Purely numerical compensation for microscope objective phase curvature in digital holographic microscopy: Influence of digital phase mask position

Article (PDF Available)inJournal of the Optical Society of America A 23(11):2944-53 · December 2006with46 Reads
DOI: 10.1364/JOSAA.23.002944 · Source: PubMed
Abstract
Introducing a microscope objective in an interferometric setup induces a phase curvature on the resulting wavefront. In digital holography, the compensation of this curvature is often done by introducing an identical curvature in the reference arm and the hologram is then processed using a plane wave in the reconstruction. This physical compensation can be avoided, and several numerical methods exist to retrieve phase contrast images in which the microscope curvature is compensated. Usually, a digital array of complex numbers is introduced in the reconstruction process to perform this curvature correction. Different corrections are discussed in terms of their influence on the reconstructed image size and location in space. The results are presented according to two different expressions of the Fresnel transform, the single Fourier transform and convolution approaches, used to propagate the reconstructed wavefront from the hologram plane to the final image plane.
Purely numerical compensation for microscope
objective phase curvature in digital
holographic microscopy: influence of digital phase
mask position
Frédéric Montfort, Florian Charrière, and Tristan Colomb
Ecole Polytechnique Fédérale de Lausanne (EPFL), Institut d’Optique Appliquée, CH-1015 Lausanne, Switzerland
Etienne Cuche
Lyncée Tec SA, PSE-A, CH-1005 Lausanne, Switzerland
Pierre Marquet
Centre de Neurosciences Psychiatriques, Département de Psychiatrie DP-CHUV, Site de Cery,
1008 Prilly-Lausanne, Switzerland
Christian Depeursinge
Ecole Polytechnique Fédérale de Lausanne (EPFL), Institut d’Optique Appliquée, CH-1015 Lausanne, Switzerland
Received March 17, 2006; revised June 1, 2006; accepted June 3, 2006; posted June 8, 2006 (Doc. ID 69123)
Introducing a microscope objective in an interferometric setup induces a phase curvature on the resulting
wavefront. In digital holography, the compensation of this curvature is often done by introducing an identical
curvature in the reference arm and the hologram is then processed using a plane wave in the reconstruction.
This physical compensation can be avoided, and several numerical methods exist to retrieve phase contrast
images in which the microscope curvature is compensated. Usually, a digital array of complex numbers is in-
troduced in the reconstruction process to perform this curvature correction. Different corrections are discussed
in terms of their influence on the reconstructed image size and location in space. The results are presented
according to two different expressions of the Fresnel transform, the single Fourier transform and convolution
approaches, used to propagate the reconstructed wavefront from the hologram plane to the final image plane.
© 2006 Optical Society of America
OCIS codes: 090.1000, 090.1760, 100.3010, 110.0180
.
1. INTRODUCTION
Because of the limited sampling capacity of the electronic
camera compared with the one of photosensitive materi-
als such as photographic plates, the spatial resolution of
reconstructed images in digital holography was formerly
limited compared with classical holography. Different ap-
proaches exist to achieve microscopic imaging with digital
holography. One can, for example, use spherical diverging
waves for the hologram recording, which allows a numeri-
cal enlargement of the object during the reconstruction
process, without any image-forming lens, as described in
Chap. 5 of Ref. 1. By the introduction of a microscope ob-
jective (MO), Cuche et al.
2
have demonstrated that digital
holographic microscopy (DHM) allows one to reconstruct,
with a lateral resolution below micrometers, the optical
topography of specimens with a nanometric accuracy.
Nevertheless, the introduction of a MO increases the com-
plexity of the reconstruction process. Indeed, the MO in-
troduces a phase curvature to the object wave that should
be compensated perfectly to perform accurate measure-
ment and imaging of the phase delay induced by the
specimen. There are two different main possibilities to
compensate for this phase curvature, either physically by
introducing the same curvature in the reference wave or
digitally as presented in Refs. 2–6.
The physical compensation, in standard interference
microscopy like the Linnik configuration (see, for ex-
ample, Chap. 20 in Ref. 7), is done experimentally by in-
serting the same MO in the reference arm, at equal dis-
tance from the exit of the interferometer. The curvature of
the object wave is then compensated by the reference
wavefront during interference. Nevertheless, this method
requires a precise alignment of all the optical elements.
Moreover, each modification in the object arm needs to be
precisely reproduced in the reference arm.
In the present paper, we call digital phase mask (DPM)
a complex numbers array, by which the reconstructed
wavefront is multiplied during the hologram processing.
