Super-resolution imaging via spatiotemporal frequency shifting and coherent detection

1.8 MB·English

by  Leonid Alekseyev

Super-resolution imaging via
spatiotemporal frequency shifting and
coherent detection
LeonidAlekseyev1,EvgeniiNarimanov1,andJacobKhurgin2
1DepartmentofElectricalandComputerEngineering,
PurdueUniversity,WestLafayette,IN47907
2
1 2DepartmentofElectricalandComputerEngineering,
0 JohnsHopkinsUniversity,Baltimore,MD21218
2 [email protected]

n
a
J Abstract: Diffraction limit is manifested in the loss of high spatial

1 frequency information that results from decay of evanescent waves. As a
2 result,conventionalfar-fieldopticsyieldsnoinformationaboutanobject’s

subwavelength features. Here we propose a novel approach to recovering
]
s evanescentwavesinthefarfield,therebyenablingsubwavelength-resolved
c
imaging and spatial spectroscopy. Our approach relies on shifting the
i
t frequency and the wave vector of near-field components via scattering on
p
o acoustic phonons. This process effectively removes the spatial frequency
. cut-off for unambiguous far field detection. This technique can be adapted
s
c for digital holography, making it possible to perform phase-sensitive
i subwavelengthimaging.Wediscusstheimplementationofsuchasystemin
s
y themid-IRandTHzbands,withpossibleextensiontootherspectralregions.
h
p © 2012 OpticalSocietyofAmerica
[
OCIScodes:230.1040Acousto-opticaldevices;300.6340Spectroscopy,infrared

1
v
7 Referencesandlinks
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1. Introduction
Microscopic imaging is the oldest and one of the most important non-invasive analysis tech-
niques.Ithasbeenimmenselysuccessfulinuncoveringthestructure,composition,anddynam-
icsofmicro-andnanoscalechemicalandbiologicalsamples.Contemporaryinvestigationsin
the life sciences demand ever-increasing resolution in a variety of spectral bands, with much
attention given to IR and THz. This task is made complicated by the diffraction limit, which
setsafundamentalupperboundonthemaximumspatialfrequencyconveyedbyconventional
refractiveoptics.
Between the initial studies of optical resolution by Rayleigh and Abbe in the 19th century
andthepresentday,amultitudeofsuper-resolutionsystemsandmethodshavebeenproposed
and demonstrated [1]. Extracting information beyond the diffraction limit remains an active
area of research. One general strategy for achieving sub-diffraction-limited images originates
in the fact that it is possible to increase the spatial bandwidth of an optical system by sacri-
ficing certain other characteristics (e.g. field of view, temporal bandwidth, acquisition time).
ThisideawaspioneeredbyLukoszinhisseminal1967paper[2].Indeed,manycontemporary
super-resolutiontechniquescanbeviewedasimplementationsoftheLukoszapproach[1,2],
includingoff-axisillumination[3],structuredillumination[4],spatialfrequency-shiftinggrat-
ings[5,6],andevennear-fieldscanningmicroscopy[7,8,9].
The idea of improving spatial resolution by using temporal degrees of freedom is partic-
ularly appealing for two reasons. First, time multiplexing can help resolve ambiguities that
arise when high spatial frequencies are scattered into the optical passband, as in the case of
evanescent waves diffracting off a subwavelength grating [6]. Second, in certain frequency
bands outside the visible spectrum (e.g. far-IR or THz) it might be easier to manipulate sig-
nals in the time domain than in the spatial frequency domain. The potential utility of time-
domain (or temporal-frequency domain) for super-resolution has long been recognized; in
fact, a frequency-multiplexing scheme involving conjugate moving gratings was described in
Lukosz’s original paper [2] and demonstrated several decades later [10, 11]. However, owing
tothecomplexityoftheexperimentalsetup,thisschemehasnotseenwideadoption.
acoustic
grating ! − Ω detector array
k x − q f
laser
object
RF Lock-in
x Computer
generator amplifier
z
Fig.1.Schematicsoftheproposedsystem.
