Formation of polarization-dependent optical vortex beams via an engineered microsphere

Yan Zhou; Minghui Hong

doi:10.1364/OE.422542

1. Introduction

Optical focal field shaping by engineering amplitude, phase and polarization on the exit pupil of an optical system [1,2], especially assisted by spatial light modulators [3–5], has attracted scores of attentions in recent years as it finds applications in numerous fields. For instance, a beam with azimuthal polarization or phase vortices on the exit pupil can form hollow focus with null intensity in the center [6,7], which can be used to trap absorbing micro- or nano-particles, and can also facilitate high resolution stimulated emission depletion (STED) microscopy [8,9]. Controlling the local electric/magnetic field distribution in the focal region enables researchers to excite specific resonances of a quantum emitter or to determine the orientation of an individual optical emitter [10,11]. There are numerous methods to shape the focal field of a radially or azimuthally polarized pupil, including specially designed binary phase masks [12] or reversed electrical dipole array radiations [13–15]. Investigation on sharp focusing of polarized phase vortex beams [16,17] has been carried out for enhanced resolution and to demonstrate spin-to-orbital angular momentum conversion [18,19].

An optical vortex (OV) is a singular structured optical field that occurs in paraxial beams [20,21]. As an important property of light, polarization is present in many forms including the well-known linear, circular, and elliptical polarizations, light beams with these polarizations are spatially homogeneous [22,23]. In the past two decades there has been increasing interest in light beams with spatially variant polarizations. One typical example is a light beam with cylindrically symmetric polarization. Such a cylindrical vector beam is called as a polarization vortex because its polarization is undetermined at the beam center (polarization singularity), leading to null intensity. Polarization vortex can take different orthogonal states described by spatially variant polarization φ(θ) = pθ + φ₀, where p is the polarization order, θ the azimuthal angle, and φ₀ the initial polarization orientation for θ = 0 [22]. Two conventional simple polarization vortex beams are radially polarized beam (TM₀₁) and azimuthally polarized beam (TE₀₁). Both present donut-shaped intensity profiles due to the polarization singularity.

The cylindrically symmetric pattern depicted in literatures show an ideal configuration characteristic for polarization OV beams [19,22,24,25]. In many real situations, this ideal pattern is distorted [19,26]. Under external disturbances, for example, when the beam is propagating through an inhomogeneous and dynamic medium, parameters like position and morphology may change, but the singularity “per se” with all its qualitative attributes is of the topological nature and thus stable against perturbations. For this reason, OV beams are promising for the information transmission in noise conditions [26]. The intriguing dynamical properties of the OV fields are especially attractive for optical manipulation techniques [27,28].

Spatial tuning of structured light at focus has attracted increasing interest in recent years [3–5,27,28]. Conventional ways to achieve such performance largely rely on using encoded spatial light modulators (SLM), high numerical aperture (NA) objectives and other mechano-optical components. Those arrangements are usually bulky and complex and can also be expensive when premier quality optics are required. Specially designed micro-optics, such as metalenses and metasurfaces, can replace SLM and objective lens in order to reduce the system complexity and size. However, at the present stage, fabrication difficulties, high costs and losses hinder mass applications at high efficiency. A promising alternative candidate with low costs is transparent microspheres, which possess intrinsic strong light focusing with low loss, and have proved to achieve promising application performances, especially in nano-patterning and nano-imaging [29–32]. Moreover, engineered nanostructures on the microsphere have gradually shown their functions on the configuration of focused light field, achieving various types of beam engineering, from super-resolution focusing to vortex beam generation [33–37].

