CORRELATION THEORY OF ELECTRICAL FLUCTUATIONS AND THERMAL RADIATION

S. M. Rytov

1. INTRODUCTION

Methods of spectral description of thermal fluctuations of quantities characterizing the state of a dissipative system have undergone substantial development in recent years. An extremely effective device in solving this problem has proved to be one that goes back to the work of P. Langevin[^1] and de Haas–Lorentz[^2] on the theory of Brownian motion, and consists in introducing certain “extraneous” (external) fluctuating forces connected with the quantities under consideration in the same energetic sense in which generalized coordinates and forces are related to one another. Since the principal problem here is the description of the spectrum of fluctuations, the required statistical characteristics of the external forces reduce to their correlation functions.

In this formulation the problem was first solved in 1927 by Nyquist[^3] as applied to current fluctuations in an electrical circuit with lumped constants. Introducing a random external e.m.f. \(\mathcal{E}(t)\), he proceeded, in contrast to the works cited above, to a spectral representation of this random e.m.f. and, on the basis of thermodynamics and the theorem of equipartition of energy, obtained a formula for the spectral intensity \(\overline{\mathcal{E}_{\omega}^{2}}\):

\[ \overline{\mathcal{E}_{\omega}^{2}}=\frac{2}{\pi}\Theta R . \tag{1} \]

Here \(R\) is the active resistance of the circuit, \(\Theta=kT\) is its energy temperature \((k=1.38\cdot10^{-16}\ \text{erg/degree}\) is Boltzmann’s constant, \(T\) is the absolute temperature).

Formula (1) is restricted to the classical frequency range \((\hbar\omega\ll\Theta)\) and by the condition of quasistationarity, which, incidentally, can to some extent be circumvented in those cases where thermal fluctuations in lines are in question by introducing a distributed (per-unit-length) e.m.f. In addition to these well-known restrictions, formula (1) also assumes that the conductivity of the material of the noisy elements of the circuit is so high that displacement currents in them may be neglected. In other words, (1) does not include the limiting transition \(R\to\infty\).

Comparatively recently, Nyquist’s fundamental theorem was substantially generalized in a series of papers by Callen and coauthors[^4–^8]. In these works the indicated spectral approach was extended to the case of an arbitrary dissipative system in which the fluctuations are described by any number \(n\) of discrete random functions \(\xi_j(t)\). Correspondingly, \(n\) generalized external forces \(f_j(t)\) are introduced, and it is established

form of the correlation matrices for the spectral amplitudes \(\xi_{j\omega}\) and \(f_{j\omega}\). Namely, if the spectral amplitudes are related to one another by the algebraic equations

\[ \begin{aligned} \xi_{j\omega} &= \sum_k a_{jk}(\omega) f_{k\omega},\\ f_{j\omega} &= \sum_k a^{-1}_{jk}(\omega)\xi_{k\omega}, \end{aligned} \tag{2} \]

where \(a^{-1}\) is the matrix inverse to \(a\)*), then the correlation matrix for \(\xi_{j\omega}\) is

\[ \overline{\xi_{j\omega}\xi^*_{k\omega'}} = i C(\omega)\left(a_{jk}-a^*_{kj}\right)\delta(\omega-\omega'), \tag{3} \]

and for \(f_{j\omega}\)

\[ \overline{f_{j\omega}f^*_{k\omega'}} = i C(\omega)\left(\alpha^{-1*}_{kj}-\alpha^{-1}_{jk}\right)\delta(\omega-\omega'), \tag{4} \]

where

\[ C(\omega)=\frac{\hbar}{4\pi}\operatorname{cth}\frac{\hbar\omega}{\Theta}. \tag{5} \]

In the classical frequency region \((\hbar\omega \ll \Theta)\)

\[ C(\omega)=\frac{\Theta}{2\pi\omega}. \tag{5'} \]

As is not difficult to see, in this latter case, for \(n=1\), the original Nyquist formula follows from (4). The indication that the quantum-mechanical generalization of (1) reduces to replacing \(\Theta\) by the quantum expression for the mean energy of an oscillator was already made by Nyquist\(^3\), but a rigorous justification of this quite natural circumstance was given, of course, only as a result of the quantum-mechanical derivation of formulas (3) and (4). The dependence of the spectral intensities on the admittance of the system, given by these formulas, should be quite obvious to the radio physicist or electrical engineer. Precisely such a dependence follows from Kirchhoff’s laws for branched circuits and the Nyquist formula for each separate branch, if one also takes into account the statistical independence of the fluctuating e.m.f.’s acting in nonoverlapping sections of the circuit.

Formulas (3) and (4) give a complete solution of the problem of the spectral description of thermal fluctuations in any dissipative linear system with lumped parameters. Often, however, we are interested not in lumped but in distributed (in the general case, three-dimensional) systems, in which the fluctuations are described by a set of random fields \(\xi_j(t,\mathbf r)\) satisfying partial differential equations. This may concern both thermal fluctuations of the electromagnetic field, including thermal radiation, and fluctuations of the mechanical and thermal parameters characterizing the state of a continuous medium. In the present article a cycle of works is summarized, carried—

*) The matrix \(Y_{jk}=i\omega a_{jk}\) is called in works\(^7,8\) the admittance matrix of the dissipative system under consideration. It should be noted that the correlation matrix (3) was obtained by M. A. Leontovich\(^9\) as early as 1941, but its connection with admittance was not traced to the end. The fluctuation-dissipation theorem (3) and (4), obtained in \(^7,8\), is written here in the form given to it by L. D. Landau and E. M. Lifshitz\(^ {10}\).

...by the author and a number of other Soviet physicists in the period 1952–1956, and concerning specifically electromagnetic fluctuations of thermal origin*).

