Theoretical description of the photorefractive detection of the ultrasound modulated photons in scattering media

: Acousto-optic imaging of thick biological tissues can be obtained in real-time with an adaptive-wavefront holographic setup, where the holographic media is a self-developping photorefractive crystal. As a consequence, the interference signal resulting from the acousto-optic effect can be easily collected with a high etendue and fast single photodetector. We present a statistical model of the ﬁeld propagating through the scattering media and show why the various acoustic frequency components contained in the speckle output pattern are uncorrelated. We then give a detailed description of the signal measured through the photorefractive effect, in order to explain the quadratic pressure response observed for the two commonly used conﬁgurations setup e.g an amplitude or a phase modulation of the ultrasound.


Introduction
Ultrasound modulated optical imaging [1] combines light and ultrasound to image optical contrast in thick and highly scattering media with the spatial resolution of ultrasound imaging. This technique is promising for biomedical applications, for example breast imaging, for which optical imaging techniques (such as Diffuse Optical Tomography [2]) can provide significant functional information but are generally limited to a poor spatial resolution. The principle is the following. A coherent (laser) light source illuminates a highly scattering sample. Due to the scattering, the photons will travel along of great number of random paths, producing a speckle pattern. At the same time, an ultrasound (US) beam is focused in the sample. Some of the travel paths, those crossing the US focal area, will be modulated due to the movement of scatterers [3][4][5] and to the local change in the refractive index [5]. This partial phase-modulation of the outcoming field leads to a modulation of the intensity of the speckle pattern at the US frequency. This corresponds to the "tagged photons", which are emitted, in the diffused light frequency spectrum, within acoustic sidebands. Measuring the weight of the tagged photon component as a function of the location of the US focal point allows to get 3D images of the scattering medium.
The detection of the tagged photon is difficult. The signal is weak, and spatially incoherent (speckle). Moreover, the parasitic motion of the scatterers, related to the blood flux in the in vivo case, decorrelates the speckle in a relatively short time. This corresponds in vivo to a spectral broadening in the kHz range [6,7]. Many techniques have been used to detect the tagged photons. First, experiments have been done with a photodiode which records the modulation of the speckle at the US frequency [1,8]. The optical etendue (product of the detector area by the detection solid angle) is low because one detects only one or a few speckle grains, but since the detector is fast, all the tagged photons in the optical etendue can be detected, whether the spectrum is broad or not. To increase the etendue, a CCD camera can be used to detect many speckle grains in parallel. To get a slow modulation component compatible with CCD detection, Leveque et al [9] use a stroboscopic illumination technique, while Gross et al [10] perform heterodyne detection with the CCD. The latter allows to get optimal detection sensitivity (shot noise). However, both Leveque and Gross techniques are slow and narrow band, and they cannot detect all the tagged photons when the spectrum is broad, like in the in vivo case.
A novel technique, based on two wave mixing in a photorefractive (PR) crystal has been introduced recently [11][12][13][14][15]. This technique, which allows to detect the tagged photons with a higher etendue than with the CCD parallel detection, is potentially able to be fast enough to detect all the tagged photons in an in vivo sample. In the Murray et al variant of the technique [11], a beam splitter on the main laser provides the pump beam of the PR crystal. The pump is thus resonant with the "un tagged" photons, which are detected by PR effect. The "tagged photon" are seen indirectly, like in a spectroscopic absorption experiment, through a decrease of the "un tagged" photon signal DC component [14]. To get resolution along the US propagation z axis, the US beam is pulsed, and the signal is detected by a fast avalanche photodiode (APD). By scanning the US focus spot along the transverse x axis, tomographic 2D images are obtained [13,15]. In the Ramaz et al variant of the technique [12], acousto optic modulators allow to freely adjust the amplitude, the phase and the frequency of the pump beam. By tuning the pump frequency resonant with the "tagged", or with the "un tagged" photons, it is possible to detect with the same setup either the "tagged" or the "un tagged" photons. In the first experiment [12], the "tagged photon" are detected, and the photodiode (PD) photoelectrons are amplified by a high impedance (R = 10 6 Ω), and thus low noise, transimpedance amplifier, which is quite slow. No attends is made to get z resolution, and 1D "image" (or plot) are presented (like in the first Murray's experiment [11]  detector, and do experiment with phantoms. Ramaz et al use a faster (τ PR = 5 ms) GaAs crystal (which is still slow with respect to the need for in vivo imaging: τ PR < 1ms) with a slow photodetector (PD), whose noise is low (because the impedance R is high), and do experiment with chicken sample (2cm and 4cm width), closer to the breast imaging need. Although many theoretical works have been done to describe both the acousto-optic imaging [3][4][5]17], and the PR effect [18][19][20], there is no theoretical attends to describe the PR effect in the context of the acousto optic imaging (except of the very recent Blonigen et al. study [14]). The purpose of this paper is to give such a global description. Moreover, to get better understanding of the physical phenomena, the model must be able to describe both the Murray, and Ramaz experiments, and to deal with new possible configurations of these setups, not yet implemented.
