. INTRODUCTIONuperresolution is one of the most applicable topics in op-ical data processing and imaging, since it can substan-ially reduce the cost of imaging systems and provide im-roved performance in exchange for payment in domainshat are a priori available, i.e., in features that are a pri-ri known as not being occupied with information [1,2].uch domains can be time [3], wavelength [4], polariza-ion [5], field-of-view [6], and gray-level coding [7]. Theseethods used to improve the spatial resolution of optical
maging systems have been unified in the Wigner domainy means of space–bandwidth product adaptation [8–10].One of the most appealing approaches used to obtain
uperresolution is related to the temporal degree of free-om; that is, the one in which the a priori knowledge in-olves temporally restricted objects. But time multiplex-ng can be implemented in a wide variety of ways [1–3]nd can be applied to several fields in which digital holog-aphy plays an important role. Examples of these fieldsan be found in lithography [11], microscopy [12,13], re-ote sensing [14], far-field imaging [15,16], and three-
imensional (3-D) coherence tomography [17,18]. Two in-eresting configurations previously discussed in the fieldf superresolution of temporally restricted objects involvewo gratings that are transversally shifted: One is at-ached to the object and the other to the detection plane3,19,20]. The fact that both gratings are in contact withhe two planes sometimes caused problems in applyinghis approach to some applications where the objectlane, for instance, was not accessible. The attachment ofratings to both object and image planes can be solved byrojecting the grating onto the object [21] and by com-uter decoding with a digitally generated grating [22],espectively.
Another interesting approach to superresolution in-olves two or three fixed gratings, and the superresolved
mage is obtained by payment in the field of view [23–25].he advantage of this concept is the fact that these grat-
ngs are in a noncontact position with the object/detector.n the case when only two gratings are used [23], one ofhem is not positioned between the object and the detectorut rather outside this volume, which imposes severalroblems on the practical realization of such a setup. Theolution for this problem required three fixed gratings24,25], all of them positioned between the object and themage planes.
In this paper we propose an approach that combineshe advantages of the previously mentioned two conceptsr, in other words, minimizes their disadvantages. We usewo rather than three gratings. Furthermore we do notay with the field of view but rather do time averaging,hich is a more acceptable payoff, and yet our gratingsre not in contact and both are located between the objectnd the image planes. In order to achieve this outcome,he two gratings are shifted in axial rather than transver-al movement. However, obviously the superresolving ef-ect is transversal. Another very important advantage ofhe proposed configuration is that it provides not an ap-roximation but rather an exact superresolution also forhe spatially coherent case, something that is not ob-ained in the previously developed time multiplexingpproaches.In Section 2 we present a full mathematical derivation
f the proposed approach. Section 3 includes a numericalnvestigation and a proof of concept. The paper is con-luded in Section 4.
. MATHEMATICAL DERIVATIONn our mathematical analysis we will assume 1-D objectsn order to simplify the mathematical treatment. How-ver, the expansion for the 2-D case is straightforward.
007 Optical Society of America
T4ift
iijodamdrsae
ttdpfmts
e
waitdF
Ttfe
Aaaaubg
fftffwp
wiEllErowtu
tggitgidst
Afi
T
Fai
Zalevsky et al. Vol. 24, No. 10 /October 2007 /J. Opt. Soc. Am. A 3221
he suggested configuration may be seen in Fig. 1. TheF system having a band-limited aperture in the 2F planemitates a regular imaging system with a magnificationactor of 1. The aperture placed in the 2F plane imitateshe aperture of the imaging lens.
The first grating, called the encoding grating, is placedn the object space just between the object and the firstmaging lens at an arbitrary distance of z0 from the ob-ect. The same happens at the image plane where a sec-nd grating, called the decoding grating, is placed at aistance of z0 in front of the image plane. Both gratingsre moved axially instead of transversally, and this move-ent is produced in a synchronous way but in different
irections. Figure 1 shows this effect by means of two ar-ows that point toward each other. For this reason, we as-ume that z1�t�=z0+vt=−z2�t�, z1�t� and z2�t� being thexial distances from the object and image planes to thencoding and decoding gratings, respectively.
