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1.3 Physical Optics 23

This equation represents the Shadow Contour Theorem:

The Shadow radiation does not depend on the whole shape of a scattering object and it

is completely determined only by the size and geometry of the shadow contour.

Now let us evaluate the total power of the reflected field and shadow radiation. The total power of the reflected field can be written as

Prefl

1

Re

 

 

( prefl) (vrefl

n)ds

(1.77)

 

 

 

s,h =

2

 

Sil

s,h

s,h

· ˆ

 

where, according to Equations (1.2), (1.41), (1.64), and (1.69),

 

psrefl = −pinc = −iωρu0eikz,

vsrefl · nˆ = vinc · nˆ = ikuoeikz(zˆ · nˆ),

(1.78)

phrefl = pinc = iωρu0eikz,

 

vhrefl · nˆ = −vinc · nˆ = −ikuoeikz(zˆ · nˆ).

 

Substitution of these quantities into Equation (1.77) results in

 

Psrefl = Phrefl =

1

k2ZA|u0

|2 = Pinc · A

(1.79)

 

2

and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

σ refl,tot = A

 

 

(1.80)

 

 

 

 

 

 

 

 

 

where A is the area of the shadow region cross-section (Fig. 1.4). These equations show that the total power of the reflected field exactly equals the power of the intercepted incident rays. The scattering object only distributes them in the surrounding space. By changing the shape of the illuminated surface Sil, one can significantly decrease the backscattering by deflection of the reflected rays from the direction back to the source of the incident wave. This is the first basic idea used in stealth technology.

The total power of the shadow radiation is determined by

Psh,tot

 

1 Re

( pinc)

vinc

·

(

n)

ds,

(1.81)

 

=

 

 

 

[

 

− ˆ ]

 

 

2sil

The minus sign in front of n is chosen because on radiation/reflection in the positive normal direction (1.41) and (1.78)

the surface of black bodies no exists. According to Equations

pinc = iωρu0eikz, and (vinc · nˆ) = iku0eikz(zˆ · nˆ ).

(1.82)

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24 Chapter 1 Basic Notions in Acoustic and Electromagnetic Diffraction Problems

Substitution of Equation (1.82) into Equation (1.81) leads to

Psh,tot =

1

k2ZA|u0

|2 = PincA

(1.83)

 

2

and

 

 

 

 

 

 

 

 

 

 

 

σ sh,tot = A.

(1.84)

 

 

 

 

 

 

Thus, the shadow radiation power equals the reflected power, and their sum exactly equals the total scattered power (1.42). This result illustrates the physics behind the fundamental diffraction law (1.45). It shows that objects with soft and hard boundary conditions reveal a dual nature. They can be interpreted as if they are simultaneously perfectly reflecting (with reflection coefficients Rs = −1 and Rh = 1) and perfectly absorbing (with R = 0); that is, black. This law can now be written in the form

σ tot = σ refl,tot + σ sh,tot = 2A,

(1.85)

 

 

where A is the area of the shadow region cross-section.

From the equation σ refl,tot = A, it is also clearly seen that the total power of the reflected waves does not depend on the object shaping if the area of shadow cross-section remains constant. However, this power can be decreased by absorbing coatings – this is the second basic idea of stealth technology. In contrast, the shadow radiation cannot be decreased by any absorbing coatings and it can be used for bistatic detection of large opaque objects with small backscattering cross-section (Ufimtsev, 1996).

1.3.6 Summary of Properties of Physical Optics

Approximation

The PO approximation describes properly both all reflected rays away from the geometrical optics boundaries and the diffracted field near these boundaries, as well as near foci and caustics. Reflected rays are revealed by the asymptotic evaluation of the PO integrals. These integrals correctly predict the magnitude and position of main and near side lobes in the directivity pattern of the scattered field. However, these surface integrals are computer time consuming. Their transformation into line integrals reduces computer time, and is the subject of continuing research (Asvestas, 1985a,b, 1986, 1995; Gordon, 1994, 2003; Gordon and Bilow, 2002; Maggi, 1888; Meincke et al., 2003; Rubinowicz, 1917).

