A general framework for multiple scattering of polarized waves including anisotropies and Berry phase
We develop a framework for the multiple scattering of a polarized wave. We consider particles with spin propagating in a medium filled with scatterers. We write the amplitudes of each spin eigenstate in a local, mobile frame. One of the axes is in the direction of propagation of the particle. We use this representation to define a directional Green’s operator of the homogeneous medium and also to write the spin-dependent scattering amplitudes. We show that this representation reveals a Berry phase. We establish a generalized Green-Dyson equation for the multiple scattering problem in this framework. We show that the generalized Green-Dyson equation can be solved by linear algebra if one uses a representation of the rotations based on Wigner D-matrices. The properties of light scattering are retrieved if we use spin 1 particles. Our theory allows to take into account several kinds of anisotropies like circular or linear dichroism and birefringence, Faraday effects and Mie scattering within the same formalism. Several anisotropies can be present at the same time.
In optics, several phenomena involve the polarization of light. Such phenomena, like birefringence, dichroism and Faraday effect, are well understood and described in textbooks Born and Wolf (1999), usually in the case where the medium is homogeneous and linear. When filled with particles, the medium is no longer homogeneous and in addition to the transport of light, one has to study its scattering by the particles. A century ago, Mie discovered that the scattering of light by small dielectric spheres is linear and also affects the polarization Mie (1908); Jackson (1972). The theory of light scattering has been extended for several different particles and studied in the context of multiple scattering Van de Hulst (1957); Newton (1982); Mischchenko et al. (2006). While the polarization is well studied for a single scattering, the problem becomes more difficult in multiple scattering and a scalar approximation is often used. Such an approximation makes it simpler to investigate the coherent effects using sophisticated diagram techniques van Rossum and Nieuwenhuizen (1999); Akkermans and Montambaux (2006). As a consequence, polarization effects are neglected whereas they may, in certains situations, be of interest. In magneto-chiral birefringence, for instance, a residual degree of circular polarization persists is chiral media Pinheiro and van Tiggelen (2003). Recent experiments of coherent backscattering of light by cold atoms exhibit polarization-dependent enhancement factors Labeyrie et al. (1999); this result is interpreted thanks to the quantum, nonlinear, nature of the cold atoms Jonckheere et al. (2000). Under certain conditions, it was even forecast that an anti-localization effect could be observed Kupriyanov et al. (2006). Quantum effects of light were also introduced, in second quantization, for the study of dense cold atomic gases Sokolov et al. (2009).
Remarkably, most of the scalar approximation techniques are applicable not only to light but also to microwaves, ultrasound and non interacting electrons, as emphasized in Ref. van Rossum and Nieuwenhuizen (1999). The existence of a polarization, alike, is not restricted to light; other waves display polarization effects: The other electromagnetic waves, elastic waves, electrons. More generally every particle with a spin is, from a quantum point of view, a polarized wave. The classical light polarization is a consequence of the vectorial nature of the electromagnetic field whereas in the quantum-mechanical picture, photons have two states called helicities. The origin of the helicity of a particle follows from the spin. Relativity imposes that the spin 0 state of massless particles does not exist in the direction of propagation, photons therefore posses only two helicities. The freedom of the relative phase between the two helicity states is at the origin of the phenomenon of polarization as we experience it in optics. All quantum particles with a non-zero spin also have helicities and hence may display polarization effects. Therefore, the problem of polarization transport in multiple scattering does not only arise in optics: Polarization transport in multiple scattering is a more general problem and optics can be seen as the most investigated particular case.
According to the quantum duality of particles and waves, we may describe the propagation of particles as wave amplitudes from a statistical, or probabilistic, point of view. Reciprocally, the propagation of a wave can be described by the statistics of particle paths, through techniques like Feynman’s path integration. Following this picture, many theoretical calculations related to the multiple scattering of polarized waves are performed by Monte-Carlo techniques. Such techniques have been developped for light, even in the presence of quantum scatterers Labeyrie et al. (2004), and for elastic waves Margerin et al. (2000). Monte-Carlo techniques are a pragmatic solution to the problem of taking into account each trajectory as a whole. Is it possible to consider each trajectory entirely and perform statistics in a more formal way ? A pioneering work on trajectory statistics was made by Sato Sato (1995) with the introduction of a directional Green’s function for the elastic energy (equivalent to the optical intensity) in a three dimensional medium with nonisotropic scatterers. However, as his study concerned energy, polarization was not included in the model.
