wave-particle duality of broadband light

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Wave-particle duality of broadbandlightR. J. Rostron a , A. Homer a & G. Roberts aa Department of Physics , University of Newcastle , Newcastleupon Tyne, NE1 7RU, UKPublished online: 19 Aug 2006.

To cite this article: R. J. Rostron , A. Homer & G. Roberts (2006) Wave-particle duality ofbroadband light, Journal of Modern Optics, 53:11, 1647-1661, DOI: 10.1080/09500340600581983

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Wave-particle duality of broadband light

R. J. ROSTRON, A. HOMER and G. ROBERTS*

Department of Physics, University of Newcastle,Newcastle upon Tyne, NE1 7RU, UK

(Received 1 August 2005; in final form 26 December 2005)

We describe and analyse a simple experiment based on interference of thermallydistributed broadband photons which demonstrates quantitatively the gradualtrade-off between knowledge of the trajectory of a quantum particle and itswave-like character as expressed via the inequality V 2

þK 2� 1. The experiment

relies upon colour-selective detection of light passing through each slit ofa Young’s double-slit arrangement: path information K is deduced from theoverlapping bandwidths of the interfering light and the classical first-orderdegree of coherence, and wave information V is obtained from the fringevisibility. A classical Fourier analysis of the experiment is given which reproducesthe observed interferograms.

1. Introduction

The paradigmatical value of Young’s double slit experiment in optics, atomic physicsand quantum mechanics has been stressed on numerous occasions, for exampleby Feynman [1], Bell [2] and Holland [3] to name but a few of many authors.In his Gifford lectures, Heisenberg invoked Young’s double slit (YDS) to constructan argument against the presence of subjective elements in the Copenhageninterpretation of quantum mechanics [4], while the arrangement also serves asa framework for discussions of ‘delayed-choice’ issues and the pilot-wave formula-tion of quantum theory [2, 3, 5]. In relation to issues of wave-particle duality,the observation of an interference pattern behind a pair of slits illuminated by abeam of light or matter confirms the non-local, wave-like character of theincident radiation; contrarily, the assignment of a definite trajectory to a particle,e.g. by blocking one slit and observing the build-up of intensity behind the openslit, confirms the localized nature of the incoming radiation which can then beinterpreted—neglecting diffractive effects—as a stream of discrete particles. For thisreason, YDS arrangements have been adopted by many workers [6] as a vehiclefor demonstrating the dual wave and particle characteristics of matter predicted

*Corresponding author. Email: [email protected]

Journal of Modern OpticsVol. 53, No. 11, 20 July 2006, 1647–1661

Journal of Modern OpticsISSN 0950–0340 print/ISSN 1362–3044 online # 2006 Taylor & Francis

http://www.tandf.co.uk/journalsDOI: 10.1080/09500340600581983

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by de Broglie [7]. The implementation of YDS schemes in the time–energy domainhas found application in atomic spectroscopy, for example in photoionizationby phase-controlled laser fields [8].

The dual nature of light has been probed in a series of experiments [9–11].Recent work has examined the inequality [12–14]

V 2 þ K 2 � 1, ð1Þ

which links the wave information V, expressed via the visibility of interferenceas conventionally defined [15], and path information K expressed via the welcherWeg predictability. Inequality (1) implies that under certain conditions it is possibleto be in simultaneous possession of partial knowledge of the wave-like andparticle-like behaviour of a system without completely suppressing its non-localconnectedness. Schwindt et al. quantitatively verified inequality (1) by single-photoninterferometry in experiments in which the polarization state of light was usedto label photon paths through an asymmetric Mach–Zehnder interferometer [10].The strictness of the exclusivity of wave and particle information has been probedby parametric down-conversion of a single photon and detection of correlatedphoton pairs selected according to their polarization state in experiments by Bridaet al. [11]. A gedanken experiment involving interference of de Broglie matter wavesin a YDS arrangement equipped with resonators for slit-specific photon–atominteractions has been analysed [16]. Durr et al. subsequently performed such anexperiment in which single-atom interference was modulated by encodingwelcher Weg information through interactions with a microwave field [17]. Theinterconversion between fringe visibility and knowledge of path of a quantumparticle can also be interpreted in terms of a smudging parameter whichcharacterizes the suppression of the interference [18].

The application of inequality (1) to broadband light is examined here by meansof a simple experiment, based on a YDS arrangement, in which welcher Weginformation is obtained by tagging photons according to their colour. The natureof the experiment is portrayed schematically in figure 1(a) and involves a standardYDS set-up modified only by mounting colour-sensitive filters immediately before orbehind each slit so as to select or detect the colour of light passing through eachslit from a white-light primary source. The experiment is a variation of a gedankenexperiment envisioned by Jordan [19] in which photon trajectories are tagged bytheir colour rather than their polarization state to derive welcher Weg informationabout the light incident on YDS. The present work was also motivated by apaper by Garcıa et al. [20], where interference between the frequency componentsof a broadband laser pulse as they traverse different, colour-sensitive paths wasconjectured. The proposal of these workers was not formulated in terms ofrelation (1), however, while complete knowledge of the path of a broadband photonat the same time as observation of an interference pattern would stand at odds withthe time–energy uncertainty principle.

