9/13/2016
EE 5303
Electromagnetic Analysis Using Finite‐Difference Time‐Domain
Lecture #26
Final Lecture
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Lecture 26
Slide 1
Lecture Outline
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•
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Key points from course
TE vs. TM
Resonance hunting
Photonic crystals fabricated by multi‐photon direct laser writing
Lecture 26
Slide 2
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Key Points From Course
Lecture 26
Slide 3
Key Steps in the Art of Computational EM
• Formulation of the method
– Maxwell’s equations  numerical equations
• Implementation of the method
– Numerical equations  computer code
• Using the modeling tool
– Device  grid  extract data
– Assumptions for model
• Reconciling the model with experimental results
– Model  reality
Lecture 26
Slide 4
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Material Covered This Semester
• MATLAB
– Basic commands
– Graphics
– Putting devices on grids
• 1D FDTD
– Formulation: Yee grid, finite‐differences, update equations, update coefficients
– Implementation: PAB, TF/SF, Fourier transforms, transmittance and reflectance, grid resolution
– Generalizations: loss, dispersion
• 2D FDTD
– Formulation: boundary conditions, PML, update equations, update coefficients
– Implementation: PML, TF/SF, Fourier transforms, transmittance, reflectance
– Generalizations: stretched coordinate PML, alternate grid schemes, waveguides, metals, periodic devices
Lecture 26
Slide 5
Other Key Points
• Rhythm of deriving update coefficients
• Order of code development
• Art of representing devices on grids
Lecture 26
Slide 6
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Questions for an In‐Class Final Exam
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Write Maxwell’s equations
Draw and label a 3D Yee cell
Discuss the benefits and drawbacks of FDTD
Rules of finite‐difference equations
Deriving an update equation from a simple finite‐
difference equation
• Outline the FDTD algorithm
Lecture 26
Slide 7
TE vs. TM Modes in 2D Simulations
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Slide 8
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Reduction from 3D to 2D
Uniform Device
Effective Index Method
Lecture 26
Slide 9
Two Distinct Modes in 2D
For 2D devices, d/dz=0 and Maxwell’s equations separate into two distinct modes.
E
H z H y
  xx x

t
y
z
E y
H x H z

  yy
z
x
t
H y H x
E

  zz z
x
y
t
Lecture 26
x
E y
x
H x
E
  zz z
y
t
H x
Ez
   xx
y
t
Hy

E
 z    yy
x
t
Ex
H z
   zz
y
t
Ex
H z
  xx
y
t
Ey

H z

  yy
x
t
Hz Mode
H y
Ex Ez

   yy
z
x
t
E y Ex
H z

   zz
x
y
t
H y
Ez Mode
H x
Ez E y
   xx

t
y
z


Slide 10
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Framework #1 for the Definition of TE and TM
E orientation relative to direction of propagation
For the Ez mode, the electric field is always transverse to waves propagating in the x‐y plane.
Ez Mode = TE Mode
For the Hz mode, the magnetic field is always transverse to waves propagating in the x‐y plane.
Hz Mode = TM Mode
Lecture 26
Slide 11
Framework #2 for the Definition of TE and TM
E orientation relative to the plane of incidence
TE – the electric field is polarized perpendicular to the plane of incidence.
TM – the electric field is polarized parallel to the plane of incidence.

