Nonimaging metaoptics

2744 Letter Vol. 45, No. 10 / 15 May 2020 / Optics Letters Nonimaging metaoptics Ivan Moreno,1, * Maximino Avendaño-Alejo,2 AND C. P. Castañeda-Almanza1,3 1 Unidad Académica de Ciencia y Tecnología de la Luz y la Materia, Universidad Autónoma de Zacatecas, 98060 Zacatecas, Mexico Instituto de Ciencias Aplicadas y Tecnología, Universidad Nacional Autónoma de México, 04510 Cd. de Méx., Mexico 3 UPIIZ, Instituto Politécnico Nacional, 98160 Zacatecas, Mexico *Corresponding author: imorenoh@uaz.edu.mx 2 Received 25 February 2020; revised 28 March 2020; accepted 29 March 2020; posted 1 April 2020 (Doc. ID 391357); published 7 May 2020 So far, metalenses have only been studied in imaging optics, where a point from the object space is mapped to a corresponding point in the image space. Here we explore metalenses and metamirrors for dealing with the optimal transfer of light energy. Owing to its compactness and high design flexibility, metasurface-based flat optics may open new opportunities in the nonimaging field, which deals with light concentration and illumination. The building blocks of metalenses are subwavelength-spaced scatterers. By suitably adjusting their shape, size, position, and orientation, one can control the light spatial distribution, as is desired in nonimaging problems. In this Letter, we introduce nonimaging metaoptics, review its basics, and briefly explore three cases: the compound-metasurface concentrator [analogous to the compound parabolic concentrator (CPC)], the total internal reflection (TIR) metalens (analogous to the TIR lens), and a simple condensing metalens. © 2020 Optical Society of America https://doi.org/10.1364/OL.391357 Recent progress in metasurfaces has led to the design and fabrication of ultrathin, lightweight, and flat lenses with novel functionalities [1,2]. Metasurfaces are clusters of subwavelength-spaced optical scatterers (metallic or dielectric) at an interface [3,4], which locally redirect the incident light by shaping the traversing wavefront in function of the spatial distribution of the scatterers. These subwavelength-sized scatterers are called unit cells, meta-atoms, or metasurface building blocks (MBBs). The MBBs are able to alter all aspects of transmitting and reflecting light beams, and they may be optical antennas, resonators, etc. Metalenses consist of carefully arranged MBBs with specific patterns at an ultrathin optical metasurface on a flat substrate. Their operation principle is related to the collective scattering of light by the array of MBBs to focus light. By spatially adjusting the MBB geometrical parameters (such as size, shape, orientation, and position across the metasurface), one can control either reflected or transmitted light spatial distribution with high spatial resolution. The recent realization of polarization-independent, high transmittance, and high numerical aperture metalenses makes them very suitable for nonimaging optics. These remarkable properties may have far-reaching implications in nonimaging optics, which deals 0146-9592/20/102744-04 Journal © 2020 Optical Society of America with the optimal transfer of radiant energy from a light source onto a target [5–8]. There are a few approaches of metalenses for light concentration, but not in the realm of nonimaging optics [9–11]. The two main problems that nonimaging optics solves are light concentration and illumination. In light concentration, the rays from the edges of the source are directed towards the edges of the target, thus ensuring that all light emitted from the source will fall on the target (edge-ray principle). In illumination problems, the rays from the source are directed to desired points of the target and produce a desired light spatial distribution. Nonimaging refractive and reflective lenses such as those used in collimators and concentrators, have useful properties but are often bulky and heavy to be used in small optical systems. Their design and fabrication rely on the shape optimization of their refractive or reflective curved surfaces. On the other hand, metalenses are very compact, and their manufacturing is becoming increasingly easier. For example, with the use of appropriate fabrication methods, the metalens phase profile can be digitized with MBBs that are manufactured with standard lithographic techniques employed in the integrated circuits industry [12]. The flatness and high design flexibility of metasurfaces are ideal properties for nonimaging optics. The two-dimensional (2D) phase profile of a metasurface has the equivalent effect of the three-dimensional (3D) surface shape in refractive or reflective optics. Optical designs may be performed by finding the optimal 2D phase profile of the metasurface. For example, for normally incident light, an imaging metalens must have a metasurface with the following phase profile [1]:  p 8 (r ) = ±k r2 + f 2 − f , (1) where k is the wave number, r is the radial position (r 2 = x 2 + y 2 ), f is the focal length, and the plus or minus sign indicates a diverging or converging lens [Fig. 1(a)]. This is a hyperbolic phase profile that focuses collimated light