We present a new technique, differential ghost imaging (DGI), which dramatically enhances the signal-to-noise ratio (SNR) of imaging methods based on spatially correlated beams. DGI can measure the transmission function of an object in absolute units, with a SNR that can be orders of magnitude higher than the one achievable with the conventional ghost imaging (GI) analysis. This feature allows for the first time, to our knowledge, the imaging of weakly absorbing objects, which represents a breakthrough for GI applications. Theoretical analysis and experimental and numerical data assessing the performances of the technique are presented

A., Gatti

D., Magatti

Ferri, Fabio

Lugiato, Luigi

FERRI, FABIO

D. Magatti

LUGIATO, LUIGI

A. Gatti

Archivio istituzionale della ricerca - Università dell'Insubria

Differential Ghost ImagingF. Ferri,* D. Magatti, L. A. Lugiato, and A. GattiCNR-INFM-CNISM, Dipartimento di Fisica e Matematica, Universita` dell’Insubria, Via Valleggio 11, 22100 Como, Italy(Received 5 February 2010; published 25 June 2010)We present a new technique, differential ghost imaging (DGI), which dramatically enhances the signal-to-noise ratio (SNR) of imaging methods based on spatially correlated beams. DGI can measure thetransmission function of an object in absolute units, with a SNR that can be orders of magnitude higherthan the one achievable with the conventional ghost imaging (GI) analysis. This feature allows for the firsttime, to our knowledge, the imaging of weakly absorbing objects, which represents a breakthrough for GIapplications. Theoretical analysis and experimental and numerical data assessing the performances of thetechnique are presented.DOI: 10.1103/PhysRevLett.104.253603 PACS numbers: 42.50.ArGhost imaging (GI) is a novel imaging technique whichhas become increasingly popular over the last decade, [1–17]. In GI the object image is retrieved by using twospatially correlated beams: the reference beam, whichnever interacts with the object and is measured with apixelated detector, and the object beam, which, after illu-minating the object is collected with a bucket detector,with no spatial resolution. By correlating the referencebeam with the bucket signal, the ‘‘ghost’’ image is re-trieved. The GI technique offers great potentialities withrespect to standard imaging because it allows the imagingof objects located in optically harsh (difficult to bereached) or noisy environments, where standard techniquesare likely to fail. Other potentialities are in the direction ofcompressive sensing [18], where the number of averagesneeded to recover an object may be lower than the numberof pixels needed to resolve it.In spite of these remarkable features, studies on GI haveso far mainly focused on the fundamental debate ofwhether GI is an intrinsic quantum phenomenon requiringthe entanglement of photon pairs, or it can be realized withclassically correlated light beams [3–15]. This debateseems to have arrived at an end point where a substantialequivalence between the classical and quantum approacheswas demonstrated [4–10,12,14,15], with some distinguish-ing features arising, e.g., in the signal-to-noise ratio (SNR)[17]. However, the possibility of implementing GI withclassical sources still seems a pure academic achievement,because so far, with only few exceptions [16,18], only theimaging of simple binary objects (transmission 0 or 1) suchas a double slit have been carried out. When it comes toreal sensing and/or imaging applications, such as the imag-ing of a low contrast object or the location and possiblysizing of a small absorbing obstacle in a large beam, theSNR of conventional GI methods is remarkably low, andvery long measurement times would be necessary [17,18].As a consequence, the great potentiality offered by GIcannot be exploited.In this Letter we focus