Shape-morphable origami electromagnetic waveguides

Abstract

Electromagnetic waveguides are widely used in spacecraft, naval, electrical, and communication systems for transferring microwave energy. Conventional designs are typically rigid and bulky, offering little shape adaptability. This limits their use in confined spaces or systems requiring compact storage and high adaptability. Here we report highly shape-morphable origami electromagnetic waveguides that fold, deploy, and change shape. Inspired by origami folding techniques, including shopping bags and bellows designs, these waveguides demonstrate low-loss, robust microwave energy transmission in numerical and experimental studies. Results demonstrate that highly shape-morphable electromagnetic waveguides may effectively replace rigid counterparts including straight, twisted, and bent waveguides. This research employs a combined analytical and experimental framework to study the kinematics and mechanics of key geometries, particularly rectangular straight and twist bellows designs, offering rigorous structural design guidance. Engineering advancements and fundamental studies in this work lay the foundation for future adaptive microwave energy delivery waveguides.

Introduction

Electromagnetic (EM) waveguides are ubiquitous devices in spacecraft, naval, electrical, and communication systems to deliver microwave energy from their inlets to outlets¹. Most EM waveguides are made from rigid, bulky structures with no or minimal shape adaptability. Those existing designs occupy considerable volumes and offer limited flexibility in adapting to varying needs. This disadvantage creates challenges in using EM waveguides, particularly in highly confined spaces or systems requiring small storage volumes and high shape adaptability. Space exploration is a typical engineering field in which spacecraft components often need to be foldable and deployable due to the extreme volume constraints, particularly when stowed for launch. Ships and underwater vehicles present similar challenges due to their highly confined spaces for electrical and energy subsystems. There is a strong engineering need for EM waveguides that can be folded, deployed, and reconfigured to adapt to small volumes and/or variable applications. To address this need, this study employs origami principles in the design of EM waveguides.

The principles of origami have enabled numerous shape-morphable structures². However, using origami techniques for EM waveguide and microwave energy delivery has not been explored. Deployable and transformable origami structures have been developed such as shelters³, bridges⁴, large-scale architectural installations⁵, ship hulls⁶, self-folding robots⁷, reprogrammable and adaptive metamaterials^8,9, deformable energy storage devices¹⁰, aerospace mechanisms¹¹, and micron-scale structures¹². Robotics also adopts origami techniques to develop flexible and foldable designs, leveraging the benefits of lightweight and compact properties¹³. In the field of biomedical engineering, minimally invasive devices such as stents have been developed using origami principles¹⁴. Origami techniques have greatly advanced spacecraft structures, enabling designs such as solar arrays, foldable telescope mirrors, space habitats, and many others^15,16,17 with developments addressing the challenges of folding thick panels that are more representative of real engineering structures¹⁸. Research and development of origami devices for EM applications have focused on foldable and deployable communication antennas coupled with traditional EM waveguides for energy delivery. Previous studies by Yao et al.^19,20 have developed reconfigurable antennas, including origami accordion and spring antennas. These antennas feature operating frequencies that vary with height. The study by Liu et al.²¹ explores a reconfigurable axial mode helical origami antenna. The antenna features the ability to fold and unfold, operating at different resonant frequencies. Furthermore, Hayes et al.²² investigate a self-folding origami antenna that transforms two-dimensional structures into functional three-dimensional (3D) shapes by utilizing shape memory polymers and conductive foils. While Babaee et al.²³ have successfully developed reconfigurable origami acoustic waveguides, this work cannot be applied to origami EM waveguides.

EM waveguides typically feature hollow metal tubes with flanges at both ends to facilitate interconnection with other components. Utilizing the principle of total internal reflection along the conducting walls, these waveguides function within specific frequency ranges above the cutoff and operate predominantly in the fundamental (lowest order) mode. This study focuses on rectangular cross-sectional waveguides due to their widespread use, aiming to transform them into highly shape-morphable structures and investigate their EM transmission capabilities and mechanical behavior. To ensure the origami EM waveguides maintain a rectangular cross-section at the ends throughout all stages of shape-morphing, we developed various designs inspired by shopping bags and rectangular accordion bellows. Yasuda et al.²⁴ explored the kinematics of the Tachi–Miura polyhedron, a bellows-like 3D origami structure based on Miura-ori cells. However, the Tachi–Miura bellows dramatically changes the cross-section at their ends as they deploy, making them infeasible for EM waveguide applications. The kinematics and mechanics of near-circular cylindrical origami bellows based on Miura-ori cells and Kresling patterns have also been extensively studied^25,26,27. However, these cylindrical bellows cannot be used as rectangular EM waveguides. Origamists have designed rectangular accordion-like bellows and twist bellows^28,29,30. However, there has been a lack of fundamental understanding of their kinematics and mechanics, which are critical to the rigorous design of origami EM waveguides. Therefore, this study uses a combined analytical and numerical approach to address this knowledge gap.

In this study, we report highly shape-morphable origami EM waveguides for microwave energy transmission that can fold, deploy, and change shapes, as well as fundamental studies on their EM behavior and mechanics. Numerical simulations and experiments demonstrate low-loss, robust performance in straight, twisted, and bent waveguide configurations. Analytical and experimental studies further elucidate the kinematics and mechanics of rectangular accordion and twist bellows. These findings establish origami EM waveguides as promising alternatives to rigid counterparts and provide guidance for future adaptive microwave energy delivery systems.

Results

Foldable and deployable origami straight EM waveguide

Figure 1 presents the designs and working principles of a standard rigid straight EM waveguide and its origami counterpart. A WR-284 straight EM waveguide, as shown in Fig. 1a, serves as a benchmark design in this study. WR stands for “Waveguide Rectangular,” referring to its hollow rectangular cross-section that conforms to the WR-284 standard³¹. This rigid aluminum waveguide has two flanges (standard UG-584/U Round Cover) at both ends with a length of 12.00 inches (304.8 mm) from flange face to face. It operates in the frequency range of 2.60–3.95 GHz (S-band). Its standard mechanical designs and wide industrial applications make it a suitable baseline for developing and evaluating origami EM waveguides.

Fig. 1: Foldable and deployable origami straight electromagnetic waveguides.

a A standard WR-284 rigid straight waveguide for transmitting electromagnetic (EM) microwave energy. b Schematic of the working principle of EM waveguide. c The power loss of the EM waveguide at a wide range of microwave frequencies. d, e Images of an origami EM waveguide inspired by shopping bags in fully deployed and folded configurations, respectively. f A computer-aided design (CAD) model of the shopping-bag waveguide and its basic design. g CAD models showing the shopping-bag waveguide in fully and partially folded shapes. Scale bars: 60 mm.

