Abstract
Perfect optical vortex beams (POVBs) carrying orbital angular momentum (OAM) possess annular intensity profiles that are independent of the topological charge. Unlike POVBs, perfect vectorial vortex beams (PVVBs) not only carry orbital angular momentum but also exhibit spin angular momentum (SAM). By incorporating a Dammann vortex grating (DVG) on an all-dielectric metasurface, we demonstrate an approach to create a pair of PVVBs on a hybrid-order Poincaré sphere. Benefiting flexible phase modulation, by engineering the DVG and changing the input-beam state we are able to freely tailor the topological OAM and polarization eigenstates of the output PVVBs. This work demonstrates a versatile flat-optics platform for high-quality PVVB generation and may pave the way for applications in optical communication and quantum information processing.
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1. Introduction
Amplitude, phase, and polarization are fundamental properties of light beams that have been subject to extensive studies. The progress in controlling these quantities has resulted in the introduction of beams with novel features. For instance, vortex beams with helical phase wavefronts carrying orbital angular momentum (OAM) [1] and vector beams of spatially varying states of polarization (SoP) [2,3] have been generated. A convenient description for the polarization and spatial degrees of freedom of optical beams (spin-orbit interaction [4]) is furnished by the so-called hybrid-order Poincaré sphere (HyOPS) [5], developed from a generalized Poincaré sphere (PS) [6,7] and the earlier higher-order Poincaré sphere (HOPS) [8]. The poles of HyOPS represent circularly polarized light beams of opposite handedness, and therefore opposite spin angular momentum (SAM), while simultaneously carrying different amounts of OAM. Elsewhere on the sphere, the beam states correspond to varying measures of OAM and SAM. Coherent superpositions of two (polar) eigenstates of HyOPS are called vectorial vortex beams (VVBs) [9]. Sharing characteristics of both vector and vortex beams, VVBs have led to a number of applications, for instance, in optical communications [10], microparticle manipulation [11], and quantum optics [12].
However, VVBs have a major drawback: their creation requires two beams of opposite circular polarizations and different OAM states, and the intensity profiles of such beams do not generally coincide due to different divergences. This results in undesired polarization state mixtures. In addition, the ensuing VVBs on propagation develop spatially random SoPs [13], which further leads to zeroes in the intensity distribution [9]. To overcome the limitations of VVBs, perfect optical vortex beams (POVBs), which show equal divergences for all vortex orders, have been introduced [14,15]. By applying this principle, a perfect vectorial vortex beam (PVVB) can be generated from two circularly polarized POVBs of opposite handedness. Conventional methods of realizing POVBs and PVVBs require a combination of optical elements, such as axicons, q-plates, phase masks, spatial light modulators, and Fourier transform lenses [16–18]. This in turn increases the light-path complexity, makes the system bulky, and renders it incompatible with integrated photonics platforms.
Recently, metasurfaces have provided an alternative for conventional optical elements. Metasurfaces consist of nanostructures that enable one to control the light field’s polarization state, phase, and intensity profiles within subwavelength dimensions [19]. Due to the flexible phase modulation capabilities and several degrees of freedom in design, a variety of functional planar optical elements have been fabricated on the basis of metasurfaces, including lenses [20], holograms [21], and structured beam generators [22,23]. Also, metasurfaces for POVB and PVVB implementation have been proposed with metallic or dielectric nanostructures. Due to absorption, plasmonic metasurfaces however tend to exhibit a low efficiency for beam generation [24]. On the other hand, the dielectric approaches utilizing silicon (Si) or titanium dioxide (TiO$_2$) metasurfaces either require a complicated oblique incidence scheme [25] or are limited to generating only one type of PVVB [26].
In this paper, we put forward an approach to generate a pair of PVVBs by means of a single-layer all-dielectric metasurface and demonstrate the method by direct numerical simulations. The key element in our system is a Dammann vortex grating (DVG) which, through appropriate designs, enables the tailoring of the topological OAM and SAM properties of the ensuing PVVBs. The metasurface we consider is composed of Si subwavelength nanopillars that support efficient resonant beam excitation at the operation wavelength of 1550 nm [27], allowing for simultaneous modulation of both the geometric and propagation phase [28]. The distinguishing feature of the method is that it creates, in a controlled manner, both diffraction-related and spin-dependent topological charges in the output beams. In particular, the generated PVVBs carrying OAM are shown to depend on the diffraction order, which allows for an important, additional degree of freedom for beam engineering. The simulation results indicate that the emerging PVVBs have a high beam quality of up to 99.9${\% }$ mode purity. We also show that the structure can operate within a broad wavelength range in infrared, thus leveraging potential for applications in various laser systems. The versatility, compact size, and high efficiency indicate that these metasurfaces could be used in many fields including optical communications, quantum information science, and integrated nanophotonics.
