Focusing properties of Azimuthally Polarized Lorentz Gauss Vortex Beam through a Dielectric Interface

—Tight focusing properties of azimuthally polarized Lorentz Gaussian vortex beam through a dielectric interface are numerically studied by vector diffraction theory. The focusing properties, such as spot size, depth of focus, and maximum intensity position, are numerically calculated by properly manipulating the Lorentz parameter with/without annular obstruction values. Thus, using annular obstruction, one can generate a highly confined focal spot of long focal depth when using an azimuthally polarized Lorentz Gaussian vortex beam.

of a Lorentz-Gauss vortex beam over the Lorentz-Gauss beam is that it has a twisted phase front and zero intensity in the center region of the beam profile.Owing to carrying the orbital angular momentum, a Lorentz-Gauss vortex beam has potential applications in optical micromanipulation, nonlinear optics, quantum information processing, etc. [12][13][14][15].However, to the best of our knowledge, there are no papers studying the tightly focusing properties of azimuthally polarized Lorentz vortex beams through a dielectric interface.
A schematic diagram of the suggested method is shown in Fig. 1.The azimuthally polarized Hollow Gaussian beam is focused through a high NA lens system.Assume the interface between two dielectric media of refractive indices n1 = 1 and n2 = 3.55, such as focusing in air onto silicon substrate in the application of semiconductor inspection.The geometric focus of the objective without the interface is located at the origin O of the coordinate system.The distance between the interface and the geometric focus d is called probe depth [16].In the focusing system, we investigated, the incident beam is the Lorentz beam, whose amplitude distribution of electric field is in the form of [12][13][14][15]: where ωx, ωy are parameters related to the beam width, C is chosen as constant, and NA is the numerical aperture of the incident beam.For azimuthally polarized beams, the Cartesian components of the electric field vector in the focal region then could be written as [16]: ( , , ) sin sin cos ( ) [ cos sin cos( )] cos , where ki=nik0 is the wave number, Jn(x) is the Bessel function of the first kind of order n, α = arcsin(NA) is the maximal angle determined by the NA of the objective; tp is the amplitude transmission coefficients for parallel polarization states, which is given by the Fresnel equations: 2 sin cos .sin( ) cos( ) The function Φ(θ1,θ2) is given by equation     Figure 3 shows the same characteristics as in Fig. 2 but obtained for δ=0.75.As evident from the 3D plots shown in Fig. 3, obtained for δ=0.75 and ωy=0.3,generated a much confined focal spot with a large DOF.It is also observed that the position of the maximum intensity shifted to 6.4λ.The radial intensity calculated at the position of maximum intensity shows that the FWHM of the generated focal spot is reduced to 0.402λ with a much dominating y-component with a large side lobe about 50% of the main lobe, and is shown in Fig. 3(c).The onaxial intensity calculated shows that the DOF of the generated focal spot is much improved to 25.2.4λ, as demonstrated in Fig. 3(d).However, for ωy =0.6, the focal spot is 0.592λ, and DOF is around 24.7λ, as shown in Figs.3(g) and 3(h), respectively.However, further increasing of ωy to 0.9 and 1.2 slightly increased the focal spot to 0.686λ and 0.687λ, as shown in Figs.   Figure 4 shows the spot size obtained corresponding to various values of the Lorential parameter.It is evident from Fig. 4 that the increase of the Lorential parameter increases the spot size.However, we get a minimum spot size when using annular obstruction with δ=0.75. Figure 5 shows the variation of DOF corresponding to different values of the Lorential parameter.It is visible that the DOF remains constant irrespective of the increase in the Lorential parameter.However, the DOF for the case with δ=0.75 was found to be maximum.Figure 6 shows the shift in the position of maximum intensity.It is also noted that the shift remains constant for all Lorential parameters and is found to be maximum for δ=0.75.Thus, on the whole, for azimuthally polarized vortex beam with different Lorential parameter, the spot size and the depth of focus increases with an increase in Lorential parameter.In contrast, the position of maximum intensity remains the same.In the case of using annular obstructions of δ=0.5 and δ=0.75, the focal spot increases with an increase in the Lorential parameter.However, it is found to be much less than the unobstructed case.Thus, using annular obstruction, one can generate a highly confined focal spot of long focal depth when using an azimuthally polarized Lorentz Gaussian vortex beam.
representing the so- called aberration function caused by the mismatch of the refractive indices n1 and n2.Here θ1 and θ2 are related by the well-known Snell law.

Fig. 2 .
Fig. 2. Panels (a, e, i, m) show the r-z plot corresponding to =0, NA = 0.95, wx = 0.3, and wy = 0.3, 0.6, 0.9 and 1.2 respectively.Panels (b, f, j, n) show the corresponding intensity in the x-y plane.Panels (c, g, k, o) show the 2D intensity in the radial direction, and panels (d, h, i, p)show the corresponding axial intensity.

Figure 2
Figure2shows the focal structure generated for the azimuthally polarized Lorentz vortex beam with different Lorential parameters.It is clear from Fig.2(a) that when ωx=0.3 and ωy =0.3, the generated focal structure is found to be not at focus (z=0), but it is shifted to 4λ.The FWHM of the focal spot is determined as 0.662λ from the radial intensity distribution measured at the point of maximum on axial intensity, and it is shown in Fig.2(c).It is also observed from Fig.2that the Ey component is

Figure 2 (
e) shows that for ωx =0.3 and ωy =0.6, the focal spot further broadens in the radial direction due to the fact that the broader Ex component starts dominating Ey component, as shown in Fig. 2(g).The calculated FWHM of the focal spot around 0.804λ, and DOF of the generated focal spot around 7.22 λ are shown in Figs.2(g) and 2(h), respectively.The intensity distribution in the x-y plane shows that the electric field vectors at the focus are more oriented along the x-direction, as shown in Fig. 2(f).Figures2(i, j, k, l) represent the same characteristics as in Figs.2(e, f, g, h), but for ωy =0.9.It is observed that further increasing the Lorential parameter makes the Ex component more dominating and leads to a further increase in focal spot size.The FWHM of the focal spot is measured to be 0.867λ, and DOF is measured as 7.34λ.The same trend is observed for ωy=1.2, and Fig.2 (m)shows the enlargement of the focal spot in the radial direction with FWHM of the focal spot around 0.924λ and DOF around 7.46λ.Thus, increasing the Lorential parameter increases the foal spot size due to the dominating Ex component.It is also noted that the position of maximum intensity is located at 4λ for all the Lorential parameters considered.
3(k) and 3(o), respectively.The corresponding focal depths are 25.2.28λ and 25.2.22λ, as shown in Figs.3(l) and 3(p), respectively.Thus, increasing the annular obstruction increases the focal depth and confines the focal spot.It is noted that the DOF and positional shift have little effect on the Lorential parameter.