Keywords:

come

Speckle decorrelation in Ultrasound-modulated optical tomography made by heterodyne holography

Speckle decorrelation in Ultrasound-modulated optical tomography made by heterodyne holography M. Gross Ultrasound-modulated optical tomography (UOT) is a technique that images optical contrast deep inside scattering media. Heterodyne holography is a promising tool able to detect the UOT tagged photons with high efficiency. In this work, we describe theoretically the detection of the tagged photon in heterodyne holography based UOT, show how to filter the untagged photon discuss, and discuss the effect of speckle decorrelation. We show that optimal detection sensitivity can obtain, if the frame exposure time is of the order of the decorrelation time.

OCIS codes: 170.1650, 170.3660, 290.7050, 090.0090, 170. 7050 Light scattering prevents optical imaging from achieving high resolution inside scattering media deeper than about 1 mm in tissue. Ultrasound-modulated optical tomography (UOT) [START_REF] Wang | Ultrasound-modulated optical tomography of absorbing objects buried in dense tissue-simulating turbid media[END_REF][START_REF] Daniel S Elson | Ultrasound-mediated optical tomography: a review of current methods[END_REF] also called acousto-optic imaging [START_REF] Steffen G Resink | State-of-the art of acousto-optic sensing and imaging of turbid media[END_REF], has been developed to overcome this limit by combining ultrasonically defined spatial resolution and optical contrast (i.e. sensitivity to the bulk optical properties like absorption). One of the purpose of the technique is to use the optical contrast to detect breast tumors that cannot be seen with ultrasound, because the ultrasonic contrast is too low. In an UOT experiment, the light scattered through a diffusing sample crosses an ultrasonic beam, and, due to the acousto-optic effect, undergoes a frequency shift equal to the ultrasonic frequency [START_REF] Leutz | Ultrasonic modulation of multiply scattered light[END_REF][START_REF] Lihong | Mechanisms of ultrasonic modulation of multiply scattered coherent light: an analytic model[END_REF]. By detecting the frequency-shifted photons, called tagged photons, and by plotting their weight as a function of the ultrasonic beam geometry, 2D or 3D images of the sample can then be obtained with ultrasonic spatial resolution.

Various methods have been developed to detect the very low tagged photons signal out of a large background of untagged photons [START_REF] Daniel S Elson | Ultrasound-mediated optical tomography: a review of current methods[END_REF][START_REF] Steffen G Resink | State-of-the art of acousto-optic sensing and imaging of turbid media[END_REF]. First experiments use singlepixel detector and detection of the tagged photon AC modulation at the ultrasonic frequency [START_REF] Leutz | Ultrasonic modulation of multiply scattered light[END_REF][START_REF] Wang | Continuous-wave ultrasonic modulation of scattered laser light to image objects in turbid media[END_REF][START_REF] Kempe | Acousto-optic tomography with multiply scattered light[END_REF]. Since each speckle grain oscillates with a different phase, the single pixel method detects, with a good efficiency, no more than one speckle grain. This severely limits the detection etendue (defined as the product of the detection area and the acceptance solid angle). To increase the detection etendue without reducing the modulation depth, three types of methods have been developed. The first type relies on incoherent detection with a narrow spectral filter (∼MHz) that filter out the untagged light. A large-area single-pixel detector can be used. Examples include Fabry-Perot interferometers [START_REF] Sakadžić | High-resolution ultrasound-modulated optical tomography in biological tissues[END_REF][START_REF] Kothapalli | Ultrasound-modulated optical microscopy[END_REF][START_REF] Rousseau | Ultrasound-modulated optical imaging using a high-power pulsed laser and a double-pass confocal fabry-perot interferometer[END_REF] and spectralhole burning [START_REF] Li | Detection of ultrasoundmodulated diffuse photons using spectral-hole burning[END_REF][START_REF] Li | Pulsed ultrasound-modulated optical tomography using spectral-hole burning as a narrowband spectral filter[END_REF][START_REF] Zhang | Slow light for deep tissue imaging with ultrasound modulation[END_REF] based methods. These techniques require bulky and expensive equipment. The second and third types of method use interferences and are thus sensitive to the signal phase decorrelation due to the living tissue inner motions, and to the corresponding Doppler broadening. For breast, this broadening is 1.5 kHz [START_REF] Gross | Heterodyne detection of multiply scattered monochromatic light with a multipixel detector[END_REF]. The second method is based on a photorefractive crystal, which records the volume hologram of the sample scat-tered field. This hologram can be then used to generate a diffracted field able to interfere with the scattered field on a large area single-pixel detector [START_REF] Todd W Murray | Detection of ultrasound-modulated photons in diffuse media using the photorefractive effect[END_REF][START_REF] Ramaz | Photorefractive detection of tagged photons in ultrasound modulated optical tomography of thick biological tissues[END_REF][START_REF] Gross | Theoretical description of the photorefractive detection of the ultrasound modulated photons in scattering media[END_REF][START_REF] Lai | Ultrasoundmodulated optical tomography at new depth[END_REF]. The method has a large optical etendue (∼ 10 8 speckle), but is somewhat sensitive to decorrelation, since the response time of the crystal is usually much longer than the speckle correlation. Promising results are expected with Sn2P2S6:Te and Nd:YVO4 crystals, because of their short response times [START_REF] Farahi | Photorefractive acousto-optic imaging in thick scattering media at 790 nm with a sn 2 p 2 s 6: Te crystal[END_REF][START_REF] Jayet | Fast wavefront adaptive holography in nd: Yvo 4 for ultrasound optical tomography imaging[END_REF].

