Imaging with multiply scattered light
Imaging techniques working with ballistic or single-scattered photons have the great advantage of having optical resolution. Yet they are limited to a few transport mean free paths which corresponds to a few millimetres in tissues. If one wants to image deeper in biological sample, several techniques using multiply scattered light exist. However, because of the nature of multiply-scattered photons, extracting the information from the outgoing light is not as straightforward.
The imaging techniques roughly fall into two categories: the techniques that use detailed model of the light propagation to reconstruct optical contrast inside a scattering medium (NIRS, DOT…) and methods that combine light with another type of wave to extract the information (Photoacoustics, Acousto-optics…).
Near Infra-Red Spectroscopy
Near Infra-Red Spectroscopy (NIRS) is a technique which can probe the optical absorption at a few centimetres depth inside multiply scattering medium. A sample is illuminated by a near infra-ref source and light is collected on the same side by a detector. The most likely trajectories form a « banana shape » between the source and the detector which probes the medium at a depth related to the distance between source and detector, as shown on 1.8(a). NIRS is not properly speaking an imaging technique since it sensitive to the average absorption on a wide area, but it is possible to probe at different depths by changing the source-detector distance. NIRS is also used to perform spectroscopy by changing the wavelength of the input source. Since it has been proposed in 1977  is it commonly used to measure the oxygenation of blood, and a variety of modalities have been developed (e.g. NIRS-CW (continuous wave), NIRS-TD (time domain) or NIRS-FD (frequency domain)).
Diffuse Optical Tomography
The recent advances in modelling light propagation in highly scattering media lead to the development of Diffuse Optical Tomography (DOT) . The basic principle is to surround the medium with a high number of near infra-red sources and detectors, as shown on figure 1.8(b), extracted from  and to detect the multiply scattered photons on the boundaries of the medium. Then the propagation of light between sources and detectors is simulated using the Radiative Transfer Equation or one of its approximations and the collected data are used with a model based inversion to reconstruct the optical properties .
DOT has been implemented in several commercial devices and can be used for breast cancer imaging  or mapping of the brain haemodynamics for functional imaging . DOT can give 3D images of the absorption or scattering coefficient with a resolution which is usually of the order of 5 to 10 mm which is sometimes too large to detect early stage tumours.
Principle of acousto-optic imaging
The acousto-optic effect was demonstrated in transparent crystals at the same time in 1932 by two independent teams, Debye and Sears , and Lucas and Biquard . They showed that when an ultrasonic wave propagates inside a crystal, it modulates the density and thus the refractive index of the medium at the frequency of the ultrasound, fUS. Since the wave is periodic, it will create a refractive index grating with a period equal to the wavelength of the ultrasound, Λ. Consequently, if the crystal is illuminated, light will be diffracted on this grating. If the grating can be considered as infinitely thin, the diffraction directions are given by: sin θp = p λ Λ , p = 0,±1,±2, …, (2.1).
where λ is the wavelength of light, and p is an integer called the diffraction order. This regime is called the Raman-Nath diffraction regime and in addition to the light diffraction, the frequency of each diffracted beam is shifted by harmonics of the ultrasound frequency: fp = fL + pfUS.
Light modulation in scattering media
In a scattering medium, the interaction between light and ultrasound is different from a case without scattering. Since the trajectories of photons are complex and random, it doesn’t make sense to talk about beam deflection any more. However, phase modulation of light still occurs and is driven by two effects which both contribute to the modulation of the optical path: the modulation of the scatterers’ position and the modulation of the refractive index of the medium. The study and modelling of the acousto-optic modulation only started in the 1990s. At first only one of the phenomena was taken into account, usually the vibration of the scatterers. The first theory was published by Leutz and Maret in 1995 . They modelled the light modulation by considering Brownian motion of the scatterers and a collective motion due to ultrasound. This model was completed by Kempe et al. in 1997  by taking into account the inhomogeneity of the ultrasound field. In 2001, Wang analytically modelled the modulation of light by considering both the scatterers movements and the modulation of the refractive index .
Modulation of the scatterers positions
The first phenomenon contributing to the modulation of light is the modulation of the scatterers’ positions. When an acoustic wave propagates in a scattering medium, the scatterers will oscillate around their rest position, thus shifting the frequency of the light by Doppler effect. Let us consider an ultrasonic wave propagating in the sample. For simplification purposes, this wave is assumed to be a continuous plane pressure wave insonifying the whole medium. PUS(r, t) = AUS sin(KUS · r − ωUSt), (2.4). where AUS represents the amplitude of the acoustic wave, KUS is the acoustic wave vector, r is a vector representing the spatial coordinates, ωUS is the angular frequency of the ultrasound, and t is the time variable.
