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Optics Communications 249 (2005) 397–405
www.elsevier.com/locate/optcom
Fresnel similarity
Adolf W. Lohmann, Jorge Ojeda-Castaneda *
Lehrsthul fur Multimediakommunikation und Signalverarbeitung,Universitat Erlangen-Nurnberg Cauerstrabe 7,
D-91058 Erlangen, Germany
Department of Physics and Mathematics, School of Science, University of las Americas, Apartado Postal 100, Cholula 72820,
Puebla, Mexico
Received 2 August 2004; received in revised form 17 December 2004; accepted 26 January 2005
Abstract
We study Fresnel diffraction, in particular the similarity between an object, at plane z = 0, and its diffraction pattern,
at z > 0. We define a degree of similarity, r(z), which depends on the object and the distance z. Perfect self-imaging is
associated with a degree of similarity equal to unity, while imperfect self-imaging is classified by a degree between zero
and one. We derive the theory that allows us to find r(z), if the object is known. And we can synthesize an object that
exhibits a wanted r(z). We describe some novel applications.
� 2005 Published by Elsevier B.V.
Keywords: Fresnel diffraction; Self-imaging; Extended depth of field; Axial imaging; Optical correlations; Apodizers
1. Introduction
For several applications in optical instrumenta-
tion, such as Talbot and Lau interferometry [1,2],
array illuminators [3,4], photolithography [5,6],
and for compression and expansion of the depth
of field [7–13], it is relevant to compare an input
transmittance with its Fresnel diffraction pattern.
0030-4018/$ - see front matter � 2005 Published by Elsevier B.V.
doi:10.1016/j.optcom.2005.01.042
* Corresponding author. Tel.: +522222292045; fax:
+522222292066.
E-mail address: jorge.ojeda@udlap.mx (J. Ojeda-Castane-
da).
Here, we aim to analyze quantitatively the blur-
ring due to Fresnel diffraction. To that end, we for-mulate a Fresnel diffraction theory in a way that
tells us how to manipulate the diffraction blur.
Our theoretical tool is a modified cross correlation
between the input object u0(x), at plane z = 0, and
its Fresnel diffraction pattern u(x,z), at plane
z > 0. This cross correlation is denoted as the sim-
ilarity function.
In Section 2, we begin by describing the under-lying fundamentals, and by standardizing the basic
terms. We also study some basic properties and
symmetries. We show how the power spectrum
SOURCE
(a)
398 A.W. Lohmann, J. Ojeda-Castaneda / Optics Communications 249 (2005) 397–405
of the object influences the degree of similarity.
Next, in Section 3, we analyze similarity propaga-
tion behind aperiodic objects. In Section 4, we
consider the Montgomery effect, which occurs
when studying the Fresnel diffraction of certainquasi-periodic objects. In Section 5, with the pur-
pose of understanding the impact of periodicity
on the similarity, we evaluate the similarity func-
tion of periodic structures. In Section 6, we discuss
some optical methods for measuring the similarity
function. And in Section 7, we consider 2-D ob-
jects with radial symmetry. We show that for this
type of objects the similarity function is a 1-DFourier transform of the angularly averaged
power spectrum. This latter result is related to
the design of annular apodizers for tailoring focal
depth.
(b)
Fig. 1. Optical setup for: (a) Fresnel diffraction with input
object u0(x), and Fresnel diffraction pattern u(x,z), (b) the
optical setup for measuring the similarity function. A pinhole at
the rear focal plane of the last lens implements the integration
of the product uðx; zÞu�0ðxÞ.
2. Underlying fundamentals
The optical setup that we have in mind is de-
picted in Fig. 1(a). A monochromatic plane wave
illuminates the object with complex amplitude
transmission u0(x). The second transversal coordi-nate y is of no interest for now. The complex
amplitude distribution, u(x, z), just behind the ob-
ject (at z = 0) is
uðx; 0Þ ¼ u0ðxÞ: ð1Þ
The complex amplitude in Eq. (1) acts as the refer-
ence signal for the definition of similarity functionas
SðzÞ ¼Z 1
�1uðx; zÞu�ðx; 0Þdx
¼Z 1
�1uðx; zÞu�0ðxÞdx; ð2Þ
where the limits of integration are ±1 for ape-
riodic objects and quasi-periodic objects, while
the limits are ±d/2 for periodic objects with per-
iod d.
In Fig. 1(b), we show an optical setup for imple-
menting the operation in Eq. (2) by using twomasks. In Section 6, we discuss other approaches
for evaluating the similarity function with a single
mask.