The trajectories of energy, or intensities, have been studied in a multiple scattering context using Stokes parameters. These parameters are preferentially used by experimentalists because they are, as intensities, directly measurable quantities. They depend on the choice of the reference frame used for their observation. Their ability to take coherence effects into account is limited to intensity correlations. In order to dispose of the reference frame indermination, Kuščer and Ribarič introduced harmonical functions to describe the Stokes parameters Kuščer and Ribarič (1959). Based on Wigner’s work concerning group theory Wigner (1959), these functions have also been used under the name “spin-weighted spherical harmonics”, though with a different phase factor Newman and Penrose (1966). They are also implicitely used in light scattering studies under the name “vector spherical harmonics” Newton (1982); Jackson (1972). The seminal work of Kuščer and Ribarič motivated a large number of studies on the phase matrix Domke (1975); Siewert (1982); Lacoste et al. (1998) and on the radiative transfer equation Hovenier and van der Mee (1983); Garcia and Siewert (1986).
The role of trajectories appears in a surprising effect involving polarized light: A four-foiled pattern is observed in backscattering experiments of polarized light from a medium containing anisotropic scatterers Pal and Carswell (1985); Hielscher et al. (1997); Cameron et al. (1998). In these experiments, the source is not a planar, infinite wave, but a localized beam. The effect does not exist if the source is extended on a scale of the order of the four-foiled patterns. The same medium does not exhibit such patterns if the observations are made in transmission rather than in reflection, so that the properties of polarization depend not only on the detailed trajectory, but also on the direction of observation, or more precisely on the frame of observation. It has been suggested that the appearance of these patterns is due to the existence of a Berry phase for the photon Rossetto and Maggs (2002a). In quantum mechanics, Berry phases have been studied already in many situations Berry (1984); Shapere and Wilczek (1989). In optics, the literature is abundant in experimental Tomita and Chiao (1986) and theoretical considerations Berry (1987); Bhandari (1997). If light is not guided in an optical fiber, but rather multiply scattered, the Berry phase still exists Maggs and Rossetto (2001). The probability distribution of the phase depends on the statistics of the paths followed by the photons.
The purpose of the present article is to introduce a general framework for the multiple scattering of any polarized wave, as simply and as pedagogically as possible. We will base our construction on the concept of path statistics, in the spirit of the Monte-Carlo calculations, or Feynman’s path integral, to account for local interactions and path-dependent effects. We express the formalism for particles with an arbitrary spin , although we consider only linear media and scatterers and thus leave the other quantum properties to further investigations. Our objective is not to provide a complete theory for multiple scattering of polarized waves but rather to set up a framework in which such a theory could be developed. On that account we assume the simplest possible point of view and make several assumptions, that are briefly discussed.
We have organized our work as follows: The representation of the field and the geometry of rotations are discussed in Section II. A directional Green’s operator is discussed in Section III, it describes the transport in a homogeneous medium. Using the same formalism, we introduce the scattering operator between incoming and scattered state in Section IV. These first three sections constitute the elements of the theory. We demonstrate that our theory takes into account the Berry phase without requiring any extra tool in Section V. We derive a generalized Green-Dyson equation for multiple scattering of polarized waves in Section VI and we show that a solution of this equation is obtained by means of the rotational harmonics, introduced in Section VII. We discuss the particular case of rotational invariant scatterers in a rotational invariant medium in section VIII and conclude. Fundamental formulæ concerning the rotational harmonics are provided in appendix A, and some of the several notations used for them are listed in appendix B.
Ii Geometry and polarization
We begin the presentation of our work with the introduction of the geometry elements used throughout this article. Mainly, the novelty of our work relies on the representation of the state of waves in local frames. Polarization is introduced as a transformation property of the wave state under local rotations. We express all space coordinates in a fixed reference frame . Any local frame is therefore obtained after a rotation of the reference frame. The coordinates of the local frame unit vectors, written in the reference frame, form a matrix, which the unique rotation matrix mapping into and preserving orientation. The group of rotations is . Considering the rotation associated to a frame is mathematically equivalent to considering the frame itself. We henceforth only use the description in terms of matrices for mathematical convenience and use exclusively the Euler representation of rotation matrices, such that any rotation of is decomposed into
where is the rotation around the -axis of angle . Similarly is a rotation around the -axis. The decomposition (1) is unique if and . For or , the decomposition is not unique.
