Theoretical concept

As illustrated in Fig. 1(a), the design incorporates foldable side walls, which are bent and folded into deformable upper and lower sections. The sidewall’s fold line is located at the midpoint of dimension b, so the upper and lower sections are equal in size and tilt symmetrically about the fold line, thereby enabling the vertical folding mechanism. Note that the fold line can be either a mountain fold or a valley fold, allowing the waveguide to expand outward or contract inward.

Figure 1.

(a) Proposed vertical folded waveguide, (b) Simulated cutoff frequencies for various folded angles.

Figure 1 Long description

A three dimensional waveguide style shape with a folded angle theta marked at the lower edge. Dimension arrows label a equals 100 millimeter and b equals 50 millimeter. Another dimension label shows L equals 187.5 millimeter. A small coordinate marker is drawn on the top face. The image B showing three separate three dimensional folded shapes stacked vertically. Each shape has a fold angle label next to it: 60 degree, 90 degree and 120 degree. Each shape has a small coordinate marker on the top face. The image C showing a line graph titled, Cutoff Frequency. The x-axis label is, Frequency (GHz), ranging from 1.2 to 2.0 with tick marks at 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9 and 2.0. The y-axis label is, S21 (dB), ranging from minus 6 to 1 with tick marks at minus 6, minus 5, minus 4, minus 3, minus 2, minus 1, 0 and 1. Six step like curves rise from about minus 6 to about 0 and then remain near 0. Each curve is labeled near its transition: 120 degree, 110 degree, 100 degree, 90 degree, 80 degree and 60 degree. The transition frequencies appear at about 1.30 for 120 degree, about 1.35 for 110 degree, about 1.40 for 100 degree, about 1.50 for 90 degree, about 1.60 for 80 degree and about 1.70 for 60 degree.

To investigate the influence of the deformed structure on the waveguide’s performance, this study defines the folded angle θ as the inward bending angle of the waveguide’s inner walls. During the deformation, different folded angles result in varying waveguide cavity sizes, leading to the change of the operating frequency. The electromagnetic (EM) simulations in Ansys HFSS (High Frequency Structure Simulator) were performed for folded angles ranging from 60° to 120° in 10° increments. The simulation results, shown in Fig. 1(b), demonstrate that as the folded angle increases, the cutoff frequency of the fundamental TE₁₀ mode in the waveguide gradually shifts toward lower frequencies.

From a structural perspective, the dimension a (the distance between the side walls) changes with the folded angle and thereby influences the cutoff frequency. Although dimension b changes as well, it has little effect on the TE₁₀ mode according to (2). Further analysis reveals that the effective broad dimension of a is equivalent to the average distance between the inclined sidewalls, denoted here as a _avg, can be defined as (3). In this case, the fold line is located at midpoint of b so that the effective broad dimension is a _avg = a[1-(b/2)cosθ]:

(2)\begin{equation}{f_{c\_{\text{T}}{{\text{E}}_{10}}}} = \frac{c}{{2a}},\end{equation}

(3)\begin{equation} a_{avg} = \left( \frac{1}{n}\right)\sum_{i=1}^n {a_i}. \end{equation}

As shown in Fig. 2, the calculated cutoff frequencies based on a _avg exhibit good agreement with the simulation results. This observation indicates that the cutoff frequency of a vertically folded waveguide can be effectively tuned through structural deformation. Such tunability offers significant advantages for the design of multiband antenna systems.

Figure 2.

Comparison of the simulated and calculated TE₁₀ cutoff frequency.

The fabricated prototype is shown in Fig. 3. Each sidewall was realized by assembling two separate FR4 panels, which were mechanically joined and electrically connected using copper foil tape. This configuration allows the sidewalls to be foldable, thereby enabling the intended structural deformation.

Figure 3.

Fabricated prototype of the vertically folded waveguide.

Simulations and measurements

This vertically folded waveguide was treated as an open-ended waveguide antenna for performance evaluation. Figure 4 shows both simulated and measured return loss (S₁₁) at three folded angles (60°, 90°, and 120°). The simulation shows that the corresponding resonant frequencies are 2.81 GHz (60°), 2.74 GHz (90°), and 2.66 GHz (120°). As the folded angle increases, the effective cavity dimension expands, resulting in a downward shift of approximately 150 MHz in the resonant frequency. The measured results, 2.80 GHz (60°), 2.74 GHz (90°), and 2.66 GHz (120°), exhibit excellent agreement with the simulations.

Figure 4.

Return loss with different folded angles.

Figure 4 Long description

Two line graphs titled Simulation and Measurement. Simulation. Horizontal axis label: Frequency (gigahertz). Range: 2.5 to 3.0. Vertical axis label: S subscript 1 1 (decibel). Range: negative 35 to 0. Angle labels near curves: 60 degree, 90 degree, 120 degree. Line identification by appearance: one curve is drawn with circular markers, one curve is drawn as a solid line without markers and one curve is drawn as a dashed line without markers. Visible minima by curve label: 60 degree curve: minimum near (2.66, negative 15). 90 degree curve: minimum near (2.74, negative 15). 120 degree curve: minimum near (2.81, negative 33). Trend summary within this plot: the 120 degree curve has the deepest dip; the 60 degree and 90 degree curves have shallower dips. Measurement. Horizontal axis label: Frequency (gigahertz). Range: 2.5 to 3.0. Vertical axis label: S subscript 1 1 (decibel). Range: negative 35 to 0. Angle labels near curves: 60 degree, 90 degree, 120 degree. Line identification by appearance: one curve is drawn with circular markers, one curve is drawn as a solid line without markers and one curve is drawn as a dashed line without markers. Visible minima by curve label: 60 degree curve: minimum near (2.66, negative 22). 90 degree curve: minimum near (2.74, negative 33). 120 degree curve: minimum near (2.60, negative 10). Trend summary within this plot: the 90 degree curve has the deepest dip; the 120 degree curve stays closer to negative 10 to negative 15 across much of the range. Purpose stated by the titles and labels: comparison of Simulation and Measurement curves for S subscript 1 1 across the three folded angles.

