Bit error rate of a Gaussian beam propagating through biological tissue

ABSTRACT

The scintillation index and bit error rate (BER) of a Gaussian beam propagating in a weakly turbulent soft tissue are formulated and analysed numerically. The scintillation indices are plotted against half of the measured slope in the range of power-law scaling at different tissue parameters, such as the random variations in the refractive index of the tissue, outer scale of the tissue turbulence and the tissue length between the optical source and the detector. Moreover, BERs of Gaussian beams against the signal to noise ratio (SNR) are examined for different tissue parameters. Our graphical results show that the scintillation index and BER increase with larger outer scales, longer tissue lengths and larger random variations in the refractive index of the tissue. In comparison with the spherical wave propagation, it was found that Gaussian beam yields larger scintillation index and BER values.     

Scintillation index of flat-topped Gaussian beams

The scintillation index is formulated for a flat-topped Gaussian beam source in atmospheric turbulence. The variations of the on-axis scintillations at the receiver plane are evaluated versus the link length, the size of the flat-topped Gaussian source, and the wavelength at selected flatness scales. The existing source model that represents the flat-topped Gaussian source as the superposition of Gaussian beams is employed. In the limiting case our solution correctly matches with the known Gaussian beam scintillation index. Our results show that for flat-topped Gaussian beams scintillation is larger than that of the single Gaussian beam scintillation when the source sizes are much smaller than the Fresnel zone. However, this trend is reversed and scintillations become smaller than the Gaussian beam scintillations for flat-topped sources with sizes much larger than the Fresnel zone.     

Effect of anisotropy on intensity fluctuations in oceanic turbulence

For an optical spherical wave propagating in an oceanic turbulent medium, the effect of anisotropy on the received intensity fluctuations is investigated. For different anisotropy factors, the variations of the scintillation index vs. the ratio that determines the relative strength of temperature and salinity in the index fluctuations, the rate of dissipation of the mean squared temperature, the rate of dissipation of the turbulent kinetic energy, viscosity, link length and the wavelength are plotted. It is found that, for all the oceanic turbulence and the link parameters of interest, as the medium becomes more anisotropic, the intensity of the optical spherical wave fluctuates less. It is concluded that the performance of an optical wireless communication systems (OWCS) operating in anisotropic oceanic turbulence is better than the performance of OWCS operating in isotropic oceanic turbulence.     

Scintillation index of flat-topped Gaussian laser beam in strongly turbulent medium

In a strongly turbulent medium, the scintillation index of flat-topped Gaussian beams is derived and evaluated. In the formulation, unified solution of Rytov method is utilized. Our results correctly reduce to the existing strong turbulence scintillation index of the Gaussian beam, and naturally to spherical and plane wave scintillations. Another checkpoint of our result is the scintillation index of flat-topped Gaussian beams in weak turbulence. Regardless of the order of flatness, scintillations of flat-topped Gaussian beams in strong turbulence are found to be determined mainly by the small-scale effects. For large-sized beams in moderate and strongly turbulent medium, flatter beams exhibit smaller scintillations.     

Scintillations of cos-Gaussian and annular beams

Based on the generalized beam formulation, we derive the scintillation index and selectively evaluate it for cos-Gaussian and annular beams propagating in weak atmospheric turbulence. Dependence of the scintillation index on propagation length, focusing and displacement parameters, wavelength of operation, and source size are individually investigated. From our graphical outputs, it is observed that a cos-Gaussian beam exhibits lower scintillations and thus has a tendency to be advantageous over a pure Gaussian beam particularly at lower propagation lengths. It is also found that at longer propagation lengths, this advantage switches to the side of the annular beam. Furthermore, the scintillation index of a focused annular beam will be below those of both Gaussian and cos-Gaussian beams starting at earlier propagation distances. When analyzed against source sizes, it is seen that cos-Gaussian beams will offer advantages at relatively large source sizes, while the reverse will be applicable for annular beams.     

