Using light scattering to measure the temporal coherence of optical sources

We demonstrate a simple and robust method for characterizing the temporal coherence of statistically stationary optical sources by using dynamic light scattering. Measurement of the contrast of the fluctuating speckle pattern produced by two counterpropagating beams incident on a scattering medium is used to evaluate their mutual coherence. Important features of this method are high statistical accuracy, the ability to compensate for imperfect spatial coherence, and the possibility of characterizing milliwatt-level optical beams with a wide range of spectral widths. As an example, the squared magnitude of the field autocorrelation function for light emitted by a broadband argon-ion laser is obtained.     

Attenuation and impulse response for multiple scattering of light in atmospheric clouds and aerosols

Model phase functions for atmospheric clouds and aerosols typically comprise a narrow forward lobe (corona), a broad diffuse background, and a narrow backscattering peak (glory), which can reach relatively high values, especially for polyhedral scattering particles, such as hexagonal ice columns and plates. The influence of these three major components on the asymptotic and transient attenuation of the scattered light is compared for several analytic phase functions to assess the dependence of radiative transfer in clouds and aerosols on the choice of phase function. The impulse response (temporal evolution of the angular intensity distribution) is sensitive to the higher moments of the phase function and could prove to be a useful technique for inferring the optical scattering parameters of clouds and aerosols.     

Measurement of radio-frequency temporal phase modulation using spectral interferometry

We present an optical method to measure radio-frequency electro-optic phase modulation profiles by employing spectrum-to-time mapping realized by highly chirped optical pulses. We directly characterize temporal phase modulation profiles of up to 12.5 GHz bandwidth, with temporal resolution comparable to high-end electronic oscilloscopes. The presented optical set-up is a valuable tool for direct characterization of complex temporal electro-optic phase modulation profiles, which is indispensable for practical realization of deterministic spectral-temporal reshaping of quantum light pulses     

On Two Numerical Techniques for Light Scattering by Dielectric Agglomerated Structures

Smoke agglomerates are made of many soot sphcres, and their light scattering response is of interest in fire research. The numerical techniques chiefly used for theoretical scattering studies are the method of moments and the coupled dipole moment. The two methods have been obtained in this tutorial paper directly from the monochromatic Maxwell curl equations and shown to be equivalent. The effects of the finite size of the primary spheres have been numerically delineated.     

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.     

Excitation with a focused, pulsed optical beam in scattering media: diffraction effects

To gain a better understanding of the spatiotemporal problems that are encountered in two-photon excitation fluorescence imaging through highly scattering media, we investigate how diffraction affects the three-dimensional intensity distribution of a focused, pulsed optical beam propagating inside a scattering medium. In practice, the full potential of the two-photon excitation fluorescence imaging is unrealized at long scattering depths, owing to the unwanted temporal and spatial broadening of the femtosecond excitation light pulse that reduces the energy density at the geometric focus while it increases the excitation energy density in the out-of-focus regions. To analyze the excitation intensity distribution, we modify the Monte Carlo-based photon-transport model to a semi-quantum-mechanical representation that combines the wave properties of light with the particle behavior of the propagating photons. In our model the propagating photon is represented by a plane wave with its propagation direction in the scattering medium determined by the Monte Carlo technique. The intensity distribution in the focal region is given by the square of the linear superposition of the various plane waves that arrive at different incident angles and optical path lengths. In the absence of scattering, the propagation model yields the intensity distribution that is predicted by the Huygens-Fresnel principle. We quantify the decrease of the energy density delivered at the geometric focus as a function of the optical depth to the mean-free-path ratio that yields the average number of scattering events that a photon encounters as it propagates toward the focus. Both isotropic and anisotropic scattering media are considered. Three values for the numerical aperture (NA) of the focusing lens are considered: NA = 0.25, 0.5, 0.75.     

Direct calculation of the moments of the distribution of photon time of flight in tissue with a finite-element method

Modeling of the full temporal behavior of photons propagating in diffusive materials is computationally costly. Rather than deriving intensity as a function of time to fine sampling, we may consider methods that derive a transform of this function. To derive the Fourier transform involves calculation in the (complex) frequency domain and relates to intensity-modulated experiments. We consider instead the Mellin transform and show that this relates to the moments of the original temporal distribution. A derivation of the Mellin transform given the Fourier transform that permits closed-form derivations of the temporal moments for various simple geometries is presented. For general geometries a finite-element method is presented, and it is demonstrated that the computational cost to produce the nth moment is the same as producing the first n temporal samples of the original function.     

Efficient numerical representation of the optical field for the propagation of partially coherent radiation with a specified spatial and temporal coherence function

We propose a method to narrow the gap between the rigorous methods for the propagation of partially coherent light, which require excessive computational capacity, and the numerical methods used in practical engineering applications, where it is not clear how to handle spatial and temporal coherence in a statistically correct manner. As is the case for the latter methods, the numerical method described can deal with fields with a large spatial and temporal extent, which is necessary in practical applications such as laser fusion or optical lithography. However, the method also takes a few steps toward a more rigorous, yet efficient, representation of the optical field, which depends on detailed specified coherence properties of the radiation. The described method uses a set of independent monochromatic fields at different oscillation frequencies. The frequencies are chosen such that the statistical properties of the integrated intensity closely resemble those from a full-time trace treatment. Finally, we demonstrate the capabilities and limitations of the method with a few numerical examples of the propagation of a large field with a specified spatial and temporal coherence.     

