. INTRODUCTIONigital holography permits reconstruction of both ampli-
ude and phase of imaged objects. The amplitude distri-ution of the imaging beam is added in the hologramlane with a reference wave and the hologram is recordedy using a CCD camera. Then the object wavefront is re-onstructed numerically by simulating the backpropaga-ion of the complex amplitude of the optical beam usinghe Kirchhoff–Fresnel propagation equations.1–4 How-ver, for both off-axis4 and on-axis5 holography, the finiteumber of recorded pixels and the size of the CCD limithe resolution of the digital holographic approach. Someechniques have been proposed in the past to overcomehis limitation. One can classify them into two groups:hase-shifting digital holography (PSDH) techniques andolographic synthetic aperture generation methods.PSDH uses both an in-line setup to decrease the fringe
pacing and a phase shifting of the reference beam tovaluate directly the complex amplitude at the CCDlane and to eliminate the conjugate images completely.6,7
SDH has also been applied to three-dimensionalicroscopy,8,9 encryption,10 and wavefront
econstruction.11 PSDH improves the number of resolvedbject points contained in the final image by approxi-ately a factor of 2 in comparison with conventional digi-
al holography.A significant resolution improvement is obtained using
olographic synthetic aperture methods. Some of theseethods are based on the generation of a synthetic aper-
ure by combining different holograms recorded at differ-
nt camera positions to construct a larger digitalologram.12,13 The resolution improvement factor is equalo the number of recorded holograms. Other approacheso generate synthetic apertures are based on superresolu-ion techniques.14–25 A synthetic enlargement of the sys-em aperture is a well-known and widely used tech-ique16,20,21 to improve the limited resolving power of op-ical systems. Because of the wave nature of light, eachptical imaging system is limited in resolution linked to aow-pass filtering in the frequency space. The cutoff fre-uency in the spatial-frequency domain is defined inerms of its numerical aperture (NA) and the wavelengthf the illumination light. One can realize that both im-roving the NA of the imaging system and/or decreasinghe wavelength result in a resolution enhancement of theptical system. But in many cases these options are diffi-ult, complex, and not always possible to achieve. Thus,he basis of superresolution is to produce a synthetic en-argement in the system aperture without changing thehysical dimensions of the lenses or the illuminationavelength.Many attempts had been proposed over the years for
uperresolution imaging based on a certain a priorinowledge about the object as its time,20–22
olarization,23,24 or wavelength independence.25 All ofhese parameters are involved in the information capacityheory,16,17,19 which gives an invariance theorem for theumber of degrees of freedom of an optical system. Thisheorem states that it is not the spatial bandwidth buthe information capacity of an imaging system that re-
006 Optical Society of America
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Mico et al. Vol. 23, No. 12 /December 2006 /J. Opt. Soc. Am. A 3163
ains constant. Thus, it is possible to extend the spatialandwidth by encoding or decoding the additional infor-ation onto unused parameters of the imaging system.In the past years, Chen and Brueck26 and Schwarz et
l.27 have implemented an interferometric approach ap-lied to lithography and microscopy imaging, respectively.hey used off-axis illumination to downshift the high-
requency components of the object spectrum in such aay that they are transmitted through the system aper-
ure. Then, by means of an optical interferometric record-ng, these transmitted components are shifted back to-ard their original position at the object spectrum and a
ynthetic enlargement of the system aperture is done byncoherent addition of the individual recorded intensities.
ico et al.28 obtained the same effect of incoherent addi-ion by using an array of mutually incoherent lightources [a vertical-cavity surface-emitting laser (VCSEL)rray] for recording all the spatial-frequency bands in aach–Zenhder interferometric configuration. In contrast
o the setups reported in Refs. 26 and 27, the transmis-ion of all the spatial frequencies is done at once, in termsf spatial multiplexing of all the incoherent illuminationources. In the method of Mico et al.,28 a large factor ofmprovement is obtained without penalty for the complex-ty of the system. In fact, the authors have used fiveources simultaneously (fivefold increase of the systempatial-frequency bandwidth) as compared with the threellumination sources for the system presented in Refs. 26nd 27.Although other authors have implemented incoherent
llumination sources to increase the resolution of opticalmaging systems,29,30 the approach presented in Ref. 28rovides higher light efficiency due to the high opticalower of the VCSEL array. Moreover, no theoretical limitegarding the limited size of the extended incoherentource restricts the system because of the great number ofingle VCSELs that can be present in the VCSEL array.n addition, the fact that the VCSEL elements can beemporally modulated up to several gigahertz implieshat any synthetic transfer function can be synthesized byemporally varying the relative amplitudes of each sourcen the array. This can be done because the synthetic ap-rture generated by the suggested approach28 is a convo-ution operation between the VCSEL line array and theoherent transfer function (CTF) of the system.
