1. Introduction

Detection and imaging of small heterogeneities or functional changes in biological tissues such as breast cancer detection or functional brain imaging has attracted interest from physical and biomedical research fields for decades^1,2. Diffuse Optical Spectroscopy and Imaging (DOS and DOI), due to their sensitivity to intrinsic contrast (i.e. oxy- and deoxy-hemoglobin) and extrinsic contrast (i.e. Indocyanine Green (ICG) or other fluorophores) in the near-infrared region (700–900 nm), have been developed rapidly in the clinical and research fields of biomedicine as potential complements to other imaging modalities such as MRI, PET, X-ray CT and Ultrasound^3–8.

In DOS and DOI, frequency domain techniques can provide highly portable and economic systems for clinical usage^9,10. The sensitivity or detection limit of these optical systems is one of the most interesting topics for research and discussion, as well as in the evaluation of the system performance^11,34. One of the promising methods to achieve high sensitivity in detecting small objects embedded in the scattering medium is the use of dual interfering sources in a phased array configuration. Previous studies have explored this issue experimentally and theoretically^12–18. The experimental results reported that the phased array (dual-source) system can detect 20 pmol ICG in a 3-mm diameter tube in the scattering medium¹³ or 1 mm displacement of a small object embedded 10 mm deep¹⁶. The analytical solution of the dual-source system can be obtained by applying the summation theorem to the single source system because of the linearity of this problem^14,27, and can be extended to tomographic image reconstruction²⁹. The phased array system has been applied to breast tumor phantom²⁰, functional brain imaging^23,24 and fluorescence detection^21,22 experimentally. Many investigators have also compared the sensitivity of single- and dual-source systems^15–19. For example, Erickson et al. demonstrated that a dual-source system could provide higher detection sensitivity than a single-source configuration from numeric simulations¹⁷, while Papaioannou et al. observed a comparable sensitivity for an optically scanned phased array system and a continuous wave system¹⁸, which suggests that this topic needs further detailed analysis.

Noise is an important factor to consider in detection with dual interfering sources since this technique is highly sensitive to small perturbations. Especially when the detector is placed in the amplitude null position, the noise from the detector will affect the detection accuracy²⁵. For DOS and DOI, the data measurements are usually made in the near field region (within one wavelength of diffuse photon density wave, which is around 10 cm for most physiological conditions¹³), thus the signal-to-noise ratio will determine the detection resolution¹¹. Several investigators have analyzed the detection sensitivities for absorption, scattering and fluorescence with different contrasts under single source-detector configuration in details^11,34, while similar studies for dual-source system have not been investigated yet.

In this paper, we compare the detection sensitivity of single- and dual-source systems with a similar experimental setup and object perturbation through analytical simulations. Our purpose is to better understand the relationship between the signals obtained from the dual-source system and those from the single-source system, and to optimize the experimental design based upon the combination of the advantages of both systems.

2. Methods

2.1 Interference of DPDW

When light enters into a turbid medium, the photons travel in the form of a random walk. The propagation of the photon can be modeled by the diffusion equation when the scattering effect is much more predominant than the absorption^26–28:

where U(r,t) is the photon fluence [W/cm²], v is the speed of light in the medium, µ _a is the absorption coefficient [cm^-1], µ’ _s is the reduced scattering coefficient [cm^-1], D is the diffusion coefficient ([cm^-1]) defined by D=[3µ’ _s ]^-1 and S(r,t) is the source term [W/m³]. When the source intensity is modulated by an rf sinusoidal wave, i.e., S(r,t)=δ(r)S{1+Ae-^iωt}, where S is the source strength, A is the modulation and ω is the modulation frequency. In the homogeneous medium, the ac component of Eq. (1) will be rewritten as

where k ²=(-vµ _a +iω)/D. This Helmholtz equation indicates that the macroscopic behavior of the photon propagation in highly scattering medium is actually the wave phenomena, hence U(r,t) is referred to as the diffuse photon-density wave (DPDW)^27,28.

The analytical solution of the diffusion equation with the presence of a spherical inhomogeneity has been derived by Boas et al ²⁷. The DPDW generated from two phased sources will interfere with each other. In the measurement with a dual-interfering-source system, the phased source includes two oscillating light sources with certain amplitude and phase relations. Theoretically, this problem can be modeled by the summation theorem^14,27. We can use the complex source term (ac part) to represent the pair of sources, i.e., A₁ exp(-iωt) and A₂ exp(-i(ωt+Δφ ₀ )), where Δφ ₀ is the phase offset between two sources. The total field is equal to the superposition of two independent solutions of those source terms based on Eq. (2).

