* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Multiply scattered waves through a spatially random medium
Fluorescence correlation spectroscopy wikipedia , lookup
Surface plasmon resonance microscopy wikipedia , lookup
Anti-reflective coating wikipedia , lookup
Optical rogue waves wikipedia , lookup
Retroreflector wikipedia , lookup
Harold Hopkins (physicist) wikipedia , lookup
Vibrational analysis with scanning probe microscopy wikipedia , lookup
Ultraviolet–visible spectroscopy wikipedia , lookup
Optical coherence tomography wikipedia , lookup
Thomas Young (scientist) wikipedia , lookup
Sir George Stokes, 1st Baronet wikipedia , lookup
Resonance Raman spectroscopy wikipedia , lookup
Ellipsometry wikipedia , lookup
Magnetic circular dichroism wikipedia , lookup
Atmospheric optics wikipedia , lookup
Birefringence wikipedia , lookup
Rutherford backscattering spectrometry wikipedia , lookup
Multiply scattered waves through a spatially random medium : entropy production and depolarization Dominique Bicout, C. Brosseau To cite this version: Dominique Bicout, C. Brosseau. Multiply scattered waves through a spatially random medium : entropy production and depolarization. Journal de Physique I, EDP Sciences, 1992, 2 (11), pp.2047-2063. <10.1051/jp1:1992266>. <jpa-00246685> HAL Id: jpa-00246685 https://hal.archives-ouvertes.fr/jpa-00246685 Submitted on 1 Jan 1992 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. J. Phys. f France (1992) 2 2047-2063 1992, NOVEMBER 2047 PAGE Classification Physics Abstracts 05.701 42.20G 05.40 Multiply scattered entropy production Dominique (') (2) (Received par February J2 Dans interaction d'entropie Joseph avec incident, in Magnet Hochfeld ddduisons nous 87, CNRS, Labor, B-P-166, les forrne July de Par la France J992) phdnom~nes de polarisation pur de ddsordonn£. la 3J Cedex, Saint-Martin-d'Hkres 38402 final form d'£tat diffusant milieu un B-P. considbre on planes maximum, Fourier, accepted Etude, cette : France J992, d'ondes faisceau medium random (2) Brosseau Festkorperforschung, fur 09, Universit6 R4sum4. d'un Cedex Christian and Institut Grenoble CERMO, and (I) Bicout Planck Max 38042 through a spatially depolarization waves des d6polarisation d'6tat et arguments matdce de de et de d£coh6rence arbitraire de sym£trie et Mueller cohdrence, principe un caract6risant Ie milieu qui est en accord le calcul explicite base sur I'£quation de Bethe-Salpeter trait6e avec dans l'approximation de la diffusion. Le r£sultat principal exprime les degr6s de polarisation et de coherence spatiale en fonction du nombre de diffusions. Deux forts saillants h Le sont noter. exponentieIIe'de la production d'entropie due h l'irrdversibilitd du premier exprime la d6croissance diffusions. indique que la de ddpolarisation, fonction du nombre de Le second processus en d6polarisation complkte d'un incident lin6airement faisceau polarisd diffuseurs de type par des Rayleigh ndcessite davantage de diffusions (facteur 2) que pour une polarisation circulaire. diffusant This Abstract. of state paper deals and of polarization changes randomly deduce the with the arbitrary position. Using depolarization state of decoherence and by coherence effects linear a of an scattering incident medium pure which entropy principle we is scattering medium which consistent with the explicit computation done in the of the Bethe-Salpeter equation handled context diffusion in the approximation. The main result the output degree of polarization and expresses degree of spatial function of coherence of the number scattering From these results, events. as a main conclusions be drawn. The first is that the entropy production per scatteRng due to two can the irreversible of depolarization is an exponentially decreasing function of the number of process scattering second result obtained is that full depolarization of linearly polarized light The events. by Rayleigh requires more scattering events (typically a factor-of-2) than are required for scatterers circularly polarized lightwave. a 1. general with form of symmetry Mueller the matrix arguments and describing a maximum the Introduction. Studies of localization phere [3], In this scattering interest, research [1-2]), and respect, of in waves connection propagation numerous the optical others subject of by with of inhomogeneous condensed electromagnetic media matter waves currently attracting a wide are physics (e.g. weak Anderson under the sea or in the atmos- (e.g, image reconstruction information [4], retrieval [5]). polarization correlations if of pRmary importance. Multiple fields 2048 JOURNAL PHYSIQUE DE N° I I1 inhomogeneities in optically dense randomizes media the of scattering of light from state effect evidenced (both polarization, however the polarization that has been recently memory confirmed experimentally [8] soon afterwards) theoretically [6-7] and that suggests some of polarization incident irretrievably information about the of the beam is lost by state not characterize depolarization of optical multiple scattering. A point of further interest is to the propagating through random multiple scattering medium. propagating in such A a wave waves rapidly depolarized the characteristic medium but knows little about spatial becomes one a proceeds. This question is important as regards the validation of the scale on which this process theory [2] to describe the transport properties of multiply scattered light. This scalar diffusion of depolarization relation mechanism and its scattering raises also the question of the to the depolarization, it is useful to consider two main approaches. understand medium. In trying to absorption of polarization states is the first approach [9]. The second approach The selective induces depolarization by decorrelation of the phases and amplitudes total flux and preserves field [9] it is essentially of the electric entropic effect arising from the components an of the polarization evolution during scattering. The present paper is mostly irreversible state latter approach and we will argue that depolarization of light by multiple devoted this to scattering is connected to a process of entropy