Digitally, the definition and the position of the DPMs used
to compensate the phase curvature can be different. Fer-
raro et al. make the compensation in the image plane by
subtracting the reconstructed phase of a hologram ac-
2944 J. Opt. Soc. Am. A /Vol. 23, No. 11 / November 2006 Montfort et al.
1084-7529/06/112944-10/$15.00 © 2006 Optical Society of America
quired without a specimen.
3
Cuche et al. define a numeri-
cal quadratic curvature model
2
that could be automati-
cally computed in the image plane.
4
Finally, two recent
papers show that the compensation for the MO curvature
(and for phase aberrations) can be done in the hologram
plane by using a reference hologram
5
or polynomial DPM
models (standard or Zernike) computed automatically.
6
In
spite of the numerous phase retrieval techniques pro-
posed in the literature, no systematic study of the behav-
ior of the phase images obtained through these recon-
struction methods has, to our knowledge, been performed
yet. In this paper, we present analytically the influence of
the DPMs’ position (hologram or image plane) in the re-
construction process, in particular in terms of position
and size of the reconstructed specimen region of interest
(ROI).
2. BASES OF DIGITAL HOLOGRAPHY
A. Principle and Reconstruction
Digital holography allows one to retrieve the original
complex wavefront from an amplitude image, called a ho-
logram, recorded on an electronic camera such as a CCD
or complementary metal-oxide semiconductor camera.
This hologram is created by the interference, in off-axis
geometry, between two coherent waves: on one side the
wave of interest, called object wave O, coming from the
object, and on the other a reference wave R. In the holo-
gram plane, the two-dimensional recorded intensity dis-
tribution I
H
x ,y can be written
8
as
I
H
x,y = O + R
2
= O
2
+ R
2
+ RO
*
+ R
*
O, 1
where R
*
O and RO
*
are the interference terms with R
*
and O
*
denoting the complex conjugate of the two waves.
After hologram apodization
9
and spatial filtering,
10
the
virtual interference term R
*
O (the same procedure can be
done with the real interference term RO
*
) is multiplied in
the hologram plane by a DPM
H
,
5
which should be ide-
ally equal to R, to reproduce the original wavefront
H
=RR
*
O=
H
R
*
O in the hologram plane. Once the wave-
front
H
has been retrieved, it has to be propagated to the
image plane to have a focused image. This propagation of
a monochromatic reconstructed wavefront
H
at wave-
length =2
/k from the hologram plane to the image
plane over a distance d is done in the Fresnel
approximation,
1–6,9–11
which allows one to implement nu-
merically the propagation by simple fast Fourier trans-
forms (FFTs), as will be pointed out further:
I
x,y =
expikd
id
冕冕
H
,
exp
i
d
关共
x
2
+
y
2
d
d
, 2
where
I
is the corresponding wavefront propagated to
the image plane. Let us define the two-dimensional
Fresnel transform (FT) of parameter
=
d of a given
function fx, y as
F
fx,y兲兴 =
1
2
冕冕
f
,
exp
i
d
关共
x
2
+
y
2
d
d
. 3
Using this definition, Eq. (2) can be written as
I
x,y =−i expikdF
d
H
x,y兲兴. 4
This analytical expression of propagation can be digi-
tized by using two different formulations: the single FT
and the convolution formulations.
1. Single Fourier-Transform Formulation
The propagation in the Fresnel approximation [Eq. (4)]
can be written using a single FT:
I
x,y =
expikd
id
exp
i
d
x
2
+ y
2
FT
H
,
exp
i
d
2
+
2
. 5
In its discrete formulation, the low time-consuming FFT
algorithm can be employed.
2
In the following text, this
formulation will be referred to as FT formulation.
In this case the sampling step of the propagated image
is not the same as the initial one. If the initial image is
given by N
pts
N
pts
points with a sampling step T
T
,
the image propagated over a distance d is sampled with
the same number of points but with a sampling step given
by
T
x
=
d
N
pts
T
, T
y
=
d
N
pts
T
. 6
2. Convolution Formulation
The Fresnel propagation given by Eq. (4) can be written
using a convolution formulation:
I
x,y =
expikd
id
H
x,y兲兴 exp
i
d
x
2
+ y
2
. 7
Its discrete form is a little more time-consuming than the
FT formulation when computed.
1,12
The convolution ex-
pression of the propagation in the Fresnel approximation
has the same sampling step before and after the propaga-
tion. Thus if the image in the hologram plane is sampled
in N
pts
N
pts
points with a sampling step T
T
, the
propagated image is sampled with the same number of
points and a sampling step T
x
T
y
=T
T
.
