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 formula 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).

Fig. 1.

a) spectrum of a nonlinear signal (solid line) and hydrophone frequency-dependent sensitivity (dash line). b) unfiltered waveform (dotted line) and waveform distorted by hydrophone (solid line). The hydrophone boosts harmonics relative to the fundamental, resulting in a waveform with reduced rarefactional pressure and exaggerated high-frequency content, which is manifested as sharper, taller, compressional peaks.

Fig. 2.

a) Theoretical sensitivity filter, M_L(f), (red dashed line) and spatial averaging filter, S_p(f) (solid blue line), b) Model true spectrum (red dotted line) and harmonic approximation (blue solid line).

Fig. 3.

Gaussian and parabolic approximations to radial diffraction pattern for the fundamental frequency component where k₁ = 2π/λ₁, λ₁=c/f₁, c = speed of sound, a_s = the radius of the source, and D = the focal length of the source. Both approximations are accurate for radial coordinate < half width half maximum (HWHM) of the beam.

Fig. 4.

Scatter plot of ka_eff estimated from RB model directivity, 2J₁(ka sinθ) / ka sinθ vs. a_g for a needle hydrophone that obeys the RP model, H(ρ, f) with a₁ = a₂ = a_g. The effective radius, a_eff, was chosen by selecting the value of a that minimized RMSD between the two models. The angle range for the directivity fit was −60° < θ < 60°.

Fig. 5.

Relative difference between a_eff and a_g plotted as a function of ka_g. The root mean square difference (RMSD) between fit and data is lower for the exponential fit than the inverse fit. The angle range for the directivity fit was −60° < θ < 60°.

Fig. 6.

Magnitude and phase of directivity functions for four values of ka_g for the rigid piston (RP) model evaluated for a₁ = a₂ = a_g and two versions of the rigid baffle (RB) model evaluated for a = a_g and a_eff. The angle range for the directivity fit was −60° < θ < 60°.

Fig. 7.

Differences in magnitude and phase between the RP model directivity evaluated for a₁ = a₂ = a_g and RB model directivity evaluated for a = a_eff(f).

Fig. 8.

−6 dB beam width (FWHM) vs harmonic number for six transducers. The title above each plot gives center frequency, diameter, and focal length. The symbol σ_m refers to the nonlinear propagation parameter. The threshold for approximate power law dependence of FWHM vs. harmonic number (straight line on log log plot) is σ_m < 2.4 (σ_q < 3) for the six transducers investigated.

Fig. 9.

Simulation and experimental results for exponent for power law dependence (based on the first 5 harmonics) of FWHM vs nonlinear propagation parameter σ_m. A least-squares linear fit is shown (red line).

Fig. 10.

Hydrophone measurements at the focus of the Blatek 3.5 MHz transducer. a) Spectrum. Measurement is shown in the blue line. Fits to harmonic strengths based on spectra with relative harmonic strengths according to Blackstock’s theory are shown in red x’s. b) time-domain radio-frequency (RF) signal. Measurement is shown in the blue line. A best-fit signal with relative harmonic strengths corresponding to Blackstock’s theory with σ = 0.35 is shown in the red dashed line.

Fig. 11.

Lateral beam scans for harmonics of Blatek transducer operated at 3.5 MHz. The solid black line shows the theoretical lateral profile at the fundamental frequency. The dotted lines show Gaussian fits for the fundamental and harmonics based on a model in which HWHM is proportional to 1 / n^q and n = harmonic number. The harmonics show approximately Gaussian behavior at least up to radial distances of 0.5 mm, corresponding to a maximum hydrophone diameter of 1 mm and therefore suggesting relevance to most hydrophones used in ultrasonics.

Fig. 12.

RP and RB hydrophone transfer functions H_RP(f) (with a₁=a₂=a_g) and H_RB(f) with a= a_eff(f), RP spatial averaging filter S_RP(f) = H_RP(f) / M_L(f), and numerical, Gaussian, and quadratic forms for the spatial averaging filter, S_Numericall(f) (17), S_Gaussian(f) (20), and S_Quadratic(f) (30). Transducer parameters are for the Sonic Concepts H101 transducer operated at its lower frequency option of 1.05 MHz. The hydrophone sensitive element geometrical diameter was 200 μm. The vertical chain-dash line gives the limit predicted by (33) of the validity of the quadratic approximation for the spatial averaging filter given by (30).

Fig. 13.

Simulated hydrophone output voltage for a tone burst produced by the Sonic Concepts H101 transducer operated at its lower frequency option of 1.05 MHz. The hydrophone sensitive element geometrical diameter was 200 μm. Voltage is in arbitrary units.

Fig. 14.

RP and RB hydrophone transfer functions H_RP(f) (with a₁=a₂=a_g) and H_RB(f) with a= a_eff (f), RP spatial averaging filter S_RP(f) = H_RP(f) / M_L(f), and numerical, Gaussian, and quadratic forms for the spatial averaging filter, S_Numericall(f) (17), S_Gaussian(f) and S_Quadratic(f) (30). Transducer parameters are for the Sonic Concepts H101 transducer operated at its higher frequency option of 3.30 MHz. The hydrophone sensitive element geometrical diameter was 200 μm. The vertical chain-dash line gives the limit predicted by (33) of the validity of the quadratic approximation for the spatial averaging filter given by (30).

Fig. 15.

Simulated hydrophone output voltage for a tone burst produced by the Sonic Concepts H101 transducer operated at its higher frequency option of 3.30 MHz. The hydrophone sensitive element geometrical diameter was 200 μm. Voltage is in arbitrary units.

Fig 16.

Graphical Guide for hydrophone spatiotemporal response. M: hydrophone sensitivity. S: spatial averaging filter. Total spatiotemporal transfer function = MS. d_g: geometrical sensitive element diameter, λ₁: wavelength of fundamental component. F#: ratio of transducer focal length to diameter. M is normalized to its value at the fundamental frequency. Note that M can vary from theory, especially for frequencies above the frequency corresponding to the maximum theoretical sensitivity, due to hydrophone design complexities not captured by the RP model. Spatial averaging filter depends on q, which determines the dependence of harmonic beam width on harmonic beam number. See text for details.

Fig 17.

Graphical Guide for hydrophone spatiotemporal response. M: hydrophone sensitivity. S: spatial averaging filter. Total spatiotemporal transfer function = MS. d_g: geometrical sensitive element diameter, λ₁: wavelength of fundamental component. F#: ratio of transducer focal length to diameter. M is normalized to its value at the fundamental frequency. Note that M can vary from theory, especially for frequencies above the frequency corresponding to the maximum theoretical sensitivity, due to hydrophone design complexities not captured by the RP model. Spatial averaging filter depends on q, which determines the dependence of harmonic beam width on harmonic beam number. See text for details.

Fig 18.

Graphical Guide for hydrophone spatiotemporal response. M: hydrophone sensitivity. S: spatial averaging filter. Total spatiotemporal transfer function = MS. d_g: geometrical sensitive element diameter, λ₁: wavelength of fundamental component. F#: ratio of transducer focal length to diameter. M is normalized to its value at the fundamental frequency. Note that M can vary from theory, especially for frequencies above the frequency corresponding to the maximum theoretical sensitivity, due to hydrophone design complexities not captured by the RP model. Spatial averaging filter depends on q, which determines the dependence of harmonic beam width on harmonic beam number. See text for details.