Recently, Mico et al.31 have extended the optical systemf Ref. 28 to two-dimensional (2D) objects. However, theain disadvantage of this setup is that, owing to the dif-
erence between the imaging and the reference arm in thenterferometer, the holograms for the different band-asses are incorrectly overlapped, and so it impedes theecording of all frequency slots in a single exposure. More-ver, the stability of the system, as well as that for Ref.8, suffers from the splitting of light for the referenceeam before illuminating the object.In this paper we present a new experimental configu-
ation that significantly increases the robustness of theystem as well as opens the possibility of performing themaging with enhanced resolution in a single exposure.he system uses a collection of mutually incoherent pointources, at different lateral positions, which serve aspherical and tilted illuminations for the object (i.e., every
oint source gives a spherical wave with a different ori-in). The imaging and reference beams are separated by aeam splitter (BS) after the low-NA microscope lens.herefore, the full interferometer is after the lens, reduc-
ng the sensitivity of the system to vibrations and/or ther-al changes. One of the arms is filtered using a pinhole
rray so that the dc of each tilted illumination beam is se-ected and in addition serves as the set of tilted referenceeams. With this new optical system the frequency slotsf the spectrum of the superresolved image can be sepa-ated, in contrast with the system shown in Ref. 31. Thus,he superresolution system can be implemented by meansf a physical 2D array of sources (such as a VCSEL array)n a single exposure. However, as an alternative to theCSEL array, the implementation can be made sequen-
ially displacing a single coherent source and recording aet of exposures for a later digital combination.
The paper is organized as follows: Sections 2 and 3 givepreliminary and theoretical analysis of the experimen-
al setup, respectively. In Section 4 experimental resultsnd the reconstruction process are presented. Section 5oncludes the paper.
. PRESENTED APPROACH: SOME SYSTEMONSIDERATIONShe optical setup is shown in Fig. 1. The input object is
lluminated by a collection of spherical waves, each oneoming from different point sources in a 2D array configu-ation. A microscope objective images the object onto aCD. Behind the microscope objective, the first beamplitter (BS1) splits the imaging beam into two beams.ne of them (right branch in Fig. 1) allows image forma-
ion of the object onto the CCD after reflection in mirror1. We call this branch as imaging arm. In the other op-
ical beam path (left branch in Fig. 1), some elements arentroduced to perform optical image processing. Thisranch is called the reference arm. In the following we ex-lain in detail the configuration of the reference arm inhis new superresolution technique.
Diffraction theory applied to optical imaging showshat the object spectrum (Fourier transformation of thebject amplitude distribution) is obtained in the imagelane of the illumination source through the imaging op-ical system. In the on-axis illumination case, a pinholelaced at the Fourier plane in the center of the referencerm, coinciding at the axial position of the source image,ill transmit only radiation representing the dc term.his procedure gives a uniform reference beam that cane used to perform the interferometric recording in theCD plane.When off-axis illumination is used, the full object spec-
rum is displaced at the Fourier plane and the zero fre-uency no longer coincides with the pinhole location and,n general, the amount of light passing the pinhole will beegligible. To optimize the intensity of the filtered refer-nce beam, the transmission of the central part in the ob-ect spectrum can be performed by moving the pinhole tohe off-axis position defined by the object spectrum center.he same effect can be obtained using a mask with an ar-ay of pinholes when a certain off-axis illumination con-guration is given, instead of shifting a single pinhole.
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3164 J. Opt. Soc. Am. A/Vol. 23, No. 12 /December 2006 Mico et al.
his pinhole mask must have a pinhole corresponding toach of the 2D illumination point-source arrays. The loca-ion of each pinhole is determined by the magnification ofhe microscope objective (see Fig. 1). Thus, for the case ofCSEL source illumination, if several off-axis sources areperated at the same time, the previously described pin-ole mask placed at the Fourier plane will transmit inarallel each of the reference beams corresponding to theifferent replicas of the object spectrum transmitted byhe imaging system.
