Using the formalism of the Green's function one can obtain the expression of the resultant electric field around of tip [13]

$$E\left(r,\omega \right)={E}_{0}\left(r,\omega \right)+\frac{{\omega }^{2}}{{\epsilon }_{0}{c}^{2}}{\stackrel{⃡}{G}}_{0}\left(r,{r}_{0};\omega \right){\alpha }_{eff}{E}_{0}\left({r}_{0},\omega \right)$$

1

Where \({\stackrel{⃡}{G}}_{0}\) is the dyadic Green’s function and \({E}_{0}\) represents the initial electric field for the plane wave laser,\({\alpha }_{eff}\) is Polarizability.

simulations of enhancement intensity around the apex for the bare and corrugated tapered gold tips is made using FDTD method The Finite Difference Time Domain (FDTD) method is a numerical method to electrodynamics problems based on the discretization of Maxwell's equations by the finite difference method.

We have proposed a special an optical Nano antenna. Two modes for this optical Nano antenna considered, bare tip and corrugated tip. The design and simulation of these Nano antennas is done with the help of software lumerical and can be repeated. For example a gold tip Nano antenna is made with dc-pulsed low-voltage electrochemical etching of a gold wire.

In the first stage, by applying a plane wave laser, a laser that emits a plane wave, wave front of which consists of parallel flat plates perpendicular to the direction of propagation of the wave, with a wavelength of 400 to 1000 nm and polarization parallel to the axis of the cone, the apex of tip of the bare optical antenna is directly illuminated. The radius of the cone is 10 nm and its angle is 30 degrees. distribution around the apex.mechanisms for electric field enhancement at the apex of a sharp metallic tip is as a result of lightning-rod effect and localized surface Plasmon resonance of the apex.

In the next step, we change the geometry of the antenna. We apply a number of concentric circular gratings on the body of the Nano antenna. The desired antenna geometry is shown in the Fig. 3. In practice grating is made by focused ion beam, FIB on the smooth surface of a gold tip's shaft.

For exciton surface Plasmon with a wave number\({\text{k}}_{\text{S}\text{P}\text{P}}\) on a corrugated surface of the tip apex [14]

$${k}_{\text{S}\text{P}\text{P}}={k}_{0}\sqrt{{\epsilon }_{d}}\text{sin}\left(\phi \right)+\frac{2\pi m}{{\Lambda }}={k}_{0}\sqrt{\frac{{\epsilon }_{m}{\epsilon }_{d}}{{\epsilon }_{m}+{\epsilon }_{d}}}$$

2

Here \({\Lambda }\) is period of grating, m = 0; ±1; ±2, \({\epsilon }_{m} and {\epsilon }_{d}\)are metal and dielectric .

The binding efficiency

is understood as the ratio of the intensity of the excited plasmon to

the intensity of the incident field:

$$\text{η=}\frac{{W}_{\text{sp}}}{W}=\frac{2{k}_{\text{sp}}\text{''}l}{N}{\text{sinc}}^{2}\left[\frac{\left({k}_{\text{sp}}{\prime }-{k}_{0x}\right)l}{2}\right]\frac{1-\text{cosNd}\left({k}_{\text{sp}}{\prime }-{k}_{0x}\right)}{1-\text{cos}d\left({k}_{\text{sp}}{\prime }-{k}_{0x}\right)}$$

3

Here, N is the number of strokes of the subwavelength metal lattice, \(l\) is the stroke width, a is the lattice period, \({k}_{0x}{k}_{\text{sp}}{\prime }\), \({k}_{\text{sp}}\text{''}\) is the spatial frequency of the incident light, the real and imaginary parts of the plasmon wavenumber, respectively.

At this stage, we first apply the laser light to the location of these gratings. Applying grating on the cone body provides a necessary condition for the excitation of surface Plasmons. These surface Plasmons often move tangentially to the body towards the top of the cone and meet at the top of the cone and become localized there. In this case, we are also faced with an increase in the intensity of the field, and compared to the case without grating; this increase in the field is more. The distribution of the intensity of the field around the subject for this condition is shown in figure number 2 .This increase in intensity occurs in a more compact area around the apex.

To calculate the thermal response of the sample to the plasmonic excitations, we consider loss dominated by Ohmic heating.

Contributions from dielectric loss, arising from the imaginary part of the dielectric constant, are ignored because, for metallic samples, they are insignificant compared to conductive losses as long as the frequency is away from a resonance[4].

The spatial distribution of the electromagnetic loss determines the thermal gradient in near-field heating. Depending on the geometries of tip ,this gradient is different.

In the Fig. 3,simulation of this temperature gradient after illumination of the corrugated tip ,is shown.

In the Figs. 4 and 5,distributions of temperature gradient around the apex are shown.

By comparing the graphs of Figs. 4 and 5, see that the temperature gradient is greater in the case with grating.

Using the concepts of plasmonic optical heating, and plasmonic nanofocusing by corrugated tips, we can develop a tool for sorting of different particles. consider a corrugated gold tip, when irradiating the vertex, a localized plasmon and a near field around the vertex will first be created. due to the high gradient of the electric field, temperature gradient and photophoresis, gas particles move to the top of the antenna. particles with a higher mass will gather far from each other, and the core-shell of the particles will be made.