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1.2 Scattered Field in the Far Zone 9

According to Equations (1.2), (1.3) and (1.4), the power flux density of the scattered field is determined by

Psc = iωρu u.

(1.20)

In the far field,

 

 

u ≈ iku · Rˆ ,

with Rˆ = R.

(1.21)

Therefore, the power flux density averaged over the oscillation period T = 2π/ω equals

av =

2

 

[

] =

2

[

 

] · ˆ

=

2

|

|

 

· ˆ

 

Psc

 

1

Re

 

p v

1

Re

(iωρu) (iku)

R

 

1

k2Z

u

2

R,

(1.22)

 

 

 

 

 

 

 

where

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Z = ρc

 

 

 

 

 

 

 

(1.23)

is the characteristic impedance of the medium.

Usually, the far field is characterized by the bistatic cross-section σ introduced through the relation

Pavsc

=

σ · Pavinc

,

 

(1.24)

 

 

4π R2

 

 

 

where

 

 

 

 

 

 

 

 

 

 

inc

 

 

1

 

2

 

inc

2

 

Pav

=

 

k

 

Z|u

 

|

 

(1.25)

2

 

 

 

is the power flux density of the incident wave. This definition suggests the interpretation given in the following paragraphs.

The bistatic cross-section is the area σ of a hypothetical plate perpendicular to the direction of the incident wave. This plate intercepts the incident power Pavinc · σ and distributes it uniformly into the whole surrounding space with the power flux density that is equal to the actual one scattered by the object in the direction of observation. Because the scattered power depends on the direction of scattering, the scattering cross-section σ is a function of this direction. The term bistatic means that the direction of scattering can be arbitrary. In the particular case when the scattering direction coincides with the direction to the source of the incident wave, the quantity σ is called the backscattering cross-section or monostatic cross-section. Thus, according to Equations (1.22) and (1.24)

σ

 

4π R2

Pavsc

 

4π R2

|usc|2

.

(1.26)

=

Pavinc =

 

 

 

|uinc|2

 

In the directions where the field scattered from a smooth convex surface has a ray structure, the bistatic cross-section is predicted by Geometrical Optics (Geometrical

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10 Chapter 1 Basic Notions in Acoustic and Electromagnetic Diffraction Problems

Acoustics) and equals

σ = πρ1ρ2.

(1.27)

Here ρ1 and ρ2 are principal radii of curvature of the scattering surface at the reflection point. It is also assumed that this surface is perfectly reflecting (soft or hard). Two interesting features of this quantity should be emphasized.

First, the expression (1.27) is universal. It is applicable both for acoustic and electromagnetic waves. The reason for this is that the ray structure does not depend on the nature of the waves, and it is totally determined by the geometry of the scattering surface. If the geometry is the same, the divergence of reflected rays will be the same for both acoustic and electromagnetic rays. Also, the modulus of reflection coefficient for any perfectly reflecting surfaces (soft or hard for acoustic waves, or perfectly conducting for electromagnetic waves) equals unity. However, just these two factors, the ray divergence and the reflection coefficient, totally determine the amplitude of reflected rays, and eventually the bistatic cross-section.

Equation (1.27) can be generalized for imperfect reflecting surfaces:

σ = |R|2πρ1ρ2,

(1.28)

where R is the reflection coefficient, which can be different for acoustic and electromagnetic waves.

The second interesting and not obvious feature of Equation (1.27) is the following. This expression does not depend on the angle between the incident and reflected rays at the same reflection point (Fig. 1.3). In other words, it is constant for any bistatic angles, including zero angle related to backscattering. This property of scattering from perfectly reflecting objects follows from the theory of Fock (1965) as it was shown in Ufimtsev (1999).

The theory of Fock (1965) is more general. It is also valid for imperfect scattering surfaces characterized by the reflection coefficient R. In this case, Fock’s theory leads straight to Equation (1.28), where R depends on the bistatic angle as well as on the boundary conditions.

Figure 1.3 Scattering from the same reflection point (at the same reflecting object) for

different bistatic angles. Bistatic cross-section σ of this perfectly reflecting object is constant for all of these angles and equals the monostatic cross-section.

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1.3 Physical Optics 11

1.3PHYSICAL OPTICS

This high-frequency approach is widely used in acoustic and electromagnetic diffraction

problems.

1.3.1Definition of the Physical Optics

Physical Optics (PO) was suggested by Macdonald (1912), and since then it has been successfully applied in the theory of diffraction. In particular it is often used in the analysis of electromagnetic waves scattered from large metallic objects. Basic features of this approach in the study of electromagnetic diffraction are exposed in the article by Ufimtsev (1999). The scalar version of PO is applicable for acoustic waves and it is known in acoustics as the extended Kirchhoff approximation (Brill and Gaunaurd, 1993; Menounou et al., 2000; Moser et al., 1993). Physical Optics is a constituent part of the Physical Theory of Diffraction developed in the present book. According to this approximation, the field induced on the surface of the object is determined by Geometrical Optics (Geometrical Acoustics).

