epl draft

Classical light analogue of the nonlocal Aharonov-Bohm effect

Nandan Satapathy1, Deepak Pandey1, Poonam Mehta1,2, Supurna Sinha1,Joseph Samuel1 and Hema Ramachandran1

1 Raman Research Institute, C. V. Raman Avenue, Sadashivanagar, Bangalore, India 560 080.2 Presently at the Department of Physics and Astrophysics, University of Delhi, Delhi, India 110 007.

PACS 03.65.Vf – Phases: geometric; dynamic or topologicalPACS 25.75.Gz – Particle correlations and fluctuationsPACS 95.75.Kk – Interferometry

Abstract – We demonstrate the existence of a non-local geometric phase in the intensity-intensitycorrelations of classical incoherent light, that is not seen in the lower order correlations. This two-photon Pancharatnam phase was observed and modulated in a Mach-Zehnder interferometer.Using acousto-optic interaction, independent phase noise was introduced to light in the two armsof the interferometer to create two independent incoherent classical sources from laser light. Theexperiment is the classical optical analogue of the multi-particle Aharonov-Bohm effect. As thetrajectory of light over the Poincare sphere introduces a phase shift observable only in the intensity-intensity correlation, it provides a means of deflecting the two-photon wavefront, while having noeffect on single photons.

Introduction: Two classic interference experiments areYoung’s double-slit experiment and the Hanbury Brownand Twiss (HB-T) experiment. The former measures theamplitude-amplitude correlation and demonstrates the in-terference of a photon with itself. The latter experi-ment measures the intensity-intensity correlation and is ademonstration of the interference of a pair of photons withitself. While the HB-T intensity-interferometry experi-ment was historically important in leading to our presentunderstanding of quantum optics and coherence, one doesnot need to invoke quantum mechanics to understand it;it can quite simply be understood entirely in terms of clas-sical electric field fluctuations. In recent times, with thedevelopment of different types of light sources and detec-tion techniques, intensity-interferometry has, once again,become a topic of great interest. It has led to many tanta-lizing ideas and interesting applications [1–5]. There hasbeen enormous debate [6–9] on the similarities and differ-ences between bi-photon interferometry and two-photoninerferometery. Bi-photon interferometry [10–12] uses en-tangled photon pairs, i.e., pairs of photons that are relateddue to a conservation principle, for example, product pho-tons in parametric down-conversion. Two-photon inter-ferometry [13], on the other hand, involves photons thatare statistically correlated, as from a thermal source; ourpresent experiment belongs to this category. Intensity-interferometry with classical light is regaining importance

with the demonstration of ghost imaging with thermallight [14–18].

A classical HB-T experiment consists of two slits, eachilluminated by an independent, incoherent light source.Two detectors measure the intensity of light falling onthem; at each detector, the intensity has contributionsfrom both slits. The cross correlation of the intensities atthe two detectors varies in a sinusoidal fashion as a func-tion of separation between the detectors. In the conven-tional HB-T experiment, the appearance of these fringeswas a purely dynamical effect, arising from the change inthe path difference of the two detectors from the two slits.Recently it was shown theoretically [19] that even withfixed detectors, one can have such fringes if the sourcesand detectors are polarised and the polarisation is var-ied. This is purely a geometric or Pancharatnam phaseeffect [20,21] and arises due to the closed trajectory of po-larised light on the Poincare sphere. While such geometricphase effects are well known in amplitude interferometry,their counterparts in intensity interferometry have beenless studied. The geometric phase is given by half the solidangle subtended by a closed path traced out by polarisingelements on the Poincare sphere (see Fig. 1). Unlike thedynamical phase shift that is restricted by the spatial andtemporal coherence of the sources, the geometric phase, isachromatic and unbounded, as will be discussed shortly.

In this Letter, we wish to convey several points. We

p-1

arX

iv:1

202.