Scintillation characteristics of cosh-Gaussian beams

By using the generalized beam formulation, the scintillation index is derived and evaluated for cosh-Gaussian beams in a turbulent atmosphere. Comparisons are made to cos-Gaussian and Gaussian beam scintillations. The variations of scintillations against propagation length at different values of displacement and focusing parameters are examined. The dependence of scintillations on source size at different propagation lengths is also investigated. Two-dimensional scintillation index distributions covering the entire transverse receiver planes are given. From the graphic illustrations, it is found that in comparison to pure Gaussian beams cosh-Gaussian beams have lower on-axis scintillations at smaller source sizes and longer propagation distances. The focusing effect appears to impose more reduction on the cosh-Gaussian beam scintillations than those of the Gaussian beam. The distribution of the off-axis scintillation index values of the Gaussian beams appears to be uniform over the transverse receiver plane, whereas that of the cosh-Gaussian beam is arranged according to the position of the slanted axis.     

On the scintillation index of a multiwavelength Gaussian beam in a turbulent free-space optical communications channel

The scintillation statistics of a multiwavelength Gaussian optical beam are characterized when the beam is subjected to a turbulent optical channel. It is assumed that the level of turbulence in the atmosphere ensures a weak-turbulence scenario and that fluctuations in the signal intensity are due to variations in the refractive index of the medium, which in turn are caused by regional temperature variations due to atmospheric turbulence. Furthermore, it is assumed that the propagation path is nearly horizontal and that the heights of the transmitter and receiver justify a near-ground propagation assumption. The Rytov approximation is used to arrive at the desired results. Furthermore, it is assumed that the first- as well as second-order perturbation terms are present in modeling the impact of atmosphere-induced scintillation. Numerical results are presented to shed light on the performance of multiwavelength optical radiation in weak turbulence and to underscore the benefits of the proposed approach as compared with its single-wavelength counterpart in combating the effect of turbulence. Furthermore, it is shown that if the separation of wavelengths used is sufficiently large, wavelength separation affects the scintillation index in a measurable way.     

Scintillation index and bit error rate of hollow Gaussian beams in atmospheric turbulence

Based on the Huygens–Fresnel principle and Rytov method, the on-axis scintillation index is derived for hollow Gaussian beams (HGBs) in weak turbulence. The relationship between bit error rate (BER) and scintillation index is found by only considering the effect of atmosphere turbulence based on the probability distribution of intensity fluctuation, and the expression of the BER is obtained. Furthermore, the scintillation and the BER properties of HGBs in turbulence are discussed in detail. The results show that the scintillation index and BER of HGBs depend on the propagation length, the structure constant of the refractive index fluctuations of turbulence, the wavelength, the beam order and the waist width of the fundamental Gaussian beam. The scintillation index, increasing with the propagation length in turbulence, for the HGB with higher beam order increases more slowly. The BER of the HGBs increases rapidly against the propagation length in turbulence. For propagating the same distance, the BER of the fundamental Gaussian beam is the greatest, and that of the HGB with higher order is smaller.     

Common omissions and misconceptions of wave propagation in turbulence: discussion

This review paper addresses typical mistakes and omissions that involve theoretical research and modeling of optical propagation through atmospheric turbulence. We discuss the disregard of some general properties of narrow-angle propagation in refractive random media, the careless use of simplified models of turbulence, and omissions in the calculations of the second moment of the propagating wave. We also review some misconceptions regarding short-exposure imaging, propagation of polarized waves, and calculations of the scintillation index of the beam waves.     

Modelling optical properties of soft tissue by fractal distribution of scatterers

Abstract

A knowledge of the local refractive index variations and size distribution of scatterers in biological tissue is required to understand the physical processes involved in light-tissue interaction. This paper describes a method for modelling the complicated soft tissue, based on the fractal approach, permitting numerical evaluation of the phase functions and four optical properties of tissue—scattering coefficient, reduced scattering coefficient, backscatter-ing coefficient, and anisotropy factor—by the use of the Mie scattering theory. A key assumption of the model is that refractive index variations caused by microscopic tissue elements can be treated as particles with size distribution according to the power law. The model parameters, such as refractive index, incident wavelength, and fractal dimension, that are likely to affect the predictions of optical properties are investigated. The results suggest that the fractal dimension used to describe how biological tissue can be approximated by particle distribution is highly dependent on how the continuous distribution is discretized. The optical properties of the tissue significantly depend on the refractive index of tissue, implying that the refractive index of the particles should be carefully chosen in the model in order accurately to predict the optical properties of the tissue concerned.     

