Tactile device

Abstract

The tactile device creates a mechanical energy field that can be felt but not heard. Since the tactile device includes an annular array transducer that is substantially transparent to visible light, when combined with a wavefront display, a projected visual object can be felt as a hologram. By controlling the transparent array of the annular array transducer and using the principles of interference and superposition, a steerable beam of the mechanical energy field can be provided at a point in space.

Description

Technical Field

[0001] The present disclosure relates to a tactile device, and more specifically, to a tactile device for outputting mechanical energy capable of inducing a tactile sensation into a three-dimensional space. Background Next-generation optics and acoustics typically require more attention than received tactile technology. This is largely due to the complexity of human senses. The somatosensory system includes various information such as proprioception, kinesthetic information, temperature, pain, texture, etc. Some can be stimulated by concentrated mechanical energy such as ultrasonic waves. Summary A tactile device creates sounds that can be felt rather than heard. Since the tactile device includes a transducer that is substantially transparent to visible light, it can be combined with a visual display so that a human can feel a projected visual object. In one embodiment, the tactile device is provided by an annular transducer array. The annular transducer array has a plurality of concentric transducer rings disposed on a surface, and these rings have an outermost transducer ring and an innermost transducer ring and are centered on a point. The outermost transducer ring and the innermost transducer ring are each driven by a circuit operable to cooperatively provide a mechanical energy field toward a volume in space. In one embodiment, a plurality of additional concentric transducer rings are disposed between the outermost transducer ring and the innermost transducer ring. In another embodiment, the tactile device is provided by an array of annular arrays, whereby mechanical energy fields from different transducer rings within different annular arrays are selectively controlled to provide a mechanical energy field concentrated and / or directed toward a point in space. In another embodiment, the haptic device includes a substrate, sidewalls extending from the substrate that define a cavity, a membrane coupled to the sidewalls and disposed over the cavity, and an electrode operable to apply a voltage to the membrane. The membrane is operable to vibrate in response to a drive signal and generate a mechanical energy field toward a focus in space. In an embodiment, the electrode forms an annular array pattern on the membrane. In an embodiment, the haptic device is substantially transparent to transmitted light. Brief Description of the Drawings FIG. 1 is a schematic diagram showing a cross-sectional view of the structure of a capacitive micromachine ultrasonic transducer. , is a schematic diagram showing a cross-sectional view of the structure of a piezoelectric micromachine ultrasonic transducer. FIG. 3 is a photograph showing an embodiment of an array of concentric ring-shaped transducers. FIG. 4 shows a computational model of a 20-ring annular array. FIG. 5 is an exploded perspective schematic view of an exemplary PMUT annular array including a plurality of annular transducer elements arranged concentrically. FIG. 6 is a schematic diagram showing a top view of an exemplary phased array showing a 1x6 array of an annular array. FIG. 7 is a schematic diagram showing an example of a 3x3 grid array of annular transducers in which each transducer ring is individually connected to a controller and controlled by the controller. FIG. 8 shows a perspective cross-sectional view of the sidewall of an exemplary annular array. FIG. 9 is a graph 900 of a computer-modeled total mechanical energy field versus distance from a transducer annular array when operating outside and inside an annular transducer ring of the annular transducer array design shown in FIG. 8. FIG. 10 is a graph 1000 of an exemplary model design plotting the total mechanical energy field in decibels on the Y-axis and the drive frequency in kilohertz on the X-axis. FIGS. 11A and 11B are perspective schematic views of a 3×3 array of an annular array 1100 showing three different exemplary foci in the space where haptic feedback is transmitted. Figure 12 shows a perspective schematic view of a 3x3 annular array, and three other exemplary foci within a space that can transmit haptic feedback in spatial and temporal correspondence with a holographic image superimposed thereon. Detailed Description The haptic device can be designed based on the basic principles of constructive and destructive interference of waves. Multiple waves in phase interfere constructively and are added linearly. Waves in opposite phase interfere negatively and cancel each other out. By controlling the phase of a series of waves, it is possible to make them interfere only in a specific direction. This is called beam steering (or beamforming). Closely related to this is the creation of a focus by guiding the waves to a single point. Therefore, the purpose of creating a visible focus is to control the phases of a large number of sound waves. This can be achieved by an ultrasonic phased array, that is, a series of acoustic transducers whose phases can be controlled individually. This has some similarities with the technology underlying holograms. For example, a hologram functions by transmitting or reflecting light through a grooved surface, and the phase of the electromagnetic wave is retarded by the difference in the surface. The resulting light waves interfere to reconstruct a 3D wavefront. The elements of the ultrasonic phased array can be configured to have the necessary phase delays to focus on a single point. The time difference between the elements is expressed as follows. t j = (d! - d") / c Here, dj is the distance between the j-th transducer and the focus, do is the distance between the center and the focus, and c is the speed of sound. Although it is a simple calculation to convert these time differences into phase differences, in the example of digital implementation, the phase delays are input as simple time values as described later, so it remains as above. The tactile device is configured according to the principles of this specification and can stimulate the texture sensation of the skin, the so-called touch or vibration touch, through focused ultrasound. The focused ultrasound solution enables non-contact interaction, no moving parts, and high resolution compared to other methods using air pumps or laser heating of the skin. Furthermore, the ultrasonic phased array can also integrate visual information and auditory information. The earliest research in this field was conducted in the 1970s using focused ultrasound to include the sensitivity thresholds of various sensations of the hand. This research has been further refined, and it has been determined that touch is the effect of the acoustic radiation force acting on the mechanical receptors of the skin. What is important is that these mechanical receptors are sensitive to changes. Therefore, signals modulated by simple amplitude modulation (100 - 200 Hz) of the focus or movement of the focus (referred to as spatio-temporal modulation) may lower the detection threshold. The spatial modulation method can generate sensations based on the user's position relative to the standing wave, thus eliminating the noise associated with amplitude modulation and spatio-temporal modulation techniques. Other modulation methods may also improve the intensity of vibration touch stimulation. In some embodiments, it can be concluded that the minimum detection threshold of vibration touch stimulation using simple amplitude modulation is 0.3 - 0.4 mN. Ultrasonic phased array haptics has been commercialized by companies such as Ultrahaptics (now Ultraleap), but the known solutions have limitations in that they are opaque transducers (i.e., they do not allow light to pass through), and also, due to the single-channel transducer architecture, they are composed of an array of single-ring or single-drum transducers that do not provide constructive interference from a single transducer unit. Ultrasonic annular array. In one embodiment, the ultrasonic array can be configured as an annular array. In one embodiment, the annular array includes a single array of nested concentric ring-shaped transducers. The annular array can generate a single tactile focus because it has a higher fill factor and focusing ability compared to other array designs such as a grid packed in a hexagonal shape. Parameters of the ultrasonic annular array. It is necessary to understand that there are a huge number of parameters regarding the manufacture of the ultrasonic annular array according to the present disclosure. These include the thickness of the film, the size of the drum, the depth of the drum, the shape of the electrodes, the shape of the array, and material science considerations. These will be described throughout this specification. A MATLAB script can be created to calculate the fill factor and phase delay of the annular array. Using this, the shape of the annular array can be determined and the fill factor can be optimized. In an exemplary embodiment, for a 7-ring annular array design with individually phase-controllable rings, the width of each ring can be selected to be 6 mm and the width between the rings can be 1.1 mm. The electrode coverage of the annular array transducer may be smaller than the transducer drum area. The exact size is usually determined by finite element modeling. In an embodiment, the electrodes can be configured with 67% coverage. Annular array transducer manufacturing technology. To create the annular array of the present disclosure, several transducer manufacturing technologies can be used. In one embodiment, a capacitive transducer, so-called capacitive micromachine ultrasonic transducer ("CMUT") can be used. CMUT performs a function somewhat similar to that of a parallel plate capacitor. FIG. 1 is a schematic diagram showing a cross-sectional view of the structure of a capacitive micromachine