##### Document Text Contents

Page 2

Designer Surfaces

Page 175

158 One-Dimensional Surfaces

In these equations θ(z) is the Heaviside unit step function, and ds is the element of arc

length along the curve x3 = ζ(x1), which we denote by s. The incident field Fν(x1, x3|ω)inc

is a solution of Eq. (2.8.1), and we assume that it has the plane wave form

Fν(x1, x3|ω)inc = exp

�

ikx1 − iα1(k)(x3 − d)

�

. (2.8.13)

Since we have assumed that the surface profile function ζ(x1) is a single-valued function

of x1, we can express the element of arc length ds in Eqs. (2.8.11) and (2.8.12) as in

Eq. (2.1.15). If we also use the boundary conditions (2.8.4b) and (2.8.5b) at the interfaces

x3 = d and x3 = ζ(x1), then Eqs. (2.8.10), (2.8.11), and (2.8.12) can be rewritten as

θ(x3 − d)F (1)ν (x1, x3|ω)

= Fν(x1, x3|ω)inc +

1

4π

∞

−∞

dx

′

1

�

∂

∂x′3

G1(x1, x3|x′1, x′3)

�

x′3=d

H

(1)

ν (x

′

1|ω)

− �G1(x1, x3|x′1, x′3)�x′3=dL(1)ν (x′1|ω)

�

, (2.8.14)

θ

�

x3 − ζ(x1)

�

θ(d − x3)F (2)ν (x1, x3|ω)

= − 1

4π

∞

−∞

dx

′

1

�

∂

∂x′3

G2(x1, x3|x′1, x′3)

�

x′3=d

H

(1)

ν (x

′

1|ω)

− λν

κν

�

G2(x1, x3|x′1, x′3)

�

x′3=d

L

(1)

ν (x

′

1|ω)

�

+ 1

4π

∞

−∞

dx

′

1

�

∂

∂N ′

G2(x1, x3|x′1, x′3)

�

x′3=ζ(x′1)

H

(2)

ν (x

′

1|ω)

− �G2(x1, x3|x′1, x′3)�x′3=ζ(x′1)L(2)ν (x′1|ω)

�

, (2.8.15)

θ

�

ζ(x1) − x3

�

F

(3)

ν (x1, x3|ω)

= − 1

4π

∞

−∞

dx

′

1

�

∂

∂N ′

G3(x1, x3|x′1, x′3)

�

x′3=ζ(x′1)

H

(2)

ν (x

′

1|ω)

− μν

λν

�

G3(x1, x3|x′1, x′3)

�

x′3=ζ(x′1)

L

(2)

ν (x

′

1|ω)

�

, (2.8.16)

Page 176

2.8. Surfaces that Produce a Prescribed Angular Dependence 159

where we have introduced the source functions

H

(1)

ν (x1|ω) = F (1)ν (x1, x3|ω)

��

x3=d, (2.8.17a)

L

(1)

ν (x1|ω) =

∂

∂x3

F

(1)

ν (x1, x3|ω)

����

x3=d

, (2.8.17b)

H

(2)

ν (x1|ω) = F (2)ν (x1, x3|ω)

��

x3=ζ(x1), (2.8.17c)

L

(2)

ν (x1|ω) =

∂

∂N

F

(2)

ν (x1, x3|ω)

����

x3=ζ(x1)

. (2.8.17d)

The scattered field in the region x3 > d is given by the second term on the right-hand side

of Eq. (2.8.14),

Fν(x1, x3|ω)sc =

∞

−∞

dq

2π

Rν(q|k) exp

�

iqx1 + iα1(q)(x3 − d)

�

, (2.8.18a)

where

Rν(q|k) =

1

2α1(q)

∞

−∞

dx1

�

α1(q)H

(1)

ν (x1|ω) − iL(1)ν (x1|ω)

�

exp(−iqx1). (2.8.18b)

The dependence of the scattering amplitude Rν(q|k) on the wave number k arises

from the dependence of the source functions H(1)ν (x1|ω) and L(1)ν (x1|ω) on the inci-

dent field, Eq. (2.8.13). In obtaining Eq. (2.8.18b) we have used the representation of

G1(x1, x3|x′1, x′3) given by Eq. (2.8.7b) in the case that x3 > d .