B. Microscope Objective Introduction
The introduction of a MO in the object arm opens the pos-
sibility of imaging at the submicrometer scale. As shown
in Fig. 1, the optical arrangement in the object arm is that
of an ordinary single-lens system producing a magnified
image of the specimen in an image plane. In comparison
with classical microscopy, the difference is that the CCD
camera is not in the image plane but is in the hologram
plane that is located between the MO and the image
plane, at a distance d from the image. This situation can
Montfort et al. Vol. 23, No. 11 / November 2006 /J. Opt. Soc. Am. A 2945
be considered to be equivalent to a holographic configura-
tion without a MO with an object wave emerging directly
from the image and not from the object itself.
The MO produces a curvature of the wavefront in the
object arm. This deformation affects only the phase of the
object wave and does not disturb amplitude contrast im-
aging. However, to perform an accurate measurement of
the phase delay induced by the specimen only, the phase
curvature induced by the MO must be perfectly compen-
sated. This compensation can be done in the hologram
plane by a DPM
H
and/or in the image plane by a DPM
I
. Therefore we can write the corrected wavefront gener-
ally as
I
x,y =−
I
i expikdF
d
H
H
x,y兲兴. 8
3. HOLOGRAM RECONSTRUCTION: THE
IDEAL CASE
Let us first express analytically the reconstruction pro-
cess that exactly reproduces the image resulting from the
object through the MO, as it would be performed on an op-
tical bench, without any scaling or lateral or axial shift-
ing. This ideal case formulation will serve as a gauge im-
age for comparison with the images obtained by the
different digital reconstruction methods.
The hologram is multiplied by an ideal DPM
id
H
corre-
sponding to a replica of the reference wave:
id
H
=
id
H
R
*
O = RR
*
O = O, 9
where the reference wave amplitude has been assumed to
be equal to one. The propagation over a distance d
id
ex-
pressed using Eq. (4) is given by
id
I
=−i expikd
id
F
d
id
id
H
, 10
=− i expikd
id
F
d
id
O, 11
where
id
I
corresponds to the exact initial object wave-
front.
In a general approach, we can consider an off-axis mi-
croscopy setup (angle
between the propagation direction
of the reference and object waves), in which the curva-
tures of the reference and object waves at the hologram
plane are different (Fig. 2). Let us define the centers of
the spherical reference and object waves as
S
R
= S
Rx
,S
Ry
,h
r
2
S
Rx
2
S
Ry
2
1/2
, 12
S
O
= 0,0,h
o
, 13
where h
r
and h
o
are, respectively, the distances between
the source points of the reference and object waves and
the recombining location of the two beams. Note that the
source point of the spherical object wave is located at the
back focal plane of the MO. The reference wavefront in
the hologram plane is thus given by
8
Rx,y = exp
i
h
r
关共x S
Rx
2
+ y S
Ry
2
. 14
Let us now define a blank object wave O
0
(without a
specimen in the transmission configuration and with a
flat surface in the reflection configuration).
4
Because we
assumed that only phase curvature is induced by the MO,
the wavefront of the blank object wave at the hologram
plane is
O
0
x,y = exp
i
h
o
x
2
+ y
2
. 15
To recover the phase delay induced by the object only,
the phase curvature induced by both the MO and the ref-
erence beam curvatures can be compensated by multiply-
ing
id
I
by a second DPM
id
I
introduced in the image
plane. The latter is determined by the complex conjugate
of the blank wave O
0
propagated to the image plane:
id
I
= F
d
id
id,0
H
兴其
*
, 16
=F
d
id
O
0
兴其
*
. 17
The corrected wavefront
id
I
becomes
id
I
=
id
I
id
I
, 18
=F
d
id
O
0
兴其
*
F
d
id
O. 19
Inserting Eqs. (14) and (15), Eqs. (17) and (18) can be
written as
Fig. 1. Standard configuration in holographic microscopy: The
CCD defining the hologram plane is placed in front of the image
obtained though the microscope objective (MO). d is the recon-
struction distance.
Fig. 2. Schema of the used notations.
2946 J. Opt. Soc. Am. A /Vol. 23, No. 11 / November 2006 Montfort et al.
id
I
x,y = exp
i
h
o
+ d
id
x
2
+ y
2
, 20
id
I
x,y =−i expikd
id
exp
i
h
o
+ d
id
x
2
+ y
2
F
d
id
id
H
. 21
id
I
logically corresponds to the spherical wavefront cen-
tered in S
O
at a distance h
o
+d
id
from the image plane.
This general development expresses the retrieval of the
object wavefront, in which the MO curvature is corrected.
Potentially this approach may also be used for correction
of optical aberrations of the holographic setup,
4
but the
development will be restricted to the case without aberra-
tions, focusing on the effect of the phase curvature com-
pensation.
