Amplification of light from an undulatory point of view

Volume 120, number 8

PHYSICS LETTERS A

16 March 1987

AMPLIFICATION OF LIGHT FROM AN UNDULATORY POINT OF VIEW F. SELLERI Dipartimento di Fisica, Università di Ban, J.N.F.N., Sezione dl Ban, Via Amendola 173, 1-70126 Ban, Italy Received 13 October 1986; revised manuscript received 21 January 1987; accepted for publication 26 January 1987

The traditional approach to light amplification is shown to be based on the idea that photons of the incoming beam act as independent particles. If atomic stimulation is instead attributed to the action of the wave, different photon statistics (but with the same average) is obtained in the final state.

The basic idea of this letter is a dualistic model of light in which energetic particles and empty waves coexist objectively. With empty wave we denote a wave devoid of energy and momentum, but nevertheless real, propagating in space and time. It is the old idea of Einstein and, partly, of de Broglie, perhaps not very popular today, but nevertheless providing a perfectly possible alternative to the usual “picture” ofthe e.m. field. Here picture only means intuitive representation. In fact, the dualistic model discussed in this letter is probably not contradictory to QED, although it must be admitted that its precise relationship to QED should be investigated much betterthan has been done up to now. It will be shown that the presently investigated model leads to empirical predictions in contradiction with those of the usual approach for the photon distribution (not for the average photon number), and that the usual approach is not QED itself but a phenomenological model “deduced” from QED with the help of mutilating additional assumptions. The basic property of empty waves is assumed to be their ability to stimulate emissions when interacting with excited atoms. The waves are empty and thus do not contribute to the energy—momentum balance which can of course be saved bykeeping into account only the atomic variables. When the wave generates instead atomic excitation, the absorbed energy is always that of an energetic quantum present in the wave. In the stimulated emission the quantum can be totally absent from the wave, or perhaps present 0375-9601/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

but not near the stimulated atom: It is in all casese assumed to be irrelevant to the stimulation process. In published papers [11 it has been pointed out that such a model with empty waves is perfectly “physical” since it is in principle capable of falsifiable predictions. This conclusion is demonstrated to be correct in the present letter, since it is shown that physical phenomena exist in which the predicted photon distribution is different from the one deduced from the usual approach to stimulated emission. Moreover the dualistic model will be shown to be superior to the usual approach by explaining the results ofthe Blake—Scarl experiment. It was shown in ref. [11 that an experiment is possible in which an empty wave (deprived of its energetic quantum by the crossing of a beamsplitter) is revealed by detecting atomic emissions stimulated by its action. Other proposals for the detection of the empty waves have been presented [2] and discussed [3] by several authors. These proposals were centered on the idea that coincidence detections should be made of primary photons which were born together with the empty wave, and of photons whose emission was stimulated by the empty wave itself. In the present paper it will be pointed out that even simpler experiments are possible since the photon statistics of light which has crossed an amplifier is different if amplification is generated by the wave, rather than by the energy quanta. It will also be shown that the attribution of amplifications to the energy content ofthe e.m. field is implicit in the usual treat371

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differential equations of the first order. It can be

~+r -.--,.-,,.

~

______________________________ x x+d~

Fig. 1. Photon amplifier of length L leading from n incoming to n+routgoing photons.

ment of these phenomena. Consider the amplifier of fig. 1 and suppose that a pulse containing n photons enters from the left. What istheprobabilityp~~,(x+dx)offinding n+rphotons at depth x+dx inside the amplifier? One can obviously write

solved with depends standard thewith initial methods condition and that solution, which should beon found certainty forthe x=0, isn photons p,~(x)= (n+y)(n+ 1 +y)...(n+r— 1 +y) (“

—

x

—

r!

(3)

n”,

(1 —e

where y P/S. From the previous result it is not difficult to obtain: ~ p,~(x)=l,

(4)

showing that probabilities are correctly normalized, and

,( x + dx)

±

=

(n+r)p,~(x)

~

=pn±r(x){l [(n+r)P+S]dx} —

+p,,.~_

(x)[(n+r— 1 )P+SJdx,

(I )

where P dx is the probability that a photon stimulates the emission of another photon, in the interval [x,x+ dx]; S dx is the probability that a photon, traveling in the same direction as the incoming photons, is spontaneously emitted in the same interval (naturally,P~0andS~0). In eq. (1) it has been assumed that if an incoming photon has a probability P dx to stimulate the emiSsion of a photon, then n + r incoming photons have a probability (n + r)P dx to stimulate the same emission. In this way photons give rise to additive effects, that is they are treated as independent systems which add incoherently their actions of stimulating emissions. This is obviously consistent with a purely corpuscular description of photons, treated as independent particles interacting with the lasing medium, but is very difficult to reconcile with an undulatory description, since in the latter case one could expect some coherence of the generated effect. This observation will be useful later on. From eq. (1) one gets P~±r(X)=

+

—

[(n+r)P+SJp~±~(x)

[(n+r—l )P+S]pn±r.i(X),

The previous calculation refers to the case of an exact number (n) of incoming photons. It is the basis for the study of the amplification of an incoming “coherent state”, characterized by the frequency fl

f(n)= —e<’>. n!