The transmitted field in the region x3 < ζ(x1) is given by the right-hand side of

Eq. (2.8.16), and can be represented in the form

Fν(x1, x3|ω)tr =

∞

−∞

dq

2π

Tν(q|k) exp

�

iqx1 − iα3(q)x3

�

, (2.8.19a)

where

Tν(q|k) =

1

2α3(q)

∞

−∞

dx1 exp

�−iqx1 + iα3(q)ζ(x1)�

×

��

qζ

′

(x1) + α3(q)

�

H

(2)

ν (x1|ω) + i

μν

λν

L

(2)

ν (x1|ω)

�

. (2.8.19b)

Page 350

Subject Index 333

transverse correlation length of the surface

roughness, 22, 298

trapezoidal grooves, 42

turbulent atmosphere, 286

two-dimensional Collett–Wolf source, 294

two-dimensional Dirichlet surface, 222

two-dimensional Gaussian random surface, 191,

298

two-dimensional randomly rough Dirichlet

surface, 220, 221, 241, 255, 260, 263, 269,

286, 294, 299, 301

– designed to act as a band-limited uniform

diffuser, 259

– within a circular domain of scattering angles,

265

– within a rectangular domain of scattering angles,

272

– within a triangular region of scattering angles,

276

two-dimensional randomly rough surface, 5, 137,

201, 202, 215, 224, 234, 275, 284, 296, 302

– statistical properties of, 241

– that obeys Gaussian statistics with a Gaussian

surface height autocorrelation function, 309

– that produces a scattered field whose intensity is

constant within an elliptical domain of

scattering angles, 265

– that produces a scattered field with a mean

intensity that has a specified spatial

dependence, 276

two-dimensional rough surface, 101, 213, 214,

251, 307

two-dimensional surface profile, 307

– with rectangular symmetry, 223

two-layer optical elements, 191

two-point amplitude correlation function, 284

two-point correlation function, 85, 86

U

uniform diffuser, 2, 42

uniform mean intensity

– in a region of the optical axis, 303

uniform rectangular diffractive optical diffuser,

225

uniform rectangular optical diffuser, 224, 225

universal attenuation limit, 119

V

vacuum, 50, 72

visible region of the optical spectrum, 43

volume, 1

volume electromagnetic wave, 71, 74, 84, 96

W

wave number

– k, 159

wavelength dependence

– of the intensity of the scattered field, 111

weight function

– F(k), 138

white light illumination, 223, 283, 284

Z

zero-mean Gaussian function, 297

zero-mean Gaussian random process, 22

zero-mean random surface, 297

Designer Surfaces

Page 175

158 One-Dimensional Surfaces

In these equations θ(z) is the Heaviside unit step function, and ds is the element of arc

length along the curve x3 = ζ(x1), which we denote by s. The incident field Fν(x1, x3|ω)inc

is a solution of Eq. (2.8.1), and we assume that it has the plane wave form

Fν(x1, x3|ω)inc = exp

�

ikx1 − iα1(k)(x3 − d)

�

. (2.8.13)

Since we have assumed that the surface profile function ζ(x1) is a single-valued function

of x1, we can express the element of arc length ds in Eqs. (2.8.11) and (2.8.12) as in

Eq. (2.1.15). If we also use the boundary conditions (2.8.4b) and (2.8.5b) at the interfaces

x3 = d and x3 = ζ(x1), then Eqs. (2.8.10), (2.8.11), and (2.8.12) can be rewritten as

θ(x3 − d)F (1)ν (x1, x3|ω)

= Fν(x1, x3|ω)inc +

1

4π

∞

−∞

dx

′

1

�

∂

∂x′3

G1(x1, x3|x′1, x′3)

�

x′3=d

H

(1)

ν (x

′

1|ω)

− �G1(x1, x3|x′1, x′3)�x′3=dL(1)ν (x′1|ω)

�

, (2.8.14)

θ

�

x3 − ζ(x1)

�

θ(d − x3)F (2)ν (x1, x3|ω)

= − 1

4π

∞

−∞

dx

′

1

�

∂

∂x′3

G2(x1, x3|x′1, x′3)

�

x′3=d

H

(1)

ν (x

′

1|ω)

− λν

κν

�

G2(x1, x3|x′1, x′3)

�

x′3=d

L

(1)

ν (x

′

1|ω)

�

+ 1

4π

∞

−∞

dx

′

1

�

∂

∂N ′

G2(x1, x3|x′1, x′3)

�

x′3=ζ(x′1)

H

(2)

ν (x

′

1|ω)

− �G2(x1, x3|x′1, x′3)�x′3=ζ(x′1)L(2)ν (x′1|ω)

�

, (2.8.15)

θ

�

ζ(x1) − x3

�

F

(3)

ν (x1, x3|ω)