2685

v1 [

quan

t-ph

] 1

3 Fe

b 20

12

N. Satapathy et al.

Fig. 1: (Colour online) The Poincare sphere. Figure showsfour points on the Poincare sphere representing two circularpolarisation states (R,L) and two linear polarisation states ofthe two polarisers in front of the two detectors (3, 4). Thegeometric phase is given by half the solid angle subtended atthe centre, by the geodesic polygon R-4-L-3-R.

demonstrate a new way of producing classical light withexactly tailored characteristics. This is achieved by cre-ating radio-frequency electrical signals of the desired fea-tures and imprinting this onto light via acousto-optic in-teraction. We use this technique to create light that hasrandom phase noise only. Next, we show that a smallmodification to the HB-T intensity interferometry exper-iment leads to an optical analogue of the multiparticleAharonov-Bohm (AB) effect [22]. This manifests as anon-local geometric phase in the intensity-intensity cor-relations of classical incoherent light, that is not seen inthe lower order correlations. Finally, we show that thisnon-local two-photon cross-correlation can be modulatedvia the geometric phase. This suggests a way of deflectingthe cross-correlated photon pairs.

The AB effect [23, 24] shows that the electromagneticpotential potential affects charged particles even in regionswhere no field exists. It is seen as a phase acquired by thewave function of an electron when it moves in a path en-closing a solenoid. The path is entirely in a region wherethe magnetic field is zero and the acquired phase is at-tributed to the enclosed flux [25]. The AB effect led to awide appreciation of the role of the electromagnetic poten-tial, in particular its interpretation [26] as a “connection”,a structure well studied by mathematicians in differentialgeometry. A connection is a rule for comparing quantitiesdefined at different points1. The AB effect can be viewedas a physical manifestation of a flat connection, since thereis no magnetic field in the region accessible to the electron.The AB effect has both geometric and topological aspectsto it. It is insensitive to small deformations of the inter-fering paths and therefore may be viewed as topological.

1A connection need not be integrable along a closed curve andthis lack of integrability is called a holonomy. If the connection islocally integrable it has no curvature and is said to be flat.

At the same time, it is sensitive to small changes of themagnetic flux and in this sense is geometric.

Essentially the same mathematical structure underliesthe Pancharatnam phase, seen, for example [27], in aSagnac interferometer, where the dynamic path differenceis zero. This geometric phase is a consequence of the solidangle enclosed on the Poincare sphere. It too is insensitiveto small changes in the paths of the interfering beams andhence is topological, while being sensitive to small changesin the solid angle enclosed on the Poincare sphere is alsogeometric. There is thus a close analogy between the Pan-charatnam phase and the AB phase.While the Pancharatnam phase predates the AB phase,the latter has recently been extended to non local mul-tiparticle effects which intriguingly, are manifest only inthe cross correlations, and not in the self correlations orin the lower order correlations. Considered theoreticallyfor a pair of electrons by Samuelsson et al [22], this two-particle non-local AB effect was observed experimentallyby Neder et al [28]. A pair of electrons together enclosedan AB flux due to an applied magnetic field, which wasthen used to modulate the two particle cross correlationvia coupling to the orbital degree of freedom of electronswhile the spin remained frozen. The effects of the fluxwere absent in the individual currents and in their selfcorrelations while the cross correlation between electroncurrents revealed the dependence on the AB flux.

The experiment of Neder et al was essentially quantumin nature, as electrons, being fermions, do not admit thelimit of large particle numbers. In contrast to fermions,bosons permit macroscopic occupancy of a single state,admitting a classical limit and permitting a classical fieldtheory description. The experiment that we report hereis the classical bosonic analogue of the two-particle non-local AB effect. Classical light beams from two sourcesmeet after having traversed two independent paths overthe Poincare sphere and together enclose a solid angle onthe sphere. One of the polarising elements was varied soas to modulate the enclosed solid angle, and the first ordercorrelation (interference visibility), the second order self-correlation and the second order cross-correlations weremeasured. The effect of the modulation of the geometricphase was seen only in the second order cross correlationand not in the other measured quantities. Thus, apartfrom the sign differences owing to the different statisticsof the particles involved, the effect that we report here andthe two-particle non-local AB effect are conceptually verysimilar.

