Sharpness of range focus in the Fresnel region for a linear aperture

Large aperture arrays, designed for greater angular resolution and beamforming gain, require focusing in the Fresnel region for some applications in acoustics, seismics and electromagnetics. An important characteristic is the sharpness of the range-response around the focal range. It can be quantified by the "M-dB depth of field" defined as the range window around the focal range for which the loss magnitude is less than M decibels. Similarly, "focal range mismatch loss" is defined as the number of decibels the response is below its peak at the focal range. The response of a focused linear aperture vs range can be expressed as a function of Fresnel integrals. This paper introduces a simple and accurate approximation that quantifies the range response around the focal range to an accuracy of a small fraction of 1 dB, down to the -10 dB level. The non-linear function of Fresnel integrals is replaced by a simple algebraic expression that relates performance measures, focal range mismatch loss, and depth of field, to the physical parameters, array length, wavelength, focal range, and direction. Results, presented in terms of dimensionless ratios of physical quantities, are shown to be valid for all steering angles, not limited to broadside.