We will denote the input object by g�x�. To fully denotehe effect of the grating positioned a given free-space dis-ance from the object, we free-space propagate the fieldistribution, multiply it by the grating, and then back-ropagate the light to the original plane. Now that the ef-ect of the grating has already been taken into account, itay be eliminated from the analytical course of the sys-
em’s analysis. We will start with the derivation of thepatially coherent case and then go to the incoherent one.
The field distribution after free-space propagation of z1quals
Uz1�x� =� G���exp�j��z1�2�exp�j2�x��d�, �1�
here G��� is the Fourier transform of the input objectnd � is the spatial frequency. The quadratic phase factors the transfer function of free-space propagation by dis-ance z1 under Fresnel approximation. We multiply thisistribution by the encoding grating, expressed as theourier series:
�n
An exp�− j2��0nx�. �2�
hen we perform free-space backpropagation of −z1 to ob-ain the field distribution in the input plane while the ef-ect of the grating is included. A simple but lengthy math-matical derivation provides the following:
ig. 1. (Color online) Schematic sketch of the setup. The grayrrows (red online) show the axial movement of both the encod-ng and the decoding gratings.
U0�x� = �n
An exp�− j��z1n2�02�
�exp�− j2�n�0x�g�x + �z1n�0�. �3�
s expected, the combined grating–object is equivalent toset of displaced copies of the object, each one traveling
t a different angle (as given by the linear phase factors)nd with a diffraction-order-dependent phase. We havesed the hat over the amplitude to stress that this distri-ution is a virtual one, which replaces the object and therating for any plane after the grating.
The imaging from the input plane to the output is per-ormed in three steps. First, the input is Fourier trans-ormed (with scaling �F), which gives the distribution athe aperture plane. Then we multiply by the apertureunction, which is assumed to be a rectangle function withull width �. Finally, a new Fourier transformation (alsoith scaling �F) provides the output plane complex am-litude:
U4F�x�� = �n
An exp��i�z1n2�02� ·� rect��F�
��G�− � + n�0�
�exp�− j2�n�z1�0��exp�j2��x��d�, �4�
here x� is the spatial variable in the output plane andrrelevant amplitude terms ��F� have been dropped.quation (4) shows that the spectrum of the input is rep-
icated by the diffraction orders of the grating, each rep-ica being affected by a different tilt and global phase.ach replica of the spectrum is displaced, allowing theectangular aperture to pass a different bandpass for eachrder. The purpose of the second grating in the systemill be to reposition these bandpasses in the proper loca-
ions, and the movement of the grating will remove thendesired cross terms.Equation (4) represents a virtual distribution, as the
rue amplitude should consider the effect of the secondrating. For this purpose, we perform a free-space propa-ation of distance z2 in order to reach the decoding grat-ng plane. As in the propagation between the input andhe first grating planes, this operation is done in the an-ular spectrum of the distribution in Eq. (4) by multiply-ng by the free-space propagation associated with axialistance z2. The resulting distribution is multiplied by theecond grating, whose fundamental frequency is assumedo coincide with that of the first grating:
�m
Bm exp�− j2��0mx�. �5�
new propagation by distance −z2 gives the final outputeld distribution:
U4F�x�� = �n,m
AnBm exp�j���02�z1n2 − z2m2
− 2z1nm�� ·� rect�� + m�0
�/�F �G�− � + �n − m��0�
�exp�− j2����0�z1n + z2m��exp�j2��x��d�. �6�
his equation is significantly simplified if we assume that
tc
T
Irsdtd
Ttttlgtts
Tson
wi
ltscawd
Ettwt
ictht
fcht
Ftwp�orqw
3TaaagF
wwlaINtsie2tFfswlh(tlIp
3222 J. Opt. Soc. Am. A/Vol. 24, No. 10 /October 2007 Zalevsky et al.
he gratings are moving in opposite directions and syn-hronously in time:
z1 = − z2 = z�t�. �7�
hen the time-dependent output distribution is
U4F�x�,t� = �n,m
AnBm exp�j���02z�t�
��n − m�2� ·� rect�� + m�0
�/�F �G�− � + �n − m��0�
�exp�− j2����0z�t��n − m��exp�j2��x��d�. �8�
nspection of Eq. (8) shows that in all terms, except in theectangle function, the indices are paired. This corre-ponds to a cancelling of both quadratic and linear order-epending terms. To achieve this cancellation, we performime integration. The integration over the time-ependent terms gives
1
T�0
T
expj���0z�t���0�n − m�2 − 2��n − m��dt = �n,m.
