Irradiance Probability Density Function Research Articles

A modification to the extended Rytov theory

This paper proposes a heuristic modification to the extended Rytov theory of strong scintillation [Andrews L, Phillips R. Laser beam propagation through random media. 2nd ed., Washington, USA: SPIE Press; 2005, Andrews LC, Al-Habash MA, Hopen CY, et al. Theory of optical scintillation: Gaussian-beam wave model. Waves Random Media. 2001;11(3):271–291]. This is done by dividing the large and small scales in the extended Rytov theory into further sub-scales. Expressions for the induced irradiance variance, and log-irradiance, due to each sub-scale are presented. The modified theory is used to derive and justify recent results in irradiance fluctuation modeling and hence answer a question initially proposed in [Shishter YM, Young RC, Ali FH. Irradiance probability density function for turbulence induced fading in free space optics. JOSA A. 2022;39(7):1267–1274, Shishter YM, Young RC, Ali FH. General statistical model of irradiance fluctuations in free space optics. JOSA A. 2023;40(1):53–63]. Plots of the irradiance pdf under the modified theory are in support of the theory. The validity of the theory presented would result in modifications to the classical scintillation analysis of waves in random media, with applications to the design of optical systems. Moreover, it is observed that the mathematical framework developed also follows from the assumption of variations in the turbulence along a horizontal path. The work presented in this paper suggests the need of further theoretical and experimental investigations into the effect of variations in atmospheric turbulence characteristics along horizontal paths in the received signal irradiance distribution.

Irradiance probability density function for turbulence induced fading in free space optics.

A model for the irradiance probability density function for turbulence induced fading in free space optical communications is derived. The derivations are based on expanding on the extended Rytov method by associating a doubly stochastic process to model large scale scintillation. In particular, the small scale induced irradiance fluctuations are modeled by a single gamma distribution, while the large scale induced fluctuations are modeled as a doubly stochastic process of gamma and inverted gamma distributions, thus better approximating the lognormal distribution. The resultant distribution's probability density and cumulative density functions are both given in closed forms. Moreover, the resultant model parameters are given based on strong scintillation theory. Through comparison with previously published results and experimental data, including lognormal and gamma-gamma distributions, it is concluded that the proposed model agrees well with measurements for weak and strong scintillation conditions in cases of both aperture averaging and a point like receiver. Therefore, the proposed model can be used for the performance analysis of optical wireless systems.

FSO system performance analysis based on novel Gamma–Chi-square irradiance PDF model

In order to provide a novel analytically traceable free space optics (FSO) channel model for describing turbulence based irradiance fluctuations, following basic scintillation theory principles, we have derived closed-form expression for probability density function (PDF) of a new statistical Gamma–Chi-square model. Further, capitalizing on provided model, error performances of FSO system over on–off keying (OOK) transmission scheme both in the presence of atmospheric turbulence and misalignment fading (pointing error) is investigated. For both cases, the average bit error rate (ABER) at the receiving side of the system is determined in an analytically closed form. The results are graphically presented in order to analyze the impact of different levels of turbulence, as well as other relevant parameters, on the quality of the received signal in the OOK modulated FSO system.

Exponentiated Weibull model for the irradiance probability density function of a laser beam propagating through atmospheric turbulence

Many distributions have been proposed to model the probability density function of irradiance fluctuations. The most widespread models nowadays are the Lognormal (LN) and Gamma–Gamma (GG) distributions. Albeit these models comply with the actual PDF data most of the time, neither of them works in all scenarios and, depending on the conditions, one of the two have to be chosen. In this paper, a new model is presented resulting in the exponentiated Weibull (EW) distribution, along with a physical justification for the appearance of the model. Previously published data are used to compare the new model with the LN and GG distributions. Results suggest that the EW distribution is the better fit for data under all aperture averaging conditions and weak-to-strong turbulence regime.

