Origami-Inspired Frequency Selective Surface with Fixed Frequency Response under Folding

Abstract

Filtering of electromagnetic signals is key for improved signal to noise ratios for a broad class of devices. However, maintaining filter performance in systems undergoing large changes in shape can be challenging, due to the interdependency between element geometry, orientation and lattice spacing. To address this challenge, an origami-based, reconfigurable spatial X-band filter with consistent frequency filtering is presented. Direct-write additive manufacturing is used to print metallic Archimedean spiral elements in a lattice on the substrate. Elements in the lattice couple to one another and this results in a frequency selective surface acting as a stop-band filter at a target frequency. The lattice is designed to maintain the filtered frequency through multiple fold angles. The combined design, modeling, fabrication, and experimental characterization results of this study provide a set of guidelines for future design of physically reconfigurable filters exhibiting sustained performance.

1. Introduction

Physically reconfigurable electromagnetic (EM) devices undergo shape change to attain electromagnetic performance tuning or tailored structural properties. The former is a type of passive EM signal control achieved through physical rearrangements of active EM components. In the latter, the primary motivation of the shape change comes from considerations other than EM functionality, such as structural deformations for deployable or adaptive structures, wearable devices, and conformal filters, while the EM performance is expected to remain fixed. In particular, exploration of EM device designs that maintain their functionality throughout a large structural deformation is identified to be one of the critical areas of scientific research according to the 2016 Multidisciplinary University Research Initiative solicitation entitled “4D Electromagnetic Origami” by the Air Force Office of Scientific Research; however, limited investigations on this subject are seen in literature. This article introduces a novel design of frequency selective surface (FSS) that combines a well-studied origami pattern called Miura-ori and densely packed circularly polarized four-arm Archimedean spirals to achieve a deployable structure that maintains frequency response. To the best of the authors’ knowledge, this is the first study on an origami-inspired physically reconfigurable EM device targeted at fixed frequency response.

Physically reconfigurable EM designs inspired by the intricate art of origami generate complex, functionally relevant 3D surfaces from easily manufactured 2D sheets, while being simply described by a small number of parameters within well-established mathematics. The tuning of EM performance through folding has been previously demonstrated in frequency selective surfaces (FSSs) [1,2,3] and antenna [4,5,6,7,8] applications. For example, linearly polarized FSSs using dipoles and the Miura-ori folding pattern offer frequency tuning through relative re-arrangement [1], relative positioning, and folding dipole elements [2,3] exhibiting frequency shifts on the order of 200 MHz for a 36° fold increase in [2] and 1 GHz for a 60° fold increase for a single layer FSS in [3]. However, there are few examples of the opposite case: Electromagnetic devices that maintain EM performance frequency while undergoing physical reconfiguration, particularly via folding. Devices with these criteria are of interest for flexible and conformal electronics applications, where decreased size, weight, and integrated functionality are important. Application spaces may include antenna arrays and FSSs embedded in adaptive wings where the structure changes due to mechanical constraints or deployable EM structures such as radomes or FSSs.

The conductive spiral trace shape and origami pattern of the FSS were selected for consistent frequency response over a wide range of polarization angles and deformations. Miura-ori is an origami pattern that reduces to a highly compact state when folded [9] and provides diverse deformations in intermediate folded states involving in-plane folding, bending and twisting [10]. These deformations can be tailored to produce a folded surface conforming to an arbitrary curvature. In addition, the facets may be constructed using a fairly rigid material and be subject to slight bending or twisting; most local deformations occur along the folds. This combination of structural characteristics make this an advantageous folding pattern for mechanical designs in large scale reconfigurable systems.

The fold repeatability and structural integrity of origami devices have been addressed using various materials that ease challenges in folding and flexibility. This includes bridging EM elements over fold lines [3] to mitigate challenges with folding and flexibility of metallic components and using shape memory alloys [11] and polymers [12,13] for fold actuation. These strategies can be directly applied to future implementation of this work with a rugged substrate design for a specific structural application but are out of the scope of this study.

Spiral elements are circularly-polarized [14,15] and are utilized in this study to mitigate detuning and response degradation caused by angular and polarization dependency, observed in linearly-polarized elements on folded surfaces [1,2]. A densely packed spiral lattice is printed on each facet of the Miura-ori pattern, to further reduce the angular dependency. An advanced additive manufacturing technique is employed to fabricate this device for an X-band application, with a high level of precision. The origami FSS is fabricated through extrusion printing of a polyurethane-based, silver ink [16,17] on a polypropylene (PP) substrate. The PP sheet is laser-scored along fold lines prior to spiral printing and folded into a Miura-ori pattern post-printing. The following includes details of the design, simulation, fabrication method and the experimental characterization of the conformal Miura FSS device.

