Nanofocusing of Surface Plasmons at the Apex of Metallic Tips and at the Sharp Metallic Wedges. Importance of Electric Field Singularity

Nanofocusing of light is localization of electromagnetic energy in regions with dimensions that are significantly smaller than the wavelength of visible light (of the order of one nanometer). This is one of the central problems of modern near-field optical microscopy that takes the resolution of optical imaging beyond the Raleigh’s diffraction limit for common optical instruments [Zayats (2003), Pohl (1984), Novotny (1994), Bouhelier (2003), Keilmann (1999), Frey (2002), Stockman (2004), Kawata (2001), Naber (2002), Babadjanyan (2000), Nerkararyan (2006), Novotny (1995), Mehtani (2006), Anderson (2006)]. It is also important for the development of new optical sensors and delivery of strongly localized photons to tested molecules and atoms (for local spectroscopic measurements [Mehtani (2006), Anderson (2006), Kneipp (1997), Pettinger (2004), Ichimura (2004), Nie (1997), Hillenbrand (2002)]). Nanofocusing is also one of the major tools for efficient delivery of light energy into subwavelength waveguides, interconnectors, and nanooptical devices [Gramotnev (2005)]. There are two phenomena of exceptional importance which make it possible nanofocusing. The first is the phenomenon of propagation with small attenuation of electromagnetic energy of light along metal-vacuum or metal-dielectric boundaries. This propagation exists in the form of strictly localized electromagnetic wave which rapidly decreases in the directions perpendicular to the boundary. Remembering the quantum character of the surface wave they say about surface plasmons and surface plasmon polaritons (SPPs) as quasi-particles associated with the wave. The dispersion of the surface wave has the following important feature [Economou (1969), Barnes (2006)]: the wavelength tends to zero when the frequency of the SPPs tends to some critical (cut off) frequency above which the SPPs cannot propagate. For SPPs propagating along metal-vacuum plane boundary this critical frequency is equal to 2 p ω (we use Drude model without absorption in metal). For spherical boundary this critical frequency [Bohren, Huffman (1983)] is equal to 3 p ω . So, the SPP critical frequency depends on the form of the boundary. By changing the frequency of SPPs it is possible to decrease the wavelength of the SPPs to the values substantially


