Optical vector fields with kaleidoscopic quasicrystal structures by multiple beam interference

Pi-Hui Tuan; Ling-Qi Huang

doi:10.1364/OE.498717

1. Introduction

Sculpting optical beam structures on transverse dimensions as well as the longitudinally propagation evolution [1–3] have received much attention because of its myriad applications ranging from optical trapping [4], high-resolution imaging [5], material processing [6], optical communication [7], to quantum information [8]. Demonstrated by Durnin in 1987 [9], the pioneering work on Bessel beam with the propagation-invariant feature have launched an important branch of structured lights on generating nearly diffraction-free beams for micromanipulation [10], biophotonics [11], high-aspect-ratio laser machining [12], optical tomography [13], and so forth. Following Durnin’s seminal concepts, various kinds of pseudo-nondiffracting waves with additional characteristics including self-healing, self-accelerating, or auto-focusing have been widely explored over the past decades [14–17]. Among the families of pseudo-nondiffracting beams, the Cosine beam can not only possess a long propagation-invariant distance but exhibit a variety of periodic or quasiperiodic transverse patterns with high spatial degrees of freedom (DOFs) [18,19]. Utilizing multibeam interference techniques, Cosine beams with transverse morphologies from low-order lattices to kaleidoscopic quasicrystal structures have been flexibly generated [20–22]. The generation of quasicrystal beams can essentially provide high-order vortex lattices [21,23] for singular optics applications as well as offer promising light sources for holographic lithography to fabricate photonic quasicrystals [24,25] for developing functional materials.

Apart from merely tailoring amplitude distributions for optical fields, controlling the spatially-dependent polarization states to produce vector structured lights has also become a mainstream issue in beam shaping because of their superior functionalities in comparison with scalar fields. Vector beams with customized polarization and intensity distributions can be used to achieve better focusing properties for super-resolution microscopy [26], to improve the trapping efficiency for optical tweezer [27], and even to offer the classical entanglement for secure communication [28]. By superposing two orthogonally polarized fields prepared via the spatial light modulator [29] or imprinting polarized states on the input beam through the specially-designed polarization optics such as q-plates or S-plates [30], nondiffracting vector Bessel beams with the unique high-numerical-aperture focusing property have been demonstrated [31]. Nevertheless, generating pseudo-nondiffracting quasicrystal beams with spatially inhomogeneous polarizations is seldom discussed so far [32].

In this work, an easily accessible approach based on multibeam interference is proposed to create vector quasicrystal beams with kaleidoscopic intensity and polarization structures. By focusing Q collimated and circularly distributed beams with azimuthally-dependent polarizations encoded using a commercial vortex retarder plate, we create quasicrystal fields exhibiting Q-fold rotational symmetries and polarization-dependent intensities in a flexible manner. To systematically characterize spatial structures of the vector quasicrystal fields, an analytical wave function by interfering multiple vector beams is derived. This wave function allows us to nicely reconstruct the polarization-resolved beam morphologies in the experiment. Demonstrating good agreement between the patterns obtained from experimental measurements and simulated results for both polarization-resolved intensities and distributions of Stokes parameters, we further characterize the states of polarization (SoPs) for vector quasicrystal fields through numerical calculations. Finally, we examine the longitudinal evolution of vector quasicrystal beams to provide fundamentally important information about propagation-variant SoPs. Our findings confirm that the vector quasicrystal fields exhibit SoPs with not only abundant nontrivial polarization singularities but also a feature of continuous evolution on the Poincaré sphere during propagation [33]. As a result, the proposed approach offers a simple yet valuable configuration for generating pseudo-nondiffracting complex light, which hold potential for applications in information processing and high-capacity data transmission.