Tomographic image reconstruction from optical projections in light-diffusing media

The recent developments in light generation and detection techniques have opened new possibilities for optical medical imaging, tomography, and diagnosis at tissue penetration depths of ~10 cm. However, because light scattering and diffusion in biological tissue are rather strong, the reconstruction of object images from optical projections needs special attention. We describe a simple reconstruction method for diffuse optical imaging, based on a modified backprojection approach for medical tomography. Specifically, we have modified the standard backprojection method commonly used in x-ray tomographic imaging to include the effects of both the diffusion and the scattering of light and the associated nonlinearities in projection image formation. These modifications are based primarily on the deconvolution of the broadened image by a spatially variant point-spread function that is dependent on the scattering of light in tissue. The spatial dependence of the deconvolution and nonlinearity corrections for the curved propagating ray paths in heterogeneous tissue are handled semiempirically by coordinate transformations. We have applied this method to both theoretical and experimental projections taken by parallel- and fan-beam tomography geometries. The experimental objects were biomedical phantoms with multiple objects, including in vitro animal tissue. The overall results presented demonstrate that image-resolution improvements by nearly an order of magnitude can be obtained. We believe that the tomographic method presented here can provide a basis for rapid, real-time medical monitoring by the use of optical projections. It is expected that such optical tomography techniques can be combined with the optical tissue diagnosis methods based on spectroscopic molecular signatures to result in a versatile optical diagnosis and imaging technology.     

Vesicle sizing by static light scattering: a Fourier cosine transform approach

A Fourier cosine transform method, based on the Rayleigh-Gans-Debye thin-shell approximation, was developed to retrieve vesicle size distribution directly from the angular dependence of scattered light intensity. Its feasibility for real vesicles was partially tested on scattering data generated by the exact Mie solutions for isotropic vesicles. The noise tolerance of the method in recovering unimodal and biomodal distributions was studied with the simulated data. Applicability of this approach to vesicles with weak anisotropy was examined using Mie theory for anisotropic hollow spheres. Aprimitive theory about the first four moments of the radius distribution about the origin, excluding the mean radius, was obtained as an alternative to the direct retrieval of size distributions.     

On tomographic reconstruction of the quantum state of a two-mode light field

Abstract

We consider the state reconstruction of an optical two-mode light field from sum quadrature distributions measured with a single balanced homodyne detector. New explicit formulas for the pattern functions necessary to reconstruct the density matrix of the two-mode field in the photon-number basis are derived. Moreover, an expression of the measured quadature distribution in terms of the two-mode normally ordered moments is given and the determination of the moments from it is discussed.     

Modeling of scattering and depolarizing electro-optic devices. II. Device simulation

We describe a simple method for performing accurate computer simulation and modeling of arbitrary-geometry electro-optic (EO) devices. We use a material EO model that includes the effects of scattering and depolarization as well as the change in the index of refraction. Finite-element analysis is used to determine the electrostatic field distribution for EO device designs. Attenuation of the transmitted light intensity as a result of scattering is modeled as an exponential function, and the intensity of transmitted depolarized light is shown to be a function of the scattering intensity. The total optical transmittance is determined by integration of these values over all the elements in the path of the propagating light. Lanthanum-modified lead zirconate titanate-based surface-electrode and transverse-electrode EO devices are designed and fabricated. Their experimentally measured performance is found to be in excellent agreement with our computer-simulation results.     

Applications of short-coherence digital holography in microscopy

We present an optical system based on short-coherence digital holography suitable for the imaging of three-dimensional microscopic objects. The short temporal coherence properties of the light source allow optical sectioning of the sample. Proper reconstruction of different layers within biological samples is possible up to a depth of a few hundred micrometers, but multiple scattering and inhomogeneities in the refractive index reduce the imaging quality for deeper layers. We have studied the possibility of numerically correcting sample-induced aberrations, and we now propose a method of improving image quality. Numerical simulations and preliminary experimental results show that compensation of these aberrations is possible to some extent.     

Quantitative spectrophotometry of scattering media via frequency-domain and constant-intensity measurements

The attenuation of light by scattering and absorbing media is nonlinearly dependent upon the absorption coefficient, since detected light has experienced many different flight times. The frequency response of such a sample to modulated light is also nonlinear. We derive an expression for the attenuation that includes both the absorption coefficient and the modulation frequency. Its form is a power series whose coefficients are the cumulants of the temporal point-spread function (TPSF). Recasting this expression in terms of intensity leads to a similar expression with the cumulants replaced by the moments of the TPSF. A means of exploiting this relationship to produce estimates of the absolute concentration of the absorbing species is suggested.     