It is important to note that the value of the sim-ilarity function at the object plane, S(0), is the total
energy of the optical signal, which is a finite num-
ber. That is, we consider optical signals that are
bounded either in the space domain (windowed
signals), or in the frequency domain (band-limited
signals). In mathematical terms,
A.W. Lohmann, J. Ojeda-Castaneda / Optics Communications 249 (2005) 397–405 399
Sð0Þ ¼Z 1
�1juðx; 0Þj2 dx ¼
Z 1
�1ju0ðxÞj2 dx
¼Z 1
�1j~u0ðvÞj2 dv ¼ E: ð3Þ
Next, we show that when dealing with optical sig-
nals that are infinitely extended, the bias term must
be zero. That is, if
u0ðxÞ ¼ ½hui þ DuðxÞ�; ð4Þ
with
hui ¼Z 1
�1u0ðxÞdx; ð5Þ
then, the condition in Eq. (3) implies that
E ¼Z 1
�1½jhuij2 þ jDuðxÞj2 þ 2Refhui�DuðxÞg�dx:
ð6ÞAnd therefore, to avoid an undefined value, we
must set
hui ¼Z 1
�1u0ðxÞdx ¼ 0; ð7Þ
and consequently
E ¼Z 1
�1jDuðxÞj2 dx: ð8Þ
Now, if the definition of similarity, in Eq. (1), is
normalized to stay within unity, 0 6 |r(z)| 6 1, wespeak of the ‘‘degree of similarity’’. For this pur-
pose we divide the original S(z) by S(0) = E. That
is, since |S(z)| 6 S(0) then
rðzÞ ¼ SðzÞ=Sð0Þ: ð9Þ
In Sections 3 and 4, we show that for quasi-
periodic and periodic signals the similarity func-tion is a periodic function, and that the degree of
similarity has the following property: r(nZT) =
r(0) = 1, where n is an integer number, and
zT = 2d2/k is the so-called Talbot distance.
Next, we recognize first that for the input
u0ðxÞ ¼Z 1
�1~u0ðvÞ expði2pxvÞdv; ð10Þ
its Fresnel diffraction pattern can be expressed in
terms of a Fourier transform as follows
uðx; zÞ ¼Z 1
�1~u0ðvÞ exp½i2pz
pðk�2 � v2Þ�
� expði2pxvÞdv: ð11Þ
Since in the paraxial regime one can approximatep(k�2�v2) � k�1 � k v2/2 then except for the
phase factor exp(i2pz/k), in the paraxial approxi-
mation equation (11) reduces to
uðx; zÞ ¼Z 1
�1~u0ðvÞ expð�ipkzv2Þ expði2pxvÞdv:
ð12Þ
Then, by substituting either Eq. (11) or Eq. (12) inEq. (2), one has that
SðzÞ ¼Z 1
�1j~u0ðvÞj2 exp½i2pz
Z 1
�1ðk�2 � v2Þ�dv
ð13Þ
or, except for the phase factor exp(i2pz/k), that inthe paraxial regime
SðzÞ ¼Z 1
�1j~u0ðvÞj2 expð�ipkzv2Þdv: ð14Þ
It is apparent from Eqs. (13) and (14) that the sim-ilarity function describes the Fresnel propagation
of the power spectrum |u0(v)|2.
The proposed measurement of similarity is
independent of the phase of the Fourier spectrum
u0(v). It is surprising because S(z) is not an auto-
correlation, but a cross correlation of two different
functions u(x, 0) and u(x, z).
We recognize that another important feature, ofEqs. (10) and (11), is that they allow for designing
similarity values by proper selection of the power
spectrum |u0(v)|2. A similar problem occurs when
synthesizing the optical transfer function by using
computer-generated holography; as is discussed in
Section 7.
Either from Eq. (13), or from Eq. (14), we can
identify the following symmetry
SðzÞ ¼ S�ð�zÞ: ð15Þ
In other words, the similarity function is a hermi-tian function both in the paraxial approximation
and beyond the paraxial approximation. Eq. (15)
also shows that |S(z)| 6 S(0).
