Polarization Modulation for FDTD Simulations 98 

As mentioned earlier, the polarization of the incident laser plays an essential role in the intensity and the distribution of the enhanced electric field at the tip apex.1 To further investigate this impact, linearly and radially polarized light in different TERS

configurations have been considered in this section.

4.3.1.1 Radial Polarization

A radially polarized light has polarization vectors oriented radially in the transverse plane with respect to the propagation direction as it is shown in Figure 4.2a. Under a tight focusing of a radially polarized light by a high numerical aperture lens, the focal longitudinal electric field (Ez) and the focal transverse electric field (Etr) could be expressed by the following equations in polar coordinate:39



 z



i P

 



 



  

 J  





iz 



d E_z , 2 cos sin2 ₀ sin exp cos

0

2 1





 z



 P

 



    

  J  





iz 



d E_tr , cos sin 2 ₁ sin exp cos

0

2 1



 (4.19)

 is the transverse size of the beam which is usually defined by the value of 2_=x2_+y2_,



is the maximum focusing angle and could be calculated by sin-1(N.A./n), n is the index of refraction.,is an angle between  and z at each point in space and J0and J1denote the Bessel functions of the first kind with the orders of zero and one. P(is the pupil function of a Bessel Gaussian beam:

P

 

 J1



20sin sin



exp



02sin2 sin2



(4.20) Where is the ratio of the pupil radius to the beam waist.40

Herein, a script that solved the equations (4.18) and (4.19) was created to describe a focused radially polarized light inside the simulation area. The script was written in Matlab programming language and integrated into FDTD. This light source has been utilized for the corresponding calculations that are reported here. The transverse and longitudinal electric field components of the described beam are shown in Figure 4.2a

and Figure 4.2b in 3D and 2D presentations.

Figure 4.2 (a) Transverse component of a focused radially polarized beam (b) longitudinal component of a focused radially polarized beam in 3D and 2D presentations

The transverse component of a radially polarized beam consists of several concentric rings with variable intensities and a minimum intensity at the middle (Figure 4.2a). The

longitudinal component (Figure 4.2b) of this beam is another set of concentric rings with overall intensities lower than the transverse component and with a maximum intensity at the center. The enhancement phenomenon that occurs in TERS originates from the interaction of the longitudinal component of the focused light with the apex of the sharp metallic tip.

It is noteworthy that the integration of a radially polarized light for calculation of the electric field distribution around a silver nanosphere at 532 nm was previously done by Du et al. in 2011. Using Full-Wave software, they created a homemade radially polarized light by the superposition of a left hand circularly polarized beam with a right hand one.41 They focused the radially polarized beam by setting an initial equivalent phase function of lens in the source file instead of the objective lens to save computational memory. However, this approach requires the integration of circularly polarized light which is not provided in the Lumerical software that is used in this thesis for conducting FDTD simulations.

4.3.1.2 Linear Polarization

In a linearly polarized electromagnetic wave, the electric field vectors are oriented along a single direction inside a plane perpendicular to the propagation direction of the wave. The top and side view of the total electric field of a focused linearly polarized light along with their corresponding longitudinal components are shown in Figures 4.3a, 4.3b and

4.3c, 4.3d respectively. Here the Gaussian light has been focused by passing a 1 mm diameter beam consisting of 1500 plane waves through a thin lens of 5 mm diameter and 1.2 numerical aperture. For the presented simulations where the structures of a few nanometers (less than 20 nm) are studied, a focused Gaussian beam was replaced with a plane wave that is provided in the FDTD software. Here, a special case of a plane wave known as total field scattered field (TFSF) was used to prevent the possible couplings with the boundaries of the simulation area. TFSF separates the computation region into two distinct regions; one contains the total field which is the sum of the incident field and the scattered field while the second region contains only the scattered field.

Figure 4.3 Electric energy density of the total field of a focused linearly x polarized

light in (a) top and (b) side view (c) longitudinal field of a linearly x polarized light in top and (d) side view.

The top view of a focused Gaussian beam consists of several concentric rings in xy plane with a maximum intensity in the middle (Figure 4.3a). From the side view, the beam would be relatively elongated in the direction of propagation along z (Figure 4.3b). The z component of the electric field in a focused Gaussian beam has two lobes in the direction of propagation with zero intensity in the middle (Figure 4.3c and 4.3d). This minimum intensity results in the absence of significant excitation of plasmon resonances at the tip apex if the tip is located in the center of the focal region. However, tip-enhanced Raman should be observed more significantly in case the tip is located inside one of the two lobes of the excitation laser.

In document Tip-Enhances Raman Spectroscopy, Enabling Spectroscopy at the Nanoscale (Page 124-127)