2.2 Relationship between single-source and dual-interfering-source signals

Since the dual-interfering-source data is the summation of the DPDW (scalar wave) from the two phased sources, we can analyze the relationship between the signals coming from signal-source and dual-source schemes using the vector diagram (Fig. 1(a)). In the dual-source system, we sum up two vectors, each of which represents the signals obtained from each single-source measurement. As shown in Fig. 1(a), vectors $\overrightarrow{AB}$ and $\overrightarrow{\mathrm{BC}}$ represent the signals from two sources, S1 and S2 respectively, with a phase difference of (180°-α). The length of the vector represents the amplitude and the orientation indicates the relative phase. Usually there is some phase mismatch in the practical system so that the two sources are not exactly 180° out of phase (i.e., α=1°~2°). In Fig. 1(a), we exaggerate the phase deviation α for better visualization.

 

Fig. 1. (a) Vector diagram for summing up two single-source signals, AB and BC, and resulting in the dual-source signal, AC. When the DPDW from single-source has been perturbed, i.e., BC changes to BC’, then the result of dual-source also changes to AC’ (b) Phase shift (absolute value) and (c) Amplitude variation ratio (absolute value) for dual-source signal vs. the perturbation in single-source signal, with different phase offset (180°-α).

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In the homogeneous background, the amplitudes from equidistant sources should be the same, i.e., $\mid \overrightarrow{AB}\mid =\mid \overrightarrow{BC}\mid ,$, so their sum will be the vector $\overrightarrow{AC}$. Assuming that there is a small absorbing object present in the optical path between S2 and the detector, the vector $\overrightarrow{BC}$ will change (supposing the phase change is negligible) to $\overrightarrow{B{C}^{\prime }},$, while $\overrightarrow{AB}$ remains the same (supposing the perturbation is small enough that the changes in $\overrightarrow{AB}$ can be neglected), the result will be that the sum of the vectors will also change from $\overrightarrow{AC}$ to $\overrightarrow{\mathrm{AC}}\prime$, which is measured by the amplitude variation $\overrightarrow{A{C}^{\prime }}-\overrightarrow{AC}$ and phase shift ϕ′. Applying the relations of trigonometry, we can express the phase shift ϕ′ and amplitude variation $\delta I=(\mid \overrightarrow{A{C}^{\prime }}\mid -\mid \overrightarrow{AC}\mid )$ by the amplitude from signal-source, $\text{M}(=|\overrightarrow{\text{BC}}|)$ and its variation $\delta \text{M}(=|\overrightarrow{\text{C}\text{'}\text{C}}|):$:

where Δ=δM/M. And for amplitude, from

Here δI/δM stands for the ratio of change in the dual-source amplitude to the variation in single-source amplitude. From the equations above, we can analyze how the phase shift ϕ’ and amplitude change δI in the dual-source scheme is related to the amplitude change in a single source-detector channel, δM, with different perturbation ratio Δ (=δM/M).

Figs. 1(b) and (c) plot the relations between the dual-source signal (phase and amplitude) variations and the perturbation ratio in single-source, under different phase offsets (180° - α). Fig. 1(b) shows the response of the dual-source phase shift to different perturbations. For instance, in the case of phase difference of 178° (α=2°), we see that the phase shift increases with the increase in perturbation ratio δM/M, and the rate of increase is very rapid especially when the perturbation is small, resulting in a 30° phase shift on a 2% perturbation, while the rate of increase slows down when the perturbation gets bigger. This trend agrees with the experiment and simulation results³⁷. Also, if we vary the phase offset α, the sensitivity of the dual-source phase shift will change. When the phase difference is close to 180° (perfect cancellation), for example, α=0.5°, there will be a large response to a small perturbation (70° for 2% perturbation), while it asymptotically approaches 85° under a larger perturbation. In this case, the system is very sensitive to the presence of small perturbations, but it could not discriminate the intensity difference for larger perturbations. On the contrary, if the phase offset is larger (α=5°), the response of the dual-source phase shift is smaller (only 10° for 2% perturbation) compared with the case of 179.5°, but the sensitivity to perturbation is more evenly distributed so as to be able to indicate the intensity of perturbation. The result suggests that by adjusting the phase difference offset, we can change the system’s sensitivity to probing different perturbation intensities. This trend agrees with the experimental results reported by Morgan et al ³³. And from Fig. 1(c), we can see that the dual-source amplitude behaves similarly in response to different perturbations. While a slight difference is that the zero phase shift will occur only when the perturbation δM/M=0, but the amplitude variation of zero will happens at two positions, corresponding to the vectors $\overrightarrow{AC}$ and $\overrightarrow{A{C}^{\prime }}$ in Fig. 1 (a) where $\overrightarrow{AC}=\overrightarrow{A{C}^{\prime }}$. This can be seen clearly in the green plot of Fig. 1 (c) when α is large.

