Correct self-assembling of spatial frequencies
in super-resolution synthetic aperture
Melania Paturzo and Pietro Ferraro*
CNR–Istituto Nazionale di Ottica Applicata, via Campi Flegrei 34, 80078 Pozzuoli (NA) Italy
* Corresponding author: firstname.lastname@example.org
Received June 12, 2009; revised October 14, 2009; accepted October 14, 2009;
posted October 27, 2009 (Doc. ID 112752); published November 20, 2009
Synthetic aperture enlargement is obtained, in lensless digital holography, by introducing a diffraction grat-
ing between the object and the CCD camera with the aim of getting super-resolution. We demonstrate here
that the spatial frequencies are naturally self-assembled in the reconstructed image plane when the NA is
increased synthetically at its maximum extent of three times. By this approach it possible to avoid the use
of the grating transmission formula in the numerical reconstruction process, thus reducing significantly the
noise in the final super-resolved image. Demonstrations are reported in 1D and 2D with an optical target
and a biological sample, respectively. © 2009 Optical Society of America
OCIS codes: 100.6640, 110.0180, 050.1950, 070.6020, 090.0090, 090.1760.
Digital holography (DH) in microscope configuration
have been extensively adopted in many fields such as
biology [1,2], 3D imaging , and particle analysis
[4,5]. An important configuration is the lensless one,
which offers high simplicity with a reasonably mag-
nification, which is useful in microscopy . The res-
olution of an optical system is limited by its NA that
in lessless DH is given by the lateral dimension of the
detector. The finite aperture of the imaging system
prevents the sensor from collecting light scattered at
large angles. Therefore, depending on the object, a
certain amount of information is missed into the re-
corded holograms, thus leading to a reconstructed
image with a band-limited spatial frequency domain.
Various strategies have been tested to synthetically
increase the NA in order to get super-resolution [6,7].
The NA was increased recording nine holograms with
a CCD in different positions and recombining them
in a synthetic hologram . Super-resolution was
achieved by rotating the sample in its plane  or in
respect to the optical axis  and recording a digital
hologram for each position in order to capture the dif-
fraction field along different directions. Recently new
approaches to get super-resolved images has been
demonstrated [11–14]. Super-resolved images can be
obtained simply by using 1D diffraction grating 
or a 2D tuneable phase grating  that allows one
to collect parts of the light diffracted from the object
that otherwise would fall outside the CCD. Different
digital holograms, according to the grating geometry,
are spatially multiplexed onto the same CCD array,
and the super-resolved images are obtained by the
numerical reconstruction, taking into account the
transmission function of the grating (GTF). The main
advantage of this method in respect to the others is
that, by introducing the correct GTF into the recon-
struction diffraction integral, the spatial frequencies
of the objects are assembled together automatically
without tedious numerical superimposing operations
[8–13]. Nevertheless, the introduction of the GTF
produces some deleterious numerical noise that af-
fects the final super-resolved image.
Here we demonstrate that, when the NA improve-
ment is exactly equal to three, it is not necessary to
introduce the GTF in the numerical reconstruction.
In fact, in this case the result is that the recon-
structed images corresponding to the different dif-
fraction orders are automatically and precisely su-
perimposed. We show that it is possible to exploit the
typical wrapping effect in the reconstructed image
plane when the bandwidth of the image does not fit
entirely into the reconstructed window. This method
has two significant advantages. First, avoiding the
use of the GTF the noise is reduced considerably. Sec-
ond, there is no restriction in the object field of view.
Strong field of view restrictions were necessary in our
previous approach so that the different bandpass im-
ages, coming from the different diffraction orders,
will not overlap in the image plane . As to the
used grating, in the first part of the Letter, we
present the experimental validation of the proposed
approach through a 1D resolution improvement, em-
ploying an amplitude grating with a pitch of 25 ?m.
In the second part, where we show a 2D super-
resolved image of a biological sample, we used 2D
amplitude gratings with the same pitch.
The sketch in Fig. 1 describes the object wave op-
tical path during the recording [Fig. 1(a)] and the re-
construction processes [Fig. 1(b)]. The CCD records
three spatially multiplexed holograms corresponding
to three diffraction orders. Each hologram carries dif-
ferent information about the object. The hologram
corresponding to the zero order contains the low-
object frequencies, while the holograms correspond-
ing to the first orders collects the rays scattered at
wider angles carrying information about higher fre-
quencies. Three distinct images corresponding to
each hologram are numerically reconstructed, as
shown in Fig. 1(b). Our aim is to assemble these
three images in an appropriate way in order to get
super-resolution. In Fig. 1 the black (blue online)
squares in the CCD plane indicate the center of the
OPTICS LETTERS / Vol. 34, No. 23 / December 1, 2009
0146-9592/09/233650-3/$15.00 © 2009 Optical Society of America
three holograms, while the gray (red online) squares
in the reconstruction plane are the center of the re-
constructed images. The distance between two con-
tiguous holograms is NxPCCD, where PCCDis the CCD
pixel size and Nxis a real number, while the distance
between two reconstructed images is NxPr, where Pr
is the reconstruction pixel. When Nxis equal to N,
that is, the number of CCD pixels, the relative dis-
tance among the three holograms is exactly equal to
the lateral dimension of the CCD array (Fig. 1).
