ABH Wedge Radiant Plate Research

• The paper demonstrates that ABH wedge radiant plates convert launch angles into controlled exit positions via carefully designed thickness profiles, validated through theory and experiments.
• ABH wedge radiant plates are tapered structures that manipulate ray kinematics by leveraging total internal reflection in optics and flexural energy concentration in ultrasonics.
• An iterative design process using thin-model scaling, polynomial smoothing, and transfer matrix analysis ensures precise mode conversion and effective wave diffraction control.

In the available arXiv literature, an ABH wedge radiant plate denotes a wedge-profile radiating structure whose thickness variation is exploited to control wave propagation and radiation. In one formulation, it is a tapered dielectric light guide that “folds” the space in front of a projector or camera into a thin optical element by total internal reflection (TIR) and a carefully chosen wedge thickness profile (Travis, 8 Dec 2025). In another, it is the radiating element of an ultrasonic mode-conversion transducer, where an acoustic black hole (ABH) wedge-shaped plate converts longitudinal drive into flexural vibration, concentrates energy toward truncated thin ends, and produces a spatially varying near-field sound-pressure distribution (Wang et al., 21 Jul 2025). Across these uses, the central technical theme is that wedge geometry governs ray or wave kinematics, reflection or mode-conversion history, and the eventual radiation field.

1. Configurations and terminology

The optical source treats the ABH wedge radiant plate as a transparent wedge that works because rays launched into the thick end bounce between the two surfaces by TIR. Since the surfaces are not parallel, each reflection slightly changes the ray’s angle relative to the guide axis. A ray whose angle is near the critical angle will eventually cross that threshold after enough reflections and then escape into air. A ray launched at a different angle will need a different number of reflections before it reaches the escape condition. The wedge therefore converts launch angle into distance traveled before exiting, and converts spatial extent in front of the device into a folded optical path inside a thin guide (Travis, 8 Dec 2025).

The ultrasonic source defines the ABH wedge radiant plate as the radiating element attached to the horn of a Langevin transducer. It is a steel plate with a wedge-like variable thickness on both sides and a uniform-thickness central section. The overall transducer comprises a bolt, back mass, piezoelectric ceramic stack (PZT-4), flange, horn, and ABH wedge radiant plate. The design principle is to match the first-order longitudinal resonance of the Langevin transducer with a symmetrical flexural resonance of the radiant plate (Wang et al., 21 Jul 2025).

Context	Structure	Governing mechanism
Optical folded-space guide	Slab waveguide followed by a tapered dielectric wedge	TIR; angle-to-exit-position mapping
Ultrasonic mode-conversion transducer	Langevin transducer, horn, and ABH wedge radiant plate	Longitudinal-to-flexural conversion; flexural energy concentration

A recurrent clarification is that a wedge alone expands only along the wedge axis. To get a 2D image, the optical configuration is usually preceded by a slab waveguide of constant thickness, which lets rays fan out sideways before the wedge begins. In effect, the slab manages one dimension of spreading, while the wedge manages the other by converting angle into exit position (Travis, 8 Dec 2025).

2. Optical invariant and local design rule

The central design principle for the optical wedge is that, in a smoothly varying guide, the product of guide thickness and the sine of the ray angle is constant:

$t\sin\theta=\text{constant}$

The source presents this as the wedge analogue of an invariant such as the Lagrange invariant or numerical-aperture-type conservation (Travis, 8 Dec 2025).

The derivation is based on a small incremental section of a slowly varying guide. Let the thickness change by $\delta t$, and let the ray make angle $\theta$ to the guide axis. From the geometry of the opposite surface,

$$\frac{\delta\theta}{2}=-\frac{\delta t}{\delta z} \qquad (1)$$

and from ray trigonometry in the guide,

$\delta z=\frac{2t}{\tan\theta} \qquad (2)$

so that

$$\frac{\delta t}{t}=\frac{\delta\theta}{\tan\theta} \qquad (3)$$

Integrating gives

$$-\int \frac{dt}{t}=\int \cot\theta\,d\theta + C \qquad (4)$$

which yields

$\ln t=\ln|\sin\theta|+C \qquad (5)$

and therefore

$t\sin\theta=\text{constant} \qquad (7)$

This rule matters because it lets the thickness profile be designed locally: if the angle a ray should have at one point is known, the required local wedge thickness can be inferred so that the ray behaves consistently. That rule is the basis of the paper’s thin model design procedure (Travis, 8 Dec 2025).

