Biomedical Engineering Reference
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k i D k SPP C k zi D " i k 0 ;
(2.64)
where k i , i D m; d, denotes the wavenumber in media i,or
k SPP D k 0 Œ" m " d =." m C " d / 1=2
D n SPP k 0 ;
(2.65)
where n SPP is the effective index of the SPP, and k 0 D !=c is the wavenumber
in vacuum. If the dielectric has positive permittivity, " d >0, and the metal has
a complex dielectric permittivity " m D " 0 m C i" 0 m where " 0 m < j " 0 m j , the SPP
wavenumber is imaginary:
k SPP D k 0 SPP C ik 0 SPP :
(2.66)
The real and imaginary parts of the SPP wavenumber are then
k 0 SPP D k 0 Œ" d " 0 m =." d C " 0 m / 1=2 ;
(2.67)
k 0 SPP D k 0 Œ" d " 0 m =." d C " 0 m / 3=2 ." 0 m =2" 0 m /:
(2.68)
The SPP wavelength, defined as SPP D 2=k 0 SPP , is thus found to be
s " d C " 0 m
" d " 0 m
SPP D 0
;
(2.69)
where 0 D 2c=!, value that satisfies the essential relation
SPP < d ;
(2.70)
where d D 0 " 1=2
d is the wavelength of the electromagnetic wave incident from
the dielectric side of the interface. The ratio SPP = d varies typically between 0.5
and 0.9 as a function of the dielectric permittivity of metal, excitation wavelength,
etc. Inequality ( 2.70 ) is the basic relation for subwavelength optics.
It is important also to estimate the capabilities of SPP to propagate along certain
distances. Thus, if the intensity of the electromagnetic radiation depends on x as
I.x/ Š I 0 exp. x=ı x / Š I 0 exp. 2 j k 0 SPP j x/;
(2.71)
the parameter ı x Š 1=2k 0 SPP can be regarded as the propagation length of the SPP.
Considering that the metal has low losses, we obtain
ı x D .1=k 0 /." 0 m =" 0 m /:
(2.72)
Equation ( 2.72 ) suggests that a long propagation length can be obtained in a metal
with a large real part of the electrical permittivity, which is negative at the working
wavelength, and a very low value of its imaginary part. In such case, the SPP
 
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