kinc
In the limit as 90°, we have

Ez Mode = TM Mode
TM
TE

Lecture 26
Hz Mode = TE Mode
This is exactly the opposite of Framework #1 when propagation is restricted to be in the x‐y plane.
Slide 12
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Resonance Hunting
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Slide 13
Device for This Example
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Slide 14
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Results from a Frequency‐Domain Method
Frequency‐domain methods compute the field at one precise frequency from which transmittance and reflectance is calculated. If frequency points selected for the sweep are two widely spaced, narrow resonances are easily missed. Small frequency steps are very time consuming because many simulations have to be performed.
Lecture 26
Slide 15
Results from a Time‐Domain Method
Time‐domain methods typically excite devices with all frequencies over a broad range. If a resonance exists, a time‐domain method will detect it in a single simulation. It may take many iterations to resolve how narrow the resonance is, but the method will detect it almost immediately.
increasing number of iterations
Lecture 26
Slide 16
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Conclusions
• Time‐domain methods are excellent for identifying the existence of resonances.
• Time‐domain methods are poor for resolving the shape of narrow resonances.
• Frequency‐domain scattering methods are not reliable for finding narrow resonances. Visualizing the fields during a sweep can help this.
• Frequency‐domain methods are excellent for resolving the shape of resonances.
Lecture 26
Slide 17
Photonic Crystals Fabricated by Multi‐Photon Direct Laser Writing
Lecture 26
Slide 18
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Multi‐Photon Direct Laser Writing
This is a scanning electron microscope image of a photonic crystal designed to operate in the infrared.
Lecture 26
Slide 19
Fourier Transform Infrared Spectrometer
The cassegrain optics illuminates samples with a “hollow” Gaussian beam.
Reflected infrared power at near normal reflection angle is collected and measured.
R. C. Rumpf, A. Tal, S. M. Kuebler, "Rigorous Electromagnetic Analysis of Volumetrically Complex Media Using the Slice Absorption Method," J. Opt. Soc. Am. A 24(10), 3123–3134 (2007).
Lecture 26
Slide 20
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Predicting Accurate Geometry
To predict the geometry of the photonic crystal more accurately, the DLW process was simulated in MATLAB.
R. C. Rumpf, A. Tal, S. M. Kuebler, "Rigorous Electromagnetic Analysis of Volumetrically Complex Media Using the Slice Absorption Method," J. Opt. Soc. Am. A 24(10), 3123–3134 (2007).
Idealized Geometry
Lecture 26
Realistic Geometry
Slide 21
Importance of Realistic Geometry
R. C. Rumpf, A. Tal, S. M. Kuebler, "Rigorous Electromagnetic Analysis of Volumetrically Complex Media Using the Slice Absorption Method," J. Opt. Soc. Am. A 24(10), 3123–3134 (2007).
Lecture 26
Slide 22
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Metallo‐Dielectric Photonic Crystals
Before copper plating
After copper plating
A. Tal, Y. Chen, H. Williams, R. C. Rumpf, S. Kuebler, “Fabrication and characterization of three‐
dimensional copper metallodielectric photonic crystals,” Opt. Express 15(26), 18283‐18293 (2007)
Lecture 26
Slide 23
“State‐of‐the‐Art” Simulation of Reflectance
Results obtained by Lumerical and Misawa’s research group.
V. Mizeikis, S. Juodkazis, R. Tarozaite, J. Juodkazyte, K. Juodkazis, H. Misawa, “Fabrication and properties of metalo-dielectric photonic
crystal structures for infrared spectral region,” Opt. Express 15, 8454-8464 (2007)
Lecture 26
Slide 24
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A Better Simulation of Reflectance
Results obtained by UCF/Rumpf team…
A. Tal, Y. -S. Chen, H. E. Williams, R. C. Rumpf, and S. M. Kuebler, "Fabrication and characterization of threedimensional copper metallodielectric photonic crystals," Opt. Express 15, 18283-18293 (2007)
Lecture 26
Slide 25
Side‐by‐Side Comparison
Results obtained
by UCF/Rumpf
reflectance
Results obtained
with Lumerical’s
FDTD software
V. Mizeikis, S. Juodkazis, R. Tarozaite, J. Juodkazyte, K. Juodkazis, H.
Misawa, “Fabrication and properties of metalo-dielectric photonic crystal
structures for infrared spectral region,” Opt. Express 15, 8454-8464 (2007)
Lecture 26
A. Tal, Y. -S. Chen, H. E. Williams, R. C. Rumpf, and S. M. Kuebler,
"Fabrication and characterization of three-dimensional copper metallodielectric
photonic crystals," Opt. Express 15, 18283-18293 (2007)
Slide 26
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What May Have Been Their Mistake?
Below 10 m (or so), this photonic crystal is a diffracting structure. The optical configuration inside the FTIR cuts off the higher order modes. Essentially, it is only the zero‐order diffracted mode that gets detected.
Lecture 26
Slide 27
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