production which falls off exponentially with The key observation underlying the work reported here is that of scattering the number events. this result derived be can from maximum a investigate principle. entropy scattering of not inquire into spatially random the electromagnetic of pure of own state waves polarization and arbitrary state of from a coherence medium fluctuates which in space. Example of such medium would be a collection of randomly oriented scattering particles suspended in a liquid. The purpose of the present work is to calculate the dependence of the polarization and coherence characteristics (Stokes vector, degree of polarization and degree of spatial coherence) upon the number of scattering events. Besides, we will briefly consider an application of this formalism technique of photon-correlation known to a spectroscopy as diffusing wave (DWS) [10] which is relevant to probe the nature of dynamic spectroscopy within correlations dense dielectric random media (e,g, colloidal suspensions). Our emphasize We propagation on al. [I I ], To this on here is of of the different of : us the defined field intensity following than that the is both simply Von the the Neumann radiation spectral entropy energy of trace is been has backscattering derived approach Such an Wolf [12-13]. transport and coherency the analytically entropy of treated [I]. scatterers coherence. the light arising by Wolf et polarized light from of spectrum in the which largely by initiated been characterize to in the Rayleigh theory of partial has change medium localization distributed [9]. Note nature the weak the use processes field entropy polarization we permits matrix a randomly problem, approach electromagnetic will we effects the stochastic of radiation that light through containing treat theory to here of neither medium a is interest with related which from we Planck's based This the entropy matrix on the coherency of while the the degree to the shall be concemed formula [14]. of organized as follows. Section 2 is devoted formulation paper to the of the outlines problem and the elementary of which the problems of concepts in terms depolarization and decoherence will be approached throughout the paper. Section 3 outlines the principle of our entropic approach. Then, section 4 investigates the production entropy associated with multiple scattering of light using the Bethe-Salpeter equation and deals with dependence of polarization the characteristics (I.e. degrees of polarization and of spatial light-intensity autocorrelation function) with the number of scattering coherence, Some events. Freund's about prediction [5] that the Stokes of incident of comments vector state an pure polarization can be fully after reconstructed total depolarization by multiple scattering are discussed section 5. Finally, in several concluding remarks presented in section 6. are The remainder of the is DEPOLARIZATION N° II 2. Preliminary Let us for the OF LIGHT MULTIPLE BY 2049 SCATTERING considerations. begin with a brief development of outline assumptions main of the background and required information theory. the Ej(r, orthogonal of the t) Q two components x, y) the plane perpendicular to the direction of propagation (of unit ez), r is a position vector of a typical point in space and t is the time. Then, the vector electric writes : E E~ e~ + E~ e~, where e~ and e~ are orthogonal unit vectors transverse vector triad in said plane (e~ ez (i, j 0, e~ e~ x, y)) with (e~ e~, ez) forming a right-handed ~~ quasimonochromatic of real field is of Suppose next that this plane-wave vectors. narrow homogeneous that the field is statistically spectral range centered around assume wo. We also spatial We characterize the second-order and stationary, at least in the wide sense. may coherence properties of the field by its cross-spectral density tensor W (also termed the (E, (rj E~*(r~)) at two coherency matrix) [12] of components : Wij(rj, r~) r~) W~j(rj points whose location is specified by position vectors rj and r~ the asterisk denoting complex conjugation. the taken the ensemble that The angular brackets denote average over and characterizes the statistical properties of the incident field. By definition W is Hermitian recall basic polarization properties of the non-negative definite. For later purpose, we some light is given by field. polarization P of The degree of the : 2.I THE INCIDENT electric Let FIELD. of the vector incident in wave = a = = = = = = P as function a from of the rotational two (S~) with: W = of W. «~ is two-level The and wave write to = choose we the of axes (lExl~+ lEyl~), (Sol (Sil (S~) = four These discussed parameters form length references at interpretation in in Z (Sk) + of terms similarities and coordinates rather spatial. than under Note 3 Stokes parameters indicating tion of the the is degree Stokes : to ; £ P which parameters of Barakat that an (s~) [16]. Stokes the altemative then is the It also Stokes in bear should E~* j - interesting with of P in terms E~ E~) parameters mind be parameters formulation (E~ Efl are physical emphasized that their = x, y whereas comments respect of the on to the time average 1/2 2 wavefield given by I vector. to operation E~ [16] made It the is e~, = useful intensities ~~0) = and (S~) Stokes I x it is conjugation the considers one k=1~ e~ , combinations when along E~ El ) + of the 4 transformation. differences (E~ El however invariant are (2) "k system components linear (So), (Sj), (S~), are (53) changes sign by this which i [12, 15] the matrices polarized (lExl~- lEyl~), = = obtained are Pauli ~ "o coordinate Cartesian our the : ~= If parameters for decomposition [9] of any partially completely polarized wave a j j (Sol W (1) ' Stokes average usual notation the where («o being the 2 x 2 unit matrix). The light into a completely unpolarized independent of each other allows us 1'2 (w))2 (tr invariants (W«~), tr (W ) dot 4 (1 = ordered. so-called viewed be may A as well-known Poincar£ sphere an order parameter geometric representa3( [9, 12] of radius JOURNAL 2050 SO(3) (I.e. P ~~~~. (So) similarity to possesses two the inside the the and Further, of