It is important to note that the input object could be il-uminated by off-axis illumination angles higher thanhat defined by the NA of the microscope objective.30,32 Asgeneral rule, we can say that the simpler the lens, the
igher the off-axis illumination angle. A resolution im-rovement by a factor of 2 is always achievable using off-xis illumination with a maximum illumination angle
ig. 1. (Color online) Experimental setup. The reference andmaging arms are marked with dashed and dotted frames, re-pectively. The divergence distance d for the illumination in themaging arm is depicted by broken segments ending in arrows.
qual to the NA of the imaging lens and a postprocessingtage.31,33 But in our case, when a low-NA objective mi-roscope is used (that means a simpler lens system inomparison with higher-NA objectives), a resolution im-rovement factor higher than 2 is possible. In fact, for theetup presented in this paper, the resolution improve-ent factor is 3.Assuming mutually incoherent point sources, each
ransmitted reference beam is coherent with its corre-ponding frequency band transmitted by the imaging armnd incoherent with the others. This property permits theecording process of different frequency bands at theame instant by mixing both the reference and the imag-ng beams using a second beam splitter (BS2). Moreover,ach filtered reference beam has a slightly different car-
ig. 2. Off-axis illumination case. The chief ray for each off-axisource (thick solid line) impinges on the CCD center. An approxi-ate size and location of the virtual image of the pinhole mask
re shown.
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Mico et al. Vol. 23, No. 12 /December 2006 /J. Opt. Soc. Am. A 3165
ier frequency that shifts back the corresponding fre-uency band to its original position in the object spec-rum. A bias frequency is introduced by tilting mirrors M1r M2 to avoid the overlapping of the hologram diffractionrders with the dc term. Simple digital postprocessing iserformed to obtain the reconstruction. This digital post-rocessing involves an inverse Fourier transformation ofhe recorded multiple hologram, a filtering and centeringrocess of the first diffraction hologram order, and finallydirect Fourier transformation of the previous centered
pectrum.For the previously mentioned system, the recording of
he hologram at the CCD requires long integration timesue to the low intensity in the reference beams, owing toheir divergence and angular shift. To eliminate this ef-ect, a lens is placed in the reference arm (see Fig. 1). Theurpose of this lens is to redirect the light from each ref-rence beam onto the CCD, allowing a more efficient ho-ographic recording (Fig. 2). In the ideal case, the lens islaced at the same plane as the pinhole mask, acting as aeld lens. For this purpose, the focal length of the lens ishosen so that the image of the exit pupil of the micro-cope objective lies on the CCD. Under this condition, thehief rays for every source (and therefore the referenceeams) are centered on the CCD. The position of the pin-ole array through the field lens is not affected. Thus, thearrier frequency of the holograms for each source is theame as it would be without the lens, providing the properverlapping of the frequency bands [see Fig. 3(a)].
For a real situation the field lens cannot be attached tohe pinhole mask, but an axial displacement is needed.he effect of the lens can be easily interpreted consideringhe image of the pinhole mask through this lens. The im-ge is virtual and it is located prior to the original maskosition and with lateral magnification (Fig. 2). The axialhift of the pinhole array image implies a different curva-ure of the reference and the imaging beams (associatedith divergence distances d and d� shown in Figs. 1 and, respectively), and the recorded holograms result in aonuniform carrier frequency along the hologram, equiva-
ent to a defocusing in the Fourier transform of the holo-ram. The defocusing can be compensated by digitallyultiplying the recorded hologram by the appropriate
pherical phase factor. On the other hand, the scalehange produces an outward lateral shift of the position ofhe pinhole images, and thus the carrier frequencies ofhe holograms for each source increase. The modificationf the carrier frequencies generates an incorrect overlap-ing of the frequency bands in the spectrum [Fig. 3(b)]. In
ig. 3. Lens selection process: (a) Synthetic aperture generatedhe different frequency bands (reconstruction is not possible), (c)he separated processing of the frequency bands and correct relo
his situation the reconstruction of the hologram would berong. Moreover, a separate processing of every bandpass
s not possible due to overlapping. To cope with this prob-em, the axial shift of the lens (that changes the pinhole
ask magnification) can be adjusted such that the fre-uency bands are not overlapped [Fig. 3(c)].Thus, the lens placed in the reference beam needs to ac-
omplish two conditions. It must image the exit pupil ofhe microscope objective onto the CCD to optimize lightntensity of the reference. And on the other hand, it mustave a magnification that avoids the overlapping of theifferent frequency bands. Notice that the second condi-ion is only necessary to work in a single exposure, that is,hen all the point sources operate simultaneously. Other-ise, if the process is done sequentially, the overlappingoes not generate any implementation problem.
