Atmospheric turbulence is usually simulated at the laboratory by generating convective free flows with hot
surfaces, or heaters. It is tacitly assumed that propagation experiments in this environment are comparable to
those usually found outdoors. Nevertheless, it is unclear under which conditions the analogy between convective
and isotropic turbulence is valid; that is, obeying Kolmogorov isotropic models. For instance, near-ground-level
turbulence often is driven by shear ratchets deviating from established inertial models. In this case, a value for
the structure constant can be obtained but it would be unable to distinguish between both classes of turbulence.
We have performed a conceptually simple experiment of laser beam propagation through two types of artificial
turbulence: isotropic turbulence generated by a turbulator [Proc. SPIE 8535, 853508 (2012)], and convective
turbulence by controlling the temperature of electric heaters. In both cases, a thin laser beam propagates across
the turbulent path, and its wandering is registered by a position sensor detector. The strength of the optical
turbulence, in terms of the structure constant, is obtained from the wandering variance. It is expressed as a
function of the temperature difference between cold and hot sources in each setup. We compare the time series
behaviour for each turbulence with increasing turbulence strength by estimating the Hurst exponent, H, through
detrended fluctuation analysis (DFA). Refractive index fluctuations are inherently fractal; this characteristic is
reflected in their spectra power-law dependence—in the inertial range. This fractal behaviour is inherited by time
series of optical quantities, such as the wandering, by the occurrence of long-range correlations. By analyzing
the wandering time series with this technique, we are able to correlate the turbulence strength to the value of
the Hurt exponent. Ultimately, we characterize both types of turbulence.
We propose the use of multifractal detrended fluctuation analysis (MF-DFA) to measure the influence of atmospheric turbulence on the chaotic dynamics of a HeNe laser. Fit ranges for MF-DFA are obtained with goodness of linear fit (GoLF) criterion. The chaotic behavior is generated by means of a simple interferometric setup with a feedback to the cavity of the gas laser. Such dynamics have been studied in the past and modeled as a function of the feedback level. Different intensities of isotropic turbulence have been generated with a turbulator device, allowing a structure constant for the index of refraction of air adjustable by means of a temperature difference parameter in the unit. Considering the recent interest in message encryption with this kind of setups, the study of atmospheric turbulence effects plays a key role in the field of secure laser communication through the atmosphere. In principle, different intensities of turbulence may be interpreted as different levels of white noise on the original chaotic series. These results can be of utility for performance optimization in chaotic free-space laser communication systems.
We have previously introduced the Differential Laser Tracking Motion Meter (DLTMM) [Proc. SPIE 7476, 74760D (2009)] as a robust device to determine many optical parameters related to atmospheric turbulence. It consisted of two thin laser beams—whose separations can be modified—that propagate through convective air, then each random wandering was registered with position detectors, sampled at 800 Hz. The hypothesis that the analysis of differential coordinates is less affected by noise induced by mechanical vibration was tested. Although we detected a trend to the Kolmogorov’s power exponent with the turbulence increasing strength, we were unable to relate it to the Rytov variance. Also, analyzing the behaviour of the multi-fractal degree estimator (calculated by means of multi-fractal detrended fluctuation analysis, MFDFA) at different laser-beam separations for these differential series resulted in the appreciation of characteristic spatial scales; nevertheless, errors induced by the technique forbid an accurate comparison with scales estimated under more standard methods. In the present work we introduce both an improved experimental setup and refined analyses techniques that eliminate many of the uncertainties found in our previous study. A new version of the DLTMM employs cross-polarized laser beams that allows us to inspect more carefully distances in the range of the inner-scale, thus even superimposed beams can be discriminated. Moreover, in this experimental setup the convective turbulence produced by electrical heaters previously used was superseded by a chamber that replicates isotropic atmospheric turbulence—anisotropic turbulence is also reproducible. Therefore, we are able to replicate the same state of the turbulent flow, specified by Rytov variance, for every separation between beams through the course of the experience. In this way, we are able to study the change in our MFDFA quantifiers with different strengths of the turbulence, and their relation with better known optical quantities. The movements of the two laser beams are recorded at 6 kHz; this apparent oversampling is crucial for detecting the turbulence’s characteristics scales under improved MFDFA techniques. The estimated characteristic scales and multi-fractal nature detected by this experiment provides insight into the non-Gaussian nature of propagated light.