The problem of a spectral description of thermal fluctuations of the electromagnetic field was solved in 1952, initially for the quasistationary frequency range, by M. A. Leontovich and the author \(^{11}\), and then also for the general case of a system of Maxwell equations \(^{12}\). The starting point in these works was only the Nyquist theorem, so that the introduction of extraneous fluctuation fields and the establishment of correlation functions for their spectral amplitudes was done on the basis of a number of physical considerations, not yet formalized in the form of any regular method. Nevertheless, these basic elements were found correctly and made it possible to construct a developed theory of thermal electric fluctuations \(^{13}\), which includes as limiting cases:

a) the classical theory of thermal radiation (the geometrical-optics approximation, i.e. body dimensions \(l \gg \lambda\)) and

b) the Nyquist theory (quasistationary region, \(l \ll \lambda\)).

At the present time there is no longer any need to repeat the guiding considerations that initially made it possible to establish the form of the correlation functions for extraneous fluctuation electromagnetic fields. After the work of Callen and his coauthors, the possibility appeared for a quite general and regular application of the fluctuation-dissipation theorem to distributed systems. L. D. Landau and E. M. Lifshitz \(^{10}\) did this for Maxwell’s equations, operating directly with the “discrete” formulas (3) and (4). To make it possible to apply these formulas, they used a subdivision of the continuous system into small volumes with the corresponding replacement of differential equations by difference equations. As a result they obtained, for extraneous fluctuation fields, precisely those correlation functions which had earlier been found in works \(^{11-13}\). Although the indicated method in principle solves the question, in practice it is not especially convenient for applications. It is simpler to use formulas that directly generalize the fluctuation-dissipation theorem to arbitrary distributed systems. These formulas can be obtained by expanding random fields in some (auxiliary) complete system of orthonormal functions and applying theorems (3) and (4) to the coefficients of such expansions \(^{14}\). We shall present these formulas, omitting their derivation.

2. FLUCTUATION-DISSIPATION THEOREM FOR DISTRIBUTED SYSTEMS

Let the thermal fluctuations in the distributed system under consideration be described by a collection of random fields \(\xi^{(j)}(t,\mathbf r)\), \(j=1,2,\ldots\). To them correspond fluctuating extraneous forces with volume densities \(f^{(j)}(t,\mathbf r)\), such that the change of the mean energy in some volume \(V\), associated with their work, proceeds at the rate

\[ \frac{d\overline W}{dt} = \sum_j \int_V f^{(j)} \frac{\partial \overline{\xi}^{(j)}}{\partial t}\,dV = -\sum_j \int_V \overline{\xi}^{(j)} \frac{\partial f^{(j)}}{\partial t}\,dV . \tag{6} \]

*) It should be emphasized that only works concerning the correlation theory and its applications are meant; therefore a number of other interesting works on thermal electric fluctuations are not touched upon. Some of these works are listed in the bibliography \(^{26-31}\).

Let further the spectral amplitudes of the fields \(\xi^{(j)}\) and \(f^{(j)}\), i.e., the amplitudes in the Fourier expansions

\[ \xi^{(j)}(t,\mathbf r)=\int_{-\infty}^{+\infty}\xi_\omega^{(j)}(\mathbf r)e^{i\omega t}\,d\omega \]

and similarly for \(f^{(j)}\), be related to one another by linear equations containing linear spatial operators:

\[ \xi_\omega^{(j)}(\mathbf r)=\sum_k A_{jk} f_\omega^{(k)}(\mathbf r),\qquad f_\omega^{(j)}(\mathbf r)=\sum_k A_{jk}^{-1}\xi_\omega^{(k)}(\mathbf r). \tag{7} \]

In applications one is usually given the matrix of inverse operators \(A_{jk}^{-1}\), which are differential, i.e., \(A_{jk}^{-1}=A_{jk}^{-1}(\nabla)\). Then, by extrapolating the theorems (3) and (4) to a countable set of variables, it is not difficult to obtain the matrices of the spatial correlation functions for \(\xi_\omega^{(j)}\) and \(f_\omega^{(j)}\). For brevity we restrict ourselves to the formula only for \(f_\omega^{(j)}\):

\[ \overline{f_\omega^{(j)}(\mathbf r+\boldsymbol\rho)f_{\omega'}^{(k)*}(\mathbf r)} =iC(\omega)\delta(\omega-\omega') \left\{A_{kj}^{-1*}(-\nabla_\rho)-A_{jk}^{-1}(\nabla_\rho)\right\}\delta(\boldsymbol\rho). \tag{8} \]

The correlation functions of the parameters \(\xi_\omega^{(j)}\) are expressed in an analogous way through the operators \(A_{jk}\). The introduction of fluctuation forces is convenient precisely because (8) contains the operators \(A_{jk}^{-1}\), which are given directly by the differential equations of the system.

3. APPLICATION TO THE ELECTROMAGNETIC FIELD IN A HOMOGENEOUS MEDIUM

Let the medium be described by tensors of complex dielectric and magnetic permeabilities \(\varepsilon_{jk}\) and \(\mu_{jk}\). Following \({}^{10}\), we introduce fluctuation forces in the form of “extraneous” inductions \(\mathbf K\) and \(\mathbf M\), so that the corresponding increment of electromagnetic energy in a volume \(V\) will be

\[ \frac{d\overline W}{dt} =-\frac{1}{4\pi}\int_V\left(\mathbf E\frac{\partial\mathbf K}{\partial t} +\mathbf H\frac{\partial\mathbf M}{\partial t}\right)dV, \tag{9} \]

and the equations for the spectral amplitudes (we omit the index \(\omega\)) are written in the form

\[ \begin{aligned} K_j&=-\varepsilon_{jk}E_k+\frac{c}{i\omega}\operatorname{rot}_j\mathbf H,\\ M_j&=-\mu_{jk}H_k-\frac{c}{i\omega}\operatorname{rot}_j\mathbf E. \end{aligned} \tag{10} \]

