Ultrasound tagged light imaging in turbid media in a reflectance geometry

Self-synchronized reflection-mode acousto-optic imaging system utilizing nanosecond laser pulses

We present an acousto-optic imaging system operating in reflection-mode and utilizing a pair of compact, triggerable lasers with 532 and 1064 nm wavelength and nanosecond pulse duration. The system maps the fluence rate distribution of light transmitted through optically scattering samples. The imaging is performed using an acousto-optic probe comprising an ultrasound linear array with attached optical fiber on one side and a camera on the other. The described hardware configuration images samples with access restricted to one side only and ensures mobility of the entire setup. The major challenge of the introduced approach is mitigating the effects of laser parameter instabilities and precise synchronization of ultrasound and laser pulses. We solved this issue by developing an electronic feedback circuit and a microcontroller-based synchronization and control system triggering the ultrasound scanner. Schematics and details regarding control algorithms are introduced. The imaging performance of the system is demonstrated on examples of results obtained for solid, acoustically-homogeneous and optically scattering phantoms with and without light absorbing inclusions present. Adjusting the size and location of the region of interest within the camera sensor matrix and the number of laser pulses illuminating every frame allows for significant improvements in terms of the achievable peak signal to noise ratio. We demonstrate that the developed synchronization algorithm and system play a crucial role in ensuring imaging quality and accuracy.

Reflection-mode acousto-optic imaging using plane wave ultrasound pulses

SIGNIFICANCE

Performance of an acousto-optic imaging system is limited by light fluence rate and acoustic pressure field distributions characteristics. In optically scattering media, the former determines the achievable contrast, whereas the latter the imaging resolution. The system parameters can be shaped by changing relative positions of ultrasound (US) transducer array and optodes. However, in the case of many potential clinical applications, optimization possibilities in this regard are limited, as a sample is accessible from one side only and using a water tank for coupling is not feasible.

AIM

We investigate the possibilities of improving performance of an acousto-optic imaging system operating in reflection mode geometry with linear US array in direct contact with a sample using plane wave instead of focused US pulses.

APPROACH

Differences in acoustic pressure field distributions for various transducer excitation patterns were determined numerically and experimentally. Acousto-optic images of phantoms with and without optically absorbing inclusions were acquired by measuring laser speckle contrast decrease due to the light modulation by plane wave and focused US pulses with different apodization patterns.

RESULTS

The residual acoustic pressure field components occupy relatively large volume and contribute to light modulation. Using nonsteered plane wave US pulses instead of focused ones allows one to mitigate their influence. It also allows one to obtain clear two-dimensional reconstructions of light fluence rate maps by shifting transducer apodization along the lateral direction.

CONCLUSIONS

Using nonsteered plane wave US pulses allows one to achieve better imaging performance than with focused pulses in the assumed system geometry.

Reflection-mode acousto-optic imaging using a one-dimensional ultrasound array with electronically scanned focus

SIGNIFICANCE

Practical implementation of acousto-optic imaging (AOI) encounters difficulties that prevent it from rapid adoption in clinical use. In many practical medical applications, the region of interest may be accessed only from one side, and using a water tank for coupling is not feasible. The solution might be to use reflection-mode imaging with an electronically scanned ultrasound (US) focus. Such an approach, however, entails considerable challenges.

AIM

The possibilities of detecting and localizing light-absorbing inclusions inside turbid media by combining reflection-mode AOI conducted using a one-dimensional US array with electronic scanning of the US focus are investigated experimentally and signal processing algorithms that could be used for this purpose are introduced.

APPROACH

We determine the speckle contrast decrease due to the acousto-optic effect as a function of the US focal point coordinates. Different signal postprocessing techniques are investigated.

RESULTS

A significant decrease in the determined speckle contrast difference values is observed due to the presence of light-absorbing inclusions. However, local minima occur in the plots only under specific conditions. Subtracting individual distributions and determining symmetry deviations allow for localizing the inclusions.

CONCLUSIONS

Detection and localization of optically distinct regions are possible using the introduced approach. Signal postprocessing is required in a general case.