2. Perfect vectorial vortex beams
We recall first that a scalar Bessel beam (of unit amplitude) in the cylindrical coordinate system ($r$, $\varphi$, $z$) is expressed in the form
where $J_l$ is the $l$th-order Bessel function of the first kind and $k_r$, $k_z$ are the radial and longitudinal wave vectors, which satisfy $(k_r^2 + k_z^2)^{1/2} = k = 2\pi /\lambda$, with $k$ being the wave vector and $\lambda$ the wavelength. Then, the most general Bessel-type VVB can be written as [9]To create a PVVB as we desire, one simple way is employing a lens to perform an optical Fourier transformation of the Bessel-type VVB. Making use of Eq. (2), on the focal plane of the lens this transformation can be expressed as [29]
3. Principles of dual PVVB generation
The focal plane field given by Eq. (5) is an idealization containing a radial delta function, which cannot be realized in practise. Thus, we consider a Bessel-Gaussian (BG) beam as a feasible approximation. By loading three optical functionalities onto a single metasurface, we can convert a Gaussian beam into two different BG-type PVVBs, as illustrated in Fig. 2. We take the incident field at plane $z = 0$ to be of the form [cf., Eq. (2)]
Next we analyze the complex field amplitude of the beams emerging from the trifunctional device, when the illumination is as specified in Eq. (7). We consider first a single diffraction order. In the coordinate frame of the (paraxial) diffraction order $u$, the output beam in the DVG near field reads
To obtain the far field, we again perform an optical Fourier transformation with a lens, as in Eq. (4). In the focal plane, the beam field of diffraction order $u$ becomes
Likewise, for the diffraction order $v$, the complex Fourier-plane field reads
Figure 3 illustrates PVVB beams generated by both symmetric and asymmetric DVGs under $x$-polarized incidence (the diffraction orders are $u = -1$, $v = +1$ and $u = -1$, $v = +2$, respectively, while the other parameters are set as $l = 2$ and $q = 4$). The Fourier-plane intensity and phase profiles for the symmetric and asymmetric DVGs are calculated and shown in Fig. 3. According to the intensity profiles in Fig. 3(a), the output PVVBs are symmetrically located at diffraction orders $\pm 1$, while in Fig. 3(b) the beams are refracted by the asymmetric DVG at different angles into orders $-1$ and $2$. Moreover, one can confirm that the phase profiles shown in Figs. 3(a) and (b) are consistent with the mathematical expressions of PVVBs consisting of eigenstate pairs $m = -6$, $n = -2$ and $m' = 2$, $n' = 6$, as well as $m = -6$, $n = -2$ and $m' = 6$, $n' = 10$, respectively. Hence symmetric DVGs create eigenstates whose topological charges obey $m^\prime = -n$ and $n^\prime = -m$, whereas the eigenstates of asymmetric DVGs are not bound to any specific relationship.
Our unique approach allows taking advantage of diffraction orders and DVGs for generating PVVBs with desired topological OAM and polarization properties. For a Dammann grating the absolute values $|c_{u}|$ and $|c_{v}|$ are equal [34], which guarantees the uniformity of the output beams. Moreover both symmetric and asymmetric DVGs can be applied, offering extra degrees of freedom for beam engineering.
4. Dual PVVB metasurface design
In this section, we introduce the design procedure for the metasurface. We consider the total phase modulation that includes the axicon, q-plate, and DVG for the POVB or PVVB generation and implement the associated phase maps through the different design dimensions of the metasurface (e.g., nanopillar size, shape, and rotation angle). For specificity, we present the metasurface design in detail for a symmetric DVG and briefly comment on the differences in the case of an asymmetric DVG.
A paraxial beam of an arbitrary polarization state can be decomposed into two spin eigenstates $|L\rangle$ and $|R\rangle$. When considering the beam’s passage into the far field, which amounts to a Fourier transformation of the near field, our metasurface is required to perform the conversion $|L\rangle \rightarrow |{\rm POVB}, R_m\rangle$ and $|R\rangle \rightarrow |{\rm POVB}, L_n\rangle$ by providing two independent phase profiles $\varphi _{\rm R}(x, y)$ and $\varphi _{\rm L}(x, y)$. The output beams are of opposite handedness and can have different topological charges $m$ and $n$, respectively.
