Reciprocity (electromagnetism)

Specifically, suppose that one has a current density ${\displaystyle \mathbf {J} _{1}}$ that produces an electric field ${\displaystyle \mathbf {E} _{1}}$ and a magnetic field ${\displaystyle \mathbf {H} _{1}}$, where all three are periodic functions of time with angular frequency ω, and in particular they have time-dependence ${\displaystyle \exp(-i\omega t)}$. Suppose that we similarly have a second current ${\displaystyle \mathbf {J} _{2}}$ at the same frequency ω which (by itself) produces fields ${\displaystyle \mathbf {E} _{2}}$ and ${\displaystyle \mathbf {H} _{2}}$. 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:

${\displaystyle \int _{V}\left[\mathbf {J} _{1}\cdot \mathbf {E} _{2}-\mathbf {E} _{1}\cdot \mathbf {J} _{2}\right]dV=\oint _{S}\left[\mathbf {E} _{1}\times \mathbf {H} _{2}-\mathbf {E} _{2}\times \mathbf {H} _{1}\right]\cdot \mathbf {dS} .}$

Equivalently, in differential form (by the divergence theorem):

${\displaystyle \mathbf {J} _{1}\cdot \mathbf {E} _{2}-\mathbf {E} _{1}\cdot \mathbf {J} _{2}=\nabla \cdot \left[\mathbf {E} _{1}\times \mathbf {H} _{2}-\mathbf {E} _{2}\times \mathbf {H} _{1}\right].}$

This general form is commonly simplified for a number of special cases. In particular, one usually assumes that ${\displaystyle \mathbf {J} _{1}}$ and ${\displaystyle \mathbf {J} _{2}}$ are localized (i.e. have compact support), and that there are no incoming waves from infinitely far away. In this case, if one integrates throughout space then the surface-integral terms cancel (see below) and one obtains:

${\displaystyle \int \mathbf {J} _{1}\cdot \mathbf {E} _{2}\,dV=\int \mathbf {E} _{1}\cdot \mathbf {J} _{2}\,dV.}$

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:

${\displaystyle \oint _{S}(\mathbf {E} _{1}\times \mathbf {H} _{2})\cdot \mathbf {dS} =\oint _{S}(\mathbf {E} _{2}\times \mathbf {H} _{1})\cdot \mathbf {dS} .}$

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 ${\displaystyle \mathbf {J} }$ above only consisted of external "source" terms introduced into Maxwell's equations. We now denote this by ${\displaystyle \mathbf {J} ^{(e)}}$ 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 ${\displaystyle \mathbf {E} ^{(e)}}$ where, by definition of σ:

${\displaystyle \mathbf {J} ^{(e)}=\sigma \mathbf {E} ^{(e)}.}$

Moreover, the electric field ${\displaystyle \mathbf {E} }$ above only consisted of the response to this current, and did not include the "external" field ${\displaystyle \mathbf {E} ^{(e)}}$. Therefore, we now denote the field from before as ${\displaystyle \mathbf {E} ^{(r)}}$, where the total field is given by ${\displaystyle \mathbf {E} =\mathbf {E} ^{(e)}+\mathbf {E} ^{(r)}}$.

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 ${\displaystyle \mathbf {J} ^{(e)}}$ to the response field terms ${\displaystyle \mathbf {E} ^{(r)}}$, and also adding and subtracting a ${\displaystyle \sigma \mathbf {E} _{1}^{(e)}\mathbf {E} _{2}^{(e)}}$ term, to obtain the external field multiplied by the total current ${\displaystyle \mathbf {J} =\sigma \mathbf {E} }$:

${\displaystyle {\begin{aligned}&\int _{V}\left[\mathbf {J} _{1}^{(e)}\cdot \mathbf {E} _{2}^{(r)}-\mathbf {E} _{1}^{(r)}\cdot \mathbf {J} _{2}^{(e)}\right]dV\\={}&\int _{V}\left[\sigma \mathbf {E} _{1}^{(e)}\cdot \left(\mathbf {E} _{2}^{(r)}+\mathbf {E} _{2}^{(e)}\right)-\left(\mathbf {E} _{1}^{(r)}+\mathbf {E} _{1}^{(e)}\right)\cdot \sigma \mathbf {E} _{2}^{(e)}\right]dV\\={}&\int _{V}\left[\mathbf {E} _{1}^{(e)}\cdot \mathbf {J} _{2}-\mathbf {J} _{1}\cdot \mathbf {E} _{2}^{(e)}\right]dV.\end{aligned}}}$

For the limit 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:

${\displaystyle \sum _{n}V_{1}^{(n)}I_{2}^{(n)}=\sum _{n}V_{2}^{(n)}I_{1}^{(n)}\!}$

where V and I denote the complex 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 ${\displaystyle V_{1}}$ and ${\displaystyle V_{2}}$.

