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12.3 Multiple Diffraction of Electromagnetic Waves 267

with

 

 

 

 

 

 

 

 

f (1)(3π/2, π/2, α) =

1

 

∂f (3π/2, 0, α)

1

 

 

 

,

 

=

√

 

.

(12.36)

2

 

∂ϕ0

2

The substitution of Equation (12.35) into Equation (11.8) determines the total crosssection

 

 

 

 

 

sin[m(2ka + π/4)]

 

 

σ

 

2π a2

1

 

2

∞

.

(12.37)

 

s =

 

 

+ ka m=1 2m−1(8ka√

π ka

)m

 

 

This is the incomplete asymptotic expression, which includes only the first term of the total asymptotic expansion for every edge wave. It can be verified by comparison with the exact asymptotic expression (14.54) in Bowman et al. (1987), which contains the asymptotic terms up to the order of (ka)−4. According to Equation (12.37),

σ

 

2π a2

1

 

 

cos(2ka − π/4)

 

 

cos(4ka)

 

O

(ka)

 

11/2

.

(12.38)

 

 

 

 

 

 

 

 

 

 

 

 

s =

+

 

+

 

4√

π

(ka)5/2

+

64π(ka)4

+

[

 

−

 

],

 

All these three terms are identical to the exact ones. Thus, the comparison of asymptotics (12.18) and (12.37) with known exact results proves that PTD correctly predicts the first term in the total asymptotic expansion for every multiple edge wave.

12.3 MULTIPLE DIFFRACTION OF ELECTROMAGNETIC WAVES

Here, we investigate the diffraction of a plane wave

 

Exinc = Z0Hyinc = eikz

(12.39)

at a perfectly conducting disk (Fig. 12.1). The basic features of this problem are essentially the same as those in the acoustical problems above. For this reason, we will not repeat them here and only briefly discuss a new specific feature caused by the vector nature of electromagnetic waves. Due to this nature and to the axial symmetry of the problem, one can separate the diffracted waves (of the second and higher orders) into two independent groups, with Eϕ - and Hϕ -polarizations. Multiple diffraction of the Eϕ -waves (Hϕ -waves) is calculated just like the diffraction of acoustic waves at a soft (hard) disk. This observation significantly facilitates the investigation, which results in the following approximations for the focal field on the z-axis (z kR2). The focal fields generated by all the multiple Eϕ -waves and Hϕ -waves are equal to

 

 

a

 

ikz

 

eim(2ka+π/4)

 

 

 

Ex = Z0Hy =

z

e

 

m=1

24mπ m/2(ka)3m/2

 

(Eϕ -waves)

(12.40)

and

 

 

 

 

 

 

 

 

 

 

a

 

 

 

m eim(2ka−π/4)

 

 

 

Ex = Z0Hy =

ikz

∞

(−1)

 

 

 

 

e

 

 

 

 

 

 

(Hϕ -waves).

(12.41)

z

 

m=1

 

2m(π ka)m/2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

TEAM LinG

268 Chapter 12 Focusing of Multiple Edge Waves Diffracted at a Disk

The focal field includes these fields plus the contributions generated by the current

j(0) (PO contribution) and the current j(1) (related to the primary edge diffraction):

 

 

Ex(0) = Z0Hy(0) =

ika2 eikz

 

 

 

 

 

 

 

 

 

 

 

 

2 z

 

 

and

 

 

 

 

 

 

 

 

Ex(1) = Z0Hy(1) =

a

[ f (1)(3π/2, π/2, 2π ) + g(1)(3π/2, π/2, 2π )]

eikz

 

 

.

2

z

However,

f (1)(3π/2, π/2, 2π ) = −g(1)(3π/2, π/2, 2π ) = − 1 2

(12.42)

(12.43)

(12.44)

and so Ex(1) = Hy(1) =

0. Therefore, the total focal field equals

 

.

(t)

(t)

 

eikz

 

ika

 

 

 

∞

m eim(2ka−π/4) ∞

eim(2ka+π/4)

Ex

= Z0Hy

= a

z

 

2

+ m=1(−1)

 

2m(π ka)m/2

+ m=1

24mπ m/2(ka)3m/2

 

 

 

 

 

 

 

 

 

 

 

(12.45)

Now, according to Equation (11.8), which is also valid for electromagnetic waves (with the replacement of u(t) by Ex(t)), one obtains the total scattering cross-section:

σ

=

2π a2

1

 

2

∞ (

−

1)m

sin[m(2ka − π/4)]

 

2

∞

sin[m(2ka + π/4)]

.