Increased SNR in acousto-optic imaging via coded ultrasound transmission

Acousto-optic imaging (AOI) is a non-invasive method that uses acoustic modulation to map the light fluence inside biological tissue. In many AOI implementations, ultrasound pulses are used in a time-gated measurement to perform depth-resolved imaging without the need for mechanical scanning. However, to achieve high axial resolution, it is required that ultrasound pulses with few cycles are used, limiting the modulation strength. In this Letter, we develop a new approach to pulse-based AOI in which coded ultrasound transmission is used. In coded-transmission AOI (CT-AOI), one may achieve an axial resolution that corresponds to a single cycle, but with a signal-to-noise ratio (SNR) that scales as the square root of the number of cycles. Using CT-AOI with 79 cycles, we experimentally demonstrate over four-fold increase in SNR in comparison to a single-cycle AOI scheme.

Coherent acousto-optic tomography with diffuse light

We propose a method to reconstruct the optical properties of a highly scattering medium from acousto-optic measurements. The method is based on solving an inverse problem with internal data for a system of diffusion equations.

Two-color interpolation of the absorption response for quantitative acousto-optic imaging

Diffuse optical tomography (DOT) is a reliable and widespread technique for monitoring qualitative changes in absorption inside highly scattering media. It has been shown, however, that acousto-optic (AO) imaging can provide significantly more qualitative information without the need for inversion algorithms due to the spatial resolution afforded by ultrasound probing. In this Letter, we show how, by using multiple-wavelength AO imaging, it is also possible to perform quantitative measurements of absorber concentration inside scattering media.

Denaturation process of laccase in various media by refractive index measurements

In this work, we are interested in the denaturation process of a laccase from Tramates versicolor via the determination of the refractive index, the refractive index increment and the specific volume in various media. The measurements were carried out using an Abbe refractometer. We have shown that the refractive index increment values obtained from the slope of the variation of the refractive index vs. Concentration are outside the range refractive index increments of proteins. To correct the results, we have followed the theoretical predictions based on the knowledge of the protein refractive index from its amino acids composition. The denaturation process was studied by calculating the specific volume variation where its determination was related to the Gladstone-Dale and the Lorentz-Lorentz models.

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Refractive index measurements to follow the denaturation process in various medium.

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The denaturation process was studied by using the Gladstone-Dale and the Lorentz-Lorentz models.

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[Morph][F] denatures laccase faster than [pyrr][F].

Acousto-optic effect in random media

We consider the acousto-optic effect in a random medium. We derive the radiative transport equations that describe the propagation of multiply scattered light in a medium whose dielectric permittivity is modulated by an acoustic wave. Using this result, we present an analysis of the sensitivity of an acousto-optic measurement to the presence of a small absorbing inhomogeneity.

Acousto-optical detection of hidden objects via speckle based imaging

Optical detection of objects hidden behind opaque screening layers is a challenging problem. We demonstrate an optically detected echographic-like method that combines collimated acoustic and laser beams. The acoustic waves cross the screening layers, and their back-reflection from the hidden objects is detected through the analysis of a dynamic laser speckle pattern created at the outer surface of the screening layer. Real-time remote detection of moving targets 15 meters away, with a few mm resolution is demonstrated using a very sensitive camera detection scheme.

State-of-the art of acousto-optic sensing and imaging of turbid media

Acousto-optic (AO) is an emerging hybrid technique for measuring optical contrast in turbid media using coherent light and ultrasound (US). A turbid object is illuminated with a coherent light source leading to speckle formation in the remitted light. With the use of US, a small volume is selected,which is commonly referred to as the "tagging" volume. This volume acts as a source of modulated light, where modulation might involve phase and intensity change. The tagging volume is created by focusing ultrasound for good lateral resolution; the axial resolution is accomplished by making either the US frequency, amplitude, or phase time-dependent. Typical resolutions are in the order of 1 mm. We will concentrate on the progress in the field since 2003. Different schemes will be discussed to detect the modulated photons based on speckle detection, heterodyne detection, photorefractive crystal (PRC) assisted detection, and spectral hole burning (SHB) as well as Fabry-Perot interferometers. The SHB and Fabry-Perot interferometer techniques are insensitive to speckle decorrelation and therefore suitable for in vivo imaging. However, heterodyne and PRC methods also have potential for in vivo measurements. Besides measuring optical properties such as scattering and absorption, AO can be applied in fluorescence and elastography applications.

