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Hypothetical dispersion quantum effects for coherent forward propagating radiation in transparent and semitransparent medium
Journal of Quantitative Spectroscopy & Radiative Transfer 65 (2000) 821}834
Hypothetical dispersion quantum e!ects for coherent forward propagating radiation in transparent and semitransparent medium V.F. Golovko* Laboratory of Theoretical Spectroscopy, Institute of Atmospheric Optics, Akademicheskii Av., 1, Tomsk, 634055, Russia
Abstract An attempt to establish relations between the continuum absorption problem and the dispersion theory is made with the help of elements of the quantum representation. A new technique for explanation of a new dispersion formula has been obtained as a consequence of the delay of the coherent radiation propagating in a random optical medium. A hypothesis of arising of pairs of coherent and anticoherent photons is assumed as some quantum e!ect on a molecule for the interaction of two photons from the forward propagating radiation and a paradox accompanying this consideration is discussed. Self-interacting e!ects of coherent wave trains are discussed and a quantization procedure has been suggested to account for transfer of the supplement e!ective photon number at the propagation of the coherent wave train within a randomly discrete medium. The ideas of this approach were recently used for modeling the continuum absorption of water vapor. ( 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction This work was stimulated by our recent studies [1] or the continuum absorption in the infra-red and microwave regions. The modeling of the spectra showed that continuum absorption of the water vapor is caused by contributions from far line wings rather than, e.g., by the dimer absorption. Nevertheless, the dimer hypothesis, in particular, and the #uctuation theory, in general, both could be important in understanding of mechanisms of the continuum absorption. Because of a great number of lines, the continuum may be referred to a problem of systems with in"nite
* Tel.: #7-3822-258794; fax: #7-3822-259086. E-mail address: [email protected] (V.F. Golovko) 0022-4073/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 4 0 7 3 ( 9 9 ) 0 0 1 4 2 - 9
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freedoms and there is some sense in describing them by an approach that is distinguished, e.g., from a method of detailed wave equations of the wave mechanics. In our studies on the continuum absorption, it has been found that the spectral density of its intensity I is proportional to a cubic power of the wave number u of the monochromatic radiation, i.e., I&u3 (u"l/c, where l is the frequency; c is the vacuum light velocity). Note that at conventional absorption of a photon, I&u, in accordance with the classical quantum nature of the radiation, and, at Rayleigh's single-scattering of light by independent sources, I&u5, it has followed a Herz emission law, which, in turn, is considered as a solution of the Maxwell equations in the so-called far wave zone and, in fact, it is a manifestation of the causality principle, due to a "nite velocity of the interaction propagation. However, Feynman [2] had demonstrated in detail a case, where the radiation, I&u3, would be observed in the far wave zone, if it was caused by emission from coherent oscillators, homogeneously positioned on an almost in"nite plane. Furthermore, Feynman [2, p. 82], noted that this emission would be observed at any separations from the plane even less than a wave length. A derivation to con"rm the Feynman's evident notion is not presented here, but it should not be di$cult to obtain near the plane emitting, if, e.g., according to a method by Feynman, the Maxwell equations will be used in the form suggested by Feynman (p. 38), when all electromagnetic terms can be separately presented as contributions from static and moving charges due to a time delay. Then, all terms originating at big separations from the sources must be omitted, by taking into consideration the fact that contributions from terms dependent on the static charges positive and negative must compensate each other at any separations from the neutral plane. Contributions from the other terms remained and depending on velocities of the oscillating charges determined the dependence needed. The example analyzed points out that a ray representation and other de"nitions of the geometric optics may be conventionally applied to propagation of the coherent radiation within the scales less than a wave length. In turn, a ray deviation from the straightforward propagation may be interpreted as a development of #uctuations or discrete properties of matter noticeable at short distances. A similar technique will be often in further applied henceforth without citing the previous basis. The approach used by Feynman may be transformed to a case of #uctuations of the coherent sources taking into account distributions of their inhomogeneity and the surface homogeneous density of #uctuations of di!erent kinds. This emission could be investigated, e.g., in random transparent and semitransparent mediums at the scattering for either the coherent radiation or the forward propagating heat radiation from a chaotic source that is being coherent, nevertheless, within some coherence radius. Curiously, the continuum absorption [1] or, in accordance with the detailed balance law, the continuum emission examined as a spectral density function in general coincides with the spectral density of the heat radiation, that is, I&u3. Thus, the ideas that were used to comprehend the heat radiation, which can oscillate within a resonant cavity at the heat balance, can be transformed to studies of the continuum absorption as well, inferring that the randomly discrete medium may be always presented as a number of the resonant cavities. 2. Dispersion formula in random optical mediums Let us consider a random transparent medium with density #uctuations as a partial randomly inhomegeneous structure, where volumes with di!erent optical densities can alternate as in
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Fig. 1. The scheme of the radiation propagation in the random medium with density #uctuations.
Fig. 1. Assume that the optical density of the ith volume is described by the refraction index m i and a radiation in the form of the unpolarized plane wave is propagating along the axis X. Two directions along the axes > and Z at the normal incidence of the ray are degenerated and become the equivalent. A ray normally incident on the boundary of two cells at the points i, i#1,2, k!1, k,2 (Fig. 1) is passed through a succession of great number of cells i, i#1,2, k!1, k,2 and it is partially re#ected on the boundaries in accordance with the Fresnel's law (see, for example, Ref. [3]). Since the semitransparent cells are symbolized su$ciently small #uctuations in a su$ciently dilute medium, the double interaction of the whole #ux of photons is accomplished just at cell k. For further convenience, one assumes that the interaction e!ect of the radiation with the matter will be concentrated just at the points i and k. Then, an electric "eld strength component of the amplitude of the radiation to be penetrated into the cell k, for example, along the axis Z, denoted as Ep,k,i, is related to a component E of the incident z z radiation at the point i by the following expression:
A
1 Ep,k,i"4 z n #1 k
BA
B
1 E , n #1 z i
(1)
where n "m /m and E is a strength amplitude of some casual electric "eld. Relative refraction i i~1 i z indexes n are random values and, moreover, we think that there should be many cells between two cells i and k in order to exclude any correlation between values n and n , if iOk, this should exist, i k as one should assume, because of the existence of some molecular "elds in some partially ordered structure (Fig. 1). On the other hand, their correlation as optical values is determined by external properties of the radiation and depends on the radiation frequency in accordance with the dispersion laws. Any boundary from a large set in Fig. 1 may be found on place of the boundaries i and k. Making the averaging procedure in Eq. (1) over an assemble of possible #uctuations (Fig. 1), i.e., the cells i, one can obtain the following equation: Ep"4(1/(n #1))(1/(n #1))E . z k i z
(2)
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The casual "eld E does not correlate with other values in Eq. (1), since it depends only on external z charges. If we take into consideration that, nN "n6 "1, then after the averaging over the whole i ensemble of the cells, all of which have not been correlated, if iOk, one "nds that n n "n2"n6 2$*n2"1$*n2. (3) k i For #uctuations of the relative refraction index su$ciently small, one can obtain the following: n2 Ep"4 E . z (n#1)2 z
(4)
On the other hand, Eq. (1) may be written in the form (n #1)(n #1)Ep,k,i"4E "const. (5) k i z z The value Ep,k,i cannot be extracted from the averaging over an assembly of the possible z #uctuations in the equation (n #1)(n #1)Ep,k,i"4E . k i z z Nevertheless, a new value E@ is assumed so that the relationship: z 4E z E@ " z (n #1)(n #1) k i is valid. Substituting Eq. (7) into (4) taking into account Eq. (3), one has
(6)
(7)
Ep"n2E@ . (8) z z The value n2 can be a speci"c constant of the dielectric permittivity e@ of the random medium and the values Ep and E@ are amplitudes of the electric "eld strengths in the medium with #uctuations z z and without, respectively. Subtracting the value E@ from bothsides of Eq. (4) leads to the "nal z expression e@!1 Ep!E@ "4 pP@ "4 E . z z z e@#3 z
(9)
Thus, the electric "eld strength will increase on average, if the radiation is passed through the random #uctuation. This conclusion followed Eq. (4), if the latter is written, with, n6 "1, in the form e@ Ep"4 E . z e@#3 z
(10)
It is seen from Eq. (10) that the inequality Ep 'E is valid for absolute values, when the standard z z case, e@'1, is provided. This inequality may be explained by the re#ected radiation as in the following speculations. The rays incident and re#ected can interfere, though a structure constant (Fig. 1) should be signi"cantly less in order that the classical standing waves could appear. The increase of the "eld strength points out the fact that the dielectric permittivity relates to the density #uctuations and
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Fig. 2. The 4th photon scheme of the propagation of the coherent radiation within a vacuum cavity in the matter with the refraction index m.