into a diffraction-limited spot [Fig. 1(b)]. In some way, the phase profile 8(x , y ) is the equivalent of the surface profile in traditional optics. Such a phase profile is constructed using a dense pattern of MBBs, each of which acts as a miniature antenna to locally impart a desired phase shift. A high-performance MBB is the cylindrical post with subwavelength diameters and heights [Fig. 1(c)], which has a polarization-independent response and high transmittance. By varying their size, the MBBs are Letter Vol. 45, No. 10 / 15 May 2020 / Optics Letters 2745 general, ideal focused spots are not needed for solving nonimaging problems, where the target or detector is much larger than the Airy disk, and then nonimaging metaoptics design may be based on the generalized laws of reflection and refraction for metasurfaces. In nonimaging metaoptics, the use of MBBs to impose a phase shift 8(x , y ) to an incoming set of wavefronts at different positions of a flat surface, has the equivalent effect of the surface shape in refractive or reflective nonimaging optics. This offers a promising and rich platform for nonimaging optics, which is characterized by the use of complex 3D surface shapes. In general, the required phase profile is deduced from the generalized laws of reflection and refraction. If the metasurface has only radial phase variation, it is [14] 1 d8 = n 2 sin θ2 − n 1 sin θ1 , ko dr Fig. 1. Optical characteristics of metalenses. (a) Schematic diagram of an imaging metalens. (b) Phase profile in an imaging metalens with radius R, which determines the subsequent light focusing. (c) Metasurface constructed by digitizing an analog optical phase profile on a flat surface into discrete cells, each of which contains a MBB that locally imparts the required phase shift to the incident beam to focus the light. (d) Typical phase shift and transmittance of light through a metasurface versus post diameter. (f ) Typical simulated intensity profiles (x z-plane) in the focal region at nanometric scale. (g) Schematic representation of the generalized refraction of light in 3D at a metasurface with arbitrary phase profile 8(x , y ). able to produce a 2π phase range and high, nearly uniform transmittance or reflectance [Fig. 1(d)]. The optical efficiency, cross-interactions, and fabrication limitations depend on the MBB refractive index [13], and on MBB-to-MBB separation. By finding the appropriate phase profile 8(x , y ), a metalens is optically designed, for example, that of Eq. (1). The design for manufacturing is achieved by selecting the MBBs and their spatial distribution to render the desired phase profile. In imaging metaoptics, the focused light field is verified by numerical simulations at the nanometric scale [Fig. 1(f )], which use the finite difference time domain method (FDTD). In nonimaging metaoptics analysis, FDTD calculations could be necessary for very precise designs, and mandatory if the detector or metalens size are at the micrometric scale. In general, imaging metaoptics performance is analyzed under wave optics by FDTD calculations. However, its design is primarily performed under geometric optics by using the generalized laws of reflection and refraction [14,15], which is particularly useful for correcting aberrations [16]. In both imaging optics and imaging metaoptics, if aberrations are larger than the ideal focused spot (diffraction-limited Airy disk), the primary design is based on ray tracing optimization [16]. In (2) where n 1 , n 2 are the refractive indices of the medium where light is incident and transmitted (or reflected), respectively. Additionally, θ1 , θ2 are the angles of incidence and refraction (or reflection), respectively; and ko is the vacuum wave number. Using Eq. (2), the required phase profile of a metalens can be deduced. For example, Eq. (1) can be derived by solving Eq. (2) with sin θ1 = 0, and sin θ2 = r (r 2 + f 2 )−1/2 , which is the geometrical condition for focusing a light beam. However, in general, the metasurface may have arbitrary phase variation 8(x , y ), and the generalized reflection and refraction laws at an interface with 2D phase gradient are given by two coupled equations [15]: ( 1 ∂8 = n 2 cos θ2 sin ϕ2 ko ∂ x , (3) 1 ∂8 = n 2 sin θ2 − n 1 sin θ1 ko ∂ y where ϕ2 is the azimuthal angle of refraction (or reflection) in the plane x z [Fig. 1(g)], θ1 is the angle of incidence in the plane y z (if ϕ1 = 0), and θ2 is the angle between the light refracted (or reflected) and its projection on the x z-plane [15]. The parameters ko , n 1 , and n 2 are the same parameters as in Eq. (2). Equation (3) is valid for incident light that lies in the y z-plane; that is, ϕ1 = 0. A suitable equation for nonimaging analysis should consider arbitrary incidence direction, i.e,. ϕ1 6= 0. After some mathematical treatments, it can be shown that only the first expression of Eq. (3) changes for the general case, giving 1 ∂8 = n 2 cos θ2 sin ϕ2 − n 1 cos θ1 sin ϕ1 , ko ∂ x (4) where ϕ1 is the azimuthal angle of incidence in the plane x z. Equations (2)–(4) represent a basis for nonimaging metaoptics analysis and design, which can be applied to