(6)

with which n photons are present in the incoming pulse with average photon number . One can thus say that in the present approach the amplification of a coherent state leads from an average initial photon number to an average photon number at depth x (n(x)> = +( (n> +y)(e” —1), (7) obtained by averaging (5) with the weights (6). The previous considerations can be carried out by using time 1, instead of the space variable x: it is enough to substitute c’ I for x, where c’ is the velocity of light in the lasing medium. Thus eq. (2) can be written —

(2)

(5)

—

dp,,÷~(t)/dt=(n+r+

which, with fixed n and varying r, is a set of coupled 372

_—n+ (n+~)(e” 1).

y)c’Pp~.~ ~(t)

+ (n+r— 1 +7)c’Pp~~~

1(t).

(8)

In the previous considerations the problem of pho-

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PHYSICS LETTERSA ri~r,l

I

GN (n.r~1)p 1 r,+r+1

‘1’—GN

2

(n.r~.Jjp

12~~n*r_1

n.r n+r-1

16 March 1987

absorption terms. It can be compared with eq. (8) only by setting such terms equal to zero: This can be done by taking N1 = 0, so that dPn+r(t)ldt (n+r+ 1 )GN2Pn+r(t) —

GN(n+r)P

Fig. 2. Energy-level diagram for the e.m. field with given fre-

+ (n

+ r) GN 2p~+

quency. Transition rates shown are contributions to dp~±,(t)/dt. On the right, photon numbers in corresponding levels are indicated.

ton amplification was studied by assuming that photons can be emitted by the lasing medium, but that they cannot be absorbed (total population inversion). It will next be shown that the usual treatment of photon amplification gives results identical to those obtained above, if the same physical conditions are considered. The time-dependent changes of the probabilities Pn+r( t) are usually obtained from “rate equations”, which can be deduced in the following way. Consider a cavity containing two-level atoms and let N1 and N2 be the number ofatoms in the ground and excited state, respectively. The energy-level diagram for the e.m. field consists of equidistant lines, as in fig. 2: the level spacing is hv if xi is the transition frequency. Photon emission from an atom corresponds to an increase of the field photon number, therefore to an arrow pointing upwards in fig. 2. Similarly, atomic absorption of photons corresponds to downward pointing arrows. The absorption rateis assumed proportional both to the number of atoms in the lower level (N1) and to the number of photons in the e.m. field. The emission rate is proportional to the number ofatoms in the upper level (N2), and to the number of photons in the field plus one, the “plus one” term corresponding to spontaneous emission. In conclusion one can write: dpn+r(t)/dt

.

.

This equation is identical to eq. (8) y = 1 and c’ P= GN2.

(11)

~

2

Pr+i(t)

—GN 2(n+r+l)pn+r+GN2(n+r)pn+r_i, (9) where the minus sign can be justified by noting that arrows pointing away from the (n + r)th level of fig. 2 correspond to a decrease of the probability Pn-I-r. As one can see eq. (9) contains also photon-

if

The first condition corresponds to S=P. This means that spontaneous emission and stimulated emission from a single photon contribute equally to the transition rate, a well known result of atomic The second condition (11) allows one to obtain Pin terms of the constant G, which has a known value (see eq. (10.6) at p. 233 of ref. [5]). It must be stressed that eqs. (9) and (10) are considered the standard “quantum mechanical” formulae for photon distribution, in spite of the simple phenomenological argument which can be used in order to deduce them. This was stressed for instance by Loudon (see p. 245 of ref. [5]), who wrote: “The rate-equation method used here is a simple technique for determining the laser photon distribution P,. It is not so general or so rigorous as other techniques which have been used to find P~.It is, however, equivalent to some of the more sophisticated calculations for the particular result which it yields, including the detailed form of the photon distribution”. This practical equivalence exists, for instance, in the cases of the results obtained by Scully [6], and by Haken and Weidlich [7]. It exists also for the more recent results of Friberg and Mandel [8] who found that the probability that one incoming photon generates r outgoing photons by crossing a linear amplifier is

=GN 1(n+r+l)pn+r+i —GN1(n+r)pn+.