Effects of modulation phase of ultrasound-modulated light on the ultrasound-modulated optical image in turbid media

In this paper, our investigations suggest that the modulation phase of ultrasound-modulated light escaping from the different locations in the ultrasonic field is different. In turbid media, the modulation phase causes the ultrasound-modulated light intensity collected outside the media to fluctuate. However, the ultrasound-modulated optical technology uses the ultrasound-modulated light signals to image. Consequently, the modulation phase affects the quality of ultrasound-modulated optical imaging.

Ultrasound-mediated optical tomography: a review of current methods

Ultrasound-mediated optical tomography (UOT) is a hybrid technique that is able to combine the high penetration depth and high spatial resolution of ultrasound imaging to overcome the limits imposed by optical scattering for deep tissue optical sensing and imaging. It has been proposed as a method to detect blood concentrations, oxygenation and metabolism at depth in tissue for the detection of vascularized tumours or the presence of absorbing or scattering contrast agents. In this paper, the basic principles of the method are outlined and methods for simulating the UOT signal are described. The main detection methods are then summarized with a discussion of the advantages and disadvantages of each. The recent focus on increasing the weak UOT signal through the use of the acoustic radiation force is explained, together with a summary of our results showing sensitivity to the mechanical shear stiffness and optical absorption properties of tissue-mimicking phantoms.

Real-time monitoring of high-intensity focused ultrasound lesion formation using acousto-optic sensing

High-intensity focused ultrasound (HIFU) is a promising modality that is used to noninvasively ablate soft tissue tumors. Nevertheless, real-time treatment monitoring with diagnostic ultrasound still poses a significant challenge since tissue necrosis, in the absence of cavitation or boiling, provides little acoustic contrast with normal tissue. In comparison, the optical properties of tissue are significantly altered accompanying lesion formation. A photorefractive crystal-based acousto-optic (AO) sensing system that uses a single HIFU transducer to simultaneously generate tissue necrosis and pump the AO interaction is used to monitor the real-time optical changes associated with thermal lesions induced in chicken breast ex vivo. It is found that the normalized change in AO response increases proportionally with the volume of necrosis. This study demonstrates AO sensing can identify the onset and growth of lesion formation in real time and, when used as feedback to guide exposures, results in more predictable lesion formation.

Parallel detection of amplitude-modulated, ultrasound-modulated optical signals

We investigated the effect of amplitude-modulated (AM) ultrasound (US) on acousto-optic (AO) signals. A phantom was exposed to both AM US and a green laser, and CCD measurements of speckle contrast were made with various exposure times. The results show that the AO signal oscillates at the AM frequency when the CCD exposure time is a fraction of the AM period and stops oscillating when the CCD exposure time is a multiple of the AM period. The AO signal decreases quickly as the AM frequency increases or peak-peak (pk-pk) amplitude decreases. With 4ms exposure time, 250Hz AM frequency and 1.27MPa pk-pk acoustic pressure, there is an ~30% increase in the AO signal compared with that of CW US. The increase in the signal is likely to be due to the particle oscillation and the induced shear wave as a result of the radiation force generated by the AM US.

Inverse scattering and acousto-optic imaging

We propose a tomographic method to reconstruct the optical properties of a highly scattering medium from incoherent acousto-optic measurements. The method is based on the solution to an inverse problem for the diffusion equation and makes use of the principle of interior control of boundary measurements by an external wave field.

Photoacoustics, thermoacoustics, and acousto-optics for biomedical imaging

Recently there have been significant advances in developing hybrid techniques combining electromagnetic waves with ultrasound for biomedical imaging, namely photoacoustic, thermoacoustic, and acousto-optic (or ultrasound modulated optical) tomography. All three techniques take advantage of tissue contrast offered by electromagnetic (EM) waves, while achieving good spatial resolution in deeper tissue facilitated by ultrasound. In this review the principles of the three techniques are introduced. A description of existing experimental and image reconstruction techniques is provided. Some recent key developments are highlighted and current issues in each of the areas are discussed.

Detection of the tagged or untagged photons in acousto-optic imaging of thick highly scattering media by photorefractive adaptive holography

We propose an original adaptive wavefront holographic setup based on the photorefractive effect (PR), to make real-time measurements of acousto-optic signals in thick scattering media, with a high flux collection at high rates for breast tumor detection. We describe here our present state of the art and understanding on the problem of breast imaging with PR detection of the acousto-optic signal.