In section 2, we will first model the scattered light outgoing the sample, and discuss its statistical properties. We will then show, in section 3, that it is possible to separate the so-called "tagged" field, which results from the interaction of the photons with ultrasound, from the "un tagged" field, since these fields do not oscillate at the same frequency. Moreover these fields are spatially incoherent. In section 4, we will consider the photorefractive effect performed with these two fields, and in section 5, the detection of the photorefractive signal on a large area photodiodes. We will got then a general expression for the PR signal (Eq. (31)). In order to verify the validity of our theoretical model, we will consider different types of ultrasonic modulations, e.g a phase or an amplitude modulation. They are discussed in detail in section 6. Finally, experimental validation of the model is presented in section 7 and we give as a comparison a Fourier spectrum of the signal in the case of amplitude and phase modulation of the US at 1500Hz.

The fields observed with and without ultrasound
Let us call E P and E S the fields coming into and out of the sample. Consider the point (x, y) located after the sample output interface. E S is a quasi monochromatic wave at the frequency ω 0 of the incoming laser. Let us introduce the complex field amplitude E S defined as: Here Re is the real part operator. E S results from the sum (or the interference) of the field components E S,i scattered through the sample along many travel paths i from input plane (z = 0) to the point (r = (x, y, z)) (see Fig. 1).
It is out of the scope of the paper to define more precisely the travel paths, and the summation over them. One can model, for example, the travel pathes by a Monte Carlo simulation. For the travel path i, let us call a i the weight of the scattered field component, and s i0 the travel path length. In s i0 , the zero index means that we do not yet consider the possible motions of the scatterers. Thus s i0 does not depend on time, and neither do E S,i and E S . To simplify the description, without losing generality, we will consider that the phase of E S,i depends only on travel path length (the change of phase associated to each scattering event is either neglected or included within s i0 ): where mod is the modulo operator, and k = 2π/λ the optical wave vector for the scattering medium. Since the travel paths length s i0 is large (s i0 > z λ ), and since the s i0 distribution is broad, θ i is random and evenly distributed over the [0...2π] interval from one travel path to the next. The travel pathes summation ∑ i in Eq. (2) is thus equivalent to the step summation made to describe a random walk in the complex plane. The field E S that results of this random walk is then a random quantity that obeys to a gaussian statistic.
We must notice that s i0 varies with the distance from the location r i,∞ of the last scatterer to the considered point r following: Substituting Eq. (4) into Eq. (3), we get that both E S,i (r) and θ i (r) vary rapidly with r. The resulting E S also varies fast with r. This illustrates the speckle character of E S : θ i varies thus in the x and y direction over distances that correspond to the speckle grain size. This property is important for θ i , because it will allow us later to simplify Eq. (28) by making i = i . Let us now apply a CW (Continuous Wave) ultrasonic (US) wave to the system by using an ultrasonic piezoelectric (PZT) device. The PZT excitation voltage is simply: Here U PZT is the PZT voltage complex amplitude with respect to the ω US frequency. To simplify the description, we will consider in the following that the PZT voltage U PZT is low enough to have an acoustic pressure proportional to U PZT . We have verified experimentally (with an hydrophone) that all the results presented here correspond to this regime.
To account possible schemes of US modulation U PZT may depend on time, but varies slowly with respect to ω US . Under the combined effect of the US motion of the scatterers, and the US modulation of the medium refractive index, the length s i of the travel path i change periodically at the US frequency.
where arg is the complex argument. Here, On each scatterer, the phase of s i, j , which is slowly varying with time, depends on the phase of the US beam at r j . Because of the "banana" shape of the photon density inside the scattering medium [21], the extension Δx of the travel pathes in the x direction is much larger than the US wavelength λ US (Δx λ US ). The phase of the US, and the phase of the local modulation δ s i, j explore thus all phases of the [0, 2π] interval. As a consequence, the summation made in Eq. (8) is equivalent to a 2D random walk in the complex plane, and δ s i is a random complex quantity. The phase φ i is thus random, and is equally distributed over the [0, 2π] interval. Because the last scatterer, which is located near the sample output surface in r j,∞ , does not move (or because its motion can be neglected), φ i do not depend on r (contrarily to θ i (r)). This will simplify the final expression of the detected signal.