Laser beam propagation in atmospheric turbulence

The optical effects of atmospheric turbulence on the propagation of low power laser beams are reviewed in this paper. The optical effects are produced by the temperature fluctuations which result in fluctuations of the refractive index of air. The commonly-used models of index-of-refraction fluctuations are presented. Laser beams experience fluctuations of beam size, beam position, and intensity distribution within the beam due to refractive turbulence. Some of the observed effects are qualitatively explained by treating the turbulent atmosphere as a collection of moving gaseous lenses of various sizes. Analytical results and experimental verifications of the variance, covariance and probability distribution of intensity fluctuations in weak turbulence are presented. For stronger turbulence, a saturation of the optical scintillations is observed. The saturation of scintillations involves a progressive break-up of the beam into multiple patches; the beam loses some of its lateral coherence. Heterodyne systems operating in a turbulent atmosphere experience a loss of heterodyne signal due to the destruction of coherence.     

Inferring path average Cn2 values in the marine environment

Current mathematical scintillation theory describing laser propagation through the atmosphere has been developed for terrestrial environments. Scintillation expressions valid in all regimes of optical turbulence for propagation in the maritime environment, based on what we believe to be a newly developed marine atmospheric spectrum, have been developed for spherical waves. Path average values of the structure parameter, C(n)(2), were inferred from optical scintillation measurements of a diverged laser beam propagating in a marine environment, using scintillation expressions based on both terrestrial and marine refractive index spectra. In the moderate-to-strong fluctuation regime, the inferred marine C(n)(2) values were about 20% smaller than inferred terrestrial C(n)(2) values, but a minimal difference was observed in the weak fluctuation regime. Measurements of angle-of-arrival fluctuations were used to infer C(n)(2) values in the moderate-to-strong fluctuation regime, resulting in values of the structure parameter that were at least an order of magnitude larger than the two scintillation-inferred C(n)(2) values.     

Effect of atmospheric spectrum models on scintillation in moderate turbulence

Experimental studies have shown that a ‘bump’ occurs in the atmospheric spectrum just prior to turbulence cell dissipation. In weak optical turbulence, this bump affects calculated scintillation. The purpose of this study was to determine if a simpler non-bump atmospheric power spectrum can be used to model scintillation for plane waves and spherical waves in moderate to strong optical turbulence regimes. Scintillation expressions were developed from an ‘effective’ von Karman spectrum using an approach similar to that used by Andrews et al. in developing expressions from an effective modified (bump) spectrum. The effective spectrum extends the Rytov approximation into all optical turbulence regimes using filter functions to eliminate mid-range turbulent cell size effects to the scintillation index. Filter cutoffs were established by matching to known weak and saturated scintillation results. The resulting new expressions track those derived from the effective bump spectrum fairly closely. In extremely strong turbulence, differences are minimal.     

Scintillation of astigmatic dark hollow beams in weak atmospheric turbulence

The scintillation properties of astigmatic dark hollow beams (DHBs) in weak atmospheric turbulence were investigated in detail. An explicit expression for the on-axis scintillation index of an astigmatic DHB propagating in weak atmospheric turbulence was derived. It was found that the scintillation index value of an astigmatic DHB with suitable astigmatism (i.e., ratio of the beam waist size in the x direction to that in the y direction), dark size, beam waist size, and wavelength can be smaller than that of a stigmatic DHB and that of stigmatic and astigmatic flat-topped, annular, and Gaussian beams in weak atmospheric turbulence particularly at long propagation ranges. Our results will be useful in long-distance free-space optical communications.     

Optical scintillations and fade statistics for a satellite-communication system

Estimates of the scintillation index, fractional fade time, expected number of fades, and mean duration of fade time associated with a propagating Gaussian-beam wave are developed for uplink and downlink laser satellite-communication channels. Estimates for the spot size of the beam at the satellite or the ground or airborne receiver are also provided. Weak-fluctuation theory based on the log-normal model is applicable for intensity fluctuations near the optical axis of the beam provided that the zenith angle is not too large, generally not exceeding 60°. However, there is an increase in scintillations that occurs with increasing pointing error at any zenith angle, particularly for uplink channels. Large off-axis scintillations are of particular significance because they imply that small pointing errors can cause serious degradation in the communication-channel reliability. Off-axis scintillations increase more rapidly for larger-diameter beams and, in some cases, can lead to a radial saturation effect for pointing errors less than 1 μrad off the optical beam axis.     

