STOKES VECTOR-PARAMETER

(Matrix Methods for Accounting for Radiation Polarization in the Ray Optics Approximation)

G. V. Rozenberg

§ 1. INTRODUCTION

The ray-optics approximation, as is well known, is applicable in considering a great variety of optical problems, whose broad range is by no means exhausted by the problems of instrumental optics, but includes all cases in which it is possible to give sufficient definiteness to the concept of a light beam and to trace the fate of individual beams in the acts of their interaction with matter. In doing so, one should distinguish phenomena of two types. First, there are phenomena of propagation in quasi-homogeneous media (refraction, birefringence, absorption, interference, etc.), when the parameters of the light beam change smoothly along its length and the problem reduces to determining the character of these changes as a function of the properties of the medium. Second, there are phenomena of scattering, reflection, refraction, etc., in which the light beam undergoes sharp local transformations and gives rise to light beams in other directions. In the latter case the ray-optics approximation is applicable to the radiation before and after its interaction with matter, but only outside the wave zone (at infinity), while the problem of formulating the laws of local transformation of light beams proves analogous to that which we encounter in quantum mechanics when considering collision phenomena by means of the scattering matrix.

As is not difficult to see, such a formulation of the problem is adequate to an appeal to the corpuscular aspect of radiation, which, in particular, expresses the well-known analogy between ray optics and classical mechanics. Following this analogy, however, it is necessary to note that it extends not only to questions connected with the trajectory of the photon flux, but also to the dynamical characteristics of the latter (energy flux, spin), thereby going far beyond the framework of directly geometrical optics.

optics. The transition to semiclassical photon representations, corresponding to the transition to the approximation of geometrical optics, necessarily requires taking into account both the intensity and the polarization of the light beam, since the result of the interaction of the latter with matter depends essentially on these parameters. The organic inclusion of the named characteristics in the system of geometrical optics has not yet been achieved. In this direction only the first steps are being taken, one of which is the development of matrix methods for taking polarization into account, which are the subject of the present article.

At the present time matrix methods have been developed in sufficient detail only for two limiting cases—completely coherent and completely incoherent monochromatic radiation. This makes it possible to use them effectively for solving the most essential problems concerning a quasi-stationary field, i.e., under conditions in which the incoherence of the different spectral components of the radiation is preserved. The more general case of partially coherent and, in particular, pulsed radiation still requires detailed investigation and is left outside the scope of our consideration. Likewise, we shall pass over in silence certain other, very interesting and important problems connected with the introduction into the system of geometrical optics of the dynamical characteristics of a light flux (for example, questions of the correct transition to the geometrical-optics approximation with allowance for the vector character of the electromagnetic field). The purpose of the article is not so much a rigorous mathematical substantiation of the matrix method itself as to acquaint a wide circle of readers with its foundations and with the ways of applying it, in view of the great practical significance that it has acquired in recent years, especially in connection with scattering problems, where its use makes it possible to pose and solve a number of previously inaccessible problems (for example, the formulation of a radiation-transfer equation with allowance for the polarization of the radiation). Such a survey is all the more necessary because in the domestic literature matrix methods for taking the polarization of radiation into account have not been covered at all, while in the foreign literature information about them is scattered over a rather large number of articles devoted to quite diverse questions and, in essence, has still not been systematized in a proper way, which seriously hinders their practical use.

§ 2. DECOMPOSITION OF THE ELECTRIC-FIELD STRENGTH VECTOR OF A WAVE INTO ELLIPTICALLY POLARIZED COMPONENTS AND THE MATRIX METHOD FOR COHERENT RADIATION

Consider a plane electromagnetic wave of angular frequency \(\omega\), propagating in some direction \(l\), the complex vector of the electric-field strength of which is equal to
\(\mathbf{E}=(E_1^0\mathbf{e}_1^0+E_2^0\mathbf{e}_2^0)e^{i\omega t}\), where \(E_i^0\) are quantities independent (or weakly dependent-

depending on time \(t\) complex amplitudes, and \(\mathbf e_1^0\) and \(\mathbf e_2^0\) are unit vectors orthogonal to the direction of propagation of the wave. The most general decomposition of \(\mathbf E\) into components satisfying the requirements of orthogonality and normalization is the decomposition into two elliptic oscillations with mutually opposite directions of rotation, mutually orthogonal major axes of the ellipses, and the same ratio of their semiaxes. Without loss of generality one may assume both oscillations to be in phase, since the phase difference is automatically taken into account by the complex coefficients of the decomposition. Let us denote by \(\operatorname{tg}\gamma\) the ratio of the minor semiaxis of an ellipse to its major semiaxis, and by \(\psi\) the angle formed with the unit vector \(\mathbf e_1^0\) by the major semiaxis of the ellipse having right-hand rotation. The plane \(Q\), determined by the direction of the unit vector \(\mathbf e_1^0\) and by the direction of propagation of the wave, will be called the reference plane.

It is not difficult to verify\({}^{31}\) that the transformation of the vector \(\mathbf E\) from the representation \(\gamma,\psi\) to the representation \(\gamma',\psi'\) is then given by the expression

\[ E(\gamma',\psi')=L(\gamma',\psi',\gamma,\psi)E(\gamma,\psi), \tag{1} \]

where \(L\) is a matrix of the form:

\[ \begin{aligned} L(\gamma',\psi',\gamma,\psi) &= \left| \begin{array}{cc} \cos(\gamma'-\gamma)\cos(\psi'-\psi)-& -\cos(\gamma'+\gamma)\sin(\psi'-\psi)+\\ {}-i\sin(\gamma'+\gamma)\sin(\psi'-\psi)& {}+i\sin(\gamma'-\gamma)\cos(\psi'-\psi)\\[4pt] -\cos(\gamma'+\gamma)\sin(\psi'-\psi)+& \cos(\gamma'-\gamma)\cos(\psi'-\psi)+\\ {}+i\sin(\gamma'-\gamma)\cos(\psi'-\psi),& {}+i\sin(\gamma'+\gamma)\sin(\psi'-\psi) \end{array} \right|. \end{aligned} \tag{2} \]

Owing to the linearity and homogeneity of the equations of electrodynamics,\({}^{*}\) the electric-field strength \(\mathbf E'\) in a light beam which has interacted with matter and is propagating at some point \(\mathbf r'\) in the direction \(\mathbf l'\) will be a linear homogeneous function of the vector \(\mathbf E\) in the light beam before the interaction, propagating at some point \(\mathbf r\) in the direction \(\mathbf l\). Consequently, the result of the interaction of radiation with matter may be represented in the form

\[ E'_i(\mathbf r',\mathbf l')=\sum_k \mu_{ik}(\mathbf r',\mathbf l',\mathbf r,\mathbf l)E_k(\mathbf r,\mathbf l) \tag{3} \]

\[ (i,\ k=1,\ 2), \]

where the components \(\mu_{ik}\) of the matrix \(\mu\), describing the action of matter

\({}^{*}\) In the case of violation of the linearity of the equations of electrodynamics (say, in the phenomena of saturation of absorption and induced radiation observed in the microwave range), everything that follows loses its validity.

upon the light beam, are, generally speaking, complex. Since a quasi-stationary case is being considered, \(\mu_{ik}\) do not depend on time. The common time factor \(e^{i\omega t}\) will everywhere henceforth be omitted.

It is always possible to find representations \(\gamma, \psi\) for the transformed beam and \(\gamma', \psi'\) for the transformed beam such that the matrix \(\mu\) is diagonalized. This means that the light beam can be decomposed into two alternatively polarized components which, in the given act of interaction with matter, are transformed independently of one another. Since (see below) the decomposition into elliptically polarized components given by the matrix \(L\) is, in essence, a decomposition with respect to linear forms relative to the proper functions of the photon spin operator, every interaction of radiation with matter therefore constitutes a spectral decomposition of the light beam into alternative spinor components and an independent transformation of these components.

In some cases it proves possible to find a representation \(\gamma, \psi\), common to the transformed and the transformed beams, in which the matrix \(\mu\) is diagonalized. A necessary and sufficient condition for this is the relation

\[ \mu\mu^{+}=\mu^{+}\mu, \tag{4} \]

where \(\mu_{ik}^{+}=\mu_{ki}^{*}\) (the asterisk, as usual, denotes complex-conjugate quantities), or, what is the same thing,

\[ \left. \begin{aligned} |\mu_{21}|&=|\mu_{12}|,\\ \arg \mu_{12}+\arg \mu_{21}&=2\arg(\mu_{22}-\mu_{11}). \end{aligned} \right\} \tag{5} \]

A necessary and sufficient condition for reducing the matrix \(\mu\) to diagonal form with real components is its Hermiticity.

Thus, every action of matter on a light beam of a given frequency may be regarded as a linear, generally complex, transformation of the complex vector of the electric-field strength, possibly accompanied by a change in the direction of the beam and depending on the coordinates of the initial \((\mathbf r)\) and final \((\mathbf r')\) points to which the description of its state refers. The result of a series of successive transformations is obtained by successive application of the corresponding matrices \(\mu\), i.e. the matrix of the complete transformation is the product of the matrices of the partial transformations. The matrix corresponding to a series of parallel (simultaneous) coherent transformations is the sum of the matrices of the partial transformations.

When a beam propagates in a quasi-homogeneous medium, the change of its parameters occurs continuously as a result of dispersion phenomena (double refraction, absorption). In this case a differential transformation matrix \(\nu\) may be introduced,

determining the change of the beam over an element of its length \(dl\):

\[ dE=-\nu E\,dl. \tag{6} \]

The action of a quasi-homogeneous medium on a light beam passing through it reduces, in addition to refraction, first, to the removal from the beam of a certain number of quanta of a definite polarization and, second, to a change in the polarization of the remaining flux owing to the difference in the propagation velocities of two alternatively polarized components. Therefore the matrix \(\nu\) decomposes into the sum of two matrices corresponding to the two indicated parallel transformations of the beam:

\[ \nu=\nu^{\mathrm e}+\nu^{\phi}. \tag{7} \]

One can always find such representations \(\gamma,\psi\) in which either the extinction matrix is diagonalized,

\[ \nu^{\mathrm e}=\nu^{\mathrm e}_{ik}\delta_{ik}, \tag{8} \]

corresponding to dichroism, or the phase matrix

\[ \nu^{\phi}= \left| \begin{array}{cc} i\varphi & 0\\ 0 & -i\varphi \end{array} \right| +i\varphi_0\delta_{ik}, \tag{9} \]

corresponding to birefringence. Let us note that the diagonalized extinction matrix must have real components, i.e. in an arbitrary representation \(\gamma,\psi\) it will be Hermitian, as will also the matrix \(i\nu^{\phi}\), since \(\varphi\) and \(\varphi_0\) in (9) are real.