From comparison of (9) with (6) it is evident that if the parameters \(\xi^{(j)}\) are taken to be \(\mathbf E/4\pi\) and \(\mathbf H/4\pi\), then the generalized forces \(f^{(j)}\) will be precisely \(\mathbf K\) and \(\mathbf M\). In turn, comparison of (10) with (7) shows that the matrix of operators \(A^{-1}\) is

\[ A_{K_jK_k}^{-1}=-4\pi\varepsilon_{jk},\qquad A_{M_jM_k}^{-1}=-4\pi\mu_{jk}, \]

\[ A_{K_jM_k}^{-1}=A_{M_kK_j}^{-1} =-\frac{4\pi c}{i\omega} \begin{pmatrix} 0 & \nabla_3 & -\nabla_2\\ -\nabla_3 & 0 & \nabla_1\\ \nabla_2 & -\nabla_1 & 0 \end{pmatrix}. \]

As a result, from formula (8) we find:

\[ \begin{aligned} \overline{K_{j\omega}(\mathbf r+\boldsymbol\rho)K^*_{k\omega'}(\mathbf r)} &=4\pi i C(\omega)(\varepsilon_{jk}-\varepsilon^*_{kj})\hat\delta(\boldsymbol\rho)\delta(\omega-\omega'),\\ \overline{M_{j\omega}(\mathbf r+\boldsymbol\rho)M^*_{k\omega'}(\mathbf r)} &=4\pi i C(\omega)(\mu_{jk}-\mu^*_{kj})\hat\delta(\boldsymbol\rho)\delta(\omega-\omega'),\\ \overline{K_{j\omega}(\mathbf r+\boldsymbol\rho)M^*_{k\omega'}(\mathbf r)} &=0. \end{aligned} \tag{11} \]

Even before the development of the regular method set forth here, these formulas were obtained by F. V. Bunkin\(^{15}\), who generalized, on the basis of indirect physical considerations, the formulas for an isotropic medium found earlier by the author\(^{12,13}\). For an isotropic medium \((\varepsilon_{jk}=\varepsilon\delta_{jk},\ \mu_{jk}=\mu\delta_{jk})\) we have

\[ \overline{K_{j\omega}(\mathbf r+\boldsymbol\rho)K^*_{k\omega'}(\mathbf r)} =4\pi i C(\omega)(\varepsilon-\varepsilon^*)\delta_{jk}\hat\delta(\boldsymbol\rho)\delta(\omega-\omega') \tag{12} \]

and an analogous expression for \(\mathbf M_\omega\) through the imaginary part of \(\mu\).

According to (11), a medium possessing no losses is not a source of thermal fluctuations. In this case the extraneous electric field is connected only with electric losses, and the extraneous magnetic field only with magnetic losses. These fields are not correlated with each other. Of course, the intensity of thermal radiation may be different from zero even where there are no extraneous fields, i.e. in completely transparent media, since transparency makes possible the arrival of radiation from remote bodies.

As follows from (12), the spectral intensity of the field strength \(K/\varepsilon\) of the extraneous fluctuation field is proportional to \(i\left(\frac{1}{\varepsilon^*}-\frac{1}{\varepsilon}\right)\). If one takes into account that \(\varepsilon=\varepsilon'-i\frac{4\pi\sigma}{\omega}\), then this factor is equal to

\[ \frac{8\pi\sigma\omega}{(\varepsilon'\omega)^2+(4\pi\sigma)^2}, \]

i.e. it vanishes both for \(\sigma\to\infty\) and for \(\sigma\to0\). It is not difficult to understand that, in passing from \(K/\varepsilon\) to the integral emf of a dipole, the expression then obtained for the spectral intensity of the emf is proportional not to \(R\), but to \(R/(1+\omega^2 C^2R^2)\), where \(C\) is the equivalent capacitance of the dipole, shunting its active resistance \(R\). Thus, in contrast to (1), the fluctuation emf disappears not only in passing to an ideal conductor \((R\to0)\), but also in passing to an ideal dielectric \((R\to\infty)\).

The extraneous inductions appearing in (10) can, of course, be expressed through the corresponding extraneous currents, which is often convenient. Thus, for example, instead of \(\mathbf K_\omega\) one may introduce an extraneous electric current with density \(\mathbf j_\omega=\frac{i\omega}{4\pi}\mathbf K_\omega\). From (11) and (5′) it follows that in the classical frequency region \((\hbar\omega\ll\Theta)\) the correlation function of the components \(\mathbf j_\omega\) will be

\[ \overline{j_\omega(\mathbf r+\boldsymbol\rho)j^*_{\omega'}(\mathbf r)} =\frac{\Theta\sigma_{jk}}{\pi}\hat\delta(\boldsymbol\rho)\delta(\omega-\omega'), \tag{13} \]

where \(\sigma_{jk}\) is the conductivity tensor.