In this work, for the first time, we proposed a method that can realize switchable spatial arrangement of the donut-shaped focusing beams through an engineered microsphere, controlled by changing the polarization state of an incident light. This specific structured beam is formed upon radially polarized wave emitted after a linear-to-radial converter, i.e. S-waveplate. We simulated and experimentally characterized the focus performance of the engineered microsphere. Under the light incidence with radial polarization, multiple focused donut beams are formed along the optical axis. By adding one more linear polarizer with a rotatable relative angle, the pair of donut beams can be re-arranged in the same transverse plane and stay close to each other. Switching in between the lateral and axial arrangements can be controlled by the addition or removal of the linear polarization to the manipulated light field. Experimental results and numerical simulation are in good agreement. Moreover, in such conditions, the central holes increase its lateral size up to several optical wavelengths in axial arrangement, while maintaining subwavelength transverse dimensions in lateral arrangements. Such a tunable light manipulation micro-optics is suitable for various applications, including multiplane imaging, multiplexing communication and particle trapping. Furthermore, it can be extended into array form for scalable light manipulations.

2. Design and methods

Zernike polynomials forms a complete and orthogonal set of polynomials on a unit circle. By pre-calculating Zernike polynomials, the focal field can be optimized by only tuning the Zernike coefficients. The complex Zernike polynomials $Z_n^m({\rho ,\varphi } )$ form a complete set of orthonormal polynomials defined on a unit disc:

(1)$$Z_n^m({\rho ,\varphi } )= R_n^{|m |}(\rho ){e^{im\varphi }},$$

(2)$$R_n^m(\rho )= \sum\limits_{s = 0}^p {\frac{{{{({ - 1} )}^s}({n - s} )!}}{{s!({q - s} )!({p - s} )!}}{\rho ^{n - 2s}}} ,$$

Where $p = 1/2({n - |m |} ),q = 1/2({n + |m |} ),n - |m |\ge 0$ and even, $({\rho ,\varphi } )$ are the normalized polar coordinates on the exit pupil plane. $Z_n^m$ is a phase vortex with the topological charge m.

The polarized field distribution on the exit pupil plane can be decomposed into a series of radially polarized Zernike polynomials:

(3)$${\vec{E}_s}({\rho ,\varphi } )= \sum\limits_{n,m} {{{\hat{e}}_s}({\rho ,\varphi } )\beta _n^mZ_n^m} ({\rho ,\varphi } ),$$

The electric fields are polarized within the exit pupil plane, as such they contain no z-component. Radial polarization is position-dependent, defined by unit vector ${\hat{e}_s}({\rho ,\varphi } )$ at position $({\rho ,\varphi } )$. For radial polarization, the electric field is polarized that ${\hat{e}_s}({\rho ,\varphi } )= {\hat{e}_\rho }({\rho ,\varphi } )= ({\cos \varphi ,\sin \varphi } )$. Zernike polynomials have been used in the Extended Nijboer Zernike theory, in which a semi-analytical solution of the focal field of each Zernike polynomial on the exit pupil is derived [38,39].

The focal field of a single radially polarized Zernike polynomials, written in polar coordinates, is:

(4)$$\begin{aligned}\vec{E}_f^{nm}({{\rho_f},{\varphi_f},{z_f}} )&={-} \frac{{iR{s_0}^2}}{\lambda }\int_0^1 {{{({1 - {\rho^2}{s_0}^2} )}^{1/4}}{e^{ - i{k_0}{z_f}\sqrt {1 - {\rho ^2}{s_0}^2} }}} \\ &\quad \times R_n^{|m |}(\rho )\rho d\rho \int_0^{2\pi } {{{\hat{e}}_f}{e^{im\varphi }}{e^{i2\pi \rho {\rho _f}\cos ({{\varphi_f} - \varphi } )}}d\varphi } , \end{aligned}$$

where, $({{\rho_f},{\varphi_f},{z_f}} )$ are the cylindrical coordinates of a point in the focal region.