Ultrasound-modulated optical microscopy

We demonstrate that microscopic imaging is feasible in ultrasound-modulated optical tomography (UOT) of soft biological tissues, using a high-frequency focused ultrasound transducer with a 75-MHz central frequency. Our experiments in tissue mimicking phantoms show that at an imaging depth of about 2 mm, an axial resolution better than 30 microm can be achieved, whereas the lateral resolution is 38 microm. A long-cavity scanning confocal Fabry-Perot interferometer (CFPI) is used for real-time detection of multiply scattered light modulated by high-frequency ultrasound pulses propagating in an optically scattering medium. We also compare the performances of various high-frequency focused ultrasound transducers with central frequencies of 15 MHz, 30 MHz, 50 MHz, and 75 MHz. The comparison is based on two-dimensional (2-D) images of optically absorbing objects positioned at a few millimeters depth below the surface of both optically scattering phantoms and soft biological tissue samples. Our experimental results show that modulation depth and image contrast decrease with an increase in ultrasound frequency. In addition, we use analytical calculations to show that modulation depth decreases with increasing ultrasound frequency.

Correlation transfer equation for multiply scattered light modulated by an ultrasonic pulse

We develop a temporal correlation transfer equation (CTE) and a Monte Carlo algorithm (MC) for multiply scattered light modulated by an ultrasonic pulse propagating in an optically scattering medium, where the ultrasound field can be nonuniform and the medium can have spatially heterogeneous distribution of optical parameters. The CTE and MC can be used to obtain the time-varying specific intensity and the spatial distribution of the time-dependent power spectral density, respectively, of ultrasound-modulated light. We expect the CTE and MC to be applicable for estimation of contrast and resolution in a wide spectrum of conditions in ultrasound-modulated optical tomography of soft biological tissues.

Correlation transfer equation for ultrasound-modulated multiply scattered light

In this paper, we develop a temporal correlation transfer equation (CTE) for ultrasound-modulated multiply scattered light. The equation can be used to obtain the time-varying specific intensity of light produced by a nonuniform ultrasound field in optically scattering media that have a heterogeneous distribution of optical parameters. We also develop a Monte Carlo algorithm that can provide the spatial distribution of the optical power spectrum in optically scattering media with focused ultrasound fields, and heterogeneous distributions of optically scattering and absorbing objects. Derivation of the CTE is based on the ladder diagram approximation of the Bethe-Salpeter equation that assumes moderate ultrasound pressures. We expect the CTE to be applicable to a wide spectrum of conditions in the ultrasound-modulated optical tomography of soft biological tissues.

Correlation transfer and diffusion of ultrasound-modulated multiply scattered light

We develop a temporal correlation transfer equation (CTE) and a temporal correlation diffusion equation (CDE) for ultrasound-modulated multiply scattered light. These equations can be applied to an optically scattering medium with embedded optically scattering and absorbing objects to calculate the power spectrum of light modulated by a nonuniform ultrasound field. We present an analytical solution based on the CDE and Monte Carlo simulation results for light modulated by a cylinder of ultrasound in an optically scattering slab. We further validate with experimental measurements the numerical calculations for an actual ultrasound field. The CTE and CDE are valid for moderate ultrasound pressures and on a length scale comparable with the optical transport mean-free path. These equations should be applicable to a wide spectrum of conditions for ultrasound-modulated optical tomography of soft biological tissues.

Modulation of multiply scattered coherent light by ultrasonic pulses: an analytical model

We present an analytical solution for the acousto-optical modulation of multiply scattered light in a medium irradiated with a train of ultrasound pulses. Previous theory is extended to cases where the ultrasound-induced optical phase increments between the different scattering events are strongly correlated, and it is shown that the approximate similarity relation still holds. The relation between the ultrasound induced motions of the background fluid and the optical scatterers is generalized, and it is shown that correlation exists between the optical phase increments that are due to the scatterer movement and the optical phase increments that are due to the modulation of the optical index of refraction. Finally, it is shown that compared with the spectrum of ultrasound pulses, the power spectral density of acousto-optically modulated light is strongly attenuated at the higher ultrasound frequencies.

Ultrasound-modulated light tomography assessment of osteoporosis

We present a preliminary series of clinical experiments showing that ultrasound modulation of light in tissues allows tissue properties to be determined well inside the tissue. In this series of clinical experiments the optical scattering coefficient determined by the optical technique is compared with the bone density obtained by dual x-ray absorption. A correlation of 0.84 (p = 0.005) was found for a limited number of patients, showing the potential of this technique for the assessment of osteoporosis.

