218 Chapter 8 Ray and Caustics Asymptotics for Edge Diffracted Waves
8.1.2Electromagnetic Waves
According to Equations (7.130) and (7.131), the EEWs diverging from a scattering edge L create the combined wave
|
|
E |
(1,t) |
= |
1 |
(1,t)(ζ ) |
eikR(ζ ) |
dζ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
2π |
L E |
|
R(ζ ) |
|
|
|
and |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
H(1,t) = |
|
1 |
|
L [ R(ζ ) |
× E(1,t)(ζ )] |
eikR(ζ ) |
dζ , |
|
|
|
|
|
|
|
2π Z0 |
R(ζ ) |
with |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
(1,t)(ζ ) |
= [ |
E0t (ζ )F(1,t)(ζ ) |
+ |
Z0H0t (ζ )G(1,t)(ζ ) |
eikφi (ζ ). |
E |
|
|
|
|
|
|
|
|
|
|
|
] |
|
|
integrands contain the fast oscillating factor exp |
ik (ζ ) |
, where |
Here, the |
i |
[ |
] |
|
(ζ ) = R(ζ ) + φ |
(ζ ). Therefore, the application of the stationary-phase technique |
to these integrals results in the following ray asymptotics:
Eϑ(1) |
= |
Z0Hϕ(1) |
|
|
|
Einc(ζst ) |
|
f |
(1)(ϕ, ϕ , α) |
|
|
|
|
eiπ/4 |
|
eikR |
|
|
|
|
|
t |
|
|
f (ϕ, ϕ |
,0α) |
|
|
|
|
|
|
|
|
|
|
|
(t) |
|
(t) |
= − |
sin γ0 |
|
|
|
|
|
|
|
|
R |
|
|
|
|
|
|
|
|
Eϑ |
|
= |
Z0Hϕ |
|
0 |
√2π k (ζst ) |
|
|
and |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Eϕ(1) |
= − |
Z0Hϑ(1) |
|
|
Z0Hinc(ζst ) |
(1)(ϕ, ϕ , α) |
|
|
|
|
eiπ/4 |
|
|
|
eikR |
|
|
|
|
t |
|
|
gg(ϕ, ϕ ,0α) |
|
|
|
|
|
|
|
|
. |
(t) |
(t) |
= |
|
sin γ0 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Eϕ |
= − |
Z0Hϑ |
|
|
|
0 |
√2π k (ζst ) R |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
They can also be written in the form of Equation (8.29):
Eϑ(1) |
|
Z0Hϕ(1) |
|
|
|
Einc(ζ |
) |
|
|
f (1)(ϕ, ϕ0, α) |
|
|
eiπ/4 |
|
|
|
eikR |
|
|
= |
|
|
|
|
t |
|
st |
|
f (ϕ, ϕ0, α) |
|
|
|
|
|
|
|
|
|
|
|
|
, |
|
(8.35) |
(t) |
|
(t) |
|
|
|
2 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Eϑ |
|
= |
Z0Hϕ |
|
|
|
|
|
γ0 |
|
√2π k √R(1 + R/ρ) |
|
|
|
= − sin |
|
|
|
and |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Eϕ(1) |
= − |
Z0Hϑ(1) |
|
|
Z0Hinc(ζst ) |
|
(1)(ϕ, ϕ , α) |
|
|
eiπ/4 |
|
|
|
eikR |
|
|
|
|
|
t |
|
|
|
|
|
|
gg(ϕ, ϕ ,0α) |
|
|
|
|
|
|
|
|
|
|
. |
(8.36) |
(t) |
(t) |
|
|
|
2 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
= |
|
|
γ0 |
|
√2π k |
|
Eϕ |
= − |
Z0Hϑ |
|
|
|
sin |
0 |
|
√R(1 + R/ρ) |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
8.1 Ray Asymptotics 219
Taking into account Equations (7.149) and (7.150), one can represent these approximations in terms of the field components tangential to the scattering edge:
|
E(1) |
= Etinc(ζst ) |
|
|
(1)(ϕ, ϕ , α) |
|
|
|
|
eiπ/4 |
eikR |
|
|
|
Et(t) |
f f (ϕ, ϕ0 |
,0α) |
|
√ |
|
|
|
|
|
|
, |
(8.37) |
|
|
|
R |
|
|
2π k (ζst ) |
|
|
t |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
H(1) |
= Htinc(ζst ) |
|
(1)(ϕ, ϕ , α) |
|
|
|
|
eiπ/4 |
|
eikR |
|
|
|
Ht(t) |
gg(ϕ, ϕ0,0α) |
|
√ |
|
|
|
. |
(8.38) |
|
R |
|
|
2π k (ζst ) |
|
t |
|
|
|
|
|
|
|
|
|
|
|
|
|
8.1.3Comments on Ray Asymptotics
•A comparison of Equations (8.12), (8.13) and (8.37), (8.38) reveals the following relationships between acoustic and electromagnetic diffracted rays:
us = Et , |
if uinc(ζ ) = Etinc(ζ ) |
(8.39) |
uh = Ht , |
if uinc(ζ ) = Htinc(ζ ), |
(8.40) |
|
|
|
where ζ is the diffraction point on a scattering edge. These relationships (together with Equations (7.149), (7.150)) allow one to completely determine the field of electromagnetic rays diffracted at a perfectly conducting object, if one knows the acoustic rays diffracted at soft and rigid objects of the same shape and size. Notice that these relationships were established earlier, in the paper by Ufimtsev (1995).
