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Previous phased array research using anti-phase sources has shown that the phase response to an object scanned through a heavily scattering medium provides information about object position but none about size. In this paper it is demonstrated that controlling the relative phase between the sources enables different phase gradients to be set within the medium. The consequence of this is that the phase response is dependent on the size of an object whilst still maintaining localization information. Furthermore, it is demonstrated that the phase response can be tuned to be most sensitive to the object size under investigation.
©2000 Optical Society of America
1. Introduction
Frequency domain optical instrumentation [1,2] is becoming more widely used for characterizing and imaging heavily scattering media such as body tissue. An alternative to single source frequency domain systems is to use interfering diffuse photon density waves in a phased array configuration [3–9]. This has been demonstrated to be a highly sensitive method of detecting and localizing objects, producing large phase changes in the presence of inhomogeneities. The phased array performs a differential measurement and so has the additional advantage over a single source system of being less sensitive to movement artefacts. Furthermore, it has been shown [6] that if the sources are generated from a single source then noise due to amplitude fluctuations, phase drift and phase jitter are also eliminated. It has also been suggested [7,8] that varying the relative amplitude or phase between the two sources will enable the null plane to be scanned without physically moving the sources.
The sources are conventionally modulated in anti-phase to produce a null plane equidistant to the sources (fig. 1). Several researchers [3,4,6] have demonstrated that the response to an object scanned through the medium is always 180° regardless of object size. A typical response is shown in fig. 2. As the object is scanned through the medium the imbalance increases resulting in an increase in the amplitude signal. When the object is at the null plane balance is restored and there is again no amplitude signal. The 180° phase transition that occurs when the object crosses the null plane can be used to accurately locate the object position but provides no information about the size. This paper is motivated by the research of Chance and co-workers [7–9] in which anti-phase modulated sources are used to obtain a phase response that depends on both object position and size. This system has provided useful results in functional brain imaging experiments [9] where a sensitive phase response is obtained for different stimuli. The signals obtained provide information about the location of the blood flow change and are also proportional to the size of the change. It is suggested that the sensitivity is obtained by operating the system on the steep phase gradient of the transition. However, this is not possible if the sources are modulated in anti-phase as adding two anti-phase vectors can only produce a 0° or 180° resultant. In this paper it is demonstrated that this response is due to the fact that the sources are not modulated in perfect anti-phase, but at a slightly different relative phase. Furthermore, we demonstrate that if the relative phase is controlled then the phase response to different objects can be controlled. For example, the phase response can be made most sensitive to small objects by operating at a relative phase difference close to 180° whereas for larger objects it is more appropriate to use a smaller relative phase difference.
These effects are investigated both experimentally and theoretically, using an approximate diffusion model. The paper is set out as follows; the next section describes the experimental set up and section 3 briefly describes the model used. Results are presented in section 4 with discussions and conclusions following in section 5.
Fig. 1. Typical phased array system. Two sources modulated in anti-phase destructively interfere at the null plane. The system is highly sensitive to any imbalance in the medium.
Fig. 2. Typical amplitude and phase response to an object scanned through the phased array system. Relative phase difference between the sources=180°.
2. Experimental configuration
The phased array system is shown in fig. 3. Two laser diodes (λ=670nm, optical power=5mW) are modulated in anti-phase at a frequency of 100MHz using a signal generator and a 0&180° power splitter. The relative phase between the sources is controlled with different lengths of coaxial cable. Light is incident on a scattering medium at a source separation S and detection is performed using a photomultiplier tube positioned equidistant to the sources. The detected RF signal is amplified, mixed down and fed to a lock-in amplifier where it is compared to a reference signal, which has not passed through the medium. In all experiments the medium is a cuvette (h=100mm, w=200mm, d=40mm) filled with a solution of polystyrene microspheres (diameter=1.6µm) and absorbing dye to obtain a medium with optical properties of µ′s=0.73mm-1 and µa=0.01mm-1 which is comparable to that of body tissue [10]. Different sized totally absorbing rods (width range 1–30mm) are scanned through the mid plane of the medium and the amplitude and phase response is measured at a source separation S=20mm, for relative phases of 178°, 172°, 163°, 153°, 130°, 80° and 50°. Single source measurements are also included for comparison.
Fig. 3. Two sources are modulated with a relative phase difference set by varying the length of the coaxial drive cable. The detected signal is amplified and fed to a lock-in amplifier.
3. Model
The model has been described previously [11,12] and so only a brief description will be provided here. It is based on a two-stage propagation of the Green’s functions of the diffusion approximation (fig. 4). The diffusive wave is calculated at each position on the object plane using Green’s function G1, is perturbed by the object and then propagates onto the detector using Green’s function G2. An approximation is made, which is analogous to classical diffraction theory, that the object plane distribution is only affected at the location of the object. In all other positions the distribution is the same as if no object were present. Qualitatively useful results are obtained, predicting the trends of the phased array experiments for different system configurations.
Fig. 4. Two-stage propagation model. The object plane distribution due to the two sources is calculated using Green’s function G1. The object plane distribution is perturbed by the object and then propagates to the detector plane using Green’s function G2.
4. Results
The system response for anti-phase modulation has been shown in fig. 2. The amplitude response is dependent on the object size but the phase transition observed when the object passes through the null plane is always 180° regardless of object size.
A set of phase responses to the scan of a 30mm wide absorbing rod is shown in fig. 5 for different relative phases between the sources. As the object is scanned through the medium it causes an imbalance in the system, which creates a change in the detected phase. It can be observed that a larger relative phase difference between the sources results in a larger phase response with a steeper gradient. Notice also in fig. 5a and b that as the object introduces more imbalance the phase response cannot increase indefinitely but effectively ‘saturates’ at a value equal to the relative phase difference between the sources. Fig. 5e demonstrates the relatively small phase response of a single source system.