An EM waveguide acts as a conduit to transfer microwave energy from the inlet to the outlet (Fig. 1b). Scattering parameters ($S$-parameters) are the key metrics to evaluate the transmission performance of EM waveguides³². ${S}_{21}$, plotted in Fig. 1c, represents the forward transmission coefficient or the forward gain of a two-port network. It measures the waveguide’s response at the outlet port (Port 2) caused by the microwave energy injected at the inlet port (Port 1) and is derived from the voltage wave measurements at these ports. The general formula to compute ${S}_{21}$ in decibel (dB) is given by:

$${S}_{21}\left({dB}\right)=20{log }_{10}\left(\left|\frac{{b}_{2}}{{a}_{1}}\right|\right)$$

(1)

where ${a}_{1}$ is the voltage wave incident on Port 1 from the source and ${b}_{2}$ is the voltage wave that exits Port 2³³. Lower $\left|{S}_{21}\right|$ values indicate a smaller loss in microwave energy transmission. Figure 1c plots the experimental ${S}_{21}$ vs. frequency curve for the straight WR-284 waveguide. The range for ${S}_{21}$ is set between −10 to 10 dB as it is a conventional approach to evaluating the transmission efficiency of standard waveguides. Waveguides with a near-flat line around 0 dB, as shown in Fig. 1c, have minimal transmission loss. The average $\left|{S}_{21}\right|$ over the entire frequency range of operation is the numerical metric to evaluate and compare the performance of different waveguides. The differences between the average $\left|{S}_{21}\right|$ of the baseline rigid waveguide and the origami counterparts determine the transmission loss of the origami designs. If this transmission loss is less than 1 dB per feet or 0.083 dB per inch, it is acceptable for many microwave applications³⁴.

We use a foldable and deployable design inspired by origami techniques employed in conventional paper shopping bags to showcase the concept of origami EM waveguides (Fig. 1d–g). Other origami EM waveguide designs will be discussed in the following sections. The inlet and outlet of the EM waveguide are required to maintain the rectangular cross-section during folding and deployment for attachment to other rigid components (e.g., flanges) in a system. The conventional paper shopping bag can keep a rectangular shape at its base and be folded flat for storage. The origami EM waveguide in Fig. 1d leverages this feature and consists of two mirror-symmetric shopping-bag sections. Figure 1f presents the computer-aided design (CAD) models of the fully deployed EM waveguide and a zoomed-in view of a shopping-bag section in extended and partially compressed configurations. We create additional creases along the tube’s transverse direction to achieve axial zigzag folding as depicted in Fig. 1e, g. The deployed waveguide has the same length and cross-sections at the ends as the standard rigid WR-284 waveguide. Its total length, including the two flanges, reduces to 85 mm in the fully folded state, resulting in a 72% length reduction compared to the 12-inch WR-284 waveguide. In this study, we use laminates of print paper and aluminum foil to fabricate the origami waveguides to demonstrate the overall concept and function of origami EM waveguides. Future work will explore other materials and their manufacturing techniques to fabricate the waveguides.

Supplementary Fig. S1a presents the actuation force of the shopping-bag waveguide measured in a uniaxial tension test during deployment, and Supplemental Video S1 shows the deployment process. The deployment includes two stages. The waveguide first axially deploys with a low actuation force (~0.44 N) by extending the zigzag folds. Once the waveguide fully extends in its axial direction, further stretching will open the cross-section to create a hollow tube for microwave propagation. Opening the cross-section requires higher forces than the axial expansion stage. The structure is deployed at 5 N, and an 85 N force leads to its final failure due to tearing.

The deployed shopping-bag waveguide differs from the straight WR-284 waveguide due to the opposing narrow walls projecting slightly inwards. The CAD model in Fig. 1f illustrates the inward projecting walls, denoted as the concave configuration that resembles a conventional shopping bag. Modifying the crease pattern can direct these projections outward to obtain a convex configuration. Completely flattening these projections after deployment can be challenging as it requires a high actuation force that may lead to structural failure. However, the microwave transmission performance of these waveguides shows a negligible difference from the standard rigid WR-284 waveguide, as discussed in the next section.

Performance of straight shopping-bag origami EM waveguide

We experimentally and numerically compare the transmission loss (i.e., ${S}_{21}$) of the straight shopping-bag origami EM waveguide to that of the WR-284 waveguide. All of the origami EM waveguides in this study are made from laminates of print paper and thin aluminum foil. Spray gluing aluminum foil onto paper forms the laminate, which is then folded to create various deployable waveguides with the aluminum foil adhered on the inner walls. The transmission capability of this laminate is determined primarily by the skin depth ($\delta$) of the aluminum foil. Skin depth refers to the distance into a conductor at which the EM wave amplitude decreases to ${e}^{-1}$ (∼37%) of its value at the surface³⁵. The equation for skin depth is given by:

$$\delta =\sqrt{\frac{2}{\mu \sigma \omega }}$$

(2)

Here, $\mu$ and $\sigma$ are the permeability and conductivity of the aluminum foil, respectively, and $\omega =2{{{\rm{\pi }}}}f$ is the angular frequency of the EM wave, where $f$ is the frequency³⁶. The conductivity of the aluminum foil determined using a 4-point probe test is 3.35 × 10⁶ Sm⁻¹ and, since aluminum is a non-magnetic material, its permeability is approximately that of free space. The maximum skin depth corresponds to the lowest frequency (2.60 GHz) and is calculated as 5.39 μm. When the thickness of the waveguide wall is at least $5\delta$, more than 99% of the EM wave is contained within it³⁵, ensuring efficient transmission with minimal loss. The mean thickness of the aluminum foil, 35 μm, is greater than the required depth of $5\delta$. Hence, effective transmission can be achieved by adopting this laminate.

A vector network analyzer (VNA) is used to experimentally validate the waveguide’s transmission performance. Figure 2a, b show the experimental setups utilizing the VNA to measure ${S}_{21}$. The VNA generates a known RF signal and transmits it through the waveguide. The signal propagates through the waveguide, and the VNA’s receivers capture both the transmitted signal (at the output port) and the reflected signal (at the input port). The ${S}_{21}$ value is calculated by comparing the transmitted signal to the original input signal. The VNA sweeps through a specified frequency range, performing these calculations for each frequency. This allows the generation of the ${S}_{21}$ vs. frequency plot.

Fig. 2: Microwave energy transmission of the shopping-bag EM waveguide.

a, b Test setups using a vector network analyzer (VNA) to measure the energy transmission losses of the WR-284 rigid and shopping-bag waveguides. c Simulation results of the S₂₁ vs. frequency plot as a comparison between the WR-284 rigid and the shopping-bag waveguides. d, e Simulated electric field distributions at the central frequency for the WR-284 rigid and the shopping-bag waveguides, respectively. f Average experimental S₂₁ vs. frequency plot as a comparison between the WR-284 rigid and the fabricated shopping-bag waveguides with deployment tests repeated ten times. Scale bars: 60 mm.