In this work, values of V and K are determined for colour-selected pathsof different centre wavelengths and bandwidths irradiated by a thermal field withan occupation number much less than one. These data, like those of Schwindtet al. [10], concur with inequality (1). The variation of the interference pattern

1648 R. J. Rostron et al.

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with the frequency overlap of the tagged photon trajectories is accounted for by thefirst-order field correlation and the bandwidths of frequency selection in each path.Consideration of the quantum mechanical expectation value of the intensity in theimage plane confirms that for the thermal multimode fields considered here, wave-like and particle-like behaviour can be described in terms of classical notions ofcoherence. An interpretation of the spatial distribution of intensity is provided bya Fourier calculation of the diffraction of a classical input field with binary colourselection, which shows how the fringe visibility depends on the difference in centrewavelength and bandwidth of the tagged photons. We also calculate interferogramsfor colour-selected components of an incident multimode (femtosecond) pulse.Section 2.1 describes the experimental arrangement and procedure for calculatingV and K, while the observations are reported and analysed in section 2.2. TheFourier analysis of the interferograms is presented in section 3 and concludingremarks are offered in section 4.

P1

P2

r1

r2

r;t

frequencyselectors

500 550 600400 650450

0.6

0.8

0.4

0.0

1.0

0.2

inte

nsity

(ar

b un

its)

λ (nm)

wl

(a)

(c)

(b)

diffractionplane

diffractionplane

a

focussinglens

image plane

image plane

dcollimatinglens

collimatinglens

lightsource

lightsource

S2S1

yx

fpolarizer polarizer

Figure 1. (a) Schematic diagram of a Young’s double slit experiment with simultaneousselection of the colour of a plane-polarized photon as it passes through slits S1 and S2. In theexperiments described in section 2, slits of width a¼ 200 mm were separated by a distanced¼ 0.93mm and the emerging interference pattern was focused into a CCD camera by a f/16lens. (b) Quantities involved in a theoretical discussion of the interference of spherical wavesemitted from pinholes P1 and P2: the distances r1 and r2 connect the two pinholes to a pointr on the image plane at time t. (c) The spectral intensity distribution of the input field labelled‘wl’ and example spectra of bandpass filters centred at 491 and 576 nm. (The colour version ofthis figure is included in the online version of the journal.)

Wave-particle duality of broadband light 1649

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2. Experimental set-up and procedures

2.1 Apparatus

A simple double-slit interference experiment was constructed as shown in figure 1(a).White light from a 150W incandescent tungsten filament was directed via a f/16collimating lens and 1mm diameter pinhole onto a diffraction screen, typicallylocated 15 cm from the primary source defined by the pinhole; plane-polarized lightwas directed onto the YDS arrangement in order that different polarization statesneed not be invoked in considerations of V and K [10, 13]. The maximum spectralradiant power incident on the diffraction plane was 1.0�10�18WHz�1 distributedover wavelengths from 400 to 900 nm. For single-photon measurements, the lampoutput was attenuated such that, for a transit time from source to detector of 5 ns,there was on average no more than 3� 10�3 photons in the apparatus at any onetime. These intensity conditions are comparable with those in the experiments basedon polarization-state detection of photon paths performed by Schwindt et al. [10].Light emergent from the double slits was weakly focused by another f/16 lens ontothe active element of a CCD camera whose output was transferred to a PC fordata storage and subsequent analysis. The active area of the camera comprised 752(horizontal)� 582 (vertical) pixels of size 8.6� 8.3 mm respectively. Strips ofcoloured filters (mainly gelatine or plastic) fixed before or behind the individualslits were applied to construct secondary sources of different bandwidth and centrewavelength across the range 435–680 nm. The spectrum of each filter wasrecorded photoelectrically in separate measurements using a monochromator witha resolution of �1 nm; typical bandwidths were ’25 nm (FWHM) within the range21–37 nm. To mitigate against a large dc background in visibility measurements,care was taken to select pairs of filters with similar maximum amplitudes andbandwidths. The maximum amplitudes of the filters applied to the slits in this workdiffered by a factor of no more than 1.25 and their bandwidths by no more than 1.5.Numerical simulations based on equation (10) (see section 3) indicate that suchdiscrepancies result in reductions in visibility of 1% and 6% respectively for thewavelengths, slit parameters and focusing conditions used in the experiments.The optical geometry of the arrangement, which was constrained by the maximumslit separation at which single-colour interference fringes could be detected onthe active area of the CCD array, restricted the range of distances at which thecolour-sensitive filters could be mounted either side of the diffraction plane to afew millimetres. The width of the slits was measured with a travelling microscopeto be no larger than 200 mm. For white light centred at �’ 600 nm, sharp fringes(without frequency restrictions) were obtained with a slit separation of d¼ 0.930mm.