(10)

1(t).

.

+

m(t)’~’ 2 ~l ~ [m(t)+1]T+

2’\ (r+2)IG(t)1 m( t) F m( t) + 1])’

—

IG(t)1 m(t)

(12)

lowe the understanding ofthis fact to Professor V. Degiorgio who is preparing a paper on photon amplification [4].

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where G(t) and m(t) are respectively given by eqs. (4) and (5) of ref. [7]. In the case N1 = 0 considered here those equations reduce to (13)

2~2fh,

m(t) = e so that+ itI =is G(t) now a 2simple task to show that eq. (12)

becomes

Pr± 1(t) =

(~

)

.

(14)

(r+ 1) G(t) 4 G(t) 2 This result coincides with our eq. (3) if n =coincides I and if 2=exp(Px). The latter condition with(13). IG(t) I —

Our conclusion is thus that the photon amplification treatments found in textbooks and in scientific papers deal essentially with corpuscular amplification processes and neglect the possibility that photonic waves be instead responsible for the stimulated emissions. The opposite point ofview will be investigated in the following by assuming that all stimulated emissions are due to the action of the wave, independently ofthe presence ofenergetic quanta. It will furthermore be assumed that there is an intensity of the wave which can be written 1(x) at depth ~ inside the amplifier (see fig. 1). The precise definition ofl(x) is not of interest here, although it can be expected to be given by the squared amplitude of the undulatory phenomenon. A further feature of 1(x) must be some connection with the energy content of the e.m. field, that is with the average number of quanta present inside the incoming wave packets. That the average photon number is the relevant parameter can be seen as follows. In the present (Einstein—de Broglie) approach ~2 excited atoms emit continuously and deterministically a wave in which they free an energetic quantum hi’ at random times. Also the action of optical devices for the attenuation of an incoming e.m. wave has a deterministic effect on the wave, but a random effect on quanta: For instance, a semitransparent mirror reduces the intensity of the wave of an incoming single-atom emission exactly by a factor 1/2, but transmits or reflects the quantum with a 50—50 probability in a random way. Beyond the semitransparent mirror one will then find in some cases an empty wave and, in U

This approach has been called “the only non-contradictory explanation of quantum phenomena” [9].

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other cases, a wave reduced in intensity but associated with a full quantum. In such a picture of the e.m. field it is therefore not possible to keep a strict proportionality wave later intensity content. It willbetween be shown that and ourenergy approach guarantees instead a proportionality between wave intensity 1(x) and average number ofquanta at depth x inside the amplifier. This is enough for all practical purposes. If the wave propagating in the amplifier can stimulate its then own the emission with a strength to 1(x), equation must hold proportional 1(x)=P 01(x)+S0, (15) where P0 and S0 are positive parameters describing the stimulated and spontaneous emission processes of the wave. The solution of (1 5) satisfying the mitial condition 1(0) =I~is 1(x) = (10+y0)(e”°’— 1) +1~,

(16)

where y~=S0/P1.

(17)

Attributing the stimulated emission of energj~’ quanta to 1(x), the equation replacing (1) is p,,+~(x+dx)=p,~±~(x){l[cI(x)+S]dx} —

+p,,~,~1(x)[cI(x)+S]dx.

(18)

where c is a positive constant which will be determined later. Keeping into account (16), eq. (18) becomes: p,±~(x)=_p~±~(x)(aePo>k+b) ~ where

(x)(a e”°’+b),

(19)

b _c~’0+S. (20) If one keeps n fixed and r varying, (19) is a set of coupled differential equation of the first order. The solution can be found with standard methods and turns out to be a=c(10 +yo).

(-s-

(e”°” 1) + bx) r! \P0 x exp (7a (1 e”°~) bx) \P 0

pn±r(x) = ~

—

—

—

—

.

(21)

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16 March 1987

From this result it is not difficult to show that eq. (4) is still valid and that

ing only the (average) energy of the outgoing e.m. field. This is the same as saying that present experimental evidence cannot rule out our proposal. (This

= ~ (n + r)p~± r(X)

conclusion agrees with the results obtained in ref. [10].) But one can say more: the experiment carried out by Blake and Scarl [11] on two-photon correla-

r=O

=

n + (a/P0 ) ( ~

—

1) + bx.

