HOME

TheInfoList



OR:

In
classical electromagnetism Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model; It is, therefore, a classical fie ...
, reciprocity refers to a variety of related theorems involving the interchange of time-
harmonic A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', t ...
electric current densities (sources) and the resulting
electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical ...
s in
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits ...
for time-invariant linear media under certain constraints. Reciprocity is closely related to the concept of symmetric operators from
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...
, 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 Hendrik Antoon Lorentz (; 18 July 1853 – 4 February 1928) was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect. He also derived the Lorent ...
in 1896 following analogous results regarding
sound In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by ...
by
Lord Rayleigh John William Strutt, 3rd Baron Rayleigh, (; 12 November 1842 – 30 June 1919) was an English mathematician and physicist who made extensive contributions to science. He spent all of his academic career at the University of Cambridge. A ...
and
light Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 t ...
by
Helmholtz Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Associatio ...
(Potton, 2004). Loosely, it states that the relationship between an oscillating current and the resulting
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical 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 An electrical network is an interconnection of electrical components (e.g., batteries, resistors, inductors, capacitors, switches, transistors) or a model of such an interconnection, consisting of electrical elements (e.g., voltage sources ...
, it is sometimes phrased as the statement that
voltage Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to ...
s 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 Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultrav ...
, which (apart from quantum effects) can be expressed in terms of classical electromagnetism, but also in terms of
radiometry Radiometry is a set of techniques for measuring electromagnetic radiation, including visible light. Radiometric techniques in optics characterize the distribution of the radiation's power in space, as opposed to photometric techniques, which ch ...
. There is also an analogous theorem in
electrostatics Electrostatics is a branch of physics that studies electric charges at rest ( static electricity). Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for a ...
, known as Green's reciprocity, relating the interchange of
electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
and
electric charge density In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in co ...
. Forms of the reciprocity theorems are used in many electromagnetic applications, such as analyzing electrical networks and
antenna Antenna ( antennas or antennae) may refer to: Science and engineering * Antenna (radio), also known as an aerial, a transducer designed to transmit or receive electromagnetic (e.g., TV or radio) waves * Antennae Galaxies, the name of two collid ...
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 Impedance parameters or Z-parameters (the elements of an impedance matrix or Z-matrix) are properties used in electrical engineering, electronic engineering, and communication systems engineering to describe the electrical behavior of linear electri ...
and
scattering matrix In physics, the ''S''-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT). More forma ...
, symmetries of
Green's function In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differenti ...
s for use in boundary-element and transfer-matrix computational methods, as well as
orthogonality In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
properties of
harmonic mode A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', the ...
s in
waveguide A waveguide is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting the transmission of energy to one direction. Without the physical constraint of a waveguide, wave intensities de ...
systems (as an alternative to proving those properties directly from the symmetries of the eigen-operators).


