OPTICAL IMAGING AND SPECTROSCOPY Phần 3 potx

3.9

B-SPLINES

A function ff (x) [ V(f) may be represented as
X
cn fn (x)
ff (x) ¼

91

(3:124)

n[Z

In contrast with the sampling theorem and with the Haar wavelet expansion, the
expansion coefficients are not samples of ff or inner products between ff and the
basis vectors. For the B-splines it turns out that we can derive complementary
functions fn (x) for each fn (x) ¼ bm (x À n) such that hfn jfn0 i ¼ dnn0 . The complementary functions can be used to produce a continuous estimate for f (x) that is
completely consistent with the discrete measurements. This interpolated function is
X
hfn j f ifn (x)
(3:125)
fest (x) ¼
n[Z

Given the orthogonality relationship between the sampling functions and the complementary functions, fest is by design consistent with the measurements. We can
further state that fest ¼ ff if the complementary functions are such that f [ V(f),
in which case there exist discrete coefficients p(k) such that
X
f (x) ¼

p(k)f(x À k)
(3:126)
k[Z

Using the convolution theorem, the Fourier transform of f (x) is
"
#
^ (u) ¼ f(u) X p(k)eÀi2pku
^
f

(3:127)

k[Z

The orthogonality between the dual bases may be expressed as
h fn jfn0 i ¼ dnn0
X
¼
p(k)a(n0 À k À n)

(3:128)

k[Z

where a(n) ¼ hf(x)jf(x À n)i. Without loss of generality, we set n ¼ 0 and sum both
0
sides of Eqn. (128) against the discrete kernel eÀi2pn u to obtain
X

0

d0n0 ei2pn u ¼ 1

n0 [Z

¼

XX

0

p(k)a(n0 À k)eÀi2pn u

k[Z n0 [Z

"
¼

X
k[Z

#"
Ài2pku

p(k)e

X
n00 [Z

where we use the substitution of variables n00 ¼ n0 À k.

#
00

Ài2pn00u

a(n )e

(3:129)


92

ANALYSIS

Poisson’s summation formula is helpful in analyzing the sums in Eqn. (3.129).
The summation formula states that for g(x) [ L1 (R)
X

X

g(n)ei2pnu ẳ

n[Z

^(u ỵ k)
g

(3:130)

k[Z

where ^(u) is the Fourier transform of g(x). To prove the summation formula,
g
note that
h(u) ẳ

X

^(u ỵ k)
g

(3:131)

k[Z

is periodic in u with period 1. The Fourier series coefficients for h(u) are
1


^n ẳ h(u)e2pinu du
h
0

X
1

ẳ

k[Z

ẳ

0

kỵ1
X
k[Z

ẳ

^(u ỵ k)e2pinu du
g

1


^(u)e2pinu du
g

k

^(u)e2pinu du
g

À1

¼ g(n)

(3:132)

^
Since a(x) is the autocorrelation of f, its Fourier transform is jf(u)j2 . Thus by the
Poisson summation formula
X

a(n)eÀi2pnu ẳ

n[Z

X

^
jf(u ỵ k)j2

(3:133)

k[Z

Reconsidering Eqn. (3.129), we nd
X
k[Z

1
a(n)ei2pnu
n[Z

p(k)ei2pku ẳ P

1
^ (u þ k)j2
k[Z jf

¼P

(3:134)


3.9

B-SPLINES

93

Substitution in Eqn. (3.127) yields
^
f(u)
^
f(u) ẳ P
^
jf(u ỵ k)j2

(3:135)

k[Z

P
^
^

We can evaluate Eqn. (3.135) to determine f(u) and f(x) if k[Z jf(u ỵ k)j2 is
nite. The requirement that there exist positive constants A and B such that
A

X

^
jf(u ỵ k)j2

B

(3:136)

k[Z

is the dening feature of a Riesz basis. A Riesz basis may be considered as a generP
^
alization of an orthonormal basis. In the case that k[Z jf(u ỵ k)j2 ẳ 1, Eqn. (135)
^
^
reduces to f(u) ¼ f(u) and an orthonormal basis may be obtained.
The Fourier transform of the mth-order B-spline is
^m
b (u) ¼ [sinc(u)](mỵ1) eipju
h 0 i(mỵ1)
^
ẳ b (u)
eip(mỵ1j)u

(3:137)

where j ẳ 0 if m is odd and j ¼ 1 if m is even. For the B-spline basis, we obtain
Qm (u) ẳ

X

^
jf(u ỵ k)j2 ẳ

k[Z

X

jsinc(u ỵ k)j2(mỵ1)

(3:138)

k[Z

Since the zeroth-order B-spline produces an orthogonal basis, we know that
Q0 (u) ¼ 1. For higher orders we note that jsinc(u ỵ k)j2(mỵ1) jsinc(u ỵ k)j2 ,
meaning that Qm (u) Qo (u). Thus, 0 , Qm (u) , 1 and the B-spline functions of
all orders satisfy the Riesz basis condition.
In contrast with the B-splines themselves, the complementary functions f(x) do
not have finite support. It is possible, nevertheless, to estimate f(x) over a finite
interval for each B-spline order by numerical methods. Estimation of Qm (u) from
Eqn. (3.138) is the first step in numerical analysis. This objective is relatively
easily achieved because Qm (u) is periodic with period 1 in u. Evaluation of the
sum over the first several thousand orders for closely spaced values of 0 ! u 1
takes a few seconds on a digital computer.

Given Qm (u), we may estimate f(x) by using a numerical inverse Fourier transform of Eqn. (3.135) or by calculating p(k) from Eqn. (3.134). Since p(k) must be
real and since Qm (u) is periodic, we obtain
p(k) ¼

1
ð

cos (2pku)
du
Qm (u)

0

Estimation of p(k) was the approach taken to calculate f (x) for Fig. 3.16.