400 A.W. Lohmann, J. Ojeda-Castaneda / Optics Communications 249 (2005) 397–405
We note also that the similarity function remains
invariant under the following transformations:
(a) S(z) does not change if one displaces later-
ally the input, namely
u0ðxÞ ! u0ðx� x0Þand
uðx; zÞ ! uðx� x0; zÞ:Since
~u0ðv; x0Þ ! ~u0ðvÞ expð�i2pvx0Þand
~uðv; x0Þ ! ~uðvÞ expð�i2pvx0Þ expð�ipkzv2Þ:Thus,
Sðz; x0Þ ¼Z 1
�1j~u0ðv; x0Þj2 expð�ipkzv2Þdv ¼ SðzÞ:
ð16Þ(b) If one takes the complex conjugate of the in-
put, namely
u0ðxÞ ! u�0ðxÞ;then
~u0ðvÞ ! ~u�0ð�vÞ;and then for the paraxial regime
SðzÞ ¼Z 1
�1j~u�0ð�vÞj2 expð�ipkzv2Þdv ¼ S�ð�zÞ:
ð17Þ(c) If one displaces longitudinally, say by z0, the
input as well as the second mask the function S(z)
does not change
u0ðx; z ¼ 0Þ ! u0ðx; z0Þ; and uðx; zÞ ! uðx; zþ z0Þ:ð18Þ
In other words, the similarity function depends
only on the difference of the axial coordinate z.
The similarity function is next applied in thefollowing sections, for analyzing some illustrative
cases.
3. Aperiodic objects
A general treatment of aperiodic objects is be-
yond our present scope. Here, we discuss a simple
example, within the paraxial approximation,
which illustrates the concept of similarity.
We consider a suitably modified Gaussian
beam. Its power spectrum is
j~u0ðvÞj2 ¼ jvj expð�pv2=a2Þ: ð19ÞThe scaling factor a�2 defines the width of the
Gaussian beam. For this example, if one considers
that z � k, then the similarity function is
SðzÞ ¼Z Na
�Najvj exp½�pða�2 þ ikzÞv2�dv;
SðzÞ ¼ 1=2
Z N2a2
0
exp½�pða�2 þ ikzÞv2�dðv2Þ
þ 1=2
Z N2a2
0
exp½�pða�2 þ ikzÞv2�dðv2Þ;
¼Z N2a2
0
exp½�pða�2 þ ikzÞv2�dðv2Þ:
ð20ÞIn Eq. (20) we set the upper limit equal to (Na)2
where N is a large integer number, such that
exp(�pN2)� 1. Then, within the approximation
that z � k, it is straightforward to evaluate Eq.
(20) to obtain
SðzÞ � 1=½a�2 þ ikz� � ½a�2 � ikz�=½a�4 þ k2z2�:ð21Þ
Now, since S(0) = a2 then the modulus of the de-
gree of similarity can be written as
jrðzÞj � 1=pð1þ a4k2z2Þ: ð22Þ
Eq. (22) represents a monotonically decreasing
function.
4. Quasi-periodic objects
Quasi-periodic objects have a power spectrumof the form
j~u0ðvÞj2 ¼ ð1=MÞXM�1
m¼0
jamj2d½v� vm�: ð23Þ
A subset of these quasi-periodic objects is the setof ‘‘Montgomery objects’’. Their power spectrum
peaks are located at
A.W. Lohmann, J. Ojeda-Castaneda / Optics Communications 249 (2005) 397–405 401
vm ¼ v1pm; and v1 ¼ 1=d: ð24Þ
We denote aspm the square root of m. Montgom-
ery identified this set, when describing the neces-
sary and sufficient conditions for self-imaging
[14]. The similarity function for the ‘‘Montgomery
objects’’ is
SðzÞ ¼ ð1=MÞXM�1
m¼0
jamj2 expð�ipkzm=d2Þ
¼ ð1=MÞXM�1
m¼0
jamj2 expð�i2pzm=ZTÞ; ð25Þ
which is to be recognized as a finite Fourier serieswith fundamental period equal to the Talbot
length ZT = 2d2/k. Hence, the similarity function,
as well as the degree of similarity, is a periodic
function. In mathematical terms,
rðzÞ ¼XM�1
m¼0
jamj2 exp �i2pzm=ZTð Þ" #,XM�1
m¼0
jamj2
ð26Þis a periodic function
rðzþ nZTÞ ¼ rðzÞ; ð27Þwith fundamental period ZT. Specifically, for
z = 0, we have that r(n ZT) = r(0) = 1.
