Page 400 - Electromagnetics

Other important relationships include the orthogonality of the longitudinal fields,

ˇ
ˇ
E zm E zn dS = 0, m
= n, (5.149)
CS

ˇ
ˇ
H zm H zn dS = 0, m
= n, (5.150)
CS
and the orthogonality of transverse fields,

ˇ
ˇ
E tm · E tn dS = 0, m
= n,
CS

ˇ
ˇ
H tm · H tn dS = 0, m
= n.
CS
These may also be combined to give an orthogonality relation for the complete fields:

ˇ
ˇ
E m · E n dS = 0, m
= n, (5.151)
CS

ˇ
ˇ
H m · H n dS = 0, m
= n. (5.152)
CS
For proofs of these relations the reader should see Collin [39].
Power carried by time-harmonic waves in lossless waveguides. The power car-
ried by a time-harmonic wave propagating down a waveguide is defined as the time-
average Poynting flux passing through the guide cross-section. Thus we may write
1
ˇ
ˇ ∗
P av = Re E × H · ˆ z dS.
2 CS
The field within the guide is assumed to be a superposition of all possible waveguide
modes. For waves traveling in the +z-direction this implies
ˇ " − jk zm z ˇ " ˇ ˇ − jk zn z
E = (ˇ e tm + ˆ zˇ e zm ) e , H = h tn + ˆ zh zn e .
m n
Substituting we have
# $
1 " − jk zm z " jk zn z
∗
ˇ ∗
P av = Re (ˇ e tm + ˆ zˇ e zm ) e × h + ˆ zh ˇ ∗ zn e · ˆ z dS
tn
2 CS m n

1 " " − j(k zm −k zn )z
∗
ˇ ∗
= Re e ˆ z · ˇ e tm × h tn dS .
2 CS
m n
By (5.148) we have

1 " − j(k zn −k zn )z
∗
P av = Re e ˆ z · ˇ e tn × h ˇ ∗ tn dS .
2 CS
n
For modes propagating in a lossless guide k zn = β zn . For modes that are cut off k zn =
− jα zn . However, we find below that terms in this series representing modes that are cut
off are zero. Thus
1
" "
ˇ ∗
P av = Re ˆ z · ˇ e tn × h tn dS = P n,av .
2
n CS n
© 2001 by CRC Press LLC