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Statistics, Correlation Function and Spectral Density 392 Small Perturbation Method Dirichlet Problem for One-Dimensional Surface Neumann Problem for One-Dimensional Surface Kirchhoff Ap

Statistics, Correlation Function and Spectral Density 392

Small Perturbation Method

Dirichlet Problem for One-Dimensional Surface

Neumann Problem for One-Dimensional Surface

Kirchhoff Approach

Dirichlet Problem for One-Dimensional Surface

Neumann Problem for One-Dimensional Surface

References and Additional Readings

Scattering of Electromagnetic Waves: Theories and Applications

Leung Tsang, Jin Au Kong, Kung-Hau Ding Copyright  2000 John Wiley & Sons, Inc ISBNs: 0-471-38799-1 (Hardback); 0-471-22428-6 (Electronic)

390 9 1-D RANDOM ROUGH SURFACE SCATTERING

Imagine now that the surface is rough It is clear from Fig 9.1.2 that the two reflected waves have a path length difference of 2h cos & This will give a phase difference of

If h is small compared with a wavelength, then the phase difference is in- significant However, if the phase difference is significant, the specular re- flection will be reduced due to interference of the reflected waves that can partially “cancel” each other The scattered wave is “diffracted” into other directions A Rayleigh condition is defined such that the phase difference is 90’ [Ishimaru, 1978; Ogilvy, 19911 Thus for

x

h < scosOi the surface is regarded as smooth, and for

x h> gcosBi

(9.1.2)

(9.1.3) the surface is regarded as rough For random rough surface, h is regarded as the rms height

We will consider rough surface scattering Both the cases of scalar waves and vector electromagnetic waves will be treated

For the case of scalar wave, let us consider an incident wave $&‘5;) im- pinging upon a rough surface The wave function $J obeys the wave equation:

The rough surface is described by a height function x = f(z, y) For a flat surface, f(z, y) = 0 Two common boundary conditions are those of the Dirichlet problem and the Neumann problem For the Dirichlet problem, at the rough surface boundary z = f (z, y) we have

For the Neumann problem, the boundary condition is at z = f(z, y)

w -

5 1 Introduction 391

d

where & is the normal derivative

For electromagnetic wave scattering by one-dimensional rough surface

z = f(z), the D’ rrrc e problem l hl t corresponds to that of a TE wave impinging upon a perfect conductor and $J is the electric field The Neumann prob- lem corresponds to that of a TM wave impinging upon a perfect electric conductor and + is the magnetic field

For acoustic waves, $J corresponds to the linearized pressure The lin- earized pressure pl is -iwp&, where po is the average density The linearized pressure pl is related to the linearized density pr by

to a pressure-free surface The boundary condition that g = 0 corresponds

to normal component of the linearized velocity equal to zero at the boundary

392 9 1-D RANDOM ROUGH SURFACE SCATTERING

2 Statistics of a Random Rough Surface

2.1 Statistics, Correlation Function and Spectral Density

For a one-dimensional random rough surface, we let z = f(z), and f(z) is a random function of x with zero mean [Papoulis, 19841

The Fourier transform of the rough surface height function is

Strictly speaking, if the surface is infinite, the Fourier transform does not exist To circumvent the difficulty, one can use Fourier Stieltjes integral [Ishi- maru, 19781 Or one can define the truncated function

F(k,) = $ JW dxe-ikxxf(x) (9.2.2)

-00

f0 Lx= { f(x) 0 1x1 5 L/2 1x1 2 L/2 (9.2.3) The Fourier transform then becomes

and the exponential correlation function

In (9.2.6)-(9.2.7), I is nown as the correlation k length

In the spectral domain we have

fj2.1 Statistics and Spectral Density 393

where W (Ic,) is known as the spectral density Since f(z) is real, we have used in (9.2.10) the relation

s

00 4z> - dkx ik, eikxxF(lcx)

-00 Hence if we let

(9.2.16)

(9.2.17)

(4~1>4~2>> = s 2 G&l - x2) (9.2.18) where s is the rms slope and CQI is the correlation function for the slope function so that Ca(0) = 1 We let Wa(kx) be the spectral density of the slope Then

s2C,(x) = dk, eikxx Wa (k,)

-00 From (9.2.10), (9.2.17), and (9.2.19) we obtain

(9.2.19)

Wa(Ic2) = k: w(lcx) (9.2.20)

394 9 1-D RANDOM ROUGH SURFACE SCATTERING

(9.2.22)

(9.2.23) For the exponential correlation function of (9.2.7), the rms slope does not exist For example, the integrand of (9.2.21) is not integrable This is because

of the sharp edges that may exist in the rough surface profiles [Barrick, 19701

To calculate the rms value of a’ where

fj2.1 Statistics and Spectral Density 395

In the spectral domain we obtain

F(LI = &J* dTle -&-FL f (Q

= @ll + &1)W(&l)b(x31 + z41)w(x31) + @IL + ~3L)w(&)@2L + E41)w(E21) + 6(&J- + &L)w(&l)@21 + ~31)w(~2I) (9.2.40)

