Physics Letters A 306 (2003) 265–270

www.elsevier.com/locate/pla

A unified view to Cologne and Florence experimentson superluminal photon propagation

Fabio Cardonea,b, Roberto Mignanic,∗

a Dipartimento di Fisica, Università degli Studi dell’Aquila, Via Vetoio, 67010 Coppito, L’Aquila, Italyb Istituto di Igiene, Facoltà di Medicina, Università di Roma “La Sapienza”, and INDAM-GNFM, Roma, Italy

c Dipartimento di Fisica “E. Amaldi”, Università degli Studi di Roma “Roma Tre”, and INFN, Sezione di Roma III,Via della Vasca Navale 84, I-00146 Roma, Italy

Received 8 September 2002; received in revised form 5 October 2002; accepted 10 October 2002

Communicated by V.M. Agranovich

Abstract

We show that two of the first performed experiments on superluminal photon propagation, namely, the 1992 Cologneexperiment on the tunneling of evanescent waves in an undersized waveguide, and the 1993 Florence experiment on themicrowave propagation in vacuum between two horn antennas, do admit a common interpretation. Precisely, both experimentaldevices behave as a high-pass filter. We get this result by two different methods, one based on the Friis law (which yieldsthe efficiency of a transmitting device), and the other on the deformation of the Minkowski space–time. This allows us to setintriguing connections between these two (a priori different) classes of experiments. In particular, in either case the superluminalpropagation can be described as a tunneling and is related to evanescent waves. 2002 Elsevier Science B.V. All rights reserved.

1. Introduction

The subject of faster-than-light propagation of elec-tromagnetic waves received in the last years a greatdeal of attention, both from the experimental [1–10]and the theoretical side (see Refs. [11–13] for re-views). Propagation at a group velocity greater thanthe light velocity has been experimentally demon-strated not only for evanescent (tunneling) waves[1–8], but also for non-evanescent ones (like X-shapedwaves) [9,10].

* Corresponding author.E-mail address:mignani@fis.uniroma3.it (R. Mignani).

One of the main problems for a theoretical treat-ment of the superluminal photon propagation is due tothe fact that it was observed indifferentkinds of exper-iments [11–13], which are not easily comparable. It isso quite impossible to state if the results of differentexperiments are compatible with each other.

In this Letter, we want to show that two of the firstperformed experiments, namely, the 1992 Cologneexperiment [1,2] on the tunneling of evanescent wavesin an undersized waveguide, and the 1993 Florenceexperiment [3] on the microwave propagation in airbetween two horn antennas, do admit a commoninterpretation. This will allow us to set intriguingconnections between these two (a priori different)classes of experiments.

0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0375-9601(02)01472-X

266 F. Cardone, R. Mignani / Physics Letters A 306 (2003) 265–270

2. Superluminal propagation and the Friis law

Schematic views of the Cologne and Florenceexperimental devices are given in Figs. 1 and 2,respectively. We denote by the same symbolL thelength of the reduced-section waveguide in the formercase and the distance between the two horn antennasin the latter.

The main tool we shall exploit is the formula whichgives the efficiencyη of a transmitting device, i.e., theratio between the received and the emitted power:

(1)η = PR

PE

= ARAE

λ2L2

(also known asFriis law [14]). Here,PR(E) is thereceived (emitted) power,AR(E) the (effective) area

Fig. 1. Rectangular waveguide with variable section used in theCologne experiment.L = length of the narrow part of the waveguide(“barrier”); h = height of the guide;a = tickness of the guide;AE = area of the large section (“emitter”);AR = area of the smallsection (“receiver”).

Fig. 2. Schematic view of the two horn antennas used in theFlorence experiment.L = distance between antennas; = distancebetween the upper border of the emitter and the lower border of thereceiver;S = distance between the centers of the antenna surfaces;δ = normal distance between the receiver lower border and theemitter upper border;d = distance between the axes of antennas;AE = area of the emitting antenna;AR = area of the receivingantenna.

of the surfaceΣR(E) of the receiver (emitter),λ thewavelength of the transmitted electromagnetic signal,andL the distance between the surfacesΣR andΣE .

