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TitleDesigner Surfaces
Author
LanguageEnglish
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Total Pages351
Table of Contents
                            Front cover
Designer Surfaces
Copyright page
Dedication
Contents
Preface
Chapter 1. Introduction
	References
Chapter 2. One-Dimensional Surfaces
	2.1. Perfectly Conducting Surfaces
	2.2. Penetrable Surfaces
	2.3. A Random Surface that Suppresses Leakage
	2.4. Surfaces that Display Enhanced Backscattering for Only a Single Specified Angle of Incidence
	2.5. Surfaces that Synthesize Infrared Absorption Spectra
	2.6. Surfaces that Produce Specified Thermal Emissivities
	2.7. Control of the Coherence of the Light Scattered
	2.8. Surfaces that Produce a Prescribed Angular Dependence
	2.9. Replacement of Ensemble Averaging by the Use of a Broadband Incident Field in Calculations
	2.10. Fabrication of One-Dimensional Surfaces
	2.11. Experimental Results
	References
Chapter 3. Two-Dimensional Surfaces
	3.1. The Design of Two-Dimensional Randomly Rough Surfaces
	3.2. Random Diffusers that Extend the Depth of Focus
	3.3. A Two-Dimensional Randomly Rough Surface that Acts as a Gaussian Schell-Model Source
	3.4. Fabrication of Circularly-Symmetric Surfaces
	References
Chapter 4. Conclusions and Outlook
	References
Appendix A. The Kernels in the Integral Equations (2.8.77)–(2.8.80)
Appendix B. The Matrix Elements Entering Eqs. (2.8.77)–(2.8.80)
Appendix C. The Singularity in Gl(r|r')
	References
Subject Index
                        
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




−∞

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



−∞

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



−∞

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



−∞

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


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


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)�

×
��



(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

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