COMSOL Multiphysics® Software as a Metasurfaces Design Tool for Plasmonic-Based Flat Lenses
Bryan M. Adomanis*1, D. Bruce Burckel2, and Michael A. Marciniak1 1Department of Engineering Physics, Air Force Institute of Technology, 2950 Hobson Way, Wright- Patterson Air Force Base, OH 45433-7765 2Sandia National Laboratories, P.O. Box 5800, Albuquerque, New Mexico 87185 *Corresponding author: [email protected]
Abstract: COMSOL Multiphysics® was used to simulate the response of parametrized gold V- antennas on a silicon substrate, which served as a basis for the design of several plasmonics-based metasurface lenses. The simulated phase and amplitude of the scattered, cross-polarized far- field of the scatterer was parametrized over the dipole length /2 and vertex angle Δ to determine the optimal geometries needed to access the full 2 phase space. These elements were then arrayed linearly according to Fresnel lens phase calculations and formed the basis of a cylindrical focusing optic. Nineteen lenses and eight metasurfaces were fabricated with varying wavelengths, focal lengths, diameters, number of unique elements and packing densities. By Figure 1. Two point scatterers, with a relative phase validating the lens design using a non-optimized difference generating a broadside plane wave. scatterer made of a simple geometry, this effort provided confidence in the future use of COMSOL Multiphysics® (COMSOL) has COMSOL for a comprehensive investigation of been proposed to accomplish the numerical lens design optimization using more complex modeling for these designs, which will serve as a structures and architectures. good comparison to previous work done in finite-difference time domain (FDTD) [2-4, 7, Keywords: metasurfaces, flat lens, plasmonics 11, 12] and other finite element method (FEM) [9] softwares. In order to establish confidence 1. Introduction before proceeding with complex designs, COMSOL was used to generate several simple Flat lenses require precise control of a phase plasmonics-based flat lens designs, which were gradient across an interface, which is enabled then fabricated and measured for validation of through the application of engineered surfaces, the simulated results. These lenses were assessed such as metasurfaces [1]. Periodic arrays of for characterization of their depth-of-field in the plasmonic antennas have been utilized to mid- and long-wave infrared (M/LWIR) regime generate this desired phase gradient [2-8], which under several changing parameters, such as a dictates the angle of “anomalous” refraction of changing focal ratio ⁄ # , packing density and the cross-polarized field scattered from a normal- number of unique discrete elements used to incidence beam. comprise the resulting phase front. However, the designs utilized to-date have been simple geometries and architectures due to 2. Flat Lens Theory the challenges of numerically modeling more complex forms, and as a result, scattering 2.1 Generalized Snell’s Law efficiencies have been unacceptable for real- world application in the infrared (IR) [4, 7, 9, Refractive optics of a plane wave at the 10]. Therefore, an optimization effort is homogeneous interface of two differing bulk necessary to improve the lens performance. media is well understood: in short, phase-
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Excerpt from the Proceedings of the 2016 COMSOL Conference in Boston theory to a greater scale and customizability introduces the possibility for lensing, but also requires a look into the structures needed to create such a function.
2.2 Metasurface Lenses
Metasurfaces are arrays of periodic subwavelength structures which can be engineered for a specific electromagnetic response, such as a particular resonance wavelength, bandwidth or scattering phase and amplitude. These factors become the impetus for designing a real-world flat lens, as an individual scattering element can be tailored to give a specific phase delay, and thus, correspondingly, an array of elements can be tailored to give a Figure 2. Expanding the scatterer concept of Figure 1 to discrete regions of constant phase for lensing. determinable phase gradient along an interface. With this in mind, there are two means to matching conditions at the interface demand that array the metasurface elements to generate a an electromagnetic wave bend according to lensing function. Looking back to Figure 1, the Snell’s law, where sin sin 0. black semicircle represents the curve spanned by However, for inhomogeneous interfaces, the a circle with focal length as the radius and phase delay ϕ imposed upon a plane wave is centered at 0, . The phase matching condition no longer translationally invariant across the for neighboring elements to converge surface, and thus Snell’s law has a more constructively at 0, is: generalized solution, now consisting of a term describing the gradient of the phase discontinuity 2 ∆ , 2 along the surface [2]. In 1-D this is:
where is the Fresnel lens phase formula sin sin Φ x , 1 2 2 which for a linear gradient Φ ⁄ 0 has , 3 been proposed simply as a reimagining of diffraction from a blazed grating [13]. is the number of scatterers, is the scatterer The result is that a normally-incident plane index of the range 1, 1 , is the freespace wave 0° will couple to a propagating wavelength and is the phase of the zeroth plane wave at some oblique angle 0° (central) scatterer. For equally spaced elements into one of the diffraction modes supported via of a constant phase gradient—i.e. the the linear phase discontinuity. Thus, changing metasurface which only bends a plane wave— the slope of the gradient will change the this condition cannot be met, and so either the broadside angle of propagation, and so an phase shift or the element spacing must be non- assortment of optical functions can be realized. linear. A demonstration of this is shown in Figure 1. The first published metasurface lens Two point scatterers, with a relative phase implemented the latter approach, in a manner difference generating a broadside plane shown in Figure 2, where discrete regions of wave.Figure 1, where two point dipoles radiating constant phase spaced non-uniformly were at different phases (but equal amplitudes) used to meet the phase matching conditions [4]. generate a broadside planar wavefront. As well V-antennas (bottom) were used to provide a known in antenna array theory, changing either known constant phase in the regions and to the separation or the relative scattered phase will generate a strong enough scattered response. change the broadside angle. Expanding this
Excerpt from the Proceedings of the 2016 COMSOL Conference in Boston Figure 3. The field components (a), unit cell model (b) and geometries of interest (c) for the 3-D COMSOL V-antenna simulations used to obtain the scattered cross-polarized field over a large parameter space.
