Evaluación del diagnóstico interno y externo

From here onward in this Chapter, I will consider the frequency dependent dielec-

tric function for the metal as

εm(ω) = Re[εm(ω)] +iIm[εm(ω)], (3.30)

and permittivity for the dielectric medium is real (negligible losses), i.e., εd≡Re[εd]. To accommodate losses associated with electron scattering the term Im[εm(ω)] has

been introduced. Using the real values forωand εdwith the condition|Re[εm(ω)]| Im[εm(ω)], one can rewrite the Eq. (3.19) as complex wave vector:

ky(ω) = Re[ky(ω)] +iIm[ky(ω)] (3.31)

in particular in more details manor,

ky(ω)≈ ω c εdRe[εm(ω)] εd+ Re[εm(ω)] ₁/2 +i ω c εdRe[εm(ω)] εd+ Re[εm(ω)] ₃/2 _Im[ε m(ω)] 2(Re[εm(ω)])2 . (3.32)

The Re[ky(ω)] determines the surface plasmon polariton wavelength, while the Im[ky(ω)]

contributes the losses associated with the damping of the surface plasmon polariton

as it propagates along the interface. As I mentioned earlier, for Re[ky(ω)] one needs

Re[εm(ω)] < 0 and |Re[εm(ω)]| > εd, which can be fulfilled in a metal and also in a doped semiconductor.

From Eq. (3.19), we can see that under conditions (3.25) and (3.26) the surface

plasmon polariton dispersion relation lies outside the light cone for the surrounding

dielectric, which means that

ky(ω)>

ω c

√

εd . (3.33)

The dispersion relation [see Fig. 3.3] approaches the light line √εd ω/c at small

ky, but remains larger than √εd ω/c. Therefore the surface plasmon polaritons are dark modes/nonradiative, i.e., they do not couple to light waves in the surrounding

dielectric and cannot be either emitted or excited by electromagnetic radiation from

the far zone.

From Eq. (3.19) it is obvious that surface plasmon polariton wave vector ky be-

comes very large compared to that of wave vector in free space when the real part of

the denominator becomes small and vanishes at a particular frequency, ωsp, which is

the so-called surface plasma frequency for flat surfaces. On the other hand this is the

upper limit of the plasmonic region, since at higher frequencies Eq. 3.23 is violated.

The ωsp satisfies the following condition,

0.2 0.6 1 1 2 3 Re[k]×10 ƫȦ (eV) ƫȦ (eV) 6 (cm )-1 (b) (a) 0.2 0.6 1 1 2 3 k×10y 6(cm )-1 _y

Figure 3.3: The relationship [see Eq. (3.19)] between ω vs. ky for silver (εm) and a dielectric (εd) interface. Where εd= 1 for the solid curve and εd= 10 for the dashed curve. (a) Calculated real part of the permittivity for silver. Here horizontal lines represent the surface plasmon frequencies. (b) Same plot as (a), but including the imaginary part of the permittivity for silver.

The surface plasmon polariton wavelength, λSP P, can be defined as:

λSP P = 2π Re[ky(ω)] ≈ λ Re[εm(ω)] +εd Re[εm(ω)]εd , (3.35)

where λ is the wavelength of the excitation light in vacuum.

Under the condition Re[εm(ωsp)] → −εd, the wavelength λSP P of the surface plasmon polariton modes becomes very small. In other words, under the condition

(3.34) the effective refractive index n_eff = λ/λSP P for surface plasmon polaritons

becomes very large. This may reduce the theoretical diffraction limit of resolution

with surface plasmon polaritons, which is determined byλSP/2. This value may reach

a scale of a few nanometers. Note that, a wave cannot be localized much shorter than

half of its wavelength λ/2 in vacuum. However, this has to be corrected as λ/2n for

medium with refractive indexn. From the practical point of view, the Ohmic losses in

a metal limit the wavelength and propagation length of the short wavelength surface

plasmon polaritons, therefore, this will severely limit the resolution.

The behavior of the surface plasmon polariton dispersion relation given in Eq. (3.19)

is illustrated in Fig. 3.3. The panel (a) shows the relations for an ideal metal, i.e.,

no losses are present. The solid curves represent the results for the silver-vacuum

interface, where εd = 1 and for silver, the experimental dielectric values of the bulk

silver have been used [17]. However, the imaginary part of the experimental dielec-

tric values of the bulk silver have been ignored. The results in Fig. 3.3(a) and the

previous discussion of the properties of surface plasmon polariton dispersion relation

are in good agreement. Here, the horizontal solid line represents the upper limit of

the plasmonic frequency, ω →ωsp, for the silver-vacuum interface ωsp ≈3.67 eV. In other words, at this frequency range the wave vector related to the surface plasmon

polariton becomes infinity, i.e., k → ∞. As a result, both the phase velocity, vp, and group velocity, vg, of surface plasmon polaritons tend to zero, which corresponds to

standing surface plasmons [87]. Note that the phase velocity and group velocity of

surface plasmon polaritons are defined as for any other wave by the following relations

[24], vp = Re ω k, vg = Re ∂ω ∂k (3.36)

The dashed curve in Fig. 3.3(a) is similar to the one we discussed above, i.e., silver-

panel (b) shows the relations for an actual metal, i.e., when losses are present, where

the dispersion relation is significantly different. In this case, the wave vector ky is

complex, therefore, Re[ky] has been shown in Fig. 3.3(b). As shown in panel (b) the

dispersion curve ends at some maximum value of k, with a cusp which is limited

by the dielectric losses at the surface plasmon frequency. The maximum value of k,

which is related to the dielectric losses, defines the minimum possible localization size

of the surface plasmon polaritons in a nanosystem.