Page 273 - Electromagnetics

producing
2 2
E 0 (t) = E 0 e −a t cos(ω 0 t). (4.230)
We think of f (t) as modulating the single-frequency cosine carrier wave, thus providing
the envelope. By using a large value of a we obtain a narrowband signal whose spectrum
is centered about ±ω 0 . Later we shall let a → 0, thereby driving the width of f (t) to
infinity and producing a monochromatic waveform.
By (1) we have
˜ 1 ˜ ˜
E 0 (ω) = E 0 F(ω − ω 0 ) + F(ω + ω 0 )
2
˜
where f (t) ↔ F(ω). A plot of this spectrum is shown in Figure 4.2. We see that
the narrowband signal is centered at ω =±ω 0 . Substituting into (4.229) and using
ˆ
k = (β − jα)k for a uniform plane wave, we have the frequency-domain field
1
˜ ˜ − j[β(ω)− jα(ω)] ˆ k·r ˜ − j[β(ω)− jα(ω)] ˆ k·r
E(r,ω) = ˆ eE 0 F(ω − ω 0 )e + F(ω + ω 0 )e . (4.231)
2
The field at any time t and position r can now be found by inversion:
1 ∞ 1 − j[β(ω)− jα(ω)] ˆ k·r
˜
ˆ eE(r, t) = ˆ eE 0 F(ω − ω 0 )e +
2π −∞ 2

˜
+ F(ω + ω 0 )e − j[β(ω)− jα(ω)] ˆ k·r e jωt dω. (4.232)
˜
We assume that β(ω) and α(ω) vary slowly within the band occupied by E 0 (ω). With
this assumption we can expand β and α near ω = ω 0 as
1 2

β(ω) = β(ω 0 ) + β (ω 0 )(ω − ω 0 ) + β (ω 0 )(ω − ω 0 ) +· · · ,
2
1 2

α(ω) = α(ω 0 ) + α (ω 0 )(ω − ω 0 ) + α (ω 0 )(ω − ω 0 ) +· · · ,
2
2
2
where β (ω) = dβ(ω)/dω, β (ω) = d β(ω)/dω , and so on. In a similar manner we can

expand β and α near ω =−ω 0 :
1 2

β(ω) = β(−ω 0 ) + β (−ω 0 )(ω + ω 0 ) + β (−ω 0 )(ω + ω 0 ) +· · · ,
2
1 2

α(ω) = α(−ω 0 ) + α (−ω 0 )(ω + ω 0 ) + α (−ω 0 )(ω + ω 0 ) +· · · .
2
Since we are most interested in the propagation velocity, we need not approximate α
with great accuracy, and thus use α(ω) ≈ α(±ω 0 ) within the narrow band. We must
consider β to greater accuracy to uncover the propagating nature of the wave, and thus
use
β(ω) ≈ β(ω 0 ) + β (ω 0 )(ω − ω 0 ) (4.233)

near ω = ω 0 and

β(ω) ≈ β(−ω 0 ) + β (−ω 0 )(ω + ω 0 ) (4.234)
near ω =−ω 0 . Substituting these approximations into (4.232) we find
1 ∞ 1 − j[β(ω 0 )+β (ω 0 )(ω−ω 0 )] ˆ k·r −[α(ω 0 )] ˆ k·r

˜
ˆ eE(r, t) = ˆ eE 0 F(ω − ω 0 )e e +
2π 2
−∞

˜

+ F(ω + ω 0 )e − j[β(−ω 0 )+β (−ω 0 )(ω+ω 0 )] ˆ k·r −[α(−ω 0 )] ˆ k·r e jωt dω. (4.235)
e
© 2001 by CRC Press LLC