This page is about reciprocity theorems in
classical electromagnetism. See also Reciprocity
theorem (disambiguation) for unrelated reciprocity
theorems, and Reciprocity (disambiguation)
for more general usages of the term.In classical
electromagnetism, reciprocity refers to a
variety of related theorems involving the
interchange of time-harmonic electric current
densities (sources) and the resulting electromagnetic
fields in Maxwell's equations for time-invariant
linear media under certain constraints. Reciprocity
is closely related to the concept of Hermitian
operators from linear algebra, applied to
electromagnetism.
Perhaps the most common and general such theorem
is Lorentz reciprocity (and its various special
cases such as Rayleigh-Carson reciprocity),
named after work by Hendrik Lorentz in 1896
following analogous results regarding sound
by Lord Rayleigh and light by Helmholtz (Potton,
2004). Loosely, it states that the relationship
between an oscillating current and the resulting
electric field is unchanged if one interchanges
the points where the current is placed and
where the field is measured. For the specific
case of an electrical network, it is sometimes
phrased as the statement that voltages and
currents at different points in the network
can be interchanged. More technically, it
follows that the mutual impedance of a first
circuit due to a second is the same as the
mutual impedance of the second circuit due
to the first.
Reciprocity is useful in optics, which (apart
from quantum effects) 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 (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
J
1
{\displaystyle \mathbf {J} _{1}}
that produces an electric field
E
1
{\displaystyle \mathbf {E} _{1}}
and a magnetic field
H
1
{\displaystyle \mathbf {H} _{1}}
, where all three are periodic functions of
time with angular frequency ω, and in particular
they have time-dependence
exp
⁡
(
−
i
ω
t
)
{\displaystyle \exp(-i\omega t)}
. Suppose that we similarly have a second
current
J
2
{\displaystyle \mathbf {J} _{2}}
at the same frequency ω which (by itself)
produces fields
E
2
{\displaystyle \mathbf {E} _{2}}
and
H
2
{\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:
∫
V
[
J
1
⋅
E
2
−
E
1
⋅
J
2
]
d
V
=
∮
S
⁡
[
E
1
×
H
2
−
E
2
×
H
1
]
⋅
d
S
.
{\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):
J
1
⋅
E
2
−
E
1
⋅
J
2
=
∇
⋅
[
E
1
×
H
2
−
E
2
×
H
1
]
.
{\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
J
1
{\displaystyle \mathbf {J} _{1}}
and
J
2
{\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:
∫
J
1
⋅
E
2
d
V
=
∫
E
1
⋅
J
2
d
V
.
{\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:
∮
S
⁡
(
E
1
×
H
2
)
⋅
d
S
=
∮
S
⁡
(
E
2
×
H
1
)
⋅
d
S
.
{\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} .}
== Reciprocity for electrical networks ==
Above, Lorentz reciprocity was phrased in
terms of an externally applied current source
and the resulting field. Often, especially
for electrical networks, one instead prefers
to think of an externally applied voltage
and the resulting currents. The Lorentz reciprocity
theorem describes this case as well, assuming
ohmic materials (i.e. currents that respond
linearly to the applied field) with a 3×3
conductivity matrix σ that is required to
be symmetric, which is implied by the other
conditions below. In order to properly describe
this situation, one must carefully distinguish
between the externally applied fields (from
the driving voltages) and the total fields
that result (King, 1963).
More specifically, the
J
{\displaystyle \mathbf {J} }
above only consisted of external "source"
terms introduced into Maxwell's equations.
We now denote this by
J
(
e
)
{\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
E
(
e
)
{\displaystyle \mathbf {E} ^{(e)}}
where, by definition of σ:
J
(
e
)
=
σ
E
(
e
)
.