4. PHASE CURVATURE COMPENSATION
DURING NUMERICAL RECONSTRUCTION
To illustrate the different reconstruction approaches and
propagation formulations, a hologram of a quartz micro-
lens recorded in a transmission DHM setup with a 20
MO (numerical aperature of 0.5) is used. This microlens
has a diameter of 240
m and a height of 21.15
m. Fig-
ures 3(a)–3(c) present the raw phase reconstruction, the
two-dimensional unwrapped phase image, and its three-
dimensional representation, respectively. The different
notations for the approaches are done with a subscript
letter: no letter, general; i, image plane; h, hologram
plane; and m, mixed.
A. General Approach
Let us develop the general approach in which a DPM is
introduced both in the hologram plane and in the image
plane. The application of this general approach is devel-
oped in detail and illustrated with examples in Ref. 6. We
first apply a DPM
H
to the interference term R
*
O in the
hologram plane,
H
=
H
R
*
O. 22
Propagating the resulting wave over a distance d to the
image plane yields
I
;
I
=−i expikdF
d
H
, 23
=− i expikdF
d
H
R
*
O. 24
Then a second DPM
I
is applied in the image plane to
compensate for the curvature of the propagated wave-
front. The corrected wavefront
I
is thus given by
I
=
I
I
, 25
=− i expikd
I
F
d
H
R
*
O. 26
To determine the effects of
H
on the propagation, we will
compare
I
with
id
I
obtained in the ideal case [see Eqs.
(19) and (21)].
The DPM applied in the image plane for the phase cur-
vature compensation will of course depend on the DPM
introduced in the hologram plane. We defined the digital
reference wave using the same notation as in the ideal
case in Eqs. (14) and (12). In this way we can define a gen-
eral DPM in the hologram plane. As it is supposed to be a
curvature correction term, it is defined as the conjugate of
a spherical wave centered in S
D
:
H
= exp
i
h
d
关共x S
Dx
2
+ y S
Dy
2
, 27
S
D
= S
Dx
,S
Dy
,h
d
2
S
Dx
2
S
Dy
2
1/2
. 28
The DPM
I
in the image plane that compensates the re-
sulting propagated wavefront is then given by
I
x,y = exp
i
关共S
Dx
S
Rx
2
+ S
Dy
S
Ry
2
h
d
h
r
exp
i
h + d/M
1
M
2
x h
S
Rx
h
r
S
Dx
h
d
2
+
1
M
2
y h
S
Ry
h
r
S
Dy
h
d
2
exp
i
h
0
+ d/M
1
M
2
x d
S
Rx
h
r
S
Dx
h
d
2
+
1
M
2
y d
S
Ry
h
r
S
Dy
h
d
2
, 29
where M and h are defined as
h =
h
d
h
r
h
d
h
r
, M =
h d
h
. 30
Finally, the phase curvature-corrected wavefront in the
image plane
I
is given by
Fig. 3. (a) Phase reconstruction of the microlens recorded in a
transmission DHM setup (diameter 240
m, height 21.15
m),
(b) two-dimensional unwrap of (a), (c) perspective representation
of (b).
Montfort et al. Vol. 23, No. 11 / November 2006 /J. Opt. Soc. Am. A 2947
I
x,y =
I
x,y
I
x,y
=−i expikd
1
M
F
d/M
x
,y
exp
i
h + d/M
1
M
2
x h
S
Rx
h
r
S
Dx
h
d
2
+
1
M
2
y h
S
Ry
h
r
S
Dy
h
d
2
= exp
ik
d
d
M
1
M
id
H
x
,y
, 31
where x
=
xd
S
Rx
h
r
S
Dx
h
d
M and y
=
yd
S
R4
h
r
S
D4
h
d
M. The propagation direction
is no longer parallel to the optical axis, but is given by the
angle
:
sin
=
S
Rx
2
+ S
Ry
2
h
r
S
Dx
2
+ S
Dy
2
h
d
. 32
By analyzing the image plane DPM given by Eq. (29),
we can find that the first term is a phase constant of no
particular interest and can be suppressed. The second
term is compensating for the phase deformation induced
by the reference wave and the DPM in the hologram
plane. Finally the third term is the correction term of the
object wavefront curvature. The final image
I
is a replica
of the ideal case image scaled by a factor M and laterally
shifted.
We note that one can retrieve the results of the ideal
approach by setting
H
=R:
H
= R S
D
= S
R
, h
d
= h
r
lim
h
d
h = h
r
, M =1,
which gives the well-known results of Eqs. (10), (20), and
(21):
lim
H
R
I
x,y =
id
I
x,y, 33
lim
H
R
I
x,y = exp
i
x
2
+ y
2
h
o
+ d
, 34
lim
H
R
I
x,y =−i expikd
exp
i
x
2
+ y
2
h
o
+ d
id
I
x,y. 35
B. Image Plane Approach
In the case of the image plane approach, no DPM is ap-
plied in the hologram plane and the propagating term is
R
*
O, the illumination wave being considered of unit in-
tensity. This can be seen as if the hologram would be re-
constructed with a plane wave propagating along the op-
tical axis (Fig. 4).