396 9 1-D RANDOM ROUGH SURFACE SCATTERING

2.2 Characteristic Functions

For a Gaussian random process, the characteristic functions can be read- ily calculated For a Gaussian rough surface f(z), the probability density function (pdf) for f is

1

df > = $j h exP f 2

[ 1 2h2

The joint pdf for f(zl) = fi and f(22) = fi is

= (fl, f27 l , fn), where t = transpose - and f is the corresponding column vector The covariance matrix is K of dimension n x n

is the inverse of E and A is the determinant of 71

The characteristic function is

( eiu’fl+iY2f2+.*.+iunfn

> = exp

[

1

53 Small Perturbation Method

and A = h4(1 - C2) Thus

( @4f1+ivzf2 > = exp -2h 12 (q + u; + 2 2v&)]

39’7

(9.2.51)

3 Small Perturbation Method

The scattering of electromagnetic waves from a slightly rough surface can

be studied using a perturbation method [Rice, 19631 It is assumed that the surface variations are much smaller than the incident wavelength and the slopes of the rough surface are relatively small The small perturbation method (SPM) makes use of the Rayleigh hypothesis to express the reflected and transmitted fields into upward- and downward-going waves, respectively The field amplitudes are then determined from the boundary conditions The extended boundary condition (EBC) method may also be used with the perturbation method to solve for the scattered fields In the EBC method, the surface currents on the rough surface are calculated first by applying the extinction theorem The scattered fields can then be calculated from the diffraction integral by making use of the calculated surface fields Both perturbation methods yield the same expansions for the scattered fields, because the expansions of the amplitudes of the scattered fields are unique within their circles of convergence

3.1 Dirichlet Problem for One-Dimensional Surface

We first illustrate the method for a one-dimensional random rough surface with height profile x = f(z) and (f(z)) = 0 The scattering is a two- dimensional problem in 113-x without y-variation

Consider an incident wave impinging on such a surface with Dirichlet boundary condition (Fig 9.3.1) This is the same as a TE electromagnetic wave with an electric field in the c-direction impinging upon a perfect electric conductor Let

+ inc =e iki,x-iki, z (9.3.1) where Ici, = k sin&, kiz = k cos & In the perturbation method, one uses the height of the random rough surface as a small parameter This is based on the assumption that kh < 1, where h is the rms height

For the scattered wave, we write it as a perturbation series

(9.3.2)

9 1-D RANDOM ROUGH SURFACE SCATTERING

The boundary condition at x = f (rc) is

&-&c + +s = 0 (9.3.3) The zeroth-order scattered wave is the reflection from a flat surface at z = 0 Thus

1c) s (0) = -eiki,x+iki,z

Perturbation theory consists in expanding the exponential functions of

In the spectral domain, we also let

53.1 Dirichlet Problem for 1-D Surface 399

Balancing (9.3.8) to zeroth order gives

If we substitute (9.3.10) into (9.3.5), we get back the zeroth-order solution

in the space domain as given by (9.3.4)

Balancing (9.3.8) to first order gives

terms of the zeroth-order solution Substituting the zeroth-order solution of

(9.3.3) Thus

The result of (9.3.13) can be interpreted as follows In order for the wave to

be scattering from incident direction ki, to scattered direction k,, the surface

has to provide the spectral component of k, - ki, This is characteristic of

Bragg scattering Balancing (9.3.8) to second order gives

Using (9.3.10) in (9.3.14), it follows that the first two terms in (9.3.14) cancel

each other Substituting (9.3.13) and (9.3.10) in (9.3.14) gives

k, - Ic$ of the rough surface Since k& is an arbitrary direction, an integration over all possible k; is needed in (9.3.16)

The coherent wave is obtained by calculating the stochastic average of &

We note that, in view of (9.2.8) and (9.3.13),

of the flat surface case

To get power conservation, we note that the incident wave has power per unit area

-

S inc l z h = 1

- - 2rl cos 8i (9.3.22)

flowing into the rough surface The negative sign in (9.3.22) indicates that the Poynting vector has a negative %component

53.1 Diricblet Problem for 1-D Surface 401

The power per unit area outflowing from rough surface is

(9.3.23)

Suppose we include only terms up to q!J”‘$~o)* + (&‘&‘*) + q!$“‘(q!&2)*) + (p)+‘o’*

S S in (9.3.23) That is, we include the intensity due to the product

of first-order fields and the product of the zeroth-order field and the second- order field Thus the power per unit area outflowing from the rough surface that is associated with the coherent field is

