ongoing work in Birmingham, Glasgow,
Hannover and Jena
 Experimental work in Birmingham attempting to measure
effects of this phase noise
 Previous work done at Birmingham by Daniel Wolliscroft on
Transmission Line method for simulating EM propagation[1]
[1] Daniel Wolliscroft. Visualising the effects of mirror surface distortions, 2009
School of Physics and Astronomy, University of Birmingham
6
Finite-Difference Time-Domain
 Is an accurate and well proven solver
of Maxwell’s equations in the for
many different applications
 Can solve Maxwell’s equations
exactly, of course in reality numerical
problems stop this
7
Finite-Difference Time-Domain
 …also been used for the very small, such as in photonics
and nanophysics(waveguides and circular resonators)
8
Finite-Difference Time-Domain
 Need to solve Maxwell
equations to simulate
propagation of EM waves
 Need to be able to
simulate complicated
problem spaces(i.e.
various 𝜖𝑟 , 𝜇𝑟 , σ)
 Faraday and Ampere’s
laws are only really
needed for the simulation
9
Finite-Difference Time-Domain
 Expand Faraday and
Ampere laws
 Change in time of
each fields component
is dependant on only
the other field and it’s
previous value
 Ignoring magnetic
and current sources
here
10
Finite-Difference Time-Domain
 The Yee Algorithm(1966)
 Defined the Yee Grid
Taflove, Allen and Hagness, Susan C. Computational Electrodynamics: The Finite-Difference. Time-Domain Method, Third Edition
11
Finite-Difference Time-Domain
 The Yee Algorithm
 Leapfrog method
 Input initial values at 𝑡 = 0
 No need to solve simultaneous equations as 𝐸 and 𝐻 are
known when needed
𝐸𝑧(0)
𝐸𝑧(1)
𝐻𝑦(1/2)
𝐸𝑧(0)
𝐻𝑦(3/2)
𝐸𝑧(1)
𝐸𝑧(2)
𝐻𝑦(5/2)
𝐸𝑧(2)
𝐸𝑧(3)
𝐻𝑦(7/2)
𝐸𝑧(3)
𝐸𝑧(4)
𝐻𝑦(9/2)
𝐸𝑧(4)
Taflove, Allen and Hagness, Susan C. Computational Electrodynamics: The Finite-Difference
Time-Domain Method, Third Edition
𝑡=0
𝑡=
Δ𝑡
2
𝑡 = Δ𝑡
12
Finite-Difference Time-Domain
 Using central finite difference operator we can approximate
first order derivatives
 Can apply this process to all derivatives
13
Finite-Difference Time-Domain
 Partial derivative approximations are then used in the six
equations:
 Update equations for each field component can then be
found and run in code
14
Numerical Stability
 The grid size and time step variables need to be
chosen to satisfy S (stability factor)
 Choosing dimensions depends on the problem
 Need balance between accuracy and computation speed
 Major limitation is computational requirements, they increase very
rapidly with finer grid sizes
15
Phase Errors
 Velocity Anisotropy Error
 Wave has different phase velocity
depending on their direction
 Effects greatly reduced by using other
algorithms or more samples per
wavelength
 Always a problem for larger simulation
spaces
16
Phase Errors
 Physical Phase Velocity Error
 Phase lead or lag that is picked up when wave passes
through a cell at a given angle
 Accumulates as wave propagates
Blue - Measured phase, Red - Corrected Phase
 Easy to take into account in straight
lines, not so much in complicated
problems
-1.58
-1.6
Calculated Phase
-1.62
-1.64
-1.66
 Is a function of the time sampling
chosen; more samples, less error
-1.68
-1.7
-1.72
-1.74
1
2
3
4
5
6
7
8
9
10
11
Number of wavelengths from the source
17
FDTD Simulation
 Initial version implemented in Java using Processing[1] for
graphics for quick development time
 FDTD simulation features implemented so far…
 2D TM and TE polarisations
 Perfectly matched layers for boundary absorption
 Ability to model materials with specified permittivity, permeability and
conductivity
 Plane wave/Gaussian beam source injection
 Tested to work for total internal reflection, reflection and
transmission coefficients, Brewster's angle, basic
diffraction, etc.