Lorentz reciprocity

Specifically, suppose that one has a current density \mathbf_1 that produces an
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field ...
\mathbf_1 and a
magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
\mathbf_1\, , where all three are periodic functions of time with
angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit ti ...
, 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 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 enclosing a volume : :\int_V \left \mathbf_1 \cdot \mathbf_2 - \mathbf_1 \cdot \mathbf_2 \right\mathrmV = \oint_S \left \mathbf_1 \times \mathbf_2 - \mathbf_2 \times \mathbf_1 \right\cdot \mathbf\ . Equivalently, in differential form (by the
divergence theorem In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the '' flux'' of a vector field through a closed surface to the ''divergence'' of the field in the ...
): : \mathbf_1 \cdot \mathbf_2 - \mathbf_1 \cdot \mathbf_2 = \nabla \cdot \left \mathbf_1 \times \mathbf_2 - \mathbf_2 \times \mathbf_1 \right . 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 In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smalle ...
), 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: : \int \mathbf_1 \cdot \mathbf_2 \, \mathrmV = \int \mathbf_1 \cdot \mathbf_2 \, \mathrmV\ . 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 Carson (1924; 1930) to applications for
radio frequency Radio frequency (RF) is the oscillation rate of an alternating electric current or voltage or of a magnetic, electric or electromagnetic field or mechanical system in the frequency range from around to around . This is roughly between the up ...
antennas. Often, one further simplifies this relation by considering point-like
dipole In physics, a dipole () is an electromagnetic phenomenon which occurs in two ways: *An electric dipole deals with the separation of the positive and negative electric charges found in any electromagnetic system. A simple example of this system ...
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 entirely contains ''both'' of the localized sources (or alternatively if 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 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 , then it corresponds to an externally applied electric field \ \mathbf^\ where, by definition of : :\ \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 rewritten by moving the 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\ : :\begin &\int_V \left \mathbf_1^ \cdot \mathbf_2^ - \mathbf_1^ \cdot \mathbf_2^ \right\operatornameV \\ = &\int_V \left \sigma \mathbf_1^ \cdot \left(\mathbf_2^ + \mathbf_2^\right) - \left(\mathbf_1^ + \mathbf_1^\right) \cdot \sigma\mathbf_2^ \right\operatornameV \\ = &\int_V \left \mathbf_1^ \cdot \mathbf_2 - \mathbf_1 \cdot \mathbf_2^ \right\operatornameV\ . \end 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: :\ \sum_n \mathcal_1^ I_2^ = \sum_n \mathcal_2^ I_1^ where \ \mathcal\ and denote the complex amplitudes of the AC applied voltages and the resulting currents, respectively, in a set of circuit elements (indexed by ) for two possible sets of voltages \ \mathcal_1\ and \ \mathcal_2\ . Most commonly, this is simplified further to the case where each system has a ''single'' voltage source \ \mathcal_\text\ , at \ \mathcal_1^ = \mathcal_\text\ and \ \mathcal_2^ = \mathcal_\text\ . Then the theorem becomes simply : I_1^ = I_2^ 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).''


