1. Introduction

Recent advances in flat optics using metasurfaces and diffractive optical elements has paved the path for a wide variety of compact singular or compound optical components capable of achieving arbitrary wave-front control [1–3]. Flat optic structures such as metalenses have drawn particularly significant interest within the research community and have rapidly advanced in terms of performance and functionality [4]. Metasurfaces with carefully engineered phase profiles have also recently been used for the direct generation of two dimensional (2D) and three dimensional (3D) accelerating light beams such as finite energy Airy beams [5–8] and half-Bessel beams [9]. These accelerating beams exhibit several distinct properties, perhaps most notably their ability to propagate in a quasi-non-diffracting manner along a curved path while self-healing [10]. A variety of non-paraxial accelerating beams have now been classified from wave solutions to Maxwell’s equations, including higher-order Bessel functions, Mathieu beams, and Weber beams; which follow circular, elliptical, and parabolic trajectories respectively [11,12]. It has also been shown that beams with arbitrary convex trajectories can be derived in both paraxial and non-paraxial regimes from a caustics based design approach [13]. This class of beams opens a variety of novel capabilities in applications ranging from particle manipulation, signaling through turbulent media, laser machining, microscopy, and extended depth of field imaging [14].

Metasurfaces offer an attractive chip-scale form factor for the direct generation of accelerating light beams, which are otherwise typically generated via Fourier transform techniques that employ discrete lenses, phase plates, and/or spatial light modulators. Direct generation of Airy beams, for example, has been achieved with a $\phi \propto {x^{3/2\; }}$phase profile [6,15]; and form birefringence has been exploited to flip the beam trajectory according to the input polarization handedness [5,6]. Aside from their high performance, compactness, and compatibility with wafer-scale manufacturing, metasurfaces attractively offer the prospect of reconfigurable and/or multi-functional devices [2,16]. A multifunctional device, for example, can be realized by tailoring the structure to yield distinct phase responses when specified degrees of freedom are varied (e.g., polarization, angle, wavelength, etc.) [4,17,18].

In this work, we report how a simple 1D metalens can readily serve as a multi-functional device which can not only focus light but also generate paraxial and non-paraxial 2D accelerating beams with widely tunable trajectories. Our approach relies on intentionally introducing and modulating coma, which is generally considered an undesirable source of aberration in imaging optics. When coma is present in an imaging system, the focal spot becomes distorted along a caustic trajectory and imaged objects appear to exhibit comet-like tails, hence the origin of the term. In the context of geometrical optics, correspondence between paraxial Airy beams and comatic beams has previously been demonstrated, as the requisite cubic phase profile used to construct an Airy beam can be decomposed into a linear combination of Seidel coma and higher order Zernike trefoil [19,20]. These prior works suggest that coma offers a promising degree of freedom for tailoring accelerating light beams. It should be noted, however, that Airy beams are restricted to exist in the paraxial approximation and cannot in practice be synthesized with infinite energy, hence limiting their achievable range of bending angles and the extent of their quasi-non-diffracting character. To date, the use of intentionally engineered coma has only been explored while relying on multiple discrete optical components such as a beam-splitter alongside a deformable mirror, or multiple cylindrical lenses carefully aligned in series with a Fourier transform lens [19,20].

Similar to conventional lenses, the wavefronts emerging from phase-gradient metasurfaces also exhibit non-negligible sensitivities to input angle, and hence metasurfaces typically require operation at a specified optimal angle of incidence [21]. This is an issue for a variety of applications such as holographic displays, which can be alleviated by pursuing metagrating based designs which preserve the intended wavefront shape across a wide range of incidence angles [22,23]. Alternatively, the angular and chromatic dispersions of metasurfaces can be exploited to achieve multifunctional devices [17,18]. Here, we embrace the angle-sensitivity of a simple 1D metalens and show how it can be utilized to readily achieve not only diffraction limited focusing, but also to generate paraxial and non-paraxial accelerating beams with dynamically tunable trajectories.