ultrasonic transducer (CMUT) 100. The CMUT 100 includes an upper electrode 110 attached to a membrane 108 and a lower electrode 104 attached to a substrate 102, and is arranged as shown in the figure. When a voltage is applied between the upper electrode 110 and the lower electrode 104 from a modulation voltage source 112 (for example, a frequency generator), the membrane 108 is deflected by the generated electrostatic force, and ultrasonic waves are generated. Manufacture of CMUT. The technology for manufacturing CMUT can be based on sidewalls defined by lithography made of SU-8 (a transparent epoxy-based negative photoresist) on a glass or PET substrate coated with indium tin oxide. A thin sheet of SU-8 can be laminated onto the sidewalls using a heated electron laminator. In certain embodiments, an opaque electrode can be deposited onto the membrane using sputtering. The formula for the electrostatic force of a parallel plate capacitor or CMUT is as follows. F_elec = (∈_o AV^2) / d^2 Here, ∈ " is the dielectric constant in the cavity, A is the transducer area, V is the applied voltage, and d is the distance between the electrodes. Since d is inversely proportional to the electrostatic force, an increase in d has a dramatic effect on the generated electrostatic force. To generate a significant pressure output, a very high voltage of several hundred or several thousand volts is required. In one embodiment, after successfully manufacturing the sidewalls on the wafer, the sidewalls are fabricated on indium tin oxide - polyethylene terephthalate (ITO - PET). The ITO - PET sheet is taped to the wafer using Kapton tape and the same process as before is executed. Initially, SU - 8 did not adhere to the ITO - PET and peeled off the film. When the ITO - PET is plasma etched for 2 minutes at maximum power, the surface roughness of the ITO - PET increases and adhesion between the SU - 8 sidewalls and the film becomes possible. Once the sidewalls are fabricated, an upper film can be created to encapsulate the sidewalls. Roll lamination is a technique that uses pressure and heat to bond a sheet of film to another sheet. For proper lamination, a pressure of 0.35 MPa or 50 psi can be used. An electric laminator (e.g., SKY - 335R6) can be used to appropriately control the heat and speed. Roll lamination can cover the cavity without filling it with re - flowed SU - 8. Also, this allows the film to be laminated to the sidewalls. In many lamination experiments, poor adhesion, non - uniform coating, compression of the sidewalls, and reverse lamination were confirmed. Reverse lamination is when the sidewalls are removed from the PET base and adhere to the film. That is, SU - 8 adheres preferentially to the film rather than the PET. Note that the lamination problems can be solved by changing the bake time, the order of the process steps, the lamination speed, and the lamination temperature. Operation of the CMUT. During operation, in one embodiment, the CMUT is driven with a large DC bias of 200 V to 2000 V and an oscillating voltage of 100 V to 300 V. Using a bias tee, an operational amplifier, and various circuits, an existing power supply can be modified to create a large AC power supply to which a DC bias can also be applied. In one embodiment, a low - power power supply using a 40 V DC bias with an AC voltage of + - 10 V can be used. To determine the capacitance of a CMUT with a diameter of 100 μm, the capacitance equation can be used. C = (?_0 k A) / d Here, the separation distance d is 2 μm, and the relative permittivity k of air is 1. In one embodiment, the capacitance is determined to be 0.034 pF. A Polytec PSV-400 laser Doppler vibrometer (LDV) can be used to measure the vertical displacement. Using such a device, the maximum displacement was about 1 nm. The breakdown voltage of the measured sample was about 80 V, and the resonance frequency was higher than the frequency that the LDV could measure. Note that driving the sample at the resonance frequency near the breakdown voltage may achieve a larger displacement. Design of the PMUT. In one embodiment, the transducer of the present disclosure can be manufactured using a piezoelectric transducer design or a piezoelectric micromachine ultrasonic transducer ( "PMUT"). FIG. 2 is a schematic diagram showing a cross-sectional view of the structure of a piezoelectric micromachine ultrasonic transducer (PMUT) 200. The PMUT 200 is particularly different from the CMUT 100 in that the CMUT membrane is replaced by a piezoelectric element 208 or the piezoelectric element is coupled to the CMUT membrane. In this exemplary embodiment, the PMUT 200 can include a piezoelectric element 208 between an upper electrode 210 and a lower electrode 204, as shown in the figure. When voltages are applied between the upper electrode 210 and the lower electrode 204 from a modulation voltage source 112 (e.g., a frequency generator), respectively, the piezoelectric element 208 deflects by an electrostatic force that generates ultrasonic waves. In a PMUT, since voltages are applied to the electrodes 204 and 210 directly attached to the piezoelectric element 208, the cavity depth is not as much of a problem (compared to a CMUT). This design results in lower drive voltage requirements and eliminates the need for a DC bias. The resonance frequency of the disk is as follows. f=(α√(D_E / ρh)) / (2πr^2 ) Here, \(D_E=\frac{Eh^3}{1 - \nu^2}\), where \(D_E\) is the bending stiffness, \(\alpha\) is the resonance mode constant, \(r\) is the radius of the diaphragm, \(E\) is the effective Young's modulus, \(\rho\) is the effective density of the PMUT diaphragm, \(\nu\) is the Poisson's ratio, and \(h\) is the thickness of the diaphragm. Another way to increase the pressure output of the PMUT is to reduce the thickness of the diaphragm. As the diaphragm becomes thinner, the displacement of the PMUT increases. Reducing the thickness also increases the bandwidth according to the following equation. \(BW = \frac{Z_{air}}{2\pi\rho_mt}\) Here, \(Z_{air}\) is the impedance of air and \(t\) is the thickness. A major advantage of micromachine transducers is that they can be monolithically integrated into the electronic manufacturing process. Furthermore, the size achievable with cleanroom technology enables high-density, high-frequency arrays, which can generate finer foci and create higher-resolution tactile holograms. There are many piezoelectric elements. Commercially available transducers typically use opaque materials such as lead zirconate titanate (PZT). However, transparent piezoelectric materials are not generally known. In one embodiment, it has been found that polyvinylidene fluoride (PVDF) can be used as a piezoelectric element. PVDF is a piezoelectric polymer that can be used for this purpose because it can be formed into a thin transparent sheet that can be used as the piezoelectric film element of a PMUT. In one embodiment, the thin film of PVDF can be manufactured by spin coating. In this process, the powdered PVDF resin is dissolved in N,N-dimethylformamide (DMF) and spin-coated at various speeds. PVDF exhibits polymorphism, and the β-phase provides the best piezoelectric properties due to its relatively high d33 constant that determines the piezoelectric response in the plane of interest. The spin speed can be controlled so that this phase is generated. Temperature and humidity are environmental factors that can be controlled to achieve the desired phase. When PVDF is produced, a poling process of several kilovolts is performed to align and polarize the dipoles within the material. Therefore, this aspect can be considered in the manufacture of PVDF films. Furthermore, since PVDF has a high dielectric constant, an upper limit of the driving voltage is given. In one embodiment, the breakdown electric field of PVDF is 100 V / μm, and when this voltage is exceeded, PVDF will re-polarize. For a 50-μm-thick PVDF, excluding the consideration of the electrodes, the driving upper limit is 5000 V. Creation of the sidewalls of the transducer. As shown in FIGS. 1 and 2, PMUTs and CMUTs can be suspended over cavities that define the drums of the transducers. The walls that define the depth of these drums are called sidewalls, and various transparent sidewall manufacturing methods can be used, such as ultraviolet photolithography, 3D printing, and laser cutting. As described above, SU-8 2000 photoresist can be used to create the sidewalls of CMUTs. A grid array packed in a hexagonal shape with a size ranging from several hundred microns to 1 mm in diameter was manufactured. Alternatively, a slightly more viscous SU-8 2008 photoresist can be used to create higher sidewalls. Thin transducers with good transparency can also be created using photolithography. Moreover, since it can be integrated with other cleanroom manufacturing processes, it is suitable for the integration of commercial electronic devices. In one embodiment, it is suitable for drum sizes of 1 mm or less. The 3D printed sidewalls can achieve a size much less than 1 mm and good transparency according to the thickness of the substrate. Also, the manufacturing turnaround time is shorter compared to lithography. In one embodiment, an SLA resin printer has the drawback that a flat substrate is prone to warping. Further, the added support struts may leave pockmarks and there is a possibility of reduced transparency. Transparency can be improved by spraying or dipping the final print with a transparent resin. 