2.3 Noise Model

In the detection of photons using a single-source system, the noise level will determine the detection threshold. There are some relevant noise sources in experimental and clinical situations. Shot noise is the dominant factor for an ideal experiment system, which is due to the randomness in photon multiplication and the fluctuation of the dark current, and is related to the square root of the number of photons detected. For a clinically relevant system (1 Hz bandwidth), the estimated shot noise level is about 0.1% in amplitude and 0.05° in phase¹¹. For the input light power (~3 mW) used clinically and the experimental temperature, the shot noise is the major source from the ideal electronic circuit³⁵, and is determined mostly by the detector. The signal current from a photon detector is:

where η is the quantum efficiency of the detector, q is the elementary charge, hν is the energy of a single photon, U is the photon fluence given by Eq. (1), R is the detecting area (cm²) and G is the internal gain of the detector.

The shot noise can be expressed as³⁸:

where i_shot is the shot noise current from the signal isig, A is the modulation of the source, B is the system bandwidth (we choose 1 Hz).

Thermal noise and other signal-independent noise can be approximated by the Noise Equivalent Power of the detection system^35,38. The expression for signal-independent noise is:

where K is photoelectric conversion efficiency and B is the system bandwidth.

When put together, the noise from the photoelectric measurement in fractional amplitude can be expressed as:

While in practice, there are other sources for the noise, such as the variation in source amplitude due to the fluctuation in RF power and laser light intensity, and the position error during the scanning of source and detector fibers. Here we estimate those effects as random error N₂=0.5% from our experimental calibration.

Phase noise is composed of the amplitude independent part (phase noise floor) and the amplitude dependent part³⁸. In our system, the phase is detected by the heterodyne method³⁰, using a zero-crossing phase meter (Krohn-Hite Corp.)³⁷. The phase meter measures the phase angle by measuring the time ratios (Fig. 2)³⁹:

where T_T is the time interval between the positive going zero-crossing of V_R (reference signal) and V_T (measured signal), and T_R is the period of V_R.

 

Fig. 2. Phase measurement through zero-crossing time interval

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When the tested signal contains amplitude variation with a standard deviation ΔS from the ideal intensity S (the peak value of V_T), as shown in Fig.2, the variance of the amplitude will cause the shift of the zero-crossing point, and hence the phase reading. We can calculate the related phase standard deviation in terms of the fractional amplitude variation Ns=ΔS/S (see Appendix I):

From Eq. (10), the phase noise is inversely proportional to the signal-to-noise ratio when the signal-to-noise ratio is larger than 10, and increases more rapidly when the signal gets even smaller³⁸. When the signal is less than or equal to the noise level, the phase will vary randomly in a range of 180°. We verified the Eq. (10) with experiments (data not shown here). The standard deviation for total phase noise is³⁸:

where σ₀=0.05° is the phase noise floor according to the specifications.

2.4 Simulation Description

The geometry for the simulation model is illustrated in Fig. 3. The background medium is an infinite slab with a thickness of 5 cm, and its optical properties are chosen to mimic the human breast tissue (µ_a=0.05 cm^-1, µ’_s=10.0 cm^-1)³⁶. We considered two clinical relevant setups: transmission mode and remission mode. For the transmission mode, as shown in Fig. 3, the source and detector face each other in the single-source configuration, and in the dual-source configuration, two out-of-phase sources (0° and 179°) are separated by 2 cm and the detector is placed in the middle line of two sources. The reason we choose 179° is that if the two sources are exactly 180° out of phase, the signal in homogenous medium will be zero, so that the phase noise will be 180°, which will be too noisy for detection. By selecting 179° between the two sources, we bring the amplitude signal down about 40dB, but can still detect the signal. The modulation frequency is 50 MHz, which corresponds to our current apparatus³⁷. For the remission mode, sources and detector are placed on the same side of the slab, with a constant source-detector separation of 5 cm to ensure the depth sensitivity. The source separation for dual-source configuration is also 2 cm. In each case, we scan an absorbing object (with scattering being the same as the background) to create the perturbation. The image source-object pair technique is used, with the extrapolated zero boundary condition^31,32.

