Revisited theory of selective reflection from a dilute Fabry-Perot interferometer
aa r X i v : . [ phy s i c s . op ti c s ] A ug Revisited theory of selective reflection from a dilute Fabry-Perot interferometer
Davit Khachatryan
Institute for Physical Research, NAS of Armenia, Ashtarak-2, 0203 [email protected] 17, 2018
Abstract
In this paper we revisited the theory of selective reflectionfrom a dilute vapor cell. A self-consistent theory was devel-oped for reflection spectrum for Fabry-Perot interferometer.Formulas for single and multiple reflections were obtained.We also obtained the effective refractive index for a singleselective reflection. The results of the paper are in a goodagreement with existing experimental results.
Reflection of radiation from the boundary between a di-electric and atomic vapor, when laser field is detuned inthe vicinity of atomic transition frequencies, is termed asselective reflection (SR) [1, 2]. SR has many applications,such as locking a diode laser frequency to atomic resonancelines [3–5], retrieval of group refractive index [6], markingatomic transition resonance lines [7,8], study of the van-der-Waals interaction of atoms with a dielectric surface [9–11],determination of the homogeneous width and the shift ofresonance lines [11–14] and cross-sections of resonant col-lisions [15], study of coherent and magneto-optical pro-cesses [16–21].In the recent decades the theory of SR was investigated bymany authors [2,22–27]. Schuurmans in his paper [2] devel-oped a theory, where he obtained the spectral narrowing of aSR signal. In [22] a theory of frequency modulated SR wasdeveloped, that allows one to obtain Doppler free spectrallines. The problem of different frequency shifts existing ina SR signal was discussed by [23–25] (e. g. caused by local-field correction, atom-wall interaction, non-exponential at-tenuation of the field in the vapor). In [26,27] theories weredeveloped for the dilute thin vapor cells.The sub-Doppler reflection spectrum in a SR signal is dueto atom-wall collisions. After a collision atoms leave thewall in the ground state. This creates a spatial transient re-gion, where the polarization has dependence on the spatialcoordinate. The existence of this transient region was exper-imentally demonstrated by several authors [28,29] by using evanescent wave fluorescence spectrum from the atoms thatare near the dielectric wall.In [30] it was shown that by changing the length of thecell’s highly parallel window one can change the SR sig-nal shape, because of the Fabry-Perot interferometer effect.Also, one can change the SR signal shape by changing thelength of the thin (nanometric) vapor cell [8]. In this paperwe demonstrated that the SR line shape can be changed alsofor a thicker (cm order) cell.This paper is an extension of the theory presented in [6].We developed a self-consistent theory by using the densitymatrix formalism and Maxwell equations. In the calcula-tions we used Laplace transformation that let us obtain for-mulas for single and multiple selective reflection. Also, theeffective complex refractive index for a single selective re-flection was obtained. We assume that atom-wall collisionsare diffusive, that is, after a collision all atoms lose theirpolarization [26, 27]. Since light is directed normally to thecell boundary, we consider the one-dimensional problem.In section 2 we will show some classical formulas for re-flection from a Fabry-Perot interferometer and for the laserfield inside the cell by taking into account the steady-statesolution. Then, in section 3 we will take into account thetransient behavior of polarization of a dilute vapor cell. Wewill obtain new formulas for single and multiple reflectionspectra and discuss the considered approximations. Addi-tionally, we will compare the new formulas with the classi-cal ones. In section 4 we will present the effective refractiveindex and discuss its properties. Finally, in section 5 we willshow the results obtained from our formulas and comparethem with existing experimental results.