2.2 Experimental results

Figure 2 shows interference patterns recorded when light of different coloursemerges from a double-slit arrangment: the observed interferograms are givenin figure 2(a)–(c), while figure 2(d )–( f ) show cross-sections through the regionsof greatest intensity variation. In figure 2(a), where the light diffracted by theslits has the same centre wavelength and bandwidth, distinct interference fringes


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modulated by the diffraction envelope are observed. When the secondary

sources have different centre wavelengths but overlapping bandwidths, figure 2(b)

and (e) indicate that interference fringes with reduced visibility are obtained.

In qualitative terms, figure 2(b) demonstrates that it is possible to affect a fairly

precise determination of the slit through which light of different colour passes

without causing overwhelming disruption to the concomitant interference pattern.

(a)

(b)

(c)

(d)

(e)

(f )

Figure 2. Examples of single-photon interference fringes obtained from a double slitwith d¼ 0.93mm and a¼ 200mm. Diagrams (a)–(c) display the fringe systems recorded bya CCD camera, while diagrams (d )–( f ) show cross-sections through the interferograms;the abscissae can be converted to distance by multiplying pixel number by 8.6 mm. Thecentre wavelengths and bandwidths of the colour filters at each slit are: (a) �1¼ �2¼ 630 nmand ��1¼��2¼ 25 nm; (b) �1¼ 576 nm, �2¼ 630 nm, ��1¼ 22 nm and ��2¼ 25 nm;(c) �1¼ 491 nm, �2¼ 630 nm, ��1¼ 37 nm and ��2¼ 25 nm.


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When the bandwidths of the secondary sources span detectably different frequencyranges, the interference pattern in figure 2(c) is seen to disappear since it is nowpossible to assign a definite, different colour to a photon which propagates alongdifferent paths between diffraction and detection planes. In the limit of unambiguousknowledge of the trajectories of individual photons through one or other of a pairof slits, the observation of an interference pattern is rendered impossible andvice versa. Whichever side of the diffraction plane the filters are situated, it alwaysturns out that no interference is observed in the image plane behind the slitswhen the bandwidths of the detectors allow photons of detectably different coloursto pass through the slits or to be identified afterwards. In addition to the resultsdisplayed in figure 2, more than thirty analogous measurements were performedusing a selection of filters within the visible spectrum from 435 to 680 nm.

To quantify the wave-like content of the diffracted light, the visibility of eachinterference pattern, defined as usual in terms of maximum and minimum averageintensities via

V ¼hImaxðrÞi � hIminðrÞi

hImaxðrÞi þ hIminðrÞi, ð2Þ

was recorded in the neighbourhood of the centre of each interferogram. The centrepoint of the interference pattern was chosen to avoid uncertainties in the determina-tion (by geometrical measurement) of the delay time T¼ (r1� r2)/c equivalent to thepath difference between r1 and r2 shown in figure 1(b) (see equations (2) and (4)).Here, for ease of discussion, we consider the arrangement portrayed in figure 1(b)in which the slits S1 and S2 are replaced by pinholes P1 and P2 at vector locations r1and r2 respectively; P1 and P2 act as secondary sources of spherical waves at timestj¼ t� rj/c ( j¼ 1, 2) in accordance with Huygens’ construction. (The geometry of thediffracting elements is accounted for in calculations of the spatial distributions of theimage field discussed in section 3, and their replacement by pinholes does not affectthe determination of V and K.) The particle-like content of the diffracted beamwas assessed in terms of the welcher Weg knowledge of the incident field, which itselfwas deduced from the spectral overlap of the bandpass filters mounted on each slit.If a detector is mounted on each slit, and the detector states are describedby a quantum mechanical state vector jDk(tj)i (k¼ 1 or 2), then the quantum stateof the combined fieldþ detector system becomes

Eðr; tÞ ¼1

21=2Eþ

ðr1; t1ÞjD1ðt1Þi þ Eþðr2; t2ÞjD2ðt2Þi

� �þ hc, ð3Þ

where ‘hc’ signifies hermitian conjugate. Equation (3) embodies the fact thatassignment of a definite trajectory to a photon is directly connected to a propertyof the detectors [21]. We adopt this proposal to establish a metric, based on theclassical first-order degree of coherence of the field and the spectral width andoverlap of the detectors, of welcher Weg knowledge of the passage of a photonthrough either one of the two slits. Accordingly, to determine K we employ therelation

K ¼ exp½�T 2�21�

22=2ðn1�

22 þ n2�

21Þ� �NAjgð1Þðr1, r2;T Þj, ð4Þ


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where �1 and �2 are bandwidths (FWHM) of the colour selectors mounted on eachslit and n1 and n2 are numerical factors which depend on the shape of the bandpassspectra. The first term of equation (4) measures the deletorious effect of finitebandwidth on the perfection of welcher Weg knowledge, while the second reflectsthe likelihood of correctly guessing the path of a particular quantum through eitherof the slits. A ‘guess’ is determined by the spectral overlap A of the bandpassfilters mounted on each slit, normalized by N with respect to the bandpass oflargest bandwidth, and by g(1)(r1, r2; T ), the normalized quantum first-order degreeof coherence of the light. To determine K, the spectral overlaps of the differentpairs of bandpass filters used in experiments and the normalization constant werecomputed numerically from the experimentally recorded spectra.