In our discussion, we use the concept of trajectory, or path of a particle. Studying the propagation of a wave using the particle picture would not be fully general if we were concerned by the behavior of single particles. But as our goal is to describe all possible trajectories as a whole, it is known that this way of addressing the problem of multiple scattering is correct. As a consequence, we will simultaneously use the wave picture, as equivalent to the statistics of particle trajectories, and the particle picture of the propagating wave. The spin of the particle emerges as an essential ingredient of the theory, after geometry considerations.
We study a medium filled with scatterers and write for the position of the scatterer . The trajectory of the particle is a succession of displacements between points and changes of direction at the points . We call such a change of direction a scattering event. We consider the momentum after scattering events and decompose it into
where is the modulus of , is a rotation matrix (expressed in the reference frame) and is the unit vector of the reference frame along the third coordinate. We choose the local frame for the segment of the path after the scattering event and before the next one. The third coordinate of the local frame is therefore always pointing in the direction of motion.
We remark immediately that the decomposition (2) of is not unique: The Euler angle of can be taken arbitrarily, it is the spin gauge freedom. Here stands a crucial point in our model: The introduction of the local frame is a natural way of introducing the spin, and hence the polarization, into a multiple scattering formulation. If the wave has a spin , there are possible values for the spin component along the direction of propagation, that we label with “” (). At a given point , the field is a superposition of partial fields, which we consider to be plane waves, with different momenta. For simplicity, we may consider all directions of propagation for a fixed absolute value of the momentum. The amplitude of probability for observing the spin state in the frame at point and time is written . In this article, we regularly use the bracket notation for functions: The functions of a variable are written as , instead of . There is no consequence of this notation concerning the physics itself, but we find it more convenient for our purpose, in particular for the introduction of the rotational harmonics in section VI. We then write the amplitude as a bracket product
Polarization is related to the frame of observation and if this frame is changed, polarization is modified according to certain rules. Light polarization, for instance, is turned by an angle if the frame of observation is turned by an angle around the direction of propagation. It is therefore natural to change the basis of representation and use the eigenbasis of rotation along the direction of propagation. Light has two circular polarization states, corresponding to the helicities of the photon. A rotation of angle creates a phase shift for each circular eigenstate. If is an eigenvalue of the spin operator along the direction of propagation, rotations along this axis commute with the spin operator. Unitarity implies that
on the on-shell states. After these remarks, we shall name the third Euler angle, usually noted , the spin angle.
We have presented a new description of the field at a given point that depends on the local frame and thereby taking into account the spin. We still have to check the completeness of the ket representation . As we have remarked, the direction of propagation is , so that and are orthogonal if . From relation (1) we conclude that non orthogonal and have the same Euler angles except the spin angle, which is arbitrary. Let and be two reference frames, we introduce the function to denote
where and are the first two Euler angles of and and the first two Euler angles of . is a kind of Dirac function for rotation matrices, which select frames having the same third unit vector. Using this notation, the condition is imposed by defining proportional to . We remark that . Finally, the orthogonality of the spin eigenstates yields . From the expression (4) we get the product
where and are the spin angles of and respectively.
The superposition of the partial fields for all directions of propagations is expressed by
The presence of the term is essential. Its role is to add the amplitudes in such a way that the amplitudes in two distinct frames with the same direction of propagation are in phase and do not cancel out. Forgetting this term would make all vanish except , because the exponential term from (4) would be integrated to give . Multiple scattering theories for scalar waves actually only consider the term . The completeness of the Hilbert basis is expressed by the closure formula ( is the identity operator)
The integral over has been defined in Eq. (7). The description we have introduced in this section will be used in the formulation of the multiple scattering for polarized waves. In the next section we introduce the Green’s operator for the polarized states and its space dependence in the absence of scatterers.