= − 1

4π

∞

−∞

dx

′

1

�

∂

∂N ′

G3(x1, x3|x′1, x′3)

�

x′3=ζ(x′1)

H

(2)

ν (x

′

1|ω)

− μν

λν

�

G3(x1, x3|x′1, x′3)

�

x′3=ζ(x′1)

L

(2)

ν (x

′

1|ω)

�

, (2.8.16)

Page 176

2.8. Surfaces that Produce a Prescribed Angular Dependence 159

where we have introduced the source functions

H

(1)

ν (x1|ω) = F (1)ν (x1, x3|ω)

��

x3=d, (2.8.17a)

L

(1)

ν (x1|ω) =

∂

∂x3

F

(1)

ν (x1, x3|ω)

����

x3=d

, (2.8.17b)

H

(2)

ν (x1|ω) = F (2)ν (x1, x3|ω)

��

x3=ζ(x1), (2.8.17c)

L

(2)

ν (x1|ω) =

∂

∂N

F

(2)

ν (x1, x3|ω)

����

x3=ζ(x1)

. (2.8.17d)

The scattered field in the region x3 > d is given by the second term on the right-hand side

of Eq. (2.8.14),

Fν(x1, x3|ω)sc =

∞

−∞

dq

2π

Rν(q|k) exp

�

iqx1 + iα1(q)(x3 − d)

�

, (2.8.18a)

where

Rν(q|k) =

1

2α1(q)

∞

−∞

dx1

�

α1(q)H

(1)

ν (x1|ω) − iL(1)ν (x1|ω)

�

exp(−iqx1). (2.8.18b)

The dependence of the scattering amplitude Rν(q|k) on the wave number k arises

from the dependence of the source functions H(1)ν (x1|ω) and L(1)ν (x1|ω) on the inci-

dent field, Eq. (2.8.13). In obtaining Eq. (2.8.18b) we have used the representation of

G1(x1, x3|x′1, x′3) given by Eq. (2.8.7b) in the case that x3 > d .

The transmitted field in the region x3 < ζ(x1) is given by the right-hand side of

Eq. (2.8.16), and can be represented in the form

Fν(x1, x3|ω)tr =

∞

−∞

dq

2π

Tν(q|k) exp

�

iqx1 − iα3(q)x3

�

, (2.8.19a)

where

Tν(q|k) =

1

2α3(q)

∞

−∞

dx1 exp

�−iqx1 + iα3(q)ζ(x1)�

×

��

qζ

′

(x1) + α3(q)

�

H

(2)

ν (x1|ω) + i

μν

λν

L

(2)

ν (x1|ω)

�

. (2.8.19b)

Page 350

Subject Index 333

transverse correlation length of the surface

roughness, 22, 298

trapezoidal grooves, 42

turbulent atmosphere, 286

two-dimensional Collett–Wolf source, 294

two-dimensional Dirichlet surface, 222

two-dimensional Gaussian random surface, 191,

298

two-dimensional randomly rough Dirichlet

surface, 220, 221, 241, 255, 260, 263, 269,

286, 294, 299, 301

– designed to act as a band-limited uniform

diffuser, 259

– within a circular domain of scattering angles,

265

– within a rectangular domain of scattering angles,

272

– within a triangular region of scattering angles,

276

two-dimensional randomly rough surface, 5, 137,

201, 202, 215, 224, 234, 275, 284, 296, 302

– statistical properties of, 241

– that obeys Gaussian statistics with a Gaussian

surface height autocorrelation function, 309

– that produces a scattered field whose intensity is

constant within an elliptical domain of

scattering angles, 265

– that produces a scattered field with a mean

intensity that has a specified spatial

dependence, 276

two-dimensional rough surface, 101, 213, 214,

251, 307

two-dimensional surface profile, 307

– with rectangular symmetry, 223

two-layer optical elements, 191

two-point amplitude correlation function, 284

two-point correlation function, 85, 86

U

uniform diffuser, 2, 42

uniform mean intensity

– in a region of the optical axis, 303

uniform rectangular diffractive optical diffuser,

225

uniform rectangular optical diffuser, 224, 225

universal attenuation limit, 119

V

vacuum, 50, 72

visible region of the optical spectrum, 43

volume, 1

volume electromagnetic wave, 71, 74, 84, 96

W

wave number

– k, 159

wavelength dependence

– of the intensity of the scattered field, 111

weight function

– F(k), 138

white light illumination, 223, 283, 284

Z

zero-mean Gaussian function, 297

zero-mean Gaussian random process, 22

zero-mean random surface, 297