3D printing has an excellent balance of manufacturing time, transparency, and resolution. In the case of large transducers where resolution is not so important, the laser cutting method may be used. One way to manufacture the sidewalls relatively quickly is by laser cutting of acrylic. The resolution is limited to 1 mm or less, but it is effective when the size of the transducer drum is large. However, laser cut acrylic may have reduced transparency depending on the relative thickness. Electrical connection to the transducer. Two conductive electrodes are used to apply a voltage to each transducer element. In one embodiment, a common ground electrode is used and the respective electrodes of each annular transducer are patterned accordingly (for example, as described in FIG. 3). As an example for manufacturing the transducers of the present disclosure, methods for depositing and patterning transparent electrodes of three materials, silver nanowires (AgNW), poly(3,4-ethylenedioxythiophene) polystyrene sulfonate (PEDOT:PSS), and indium tin oxide (ITO), are disclosed. Silver nanowires (AgNWs) are aqueous solutions of silver rods that are only a few nanometers in length. A collective mesh of small wire rods forms a conductive network. Thin electrode films with excellent resistance can be deposited by spray coating or dip coating. To increase conductivity, the density of silver nanowires was increased, but as a result, transparency decreased. To increase transparency, the density was decreased, but as a result, sheet resistance increased. Silver nanowires have been shown to create electrodes with an average transmittance of 85%, and the sheet resistance can be lowered to 20 Ω / sq. In one embodiment, the optimal dimensions of the silver nanowires were diameter × length 30 nm (±5 nm) × 20 μm (±3 μm). The optimal concentration was 0.3468 mg / ml of silver nanowires dispersed in isopropyl alcohol (IPA), and the most effective method for obtaining a uniform crosslinked coating was ultimately spray coating. The coated sample had one layer of SU-8 pre-rolled laminated on the sidewalls. In one embodiment, production was carried out by dip coating polyethylene terephthalate (PET) and glass pieces in an AgNW solution. Dip coating functions by immersing the substrate in the solution and slowly pulling it out. When pulled out, the solution dries with a liquid-air barrier, leaving a thin conductive film. In one embodiment, dip coating of silver nanowires results in electrodes with low sheet resistance, but usually only in local areas. The sheet resistance was infinite in some areas and non-uniform in most other areas. In one embodiment, a spray coating machine can be used to uniformly disperse silver nanowires on the substrate, prevent short circuits between the upper and lower electrodes, prevent the fluid from filling the cavity, reduce exposure to IPA, and increase transparency and uniformity. In one embodiment, the sample can be crosslinked to create a flexible and conductive uniform electrode. Crosslinking means that the silver nanowires do not align perfectly, overlap and stack on top of each other, creating many electrical paths and good electrical conductivity. This is similar to the microgame of a pickup stick. The additional variables of the spray coater were the number of raster lines, injection speed, scan speed, and scan size. The larger the number of raster lines, the more the silver nanowires overlap, and the lower the sheet resistance. However, this sacrificed transparency. The smaller the number of raster lines, the higher the sheet resistance and the lower the transparency. The injection speed used was 5 mL / hour. Usually, AgNW films are annealed to fuse the wire mesh and reduce the sheet resistance. This may not be suitable for temperature-sensitive PVDF films. Furthermore, it is relatively costly and the adhesion to a vibrating substrate may not be reliable. PEDOT-PSS is an attractive conductive polymer that is relatively inexpensive, highly transparent, and can be gelled. It can also be used to manufacture thin conductive films. There are various deposition methods such as spray coating and inkjet printing. There is also a lithography method, but usually a high-temperature annealing process is required, which is not suitable for PVDF films. PEDOT:PSS is usually available as a colloidal solution of fine PEDOT:PSS particles suspended in water. To deposit this, a spray coating device can be used. In one embodiment, the spray coating device extrudes the liquid through an ultrasonic nozzle that evenly disperses the solution. The PEDOT:PSS solution is provided in water, but a diluent such as IPA can be added to lower the viscosity and enable free spraying. Depending on the situation, this process can become unstable and the spray coating device may become clogged due to a state called aggregation. If the liquid ratio of the solution is not appropriate, an unstable colloid is generated and the PEDOT:PSS particles aggregate and fall out of the solution. The stability of the colloidal solution is determined by the zeta potential, which represents the potential difference between the dispersion and the particles in it. ζ =η / ? μ Here, ζ is the zeta potential, η is the viscosity, ε is the dielectric constant of the solution, and μ is the electrophoretic mobility of the particles. It has been found that in order to maintain a high zeta potential and reduce aggregation, it is necessary to make the viscosity relatively high and the dielectric constant relatively low. However, if the viscosity is too low, the solution will not be sprayed. In one embodiment, a solution of 1-1.3 wt% PEDOT:PSS and 10 wt% IPA in water can effectively spray a high-quality film and a practical transducer array using PVDF. PEDOT:PSS is inherently bluish, but by using various dispersions and surfactants, both transparency and sheet resistance can be improved. Due to its flexibility, it is also optimal as the electrode of the diaphragm. A shadow mask was used to pattern the electrode by spray coating. Although the manufacturing is simple, it is difficult to manufacture small feature sizes with this technology. When spin coating PEDOT:PSS on a PVDF film, it should be recognized that several manufacturing techniques can be applied to maximize transparency and minimize sheet resistance. Such manufacturing techniques include 1) layer change, 2) plasma etching, 3) post-treatment immersion in an ethylene glycol (EG) bath, and 4) addition of EG and surfactant to the PEDOT:PSS solution. In one embodiment, the sheet resistance of PEDOT:PSS decreases from above 3 M.Ohms / sq to 600 - 150 ohms / sq depending on the number of layers. In one embodiment, the sheet resistance of a 1-layer sample is 600 ohms / sq and the optical transparency is about 75%, and the sheet resistance of a 5-layer sample is 150 ohms / sq and the optical transparency is about 50%. Indium tin oxide is a transparent metallic alloy commonly used in LCD displays because it is easy to deposit a transparent thin conductive film. It can be deposited by magnetron sputtering. PVDF film can also be used even when an ITO film has been previously deposited. The patterning of the electrodes is a subtractive process. In one embodiment, lithography and hydrochloric acid (HCL) can be used to selectively etch the ITO film. In one embodiment, laser cutting and tape transfer technology can be used to apply a mask using a polyimide film to the ITO-PVDF film. The structure is taped to a silicon wafer and immersed in an HCL bath to remove the film. This process results in a manufacturing turnaround time of only a few hours, with good transparency and sheet resistance. The main drawback of ITO electrodes is their brittleness. After fabricating the electrodes patterned on the piezoelectric film, various adhesion techniques suitable for enabling the vibration of the film can be used to adhere to the sidewalls. Polydimethylsiloxane (PDMS) spin coating and laser cut double-sided tape are two examples of this. Spin-coated (PDMS) can be used for small transducers less than 1 millimeter. First, mix the base and curing agent in a ratio of 10:1. Next, spin coat the PDMS onto the PVDF piece taped to the silicon wafer at 5000 rpm for 60 seconds. Next, place the sidewalls on top of the PVDF and the thin PDMS film and cure at 80 °C for 2 hours. PDMS functions not only as an adhesive layer but also as a film layer bonded to the PVDF on the cavity. The main drawbacks of PMDS are that it is not suitable for transducers larger than the size of the silicon wafer and has relatively weak adhesion. For larger transducers, laser-cut double-sided tape can be used. Such tape has strong adhesion, but the air bubbles trapped beneath it may have an adverse effect on transparency. The tape needs to be cut so that it does not adhere to the PVDF on the cavity. Otherwise, the vibration may be reduced. Due to the limitations of laser cutting, it may not be suitable for transducers less than about 1 mm. In an embodiment, this tape treatment can be used with laser-cut acrylic sidewalls. A digital solution can be used to drive a piezoelectric transducer array, and the phase delay can be implemented in software. Thus, the interconnection is based on metal-oxide-semiconductor field-effect transistors (MOSFETs) and gate drivers. For example, in one embodiment, low-voltage MOSFETs can be used to implement a 12-element linear array. The same design can be scaled up to a larger array that runs on the digital output of an FPGA. FIG. 3 is a photograph showing an embodiment of an array of concentric ring transducers. FIG. 3 shows a top view of a 1×2 array embodiment of an annular transducer array 300 according to the present disclosure. This 1×2 array 300 has two functionally similar annular transducer arrays 350 and 360. For the sake of brevity, the first transducer array 350 will be described, and the second transducer array 360 should be considered functionally similar to the array 350. As shown here, the arrays 350 and 360 are processed on a single substrate, and the electrical connections at the bonding pads 352a?352n and 362a?362n can be arranged to conveniently and efficiently bond the electrical connections to a controller (not shown). The annular transducer 350 includes a plurality of concentric transducer rings disposed on the surface, with the center being a point, and there are the outermost transducer ring 302n and the innermost transducer ring 302a (n is an integer). As can be seen here, the outermost transducer ring and the innermost transducer ring may each be driven by a circuit operable to cooperatively provide the mechanical energy field sound pressure level towards the volume in space. As shown here, there may be additional transducer rings between the innermost transducer ring 302a and the outermost transducer ring 302n. In one embodiment, the array of annular transducers can be implemented as a zone plate that generates a focus of mechanical energy in three-dimensional space. In one embodiment, implementing it as a 40 kHz zone plate with a 150 Vpp 40 kHz signal with 200 Hz amplitude modulation allows for a high mechanical energy field level and, in some cases, can provide