The time evolution of the output of a semiconductor laser subject to optical feedback can exhibit high-dimensional
chaotic fluctuations. In this contribution, our aim is to quantify the complexity of the chaotic time-trace generated
by a semiconductor laser subject to delayed optical feedback. To that end, we discuss the properties of two
recently introduced complexity measures based on information theory, namely the permutation entropy (PE)
and the statistical complexity measure (SCM). The PE and SCM are defined as a functional of a symbolic
probability distribution, evaluated using the Bandt-Pompe recipe to assign a probability distribution function to
the time series generated by the chaotic system. In order to evaluate the performance of these novel complexity
quantifiers, we compare them to a more standard chaos quantifier, namely the Kolmogorov-Sinai entropy. Here,
we present numerical results showing that the statistical complexity and the permutation entropy, evaluated at
the different time-scales involved in the chaotic regime of the laser subject to optical feedback, give valuable
information about the complexity of the laser dynamics.
The Differential Image Motion Monitor (DIMM) is a standard and widely used instrument for astronomical
seeing measurements. The seeing values are estimated from the variance of the differential image motion over
two equal small pupils some distance apart. The twin pupils are usually cut in a mask on the entrance pupil
of the telescope. As a differential method, it has the advantage of being immune to tracking errors, eliminating
erratic motion of the telescope. The Differential Laser Tracking Motion (DLTM) is introduced here inspired
by the same idea. Two identical laser beams are propagated through a path of air in turbulent motion, at the
end of it their wander is registered by two position sensitive detectors-at a count of 800 samples per second.
Time series generated from the difference of the pair of centroid laser beam coordinates is then analyzed using
the multifractal detrended fluctuation analysis. Measurements were performed at the laboratory with synthetic
turbulence: changing the relative separation of the beams for different turbulent regimes. The dependence, with
respect to these parameters, and the robustness of our estimators is compared with the non-differential method.
This method is an improvement with respect to previous approaches that study the beam wandering.
We have previously shown that the Levy fractional Brownian field family accounts for a complete statistical and
analytical description of non-Kolmogorov wavefront phase [Opt. Lett. 33(6), 572 (in press, 2008)]. This is a nonstationary
process having zero mean and stationary increments; then, replicating the well-known properties of the
turbulent phase. Opposite to traditional models relying in the stationary (spectral) approximation of the phase,
that ultimately leads to non-physical divergences. Our model avoids these pitfalls and gives exact analytical
results to many observable quantities: Strehl ratio,
angle-of-arrival variance, seeing and Zernike coefficients, and
also, a generalized DIMM theory. Nevertheless, some coefficients are slightly below (~ 5-10%) when compared
to other estimates in the occurrence of Kolmogorov turbulence. In the present work we show that this is due
to the mono-fractal nature of this model; that is, the absence of inner- and outer-scales. To address this issue
we introduce a Gaussian stochastic process whose realizations are multi-fractals: the multi-scale Levy fractional
Brownian field.
In Perez et al. [J. Opt. Soc. Am. A 21 (10), 2004] we have given a general formalism to model the turbulent
wave-front phase by using fractional Brownian motion processes. Moreover, it extends classical results to non-
Kolmogorov turbulence: the Strehl ratio and the angle-of-arrival variance are shown to be dependent on the
dynamic state of the turbulence. Nevertheless, this model has its drawbacks as it is unable of handling the
stationarity of the phase increments over the full inertial range. The Levy fractional Brownian motion (LfBm)
family is then introduced here in order to overcome this problem.
KEYWORDS: Turbulence, Motion models, Telescopes, Atmospheric optics, Motion measurement, Lanthanum, Point spread functions, System on a chip, Americium, Quality measurement
We have previously modeled the turbulent wave-front phase by using a fractional Brownian motion (J. Opt. Soc. Am.
A, Vol. 21, N° 10, pp. 1962-1969, October 2004). Non-Kolmogorov turbulence is primarily considered within this
approach. Now, in this work we study the relationship between seeing, usual measure of quality associated to a groundbased
telescope, and Hurst exponent, characteristic parameter of a fractional Brownian motion. The theory behind the
differential image motion monitor (DIMM), a standard and widely used instrument for seeing measurements, is
reviewed by us. It is shown that there is a direct connection between both parameters. Thus, it is concluded that Hurst
exponent is a quantifier of the atmospheric turbulent state.
