# Most generic form of refractive index tensors

The refractive index of a material is in general a $$3x3$$ tensor (as in the case of birefringent crystals). From literature, it seems that in the case of transparent crystals, this tensor is in general real and symmetric. Additionally, in the case of absorbing or lossy materials, the refractive index is often modeled as a complex number $$n+i\kappa$$, where the imaginary part $$\kappa$$ describes the loss rate.

I am wondering what is the most generic form of a refractive index tensor. For example, is the real part of the tensor necessarily symmetric? What constraints on the tensor are imposed by the laws of physics?

Symmetry: In non-magnetic media, the real part of the refractive index tensor is typically symmetric due to energy conservation principles. This symmetry is related to the reciprocity theorem in electromagnetism. However, in the presence of strong magnetic fields or in certain non-reciprocal materials, the tensor can become asymmetric.

Complex nature: As you correctly noted, for absorbing or lossy materials, the refractive index is often represented as a complex quantity n + iκ, where κ describes the attenuation or loss. This concept extends to the tensor form, where both real and imaginary parts can be tensors.

Constraints from physics: The laws of physics, particularly the principles of thermodynamics and causality, impose certain constraints on the refractive index tensor: a. Passivity: For passive materials, the imaginary part of the dielectric tensor (related to the refractive index tensor) must be positive definite to ensure energy dissipation rather than gain. b. Kramers-Kronig relations: These relations, which stem from causality, connect the real and imaginary parts of the refractive index, ensuring physical consistency across all frequencies.

Lorentz invariance: The general principle of relativity suggests that the laws governing the refractive index tensor should be formulated in a way that is independent of the choice of coordinate system. This requirement places additional constraints on how the tensor can transform under different reference frames.

Frequency dependence: The refractive index tensor is generally frequency-dependent, a phenomenon known as dispersion. This dependence must be consistent with the Kramers-Kronig relations mentioned earlier.

The most generic form of the refractive index tensor would thus be a 3x3 complex matrix that satisfies the constraints imposed by thermodynamics, causality, and Lorentz invariance. Its exact form and properties would depend on the material's microscopic structure, external fields, and the frequency of the electromagnetic waves interacting with it.

• As you mentioned, with magnetic fields the index tensor is asymmetric. I know magneto-optic materials have a hermitian permittivity. Your response makes sense but does not prescribe concrete mathematical relations. For example, can I definitively say the real part must be symmetric? The imaginary part must be positive semi definite? Is the imaginary part antisymmetric off the diagonal? I am only concerned with single frequencies at a time so I am disregarding constraints like Kramer's Kronig. Commented Jul 4 at 7:43