4. GENERAL CHARACTERIZATION OF THE THEORY

The inhomogeneous Maxwell equations (10) make it possible to formulate any problem on thermal fluctuations of field strengths (or of any other quantities linearly connected with these strengths—charge and current densities, integral quantities such as the total current strength, etc.) as an ordinary boundary-value problem of electrodynamics. The solution of the boundary-

the problem constitutes the first—purely electrodynamic—stage and leads to expressions for \(\mathbf E\) and \(\mathbf H\) (and other electric and magnetic quantities) in the form of linear integral operators of the external fields or currents. But we are usually interested in the intensities of the fluctuations, i.e. statistically averaged bilinear functions of the components \(\mathbf E\) and \(\mathbf H\), such as, for example, the spectral densities of the electric and magnetic energy\(^*\):

\[ w_\omega^{el}=\frac{\varepsilon+\varepsilon^*}{8\pi}\,\overline{\mathbf E_\omega \mathbf E_\omega^*}, \qquad w_\omega^{mag}=\frac{\mu+\mu^*}{8\pi}\,\overline{\mathbf H_\omega \mathbf H_\omega^*}, \tag{14} \]

the Poynting vector:

\[ \mathbf S_\omega=\frac{c}{4\pi}\left\{\overline{[\mathbf E_\omega,\mathbf H_\omega^*]}+\overline{[\mathbf E_\omega^*,\mathbf H_\omega]}\right\}, \tag{15} \]

average Maxwell stresses, etc. The derivation of such quantities constitutes the second—statistical—stage in the solution of any fluctuation problem and is carried out with the aid of the already known correlation functions of the external fields (11).

Let us emphasize once more: since the electrodynamic part of the problem is solved on the basis of the general field equations, the relation between the dimensions of the bodies \((l)\) and the wavelength \((\lambda)\) is in no way restricted, i.e. the solution covers all diffraction phenomena occurring under the given geometrical conditions. In the limiting cases \(l\gg\lambda\) or \(l\ll\lambda\), respectively, the approximation of geometrical optics (the classical theory of thermal radiation) or the quasistationary approximation (Nyquist’s theory for fluctuations of integral quantities in circuits with lumped parameters) come into force. Obviously, the greatest interest lies in applying the theory to problems in which \(l\) is of the same order as \(\lambda\) (as is for the most part the case in the microwave range), since precisely in such cases the question cannot be approached either with the classical theory of thermal radiation or with the quasistationary theory, which makes use of Nyquist’s integral emf. Before giving examples of the solution of concrete problems, we shall indicate one variety of the method described that is of known practical interest.

5. THE CASE OF WELL-CONDUCTING BODIES \(^{13}\)

If the skin effect is sufficiently strong, then, as is known, the electromagnetic field outside the bodies is only slightly perturbed in comparison with what it would be in the case of ideally conducting bodies of the same shape and arrangement. This makes it possible in general to dispense with explicit accounting of the field inside the bodies and to consider only the external field of interest to us, subjecting it to certain approximate boundary conditions close to the conditions on the surface of an ideal conductor. The indicated approximate relations, whose possible use as boundary conditions was pointed out in 1940 by M. A. Leontovich \(^{16}\), can be obtained both by solving the skin-effect problem by the perturbation method \(^{17}\), and from simple intuitive considerations. These conditions relate the values of the tangential components of the external electric and magnetic fields on the surface of the body:

\[ \sqrt{\mu}\,H_t=-\sqrt{\varepsilon}\,[\mathbf N,\mathbf E_t], \]

where \(\mathbf N\) is the unit vector of the normal to the surface.

\(^*\) Formulas (14) refer to the case of a nondispersive medium.

It is natural to use the advantages of the indicated formulation of the problem also in the question of electromagnetic fluctuations. For this it proves sufficient to introduce into M. A. Leontovich’s boundary conditions the surface extraneous fields, namely:

\[ \sqrt{\mu}\,(\mathbf H_t+\mathfrak B)=-\sqrt{\varepsilon}\,[\mathbf N,\mathbf E_t+\mathfrak R]. \tag{16} \]

The problem of finding the fluctuation fields \(\mathbf E\) and \(\mathbf H\) is now posed with homogeneous Maxwell equations, but with inhomogeneous conditions (16) on the surfaces of conductors. To obtain average energy quantities, the correlation functions of the surface extraneous fields \(\mathfrak B\) and \(\mathfrak R\) are needed, but they follow uniquely from the already known correlation functions (11).

The described modification of the theory considerably simplifies the solution of a number of problems and in many cases proves applicable in the microwave range.

6. SOME RESULTS OF THE APPLICATION OF THE THEORY

a) Radiation into free space

Let the radiating body be an infinite circular cylinder, whose radius \(a\) may be in any ratio to the wavelength \(\lambda\) in the surrounding transparent medium (for simplicity, in vacuum). The material of the cylinder is nonmagnetic \((\mu=1)\). We are interested in the power \(P_\omega\) of thermal radiation from a unit length of the cylinder in a unit frequency interval about \(\omega\). The solution has the form (13)

\[ P_\omega=\frac{\Theta}{2\pi}\,f\left(\varepsilon,\frac{a}{\lambda}\right). \]

Without giving the rather cumbersome general expression for the function \(f\), we shall restrict ourselves to three special cases.

Let us denote by \(d\) the thickness of the skin layer in the material of the cylinder:

\[ d=\frac{c}{\sqrt{2\pi\sigma\omega}}. \]

For \(a\gg \lambda\gg d\), i.e. for a thick and well-conducting cylinder, the power \(p_\omega\) radiated from a unit surface area \((p_\omega=P_\omega/2\pi a)\) is

\[ p_\omega=\frac{2\Theta}{3\pi^2}k^3d,\qquad \left(k=\frac{\omega}{c}=\frac{2\pi}{\lambda}\right). \]

Such power is radiated from a unit area of any well-conducting surface in the case of wavelengths so short that the geometrical-optics approximation is valid. The formula written is a direct consequence of Kirchhoff’s law. The two following formulas refer to the opposite case of a thin cylinder \((\lambda\gg a)\).

