Reciprocity (electromagnetism)

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This page is about reciprocity theorems in classical electromagnetism. See also Reciprocity (mathematics) for unrelated reciprocity theorems, and Reciprocity for more general usages of the term.

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 Hermitian 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 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.

Forms of the reciprocity theorem 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 with the same directionality. Reciprocity is also a basic lemma that is used to prove other theorems about electromagnetic systems, such as the symmetry of the mutual-impedance matrix, symmetries of the scattering matrix or 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).

Contents

1 Lorentz reciprocity
2 Reciprocity for electrical networks
3 Conditions for reciprocity

3.1 Surface-term cancellation
3.2 Reciprocity and the Green's function
3.3 Lossless and magneto-optic materials
3.4 Generalization to non-symmetric materials
3.5 Exceptions to reciprocity

4 Feld-Tai reciprocity
5 References

Lorentz reciprocity

Specifically, suppose that one has a current density [\mathbf_1] that produces an electric field [\mathbf_1] and a magnetic field [\mathbf_1], where all three are periodic functions of time with angular frequency ω, and in particular they have time-dependence [\exp(-i\omega t)]. Suppose that we similarly have a second current [\mathbf_2] at the same frequency ω which (by itself) produces fields [\mathbf_2] and [\mathbf_2]. The Lorentz reciprocity theorem then states, under certain simple conditions on the materials of the medium described below, that for an arbitrary surface S enclosing a volume V:

[\int_V \left[ mathbf_1 cdot mathbf_2 - mathbf_1 cdot mathbf_2 right] dV = \oint_S \left[ mathbf_1 times mathbf_2 - mathbf_2 times mathbf_1 right] \cdot \mathbf]

This general form is commonly simplified for a number of special cases. In particular, one usually assumes that [\mathbf_1] and [\mathbf_2] are localized (i.e. have compact support), and that there are no incoming waves from infinitely far away. In this case, if one integrates over all space then the surface-integral terms cancel (see below) and one obtains:

[\int \mathbf_1 \cdot \mathbf_2 \, dV = \int \mathbf_1 \cdot \mathbf_2 \, dV]

This result (along with the following simplifications) is sometimes called the Rayleigh-Carson reciprocity theorem, after Lord Rayleigh's work on sound waves and an extension by John R. Carson (1924; 1930) to applications for radio frequency antennas. Often, one further simplifies this relation by considering point-like dipole sources, in which case the integrals disappear and one simply has the product of the electric field with the corresponding dipole moments of the currents. Or, for wires of negligible thickness, one obtains the applied current in one wire multiplied by the resulting voltage across another and vice versa; see also below.

Another special case of the Lorentz reciprocity theorem applies when the volume V entirely contains both of the localized sources (or alternatively if V intersects neither of the sources). In this case:

[\oint_S (\mathbf_1 \times \mathbf_2) \cdot \mathbf = \oint_S (\mathbf_2 \times \mathbf_1) \cdot \mathbf ]

Reciprocity for electrical networks

Above, Lorentz reciprocity was phrased in terms of an externally applied current source and the resulting field. Often, especially for electrical networks, one instead prefers to think of an externally applied voltage and the resulting currents. The Lorentz reciprocity theorem describes this case as well, assuming ohmic materials (i.e. currents that respond linearly to the applied field) with a 3×3 conductivity matrix σ that is required to be symmetric, which is implied by the other conditions below. In order to properly describe this situation, one must carefully distinguish between the externally applied fields (from the driving voltages) and the total fields that result (King, 1963).

More specifically, the [\mathbf] above only consisted of external "source" terms introduced into Maxwell's equations. We now denote this by [\mathbf^] to distinguish it from the total current produced by both the external source and by the resulting electric fields in the materials. If this external current is in a material with a conductivity σ, then it corresponds to an externally applied electric field [\mathbf^] where, by definition of σ:

[\mathbf^=\sigma\mathbf^]

Moreover, the electric field [\mathbf] above only consisted of the response to this current, and did not include the "external" field [\mathbf^]. Therefore, we now denote the field from before as [\mathbf^], where the total field is given by [\mathbf = \mathbf^ + \mathbf^].