(3:139)


94

ANALYSIS

Given f(x) ¼ bm (x À n) and f(x), we can calculate ff (x) for target functions. For
example, Fig. 3.17 shows the signals of Figs. 3.8 and 3.9 projected onto the V(f) subspaces for B-splines of orders 0 – 3. Higher-order splines smoothly represent signals
with higher-order local polynomial curvature. Note that higher-order splines are not
more localized than the lower-order functions, however, and thus do not immediately
translate into higher signal resolution. Notice also the errors at the edges of the signal
windows in Fig. 3.17. These arise from the boundary conditions used to truncate the
infinite time signal f (x). In the case of these figures, f (x) was assumed to be periodic
in the window width, such that sampling and interpolation functions extending

beyond the window could be wrapped around the window.
The interpolated signals plotted in Fig. 3.17 are the projections ff (x) [ V(f) of
f (x) onto the corresponding subspaces V(f). The consistency requirement designed
into the interpolation strategy means that these functions, despite their obvious discrepancies relative to the actual signals, would yield the same sample projections.
Corrections that map the interpolated signals back onto the actual signal lie in
V? (f). Strategies for sampling and interpolation to take advantage of known
constraints on f (x) to so as to infer correction components f? (x) are discussed in
Chapter 7.

Figure 3.16 Complementary interpolation functions f(x) for the B-splines of orders 0–3.
The zeroth-order B-spline is orthonormal such that f(x) ¼ b0 (x).


3.9

B-SPLINES

95

Figure 3.17 Projection of f (x) ¼ x2 =10 and the signal of Fig. 3.9 onto the V(f) subspace for
B-splines of orders 0 –3.

Use of Eqn. (3.125) to estimate f (x) is somewhat unfortunate given that fn (x) does
not have finite support. A primary objection to the use of the original sampling
theorem [Eqn. (3.92)] for signal estimation is that sinc(x) has infinite support and
decays relatively slowly in amplitude. While f(x) is better behaved for low-order
B-splines, it is is still true that accurate estimation of f (x) may be computationally
expensive if a large window is used for the support of f. As the order of the
B-spline tends to infinity, f(x) converges on sinc(x) [235]. If we remove the requirement that f(x) [ V(f), it is possible to generate a biorthogonal dual basis for bm (x)
with compact support [49]. The compactly supported biorthogonal wavelets in this

case introduce a complementary subspace V spanned by f(x).
The goal of the current section has been to consider how one might use a set of
discrete B-spline inner products to estimate a continuous signal. This problem is
central to imaging and optical signal analysis. We have already encountered it in
the coded aperture and tomographic systems considered in Chapter 2, and we will
encounter it again in the remaining chapters of the text. We leave this problem for
now, however, to consider the use of sampling functions and multiscale representations in signal and system analysis. One may increase the resolution and fidelity


96

ANALYSIS

of the reconstructions in Fig. 3.17 by increasing the resolution of the sampling function in a manner similar to the wavelet approach taken in Section 3.8.

3.10 WAVELETS
As predicted in the Section 3.1, this chapter has developed three distinct classes of
mathematics: transformation tools, sampling tools, and analysis tools. In the first
several sections we considered fields and field transformations. We have just completed three sections focusing on sampling. Section 3.9 describes a method for
representing a function f (x) on the space V(f) spanned by the scaling function
f(x) ¼ bm (x). This section extends our consideration of B-splines to wavelets,
similar to our extension of Haar analysis in Section 3.8. We have already considered
mathematical bases suitable for field analysis in terms of the Fourier transform and
Hermite – Gaussian functions. In fact, many functional families could be used to
analyze fields. The choice of which family to use depends on which family arises
naturally in the physical specification of the problem (e.g., Laguerre – Gaussian functions arise naturally in the specification of cylindrically symmetric fields), which
family arises at sampling interfaces, and which family enables the most computationally efficient and robust analysis of field transformations.
Wavelet theory is a broad and powerful branch of mathematics, and the student is
well advised to consult standard courses and texts for deeper understanding [53,164].
Wavelets often describe images and other natural signals well. The intuitive match

between wavelets and images arises from the assumption that “features” in natural
signals tend to cluster, meaning that higher resolution is desirable in the vicinity of
a feature than elsewhere in the signal. Multiscale clustering enables wavelet representations to estimate signals with fewer samples than might be used with uniform
regular sampling. Under the Whittaker– Shannon sampling strategy, functional
samples are distributed uniformly in space even in regions with no significant
image features. Wavelets enable samples to be dynamically assigned to regions
with interesting features. This dynamic resource allocation is the basis of natural
signal compression.
B-splines may be used to generate semiorthogonal bases as in Section 3.9, biorthogonal spaces and orthogonal wavelet bases. As before, we imagine a hierarchy
of spaces
{0} , Á Á Á , V2 , V1 , Vo , VÀ1 , VÀ2 , Á Á Á , L2 (R)

(3:140)

Semiorthogonal bases are spanned by sets of functions that are not themselves
orthogonal but are orthogonal to a complementary set of functions. Biorthogonal
bases generate complementary spaces spanned by complementary sets of functions.
Orthogonal bases generate a single hierarchy of spaces spanned by a single set of
orthogonal functions. We have already encountered an orthogonal wavelet basis in
the form of the Haar wavelets of Section 3.8. In this section we extend the Haar analysis to orthogonal bases based on higher-order B-splines.