2. Approach

Let us consider a flat lens with a phase profile$\; \phi (x )$ that is independent of y which molds wavefronts in the xz plane to achieve focusing. At the surface of the flat lens, the transverse wavevector ${k_x}$ of the local anomalously refracted wave can be determined according to the generalized law of refraction [24]:

For collimated and on-axis illumination of the lens (${\theta _1} = 0^\circ $), ${k_x}$ is entirely described by the phase gradient term of Eq. (1) and all refracted rays converge constructively at the lens focus as illustrated in Fig. 1(b). However, for off-axis illumination $({\theta _1} \ne 0^\circ )$, the transverse momentum of the refracted rays are shifted according to the non-zero second term in Eq. (1) and they no-longer converge to a single focal point leading to wavefront aberrations including signficant coma. In the comatic case, the ray picture [Fig. 1(b)] reveals the formation of an enveloping caustic curve $c(z )\; $which has a slope $c^{\prime}(z )\; $described by $tan({{\theta_2}} )$ of the locally tangent ray [13]. With a monochromatic light source, constructive interference near and along this caustic enables the formation of a non-paraxial accelerating light beam.

 

Fig. 1. (a) Transverse wavevector k_x for θ₁ = 0° (black) and θ₁ = 6° (blue) incident angles vs x. (b) Raytracing for incident angles θ₁ = 0° (black) and θ₁ = 6° (blue), and caustic curve c(z) (red) showing the comatic accelerating beam trajectory (c) FDTD simulation showing light propagation at various NA and incident angles at λ = 600 nm.

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3. Results/discussion

Figure 1(c) presents numerical simulation of light propagation from flat lenses with varying numerical aperture (NA) and constant focal length $f = 35\; \mu m$ under monochromatic illumination at $\lambda = \; 600\; nm$. Modelling is performed via a combination of 2D finite difference time domain (FDTD) simulation and application of Kirchoff’s integral, where we treat the metalens as a patterned refractive index film in accordance with the capabilities of our recently developed nanomanufacturing technique [25]. We should emphasize however, the phase profile $\phi (x )$ is the defining feature of the metasurface and the exact mechanism by which the gradient metasurface achieves its phase profile is not crucial. In the absence of any incident beam tilt (${\theta _1} = 0^\circ $), all three lenses show diffraction limited focusing at the desired focal length.

For moderate NA = 0.4 or lower, a small incident beam tilt of ${\theta _1} = 2^\circ $ from the substrate (${n_1} = 1.45$) begins to tilt and shift of the focal spot, indicative of non-zero tilt and field curvature, while the simultaneously introduced comatic aberration remains small. For high NA = 0.8 and ${\approx} {\; }$0.99 however, ${\mathrm{\theta }_1} = 2^\circ $ is sufficient to produce significant coma and an accelerating beam which preserves its shape especially well in the vicinity of $z \approx f$ is produced. For finite energy beams which correspond to finite aperture sizes and NA < 1 for propagation in air, shape-preserving accelerating beam formation along the caustic trajectory is stronger and longer lived, corresponding to a larger depth of focus, as ${\theta _1}$and NA are increased. Moreover, tuning the incidence angle ${\theta _1}$offers a direct means for modulating both the curvature and trajectory of the accelerating beam. We also note that that the peak field intensity necessarily decreases as power is distributed into the sidelobes, which is a feature common to all accelerating beams. For example, in a typical finite energy accelerating beam [e.g. Figure 1(c)], the primary lobe might contain on the order of ∼20% of the total power crossing a plane near the beam apex or focus.

