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The focal point of a high-numerical-aperture (NA) ultrasonic transducer can be used as a virtual point detector. This virtual point detector detects omnidirectionally over a wide acceptance angle. It also combines a large active transducer surface and a small effective virtual detector size, thus the sensitivity is high compared with that of a real point detector, and the aperture effect is small compared with that of a finite size transducer. We present two kinds of high-NA-based virtual point detectors and their successful application in photoacoustic tomography (PAT). They can also be applied in other ultrasound-related fields.
Photoacoustic (PA) tomography (PAT) is a promising non-ionizing, non-invasive biomedical imaging modality [1, 2]. It combines sensitive optical contrast and high ultrasonic resolution, and has been successfully applied in imaging both small animal and human tissues [3–5]. However, the ultrasonics transducer’s finite aperture limits the image resolution , and due to its limited acceptance angle, also limits the field of view of the detection system. In practice, point-like ultrasonic detectors have never been used in PAT because of its poor sensitivity, caused by their high thermal noise level.
Previously, a ring-based virtual point detector was introduced for PAT , which works like an ideal point detector with high sensitivity. However, the ring-based virtual point detector has a limited imaging region, and it also requires that the object be smaller than the ring. Another virtual point detector method, using a positively focused ultrasonic transducer, was previously introduced in  for photoacoustic microscopy (PAM), and its possible application in PAT was also numerically studied. In this letter, we extend the second method to include negatively focused transducers, and we are able to construct NA = 1 transducers to demonstrate their power to be used as virtual point detectors.
The radiation pattern of a focused transducer, at a certain frequency f, is shown in Fig. 1. If the transducer’s active surface is a part of a spherical shell with a radius of R, the focal point F is at the sphere’s center. Owing to diffraction, emitted ultrasound waves are not focused to an infinitesimal point at F; however, they are focused to a small region with a finite beam diameter (BD). From there, the waves diverge into a region limited by the beam spread angle θb. From the principle of reciprocity, the radiation pattern also describes the detection pattern.
The beam diameter describes how narrowly the ultrasound can be focused at the focal point, and it is determined by the numerical aperture and wavelength. In this letter, we use the full width at half maximum (FWHM) of the field magnitude, where the field magnitude at the beam boundary is half of that at the focal point, to determine the beam diameter. When the wavelength is much less than the size of the transducer, the amplitude of the emitted field at location P in Fig. 2(a), Δl away from the focal point F, can be approximately calculated by Rayleigh integral as 
where θ, are the polar and azimuthal angles, respectively, J0 is the Bessel function of the first kind, k = 2π/λ, and λ is wavelength. By numerically solving the solution of |A(Δl)| = |A(0)|/2(“|···|” represents absolute value), we obtained the relation between BD and NA, as seen in Fig. 2(b). This relation can be approximately fitted by function
Eq. (2) tells us that the beam diameter decreases as the transducer’s NA increases. To achieve a “point-like” virtual detector by using the focal point of a positively focused transducer, high-NA transducers are required. In addition, by using a high-NA transducer, the effective size of the virtual point detector is less than the wavelength, leading to a wide omnidirectional detection angle θb over a broad bandwidth.
Similarly, a negatively focused ultrasonic transducer can also be used as a virtual point detector. It differs from a positively focused transducer in that the virtual point detector is located behind the real detection surface. Figure 3(a) represents a virtual point detector by using a negatively focused transducer, and (b) is a positively focused counterpart.
The image reconstruction method used with the virtual point detector is almost the same as the one used with real point detectors, except for minor modifications in the signal time delay, which is due to the spatial difference between the locations of the virtual point detector and the transducer’s active surface. The modifications are, respectively,
where t is the time delay used in image reconstructions, td is the real detection time, R is the radius of the positively or negatively focused ultrasonic transducer, and c is the acoustic speed.