To illustrate the operation of equations (2) and (4), we consider brieflyDoppler-broadened secondary emission with the power spectrum [22]

jEkð!Þj2 ¼ pjEk0j

2�2k exp

�ð!� !kÞ2�2

k

2

� �,

where �k is the coherence time and !k is the centre frequency of the light from thekth secondary source. The normalized first-order correlation function is then

gð1Þðr1, r2;T Þ ¼ jgð1Þðr1, r2;T Þj exp�ið�2

1!1 þ �22!2ÞT Þ

�21 þ �2

2

� �ð5Þ

with

jgð1Þðr1, r2;T Þj ¼ exp�½T 2 þ ð�!�1�2Þ

2�

2ð�21 þ �2

2Þ

� �, ð6Þ

where �!¼!2�!1 is the difference in centre frequencies. Accordingly, equation (4)becomes

K ¼ exp�T 2

2ð�21 þ �2

2Þ

� �1�

2�2s

�21 þ �2

2

� �1=2

exp�ð�!�1�2Þ

2

�21 þ �2

2

� �( ), ð7Þ

where �s is the smaller of �1 and �2. The visibility at position r in the image planeis given as usual [22] by

V ¼2 hI ð1ÞðrÞihI ð2ÞðrÞi� �1=2hI ð1ÞðrÞi þ hI ð2ÞðrÞi

jgð1Þðr1, r2;T Þj, ð8Þ

where hI (k)(r)i is the expectation value of the intensity at r due to the light fieldat pinhole Pk. For white light at P1 and P2, equations (7) and (8) indicatethat V¼K¼ 0 as expected; whereas, for overlapping �-function bandpasses, V¼ 1and K¼ 0, and for non-overlapping �-functions, V¼ 0 and K¼ 1. We noteparenthetically that the intensity at point r in the image plane is

hIðr; tÞi ¼ hI ð1ÞðrÞi þ hI ð2ÞðrÞi þ 2 hI ð1ÞðrÞihI ð2ÞðrÞi� �1=2

� gð1Þðr1, r2;T Þ exp �

i ½!1 þ ð�1=�2Þ2!2�T

1þ ð�1=�2Þ2

�: ð9Þ


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The form of equation (5) for gð1Þðr1, r2;T Þ for thermal multimode radiation maybe confirmed from considerations of the expression hIðr; tÞi ¼ Tr �E�

ðr; tÞEþðr; tÞ

� for the expectation value of the quantum mechanical intensity operator. Whenthe density operator � is expressed in a basis of number states jfnkgi for mode k,it is found that each term on the right-hand side of equation (9) is directlyproportional to half the number of incident photons, as for single-mode light [23];that the centre frequencies of the secondary sources at P1 and P2 are different has nobearing on terms such as

Qk pnkhfnkgja

y

1a2jfnkgi which contribute to hIðr; tÞi, wherepnk is (in this case) the Boltzmann occupation probability of jnki and ay1 and a2 arerespectively the creation and annihilation operators corresponding to the secondaryfield operators E�

ðr1; t1Þ and Eþðr2; t2Þ. Equations (5) and (6) are then recovered

when the termsQ

k pnkð!i!jÞ1=2

hfnkgjay

i ajjfnkgi arising from the operator productsE�

ðri; tiÞEþðrj; tjÞ are ascribed a Gaussian distribution of frequencies. Although

a basis of number states for � is appropriate for the measurements reported here,for an experiment in which a multimode coherent state jf�kgi were incident onfrequency-selective pinholes (slits), evaluation of

Tr �E�ðri; tiÞE

þðrj; tiÞ

� ¼

ðd2

f�kgPðf�kgÞhf�kgjE�ðri; tiÞE

þðrj; tjÞjf�kgi

with a Gaussian positive P-distribution [24] leads to terms Pðf�kgÞhf�kgjay

i ajjf�kgi ¼

Pðf�kgÞQ

kðnk=2Þ, i.e. a weighted product of half the number of photons in eachmode; in this case also, no phase information about the jf�kgi is derivable from thefringes of the interferogram. For a thermal radiation field, a quantitative appraisalof its wave-like and particle-like characteristics can be obtained from a classicalfirst-order degree of temporal coherence.