In the present paper, we propose a time-multiplexed super-resolution system that requires
nomovingpartsandisbasedoncoherentdetectionofafrequency-shiftedsignal.Thisscheme
lendsitselfparticularlywelltosuper-resolvedimaginginIRandTHz.Ourapproachisbased
on a device that converts evanescent waves to propagating waves via diffraction on acoustic
phonons.Thescatteredandfrequency-shiftedwavescanbeeasilydecoupledfromtheexisting
propagatingspectrumthatformstheregulardiffraction-limitedimage(thus,theimageisfree
fromaliasing).Withminimalprocessing,theseshiftedcomponentscanbeusedtodistinguish
subwavelengthfeatures.
Wewilldiscusstwovariationsofthisapproach.Bothrelyonmixingthefrequency-shifted
fields scattered from the object with a reference wave, creating, at the detector, a beat note
photocurrent which can be isolated through lock-in techniques. We will see that detection of
high spatial frequencies is enabled by the spatial frequency offset of the scattered signal, and
that a true super-resolved imaging configuration can be attained with an additional temporal
frequencyshiftofthereferencesignal.
2. Scatteringfromaphonongrating:generaldescription
The proposed super-resolution microscope/sensing system is shown in Fig. 1. The object is
placed in the near field of an acousto-optic modulator (AOM) and illuminated with a plane
wavefromamid-IRorTHzsource.Wavesscatteredfromtheobjectstrikethephonongrating
set up in the AOM by a running acoustic wave with frequency Ω. Due to scattering on the
phonons,thetransversewavevectork oftheincidentradiationisshiftedbyintegermultiples
x
ofthephononwavevectorq,whileitscorrespondingfrequencyisshiftedbyintegermultiples
of Ω. For a sufficiently large q, the evanescent components of the object’s spatial spectrum
(|k |>ω/c)canbescatteredintothepropagatingwaveswith|k(cid:48)|≡|k −q|<ω/c.Thevarious
x x x
spatialfrequencycomponentscanbemeasuredusingaFourieropticssetup(e.g.alenswitha
detectorarrayinitsfocalplane).
Wemodeloursystembyconsideringarectangularsoundcolumn(i.e.planaracousticwave-
fronts propagating in the x direction) interacting with a spectrum of incident plane waves in
a dielectric medium. We neglect the diffraction of the sound field or multiple reflections, and
assumeweakinteraction.Duetophotoelasticeffect,thesoundfieldproducesasinusoidalmod-
ulationofthedielectricpermittivity[12],
ε(x)=ε+∆εcos(qx−Ωt), (1)
whichcorrespondstoaspatiotemporalvolumegrating.Wemaywritethegeneralformofthe
fieldinsidethegratingasasumoverthediscretediffractedorders,
E=∑A (z)exp[i(k +jq)x−i(ω+jΩ)t]. (2)
j x
j
The scattered plane wave components A (z) are then governed by the Raman-Nath equa-
j
tions[13],
∆ε
A(cid:48)(cid:48)(z)+k2A (z)=− ω2[A (z)+A (z)], (3)
j zj j 2c2 j j−1 j+1
(cid:104) (cid:105)1/2
wherek = εω2 −(k +jq)2 ,andω (cid:39)ω.
zj c2 x j
The amplitude of the jth diffracted order, A , is proportional to (∆ε)j, with ∆ε/ε (cid:28)1, al-
j
lowingtoignorehigherorderterms(j≥2).Wecan,furthermore,concludethattheamountof
energyscatteredintotheshiftedwavesissmall,therebypermittingtoneglectthevariationof
0thdiffractedorderA (theundepletedpumpapproximation)[12].Wenotethatforpropagating
0
waves,thisconclusionisvalidinsofarasthereexistsnoBraggmatchingbetweentheincident
and diffractedwaves. Since thephonon wave vector q is atunable parameter inour model, it
is always possible to pick a range of q values to ensure minimal energy loss in the incident
wave.Fortheevanescentwaves,theundepletedpumpapproximationisjustifiedbythesmall
interactionlength.