Experimental Setup: Our experiment employs twophase-incoherent sources S1 and S2 in a Mach-Zehndersetup with birefringent elements in the two arms, and po-larisers preceding two detectors D3 and D4 (see Fig. 2).A variation of the relative polarisation angle of the de-tectors introduces a geometric phase equal to half thesolid angle enclosed by the two interfering paths onthe Poincare sphere (see Fig. 1). Light from a diodelaser at 767nm was incident on a non-polarising beam-

p-2

Classical light analogue of the nonlocal Aharonov-Bohm effect

Fig. 2: (Colour online) Figure shows a schematic diagram ofthe experimental setup. Two sources S1 and S2 illuminatetwo detectors D3 and D4 as explained in the text. BS1 andBS2 are non-polarising beam splitters, Q1 and Q2 are quarterwave plates, P3 and P4 are linear polarisers and BD are beamdumps.

splitter depicted as BS1 in Fig. 2, and the two emergentbeams were passed through two acousto-optic-modulators(AOM) shown in Fig. 2 as AOM1 and AOM2, fed by tworadio frequency sources (RF1 and RF2) respectively. Thefirst order diffracted beams emerging from the two AOMswere passed through two quarter-wave plates (Q1, Q2), torender one beam right circularly polarised (R) and theother left-circularly polarised (L). The two beams werethen combined at a non-polarising beam splitter (BS2).At each exit face of the final beam splitter, a polariser(P3, P4) was kept, followed by a detector (D3, D4). Theundiffracted beams terminated in beam dumps (BD) asshown in Fig. 2. By keeping one of the polarisers (e.g., P3) fixed and changing the orientation of the other polariser(P4), the relative angle between the polarisers, φ34 couldbe varied continuously from 0 to 360. This results ina corresponding variation in the geometric phase. As weshow below, this also causes a modulation of the intensitycross-correlation of the light reaching the two detectors.In order to measure this, the intensities of light reach-ing detectors D3 and D4, for each orientation of P4, wasrecorded for a certain length of time on a digital storageoscilloscope and cross correlated offline. Thus, from therecorded time series, one may determine first and secondorder correlations.

In striking contrast to the other interference experi-ments which require a source of coherent light, the two-photon intensity-interferometery needs incoherent sourcesas HB-T correlations vanish for a laser light source. Onecould use a thermal source such as a mercury vapourlamp. However, the coherence time of natural thermallight sources is too short for bunching effects to be dis-cernible by present day solid-state photo-detectors. Mosttwo-photon interferometry experiments simulate thermallight by passing laser light through a rotating ground-glass

plate. This introduces, random intensity and phase fluctu-ations that occur at a time scale detectable by present daysolid-state detectors. We employed a different techniquethat utilises acousto-optic interaction.

In an AOM, a radio-frequency (RF) electrical signal isapplied to create a travelling acoustic-wave grating thatcan diffract light. The frequency, intensity, and phase oflight can be manipulated through acousto-optic interac-tion by suitably tailoring the RF input to the AOM. Inthis experiment, however, only phase fluctuations were in-troduced. As demonstrated recently [29], an incoherentsource may be created by electronically introducing ran-dom phase jumps to the RF and imprinting these ontolaser light. The phase evolves undisturbed for a time T ,and then is changed by a random amount δ. The phasejump δ is uniformly randomly distributed over the circleand T has the distribution of a truncated exponential:

P (T ) =1

Tce−

TTc , (1)

where 1 µs < T < 100 µs, and Tc = 10 µs. Tc representsthe timescale over which the coherence of the light beam islost. Such a distribution (in untruncated form) has beendiscussed [30] in connection with the emission of a singleatom interrupted by collisions. Each AOM in the setupwas independently phase modulated in this manner to de-rive two independent phase-incoherent sources (S1 and S2in Fig. 2) from the same laser light.

Theory: We now examine E3 and E4, the light fieldsreaching detectors D3 and D4, in terms of E1 and E2, thelight fields emerging from Q1 and Q2. Using the helic-

ity basis vectors |R 〉 =

(10

)and |L 〉 =

(01

)we may

represent linear states by |ϕi 〉 = 1√2(e−iϕi |R 〉+eiϕi |L 〉).

