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Abstract
In this paper, we propose a method for the generation of a two-dimensional spin-orbit beam lattice using a Dammann grating. A Dammann grating is fabricated and is illuminated by ellipse field/vector filed singular beam that contains a polarization singularity. Since, Dammann grating is used to produce equal-intensity light spots, each of the spin-orbit beams in the lattice has equal intensity distribution. Interestingly, they also have the same polarization distribution as that of the input light. Unlike the interferometric methods of lattice generation, the diffractive method proposed here produces lattices of the same index singularities. Simulation and experimental results are provided to demonstrate the concept.
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1. Introduction
Optical lattices are periodic structures of the electromagnetic fields that can be generated by various methods like interference and diffraction. These periodic structures can be in intensity [1,2], phase [3,4], or in polarization distributions [5,6]. A lattice generator is an optical system that divides a single incident light beam into a one- or two-dimensional lattice of beamlets. The Research community has shown lot of interest in lattice generators because of its uses in many fields such as optical data storage [7], multiple image generator [8], and coherent summation of laser beams [9]. Several techniques for lattices generation have been proposed, including Fraunhofer diffraction with special computer-generated gratings [1], Fresnel diffraction with the Talbot effect [10], spatial filtering with a phase-contrast method [11], and a waveguide optical technique with grating couplers [12]. Dammann grating (DG) is a type of binary surface-relief grating that can be used for lattice illumination in the Fourier-transform plane [1,13]. Much progress in the use of binary-phase gratings for lattice illumination has been made in its design, fabrication, and applications. DG is a pure-binary-phase-modulation grating with phase transition points optimized to produce equal-intensity spots with high efficiency at desired diffractive orders. DG is a key approach to uniformly distribute energies among the designed diffraction orders without changing the mode of generation. It is important to note that previously designed DGs are aimed at distributing equal intensity or energy in all the desired diffracted orders. Attention was never aimed at preserving the phase and polarization distribution of beams in its diffracted orders. We have recently shown that although the design optimization is done taking into account equal intensity at every diffraction order, the phase and polarization structure of the incident beam is also preserved in all the diffraction orders [14]. Earlier reported study was restricted to V-points lattice generation which is a specific case of Higher-order Poincarè sphere (HOPS) beams. The DG is used here to realize lattice of spin-orbit beams containing polarization singularities. In this paper, we report the generation of lattice of spin-orbit beams which include beams represented by points on HOPS as well as Hybrid order Poincarè Sphere (HyOPS). It is important to note that, unlike interference based methods, here the lattice consists of singularities with the same topological index. In interferometric methods, lattices with both positive and negative index polarization singularities are generated.
Polarization singularity is a point azimuth defect at which the azimuth ($\gamma$) of a polarization ellipse/linear polarization becomes indeterminate [15,16]. In the immediate neighbourhood of a polarization singularity, the azimuth undergoes rotation and thus, the gradient of the azimuth has a non-zero curl around a polarization singularity, $\nabla \times \nabla \gamma \neq 0$. Polarization singularities can be created by superimposing two beam carrying orthogonal orbital angular momentum in orthogonal spin states [17]. Due to the presence of both the spin and orbital angular momenta these beams are also known as spin-orbit beams and have applications in diverse areas. Spin-orbit beams have found applications in optical chirality measurement [18], edge enhancement [19], turbulence-free propagation in atmosphere [20]. C-point, V-point, and L-line are the three types of polarization singularities. V-points are surrounded by spatially varying polarization distribution of linear polarization, whereas C-points and L-lines are surrounded by inhomogeneous distribution of elliptical polarization. In an ellipse field distribution, the L-lines separate the regions of the right and left-handed polarization states. At V-point polarization singularity, azimuth of linear polarization can not be defined. V-points occur at the intensity null point. Beams containing V-point polarization singularities are represented by equatorial points on HOPS [21,22]. North and South poles in HOPS represent right and left circularly polarized vortex beams with topological charges $p$ and $-p$, respectively. Except for the polar points, each point on the HOPS represents a vector vortex beam (VVB) with constant ellipticity and varying azimuth. On the other hand, at a C-point polarization singularity, the state of polarization is circular. C-points can be found at any intensity level. The beams containing C-point polarization singularities are represented by points on HyOPS [23,24]. A non-polar point on the HyOPS can serve as a representation for each C-point polarization singularity distribution. North and South poles in a HyOPS represent right and left circularly polarized vortex beams, respectively, with topological charges $p$ and $q$ that differ from each other. In the initial seminal papers by Freund [15] and Dennis’ [25], C-points are referred to as ellipse field singularities and the SOP distribution in the immediate neighbourhood are primarily ellipses, while V-points are referred as vector field singularities and SOP distributions are primarily linear. But in recent years some authors refer to all polarization singularities that are represented by both HOPS and HyOPS as VVB, since all these polarization singularities can be represented as the superposition of vortex beams in orthogonal polarization state (vector states). The C-points are specific to HyOPS whereas V-points form a subset of beams represented by HOPS.