The Cartesian components of the focal plane electric field of the radially polarized pupil field are thus given by [40]:

(5)$$\begin{aligned}E_{f,x}^{nm}({{\rho_f},{\varphi_f},{z_f}} )&={-} \frac{{i\pi R{s_0}^2}}{\lambda }{({ - i} )^{m + 1}}{e^{im{\varphi _f}}}\int_0^1 \rho d\rho \\ &\quad \times {({1 - {\rho^2}{s_0}^2} )^{1/4}}{e^{ - i{k_0}{z_f}\sqrt {1 - {\rho ^2}{s_0}^2} }}R_n^{|m |}(\rho )\\ &\quad \times [{{e^{i{\varphi_f}}}{J_{m + 1}}({2\pi \rho {\rho_f}} )- {e^{ - i{\varphi_f}}}{J_{m - 1}}({2\pi \rho {\rho_f}} )} ], \end{aligned}$$

(6)$$\begin{aligned}E_{f,y}^{nm}({{\rho_f},{\varphi_f},{z_f}} )&={-} \frac{{i\pi R{s_0}^2}}{\lambda }{({ - i} )^{m + 2}}{e^{im{\varphi _f}}}\int_0^1 {\rho d\rho } \\ &\quad\times {({1 - {\rho^2}{s_0}^2} )^{1/4}}{e^{ - i{k_0}{z_f}\sqrt {1 - {\rho ^2}{s_0}^2} }}R_n^{|m |}(\rho )\\ \times [{{e^{i{\varphi_f}}}{J_{m + 1}}({2\pi \rho {\rho_f}} )+ {e^{ - i{\varphi_f}}}{J_{m - 1}}({2\pi \rho {\rho_f}} )} ], \end{aligned}$$

(7)$$\begin{aligned}E_{f,z}^{nm}({{\rho_f},{\varphi_f},{z_f}} )&={-} \frac{{i2\pi R{s_0}^2}}{\lambda }{({ - i} )^m}{e^{im{\varphi _f}}}\int_0^1 {\frac{{{s_0}\rho }}{{{{({1 - {\rho^2}{s_0}^2} )}^{1/4}}}}} \\ &\quad\times {e^{ - i{k_0}{z_f}\sqrt {1 - {\rho ^2}{s_0}^2} }}R_n^{|m |}(\rho ){J_m}({2\pi \rho {\rho_f}} )\rho d\rho , \end{aligned}$$

where J_m is a Bessel function of the first kind of order m. It is readily verified that, just with the paraxial solutions, a conventional focusing lens under radial polarization illumination produces a focal region with an on-axis null at the paraxial focus.

It is noted that in the proposed experiments, the radially polarized incident light is generated by using an S-waveplate. The S-waveplate is a spatially variant half-wave plate fabricated through the femtosecond laser writing of self-assembled nano-gratings in silica glass, and it can directly convert linear polarization to cylindrical polarization with high efficiency [41]. Such converters can be simply modeled with the following Jones matrix [42]:

(8)$${\rm M} = \left( {\begin{array}{{cc}} {\cos \theta }&{\sin \theta }\\ {\sin \theta }&{ - \cos \theta } \end{array}} \right),$$

where θ is a polar angle in the polar coordinate system. Then, assuming the horizontally polarized incident light matches the fast axis of the S-waveplate, the radial polarization is derived and it reads

(9)$${e_r} = {\rm M} \cdot \left( {\begin{array}{{c}} 0\\ 1 \end{array}} \right) = \left( {\begin{array}{{c}} {\sin \theta }\\ { - \cos \theta } \end{array}} \right).$$

So we can easily obtain cylindrical polarization states by rotating the incident beam polarization with respect to the S-waveplate.