explore this new field. With the aim of showing the potential of nonimaging metaoptics, in the following we briefly explore three nonimaging basics: a meta concentrator, a TIR metalens, and a condensing metalens. Let us begin with a concentrator. The basic nonimaging optical design for light concentration is the CPC. The CPC is a concentrator designed for capturing and concentrating a beam of light with a given angular incidence onto a flat receiver [Fig. 2(a)]. It consists of two parabolic mirrors, A and B, the axes of which are inclined at the collector half-acceptance angle θc (angle through which the beam of light can be moved from 2746 Letter Vol. 45, No. 10 / 15 May 2020 / Optics Letters Fig. 2. Light concentrator: (a) schematic diagram of a classic CPC and (b) schematic diagram of a CMC. the normal to the CPC axis and converge at the receiver). The 2D CPC achieves the ideal concentration limit C = (sin θc )−1 [5,6]. A basic metaoptics design for light concentration could be the compound-metasurface concentrator (CMC). The CMC could be designed for capturing and concentrating a beam of light with angular divergence [Fig. 2(b)]. It consists of two flat metamirrors, A and B, the phase profiles 8(ℓ) of which could be designed to focus the beam at angle θc at the edge of the receiver. We derived the phase profile to focus the reflected edge rays of the CMC, which is given by  p 8 (ℓ) = −k ℓ2 + a 2 + 2a ℓ cos β − a − ℓ sin (β − θc ) , (5) where l is the position along the metamirror, with l = 0 at the receiver corner. Parameter a is the receiver size, and β is the metamirror angle [Fig. 2(b)]. The phase profile is a function of the CMC half-acceptance angle θc . At first glance, it seems that the CMC may achieve the concentration ratio C without angle θc restriction, but all efficiency losses should be examined. Additionally, the CMC size seems to have no severe restrictions such as those found in the CPC due to the geometric properties of parabolas, but could be an optimal β value. The string method requires that all light rays be the same length, which is what gives the CPC its curved surface [5–7]. Interestingly, Fig. 2(b) shows that the rays in the CMC perform as they do in the CPC, but with rays of different lengths. This may be explained by the phase shift due to the metamirror phase profile, which produces an equivalent optical path length for each ray. A CMC without metamirrors is a typical “cone” or “flat mirror” concentrator, which is not ideal based on the laws of thermodynamics [5–7]. However, the metasurfaces remove this inefficiency to produce the CPC throughput. These observations open new questions for future research. The second basic example is a collimator. An optical design for light collimation is the TIR lens, in which light undergoes refraction and TIR. The basic TIR lens is a collimator designed for efficiently projecting a point light source into a collimated beam [Fig. 3(a)]. It consists of a front lens and a parabola, the source is placed at the focal point of these optical elements. This collimator is a popular solution to maximally collect the light emitted from LEDs and redirect it into a narrow beam [17]. A basic metaoptics design for light collimation could be the TIR metalens. The TIR metalens could be designed for efficient projection of a point light source into a collimated beam by means of metasurfaces [Fig. 3(b)]. It consists of a flat plate with Fig. 3. Light collimator: (a) schematic diagram of a classic TIR lens and (b) schematic diagram of a TIR metalens. a front metasurface for direct collimation and a flat metamirror for collimating light after TIR. The phase profile of the front surface is hyperbolic, given by 8(r ) = −nk[(r 2 + t 2 )−1/2 − t]. On the other hand, the light after TIR is collimated by a metamirror with a phase profile that may be determined to be 8(r ) = −nk[(r 2 + 4t 2 )−1/2 − 2t]. Here r is the radial distance from the light source along the metamirror, t is the plate width, and n is the refractive index of the plate. The front metasurface radius may be determined by TIR as r L = t 2 (n 2 − 1)−1/2 , and the metamirror starting at radial distance r M = 2r L . Ideally, the TIR metalens may achieve light collimation as a thin plate with high compactness, without volume restrictions present in the geometric characteristics of parabolas in typical TIR lenses, which opens new challenges and approaches for future research. Finally, we explore the optimization of a condensing metalens. Nonimaging optics design methods of free-form lenses are complex to describe and to implement. Thus, unless advanced methods are used [18], the optimization techniques can get lost in the infinitude of local minima or the large number of optical parameters to be optimized in 3D surfaces. On the other hand, nonimaging metaoptics design is 2D, which may be easier to describe and to implement. For instance, consider a condensing metalens with the goal of maximum light transfer efficiency between a set of light beams and a target. For simplicity, let us analyze a metalens with only one metasurface, whose purpose is to condense a cone of beams