In the general case the matrices \(\nu^{\mathrm e}\) and \(\nu^{\phi}\) may turn out to be simultaneously irreducible to diagonal form. If, however, \(\nu\) satisfies condition (4), then the medium is dichroic and birefringent for the same alternatively polarized (elliptic) components, corresponding to some representation \(\gamma,\psi\). Thus there arises the possibility of separately tracing the fate of each of the components individually, and at the same time of introducing refractive indices for them. Indeed, in the corresponding representation \(\gamma,\psi\) the matrix \(\nu\) assumes, in the case under consideration, the form

\[ \nu= \left| \begin{array}{cc} i(\varphi_0+\varphi) & 0\\ 0 & i(\varphi_0-\varphi) \end{array} \right|, \tag{10} \]

where \(\varphi_0\) and \(\varphi\) are complex quantities with nonnegative imaginary part (there is no amplification of the ray), whence

\[ E_i(l)=E_i(0)e^{-i(\varphi_0\pm\varphi)l}=E_i(0)e^{-ik_0 n_i l}, \]

where the refractive index for the \(i\)-th component is determined by the relation

\[ n_i=\frac{1}{k_0}(\varphi_0\pm\varphi) \tag{11} \]

(\(k_0\) is the wave number for vacuum).

In the case when the coherent interaction of the particles scattering the radiation may be neglected, i.e., under conditions when the distances between the scattering particles exceed the wavelength of the radiation, the differential matrix \(\nu\) takes the form[^31]

\[ \nu_{ik}=\frac{2\pi N}{k^2}\,\overline{g}_{ik}(1,1), \tag{12} \]

where \(N\) is the number of particles per unit volume, \(k\) is the wave number (generally speaking, complex) for the medium in which the particles are located, and \(\overline{g}_{ik}(1,1)\) are the expansion coefficients of the components of the amplitude scattering matrix of an individual particle (for example, the amplitude scattering coefficients in the Mie problem) in plane waves, corresponding to a plane scattered wave propagating in the same direction \(1\) as the incident wave, and averaged over all possible orientations of the scattering particles.

The considerations set forth pertain to the transformation of exclusively coherent beams (dispersion, reflection, refraction, scattering, interference, etc.). In particular, they may be made the basis for taking polarization into account in calculations of various optical instruments, especially polarizing ones (compensators, analyzers, etc.) and interferometric ones. In this case the problem is reduced to determining the form of the matrices \(\mu\) for each of the elements of the instrument and finding the complete matrix corresponding to the entire complex of successive transformations undergone by the light beam in passing through the instrument. In the case of incoherent beams (for example, “natural” light), the fate of each of them is traced independently. We note that the mathematical side of the question for the representation \(\gamma=0\) (expansion in linearly polarized components) has been partly considered in the works of Jones[^1] and Parke[^2], while examples of practical application of the method described (also for the representation \(\gamma=0\)) may be found in the works of Chien Yu-tzu, Richards and Yung Kang Liang[^3], Billings and Land[^4], Richards and Chien Yu-tzu[^5], and some others (see also A. V. Shubnikov’s book Optical Crystallography).

§ 3. STATISTICAL PARAMETERS FOR DESCRIBING INCOHERENT LIGHT BEAMS

The matrix method described above, intended for analyzing the transformations undergone by completely polarized coherent light beams in their interaction with matter or interference with one another, proves unsuitable when we encounter partially or completely depolarized light and the statistical processes of its interaction with matter, for example in phenomena of multiple scattering. Indeed, every light beam propagating in a turbid medium,

constitutes a complex of mutually incoherent light beams having different prehistories and, consequently, different intensities, phases, and polarization character. At the same time, in the transformations undergone by a light beam in its interaction with matter, the separate, generally speaking elliptically polarized “pure” components of this beam behave independently of one another. The transformations of each of these components are given by the laws of local transformations, identical for all components. However, the result of the transformations for different components of the “mixture” will be different, and the result of the transformation of such a “mixed” light beam will obviously be determined by its “composition,” the determination of which is a statistical problem. Thus there arises the necessity of abandoning the description of the light flux by means of field-strength vectors and of turning to statistical parameters corresponding to the essence of the phenomena under consideration. The latter must, on the one hand, ensure the completeness of the description of the properties of the light beam and, on the other hand, be additive for incoherent components of a “mixed” light flux.

Below we shall see that such parameters may be quadratic and bilinear forms with respect to \(E_i\), or their linear combinations. As is known, the set of four linearly independent time-averaged quadratic and bilinear forms with respect to \(E_i\) corresponds to the complete set of observable quantities (in the quantum-mechanical sense of the word), provided that the receivers of radiation are energy (i.e. quadratic) devices—the only ones that exist for radiation of sufficiently high frequency (including light)\(^{6-8}\). The corresponding mean quadratic and bilinear forms with respect to \(E_i\) may be introduced as follows (Wiener\(^{6}\), Parke\(^{7}\)).

Let us consider a completely (elliptically) polarized light beam, for which \(E_i\) are functions of coordinates and time, and find the interference matrix for the quantities \(E_i\) and \(E_j^*\) at some point \(\mathbf r\) of the beam*):

\[ \varphi_{ij}(\mathbf r,\tau)=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{+T} E_i(\mathbf r,t+\tau)E_j^*(\mathbf r,t)\,dt . \tag{13} \]

Carrying out, further, Fourier transformations, we obtain a spectral—

) Establishing the correlation between \(E_i\) and \(E_j^*\) at different* points of a quasistationary light field should apparently make it possible to formulate the laws of optics directly in terms of observable quantities; see, for example,\(^{9}\).

matrix

\[ S_{ij}(\mathbf r,\omega)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\varphi_{ij}(\mathbf r,\tau)e^{i\omega\tau}\,d\tau, \tag{14} \]

the totality of whose components exhausts the totality of observable quantities. If the radiation is monochromatic, i.e., if

\[ E_i(\mathbf r,t)=E_i(\mathbf r)e^{-i\omega_0 t}, \tag{15} \]

then

\[ \varphi_{ij}(\mathbf r,\tau)=E_i(\mathbf r)E_j^*(\mathbf r)e^{-i\omega_0\tau} \tag{16} \]

and

\[ S_{ij}(\mathbf r,\omega)=\frac{1}{2}E_i(\mathbf r)E_j^*(\mathbf r)\delta(\omega-\omega_0), \tag{17} \]

whence the integral spectral matrix, i.e., the complete set of mean bilinear forms, takes the form

\[ S_{ij}(\mathbf r)=\int_0^\infty S_{ij}(\mathbf r,\omega)\,d\omega =\frac{1}{2}E_i(\mathbf r)E_j^*(\mathbf r). \tag{18} \]

The following four parameters are apparently the most convenient for practical use:

\[ \left. \begin{aligned} S_1&=2(S_{11}+S_{22})=E_1E_1^*+E_2E_2^*,\\ S_2&=2(S_{11}-S_{22})=E_1E_1^*-E_2E_2^*,\\ S_3&=2(S_{12}+S_{12}^*)=E_1E_2^*+E_1^*E_2,\\ S_4&=-i2(S_{12}-S_{12}^*)=-i(E_1E_2^*-E_1^*E_2), \end{aligned} \right\} \tag{19} \]

forming the components of a certain four-dimensional vector \(\vec S\) in the corresponding four-dimensional functional space*). In what follows, for brevity we shall call the vector \(\vec S\)

*) For a completely (elliptically) polarized beam, the decomposition of \(\mathbf E\) into alternative linearly polarized components \((\gamma=0)\) has (up to an overall phase factor) the form:

\[ E_1=a_1;\qquad E_2=a_2e^{i\delta}, \]

where \(a_1\) and \(a_2\) are real numbers and \(\delta\) is the phase difference of the components, depending on the angle \(\varphi\) of inclination of the major axis of the ellipse to the reference plane \(Q\) (for \(\varphi=0\), \(\delta=\frac{\pi}{2}\)). Accordingly,

\[ S_1=a_1^2+a_2^2;\quad S_2=a_1^2-a_2^2;\quad S_3=2a_1a_2\cos\delta;\quad S_4=-2a_1a_2\sin\delta. \]

vector-parameter of a light beam. In the representation \(\gamma=0\) (decomposition into linearly polarized components), the components of the vector-parameter coincide (to within a constant factor) with the parameters introduced by Stokes\(^{10}\) and bearing his name\(^*\). Therefore \(\vec S\) could be called the generalized Stokes vector-parameter.

Let us note that expression (19) for the components of the vector-parameter can be written in the form:

\[ S_j=E\cdot\sigma_j E^*, \]

where \(\sigma_1\) is the unit matrix \(\hat e_{ik}\), while \(\sigma_2=\sigma_x,\ \sigma_3=\sigma_y,\ \sigma_4=\sigma_z\) are the Pauli spin matrices.

Since for incoherent light beams the cross terms of the interference matrix \(\varphi_{ij}\) vanish, the vector-parameters of incoherent beams are additive. From the point of view of quantum mechanics, a completely polarized beam is a “pure case,” whereas a partially polarized beam, being an incoherent aggregate of completely polarized beams, forms a “mixture”\(^{6-8}\). Consequently, the vector-parameter of a mixture is formed from the vector-parameters of its pure components by simple summation. Thus, the state of a light beam is completely determined if its direction \(\mathbf l\), frequency \(\omega\), and vector-parameter \(\vec S\) are specified.

§ 4. PROPERTIES OF THE VECTOR-PARAMETER

As follows from the way in which the vector-parameter was introduced, its components depend essentially on the choice of the reference plane \(Q\) (which includes the direction of the ray \(\mathbf l\) and the direction of the unit vector \(\mathbf e_1^0\)) and of the representation \(\gamma,\psi\). It is therefore important to find the transformation law of the vector-parameter from one representation to another, as well as the invariants of this transformation, i.e. quantities immanent to the given beam and determining its properties independently of the reference system. Obviously, finding the transformation law will also make it possible to answer the question of the intensity of the beam if the latter is considered through an analyzer transmitting one or another of the elliptically polarized components, i.e. to find the probability of detecting photons with the given polarization.

