Distant small spot presentation in midair haptics using polyhedral reflector Open Access

Airborne ultrasound phased arrays^1–3 are widely utilized in many kinds of research in the realm of midair haptics for remarkable spatiotemporal controllability. For instance, it is feasible to generate complex sound fields^4,5 and provide perceptible sensations to humans with amplitude modulation (AM), lateral modulation (LM),^6–8 and spatiotemporal modulation (STM).^9,10 However, the effectiveness of ultrasonic phased arrays is contingent upon several factors, including the specifications of the sound source, the placement of the sound sources, and the physical shape of the device. A limitation of these displays is the restricted presentation range, with the size of the focal point being influenced by the aperture of the device and its distance from the aperture.^2,11–14 To increase the presentation distance and expand the workspace, the primary strategy is to increase the array size through the implementation of multiple devices^12,13,15 or physically relocating the devices.^16–19 

In our previous works,^14,20 we proposed a method to converge and reflect irradiated sound waves using a curved reflector, allowing for the creation of tactile sensations beyond the presentation range limit of airborne ultrasound phased arrays. This results in forming a small-diameter focal point, even from a distance that exceeds several times the array aperture size.²⁰ The authors also evaluated the workspace for long-distance ultrasound tactile presentation utilizing a curved reflector; however, they assumed only an ideal concave reflector with a certain focal length. Using an ideal concave reflector is a very strong constraint practically, and no consideration has been given to whether a small spot can be formed in a more practical case: for instance, where multiple reflecting surfaces are joined.

In this paper, we investigate the convergence of the focal point by substituting the concave reflector with an approximated surface comprised of multiple planes in long-distance midair haptic presentations. Compared to an ideal concave reflector, a polyhedral reflector is practical and very easy to implement. Our proposed method is expected to be applied to large-scale midair haptic applications, such as those in education,^21,22 museums,^23,24 and potentially vehicle interfaces.²⁵ 

In Sec. II, we examine the sound pressure distribution at the focal plane, considering the imaginary sound sources provided by an approximated reflector. We discuss the potential for interference resulting from the in-phase-driven mirrored sources created by the split surface, and we derive formulas for the feature values. In Sec. III, we conduct numerical simulations and verify whether the feature values observed in the sound pressure distribution are equal to those calculated in Sec. II. In Sec. IV, we conduct an experimental campaign using an approximated concave surface to verify the convergence of the actual reflector.

We consider a reflector approximated by multiple planes so as to equally divide a spherical reflector on a two-dimensional (2D) plane. For the sake of simplicity, let us first consider the case where the reflector is composed of two planes. As shown in Fig. 1, the reflector is symmetrical with respect to the optical axis, the y axis, and the phased array is also symmetrically arranged with respect to the axis, facing the reflector.

By using the method of images, it appears that an imaginary phased array exists at each mirror image position, with the plane reflecting surface serving as the plane of symmetry. We also assume that the focal plane is beyond the Fresnel distance of the imaginary phased array. When focusing on a mirrored phased array, this array forms its Fraunhofer diffraction pattern on the focal plane through phase-shift control like a lens.^11,26 However, when concurrently focusing on multiple imaginary phased arrays, it is considered that interference occurs between the different imaginary sources of the same actual source. Therefore, when the focal point is formed by the reflected waves, the Fraunhofer diffraction pattern is formed by the mirrored phased array on the focal plane, while the effect of interference causes constructive and destructive effects in the distribution.

We consider the interference between adjacent two planar segments of the approximated reflector. We assume that the two planar segments of the reflector, the mirrored phased array, and the focal plane can be regarded as almost parallel. As illustrated in Fig. 1, let L and D denote the distance between the center of the imaginary phased array and the focal plane and the center-to-center distance of the imaginary phased arrays, respectively. Additionally, it is assumed that $L ≫ λ$⁠, where λ is the wavelength.