Most commonly, this is simplified further to the case where each system has a single voltage source V, at ${\displaystyle V_{1}^{(1)}=V}$ and ${\displaystyle V_{2}^{(2)}=V}$. Then the theorem becomes simply

${\displaystyle I_{1}^{(2)}=I_{2}^{(1)}}$

or in words:

The current at position (1) from a voltage at (2) is identical to the current at (2) from the same voltage at (1).

The Lorentz reciprocity theorem is simply a reflection of the fact that the linear operator ${\displaystyle {\hat {O}}}$ relating ${\displaystyle \mathbf {J} }$ and ${\displaystyle \mathbf {E} }$ at a fixed frequency ${\displaystyle \omega }$ (in linear media):

${\displaystyle \mathbf {J} ={\frac {1}{i\omega }}\left[{\frac {1}{\mu }}\left(\nabla \times \nabla \times \right)-\;\omega ^{2}\varepsilon \right]\mathbf {E} \equiv {\hat {O}}\mathbf {E} }$

is usually a symmetric operator under the "inner product" ${\displaystyle (\mathbf {F} ,\mathbf {G} )=\int \mathbf {F} \cdot \mathbf {G} \,dV}$ for vector fields ${\displaystyle \mathbf {F} }$ and ${\displaystyle \mathbf {G} }$. (Technically, this unconjugated form is not a true inner product because it is not real-valued 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 permittivity 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 ${\displaystyle \varepsilon \rightarrow \varepsilon +i\sigma /\omega }$)—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 ${\displaystyle {\hat {O}}}$ under an inner product ${\displaystyle (f,g)\!}$, we have ${\displaystyle (f,{\hat {O}}g)=({\hat {O}}f,g)}$ by definition, and the Rayleigh-Carson reciprocity theorem is merely the vectorial version of this statement for this particular operator ${\displaystyle \mathbf {J} ={\hat {O}}\mathbf {E} }$: that is, ${\displaystyle (\mathbf {E} _{1},{\hat {O}}\mathbf {E} _{2})=({\hat {O}}\mathbf {E} _{1},\mathbf {E} _{2})}$. 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 ${\displaystyle \mathbf {F} }$ and ${\displaystyle \mathbf {G} }$, integration by parts (or the divergence theorem) over a volume V enclosed by a surface S gives the identity:

${\displaystyle \int _{V}\mathbf {F} \cdot (\nabla \times \mathbf {G} )\,dV\equiv \int _{V}(\nabla \times \mathbf {F} )\cdot \mathbf {G} \,dV-\oint _{S}(\mathbf {F} \times \mathbf {G} )\cdot \mathbf {dA} .}$

This identity is then applied twice to ${\displaystyle (\mathbf {E} _{1},{\hat {O}}\mathbf {E} _{2})}$ to yield ${\displaystyle ({\hat {O}}\mathbf {E} _{1},\mathbf {E} _{2})}$ plus the surface term, giving the Lorentz reciprocity relation.

Conditions and proof of Lorenz reciprocity using Maxwell's equations and vector operations[1]

We shall prove a general form of the electromagnetic reciprocity theorem due to Lorenz which states that fields ${\displaystyle \mathbf {E} _{1},\mathbf {H} _{1}}$ and ${\displaystyle \mathbf {E} _{2},\mathbf {H} _{2}}$ generated by two different sinusoidal current densities respectively ${\displaystyle \mathbf {J} _{1}}$ and ${\displaystyle \mathbf {J} _{2}}$ of the same frequency, satisfy the condition ${\displaystyle \int _{V}\left[\mathbf {J} _{1}\cdot \mathbf {E} _{2}-\mathbf {E} _{1}\cdot \mathbf {J} _{2}\right]dV=\oint _{S}\left[\mathbf {E} _{1}\times \mathbf {H} _{2}-\mathbf {E} _{2}\times \mathbf {H} _{1}\right]\cdot \mathbf {dS} .}$