We introduce projection matrices PK = |K〉〈K| onto po-larisation states |K 〉, where K= R, L, 3, 4. More explic-itly,

PR =

(1 00 0

); PL =

(0 00 1

); Pi =

1

2

(1 e−i2ϕi

ei2ϕi 1

)(2)

Representing polarisation vectors with Greek incides andusing the Einstein summation convention (for the Greek,but not for the Latin indices), we can write

Eαi =1√2Pαβi [ εiP

βγL Eγ2 ui2 + P βγR Eγ1 ui1 ]

Eαi =1√2

[ ui2 Eγ2 P

γβL εi + ui1 E

γ1 P

γβR ] P βαi , (3)

where the overbar stands for complex conjugation; εi is apure phase ε3 = 1, ε4 = −1 due to the geometry of theMach-Zehnder setup. The quantities of interest for theAB effect are the different correlations at detectors D3and D4. The first order correlations or intensities are

I1i (τ = 0) = 〈 Ii 〉 = 〈 Eαi Eαi 〉; i = 3, 4 (4)

p-3

N. Satapathy et al.

Note that each detector i has contributions from both E1

and E2. In our experiment, the light intensities were ad-justed so that 〈 E1E1 〉 = 〈 E2E2 〉, i .e.,〈 I1 〉 = 〈 I2 〉. Thesecond order cross correlations are

G234(τ) =

〈 I3(0) I4(τ) 〉〈 I3 〉〈 I4 〉

=〈 Eα3 (0)Eα3 (0)Eβ4 (τ)Eβ4 (τ) 〉

〈 Eα3 Eα3 〉〈 Eβ4E

β4 〉

(5)and self correlations

G2ii(τ) =

〈 Ii(0) Ii(τ) 〉〈 Ii 〉2

=〈 Eαi (0)Eαi (0)Eβi (τ)Eβi (τ) 〉

〈 Eαi Eαi 〉2

(6)where I3(0) and I4(τ) are the intensities measured at D3at time 0 and at D4 at time τ respectively and the average〈. . .〉 is a time average over the integration time Tint, fora given setting of the polarisers P3 and P4 :

〈 f 〉 =1

Tint

∫ Tint

0

f(t)dt (7)

Substituting in Eqs. 5 and Eq. 6 from Eq. 3 andmultiplying the terms explicitly yields 16 terms, of which10 vanish on time averaging. Use is made of the fact thatthe 2×2 Hermitean projection matrices P satisfy P 2 = P .Two of the surviving terms have a sequence of projectionsof the form PRP3PLP4PR, which define a closed trajectoryon the Poincare sphere. This gives rise to the geometricphase ΦG = Ω/2, that is, half the solid angle enclosed.Simple algebra yields, for the cross correlation

G234(τ = 0) = 1− 1

2cos

(φD +

Ω

2

), (8)

.and for the self correlation

G2ii(τ = 0) = 1 +

1

2cos (φD) , (9)

where φD represents the dynamical phase (which, in ourMach-Zehnder interferometer has been adjusted to zero)and Ω

2 is the geometric phase. The negative sign for thesecond term (in Eq. 8) is due to the Mach-Zehnder con-figuration.

On similar lines, using Eq. 3 in Eq. 4, the first or-der correlation may be evaluated. Of the four terms inthe product, only two survive, with products of the formPiPRPi, (i = 3, 4), which depend only on the intensities ofthe sources and not the geometric phase. Thus,

I1i =〈I1 + I2〉

4, i = 3, 4 (10)

The geometric phase appears only in the second orderintensity cross correlation (Eq. 8), and is absent both inthe second order intensity self correlation (Eq. 9) and thefirst order correlation (Eq. 10). Thus, this effect is analo-gous to the two-particle non-local AB effect.

Fig. 3: (Colour online) Figure shows the experimentally deter-mined cross correlation, G234, as a function of φ34 the relativeangle between the polarisers P3 and P4, and as function oftime delay τ . Clearly, the cross correlation G234 is modulatedby the geometric phase.