3.10

WAVELETS

97

The orthonormal basis for spaces spanned by discretely shifted B-splines was
introduced by Battle [15] and Lemarie [150]. For the Battle – Lemarie basis, f(x) is

a scaling function on the space V(bm (x)) spanned by the mth-order B-spline. Since
f(x) [ V(bm (x)) there exist expansion coefficients p[n] such that
X
f(x) ¼
p[n]bm (x À n)
(3:141)
The Fourier transform of Eqn. (3.141) yields
m
^
f (u) ¼ ^(u)b (u)
p ^

(3:142)

Our goal is to select f(x) to be an orthonormal scaling function such that

hf(x À n), f(x À m)i ¼

1
ð

fà (x À n)f(x À m)dx

À1

¼ dnm

(3:143)

We may apply the Poisson summation formula as in Section 3.9 to derive a simple

identity from Eqn. (3.143). Again letting a(x) ¼ hf(x0 ), f(x0 À x)i, we note from
Eqn. (3.130) that
X

a(n)eÀi2pnu ¼

n[Z

X

^(u þ k)
a

(3:144)

k[Z

P
Ài2pnu
¼ 1 and
For an orthonormal scaling function, however,
n[Z a(n)e
2
^ (u)j , which yields the identity for orthonormal scaling functions
^(u) ẳ jf
a
X

^
jf (u ỵ k)j2 ẳ 1

(3:145)

k

^
Referring to Eqn. (142), we see that f (u) satisfies Eqn. (145) if we select
1
^(u) ẳ q
p
P
2
^m
k jb (u ỵ k)j

(3:146)

where ^(u) is nite and well defined because the B-splines form a Riesz basis, as
p
discussed in Section 3.9. Since ^(u) is periodic with period 1 in u, it generates a
p
^m
discrete series p[n] for use in Eqn. (3.141). Substituting b (u) from Eqn. (3.137)
in Eqns. (3.146) and (3.142) yields
^
f (u) ẳ

umỵ1

eipju

p
S2 mỵ2 (u)

(3:147)


98

ANALYSIS

where
Sn (u) ẳ

X
k[Z

1
(u ỵ k)n

(3:148)

We know that the m ẳ 0 spline produces the Haar scaling function
sin (pu)
^0
f (u) ¼ eÀipu
pu

(3:149)

Comparing Eqns. (3.147) and (3.149), we see that

S2 (u) ¼

p2
sin2(pu)

(3:150)

Higher orders of Sn (u) are obtained by noting that Snỵ1 (u) ẳ S0n (u)=n. This yields
S4 (u) ẳ

p4 (2 ỵ cos (2pu))
6 sin4(pu)

(3:151)

S6 (u) ẳ

p6 (33 ỵ 26 cos (2 pu) ỵ cos (4pu))
180 sin6(pu)

(3:152)

and
S8 (u) ẳ

p8 (1208 ỵ 1191 cos (2pu) ỵ 120 cos (4pu) ỵ cos(6pu))
10,080 sin8(pu)

(3:153)

To satisfy the requirement that Vj , V jÀ1 , we require that f j,n (x) [ V jÀ1 , which
means that there exist expansion coefficients h[n] such that
 x

 X 1
1 x
pffiffiffiffi f j À n ¼
pffiffiffiffiffiffiffiffiffi h[n0 À n]f jÀ1 À n0
2
2j 2
2 jÀ1
n0

(3:154)

Equation (3.154) reduces without loss of generality to
1 x X
p f ẳ
h[n]fx nị
2 2
n

(3:155)

The Fourier transform of Eqn. (155) yields
pffiffiffi
^
2f (2u) ¼ ^ f (u)
h(u) ^

(3:156)


3.10

WAVELETS

99

where
^ ¼
h(u)

X

h[n]eÀ2pinu

(3:157)

n[Z

For the Battle – Lemarie scaling functions
^
pffiffiffi f (2u)
2
^
f (u)
s
S2 mỵ2 (u)
ipju

ẳe
22 mỵ1 S2 mỵ2 (2u)

^
h(u)ẳ

(3:158)

As with the Haar scaling function, we are interested in obtaining orthogonal wavelets spanning the spaces Wj such that VjÀ1 ¼ Vj È Wj . Such wavelets are immediately
obtained using the conjugate mirror filter ^
h(u). The wavelet corresponding to the
scaling function f(x) has the Fourier transform

  
1
uỵ1
^ u
^
c(u) ẳ p eipu ^
f
h
2
2
2

(3:159)

The Battle – Lemarie scaling function and wavelet can be reconstructed by inverse
Fourier transforming Eqns. (3.147) and (3.159). These functions satisfy the same
orthogonality and scaling rules as the Haar wavelets discussed earlier; specifically


1 x
f j,n (x) ¼ pffiffiffiffiffi f j À n
2j 2

1 x
c j,n (x) ¼ pffiffiffiffiffi c j À n
2j 2
hf j,n jf j0 ,n0 i ¼ d jj0 dnn0


c j,n jc j0 ,n0 ¼ d jj0 dnn0


c j,n jf j0 ,n0 ¼ 0

(3:160)
(3:161)
(3:162)
(3:163)
(3:164)

As with the Haar wavelets, the Battle – Lemarie functions span L2 (R2 ) in the hierarchy of spaces described by Eqn. (3.140). The Battle – Lemarie wavelets are presented here to provide an accessible introduction to wavelet theory. Many other
wavelet families have been developed [164]; the selection of which family to use
for a particular class of signals is application-specific. Some wavelets are attractive
because they have compact support, which the Shannon wavelet famously does
not. Other wavelets, such as the Haar and B-splines, arise naturally from physical


100

ANALYSIS

or system design considerations. In still other cases, a particular basis may prove more
amenable to compact support of a particular signal class.

PROBLEMS
3.1

2

Fourier Uncertainty. Show that for f (x) ¼ aeÀb(xÀxo )
2
sf2 s ^ ¼
f

1
16p 2

(3:165)

3.2
3.3

Fourier Rotation. Derive Eqn. (3.39).
Fresnel Identities:
(a) Derive Eqn. (3.63).
(b) Derive Eqn. (3.64).