If it conducts well \((a\gg d)\), then

\[ p_\omega=\frac{\Theta k}{4\pi^2 a}\sqrt{\frac{d}{a\{\ln(ka/2)\}^3}}. \]

This formula covers the transition to an ideal conductor, when \(\sigma\to\infty\)

and \(d \to 0\). If, on the contrary, \(|\varepsilon|^2\) is small, so that \(\lambda \gg a|\sqrt{\varepsilon}|\), then

\[ p_\omega=\frac{\Theta k^3}{6\pi^2}\,a\varepsilon'', \qquad \left(\varepsilon''=\frac{4\pi\sigma}{\omega}\right). \]

Here the transition to an ideal dielectric is covered (i.e., the case \(\varepsilon'' \to 0\)). The dependence of \(p_\omega\) on \(\omega\) and \(a\) is completely different in all three cases.

The analogous problem for a sphere of radius \(a\), made of a well-conducting material \((a \gg d)\), was solved by the method set forth in Section 5[^13]. Figure 1 shows the dependence of the power \(p_\omega\), radiated from a unit surface area of the sphere, on \(ka\), with the radius \(a\) varied. The curve is constructed for \(kd \le 0.001\) (for \(ka < 3\) the curves corresponding to values \(kd \le 0.001\) coincide).

For \(ka \ll 1\) (but, of course, \(ka \gg kd\)) \(p_\omega\) assumes the value

\[ p_\omega=\frac{3\Theta}{4\pi^2}\,k^3 d. \]

This result can also be obtained with the aid of Nyquist’s formula, starting from the expression for the radiation of a fluctuation current flowing in a small (in comparison with \(\lambda\)) ring turn[^13]. For large \(ka\), the specific power \(p_\omega\) approaches the same value as was obtained above for a thick, well-conducting cylinder and as follows from Kirchhoff’s law. The fact that, for \(ka \sim 1\), \(p_\omega\) exceeds by one and a half to two times the value ensuing from Kirchhoff’s law, and also the oscillatory behavior of \(p_\omega\) for small \(kd\) (the maxima lie approximately at those values of \(ka\) which correspond to natural oscillations of the electric type), is the result of diffraction of the emitted waves around the sphere.

Fig. 1.

In the problems considered, the conditions are simplified by the symmetry of the bodies: the flux of the energy of fluctuation radiation is distributed uniformly over all radial directions. Of greater interest are cases in which the radiating body is not so symmetric, but is, for example, a plate or an aperture in a screen, a wire or a slot antenna. In such cases the radiation will no longer be uniform, but will possess a directivity pattern depending on the shape and dimensions of the radiator and on the spectral interval under consideration, i.e., on the wavelength.

Figure 2 gives the directivity patterns of the thermal radiation of a thin rectilinear antenna of finite length \(2L\), calculated for \(\beta=2kL=5\) (upper quadrant) and \(\beta=2.5\pi\) (lower quadrant)[^18]. For comparison, dashed lines show the patterns corresponding to a concentrated emf applied at the middle of the antenna. The theory makes it possible to calculate not only the directivity of the radiation, but also the distribution of the noise current along the antenna, the dependence of the radiation resistance on \(\beta\), etc.[^18]

In this problem there is also a simplifying circumstance, affecting both its electrodynamic part and its statistical part. This is the предпо-

assumption of the thinness of the wire, which, first, makes it possible to pass from the general field equations to the equations of the theory of thin-wire antennas, developed several years ago by M. A. Leontovich and M. L. Levin19, and

Fig. 2.

second, makes it possible to introduce, instead of an extraneous field distributed over the volume of the wire, Nyquist’s electromotive force per unit length, distributed only along the length of the antenna13.

b) Radiation in lines and volume resonators.

Another class of questions where the general theory finds a natural and practically interesting application is thermal radiation not into free space, but into cavities and channels with metallic walls—volume resonators, waveguides, and coaxial lines, i.e., thermal radiation in ultra-high-frequency radio apparatus. Here too a number of specific problems have been solved, for example, the thermal radiation of a partition in a waveguide, the radiation of the walls of the waveguide itself, etc.13.

The general result here is a distinctive form of Kirchhoff’s law, which it is appropriate to call the waveguide form, and which relates the power of thermal radiation passing through the cross-section of a channel to the absorption coefficients of the radiating body for the various types of waves possible in the given channel13. Namely, the total spectral power \(P_\omega\) entering a waveguide from a certain radiator satisfying the reciprocity principle and having temperature \(\Theta\) is

\[ P_\omega=\frac{\Theta}{2\pi}\sum_{m,n}^{\prime}\left\{A_{mn}^{(E)}+A_{mn}^{(H)}(\omega)\right\}, \tag{17} \]

where \(A_{mn}^{(E,H)}(\omega)\) are the absorption coefficients of the radiator for \(TM\)- and \(TE\)-waves of numbers \(m,n\). The prime by the summation sign indicates that the summation is carried out only over those values of \(m\) and \(n\) which correspond to propagating (subcritical) waves. If, for the radiating body, the reciprocity principle is not satisfied owing to the presence of gyrotropy (we exclude nonlinear properties), then the differences of the corresponding transparency coefficients for waves of the forward and reverse directions of propagation20 are added to the absorption coefficients in (17).

Formula (17) is valid not only for feeders, but also for any set of normal waves propagating in one dimension. In particular, the power radiated by a cylinder, sphere, or any other body into free space can be reduced to the same form.

Therefore only the “one-dimensionality” of result (17) gives grounds for calling it the waveguide form of Kirchhoff’s law. However, this is also justified by the fact that waveguides precisely provide a real possibility of separating waves of individual types and numbers, whereas in radiation into free space only the entire sum entering into (17) is of interest, and moreover the sum is infinite because of the absence of critical frequencies.

