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Abstract
Dispersion management is crucial in constructing spectrometers, superprisms, and achromatic lens systems. Unfortunately, the dispersion of natural materials is determined by the molecular energy levels with limited tunability, and thus conventional methods of dispersion controlling are complex and need to trade off other aberration. Metasurface offers an alternative method to overcome those limits via utilizing dedicatedly designed nanostructures that response to special wavelength, which results in well-engineered dispersions. As proof of the concept, we design a series of flat dielectric metasurface lenses, which are able to steer the dispersion arbitrarily for three wavelengths at visible frequency (473, 532, and 632.8 nm). Based on the unique dispersion engineering ability of metasurface, the achromatic meta-lens and the super-dispersion meta-lenses are realized. Furthermore, the light of different wavelengths can be focused on any desired spatial positions.
© 2017 Optical Society of America
1. Introduction
Dispersion is a ubiquitous phenomenon can be found in electromagnetic wave, acoustic wave and matter wave, which has far-reaching implications ranging from optical imaging systems to communication systems. It is a long-term pursuit to efficiently control the dispersion of natural materials at will. Commonly, dispersion controlling can be divided into two aspects: elimination of dispersion and expansion of dispersion. For instance, in optical imaging systems (e.g. microscopic imaging system and telescope system), dispersion should be eliminated since it causes the chromatic aberration and thus decrease image quality. On the other hand, extraordinary strong dispersion is anticipated for the spectroscopic analysis system and dense wavelength division multiplexing (DWDM). Moreover, proper dispersion can be utilized to suppress the nonlinear effects (e.g. harmonic wave generation and four-wave mixing) in fiber communication systems [1]. Unfortunately, the dispersion of natural materials is determined by their electronic and molecular energy levels, with limited tunability [2]. With conventional methods, rectifying or compensating chromatic aberrations can be realized by multi-lens combinations [3], which is bulky and need complex design and fabrication to obtain optimized performances. Furthermore, the wavelength resolving power of tradition spectroscopic analyzer is stringently limited by dispersion of grating [4] and prism [5]. Therefore, the ultrathin, ultralight, planar and integrated optical elements with flexible dispersion engineering ability are expected.
In the last decades, metasurface has emerged as revolutionary material offering unprecedented superiority for dispersion engineering, while its dispersion property is mainly determined by the specific geometry and arrangement of artificial meta-atom. With spatially varied metal/dielectric nanostructures as resonant optical antennas along the surface, metasurfaces provide a new approach for amplitude, phase and polarization manipulation [6]. Distinct from the conventional optical components that rely on gradual phase shifts accumulated during light propagation to shape light beams, metasurfaces can provide abrupt phase changes coverage 2π over the sub-wavelength scale, which have found widespread applications in vortex beam generating [7–9], light bending [10–12], focusing [13–28], holograms [10, 23, 26, 29-31], spin Hall effect [32] and split ring resonator (SRR) [33, 34].
Despite of the great progress that have been made in the past few years, it should be noted that, the highly resonant nature of metasurface that forces the electromagnetic waves undergo a phase change ultimately causes a small bandwidth around their design frequency, as a result of the general Kramers-Kronig relations [2]. The concept of dispersion management both one-dimensional and two-dimensional has been utilized to extend the bandwidth of metasurfaces, which are exploited to construct ultra-broadband absorbers and polarization converters [2, 34–39]. The later has been utilized to construct broadband phase-only devices based on the photonic spin-orbit interaction (SOI).
Although the SOI accompanied Pancharatnam–Berry (P-B) phase shift (i.e. geometric phase) is frequency-independent, the flat optical devices based on metasurface also possess chromatic aberration, which results in a wavelength-dependent focal length or deflection angle. Quite recently, several works about achromatic [13, 19, 25, 26, 35, 38–40] and super-dispersive [13, 21, 41] devices based on metasurfaces have been reported in near-infrared and terahertz spectrum. However, the demonstrated devices cannot engineer the dispersion discretionarily with simple design method and high numerical aperture. For example, Capasso’s group achieved multiwavelength achromatism by dispersive phase compensation with an aperiodic arrangement of coupled rectangular dielectric resonators in [19]. But it requires heavy workload to optimize structure and can only apply to one-dimension with aperiodic silicon bars. Besides achromatism, they have also demonstrated the super-dispersive off-axis meta-lenses for compact high resolution spectroscopy with silicon nanofins by enhancing the intrinsic dispersion [21]. However, the distribution of the focal points is still related to the wavelengths.
