Sensing of Biomolecules - Bionanoelectronics: Bioinquiring and Bioinspired Devices - page 90

propagation length is much longer than the optical wavelength. If the metal is silver,
the propagation length ranges between 50 and 300m in the wavelength interval
500-1,500 nm.
It is also necessary to determine the penetration depths of the electromagnetic
field in both dielectric and metal. These parameters are given by

k zd D k 0 " d =." d C " 0 m / 1=2 ; (2.73)

k zm D k 0 " 0 m =." d C " 0 m / 1=2

(2.74)

and are purely imaginary since ." d C " 0 m / is always negative in plasmonic devices.
Under these conditions, the field component perpendicular to the interface is
evanescent, its value decreasing with 1=e in the dielectric and metal after a distance

ı zi D 1= j k zi j ; (2.75)

with i D m; d. More precisely, the penetration depth of the SPP wave in the
dielectric is

ı zd D .1=k 0 / j ." 0 m C " d /=" d j

1=2 :

(2.76)

For example, the penetration depth in air at the air/silver interface in the wavelength
range 500-1,500 nm varies between 450 and 2,500 nm and increases as a function
of the excitation wavelength. This parameter is important for excitation and
manipulation of SPP modes, especially in sensing applications.
Analogously, in metal, the penetration depth,

ı zm D .1=k 0 / j ." 0 m C " d /=" 0 m j

1=2 ;

(2.77)

is about 26 nm for the air/silver interface in the wavelength range 500-1,600 nm
and is nearly independent of the wavelength. This penetration depth dictates the
thickness of metal deposition needed for SPP launching and propagation. Generally,
the penetration depth in metal is much smaller than in dielectric.
Further, by comparing the SPP dispersion relation,

k SPP =k 0 D Œ" d " m =." d C " m / 1=2

D n SPP ;

(2.78)

with that of a photon propagating in the dielectric (often free space),

k d =k 0 D ." d / 1=2 ; (2.79)

we obtain

k d <k SPP : (2.80)