where
- t is time (in seconds),
- ω is the angular frequency (in radians per second),
- k = (kx, ky, kz) is the wave vector (in radians per meter), and
is the phase angle (in radians).
The wave vector is related to the angular frequency by
where k is the wavenumber and λ is the wavelength.
The electromagnetic spectrum is a plot of the field magnitudes (or energies) as a function of wavelength.
Multipole expansion
Assuming monochromatic fields varying in time as is the wavenumber and λ is the wavelength.
The electromagnetic spectrum is a plot of the field magnitudes (or energies) as a function of wavelength.
Multipole expansion
Assuming monochromatic fields varying in time as 
, if one uses Maxwell's Equations to eliminate B<The electromagnetic spectrum is a plot of the field magnitudes (or energies) as a function of wavelength.
Assuming monochromatic fields varying in time as , if one uses Maxwell's Equations to eliminate B, the electromagnetic wave equation reduces to the Helmholtz Equation for E:
A generic electromagnetic field with frequency ω can be written as a sum of solutions to these two equations. The three-dimensional solutions of the Helmholtz Equation can be expressed as expansions in spherical harmonics with coefficients proportional to the spherical Bessel functions. However, applying this expansion to each vector component of E or B will give solutions that are not generically divergence-free (∇ · E = ∇ · B = 0), and therefore require additional restrictions on the coefficients.
The multipole expansion circumvents this difficulty by expanding not E or B, but r · E or r · B into spherical harmonics. These expansions still solve the original Helmholtz equations for E and B because for a divergence-free field F, ∇2 (r · F) = r · (∇2 F). The resulting expressions for a generic electromagnetic field are:
- or B, but r · E or r · B into spherical harmonics. These expansions still solve the original Helmholtz equations for E and B because for a divergence-free field F, ∇2 (r · F) = r · (∇2 F). The resulting expressions for a generic electromagnetic field are:
where 
and 
are the electric multipole fields of order (l, m), and 
and 
are the corresponding magnetic multipole fields, and aE(l, m) and aM(l, m) are the coefficients of the expansion. The multipole fields are given by
![\mathbf{B}_{l,m}^{(E)} = \sqrt{l(l+1)} \left[B_l^{(1)} h_l^{(1)}(kr) + B_l^{(2)} h_l^{(2)}(kr)\right] \mathbf{\Phi}_{l,m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bed6d25c3f5817c694944a905007f775335d96c7)