In this section we are going to focus mainly on basic con-cepts and classical formulas to be able to compare themwith our obtained results. First of all, we are going tostudy the properties of light-two level vapor interactions.Then we will generalize our results for multilevel systemsin section 5. The cell consists of three layers (glass, vapor,glass). Both boundaries are parallel to each other and light1s directed normally to the first boundary. The constants(e.g. dipole moment, resonance frequency) that we use hereare taken from [31], where one can find physical parametersfor rubidium atoms. The static magnetic field is set to zeroand the cw laser field is considered to be weak to neglect allnonlinear effects associated with it.We describe light-medium interactions by Maxwell equa-tions using the density matrix formalism, d Edx + k E = − πk P,dρdt = − i ~ [ H ρ ] + Λ ,P = N Sp ( dρ ) , (1)where E is the amplitude of the electric field, P is the po-larization of the medium, N is the density of atoms, d is thedipole moment of resonant atoms, H is the Hamiltonian ofthe system, and Λ is the dissipation matrix, which describesall the relaxation processes, as well as the laser radiationlinewidth. In the presence of spatial dispersion P also is afunction of the spatial coordinate (this case we will discussin section 3).From the continuity of the electromagnetic field at the bor-ders of the medium ( x = 0 and x = L ) we have [27], E (0) = E I + E R ,E ′ (0) = ikn ( E I − E R ) ,E ( L ) = E T ,E ′ ( L ) = ikn E T , (2)where E I is the amplitude of the incident light, E R is theamplitude of the reflected light, E T is the amplitude of thetransmitted light, n , are the refractive indexes of the win-dows, k = ω/c is the wave vector, ω is the frequency of thefield and c is the speed of light in vacuum.In our calculations we will use the linear interaction ap-proximation. This statement in density matrix formalismcorresponds to ρ ≈ approximation. Namely, we as-sume that atoms are mainly in the ground state. For ρ component of the density matrix we will have the followingequation, u ∂ρ ∂x = i Ω e − i ( ωt − ϕ ) − (Γ + iω ) ρ , (3)where Ω = | E || d | ~ is the Rabi frequency, Γ = γ/ l +Γ c + ... is the transverse decay rate, γ is the natural decayrate of the excited state, Γ l is the laser spectral width, Γ c isa phenomenological decay rate that models the collisions, u is the velocity of atoms, ω is the resonant frequency and ϕ is the phase of the field.For a very dense vapor, we can assume that the homoge-neous width Γ >> ku T , where ku T is the Doppler widthand u T is the most probable thermal velocity of atoms. Therefore, the u ∂ρ ∂x term in (3) can be neglected [26]. So,from equations (1), (3) one can obtain classical formulas forsusceptibility χ and refractive index n : χ = iq Γ − i ∆ ,n = p πχ, (4)where ∆ = ω − ω is the detuning from the resonance fre-quency ω , q = N | d | ~ is a parameter of the medium.In order to take into account Doppler shift for a dilute gasin conventional theory one assumes that Doppler shift canbe accounted for by simply replacing ∆ in (4) by ∆ − ku T [26]. In this manner, we will obtain the steady-state solu-tion. By using equations (1), (2) and (4) we can derive wellknown formulas for the field inside the medium and the re-flection coefficient for Fabry-Perot interferometer, E ( x ) = E I Ae − iknx + E I Be iknx , (5)where A = e R − r n ( n + n ) ,B = 1 − e Rr n ( n + n ) , e R = r − r e iknL − r r e iknL ,r , = n , − nn , + n , (6)where L is the length of the medium, R = (cid:12)(cid:12)(cid:12) e R (cid:12)(cid:12)(cid:12) is the re-flection coefficient for the Fabry-Perot interferometer. If weset n = n = n (and, therefore, r = r = r ) we willobtain the following well-known classical formula for thereflection coefficient: R = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r − re iknL − r e iknL (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (7)The formulas presented in (6) and (7) are well known fromclassical theory of the Fabry-Perot interferometer [32]. It iseasy to show, that when x → L the first term in (5) is ex-ponentially increasing ( Im ( n ) > ), while the second oneis exponentially decreasing. However, the increasing termis not a problem. In order to show that we should repre-sent the first term in the following way by substituting theexpression of e R into the expression of A from (6): E I Ae − iknx ∝ e ikn ( L − x/ . (8)As can be seen from (8), the first term of (5) is exponen-tially decreasing when L → ∞ . For long enough cell from26) we will have e R = r and, consequently, A = 0 . There-fore, for such cells we can neglect the first term.For a more rigorous solution of (3) one should not neglect u ∂ρ ∂x and find a solution that takes into account the tran-sient behavior of the medium. This problem will be solvedin the next section and we will compare our obtained for-mula for the reflection coefficient (22) with the classical one(7). Also, we will show that the first term of our approxi-mate solution (20) has the same property as mentioned in(8). Here we assume that the atoms are loosing their polariza-tion after atom-wall collisions. In other words, atoms areleaving the wall in the ground state. This assumption canbe described by boundary conditions for the medium polar-ization (see for example [26, 27]): P ( x = 0 , u >
0) = 0 , P ( x = L, u <
0) = 0 . (9)The effect of loosing the polarization gives rise to the spa-tial dispersion. That is to say, the susceptibility dependsalso on the spatial coordinate. From equations (1) and (9)one can derive P ( x ) = Z x E ( y ) h χ ( x − y ) i u> dy + Z xL E ( y ) h χ ( x − y ) i u< dy, (10)where χ ( x ) = iqu e − Γ − i ∆ u x is the linear susceptibility for themedium with a spatial dispersion and h ... i u denotes averag-ing over velocities (Here we assume Maxwellian distribu-tion).It’s hard to solve the differential equation in (1) with thepolarization from (10). First of all let’s find the asymptoticsolution when x → L . In this solution we will neglect thesecond integral, because it vanishes, when x → L . Theresulting differential equation will be d E ( x ) dx + k E ( x ) = − πk Z x E ( y ) h χ ( x − y ) i u> dy. (11)After a Laplace transformation we will obtain e E ( s ) = sE (0) + E ′ (0) s + k (1 + 4 π e χ ( s ) u> ) , (12)where e χ is the Laplace transform of h χ ( x − y ) i u> , e E ( s ) isthe Laplace transform of E ( x ) and s is the Laplace variable.For two level system the explicit form of e χ is e χ ( s ) = h iq Γ + su − i ∆ i u> . (13) Here we will do the following approximation. We willrepresent denominator of (12) in this way s + k (1 + 4 π e χ ( s ) u> ) = ( s − s )( s − s ) . (14)where s and s are given by the following iteration proce-dure: s ( n )1 = − ik (1 + 2 π e χ ( s n − )) = − ikn ( s n − ) ,s ( n )2 = ik (1 + 2 π e χ ( s n − ))) = ikn ( s n − ) ,s (0)1 = − ik, s (0)2 = ik, (15)here n is the iteration step. With each iteration step wewill obtain more precise values for s and s . This itera-tion procedure works when π (cid:12)(cid:12)e χ ( s , ) u> (cid:12)(cid:12) ≪ . Note that s (0)1 and s (0)2 are the roots of the denominator of (12), when χ ( s ) u> = 0 . If π (cid:12)(cid:12) χ ( s , ) u> (cid:12)(cid:12) ≪ , the iteration pro-cedure will give the approximate roots of the denominatorof (12). Convergence of the iteration procedure is demon-strated in Fig.1. R e ( s ) Real part of s -9-8.5-8-7.5 I m ( s ) Imaginery part of s -1 -0.5 0 0.5 1
Detuning (GHz) -2-1.5-1-0.50 R e ( s ) Real part of s -1 -0.5 0 0.5 1
Detuning (GHz) I m ( s ) Imaginery part of s b)a)c) d)
Figure 1: Convergence of the iteration procedure. a) andb) correspond to real and imaginary parts of s and c) andd) correspond to real and imaginary parts of s . In all fourplots the red solid lines correspond to 10th iteration from(15), blue solid horizontal lines correspond to n = 0 itera-tion step, and the rest blue solid lines correspond to n = 1 , iterations. The density of atoms is N = 10 cm − and Γ c = 2 π · M Hz .3rom Fig.1 one can see that only three iterations areneeded in order to find s and s . We should note that fordensities N > cm − the iteration procedure doesn’twork, because π (cid:12)(cid:12)e χ ( s , ) u> (cid:12)(cid:12) ≪ is not true for thesedensities. The dependence of π (cid:12)(cid:12)e χ ( s , )) u> (cid:12)(cid:12) on the den-sity of atoms is shown in Fig.2. | ( s ) | Iteration approximation