�9�
hat is, the time integral is nonzero only for n=m when itakes a unity value. From a physical point of view, theime integration gets rid of the cross terms, owing to theime variation of phase of these terms. Eventually, theight diffracted in a given diffraction order by the firstrating is coupled only with the corresponding order ofhe second grating. Then the final output consists in theemporal averaging of the field given by Eq. (8), which re-ults in
�U4F�x�,t�� =� �n
AnBn rect�� + n�0
�/�F ��G�− ��
�exp�j2��x��d�. �10�
his equation gives the superresolved output. Note thatimilar to the regular approach of time multiplexing, inrder to realize the time averaging properly here, oneeeds the time of integration, T, to fulfill
T �1
�V�02 , �11�
here V is the velocity of movement of the grating and Ts the integration period.
Equation (10) is the classical expression for superreso-ution [1], representing the Fourier transform of the spec-rum of the object multiplied by a synthetic aperture. Theynthetic aperture [in square brackets in Eq. (10)] in-reases the width with respect to the original rectangularperture by a number equal to the highest index n forhich the coefficients AnBn are not zero, i.e., the highestiffraction order of the encoding/decoding gratings.It is worth noting that the output field expression in
q. (10) gives a closed form for the output where the syn-hetic aperture can be factorized. This is in contrast withhe case when lateral movement of the gratings is used,here it is needed to assume the approximation of a syn-
hetic aperture much larger than the spectral width of the
nput [1,2]. This fact constitutes a great advantage in thease of objects that are not band limited (like many prac-ical cases with abrupt borders). These cases can beandled exactly and precisely by the method proposed inhis paper.
We have obtained a well-defined coherent transferunction (CTF). Thus for the fully spatially incoherentase, when the system becomes linear in intensity, the be-avior is governed by the optical transfer function (OTF)hat is obtained as the autocorrelation of the CTF:
OTF��� =� �n,n�
AnBnAn�* Bn�
* rect�� + �/2 + n�0
�/�F ��rect�� − �/2 + n��0
�/�F �d�. �12�
or both coherent and incoherent cases, the most advan-ageous situation for increasing the system’s bandpass ishen the frequency of the grating is such that the band-asses of the synthetic aperture are touching side by side�=�F�0�. As a particular case of interest, a grating withrders from −N to N with equal amplitude will result in aectangular synthetic aperture of size �2N+1��. Conse-uently, the incoherent OTF is just a triangle functionith a total width of 2�2N+1��.
. NUMERICAL INVESTIGATIONhe schematic configuration of Fig. 1 was numerically re-lized for the 2-D case. A superresolution factor of 5 wasimed for. The movement of the grating was between 0.1nd 0.9 of the critical distance Zc, beyond which the an-ular spectrum approach for numerical computing of theresnel–Kirchoff discrete convolution is not accurate [26]:
Zc =�x2N
�=
Lx�x
�, �13�
here �x is the spatial resolution of the object, � is theavelength, N is the number of pixels used for the simu-
ation, and Lx is the size of the object (i.e., the size of thexis of the matrix that is being free-space transformed).n the simulation we chose �x=10 �m, �=0.5 �m, and=256 (i.e., the matrix was N�N). Therefore the dis-
ance Zc in this case was 5.12 cm. The aperture was aquare aperture of 1/16 of the matrix dimensions, and wentended to improve the resolution for a factor of 5 in ev-ry axis. The obtained results are seen in Fig. 2. In Fig.(a) we present the intensity point-spread function ob-ained without applying the superresolving approach. Inig. 2(b) we present the point-spread function obtained
or various encoding/decoding gratings designed to have auperresolved factor of 5. On the left-hand side of Fig. 2(b)e see the point-spread function obtained with the gray-
evel (phase and amplitude) grating, and on the right-and side it was obtained with the binary phase gratingphase of zero and �). In Fig. 2(c) we present the cross sec-ion of the OTF. The left panel is obtained with the gray-evel grating, and right is with the binary phase grating.n both images one may see the OTF with and without ap-lying the superresolving approach. In the OTF charts
ot
pit3asm
ac
4Iaagp
Ftpg
Zalevsky et al. Vol. 24, No. 10 /October 2007 /J. Opt. Soc. Am. A 3223
ne may indeed see an improvement factor of 5 times inhe spectral band width of the imaging system.
