Ray-tracing model

The inputs to the ray-tracing model consisted of: a set of targeted focal points r_k, the dimensions and shape of the absorber and its position relative to the focal points, the set of coordinates making up each absorbing layer (x_i, y_j, d_l), and the sound speed and density of the absorber (c_h, ρ_h) and the coupled medium (c_m, ρ_m).

To select the ray-trajectories, one approach would be to select a set of angles φ ∈ [0, π ] and θ ∈ [0, 2π ] that are evenly distributed on the surface of a sphere, then restrict these to those that intersect with the absorber. However, due to

wave-spreading, it was found that a large number of trajectories were required to suffi-ciently sample and approximate the pressure in pixels at higher transmitted angles.

Therefore, the ray-trajectories were selected by evenly sampling between 0-^π₂ in the transmitted angle (φ_t). These angles were then transformed into a set of incident angles (φ_i) using Snell’s law:

φ_i= sin⁻¹ c_m c_h sin φ_t



. (5.1)

For simplicity, the problem was orientated such that normal incidence to the ab-sorber corresponded to φ_i= 0 (Fig. 5.2) and the absorbing layers were placed in the (x, y) plane. For each φi, the number N of radial angles θ used was weighted based on the interaction radius with the absorber surface. This was calculated using

N= Ah tan φ_i+ B. (5.2)

Here h is the distance from the plane of the target point to the top layer of the absorber indicated in Fig. 5.2, and A and B are constants. The constant B was used to ensure that φ_i = 0 was sampled, and to ensure a minimum sampling for small values of φi. The constant A was used to scale the number of rays. The set of values for θ for each φ_iwere chosen using

θn= 2πn

N +2πC

N . (5.3)

Here C is a random number between 0 − 1 and n is an integer between 1 − N. This ensures the whole range 0 − 2π is sampled. The random factor was introduced for practical reasons to prevent pixels at certain radial angles from being systematically missed.

The propagation of each ray was modelled using Snell’s law to transmit into the absorber and the Fresnel equations to account for subsequent reflection losses.

Since the set of ray-trajectories used was uneven, the amplitude was calculated us-ing an analytic solution introduced by Young [176]. This gives the pressure ampli-tude p at a particular depth for a point source above an infinite half-space as

p= Tcos φ_i (h + q_c^ch

mcos φ_isec φ_t)¹²(h + q_c^ch

mcos³φ_isec³φ_t)¹²

. (5.4)

Here q is the distance of the point below the interface shown in Fig. 5.2, and T is the pressure transmission coefficient for the incident angle calculated using

T = 2ρ_hc_hcos φ_i

ρ_hc_hcos φ_i+ ρmc_mcos φ_t. (5.5) For each intersection of a ray with a pixel (x_i, y_j, d_l), the time of arrival t, the amplitude, and the original trajectory (φi, θi) of the ray were stored. The ampli-tude was calculated using Eq. 5.4, accounting separately for losses caused by any subsequent reflections within the absorber. These were calculated using

R=ρ_mc_mcos ψ_i− ρ_hc_hcos ψ_t

ρ_mc_mcos ψ_i+ ρhc_hcos ψ_t, (5.6) where ψ is the angle between the boundary normal and the incident ray. The ray was judged to have intersected a pixel if it passed through the square defined by {(x_i−^∆₂, y_j−^∆₂, d_l), (x_i−^∆₂, y_j+^∆₂, d_l), (x_i+^∆₂, y_j−^∆₂, d_l), (x_i+^∆₂, y_j+^∆₂), d_l)}. To avoid over or underestimating the pressure in a given pixel, the properties of rays with similar initial trajectories incident on the same pixel were averaged. For two rays (φ₁, θ1) and (φ2, θ2) the properties were averaged if

|φ1− φ2| < 1.5^oor |θ₁− θ2| < 1.5^o (5.7)

The output of the ray-tracing was a set of impulse responses I(r_k,t; x_i, y_j, d_l).

To get the acoustic signal p(r_k,t; x_i, y_j, d_l) generated by each pixel, these were con-volved with the signal generated optically from a pixel in the absorber p(t). This was approximated by using k-Wave to simulate a point source on a 3-D grid, with a spacing ∆, driven by a low-pass filtered impulse.

The ray-tracing model was validated by comparison with k-Wave. This com-parison used a cylindrical absorber with a radius of 11 mm, a height of 4 mm, a sound speed of 2450 m s^-1, and a density of 1180 kg m^-3. A layer 2 mm below the

0 4 Time ( s)

-1 0 1 2 3 4 5

Amplitude (arb units) Simulation

Ray-tracing

3 2

1

Figure 5.3: Comparison of the acoustic response calculated at three different points using the ray-tracing approach (dashed line) and simulation (continuous line). The time series for the ray-tracing and simulation are each normalised to the maxi-mum pressure across all three points for the respective method. At all 3 points good agreement is found over the first 4 µs. Figure reprinted from [175] ^© IEEE 2016.

top surface was used for the comparison. The target point was placed 1.8 cm above the cylinder displaced 2 mm from its centre in both the x and y directions. The sur-rounding medium was assumed to be water with a sound speed of 1500 m s^-1 and a density of 1000 kg m^-3. The ray-tracing was run for these inputs and the output compared against a k-Wave simulation.

The k-Wave simulation was carried out on a 256×256×288 domain with a grid spacing of 0.1 mm. The absorber was inserted at one end of the domain as a cylinder of matching dimensions, with acoustic properties matching those of the absorber.

The rest of the medium was set to a sound speed of 1500 m s^-1 and density 1000 kg m^-3. The target point was inserted as a source point 180 grid points above the cylinder, displaced by 20 grid points from its centre in both x and y. The pressure was recorded across a plane 20 grid points into the cylinder. A bipolar pulse low-pass filtered at 6 MHz was used as the acoustic signal p(t) for both the simulation and the ray-tracing. A comparison of the ray-tracing and k-Wave outputs from 3 separate positions is shown in Fig. 5.3. The two models show good agreement over the first 4 µs or, equivalently, the first 2 reflections.

5.2.3 Multi-layer optimisation