one : = the case This derived and I, s(x) OS that with here other (52) (q~( ( (So) <P extreme w + ~~~ (Si) ) ~~~ ~ The I. ~~~ , extreme I q~ case usually is W a = when case spatial represents taken be to Von the simple analytical expression [9] q~ 0 = coherence Neumann when tr (W) measure I which is = : (s(P )), In = with (x) s = (I i x)~ + x)~ (l -x (4) decomposition and was above-mentioned two-level decomposition theorem [9]. For x varying between 0 1/2 and I, s(x) is a bijective strictly increasing values between function : its is noted Two remarks importance. of First, the in (4) s~ '. entropy are takes exists and depends only both on linear (P is expression is reminiscent the to from equation (2) via the spectral inverse S polarization of : 1+x S(P axes the convexity [9] bears of property states on some analysis of spin1/2 [17]. The of set states convex 2 corresponding to mixed (partially polarized) located states (totally polarized) on the rank I corresponding to pure states of the coherence the complex degree of spatial components the It takes considered be to of ; degree the with space components. (W)). In where ~~/~~o~~i/2 (w~~(o))~'~ associated (W polarization dimensional three ii the ensures field in W~(r) incoherence entropy tr rank = (3) in parameter) N° I based in expression the normalization spatial represents between the two The of other qxy(r) S(W) states introduce we E~ and E~ [12] by the order description strata El ball the encountered that 3(. surface of set geometrical This P. is this of symmetry PHYSIQUE DE I # ) and P not S S S (P S the on detailed state polarizations. circular and of polarization Second, for : instance, satisfies it it is the the same for inequalities: 0 ). = We will medium is linear, that the scattering assume occupying a finite volume il (of arbitrary shape and orientation) fluctuations in free and that the spatial dielectric susceptibility q~~ (r) tensor are of its space, statistically homogeneous and stationary (at least in the wide sense). We restrict our treatment considering a non-absorbing medium. The effect due to a weak absorption can be easily to formalism introduced into our and will be considered in a subsequent work. Typical realization of of such medium would be a collection pointlike scattering discrete sizes is whose centers small compared to the wavelength. We also the time fluctuations of that the scatterers assume sufficiently slow relative to the period of the field oscillations that the scattering medium are so essentially time-invariant approximation). The usual boundary behaves (I,e, adiabatic as it is conditions require continuity of the magnetic field H and tangential electric field at every discontinuity surface. From assumptions, characterize dielectric susceptibility of the the above the we may (r) being medium by q~~ (r) the Kronecker symbol), of white-noise and zero mean q (&~~ 2.2 THE SCATTERING isotropic, and MEDIUM. non-magnetic &~~ = correlation function : (R (ri) R (r2)) = u 3 (ri r2) 0 when rj , where u is a constant and 3 (r) the Dirac delta function. e i2, r~ e i2 ~~~ otherwise N° DEPOLARIZATION II LIGHT OF assumption is made regarding the thermodynamic disorder (far from the Anderson consider only weak No will ~ fm path free ( is larger much than of state the transition) wavelength the 2051 SCATTERING MULTIPLE BY of scattering medium. that elastic such the mean kof (I,e, radiation the We ml), uko ko w = light o/c is in vacuo diffusion number wave vacuum : process of the fluctuations 2.3 the consequently, [2]. Finally, SYMMETRY are RELATIONSHIPS frequency the c being the speed that described by a may be medium fluctuations of the the statistically independent. SCATTERING OF FOR wo, propagation assume we field incident with associated wavefield the by related matrix = linear a the and input The LIGHT. POLARIzED medium S~ are vector S~ and output Stokes vector to the scattering the form : S~ MS,, where M is the 4 x 4 Mueller real-valued of classical Stokes of relation characterizing the scattering. A number of restrictions the form of the M-matrix placed at the outset are on depending upon the symmetry and reciprocity requirements. Perrin [19] was the first to give a detailed study of the of independent number of matrix elements parameters (among the sixteen M) which are for specifying the polarization of characteristics light scattered by necessary an arbitrary medium. Following the analysis made by Perrin, the forward (or backward) axial scattering by a symmetrical medium (e,g, identical particles having spherical symmetry) involves coefficients only three Mueller matrix is diagonal. We note in passing the moreover ; further for a variety of symmetry that refinements properties of the matrices M describing the of polarized radiation by a slab of randomly reflection and transmission oriented particles have Hovenier [21]. In a statistically isotropic threebeen treated by Van de Hulst [20] and later by medium, dimensional dimensions, in 3. In section, the symmetry Consequently proved by Freund arguments the elements the presented in general form of is In two- not diagonal the multiple [22]. of production per AS (n S = section and the / '~~ scattering proceed 0 0 reads + matrix M as fact that as view In multiple written be can follows. of scattering : ~~ 0 (6) /~ 0 (P (n describing matrix We Mueller the l Mueller entropy. previous the 0 entropy the maximum of argument an M= The direction. matrix Mueller the ~. as propagation the around symmetry C~ to diagonal evaluate by dissipative, non reduced principle. we medium C~~ full is is block but entropy Maximum this there symmetry dimensions two scattering is the as : i(n) S(P (n)) )) I (7) = , degree of polarization after n + I scattering events, S(P ) being given particular state of indicates that ( depends the by equation (4) and the superscript on production after n + 2 scatterings is given by : polarization. Then the total entropy where P (n ) denotes the ' S (P (n AS = + I )) jj S(P (0)) IQ = = )~~ ~~)~~ In ~ j The this function is to ( (x) expected is taken from to the be a theory monotonically of irreversible l~ (8) + =o decreasing