a tactile sensation to some individuals. Figure 4 shows the calculation model of a 20-ring annular array. The calculation model was created in MATLAB. To simulate the annular array, concentric circles of point sources were used to simulate each annular part. Each circle contains 100 point sources. Other variables are the number of rings, the distance between rings, the frequency, and the focal height. The time delay was calculated using the following formula. t_d=(d_fp-d_ref) / c The distance from the ring to the focus is d_ref, and the distance from the center ring d_fp to the focus is c. The speed of sound is 343 m / s. The pressure at any coordinate (r, θ) is calculated using the following formula. p(r, θ, t)=e^(i(ω(t+it)-kd)) / d Here, ω is the wave speed, t is the time delay in seconds of the point source, k is the wave number, and d is the distance from the center ring to the coordinate (r, θ). In the modeling, the pressure was calculated in the xy plane of the selected focus. Each time a ring is added, a narrower focus with a higher mechanical energy field is created on the annular array. Driving of the annular array. In one embodiment, to connect and drive the annular array, an 8-channel STMicroelectronics driver board can be used to drive the annular array. The phase signal is sent to each annular part of the array. This can be programmed using the STM's Ultrasonic Pulser Waveform User Interface (UPWUI). The phase delay is calculated in MATLAB and manually programmed into the software. Each annular part can be connected from the driver board to an impedance matching network. Transparency test. The measurement of transparency can be performed using a JDS Uniphase 633 nm laser. Such a laser can be directed through a Thor Labs IS236A integrating sphere. The average output can be recorded with a Rigol oscilloscope. The sample can be placed on a Thor Labs ELL20 60 mm linear stage placed in front of the laser. When performing the test, it is desirable to turn off the light to create complete darkness. Using such exemplary test equipment and techniques, the transparency of an exemplary ITO 36 - element array was measured to be 91.6%. To collect maximum mechanical energy field data, the transducer is preferably driven at its resonant frequency. To find the resonant frequency, a frequency sweep can be performed for each sample from 100 Hz to 55 kHz. In one embodiment, a PMUT sample including a 20 mm - diameter aluminum electrode single - electrode with 12 μm PVDF was measured to have a resonant frequency of 27 kHz and a maximum mechanical energy field of 92.8 dB. In one embodiment, a PMUT sample including a 15 mm - diameter aluminum electrode single - electrode with 12 μm PVDF was measured to have a resonant frequency of 33 kHz and a maximum mechanical energy field of 89 dB. In one embodiment, an aluminum 36 - element array including individual 15 mm - diameter elements was measured to have a resonant frequency of 33 kHz and a maximum mechanical energy field of 88.2 dB. When the pressures from two sound sources combine at the focus, the pressure from the two sound sources doubles, and when two more sound sources, four sound sources, eight sound sources are added, the pressure doubles again, and this repeats. For 32 elements, the pressure doubles 5 times. Each time the pressure doubles, it increases by +6 dB. For an array of 36 elements, it is considered to reach about 120 dB at the focus. In one embodiment, a boundary experiment was conducted to determine what percentage of the opening produced the best transducer. The sweep of the perforated sample was performed for 10 seconds on a 15 mm diameter sample, and the resonance frequency was found to be 49.1 kHz. Longer measurements were taken, and the maximum mechanical energy field recorded was found to be 86.1 dB at 49 kHz. The 15 mm sample that was 100% closed showed the best performance with a mechanical energy field of 89 dB, and the 75% closed sample, i.e., the perforated sample, showed the second best performance with a mechanical energy field of 86.1 dB. The 25% closed sample showed the worst performance with a maximum mechanical energy field of 77.7 dB. According to this trend line, the closer the membrane of the sample is closed at the boundary of the drum, the higher the maximum mechanical energy field. In one embodiment, an impedance matching experiment was conducted to determine its effect on the mechanical energy field. According to the output data, when there was no impedance matching network, the maximum mechanical energy field was 49.2 dB, whereas when there was an impedance matching network, the maximum mechanical energy field was shown to be 66.9 dB. In one embodiment, acoustic measurements were performed using an array of 48 microphones to show factors affecting the width of the focus of the acoustic array. The diameter of the transducer greatly affects the resolution of the array. The smaller the diameter, the higher the resonance frequency and the smaller the focus. Other factors include the gain pattern of the individual transducers, the overall dimensions of the array, and the distance from the array to the focus. The gain pattern depends on the frequency and the mechanical characteristics of the transducer itself. If there are no factors affecting the intensity of the focus other than whether the array is being driven at the resonance frequency, the same level of tactile feedback should be felt at all frequencies other than the resonance frequency, but the gain pattern of the transducer changes with frequency. The higher the frequency, the higher the directivity, so most medical acoustic arrays are typically at ultrasonic frequencies in the MHz range. As the frequency decreases, it becomes more difficult to feel the tactile feedback according to the power distribution of the gain pattern. It should be understood that there are both minimum and maximum intensities of the mechanical energy field suitable for generating touch. Sufficient force is required to generate touch, but not so much force that temperature-based sensations begin to be felt. Furthermore, this sensitivity threshold depends on frequency. In fact, different frequencies correspond to different "types" of touch. In one embodiment, for 40 kHz, the possible level of safe ultrasonic exposure to the skin is 100 W / cm2 or a total force of 428 mN. In one explanation, touch is caused by the non-linear effect of focused ultrasound called acoustic radiation force. The radiation force induces shear waves in the skin tissue, creates displacement, and stimulates the mechanical receptors in the skin. The maximum displacement u'() of the medium induced by the radiation force from the pulse of focused ultrasound can be defined as follows. Here, a is the radius of the focus region, t 0 is the pulse duration, c t is the propagation speed of the shear wave, c l is the speed of sound, μ is the shear modulus, α is the absorption coefficient, I and W are the intensity and acoustic power (both averaged over the pulse duration), and k is a composite constant. Based on the above, the larger the focal region and the longer the pulse, the deeper the skin displacement. Acoustic power (W) has been shown to be an important factor affecting skin displacement and thus tactile sensation. Since acoustic power is directly related to the mechanical energy field, this becomes one of the indicators for optimization. This displacement equation also introduces a basic trade-off between resolution and the mechanical energy field. The larger the focal region (a), the larger the displacement on the skin. However, the larger the focal point, the worse the resolution of the generated tactile image. Note, however, that since mechanoreceptors are sensitive to changes, modulation methods that change this radiation force over time are also relevant. To optimize the mechanical energy field for tactile sensation, it may be necessary to consider several design parameters. To derive the pressure generated by a vibrating membrane at a specific point, something called the Rayleigh integral can be used. Based on this, the higher the frequency and the larger the displacement, the faster the velocity of the membrane. The larger this velocity, the higher the pressure. Therefore, the larger the size of the PMUT, the larger the displacement and the higher the pressure. However, beyond a certain size, the PMUT begins to bend. This bending prevents the expected large displacement. Therefore, there is an upper limit to the diameter. In one embodiment where 500 nm PVDF-TrFE is used, the ideal size was found to be 600 um. Although there are differences such that the actual results are lower, each element of the annular array contributes almost linearly to the final pressure output. Therefore, the more elements phase - adjusted to the same spot, the higher the mechanical energy field. However, to reduce the formation of side lobes, the pitch between adjacent elements needs to be made smaller than half the wavelength of the resonant frequency. Generally, as the density increases, the diameter of each individual element becomes smaller. This is a trade - off that can be considered. The thickness of the piezoelectric film of the annular array is a property that can be optimized. This is one of the factors that determine the resonance frequency. Generally, the thinner the film, the greater the electric field across the entire film and the greater the deflection, resulting in a higher output pressure. Also, the larger the piezoelectric coefficient (d33), the greater the deflection and output pressure. The breakdown voltage is a limiting factor for the supply voltage. This is also limited by the thickness of the film. In other words, the thinner the film, the lower the voltage that can be applied before reaching the breakdown voltage. However, since the breakdown voltage of PVDF is very high, on the order of thousands of volts, a high drive voltage may be an option. In one embodiment, the radius of the upper electrode may be smaller than the radius of the film. The annular array has a high filling rate and is focused, resulting in a high mechanical energy field, but other shapes can also be implemented for various optimizations. For example, in one embodiment, the pressure is increased by a bimorph design. The bimorph design includes two