Measurement of seeing and the atmospheric time constant by differential scintillations

A simple differential analysis of stellar scintillations measured simultaneously with two apertures opens the possibility to estimate seeing. Moreover, some information on the vertical turbulence distribution can be obtained. A general expression for the differential scintillation index for apertures of arbitrary shape and for finite exposure time is derived, and its applications are studied. Correction for exposure time bias by use of the ratio of scintillation indices with and without time binning is studied. A bandpass-filtered scintillation in a small aperture (computed as the differential-exposure index) provides a reasonably good estimate of the atmospheric time constant for adaptive optics.     

BER evaluations for multimode beams in underwater turbulence

Abstract

In underwater optical communication links, bit error rate (BER) is an important performance criterion. For this purpose, the effects of oceanic turbulence on multimode laser beam incidences are studied and compared in terms of average BER (<BER>), which is related to the scintillation index. Based on the log-normal distribution, <BER> is analysed for underwater turbulence parameters, including the rate of dissipation of the mean squared temperature, the rate of dissipation of the turbulent kinetic energy, the parameter that determines the relative strength of temperature and salinity in driving index fluctuations, the Kolmogorov microscale length and other link parameters such as link length, wavelength and laser source size. It is shown that use of multimode improves the system performance of optical wireless communication systems operating in an underwater medium. For all the investigated multimode beams, decreasing link length, source size, the relative strength of temperature and salinity in driving the index fluctuations, the rate of dissipation of the mean squared temperature and Kolmogorov microscale length improve the <BER>. Moreover, lower <BER> values are obtained for the increasing wavelength of operation and the rate of dissipation of the turbulent kinetic energy in underwater turbulence.     

Measurement of air refractive index fluctuation based on interferometry with two different reference cavity lengths

A measurement method based on interferometry with two different reference cavity lengths is presented and applied in air refractive index measurement in which the two cavity lengths and a laser wavelength are combined to generate two wavelength equivalents of cavity. Corresponding calculation equations are derived, and the optical path configuration is designed, which is inspired by the traditional synthetic wavelength method. Theoretical analyses indicate that the measurement uncertainty of the determined index of refraction is about 2.3×10^-8, which is mainly affected by the length precision of the long vacuum cavity and the ellipticity of polarization components of the dual-frequency laser, and the range of nonambiguity is 3.0×10^-5, which is decided by the length difference of the two cavities. Experiment results show that the accuracy of air refractive index measurement is better than 5.0×10^-8 when the laboratory conditions changes slowly. The merit of the presented method is that the classical refractometry can be also used without evacuation of the gas cavity during the experiment. Furthermore, the application of the traditional synthetic wavelength method may be extended by using the wavelength equivalents of cavity, any value of which can be easily acquired by changing cavity length rather than using actual wavelengths whose number is limited.     

Light propagation through anisotropic turbulence

A wealth of experimental data has shown that atmospheric turbulence can be anisotropic; in this case, a Kolmogorov spectrum does not describe well the atmospheric turbulence statistics. In this paper, we show a quantitative analysis of anisotropic turbulence by using a non-Kolmogorov power spectrum with an anisotropic coefficient. The spectrum we use does not include the inner and outer scales, it is valid only inside the inertial subrange, and it has a power-law slope that can be different from a Kolmogorov one. Using this power spectrum, in the weak turbulence condition, we analyze the impact of the power-law variations α on the long-term beam spread and scintillation index for several anisotropic coefficient values ?. We consider only horizontal propagation across the turbulence cells, assuming circular symmetry is maintained on the orthogonal plane to the propagation direction. We conclude that the anisotropic coefficient influences both the long-term beam spread and the scintillation index by the factor ?(2-α).     

An Approximation Method for the Backward Scattering of Light by a Soft Spherical Obstacle

Abstract

The optical analogue of a formula by Reading and Bassichis for the backscattering of a high-energy scalar wave by a square-well potential has been examined. In optical scattering, this corresponds to the problem of the back-scattering of light by a homogeneous spherical particle. Numerical checks with the Mie theory are presented for various values of the refractive index and size parameter. The formula is found to reproduce some scattering features extremely well for intermediate-size soft particles.