# Beam Propagation Factor of a Cosh-Airy Beam
## 1 Introduction  
The Airy wave packet was first proposed in 1979 [1]. An Airy beam has the ability to resist diffraction over long distances and can freely accelerate during beam propagation [2; 3]. In spite of the severity of the imposed perturbations, the Airy beam has a self-healing property and tends to reform [4]. By using the method of geometrical optics and the Kirchhoff-Huygens integral, as well as the Wigner distribution function, Airy beams have been analyzed to provide a cogent physical explanation of the intriguing features [5; 6]. The propagation of an Airy beam in free space [7], in a uniaxial crystal [8], in a photorefractive medium [9], in a Kerr medium [10], in a strongly nonlocal nonlinear media [11], in water [12], in turbulence [13], through an apertured and misaligned paraxial \(ABCD\) optical system [14], through the system of the fractional Fourier transform [15], and from a right-handed material to a left-handed material [16] have been investigated, respectively. The evolution of the Poynting vector and angular momentum of the Airy beam has been investigated in free space [17]. Based on the method of the stationary phase, the far-field divergent properties of an Airy beam have been examined [18]. The analytical expression of the beam propagation factor of an Airy beam has been derived on the basis of the Ermakov-Pinney equation [19]. Under the illumination of a one-dimensional Airy beam, the optical force on a dielectric cylindrical particle of an arbitrary size has been calculated [20]. Airy beams can be created by employing a transmissive spatial light modulator [21]. A novel family of finite Airy array beams has been produced by an optical Airy transform system illuminated by Gaussian array beams [22]. Although Airy beams have been studied more in the optics domain, they have also been investigated in the millimeter wave and radio frequency domains [23; 24].  
Cosh-Airy beams can be considered as a superposition of two Airy beams with different decay factors. Cosh-Airy beams have similar propagation properties to Airy beams, and they possess moremanipulation degrees of freedom than the corresponding Airy beams [25]. The propagation properties of cosh-Airy beams have been investigated in free space [25] and uniaxial crystals orthogonal to the optical axis [26], respectively. The periodic phase transition of a cosh-Airy beam has been found in a quadratic-index inhomogeneous medium [27]. The self-healing ability of the cosh-Airy beam is verified to be higher than that of the corresponding Airy beam [28]. The above studies on cosh-Airy beams reveal certain characteristics of the beams, but they are not thorough. For example, how can the quality of a cosh-Airy beam be measured? The beam propagation factor, which is defined by the second-order moments, plays an important role in the study of beam propagation characteristics. The international standardization organization committee suggests the beam propagation factor to be a unified standard to evaluate the quality of an arbitrary optical beam. The beam propagation factor can be defined by the ratio of the product of the beam waist and the divergence of the optical beam to the product of the beam waist and the divergence of the fundamental Gaussian beam. The fundamental Gaussian beam is regarded as the comparing standard. In the rest of this article, therefore, the beam propagation factor of a cosh-Airy beam is investigated. The purpose of this paper is to further reveal the propagation properties of a cosh-Airy beam from the second-order moments.