Figure 5 presents both simulated and measured radiation patterns of this open-ended waveguide antenna at 2.75 GHz. It can be observed that the radiation characteristics in the XZ-plane (H-plane) exhibit obvious variations as the folded angle increases. When the folded angle θ changes from 60° to 120°, the main beam in the H-plane gradually becomes narrower, accompanied by an improvement in directivity. Correspondingly, the antenna gain increases slightly from 6.8 to 7.9 dBi, while the half-power beamwidth is reduced from 66° to 52°. In contrast, the radiation patterns in the YZ-plane (E-plane) remain relatively stable with less significant variation.

Figure 5.

Radiation patterns of the vertically folded waveguide antenna.

Figure 5 Long description

Four polar plots comparing simulated and measured radiation patterns of a vertically folded waveguide antenna. The radial axis is labeled gain in decibel, ranging from 10 to minus 20. The angular axis represents azimuth angle in degrees, ranging from 0 to 360. Three curves are shown for angles 60 degrees, 90 degrees and 120 degrees, representing different folded angles. Simulation, XZ-plane: The main lobe is directed near 0 degrees, with peak gains of approximately 7, 8 and 9 decibel for 60, 90 and 120 degrees respectively. The back lobe near 180 degrees shows gains of minus 10, minus 12 and minus 14 decibel. Simulation, YZ-plane: The main lobe is broad, centered near 0 degrees, with peak gains similar to the XZ-plane. The back lobe is less pronounced, maintaining similar gain values. Measurement, XZ-plane: The main lobe direction remains near 0 degrees, with peak gains slightly lower than simulation. The back lobe is more pronounced, showing gains of minus 20 decibel across all angles. Measurement, YZ-plane: The main lobe is broad with additional side lobes visible, indicating deviations from simulation. Gains are consistent with the XZ-plane, showing more pronounced back and side lobes. Overall, simulation and measurement show similar main-lobe directions, but measurement exhibits stronger back and side lobes, particularly in the YZ-plane.

The measured results reveal a similar trend. Due to the handcrafted nature of the prototype, the measured gain was slightly lower than expected. Specifically, the main-lobe gain in the H-plane was approximately 4.8 dBi at θ = 60°, increasing to about 7 dBi at θ = 120° as the beam gradually narrowed with larger folded angles. This behavior can be attributed to the structural changes induced by sidewall folding. The effective broad dimension a _avg varies significantly, from 75 mm at 60° to 125 mm at 120°, while the narrow dimension b undergoes only a minor change, ranging from 43 mm (60°, 120°) to 50 mm (90°). Consequently, the radiation patterns exhibit more obvious variations in the H-plane compared with the E-plane. Furthermore, the larger effective aperture at a 120° folded angle accounts for the observed enhancement in antenna gain.

In brief summary, the proposed vertically folded waveguide antenna demonstrates that structural folding is an effective approach to control the aperture dimension, enable beam shaping as well as gain adjusting.

Theoretical concept

Unlike the conventional rectangular waveguides have orthogonal boundaries, this proposed design maintains a constant separation between the parallel broad walls, while the waveguide’s lateral sidewalls are tilted with respect to the propagation axis. This geometric manipulation creates an inherently asymmetric cavity, resulting in a parallelogrammatic cross-section. The sidewall’s tilt angle, which is defined as shear angle α, is the angle between the tilted lateral sidewall and the waveguide’s broad plane, as shown in Fig. 6.

Figure 6.

Proposed tilted waveguide.

The tilted structure forces the energy distribution within the cavity to become non-uniform and asymmetrical. When the geometry of the waveguide departs from orthogonal symmetry, as in this case, higher-order modes can interact in non-trivial ways.

Above cutoff, the mode propagates with propagation constant β _mn:

(4)\begin{equation}{\beta _{mn}} = \sqrt {{k_0}^2 - {{\left( {\frac{{m\pi }}{a}} \right)}^2} - {{\left( {\frac{{n\pi }}{b}} \right)}^2}} ,\end{equation}

where k ₀ = 2πf /c is the free-space wavenumber.

When multiple propagating modes coexist, the total field is a superposition, as shown in (5). Since different modes propagate with different β values, the interference of these terms creates a composite field distribution with relative phase differences Δϕz = (β ₁–β ₂)z:

(5)\begin{equation}E\left( {x,y,z} \right) = {A_1}{E_1}\left( {x,y} \right){e^{ - j{\beta _1}z}} + {A_2}{E_2}\left( {x,y} \right){e^{ - j{\beta _2}z}}.\end{equation}

If the aperture is tilted, the boundary conditions project the superimposed fields onto different directions in space, effectively coupling modal interference to radiated beam steering. The radiation beam direction θ can be approximated by a phase-matching condition k ₀sinθ ≈((β ₁–β ₂). Thus, the beam tilt is not a consequence of frequency alone, but of multi-mode interference in a tilted boundary geometry.