{\displaystyle \mathbf {J} ^{(e)}=\sigma \mathbf
{E} ^{(e)}.}
Moreover, the electric field
E
{\displaystyle \mathbf {E} }
above only consisted of the response to this
current, and did not include the "external"
field
E
(
e
)
{\displaystyle \mathbf {E} ^{(e)}}
. Therefore, we now denote the field from
before as
E
(
r
)
{\displaystyle \mathbf {E} ^{(r)}}
, where the total field is given by
E
=
E
(
e
)
+
E
(
r
)
{\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
J
(
e
)
{\displaystyle \mathbf {J} ^{(e)}}
to the response field terms
E
(
r
)
{\displaystyle \mathbf {E} ^{(r)}}
, and also adding and subtracting a
σ
E
1
(
e
)
E
2
(
e
)
{\displaystyle \sigma \mathbf {E} _{1}^{(e)}\mathbf
{E} _{2}^{(e)}}
term, to obtain the external field multiplied
by the total current
J
=
σ
E
{\displaystyle \mathbf {J} =\sigma \mathbf
{E} }
:
∫
V
[
J
1
(
e
)
⋅
E
2
(
r
)
−
E
1
(
r
)
⋅
J
2
(
e
)
]
d
V
=
∫
V
[
σ
E
1
(
e
)
⋅
(
E
2
(
r
)
+
E
2
(
e
)
)
−
(
E
1
(
r
)
+
E
1
(
e
)
)
⋅
σ
E
2
(
e
)
]
d
V
=
∫
V
[
E
1
(
e
)
⋅
J
2
−
J
1
⋅
E
2
(
e
)
]
d
V
.
{\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:
∑
n
V
1
(
n
)
I
2
(
n
)
=
∑
n
V
2
(
n
)
I
1
(
n
)
{\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
V
1
{\displaystyle V_{1}}
and
V
2
{\displaystyle V_{2}}
.
Most commonly, this is simplified further
to the case where each system has a single
voltage source V, at
V
1
(
1
)
=
V
{\displaystyle V_{1}^{(1)}=V}
and
V
2
(
2
)
=
V
{\displaystyle V_{2}^{(2)}=V}
. Then the theorem becomes simply
I
1
(
2
)
=
I
2
(
1
)
{\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).
== Conditions and proof of Lorentz reciprocity
==
The Lorentz reciprocity theorem is simply
a reflection of the fact that the linear operator
O
^
{\displaystyle {\hat {O}}}
relating
J
{\displaystyle \mathbf {J} }
and
E
{\displaystyle \mathbf {E} }
at a fixed frequency
ω
{\displaystyle \omega }
(in linear media):
J
=
1
i
ω
[
1
μ
(
∇
×
∇
×
)
−
ω
2
ε
]
E
≡
O
^
E
{\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"
(
F
,
G
)
=
∫
F
⋅
G
d
V
{\displaystyle (\mathbf {F} ,\mathbf {G} )=\int
\mathbf {F} \cdot \mathbf {G} \,dV}
for vector fields
F
{\displaystyle \mathbf {F} }
and
G
{\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
ε
→
ε
+
i
σ
/
ω
{\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
O
^
{\displaystyle {\hat {O}}}
under an inner product
(
f
,
g
)
{\displaystyle (f,g)\!}
, we have
(
f
,
O
^
g
)
=
(
O
^
f
,
g
)
{\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
J
=
O
^
E
{\displaystyle \mathbf {J} ={\hat {O}}\mathbf
{E} }
: that is,
(
E
1
,
O
^
E
2
)
=
(
O
^
E
1
,
E
2
)
{\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
F
{\displaystyle \mathbf {F} }
and
G
{\displaystyle \mathbf {G} }
, integration by parts (or the divergence
theorem) over a volume V enclosed by a surface
S gives the identity:
∫
V
F
⋅
(
∇
×
G
)
d
V
≡
∫
V
(
∇
×
F
)
⋅
G
d
V
−
∮
S
⁡
(
F
×
G
)
⋅
d
A
.
{\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
(
E
1
,
O
^
E
2
)
{\displaystyle (\mathbf {E} _{1},{\hat {O}}\mathbf
{E} _{2})}
to yield
(
O
^
E
1
,
E
2
)
{\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 operationsWe
shall prove a general form of the electromagnetic
reciprocity theorem due to Lorenz which states
that fields
E
1
,
H
1
{\displaystyle \mathbf {E} _{1},\mathbf {H}
_{1}}
and
E
2
,
H
2
{\displaystyle \mathbf {E} _{2},\mathbf {H}
_{2}}
generated by two different sinusoidal current
densities respectively
J
1
{\displaystyle \mathbf {J} _{1}}
and
J
2
{\displaystyle \mathbf {J} _{2}}
of the same frequency, satisfy the condition
∫
V
[
J
1
⋅
E
2
−
E
1
⋅
J
2
]
d
V
=
∮
S
⁡
[
E
1
×
H
2
−
E
2
×
H
1
]
⋅
d
S
.