We compare the WR-284 waveguide performance against a replica made from the laminate material to evaluate the additional loss from using this laminate. The ${S}_{21}$ vs. frequency data of the WR-284 waveguide and the laminate replica are in Figure S1b, indicating comparable EM transmission properties. The average $\left|{S}_{21}\right|$ of the laminate waveguide differs by only 1.08 × 10⁻²dB per inch from the baseline, as shown in Table S1. Imperfections such as the wrinkles of the aluminum foil in the laminate can account for this deviation. Nevertheless, these results demonstrate that the laminate is a viable material to assess the proof-of-concept energy transmission capabilities of origami EM waveguides.

We use finite element analyses (FEA) in COMSOL Multiphysics to numerically compare the transmission performance of the shopping-bag waveguide and the baseline waveguide. The simulated ${S}_{21}$ curves in Fig. 2c show comparable performance although the shopping-bag waveguide has concave, inward-projecting walls as seen in Fig. 1f. The average numerical $\left|{S}_{21}\right|$ of the shopping-bag waveguide is only 3.3 × 10⁻³dB per inch higher than the baseline (Supplementary Table S2). Figure 2d, e are the simulated electric field norms of the baseline and shopping-bag waveguides. Compared to the baseline waveguide, the electrical field of the shopping-bag waveguide is less uniform due to imperfections such as the slightly concave walls.

Figure 2f presents the experimental ${S}_{21}$ curves. The shopping-bag waveguide is folded, deployed, and then tested with the setup shown in Fig. 2b to measure its ${S}_{21}$. This process is repeated 10 times, and the plot in Fig. 2f is its averaged ${S}_{21}$. The experimental ${S}_{21}$ data show that the origami waveguide has a comparable microwave energy transmission loss to that of the baseline waveguide. The origami waveguide’s experimental average value of $\left|{S}_{21}\right|$ is only 1.17 × 10^-2dB per inch higher than the baseline waveguide (Supplementary Table S1). The standard deviation of the 10 experimental $\left|{S}_{21}\right|$ values is 1.7 × 10⁻³dB per inch, showcasing the shopping-bag origami waveguide’s repeatability and stable performance. The numerical and experimental results demonstrate that the shopping-bag origami design provides an approach to foldable and deployable straight EM waveguides with comparable performance as the standard, rigid, bulky ones.

The crease pattern of the shopping-bag design can be altered to have convex, outward-projecting narrow walls in the structure (Supplementary Fig. S1c). The simulations and experiments indicate that the waveguide with convex narrow walls also has an energy transmission loss comparable to the baseline WR-284 waveguide (Supplementary Figs. S1d, e, Tables S1, S2). Therefore, the concave or convex narrow walls have negligible effects on the origami waveguide’s performance.

Shape-reconfigurable origami EM waveguide

Shape-reconfigurable EM waveguides capable of functioning under different geometries can meet multiple needs with one structure, providing a versatile solution for microwave energy delivery. Existing rigid EM waveguides, such as the straight WR-284 waveguide, have fixed geometries and cannot meet different waveguide lengths or energy delivery paths. Current flexible EM waveguides utilize plastic or elastic strains in their materials to enable structural flexibility³⁷. However, these waveguides can only achieve limited deformations due to the constraints in structural stiffness and material strength. Although the shopping-bag EM waveguide achieves excellent foldability, its energy transmission capability in partially or fully folded configurations is limited. There is a strong need for versatile, shape-reconfigurable EM waveguides beyond the capabilities of foldable and deployable designs.

We introduce a shape-reconfigurable straight EM waveguide inspired by origami bellows with accordion-like ridges. This bellows EM waveguide is not only foldable and deployable but also shape-reconfigurable, functioning effectively at various lengths and bends. The origami bellows derived from accordions³⁸ is a classic origami design with potential engineering applications such as inflatable booms³⁹ and shields for space exploration instruments⁴⁰. Here, we adopt this classic origami to design shape-reconfigurable EM waveguides. In addition, we also employ analytical models, simulations, and experiments to study the fundamental mechanical and EM behavior of origami-based waveguides.

The basic shape of the bellows EM waveguide is a straight, hollow, accordion-like tube consisting of identical units that can be axially deployed or folded (Fig. 3a, b). The CAD models in Fig. 3c, d show its transition from a deployed state to a fully packaged configuration with a zoomed-in view of the constituent units. The nominal deployment length is the same as the length of the baseline straight WR-284 waveguide. The fully packaged bellows EM waveguide with the two flanges is 126 mm, resulting in a 59% length reduction compared to the baseline WR-284 waveguide (longer waveguide deployed lengths would result in an even greater length reduction percentage). Supplementary Video 1 presents the deployment process of this waveguide. The bellows EM waveguide is folded from a thin laminate of print paper and aluminum foil using the crease pattern in Supplementary Fig. S2. The lengths of the crease lines are configured such that the inner sizes of the cross-section of the resulting bellows are the same as the rigid, straight WR-284 waveguide.

Fig. 3: Structure, folding, and deployment of a reconfigurable bellows origami EM waveguide.

a, b Optical images of a bellows origami EM waveguide in extended and compressed configurations, respectively. c CAD model and a zoomed-in view of the constituent units of the bellows EM waveguide. d CAD illustrations of the reconfigurable lengths of the bellows EM waveguide. Scale bars: 60 mm.

While the waveguide cross-section determines the crease dimensions, the number of units can vary to tune the maximum length of the deployed waveguide. It is found that the bellows waveguide can deploy nearly to the maximum length with a low actuation force before the required force sharply increases. In this high-force stage, the bellows waveguide only extends slightly but quickly reaches failure. The waveguide in Fig. 3 includes 26 identical units to ensure it reaches the nominal deployment length with low actuation force. Although previous work has developed applications of bellows-like origami, their mechanics and deployment behavior have not been studied in depth, leaving a knowledge gap that limits the rigorous structural design.

This work uses an analytical model and experiments to investigate the fundamental mechanical behavior of the bellows waveguides to address that knowledge gap. The analytical model relates the actuation force to the deployment distance by analyzing only one-eighth of a unit, i.e., panels 1 and 2 in Fig. 4a, which is possible due to the structure’s repetitive and symmetric construction. Supplementary Note 1 includes the detailed method to construct the analytical model. Here, we summarize the major steps and key findings. Panels 1 and 2 rotate around their center lines C1 and C2 when the bellows waveguide folds or deploys. The edges EF and CD stay on the horizontal and vertical symmetric planes. AE and BF remain vertical, and AD and BC stay horizontal.