\(^*\) In the literature there are no generally accepted notations for the Stokes parameters. In particular, they are denoted either\(^{10,11}\) \(S_1=I,\ S_2=M,\ S_3=C,\ S_4=S\), or\(^{12}\) \(S_1=I,\ S_2=Q,\ S_3=U,\ S_4=V\), or \(S_1=I,\ S_2=P_1,\ S_3=P_2,\ S_4=P_3\). For reasons that will be clear from what follows, the notation adopted here is the most convenient.

It is easy to show[^31] that the transformation of the vector-parameter from the representation \(\gamma, \psi\) to the representation \(\gamma', \psi'\) is a homogeneous linear operation

\[ \vec S(\mathbf r,\gamma',\psi')= K(\gamma',\psi',\gamma,\chi)\,\vec S'(\mathbf r,\gamma,\psi), \tag{20} \]

where the matrix \(K\) has the form

\[ K(\gamma',\psi',\gamma,\psi)= \tag{21} \]

\[ = \left| \begin{array}{cccc} 1 & 0 & 0 & 0\\[2mm] 0 & \cos 2\gamma'\cos 2\gamma \times \cos 2(\psi'-\psi)+ \sin 2\gamma'\sin 2\gamma & \cos 2\gamma'\sin 2(\psi'-\psi) & \cos 2\gamma'\sin 2\gamma \times \cos 2(\psi'-\psi)- \sin 2\gamma'\sin 2\gamma \\[2mm] 0 & \cos 2\gamma \sin 2(\psi'-\psi) & \cos 2(\psi'-\psi) & -\sin 2\gamma \sin 2(\psi'-\psi) \\[2mm] 0 & \sin 2\gamma'\cos 2\gamma \times \cos 2(\psi'-\psi)- \cos 2\gamma'\sin 2\gamma & \sin 2\gamma'\sin 2(\psi'-\psi) & \sin 2\gamma'\sin 2\gamma \times \cos 2(\psi'-\psi)+ \cos 2\gamma'\cos 2\gamma \end{array} \right|. \]

The invariants of this transformation are:

\[ I=S_1 \tag{22} \]

\[ M=qS_1=\sin 2\gamma S_2-\cos 2\gamma S_4, \tag{23} \]

\[ p^2S_1^2=(\cos 2\gamma S_2+\sin 2\gamma S_4)^2+S_3^2, \tag{24} \]

\[ U^2=r^2S_1^2=(p^2+q^2)S_1^2=S_2^2+S_3^2+S_4^2, \tag{25} \]

and, finally,

\[ (1-r^2)S_1^2=S_1^2-(S_2^2+S_3^2+S_4^2). \tag{26} \]

The physical meaning of the invariant \(I=S_1\) is obvious. This is the intensity of the beam, i.e. the modulus of the Poynting vector (in what follows the term “intensity” is used everywhere in this sense). No less obvious is the physical meaning of the invariant \(M\): it is the flux density of angular momentum (spin) of a plane electromagnetic wave, multiplied by the radiation frequency. Indeed, it is not difficult to show that

\[ M=\pm \omega |[\mathbf E\mathbf A]|, \tag{27} \]

where \(\mathbf A\) is the vector potential of the electromagnetic field of the wave; the expression standing on the right, as is known[^14–^16], is the action integral for the propagation of electromagnetic waves in vacuum and corresponds to the spin angular momentum of the wave. The validity of relation (27), first indicated (in implicit form) by A. A. Sadovskii[^17], is confirmed, for example, by the experiments of Beth and Carradi[^19].

The quantity \(q\), which characterizes the ratio of the flux of rotational (spin) momentum of the beam to the energy flux, may be called

degree of ellipticity. \(q>0\) corresponds to right-hand rotation, \(q<0\) to left-hand rotation, \(q=\pm 1\) to circular polarization; in the pure case \(q=0\) corresponds to linear polarization. The invariant \(|p|\) is nothing other than the ratio of the difference of the extreme values of the beam intensity, when it is viewed through an analyzer oriented in various ways, to their sum:

\[ |p|=\frac{I_{\max}-I_{\min}}{I_{\max}+I_{\min}}, \tag{28} \]

i.e., what is usually called the degree of polarization. \(p=1\) corresponds to linear polarization, and \(p=0\), in the pure case, to circular polarization. As for the invariant \(r^{2}=p^{2}+q^{2}\), for the pure case \(r^{2}=1\). For a mixture, however, this relation no longer holds. In particular, for “natural” light \(r^{2}=0\). Thus the parameter \(r\) characterizes the degree of difference in the polarization of the separate pure components forming the mixture. Since a mixture of identically polarized components (a “homogeneous” mixture) is indistinguishable from the corresponding pure case (which is an expression of their experimental indistinguishability), the invariant \(r\) should be regarded as a measure of the “degree of homogeneity” of the mixture. It is known, however, that partially polarized light can be regarded as a mixture, in a certain proportion, of completely polarized and natural light\(^{20}\). Obviously, the parameter \(r\) characterizes precisely this proportion, i.e., it may be interpreted (in the indicated sense) as a measure of the degree of polarization (as is sometimes done). However, such terminology, although it corresponds more to the essence of the matter, seems inadvisable, since it introduces confusion with the generally accepted definition of the degree of polarization, corresponding to the quantity \(p\).

Let us give several relations connecting the components of the vector-parameter and the transformation invariants (21) with certain other quantities used to characterize the beam. These relations prove useful in practical calculations.

For the pure case one can find such a representation \(\gamma_0,\ \psi_0\), in which only one of the two alternatively polarized components remains (for example, \(E_1;\ E_2=0\)). Then\(^{31}\)

\[ q=\sin 2\gamma_0, \tag{29} \]

\[ p=\cos 2\gamma_0, \tag{30} \]

\[ I_{\max}=S_1\cos^2\gamma_0, \tag{31} \]

\[ I_{\min}=S_1\sin^2\gamma_0, \tag{32} \]

\[ \operatorname{ctg} 2(\psi_0-\psi)= \frac{S_2\cos 2[[unclear: subscripted angle]]+S_4\sin 2[[unclear: subscripted angle]]}{S_3}. \tag{33} \]

Relation (33), in contrast to (29)—(32), is also preserved for a mixture; here by \(\psi_0\) one should understand the value of \(\psi\) corresponding to the extremal intensity of the radiation transmitted by an analyzer that selects the component linearly polarized in the plane making, with the reference plane \(Q\), the angle \(\psi+\frac{\pi}{2}\) (the so-called directions of “greatest” and “least” polarization).

For a mixture the following relations are also valid:

\[ S_3=Ip\sin 2(\psi_0-\psi), \tag{34} \]

\[ S_2\cos 2\gamma+S_4\sin 2\gamma=Ip\cos 2(\psi_0-\psi), \tag{35} \]

\[ S_2\sin 2\gamma-S_4\cos 2\gamma=Iq. \tag{36} \]

§ 5. THE MATRIX OF INTERACTION OF RADIATION WITH MATTER AND THE EXPERIMENTAL DETERMINATION OF THE VECTOR-PARAMETER

Owing to the linearity and homogeneity of Maxwell’s equations, any interaction of radiation with matter can be described as a linear homogeneous transformation of the vector-parameter of a light beam as a function of its coordinates and direction. Indeed, we have seen that for each pure component of a beam relation (3) holds. Let us now form, in accordance with (13)—(18), the spectral matrix of the transformed beam. Since we are considering a quasi-stationary field, the coefficients of the matrix \(\mu\) do not depend on time, and for the interference matrix we have:

\[ \varphi_{ik}(\mathbf r',\tau)=\sum_{\rho,\sigma}\mu_{i\rho}\mu^{*}_{k\sigma}\varphi_{\rho\sigma}(\mathbf r,\tau), \tag{37} \]

whence

\[ S_{ik}(\mathbf r',l')=\sum_{\rho,\sigma}\mu_{i\rho}\mu^{*}_{k\sigma}S_{\rho\sigma}(\mathbf r,l). \tag{38} \]

Passing further, according to (19), to the components of the vector-parameter \(S_l\), we find:

\[ \vec S(\mathbf r',l')=D(\mathbf r',l',\mathbf r,l)\vec S(\mathbf r,l), \tag{39} \]

where the transformation matrix \(D\) has the form

\[ D=\frac{1}{2} \left| \begin{aligned} &(\mu_{11}\mu_{11}^{*}+\mu_{12}\mu_{12}^{*}) +(\mu_{11}\mu_{11}^{*}-\mu_{12}\mu_{12}^{*}) +(\mu_{11}\mu_{12}^{*}+\mu_{12}\mu_{12}^{*}) +i(\mu_{11}\mu_{12}^{*}-\mu_{11}\mu_{11}^{*}) \\ &+(\mu_{21}\mu_{21}^{*}+\mu_{22}\mu_{22}^{*}) +(\mu_{21}\mu_{21}^{*}-\mu_{22}\mu_{22}^{*}) +(\mu_{21}\mu_{22}^{*}+\mu_{22}\mu_{22}^{*}) +i(\mu_{21}\mu_{22}^{*}-\mu_{21}\mu_{22}^{*}) \\ &(\mu_{11}\mu_{11}^{*}+\mu_{12}\mu_{12}^{*}) -(\mu_{11}\mu_{11}^{*}-\mu_{12}\mu_{12}^{*}) -(\mu_{11}\mu_{12}^{*}+\mu_{12}\mu_{12}^{*}) -i(\mu_{11}\mu_{12}^{*}-\mu_{11}\mu_{11}^{*}) \\ &-(\mu_{21}\mu_{21}^{*}+\mu_{22}\mu_{22}^{*}) -(\mu_{21}\mu_{21}^{*}-\mu_{22}\mu_{22}^{*}) +(\mu_{21}\mu_{22}^{*}+\mu_{22}\mu_{22}^{*}) -i(\mu_{21}\mu_{22}^{*}-\mu_{21}\mu_{22}^{*}) \\ &(\mu_{11}\mu_{11}^{*}+\mu_{11}\mu_{21}^{*}) +(\mu_{11}\mu_{11}^{*}+\mu_{11}\mu_{21}^{*}) +(\mu_{11}\mu_{22}^{*}+\mu_{11}\mu_{22}^{*}) +i(\mu_{11}\mu_{22}^{*}-\mu_{11}\mu_{22}^{*}) \\ &+(\mu_{12}\mu_{12}^{*}+\mu_{12}\mu_{22}^{*}) +(\mu_{12}\mu_{12}^{*}+\mu_{12}\mu_{22}^{*}) +(\mu_{12}\mu_{21}^{*}+\mu_{12}\mu_{21}^{*}) -i(\mu_{12}\mu_{21}^{*}-\mu_{12}\mu_{11}^{*}) \\ &i(\mu_{11}\mu_{11}^{*})-i(\mu_{11}\mu_{21}^{*}) +i(\mu_{11}\mu_{22}^{*}-\mu_{11}\mu_{22}^{*}) +(\mu_{11}\mu_{22}^{*}+\mu_{11}\mu_{22}^{*}) \\ &+i(\mu_{12}\mu_{12}^{*})-i(\mu_{12}\mu_{22}^{*}) +i(\mu_{12}\mu_{21}^{*}-\mu_{12}\mu_{21}^{*}) -(\mu_{12}\mu_{21}^{*}+\mu_{12}\mu_{21}^{*}) \end{aligned} \right| \tag{40} \]