We discuss next the behavior of the similarity
function at fractions; say 1/M, of the Talbot
length. Since we assume that z = n(ZT/M), forn = 0,1,2,. . .,M � 1, then Eq. (25) becomes
SðnZT=MÞ ¼ ð1=MÞXM�1
m�0
jamj2 expð�i2pmn=MÞ;
ð28Þwhich is to be recognized as the finite Fourier
transform of the coefficients |am|2. And therefore,
if the finite Fourier transform of the coefficient
am is the coefficient An,
An ¼ ð1=MÞXM�1
m�0
am expð�i2pmn=MÞ; ð29Þ
then the similarity function in Eq. (28) can be writ-
ten as the autocorrelation of the Am coefficients.
That is,
SðnZT=MÞ ¼XM�1
m�0
A�mAmþn: ð30Þ
A nice application of Eq. (30) is the so-called
pseudo-random sequence. By using a pseudo-ran-
dom sequence for Am one can shape (SnZT/M) to
be equal to 1 only if n = 0 and multiple numbers
of M. Otherwise S(nZT/M) should be equal tozero. And consequently, the degree of similarity
can be manipulated to be a Dirac�s comb, with per-
iod ZT. Next, we discuss the periodic case.
5. Periodic structures
Optical gratings are good examples of periodicinputs. Their power spectrum is represented by
j~u0ðvÞj2 ¼X1
m¼�1jamj2d½v� m=d�: ð31Þ
In Eq. (31) we disregard the evanescent waves by
limiting the summation index m to |m|<d/k = M.
In other words, we restrict the term ‘‘similarity’’
to distances beyond the reach of evanescent waves.
Often the grating may deflect only into a narrow
angular range |sinam|. In that case the z-dependent
part of the exponent (1 � cosa)z/k maybe approx-imated by (z/2k)sin2am = m2kzv2/2. In other words,
we invoke the paraxial approximation that simpli-
fies the analysis.
Hence, within the paraxial regime, the similarity
function is the finite Fourier series
SðzÞ ¼XMm¼�M
jamj2 expð�ipkzm2=d2Þ;
¼XMm¼�M
jamj2 expð�i2pzm2=ZTÞ: ð32Þ
We note that Eq. (32) agrees well with the predic-
tion made on symmetries in Eq. (15). Again, as in
the case of the ‘‘Montgomery objects’’, for a peri-
odic input the degree of similarity is unity at the
self-imaging planes, as was expected.
It is worth noting that for gratings with finiteextent, say with a width that is equal to Nd, one
has to consider the walk-off effect, as is depicted
in Fig. 4, see Ref. [15]. In this case there is an upper
limit for the value of z. It is straightforward to
show that the upper limit is
u0(x)
z0
.dx |.|2ADD
z + z(a)
(b)
0
402 A.W. Lohmann, J. Ojeda-Castaneda / Optics Communications 249 (2005) 397–405
Z ¼ Nðd2=kÞp½1� ðk=dÞ2�Z¼ ½MN
pð1�M�2Þ�d: ð33Þ
If one neglects the diffraction produced by the fi-
nite size of the grating, then at z = Z the non-zero
diffracting orders have left the transverse region of
interest. Under these conditions, the degree of sim-
ilarity is equal to unity. However, roughly speak-
ing, the walk-off effect is harmless if the numbers
of grating periods is much larger than the numberof diffraction orders.
u(x,z+z0)
z0
u0(x) u0(x,z0)
Fig. 3. (a) Block diagram for measuring interferometrically the
similarity function, (b) the optical setup for the previous
diagram.
6. Experimental measurement of similarity
The optical setup in Fig. 1(b) needs two masks.
One mask is for u0(x) and the other mask is for
u�0ðxÞ.In Fig. 2, we show schematically another setup,
which uses a similar arrangement as the joint
transform correlator [16,17]. As depicted in Fig.
2(a), the input is split into two arms. Along one
arm, the input is allowed to propagate an addi-
tional distance z. Then, we add the two beams.
And by using a square law detector, we record
the interference pattern to obtain
Iðx; zÞ ¼ ju0ðxÞexpð�i2pv1xÞþ uðx; zÞexpði2pv1xÞj2:ð34Þ
Under suitable holographic reconstruction, we can
obtain the cross product term Cðx; zÞ ¼
FIRST ORDER
ZERO ORDER
N
z = 0 z = Z
tan(β
β
) = Nd/Z
sin(β) = λ/d
Fig. 2. Schematic diagram of the walk-off effect, for identifying
the upper limit of z, when describing finite size gratings.
u�0ðxÞuðx; zÞ. Its Fraunhofer diffraction pattern, at
zero value, is the similarity function.For the above purpose almost any two-arm
interferometer can be employed. In Fig. 2(b), we
depict a Sagnac interferometer, where the carrier
frequency v1 = 0.