Fig. 4: Deployment mechanics of a reconfigurable bellows origami EM waveguide.

a Unit bellows model with one-eighth highlighted portion under study. b Plots of non-dimensionalized force (F_n) vs. deployment angle (Ψ) for various natural deployment angles (Ψ₀) with a zoomed-in view highlighting the detailed behavior below. c Plots of non-dimensionalized force (F_n) vs. non-dimensionalized displacement (D_n) for various natural deployment angles (Ψ₀) with a zoomed-in view highlighting the detailed behavior below. d 3D CAD model of a bellows model composed of four repeating units with one corner highlighted (top), and load vs. displacement curves from the analytical model and experimental results (bottom). e 3D DIC results of the four-unit bellow model captured at four different stages of deployment corresponding to points marked in (d).

The bellows waveguide is not a rigid origami structure with non-zero strains on panels 1 and 2 during folding or deployment. The interface between panels 1 and 2 becomes to a complex, curved, variable surface instead of a sharp and straight edge when the structure deforms, making analytical investigations challenging. To build the analytical model, we assume panel 1 is rigid, panel 2 is elastic, and both panels remain flat. The rotation of panel 1 creates in-plane compressive and tensile stresses in panel 2 along the C2 direction, which leads to a twisting moment applied panel 1 around its center line C1. The twisting moment is balanced with the moment by the axial actuation force F, which is found as:

$$F=\frac{1}{12}{Et}\frac{1}{L}{W}^{2}\left(cos {\varPsi }_{0}-cos \varPsi \right)\frac{sin \varPsi }{\sqrt{1+{sin }^{2}\varPsi }}\frac{1}{cos \varPsi }$$

(3)

where $E$, $t$, and $W$ are the homogenized modulus, thickness, and facet width of the panel, respectively. $L$ is the length of panel 2. $\Psi$ denotes the deployment angle of the panel 1, which is the dihedral angle between panel 1 and $y$–$z$ plane. $\Psi ={0}^{{{\circ}}}$ corresponds to a fully folded bellows, while $\Psi ={90}^{{{{\rm{o}}}}}$ indicates the structure is stretched to the maximum. ${\Psi }_{0}$ is the natural deployment angle after creasing the fold lines when no external force is applied. The actual structure always has non-zero ${\Psi }_{0}$. The result shows that the bellows with larger stiffness, thickness, and facet width requires a higher force to deploy, where the facet width has more impact since it quadratically change the force. The non-dimensionalized form of the force ${F}_{{{{\rm{n}}}}}$ is given by:

$${F}_{{\rm{n}}}=\frac{F}{\frac{1}{12}{Et}{\frac{1}{L}W}^{2}}=\left({cos \varPsi }_{0}-cos \varPsi \right)\frac{sin \varPsi }{\sqrt{1+{sin }^{2}\varPsi }}\frac{1}{cos \varPsi }$$

(4)

The opening length, $D$, due to the rotation of panel 1 is defined as the deployment length:

$${{D}}=W\sin \Psi$$

(5)

Its normalized form ${D}_{{{{\rm{n}}}}}$ is:

$${{{D}}}_{{{{\rm{n}}}}}={\frac{D}{W}}=\sin \Psi$$

(6)

Figure 4b plots the relations between ${F}_{{{{\rm{n}}}}}$ and $\Psi$ with various initial natural angle ${\Psi }_{0}$ of 0°, 5°, 10°, and 15°. The plots show that ${F}_{{{{\rm{n}}}}}$ remains a low value from $\Psi ={0}^{{{{\rm{o}}}}}$ to nearly 60°, followed by a sudden increase range. ${F}_{{{{\rm{n}}}}}$ is negative before reaching the initial natural angle, indicating that a small compressive force is required to compress the waveguide to the fully folded state from the natural state. However, the initial natural angle has negligible impact on the ${F}_{{{{\rm{n}}}}}$ vs. $\Psi$ characteristics. Figure 4c plots the relationships between ${F}_{{{{\rm{n}}}}}$ and deployment length ${D}_{{{{\rm{n}}}}}$ with various initial natural angle ${\Psi }_{0}$ of 0°, 5°, 10°, and 15°. Like the ${F}_{{{{\rm{n}}}}}$ vs. $\varPsi$ plots, ${F}_{{{{\rm{n}}}}}$ remains low until ${D}_{{{{\rm{n}}}}}$ is ~0.87 ($\Psi ={60}^{{{{\rm{o}}}}}$), followed by a sharp increase in the load value. Therefore, this analytical model shows that a low-force actuation can deploy the bellows waveguide to nearly 87% of its maximum length. Therefore, for the bellows waveguide in Fig. 3 with $W$ = 6 mm, the maximum opening distance of panel 1 at the low-force actuation state is $W\sin (60^\circ )$ that is 5.2 mm, corresponding to 10.4 mm for a full unit. The waveguide is 212.8-mm long at its nominal length. Therefore, it requires at least 21 units. The fabricated model has 26 units to ensure the waveguide is well within the low-force actuation stage ($\Psi < {60}^{{{{\rm{o}}}}}$), when it reaches the nominal deployment length.

Experimental investigations on a 4-unit bellows by uniaxial tension tests and digital image correlation (DIC) provide insights into the deployment behavior of the bellows EM waveguide (Fig. 4d, e). The test article’s units are identical to those in Fig. 3a. The 4-unit waveguide serves as a scaled model to ensure the fully stretched test article stays within the field of view of the DIC cameras. Figure 4d shows that an actuation force below 10 N can deploy 90% of the maximum length. The force increases sharply to 120 N at the maximum where the model fails. The experimental behavior largely agrees with the results from the analytical model. The experimental force presents a more gradual transition between the low- and high-force stages than the analytical results.

The 3D DIC measurements reveal the reasons for the discrepancies between the analytical and experimental results. The DIC images in Fig. 4e illustrate the measured ridge shapes of a bellows edge in compressed, extended, and fully stretched states. The origami folds and facets remain straight and flat at the low-force stage while they rotate to expand the structure (images 1 and 2). Further deployment starts to flatten the origami folds (image 3), leading to a sharply increased actuation force until the structure ultimately fails (image 4). Supplementary Video 2 presents the deployment process and 3D shape of the ridge. In the analytical model, we assume the interface between panels 1 and 2 is sharp and straight. The DIC measurements show that the origami edges and facets near the vertices cannot remain straight or flat when the actuation force is high. As those edges and facets deform, more extension is allowed, resulting in a more compliance force-displacement relation at the high-force actuation stage compared to the analytical results.

We perform simulations and experiments to compare the microwave transmission performance of the bellows and baseline WR-284 rigid waveguide. Figure 5a presents the simulated electric field of the bellows waveguide deployed at the same length as the WR-284 waveguide. The simulated ${S}_{21}$ curves of the bellows and baseline waveguides are in Fig. 5b, showing comparable microwave energy transmission loss. The simulated average $\left|{S}_{21}\right|$ is only 1.7 × 10⁻³dB per inch higher than the baseline WR-284 waveguide (Table S2). These results show that the accordion-like cross-section with a non-smooth ridge surface has a negligible effect on the waveguide’s microwave energy transmission performance.