this may be explained by the emergence of a wave package after, for example, the double re#ections of a #ux portion within the chain of cells (Fig. 1). Moreover, the cell boundaries can be semitransparent because of the discrete matter and the re#ection should occur initially on the boundary surface of a far cell and then on the surface of a near cell. Using Eq. (7) as an analog, we may derive Eq. (9) by a simpler, but less obvious way. Indeed, without repeating the previous considerations taken place priori to obtaining Eqs. (1) and (6), one can write the following equations: (n #1)(n #1)Er,i,k"AE (n !1)(n !1)"4pP@ (n #1)(n #1) i k z z i k z i k
(11)
with the amplitude of the double re#ected radiation Er,i,k. Introducing the e!ective value of the z electric "eld amplitude as 4pP@ instead of the value Er,i,k and averaging over the ensemble of z #uctuations, lead to Eq. (9), if the coe$cient A equals 4. The value pP@ in electrostatics is performed by surface charges polarized inside the ellipsoid cavity (Fig. 2) positioned within the dielectric which, in turn, is inserted into an external electric "eld. Thus, radiation passed through the #uctuation and the radiation delayed inside the #uctuation at propagating in the randomly discrete medium produces the following two e!ects. These, in the average stochastic mediums, are the presence of e!ective rotation ellipsoid (Eq. (7)) and the presence of e!ective ellipsoid cavity (Eq. (9)), respectively. The plane wave produces the polarized surface charges outside and inside the #uctuation. The total local "eld Ep is represented by Eq. (10) z and it is equal to E@ #4pP@ in the randomly discrete medium. Increase in the local electric "eld z z (Eq. (10)) that has its e!ects, e.g., on one molecule does not yield the breakdown of the conservation of the energy to be transferred by the radiation #ux because of the presence of an accompanying e!ect of volume contracting of a wave package; this will be discussed later.
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Eq. (9) is close to a dispersion formula, that was used in Ref. [1] for studies on the continuum absorption. However, it is necessary to assume that, "rstly, in the following de"nition: n2"(1#n@)2"m@2,
(12)
the value m@ is the average absolute refraction index. Secondly, the value E as a casual electric "eld z in Eq. (10) may be assumed as an amplitude of the electric "eld strength in vacuum. Eventually, a component P@ relates to classical polarization vector P per volume unit in the CGS system, z where, conventionally, this polarization does not depend on the sample form and characterizes itself only internal properties of matter. Thus, the randomly discrete medium can be considered as random alternation of vacuum cavities with matter. These assumptions, as a whole, mean the transformation of our consideration to a deeper level of discreteness such as a molecular level, where the polarization vector becomes as an additive function of molecules. For in"nitesimal sizes of a cell in Fig. 1, these assumptions cause a quantum consideration of the scattering phenomena on one molecule. In this case, a model of cavity should be a classical system of two energy levels and we will return to the discussion of this case in Section 4. The consideration of the optical medium, originally, as the random one is attractive for the following main reason. The radiation is described by its group velocity that must be always decreased as the radiation is transferred from vacuum to the medium. Quasiheterogeneous optical properties of the medium (Fig. 1) assume that a time delay is observed because of the double radiation re#ections on #uctuation boundaries, when the radiation #ux slows down. The radiation partitions propagated forward or re#ected backward, that are moving with initial vacuum velocity, are fast extinguished due to many re#ections. Then, the radiation, as a whole, moves with a group velocity inherent for the medium. In classical optics, the phase velocities change according to the Ewald}Oseen theorem (see, e.g., Ref. [3]) in order for it to be a phase velocity inherent for the given medium.
3. Hypothesis of the photon correlation Absolute refraction index m@ was used since it plays an important role in classical optics of the determinate homogeneous medium. Also, de"nition (12) was used to study the radiation propagation in turbulent mediums [4]. Conventionally, one suggests that n@ is a random value and, for dilute gases, m@2"e"1#2n@, where e is one of possible values of the permittivity constant of the dielectric and, because value m@ is close to unity, one assumes that [4], m@+1. For enforcing the physical meaning of Eqs. (3) and (12), let us perform the following steps. 1. In accordance with Eqs. (3) and (12), the following relationship is valid: n@"J1$*n2!1+$*n2/2.
(13)
Procedures (3) and (12), in fact, are methods to extract the statistically independent contributions from a product of two stochastic values. 2. A statistical expectation of a product of two stochastic values is not always equal to the product of their average values and equivalence takes place only for statistically independent values. In general case, the stochastic values correlate with each other and Eq. (12) separates the
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correlations from independent contribution, and this, though for another reason, the summing of the intensities of two partial coherent waves, that separate the coherent and incoherent contributions. These two approaches demonstrate two sides of the dualistic view in the quantum wave mechanics. In order to present a quantum point of view, some particles, evidently, photons, must be taken into consideration. If the in#uence of molecular "elds on correlation of the refraction index is neglected, one can discuss correlation phenomena from a side of the electromagnetic "eld as a correlation of the photon density within a #ux as photon interactions, in our case, through a transparent medium. 3. This hypothesis brings into light the #ux of the compressible liquid and the requirements of continuity are expected, in turn, for the photon #ux. If a section of the medium as a waveguide is constant, then the photon energy density u should be proportional to the wavenumber following the evident relationships, u&l~1&j~1&u@, where l is the phase velocity of the radiation with the wavelength j or with the wavenumber u@ in the medium. However, spatial e!ects cannot be removed from these studies and the dependence u(u@) is thought to have a more complex form. At least, one should expect that, according to quantum representations, any radiation density captured within some resonant cavity must depend as follows: &u@ 3 or &1/j3. 4. Vacuum photons described by the coherent plane wave do not correlate and re#ection phenomena presented by Eqs. (1) and (9) are impossible. Certainly, the latter is valid for small intensities, when the vacuum polarization can be neglected. With the radiation propagation in a medium, the photons begin to interact with each other or to correlate, for the forward propagating radiation is coherent even for the heat source within a coherence radius. The sign (!) in Eqs. (3) and (12) may correspond to the photon anticorrelation, i.e., `pushing outa of the photons from a volume. This e!ect is seen in the medium having the phase velocity higher than the vacuum velocity of light and it may be possible for speci"c frequencies, in general, for any mediums. For the standard case of correlation, the photons of a coherent beam are `pulling intoa the volume. In two mentioned alternative cases, the volume of the coherent train is either extending or contracting. The term `traina will be in use instead of the mentioned `volume of the coherent wavesa. Since a transit time of the train is small (&10~11 c), direct observation of these e!ects is practically impossible. 5. The photons correlate through their wave properties and a coherence function of the "rst order (see, e.g., Ref. [5]) is implied here under the photon correlation function. The mentioned dispersion formula as the supplement contribution to the electric "eld strength is represented as m2!1 pP" E. m2#3
(14)
The form of Eq. (14) emphasizes the coherent properties of the forward propagating radiation by means of the successive re#ections within #uctuations. The coherent radiation separated by `a speci"c "ltera in the right-hand side of Eq. (14) enables us in principal to discuss the self-acting problem, by methods of wave theories, on a level of deep discreteness such as a molecule level and the coherent radiation is used as a basis for description of the radiation interaction with a single molecule. The left side of Eq. (14) to be expressed through the susceptibilities becomes an additive function of the molecules and one said that dispersion formulas similar Eq. (14) would link the micro- and macroscopic physical properties.
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For optically active medium, the refraction index m and the polarization vector P are convenient to be represented in the complex form, respectively, as m"m@#is,
(15)
4pP"(M#iK)E.
(16)
Dispersion relationship (14) that is distinguished from the classical Lorentz}Lorenz formula was applied by us in Ref. [1]. Curiously, when the M is small, the values m@ is close to unity in the transparent mediums and the value n@ is determined from classical formula with exactness up to the terms of the second order of the smallness parameter; however, for the Eq. (14), the exactness of this determination has the third order. Only from Eq. (14), in the case of the small refraction and absorption functions M and K (M2, K2;1), the dimensionless extinction coe$cient s is presented by the simple expression [1]
A
B
K 3 1 s" 1# M2! K2 . 2 16 16
(17)
The "rst term in Eq. (17) describes the ordinary linear absorption. A nonlinear absorption is represented by the second term that may be interpreted as a mixing of absorption with scattering. The appearance of the third term, in Eq. (17), with a negative sign is not obvious. If a stable sign of the function M is determined by the di!erences between radiation and resonance frequencies, the sign of the function K has a random character even for the pair of the energy terms and, depending on the sign, a photon absorption or emission is observed. For example, the third term in Eq. (17) may be considered the following quantum acts: (1) photon absorption (K'0); (2) photon emission (K(0); (3) photon absorption (K'0) once again. Because of a speci"c line mixing [1], all three elementary acts are carried out for transitions between energy levels of di!erent lines. The second act may be provided by the spontaneous emission. Using Eq. (17), one can obtain the relation for `nonlineara absorption coe$cient [1] as a&u3KM2
(18)
that allows us to mention at the begining of the paper that the intensity of the continuum absorption or emission is proportional to a cubic power of the wavenumber. The elegance of Eqs. (14) and (17) compels us to believe that the search for the dispersion formula, that connects the micro and macro phenomena, is not completed by a classical derivation of the Lorentz}Lorenz formula through the Lorentz sphere. Using the Lorentz sphere in the electrostatic "eld for solids assumes that its volume is big in comparison with the volume of the elementary cell. In addition, the cubic symmetry of the grid must be presented in order to compensate the electric "elds of the polarized dipoles positioned within the Lorentz sphere (see, e.g., Ref. [6]). Being in the framework of the electrostatics, it is di$cult to explain the negative polarization of matter and this compels the transformation to electromagnetic "elds. However, a strong derivation of the dispersion formula in the framework of the Maxwell theory [3, (pp. 128 and 830)] supposes that a radius of the similar sphere must approach to zero, that leads to contradictions with the quantum representations such as the Heisenberg indeterminacy principle. Thus, the presence of the random optically active medium, consideration of the "nite velocity of light, relations with quantum and wave phenomena, all the above numerated requirements make
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this problem as an extremely di$cult one. Preliminary derivation of the dispersion formula indicates that we do not escape the application of nonlinear theoretical models, in which the wave train or the wave package self-acts through the medium similar to that at the derivation of Eq. (9).