Some remarks are needed concerning the validitylimits of the Friis law. It strictly holds true for largedistance between emitter and receiver. However, itcan be shown that it is validapproximatelyeven forsmall distances (it can be derived indeed from thebehaviour of the near field of the emitter1). Moreover,Eq. (1) is usually interpreted as strictly holding fortwo faced antennas. But by its very derivation [14]it holds too if the antennas are shifted (as in theFlorence experiment [3]), provided the areasAR , AE

are regarded aseffective. In this case, their values areobtained by suitably projecting the real areas alongthe direction of propagation of the wave (i.e., of thePoynting vector) (see below).

As is well known, in the Cologne experiment therectangular waveguide with reduced section behaveslike a high-pass filterwith a cutoff frequencyνc =9.49 GHz. The experiment was carried out in un-dersized conditions, i.e., at an under-cutoff frequencyν = 8.70 GHz. In the narrow part of the waveguide(namely, “under barrier” in the particle tunneling anal-ogy), of lengthL = 4 cm, the evanescent waves exhibita decaying behaviour of the type

(2)f ∼ e−z/Lc,

where thez-axis is along the axis of the guide, and thecutoff lengthLc (the penetration distanceof the wave)is given by

(3)Lc = c

2π

(ν2c − ν2)−1/2 = 1.259 cm.

The propagation of such evanescent waves was foundto occur at superluminal group velocity (u = 1.03c).The energy of the wave under the barrier decaysaccording to [15–17]

(4)E = Eine−z/L0,

whereEin is the input energy, i.e., the energy enteringthe barrier, andL0 is the penetration lengthof theenergy. It follows obviously from Eq. (2)

(5)L0 = Lc

2.

1 We are grateful to T.F. Arecchi for his kind advice on this point.

F. Cardone, R. Mignani / Physics Letters A 306 (2003) 265–270 267

We want to show that the application of the Friislaw (1) to the Florence experiment allows us tostate that the two-horn antennas device, too, behaveslike a high-pass filter. The frequency used in suchan experiment wasν = 9.50 GHz. The area of the(rectangular) surface of either the emitting and thereceiving antenna wasA = a × h = 9 cm× 8 cm=72 cm2 (with a andh being the width and the heightof antenna surface, respectively). If the antennas faceeach other, it is obviouslyA = AR = AE . Supposenow to shift the antennas orthogonally so that thedistance between their axes isd . Let be the distancebetween theupperborder of the emitter and thelowerborder of the receiver andS the distance betweenthe centers of the antenna surfacesΣ (see Fig. 2)(S = √

L2 + d2 ). In the following, we shall also needthe normal distance between the receiver lower borderand the emitter upper borderδ = d − h (whence =√

L2 + δ2 ).The measurement runs were carried out for three

different values of the distanceL between the anten-nas and for different (integer) values ofd . Superlumi-nal propagation of the signal was observed only forthe lowest value ofL, L = 21 cm. In this case, thetime taken by a light signal to travel such a distanceis t0 = L/c = 0.7 ns. For four values ofd (namely,d = 12,13,14,15 cm) the measured time wast =0.6 ns< t0. Notice that such values ofd correspondto a lack of space coherence between the two anten-nas[15] (i.e., the projection of the emitter surface onthe position of the receiver does not intersect the ac-tive surface of the receiver2). We shall consider thesevalues as corresponding to aninescapable superlumi-nality, namely, as a superluminal condition wholly in-dependent of the distance between antennas.

Let us estimate, by means of the Friis law, thecutoff frequency of the Florence experiment in sucha condition. One gets from (1)

(6)ν = η1/2 L

(ARAE)1/2c.

In order to apply Eq. (6) to the two-antenna system,we have to introduce two corrective factors due to thevalidity limits of the Friis law. First, since the antennas

2 Such a definition of space coherence is therefore a measure ofthe facing of the antenna surfaces, and is correlated to the geometricefficiency of an emitter–receiver system.

are shifted, the effective areaAR of the receiveris obtained by projecting the surfaceΣR along thedirection ofS, i.e.,AR = Acosα = A(L/S).

Furthermore, the near-field effect must be explicitlytaken into account. It results in aneffective reductionof the emitter surfaceAE , due to the reduced fieldamplitude at small distances. Such an effect wasexperimentally observed and measured for a hornantenna operating atν � 10 GHz [16]. The near-fieldamplitude as a function of the displacement along theheight h of the antenna surface exhibits an almostGaussian behaviour3 (see Fig.1 of Ref. [16]). The fieldis significantly different from zero over a distanceheffthat is roughly halfh: heff � h/2. It follows that theeffect of the near field is to reduce the areaA of theemitter by the same factor, i.e., in the Friis law we haveto putAE � A/2.

