Acquisition of the transfer matrix with the FDORT method. It differs from the conventional DORT method by the use of focused transmit. The element of the matrix is the signal received by element after the focused beam has been transmitted.
[(a) and (c)] Wave front resulting from a focused transmit, before (c) and after (a) it reaches the focal point . For , the wave is propagating inward, converging toward . For , the wave is propagating outward, diverging from . An observer located in the sound cone hears the wave front as if it were coming from a virtual transducer located at . (b) Wave front for an equivalent virtual transducer (right) located at , with a directivity angle , transmitting a pulse at . For the observer , fields in (a) and (b) are very similar. (d) Wave front for an equivalent virtual transducer (right) located at and facing the array. For the observer , the field in (c) appears similar to the time-reversed version of the field in (d).
Left: each virtual transducer has a directivity function corresponding to the sound cone of the beam. Therefore, only the scatterer inside the cone (scatterer S1) will hear and be heard by the virtual transducer. The scatterer S2 does not hear the virtual transducer. Right: because of this directivity effect, a scatterer hears only a limited number of virtual transducer. The virtual array that insonifies a given point has then a limited size, which limits the resolution of the FDORT method with linear scan. It can be seen geometrically that the angle between the scatterer and the virtual array is the same as the angle of the beams. Therefore, the resolution for any point in the medium, independent of its depth, is the same and is equal to the resolution of the focus, . The angle that would be obtained with the full aperture is shown in dash line.
Example of sector scan. The foci are located on a circle, and the beams are steered. The array of the three virtual transducers insonifiing the scatterer is shown. In this geometry, the virtual array sees the scatterer with the same angle as the full physical array. The resolution depends then only on the size of the physical array.
Temporal singular vectors expressed in the real array. They correspond to each scatterer’s wave front.
Temporal singular vectors expressed in the virtual array at depth [(a) and (b)], [(c) and (d)], and [(e) and (f)]. The scatterers were located at . The temporal singular vectors are obtained by the Fourier transform of the frequency domain vectors . [(a) and (b)] The wave fronts are truncated on the edges compared to figure because of the directivity of the virtual transducers. The scatterers do not hear the virtual transducers on the edges. [(c) and (d)] The virtual array is in the scatterer plane. In this plane, the virtual transducers, or transmit beams, are sinc functions, with a narrow main lobe, and only a few insonify significantly the scatterer. [(e) and (f)] The curvature of the wave front and of the phase are inversed compared to (a) and (b), because the scatterer is now shallower than the focal depth, and as explained in Sec. ???, the signals are time reversed in this case.
Simulation setup. A virtual transducer is shown, with its directivity pattern in dashed line.
Top: Temporal Green’s function of two scatterers (respectvely and ) in the physical array. Because of the far-field phase screen, the signals are distorted in time, and there are important variations of amplitude across the array. Moreover, the effect of the aberration depends on the position of the scatterer. For example, the amplitude peaks are at different position for each scatterer. Bottom: Same in virtual array, located just behind the phase screen. The wave fronts are merely delayed by the aberration. There is no distortion of the pulse or variation of amplitude. For each scatterer, the signal is received only by about 80 virtual elements, because of the directivity of the virtual transducers.
Focusing from the physical array, respectively, from left to right: and . Green’s function for the scatterer at was used to correct the aberration. Therefore, the focusing is perfect at . However, it degrades very rapidly, as the aberration varies with the position.
Focusing from the virtual array at, respectively, from left to right, and . Reference Green’s function was at in the first case, and and in the second case.
Estimated delay profile of the phase screen (circles) compared to the true delay profile (solid line).
To compute Green’s function of any point in the medium in the physical array, one merely has to add the wave front corresponding to each virtual transducer that insonify the point with the proper time delays. The virtual transducers that do not insonify the point (because of the directivity) are not summed.
Temporal Green’s function of the point at , measured (left) and synthesized using the method described in this paper (right).
From left to right: image of an unaberrated point-scatterer phantom, image of the same phantom with a rms phase screen located at ; image of the aberrated phantom corrected using the FDORT method. The corrected image is not perfect but much better than the aberrated image.
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