**3.4 Surface integral methods**

Surface integral methods restrict an electromagnetic scattering problem with open boundary conditions to the surface limits of the substance. As a consequence, commonly used methods to piecewise homogeneous media, such as the boundary element method (BEM) [32] and the SIE method [20, 33], are best suited. The SIE process, like the FE and VIE methods, will specifically treat dispersive materials as a frequency-domain method [34].

## **3.5 Other methods**

The hybrid differential–integral approaches are particularly useful in dealing with inhomogeneous scatters or anisotropic [27], and are an alternative to the integral and differential methods to solve Maxwell's equations. Brief overview of other methods used for simulation of nanophotonics includes;

## *3.5.1 Finite integration technique*

The finite integration technique (FIT) is an integral form discretization scheme of Maxwell's Equations [35]. Unlike integral and differential approaches, which may cover a wide variety of structures, few other methodologies are considered to be very effective for unique conditions and geometries since they depend on electromagnetic field expansions on basis functions with specific symmetries [23].

### *3.5.2 T-matrix system*

The incident and scattered electromagnetic fields are extended on a series of spherical basis functions in the T-matrix system, with boundary conditions imposed at the interfaces of the different materials. This approach excels at measuring the scattering of spherical and/or quasi-spherical particles [36] and extended to plasmonics, layered particles as well as particle-substrate interactions with applications in sensing, plasmonic trapping, and SERS [37, 38].

## *3.5.3 Multiple multipole approaches*

The generalized multipole method is a semi-analytical technique, also known as the multiple-multipole approach that is used to extend the electromagnetic field in multipoles, allowing it to handle a wide variety of symmetries [39, 40]. In this method, only the domain boundaries are discretized and no integral is numerically solved. Multiple elastic scattering of multipole expansions disintegrates dispersed

*Nanophotonics: Fundamentals, Challenges, Future Prospects and Applied Applications DOI: http://dx.doi.org/10.5772/intechopen.98601*

fields to multipoles with respect to centers near each cluster object, and this occurs until cluster convergence [41, 42].