Figure 7 simulated electric-field distributions and radiation patterns of the 60°-tilted waveguide antenna at different operating frequencies. At 2.0 GHz, only the fundamental mode propagates in the waveguide, producing a symmetric field and a purely longitudinal propagation constant. Consequently, the radiated beam aligns with the aperture normal without noticeable steering. At 3.3 GHz, the operating frequency exceeds the second-mode cutoff (3 GHz). The combined effect of internal reflections and dual-mode interactions produces an asymmetric aperture field, which in turn leads to the observed beam steering at the waveguide opening. At 3.8 GHz, multiple higher-order modes are excited. The aperture field thus becomes a superposition of three or more modes, resulting in a much more complex distribution. The accumulated phase differences increase rapidly at higher frequencies, producing a multi-peaked phase front at the opening and leading to beam-splitting phenomena.

Figure 7.

E-field distributions (left) and radiation patterns (right) of the tilted waveguide antenna.

Figure 7 Long description

Two groups of simulation outputs for a tilted waveguide antenna are presented. The first group shows false-color electric-field magnitude distributions on the waveguide structure at 2.0 GHz and 3.3 GHz. The second group shows 3D far-field radiation patterns at 2.0 GHz, 3.3 GHz and 3.8 GHz. The waveguide structure is rendered as a 3D rectangular solid with a tilt angle of 60 degrees marked on the first field distribution panel. No axis labels, scale bars, units, or normalization indicators are visible on the field distribution panels. The field component plotted and whether the values are normalized are not labeled in the figure. At 2.0 GHz, the field distribution appears symmetric across the waveguide body. At 3.3 GHz, the field distribution becomes visibly asymmetric. The 3D far-field radiation pattern panels each show a closed surface plotted in 3D space with a reference ring indicating the 0 degree direction. No radial scale, gain, directivity, or power units are labeled on any radiation pattern panel. Whether the patterns represent realized gain, directivity, or normalized power is not stated in the figure. At 2.0 GHz, the radiation pattern shows a single dominant lobe directed along the aperture normal. At 3.3 GHz, the lobe shifts from the aperture normal direction, indicating beam steering. At 3.8 GHz, the pattern shows multiple lobes, indicating beam splitting. No numeric beam angles, peak gain values, sidelobe levels, or null locations are labeled in the figure.

Because different modal components coexist and interfere, causing the field to bounce between the inclined walls, the length of the tilted waveguide directly influences the radiation pattern at the opening since the wave progresses along the structure. Figure 8 shows the simulated electric-field distributions for a 60°-tilted waveguide with different waveguide lengths. It can be observed that the waveguide length determines the direction of the radiated beam at the opening. Figure 8 also compares the far-field radiation patterns for L = 187.5 mm (1.25 λg) and L = 337 mm (2.25 λg), revealing that the beam directions in H-plane are almost reversed between those two cases. These results confirm that, in addition to the shear angle, the waveguide length plays a critical role in determining the beam-steering angle of the tilted waveguide antenna. To align the structure’s inclination with the beam steering direction for more intuitive operation, L = 187.5 mm was selected in this work.

Figure 8.

Electric-field distributions and radiation patterns with different waveguide length.

Figure 8 Long description

The composite figure contains three panels describing electric-field distributions and radiation patterns for different waveguide lengths. The left panel is a 2D contour heatmap showing electric-field distributions inside a waveguide structure. The vertical dimension is labeled L subscript 1 and L subscript 2, representing two distinct waveguide lengths. The contours show field variation along the waveguide interior, with concentrated field regions visible at multiple points along the length. No numeric scale or unit label is directly readable on this panel. The middle panel is a polar radiation pattern plot. The angular axis spans 0 to 360 degrees, with visible degree markers at 0, 90, 180 and 270 degrees. The radial axis shows values at labeled increments of 0, minus 10, minus 20 and minus 30, extending outward from the center. Two curves are plotted. The solid line corresponds to a waveguide length of 187.5 millimeter, equal to 1.25 times lambda subscript g, where lambda subscript g denotes guided wavelength. The dash-dot line corresponds to a waveguide length of 337 millimeter, equal to 2.25 times lambda subscript g. The solid line shows a prominent main lobe directed toward the upper region of the plot, with a smaller secondary lobe visible in the lower region. The dash-dot line shows a main lobe directed toward the lower region of the plot, with a secondary lobe in the upper region, indicating the two beam directions are oriented in opposing directions. The right panel is a schematic of the waveguide structure. It shows a rectangular waveguide section labeled with length L and an inclination angle of 60 degrees.

Figure 9 shows the fabricated prototypes. To enhance the radiation performance, two metallic plates were added to the aperture of the tilted waveguide, forming a flared extension. These tapered plates act as a gradual transition from the waveguide to free space, increasing the effective aperture area. The flared angle and flared length are 30° and 60 mm, respectively.

Figure 9.

Fabricated prototypes of the tilted waveguide without (left) and with (right) flared extension.

Simulations and measurements

Figure 10 presents the simulated and measured return loss (S ₁₁) of the proposed tilted waveguide antenna at 3.3 GHz. In the simulations, all angle pairs, including 60°/120°, 70°/110°, and 80°/100°, exhibit nearly identical S ₁₁ responses. This similarity arises from the intrinsic structural symmetry of the waveguide. For each paired configuration, the internal cavity geometry is effectively mirrored across the central axis, resulting in symmetrical electromagnetic characteristics. The measured results for these symmetric angle pairs show only minor deviations, attributable to fabrication tolerances and experimental inaccuracies. Nevertheless, the overall measurement trends agree well with the simulations.