In Fig. 3(a), we depict another setup that resem-
bles the setup used to implement optically the Wig-
ner distribution function [18]. The input u�0ðxÞ is
allowed to propagate an additional distance z.Half the way, we use a retro-reflecting device to
impinge u*(�x,z) = u(x,z) on the input, and in this
manner we implement the product u�0ðxÞuðx; zÞ. ItsFraunhofer diffraction pattern, at zero value, is the
similarity function as in the previous case.
7. Inputs with radial symmetry
We consider 2-D inputs with radial symmetry.
The definition of similarity propagation, for 2-D
inputs is
u0(x)
.dxMULT
z0
z + z(a)
(b)
0
z = 0
Fig. 4. (a) Block diagram for measuring as a product the
similarity function, (b) the optical setup for the previous
diagram.
A.W. Lohmann, J. Ojeda-Castaneda / Optics Communications 249 (2005) 397–405 403
SðzÞ ¼Z 1
�1
Z 1
1uðx; y; zÞu�ðx; y; 0Þdxdy: ð35Þ
Since we are interested in radial symmetric objects,
we employ cylindrical coordinates
x ¼ r cosð/Þ; y ¼ r sinð/Þ;v ¼ q cosðhÞ; l ¼ q sinðhÞ; z ¼ z: ð36Þ
By using the above change of variables, we can re-
write Eq. (35) as
SðzÞ¼Z 1
0
Z 2p
0
uðr;/;zÞu�ðr;/;0Þrdrd/
¼Z 1
0
Z 1
2pj~u0ðq;hÞj2 exp½i2pz
pðk�2�q2Þ�qdqdh:
ð37Þ
For the paraxial regime, except for the phase fac-
tor exp(i2pz/k), Eq. (37) becomes
SðzÞ ¼Z 1
0
Z 2p
0j~u0ðq; hÞj2 expð�ipkzq2Þqdqdh:
ð38ÞNow, it is convenient to rewrite Eq. (38) as follows
SðzÞ ¼ pZ 1
0
½ð2pÞ�1
Z 2p
0
j~u0ðq; hÞj2 dh�
� expð�ipkzq2Þdðq2Þ: ð39Þ
It is apparent from Eq. (39) that, for the paraxial
regime, the similarity function is the 1-D Fourier
transform of the angular average power spectrum.
We can perform the angular averaging now. But
we postpone the averaging because the angular
structure of ~u0ðq; hÞ provides valuable design
parameters, as we describe next.
One can tailor the similarity function by usingannular shade binary apertures; as it happens
when designing apodizers for increasing the depth
of field [19].
If we consider the cut-off spatial frequency, X,of the radially symmetric pupil aperture, it is con-
venient to employ the change of variable
f ¼ ðq=XÞ2 � 1=2;
HðfÞ ¼ ð2pÞ�1
Z 2p
0
j~u0ðq; hÞj2 dh ð40Þ
for writing Eq. (39) as
SðzÞ¼ pX2 expðiWÞZ 1=2
1=2
HðfÞexp½�i2pðkX2=2Þzf�df;
ð41Þwhere W = �(pkX2/2)z. Hence, the degree of simi-
larity can be written as the modulus of a 1-D point
spread function, which results from taking the 1-DFourier transform of the effective optical transfer
function HðfÞ
jrðzÞj ¼Z 1=2
1=2
HðfÞ exp½�i2pðkX2=2Þzf�df����
����, Z 1=2
1=2
HðfÞdf����
����; ð42Þ
Now, we perform the angular average operation
that is expressed in Eq. (40). In Fig. 5 we depict
the following procedure. At every radius q, be-
Fig. 5. Schematic diagram for setting annular apodizers that manipulate the similarity function.
404 A.W. Lohmann, J. Ojeda-Castaneda / Optics Communications 249 (2005) 397–405
tween zero and X, we select a ring. Dividing the
ring into N angular sectors controls the light
throughput. Each angular sector is center at the
point (q,hn), where hn = (2n + 1)p/N. And the
width of the angular sector is made to be propor-
tional to the function HðfÞ. In mathematical terms,
j~u0ðq; hÞj2 ¼ðNÞ�1XN�1
n¼0
rect½ðh� hnÞ
=ð2p=NÞHððq=XÞ2 � 1=2; Þ�: ð43Þ
It is straightforward to show that indeed the angu-
lar average of the function j~u0ðq; hÞj2, in Eq. (43), is
the effective transfer function HðfÞ.In Table 1, we evaluate three examples of the
use of Eqs. (39) and (43). The first example is a cir-
cular pupil aperture that is diffraction limited. In
this example, it is easy to show that the degree of
similarity is a sinc function.