2. Design of the Conformal FSS

Figure 1a shows a schematic of the four-armed Archimedean spiral used in this work [14]. Each arm is constructed based on the expressions in Equation (1) and thickened to obtain width w. The arm is capped at the end by a semi-circle of diameter w to match the arm width, and a circle of radius r is inserted at the center where the four arms converge. In this work, $a=0.424$ and $b=\{0,\frac{\pi }{2},\pi ,\frac{3\pi }{2}\}$. The number of turns in each spiral arm are 0.80. These parameters result in 4.4 mm diameter spiral elements.

$$\begin{array}{c}X=atcos(t+b)\\ Y=atsin(t+b)\\ -2\pi <t<2\pi \\ b=0,\frac{\pi }{2},\pi ,\frac{3\pi }{2}\end{array}$$

(1)

The spirals are arranged into a lattice with 5 mm center-to-center spacing. The spirals are tessellated to fill each parallelogram facet of a Miura-ori origami pattern. The Miura-ori fold pattern lends well to triangular and hexagonal packing when the parallelogram facets are rhombic with 60° vertex angles which allows for elements to have equilateral triangle spacing between centers. This triangular lattice is the tightest packing possible for the spiral elements on the Miura-ori substrate. Two lattices were explored in simulation and fabrication, a densely packed triangular lattice and a loosely packed hexagonal lattice. Figure 1b shows a triangular lattice and its hexagonal unit cell. In this work, samples are fabricated with a triangular lattice on the facets and a hexagonal lattice at the fold lines to accommodate folding.

Figure 2 shows the folding progression of the Miura-ori fold pattern. Miura-ori unit cells consist of four parallelogram facets tessellated into a doubly corrugated pattern. The fabricated samples in this work are comprised of a sheet of polypropylene containing a grid of 7 × 5 parallelograms each with vertex angle $\alpha$ = 60°, fold angle $\beta$, and side length $l=80$ mm.

3. Design Study

A triangular lattice is simulated using Floquet ports in Ansys HFSS to create a periodic structure based on the unit cell shown in Figure 1b. The lattice is simulated on a 0.75 mm thick polypropylene (PP) substrate (${ϵ}_r=2.2,tan\delta =0.01$). Spirals are assigned conductivity corresponding to measured values of the conductive ink (22,500 $\raisebox{1ex}{$\mathrm{S}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{cm}$}\right.$). Ink formulation is explained more in Section 4. Additional studies of material conductivity impact on RF performance are found in Appendix A.4.

The lattice is subjected to incident plane waves at normal and oblique incidence. Figure 3 shows the corresponding coordinate system for oblique incidence sweeps. The oblique incidence results can be extrapolated to predict the impact folding Miura-ori facets will have on the spiral lattice frequency response. However, it should be noted there are additional scattering parameters in the origami geometry not present in this model.

Figure 4 and Figure 5 show simulation results from the angle of incidence and polarization studies. Figure 4 shows the simulated transmission response ($S_{21}$) of a triangular spiral lattice while varying the incident angle ($\theta$). The frequency response of the lattice remains relatively constant through varying the incident angle of the plane wave up to $\theta =30$°. Filtering bandwidth increases for incidence angles greater than $\theta =45$°, from 1040 MHz at $\theta =30$° to 1730 MHz at $\theta =60$°; however, resonance is maintained at around 11.8 GHz (between 11.74 GHz and 11.87 GHz). At $\theta =75$° we observe a rightward shift in center frequency to 12.07 GHz and a significant broadening of the bandwidth of the filter (3.54 GHz). At $\theta =90$° (grazing angle) the device is non-functional, as would be expected for this planar geometry. The incidence angle parameter sweep on a flat substrate ($\beta =0$°) serves as an approximation for the transmission behavior of a folded substrate, as the facet orientations will alter the effective incidence of the incoming EM signal. The results of Figure 4 suggest filtering performance should be stable for a fold angle range of $\beta =0$° and 60°.

Figure 5 shows the simulated $S_{21}$ response of a triangular spiral lattice while varying polarization of the incident wave. Due to the four-fold symmetry of the spiral geometry and the densely packed FSS design, the frequency response remains constant despite varying polarization at any $\varphi \in [0^{\circ },{360}^{\circ }]$, while tunable FSSs based on linear elements in previous works [2,3] are operational at two orthogonal linear polarization directions only. The lattice maintains its center frequency and its bandwidth through all polarization angles. This is to be expected as the spiral geometry is designed to be polarization independent.

All results imply that the spiral lattice is suited to maintain a notch filter at a constant frequency with an $S_{21}$ below $-10$ dB for fold angles ranging from $\beta =0$° to 60°. These results can be extrapolated to infer how the response of the spirals will be impacted by the origami folds of the structure at varying fold angles.