In Fig. 3 we applied the proposed superresolving ap-roach over the USAF resolution target image. The targetmage is seen in Fig. 3(a). The low-resolution image ob-ained after imaging through the imager is seen in Fig.(b). Figure 3(c) presents the reconstructed image afterpplying the axial movement of the gratings. The pre-ented results were obtained for spatially incoherent illu-ination. The reconstruction of Fig. 3(c) is obtained in an
ig. 2. (Color online) Numerical simulations. (a) Point-spread fuhe superresolved system designed for a superresolution factor ofhase grating (right). (c) Cross section of the optical transfer furating; right, the binary phase grating.
ll-optical way. No image postprocessing was applied ex-ept for subtracting the low-resolution image of Fig. 3(b).
. CONCLUSIONSn this paper we have presented a novel superresolvingpproach based upon time integration of light captured indetector during synchronized axial movement of two
ratings positioned between the object and the imagelanes. In many methods that rely on encoding of the an-
for intensity for the regular system. (b) Point-spread function forgray-level (phase and amplitude) grating (left) and with binary(OTF) for superresolution factor of 5. Left, with the gray-level
nction5 withnction
gmocpm
fmlt
esavlrpwttsa
ATE
R
1
1
1
1
1
FL
3224 J. Opt. Soc. Am. A/Vol. 24, No. 10 /October 2007 Zalevsky et al.
ular information of the input object, an encoding ele-ent (a grating) must be located close to the input
bject plane. This condition limits in principle the appli-ation fields to those situations where the object can behysically accessed, discarding those cases such as re-ote sensing or astronomy where the imaging system is
ar from the object (practical examples where thoseethods can be successfully applied are microscopy and
ithography). In the proposed approach we do not havehis limitation.
The suggested approach has several advantages overxisting configurations. First, the superresolution for thepatially coherent case is obtained without any kind ofpproximation, in contrast to the approach of two trans-ersally moving gratings [27]. Second, both gratings areocated between the object and the image planes withoutequiring physical contact with those planes, which sim-lifies significantly the experimental setup in comparisonith other approaches [3,19–22]. Third, the superresolu-
ion effect is produced without the field-of-view restric-ions seen in other approaches [23–25]. This paper pre-ents a mathematical derivation of the approach as wells its numerical validation.
CKNOWLEDGMENThis work was supported by the Spanish Ministerio deducación y Ciencia under project FIS2007–6026.
EFERENCES1. Z. Zalevsky, D. Mendlovic, and A. W. Lohmann, “Optical
systems with improved resolving power,” in Progress inOptics, Vol. XL, E. Wolf, ed. (Elsevier, 1999).
2. Z. Zalevsky and D. Mendlovic, Optical Super Resolution(Springer, 2002).
3. W. Lukosz, “Optical systems with resolving powersexceeding the classical limit,” J. Opt. Soc. Am. 56,1463–1472 (1966).
4. A. I. Kartashev, “Optical systems with enhanced resolvingpower,” Opt. Spectrosc. 9, 204–206 (1960).
5. W. Gartner and A. W. Lohmann, “Ein experiment zuruberschreitung der abbeschen auflosungsgrenze,” Z. Physik174, 18 (1963).
6. G. Toraldo di Francia, “Super-gain antennas and opticalresolving power,” Nuovo Cimento, Suppl. 9, 426–428(1952).
7. Z. Zalevsky, P. García-Martínez, and J. García,“Superresolution using gray level coding,” Opt. Express 14,5178–5182 (2006).