where all its components are real quantities. In the case when the beam propagates in a quasi-homogeneous medium, it appears possible to introduce the differential matrix \(\chi\) of the transformation of the vector-parameter:

\[ d\vec S=-\chi(\mathbf r,l)\vec S\,dl, \tag{41} \]

related to the matrix \(\nu\) (see p. 81) by the relation \({}^{31}\)*)

\[ \chi=\frac{1}{2} \left| \begin{array}{rrrr} \operatorname{Re}(\nu_{11}+\nu_{22}) & \operatorname{Re}(\nu_{11}-\nu_{22}) & \operatorname{Re}(\nu_{12}+\nu_{21}) & -\operatorname{Im}(\nu_{12}-\nu_{21}) \\[2mm] \operatorname{Re}(\nu_{11}-\nu_{22}) & \operatorname{Re}(\nu_{11}+\nu_{22}) & \operatorname{Re}(\nu_{12}-\nu_{21}) & -\operatorname{Im}(\nu_{12}+\nu_{21}) \\[2mm] \operatorname{Re}(\nu_{12}+\nu_{21}) & -\operatorname{Re}(\nu_{12}-\nu_{21}) & \operatorname{Re}(\nu_{11}+\nu_{22}) & \operatorname{Im}(\nu_{11}-\nu_{22}) \\[2mm] \operatorname{Im}(\nu_{12}-\nu_{21}) & \operatorname{Im}(\nu_{12}+\nu_{21}) & -\operatorname{Im}(\nu_{11}-\nu_{22}) & \operatorname{Re}(\nu_{11}+\nu_{22}) \end{array} \right|. \tag{42} \]

It is not difficult to see that the components of the matrix \(\chi\) satisfy the conditions

\[ \begin{gathered} \chi_{11}=\chi_{22}=\chi_{33}=\chi_{44};\quad \chi_{12}=\chi_{21};\quad \chi_{13}=\chi_{31};\quad \chi_{14}=-\chi_{41};\\ \chi_{32}=-\chi_{23};\quad \chi_{24}=-\chi_{42};\quad \chi_{34}=-\chi_{43}, \end{gathered} \tag{43} \]

which is necessary and sufficient for a beam characterized by the value \(r=1\) (the pure case or a homogeneous mixture) to be transformed with preservation of the value of this invariant, i.e. for a homogeneous mixture to be transformed again into a homogeneous mixture (preservation of coherence).

Since for dichroism, i.e. the process of removal of photons from a beam passing through a medium (as a result of their absorption or scattering), the extinction matrix \(\nu^{\mathrm{э}}\) is Hermitian, in an arbitrary representation the corresponding extinction matrix \(\chi^{\mathrm{э}}\) must have the form

\[ \chi^{\mathrm{э}}=\frac{1}{2} \left| \begin{array}{rrrr} \nu^{\mathrm{э}}_{11}+\nu^{\mathrm{э}}_{22} & \nu^{\mathrm{э}}_{11}-\nu^{\mathrm{э}}_{22} & \nu^{\mathrm{э}}_{12}+\nu^{\mathrm{э}}_{21} & i(\nu^{\mathrm{э}}_{12}-\nu^{\mathrm{э}}_{21}) \\[2mm] \nu^{\mathrm{э}}_{11}-\nu^{\mathrm{э}}_{22} & \nu^{\mathrm{э}}_{11}+\nu^{\mathrm{э}}_{22} & 0 & 0 \\[2mm] \nu^{\mathrm{э}}_{12}+\nu^{\mathrm{э}}_{21} & 0 & \nu^{\mathrm{э}}_{11}+\nu^{\mathrm{э}}_{22} & 0 \\[2mm] -i(\nu^{\mathrm{э}}_{12}-\nu^{\mathrm{э}}_{21}) & 0 & 0 & \nu^{\mathrm{э}}_{11}+\nu^{\mathrm{э}}_{22} \end{array} \right|. \tag{44} \]

In the representation in which the matrix \(\nu^{\mathrm{э}}\) is reduced to diagonal form (i.e. \(\nu_{12}=\nu_{21}=0\)), all off-diagonal terms vanish in the matrix \(\chi^{\mathrm{э}}\), with the exception of \(\chi_{12}=\chi_{21}\). The disappearance of the latter corresponds to the independence of absorption from the state of polarization of the radiation (isotropy).

* In the absence of coherent interaction between the scattering particles, the expression (12) should be used as \(\nu_{ik}\) for determining \(\chi_{ik}\).

In exactly the same way, the phase matrix \(\chi^\phi\) is related to the matrix \(\psi^\phi\) by the relation

\[ \chi^\phi=\frac{1}{2} \left| \begin{array}{cccc} 0 & 0 & 0 & 0\\ 0 & 0 & (\psi^\phi_{12}-\psi^\phi_{21}) & i(\psi^\phi_{12}+\psi^\phi_{21})\\ 0 & -(\psi^\phi_{12}-\psi^\phi_{21}) & 0 & -i(\psi^\phi_{11}-\psi^\phi_{22})\\ 0 & -i(\psi^\phi_{12}+\psi^\phi_{21}) & i(\psi^\phi_{11}-\psi^\phi_{22}) & 0 \end{array} \right| . \tag{45} \]

If the matrix \(\psi^\phi\) is diagonalized, i.e., if it is possible to introduce refractive indices (see above), then in the matrix \(\chi^\phi\) all terms disappear except \(\chi_{34}=-\chi_{43}\). In the representation \(\gamma=0\) this corresponds to double refraction, and in the representation \(\gamma=\dfrac{\pi}{4}\) to rotation of the plane of polarization, i.e., to the optical activity of the medium.

In the general case, obviously,

\[ \chi=\chi^{\mathrm{a}}+\chi^\phi . \tag{46} \]

Returning to the matrix of the finite transformation \(D\), let us note some special cases that are especially important from the point of view of practical applications.

a) Reflection and refraction at the boundary of two media is described by the matrix \(D\), which is easily obtained (in the representation \(\gamma=0,\ \psi=0\); the reference plane \(Q\) coincides with the plane of incidence), if in expression (40) one sets \(\mu_{11}=r_p,\ \mu_{22}=r_s,\ \mu_{12}=\mu_{21}=0\), or \(\mu_{11}=\tau_p,\ \mu_{22}=\tau_s,\ \mu_{12}=\mu_{21}=0\), where \(r_p,\ r_s\) and \(\tau_p,\ \tau_s\) are the Fresnel amplitude coefficients of reflection \((r)\) and refraction \((\tau)\) for the \(p\)- and \(s\)-components, respectively.

b) A polarizer that transmits oscillations in the plane \(\psi\) without absorption, i.e., that transmits light without change, characterized in the representation \(\gamma=0,\ \psi=0\) by the vector-parameter \(\vec S(1,1,0,0)\), corresponds in the same representation to the matrix

\[ \mu= \left| \begin{array}{cc} 1 & 0\\ 0 & 0 \end{array} \right|, \tag{47} \]

i.e.

\[ D=\frac{1}{2} \left| \begin{array}{cccc} 1 & 1 & 0 & 0\\ 1 & 1 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{array} \right| . \tag{48} \]

c) A compensator that introduces a phase difference \(\tau\) between linearly polarized components in the representation \(\gamma=0,\ \psi=0\),

corresponds to the matrix

\[ \mu = \begin{vmatrix} e^{i\frac{\tau}{2}} & 0\\ 0 & e^{-i\frac{\tau}{2}} \end{vmatrix}, \tag{49} \]

i.e.

\[ D = \begin{vmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & \cos\tau & -\sin\tau\\ 0 & 0 & \sin\tau & \cos\tau \end{vmatrix} \tag{50} \]

(for a quarter-wave plate \(\tau=\pi/2\)).

c) An optically active crystal, rotating the plane of polarization through an angle \(\theta\), is characterized in the representation \(\gamma=0,\ \psi=0\) by the matrix

\[ D = \begin{vmatrix} 1 & 0 & 0 & 0\\ 0 & \cos 2\theta & -\sin 2\theta & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{vmatrix}, \tag{51} \]

and in the representation \(\gamma=-\dfrac{\pi}{4}\) by the matrix (50).

d) To a dichroic plate, attenuating the alternatively polarized components \(E_1\) and \(E_2\) by factors \(\mu_1\) and \(\mu_2\), corresponding to an arbitrary representation \(\gamma,\ \psi\), there corresponds (in the same representation \(\gamma,\ \psi\)) the matrix

\[ D=\frac{1}{2} \begin{vmatrix} \mu_1^2+\mu_2^2 & \mu_1^2-\mu_2^2 & 0 & 0\\ \mu_1^2-\mu_2^2 & \mu_1^2+\mu_2^2 & 0 & 0\\ 0 & 0 & 2\mu_1\mu_2 & 0\\ 0 & 0 & 0 & 2\mu_1\mu_2 \end{vmatrix}. \tag{52} \]

e) If a compensator is placed in the path of the beam, introducing a phase difference \(\tau\) between the components \(E_1\) and \(E_2\) (in the representation \(\gamma=0,\ \psi=0\)), and after it an analyzer, selecting the component linearly polarized in the plane making an angle \(\varphi+\pi/2\) with the reference plane \(Q\) (i.e. with the vector \(\mathbf E\) lying in the plane \(Q\)), then the intensity of the light passing through the system will be a function of \(\tau\) and \(\varphi\), and is determined by the relation

\[ I(\tau,\varphi)=\frac{1}{2}\,[S_1+(\cos 2\gamma\cos 2\varphi-\sin 2\gamma\sin 2\varphi\sin\tau)S_2+ \]

\[ +\sin 2\varphi\cos\tau\,S_3+(\sin 2\gamma\cos 2\varphi+\cos 2\gamma\sin 2\varphi\sin\tau)S_4], \tag{53} \]

where the vector-parameter is specified in the representation \(\gamma, \psi = 0\). This relation makes it possible to indicate a method for the experimental determination of the components of the vector-parameter. Obviously, \(S_i\) can be found by measuring \(I\) for four different values of \(\tau\) and \(\varphi\). In particular, in the representation \(\gamma = 0, \psi = 0\) we have:

\[ \begin{aligned} S_1 &= I(0,0) + I\left(0,\frac{\pi}{2}\right); & S_2 &= I(0,0) - I\left(0,\frac{\pi}{2}\right); \\ S_3 &= 2I\left(0,\frac{\pi}{4}\right) - S_1; & S_4 &= 2I\left(\frac{\pi}{2},\frac{\pi}{4}\right) - S_1 . \end{aligned} \tag{54} \]

Thus the determination of the components of the vector-parameter is reduced to measuring the intensity of a light beam prepared by passing it through an analyzer (for three positions of the latter: \(\psi = 0,\ \pi/4\) and \(\pi/2\)), and also through a quarter-wave plate \((\gamma = \pi/2)\) and an analyzer in the position \(\psi = \pi/4\).