The vectorial light field is then simulated using a finite-difference time-domain solver (Lumerical FDTD). The special optical element is set to be a truncated dielectric microsphere, with the shadow side surface to be flat. As seen in Fig. 1(a), a set of modified Fresnel ring structures is patterned on the flat surface to form the so-called Fresnel zone microsphere (FZMS) [36]. The major difference as compared to the previous literature is the material of the microsphere used is changed from silicon dioxide (n = 1.4696) to borosilicate glass (n = 1.56). The illumination light incidents along the optical axis and towards the spherical side of the microsphere (or the illumination side surface). The illumination source is pre-set to be a radially polarized light (wavelength = 405 nm) focused by a converging lens with low numerical aperture (NA = 0.05), which is close to the experimental condition, where a thin lens with low NA is used to focus the converted light from an S-waveplate. The low NA value is chosen to generate a suitable size of incident radial polarization beam, which is about twice the diameter of the engineered microsphere. Such a pre-set is achieved by generating a linearly polarized coherent plane wave source, and the field data from specified FDTD monitor is exported to MatLab (MathWorks). By using the previously reported vector diffraction theory [1,43,44], a script that solved Eqs. (5)–(7) was created to describe a weakly focused radially polarized light inside the simulation area. The script was written with MatLab and integrated into FDTD. The boundary condition in the FDTD region is set to be perfectly matched layers (PMLs) and a mesh size of λ/10 is applied within the entire simulation region.

Fig. 1. (a) Conceptual sketch design of the engineered microsphere. D: diameter of microsphere, w: etched annular zone width, d: etching depth. (b) Conceptual scheme to generate double donut beams along optical axis. Two donut beams are formed after a single focus spot under the illumination wavelength of 405 nm, light field distributions are obtained with simulation. F1 – F3: focal planes 1-3. FZMS: Fresnel zone microsphere. Scale bars: 2 μm.

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Simulation data is visualized in Fig. 1(b). The output light field is examined as the square of electric field, |E|², and YZ-plane data along the optical axis is shown. There are three typical “focused regions” as seen in the figure. The first one is a tightly focused point with a transverse full width at a half maximum (FWHM) of 266 nm (0.66λ). The second and the third are two focused donut beams with a hollowed center. The central holes of the two donut beams possess sizes (measured by FWHM of the valley) are 1.41λ (570 nm) and 0.71λ (286 nm), respectively.

A more insightful investigation is conducted to see the effects of the fabrication parameters on light focusing properties for the engineered microsphere, and simulated results are summarized in Fig. 2. Figures 2(a) – (c) show the simulated focal intensity profiles at all the three focal points at an etching depth of 0.941λ, 0.765λ and 0.652λ. Considering the uniformity of peak power distribution at all the three focusing planes, 0.652λ should be chosen as it gives the smallest peak power variations. This point is also displayed in Fig. 2(d), as it depicts the variations of the first donut-hole size as well as minimum/maximum intensity difference ratios over four different etching depths of 0.496λ, 0.652λ, 0.765λ and 0.941λ, respectively. If only the first donut beam hole size is concerned, a deeper etching depth should be chosen for a smaller hole size. Figure 2(e) displays the focal positions of all the three focal points F1 – F3 at different etching depths indicated in Fig. 2(d). It can be clearly seen that the positions for all the focal regions are robust with small variations. To observe it in more details on the focal distances, variation due to the etching depth gets smaller from F1 to F3. This is because the first peak is heavily contributed by the outer rings on the engineered microsphere, changing in etching depth alters the refraction angle of light coming from these annular zones more significantly. These analyses serve as a guide to design the positions and sizes of all the donut focal spots generated by the engineered microsphere under a single-wavelength radial illumination.

Fig. 2. Tuning of relative intensity and focal position by changing etching depth of the pattern on the engineered microsphere. (a) – (c) Simulated focal intensity profiles at all the three focal points with an etching depth of 0.941λ, 0.765λ and 0.652λ, respectively. (d) Variation of the first donut hole size and minimum/maximum intensity difference ratios over different etching depths of 0.496λ, 0.652λ, 0.765λ and 0.941λ. (e) Focal positions of all the three focal points F1 – F3 at different etching depths indicated in (d), each of the curves indicates the normal distribution of focal positions at each focal point, and the number aside marks the mean value (unit: λ).