function from ( (0 thermodynamics [18]. It can down be to shown zero that ; candidate the ("(x)~0 where postulate ((x) (P (n exp(- ~k = + ) xx) = ~~~~~ expression equivalent An ~~~ get we ~ ~° It x. work which the is necessary to function the out polarized linearly is 11 condition I-i)tf)~ ~ ~~~ to to requirements. of polarization state pure equation (8), respect chosen satisfy and L space with have we above the meets From differentiation N° I metric the to ((x): of incident an mjj (n)). indicates form which consider we I prime the belong also must functional the Next, h(x) function PHYSIQUE DE JOURNAL 2052 ~~~ is n-1 mu(n) where B have we I = i~~~ (lo) , = ~k) and be can fully B is exp(- A B that the to s- = convenience notational for set important fact S(P (I ))/In (2) : consequently attention AJ by determined exp(-x). B = written We call equation(7) from double scattering. require that A as Moreover i when maximum entropy (2) (1 exp(- In ~k = (I,e, in the achieved is x)). The final result mj j (n This equation parameter B. It allows is successive the worth noting circularly method us what ((x) polarized (I,e. P (n + I ) is quite general and may be functions ( (x) kind of trial that satisfies physical the Light depolarization approach 4. = for used are be to we ~ = ~ i.e, ; iteration with a to will change involved more used. (l ~) formula same m~~(n)) tx~), - (2~~~ of orders n : = This expressed apply for a be in the value can of the terms pure of B. matrix, Mueller simulation in state which Note that but it does incorporate any not I sole is this tell function constraints. by decoherence and is s~ the that limit a scattering medium : the Bethe.Salpeter equation calculation of the dependence of the degrees of problem now shifts the explicit to polarization and of spatial coherence in function of the number of scattering events when the fulfills described in section 2 and the medium the incident wavefield plane is of the type evaluate exactly the Mueller matrix elements Indeed, assumptions stated above. can one approximation (see Appendix A). handled in the ladder mii (n) by the Bethe-Salpeter equation entropic approach developed in the above justify of the This will the to use serve our purpose distances derivation is valid section. We emphasize that this greater than the elastic over mean by Stephen and Cwilich [23], the problem of evaluating the free path. As demonstrated coherency matrix W reduces to a matrix eigenvalue problem. We leave the details of formulae in the Appendices A and B. The 4, I DEPOLARIZATION (labeled I) pure normally on the TIME-INVARIANT BY A SCATTERING MEDIUM. We consider polarization and arbitrary degree of spatial coherence, half-space z ~ 0 its Stokes writes follows : vector as state of (So ) ~ ' (Si I (S~) (S~) = E~ ~) = Ex 1~) = = + E~ ~ Ey 1~) (Efl E~ + E~ Ef) I (Efl E~ E~ Ef) an of unit incident intensity l = ~~ DEPOLARIZATION II N° limit In the by ensemble derivation of techniques expression scattering limit, covariance averaged weak of the the correlation has been for the the discussed (labeled o) output of the scattering medium is determined Bethe-Salpeter equation (Eq. (Al )). The Green's functions of the field by time-independent [23, 30, 41-42]. Following this method, we obtain the linear Stokes (S21 ( ° (S31 ( where the ignored, with respect normalize we write to the denote G~~'s components S~ in the to vector (tit (Gt Gxxl : Cartesian respect (lGxyl~) (tit G~yl + (Gi G~X ~~~~ ) ' G~yl ) G~ ~). + and the suffices x, y label has been absorption equations (A9) Since chosen. system ~) ) (Eq. (A6)) function coordinate to (lG~yl~) + Gxxl Green's retarded the S~ with form response the (lGxxl~) (Sil ((lGxxl~) ~ 2053 SCATTERING MULTIPLE BY satisfying function largely LIGHT OF Then, allow us : ~l~~~ ~)~ ~° ~~~~ ' (53) ~~~~~~~~ f(n) with = 2 ~~~~~~~ g(n)= and (7/10)~ + expressions g(n) 2 n (7/10)~ + + being I number the of scattering derived by Freund [6] (his Eq. (I I)), albeit equations (12, 14) that the Mueller matrix of the scattering medium is of the form given by equation (6). The fact that mjj m~~ is further substantiated by the property of rotational invariance of the degree of polarization. Having specified S~, it follows from equation (14) that the output degree of polarization is given by : These events. in a different fashion. very similar readily verified are It is those to from = ~o which degree involves of three spatial ~ ~~l) ~ f(~) independent coherence ~~~~~~~ 5 ~~3) ~ + 2n f Combining parameters. writes equations (3) ~~3) +1 (S~) + I j~ l (53) = For f~~ 0 0 it is concreteness, instance, degree , of an worthwhile input polarization specialize polarization formulae to linear P~ and (14), the output = f(n) which is state a (Sj)~ l/2 ~~~~ (Sj)~ ~f(n))~ l they stand, equations (15-16) show that the output degree of will, in general, differ from input degree of coherence) the coherence) because of the effect of the scattering medium. These the fact single scattering (I,e. n 0) by particles with that the and degree of polarization [19-21]. state preserves For (~~) ' As interest. I 1/2 : ~~2) ~~~ ~~2) ~ ~ (E = polarization (I,e, polarization (I.e. equations are in having spherical of of accordance symmetry (15-16) to some special e~) has for output Stokes monotonically decreasing spatial spatial function cases of vector of the JOURNAL 2054 number of scattering (see Fig. I) and PHYSIQUE DE degree of spatial a I N° equal coherence to Similarly, zero. ii for I input an right-handed P~ g(n) meanings. = Numerical thjt see we (q(~( (Mf~( and = ~~~ polarization circular / = ie~) (e~ gets one ° : ° , g P~. calculations jjnctions these E state Then, have not f and g have clear physical Upon comparing f and g in figure I, f(n)~g(n)). decay (I,e. The parameter functions the two in figure shown are the same , (n) I. ~~~~ ~ which polarization has been represents the ratio of the linear to circular ~~3) the exponential indicates that as the degree of polarization also displayed in figure I increase decreases, the polarization ellipse flattens towards This fact is also expressed the major axis. by saying that depolarization of linearly polarized light requires more scattering events than are required for a circularly polarized light. Bourret [24] made a similar in his study of comment propagation of light in a medium with a stochastic refractive index. In this insightful paper, he found by using a random perturbation depolarization characteristic that the length treatment associated with multiple scattering is for linearly polarized light exactly twice the value for circularly polarized light. From the numerical expressions of the functions f and g we find this v = ratio by found For the be in to n ~ the In ~~~~ (0.7) of depolarization ~~ range (n l.94 = - ), 2, lo (low tx~ n)j, I,e, very close to the value Bourret. l, Poincar6 the process 3( sphere ~~ ~~ but induces be cannot symmetry a assimilated breaking to (I,e. an the isotropic contraction of symmetry SO(3) of is i loco 0. 