The third type of method uses a pixel array, i.e., a camera, to detect the UOT tagged photons [START_REF] Leveque | Ultrasonic tagging of photon paths in scattering media:? parallel speckle modulation processing[END_REF][START_REF] Yao | Frequency-swept ultrasound-modulated optical tomography in biological tissue by use of parallel detection[END_REF][START_REF] Li | Methods for paralleldetection-based ultrasound-modulated optical tomography[END_REF][START_REF] Li | Ultrasoundmodulated optical tomography of biological tissue by use of contrast of laser speckles[END_REF]. The optical etendue (∼ 10 5 to 10 6 speckle) is then related to the number of pixels of the camera. The camera method has been improved by adapting the heterodyne holography technique [START_REF] Le Clerc | Numerical heterodyne holography with two-dimensional photodetector arrays[END_REF] to the tagged photon detection [START_REF] Gross | Shot-noise detection of ultrasound-tagged photons in ultrasound-modulated optical imaging[END_REF]. By tuning the LO beam frequency near the ultrasonic sideband, and by using a properly adjusted spatial filter, the tagged photons were detected selectively. Moreover, optimal noise detection was obtained, since shot noise is the dominant noise in heterodyne holography [START_REF] Gross | Digital holography with ultimate sensitivity[END_REF][START_REF] Verpillat | Digital holography at shot noise level[END_REF][START_REF] Lesaffre | Noise and signal scaling factors in digital holography in weak illumination: relationship with shot noise[END_REF]. The reference [START_REF] Gross | Shot-noise detection of ultrasound-tagged photons in ultrasound-modulated optical imaging[END_REF] experiment was nevertheless performed with a phantom sample, whose decorrelation is low, and it is generally considered that the heterodyne holography UOT method cannot be used with a living sample, whose speckle decorrelation time τ c is shorter than the time needed to record four camera frames (where τ c = 0.1 ms for the light scattered "in vivo" through a woman's breast [START_REF] Gross | Heterodyne detection of multiply scattered monochromatic light with a multipixel detector[END_REF]). Resink et al. [START_REF] Steffen G Resink | State-of-the art of acousto-optic sensing and imaging of turbid media[END_REF] wrote, for example, "all frames of the one to four phases [i.e. the four frames of the camera] should be taken within the speckle decorrelation time".

In this work, we analyzed theoretically the heterodyne holography UOT detection scheme, and we calculated how untagged photons, speckle noise, shot noise, decorrelation and etendue, affect the UOT signal. By adjusting the calculation parameter, we got results very similar to the ones of [START_REF] Gross | Shot-noise detection of ultrasound-tagged photons in ultrasound-modulated optical imaging[END_REF]. By comparing results obtained with and without decorrelation, we showed that the Resink et al. remark is not valid, and that heterodyne holography remains, with decorrelation, an optimal detection scheme of the tagged photons. Note that this point was already demonstrated for the detection of the untagged photon in experiments done without ultrasound [START_REF] Michael Atlan | Frequency-domain wide-field laser doppler in vivo imaging[END_REF][START_REF] Atlan | Laser doppler imaging, revisited[END_REF][START_REF] Lesaffre | Effect of the photon's brownian doppler shift on the weaklocalization coherent-backscattering cone[END_REF][START_REF] Michael Atlan | Cortical blood flow assessment with frequencydomain laser doppler microscopy[END_REF][START_REF] Atlan | Spatiotemporal heterodyne detection[END_REF][START_REF] Michael Atlan | High-speed wavemixing laser doppler imaging in vivo[END_REF].