3. Use of COMSOL Multiphysics® ports and PBCs with perfectly-matched layers Software (PMLs) to solve for the true scattered field using from the first environment as the The focus of this work was to reproduce background field of the second, such that variants of the first metasurface lens design, . using COMSOL to generate a large trade space Note that this second environment does not from which to choose elements that are, unlike include any periodicity, which is acceptable for previous work, optimized in amplitude [2, 4]. this work. The assumption is that each scatterer While analytical solutions of the scattered field will be placed far enough from nearest neighbors for a simple V-antenna geometry are available, so that mutual coupling can be ignored. As a the intent of this effort was to validate a simple result, the unit cell periodicity was set to geometry before progressing to more complex 6 m to be large enough to not interact with geometries. the PML, but small enough to limit the The goal of the simulations was to obtain the diffraction orders to 1. scattered cross-polarized field of the V- The incident field was launched from the Si antennas as a function of the vertex angle ∆ base into the ̂ direction towards the listening and dipole length . From this, the phase port at the top of the air domain. As shown in Figure 3, the central axis of the V-antenna was and amplitude were extracted to map the scatterer response aligned to 45° in the x-y plane and the field was from a full range of parametrized V-antenna aligned in direction to equally excite the two geometries. fundamental modes of the V-antenna: a high- The field components, full COMSOL model frequency symmetric mode which oscillates and scatterer geometries of interest are shown in along the half-length of the dipole, and a lower- Figure 3(a-c), respectively. The model consists frequency antisymmetric mode which oscillates of a 65nm-thick gold (Au) V-antenna scatterer the length of the dipole. This way, when the with a trace width of 250nm on a 10 m- antenna undergoes a 90° rotation to induce a thick silicon (Si) substrate embedded in 24 m- 180° phase shift [2], the field components thick air domain. The setup mimics the remain equally applied and there is no need to re- “Scatterer on Substrate” COMSOL library calibrate the incident field strength. model, utilizing the Wave Optics module and Both near-field (NF) and far-field (FF) data two Electromagnetic Waves, Frequency Domain was collected. The NF was sampled using a (ewfd) physics environments. The first Domain Point Probe located directly above the environment solves for the total field of the scatterer at 20 m away, which was verified as being an appropriate distance to approximate a undecorated air/Si interface, , using periodic boundary conditions (PBCs) and periodic ports. plane wave. The Far-Field Calculation was used The second adds in the scatterer and replaces the on the internal air/PML boundaries surrounding the model. This process projects the NF out to a
Excerpt from the Proceedings of the 2016 COMSOL Conference in Boston
Figure 4. Interpolated COMSOL results for the parametric profiles and their nominal contours for phase (a,b) and amplitude (c,d) at designed operating wavelengths of 8 m (a,c) and 5 m (b,d). distance where only the radiation fields compressed mesh was needed in a region about (transverse to the radial direction of propagation) 1.5 m around the interface to accurately are important. The governing equation is termed resolve the scattered field, as shown in Figure the Stratton-Chu formulation, and COMSOL’s 3(b). Finally, the PML mesh was assessed for form is given by: variations in due to the PML thickness and mesh density, indicating that a PML , [4] thickness greater than /3 and mesh density less
⋅ than 12/ resulted in unacceptable variance in the FF amplitude—though the NF remained relatively unchanged. where , are the fields on the surface S, Solutions were calculated separately at 5 m (which is acting as the aperture), is unit vector and 8 m for preparation of lenses with different from origin to the observation point, is the wavelength responses. The vertex angle ∆ and radius vector of S, and is a unit vector normal half-dipole length /2 were scanned in steps to S. For COMSOL, is set at 1m, which is of ∆ 4° and 0.05 m over the respective important for understanding how it determines the FF phase. ranges of 25° 180° and 0.7 m 1.6 m for Meshing was a primary concern in several the 8 m solution and 40° 180° and 0.3 m regions of the model. Since the V-antenna was 1.5 m for the 5 m solution. To effectively only 65nm thick, which is only about three skin collect the data from this large parameter space, depths into an Au layer for a field propagating in a Batch Sweep study was accomplished. Each the M/LWIR, a Boundary Layer mesh of two solution was on the order of 40GB, with ~200K layers was applied to the top and bottom of the domain meshes and ~2.2M degrees of freedom. element. Due to the nature of large field Using a 2 CPU/16-core/128GB workstation, this confinement in subwavelength plasmonic permitted three concurrent solutions running on materials in this regime, an additional highly- 5 cores each, and each solution (encompassing both physics environments) took ~15 minutes.
