Electromagnetic phenomena have excited researchers and scientists for centuries.
The contributions of Maxwell, Faraday, Coulomb, and Ampere in the field of electro- magnetic theory helped the scientific community to get a deeper insight into the fun- damentals of electromagnetic phenomena. The most crucial scientific breakthrough can be attributed to establishing the relation between electric and magnetic fields in the 19th century [127]. Further confinement of these relationships into physical laws by Faraday, Ampere, and Gauss paved the way for James Clerk Maxwell’s seminal paper, presented in 1865, which is now well known as the Maxwell’s equations of elec- tromagnetics [128]. It is incredible to know that even today, after 150 years, Maxwell’s equations are valid in the classical domain, relevant to the field of nanophotonics and metamaterials!
2.3 Relevant Theoretical Background
2.3.1 Maxwell’s Equations
The institution of Maxwell’s equations still remains one of the paramount achieve- ments in the field of modern physics. The set of four Maxwell’s equations is self- sufficient to define the dynamics of the electric and magnetic fields in media, either due to the presence of sources (such as charges or currents) or due to incident fields elsewhere. These equations have been extensively explored to study the light–matter interactions over a whole range of electromagnetic spectrum. However, Maxwell’s equations are valid in the classical domain and cannot accurately predict quantum phenomena. The Maxwell’s equations determining the relationship between the elec- tric, E, magnetic, H, electric displacement, D, and the magnetic induction, B, fields, and are given by:
∇ ·D=ρ (2.1a)
∇ ·B= 0 (2.1b)
∇ ×E=−∂B
∂t (2.1c)
∇ ×H= ∂D
∂t +J (2.1d)
whereρis the free electric charge density andJis the electric current density. These four equations are sufficient to describe electric and magnetic fields in all space and time. However, while considering the macroscopic media where every atom acts as a source, it is impractical to solve the microscopic Maxwell’s equations. More often than not, we are much more interested in examining the response of a collective media rather than the response from an individual atom. For the case of microscopic media, EandHare now representing the averaged fields. TheDandBfields are derived from the relation given by:
D =0E+P (2.2a)
B =µ0H (2.2b)
wherePbeing the polarization response of the medium,0andµ0is the permittivity and permeability of free space, respectively. Since for the case of vacuum there is no polarization response, the above equation becomesD = 0EandB = µ0H. We would like to emphasize that we are considering linear, isotropic, and non-magnetic media for which the polarization is given by:
P=0χeE (2.3)
where χe is the electric susceptibility of the medium. The above equations can be used to rewrite Maxwell’s equation to derive the wave equation given by:
∇2E=µ00r∂2E
∂t2 (2.4a)
∇2H=µ00r∂2H
∂t2 (2.4b)
One of these wave equations can be used to solve for either the electric or mag- netic field for a particular region, and the other can be obtained by utilizing Maxwell’s equation. While using frequency domain solvers, we assume the time dependence to be harmonic, such that∂/∂t→ −iω. In that case, the wave-equation becomes the well- known Helmholtz equation. On the contrary, the time domain solvers use Maxwell’s curl equations.
2.3 Relevant Theoretical Background
2.3.2 Boundary Conditions
In most practical situations, often one has to solve equations for electromagnetic wave propagation across different media. However, to establish a relation between wave equations in two different media, one needs to solve for boundary conditions at the interface. The Stokes and divergence theorems can be used to find out these boundary conditions using Maxwell’s equations, as given below:
(D2−D1)·n=ρs (2.5a)
(B2−B1)·n= 0 (2.5b)
(E2−E1)×n= 0 (2.5c)
(H2−H1)×n=Js (2.5d)
where ρs and Js are the surface charge density and the surface current density, re- spectively, at the interface between the two media (1 and 2) and n being the surface normal [129]. It is important to note that surface charges and currents are absent as the charge carriers’ positions are fixed within dielectric media.
2.3.3 Metals and Dispersive Media
The Lorentz oscillator model considers atoms inside metal to be composed of nu- cleus and electron where the mass of the electron is very small compared to the nu- cleus. So, one can assume the system to be a harmonic oscillator with the nucleus having infinite mass being fixed and an electron being attached to the nucleus via a spring. The equation of motion is given by:
md2x(t)
dt2 +γmdx(t)
dt +Kx(t) = −eE(t) (2.6) where the second term represents the resultant damping force, the third term rep-
resents the restoring force, and the fourth term represents the driving forces. Also the polarizationPas a function of Electric fieldEis represented by:
P(ω) = ne2 m
E(ω)
K
m −iγω−ω2 =0χeE(ω) (2.7) Now, one can assume the electrons in the metals are free (unbounded) i.e. there is no restoring force, K = 0. This is called the Drude model for metals, and the Drude permittivity can be extracted from the expression shown below:
Drude = 1− ωp2
ω2+iωγ (2.8)
where the plasma frequency is given by ωp = p
ne2/m0. The Drude model still gives an accurate electromagnetic response of metals despite neglecting both the elec- tron–electron and electro–ion interactions. The modified form of Drude model is given by:
Drude =∞− ωp2
ω2+iωγ (2.9)
where ωp, ∞and γ represent the plasma frequency, high-frequency dielectric con- stant, and collision frequency, respectively. One can use these parameters to fit the experimental data of novel plasmonic metals such as gold and silver, as shown in Fig. 2.11. However, the interband transitions limit the validity of the Drude model at visible and higher frequencies [18]. For the case of dielectrics, their permittivity can be derived from the Sellmeier equation, which gives an empirical relationship between refractive index and wavelength for a particular medium.
To summarize, Maxwell’s equation and boundary conditions can be applied to metal and dispersive media for designing nanophotonics and metamaterials based tunable optical devices for smart window design and solar energy harvesting. We now move to the analytical modeling section to discuss the different relevant theoreti-