Fig. 5. Typical phase responses to a scanned object. Object size=30mm, relative phase between sources=a) 178°, b) 153°, c) 130° d) 80° e) 50° f) single source.
Fig. 6. Phase shift of linescans for a range of object sizes and relative phases between sources (a) modelled (b) experiment.
Fig. 6a demonstrates the phase shift of the response for a range of object sizes and relative phases between the sources obtained from the model. For each relative phase a set of different sized objects are scanned through the medium and the maximum phase shift across the scan measured. As expected, when the sources are in perfect anti-phase the response is large, but insensitive to object size. For the single source system the phase response is small and relatively insensitive to object size. The phase shift due to a given object is always larger for bigger relative phase differences between the sources. However, to obtain the maximum sensitivity to a particular object size the system should be configured to operate in the region of steepest gradient. For small objects (width<7mm) a relative phase of greater than 170° provides the highest sensitivity. However, as the object size increases the object sensitivity decreases i.e. the phase effectively ‘saturates’ as discussed earlier. For larger objects a smaller relative phase (e.g. 130°) has a steeper gradient and provides the greatest sensitivity to object size.
The experimental results (fig. 6b) demonstrate similar trends. The single source results show small phase shifts and a relative insensitivity to object size. The magnitude of the phase shift increases with increasing relative phase shifts. The maximum object sensitivity for small objects is obtained with a 163° relative phase and for larger objects greater sensitivity is observed when the relative phase is set to 130°. The response of phases >170° demonstrate large phase shifts but do not show the same sensitivity to object size predicted by the model. This is due to the approximations of the model discussed in section 3 and for objects smaller than 1mm it is expected that the maximum sensitivity will be obtained for these phases.
5. Discussion and conclusions
It has been demonstrated in this paper that by replacing the anti-phase sources conventionally used in phased array systems with a controlled relative phase difference the phase response of the system can be controlled. When anti-phase sources are used the phase transition is insensitive to object size. Controlling the phase response has the advantage of maintaining the sensitivity of the system while obtaining additional information about the object size or functional response. Relative phase differences above 160° offer the greatest sensitivity to small objects. However, these phase responses ‘saturate’ as the object size increases and it is more appropriate to use a smaller relative difference to maximise the sensitivity to object size. This has applications in functional brain imaging where one is not only interested in locating the blood flow change but also the magnitude of the change. For small blood flow changes the maximum sensitivity should be set with a large relative phase difference, whereas larger changes will require a smaller relative phase difference.
A simple way of explaining the responses observed in this paper is to consider the detector plane phase distribution. In the presence of an object the detector plane distribution shifts in the direction of the object a distance Δx, which depends on the size of the object1. When the sources are perfectly in anti-phase the phase gradient is infinitely steep so it is not possible to make measurements on this slope and the phase has the value of 0 or 180° only. The detector plane phase distribution for a range of different relative phase shifts are shown in fig. 7 (modeled data) with and without a 10mm wide object situated at the mid-plane of the medium, 10mm off axis (source separation=20mm). In all cases, when the object is inserted into the medium the imbalance introduced causes the detector plane distribution to move in the direction of the object. A detector at the null plane measures the change in phase due to the shift of the detector plane distribution. For example, when the relative phase is set to 178° (fig. 7) the phase gradient is steep. For small objects the displacement Δx is small so sensitive measurements can be made on the steep slope. However as the object size increases the detector plane distribution shifts a sufficient amount that the measurement is no longer made on the slope and the phase response has saturated. In this situation it is better to set a phase gradient that is sensitive over a larger range of Δx e.g. 153° (fig. 7).
When using diffusive wave interference to increase phase sensitivity one drawback is the reduction in signal to noise ratio of the amplitude response. We have observed (unpublished data) that the maximum change in the amplitude response during an object scan is obtained with the single source arrangement, approximately a factor of two greater than the response from the phased array configuration. Clearly there is a trade off between choosing the optimum phase response and obtaining the maximum amplitude signal. However, due to tissue surface effects amplitude signals are generally considered as a less reliable measurement when characterizing tissue. Therefore provided there is sufficient amplitude for detecting the signal it is desirable to optimize the phase response.
Noise in phased array systems is also an important consideration. The high degree of sensitivity means that the system is also sensitive to noise. Phase noise caused by the detection electronics becomes insignificant when the system is configured to introduce large phase changes i.e. at a relative phase close to 180°. Phase drift between the sources will cause the phase gradient to change, although the drift problem is reduced over the single source case as the lasers suffer similar environmental changes. Amplitude noise and phase jitter also cause phase noise that increases as the relative phase increases. However we have demonstrated previously [6] that if the two sources are generated from a single source then these sources of noise can be virtually eliminated. Thus, it is possible to increase the sensitivity of the phased array without increasing the phase noise.
Fig. 7. Phase distribution at the detector plane for relative phase=178°, 153°, 130°, 80°, 50° and single source. The distribution becomes shallower and wider as the relative phase between the sources decreases. The two curves shown are with (red) and without (black) the presence of a 10mm wide object situated at the mid- plane of the medium, 10mm off axis.
Acknowledgements
The authors would like to thank the EPSRC (U.K.) and the Royal Society for support.
Footnotes
| 1 | There are also small changes in the shape of the detector plane distribution but this does not affect the trends in the results. |
References and links
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