{\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:
∇
×
E
=
−
∂
B
∂
t
,
∇
×
H
=
J
+
∂
D
∂
t
.
{\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:
∇
×
E
=
−
j
ω
B
,
∇
×
H
=
J
+
j
ω
D
.
{\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
e
j
ω
t
{\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
e
j
ω
t
{\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
H
⋅
(
∇
×
E
)
−
E
⋅
(
∇
×
H
)
=
∇
⋅
(
E
×
H
)
{\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
E
{\displaystyle \mathbf {E} }
and
H
{\displaystyle \mathbf {H} }
.
If we apply this equivalence to
E
1
{\displaystyle \mathbf {E} _{1}}
and
H
2
{\displaystyle \mathbf {H} _{2}}
we get:
H
2
⋅
(
∇
×
E
1
)
−
E
1
⋅
(
∇
×
H
2
)
=
∇
⋅
(
E
1
×
H
2
)
{\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,
−
H
2
⋅
j
ω
B
1
−
E
1
⋅
j
ω
D
2
−
E
1
⋅
J
2
=
∇
⋅
(
E
1
×
H
2
)
{\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,
∫
V
(
H
2
⋅
j
ω
B
1
+
E
1
⋅
j
ω
D
2
+
E
1
J
2
)
d
V
=
−
∫
V
∇
⋅
(
E
1
×
H
2
)
d
V
{\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
d
i
v
(
E
1
×
H
2
)
{\displaystyle div(\mathbf {E} _{1}\times
\mathbf {H} _{2})}
equals the surface integral of
E
1
×
H
2
{\displaystyle \mathbf {E} _{1}\times \mathbf
{H} _{2}}
over the boundary.
∫
V
(
H
2
⋅
j
ω
B
1
+
E
1
⋅
j
ω
D
2
+
E
1
⋅
J
2
)
d
V
=
−
∮
S
⁡
(
E
1
×
H
2
)
⋅
d
S
^
{\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,
ϵ
{\displaystyle \epsilon }
is a scalar independent of time. Then generally
as physical magnitudes
D
=
ϵ
E
{\displaystyle \mathbf {D} =\epsilon \mathbf
{E} }
and
B
=
μ
H
{\displaystyle \mathbf {B} =\mu \mathbf {H}
}
.
Last equation then becomes
∫
V
(
H
2
⋅
j
ω
μ
H
1
+
E
1
⋅
j
ω
ϵ
E
2
+
E
1
⋅
J
2
)
d
V
=
−
∮
S
⁡
(
E
1
×
H
2
)
⋅
d
S
^
{\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
E
2
{\displaystyle \mathbf {E} _{2}}
and
H
1
{\displaystyle \mathbf {H} _{1}}
the following expression:
∫
V
(
H
1
⋅
j
ω
μ
H
2
+
E
2
⋅
j
ω
ϵ
E
1
+
E
2
⋅
J
1
)
d
V
=
−
∮
S
⁡
(
E
2
×
H
1
)
⋅
d
S
^
{\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
∫
V
[
J
1
⋅
E
2
−
E
1
⋅
J
2
]
d
V
=
∮
S
⁡
[
E
1
×
H
2
−
E
2
×
H
1
]
⋅
d
S
.
{\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
J
1
⋅
E
2
−
E
1
⋅
J
2
=
∇
⋅
[
E
1
×
H
2
−
E
2
×
H
1
]
{\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.
=== Surface-term cancellation ===
The cancellation of the surface terms on the
right-hand side of the Lorentz reciprocity
theorem, for an integration over all space,
is not entirely obvious but can be derived
in a number of ways.