Now, the equation on the left-hand side of the Lorentz reciprocity theorem can be re-written by moving the σ from the external current term [\mathbf^] to the response field terms [\mathbf^], and also adding and subtracting a [\sigma\mathbf_1^\mathbf_2^] term, to obtain the external field multiplied by the total current [\mathbf=\sigma\mathbf]:

[\int_V \left[ mathbf_1^ cdot mathbf_2^ - mathbf_1^ cdot mathbf_2^ right] dV= \int_V \left[ sigma mathbf_1^ cdot (mathbf_2^ + mathbf_2^) - (mathbf_1^ + mathbf_1^) cdot sigmamathbf_2^ right] dV]

:[= \int_V \left[ mathbf_1^ cdot mathbf_2 - mathbf_1 cdot mathbf_2^ right] dV]

For the case of thin wires, this gives the product of the externally applied voltage (1) multiplied by the resulting total current (2) and vice versa. In particular, the Rayleigh-Carson reciprocity theorem becomes a simple summation:

[\sum_n V_1^ I_2^ = \sum_n V_2^ I_1^ \!]

where V and I denote the amplitudes of the AC applied voltages and the resulting currents, respectively, in a set of circuit elements (indexed by n) for two possible sets of voltages [V_1] and [V_2].

Most commonly, this is simplified further to the case where each system has a single voltage V, at [V_1^=V] and [V_2^=V]. Then the theorem becomes simply [I_1^=I_2^]: the current at position (1) from a voltage at (2) is identical to the current at (2) from the same voltage at (1).

Conditions for reciprocity

The Lorentz reciprocity theorem is simply a reflection of the fact that the linear operator [\hat] relating [\mathbf] and [\mathbf] at a fixed frequency (in linear media):

[\mathbf = \frac \left[ left( nabla times frac nabla times right) - ; frac varepsilon right] \mathbf \equiv \hat\mathbf]

is usually a Hermitian operator under the inner product [(\mathbf, \mathbf) = \int \mathbf \cdot \mathbf \, dV] for vector fields [\mathbf] and [\mathbf]. (Technically, this unconjugated form is not a true inner product because e.g. it is not positive-definite for complex-valued fields, but that is not a problem here. In this sense, the operator is not truly Hermitian but is rather complex-symmetric.) This is true whenever the dielectric function ε and the magnetic permeability μ, at the given ω, are symmetric 3×3 matrices (symmetric rank-2 tensors) — this includes the common case where they are scalars (for isotropic media), of course. They need not be real—complex values correspond to materials with losses, such as conductors with finite conductivity σ (which is included in ε via [\varepsilon \rightarrow \varepsilon + i\sigma/\omega])—and because of this the reciprocity theorem does not require time reversal invariance. The condition of symmetric ε and μ matrices is almost always satisfied; see below for an exception.

For any Hermitian operator [\hat] under an inner product [(f,g)\!], we have [(f,\hatg) = (\hatf,g)] by definition, and the Rayleigh-Carson reciprocity theorem is merely the vectorial version of this statement for this particular operator [\mathbf = \hat \mathbf]: that is, [(\mathbf_1, \hat \mathbf_2) = (\hat \mathbf_1, \mathbf_2)]. The Hermitian property of the operator here can be derived by integration by parts. For a finite integration volume, the surface terms from this integration by parts yield the more-general surface-integral theorem above. In particular, the key fact is that, for vector fields [\mathbf] and [\mathbf], integration by parts (or the divergence theorem) over a volume V enclosed by a surface S gives the identity:

[\int_V \mathbf \cdot (\nabla\times\mathbf) \, dV = \int_V (\nabla\times\mathbf) \cdot \mathbf \, dV - \oint_S (\mathbf \times \mathbf) \cdot \mathbf]

This identity is then applied twice to [(\mathbf_1, \hat \mathbf_2)] to yield [(\hat \mathbf_1, \mathbf_2)] plus the surface term, giving the Lorentz reciprocity relation.

Surface-term cancellation

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.

