Focusing properties of circularly polarized annular multi-Gaussian beams through a uniaxial birefringent crystal is investigated numerically by vectorial Debye theory. Results show that the optical intensity in focal region of circularly polarized annular multi-Gaussian beams can be altered considerably by the topological charge [n], beam order [m] and birefringence [Δn]. Many novel focal structures such as focal spot, focal hole and flat top profile of sub wavelength structure are achieved.

Uniaxial birefringent crystal, Circularly polarized laser, Annular multi-Gaussian Beam

In Recent years, focal shaping of tight-focusing of laser beams such as highly confined focal spot/hole with extended focal depth, multiple focal spots, optical bubble etc., has important applications in high-density optical data storage [1], laser micro/nano-processing [2], particle and electron acceleration [3], super-resolution scanning confocal microscopy [4] and optical trapping [5]. The intensity distribution of the focal structure in the focal region has been frequently discussed [6-13]. To produce different types of intensity distribution in the focal plane one may insert a physical mask in the pupil plane of the objective [14-18]. However, fabricating those physical masks are difficult and their efficiency is also low. On the other hand, optically, uniaxial crystals are performing the same operations without additional devices [19]. Alejo and Deng, et al. studied the properties of light beams propagating in birefringent material [20,21]. The brief explanation of focusing into a birefringent medium is given by Stamnes, Jiang and Dhayalan, et al. [22-26]. A. Ciattoni, et al. analyzed the propagation of radial and azimuthal vortex beam in the uniaxial crystal system [27-29]. Stallinga, et al. briefly explained the vectorial approach of focusing into the uniaxial crystal [30]. Radially and azimuthally polarized vortex beams through a uniaxial crystal system are studied numerically and experimentally [31,32]. Recently properties of circularly polarized beam have explained by many researchers [33-37]. Rao Lian-Zhou, et al. extends the analysis of tightly focusing circularly polarized vortex beams through uniaxial crystals with small axial birefringence [38]. Recently, Murugesan, et al. numerically demonstrated that focal structures such as splitting of single focal spot on to multiple spots and focal shift of maximum intensity in the axial direction can be obtained by tightly focused circularly polarized double ring-shaped beam and axial birefringence [39]. Anthony A. Tovar, et al. introduced a new class of beam called multi-Gaussian beam which consist of a small sum of finite-width Gaussian beams side by side each of which represents an intuitive component of the entire beam. Unlike the flattened Gaussian beams, each of the multi-Gaussian beam components can be traced individually to resort to further series expansion. They proposed multi-Gaussian beam has the advantages of the super-Gaussian while having diffraction characteristics that are analytically solvable [40]. Chen Jian-Nong, et al. proposed annular multi-Gaussian beam mode for sharp focusing [41]. Followed by this, tight focusing properties of annular multigaussian beam are investigated in several works [42-45]. In this paper, we demonstrate the tight focusing properties of circularly polarized annular multi-Gaussian beam through Uniaxial Birefringent Crystal. The intensity distribution of the light beam in focal plane is investigated in detail using numerical simulation.

The schematic of the focusing system is shown in Figure 1. Here medium 1 is isotropic whereas medium 2 is a Uniaxial birefringent and d is distance between the interface and geometrical focus. Here kˆ1 and kˆ2 are the wave vectors and sˆ1 pˆ1 are the polarization vectors in parallel and perpendicular direction to the plane of incidence of medium 1 and sˆ2 pˆ2 are the corresponding polarization vectors in medium 2. Adopting the theoretical model developed by Stallinga and for the approximation of small axial birefringence of the order of 10-3, the electric field in the focal region is expressed as [30,46].

$\left.\left\{\begin{array}{l}{E}_{x}(r,\psi ,z)\text{}=\text{}\frac{-i{E}_{o}}{\pi}\underset{0}{\overset{\alpha}{{\displaystyle \int}}}\underset{0}{\overset{2\pi}{{\displaystyle \int}}}\mathrm{sin}{\theta}_{1}\sqrt{\mathrm{cos}{\theta}_{1}}P({\theta}_{1})\mathrm{exp}[i{k}_{2}z\mathrm{cos}{\theta}_{2}+i{k}_{1}r\mathrm{sin}{\theta}_{1}\mathrm{cos}(\psi -\varphi )]\hfill \\ [{t}_{p}\mathrm{cos}{\theta}_{2}\mathrm{cos}\varphi \mathrm{exp}[i{k}_{o}(W+\Delta W)]\pm i{t}_{s}\mathrm{sin}\varphi \mathrm{exp}(iW)]exp(-i\varphi )d\varphi d{\theta}_{1}\hfill \end{array}\right.\right\}\text{(1)}$

