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, ux, vy).

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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