piezoelectric elements with four differentially driven electrodes. In one embodiment, a PMUT with air holes, slits, and rings can also be implemented. FIG. 5 is an exploded perspective view of an exemplary PMUT annular array 500 that provides a plurality of annular transducer elements arranged in concentric circles in accordance with the present disclosure. In this example, the PMUT annular array 500 has an innermost sidewall 502a that extends to the outermost sidewall 502n. In this example, the innermost sidewall 502a and the sidewall 502b define the innermost cavity 504a. The outermost sidewall 502n and the sidewall 502c define the outermost cavity 504n. The sidewalls 502b and 502c define the intermediate cavity 504b. As will be appreciated and as described elsewhere in this specification, in other embodiments, there may be additional sidewalls that define more than three cavities as described herein. The bottom electrode layer 506 may be provided with a radius that is approximately equal to the radius of the outermost sidewall 502n. The lower electrode layer 506 is conductive and can provide a common electrical ground connection. The patterned electrodes 508a, 508b, 508n are shaped to provide concentric electrodes that align with the innermost cavity 504a, the intermediate cavity 504b, and the outermost cavity 504n, respectively. A piezoelectric film 510 is sandwiched between the patterned electrodes 508a, 508b, 508n. The radius of the piezoelectric film 510 is approximately the same as the radius of the lower electrode layer 506. Since the patterned electrodes 508a, 508b, 508n are electrically separated from each other, different potential voltages can be applied to each of them to independently operate each piezoelectric annular transducer element within the annular array 500. Bonding pads (not shown here) are patterned and electrically connected to electrical traces 512a, 512b, 512n that electrically connect the patterned electrodes 508a, 508b, 508n, respectively. As shown in this exemplary embodiment, the concentric ring array can have the same thickness for each ring. In other embodiments, the thickness of each ring may be different for the purpose of designing optimal parameters for tactile performance. Such physical design differences can be calculated using mathematical modeling such as MATLAB. Directivity. In some embodiments, the focus can be optimized based on the dimensions, thickness, depth, etc. of the concentric sidewall rings, so that larger rings can be used for closer points, smaller rings can be used for farther points, or a combination thereof can be used, improving the accuracy and adjustment ability of the array. FIG. 6 is a top view of an exemplary phased array showing a 1x6 array of the annular array 600. Similar to the embodiment shown in FIG. 5, each transducer ring is independently controlled and electrically connected to the controller 650 via the data bus 630. The electrical traces extending outward from each annular array 620a, 620b, 620c, 620d, 620e, 620f show the individual connections of each annular transducer. Similar to the above description of the exemplary transducer in FIG. 5, it is advantageous for each transducer array to be substantially transparent to the visible light passing through. FIG. 7 shows an example of a 3x3 grid array of the annular transducer 700, where each transducer ring is individually connected to and controlled by the controller. Although a 3x3 array of annular transducer arrays is shown as an example, it should be understood that any configuration of an mxn grid array can be used. Here, m and n are integers greater than or equal to 1, and mxn is greater than or equal to 2. However, m may or may not be equal to n. Filling factor. The annular array has the advantage that it can output a higher mechanical energy field for the occupied space footprint compared to known single-drum transducers because of its high filling factor, i.e., the transducer number density per unit area. Furthermore, acoustic modes available at high output levels (e.g., mode 1 of the drum head) can be used. Since more energy may be lost in the system in higher modes, it is desirable for the transducer to use a simple first mode. Also, since each ring can be clamped at the ends, there is an advantage that higher output is possible in the first mode. Figure 8 shows a perspective cross-sectional view of the sidewall of an exemplary annular array 800 according to the present disclosure. This is a simple annular transducer array with the innermost transducer 802a and the outermost transducer 802b. As can be seen from the figure, it is desirable for the sidewall to be perpendicular (vertical) to the ground plane of the lower electrode. However, during the process of creating the sidewall, the sidewall may be slightly inclined by an etching or 3D printing process. Such processing inclination characteristics are known in the art. The physical parameters of the sidewall can be modeled together with the modulation technique to generate a plot of the mechanical energy field and the distance from the transducer. Figure 9 is a graph 900 of the total mechanical energy field versus the distance from the transducer annular array when operating with the outer and inner annular transducer rings 802b, 802a of the design shown in Figure 8. As expected, it can be seen that the inner and outer transducer rings 802a, 802b interfere with each other and the mechanical energy fields overlap at points in space. Figure 10 is a plot 1000 of the total mechanical energy field represented in decibels on the y-axis and the drive frequency represented in kilohertz on the x-axis. As can be seen from plot 1000, in this modeled design, the maximum mechanical energy field is obtained at a resonance frequency of about 36 KHz. The transducer can operate at any frequency from 1 Hz to 100 MHz or higher, but when the mechanical energy field is less than 20 kHz, humans can hear it, so it is desirable to keep it above 24 kHz and less than 100 kHz. The human finger cannot feel the selected resonance frequency of 36 KHz, but can feel 200 Hz. Therefore, the resonance frequency of 36 KHz driving the annular transducer element can be modulated at 200 Hz so that the tactile sensation is recognized by the human tactile receptor. Figure 11A shows a perspective schematic view of a 3x3 annular array 1100 that depicts three exemplary foci within the space in which tactile feedback is transmitted. Here, for example, the innermost annular transducer ring within each annular array can be controlled to provide a tactile focus in volume space f1 (1102). The volume spaces f1, f2, f3 can be 1 - 5 mm wide in the space at a distance z and (x,y) position from the origin in the plane of the array of annular arrays. The central annular transducer ring within each annular array can be controlled to provide a second tactile focus at a second (x,y,z) position from the origin. This is shown as the tactile focus in volume space f2 (1104). Also, the outermost annular transducer ring within each annular array may be controlled to provide a third tactile focus at a third (x,y,z) position from the origin in volume space f3 (1106). Of course, in other embodiments, rings of different diameters can interact with rings of other diameters to provide different tactile foci and mechanical energy fields of different sizes. Figure 11B shows a perspective schematic view of a 3x3 array of an annular array 1100 similar to that of Figure 11A, with three other exemplary foci, f4 (1112), f5 (1114), f6 (1116), within the space in which tactile feedback is transmitted, illustrated. As will be understood by those skilled in the art, different combinations of inner, outer, and intermediate rings can be activated at different frequencies at any given time to cooperatively provide the desired constructive interference that results in a desired mechanical energy field at the focus position in (x,y,z) space. Here, this is the case, resulting in three different focus positions f4 (1112), f5 (1114), f6 (1116) in (x,y,z) space. FIG. 12 shows a perspective schematic view of a 3×3 array of an annular array, shows illustrations of three other exemplary foci in space, and a holographic image of the chameleon 1250 superimposed thereon with tactile feedback transmitted spatially and temporally coincident therewith. Foci f7(1202), f8(1204), f9(1206) are here in coincidence with the exemplary wavefront image shown as the chameleon 1250. Thus, when a person reaches out a hand as if to touch the chameleon in space, the person can observe tactile feedback that coincides with the point in space where the lizard is visually observed. Thus, vision and kinesthesia are stimulated and a person can perceive the holographic object in space. Some exemplary systems for providing such visual wavefront images are described in U.S. Patent Applications 17 / 724,815, 17 / 776,130, 17 / 990,258, which are hereby incorporated by reference herein. Further, the directional audio output (within the human audible range) can also be directed to a specific point in space that coincides with the image and sensory output using known techniques for providing directional audio. Such a presentation system can provide sensory stimuli of vision, voice, and kinesthesia simultaneously. The term "light wavefront" and related terms used in this book are explained as follows. All optical images are made of light, which is a form of electromagnetic radiation. Light waves emitted from a point source spread out concentrically and propagate as a vibrating energy field. It is convenient to represent one cycle of the wave's vibration, and one cycle is 360 degrees, or 2π radians. The phase of the wave's vibration, in the case of a harmonic sine wave, is defined as o = Asin(2πx / λ). Here, A is the amplitude of the wave, defined as the maximum value of the wave's vibration, x is the length of the wave's path from the origin, and λ is the wavelength of light. A two-dimensional EM wave is usually represented as a continuous series of vibrations of the electric and magnetic fields, with each vibration vibrating perpendicular to each other in a plane (left figure). If the plane of vibration does not change, the light is linearly polarized (in the case of circular polarization, the plane of the electric field (the plane of the magnetic field