$\left\{\left.\begin{array}{l}{E}_{y}(r,\psi ,z)\text{}=\text{}\frac{-i{E}_{o}}{\pi}\underset{0}{\overset{\alpha}{{\displaystyle \int}}}\underset{0}{\overset{2\pi}{{\displaystyle \int}}}\mathrm{sin}{\theta}_{1}\sqrt{\mathrm{cos}{\theta}_{1}}P({\theta}_{1})\mathrm{exp}[i{k}_{2}z\mathrm{cos}{\theta}_{2}+i{k}_{1}r\mathrm{sin}{\theta}_{1}\mathrm{cos}(\psi -\varphi )]\hfill \\ [{t}_{p}\mathrm{cos}{\theta}_{2}\mathrm{sin}\varphi \mathrm{exp}[i{k}_{o}(W+\Delta W)]\pm i{t}_{s}\mathrm{cos}\varphi \mathrm{exp}(iW)]exp(-i\varphi )d\varphi d{\theta}_{1}\hfill \end{array}\right\}\right.\text{(2)}$

$\left\{\left.\begin{array}{l}{E}_{z}(r,\psi ,z)\text{}=\text{}\frac{-i{E}_{o}}{\pi}\underset{0}{\overset{\alpha}{{\displaystyle \int}}}\underset{0}{\overset{2\pi}{{\displaystyle \int}}}\mathrm{sin}{\theta}_{1}\sqrt{\mathrm{cos}{\theta}_{1}}P({\theta}_{1})\mathrm{exp}[i{k}_{2}z\mathrm{cos}{\theta}_{2}+i{k}_{1}r\mathrm{sin}{\theta}_{1}\mathrm{cos}(\psi -\varphi )]\hfill \\ [-{t}_{p}\mathrm{sin}{\theta}_{2}\mathrm{exp}[i{k}_{o}(W+\Delta W)]]exp(-i\varphi )d\varphi d{\theta}_{1}\hfill \end{array}\right\}\right.\text{(3)}$

Where r,ψ and z are the cylindrical coordinates and Eo is a constant. α = sin−1(NA) is the maximal angle determined by the NA of the objective; tp and ts are the amplitude transmission coefficients for parallel and perpendicular polarization states respectively, which are given by the Fresnel equations [47].

${t}_{s}\text{=}\frac{2\mathrm{sin}{\theta}_{2}\mathrm{cos}{\theta}_{1}}{\mathrm{sin}({\theta}_{1}+{\theta}_{2})},\text{}{t}_{p}\text{=}\frac{2\mathrm{sin}{\theta}_{2}\mathrm{cos}{\theta}_{1}}{\mathrm{sin}({\theta}_{1}+{\theta}_{2})\mathrm{cos}({\theta}_{1}-{\theta}_{2})}\text{(4)}$

Here W is the aberration function caused by the mismatch of the refractive indices medium 1 and medium 2, Where ΔW is the phase difference between the ordinary and extraordinary modes in the uniaxial birefringent medium 2.W and ΔW are expressed as [30].

$W\text{=}{k}_{o}d({n}_{2}\mathrm{cos}{\theta}_{2}-{n}_{1}\mathrm{cos}{\theta}_{1}),\text{}\Delta W\text{=}{k}_{o}(d+z)\Delta n{\mathrm{sin}}^{2}{\theta}_{2}/\mathrm{cos}{\theta}_{2}\text{(5)}$

Where Ko = 2π/λ is the wave numbers in vacuum and Δn = ne-no represents the difference between the refractive indices of ordinary and extraordinary modes in the uniaxial crystal. It is assumed that the focusing lens is corrected for aberrations introduced by anisotropic cover layer of thickness d and refractive index n2 = n0. As a result, W = 0. Here P(θ1) is the pupil function of the beam; the pupil function of annular multi-Gaussian beams is [41].

$P({\theta}_{1})\text{=}{\left(\frac{{\theta}_{1}}{{\theta}_{o}}\right)}^{m}{\displaystyle \sum}_{n\text{}=\text{}-N}^{N}\mathrm{exp}\left[-{\left(\frac{{\theta}_{1}-{\theta}_{c}-n{\omega}_{o}}{{\omega}_{o}}\right)}^{2}\right]\text{(6)}$

Here, θ1 is the converging semi-angle. We denote the maximum converging semi-angle as θmax which is related to objective numerical aperture by θmax = arcsin (NA). θo be an angle which, along with integer m, determines the shape of the modulation function. θo is usually choosing to be slightly smaller than θmax. θc determine the radial position translation. Here we take θc = θmax2/.

The focal properties are evaluated numerically by solving the above equations numerically using the parameters NA = 0.85, n1 = 1, n2 = 1.5, λ = 400 nm, d = 100 μm. Figure 2 shows the focal pattern evolved for different beam order (m) of the incident circularly polarized multi-Gaussian beam in the absence of axial birefringence .The Figure 2a, Figure 2b, Figure 2c and Figure 2d shows the 3D intensity in the r-z plane corresponding to m = 2,4,6 and 8 respectively. Figure 2e, Figure 2f, Figure 2g, and Figure 2h shows the respective 2D intensity measured in the radial axis corresponding to the position of maximum on axial intensity. Figure 2i, Figure 2j, Figure 2k and Figure 2l shows the 2D axial intensity distribution. It is noted from the Figure, that increasing the beam orders m increases the focal depth and reduced the FWHM of the generated focal spot.

