The Lorentz reciprocity theorem is simply a reflection of the fact that the linear operator \operatorname{\hat{O}} relating \mathbf{J} and \mathbf{E} at a fixed frequency \omega (in linear media): \mathbf{J} = \operatorname{\hat{O}} \mathbf{E} where \operatorname{\hat{O}} \mathbf{E} \equiv \frac{1}{i\omega} \left[ \frac{1}{\mu} \left( \nabla \times \nabla \times \right) - \; \omega^2 \varepsilon \right] \mathbf{E} is usually a
symmetric operator under the "
inner product" (\mathbf{F}, \mathbf{G}) = \int \mathbf{F} \cdot \mathbf{G} \, \mathrm{d}V for
vector fields \mathbf{F} and \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 and the
magnetic permeability , 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. The condition of symmetric and matrices is almost always satisfied; see below for an exception. For any Hermitian operator \operatorname{\hat{O}} under an inner product (f,g)\!, we have (f,\operatorname{\hat{O}}g) = (\operatorname{\hat{O}}f,g) by definition, and the Rayleigh-Carson reciprocity theorem is merely the vectorial version of this statement for this particular operator \mathbf{J} = \operatorname{\hat{O}} \mathbf{E}\ : that is, (\mathbf{E}_1, \operatorname{\hat{O}}\mathbf{E}_2) = (\operatorname{\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 \mathbf{F} and \mathbf{G}\ , integration by parts (or the
divergence theorem) over a volume enclosed by a surface gives the identity: \int_V \mathbf{F} \cdot (\nabla\times\mathbf{G}) \, \mathrm{d}V \equiv \int_V (\nabla\times\mathbf{F}) \cdot \mathbf{G} \, \mathrm{d}V - \oint_S (\mathbf{F} \times \mathbf{G}) \cdot \mathrm{d}\mathbf{A}\ . This identity is then applied twice to (\mathbf{E}_1, \operatorname{\hat{O}} \mathbf{E}_2) to yield (\operatorname{\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 We shall prove a general form of the electromagnetic reciprocity theorem due to Lorenz which states that fields \mathbf {E}_1, \mathbf {H}_1 and \mathbf {E}_2, \mathbf {H}_2 generated by two different sinusoidal current densities respectively \mathbf {J}_1 and \mathbf {J}_2 of the same frequency, satisfy the condition \int_V \left[ \mathbf{J}_1 \cdot \mathbf{E}_2 - \mathbf{E}_1 \cdot \mathbf{J}_2 \right] \mathrm{d}V = \oint_S \left[ \mathbf{E}_1 \times \mathbf{H}_2 - \mathbf{E}_2 \times \mathbf{H}_1 \right] \cdot \mathbf{\mathrm{d}S} . 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{array}{ccc} \nabla\times\mathbf E & = & - \frac{\partial}{\partial t}\mathbf B\ ,\\ \nabla\times\mathbf H & = & \mathbf J + \frac{\partial}{\partial t}\mathbf D\ . \end{array} Under steady constant frequency conditions we get from the two curl equations the Maxwell's equations for the Time-Periodic case: \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 e^{j\omega t} , giving the in-phase and out-of-phase parts with respect to the chosen reference. The complex vector multipliers of 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 \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 {E}_1 and \mathbf {H}_2 we get: \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, -\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, \int_V \left(\mathbf{H}_2 \cdot j \omega \mathbf{B}_1+\mathbf{E}_1 \cdot j\omega \mathbf{D}_2+\mathbf{E}_1\mathbf{J}_2\right) \mathrm{d}V = -\int_V \nabla \cdot (\mathbf{E}_1 \times \mathbf{H}_2) \mathrm{d}V\ . From the divergence theorem the volume integral of \operatorname{div}(\mathbf{E}_1\times\mathbf{H}_2) equals the surface integral of \mathbf{E}_1\times\mathbf{H}_2 over the boundary. \int_V \left(\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\right) \mathrm{d}V = -\oint_S(\mathbf{E}_1 \times \mathbf{H}_2)\cdot \widehat{\mathrm{d}S}\ . 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{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\right) \mathrm{d}V = -\oint_S(\mathbf{E}_1\times\mathbf{H}_2) \cdot \widehat{\mathrm{d}S}\ . In an exactly analogous way we get for vectors \mathbf{E}_2 and \mathbf{H}_1 the following expression: \int_V \left(\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\right) \operatorname{d}V = -\oint_S(\mathbf{E}_2\times\mathbf{H}_1) \cdot \widehat{\mathrm{d}S}\ . Subtracting the two last equations by members we get \int_V \left[ \mathbf{J}_1 \cdot \mathbf{E}_2 - \mathbf{E}_1 \cdot \mathbf{J}_2 \right] \operatorname{d}V = \oint_S \left[ \mathbf{E}_1 \times \mathbf{H}_2 - \mathbf{E}_2 \times \mathbf{H}_1 \right] \cdot \mathbf{\mathrm{d}S}\ . and equivalently in differential form \ \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. 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 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{\hat{O}}{\mathbf{r}} direction) with \operatorname{\hat{O}}{\mathbf{r}} \cdot \mathbf{E} = 0 and \mathbf{H} = \hat{\mathbf{r}} \times \mathbf{E} / Z where is the scalar
