The first of these is based on the electric field is used in The Electromagnetic Waves, Frequency Domain Interface.

Using the relation εr = n2, where n is the refractive index, the equation can alternatively be written

The wave number in vacuum k0 is defined by

where c0 is the speed of light in vacuum.

When the equation is written using the refractive index, the assumption is that μr = 1 and σ = 0 and only the constitutive relations for linear materials are available. When solving for the scattered field the same equations are used but E = Esc + Ei and Esc is the dependent variable.

When solving the frequency domain equation as an eigenfrequency problem the eigenvalue is the complex eigenfrequency λ = jω + δ, where δ is the damping of the solution. The Q factor is given from the eigenvalue by the formula

The spatial parameter, α = δz + jβ = −λ, can have a real part and an imaginary part. The propagation constant is equal to the imaginary part, and the real part, δz, represents the damping along the propagation direction. When solving for all three electric field components the allowed anisotropy of the optionally complex relative permittivity and relative permeability is limited to:

In 2D, the electric field varies with the out-of-plane wave number kz as

where z is the unit vector in the out-of-plane z direction.

Similarly, in 2D axisymmetry, the electric field varies with the azimuthal mode number m as

where is the unit vector in the out-of-plane φ direction.

referred to as the covariant formulation. Here, Ψ is the dependent variable and the out-of-plane electric field component is calculated as

The TM waves polarization has only one magnetic field component in the z direction, and the electric field lies in the modeling plane. Thus the time-harmonic fields can be obtained by solving for the in-plane electric field components only. The equation is formally the same as in 3D, the only difference being that the out-of-plane electric field component is zero everywhere and that out-of-plane spatial derivatives are evaluated for the prescribed out-of-plane wave vector dependence of the electric field.

As the field propagates in the modeling xy-plane a TE wave has only one nonzero electric field component, namely in the z direction. The magnetic field lies in the modeling plane. Thus the time-harmonic fields can be simplified to a scalar equation for Ez,

To be able to write the fields in this form, it is also required that εr, σ, and μr are nondiagonal only in the xy-plane. μr denotes a 2-by-2 tensor, and εrzz and σzz are the relative permittivity and conductivity in the z direction.

Solving for all three components in 2D is referred to as “hybrid-mode waves”. The equation is formally the same as in 3D with the addition that spatial derivatives with respect to φ are evaluated for the prescribed azimuthal mode number dependence of the electric field.

A TM wave has a magnetic field with only a φ-component and thus an electric field with components in the rz-plane only. The equation is formally the same as in 3D, the only difference being that the φ-component is zero everywhere and that spatial derivatives with respect to φ are evaluated for the prescribed azimuthal mode number dependence of the electric field.

A TE wave has only an electric field component in the φ direction, and the magnetic field lies in the modeling plane. Given these constraints, the 3D equation can be simplified to a scalar equation for Eφ. To write the fields in this form, it is also required that εr and μr are nondiagonal only in the rz-plane. μr denotes a 2-by-2 tensor, while εrφφ and σφφ are the relative permittivity and conductivity in the φ direction.

The frequency domain equations allow for several ways of introducing electric losses. Finite conductivity results in a complex permittivity,

where ε' is the real part of εr, and all losses (dielectric and conduction losses) are given by ε''. The dielectric loss model can also single out the losses from finite conductivity (so that ε'' only represents dielectric losses) resulting in:

In optics and photonics applications, the refractive index is often used instead of the permittivity. In materials where μr is 1, the relation between the complex refractive index

The parameter κ represents a damping of the electromagnetic wave. When specifying the refractive index, conductivity is not allowed as an input.

In the physics and optics literature, the time harmonic form is often written with a minus sign (and “i” instead of “j”):

This makes an important difference in how loss is represented by complex material coefficients like permittivity and refractive index, that is, by having a positive imaginary part (material parameters ε'' and κ) rather than a negative one. Therefore, material data taken from the literature might have to be conjugated before using it in a model.

The frequency domain equations allow for magnetic losses to be introduced as a complex relative permeability.