Each imaginary phased array emits sound waves with an identical phase-shift distribution on the aperture. Therefore, this problem we have dealt with can be understood as multiple-slit interference by regarding a single imaginary phased array as a slit. Since the multiple-slit diffraction pattern on the presentation surface is composed of the interference pattern and a single-slit diffraction envelope,^27,28 the sound pressure distribution is also composed of the interference pattern by the imaginary phased array and its aperture diffraction envelope.

Figure 2 shows an example of the approximated sound pressure distribution on the focal plane expressed by Eq. (12) and the aperture diffraction envelope. As seen in Fig. 2, the sound pressure distribution has some peaks at intervals of d_x caused by constructive interference. In addition, the radius of the mainlobe of the aperture diffraction envelope is w_f, where the approximated sound pressure becomes zero.

In Sec. II, the sound pressure distribution and single-point convergence condition equations are derived by using some assumptions and approximations. In this section, we confirm the validity of the single-point convergence condition. Additionally, we confirm the robustness against noise deformation of the reflector due to shape errors during manufacturing.

Figure 3 shows the simulation condition. The background concave reflector, which was part of a 2D sphere, was symmetrical with respect to the y axis, and its bottom was located at the origin. The radius of the background 2D sphere was 1440 mm, where the focal length was half of the radius, 720 mm. The planar segments of the reflector were the chords obtained by equally dividing the background 2D sphere.

The phased array was comprised of 36 monopole sound sources, which were spaced by a regular interval and whose size was the product of the number of the sources and the interval. The phased array was centered at $( 0 , 2000 mm )$⁠, whose central axis was aligned with the y axis and directed in the negative direction.

A focal point was formed at $( 0 , 720 mm )$⁠, the focus of the background 2D sphere, by synchronizing the reflected waves from the approximated reflector. The velocity of sound was 340 m/s, the frequency of the sound source was 40 kHz, and the wavelength, λ, was 8.5 mm. The size of the phased array, the aperture of the background 2D sphere, and the number of segments of the reflector were varied.

We investigated the validity of the single-point convergence conditions. Table I shows the parametric variations that were performed in the following simulations.

The reflector aperture was 720 mm, and it was separated into 2 and 8 planar segments. The distance between sound sources comprising the phased array was 10 mm, where the array size was 360 mm. The 8-segment reflector satisfied the single-point convergence condition of Eq. (14); the theoretical value of half the width of the mainlobe in the diffraction pattern, w_x, was 64 mm, and the theoretical values of the constructive interference interval derived from Eq. (5), d_x, were 23 and 91 mm for 2 and 8 segments, respectively.

Figure 4 illustrates the normalized sound pressure distribution where the numbers of segments were 2 and 8. The positions of the first secondary maxima were x = 22 and 23 mm for 2 and 8 segments, respectively. The 8-segment reflector formed a single-point focus, where the sidelobes were less than 20% of the mainlobe amplitude. For each focal radius, the values for half the width of the mainlobe were 11 and 16 mm for 2 and 8 segments, respectively. As shown in Fig. 4, the approximated reflector by a sufficient number of planar segments can form a single-point focus if the convergence condition is satisfied.

Figure 5 illustrates the Strehl ratios²⁹ when the segment number of the reflector varied from 2 to 20. Here, we use the “Strehl ratio” as the ratio of the peak acoustic pressure at the focus of a polyhedron reflector to the maximum attainable value for an ideal reflection surface with the same aperture. The reflector where the segment number was more than 6 satisfied the convergence condition of Eq. (14); the constructive interference interval derived from Eq. (5) in the 6-segment reflector was 69 mm. As shown in Fig. 5, as the number of segments increases, the focal sound pressure when using the approximated reflector approaches the focal sound pressure when using the background concave reflector. Moreover, all the reflectors that satisfy the single-point convergence condition have achieved Strehl ratios of 0.8. When the reflectors satisfy the single-point convergence condition, the Strehl ratios are ≥0.8. However, the opposite is not always true. For example, the Strehl ratio for the 4-segment case is more than 0.8 in Fig. 5, while it does not satisfy the single-point convergence condition.

