Diffraction catastrophe theory is the theoretical foundation for generating accelerated beams such as Airy beams which are derived from fold diffraction catastrophe,^[1] Pearcey beams from cusp diffraction catastrophe,^[2] regular polygon beams from Thom’s elliptic umbilics,^[3,4] and other accelerated beams that were recently introduced.^[5–7] Hence, based on the relation between diffraction integrals and catastrophe polynomials,^[8,9] a general canonical catastrophe integral can be obtained.^[6] So, to realize a diffraction catastrophe, one of the methods is to use phase-only diffraction patterns. The key task of such a method is to produce an appropriate phase mask. Hence, the corresponding phase profile masks can be obtained on the basis of caustic characteristic diffraction pattern. For example, as is well known, 3/2 phase only pattern can generate Airy beams,^[10] observed experimentally for the first time by Siviloglou et al.,^[11] and the quartic polynomial exponent phase can generate Pearcey beams.^[12] These last new kinds of paraxial beams were formulated by Ren et al.,^[13] based on the Pearcey function of catastrophe theory.

The propagating form of the Pearcey beam has several noteworthy properties, some of which are reminiscent of not only Airy beams, but also Gaussian and Bessel beams. Like the Bessel beam and Airy beam, a pure Pearcey beam has infinite energy, but can be made finite by modulating the Pearcey function with a Gaussian one in real space, which does not significantly change the beam properties. Earlier, it was shown that the Pearcey beams are auto-focusing and self-healing after being distorted by obstacles.^[12] In a recent paper, a virtual source was proposed to generate the Pearcey beams.^[14] In Ref. [15] Pearcey beams were generalized into half Pearcey beams and it is shown that such beams, which are form-invariant,^[12,15,16] propagate with acceleration and auto-focusing.

Recently, a lot of attention was paid to studying the propagation of some laser beams in a turbulent atmosphere^[17–28] because of their large applications in optical communications, imaging systems and remote sensing.^[29–32] As is well known, when a laser beam propagates in the atmosphere, it produces random variations in amplitude and phase of the electric field caused by random changes of the refractive index. In our research group, the propagations of Li’s flat-topped beams,^[33] Bessel-modulated Gaussian beams,^[34] Superposition of Kummer beams,^[35] truncated modified Bessel modulated Gaussian beams with quadrature radial dependence (MQBG),^[36] and Li’s flattened Gaussian beams^[37] in turbulent atmosphere were investigated. In Ref. [12], the propagation of PG beams in free space has been given. But to our knowledge, the propagation of PG beams in turbulent atmosphere has not been studied elsewhere.

In this paper, our interest is to study the propagation characteristics of PG beams through a turbulent atmosphere. The rest of this paper is organized as follows. In Section 2, an analytical expression of the average intensity of the considered beams is given based on the extended Huygens–Fresnel integral formula in the paraxial approximation and on the Rytov theory. An approximate analytical axial intensity distribution is derived in Section 3. We present some numerical simulations in Section 4. Finally, a conclusion is drawn from the present work in Section 5.

In this section, we give the expression of the one-dimensional average intensity of the PG beams after propagating in turbulent atmosphere. The Pearcey function is defined by an integral representation^[38,39] as

The field distribution of the PG beams in the Cartesian coordinates (x₁,y₁, z) at the plane z = 0 is given by^[12]

We illustrate in Figs. 1–4 the two-dimensional intensity of PG beams at the input plane (z = 0).

Fig. 1.

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	Fig. 1. Incident transverse intensities of PG beams in source plane (z = 0) for ω0 = 0.5 mm and λ = 1550 nm with (a) x0 = y0 = 1 × 10−2 mm, (b) x0 = y0 = 1.2 × 10−2 mm, (c) x0 = y0 = 1.6 × 10−2 mm, and (d) x0 = y0 = 3 × 10−2 mm, respectively.

Fig. 2.

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	Fig. 2. Incident transverse intensities of PG beams in source plane (z = 0) for ω0 = 1 mm and λ = 1550 nm with (a) x0 = y0 = 1 × 10−2 mm, (b) x0 = y0 = 2 × 10−2 mm, and (c) x0 = y0 = 3 × 10−2 mm, respectively.

Fig. 3.

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	Fig. 3. Incident transverse intensities of PG beams in source plane (z = 0) for λ = 1550 nm, x0 = y0 = 1.064 × 10−2 mm with (a) ω0 = 5 × 10−2 mm, and (b) ω0 = 0.1 mm, and (c) ω0 = 3 mm, respectively.

Fig. 4.

