2.1 Electromagnetic Fields at Interface
15
polarization P (2) (r, t). This source polarization is equivalent to the following
charge density ρ(r, t) and current density j (r, t) in the Maxwell equations,
ρ(r, t) = −∇ · P
(2) (r, t),
j (r, t) =
∂P (2) (r, t)
∂t
(2.1)
We assume that no other source of charge or current is present in Fig. 2.1. Then the
Maxwell equations are given in the cgs Gauss unit system [1, 3], 1
(a)
∇ · D = 4πρ = −4π ∇ · P
(2) ,
(2.2)
(b)
∇ × E +
1
c
∂B
∂t
= 0,
(2.3)
(c)
∇ · B = 0,
(2.4)
(d)
∇ × H −
1
c
∂D
∂t
=
4π
c
j =
4π
c
∂P (2)
∂t
,
(2.5)
where c is the light velocity in vacuo.
The nonlinear source polarization P (2) is distributed in the interface layer at
around z = 0 in Fig. 2.1, where the z axis is normal to the interface. In the
macroscopic description of electromagnetic fields by Eqs. (2.2), (2.3), (2.4), and
(2.5), the thickness of the interface layer is considered to be much thinner than the
order of light wavelength (∼100 nm), and consequently the spatial distribution of
P (2) can be represented with a delta function,
P
(2) (r, t) = P
S (x, y, t) δ(z).
(2.6)
Equations (2.2), (2.3), (2.4), (2.5) and (2.6) with proper boundary conditions (at z =
0 and z → ±∞) determine the electromagnetic fields, on condition that the surface
polarization P S (x, y, t) is given. Note that Eq. (2.6) using the delta function is a
macroscopic description of the interface polarization. The microscopic distribution
of the induced polarization along the z axis will be treated in Chap. 7.
2.1.2 Boundary Conditions at Interface
The boundary conditions for electromagnetic fields between two different dielectric
media are derived with the help of the Gauss and Stokes theorems [1, 3]. We derive
the boundary conditions for B, D, E, and H at the interface z = 0 in Fig. 2.1
by taking account of the source polarization P s at z = 0. The present boundary
conditions at z = 0 are somewhat different from the conventional ones due to the
source polarization [2].
1 The cgs Gauss units are used throughout for the electrodynamics in this chapter, except for
otherwise noted.
15
polarization P (2) (r, t). This source polarization is equivalent to the following
charge density ρ(r, t) and current density j (r, t) in the Maxwell equations,
ρ(r, t) = −∇ · P
(2) (r, t),
j (r, t) =
∂P (2) (r, t)
∂t
(2.1)
We assume that no other source of charge or current is present in Fig. 2.1. Then the
Maxwell equations are given in the cgs Gauss unit system [1, 3], 1
(a)
∇ · D = 4πρ = −4π ∇ · P
(2) ,
(2.2)
(b)
∇ × E +
1
c
∂B
∂t
= 0,
(2.3)
(c)
∇ · B = 0,
(2.4)
(d)
∇ × H −
1
c
∂D
∂t
=
4π
c
j =
4π
c
∂P (2)
∂t
,
(2.5)
where c is the light velocity in vacuo.
The nonlinear source polarization P (2) is distributed in the interface layer at
around z = 0 in Fig. 2.1, where the z axis is normal to the interface. In the
macroscopic description of electromagnetic fields by Eqs. (2.2), (2.3), (2.4), and
(2.5), the thickness of the interface layer is considered to be much thinner than the
order of light wavelength (∼100 nm), and consequently the spatial distribution of
P (2) can be represented with a delta function,
P
(2) (r, t) = P
S (x, y, t) δ(z).
(2.6)
Equations (2.2), (2.3), (2.4), (2.5) and (2.6) with proper boundary conditions (at z =
0 and z → ±∞) determine the electromagnetic fields, on condition that the surface
polarization P S (x, y, t) is given. Note that Eq. (2.6) using the delta function is a
macroscopic description of the interface polarization. The microscopic distribution
of the induced polarization along the z axis will be treated in Chap. 7.
2.1.2 Boundary Conditions at Interface
The boundary conditions for electromagnetic fields between two different dielectric
media are derived with the help of the Gauss and Stokes theorems [1, 3]. We derive
the boundary conditions for B, D, E, and H at the interface z = 0 in Fig. 2.1
by taking account of the source polarization P s at z = 0. The present boundary
conditions at z = 0 are somewhat different from the conventional ones due to the
source polarization [2].
1 The cgs Gauss units are used throughout for the electrodynamics in this chapter, except for
otherwise noted.