KeepingtermsuptofirstorderinEq.(2),weobtain:
∆ε
A(cid:48)(cid:48)(z)+k2A (z)=− ω2A (z). (4)
± zj ± 2c2 j 0
The scattering amplitudes of “upshifted” and “downshifted” waves can be obtained from this
expression.Sincetheinputfieldatspatialfrequencies(k ∓q)contributestotheoutputfieldat
x
spatialfrequencyk ,wemaywrite:
x
E (k )=(cid:2)A˜ exp(iΩt)+A˜ exp(−iΩt)+A˜ (cid:3)exp[i(k x−ωt)], (5)
out x − + 0 x
withA˜ ≡t±E (k ∓q),A˜ ≡t E (k ).Scatteringcoefficientst± are,forthecaseofevanes-
± in x 0 0 in x
centwaves,givenby
∆ε (cid:16)ω(cid:17)2 1 ∆ε (cid:16)ω(cid:17)2 1
t±=t ≈t , (6)
0 2 c kz±(cid:112)(kz±)2+κ2 0 2 c kz±(cid:112)q(q∓2kx)
(cid:112) (cid:112)
wherek±= ε(ω/c)2−(k ∓q)2,κ=ik = k2−ε(ω/c)2,andt isaFresneltransmission
z x z x 0
coefficient. We see that the conversion of incident evanescent waves into propagating signals
dependscriticallyontheacoustoopticindexcontrast∆ε andontheeffectiveinteractionlength
1/κ.Wenote,also,thatEq.(6)describesalsothegenerationofshiftedspatialfrequenciesfor
thecasewheretheincidentwaveispropagating(providedwemaketheassociationκ=ik ).In
z
thiscase,thedivergenceofEq.(6)aroundq≈0,aswellasq≈±2k signifiesthebreakdown
x
oftheperturbativetreatmentofEq.(3)duetotheonsetofBragg-matching.
FromEq.(6)wecanestimatethediffractionefficiencyofhighspatialfrequencyinputsignal
E (kin)as
in x
(cid:12)(cid:12)t±(cid:12)(cid:12)≈ ω/c ∆ε , (7)
2kin n(1+n)
x
√ √
withn= ε (therefractiveindexoftheacousticmedium),and∆ε ∝ F,thefluxofacoustic
energyperunitarea.
Wethusobtaintheamplitudesofthefrequency-shiftedwaves.WedefineA˜ =t±E (k ∓q),
± in x
A˜ =t E (k ) (with linear coefficients t , t± describing the generation of phonon-scattered
0 0 in x 0
and/ordevicetransmissioncharacteristics)andassumeA˜ (cid:29)A˜ (cid:29)A˜±,whereA˜ isthedetected
i 0 i
amplitude of the illuminating wave Aieik0z. Averaging out the signal over the finite detector
apertureandsubtractingthebackground(whichcanbedoneelectronically),wecanwritethe
intensitydetectedbythesystemofFig.1as
I =|A˜ |2+2(cid:0)|A˜ A˜ |2+|A˜ A˜ |2(cid:1)1/2cos(Ωt+γ). (8)
out 0 i − i +
The two terms in this equation can be decoupled using standard techniques: the DC term is
isolated with the aid of a low-pass filter, while the term oscillating at the acoustic frequency
Ωisrecoverableusingstandardlock-indetection.Foranygivenk ,thissecondtermcontains
x
contributions from both A˜ =t+E (k −q) and A˜ =t−E (k +q). Although the coupling
+ in x − in x
betweenthesetwoquantities,togetherwiththelackofphaseinformation,makesitdifficultto
recoverthespatialspectrum,theinformationcollectedcanbeusedindetectingsubwavelength
morphologicalchangesbetweendifferentsamples.
