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I am interested in modeling a periodic device via 2D Finite Difference Time Domain method (FDTD), in order to calculate the reflection and transmission from a single-frequency plane wave incident at a variety of angles. I implemented a 2D FDTD in Matlab with:

  • Bloch periodic boundary conditions along the x-boundaries (see EMPossible lecture 3i Slide 26)
  • Perfectly Matched Layer (PML) at the y-boundaries
  • Light propagating in the y-direction (normal incidence)

I tested the simulation by comparing simulated results to theory in these devices:

In all these tests my simulation results matched well to the theory. I then adapted my program to allow for oblique incidence (or so I thought), and compared results to Fresnel equations (interface between two dielectrics) at various angles of incidence. My results did NOT match up with theory.

I used this paper to formulate my equations for calculating reflectivity and transmission: https://studylib.net/doc/9703602/calculating-power-flow-in-fdtd.
What do I need to change in the equations to properly allow for oblique incidence?

Simulation Boundary Conditions

FDTD Results vs. Fresnel Equations

Benjamin Dalin
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1 Answers1

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I found the answer - the intensity should be calculated as the y-value (i.e. the direction of power flow) of the Poynting Vector: $\qquad S_y = E_zH_x - E_xH_z$

For TE ($H_z=0$): $\quad S_y = E_zH_x$

For TM ($E_z=0$): $\quad S_y = -E_xH_z$

Since the $E$ and $H$ fields in the Yee grid are staggered in time by half a time step, and in space by half a grid cell, the H field needs to be averaged:

$H_x(i,j) = [H_x(i,j-1/2) + H_x(i,j+1/2)]/2$

$H_z(i,j) = [H_z(i,j-1/2) + H_z(i,j+1/2)]/2$

where $i$,$j$ are the respective $x$,$y$ grid coordinate indices.

See EMPossible Lecture 3G Power-Flow-and-PML-Placement:

EMPossible Lecture 3G Power-Flow-and-PML-Placement slide 16

Benjamin Dalin
  • 11