An illustration of formula (17) is given by Fig. 3, in which, on a conventional scale, the dependence of the power \(P_\omega\), radiated into a rectangular waveguide with a well-conducting partition, on the parameter \(\xi = 2a/\lambda\), is shown, where \(a\) is the smaller side of the rectangular cross-section. The sharp peaks of the absorption coefficients of \(TM\)-waves, situated at the critical frequencies of these waves, produce surges of \(P_\omega\), which on the adopted scale go far beyond the limits of the drawing. The dashed line shows that course of \(P_\omega(\xi)\) which is obtained when the classical Kirchhoff law is extrapolated into the region of small \(\xi\).

Fig. 3.

It should be noted that so long as the radiators are uniformly heated bodies, the waveguide form of Kirchhoff’s law can be obtained, in addition to the general theory of electrical fluctuations, on the basis of energy considerations alone and the theorem on the distribution of energy over degrees of freedom. But as soon as the question concerns nonuniform heating and (or) inhomogeneous bodies (which is more interesting from the practical point of view), the solution of the problem requires the use precisely of the general theory. Its regular method is applicable also under these conditions, since with macroscopic inhomogeneities of material and temperature the spatial \(\delta\)-correlation of the extraneous field can be preserved. The result is the waveguide form of Kirchhoff’s law, generalized in accordance with the stated formulation of the problem \(^{20,21}\).

7. USE OF THE RECIPROCITY PRINCIPLE

If what is of interest is not only the total flux of thermal radiation given by formula (17), but also its spatial distribution and, in general, the entire structure of the thermal fluctuation field, then energy considerations and the theorem on the distribution of energy over degrees of freedom are no longer sufficient. The corresponding results can be obtained only on the basis of the general theory of electrical fluctuations.

As was noted, the first stage here is the solution of the electrodynamic boundary-value problem. Thus all interference and diffraction phenomena taking place under the given geometrical conditions are already included in the answer. Suppose now that, for the body under consideration, a ready solution of the diffraction problem is already available; for example, the diffraction field is known which is formed when the given body is irradiated by a wave emitted by a dipole located somewhere in the external space. Then, as M. L. Levin \(^{22}\) showed, with the aid of the reciprocity theorem one can obtain very simple formulas in which the intensity of the thermal radiation of the body at some external point is expressed through an auxiliary field produced by a dipole placed

to this point. The determination of the energy of thermal radiation and of other mean quadratic quantities is thereby reduced to quadratures; moreover, for good conductors the calculation is still further simplified, and in the limiting cases of short and long waves becomes elementary.

Let a dipole with moment \(\mathbf{p}\) (oscillating with frequency \(\omega\)) be located at some point \(P\) and, in the presence of the radiating body, produce a diffraction field \(\mathbf{E}_0\). This field causes, in the volume element \(dV\) of the body under consideration (for simplicity, isotropic), thermal losses

\[ dQ_0=\frac{\sigma}{2}\,|\mathbf{E}_0|^2\,dV . \]

Application of the reciprocity theorem and of the correlation function (13) then gives, for the mean square of the \(\mathbf{p}\)-component of the thermal field produced by the given body at the point \(P\), the expression

\[ \overline{|E_p|^2}=\frac{2}{\pi\omega^2|\mathbf{p}|^2}\int \Theta\,dQ_0, \tag{18} \]

where the integration extends over the volume of the body. The intensity of the thermal magnetic field \(\mathbf{H}\), as well as the mean values of products of different components of \(\mathbf{E}\) and \(\mathbf{H}\), are expressed by analogous quadratures. Formula (18) does not presuppose uniform heating of the body. Introducing the effective temperature (with a weight equal to the density of thermal losses):

\[ \widetilde{\Theta}=\frac{1}{Q_0}\int \Theta\,dQ_0, \]

one may rewrite (18) in the form

\[ \overline{|E_p|^2}=\frac{2\widetilde{\Theta}Q_0}{\pi\omega^2|\mathbf{p}|^2}. \tag{18'} \]

If the body is in free space and the point \(P\) is at a distance \(R\gg\lambda\) (the wave zone), then (18′) is transformed into the formula

\[ \overline{|E|^2}=\frac{\widetilde{\Theta}A}{c k^2 R^2}, \]

where

\[ A=\frac{Q_0}{\dfrac{c}{8\pi}\,|\mathbf{E}_0|^2} \]

is the effective absorption cross section of the body for a plane wave of amplitude \(\mathbf{E}_0\), incident in the direction \(\mathbf{R}\).

By the method indicated, M. L. Levin\(^{23}\) obtained formulas for short-wave thermal radiation of well-conducting plane plates and of bodies of revolution, as well as formulas for the radiation of spheroids in the opposite case of long waves. He further found the fluctuation field near well-conducting surfaces.

In particular, he showed that, with approximate allowance for the Sommerfeld attenuation function, the following expressions are obtained for the densities of electric and magnetic energy of the fluctuation field at a distance \(z\) from a conducting plane:

\[ w_{\omega}^{\mathrm{el}}\approx \frac{\eta w_0}{2}\left(\ln\frac{L}{z}+\frac{3}{4k^2z^2}\right), \]

\[ w_{\omega}^{\mathrm{mag}}\approx \frac{\eta w_0}{2}\left(\ln\frac{L}{z}+\frac{1}{4k^2z^2}+\frac{1}{2k^4z^4}\right). \tag{19} \]

Here

\[ k=\frac{\omega}{c}=\frac{2\pi}{\lambda},\qquad w_0=\frac{\Theta k^2}{\pi^2 c} \]

is the energy density of equilibrium thermal

radiation,

\[ \eta=\sqrt{\frac{\mu\omega}{8\pi\sigma}}=\frac{\mu k d}{2} \]

(\(d\) is the thickness of the skin layer) and

\[ L\sim \frac{\sigma \mu \lambda^{2}}{c}\sim r/r_{\text{num}} \]

(\(r_{\text{num}}\) is the so-called numerical distance). The distance \(z\) of the point under consideration from the plane is subject in these formulas to the conditions \(L\gg z\gg d\). From (19) it follows that for \(z\ll \lambda\) the magnetic energy considerably exceeds the electric energy, so that at such distances

\[ \frac{w}{w_{0}}\simeq \frac{w^{\text{mag}}}{w_{0}}\simeq \frac{\eta}{4k^{4}z^{4}} . \tag{20} \]

In the same work, in the approximation of geometrical optics, the fluctuation field was calculated at the focus of a parabolic mirror and at the center of a spherical one.

