14
2 Electrodynamics at Interface
2.1 Electromagnetic Fields at Interface
In the macroscopic description of electromagnetic fields, the interface region
between two bulk media is often modeled with three layers, consisting of two bulk
media and the interface, illustrated in Fig. 2.1. The two bulk media are assumed
to be centrosymmetric, and their optical properties are represented with different
dielectric constants, ε α (ω) and ε β (ω). (The dispersion with the frequency ω is taken
into account.) The interface layer represents a transient region between the two bulk
media, and accordingly its dielectric constant is given with a phenomenological
parameter ε (ω). The thickness of the transient region is usually much smaller
than the typical scale of light wavelength. We discuss spatial configuration of
electromagnetic fields near the interface on the basis of the three-layer model in
Fig. 2.1.
In this chapter, we assume that the nonlinear source polarization P (2) is generated
only in the interface layer. The oscillating source polarization P (2) emits the sum
frequency signal of electromagnetic field. This sum frequency field is related to
the source polarization by solving the Maxwell equations under proper boundary
conditions in Sect. 2.1 [2]. In Sect. 2.2, the nonlinear source polarization P (2) is
derived from the incident visible and infrared electromagnetic fields with nonlinear
susceptibility of the interface χ (2) . The combined discussion of the two subsections
describes the whole SFG process that the incident lights induce the nonlinear
polarization (in Sect. 2.2), which in turn generates the sum frequency signal (in
Sect. 2.1). We also note that the following discussion will be expanded in Chap. 7 to
incorporate quadrupole contributions in the bulk region.
2.1.1 Maxwell Equations
First we present the Maxwell equations to describe the electromagnetic fields
in Fig. 2.1. The electromagnetic fields are generated from the nonlinear source
Fig. 2.1 Spatial
configuration of lights near
the interface. The two
incident fields of visible
(green) and infrared (red)
frequencies and the sum
frequency signals (purple) of
reflected and transmitted
directions are illustrated
0
z
'
x
k ( )
2
k ( )
1
k ( )
k ( )
p p s
1
2
I
I
2 Electrodynamics at Interface
2.1 Electromagnetic Fields at Interface
In the macroscopic description of electromagnetic fields, the interface region
between two bulk media is often modeled with three layers, consisting of two bulk
media and the interface, illustrated in Fig. 2.1. The two bulk media are assumed
to be centrosymmetric, and their optical properties are represented with different
dielectric constants, ε α (ω) and ε β (ω). (The dispersion with the frequency ω is taken
into account.) The interface layer represents a transient region between the two bulk
media, and accordingly its dielectric constant is given with a phenomenological
parameter ε (ω). The thickness of the transient region is usually much smaller
than the typical scale of light wavelength. We discuss spatial configuration of
electromagnetic fields near the interface on the basis of the three-layer model in
Fig. 2.1.
In this chapter, we assume that the nonlinear source polarization P (2) is generated
only in the interface layer. The oscillating source polarization P (2) emits the sum
frequency signal of electromagnetic field. This sum frequency field is related to
the source polarization by solving the Maxwell equations under proper boundary
conditions in Sect. 2.1 [2]. In Sect. 2.2, the nonlinear source polarization P (2) is
derived from the incident visible and infrared electromagnetic fields with nonlinear
susceptibility of the interface χ (2) . The combined discussion of the two subsections
describes the whole SFG process that the incident lights induce the nonlinear
polarization (in Sect. 2.2), which in turn generates the sum frequency signal (in
Sect. 2.1). We also note that the following discussion will be expanded in Chap. 7 to
incorporate quadrupole contributions in the bulk region.
2.1.1 Maxwell Equations
First we present the Maxwell equations to describe the electromagnetic fields
in Fig. 2.1. The electromagnetic fields are generated from the nonlinear source
Fig. 2.1 Spatial
configuration of lights near
the interface. The two
incident fields of visible
(green) and infrared (red)
frequencies and the sum
frequency signals (purple) of
reflected and transmitted
directions are illustrated
0
z
'
x
k ( )
2
k ( )
1
k ( )
k ( )
p p s
1
2
I
I