Fig. 5: Transmission characteristics of the straight bellows origami EM waveguide.

a Simulated electric field distribution at the central frequency. b Simulation results of the S₂₁ vs. frequency plot as a comparison between the WR-284 rigid and the bellows EM waveguides. c Experimental S₂₁ vs. frequency plot as a comparison between the WR-284 rigid and the fabricated bellows EM waveguides. d 3D CAD model of the bellows waveguide when partially (top) and fully deployed (bottom). e, f Simulation and experimental results illustrating the S₂₁ vs. frequency plot providing a comparative analysis at the two stages of deployment. g 3D CAD model of the bellows model when number of units (N) is 21 (top) and 31 (bottom). h Simulation S₂₁ vs. frequency plots, presenting a performance comparison between three bellows waveguides with different number of units and constant facet width (W = 6 mm). i Simulation S₂₁ vs. frequency plots, presenting a performance comparison between three bellows waveguides with different facet widths and constant number of units (N = 26).

Figure 5c includes the experimental ${S}_{21}$ curves of the bellows and baseline waveguides. The bellows waveguide is folded, deployed, and tested to measure its ${S}_{21}$ characteristics. This process was repeated 10 times, and the result in Fig. 5c is the averaged values over these 10 measurements. The ${S}_{21}$ curve of the bellows waveguide overlays with the rigid waveguide at the mid-frequency range from 3.08 GHz to 3.72 GHz, while deviating at the low- and high-frequency ranges. The experimental average $\left|{S}_{21}\right|$ over the full frequency range is 7.08 × 10⁻²dB per inch higher than the baseline waveguide (Table S1). The discrepancy between the simulated and experimental results indicates that fabrication imperfections are the key factor attributing to the higher loss. The complex crease pattern and the manual folding process make the bellows waveguide prone to imperfections. In the future, using advanced manufacturing techniques such as 3D printing could improve the bellows waveguide’s performance. Nevertheless, the average loss of this model remains lower than 0.083 dB per inch and is generally acceptable in engineering applications. The experimental standard deviation of $\left|{S}_{21}\right|$ is 3.3 × 10⁻³dB per inch, indicating a repeatable performance of the bellows waveguide.

The bellows waveguide can effectively transmit microwave energy at other lengths. Figure 5d shows the CAD models of a bellows waveguide at partially and fully deployed configurations. The simulated ${S}_{21}$ curves in Fig. 5e show that the waveguide has comparable performance in these configurations. The simulated average $\left|{S}_{21}\right|$ of these configurations is only 1.7 × 10⁻³dB per inch higher than the baseline rigid WR-284 waveguide (Table S2). The experimental ${S}_{21}$ curves also show the transmission performance at different configurations is comparable (Fig. 5f). The experimental average $\left|{S}_{21}\right|$ of the partially deployed waveguide over the full frequency range is 6.67 × 10⁻²dB per inch higher than the baseline waveguide with a 3.3 × 10⁻³dB per inch standard deviation across 10 measurements (Table S1). These results showcase that the bellows waveguide can be reconfigured into different lengths to meet the need for tunable distances for microwave energy transmission.

We perform parametric studies to numerically investigate the energy transmission loss of bellows waveguides constructed with other design parameters. In this work, we vary the number of units ($N$) and the facet width ($W$) to obtain different waveguide designs. Figure 5g includes two CAD models with the same length of 212.8 mm and facet width of 6 mm but different numbers of units. The plots in Fig. 5h illustrate the transmission response ${S}_{21}$ for bellows waveguides with different numbers of units. The values in Table S2 reveal that the rise in transmission loss with an increased $N$ is marginal. The simulated average $\left|{S}_{21}\right|$ values of all these designs are below 8.3 × 10⁻³dB per inch, showcasing a strong EM energy transmission performance for all designs. Although the simulations show the effects of $N$ are marginal, it should be noted that smaller N leads to smaller maximum deployment length, and larger $N$ increases the complexity of fabrication. We also simulate the models by varying $W$ while fixing the bellows units to $N=26$, resulting in the curves in Fig. 5i. The waveguide with the smallest facet width (6 mm) exhibits the lowest transmission loss (Table S2). Larger $W$ leads to higher loss. This trend aligns with the expected behavior as a reduction in facet width causes the design to approach closer to the standard rectangular waveguide design. Therefore, the shape or diameter of the hollow space can slightly influence the supported frequency range, particularly for the TE₁₀ (fundamental) mode. However, as long as the operating frequency remains away from the cutoff edge, these effects are negligible, and the waveguide continues to perform reliably.

The bending flexibility of the bellows origami design enables the reconfiguration of the straight waveguide into two bent shapes, making them suitable for use as waveguide bends. Figure 6a shows a standard rigid WR-284 E-bend waveguide and the origami counterpart reconfigured from the straight bellows waveguide. The WR-284 E-bend waveguide is an L-shaped device, consisting of two straight rectangular tubes and a curved transition section in between bent along the wide side walls. Figure 6b presents the simulated electric field norm of the bellows waveguide bent to 90 degrees. Supplementary Video 1 presents the bending process of this waveguide. The simulated ${S}_{21}$ curves of the baseline rigid E-bend and origami bellows waveguides are in Fig. 6c, showing comparable performance. The simulated average $\left|{S}_{21}\right|$ in Table S2 indicates that the bellows waveguide slightly outperforms the baseline. Compared to the sharp directional transition in the L-shaped rigid E-bend, the origami waveguide exhibits a gradual change that can improve transmission performance. The experimental results in Fig. 6d resemble those of the straight bellows waveguide, showing comparable loss in the mid-frequency range and higher loss at low and high frequencies. The simulation results showing low loss over the entire frequency range indicate that fabrication imperfections are the key factor for the discrepancy. The experimental average $\left|{S}_{21}\right|$ in Table S1 shows a 7 × 10⁻²dB per inch loss, comparable to the straight bellows configuration. The standard deviation of 10 measurements is 3.3 × 10⁻³dB per inch, showing the bent bellows waveguide’s excellent repeatability.

Fig. 6: Reconfiguration of the bellows origami EM waveguide into bendable waveguides.

a A WR-284 E-bend waveguide (left) as the baseline model and the fabricated bellows waveguide in bent (center) and compressed configurations (right). b, c Simulation results of electric field distributions at the central frequency and the S₂₁ vs Frequency, comparing the baseline WR-284 E-bend waveguide with the bellows waveguide. d Experimental S₂₁ vs frequency plot, comparing the performance of the WR-284 E-bend waveguide with the fabricated bellows bend waveguide. e A WR-284 H-bend waveguide (left) as the baseline model and the fabricated bellows waveguide in bent (center) and compressed configurations (right). f, g Simulation results illustrating electric field distributions at the central frequency (left) and the S₂₁ vs frequency plot (right), comparing the baseline WR-284 H-bend waveguide with the bellows bend waveguide. h Experimental S₂₁ vs frequency plot for a comparison between the WR-284 bend waveguide and the fabricated bellows bend waveguide. Scale bars: 60 mm.