4. Fabrication Method

An extrusion print method was employed for sample fabrication. This method produces spirals with high repeatability while also allowing for rapid translation from design to print. Conductive ink is dispensed via pressure driven syringe. The print path is controlled directly via g-code or Python code to facilitate higher levels of control over the print process. Prior to printing the spiral traces, the printer measures the height of the substrate at each spiral position and then adjusts the height of the syringe tip to accommodate for the variances in the thickness and flatness of the polypropylene. This process results in accuracy on the order of 1 $\mathsf{\mu }$m. Figure 6 shows the gantry printer setup. The impact of the print process on spiral geometries is addressed in Appendix A.

A silver thermoplastic polyurethane (Ag-TPU) ink formulation [16] was used to print the conductive traces, consisting of a 40:60 volume ratio of silver flakes (procured from Inframat Inc) to TPU (BASF Elastollan) dispersed in N-methyl-2-pyrrolidone (NMP). This ink was used for its flexibility, stability, and repeatability in the extrusion print process. Importantly, it can adhere to a multitude of materials, including the polypropylene used in this work.

Prior to printing the Ag-TPU ink on the PP substrate, the PP was laser-scored with the Miura-ori fold pattern. The spiral traces were then printed in Ag-TPU ink on the flat (unfolded) substrate using two syringes printing two spirals at once to increase print throughput. Post-printing, the sample was annealed at 125 °C in an oven to evaporate the residual NMP solvent, leaving behind silver particles encased in TPU. Figure 7 shows a fabricated prototype of this Miura-ori spiral FSS with a triangular lattice.

5. Experimental Setup

The origami FSSs are tested in a free-space setup with the sample placed between an incident horn and a receive horn. The test setup is calibrated using a gated-reflect-line (GRL) technique to mitigate the effects of external signals in the testing area. An 8 ns time gate was applied to the measurements. This setup measures the impact of the origami FSS acting as a spatial filter by comparing the incident and received field at the two horns.

As pictured in Figure 8, the setup consists of two 2–18 GHz horn antennas encased in a shielding barrel to obtain an acceptable noise floor for testing. The beams of these horns are collimated with RF lenses to focus the spot size at the GRL reference plane where the sample resides. This setup can accommodate samples up to 24 × 24 inches with a maximum thickness of 0.375 inches.

The FSS is subjected to deformations through folding and its spatial filtering impact is measured at each folded state. The FSS is folded by hand and the distance between mountain folds are measured to verify a precise fold angle. The structure is held in place by a test fixture between the two horn antennas. Due to the polarization independence exhibited by the spiral elements, the samples were tested under a horizontally polarized, normal incidence wavefront after determining the vertically polarized case maintained comparable results at all fold angles.

6. Experimental Results

A Miura-ori spiral FSS was tested using the experimental setup described in Section 5. Figure 9 shows the frequency response over the X-band (8–12 GHz) for the sample at multiple fold angles.

These results show that the origami FSS that maintains a fixed frequency response through a large shape deformation when compared to previous origami studies using dipoles. The structure maintains an acceptable filtering response below –10 dB at 11.5 GHz from a flat state to a 60° fold. The measurements agree with the incidence angle simulations from Figure 4, which predicted consistent filtering frequency up to $\theta =30$° ($\beta =30$°) and a rightward frequency shift at higher incidence angles. However, deviances due to test setup and sample fabrication should be noted. The center frequency of the filter differs by approximately 250 MHz between the simulated (11.80 GHz) and measured (11.55 GHz) average results. The slight quantitative deviation of the center frequency is likely the consequence of variances in printed spiral geometry and the conductivity of the Ag-TPU ink, which is dependent on ink formulation and the sintering processes. The geometric and material property variances are further discussed in studies found in Appendix A. The trend of the measured FSS shows a decrease in filtered $S_{21}$ response as fold angles increase, this is attributed to many spirals folding out of plane during the experiment thereby changing the number of spirals in the GRL measurement space. Additionally, spirals are simulated in an infinite plane rather than a full Miura-ori unit cell due to computational limitations in simulating 784 spirals per each 4-facet unit cell. This can impact the simulated effects of angles of incidence on measured $S_{21}$ values.

7. Conclusions

An origami-based FSS demonstrating consistent stop-band spatial filtering under large deformation was designed, fabricated and experimentally demonstrated in this study. Through systematic simulations, the geometry and close packing of the spirals demonstrated robust tolerance to changes in polarization and incident angles. The Miura-ori FSS leverages this quality to maintain filter performance at a fixed frequency, while undergoing a 50% decrease in size. The FSS design concepts from this study provide guidelines that can be applied to areas requiring physical reconfiguration, such as aerospace systems, wearables, and satellite operations.