Considerations for Choosing Sensitive Element Size for Needle and Fiber-Optic Hydrophones—Part I: Spatiotemporal Transfer Function and Graphical Guide

Abstract

The spatiotemporal transfer function for a needle or reflectance-based fiber-optic hydrophone is modeled as separable into the product of two filters corresponding to frequency-dependent sensitivity and spatial averaging. The separable hydrophone transfer function model is verified numerically by comparison to a more general rigid piston spatiotemporal response model that does not assume separability. Spatial averaging effects are characterized by frequency-dependent “effective” sensitive element diameter, which can be more than double the geometrical sensitive element diameter. The transfer function is tested in simulation using a nonlinear focused pressure wave model based on Gaussian harmonic radial pressure distributions. The pressure wave model is validated by comparing to experimental hydrophone scans of nonlinear beams produced by three source transducers. An analytic form for the spatial averaging filter, applicable to Gaussian harmonic beams, is derived. A second analytic form for the spatial averaging filter, applicable to quadratic harmonic beams, is derived by extending the spatial averaging correction recommended by IEC 62127-1 Annex E to nonlinear signals with multiple harmonics. Both forms are applicable to all hydrophones (not just needle and fiber optic hydrophones). Simulation analysis performed for a wide variety of transducer geometries indicates that the Gaussian spatial averaging filter formula is more accurate than the Quadratic form over a wider range of harmonics. Additional experimental validation is provided in Part II. Readers who are uninterested in hydrophone theory may skip the theoretical and experimental sections of this paper and proceed to the Graphical Guide for practical information to inform and support selection of hydrophone sensitive element size (but might be well-advised to read the Introduction).

Correction for Spatial Averaging Artifacts in Hydrophone Measurements of High-Intensity Therapeutic Ultrasound: An Inverse Filter Approach

“…The subscript p is a reminder that the function depends on the particular pressure field distribution incident upon the hydrophone. The separability assumption was previously shown to be valid for a wide variety of transducers and hydrophones [50,51].…”

“…The ratio of this integral to the area of the sensitive element quantifies the effects of spatial averaging [62][63][64][65][66]. The numerator and denominator of the ratio should be calculated based on the hydrophone effective sensitive element radius a eff (f), which is a function of frequency and can differ from the geometrical radius, a g [50]. Functional forms for a eff (f) have been reported for needle and fiber optic hydrophones [50,67,68].…”

“…The numerator and denominator of the ratio should be calculated based on the hydrophone effective sensitive element radius a eff (f), which is a function of frequency and can differ from the geometrical radius, a g [50]. Functional forms for a eff (f) have been reported for needle and fiber optic hydrophones [50,67,68]. The spatial averaging filter may be written as [50]…”

“…These forms of distortion are relevant to HITU signals, which can be highly nonlinear and can therefore have very broad bandwidths due to the presence of multiple harmonics [28,29,[40][41][42][43]. Nonlinearity affects spatial averaging because as harmonic frequency increases, harmonic beam width decreases [40,[44][45][46][47][48][49] and the potential for harmonic spatial averaging increases [50,51].…”

“…The distorting effects of frequency-dependent sensitivity can be suppressed by performing sensitivity deconvolution of hydrophone measurements [27][28][29][30][52][53][54][55][56][57][58][59][60][61]. Recently, an inverse-filter method for correction for spatial averaging artifacts in nonlinear ultrasound signals was derived and validated for signals in the diagnostic pressure range [50,51]. The objective of the present paper is to extend that validation to HITU pressure signals.…”