. THEORETICAL SYSTEM ANALYSISn this section we review the mathematical foundations ofhe experimental setup diffraction analysis. Let us firstnalyze the imaging arm of the system (Fig. 1). An inputbject f�x ,y� will produce a scaled image in the outputlane, convolved with the point-spread function (PSF) ofhe imaging system. We will assume that the CFT of theystem is a circular aperture with radius �� , circ�� /���, �eing the radius in the spatial-frequency domain. Thus,he PSF is its inverse Fourier transform, related to therst-order Bessel function, that we will denote asisk��� r�, where r is the radial coordinate in the outputlane. The complex distribution at the image plane whenhe object is illuminated by a centered source, underresnel approximation, is
U�x,y� = �f�−x
M,−
y
M�exp�jk
2d�x2 + y2� � disk���r�,
�1�
here �x ,y� are the spatial coordinates on the outputlane and k=2� /� is the wavenumber. Note that the im-ge is scaled according to the image magnification �M�nd the phase factor multiplying the image diverges fromdistance d from the output plane, that is, from the in-
ermediate image of the source (see Fig. 1).An m�n 2D source array is used to illuminate the in-
ut object. For a single off-axis source, the only modifica-ion is a shift in the spatial coordinates in the quadratichase factor. Naming �xm ,yn� the coordinates of VCSELources in the source plane, their images in the interme-
t the field lens (correct overlapping), (b) incorrect overlapping ofted frequency bands obtained using a suitable lens that permitsto obtain the desired synthetic aperture shown in (a).
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3166 J. Opt. Soc. Am. A/Vol. 23, No. 12 /December 2006 Mico et al.
iate aerial image will be displaced according to the mag-ification given by the lens between the source and theource image plane �Ms�. Thus for a single source the am-litude distribution on the CCD through the imaging armf the setup is given by
Um,nI �x,y� = �f�−
x
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k
2d��x − Msxm�2 + �y − Msyn�2�
� disk���r�, �2�
here the superscript I stands for the imaging arm andhe subscripts give the indices of the source in the array.
Concerning the reference arm of the system (Fig. 2), aingle source will give an aerial intermediate image, justs in the imaging arm. In addition, the lens close to theinhole mask will axially shift and scale the intermediatemage. As a whole, the light impinging on the CCD williverge from a distance d� and a lateral position affectedy a magnification M�s (both parameters depend on theens layout in the experimental setup). With a the dis-ance between the mask and the lens and F the focalength of the lens, the distance d� is
d� =a�a − d� + F2
F − a. �3�
nd the magnification is modified as follows:
Ms� = Ms
F
F − a. �4�
he amplitude distribution incoming onto the CCD fromhe reference arm for the �m ,n� source is
Um,nR �x,y� = exp�j
k
2d���x − Ms�xm�2 + �y − Ms�yn�2� . �5�
ote that we assume the pupil function of the field lensoes not trim the reference beam. An additional linearhase factor, exp�j2�Qx�, playing the role of a carrier withrequency Q, can be introduced by tilting one of the mir-ors in the reference arm.
Thus, the overall amplitude that impinges on the CCDomes from the addition of Eqs. (2) and (5), and it giveshe following intensity distribution:
Im,n�x,y� = Um,nI �x,y� + Um,n
R �x,y�exp�j2�Qx� 2. �6�
ote that the carrier introduced in the x axis is the sameor all sources. Equation (6) can be split into four terms:1�x ,y� ,T2�x ,y� ,T3�x ,y�, and T4�x ,y�.
Im,n�x,y� = 1 + Um,nI �x,y� 2 + Um,n
I �x,y��Um,nR �x,y��*
�exp�− j2�Qx� + �Um,nI �x,y��*Um,n
R �x,y�ej2�Qx
= T1�x,y� + T2�x,y� + T3�x,y� + T4�x,y�. �7�
Equation (7) represents the hologram recorded on theCD for a single source the centered at the �xm ,yn� posi-
ion. In the reconstruction procedure, we perform digi-ally an inverse Fourier transformation of Eq. (7) to ana-
yze each term separately. The first term, T1�x ,y�=1, isonstant and its Fourier transform �T1�u ,��� is just aelta function centered at the origin. The second term,2�x ,y�, is the intensity of a low-pass version of the images given by the system. Thus, its Fourier transformT2�u ,��� is also centered at the origin, with a width thatoubles the bandpass of the system.The third and fourth terms in Eq. (7) contain the infor-ation about the phase and the amplitude of the object.he third term is
T3�x,y� = ��f�−x
M,−
y
M�exp�jk
2d��x − Msxm�2
+ �y − Msyn�2�� � disk���r���exp�− j
k
2d��x − Ms�xm�2 + �y − Ms�yn�2�
�exp�− j2�Qx�. �8�
The Fourier transform of the third term, �T3�u ,���, is aandpass of the object spectrum shifted by the carrier fre-uency Q and placed at the left position of the central au-ocorrelation term (−1 diffraction order for the holo-raphic recording process).