4. Quantization of the polarization 5eld For quantization of the supplement "elds emerging at the medium polarization, there may be a standard scheme of quantization of the "eld energy, presented via a square of its electric and magnetic "eld strengths. However, it is di$cult to use the wave train volume to calculate the density of the quantum modes, for such resonant cavity moves together with the train. Also, the linear sizes of a vacuum cavity within the matter, as a rule, are signi"cantly less than the wavelength. Therefore, the quantization procedure will be discussed through the energy transfer within the self-interacting wave train, since the transfer phenomena cause changes of the train photon density and an inherent quantum character of the energy transfer is obeyed by its determination. Emphasizing the vacuum as a speci"c medium with speci"c absolute properties in accordance with Eq. (12) assumes the consideration of a discrete and, in our case, randomly discrete medium. Let us assume that a monochromatic plane wave is propagating within a homogeneous medium with the absolute refraction index m and passing through the cavity (Fig. 2). The cavity sizes are su$ciently small that rays 2 and 4 do not interfere after the di!raction at points 1 and 2, and they can di!ract once again at points 3, 4 on the opposite surface. There is always a probability that two photons simultaneously follow the paths drawn by rays 2 and 4 and then, with a time delay, meet two photons 1 and 3 at points 3, 4. Nothing is changed within the photon beam of the forward propagating radiation (Fig. 2) except that two photons have interexchanged their positions after two symmetrical deviations. However, photons 1 and 3 reach points 3 and 4, in general, with some phase delay in comparison with photons 2 and 4. This delay comprises the contribution from di!erent paths and from di!erent interactions with the molecules. The assumed obligatory interference of these four rays leads to decrease of the intensity of the forward propagating radiation. According to the energy conservation, this energy damage either must be absorbed in optically active mediums, that had been discussed in Ref. [1], or has to be re#ected at points 3 and 4 in a case of transparent mediums, or must be shifted to plane perpendicular to the axis X. Let us temporarily omit the consideration of the latter that yields the breakdown of the symmetrical scheme presented in Fig. 2. Then, let us discuss in detail an act of the simultaneous scattering of two coherent photons on one particle (molecule). Minimum photon number, participating simultaneously at the scattering on one molecule in order to take into account the interference e!ect, must be equal to two. Inferring that the energy of the photon pair has been increased four times after the coherent summing of their amplitudes, paradoxically four photons instead of two arise during this scattering. This paradox, in our opinion, is impossible to explain by the Heisenberg indeterminacy principle, since we deal with a quantum event on one molecule with determinate energies of indivisible particles. There should be, in our opinion, two explanations for this fact, if the molecular absorption is not presented. 1. Since the vacuum polarization for small photon energies is not taken into account and, therefore, the particle transformation is removed from the consideration, in order to save the
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conservation of the particle number, one has to assume that one of these photons is always polarizing in a plane perpendicular to the polarization plane of another photon. The term `polarizationa is distinguished here from the above-mentioned `medium polarizationa. The photon depolarization may be observed at the scattering of the coherent photons and they has been transformed to the incoherent ones through depolarization. However, a second interpretation is needed to escape a situation, when two polarization states have been `occupieda. 2. If for some interference experiment instead of a screen, where the interference picture is "xed, a transparent discrete medium is selected, there is always a probability that after di!raction on medium particles the waves from two interfering beams again go together into a forward propagating plane wave. The new superposition of two beams may be also carried out by means of semitransparent mirrors. The intensity distribution is obtained on a front plane as a result of the interference. This should mean in the quantum language that the existence of two coherent photons is possible due to two neighborhood or spatially anticoherent photons. One should generally expect, that both mechanisms are probable, but only the second interpretation is made use of in the present paper. The 4th photon #uctuation represented in Fig. 2 can be called a structure of the two coherent photons. The forward scattering of two anticoherent photons can be explained as a re#ection of two coherent photons. An intermediate state of the partial coherence due to for two photons at the simultaneous forward scattering is forbidden by the energy conservation law, the quantum hypothesis of the photon indivisibility and the negligibly small Compton's e!ect for heavy molecules. The act of the simultaneous forward scattering on molecules has a special quantum character and the relation between the propagating and re#ecting radiation may be examined as a relationship between a probability of arising of the coherent and anticoherent pairs. If two anticoherent forward propagating photons are perished at one point 3 and 4, two coherent photons must be appear after re#ection with a transfer of the 8th-multiple impulse of one photon to a molecule or molecules. The re#ection amplitude of two coherent photons is two times greater than that of one photon. The four photons 1, 2, 3, 4 after the re#ection can be directed, with the equal probability, along four inverse paths depicted in Fig. 2 and they may be re#ected by the same means with some probability at points 1 and 2. If medium properties lead to a phase delay after elementary interactions of the radiation with molecules, then the phase di!erence between two waves re#ected and main should vanish and the electric "eld should be enforced in the cavity in Fig. 2, and the wave train has been contracted. On the contrary, the advanced phase e!ect enforces the phase di!erence between the two re#ected rays and main and a smaller portion of the radiation is re#ected at points 1 and 2 (Fig. 2). The negative polarization of the medium is observed and, in turn, the wave train should be extended. The probability of the return of the scattered coherent photon to meet other time-coherent photons from the coherent beam is the main idea of the present consideration. The scheme shown in Fig. 2 gives a new explanation of the negative polarization. A choice of the ellipsoid cavity with relationships of the semimajor/semiminor axes as 3 is conditioned by the 2 correspondence of the dispersion formula (14) with those that can be derived in the electrostatics (see, e.g., Ref. [6]). The cavity form is not signi"cant here and can be replaced (Fig. 2) by a density #uctuation. The necessary condition is only a cavity symmetry according to the axis X, that is on average valid in a random medium.