on the thermal case, and wepresent a new method that we name differential ghostimaging (DGI) that overcomes this limitation and allowsus to apply, for the first time to our knowledge, the GIprotocol to realistic and potentially interesting problems.The technique can measure the transmission function of anobject in absolute units, with a SNR that, depending on theobject transmission relative variance, can be orders ofmagnitude higher than the one achievable with conven-tional GI. Thus DGI paves the way to real applications ofthe GI protocol, with the possibility of exploiting all theadvantages of GI against standard imaging.The optical setup is shown in Fig. 1 and is identical(except for the bucket detection, see below) to the onereported in Ref. [15]. The pseudothermal source operatesat  ¼ 0:532 m and produces a collimated beam of deep-Fresnel speckles [19–21] whose size 0  84:7 m doesnot change upon propagation. The area of the beam isAbeam  32:9 mm2 and contains Nspeckle ¼ Abeam=Acoh 4590 speckles (Acoh ¼ 20). The speckle beam is dividedby a cube beam splitter in two spatially correlated beams,the object and the reference beams, whose intensity dis-tributions are I1ðx1Þ and I2ðx2Þ, respectively. The objectbeam hits the object at a distance z1 ¼ 130 mm from thesource, and the bucket signal S1 is collected with a photo-diode. The reference beam I2ðx2Þ is detected at the samedistance z1 ¼ z2 with a CCD sensor whose square pixelshave a size 6:67 m 0. The reference bucket S2 isobtained by summing the signals out from all the CCDpixels.<S2I2(x2)>S2I2(x2)OBJz1z2S1REFT(x)<S1I2(x2)>pseudo-thermal sourceFIG. 1. Schematic diagram of the experimental setup.PRL 104, 253603 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending25 JUNE 20100031-9007=10=104(25)=253603(4) 253603-1  2010 The American Physical SocietyIn the following we shall adopt a classical formalism, inwhich all the detected photon numbers are so large that theshot noise is always negligible. Thus, the reference andobject beams are perfectly correlated stochastic variables,and we can set I1ðx1Þ ¼ I2ðx2Þ, where  is a factor whichtakes into account any unbalancing (beam splitter anddetectors) between the two arms.In the conventional GI analysis the ghost image is re-trieved by correlating I2ðx2Þ with the bucket signal S1S1 ¼ZAbeamI1ðx1ÞTðx1Þd2x1 (1)where TðxÞ denotes the (intensity) transmission function ofthe object. The measured quantity in GI is [5]:hOðx2Þi ¼ hS1I2ðx2Þi (2)where h  i is an average over independent speckle con-figurations and S1 ¼ S1  hS1i, I2ðx2Þ ¼ I2ðx2ÞhI2ðx2Þi. Note that the procedure defined by Eq. (2) isequivalent to the removal of the dc photocurrent compo-nent done in [11,13]. Equation (2) can be worked out interms of Tðx2Þ obtaining [5,15]hOðx2Þi ¼ AcohhI1ðx2ÞihI2ðx2Þi½T  ðx2Þ AcohhI1ðx2ÞihI2ðx2ÞiTðx2Þ (3)where Acoh is the speckle coherence area defined in termsof the complex coherence factor ðxÞ [22] as Acoh ¼R jðxÞj2d2x. The symbol  denotes convolution betweenTðxÞ and the function ðxÞ ¼ jðxÞj2=Acoh, which repre-sents the point spread function (PSF) of the technique(RðxÞd2x ¼ 1). The second line of Eq. (3) is valid underthe assumption of ‘‘perfect resolution’’, i.e., when thelinear width 0 ﬃﬃﬃﬃﬃﬃﬃﬃﬃAcohpof ðxÞ is much smaller than thelength scales characterizing the variations of TðxÞ. Sinceall the factors in front of TðxÞ are accurately measurable,Eq. (3) provides, in principle, a way of measuring TðxÞ inabsolute units. However, as reported in [23], the presenceof extra sources of noise (as fluctuations in the emittingpower of the thermal source), introduces a spurious offsetin TðxÞ, which prevents from retrieving it in absolute units.The noise associated to the stochastic variable Oðx2Þ ishO2ðx2Þi ¼ hS21I22ðx2Þi  hS1I2ðx2Þi2 (4)whose exact computation is rather cumbersome because itinvolves