High-intensity focused ultrasound (HIFU) is a non-invasive medical thermal ablation technology that can realize the “non-invasive” treatment of diseases. It is developing into a standard tumor treatment method. The accurate measurement of the HIFU sound pressure is very important for formulating clinical schemes involving HIFU treatment and ensuring the safety and effectiveness of the treatment. Laser interferometry has been adopted by new-generation sound pressure benchmarks because of its high precision, non-contact characteristic, and direct traceability to the length benchmark. The bandwidth of a laser interference system directly affects the upper limit of sound pressure measurements. Current methods of estimating system bandwidth are based on linear sound field conditions; however, with an increase in the measured sound pressure, the HIFU field will become nonlinear, leading to a deviation between the system bandwidth requirements and the model solution results. Aiming to address the problem of bandwidth calculation of interference systems for HIFU sound pressure measurement, this study theoretically analyzes a mathematical model of interference system bandwidth and vibration signal in HIFU sound pressure measurement, refers to the measured sound pressure value, and establishes the relationship between interference system bandwidth and sound pressure by numerical simulation.

According to the mathematical relationship between sound particle vibration and interference signal, a bandwidth model of interference signals under the condition of a nonlinear sound field is established. Because the interference signal is expressed as a nonelementary function that cannot be expanded by the Bessel function, the bandwidth of the interference signal under the condition of a nonlinear sound field is analyzed by numerical simulation. The simulation results show that compared with that in a linear sound field, the interference signal spectrum in a nonlinear sound field is not symmetrically distributed centered on the carrier frequency but widened and shifted. In order to obtain the real HIFU sound pressure and ensure that the simulation results are closer to the actual values, the HIFU sound field is measured using the optical fiber hydrophone, and the sound pressure waveforms and frequency domain distributions corresponding to different sound pressures are obtained. According to the waveform parameters of the sound pressure, the simulation conditions of the interference signal bandwidth are set, and the bandwidths of the interference systems required by different sound pressures are obtained.

Using the measured sound pressure data of the HIFU sound field, the relationship between the bandwidth of the heterodyne interference system and HIFU sound pressure is simulated. Under the condition that the relative error of the sound pressure measurement caused by the system bandwidth is less than 2%, the corresponding relationship between the lowest bandwidth of the heterodyne interference system and the peak value of the measured sound pressure is determined. Through curve fitting, it is deduced that quadratic fitting has a smaller fitting error than linear fitting. Therefore, it is inferred that under the condition of a nonlinear sound field, the bandwidth requirement of the heterodyne interferometry system changes via a quadratic law with the peak-to-peak of the measured sound pressure, which means that the bandwidth requirement of the heterodyne interferometry system will increase sharply with a higher intensity sound pressure measurement. When the peak value of the measured HIFU sound pressure reaches 60 MPa, the bandwidth of the heterodyne interferometry system should not be less than 473 MHz; thus, the sound pressure measurement error introduced by the system bandwidth can be less than 2%. Here, if the calculation is carried out according to linear sound field conditions, the system bandwidth in the same case is not less than 130 MHz, which obviously differs considerably from the actual situation and causes a large measurement error.

This study focuses on the problem that the bandwidth solution of the interference system in HIFU sound pressure measurement by laser interferometry depends on the linear sound field conditions, which causes a large difference between the solution results and reality. First, through theoretical analysis, a mathematical model of the interference signal for HIFU sound pressure measurement is established. The frequency domain of the interference signal is analyzed by numerical simulation. It is inferred that the spectrum of the interference signal under the condition of a nonlinear sound field is widened and shifted compared with that under the condition of a linear sound field. It is proved that existing methods for estimating the interference system bandwidth under the condition of a linear sound field are not suitable for HIFU sound pressure measurements. Combined with the HIFU sound pressure measurement data, the relationship between the bandwidth requirements of the heterodyne interference system and the peak value of the sound pressure is simulated and analyzed. The analysis results show that under the condition of a nonlinear sound field, the bandwidth requirements of the heterodyne interference system change via a quadratic law with the peak-to-peak value of the sound pressure.