is perpendicular to the plane of the electric field) rotates around its direction axis with time). Actual waves are generally unpolarized. That is, because the positions and directions of the emitting atoms are constantly changing, the electric field randomly changes direction in space. Although there is no actual circular motion, as already mentioned, it is convenient to represent the vibration in that way. In this case, the complete phase of the wave (corresponding to the spatial period between the two closest points in the same phase) is equal to 2π radians. The virtual surface connecting the points of the same vibration motion, that is, the wave of the same phase, is called the phase surface. The geometric approximation of the phase surface based on the same optical path length (OPL) of the light rays from the light source is called the light wavefront, or simply the wavefront. On the other hand, a light ray is a straight line with the point source as the origin and perpendicular to the wavefront. Light rays are convenient for representing the geometric aspects of optical phenomena, but they represent only a small part of the total energy that propagates the energy field. The wavefront, although it is itself a geometric category, is more directly related to the underlying physics. The wavefront identifies the positions of the in-phase wave sources and forms the basis for calculations that determine the characteristics of the interaction of the waves at the focus and around it. Therefore, the importance of the wavefront lies in the fact that its shape directly determines the quality of the optical image of the telescope. Obviously, the shape of the wavefront and the geometric characteristics of the light rays are directly related to each other, but the shape of the light rays is only loosely related to the interactions that occur within the energy field. The wave equation defines the analytical framework for the propagation of electromagnetic waves. The analytical solution of the wave equation describes the wave field of the wavefront propagating at any location in space. However, it is complex and almost impossible to obtain the analytical solution for any object with arbitrary shape and size. Instead, the numerical solution of the wave equation can be determined by calculation, but even the determination by calculation takes time and is unrealistic. Therefore, various approximate models can be derived and used to simplify the determination by calculating the numerical solution of the wave equation. In one approach, the solution of the vector wave equation is approximated by making a specific assumption of reducing the wave equation to scalar components. Since these integral equations are essentially scalar, they are also called scalar diffraction formulas. These approximate solutions based on scalar diffraction are called "scalar theory" and can simplify the determination by calculating holograms with sufficient resolution and accuracy in holographic display applications. The starting point of scalar diffraction theory is the Maxwell equations that describe electromagnetic energy waves. Assuming that the electromagnetic energy wave is propagating in a linear, homogeneous, isotropic, homogeneous, and non-dispersive material, the following vector wave equation 1.1 can be derived from the Maxwell equations to describe the electric field component. ∇ 2 E =?μ_0??_0?_r d^2 / ?dt?^2 (1.1 ) Here,?_0 is the permittivity of vacuum,?_r is the relative permittivity, and μ_0 is the permeability of vacuum. Since light behaves like a wave, the vector wave equation 1.1 can be expressed in the form of the scalar wave equation 1.2. ∇ 2 u ( p, t ) -1 / c^2 (d^2 u(p, t)) / (dt^2 )= 0 (1.2) Where c is the speed of the wave in the dielectric medium. Comparing the equations 1.1 and 1.2 of the vector wave and scalar wave, the speed of light can be expressed as follows. c = 1 / √(?μ_0??_0?_r )=c_0 / n (1.3) Here, \(c_0\) is the speed of light in vacuum, equal to \(1 / \sqrt{\mu_0\epsilon_0}\), and \(n\) is the refractive index, equal to \(\sqrt{\epsilon_r}\). The component \(u(p, t)\) can be regarded as a wave function that defines the scalar field component at a specific position \(p\) and time \(t\) within a material with refractive index \(n\). In the case of a monochromatic wave, the scalar field function can be expressed as a complex amplitude according to Equation 1.4. \(u(p, t)=\text{Re}\{U(p)e^{−i\omega t}\}\) (1.4) Equal to \(\nu\). The function \(U(p)\) is called the complex amplitude given by Equation 1.5. \(U(p) = A(p)e^{i\varphi(p)}\) (1.5) Here, \(A(p)\) is a real-valued quantity that can be understood as the amplitude, and \(\varphi(p)\) can be understood as the phase of the complex amplitude. The complex amplitude \(U(p)\) is a three-dimensional function, and its two-dimensional distribution in a given plane is called the wave field. Substituting Equations 1.4 and 1.5 into the wave equation of Equation 1.2, the wave equation can be rewritten in the same form as the Helmholtz equation. \(\nabla\) 2 \(U(p)+k\) 2 の \(U(p) = 0\) (1.6). Here, \(\lambda\) is the wavelength of light, \(k\) is equal to \(2\pi / \lambda\), and is called the wave number. In the context of rendering wavefronts, the object wave, reference wave, and signal wave can be modeled as plane waves or spherical waves based on the above wave equation. A plane wave has a plane wavefront, has a constant frequency and amplitude, and extends infinitely. However, in a specific space, a complex wavefront can be modeled as one or more local plane waves. The solution of the plane wave of the above wave equation is as follows. \(u(p, t)=A\cos(k\cdot p-\omega t+\varphi\) 0 ) (1.7) Here, \(\varphi\) 0is a constant that defines the phase of the cosine function at time t = 0 and r = 0, p is the position vector, and the length k is the wavenumber vector pointing in the propagation direction of the light in vacuum. (|k|) = 2πn / λ0 0 defines the wavenumber vector pointing in the propagation direction, where λ is the wavelength of light in vacuum. (|k|) = 2πn / λ0 Based on Equations 1.4 and 1.5, the complex amplitude of the plane wave represented by Equation 1.7 can be expressed as follows. U(p) = A e^(i(k·p + φ 0) ) (1.8). 0 The wave field of the plane wave at is as follows. U(x, y; z 0 ) = A e^(i(k_x x + k_y y + k_z z_0 + φ 0) ) (1.9) Spherical waves are radiated from a single point source and have spherical wavefronts. The solution of the above wave equation for plane waves is as follows. u(p, t) = A / r cos(kr ± ωt + φ 0 ) (1.10) Here, r is the radial distance in the spherical coordinate system and can be determined based on the Cartesian coordinates x, y, z as shown in Equation 1.11. r = |p| = √(x^2 + y^2 + z^2) (1.11) Based on Equations 1.4 and 1.5, the complex amplitude of the spherical wave represented by Equation 1.10 can be expressed as follows. U(p) = A / r e^(±i(kr + φ 0) ) (1.12). 0 The wave field of the spherical wave at is as follows. U(x, y; z 0 ) = A / r e^(±i(kr + φ 0) ) (1.13) where r = √(x^2 + y^2 + z_0^2) (1.14) For each of the plane wave or spherical wave, z = z 0平面The wave field can be sampled at various (x, y) positions above. The energy waves in free space under the assumptions of scalar theory are described by wave equations 1.2 and 1.6, and the plane waves and spherical waves described above are particular solutions of wave equations 1.2 and 1.6. The boundary conditions help solve wave equations 1.2 and 1.6 to provide more general solutions for waves propagating in free space. When light is diffracted by an aperture, the boundary conditions can be understood as a binary function where the value of the function is 1 inside the aperture and 0 outside the aperture. In one embodiment, by using the wave field defined by the first complex amplitude of the first plane as the boundary condition and solving wave equations 1.2 and 1.6, a diffracted wave field defined by the second complex amplitude of the second plane at any location in 3D space can be obtained. In one embodiment, the computational implementation of this process is called numerical field propagation, which includes sampling the source wave field (i.e., the first amplitude) and using it as the boundary condition to numerically solve the wave equation and simulate the destination / diffracted field (i.e., the second complex amplitude). The propagation of the wave field can be modeled according to several diffraction formulations known in the art, such as Fresnel-Kirchhoff diffraction and Rayleigh-Sommerfeld diffraction. Since both the Fresnel-Kirchhoff formulation and the Rayleigh-Sommerfeld formulation have been widely derived and explained in the art, detailed derivations will not be repeated here. The Fresnel-Kirchhoff diffraction formulation utilizes Green's identity to express the disturbance at any point P as a function of the solutions of the wave equation and the values of its first derivative at all points on any surface surrounding P. The most general form can be expressed as Equation 1.15 below. Here, U is the complex amplitude of the disturbance at the surface, and s is the distance from P to the surface. In the coordinate system of Figure 8, the Fresnel-Kirchhoff formulation can be expressed as Equation 1.16 below. U(x, y; z) = -i / λ ∫??U(x^',y^';0) e^ikr / r ((1+cos?χ)) / 2 dx'dy'? (1.16) where r = [ (x - x^' )^2+(?y - y?^' )^2+z 2 1 / 2 であり and χ is the diffraction angle at the point (x', y', 0) between the diffracted wave and the normal to the z = 0 plane. The Fresnel-Kirchhoff formulation is an approximate scalar solution of the Helmholtz equation and is usually accurate except in cases very close to the aperture. The component ((1+cos?χ)) / 2 of Equation 1.16 is also called the tilt coefficient and can be approximated to 1 when the diffraction angle is small. Finding the solution of the integral of Equation 1.16 can be difficult for most applications, but by making specific assumptions about r in Equation 1.16, an approximate diffraction model known in the art that is easy to solve analytically and numerically can be derived. For example, the Fresnel approximation is a known approximate model that can simplify Equation 1.16 by making several approximations. The first approximation is based on the assumption that the diffraction angle is small, and thus the component ((1+cos?