The phase curvature compensation process is therefore
applied to the propagated interference term R
*
O. The
DPM can be computed from known flat areas on the speci-
men with the procedure described in Ref. 4 or from the
propagation of a blank hologram as described in Ref. 3.
This DPM is given by
i
I
= F
d
i
R
*
O
0
兴其
*
. 36
The flattened wavefront
i
I
can be written as
i
I
=−i expikd
i
兲兵F
d
i
R
*
O
0
兴其
*
F
d
i
R
*
O. 37
The condition
H
=1 imposes the following:
S
D
= 0, lim
H
1
h
d
= ,
S
D
h
d
=0,
Fig. 4. (a) Reconstruction in the image plane approach: The illumination beam is a plane wave propagating along the optical axis. The
phase curvature is compensated in the image plane. The reconstructed image is not the image of the object through the MO (shown by
a dashed line). (b) Phase image in the hologram plane. (c) and (d) Phase images in the image plane in convolution and FT formulations,
respectively.
2948 J. Opt. Soc. Am. A /Vol. 23, No. 11 / November 2006 Montfort et al.
lim
h
d
h = h
r
, lim
h
d
M =
h
r
d
h
r
= M
i
.
Introducing these results in Eqs. (29) and (31), we can ex-
press the DPM
i
I
that expresses the phase curvature cor-
rection leading to the expression of the corrected image
wavefront
i
I
:
i
I
x,y = lim
H
1
I
x,y = exp
i
h
r
+ d
i
/M
i
1
M
i
2
x S
Rx
2
+
1
M
i
2
y S
Ry
2
exp
i
h
0
+ d
i
/M
i
1
M
i
2
x
d
i
h
r
S
Rx
2
+
1
M
i
2
y
d
i
h
r
S
Ry
2
, 38
i
I
x,y = lim
H
1
I
x,y =−i exp
ik
d
i
d
i
M
1
M
i
id
I
x
d
i
h
r
S
Rx
M
i
,
y
d
i
h
r
S
Rx
M
i
. 39
Equation (39) shows that the correction in the image
plane approach also introduces a resizing of the image in
comparison with the ideal case. The scale factor is a func-
tion of the reference beam curvature h
r
. This scaling is
due to the fact that, compared with the ideal solution, the
correction of the reference curvature is not performed in
the hologram plane as it is when the hologram is pro-
cessed with exactly the same reference wave used during
acquisition.
In the image plane approach, the image is also laterally
shifted in space, as mentioned in the general approach
and shown in Fig. 4. The shift is due to the fact that the
propagation direction is modified by an angle
from the
optical axis of the object beam.
is given by
sin
=
S
Rx
2
+ S
Ry
2
h
r
. 40
This inclination of the propagation direction arises from
the fact that the illumination wave propagates along the
optical axis, which is precisely inclined of an angle
from
the correct reference wave. This induced error corre-
sponds to a tilt of the wavefront that is not corrected in
the hologram plane and induced this propagation devia-
tion. The lateral shift is thus given by
L
shift
=
d
i
h
r
S
Rx
2
+ S
Ry
2
1/2
= d
i
sin
. 41
This shift is not convenient for the numerical propaga-
tion. Indeed, the image is no longer centered in the recon-
struction window. In a convolution formulation of the
propagation (see subsection 5.B), this results in a tailed
image [Fig. 4(c)]. In the case of the FT formulation, it may
not be a problem if the sampling step is small enough so
that the field of view of the window is large enough to
cover the off-axis propagating wavefront [Fig. 4(d)]. The
mixed approach will give a solution in which the recon-
structed image has the same size and sampling step as
the reconstructed image in the image plane approach, but
without lateral shift (Fig. 5).
C. Hologram Plane Approach
In this second digital approach, a single DPM is applied
in the hologram plane.
5
Thus the considered wavefront is
directly R
*
O. Let us suppose a recording of a reference
hologram, where no object is present in the object beam.
The recorded term is then given by R
*
O
0
. Its conjugate
defines perfectly the DPM to be applied in the hologram
plane:
h
H
= RO
0
*
. 42
By multiplying the interference term by the DPM and ex-
pressing the result as a function of the ideal retrieved
wavefront, we obtain [Fig. 6(b)]:
h
H
=
h
H
R
*
O = O
0
*
id
H
=
h
H
. 43
Thus the propagation of this resulting wavefront over a
distance d
h
to the image plane can be expressed as
h
I
=−i expikd
h
F
d
h
h
H
=−i expikd
h
F
d
h
O
0
*
O.
