Numerical Methods in Photonics

2	Maxwell’s Equations

2.1    NOTATION

This book employs the following notations:

Object	Notation	Example
Vector in 3D space	Bold	r, E
Unit vector	Bold with hat	x̂, ŷ, ẑ
Vector of expansion coefficients	Plain, single bar	c̅
Matrix	Plain, double bar	O̅̅
Dyadic tensor	Bold	G
Scalar (dot, inner) product	Single dot	x̂ · E
Vector (cross) product	Single cross	x̂ × E
Tensor (dyadic, outer) product or dyad	No dot	RR

Furthermore, we introduce the shortened designation of partial differential operators:

${\partial }_x\equiv \frac{\partial }{\partial x},{\partial }_y\equiv \frac{\partial }{\partial y},{\partial }_z\equiv \frac{\partial }{\partial z},{\partial }_t\equiv \frac{\partial }{\partial t}.$

(2.1)

2.2    MAXWELL’S EQUATIONS

The basic differential form of Maxwell’s equations reads [1,2]

$\nabla \times E(r,t)=-{\partial }_{t}B\left(r,t\right),$

(2.2)

$\nabla \times H\left(r,t\right)={\partial }_{t}D\left(r,t\right)+J\left(r,t\right),$

(2.3)

$\nabla .D\left(r,t\right)=\text{ρ}($