Beam Propagation Factor of a Cosh-Airy Beam

Investigation of the effect of chirped factors on the interference enhancement effect of an Airyprime beam propagating in free space

The first-order and the second-order chirped factors are imposed on the Airyprime beam, and the analytical expression of the chirped Airyprime beam propagating in free space is derived. The phenomenon that the peak light intensity on observation plane other than initial plane is greater than that on initial plane is defined as the interference enhancement effect, which is caused by the coherent superposition of the chirped Airyprime and the chirped Airy-related modes. The effects of the first-order and the second-order chirped factors on the interference enhancement effect are theoretically investigated, respectively. The first-order chirped factor only affects the transverse coordinates where the maximum light intensity appears. The strength of interference enhancement effect of the chirped Airyprime beam with any negative second-order chirped factor must be stronger than that of the conventional Airyprime beam. However, the improvement of the strength of interference enhancement effect caused by the negative second-order chirped factor is realized at the expense of shortening the position where the maximum light intensity appears and the range of interference enhancement effect. The chirped Airyprime beam is also experimentally generated, and the effects of the first-order and the second-order chirped factors on the interference enhancement effect are experimentally confirmed. This study provides a scheme to improve the strength of interference enhancement effect by controlling the second-order chirped factor. Compared with traditional intensity enhancement methods such as using lens focusing, our scheme is flexible and easy to implement. This research is beneficial to the practical applications such as spatial optical communication and laser processing.

Generation of finite energy Airyprime beams by Airy transformation

In this paper, the lone generation of a new kind of beam named finite energy Airyprime (FEA) beam through the Airy transformation of the coherent superposition of four different elegant Hermite-Gaussian modes is reported for the first time. Analytical expressions of the centroid, the r.m.s beam width, the divergence angle, and the beam propagation factor of the FEA beam are derived in the output plane of Airy transformation, respectively. The effects of the Airy control parameters on the intensity distribution, the centroid, the r.m.s beam width, and the beam propagation factor are examined in detail through numerical examples. Unlike the Airy beam, the FEA beam upon free space propagation will be associated with an additional Airy mode, and the beam pattern of the FEA beams propagating in free space will evolve into a solid beam spot with two tails along two transverse directions, as well as the the intensity of main lobe of the FEA beam decays much slowly during free space propagation. Further, an experiment setup is established to generate the FEA beam via Airy transformation of four mixed elegant Hermite-Gaussian modes. The propagation characteristics such as the intensity distribution, the r.m.s beam width and the beam propagation factor are measured. The experimental results agree well with the theoretical predictions. Our study affords an effective and novel approach to generate the FEA beam, and is beneficial to expand the potential application of the FEA beam.

Quality factor of partially coherent Airy beams in a turbulent atmosphere

The quality factor of a partially coherent Airy beam in a turbulent atmosphere is investigated with the help of the extended Huygens-Fresnel integral formula and the Wigner distribution. From the numerical results, based on the analytical formula, we find that increasing the inner scale or decreasing the structure constant of the refractive index fluctuations of the turbulent atmosphere tends to decrease the quality factor of a partially coherent Airy beam in a turbulent atmosphere. Moreover, it is found that increasing the scale factor and wavelength or decreasing the coherent length of a partially coherent Airy beam can also be used to decrease the quality factor of a partially coherent Airy beam in a turbulent atmosphere. However, the quality factor of a partially coherent Airy beam in a turbulent atmosphere has a maximum when the exponential truncation factor of a partially coherent Airy beam is 0.63. Our results will be useful in long-distance free-space optical communications and laser defense.

Transformation of a Hermite-Gaussian beam by an Airy transform optical system

Analytical expression of the Airy transform of an arbitrary Hermite-Gaussian beam is derived. The optical field in the x-direction of the Airy transform of Hermite-Gaussian beams with transverse mode number m is the sum of the zero-order derivative to mth-order derivative of the Airy function with different weight coefficients. The analytical expressions of the centre of gravity and the beam spot size of an arbitrary Hermite-Gaussian beam passing through an Airy transform optical system are also presented, which are very concise. Because the Airy transform of a Hermite-Gaussian beam has the same evolution law in the two transverse directions, only the effects of the control parameter α and the transverse mode number m on the normalized intensity distribution, the centre of gravity, and the beam spot size in the x-direction are theoretically investigated, respectively. The Airy transform of Hermite-Gaussian beams is also realized in the experiment. The influence of the control parameters on the normalized intensity distribution, the centre of gravity, and the beam spot size is experimentally investigated, respectively. The experimental results are consistent with the theoretical simulation results. When Hermite-Gaussian beams pass through an Airy transform optical system, the number of lobes may change, and the importance of lobes with the same status in the input plane may become different. By using the Airy transform of Hermite-Gaussian beams, the practical applications of Hermite-Gaussian beams can be extended.