 

Fig. 3. Transmission and remission geometry for single- and dual-source configurations

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The forward simulation was performed using the photon migration imaging software (PMI) developed by D. A. Boas, M. A. O’Leary and Xingde Li at the University of Pennsylvania⁵. For both single-source and dual-source cases, we calculated the signal variation due to the scanning of the absorbing object as shown in Fig. 3. Then we compared the perturbation with the noise floor in the homogeneous background. If the perturbation in amplitude or phase is larger than the noise floor in the homogeneous background, then the presence of the object will be detected^11,34.

For single source, we calculate the noise level with:

and the phase noise from the circuit:

For the dual-source case, we first obtained the scattered waves from each single source (amplitude and phase), and then synthesized them by vector summation. From the noise-free data, we obtained the difference between with and without the presence of the object, which is the signal perturbation we would like to detect. The noise level for the homogeneous background in dual-source detection can be obtained through the standard deviation of the summation vector from single source data added with the noise from Eq.s (12) and (13) (see Fig. 4).

 

Fig. 4. Noise model for summation of two vectors. The vectors are the average signals and the dotted circles are the distribution of sampling values. The standard deviation of the green spots is the noise distribution for single-source. The standard deviation of the red spots gives the amplitude and phase noise of the dual-source measurement.

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Suppose we sample the signal from source a, the vector $\overrightarrow{A1}$ can be expressed by the normal distribution with the mean value of A1 (including amplitude and phase) and the amplitude standard deviation ${\mathrm{N}}_{\mathrm{s}-\mathrm{s}}^{\mathrm{a}}$ and the phase standard deviation ${\mathrm{\sigma }}_{\mathrm{s}-\mathrm{s}}^{\mathrm{a}}$:

as shown in Fig. 4. Similarly, for source b, we have:

Note that when we calculate the shot noise in (14) and (15), there is a √2 factor due to the increase of the dc component. If we sum them together, we will get the dual-source signal:

which can also be approximated as a normal distribution where N_d-s and σ_d-s represent the fractional amplitude and phase noise for the dual-source system that can be calculated if we have enough samplings.

For detection and location of the heterogeneity, we compare the signals (fractional amplitude [|Φ_hetero - Φ_homo|/Φ_homo] and phase [|Arg(Φ_hetero) - Arg(Φ_homo)|]) to the noise threshold from the homogeneous background. In other words, suppose we scan an object with different optical properties from the background, there will be some variations or perturbations in the signals (amplitude and phase deviate from the values in homogeneous case), and the perturbation will be related to the level of heterogeneity. The smaller the object’s optical properties deviate from the background, the less perturbation signals we can detect. If Δµ_a and Δµ’_s are small enough, the perturbation signals obtained from the detector will be less than the detection noise level, then the object is beyond our system’s detection limit that we could not detect the object.

3. Results

3.1 Noise Level for Single- and Dual-Source

Table 1 lists the noise level for single- and dual-source systems from both simulation and experiment. The experimental results are obtained from the system described previously³⁷ and the bandwidth is 1 Hz.

Table 1:. Noise Level for Single- and Dual-Source Systems

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Both the experiment and simulation results suggest that due to the increase of the sensitivity, the noise level of the dual-source system also becomes larger than single-source, which is 10- to 100-fold (see the Ratio column in Table 1). The ratio from the simulation is larger than the experiment because the cancellation between two single sources is better in the simulation, which will result in a smaller background signal, hence the larger percentage in amplitude variation and phase noise. For the experiments, the balance of two sources (equal amplitude and 179° phase difference) are not ideal due to the resolution of the RF attenuator and the slight difference between the two cables’ length, which reduces the sensitivity of the system.

In the object detection, the criterion we consider is the signal-to-noise ratio. We can calculate the signals in the dual-source condition and compare it with the noise threshold. If the SNR is larger than 1, we will be able to detect the object, otherwise the observation from the detector can not tell the difference from the homogeneous background.