2. Experimental setup and results

Figure 1 shows the experimental setup to generate vector quasicrystal beams. The light source was a 532-nm green laser followed by a linear polarizer to control the polarization of the input beam. The combination of a spatial filter and a biconvex lens with a focal length of 250 mm was used to eliminate the high-frequency noise while collimating the laser beam with a divergence angle < 0.1 mrad after beam expansion. Stainless-steel sheets, each patterned with a varying number of Q circular apertures evenly distributed on a ring, was fabricated using a laser stencil-cutting machine to serve as masks. The radii w_o of the circular aperture and r_c of the ring were 0.1 mm and 6.5 mm, respectively. To independently adjust the linearly polarized states for the Q collimated beams after passing through the mask, commercial vortex retarders (zero-order vortex half-wave plates for 532 nm, Thorlabs) with different azimuthally-dependent fast-axis orientations were used. Subsequently, a focusing lens with a focal length of f_c= 150 mm was employed to achieve the interference of Q vector beams. Note that both the vortex retarder and the focusing lens were placed as close to the mask as possible for minimizing the slight beam divergence before focusing. For clearer examining the detailed structures of quasicrystal fields formed near the focal region, a reimage lens of a focal length f_r= 50 mm was used to enlarge the field of interest. The distance between the focusing lens and the reimage lens was set to be L≅200 mm in a confocal configuration. To characterize the SoPs of vector quasicrystal fields, a quarter-wave plate and a linear analyzer were used to measure the spatially-dependent Stokes parameters (referred to as Stokes fields) [34,35]. By scanning the image screen along the optical axis at a distance z behind the reimage lens, the polarization-resolved and propagation-evolved intensity patterns of the interfered fields were captured using a digital camera.

Fig. 1. The experimental setup for generating vector quasicrystal fields. The insets show the schematic diagrams for the input polarization at the mask |E_mask> and the fast-axis orientations θ_s for vortex retarders with q = 1/2 and 1.

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The Jones formalism can be conveniently utilized to analyze the transformation for the polarized state by the vortex retarder on the input beam. Without voltage control on the phase retardance, the vortex retarder can be viewed as a q-plate with a fix phase retardance at a specified wavelength [36]. The Jones transformation matrix M for a q-plate can be expressed as [37]

(1)$${\mathbf M} = \left[ {\begin{array}{{cc}} {{t_x}{{\cos }^2}\theta + {t_y}{e^{i{\kern 1pt} \delta }}{{\sin }^2}\theta }&{\sin \theta \cos \theta ({{t_x} - {t_y}{e^{i{\kern 1pt} \delta }}} )}\\ {\sin \theta \cos \theta ({{t_x} - {t_y}{e^{i{\kern 1pt} \delta }}} )}&{{t_x}{{\sin }^2}\theta + {t_y}{e^{i{\kern 1pt} \delta }}{{\cos }^2}\theta } \end{array}\,} \right]{\kern 1pt} \,, $$

where θ=q·φ+φ_o is the fast-axis orientation with respect to the horizontal axis, and φ is the azimuthal angle on the waveplate, φ_o is the reference orientation, t_x and t_y are the transmission coefficients respectively for the polarized lights parallel and perpendicular to the fast axes, and δ is the retardation of the waveplate. For the commercial vortex retarder used in the experiment, the fixed retardance δ=π and the transmission coefficients can be assumed to be t_x= t_y= 1 at 532 nm. By setting φ_o= 0 to be aligned with the horizontal axis of experimental coordinates, the transformation matrix for the used vortex retarder can be simply given by

(2)$${\mathbf M} = \left[ {\begin{array}{{cc}} {\cos 2\theta }&{\sin 2\theta }\\ {\sin 2\theta }&{ - \cos 2\theta } \end{array}\,} \right]\,. $$

Two vortex retarders with different azimuthally-dependent fast-axis orientations corresponding to q = 1/2 and 1 were used in the experiment as shown by the insets of Fig. 1. For convenience, we set the input laser beam at the mask to have a uniformly horizontal polarization throughout the experiment. The Jones vector for the electric field at mask can be written as |E_mask>=[1, 0]^T, where T stands for the matrix transpose. Based on Eq. (2), the polarization states of Q collimated beams will be transformed to be |E_Q>=[cos(2θ_s), sin(2θ_s)]^T after passing through the vortex retarder. Here, the corresponding fast-axis orientations for collimated beams on the vortex retarder are θ_s= q·2πs/Q, where s = 0, 1, 2, …, Q−1.