The transformation can be expressed as
In order to design a metasurface whose phase modulation simultaneously satisfies Eqs. (15) and (16), we first describe the metasurface with a single Jones matrix $J(x,y)$ in the circular polarization basis [23]
Making use of the phase profiles given in Eqs. (17) and (18), the phase shifts $\delta _x$ and $\delta _y$ and the rotation angle $\theta$ are calculated as
Here, $\delta _x$ and $\delta _y$ are dynamic (propagation) phases and $\theta$ is viewed as a geometric phase imposed by the metasurface onto the incident field.In this work we implement three different metasurfaces (Meta1, Meta2, Meta3), each generating a pair of dual PVVBs. The design wavelength is $1550$ nm. The metasurface radius $R \approx 59\,\mu$m and the spot size of the incident Gaussian beam is $w_g = 60.0\,\mu$m. In the designs, the axicon radial period $d_a = 9.75\,\mu$m and the Dammann grating period $d = 5.85\,\mu$m. The ensuing $\delta _x$ and $\theta$ phase maps for Meta2, for which the vortex phase order $q = 4$ and the q-plate topological charge $l = 2$, are illustrated as characteristic examples in Fig. 4.
The expressions in Eqs. (20)–(22) assign values of $\delta _x$, $\delta _y$, and $\theta$ to all points $(x,y)$ on the metasurface and we design a planar nanostructure to implement the corresponding phase map. A schematic diagram of the metasurface is shown in Fig. 5(a). The metasurface is composed of elliptical silicon nanopillars periodically arranged on a fused silica substrate. Each unit has two cross-sectional parameters – the long and short radii ($R_{\rm L}$ and $R_{\rm S}$), whose variation allows us to control the phase shifts. The rotation angle $\theta$, i.e., the angle between the $x$ axis and the $R_{\rm L}$ direction, is another parameter that can be independently varied to satisfy Eq. (22). All nanopillars are of the same height. Since $\delta _x$ and $\delta _y$ have a constant $\pi$ phase difference, the nanopillars may be viewed as acting like a half-wave plate.
To characterize the nanostructure, finite-difference time-domain (FDTD) simulations (software by Lumerical) are employed at the central wavelength of 1550 nm. The periods in the $x$ and $y$ directions are optimized at $P_x = P_y = 650$ nm to reduce coupling effects between the units. The height is chosen as $H = 800$ nm to cover a $2\pi$ phase modulation. The dimensions of eight selected nanostructures, from which the metasurfaces can be constructed, are listed in Table 1 and their simulated phase shifts and transmittance are shown in Fig. 5(b). The chosen elliptical nanopillars are seen to exhibit accurate phase control combined with high transmission, which ensures efficient generation of dual PVVBs.
The rotation angle $\theta$ is illustrated in Fig. 5(c) in two selected cases. Due to the high refractive index of the material, the nanopillars exhibit strong resonances [27] resulting in excellent polarization conversion rate $PCR = T_{{\rm cross}}/(T_{{\rm cross}}+T_{{\rm co}}$). Here $T_{{\rm cross}}$ and $T_{{\rm co}}$ are the cross- and co-polarization transmittances (calculated for an $x$-polarized incident plane wave). We observe that $PCR$, demonstrated Fig. 5(d), is close to unity over a wide wavelength range, which may enable broadband operation.