Figure 10.

Return loss with different shear angles.

Figure 10 Long description

Two sub graphs titled, Simulation and, Measurement. Simulation sub graph. Horizontal axis label: Frequency gigahertz. Range: 2.6 to 3.6. Vertical axis label: S subscript 11 decibel. Range: 0 to minus 40. Four lines are labeled on the plot as, 90 degree, 60 slash 120, 80 slash 100 and 70 slash 110. The 60 slash 120 line has a deep dip near 2.8 gigahertz, reaching close to minus 40 decibel. The 80 slash 100 line has a dip near 3.0 gigahertz, reaching below minus 20 decibel. The 70 slash 110 line has a dip near 3.1 gigahertz, reaching below minus 20 decibel. The 90 degree line stays closer to 0 decibel than the other lines across most of the frequency range, with a shallow dip near about 3.4 to 3.5 gigahertz. Measurement sub graph. Horizontal axis label: Frequency gigahertz. Range: 2.6 to 3.6. Vertical axis label: S subscript 11 decibel. Range: 0 to minus 40. Four lines are labeled on the plot as, 90 degree, 60 slash 120, 80 slash 100 and 70 slash 110. The 60 slash 120 line shows a dip near about 3.5 gigahertz, reaching close to minus 40 decibel. The 80 slash 100 line shows a dip near about 3.4 to 3.5 gigahertz, reaching below minus 20 decibel. The 70 slash 110 line shows a dip near about 3.3 to 3.4 gigahertz, reaching below minus 20 decibel. The 90 degree line stays closer to 0 decibel than the other lines across most of the frequency range, with smaller dips than the other labeled lines.

The simulated and measured XZ-plane (H-plane) radiation patterns are shown in Fig. 11. In simulation, the peak gain remains relatively stable between 10.3 and 11.5 dBi across all shear angles, while the main beam steering gradually shifts from +23° at α = 60° to (−22)° at α = 120°. The measurements show a similar trend, with peak gains ranging from 8.5 to 11.5 dBi and beam steering moving from +28° at α = 60° shear to (−28)° at α = 120°. Because the shear angles were adjusted manually, minor deviations from the intended shear angles may lead to modest differences in the beam-steering direction between simulation and measurement. Nevertheless, the trends of peak gain, beam-steering angle, and the half-power beamwidth (HPBW) show strong agreement between the simulated and measured results, confirming the validity of the proposed design concept. The simulated and measured performance of the proposed tilted waveguide antenna are summarized in Table 1.

Figure 11.

Radiation patterns (XZ-plane) of the tilted waveguide antenna.

Figure 11 Long description

Four polar plots illustrate radiation patterns of a tilted waveguide antenna at various shear angles. The top plots represent simulation results, while the bottom plots show measurement results. Each plot has a radial axis labeled in decibels, representing realized gain, ranging from minus 30 to 15. The angular axis spans 0 to 360 degrees, indicating azimuth angles. The shear angles are represented by different colored lines: 60 degrees in blue, 70 degrees in orange, 80 degrees in red, 90 degrees in black, 100 degrees in purple, 110 degrees in green and 120 degrees in dashed red. The simulation plots show peak gains between 10.3 and 11.5 decibels, with beam steering shifting from plus 23 degrees at 60 degrees shear to minus 22 degrees at 120 degrees shear. Measurement plots display similar trends, with peak gains ranging from 8.5 to 11.5 decibels and beam steering moving from plus 28 degrees at 60 degrees shear to minus 28 degrees at 120 degrees shear. Notable discrepancies include reduced peak levels and increased ripple in measurements compared to simulations. The tilted waveguide antenna configuration varies shear angles to observe changes in main lobe direction and sidelobe behavior.

Table 1.

Performance comparison of the tilted waveguide antenna

Table 1 Long description

Rows list shear angle alpha in degrees, with columns for simulation and measurement: peak gain in dBi, beam steering in degrees, and HPBW in degrees. At 60 degrees, simulation gives 11.5 dBi peak gain, 23 degrees beam steering, and 42.1 degrees HPBW; measurement gives 10.1 dBi, 28 degrees, and 33.4 degrees. At 70 degrees, simulation is 11.2 dBi, 20 degrees, 48.1 degrees; measurement is 9.7 dBi, 26 degrees, 36.7 degrees. At 80 degrees, simulation is 10.3 dBi, 14 degrees, 74.2 degrees; measurement is 9.1 dBi, 23 degrees, 51.6 degrees. At 90 degrees, simulation is 11.6 dBi, 0 degrees, 53.4 degrees; measurement is 8.5 dBi, minus 2 degrees, 65.7 degrees. At 100 degrees, simulation is 10.3 dBi, minus 14 degrees, 74.2 degrees; measurement is 9.1 dBi, minus 22 degrees, 55.2 degrees. At 110 degrees, simulation is 11.2 dBi, minus 20 degrees, 49.9 degrees; measurement is 9.8 dBi, minus 27 degrees, 41.9 degrees. At 120 degrees, simulation is 11.5 dBi, minus 22 degrees, 41.3 degrees; measurement is 11.5 dBi, minus 28 degrees, 36.2 degrees.