The planar segment size and the center angle of the background 2D sphere were fixed, corresponding to the case of the reflector with a 720 mm aperture and 3 planar segments. The reflector was separated into 2 and 6 planar segments, and the distance between sound sources comprising the phased array was 10 mm. Neither reflector satisfied the single-point convergence condition: the theoretical values of w_x and d_x were 64 and 34 mm, respectively.

Figure 6 illustrates the normalized sound pressure distributions, where the numbers of segments were 2 and 6 and the reflector apertures were 483 and 1394 mm, respectively. The positions of the first secondary maxima were x = 34 mm in both cases, where the first secondary maxima were more than 50% of the mainlobe amplitude. Each focal radius was 19 mm for 2 segments and 10 mm for 6 segments.

As shown in Fig. 6, the mainlobe width decreased when the reflector aperture enlarged. Because of not satisfying the convergence condition, however, neither reflector forms a single-point focus, and the sidelobe amplitudes are not ignorable.

The phased array size was adjusted by modifying the distance between the sources. The distances between sound sources were 5 and 30 mm, yielding array sizes of 180 and 1080 mm, respectively. The reflector aperture was 720 mm, and it was separated into 2 planar segments. In that case of the 1080 mm phased array size, the single-point convergence condition was satisfied: the constructive interference interval, d_x, has been calculated to be x = 23 mm, and the theoretical values of w_x were 128 and 21 mm for array sizes of 180 and 1080 mm, respectively.

Figure 7 illustrates the normalized sound pressure distribution where the phased array sizes were 180 and 1080 mm. The positions of the first secondary maxima were x = 23 and 16 mm, and the focal radii were 12 and 11 mm for the 180 and 1080 mm array sizes, respectively. The 1080 mm array formed a single-point focus, where the first secondary maxima were around 20% of the mainlobe amplitude. As shown in Fig. 7, a sufficient-size phased array can form a single-point focus using a given approximated reflector if the convergence condition is satisfied.

We investigated the robustness against noise deformation caused by shape errors in the planar segments. Figure 8 shows the reflector after deformation. Based on the simulation condition shown in Fig. 3, the reflector was deformed so that the position of each vertex of the reflector was randomly displaced within a colored circle of radius r_e with uniform probability. The reflector aperture, the number of planar segments, and the phased array size before deformation were set to satisfy the single-point convergence condition: the reflector aperture was 720 mm, and it was separated into 8 planar segments, and the distance between sound sources comprising the phased array was 10 mm. The planar segment size was 91 mm. A focal point was formed at $( 0 , 720 mm )$ by synchronizing the reflected waves from the deformed reflector.

Figure 9 illustrates the mean and standard deviations of Strehl ratios in 200 iterations when segment numbers and the deformation noise varied. As in the case of Fig. 4, the reflector where the segment number was more than 6 satisfied the convergence condition. As shown in Fig. 9, the reflectors that satisfy the single-point convergence condition allowed noise deformations of 5% of the segment size and achieved Strehl ratios of about 0.8.

We also measured the sound pressure distribution when ultrasound waves were transmitted to a polyhedral reflector to verify the convergence to a small spot.

Figure 10 illustrates the measurement setup. As shown in the figure, ultrasound waves were emitted from the phased array to the approximated concave reflector, and the reflected sound waves formed a focal point.

The coordinate axes were taken as shown in Fig. 10: the xz-plane was set to coincide with the reflector aperture, and the y axis was set to coincide with both the center axes of the reflector and the sound source array.

A 2 × 2 matrix of airborne ultrasound tactile displays (AUTDs) was utilized, whose aperture was $x × z = 375 × 294$ mm². An AUTD comprises 249 T4010A1 transducers (developed by Nippon Ceramics Co., Ltd., Tottori, Japan) and produces a focused sound pressure of 6.13 kPa when forming a focus at a distance of 180 mm from the aperture.²⁰ The phased array was positioned facing the reflector at a distance of y = 150 cm.

A Paperdome HD150 (developed by Soupstudio Architect, Inc., Nagai, Japan) was utilized for the approximated concave reflector, which was a cardboard 4V geodesic dome with a radius of approximately 75 cm.