To introduce our theoretical discussion, let us consider To analyze theoretically the Fig. 1 experiment, let us define, in plane A and C, the untagged, tagged and LO fields and their respective complex amplitudes, which are slow varying with time t:

E A,U (X, Y, t) = E A,U (X, Y, t) e jωLt + c.c. (1) E A,T (X, Y, t) = E A,T (X, Y, t) e j(ωL+ωUS)t + c.c. E C,U (x, y, t) = E C,U (x, y, t) e jωLt + c.c. E C,T (x, y, t) = E C,T (x, y, t) e j(ωL+ωUS)t + c.c. E C,LO (x, y, t) = E LO e jωLOt + c.c.
where c.c. is the complex conjugate. Here, X, Y are the coordinates in plane A, and x, y in plane C. To simplify theory, we have considered here that E LO do not depend on x, y and t. In plane A, the tagged and untagged photon fields are fully developed speckle. The complex fields E A,T /U (X, Y, t m ) are thus random Gaussian complex quantities uncorrelated from one pixel X, Y to any other X ′ , Y ′ . The random amplitudes E A,T /U (X, Y, t m ) do not depend on t m without decorrelation, and are uncorrelated from one frame (i.e. m) to the next (i.e. m+1) with decorrelation. A lens L, which is located near the camera, and whose focal plane is close to plane A, collects the fields. Because of L, the tagged and untagged fields in planes C and A are related by a Fourier transform

E A,U/T (X, Y ) = ẼC,U/T (k x , k y ) = FFT(E C,U/T (x, y))
where (X, Y ) = (k x , k y )×|CA|/k with k = 2π/λ. To simplify calculations, the discrete Fourier transform (FFT) is made within a calculation grid that fits with the camera pixels in plane C. The pitch ∆x of the discrete coordinates x, y is thus equal to the size of the pixel of the camera. Because of the FFT, the pitch ∆X in plane A is

∆X = 2π|CA| /(N k∆x) ( 2
)
where N is the number of pixels of the camera (N = 1024 typically). The detection etendue is thus

G = S A S D /|CA| 2 = N 2 λ 2
, where S A = |N ∆X| 2 and S C = |N ∆y| 2 are the areas of the calculation grid in plane A and C. The number of modes or speckle grains that can be detected is thus equal to the number of pixels of the camera N 2 . The frame signal I m correspond to the sum of the tagged, untagged and LO photons. To detect the tagged photons, ω LO is made close to the tagged photon frequency ω L + ω US . The LO and the tagged photons thus interfere (and are summed in fields), while the untagged photons do not interfere (and are summed in intensities). We have thus:

I m (x, y) = |E C,T (x, y, t m ) + c m E LO | 2 (3)
+ |E C,U (x, y, t m )| 2
where c = e j(ωLO-ωUS-ωL)∆t is the LO versus tagged photon shift of phase. On the other hand, to detect the untagged photons, ω LO is close to ω L , and I m is given by an equation similar to Eq. 3, where the indexes U and T are exchanged, and where c = e j(ωLO-ωL)∆t . Because of the random nature of light emission and camera photo conversion, the frame signal I m is affected by shot noise yielding I ′ m :

I ′ m (x, y) = I m (x, y) + s(x, y, m) I m (x, y) ) (4)
where the term s √ I m accounts for shot noise. Here, I m must be expressed in photo electron Units per pixel and per frame, while s is a real Gaussian random variable of variance s 2 = 1 uncorrelated with pixels (i.e. X, Y ) and with frames (i.e. m).