(22)

The previous calculation refers to an exact number (n) of incoming photons, as it is clear from the fact that the results (21) satisfy the initial conditions p,,( 0) = 1, and p~,+ ~(0) = 0 if ritr 0. It can be used for studying the amplification ofan incoming “coherent state”, characterized by the frequencyf(n) ofeq. (6). One can easily show that in the present approach the amplification ofa coherent state leads from an average initial photon number to an average photon number at depth x =

+ (a/Po)(e~’0x 1) + bx. —

(23)

It is then clear that (21) is a Poisson distribution for the r additional photons. By comparing (23) with (16) one can see that the possibility exists of keeping the wave intensity and the average photon number constantly proportional to one another, as they must be if the present (quantum) theory has to reproduce, at least on the average, the predictions of classical electrodynamics. In fact, the condition I(x)=

10+yo=ka/P0.

(25)

(26)

In conclusion eq. (23) can be written
>

=

+ ( +S/Po)(e~°0x —1).

(27)

Comparison with eq. (7) (case of corpuscular stimulation) shows that the two results are identical, provided that P0=P.

U

Detailed calculations, omitted here for reasons of space, are contained in ref. [12].

References

These two results lead immediately to c_—P0/k, S0 =kS.

The author is grateful to Professor V. Degiorgio, to the staff of CISE (Milan), to Professor R. Giovanelli and to Dr. F. Ponti for several helpful and stimulating discussions. Warm thanks are also due to Dr. A. Garuccio for providing the author with helpful criticisms and suggestions. The thesis of Dr. N. Conenna has been a source of help and stimulation.

(24).

(k> 0) can easily be shown to imply b=0,

tions has shown that no time correlations exist in a beam of amplified laser light, a result strongly indicative of the validity of the poissonian distribution which was deduced here from the idea of wavestimulation. Anyway, the present proposal can easily be checked by measuring the photon statistics in pulses ofamplifled laser light, given the average number ofphotons per pulse in the incoming beam. Eqs. (3) and (21) predict in fact considerably different results for such an experiment 13

(28)

This is an extremely important result: particle-generated stimulation and wave-generated stimulation give rise to the same average photon number and cannot be discriminated with experiments measur-

[I] F. Selleri, Lett. Nuovo Cimento 1 (1969) 908; Found. Phys. 12 (1982) 1087; Gespensterfelder, in: The wave—particle dualism, eds. S. Diner et al. (Reidel, Dordrecht, 1984). [2] J. Andrade de Silva and M. Andrade de Silva, C.R. Acad. Sci. 290 (1980) 501; A. Garuccio, V. Rapisarda and J.-P. Vigier, Phys. Lett. A 90 (1982) 17; G. Tarozzi, Lett. Nuovo Cimento 36 (1983) 503. [3] P.W. Milonni and M.L. Hardies, Phys. Lett. A 92 (1982)

321; R. Loudon, Opt. Commun. 45(1983) 361; L. Mandel, Phys. Lett. A 103 (1984) 416; A. Heidemannand S. Reynaud, I. Phys. (Paris) 45 (1984) 873; D.K. Umberger, P.W. Milonni and M. Lieber, Nuovo Cimento B 83 (1984) 23; A. Gozzini, in: The wave—particle dualism, eds. S. Diner et al. (Reidel, Dordrecht, 1984); CE. Engelke and C.W. Engelke, Found. Phys. 16 (1986) 905;

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C.E. Engelke, Found. Phys. 16 (1986) 917; H.P. Yuen, Phys. Lett. A 113 (1986) 401, 405; L.C.B. Ryff, Phys. Lett. A 119(1986)1: R. Giovanelli, in: Microphysical reality and quantum formalism, eds. G. Tarozzi and A. van den Merwe (Reidel, Dordrecht),to be published; C. Fenech, preprint. [4] V. Degiorgio, CISE Report. [5] R. Loudon, The quantum theory of light (Clarendon, Oxford, 1973). [6] M. Scully, in: Proc. Intern. School of Physics Enrico Fermi, Quantum optics (Academic Press, New York, 1967).

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17] H. Haken and W. Weidlich, in: Proc. Intern. School of Physics Enrico Fermi. Quantum optics (Academic Press, New York, 1967). [81 5. Friberg and L. Mandel, Opt. Commun. 46 (1983) 141. [9] A.A. Tyapkin, Filosofskie voprosi kvantovoi fisiki (Nauka. Moscow, 1970) p. 143. [10] M. Cray, M. Shih and P.W. Milonni, Am. J. Phys. 50 (1982) 1016. [Ill G.D. Blake and D. Scarl, Phys. Rev. A 19 (1979) 1948. [121 F. Selleri, Coherence properties ofphoton amplifiers, Found. Phys. (1987), to be published.