Modified moment-matching method for estimating pointing parameters in the presence of atmospheric turbulence

An accurate pointing system is required in free-space optical (FSO) communication links. Low energy-transmission efficiency caused by pointing errors would decline the communication system's performance. The statistics of the detected signal or return signal values could be used to estimate the pointing parameters, whereas atmospheric turbulence brings in serious challenges. A modified moment-matching estimation method is presented in this paper. The irradiance fluctuation caused by the atmospheric turbulence is considered, and the probability density function (PDF) in a weak turbulence condition is assumed to be lognormal. This modified approach is evaluated with wave-propagation simulation data and shows significant improvement over the conventional approach. The estimation accuracy and the properties of this new approach are also discussed. Although our method is based on lognormal irradiance PDF under a weak turbulence condition, the irradiance PDF would tend to be lognormal with aperture averaging effect under moderate to strong turbulence, and the ideas can be extended with appropriate PDF models to satisfy different conditions.

Improved bit error rate evaluation for optically pre-amplified free-space optical communication systems in turbulent atmosphere

The use of an optical pre-amplifier at the receiver of free-space optical (FSO) communication systems necessitates the consideration of both turbulence and amplified spontaneous emission (ASE) noise in performance modelling. Until now, this problem has been typically approached by the use of a Gaussian approximation (GA) for the conditional probability of error with averaging then performed using the received irradiance probability density function (PDF) (governed by scintillation). However, the GA is deficient as it only uses the first two moments. Moment generating function (MGF) techniques, notably the Chernoff bound (CB) and modified Chernoff bound (MCB), are used to evaluate the bit error rate performance of an optically pre-amplified FSO system in the presence of both atmospheric turbulence and ASE noise. The MGF-based methods incorporate a fuller statistical description of the signal and noise processes encountered in the optically pre-amplified case when compared to the GA. The lognormal, gamma–gamma, K and negative exponential distributions have been used to characterise the weak, moderate, strong and saturated turbulence regimes. The MCB gives the tightest bound upon the BER compared to the CB, particularly at lower gains, and, as it also can be exceeded by the GA at higher gains, it is a logical method to use in general.

Average capacity for non-Kolmogorov turbulent slant optical links with beam wander corrected and pointing errors

The effects of beam wander corrected and pointing errors on the average capacity of a non-Kolmogorov turbulent atmosphere communication channel are studied. The gamma–gamma irradiance probability density function model has been considered to include the effects of non-Kolmogorov turbulence. An expression for the average capacity under beam wander corrected, pointing errors and the spectral index of refractive-index fluctuation. Simulation results for effects of circular detection aperture of radius, atmosphere turbulence strength and wavelength on average capacity are presented. Our results show that the average capacity has an oscillation in the case of short propagation distance, but in the case of long propagation distance, it has gradually reduced curves. And the turbulence strength has a strong influence on average capacity. In general, turbulence reduces the average capacity; it can reduce average capacity by increasing of the power law α. We can ignore the little effects of wavelength increases on the average capacity.

Pointing Error Effects on Free-Space Optical Communication Links in the Presence of Atmospheric Turbulence

The effects of pointing errors on the performance of a free-space optical laser communication system are studied. The log-normal and gamma-gamma irradiance probability density function models have been considered to include the effects of optical turbulence. An expression for the mean intensity under pointing fluctuations is derived. Results for ergodic and outage channel capacities are presented. Since the irradiance statistics vary with pointing fluctuations, a wave-optics-based approach is proposed to evaluate the capacity expressions. Although, in general, turbulence degrades communications link performance, it can reduce sensitivity to pointing errors by increasing the effective beam radius. Finally, we present analytical expressions for evaluating the bit-error-rate performance of the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">on</i> - <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">off</i> keying system in the presence of pointing errors, and provide error performance results.

Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media

We develop a model for the probability density function (pdf) of the irradiance fluctuations of an optical wave propagating through a turbulent medium. The model is a two-parameter distribution that is based on a doubly stochastic theory of scintillation that assumes that small-scale irradiance fluctuations are modulated by large-scale irradi- ance fluctuations of the propagating wave, both governed by indepen- dent gamma distributions. The resulting irradiance pdf takes the form of a generalized K distribution that we term the gamma-gamma distribution. The two parameters of the gamma-gamma pdf are determined using a recently published theory of scintillation, using only values of the refractive-index structure parameter C n (or Rytov variance) and inner scale l 0 provided with the simulation data. This enables us to directly calculate various log-irradiance moments that are necessary in the scaled plots. We make a number of comparisons with published plane wave and spherical wave simulation data over a wide range of turbu- lence conditions (weak to strong) that includes inner scale effects. The gamma-gamma pdf is found to generally provide a good fit to the simu- lation data in nearly all cases tested. © 2001 Society of Photo-Optical Instrumen-