Density 10 (cm -3 ) | ( s ) | b)a) Figure 2: Dependence of π e χ ( s k )) u> (where k = 1 , cor-responds respectfully to sub-figures a) and b) on the densityof atoms. The red lines correspond to ∆ = 0 , the yellowlines to ∆ = 0 . GHz and the blue ones to ∆ = − . GHz .So, by taking into account (14) we can rewrite (12) as fol-lows: e E ( s ) = sE (0) + E ′ (0)( s − s )( s − s ) . (16)After an inverse Laplace transformation from (16) one canobtain an expression for E ( x ) , E ( x ) = s E (0) + E ′ (0) s − s e s x − s E (0) + E ′ (0) s − s e s x . (17)Here we should remember that (17) is an approximativesolution, hence, it is correct only for x → L . With the useof (17) we can find E ( L ) and E’(L): E ( L ) = s E (0) + E ′ (0) s − s e s L − s E (0) + E ′ (0) s − s e s L ,E ′ ( L ) = s E (0) + s E ′ (0) s − s e s L − s E (0) + s E ′ (0) s − s e s L . (18)From (18) together with the conditions from (2) one canobtain an expression for e R = E R /E I : e R = r ( s ) − Dr ( s ) e φ − Dr ( s ) r ( s ) e φ ,D = ( n + n ( s ))( n + n ( s )( n + n ( s ))( n + n ( s )) ,r l ( s m ) = n l − n ( s m ) n l + n ( s m ) ,φ = 2 ikn avg L, (19)where l, m = 1 , are integers and n avg = ( n ( s ) + n ( s )) / .In (17) one can notice two exponential expressions, oneof which is increasing ( Re ( s ) > ) and the other one isdecreasing ( Re ( s ) < ), when x → ∞ . So, a naturalquestion arises: when L → ∞ and, consequently, x canincrease to infinity, can the first term become infinite? Ifso, this solution will be unphysical (the field should be zeroat infinity). To show that there is no problem with the firstterm, we will rewrite (17) by using the conditions from (2)in the following way: E ( x ) = E I Ae s x + E I Be s x ,A = e R − r ( s )( n ( s ) + n ( s ))( n + n ( s )) ,B = 1 − e Rr ( s )( n ( s ) + n ( s ))( n + n ( s )) . (20)By substituting the expression of e R from (19) to the firstterm of (20) we will obtain E I Ae s x ∝ e s L − s ( L − x ) . (21)From (21) one can notice that when L → ∞ that expressiontends to zero for every x ∈ (0 , L ) . So, when we have onlythe first border (or the second border is far enough) we canneglect the exponentially increasing term in (20). There-fore, here, like in (5), there are no problems with infinities.If we set n = n = n in the expression for e R from (19)we can obtain a more simpler expression for the reflectioncoefficient R = (cid:12)(cid:12)(cid:12) e R (cid:12)(cid:12)(cid:12) : R = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r ( s ) − Dr ( s ) e φ − Dr ( s ) e φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (22)4n section 4 this result will be compared with the classicalformula from (7). Also, in the same section we will discussphysical meaning of n ( s ) and n avg . Diffusive atom collisions with the wall create a transientspatial region near the boundaries of the cell. In this tran-sient region we do not have one uniform vapor, thus, wecan not describe our medium with the linear refractive in-dex [33]. From (10) one observes nonlocal dependence ofthe polarization on the spatial coordinate x . Thus, the re-fractive index should have dependence on x . Although, itis hard to derive the expression of the refractive index forthis kind of medium, we are able to attribute the concept ofeffective refractive index to the medium.First of all, let us compare our obtained formula from (22)with the classical formula for the reflection coefficient pre-sented in (7). In our formula, instead of r we have r ( s ) and r ( s ) . Also, we have an additional term D . Actually D like all terms of (22) has a dependence on the detuning ∆ , but it is always close to one and can be neglected. Fi-nally, notice that the refractive index n in (7) and n avg from(22) are both in the phases of exponents in the correspond-ing formulas. So, n avg in (22) plays the same role as n in(7). This comparison can lead to an assumption that n avg can play the role of the refractive index in our medium, butit can not be regarded as actual refractive index, because itdoesn’t depend on the spatial coordinate (the refractive in-dex of the medium with the spatial dispersion should havea dependence on x ). The real and imaginary parts of n avg are presented in Fig.3.As you can see from Fig.3 the curve of the real part of n avg has a dispersive profile and the curve of the imaginary partof n avg has a absorptive profile, like the refractive index inthe conventional theory [34]. This is another argument that n avg has a physical meaning of the refractive index insidethe medium.From (22) we can derive another interesting formula, if weassume that the cell is long enough ( L → ∞ ). Notice that e φ → , when L → ∞ ( Im ( n avg ) > ). Hence, we willobtain the following formula: R s = (cid:12)(cid:12) r ( s ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) n − n ( s ) n + n ( s ) (cid:12)(cid:12)(cid:12)(cid:12) . (23)In this formula only the first reflection from the cell istaken into account, thus, it represents the formula for thesingle selective reflection. Notice that (23) is similar to theFresnel equation for normal incidence. Therefore, we cansay that n ( s ) plays the role of the refractive index. So,we will call n ( s ) the effective complex refractive index fora single reflection. Also, by analogy to the Fresnel equa-tion, we will attribute Im ( n ( s )) as the effective absorp- The real and imaginary parts of n avg R e ( n a vg ) -1 -0.5 0 0.5 1 Detuning (GHz) I m ( n a vg ) - b)a) Figure 3: The spectra of a) real and b) imaginary parts of n avg . The density of atoms is N = 4 · cm − .tion, and Re ( n ( s )) as the effective real refractive index.Moreover, n ( s ) can also be referred to as the surface ad-mittance M = E ′ (0) / ( ikE (0)) as defined in [2]. In Fig.4we show the dependence of Im ( n ( s )) and Re ( n ( s )) onthe detuning from the resonance line.In the effective absorption curve in Fig.4 b one can see thatthe absorption curve is red-shifted. This means that theatoms with positive velocities (their Doppler shift is red-shifted) have dominant contribution in the absorption curve.It is interesting to recall the experimental results obtained byBurgmans et al. [28], where the authors observed fluores-cence radiation from the transient region of the Na vapor.They showed that the spectrum of fluorescence of atomsnear the wall has a decrease in the blue-shifted sides of res-onance lines. The reason is the possibility for the atoms thathave a polarization and are moving towards the boundary tolose their polarization non-radiatively by quenching to thewall. In Fig.4 one can notice that the effective absorptioncurve is different from the absorption curve described bythe conventional theory [34]. The difference, like in [28]experiment, is in the blue-shifted side from the resonanceline. So, in this sense, we can claim that our result is con-sistent with that experiment.5 .9980.9991.0001.001 R e ( n ( s )) The real and imaginary parts of n(s ) -1 -0.5 0 0.5 1 Detuning (GHz) I m ( n ( s )) - a)b) Figure 4: The spectra for a) the effective real refractive in-dex and b) the effective absorption. The density of atoms is N = 4 · cm − . In Fig.5 we show the dispersion of the single selective re-flection calculated from (23). -1 -0.5 0 0.5 1Detuning (GHz)6.686.746.86.86 R ( % ) Selective reflection
Figure 5: The selective reflection spectrum in the vicinity ofthe resonance line. The density of atoms is N = 10 cm − .The result presented in Fig.5 is similar to the well knownprofile of the selective reflection presented, for example, infigure 8 from [35]. Of course, it is a theoretical simplifi-cation to assume that we have only a single resonance line. In real experiments it is useful to consider multilevel sys-tems. To generalize our theory to the multilevel system weneed to make an assumption that the light-medium interac-tion is linear. In this case we should leave the same all thecalculations mentioned above, but we need to change theexpression for e χ ( s ) presented in (13). So, for example, thesusceptibility of D line of rubidium vapor will be as fol-lowing: e χ ( s ) = X k =1 h iq k su + Γ k − i ∆ k i u> , (24)where q k = N | d k | / ~ is a parameter, d k are the dipolemoments, ∆ k = ω − ω k are the detunings from the cor-responding resonance frequencies ω k , Γ k are the homoge-neous widths of the corresponding F → F’ hyperfine transi-tions for rubidium atomic vapor. The calculated spectrumof the rubidium D line is shown in Fig.6. Detuning (GHz) R ( % ) Selective reflection
Figure 6: The selective reflection spectra from all Rb ’sD lines (including both Rb and Rb ). The density ofatoms is N = 10 cm − .In Fig.6 we assume natural abundance for rubidium atoms( . Rb and . Rb ). Here we also should men-tion that in (24) we don’t take into account the depopulationof levels by assuming that it has a small effect on our model.The result can be compared to the experimental results ob-tained by Wang et al. [13] and Badalyan et al. [36] presentedin the corresponding figures for rubidium D lines. Ourresult is consistent with the experimental curves presentedthere, although in these two papers the order of the densityof the vapor is N ≈ cm − .Another interesting spectrum can be obtained by increas-ing the scale of detuning for the selective reflection. In thiscase we can see from Fig.7 that we have oscillations in the”wings” of the resonance line.In Fig.7 the selective reflection profile presented in Fig.5also exists. Selective reflection profile is ”hidden” in the6 Detuning (GHz) R ( % ) Reflection