Excerpt from the Proceedings of the 2016 COMSOL Conference in Boston Figure 5. COMSOL demonstration of how a linear phase gradient along a metasurface made of 8 discrete elements phased / apart scatters an oblique plane wave (thick dashed) from a normally-incident pane wave.
to the lack of phase and amplitude accessible by 4. COMSOL Simulation Results the given parameter space. The good news is, as referenced in Section 3, the symmetry in the V- The parametric profiles and their nominal antenna dictates that under 90° rotation it should contours are shown in Figure 4(a-d) for (a,b) produce an identical amplitude as the original with only a 180° phase shift. In this new and (c,d) at 8 m (a,c) and 5 m (b,d). These profiles were interpolated in MATLAB configuration, the dipole arms are aligned with out to 500 data points, generated from the the field components such that the current flows original COMSOL output, and the amplitudes the same distance along the same polarization were normalized to the maximum of the vectors, but at differing times than the original, combined data set. A characteristic dipole causing the phase difference. resonance, where the scattered field amplitude From here, the COMSOL profiles were used drops, can be seen at regions of strong phase as the input for algorithms used to produce the gradients. The steeper the phase shift, the larger specific designs for the fabricated metasurfaces the loss; this indicates areas to avoid in the and lenses, which will be described in the next selection process. As expected, the 5 m Section. amplitude is lower on average than at 8 m due to the smaller scattering cross-section. This will 5. Lens Fabrication lead to a less efficient lens unless the smaller characteristic length permits a larger packing While the importance of having an equally density. spaced phase shift is evident, it is also important The scattered cross-polarized electric field to not just simply bin those elements that meet a from the COMSOL results can also be used to given phase value and pick the one which demonstrate the anomalous refraction generated possesses the maximum amplitude in that bin, as from a linear phase gradient, as seen in Figure 5. this maximum would be different for each Here, by selecting eight V-antenna geometries chosen element, leading to variations in the (bottom) as dictated by the profiles in Figure 4, a scattered phase fronts due to incomplete broadside plane wave can form from neighboring constructive/destructive interference. Therefore, elements equally spaced one period apart and element selections for the fabricated devices separated in phase by /4. However, the were chosen based on an optimization algorithm, elements are not all unique; elements 5-7 are a which finds for a given the required relative 90° CCW rotation of elements 2-4. This is due phase difference ∆ and maximum average
Excerpt from the Proceedings of the 2016 COMSOL Conference in Boston
°
Figure 6. Phase calculation for example lens designs 5 m, 8, 100mm (left) and 8 m, 16, 50mm (right) used to identify the physical cut-offs of the phase regions. Black dots represent the midpoint values of Equation [3] which were used to better represent the deviation of the phase in a given region. amplitude Max Mean | | to within desired lenses and eight unique metasurfaces were tolerances, if it exists. If it does not exist, then fabricated, varying several parameters, to the 90° rotated element of an existing include: corresponding element is used. Additionally, the code assessed the best starting phase value— Operating wavelength, 5/8 m representing the zero phase point—for all /# 5 40 possible starting values from 0°→2 / . Number of unique elements, 8/16 This algorithm provided a series of elements Packing density (intracell periodicity) which had an amplitude deviation much smaller than previous published works [2, 4], while The investigation of a low /# is desired for maintaining a deviation in relative phase shift of “fast” lenses. The increase in pushes the less than 1%. discretization to a much better approximation of The last piece of the puzzle was to determine the continuous phase gradient, and thus should the physical boundaries of the phase regions, provide improved focus. The higher packing which is given by solving for in Equation [3], density should improve the efficiency of knowing all other parameters and setting 0, coupling energy into the anomalous refractive since this is a cylindrical lens. Examples of mode. these phase calculations for two lenses of SEM images in Figure 7 show (a) a full lens different focal lengths and different are shown at diameter 5mm with design parameters in Figure 6. The green line represents the direct phase calculation [3], while the blue line represents the normalization of this calculation to 0, which is entirely valid, since the relative difference is what matters. The colored dots are the points 2 / , while the black dots have been shifted over in to the midpoint position. The midpoints were selected as the region boundaries as they better represented the discretization of the constant phase regions1. The lens fabrication was accomplished at Sandia National Laboratories using a deep-UV photolithographic process. Nineteen unique
Figure 7. SEM image of a fabricated lens for 5 m, 1 Analogous to the improvement gained from 8, 100mm (a) and a 250x close-up of a using the Midpoint Rule for a Riemann Sum. similar lens but with 16, showing the gradient of regions with constant phase (b).