Decomposition of the transmitted field in Fourier components (CW ultrasonic excitation)
The periodic change of the travel path length s i at the US frequency (Eq. (6)) can be written as : Substituting s i (r,t) in Eq. (1) and Eq. (3), we get : Using the Fourier's expansion for the e jβ i sin(ω US t...) factor, we can develop the field E S,i , propagating along the travel path i, in Fourier components E S,i,ω n (with respect to the ultrasonic modulation): In typical experimental situation, the US excitation is quite low, and β i 1. The field E S is then a sum of Fourier components E S,ω n of frequency ω n = ω 0 + nω US : The Eq. (13) will be used in the following to develop of the PR signal in Fourier components in Eq. (28). The general expression for the travel path field Fourier components of Eq. (12) will be used then to simplify Eq. (28) in order to get the final general expression for the signal (i.e. Eq. (31)). For practical use, we will consider mainly the n = 0 and n = 1 Fourier components, i.e. the "carrier" component (n = 0), which corresponds to the "un tagged" photons, and the first "sideband" component (n = 1), which corresponds to the "tagged" photons. We get: Compare the phases of the i th terms of the Eq. (14) and Eq. (15) summations. These phases differ by a factor e j(φ i +π/2) which is random (φ i is random) from one travel path to the next. As a consequence, the fields E S,ω 0 and E S,ω 1 , which results from 2 statistically independent random walks (Eq. (14) and Eq. (15)), are uncorrelated (i.e. optically spatially incoherent).

The photorefractive (PR) effect on the tagged and un-tagged photons
The principle of the photorefractive (PR) interferometer is shown on Fig. 2. The signal beam E S (tagged or un-tagged by the ultrasound) issued from the scattering media is collected by large numerical aperture (NA) lenses, and focused into a PR crystal. The E S wavefront is highly distorted. In order to extract the desired information, a coherent reference beam E R , called also the pump beam, interferes with the signal beam into the PR crystal. The reference beam E R will be considered as a plane wave (this condition is not necessary but it simplifies the notations). The reference and the signal beams, which interfere within the photorefractive crystal, create through the photorefractive effect an hologram (or a grating) corresponding to a little change δ n of the crystal complex refractive index (i.e. refractive index, and/or absorption) [22,23]. The PR effect having a finite response time, only the static components of the interference pattern (i.e. moving with frequency smaller than the inverse of the response time τ PR of the crystal) will participate to the hologram inscription process. If the signal beam E S has several frequency components, the PR effect selects the components whose frequencies are close the frequency of E R . The pump beam is then diffracted by the PR holographic grating. The diffracted beam E D has then the same wavefront than the signal beam E S , which has written the hologram. E D goes out the crystal, and propagates in the same direction as E S [18-20, 24-27].
To make a quantitative analysis of the PR detection, we have to introduce the complex field amplitudes ε R and ε S with respect to the frequency ω of the reference pump beam E R , which plays here the central role (and may differ from ω 0 depending on the implementation [11,12]).
ε R is constant, and ε S , which contains all the ω n frequency components, is time varying, we have from Eq. (13).
Two wave mixing in a photorefractive media with a constant pump beam ε R , and a time varying (in amplitude and/or phase) optical signal beam ε S , has been described in [18,28]. The signal field ε S (x, y, z) within the crystal, i.e. for z 1 < z < z 2 (where z = z 1 and z = z 2 are the crystal input and output plane), can be written as a function of the crystal incoming field ε S (x, y, z 1 ), which is supposed to be known: G(z,t) is given in [28]. In the condition of low efficiency of the grating and low absorption α of the crystal the transfer function G(z,t) takes the simple form: Here τ PR is the response time of the photorefractive crystal and γ the photorefractive gain (that can be complex values in the more general modelling, but that are taken real in the present calculation, in accordance with the experimental conditions we use). In Eq. (18), t = 0 is the time origin, at which it is supposed that no hologram is written. Equation (18) is established using several approximation that we will recall here [18]. The first one, is that we suppose that the pump beam is a constant monochromatic wave of pulsation ω (i.e. it is not time modulated). It is not perturbed by the grating, but it can lose its energy due to the crystal absorption. The pump beam intensity is also supposed to be much higher than signal beam intensity. The total incident illumination that governs the response time of the photorefractive crystal [22,23] is thus equal to the pump beam intensity. The response time that appears in the expression of G(z,t) is the response time of the crystal taken at the entrance of the photorefractive crystal at position z = z 1 . The model is established for two incident plane waves but it is immediately generalized to a speckled signal wave, that will be written into its plane wave decomposition, each plane wave being treated independently and the transmitted wave being the linear recomposition of the different transmitted plane wave components.