8 j ~ j~~ V (0) 100 ~ 0. 6 i °.4 ~, ' o 4 I >6 i~ ~ ' 0.2 ' , ' 0 ' 4 0 8 16 12 n Fig. I.- Degree of polarization (15)) for an input pure state n (Eq. (dashed line). The inset depicts the of of scattered linear of scatterings function of the number light as a polarization (solid line), right circular polarization "~~~ parallel dependence of the normalized parameter v (0) on n. N° DEPOLARIZATION Ii LIGHT OF 2055 SCATTERING MULTIPLE BY significance of these results, we have also plotted (Fig. 2) the variation of polarization using equation (4). The of the states production AS(n) S(P(n + I)) S(P (n)) with the number of scattering events entropy with ~k and (Fig. 3) is well represented by an exponential decay AS(n) ~kexp(-xn), consistent with depending on the particular state of polarization. This form is results x obtained from equation (11). For example, mjj (resp. m~~) given by equation (11) was numerically tested assuming that the exact value is given by f(n) (resp, g(n)). From such calculations, find that the fitting parameter B was equal to within 4 fb of the value. exact we This irreversible of behavior of the entropy production already expresses in a way the nature radiation of n (say, lo), the entropy of is the considered. For large values process nm defines of further the maximum unaffected by scattering, it steady state entropy (S(P 0) attainable standard of In (2)), by multiple scattering. This is of course the sort transition from physics about this from statistical [14, 25]. Relatively litle is known argument broken). entropy discuss To associated the physical with different = = = = ~j~~ o.~ ' / 0.6 ' / / / 0.4 i 0.2 0 0 6 4 2 8 lo n Fig. 2. Sarne Entropy (Eqs. (4, 15)) symbols as in figure I. plotted radiation of as function a of the number of scattering events, Asin) 10-~ , , , , , 10-~ ' , , , , , , 10-~ ' , , , 10~? 0 6 4 2 8 lo n Fig. 3. events. Entropy production AS (n ) Same symbols as in figure1. = S (P (n + I )) S(P (n )) as a function of the number of scattering JOURNAL 2056 PHYSIQUE DE N° I ii AS considered by several 0, but some problems of this type have been briefly [26-27] (e,g, loading of a spring under the action of a gravitational force, charging of a capacitor, compression of a perfect gas). They found that the irreversibility involved in such physical processes in transforming the system from an initial state to a terminal bears an state relationship to the number of discrete steps in which it is carried out [26, 27]. inverse AS 0 ~ and = authors 4.2 POLARIZATION DEPENDENCE autocorrelation of the OF TIME THE temporal of the AUTOCORRELATION in the motion of the scattering termed is centres technique is described in details in [lo, 29] and was devised to light scattering to the multiple scattering regime for probing the of dense scattering media. Light scattering experiments measure the correlation field as defined ~~~~~ ~~ ~°~~ = m I + correlation intensity. The evidenced was gl'l (t form of scattering the for which [30], al. Treating time. the the of times, short the of memory field theorem field the auto- polarization leads to correlation of function is state of incident their [30]. decay a : of the transport of state path diffusion each the factorization diffusive the of functions measurement very have lost waves delay a Siegert the For events. the the describing depends on functions auto-correlation et related function Green's MacKintosh depends on by long paths [10] Wolf is auto-correlation through related t multiply-scattered light by Spectroscopy (DWS). This extend the usual quasielastic dynamic structural properties the intensity autoscattered of , the process, are and the to these by which dominated Maret gl'~ (t) function As intensity where ~~§~~ (f(0)) = Gaussian and (f(0)) [gl~~ (t)[~. ~~~~~ gl~l (t) as complex zero-mean a gl~~ (t) gl~~ (t) function Measurement FUNCTION. intensity Diffusing Wave fluctuations polarization. In the following, we derive an expression of the degree of polarization for multiply scattered light from a half-space. For the incident pure state of polarization defined by equation (6), the writes (for t* l) : output degree of polarization ~ P~(t*) ((Sj)~P((t*) P(t*) (S~)~P((t*) + = (S~)~P((t*))~~~, + (17) ~'~ , / 0.4 ' / / / / / / 0.2 1' / / / / / 0 0.2 0 0.4 0.6 0.8 @ Fig. 4. Polarization dimensionless (dashed line). time t*. dependence of The are curves the linear equation (17) functions parallel polarization (solid line), right autocorrelation as a function circular of the polarization N° DEPOLARIZATION II LIGHT OF 2057 SCATTERING MULTIPLE BY in the respectively the where the P~(tl's Q defined Appendix B and denote 1, 2, 3) are polarization at 45° and circular degree of polarization for linear polarization, linear polarization, while t* is the dimensionless time (t/r~ with r~ being the time required by a optical wavelength). to scatterer one move depolarized at very times As displayed in figure4, the light is completely short * diffusion of contributions (t « I corresponds to long paths). Because different due to long the paths, the P~ (t*) versus (t *)~'~ differ in slope according to the input state of polarization. As before, we find that complete depolarization of circularly polarized incident light requires less (short scattering events than for the linearly