Resolution limits in practical digital holographic
systems
Damien P. Kelly
Bryan M. Hennelly,
MEMBER SPIE
Nitesh Pandey, MEMBER SPIE
National University of Ireland, Maynooth
Department of Computer Science
Maynooth, Kildare County
Ireland
E-mail: damienpk@cs.nuim.ie
Thomas J. Naughton,
MEMBER SPIE
National University of Ireland, Maynooth
Department of Computer Science
Maynooth, Kildare County
Ireland
and
University of Oulu
RFMedia Laboratory
Oulu Southern Institute
Vierimaantie 5
84100 Ylivieska
Finland
William T. Rhodes,
FELLOW SPIE
Florida Atlantic University
Imaging Technology Center
777 Glades Road
Building 43, Room 486
Boca Raton, Florida 33431
Abstract. We examine some fundamental theoretical limits on the abil-
ity of practical digital holography 共DH兲 systems to resolve detail in an
image. Unlike conventional diffraction-limited imaging systems, where a
projected image of the limiting aperture is used to define the system
performance, there are at least three major effects that determine the
performance of a DH system: 共i兲 The spacing between adjacent pixels on
the CCD, 共ii兲 an averaging effect introduced by the finite size of these
pixels, and 共iii兲 the finite extent of the camera face itself. Using a theo-
retical model, we define a single expression that accounts for all these
physical effects. With this model, we explore several different DH record-
ing techniques: off-axis and inline, considering both the dc terms, as well
as the real and twin images that are features of the holographic record-
ing process. Our analysis shows that the imaging operation is shift vari-
ant and we demonstrate this using a simple example. We examine how
our theoretical model can be used to optimize CCD design for lensless
DH capture. We present a series of experimental results to confirm the
validity of our theoretical model, demonstrating recovery of super-
Nyquist frequencies for the first time.
© 2009 Society of Photo-Optical Instrumen-
tation Engineers. 关DOI: 10.1117/1.3212678兴
Subject terms: Holography; digital imaging; interference.
Paper 080958RR received Dec. 10, 2008; revised manuscript received Jun. 27,
2009; accepted for publication Jul. 7, 2009; published online Sep. 4, 2009.
1 Introduction
Holography is a technique that enables the magnitude and
phase of an optical field to be recorded.
1,2
Because record-
ing media, in general, are sensitive only to the intensity of
the light field incident upon them, a more complex optical
setup is required to record the phase of the incident field.
To record the phase information of the object field, it must
be interfered with a so-called reference beam at the record-
ing plane, thereby encoding the phase information as inten-
sity variations.
1,3–6
In traditional holography, these intensity
variations are recorded on a photosensitive material. Digital
holography 共DH兲 is an extension of this technique where
the photosensitive material is replaced with a digital
camera.
7–9
Recording the hologram electronically means
that this information can be stored and manipulated in real
time. The flexibility this allows the user in the recording,
processing, and remote replaying of holograms is one of the
primary advantages of a digital holographic approach. DH
has growing applications in 3D recording and display
devices,
10–12
as well as industrial metrology systems
13
with
potentially far-reaching implications for cellular and micro-
electromechanical systems imaging.
14,15
One significant
drawback of this technique however is the limited space-
bandwidth product
16
of the recorded digital holograms, due
in large part to the limited number of pixels in a typical
CCD camera. These cameras are discrete devices, and
therefore, sampling theory plays an important role in mod-
eling and optimizing DH setups. Other factors, such as the
finite size of the camera and the finite pixel size of the
individual detectors, both act to significantly limit the per-
formance of DH systems. This is not unexpected and has
been discussed by other authors 共see, for example, Refs.
17–21兲. What perhaps is not as well appreciated in the DH
community is that it is not necessary for the field in the
recording plane to be sampled at the Nyquist limit.
17–21
In
the following analysis, we examine lensless DH exclu-
sively.
In Fig. 1, a typical 共in-line兲 optical setup for performing
DH is illustrated. At the camera face, a reference wave and
a field that has been scattered from the object interfere and
the resulting intensity pattern is recorded by the camera.