High-resolution ultrasound-modulated optical tomography in biological tissues

We present a novel implementation of high-resolution ultrasound-modulated optical tomography that, based on optical contrast, can image several millimeters deep into soft biological tissues. A long-cavity confocal Fabry-Perot interferometer, which provides a large etendue and a short response time, was used to detect the ultrasound-modulated coherent light that traversed the scattering biological tissue. Using 15-MHz ultrasound, we imaged with high-contrast light-absorbing structures placed >3 mm below the surface of chicken breast tissue. The resolution along the axial and the lateral directions with respect to the ultrasound propagation direction was better than 70 and 120 microm, respectively. The resolution can be scaled down further by use of higher ultrasound frequencies. This technology is complementary to other imaging technologies, such as confocal microscopy and optical-coherence tomography, and has the potential for broad biomedical applications.

Shot-noise detection of ultrasound-tagged photons in ultrasound-modulated optical imaging

We propose a new detection method for ultrasound-modulated optical tomography that allows us to perform parallel speckle detection with optimum shot-noise sensitivity, using a CCD camera. Moreover, we show that making use of a spatial filter system allows us to fully filter out speckle decorrelation noise. This method is confirmed by a test experiment.

In vivo demonstration of the ultrasound-modulated light technique

We present, to our knowledge for the first time, results of ultrasound-modulated light signals on living tissues. In particular, we analyze, both theoretically and experimentally, the effect of speckle fluctuations on the signal. We find that two different kinds of noise compete--shot noise and speckle noise--and are present at different levels in static phantoms and ex vivo tissue samples on the one hand and in dynamic phantoms and living tissues on the other hand.

Pulsed ultrasound-modulated light tomography

Ultrasound-modulated light tomography is a new technique that combines laser light and ultrasound to provide a representation of the light density inside turbid media. We present a method that can produce two- or three-dimensional light density representations with standard ultrasonic pulses. This technique should allow simple, direct fusion of ultrasonic images with optical tomography.

Transmission- and side-detection configurations in ultrasound-modulated optical tomography of thick biological tissues

Ultrasound-modulated optical tomography of thick biological tissues was studied based on speckle-contrast detection. Speckle decorrelation was investigated with biological tissue samples of various thicknesses. Images of optically absorbing objects buried in biological tissue samples with thicknesses up to 50 mm were obtained in a transmission-detection configuration. The image contrast was more than 30%, and the spatial resolution was approximately 2 mm. In addition, a side-detection scheme along with two specific configurations were examined, and the advantages were demonstrated. Experimental results implied feasibility of applying the ultrasound-modulation technique to characterize optical properties in inhomogeneous biological tissues.

Ultrasound detection through turbid media

Optical coherence-domain reflectometry and laser-based ultrasound detection have been combined with the use of adaptive optics to detect ultrasound through turbid media. The dynamic hologram in a photorefractive quantum-well device performs as a coherence gate that eliminates multiply scattered background. Quadrature homodyne detection conditions are selected by the choice of center wavelength of the pulse spectrum, requiring no active stabilization or feedback. A depth resolution of 30 microm was achieved, with a pulse duration of nominally 120 fs for ultrasound detection through turbid media up to optical thicknesses of 11 mean free scattering lengths.

Ultrasound-modulated optical tomography of biological tissue by use of contrast of laser speckles

Ultrasound-modulated optical tomography based on the measurement of laser-speckle contrast was investigated. An ultrasonic beam was focused into a biological-tissue sample to modulate the laser light passing through the ultrasonic column inside the tissue. The contrast of the speckle pattern formed by the transmitted light was found to depend on the ultrasonic modulation and could be used for imaging. Variation in the speckle contrast reflected optical inhomogeneity in the tissue. With this technique, two-dimensional images of biological-tissue samples of as much as 25 mm thick were successfully obtained with a low-power laser. The technique was experimentally compared with speckle-contrast-based, purely optical imaging and with parallel-detection imaging techniques, and the advantages over each were demonstrated.