The sound pressure was measured by a standard microphone (Brüel & Kjaer 4138-A-015; Brüel & Kjaer, Naerum, Denmark), placed parallel to the y axis at a distance of y = 43 cm and facing in the negative direction of the y axis. The microphone was moved in the xz-direction using a 2-axis stage.

A focus was formed at $( x , y , z ) = ( 0 , 43 , 0 cm )$ through synchronizing phases of the observed data. The driving phases were determined by calculating the phase difference between the transducers where the sound pressure at the target point was highest in all the phase differences.²⁰ The sound pressure was measured within the range of $− 50 ≤ x , z ≤ 50 mm$ at 2 mm intervals.

Figure 11 shows the measured sound pressure distribution. A focal sound pressure was 4.14 kPa at $( x , z ) = ( 0 , 0 mm )$⁠, and the focal radius was 4 mm.

For comparison, Fig. 12 shows the normalized sound pressure distribution on the presentation plane simulated through the method of images with the same device arrangement. The simulation was calculated taking into account the directivity and the sound pressure level of a T4010A1 transducer.⁴ The angle θ between the bottom equilateral triangle of the 4V geodesic dome and the three adjacent triangles was 0.18 radian (rad), and $A / 2 H$ in Eq. (16) was 0.12 rad. As shown in Fig. 12, the simulated sound pressure peaked at $( x , z ) = ( 0 , 0 mm )$⁠. The focal radius was 4 mm, approximately corresponding to the measured focal radius in Fig. 11. The difference in the sidelobes near the focus between Fig. 11 and Fig. 12 is considered due to noises in the observed sound pressure and the method of the images in the simulation.

In the numerical simulations, the validity of the single-point convergence condition has been confirmed. Under the experimental conditions that have not satisfied the convergence condition, a non-negligible peak due to constructive interference has been confirmed at the position derived by Eq. (5). On the other hand, under the experimental conditions such as Fig. 4(b) and Fig. 7(b), which have satisfied the convergence condition, the interference peak that should have been at the position derived by Eq. (5) has been suppressed to the order of the sidelobe amplitudes.

According to the result shown in Fig. 5 in Sec. III B 1, all of the reflectors that satisfy the single-point convergence condition have achieved Strehl ratios of 0.8. Because the focal sound pressure increases as the phase error of the incident sound wave decreases, the high normalized focal sound pressure indicates a high level of convergence. However, some cases are observed where the Strehl ratios of 0.8 are not achieved, even if the convergence condition was satisfied. The Strehl ratio decreases as the wavelength becomes shorter since the wavefront aberrations increase.³⁰ The single-point convergence condition is an equation for the sidelobe amplitudes to be ignorable compared to that of the mainlobe. Though the Strehl ratios are not sufficiently high in all the cases where the conditional expression is satisfied, the approximated reflectors can form a single focal point if the Strehl ratios are sufficiently high, such as exceeding 0.8. As the Strehl ratio increases, the acoustic energy distributed to areas other than the mainlobe decreases, as well as the secondary maxima.

According to the result shown in Fig. 5, the Strehl ratio oscillates for odd versus even segment numbers of the reflector. This is caused by the fact that the shape of the bottom surface of the reflector differs between odd and even segment numbers. If the segment number is even, the vertex of the reflector is on the central axis, creating a V-shaped bottom, as shown in Fig. 1 and Fig. 3. On the other hand, if the segment number is odd, the bottom segment is perpendicular to the axis, creating a U-shaped bottom. Since the distance from the bottom segment of a reflector with a U-shaped bottom to the focal plane is different from the others, the standard deviation of the reflected wave aberration is greater than that of a reflector with a V-shaped bottom, leading to the lower Strehl ratio. The Strehl ratio varies by shifting the focal point apart from the central axis or rotating the reflector.

According to the experiment in Sec. III B 2, the focal diameter has decreased as the reflector aperture has enlarged. However, Eq. (13) for the focal radius does not hold strictly. According to Eq. (13), the focal diameter is inversely proportional to the number of divisions, but the results of the experiment showed that the focal diameter was about half despite the tripled segment number. This is because the contributions to the focus of the outer segments of the reflector decrease as the aperture size increases. In addition, although the focal diameter can be made arbitrarily small in Eq. (13), the focal size is physically limited up to half the wavelength and cannot be smaller.