a series of moments up to the order 4 in theintensities [17,23]. However, for Gaussian speckle fields,it is possible to show [23] that the dominant term is givenby the ‘‘uncorrelated’’ contributionhO2ðx2Þi ¼ hS21ihI22ðx2Þi: (5)To extract the dependence of Eq. (5) on TðxÞ, we can makethe simplifying assumption of uniform illumination, i.e.,that the average speckle beams are constant over their crossarea [hI1ðx1Þi ¼ hI1i, hI2ðx2Þi ¼ hI2i]. By assuming againperfect resolution (T   ), after some calculations, weget to the result [23]hO2i ¼ AcohAbeamhI1i2hI2i2T2 (6)where T2 ¼ RAbeamhIðxÞiT2ðxÞd2x=RAbeamhIðxÞid2x RAbeamT2ðxÞd2x=Abeam is the average quadratic transmis-sion function of the object. If we define the minimumvariation of hOðx2Þi as the signal that needs to be de-tected, and average over m independent measurements, weobtain from Eqs. (3) and (6) that the SNR of the conven-tional GI analysis isðSNRÞGI ¼ ½hOðx2Þi2minhO2i ¼mNspeckleT2minT2(7)where Nspeckle ¼ Abeam=Acoh is the number of speckles inthe beam and Tmin is the minimum variation of the objecttransmission function to be detected. Equation (7) showsnot only the well-known result [17] that ðSNRÞGI scales as1=Nspeckle, but also as 1=T2. Thus, quasitransparent objects(T2  1) are much more difficult to retrieve than highlyabsorbing objects (T2  1) and may require an exceed-ingly high number of measurements.To implement the ghost imaging technique on quasi-transparent objects, we developed a new method that wename differential ghost imaging (DGI). The idea is toimprove the sensitivity of GI by using a new differentialbucket signal S, which is sensitive only to the fluctuatingpart TðxÞ ¼ TðxÞ  T. ThusS ¼ZAbeamI1ðx1ÞTðx1Þd2x1 (8)where T ¼ RAbeamhIðxÞiTðxÞd2x=RAbeamhIðxÞiTðxÞd2x isthe average transmission function of the object. To findan operative procedure for measuring S, we define thereference bucket signal S2S2 ¼ZAbeamI2ðx2Þd2x2 (9)and notice that T can be expressed in terms of hS2i asT ¼ 1hS1ihS2i : (10)Thus S can be put in the operative formS ¼ S1  hS1ihS2iS2 (11)which makes it easily measurable. The DGI techniqueworks by substituting S1 with S and performing thestandard correlation analysis [Eq. (2)]. Therefore, thequantity measured in DGI ishOðx2Þi ¼ hSI2ðx2Þi¼ hS1I2ðx2Þi  hS1ihS2i hS1I2ðx2Þi (12)where we used Eq. (11) and the property hSi ¼ 0.PRL 104, 253603 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending25 JUNE 2010253603-2Equation (12) can be worked out in terms of Tðx2Þ and,under the same assumptions used for Eq. (3), it becomeshOðx2Þi ¼ AcohhI1ðx2ÞihI2ðx2Þi½T  ðx2Þ AcohhI1ðx2ÞihI2ðx2ÞiTðx2Þ (13)which is identical to Eq. (3), but with Tðx2Þ instead ofTðx2Þ. Thus DGI works by retrieving the fluctuating partTðxÞ and, by combining Eq. (13) with Eq. (10), the entireobject transmission function TðxÞ can be recovered. Weemphasize that, differently from what happens in GI, DGIis immune from extra sources of noise [23], so that iteffectively allows us to retrieve TðxÞ in absolute units.The noise associated to the variable Oðx2Þ ishO2ðx2Þi ¼ hS2I22ðx2Þi  hSI2ðx2Þi2 (14)whose computation can be carried out as for Eq. (4),obtaining, under the same assumptions,hO2i ¼ AcohAbeamhI1i2hI2i2T2 (15)where T2 ¼ T2  ðTÞ2 is the variance of TðxÞ. FromEqs. (13) and (15) we arrive at the final result for theSNR after m independent measurementsðSNRÞDGI ¼ ½hOðx2Þi2minhO2i¼ mNspeckleT2minT2(16)which is identical to Eq. (7), but with T2 in place of T2.The ratio between Eqs. (16) and (7)ðSNRÞDGIðSNRÞGI¼ 1þ 12rel(17)is always >1 and shows that the enhancement of the SNRbetween the two techniques depends only on the objectrelative variance 2rel ¼ T2=ðTÞ2. Thus, for highly trans-parent objects [T2  ðTÞ2, 2rel  1], the DGI techniqueworks much better than the conventional one. Conversely,for highly absorbing objects [ðTÞ2  T2  1, 2rel ! 