Foundations o f Physics Letters, VoL 5, No. 4, 1992

P R O P O S E D E X P E R I M E N T S ON T H E "WAVE-PARTICLE D U A L I T Y OF L I G H T

C. Cormier -De lanoue

Fondation Louis de Broglie 23 Quai de Conti, 75006 Paris, France

Received January 12, 1992

The observed wave-particle duality of light suggests such represen- tations as permanent corpuscular photons, and waves, either closely associated with these particles, or empty of any observable energy, or, still otherwise, propagating intrinsic electromagnetic energy. Two experimental schemes are proposed in the present study to decide between these various concepts and eventually to reach a clearer de- scription of light.

Key words: wave-particle duality, corpuscular photons, empty waves.

1. I N T R O D U C T I O N

Since the famous 1905 paper by Einstein [1], first suggesting the idea of corpuscular light quanta, later to be called photons, the wave-corpuscle duality of light has been puzzling many physicists.

With a deep belief in unity of nature, Louis de Broglie [2], ex- tended this wave-corpuscIe duality to all material entities, only con- sidering the photon as one among other elementary particles.

If the energy of light is exclusively concentrated in corpuscles, waves have nonetheless to be maintained to explain all interference effects, for instance, and the coexistence of corpuscular energy and waves makes these latter devoid of energy and momentum.

Fotlowing this oversimple representation, many recent publica- tions still refer to the concept of photons as corpuscles of permanent existence between emission and absorption, more or less closely asso- ciated with "empty" waves having no proper energy whatsoever, hut yet, observable through various effects.

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Such descriptions being, on the other hand, totally absent from modern radiation theories, it appears that the wave-particle duatity of light is far from being explained to everyone's satisfaction.

As physics is primarily a science of observables, it does seem relevant to test these concepts by further experiment.

Some points of different theories of light, specifically related to this problem, will first be briefly recalled. Two experimental schemes will then be proposed, allowing a discrimination among these theories.

2. D U A L I S M IN V A R I O U S T H E O R I E S OF L I G H T

2.1. Maxwel l ' s E l e c t r o m a g n e t i c T h e o r y

In this theory, generally refered to as classical, waves propagate according to d'Alembert's equation, satisfied by both fields, E and H.

Diffuse energy is defined in every point by its flux, Poynting's vector.

There are no transverse waves, devoid of energy, which might be called "empty," nor are there any instantaneous actions at a distance.

2.2. Einstein-de Broglie Theory

After having suggested the concept of corpuscular quanta, Ein- stein, later on [3], had to admit a deeper form of dualism, deeper and so troublesome that he refered to light waves as ghost waves, "Gespensterwellen," in his own terms.

A much more precise theory was, later on, elaborated by L. de Broglie who named it the "Double Solution Theory" [4].

In this representation, corpuscular photons are associated with two distinct kinds of waves. The ¢ wave, which determines the prob- ability of observing a photon, is a subjective mathematical construc- tion. The v wave, on the other hand, is a real electromagnetic wave of very weak intensity [5]. Both waves are related by ¢ = Cv, with C » 1 as a normalization constant such that fv I¢12dv = 1 in the volume occupied by ¢. The corpuscular photon is a singularity of the regular v wave, fields only having very high amplitudes in a limited region of space, the photon, precisely.

A photon is permanently guided by a physically real v wave. It is possible to separate this wave in parts, with a beamsplitter for instance, and, if one part carries the corpuscule, the other part may be considered as "empty," as it propagates no observable energy.

Wave-Particle Duality of Light 35"/

2.3. Q u a n t u m T h e o r y of R a d i a t i o n

Fundamental quantum indeterminism limits this theory in its descriptions, and particularly, it does not give a precise image of ra- diation between emission and absorption.

A photon is no more than an elementary quantized excitation of a particular mode of the global system considered. A well-known formalism rules the annihilation and creation of photons.

It must be emphasized that there is no position operator related to a massless particle such as a photon [6].It may therefore, in no way be considered as a permanent corpuscle following a definite trajectory.

There are no empty waves, but only empty states 10k > of a given mode k.

2.4. Semi-Class ica l H y p o t h e s i s of E l e c t r o m a g n e t i c Wave R e d u c t i o n

This semi-classical hypothesis was first advanced to describe electromagnetic standing waves [7].

According to this description, light is formed by electromagnetic waves following Maxwell's equations. Photons as corpuscles do not exist. The detection of an energy quantum hv is the reduction of diffuse wave energy in a limited space region,reached by the radiation, and this, instantly, and at a distance.

The necesssary conservation of a zero divergence for local dis- placement currents in an electromagnetic wave is the basis of this instant reduction process by quantized absorbers.

Quite evidently, there are no empty waves in this theory either.

3. E X P E R I M E N T 1

A monochromatic and parallel light beam, of weak intensity, from a well stabilized laser, enters the system in I. (Fig. 1.)