In this paper, the concept of spatial multiplexing is adopted for three dedicatedly designed silicon nanocuboids that response to the specific wavelengths (473, 532, and 632.8 nm), respectively. With PB phase, the chromatic dispersion among different wavelengths can be engineered independently. Subsequently, a series of flat optical devices with achromatic and super-dispersive (positive or negative) focusing properties are demonstrated.
2. Principle and unit cell design
As illustrated in Fig. 1(a), the proposed meta-lenses are composed of rotational silicon nanocuboids on a silicon dioxide wafer, where x-y coordinates denote the lattice direction, ξ-η coordinates denote the spatial orientation of the Si nanocuboids, and θ defines the orientational angle of nanocuboid. Considering nanocubiod as a truncated waveguide, its anisotropic geometry will cause a phase difference along the two main axes, which can approximate π by adopting appropriate geometric parameters. In this circumstance, the whole metasurface can be taken as a space-variant waveplate, thus subjected to the famous SOI. By applying the Jones matrix theory, the scattering effect can be explained clearly [10, 20, 39–42]. For circularly polarized incident light EiR/L (R/L means right/left circularly polarized light), the scattering light EsR/L will be written as [8, 10]
where to and te represent the complex scattering coefficients along the optical fast axis and slow axis, respectively. According to Eq. (1), the scattering light can be divided into two parts. The first term represents the co-polarization without phase modulation. The second term represents the opposite handedness with modulated amplitude (to-te)/2 and abrupt phase shift .
Fig. 1 The basic unit cell. (a) 3D view of the basic unit cell. The pixels are arranged with a period of P = 200 nm and a height of h = 400 nm. (b) The simulation transmissions (conversion efficiency) results of three individual designed nanocuboids illuminated by a normally-incident LCP (left circularly polarized) light beam. (c) The transmission (conversion efficiency) coefficient and phase shift of transmitted light as a function of θ for three nanocuboids. (d) Top view of a small metasurfaces with a radius of 3 μm.
It is well known that PB phase exhibits inherent broadband operation. Here, in order to realize the wavelength independent wavefront shaping, three dedicatedly designed nanocuboids with different geometries, denoted by Sb (l = 80 nm, w = 42 nm), Sg (l = 91 nm, w = 67 nm), Sr (l = 131 nm, w = 102 nm), are utilized to filter out the three wavelengths (473, 532, and 632.8 nm), respectively. The corresponding cross-polarization transmissions (conversion efficiency) of three nanocuboids are presented in Fig. 1(b). Obviously, the cross-talk between them can be neglected. Therefore, the anticipated wavefront can be independently constructed by changing the orientation of them. Note that, owing to the periodicity of unit cells is identical to each other, the whole covering area of each unit cells are almost same in our design, the efficiency can only be changed by the geometries, which results in that the max efficiency difference of our design is ~16%, slightly smaller than that (i.e. 20%) in the reference [43] mentioned above, where the whole covering area of each unit cells is unequal. The relationship between the transmission/phase and the orientation is simulated by the commercial software CST Microwave Studio. The complex permittivity of silicon is taken from data from Palik [44]. Since silicon and air have high refractive index contrast, each silicon nanopillar can be considered a waveguide which is truncated on both sides and operates as a low quality factor Fabry-Perot resonator. Light is mainly confined inside the high refractive index nanopillar, which behave as weakly coupled low quality factor resonators and thus cause a high transmission. Similar high transmission has been reported in the previous reference [20].
The design process of the metasurface is simplified to two steps: (1) the design of three meta-lenses by three nanocuboids separately and (2) the interleaving of three different nanocuboids Sb, Sg, Sr. In the first step, by molding incident planar wavefronts into spherical ones, we can realize a flat lens. As illustrated in Fig. 2(a), the total phase shift ΔФtot(r, λ) from incident light wavefront to desired wavefront can be divided into the phase shift of metasurface Фm(r, λ) and the phase accumulated through propagation path Фc(r, λ) [19]. They relate to the position r at interface and incident light wavelength λ:
Fig. 2 The design principle of meta-lenses and five dispersion controllable meta-lenses. (a) The sketch map is used for phase distribution analyzing and illustrating. The circular flat meta-lens with the focal point F lens converts plane wave into spherical wave, where incident light travels along the z-aixs. Green semi-spherical surface is the desired equiphase surface, and the straight line intersects with the equiphase surface at point B. (b) Schematic and (c) phase distribution of achromatic meta-lens M1. (d) Schematic and (e) phase distribution of super-dispersion meta-lens M2 designed to separate different wavelength light in order of normal dispersion. (f) Schematic and (g) phase distribution of super-dispersion flat meta-lens M3 with anomalous dispersion. The distribution of focal points is opposite to M3. (h) Schematic and (i) phase distribution of super-dispersion flat meta-lens M4 with off-axis colors separation. (j) Schematic and (k) phase distribution of super-dispersion flat meta-lens M5 with different off-axis colors separation.