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:

${\displaystyle {\begin{array}{ccc}\nabla \times \mathbf {E} &=&-{\frac {\partial \mathbf {B} }{\partial t}},\\\nabla \times \mathbf {H} &=&\mathbf {J} +{\frac {\partial \mathbf {D} }{\partial t}}.\end{array}}}$

Under steady constant frequency conditions we get from the two curl equations the Maxwell's equations for the Time-Periodic case:

${\displaystyle {\begin{array}{ccc}\nabla \times \mathbf {E} &=&-j\omega \mathbf {B} ,\\\nabla \times \mathbf {H} &=&\mathbf {J} +j\omega \mathbf {D} .\end{array}}}$

It must be recognized that the symbols in the equations of this article represent the complex multipliers of ${\displaystyle e^{j\omega t}}$, giving the in-phase and out-of-phase parts with respect to the chosen reference. The complex vector multipliers of ${\displaystyle e^{j\omega t}}$ 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

${\displaystyle \mathbf {H} \cdot (\nabla \times \mathbf {E} )-\mathbf {E} \cdot (\nabla \times \mathbf {H} )=\nabla \cdot (\mathbf {E} \times \mathbf {H} )}$ for every vectors ${\displaystyle \mathbf {E} }$ and ${\displaystyle \mathbf {H} }$.

If we apply this equivalence to ${\displaystyle \mathbf {E} _{1}}$ and ${\displaystyle \mathbf {H} _{2}}$ we get:

${\displaystyle \mathbf {H} _{2}\cdot (\nabla \times \mathbf {E} _{1})-\mathbf {E} _{1}\cdot (\nabla \times \mathbf {H} _{2})=\nabla \cdot (\mathbf {E} _{1}\times \mathbf {H} _{2})}$.

If products in the Time-Periodic equations are taken as indicated by this last equivalence, and added,

${\displaystyle -\mathbf {H} _{2}\cdot j\omega \mathbf {B} _{1}-\mathbf {E} _{1}\cdot j\omega \mathbf {D} _{2}-\mathbf {E} _{1}\cdot \mathbf {J} _{2}=\nabla \cdot (\mathbf {E} _{1}\times \mathbf {H} _{2})}$.

This now may be integrated over the volume of concern,

${\displaystyle \int _{V}(\mathbf {H} _{2}\cdot j\omega \mathbf {B} _{1}+\mathbf {E} _{1}\cdot j\omega \mathbf {D} _{2}+\mathbf {E} _{1}\mathbf {J} _{2})dV=-\int _{V}\nabla \cdot (\mathbf {E} _{1}\times \mathbf {H} _{2})dV}$.

From the divergence theorem the volume integral of ${\displaystyle div(\mathbf {E} _{1}\times \mathbf {H} _{2})}$ equals the surface integral of ${\displaystyle \mathbf {E} _{1}\times \mathbf {H} _{2}}$ over the boundary.

${\displaystyle \int _{V}(\mathbf {H} _{2}\cdot j\omega \mathbf {B} _{1}+\mathbf {E} _{1}\cdot j\omega \mathbf {D} _{2}+\mathbf {E} _{1}\cdot \mathbf {J} _{2})dV=-\oint _{S}(\mathbf {E} _{1}\times \mathbf {H} _{2})\cdot {\widehat {dS}}}$.

This form is valid for general media, but in the common case of linear, isotropic, time-invariant materials, ${\displaystyle \epsilon }$ is a scalar independent of time. Then generally as physical magnitudes ${\displaystyle \mathbf {D} =\epsilon \mathbf {E} }$ and ${\displaystyle \mathbf {B} =\mu \mathbf {H} }$.

Last equation then becomes

${\displaystyle \int _{V}(\mathbf {H} _{2}\cdot j\omega \mu \mathbf {H} _{1}+\mathbf {E} _{1}\cdot j\omega \epsilon \mathbf {E} _{2}+\mathbf {E} _{1}\cdot \mathbf {J} _{2})dV=-\oint _{S}(\mathbf {E} _{1}\times \mathbf {H} _{2})\cdot {\widehat {dS}}}$.

