Sensing of Biomolecules - Bionanoelectronics: Bioinquiring and Bioinspired Devices - page 91

Fig. 2.30 The SPP
wavenumber mismatch in the
dispersion relation

w

light in dielectric

SPP

k

From this relation, it follows that there is a wavenumber mismatch between SPP
and the incoming electromagnetic wave excitation (see Fig. 2.30 ), so that SPP is a
nonradiative wave that propagates on a smooth interface.
The matching between the incoming electromagnetic wave moment and that of
the SPP is achieved by several methods. One of these methods relies on a prism,
which increases the moment of incoming photons, while others change the surface
of the metallic medium from a smooth one into a nonsmooth surface, with gratings,
perforations, or holes.
If the incoming optical field illuminates a half-cylindrical prism with permittivity
" p , its wavenumber is modified from k d to k 0 ." p / 1=2 , being thus enhanced with
a factor of ." p =" d / 1=2 . It then becomes possible for the incoming field with an
excitation frequency ! exc to match the SPP wavevector component along the
interface for a specific incident angle ' 0 for which

" 1= p ! exc sin ' 0 D ck SPP : (2.81)

SPPs can only be excited if the matching condition ( 2.81 ) is satisfied. The incident
angle for which this requirement is met and SPP propagation is enabled is called
resonant angle. At resonance, the reflectivity of the entire structure is zero, since the
SPP is entirely localized along the interface.
Based on this principle, there are two main matching configurations named Otto
and Kretschmann configurations, respectively, which make use of prisms that can
have a variety of shapes besides the half cylinder. These two configurations for
prism-SPP matching are illustrated in Fig. 2.31 .
If we use a metallic grating with period ƒ embedded in a dielectric with
permittivity " d to excite SPPs by exciting the grating with an electromagnetic field
incident under an angle ' 0 , as shown in Fig. 2.32 , the matching condition becomes

k SPP D k d sin ' 0 C k grat D k 0 " 1=2
d

sin ' 0 ˙ m2=ƒ; (2.82)