**4. Discussion**

Although the analysis was carried out with a single element transducer with no aberration correction, and a specific skull geometry designed to approximate the varied shaped of the skull, it is expected that the relative influence of different medium properties and aspects of medium geometry will be maintained. Phantom geometry is the major material influence on the intracranial field and sound speed is shown to be the most influential acoustic property in focus pressure, position, and volume. From the experimental beam patterns shown in Figure 11, the unexpected focusing property of the skull in axial axis may be described as a nonlinear effect that causes the beam to rotate back toward the skull insertion point, creating a more compact pressure cigar-shaped acoustic field. This was also observed using segmented-sphere transducers [16,17]. Thus, the skull is not an obstacle for transcranial focusing of US and may exert an additional acoustic lensing effect to enhance spatial resolution under certain conditions. Out of focus, the sound pressure decreases with a very steep slope. From the comparison in Table 4, there is a good coincidence between the simulation and the values obtained experimentally in water for skull thickness of 5 mm, separated 2 mm from the lens. Regarding the influence of the thickness skull in the focus of axicon lenses, since on both sides of the skull there is a discontinuity in the acoustic impedance, with different velocities of sound propagation, and the average sound speed is the acoustic property that most influences the focal distance, we observe as expected, small variations in the position, diameter, and depth of focus, for different thicknesses of the cranial bone, resulting in the average deviations of the focus less than 1 mm.

On the other hand, transmitting FUS through human cranial bone caused an approximately 40% loss in lateral resolution of the acoustic beam, estimated by the intensity full width at half maximum. However, this loss of resolution is compensated by adequate stand-off of axicon lens. In addition, we

find that the PDMS provides a smaller focal zone, which is desirable for neurostimulation [6]. All this allows a higher resolution, comparable to spherical transducers (the most commonly used for ultrasonic brain therapy), but with the advantage that the near field is eliminated, and the focus distance is shortened. For other applications, a wider focal zone and a line focus such as that obtained with glycerin or ethylene glycol may be desirable [7]. From Equation (5), the elimination of the near field, by means of an appropriate axicon lens, enables transducers featuring the same wavelength/diameter ratio to produce the same focal spot size. Thus, large diameter, low-frequency transducers may be used. This is useful for brain stimulation where low frequencies are required for penetration of the skull.

One problem of devices with axicon lenses are the relatively high sidelobes [18]. How much this will affect will depend on the proposed applications. With the calculation of the optimum value of δ, by Equation (6), a better lateral resolution is achieved. This relation between axicon lens stand-off and the value of F/N is applicable to high-resolution epoxy resin/PDMS lenses or other similar combination.