3. Flat-wedge and curved-wedge synthesis

The optical design method recommends first designing the guide as a very thin wedge, and only later scaling it up to a practical thickness. For a thin guide, the local geometry is easier to analyze and the constant-$t\sin\theta$ rule works well. The procedure is iterative: start at the critical-angle ray in the slab; count how many reflections it undergoes before exit; consider rays at slightly smaller launch angles; for each smaller angle, add just enough wedge length so that the ray gets the same total number of reflections as the critical ray; determine the needed local thickness from the invariant; and repeat until the wedge tapers to a tip. The resulting wedge is not a simple straight taper: it becomes slightly bulged relative to a straight line, and the final profile is not a standard geometric curve (Travis, 8 Dec 2025).

For a flat wedge, the algorithm defines a slab of constant thickness $\delta t$0, computes the critical angle

$\delta t$1

computes how many bounces the critical ray makes in the slab, and then extends the guide segment-by-segment so that every ray exits after the same number of total bounces. The thickness at each new point is determined by

$\delta t$2

where $\delta t$3 is the launch angle of the ray at the wedge entry. The shape is therefore built from the ray-angle history rather than from a predefined curve (Travis, 8 Dec 2025).

For a curved wedge, the guide curvature must be included explicitly. The source introduces

$\delta t$4

for one bounce and

$\delta t$5

for $\delta t$6 bounces, together with the right-triangle relation

$\delta t$7

from which the arc-angle per bounce is obtained as

$\delta t$8

and the number of bounces as

$\delta t$9

The design principle $\theta$0 still serves as the small-thickness approximation, but the actual curved design is built with a curvature-aware bounce algorithm. The source explicitly warns that if the application needs a bend or a more compact folded layout, a curved wedge can be used, but simply bending a flat design afterward gives poor results (Travis, 8 Dec 2025).

4. Scaling, smoothing, and practical profile realization

Once the thin optical wedge is designed, the source states that thickness and length can be scaled linearly without changing the fundamental bounce-count property. If the thin design has thicknesses and lengths $\theta$1, then a scaled design uses

$\theta$2

In the worked example, the thin output is scaled by

$$-\int \frac{dt}{t}=\int \cot\theta\,d\theta + C \qquad (4)$$0

which makes the real wedge thicker and shorter in the transverse direction while preserving the intended ray behavior (Travis, 8 Dec 2025).

Scaling, however, introduces a practical difficulty: a kink at the slab/wedge boundary unless the profile is smoothed. The source notes that this kink can produce banding in the projected image. To obtain a manufacturable surface, the calculated thickness profile is often fit with a polynomial, typically 6th order:

$\theta$3

The worked example uses polyfit(z,t,6), and the coefficient list is interpreted so that the highest-order term controls global curvature, the linear term controls slope at the wedge entry, and the constant term controls thickness at the entry. The resulting polynomial is then used as the practical surface profile in ray-tracing software (Travis, 8 Dec 2025).

At finite thickness, the ideal thin-model profile is described as good but not perfect. Finite thickness can cause slight changes in reflection count, banding, imperfect collimation, and entry/exit slope mismatch. Two main remedies are identified: polynomial smoothing of the slab-plus-wedge profile and iterative adjustment using ray tracing and profile updates. A particularly important observation is that a slightly bulged slab can improve continuity of slope, curvature, and even curvature derivative at the transition into the wedge. The paper’s “best collimation” algorithm constructs a slab profile as a polynomial so that at the slab exit / wedge entry the following are continuous: thickness, slope, curvature, and rate of change of curvature. This improves optical smoothness and helps the wedge act more like a lens (Travis, 8 Dec 2025).