3.4

Hermite –Gaussian Eigenfunctions. The Hermite polynomial Hn (x) is
defined as
dn Àx2
e
dxn

(3:166)

pffiffiffiffiffiffi
2
fn (x) ¼ eÀpx Hn ( 2px)

(3:167)

pffiffiffiffiffiffi
d
2pfn (x) ¼ 2pxfnÀ1 (x) À fnÀ1 (x)
dx

(3:168)

Hn (x) ¼ (À1)n ex

2

Defining

show that for n . 0

Combine this relationship with Eqns. (3.13) and (3.57) to show by recursion
that
F {fn (x)} ¼ in fn (u)
3.5

(3:169)

One-dimensional Numerical Analysis:
(a) Plot sin (2pux) on [0, 1] using 1024 uniformly spaced samples for
u ¼ 16,32,64,128,256. At what point does aliasing become significant?
Can you describe the structure of the aliased signal?
(b) Plot the discrete Fourier transform of sin (2pux) on [0, 1] using 1024 uniformly spaced samples for u ¼ 16, 32, 64. Label the plot in frequency
units. What is the width of the Fourier features that you observe? What
causes this width?
(c) Plot the discrete Fourier transform of b0 (x) sin (2p ux) on [À1:5, 2:5]
using 4096 uniformly spaced samples for u ¼ 16, 32, 64. Label the plot
in frequency units and explain the plot.


PROBLEMS

101

3.6

Fourier Analysis of a Hermite – Gaussian:
(a) Plot the Hermite – Gaussian f5 (x) over the range of the function.
~
(b) Plot f5t (x) over a representative range of t.

3.7

Transformations:
2
(a) Plot the Fourier transform of the function eÀ2(xÀ:1) circ(x/0.3) cos(20px).
Mark units on your plot.
(b) Plot the Fresnel transform for t ¼ 1, 3, 10 for the function from part (a).

3.8

Fresnel Transformation of the Hermite – Gaussian Functions. Prove
Eqn. (3.76).

3.9

Fresnel Transformation of the Laguerre – Gaussian Functions. Use the convolution theorem and the fast Fourier transformation to numerically calculate
the Fresnel transformation of the Laguerre – Gaussian modes for m, n equal to
0,0, 1,0, 1,1, 2,0, 2,1, and 4,3 for t ¼ 0:5, t ¼ 1, and t ¼ 2. Use your computational result and the analytic result given by Eqn. (3.83) to plot the absolute value and phase of the mode distribution at t ¼ 0 and for the transform
values of t in each case. Submit your code, plots, and comments regarding
features of the modes or discrepancies between the computational methods.

3.10

Haar Analysis:
(a) Generate and plot a function of similar complexity to f (x) in Fig. 3.9.
(b) Replicate Figs. 3.9, 3.12, and 3.17 for your function.

3.11

2D Wavelet Analysis. Replicate Figs. 3.13 and 3.14 for an image of your

choosing.

3.12

Spline Interpolation:
(a) Show that a one-dimensional pinhole imaging system produces
measurements
ð

gn ¼ fn ¼ f (x)b

1



x À nD
D


dx

(3:170)

(b) Plot the values of fn for

x  À(x=30D)2
e
f (x) ¼ cos 2p
5D

(3:171)

(c) Use Eqn. (3.125) to estimate f (x) from fn . Plot f (x) at a sampling period of
D=10. Compare your plots.
3.13 Wavelets. For the Battle – Lemarie bases, plot the scaling function f(x) and
wavelet c(x) for orders 0, 1, 2, and 3.


4
WAVE IMAGING
The physical phenomenon called diffraction is of the utmost importance in the theory of
optical imaging systems.
—J. W. Goodman [100]

4.1

WAVES AND FIELDS

The optical field is an electromagnetic field. The physical nature of the field is determined by the laws of electromagnetic propagation and by quantum mechanical and
thermal laws describing the interaction between the field and materials. In the
design and analysis of optical systems we consider
†

How the field is generated. Common mechanisms include
Thermal radiation generated, for example, by the Sun, a flame, or an incandescent lightbulb
Electrical discharge by gases such as neon or mercury vapor
Fluorescence
Electrical recombination in semiconductors

While we do not consider light generation in detail in this text, differences in the

coherence properties of the source are central to our discussion. Coherence theory,
which relates the electromagnetic nature of the field to statistical properties of
quantum (e.g., photonic) processes, is the focus of Chapter 6.
†

How the field is detected. The field may be detected by optically induced
chemical, physical, thermal, and electronic effects. Optoelectronic detection
interfaces for imaging and spectroscopy are the focus of Chapter 5.

Optical Imaging and Spectroscopy. By David J. Brady
Copyright # 2009 John Wiley & Sons, Inc.

103


104
†

WAVE IMAGING

How the field propagates and how propagating fields are modulated by
materials. Field propagation is described by the Maxwell equations for electromagnetic waves, and field – matter interactions are described by materials
equations. The electromagnetic description of optical waves and optical interactions is the focus of this chapter.

In view of the peculiarly quantum mechanical nature of optical field generation and
detection, it is important to understand that the conventional electromagnetic field of
the Maxwell equations is not a sufficient description of optical fields. The description
derived in this chapter provides a basis for optical analysis, but complete understanding of optical field propagation and field properties must incorporate the detection
and coherence processes discussed in Chapters 5 and 6. In short, the student must
understand the next three chapters as a group to have a vision for the peculiar and

beautiful nature of optical fields.

