Optical turbulence distorts beam amplitude and phase, causing spreading, wandering, and irradiance fluctuations. Reconstructing perturbed beams’ complex fields is experimentally challenging due to these dynamic effects. Our complex phase retrieval technique, using binary amplitude modulation with a DMD and high-speed camera, characterizes collimated beams through turbulence and overcomes interferometric limitations. Conventionally, phase retrieval modulates optical fields via random coded apertures (RCA) to recover amplitude and phase without prior knowledge, solving ill-posed problems with phase-lift algorithms. Our previous approach required ≥20 apertures, increasing acquisition time and complexity. We designed a new coded aperture, reducing time and enhancing quality over traditional RCA. Then we apply a novel deep-learning phase unwrapping algorithm enabling efficient unwrapping of phases with turbulence-induced branch point singularities manifesting as vortices. This is the first experimental observation of turbulence complex wavefronts reconstructed with high spatial resolution and sampling rate. We discuss observed statistical properties and compare with current models.
The explosive growth of satellites in low Earth orbit (LEO) demands advanced surveillance and communication capabilities. However, atmospheric turbulence hinders high-resolution imaging and high-speed communication through optical wavelengths, which remains the only viable option. SEETRUE (Sharp wavefront sEnsing for adaptivE opTics in gRound-based satellite commUnications and spacE surveillance), proposes a game-changing solution: cost-effective, AI-driven wavefront sensing for Adaptive Optics (AO) in optical ground stations. It features a unique ground station equipped with a 50 cm robotic telescope with AO capability, a multi-purpose 38 cm binocular telescope, and an atmospheric profiling system. AI-powered wavefront sensors (WFS) within the system leverage novel turbulence models and a revolutionary ”end-to-end” design approach to maximize information extraction. This enables compact and low-cost AO solutions, overcoming a major barrier to widespread adoption. Paving the way for a future with accessible and affordable space communication and surveillance for all.
Taylor’s Frozen Turbulence Hypothesis (TFTH) has been used extensively in theoretical studies to model the temporal fluctuations of optical quantities affected by atmospheric turbulence. It has been relied upon to provide temporal-frequency spectra under varying propagation conditions and for different atmospheric refractive index models. However, experimental works have revealed its limitations, such as systematic inaccuracies in estimating cross winds during calm nights in scintillation measurements at astronomical sites, scintillation discrepancies in ground-layer measurements, and broad estimates of the coherence time in phase fluctuation measurement techniques. This highlights the need to recognize the limitations of the TFTH and seek alternatives that can provide a more reliable description of atmospheric turbulence’s temporal fluctuations. Here, we propose a spatio-temporal statistics for refractive index fluctuations through fluid dynamics models and evaluate the complex phase propagation under weak turbulence. Then, we test its ability to reproduce experimental observations under different ground-layer turbulence conditions.
Any beam that propagates through optical turbulence will experience distortions in both its amplitude and phase, leading to various effects such as beam wandering, beam spreading, and irradiance fluctuations. Reconstructing the complete field of a perturbed beam is a challenging task due to the dynamic nature of these effects. Interferometric wavefront reconstruction techniques—such as those based on holography—are commonly used but are hindered by their sensitivity to environmental disturbances and alignment errors. However, new complex phase retrieval methods based on propagation equations have emerged, which do not require prior knowledge of the beam to be reconstructed and are suitable for amplitude or phase objects, or both. We propose an experimental implementation of a complex phase retrieval technique for characterizing Gaussian beams propagating through optical turbulence, using binary amplitude modulation with a digital micro-mirror device (DMD). This approach is ideal for dynamic applications and has enabled us to achieve experimental high-speed complex wavefront reconstruction of optical beams through controlled real turbulence. This experiment corresponds to the initial step in our research focused on gaining a deeper understanding of optical turbulence from an experimental perspective.
Optical turbulence induces distortions in amplitude and phase in any beam propagating through it, resulting in beam spreading, beam wandering, and irradiance fluctuations among other effects. Due to the dynamic nature of these effects, the complex field reconstruction of a perturbed beam presents a great experimental challenge. Interferometric wavefront reconstruction techniques require very sophisticated assemblies prone to alignment errors due to their high sensitivity to environmental disturbances. This hinders its experimental implementation. New complex phase retrieval methods overcome most of the limitations of interferometric methods: they are suitable for amplitude or phase objects (or both) and their reconstruction algorithms—based on propagation equations—make unnecessary any a-priori knowledge of the beam to be reconstructed. We propose an experimental implementation of a complex phase retrieval technique for the characterization of Gaussian beams propagating through turbulence. This technique is based on binary amplitude modulation using a digital micro-mirror device (DMD) which has proven to be suitable for dynamic applications. To our knowledge, this is the first experimental high-speed complex wavefront reconstruction of optical beams—by binary amplitude modulation—through controlled real turbulence. This experiment represents the first step in our research focused on understanding optical turbulence from an experimental point of view.
In the last decade, a nascent trend of characterizing turbulence from observing features of distant targets through ground-layer turbulence have been relentless growing. Either from observing regular geometrical features of buildings or arrays of LEDs, it is possible to retrieve the structure constant of the refractive index fluctuations. On the other hand, because of the lack of a definitive theoretical model describing anisotropic or inhomogeneous turbulence, most experimental observations have been reduced to mere descriptions in the event of deviations from expected Obukhov-Kolmogorov predictions. Our group has been able to retrieve power-spectrum exponents, without a prior knowledge of a subjacent model, and henceforth determine anisotropic behavior in controlled optical turbulence; furthermore, under convective turbulence, an exponent can be obtained from time series of the occurrence of power drops in optical communication links: extreme events.
In this manuscript, we present a technique identifying as extreme events sudden changes in morphological characteristics of an array of point sources observed through real controlled anisotropic turbulence assisted by a deep-learning ad-hoc. This approach provides an effective approach to reduce high-volume data from imaging targets into a real-time stream of parameters to fully characterize optical turbulence.
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 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, 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.
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.
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.
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