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To conduct experimental analyses, a microsphere (17 μm diameter) is truncated to become a hyper-hemisphere using a focused ion beam (FIB) system (GAIA, TESCAN). The designed pattern described in Ref. [36] is then etched on the flat surface of the truncated microsphere using the same FIB. The radius of curvature for the unstructured surface of the FZMS is ∼8.5 μm. Compared to the previous works, the material of microsphere used is changed from silica to borosilicate glass (Thermal Fischer, n = 1.56 @ 589 nm). The fabricated engineered microsphere is positioned within a grid on a gold mesh film (thickness ∼5 μm), with its lower waist mechanically held and supported.

3. Results and discussion

The FIB fabricated engineered microsphere as seen in Fig. 3(d) was characterized with the experimental setup as described in previous literatures using a commercial optical microscope system (Nikon Eclipse Ni-E) [36,37]. As mentioned before, the key modification is to place an S-waveplate which converts the linearly polarized illumination laser (532 nm, MGL-FN-532, CNI Lasers) into the radial polarization. The radially polarized laser beam is then directed to the spherical side of the engineered microsphere, focused by the microsphere, and exits from the patterned side to generate a pair of center-hollowed focused beams along the optical axis. Taking the propagation direction or optical axis to be along Z axis, the XZ-plane of measured light field intensity is displayed in Fig. 3(a). The two yellow dotted lines mark the two Z-positions where each of the focused donut beams has the highest light intensities. Intensity profiles along the two dotted lines are measured, and normalized intensity profiles are plotted in Fig. 3(b), where the dotted curve with solid circles (red) marks the first focused donut beam (B1), and the solid curve with hollow circles (black) marks the second focused donut beam (B2). Their central-hole sizes are then measured by taking the full width at half minimum values for the valleys in the two graphs. As seen from Fig. 3(b), the central-hole sizes of B1 and B2 are 1.485λ (790 nm) and 0.414λ (220 nm), respectively. Figure 3(c) shows the simulated light field profile under illumination by 532 nm laser (radially polarized and weakly focused at NA = 0.05). Figure 3(d) displays the scanning electron microscope (SEM, TESCAN) image of the fabricated engineered microsphere.

Fig. 3. Experimental characterization for double donut beams formed under 532 nm illumination wavelength. (a) Measured light intensity of the output beams in XZ-plane. (b) Plotted profiles for focused donut beams B1 and B2, with central hole sizes of 790 nm and 220 nm, respectively. (c) Simulated light intensity field profile. (d) SEM image of the fabricated engineered microsphere. Scale bars: (a) & (c) 2 μm for x & z, (d) 5 μm.

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Spatial rearrangement of the focal field can be achieved by the partially impaired cylindrical symmetry incurred when an additional linear polarization is applied. In the experimental setup, the S-waveplate has an alignment line mark to help with proper operations. The mark should be aligned in parallel to incident linear polarization orientation to get radial polarization, and perpendicular to get azimuthal polarization. In both cases, the output beam has cylindrical symmetry. With the additional linear polarization, say an x-polarized case, the resulted beam has its E_x component enhanced and its E_y component suppressed, while the longitudinally polarized E_z component is sustained. With the presence of the engineered microsphere (with fabricated diffractive patterns), these electric field components with comparable strengths are found to be directed to the same focal region. As such, the amplitude sum of the remained |E_z|² component at the center and the enhanced |E_x|² component results in a pair of hollow or donut beams at the same focal plane. In the meantime, the impaired cylindrical symmetry deteriorates strengths at other focal fields, and they have become relatively negligible as compared to the paired donut beam region.

The effect of additional linear polarization after the engineered microsphere is first investigated with FDTD simulation. The simulated light fields in XZ-plane is shown in Fig. 4(a), characterized by the square of the electric field |E|². Plotted fields for x, y and z components are also displayed. When a linear polarization is added after the exit surface to manipulate the light field, and when it is matched to either x or y axis of the original illumination, the resulted light field displays either x or y component field. Note that all light fields in Fig. 4(a) are normalized with respect to the maximum intensity value in each of the fields. Assuming the analyzer polarization matches with x-component, the intensity and phase distributions at the three focused beams are demonstrated in Fig. 4(b). Note that the illumination wavelength (405 nm) in Fig. 4 is identical to that in Figs. 1 and 2, which generates three separate focused donut beams along the optical axis, while in Fig. 3, in which the experimental illumination wavelength is from a 532 nm continuous wave (cw) laser, so only two axially located focused donut beams are formed. Such a change in beam profile formation is due to the altered intensity distribution at different diffraction orders. As the wavelength increases, light field intensity at higher diffraction orders are suppressed, so the number of focused donut beams decreases. When the wavelength increases to a certain value, only the zeroth order is significant and thus only one axially elongated focused donut beam is present.