3.2 Detecting the Object with Different Depth and Size

We study the detection sensitivity for single- and dual-source systems in both transmission and remission geometries (see Fig. 3). The background medium and object’s optical properties are kept the same: ${{\mu }_{\mathrm{a}}}^{\mathit{\text{Background}}}$=0.05 cm^-1, µ${’}_{\mathrm{s}}^{\mathit{\text{Background}}}$=10.0 cm^-1 and ${{\mu }_{\mathrm{a}}}^{\mathit{\text{Object}}}$=0.20 cm^-1, µ${’}_{\mathrm{s}}^{\mathit{\text{Object}}}$ =10.0 cm^-1. The objects are scanned with i) different sizes; ii) different depths from the source plane (source-detector plane in remission case).

In Fig. 5 we plot the contours for the detection threshold from amplitude and phase signals in both single- and dual-source configurations. For the transmission case with single-source conditions, the amplitude signal gives better sensitivity than the phase signal to detect the absorption heterogeneity, and the threshold contour is symmetrical to the central plane between the source and detector, which is due to the reciprocity of the source and detector in single-source setup. The detection sensitivity becomes worse when the object is closer to the middle plane (depth=2.5 cm). Generally, the detection limit in this case is about 3 to 4 mm, which is similar to the results given by Boas et al ¹¹. For dual-source geometry, we can see that the phase signal gives a better sensitivity than the amplitude signal, which is due to the larger transition of phase signal when the object scans through the null line. Note that the detection ability for the dual-source system is non-symmetrical from the higher detection sensitivity (2.3 mm) when the object is closer to the source plane, to the lower sensitivity (4 mm) when the object is closer to the detector plane. This can be explained by the non-symmetrical spatial sensitivity profile created by the interference of two sources²⁹. And when the object is closer to the source plane, the dual-source system provides better detection sensitivity than the single-source system, but when the object is far from the sources, the sensitivity for dual-source actually becomes weaker than the single-source case. The reason is that the photon paths from source 1 to the detector and source 2 to the detector get closer when approaching the detector, hence the scanning of object will affect the signals from both sources, which decreases the perturbation signal.

 

Fig. 5. Contour plot of the signal-to-noise ratio equals to one for amplitude and phase signals in single- and dual-source configurations. To the right side of the curve indicates a signal-to-noise ratio larger than 1 and the object with those parameters will be detected

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For the remission case, both single- and dual-source contours are non-symmetrical, and could not detect the object depth larger than 3 cm for a 1 cm diameter object, which is limited by the source-detector separation (here it is 5 cm). While in this case, the dual-source system always shows higher sensitivity than the single-source system.

From the two charts in Fig. 5, we can see that different systems give different detectability for the same model, which might be helpful when we design our detection systems in term of the selection of either transmission or remission geometry, and single- or dual-source configuration.

3.3 Detecting the Object with Different Absorption Contrast

Fig. 6 shows the contours indicating the smallest detectable absorber diameters as a function of absorption of the outside background, ${{\mu }_{\mathrm{a}}}^{\text{out}}$ and inside the object, ${{\mu }_{\mathrm{a}}}^{\text{in}}$ under transmission mode. In this case, the object depth is fixed to 2.5 cm, which is the central plane. Since the central plane has the worst detection sensitivity for single-source, for dual-source, it can also be considered as the lowest detection sensitivity since we can replace the sources and detector plane to overcome the poor sensitivity near the detector part. From the contour plots, we can see that for the single-source results, the sensitivity of detection decreases with the increase of background absorption, which will give higher shot noise and lower absorption contrast. The trend of the contour agrees with the results reported in reference 11, though the numbers are not exactly the same since we use different modulation frequency and slab thickness. When we compare it with the dual-source’s contour, the pattern is similar but the sensitivity has increased. This is indicated by the shifting of the contours towards the upper-left direction. And the smaller the object diameter, the larger the difference. This is due to a³${{\mu }_{\mathrm{a}}}^{\text{in}}$, which is what really counts for the perturbation signal, so that for smaller object, the dual-source detection will be more sensitive in terms of absorption contrast.