The analysis of interferograms such as those shown in figure 2 is summarizedby figure 3, which plots the values of V2 and K2 derived from the interferograms,guided by the Gaussian expectation of equations (7) and (8), as a function of �!.As anticipated from figure 2 and the above discussion, the passage of light ofidentical colour and amplitude through the two slits leads to values of V’ 1(governed by the slit bandwidths) and little welcher Weg knowledge, whereaswhen the bandpass filters sample different frequency regimes, the visibilitydiminishes to zero and K’ 1 predominates. The error bars associated with V wereobtained from repeated measurements of hImaxðrÞi and hIminðrÞi; those associatedwith K arise from the determination of the centre frequencies and bandwidthsof the colour selectors. Figure 3 demonstrates that the measurements conform toinequality (1).

3. Fourier analysis

In this section we calculate the image field generated by the YDS arrangementof figure 1(a) for collisionally broadened thermal light and a broadband laser pulse.The aim of the calculations is to examine how the interference pattern varieswith colour selection, in support of the experimental measurements, rather than


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to probe the properties of a single photon per se as in the analysis of section 2.2.To simulate the experimental interferograms, numerical calculations using theexperimental slit dimensions and spectral parameters were performed. Our approachfollows that of [25] and [26], modified to account for frequency selection at the twoslits in the diffraction plane, and assumes the validity of the Fresnel approximationover the distances of the experiment. To conform with the quantum descriptionof the situation expressed by equation (3), the intensity distribution in the imageplane is calculated from the superposition integral

Sðx, y;!1,!2Þ ¼

ð1�1

eU ð1Þim ðx, y;!1ÞeE1ð!Þ þ eU ð2Þ

im ðx, y;!2ÞeE2ð!Þ 2 d!, ð10Þ

which relates the emerging interference pattern to the field amplitudes eU ðkÞim ðx, y; !kÞ

in the image plane (see figure 1a). The input parameters required for numericalsimulations based on equation (10) are the slit dimensions, their separation, thebandpass frequency distributions and the spectral intensity variation of the inputfield given in figure 1(c) [25, 26]. Figure 4 plots interferograms calculated fromequation (10) for combinations of !k and �k chosen to match the experimentalparameters of figure 2. We again see that the fringe visibility of the interferogramsis maximal when the secondary sources share the same frequency space (subjectto the experimental requirement that the ac contributions to Sðx, y; !1,!2Þ can be

∆ω (1014 rad/s)

0.0

0.4

0.2

0.8

0.6

1.0

3.02.52.01.51.00.50.0

visibility

welcher Wegknowledge

Figure 3. Plots of V2 (magenta points) and K 2 (light blue points) as a function of thedifference �! between centre frequencies of the bandpass filters mounted on the slits ofthe YDS arrangement. Values of V 2

’ 0 and K 2’ 1 at �!>3.4� 1014 rad s�1 and multiple

values of V 2’ 1 and K 2

’ 0 at �!¼ 0 are not shown to highlight more clearly the variationof V 2 and K 2 with �!. The error bars correspond to 2� uncertainties determinedfrom measurements of the diffracted and spectral intensities for V and K respectively.The red and blue lines show K 2 and V 2 calculated from equations (7) and (8) forDoppler-broadened secondary sources with bandwidths of ��1¼��2¼ 26 nm (averageof experimental values); they are not fits to the data points. (The colour version of this figureis included in the online version of the journal.)


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monitored relative to the dc signal and above the noise limit set by the detector)

and reduces when the overlap is decreased.An interpretation of the change in fringe pattern between figure 4(a)–(c)

is aided by an analytical calculation of Sðx, y; !1,!2Þ: to this end we consider

constant-amplitude white-light illumination of the diffraction plane at normal

incidence with intensity I0 ¼ jeE0j2 and assume that the slits diffract frequencies

x (mm)

0.4

1.0

0.6

0.8

0.2

0.0

0.0 1.00.5−0.5−1.0

0.4

1.0

0.6

0.8

0.2

0.0

(a)

(b)

(c)

0.4

1.0

0.6

0.8

0.2

0.0S

(x,y

;ω1,

ω2)

(ar

b un

its)

Figure 4. Calculated image field distributions S(x, y; !1,!2) due to double-slit diffraction ofa constant-amplitude input field with frequency selection at the slits. The centre wavelengthsand bandwidths of the secondary fields are chosen to match the experimenal values given inthe caption to figure 2. The maximum passband amplitudes D10 in arbitrary units relative toD20¼ 1.00 are: (a) D10¼ 1.00; (b) D10¼ 1.21; (c) D10¼ 1.10. The double slit parameters ared¼ 0.93mm and a¼ 200mm and the images are focused by a lens with a nominal focal lengthf¼ 400mm.


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with a Gaussian distribution about !1 and !2. Evaluating equation (10) with

Dkð!� !jÞ ¼ Dk0 exp½��2j ð!� !jÞ=2� leads to the result

Sðx, y ! 1; !1,!2Þ ¼I0

2p3=2x2D2

10=�1 þD220=�2

�� ðD2

10=�1Þ exp½�ðax=cf Þ2=4�21 � cos½!1ax=cf �

� ðD220=�2Þ exp½�ðax=cf Þ2=4�2

2 � cos½!2ax=cf �

�D10D20ð2=s212Þ

1=2

� exp½�ð�21!