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	Fig. 4. Incident transverse intensities of PG beams in source plane (z = 0) for λ = 1550 nm, x0 = y0 = 2 × 10−2 mm with (a) ω0 = 5 × 10−2 mm, (b) ω0 = 0.1 mm, and (c) ω0 = 3 mm, respectively.

Based on the extended Huygens–Fresnel integral formula in the paraxial approximation and on the Rytov theory, the expression of the electric field of the PG beams after propagating in a turbulent atmosphere is written as^[40]

Fig. 5.

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	Fig. 5. A schematic propagation of the PG beams in a turbulent atmosphere optical system.

In the following, we only consider a one-dimensional expression of the electric field of the PG beams at the z = 0 plane, which is expressed as

In this section, we derive the axial intensity distribution of PG beams in turbulent atmosphere using Eq. (5). Let x₂ = 0, then one will obtain

Finally, after tedious calculations, the analytical expression of the axial intensity distribution of PG beams in turbulent atmosphere is written as

We illustrate in Fig. 6 the variations of on-axis average intensity with propagation distance z of PG beams for three values of turbulent strength two values of the specified scaling length x₀ of the beam, λ = 632.8 nm and ω₀ = 1 mm. The curves show that the intensity is equal to zero within the first several hundred meters of the near field when the turbulent strength increases and that for each value of the average intensity first increases, and then decreases with the increase of the propagation distance. We note that the axial intensity increases with increasing and also with x₀ decreasing. From these plots, we can also see that the position of the maximum depends on the turbulent strength i.e., the axial intensity vanishes quickly for the large values of turbulent strength. Moreover, the position of the maximum moves towards small value of the propagation distance when the specified scaling length x₀ increases, which means that the Pearcey–Gaussian beam is affected by the turbulent atmosphere and the specified scaling length x₀, i.e., the propagation is shorter when the atmosphere is very turbulent and x₀ is greater.

Fig. 6.

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	Fig. 6. Plots of on-axis average intensity versus the propagation distance z of PG beam for (a) x0 = 2 × 10−2 mm and (b) x0 = 6 × 10−2 mm, three values of the turbulent strengths, λ = 632.8 nm, and ω0 = 1 mm, respectively.

To show how the beam wavelength affects the on-axis average intensity, we illustrate the variations of on-axis average intensity with propagation distance z in Fig. 7, for the chosen values x₀ = 2 × 10⁻² mm, and two values of the beam waist: ω₀ = 1 mm and ω₀ = 0.5 mm, respectively. We can see from this figure that the axial intensity is equal to zero beyond the first several kilometers and it increases with wavelength decreasing. We note from the illustrations that the position of the maximum depends on the wavelength and the axial intensity vanishes rapidly for the small values of λ. From these illustrations, we can see that the position of the maximum moves with propagation distance decreasing when the beam waist increases. Our numerical results show that the propagation in turbulent atmosphere of PG beam with smaller wavelength λ is shorter than that with larger wavelength.

Fig. 7.

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	Fig. 7. Plots of on-axis average intensity versus propagation distance z of PG beam for (a) ω0 = 1 mm and (b) ω0 = 0.5 mm, three values of λ, and x0 = 2 × 10−2 mm, respectively.

Figure 8 shows the variations of on-axis average intensity with propagation distance z of PG beam for λ = 632.8 nm, x₀ = 2 × 10⁻² mm, and two values of turbulent strength respectively. For each value, three curves are plotted for different waist values. It is shown that the intensity profile keeps unchanged and the axial intensity is equal to zero beyond the first thousand meters in Fig. 8(a) and the first thousand kilometers in Fig. 8(b). From this figure, we also observe that the axial intensity increases with ω₀ increasing and it vanishes very quickly for the large values of the beam waist. We note that the position of the maximum depends on the waist and moves with propagation distance decreasing as turbulent strength increases. This phenomenon can be explained physically by the fact that the PG beam with larger beam waist ω₀ spreads more rapidly than with smaller beam waist propagating in turbulent atmosphere.

Fig. 8.

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	Fig. 8. Plots of on-axis average intensity versus the propagation distance z of PG beams with three values of ω0, λ = 632.8 nm, x0 = 2 × 10−2 mm for (a) and (b) respectively.

In this work, we derive an analytical expression of the axial intensity for PG beams, based on the extended Huygens–Fresnel integral formula in the paraxial approximation and on the Rytov theory, in turbulent atmosphere. Our numerical simulations shown that the PG beams with larger wavelength and smaller waist are less affected by the atmospheric turbulence, which is useful for applications in the field of optical communication for a long distance.