5. Results and discussion
To characterize the generation of dual PVVBs with hybrid polarization states, three metasurfaces (Meta1, Meta2, Meta3) were designed and simulated. Besides the axicon, Meta1 is composed of DVG of charge value $q = 2$ and a 1st-order q-plate ($l = 1$ spiral phase plate). Thus, at diffraction orders $u = -1$ and $v = +1$ Meta1 generates PVVBs that are superpositions of two POVBs of topological charges $(m,n)_{+1} = (1,3)$ and $(m,n)_{-1} = (-3,-1)$. In view of Eq. (9), the PVVBs at diffraction orders $+1$ and $-1$ can therefore be expressed as: $|\mathrm {PVVB}\rangle = \cos \alpha e^{i\beta } |\mathrm {POVB},R_1\rangle + \sin \alpha e^{-i\beta } |\mathrm {POVB},L_3\rangle$ and $|\mathrm {PVVB}\rangle = \cos \alpha e^{i\beta } |\mathrm {POVB},R_{-3}\rangle + \sin \alpha e^{-i\beta } |\mathrm {POVB},L_{-1}\rangle$, respectively. Different input polarization states (governed by $\alpha$ and $\beta$) lead to different spatial polarization distributions and phase vortices for the emerging beams. Meta2 consists of DVG of charge $q = 4$ and a 2nd-order q-plate ($l = 2$), hence the POVB topological charges are $(m,n)_{+1} = (2,6)$ and $(m,n)_{-1} = (-6,-2)$. Likewise, Meta3 corresponds to DVG of charge $q = 8$ and a 4th-order q-plate ($l = 4$), whereby $(m,n)_{+1} = (4,12)$ and $(m,n)_{-1} = (-12,-4)$.
Figure 6 shows simulated far-field diffraction patterns for metasurfaces Meta1, Meta2, and Meta3 with an $x$-polarized ($2\alpha ^\prime = \pi /2$, $2\beta ^\prime = 0$) incident field. The results are plotted in the $xy$ plane (perpendicular to the beams) at a constant propagation distance of $L = 1.4$ cm. In Fig. 6(a) annular intensity distributions of PVVBs at both $u = -1$ and $v = +1$ diffraction orders are presented for Meta1 ($|H_{-3,-1}\rangle$, $|H_{1,3}\rangle$), Meta2 ($|H_{-6,-2}\rangle$, $|H_{2,6}\rangle$), and Meta3 ($|H_{-12,-4}\rangle$, $|H_{4,12}\rangle$). As expected, the polarization states of PVVBs generated by Meta1, Meta2, and Meta3 exhibit azimuthal periodicities, respectively, of $2\pi /p$ with $p =$ 1, 2, and 4, in full accordance with the relations given in Sect. 2. We can clearly see that although the spin states and the topological charges are different for these PVVBs, their intensity profiles have equal radii (of approximately 0.34 mm). To explore the beams further, the cross-sectional intensity distributions of the PVVBs are simulated and displayed in Fig. 7. The theoretical radii of PVVBs obtained from Eq. (13) are $R_r = 0.37$ mm, which are close to the simulated values. Notably, both the ring radii and the transverse intensity profiles are essentially independent of the type of PVVB.
To verify the performance of these metasurfaces to generate PVVBs of varying states, we choose Meta2 as an example for a detailed study. Outputs from Meta2 can be represented by points on the HyOPS, as shown in Fig. 8(a). Depending on the order of diffraction, we encounter specific HyOPSs with different polar eigenstates. At order $u = -1$ the eigenstates that determine the HyOPS are $|\mathrm {POVB}, R_{-6}\rangle$ and $|\mathrm {POVB}, L_{-2}\rangle$, while at order $v = +1$ they are $|\mathrm {POVB}, R_{2}\rangle$ and $|\mathrm {POVB}, L_{6}\rangle$. For $u = -1, v = +1$, the PVVB states located on the HyOPS equator are characterized by polarization order [9] $p_{u(v)} = (n_{u(v)}-m_{u(v)})/2 = 2$ and by topological Pancharatnam charge [26] $s_{u(v)} = (m_{u(v)}+n_{u(v)})/2 = \pm 4$, respectively. At other HyOPS points, the PVVB polarization and topological properties are different. We have selected six points on HyOPS whose coordinates are labeled by I–VI in Fig. 8(a). The corresponding polarization states of the incident light are depicted in Fig. 8(b) and we analyze by numerical simulations the intensity patterns and phase distributions of the ensuing PVVBs in the $xy$ plane at wavelength 1550 nm.
In particular, Figs. 8(c), (d), and (e) show the annular intensity profiles, the intensity patterns after passing through a horizontal linear polarizer, and the beam phase distributions, respectively, corresponding to HyOPS points I–VI. In Fig. 8(c), the intensities are rings of identical radii, indicative of the perfect vortex nature of the beam. Notably, cases III and IV corresponding to $y$- and $x$-polarized incident light demonstrate that the intensity patterns show anisotropic polarization distributions with 4 lobes, which is consistent with polarization order $p=2$. This means that the polarization rotates by $4\pi$ in a full circle. In Fig. 8(e), cases III and IV show the simulated phase patterns with $s_{-1} = -4$ and $s_{+1} = 4$, which further confirm the four-fold polarization winding around the perimeter of the beam. This is in full agreement with the theory above. For cases I and VI (LCP and RCP incidence) in Fig. 8(e), the phases around the azimuth undergo $2\pi$ rotations $m_{-1} = -6$, $n_{-1} = -2$ and $m_{+1} = 2$, $n_{+1} = 6$ times, respectively. The simulation results are entirely consistent with the theoretical predictions for each HyOPS and verify that the metasurface performs as expected.