Introduction
Nanofocusing of light is localization of electromagnetic energy in regions with dimensions that are significantly smaller than the wavelength of visible light (of the order of one nanometer). This is one of the central problems of modern near-field optical microscopy that takes the resolution of optical imaging beyond the Raleigh's diffraction limit for common optical instruments [Zayats (2003), Pohl (1984), Novotny (1994), Bouhelier (2003), Keilmann (1999), Frey (2002), Stockman (2004), Kawata (2001), Naber (2002, Babadjanyan (2000), , Novotny (1995), Mehtani (2006), Anderson (2006)]. It is also important for the development of new optical sensors and delivery of strongly localized photons to tested molecules and atoms (for local spectroscopic measurements [Mehtani (2006), Anderson (2006), Kneipp (1997), Pettinger (2004), Ichimura (2004), Nie (1997), Hillenbrand (2002)]). Nanofocusing is also one of the major tools for efficient delivery of light energy into subwavelength waveguides, interconnectors, and nanooptical devices [Gramotnev (2005)]. There are two phenomena of exceptional importance which make it possible nanofocusing. The first is the phenomenon of propagation with small attenuation of electromagnetic energy of light along metal-vacuum or metal-dielectric boundaries. This propagation exists in the form of strictly localized electromagnetic wave which rapidly decreases in the directions perpendicular to the boundary. Remembering the quantum character of the surface wave they say about surface plasmons and surface plasmon polaritons (SPPs) as quasi-particles associated with the wave. The dispersion of the surface wave has the following important feature [Economou (1969), Barnes (2006)]: the wavelength tends to zero when the frequency of the SPPs tends to some critical (cut off) frequency above which the SPPs cannot propagate. For SPPs propagating along metal-vacuum plane boundary this critical frequency is equal to 2 p ω (we use Drude model without absorption in metal). For spherical boundary this critical frequency [Bohren, Huffman (1983)] is equal to 3 p ω . So, the SPP critical frequency depends on the form of the boundary. By changing the frequency of SPPs it is possible to decrease the wavelength of the SPPs to the values substantially www.intechopen.com Wave Propagation 194 smaller than the wavelength of visible light in vacuum and use the SPPs for trivial focusing by creation a converging wave [Bezus (2010)]. In this case there is no breaking the diffraction Raleigh's limitation and the energy of the wave is focused into the region with dimensions of the order of wavelength of the SPPs. These dimensions may be substantially smaller than the wavelength of light in vacuum corresponding to the same frequency. As a result we have nanofocusing of light energy. The second phenomenon is electrostatic electric field strengthening at the apex of conducting tip (at the apex of geometrically ideal tip there is electrostatic field singularity, i.e. the electrostatic field tends to infinity at the apex). This phenomenon exists not only in electrostatics. For alternating electric field in the region with the apex of the tip at the center (with dimensions smaller than wavelength) the quasi-static approximation is applicable and there is a singularity of the time varying electric field (if the frequency is low enough as we will see below). Surely, at the apex of a real tip there is no singularity of electrostatic field since the apex is rounded. But near the apex the electric field increases in accordance with power (negative) law of the singularity and the electric field saturation at the apex is defined by the radius of the apex. This radius may be very small, of the order of atomic size. Nanofocusing of SPPs at the apex of metal tip is considered in [Stockman (2004), De Angelis (2010)]. SPPs are created symmetrically at the basement of the tip and this surface wave converges along the surface of the metal to the tip's apex where surface wave energy is focused. But conditions for existence of electric field singularity are considered in [Stockman (2004), De Angelis (2010)] only for very sharp conical metal tips with small angle at the apex. In [Petrin (2010)] it is shown that due to frequency dependence of metal permittivity in optic frequency range the singularity of electric field at the tip's of not very sharp apex may exist in different forms. The goal of the present chapter is investigation of the factors defined the type of singular concentration of electromagnetic energy at the geometrically singular metallic elements (such as apexes and edges) as one of the important condition for optimal nanofocusing. In the next sections of this chapter we discuss the following: electric field singularities in the vicinity of metallic tip's apex immersed into a uniform dielectric medium; electric field singularities in the vicinity of metallic tip's apex touched a dielectric plate; electric field singularities in the vicinity of edge of metallic wedge.

Nanofocusing of surface plasmons at the apex of metallic probe microtip.
Conditions for electric field singularity at the apex of microtip immersed into a uniform dielectric medium.
In this section of the chapter we focus our attention on finding the condition for electric field singularity of focused SPP electric field at the apex of a metal tip which is used as a probe in a uniform dielectric medium. As we have discussed above this singularity is an important feature of optimal SPP nanofocusing.

Condition for electric field singularity at the apex
Consider the cone surface of metal tip (see Fig. 1). Let calculate the electric field distribution near the tip's apex. In spherical coordinates with origin O at the apex and polar angle θ (see Fig. 1), an axially symmetric potential Ψ obeys Laplace's equation (we are looking for singular solutions, so in the vicinity of the apex the quasistatic approximation for electric field is applicable)