Since single-mode operation is usually desired, the primary concern with the proposed tilted waveguide operating in dual modes is modal dispersion. When two or more modes propagate simultaneously, they exhibit different propagation constants and group velocities, inevitably leading to differential modal delay. For narrowband single-tone or low modulation-depth signals, this effect may be negligible. However, for broadband or high data rate signals, it can cause phase/amplitude distortion and thus degrade signal integrity. To examine this practical issue, a communication experiment using a realistically modulated 64-QAM (quadrature amplitude modulation) signal was performed. Furthermore, in the measurement setup, a misalignment was intentionally introduced between the transmitting and receiving antennas, while the tilted waveguide antenna can demonstrate its beam-steering capability to capture signals arriving from off-axis directions. The results indicate that, within the beam-steering range of ±28°, the measured error vector magnitude (EVM) stays below 3.5%, demonstrating that good signal quality is maintained over the tested steering angles despite multi-mode propagation.

Bellowed tilted waveguide antenna

As noted above, the tilted waveguide is sensitive to its length. To increase the degree of tunability, an origami-inspired bellow structure is incorporated into the design to allow fine adjustment of the waveguide length, as shown in Fig. 12(a). However, tuning the bellowed section alone does not provide smooth or continuous beam steering. As seen in Fig. 8, changing only the length causes the main beam to jump between a few extreme angles rather than producing evenly spaced or wide-angle scanning. Thus, length tuning is better suited for discrete beam switching or minor performance adjustments, while the shear angle remains the primary and more reliable parameter for achieving controllable and well-distributed beam steering across the angular range.

Figure 12.

(a) Bellowed tilted waveguide with fold patterns and (b) E-field distributions with different corrugation depths.

Figure 12 Long description

Two main parts labeled as A and B. The image A shows a bellowed tilted waveguide with fold patterns. On the left, a three-dimensional schematic of the waveguide is depicted, featuring a series of parallel lines representing the folds. Above this, a diagram illustrates the fold patterns, with blue lines indicating valley folds and red dashed lines indicating mountain folds. To the right, a sequence of diagrams shows the transformation of the waveguide structure, progressing from a flat sheet to a folded bellowed form. The image B displays field distributions at different corrugation depths. Two separate field distribution maps are shown side by side. The left map is labeled with a depth of 5 millimeters, while the right map is labeled with a depth of 12 millimeters. Both maps use color gradients to represent field intensity, with areas of higher intensity marked by warmer colors. The corrugation patterns are visible along the sides of each map, indicating the structural changes at different depths.

The bellowed waveguide produces non-smooth sidewalls that alter the original boundary conditions. To evaluate its impact, electric-field distributions were simulated for corrugation depths of 5 and 12 mm, as seen in Fig. 12(b). Results show that a smaller corrugation depth yields a smoother structure, while 12-mm corrugation chaotic energy distribution due to highly irregular sidewalls. Taking both waveguide performance and folding difficulty into account, a corrugation depth of 5 mm was selected in this work. Simulation results also show that, for the same length and the same shear angle, the radiation patterns of structures with and without corrugation exhibit very close agreement, as seen in Fig. 13. This further confirms that the corrugation has negligible impact.

Figure 13.

Radiation patterns with and without corrugation.

Figure 13 Long description

The polar plot displays radiation patterns for three waveguide configurations, with angles marked at 0, 30, 60, 90, 120, 150, 180, 210, 240, 270, 300 and 330 degrees. Radial grid rings are labeled from 0 to 30, representing gain or directivity in decibels. Three curves are overlaid, each corresponding to different waveguide configurations, as indicated by the legend with red, blue and black lines. The red curve shows the highest main lobe around 90 degrees, while the blue and black curves have broader beamwidths and lower sidelobes. Key features include peak levels near 30 decibels and nulls at approximately 180 degrees. Two waveguide schematics are shown to the right, illustrating designs with and without corrugation. Each schematic includes a marked point on the top surface and a dimension line labeled 187.5 millimeters. The lower schematic features a grid pattern, indicating corrugation. The plot and schematics together compare the impact of corrugation on radiation patterns, highlighting differences in main lobe strength and beamwidth.

Figure 14 illustrates the implementation of the proposed bellowed tilted waveguide antenna. Because the bellowed structure is more complex, a crease layout with mountain and valley fold lines was first generated using drawing software. This layout was then printed at a 1:1 scale onto transparent adhesive film, placed over the FR4 sheet, and used as a guide to cut the crease lines before folding. The bellowed section uses 0.4-mm-thick FR4 for easier folding, while the remaining parts use 0.8-mm-thick substrate.

Figure 14.

Fabricated prototypes of the bellowed tilted waveguide antenna.

While the material thickness limits the bellowed section’s deformation, the achievable range from 131 to 206 mm is adequate for optimization. By tuning the structure, the length was reduced to L = 169 mm, resulting in improved performance. As illustrated in Fig. 15 and Table 2, the bellowed version, compared with the previous tilted waveguide antenna without the bellowed section, achieves a slightly smaller steering range of ±25° but retains a higher and more stable peak gain (gain variation <1 dB). In addition, the beam directions at different shear angles are more evenly distributed in space, significantly enhancing its practical applicability. Note that the slight differences in performance arise from the small discrepancy in physical length between the two structures. Nevertheless, the overall performance trends and tuning capabilities remain consistent.

Figure 15.

Measured return loss and radiation patterns of the bellowed tilted waveguide antenna.

