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Ultrasound modulation of coherent light in a multiple-scattering medium: experimental verification of nonzero average phase carried by light

We demonstrate the phase fluctuation introduced by oscillation of scattering centers in the focal volume of an ultrasound transducer in an optical tomography experiment has a nonzero mean. The conditions to be met for the above are: (i) the frequency of the ultrasound should be in the vicinity of the most dominant natural frequency of vibration of the ultrasound focal volume, (ii) the corresponding acoustic wavelength should be much larger than ℓn*, a modified transport mean-free-path applicable for phase decorrelation and (iii) the focal volume of the ultrasound transducer should not be larger than 4 – 5 times (ℓn*)3. We demonstrate through simulations that as the ratio of the ultrasound focal volume to (ℓn*)3 increases, the average of the phase fluctuation decreases and becomes zero when the focal volume becomes greater than around 4(ℓn*)3; and through simulations and experiments that as the acoustic frequency increases from 100 Hz to 1 MHz, the average phase decreases to zero. Through experiments done in chicken breast we show that the average phase increases from around 110° to 130° when the background medium is changed from water to glycerol, indicating that the average of the phase fluctuation can be used to sense changes in refractive index deep within tissue.


Introduction
Ultrasound modulated optical tomography (UMOT) combines the advantages of an optical property recovery in a scattering medium with the high resolution available with focused ultrasound (US) [1,2]. Focused US creates a perturbation in mean position of the scattering centers and refractive index (n) in the focal volume of the focusing US transducer, which is the region of interest (ROI) for imaging. Coherent light picks-up a phase modulation from the ROI which is observed as an intensity modulation in the speckle formed through beating of exiting light. The modulation depth (M), obtained from the speckle intensity or field autocorrelation, is the measurement, from which both the optical (for example, absorption coefficient (μ a )) and mechanical properties (for example, the shear modulus (G)) pertaining to the ROI are ascertained.
The parent discipline whose extension UMOT is, is diffusing wave spectroscopy (DWS) wherein coherent light is employed to study multiple scattering media undergoing temperatureinduced Brownian motion. The measurement is a correlation function (field-or intensity autocorrelation; g 1 (τ) or g 2 (τ) respectively) whose decay is used to quantitatively measure the mean complex modulus of elasticity of the medium. In UMOT we add a local, deterministic sinusoidal perturbation to the stochastically driven multiple scattering medium at the focal region of the US transducer. This perturbation gives rise to a sinusoidal modulation on the correlation fluctuation. The speckle modulation mentioned earlier is a measure of the modulation in g 1 (τ) whose depth M is related to μ a and G of the US focal volume.
A general assumption under which quantitative expressions connecting g 1 (τ) to the properties of the object is derived both in DWS and UMOT is that the optical wavelength λ • is very small compared to the scattering mean-free-path ( s ) [3,4], which is the so-called weak scattering condition. This would imply that correlation of light reaching the detector along different paths is negligibly small and only light traveling the same path contributes to g 1 (τ). Another assumption in regard to UMOT is that the phase increments (Δφ ) introduced by US forcing of scattering centers within the ROI are uncorrelated. This is true when λ a << * where λ a is the acoustic wavelength and * is the transport mean-free-path of light in the medium. In addition, one also assumes that the random process associated with the overall phase fluctuation picked-up within the object is wide-sense stationary which is true when the object dimensions are large compared to λ a .
The above assumptions, particularly, the uncorrelatedness of Δφ within the ROI makes the phase measured in the g 1 (τ) modulation (i.e., the speckle interference term) a random variable with zero mean and therefore, carrying no useful information. On the other hand, when the acoustic excitation is of low frequency (say < 1 kHz) the assumption λ a << * is violated, and more importantly when the acoustic excitation is in the neighborhood of one of the dominant natural frequencies of vibration of the insonified US focal region, the movements introduced by the US forcing within a volume ( * ) 3 are correlated and fall quite within one acoustic wavelength, and the corresponding phase increments picked up by light in that volume do not add up to a zero in the mean. The mean, Δφ av obtained from g 1 (τ) (along with M) enriches the UMOT data enabling recovery of additional parameters such as refractive index, n, and the scattering coefficient, μ s . For Δφ av to be nonzero, in addition to demanding that λ a >> * , we should have a small enough ROI so that the intercept length of photon trajectory with the insonified ROI should have a good chance of being within * . For this, our experience has been that if the ROI (which is usually an elongated hyperboloid with a narrow waist region) is bounded by 4 − 5 times ( * ) 3 , Δφ av will have a consistent nonzero value. In addition, we have shown in Sec. 2 that the length scale involved in phase decorrelation is * n = s (1 − cos θ 2 ) which is larger than * .