Fig. 4.

Applying the same method, M. L. Levin clarified the question of thermal noises induced in an arbitrary wire antenna by external radiators, in particular by the surface of the earth, above which the antenna is located\(^ {24}\).

The spectral intensity (over positive frequencies \(\omega\)) of the thermal e.m.f. induced in the antenna by the radiation of surrounding bodies, in the case where these are nonmagnetic bodies (\(\mu=1\)) possessing a strongly pronounced skin effect (\(kd\ll 1\)), turns out to be equal to

\[ \overline{\mathcal{E}_{\omega}^{2}}=\frac{4\Theta}{\pi}\,\frac{kd}{c}\,F, \]

where \(F\) is a dimensionless factor depending on the mutual arrangement of the antenna and the bodies, on their shape and on the wavelength. For a half-wave vibrator located: a) at the focus of a parabolic mirror, b) at the center of a spherical mirror, and c) on the axis of a round plane mirror (Fig. 4), the factor \(F\), calculated in the approximation of geometrical optics, varies as a function of the aperture angle \(\Phi\) as shown in Fig. 5, \(a, b, c\).

Fig. 5.

From the solutions of diffraction problems available in the literature,*) the method described will make it possible in many further cases to obtain fluctua-

*) Let us point out, in particular, the recently published collection of articles Diffraction of Electromagnetic Waves by Certain Bodies of Revolution. Sov. Radio Publishing House, 1957.

tional electromagnetic field for the corresponding bodies. One may mention, for example, the radiation of a disk, an elliptic cylinder, the fluctuational field in the near zone for a circular cylinder and a sphere, etc.

8. THE THERMAL ELECTROMAGNETIC FIELD NEAR RADIATING BODIES

The distinction between the fluctuational field of the near zone and fluctuational radiation is, of course, not accidental. The classical theory of thermal radiation, relying on geometrical optics, deals only with the wave field. If, as classical theory assumes, the wavelength is vanishingly small, then at any finite distance from the source we are already in its wave zone. General electrodynamics broadens the notion of an electromagnetic field of thermal origin. Random elementary sources, which are distributed in the volume of a body and are described with the aid of an external field, create not only electromagnetic waves but also a quasistationary field, which falls off more rapidly with distance. Inside the body, i.e. in an absorbing medium, the wave and quasistationary fields are intermixed and, strictly speaking, inseparable. But outside the body, in the surrounding nonabsorbing medium or in vacuum, they behave differently. The elementary wave fields, when superposed, give the radiation of the given body, i.e. the total chaotic wave field. The elementary quasistationary fields, however, do not create an energy flux, but form a total quasistationary thermal field that decreases rapidly with distance from the surface of the body. This field, as though lining the surfaces of bodies, is alien to the classical theory of thermal radiation to the same extent as it is completely natural and comprehensible from the point of view of fluctuational electrodynamics[^13].

The energy density of the quasistationary thermal field increases as one approaches the surface of the body and, beginning at distances of the order of a wavelength, predominates over the energy density of the radiation. This fact is reflected in formulas (19), (20). It follows from this that cavities whose dimensions are smaller than the wavelength are filled mainly by precisely the quasistationary thermal field. From this one may also conclude that any device possessing frequency-selective absorption, when brought to the surface of a heated body closer than one wavelength, must register a strong increase in the fluctuational field strengths.

But this is not the only manifestation of the quasistationary thermal field. It appears in an essential way in a quite different phenomenon, which was investigated on the basis of the theory set forth here by E. M. Lifshitz[^25]. He considered the ponderomotive forces of interaction of bodies caused by the fluctuational electromagnetic field in the space between the surfaces of these bodies. Suppose, for example, that two half-spaces, filled for simplicity with one and the same substance at the same temperature, are separated by a plane gap. Fluctuational electrodynamics makes it possible to find the thermal electromagnetic field in this gap and then to calculate the mean value of the corresponding component of the Maxwell stress tensor, i.e. the force of attraction per unit surface of the bodies.

Having done this, E. M. Lifshitz thus obtained an expression for the force of “molecular cohesion” between macroscopic bodies, applicable for any densities of the bodies and any temperatures. This expression automatically takes into account the predominance of the quasistationary thermal field in the case of thin gaps and, conversely, the predominance of the wave field, i.e. retardation effects, for a large width of the gap. At low temperatures

at temperatures \((\Theta \ll \hbar \omega)\), the only scale for the width of the gap is given by those wavelengths that are most strongly absorbed by the substance under consideration. Let us denote such a characteristic wavelength by \(\lambda_a\). The adhesion force \(F\) turns out to be inversely proportional to the cube of the distance \(l\) between the surfaces if the latter is small in comparison with these wavelengths \((l \ll \lambda_a)\), and to the fourth power of the distance if it is large \((l \gg \lambda_a)\). In this latter case the adhesion force for two metals is, in first approximation, equal to

\[ F=\frac{\pi^2}{240}\,\frac{\hbar c}{l^4}, \]

i.e., in general it does not depend on the kind of metals. When two dielectrics are brought into contact (or a dielectric with a metal), however, a factor enters which depends only on the static values of the dielectric permittivities. What has been said is valid, however, under the condition that the width of the gap \(l\), while large in comparison with \(\lambda_a\), is at the same time small in comparison with the correlation radius of the thermal-radiation field \(\lambda_r \sim \dfrac{\hbar c}{\Theta}\), i.e. in the region

\[ \lambda_a \ll l \ll \lambda_r . \]

