Ultrasound Haptic Holographic Display

The holographic haptic display (UHEV1, Ultrahaptics Ltd.) comprised a planar array of N ultrasound emitters. Such a display focuses sound in air by applying a phase delay to each transducer signal. Constructive interference yields localized regions of high differential pressure, sufficient to impart time-varying displacements to the skin (1 (link), 2 ). The differential acoustic pressure field may be described by $$p(\mathbf{x},t)={\displaystyle \sum _{n-1}^N}\frac{1}{4\mathrm{\pi }{r}_n}{a}_n\exp [j({kr}_n-\mathrm{\omega }t-{\mathrm{\varphi }}_n)]$$ where x is a position within the Fresnel zone of the array, a_n is the amplitude of the acoustic pressure wave emitted from transducer n, k = ω/c is the wave number, f = ω/(2π) = 40 kHz is the ultrasound carrier frequency, c is the speed of sound, r_n = ‖x − x_n‖ is the distance from x to the location x_n of transducer n, and ϕ_n is the corresponding phase delay. Focusing at a location x_f is achieved by matching the phase delays, ϕ_n, to the propagation time for a wavefront to reach the focus ϕ_n = ω‖x_f − x_n‖/c, thus yielding constructive interference. The focal width, δ, satisfies a Rayleigh diffraction limit, with δ ⪆ c/(2f).
Ultrasound frequency oscillations cannot be directly felt via touch. In haptic holography, palpable low-frequency mechanical signals are produced via a nonlinear phenomenon known as acoustic radiation pressure. Neglecting viscosity, the Langevin acoustic radiation force F_L imparted to an object (here, the skin) at a focus location is, to second order, given by $${\mathbf{F}}_L=-{\iint }_{S}dS(⟨p_2⟩\mathbf{I}+{\mathrm{\rho }}_0⟨{\mathbf{u}}_1\otimes {\mathbf{u}}_1⟩)\cdot \mathbf{n}$$ where p, ρ, and u are the fluid pressure, density, and velocity fields. The angular brackets denote time-averaged quantities, n is the surface normal, I is the unit tensor, and S is a surface region containing the focus location. Subscripts 0, 1, and 2 refer to successive terms in a perturbation expansion about a quiescent fluid configuration (33 –36 ). Applying the same expansion to the Navier-Stokes equation yields an expression for 〈p₂〉 in terms of lower-order quantities $$⟨p_2⟩=\frac{1}{2}\frac{1}{{\mathrm{\rho }}_0{c}_0^2}⟨p_1^2⟩-\frac{1}{2}{\mathrm{\rho }}_0⟨{\mid {\mathbf{u}}_1\mid }^2⟩$$
Averaged over 1 cycle of oscillation, only the quadratic terms are nonzero. Together, they yield a nonvanishing, low-frequency force F_L(x, t) that elicits vibrations in the skin (see Supplementary Text). The time evolution of these vibrations is governed by a driven elastic wave equation $${\mathbf{F}}_L(\mathbf{x},t)=\{-m\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}+\mathrm{\mu }{\mathrm{\nabla }}^2+[(K+\mathrm{\mu }/3)\mathrm{\nabla }]\mathrm{\nabla }\cdot \}\mathrm{\xi }(\mathbf{x},t)$$
Here, ξ(x, t) is tissue displacement, and m, μ, and K are the mass density, shear modulus, and bulk modulus, respectively. Plane wave solutions, $\mathrm{\xi }(\mathbf{x},t)=A\hat{\mathbf{r}}{e}^{j(\mathbf{k}\cdot \mathbf{x}-\mathrm{\omega }t)}$ , describe oscillations along polarization directions $\hat{\mathbf{r}}$ . Because of the low-frequency content of the acoustic radiation force and high propagation speed of compression waves, most acoustic energy is transferred to shear (transverse) wave components, $\hat{\mathbf{k}}\cdot \hat{\mathbf{r}}=0$ . For skin or soft tissues, shear wave speeds, $c=\sqrt{\mathrm{\mu }/m}$ , are frequency dependent and can range from 1 to 10 m/s at the tactile frequencies applicable to this study (22 (link), 23 ). Because of viscoelasticity, wave amplitudes are also attenuated in a frequency-dependent manner with increasing propagation distance (30 (link)).