Conditions and proof of Lorentz reciprocity

The Lorentz reciprocity theorem is simply a reflection of the fact that the linear operator \operatorname relating \mathbf and \mathbf at a fixed frequency \omega (in linear media): \mathbf = \operatorname \mathbf where \operatorname \mathbf \equiv \frac \left \frac \left( \nabla \times \nabla \times \right) - \; \omega^2 \varepsilon \right\mathbf is usually a symmetric operator under the "
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
" (\mathbf, \mathbf) = \int \mathbf \cdot \mathbf \, \mathrmV for vector fields \mathbf and \mathbf\ . (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 In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' (epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in ...
and the
magnetic permeability In electromagnetism, permeability is the measure of magnetization that a material obtains in response to an applied magnetic field. Permeability is typically represented by the (italicized) Greek letter ''μ''. The term was coined by Willi ...
, at the given , 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 (which is included in via \varepsilon \rightarrow \varepsilon + i\sigma/\omega\ ) – and because of this, the reciprocity theorem does ''not'' require
time reversal invariance T-symmetry or time reversal symmetry is the theoretical symmetry of physical laws under the transformation of time reversal, : T: t \mapsto -t. Since the second law of thermodynamics states that entropy increases as time flows toward the future ...
. The condition of symmetric and matrices is almost always satisfied; see below for an exception. For any Hermitian operator \operatorname under an inner product (f,g)\!, we have (f,\operatornameg) = (\operatornamef,g) by definition, and the Rayleigh-Carson reciprocity theorem is merely the vectorial version of this statement for this particular operator \mathbf = \operatorname \mathbf\ : that is, (\mathbf_1, \operatorname\mathbf_2) = (\operatorname \mathbf_1, \mathbf_2)\ . The Hermitian property of the operator here can be derived by
integration by parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivat ...
. 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 In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the '' flux'' of a vector field through a closed surface to the ''divergence'' of the field in the ...
) over a volume enclosed by a surface gives the identity: \int_V \mathbf \cdot (\nabla\times\mathbf) \, \mathrmV \equiv \int_V (\nabla\times\mathbf) \cdot \mathbf \, \mathrmV - \oint_S (\mathbf \times \mathbf) \cdot \mathrm\mathbf\ . This identity is then applied twice to (\mathbf_1, \operatorname \mathbf_2) to yield (\operatorname \mathbf_1, \mathbf_2) 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 \mathbf _1, \mathbf _1 and \mathbf _2, \mathbf _2 generated by two different sinusoidal current densities respectively \mathbf _1 and \mathbf _2 of the same frequency, satisfy the condition \int_V \left \mathbf_1 \cdot \mathbf_2 - \mathbf_1 \cdot \mathbf_2 \right\mathrmV = \oint_S \left \mathbf_1 \times \mathbf_2 - \mathbf_2 \times \mathbf_1 \right\cdot \mathbf . 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: \begin \nabla\times\mathbf E & = & - \frac\mathbf B\ ,\\ \nabla\times\mathbf H & = & \mathbf J + \frac\mathbf D\ . \end Under steady constant frequency conditions we get from the two curl equations the Maxwell's equations for the Time-Periodic case: \begin \nabla\times\mathbf E & = & - j\omega\mathbf B\ ,\\ \nabla\times\mathbf H & = & \mathbf J + j\omega\mathbf D\ . \end It must be recognized that the symbols in the equations of this article represent the complex multipliers of e^ , giving the in-phase and out-of-phase parts with respect to the chosen reference. The complex vector multipliers of e^ 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 \mathbf H\cdot(\nabla \times \mathbf E) - \mathbf E \cdot (\nabla \times \mathbf H) = \nabla \cdot (\mathbf E \times \mathbf H) for every vectors \mathbf E and \mathbf H\ . If we apply this equivalence to \mathbf _1 and \mathbf _2 we get: \mathbf _2 \cdot (\nabla\times\mathbf _1)-\mathbf _1\cdot(\nabla\times\mathbf _2) = \nabla\cdot(\mathbf _1 \times\mathbf _2)\ . If products in the Time-Periodic equations are taken as indicated by this last equivalence, and added, -\mathbf_2\cdot j\omega \mathbf_1 - \mathbf_1 \cdot j\omega \mathbf_2 - \mathbf_1 \cdot \mathbf_2 = \nabla \cdot(\mathbf_1 \times \mathbf_2)\ . This now may be integrated over the volume of concern, \int_V \left(\mathbf_2 \cdot j \omega \mathbf_1+\mathbf_1 \cdot j\omega \mathbf_2+\mathbf_1\mathbf_2\right) \mathrmV = -\int_V \nabla \cdot (\mathbf_1 \times \mathbf_2) \mathrmV\ . From the divergence theorem the volume integral of \operatorname(\mathbf_1\times\mathbf_2) equals the surface integral of \mathbf_1\times\mathbf_2 over the boundary. \int_V \left(\mathbf_2 \cdot j\omega\mathbf_1+\mathbf_1\cdot j\omega\mathbf_2+\mathbf_1\cdot\mathbf_2\right) \mathrmV = -\oint_S(\mathbf_1 \times \mathbf_2)\cdot \widehat\ . 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 \mathbf D = \epsilon\mathbf E and \mathbf B = \mu \mathbf H\ . Last equation then becomes \int_V \left(\mathbf_2 \cdot j \omega\mu\mathbf_1+\mathbf_1 \cdot j \omega \epsilon\mathbf_2 + \mathbf_1 \cdot \mathbf_2\right) \mathrmV = -\oint_S(\mathbf_1\times\mathbf_2) \cdot \widehat\ . In an exactly analogous way we get for vectors \mathbf_2 and \mathbf_1 the following expression: \int_V \left(\mathbf_1 \cdot j \omega \mu \mathbf_2+\mathbf_2 \cdot j \omega \epsilon\mathbf_1 + \mathbf_2 \cdot \mathbf_1\right) \operatornameV = -\oint_S(\mathbf_2\times\mathbf_1) \cdot \widehat\ . Subtracting the two last equations by members we get \int_V \left \mathbf_1 \cdot \mathbf_2 - \mathbf_1 \cdot \mathbf_2 \right\operatornameV = \oint_S \left \mathbf_1 \times \mathbf_2 - \mathbf_2 \times \mathbf_1 \right\cdot \mathbf\ . and equivalently in differential form \ \mathbf_1 \cdot \mathbf_2 - \mathbf_1 \cdot \mathbf_2 = \nabla \cdot \left \mathbf_1 \times \mathbf_2 - \mathbf_2 \times \mathbf_1 \right
Q.E.D. Q.E.D. or QED is an initialism of the Latin phrase , meaning "which was to be demonstrated". Literally it states "what was to be shown". Traditionally, the abbreviation is placed at the end of mathematical proofs and philosophical arguments in pri ...


