Reciprocity (electromagnetism)

Theorem in classical electromagnetism
reciprocity theorems in classical electromagnetism

In classical electromagnetism, reciprocity refers to a variety of related theorems involving the interchange of time harmonic electric current densities (sources) and the resulting electromagnetic fields in Maxwell's equations for time invariant linear media under certain constraints. Reciprocity is closely related to the concept of symmetric operators from linear algebra, applied to electromagnetism. Perhaps the most common and general such theorem is Lorentz reciprocity (and its various special cases such as Rayleigh Carson reciprocity), named after work by Hendrik Lorentz in 1896 following analogous results regarding sound by Lord Rayleigh and light by Helmholtz (Potton, 2004). Loosely, it states that the relationship between an oscillating current and the resulting electric field is unchanged if one interchanges the points where the current is placed and where the field is measured. For the specific case of an electrical network, it is sometimes phrased as the statement that voltages and currents at different points in the network can be interchanged. More technically, it follows that the mutual impedance of a first circuit due to a second is the same as the mutual impedance of the second circuit due to the first. Reciprocity is useful in optics, which (apart from quantum effects) can be expressed in terms of classical electromagnetism, but also in terms of radiometry. There is also an analogous theorem in electrostatics, known as Green's reciprocity, relating the interchange of electric potential and electric charge density. Forms of the reciprocity theorems are used in many electromagnetic applications, such as analyzing electrical networks and antenna systems. For example, reciprocity implies that antennas work equally well as transmitters or receivers, and specifically that an antenna's radiation and receiving patterns are identical. Reciprocity is also a basic lemma that is used to prove other theorems about electromagnetic systems, such as the symmetry of the impedance matrix and scattering matrix, symmetries of Green's functions for use in boundary element and transfer matrix computational methods, as well as orthogonality properties of harmonic modes in waveguide systems (as an alternative to proving those properties directly from the symmetries of the eigen operators).

The cancellation of the surface terms on the right hand side of the Lorentz reciprocity theorem, for an integration over all space, is not entirely obvious but can be derived in a number of ways. A rigorous treatment of the surface integral takes into account the causality of interacting wave field states: The surface integral contribution at infinity vanishes for the time convolution interaction of two causal wave fields only (the time correlation interaction leads to a non zero contribution). Another simple argument would be that the fields goes to zero at infinity for a localized source, but this argument fails in the case of lossless media: in the absence of absorption, radiated fields decay inversely with distance, but the surface area of the integral increases with the square of distance, so the two rates balance one another in the integral. Instead, it is common (e. g. King, 1963) to assume that the medium is homogeneous and isotropic sufficiently far away. In this case, the radiated field asymptotically takes the form of planewaves propagating radially outward (in the direction) with and where Z is the scalar impedance of the surrounding medium. Then it follows that which by a simple vector identity equals Similarly, and the two terms cancel one another. The above argument shows explicitly why the surface terms can cancel, but lacks generality. Alternatively, one can treat the case of lossless surrounding media with radiation boundary conditions imposed via the limiting absorption principle (LAP): Taking the limit as the losses (the imaginary part of ε) go to zero. For any nonzero loss, the fields decay exponentially with distance and the surface integral vanishes, regardless of whether the medium is homogeneous. Since the left hand side of the Lorentz reciprocity theorem vanishes for integration over all space with any non zero losses, it must also vanish in the limit as the losses go to zero. (Note that the LAP implicitly imposes the Sommerfeld radiation condition of zero incoming waves from infinity, because otherwise even an arbitrarily small loss would eliminate the incoming waves and the limit would not give the lossless solution.)

The inverse of the operator i. e., in (which requires a specification of the boundary conditions at infinity in a lossless system), has the same symmetry as and is essentially a Green's function convolution. So, another perspective on Lorentz reciprocity is that it reflects the fact that convolution with the electromagnetic Green's function is a complex symmetric (or anti Hermitian, below) linear operation under the appropriate conditions on ε and μ. More specifically, the Green's function can be written as giving the n th component of at from a point dipole current in the m th direction at (essentially, gives the matrix elements of ), and Rayleigh Carson reciprocity is equivalent to the statement that Unlike it is not generally possible to give an explicit formula for the Green's function (except in special cases such as homogeneous media), but it is routinely computed by numerical methods.

One case in which ε is not a symmetric matrix is for magneto optic materials, in which case the usual statement of Lorentz reciprocity does not hold (see below for a generalization, however). If we allow magneto optic materials, but restrict ourselves to the situation where material absorption is negligible, then ε and μ are in general 3×3 complex Hermitian matrices. In this case, the operator is Hermitian under the conjugated inner product and a variant of the reciprocity theorem still holds:

where the sign changes come from the in the equation above, which makes the operator anti Hermitian (neglecting surface terms). For the special case of this gives a re statement of conservation of energy or Poynting's theorem (since here we have assumed lossless materials, unlike above): The time average rate of work done by the current (given by the real part of ) is equal to the time average outward flux of power (the integral of the Poynting vector). By the same token, however, the surface terms do not in general vanish if one integrates over all space for this reciprocity variant, so a Rayleigh Carson form does not hold without additional assumptions. The fact that magneto optic materials break Rayleigh Carson reciprocity is the key to devices such as Faraday isolators and circulators. A current on one side of a Faraday isolator produces a field on the other side but not vice versa.

For a combination of lossy and magneto optic materials, and in general when the ε and μ tensors are neither symmetric nor Hermitian matrices, one can still obtain a generalized version of Lorentz reciprocity by considering and to exist in different systems. In particular, if satisfy Maxwell's equations at ω for a system with materials and satisfy Maxwell's equations at ω for a system with materials where denotes the transpose, then the equation of Lorentz reciprocity holds. This can be further generalized to bi anisotropic materials by transposing the full 6×6 susceptibility tensor.

For nonlinear media, no reciprocity theorem generally holds. Reciprocity also does not generally apply for time varying ("active") media; for example, when ε is modulated in time by some external process. (In both of these cases, the frequency ω is not generally a conserved quantity.)

In 1992, a closely related reciprocity theorem was articulated independently by Y. A. Feld and C. T. Tai,
and is known as Feld Tai reciprocity or the Feld Tai lemma. t relates two time harmonic localized current sources and the resulting magnetic fields:

However, the Feld Tai lemma is only valid under much more restrictive conditions than Lorentz reciprocity. It generally requires time invariant linear media with an isotropic homogeneous impedance, i. e. a constant scalar μ/ε ratio, with the possible exception of regions of perfectly conducting material. More precisely, Feld Tai reciprocity requires the Hermitian (or rather, complex symmetric) symmetry of the electromagnetic operators as above, but also relies on the assumption that the operator relating and is a constant scalar multiple of the operator relating and which is true when ε is a constant scalar multiple of μ (the two operators generally differ by an interchange of ε and μ). As above, one can also construct a more general formulation for integrals over a finite volume.

Whereas the above reciprocity theorems were for oscillating fields, Green's reciprocity is an analogous theorem for electrostatics with a fixed distribution of electric charge (Panofsky and Phillips, 1962). In particular, let denote the electric potential resulting from a total charge density The electric potential satisfies Poisson's equation, , where is the vacuum permittivity. Similarly, let denote the electric potential resulting from a total charge density , satisfying In both cases, we assume that the charge distributions are localized, so that the potentials can be chosen to go to zero at infinity. Then, Green's reciprocity theorem states that, for integrals over all space:

This theorem is easily proven from Green's second identity. Equivalently, it is the statement that

i. e. that is a Hermitian operator (as follows by integrating by parts twice).