In order to analyze the SoP of interfering fields by Q vector beams, the measurement for Stokes parameters was performed [34,35]. According to polarization optics, the Stokes parameters (S₀, S₁, S₂, S₃) for optical fields under the paraxial condition can be determined by six measurable intensities as

(3)$$\left\{ \begin{array}{l} {S_0} = {I_H} + {I_V}\\ {S_1} = {I_H} - {I_V}\\ {S_2} = {I_{ + 45^\circ }} - {I_{ - 45^\circ }}\\ {S_3} = {I_R} - {I_L} \end{array} \right.,$$

where I_H, I_V, I ₊_45°, I_−45°, I_R, and I_L respectively correspond to the horizontally, vertically, linearly +45°, linearly −45°, right-handed circularly, and left-handed circularly polarized components. With the Stokes parameters, the orientation angle ψ and ellipticity angle χ for the polarization ellipse of a fully polarized optical field can be evaluated as ψ=tan^-1(S₂/S₁)/2 and χ=sin^-1(S₃/S₁)/2. Figures 2(a)–2(c) show total (I_o) and polarization-resolved intensity patterns of vector quasicrystal fields observed at z = 1250 mm generated by Q = 8, 12, and 21 when using a vortex retarder with q = 1/2. As shown by the first columns of Figs. 2(a)–2(c), the polarized states |E_Q> for the Q beams are radially polarized for the three cases according to previous discussion. The total intensities I_o shown in the 2^nd columns of Figs. 2(a)–2(c) present Q-fold quasiperiodic structures similar to the results by interfering Q scalar beams [20,21]. However, it is noticeable that the vector quasicrystal patterns demonstrate higher spatial frequencies compared to the scalar quasicrystal fields under the same Q. On the other hand, because of the azimuthal symmetry for SoPs of the Q vector beams, the linearly polarization-resolved intensities show mirror symmetries along the axis perpendicular to the transmission axis of analyzer. The patterns of I_R and I_L for Q = 8 and 12 exhibit the same structures as I_o, owing to the fact that the vector beam sources for interference appear in pairs along the symmetric axes (indicated by gray dashed lines in the first columns of Figs. 2(a)–2(c)). However, for Q = 21, where vector beam sources are not present in pairs, I_R and I_L can be seen to have structures different from that of I_o.

Fig. 2. Total (I_o) and polarization-resolved intensity patterns of vector quasicrystal fields observed at z = 1250 mm for the cases of (a) Q = 8, (b) 12, and (c) 21 when using a vortex retarder with q = 1/2. 1^st columns show SoPs for Q beams before interference. Gray dashed lines mark the symmetric axes for Q vector sources. I_H, I_V, I ₊_45°, I_−45°, I_R, and I_L respectively correspond to horizontally, vertically, linearly +45°, linearly −45°, right-handed circularly, and left-handed circularly polarized intensities.

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Figures 3(a)–3(c) display the results of vector quasicrystal fields generated under experimental parameters similar to those of Figs. 2(a)–2(c) but with the replacement of vortex retarder from q = 1/2 to q = 1. The SoPs of Q vector beams, illustrated in the first columns of Figs. 3(a)–3(c), distinctly indicate that the total rotation angle for the initially horizontal polarization evolved along the azimuthal direction becomes twice than that of the case with q = 1/2. For Q = 8, the polarized states of 8 beams can be divided into two orthogonal groups of horizontal and vertical polarizations with relative azimuthal angle of π/4. Consequently, I_H and I_V can be seen to reveal 4-fold crystalline structures. Similarly, for Q = 12, there are three groups of polarization orientations, i.e. 0, π/3, and 2π/3, with relative azimuthal angle of π/6. Hence, linearly polarized intensities resemble to the 4-fold superlattice structures, especially evident in I_V [38]. As mentioned earlier, for Q = 21, where the polarization orientations of the vector beams cannot be paired because of the incommensurate θ_s, both I_R and I_L exhibit clearly distinct structures from I_o. By comparing the transverse patterns between I_o and the linearly polarized intensities, it can be seen that the central Bessel ring-like structures split into four lobes after the analyzer, while the outer dendrite structures present increased intricacy compared to the situation with q = 1/2. Notably, the overall spatial frequencies of the interfered fields in Figs. 3(a)–3(c) are higher than those in Figs. 2(a)–2(c). Though the effective observation range for the interfered field is mainly determined by the size of circular aperture, critical dimensions for quasicrystal fields can be seen to vary with Q. When interfering multiple plane waves to generate lattice-like optical fields, it can be shown that the lattice period Λ is determined by the value of transverse wave vector k_T for the partial waves as Λ=π/k_T based on the concept of Fourier optics [22]. Therefore, the critical dimensions and thus the spatial frequencies for the quasicrystal fields generated by multibeam interference can be deduced to be positively correlated to π/[2·r_c·sin(2π/Q)] in current experiment. Consequently, as the azimuthal gradients for the illuminating beams increase, the spatial frequencies of interfered fields also increase.