Eq. (6) becomes therefore

(7)ν = η1/2 (2SL)1/2

Ac.

The cutoff frequencyνc is obtained forη = 1:

(8)νc = (2SL)1/2

Ac.

The values ofνc are given in Table 1.Since the operating frequency of the Florence ex-

perimentν = 9.50 GHz isunder the cutoff frequencyfor every configuration of the apparatus,we can con-clude that the whole system is working as a high-passfilter in the evanescent mode,like the Cologne device.

Let us attempt to treat the Cologne experiment interms of law (1), too. In this case, we can consider thewaveguide with reduced section as a system of twoantennas with emitter surfaceΣE given by the largersection of the waveguide, and the reduced section asreceiver surfaceΣR (large-narrow antenna system).Then, we have [1,2]AE = hE × aE = 2.296 cm×1.016 cm= 2.33 cm2, AR = hR × aR = 1.58 cm×0.79 cm= 1.25 cm2 (wherehi , ai , i = E,R, denotethe height and the width of the waveguide). FromEq. (1), with η = 1, we can evaluate the effectivelength L0 of the two-antenna system correspondingto the undersized waveguide at the cutoff frequency

3 Actually, the field amplitude displays a structure (characteristicof the interference processes), but this is ininfluent to our presentaims. See Ref. [16].

268 F. Cardone, R. Mignani / Physics Letters A 306 (2003) 265–270

Table 1Data and parameters of the Florence experiment (withνc from the Friis law)

d (cm) S (cm) δ (cm) (cm) τ (ns) t (ns) νc (GHz)

12 24.2 4 21.4 0.71 0.55 13.1013 24.7 5 21.6 0.72 0.6 13.2414 25.2 6 21.8 0.73 0.6 13.3715 25.8 7 22.1 0.74 0.6 13.54

νc = 9.49 GHz. One has

(9)L0 = νc

c

√ARAE = 0.542 cm.

Such a value is about half the cutoff lengthLc:

(10)L0 � 1

2Lc

in agreement with Eq. (5).On the other hand, let us evaluate by the same

approach the efficiencyη of the undersized waveguideat the cutoff lengthLc and at the operating frequencyν = 8.70 GHz. We get from Eq. (1)

(11)ηc = ARAE

(ν

cLc

)2

= 0.156,

in good agreement with the value derived from theenergy decay law (4) (ηc = exp(−Lc/L0) = e−2 =0.135).

3. Two-antenna system as a barrier

On the basis of the analogy between the twoexperiments, we can therefore assume that (in theoperating conditions of the Florence experiment)thetwo-antenna system behaves as a barrier. The roleof L, namely, the length of the reduced portionof the waveguide, is played in this case by theminimal distance = √

L2 + δ2 between the antennas.The lengthδ represents the space extension of thebarrier. The minimal value of for which superluminalpropagation is observed is = 21.4 cm (see Table 1),with δ = 4 cm.

On the basis of the previous arguments, such abarrier behaves as a high-pass filter at an operatingunder-cutoff frequency. Therefore, we can assume forthe energy a decaying behaviour similar to Eq. (4) forthe undersized waveguide, namely,

(12)E = Ein exp(− / 0).

Here,Ein is the initial energy of the beam (namely, theemitted energy), which for a monochromatic beam ofN photons with frequencyν reads

(13)Ein =Nhν,

and 0 is the critical length.In order to evaluate 0 for the Florence experiment,

we shall exploit the formalism of deformation of theMinkowski space we developed in the last years [18].We recently discussed in such a framework [17,19,20] the Cologne experiment, by assuming that, insidethe barrier, the space–time is no longer Minkowskianbut is endowed with an energy-dependent (spatiallyisotropic), “deformed” metric of the type

ds2 = c2dt2 − b2(E) dx2 − b2(E) dy2 − b2(E) dz2

= ηµν(E)dxµ dxν,

(14)ηµν = diag(1,−b2(E),−b2(E),−b2(E)

)that is to be regarded as a phenomenological descrip-tion of the e.m. interaction under nonlocal conditions[19]. The fit to the experimental data yields [17,18]

(15)

b2(E) =

(E/E0,e.m.)1/3,

0 � E < E0,e.m. = 4.5± 0.2 µeV,1, E0,e.m. � E.