3. Super-resolvedfingerprinting:numericalsimulation
We nowillustrate the abilityof the proposedsystem to distinguishedbetween subwavelength
spatial features of different objects. In particular, we utilize Eq. (8) to perform a comparison
between the standard optical target (USAF test chart) and a modified target, where the label
ofevery6thlinegrouphasbeenrandomlyreplaced.Thefirstreplacementcorrespondstothe
last resolvable line group (λ/2.5 line separation); the subsequent replacements correspond to
halvingthesizeofthelinegroups(λ/5,...,λ/40).Weassumethemeasurementisperformedby
selectinganelementofaphotodetectorarrayintheobservationplaneandusingtwoorthogonal
acoustictransducerstoscantheacousticwavevectorwithintherangeq ∈[−25ω/c,25ω/c].
x,y
(Wechoosethesevalueswiththeaimtoimprovebyafactorof∼20ontheAbbeλ/2resolution
limit,therebycollectingmeaningfulinformationabouttheλ/40linegroup.)
Inourcomputations,weassumetheoperatingwavelengthof10µmwithgermaniumasthe
acousticmedium.Wetake∆ε=10−3andrestrictthemagnitudeoftheacousticwavevectorqto
25ω/c.Sinceforhighspatialfrequencieskin≈q,acousticdrivingfrequenciesupto8.75GHz
x
arerequiredtoretrievekin≈25ω/c.Theseparametersarewithinreachofmodernultrasonic
x
transducers[14],aswellassurfaceacousticwavedevices[15].
It should be emphasized that any method that relies on digital processing of raw data can
suffer form rapid – sometimes exponential [16] – accumulation of noise. (Indeed, some pro-
posedsuper-resolutionmethodsevenstipulatetheneedforanexponentiallystronginputsignal
toovercomenoiseandlosses[17].)Toshowthatthisisnotthecasehere,inourcomputations
weaddanormally-distributedrandomtermtotheACamplitudeofEq.(8)inordertosimulate
noiseinthesystem.BecauseSNRisexpectedtobelowestformaximumvaluesoftheacoustic
wavevectorq,weconsiderSNR=10forq=25ω/c1.Assuminga20×20elementphotodetec-
torarray,wecomputethesignalgivenbyEq.(8)forthestandardtarget,aswellasthemodified
target[Fig.2(a)].Fig.2(b)showstheresultofsubtractingthetwodatasetsandperformingan
1Acoustoopticdiffractionefficiency,andhencethesignal-to-noiseratiovariesas1/q.
(A) (B)
Fig. 2. (a) Optical test target and its modified version (inset). In the modified target, the
“5”labelofeverycolumnhasbeenreplacedbyanotherdigit.(b)Computedoutputofthe
systeminthepresenceofnoise(showningrayscale)assumingarealistic,noisydetector
with 400 active photocells. The modified optical target is superimposed for illustration
purposes.Theoutputofthesystemclearlyidentifiesthelocationofeverymodifieddigit,
evenforregionsfarbelowthediffractionlimit.
inverseFouriertransform,withtheresultingplotsuperimposedontothemodifiedopticaltarget.
Evidently,everychangeintheoriginalimageismanifestedinthisdifferencediagram.Further-
more,itislargelylocalizedinthevicinityoftheactualchangedpixels.Itispossibletodiscern
thedifferencesignalevenfromtheλ/40linegrouplabel.
The ability to distinguish between fine spatial features of optical targets makes the system
described above uniquely suited for identifying objects based on their subwavelength spatial
features.Asaresult,itmayfindapplicationsinfingerprintingand/ordetectionofchemicaland
biologicalstructures.
4. Super-resolveddigitalholography
A straightforward modification of the setup described above not only allows to measure the
“downshifted” A˜ component directly, but also provides a method for retrieving phase infor-
−
mation,makingitpossibletoperformphase-contrastmicroscopy,andpotentiallyenabling3D
imagingonsubwavelengthscales.