Our aim in this work is to demonstrate by simulations and experiments that Δφ av measured from the speckle modulation can be nonzero if the above mentioned conditions are met. A brief summary of the rest of the paper is as follows: In Sec. 2 we put forth an expression for Δφ in terms of displacements of scattering centers and the local n. Here we also discuss in detail the conditions under which Δφ av is nonzero. Sec. 3 discusses the details of our simulation studies. Here we vary the volume of the ROI and λ a (in effect the US frequency), and obtain histograms of Δφ measured at 256 × 256 elements in the CCD detector array. These histograms reveal a nonzero mean for Δφ when λ a is large and the US frequency is in the neighborhood of a dominant natural frequency of vibration of the ROI, and the ROI volume is small. Sec. 4 describes an UMOT experiment wherein poly-Vinyl Alcohol (PVA) phantoms and chicken breast-tissue slabs are used as samples. From the histograms obtained (from the measured Δφ in all the elements in the detector array) we obtain Δφ av and demonstrate the sensitivity of the measured Δφ av to the local n in the ROI. Our concluding remarks are left to the last section (Sec. 5).

Theoretical considerations
Let N p be the number of scattering events experienced by photons in their passage through the insonified ROI. The scattering centers are themselves vibrating because of sinusoidal US forcing. The typical displacement of the j th scattering center about its mean position is given by Here u(r j ) is the maximum displacement suffered by the j th scattering center, k a is the acoustic wave vector, r j is the position vector (at rest) of the j th scattering center and ω a is the acoustic frequency. Fig. 1 shows a typical representation of scattering of light by a vibrating scattering center, with unit vectors in the incident and scattered photon directions (in a typical scattering event) represented byk i andk s (k i and k s are the corresponding wave vectors) and the displacement by u (taken as alongk i as shown in Fig. 1). The phase fluctuation experienced by the photon following path p is given by [5] δ φ p (t) = Here q j = k s − k i . Since we consider electric field arriving along the same photon path alone to be correlated, the phase of the field autocorrelation of light reaching the detector along path p will have value: To find the average Δφ (τ) av which influences g 1 (τ) we take the average of Δφ p (τ) 's evaluated over all the photon paths reaching the detector which have nonzero intercepts with the ROI, weighted by the probability density for the photon paths: Here s is the photon path length, and s 1 and s 2 are limits on path lengths s which pertain to those with nonzero intercepts with the ROI. The integral in Eq. 4 can also be evaluated, un- der weak scattering approximation, by first summing over Δφ (τ) 's pertaining to all paths passing through a particular scattering center and then summing the result over all the scattering centers in the ROI [5]. When this is done Δφ (τ) av can be nonzero only when there is scattering anisotropy for the medium to such an extent that ( * ) 3 is large enough to have the intercept length of the photon trajectory with the ROI ≤ * . Moreover, while considering correlation of Δφ (τ) one should use a scattering anisotropy factor g n = cos θ 2 to compute * instead of the usual g = cosθ . This results in a larger * (denote it by * n ) with which we can put limits on the volume of ROI so that Δφ (τ) av is nonzero (as demonstrated in Sec. 3). (This can be seen from the diagram of Fig. 1. The phase, Δφ picked-up in a typical scattering event where k 0 is the modulus of light propagation vector. Therefore the uncertainty in cos θ 2 , and not cos θ , determines the decorrelation of the overall phase fluctuation; and we use a modified g n = cos θ 2 as the anisotropy factor to arrive at * n ). As we demonstrate through simulations, Δφ (τ) av remains nonzero so long as the US focal volume V is bounded by 4 − 5 times ( * n ) 3 . The reason for this persistence of correlation in Δφ (τ) av over a larger volume compared to ( * n ) 3 is that for the rather elongated shape of the ROI, the probability of a number of the intercept length of the photon trajectory with the ROI remaining within * n is large for ROI volumes considerably larger than ( * n ) 3 . However, in our simulations we have used a cuboid ROI which is not elongated along any particular direction. The consequence of using this 'symmetric' object is that Δφ (τ) av decays rapidly to zero as V increases from ( * n ) 3 to ∼ 5( * n ) 3 (See Fig. 2). In addition to the above there should also be 'consonance' (i.e., directional bias along the US forcing) of the movement of scattering centers in the insonified region introduced by the ultrasound forcing, for Δφ (τ) av to be nonzero. If λ a << * n , there will be many 'periods' of US-induced oscillations within the insonified volume, which would mean that the resulting oscillations (or displacements) of the scattering centers will have equal propensity to be in a certain direction and its opposite. Therefore for 'consonance' of the movement of scattering centers one should have λ a >> * n . This, however, is not enough to ensure that the movements will have correlatedness in direction (and not randomly distributed) so that phase picked up is not averaged to zero. The jelly-like material, which is the model used to represent tissue, for the particular shape of the insonified volume, has natural frequencies of vibrations in the range of 100 − 400 Hz [6] corresponding to the axial translational mode. When the ultrasound forcing is at the natural frequency of vibration a resonance is observed in the oscillating movement of particles in the volume. In the vicinity of this resonant frequency, primarily only the prominent axial mode is excited (which sets up vibration along the US transducer axis) and the eigen values corresponding to the other modes such as non-axial translation, dilatation etc. are small. However when the forcing frequency is away from the resonant frequency of vibration, the input energy is more evenly distributed amongst all possible degrees of freedom with the result that oscillation amplitudes are more randomly oriented. Therefore when the forcing is at 1 − 2 kHz, even though one wavelength (∼ m) generously covers the insonified region, owing to the apparent randomness of the apparent displacements the average phase picked-up is seen to be zero.
Therefore, for the average phase to be nonzero we should have V ∼ 4 − 5 times ( * n ) 3 and λ a >> * n (for typical soft biological tissue, * ∼ 1.00 mm and s ∼ 0.10 mm [7]), with the corresponding frequency close to the dominant natural frequency of vibration of the insonified region. In our experience (see Sec. 3) when the US frequency is below 1 kHz, i.e., λ a ∼ 1.5 m, Δφ (τ) av evaluated over a cuboidal ROI of volume 3 × 7 × 3 mm 3 = 0.063 cm 3 is nonzero when g n ∼ 0.97 and * n = 3.58 mm. If we take V = 5( * n ) 3 this works out to be 5 × 0.0461 cm 3 = 0.2305 cm 3 , much larger compared to the volume of the ROI. Using Eqs 3 and 4 one can obtain computable expressions for Δφ (τ) av in which the optical properties of the ROI and its size enter as parameters. In Sec. 3 we demonstrate through Monte Carlo (MC) simulations how Δφ (τ) av varies when (i) V ( * n ) 3 is varied and (ii) Δν the US forcing frequency is varied. We also demonstrate variation of Δφ (τ) av when n is varied in the neighborhood of the ROI.

Numerical simulations
For numerical simulations we consider a cylindrical object of diameter 3 cm and height 5 cm (to represent PVA phantom), and a slab 3 × 3 × 5 cm 3 (to represent chicken breast sample). For PVA phantom, the optical properties used in the simulations are μ a = 2.85 × 10 −4 mm −1 , μ s = 10.00 mm −1 , g = 0.89, and n = 1.35 [8], which is also varied to demonstrate the shift of Δφ (τ) av . For chicken breast tissue (slab of dimension 3 × 3 × 5 cm 3 ) μ a , μ s and g are assumed to be 0.01 mm −1 , 23 mm −1 and 0.90 respectively in accordance with the properties of chicken breast tissue measured and reported in [9]. We use two identical focusing US transducers with focal regions intercepting in the object giving the ROI (for details, please see Section 4). The US focal volume of the individual transducer is assumed nearly hyperboloidal with axial length of ∼ 1.2 cm and diameter across the waist (the thinnest middle region) 0.25 cm. The intersection of two such focal volumes which is labeled the ROI is assumed to be a cuboid of volume V . In fixing these parameters we took recourse to computing the acoustic pressure (by solving the Westervelt equation as is done in [10]) inside the focal volume of the hemispherical PZT transducers that we, in fact, employed in the experiments. The US focal volume is fixed as the region where the pressure is greater than 100 th of its maximum pressure along the axis. The insonified volume, V , is varied in the simulations to demonstrate the dependence of Δφ (τ) av on it.