C-point and V-point polarization singularities are characterized by topological indices $I_c$ and $\eta$, and are defined as:
The polarization azimuth $\gamma$ can be found using the Stokes phase. It can be obtained by using the Stokes parameters $S_0, S_1, S_2$ and $S_3$. Using $S_1$ and $S_2$ a complex Stokes field $S_{12}=S_1+iS_2$ can be constructed. The Stokes phase distribution is the phase distribution of the complex field $S_{12}$ and is given by $\phi _{12}=Arg\{S_{12}\}$. Stokes phase and azimuth are related by relation $\phi _{12} = 2\gamma$. Since polarization singularities are singularities in azimuth distribution, phase singularities of $S_{12}$ are actually polarization singularities.
Spin-orbit beams can be generated using q-plates [26,27], metasurfaces [28], interferometric [16,17] and diffractive methods. In interferometric setups, the use of spatial light modulators (SLMs) and digital micromirror devices have increased the flexibility of the generation methods [29–31], but at the cost of decreased generation efficiency. Many of these methods mentioned above produce beams with isolated singularities. However, it is sometimes necessary to generate multiple spin-orbit beams at the same time. In laser processing, radially and azimuthally polarized lights are used for laser cutting and punching respectively [32]. In optical trapping, an array of vector beams is critical for faster sorting of micro/nanoparticles [33]. Spin-orbit beam lattice could be useful in multichannel free-space optical communication [34]. In this paper, we describe how the unique property of DG in producing equal intensity light spots can be extended to realize lattice of the spin-orbit beam where phase and polarization distributions are also found to be preserved.
This article is structured as follows. Section 2 describes some of the features of DG. In section 3 with the help of simulations, we present the study of DG under spin-orbit beam illumination. Section 4 contains the experimental scheme and results. Finally, in section 5, concluding remarks are made.
2. Dammann grating
Dammann and Gortler in $1972$ proposed the use of 2D phase structure to create multiple images of an object with high efficiency and uniformity. It was discovered as a result of their attempts to make multiple copies of an object using a holographic structure [1,35,36]. DGs are binary phase structures whose transmission function can be written as $T = T_0 \exp (i\Phi )$, where phase ($\Phi$) is spatially varying and has values, 0 or $\pi$ and amplitude $T_0$ is constant. The transmission function has only two values: 1 and - 1, which correspond to the phase values of 0 and $\pi$, respectively.