§ 6. SPECIAL FORMS OF THE SCATTERING MATRIX

Since scattering (if it is not a question of the scattering by individual particles, but of the scattering by the medium as a whole) is always completely incoherent\(^{31}\), a vector-parametric description of light beams should be used for its study. The applicability of the approximation of ray optics here is ensured by the fact that we are always interested in the field of scattered radiation at distances large in comparison with the dimensions of the wave zone. Therefore we may neglect the radial field, which decreases rapidly with distance, and write the scattering law on a unit inhomogeneity (atom, fluctuation, large particle) in the form

\[ E_i^{\mathrm{scat}}(\mathbf r,\mathbf l) = \frac{1}{r} F_{ik}(\mathbf l,\mathbf l_0) E_k^0(\mathbf l_0)e^{-ikr}, \tag{55} \]

where \(\mathbf l_0\) and \(\mathbf l\) are the directions of propagation of the incident and scattered beams, \(r\) is the distance from the scattering inhomogeneity to the observation point, and the indices \(i\) and \(k\) \((i,k=1,2)\) refer respectively to the longitudinal and latitudinal components of the radiation field\(*\).

\(*\) We emphasize that, generally speaking, the scattering indicatrix \(F_{ik}\) corresponds not to a single scattering inhomogeneity, but to an element of the volume of the scattering medium, i.e. to a complex of inhomogeneities (fluctuations, large molecules, isolated particles), whose sizes and mutual distances are at least comparable with the wavelength of light and which scatter completely incoherently relative to one another. Strictly speaking, only in this case should one speak of scattering, and not of the phenomenon of diffraction, and use the Stokes vector-parameter for describing light beams.

The scattering matrix \(\dfrac{1}{r^{2}}D(\mathbf l,\mathbf l_{0})\) for this case is obtained directly from (40), if one sets

\[ \mu_{ik}(\mathbf l,\mathbf l_{0})=F_{ik}(\mathbf l,\mathbf l_{0}) \tag{56} \]

(the factor \(e^{-ikr}\) disappears, since only products of the form \(\mu_{ik}\mu_{js}^{*}\) enter into \(D\)). It is obvious that the matrix \(\dfrac{1}{4\pi}D(\mathbf l,\mathbf l_{0})\,d\Omega\) is nothing other than the differential transverse scattering cross section in the given direction \(\mathbf l\) within the solid angle \(d\Omega\).

We give particular forms of the matrix \(D\) for scatterings of various types.

a) Molecular scattering\(^{27}\).

In the representation \(\gamma=0,\ \psi=0\)

\[ \frac{1}{r^{2}}D= \frac{\omega^{4}}{16\pi^{2}c^{4}r^{2}}\cdot \frac{(1-\Delta)}{1-\dfrac{4}{3}\Delta}\cdot \frac{(n^{2}-1)}{N}\times \tag{57} \]

\[ \times \left| \begin{array}{cccc} (1+\cos^{2}\vartheta)+\dfrac{4\Delta}{1-\Delta} & -\sin^{2}\vartheta\cos 2\varphi & \sin^{2}\vartheta\sin 2\varphi & 0\\ -\sin^{2}\vartheta & (1+\cos^{2}\vartheta)\cos 2\varphi & -(1+\cos^{2}\vartheta)\sin 2\varphi & 0\\ 0 & 2\cos\vartheta\sin 2\varphi & 2\cos\vartheta\cos 2\varphi & 0\\ 0&0&0&2\cos\vartheta \end{array} \right|, \]

where \(\theta\) is the scattering angle, \(\varphi\) is the angle between the scattering plane (coinciding with the reference plane \(Q'\) for the scattered beam) and the reference plane \(Q\) for the incident beam, and \(\Delta\) is the depolarization of the scattering in the case of illumination by linearly polarized light \((E^{0}_{2}=0)\) and observation in the direction \(\theta=\dfrac{\pi}{2}\), \(\varphi=\dfrac{\pi}{2}\).

b) Scattering by spherical particles (Mie)\(^{27}\).

In the representation \(\gamma=0,\ \psi=0\)

\[ \frac{1}{r^{2}}D= \frac{1}{r^{2}} \left| \begin{array}{cccc} d_{1} & d_{2}\cos 2\varphi & -d_{2}\sin 2\varphi & 0\\ d_{2} & d_{1}\cos 2\varphi & -d_{1}\sin 2\varphi & 0\\ 0 & d_{3}\sin 2\varphi & d_{3}\cos 2\varphi & d_{4}\\ 0 & -d_{4}\sin 2\varphi & -d_{4}\cos 2\varphi & d_{3} \end{array} \right|, \tag{58} \]

where

\[ \left. \begin{aligned} d_1 &= \frac{1}{2}\sum_{l,m}^{\infty}\left[(G_{lm}^{(1)}+G_{lm}^{(2)})(\Theta_l'\Theta_m' + \Theta_l\Theta_m) + \right.\\ &\qquad\qquad\left. + 2G_{lm}^{(3)}(\Theta_l\Theta_m' + \Theta_l'\Theta_m)\right],\\[6pt] d_2 &= \frac{1}{2}\sum_{l,m}^{\infty}\left[(G_{lm}^{(1)}+G_{lm}^{(2)})(\Theta_l'\Theta_m' - \Theta_l\Theta_m) + \right.\\ &\qquad\qquad\left. + 2G_{lm}^{(3)}(\Theta_l\Theta_m' - \Theta_l'\Theta_m)\right],\\[6pt] d_3 &= \sum_{l,m}^{\infty}\left[(G_{lm}^{(1)}+G_{lm}^{(2)})\Theta_l'\Theta_m + \right.\\ &\qquad\qquad\left. + G_{lm}^{(3)}(\Theta_l'\Theta_m' + \Theta_l\Theta_m)\right],\\[6pt] d_4 &= \sum_{l,m}^{\infty}\left[G_{lm}^{(4)}(\Theta_l'\Theta_m - \Theta_l\Theta_m') + \right.\\ &\qquad\qquad\left. + G_{lm}^{(5)}(\Theta_l'\Theta_m' - \Theta_l\Theta_m)\right] \end{aligned} \right\} \tag{59} \]

are real functions of the scattering angle \(\theta\), with

\[ \left. \begin{aligned} \Theta_l(\theta) &= P_l^{(1)}(\cos\theta)\,\frac{1}{\sin\theta}; \qquad \Theta_l'(\theta) = P_l^{(1)\prime}(\cos\theta)\,\sin\theta;\\ \alpha_l &= \operatorname{Re}\left(\frac{1}{k}C_l\zeta_l\right); \qquad \alpha_l' = \operatorname{Im}\left(\frac{1}{k}C_l\zeta_l\right);\\ \beta_l &= \operatorname{Re}\left(\frac{1}{k}B_l\zeta_l\right); \qquad \beta_l' = \operatorname{Im}\left(\frac{1}{k}B_l\zeta_l\right);\\ G_{lm}^{(1)} &= \alpha_l\alpha_m + \alpha_l'\alpha_m'; \qquad G_{lm}^{(2)} = \beta_l\beta_m - \beta_l'\beta_m';\\ G_{lm}^{(3)} &= \alpha_l\beta_m' + \alpha_l'\beta_m; \qquad G_{lm}^{(4)} = \alpha_l\alpha_m' - \beta_l\beta_m';\\ G_{lm}^{(5)} &= \alpha_l\beta_m - \alpha_l'\beta_m' \end{aligned} \right\} \tag{60} \]

and the functions \(P_l^{(1)}\), \(\zeta_l\), \(C_l\), and \(B_l\) have the usual meanings in Mie scattering theory.

It is not difficult to verify that (57) satisfies the reciprocity principle (cf. \(^{11}\)).

If in (56) one sets \(\Delta = 0\), then the matrix \(D\) for molecular scattering takes the form (57), with \(d_4 = 0\) (cf. \(^{12,21}\)).

c) Compton scattering.
Starting from the Klein–Nishina formula, Fano \(^{7}\) showed that the scattering matrix of \(\gamma\)-radiation by electrons in the representation \(\gamma = 0\), \(\psi = 0\)

(the reference planes \(Q\) for the incident and scattered beams coincide with each other and with the scattering plane) has the form

\[ D_{\mathrm{Compt}}=\frac{1}{2}\frac{e^2}{mc^2}\left(\frac{k}{k_0}\right)^2 \times \tag{61} \]

\[ \times \left| \begin{array}{cccc} (1+\cos^2\theta)+ (k_0-k)(1-\cos\theta) & -\sin^2\theta & 0 & (1-\cos\theta)\times (k_0\cos\theta+k)\mathbf{s} \\[4pt] -\sin^2\theta & 1+\cos^2\theta & 0 & (1-\cos\theta)\times [\mathbf{n}\mathbf{n}_0]\,[k_0\mathbf{s}] \\[4pt] 0 & 0 & 2\cos\theta & (1-\cos\theta)\times [\mathbf{k}\mathbf{n}_0]\,\mathbf{s} \\[4pt] (1-\cos\theta)\times (\mathbf{k}\cos\theta+k_0)\mathbf{s} & (1-\cos\theta)\times [\mathbf{n}_0\mathbf{n}]\,[\mathbf{k}\mathbf{s}] & (1-\cos\theta)\times (k_0\mathbf{n})\mathbf{s} & 2\cos\theta+(k_0-k)\times (1-\cos\theta)\cos\theta \end{array} \right|, \]

where \(\theta\) is the scattering angle, \(\mathbf{k}_0\) and \(\mathbf{k}\) are the momenta of the incident and scattered photons, expressed in units \(mc\), \(\mathbf{n}_0\) and \(\mathbf{n}\) are unit direction vectors of the incident and scattered beams, and \(\mathbf{s}\) is the mean initial spin of the scattering electrons.

§ 7. RELATION OF THE SCATTERING MATRIX TO THE DISPERSION MATRIX

Let us now turn to the relation of the scattering matrix \(D\) to the dispersion matrix \(\chi\). For this purpose, let us single out a portion of a layer of the medium of unit area and thickness \(dl\), and compute the flux of the Poynting vector through the closed surface enclosing this portion. That part of this flux which is determined by the interference cross-terms for the fields of the scattered and incident waves gives the total amount of energy removed per unit time from the incident wave illuminating the layer, and is fully taken into account by the dispersion matrix \(\chi\):

\[ (dS_1)_{\mathrm{pass}}=\sum_j \chi_{1j}S_j\,dl . \tag{62} \]

This energy removed from the incident wave is expended, on the one hand, on scattering:

\[ (dS_1)_{\mathrm{scatt}}=\frac{1}{4\pi}\sum_j S_j \int D_{1j}\,d\Omega\,dl, \tag{63} \]

where \(d\Omega\) is the solid angle with vertex in the scattering volume, and, on the other hand, on absorption of radiation by the inhomogeneity itself. In particular, if absorption is absent, then

\[ \chi_{1j}=\chi_{j1}=\frac{1}{4\pi}\int D_{1j}\,d\Omega\,dl. \tag{64} \]

Similar relations can be obtained for the components \(x_{4j}\) and \(D_{4j}\) from the law of conservation of angular momentum (cf. 27). Thus, the use of the conservation laws makes it possible (in the absence of absorption) to relate to \(D_{ij}\) all the components of the dispersion matrix \(x\), except \(x_{23}=-x_{32}\), corresponding to rotation of the plane of polarization. The question of the connection of \(x_{23}\) with \(D_{ij}\) remains open for the present.

In the presence of absorption, comparison of (62) and (63), as well as of the corresponding expressions for \(dS_4\), makes it possible to determine not only the amount of energy absorbed by the medium, but also the momentum absorbed by the medium. If, in addition, the law of absorption by isolated inhomogeneities is known, this makes it possible to determine both the magnitude and the polarization of the effective field in which the inhomogeneities are found, since the magnitude of the absorption is determined precisely by the effective field.

§ 8. THE STOKES VECTOR-PARAMETER AND THE POINCARÉ SPHERE

From the definition of the components of the vector-parameter it is evident that

\[ S_1=I,\quad S_2=IP_1,\quad S_3=IP_2,\quad S_4=IP_3, \tag{65} \]

where \(P_i\) do not depend on the intensity of the beam \(I\) and, according to (25), satisfy the relation

\[ P_1^2+P_2^2+P_3^2=r^2. \tag{66} \]

For both completely and partially polarized beams, in the representation \(\gamma=0\) (cf. (34)—(36))

\[ \begin{aligned} P_1&=p\cos 2(\psi_0-\psi),\\ P_2&=p\sin 2(\psi_0-\psi),\\ P_3&=-q. \end{aligned} \tag{67} \]

On the other hand, in accordance with (66), one may set

\[ \begin{aligned} p&=r\sin 2(\beta_0-\beta),\\ q&=-r\cos 2(\beta_0-\beta) \end{aligned} \tag{68} \]

and regard \(P_i\) as the components of a certain three-dimensional vector \(\mathbf P\), whose modulus is equal to \(r\). It is not difficult to see that, for completely (elliptically) polarized light, \((\beta_0-\beta)\) means the phase difference of two alternative components linearly polarized along the principal axes of the polarization ellipse, while \((\psi_0-\psi)\) is the angle of rotation of the major axis of the polarization ellipse relative to the reference plane \(Q\). Moreover, for completely polarized light \(r=1\), i.e. \(\mathbf P\) is a unit vector. For partially polarized light the quantities \((\psi_0-\psi)\) and \((\beta_0-\beta)\) retain their values, but only for the completely polarized component (for the completely depolarized component \(r=0\)).

Thus, the state of polarization of the beam is completely specified if the vector \(\mathbf P\) is specified; this vector has been called the polarization vector. In this case the four-dimensional vector-parameter \(\vec S\) may be written in the form

\[ \vec S = I(1,\mathbf P). \tag{69} \]

or, introducing the unit polarization vector \(\mathbf Q\), coinciding in direction with \(\mathbf P\), i.e., putting

\[ \mathbf P = r\mathbf Q, \tag{70} \]

we have:

\[ \vec S = I(1,r\mathbf Q). \tag{71} \]

The introduction of the unit polarization vector \(\mathbf Q\) leads us directly to a very visual and, in some cases (far from all), convenient method of describing polarization, proposed by Poincaré\({}^{22}\).

Let us associate three orthogonal coordinate axes with the components \(Q_i\). In doing so, we assign axis 1 to light linearly polarized in the reference plane, axis 2 to light linearly polarized at an angle \(\frac{\pi}{4}\) to the reference plane, and axis 3 to circularly (right-hand) polarized light. Next, construct a sphere of unit radius (the Poincaré sphere) with its center at the origin (Fig. 1), and introduce polar coordinates of the end of an arbitrary vector \(\mathbf Q\):

\[ \left. \begin{array}{l} \text{latitude } \theta = 2(\beta_0-\beta),\\ \text{and longitude } \varphi = 2(\psi_0-\psi). \end{array} \right\} \tag{72} \]

Then the state of polarization (i.e., the vector \(\mathbf P\)) is completely determined by specifying the coordinates \(\theta\) and \(\varphi\) of a point on the Poincaré sphere and the modulus \(r\) of the vector \(\mathbf P\). Passage to any other representation means a rotation of the polarization coordinate system: a change in \(\psi\) corresponds to its rotation about the polar axis, and a change in \(\gamma\) to a displacement of the poles.

Suppose now that there is a light beam with vector-parameter \(\vec S = (1,r\mathbf Q)\), and it is required to find the intensity \(I'\) which this beam will acquire if it is passed through an optical device selecting the completely polarized component with polarization \(\mathbf Q'\), i.e., the probability of finding photons with polarization \(\mathbf Q'\) in the beam \(\vec S\). Using the matrix characterizing the action of such a device, we obtain (in an arbitrary representa-

... direction), that \(I'\) is determined by the scalar product of the vectors \((1,\mathbf Q')\) and \(\vec S\):

\[ I'=\frac{1}{2}I(1,\mathbf Q')(1,r\mathbf Q), \tag{73} \]

or

\[ I'=\frac{1}{2}I(1+r\mathbf Q'\mathbf Q). \tag{74} \]

Since (Fig. 1)

\[ \mathbf Q'\mathbf Q=\cos\Theta, \tag{75} \]

we have:

\[ I'=\frac{1}{2}I(1+r\cos\Theta). \tag{76} \]

Let us note that, alongside the vector-parametric description of a light beam and its description by means of the Poincaré sphere, one often resorts directly to the spectral matrix \(S_{ik}\), or to linear combinations of its components different from \(S_i\) (see, for example,\(^{21}\)). Since this introduces nothing fundamentally new and provides only certain technical conveniences in particular concrete cases, we shall not dwell on the description of all particular variants of the matrix method and refer the reader to the original works\(^{6-13,\,21-29}\), which contain numerous examples of applications of the Stokes vector-parameter and analogous parameters to a wide variety of concrete problems, as well as descriptions of various methods for introducing and measuring the components of the vector-parameter.

Fig. 1.

§ 9. THE VECTOR-PARAMETER

AND THE QUANTUM-MECHANICAL DENSITY MATRIX

From the quantum-mechanical point of view, a completely polarized \((r = 1)\) light beam is determined if its wave function is known; moreover, the variables should be taken to be the numbers of photons \(N\) in each of the quantum states that differ in frequency, direction of propagation, and state of polarization. For a beam of a given direction and definite frequency,

\[ \psi = \sum_i a_i \psi_i \qquad (i = 1, 2), \tag{77} \]

where \(\psi_i\) are mutually orthogonal eigenfunctions corresponding to the two alternative states of polarization. The probability of finding a photon in the state \(\psi\) is then determined by the well-known relation

\[ w_\psi = \sum_{i,j} \rho_{ij}\psi_i\psi_j^*, \tag{78} \]

where

\[ \rho_{ij} = a_i a_j^*, \tag{79} \]

the so-called density matrix (see, for example, \(^{30}\)). Obviously, if the function \(\psi\) is normalized, then the trace of the matrix \(\rho_{ij}\)

\[ \operatorname{Sp}\rho_{ij} = \sum_i \rho_{ii} = 1, \tag{80} \]

which corresponds to the certainty of finding the photon in one of the alternative states of polarization. Of very great importance is the fact that the density matrix for a quantum-mechanical mixture is the sum of the density matrices for its individual pure components, i.e. the density matrix is additive for incoherent beams.

In the case of electromagnetic radiation,

\[ \mathbf{E} = E_1\mathbf{e}_1 + E_2\mathbf{e}_2, \tag{81} \]

where \(\mathbf{e}_1\) and \(\mathbf{e}_2\) are unit vectors (generally speaking, complex) playing the role of the eigenfunctions of the polarization operator, while \(|E_1|^2\) and \(|E_2|^2\) are the probabilities of finding the photon in each of the alternative polarization states.

Thus, in the case of electromagnetic waves the density matrix coincides with the spectral matrix: \(\rho_{ij} = \dfrac{2}{I} S_{ij}\), where \(I\) is the intensity of the light beam.