This beam is first circularly polarized by a plane polarizer P1 followed by a quarter wave plate Q, so oriented that its neutral axis is at 45 ° to the polarizer axis.

When so prepared, the beam then enters a polarizing prism BS1 by which it is classically divided into ordinary and extraordinary rays o and e.

These two rays recombine in another prism BS2, identical to BS1, after respectively following the paths BS1, M1, M2, BS2, for the ordinary ray, and BS1, M3, Ma, Ms, M6, BS2, for the extraordinary ray. Furthermore, a Bravais compensator C, of variable thickness, is inserted between M1 and M2, so as to finely adjust the two optical pathlengths difference, or the relative phase A~ of the rays.

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The recombined beam emerging from BS2 is then analysed by a plane polarizer P2, before eventually emerging from the system in O and being measured.

The orientable plane polarizers P1 and P2, represented as disks on the figure, are such that only the component of incident licht parallel to their axis may emerge, white the perpendieular eomponent is eliminated by absorption.

Polarizing prisms BS1 and BS2 are represented as cubic, but could be of any type, Glan-Foucault, for instance. Absorption and stray reflections are supposed negligible.

The fact that light incident upon BS1 is circularly polarized en- tails identical classical amplitudes for the o and e rays, or equivalently, identical mean intensities.

Although the present system is far more elaborate, both in the A~ relative phase determination and the BS2 recombination deviee, the general prineiple of this experiment is closely related to the setup used by Presnel and Arago [8] to study eventual interferences by su- perposition of two light rays wich crossed plane polarizations. The very classical results obtained in this old experiment readily apply to the present oase.

If the two paths, followed by the ordinary and extraordinary rays o and c, are of equal geometrical leng~h, the C compensator will allow a precise adjustment of the relative phase AT. For A~ = (n + 1/2)~r, with Inl = 0,1, 2,. . , the beam emerging from BS2 will have richt or leff circular polafization. At the other extreme, if A~ = n~r, plane polarization of the outgoing beam will be obtained.

It is to be noticed that with optical paths of almost equal length, the coherence properties of the incoming light I are of no importance, the relative phase/k~o being eonstant for a given setting of C, even if the phase of the incoming beam I fluctuates.

With the particular condition/k~o = 0, the P2 polarizer is at first oriented with its axis at 45 ° of the plane defined by o and e, i.e. at 45 ° of the BS2 polarizing plane, and furthermore, parallel to the potarization plane of the recombined beam emerging from BS2.

According to classical electromagnetic theory, this outgoing re- combined ray will get across P2 unaltered.

The superposition of classical waves makes the intensity of the global beam coming out of BS3 equal to the sum of the initial inten- sities of o and e, whatever their phase difference A~, and thus equal to the incident intensity on BS1.

What can be said, if according to the Einstein-de Broglie the- ory, all radiant energy is concentrated, by units of h~, in corpuscular photons of permanent existence between creation and annihilation, and if further, these photons enter BS1 only one by one?

Such a corpuscle, accompanied by its anenergetic wave, will have to follow one or the other of the o or e paths, if the naive in- terpretation of Clauser's experiment [9] is remembered. Along the

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other path, e or o will then propagate a part of the wave which may logically be qualified as "empty."

In BS2 though, this empty partial wave will recombine with the coherent other part, which remained directly associated to the corpuscular photon. The total wave, resulting from the superposition, which is plane polarized at 45 ° to the BS2 polarization plane, will then reorient the corpuscular photon to give it a 100% probability of going through P2. This must be so if the corpuscular theory has the classical theory as a limit for a large number of photons.

Following this theory, the so-called empty waves therefore do influence the behaviour of a corpuscular photon with respect to a plane polarizer.

If it were not so, each e or o photon, and its associated wave, would only have a 50% probability of passing through the P2 analyser. The outgoing beam O would only have a mean intensity half that of the beam incident on BS1, at variance with classical ware theory and experiment.

In this particulav case at least, corpuscular photons, if they really exist, behave in polarization analysis as the global coherent wave with which they are associated of, in other terms, the total coherent wave commands the polarization of its eventually associated photons.

These waves, possibly guiding corpuscular photons, may be de- composed and recombined, just as classical electromagnetic waves.

In another disposition of the system, the P2 polarizer is oriented so that its axis is parallel to the polarization plane of one of the rays o or e as it emerges from BS2, o, for instance. The beam going out towards O will thus have its intensity reduced by 50% compared to the intensity of the beam entering BS»

According to the above analysis, this can get two very different explanations.

1. All corpuscular photons coming out of BS2 are plane polar- ized as their globally associated waves, at 45 ° to the P2 polarizer's axis, and therefore only have a 50% probability of crossing through P2 towards O.

2. If there are no corpuscular photons, but only waves with intrinsic energy, only the o component of the total wave will cross through P2 with certainty, the e component always being prevented from reaching O by the crossed polarizer P2.