44
The interference term has been at the same time multi-
plied by the illumination wave R and by the correction
term that compensates for the object wavefront curva-
ture. The result is a plane wave modulated by the object-
related phase variations. Its propagation will therefore be
a plane wave and no curvature compensation will be
needed at any reconstruction distance, in particular in
the focused image plane (Fig. 6).
Nevertheless, the multiplication, in the hologram plane
already, of the interference term by the curvature com-
pensation term has an influence on the image. Indeed, the
propagated wavefront is O
0
*
O instead of O in the ideal
case, which influences the focus distance, image size, etc.
In the case of a microscope without aberrations, the
term O
0
*
compensating the curvature of the MO corre-
sponds to the transfer function of a lens. The corrected
wavefront in the image plane
h
I
is given by
h
I
x,y =−i expikd
h
F
d
h
exp
i
h
o
x
2
+ y
2
id
H
x,y
45
=exp
ik
d
h
d
h
M
h
1
M
h
id
I
x
M
h
,
y
M
h
. 46
The algorithm compensating for the phase curvature is
thus equivalent to the insertion of a numerical lens in the
hologram plane. We note that the focal length is deter-
mined only by the object wave shape and is totally inde-
pendent of the reference wavefront, which has been com-
pensated by the DPM.
Montfort et al. Vol. 23, No. 11 / November 2006 /J. Opt. Soc. Am. A 2949
The resulting image
h
H
is thus focused at a distance d
h
and magnified by a factor M
h
given by the thin-lens rela-
tion:
1
h
o
=−
1
d
id
+
1
d
h
, M
h
=
d
h
d
id
, 47
where h
o
is the focal length of the introduced numerical
lens, and d
id
is the focus distance of the reconstructed im-
age in the ideal case (equal to the distance between the
image of the object through the MO and the hologram
plane). We note that h
o
corresponds to the distance be-
tween the back focal plane of the MO and the hologram
plane.
D. Mixed Approach
The mixed approach is a method combining both the ho-
logram and the image plane approaches. It consists in de-
fining the DPM in the hologram plane keeping account of
only some selected polynomial orders for a partial holo-
gram plane correction. After propagation, an image plane
DPM is defined and the remaining polynomial orders are
corrected. Several combinations are possible depending
on which orders are corrected in the hologram plane. Nev-
Fig. 5. (a) Reconstruction in the mixed approach: The illumination beam is a plane wave propagating along the reference wave axis. The
first-order DPM is applied in the hologram plane and a second of higher orders in the image plane. The reconstructed image is not the
image of the object through the MO (shown by a dashed line). (b) Phase image in the hologram plane. (c) and (d) Phase images in the
image plane in convolution and FT formulations, respectively. The white lines define the diameter of the microlens diameter to be com-
pared with Fig. 6
Fig. 6. (a) Reconstruction in the hologram plane approach: The illumination beam is a replica of the reference beam. The phase cur-
vature is compensated in the hologram plane. The reconstructed image is not the image of the object through the MO (shown by a dashed
line). (b) Phase image in the hologram plane. (c) and (d) Phase images in the image plane in convolution and FT formulations, respec-
tively. The white lines define the diameter of the microlens reconstructed with the mixed approach (Fig. 5).
2950 J. Opt. Soc. Am. A /Vol. 23, No. 11 / November 2006 Montfort et al.
ertheless, only the case of the first-order phase correction,
i.e., planar phase correction, in the hologram plane will be
discussed. This corresponds to illuminating the hologram
with a plane wave having the same propagation direction
as the reference wave. It is thus similar to the image
plane approach, except that the illumination wave has
the same propagation direction as the reference wave in-
stead of the object wave (Fig. 5).
In the hologram plane, the interference term R
*
O is
multiplied by a first-order DPM corresponding to a plane
wave PW
m
H
[Fig. 5(b)]:
m
H
= PW
m
H
= exp
i
2
S
Rx
h
r
x +
S
Ry
h
r
y
. 48
The determination of
m
I
results in
m
I
x,y =
PW
m
H
x,y
F
d
m
R
*
O
x +
d
m
h
r
S
Rx
,y +
d
m
h
r
S
Ry
*
,
49
and the corrected wavefront
m
I
in the image plane is
given by
m
I
x,y =−i expikd
m
i
I
x +
d
m
h
r
S
Rx
,y +
d
m
h
r
S
Ry
F
d
m
R
*
O
x +
d
m
h
r
S
Rx
,y +
d
m
h
r
S
Ry
.