Figure 15 Long description

A line graph with the y-axis labeled, S one one in decibel and the x-axis labeled, Frequency in gigahertz. The x-axis runs from 2.9 to 3.6 with tick labels at 2.9, 3.0, 3.1, 3.2, 3.3, 3.4, 3.5 and 3.6. The y-axis runs from 0 to minus 50 with tick labels at 0, minus 10, minus 20, minus 30, minus 40 and minus 50. Multiple colored curves are plotted. Several curves show a pronounced minimum between about 3.2 and 3.3 gigahertz. One curve reaches close to minus 40 decibel in that region. Across most curves, values are generally between about minus 10 and minus 30 decibel outside the deepest dip and many curves rise toward about minus 10 to minus 20 decibel by about 3.6 gigahertz. A small inset drawing of a rectangular antenna structure is shown near the lower right of this plot. The image B showing a polar plot with angular tick labels around the circle at 0, 30, 60, 90, 120, 150, 180, 210, 240, 270, 300 and 330. Concentric rings are shown with radial tick labels including minus 30, minus 20, minus 10 and 0. Multiple colored radiation pattern curves are overlaid. A legend lists several angle settings in degrees, each corresponding to one curve. The curves form a main lobe centered near 0 degrees for several traces, with additional smaller lobes visible near 180 degrees and at intermediate angles. A vertical reference line passes through 0 degrees and 180 degrees and a small marker is placed on that line near the outer rings.

Table 2.

Comparison of bellowed and non-bellowed tilted waveguide antennas

Table 2 Long description

Measurements are listed by shear angle for two designs: bellowed tiled waveguide antenna with length 169 mm, and tiled waveguide antenna with length 187.5 mm. For each design, columns report peak gain in dBi, beam steering in degrees, and HPBW in degrees. At 60 degrees shear angle, bellowed values are peak gain 10, beam steering 25 degrees, and HPBW 43.4 degrees; tiled values are peak gain 10.1, beam steering 28 degrees, and HPBW 33.4 degrees. At 70 degrees, bellowed is 10.1, 19 degrees, and 45.2 degrees; tiled is 9.7, 26 degrees, and 36.7 degrees. At 80 degrees, bellowed is 10.1, 7 degrees, and 59.9 degrees; tiled is 9.1, 23 degrees, and 51.6 degrees. At 90 degrees, bellowed is 10.4, minus 2 degrees, and 49.9 degrees; tiled is 8.5, minus 2 degrees, and 65.7 degrees. At 100 degrees, bellowed is 9.9, minus 9 degrees, and 56.9 degrees; tiled is 9.1, minus 22 degrees, and 55.2 degrees. At 110 degrees, bellowed is 10.1, minus 16 degrees, and 55.8 degrees; tiled is 9.8, minus 27 degrees, and 41.9 degrees. At 120 degrees, bellowed is 10.5, minus 25 degrees, and 42.0 degrees; tiled is 11.5, minus 28 degrees, and 36.2 degrees.

Theoretical concept

To implement this concept, an H-plane sectoral horn configuration was adopted in this work. This arrangement expands the waveguide solely in the H-plane while preserving the E-plane dimension, producing a highly directional, fan-shaped beam whose width can be tailored in the H-plane while remaining nearly constant in the E-plane, as shown in Fig. 15. Closed-form expressions for the directivity of an H-plane sectoral horn are given in (6)–(8). In these formulas a and b denote the cross-sectional dimensions of the feeding waveguide, a ₁ is the aperture dimension along the wider side of the horn opening, and ρ₂ denotes the flare length that is the distance from the point where the guide begins to expand to the edge of the aperture in the H-plane. The Fresnel integrals C(u), C(ν), S(u), and S(ν) appear in these expressions to account for the aperture phase distribution:

(6)\begin{equation}{D_H} = \frac{{4\pi b{\rho _2}}}{{\lambda {a_1}}}\left\{ {{{\left[ {C\left( u \right) - C\left( \nu \right)} \right]}^2} + {{\left[ {S\left( u \right) - S\left( \nu \right)} \right]}^2}} \right\},\end{equation}

(7)\begin{equation}u = \frac{1}{{\sqrt 2 }}\left( {\frac{{\sqrt {\lambda {\rho _2}} }}{{{a_1}}} + \frac{{{a_1}}}{{\sqrt {\lambda {\rho _2}} }}} \right),\end{equation}

(8)\begin{equation}\nu = \frac{1}{{\sqrt 2 }}\left( {\frac{{\sqrt {\lambda {\rho _2}} }}{{{a_1}}} - \frac{{{a_1}}}{{\sqrt {\lambda {\rho _2}} }}} \right).\end{equation}

Figure 16 also shows the horn aperture constructed with a fan-fold origami structure together with its detailed dimensions. In this configuration, alternating mountain and valley folds are arranged in an interleaved pattern, allowing the controlled expansion and contraction of the horn walls. The fold geometry provides a compliant mechanism that enables real-time adjustment of the aperture while preserving the intended shape. Each fan blade measures 150 mm in length with 8.6 mm spacing between adjacent blades, and the two opposing sets of blades are positioned 10 cm apart. With the flare length held constant, this folding mechanism enables mechanical adjustment of the flare angle θ to tune the antenna aperture while simultaneously preserving mechanical stability during operation.

Figure 16.

H-plane sectoral horn antenna with origami fan-fold aperture.