These two explanations are equivalent as regards the final result and so remain undistinguishable at this stage. Another experiment is needed to reach a proper discrimination.

Wave-Parlicle Duality of Light 361

4. E X P E R I M E N T 2

In this second experiment, two sources are now used, instead of one (Fig. 2).

A monochromatic parallel light be_am/1, of weak intensity, is circularly polarized, as usual, by a plane polarizer P1, followed by a quarter wave plate Q1, with adequate mutual orientations. It is further split into ordinary and extraordinary rays Ol and el by a polarizing prism BS1.

A second beam /2, identical to Il , particularly as regards its frequency, is prepared in the same way by a polarizer P2 and a quarter wave plate Q2. It is then split by a polarizing prism BS2 into ordinary and extraordinary rays 02 and 02.

Rays el and02 are incident on the adequate faces of a third polarizing prism BS3 which allows their superposition. The Ol and e2 rays are respectively monitored by detectors D1 and D2.

The global beam emerging from BS3 is analyzed by a third plane polarizer P3 before eventually reaching detector D3.

The axis of P3 is oriented orthogonally to the polarization plane of the el ray. With this setting of P3, el alone will never pass through to D3, while 02 will always get through unaltered.

The sources for the incoming bearns/1 and/2 are two identical and well stabilized lasers so that their respective phases ~l(t) and 92 (t), although random, may be related by a definite A~ during very short instants shorter than their diffusion times. Both beams may be considered to be in coherent states la1 > and [a2 >, with no bunching of detected photons to be expected.

Following the Einstein-de Broglie corpuscular theory, it is fur- ther assumed that a2 is so weak that there is a negligible probability that more than one photon at a time propagates from beam/2 through the system.

If a corpuscular photon then follows the e2 path towards D2, an empty wave will propagate along 02 towards BS3. This empty wave should therefore become superposed on any el wave, and particularly on an el wave carrying a photon from the /1 beam through P1, Q1, and BS1.

For certain values of the relative phase Ap, the global wave resulting from the superposition of el and 02 (empty) will be plane or elliptically polarized. If, as previously shown in Experiment 1, a corpuscular photon behaves with respect to an analyser as the total wave guiding it, then, according to the Malus law, there may be a real possibility for a photon from the el ray to get through P3.

Of course, in Experiment 1, a photon was guided, after going through BS2, by a superposition of two coherent waves which had been previously split from one single wave initially associated to the

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particle. In this second case, the superposition is between the wave still bound to a photon, and part of the wave formerly associated to another photon. Mutual coherence of such waves, from two separate sources, at least for short observation times, was considered in an explanation [10] of the Pfleegor-Mandel experiment [11].

The elements of the system are therefore set so that the opticat path from BS2 to D2 is very slightly shorter than the path from BS2 to BS3, with the distance from BS3 to Da as short as possible. The outputs from detectors D2 and D3 go to a coincidence counter C (Fig. 2). According to the corpuscular theory, and the explanations of Experiment 1, coincidences between detections in 1:)2 and D3 should be observable in these conditions.

On the other hand, if the hypothesis of real electromagnetic wave reduction is followed, 02 and e2 are only branches of a whole set of electromagnetic waves filling up the entire system between BS2 and D2, BS2 and D3, and eventually reducible, instantly and at a distance, by detection of a photon in either D2 or D3. If this to- tal electromagnetic wave statistically only propagates one hu unit of energy, the predicted results will be quite different.

In this case, the el ra), considered as one component of the total wave coming out of BS3, can absolutely not go through P3- Inversely, the 02 component of the global wave from BS3 will ger through P3, exclusively and unaltered.

In other words, quantum states ]¢1 > and 1¢2 > of the two beams reaching BS3, pertaining to orthogonally polarized fields, can have no mutual influence whatsoever.

Following the results of Clanser's experiment [9], and in ac- cordance with quantum theory, no coincidences whatsoever can be expected between photon detections by D2 and D3.

5. C O N C L U S I O N

Interpretation of Experiment 1 left open the c~loice between two different, albeit equally plausible, explanations. The second experi- ment does resolve this dilemma.

If coincidences were observed between photon detections by D2 and D3 in Experiment 2, this would seem to confirm the corpuscular theory. Photons, as distinct corpuscles of permanent existence, could be seen as propagating along with anenergetic waves, with these latter separable, and eventually empty, but even so, active on other photons, following de Broglie's theory.

Conversely, if no such coincidences were observable between sig- nals from detectors D2 and D3, the mode occupation by extended energetic waves, instantly reducible as pointlike quanta, would be favoured at the expense of the corpuscular theory of radiation.

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Whatever the result of these tests, the above analysis is in the spirit of a definition given by Louis de Broglie [11]: "Physics, after all, is an experimental science."

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