This pattern contains the dc terms, the real and virtual im-
age. Typically, the user is interested in one of the image
terms and the other elements only act to degrade the quality
of the reconstructed hologram. Removal of the dc terms
and the virtual 共or real兲 image is thus an important practical
consideration in all holographic systems, and accordingly,
many different techniques have been proposed to get rid of
these unwanted terms.
9,22–24
Let us assume for the moment0091-3286/2009/$25.00 © 2009 SPIE
Optical Engineering 48共9兲, 095801 共September 2009兲
Optical Engineering September 2009/Vol. 48共9兲095801-1
that the complex field 共the real image term兲, u
z
共x兲, is avail-
able to us 共i.e., that we have somehow removed the dc and
virtual terms兲. We note then that once the complex field,
u
z
共x兲, at the camera plane has been obtained, the field at the
object plane, u 共X兲共the reconstructed image兲, can be calcu-
lated digitally using numerical techniques by a computer.
Suppose that this field u共X兲 has a maximum spatial fre-
quency given by f
max
. u共X兲 now propagates 共described un-
der paraxial conditions with the Fresnel transform兲 to the
camera face, where its complex amplitude, u
z
共x兲,isre-
corded. It is a property of the Fresnel transform that the
magnitude of the field’s spectral distribution remains in-
variant under propagation and so u
z
共x兲 must also have a
maximum spatial frequency f
max
共see Refs. 25 and 26兲.A
camera’s spatial frequency bandwidth, B
c
=2f
c
, and thus, its
ability to resolve a spatial frequency at the camera plane is
determined by the distance between the centers of adjacent
pixels on the camera face. Therefore, on initial consider-
ation, one might conclude that the camera must be able to
resolve f
max
共i.e., f
c
艌 f
max
兲 if the object field is to be recon-
structed properly. However, we note that several
authors
27–32
have shown that this sampling criterion may be
too strict and that it is possible to recover the object signal
when its Fresnel transform is sampled in the camera plane
共over an infinite extent兲 at a rate lower than the Nyquist
limit. These results are clearly of interest in DH and may
have practical implications: the camera could be placed
closer to the object with a resultant increase in numerical
aperture and, thus, an improvement in 3D perspective. In
Ref. 30, the authors describe recovery of these super-
Nyquist frequencies in terms of a generalized sampling
theory 共GST兲. In Ref. 29, the authors apply the GST to DH
systems however do not examine the effect of the finite
pixel size. In a later paper,
33
these authors do consider the
effect of pixel size, concluding that the maximum recover-
able frequency, f
max
=1/ 2
, where 2
is the width of the
pixel 关see Eq. 共7兲 in Ref. 33兴. As we shall see, however, by
suitably designing a camera, this limitation can be over-
come. In Refs. 20 and 21 resolution limits in DH systems
are also examined, however only for signals that are con-
sidered well sampled in the Nyquist sense. In this paper, we
develop a theoretical model that describes the limitations
on resolution that are imposed by 共i兲 the finite extent of the
camera, 共ii兲 the sampling rate, and 共iii兲 the finite extent of
the pixels. We note that if the camera pixels can be consid-
ered point detectors 共described by a comb of Dirac
␦
func-
tions兲, then the analysis we present for the real image term
reduces to that discussed in Ref. 34. Also, in Refs. 20 and
21 the authors derive a similar expression to that discussed
in Ref. 34. In our model, we treat the averaging effect in-
troduced by finite-size pixels in a different manner to that
discussed in Refs. 20, 21, and 34, providing an alternative
means of understanding this effect.