Iii Transport in a homogeneous medium
Before studying multiple scattering, it is necessary to investigate the transport in a homogeneous medium. By the word transport, we mean the response at a position and time in the frame (see previous section for the introduction of the frames) to a source at position and time in the frame . It is advantageous to use the Green’s functions because the two situations of massive and massless particles can both be handled with the same formalism. In the case of a massless particle, like the photon, transport follows from Maxwell’s equations and is often formulated by the Helmholtz equation. Massive particles transport is of a different nature and the dynamics of their wavefunction responds to the Schrödinger equation.
We denote by the Green’ function of the operator or of the operator to describe the transport of the electromagnetic field or of the wavefunction respectively. The equivalence between these two operators is obtained through the relation
so that we shall use in our formulæ only the “massless” notation . The results we present can be extended to massive particles of arbitrary spin by using equation (9) and changing the sign of . If the medium is invariant under translation and in time depends on and . In this section, we construct a Green’s operator which depends on the direction of motion at and , and we call it the directional free Green’s operator.
We introduce the free Green’s operator as the operator transforming the wave function along the propagation of the wave if no scattering events occur. The transition amplitudes characterizing the response may depend on the spin. If we note the free Green’s operator, transport is described by the matrix elements . We have introduced the position ket and denoted . Naturally, it is not physical in quantum mechanics to consider the position and the direction of motion of a particle simultaneously. The directional Green’s operator that we need for our theory can be seen as an intermediate element of computation: The physical Green’s function is the superposition of the directional Green’s operators for all initial and all final directions of motion. In Feynman’s picture of path integrals, it corresponds simply to decompose the path integral formulation of into path integrals over trajectories with a constraint on the direction of motion at the initial and final point. We have illustrated this feature in figure 1.
In a medium with translational invariance, the momentum is conserved so that has a factor . Between two scattering events, the wave travels in space from to along the direction . The direction of the momentum, is the same as the direction because of the relation , so that we have to impose this constraint to the free Green’s operator. To express the directional constraint on , we write a similar equation as (2) :
so that the dependence of the Green’s operator on is simply reduced to . Finally, we have the expression for the dynamics Green’s operator
The factor in expression (11) is a very strong restriction imposed by rotational invariance. It states that a rotation of the medium must be accompanied by the same rotation of the frames and (through the term ) to leave the Green’s operator unchanged.
In the presence of scatterers, it has been observed in backscattering configurations the transport of polarization depends on the relative directions of the incident polarization beam and the vector Hielscher et al. (1997). Moreover, in some cases, like in the presence of linear birefringence or dichroism, the direction of propagation has a strong influence on the transport. Such complex geometrical dependences will be described by the dependence on , and of the generalized Green’s operator.
We now consider the evolution of the spin amplitudes during the transport. In the simple case where the time evolution of the spin does not depend on position or time, it is described by
where is the evolution operator of the spin. If there is no anisotropy associated with the spin and if the spin is conserved during transport, then the terms form a diagonal unitary matrix. Effects depending on the spin, like circular birefringence or dichroism, modify the diagonal terms of such that it is not necessarily unitary anymore. Consider for instance a medium with absorption length and index for the spin . In this case, we have
In this expression, the index denotes the ratio of wave celerity for a spin eigenstate compared to the celerity contained in . Dichroism appears when the values of depend on , birefringence when the values of depend on . It is also possible to introduce spin-flips.
To summarize our construction of the Green’s operator of the homogeneous medium we have the following formula for
The dependence of the Green’s function as a function of the distance is, according to the expression (14), the product of the scalar Green’s function and the Green’s function of the spin . If there is no absorption, the Green’s function follows the well known decrease of the Green’s function in three dimensions. In this situation, the enhanced Green’s function (14) verifies the conservation of energy by construction.
Iv Description of a scattering event
In the previous section we have constructed the Green’s function for a homogeneous medium. In this section, we include the description of the scatterers into the same formalism. The problem of scattering with spin particles was originally discussed by Jacob and Wick Jacob and Wick (1959) when two particles with spins are colliding. We consider more general scatterers in our formalism and describe an arbitrary interaction in the far-field. In this situation, we only need to know the so-called on-shell “-matrix”, or transition matrix, in a theory of multiple scattering.