Here, we adopt the term ‘comatic accelerating beams’ (CABs) to denote the type of accelerating beams which are formed along the curved caustics of a strongly comatic beam emerging from a focusing lens. We should note that the presence of coma in no way precludes the potential presence of other aberrations, such as tilt or field curvature, which may grow especially strong as the beam is spatially or angularly displaced from its ideal focal point and/or plane. Unlike paraxial Airy beams, which could be considered a sub-type of CAB when generated from cubic phase modulation and a Fourier transform lens, we find that CABs more generally can exhibit sustained shape preserving qualities around large bending angles similar to non-paraxial Mathieu and Weber beams [12]. Unlike Mathieu or Weber beams which follow elliptical and parabolic trajectories respectively, the CABs considered in this work follow hyperbolic trajectories where the marginal anomalously refracted rays serve as asymptotes. The conic trajectories of these beams in the $xz$ plane can be generalized under the general second degree equation $A{x^2} + Bxz + C{z^2} + Dx + Ez + F = 0$; where elliptical, parabolic, and hyperbolic trajectories occur under the conditions $AC > 0$, $AC = 0$, and $AC < 0$ respectively. It should also be noted that in this general form B can be non-zero, which corresponds to the transverse axis of the hyperbola being rotated relative to the x-axis by an angle $\alpha = \frac{1}{2}{\cot ^{ - 1}}\left( {\frac{{A - C}}{B}} \right)$. Hence, our CAB trajectory may be most accurately fitted to a traditional hyperbolic profile of the form: $c({z^{\prime}} )= \frac{b}{a}\sqrt {{{({x^{\prime} - x{^{\prime}_0}} )}^2} - {a^2}} - z{^{\prime}_0}$ where b and a are constants and the coordinate system $({x^{\prime},z^{\prime}} )$ is rotated by angle $\alpha $ relative to $({x,z} ).$

As illustrated in Fig. 1(c) for the case of NA = 0.8, ${\theta _1} = 6^\circ $ the observed trajectory of the accelerating beam, which bends ∼90$^\circ $, agrees very well with the predicted caustic in Fig. 1(b). In principle, the hyperbolic trajectory of a CAB derived from an otherwise ideal 1D metalens can be tailored through various degrees of freedom including the NA, focal length, and tilt angle ${\theta _1}$. Moreover, it is possible to extend the CAB concept using customized flat lens or metasurface designs which achieve complete control over the accelerating beam trajectory even for on-axis illumination (${\theta _1} = 0^\circ $) by using a purposefully engineered phase profile:

In Eq. (3) the first term describes a hyperbolic lens designed to achieve a diffraction limited focus either on-axis or tilted off-axis at a prescribed location $({x = {x_0},z = f} )$:

With this design approach, control over the hyperbolic trajectory, including its rotation $\alpha $ relative to the x-axis and $({x,z} )$ positioning of its vertex, can be achieved with the appropriate choice of the terms ${x_0}$ and ${\theta _{tilt}}$ in combination with the available freedom over f and NA.

In Fig. 2, we illustrate how a CAB with a tailored long-range (∼10 mm) hyperbolic trajectory can be realized from on-axis illumination ${\theta _1} = 0^\circ $. In this example, we engineered a shape preserving accelerating beam at $\lambda = \; 1.31\; \mu m$ which could be relevant to an optical probing application such as optical coherence tomography where both an extended depth of field and compact device form factor are preferred [26]. Here, the metalens is implemented using $\phi (x )= {C_0}mod({{\phi_{CAB}},2\pi } )$ where the constant ${C_0} = 1.56$ is optimized by parameter sweep (and observation of the field profile) to suppress coupling into undesired diffraction orders; $f = 1$ mm; ${x_0} = 1$ mm; ${\theta _{tilt}} ={-} 48.9^\circ $; and the diameter is set to 2.66 mm. Here, the design angle ${\theta _{tilt}} ={-} 48.9^\circ $ is selected to approximately counter the off-axis focus angle (∼45°) of the otherwise ideal lens and skew the accelerating beam toward an on-axis orientation. The ∼mm scale focal length of this nominal lens design promotes a significantly larger depth of focus than achieved in Fig. 1, which utilized a lens with a much shorter focal length. As shown in Figs. 2(b)–2(d), where the x-axis is magnified for clarity, this modified flat lens generates an accelerating beam which exhibits shape preserving character over many millimeters and is defined along a hyperbolic trajectory which closely agrees with the predicted caustic in Fig. 2(a). The simulated transmission efficiency of the metasurface is 84.6%.