In an exactly analogous way we get for vectors ${\displaystyle \mathbf {E} _{2}}$ and ${\displaystyle \mathbf {H} _{1}}$ the following expression:

${\displaystyle \int _{V}(\mathbf {H} _{1}\cdot j\omega \mu \mathbf {H} _{2}+\mathbf {E} _{2}\cdot j\omega \epsilon \mathbf {E} _{1}+\mathbf {E} _{2}\cdot \mathbf {J} _{1})dV=-\oint _{S}(\mathbf {E} _{2}\times \mathbf {H} _{1})\cdot {\widehat {dS}}}$.

Subtracting the two last equations by members we get

${\displaystyle \int _{V}\left[\mathbf {J} _{1}\cdot \mathbf {E} _{2}-\mathbf {E} _{1}\cdot \mathbf {J} _{2}\right]dV=\oint _{S}\left[\mathbf {E} _{1}\times \mathbf {H} _{2}-\mathbf {E} _{2}\times \mathbf {H} _{1}\right]\cdot \mathbf {dS} .}$

and equivalently in differential form

${\displaystyle \mathbf {J} _{1}\cdot \mathbf {E} _{2}-\mathbf {E} _{1}\cdot \mathbf {J} _{2}=\nabla \cdot \left[\mathbf {E} _{1}\times \mathbf {H} _{2}-\mathbf {E} _{2}\times \mathbf {H} _{1}\right]}$ q.e.d.

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:

${\displaystyle \int \mathbf {J} _{1}\cdot \mathbf {H} _{2}\,dV=\int \mathbf {H} _{1}\cdot \mathbf {J} _{2}\,dV.}$

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 ${\displaystyle \mathbf {E} }$ and ${\displaystyle i\omega \mathbf {J} }$ is a constant scalar multiple of the operator relating ${\displaystyle \mathbf {H} }$ and ${\displaystyle \nabla \times (\mathbf {J} /\varepsilon )}$, 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.

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:

"A ray of light proceeding from point A arrives at point B after suffering any number of refractions, reflections, &c. At point A let any two perpendicular planes a₁, a₂ be taken in the direction of the ray; and let the vibrations of the ray be divided into two parts, one in each of these planes. Take like planes b₁, b₂ in the ray at point B; then the following proposition may be demonstrated. If when the quantity of light J polarized in the plane a₁ proceeds from A in the direction of the given ray, that part K thereof of light polarized in b₁ arrives at B, then, conversely, if the quantity of light J polarized in b₁ proceeds from B, the same quantity of light K polarized in a₁ will arrive at A."[3]

This is sometimes called the Helmholtz reciprocity (or reversion) principle.[4][5][6][7][8][9] When the wave propagates through a material acted upon by an applied magnetic field, reciprocity can be broken so this principle will not apply.[3] 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.[10][11][12]

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.[13][14]

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 (BRDF) outcome.[14]

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 ${\displaystyle \phi _{1}}$ denote the electric potential resulting from a total charge density ${\displaystyle \rho _{1}}$. The electric potential satisfies Poisson's equation, ${\displaystyle -\nabla ^{2}\phi _{1}=\rho _{1}/\varepsilon _{0}}$, where ${\displaystyle \varepsilon _{0}}$ is the vacuum permittivity. Similarly, let ${\displaystyle \phi _{2}}$ denote the electric potential resulting from a total charge density ${\displaystyle \rho _{2}}$, satisfying ${\displaystyle -\nabla ^{2}\phi _{2}=\rho _{2}/\varepsilon _{0}}$. 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:

${\displaystyle \int \rho _{1}\phi _{2}dV=\int \rho _{2}\phi _{1}dV.}$

This theorem is easily proven from Green's second identity. Equivalently, it is the statement that ${\displaystyle \int \phi _{2}(\nabla ^{2}\phi _{1})dV=\int \phi _{1}(\nabla ^{2}\phi _{2})dV}$, i.e. that ${\displaystyle \nabla ^{2}}$ is a Hermitian operator (as follows by integrating by parts twice).

• L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Addison-Wesley: Reading, MA, 1960). §89.
• 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).
• Wolfgang K. H. Panofsky and Melba Phillips, Classical Electricity and Magnetism (Addison-Wesley: Reading, MA, 1962).
• Viktar Asadchy, Mohammad S. Mirmoosa, Ana Díaz-Rubio, Shanhui Fan, Sergei A. Tretyakov, Tutorial on Electromagnetic Nonreciprocity and Its Origins, arXiv:2001.04848 (2020).