50
The comparison of
m
I
with
i
I
given by Eq. (37) indicates
that the reconstructed images are exactly the same in
both cases, but spatially located at different positions. In-
deed, the propagation in the mixed approach deviates by
an angle
from the image plane approach, where
is
given by Eq. (40), which means that the reconstructed
wavefront is again propagating along the optical axis. In
the case without aberrations, the mixed approach is a
particular case of the general approach in which
H
=PW
m
H
:
H
= PW
m
H
S
D
, h
d
,
S
D
h
d
=
S
R
h
r
,
lim
h
d
h = h
r
, lim
h
d
M =
h
r
d
h
r
= M
i
.
Using Eqs. (29) and (31), we can express the DPM
m
I
for
phase correction:
m
I
x,y =
I
x,y兲兩
H
=PW
m
H
= exp
i
h
r
+ d
m
/M
i
x
2
M
i
2
+
y
2
M
i
2
exp
i
h
0
+ d
m
/M
i
x
2
M
i
2
+
y
2
M
i
2
, 51
which leads to the expression of the curvature-
compensated image wavefront
m
I
:
m
I
x,y =
I
x,y兲兩
H
=PW
m
H
52
=exp
ik
d
m
d
m
M
i
1
M
i
id
I
x
M
i
,
y
M
i
. 53
These results show that the reconstructed image is the
same as the one issued from the image plane approach,
except that it is centered on the optical axis. The scaling
factor and propagation distance are the same. The effect
of the first-order correction in the hologram plane is to
center the image on the optical axis. This mixed approach
is thus interesting in the sense that it can be applied to
the convolution approach of the Fresnel propagation.
5. DISCUSSION
A. Analytical Formulation
It has been shown that in the image plane approach, the
reference wave may induce some differences compared
with the ideal case, as the phase curvature correction is
not performed in the hologram plane, but only in the im-
age plane. In the hologram plane approach, the difference
with the ideal case is due to the correction of the object
wave curvature, already performed in the hologram
plane, the consequences therefore depending on the shape
of the object wavefront. Finally, the mixed approach re-
duces the difference between the image approach and the
ideal case by correcting the propagation direction. Table 1
summarizes quantitatively the consequences in terms of
magnification and shift for each approach.
Each of these approaches has its own particularities.
The hologram plane approach has the advantage of
propagating, along the optical axis, a wave containing
only the phase deformations due to the object. The phase
modulations induced by the object are most often weak
and the wave propagates quite like a plane wave, mean-
ing that the phase curvature is compensated for any
propagation distance. As the DPM is applied in the fixed
hologram plane, it does not depend on the reconstruction
distance like in the image plane approach. Thus the DPM
can be determined once for a given setup, which is of
great interest in automated reconstruction processes. The
drawback of this solution is that the image is not focused
in the hologram plane. Thus, the areas used for the fit of
the DPM
6
are disturbed by the diffraction pattern of the
object. The DPM determined in the hologram approach
may thus be approximative in some cases, and may need
a minor adjustment in the image plane, creating a par-
ticular mixed approach.
The image plane approach has the opposite arguments.
It has the advantage of a focused image, and therefore
clear constant phase areas are available around the object
to perform the phase compensation procedure. This ad-
vantage is balanced with the fact that the phase curva-
ture is not compensated for any reconstruction distance
Montfort et al. Vol. 23, No. 11 / November 2006 /J. Opt. Soc. Am. A 2951
and that the image is not centered in the reconstruction
window. Only the presented mixed approach corrects this
last disadvantage.
B. Discrete Formulation
All the considerations on the reconstruction approach are
done considering continuous functions. Nevertheless, the
sampling of the image and the propagation method have
an influence on the size of the reconstructed images.
The continuous expression of the resulting image
I
in
the different propagation methods can be summarized by
the expression
I
x,y = exp
ik
d
d
M
1
M
id
H
x a
M
,
y b
M
. 54
The ideal case corresponds to d =d
id
, M =1, a= b =0. As de-
fined in the different reconstruction approaches, the fo-
cusing distance d and the magnification ratio M are given
by
M =
d
d
id
. 55
Considering these expressions, let us define the image
sizes in the discrete formulation using both the convolu-
tion and FFT expression of the propagation.
1. Fourier Transform Formulation
Using the FT formulation of the propagation [Eq. (5)],
I
is expressed as
I
n,m = exp
ik
d
d
M
1
M
⫻⌼
id
H
n a
d
N
pts
T
1
M
, m b
d
N
pts
T
1
M
.