Therefore, only when Nx=N, we obtain a resolution
enhancement equal to three. By a simple geometrical
computation, the distance between the zero and the
first diffracted orders in the CCD plane is ?x
=?d2/p, where p is the grating pitch and d2is the dis-
tance between the grating and the CCD. To obtain an
NA improvement equal to three it is necessary to ful-
fill the condition ?x=NPCCD; the distance d2has to be
If d2does not match Eq. (1) exactly, we have ?x
=NxPCCD, and the improvement of the NA is given by
1+2?Nx/N? with Nx?N. Only if we get the resolution
improvement of three is it possible to obtain at the
same time an automatic self-assembling of the vari-
ous spatial frequency of the objects in the reconstruc-
tion image plane. In Fig. 1(c) the geometric diagram
is shown, concerning the reconstructed images, to ex-
plain how it happens. It results that the lateral dis-
tance between the three reconstruction images is
?? = NxPr,
where Pr=d1/d2PCCDis the reconstruction pixel for
the double-step reconstruction
[16,17]. In fact, all the reconstructions are obtained
by the two-step process computing two Fresnel inte-
grals: first we reconstruct the hologram in the grat-
ing plane, and then we propagate this complex field
up to the object plane . In the double-step recon-
struction, the Pr depends only on the ratio d1/d2,
while it does not depend on the distance D=d1+d2
between the object and the CCD, different from what
occurs in a typical single-step Fresnel reconstruction
process, where Pr=?D/?NPCCD?. It is worth noting
that in DH, when the reconstruction window is not
large enough to contain the entire spatial frequency
band of the object, the object signal is wrapped
within the reconstruction window. In our case, the
object lateral dimension is ?N+2Nx?Pr, while the re-
construction window is only NPr. In Fig. 1(c) (I) only
the central and right reconstruction image windows
are shown, corresponding to ?N+Nx?Pr. The left re-
constructed image has been omitted for clarity. Letter
A represents the central point of the field of view of
each reconstructed hologram. In Fig. 1(c) (I), the por-
tion of the right image that includes the gray (red on-
line) letter A falls outside the reconstruction window
and, therefore, is wrapped and re-enters in the recon-
struction window from the left side. In this case the
multiple reconstructed images are incorrectly super-
imposed in the reconstruction plane. Only when Nx
=N [as in Fig. 1(c) (II)], when the gray letter is per-
fectly superimposed to the black one, the recon-
structed images, corresponding to the different dif-
fraction orders, are automatically and perfectly
superimposed. The fulfillment of the condition Nx
=N assures the self-assembling of the object spatial
frequencies in the reconstructed plane and allows one
to enhance the resolution with a factor of 3 without
introducing the GTF in the diffraction propagation
In Fig. 2 are shown different numerical reconstruc-
tions obtained for different values of the distance d1
and d2. The grating has a pitch of 25 ?m; therefore,
to fulfill the condition of Eq. (1), d2has to be 21 cm.
Looking at Fig. 2, it is clear that only when d2
=21 cm the three reconstructions are superimposed.
Moreover, we can assert that the value of distance d1
does not affect the self-assembling properties but
Spatial frequencies are wrongly assembled in (a) and (b)
and are correct in (c) and (d).
Reconstructions adopting different values ?d1,d2?.
assembling mode, (c) geometric scheme to show the automatic superimposition in the self-assembling mode.
(Color online) (a) DH setup with a grating, (b) geometric scheme of the reconstruction without a grating in self-
December 1, 2009 / Vol. 34, No. 23 / OPTICS LETTERS
only the value of the pixel of reconstruction and,
therefore, the field-of-view width. Fixing the values
?d1d2?=?32,21? (units are centimeters), we measured
the effective resolution improvement. Figures 3(a)
and 3(b) show the numerical reconstructions without
and with the grating in the setup. The profiles corre-
sponding to the reticule with pitch of 100 ?m (line 1)
are shown in Figs. 3(c) and 3(d), respectively. Figures
3(e) and 3(f) show the profiles of the reticules with
pitch of 50.14 ?m (line 2) and 31.63 ?m (line 3) ob-
tained by the hologram recorded with the grating in
the setup. The last one has approximately the same
contrast of the grating with a pitch of 100 ?m ob-
tained by the reconstruction of the hologram ac-
quired without the grating in the setup. Therefore,
the resolution enhancement is effectively equal to
three. Moreover, to show how the use of the GTF in
the reconstruction process affect the image quality, a
magnified view of a portion of the reconstructed tar-
get using the method that adopts the GTF  and
by the self-assembling approach proposed here are
shown in Figs. 4(a) and 4(b), respectively. It is clear
that the noise introduced by the GTF hinders some
relevant information in the super-resolved image.
Finally, we recorded a hologram of a biological
sample, a slice of a fly’s head. The image of the slice
as it appears at an optical microscope is shown in
Fig. 5(a). In this case a 2D grating was used. There-
fore, the resolution increases three times along two
directions. Figures 5(b) and 5(c) show the reconstruc-
tions of the holograms with and without the grating,
respectively. The insets in Figs. 5(b) and 5(c) show
how fine details of the object are clearly unresolved in
the optical configuration without a grating, while
they are visibly resolved with the 2D grating and
Research funded from the EU FP7/2007–2013,
grant 216105. The authors thank A. Finizio and V.
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100 ?m along the line in the inset of (a) showing that the grating is not well resolved. (d), (e), and (f) Profiles of lines 1, 2,
and 3 indicated in (b).
(Color online) Reconstructions (a) without and (b) with the grating (super-resolved image). In (c) profile of the
in [15,16] and (b) by using self-assembling approach.
(a) Super-resolved image obtained by the method
microscope, (b) DH reconstructed image without super-
resolution, (c) super-resolved image.
(Color online) (a) Slice of fly’s head eye at optical
OPTICS LETTERS / Vol. 34, No. 23 / December 1, 2009