Nominal Versus Actual Spatial Resolution: Comparison of Directivity and Frequency-Dependent Effective Sensitive Element Size for Membrane, Needle, Capsule, and Fiber-Optic Hydrophones
Keith A. Wear, Anant Shah
Abstract
Frequency-dependent effective sensitive element radius <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\boldsymbol{a}_{\text {eff}}{({f})}$ </tex-math></inline-formula> is a key parameter for elucidating physical mechanisms of hydrophone operation. In addition, it is essential to know <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\boldsymbol{a}_{\text {eff}}{({f})}$ </tex-math></inline-formula> to correct for hydrophone output voltage reduction due to spatial averaging across the hydrophone sensitive element surface. At low frequencies, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\boldsymbol{a}_{\text {eff}}{({f})}$ </tex-math></inline-formula> is greater than geometrical sensitive element radius <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\boldsymbol{a}_{g}$ </tex-math></inline-formula> . Consequently, at low frequencies, investigators can overrate their hydrophone spatial resolution. Empirical models for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\boldsymbol{a}_{\text {eff}}{({f})}$ </tex-math></inline-formula> for membrane, needle, and fiber-optic hydrophones have been obtained previously. In this article, an empirical model for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\boldsymbol{a}_{\text {eff}}{({f})}$ </tex-math></inline-formula> for capsule hydrophones is presented, so that models are now available for the four most common hydrophone types used in biomedical ultrasound. The <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\boldsymbol{a}_{\text {eff}}{({f})}$ </tex-math></inline-formula> value was estimated from directivity measurements (over the range from 1 to 20 MHz) for five capsule hydrophones (three with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${d}_{g}= {2} \boldsymbol{a}_{g}= {85}\,\,\mu \text{m}$ </tex-math></inline-formula> and two with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${d}_{g}= {2} \boldsymbol{a}_{g}= {200}\,\,\mu \text{m}$ </tex-math></inline-formula> ). The results suggest that capsule hydrophones behave according to a “rigid piston” model for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${k}~{\boldsymbol {a}}~_{g} \ge {0.7}$ </tex-math></inline-formula> ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${k} = {2}\pi $ </tex-math></inline-formula> /wavelength). Comparing the four hydrophone types, the low-frequency discrepancy between <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\boldsymbol{a}_{\text {eff}}{({f})}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\boldsymbol{a}_{g}$ </tex-math></inline-formula> was found to be greatest for membrane hydrophones, followed by capsule hydrophones, and smallest for needle and fiber-optic hydrophones. Empirical models for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\boldsymbol{a}_{\text {eff}}{({f})}$ </tex-math></inline-formula> 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\boldsymbol{a}_{\text {eff}}$ </tex-math></inline-formula> at the center frequency (which may be estimated from the models presented here) in addition to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\boldsymbol{a}_{g}$ </tex-math></inline-formula> .