21 þ �2

2!22Þ=2þ ð�2

1!1 þ �22!2Þ

2=2s212�

� exp½�ð½dþ a�x=cfþ�nl=cÞ2=2s212��

� cos½ð½dþ a�x=cfþ�nl=cÞð�21!1 þ �2

2!2Þ=s212�

þ exp½�ð½d� a�x=cfþ�nl=cÞ2=2s212�

� cos½ð½d� a�x=cfþ�nl=cÞð�21!1 þ �2

2!2Þ=s212�

� 2 exp½�ðdx=cfþ�nl=cÞ2=2s212�

� cos½ðdx=cfþ�nl=cÞð�21!1 þ �2

2!2Þ=s212�

��ð11Þ

for slits which are infinitely long in the y direction, where s212 ¼ �21 þ �2

2 and

�n ¼ nð!1Þ � nð!2Þ is the difference in the indices of the glass of the focusing

lens of thickness l at !1 and !2. The image predicted by this expression

comprises dc terms, two ac terms that arise from the individual slits, and three

ac terms which depend on d and d� a whose amplitudes are damped by exponential

factors containing the centre frequencies and bandwidths of the two pulses.

Inspection of equation (11) reveals that it is the term D10D20ð2=s212Þ

1=2

exp½�ð�21!

21 þ �2

2!22Þ=2þ ð�2

1!1 þ �22!2Þ

2=2s212� which pre-multiplies the last three

ac contributions which determines whether or not an interference pattern can

in principle be observed. The dependence of the exponent on the dissimilarity of

the two centre frequencies is most apparent when �1 ¼ �2 ¼ � and D10 ¼ D20 ¼ D0,

in which case the same exponential term becomes ðD20=�Þ exp½��2ð!1 � !2Þ

2=4�;for !1 6¼ !2 (and � 6¼ 0) we see that the interferometric contribution to

Sðx, y ! 1; !1,!2Þ ! 0 in the limit that j!1 � !2j ! 1, leaving the effect

of diffraction through the two slits superimposed upon a dc background as the

experimentally monitorable signature of welcher Weg knowledge. Equation (11)

also indicates that a large dissimilarity between D10 and D20 or between �1 and �2can cause one dc and diffractive term to dominate the ac interference, even when

!1¼!2; care was taken to avoid this situation in the experiments, as described

in section 2.1.


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To conclude this section it is of interest to consider diffraction by YDS ofan optical pulse whose bandwidth is spatially chirped to allow selection at the twoslits of different colours from within the spread of frequencies supported by thepulse width. For analytical simplicity we choose a primary field with a Gaussianenvelope function of the form [27]eEð!Þ ¼ E0 exp½i�ð!Þ � ð!� !0Þ

2�2G=2ð1þ �2Þ� ð12Þ

with �ð!Þ ¼ �ð!� !0Þ2�2

G=2ð1þ �2Þ � ð1=2Þ tan�1 � and pulse width �p ¼�Gð4 ln 2Þ

1=2 to illustrate this situation. Here, � is a linear (temporal) chirpparameter which imparts a quadratic phase dependence onto the local frequencywithin the pulse. YDS diffraction of Gaussian fields with higher-order chirp hasto be evaluated numerically [25]. We again select Gaussian functions Dk(!�!k)to represent the passband functions of the two slits: the secondary fields arethus products of Dk(!�!k) with eEð!Þ given by equation (12), modulated by the

image distributions eU ðkÞim ðx, y ! 1;!kÞ. Substituting for Dkð!� !kÞeEð!Þ with k¼ 1

and 2 into equation (10) gives

Sðx, y ! 1; !1,!2Þ ¼I0

2p3=2x2ðD2

10=s1GÞ exp½��21�

02Gð!1 � !0Þ

2=s21G��

� 1� exp½�ðax=2cfs1GÞ2� cos½ð�2

1!1 þ �02G!0Þðax=cfs

21GÞ�

� �þ ðD2

20=s2GÞ exp½�ð�22�

02Gð!2 � !0Þ

2=s22G�

� 1� exp½�ðax=2cfs2GÞ2� cos½ð�2

2!2 þ �02G!0Þðax=cfs

22GÞ�

� ��D10D20ð2=s

212GÞ

1=2 exp½�ð�21!

21 þ �2

2!22 þ 2�02

G!20Þ=2

þ ð�21!1 þ �2

2!2 þ 2�02G!0Þ

2=2s212G�

� ðexp½�ðSþ=cþ TgÞ2=2s212G�

� cos½ðSþ=cþ TgÞð�21!1 þ �2

2!2 þ 2�2G!0Þ=s

212G�

þ exp½�ðS�=cþ TgÞ2=2s212G�

� cos½ðS�=cþ TgÞð�21!1 þ �2

2!2 þ 2�2G!0Þ=s

212G�

� 2 exp½�ðS0=cþ TgÞ2=2s212G�

� cos½ðS0=cþ TgÞð�21!1 þ �2

2!2 þ 2�2G!0Þ=s

212G�

��, ð13Þ

where s2kG ¼ �2k þ �02

G (k¼ 1 or 2), s212G ¼ �21 þ �2

2 þ 2�02G and �0

G ¼ �G=ð1þ �2Þ1=2.