Experimental Results : The main results of our experi-ment are displayed in the three-dimensional plot of Fig. 3,which shows the quantity of interest, G2

34, as a function ofthe time delay, τ , and the relative angle, φ34, between thelinear polarisers P3 and P4 in front of the two detectorsD3 and D4 respectively. In the experiment P3 was keptfixed and P4 was rotated. In effect, φ34 is a measure ofthe geometric phase.

For zero delay (τ = 0) between the two detectors, thecross correlation G2

34 was found to vary sinusoidally from∼ 0.5 to ∼ 1.5 as φ34 is varied. Clearly, this is due to thegeometric phase ΦG = Ω/2 = 2φ34. For larger values ofτ the amplitude of the sinusoidal variation is found to beprogressively diminished, till finally, for τ > TC , G2

34 re-mains nearly constant at 1, i.e., correlations that are max-imum for zero delay gradually decrease, and are absent fordelays larger than the coherence time. At the regions ofconstructive intereference (e.g. φ34 = 90), the value ofG2

34 starts from 1.5 for τ = 0 as is expected for a sourcewith pure phase fluctuations [31]. Note that in regions ofdestructive interference, the cross correlation starts from0.5, a value lower than that for a coherent source. Thishas been discussed in various contexts earlier [32].

In order to illustrate that it is a purely nonlocal effect wealso plot in Fig. 4, the second order self correlation. In con-trast to the cross-correlation, the self-correlation remainsunaltered as the relative angle between the polarisers isvaried. This may be easily understood by visualising thetrajectory on the Poincare sphere. For example, for G2

44

the trajectory is obtained by the sequence PRP4PLP4PRand defines an arc connecting the three points R, 4, L anddoes not enclose any solid angle; hence the absence of ageometric phase [20].

Having established the non-local nature of this effect ,we next verify its absence in the lower order correlation.Using the same experimental data that was used above,

p-4

Classical light analogue of the nonlocal Aharonov-Bohm effect

Fig. 4: (Colour online) Figure shows the self correlation G244as a function of φ34 the relative angle between the polarisationangles of the two detectors, and of the time delay τ . Note thatthe self correlation (G244 is plotted) is independent of φ34 . φ34

is varied by turning the polaroid P4 in the observed channel.Thus the geometric phase effect of Fig. 3 is a purely nonlocaleffect.

we now examine the quantity I14 . In Fig. 5 we display the

quantity I1i (τ = 0) as a function of the setting φ4 of the

polaroid P4 in front of detector D4. Note that this curveis practically constant, showing that turning the polaroidP4 does not affect the lower order correlation function,that is, the geometric phase is not seen in the lower ordercorrelations.Discussion: The experiment and the theoretical analy-sis presented here are purely classical, in contrast to theearlier quantum mechanical treatment [19] that predictedsimilar results, and the recent photon-counting intensityinterferometry experiments [33] that verified it. In thisLetter we develop the analogy between classical polarisedlight and the quantum nonlocal AB effect. Further, we ex-plicitly show its non-local nature, and its absence in lowerorder correlations.

The classical approach has the advantage that the phys-ical ideas are easy to grasp. In fact, Hanbury Brown andTwiss were originally motivated by their classical experi-ence with radio waves to propose the corresponding opticalexperiment. The quantum interpretation of this classicallysimple experiment led to profound changes in our under-standing of quantum optics and coherence.

In addition to being completely in the classical domain,our experiment has another novelty - the use of a sourcethat has phase fluctuations only. In the case of thermal orpseudothermal light (i.e., one with both phase and inten-sity fluctuations), G2 varies between 1 and 2. In contrast,for pure phase modulation, one expects a variation from0.5 to 1.5. Baym, in his review article [31] that examinesthe HB-T effect in a wide variety of physical contexts rang-ing from nuclear physics to astronomy, had reached thisconclusion in his discussion of the case of pure phase fluc-tuations. However, this has hitherto not been verified, as

Fig. 5: (Colour online) I14 as a function of φ34 as obtainedfrom the experiment. φ34 = φ3 − φ4 is varied by turning thepolaroid P4 in the observed channel while keeping P3 fixed.

pure ”phase-only” fluctuating sources have not been avail-able. In our experiment, such a source has been realisedby tailoring fluctuations in light by the acousto-optic tech-nique, and the result of Baym verified.