To this end, a portion of the illuminating radiation is shifted in frequency by Ω using
b
a second AOM. Unlike the modulator that interacts with light scattered from the sample in
the Raman-Nath regime [12], this second AOM utilizes an appropriately oriented and longer
cell to produce Bragg scattering. This results in a strong optical signal at frequency ω+Ω ,
b
|A˜ |exp[i(k ·r−(ω+Ω )t)], which is projected onto the detector [see Fig. 3(a)]. Interfer-
b b b
encebetweenthetwoopticalsignalsproducesbeatnotephotocurrentswithfrequenciesΩ,Ω ,
b
Ω +Ω,Ω −Ω:
b b
Iout(kx)= (cid:12)(cid:12)Eiexp(ik0z)+A˜bexp(ik·r)+[A˜−exp(iΩt)+A˜+exp(−iΩt)+A˜0]exp(ik·r)(cid:12)(cid:12)2
= ...+2|A˜−A˜ |cos[(Ω +Ω)t+∆Φ−]+2|A˜+A˜ |cos[(Ω −Ω)t+∆Φ+]+..., (9)
b b b b
where∆Φ±=(k −k)·r−φ± isthephasedifferencebetweenthesignalfromtheBraggcell,
b
|A˜ |exp(ik ·r),andtheRaman-Nath-scatteredsignalA˜±exp(ik·r)=|A˜±|exp[i(φ±+k·r)].
b b
(А) (B)
Fig.3.(a)Schematicsoftheproposedsystem.NotetheBragg-shiftedreferencebeamthat
aidsinprovidingphaseinformation.(b)Thecomputedoutputofthesystemwithoptical
testtargetastheobjectinthepresenceofnoise.
Of special interest is the component at frequency Ω+Ω , which carries the high spatial
b
frequencyinformationcontainedinitsmodulusanditsphase∆Φ−(cid:39)(kb−k )x−φ−.Bothof
x x
thesequantitiescanberetrievedusinglock-intechniques.Toproducethelock-inreference,the
RFsignalsdrivingthetwoacousticcellscanbemixedusinganonlinearelement(e.g.adiode)
andappropriatelyfilteredtoproducethesumfrequency.Asaresult,completeinformationcan
beobtainedaboutthecomplexhighspatialfrequencyFouriercomponentA˜−,fromwhichitis
straightforward todeduce the field E (k +q). Bycollecting data frommultiple CCDpixels,
in x
aswellasbyvaryingtheacousticwavevectorq,informationcanbecollectedabouttheentire
spatial spectrum of the object. The data can then be digitally processed to produce a spatial-
domainimagecontainingsubwavelengthdetails,aswellasphasecontrast.
BecausetheBragg-shiftedsignalweusetodecoupletheA˜ andA˜ termsservesasarefer-
+ −
enceneededtorecordphaseinformation,andbecausetheimageisreconstructeddigitally,our
techniquebelongsinthecategoryofdigitalFourierholography(DFH)[18,19,20].However,
theproposedcoherentdetectionmethodisdifferentfromtraditionalDFHsetupsinthatiteffec-
tively converts sample spatial frequencies to temporal ones. In conventional holography, care
has to be taken to isolate the target signal both in real and Fourier space. This translates into
limitationonthefieldofview,aswellasmaximumattainableresolution[20].Therequirement
that the CCD pixel spacing must allow for imaging the reference wave fringes further limits
the resolution. As a result, simple digital holography setups suffer from very narrow spatial
frequency passbands (N.A.∼0.1). Synthetic aperture techniques based on gratings [21, 22, 5]
have been successfully shown to increase the effective numerical aperture, as has heterodyne
detection[23].Ourmethodcombinesthebenefitsoftheseapproaches:dynamicacousticgrat-
ingallowstoscanthespatialfrequencyspacewhilealwaysremainingintheCCD’spassband,
while the temporal frequency shifts multiplex the data, effectively increasing the number of
informationchannelsinagivenspatialfrequencyband.