Putting (9.3.19), (9.3.20) and (9.3.21) into (9.3.24) gives

dk,(Rek,)W(Ici, - kz) (9.3.25)

Since k, is imaginary for evanescent waves and W is real, the integration limits of (9.3.25) are to be replaced by -k to k because evanescent waves do not contribute to the power outflow Thus

b (9.3.26)

For the incoherent wave power flow, we use the first-order scattered fields

(ss 4)incoh = Re [& (&‘)e)]

Using the spectral domain in (9.3.27), we have

402 9 1-D RANDOM ROUGH SURFACE SCATTERING

(cps(kc)m~>) = ~(w(k, - k;) (9.3.33) From (9.3.30), we define the power flow per unit area of the incoherent wave

Thus if we divide (9.3.37) by the incident power per unit area of (9.3.22),

we can define the incoherent bistatic scattering coefficients a(&) as

a(e,) = k cos2 8, cos8 Z(k, = ksin0,)

Note that (9.3.38) is defined in such a manner that the integration of a(&) over 0, will combine with the reflected power of the coherent wave to give

an answer that obeys energy conservation

For first-order scattering, &) (Ic,) = q!&” (kz), so that from (9.3.33) and (9.3.30) we obtain

53.2 Neumann Problem for 1-D Surface 403

and (9.3.38) assumes the form

4°s> = 4k3 cos2 0, cos O$V(k sin 0, - k sin e,> (9.3.40) The results for Gaussian and exponential correlation functions can be readily obtained by using the spectral densities as listed in (9.2.14) and (9.2.15) The backscattering coefficient is, for 0, = -Oi,

a(-Oi) = 4k3 cos3 O$V(-2k sin Oi)

3.2 Neumann Problem for One-Dimensional Surface

Let the incident wave be

+ inc =e iki,x-iki, z

(9.3.41) where ki, = ksinOi, ki, = k cos Oi The scattered wave is written as a per- turbation series:

dX ( w

inc + w

S

> ( + w inc

&(F) = /OO dkxeikxx+ikzz~s(kx)

we match bzndary condition of (9.3.45):

(9.3.46)

& ik eikxX+“zf C”I$s (k,) + x x dk ik eikxX+‘kxf (XI+ (k x z s x >

-

404 9 1-D RANDOM ROUGH SURFACE SCATTERING

In a spectrtil domain, we let

$b(kx) = &O)(kx) + 1cly(kx> + +:2)(lc,) + l * l

Also we expand exp(ik,f (z)) in a power series

,ikfb) = 1 + ik,f (x) _ $;f” + Putting (9.3.48) and (9.3.49) into (9.3.47) gives

53.2 Neumann Problem for I-D Surface 405

Substituting (9.3.54) and (9.3.55) in (9.3.53) gives

I

O” dkx2kxkixF(kx)ei(kx+kix)x -

-00 + I 00 dk ik x z eikxx@)(k S x - ) - 0 (9.3.56)

In the firsGwo integrals, we next change dummy variable of k, + ki, + k, This gives

I -00 * dk, [2(k, - k,)kix - 2kfz] F(k, - kiz)eikxx

I

00 + dk,ik z e’kxx+‘l’(k S x - ) - 0 (9.3.57) Taking the inverse F’ourierGYansfer of (9.3.57) gives

k [k k 2 ix - k2]F(kx - kix) Next we balance (9.3.50) tof second order:

1 F(kk)e”(kx+k:)xf- z [Ic,ki, - k2] F(k, - ki,)

(9.3.61)

406 9 1-D RANDOM ROUGH SURFACE SCATTERING

On the right hand side of (9.3.61), let kk + k, = kg and then k, -+ kk,

kg + k, We get

= SP,dkx~~kxx~~dk~[-khlc,+k2]F(kx-k~)~[klbi,-k2]F(k~-k~x)

(9.3.62) Taking the inverse Fourier transfer of (9.3.62) gives

$L2’(kx) = $ Jm dk$(k2 - k;kx)(k;kix - k2)F(kx - k;)F(k; - kix)

(9.3.63)

Taking the average of (9.3.58) gives

The coherent reflected wave is in the specular direction and is less than that

of the flat surface for the case of Neumann boundary condition

Transforming to that of the angles kx = k sin 8, and k, = k cos 8,, we obtain

7r

at any point on the surface are approximated by the fields that would be present on the tangent plane at that point For that to be valid, it is required that every point on the surface has a large radius of curvature relative to the wavelength of the incident field The Kirchhoff approach casts the form of the solution in terms of diffraction integral We first illustrate the approximation

by considering the Dirichlet problem of a one-dimensional surface

the

Consider an incident wave impinging on a rough surface (Fig 9.3.1) Let incident wave vector be ki = kixii - k&