Two salient features of the geometric phase measuredin our experiment are its purely non-local nature and itsabsence in lower order correlations, in contrast to earlierwork on geometric phase and intensity interferometry [34].These specific features give rise to the possibility of theapplication of this nonlocal AB effect as a two-photon de-flector. We refer again to Fig. 2, where we now replace thepolariser P4 by a “graded polariser” - one where the axis ofpolarisation changes in orientation gradually as one movesacross the polariser. In intensity interferometry this wouldintroduce a geometric phase gradient. As is well known,a phase gradient in a wave is equivalent to a deflectionor a change of wave vector. In the present experiment,the phase change appears only in the intensity cross cor-relations. Thus, we may generate a “correlated-photondeflector” which affects the intensity-intensity correlation,leaving individual photons unaffected. Such a device mayhave applications in creating and manipulating correlatedphotons.

We close with a few classical remarks which may haveinteresting quantum interpretations. Let us note thatthe geometric phase is unbounded. As one turns the po-lariser P4, the geometric phase keeps accumulating with-out bound and the visibility of the interference patternremains undiminished. This is in sharp contrast to thedynamical phase: as one increases the dynamical phase,by separating the detectors, the interference pattern iswashed out when their spatial (temporal) separation ex-ceeds the coherence length (time). This difference is due tothe fact that the geometric phase is achromatic and affectsall frequencies equally. Thus the geometric phase effectsare less susceptible to decoherence than the correspondingdynamical effects. This illustrates a point often made inthe quantum computation literature that geometric andtopological effects are robust against decoherence. This is

p-5

N. Satapathy et al.

crucial to the development of quantum computers.At the level of classical information theory, our exper-

iment can be interpreted as a way of delocalising infor-mation. Consider the experimental setup above and makethe following minor changes: We choose P3 and P4 to beorthogonal linear polarisations represented by antipodalpoints on the equator of the Poincare sphere. We take thesources to be elliptically polarised and lying on the greatcircle orthogonal to the line joining the antipodal points3 and 4. If the angle φ between the elliptical polarisa-tions of the two sources is varied in time φ(t), one wouldfind that there is a corresponding modulation in the cross-correlations of the intensities detected at D3 and D4. Thesignal φ(t) can be viewed as carrying information. How-ever, the two beams emerging from the experiment are offixed linear polarisation, and each beam by itself appearsthermal in its self correlation. It is only by looking attheir cross correlations that one can recover the originallyimpressed signal φ(t). Thus, in this example, informationis stored in a completely delocalised manner. In quantuminformation theory, one expects that it would be similarlypossible to have apparently thermal beams carrying infor-mation entirely in their quantum cross correlations.

To summarise, we have demonstrated the existence ofa non-local geometric phase in the intensity-intensity cor-relations of classical incoherent light, that is not seen inthe lower order correlations. This two-photon Pancharat-nam phase is the classical optical analogue of the multi-particle AB effect [28]. As the trajectory of light over thePoincare sphere introduces a phase shift observable onlyin the intensity-intensity correlation, it provides a meansof deflecting the two-photon wavefront, while having noeffect on single photons. The experiments were performedusing sources that had pure phase fluctuations. We ex-pect that the results presented here will be of interest forapplications in the realm of classical and quantum com-munication and cryptography.

∗ ∗ ∗

REFERENCES

[1] Strekalov D. V., Stowe M. C., Chekova M. V. andDowling J. P., J. Mod. Opt. , 49 (2002) 2349.

[2] Saleh B. E., Abouraddy A. F., Sergienko A. V. andTeich M. C., Phys. Rev. A , 62 (2000) 043816.

[3] Pittmann T., Shih Y., Strekalov D. and A.V. S.,Phys. Rev. A , 52 (1995) R3429.