Ultrashort pulse propagation through a strongly scattering medium: simulation and experiments

Forward light scattering of femtosecond pulses through strongly scattering media is investigated experimentally and numerically. Computations are based on a semi-Monte Carlo method including polarization effects when experiments depend on a Ti:sapphire regenerative amplifier (100 fs, 1 kHz, 1 mJ@ 800 nm). The temporal separation between ballistic light and scattered light is exhibited and used to perform optical depth measurements up to 22 (transmission factor of approximately 10(-10)). Quantitative comparisons between experiments and Monte Carlo simulations show a good agreement. Temporal forward scattered light evolutions with concentration and particle size are presented. Numerical results show that the early scattered light contains information on particle size, opening the way to particle sizing in strongly scattering media.     

Perturbation theory for the diffusion equation by use of the moments of the generalized temporal point-spread function. I. Theory

We approach the perturbative solution to the diffusion equation for the case of absorbing inclusions embedded in a heterogeneous scattering medium by using general properties of the radiative transfer equation and the solution of the Fredholm equation of the second kind given by the Neumann series. The terms of the Neumann series are used to obtain the expression of the moments of the generalized temporal point-spread function derived in transport theory. The moments are calculated independently by using Monte Carlo simulations for validation of the theory. While the mixed moments are correctly derived from the theory by using the solution of the diffusion equation in the geometry of interest, in order to obtain the self moments we should reframe the problem in transport theory and use a suitable solution of the radiative transfer equation for the calculation of the multiple integrals of the corresponding Neumann series. Since the rigorous theory leads to impractical formulas, in order to simplify and speed up the calculation of the self moments, we propose a heuristic method based on the calculation of only a single integral and some scaling parameters. We also propose simple quadrature rules for the calculation of the mixed moments for speeding up the computation of perturbations due to multiple defects. The theory can be developed in the continuous-wave domain, the time domain, and the frequency domain. In a companion paper [J. Opt. Soc. Am. A23, 2119 (2006)] we discuss the conditions of applicability of the theory in practical cases found in diffuse optical imaging of biological tissues.     

Spatial coherence effect on layer thickness determination in narrowband full-field optical coherence tomography

Longitudinal spatial coherence (LSC) is determined by the spatial frequency content of an optical beam. The use of lenses with a high numerical aperture (NA) in full-field optical coherence tomography and a narrowband light source makes the LSC length much shorter than the temporal coherence length, hence suggesting that high-resolution 3D images of biological and multilayered samples can be obtained based on the low LSC. A simplified model is derived, supported by experimental results, which describes the expected interference output signal of multilayered samples when high-NA lenses are used together with a narrowband light source. An expression for the correction factor for the layer thickness determination is found valid for high-NA objectives. Additionally, the method was applied to a strongly scattering layer, demonstrating the potential of this method for high-resolution imaging of scattering media.     

Stray-light corrections in integrating-sphere measurements on low-scattering samples

A method for correcting integrating-sphere signals that considers differences in the angular distribution of scattered light is extended to sources of errors that are due to stray light from imperfect optical components. We show that it is possible to measure low levels of scattering, below 1%, by using a standard integrating sphere, provided that the various contributions to stray light are taken into account properly. For low-scattering samples these corrections are more important than those from the angular distribution of the scattering. A procedure for the experimental determination of stray-light components is suggested. Simple, easy to use, compact equations for the diffuse and specular reflectance and transmittance values of the sample as functions of the recorded signals are presented.     

Absolute interferometric distance measurement using a FM-demodulation technique

We propose an interferometric method for measuring absolute distances larger than the wavelength. A laser diode is used as a light source. The principle of operation is based on multiple-wavelength interferometry that uses a modulated light source. This method uses the fact that the wavelength of light emitted by the laser diode can be varied by means of the injection current. The modulation of the injection current in combination with the optical heterodyne technique causes a high-frequency phase-modulated detector signal. The phase deviation of the signal is a measure of the optical path difference in the interferometer. By FM demodulation of the detector output with a phase-locked loop demodulator, the optical path difference can be determined directly without the classical ambiguity problem of interferometry. The measuring range in the experiments was limited to 50 mm by the maximum travel range of the used specimen translation stage. Because of the inherent light sensitivity of the method described, the rangefinder can be used for three-dimensional profile measurements on a wide variety of objects, even on diffuse scattering surfaces.     

Two-dimensional quantitative photoacoustic image reconstruction of absorption distributions in scattering media by use of a simple iterative method

Photoacoustic imaging is a noninvasive biomedical imaging modality for visualizing the internal structure and function of soft tissues. Conventionally, an image proportional to the absorbed optical energy is reconstructed from measurements of light-induced acoustic emissions. We describe a simple iterative algorithm to recover the distribution of optical absorption coefficients from the image of the absorbed optical energy. The algorithm, which incorporates a diffusion-based finite-element model of light transport, converges quickly onto an accurate estimate of the distribution of absolute absorption coefficients. Two-dimensional examples with physiologically realistic optical properties are shown. The ability to recover optical properties (which directly reflect tissue physiology) could enhance photoacoustic imaging techniques, particularly methods based on spectroscopic analysis of chromophores.