Like the result of equation (11) for a dc primary field, equation (13) comprises dc,slit and interference terms. The group delay time Tg ¼ Tð!2Þ � Tð!1Þ which appearsin the latter takes into account the different arrival times of the front of maximumamplitude of a tilted primary field at the two slits; the mismatch of phase andpulse fronts over the dimensions of a slit is not taken into account in equation (13)because a� d.


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For a distortion-free, pulsed primary field, figure 5 shows the effect of selectingparticular colours from the broadband input at the double slit: figure 5 (a)–(c) showinterferograms for pulses centred at �0¼ 800 nm with a bandwidth of ��0¼ 50 nm,equivalent to a Fourier-limited pulse width of ’27 fs; figure 5(d )–( f ) give theresults for an 800 nm input field linearly chirped to give a bandwidth five timesbroader. The values of �0¼ 800 nm and ��0¼ 50 nm correspond to the outputof a standard Ti:sapphire ultrafast laser. Comparison of figure 5(a) and (d )with (b), (c), (e) and ( f ) shows how quickly the interferometric contribution to

x (mm)

0.4

1.0

0.6

0.8

0.2

0.0

0.4

1.0

0.6

0.8

0.2

0.0

0.4

1.0

0.6

0.8

0.2

0.0

(a) (d)

(b) (e)

(c) (f )

0 2 6 84−2−4−10 10−8 −6 0 2 6 84−2−4−10 10−8 −6

S(x

,y←

8; ω

1, ω

2) (

arb

units

)

Figure 5. Calculated image intensity distributions due to double-slit diffraction of atime-limited, Gaussian input field centred at �0¼ 800 nm with bandwidths of ��0¼ 50 nm(diagrams (a)–(c)) and 250 nm (diagrams (d )–( f )); the group time delay is zero in all cases.The Gaussian passband frequency distributions at each slit have centre wavelengths andbandwidths: (a), (d ) �1¼ �2¼ 800 nm and ��1¼��2¼ 50 nm; (b), (e) �1¼ 790 nm,�2¼ 810 nm and ��1¼��2¼ 20 nm; (c), ( f ) �1¼ 790 nm, �2¼ 810 nm and ��1¼��2¼ 5 nm.The relative maximum amplitudes of the passbands are D10¼D20 in each case. The double slitparameters are d¼ 1.0mm and a¼ 10mm and the images are focused by a lens of nominalfocal length f¼ 500mm.


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Sðx, y ! 1;!1,!2Þ is diminished when two distinct frequencies present in the inputfield are selected as secondary fields. When the passband frequencies spectrallyoverlap, interference fringes may still be discerned within a distribution of dimin-ished light intensity in the image plane; but when the passbands do not overlapto any meaningful extent, the resulting image exhibits only a dc signal. That amismatch between the centre frequencies of the secondary and primary fieldsresults in suppression of the diffraction and interference contributions toSðx, y ! 1;!1,!2Þ can be readily appreciated when equation (13) is re-written fora Fourier-limited, undistorted incident field (�¼ 0,Tg¼ 0), from which secondaryfields with frequencies !1 ¼ !2 ¼ ! 6¼ !0 are selected at the double slit. With theassumption of Gaussian passbands characterized by D10¼D20¼D0 and�1¼ �2¼ �G, equation (13) takes on the abbreviated form

Sðx, y ! 1;!1,!2Þ ¼21=2D2

0

�Gx2exp½��2

Gð!� !0Þ2=2�

�

n1� exp½�ðax=cf Þ2=8�2

G�

� cos½ðax=cf Þð!þ !0Þ=2� � exp½�ðdx=cf Þ2=8�2G�

� cos½ðdx=cf Þð!þ !0Þ=2�

þ1

2exp½�ð½dþ a�x=cf Þ2=8�2

G� cos½ð½dþ a�x=cf Þð!þ !0Þ=2�

þ1

2exp½�ð½d� a�x=cf Þ2=8�2

G� cos½ð½d� a�x=cf Þð!þ !0Þ=2�o,

from which it is apparent that the difference between ! and !0 acting throughthe agency of the exp½��2

Gð!� !0Þ2=2� term suppresses the dc term as well as the

diffractive and interference contributions to Sðx, y ! 1;!1,!2Þ.