In Fig. 6e, the bellows waveguide design showcases its versatility by replicating the rigid WR-284 H-bend waveguide. The H-bend waveguide directs microwave energy via a 90-degree bend through a hollow tube bent on its narrow side walls. The flexibility of the bellows waveguide allows its reconfiguration from a straight shape to a 90-degree bend. Supplementary Video 1 presents the bending process of this waveguide. Figure 6f presents the simulated electric field norm of the bent bellows waveguide. The simulated ${S}_{21}$ curves of the baseline rigid H-bend and origami bellows waveguides show comparable performance. The simulated average $\left|{S}_{21}\right|$ of the bent bellows waveguide has a slightly higher loss of 1.7 × 10⁻³dB per inch than the rigid WR-284 H-bend waveguide (Table S2). The experimental ${S}_{21}$ curves are similar to those of the straight and E-bend bellows waveguides with an average |S₂₁| of 7.83 × 10⁻²dB per inch loss. The standard deviation of 10 measurements is 3.3 × 10⁻³dB per inch, showing the H-bend bellows waveguide’s excellent repeatability. This study highlights the shape-reconfigurability for versatile practical applications using origami waveguide designs.

Origami twist EM waveguide

Twist waveguides rotate the polarization of microwaves are commonly used devices for microwave energy transmission. Figure 7a (left) shows a rigid twist waveguide with a rectangular cross-section conforming to the WR-284 standard. This waveguide features a helical twist along the waveguide length, which enables the controlled rotation of the polarization of the EM signals by 90° from the inlet to the outlet. In this work, we introduce a twist origami waveguide derived from the straight bellows design that can fold, extend, and twist, offering a solution to reconfigurable twist waveguides. Figure 7a (center and right) shows the twist bellows waveguide in extended and compressed configurations. The edges of the twist bellows waveguide deform from straight paths in the fully compressed state to helical curves during deployment. Figure S4a presents the CAD models of the extended twist bellows waveguide and a zoomed-in view of its constituent units. The twist waveguide is fabricated from laminates of print paper and aluminum foil, the same as the origami waveguides discussed above. The nominal deployment length of the waveguide, including the flanges, is 11.00 inches (279.40 mm), the same as the rigid twist waveguide. The size changes to 114 mm when fully compressed, resulting in a 59% reduction in packaging length.

Fig. 7: Structural design, analyses, and microwave energy transmission performance of a twist bellows EM waveguide.

a A twist WR-284 EM waveguide (left) as the baseline model and the fabricated bellows waveguide in extended (center) and compressed configurations (right). b 2D crease pattern adopted to fabricate the twist bellow model. c Plot of rotation per half-unit (ρ/M) vs. deployment angle (ψ) for different cut angles (α). d Plot of normalized deployment length per half-unit (D_n/M) vs. deployment angle (ψ) for different cut angles (α). e Plot of rotation per half-unit (ρ/M) vs. normalized deployment length per half-unit (D_n/M) for different cut angles (α). f Comparison of analytical and experimental curves showing the relationship between rotation angle and deployment length. g The relationship between the cut angle (α) and the required number of half units (M) to reach 90˚ twist at its nominal length. h, i Simulated electric field distributions at the central frequency (left) and the S₂₁ vs. frequency relation, comparing the baseline WR-284 twist waveguide with the twist bellows waveguide. j Experimental S₂₁ vs. frequency relations as a comparison between the WR-284 twist waveguide and the fabricated twist bellows waveguide. Scale bars: 60 mm.

The twist bellows waveguide rotates while it deploys in the axial direction. Its origami folding pattern needs to be designed to ensure its end rotates by 90° at the nominal deployment length so that it may be used to replace the rigid twist waveguide. The twist bellows waveguide is folded from a 2D crease pattern derived from the straight bellows waveguide using the same crease and facet dimensions but with a rotation. Specifically, the crease pattern is obtained by cutting the straight bellows waveguide’s pattern along two inclined parallel lines at an angle $\alpha$ as shown in Fig. 7b. The cross-sectional dimensions of the waveguide define the pattern’s width ${L}_{0}$. Supplementary Fig. S4b includes the dimensions of the creases. Joining line PS to QR forms a cylinder, and PS or QR becomes helical. Folding the crease lines creates the twist bellows. The bellows length at which a 90° rotation occurs depends on the cut angle $\alpha$ and the number of half-units $M$. Therefore, modeling the coupled rotational and axial motions of the twist bellows is key to the waveguide design. However, this fundamental behavior has not been previously studied. In this work, we develop an analytical model to address this knowledge gap to guide the twist bellows waveguide design.

The model relates the rotation angle ρ per half unit (ρ/M) and the normalized deployment distance ${D}_{{{{\rm{n}}}}}$ per half unit (${D}_{{{{\rm{n}}}}}/M$) to the facet deployment angle Ψ° for a given pattern cut angle $\alpha$ by studying the cross-section of a single half-unit. ${D}_{{{{\rm{n}}}}}$ is the deployment distance $D$ normalized by the width of the crease pattern ${L}_{0}$, i.e., ${D}_{{{{\rm{n}}}}}\,={D}/{L}_{0}$. The definition of Ψ is the same as the one used for the straight bellows waveguide. Ψ$\,=\,90$° indicates the bellows unit is fully extended, and Ψ$\,=\,0$° means it is fully compressed. The crease dimensions and facet width $W$ are constant. We focus on Ψ from 0° to 60° because only low force is needed for this range as discussed above. The details of the method and illustrations to construct the model are in Supplementary Note 2 and Figure S5. Figure 7c presents the relation between $\rho /M$ and Ψ for cut angles $\alpha$ from 5° to 30°. The bellows unit rotates as the origami facets open with increasing deployment angle. A higher cut angle $\alpha$ leads to more rotation for the same deployment angle. Figure 7d shows the relations between ${D}_{{{{\rm{n}}}}}/M$ and Ψ for cut angles $\alpha$ from 5° to 30°. In contrast to the rotating behavior, a higher cut angle leads to a lower deployment distance per half unit for the same deployment angle.