The simplest general argument comes from a
straightforward application of the divergence
theorem: For localized sources, one can choose
the bounding surface
S
{\displaystyle S}
such that it contains all sources. This bounding
surface is also a bounding surface (reversing
the unit-normal vector) for the complementary
region of space going out to infinity,
V
c
{\displaystyle V_{c}}
, containing no sources. The reciprocity relation
thus still holds:
∫
V
c
[
J
1
⋅
E
2
−
E
1
⋅
J
2
]
d
V
=
−
∮
S
⁡
[
E
1
×
H
2
−
E
2
×
H
1
]
⋅
d
S
,
{\displaystyle \int _{V_{c}}\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} ,}
with the replacement
V
→
V
c
{\displaystyle V\rightarrow V_{c}}
and a negative sign for the unit normal. The
left-hand side of the expression is zero,
because there are no sources in
V
c
{\displaystyle V_{c}}
, and thus the right-hand side is zero as
well.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
r
^
{\displaystyle {\hat {\mathbf {r} }}}
direction) with
r
^
⋅
E
=
0
{\displaystyle {\hat {\mathbf {r} }}\cdot
\mathbf {E} =0}
and
H
=
r
^
×
E
/
Z
{\displaystyle \mathbf {H} ={\hat {\mathbf
{r} }}\times \mathbf {E} /Z}
where Z is the impedance
μ
/
ϵ
{\displaystyle {\sqrt {\mu /\epsilon }}}
of the surrounding medium. Then it follows
that
E
1
×
H
2
=
E
1
×
r
^
×
E
2
/
Z
{\displaystyle \mathbf {E} _{1}\times \mathbf
{H} _{2}=\mathbf {E} _{1}\times {\hat {\mathbf
{r} }}\times \mathbf {E} _{2}/Z}
, which by a simple vector identity equals
r
^
(
E
1
⋅
E
2
)
/
Z
{\displaystyle {\hat {\mathbf {r} }}(\mathbf
{E} _{1}\cdot \mathbf {E} _{2})/Z}
. Similarly,
E
2
×
H
1
=
r
^
(
E
2
⋅
E
1
)
/
Z
{\displaystyle \mathbf {E} _{2}\times \mathbf
{H} _{1}={\hat {\mathbf {r} }}(\mathbf {E}
_{2}\cdot \mathbf {E} _{1})/Z}
and the two terms cancel one another.
The above argument shows explicitly why the
surface terms can cancel, but lacks generality.
Alternatively, one can treat the case of lossless
surrounding media by taking the limit as the
losses (the imaginary part of ε) go to zero.
For any nonzero loss, the fields decay exponentially
with distance and the surface integral vanishes,
regardless of whether the medium is homogeneous.
Since the left-hand side of the Lorentz reciprocity
theorem vanishes for integration over all
space with any non-zero losses, it must also
vanish in the limit as the losses go to zero.
(Note that we implicitly assumed the standard
boundary condition of zero incoming waves
from infinity, because otherwise even an infinitesimal
loss would eliminate the incoming waves and
the limit would not give the lossless solution.)
=== Reciprocity and the Green's function ===
The inverse of the operator
O
^
{\displaystyle {\hat {O}}}
, i.e. in
E
=
O
^
−
1
J
{\displaystyle \mathbf {E} ={\hat {O}}^{-1}\mathbf
{J} }
(which requires a specification of the boundary
conditions at infinity in a lossless system),
has the same symmetry as
O
^
{\displaystyle {\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 ε and μ. More specifically,
the Green's function can be written as
G
n
m
(
x
′
,
x
)
{\displaystyle G_{nm}(\mathbf {x} ',\mathbf
{x} )}
giving the n-th component of
E
{\displaystyle \mathbf {E} }
at
x
′
{\displaystyle \mathbf {x} '}
from a point dipole current in the m-th direction
at
x
{\displaystyle \mathbf {x} }
(essentially,
G
{\displaystyle G}
gives the matrix elements of
O
^
−
1
{\displaystyle {\hat {O}}^{-1}}
), and Rayleigh-Carson reciprocity is equivalent
to the statement that
G
n
m
(
x
′
,
x
)
=
G
m
n
(
x
,
x
′
)
{\displaystyle G_{nm}(\mathbf {x} ',\mathbf
{x} )=G_{mn}(\mathbf {x} ,\mathbf {x} ')}
. Unlike
O
^
{\displaystyle {\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 ε is not a symmetric matrix
is for magneto-optic materials, in which case
the usual statement of Lorentz reciprocity
does not hold (see below for a generalization,
however). If we allow magneto-optic materials,
but restrict ourselves to the situation where
material absorption is negligible, then ε
and μ are in general 3×3 complex Hermitian
matrices. In this case, the operator
1
μ
(
∇
×
∇
×
)
−
ω
2
c
2
ε
{\displaystyle {\frac {1}{\mu }}\left(\nabla
\times \nabla \times \right)-{\frac {\omega
^{2}}{c^{2}}}\varepsilon }
is Hermitian under the conjugated inner product
(
F
,
G
)
=
∫
F
∗
⋅
G
d
V
{\displaystyle (\mathbf {F} ,\mathbf {G} )=\int
\mathbf {F} ^{*}\cdot \mathbf {G} \,dV}
, and a variant of the reciprocity theorem
still holds:
−
∫
V
[
J
1
∗
⋅
E
2
+
E
1
∗
⋅
J
2
]
d
V
=
∮
S
⁡
[
E
1
∗
×
H
2
+
E
2
×
H
1
∗
]
⋅
d
A
{\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 {dA} }
where the sign changes come from the
1
/
i
ω
{\displaystyle 1/i\omega }
in the equation above, which makes the operator
O
^
{\displaystyle {\hat {O}}}
anti-Hermitian (neglecting surface terms).