Since Фtot(r, Si) is phase shift between two equiphase surfaces (i.e. it’s a constant), one direct way to construct dispersion controlling devices is the so-called dispersion compensation between the structural dispersion stemming from subwavelength patterns Фm(r, λ) and the chromatic dispersion that accumulated along the optical path Фc(r, λ), where Фtot(r, Si) is considered to be zero [31, 34]. Therefore, for arbitrary focal point F (xf, yf, zf), the phase modulation of metasurface Фm(r, λ) is given as:
In the second step, we construct interleaved rotating nanocuboids as shown in Fig. 1(d), where the red, green and blue colored rectangles represent the three nanocuboids responding to the blue (473 nm), green (532 nm) and red (632.8 nm) light, respectively. A series of meta-lenses (denoted by M1 – M5, respectively) are dedicatedly designed to validate the dispersion controlling ability. Figures 2(b)-2(k) show the layout of the schematics and phase profiles of five flat meta-lenses with a radius of 8.1μm.
The first meta-lens M1 is a flat achromatic lens with f = 10 μm for all the three wavelengths and the corresponding numerical aperture (NA) is calculated as 0.6294. The schematic and phase profiles of M1 are shown in Figs. 2(b) and 2(c). Besides, two super-dispersion meta-lenses M2, M3 with focal points located separately along the optic axis are designed. As observed in Figs. 2(d) and 2(e), M2 is similar to the lens with normal dispersion, where the focal lengths of red, green, and blue light are 6 μm, 10 μm, 17 μm, respectively. On the contrary, M3 is similar to the lens with anomalous dispersion, where the focal points of red, green, and blue light are in reverse order. The schematic and phase distribution of M3 are shown in Figs. 2(f) and 2(g). M4 and M5 are off-axis super-dispersion meta-lenses with radially separated focal points. Both of meta-lenses are arranged to converge three wavelengths at (4,0,10), (0,0,10) and (−4,0,10), respectively. The schematic and the phase profiles of M4 and M5 are shown in Figs. 2(h)-2(i) and 2(j)-2(k), respectively.
3. Performance analysis of the designed meta-lenses
To show the difference between normal dispersion and dispersion-free, a normal flat meta-lens which can converge green light (532 nm) at a distance of 10µm behind the meta-lens is proposed. Figure 3 shows the simulated results of normal dispersion and dispersion-free meta-lenses at the wavelengths of 473 nm, 532 nm and 632.8 nm.
Fig. 3 The comparison of normal flat meta-lens and achromatic meta-lenses. (a)-(c) Simulative normalized intensities distribution of normal flat meta-lenses in xz-plane upon illumination with λ = 473 nm, 532 nm and 632.8 nm wavelength light, respectively. (d) Comparison of the normalized intensity curves of three light along the z-axis. (e)-(g) Simulative normalized intensities distribution of achromatic flat meta-lenses M1 in xz-plane and (inset) the intensity profiles across the focal plane. (h) Comparison of the normalized intensity curves of three light along the z-axis.
The simulation results of normal meta-lens are given by Figs. 3(a)-3(d), which show that the focal lengths of three lights are 11.62 μm (blue), 9.89 μm (green), 7.74 μm (red), respectively. Figures 3(e)-3(h) shows the normalized intensities distribution of achromatic meta-lenses M1 in xz-plane and the intensity profiles across the focal plane, where the intensities are normalized with respect to their maximum light intensities (98 for blue, 143 for green and 105 for red). Obviously, achromatic meta-lens M1 is able to focus three lights on the same plane at designed focal length (f = 10μm). The exact focal lengths of three lights are 10.19 μm (blue), 10.19 μm (green) and 9.89 μm (red).
