An Introduction to Computational Electromagnetics using FDTD

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• An Introduction to Computational Electromagnetics
  using FDTD
• R. E. Diaz

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Finite Difference Time Domain is a Partial
Differential Equation method

• The DDSURF/SUB/FILM family is an Integral
  Equation method.
• Advantage Need to dicretize only scatterer.
• If The Green function of the environment is
  known.
• Disadvantage Must invert a huge matrix.
  Unstable if egtgt1.

• PDE methods
• Advantage No Green function, no inversion.
• If You have time (computing power) if egtgt1.
• Disadvantage Must discretize all of space.

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FDTD encodes Maxwells curl equations over all
space.

• Maxwells first curl equation
• where e is in general an operator e(w)
• Take its Fourier Transform
• Thus,
• And

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This is an Initial Value PDE problem that can be
solved from time t to tdt

• To solve the inhomogeneous PDE in discretized
  time, set up a leapfrog scheme

• If H is evaluated at the half-integer steps while
  E is evaluated at the integer steps, the curl
  acts as a source term.

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We therefore have a PDE with constant
coefficients and a constant inhomogeneous term.

• We have two alternatives
• (a) solve the initial value problem (gives an
  exponential characteristic solution) or
• (b) turn the equation into finite difference form
  using the fundamental theorem of calculus

• But at what time must E(?) be evaluated?

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For the equation to be valid at the same instant
everything must be evaluated at the half-integer
time step.

• The time derivative of E is clearly evaluated at
  the half-integer step.
• So is the curl of H.
• Therefore so must be E(?)(E(tdt)E(t))/2

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In the same way, Maxwells second curl equation
is reduced to an update equation.

• Now, since curl of E is evaluated at t, and the
  time derivative of H also occurs at t,
• H(?)(H(tdt/2)H(t-dt/2))/2
• And
• becomes

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Thus, we have a leapfrog algorithm, from H to E
and from E to H, and back again...

• Now lets talk about discretization in space.
• The Yee lattice intercalates E and H in space,
  making the definitions of the curl operators
  straight-forward.
• The Yee unit cell
• At the (i,j,k) point E is on the edges, H is on
  the faces.

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Consider the x component of the curl of H
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Similarly, the x component of the curl of E...
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The FDTD algorithm marches E and H in time
throughout the grid the way an EM wave propagates.

• All interactions in materials and satisfaction of
  boundary conditions across material boundaries
  occur automatically.
• Because the computational domain must be
  truncated, need Absorbing Boundary Conditions.
• The finite grid size introduces grid dispersion
  that limits the upper frequency at which a
  Fourier Transform of the time domain result is
  valid.
• The time domain solution is multi-frequency by
  nature. Not necessarily overkill because it can
  be used to discern phenomena.

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Consider a 108 nm Si sphere sitting on a 75 nm
SiO2 film on top of Si
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Conclusions and Future Work

• FDTD is a very stable PDE computational
  electromagnetics tool that is rapidly becoming a
  standard in Optics as well as RF research.
• It serves well as a complement to the IE method
  family of codes DDSURF/SUB/FILM.
• The latter can be extremely fast for moderate
  scenes But when material parameters are extreme
  and/or the matrix just cannot be inverted, all
  that FDTD needs is memory and time.
• Ongoing work to improve its efficiency for
  optical scattering focuses on dispersive material
  modeling, the absorbing boundary conditions, and
  the input and output of incident plane waves.