The main difference is that the electric conductivity is for a static field, i.e. it has a specific direction and a specific time independent value. Electrical conductivity is the measure of a material's ability to allow the transport of an electric charge. But the optical conductivity is a time dependent value. Light has an electric field component that varies in amplitude and direction over time, and that is why it is called optical conductivity. The optical conductivity arises under optical excitation without the presence of applied electrical field. Such conductivity appears in result of the presence of the gradient in concentration of carriers in different regions of semiconductor material. Thus the optical conductivity is caused with the diffusion current in the material. The value of optical conductivity depends on the power of the excited light. The optical conductivity is also closely related to the dielectric function, the generalization of the dielectric constant to arbitrary frequencies.
However, Nurul Islam I know this theoretical background, but I asked for reference papers which would be great for your M.Sc. thesis. You are also doing great as a M.Sc student. Keep it up.
In case, we travel, It is helpful to advice a map; we find a route map for a vehicle (bicycle, car, train, airplane, etc.). For our genuine, actual, way(s), we select, often, to percolate, even on a mixed route; if possible, we select the lowest percolation threshold for our traveling requirements.
In traveling cases of the electrical carriers, either in optical or DC electrical conductivity, they follow their own suitable route maps. The carriers follow the easiest nornalized route topology (sometimes, re-)strained by the electrical field's direction (optical polarization effects). In other words, their routes are (re-)scalled, using the nornalized units of their diffusion length[1].
The optical and electrical conductivity in bulk crystals, with a high diffusion length(s) of the carriers, are, usually, very close, nearly the same. However, the effective diffusion length(s) of the carriers in new materials, e.g. in films, composite bulk (nano)materials, etc., correlates, sometimes very strongly, with the materials' topological length's, size effects in nanophysics. So, a topological (conductivity) size constraint(s) might limit[2] the DC electrical conductivity, say a case of the grains' size or the (thin) films' thickness[3]; these direct size limitation(s) might be, stepwise, less significant, as the frequency increases, because the nornalized length, becomes[4-6], not any more, a very important conductivity limit(er).
The effective conductivity S, in nanophysics[5,6], scales with the frequency F; so, S scales with F; S is following, usually, in size, with (the increasing) frequency: (high freq.) AC electrical conductivity, microwaves, ir and optical, electrical conductivity.
3. Electrical Conductivity of Thin-Film Composites Containing Silver Nanoparticles Embedded in a Dielectric Teflon® AF Matrix https://arxiv.org/pdf/1105.2606.pdf
4. Characteristics of the electrical percolation in carbon nanotubes/polymer nanocomposites http://www3.ntu.edu.sg/home/zhaoyang/percolation.pdf
5. Percolation threshold and electrical conductivity of graphene... https://aip.scitation.org/doi/abs/10.1063/1.4928293?journalCode=jap
6. Inter-cluster distance dependence of electrical conduction in nanocluster assembled films of silver... https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4286756/
It is not quite a very recent book but is a pretty good semi-advanced text.
When you think about the electrical conductivity, it describes the response of the material to a voltage applied. Current will flow along the eletrical field, and this is being called a longitudinal conductivity, therefore. The applied voltage may be DC or AC in character, so both static and frequency dependent conductivity can be probed. It is a matter of technology to see how far up in frequency you can go up with that.
The optical conductivity is essentially the optical response of the material, written in a form such as to be analogous to the longitudinal conductivity expression. Since the electric field is perpendicular to the propagation direction it is said to be the transverse conductivity. This distinction is particularly relevant in optically anisotropic materials.
In practice, optical conductivity measurements will normally be conducted at higher frequencies than electrical conductivity measurements. By definition, however, they're both defined for all frequencies. In optics, knowing the complex refractive index, the complex dielectric function or the complex optical conductivity amounts to the same information.