First we carried out two sets of simulations to demonstrate the effect of V ( * n ) 3 and Δν on Δφ (τ) av . In the first, through MC simulation, we compute the set of Δφ (τ) av 's from which histograms of Δφ (τ) av are plotted for V going from ∼ 1.5( * n ) 3 to ∼ 6( * n ) 3 . The results plotted in Fig. 2 show that the mean obtained from the histogram steadily goes to zero as V is increased from ∼ 1.5( * n ) 3 to ∼ 6( * n ) 3 . In the second set of simulations we varied Δν from 100 Hz to 2 kHz in the steps of 250 Hz and computed Δφ (τ) av . For Δν > 1 kHz the average phase is close to zero, verified also for a high frequency of 1 MHz. Here, the dimensions of the ROI is kept at 3 × 7 × 3 mm 3 and the other optical properties are kept at values same as those used in the first set. The plot of the Δφ (τ) av .vs. Δν obtained through simulations is shown in Fig. 3, which also shows results from the experiments described later in Sec. 4. For arriving at Δφ (τ) av we proceeded as follows: 5 × 10 7 photons are propagated using Monte Carlo simulation to compute the photon path distribution in the object in the transmission geometry. We kept track of, primarily, the optical path length of photons in the insonified ROI and the photon weight from which the distribution of photon phase fluctuation is found, using Eqs 3 and 4 at various exit points corresponding to the 256 × 256 elements in the CCD camera. (The details of MC simulation used are given in [11,12]). From this we have plotted histograms of Δφ (τ) av for two US beat frequencies of Δν = 1 MHz and 100 Hz. These are shown in Fig. 4(a) and (b) respectively for the cylindrical PVA phantom described earlier with n around the ROI assumed to be 1.33. Histograms obtained from experimentally generated phase distribution (details in Sec. 4) are also shown in the figure for comparison. When we repeated the simulations for the slab of chicken breast (with water, n = 1.33) in the ROI the histograms shown in Fig. 5(a) and 5(b) are obtained for Δν = 1 MHz and 100 Hz respectively. It is seen that when frequency is 100 Hz the evaluated average phase in the two cases are 145 • (for PVA) and 105 • (chicken breast). When the simulation is repeated for chicken breast slab with glycerol in the ROI (n = 1.47) with frequency of 100 Hz we obtained the histogram shown in Fig.  6 with Δφ (τ) av extracted as 120 • . It is seen that the average phase fluctuation shifts when the n in the ROI changes. As stated earlier the above set of figures come with corresponding experimental results overlaid which will be explained in the next section. Fig. 6. Same as 5(b) but with glycerol injected around the ROI. It is seen that owing to the higher n of glycerol the average phase has increased to ≈ 130 • (for experiment) and 120 • (for simulations). The percentage error in phase extracted through measurements is 1 − 7%.

Experiments
Our experimental set-up is similar to the one introduced in [13,14] for UMOT which uses a source-locking technique to measure speckle modulation (of frequency as high as 1 MHz) with a slow 50−60 Hz CCD camera. For more details see [11]. It adapts the 4-or 3-intensity method of phase shift interferometry to compute the modulation and phase of interference patterns by recording 4 frames of intensities with preset phase differences between reference and object beams [15]. The system essentially uses a modulated laser beam locked to the driving oscillator of the US transducer. A low frequency acoustic excitation in the range of 100 Hz to 1 kHz is required to obtain the resonance of vibrating ROI of jelly-like soft tissue-mimicking phantom [6] while US frequency of the order of MHz is demanded to achieve penetration depth of the order of cm s in soft tissue. Here, we employed two identical, focusing US transducers (Panametrics, V394-SU-F50MM-PTF) operated in continuous mode with a small and adjustable frequency difference; one at ν 1 = 1 MHz and the other at ν 2 = ν 1 + Δν (with the possibility of Δν to be fixed accurately in the range 100 − 1200 Hz). The transducers are aligned, with the angle of inclination 60 • (at the depth 1.5 cm inside the tissue samples), to have an intersecting common focal region and US induced low frequency acoustic wave at the beat frequency Δν at the region of intersection [11]. The experimental set-up is schematically shown in Fig. 7. During