4.2

WAVE MODEL FOR OPTICAL FIELDS

The Maxwell equations for electromagnetic propagation are
@
B
@t
@
rHẳJỵ D
@t
rEẳ

rDẳr
rBẳ0

(4:1)
(4:2)
(4:3)
(4:4)

where E is the electric field, D is the electric displacement, B is the magnetic
induction, H is the magnetic field, J is the current density, and r is the charge density.
Equation (4.1), which expresses the tendency of a moving magnet to generate an
electromotive force, is called Faraday’s law. Equation (4.2), which expresses the
tendency of an electric current to generate a magnetic flux, is called Ampere’s
law. The electric displacement current @D/@t in Ampere’s law was added by Maxwell
as a means of explaining electrodynamics. Equation (4.3), which expresses the

Coulomb attraction of electromagnetic charge, is called Gauss’ law. Equation (4.4), is
called Gauss’ law for magnetism and expresses the absence of magnetic monopoles.
The fields are further related by the material equations
D ẳ 10 E ỵ P

(4:5)

B ẳ m0 H ỵ M

(4:6)

where P is the polarization of the material and M is the magnetization. In most optical
materials, M ¼ 0 and P is a function of E. The simplest and most common case is the


4.2

WAVE MODEL FOR OPTICAL FIELDS

105

linear dielectric relationship
P ¼ 10 xe E

(4:7)

D ẳ 1E

(4:8)

1 ẳ (1 ỵ xe )10

(4:9)

such that

where

Since charge dynamics at optical frequencies are described by quantum mechanical
processes that cannot be accurately analyzed by continuous models, the space charge
density r and the current density J are generally neglected in optical analysis. A
nonzero current density is sometimes applied to formally account for optical absorption. We also note that E and D need not be collinear, meaning that 1 is in general
tensor-valued. Materials in which 1 is a scalar are called isotropic. Optical glasses
are isotropic, but optical crystals are often anisotropic. While most of the optical
systems discussed in this text utilize isotropic materials, we consider the application
of anisotropic materials to tunable filters in Section 9.7.
Using the material relations to eliminate B and D from the Faraday and Ampere
relationships yields
r  E ¼ Àm 0
rÂH¼1

@
H
@t

@
E
@t

(4:10)

(4:11)

Operating on Eqns. (4.10) and (4.11) with the curl yields the wave equations
r  r  E ¼ Àm 0 1

@2
E
@t 2

(4:12)

r  r  H ¼ Àm 0 1

@2
H
@t 2

(4:13)

The equations are reduced to a simpler form by the vector identity
r  r  A ¼ r ðr Á A Þ À r 2 A

(4:14)

From Gauss’ law we know that
r 1Eị ẳ E r1 ỵ 1r Á E ¼ 0

(4:15)

where we have assumed for the moment that 1 is a scalar. We can reexpress

Eqn. (4.15) as
r E ẳ E r log1ị

(4:16)


106

WAVE IMAGING

Substituting Eqn. (4.16) in the wave equation yields
r2 E rE r log1ịị ẳ m0 1

@2
E
@t 2

(4:17)

A medium in which r 1 ¼ 0 is termed homogeneous. Most optical dielectrics (i.e.,
transparent glasses and crystals and liquids) are homogeneous. Optically interesting
inhomogeneous media include photonic crystals, optical fiber, graded-index lenses,
and volume holograms.
In isotropic homogeneous media, the wave equations reduce to
r 2 E À m0 1

(4:18)

r 2 H À m0 1

4.3

@2
E¼0
@t 2
@2
H¼0
@t 2

(4:19)

WAVE PROPAGATION

Solutions to Eqn. (4.12) take many forms, but the linearity of the equations and the
natural harmonic nature of optical sources make harmonic solutions particularly
attractive. The basic harmonic solution is the plane wave described by
E ¼ E0 ei2p(ntÀuÁr)

(4:20)

Substituting this solution into Eqn. (4.13) yields
u  u  E0 ¼ Àm0 1n2 E0

(4:21)

If 1 is a scalar, this equation has solutions E 0 only if juj2 ¼ m01 n2. Allowing for the
possibility that 1 is a tensor, solutions correspond to values of u such that


u u ỵm0 1n2  ¼ 0

(4:22)

Equation (4.22) reduces the range of u from three dimensions to two. The surface
defined by this equation is called the wave normal surface. In isotropic materials,
pffiffiffiffiffiffiffiffi
the wave normal surface is a sphere in u space of radius n m0 1, as sketched in
Fig. 4.1. In anisotropic materials (crystals), the wave normal surface splits into two
sheets, so that there are two solutions for u in almost every direction. The relationship
between u and n (or radial coordinates k ¼ 2p u and v ¼ 2pn) expressed by the wave
normal surface is called the dispersion relationship, which reduces the fourdimensional u, n space to a three-dimensional manifold of allowed solutions for
wave propagation. Each solution for u corresponds to an eigenvector E 0. The direction of E 0 is the polarization. For the isotropic case, two possible polarizations exist
for each u. In the general case, each eigenvector corresponds to a different value of u.


4.3

Figure 4.1

WAVE PROPAGATION

107

The wave normal surface in free space.

The primary attraction of plane wave analysis is that more general solutions can be
expressed as superpositions of plane waves. In the remainder of this chapter we
restrict our attention to isotropic materials, in which case 1 is a scalar and juj2 ¼
m01 n2. A general solution to Eqn. (4.18) in this case is
ððð

E(r, t) ¼

À

Ài2p ntÀuxÀvyÀ

F(u, v, n)p(u, v, n)e

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÁ
2
2
2
m1n Àu Àv z

du dv dn

(4:23)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
where (u, v, w ¼ m1n2 À u2 À v2 ) corresponds to u, p(u, v, n) Á u ¼ 0 and
jp(u, v, n)j ¼ 1.
As discussed in some detail in subsequent chapters, the space – time field E(r,t) is
not generally measurable at optical frequencies. It will take us a while to introduce
functions that are measurable over the optical band; for the present purposes we
prefer to analyze the field using the temporal Fourier transform of E(r, t):
ð

E(r, n) ¼ E(r, t)ei2pnt dt

(4:24)

In this chapter n is treated as an implicit variable in the function E(r) ¼ E(r, n).
We according drop the harmonic time dependence e 2i2p nt from Eqn. (4.23) and
describe spatial distribution of the field amplitude according to
ðð
E(r) ẳ