The hologram H C is then calculated from I ′ m . Without decorrelation, four phase detection of the tagged photons is made. We have thus: LO (where * is the complex conjugate operator), is proportional to the field E C,T . Because of L, the holograms H A and H C in planes A and C are related by a Fourier transform

H C = M m=0 j m I m (
H A (X, Y ) = HC (k x , k y ) = FFT ( H C (x, y) ) (5)
We have calculated, without decorrelation, the tagged photon hologram H A for M = 12 frames (like in [START_REF] Gross | Shot-noise detection of ultrasound-tagged photons in ultrasound-modulated optical imaging[END_REF]). The calculation is made with

|E LO | 2 = 10 4 , |E A,T | 2 = 1.33 and |E A,U | 2 = 3 × 10 3 photo
electron per frame and per pixel, where is the average over X and Y within the aperture. Note that tagged and untagged energies ( pixels |E| 2 ) are the same in planes A and C, because Eq.2 conserves energy. The coordinates of the upper left and bottom right aperture corners were (125, 50) and (300, 974).

We have displayed in Fig. 2 the arbitrary logarithmic scale intensity image |H A (X, Y )| 2 obtained by calculation (a), and in experiment (b) [START_REF] Gross | Shot-noise detection of ultrasound-tagged photons in ultrasound-modulated optical imaging[END_REF]. Note that the calculation parameters were chosen here to fit with [START_REF] Gross | Shot-noise detection of ultrasound-tagged photons in ultrasound-modulated optical imaging[END_REF]. To further compare our calculation with [START_REF] Gross | Shot-noise detection of ultrasound-tagged photons in ultrasound-modulated optical imaging[END_REF], we have

|H A (X)| 2 = (1/N ) X |H A (X, Y )| 2 (6)
Figure 2 (c,d) show the curves |H A (X)| 2 obtained by calculation (c) and from [START_REF] Gross | Shot-noise detection of ultrasound-tagged photons in ultrasound-modulated optical imaging[END_REF] (d). The curves are normalized with respect to the background that is obtained without tagged and untagged photons and that corresponds to shot noise. The good agreement with [START_REF] Gross | Shot-noise detection of ultrasound-tagged photons in ultrasound-modulated optical imaging[END_REF] validates our theoretical calculation.

In figure 2, the tagged photon signal corresponds to the image of the aperture, i.e. to the bright rectangular zone 1, which is located in the left of images (a,b), because the aperture is located of axis. The aperture corresponds also to the rectangular walls 1, in curves (c,d). On the other hand, the blurred bright zone 2, in the center of (a,b), and the triangular wall 2, in (c,d), corresponds to a parasitic detection of the untagged photon signal, which does not cancel here because of decorrelation (in experiment) and shot noise (in experiment and calculation). The parasitic detection of the LO fields yields a very narrow peak located in the center of the calculation grid, which is only visible on the curves (arrow 3). To the end, shot noise yields a flat background in all points of the images and the curve (zones 4).

To confirm this analyse of Fig. 2, and to evaluate how untagged photons, shot noise and decorrelation affect the UOT signal, we have calculated the curves |H A (X)| 2 without and with decorrelation by switching on and off the tagged and untagged photons. To better compare results obtained without and with decorrelation, the tagged and untagged photon energies were measured Figure 3 shows the curves obtained without (a) and , where α is the number of tagged photons per pixel with α = 1, 0.5, 0.25, 0.125 and 0.0625 for curves 1 to 5. To better visualize them, the curves were plotted in log scale, and the curves were arbitrarily shifted up or down to better separate them from each other. The results of Fig. 4 show that heterodyne holography UOT exhibits roughly the same sensitivity for the detection of the tagged photon with and without decorrelation. The key parameter is the tagged photon energy per pixel during the coherent measurement time with is equal to τ c with decorrelation, and to time M 2π/ω C needed to record the sequence of M frames without decorrelation. A signal versus background ratio of 1 corresponds, with and without decorrelation, to one photo electron per pixel. By averaging over the about 10 5 pixels of the rectangular aperture, the sensitivity limit is improved down to about 1/ √ 10 5 ∼ 1/300 photo electron. This result agrees with what observed experimentally for the detection of the untagged photons [START_REF] Gross | Heterodyne detection of multiply scattered monochromatic light with a multipixel detector[END_REF].

In this letter, we have proposed a theoretical model to describe the detection of the tagged photons in heterodyne holography UOT. This model, which agrees with the results of [START_REF] Gross | Shot-noise detection of ultrasound-tagged photons in ultrasound-modulated optical imaging[END_REF], has been used to calculate how untagged photons, speckle noise, shot noise, decorrelation and etendue, affect the UOT signal. For a given coherent measurement time, which is M 2π/ω C and τ c without and with decorrelation, the model yields the same detection sensitivity, and the same noise floor (one photon electron per pixel). Heterodyne holography UOT is thus shot noise limited. By averaging over the K ∼ 10 5 pixels of the image of the aperture, the detection sensitivity becomes 1/ √ K photo electron per speckle (i.e per etendue λ 2 ). We hope this work will stimulate further UOT development.

