Spatially offset optical coherence tomography: Leveraging multiple scattering for high-contrast imaging at depth in turbid media.

The penetration depth of optical coherence tomography (OCT) reaches well beyond conventional microscopy; however, signal reduction with depth leads to rapid degradation of the signal below the noise level. The pursuit of imaging at depth has been largely approached by extinguishing multiple scattering. However, in OCT, multiple scattering substantially contributes to image formation at depth. Here, we investigate the role of multiple scattering in OCT image contrast and postulate that, in OCT, multiple scattering can enhance image contrast at depth. We introduce an original geometry that completely decouples the incident and collection fields by introducing a spatial offset between them, leading to preferential collection of multiply scattered light. A wave optics-based theoretical framework supports our experimentally demonstrated improvement in contrast. The effective signal attenuation can be reduced by more than 24 decibels. Notably, a ninefold enhancement in image contrast at depth is observed in scattering biological samples. This geometry enables a powerful capacity to dynamically tune for contrast at depth.

Digital simulation of two-dimensional random fields with arbitrary power spectra and non-Gaussian probability distribution functions.

Methods for simulation of two-dimensional signals with arbitrary power spectral densities and signal amplitude probability density functions are disclosed. The method relies on initially transforming a white noise sample set of random Gaussian distributed numbers into a corresponding set with the desired spectral distribution, after which this colored Gaussian probability distribution is transformed via an inverse transform into the desired probability distribution. In most cases the method provides satisfactory results and can thus be considered an engineering approach. Several illustrative examples with relevance for optics are given.

Speckle and fringe dynamics in imaging-speckle-pattern interferometry for spatial-filtering velocimetry.

This paper analyzes the dynamics of laser speckles and fringes, formed in an imaging-speckle-pattern interferometer with the purpose of sensing linear three-dimensional motion and out-of-plane components of rotation in real time, using optical spatial-filtering-velocimetry techniques. The ensemble-average definition of the cross-correlation function is applied to the intensity distributions, obtained in the observation plane at two positions of the object. The theoretical analysis provides a description for the dynamics of both the speckles and the fringes. The analysis reveals that both the magnitude and direction of all three linear displacement components of the object movement can be determined. Simultaneously, out-of-plane rotation of the object including the corresponding directions can be determined from the spatial gradient of the in-plane fringe motion throughout the observation plane. The theory is confirmed by experimental measurements.

Low Earth orbit satellite-to-ground optical scintillation: comparison of experimental observations and theoretical predictions.

Scintillation measurements of a 1064 nm laser at a 5 kHz sampling rate were made by an optical ground station at the European Space Agency observatory in Tenerife, Spain while tracking a low Earth orbit satellite during the spring and summer of 2010. The scintillation index (SI), the variance of irradiance normalized to the square of the mean, and power spectra measurements were compared to theoretical predictions based on the Kolmogorov spectrum, the Maui3 nighttime turbulence profile, weak scintillation finite-beam wave theory, included receiver, and source aperture averaging with no free-fitting parameters. Good agreement was obtained, not only for the magnitude of the observed fluctuations, but also for the corresponding elevation angle dependence and shape of the power spectra. Little variation was seen for the SI between daytime and nighttime links. For all elevation angles, ascending and descending, the observed scintillation over extensive regions of the atmosphere is consistent with log-normal statistics. Additionally, it appears from the results presented here that the nighttime turbulence profile for the atmosphere above the observatory in Tenerife is similar to that above Haleakala in Maui, Hawaii.

Level crossing statistics for optical beam wander in a turbulent atmosphere with applications to ground-to-space laser communications.