 

 

2m(π ka)m/2

 

 

 

 

 

+ ka m=1

 

+ ka m=1 24mπ m/2

(ka)3m/2

 

 

 

 

 

 

 

 

 

 

(12.46)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

It turns out that this quantity is connected by the relation

 

 

 

 

 

 

 

 

 

 

 

σ = 21 (σh + σs)

 

 

 

 

(12.47)

with the similar quantities (11.9) and (11.18) found for acoustic waves.

PROBLEMS

12.1Prove Equation (12.14) for the focal field generated by all of the acoustic edge waves scattered at a hard disk. Explain all of the details, including caustic parameters, phase shifts, directivity factors, and fractional coefficients.

12.2Prove Equation (12.34) for the focal field generated by all of the acoustic edge waves scattered at a soft disk. Explain all of the details, including caustic parameters, phase shifts, directivity factors, and fractional coefficients.

12.3Prove Equation (12.40) for the focal field generated by the Eϕ -group of the electromagnetic edge waves scattered at a perfectly conducting disk. Explain all of the details, including caustic parameters, phase shifts, directivity factors, and fractional coefficients.

12.4Prove Equation (12.41) for the focal field generated by the Hϕ -group of the electromagnetic edge waves scattered at a perfectly conducting disk. Explain all of the details, including caustic parameters, phase shifts, directivity factors, and fractional coefficients.

TEAM LinG

Chapter 13

Backscattering at a

Finite-Length Cylinder

13.1 ACOUSTIC WAVES

The geometry of the problem is shown in Figure 13.1. A solid circular cylinder with flat bases is illuminated by the incident plane wave

uinc = u0eik(y sin γ +z cos γ ),

with 0 ≤ γ ≤ π/2.

(13.1)

The total length of the cylinder and its diameter are denoted by L = 2 and d = 2a, respectively. The scattered field is evaluated for the backscattering direction ϑ =

π − γ , ϕ = 3π/2.

13.1.1PO Approximation

According to Equation (1.37), the PO fields backscattered by the acoustically hard and soft objects differ from each other only in sign. Hence, it is sufficient to exhibit the PO calculations only for the case of scattering at a hard cylinder.

First we calculate the far field scattered by the left base/disk of the hard cylinder. In this case, the application of Equation (1.37) leads to the expression

(0)disk

 

ik

i2kl cos ϑ eikR

a

 

2π

 

i2kr

sin ϑ sin ϕ

 

uh

= u0

 

cos ϑ e

 

 

 

r dr

 

e

 

dϕ ,

(13.2)

2π

 

R

0

0

 

where ϑ = π − γ . In view of Equation (6.55) we have

(0)disk

ia cos ϑ

i2kl cos ϑ eikR

 

uh

= u0

 

 

 

J1(2ka sin ϑ )e

 

 

.

(13.3)

2

sin ϑ

R

The application of Equation (1.37) to the field scattered by the cylindrical part of the object leads to the integral expression

(0)cyl

 

ika

eikR l

 

i2kz

cos ϑ

 

2π

i2ka sin ϑ sin ϕ

uh

= u0

 

sin ϑ

 

 

e−

 

 

dz

e

sin ϕ dϕ . (13.4)

2π

R

−l

 

 

 

 

 

 

 

 

 

 

 

π

 

Fundamentals of the Physical Theory of Diffraction. By Pyotr Ya. Ufimtsev

Copyright © 2007 John Wiley & Sons, Inc.

269

TEAM LinG

270 Chapter 13 Backscattering at a Finite-Length Cylinder

Figure 13.1 Cross-section of the cylinder by the y0z-plane. Dots 1, 2, and 3 are the stationary phase points visible in the sector π/2 < ϑ < π .

Here, the integration is performed over the illuminated part of the scattering surface where π ≤ ϕ ≤ 2π under the condition π/2 + 0 ≤ ϑ ≤ π − 0. In the limiting case when ϑ = π , the integration encompasses the whole cylindrical surface (0 ≤ ϕ ≤ 2π ) and results in zero scattered field. However, Equation (13.4) is also valid in this case as it equals zero due to the factor sin ϑ . Thus,

uh(0)cyl = 0,

if ϑ = π .