The threshold energyE0,e.m. is the energy value atwhich the metric parameters become constant, i.e.,the electromagnetic metric attains the Minkowskianlimit.4

4 The use of such a formalism allowed us, among the others, todescribe the nonlocal behaviour of the barrier in terms of a “space–time deformation” tensor (analogous to the Cauchy stress tensorin a continuous medium) [19]. Moreover, a similar description ispossible in the case of total internal reflection of light, too [20].

Such an approach to faster-than-light propagation is similar, insome respects, to that where superluminal propagation is connectedto vacuum effects [22]. In this case, the influence of the structured

F. Cardone, R. Mignani / Physics Letters A 306 (2003) 265–270 269

According to the formalism of the deformed Min-kowski space (see [18] and the references thereinquoted), the maximal causal speed of lightin anydirection for metric (14) is expressed in terms of thedeformation coefficient as

(16)u(E) = c

b(E).

Then, by Eq. (15)

(17)u(E) = c

(E0,e.m.

E

)1/6

.

On the other hand, the (superluminal) speedu inour case is given by

(18)u = /t,

where t is the measured time (see Table 1). ByassumingN = 1 in Eq. (13),5 we get therefore fromEqs. (12), (13), (17), (18) the following expression for 0:

(19) 0 =

ln[(hν/E0,e.m.)(ct/ )6

] .This allows us to evaluate in an independent way,

by a completely different approach, the cutoff fre-quency νc, which in this framework is given by

vacuum is described in an effective way in terms of a refractiveindex (as pioneered by Sommerfeld). In General Relativity, too, thedeflection of light in a gravitational field can be considered as apropagation in an Euclidean space, filled with a refractive medium[23]. In some cases, such a propagation—due to the influence ofthe gravitational vacuum—turns out to be superluminal [24]. Thedeformed metric approach can be therefore regarded asdual to thegeneral relativistic one, in the sense that the vacuum or nonlocaleffects which affect propagation in the waveguide are described interms of a space–time deformation (and the role of the refractiveindex is played by a deformation tensor: see Ref. [19]).

5 Notice that the corresponding expression for the energy

E = hν exp(− / 0)

does not mean at all that we are considering fractions of the photonenergy. The meaning of the above expression is simply that theenergy lost by the signal after travelling a distance from theemitting antenna

%E = Ein − E = hν(1− e− / 0

)can be considered as the energy needed to deforming the space–timebetween the antennas.

Table 2Data and parameters of the Florence experiment (withνc from thebarrier law)

(cm) 0 (cm) t (ns) νc (GHz)

21.4 8.80 0.55 9.51521.6 9.18 0.6 9.51421.8 9.23 0.6 9.51422.1 9.30 0.6 9.514

(cf. Eq. (3))

(20)νc =√(

c

2π c

)2

+ ν2,

where the cutoff length c (related to the decay of thewave) is related to 0 by

(21) c = 2 0.

The values ofνc are reported in Table 2.We recover then, by a completely different ap-

proach, the result that the two-horn antennas devicebehaves as a high-pass filter. The different value found,in this framework, for the cutoff frequency, is dueto the fact that our treatment is based on the spa-tially isotropic metric (14). The hypothesis of spaceisotropy, valid in the Cologne case, does no longerhold for the Florence experiment, due to the edge ef-fects of the emitting antenna and the near-field behav-iour. A more sound treatment requires to take into ac-count both effects in the functional form of the spacemetric coefficientsb2

i (E) (i = 1,2,3) (now to be as-sumed different). This topics will be discussed else-where.

4. Conclusions

We have therefore shown, by two different andindependent approaches—one based on the Friis lawand the other on the deformation of the space–time—that the two-antenna system of the Florenceexperiment can be considered as a high-pass filter. Theformalism of the deformed Minkowski space permitsalso to describe the behaviour of the Florence deviceas a barrier, with a decaying law for the energy ofthe evanescent-wave type, and therefore to interpret

270 F. Cardone, R. Mignani / Physics Letters A 306 (2003) 265–270

the experiment as a genuine tunneling one,6 in fullanalogy with the Cologne case. By borrowing theresults of Refs. [17–21], we can state that (as inthe Cologne experiment) such a wave (evanescent inthe ordinary space–time) propagates at superluminalspeed in the deformed Minkowski space with arealwavevector. We shall refer in our future work to a wavepropagating in a deformed space–time (whatever theinteraction responsible of the deformation) as anon-Lorentzian wave.