Airy transform of Laguerre-Gaussian beams

Airy transform of Laguerre-Gaussian (LG) beams is investigated. As typical examples, the analytic expressions for the Airy transform of LG₀₁, LG₀₂, LG₁₁, and LG₁₂ modes are derived, which are special optical beams including the Airy and Airyprime functions. Based on these analytical expressions, the Airy transform of LG₀₁, LG₀₂, LG₁₁, and LG₁₂ modes are numerically and experimentally investigated, respectively. The effects of the control parameters α and β on the normalized intensity distribution of a Laguerre-Gaussian beam passing through Airy transform optical systems are investigated, respectively. It is found that the signs of the control parameters only affect the location of the beam spot, while the sizes of the control parameters will affect the characteristics of the beam spot. When the absolute values of the control parameters α and β decrease, the number of the side lobes in the beam spot, the beam spot size, and the Airy feature decrease, while the Laguerre-Gaussian characteristic is strengthened. By altering the control parameters α and β, the performance of these special optical beams is diversified. The experimental results are consistent with the theoretical simulations. The Airy transform of other Laguerre-Gaussian beams can be investigated in the same way. The properties of the Airy transform of Laguerre-Gaussian beams are well demonstrated. This research provides another approach to obtain special optical beams and expands the application of Laguerre-Gaussian beams.

Propagation of Cosh-Airy and Cos-Airy Beams in Parabolic Potential

The analytical expressions of one-dimensional cosh-Airy and cos-Airy beams in the parabolic potential are derived in the general and the phase transition points. The expression in the phase transition point shows a symmetric Gaussian intensity profile and is independent of any Airy features, which is completely different from that in the general point. The intensity, the center of gravity, and the effective beam size of the cosh-Airy and cos-Airy beams in the parabolic potential are periodic and have the same period. The effects of the transverse displacement, the cosh factor, and the cosine factor on these periodic behaviors are also investigated. The direction of self-acceleration reverses every half-period. The phase transition point is also the inversion point of the intensity distribution, which indicates that the intensity distributions before and after the phase transition point are mirror symmetrical. The periodic behaviors of the normalized intensity, the center of gravity, and the effective beam size of the cosh-Airy and cos-Airy beams in the parabolic potential are attractive and well displayed. The results obtained here may have potential applications in particle manipulation, signal processing, and so on.

Beam propagation factor and kurtosis parameter of hollow vortex Gaussian beams: an alternative method

Based on the second-order moments, an analytical and concise expression of the beam propagation factor of a hollow vortex Gaussian beam has been derived, which is applicable for an arbitrary topological charge $m$m. The beam propagation factor is determined by the beam order $n$n and the topological charge $m$m. With increasing the topological charge $m$m, the beam propagation factor increases. However, the effect of the beam order $n$n on the beam propagation factor is associated with the topological charge $m$m. By using the transformation formula of higher-order intensity moments, an analytical expression of the kurtosis parameter of a hollow vortex Gaussian beam passing through a paraxial and real $ABCD$ABCD optical system has been presented. The kurtosis parameter is determined by the beam order $n$n, the topological charge $m$m, and the position of observation plane $ \eta $η. The influence of the beam order $n$n on the kurtosis parameter is related with the topological charge $m$m and the position parameter $ \eta $η. When the beam order $n$n is larger than 1, the kurtosis parameter in different observation $\eta$η-planes decreases and tends to a stable value with increasing the topological charge $m$m. When $m = \pm {2}n$m=±2n, the kurtosis parameter is independent of the position parameter $ \eta $η and keeps unvaried during the beam propagation. Regardless of the values of $n$n and $m$m, the kurtosis parameter must tend to a saturated value or a stable value as the position parameter $ \eta $η increases to a sufficiently large value. This research is beneficial to the practical application of a hollow vortex Gaussian beam.