The paper is organized as follows; In Sec. 2, we develop
a theoretical model describing the imaging process that in-
corporates 共i兲 the finite extent of the CCD, 共ii兲 the reduction
of power in higher spatial frequencies due to averaging
introduced by rectangular size pixels, and 共iii兲 the effective
sampling rate imposed by the spacing between adjacent
pixels on the CCD face. We specifically examine the dc
terms and the twin image. Using this theoretical framework
in Sec. 3, we examine the predictions of our model for
several different recording architectures, off-axis and in-
line, and show that recovery of super-Nyquist frequencies
is not possible using an off-axis configuration. We then
examine how to optimally design CCD sensors to resolve
frequencies much higher than the Nyquist limit and that
limit defined in Ref. 33, by balancing these three different
effects 关i.e., 共i兲, 共ii兲, and 共iii兲兴. In Sec. 4, we demonstrate
that the imaging operation performed by a DH system is a
shift variant using a simple example. In Sec. 5, we provide
a series of experimental results that clearly demonstrate the
limitations on resolution in a DH system imposed by the
three different factors identified in our theoretical model.
We provide what we believe is the first experimental evi-
dence demonstrating that we can recover frequencies above
the Nyquist limit of the CCD camera. A fast reconstruction
algorithm based on the fast Fourier transform 共FFT兲 is dis-
cussed. Finally, we close the paper with a brief conclusion.
2 Theoretical Analysis
In Fig. 1, an object is illuminated with a temporally and
spatially coherent monochromatic plane wave. We describe
the resulting scattered field at plane X 共see Fig. 1兲 by the
function u共X兲. This field then propagates to the camera
plane 共located in the plane z=z
c
兲, where it interferes with a
reference wavefield u
R
共x兲, and the resulting intensity is re-
corded by the CCD. Through a numerical reconstruction
process, where we simulate free-space propagation back to
the object plane 共plane X, see Fig. 1兲, we can approxi-
mately recover u共X兲. There are several features of the re-
cording process, however, that limit the accuracy of our
recovered signal: 共i兲 The finite extent of the camera, 2D,
共ii兲 the spacing between the centers of adjacent pixels, T,
and 共iii兲 the finite extent, 2
of the pixels themselves. In
this section, we investigate each of these effects. We begin
by writing the continuous and instantaneous field intensity,
I
c
共x ; t兲, at the camera face as
17,18,35
I
c
共x;t兲 = 兩u
z
共x兲 + u
R
共x兲兩
2
= I
z
共x兲 + I
R
共x兲 + u
z
*
共x兲u
R
共x兲 + u
z
共x兲u
R
*
共x兲, 共1兲
where I
z
共x兲 and I
R
共x兲 are the intensities of the object and
reference fields, respectively, and are referred to as the dc
terms. The two latter terms in Eq. 共1兲 contain the virtual
Fig. 1 Schematic depicting a typical inline DH setup: M, mirror; P,
polarizer; BS, beamsplitter; Ph, pinhole; L lens; and MO, microscope
objective.
Kelly et al.: Resolution limits in practical digital holographic systems
Optical Engineering September 2009/Vol. 48共9兲095801-2
and real images, respectively, and the superscripted asterisk
denotes the complex conjugate operation. Under paraxial
conditions, we may relate the field u
z
共x兲 to u共X兲 using a
Fresnel transform,
3
which we define as
u
z
共x兲 =
z
兵u共X兲其共x兲,
u
z
共x兲 =
1
冑
jz
冕
u共X兲exp
冋
j
z
共x − X兲
2
册
dX, 共2兲
u共X兲 =
1
冑
− jz
冕
u
z
共x兲exp
冋
− j
z
共X − x兲
2
册
dx, 共3兲
where
is the Fresnel transform operator. In Eq. 共3兲, the
