Page 393 - Electromagnetics

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Because E 1 has no u-component, E 1 and H 1 are termed transverse electric (or TE)to
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the u-direction; H 2 has no u-component, and E 2 and H 2 are termed transverse magnetic
(or TM ) to the u-direction. 2 We see that in a source-free region any electromagnetic
field can be decomposed into a set of two fields that are TE and TM, respectively, to
some fixed u-direction. This is useful when solving boundary value (e.g., waveguide
and scattering) problems where information about external sources is easily specified
using the values of the fields on the boundary of the source-free region. In that case
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E u and H u are determined by solving the homogeneous wave equation in an appropriate
coordinate system, and the other field components are found from (5.115)–(5.116). Often
the boundary conditions can be satisfied by the TM fields or the TE fields alone. This
simplifies the analysis of many types of EM systems.
5.4.2 TE–TM decomposition in terms of Hertzian potentials
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We are free to represent E and H in terms of scalar fields other than E u and H u .In
doing so, it is helpful to retain the wave nature of the solution so that a meaningful
physical interpretation is still possible; we thus use Hertzian potentials since they obey
the wave equation.
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˜ i
For the TM case let Π h = 0 and Π e = ˆ u e . Setting J = 0 in (5.64) we have
2 2 ˜
(∇ + k )Π e = 0.
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Since Π e is purely longitudinal, we can use (B.99) to obtain the scalar Helmholtz equation
˜
for e :
˜
2 2
(∇ + k ) e = 0. (5.119)
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Once e has been found by solving this wave equation, the fields can be found by using
˜ i
(5.62)–(5.63) with J = 0:
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˜
E =∇ × (∇× Π e ), (5.120)
c
˜
˜
H = jω˜ ∇× Π e . (5.121)
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We can evaluate E by noting that Π e is purely longitudinal. Use of property (B.98) gives
˜
˜ ∂ e 2 ˜
∇× ∇× Π e =∇ t − ˆ u∇ e .
t
∂u
Then, by property (B.97),
˜ 2 ˜
˜ ∂ e 2 ˜ ∂ e
∇× ∇× Π e =∇ t − ˆ u ∇ e − 2 .
∂u ∂u
By (5.119) then,
˜
∂ 2
˜ ∂ e 2 ˜
E =∇ t + ˆ u 2 + k e . (5.122)
∂u ∂u
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The field H can be found by noting that Π e is purely longitudinal. Use of property
(B.96) in (5.121) gives
c
˜
˜
H =− jω˜ ˆ u ×∇ t e . (5.123)
2 Some authors prefer to use the terminology Emode in place of TM, and Hmode in place of TE,
indicating the presence of a u-directed electric or magnetic field component.
© 2001 by CRC Press LLC