**Significance **

Photonics is simply defined as the science of radiant energy. Specifically, it is the scientific study of the properties and applications of light and other forms of radiant energy, including the generation of energy and information processing. In current industrial regimes, the same can be defined as the design of material structures that support desired electromagnetic field distributions. Overtime, it has been established that creating fields in a material is contingent on the condition that the plane waves composing the fields are supported by the material. In addition, it is known that a particular electromagnetic plane wave is allowed only if it follows the Maxwell’s equations in *k*-space. Conventional materials correspond to dispersive, local, and isotropic material relationships.

Following increasing interest in metamaterials with extreme and unconventional properties for the growing markets of industries, marketplaces and security, it is imperative that a formula to solve the inverse problem of quartic photonics, or photonics of metamaterials, whose *k*-surfaces are quartic surfaces, be developed.

Recently, Georgia Southern University researchers: Thomas Mulkey, Dr. Jimmy Dillies and Dr. Maxim Durach developed a novel approach to engineer the effective parameters of metamaterials starting from the desired plane waves. They focused on demonstrating that one ought to specify the *k*-vectors and *E _{k}*,

*H*amplitudes of six arbitrary plane waves to fully define the required 36 material parameters of the material that could support the waves and, correspondingly, all the other fields possible in the bulk of the used material. Their work is currently published in the research journal,

_{k}*Optics Letters*.

In brief, the research method employed entailed formulation of the inverse problem by imagining a necessity case that required one to create a metamaterial that had the capability to propagate a set of desired plane waves whose *k*-vectors as well as the field were given. In their case, since they already had the 36 unknown material parameters, they progressed to specify the characteristics of six plane waves so as to form a complete system of equations. Lastly, the *k*-space with six points selected for the inverse problem; vector amplitudes of the electric and magnetic fields were then resolved.

The authors observed that by solving the resultant equations, they arrived at a quartic surface from which they were able to select six points. Following this success, the researchers applied their technique to show how it worked and noted that, for isotropic materials the* k*-surfaces were quadratic and corresponded to pairs of spheres.

In summary, the study by Georgia Southern University scientists presented the solution of an inverse problem of photonics that could enable one to find effective material parameters of a quartic or quadratic metamaterial that supports a set of desired plane waves with customizable *k*-vectors and field amplitudes. Altogether, they showed that their approach was exemplary efficient by studying the high-*k* limit of quartic metamaterials, extreme non-reciprocity, and metamaterials with complex bi-anisotropic parameters.

**Reference**

Thomas Mulkey, Jimmy Dillies, Maxim Durach. **Inverse problem of quartic photonics. **Volume 43, Number 6 / 15 March 2018 / Optics Letters.