Figure 7: The multiple reflection spectrum from the Fabry-Perot interferometer in the far wings of the resonance line.In the insertion we show the zoomed image of the horizontalline from the reflection spectrum. The density of atoms is N = 3 · cm − .region where we have a horizontal line in Fig.7. To see thisone should zoom in this region and a picture like in Fig.5will emerge (see the zoomed image). Fig.7 is interesting,because from this spectrum it is easy to straightforwardlyobtain the group refractive index as presented in [6].In regions near the resonance in Figs.5, 6 and in the hori-zontal region of Fig.7 we have only a single reflection fromthe first boundary. For the multiple reflection spectrum thelight should be able to reach the second boundary, to reflectfrom it and, finally, to be able to reach and to pass the firstboundary of the cell. If this doesn’t happen, because the ab-sorption is high in the vicinity of the resonance line, therewill be only a single reflection. To see a multiple reflectionone should either decrease the length of the cell, or decreasethe density of the vapor. In Fig.8 we show the selective re-flection from the cell with length L ≈ . cm and density N = 10 cm − , calculated from (22).In Fig.8 one can see different profiles of the selective re-flection that correspond to different lengths of the medium.The differences arise from the fact that reflection is not gen-erated from a single reflection, but from multiple reflectionsfrom the cell. Consequently, we will have interference pic-ture from all those reflections. When we vary the lengthof the cell the interference pattern changes and the profileof reflection also changes by its shape and amplitude. Weshould note here that after adding . λ to L the same pic-ture will be obtained. So, the pattern repeats itself afterevery . λ . This repeating pattern is well known from theliterature [32]. R ( % ) L = 0.5 + 0.1 cm
L = 0.5 + 0.2 cm -1 -0.5 0 0.5 1
Detuning (GHz) R ( % ) L = 0.5 + 0.3 cm -1 -0.5 0 0.5 1
Detuning (GHz)
L = 0.5 + 0.4 cm c)a) b)d)
Figure 8: The selective reflection spectrum for different celllengths. The subfigures a-d are the spectra from the cellswith length varying from L ≈ . . λ cm to L ≈ . . λ cm . The density of atoms is N = 10 cm − . In this paper we developed a self-consistent theory for theselective reflection from a dilute vapor. We obtained for-mulas for single and multiple reflections. We obtained thespectrum for the effective refractive index that is also knownas the surface admittance [2]. We can say that our results arewell consistent with the above mentioned experimental re-sults. Here we should notice that our theory is developed fora dilute vapor. When the medium is dense we should takeinto account effects like radiation trapping, the Dicke nar-rowing [37] and etc. Also, in our model we did not take intoaccount various shifts that are present in the selective reflec-tion spectrum (e.g. caused by local-field correction, atom-wall interaction, non-exponential attenuation of the field inthe vapor [25]).Our developed model can be applied in a wide range ofphysical problems. For example, it can be used for determi-nation of the density of unwanted (or in other cases desired)vapors by the selective reflection spectrum. If unwantedatoms have high enough density, the signal from Fabry-Perot interferometer will be a single selective reflection thatcan be compared with the spectrum calculated from (23)and used to find the density of unwanted atoms. In the casewhen Fabry-Perot interferometer has a small length and the7ensity of atoms is low enough that there will be multiplereflections from the boundaries of the cell, (22) can be used.Also, with this technique one can change the spectral profileof radiation by manipulating the length and the density ofthe dilute vapor cell, as presented in Fig.8. This theory canbe generalized for multiphoton interactions that will pro-vide an opportunity to address problems like, for example,selective reflection in EIT configuration [38].
The author is grateful to Gayane Grigoryan for stimulatingdiscussions and valuable advices.
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