χ)) / 2 of Equation 1.16 (also called the tilt coefficient) can be approximated to 1. Based on the assumption that the diffraction angle is small, the denominator e^ikr / r of can also be approximated by z. However, in the exponent of, a small change in e^ikr / r can significantly change the value of. Therefore, e^ikr r = [ z 2 1 / 2 By assuming that the expansion of is truncated at z, a more accurate approximation of (x - x^' )^2+(?y - y?^' )^2+r can be made. ≫(x - x^' )^2+(?y - y?^' )^2 r?z+ ((x - x^' )^2+(?y - y?^' )^2) / 2z (1.17) The Fresnel diffraction integral of Equation 1.17 is derived from the above assumptions. ​​U(x, y; z) = e^ikz / iλz ∫??U(x',y';0) e^(ik / 2z[(x-x')^2+(?y-y?^' )^2]) dx'dy'? (1.18) The Fresnel diffraction integral can be solved using various numerical methods known in the art, including the fast Fourier transform. The Fraunhofer approximation is another approximation that can simplify the Fresnel-Kirchhoff diffraction formula. The diffraction pattern evolves continuously along the z-direction at a distance from the aperture and ultimately evolves into the final diffraction pattern. This diffraction pattern maintains itself while propagating (although its size increases proportionally to the distance). This far-field diffraction pattern in the far field is described by the Fraunhofer approximation. This is a limiting case of the Fresnel approximation when the field is observed at a distance far from the aperture. Equation 1.18 can be expanded into the following Equation 1.19. U(x, y; z) = e^ikz / iλz ∫??U(x',y';0) e^(ik / 2z[?(x?^2+y^2)-2(xx^'+yy^')?+(x?^'2+y^'2)]) dx'dy'? (1.19) Under the far-field condition (z ≫ k / 2), the exponential component of Equation 1.19 can be approximated as e^(ik / 2z?(x?^'2+y^'2))1, and the Fraunhofer diffraction integral becomes as shown in Equation 1.20. U(x, y; z) = (e^ikz e^(ik / 2z[?(x?^2+y^2))) / iλz ∫??U(x',y';0) e^(ik / z(xx^'+yy^')) dx'dy'? (1.20) The Fraunhofer integral can be interpreted as the two-dimensional (inverse) Fourier transform of the source wave field, where kx / z and ky / z can be regarded as spatial frequencies. U(x',y';0) The Fresnel approximation and the Fraunhofer approximation described above can also be applied to the Rayleigh - Sommerfeld diffraction formulation. Compared with the Fresnel - Kirchhoff formulation, in the Rayleigh - Sommerfeld formulation, two different boundary conditions are used to obtain more accurate solutions in equations 1.21 (the first solution) and 1.22 (the second solution). And In the coordinate system of Figure 8, the first solution of the Rayleigh - Sommerfeld formulation can be expressed as Equation 1.23 below. U(x, y; z) = ∫??U(x^',y^';0) e^ikr / r z / r(1 / 2πr+1 / iλ)dx'dy'? (1.23) Applying the same assumptions as the Fresnel approximation described above (i.e., r for the non - exponential component of?z, r?z+ ((x - x^' )^2+(?y - y?^' )^2) / 2z for the exponential component of r), Equation 1.23 can be simplified as Equation 1.24. U(x, y; z) =e^ikz / iλz ∫??U(x^',y^';0) e^[iπ / λz (x - x^' )^2+(?y - y?^' )^2 ] dx^' dy^'? (1.24) From the Fresnel approximation of Equation 1.24, applying the Fraunhofer approximation in the same way to further simplify Equation 1.24, U(x, y; z) = (e^ikz e^([ik / 2z?(x?^2+y^2))) / iλz ∫??U(x',y';0) e^(-ik / z(xx^'+yy^')) dx'dy'? (1.25) The Rayleigh - Sommerfeld formulation is a more accurate model than the Fresnel - Kirchhoff formulation due to the former's mathematical consistency and the ability to faithfully reproduce the diffraction field just behind the aperture of the latter. However, the Rayleigh - Sommerfeld formulation has limitations because it assumes a plane, while the Fresnel - Kirchhoff formulation can handle surfaces of any shape, enabling more accurate propagation in optical applications. While various embodiments in accordance with the principles disclosed herein have been described above, it should be understood that these are presented by way of example only and are not limiting. Accordingly, the scope of the invention should not be limited by any of the exemplary embodiments described above, but should be defined only in accordance with the claims arising from the present disclosure and their equivalents. Further, the above advantages and features are provided in the described embodiments, but do not limit the application of such issued claims to processes and structures that achieve any or all of the above advantages. It will be understood that the main features of the present disclosure can be used in various embodiments without departing from the scope of the present disclosure. Those skilled in the art will recognize, or be able to ascertain using no more than routine experimentation, numerous methods equivalent to the specific procedures described herein. Such equivalent methods are considered to be within the scope of the present disclosure and are encompassed by the claims. Furthermore, the section headings provided herein are provided to maintain consistency with the proposals of 37 CFR 1.77 or to provide organizational cues. These headings do not limit or characterize the inventions described in any claims that may issue from this disclosure. Specifically, by way of example, the heading refers to the "Field of the Invention," but such claims should not be limited by the language under this heading to describe the so-called technical field. Further, the description of the technology in the "Background of the Invention" section should not be construed as an admission that the technology is prior art to the inventions of this disclosure. Also, the "Summary" should not be regarded as characterizing the inventions described in the issued claims. Further, the use of the singular "invention" in this disclosure should not be used to claim that there is only one novelty in this disclosure. Multiple inventions may be described in accordance with the limitations of the multiple claims arising from this disclosure, and such claims define the inventions and their equivalents protected thereby. In all cases, the scope of such claims is to be considered on its own merits in light of this disclosure and should not be limited by the headings set forth herein. The use of the words "a" or "an" used in combination with the term "comprising" in the claims and / or the specification may mean "one", but is also consistent with the meaning of "one or more", "at least one", "one or more". The use of the term "or" in the claims is used to mean "and / or" unless it is explicitly indicated that only alternatives are being referred to, or the alternatives are mutually exclusive, although the disclosure supports a definition that refers only to alternatives and "and / or". Throughout this application, the term "about" is used to indicate that a value includes the inherent error variability of the device, the method used to determine the value, or the variability that exists among subjects being studied. Generally, but subject to the foregoing explanation, numerical values in this specification modified by words of approximation such as "about" may vary by at least ±1, 2, 3, 4, 5, 6, 7, 10, 12, or 15% from the recited value. As used herein and in the claims, the terms "comprising" (and any form of "comprising" such as "comprises" or "comprise"), "having" (and any form of "having" such as "has" or "have"), "including" (and any form of "including" such as "includes" or "include") or "containing" (and any form of "containing" such as "contains" or "contain") are inclusive or open-ended and do not exclude additional, unrecited elements or method steps. Words indicating comparison, measurement, timing, such as "at that point", "equivalent", "during that time", "complete", should be understood to mean "substantially at that point", "substantially equivalent", "substantially during that time", "substantially complete", etc., where "substantially" means that such comparison, measurement, timing is feasible to achieve the desired result stated either implicitly or explicitly. Words regarding the relative position of elements, such as "close", "closest", "adjacent", etc., shall mean close enough to substantially affect the interaction of the respective system elements. Similarly, other words representing approximation, when changed in such a way, are not necessarily understood to be absolute or complete, but refer to a state that is considered close enough by those skilled in the art to specify that such a state exists. The degree to which the description changes depends on the degree of change that is feasible and whether those skilled in the art can recognize that the changed feature still has the desired characteristics and functions of the unchanged feature. The term "substantially transparent" as used herein means that 90% of the light incident on a substantially transparent object can pass through. The term "or combinations thereof" as used herein refers to all permutations and combinations of the listed items preceding the term. For example, "A, B, C, or combinations thereof" is intended to include at least one of A, B, C, AB, AC, BC, or ABC, and when the order is important in a particular context, BA, CA, CB, CBA, BCA, ACB, BAC, or CAB are also included. Continuing with this example, combinations including repetitions of one or more items or terms such as BB, AAA, AB, BBC, AAABCCCC, CBBAAA, CABABB, etc. are explicitly included. Those skilled in the art will generally understand that, unless otherwise apparent from the context, there is no limit to the number of items or terms in a combination. All of the compositions and / or methods disclosed and claimed herein can be made and executed without undue experimentation in light of the present disclosure. Although the compositions and methods of the present disclosure have been described with respect to preferred embodiments, it will be apparent to those skilled in the art that variations can be applied to the compositions and / or methods and to the steps or the order of the steps of the methods described herein without departing from the concept, spirit, and scope of the present disclosure. All such similar alternatives and modifications apparent to those skilled in the art are considered to be within the scope of the spirit, scope, and concept of the present disclosure as defined by the appended claims.