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1.4 Nonuniform Component of Induced Surface Field 25

The backscattered field in the PO approximation possesses a special property. According to Equation (1.37), the fields scattered by the soft and hard objects (of the same shape and size) differ only in sign.

Separation of the PO field into the reflected field and the shadow radiation elucidates the scattering physics. In particular, it explains the fundamental law of diffraction theory, according to which the total scattering cross-section of large perfectly reflecting objects is double the area of their shadow cross-section. Well-known manifestations of shadow radiation are Fresnel diffraction and forward scattering.

The PO drawbacks are the following. It is not self-consistent. When the observation point approaches the scattering surface, the PO integrals do not reproduce the initial Geometrical Acoustics (GA) values for the surface field. Also, the PO field does not satisfy rigorously the boundary conditions and the reciprocity principle. The reason for these shortcomings is the Geometrical Optics (GO) approximation for the surface field, which does not include its diffraction components. The PO shortcomings are overcome in the PTD (Ufimtsev, 1962, 1991, 2003), which improves PO by taking into account the diffracted surface field.

1.4 NONUNIFORM COMPONENT OF INDUCED SURFACE FIELD

Surface fields u or ∂u/∂n induced by the incident wave on the scattering object can be considered as the sources of the scattered field. As noted in the Introduction, the central and original idea of PTD is the separation of these sources into uniform and nonuniform components:

js,h = js,h(0) + js,h(1).

(1.86)

The uniform component js,h(0) is defined by Equation (1.31) and represents the surface field induced on the infinite plane tangent to the object (Figs. 1.4 and 1.12). In the case of the incident plane wave, this field is uniformly distributed over the tangent plane. Its amplitude is constant and its phase is a linear function of the plane coordinates. That is why this component is called uniform. According to the definition (1.31), it can also be called the geometrical optics or ray component.

Figure 1.12 Uniform components of the field at points Q1, Q2, and Q3 on the scattering objects are identical to those on the infinite tangential planes shown by the dotted lines.

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26 Chapter 1 Basic Notions in Acoustic and Electromagnetic Diffraction Problems

Figure 1.13 Different shapes and structures where the incident wave generates the nonuniform scattering sources.

In contrast, the nonuniform component js,h(1) is the diffraction part of the surface field. It is caused by diffraction due to any deviation of the scattering surface from that of the tangential infinite plane. These deviations can be a smooth or sharp bending, sharp edges, discontinuity of curvature, discontinuity of material properties, apertures, small bumps and dips, and so on. (Fig. 1.13).

If the scattering object is convex and smooth and its dimensions and radii of curvature are large compared to the wavelength, then the induced nonuniform component concentrates near the boundary between the illuminated and shadowed surfaces (Fig. 1.4). This component is described asymptotically by the Fock functions (Fock, 1965). From the physical point of view it represents creeping waves that radiate surface diffracted rays. If the object possesses sharp edges, the nonuniform component concentrates in their vicinity (Fig. 1.14) and it is described asymptotically by the Sommerfeld functions (Sommerfeld, 1935) presented in Chapter 2. This form of the nonuniform components radiates the edge waves that are often called fringe waves. Similarly the nonuniform sources near the vertices radiate vertex waves.

Taking into account the diffraction/nonuniform components of the surface fields, PTD overcomes the PO shortcomings and provides more accurate asymptotic results for high-frequency scattered fields. The PTD separation of the surface fields into uniform and nonuniform components has proved to be very productive and is often used

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1.5 Electromagnetic Waves 27

Figure 1.14 Nonuniform components of the surface field induced by the incident wave near edges A and B are asymptotically identical to those at the tangential wedges with infinite faces shown by dotted lines.

in diffraction theory. This concept is quite flexible. It can be extended for objects with other boundary conditions. It is also successfully used in hybrid techniques in combination with direct numerical methods. A proper choice of the uniform component depends on specific properties of the problem under investigation and can essentially facilitate its solution. See for example Ufimtsev (1998) and related references given in Ufimtsev (1996, 2003), as well as in the section “Additional references related to the PTD concept: Applications, modifications, and developments” shown at the end of this book.