For the special case of
J
1
=
J
2
{\displaystyle \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
−
J
∗
⋅
E
{\displaystyle -\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
(
J
1
,
E
1
)
{\displaystyle (\mathbf {J} _{1},\mathbf {E}
_{1})}
and
(
J
2
,
E
2
)
{\displaystyle (\mathbf {J} _{2},\mathbf {E}
_{2})}
to exist in different systems.
In particular, if
(
J
1
,
E
1
)
{\displaystyle (\mathbf {J} _{1},\mathbf {E}
_{1})}
satisfy Maxwell's equations at ω for a system
with materials
(
ε
1
,
μ
1
)
{\displaystyle (\varepsilon _{1},\mu _{1})}
, and
(
J
2
,
E
2
)
{\displaystyle (\mathbf {J} _{2},\mathbf {E}
_{2})}
satisfy Maxwell's equations at ω for a system
with materials
(
ε
1
T
,
μ
1
T
)
{\displaystyle \left(\varepsilon _{1}^{T},\mu
_{1}^{T}\right)}
, 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 ("active")
media; for example, when ε is modulated in
time by some external process. (In both of
these cases, the frequency ω is not generally
a conserved quantity.)
== Feld-Tai reciprocity ==
A closely related reciprocity theorem was
articulated independently by Y. A. Feld and
C. T. Tai in 1992 and is known as Feld-Tai
reciprocity or the Feld-Tai lemma. It relates
two time-harmonic localized current sources
and the resulting magnetic fields:
∫
J
1
⋅
H
2
d
V
=
∫
H
1
⋅
J
2
d
V
.
{\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
E
{\displaystyle \mathbf {E} }
and
i
ω
J
{\displaystyle i\omega \mathbf {J} }
is a constant scalar multiple of the operator
relating
H
{\displaystyle \mathbf {H} }
and
∇
×
(
J
/
ε
)
{\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.
== 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:
"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."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. 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 most extremely
simple 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.
== 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
(Panofsky and Phillips, 1962).
In particular, let
ϕ
1
{\displaystyle \phi _{1}}
denote the electric potential resulting from
a total charge density
ρ
1
{\displaystyle \rho _{1}}
. The electric potential satisfies Poisson's
equation,
−
∇
2
ϕ
1
=
ρ
1
/
ε
0
{\displaystyle -\nabla ^{2}\phi _{1}=\rho
_{1}/\varepsilon _{0}}
, where
ε
0
{\displaystyle \varepsilon _{0}}
is the vacuum permittivity. Similarly, let
ϕ
2
{\displaystyle \phi _{2}}
denote the electric potential resulting from
a total charge density
ρ
2
{\displaystyle \rho _{2}}
, satisfying
−
∇
2
ϕ
2
=
ρ
2
/
ε
0
{\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:
∫
ρ
1
ϕ
2
d
V
=
∫
ρ
2
ϕ
1
d
V
.
{\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
∫
ϕ
2
(
∇
2
ϕ
1
)
d
V
=
∫
ϕ
1
(
∇
2
ϕ
2
)
d
V
{\displaystyle \int \phi _{2}(\nabla ^{2}\phi
_{1})dV=\int \phi _{1}(\nabla ^{2}\phi _{2})dV}
, i.e. that
∇
2
{\displaystyle \nabla ^{2}}
is a Hermitian operator (as follows by integrating
by parts twice