5. Ultrasonic ABH wedge radiant plate as a mode-conversion transducer element

In the ultrasonic implementation, the ABH wedge radiant plate is designed to convert the high-amplitude longitudinal vibration of a Langevin transducer into a flexural vibration of a wedge-shaped plate whose thickness varies according to an ABH profile. Because an ABH structure slows flexural waves as the thickness tapers down, energy accumulates toward the thin end, producing strong vibration concentration and a nonuniform radiated sound field. The source states that the ABH structure is used not for damping, but for enhancing useful radiation; it amplifies vibration toward the plate ends, converts longitudinal motion into flexural motion efficiently, creates a spatially varying near-field sound pressure, and enables gradient acoustic fields useful for levitation and particle manipulation (Wang et al., 21 Jul 2025).

The reported design parameters for the radiant plate are specific:

Parameter	Value	Description
$\theta$4	$\theta$5	ABH section length on each side
$\theta$6	$\theta$7	Uniform-thickness section length
$\theta$8	$\theta$9	Total length
$$\frac{\delta\theta}{2}=-\frac{\delta t}{\delta z} \qquad (1)$$0	$$\frac{\delta\theta}{2}=-\frac{\delta t}{\delta z} \qquad (1)$$1	Width
$$\frac{\delta\theta}{2}=-\frac{\delta t}{\delta z} \qquad (1)$$2	$$\frac{\delta\theta}{2}=-\frac{\delta t}{\delta z} \qquad (1)$$3	Uniform thickness
$$\frac{\delta\theta}{2}=-\frac{\delta t}{\delta z} \qquad (1)$$4	$$\frac{\delta\theta}{2}=-\frac{\delta t}{\delta z} \qquad (1)$$5	Truncation thickness

The steel plate material properties are reported as density $$\frac{\delta\theta}{2}=-\frac{\delta t}{\delta z} \qquad (1)$$6, Young’s modulus $$\frac{\delta\theta}{2}=-\frac{\delta t}{\delta z} \qquad (1)$$7, and Poisson’s ratio $$\frac{\delta\theta}{2}=-\frac{\delta t}{\delta z} \qquad (1)$$8. The horn is aluminum, the gray parts of the overall transducer are steel, and the piezoelectric stack is PZT-4 (Wang et al., 21 Jul 2025).

The flexural vibration model is established using Timoshenko beam theory and the transfer matrix method. The ABH section is discretized into $$\frac{\delta\theta}{2}=-\frac{\delta t}{\delta z} \qquad (1)$$9 small segments on each side, and the uniform-thickness center is treated as one element. For the $\delta z=\frac{2t}{\tan\theta} \qquad (2)$0-th element, the left and right end quantities are related by a $\delta z=\frac{2t}{\tan\theta} \qquad (2)$1 transfer matrix,

$\delta z=\frac{2t}{\tan\theta} \qquad (2)$2

and the overall matrix is

$\delta z=\frac{2t}{\tan\theta} \qquad (2)$3

Because both ends are free, the boundary conditions are

$\delta z=\frac{2t}{\tan\theta} \qquad (2)$4

which leads to the frequency equation

$\delta z=\frac{2t}{\tan\theta} \qquad (2)$5

The source reports that the calculated flexural frequencies from the transfer-matrix model and finite element simulation (FES) agree within less than 4% relative error, and that the third-order symmetrical flexural mode of the radiant plate is the one used for matching to the transducer’s first longitudinal resonance near 26.055 kHz (Wang et al., 21 Jul 2025).

6. Radiated-field behavior, experimental validation, and wedge-diffraction context

For the assembled ultrasonic transducer, the resonance alignment is reported with unusual precision. The FES resonance frequency is $\delta z=\frac{2t}{\tan\theta} \qquad (2)$6, the Langevin first-order longitudinal frequency is $\delta z=\frac{2t}{\tan\theta} \qquad (2)$7, and the relative error is 0.30%. Experimentally, the impedance-measured resonance frequency is $\delta z=\frac{2t}{\tan\theta} \qquad (2)$8, with a relative error versus FES of 0.23%. The source attributes the residual discrepancy to real material parameter deviations, unmodeled mechanical or dielectric losses and prestress, and machining and assembly errors. It also reports that the normalized transverse displacement is smallest near the center, increases toward both ends, and reaches its maximum at the ends; the abstract summarizes this as a stepwise amplitude increase (Wang et al., 21 Jul 2025).