We use the following definition of the on-shell -matrix, in the case of light scattering: Consider an incoming field and a scatterer at position . At a far-field position, the scattered field depends linearly on
The -matrix can be computed or measured in several ways that we will not discuss. In general, the -matrix depends on the incoming and outgoing directions and on the polarization. These dependences will be rigorously taken into account in the present formalism.
The position in space of the scatterers are denoted by . A scattering event corresponds to a change in the direction of propagation at from to . In the laboratory frame, the rotation matrix corresponding to a scattering event is . However, it is the local rotation which is physically relevant in a local description of scattering. The expression of this local rotation is . This is the local rotation experienced from the point of view of the particle during the scattering event. Using the decomposition of into Euler angles ( (Eq. (1)), we rewrite as
is the scattering angle and and will act as spin angles in (4). As a consequence of this relation, the modifications of the spin angles of and transform according to
and cast the rotation into local frames adapted to the scattering event (see figure 2).
If there is a spherical scatterer at position , the scattering operator between incoming and outgoing states is
( is a rotation around the -axis of the reference frame) We have used the spherical symmetry of the scatterer to bring the incoming and outgoing frames back into the incoming one, so that the outgoing frame is simply . In this formulation, is just the usual -matrix written in the spin eigenstates basis, which we write . We impose that there is a scatterer at position with the density operator , with
If there is no scatterer at position the scattered field vanishes. The scattered field is obtained using the the closure relation (8)
To obtain the last equality, we have used the fact that the spin angle of the rotation matrix is arbitrary and does not play any role in the final expression and we have therefore used the spin angle of to simplify the formula. The angles ( are the Euler angles of .
The operator in formula (21) stands for a single spherical scatterer, correlations or the influence of the other scatterers are, so far, ignored. The imaginary part of stands for extinction. The extinction cross-section for spin is then
It is constructive to compare with the scattering cross section for spin
The total extinction and scattering cross sections are obtained by summing over the spin eigenstates: and . By conservation of energy the ratio cannot exceed . The ratio is called the albedo. If , all energy captured by the particle is scattered. This is known as the optical theorem. Let us remark that the inequality valid for the total albedo is not necessarily true for the albedoes at fixed spin.
V The Berry phase
At this point of the discussion, it is interesting to point out that our description of the trajectory is able to keep track of the Berry phase of the wave. Originally, the Berry phase was proposed as the phase factor that can appear after a cycle during an adiabatic time evolution of a non-degenerated quantum state Berry (1984). Later it was discovered that the concept applies to light polarization as well Tomita and Chiao (1986); Kwiat and Chiao (1991) and that the time evolution of the system does not need to be cyclic or unitary Samuel and Bhandari (1988). More recently, it was shown that the Berry phase also shows up in multiply scattered light Maggs and Rossetto (2001). The expression for the Berry phase involves two factors: The spin of the particle and a solid angle (which we call the geometric phase), or more generally the curvature of the phase space of the system enclosed by the evolution of the system during one cycle. The Berry phase expression is Berry (1984)
In polymer physics, time is replaced by a curvilinear space coordinate and the geometric phase is called writhe Fuller (1971). The fluctuations of the writhe induced by thermal fluctuations of the polymer’s shape have been studied numerically using a Monte-Carlo approach Rossetto and Maggs (2002b). For long polymers, the distribution of the Berry phase was found Gaussian. There is no exact approach for the statistics of the Berry phase for random paths.
For simplicity, we consider the equivalent of a nearly adiabatic time evolution and compare the phases of the field before and after a path for which the initial and final frames are equal. Such a path corresponds to a closed circuit in the phase space Tomita and Chiao (1986). We also assume that the scattering is elastic and that the spin is conserved in both scattering and propagation. Between two scattering events, the state of a particle is simply defined by the direction of its momentum, and the phase space is thus the unit sphere. The adiabatic evolution would corresponds to a continuous and slow movement of the direction of the momentum on the sphere. In case of strong forward scattering each scattering event corresponds to a small change in the direction of propagation, which nearly satisfies the adiabatic condition for the Berry phase to occur. The geometric phase is the solid angle enclosed by the unit wave vector along its trajectory on the unit sphere.