Figure 7e presents the relation between the rotation per half unit ($\rho /M$) and the normalized deployment distance per half unit (${D}_{{{{\rm{n}}}}}/M$). As expected, the twist bellows rotates monotonically during deployment. A higher cut angle $\alpha$ results in more rotation at the same deployment distance. However, the maximum ${D}_{{{{\rm{n}}}}}/M$ at Ψ$\,=\,60$° at the ends of the curves decreases as the cut angle increases. This indicates that a higher cut angle reduces the maximum reachable deployment distance per half-unit at the low-force actuation stage. The experimental $\rho /M$ vs. ${D}_{{{{\rm{n}}}}}/M$ relation is in Fig. 7f. The “Materials and Methods” section and Supplementary Fig. S6 include the details of the experimental methods and setup. The tested twist bellows waveguide has 116 half units in total and a cut angle of 29°. The corresponding analytical relation is also included for comparisons. Unlike the analytical model, the twist bellows cannot be physically compressed to zero length. Therefore, the experimental curve begins from the compressed position rather than zero, with rotation recorded as the model deploys. The comparison shows that the analytical results of the coupled axial deployment and rotation motions match the measurements.

The combination of the cut angle $\alpha$ and the number of half-units $M$ must ensure the twist bellows waveguide rotates by ${\rho }_{0}=\,90$° at the nominal length (${D}_{0}=\,$219.4 mm) without the flanges, the same as the rigid twist waveguide. The required rotation per half-unit and normalized deployment distance per half-unit are ${\rho }_{0}/M$ and ${D}_{0}/({L}_{0}M)$, respectively. The twist bellows designs with different $M$ satisfying this requirement fall on the straight dash line, ${\rho }_{0}/M$ vs. ${D}_{0}/({L}_{0}M)$, in Fig. 7e. Therefore, for a given cut angle $\alpha$, the intersection between the straight dash line and the solid curve can satisfy both this requirement and the coupled rotation-deployment relation. The value of ${\rho }_{0}/M$ at the intersection can be used to calculate $M$ to obtain a 90° rotation at the nominal length for the given cut angle $\alpha$. Figure 7g presents the relation between the cut angle $\alpha$ and the required number of half-units $M$. When the cut angle is small ($\alpha \, < \,10$°), the intersection does not exist, indicating that no twist bellows can reach 90° rotation at the nominal length with low-force actuation ($\psi \, < \,60$°). As the cut angle increases, the required number of half-units also increases as the value of ${\rho }_{0}/M$ at the intersection decreases. The twist bellows waveguide fabricated and tested in this work has a cut angle of 29°, which needs 116 half-units to obtain 90° rotation at the nominal length according to the analytical model. Therefore, our model includes 116 half-units, and the experimental results in Fig. 7f confirm the accuracy of the analytical model.

We use both simulations and experiments to evaluate the EM transmission capability of the twist bellows waveguide. Figure 7h and i display the simulated electric field and the ${S}_{21}$ curves of the twist bellows waveguide. The close alignment of the simulated ${S}_{21}$ curves of the baseline and origami waveguides shows they have comparable transmission capabilities with an average $\left|{S}_{21}\right|$ of 1.8 × 10⁻³dB per inch (Table S2). The experimental ${S}_{21}$ curves in Fig. 7j indicate the twist bellows design has a higher loss than the rigid twist waveguide, but their transmission losses are equivalent at higher frequencies from 3.45 to 3.95 GHz. The measured average $\left|{S}_{21}\right|$ of the bellows twist waveguide is only 6.36 × 10⁻²dB per inch higher than the value of the rigid twist waveguide. The standard deviation of 10 measurements is 2.7 × 10⁻³dB per inch, indicating excellent repeatability.

Demonstration of microwave power transmission

Microwave power delivery experiments demonstrate the capability of the origami waveguides to transmit microwave power (Fig. 8). The extended origami waveguides are connected to a microwave source at their right ends and transmit the microwave power to an LED at a distance. The circuitry of the LED harvests the microwave power and rectifies it to illuminate the LED. All three origami waveguides successfully illuminate the LED at 10-cm and 20-cm distances. Figure 8 also shows that transmission attenuation at longer distances reduces LED light intensity. Supplemental Video S3 shows the experimental processes. It should be noted that EM waveguides are usually employed to transmit microwave energy from the inlet to the outlet through contact connections at the flanges. The experiments of remote power transmission here are not the typical way of using EM waveguides; rather, they serve as demonstrations to visually showcase and further confirm the power transmission capability of the origami waveguides.

Fig. 8: Demonstration of microwave energy transmission at distances.

In an anechoic chamber, an LED is illuminated at 10 cm (left) and 20 cm (right) distances by microwave energy delivered through a (a) shopping-bag waveguide, b straight bellows waveguide, and c twist bellows waveguide. Scale bars: 100 mm.

Discussion

This paper introduces an innovative approach to highly shape-morphable EM waveguides based on origami-inspired designs. The origami EM waveguides leverage the folding techniques of shopping bags and accordion-like rectangular straight or twist bellows to realize folding, deployment, bending, and twisting motions with minimal actuation forces. Folding the origami EM waveguides achieves a 59% to 72% reduction in structural lengths compared to their conventional rigid counterparts. This substantial packaging efficiency provides a practical solution for using EM waveguides in highly confined spaces such as spacecraft, sea-surface, and underwater vehicles. Bending the straight origami bellows waveguide reconfigure it into H- and E-bend waveguides. The design of twist bellows waveguide obtains a coupled motion between its axial deployment and rotation to reach a 90-degree twist at the nominal length. These results demonstrate the origami waveguide’s adaptability to variable applications without changing waveguide structures.

This study combines simulations, analytical models, and experiments to investigate the EM and mechanical behavior of origami-based EM waveguides. COMSOL Multiphysics simulations and VNA transmission measurements confirm that the energy losses of the origami waveguides are in the range from minimum to less than 0.083 dB per inch with excellent repeatability, indicating the potential to replace conventional rigid waveguide designs for applications where flexibility and deployability are needed. These losses (<0.083 dB per inch) align with state-of-the-art flexible waveguides such as He et al.’s⁴¹ S-band design, which demonstrated similar insertion loss levels and successfully handled microwave power up to ~2.7 kW despite fabrication-induced bandwidth reductions. This comparison suggests that with further optimization of materials and fabrication methods, our origami waveguides could achieve comparable power handling and efficiency. An analytical model on the deployment of the straight bellows waveguide reveals the mechanics leading to the sharp transition from low-force to high-force actuations. The mechanical loading experiments assisted by 3D DIC measurements align well with the analytical modeling. The mechanical tests also reveal that the sharp origami ridges are fully flattened or stretched into a smooth curved surface at the bellows corners, which is the fundamental reason causing the maximum deployment length and the final structural failure. An analytical model for the kinematics of twist bellows waveguides provides the relationships between the deployment length and the end-rotation angles as functions of the parameters of crease pattern designs. Experiments on the twist bellows motions confirm the analytical insights. The model serves as a design guideline for the rigorous design of twist bellows structures with rectangular cross-sections. Overall, this work addresses the engineering needs for and the lack of fundamental knowledge of highly shape-morphable, origami-based EM waveguides.