Fig. 3. Total (I_o) and polarization-resolved intensity patterns of vector quasicrystal fields observed at z = 1250 mm for the cases of (a) Q = 8, (b) 12, and (c) 21 when using a vortex retarder with q = 1. 1^st columns show SoPs for Q beams before interference. Gray dashed lines mark the symmetric axes for Q vector sources. I_H, I_V, I ₊_45°, I_−45°, I_R, and I_L respectively correspond to horizontally, vertically, linearly +45°, linearly −45°, right-handed circularly, and left-handed circularly polarized intensities.

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3. Numerical reconstruction and analyses

To explicitly characterize the states of polarization (SoPs) for the generated vector quasicrystal fields, an analytical wave function for the interference of multiple Gaussian beams with azimuthally-dependent polarizations is further derived to reconstruct the experimental results. Generalized from the current model [21], the electric field by Q collimated vector Gaussian beams before focusing can be expressed as

(4)$${{\mathbf \Psi }_o}({\mathbf r^{\prime}}) = \left[ {\begin{array}{{c}} {{\Psi _{ox}}({\mathbf r^{\prime}})}\\ {{\Psi _{oy}}({\mathbf r^{\prime}})} \end{array}} \right] = \sqrt {\frac{1}{Q}} \sqrt {\frac{\textrm{2}}{{\pi w_o^2}}} \sum\limits_{s = 0}^{Q - 1} {{e^{i{\kern 1pt} s{\kern 1pt} \alpha }}\exp \left( { - \frac{{{{({{\mathbf r^{\prime}} - {{\mathbf r}_s}} )}^2}}}{{w_o^2}}} \right)} \left[ {\begin{array}{{c}} {\cos 2{\theta_s}}\\ {\sin 2{\theta_s}} \end{array}\,} \right]\,\,, $$

where α is the global phase shift between adjacent beams, r=(x, y) is the coordinates of near-field plane, r_s= (r_ccosϕ_s, r_csinϕ_s), ϕ_s= 2πs/Q, and the Gaussian beam waist w_o can be approximated by the radius of the circular apertures. The transverse wave function for optical field at the plane z after a paraxially optical system described by the ABCD transfer matrix can be evaluated from the Fresnel-Huygens integral as

(5)$${\mathbf \Psi }({\mathbf r},z) = {e^{i{\kern 1pt} k{\kern 1pt} z}}\,\int_{all} {{{\mathbf \Psi }_o}({\mathbf r^{\prime}})h({\mathbf r^{\prime}},{\mathbf r}){d^2}{\mathbf r^{\prime}}}$$

with the kernel function to be

(6)$$h({\mathbf r^{\prime}},{\mathbf r}) = \frac{1}{{i\lambda B}}\exp \left[ {i\frac{k}{{2B}}({A{{|{{\mathbf r^{\prime}}} |}^2} + D{{|{\mathbf r} |}^2} - 2{\mathbf r^{\prime}}\cdot {\mathbf r}} )\,} \right]\,, $$

where k = 2π/λ is the wave number of wavelength λ, and r = (x, y) is the coordinates of observation plane. Considering no astigmatism in the optical system, the x- and y-parts of the integral in Eq. (5) can be separately carried out [21]. After some algebra, the interfered wave function of optical vector field at the observation plane z can be analytically derived as

(7)$${\mathbf \Psi }({\mathbf r},z) = \left[ {\begin{array}{{c}} {{\Psi _x}({\mathbf r},z)}\\ {{\Psi _y}({\mathbf r},z)} \end{array}\,} \right] = {e^{i{\kern 1pt} \xi ({\mathbf r},z)}}\,{e^{ - i{{\tan }^{ - 1}}\left( {\frac{B}{{A{\kern 1pt} {z_R}}}} \right)}}\sqrt {\frac{2}{{Q{\kern 1pt} \pi {w^2}(z)}}} \left( {\sum\limits_{s = 0}^{Q - 1} {{\Phi _s}({\mathbf r},z)\left[ {\begin{array}{{c}} {\cos 2{\theta_s}}\\ {\sin 2{\theta_s}} \end{array}\,} \right]} \,} \right)$$