inverse operation is defined. In what follows, we drop the
constant term 1 /
冑
⫾jz from Eqs. 共2兲 and 共3兲. The CCD
has a finite number of pixels, with an assumed uniform
structure, that average the light energy incident upon them
共see Chap. 9 of Ref. 36兲. The measured quantity is referred
to as the integrated intensity and has units of energy. A
camera consists of an array of N ⫻ N pixels separated from
each other by a distance T. We can represent 共in one dimen-
sion兲 the integrated intensity array using the vector
W
n
= 关W
1
,W
2
, ... ... . . W
N
兴 = W共x
⬘
;t兲
␦
T
共x
⬘
兲p
D
共x
⬘
兲, 共4兲
where p
D
共x
⬘
兲=1, when 兩 x
⬘
兩⬍D and is 0 otherwise. Thus,
p
D
共x
⬘
兲 is an aperture function that defines the extent of the
camera face. The function
␦
T
共x兲 in Eq. 共4兲 is a train of
Dirac
␦
functions
37
and is defined as
␦
T
共x兲=兺
n=−⬁
⬁
␦
共x
−nT兲. Because the function
␦
T
共x兲 is periodic, it may also be
expressed mathematically using a Fourier series
representation,
38,39
␦
T
共x兲 =
1
T
兺
n=−⬁
⬁
exp
冉
j2
nx
T
冊
. 共5兲
The function W共x
⬘
;t兲 is continuous and is related to the
temporally and spatially varying intensity I
c
共x ; t兲 at the
camera face by
W共x
⬘
;t兲 =
1
2
冕
t
t+⌬t
冕
x
⬘
−
x
⬘
+
I
c
共x;t兲dxdt, 共6兲
where ⌬t is the integration time of the camera and 2
is the
width 共area兲 of the pixel that is sensitive to light, and is
related to the fill factor of the camera;
艋T/ 2.
17,18,35,40
If
we assume a stationary object and note that the illumination
source is coherent and monochromatic, then the intensity of
the light field will not vary over the integration time of the
camera. Thus, we need only consider the spatial variation
of intensity over each pixel area, and thus, we rewrite Eq.
共6兲 as
W共x
⬘
兲 =
C
2
冕
x
⬘
−
x
⬘
+
I
c
共x兲dx, 共7兲
where C is an unimportant constant. We now reinterpret Eq.
共7兲 as a convolution relation;
W共x
⬘
兲 =
C
2
冕
−⬁
⬁
p
共x − x
⬘
兲I
c
共x兲dx
=
1
2
I
c
共x
⬘
兲 ⴱ p
共x
⬘
兲, 共8兲
where p
共x
⬘
兲=1, when 兩x
⬘
兩⬍
and is 0 otherwise, and
where ⴱ indicates a convolution operation. Substituting
Eqs. 共1兲 and 共8兲 into Eq. 共4兲 and dropping the unimportant
scaling constant C, we arrive at the following result:
20,21,34
W
n
=
1
2
p
D
共x兲
␦
T
共x兲兵 p
共x兲 ⴱ 关I
z
共x兲 + I
R
共x兲 + u
z
*
共x兲u
R
共x兲
+ u
z
共x兲u
R
*
共x兲兴其, 共9兲
where for notational simplicity, we set x
⬘
→ x. Thus, we see
that the discrete vector of values returned by the camera
arise due to contributions from four separate sources: the
two dc terms and both the real and twin image. If we sim-
ply apply an inverse Fresnel transform to Eq. 共9兲, then we
find that we will indeed arrive at an approximation to u共X兲,
which arises due to the contribution of the real image 关i.e.,
the fourth term in Eq. 共9兲兴; however, this result will, in
general, be effected by the contributions of both the dc and
twin image terms. We note that the Fresnel operation is
linear, and thus, in the following subsections, we consider
the terms in Eq. 共9兲, separately.
2.1 DC Terms
The dc terms in Eq. 共9兲 can be removed by recording the
intensities of the reference and object fields separately and
then subtracting them digitally from the captured hologram.
We note that other numerical approaches can also be used
to suppress these terms.
8,41
Phase-shifting interferometric
techniques can also be used to remove both the twin image
and the dc terms. Although these dc terms do effect the
quality of the reconstructed hologram, they are relatively
unimportant compared to the behavior of the twin and real-
image terms.
