Fourier expansion electromagnetic field

The electromagnetic (EM) field is of importance as a carrier of solar energy and electronic signals (radio, TV, etc.). As its name suggests, it consists of two tightly coupled vector fields, the electric field E and the magnetic field B. The Fourier expansion of the electromagnetic field is used in the quantization of the field that leads to photons, light particles of well-defined energy and momentum. Further the Fourier transform plays a role in theory of wave propagation through different media and light scattering.

In the absence of charges and electric currents, both E and B can be derived from a third vector field, the vector potential A. In this article the Fourier transform of the fields E, B, and A will be discussed. It will be seen that the expansion of the vector potential A yields the expansions of the fields E and B. Further the energy and momentum of the EM field will be expressed in the Fourier components of A.

Fourier expansion of a vector field

For an arbitrary real vector field F its Fourier expansion is the following:

$\mathbf {F} (\mathbf {r} ,t)=\sum _{\mathbf {k} }\left(\mathbf {f} _{k}(t)e^{i\mathbf {k} \cdot \mathbf {r} }+{\bar {\mathbf {f} }}_{k}(t)e^{-i\mathbf {k} \cdot \mathbf {r} }\right)$

where the bar indicates complex conjugation. Such an expansion, labeled by a discrete (countable) set of vectors k, is always possible when F satisfies periodic boundary conditions, i.e., F(r + p,t) = F(r,t) for some finite vector p. To impose such boundary conditions, it is common to consider EM waves as if they are in a virtual cubic box of finite volume V = L³. Waves on opposite walls of the box are enforced to have the same value (usually zero). Note that the waves are not restricted to the box: the box is replicated an infinite number of times in x, y, and z direction. The vectors k are,

$\mathbf {k} ={\frac {2\pi }{L}}(n_{x},\;n_{y},\;n_{z})\quad {\hbox{with}}\quad n_{x},\,n_{y},\,n_{z}=0,\;\pm 1\;\pm 2,\ldots$

Vector potential

The magnetic field B satisfies the following Maxwell equation:

${\boldsymbol {\nabla }}\cdot \mathbf {B} (\mathbf {r} ,t)=0,$

that is, the divergence of B is zero. This equation expresses the fact that magnetic monopoles (charges) do not exist (or, rather, have never been found in nature). A divergence-free field, such as B, is a also referred to as a transverse field. By the Helmholtz decomposition, B can be written as

$\mathbf {B} (\mathbf {r} ,t)={\boldsymbol {\nabla }}\times \mathbf {A} (\mathbf {r} ,t),$

in which the vector potential A is introduced though the curl ∇×A.

The electric field obeys one of the Maxwell equations, in electromagnetic SI units it reads,

${\boldsymbol {\nabla }}\cdot \mathbf {E} (\mathbf {r} ,t)={\frac {\rho }{\epsilon _{0}}}=0$

because it is assumed that charge distributions ρ are zero. The quantity ε₀ is the electric constant. Hence, also the electric field E is transverse. Since there are no charges, the electric potential is zero and the electric field follows from A by,

$\mathbf {E} (\mathbf {r} ,t)=-{\frac {\partial \mathbf {A} (\mathbf {r} ,t)}{\partial t}}.$

The fact that E can be written this way is due to the choice of Coulomb gauge for A:

${\boldsymbol {\nabla }}\cdot \mathbf {A} (\mathbf {r} ,t)=0.$

By definition, a choice of gauge does not affect any measurable properties (the best known example of a choice of gauge is the fixing of the zero of an electric potential, for instance at infinity). The Coulomb gauge makes A transverse as well, and clearly A is in the same plane as E. (The time differentiation does not affect direction.) So, the vector fields A, B, and E are all in the same plane.

