As a consequence of the causality principle, the real part (dispersion) and the imaginary part are related by a Hilbert transform.

https://en.wikipedia.org/wiki/Kramers%E2%80%93Kronig_relations

Let me explain this question in general terms for periodic structures on the basis of our paper 

https://www.researchgate.net/publication/45899426_Polarization_transformations_by_a_magneto-photonic_layered_structure_in_vicinity_of_ferromagnetic_resonance

For fiber modes it is the same. 

By definition, the eigenwaves of a periodic structure are non-trivial solutions of the homogeneous Maxwell equations that satisfy the conditions of Floquet’s quasi-periodicity:

Psi(x, y, z) = exp(iγ L)Psi (x, y, z + L)               (1)

where Psi is some linear function of the field components and L is the structure period. Condition (1) expresses the intuitive notion about waves in a periodic structure: in the neighboring periods the field differs only by a certain phase factor γ. This parameter γ is a wavenumber related to a certain eigenwave. They are called the Bloch wavenumber and the Bloch wave, respectively. It is obvious that the Bloch wave exists in any section of an infinite periodic medium. For lossless structures, the parameter γ takes either purely real or purely imaginary values in the passbands and stopbands, respectively. 

When the material losses are taken into consideration, the formal solution of the dispersion equation leads to complex values of γ (γ = γ'  + iγ''). In this case an exponential decay of the field follows from condition (1), which gives a contradiction to the definition of the Bloch waves in an infinite periodic structure. Thus it is necessary to initially refuse the imposition of the condition (1). Instead, the standard method of the theory of irregular waveguides can be applied. In the context of this theory it is assumed that the eigenwaves of the irregular waveguide with impedance sidewalls are orthogonal in energy terms. It means that every eigenwave propagates independently from the others inside the area free of sources. Thus the eigenwaves have clear physical meaning: it is the field that can be excited in the waveguide outside the area occupied by the sources. 

Usually the solution of dispersion equation has multiple roots. In order to select physically correct roots we need to consider the imaginary part of propagation constants to find those ones which satisfy the condition of the wave decay as the wave propagates from the source.  So it is the essence of imaginary part of propagation constant in dispersion curves.

Article Polarization transformations by a magneto-photonic layered s...

What is the relationship between dispersion and the imaginary part of the propagation constant?

Ans:

A propagating wave has a solution

F(x)=A exp(-iKx)

In case propagation constant K is complex K=K1 + i K2 ,it is obvious that only the real part allows free propagation of the wave and the imaginary part leads to attenuation of the wave.

When we solve Maxwell equation for propagation in free space, we get a dispersion free situation, where the phase velocity is

V=ω/K =  , where µ is the permeability and ε is the dielectric constant .In this case ωvsk curve is linear.

In case of propagation through a glass fibre, where the core has refractive index n2 and the cladding has refractive index n1  for the allowed modes, ω vs  k curve will be nonlinear. It will vary and remain between two asymptotes ω= cK/n1  and ω= cK/n2 , where k is the real part of the propagation constant.

Thus the phase velocity V=ω/K is not a constant but varies leading to dispersion.

Kindly see “Optical Physics” by S.G.Lipson, H.Lipson and D.S.Tannhausan, Cambridge Univ Press ,3rd edition, 1996.

Dear Xing Luo,

My idea for your question is described in the attached file ‘MyIdea.pdf’.

Please find the attached file ‘MyIdea.pdf’

Sincerely,

Zung-Shik Suh, Gumi University

Dispersion relation contains both imaginary and real part. Real part is responsible for free propagation while imaginary part is responsible for attenuations/absorption. This is applicable for time and space both.

Thank you all very much. I have learned much from your answers.