The phase difference associated to the movement of the jth scatterer is then written  Φj s(t) = −n0k0(kj+1 − kj)AUS sin(KUS · rj − ωUSt), (2.5) where n0 is the optical index of the medium, kj and kj+1 are respectively the unitary wave vectors of the light after scatterers j and j + 1, and rj is the position of the jth scatterer.
Coherent acousto-optic modulation
When the sample is illuminated by a coherent light, the light exiting the sample will form a speckle pattern because of random interferences between the different paths inside the sample. This speckle will be modulated by the ultrasound but because of the random walk inside the medium is it difficult to deduce any further information. To better understand the effect of the acoustic modulation on light exiting the medium, several models have been developed and perfected over the years [53, 69, 70].
Using the auto-correlation function of the electric field and the Wiener-Khinchin theorem, the amplitude of the light modulation can be retrieved. This theorem states that the power spectral density is equal to the Fourier transform of the auto-correlation function of the electric field, G(τ ). The power of the component modulated at mωUS can thus be written: Ψp = ωUS 2 Z 2 !US 0 G(τ ) cos(mωUSτ )dτ m ∈ Z.
Image formation and resolution
We have seen that the number of tagged photons is proportional to the local light fluence in the volume of the ultrasound. The shape of this volume will thus define the resolution of the technique. Several approaches have been considered to construct AO images by sending different acoustic sequences. Each comes with a different image formation, resolution and signal to noise ratio. We will here give a brief overview of the different techniques.
Single-element transducer and continuous wave The first proof- of-concept were obtained with single-element focused transducers and continuous wave ultrasound [56, 71]. In this configuration, photons are tagged in the whole acoustic column along the direction of ultrasound propagation, meaning that the resolution in this direction is equal to the entire medium depth. The resolution in the transverse directions correspond approximately to the size of the focus (a few acoustic wavelengths). To construct an image, the transducer needs to be mechanically translated to probe different regions of the medium. A solution to obtain a longitudinal resolution is to rotate the transducer around the sample to form a tomographic image .
Pulsed ultrasound The most straightforward way to recover the longitudinal resolution is to use pulsed ultrasound instead of continuous wave. By sending a few-cycles-long pulse, the tagging volume is greatly reduced. The lateral resolution is still given by diffraction (a few acoustic wavelengths) but the longitudinal resolution is now the length of a pulse. Given the difference between the speed of light and the speed of sound, it is possible with a fast detector to follow optically the propagation of the pulse. To construct a line of the image, the number of tagged photons is recorded while the pulse travels through the medium as shown on figure 2.5. The pulse is then translated to form an new line and an image is constructed by stacking the lines together.
Table of contents :
1 Light propagation in thick scattering media
1.1 Of the interest of optical imaging
1.2 Light-matter interaction
1.2.3 Orders of magnitude in biological tissues
1.3 Light propagation in scattering media
1.3.1 Propagation regimes
1.4 Optical imaging in scattering media
1.4.1 Imaging with ballistic light
1.4.2 Imaging with multiply scattered light
2 Acousto-optic imaging
2.1 Principle of acousto-optic imaging
2.1.1 The acousto-optic effect
2.1.2 Applications of the technique
2.2 Light modulation in scattering media
2.2.1 Modulation of the scatterers positions
2.2.2 Modulation of the refractive index
2.2.3 Coherent acousto-optic modulation
2.3 Image formation and resolution
2.4 Detection of tagged photons
2.4.1 Incoherent methods
2.4.2 Coherent methods
2.5 Conclusion on acousto-optic imaging
3 Photorefractive detection of tagged photons
3.1 The photorefractive effect
3.1.2 The band-transport model
3.1.3 Characteristics of the photorefractive effect
3.2 Detection of the acousto-optic signal
3.2.1 Two-wave mixing
3.2.2 Acousto-optic signal detection
3.2.3 Experimental configuration
3.3 Choice of the crystal
3.3.1 Measurement of the crystals characteristics
3.3.2 SPS vs. ZnTe
4 Acousto-optic imaging in reflection mode
4.1 Fusion between conventional ultrasound and Acousto-Optic Imaging
4.2 Monte Carlo Simulations
4.2.1 The algorithm
4.2.2 Influence of source-detector distance
4.3 Imaging in reflection mode
4.3.1 The imaging setup
4.3.2 Influence of source-detector distance
4.3.3 Towards a handheld probe
4.3.4 A multiple detector approach
4.4 Towards In Vivo imaging
5 Inverse Problems for Quantitative Acousto Optic Imaging
5.1 Inverse problems for medical imaging
5.1.1 Definition of an inverse problem
5.1.2 Reconstruction algorithms
5.1.3 Application to imaging
5.2 Inverse problems for acousto-optic imaging
5.2.1 The inverse problem and model corrections
5.2.2 Reconstruction of the absorption coefficient
5.2.3 Limitations of the current algorithm
5.2.4 Next steps for quantitative acousto-optic imaging
Conclusions and perspectives