with

(8)$${\Phi _s}({\mathbf r},z) = {e^{i{\kern 1pt} s{\kern 1pt} \alpha }}\exp \left\{ {\,\frac{{ik}}{{2{q_o}(z)}}[{{{({x - A{x_s}} )}^2} + {{({y - A{y_s}} )}^2}\,} ]} \right\}\,, $$

where ξ(r,z)=k{z + [C|r|²/(2A)]}, w²(z)=w_o²[A²+ (B/z_R)²], R(z) = (A/B)[(Az_R)²+ B²], z_R=πw_o²/λ, and 1/q_o(z) = 1/R(z) + 2i/[kw²(z)]. Once the mask, the retarder, and the focusing lens are precisely aligned with the optical axis, the effect of global phase shift on the interfered field is small and can be ignored. For concise presentation, we assume α=0 in the numerical reconstruction and will address the influence of non-uniform α on vector quasicrystal fields in later discussion. According to the experimental configuration shown in Fig. 1, the ABCD transfer matrix T of the optical system for multi-beam focusing and propagation can be given by

(9)$${\mathbf T} = \left[ {\begin{array}{{cc}} A&B\\ C&D \end{array}\,} \right] = \left[ {\begin{array}{{cc}} {({1 - {z / {{f_r}}}} )({1 - {L / {{f_c}}}} )- {z / {{f_c}}}}&{L + z - {{zL} / {{f_r}}}}\\ {{L / {{f_c}{f_r}}} - {1 / {{f_c}}} - {1 / {{f_r}}}}&{1 - {L / {{f_r}}}} \end{array}\,} \right]\,. $$

Substituting the experimental parameters into Eqs. (7)-(9), the total intensity distribution I_o and the polarization-resolved patterns of vector quasicrystal field can be respectively calculated as follows: I_o=|Ψ_x|²+|Ψ_y|², I_H=|Ψ_x|², I_V=|Ψ_y|², I ₊_45°=|(Ψ_x+Ψ_y)/√2|², I_-_45°=|(Ψ_x−Ψ_y)/√2|², I_R=|(Ψ_x−i·Ψ_y)/√2|², and I_L=|(Ψ_x+ i·Ψ_y)/√2|². Figures 4(a)–4(c) and Figs. 5(a)–5(c) show the numerical calculations respectively corresponding to the experimental results shown in Figs. 2(a)–2(c) and Figs. 3(a)–3(c). The good agreement between the experimental observations and the calculated patterns ensures that the theoretical wave function of Eq. (7) can be further exploited to numerically analyze the polarization structures of vector quasicrystal fields. Since the size of circular aperture on the mask is far larger than the laser wavelength in current experiment, the vector quasicrystal fields close to the optical axis and near the focal plane of focusing lens can also be simply modelled by the superposition of multiple plane waves as [20]

(10)$${{\mathbf \Psi }_Q}(\rho ,\varphi ,z) = \left[ {\begin{array}{{c}} {{\Psi _Q}_x(\rho ,\phi ,z)}\\ {{\Psi _Q}_y(\rho ,\phi ,z)} \end{array}\,} \right] = \frac{1}{{\sqrt Q }}\left( {\sum\limits_{s = 0}^{Q - 1} {{e^{i{\kern 1pt} s{\kern 1pt} \alpha }}\exp \left[ { - \frac{{ik{r_c}}}{{{f_c}}}\rho \cos ({\phi - {\phi_s}} )} \right]\left[ {\begin{array}{{c}} {\cos 2{\theta_s}}\\ {\sin 2{\theta_s}} \end{array}\,} \right]} \,} \right), $$

where (ρ, ϕ) are the polar coordinates of (x, y). Note that the optical fields focused via a lens given by Eq. (10) can be interpreted as the Fourier transform of the near-field distribution, i.e. the Fraunhofer diffraction, according to Fourier optics. For clearer comparison, the final columns of Figs. 4(a)–4(c) and Figs. 5(a)–5(c) show the total intensities I_o of vector quasicrystal fields calculated by the plane-wave superposition. For Q = 8 and 12 with relatively less beam numbers, the effect from the finite aperture size can be neglected. The numerical results by Eq. (10) are almost the same as those by Eq. (7). As the input beam number becomes so large and each beam is so close to other neighboring beams, the diffraction effect by the finite-size aperture cannot be ignored. Consequently, the results by Eq. (10) show significant differences from the calculations by Eq. (7).