A metallized polyvinylidene fluoride (PVDF) film (from Measurement Specialties, Inc.), with a thickness of 110 μm, was used to construct the transducer. Because the flat PVDF film was not suitable to be stretched into a spherical shell, we constructed 2D transducers by bending strips of the film into the half circular shapes seen in Fig. 3. These 2D transducers have NA= 1. The PVDF films of the negatively focused and positively focused transducers were all 6.0 mm in width, and their radii were 12 mm and 13 mm, respectively. The PVDF films were glued to acrylic plastic surfaces, and the estimated center frequency was around 6.0 MHz. To compare with flat transducers, we further constructed a square flat PVDF transducer, 5.0 mm by 5.0 mm.
According to this construction and the center frequency, the Rayleigh integral was used to calculate the focal width , i.e., the effective virtual point detector size, to be around 120 μm, about half of the wavelength of center frequency. This sub-wavelength virtual detector also has a very broad uniform acceptance angle (close to π/2), which is another key characteristic of ideal point detectors.
Tissue phantoms were imaged by these virtual point detectors in PAT. As shown in Fig. 4, the PA source contained three black human hair crosses glued on top of optical fibers, with an interval between the hair samples of about 1.0 cm. A Nd:YAG laser (Brilliant B, Quantel) generated 6.5 ns, 532 nm laser pulses with a repetition rate of 10 Hz, which were diverged by a ground glass, so that all three hair crosses were illuminated. Both the phantom and the transducer were immersed in a tank filled with water. The virtual point detectors evenly scanned the object along a horizontal circle, stopping at 240 points, and the signals were averaged of 20 times at each stop. The PA signals were first amplified 40 dB by a pre-amplifier, and then were recorded by an oscilloscope (Tektronix TDS640A) with a sampling rate of 50 MHz. Finally, the recorded signals were sent to a PC for image reconstruction.
The hair phantoms were scanned by the two virtual point detectors and a flat transducer. Fig. 5(a) shows the result of scanning with a positively focused transducer; the scanning radius relative to its focal point was about 2.3 cm. Fig. 5(b) shows the result of scanning with a negatively focused transducer; the scanning radius relative to its focal point was about 4.2 cm. Fig. 5(c) shows the result of scanning with a flat transducer; the scanning radius was about 3.0 cm.
Although these three experiments had comparably short scanning distances, the results were clearly different. Both images in Fig. 5(a, b), which were reconstructed from data obtained by two virtual point detectors, presented complete reconstructions with a high and uniform signal-to-noise-ratio (SNR). However in Fig. 5(c), owing to the aperture effect and the limited acceptance angle, the image not only was blurred at the ends of each hair, but also almost completely failed to detect the horizontal hairs on the left and right phantoms.
To alleviate the image blurring and incompleteness, the finite size flat transducer had to be put much further away from the scanning center, at the expense of signal strength. Fig. 5(d) shows the result for the flat transducer at 8.8 cm from the center, a distance that yielded a lower SNR than in Fig. 5(c). Even at this long scanning distance, the images of the left and right horizontal hairs were still partially blurred in comparison with Fig. 5(a,b). Thus, this phantom experiment demonstrated the advantages of the virtual point detectors over the flat transducer.
In summary, both theoretical analysis and phantom experiments demonstrate that high-NA-based virtual point detectors offer significant improvements over a flat transducer in PAT. Without losing sensitivity relative to flat transducers, virtual point detectors have a negligible aperture effect and a wide acceptance angle, so they can be put close to the PA source, where the signals are strong, maximizing the SNR. Compared with the ring-based virtual point detector , the high-NA-based virtual point detector can also be used as like any other kind of single ultrasonic transducer to image arbitrarily sized and shaped objects. Moreover, they can also be used to build an ultrasonic transducer array. In addition to PAT, this idea can also be used in microwave-induced thermoacoustic tomography (TAT), where the laser is replaced with a microwave source. These virtual point detectors can also be implemented in other tomography methods or technologies that use ultrasonic detectors, such as ultrasonic tomography.
We thank Dr. Xinmai Yang for helping us manufacture the detector. This project was sponsored in part by National Institutes of Health Grant Nos. R01 NS46214 and R01 EB000712.