**Appendix D Guideline for Axicon Lens Design**

This guide is based on design Equations (4)–(6) for Epoxy/PDMS materials combination.

(1) Calculate transducer near field, as:

$$\mathbf{N} = \frac{D^2 f}{4c}$$

where *D* is transducer diameter, *f* is transducer frequency, and c is sound velocity of the material under inspection

(2) Select the desired lens focus F and check that:

$$\mathbf{0.1} \le \frac{F}{N} < \mathbf{0.4}$$

Values of 0.1 are rarely used because it gives a very near focus. Values or 0.4 give a profile similar to the transducer but remove the *N* zone.

(3) Calculate the angle of the axicon lens, as:

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

(4) Calculate the optimum value of stand-off, as:

$$\delta\_{Optimum} = \frac{\mathbf{10.921} + \ln\left(\frac{F}{N}\right)}{\mathbf{1.96} \cdot \mathbf{10^2}} \,\mathrm{[meters]}.$$

*Materials* **2019**, *12*, 3433

(5) Focus diameter is:

$$d\_F = \frac{D \cdot F}{2N}.$$

(6) Depth of focus is:

$$\mathsf{DOF}\_{\mathsf{F}} = \mathsf{2F}.$$

Figure A2 shows the relative LLR as a function of κδ for value of F/N between 0.1 and 0.3.

**Figure A2.** The relative percentage loss of lateral resolution (LLR) as a function of κδ. (**a**) F/N = 0.092; (**b**) F/N = 0.13; (**c**) F/N = 0.16; (**d**) F/N = 0.21; (**e**) F/N = 0.28; (**f**) F/N = 0.33.