Acknowledgements

We are greatly indebted to D. Mugnai and A. Ran-fagni for communicating to us the unpublished data oftheir 1993 experiment, and for very useful discussions.Stimulating discussions with T.F. Arecchi are alsogratefully acknowledged.

References

[1] A. Enders, G. Nimtz, J. Phys. I (Paris) 2 (1992) 1693;A. Enders, G. Nimtz, J. Phys. I (Paris) 3 (1993) 1089;A. Enders, G. Nimtz, Phys. Rev. E 48 (1993) 632;G. Nimtz, A. Enders, H. Spieker, J. Phys. I (Paris) 4 (1994) 1;W. Heitmann, G. Nimtz, Phys. Lett. A 196 (1994) 154.

[2] A. Enders, G. Nimtz, Phys. Rev. B 47 (1993) 9605;G. Nimtz, A. Enders, H. Spieker, J. Phys. I (Paris) 4 (1994)565.

[3] A. Ranfagni, P. Fabeni, G.P. Pazzi, D. Mugnai, Phys. Rev. E 48(1993) 1453.

6 Let us recall that actually no definite explanation of the resultsof the Florence experiment exists. The most accredited one is basedon the hypothesis that, due to the microwave diffraction out ofthe square aperture of the emitter, “leaky” or “complex” wavespropagate in the shadow region [3,11–13].

[4] D. Mugnai, A. Ranfagni, L.S. Schulman, Phys. Rev. E 55(1997) 3593;D. Mugnai, A. Ranfagni, L. Ronchi, Phys. Lett. A 247 (1998)281.

[5] A.M. Steinberg, P.G. Kwait, R.Y. Chiao, Phys. Rev. Lett. 71(1993) 708.

[6] Ch. Spielmann, R. Szipocs, A. Singl, F. Krausz, Phys. Rev.Lett. 73 (1994) 2308.

[7] Ph. Balcou, L. Dutriaux, Phys. Rev. Lett. 78 (1997) 851.[8] V. Laude, P. Tournois, J. Opt. Soc. Am. B 16 (1999) 194.[9] P. Saari, K. Reivelt, Phys. Rev. Lett. 79 (1997) 4135.

[10] D. Mugnai, A. Ranfagni, R. Ruggeri, Phys. Rev. Lett. 84(2000) 4830.

[11] G. Nimtz, W. Heimann, Prog. Quantum Electr. 21 (1997) 81.[12] R.Y. Chiao, A.M. Steinberg, Tunneling Times and Superlumi-

nality, in: E. Wolf (Ed.), Progress in Optics, Vol. 37, Elsevier,1997, p. 346.

[13] F. Cardone, R. Mignani, Superluminal effects and tachyontheory, in: Modern Nonlinear Optics, Vol. 3, in: M.W. Evans(Ed.), Adv. Chem. Phys., Vol. 119, Wiley, New York, 2002,p. 683.

[14] See H.P. Friis, IEEE Spectrum 8 (1971) 55;J. Pierce, Almost All About Waves, MIT Press, Boston, 1974,Ch. 16.

[15] See G. Bekefi, A.H. Barrett, Electromagnetic Vibrations,Waves, and Radiations, MIT Press, Boston, 1977, p. 481.

[16] A. Ranfagni, D. Mugnai, Phys. Rev. E 58 (1998) 6742.[17] F. Cardone, R. Mignani, Ann. Fond. L. de Broglie 23 (1998)

173.[18] F. Cardone, R. Mignani, Found. Phys. 29 (1999) 1735, and

references therein.[19] F. Cardone, R. Mignani, V.S. Olkhovsky, J. Phys. I (Paris) 7

(1997) 1211.[20] F. Cardone, R. Mignani, V.S. Olkhovsky, Mod. Phys. Lett. B 14

(2000) 109.[21] F. Cardone, R. Mignani, V.S. Olkhovsky, Phys. Lett. A 289

(2001) 279.[22] G. Barton, K. Scharnhorst, J. Phys. A 26 (1993) 2037, and

references therein.[23] Such a result can be traced back to T. Levi-Civita, The

Absolute Differential Calculus, Blackie & Son, 1954, p. 403.[24] I.T. Drummond, S.J. Hathrell, Phys. Rev. D 2 (1980) 345.