polarized case. This with longer times contrasts diffusion paths). = 5. Comment To close the Stokes vector useful in [14]. As investigate to Recently [14] Gudkov method. implications of the recently proposed method [5] of some complete depolarization, to the question of irreversibility. a about irreversibility in Polarization Optics which we found comment after references indicated are first reconstruction paper, we now reconstruction representative Some the Freund's on historical an off trade the been has using able should it note, between Planck's mentioned be that polarization light spectral entropy, law calculate to [14] Jones was irreversibility. and the of flux total spectral entropy of an unpolarized light across the surface of a scattering object. A similar but generalized to partially polarized light, treatment, put forward by Callies [14]. This was derived a formula for the local author entropy production rate by a single scattering using the radiative transfer theory. However these connected approaches are not easily the multiple to the scattering the situation which in sense medium and interaction a if and irreversible. a unitary operation [9, 14]. Typical tions which states (I.e. Freund random diffusive reconstructed process by of prediction of entropic approach The We show that compensation by is therefore it consequence, Mueller matrix (e,g, evolutions degree lead irreversible. filter to scattering). by Measurements scattering under the otherwise optical any indicate to a invariant reversible, is that prove useful between it is transformation selective reversible a transforma- absorption of produces polarization of to loss any of light from complete w BaS04 coating depolarization of argument It is Stokes imply that 0.03). Reference information about of parameters the process is (assumed a the a [5] incident the he beam reversible polarization incident says is very of light by in as a any input totally claims show that to state state can similar to be the filtering [15]. contradiction with the here. speckle-pattem produces that comprehensive note we a only with precise knowledge circular it is constitutes rotator) compensator, increase the depolarizing entropy are reflected of the would developed the evolution the depolarization by multiple scattering, termed speckle-pattem filtering which process determination Freund say easy which is not key point here is that the entropy tenable. The induces a loss of intensity and suggested by Freund Stokes and simple to analyze this by expanding the light (of intensity (S~)) in of orthogonal terms filtering instructive corresponding to unpolarized vector polarization states (e,g, right and left handed observations, these irreversible. diffusively not is that (e, g. multiple a does a shall scatterers) complete after we also (residual transport even usual point of light polarized and filtering) are [5] considers array Given here. occurs, of irreversible are considered direct a by is absorption no As characterized which irreversibility is featured. Consider a linear interaction plane-wave field. If and only if the remains entropy circular states each of intensity (S~)/2). Now a right- entropy (- In (2 )) and intensity (50 fb) losses [9]. Finally discussion of the data produced by Freund [5] could take place of both the determined scattering characteristics experimentally both JOURNAL 2058 of BaSo4 the coating and PHYSIQUE DE procedure normalization the N° I the in of measurement I I Stokes the parameters. Concluding 6. We remarks. details depolarization and decoherence Three non-absorbing scattering medium. pointlike scattering centers, uncorrelated disorder and considered have weakly in disordered introduced : typical realization scattering particles considerations (a) of some such of much smaller points main Maximum would size of two medium entropy a collection the wavelength. from this paper plane-wave field by a assumptions were approximation. diffusion On of basis the A spherical entropic non-interacting of : depolarized totally output an dense than drawn be can yields be a main and light beam (large vector I n) which is similar of that to light natural Stokes whose vector is of the ° form The results 0 depend on the specific nature of the but scatterers, the properties of the scattering rather medium. Now returning to the question symmetry on raised in the Introduction, found that the length scale which the depolarization and loss we over nf, with typical number of scattering events n being about ten. of take place is ( coherence for distances Then larger than (, one is justified in describing the transport properties of multiply scattered light by a radiative transfer theory which treat the radiation field as a scalar than a four Stokes rather In addition, we found that the entropy production component vector. falls off exponentially with (. derived in preceding the section do not = (b) principle The of maximum allows entropy us also deduce to the form of Mueller the describing the scattering medium. The explicit calculation of these elements from the Bethe-Salpeter derived equation permitted to test the validity of this approach. In that knowledge of the degree of polarization after a double scattering suffices to enable one case a matrix. the Mueller Specifically, it was that full depolarization of linearly construct to seen polarized light requires more scattering events (typically a factor-of-2) than are required for a circularly polarized light. It is believed that this reflects behavior the specific symmetry properties of (S~) the details thereof however have not been completely disentangled. This be put in parallel with the results of MacKintosh who investigated the effects of et al. [31] can polarization on backscattering of a laser radiation from monodisperse polystyrene spheres in These authors also pointed out that due to the large of have number statistically water. the independent contributions, scattered field is Gaussian distributed by virtue of the central matrix elements limit theorem. distribution The of Stokes for parameters a Gaussian distributed field has been spatial [36] fluctuations. is