As a second example we consider an annularly
shaped apodizer that is used for increasing focal
Table 1
Tailoring the degree of similarity: illustrative examples
Fourier spectrum Effective tran
~u0ðq; hÞ ¼ rect½ðq� X=2Þ=X� HðfÞ ¼ rectðf~u0ðq; hÞ ¼ sin ðpðq=XÞ2Þ HðfÞ ¼ cos2ð
~u0ðq; hÞ ¼ expð�2pÞ expf�8pðq=XÞ2½ðq=XÞ2 � 1�g HðfÞ ¼ expð�
depth. It is straightforward to evaluate the similar-
ity function, which contains three replicated sinc
functions, along the z-axis. An interesting feature
is that the replicas are separated by the axial dis-
tance z = 2/kX2 where kis the wavelength of the
radiation and X is the cut-off spatial frequency.Since this axial distance is equal to the width of
the sinc function, then the replication can be
thought of as applying the Whitaker–Shannon the-
orem for interpolation.
Our final example is an annularly shaped
Gaussian apodizer, or if you will a doughnut
shape. For this example the degree of similarity
function has a Gaussian shape.Of course, a challenging problem is to achieve a
pre-specified |S(z)| by synthesizing a power spec-
trum j~u0ðq; hÞj. In this case, one has to use the in-
verse Fourier transform of Eq. (41), namely
HðfÞ ¼ ð1=pX2ÞZ 1
1jSðzÞj exp½i2pzf�dz: ð44Þ
sfer function Degree of similarity
Þ r(z) = sinc((kX2/2)z)
pfÞ r(z) = sinc[(kX2/2)z] + 1/2sinc[(kX2/2)z � 1]
+ 1/2sinc[(kX2/2)z + 1]
pð4fÞ2Þ r(z) = exp{�p[(kX2/8)2z]2}
A.W. Lohmann, J. Ojeda-Castaneda / Optics Communications 249 (2005) 397–405 405
And since from Eq. (15) we know that
S(z) = S*(�z) then HðfÞ is a real function. Hence,
by properly selecting the angular widths one can
synthesis HðfÞ.
8. Final remarks
We have proposed to measure the similarity of
a complex amplitude transmittance with its Fres-
nel diffraction pattern, at a distance z, by the use
of a modified cross correlation, which is denoted
as the similarity function S(z) = S*(�z).We have described schematically three types of
experimental setups for experimentally evaluating
S(z). For the first type, the optical setup uses two
masks, one mask for u0(x) and the other mask
for u�0ðxÞ. In the second type, we use the experi-
mental scheme of the joint transform correlation.
A convenient interference pattern is recorded,
and under holographic reconstruction we obtainthe similarity function. In the third type, we take
advantage of the symmetry properties of real
transmittance objects to implement the product
with a retro-reflecting device.
We have shown that under the paraxial regime,
the similarity function is the Fresnel diffraction
pattern of the power spectrum. For radially sym-
metric objects the similarity function can be ex-pressed as the 1-D Fourier transform of the
angular average power spectrum.
We have discussed specific applications that
illustrate the use of this concept. Specifically, we
have shown that for certain apodized Gaussian
beams the similarity function can be a monotoni-
cally decreasing function.
We have indicated that for certain quasi-peri-odic objects, as well as for periodic structures,
the degree of similarity is periodic function with
fundamental period equal to the Talbot length.
The degree of similarity can be shaped as a comb
function by using quasi-periodic structures related
to the pseudo-random sequences. For periodic ob-
jects with finite extension, we employ the concept
of the walk-off effect to identify the upper limitof z.
We have shown that for diffraction limited
apertures the degree of similarity is a sinc function.
We have discussed that certain apodizers can ex-
tend the depth of focus, by replicating the sinc
function along the z-axis. Since the replicated sinc
functions are separated axially by the sinc width,
then the depth of focus increases following theWhitaker–Shannon interpolation formula.
Finally, we have shown how to synthesize a
similarity function by means of binary screens with
angular sectors.
Acknowledgements
We are grateful to the reviewers for their helpful
suggestions. One of us (J.O.C.) is indebted to the
Alexander von Humboldt Foundation for finan-
cial support.
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