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Since the transit time of the train propagation is a very small value, a device detects an integral signal from the big number of trains. Every photon within the train has a possibility of participating in the construction of the 4th photons scheme as seen in Fig. 2 as in its destruction. The radiation emissions at di!erent points 1 and 2 of the front plane are two independent events in accordance with the Huygens' principle, which is a consequence of the "nite velocity of light. The probability of the scheme construction is proportional to a square of the probability of a photon to be `alivea, namely, &(N/N )2, where N is the initial photon number in the train, The photon 0 0 destruction as a result of the catastrophic absorption or scattering on molecules in the cavity in Fig. 2 is also proportional to (N/N ). The latter breaks down the symmetrical scheme and causes 0 an intensity #uctuation along a front plane perpendicular to the axis X, therefore the absorption probability, as a whole, increases after this elementary absorption act. The absorption and deviation acts are also independent because there is no interference among four photons after the deviation of two photons within the symmetrical scheme. The nonlinear absorption law may be obtained as
A B
dNM NM 3 NM (19) "!a !a NM dx NM NM 0 0 0 that is obligated by arising of a supplement e!ective photon #ux owing to the nonlinear e!ect of the photon delay in Fig. 2. Following from Eq. (18), the supplement density *o of the photon modes arises, so that, *o&u2M2. The averaging over all wave trains in Eq. (19) is based on an assumption that photons from the di!erent trains do not correlate and this is not always valid in comparison with modern representations of the quantum optics [5]. The de"nitions of the photon numbers NM and NM are related to average total values of local and initial photon numbers, 0 respectively, crossing a given cross-section at a point x for a chosen time period. Thus, the two photon interaction in the #ux of the coherent radiation can be statistically considered as an exchange of positions of these photons, that leads to doubling of the amplitude of the electric "eld of the re#ected radiation and explains a logical appearance of factor 4 (A"4) in Eqs. (11). The mechanisms and the medium types to be discussed in the present work may both be useful to specify the physical meanings of some types of absorption and scattering. For in"nitesimal sizes inherent for molecules, the resonant cavity may represented in the form of a classical two-level system. A spontaneous emission may interrupt the two-level exchange of the coherent radiation and cause an emission of a chaotic photon that can be either absorbed in the optically active medium in accordance with the linear law or contribute to the Rayleigh's scattering of the independent molecules. In turn, the exchange of the coherent radiation between the cavity walls in Fig. 2 may be interrupted by a linear absorption or by the Rayleigh's scattering of a photon and it leads either to the nonlinear continuum absorption in the semitransparent mediums or to the opalescense by molecular #uctuations in transparent mediums. The continuum absorption seems to be transformed to the critical opalescense near a critical point of matter, and along with this the number of resonant cavities is dramatically increased near this point. In the terms of intensities, Eq. (19) may be presented in the following di!erential form: dI I3 "!aI!a . dx I2 0
(20)
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The solution of Eq. (20) will be as follows: I exp(!ax) 0 I" . (1#a/2a(1!exp(!2ax)))1@2
(21)
Eq. (21) produced [1] the selective continuum, that would arise by the interruption of the nonlinear e!ect of generating the coherent photons because of the strong linear absorption. The linear absorption coe$cient a approaches zero in the case of a transparent medium and Eq. (21) should be transformed to I 0 . I" (1#ax)1@2
(22)
The absorption coe$cients a and a may have di!erent origins and the approaching of the value a to zero does not mean the same for the coe$cient a. In the above-mentioned expressions, one assumed by default that the nonlinear absorption coe$cient a is always greater than zero. However, the physical meaning of Eq. (21) is lost in the case of big paths of x, where the inverse absorption along opposite direction of the axis X takes place. Thus, the macroscopic irreversibility for the time direction should be found out in Eq. (21). Eq. (22) may be interpreted as a case of emission for su$ciently small negative values of x, and the coe$cient a should be as a source of the backward re#ected radiation. Taking into account indeterminacy for the x interval as well as for change of the impulse p and using the quantum hypothesis by Heisenberg, Eq. (22) may be expressed in the curious form: I 0 I" , (1!*p*x/+)1@2
(23)
that might be useful to understand the increase of quantum #uctuations of the radiation in a cavity. The scheme presented in Fig. 2 leads to an energy #uctuation in a local volume taken from neighborhood volumes. Since the breakdown of either the time irreversibility principle or the casualty principle, both being in the macrocosm, is forbidden, the Heisenberg indeterminacy principle is needed to show a boundary of the microcosm. The breakdown of the irreversibility principle is possible only in the microcosm in the framework of the Heisenberg indeterminacy principle, and, perhaps, Eq. (23) can show a possible way of understanding the limits of the macroand micro- formulations of the casualty principle.
5. Summary and conclusions 1. A fact is represented the causal relationships and events by elements of the geometric optics within scale of less than a wavelength by default assumes the quantum character of the phenomena, accompanying the particle #ux with the inherent wave properties. 2. The refraction index is discussed as a result of the photon correlation within the #ux. Though the present work without a speci"c apparatus of quantum "eld theory is unlikely to be a quantum representation of the refraction index, the simple preliminary results favor this direction. The dispersion as the dependence of the refraction index on the radiation frequency is easily explained
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833
by the wave properties of the interaction of the plane wave with a single molecule; however, the dispersion relationship (14) may be explained by the quantum representations such as the photon correlation. The main conclusions of this paper are based on the probability of a coherent photon returning to a point of the initial deviation after some time period and meeting a time-coherent photon from the coherent beam or train. 3. The "nite velocity of light or casual relationships brings about nonlinear e!ects of selfinteracting wave trains. The wave train or the wave package changes its volume and hence its photon density at the train moving within random or discrete, on the small distances, medium, following the random or discrete medium #uctuations. The supplement photon #ux arises and moves through the cross-section selected and the nonlinear e!ects is concluded by Eq. (19) of the radiative transfer. 4. These nonlinear e!ects are probably impossible for one spatial dimension, for the probability to transform coherent pairs into anticoherent ones and vice versa must be always provided. Eqs. (14) and (20) themselves could have a more fundamental character than an approximation of power series over the smallness parameter. We explicitly did not use the perturbation theory for their derivation and they could describe integral e!ects of the train self-interacting through the optical medium as a consequence of the three-dimensional space similar to that when the heat radiation is locked in a three-dimensional box. The perturbation theory technique and molecular properties, as a rule, a!ected only the left}right side of Eq. (14) and they were used to study the speci"c molecular properties via susceptibilities and molecular "elds. 5. In connection with the quantum representations, one needs to underline speci"c properties of the forward propagating and backward re#ecting radiation. Let us assume that two time-coherent polarized photons are moving within the transparent medium, comprising relatively heavy molecules. One of them, falling earlier on medium, obtains a random phase after manifold collisions. Let us call this photon a chaotic photon. The chaotic photon has a probability meeting a time-coherent photon on one molecule with a time delay and they both can interfere. Following a hypothesis of two coherent photons, the chaotic photon, on average, does not change the coherence of the forward propagating radiation. The same is related to the chaotic heat photons. Thus, the information about a radiation source to be attributes of the coherent radiation has been saved, as a whole, by the forward propagating radiation and the hypothesis of the coherent photons does not contradict with the superposition principle in the wave mechanics. 6. Controversial de"nition such as the coherent wave train has been used in the present paper. The question is whether this train is as a result of an elementary act of the photon emission on one molecule or whether this is a collective phenomenon? This old problem took its origin from the begin of this century. In the line shape theory, this e!ect was used to present itself as the emission of one molecule. Experiments on the quantum optics tell us that photon is self-interfering and here is no possibility of knowing the source for this photon at this interference. It is logical to suppose that it is impossible do determine as well what is the number of photons and what are the molecules and energy levels to be the photon sources for small time periods. For conclusions of this paper, it is only essential that a probability of producing the coherent wave train must be induced for every photon. The important role of the continuum absorption is not surprising because the continuum concept has a broader interpretation than the continuum absorption of a simple molecule. Thus, any attempt to "nd a new solution of this complex problem even for one molecule must agree
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a number of classical laws and principles and this requirement dictated the subject of the present studies. The existence of the coherent photons is nearly evident, since all classical interference experiments are likely to use similar de"nition for the experiment interpretation on the quantum language. In this sense, this idea seems to be useful to examine once more, in the quantum language, the rotation of the polarization plane in mediums with mirror isomers. A quantum hypothesis to present the scattering as two-fold elementary quantum acts belongs to Pauli. This hypothesis stimulated Einstein and Ehrenfest in creating the heat balance theory of the manifold quantum transitions and led them to the detailed balance principle. A state of the heat balance itself indicates that the simultaneous quantum events would very popular in the microcosm. However, the present hypothesis of the arising of the coherent photons, in our opinion, is not innocent and it could have far-reaching consequences by considering both the heat balance and the interaction of high-energy photons on nucleus. Recently [7], an important applied problem of the excess absorption in the atmosphere had been transformed to a basic problem of the water vapor absorption. Therefore, in order to calculate the transmission function within inhomogeneous medium with water mixing ratios, it is interesting to try to apply the new nonlinear radiative transfer equation (20) that contains quantum absorptionscattering e!ects at short distances and had been performed [1] by parametrization just as a trial on the water vapor absorption.
Acknowledgements This work is based on our studies of the radiation transfer by programming the information system [8]. The work was supported by the grant RFBR No. 96-07-89007, the author deems it his duty to express the appreciative acknowledgments and gratitude to Russian Foundation of the Basic Researches.
References [1] Golovko VF. Dispersion formula and continuous absorption of water vapor. JQSRT 2000 (in press). [2] Feynman RP, Leighton RB, Sands M. The Feynman lectures on physics, Vol. 1. Massachusetts: AddisonWesley Publishing Company, Inc., 1963 [in Russian: Vol. 3, Moscow: Mir, 1966. p. 78]. [3] Born M, Wolf E. Principles of optics, 2nd ed. Oxford: Pergamon Press, 1964 [in Russian: Moscow: Fizmatgiz, 1970. p. 66]. [4] Monin AS, Yaglom AM. Statistical hydromechanics, Vol. 2. Moscow: Fizmatgiz, 1967. p. 547 [in Russian]. [5] London R. The quantum theory of light. Oxford: Clarendon Press, 1973 [in Russian: Moscow: Mir, 1976. p. 530]. [6] Kittel C. Introduction to solid state physics, 4th ed. New York: Wiley [in Russian: Moscow: Fizmatgiz, 1978]. [7] Vogelmann AM, Ramanathan V, Conant WC, Hunter WE. JQSRT 1988;60:231}46. [8] Golovko VF, Nikitin AV, Chursin AA, Tashkun AS, Mikhailenko SN. Developments of the information system, based on large-scale spectroscopic databases, for application in the atmospheric optics and molecular spectroscopy. Report RFBR No. 96-07-89007, January 1999.