Level crossing statistics is applied to the complex problem of atmospheric turbulence-induced beam wander for laser propagation from ground to space. A comprehensive estimate of the single-axis wander angle temporal autocorrelation function and the corresponding power spectrum is used to develop, for the first time to our knowledge, analytic expressions for the mean angular level crossing rate and the mean duration of such crossings. These results are based on an extension and generalization of a previous seminal analysis of the beam wander variance by Klyatskin and Kon. In the geometrical optics limit, we obtain an expression for the beam wander variance that is valid for both an arbitrarily shaped initial beam profile and transmitting aperture. It is shown that beam wander can disrupt bidirectional ground-to-space laser communication systems whose small apertures do not require adaptive optics to deliver uniform beams at their intended target receivers in space. The magnitude and rate of beam wander is estimated for turbulence profiles enveloping some practical laser communication deployment options and suggesting what level of beam wander effects must be mitigated to demonstrate effective bidirectional laser communication systems.

Digital simulation of an arbitrary stationary stochastic process by spectral representation.

In this paper we present a straightforward, efficient, and computationally fast method for creating a large number of discrete samples with an arbitrary given probability density function and a specified spectral content. The method relies on initially transforming a white noise sample set of random Gaussian distributed numbers into a corresponding set with the desired spectral distribution, after which this colored Gaussian probability distribution is transformed via an inverse transform into the desired probability distribution. In contrast to previous work, where the analyses were limited to auto regressive and or iterative techniques to obtain satisfactory results, we find that a single application of the inverse transform method yields satisfactory results for a wide class of arbitrary probability distributions. Although a single application of the inverse transform technique does not conserve the power spectra exactly, it yields highly accurate numerical results for a wide range of probability distributions and target power spectra that are sufficient for system simulation purposes and can thus be regarded as an accurate engineering approximation, which can be used for wide range of practical applications. A sufficiency condition is presented regarding the range of parameter values where a single application of the inverse transform method yields satisfactory agreement between the simulated and target power spectra, and a series of examples relevant for the optics community are presented and discussed. Outside this parameter range the agreement gracefully degrades but does not distort in shape. Although we demonstrate the method here focusing on stationary random processes, we see no reason why the method could not be extended to simulate non-stationary random processes.

Mean level signal crossing rate for an arbitrary stochastic process.

The issue of the mean signal level crossing rate for various probability density functions with primary relevance for optics is discussed based on a new analytical method. This method relies on a unique transformation that transforms the probability distribution under investigation into a normal probability distribution, for which the distribution of mean level crossings is known. In general, the analytical results for the mean level crossing rate are supported and confirmed by numerical simulations. In particular, we illustrate the present method by presenting analytic expressions for the mean level crossing rate for various probability distributions, including ones that previously were unavailable, such as the uniform, the so-called gamma-gamma, and the Rice-Nakagami distribution. However, in a limited number of cases the present results differ somewhat from the result reported in the literature. At present, this discrepancy remains unexplained and is laid open for future discussion.

Speckle dynamics for dual-beam optical illumination of a rotating structure.

An out-of-plane rotating object is illuminated with two spatially separated coherent beams, giving rise to fully developed speckles, which will translate and gradually decorrelate in the observation plane, located in the far field. The speckle pattern is a compound structure, consisting of random speckles modulated by a smaller and repetitive structure. Generally, these two components of the compound speckle structure will move as rigid structures with individual velocities determined by the characteristics of the two illuminating beams. Closed-form analytical expressions are found for the space- and time-lagged covariance of irradiance and the corresponding power spectrum for the two spatially separated illuminating beams. The present analysis is valid for propagation through an arbitrary ABCD system, though the focus for the experimental evaluation is far-field observations using an optical Fourier transform system. It is shown that the compound speckle structures move as two individual structures with the same decorrelation length. The velocity of the random speckles is a combination of angular and peripheral velocity, where the peripheral velocity is inversely proportional to the radius of the wavefront curvature of the incident beams. The velocity of the repetitive structure is a combination of angular and peripheral velocity, where the peripheral velocity is proportional to the ratio of the angle to the distance between the beams in the object plane. Experimental data demonstrate good agreement between theory and measurements for selected combinations of beam separation, angle between beams, and radius of wavefront curvature at the object.

Statistics of spatially integrated speckle intensity difference.

