*3.3. Estimation of the Optimum Stand-O*ff

We found a numerical relationship for δ based on the simulation results of 2420 angle/stand-off combinations for axicon lenses (see Appendix A). The relation between the optimum value of δ (which improves spatial resolution) and the value of F/N is shown in Figure 7. The linear regression equation is given by:

$$\delta\_{\text{Optimum}} = \frac{10.921 + \ln\left(\frac{F}{N}\right)}{1.96 \cdot 10^2} \,\left[mets\,\text{s}\right].\tag{6}$$

The coefficient of determination R<sup>2</sup> is 0.96 with a *p*-value significance level of < 0.00001.

**Figure 4.** (Top) Value of F/N and (Bottom) relative loss of lateral resolution, both, vs. stand-off given in number of wavelengths (κδ) for 445 kHz PDMS-based axicon lens with different angles. (**a**) φ = 140◦ ; (**b**) φ = 150◦ ; (**c**) φ = 160◦ ; (**d**) φ = 170◦ .

**Figure 5.** Housing of the axicon lens.

**Figure 6.** Outer case effect for 445 kHz PDMS-based axicon lens with cone angle φ = 130◦ . (Top) Relative loss of lateral resolution vs. stand-off given in number of wavelengths (κδ) and (Bottom) normalized pressure amplitudes of the lateral beam profile. (**a**) Lens housing with reflectivity of 0.8 dB down; (**b**) lens housing with reflectivity of 40 dB down.

**Figure 7.** Illustration of the relation between the axicon lens optimum stand-off δ and the ratio of F/N. The value of δ estimated by linear regression is indicated to obtain the highest lateral resolution for different lens angles.

As an example, Figure 8 shows the focusing behavior of PDMS-based 144◦ axicon lens with four different values of δ (a: 0 mm, b: 20.5 mm, c: 34 mm, and d: 45.25 mm). These different settings are indicated in Figure 9 showing the relative LLR as a function of κδ*.* With a stand-off of 34 mm, the lateral spatial resolution improves by up to 40%, compared to the same lens without stand-off.

The optimum stand-off predicted by Equation (6) was checked for different frequencies. Figure 10 compares the experimental loss of lateral resolution (LLR) for transducers with acoustic frequencies (f) of (a) 0.2225 MHz, (b) 0.445 MHz, (c) 0.890 MHz, and (d) 4.45 MHz.

**Figure 8.** 445 kHz-Cigar-shaped acoustic focus for different settings of the stand-off. (**a**) δ = 0 mm; (**b**) δ = 20.5 mm; (**c**) δ = 34 mm; (**d**) δ = 45.25 mm. (Top) Focusing behavior of the PDMS-based 144◦ axicon lens with four different values of δ. (Middle) Normalized pressure amplitudes of the lateral beam profile. (Bottom) Normalized pressure amplitudes of the axial beam profile, 0 mm indicates the focus.

**Figure 9.** Stand-off, given in number of wavelengths, vs. relative loss of lateral resolution for 445 kHz PDMS-based 144◦ axicon lens. (**a**) δ = 0 mm; (**b**) δ = 20.5 mm; (**c**) δ = 34 mm; and (**d**) δ = 45.25 mm.

**Figure 10.** Stand-off given in number of wavelengths vs. relative loss of lateral resolution for different frequencies. The value of κδ for the optimum stand-off (red circle) is directly proportional to the frequency. (**a**) f = 0.2225 MHz; (**b**) f = 0.445 MHz; (**c**) f = 0.890 MHz; (**d**) f = 4.45 MHz.