also Reference examined made to a in details the recent paper of Cohen et al. [32] who have recently some statistical distribution of the polarization of multiply scattered optical The ratio of state waves. semi-minor axis the semi-major axis polarization ellipse is the of the denoted by to [32]. They found that the distribution function of ellipticity p(e) parameter e ~the ~~~ defining the ellipse of polarization is given by p(e) indicating that the least : e characterized for both temporal [16, 34, 35] and = probable obtained There issue of state with do of how completely is polarization data [32]. circular experimental remain course these results uncorrelated are : a number a (I,e. of changed more realistic (e) = points when the to be (I 0.307). + A e ) satisfactory agreement was investigated, regarding the fundamental susceptibility of the medium is not dielectric assumption would be to allow for a finite correlation N° DEPOLARIZATION ii length Gaussian I.e. LIGHT OF distribution (rj ) q BY = (2 2059 j~ ~ (r~ )) q SCATTERING MULTIPLE exp ~ ~'~ ~~ gr« j2 where «~ 2 , positive the analysis is extended radiations when with a broad constants to or possibility of generating frequency shifts scattering. by Our method of : calculating the dependence of the Mueller matrix elements the number of scattering events on gives also rise to interesting possibilities for exploring the behavior of the above in more complicated systems (e,g, scattering from large particles for which the transport of light becomes ballistic [30], ideal particles shapes which do not extension less light to scatter isotropically and involving non-diagonal Mueller matrices). C and are « I,e, bandwidth Acknowledgments. (C.B.) is indebted to Prof. Craig. F. correspondence and for bringing some pleased to acknowledge Prof. R. Maynard for of One us useful for of the related research critical reading of the Appendix of this Appendix is to briefly purpose field that is needed in section 4. The State his to University attention. manuscript and for We G. are Maret discussions. useful the Pennsylvania Bohren earlier A outline the conveniently The following calculations are more point is the Bethe-Salpeter equation for the field-field diagrammatic expansion [23, 30, 37-42]. (Gim Gjsl G,~ GjS " + derivation done of the in correlation of Our starting space. derived from usual momentum correlation function function Gim'Gjt'w>jm'n' Gm'm Gfl (Al) n , repeated indices is presumed, G* being the complex conjugate of which coherent contribution right side of (Al represents the goes to is the incoherent contribution. In the while the other term zero very rapidly and is neglected incoherent of contributions of light weak scattering situation, the observed intensity is an sum scattered through all possible paths. For scalar shows that within the diffusion waves, one approximation (I,e, incoherent intensity can be incoherent diffusive and transport of light), the written [39, 43] : as summation where G. first The over in the term fin~ (r) ~/ i = £ W(q) ~ ) (iq exP r ~£ and be may For Green's function diffusion the successive ladder by summing field, the intensity operator Wij~~ obtained polarized a (r, n) of F ~ wijmn(~) (A2) n , i q the where (r, F = 10 £ £ (ol"I equation is diagrams. can be (Aa(~)Y ~ weight the written as of diffusion paths : (" lmnj, (A3) a the first term n [a) eigenvectors O (q~)) (A2) have and 0 = been (A3), corresponding associated and derived one to finds by a single eigenvalues the number of ~ ( : (G~~ G() = ~ authors scattering A~(q) (to [23, 30] and situation. the are The (ij la ) A~(r, n) (a [mn) ~ nine ortho-normal degree of approximation reproduced here. From not (A4) , a ~~~~ (~ ~) a , l = £ n (q)Y (A a q exp (iq r), (A5) 2060 JOURNAL and ii j ), [mn) denoting respectively the dependence of interested evaluate to number, write we the the propagation of direction (G~X Gj~) (G~~ = (G~ G() F~~~ = the and the final N° Green's function as F G() = (r, ez, where follows It the have we at (G,~ G~() F~ = ~ lo n, L~ for ~ (r, F j ~ once are set (r, n, L, = j ~~ ~~ ~~, L( L( = (r, F 3 carrying given by the that ~ formulae scattered the by a diffusion approximation, diffusion [10, 30, 43] Appendix is within light entire space, 7 7 4 lo )~ scattered ~ jnj 2 (A9c) ' l )~ jnj 2 (A9d) B of calculation the that (A9b) ' l ~ one (A9a) ~~ lo lo D~, j of the the DWS. shows, in the electric field degree Brownian For weak polarization of scattering correlation scatterers limit function for and is within given by : ~~~~~~~ ~~ n exp n) scattering paths denoting the length nf, of (- 2 nt/rq ) dn ~~~~ ~~~~~~~~ F F(r, the over , IF (r, where , n ( 2 + 4 coefficient the : 5 1 framework the (A7d) , (Ji8) L) equations (A7) of = ~ consider to L~ )j : Appendix this n, ~ ~~ L( = ~ = (r, F ~2 4 n (G~~ Gj~) (G~G() (A7a) (A7b) given by medium integration the out Lj n, Lj n, 2 eXp and 49 = = characterized are , infinite 7rL) 4 (G~G() multiply we scattering , L~ n, (r, F lo ~ an ~~'~ ~~~ of the (A6) lo 2 4 = purpose a Since of ~ l + 2 2 l = ~ (G~ G£) The function ii have we F~~~ = = ~ijmn (~, ~ 3/2 F and polarization. of states intensity mean GJ) along is (G~ Gj~) with initial I : (Gim As PHYSIQUE DE function Green's and To ki = ~ Dj of time dn equation required by a diffusion the is the (r, n) is scatterer number the to move of one N° DEPOLARIZATION II OF LIGHT SCATTERING MULTIPLE BY 2061 normalized optical wavelength. From inspection of equation (B I), we note that gl~ ~(r, t is the of the number Laplace transform of F (r, n ), giving then a one to one correspondence between and dimensionless t* scattering the variable t/ro. events n situation of reflected We consider the special light from a half-space. In this geometry, now scattering medium occupies of space z ~ 0 and that the light is that the the region assume we semi-infinite semi-infinite medium For a incident normally on the from the tx~. vacuum z from the medium, diffusion paths for scalar be obtained above the weight of waves may of images infinite-space by the method [37]. case = = a) LINEAR POLARIZATION. Considering an perpendicularly polarized along e~ (labeled