Nominal Versus Actual Spatial Resolution: Comparison of Directivity and Frequency-Dependent Effective Sensitive Element Size for Membrane, Needle, Capsule, and Fiber-Optic Hydrophones

Frequency-dependent effective sensitive element radius $\boldsymbol{a}_{\text {eff}}{({f})}$ is a key parameter for elucidating physical mechanisms of hydrophone operation. In addition, it is essential to know $\boldsymbol{a}_{\text {eff}}{({f})}$ to correct for hydrophone output voltage reduction due to spatial averaging across the hydrophone sensitive element surface. At low frequencies, $\boldsymbol{a}_{\text {eff}}{({f})}$ is greater than geometrical sensitive element radius $\boldsymbol{a}_{g}$ . Consequently, at low frequencies, investigators can overrate their hydrophone spatial resolution. Empirical models for $\boldsymbol{a}_{\text {eff}}{({f})}$ for membrane, needle, and fiber-optic hydrophones have been obtained previously. In this article, an empirical model for $\boldsymbol{a}_{\text {eff}}{({f})}$ for capsule hydrophones is presented, so that models are now available for the four most common hydrophone types used in biomedical ultrasound. The $\boldsymbol{a}_{\text {eff}}{({f})}$ value was estimated from directivity measurements (over the range from 1 to 20 MHz) for five capsule hydrophones (three with ${d}_{g}= {2} \boldsymbol{a}_{g}= {85}\,\,\mu \text{m}$ and two with ${d}_{g}= {2} \boldsymbol{a}_{g}= {200}\,\,\mu \text{m}$ ). The results suggest that capsule hydrophones behave according to a “rigid piston” model for ${k}~{\boldsymbol {a}}~_{g} \ge {0.7}$ ( ${k} = {2}\pi $ /wavelength). Comparing the four hydrophone types, the low-frequency discrepancy between $\boldsymbol{a}_{\text {eff}}{({f})}$ and $\boldsymbol{a}_{g}$ was found to be greatest for membrane hydrophones, followed by capsule hydrophones, and smallest for needle and fiber-optic hydrophones. Empirical models for $\boldsymbol{a}_{\text {eff}}{({f})}$ are helpful for choosing an appropriate hydrophone for an experiment and for correcting for spatial averaging (over the sensitive element surface) in pressure and beamwidth measurements. When reporting hydrophone-based pressure measurements, investigators should specify $\boldsymbol{a}_{\text {eff}}$ at the center frequency (which may be estimated from the models presented here) in addition to $\boldsymbol{a}_{g}$ .