At sufficiently low temperatures this region always exists, but at higher ones, for example at room temperatures, it may turn out that \(\lambda_r < \lambda_a\), and then the case \(l \gg \lambda_a\) will mean that \(l \gg \lambda_r\). Under these conditions the attraction force depends substantially on temperature. For two identical substances it is, in first approximation, equal to

\[ F \simeq \frac{\Theta}{8\pi l^3}\left(\frac{\varepsilon_0-1}{\varepsilon_0+1}\right)^2, \]

where \(\varepsilon_0\) is the static value of the dielectric permittivity.

The theory of macroscopic adhesion forces was previously constructed on the basis of the elementary law of van der Waals forces between atoms or molecules, which in advance limited the result to the case of rarefied media. A purely phenomenological theory, based on fluctuation electrodynamics, removes this limitation. Conversely, starting from the expression for the macroscopic adhesion force, one can in the case of rarefied media draw the reverse conclusion—about the law of pair interaction of individual neutral atoms and molecules. Such a path, paradoxical as it may seem at first glance, turns out to be simpler than the direct quantum-mechanical calculation for two neutral particles, in which the interaction law is obtained only in high orders of perturbation-theory calculations.

9. INTERFERENCE PHENOMENA IN THERMAL RADIATION

Interference phenomena in thermal radiation are encountered in practice in apparatus for ultrahigh radio frequencies, when the thermal radiation entering a waveguide is partially reflected at the other end because of an incompletely matched load. Standing waves are formed in the waveguide, whose contrast depends on the passband of the receiver, or, in the language of optics, on the degree of monochromaticity of the observed radiation. Similar phenomena occur whenever the dimensions of the space available to the thermal field are not sufficiently large, just as optical interference arises in thin plates. In optics, for a given monochromaticity of light, interference with increasing plate thickness

is smeared out, disappears. Likewise, in the microwave range, for a given pass band of the apparatus, interference phenomena disappear when the space filled with thermal radiation (in the example given, the length of the waveguide) becomes too large. The energy density then becomes uniform, and the asymptotic laws of the classical theory of radiation come into force.

Thus, for a sufficiently selective (narrow-band) receiver, the field of thermal radiation—even equilibrium radiation—is not homogeneous and isotropic. It contains interference maxima and minima, whose explanation lies beyond the limits of the classical theory of radiation, but which are fully taken into account by the general theory\(^ {13}\).

10. THERMAL RADIATION IN ANISOTROPIC MEDIA

Until recently, the question of thermal radiation in anisotropic (including gyrotropic) media could not be regarded as having been studied sufficiently completely. Kirchhoff’s laws, which form the basis of the classical theory of thermal radiation, were established for isotropic media. Their direct application to anisotropic media encounters certain difficulties, connected chiefly with the presence of double refraction. Meanwhile, this question has acquired interest primarily in connection with radio astronomy (thermal radiation of the solar corona with allowance for the general magnetic field of the Sun, and also thermal radiation of sunspots). Another possible field of application of the corresponding theory concerns the apparatus itself, namely the thermal radiation of ferrites used in the waveguide paths of modern receivers. Obviously, the correlation functions (10) make it possible to cover completely also the problems of radiation from anisotropic media, so that in principle, i.e. apart from an increase in purely computational difficulties, we have here as well a ready-made apparatus for obtaining the solution.

A detailed analysis carried out by F. V. Bunkin\(^ {15}\) showed that the flux of thermal radiation from a volume element of a homogeneous magnetoactive medium consists, in general, of three fluxes: two proper fluxes (ordinary and extraordinary waves) and an interference flux, characteristic precisely of a magnetoactive medium. In passing to simple anisotropy (a uniaxial crystal), the interference flux disappears, while the proper fluxes satisfy, respectively, the generalized Kirchhoff law: the emissivities for both kinds of waves \((i = 1, 2)\) in the direction making an angle \(\theta\) with the axis of symmetry are equal to

\[ \eta_{\omega i}(\theta) = \alpha_{\omega i}(\theta) I_{\omega i}(\theta). \tag{21} \]

Here \(I_{\omega i}(\theta)\) are the equilibrium intensities in a transparent uniaxial crystal, found in work\(^ {13}\), and \(\alpha_{\omega i}(\theta)\) are the absorption abilities, related in a definite way to the real and imaginary parts of the complex refractive indices for waves of both types and of the given direction of propagation.

The investigation showed that under real conditions of observation the interference flux plays no role in the case of a magnetoactive medium either, i.e. law (21) is practically valid with the corresponding expressions for \(a_{\omega i}(\theta)\).

A more detailed study was made of the more general case of quasi-homogeneous magnetoactive media, i.e. the case of the geometrical approximation, which is fully applicable to the study of the solar corona and spots. Of course, here the transfer equations prove to be valid, which could have been written down at once on the basis of the usual

energy considerations. But the transfer equations have real content only in the case where the expressions for the absorptive and emissive capacities of the medium are known. It is precisely this latter question that is solved in the best way with the aid of the regular method of fluctuation electrodynamics.

In the cited work by F. V. Bunkin there is also considered the practically interesting case of weak gyrotropy, when the effects due to anisotropy are small and can be taken into account in first approximation as corrections to the solution for an isotropic medium. The expressions for these corrections, and hence for the radiation intensities and its degree of polarization, turn out in this case to be very simple.

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