2.2 Real Image Term
We now turn our attention to numerically reconstructing an
approximation to the continuous field u共X兲. First, however,
we simplify the fourth term in Eq. 共9兲 further by assuming
an ideal unit amplitude in-line reference beam, u
R
共x兲
=exp关j2
共z − z
c
兲/ 兴. Setting z = z
c
and applying an inverse
Fresnel transform on the real-image term, we get the fol-
lowing result for the reconstructed image:
u
s
共X兲 =
1
2
冕
关u
z
共x兲 ⴱ p
共x兲兴
␦
T
共x兲p
D
共x兲
⫻exp
冋
− j
z
共X − x兲
2
册
dx. 共10兲
Using the results from Appendix A, we can rewrite Eq. 共10兲
as
Kelly et al.: Resolution limits in practical digital holographic systems
Optical Engineering September 2009/Vol. 48共9兲095801-3
u
s
共X兲 = exp
冉
− j
z
X
2
冊
冕
u共X
1
兲exp
冉
j
z
X
1
2
冊
⌿共X,X
1
兲dX
1
,
共11兲
where
⌿共X,X
1
兲 =
冕
B共x,X
1
兲
␦
T
共x兲p
D
共x兲exp
冋
j2
z
x共X − X
1
兲
册
dx
=
冕
−D
D
B共x,X
1
兲
␦
T
共x兲exp
冋
j2
z
x共X − X
1
兲
册
dx, 共12兲
and where
B共x,X
1
兲 =
1
2
冕
−
exp
冉
j
z
u
2
冊
exp
冋
− j2
z
共x − X
1
兲u
册
du.
共13兲
From Eqs. 共10兲–共13兲, we can see that there is a complex
interaction between the finite camera extent p
D
共x兲, the sam-
pling rate
␦
T
共x兲, and the averaging due to the finite pixel
extent p
共x兲. In order to gain some insight into how these
different factors effect the numerically calculated recon-
struction, u
s
共X兲, we apply a series of limiting operations to
Eq. 共10兲. Initially, we will allow the camera extent ap-
proach infinity 共i.e., D → ⬁ 兲 and the finite extent of the
pixels to approach Dirac
␦
functions 共i.e.,
→ 0兲. Following
this, we examine, separately, how the finite camera aperture
and how the finite pixel size change this initial result. Fi-
nally, we will discuss the interaction of all three factors.
2.2.1 Infinitely large camera face with infinitely
narrow pixels: D→ ⬁,
→ 0
Making use of Eq. 共5兲, letting D→ ⬁ and
→ 0 reduces Eq.
共10兲 to the following:
u
s
共X兲 =
1
T
冑
− jz
冕
u
z
共x兲
␦
T
共x兲exp
冋
− j
z
共X − x兲
2
册
dx
=
1
T
冑
− jz
兺
n=−⬁
⬁
冕
u
z
共x兲exp
冉
j2
nx
T
冊
⫻exp
冋
− j
z
共X − x兲
2
册
dx. 共14兲
We also note the shifting property
27,42
of the Fresnel trans-
form 共for an arbitrary linear phase
兲, for some analytical
signal f共X兲,
z
兵f共X兲exp共j2
X兲其共x兲
= exp
冉
− j
2
z
冊
exp共j2
x
兲
z
兵f共X兲其共x −
z兲. 共15兲
Combining the results from Eqs. 共14兲 and 共15兲, we arrive at
u
s
共X兲 =
1
T
冑
− jz
兺
n=−⬁
⬁
冕
u
z
共x兲exp
共
j2
nx/T
兲
⫻exp
冋
− j
z
共X − x兲
2
册
dx,
u
s
共X兲 =
1
T
兺
n=−⬁
⬁
−z
再
u
z
共x兲exp
冋
j2
冉
n
T
冊
x
册
冎
共X兲,
u
s
共X兲 =
1
T
兺
n=−⬁
⬁
exp
冋
− j
共n/T兲
2
z
册
exp
共
j2
Xn/T
兲
⫻
−z
兵u
z
共x兲其
冉
X −
nz
T
冊
. 共16兲
Thus, from Eq. 共16兲, we can relate u
s
共X兲 to the actual field
u共X兲. The sampling process however causes differences be-
tween the actual signal and our approximation to it. We
note several points in relation to this: 共i兲 the sampling pro-
cess creates an infinite number of replicas in the object
plane, 共ii兲 the centers of adjacent replicas are separated by a
distance z / T, 共iii兲 each of the replicas is also multiplied by
a different linear phase as well as some unimportant con-
stant phase factor.