impedance \sqrt{ \mu / \epsilon} of the surrounding medium. Then it follows that \ \mathbf{E}_1 \times \mathbf{H}_2 = \frac{ \mathbf{E}_1 \times \hat{\mathbf{r}} \times \mathbf{E}_2 }{Z}\ , which by a simple
vector identity equals \frac{ \mathbf{E}_1 \cdot \mathbf{E}_2}{Z}\ \hat{\mathbf{r}}\ . Similarly, \mathbf{E}_2 \times \mathbf{H}_1 = \frac{ \mathbf{E}_2 \cdot \mathbf{E}_1 }{Z} \ \hat{\mathbf{r}} and the two terms cancel one another. The above argument shows explicitly why the surface terms can cancel, but lacks generality. Alternatively, one can treat the case of lossless surrounding media with radiation boundary conditions imposed via the
limiting absorption principle (LAP): Taking the limit as the losses (the imaginary part of ) go to zero. For any nonzero loss, the fields decay exponentially with distance and the surface integral vanishes, regardless of whether the medium is homogeneous. Since the left-hand side of the Lorentz reciprocity theorem vanishes for integration over all space with any non-zero losses, it must also vanish in the limit as the losses go to zero. (Note that the LAP implicitly imposes the
Sommerfeld radiation condition of zero incoming waves from infinity, because otherwise even an arbitrarily small loss would eliminate the incoming waves and the limit would not give the lossless solution.)
Reciprocity and Green's function The inverse of the operator \operatorname{\hat{O}}\ , i.e., in \mathbf{E} = \operatorname{\hat{O}}^{-1} \mathbf{J} (which requires a specification of the boundary conditions at infinity in a lossless system), has the same symmetry as \operatorname{\hat{O}} 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 and . More specifically, the Green's function can be written as G_{nm}(\mathbf{x}',\mathbf{x}) giving the -th component of \mathbf{E} at \mathbf{x}' from a point dipole current in the -th direction at \mathbf{x} (essentially, G gives the matrix elements of \operatorname{\hat{O}}^{-1} ), and Rayleigh-Carson reciprocity is equivalent to the statement that G_{nm}(\mathbf{x}',\mathbf{x}) = G_{mn}(\mathbf{x},\mathbf{x}')\ . Unlike \operatorname{\hat{O}}\ , 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{1}{\mu} \left(\nabla \times \nabla \times\right) - \frac{\omega^2}{c^2} \varepsilon is Hermitian under the
conjugated inner product (\mathbf{F}, \mathbf{G}) = \int \mathbf{F}^* \cdot \mathbf{G} \, \mathrm{d}V\ , and a variant of the reciprocity theorem still holds: - \int_V \left[ \mathbf{J}_1^* \cdot \mathbf{E}_2 + \mathbf{E}_1^* \cdot \mathbf{J}_2 \right] \mathrm{d}V = \oint_S \left[ \mathbf{E}_1^* \times \mathbf{H}_2 + \mathbf{E}_2 \times \mathbf{H}_1^* \right] \cdot \mathbf{\mathrm{d}A} where the sign changes come from the \frac{1}{i\omega} in the equation above, which makes the operator \operatorname{\hat{O}}
anti-Hermitian (neglecting surface terms). For the special case of \mathbf{J}_1 = \mathbf{J}_2\ , this gives a re-statement of
conservation of energy or
Poynting's theorem (since here we have assumed lossless materials, unlike above): The time-average rate of work done by the current (given by the real part of - \mathbf{J}^* \cdot \mathbf{E} ) 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{J}_1, \mathbf{E}_1) and (\mathbf{J}_2, \mathbf{E}_2) to exist in
different systems. In particular, if (\mathbf{J}_1, \mathbf{E}_1) satisfy Maxwell's equations at ω for a system with materials (\varepsilon_1, \mu_1)\ , and (\mathbf{J}_2, \mathbf{E}_2) satisfy Maxwell's equations at for a system with materials \left(\varepsilon_1^\mathsf{T}, \mu_1^\mathsf{T} \right)\ , where {}^\mathsf{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 ("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==