We have also studied the quality of the beams generated by the metasufaces. The beam purity plays a crucial role, in particular, in the fields of optical communication and laser physics. It is defined as
where $I_m$ is the intensity of $m$th mode, and $I_l$ is the target mode intensity. Figure 9 shows the calculation results for Meta2 illuminated by circularly polarized light at wavelength 1550 nm. The quantified mode purity for the four modes $|\mathrm {POVB}, R_{-6}\rangle, |\mathrm {POVB}, R_{+2}\rangle, |\mathrm {POVB}, L_{-2}\rangle$ and $|\mathrm {POVB}, L_{+6}\rangle$ is higher than 99.9${\% }$ in all cases, ensuring excellent beam quality for potential applications. Our simulations further indicate that the mode purities are higher than $70{\% }$ for beams within the spectral range of 1500–1700 nm.6. Conclusion
In summary, we have proposed and simulated all-dielectric metasurfaces that generate dual perfect vectorial vortex beams. The metasurface is composed of a single layer of elliptic silicon nanopillars and it operates in a transmission mode at the wavelength of 1550 nm. By changing the metasurface design or the state of polarization of the incident light, the output beam can assume any state on the hybrid-order Poincaré sphere (HyOPS). The vortex order and the HyOPS eigenstates are shown to depend on the diffraction order and spin-dependent topological charges. The possibility to utilize both symmetric and asymmetric diffraction modes introduced by the DVG enable unprecedented control over the output beams. These degrees of freedom allow for a convenient and meaningful control over the topological, OAM and polarization porperties of the beams. Such metasurface elements are likely to inspire a new family of functional ultra-compact flat optical devices for various applications that require flexible phase control and high efficiency. The results presented in this work may prove important in the fields of optical communications, micro-manipulation, optical data storage, and quantum information processing.
Funding
Research Council of Finland (322002, 359450, PREIN 346518).
Disclosures
The authors declare no conflicts of interest.
Data availability
No data were generated or analyzed in this research.
References
1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, et al., “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef]
2. Y. F. Chen, K. F. Huang, H. C. Lai, et al., “Observation of vector vortex lattices in polarization states of an isotropic microcavity laser,” Phys. Rev. Lett. 90(5), 053904 (2003). [CrossRef]
3. A. M. Beckley, T. G. brown, and M. A. Alonso, “Full Poincaré beams,” Opt. Express 18(10), 10777–10785 (2010). [CrossRef]
4. K. Y. Bliokh, F. J. Rodriques-Fortunado, F. Nori, et al., “Spin-orbit interactions of light,” Nat. Photonics 9(12), 796–808 (2015). [CrossRef]