Fig. 7 compares the contours of the diameter for the smallest detectable absorber as a function of ${{\mu }_{\mathrm{a}}}^{\text{out}}$ and ${{\mu }_{\mathrm{a}}}^{\text{in}}$ under remission mode. Also in this case, the object depth is fixed at 2.5 cm. Generally, the detection with the transmission mode will give a higher sensitivity than the remission mode, if we compare the contours with the same object diameter to the corresponding one in Fig. 6. And the detection sensitivity deteriorates faster with the decrease of absorption contrast in the remission case (the contour bends towards the lower right direction in Fig. 7) due to the reason that the remission measurement is less sensitive than transmission, as shown in Fig. 5. Similar to Fig. 6, the dual-source system still gives better sensitivity than the single-source system.

 

Fig. 6. Diameter of the smallest detectable absorber plotted as a function of ${{\mu }_{\mathrm{a}}}^{\text{out}}$ and ${{\mu }_{\mathrm{a}}}^{\text{in}}$ for single- and dual-source systems in transmission mode. The background and object scattering coefficients are kept same (10 cm^-1) and the modulation frequency is 50 MHz. (Bandwidth=1 Hz)

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Fig. 7. Diameter of the smallest detectable absorber plotted as a function of ${{\mu }_{\mathrm{a}}}^{\text{out}}$ and ${{\mu }_{\mathrm{a}}}^{\text{in}}$ for single- and dual-source systems in remission mode. The background and object scattering coefficients are kept same (10 cm^-1) and the modulation frequency is 50 MHz. (Bandwidth=1 Hz)

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4. Discussion and Conclusion

In this paper we focused on the detection sensitivity analysis and hence provided information for detection and localization of absorbing objects. The signal-to-noise ratio is the limiting factor for detection. The noise threshold can be reduced by repeating the measurements or narrowing the bandwidth (increasing the signal integrating time). It is difficult to directly compare the performance of single- and dual-source systems, since there are many parameters involved regarding the optimal configuration for the dual-source system such as the sources’ separation and modulation frequency³⁷. Instead, we have studied the detectability for the same model (background and absorber) under practical experimental setups. The analysis of the amplitude and phase signals suggests that the dual-source phase component can sensitively “amplify” the small perturbation to a larger phase signal, which could yield higher signal-to-noise ratio for the purpose of detection. The sensitivity of the dual-interfering-source detection might provide complementary information for the conventional single-source detection, in both the detection and characterization, which is worth further exploration and discussion.

The vector model discussed in section 2.2 could provide an easy and qualitative analysis before launching simulations or performing experiments. And it could help in the experimental setup for optimization and calibration. Fig. 8 shows the case of dual-source system calibration on the heterogeneous model. Fig. 8(a) is calibration on a homogeneous model, so the two sources are balanced (same amplitude) and can detect the perturbation of a small absorber (the signal is depicted as the blue curve in Fig. 8(c)). While on a heterogeneous model (like calibrating on the brain, or other heterogeneous tissue, and a simplified model is represented by Fig. 8(b) with the presence of another fixed absorber), the two sources will not be balanced so that the system is working on the asymptotical region of Fig. 1(b), thus the sensitivity is low (the signal is depicted as the black curve in Fig. 8(c)). We can rebalance the sources by adjusting the amplitude for each source and restore the higher sensitivity (the signal is depicted as the red curve in Fig. 8(c)).

 

Fig. 8. Dual-source detection on homogenous medium (a) and heterogeneous medium (b). The background optical coefficients are µ_a=0.08 cm^-1, µ’_s=12.0 cm^-1. For the homogeneous case, the small scanning absorber has µ_a=0.5 cm-1and r=0.1 cm. The two sources are balanced (S1=S2=1.0). For the heterogeneous case, we introduce another larger fixed absorber (µ_a=0.5 cm^-1and r=0.5 cm) to simulate the inhomogeneous background, and the balanced is disturbed. If we increase the strength of S2, the two sources will cancel with each other again (S1=1.0, S2=1.12). The phase transition for the above three cases are plotted in (c). (Blue is for case (a); Black is for case (b) with S1=S2; Red is for case (b) with S2=1.12*S1).

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For the purpose of characterizing the optical properties in a single-source system, we can use the conventional fitting approach to minimize the chi-squared difference between the measured DPDW and the prediction from the analytical model¹¹ or apply the image reconstruction technique⁷. The higher detection sensitivity of dual-source system may provide useful information in the localization for object characterization and improve the imaging quality²⁹.