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APPROACH TO NUMERICAL SOLUTION OF EM PROBLEMS 457

electric and magnetic conductivity gradually rises thereby improving the absorption
performance. It is worth pointing out that the basic theory behind PMLs is based on the field
presentation as continues variables, and they indeed work “perfectly” (as its name implies) for
all incident angles, polarizations, and frequencies. However, the analytical perfection of PML
is no longer valid when it is implemented in the discretized world of FDTD. There are always
some reflections back into internal domain shown as a red dotted line in Figure 9.1.18. The
appropriate PMLs of 6 – 12 cell thick can reduce such reflections 10 – 10 times or 120 - 160dB
6
8
over a rather broad range of frequencies and incident angles (typically not exceeding 70 that
o
is quite enough to get the correct solution for a wide variety of practical problems).
One of the obvious PMLs drawbacks is that they absorb not only EM fields leaving the internal
domain forever in the form of radiation but as well the reactive (evanescent) fields accumulated
in the model proximity. Evidently, such effect can be diminished if the nearest PML boundaries
are shifted relatively far-far away from the model area, for example, several wavelengths’ apart
at lowest frequency. Apparently, this approach increases the discretized area and might make
the computer simulation prohibitive long. Perhaps, the best method to evaluate and control this
effect is the numerical experiment, since the forecast of near-field intensity and structure is
rather problematic. Run the model gradually increasing PML cavity sizes, animate the field
distribution, and compare results. The final goal is to bring such influence to the acceptable
level. Sometimes, this effect may be controlled by the balance energy following from
Poynting’s theorem.

9.1.4 Far-Field Calculation

As we have mentioned before, the numerical methods based on FDTD, FEM, and many others
restrict the size of computation domain thereby leaving unknown the fields over the infinite
space outside PML cavity. Meanwhile, the data about these radiated far away fields is one of
the principal purposes of the undertaken simulation while the near field distribution inside the
cavity is significant giving us the information about the model input impedance, losses, power
handling, etc., but not complete. So we need somehow to relay our knowledge of adjacent fields
with far field spreading, i.e. find the way to transform near field into the far field. As usual, the
critical support comes from Poynting’s theorem proved in Chapter 3. According to expression
(3.18), the net power flow depends only on the tangential to the surface components of

electric and magnetic field, i.e. () = ∯ x . Considering in this integral the
Σ

interior surface of PML cavity as the integration surface and taking the tangential components
from numerical simulation, we can calculate the total radiated power. Furthermore, Huygens’
Principle (see Figure 4.3.8) formulated in Section 4.3.4 of Chapter 4 provides the actual tool
for such far field calculation. Simply interpret each sport on the surface as the equivalent
Huygens’ radiator and summarize (actually integrate) the fields radiated by all of them. The
real implementation of the near-to-far-field transformation could deviate from the described in
some details but not significantly. Typically, this transformation is based on the vector and
scalar potential application, i.e. on the expression (4.4.6) from Chapter 4, and a quite
complicated mathematical procedure. You do not concern about all these issues because
commercial full-wave products prepare this task automatically as the post-processing task. We
recommend the reader to turn to comprehensive online publication [16] for more detailed
presentation.