The physics behind this is as follows. Geometrical Optics describes a wave field in the limiting case when a wavelength tends to zero. With respect to such a small wavelength, the scattering surface at the reflection point can be considered approximately as a tangential plane. Therefore, the surface field induced at the tangential infinite plane is a good high-frequency approximation for true scattering sources induced on a large scattering object. Two such planes P1 and P2 tangent at the points Q1 and Q2 are shown in Figure 1.4. These points are located at the “illuminated” side of the object. Notice that according to Geometrical Optics, the field equals zero in the shadow region, including the points on the object surface.

Thus, the reflection from a tangential plane is an appropriate canonical (“fundamental”) problem. Its exact solution can be easily found using Geometrical Optics, as well as by image theory. The total field generated by an external source above an infinite reflecting plane (Fig. 1.5) is the sum of the incident field and the reflected field, which can be interpreted as the field created by the image source. On the acoustically soft plane, the total field is zero as a result of the boundary condition (1.5), but its normal derivative equals

∂us

 

∂uinc

 

 

= 2

s

(1.29)

∂n

∂n

due to the exact solution of this problem. On the acoustically hard plane, the normal derivative of the total field is zero as a result of the boundary condition (1.6), and the field itself equals

uh = 2uhinc,

(1.30)

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12 Chapter 1 Basic Notions in Acoustic and Electromagnetic Diffraction Problems

Figure 1.4 Surface fields induced on the scattering objects at the points Q1 and Q2 are asymptotically identical to the fields induced at the tangential planes P1 and P2, respectively. Sil is the illuminated part of the object surface. A dark plate behind the object displays the cross-section of the geometrical shadow region.

as follows from the solution of this reflection problem. In the general PTD, these quantities are interpreted as the uniform components of induced sources on a smooth convex scattering surface:

js(0) = 2

∂uinc

,

jh(0) = 2uinc.

(1.31)

∂n

These expressions define the induced sources only on the “illuminated” part of the scattering object. On the shadowed part, these components are set to zero.

By substituting Equation (1.31) into Equation (1.10) one obtains expressions for the scattered field at any distance from the scatterer (Fig. 1.4):

 

PO

(0)

 

1

 

 

(0) eikr

 

 

 

 

us

≡ us

= −

 

 

 

js

 

 

 

 

ds,

4π

Sil

 

 

r

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(1.32)

 

PO

(0)

 

1

 

 

(0)

∂ eikr

 

uh

≡ uh

=

 

 

 

 

jh

 

 

 

 

 

 

ds.

4π

Sil

 

∂n r

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 1.5 Reflection from an infinite plane.

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1.3 Physical Optics 13

These expressions represent the scalar Physical Optics approximation, also known in Acoustics as the extended Kirchhoff Approximation (KA). In the present book we use the term Physical Optics for both acoustic and electromagnetic waves. The symbols us(0) and uh(0) are introduced in Equations (1.32) to emphasize that these fields are generated by the uniform component of the induced surface sources. Thus, Physical Optics, which deals with these uniform components, is a constituent part of the general PTD.

The Physical Optics of Equations (1.32) possesses a special property related to the field scattered in the direction to the source of the incident wave. According to Equations (1.16) and (1.17) the PO far field is determined as

 

u(0)

 

 

1 eikR

 

∂uinc

e−ikr cos ds

(1.33)

 

 

 

 

 

 

 

 

 

 

 

= − 2π

 

 

 

 

 

s

 

R

Sil ∂n

 

 

 

and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

u(0)

= −

ik eikR

uince−ikr

cos (m

n)ds.

(1.34)

 

 

 

 

 

 

 

h

 

2π R

 

Sil

 

 

 

ˆ

· ˆ

 

The field incident on the scattering object (being at the large distance from the source) can be represented in the form

 

 

 

 

 

= ˆ

 

 

 

uinc = const eikφi .

 

 

 

 

 

 

 

(1.35)

The unit vector

 

s

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

=

φi

i

ki indicates the direction of the incident wave, and the unit

ˆ = s

k

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

vector m

r

 

 

ˆ

k

 

is valid. Note also that

 

 

 

 

 

 

 

 

 

 

 

k

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

the equality ˆ

= −ˆ

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

∂uinc

=

 

inc

· ˆ =

 

 

inc

 

 

 

i

· ˆ

=

 

inc

ki

n).

(1.36)

 

 

 

 

∂n

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

u

 

 

iku

 

(

φ

 

iku

(ˆ

 

· ˆ

 

 

 

 

 

 

 

 

 

 

 

 

n

 

 

 

 

n)

 

 

 

 

 

The substitution of Equation (1.36) and the quantity (m

n)

 

 

 

ki

n) into Equations

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

= −

(ˆ

· ˆ

 

(1.33) and (1.38) leads to the equation

 

 

 

 

 

 

ˆ · ˆ

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

us(0)

 

 

u(0)

 

 

ik

eikR

 

 

uince−ikr cos (ki

n)ds.

(1.37)

 

 

 

 

 

 

 

 

 

 

 

 

 

= −

= − 2π R

 

 

 

 

 

 

 

 

h

 

Sil

 

 

 

 

 

ˆ

·

ˆ

 

 

Hence, in the frame of the Physical Optics approximation, the backscattered fields created by the soft and hard objects (of the same shape and size) have equal magnitudes and differ only in sign.

1.3.2 Total Scattering Cross-Section

The power flux density of the scattered waves is defined by Equation (1.22). By the integration of this quantity over the object surface, one can find the total power

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