(9.4.10)

For one-dimensional surface, the tangential H field (n x H) is proportional to N x "V x (fJ'ljJ) = N X (\7 'ljJ X fJ). Since N . fJ = 0, the tangential H field is proportional to -fJ(N . \7 E). Hence for the tangential plane approximation of Fig. 9.4.1, we have (9.4.11) The approximation is exact if we do have an infinite plane with normal n. Thus the region of validity of the Kirchhoff approximation is that the radius of curvature must be much larger than the wavelength. It also requires that shadowing and multiple scattering are negligible. For a curve given by z = f(x), the tangent vector is the rate of change of position vector with respect to the arc length s.

(9.4.12)

Using the chain rule of differentiation and ds ,--,--,---::-dx = + (J'(x))2

(9.4.13)

1 + f'() z -r==;=::=;=~ x A] VI + (J'(x))2

(9.4.14)

VI + (J'(x))2 = [1 + (J'(x))2]3/2

1f"(x)1

(9.4.15)

The radius of curvature p is the reciprocal of the curvature. (9.4.16)

For a straight line, f"(x) Kirchhoff approximation is good for a surface that can be represented by its tangent plane. To estimate p, we note that for C(x) = exp( -x 2 /1 2 ), from Section 2.1, we have

I d2 f 2V3h rms of dx 2 = -12

df rmS of - =

h V2-

(9.4.17a) (9.4.I7b)

= _1_

(9.4.18)

For example, if h = 1,)" I = 4,)" then p = 5.5,), and is much bigger than wavelength. For h = 0.5,), and l = 1), we have p = 1.06\ which is comparable to a wavelength. Besides the radius of curvature condition, the Kirchhoff approximation also ignores shadowing and multiple scattering. Since the Kirchhoff approximation ignores shadowing, the region of validity is also dependent on incident angle. Using (9.4.1) in (9.4.4), we have

(9.4.19)

From (9.4.8) and (9.4.19), the scattered field for z'

'ljJsCr')

(9.4.20)

with r = (x, f(x)). In the spectral domain, the two-dimensional Green's function is 00 i eikx(x'-x)+ikxlz'-zl (9.4.21) g(r, r') = -4 dk x k

Substituting (9.4.21) into (9.4.20) gives, for z'

'l/Js(r') =

> z = f(x),

(9.4.22)

dk x eikxx'+ikxz' 'l/Js(kx )

dx ei(kiX-kx)x-i(kx+kiX)f(x) [:~ kix + kiZ] (9.4.23) z We cast (9.4.22) and (9.4.23) in the forms that are similar to the small perturbation method of Section 3.1. Define 'l/Js (k x ) = - 2:k

(9.4.24)

(9.4.25)

To treat the df /dx term in (9.4.23), we perform integration by parts and ignore "edge" effects. Ignoring edge effects implies that

+ ikdz df ) eikdXX+ikdxf(x)

(9.4.26)

+ kixkdx)

1 _=

dx eikdxX+ikdxf(x)

(9.4.27)

It is interesting to digress to make a small height approximation in (9.4.26) and see how the results compare with SPM. Then we have

(kixkdx - kizkdz)F( -kdx) (9.4.28) z In the vicinity of the specular direction kdx ~ 0, kdz ~ -2kiz, and k z ~ kiz, from (9.4.28) we obtain 'l/Js(k x ) ~ -<5(kdx) 'l/Js(k x ) ~ -<5(kdx)

+ 2ikizF( -kdx)

(9.4.29)

We note that from the small perturbation method, Eqs. (9.3.10) and (9.3.13), we have (9.4.30) Thus we know that the Kirchhoff approximation result generally is different from that of the small perturbation method. They are close to each in the vicinity of the specular direction for the case of small height. Comparison of Monte Carlo simulation shows that the small perturbation method, within

its domain of validity (kh 1 and s 1), gives better solution than Kirchhoff approximation. The Kirchhoff approximation has a partially overlapping region of validity with SPM (for example, the correlation length l must be much larger than the wavelength). Note that (9.4.27) of the Kirchhoff approximation is applicable to large height. Taking the average of (9.4.27) to obtain the coherent field and using (9.2.44), an important consequence is that the average of (exp(ikdzf(x))) is independent of x and can be taken outside the integral sign. ('tps(k x )) =

27T zkdz = -6(kdx) exp( -2k;zh2 )