T3�u,�� = K�� f�Mu +MMs
�dxm,M� +
MMs
�dyn�
� FT−1�exp�jk
2d�x2 + y2���circ� �
����
� FT−1�exp�− jk
2d��x2 + y2��
� ��u + Q −Ms�
�d�xm,� −
Ms�
�d�yn� , �9�
here K is a constant that also includes constant phaseactors, and f is the Fourier transform of the input. Notehat the fourth term in Eq. (7) is the complex conjugate ofhe third term and has a similar meaning. The last deltaunction in Eq. (9) has the following meaning: It implies arequency shift in the transmitted spectral bandpass nec-ssary to guarantee the separation between the −1 dif-raction order from the zero order; on the other hand, itmplies additionally a slight frequency shift due to the dif-erent position of the source in the 2D array �xm ,yn� af-ected by the magnification factor Ms�. This fact will beonsidered below. Now, if we put aside the convolutionith delta in Eq. (9), the convolution between the two re-aining terms must be analyzed separately. The first one
s related to the frequency band of the object spectrumonvolved with the Fourier transform of a spherical phaseactor and is truncated by the pupil function (circ func-ion) of the microscope objective, i.e., restricted to aircular-limited frequency extension in the Fourier do-ain. Then this term is convolved with the Fourier trans-
ormation of another spherical phase factor, which has noffect outside the limited circular region. Thus, a combi-
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Mico et al. Vol. 23, No. 12 /December 2006 /J. Opt. Soc. Am. A 3167
ation of the two spherical phase factors is possible insidehe pupil extension.
T−1�exp�jk
2d�x2 + y2�� � FT−1�exp�− j
k
2d��x2 + y2��
= FT−1�exp�jk
2d� 1
d
1
d���x2 + y2�� . �10�
The combined factor presented in Eq. (10) implies a de-ocusing inside the limited region defined by the circ func-ion of the corresponding frequency band and has to be re-oved to perform a correct reconstruction. Note that if
he lens could be placed on the mask pinhole plane (actings a field lens), no defocusing effect would happened. Toemove this defocusing, we digitally multiply the recordedologram by the complex conjugate of the defocusingpherical phase factor in Eq. (10), prior to its Fourierransforming. After removing the defocusing, the thirderm becomes
T3��u,�� = K� f�Mu +MMs
�dxm,M� +
MMs
�dyn�circ� �
���
� ��u + Q −Ms�
�d�xm,� −
Ms�
�d�yn� . �11�
Equation (11) describes the information of the spectralrequency bandpass of the object spectrum transmitted byhe pupil microscope objective in the case of one sourceentered at the �xm ,yn� position. The carrier frequency Qs controlled by tilting the mirror of the reference beamnd permits the separation of the bandpasses from therigin.
Now, if we consider all the illumination sources fromhe 2D source array, the addition of the refocused thirderms gives
T�˜ 3sum�u,�� = K�
m,n�� f�M�u +
Ms
�dxm�,
M�� +Ms
�dyn��circ� �
����
� ��u −Ms�
�d�xm,� −
Ms�
�d�yn� � ��u + Q,��.
�12�
In Eq. (12), the term between square brackets repre-ents different frequency bands of the object spectrumransmitted by the circ function. Those different fre-uency bands are shifted by means of the convolution op-rations according to distance d�, the magnifying factor
s�, and the bias carrier frequency Q. Note that the fre-uency bands are centered at different positions than theisplacements introduced by the delta functions. In thease that the lens acts as a field lens, the magnificationactors and the axial distances coincide (Ms�=Ms and dd�); then the addition in Eq. (12) can be simplified to
T�3sum�u,�� = Kf�Mu,M��SA�u,�� � ��u + Q,��, �13�
here
SA�u,�� = �m,n
circ�u −Ms
�dxm
��,
� −Ms
�dym
��� . �14�
Equation (14) represents a synthetic aperture, obtainedy adding the shifted versions of the circular aperture ofhe system, analogously to Fig. 3(a). In practice, for thexperimental parameters that we have used, the term
˜3�
sum�u ,�� gives a first diffraction order frequency distri-ution similar to the representation of Fig. 3(c); by per-orming a digital stage of filtering and relocation of theifferent frequency bands it would be possible to obtainll terms overlapping at the desired locations yielding theeneration of the desired synthetic aperture. Moreover,igital processing is not time-consuming, as it involvesnly linear and simple calculations.