Table 1 shows the FWHM and focal depth of the generated focal spot corresponding to different m. It is noted that the Ex component having central maximum dominates the entire focal spot on all the m values. The Ey and Ez components having central minimum is found to be feeble. It is noted that a highly confined focal spot with FWHM of 0.48λ and extend focal depth of 8.87λ is obtained for m = 8. Here focal depth is considered as FWHM of axial intensity.

Figure 3 shows the same as Figure 2 but for the presence of axial birefringence with Δn = 6×10−3. It is noted from Figure 2a, Figure 2e and Figure 2i when m = 2,the presence of axial birefringence makes the Ez component having central minimum dominates the focal structure and the resultant focal structure appears to be an radial extended focal spot with large FWHM around 1.8λ with focal depth around 7.16λ. However it is noted from Figure 3b, Figure 3c and Figure 3d and their respective intensity in the radial and axial direction ,that further increasing of beam order to 4,6 and 8 makes the Ez component more dominating than the Ex component having central maximum and the generated focal structure has a central minimum. It is noted from Figure 2b the existence of residual intensity at the centre due to the presence of Ex component is completely vanished and resulted in formation of focal hole as shown in Figure 3d. The on axial intensity calculated at r = 0.58λ shows the depth of focus on the generated focal hole is around 8.46λ and FWHM of focal hole is around 0.65λ. such a dark channel of sub wavelength size and axially extended focal depth finds application in trapping absorbing particles, cold atoms [48] and in high resolution STED microscopy [49].

Hence by properly choosing axial birefringence and beam order of multi-Gaussian beam one can tune a focal structure from a focal spot to a focal hole. We also noted that the on axial intensity is found to be maximum at z = 1.21λ for all the beam order considered. Thus, the presence of axial birefringence generates focal shift effect apart from tuning the focal patterns.

Figure 4 Shows same as Figure 3 but for m = 2,4,6,8 and Δn = 8×10−3. It is noted from Figure 4a, Figure 4e and Figure 4i that when m = 2, the generated focal structure is an optical bubble having radial extend of 0.74λ and axial size of 3.05λ. such an optical bubble is useful in trapping and manipulating of particles having refractive index lower than the ambient [50]. Thus, using a single focusing unit, one can tune the focal structure from a focal spot to focal hole and then to an optical bubble. such a tunability in focal structure ensure the capability of trapping particles of refractive index higher or lower than the ambient with single focusing unit. Hence one can avoid the problem of mounting and aligning specially designed dedicated binary phase/amplitude filters to modulate the focal patterns. It is noted that the centre of the bubble is located at z = 1.64λ. Figure 4b, Figure 4c and Figure 4d shows that further increasing of m makes the Ex component dominating and created a non-zero central focal pattern.

Figure 5 shows the same as Figure 4 but for Δn = 10×10−3. It is noted from the Figure 5a, when m = 2, the generated focal structure is found to be axially shifted and splitted focal spots with a residual intensity between them. It is noted from Figure 5e that the Ex component starts dominating the focal structure. We also noted that further increasing m to 4, 6 and 8 results in further increasing of Ex component which results in a highly confined focal spot with FWHM around 0.48λ. However, it is noted from the on axial intensity plots shown in Figure 5j, Figure 5k and Figure 5l that the on axial bumpy structure turns to be flattop and axially extended corresponding to the beam order m = 6. Such a flat top focus is useful in applications such as improved printing filling factor, improved uniformity and quality in material processing, microlithography, medical treatment [51-53].

The depth of focus and spot size of the generated focal spots is shown in Table 2. It is noted that a smallest focal spot size around 0.48λ with maximum focal depth of 18.05λ and focal shift of 2.20λ is noted for m = 8. Such a focal spot of sub wavelength size and extended focal depth is highly useful in applications such as high-density optical data storage, particle acceleration, biomedical imaging and in optical trapping [1,54-56]. Thus, by modulating beam order of a circularly polarized annular multi Gaussian beam and axial birefringence of a uniaxial crystal one can generate many novel focal structures usable for nano scale applications such as high-density optical recording, particle trapping and material processing etc.

In conclusion, a simple method to achieve tunability of focal structures suitable for nanoscale applications using the effect of axial briefinges on tightly focused circularly polarized annular multigaussian beams is demonstrated numerically using vector diffraction theory. It is observed that many novel focal patterns such as focal spot/hole of extended focal depth, optical bubble and flat top profile of sub wavelength scale is achieved by properly modulating the axial birefringence and order of multi Gaussian beam. The authors expect such a study is useful in applications such as optical trapping, high density optical recording and in optical material processing etc.