Figure 16 Long description

A horn antenna with dimensions labeled, including a flare length of 70 millimeters, a width of 100 millimeters and a height of 50 millimeters. The flare angle theta is indicated at the center. The image B illustrates the origami fan-fold structure of the horn antenna, with alternating valley and mountain folds. Valley folds are marked in blue and mountain folds in red. Each fan blade measures 150 millimeters in length, with 8.6 millimeters spacing between adjacent blades and the two opposing sets of blades are positioned 100 millimeters apart. The flare angle theta is again indicated at the center. The image C presents a three-dimensional view of the horn antenna, showing the origami fan-fold structure from a different perspective, highlighting the interleaved pattern of folds that allows for controlled expansion and contraction of the horn walls.

To further evaluate the influence of the folded geometry on the field distribution, Fig. 17 compares the top- and cross-sectional views of the E-field in a conventional sectoral horn and fan-fold sectoral horn. The results indicate that the introduction of the folding pattern only cause a minimal perturbation of the field distribution. This behavior arises because the E-field of the dominant TE₁₀ mode inherently exhibits a cosine-like variation across the aperture width, with the maximum field intensity at the center gradually tapering toward the edges.

Figure 17.

E-field distribution in conventional and fan-fold sectoral horns.

Figure 17 Long description

Two sections compare the E-field distribution in conventional and fan-fold sectoral horns. On the left, the conventional sectoral horn is depicted with a top view showing a series of concentric circles representing the E-field intensity, with the highest intensity at the center. Below this, a cross-sectional view illustrates the field distribution across the aperture width, tapering towards the edges. On the right, the fan-fold sectoral horn is shown with a similar top view, but the geometry includes folds. The cross-sectional view below indicates minimal perturbation in the field distribution due to the folding pattern, maintaining a similar tapering effect as the conventional horn. Both sections highlight the cosine-like variation of the E-field across the aperture width.

Figure 18 presents the implemented prototype. The feeding waveguide section is fabricated from a 0.8-mm-thick FR4 substrate, whereas the fan-shaped aperture is constructed using a 0.4-mm-thick FR4 panel. However, due to the finite material thickness, the fan blades cannot be fully folded. Even in the most compact configuration, the flare angle still remains at least 10°.

Figure 18.

Fabricated fan-fold horn antenna.

Simulations and measurements

With independent actuation of each side, this structure enables two operating modes. For double-sided flaring, in which both sides of the aperture are folded or unfolded simultaneously while maintaining structural symmetry, the goal is to achieve gain tuning or beam shaping. Measurements were conducted with the flare angle beginning at 10° and increased in 5° increments up to 30°. Figure 19 presents the simulated and measured return loss as a function of flare angle. Both shows that the resonant frequency shifts toward lower values with increasing flare angle (simulation: 2.80 GHz at 10° to 2.74 GHz at 30°; measurement: 2.82 GHz at 10° to 2.73 GHz at 30°). This behavior reflects the fact that enlarging the aperture also increases the electrical volume of the antenna, thereby lowering its natural resonance.

Figure 19.

Return loss from double-sided flaring configuration.

Figure 19 Long description

Two side by side line graphs. Left graph: Title, Simulation. The x-axis label is Frequency in gigahertz, ranging from 2.5 to 3.0. The y-axis label is S subscript 1 1 in decibel, ranging from minus 50 to 0. A legend lists 10 degree, 15 degree, 20 degree, 25 degree and 30 degree. Five curves each show a single dip between about 2.7 and 2.8 gigahertz, then rise toward 0 decibel by about 3.0 gigahertz. The lowest point of each curve is between about minus 20 and minus 30 decibel. Right graph: Title, Measurement. The x-axis label is Frequency in gigahertz, ranging from 2.5 to 3.0. The y-axis label is S subscript 1 1 in decibel, ranging from minus 50 to 0. A legend lists 10 degree, 15 degree, 20 degree, 25 degree and 30 degree. Five curves each show a dip between about 2.7 and 2.8 gigahertz, with the lowest points reaching between about minus 25 and minus 40 decibel, then rising toward 0 decibel by about 3.0 gigahertz.

Figure 20 compares the simulated and measured radiation patterns at 2.7 GHz, showing good agreement between each other. In the YZ-plane (E-plane), varying the flare angle does not alter the radiation pattern. In contrast, in the XZ-plane (H-plane), increasing the flare angle enlarges the aperture dimension of that cross section, leading to a corresponding rise in gain. In the simulations, increasing the flare angle from 10° to 30° produces a 2-dB gain enhancement, reaching a maximum of 10.7 dBi, while the 3-dB beamwidth narrows from 48° to 29.9°. The measured results, although 2–3 dB lower due to fabrication tolerances, exhibit the same trend, with a maximum gain of 8.8 dBi and a 3-dB beamwidth narrowing from 46.2° to 33.0°. These results demonstrate that the proposed fan-fold horn antenna can adjust its aperture flare angle through folding actions, thereby effectively tuning its radiation characteristics. Both simulation and measurement data are summarized in

Figure 20.

Radiation patterns of the double-sided flaring configuration.