 

Fig. 2. (a) Raytracing and (b) FDTD simulation of a shape preserving long range (∼ 10 mm) accelerating beam with (c), (d) magnified view. Note: x-axis (µm) and z-axis (mm) are not shown on proportional scales. (e) Transverse slice of the beam profile at z = 3.6 mm (blue) with an incomplete Airy function fit (dashed black). (f) Normalized peak intensity and full width half maximum (FWHM) along the z-axis.

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A transverse slice of the beam profile at $z$ = 3.6 mm is depicted in Fig. 2(e). Despite the non-parabolic trajectory of our beam, we found that this beam profile ${|{E(x )} |^2}$ can be described as a finite energy Airy profile with a sharp spectral cutoff, also referred to as an incomplete Airy beam [27]. The incomplete Airy function differs from the conventional Airy function $Ai(X )$ in that it uses finite bounds of integration:

In the example shown in Fig. 2(e), the beam profile follows the form ${|{E(x )} |^2} \propto {|{Ai_{a,b}^{inc}(X )} |^2}$, where $X = \kappa ({x - {x_0}} )$ with $\kappa $ representing a characteristic inverse length which accounts for the transverse scaling of the beam and ${x_0}$ accounts for the transverse position of the caustic. In principle, the $xz$ propagation of this accelerating beam could be analytically described with the appropriate parameterization of Eq. (6). It is worth noting that the now famous $Ai(X )$ integral was originally developed in 1838 to describe the nature of light intensity in the neighborhood of a caustic [28]. This function now belongs within a wider class of diffraction catastrophe integrals which describe the intensity of light formed near caustics of various stable topologies [29,30].

The results of both Fig. 1(c) and Fig. 2(e) indicate that the point spread function of our ideally focused beam (i.e., 1D Airy disk) evolves into an incomplete Airy beam profile in the presence of coma. As described previously by Berry [29,31], this phenomena is readily predicted by catastrophe optics as an ideal point focus is structurally unstable to perturbation and in our 2D configuration can break into folds or cusps when subjected to comatic or spherical aberrations respectively. Our present results highlight however, that the diffraction catastrophe integrals should be implemented with finite bounds of integration, e.g. Eq. (6), rather than in their canonical form. As the beam asymptotically approaches the path of the marginal ray of Fig. 2(a), which is highlighted in black, the beam in the vicinity of the caustic forms from rays corresponding to a narrowing spectrum and a smaller integration window for Eq. (6). Hence, as the beam acceleration asymptotically reduces, its spectral content decreases and diffraction resumes. This transition is observable in Fig. 2(f) where the peak intensity and full width half maximum (FWHM) of the main Airy lobe are examined along the $z$-axis. A large region of shape and intensity preserving quasi-non-diffraction is observed up until $z\; \sim \; $10 mm, which is significantly larger than the ${\sim} \; $1 mm depth of field for a focused beam with a comparable spot size. For $z > \; $10 mm, diffraction becomes clearly visible as the beam exhibits an approximately linear divergence and exponentially decaying intensity.

It should be noted that the present phenomena, which uses a 1D flat lens to generate caustic sheets exhibiting 2D propagation, does not simply map to the generation of shape-preserving 3D propagating beams generated from a 2D flat lens. In 3D, coma generates a more complex hyperbolic umbilic topology rather than a fold [19,29], and an appropriate symmetry in the paraxial regime must be maintained to achieve a beam with a continuous primary lobe [32]. Moreover, off-axis illumination of a 2D flat lens would simultaneously introduce oblique astigmatism (non-existent for a 1D lens) which would further complicate the beam diffraction. If a 3D beam is desired and dynamic control over the beam trajectory can be sacrificed, one could directly add cubic phase to an otherwise ideal metalens profile to generate a more traditional Airy like beam. This approach has recently been utilized to achieve extended the depth of focus and enable broadband computational imaging [33].

