*3.1. Estimation of the Axicon Lens Angle*

The relation between axicon lens angle φ and the value of F/N produced (see Appendix A) is illustrated graphically in Figure 3. The transducer near field length N is given by *N* = *D*<sup>2</sup> *f*/4*c*. The coefficient of determination R<sup>2</sup> is 0.97 with *p*-value significance level of < 0.00001. This relation is for δ = δOptimum:

$$\phi = \frac{9.9708 + \ln\left(\frac{F}{N}\right)}{5.52 \cdot 10^{-2}} \text{ [degrees]}.\tag{4}$$

**Figure 3.** Illustration of the relation between the axicon lens angle φ and the ratio of *F*/*N*.

The following relation, based on our study summarized in Table A2 (see Appendix A), was found experimentally valid for the lens described. The optimum ratio of F/N appears to be between 0.1 and 0.3. This produces focal beam diameters (d<sup>F</sup> with the axicon lens equipped transducers and d<sup>N</sup> of a conventional transducer), and depths of focus (DOF) of similar ratio according to:

$$\frac{\text{F}}{\text{N}} = \frac{\text{d}\_{\text{F}}}{\text{d}\_{\text{N}}} = \frac{\text{DOF}\_{\text{F}}}{\text{DOF}\_{\text{N}}}.\tag{5}$$

For values of F/N > 0.4, some evidence of the original near field still remains. As the value F/N decreases below 0.4, all evidence of the original near field is rapidly suppressed in the lens system. This contrasts with the behavior of spherical lenses where some original near field is always present.

DOF DOF .

F N = d d =

We observe that there is an inversely proportional relationship between F/N and κδ values, as shown in Figure 4 for different values of angle φ. The relative percentage loss of lateral resolution (LLR) as a function of κδ is shown in the same figure, where κ is the wave number. LLR is relative to the best lateral resolution that can be achieved with κδ value that minimizes energy transmitted outside of the narrowest possible main beam. Since the speed of sound in PDMS is lower than in water, as the value of δ increases, the wavelengths-weighted average sound speed through the heterogeneous medium with step discontinuity of velocity decreases, which effectively reduces the ratio of F/N.