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Representing the solution as ( )  [Angot (1962)], the field components outside and inside the cone are, respectively,  Er α − ∼ ) will give singular solutions. So, we have interest only in these solutions. To find the lower boundary of the roots of interest it is necessary to remember that in the vicinity of a singular point the density of electric field power must be integrable value. It means that in the limit 0 r → the electric field must increase slower than 32 r − and, therefore, the density of electric field power must increase slower than 3 r − . It gives us the lower boundary for the roots. I.e. the appropriate roots must satisfy the inequity ). It follows that there are two essentially different types of the SPP nanofocusing. First type takes place when cr ω ω < and is characterized by the electric field singularity without oscillations along radius in the vicinity of the apex. Second type takes place when 2 cr p ωω ω << and is characterized by the electric field singularity with oscillations along radius and the wave number of these oscillations tends to infinity when 0 r → . Note, that in the second type of nanofocusing ( ) if there is no losses in metal. The integral of the total electric power in the vicinity of the apex diverges. But if we take into account the metal losses considering the Drude's formula with losses Note, that here there is an essential difference of the quasistatic approach to the problem from the pure static one. In the static problem physical sense have electric potentials with ( ) Re 0 α > since only under this condition the potential on the surface of conducting cone may be made constant. In the quasistatic problem the equipotentiality of the metallic cone is not a necessary condition since the metal of the cone is considered as a dielectric with frequency dependent permittivity. The dielectric surface is not obligatory equipotential. 1c o s 1c o s 1 2,1 2,1, 3 2,3 2,2, 22 1 1 cos 1 cos 12,12,1, 32,32,2, 22 In Fig. 3 the frequency 2 p ω is shown. SPPs can not exist above this frequency on the plane boundary metal-vacuum. We may see that when 90 γ →° the critical frequency tends to 2 p ω . It is absolutely unexpectable that the utmost frequency of SPP existence arises in quasistatic formulation of the problem on electric field singularity finding.  the apex was used. The wavelength of laser used in the experiment was equal to 0 532nm λ = . The excited SPPs propagated along the surface of the tip to the focus at the apex. But from the above results we may conclude that the angle is not optimal for the best focusing. Indeed, from Fig. 4 we may see that the critical angle for 0 532nm λ = is 24.7 cr γ ≈°.
So, there is electric field singularity of the second type near the apex of the tip with 12 γ ≈°.
But it is remained the following unresolved problem yet. If the metal microtip is used as a SPP probe of the surface properties it is obvious that the value of the microtip's angle depends on the dielectric constant of the surface which the microtip's apex is touched and on the dielectric constant of the surrounding medium. It is clear that it is necessary to consider the influence of the probed surface on the electric field singularity at the microtip's apex and therefore on nanofocusing. The next section of this chapter gives the answer to the following question: what happens with the considered phenomenon when the apex of metal cone touches a dielectric plane plate?

Nanofocusing of surface plasmons at the apex of metallic microtip probe touching a dielectric plane. Conditions for electric field singularity at the apex of a microtip immersed into uniform dielectric medium and touched a probed dielectric plane.
In experiments the apex of the microtip may touch the surface of a probed dielectric plane plate (see Fig. 5). In this connection the following question arises: how the dielectric constant of the plate affects the electric field singularity index at the apex? By another words, if the microtip used as a concentrator of SPPs for Raman's spectroscopy of a dielectric surface [De Angelis (2010)] how the investigated material affects the electric field singularity at the apex and therefore the efficiency of SPPs focusing?

Method of electric field singularity finding at the apex of metal microtip touching a dielectric plane
Consider the cone metal microtip touching a dielectric plane plate (see Fig. 5). The space between the metal tip and dielectric plane is filled by a uniform dielectric. The dielectric constants of the metal, the dielectric plane and the filling uniform dielectric are equal to m ε , p ε and d ε respectively. The frequency of SPP wave which is focused at the apex is equal to ω . As in the previous part, in the quasistatic formulation of the problem the electric field potential must obeys Laplace's equation and normal and tangential components of electric field must obey the following boundary conditions: at the boundary of metal cone and free space Therefore, the field components in the considering three regions may be written by the following expressions: tangential components:  At the boundaries of the considering regions (at θ πγ = − and 2 θ π = ) the mentioned above boundary conditions for tangential and normal components of electric field must be satisfied. Substituting into the boundary conditions the expressions for the field components and taking into account that ( ) cos cos π γγ −− = we have the following four equations: A nontrivial solution of the system exists when the determinant is equal to zero, Eq.
(3) is identical to the corresponding Eq.(1) for the geometry without dielectric plate. In this case the cone tip with dielectric constant m ε is immersed into the uniform dielectric with constant d ε . This problem has been solved for example in [Petrin (2007)].
The minimal root α of Eq.(2) (which corresponds to the physically correct solution) defines the character of electric field singularity in the vicinity of the cone apex. From Eq.
(2) it follows that α is a function of three independent variables: the angle γ and the ratios of dielectric constants md ε ε and p d ε ε , i.e.

( )
,, mdp d α αγε ε ε ε give the singular electric field. So, we will be interested by the solutions of Eq.(2) in the interval ( ) Re 1 α < . To define the lower boundary of the solution's interval it is necessary to remind that in the vicinity of the apex the electric field density must be integrable. It means that the electric field and density must increase slower than 32 r − and 3 r − respectively when 0 r → . So, the lower boundary of the roots interval is equal to 12 and the total roots interval of interest is Using the same approach as in the case of Fig. 3 it were calculated numerically (see Fig. 6) the dependences of the critical frequency cr ω (normalized on the plasma frequency of the metal) on the cone angle γ for 1 d ε = and several values of p ε .
These dependences may be found analytically.  absolutely unexpected that the utmost maximal frequency of SPP's existence arises in the quastatic statement of the singularity existence problem.