Since we are mainly interested by the solution of Eq. (18) with a periodic source term ε S (x, y, z 1 ,t), we may extend the lower limit of the dt integral of Eq. (18) from t = 0 to t = −∞. Our purpose is to the make the periodicity more visible within the equations. Extending the integral to t = −∞ means that any transient behaviour such as the initial build-up of the grating is neglected and that we consider only the stationnary periodic response of the crystal. Making the transformation: t → (t − t ), in the dt integral of Eq. (18), we get: Here, the PR hologram is written at the retarded time t − t , the retardation being t > 0 .
To help the following discussions, we have to notice that the first term of Eq. (20) or Eq. (18) (field E T and complex amplitude ε T ) corresponds to the signal beam transmitted (T) through the photorefractive crystal, whereas the second term (E D and ε D ) is the pump beam diffracted (D) by the PR hologram. We have thus: By developing ε S (x, y, z 1 ,t − t ) from Eq. (17), we get: Equation (25), which is equivalent to Eq. (24), shows that the diffracted field E D can be developed in a sum of Fourier components, each of them evolving at the Fourier frequency ω n . Because of the e − j(ω n −ω)t factor in the ∞ 0 dt integral, the crystal selects, through its response time that governs the evolution of G(z,t ), the frequency component ω n closest to ω.
If for example ω ω 1 , the sum over n (i.e. ∑ n in Eq. (25)) restricts to n = 1 (as ω US is generally much greater than the inverse of the response time of the PR crystal). One select thus the ω 1 frequency component, i.e. the "tagged photons". As seen in Eq. (25), the diffracted beam E D evolves then at the "tagged photons" frequency ω 1 . If ω is not exactly equal to ω 1 (with still a difference smaller thant 1/τ PR ), the E R versus E S interference pattern (and thus the PR hologram) drifts continuously in phase, but the phase of the field, which is diffracted by this moving grating, do not drift (the phase evolution is exactly e jω 1 t ) with respect to the "tagged photons" field. This is a known property of the diffraction with a PR moving grating [29].

The PR signal with a photo detector of large area
To simplify the discussion, we will consider that the signal is detected by a photodiode of large area, located in the PR crystal output plane z = z 2 . The photodiode signal S PD is thus equal to the integral of |ε S | 2 over the photodiode area. From Eq. (21) we get: (26) where c.c. means the complex conjugate.
To get a sensitive detection we will modulate the amplitude, or the phase of the ultrasonic wave, and we will consider mainly the time varying component of S PD . Because the acoustic wave do not change the total number of scattered photons (i.e. "tagged" + "un tagged"), the |ε T | 2 term of Eq. (26) do not depend on the acoustic wave modulation. This term can thus be discarded. Since the PR gain is supposed to be quite low (η PR = γ(z 2 − z 1 ) 1), the |ε D | 2 term is much lower than the ε T ε D * cross term. Moreover, the |ε D | 2 term builds up slowly, in a time τ PR (see Eq. (19)), which can be much larger than the modulation period. The |ε D | 2 term can thus be discarded too.
As in many PR experiments [18, 30, 31], we will consider thus only the ε T ε D * cross term, which is the product of the amplitude ε D * , which builds up slowly (in τ PR ), with the transmitted amplitude ε T , which can vary very fast, since it is directly related to the ε S (x, y, z 1 ,t) source term.