I ), L) polarized output light [30] : incident field (labeled (xx (n ) F xx) F (G~X = = GQ) l = 2 = 2 ii (xx [F[ yy) Fi(n) Then, obtain we a where /~ = In CIRCULAR a vector ~ (l (B2a) (n, Lj ) , )~j lo and F(n, Lj). (B2b) ; = b + t* gi~~(t) = 3 n ~~ ~~~ ~~~ ~~ (t ) j ~/6 + (84) j)(t) g POLARIZATION. similar the manner, E, is F lo parallel : P b) (G~G() = = j ~ 2 + respectively get for we = / ie~) (e~ of case and polarization circular polarization the I+ +1 ll~l i " ly~l developed [30]. is states are l~yl + I The incident unit : lyyll + (85~) ' passing, conjugation In We call helicity. In we would = like operation. F~ (resp. F_ ) like fashion )1[xxi ) 1+ to the as the remark Green's above iYx) = F_ = <+ <+ ixY) I permutation function have we F~ + I + symmetry corresponding (B5b) IYY)1 to of the these vectors positive (resp, ((G~GQ) + the negative) : + F + (B6a) + , F + (B6b) + , )~ l= under (G~ G() ± (G~G() (± ) (G~ G() 2062 JOURNAL PHYSIQUE DE I N° 11 with corresponding The temporal correlation functions ~~~~ ~ : ~ ~~~ ~ ~ are ~~~ ~ ~~, )~)~~~) (88) = ~+ with a, b (2). In 2 the aforementioned by iven b' (5/3) d'= +exp(-b')±-exp(-d') ~~~ corresponding The expressions and of polarization degree ~j~~~~~ P~(t) = Similarly, for gets one polarization a state ~ (t = gi (2 t* (2)) In + : (89) (t) ~ (t) i writes 45° at ))~~~ P = j~~~~~~ (t)+g_ g+ ~/(5/3) d + ~~ gi i (B lo) (t) , with ~~~~~~~ ii(~(t)) _~~~~~ ~~, 2+exp(-b')±~~exp(-c') gi((t) ~/(49/23) ~~~~~~~~~ (Bll) ~/(49/23 ) In (10/7). (10/7)) c' + In indicate compared with those of reference [30] : for this purpose These results we may /~ respectively of P~(t) the origin equal Pj(t) and taken that the slopes to at are vs. figure 4 also and )/2, in notations of reference [30]. From )/2 (y~ the (yi note we y_ yjj discussed in section 4,I. close to the value that the ratio of these slopes is equal to 2.2, I.e. and c (2 t* = = be References [II Scattering and 1990). For phenomena, reference SHENG P., Singapore, between [2] these A., P. LEE A., ISHIMARU two Wave is of sets Science Localization a 251 also made problems (1991) Propagation Classical of presentation of to has : some KAVEH been also Waves Random in analogies between Media electron (World Scientific, optical and Physica B 175 (1991) 1. The developed in a lucid presentation by M., wave connection FENG S,, 633. and Scattering in Random Media (Academic Press, New York, 1978). [3] See the feature (1985) [4] [5] [6] FREUND FREUND FREUND I., Opt. I., Opt. I., Opt. Fourier j7] KAVEH 206 M., issue and on wave references propagation and scattering in random media, J. Opt. Soc, Am. A 2 therein. 86 (1991) 216. (1990) 1425. Commun. 81(199I) 251. This treatment University, Grenoble, (1986). France Waves in Random Media 3 (1991) S121. Commun. Lett. IS follows AKKERMANS E., Ph. D. Thesis, N° j8] DEPOLARIZATION 11 [9] [10] also G., also BORN ROSENBLUH 88 P. E., WOLF LIGHT BY SCATTERING MULTIPLE (1991) 109. Z. Phys. B 65 (1987) 409. M., HOSHEN M., FREUND I., KAVEH M., Phys, 1142. J. T., GORI F., J. Opt. Soc. Am. 6 (1989) E., Principles of Optics (Pergamon Press, Oxford, FOLEY WOLF 2063 R., ROSENBLUH M., Phys. Rev. B 42 (1990) 2613, M., BERKOVITS R., Phys, Rev. B 39 (1989) 12403. ROSENBLUH E., M., WOLF BERKOVITS I., Optik C., BRossEAu M., KAVEH FREUND MARET See [I II [12] [13] I., FREUND See OF Rev, Lett, 58 (1987) 2754. 1980). formulated in of spatial-time correlation theory of partial coherence terms Mod, Phys, 37 (1965) 231; functions, for example, MANDEL L., WOLF E., Rev. see, coherence See also, WOLF E,, J. Opt. Soc. Am, 72 (1982) 343 for presentation of a theory of partial space-frequency domain. in the [14] JONES R. C., J. Opt, Soc. Am. 43 (1953) 138 USSR 68 (1990) 130) and See also GUDKOV N. D., Opt. Spektrosk. 68 (1990) 224 (Opt. Spectrosc. CALLIES U., Beitr, Phys. Atmosph. 62 (1989) 212. Light (North-Holland, Amsterdam, Polarized [15] AzzAM R. M. A., BASHARA N. B,, Ellipsometry and For of account an the 1977). [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] 1256. J. Opt. Soc. Am. 4 (1987) Density Matrix Theory and Applications (Plenum Press, New York, 1981). Processes (Wiley, New York, 1951). PRIGOGINE I., Thermodynamic of Irreversible PERRIN F., J. Chem. Phys. 10 (1942) 415. (Wiley, New York, 1957) p. 40. HULST H. C., Light Scattering by Small Particles VAN DE HOVENIER J. W., J. Atm. Sci. 26 (1969) 488. FREUND I., 245. Waves in Random Media 1 (1991) STEPHEN M. J., CWILICH G., Phys. Rev. B 34 (1986) 7564 See also STEPHEN M. J., Phys. Rev. B 37 (1988) 1. BOURRET R., Opt. Acta 21 (1974) 721. LYOYD S., Phys. 5378. Rev. A 39 (1989) CALKIN M. G., KRANG D., Am, J. Phys. 51 (1983) 78. See also GUPTA V. K., SHANKER G., SHARMA N. K., Am. J. Phys. 52 (1984) 945 HEINRICH F., Am. J. Phys. 54 (1986) 742. BERNE B. J., PECORA R., Dynamic Light Scattering (Wiley, New York, 1976). PINE D. J., WEITz D. A., CHAIKIN P. M., E., Phys. Rev. Lett. 60 (1988) 1134. HERBOLzHEIMER MACKINTOSH F. C., JOHN S., Phys, Rev. B 40 (1989) 2383. MACKINTOSH F. C., ZHU J. X., PINE D. J., WEITz D. A., Phys. Rev, B 40 (1989) 9342. COHEN S, M., 5748. ELIYAHU D., FREUND I., KAVEH M., Phys. Rev, A 43 (1991) BROSSEAU C., BARAKAT R., ROCKOWER E. B., Opt. Commun. 82 (1991) 204. SALEH B. E. A., Photoelectron Statistics (Springer, Berlin, 1978). BROSSEAU C., BARAKAT R., Opt. Commun. 84 (1991) 127. STEEGER P. F., FERCHER A. F., Optica Acta 29 (1982) 1395. MORSE P. M., FESHBACH H., Methods Theoretical Physics (McGraw-Hill, New York, 1953) in K., BLUM vol. [38] R., BARAKAT 1. I. E., of Quantum L. P., DzIALOSHINSKII Methods Field Theory in York, 1963). (Prentice-Hall, New [39] AKKERMANS E., WOLF P. E., France MAYNARD R., MARET G., J. Phys. 49 (1988) 77. [40] GOLUBENTzEV A. A., Zh. Eksp. JETP 59 (1984) 26). Teor. Fiz. 86 (1984) 47 (Sov. Phys. See also GOLUBENTzEV 1933 (Sov. Phys. A. A., Zh. Eksp. Fiz. 96 (1989) JETP Teor. 69 (1989) ABRIKOSOV A., A. GOR.KOV Physics Statistical 1090). [41] [42] [43] BARABANENKOV LEE J. K., BIcouT JOURNAL DE KONG D., PHYSIQUE Y. N., J. AKKERMANS t T 2, N' Phys. Usp. Opt. Soc. Am. Sov. J. A., II, E., MAYNARD NOVEMBER 1992 (1975) 673. (1985) 2171. R., J. Phys. J France 18 A 2 1 (1991) 471.