(13.5)

The integral in Equation (13.4) over the variable z is calculated in the closed form. The integral over the variable ϕ is calculated (under the condition 2ka sin ϑ 1), by the stationary-phase technique (Copson, 1965; Murray, 1984). The details of this technique have already, been considered in Sections 6.1.2 and 8.1. The stationary point ϕst = 3π/2 is found from the equation

 

 

 

d

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2ka sin ϑ sin ϕ

= 2ka sin ϑ cos ϕ = 0.

 

(13.6)

 

 

 

dϕ

 

The asymptotic expression for the integral is given by

 

 

2π

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ei2ka sin ϑ sin ϕ sin ϕ dϕ −

π

 

 

 

 

 

 

 

 

e−i2ka sin ϑ +iπ/4.

(13.7)

π

ka sin ϑ

Therefore, under the condition 2ka sin ϑ

1, the scattered field u(0)cyl

is determined

asymptotically as

 

 

 

 

 

 

 

 

 

 

 

 

 

 

h

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(0)cyl

−u0

ia sin ϑ

 

 

e−i2ka sin ϑ +iπ/4 eikR

 

 

uh

 

 

 

 

sin(2kl cos ϑ )

√

 

 

 

.

(13.8)

 

2

cos ϑ

R

 

 

π ka sin ϑ

Having Equations (13.5) and (13.8), one can construct the approximation valid in the entire region π/2 ≤ ϑ ≤ π . This can be done in a manner similar to that in Section 6.1.4. We use the asymptotic expressions for the Bessel functions with large

TEAM LinG

13.1 Acoustic Waves 271

arguments (x 1) and observe that

 

 

 

 

 

e−i2x+iπ/4

≈ J0(2x) − iJ1(2x),

 

 

 

 

 

 

√

 

 

(13.9)

 

 

 

 

 

π x

 

 

 

 

 

e−i2x+iπ/4

1

− iJ2(2x)],

 

 

 

 

 

 

√

 

 

≈

 

[J1(2x)

(13.10)

 

 

 

 

 

i

 

 

 

 

 

π x

and

 

 

 

 

 

 

 

 

 

e−i2x+iπ/4

≈ e−inπ/2[Jn(2x) − iJn+1(2x)],

n = 0, 1, 2, 3, . . . .

 

 

√

 

 

(13.11)

 

π x

Each of these combinations can be used to construct the approximation for the field in the region π/2 ≤ ϑ ≤ π . We apply and analyze the simplest ones, (13.9) and (13.10). With these approximations the total PO field can be represented in the two following forms:

uh,1(0)

 

ia eikR

+

cos ϑ

J1(2ka sin ϑ )ei2kl cos ϑ

 

= u0

 

 

 

 

 

 

 

 

 

 

 

2

R

sin ϑ

 

 

 

 

 

 

 

sin ϑ

 

 

 

 

 

 

 

 

−

 

 

 

 

sin(2kl cos ϑ ) [J0(2ka sin ϑ ) − iJ1(2ka sin ϑ )],

(13.12)

 

 

cos ϑ

and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ia eikR

 

cos ϑ

 

 

 

uh,2(0)

= u0

 

 

 

 

 

+

 

J1(2ka sin ϑ )ei2kl cos ϑ

 

2

 

 

R

sin ϑ

 

 

 

 

 

 

 

sin ϑ

 

 

 

 

 

 

 

 

+ i

 

sin(2kl cos ϑ ) [J1(2ka sin ϑ ) − iJ2(2ka sin ϑ )], .

(13.13)

 

 

cos ϑ

The scattering cross-section σ is defined by Equation (1.26). We have calculated the normalized scattering cross-section as

σnorm = σ/σd

(13.14)

where the quantity

 

σd = π a2(ka)2

(13.15)

is the PO scattering cross-section of the disk under the normal incidence (ϑ = π ). The results are shown in Figures 13.2 and 13.3. The curves PO-1 and PO-2 relate to Equations (13.12) and (13.13), respectively.

The small discrepancy between the two curves in Figure 13.2 is caused by the different higher-order terms in the asymptotic expressions (13.9) and (13.10). It takes place when the cylinder diameter is not sufficiently large and equals only one wavelength (d = 2a = λ) when the argument of the Bessel functions does not exceeds 2π . The discrepancy between the approximations PO-1 and PO-2 becomes practically

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