To reveal the Berry phase, we consider a trajectory made of scattering events, and the local frames associated to each propagation. We denote by the Euler angles of each frame after the decomposition of equation (2). An example of this construction, with , is shown on Figure 3. We consider the case of strong forward scattering so that the coordinates and of the frames and are close to one another. In Figure 3, it corresponds to the case where the points and on the unit sphere are close to each other, compared to the radius of the sphere. We call the Euler angles of the rotations in the same way as we have done in Eq. (16). The strong scattering assumption corresponds to . To ensure readability of the Figure 3, we have represented a path with large angles that do not correspond to strong forward scattering. By analogy with a spinning top, for which the instantaneous rotation around the third axis is expressed by (Landau and Lifshitz, 1993, chap. VI) we deduce that the rotation around its direction of propagation experienced by the particle is . The Berry phase for a given component of the field is, in our case, identified as the phase difference between the values of this component expressed in the two frames and . The original definition of the Berry phase demands that the evolution be cyclic. So we have to impose that the frames and are equal, which corresponds to one cycle. Therefore we have and . The extra phase for a spin state between the frames and is, according to the phase factor in equation (21),
We have used the conservation of the spin (). We rewrite the sum using the quantities and we get, up to a multiple of :
We recognize that the sum in (26) is a spherical area. It is equal to the geometric phase up to a multiple of because of the indetermination of the solid angle Rossetto and Maggs (2002b). We have retrieved the expression of the Berry phase (24). The equality between the sum in expression (26) and the geometric phase holds modulo , rather than , for topological reasons Rossetto and Maggs (2002a). The indetermination modulo plays here no role because is an integer or a half-integer, so the phase is undetermined modulo .
Remarkably, the Berry phase emerges in our local frame model without any special effort. The geometrical nature of the Berry phase appears here clearly. As it was already demonstrated, the existence of the Berry phase is not restricted to cyclic systems and can be extended to non close path and to non-unitary evolution (in our theory, if there is absorption, for instance) Samuel and Bhandari (1988). The statistics of the Berry phase for random walks is a technical and difficult problem Krishna et al. (2000); Rossetto (2005). Our approach will make it possible to obtain exact results concerning the Berry phase statistics.
A possible experiment to observe the Berry phase of light consists in setting a directional lineary polarized source at the edge of a slab of a medium filled with scatterers and observe the outgoing light through a collimator and a linear polarizer. To ensure strong forward scattering the scatterers must be large such that the anisotropy , defined as the average cosine of the scattering angle, verifies . One can for instance place the polarizer in front of the source, its plane orthogonal to the direct beam and its axis making an angle with the polarization of the source. The intensity as a function of will display deviations from the expected behavior if there were no scatterers. If the slab depth is large, the Berry phase will spread out on a width larger than and the intensity will be constant as a function of . The experiment has therefore to be performed with slab depths of a few transport mean free paths ( is defined in the next section). Different concentrations will lead to different deviations.
Vi Generalized Green-Dyson equation
The complexity in multiple scattering stems from the summation of the contributions to the field of all the paths from the source to the observer, and potentially from their interferences. In a first approximation one often neglects these interferences and assumes that the scattering events are independent from each other. In a disordered system the averaging over disorder can thus be performed without the sophisticated diagram techniques needed to preserve interference effects. The elements of the theory that we have introduced in sections III and IV are used in this section to write a generalized Green-Dyson equation for the multiple scattering of polarized waves under the assumption of independent scattering events. This equation applies to directional Green’s operators: It relates the free Green’s operator we have introduced in section III to the effective Green’s operator of the same medium filled with scatterers. In the second part of this section, we introduce the Fourier transform of the generalized Green-Dyson equation.
Consider a medium filled with scatterers at fixed positions . The presence of the scatterers is equivalent to the introduction of the perturbation in the transport equation. In the formalism of Green’s functions it corresponds to introducing a transition matrix, or -matrix. We now consider the perturbed Green’s operator developped as a Born expansion in terms of :
The first term of the right-hand side of (27) stands for the unscattered field, the second term for the single scattering and so on.