Fig. 4. Numerical reconstructions corresponding to experimental results for cases of (a) Q = 8, (b) 12, and (c) 21 shown in Figs. 2(a)–2(c). The final columns show the calculations by plane-wave superposition for comparison.

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Fig. 5. Similar presentation as Figs. 4(a)–4(c) but for reconstructing the experimental results shown in Figs. 3(a)–3(c).

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Using the polarization-resolved intensities by experimental measurement and numerical calculation, we conducted further analysis on the Stokes fields (S₀, S₁, S₂, S₃) to explicitly characterize the SoPs of vector quasicrystal fields. Figures 6(a)–6(c) show experimental and numerical results for the orientation angle ψ and ellipticity angle χ of the polarization ellipse in the regions of interest (ROIs) for vector quasicrystal fields of Q = 8, 12, and 21 by the vortex retarder of q = 1/2. Figures 7(a)–7(c) display similar presentation but for the cases by q = 1. Note that the ROIs are marked by the dashed boxes overlaid on I_o in Figs. 2(a)–2(c) to Figs. 5(a)–5(c). In the experiments of multibeam interference, tilted optical components such as the focusing lens and the mask introduce global phase shift α between the beams to be interfered [20,39]. To demonstrate the effect of global phase shift on the vector quasicrystal fields, Fig. 8 shows numerical calculations of I_o along with the corresponding ψ and χ in the ROI for the case of Q = 8 and q = 1/2, under different values of α. It can be seen that the structure of I_o is gradually distorted with broken rotational symmetry as α increases. The break of rotational symmetry causes an obvious change on the χ distribution. However, when α is controlled to remain smaller than π/18, the variations in I_o and ψ can be neglected. On the other hand, when the introduced global phase satisfies α=2mπ/Q, where m = 1, 2, …, Q−1, the rotational symmetry of the interfered field is restored, leading to vector quasicrystal fields with more abundant and higher DOF structures [22,23]. For practical applications, the challenge associated with systematically establishing the relationship between α and the misalignment of optical components is pivotal and warrant deeper investigation in the future. Despite the inherent misalignment from slightly tilted or off-axis optical components as well as the finite spatial resolution of imaging camera inevitably cause local discrepancies, the global structures still exhibit good agreement between the experimentally measured and theoretically predicted distributions for ψ and χ. Thus, the numerical calculations prove instrumental in highlighting the inherent characteristics of the SoPs within the vector quasicrystal fields.

Fig. 6. Experimental and theoretical analyses of polarization structures by the orientation angle ψ, ellipticity angle χ, and the map of polarization ellipse for the ROIs of vector quasicrystal fields in cases of Q = (a) 8, (b) 12, and (c) 21 by q = 1/2. V-points, C-points, and L-lines are marked by the green spots, black crosses, and gray dashed lines, respectively. Note that the red/blue color in the map of polarization ellipse shown in the last columns denotes the left-/right handedness of polarization.

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Fig. 7. Similar presentation as Figs. 6(a)–6(c) but for q = 1.

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Fig. 8. Numerical calculations of I_o and the corresponding ψ and χ in the ROI for the case of Q = 8 and q = 1/2 under different values of the global phase shift α.

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For vector optical fields, polarization singularities play a crucial role in characterizing the topological structures of SoPs. Compared to phase singularities, polarization singularities have greater spatial DOFs, which are beneficial for applications in optical micromanipulation, optical micromachining, and information communication [40–44]. Generic polarization singularities in the cross section of paraxial vector fields are C-points, L-lines, and V-points [34]. The C-point, surrounded by neighboring elliptically polarized points, signifies the position of circular polarization with an undefined ψ. Similarly, the V-point corresponds to a position with undefined ψ, situated within a local region composed of linearly polarized points. The L-line connects positions of linear polarization, with an undefined handedness, effectively dividing the right- and left-handed elliptically polarized points. The last columns of Figs. 6(a)–6(c) and Figs. 7(a)–7(c) show the SoPs of vector quasicrystal fields within the ROIs by the map of polarization ellipse. Since the positions of undefined orientation are singular points (marked by the white spots) of the distribution ψ, their intersection with χ=0 and ±π/4 determine the V-points and C-points, respectively. V-points are marked by green spots, while C-points are indicated by black crosses in the theoretical SoPs. Additionally, L-lines are depicted by gray dashed lines in the numerical χ distributions for clarity. Notably, the central positions for all instances of Q = 8, 12, and 21, for both q = 1/2 and 1, are clearly identified as V-points. The analyzed results reveal that the vector multibeam interference can create abundant nontrivial polarization singularities [45].