DG is a periodic structure whose period is taken as unity. The transmission function $T(x)$ for half period of the DG can be written as [1]:
In the above equation, $A_m$ is the amplitude of the plane wave diffracted in the $m^{th}$ order, which travels at an angle $\theta _m$ given by
where $f_x$ is spatial frequency corresponding to the plane wave traveling at an angle $\theta _m$ to the z-axis and $\lambda$ is the wavelength of light. Phase transition points in the grating are found with the condition that the amplitude at every desired diffraction order is same, which means $|A_m| = |A_0|$, for every $m$, where $A_0$ is the amplitude in the zeroth diffraction order.3. Dammann grating under spin-orbit beam illumination
In the last section, it is discussed that DG acts as a beam splitter under plane wave illumination and splits the input beam into multiple beams each with equal intensity. This feature of DG makes it a useful element in many applications, such as multiplexing, array illumination, and so on. In this section, we study the performance of DG when illuminated by a spin-orbit beam. The incident beam embedded with polarization singularity can be expressed as a superposition of beams in orthogonal spin and orbital angular momentum states and can be written as
Here, the diffracted light from the DG is expressed in terms of waves traveling in different directions. Let us consider the two-phase terms in Eq. (7) for each of the waves in the summation. The phase terms $\exp (i 2\pi mf_xx)$ correspond to the plane waves traveling in different directions. The second phase term appears as $\exp {(i p \theta )}$ and $\exp {(i q \theta )}$. Each of these term indicates that there is a phase vortex with topological charge $p$ or $q$ in each of the diffracted order $m$. The amplitude of the right circular polarization (RCP) and left circular polarization (LCP) component is $A_mAr^{|p|}$ and $A_mBr^{|q|}$ respectively. The phase difference between RCP and LCP components is $(q-p)\theta$. Hence the resultant beam after the interference of RCP and LCP components has the same polarization distribution as the incident beam. Different combination of vortex charges $p$ and $q$ in the RCP and LCP components result in desired polarization singularity.
Some of the outcomes of the proposed method are presented here. Figures 1–2 show the simulated $2\times 2$ lattice of C-point polarization singularities with $I_c=\pm \frac {1}{2}$ and $\pm 1$ indices. Simulated intensity distributions for bright C-point obtained upon diffraction at the image plane for lemon and star are shown in Figs. 1(a) and (d) respectively. Lemon and star are C-point polarization singularities with $I_c= +1/2$ and $-1/2$ respectively. Note that each of the intensity spots in the $2 \times 2$ lattice has equal distribution of intensity due to DG. Also in each of these figures, Stokes phase distributions are also shown to identify the presence of polarization singularities in the diffraction orders. In Fig. 1 red and blue color represent right and left handed polarization states respectively. This color convention will be followed throughout the manuscript.
Fig. 1. Simulated $2 \times 2$ equal intensity spots produced by DG each embedded with lemon (upper panel) and star (lower panel) bright C-points (HyOPS beams) of index $I_c =\pm {1/2}$. (a,d) Total intensity distribution; (b,e) Stokes phase distribution and (c,f) polarization distribution of lemon and star bright C-points. Simulation parameters of incident beams (in eqn. (5)) for lemon and star lattices are $(\frac {A}{B}=1, p=0$, $q=+1, \theta _0=0$) and ($\frac {A}{B}=1, p=0, q=-1, \theta _0=0$) respectively.
Fig. 2. Simulated $2 \times 2$ equal intensity spots produced by DG each embedded with dark C-point (HyOPS beams) of polarization index $I_c =+1$ (upper panel) and $I_c =-1$ (lower panel). (a,d) Total intensity distribution; (b,e) Stokes phase distribution and (c,f) polarization distribution dark C-points. Simulation parameters of incident beams (in eqn. (5)) for $I_c =+1$ and $I_c =-1$ lattices are $(\frac {A}{B}=1, p=-3, q=-1, \theta _0=0$) and ($\frac {A}{B}=1, p=3, q=1, \theta _0=0$) respectively.
It can be seen from Figs. 1(b) and (e) that all the diffracted spots are embedded with the same Stokes index of +1 and −1 respectively. The polarization distributions for lemon and star that are reconstructed by DG are shown in Figs. 1(c) and (f) respectively. Simulated results are given for the case when amplitude weight factors A and B are same. The difference in A and B changes the radius of the L-line in the polarization distribution. When $A=B$, the beams are represented by equatorial points on the HyOPS whereas for $A\ne B$ the beams are represented by non-equatorial points on HyOPS. The action of DG is independent of the topological index of the incident beam, and the index in each diffraction order are same as that of the incident beam. Same topological index, helicity and other attributes of the singularity in every diffraction order is due to the fact that the DG itself is not a phase vortex producing element. Therefore, the proposed grating can be used to generate a lattice of C-points of any index. Figure 2 shows the simulated results for lattice of dark C-points with $I_c =\pm {1}$. Doughnut shape intensity distribution can be observed fro Figs. 2(a), (d) as the beams are embedded with dark C-points. Similarly, simulated results for the HOPS beam represented by non-equatorial points with coordinates (latitude = $\pi /4$, longitude = $\pi /2$) are shown in Fig. 3(a–c) and 3(d–f) respectively.