[4] Eisenberg H., Hodelin J., Khoury G. andBouwmeester D., Phys. Rev. Lett. , 94 (2005)090502.

[5] Shih Y., An Introduction to Quantum Optics : Photonand Biphoton Physics (CRC Press (U.S.)) 2011.

[6] Bennick, R. S., Bentley, S. J. and Boyd, R. W.,Phys. Rev. Lett. , 89 (2002) 113601.

[7] Abouraddy A., Saleh B., Sergienko A. and TeichM., Phys. Rev. Lett. , 87 (2001) 123602.

[8] Saleh B. E. and Teich M. C., Noise in Classical andQuantum Photon-Correlation in Advances in InformationOptics and Photonics ed. A.T. Frieberg and R. Dandikerin (SPIE Press, Bellingham, WA) 2008.

[9] Erkmen B. I. and Shapiro J. H., Adv. in Optics andPhotonics , 2 (2010) 405.

[10] Klyshko D., Phys. Lett. A , 146 (1990) 93.[11] Burlakov, A. V., Chekova, M. V., Mamaeva, Yu B.,

Karabutova, O. A., Korystov, D. Y. and Kulik, S.P., Laser Phys. , 12 (2002) 1.

[12] Shih Y., Rep. Prog. Phys. , 66 (2003) 1009.[13] Scarcelli G., Valencia A. and Shih Y., EPL , 68

(2004) 618.[14] Valencia A., Scarcelli G., D’Angelo M. and Shih

Y., Phys. Rev. Lett. , 94 (2005) 063601.[15] Shirai T., Setala T. and Frieberg, A. T., Phys. Rev.

A , 84 (2011) 041801.[16] Zhou Y., Simon J., Liu J. and Shih Y., Phys. Rev. A ,

81 (2010) 043831.[17] Karmakar S., Zhai Y., Chen H. and Shih Y., The first

ghost image using sun as light source in proc. of QuantumElectronics and Laser Science Conference. (Optical Soci-ety of America) 2011 p. QFD3.

[18] Venkataraman D., Hardy N., Wong Francis, N. C.and Shapiro, Jeffrey H., Opt. Lett. , 36 (2011) 3684.

[19] Mehta P., Samuel J. and Sinha S., Phys. Rev. A , 82(2010) 034102.

[20] Pancharatnam S., Proc. Ind. Acad. Sci. A, 44 (1956)247.

[21] Ben-Aryeh Y., Journal of Optics B: Quantum and Semi-classical Optics , 6 (2004) R1.

[22] Samuelsson P., Sukhorukov E. V. and Buttiker M.,Phys. Rev. Lett. , 92 (2004) 026805.

[23] Aharonov Y. and Bohm D., Phys. Rev. , 115 (1959)485.

[24] Samuel J., Curr. Sc. , 66 (1994) 781.[25] Batelaan A. and Tonomura A., Phys. Today , 62

(2008) 38.[26] Wu T. T. and Yang C. N., Phys. Rev. D , 12 (1975)

3845.[27] Hariharan P., Ramachandran H., Suresh K. and

Samuel J., J. Modern Optics , 44 (1997) 707.[28] Neder I., Ofek N., Chung Y., Heiblum M., Mahalu

D. and Umansky V., Nature , 448 (2007) 333.[29] Pandey D., Satapathy N., Meena M. S. and Ra-

machandran H., Phys. Rev. A , 84 (2011) 042322.[30] Loudon R., The Quantum Theory of Light (Oxford Uni-

versity Press) 2010.[31] Baym G., Acta. Phys. Polonica B, 29 (1998) 1839.[32] Paul H., Reviews of Modern Physics , 54 (1982) 1061.[33] Martin A., Alibart O., Flesch J.-C., Samuel J.,

Sinha S., Tanzilli S. and Kastberg A., EPL (Euro-physics Letters) , 97 (2012) 10003.

[34] Brendel J., Dultz W. and Martienssen W., Phys.Rev. A , 52 (1995) 2551.

p-6