Hypothetical dispersion quantum e!ects for coherent forward propagating radiation in transparent and semitransparent medium V.F. Golovko* Laboratory of Theoretical Spectroscopy, Institute of Atmospheric Optics, Akademicheskii Av., 1, Tomsk, 634055, Russia
Abstract An attempt to establish relations between the continuum absorption problem and the dispersion theory is made with the help of elements of the quantum representation. A new technique for explanation of a new dispersion formula has been obtained as a consequence of the delay of the coherent radiation propagating in a random optical medium. A hypothesis of arising of pairs of coherent and anticoherent photons is assumed as some quantum e!ect on a molecule for the interaction of two photons from the forward propagating radiation and a paradox accompanying this consideration is discussed. Self-interacting e!ects of coherent wave trains are discussed and a quantization procedure has been suggested to account for transfer of the supplement e!ective photon number at the propagation of the coherent wave train within a randomly discrete medium. The ideas of this approach were recently used for modeling the continuum absorption of water vapor. ( 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction This work was stimulated by our recent studies [1] or the continuum absorption in the infra-red and microwave regions. The modeling of the spectra showed that continuum absorption of the water vapor is caused by contributions from far line wings rather than, e.g., by the dimer absorption. Nevertheless, the dimer hypothesis, in particular, and the #uctuation theory, in general, both could be important in understanding of mechanisms of the continuum absorption. Because of a great number of lines, the continuum may be referred to a problem of systems with in"nite
* Tel.: #7-3822-258794; fax: #7-3822-259086. E-mail address: [email protected] (V.F. Golovko) 0022-4073/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 4 0 7 3 ( 9 9 ) 0 0 1 4 2 - 9
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freedoms and there is some sense in describing them by an approach that is distinguished, e.g., from a method of detailed wave equations of the wave mechanics. In our studies on the continuum absorption, it has been found that the spectral density of its intensity I is proportional to a cubic power of the wave number u of the monochromatic radiation, i.e., I&u3 (u"l/c, where l is the frequency; c is the vacuum light velocity). Note that at conventional absorption of a photon, I&u, in accordance with the classical quantum nature of the radiation, and, at Rayleigh's single-scattering of light by independent sources, I&u5, it has followed a Herz emission law, which, in turn, is considered as a solution of the Maxwell equations in the so-called far wave zone and, in fact, it is a manifestation of the causality principle, due to a "nite velocity of the interaction propagation. However, Feynman [2] had demonstrated in detail a case, where the radiation, I&u3, would be observed in the far wave zone, if it was caused by emission from coherent oscillators, homogeneously positioned on an almost in"nite plane. Furthermore, Feynman [2, p. 82], noted that this emission would be observed at any separations from the plane even less than a wave length. A derivation to con"rm the Feynman's evident notion is not presented here, but it should not be di$cult to obtain near the plane emitting, if, e.g., according to a method by Feynman, the Maxwell equations will be used in the form suggested by Feynman (p. 38), when all electromagnetic terms can be separately presented as contributions from static and moving charges due to a time delay. Then, all terms originating at big separations from the sources must be omitted, by taking into consideration the fact that contributions from terms dependent on the static charges positive and negative must compensate each other at any separations from the neutral plane. Contributions from the other terms remained and depending on velocities of the oscillating charges determined the dependence needed. The example analyzed points out that a ray representation and other de"nitions of the geometric optics may be conventionally applied to propagation of the coherent radiation within the scales less than a wave length. In turn, a ray deviation from the straightforward propagation may be interpreted as a development of #uctuations or discrete properties of matter noticeable at short distances. A similar technique will be often in further applied henceforth without citing the previous basis. The approach used by Feynman may be transformed to a case of #uctuations of the coherent sources taking into account distributions of their inhomogeneity and the surface homogeneous density of #uctuations of di!erent kinds. This emission could be investigated, e.g., in random transparent and semitransparent mediums at the scattering for either the coherent radiation or the forward propagating heat radiation from a chaotic source that is being coherent, nevertheless, within some coherence radius. Curiously, the continuum absorption [1] or, in accordance with the detailed balance law, the continuum emission examined as a spectral density function in general coincides with the spectral density of the heat radiation, that is, I&u3. Thus, the ideas that were used to comprehend the heat radiation, which can oscillate within a resonant cavity at the heat balance, can be transformed to studies of the continuum absorption as well, inferring that the randomly discrete medium may be always presented as a number of the resonant cavities. 2. Dispersion formula in random optical mediums Let us consider a random transparent medium with density #uctuations as a partial randomly inhomegeneous structure, where volumes with di!erent optical densities can alternate as in
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Fig. 1. The scheme of the radiation propagation in the random medium with density #uctuations.
Fig. 1. Assume that the optical density of the ith volume is described by the refraction index m i and a radiation in the form of the unpolarized plane wave is propagating along the axis X. Two directions along the axes > and Z at the normal incidence of the ray are degenerated and become the equivalent. A ray normally incident on the boundary of two cells at the points i, i#1,2, k!1, k,2 (Fig. 1) is passed through a succession of great number of cells i, i#1,2, k!1, k,2 and it is partially re#ected on the boundaries in accordance with the Fresnel's law (see, for example, Ref. [3]). Since the semitransparent cells are symbolized su$ciently small #uctuations in a su$ciently dilute medium, the double interaction of the whole #ux of photons is accomplished just at cell k. For further convenience, one assumes that the interaction e!ect of the radiation with the matter will be concentrated just at the points i and k. Then, an electric "eld strength component of the amplitude of the radiation to be penetrated into the cell k, for example, along the axis Z, denoted as Ep,k,i, is related to a component E of the incident z z radiation at the point i by the following expression:
A
1 Ep,k,i"4 z n #1 k
BA
B
1 E , n #1 z i
(1)
where n "m /m and E is a strength amplitude of some casual electric "eld. Relative refraction i i~1 i z indexes n are random values and, moreover, we think that there should be many cells between two cells i and k in order to exclude any correlation between values n and n , if iOk, this should exist, i k as one should assume, because of the existence of some molecular "elds in some partially ordered structure (Fig. 1). On the other hand, their correlation as optical values is determined by external properties of the radiation and depends on the radiation frequency in accordance with the dispersion laws. Any boundary from a large set in Fig. 1 may be found on place of the boundaries i and k. Making the averaging procedure in Eq. (1) over an assemble of possible #uctuations (Fig. 1), i.e., the cells i, one can obtain the following equation: Ep"4(1/(n #1))(1/(n #1))E . z k i z
(2)
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The casual "eld E does not correlate with other values in Eq. (1), since it depends only on external z charges. If we take into consideration that, nN "n6 "1, then after the averaging over the whole i ensemble of the cells, all of which have not been correlated, if iOk, one "nds that n n "n2"n6 2$*n2"1$*n2. (3) k i For #uctuations of the relative refraction index su$ciently small, one can obtain the following: n2 Ep"4 E . z (n#1)2 z
(4)
On the other hand, Eq. (1) may be written in the form (n #1)(n #1)Ep,k,i"4E "const. (5) k i z z The value Ep,k,i cannot be extracted from the averaging over an assembly of the possible z #uctuations in the equation (n #1)(n #1)Ep,k,i"4E . k i z z Nevertheless, a new value E@ is assumed so that the relationship: z 4E z E@ " z (n #1)(n #1) k i is valid. Substituting Eq. (7) into (4) taking into account Eq. (3), one has
(6)
(7)
Ep"n2E@ . (8) z z The value n2 can be a speci"c constant of the dielectric permittivity e@ of the random medium and the values Ep and E@ are amplitudes of the electric "eld strengths in the medium with #uctuations z z and without, respectively. Subtracting the value E@ from bothsides of Eq. (4) leads to the "nal z expression e@!1 Ep!E@ "4 pP@ "4 E . z z z e@#3 z
(9)
Thus, the electric "eld strength will increase on average, if the radiation is passed through the random #uctuation. This conclusion followed Eq. (4), if the latter is written, with, n6 "1, in the form e@ Ep"4 E . z e@#3 z
(10)
It is seen from Eq. (10) that the inequality Ep 'E is valid for absolute values, when the standard z z case, e@'1, is provided. This inequality may be explained by the re#ected radiation as in the following speculations. The rays incident and re#ected can interfere, though a structure constant (Fig. 1) should be signi"cantly less in order that the classical standing waves could appear. The increase of the "eld strength points out the fact that the dielectric permittivity relates to the density #uctuations and
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Fig. 2. The 4th photon scheme of the propagation of the coherent radiation within a vacuum cavity in the matter with the refraction index m.