If we impose the constraint that our object 共field兲 has a
finite support ⌬ in the object plane, then this field can be
imaged without overlapping replicas provided that T
艋共z兲/ ⌬. This important result is known
27,28,30,32–34,43
and
means that, under certain conditions, it is possible to
sample the diffracted field at rates below the Nyquist limit
and to recover, through a generalized interpolation formula,
super-Nyquist frequencies. In Ref. 26, some implications of
this result are explored in more detail using a simple ana-
lytical example. As we will see, however, the effect of the
finite pixel size and camera extent impose resolutions limits
in addition to this constraint.
2.2.2 Infinitely large camera face and averaging
due to finite pixel extent: D → ⬁
In this section, we look at the effect of averaging due to the
finite extent of the pixels in the camera plane and examine
how this impacts on our reconstructed approximation u
s
共X兲.
In particular, we note the effect on the resolution obtainable
in DH systems. If we now include the effect of pixel aver-
aging in our analysis, then we replace u
z
共x兲 with
u
z
共x兲ⴱ p
共x兲 in Eq. 共16兲, to give
u
s
共X兲 =
1
T
兺
n=−⬁
⬁
exp
冋
− j
共n/T兲
2
z
册
exp共j2
Xn/T兲
⫻
−z
兵u
z
共x兲 ⴱ p
共x兲其
冉
X −
nz
T
冊
. 共17兲
We now consider the inverse Fresnel operation
Kelly et al.: Resolution limits in practical digital holographic systems
Optical Engineering September 2009/Vol. 48共9兲095801-4
−z
兵u
z
共x兲 ⴱ p
共x兲其共X兲 = u
z
共x兲 ⴱ p
共x兲 ⴱ exp
冉
− j
z
x
2
冊
= u
z
共x兲 ⴱ exp
冉
− j
z
x
2
冊
ⴱ p
共x兲
= u共X兲 ⴱ p
共X兲. 共18兲
The conclusion we draw from this result is that the averag-
ing introduced by the finite size pixels acts to degrade the
quality of the reconstructed hologram by convolving it with
a narrow rectangular function that is the same size as the
pixel. As
→ 0, this rectangular function narrows, reducing
the distorting effect that it has on the reconstruction. Pro-
vided that the distribution u共 X兲 is approximately constant
over the width of the function, p
共X兲, there will be little
distortion of u共X兲. However, if we are attempting to recover
spatial frequencies higher than the Nyquist limit, then u共X兲
will, by definition, vary significantly over the width of the
pixel and thus will act to make these distortions increas-
ingly pronounced. We also note that the effect of convolv-
ing p
共X兲 with u共X兲 is to increase the spatial extent of the
resultant signal from ⌬ to ⌬+2
. Thus, this new signal may
be recovered provided that T 艋 共z兲/ 共⌬+2
兲.
We may also examine the effect of the convolution of
p
共X兲 with u共X兲 in the spatial frequency domain. From Ref.
37 共p. 128兲, we find that the Fourier transform of p
共X兲 is
given by
F兵p
共x兲其共f
x
兲 =
冕
p
共x兲exp共− j2
xf
x
兲dx
F兵p
共x兲其共f
x
兲 = sinc关2
f
x
共
兲兴, 共19兲
where sinc共x 兲= sin共x兲/ x. Therefore, the effect of the convo-
lution operation is to multiply the spatial frequency content
of the signal, F兵u共X兲其共f
x
兲 by a sinc function. We note that
for values of f
x
such that f
x
=n/ 共2
兲, then Eq. 共19兲 is zero,
and therefore, these spatial frequencies will be entirely re-
moved from the signal. A more detailed interpretation of
this result is discussed in Ref. 26.