The insets of Figs. 3(e)-3(g) show the intensity profiles across the focal plane of the meta-lens for three wavelengths of 473nm, 532nm and 632.8nm. The simulated full-widths at half maximums (FWHMs) of blue, green and red light (380 nm, 437.4 nm, 490.6 nm) are in good agreement with the theoretical counterparts (383 nm, 430.8 nm and 502.5 nm). The deviations may be caused by the interactions between three nanocuboids and the asymmetric of meta-lenses due to the arrangement of three nanocuboids. For the on-axis achromatic focal lens, we have calculated the focusing efficiency of achromatic metalens (M1), which is defined as the ratio of the optical power of the focused beam to the optical power of the incident beam. The calculated results are 21.13% (blue), 54.66% (green) and 31.49% (red) for three unit cells.
The simulation results of super-dispersion meta-lenses M2 and M3 are given in Fig. 4. Figures 4(a)-4(c) and 4(e)-4(f) shows the normalized intensities distribution of M2 and M3 in xz-plane, where the intensities are normalized with respect to their maximum light intensities (M2: 113 for blue, 146 for green and 44 for red; M3: 137 for red, 168 for green and 38 for blue). Based on the vector diffraction method, we calculate the theoretical results (dotted line) and compare them with the simulation results (solid line) in Figs. 4(d) and 4(h), which shows that super-dispersion meta-lenses M2 and M3 are able to separate three lights completely. There are slight differences between the theory and simulation. We believe the deviations are from the interaction between the unit cells where the wavefronts that metalenses shaped aren’t the spherical wavefronts as we hoped. As shown in Figs. 4(d) and 4(h), the focal lengths of M2 and M3 are close to their designed value, and FWHMs of M2 and M3 are very close to their diffraction limits, respectively. Comparing the blue depth of focus (DOF) of M2 (2.10 μm) with blue DOF of M3 (8.50 μm), since the wavelengths are the same, we can see that the DOFs increased with decreased NA (i.e. increased focal length). And comparing the theoretical blue DOF of M2 (2.10 μm) with red DOF of M3 (2.85 μm), since the focal lengths are the same, we can see that the DOF increased with increased wavelength.
Fig. 4 Simulation results of super-dispersion meta-lenses M2 and M3. (a)-(c) Simulative normalized intensities distribution of super-dispersion meta-lenses M2 in xz-plane and (inset) the intensity profiles across the focal plane upon illumination with λ = 473 nm, 532 nm and 632.8 nm, respectively. (d) Comparison of theoretical results (dotted line) and simulation results (solid line) the normalized intensity curves of three light along the z-axis. (e)-(g) Simulative normalized intensities distribution of super-dispersion meta-lenses M3 in xz-plane and (inset) the intensity profiles across the focal plane upon illumination with λ = 473 nm, 532 nm and 632.8 nm, respectively. (h) Comparison of theoretical results (dotted line) and simulation results (solid line) the normalized intensity curves of three light along the z-axis
Simulation results of off-axis super-dispersion meta-lens M4 and M5 are exhibited in Fig. 5. It shows that meta-lenses can focus different wavelengths on arbitrary three-dimensional positions. Figures 4(a)-4(c) and 4(e)-4(g) illustrate simulative normalized intensities distribution and Figs. 4(d) and 4(h) overall images in xz-plane upon illumination with λ = 473 nm, 532 nm and 632.8 nm, respectively. The three-dimensional coordinates of the focus of M4 and M5 are (−3.997,0,10), (0,0,10), (3.965,0,10) and (0,0,10), (3.996,0,10), (−3.972,0,10), respectively. Comparing with (4,0,10), (0,0,10) and (−4,0,10) of devised focal points, simulation results have fractional radial deviations δ = 0.075%, 0%, 0.875% and 0%, 0.1%, 0.7%.
Fig. 5 The CST simulation results of off-axis super-dispersion meta-lenses M4 and M5. (a)-(c) and (e)-(g) Simulative normalized intensities distribution in xz-plane of M4 and M5 upon illumination with λ = 473 nm, 532 nm and 632.8 nm, respectively. (d) and (h) Overall images of M4 and M5.
4. Conclusion
Meatsurfaces add another dimension for multifunctional optical component designing by breaking the dependence of phase accumulation on the propagation effect. For the optical systems that require achromatism or large chromatic aberrations, their requirements can be satisfied respectively by the dispersion controlling meta-lenses we designed: the achromatic meta-lens, the super-dispersion meta-lenses which can focus different wavelengths on arbitrary special positions. The designed meta-lenses can engineer the dispersion discretionarily with simple design method and high numerical aperture and possess great potential applications for different optical systems.
Funding
973 Program of China (No. 2013CBA01700), National Natural Science Foundation of China (No. 61575201).
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