By using Eq. (22) and Eq. (23), we get for the cross term photodiode signal S PD = c.c. + dx dy ε T ε D * : To keep full generality, we have introduced here G * although G is supposed to be real. Developing ε S in Fourier components E S,ω n with Eq. (17), and E S,ω n in travel pathes field components E S,i,ω n with Eq. (13), we get: As mentioned above, due to the speckle character of the scattered field E S , θ i varies very fast with r = (x, y, z 1 ). As a consequence, in Eq. (29), the x, y integral of e − j(θ i −θ i )(r) yields zero, when i = i . In Eq. (28), one can thus restrict the i, i summation to the i = i terms. This approximation is very common in optics. In Eq. (27) the ε S (t)ε S * (t − t ) terms corresponds to a field intensity (or more exactly to a time correlation intensity), which must be summed over the photodiode area. Because of the spatial integral for the intensity, one can neglect the speckle interference terms, which correspond in Eq. (28) to i = i .
For i = i , consider now the ∑ i E S,i,ω n (...) E * S,i,ω n (...) term of Eq. (28). By developing E S,i,ω n using Eq. (12) we get: Since the phase φ i is randomly distributed with equal probability in the All these simplifications result from incoherent character of the "tagged" versus "un tagged" speckle fields E S,ω 0 and E S,ω 1 described at the end of section 3. If for example one selects the "tagged photon" by making ω ω 1 , the diffracted field E D , whose frequency is ω 1 , match the "tagged photon" field. The "un tagged photons" field, whose frequency is ω 0 , cannot thus interfere coherently with E D , since both fields are spatially incoherent. The photodiode signal at ω 1 − ω 0 = ω US is thus very low.
Making i = i and n = n , and developing E S,i,ω n with Eq. (12), Eq. (28) can be rewritten in a form that will be useful for the following discussion: Because of the time integral ∞ 0 dt with the G(z 2 ,t ) kernel, one has to tune (to get a non zero signal) the pump beam frequency ω to be nearly resonant with the one of Fourier component ω n , with practically n = 0 or n = 1. In that case, in Eq. reference field is dominant. The reference beam diffracts then a field proportional to this index grating, and the flux onto the detector exhibits such a J 1 2 (β i ) a dependence. We can now develop the interference signal in harmonic components versus ω mod , omitting the various static components. We get then : Consider now the case of an amplitude modulation (AM) of the US with a reference beam at ω = ω 0 . We detect the "un tagged" photons (n = 0 in Eq. (31)). Since β i 1, we can replace J 0 by one, in the dt integral of Eq. (31). Once again, the time-varying contribution gives then : Here, A A since (1 − J 0 ) J 1 2 for small argument β i . Here there is no dependance of the signal amplitude (maximum to minimum) with the cyclic ratio x, because the PR hologram is written by the "un tagged" photons (which remains roughly constant since β i 1). The PR signal observed here has the same order of magnitude than for PM modulation, but the signal sign is reversed. This is expected. The total number of photons ("un tagged" + "tagged") do not depend on the US. Here, we detect the "un tagged" photon (ω = ω 0 ), and the signal is maximum, when the US is off (no tagged photons). In the PM modulation previous case, we have detected the "tagged photon", and the signal was maximum, when the US was on. The two situations (ω = ω 0 or ω = ω 1 ) can be viewed as an absorption-like detection (bright background), or an emission-like detection (dark background) respectively.
By developing the signal in harmonic components, we get: Consider finally the case of an amplitude modulation (AM) with a reference beam at ω = ω 1 . We get then for the signal shape versus time: where A is given by Eq. (37). Here again the average "tagged photon" field yields to the x proportionality factor of Eq. (42), since x =< H AM (t) e jψ AM t > t . The development in harmonic components is:

Experimental results and discussion
The PR interferometric setup is similar to the one previously described [12] (see Fig. 3). The PR crystal is a single crystal of undoped GaAs grown by Liquid Uncapsulated Czochralski method. The crystal is oriented in an energy transfer configuration [22,23] with (110), (1 − 10) and (001) faces. The signal and the reference beams enter both along the (110) (co-directional type) face or enter respectively on the (110) and (001), in order to reduce pump scattering onto the detector (orthogonal-type configuration). The orthogonal configuration is surely not suitable for media that exhibit fast speckle decorrelation time : in such a case the related τ PR time establishment can be long (some tens of ms -depending on the inverse of the average flux) due to a higher spatial frequency of the fringes within the crystal. We used two single frequency  YAG : Nd 3+ lasers (1200mW , Crystal Laser corp. and 120mW , LightWave electronics corp.), vertically polarized and operating at 1064nm. The present setup is not yet fully optimized in flux, especially because of the presence of the acousto-optical modulators on the arms of the interferometer that introduce many losses. Most of the experimental results presented here have been performed with the 120mW source. The intensity used with the two sources depend on the sample, they varies typically between 10mW /cm 2 and 70mW /cm 2 , which is beneath safety regulations for invivo imaging. The intensity of the reference beam (onto the crystal) can vary between 25mW /cm 2 and 200mW /cm 2 . Evidently, the configuration using the 1200mW source is able to provide a faster built up of the photorefractive effect. The detector is a large area InGaAs (φ = 5mm) photodiode (Hamamatsu corp.) mounted on a high pass two-stages amplification in order to favor the ac component compared the dc one. The first stage is low bias current (OPA111) transimpedance amplifier (R = 10 6 Ω).