Application of the theory to a silver tip
So, as in the section 2.2, we see that if the working frequency is fixed, then there are two different types of singularity. In this case there is a critical angle cr γ which separates the regime with the first type of singularity from the regime with the second type of singularity. we neglect by losses in silver. If the angle of the cone γ is more than cr γ , then the singularity at the apex is of the first type. If the angle γ is smaller than cr γ , then the singularity is of the second type.

Nanofocusing of surface plasmons at the edge of metallic wedge.
Conditions for electric field singularity existence at the edge immersed into a uniform dielectric medium.
In this part of the chapter we focus our attention on finding the condition for electric field singularity of focused SPP electric field at the edge of a metal wedge immersed into a uniform dielectric medium. SPP nanofocusing at the apex of microtip (considered in the previous sections) corresponds (based on the analogy with conventional optics) to the focusing by spherical lens. Thus, SPP nanofocusing at the edge of microwedge corresponds to the focusing by cylindrical lens at the edge [Gramotnev (2007)]. The main advantage of the wedge SPP waveguide in nanoscale is the localization of plasmon wave energy in substantially smaller volume [Moreno (2008)] due to the electric field singularity at the edge of the microwedge. This advantage is fundamentally important for miniaturization of optical computing devices which have principally greater data processing rates in comparison with today state of the art electronic components [Ogawa (2008), Bozhevolnyi (2006)].
As it will be shown below the electric field singularity at the edge of the microwedge may be of two types due to frequency dependence of dielectric constant of metal in optical frequency range. This phenomenon is analogues to the same phenomenon for microtips which was considered in the previous parts of this chapter. The investigation of these types of electric field singularities at the edge of metal microwedge is the goal of this chapter section.

Condition for electric field singularity at the edge of metallic wedge
Let consider the metal microwedge (see Fig. 8) with dielectric constant of the metal m ε . The frequency of the SPP is ω . The wedge is immersed into a medium with dielectric costant d ε .  [Landau, Lifshitz (1982)] where α is a constant; θ is the angle from the axis OX; r is the radial coordinate from the origin O. Taking into account that the electric potential in the metal and dielectric depends on r as the same power we may write the following expressions for potential in metal and dielectric respectively cos( ) Using the above expressions for electric potential in the two media the boundary conditions may be rewritten in the following form  [Landau, Lifshitz (1982)]    From the dependences like of Fig. 9 it was numerically found cr ω (normalized on the plasma frequency p ω ) as a function of the wedge angle ψ (see Fig. 11 Fig. 11 also shows the frequency 2 p ω , above which SPP can not exist. Note, that when 180 ψ →° (the wedge turns into a plane) the critical frequency tends to 2 p ω -the utmost frequency of SPP existence on plane surface. Again it is absolutely unexpectable that the utmost frequency of SPP existence arises in the quasistatic problem of singularity existence.

Results of calculation for a silver microwegde
From the results of the previous section 4.1 we see that the problem of finding of the SPP nanofocusing properties of microwedge is the following. On the one hand the wavelength of SPP must be possibly smaller. So, the SPP frequency must be close (but smaller) to the critical frequency of SPP existence 2 p ω . On the other hand it is necessary to use the effect of additional increasing of the SPP electric field at the edge of the microwedge due to electric field singularity at the edge. As we have seen the electric field singularity at the edge exists for any frequency of SPP, but there two different types of electric field singularity. The choice of the singularity depends on the particular technical problem (in this work this problems do not discuss). Consider the following problem. Let there is a microwedge on the edge of which SPPs with frequency ω are focused (the wavelength in vacuum of a laser exciting the SPP is equal to 0 λ ). What is the value of wedge angle cr ψ which separates regimes of nanofocusing with different types of singularities? Consider a microwedge made from silver (plasma frequency of silver is equal to

Conclusion
It was obtained the fundamental result: the electric field singularity of alternative electric field of SPP wave at the apex of geometrically ideal metal conical tip or at the edge of geometrically ideal metal wedge may exist in two forms depending on the type of the index of singularity. For a given SPP frequency and surrounding media the type the singularity defines by the angle of the tip (or wedge). This remains true for the case when the apex of metal cone touches a dielectric plane plate.