Bessel beams are a family of optical beams that maintain their transverse shape as they propagate, making them non-diffractive. This property allows the beam to be rebuilt after passing through an obstacle, making it useful for applications such as light sheet microscopy, optical particle manipulation, and information transmission.

However, Bessel beams are not immune to disturbances from the medium through which they propagate. If the medium is turbulent or turbid, meaning it has random variations in the refractive index, absorption coefficients, scattering, or anisotropy parameter due to changes in temperature or pressure, the Bessel beam can suffer distortions in its shape and phase. These distortions can affect the performance and quality of the aforementioned applications.

In this article, we will explain how Bessel beams are generated, how they behave in a turbulent environment, and what techniques exist to mitigate or correct the distortions caused by turbulence. We will also mention their potential applications in studying biological media.

Generation of Bessel beams

Bessel beams can be generated from a Gaussian beam, which is the most common type of laser beam, using an axicon lens or diffraction ring. These optical elements transform the transverse profile of the Gaussian beam into a circular pattern with concentric rings corresponding to the zeros and maxima of the Bessel beam (as seen in the figure above). The main advantage of the axicon lens is that it is relatively inexpensive and easy to implement. However, the main drawback is that the Bessel beam it generates is only defined to a limited length of a few centimeters. Alternatively, computer-generated holograms (CGH) can be used. Although this is a more complex and expensive system to implement, the advantage of these systems is that they are very versatile for generating all kinds of Bessel beams or vector beams in general. The disadvantage is that they are expensive systems and not as easy to implement in the laboratory as an axicon lens.

The Bessel beam has a mathematical form that can be expressed using a special function called the Bessel function. This function depends on the order of the beam (an integer that determines the number of rings) and the conical parameter (a real number that determines the angle at which the beam opens). The Bessel function also has an imaginary part that represents the phase of the beam.

The phase of the Bessel beam can have an optical singularity called an optical vortex. An optical vortex is a point where the phase changes abruptly by 2π radians around a central axis. The optical vortex has an associated topological charge that indicates how many times the phase changes when going around the axis. Bessel beams with optical vortices are called vector Bessel beams.

Propagation in turbulent media

A turbulent medium is one that has random and isotropic fluctuations (equal in all directions) of refractive index due to local changes in temperature or pressure. These fluctuations cause the medium to act as a series of random lenses that deflect and disperse light.

The propagation of optical beams in a turbulent medium can be modeled using the stochastic differential equation of Rytov. This equation describes how the amplitude and phase of the electric field change due to medium perturbations.

The main effects that turbulence produces on optical beams are:

Angular broadening: The angle at which the beam opens increases due to random deviations caused by turbulent lenses.

Contrast reduction: The difference between the maxima and minima of the transverse profile decreases due to dispersion caused by turbulent lenses.

Loss of coherence: The correlation between two points of the electric field decreases due to random phase variations caused by turbulent lenses.

These effects depend on both the statistical properties of the turbulent medium and the characteristics of the beam, such as its wavelength and shape.

Techniques for mitigating turbulence effects

Several techniques have been developed to mitigate the effects of turbulence on optical beams, including adaptive optics, phase correction, and post-processing algorithms. These techniques aim to correct for the distortions caused by turbulence and restore the beam's original shape and phase.

Adaptive optics uses deformable mirrors or other devices to correct for the distortions caused by turbulence in real-time. This technique has been used in astronomy to correct for atmospheric turbulence and improve image quality.

Phase correction techniques use wavefront sensors to measure the distortions caused by turbulence and then apply a corrective phase to the beam. This technique has been used in free-space optical communication to improve data transmission rates. Post-processing algorithms use computational methods to correct for the distortions caused by turbulence after the beam has propagated through the medium. These algorithms can be applied to images acquired with light sheet microscopy or other imaging techniques to improve image quality.

One possible application of Bessel beams is the study of biological tissues, which are sets of specialized cells that perform a specific function in the body. Biological tissues are classified into four types: epithelial, connective, muscular, and nervous. Each type of tissue has a different structure and optical properties, which can be analyzed with Bessel beams. For example, connective tissue is made up of collagenous fibers that are arranged in parallel bundles, which can affect the propagation and diffraction of Bessel beams. The study of the effects of Bessel beams on biological tissues could have applications in the diagnosis and treatment of various diseases.

Final remarks

In conclusion, Bessel beams are a useful tool in optical applications due to their non-diffractive nature. However, they are not immune to the effects of turbulence, which can cause distortions in their shape and phase. Several techniques have been developed to mitigate these effects and restore the beam's original characteristics. Understanding the behavior of Bessel beams in turbulent media is crucial for improving the performance and quality of optical applications.

Emiliano Teran

To know more:

Philip Birch, Iniabasi Ituen, Rupert Young, and Chris Chatwin, "Long-distance Bessel beam propagation through Kolmogorov turbulence," J. Opt. Soc. Am. A 32, 2066-2073 (2015). https://doi.org/10.1364/JOSAA.32.002066

Li, S., Wang, J. Adaptive free-space optical communications through turbulence using self-healing Bessel beams. Sci Rep 7, 43233 (2017). https://doi.org/10.1038/srep43233

Nokwazi Mphuthi, Roelf Botha, and Andrew Forbes, "Are Bessel beams resilient to aberrations and turbulence?," J. Opt. Soc. Am. A 35, 1021-1027 (2018). https://doi.org/10.1364/JOSAA.35.001021

## Comments