Nevertheless, to simplify the setup and to show the ca-abilities of the presented approach, we have used a lasers a point source and moved it to the off-axis positions se-uentially. By the arrangement of the different frequencyands in a second stage, the synthetic aperture is per-ormed digitally and is depicted in Fig. 4. The size of themallest circles corresponds to the NA of the microscopebjective (�� radius). The desired synthetic pupil (dashedircle line of 6�� diameter approximately) is almost cov-red by a set of elemental apertures. The on-axis illumi-ation pupil (dark gray in Fig. 4) is complemented withight additional shifted apertures. Four of them are ac-omplished by source shifts in �X ,Y� orthogonal directionsmedium gray level in Fig. 4). The other four pupils arebtained by off-axis illuminations for each oblique direc-ion (light gray level in Fig. 4). The actual cutoff fre-uency is increased to three times the conventional cutoffrequency of the microscope objective, resulting in a no-able resolution enhancement when an inverse Fourierransformation is done to recover the superresolved ob-ect.
One important advantage of this new superresolutionystem, in comparison with those systems developed pre-
ig. 4. Synthetic aperture generation by the off-axis illumina-ion used in the present approach. The different gray levels rep-esent the frequency bandpasses. The dashed area shows a fullperture of width 6��.
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3168 J. Opt. Soc. Am. A/Vol. 23, No. 12 /December 2006 Mico et al.
iously by us in Refs. 28 and 31, is, aside from the possi-ility of obtaining a superresolution effect for a 2D objectn a single exposure, that the interferometric process iserformed after the imaging system. Thus, the systemhows an easy and simplified configuration that may bedapted for any practical microscopy systems because thenterferometric setup is located between the microscopeens and the CCD.
Further advantages are related to the possibility oforking not only for real-valued objects but also for com-lex or phase objects. Note that no assumption about theroperties of the amplitude object f�x ,y� is made; so forny complex input distribution, the reconstruction of thenput may be obtained.
. EXPERIMENTAL RESULTSor the new proposed superresolution method we havesed a 2D array source that consists of a point laserource with the possibility of lateral displacement. Weave used a Nikon microscope objective with a NA of 0.1nd 5� magnification. Note that a large working dis-ance, as needed for many practical applications, is usu-lly related to a low NA. A 532 nm wavelength laser issed as illumination. The laser beam is spatially filteredroviding the point source necessary for the experimentsnd it is placed onto a motorized mechanical platform tobtain the off-axis illuminations in a sequential mode. Inddition to the on-axis illumination, we have carried outight off-axis illuminations, as depicted in Fig. 4. The in-ut object is a negative 1951 U.S. Air Force high-esolution test. The microscope objective, as given by itsA, has a resolution spot size of 4.9 �m, which approxi-ately corresponds to group 7, element 5 resolution in the
est target, when coherent illumination is used. This situ-tion is shown in Fig. 5 where the low-resolution imagebtained with the 0.1 NA objective is compared with aigh-resolution image done by a Spindler & Hoyer micro-cope objective with a 0.65 NA. The high-resolution images taken as a reference and it is captured at very differentmaging conditions (magnification, working distance,tc.,). Clearly, many spatial frequencies are not trans-erred by the low-NA objective and a low-pass version ofhe object test is obtained.
Then we perform the superresolving approach usinghe recording of eight off-axis holograms and store this inhe computer memory. Taking into account the 0.1 NA ofhe lens, four exposures along the X and Y axes are madey shifting the source 11.5 deg with respect to the opticalxis. The four diagonal bandpasses require an angle of.75 deg on both the X and Y axes. Although the process isone sequentially, Fig. 6 depicts the image addition of theine recorded holograms in the Fourier transformationeight off-axis illuminations and one on-axis case) show-ng the ability of the present approach to work in one stephen a pinhole mask is used. Note that the different fre-uency bands are nonoverlapping due to the proper ad-ustment of the reference arm lens. A lens �f�=100 mm�ttached to the pinhole mask in the reference arm is used.spherical phase factor [see Eq. (6)] has been multiplied
o each recorded hologram to focus the −1 order. However,ecause the lens in the experimental setup is placed close
o the pinhole mask, the defocusing effect is almost negli-ible. Note that the carrier frequency has been shiftedrom zero (tilting the reference mirror) to separate thepatial-frequency slot from the zero-order terms.
