Calculation of Electromagnetic Force Density in Artificial Materials by an Effective Medium Theory

In the long wavelength limit, light propagation in artificial electromagnetic materials, such as photonic crystals, metamaterials, and colloidal suspensions, can be treated very efficiently by the effective medium theory, where the complicated scattering problem of numerous sub- units is now simplified to that of a homogeneous material.

We calculate the continuous-wave laser induced time-averaged electromagnetic force density inside artificial materials. The involvement of a substantial amount of sub-wavelength sized sub-units calls for an effective medium theory. However, there are fierce controversies on what is the appropriate stress tensor for effective media. Neither the Maxwell stress tensor, nor its trivial generalization of replacing the free space constitutive parameters with that of the effective medium, works.

In 2D photonic crystals, by incorporating electrostriction and magnetostriction, we correctly computed electromagnetic force density using effective medium theory. Surprisingly, in addition to the effective permittivity eff  and permeability eff  , one also needs /eff iku   and /eff iku   , where iku is the strain. This literally means that materials characterized by the same permittivity and permeability can be subjected to different electromagnetic force density.

The proposed works here are to generalize our 2D results to 3D, to different lattices (SC, FCC, fluidic, etc.), and to different materials (magnetic, dissipative, dispersive, etc.). We will then investigate how the electromagnetic force density affects the functionality of artificial materials.

The consequence of incorrectly calculating electromagnetic force density is more serious than it first appears to be. Aside from lacking theoretical guidance, it is very difficult to judge in experimental situations whether an effect is due to the optical force or other relevant forces. This highlights the importance of theoretical calculation.

High-Q resonators will also be considered. The resonant electromagnetic force density is expected to increase proportionally with the stored energy. For small deformation, the force per unit stored energy is proportional to the frequency shift induced by the deformation. This in turn can be calculated by an advanced perturbation theory for quasi- normal modes.

Our work will open up an unexplored aspect of artificial materials. In addition to its fundamental interest, applications can be found in, but not limited to, cell stretching, density redistribution of colloid, deformation of high-Q cavities, and mutual forces in strongly coupled metamaterial resonators. Last but not least, effective medium theory would not be complete, if the electromagnetic force density cannot be computed from it. It is our task to present a usable approach to compute it.