Figure 20 Long description

Simulation. XZ-plane. Polar line plot with five overlaid curves and a legend listing 10 (deg), 15 (deg), 20 (deg), 25 (deg) and 30 (deg). Radial axis label, Gain (dBi). Radial tick labels, 15, 10, 5, 0, minus 5, minus 10, minus 15, minus 20. Angular axis shows degree markings with 0 at the top, 90 at the right, 180 at the bottom, 270 at the left, with intermediate degree markings. The curves form a large forward lobe around 0 degrees and multiple smaller lobes around 180 degrees. Simulation. YZ-plane. Polar line plot with the same five legend entries. Radial axis label, Gain (dBi), with radial tick labels, 15, 10, 5, 0, minus 5, minus 10, minus 15, minus 20. Angular axis shows degree markings with 0 at the top, 90 at the right, 180 at the bottom, 270 at the left, with intermediate degree markings. The curves form a rounded forward lobe around 0 degrees and a smaller multi lobed region around 180 degrees. Measurement. XZ-plane. Polar line plot with the same five legend entries. Radial axis label, Gain (dBi), with radial tick labels, 15, 10, 5, 0, minus 5, minus 10, minus 15, minus 20. Angular axis shows degree markings with 0 at the top, 90 at the right, 180 at the bottom, 270 at the left, with intermediate degree markings. The curves form a forward lobe around 0 degrees and multiple smaller lobes around 180 degrees. Measurement. YZ-plane. Polar line plot with the same five legend entries. Radial axis label, Gain (dBi), with radial tick labels, 15, 10, 5, 0, minus 5, minus 10, minus 15, minus 20. Angular axis shows degree markings with 0 at the top, 90 at the right, 180 at the bottom, 270 at the left, with intermediate degree markings. The curves form a rounded forward lobe around 0 degrees and a smaller multi lobed region around 180 degrees.

Table 3.

Table 3.

Performance comparison from double-sided flaring configuration

Table 3 Long description

Columns list flare angle theta, plus simulation and measurement results for max gain in dBi and HPBW in degrees. For 10 degrees, simulation is 8.7 dBi and 48 degrees, and measurement is 6.4 dBi and 46.2 degrees. For 15 degrees, simulation is 9.4 dBi and 43.4 degrees, and measurement is 6.9 dBi and 44.6 degrees. For 20 degrees, simulation is 10.4 dBi and 34.9 degrees, and measurement is 7.4 dBi and 36.4 degrees. For 25 degrees, simulation is 10.7 dBi and 31.6 degrees, and measurement is 8.7 dBi and 32.4 degrees. For 30 degrees, simulation is 9.5 dBi and 29.9 degrees, and measurement is 8.7 dBi and 33.0 degrees.

In the single-sided flaring configuration, one side of the aperture remains fixed while the opposite side is progressively opened, creating an asymmetric aperture whose operation is aimed at achieving beam steering. In this arrangement, the fan blades on one side are kept tightly folded, whereas those on the other side are unfolded with the flare angle varied from 10° to 30° in 5° increments. Figure 21 shows the corresponding E-field distributions. With both sides set to the initial 10° angle, the antenna geometry remains symmetric and the field distribution is nearly uniform, directing most of the power forward. As the flare angle on one side increases, the aperture becomes progressively asymmetric, causing the electric field to tilt toward the side with the larger flare angle and thereby producing a beam-steering effect.

Figure 21.

E-field distribution from single-sided flaring configuration.

Figure 22 compares the simulated and measured radiation patterns in the XZ-plane, again demonstrating good agreement between those data. As the flare angle is varied, the main beam direction is shifting correspondingly. In the simulations, the maximum steering reaches 10°, while the measurements indicate a maximum of approximately 11°. In addition to the steering effect, the gain also increases as the aperture is enlarged. The current structure restricts the flare angle to 30°, thereby limiting the beam-steering capability to approximately ±10°. Increasing the number of fan blades is expected to further expand the achievable steering range. Both simulation and measurement results are summarized in Table 4.

Figure 22.

Radiation patterns from single-sided flaring configuration.

Figure 22 Long description

The polar plots compare beam steering radiation patterns for simulation and measurement. Both plots have angle markings in degrees at 0, 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330. The radial axis is labeled gain in decibel, ranging from -20 to 15. The legend indicates flare angles of 10 degrees, 15 degrees, 20 degrees, 25 degrees and 30 degrees. In the simulation plot, the main lobe is centered near 0 degrees, with a peak gain close to 10 decibel. Smaller lobes appear near 180 degrees, with gain mostly between -5 and 0 decibel. In the measurement plot, the main lobe is similarly centered near 0 degrees but shows broader distribution and slightly higher sidelobe levels. The peak gain is also close to 10 decibel, with more asymmetry in the smaller lobes near 180 degrees. The comparison highlights differences in main lobe width and sidelobe levels between simulation and measurement, indicating validation and discrepancies in beam steering performance.

Table 4.

Performance comparison from single-sided flaring configuration

Table 4 Long description

Measurements are listed by flare angle theta, with simulation and measurement columns for gain in dBi, HPBW, and steering. At 10 degrees, simulation gives gain 8.9 dBi, HPBW 47.3 degrees, steering 0 degrees, while measurement gives gain 6.7 dBi, HPBW 46.3 degrees, steering minus 2 degrees. At 15 degrees, simulation is 9.1 dBi, 44.1 degrees, 2 degrees, and measurement is 6.9 dBi, 45.0 degrees, 1 degree. At 20 degrees, simulation is 10.0 dBi, 38.4 degrees, 3 degrees, and measurement is 7.0 dBi, 44.5 degrees, 2 degrees. At 25 degrees, simulation is 10.5 dBi, 34.5 degrees, 5 degrees, and measurement is 8.1 dBi, 37.3 degrees, 6 degrees. At 30 degrees, simulation is 10.3 dBi, 35.1 degrees, 9 degrees, and measurement is 8.0 dBi, 35.3 degrees, 9 degrees.