Ultrasonic modulation of multiply scattered coherent light: an analytical model for anisotropically scattering media

In this work, we have calculated analytically the temporal autocorrelation function of the electrical field component of multiply scattered coherent light transmitted through an anisotropically scattering media irradiated with a plane ultrasonic wave. The accuracy of the analytical solution is verified with an independent Monte Carlo simulation for different values of the ultrasonic and optical parameters. The analytical model shows that an approximate similarity relation exists; if the reduced scattering coefficient is unchanged regardless of the mean cosine of the scattering angle, the autocorrelation function remains approximately the same.

Autocorrelation of scattered laser light for ultrasound-modulated optical tomography in dense turbid media

Based on measurement of the intensity autocorrelation function, a new method to determine the modulation depth of scattered laser light modulated by an ultrasonic wave in turbid media was applied to ultrasound-modulated optical tomography. Good signal-to-noise ratios and high sensitivities were demonstrated. Images of double optically absorbing objects buried in a highly optically scattering gel sample were obtained. The contrast was more than 10%, and the spatial resolution was approximately 2 mm.

Nonlinear effects in acousto-optic imaging

Acousto-optic (AO) imaging is a promising technique that is able to reveal optical properties in the millimeter range inside scattering media by tagging the photon paths with an ultrasonic beam. To increase both the contrast and the resolution of the AO images, we have explored the possibility of using the nonlinear response of the speckle modulation. Variation of the second-harmonic signal as the square of the ultrasonic amplitude has been found, and strong reduction of the tagged zone size has been demonstrated.

Direct, noninvasive detection of photon density in turbid media

Ultrasound tagging of light provides a unique way to probe photon density inside turbid media. We show that this technique allows one to probe the well-known banana-shaped photon density noninvasively, giving rise to a new tool for modeling diffusive photon propagation. Moreover, we show that this technique is quantitative and allows one to get a precise determination of the absorbing constituents inside the turbid medium.

Methods for parallel-detection-based ultrasound-modulated optical tomography

The research reported here focuses on ultrasound-modulated optical tomography based on parallel speckle detection. Four methods were investigated for signal acquisition and analysis, in which laser speckle statistics was applied. The methods were compared with the previously used four-phase method in the imaging of all-biological-tissue samples, in which the buried objects were also biological tissues. The image quality obtained with these methods was comparable with that obtained with the four-phase method; in addition, these methods have advantages in reducing acquisition time and improving the signal-to-noise ratio.

Mechanisms of ultrasonic modulation of multiply scattered coherent light: a Monte Carlo model

A Monte Carlo model of the ultrasonic modulation of multiply scattered coherent light in scattering media is provided. The model is based on two mechanisms: the ultrasonic modulation of the index of refraction, which causes a modulation of the optical path lengths between consecutive scattering events, and the ultrasonic modulation of the displacements of scatterers, which causes a modulation of optical path lengths with each scattering event. Multiply scattered light accumulates modulated optical path lengths along its path. Consequently, the intensity of the speckles that are formed by the multiply scattered light is modulated. The contribution from the index of refraction is comparable with the contribution from displacement when the acoustic-wave vector is less than a critical fraction of the transport mean free path and becomes increasingly greater than the contribution from displacement beyond this critical point. This Monte Carlo model agrees well with an independent analytical model for isotropically scattering media. Both mechanisms are coherent phenomena, requiring the use of a coherent light source.

Detection of inhomogeneities with ultrasound tagging of light

Ultrasound modulated light for optical tomography is very useful, since it can provide three-dimensional data with minimal mathematical processing. Although several experimental studies have shown the potential of this method, the link between the ultrasound location and the modulated signal intensity at the detector is not yet fully understood. We derive an analytical formula relating the position of the ultrasound transducer and the optical signal at the detector. We also derive an expression for the signal-to-shot-noise ratio as a function of the transducer position. We show that in certain conditions this ratio is only slowly decreasing as a function of the light penetration depth, which makes this technique attractive for optical tomography.

Mechanisms of ultrasonic modulation of multiply scattered coherent light: an analytic model

An analytic model of the ultrasonic modulation of multiply scattered coherent light in scattering media is developed based on two mechanisms: the ultrasonic modulation of the index of refraction and the ultrasonic modulation of the displacements of Rayleigh scatterers. In water solutions, for example, the first mechanism is slightly less important than the second mechanism when the scattering mean free path is less than a critical fraction (0.0890) of the acoustic wavelength, and it becomes increasingly more important beyond this point. This model agrees well with an independent Monte Carlo model.