Statistical treatment of the PAR variability and its application to willow coppice

For characterization of spatial variability of the global photosynthetically active radiation (PAR) Q at different depths in a plant canopy a new statistical treatment has been developed. It is based on the use of the irradiance probability density function (IPDF) which is divided into three parts: IPDF for umbra f u Q , for penumbra f p Q and for sunflecks f s Q . In umbra, IPDF was approximated by a normal distribution, in penumbra by the Beta-distribution, and in sunflecks, by a normal distribution. Statistical treatment has been applied for the fast growing willow coppice. In the umbra area the mean value Q I of global PAR decreases exponentially with the optical path length of the direct solar radiation inside the canopy τ= L/sinh with small values of the standard deviation σ u. In the sunfleck area the mean value Q II decreases linearly with τ with small values of the standard deviation. In the penumbra area agreement with experimental data is inferior to that in umbra and sunfleck areas. Inside the canopy, the sunfleck fractional area k s, fitted by a second order polynomial, decreases with τ up to τ=3.3. The umbra fractional area k u increases nearly linearly with depth τ starting from τ=0.8. The fractional area of the penumbra k p increases rapidly with τ, reaches the maximum value ( k p=0.6–0.8) at τ=3 and then decreases nearly linearly.

Photovoltaic module-site matching based on the capacity factors

In this paper, a methodology for the selection of the optimum photovoltaic module for a specific power plant site is developed. The selection is based on the capacity factors (CF) of the available PV modules. Long term irradiance data recorded for every hour of the day for 30 years are used. These data are used to calculate the probability density function of the irradiance for different hours of a typical day in a month. The irradiance probability density function and the manufacturer's specifications on PV modules are used to calculate the capacity factors for the PV modules. The PV module with the highest average capacity factor for the specific site is the optimal and recommended PV module. In this paper, the price per installed maximum peak watt is approximately the same for different modules and hence the cost is not an issue. >

Sample size influence on optical scintillation analysis 2: Simulation approach

This paper is the second part of the sample size influence analysis. We estimate this influence on different methods of irradiance probability density function (PDF) diagnostics. The estimation is achieved by a Monte Carlo-based simulation. The problem of the fastest effective sampling rate is also considered. The results indicate the low resolving power of conventional irradiance PDF integrating methods.

Probability distribution of turbulent irradiance in a saturation regime

Application of the central limit theorem to the stochastic equation of propagation suggests that the probability distribution of the complex wave amplitude defined on the geometrical phase front is approximately normal. The resulting irradiance probability density function, valid in the strong scintillation regime, is an exponential multiplied by the modified Bessel function I(0) both of argument proportional to the irradiance; it is not the Rice-Nakagami density function. Quantitative tests show that this exponential-Bessel function constitutes as good a fit as the log-normal to the irradiance probability data reported in this paper. Since the normal distribution hypothesis is consistent with the stochastic wave equation, the model proposed here should be a simple substitute to the often used but theoretically incorrect log-normal irradiance probability distribution model.

Statistical Theory of Wave Propagation in a Random Medium and the Irradiance Distribution Function

The complete statistical description of the waves in a random medium can be obtained from the characteristic functional or the cumulant functional of the wave function. The basic equations of these functionals are found first for the gaussian medium and then for the nongaussian medium. As an application of those equations, the approximate equations of the arbitrary-order coherence functions are derived, which are found to be equivalent to those recently obtained by several authors using different methods. The operational method is introduced to solve the equation of the νth-order moment of the irradiance, and an exact solution is obtained for the particular correlation function of the medium, assuming the gaussian form of the incident wave. The irradiance probability-density function is obtained without approximation by the use of the νth-order moment of irradiance for this particular medium, and is found to be exactly the Rice–Nakagami distribution with respect to the log irradiance. This distribution approaches the log-normal distribution in the outside domain of the wave beam, and is also checked in several points. Finally, the operator representation of the physical variables is introduced, and it is shown that the various equations, e.g., the wave equation and the energy-conservation equation, in the statistical system of the wave plus the random medium can be represented by the same equations as the corresponding equations in the deterministic medium. The discussion is also extended on the basis of DeWolf’s result for the irradiance-distribution function.