The authors report families of generalized nondiffracting solutions of the free-space scalar wave equation, and specifically, a subset of these nondiffracting solutions, which are called X waves. These nondiffracting X waves can be almost exactly realized over a finite depth of field with finite apertures and by either broadband or bandlimited radiators. With a 25-mm diameter planar radiator, a zeroth-order broadband X wave will have about 2.5-mm lateral and 0.17-mm axial -6-dB beam widths with a -6-dB depth of field of about 171 mm. A zeroth-order bandlimited X wave was produced and measured in water by a 10 element, 50-mm diameter, 2.5-MHz PZT ceramic/polymer composite J/sub 0/ Bessel nondiffracting annular array transducer with -6-dB lateral and axial beam widths of about 4.7 mm and 0.65 mm, respectively, over a -6-dB depth of field of about 358 mm. Possible applications of X waves in acoustic imaging and electromagnetic energy transmission are discussed. >

Nondiffracting X waves-exact solutions to free-space scalar wave equation and their finite aperture realizations

New exact solutions of the homogeneous, free‐space wave equation are obtained. They originate from complex source points moving at a constant rate parallel to the real axis of propagation and, therefore, they maintain a Gaussian profile as they propagate. Finite energy pulses can be constructed from these Gaussian pulses by superposition.

Exact solutions of the wave equation with complex source locations

The propagation of acoustic waves in isotropic/homogeneous media and electromagnetic waves in free space is governed by the isotropic/homogeneous (or free space) scalar wave equation. A zeroth-order acoustic X wave (axially symmetric) was experimentally produced with an acoustic annular array transducer. The generalized expression includes a term for the frequency response of the system and parameters for varying depth of field versus beam width of the resulting family of beams. Excellent agreement between theoretical predictions and experiment was obtained. An X wave of finite aperture driven with realizable (causal, finite energy) pulses is found to travel with a large depth of field (nondiffracting length). >

Experimental verification of nondiffracting X waves

Diffraction-free optical beams propagate freely without change in shape and scale. Monochromatic beams that avoid diffractive spreading require two-dimensional transverse profiles and there are no corresponding solutions for profiles restricted to one transverse dimension. Here, we demonstrate that the temporal degree of freedom can be exploited to efficiently synthesize one-dimensional pulsed light sheets that propagate self-similarly in free space, with no need for nonlinearity or dispersion. By introducing programmable conical (hyperbolic, parabolic or elliptical) spectral correlations between the beam’s spatiotemporal degrees of freedom, a continuum of families of propagation-invariant light sheets is generated. The spectral loci of such beams are the reduced-dimensionality trajectories at the intersection of the light-cone with spatiotemporal spectral planes. Far from being exceptional, self-similar axial-propagation in free space is a generic feature of fields whose spatial and temporal degrees of freedom are tightly correlated. These ‘space–time’ light sheets can be useful in microscopy, nonlinear spectroscopy, and non-contact measurements. One-dimensional non-diffracting sheets of light are achieved without exploiting nonlinearity. Such light sheets may be exploited in microscopy and sensing applications.

Diffraction-free space–time light sheets

A new 2D (two-dimensional) and 3D (three-dimensional) pulse-echo imaging method (Fourier method) has been developed with limited diffraction beams. In this method, a plane wave pulse (broadband) is used to transmit and limited diffraction beams of different parameters are used to receive. Signals received are processed to obtain spatial Fourier transform of object functions and images are constructed with an inverse Fourier transform. Because only one transmission is required to construct images, this method may achieve a high frame rate (up to 3750 frames/s for biological soft tissues at a depth of 200 mm). To demonstrate the efficacy of the method, both 2D C-mode and 3D images have been simulated using conditions that are typical for medical ultrasound. Results show that images of high resolutions (about 6 wavelengths at 200 mm) and low sidelobes (around -60 dB) can be constructed over a large depth of interest (30 to 200 mm) with a 50 mm diameter aperture. Experiments with the new method have also been carried out. 2D B-mode images have been constructed with conventional linear arrays. In the experiment, an ATS 539 tissue-equivalent phantom and two linear arrays were used. The first array had a center frequency of 2.25 MHz, dimension of 18.288 mm/spl times//spl times/12.192 mm, and 48 elements. The second had a center frequency of 2.5 MHz, 38.4 mm/spl times/10 mm in dimension, and 64 elements. Images of different fields of views were constructed from RF data acquired with these arrays using both the new and conventional dynamic focusing (delay-and-sum) methods. Results show that qualities of images constructed are almost identical with the two methods in terms of sidelobes, contrast, and lateral and axial resolutions. Phase aberration has also been assessed for the two methods, and results show that its influence is about the same on both methods. In addition, a practical imaging system to implement the new method is suggested and potential applications of the method are discussed.

2D and 3D high frame rate imaging with limited diffraction beams

This paper presents mathematical formulations for new, three‐dimensional, packet‐like solutions to the free‐space homogeneous Maxwell’s equations. These solutions are real, nonsingular, continuous functions which propagate in a straight line at light velocity. They remain focused for all time. The asymptotic behavior of the fields away from the moving pulse center has a magnitude which decreases as the inverse of the distance from the pulse centers.

James Neill Brittingham

Papers

Focus waves modes in homogeneous Maxwell’s equations: Transverse electric mode