into a detector. In other words, a set of light beams in a continuous angular range of angles of incidence is concentrated on a planar target (Fig. 4). The goal is to maximize the radiation flux at the target, i.e., to maximize the 2D concentration ratio of radiant flux C = 9/90 , where 9 and 90 are the detector radiation flux with and without metalens, respectively [5]. Additionally, considering that the target radiation flux is proportional to both the number of light beams through the target of size D and to the projected area of the beam at an angle of incidence θ. Then the C due to a set of incident light beams with angular range 21θ , is C= 1 2Dsin1θ Z 1θ −1θ Z R U [s (r , θ ) , D] cos θdr dθ, (6) −R where R is the radius of metalens, and U is a rectangular function associated to the light concentration at the detector placed at distance z, which is defined as U = 1 for −1/2D < s(r , θ)<1/2D and U = 0 for |s(r , θ)| > 1/2D. The light rays cross the detector plane at a distance s(r , θ) from the optical axis, which also depends on the phase derivative d8/dr of the metalens, and is obtained by Eq. (2): Letter Vol. 45, No. 10 / 15 May 2020 / Optics Letters Fig. 4. Schematic diagram of a condensing metalens with a single metasurface. The phase profile of the metasurface may maximize the concentration on the detector. z s (r , θ ) = r − r  1− 1 d8 ko dr  + sin θ 1 d8 ko dr  + sin θ 2 . (7) Hence, the optimization problem reduces to finding the metalens phase profile 8(r ) that maximizes the double integral of Eq. (6). The optimal phase profile was obtained through nonlinear optimization. Considering the symmetry imposed by the angular range of ±1θ , the phase profile was defined as an even order polynomial of the radial coordinate r as 8 (r ) = 7 X n=1 an  r 2n R . (8) The coefficients a n are optimizationPparameters. The optimization is based on phase derivative 2n R −1 a n (r /R)2n−1 . For the first optimization step, the initial conditions for a n were chosen to make (1/ko )d8/dr + sin θ < 1 to avoid TIR [14]. After multiple cycles, the optimization parameters a n were obtained such that Eq. (6) was maximized. With no constraints, the metalens phase profile 8(r) that maximizes the concentration ratio in this nonimaging problem is the hyperbolic phase profile given by Eq. (1), with a detector placed at the focal distance f = D/2 tan1θ . By adding constraints to the problem, other solutions can be reached. If using a hyperbolic metalens with a fixed focal length f > D/2 tan1θ , the detector distance z that maximizes the concentration ratio is shorter than the focal length, i.e., zo < f . The optimal detector distance zo decreases exponentially with acceptance angle 1θ , i.e., zo ∝exp[−0.048(1θ − 1θo )], where 1θo = atan(D/2 f ). For example, with 21θ = 20◦ , f = 10 mm, R = 5 mm, and D = 1 mm, the optimal distance is zo = 7 mm. The approach outlined can be applied to problems with multiple metasurfaces, or to 3D concentration ratio problems, as well as optimization problems involving more constraints. In summary, we have introduced the concept of nonimaging metaoptics and some of their basics. We explored metalenses and metamirrors that deal with the optimal transfer of light energy between a light source and a target. The compactness and high design flexibility of metasurface-based flat optics may open new opportunities in the nonimaging field for light concentration and illumination. An important feature is the high spatial control of light. By suitably designing a phase profile, one can control the light spatial distribution, as is desired in nonimaging problems. Another important feature of nonimaging metaoptics is that the 2D phase profile of the metasurface 2747 has the equivalent effect of the 3D surface shape in refractive or reflective nonimaging optics. This offers a promising and rich platform for nonimaging optics, which is characterized by the use of complex 3D surface shapes. Therefore, the typical problems in nonimaging optics such as illumination and light concentration may be approached by solving 2D problems instead of 3D problems. These features were briefly explored in three nonimaging metaoptics examples: the compoundmetasurface concentrator, the TIR metalens, and condensing metalens optimization. In these analyses, we discussed the challenges and opportunities of nonimaging metaoptics, and its basics; we also introduced Eqs. (4)–(7). Future work may include addressing chromatic effects, which are important for polychromatic light management; evaluating and quantifying the effects of fabrication imperfections [9]; and, in general, addressing advanced nonimaging optics problems [8]. Also important is the issue of the development of analysis tools in the realm of nonimaging optics [18] and a review of fundamentals such as the edge-ray principle [5–7]. In addition, in nonimaging metaoptics, ray-optics considerations are incomplete, and the traditional nonimaging optics parameters have to be adapted within a wave-optics frame. Among these parameters are the concentration ratio, throughput, and efficiency. 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