56
In the digital reconstruction, the focusing distance is
given from Eq. (56):
d = Md
id
. 57
Inserting Eq. (58) into Eq. (57), we obtain
I
n,m = exp
ik
d
d
M
1
M
⫻⌼
id
H
n a
d
id
N
pts
T
, m b
d
id
N
pts
T
=
id
I
n a
,m b
. 58
This last result shows that the multiplication of the wave-
front by a quadratic DPM in the hologram plane has no
influence on the resulting image size. The FT formulation
of the propagation is thus not sensitive to scaling induced
by the different reconstruction methods [see Figs. 4(d),
5(d), and 6(d)].
As presented extensively in Ref. 1, one seems at first
sight to lose (or gain) resolution by applying the FFT ver-
sion. On closer examination one recognizes that
d /N
pts
T
corresponds to the resolution limit given by the
diffraction theory of optical systems: The hologram is the
aperture of the optical system with side length N
pts
T
.
According to the theory of diffraction, at a distance d be-
hind the hologram a diffraction pattern develops. T
x
=d /N
pts
T
is therefore the diameter of the Airy disk (or
speckle diameter) in the plane of the reconstructed image,
which limits the resolution. This can be regarded as the
automatic scaling algorithm, setting the resolution of the
image reconstructed in the Fresnel approximation by a
FFT always to the physical limit. The numerical lens in-
serted by the reconstruction algorithm thus has no effect.
2. Convolution Formulation
It has already been shown that the convolution expres-
sion [Eq. (7)] of the propagation in the Fresnel approxi-
mation has the same sampling step before and after the
propagation. This means that the image is sampled in the
same manner in the hologram plane than in the image
plane.
I
can be written as
Table 1. Summary of the Different Reconstructed Image Properties
Approach
Magnification Shift
Ideal
id
x,y
1 (0, 0)
Hologram plane
expikd d
id
兲兴
1
M
h
id
I
x
M
h
,
y
M
h
h
o
d
h
o
(0, 0)
Image plane
expikd d
id
兲兴
1
M
h
id
I
x
d
h
r
S
Rx
M
i
,
y
d
h
r
S
Ry
M
i
h
r
d
h
r
d
h
r
S
Rx
,
d
h
r
S
Ry
Mixed
expikd d
id
兲兴
1
M
i
id
I
x
M
i
,
y
M
i
h
r
d
h
r
(0, 0)
2952 J. Opt. Soc. Am. A /Vol. 23, No. 11 / November 2006 Montfort et al.
I
n,m = exp
ik
d
d
M
1
M
id
H
n a
T
M
,
m b
T
M
,
59
which means that the size of the images is magnified the
same way as in the continuous domain. Figure 6(c) is re-
constructed using the hologram plane approach. The im-
age has the same size as the hologram. Figure 5(c) is re-
constructed using the mixed approach. The size of the
images of Figs. 5(c) and 5(d) is not the same because the
magnification factor with respect to the ideal case is dif-
ferent. Reconstruction of holograms by the convolution
approach results indeed in images with more or fewer pix-
els per unit length than those reconstructed by the FFT.
However, because of the physical limits, the image reso-
lution does not change.
6. CONCLUSION
In digital holography, the access to numerical data allows
an easy compensation of the nondesired phase curvatures
in the reconstructed images. Nevertheless, procedures are
required to determine the correction to be applied. The re-
corded data, corresponding to the interference term R
*
O,
usually do not allow the retrieval of the original image,
but only a scaled replica displaced both laterally and axi-
ally. The reconstructed image can therefore be considered
to be an image of the object through a system, composed
of a MO and an additional numerical lens, that can be
analytically characterized to get the exact properties of
the complete holographic microscope. In this paper, we
have presented for the first time to our knowledge the in-
fluence of the phase masks’ position involved in the recon-
struction process, in particular in terms of position and
size of the reconstructed specimen ROI.
We hope that this study on the reconstruction methods,
in conjunction with the remarks on the reconstructed im-
age sampling regarding the FT or convolution formula-
tion, will clarify the relations subtended between the
available hologram processing techniques, facilitating the
users choice for each specific application of DHM.
ACKNOWLEDGMENT
This work was funded through research grants 205320-
103885/1 from the Swiss National Science Foundation.
The e-mail address for F. Montfort is
frederic.montfort@a3.epfl.ch.
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Montfort et al. Vol. 23, No. 11 / November 2006 /J. Opt. Soc. Am. A 2953
    • "Holograms of microscopic objects recorded with DHM setups can be numerically reconstructed in amplitude and phase using the same Fresnel DH reconstruction techniques. The phase aberrations due to the MO and the tilt from the reference beam have to be corrected to obtain the topographic profile or the phase map of the object [25][26][27][28][29][30]. Figures 1(a) and 1(b) show a Michelson SW/MW-DHM in reflection and transmission configurations, respectively. "
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