The near acoustic field was simulated in air using the Pressure Acoustics module with a hemispherical air domain radius of 110 mm, 15 mm PML thickness, 1 V excitation voltage, and 26132 Hz excitation frequency. The resulting field has a gradient sound-pressure distribution in front of the plate: pressure is highest near the plate ends, the field contains multiple energy wells, the distribution is nonuniform and asymmetric along the length direction, and the width direction shows higher pressure in the middle and lower pressure on the sides. Experimental sound-field measurements were taken on a plane 10 mm away from the plate, with 14 points sampled along one side of the length direction, 5 points along one side of the width direction, and 22 points along the axial direction. In the levitation experiment, the transducer faced downward, an aluminum reflective block formed a standing-wave cavity, excitation was 26161 Hz, 500 mVpp, with 4 W effective output, and 4 mm EPS spheres were levitated. When the gap between the radiator and reflector was about 20.3 mm, the spheres levitated in a sequence of $\delta z=\frac{2t}{\tan\theta} \qquad (2)$9 from left to right, indicating a gradient acoustic field that facilitates precise particle sorting (Wang et al., 21 Jul 2025).

For wedge-based radiating or scattering structures, a distinct but relevant analytical framework is provided by the rigorous justification of Sobolev’s classical diffraction formula for a two-dimensional wedge in the time domain (Komech et al., 2014). In that treatment, the wedge is the angular domain

$$\frac{\delta t}{t}=\frac{\delta\theta}{\tan\theta} \qquad (3)$$0

with opening angle $$\frac{\delta t}{t}=\frac{\delta\theta}{\tan\theta} \qquad (3)$$1, and the physical domain is the exterior angular region $$\frac{\delta t}{t}=\frac{\delta\theta}{\tan\theta} \qquad (3)$$2. For causal incident waves of the form

$$\frac{\delta t}{t}=\frac{\delta\theta}{\tan\theta} \qquad (3)$$3

with $$\frac{\delta t}{t}=\frac{\delta\theta}{\tan\theta} \qquad (3)$$4 and $$\frac{\delta t}{t}=\frac{\delta\theta}{\tan\theta} \qquad (3)$$5, the scattered field is uniquely represented by the convolution formula

$$\frac{\delta t}{t}=\frac{\delta\theta}{\tan\theta} \qquad (3)$$6

and for the Heaviside pulse $$\frac{\delta t}{t}=\frac{\delta\theta}{\tan\theta} \qquad (3)$$7, this reproduces Sobolev’s wedge diffraction formula exactly. The paper also gives the Sommerfeld–Maluzhinetz integral representation of the diffracted wave,

$$\frac{\delta t}{t}=\frac{\delta\theta}{\tan\theta} \qquad (3)$$8

treats DD, NN, and DN boundary conditions, and establishes long-time stabilization and limiting-amplitude results. For a Heaviside incident wave, the field is fully explicit; the source identifies this as a benchmark for validating simulations of wedge plates, ABH tips, and corner diffraction (Komech et al., 2014).

Taken together, these sources describe the ABH wedge radiant plate as a geometry-driven radiating structure in which the thickness profile controls wave slowing, reflection count, exit condition, mode conversion, or diffraction signature. In the optical setting, the decisive rule is the invariant $$\frac{\delta t}{t}=\frac{\delta\theta}{\tan\theta} \qquad (3)$$9, used to construct a thin wedge model that is later scaled and smoothed. In the ultrasonic setting, the decisive mechanism is ABH-enabled flexural energy concentration in a wedge-shaped plate, coupled to a Langevin transducer and validated by agreement among transfer-matrix calculation, finite element simulation, impedance measurement, laser vibrometry, sound-field measurement, and levitation experiments (Travis, 8 Dec 2025, Wang et al., 21 Jul 2025).