For a known distribution of scatterers, the operator is the Green’s function of the multiply scattering system for polarized waves. If the functions , (defined in section III and IV respectively) and the operator are known, equation (27) for solves the problem. But usually we are interested in disordered systems, for which the exact position of scatterers is unknown. It will thus be impossible to make a prediction depending on the particular realization of disorder, but it is possible to average over all such realizations (ensemble average) and to write a generalized Green-Dyson equation for the average operators. We formulate the results using the simplest possible model of disorder. To perform the average over disorder, we neglect the volume of the scatterers and consider that each position is uniformly distributed in the volume . The contribution of disorder in (27) comes exclusively from the averaging of the operator . We introduce the self-energy
The line denotes the average over all positions for all the scatterers, which we will explicitely define now for the calculation of the average density. The average of an operator is simply defined for arbitrary and by . If the scatterers all have the same operator the self-energy becomes
where is the mean density. Our approach of disorder is nonetheless naive, we have neglected the possibility that a trajectory visits several times the same scatterer and thereby omitted some possible correlations in the averaging procedure. The topic of correlations created by several visits to the same scatterer is addressed abundantly in the literature. The most remarkable result is that for a sufficiently small density of scatterers, the expression of the Green-Dyson equation remains the same, but with a modified self-energy expressed as an expansion in powers of van Rossum and Nieuwenhuizen (1999). The first order of the expansion is exactly the expression we have found in formula (28), which is an encouraging fact in favor of an equivalent expansion in our framework. Such an equivalent expansion would extend the domain of validity of our results to higher densities. We will address this question in our future work.
Let us call the average Green’s operator resulting from the averaging over disorder of the operator defined in (27). In the case where all scattering events are independent, the average of the products involving several times are the products of the averages, this is sometimes called the Twersky approximation Mischchenko et al. (2006). The Born expansion (27) becomes, in this approximation the series
We have so far expressed the functions only in the direct space of position and frames. Physical problems however are easier to formulate in the reciprocal space, because convolutions become products. We denote by the reciprocal space representation of and we have the well known contraction
which gives, together with the closure relation
, the fundamental relations for the
As the system is invariant along time translations, the operators , , , noted as , depend on one frequency . Similarly, in the average medium obtained after integration over disorder, the translational invariance is restored the operators only depend on one vector . We introduce the notation for operators acting on the Hilbert space made of the elements , so that . The representation in reciprocal space for rotation and spin has different properties and is presented in the next section using an expression equivalent to (31).
The Born expansion written using the operators , and reduces to
The operator has a norm and is therefore smaller than for large or sufficiently small density . Consequently, the series converges and we can rewrite the expansion (32) in a self-consistent way
Equation (33) is a generalized Green-Dyson equation for the directional Green’s operators. The convergence of the series occurs when where is the extinction mean free path. The theory described by the generalized Dyson equation does not apply to situations where because scatterers are not in the far-field of each other anymore. In the next section, we introduce the rotational harmonics transform on the variable to solve equation (33).
In the expression of the free Green’s operator , Eq. (14), the factor modifies the usual expression of the Green’s function , where the quantity is a infinitesimaly small positive quantity. Using the identity
we get the Fourier transform
The superscript stands for the retarded and advanced Green’s operator. As we have already mentioned, the scalar theory must be retrieved if we sum our expressions over and . Here we can check that equation (34) integrated over and summed up on yields formally
The regularization by the infinitesimal imaginary term can be performed on the r.h.s term of Eq. (35) so that we retrieve the usual Green’s function for the field component with spin . The superposition of all partial directional fields cancels out because of the phases of all the spins with . The final result depends on : the directivity of the directional Green’s operator (34) is lost. We call the expression the directional factor of the Green’s operator.
The Green’s operator does not necessarily act separately on and . In optics for instance the Faraday effect in a medium with Verdet constant creates a phase shift . In this case, provided that the spin is conserved during propagation, equation (34) is modified and the Green’s operator becomes
The separation of the article into two parts to explain the harmonic transforms is intended only to keep the explanations simple. The Fourier transform presented in this section and the rotational harmonics transform that we introduce in the next section could have been presented simultaneously. This is actually mandatory in the case of a spin-orbit coupling involving and like the Faraday effect, as illustrated by formula (36).