According to the experiment in Sec. III C, the approximated reflectors have allowed noise deformations of 5% of the segment size. For the reflectors that satisfied the convergence condition, the decrease in the Strehl ratio was within 20% of the original value after the 5% noise deformation. As long as the convergence condition is satisfied and the reflector has enough planar segments, the noise deformation within this percentage is acceptable even if the segment size is changed. At the smaller segment sizes, the smaller noise deformation sizes are allowed, but the fine approximation of the concave reflector results in low sidelobe amplitude. On the other hand, if the number of segments is not sufficient, the sidelobe values may be high due to insufficient cancellation of noise effects.

In the measurement campaign, we verified the convergence of the approximated concave reflector. A spot of size comparable to the wavelength was formed, and the focal sound pressure was sufficiently high to evoke a perceptible tactile sensation. A tactile sensation was felt at the target position, and its size was as small as when forming a focus with direct incident waves, although this was a subjective evaluation. No conspicuous sound pressure peak except for the target position was confirmed. Because the imaginary sound sources have been arranged in three-diensional (3D) space, the sound pressure peaks due to constructive interference have been canceled other than the target position by the imaginary sources that have not been in the identical plane.

In the measurement experiment, we used a ready-made geodesic hemisphere dome made of cardboard because of its availability. In the audible frequency range, the surface of the cardboard is not an ideal reflector. Yet/however at 40 kHz, we assumed it was a perfect reflector, and the experimental results were consistent with this assumption. For example, if the surface is $h =$ 0.3 mm plastic board of $ρ = 1.4 g / cm 3$ with no back support, the estimated acoustic impedance at the surface is $ρ h ω = 1.1 × 10 5 kg / m 2 s$ at 40 kHz, which is 250 times larger than the air acoustic impedance.

The three equivalent Eqs. (14), (15), and (16) of a single-point convergence condition derived in Sec. II, especially Eq. (16), provide a rough guide for the system design of distant presentation in midair haptics with the approximated reflector and the phased array. There are two ways for a single point convergence: one is to increase the segmentation of the reflector into finer segments, and the other is to increase the array size. The approximation accuracy of segmentation is required to satisfy Eq. (16). For example, if the phased array is arranged at the center of the background sphere of the approximated reflector, the segment size needs to be smaller than half the size of the array.

The phased array is also required to be enlarged for a single-point convergence. However, according to the numerical simulations in Sec. III B 3, even with a limited number of sound sources, the approximated reflector can form a sufficiently small single spot by arranging the sound sources apart and enlarging the phased array size. Importantly, this indicates that the reflected sound waves can converge to a single small spot by even using a poorly accurate approximated concave surface as long as the convergence condition is satisfied. This is a remarkable factor and a great advantage for long-distance haptic presentations and applications requiring a large workspace.

In this paper, we confined our discussion to a planar segment approximation of a sphere, although, the analysis can be extended to curved surfaces with more complex shapes. The approximation of the sound pressure distribution on a given observation plane can be obtained by giving a plane approximation surface to a curved surface and then considering the focus formation and interference caused by imaginary sound sources.

This paper verifies the convergence of the focal point when approximating the concave surface onto multiple planes. A mirrored phased array produced by the planar segments forms a focus and concurrently creates interference among imaginary sources. A single-point convergence condition has been derived that has constrained the accuracy of the approximated reflector and the phased array size. As long as it satisfies the convergence condition, the approximated reflector can form a single focal point. Numerical simulations were conducted to confirm the validity of the convergence equation and the 5% tolerance of the segment size for the reflector deformation. We also conducted a measurement campaign and confirmed that the polyhedral reflector was able to form a single small focus by controlling the phase shift of the sound source.

This work was supported in part by JST ACT-X JPMJAX21KJ and CAO-NEDO SIP 23201554-0.

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.