The signal can be recorded either with a lock-in detection (EG&G instruments 7260 digital model), or with an analog to digital converter with 16bits resolution working at a rate of 48kHz : in this configuration, the signal is recorded during 1.5s and we perform a Fast Fourier Transform (FFT) in order to extract the magnitude of the signal at the modulation frequency, and to get an estimation the signal-to-noise ratio (SNR). Figure 4 shows the Fourier spectrum of the signal with intralipid solution, at maximum response for amplitude and phase modulation of the US at 1500Hz, e.g with a cyclic ratio respectively of 50% and 25%. These spectra exhibit a comparable power SNR (53dB for the amplitude modulation, and 51dB for the phase modulation), but the amplitude modulation case contains an important noise at low frequency : this is not surprising since the grating is built-up with the un-tagged photons, and the interference pattern is more sensitive to technical noise of the setup. Test experiments of the acousto-optical response have been performed either with a e = 2cm thick chicken breast sample, or with a e = 10cm liquid scattering sample (intralipid aqueous solution). We have at first measured with a lockin detection, at fixed pressure and light flux, in the intralipid sample, and for the various modulation, the response of the AO signal, which is described in section 7.2, as a function of the cyclic ratio x. Figure 5 shows the comparison between the experimental data, and the corresponding fit deduced from Eq. (38), Eq. (41) and Eq. (43). The agreement validates our theoretical analysis, and the S PD (t) expressions.
As a final test of our model, Fig. 6 shows the magnitude of the AO signal as a function of the acoustic pressure (∝ U PZT ), with the chicken sample, and the corresponding fit. Measurements are performed with amplitude and phase modulation (AM and PM) of the US excitation. For the AM case, the cyclic ratio is of x = 1/2, and the reference beam frequency is ω = ω 0 (detection of the "un tagged photons" at the US carrier frequency). For the PM case, x = 1/8 and ω = ω 1 (detection of the "tagged photons" at the US first sideband frequency). These measurements exhibit a quadratic response with a comparable magnitude, as expected by the model when the acoustic pressure excitation is weak, since J 2 1 (x) = 1 − J 0 (x) x 2 /4 for a small x argument. Moreover, we measure in this case a ratio S AM /S PM 1.74 : this is fully compatible with the results shown on Fig. 5, for the case of a weak acoustic pressure perturbation.  In these test experiments, we have not tried here to minimize τ PR , but preliminary results show that a co-directional beam coupling with a pump beam of 150mW generates a τ PR ∼ 1ms.

Conclusion
In this paper, we have presented, and verified experimentally, a model of the real-time acousto optical response for thick highly scattering media when coupled to an interferometric setup using a photorefractive (PR) crystal as recombination plate. We have shown that the various speckle patterns associated to the sidebands generated by the acousto optical effect are not correlated. By choosing the frequency of the PR pump beam, we are able to select the acousto optic component that builds up the PR hologram ("un tagged" photon for a pump beam at the carrier frequency [11], or "tagged photons" for a pump beam at the US sideband frequency [12]). The PR effect selects thus the specified acousto optic component, that can be easily detected by a proper amplitude or phase modulation scheme.
The agreement of predictions the theoretical model with the experimental results is excellent. As predicted by the model, and as seen experimentally, the modulation signals obtained for both "tagged" and "un tagged photon" detection have almost the same order of magnitude. The response of the system versus the acoustic pressure reveals also a quadratic variation (linear in acoustic power) for untagged-photons and tagged-photons shifted from the US frequency. This confirms that the perturbation brought by the US is weak.
A careful study of the PR time response needs further to be performed, in order to find suitable conditions for in vivo measurements, e.g faster than the speckle decorrelation time.