Vii Rotational harmonics representation
The linear representation of compact Lie groups provides an harmonical analysis on the group which is the equivalent of Fourier series for periodic functions. These harmonical analysis is different from the Fourier transform because is a compact, non-commutative Lie group. For our purpose there is no need to know this mathematical theory in detail. Some formulæ used in our theory are presented in the appendix A.
The reciprocal representations are labelled by a main index which can be an integer or a half integer and two sub-indices and taking the values . We denote them by and we have
with , and the Euler angles of . The closure relation,
is useful to obtain most of the result provided in the present work. We call the function the rotational harmonics. (the misleading terminology “generalized spherical harmonics” is often used). These functions are normalized Wigner -functionsWigner (1959); Gel’fand and Šapiro (1956). The indices of the sums on , and in (38) have steps equal to . , the lowest order of , depends on the spin of the wave and takes the value or if the spin is an integer or a half-integer respectively. It follows that the indices , and are either all integers or all half-integers depending on . For a given (the order of the harmonics) there are orthogonal functions, which seems a lot. We will see that for only harmonics components have to be taken into account. The functions are defined from Euler angles and are real-valued, their full expression is given in Equation (50). We have added the unconventional factor to obey the identity (that is we have suppressed the factor in this latter relation).
The rotational dependence of a state can be expressed as a function of the frame with the bracket notation or as a function of rotational harmonics indices . Our computation uses this latter representation because it is related to the spin in a fundamental way: Let us use the closure relation of the rotation representation to get the expression
According to the relations (4) and (37) the integration over the spin angle just leaves a factor , stating that the representation for a spin eigenvalue involves only the kets . In the theory of multiple scattering, is fixed and there is only one species of particle involved. It means that in the Hilbert space of physical states the equality is valid and the -index can be left out without any restriction on the physical content of the theory.
We use the rotational harmonics to expressed the products of the operators used in the generalized Dyson equation (33):
Would we have used the representation in , we would have obtained integrals instead of the discrete sums. The sums over and are infinite. However, like in Fourier series, it is possible to truncate the sum because the brackets amplitudes decrease for large like (for a certain , ) such that high order coefficients can be neglected. The sums on indices , and can be seen as sums over a single index. Equation (40) is therefore a discrete linear equation and is, up to a one-to-one transformation of into a single index, a matrix equation. If the coefficients of and are known, one can compute with standard linear algebra and get a general solution for the transport of polarization in multiple scattering by inverting the matrix of .
The directional Green’s operator describes the transport in a effective, homogeneous medium. Paths statistics and Berry phases are included in this description, which represent an improvement over the usual effective Green’s functions used in multiple scattering theories. In the presence of several different kinds of scatterers, with densities and scattering matrices , we get an effective Green’s operator by replacing by : The scattering properties of the different scatterers are averaged and their respective weights are proportional to their respective densities. If the scatterers are not spherical and have random independent orientations, the effective -matrix is the average over all orientations of the orientation-dependent -matrices. In particular, if the orientation probability is uniform, the average -matrix is equivalent to a spherical scatterer’s -matrix and the spherical symmetry can be used as is described in the next section.
To compute the rotational harmonics expansion of up to order , one needs to use approximately triplets for the incoming frame, and as much for the outgoing frame. Therefore required number of rotational harmonics coefficients for the computation of the Green’s operator scales like . The complexity of the inversion of the linear system (40) scales approximately like .
Viii The role of rotational invariance
We have considered in the previous sections the general case of an homogeneous medium invariant under translations and we have decomposed the vectors of the three-dimensional space into a radius part and an angular part to construct the general formalism for a directional Green’s operator. A special and important case shows up when the medium and the scatterers are also both invariant under rotation. In the representation, we have in this case for all frames and the equality already discussed in section IV
Using the closure relation (38) we get
By identification of the terms of the right-hand sum with the rotational harmonics expansion in and of the left-hand term (56) we get
If the medium and the scatterers are both invariant under rotations, the free Green’s operator and the Green-Dyson operator are both independent from the reference frame. These operators depend on and on the incoming and outgoing directions of propagation and , a rotation of the reference frame acts simultaneously on these three vectors. Using the decomposition
where is a rotation matrix and , we express the invariance of the Green’s operators as