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. A rigorous treatment of the surface integral takes into account the
causality Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the cau ...
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 \operatorname direction) with \operatorname \cdot \mathbf = 0 and \mathbf = \hat \times \mathbf / Z where is the scalar impedance \sqrt of the surrounding medium. Then it follows that \ \mathbf_1 \times \mathbf_2 = \frac\ , which by a simple vector identity equals \frac\ \hat\ . Similarly, \mathbf_2 \times \mathbf_1 = \frac \ \hat 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 In mathematics, the limiting absorption principle (LAP) is a concept from operator theory and scattering theory that consists of choosing the "correct" resolvent of a linear operator at the essential spectrum based on the behavior of the resolvent ...
: Taking the limit as the losses (the imaginary part of ) 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 this approach implicitly imposes the
Sommerfeld radiation condition In applied mathematics, the Sommerfeld radiation condition is a concept from theory of differential equations and scattering theory used for choosing a particular solution to the Helmholtz equation. It was introduced by Arnold Sommerfeld in 1912 an ...
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 Green's function

The inverse of the operator \operatorname\ , i.e., in \mathbf = \operatorname^ \mathbf (which requires a specification of the boundary conditions at infinity in a lossless system), has the same symmetry as \operatorname and is essentially a
Green's function In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differenti ...
convolution In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''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 and . More specifically, the Green's function can be written as G_(\mathbf',\mathbf) giving the -th component of \mathbf at \mathbf' from a point dipole current in the -th direction at \mathbf (essentially, G gives the matrix elements of \operatorname^ ), and Rayleigh-Carson reciprocity is equivalent to the statement that G_(\mathbf',\mathbf) = G_(\mathbf,\mathbf')\ . Unlike \operatorname\ , 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 magneto-optic materials

One case in which 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 and are in general 3×3 complex Hermitian matrices. In this case, the operator \ \frac \left(\nabla \times \nabla \times\right) - \frac \varepsilon is Hermitian under the ''conjugated'' inner product (\mathbf, \mathbf) = \int \mathbf^* \cdot \mathbf \, \mathrmV\ , and a variant of the reciprocity theorem still holds: - \int_V \left \mathbf_1^* \cdot \mathbf_2 + \mathbf_1^* \cdot \mathbf_2 \right\mathrmV = \oint_S \left \mathbf_1^* \times \mathbf_2 + \mathbf_2 \times \mathbf_1^* \right\cdot \mathbf where the sign changes come from the \frac in the equation above, which makes the operator \operatorname anti-Hermitian (neglecting surface terms). For the special case of \mathbf_1 = \mathbf_2\ , this gives a re-statement of
conservation of energy In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means tha ...
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 - \mathbf^* \cdot \mathbf ) is equal to the time-average outward flux of power (the integral of the
Poynting vector In physics, the Poynting vector (or Umov–Poynting vector) represents the directional energy flux (the energy transfer per unit area per unit time) or ''power flow'' of an electromagnetic field. The SI unit of the Poynting vector is the watt p ...
). 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 isolator An optical isolator, or optical diode, is an optical component which allows the transmission of light in only one direction. It is typically used to prevent unwanted feedback into an optical oscillator, such as a laser cavity. The operation ...
s and
circulator A circulator is a passive, non-reciprocal three- or four- port device that only allows a microwave or radio-frequency signal to exit through the port directly after the one it entered. Optical circulators have similar behavior. Ports are where ...
s. 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) satisfy Maxwell's equations at ω for a system with materials (\varepsilon_1, \mu_1)\ , and (\mathbf_2, \mathbf_2) satisfy Maxwell's equations at for a system with materials \left(\varepsilon_1^\mathsf, \mu_1^\mathsf \right)\ , where ^\mathsf denotes the
transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...
, 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 ("active") media; for example, when is modulated in time by some external process. (In both of these cases, the frequency is not generally a conserved quantity.)