Fig. 3. Simulated $2 \times 2$ equal intensity spots produced by DG each embedded with HOPS beams of polarization index $\eta =+1$ (upper panel) and $\eta =-1$ (lower panel) and represented by non-equatorial points on HOPS with coordinates (latitude = $\pi /4$, longitude = $\pi /2$). (a,d) Total intensity distribution; (b,e) Stokes phase distribution and (c,f) polarization distribution of HOPS. Simulation parameters of incident beams (in Eq. (5)) for $\eta =+1$ and $\eta =-1$ lattices are $(\frac {A}{B}\approx 2.4, p=-1, q=+1, \theta _0=\frac {\pi }{2}$) and ($\frac {A}{B}\approx 2.4, p=1, q=-1, \theta _0=\frac {\pi }{2}$) respectively.
4. Experimental setup and results
Electron beam lithography (EBL) is used to fabricate a DG using the standard procedure given in [14]. Indium tin oxide (ITO) coated glass plate is used as a substrate for e-beam lithography. Positive e-beam resist namely 950K poly methyl methacrylate (PMMA)-A4 is spin-coated onto the substrate. Using the procedure described in [14] binary grating pattern is transferred on to the substrate. The experimental setup used for generating the spin-orbit beam lattice using designed DG is shown in Fig. 4.
Fig. 4. Schematic of the experimental setup. He-Ne Laser, SF: Spatial filter, L: lenses, I: Iris, HWP: half-wave plate, SWP: S-wave plate, SPP: Spiral phase plate, QWP: Quarter wave plate, P: Polarizers, DG: Dammann grating, CCD: Charge-coupled device camera and PC: Computer/Laptop.
The polarization singular beams are generated by different combinations of topological charges $p$ and $q$ in orthogonal spin-orbit states using a combination of the S-wave plate [26,27] and spiral phase plate (SPP). S-wave plate is a spatially varying half-wave plate (H-SWP) or spatially varying quarter-wave plate (Q-SWP) that converts spin-angular momentum into orbital angular momentum in light beams [26,38]. SPP on the other hand has a spirally varying thickness, suitable to create an orbital angular momentum state of light [39]. H-SWP can create an HOPS beam and the combination of H-SWP and SPP can create HyOPS beam [40].
A He-Ne laser of the wavelength of 632.8nm is spatially filtered and collimated using a spatial filter (SF) and lens $L_1$. The collimated light passes through an amplitude aperture (iris, I) and illuminates the polarizer $P_1$. A half-wave plate (HWP) is used to alter the polarization plane of the input light. Depending on the input polarization state and type of S-wave plate, it can produce HOPS or HyOPS beams. Further, the HOPS beam passes through SPP also results in the HyOPS beam. For example, an SPP with charge $+1$ placed after H-SWP of order $+\frac {1}{2}$ produces a combination of vortices of charge 0 and $+2$ in RCP and LCP components respectively and their superposition corresponds to a bright C-point. Similarly, SPP with charge $+2$ in combination with the same H-SWP will produce vortices of charge $+1$ and $+3$ in orthogonal polarization states producing a dark C-point. The beam carrying polarization singularity illuminates the fabricated DG and produces spots of equal intensity each of which carry C-point singularity with the same index as that of the incident beam. The lens $L_2$ is used to obtain a Fraunhofer diffraction pattern at its back focal plane. Stokes parameters are measured by performing Stokes polarimetry using a polarizer and quarter wave-plate to plot the polarization distribution of the diffracted beams. The ellipticity $\left (\chi =\frac {1}{2} \sin ^{-1}\frac {S_3}{S_0}\right )$ and azimuth $\left (\gamma =\frac {1}{2} \tan ^{-1}\frac {S_2}{S_1}\right )$ parameters are calculated at each pixel using the experimentally measured Stokes parameters. The SOP distribution is calculated using these polarization ellipse parameters.
