Reciprocity (electromagnetism)
In classical electromagnetism, reciprocity refers to a variety of related theorems involving the interchange of time-harmonic electric current densities 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, named after work by Hendrik Lorentz in 1896 following analogous results regarding sound by Lord Rayleigh and light by Helmholtz. 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 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.
Lorentz reciprocity
Specifically, suppose that one has a current density that produces an electric field and a magnetic field, where all three are periodic functions of time with angular frequency ω, and in particular they have time-dependence. Suppose that we similarly have a second current at the same frequency ω which produces fields and. 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:Equivalently, in differential form :
This general form is commonly simplified for a number of special cases. In particular, one usually assumes that and are localized, and that there are no incoming waves from infinitely far away. In this case, if one integrates throughout space then the surface-integral terms cancel and one obtains:
This result is sometimes called the Rayleigh-Carson reciprocity theorem, after Lord Rayleigh's work on sound waves and an extension by John R. Carson 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. In this case:
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 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 and the total fields that result.More specifically, the above only consisted of external "source" terms introduced into Maxwell's equations. We now denote this by 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 where, by definition of σ:
Moreover, the electric field above only consisted of the response to this current, and did not include the "external" field. Therefore, we now denote the field from before as, where the total field is given by.
Now, the equation on the left-hand side of the Lorentz reciprocity theorem can be rewritten by moving the σ from the external current term to the response field terms, and also adding and subtracting a term, to obtain the external field multiplied by the total current :
For the limit of thin wires, this gives the product of the externally applied voltage multiplied by the resulting total current and vice versa. In particular, the Rayleigh-Carson reciprocity theorem becomes a simple summation:
where V and I denote the complex amplitudes of the AC applied voltages and the resulting currents, respectively, in a set of circuit elements for two possible sets of voltages and .
Most commonly, this is simplified further to the case where each system has a single voltage source V, at and. Then the theorem becomes simply
or in words:
Conditions and proof of Lorentz reciprocity
The Lorentz reciprocity theorem is simply a reflection of the fact that the linear operator relating and at a fixed frequency :is usually a symmetric operator under the "inner product" for vector fields and. This is true whenever the permittivity ε and the magnetic permeability μ, at the given ω, are symmetric 3×3 matrices — this includes the common case where they are scalars, of course. They need not be real—complex values correspond to materials with losses, such as conductors with finite conductivity σ —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 under an inner product, we have by definition, and the Rayleigh-Carson reciprocity theorem is merely the vectorial version of this statement for this particular operator : that is,. 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 and, integration by parts over a volume V enclosed by a surface S gives the identity:
This identity is then applied twice to to yield plus the surface term, giving the Lorentz reciprocity relation.
Conditions and proof of Lorenz reciprocity using Maxwell's equations and vector operations
We shall prove a general form of the electromagnetic reciprocity theorem due to Lorenz which states that fields and generated by two different sinusoidal current densities respectively and of the same frequency, satisfy the condition
Let us take a region in which dielectric constant and permeability may be functions of position but not of time. Maxwell's equations, written in terms of the total fields, currents and charges of the region describe the electromagnetic behavior of the region. The two curl equations are:
Under steady constant frequency conditions we get from the two curl equations the Maxwell's equations for the Time-Periodic case:
It must be recognized that the symbols in the equations of this article represent the complex multipliers of, giving the in-phase and out-of-phase parts with respect to the chosen reference. The complex vector multipliers of may be called vector phasors by analogy to the complex scalar quantities which are commonly referred to as phasors.
An equivalence of vector operations shows that
for every vectors and.
If we apply this equivalence to and we get:
If products in the Time-Periodic equations are taken as indicated by this last equivalence, and added,
This now may be integrated over the volume of concern,
From the divergence theorem the volume integral of equals the surface integral of over the boundary.
.
This form is valid for general media, but in the common case of linear, isotropic, time-invariant materials, is a scalar independent of time. Then generally as physical magnitudes and.
Last equation then becomes
In an exactly analogous way we get for vectors and the following expression:
Subtracting the two last equations by members we get
and equivalently in differential form
q.e.d.
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.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 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 with and where Z is the 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 by taking the limit as the losses 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.
Reciprocity and Green's function
The inverse of the operator, i.e. in , 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 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 , 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, but it is routinely computed by numerical methods.Lossless 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. 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. For the special case of, this gives a re-statement of conservation of energy or Poynting's theorem : the time-average rate of work done by the current is equal to the time-average outward flux of power. 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 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 T 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.
Exceptions to reciprocity
For nonlinear media, no reciprocity theorem generally holds. Reciprocity also does not generally apply for time-varying media; for example, when ε is modulated in time by some external process.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: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 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. As above, one can also construct a more general formulation for integrals over a finite volume.
Optical reciprocity in radiometric terms
Apart from quantal effects, classical theory covers near-, middle-, and far-field electric and magnetic phenomena with arbitrary time courses. Optics refers to far-field nearly-sinusoidal oscillatory electromagnetic effects. Instead of paired electric and magnetic variables, optics, including optical reciprocity, can be expressed in polarization-paired radiometric variables, such as spectral radiance, traditionally called specific intensity.In 1856, Hermann von Helmholtz wrote:
This is sometimes called the Helmholtz reciprocity principle. When the wave propagates through a material acted upon by an applied magnetic field, reciprocity can be broken so this principle will not apply. Similarly, when there are moving objects in the path of the ray, the principle may be entirely inapplicable. Historically, in 1849, Sir George Stokes stated his optical reversion principle without attending to polarization.
Like the principles of thermodynamics, this principle is reliable enough to use as a check on the correct performance of experiments, in contrast with the usual situation in which the experiments are tests of a proposed law.
The simplest statement of the principle is 'if I can see you, then you can see me.' The principle was used by Gustav Kirchhoff in his derivation of his law of thermal radiation and by Max Planck in his analysis of his law of thermal radiation.
For ray-tracing global illumination algorithms, incoming and outgoing light can be considered as reversals of each other, without affecting the bidirectional reflectance distribution function outcome.
Green's reciprocity
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.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.