We consider the statistics of the spatially integrated speckle intensity difference obtained from two separated finite collecting apertures. For fully developed speckle, closed-form analytic solutions for both the probability density function and the cumulative distribution function are derived here for both arbitrary values of the mean number of speckles contained within an aperture and the degree of coherence of the optical field. Additionally, closed-form expressions are obtained for the corresponding nth statistical moments.

Speckle dynamics resulting from multiple interfering beams.

The space-time intensity covariance function for illuminating an object giving rise to fully developed speckle is considered in the case where the object is illuminated with two spatially separated beams, or with a multitude of equidistant but spatially separated spots. Specifically, and to the best of our knowledge for the first time, we obtain the result that the larger speckles will be covered by a fine structure that, in general, translates at a different velocity from that of the larger speckles. In particular, closed-form analytical expressions are found for the space- and time-lagged covariance of irradiance as well as the corresponding power spectrum for each of the two spatially separated, N equidistant separated illuminating beams. The present analysis is valid not only for free-space propagation but also for an arbitrary real ABCD optical system. Finally, the corresponding statistical signal properties, including the power spectrum, are derived and discussed for the (practical) case where the comprehensive speckle field is spatially filtered by a gratinglike structure.

Extraction of optical scattering parameters and attenuation compensation in optical coherence tomography images of multilayered tissue structures.

A recently developed analytical optical coherence tomography (OCT) model [Thrane et al., J. Opt. Soc. Am. A 17, 484 (2000)] allows the extraction of optical scattering parameters from OCT images, thereby permitting attenuation compensation in those images. By expanding this theoretical model, we have developed a new method for extracting optical scattering parameters from multilayered tissue structures in vivo. To verify this, we used a Monte Carlo (MC) OCT model as a numerical phantom to simulate the OCT signal for heterogeneous multilayered tissue. Excellent agreement between the extracted values of the optical scattering properties of the different layers and the corresponding input reference values of the MC simulation was obtained, which demonstrates the feasibility of the method for in vivo applications. This is to our knowledge the first time such verification has been obtained, and the results hold promise for expanding the functional imaging capabilities of OCT.

Advanced modelling of optical coherence tomography systems.

Analytical and numerical models for describing and understanding the light propagation in samples imaged by optical coherence tomography (OCT) systems are presented. An analytical model for calculating the OCT signal based on the extended Huygens-Fresnel principle valid both for the single and multiple scattering regimes is reviewed. An advanced Monte Carlo model for calculating the OCT signal is also reviewed, and the validity of this model is shown through a mathematical proof based on the extended Huygens-Fresnel principle. Moreover, for the first time the model is verified experimentally. From the analytical model, an algorithm for enhancing OCT images is developed: the so-called true-reflection algorithm in which the OCT signal may be corrected for the attenuation caused by scattering. For the first time, the algorithm is demonstrated by using the Monte Carlo model as a numerical tissue phantom. Such algorithm holds promise for improving OCT imagery and to extend the possibility for functional imaging.

Derivation of a Monte Carlo method for modeling heterodyne detection in optical coherence tomography systems.

A Monte Carlo (MC) method for modeling optical coherence tomography (OCT) measurements of a diffusely reflecting discontinuity embedded in a scattering medium is presented. For the first time to the authors' knowledge it is shown analytically that the applicability of an MC approach to this optical geometry is firmly justified, because, as we show, in the conjugate image plane the field reflected from the sample is delta-correlated from which it follows that the heterodyne signal is calculated from the intensity distribution only. This is not a trivial result because, in general, the light from the sample will have a finite spatial coherence that cannot be accounted for by MC simulation. To estimate this intensity distribution adequately we have developed a novel method for modeling a focused Gaussian beam in MC simulation. This approach is valid for a softly as well as for a strongly focused beam, and it is shown that in free space the full three-dimensional intensity distribution of a Gaussian beam is obtained. The OCT signal and the intensity distribution in a scattering medium have been obtained for several geometries with the suggested MC method; when this model and a recently published analytical model based on the extended Huygens-Fresnel principle are compared, excellent agreement is found.