1.5ELECTROMAGNETIC WAVES

This section briefly presents the basic notions used in this book for the description of electromagnetic waves. This book studies the diffraction of electromagnetic waves at perfectly conducting bodies that are large compared to the wavelength. It is assumed

that the waves and scattering objects are in free space (vacuum). The electric (E) and

magnetic vectors (H) of the wave field are determined as

 

 

 

 

 

 

 

E

=

i

· [( · Ae) + k2Ae] − × Am

 

(1.87)

 

 

 

 

 

 

 

 

 

Z0

 

 

 

 

 

 

 

 

k

 

and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

H

=

 

i

[( · Am) + k2Am] + × Ae.

 

(1.88)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

kZ0

 

Here, Z0 = 1/Y0 = √

 

 

= 120π

ohms

is

the

impedance

of

free space,

μ0/ε0

k

=

ω√

 

 

=

2π/λ is the wave number, and

 

 

 

 

 

 

 

 

 

ε0

μ0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

e

 

1

 

 

e eikr

 

 

m

 

1

 

m eikr

 

 

 

 

 

 

 

A

=

 

 

j

 

 

 

 

dv

and

A

 

=

 

j

 

 

dv

(1.89)

 

 

 

 

 

4π

 

 

r

 

 

4π

 

r

are the electric and magnetic vector-potentials. They are the solutions of the equation

Ae,m + k2Ae,m = −j e,m,

(1.90)

e m

where j ( j ) is the electric (magnetic) current density of the field sources.

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28 Chapter 1 Basic Notions in Acoustic and Electromagnetic Diffraction Problems

Notice that these Equations (1.87) to (1.90) are more convenient for calculation than those usually accepted in books on engineering electromagnetics. Indeed, Equations (1.89) and (1.90) have exactly the same form both for electric and magnetic potentials. The second terms (out of brackets) in Equations (1.87) and (1.88) do not contain the factors 1/μ0 and 1/ε0, which eventually disappear in the integral field expressions.

In the far zone from a scattering object, one can use the following approximation (similar to Equations (1.16) and (1.17):

Ae,m

 

1 eikR

je,me−ikr

cos dv.

 

=

 

 

 

 

 

4π R

This leads to the field components

Eϑ = Z0Hϕ = ik(Z0Aeϑ + Amϕ )

and

Eϕ = −Z0Hϑ = ik(Z0Aeϕ − Amϑ ).

(1.91)

(1.92)

(1.93)

The radial components ER, HR are of the order 1/R2 and are neglected here. The coordinate system is shown in Figure 1.2.

The scattering cross-section is determined by Equations (1.26) and (1.9) as

 

 

 

 

2

 

 

2

 

 

σ = 4π R

 

Esc

 

 

(1.94)

 

 

 

 

 

 

 

 

 

Einc

.

 

 

 

 

 

 

 

Equation (1.45) for the total scattering cross-section is also valid for electromagnetic waves. In the case of an incident wave with a linear polarization, it can be represented in the form of Equation (1.53) as

σ tot =

4π

 

k Im[(Esc/Einc) · Re−ikR],

(1.95)

where Esc is the field scattered in the forward direction.

The PO approximation for the far field scattered by perfectly conducting objects

is defined by

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Ae

=

1

 

eikR

 

j(0)e−ikr cos ds,

Am

=

0,

(1.96)

 

 

 

 

 

 

4π

 

R

Sil

 

 

 

 

 

 

 

 

 

where Sil is the illuminated side of the scattering surface and

 

 

 

 

 

 

 

 

 

j(0)

=

2

n

Hinc

]

 

 

 

(1.97)

 

 

 

 

 

 

 

 

 

[ˆ ×

 

 

 

 

 

is the uniform component of the surface electric current induced by the incident wave on the illuminated side of a scattering object. The paper by Ufimtsev (1999) describes in detail the properties of the PO approximation (see also Ruck et al. (1970)).

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