1]the two techniques perform quite similarly.The first experiment shows the striking differences be-tween conventional GI and DGI when a small object in alarge beam has to be detected and sized. Two totally (T ¼0) absorbing particles, 820 and 400 m in diameter, wereglued on a microscope slide (T ¼ 0:92) and shined with abeam of size5:7 mm. Thus for this object (T  0:90 andrel  0:14) the SNR enhancement predicted by Eq. (17) isðSNRÞDGI=ðSNRÞGI  52. The GI and DGI analyses wereperformed on the same raw data, and the images retrievedafterm ¼ 104 averages are reported in Fig. 2. While the GIimage is almost completely blurred, the DGI image isneatly resolved, allowing the retrieval of both particle sizesand positions. The measured enhancement of the SNR was66, higher than expected (see below).The second experiment checked the validity of Eqs. (7)and (16). We used a binary (T ¼ 0 or T ¼ 1) object, madeby a dark knife obstructing the beam, so that the ratiobetween the obstacle area and the beam size  ¼Aobst=Abeam could be varied. For this object T ¼ 1 ,T2 ¼ T, T2 ¼ Tð1 TÞ. Figure 3 compares ðSNRÞDGIand ðSNRÞGI normalized to mT2min as a function of (bottom scale) and T (top scale). The experimental datafor DGI (solid circles) and for GI (solid squares) are plottedagainst theory (solid lines). The open symbols are obtainedfrom computer simulations in which2% rms fluctuationsof the power emitted by the thermal source was introduced.The crosses indicate the SNR’s associated to the data ofFig. 2. Beside demonstrating the excellent agreement be-tween DGI data and theory, Fig. 3 shows the strikingdifference between the two techniques: they behave quitesimilarly when the object is a small aperture (! 1), but ifthe object is a small obstacle (! 0), GI becomes highlyinefficient while DGI remains quite performing, beingðSNRÞDGI=ðSNRÞDGI ¼ 1=. Furthermore, for conven-tional GI, the SNR retrieved from data and simulations issystematically smaller than the expected one. This discrep-ancy is due to the presence of other sources of noise, whichdegrade ðSNRÞGI but not ðSNRÞDGI [23].(a) (b)5.7 mm 5.7 mmFIG. 2. Conventional GI (a) and DGI (b) of two totally absorb-ing particles 820 and 400 m in diameter, retrieved after m ¼104 averages. The number of speckles in the beam was Nspeckle 4590.GIDGIbeamobst A/A=εε−=1T])T/[(SNR2minm∆0.0 0.5 1.010-410-310-20.00.1FIG. 3. Comparison between normalized SNR’s of GI and DGItechniques for an obstacle object as a function of relative size ¼ Aobst=Abeam and T(top horizontal axis). Theoretical behav-iors [Eqs. (7) and (16)] are indicated by the two lines. Theexperimental data (solid symbols) and numerical simulations(open symbols), match accurately the theory only for DGI.Crosses indicate SNR values for the object of Fig. 2.PRL 104, 253603 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending25 JUNE 2010253603-3The accuracy of the DGI technique in retrieving absolutevalues of TðxÞ was ascertained by using an object made oftwo absorption filters (T ¼ 0:40) partially staggered to-gether. Thus in the region where the filters are overlappedTðxÞ ¼ 0:16, where they are not overlapped TðxÞ ¼ 0:40,and where there is no filter, TðxÞ ¼ 1. The DGI imageretrieved afterm ¼ 4 104 averages is shown in Fig. 4(a),where the four regions are clearly recognizable. The abso-lute values of TðxÞ are shown in Fig. 4(b), where theaveraged horizontal sections of the 2nd and 1st (solidline) and 3rd and 4th (dashed line) quadrants are reported.The absolute accuracies in all of the four regions were0:04 . When the same raw data of Fig. 4 were analyzedwith the GI technique, the retrieved image was qualita-tively similar to Fig. 4(a), but TðxÞ was displaced by anoffset of the same order of T (data not shown).Finally, we tested the DGI technique with a continuousvarying grayscale object, i.e., a