By a simple digital postprocessing operation, each ofhe frequency bands of the −1 diffraction order is shiftedo the correct spectral positions and then all of them areuperimposed to synthesize the desired synthetic aper-
ig. 5. (a) Image obtained with the 0.1 NA objective lens and co-erent illumination. (b) High-resolution image obtained with apindler & Hoyer microscope objective with a 0.65 NA and coher-nt illumination. Note that the resolution limit corresponds tohe rectangle shown in (a) and it implies a cutoff frequency of03.0 line pairs/mm (group 7, element 5), which means a small-st resolved detail of 4.93 �m.
ig. 6. Fourier transform of the addition of different recordedolograms. The dc has been blocked to improve contrast.
Fig. 7. Resulting synthetic aperture.
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Mico et al. Vol. 23, No. 12 /December 2006 /J. Opt. Soc. Am. A 3169
ure. An inverse Fourier transform yields the final super-esolved image. Figure 7 shows the generated syntheticperture where five of the partial bandpasses have beenarked with dashed circles for clarity. According to
heory, the resolution has been enhanced by a factor of 3n both the horizontal and the vertical directions and by aactor of 2.4 in the oblique directions. As a consequence,he resolution spot size is reduced until it is 1.64 �m forhe horizontal and vertical directions, which approxi-ately corresponds to the resolution limit given by group
, element 2 of the resolution test target (575.0 lineairs/mm frequency cutoff, 1.74 �m resolution limit). Sohe resolution improvement implies a synthetic aperturef approximately 0.32 synthetic NA. Figure 8 shows aomparison between the image with the 0.1 NA objectiveens used in the conventional imaging [Fig. 8(a)], andith the suggested superresolving approach [Fig. 8(b)]. It
s evident that the improvement of resolution and theesolution expected from the theoretical calculations areffectively obtained.
. CONCLUSIONSn this paper we have presented a superresolving ap-roach for digital holographic microscopy where the su-erresolution effect is described in terms of a syntheticperture generation. The basic idea is to superimposeultiple digital image holograms obtained using different
llumination point sources. Although the approach is dem-nstrated experimentally by shifting a single point sourcen sequential mode, the system can work in a single-xposure approach using the illumination produced by aD array of mutually incoherent sources. Thus, eachhifted illumination beam generates a shift in the objectpectrum in such a way that different spatial-frequencyands are transmitted through the objective lens. An in-erferometric setup after the microscope objective allowshe holographic recording process for each transmittedrequency band. An improvement resolution factor of 3 ischieved, in comparison with the resolution of the testedicroscope objective for a standard configuration.One of the main advantages of our method is the pos-
ibility of recording all the holograms at once without anyhanges in the optical setup. The system, as comparedith previous setups, is simple and robust because the in-
erferometric system is placed after the imaging lens andhe whole process is done on the image space. The ap-roach does not require a high degree of temporal coher-nce for the illumination system because the interfero-
ig. 8. (a) Image obtained with 0.1 NA lens and conventional ilure. The group 9, element 2 corresponding to the resolution lim
etric imaging setup can be adjusted close to zero pathifference; thus, low temporal coherent sources, such asiodes arrays and high-powered VCSEL arrays with lowpatial coherence, can be used as illumination sources.
CKNOWLEDGMENTShis work was supported by Fondo Europeo de Desarrolloegional funds and the Spanish Ministerio de EducaciónCiencia under project FIS2004-06947-C02-01.
EFERENCES1. W. S. Haddad, D. Cullen, J. C. Solem, J. W. Longworth, A.
McPherson, K. Boyer, and C. K. Rhodes, “Fourier-transform holographic microscope,” Appl. Opt. 31,4973–4978 (1992).
2. U. Schnars and W. P. O. Jüpter, “Direct recording ofholograms by a CCD target and numerical reconstruction,”Appl. Opt. 33, 179–181 (1994).
3. S. Grilli, P. Ferraro, S. De Incola, A. Finizio, G. Pierattini,and R. Meucci, “Whole optical wavefields reconstruction bydigital holography,” Opt. Express 9, 294–302 (2001).
4. U. Schnars, “Direct phase determination in holograminterferometry with use of digitally recorded holograms,” J.Opt. Soc. Am. A 11, 2011–2015 (1994).
5. A. Decker, Y. Pao, and P. Claspy, “Electronic heterodynerecording and processing of optical holograms using phasemodulated reference waves �T�,” Appl. Opt. 17, 917–921(1978).
6. I. Yamaguchi and T. Zhang, “Phase-shifting digitalholography,” Opt. Lett. 22, 1268–1270 (1997).
7. P. Guo and A. J. Devaney, “Digital microscopy using phase-shifting digital holography with two reference waves,” Opt.Lett. 29, 857–859 (2004).