F(u, v)p(u, v)ei2p



uxỵvyỵ

p
2
2
2
m1n u v z

du dv

(4:25)

In the remainder of this chapter we also assume that the field propagates paraxially.
The paraxial approximation consists of the assumption that values of u and v for
which jF(u, v)j is nonzero lie on a compact window on the wave normal sphere


108

WAVE IMAGING

centered on the w axis, as illustrated in Fig. 4.1. This window is centered on the z axis,
such that w ) u, v over the full spatial bandwidth. This means that the polarization
vector p(u, v) is nearly parallel to the (x, y) plane over the entire spatial bandwidth.
In an isotropic material p(u, v) may be represented on any basis orthogonal to u.
We select as an example a basis in which one of the polarization vectors is also
orthogonal to the y axis. The resulting orthonormal basis for p(u, v) is
À
Á
px ẳ k uix ỵ viy ỵ wiz iy

lwix luiz
ẳ p
1 l2 v2


py ẳ luix ỵ lviy þ lwiz  px

(4:26)

(1 À l2 v2 )iy þ l2 uvix ỵ l2 iz
p
1 l2 v2
p
where k is a normalization constant and l ¼ 1/n m1. Substituting the polarization
into Eqn. (4.25) separates the field into polarized components
¼

ðð

f x (r) ẳ


p
2
2
2
Fx (u, v)px ei2p uxỵvyỵ 1=l u v z du dv


f y (r) ẳ



i2p uxỵvyỵ

Fy (u, v)py e

p
2
1=l u2 v2 z

(4:27)
du dv

Equation (4.27) is an exact vector model relating the Fourier distribution of the field
in linear polarizations to the spatial field distribution in three dimensions. The inverse
relationship is
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð ð
2

2
2
f x (x, y, z)ei2p(uxỵvy) dx dy
Fx (u, v)px ei2p 1=l u v z ẳ
p
2
2
2
f y (x, y, z)ei2p(uxỵvy) dx dy
Fy (u, v)py ei2p 1=l Àu Àv z ¼

(4:28)

From Eqn. (4.28) we see that knowledge of f x(x, y, z) and f y(x, y, z) as functions
of (x, y) for any specific value of z is sufficient to calculate Fx (u, v) and Fy (u, v).
In particular, if we know f(x, y, z ¼ 0), we can then calculate
ðð
Fx (u, v)px ¼
ðð
Fy (u, v)py ¼

f x (x, y, z ẳ 0)ei2p(uxỵvy) dx dy
(4:29)
i2p(uxỵvy)

f y (x, y, z ẳ 0)e

dx dy

Once we have determined Fx (u, v) and Fy (u, v), the field at all (x, y, z) may be

calculated from Eqn. (4.27). Specification of the field on a surface, such as the


4.4

DIFFRACTION

109

plane z ¼ 0, is called a boundary condition and the evolution of the field distribution
from one boundary to another is called diffraction. Equations (4.27) and (4.28) enable
us to computationally model diffraction in homogeneous media.

4.4

DIFFRACTION

Diffraction is the process of wave propagation from one boundary to another. A
canonical example of optical diffraction, propagation of a monochromatic field
from the plane (x, y, z ¼ 0) to the plane (x0 , y0, z ¼ d ), is illustrated in Fig. 4.2.
Given the electric field distribution on the input plane, we seek to estimate the
field distribution on the output plane. Viewed as a transformation between a function
f(x, y) over the input plane and a function g(x0 , y0 ) over the output plane, diffraction is
linear and shift-invariant. Our goal in this section is to describe the transfer function
and impulse response corresponding to diffraction from one plane to another.
An arbitrary vector field f(x, y) in the plane z ¼ 0 corresponds to the Fourier space
distribution
ðð
F(u, v) ẳ

f(x, y)ei2p(uxỵvy) dx dy

(4:30)

where F(u, v) may be separated into x, y and z components Fx (u, v) ¼ F(u, v) Á
px (u, v), Fy (u, v) ¼ F(u, v) Á py (u, v), and Fz (u, v) ¼ F(u, v) Á lu. The Fz component
does not produce a propagating field.
Let Gx (u, v) and Gy (u, v) be the Fourier distributions of the field in the x0 , y0 plane
at z ¼ d. From Eqn. (4.28) we see that
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
2
Gx (u, v) ¼ ei2p 1=l Àu Àv d Fx (u, v)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
2
Gy (u, v) ¼ ei2p 1=l Àu Àv d Fy (u, v)

Figure 4.2

Diffraction between two planes.

(4:31)


110

WAVE IMAGING

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
2
the factor T(u, v) ¼ ei2p 1=l Àu Àv d is the transfer function for diffraction from the
z ¼ 0 plane to the z ¼ d plane.
Nominally, the impulse response for diffraction is the inverse Fourier transform of
the transfer function. We continue along this line with care, however, by briefly
accounting for the vector nature of the field. Using the transfer function and
Eqn. (4.27), we obtain
ðð
Â
Ã
F(u, v) Á px px þ F(u, v) Á py py
g(x0 , y0 ) ¼
À
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÁ
2
0
0
2
2
 ei2p ux ỵvy ỵ 1=l u v z du dv

(4:32)

If there are no longitudinal (nonpropagating) field components on the input boundary
then F(u, v) Á lu ¼ 0, and
F(u, v) Á px px ỵ F(u, v) py py ẳ F(u, v)

(4:33)

In this case Eqn. (4.32) simplifies considerably to yield
ðð
À
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÁ
i2p ux0 þvy0 þ 1=l2 Àu2 Àv2 z
0 0
du dv
F(u, v)e
g(x , y ) ¼
ðð
¼

f(x, y)h(x0 À x, y0 À y)dx dy

(4:34)

where
ðð
h(x, y) ẳ

ei2p



uxỵvyỵ

p

2
1=l u2 v2 z

du dv

(4:35)