[1] Processing Library – www.processing.org
18
Gaussian
Beam
Phase
probes
3 mode
grating
19
1st diffraction order
3
3
2
2
1
1
Phase
Phase
0th diffraction order
0
-1
-2
0
-1
-2
-3
-3
0
0.2
0.4
0.6
0.8
1
0
0.4
0.6
0.8
1
Offset(m)
3rd diffraction order
3
3
2
2
1
1
Phase
Phase
Offset(m)
2nd diffraction order
0.2
0
-1
-2
0
-1
-2
-3
-3
0
0.2
0.4
0.6
Offset(m)
0.8
1
0
0.2
0.4
0.6
Offset(m)
0.8
1
20
Waveguide Coating
 Thin grating layer of Ta2O5 applied to surface of a substrate
 Can adjust the grating parameters and depth to get
potentially 100% reflectivity at 1064nm
 This setup is also suggested to be immune to the phase
noise seen previously
Bunkowski et al.(2006)
21
Ta2O5
Si02
Steady state reached
Approx 1000 time steps
Simulation size 7.5um x 20um, processing time approx 300s
22
Waveguide grating - Preliminary
Normal incident Gaussian beam(TE,1064nm)
transmission through waveguide grating, d=700nm.
Normal incident Gaussian beam(TM,1064nm)
transmission through waveguide grating, d=700nm.
0.4
0.4
0.35
0.35
0.8
0.3
0.85
0.3
0.7
0.25
0.6
0.2
0.5
0.15
0.1
0.4
0.05
Grating depth(m)
Grating depth(m)
0.9
0.8
0.25
0.75
0.2
0.7
0.15
0.65
0.1
0.6
0.05
0.55
0.3
0
0
0.2
0.4
0.6
Fill factor
0.8
1
0
0.5
0
0.2
0.4
0.6
0.8
1
Fill factor
Only 40x16 samples done for this output, needs many more samples for more
accurate graph
Minimum TE transmission at fill factor ≈ 0.5 and grating depth ≈ 0.35μ𝑚
Minimum TM transmission at fill factor ≈ 0.5 and grating depth ≈ 0.40μ𝑚
23
Waveguide Grating - Preliminary
Normal incident Gaussian beam(TE,1064nm)
transmission through waveguide coating, d=700nm.
Normal incident Gaussian beam(TM,1064nm) transmission
through waveguide coating, d=700nm.
1
1
0.9
0.8
0.8
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.3
0.4
0.2
Grating depth(um)
Grating depth(um)
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.3
0.1
0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Waveguide thickness(um)
0.8
0.9
1
0.2
0
0.2
0.4
0.6
0.8
1
Waveguide thickness(um)
Only 40x40 samples done for this output, again more samples needed to find lower
transmissions
24
Waveguide Grating - Preliminary
Normal incident Gaussian beam(TM,1064nm) transmission
through waveguide coating, d=700nm.
1
0.8
0.9
Grating depth(um)
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
Comparing to A. Bunkowski et al. (2006)
TM reflectivity results.
0.1
0.2
0
0.2
0.4
0.6
0.8
1
Waveguide thickness(um)
Low transmissions are comparable to high reflectivity seen from RCWA
computations
25
Conclusion
 What has happened so far:
 Initial implementation of 2D FDTD simulation has been
developed in Java and running on Beowulf cluster
 FDTD method appears to be a suitable method for simulating
grating movements and the phase noises
 Shown that the ray picture phase noise also occurs in FDTD
simulations of TEM00 diffraction
 Have begun to model the waveguide coatings, but some
issues that need looking into
26
Conclusion
 Future plans:
 More work on verifying what is seen in FDTD simulations to
the RCWA method used by A.Bunkowski et al. (2006)
 Measure the more relevant reflection coefficient for waveguide
coatings and determine if it is immune to phase noises from
moving grating
 Looking into isotropic dispersion techniques and how they can
improve the simulation errors
 To further develop the tool for optical simulations
27