The experimentally obtained results are depicted in Figs. 5–8, that depict both positive and negative index polarization singularity lattice structures. Figure 5 shows the experimental results for bright C-point lattice. The upper and lower panels show the experimental results for the lattice of lemon and star-type singularity. The corresponding intensity, Stokes phase and polarization distributions for lemon and star are shown in Fig. 5(A,D), (B,E), and (C,F) respectively. Beam embedded with lemon is generated by illuminating Q-SWP of order $+\frac {1}{2}$ by circularly polarized light. Star-type singularity from the lemon can be generated by using HWP [41]. Figures 6 and 7 show the experimental result for dark C-point with index $I_c=\pm {1}$ and $I_c=\pm {3}$. Dark C-point with $I_c=+1$ and $I_c=-1$ are generated by passing horizontally polarized light through a combination of (H-SWP, SPP) of order ($+\frac {1}{2},-2$) and ($-\frac {1}{2},+2$) respectively. Similarly, $I_c=+3$ and $I_c=-3$ are generated using a combination of (H-SWP, SPP) of order ($+\frac {3}{2},+1$) and ($-\frac {3}{2},-1)$ respectively. Qualitatively, the total intensity $S_0$ looks similar in dark C-points, but the size of the dark core increases with $I_c$. The left and right panel of Fig. 8 depicts the results for $\eta = +1$ and $-1$ lattice of a HOPS beam. HOPS beams are generated using H-SWP of the order of $\pm \frac {1}{2}$ illuminated with elliptically polarized beam. The total intensity obtained at the camera plane is shown in Fig. 8(A,D). Note that each of the intensity spots in the lattice has an equal distribution of energy due to DG. Figure 8(B,E) show the respective obtained Stokes phases. The polarization distribution of the HOPS beam for one of the diffraction spots is shown in Fig. 8(C) and (F) respectively. The fabrication error results in the spilling over of light into undesirable diffraction orders. The simulated and experimental results show that the SOP distribution in the input and in the diffraction order matches closely.
Fig. 5. Experimentally obtained $2 \times 2$ equal intensity spots produced by DG each embedded with lemon (upper panel) and star (lower panel) bright C-points (HyOPS beams) of index $I_c =\pm {1/2}$. (A,D) Total intensity distribution; (B,E) Stokes phase distribution and (C,F) polarization distribution of lemon and star bright C-points.
Fig. 6. Experimentally obtained $2 \times 2$ equal intensity spots produced by DG each embedded with dark C-point (HyOPS beams) of polarization index $I_c =+1$ (right panel) and $I_c =-1$ (left panel). (A,D) Total intensity distribution; (B,E) Stokes phase distribution and (C,F) polarization distribution of C-points. One of the diffracted spot is taken and expanded to show the polarization distribution.
Fig. 7. Experimentally obtained $2 \times 2$ equal intensity spots produced by DG each embedded with dark C-point (HyOPS beams) of polarization index $I_c =+3$ (right panel) and $I_c =-3$ (left panel). (A,D) Total intensity distribution; (B,E) Stokes phase distribution and (C,F) polarization distribution of C-points. One of the diffracted spot is taken and expanded to show the polarization distribution.
Fig. 8. Experimentally obtained $2 \times 2$ equal intensity spots produced by DG each embedded with HOPS beams of polarization index $\eta =+1$ (right panel) and $\eta =-1$ (left panel) and represented by non-equatorial points on HOPS with coordinates (latitude = $\pi /4$, longitude = $\pi /2$). (A,D) Total intensity distribution; (B,E) Stokes phase distribution and (C,F) polarization distribution of HOPS. One of the diffracted spot is taken and expanded to show the polarization distribution.
5. Conclusion
We have demonstrated the generation of a lattice of Spin-orbit beams by a DG. The grating is basically designed to produce equal-intensity light spots, but it is shown here that it also preserves the polarization distribution of the incident beam. This will be useful in multiplexing in optical communications in which spin-orbit beams are envisaged to play a vital role in the future. The diffractive element was fabricated using electron beam lithography. The grating is illuminated with HyOPS or HOPS beams embedded with C-points or V-points of various topological index values. The experimentally obtained lattice structures closely match the simulations.
Funding
Council of Scientific and Industrial Research, India (09/086(1323)/2018-EMR-I).
Disclosures
The authors declare no conflicts of interest.
Data availability
The experimental data used to support the findings of this study are available from the corresponding author upon reasonable request.
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