Figure 2.2.2 Huygens' principle for surface scattering problem.

Hence lim EsCr') =

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::~~ [1 d5e- ik .r {(VsVs + hsh s) . n x Hs(r)

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+ ~ks

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(2.2.24)

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For the surface scattering problem (Fig. 2.2.2), the extinction theorem of (2.2.10), (2.2.15), (2.2.16), and (2.2.21) still apply. The Huygens' principle of (2.2.11) and (2.2.23) still apply with ks pointing into region O. However, the surface integral 51 is only over the half-space boundary separating region o and region 1 (Fig. 2.2.2). Assuming dissipation in medium 1, the surface integral over hemispherical surface 5 100 of lower half-space vanishes. From (2.2.21) and (2.2.24), we get the relation {

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lim Es(r')

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2 BASIC THEORY OF ELECTROMAGNETIC SCATTERING

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It is important to know that Huygens' principle does not solve the electromagnetic boundary value problem. From (2.2.10), we need to know both the tangential electric field and the tangential magnetic field on the boundary S in order to calculate the field at a point in space. On the other hand, the uniqueness theorem states that in order that the field be uniquely determined, we can specify one of the following:

(1) it x E everywhere on 5, (2) it x H everywhere on 5, (3) n x E on a part of Sand

Thus one cannot on S. For example, to specify the boundary value problem, one can specify n x E on S (say n x E = 0 on a perfect electric conductor), and then solve it x H as a boundary value problem. After n x H is solved, then together with the specified it x E, one can use Huygens' principle to calculate the field everywhere. Thus Huygens' principle is used to compute the field "output" in a rigorous electromagnetic analysis. However, in approximate electromagnetic analysis, e.g., the Kirchhoff approximation discussed in 9, one applies Huygens' principle by using approximate values of n x E and n x H on S.

n x H on the rest of S. specify both n x E and n x H

Active Remote Sensing and Bistatic Scattering Coefficients

For active remote sensing, a radar consisting of a transmitter system and a receiver system is utilized. Bistatic radar has the transmitting system and the receiving system situated at different locations, whereas in monostatic radar, these systems are located at the same place, usually sharing the same antenna system. The transmitter sends out a signal to the target and the scattered signal in a specified direction is measured by the receiver. The incident power density Si at the target can be expressed in terms of the transmitter power Pt and the gain function Gt in the direction ki (Fig. 2.3.1):

GtPt Si = 41rr 2 exp( -'Yt)

(2.3.1)

where rt is the distance between the transmitting antenna and the target and exp( -'Yt) is the numerical factor that accounts for the propagation loss in the medium between the transmitter and the target. For example, if at is the attenuation coefficient between transmitter and target, then 'Yt is the

3 Active Remote Sensing and Bistatic Scattering

transmitter

receiver

Figure 2.3.1 Bistatic scattering with incident direction (Oi, >i) and scattered direction (Os,1>s). Area A of the target is illuminated by the incident beam.

integration of at over distance

"it =

at ds

(2.3.2)

The target is characterized by the quantity OA, which has the dimension of area and is dependent on incident and scattered directions and the polarization of the incident and scattered waves

(JA=

47r 1 - 8 1m r2Ss

r->oo

(233) ..

where r is the distance between the target and the receiving antenna and 8 s is the scattered power density. The received power Pr at the receiving antenna is Pr = ArSs exp( -"it), where A r is the receiver cross section of the receiving antenna, and exp( -"it) is the attenuation factor of the medium between target and receiver. The receiver cross section can be related to the receiving antenna gain by A r = Gr )..2/(47f), where G r is the receiver antenna gain in the direction ks and)" is the wavelength [Ishimaru, 1978]. Thus,