this may be explained by the emergence of a wave package after, for example, the double re#ections of a #ux portion within the chain of cells (Fig. 1). Moreover, the cell boundaries can be semitransparent because of the discrete matter and the re#ection should occur initially on the boundary surface of a far cell and then on the surface of a near cell. Using Eq. (7) as an analog, we may derive Eq. (9) by a simpler, but less obvious way. Indeed, without repeating the previous considerations taken place priori to obtaining Eqs. (1) and (6), one can write the following equations: (n #1)(n #1)Er,i,k"AE (n !1)(n !1)"4pP@ (n #1)(n #1) i k z z i k z i k
(11)
with the amplitude of the double re#ected radiation Er,i,k. Introducing the e!ective value of the z electric "eld amplitude as 4pP@ instead of the value Er,i,k and averaging over the ensemble of z #uctuations, lead to Eq. (9), if the coe$cient A equals 4. The value pP@ in electrostatics is performed by surface charges polarized inside the ellipsoid cavity (Fig. 2) positioned within the dielectric which, in turn, is inserted into an external electric "eld. Thus, radiation passed through the #uctuation and the radiation delayed inside the #uctuation at propagating in the randomly discrete medium produces the following two e!ects. These, in the average stochastic mediums, are the presence of e!ective rotation ellipsoid (Eq. (7)) and the presence of e!ective ellipsoid cavity (Eq. (9)), respectively. The plane wave produces the polarized surface charges outside and inside the #uctuation. The total local "eld Ep is represented by Eq. (10) z and it is equal to E@ #4pP@ in the randomly discrete medium. Increase in the local electric "eld z z (Eq. (10)) that has its e!ects, e.g., on one molecule does not yield the breakdown of the conservation of the energy to be transferred by the radiation #ux because of the presence of an accompanying e!ect of volume contracting of a wave package; this will be discussed later.
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Eq. (9) is close to a dispersion formula, that was used in Ref. [1] for studies on the continuum absorption. However, it is necessary to assume that, "rstly, in the following de"nition: n2"(1#n@)2"m@2,
(12)
the value m@ is the average absolute refraction index. Secondly, the value E as a casual electric "eld z in Eq. (10) may be assumed as an amplitude of the electric "eld strength in vacuum. Eventually, a component P@ relates to classical polarization vector P per volume unit in the CGS system, z where, conventionally, this polarization does not depend on the sample form and characterizes itself only internal properties of matter. Thus, the randomly discrete medium can be considered as random alternation of vacuum cavities with matter. These assumptions, as a whole, mean the transformation of our consideration to a deeper level of discreteness such as a molecular level, where the polarization vector becomes as an additive function of molecules. For in"nitesimal sizes of a cell in Fig. 1, these assumptions cause a quantum consideration of the scattering phenomena on one molecule. In this case, a model of cavity should be a classical system of two energy levels and we will return to the discussion of this case in Section 4. The consideration of the optical medium, originally, as the random one is attractive for the following main reason. The radiation is described by its group velocity that must be always decreased as the radiation is transferred from vacuum to the medium. Quasiheterogeneous optical properties of the medium (Fig. 1) assume that a time delay is observed because of the double radiation re#ections on #uctuation boundaries, when the radiation #ux slows down. The radiation partitions propagated forward or re#ected backward, that are moving with initial vacuum velocity, are fast extinguished due to many re#ections. Then, the radiation, as a whole, moves with a group velocity inherent for the medium. In classical optics, the phase velocities change according to the Ewald}Oseen theorem (see, e.g., Ref. [3]) in order for it to be a phase velocity inherent for the given medium.
3. Hypothesis of the photon correlation Absolute refraction index m@ was used since it plays an important role in classical optics of the determinate homogeneous medium. Also, de"nition (12) was used to study the radiation propagation in turbulent mediums [4]. Conventionally, one suggests that n@ is a random value and, for dilute gases, m@2"e"1#2n@, where e is one of possible values of the permittivity constant of the dielectric and, because value m@ is close to unity, one assumes that [4], m@+1. For enforcing the physical meaning of Eqs. (3) and (12), let us perform the following steps. 1. In accordance with Eqs. (3) and (12), the following relationship is valid: n@"J1$*n2!1+$*n2/2.
(13)
Procedures (3) and (12), in fact, are methods to extract the statistically independent contributions from a product of two stochastic values. 2. A statistical expectation of a product of two stochastic values is not always equal to the product of their average values and equivalence takes place only for statistically independent values. In general case, the stochastic values correlate with each other and Eq. (12) separates the
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correlations from independent contribution, and this, though for another reason, the summing of the intensities of two partial coherent waves, that separate the coherent and incoherent contributions. These two approaches demonstrate two sides of the dualistic view in the quantum wave mechanics. In order to present a quantum point of view, some particles, evidently, photons, must be taken into consideration. If the in#uence of molecular "elds on correlation of the refraction index is neglected, one can discuss correlation phenomena from a side of the electromagnetic "eld as a correlation of the photon density within a #ux as photon interactions, in our case, through a transparent medium. 3. This hypothesis brings into light the #ux of the compressible liquid and the requirements of continuity are expected, in turn, for the photon #ux. If a section of the medium as a waveguide is constant, then the photon energy density u should be proportional to the wavenumber following the evident relationships, u&l~1&j~1&u@, where l is the phase velocity of the radiation with the wavelength j or with the wavenumber u@ in the medium. However, spatial e!ects cannot be removed from these studies and the dependence u(u@) is thought to have a more complex form. At least, one should expect that, according to quantum representations, any radiation density captured within some resonant cavity must depend as follows: &u@ 3 or &1/j3. 4. Vacuum photons described by the coherent plane wave do not correlate and re#ection phenomena presented by Eqs. (1) and (9) are impossible. Certainly, the latter is valid for small intensities, when the vacuum polarization can be neglected. With the radiation propagation in a medium, the photons begin to interact with each other or to correlate, for the forward propagating radiation is coherent even for the heat source within a coherence radius. The sign (!) in Eqs. (3) and (12) may correspond to the photon anticorrelation, i.e., `pushing outa of the photons from a volume. This e!ect is seen in the medium having the phase velocity higher than the vacuum velocity of light and it may be possible for speci"c frequencies, in general, for any mediums. For the standard case of correlation, the photons of a coherent beam are `pulling intoa the volume. In two mentioned alternative cases, the volume of the coherent train is either extending or contracting. The term `traina will be in use instead of the mentioned `volume of the coherent wavesa. Since a transit time of the train is small (&10~11 c), direct observation of these e!ects is practically impossible. 5. The photons correlate through their wave properties and a coherence function of the "rst order (see, e.g., Ref. [5]) is implied here under the photon correlation function. The mentioned dispersion formula as the supplement contribution to the electric "eld strength is represented as m2!1 pP" E. m2#3
(14)
The form of Eq. (14) emphasizes the coherent properties of the forward propagating radiation by means of the successive re#ections within #uctuations. The coherent radiation separated by `a speci"c "ltera in the right-hand side of Eq. (14) enables us in principal to discuss the self-acting problem, by methods of wave theories, on a level of deep discreteness such as a molecule level and the coherent radiation is used as a basis for description of the radiation interaction with a single molecule. The left side of Eq. (14) to be expressed through the susceptibilities becomes an additive function of the molecules and one said that dispersion formulas similar Eq. (14) would link the micro- and macroscopic physical properties.
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For optically active medium, the refraction index m and the polarization vector P are convenient to be represented in the complex form, respectively, as m"m@#is,
(15)
4pP"(M#iK)E.
(16)
Dispersion relationship (14) that is distinguished from the classical Lorentz}Lorenz formula was applied by us in Ref. [1]. Curiously, when the M is small, the values m@ is close to unity in the transparent mediums and the value n@ is determined from classical formula with exactness up to the terms of the second order of the smallness parameter; however, for the Eq. (14), the exactness of this determination has the third order. Only from Eq. (14), in the case of the small refraction and absorption functions M and K (M2, K2;1), the dimensionless extinction coe$cient s is presented by the simple expression [1]
A
B
K 3 1 s" 1# M2! K2 . 2 16 16
(17)
The "rst term in Eq. (17) describes the ordinary linear absorption. A nonlinear absorption is represented by the second term that may be interpreted as a mixing of absorption with scattering. The appearance of the third term, in Eq. (17), with a negative sign is not obvious. If a stable sign of the function M is determined by the di!erences between radiation and resonance frequencies, the sign of the function K has a random character even for the pair of the energy terms and, depending on the sign, a photon absorption or emission is observed. For example, the third term in Eq. (17) may be considered the following quantum acts: (1) photon absorption (K'0); (2) photon emission (K(0); (3) photon absorption (K'0) once again. Because of a speci"c line mixing [1], all three elementary acts are carried out for transitions between energy levels of di!erent lines. The second act may be provided by the spontaneous emission. Using Eq. (17), one can obtain the relation for `nonlineara absorption coe$cient [1] as a&u3KM2
(18)
that allows us to mention at the begining of the paper that the intensity of the continuum absorption or emission is proportional to a cubic power of the wavenumber. The elegance of Eqs. (14) and (17) compels us to believe that the search for the dispersion formula, that connects the micro and macro phenomena, is not completed by a classical derivation of the Lorentz}Lorenz formula through the Lorentz sphere. Using the Lorentz sphere in the electrostatic "eld for solids assumes that its volume is big in comparison with the volume of the elementary cell. In addition, the cubic symmetry of the grid must be presented in order to compensate the electric "elds of the polarized dipoles positioned within the Lorentz sphere (see, e.g., Ref. [6]). Being in the framework of the electrostatics, it is di$cult to explain the negative polarization of matter and this compels the transformation to electromagnetic "elds. However, a strong derivation of the dispersion formula in the framework of the Maxwell theory [3, (pp. 128 and 830)] supposes that a radius of the similar sphere must approach to zero, that leads to contradictions with the quantum representations such as the Heisenberg indeterminacy principle. Thus, the presence of the random optically active medium, consideration of the "nite velocity of light, relations with quantum and wave phenomena, all the above numerated requirements make
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this problem as an extremely di$cult one. Preliminary derivation of the dispersion formula indicates that we do not escape the application of nonlinear theoretical models, in which the wave train or the wave package self-acts through the medium similar to that at the derivation of Eq. (9).