KEYWORDS: Turbulence, Collimation, Motion models, Lanthanum, Sensors, Stochastic processes, Fractal analysis, Data modeling, System on a chip, Americium
We analyze the angle-of-arrival variance of an expanded and collimated laser beam after it has traveled through indoor
convective turbulence. A continuous position detector is set at the focus of a lens collecting the light coming from this
collimated laser beam. The effect of the different turbulent scales, above the inner scale, is studied changing the
diameter of a circular pupil before the lens. The experimental setup follows the design introduced by Masciadri and
Vernin (Appl. Opt., Vol. 36, N° 6, pp. 1320-1327, February 2004). Tilt data measurements are studied within the
fractional Brownian motion model for the turbulent wave-front phase. In a previous paper the turbulent wave-front
phase was modeled by using this stochastic process (J. Opt. Soc. Am. A, Vol. 21, N° 10, pp. 1962-1969, October 2004).
The Hurst exponents associated to the different degree of turbulence are obtained from the new D2H-2 dependence.
We experimentally study the variance of the transverse displacement (wandering) of a laser beam after it has traveled
through indoor artificially convective turbulence. In a previous paper (Opt. Comm., Vol. 242, N° 1-3, pp. 76-63,
November 2004) we have modeled the atmospheric turbulent refractive index as a fractional Brownian motion. As a
consequence, a different behavior is expected for the wandering variance. It behaves as
L2+2H , where L is the
propagation length and
H the Hurst exponent associated to the fractional Brownian motion. The traditional cubic
dependence is recovered when
H=1/2--the ordinary Brownian motion. That is the case of strong turbulence or long
propagation path length. Otherwise, for weak turbulence and short propagation path length some deviations from the
usual expression should be found. In this presentation we experimentally confirm the previous assertion.
This paper introduces a general and new formalism to model the turbulent wave-front phase using fractional Brownian motion processes. Moreover, it extends results to non-Kolmogorov turbulence. In particular, generalized expressions for the Strehl ratio and the angle-of-arrival variance are obtained. These are dependent on the dynamic state of the turbulence.
We analyze the fractal (box-counting) dimension of laser beam wandering. The wandering is due to the light travelling across a path filled by laboratory generated turbulence. The laser's centroid position is collected by a continuous position detector -- "light-tracker." We determine the box-counting dimension by means of two independent algorithms. The first method calculates the Hurst exponent of each axis, within the fractional Brownian model, and then the fractal dimension is determined applying a theoretical result. For the second one a new algorithm is proposed to estimate it directly. These results are compared.
The optical diffraction by fractal openings is increasingly being studied, because it allows to determine the properties and parameters that characterize these objects. Allain and Cloitre published the first results showing that the resulting analysis of the distribution of intensity obtained for diffraction experiments through fractal deterministic pupils permits the direct objection of the Hausdorff dimension an of other geometrical characteristic of these structures. In this work are studied, solved analytically, and characterized the lacunarity effect (epsilon) , dimension d and stage of growth k of Cantor fractal about he distribution of intensities of the diffraction pattern. In particular we make notice in the influence of lacunarity because this is one of the first works in which this geometric fractal parameter is being into consideration. The result of this study allow to say that an intimate relation exists between the distribution of the diffracted waves and the parameters that describe this kind of fractal geometry.
The van Cittert-Zernike theorem allows interpreting the coherence distribution of an electromagnetic field from simple interferential experiments, like Young experiments. The visibility of the interferential fringes is equal to the modulus of the complex degree of coherence of the optical field. In the experiments commented in this communication, an electrical bulb linear carbon filament and the focal line of a cylindrical lens illuminated by a collimated expanded He-Ne red laser light, were both assumed to be filamentary light sources in order to observe Young interferential fringes with each of them. The electric lamp is a polychromatic light source, while the laser is a monochromatic one. When a filamentary light source is parallel to the Young slits, the interferential fringes are well defined and highly contrasted. If the source is rotated in a plane parallel to that of the slits, the visibility of the interferential fringes monotonically decreases and reaches zero in case of perpendicularity. Interferential fringes were photographically recorded and the color slides were electronically scanned and processed.
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