Excerpt from the Proceedings of the 2016 COMSOL Conference in Boston 5 m, 8, 100mm and (b) a 250x 8. References magnified region for a lens of 5 m, 16, 100mm. The strong gradients [1] N. Yu and F. Capasso, "Flat optics with generated by the variations in areal density can designer metasurfaces," Nat Mater, 13, 139 be seen clearly from the macroscopic view (a), (2014) and the logical progression of V-antenna [2] N. Yu et al., "Light propagation with phase geometries to induce a variation in phase can be discontinuities: generalized laws of reflection seen in the microscopic view (b). and refraction," Science, 334, 333 (2011) [3] N. Yu et al., "Flat optics: controlling 6. Discussion wavefronts with optical antenna metasurfaces," IEEE J Quantum Elect, 19(3), 2341 (2013) The 8 m results are in good agreement with [4] F. Aieta et al., "Aberration-free ultrathin flat previous works [2, 11], with the major difference lenses and axicons at telecom wavelengths based being a shift in the pattern to higher Δ and longer on plasmonic metasurfaces," Nano Lett, 12(9), /2 due to our 30 m thicker trace width. And 4932 (2012) while the 5 m results have no direct comparison [5] F. Aieta et al., "Aberrations of flat lenses and in literature, both the phase and amplitude aplanatic metasurfaces," Opt Express, 21(25), patterns are quite similar to the 8 m profile 31530 (2013) giving a high measure of confidence to the data. [6] C. Pfeiffer and A. Grbic, "Metamaterial The glaring difference is where there is a Huygens' surfaces: tailoring wave fronts with crossover of the zero phase point in the 8 m reflectionless sheets," Phy Rev Lett, 110, 197401 profile, in the 5 m profile there is a crossover of (2013) the 180° phase point. This becomes an arbitrary [7] S. Sun et al., "Gradient-index metasurfaces as viewpoint when one considers the effects of the a bridge linking propagating waves and surface 90° rotation, which in effect reverses this waves," Nat Mater, 11, 426 (2012) contrast (recorded, but not shown). [8] F. Aieta et al., "Reflection and refraction of Measurements are currently in progress, light from metasurfaces with phase which will illuminate the samples at with a discontinuities," J Nanophotonics, 6(1), 063532 plane wave from a quantum cascade laser (QCL) (2012) while using a microbolometer array to scan along [9] W. Wan et al., "Control the dispersive the optical axis and characterize the depth-of- properties of compound plasmonic lenses," field. This will spatially resolve the focal length Optics Communications, 291, 390 (2013) and thus provide a validation of the COMSOL- [10] A. Pors et al., "Broadband focusing flat based metasurface lens design process. mirrors based on plasmonic gradient metasurfaces," Nano Letters, 13, 829 (2013) 7. Conclusions [11] R. Blanchard et al., "Modeling nanoscale V- shaped antennas for the design of optical phased Through the use of COMSOL Multiphysics arrays," Phys Rev B, 85, 155457 (2012) as a design/characterization tool, planar single- [12] F. Aieta et al., "Out-of-plane reflection and layer metasurface lenses were successfully refraction of light by anisotropic optical antenna recreated at 5 m and 8 m over the focal ratios metasurfaces with phase discontinuities," Nano /5 /40 using variants of the seminal Lett, 12(3), 1702 (2012) metasurface design. In the end, while an analytic [13] S. Larouche and D. R. Smith, approach is a more rapid solution for assessing "Reconciliation of generalized refraction with simple geometries for metasurface lenses, diffraction theory," Opt Lett, 37(12), 2391 COMSOL has demonstrated an equal ability to (2012) efficiently design these lenses, with the added benefit of an expanded functionality to more 9. Acknowledgements complex structures and architectures which analytical methods cannot address. This makes We would like to acknowledge support from COMSOL a key asset in our future work of the Air Force Office of Scientific Research. optimizing the metasurface lens performance.
Excerpt from the Proceedings of the 2016 COMSOL Conference in Boston