•The ray asymptotics (8.29), (8.35), (8.36) (for the field generated by the total scattering sources j(t) = j(0) + j(1)) were postulated in the Geometrical Theory of Diffraction (GTD) (Keller, 1962). Now it is seen that GTD can be interpreted as the ray asymptotic form of PTD for the total diffracted field. Notice that the ray asymptotics of Equation (8.29) type (but in the Kirchhoff approximation) were obtained first by Rubinowicz (1924). In the paper by Ufimtsev (1995), it is shown that the above ray asymptotics can be easily obtained by the direct extension of the Rubinowicz theory.
•In contrast to PTD, GTD is not applicable in the regions where the field does not have a ray structure and where the actual diffraction phenomena happen (GO boundaries, foci, caustics). Several ray-based techniques have been developed to overcome the deficiencies of GTD (Kouyoumjian and Pathak, 1974; James, 1980; Borovikov and Kinber, 1994). Among them, the most developed for practical applications is the Uniform Theory of Diffraction (Kouyoumjian and Pathak, 1974; McNamara et al. 1990).
220 Chapter 8 Ray and Caustics Asymptotics for Edge Diffracted Waves
Figure 8.2 Rectangular facet of the scattering edge. The edge diffracted rays (solid arrows) exist only in region A and satisfy the boundary conditions there. Individual elementary edge rays (dotted arrows) in region B do not satisfy the boundary conditions, but they asymptotically cancel each other there.
• |
The ray asymptotics (8.29), as well as (8.35), and (8.36) for the fields E(t), H(t) |
|
are invariant with respect to the permutations ϑ ↔ γ0, ϕ ↔ ϕ0, and therefore |
|
they satisfy the reciprocity principle. We note that these expressions are valid |
|
of the diffraction cone (ϑ |
= |
π |
− |
γ |
|
). Away from this |
|
only in the directionstot |
|
|
0 |
|
|
cone, the total field us,h is asympotically (with k |
→ ∞) equal to zero, due to |
|
the absence of the stationary point on the edge. In this region, the individual |
|
elementary edge waves asymptotically cancel each other. |
|
|
|
• |
The asymptotic expressions (8.29) (and E(t), H(t) in Equations (8.35), (8.36) |
|
satisfy the boundary conditions on the planar faces of the edge. A situation with |
|
these conditions is illustrated in Figure 8.2. |
|
|
|
|
|
|
•The ray asymptotics are not valid at caustics (R = 0 and R = −ρ), where they predict an infinitely large field intensity. The caustic R = 0 is located at the edge itself. The caustic R = −ρ can be real or imaginary.A real caustic occurs outside
ˆs
the scattering object in the positive direction of the vector k .An imaginary caus-
ˆs
tic is located in the direction contrary to k . In particular, imaginary caustics may be inside the scattering body. The value (ζst ) > 0 relates to the case when the edge diffracted ray has not yet reached a caustic, and the value (ζst ) < 0 corresponds to the ray that has already passed a caustic and acquired there the additional phase shift equal to −π/2 (according to Equation (8.11)).
•The theory of EEWs developed in Chapter 7 allows one to calculate the edge diffraction field in the vicinity of any caustics away from the scattering edge. An example of such a calculation is considered in the following section.
8.2CAUSTIC ASYMPTOTICS
Caustic asymptotics are presented here for both acoustic and electromagnetic waves.
These asymptotics have the same structure and differ only in coefficients.
8.2 Caustic Asymptotics 221
8.2.1 Acoustic Waves
Suppose that the edge diffracted rays form a smooth caustic C (Fig. 8.3). It is the envelope of diffracted rays, where a high-intensity field concentrates. According to Section 8.1 the diffracted field away from the caustic (and in front of the caustic) is the sum of two rays coming from the stationary points ζ1 and ζ2 on the scattering edge L:
|
u = u0(ζ1) |
√ |
eiπ/4 |
eik (ζ1) |
+ u0(ζ2) |
√ |
e−iπ/4 |
eik (ζ2) |
, |
|
|
F(ζ1) |
|
|
F(ζ2) |
|
|
R1 |
R2 |
|
2π k (ζ1) |
2π k| (ζ2)| |
(8.41)
where (ζ ) = φi(ζ ) + R(ζ ) and φi(ζ ) is the phase of the incident wave at the point ζ on the scattering edge. Depending on the type of the function F(ζ1,2), Equation (8.41) represents either the field us,h(1) generated by the nonuniform/fringe sources js,h(1) or the
field us,htot generated by the total scattering sources js,h(t) = js,h(1) + js,h(0). Specifically, the functions
Fs(1)(ζ1,2) = f (1)(ϕ1,2, ϕ01,02, α1,2), |
Fh(1)(ζ1,2) = g(1)(ϕ1,2, ϕ01,02, α1,2) |
|
|
(8.42) |
relate to the field us,h(1), and the functions |
|
|
Fs(t)(ζ1,2) = f (ϕ1,2, ϕ01,02, α1,2), |
Fh(t)(ζ1,2) = g(ϕ1,2, ϕ01,02, α1,2) |
(8.43) |
correspond to the field us,h(t) . We note that functions f , g, f (1), g(1) are defined in Equations (2.62), (2.64), (4.14), (4.15), and (3.55) to (3.57). The function (ζ1,2)
can be represented in the form of Equation (8.22).
The first term in Equation (8.41) describes the ray that did not yet reach the caustic and the second term relates to the ray that has already touched the caustic.
Figure 8.3 Edge diffracted rays at the point P in the vicinity of caustic C.