4. Conclusions

We have described a simple experiment which demonstrates the graded complemen-tarity between wave and particle descriptions of broadband light. The experimentis based on Young’s double slit and involves tagging the path of each photonaccording to its centre frequency. Measurements of the visibility and welcher Wegknowledge extracted from interferograms with different coloured components arequantitatively in accord with inequality (1). For thermal multimode light fields,wave-like and particle-like behaviour can be expressed in terms of the classicalfirst-order correlation function together with appropriate parameters to describe theaction of the detectors. Numerical calculations of the spatial intensity distributionin the image plane based on the Huygens–Fresnel principle support the experimentalobservations by showing how fringe visibility becomes more pronounced whenthe frequency distributions of the secondary light sources increasingly overlap.Analytical formulae for the spatial dependence of the colour-selected image intensityfrom Doppler-broadened thermal light and a multimode, coherent laser pulseindicate how the difference in centre wavelengths and the bandwidths of the pathfilters act to vary the amount of wave and path information in the photon beam.


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Acknowledgments

RJR thanks the Nuffield Foundation for an undergraduate bursary (URB/01249/G)and AH thanks the University of Newcastle for a vacation studentship, during thetenure of which this work was carried out.

References

[1] R.P. Feynman, R.B. Leighton and M. Sands, The Feynman Lectures on Physics, Vol. 1,Ch. 37 (Addison-Wesley, Reading, MA, 1963), pp. 37.2–37.11.

[2] J.S. Bell, Speakable and Unspeakable in Quantum Mechanics, Ch. 14 (CambridgeUniversity Press, Cambridge, 1993), pp. 111–116.

[3] P.R. Holland, The Quantum Theory of Motion, Ch. 5 (Cambridge University Press,Cambridge, 1993), pp. 173–190.

[4] W. Heisenberg, Physik und Philosophie, 6th ed., Ch. 3 (Hirzel Verlag, Stuttgart, 2000),pp. 76–78.

[5] J.A. Wheeler, in Mathematical Foundations of Quantum Theory, edited by A.R. Marlow(Academic Press, New York, 1978), pp. 9–48.

[6] See for example: H. Rauch and J. Summhammer, Phys. Lett. A 104 44 (1984); O. Carnaland J. Mlynek, Phys. Rev. Lett. 66 2689 (1991); A. Zeilinger, R. Gahler, C.G. Schull,et al., Rev. Mod. Phys. 60 1067 (1998).

[7] L. de Broglie, Nature (London) 112 540 (1923); Une Tentative d’Interpretation Causaleet Non-lineare de la Mechanique Ondulatoire, Ch. 17 (Gauthier-Villars, Paris, 1956),pp. 238–239.

[8] F. Lindner, M.G. Schatzel, H. Walther, et al., Phys. Rev. Lett. 95 040401 (2005).[9] P. Grangier, G. Roger and A. Aspect, Europhys. Lett. 1 173 (1986).[10] P.D.D. Schwindt, P.G. Kwiat and B.-G. Englert, Phys. Rev. A 60 4285 (1998).[11] G. Brida, M. Genovese, M. Gramegna, et al., Phys. Lett. A 328 313 (2004).[12] D.M. Greenberger and A. Yasin, Phys. Lett. A 128 391 (1988).[13] B.-G. Englert, Phys. Rev. Lett. 77 2154 (1996).[14] G. Bjork and A. Karlsson, Phys. Rev. A 58 3477 (1998).[15] M. Born and E. Wolf, Principles of Optics, 6th ed., Ch. 7 (Pergamon Press, Oxford, 1980),

pp. 260–261, 265–268.[16] M.O. Scully, B.-G. Englert and H. Walther, Nature (London) 351 111 (1991);

B.-G. Englert, M.O. Scully and H. Walther, Am. J. Phys. 67 325 (1999).[17] S. Durr, T. Nonn and G. Rempe, Nature (London) 395 33 (1998).[18] W.K. Wootters and W.H. Zureck, Phys. Rev. D 19 473 (1979).[19] T.F. Jordan, Am. J. Phys. 69 155 (2001).[20] N. Garcıa, I.G. Saveliev and M. Sharanov, Phil. Trans. R. Soc. 360 1039 (2002).[21] C. Brukner and A. Zeilinger, Phil. Trans. R. Soc. 360 1061 (2002).[22] M.O. Scully and M.S. Zubairy, Quantum Optics, Ch. 4 (Cambridge University Press,

Cambridge, 1997), pp. 115–119.[23] R. Loudon, The Quantum Theory of Light, 2nd ed., Ch. 6 (Clarendon Press, Oxford,

1983), pp. 211–214.[24] S.M. Barnett and P.M. Radmore, Methods in Theoretical Quantum Optics, Ch. 4

(Clarendon Press, Oxford, 1997), pp. 121–122; see also [22], Ch. 3, pp. 75–79.[25] R. Netz and T. Feurer, Appl. Phys. B 70 813 (2000).[26] G. Roberts, Phil. Trans. R. Soc. 360 987 (2002).[27] J.C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, Ch. 1 (Academic Press,

San Diego, CA, 1996), pp. 10–12.


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wave-particle duality of broadband light

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