2.2.3 Finite camera extent, neglecting averaging
due to pixels:
→ 0
We now examine the third factor that impacts on the quality
of our reconstructed hologram u
s
共X兲, the finite extent of the
camera. For simplicity, we assume that we are sampling
with point detectors 共i.e., we allow
→ 0兲. In this instance,
Eq. 共12兲 can be expressed as
34
⌿共X,X
1
兲 =
冕
−D
D
␦
T
共x兲exp
冋
j2
z
x共X − X
1
兲
册
. 共20兲
Using Eq. 共5兲 in conjunction with the Fourier shift theorem
共see Ref. 37,p.104兲, we find that
⌿共X,X
1
兲 =
1
T
兺
n=−⬁
⬁
sinc
冋
2
D
z
冉
X − X
1
−
zn
T
冊
册
. 共21兲
Subbing Eq. 共21兲 into Eq. 共11兲
u
s
共X兲 = exp
冉
− j
z
X
2
冊
1
T
兺
n=−⬁
⬁
冕
u共X
1
兲exp
冉
j
z
X
1
2
冊
⫻sinc
冋
2
D
z
冉
X − X
1
−
zn
T
冊
册
dX
1
=exp
冉
− j
z
X
2
冊
1
T
兺
n=−⬁
⬁
R共X,n兲, 共22兲
where
R共X,n兲 = u共X兲exp
冉
j
z
X
2
冊
ⴱ sinc
冋
2
D
z
冉
X −
zn
T
冊
册
. 共23兲
Thus, we can see from Eqs. 共22兲 and 共23兲 that the effect of
the finite camera extent is to reduce the resolving ability of
the DH system by convolving the product of the initial
input and a quadratic phase term, with a sinc function,
whose width is determined by the wavelength of the light,
the size of the aperture, and the distance the camera is
placed from the object plane. However, as we shall shortly
demonstrate in Sec. 4, it is important to note that this “con-
volution” relationship is not a shift-invariant operation due
to the presence of the quadratic phase factor. Nevertheless,
as a useful “rule-of-thumb” approximation that we use later
when examining experimental results, it is convenient to
reinterpret Eq. 共23兲 in the spatial frequency domain as a
low-pass filtering operation,
R共X,n兲 = F
−1
再
F
冋
u共X兲exp
冉
j
z
X
2
冊
册
P
D/z
共
v
兲
⫻exp
冉
j2
v
nD
T
冊
冎
共X兲, 共24兲
where P
D/z
共
v
兲=1, when 兩
v
兩⬍D/ z and is 0 otherwise, and
where F and F
−1
indicate Fourier and inverse Fourier trans-
form operations.
Again assuming that our input field u共X兲 has a finite
spatial extent ⌬, we can see from Eq. 共23兲 that this input
extent will be increased due to the presence of the sinc
function. Strictly speaking, a sinc function spans an infinite
spatial extent, implying that our recovered signal will in-
evitably suffer from aliasing. Practically, however, we may
assume that the sinc function can be effectively limited in
space. We therefore define the effective extent of the sinc
function, ⌬
S
, as being twice the distance from its maximum
value to its first null in keeping with the analysis presented
in Ref. 34 共i.e., ⌬
S
=z / D兲. Therefore, in order to ensure
the successive replicas do not overlap, we require that T
艋共z兲/ 共⌬ + ⌬
S
兲.
2.2.4 Finite camera extent and averaging due to
pixels
In this section, we look at how all three factors interact with
each other to limit the resolution of a practical DH system.
We begin by examining Eq. 共13兲 and discuss how we may
simplify the expression considerably. Substituting this sim-
plified expression into Eqs. 共11兲 and 共12兲, we then investi-
gate how finite pixel size and the finite extent of the camera
limit the resolution of the imaging system, identifying the
Kelly et al.: Resolution limits in practical digital holographic systems
Optical Engineering September 2009/Vol. 48共9兲095801-5