Next, we illustrate how a compound flat optic could be utilized to achieve an accelerating beam with a wavelength controlled trajectory. This structure, depicted in Fig. 3(a), relies on a similar principle as Eq. (3) except the two phase functions are isolated into different metasurface layers. The first layer consists of a multi-chrome 1D metalens with NA = 0.8 and f = 175 μm which satisfies Eq. (4) to achieve focusing while suppressing chromatic aberration for the wavelengths 450 nm, 524 nm, and 635 nm. In principle a broadband achromatic metalens could potentially also be used, however as recently detailed there are fundamental limits to the combination of NA, f, and bandwidth that can be achieved in a single thin-lens [34]. Recently, this limit has been broken in a freeform metalens working outside the thin-lens approximation [35]. In either case, it is important that our presently considered structure induces wavelength dependent coma without simultaneously introducing chromatic aberration, which could otherwise dominate the beam diffraction. Here, we choose to simply emulate an ideal multi-chrome metalens [35,36], which for our choice of high NA would currently be significantly simpler to design and manufacture than a freeform achromatic design.

 

Fig. 3. (a) Device incorporating a chromatic beam tilt (blazed grating) over a multi-chrome metalens. (b) Resultant transverse wavevector k_x after momentum contribution G from the beam tilt. At a specified angle and design wavelength the contributions cancel each other out, resulting in zero coma. Outside this condition, accelerating beams with tunable trajectories are formed. (c) FDTD simulation of the device at various wavelengths and angles of incidence θ₁, showing reconfigurable and multi-functional focusing or accelerating of light along tunable trajectories.

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The second layer of the device shown in Fig. 3(a) is designed to induce a wavelength dependent phase tilt (anomalous refraction) on the rays exiting the metalens and does not require any particular spatial alignment to the first layer. The chromatic beam tilt of the second layer can be achieved with a wrapped phase profile ${\phi _2}(x )= {C_0}mod\left( {x\frac{{2\pi }}{{{\lambda_d}}}sin({{\theta_{tilt}}} ),2\pi } \right)$ which effectively functions as a flat blazed grating where the grating period ${\mathrm{\Lambda }_\textrm{d}}$ is determined by the choice of phase tilt desired at a given design wavelength ${\lambda _d}$, and the effective blaze angle can be adjusted by the constant ${C_0}$. Rays emerging from the surface of the compound structure in Fig. 3(a) will exhibit a transverse wavevector ${k_x}$ described by a modified form of Eq. (1) which accounts for the momentum contribution from the flat blazed grating:

4. Conclusion

In this work we have shown how a simple 1D flat lens can be utilized to generate non-paraxial comatic accelerating beams with Airy-like qualities. The beam properties can be readily manipulated through various degrees of freedom including the passive lens properties (e.g., NA, f, etc.) and controlling the incident angle or wavelength. Unlike paraxial Airy beams generated under cubic phase or a $\phi (x )\propto {x^{3/2\; }}$profile, these finite energy accelerating beams propagate along tunable hyperbolic trajectories, can preserve their shape/intensity around large bending angles (>90$^\circ $), and can be reversibly unfolded into a point focus. These results highlight a unique regime wherein the incidence angle sensitivity of phase-gradient metasurfaces is in fact desirable and can be exploited to generate accelerating beams with highly tunable characteristics. Moreover, the ability to dynamically transition between a focused beam and an accelerating beam offers multifunctionality which may benefit applications ranging from optical probing and particle manipulation to laser milling.