We simulate the performance of the frequency-shifted digital holography system by first
usingEq.(9)tocomputetheresponseofthesystemtoacalibrationsignalhavingunitamplitude
forallspatialfrequencies.Inpractice,suchcalibrationsignalmightbegeneratedbyplacinga
pointsourceinthevicinityoftheAOM.Eq.(9)alsoprovidestheeffectiveamplitudeandphase
transferfunctionsthatallowtodeterminethedetectedsignalforagiveninputfielddistribution.
Gaussiannoiseisaddedtosimulatespurioussignalsinthesystem.Theinputsignalcanthenbe
obtainedbydividingoutthecalibrationquantities.InFig.3(b)weplotthesimulatedretrieved
fieldmagnitude.Zoominginonthecentralpartofthetestpatten(figureinset)itisevidentthat
everylinegroupisdistinctlyresolved,suggestingthattheeffectivenumericalapertureis>1
(duetoreconstructionofevanescentwaves).
5. Potentialimprovementsoftheproposedsystemandextensiontohigherfrequencies
Asdemonstratedearlier,theresolutionoftheproposedsystemswilldependonmanyfactors,
including the maximum attainable frequency shift, integration time, acoustooptic index con-
trast, and success in minimizing detector noise, laser linewidth, and speckle. In addition to
resolution improvement, there exist many possible ways to enhance the functionality and the
performanceoftheproposeddevices.Forinstance,becausethephaseinformationispreserved,
thefullcomplexfieldintheobjectplanecanbereconstructed,potentiallyenabling3Dimag-
ing via, e.g., phase-shifting interferometry [24]. On the performance front, the sensitivity of
the device may be improved with a subwavelength layer of highly doped semiconductor at
thefrontAOMfacet.Whenthedielectricconstantofthislayerisequalto−1,theevanescent
fieldsarestronglyenhancedduetoresonantcouplingtosurfaceplasmonsinthedopedlayer(a
phenomenonknownas“poor-man’ssuperlensing”[25]),leadingtobetterSNRatthedetector.
Another possible way to improve the scattering efficiency of evanescent waves is placing the
sample directly in the path of an acoustic wave, for instance, by running the wave through a
microchannel containing objects to be studied. This approach may find many applications in
novelintegratedbiological/chemicaldetectiondevices.
Finally, we comment on the possibility of extending the proposed approach to frequencies
otherthanthemid-IRandTHzbandsdiscussedhere.Whileimplementingthesystemforlower
frequencies is essentially trivial, near-IR and optical frequencies pose a challenge. Acoustic
phonon energies in practical devices do not approach the values necessary to produce a sub-
stantial wave vector shift in these spectral bands. However, the required shift in spatial and
temporalfrequenciescaninprinciplebeattainedbyreplacingtheacoustoopticmediumwitha
nanostructuredperiodicallymovinggrating.Inthisscenario,thespatialfrequencyshiftisde-
terminedbytheperiodicityofthegratingwhilethetemporalfrequencyshiftisdeterminedby
the speed of grating oscillation. These parameters can be adjusted independently to optimize
performance.
6. Conclusion
We have proposed a system that enables detection of sub-diffraction-limited spatial spectrum
componentsinthefarfieldbyutilizingscatteringfromanacousticgrating.Thisprocessworks
wheneverthespatialfrequenciesoftheobjectarecomparableinscaletotheacousticwavevec-
tor.Initssimplestimplementation,thesystemcouldaidin“fingerprinting”ofsamplesbasedon
theirsubwavelengthspatialfeatures.WiththeuseofanadditionalBragg-shiftedreferencesig-
nal,itisalsopossibletorecoverthephaseoftheoriginalopticalsignal.Theproposedapproach
hasthepotentialtogreatlyenhancethespecificityofmid-IRandTHzspectroscopy.
Acknowledgements
ThisworkwassupportedinpartbyUnitedStatesArmyResearchOffice(USARO)Multidis-
ciplinaryUniversityResearchInitiativeprogramawards50342-PH-MURandW911NF-09-1-
0539.