The three fields can be written as a linear combination of two orthonormal vectors, e_x and e_y. It is more convenient to choose complex unit vectors obtained by a unitary transformation,

$\mathbf {e} ^{(1)}\equiv {\frac {-1}{\sqrt {2}}}(\mathbf {e} _{x}+i\mathbf {e} _{y})\quad {\hbox{and}}\quad \mathbf {e} ^{(-1)}\equiv {\frac {1}{\sqrt {2}}}(\mathbf {e} _{x}-i\mathbf {e} _{y})$

which are orthonormal,

$\mathbf {e} ^{(\mu )}\cdot {\bar {\mathbf {e} }}^{(\mu ')}=\delta _{\mu ,\mu '}\quad {\hbox{with}}\quad \mu ,\mu '=1,\,-1.$

Expansions

The Fourier expansion of the vector potential reads

$\mathbf {A} (\mathbf {r} ,t)=\sum _{\mathbf {k} }\sum _{\mu =-1,1}\left(\mathbf {e} ^{(\mu )}(\mathbf {k} )a_{\mathbf {k} }^{(\mu )}(t)\,e^{i\mathbf {k} \cdot \mathbf {r} }+{\bar {\mathbf {e} }}^{(\mu )}(\mathbf {k} ){\bar {a}}_{\mathbf {k} }^{(\mu )}(t)\,e^{-i\mathbf {k} \cdot \mathbf {r} }\right).$

The vector potential obeys the wave equation,

$\nabla ^{2}\mathbf {A} (\mathbf {r} ,t)={\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {A} (\mathbf {r} ,t)}{\partial t^{2}}}$

The substitution of the Fourier series of A into the wave equation yields for the individual terms,

$-k^{2}a_{\mathbf {k} }^{(\mu )}(t)={\frac {1}{c^{2}}}{\frac {\partial ^{2}a_{\mathbf {k} }^{(\mu )}(t)}{\partial t^{2}}}\quad \Longrightarrow \quad a_{\mathbf {k} }^{(\mu )}(t)\propto e^{-i\omega t}\quad {\hbox{with}}\quad \omega =kc.$

It is now an easy matter to construct the corresponding Fourier expansions for E and B from the expansion of the vector potential A. For instance, the expansion for E follows from differentiation with respect to time,

$\mathbf {E} (\mathbf {r} ,t)=i\sum _{\mathbf {k} }\sum _{\mu =-1,1}\omega \left(\mathbf {e} ^{(\mu )}(\mathbf {k} )a_{\mathbf {k} }^{(\mu )}(t)\,e^{i\mathbf {k} \cdot \mathbf {r} }-{\bar {\mathbf {e} }}^{(\mu )}(\mathbf {k} ){\bar {a}}_{\mathbf {k} }^{(\mu )}(t)\,e^{-i\mathbf {k} \cdot \mathbf {r} }\right)$

Fourier-expanded energy

The electromagnetic energy density $\scriptstyle {\mathcal {E}}_{\mathrm {Field} }$, defined earlier in this article, can be expressed in terms of the Fourier coefficients. We define the total energy (classical Hamiltonian) of a finite volume V by

$H=\iiint _{V}{\mathcal {E}}_{\mathrm {Field} }(\mathbf {r} ,t)\mathrm {d} ^{3}\mathbf {r} .$

The classical Hamiltonian in terms of Fourier coefficients takes the form

$H=V\epsilon _{0}\sum _{\mathbf {k} }\sum _{\mu =1,-1}\omega ^{2}{\big (}{\bar {a}}_{\mathbf {k} }^{(\mu )}(t)a_{\mathbf {k} }^{(\mu )}(t)+a_{\mathbf {k} }^{(\mu )}(t){\bar {a}}_{\mathbf {k} }^{(\mu )}(t){\big )}$

with

$\omega \equiv |\mathbf {k} |c=2\pi \nu =2\pi {\frac {c}{\lambda }}$

and ε₀ is the electric constant. The two terms in the summand of H are identical (factors commute) and may be summed. However, after quantization (interpretation of the expansion coefficients as operators) the factors do no longer commute and according to quantum mechanical rules one must depart from the symmetrized classical Hamiltonian.

Fourier-expanded momentum

The electromagnetic momentum, P_EM, of EM radiation enclosed by a volume V is proportional to an integral of the Poynting vector (see above). In SI units:

$\mathbf {P} _{\textrm {EM}}\equiv {\frac {1}{c^{2}}}\iiint _{V}\mathbf {S} \,{\textrm {d}}^{3}\mathbf {r} =\epsilon _{0}\iiint _{V}\mathbf {E} (\mathbf {r} ,t)\times \mathbf {B} (\mathbf {r} ,t)\,{\textrm {d}}^{3}\mathbf {r} .$