4. Quantization of the polarization 5eld For quantization of the supplement "elds emerging at the medium polarization, there may be a standard scheme of quantization of the "eld energy, presented via a square of its electric and magnetic "eld strengths. However, it is di$cult to use the wave train volume to calculate the density of the quantum modes, for such resonant cavity moves together with the train. Also, the linear sizes of a vacuum cavity within the matter, as a rule, are signi"cantly less than the wavelength. Therefore, the quantization procedure will be discussed through the energy transfer within the self-interacting wave train, since the transfer phenomena cause changes of the train photon density and an inherent quantum character of the energy transfer is obeyed by its determination. Emphasizing the vacuum as a speci"c medium with speci"c absolute properties in accordance with Eq. (12) assumes the consideration of a discrete and, in our case, randomly discrete medium. Let us assume that a monochromatic plane wave is propagating within a homogeneous medium with the absolute refraction index m and passing through the cavity (Fig. 2). The cavity sizes are su$ciently small that rays 2 and 4 do not interfere after the di!raction at points 1 and 2, and they can di!ract once again at points 3, 4 on the opposite surface. There is always a probability that two photons simultaneously follow the paths drawn by rays 2 and 4 and then, with a time delay, meet two photons 1 and 3 at points 3, 4. Nothing is changed within the photon beam of the forward propagating radiation (Fig. 2) except that two photons have interexchanged their positions after two symmetrical deviations. However, photons 1 and 3 reach points 3 and 4, in general, with some phase delay in comparison with photons 2 and 4. This delay comprises the contribution from di!erent paths and from di!erent interactions with the molecules. The assumed obligatory interference of these four rays leads to decrease of the intensity of the forward propagating radiation. According to the energy conservation, this energy damage either must be absorbed in optically active mediums, that had been discussed in Ref. [1], or has to be re#ected at points 3 and 4 in a case of transparent mediums, or must be shifted to plane perpendicular to the axis X. Let us temporarily omit the consideration of the latter that yields the breakdown of the symmetrical scheme presented in Fig. 2. Then, let us discuss in detail an act of the simultaneous scattering of two coherent photons on one particle (molecule). Minimum photon number, participating simultaneously at the scattering on one molecule in order to take into account the interference e!ect, must be equal to two. Inferring that the energy of the photon pair has been increased four times after the coherent summing of their amplitudes, paradoxically four photons instead of two arise during this scattering. This paradox, in our opinion, is impossible to explain by the Heisenberg indeterminacy principle, since we deal with a quantum event on one molecule with determinate energies of indivisible particles. There should be, in our opinion, two explanations for this fact, if the molecular absorption is not presented. 1. Since the vacuum polarization for small photon energies is not taken into account and, therefore, the particle transformation is removed from the consideration, in order to save the
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conservation of the particle number, one has to assume that one of these photons is always polarizing in a plane perpendicular to the polarization plane of another photon. The term `polarizationa is distinguished here from the above-mentioned `medium polarizationa. The photon depolarization may be observed at the scattering of the coherent photons and they has been transformed to the incoherent ones through depolarization. However, a second interpretation is needed to escape a situation, when two polarization states have been `occupieda. 2. If for some interference experiment instead of a screen, where the interference picture is "xed, a transparent discrete medium is selected, there is always a probability that after di!raction on medium particles the waves from two interfering beams again go together into a forward propagating plane wave. The new superposition of two beams may be also carried out by means of semitransparent mirrors. The intensity distribution is obtained on a front plane as a result of the interference. This should mean in the quantum language that the existence of two coherent photons is possible due to two neighborhood or spatially anticoherent photons. One should generally expect, that both mechanisms are probable, but only the second interpretation is made use of in the present paper. The 4th photon #uctuation represented in Fig. 2 can be called a structure of the two coherent photons. The forward scattering of two anticoherent photons can be explained as a re#ection of two coherent photons. An intermediate state of the partial coherence due to for two photons at the simultaneous forward scattering is forbidden by the energy conservation law, the quantum hypothesis of the photon indivisibility and the negligibly small Compton's e!ect for heavy molecules. The act of the simultaneous forward scattering on molecules has a special quantum character and the relation between the propagating and re#ecting radiation may be examined as a relationship between a probability of arising of the coherent and anticoherent pairs. If two anticoherent forward propagating photons are perished at one point 3 and 4, two coherent photons must be appear after re#ection with a transfer of the 8th-multiple impulse of one photon to a molecule or molecules. The re#ection amplitude of two coherent photons is two times greater than that of one photon. The four photons 1, 2, 3, 4 after the re#ection can be directed, with the equal probability, along four inverse paths depicted in Fig. 2 and they may be re#ected by the same means with some probability at points 1 and 2. If medium properties lead to a phase delay after elementary interactions of the radiation with molecules, then the phase di!erence between two waves re#ected and main should vanish and the electric "eld should be enforced in the cavity in Fig. 2, and the wave train has been contracted. On the contrary, the advanced phase e!ect enforces the phase di!erence between the two re#ected rays and main and a smaller portion of the radiation is re#ected at points 1 and 2 (Fig. 2). The negative polarization of the medium is observed and, in turn, the wave train should be extended. The probability of the return of the scattered coherent photon to meet other time-coherent photons from the coherent beam is the main idea of the present consideration. The scheme shown in Fig. 2 gives a new explanation of the negative polarization. A choice of the ellipsoid cavity with relationships of the semimajor/semiminor axes as 3 is conditioned by the 2 correspondence of the dispersion formula (14) with those that can be derived in the electrostatics (see, e.g., Ref. [6]). The cavity form is not signi"cant here and can be replaced (Fig. 2) by a density #uctuation. The necessary condition is only a cavity symmetry according to the axis X, that is on average valid in a random medium.