Equation (4.35) integrates in closed form to yield [21]
h(x, y) ẳ

!
p
d
i
l
2
2
2
ei(2p=l) d ỵx ỵy
ỵ
l (d2 þ x2 þ y2 ) 2p (d2 þ x2 þ y2 )3=2

(4:36)

Given f(x, y), it is not difficult to numerically apply Eqns. (4.30) and (4.34) to calculate the diffracted field. In paraxial systems such that u, v ( w over the spatial
bandwidth of the field, we assume that p x % i x and p y % i y. Under this assumption
the polarization components are independent of u and v and Eqn. (4.34) reduces to
independent scalar equations for each polarization. Accordingly, we base our model
for diffraction in the remainder of the text on the scalar transformation
ðð

f (x, y)h(x0 À x, y0 À y)dx dy
(4:37)
g(x0 , y0 ) ¼
Integration of Eqn. (4.37) using h(x, y) as given by Eqn. (4.36) is a bit tricky, but
one can numerically model diffraction by applying the transfer function using the
methods described in Section 3.7. In analytic work, however, one generally


4.4

DIFFRACTION

111

chooses to work with a simplified approximate impulse response. As an example,
l ( d in essentially all optical systems, meaning that the 1/ld term in Eqn. (4.36)
dominates the 1/d 2 term. In imaging system analysis, the impulse response is
often simplified by the Fresnel (near-field) approximation or the more restrictive
Fraunhofer (far-field) approximation. Both approximations are paraxial, meaning
that we restrict our attention to field distributions over the space close to the axis
of optical propagation (the z axis in Fig. 4.2).
The Fresnel approximation is just the paraxial approximation that d ) jx À x0 j,
jy À y0 j for all x, y and x0 , y0 of interest. In this case
h(x, y) %

1 i(2pd=l) i(p=ld)x2 ỵy2 ị
e
e
ild

(4:38)

Under the Fresnel approximation, diffraction in homogeneous isotropic space is
described by a 2D version of the Fresnel transform discussed in Section 3.5 evaluated
pffiffiffiffiffiffi
at t ¼ ld. Noting from Fig. 3.7 that the Fresnel transformation produces significant
blurring for Gaussian features of width D when t/D . 1, one might expect features
of size D to blur on propagation at distances greater than d ¼ D2/l. This suggests
that wavelength scale features will blur quite rapidly on diffraction. Features with
an initial scale of 10 wavelengths blur in 100 wavelengths, while features on a
scale of 100 wavelengths blur in 10,000 wavelengths. This effect is illustrated
in Fig. 4.3, which shows diffraction of Gaussian spots of various sizes. Notice,
however, that high-frequency features reappear at 10 mm as a result of interference
between the diffracting spots. Such interference appears in the diffraction of coherent
laser fields, but is not observed in the diffraction of incoherent fields.
Figure 4.3 was generated using numerical analysis in Matlab. The figure used a
2 Â 2-mm spatial window sampled with 1024 Â 1024 pixels. The Fresnel transfer
function multiplied the DFT of the input field and an inverse DFT was used to generate the diffracted field. Of course, numerical analysis is not necessary for analysis of
diffraction of these particular sources because, as discussed in Section 3.5, Hermite –
Gaussian distributions are eigenfunctions of the Fresnel transform. According to
Eqns. (3.76) and (4.38), if the input field f (x, y) ¼ fn (x=w0 )fm ( y=w0 ) for real w0,
then the diffracted field is
2

g(x, y) ¼

ei(p=4) ei(2pd=l) ei(nỵmỵ1)arctanld=w0 ị ipẵ(x2 ỵy2 )ld=(w4 ỵl2 d2 )
0
q
e

w4 ỵ l2 d 2
0
0
1 0
1
xw0
xw0
B
C B
C
fn @qAfm @qA
2 2
4 ỵ l2 d 2
w4 ỵ l d
w0
o

(4:39)

With specic reference to the Gaussian input spots of Fig. 4.3, the diffracted field for
the input distribution f (x, y) ẳ exp (p(x2 ỵ y2 )=w2 ) is
0
g(x, y) ẳ

ei(p=4) i(2pd=l) p ẵ(x2 ỵy2 )=(w2 ỵild)
0
e
e
w2 þ ild
0

(4:40)


112

WAVE IMAGING

2

2

2

Figure 4.3 Absolute values of the diffracted field for the Gaussian spots ep ẵ(x ỵy )=20 ,
2
2
2
2
2
2
2
2
2
ep fẵ(x100) þy Š=50 g , eÀp f½(xþ200) þy Š=100 g , and ep fẵx ỵ(y200) =250 g under the Fresnel approximation for various diffraction distances. All units are in microns and l ¼ 1 mm.

Additional interesting Fresnel diffraction effects are observed in Figs. 4.4 and 4.5.
These figures were generated from the same spatial window and sampling as above,
but the images are zoomed to focus on features of interest. Figure 4.4 is a harmonic
field modulated by a Gaussian envelope. Note that the harmonic features do not blur

(features with the same frequency are present at all diffraction lengths). At d ¼ 10 mm,
the diffracted field has begun to separate horizontally into multiple images of the
Gaussian envelope and harmonic modulation is observed only in the interference
between the separating spots, not within individual spots.
Figure 4.5 is a chirped harmonic field modulated by a Gaussian envelope. For this
input, the diffracting field sharpens to a focus at d ¼ 2 mm rather than blurring on
propagation. After the focus, the field blurs. In both Figs. 4.4 and 4.5 it is interesting
to note that blur is not a fundamental process of diffraction for coherent fields. In fact,
a diffracting coherent field maintains its spatial frequency bandwidth on propagation.


4.4

113

DIFFRACTION

2

Figure 4.4 Absolute value of the diffracted field for f (x, y) ¼ eÀp [(x
cos (0:05p x)]. All units are in micrometers; l ẳ 1 mm.

