Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Silicon photonics wikipedia , lookup
Ultraviolet–visible spectroscopy wikipedia , lookup
Diffraction topography wikipedia , lookup
Interferometry wikipedia , lookup
Anti-reflective coating wikipedia , lookup
Nonlinear optics wikipedia , lookup
Powder diffraction wikipedia , lookup
Fiber Bragg grating wikipedia , lookup
Diffraction wikipedia , lookup
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