The simplest 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 [\hat}] direction) with [\hat} \cdot \mathbf = 0] and [\mathbf = \hat} \times \mathbf / Z] where Z is the impedance [\sqrt] of the surrounding medium. Then it follows that [\mathbf_1 \times \mathbf_2 = \mathbf_1 \times \hat} \times \mathbf_2 / Z], which by a simple vector identity equals [\hat} (\mathbf_1 \cdot \mathbf_2) / Z]. Similarly, [\mathbf_2 \times \mathbf_1 = \hat} (\mathbf_2 \cdot \mathbf_1) / Z] 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 by 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 we implicitly assumed the standard boundary condition of zero incoming waves from infinity, because otherwise even an infinitesimal loss would eliminate the incoming waves and the limit would not give the lossless solution.)

Reciprocity and the Green's function

The inverse of the operator [\hat], i.e. in [\mathbf = \hat^ \mathbf] (which requires a specification of the boundary conditions at infinity in a lossless system), has the same symmetry as [\hat] 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 [G_(\mathbf',\mathbf)] giving the n-th component of [\mathbf] at [\mathbf'] from a point dipole current in the m-th direction at [\mathbf] (essentially, [G] gives the matrix elements of [\hat^]), and Rayleigh-Carson reciprocity is equivalent to the statement that [G_(\mathbf',\mathbf) = G_(\mathbf,\mathbf')]. Unlike [\hat], 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.

Lossless and magneto-optic materials

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 [\nabla \times \frac \nabla \times - (\omega^2/c^2) \varepsilon] is Hermitian under the conjugated inner product [(\mathbf, \mathbf) = \int \mathbf^* \cdot \mathbf \, dV], and a variant of the reciprocity theorem still holds:

[- \int_V \left[ mathbf_1^* cdot mathbf_2 + mathbf_1^* cdot mathbf_2 right] dV = \oint_S \left[ mathbf_1^* times mathbf_2 + mathbf_2 times mathbf_1^* right] \cdot \mathbf]

where the sign changes come from the [c/i\omega] in the equation above, which makes the operator [\hat] anti-Hermitian (neglecting surface terms). For the special case of [\mathbf_1=\mathbf_2], this gives a re-statement of conservation of energy (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 [-\mathbf^* \cdot \mathbf]) 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.

Generalization to non-symmetric materials

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 [(\mathbf_1, \mathbf_1)] and [(\mathbf_2, \mathbf_2)] to exist in different systems.

In particular, if [(\mathbf_1, \mathbf_1)] solve Maxwell's equations at ω for a system with materials [(\varepsilon_1, \mu_1)], and [(\mathbf_2, \mathbf_2)] solve Maxwell's equations at ω for a system with materials [(\varepsilon_1^T, \mu_1^T)], where T denotes the transpose, then the equation of Lorentz reciprocity holds.

Exceptions to reciprocity

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.)

Feld-Tai reciprocity

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

[\int \mathbf_1 \cdot \mathbf_2 \, dV = \int \mathbf_1 \cdot \mathbf_2 \, dV]

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 [\mathbf] and [i(\omega/c)\mathbf] is a constant scalar multiple of the operator relating [\mathbf] and [\nabla\times(\mathbf/\varepsilon)], 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.

References

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Addison-Wesley: Reading, MA, 1960). §69.
Ronold W. P. King, Fundamental Electromagnetic Theory (Dover: New York, 1963). §IV.21.
C. Altman and K. Such, Reciprocity, Spatial Mapping and Time Reversal in Electromagnetics (Kluwer: Dordrecht, 1991).
H. A. Lorentz, "The theorem of Poynting concerning the energy in the electromagnetic field and two general propositions concerning the propagation of light," Amsterdammer Akademie der Wetenschappen 4 p. 176 (1896).
R. J. Potton, "Reciprocity in optics," Reports on Progress in Physics 67, 717-754 (2004). (A review article on the history of this topic.)
J. R. Carson, "A generalization of reciprocal theorem," Bell System Technical Journal 3 (3), 393-399 (1924). Also J. R. Carson, "The reciprocal energy theorem," ibid. 9 (4), 325-331 (1930).
Ya. N. Feld, "On the quadratic lemma in electrodynamics," Sov. Phys—Dokl. 37, 235-236 (1992).
C.-T. Tai, "Complementary reciprocity theorems in electromagnetic theory," IEEE Trans. Antennas Prop. 40 (6), 675-681 (1992).

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