In principle, discontinuities in waveguides, such as those introduced by folding mechanisms of our origami-inspired waveguides, can lead to the excitation of higher-order modes and additional losses. However, in our waveguide designs the operating frequency lies below the cutoff frequencies of the next higher-order modes. Simulated field distributions confirm TE₁₀-dominated propagation throughout the structure. These simulations incorporate the full geometry, including angular fold transitions, and do not show appreciable energy transfer to higher-order modes. Additionally, although the mechanical deformations could lead to small conductor shape variations, the scale and nature of the deformations introduced by our origami pattern were carefully selected to preserve cross-sectional uniformity and continuity. While a detailed mode decomposition was beyond the scope of this feasibility study, we acknowledge that more advanced mode-based modeling and measurement strategies would be valuable for quantifying the influence of more complex reconfigurations and to address losses due to deviations from ideal construction. These investigations represent a promising direction for future work.

This work focused on EM waveguides equivalent to their rigid WR-284 counterparts. Smaller origami structures could exhibit greater sensitivity to folding curvature due to the interplay between physical dimensions and operating wavelength. While folding resolution impacts frequency response, our designs maintain TE₁₀ dominance below higher-mode cutoffs. Future work will establish folding-dimension thresholds via parametric mode analysis. At reduced scales minor deviations in the geometry and irregularities caused by folding can scatter energy or create impedance mismatches, potentially increasing the loss. However, our deployable design strategy prioritizes applications where size and stowage efficiency are critical such as satellite communication arrays or large reconfigurable antennas. For these systems, the benefits of compact packaging outweigh the minor performance trade-offs associated with miniaturization. It would be less advantageous to adopt origami-inspired waveguides for smaller devices as these conventional rigid designs already achieve sufficient compactness without folding.

The promising results here warrant a few future directions to explore. Implementing more advanced fabrication methods using advanced engineering materials can further reduce transmission losses in the current designs and improve manufacturing efficiency. Additive manufacturing of conductive soft or soft-hard composite materials may be a viable approach to replace the current construction methods based on paper–aluminum foil laminates. Exploring different actuation mechanisms to facilitate the autonomous deployment of the waveguides at the service location is another important research direction. Additionally, a deeper exploration into the maximum power handling capabilities is essential to extend this concept to high-power applications. This includes thermal-deformation analysis during folding cycles, building on flexible waveguide precedents to ensure foldability under high power. In conclusion, the findings presented here set the stage for further innovation and practical implementation of highly shape-morphable origami EM waveguides.

Materials and methods

Fabrication of origami waveguide models

We constructed the origami models using a laminate made of paper and aluminum, with aluminum lining the waveguide’s inner walls. This laminate was fabricated by attaching 35-µm-thick aluminum foil onto 20-lbs paper using industrial spray adhesive. The printed pattern on the paper enabled easy folding and creasing to create different origami models. The ends of the models were attached using double-sided tape to generate the enclosed structure. Round flanges were connected to the ends of the structure to form the complete waveguide structure.

Testing waveguides for transmission performance using VNA

We extracted the ${S}_{21}$ values over the waveguide’s operating frequency range using a calibrated VNA. In this setup, the VNA generated a signal with a known value that passed through the waveguide under test. As the waveguide modified the signal, the VNA’s receivers monitored and recorded the changes, allowing us to measure the transmission characteristics across the frequency range. A key focus during testing was ensuring the repeatability of transmission performance. After fabrication, the waveguides were subjected to ten full compression and deployment cycles, with the ${S}_{21}$ values recorded after each deployment. The standard deviations of the recorded transmission curves are presented in Table S1, reflecting the consistency of the designs. Additionally, the mean ${S}_{21}$ curves for each waveguide, computed from the ten cycles, were plotted to visualize the overall performance stability.

Finite element analysis

Electromagnetic transmission performance analysis

To analyze the transmission performance of the origami waveguide relative to its rigid counterpart, FEA using COMSOL Multiphysics was adopted. EM wave propagation inside the waveguide was modeled by solving Maxwell’s Equations in the frequency domain. The 3D CAD models of different rigid and origami designs were modeled initially and then imported to COMSOL software. Since the rigid waveguide is fabricated using aluminum, the material properties of aluminum were assigned to the boundaries of the waveguide to represent the metallic walls. Impedance boundary conditions were applied to account for minor losses due to the material’s finite conductivity. The material properties of air were assigned to the inner volume representing the dielectric in which the wave propagates. The TE₁₀ mode, the dominant mode for rectangular waveguides, was used to excite the system and allow wave propagation within the waveguide. The inlet and outlet ports were defined as wave ports to allow wave propagation and measure the $S$-parameters, specifically ${S}_{21}$. A frequency sweep was conducted within the frequency range of operation of the waveguide to generate the ${S}_{21}$ vs. frequency plots.

Parametric study of the straight bellows waveguide EM performance

We conducted a comprehensive parametric study on the straight bellows using Rhino 3D for design generation and COMSOL Multiphysics for simulation analysis. First, we used a Python script to modify the design parameters, such as varying facet widths and the number of bellows units, and generated CAD models with different geometries in Rhino 3D. After generating each model, we exported it as a.step file and imported it into COMSOL Multiphysics. Once imported, we set up the necessary physics, material properties, and boundary conditions to simulate the waveguide’s transmission performance. By systematically adjusting the bellows parameters, we analyzed how changes in geometry impact the waveguide’s transmission characteristics.

Uniaxial tension test of the twist bellows model

The experimental setup, shown in Supplementary Fig. S6, for the uniaxial tension test on the twist bellows model involved attaching both ends of the waveguide to 3D-printed plates using double-sided tape. In this setup, the top plate was clamped whereas the bottom plate remained free to rotate. The load frame secured both plates with the twist bellows initially in its compressed state. As the top fixture moved upward, the bottom plate rotated, and the experiment monitored the resulting angular displacement using a Rotary Variable Differential Transformer (RVDT). To extend the range of angular measurements, the RVDT was integrated with a pulley system. The voltage output from the RVDT was recorded throughout the deployment process, and angular displacements were calculated based on previously calibrated values. The RVDT was calibrated using a biaxial extensometer calibration block to establish the rotation vs. voltage values.

Power transmission capability

We used a Traveling Wave Tube Amplifier (TWTA) from Xicom Technology for this demonstration. The TWTA generated a high-power 2.32-GHz signal, which is above the 2.07-GHz cutoff frequency of the WR-284 waveguide. We connected the TWTA to the input port of various waveguides inside an anechoic chamber, which minimizes reflections and external interference. This input port served as the signal inlet. At the output port of the waveguide, we positioned an LED soldered to a Schottky diode, starting at shorter distances and then increasing the distance to observe signal reception. We adjusted the length of the diode to configure it as a half-wave dipole antenna, allowing it to capture the high-frequency signal. The diode then generated the necessary current to light up the LED, demonstrating successful signal transmission through the origami waveguide.