2 Surface Plasmons for Chiral Sensing
31
2.2.1 SPPs at a Metal-Dielectric Interface
The propagation characteristics of SPPs and their associated effects on the SPR
have been widely discussed in many textbooks [41–43] and papers [42, 44–52]. To
find the analytical form of the SPP, one starts with solving Maxwell’s equations for
the simple case of a flat interface between two semi-infinite spaces (conductor and
dielectric), i.e. (2.2)–(2.5) with κ = 0. Although realistic systems do not involve
semi-infinite material regions, the results are directly applicable to metallic films of
finite thickness, because the field penetration inside the metallic region is usually
much smaller than the thickness of the finite metallic layer (i.e., the film is is seen
by the wave as having effectively infinite thickness).
Let us assume that the metallic layer extends along the x y-plane and SPP propagation occurs along the x-direction, with the evanescent field being confined in
the z-direction, as shown schematically in Fig. 2.2. Both regions are non-magnetic
(μ r = 1) and we may denote the relative permittivity r of the metal and dielectric
as m and d , respectively.
Due to the homogeneity of the geometry along the y-direction (∂/∂ y = 0),
Maxwell’s equations are decomposed in two sets, one involving only H x , E y , H z
components (TE mode) and one involving only E x , H y , E z components (TM mode).
As also shown in [42], the SPP is a TM mode (it cannot exist for TE polarization)
and, assuming propagation along the x-direction with propagation constant k SPP ,
H y ∼ e
ik SPP x , (∂/∂ x = ik SPP ), the y-component satisfies the wave equation in both
regions:
∂
2 H y
∂z 2 + (k
2
0 m,d − k
2
SPP )H y = 0,
(2.6)
where m,d denotes m or d , depending on the material region this equation refers
to. The solution for both sub-spaces is written in the form:
Fig. 2.2 Properties of SPPs at a metal-dielectric interface. a Geometry for SPP propagation
b Field components of SPP wave at a Au-H 2 O interface at 633 nm. The electric field components
are normalized with ζ the wave impedance in H 2 O and the penetration distance z is normalized
with λ SPP = 2π/Re(k SPP ), the SPP wavelength. The SPP is studied analytically in the configuration
shown in a, however the results are directly applicable to real SPR experiments with metallic films
of finite size, as the field penetration inside the metal is usually much smaller than its thickness
31
2.2.1 SPPs at a Metal-Dielectric Interface
The propagation characteristics of SPPs and their associated effects on the SPR
have been widely discussed in many textbooks [41–43] and papers [42, 44–52]. To
find the analytical form of the SPP, one starts with solving Maxwell’s equations for
the simple case of a flat interface between two semi-infinite spaces (conductor and
dielectric), i.e. (2.2)–(2.5) with κ = 0. Although realistic systems do not involve
semi-infinite material regions, the results are directly applicable to metallic films of
finite thickness, because the field penetration inside the metallic region is usually
much smaller than the thickness of the finite metallic layer (i.e., the film is is seen
by the wave as having effectively infinite thickness).
Let us assume that the metallic layer extends along the x y-plane and SPP propagation occurs along the x-direction, with the evanescent field being confined in
the z-direction, as shown schematically in Fig. 2.2. Both regions are non-magnetic
(μ r = 1) and we may denote the relative permittivity r of the metal and dielectric
as m and d , respectively.
Due to the homogeneity of the geometry along the y-direction (∂/∂ y = 0),
Maxwell’s equations are decomposed in two sets, one involving only H x , E y , H z
components (TE mode) and one involving only E x , H y , E z components (TM mode).
As also shown in [42], the SPP is a TM mode (it cannot exist for TE polarization)
and, assuming propagation along the x-direction with propagation constant k SPP ,
H y ∼ e
ik SPP x , (∂/∂ x = ik SPP ), the y-component satisfies the wave equation in both
regions:
∂
2 H y
∂z 2 + (k
2
0 m,d − k
2
SPP )H y = 0,
(2.6)
where m,d denotes m or d , depending on the material region this equation refers
to. The solution for both sub-spaces is written in the form:
Fig. 2.2 Properties of SPPs at a metal-dielectric interface. a Geometry for SPP propagation
b Field components of SPP wave at a Au-H 2 O interface at 633 nm. The electric field components
are normalized with ζ the wave impedance in H 2 O and the penetration distance z is normalized
with λ SPP = 2π/Re(k SPP ), the SPP wavelength. The SPP is studied analytically in the configuration
shown in a, however the results are directly applicable to real SPR experiments with metallic films
of finite size, as the field penetration inside the metal is usually much smaller than its thickness