ỵy2 )=2502

ẵ1 ỵ

Blur in the normally observed sense of optical fields is a property of partially coherent
or incoherent fields, as discussed in Chapter 6.
As illustrated in Fig. 4.4, however, Fourier components of diffracting fields tend to
separate on propagation. This effect is easily explained in the context of Fraunhofer

diffraction theory. Fraunhofer diffraction is most easily derived from the integral form
of Fresnel diffraction
g(x0 , y0 ) ẳ

ei(2pd=l)
ild



ei(p=ld) [(x x0 )2 ỵ ( y y0 )2 ] f (x, y)dx dy




ei(2pd=l) ei(p=ld) (x0 2 ỵ y0 2 )
xx0 ỵ yy0
exp i2p
ẳ
ild
ld
h p
i
exp i (x2 þ y2 ) f (x, y)dx dy
ld

(4:41)


114

WAVE IMAGING

Figure 4.5 Absolute value of the diffracted field for f (x, y) ẳ ep ẵ(x
(5p104 x2 ). All units are in micrometers; l ẳ 1 mm.

2

ỵy2 )=2502

ẵ1 ỵ cos

Assuming that x 2 ( ld and y 2 ( ld over the support of f(x, y), we may drop the
second exponential in the integrand to obtain
g(x0 , y0 ) %

ei(2pd=l) ei(p=ld)x
il d

0 2 ỵy0 2


ị 
x0
y0
^
f uẳ
,vẳ
ld
ld

(4:42)

meaning that the diffracted field is proportional to the Fourier transform of the input
field. Note that the Fraunhofer assumption is quite restrictive, however. For example,
a 100 wavelength scale input must diffract for well over 10,000 wavelengths to reach
the Fraunhofer regime and a 1000 wavelength feature, for well over 1,000,000 wavelengths. Fraunhofer diffraction is nevertheless often useful in determining the rough
size and spatial frequency structure of objects. The Fraunhofer assumption is commonly applied at opposite ends of the electromagnetic imaging frequency scale,
such as in X-ray crystallography and radio astronomy.


4.5

4.5

WAVE ANALYSIS OF OPTICAL ELEMENTS

115

WAVE ANALYSIS OF OPTICAL ELEMENTS

Chapter 2 considered the use of optical elements to shape the mutual visibility of
source points and detection points. The visibility function, renamed the impulse
response or point spread function, remains of central interest under the wave model.
The goal of optical sensor design is to use optical elements to program the impulse
response, within physical constraints, to usefully encode target object features into
detected data.
This section presents wave models for the optical elements that we described using
geometric models in Section 2.2. In addition, we consider diffractive optical
elements, which cannot be described by ray models. As in Section 2.2, analysis of
refraction and reflection at dielectric interfaces is a good starting point for optical

element analysis. In analogy with Fig. 2.4, the effect of a dielectric interface on a
plane wave is illustrated Fig. 4.6. A plane wave is incident on the interface in a
medium of index of refraction n1. The incident field is Ei (r) ¼ Eiexp(2piui . r).
The incident wave is refracted at the interface into the second medium of index of
refraction n2, and a reflected wave is returned into the first medium. The refracted
and reflected fields are Et (r) ¼ Et exp(2piut . r) and Er (r) ¼ Er exp(2pi ur . r).
Boundary conditions derived from the Maxwell equations determine the relative
amplitudes of these waves. The boundary conditions may be stated as follows:
†

†

Vector components of E and H that lie in the plane of the interface are
continuous.
Vector components of D and B normal to the plane of the interface are
continuous.

In both cases we assume that there are no surface charges or currents, which is always
the case at optical frequencies. These boundary conditions are used in standard texts

Figure 4.6 Refraction of a plane wave at a planar interface.


116

WAVE IMAGING

on optics and electromagnetics to relate the amplitudes of the refracted and reflected
waves to the amplitude of the incident wave. Typically, the power in the reflected
wave at a dielectric interface is a few percent of the incident power, and most of

the power is transmitted. Thin film layers are often used to encode the impedance
at the interface to suppress or enhance reflection.
It is not necessary to model reflection and refraction in detail to understand the
functional utility of optical elements in shaping the impulse response. The most
important features from a wave perspective are obtained simply by noting that the
functional form of the wave distribution must be maintained on both sides of the
interface for the boundary conditions to be satisfied. To satisfy the boundary
conditions in the plane of the interface, we require that [Ei (r) ỵ Er (r)] is ẳ
Et (r) Â is , for all r on the interface. is is the surface normal for the interface. To
satisfy this condition, one must require that
ui À ui Á is is ¼ ut À ut Á is is

(4:43)

In combination with the requirement that jutj ¼ n2/l, we find that
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n2 n2
ut ¼ ui ui is is ỵ is 2 1 þ ðui Á is Þ2
l2 l2

(4:44)

If we use angles relative to the surface normal to decompose ui and ut into transverse and longitudinal components, Eqn. (4.44) immediately reduces to Snell’s law
[Eqn. (2.5)]. In the the paraxial case is and ui are nearly collinear and Eqns. (4.44)
can be approximated by
ut % ui ỵ is

Dn"
n
l2 ui is

(4:45)

where Dn ẳ n2 2 n1 and n ẳ (n1 ỵ n2)/2.

As in Section 2.2, we first apply Snell’s law to the analysis of prism refraction.
As illustrated in Fig. 4.7, a prism consists of a series of two tilted planar interfaces.
The prism of Fig. 4.7 consists of a dielectric of index n2 embedded in a dielectric of
index n1. If i1 is the surface normal at the first interface of a prism and i2 the surface
normal at the second interface, recursive application Eqn. (4.45) produces an estimate
of the output wavevector u3
u 3 % u 2 À i2
% u1 ỵ

Dn"
n
l2 u2 i2

Dn"
n
(i1 i2 )
l

(4:46)