Ultrasound Physics

Ultrasound Beamformation

Interference phenomena: Sound is an acoustic wave, following the superposition principle. Acoustic wave cause pressure at each location of the medium vary with time, and thus the particles to vibrate with time. If the pressure or the displacement of the particle is recorded, in most cases, it is a sinusoidal function: y = A sin (2π f t + Ф ), A is the amplitude, f is frequency and Ф is the initial phase. Inside the acoustic medium, or the acoustic field, Ф varied with location, and amplitude and frequency are the same if we don’t consider the wave spread and attenuation. If two pressure waves propagate to a single point, the final pressure at the point will be arithmetic summation of these two: y = A₁ sin (2π f₁ t + Ф₁ ) + A₂ sin (2π f₂ t + Ф₂ ). Depends on frequency and initial phase, the combination can be stronger or weak compared to the incident waves, and can even become zero if the two have the same amplitude and frequency, but opposite initial phase. If the wave sources and acoustic field are fixed, the initial phase for each location will be fixed too, and thus vibration at some points will be enhanced while other points may be weakened consistently.
Huygen's principle: This principle of wave analysis, proposed by the physicist Christiaan Huygens (1629-1695), basically states that: Every point of a wave front may be considered the source of secondary wavelets that spread out in all directions with a speed equal to the speed of propagation of the waves.
Aperture size and wavelength: The aperture is the active area that transmits or receives acoustic wave at certain moment. For a single-element transducer, the aperture size is the transducer element size. For array transducer, the aperture are all the elements that works together simultaneously. To achieve a confined beam, the aperture size need to be much larger compared to the sound wave length. At 5MHz, the ultrasound wavelength is about 0.3mm in water, and a 5mm diameter transducer will give a decent beam. However, at normal sound frequency such as 1kHz, the wavelength is about 0.3m, it need a 5m diameter speaker to give a sound beam that propagate forward. Since most speakers are small compared to the sound wavelength, and they behave like a point source, with sound spread all the directions.
Beam field from a piston aperture: The most simple transducer shape is a piston transducer. The beam from a piston transducer is similar to a flash light beam.

Acoustic pressure along central axis have many maximums and minimums and from the last peak, it goes down monotonously.
Cross section view of the beam at different depth vary with depth.
Longitudinal section view view
Main lobe and side lobes

Near field and farfield:

At each sound field point location, the acoustic pressure is the summation of contributions from each point at transducer surface. When aperture size is much bigger than the wavelength, the points locations within the transducer area and close to the center see an unlimited aperture, at same depth, will receive the same amount of acoustic contribution from the nearly unlimited transducer surface, and thus ultrasound wave behaves like plane wave. However, the locations close to the edge still see the limited aperture, and thus the plane wave area is smaller than the aperture area. Moving away from transducer, this plane wave zone decreases quickly.
For a point at the central axis of the aperture surface, the biggest time difference for sound to travel from different points on aperture surface to it is from aperture center point compared from aperture edge point. This time difference vary lot at distance close to aperture surface, and acoustic pressure will become maximum minimum alternatively. At certain depth it became one wavelength, and from there it slowly decrease to infinitesimal when depth goes to infinity, and accordingly the acoustic pressure will decrease monotoneuosly.
The acoustic field before this depth is called near field, and beyond this depth is called farfield. Since acoustic intensity is unpredictable in near field, and strictly speaking, it should be avoid to use it for echo information. However, for imaging ultrasound, since it is wide bandwidth, the acoustic intensity is also uniform in near field, and thus near field is not so serious.

Beamwidth: Beam width is usually calculated from the cross-sectional or longitudinal section acoustic field view, and it is a parameter related with dB level. On cross-section view, draw a line through the center, or on longitudinal section view, draw a line at certain depth perpendicular to the central axis, a 1-D acoustic profile is obtained. On this profile, -xdB level bean width is the distance between the two points that have this dB level intensity. Beam width can also be represented in angle. At certain distance, normally the focal or natural focal depth, draw a half circle center at the aperture center, and along this half circle, a 1D acoustic profile can be obtained. One this 1-D profile, the horizontal axis is angle from -90 to +90 degree, the beam width will be the angle difference between the two dB level points.
Natural focus: ultrasound beam from a flat aperture will get narrow and then spread out within and angle range. The depth where beam is most narrow is the natural focus of the aperture.