In particular, for light completely (elliptically) polarized in the direction of the (complex) unit vector \(\mathbf e_1\),

\[ S_{11}=\frac{I}{2},\quad S_{12}=S_{21}=S_{22}=0, \tag{82} \]

and for completely depolarized light

\[ S_{ij}=\frac{I}{4}\,\delta_{ij}. \tag{83} \]

Since a partially polarized beam may be regarded as an incoherent mixture of “natural” and completely polarized beams\({}^{20}\), and the fraction of the energy falling on the polarized component is equal to \(r\), in the general case (in the corresponding representation) the spectral matrix is

\[ S=\frac{I}{4}(1-r) \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} +\frac{Ir}{2} \begin{pmatrix} 1 & 0\\ 0 & 0 \end{pmatrix} = \]

\[ =\frac{I}{4} \begin{pmatrix} 1+r & 0\\ 0 & 1-r \end{pmatrix}, \tag{84} \]

whence, in particular,

\[ \begin{aligned} I_{\max}&=\frac{I}{2}(1+r),\\ I_{\min}&=\frac{I}{2}(1-r), \end{aligned} \tag{85} \]

where \(I_{\max}\) and \(I_{\min}\) are the intensities of the beams transmitted through a polarizing device that either completely transmits or completely stops the polarized component.

Let us recall that the probability of finding a photon (or particle) in the polarization state corresponding to the density matrix \(\rho'\), in a beam characterized by the density matrix \(\rho\), is equal to \(w=\operatorname{Sp}(\rho\rho')\), which coincides with (73).

In conclusion we give the obvious relations between the components of the vector-parameter and the density matrix \(\rho\):

\[ \left. \begin{aligned} S_1&=I(\rho_{11}+\rho_{22})^{*}; & S_2&=I(\rho_{11}-\rho_{22}),\\ S_3&=I(\rho_{12}+\rho_{21}); & S_4&=-Ii(\rho_{12}-\rho_{21}), \end{aligned} \right\} \tag{86} \]

whence

\[ \rho=\frac{1}{2I} \begin{pmatrix} S_1+S_2 & S_3+iS_4\\ S_3-iS_4 & S_1-S_2 \end{pmatrix}. \tag{87} \]

\[ \text{*) Recall that, by the normalization condition (80), } \rho_{11}+\rho_{22}=1. \]

§ 10. THE VECTOR PARAMETER FOR AN ELECTRON BEAM \({}^{13,26}\)

For particles with spin \(\frac{1}{2}\), including electrons, two alternative values \(\left(\pm \frac{1}{2}\right)\) of the spin projection on some (arbitrary) axis are possible; to these there correspond two mutually orthogonal wave functions \(\psi_1\) and \(\psi_2\). In other words, the state of polarization of an electron beam is determined by the wave function

\[ \psi = a_1\psi_1 + a_2\psi_2 . \tag{88} \]

Consequently, the density matrix \(\rho\) will formally have the same form as in the case of electromagnetic waves, and one may introduce the vector parameter \(\vec S\), using relations (86), in which \(I\) will now denote the intensity of the electron beam.

Fig. 2.

Let us choose a representation in which \(\psi_1\) and \(\psi_2\) are eigenfunctions of the operator of the spin projection on some axis \(z\), perpendicular to the direction \(y\) in which the beam propagates; moreover, let us assume that \(\psi_1\) corresponds to spin \(+\frac{1}{2}\), and \(\psi_2\) to spin \(-\frac{1}{2}\). For the orientation of the spin in an arbitrary direction \((\theta,\varphi)\) (see Fig. 2) we have (up to an arbitrary common phase)

\[ a_1 = \cos \frac{\theta}{2}, \qquad a_2 = \sin \frac{\theta}{2}\, e^{i\varphi}, \tag{89} \]

whence, on the basis of (79) and (86),

\[ \begin{aligned} S_1 &= I, &\qquad S_2 &= I\cos\theta,\\ S_3 &= I\sin\theta\cos\varphi, &\qquad S_4 &= -I\sin\theta\sin\varphi . \end{aligned} \tag{90} \]

Let us note that in the case of an electron beam the polarization vector \(\mathbf P\) coincides with the spin vector \(\mathbf s\) (the vector \(\mathbf Q\) corresponds to the direction of spin, and the parameter \(r\) to the mean value of the spin for the given mixture).

The method of experimental determination of the components of the vector parameter is clear from the following considerations. The eigenfunctions \(\psi_1,\psi_2\) of the operator of the spin projection on the \(z\)-axis are, as is known, related to the eigenfunctions \(\psi'_1,\psi'_2\) of the operator of the spin projection

on the \(x\)-axis by the relations

\[ \psi_1=\frac{1}{\sqrt{2}}(\psi'_1+\psi'_2);\quad \psi_2=\frac{1}{\sqrt{2}}(\psi'_1-\psi'_2) \tag{91} \]

and to the eigenfunctions \(\psi''_1,\ \psi''_2\) of the operator of the spin projection on the \(y\)-axis by the relations

\[ \psi_1=\frac{1}{\sqrt{2}}(\psi''_1+\psi''_2);\quad \psi_2=\frac{1}{\sqrt{2}}(-i\psi''_1+i\psi''_2). \tag{92} \]

Let us denote the intensity of the component of the beam polarized in the direction \((\theta,\varphi)\) by \(I(\theta,\varphi)\). Then, as is easily verified,

\[ I(0,0)=I\rho_{11} \quad \left[\text{spin }+\frac{1}{2}\text{ along the }z\text{-axis}\right], \]

\[ I(\pi,0)=I\rho_{22} \quad \left[\text{spin }-\frac{1}{2}\text{ along the }z\text{-axis}\right], \]

\[ I\left(\frac{\pi}{2},0\right)=I\rho'_{11} =\frac{I}{2}(\rho_{11}+\rho_{12}+\rho_{21}+\rho_{22}) \]

\[ \left[\text{spin }+\frac{1}{2}\text{ along the }x\text{-axis}\right], \]

\[ I\left(\frac{\pi}{2},\pi\right)=I\rho'_{22} =\frac{I}{2}(\rho_{11}-\rho_{12}-\rho_{21}+\rho_{22}) \]

\[ \left[\text{spin }-\frac{1}{2}\text{ along the }x\text{-axis}\right], \]

\[ I\left(\frac{\pi}{2},\frac{\pi}{2}\right)=I\rho''_{11} =\frac{I}{2}(\rho_{11}+i\rho_{12}-i\rho_{21}+\rho_{22}) \]

\[ \left[\text{spin }+\frac{1}{2}\text{ along the }y\text{-axis}\right], \]

\[ I\left(\frac{\pi}{2},-\frac{\pi}{2}\right)=I\rho''_{22} =\frac{I}{2}(\rho_{11}-i\rho_{12}+i\rho_{21}+\rho_{22}): \]

\[ \left[\text{spin }-\frac{1}{2}\text{ along the }y\text{-axis}\right]. \]

Consequently, in accordance with (86),

\[ \begin{aligned} S_1&=I; \quad S_2=I(0,0)-I(\pi,0);\\ S_3&=I\left(\frac{\pi}{2},0\right)-I\left(\frac{\pi}{2},\pi\right);\\ S_4&=I\left(\frac{\pi}{2},\frac{\pi}{2}\right)-I\left(\frac{\pi}{2},-\frac{\pi}{2}\right). \end{aligned} \tag{93} \]

Comparison of (93) with (54) makes it possible to establish a simple correspondence between the physically meaningful components of the vector-parameter in the two cases (in the representation \(\gamma=0\)):

Components of the vector-parameter	Photons	Electrons
\(S_1\)	Beam intensity	Beam intensity
\(S_2\)	Linear polarization in the direction of some axis \(z\)	Transverse spin in the direction of the axis \(z\)
\(S_3\)	Linear polarization in a direction making an angle \(\dfrac{\pi}{4}\) with the direction of the axis \(z\)	Transverse spin in a direction perpendicular to \(z\)
\(S_4\)	Circular polarization (longitudinal spin)	Longitudinal spin

§ 11. EXAMPLES OF APPLICATION OF THE VECTOR-PARAMETRIC DESCRIPTION

The equation of radiative transfer in turbid media with allowance for its polarization

The change of a light beam propagating in some direction \(\mathbf l\) within a solid angle \(d\Omega\), when it passes through a volume element \(dV\) of a turbid medium, occurs as a result of the combined action of two causes: 1) dispersion (including extinction) and 2) scattering by the element \(dV\) into the direction \(\mathbf l\) of radiation incident on the element \(dV\) from other directions—\(\mathbf l'\). Since the transmitted and scattered waves are incoherent with one another, just as beams scattered from different directions are incoherent, there is additivity not of the components of the field strengths, but of the vector-parameters of the corresponding beams. At the same time, one cannot confine oneself (as is often done) to taking into account only the intensities of the beams, since the conditions of their transformation (scattering and dispersion) depend essentially on their character of polarization. Therefore, in order to take simultaneous account of both causes of the change of a light beam as it passes through a turbid medium, i.e. in order to formulate the transfer equation, it is necessary to turn to the vector-parametric description of beams. This makes it possible not only to correct the usually used transfer equation by adding to it terms of the same order of magnitude as the principal ones, but also to solve problems concerning the character of the polarization of scattered light (see, for example, \(^{12,\,27-29,\,31}\)).

For the purpose of establishing uniformity in the choice of representations for beams of different directions, we shall take as the plane of re-

the scattering plane as the reference plane and put \(\gamma\) and \(\psi\) the same for all beams.

The power of the radiation flux crossing the volume element \(dV\) from the direction \(-\mathbf{l}'\) within the solid angle \(d\Omega'\) is obviously equal to \(S_1(\mathbf{l}')\,d\Omega'\,d\tau'\), where \(d\tau'\) is the area of the section of the volume element \(dV\) normal to \(\mathbf{l}'\). Consequently, the power of the light flux propagating in the direction \(\mathbf{l}\) changes, in passing through the volume element \(dV\), by the amount

\[ dS_1(\mathbf{l})\,d\Omega\,d\tau = -\sum_{k=1}^{4} \chi_{1k}(\mathbf{l})\,S_k(\mathbf{l})\,d\Omega\,d\tau\,dl + \]

\[ + \frac{1}{4\pi} \int_{\Omega'} \sum_{k=1}^{4} D_{1k}(\mathbf{l},\mathbf{l}')\,S_k(\mathbf{l}')\,d\Omega\,d\Omega'\,dV, \tag{94} \]

where the first term takes account of dispersion, and the second of the scattering, in the given direction, of light propagating in all other directions.

