 5.4. FDTD/PBC FLOQUET HARMONIC ANALYSIS OF P ERIODIC STRUCTURES 73
the DFT, which does not require any extra memory compared to the conventional FDTD technique,
due to the fact that the ﬁelds are captured in the code using (5.15). This feature of the constant
horizontal wavenumber FDTD/PBC approach is considered an important advantage because of the
reduction in memory usage.
FDTD Harmonic Analysis Procedure:
1) Use constant horizontal wavenumber approach to calculate E(t, ux, vy).
2) Use (5.15) to calculate the time-domain magnitude of different harmonics in the FDTD/PBC
simulation.
3) Repeat steps 1 and 2 until all time steps in the FDTD simulation are completed.
4) Use the DFT to calculate the frequency-domain magnitude of different harmonics.
The above procedure can be used with any periodic structure to completely study the effect of
different harmonics on the cascading conﬁguration.
5.4.2 GUIDELINE FOR HARMONIC SELECTION
In this section a procedure for determining the proper gap size in order to neglect the higher
harmonics effects is described. The procedure can also be used to determine which harmonics to be
considered for speciﬁc gap size.
FDTD Gap Size Determination Procedure:
1) Specify the periodicity, the order, and the geometry of each layer:The periodicity and geometry
of the layer are important to determine the cut-off frequencies and magnitudes of different
harmonics. The layers order determines which of the reﬂected or transmitted harmonics are
to be considered.
2) Specify the frequency range of interest. The frequency range of interest is important to de-
termine whether the harmonics are propagating or evanescent harmonics in this frequency
range.
3) Specify the incident wave parameters (k
i
x
and k
i
y
).Usek
i
x
and k
i
y
to determine the cut-off
frequencies of different harmonics. Any propagating harmonics in the frequency range of
interest should be considered whatever the gap size is.
4) Use the harmonic analysis procedure to determine the magnitudes of the evanescent harmon-
ics: Calculate k
m,n
z
and use it together with the harmonic magnitudes to study the decaying
behavior of the evanescent harmonics with distance.
5) The gap size for neglecting speciﬁc harmonic effect is calculated as the distance after which
all evanescent harmonics magnitudes decay below 40 dB compared to the excitation level of 74 5. MULTILAYERED PERIODIC STRUCTURES
Start
Specify the periodicity of different layers, the order and geometry of each
layer
Specify the frequency range of interest
Specify the incident wave parameters (
i
x
k and
i
y
k
)
Using
i
x
k and
i
y
k
, determine the cut-off frequencies of different modes
Using the harmonic analysis procedure, determine which harmonics to be
included according to the frequency range of interest
Set -40 dB from the excitation electric field magnitude as threshold for
neglecting the effect of the harmonics
End
Figure 5.10: The ﬂow chart of gap size determination procedure.
the corresponding ﬁeld component magnitude. The 40 dB threshold was concluded from
different test cases for error less than 5%. Other accuracy can be achieved by c hanging the
threshold value.
If the gap size is less than that determined by this procedure, all the evanescent harmonics that have
magnitudes larger than the threshold level should be included in the cascading process for accurate
results.

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