slide representing a detailof the famous painting ‘‘Nascita di Venere’’ by S. Botticelli(Firenze, Italy, A.D. 1484). Figure 5(a) shows the imageof the original slide after convolution with the PSF corre-sponding to a speckle size of 0  85 m. For this objectT  0:086 and rel  0:72, so that the expected ratioðSNRÞDGI=ðSNRÞGI  2:9. Figures 5(b) (DGI) and 5(c)(GI) show that the images retrieved (m ¼ 105) with thetwo techniques are remarkably different with a SNR en-hancement of 6:4, higher than expected. Once again thecomparison between the DGI and GI techniques is betterthan what is predicted by Eq. (17), because DGI is morerobust against extra source of noise.In conclusion, we have presented a new technique calleddifferential ghost imaging (DGI) that enhances dramati-cally the SNR of the GI protocol when applied to realisticcomplex objects, which are of interest in real word appli-cations. We believe that this new technique will pave theway to the use of GI in many sensing or imaging problems,and in particular, in all the situations where standardtechniques appear not to be adequate, such as when theobject to be imaged is located in optically harsh or noisyenvironments.We thank M. Molteni for the critical reading of themanuscript.*Corresponding author; fabio.ferri@uninsubria.it[1] T. B. Pittman, Y.H. Shih, D.V. Strekalov, and A.V.Sergienko, Phys. Rev. A 52, R3429 (1995).[2] A. F. Abouraddy et al., Phys. Rev. Lett. 87, 123602 (2001);93, 213903 (2004).[3] R. S. Bennink, S. J. Bentley, R.W. Boyd, and J. C. Howell,Phys. Rev. Lett. 92, 033601 (2004).[4] R. S. Bennink, S. J. Bentley, and R.W. Boyd, Phys. Rev.Lett. 89, 113601 (2002).[5] A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, Phys.Rev. Lett. 93, 093602 (2004).[6] F. Ferri et al., Phys. Rev. Lett. 94, 183602 (2005).[7] A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, Phys.Rev. Lett. 94, 063601 (2005).[8] M. Bache et al., Phys. Rev. A 73, 053802 (2006).[9] A. Gatti et al., J. Mod. Opt. 53, 739 (2006).[10] J. H. Shapiro, Phys. Rev. A 78, 061802(R) (2008).[11] G. Scarcelli, V. Berardi, and Y. Shih, Phys. Rev. Lett. 96,063602 (2006).[12] A. Gatti et al., Phys. Rev. Lett. 98, 039301 (2007).[13] G. Scarcelli, V. Berardi, and Y. Shih, Appl. Phys. Lett. 88,061106 (2006).[14] L. Basano and P. Ottonello, Appl. Phys. Lett. 89, 091109(2006).[15] F. Ferri, D. Magatti, V. G. Sala, and A. Gatti, Appl. Phys.Lett. 92, 261109 (2008).[16] R. Meyers, K. S. Deacon, and Y. Shih, Phys. Rev. A 77,041801(R) (2008).[17] B. I. Erkmen and J. H. Shapiro, Phys. Rev. A 79, 023833(2009).[18] O. Katz, Y. Bromberg, and Y. Silberberg, Appl. Phys. Lett.95, 131110 (2009).[19] R. Cerbino, Phys. Rev. A 75, 053815 (2007).[20] A. Gatti D. Magatti, and F. Ferri, Phys. Rev. A 78, 063806(2008).[21] D. Magatti, A. Gatti, and F. Ferri, Phys. Rev. A 79, 053831(2009).[22] J.W. Goodman, Speckle Phenomena in Optics (Roberts-Company, Greenwood Village, CO, 2006), pp. 90–92.[23] A. Gatti et al. (to be published).FIG. 4. Panel (a): DGI of an object made of two absorptionfilters partially staggered together, so to create four regions oftransmission TðxÞ ¼ 0:16 (4th quadrant), 0.40 (1st and 3rdquadrants), 1.0 (2nd quadrant). Panel (b): averaged horizontalsections of the 2nd and 1st (solid line) and 3rd and 4th (dashedline) quadrants. Absolute accuracies were 0.04, number ofaverages m ¼ 4 104.(a) (b) (c)FIG. 5. Original slide (a),DGI (b) and GI (c) images of a detailof the famous painting ‘‘Nascita di Venere’’ by S. Botticelli(Firenze, Italy, A.D. 1484) retrieved after m ¼ 105 averages.The original image was convolved with a PSF of width equal tothe speckle size 0  85 m.PRL 104, 253603 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending25 JUNE 2010253603-4

Differential Ghost Imaging

Differential Ghost Imaging

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