8. A. W. Lohmann, R. G. Dorsch, D. Mendlovic, Z. Zalevsky,and C. Ferreira, “About the space bandwidth product ofoptical signal and systems,” J. Opt. Soc. Am. A 13, 470–473(1996).
9. D. Mendlovic and A. W. Lohmann, “Space–bandwidthproduct adaptation and its application for superresolution:fundamentals,” J. Opt. Soc. Am. A 14, 558–562 (1997).
0. D. Mendlovic, A. W. Lohmann, and Z. Zalevsky,“Space–bandwidth product adaptation and its applicationfor superresolution: examples,” J. Opt. Soc. Am. A 14,563–567 (1997).
1. X. Chen and S. R. J. Brueck, “Imaging interferometriclithography: approaching the resolution limits of optics,”Opt. Lett. 24, 124–126 (1999).
2. V. Mico, Z. Zalevsky, P. García-Martínez, and J. García,“Superresolved imaging in digital holography bysuperposition of tilted wavefronts,” Appl. Opt. 45, 822–828(2006).
3. V. Mico, Z. Zalevsky, and J. García, “Superresolutionoptical system by common-path interferometry,” Opt.Express 14, 5168–5177 (2006).
4. J. García, Z. Zalevsky, and C. Ferreira, “Superresolvedimaging of remote moving targets,” Opt. Lett. 31, 586–588(2006).
1
1
1
1
1
2
2
2
2
2
2
2
2
Zalevsky et al. Vol. 24, No. 10 /October 2007 /J. Opt. Soc. Am. A 3225
5. R. Binet, J. Colineau, and J. C. Lehureau, “Short-rangesynthetic aperture imaging at 633 nm by digitalholography,” Appl. Opt. 41, 4775–4782 (2002).
6. J. H. Massig, “Digital off-axis holography with a syntheticaperture,” Opt. Lett. 27, 2179–2181 (2002).
7. Ch. K. Hitzenberger, P. Trost, P. W. Lo, and Q. Zhou,“Three-dimensional imaging of the human retina by high-speed optical coherence tomography,” Opt. Express 11,2753–2761 (2003).
8. P. Massatsch, F. Charrère, E. Cuche, P. Marquet, and Ch.D. Depeursinge, “Time-domain optical coherencetomography with digital holographic microscopy,” Appl.Opt. 44, 1806–1812 (2005).
9. D. Mendlovic, A. W. Lohmann, N. Konforti, I. Kiryuschev,and Z. Zalevsky, “One-dimensional superresolution opticalsystem for temporally restricted objects,” Appl. Opt. 36,2353–2359 (1997).
0. D. Mendlovic, I. Kiryuschev, Z. Zalevsky, A. W. Lohmann,and D. Farkas, “Two-dimensional superresolution opticalsystem for temporally restricted objects,” Appl. Opt. 36,6687–6691 (1997).
1. A. Shemer, Z. Zalevsky, D. Mendlovic, N. Konforti, and E.
Marom, “Time multiplexing superresolution based on
2. A. Shemer, D. Mendlovic, Z. Zalevsky, J. García, and P. G.Martínez, “Superresolving optical system with timemultiplexing and computer decoding,” Appl. Opt. 38,7245–7251 (1999).
3. E. Sabo, Z. Zalevsky, D. Mendlovic, N. Konforti, and I.Kiryuschev, “Superresolution optical system using twofixed generalized Dammann gratings,” Appl. Opt. 39,5318–5325 (2000).
4. Z. Zalevsky, D. Mendlovic, and A. W. Lohmann, “Superresolution optical systems using fixed gratings,” Opt.Commun. 163, 79–85 (1999).
5. E. Sabo, Z. Zalevsky, D. Mendlovic, N. Konforti, and I.Kiryuschev, “Superresolution optical system using threefixed generalized gratings: experimental results,” J. Opt.Soc. Am. A 18, 514–520 (2001).
6. J. García, D. Mas, and R. G. Dorsch, “Fractional-Fourier-transform calculation through the fast-Fourier-transformalgorithm,” Appl. Opt. 35, 7013–7018 (1996).
7. A. Shemer, Z. Zalevsky, D. Mendlovic, E. Marom, J. Garcia,and P. G. Martinez, “Improved superresolution in coherent