This work has been carried out thanks to the supports of the LabEx NUMEV project (n • ANR-10-LABX-20) funded by the "Investissements d'Avenir" French Government program, managed by the French National Research Agency (ANR) and thanks to the ANR grand (reference n • ANR-11-BS04-0017: ICLM).

FIG. 1 .

 1 FIG. 1. Typical UOT setup: BS1, BS2: beam splitter; M: mirror; AOM1,AOM2: acousto optic modulator; PZT: piezoelectric transducer that generates the ultrasonic beam US; a: absorber embedded in the diffusing sample; A: rectangular aperture; L: lens of focal d; C: camera; ELO, ET , EU : LO, tagged and untagged fields.

FIG. 2 .

 2 FIG. 2. Tagged intensity images |HA(X, Y )| 2 (a,b) and curves |HA(X)| 2 (c,d) obtained by calculation (a,c) and from ref.[26] experiment (b,d). The images |HA(X, Y )| 2 (a,b) are displayed in an arbitrary logarithmic scale. The curves |HA(X)| 2 (c,d) are normalized with respect to the background.

  where M frames is a multiple of four) and ω LO = ω L + ω US + ω C /4 yielding c = j in Eq. 3. With decorrelation, two phase detection with two frames is made: H C = I 0 -I 1 , and ω LO = ω L + ω US + ω C /2 yielding c = -1. In H C , the holographic term of interest: E C,T E *

FIG. 3 .

 3 FIG. 3. Curves |HA(X)| 2 calculated without (a) and with decorrelation (b) by switching on and off the tagged and untagged photons signals.

  Figure2 (c,d) show the curves |H A (X)| 2 obtained by calculation (c) and from[START_REF] Gross | Shot-noise detection of ultrasound-tagged photons in ultrasound-modulated optical imaging[END_REF] (d). The curves are normalized with respect to the background that is obtained without tagged and untagged photons and that corresponds to shot noise. The good agreement with[START_REF] Gross | Shot-noise detection of ultrasound-tagged photons in ultrasound-modulated optical imaging[END_REF] validates our theoretical calculation.In figure2, the tagged photon signal corresponds to the image of the aperture, i.e. to the bright rectangular zone 1, which is located in the left of images (a,b), because the aperture is located of axis. The aperture corresponds also to the rectangular walls 1, in curves (c,d). On the other hand, the blurred bright zone 2, in the center of (a,b), and the triangular wall 2, in (c,d), corresponds to a parasitic detection of the untagged photon signal, which does not cancel here because of decorrelation (in experiment) and shot noise (in experiment and calculation). The parasitic detection of the LO fields yields a very narrow peak located in the center of the calculation grid, which is only visible on the curves (arrow 3). To the end, shot noise yields a flat background in all points of the images and the curve (zones 4).To confirm this analyse of Fig.2, and to evaluate how untagged photons, shot noise and decorrelation affect the UOT signal, we have calculated the curves |H A (X)| 2 without and with decorrelation by switching on and off the tagged and untagged photons. To better compare results obtained without and with decorrelation, the tagged and untagged photon energies were measured within a time equal to the recording time of the sequence of M frames (2πM/ω C ) without decorrelation, and to the frame exposure time, which is made equal to τ c , with decorrelation. Calculations were made with |E LO | 2 = 10 4 , M |E A,U | 2 = 10000, M |E A,T | 2 = 1 and M = 12 without decorrelation, and with |E LO | 2 = 10 4 , |E A,U | 2 = 250 and |E A,T | 2 = 1 with decorrelation. Note that the calculations are made with the same tagged photon signal with and without decorrelation (1 photo electron per pixel).Figure3shows the curves obtained without (a) and

FIG. 4 .

 4 FIG. 4. Curves |HA(X)| 2 obtained without (a) and with decorrelation (b). Calculation is made with α = 1 (1), 0.5 (2), 0.25 (3) and 0.125 (4). Plots are made in arbitrary logarithmic scale.

1 1

 1 Laboratoire Charles Coulomb -UMR 5221 CNRS-UM2 Université Montpellier Place Eugène Bataillon 34095 Montpellier, France