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Since the transit time of the train propagation is a very small value, a device detects an integral signal from the big number of trains. Every photon within the train has a possibility of participating in the construction of the 4th photons scheme as seen in Fig. 2 as in its destruction. The radiation emissions at di!erent points 1 and 2 of the front plane are two independent events in accordance with the Huygens' principle, which is a consequence of the "nite velocity of light. The probability of the scheme construction is proportional to a square of the probability of a photon to be `alivea, namely, &(N/N )2, where N is the initial photon number in the train, The photon 0 0 destruction as a result of the catastrophic absorption or scattering on molecules in the cavity in Fig. 2 is also proportional to (N/N ). The latter breaks down the symmetrical scheme and causes 0 an intensity #uctuation along a front plane perpendicular to the axis X, therefore the absorption probability, as a whole, increases after this elementary absorption act. The absorption and deviation acts are also independent because there is no interference among four photons after the deviation of two photons within the symmetrical scheme. The nonlinear absorption law may be obtained as
A B
dNM NM 3 NM (19) "!a !a NM dx NM NM 0 0 0 that is obligated by arising of a supplement e!ective photon #ux owing to the nonlinear e!ect of the photon delay in Fig. 2. Following from Eq. (18), the supplement density *o of the photon modes arises, so that, *o&u2M2. The averaging over all wave trains in Eq. (19) is based on an assumption that photons from the di!erent trains do not correlate and this is not always valid in comparison with modern representations of the quantum optics [5]. The de"nitions of the photon numbers NM and NM are related to average total values of local and initial photon numbers, 0 respectively, crossing a given cross-section at a point x for a chosen time period. Thus, the two photon interaction in the #ux of the coherent radiation can be statistically considered as an exchange of positions of these photons, that leads to doubling of the amplitude of the electric "eld of the re#ected radiation and explains a logical appearance of factor 4 (A"4) in Eqs. (11). The mechanisms and the medium types to be discussed in the present work may both be useful to specify the physical meanings of some types of absorption and scattering. For in"nitesimal sizes inherent for molecules, the resonant cavity may represented in the form of a classical two-level system. A spontaneous emission may interrupt the two-level exchange of the coherent radiation and cause an emission of a chaotic photon that can be either absorbed in the optically active medium in accordance with the linear law or contribute to the Rayleigh's scattering of the independent molecules. In turn, the exchange of the coherent radiation between the cavity walls in Fig. 2 may be interrupted by a linear absorption or by the Rayleigh's scattering of a photon and it leads either to the nonlinear continuum absorption in the semitransparent mediums or to the opalescense by molecular #uctuations in transparent mediums. The continuum absorption seems to be transformed to the critical opalescense near a critical point of matter, and along with this the number of resonant cavities is dramatically increased near this point. In the terms of intensities, Eq. (19) may be presented in the following di!erential form: dI I3 "!aI!a . dx I2 0
(20)
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The solution of Eq. (20) will be as follows: I exp(!ax) 0 I" . (1#a/2a(1!exp(!2ax)))1@2
(21)
Eq. (21) produced [1] the selective continuum, that would arise by the interruption of the nonlinear e!ect of generating the coherent photons because of the strong linear absorption. The linear absorption coe$cient a approaches zero in the case of a transparent medium and Eq. (21) should be transformed to I 0 . I" (1#ax)1@2
(22)
The absorption coe$cients a and a may have di!erent origins and the approaching of the value a to zero does not mean the same for the coe$cient a. In the above-mentioned expressions, one assumed by default that the nonlinear absorption coe$cient a is always greater than zero. However, the physical meaning of Eq. (21) is lost in the case of big paths of x, where the inverse absorption along opposite direction of the axis X takes place. Thus, the macroscopic irreversibility for the time direction should be found out in Eq. (21). Eq. (22) may be interpreted as a case of emission for su$ciently small negative values of x, and the coe$cient a should be as a source of the backward re#ected radiation. Taking into account indeterminacy for the x interval as well as for change of the impulse p and using the quantum hypothesis by Heisenberg, Eq. (22) may be expressed in the curious form: I 0 I" , (1!*p*x/+)1@2
(23)
that might be useful to understand the increase of quantum #uctuations of the radiation in a cavity. The scheme presented in Fig. 2 leads to an energy #uctuation in a local volume taken from neighborhood volumes. Since the breakdown of either the time irreversibility principle or the casualty principle, both being in the macrocosm, is forbidden, the Heisenberg indeterminacy principle is needed to show a boundary of the microcosm. The breakdown of the irreversibility principle is possible only in the microcosm in the framework of the Heisenberg indeterminacy principle, and, perhaps, Eq. (23) can show a possible way of understanding the limits of the macroand micro- formulations of the casualty principle.
5. Summary and conclusions 1. A fact is represented the causal relationships and events by elements of the geometric optics within scale of less than a wavelength by default assumes the quantum character of the phenomena, accompanying the particle #ux with the inherent wave properties. 2. The refraction index is discussed as a result of the photon correlation within the #ux. Though the present work without a speci"c apparatus of quantum "eld theory is unlikely to be a quantum representation of the refraction index, the simple preliminary results favor this direction. The dispersion as the dependence of the refraction index on the radiation frequency is easily explained
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by the wave properties of the interaction of the plane wave with a single molecule; however, the dispersion relationship (14) may be explained by the quantum representations such as the photon correlation. The main conclusions of this paper are based on the probability of a coherent photon returning to a point of the initial deviation after some time period and meeting a time-coherent photon from the coherent beam or train. 3. The "nite velocity of light or casual relationships brings about nonlinear e!ects of selfinteracting wave trains. The wave train or the wave package changes its volume and hence its photon density at the train moving within random or discrete, on the small distances, medium, following the random or discrete medium #uctuations. The supplement photon #ux arises and moves through the cross-section selected and the nonlinear e!ects is concluded by Eq. (19) of the radiative transfer. 4. These nonlinear e!ects are probably impossible for one spatial dimension, for the probability to transform coherent pairs into anticoherent ones and vice versa must be always provided. Eqs. (14) and (20) themselves could have a more fundamental character than an approximation of power series over the smallness parameter. We explicitly did not use the perturbation theory for their derivation and they could describe integral e!ects of the train self-interacting through the optical medium as a consequence of the three-dimensional space similar to that when the heat radiation is locked in a three-dimensional box. The perturbation theory technique and molecular properties, as a rule, a!ected only the left}right side of Eq. (14) and they were used to study the speci"c molecular properties via susceptibilities and molecular "elds. 5. In connection with the quantum representations, one needs to underline speci"c properties of the forward propagating and backward re#ecting radiation. Let us assume that two time-coherent polarized photons are moving within the transparent medium, comprising relatively heavy molecules. One of them, falling earlier on medium, obtains a random phase after manifold collisions. Let us call this photon a chaotic photon. The chaotic photon has a probability meeting a time-coherent photon on one molecule with a time delay and they both can interfere. Following a hypothesis of two coherent photons, the chaotic photon, on average, does not change the coherence of the forward propagating radiation. The same is related to the chaotic heat photons. Thus, the information about a radiation source to be attributes of the coherent radiation has been saved, as a whole, by the forward propagating radiation and the hypothesis of the coherent photons does not contradict with the superposition principle in the wave mechanics. 6. Controversial de"nition such as the coherent wave train has been used in the present paper. The question is whether this train is as a result of an elementary act of the photon emission on one molecule or whether this is a collective phenomenon? This old problem took its origin from the begin of this century. In the line shape theory, this e!ect was used to present itself as the emission of one molecule. Experiments on the quantum optics tell us that photon is self-interfering and here is no possibility of knowing the source for this photon at this interference. It is logical to suppose that it is impossible do determine as well what is the number of photons and what are the molecules and energy levels to be the photon sources for small time periods. For conclusions of this paper, it is only essential that a probability of producing the coherent wave train must be induced for every photon. The important role of the continuum absorption is not surprising because the continuum concept has a broader interpretation than the continuum absorption of a simple molecule. Thus, any attempt to "nd a new solution of this complex problem even for one molecule must agree
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a number of classical laws and principles and this requirement dictated the subject of the present studies. The existence of the coherent photons is nearly evident, since all classical interference experiments are likely to use similar de"nition for the experiment interpretation on the quantum language. In this sense, this idea seems to be useful to examine once more, in the quantum language, the rotation of the polarization plane in mediums with mirror isomers. A quantum hypothesis to present the scattering as two-fold elementary quantum acts belongs to Pauli. This hypothesis stimulated Einstein and Ehrenfest in creating the heat balance theory of the manifold quantum transitions and led them to the detailed balance principle. A state of the heat balance itself indicates that the simultaneous quantum events would very popular in the microcosm. However, the present hypothesis of the arising of the coherent photons, in our opinion, is not innocent and it could have far-reaching consequences by considering both the heat balance and the interaction of high-energy photons on nucleus. Recently [7], an important applied problem of the excess absorption in the atmosphere had been transformed to a basic problem of the water vapor absorption. Therefore, in order to calculate the transmission function within inhomogeneous medium with water mixing ratios, it is interesting to try to apply the new nonlinear radiative transfer equation (20) that contains quantum absorptionscattering e!ects at short distances and had been performed [1] by parametrization just as a trial on the water vapor absorption.
Acknowledgements This work is based on our studies of the radiation transfer by programming the information system [8]. The work was supported by the grant RFBR No. 96-07-89007, the author deems it his duty to express the appreciative acknowledgments and gratitude to Russian Foundation of the Basic Researches.
References [1] Golovko VF. Dispersion formula and continuous absorption of water vapor. JQSRT 2000 (in press). [2] Feynman RP, Leighton RB, Sands M. The Feynman lectures on physics, Vol. 1. Massachusetts: AddisonWesley Publishing Company, Inc., 1963 [in Russian: Vol. 3, Moscow: Mir, 1966. p. 78]. [3] Born M, Wolf E. Principles of optics, 2nd ed. Oxford: Pergamon Press, 1964 [in Russian: Moscow: Fizmatgiz, 1970. p. 66]. [4] Monin AS, Yaglom AM. Statistical hydromechanics, Vol. 2. Moscow: Fizmatgiz, 1967. p. 547 [in Russian]. [5] London R. The quantum theory of light. Oxford: Clarendon Press, 1973 [in Russian: Moscow: Mir, 1976. p. 530]. [6] Kittel C. Introduction to solid state physics, 4th ed. New York: Wiley [in Russian: Moscow: Fizmatgiz, 1978]. [7] Vogelmann AM, Ramanathan V, Conant WC, Hunter WE. JQSRT 1988;60:231}46. [8] Golovko VF, Nikitin AV, Chursin AA, Tashkun AS, Mikhailenko SN. Developments of the information system, based on large-scale spectroscopic databases, for application in the atmospheric optics and molecular spectroscopy. Report RFBR No. 96-07-89007, January 1999.