Feld-Tai reciprocity

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 field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
s: :\int \mathbf_1 \cdot \mathbf_2 \, \operatornameV = \int \mathbf_1 \cdot \mathbf_2 \, \operatornameV\ . 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 ''/'' 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 \mathbf\ is a constant scalar multiple of the operator relating \ \mathbf\ and \ \nabla\times (\mathbf/\varepsilon)\ , which is true when is a constant scalar multiple of (the two operators generally differ by an interchange of and ). 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 Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Associat ...
wrote: ::"A ray of light proceeding from point arrives at point after suffering any number of refractions, reflections, &c. At point let any two perpendicular planes , 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 , in the ray at point ; then the following proposition may be demonstrated. If when the quantity of light polarized in the plane proceeds from in the direction of the given ray, that part thereof of light polarized in arrives at , then, conversely, if the quantity of light polarized in proceeds from , the same quantity of light polarized in will arrive at ." cited by Planck. English version quoted here based on This is sometimes called the Helmholtz reciprocity (or reversion) 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 Sir George Gabriel Stokes, 1st Baronet, (; 13 August 1819 – 1 February 1903) was an Irish English physicist and mathematician. Born in County Sligo, Ireland, Stokes spent all of his career at the University of Cambridge, where he was the Luc ...
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 Gustav Robert Kirchhoff (; 12 March 1824 – 17 October 1887) was a German physicist who contributed to the fundamental understanding of electrical circuits, spectroscopy, and the emission of black-body radiation by heated objects. He ...
in his derivation of his law of thermal radiation and by
Max Planck Max Karl Ernst Ludwig Planck (, ; 23 April 1858 – 4 October 1947) was a German theoretical physicist whose discovery of energy quanta won him the Nobel Prize in Physics in 1918. Planck made many substantial contributions to theoretical p ...
in his analysis of his law of thermal radiation. For ray-tracing
global illumination Global illumination (GI), or indirect illumination, is a group of algorithms used in 3D computer graphics that are meant to add more realistic lighting to 3D scenes. Such algorithms take into account not only the light that comes directly from ...
algorithms, incoming and outgoing light can be considered as reversals of each other, without affecting the
bidirectional reflectance distribution function The bidirectional reflectance distribution function (BRDF; f_(\omega_,\, \omega_) ) is a function of four real variables that defines how light is reflected at an opaque surface. It is employed in the optics of real-world light, in compute ...
(BRDF) 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 Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons res ...
(Panofsky and Phillips, 1962). In particular, let \phi_1 denote the electric potential resulting from a total charge density \rho_1. The electric potential satisfies
Poisson's equation Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with ...
, -\nabla^2 \phi_1 = \rho_1 / \varepsilon_0, where \varepsilon_0 is the
vacuum permittivity Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric const ...
. Similarly, let \phi_2 denote the electric potential resulting from a total charge density \rho_2, satisfying -\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: :\int \rho_1 \phi_2 dV = \int \rho_2 \phi_1 \operatornameV\ . This theorem is easily proven from
Green's second identity In mathematics, Green's identities are a set of three identities in vector calculus relating the bulk with the boundary of a region on which differential operators act. They are named after the mathematician George Green, who discovered Green's ...
. Equivalently, it is the statement that : \int \phi_2 ( \nabla^2 \phi_1 ) dV = \int \phi_1 ( \nabla^2 \phi_2 ) \operatornameV\ , i.e. that \nabla^2 is a Hermitian operator (as follows by integrating by parts twice).


See also

*
Surface equivalence principle In electromagnetism, surface equivalence principle or surface equivalence theorem relates an arbitrary current distribution within an imaginary closed surface with an equivalent source on the surface. It is also known as field equivalence principle ...


References

* 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). * 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). * M. Stumpf, ''Electromagnetic Reciprocity in Antenna Theory'' (Wiley-IEEE Press: Piscataway, NJ: 2018). * M. Stumpf, ''Time-Domain Electromagnetic Reciprocity in Antenna Modeling'' (Wiley-IEEE Press: Piscataway, NJ: 2020). * Viktar Asadchy, Mohammad S. Mirmoosa, Ana Díaz-Rubio, Shanhui Fan, Sergei A. Tretyakov, ''Tutorial on Electromagnetic Nonreciprocity and Its Origins,'
arXiv:2001.04848
(2020).


Citations

{{reflist, 25em Electromagnetism Circuit theorems