Since all components of the vector-parameter are intensities of the same light beam, only after the corresponding preliminary preparation by means of compensators and analyzers, analogous relations are obtained also for the changes in the powers of such prepared beams. Taking further into account that \(dV=d\tau\,dl\), we finally obtain the system of integro-differential equations

\[ \frac{dS_i(\mathbf{l})}{dl} = \sum_{k=1}^{4} \left\{ -\chi_{ik}(\mathbf{l})\,S_k(\mathbf{l}) + \frac{1}{4\pi} \int D_{ik}(\mathbf{l},\mathbf{l}')\,S_k(\mathbf{l}')\,d\Omega' \right\}, \tag{95} \]

which is the “radiative-transfer equation taking into account the polarization of the radiation,” suitable both for isotropic and for anisotropic media\(^*\). The inclusion of the intrinsic radiation of the medium obviously reduces to adding to the right-hand side of the equation the corresponding components of the vector-parameter of the intrinsic radiation, referred to unit volume

\[ \text{\(^*\)} \]
The transfer equation with allowance for the polarization of radiation was first formulated simultaneously and independently in 1946 by Chandrasekhar\({}^{12}\) and by the author\({}^{27}\), but only for the case of an isotropic medium (when the matrix \(\chi\) is scalar and one may restrict oneself to the representation \(\gamma=0\)). The formulation of the more general equation\({}^{31}\) given here required, on the one hand, taking account of the dependence of extinction and dispersion on the character of the polarization and, on the other hand, generalizing the Stokes vector-parameter to an arbitrary representation \(\gamma,\psi\), which in 1946 had not yet been done.

and which are (like \(\chi\) and \(D\)) functions of the coordinates and directions.

In the same way one can take into account scattering accompanied by a change in frequency (the Doppler effect, Compton scattering, combination scattering, etc.). In this case the vector-parameter \(\vec S\) depends on the wavelength, and the components of the scattering matrix \(D\) should be regarded as functions of the initial and final wavelengths of the radiation; moreover, integration over the initial wavelengths of the radiation scattered by the given volume element into a certain (arbitrary) final wavelength, for which the transfer equation is formulated, is added to the integral term.31

In accordance with the results of §§ 9 and 10, the transfer equation formulated above applies not only to electromagnetic radiation, but also to the propagation, in scattering media, of fluxes of any particles with spin \(\frac{1}{2}\), including electrons.

§ 12. EXAMPLES OF THE APPLICATION OF THE VECTOR-PARAMETRIC DESCRIPTION

Compton scattering

The differential cross section for the scattering of \(\gamma\)-quanta with polarization \((1,\mathbf Q')\) under conditions in which the electrons are irradiated by \(\gamma\)-radiation with polarization \((1,r\mathbf Q)\), according to (73), is equal to

\[ \frac{d\sigma}{d\Omega} = -\frac{1}{2}(1,\mathbf Q')D_{\text{comp}}(1,r\mathbf Q), \tag{96} \]

where \(D_{\text{comp}}\) is the Compton scattering matrix, which, according to Fano,7 has the form indicated in § 6 [see (61)].

Let us suppose first that the beam of \(\gamma\)-quanta irradiating the electrons is completely depolarized, i.e. \(r=0\). Then

\[ D_{\text{comp}}(1,r\mathbf Q) = \frac{1}{2}\frac{e^2}{mc^2} \left(\frac{k}{k_0}\right)^2 \times \]

\[ \times \begin{pmatrix} 1+\cos^2\theta+(k_0-k)(1-\cos\theta)\\ -\sin^2\theta\\ 0\\ (1-\cos\theta)(k\cos\theta+k_0)s \end{pmatrix}, \tag{97} \]

i.e. the scattered beam is partially elliptically polarized, with the major axis of the polarization ellipse lying in the scattering plane. The differential scattering cross sections for the linearly polarized (parallel and perpendicular to the scattering plane) components

of the scattered radiation, according to (96), are equal to

\[ \left. \begin{aligned} \frac{d\sigma_{\perp}}{d\Omega} &= \frac{1}{2}(1,-1,0,0)\,D_{\mathrm{Comp}}(1,\mathbf r\mathbf Q) = \\ &= \frac{1}{4}\,\frac{e^2}{mc^2}\left(\frac{k}{k_0}\right)^2 \left[2+(k_\nu-k)(1-\cos\theta)\right],\\[6pt] \frac{d\sigma_{\parallel}}{d\Omega} &= \frac{1}{2}(1,1,0,0)\,D_{\mathrm{Comp}}(1,\mathbf r\mathbf Q) = \\ &= \frac{1}{4}\,\frac{e^2}{mc^2}\left(\frac{k}{k_0}\right)^2 \cdot\left[2\cos^2\theta+(k_\nu-k)\times\right.\\ &\qquad\qquad\qquad\qquad\left.\times(1-\cos\theta)\right]. \end{aligned} \right\} \tag{98} \]

Let us now suppose that the incident beam is linearly polarized \((r=1)\). We choose the scattering plane as the reference plane both for the scattered and for the incident beams. If the plane of polarization of the incident beam makes an angle \(\varphi\) with the scattering plane, then in the representation \(\gamma=0,\ \psi=0\) we obtain:

\[ (1,\mathbf Q)= \begin{pmatrix} 1\\ \cos 2\varphi\\ \sin 2\varphi\\ 0 \end{pmatrix}, \tag{99} \]

whence

\[ D_{\mathrm{Comp}}(1,\mathbf Q)= \frac{e^2}{mc^2}\left(\frac{k}{k_0}\right)^2 \times \]

\[ \times \begin{pmatrix} 1-\sin^2\theta\cos^2\varphi+(k_\nu-k)\sin^2\dfrac{\theta}{2}\\[4pt] \cos^2\theta\cos^2\varphi-\sin^2\varphi\\[4pt] 2\cos\theta\sin\varphi\cos\varphi\\[4pt] \dfrac{1}{2}(1-\cos\theta)\left[(k\cos\theta-k_\nu)S+\right.\\[4pt] \left.+[\mathbf n_\nu,\mathbf n]\,[\mathbf k\mathbf s]\cos 2\varphi +[\mathbf k_\nu\mathbf n]\,S\sin 2\varphi\right] \end{pmatrix} \tag{100} \]

and, consequently, a partial elliptic polarization of the scattered beam takes place, with the direction of the major axis of the polarization ellipse making a certain angle with the scattering plane. The component \(S_3\) of the vector-parameter of the scattered beam becomes zero if we pass to the representation \(\gamma=0,\psi\), where \(\psi\) is determined by the condition (cf. (20)):

\[ K_{32}S_2+K_{33}S_3=0, \tag{101} \]

that is, according to (21), the angle of inclination of the major axis of the polarization ellipse to the scattering plane is equal to

\[ \psi=\frac{1}{2}\operatorname{arctg} \frac{2\cos\theta\sin\varphi\cos\varphi} {\cos\theta\cos^2\varphi-\sin^2\varphi} \tag{102} \]

and does not depend on \(s\). As is evident from the form of the Compton matrix \(D_{\text{comp}}\), only the ellipticity \(q\) depends on \(s\). However, if the scattered beam is elliptically polarized (i.e., \(S_4=0\)), then all components of the vector-parameter of the scattered light, including its intensity, depend on \(s\). The differential cross sections and the degree of polarization are found without difficulty from (100) in the usual way. In particular, the total (without taking polarization into account) scattering cross section is equal to

\[ \frac{d\sigma}{d\Omega} = \frac{1}{2}(1,0,0,0)D_{\text{comp}}(1,\mathbf Q) = \]

\[ = \frac{1}{4}\frac{e^2}{mc^2} \left(\frac{k}{k_0}\right)^2 \left[ 1-\sin^2\theta\cos^2\varphi-(k_\nu-k)\sin^2\frac{\theta}{2} \right]. \tag{103} \]

If repeated Compton scattering of \(\gamma\)-radiation takes place, then the differential cross sections are found from the relation

\[ \frac{d\sigma_{\mathbf Q'}}{d\Omega} = \frac{1}{2}(1,\mathbf Q') D_{\text{comp}}^{(2)} D_{\text{comp}}^{(1)} (1,\mathbf Q), \tag{104} \]

where the superscripts on the Compton-scattering matrices denote the number of the scattering event. From the form of the matrix \(D_{\text{comp}}\) it follows directly that, whatever the primary scattered beam may be, the result of double scattering depends essentially on the state of polarization of both the primary and the secondary scattering electrons.

§ 13. CONCLUDING REMARKS

The examples presented show that the use of the vector-parametric method makes it possible not only to simplify substantially the solution of a wide range of problems, but also to solve problems practically inaccessible to other methods (say, problems connected with allowance for multiple scattering or self-reversal of lines in plasma radiation). At the same time it is quite obvious that analogous techniques can and should be used also outside optics for the analysis, for example, of scattering phenomena of electron beams or beams of any other particles with spin \(1/2\). The fact that this is almost never done in explicit form (in implicit form this method is resorted to by no means rarely, namely almost always when use is made of the density matrix) should be attributed mainly to the unfamiliarity of broad circles of physicists with the concrete mathematical apparatus of the vector-parametric description of the polarization of radiation. It is this gap in the review literature that we have sought to fill.

At the same time, we have completely bypassed an important circle of questions connected with the use of coherent matrices \(\psi\) in instrumental optics. The reason for this is that, apart from more or less trivial cases involving polarizing devices, the method in question has not yet found application, apparently for the same reason—because wide circles of physicists are unfamiliar with it. It is not difficult to see, however, that its application should substantially facilitate the consideration of numerous questions concerning changes in the polarization characteristics of light beams as they pass through optical systems (lenses, prisms, mirrors, etc.). As is well known, the corresponding sections of instrumental optics are still in an embryonic state.

In conclusion, we note that for a number of practical applications it is extremely important to have ready formulations of the laws of interaction of radiation with matter in matrix form, corresponding to the vector-parametric description of radiation. This not too difficult, but rather laborious, task has by no means been carried out, which noticeably hinders the introduction of the matrix methods described into the practice of scientific research.

In this connection, the problem of creating generalized matrix methods applicable also beyond the limits of ray optics is likewise of considerable interest. It should be assumed that the development of such generalized methods will make it possible to formulate the basic laws of optics (including Maxwell’s equations), at least for quasi-stationary periodic fields, in terms of optically observable quantities, which should substantially facilitate their application to a known class of problems.

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