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The evolution of Airy beam intensity distribution in turbulence is examined. Results show that the centroid position and skewness of an Airy beam are independent on turbulence. When the exponential truncation factor is small, an Airy beam has a long tail on the left side. When the exponential truncation factor is larger than 1, an Airy beam can be approximately expressed by an off-axis Gaussian beam. If the effect of turbulence is large enough, the Airy beam converges to a Gaussian distribution. However, the convergence is slower for a small exponential truncation factor than for a large one. That is, an Airy beam with a smaller exponential truncation factor exhibits more resilience against perturbations of turbulence than does a larger one.

An analytical expression for the optical field of an Airy beam partially blocked by an opaque obstacle is derived. The self-healing properties of the Airy beam are studied and discussed in detail. The study shows that the self-healing process of the Airy beam is affected by many factors such as the opaque obstacle size, propagation distance, wavelength, and parameters of the Airy beam. The self-healing process is caused by the convergence of energy from the side to the position of the opaque obstacle and is finished when the convergence of energy flow disappears. When the propagation distance is short, the main lobe of the Airy beam is affected by the obstruction located near the main lobe. When the propagation distance is long, the main lobe of the Airy beam can be affected by the obstruction located far away from the main lobe. The result agrees with the existing results and can be explained by the caustic of the Airy beam.

Analytical propagation expression of an Airy beam in uniaxial crystals orthogonal to the optical axis is derived. The ballistic dynamics of an Airy beam in uniaxial crystals is also investigated. The Airy beam propagating in uniaxial crystals orthogonal to the optical axis mainly depends on the ratio of the extraordinary refractive index to the ordinary refractive index. As an example, the propagation of an Airy beam in the positive uniaxial crystals orthogonal to the optical axis is demonstrated. The acceleration of an Airy beam in the transversal direction along the optical axis is more rapidly than that in the other transversal direction. With increasing the ratio of the extraordinary refractive index to the ordinary refractive index, the acceleration of the Airy beam in the transversal direction along the optical axis speeds up and the acceleration of the Airy beam in the other transversal direction slows down. The Airy beam propagating in uniaxial crystals orthogonal to the optical axis follows a ballistic trajectory. The effective beam size of the Airy beam in the transversal direction along the optical axis is always larger than that in the other transversal direction.

The propagation of a multi-Gaussian beam in turbulent atmosphere in a slant path is studied. The analytical expression for the average intensity of a general multi-Gaussian beam is derived. As special cases the average intensities of a two- and a four-Gaussian beam are investigated and numerically calculated. The investigation reveals that at lower altitude and with large sigma the intensity distribution at the receiver plane can have a shape (multiple peaks) similar to that at the source plane. But with increase in altitude or decrease in sigma, the multiple peaks gradually disappear and evolve into the profile of a fundamental Gaussian beam. From the comparisons between the different propagations we can see that the beam spreading due to wavelength and initial waist width in a slant path is much slower than that in a horizontal path.

Bessel beam is the important member of the family of non-diffracting beams and has many novel properties which can be used in many areas. However, the source of Bessel beam generated by the existing methods can be used only in a short distance due to its low power. In this paper, based on the coherent combining technology, we have proposed a method which can be used to generate a high-power Bessel beam. Even more, we give an innovative idea to form vortex phase by using discontinuous piston phase. To confirm the validity of this method, the intensity evolution of the combined beam and the Bessel-Gaussian beam at different propagation distance have been studied and compared. Meanwhile, the experimental realization has been discussed from the existing experimental result related to the coherent combining technology.

The propagation of a cosh-Gaussian beam through an arbitrary ABCD optical system in turbulent atmosphere has been investigated. The analytical expressions for the average intensity at any receiver plane are obtained. As an elementary example, the average intensity and its radius at the image plane of a cosh-Gaussian beam through a thin lens are studied. To show the effects of a lens on the average intensity and the intensity radius of the laser beam in turbulent atmosphere, the properties of a collimated cosh-Gaussian beam and a focused cosh-Gaussian beam for direct propagation in turbulent atmosphere are studied and numerically calculated. The average intensity profiles of a cosh-Gaussian beam through a lens can have a shape similar to that of the initial beam for a longer propagation distance than that of a collimated cosh-Gaussian beam for direct propagation. With the increment in the propagation distance, the average intensity radius at the image plane of a cosh-Gaussian beam through a thin lens will be smaller than that at the focal plane of a focused cosh-Gaussian beam for direct propagation. Meanwhile, the intensity distributions at the image plane of a cosh-Gaussian beam through a lens with different w(0) and Omega(0) are also studied.

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