# Linear Optical Susceptibility

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This part continues from the harmonic fields tutorial.

The susceptibility tensor **χ**(**r**, *t*) and the permittivity tensor **ε**(**r**, *t*) of space and time are always real quantities although all field quantities, including both **E**(**r**, *t*) and **E**(**k**, *ω*), can be defined in a complex form. This is true even in the presence of an optical loss or gain in the medium.

However, the susceptibility and permittivity tensors in the momentum space and frequency domain, **χ**(**k**, *ω*) and **ε**(**k**, *ω*), can be complex. If an eigenvalue, χ_{i}, of **χ** is complex, the corresponding eigenvalue, ε_{i}, of **ε** is also complex, and their imaginary parts have the same sign because **ε** = ε_{0}(1 + **χ**).

The signs of such imaginary parts of eigenvalues tell whether the medium has an optical gain or loss. In our convention, we write, for example, χ_{i} = χ_{i}' + iχ_{i}'' in the frequency domain. Then χ_{i}''(ω) > 0 corresponds to an optical loss or absorption, while χ_{i}''(ω) < 0 represents an optical gain or amplification.

The fact that **χ**(**r**, *t*) and **ε**(**r**, *t*) are real quantities leads to the following symmetry relations for the tensor elements of **χ**(**k**, *ω*) and **ε**(**k**, *ω*):

\[\tag{56}\chi_\text{ij}^*(\mathbf{k},\omega)=\chi_\text{ij}(-\mathbf{k},-\omega)\]

and

\[\tag{57}\epsilon_\text{ij}^*(\mathbf{k},\omega)=\epsilon_\text{ij}(-\mathbf{k},-\omega)\]

which are called the ** reality condition**. The reality condition implies that χ

_{ij}'(

**k**,

*ω*) = χ

_{ij}'(-

**k**, -

*ω*) and χ

_{ij}''(

**k**,

*ω*) = - χ

_{ij}''(-

**k**, -

*ω*). Similar relations also apply for the real and imaginary parts of ε

_{ij}.

Therefore, the real parts of χ_{ij} and ε_{ij} are even functions of **k** and *ω*, whereas the imaginary parts are odd functions of **k** and *ω*. Any constant contribution, independent of **k** and *ω*, in χ_{ij} and ε_{ij} is an even function of **k** and *ω*; hence it can appear only in the real parts.

As a result, the imaginary parts, if they exist, are always functions of either **k** or *ω*, or both. The loss, or gain, in a medium is associated with the imaginary parts of the eigenvalues of **χ**(*ω*); consequently, it is inherently dispersive. Any other effects that can be described by the imaginary parts of the eigenvalues of **χ**(**k**, *ω*) are also dispersive in either momentum or frequency, or both.

The momentum and frequency dependencies of an electric susceptibility, **χ**(**k**, *ω*), are due to the spatial and temporal nonlocality properties of the underlying physical mechanisms that contribute to **χ**. Spatial nonlocality causes spatially convoluted effects and results in momentum dependence of the susceptibility, and temporal nonlocality causes temporal convolution and results in frequency dispersion of the medium.

In addition to nonlocality, it is also important to consider inhomogeneity, in both space and time. In a linear medium, changes in the wavevector of an optical wave, or coupling between waves of different wavevectors, can occur only if the optical property of the medium in which the wave propagates is spatially inhomogeneous such that **χ**(**k**, *ω*) is spatially dependent.

Likewise, changes in the frequency of an optical wave, or coupling between waves of different frequencies, are possible in a linear medium only if the optical property of the medium is time varying such that **χ**(**k**, *ω*) varies with time.

Changes in the wavevector of an optical wave can take the form of changes in the wave propagation direction, as in reflection and diffraction, or in the optical wavelength, as in the case when a wave propagates from one part of the medium to another of different refractive index.

Changes in the frequency of an optical wave result in the generation of other frequencies or the conversion of the optical wave to a completely different frequency. Consequently, for practical photonic devices, it is often necessary to consider both nonlocality and inhomogeneity in both space and time, thus writing **χ**(**r**,*t*; **k**, *ω*) and, correspondingly, **ε**(**r**,*t*; **k**, *ω*).

The next part continues with the Polarization of Light tutorial.