Fig. 4. Lateral beam splitting with additional linear polarization. (a) Total electrical field and Cartesian component fields in XZ-plane, with all field values normalized; (b) X-polarized case: intensity and phase distributions at each of the focusing positions. Scale bars: 2 μm.

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To verify the hypothesis in Fig. 4, an analyzer is placed after the engineered microsphere, before the manipulated light reaches the detector, as seen in Fig. 5(a). Moreover, this arrangement can be numerically modelled in Lumerical software, as seen in Fig. 5(b). By decomposing the electric field into E_x, E_y and E_z, a script is then used to arbitrarily combine them to form a linearly polarized field with any polarization angle. As the illumination plane wave source is set to be linearly polarized, and its polarization angle can also be assigned arbitrarily, such an output light field from this configuration can well simulate the circumstance.

Fig. 5. Experimental result of lateral splitting of focused beams with a linear analyzer. (a) Experimental setup to characterize the beam profiles, illumination wavelength is 532 nm. FZMS: Fresnel zone microsphere. (b) FDTD modelled normalized focused dual-donut field with impaired symmetry due to additional linear polarization within a transverse area of 6λ in size. (c) Experimental result of dual-polarization tuning of focused beam profile to create the double-hole donut beam at the B2 focal position in Fig. 3. Note that in this case, the light field intensity at the original B1 position becomes negligible. Dashed arrows: analyzer polarization; solid arrows: illumination polarization. Scale bar: 5 μm.

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The setup is experimentally executed, and results can be seen in Fig. 5(c). After the analyzer, the axially distributed discrete focused donut beams are converted into a single plane focusing field with the same number of hollowed light field, but now they are positioned aside each other at a single transverse plane. The relative angle between the illumination polarization (linearly polarized) and the analyzer polarization determines the exact shape of transverse light field distribution at the focal plane. The measured results have good agreement with modelled result as seen in Fig. 5(b). As seen from Fig. 5(c), tuning the relative angles of the two linear polarization states leads to changes in the intensity distribution at the focal plane, and the resulted dual-donuts can have their shapes being altered as two elliptical holes, with their major axes forming different angles.

4. Conclusions

This paper proposes a novel scheme of the engineered microsphere coupled with radial polarization illumination (wavelength = 532 nm) to generate a focused light field with specially manipulated wavevectors. At the default state, the emitted light field from the exit surface of the engineered microsphere forms a pair of axially arranged focused donut beams, each with their center-hollowed field in the plane perpendicular to the optical axis. After a polarizer is added into the system, the axially positioned pair is spatially rearranged and reform a neighboring pair of partially merged donut beams in the same plane perpendicular to the optical axis. This design is investigated both numerically and experimentally, and the two groups of results agree with each other. This novel scheme paves a way of a new micro-optics for specialized functional light field generation, with an additional degree of active tuning by inclusion of a polarization during the propagation light path. Such a design can be used in the applications of structured light illumination microscopy, micro-particle manipulation and laser micro-processing.

Funding

Ministry of Education - Singapore (MOE2019-T2-2-147).

Disclosures

The authors declare no conflicts of interest.

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Formation of polarization-dependent optical vortex beams via an engineered microsphere

Abstract

1. Introduction

2. Design and methods

3. Results and discussion

4. Conclusions

Funding

Disclosures

References

Cited By

Figures (5)

Equations (9)

Optics Express