Finally, to obtain the fundamentally important information for the propagation-variant SoPs of vector quasicrystal fields, the longitudinal evolution for the polarization-resolved intensities are further examined. Figure 9 shows the comparisons between the experimental and numerical results of I_o patterns as well as ψ distributions in the ROIs marked by the square boxes for vector quasicrystal fields of Q = 21 and q = 1 at different observation plane z from 250 to 850 mm. As z is larger than the effective overlapping distance for Q focused Gaussian beams, the inner structures for interfered fields remain nearly unchanged during propagation [21]. From the experimental results, the divergence angle for the vector quasicrystal beam by the proposed configuration was estimated to be around 4 mrad. It is worthy to note that the nearly diffraction-free region for the interfered quasicrystal fields can be further increased by utilizing a focusing lens with a longer focal length [20]. With good agreement between the numerical and experimental results, the calculated fields were further used to numerically characterize the SoP evolution for vector quasicrystal fields during propagation as shown in the bottom row of Fig. 9. With a small but non-negligible divergence angle for interfered fields, the polarization ellipse and the positions of polarization singularities can be seen to locally vary during the beam propagation. To clearer examining how the SoP of vector quasicrystal field evolves along the longitudinal direction, the numerical analysis was further performed in a small range of z near the focal plane. Figure 10 shows the total intensities I_o and the corresponding maps of polarization ellipse in the first quadrant of the ROI shown in Fig. 4(b) for Q = 12 and q = 1/2 at the observation planes of z = 1130 to 1170 mm. Except for the positions on the L-lines, the polarization state at most sites can be found to continuously transform in the small-range propagation. For clearer demonstration, the evolved polarization states at the marked positions (i) and (ii) are projected on the Poincaré sphere. It can be seen that the polarization evolution smoothly follows a continuous path on the Poincaré sphere in both cases. The continuous transformation for the polarization states is important for the application of complex vector light in high-capacity information processing based on classical entanglement [41,44].

Fig. 9. Experimental and theoretical analyses for the propagation evolution of total intensity I_o and orientation angle ψ for the vector quasicrystal field of Q = 21 and q = 1 at different observation plane z. The bottom row shows the corresponding SoPs with the central V-points marked by the green spot and C-points marked by the black crosses.

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Fig. 10. The propagation-evolution of total intensities I_o and map of polarization ellipse at the 1^st quadrant of the ROI shown in Fig. 4(b) for Q = 12 and q = 1/2. The positions (i) and (ii) marked by the small square boxes demonstrate the continuous transformation of polarization states along the specific paths on the Poincaré sphere.

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4. Conclusion

In conclusion, vector quasicrystal beams with spatially-dependent polarization states have been generated by a simple approach based on multibeam interference. By changing the number Q of input collimated beams as well as their azimuthally-dependent polarizations via using different vortex retarders, various quasicrystal fields with kaleidoscopic intensity and polarization structures can be flexibly realized. More importantly, it has been verified that all polarization-resolved intensities and distributions of Stokes parameters for the generated quasicrystal fields can be well reconstructed by an analytical wave function derived from the diffraction theory for the interference of multiple vector beams. Based on good agreement between numerical calculations and experimental observations, the derived wave function has been further employed to numerically characterize the propagation-variant polarization states to provide fundamentally important information for the vector quasicrystal beams. With abundant nontrivial polarization singularities and the important feature for continuously transforming polarization states in a small-range propagation, vector quasicrystal fields generated by the proposed approach can serve as a source of pseudo-nondiffracting complex light, offering considerable potential for various applications.

Funding

National Science and Technology Council (MOST-111-2112-M-194-002-MY3).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Optical vector fields with kaleidoscopic quasicrystal structures by multiple beam interference

Abstract

1. Introduction

2. Experimental setup and results

3. Numerical reconstruction and analyses

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (10)

Equations (10)

Optics Express