5. X. Yi, Y. Liu, X. Ling, et al., “Hybrid-order Poincaré sphere,” Phys. Rev. A 91(2), 023801 (2015). [CrossRef]
6. C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).
7. M. J. Padgett and J. Courtial, “Poincaré-sphere equivalent for light beams containing orbital angular momentum,” Opt. Lett. 24(7), 430–432 (1999). [CrossRef]
8. G. Milione, H. I. Sztul, D. A. Nolan, et al., “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107(5), 053601 (2011). [CrossRef]
9. A. Niv, G. Biener, V. Kleiner, et al., “Manipulation of the Pancharatnam phase in vectorial vortices,” Opt. Express 14(10), 4208–4220 (2006). [CrossRef]
10. P. Gregg, P. Kristensen, and S. Ramachandran, “Conservation of orbital angular momentum in air-core optical fibers,” Optica 2(3), 267–270 (2015). [CrossRef]
11. M. Chen, M. Mazilu, Y. Arita, et al., “Dynamics of microparticles trapped in a perfect vortex beam,” Opt. Lett. 38(22), 4919–4922 (2013). [CrossRef]
12. M. V. Jabir, N. Apurv Chaitanya, A. Aadhi, et al., “Generation of perfect vortex of variable size and its effect in angular spectrum of the down-converted photons,” Sci. Rep. 6(1), 21877 (2016). [CrossRef]
13. G. M. Philip, V. Kumar, G. Milione, et al., “Manifestation of the Gouy phase in vector-vortex beams,” Opt. Lett. 37(13), 2667–2669 (2012). [CrossRef]
14. A. S. Ostrovsky, C. Rickenstorff-Parrao, and V. Arrizón, “Generation of the perfect optical vortex using a liquid-crystal spatial light modulator,” Opt. Lett. 38(4), 534–536 (2013). [CrossRef]
15. G. Joaquín, R. Carolina, R. Rubén, et al., “Simple technique for generating the perfect optical vortex,” Opt. Lett. 39(18), 5305–5308 (2014). [CrossRef]
16. P. Li, Y. Zhang, S. Liu, et al., “Generation of perfect vectorial vortex beams,” Opt. Lett. 41(10), 2205–2208 (2016). [CrossRef]
17. R. Xu, P. Chen, J. Tang, et al., “Perfect higher-order Poincaré sphere beams from digitalized geometric phases,” Phys. Rev. Appl. 10(3), 034061 (2018). [CrossRef]
18. D. Li, S. Feng, S. Nie, et al., “Generation of arbitrary perfect Poincaré beams,” J. Appl. Phys. 125(7), 073105 (2019). [CrossRef]
19. A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339(6125), 1232009 (2013). [CrossRef]
20. M. Khorasaninejad and F. Capasso, “Metalenses: Versatile multifunctional photonic components,” Science 358(6367), eaam8100 (2017). [CrossRef]
21. Y. Bao, Y. Yu, H. Xu, et al., “Full-colour nanoprint-hologram synchronous metasurface with arbitrary hue-saturation-brightness control,” Light: Sci. Appl. 8(1), 95 (2019). [CrossRef]
22. R. C. Devlin, A. Ambrosio, N. A. Rubin J. P. B. Mueller, et al., “Arbitrary spin-to-orbital angular momentum conversion of light,” Science 358(6365), 896–901 (2017). [CrossRef]
23. J. Yang, T. K. Hakala, and A. T. Friberg, “Generation of arbitrary vector Bessel beams on higher-order Poincaré spheres with an all-dielectric metasurface,” Phys. Rev. A 106(2), 023520 (2022). [CrossRef]
24. Y. Zhang, W. Liu, J. Gao, et al., “Generating focused 3D perfect vortex beams by plasmonic metasurfaces,” Adv. Opt. Mater. 6(4), e1701228 (2018). [CrossRef]
25. Y. Bao, J. Ni, and C. W. Qiu, “A minimalist single-layer metasurface for arbitrary and full control of vector vortex beams,” Adv. Mater. 32(6), e1905659 (2020). [CrossRef]
26. M. Liu, P. Huo, W. Zhu, et al., “Broadband generation of perfect Poincaré beams via dielectric spin-multiplexed metasurface,” Nat. Commun. 12(1), 1–9 (2021). [CrossRef]
27. A. I. Kuznetsov, A. E. Miroshnichenko, M. L. Brongersma, et al., “Optically resonant dielectric nanostructures,” Science 354(6314), aag2472 (2016). [CrossRef]
28. J. P. B. Mueller, N. A. Rubin, R. C. Devlin, et al., “Metasurface polarization optics: independent phase control of arbitrary orthogonal states of polarization,” Phys. Rev. Lett. 118(11), 113901 (2017). [CrossRef]
29. J. Goodman, Introduction to Fourier Optics (Roberts & Co. Publishers, 2005).
30. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, 2007).
31. M. Abrammowitz and I. S. Stegun, eds., Handbook of Mathmatical Functions with Formulas, Graphs, and Mathematical Tables (Dover Publications, 1965).
32. I. Moreno, J. A. Davis, D. M. Cottrell, et al., “Encoding generalized phase functions on Dammann gratings,” Opt. Lett. 35(10), 1536–1538 (2010). [CrossRef]
33. J. Albero, J. A. Davis, D. M. Cottrell, et al., “Generalized diffractive optical elements with asymmetric harmonic response and phase control,” Appl. Opt. 52(15), 3637–3644 (2013). [CrossRef]
34. J. Jahns, M. M. Downs, M. E. Prise. N. Streibl, et al., “Dammann gratings for laser beam shaping,” Opt. Eng. 28(12), 1267–1275 (1989). [CrossRef]