Claims

1. 1. An annular transducer array comprising the following elements: A plurality of concentric transducer rings are arranged on the surface, centered on a point, and have the outermost transducer ring and the innermost transducer ring. Here, the outermost transducer ring and the innermost transducer ring are each driven by a circuit operable to cooperatively provide a mechanical energy field towards a volume in space.

2. The annular transducer array according to claim 1, wherein the transducer ring is substantially transparent to electromagnetic energy including a modulated wavefront.

3. The annular transducer array according to claim 2, wherein the modulated wavefront is modulated light.

4. The annular transducer array according to claim 1, wherein the modulated wavefront is a spatially modulated wavefront.

5. The annular transducer array according to claim 4, wherein the modulated wavefront is provided by either a spatial light modulator (SLM) or a wavefront modulator.

6. The annular transducer array according to claim 1, wherein the mechanical energy field has a directionality.

7. The annular transducer array according to claim 1, wherein the surface is coplanar.

8. The annular transducer array according to claim 1, characterized in that the transducer is substantially transparent.

9. The annular transducer array according to claim 1, characterized in that more than 50% of the incident modulated wavefront light passes through without attenuation.

10. The annular transducer array according to claim 1, wherein each concentric transducer ring has a predetermined resonance frequency.

11. The annular transducer array according to claim 10, characterized in that the resonance frequencies of the innermost transducer ring and the outermost transducer ring are the same.

12. The annular transducer array according to claim 10, wherein the resonance frequencies of the innermost transducer ring and the outermost transducer ring are different.

13. The annular transducer array according to claim 1, characterized in that the resonance frequencies of the concentric transducer rings are greater than 20 kHz.

14. The annular transducer array according to claim 13, wherein the concentric transducer rings are further modulated at a frequency of 100 Hz to 250 Hz.

15. The annular transducer array according to claim 1, further comprising additional rings driven by circuits respectively between the outermost transducer ring and the innermost transducer ring.

16. The annular transducer array according to claim 1, wherein the volume in the space is a volume having a diameter in the range of 1 mm to 5 mm.

17. The annular transducer array according to claim 1, wherein the transducer ring is in an annular shape.

18. The annular transducer array according to claim 1, wherein the transducer is a micromachined ultrasonic transducer.

19. The annular transducer array according to claim 18, wherein the micromachined ultrasonic transducer is one of a capacitive micromachined ultrasonic transducer and a piezoelectric micromachined ultrasonic transducer.

20. The annular transducer array according to claim 1, wherein each transducer ring generates a mechanical energy field independently of other transducer rings in the annular transducer array.

21. The annular transducer array according to claim 20, wherein by selectively driving the transducer rings in the annular transducer array, control of the direction of the mechanical energy field is provided.

22. The annular transducer array according to claim 1, wherein by activating a plurality of rings in the annular array, control of the focus of the mechanical energy field is provided.

23. The annular transducer array according to claim 1, wherein the control parameters of each transducer ring are selected from the group consisting of changing the driving frequency, phase delay, timing, and waveform structure.

24. The annular transducer array according to claim 23, wherein the wave structure is selected from the group consisting of Bessel waves, sine waves, square waves, or superpositions of waves.

25. The annular transducer array according to claim 1, wherein each transducer ring is operable to have different output levels that can contribute to increasing the mechanical energy field in one direction and decreasing the mechanical energy field in the other direction.

26. The annular transducer array according to claim 1, wherein the spacing between each transducer ring and the thickness of each transducer ring are selected to define the mechanical energy field output and the resonance frequency.

27. The annular transducer array according to claim 1, wherein the transducer ring is driven by a spatio-temporal modulation operable to raster scan the focus of the mechanical energy field at a first point in space multiple times per second.

28. The annular transducer array according to claim 27, wherein the transducer ring is driven by a spatio-temporal modulation operable to raster the acoustic focus at a second point in space multiple times per second.

29. The annular transducer array according to claim 1, wherein the transducer ring is driven by an amplitude modulation operable to concentrate the acoustic focus at a first point in space.

30. The annular transducer array according to claim 1, wherein the transducer ring is driven by amplitude modulation and spatio-temporal modulation.

31. An annular transducer array comprising the following elements: An annular sidewall having concentric circles centered on a point. A bottom electrode attached to the annular sidewall; A plurality of patterned electrodes arranged concentrically, A piezoelectric film is provided between the lower electrode and the plurality of patterned electrodes, wherein the piezoelectric film is operable to resonate between respective concentric rings in cooperation with drive signals applied to each of the patterned electrodes.

32. A tactile device, A substrate; Sidewalls extending from the substrate and defining a cavity; A membrane coupled to the sidewalls and disposed over the cavity, and An electrode operable to apply a voltage to the membrane; The membrane is operable to vibrate by the voltage and generate mechanical energy towards a focus in space.

33. The tactile device according to claim 1, wherein the electrode forms an annular array pattern on the membrane.

34. An array of the annular transducer arrays according to claim 1, wherein the array has at least two annular transducer arrays. It further includes a controller operable to direct a mechanical energy field toward a volume within the space.

36. An array of the annular transducer arrays according to claim 1, wherein the array is substantially transparent.

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