8. I. Yamaguchi, J.-I. Kato, S. Otha, and J. Mizuno, “Imageformation in phase-shifting digital holography andapplications to microscopy,” Appl. Opt. 40, 6177–6186(2001).
9. T. Zhang and I. Yamaguchi, “Three-dimensional microscopywith phase-shifting digital holography,” Opt. Lett. 23,1221–1223 (1998).
0. S. Lai and M. A. Neifeld, “Digital wavefront reconstructionand its application to image encryption,” Opt. Commun.178, 283–289 (2000).
1. S. Lai, B. King, and M. A. Neifeld, “Wave frontreconstruction by means of phase-shifting digital in-lineholography,” Opt. Commun. 173, 155–160 (2000).
2. F. Le Clerc, M. Gross, and L. Collot, “Synthetic apertureexperiment in the visible with on-axis digital heterodyneholography,” Opt. Lett. 26, 1550–1552 (2001).
3. J. H. Massig, “Digital off-axis holography with a syntheticaperture,” Opt. Lett. 27, 2179–2181 (2002).
tion. (b) Superresolved image obtained with the synthetic aper-g the proposed method is marked with an arrow.
luminait usin
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3170 J. Opt. Soc. Am. A/Vol. 23, No. 12 /December 2006 Mico et al.
4. Z. Zalevsky and D. Mendlovic, Optical Super Resolution(Springer, 2002).
5. Z. Zalevsky, D. Mendlovic, and A. W. Lohmann, “Opticalsystems with improved resolving power,” in Progress inOptics, Vol. XL, E. Wolf, ed. (Elsevier, 1999), Chap. 4.
6. G. Toraldo di Francia, “Resolving power and information,”J. Opt. Soc. Am. 45, 497–501 (1955).
7. G. Toraldo di Francia, “Degrees of freedom of an image,” J.Opt. Soc. Am. 59, 799–804 (1969).
8. I. J. Cox and J. R. Sheppard, “Information capacity andresolution in an optical system,” J. Opt. Soc. Am. A 3,1152–1158 (1986).
9. W. Lukosz, “Optical systems with resolving powersexceeding the classical limit. II,” J. Opt. Soc. Am. 57,932–941 (1967).
0. A. Shemer, D. Mendlovic, Z. Zalevsky, J. Garcia, and P.García-Martínez, “Superresolving optical system with timemultiplexing and computer decoding,” Appl. Opt. 38,7245–7251 (1999).
1. P. C. Sun and E. N. Leith, “Superresolution byspatial–temporal encoding methods,” Appl. Opt. 31,4857–4862 (1992).
2. M. Françon, “Amélioration de la resolution d’optique,”Nuovo Cimento, Suppl. 9, 283–290 (1952).
3. A. W. Lohmann and D. P. Parish, “Superresolution fornonbirefringent objects,” Appl. Opt. 3, 1037–1043 (1964).
4. A. Zlotnik, Z. Zalevsky, and E. Marom, “Superresolution
with nonorthogonal polarization coding,” Appl. Opt. 44,3705–3715 (2005).
5. A. I. Kartashev, “Optical system with enhanced resolvingpower,” Opt. Spectrosc. 9, 204–206 (1960).
6. X. Chen and S. R. J Brueck, “Imaging interferometriclithography: approaching the resolution limits of optics,”Opt. Lett. 24, 124–126 (1999).
7. C. J. Schwarz, Y. Kuznetsova, and S. R. J. Brueck,“Imaging interferometric microscopy,” Opt. Lett. 28,1424–1426 (2003).
8. V. Mico, Z. Zalevsky, P. García-Martinez, and J. García,“Single step superresolution by interferometric imaging,”Opt. Express 12, 2589–2596 (2004).
9. E. N. Leith, D. Angell, and C.-P. Kuei, “Superresolution byincoherent-to-coherent conversion,” J. Opt. Soc. Am. A 4,1050–1054 (1987).
0. E. N. Leith, “Small-aperture, high-resolution, two-channelimaging system,” Opt. Lett. 15, 885–887 (1990).
1. 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).
2. C. J. R. Sheppard and Z. Hegedus, “Resolution for off-axisillumination,” J. Opt. Soc. Am. A 15, 622–624 (1998).
3. M. G. L. Gustafsson, “Surpassing the lateral resolutionlimit by a factor of two using structured illuminationmicroscopy,” J. Microsc. 198, 82–87 (2000).
