1 Introduction

Sinusoidal-Gaussian modes are well-known set solutions of the paraxial wave equation in the rectangular symmetry since their introduction by Casperson and Tovar (1998) and Tovar and Casperson (1998) as a generalized form of the Gaussian and sinusoidal beams. Theses beams can be written as product of sinusoidal and Gaussian functions having both complex arguments. Among the set of sinusoidal-Gaussian modes, the cosine hyperbolic-Gaussian beam has been the most exploited in laser research over the last few years. In the initial plane, this beam can be regarded as a superposition of four decentered Gaussian beams with the same waist width. The decentered beam parameter b plays a crucial role in the beam shaping properties, that is, the beam exhibits different profiles with respect to the magnitude of the parameter b. Indeed, when b is small (b < 1.5), the field profile may resemble closely to the fundamental Gaussian beam whereas for large b (\(b \ge 4\)), the beam will be a hollow dark beam-like with four lobes, and for intermediate values of b, the beam may resemble to the flat-topped model beam (Lu et al. 1999; Belafhal and Ibnchaikh 2000; Ibnchaikh et al. 2001; Hricha and Belafhal 2004, 2005a, b). This profile flexibility has stimulated the interest of many laser researchers over the last few years, so a lot of works have been devoted to this family of beams; for instance, their propagation properties through a paraxial ABCD optical system, in turbulent atmosphere, optical transforms and the self focusing in plasma have been examined (Lu et al. 1999; Belafhal and Ibnchaikh 2000; Ibnchaikh et al. 2001; Hricha and Belafhal 2004, 2005a, b; Chu 2007; Patil et al. 2009, 2012; Dong et al. 2013; Wani and Kant 2014; Ez-Zariy et al. 2018).

On the other side, a new type of laser beams, which is referred to as vortex-Gaussian beam, has been introduced by Zhou et al. (2013a) as a beam model to describe the simple Gaussian beam having an embedded vortex. This beam has a dark hollow profile and it also belongs to the vortex beams family, i.e., it has the advantage to carry the orbital angular momentum due to imbedded topological charge (Luo et al. 2014; Liu et al. 2016a, 2017; Zeng et al. 2018). During the last few years, there has been growing interest for optical vortex beams owing to their potential in practical applications, e.g., in optical microscopy, wireless communications and optical micromanipulations (Rubinsztein-Dunlop et al. 2017; Torok and Munro 2004; Wang et al. 2012; Paterson et al. 2001; Bishop et al. 2004; Ito et al. 1996; Kuga et al. 1997; Cai et al. 2003; Cai and Ge 2006), etc. The propagation properties of the vortex-Gaussian beams in free space have been investigated in details in Zhou et al. (2013a). The authors showed that the said beams have high propagation stability that makes them relevant candidates for micromanipulation and optical trapping.

In the present paper, we introduce a new vortex beam related to cosh-Gaussian mode, i.e., a cosh-Gaussian beam embedded with a vortex phase, and which we call as vortex cosh-Gaussian beam. The proposed beam has one additional control parameter more (i.e., the decentered parameter b) than to the simple vortex Gaussian beam. In the initial plane, the beam may resembles to the vortex-Gaussian field for small values of b, moreover it may possess four-lobes structure resembling to the four-petal Gaussian beams (Duan and Lü 2006; Guo et al. 2014; Liu et al. 2016b; Wang et al. 2019) when b is large. In the present work, the closed-form expression of vortex cosh-Gaussian beam propagating through a paraxial ABCD optical system is derived, and mainly, the propagation properties of the beam in free-space and in a FrFT system versus the decentered parameter b and the topological vortex charge are illustrated in details. The reminder of the paper is structured as follows: In Sect. 2, we illustrate the field pattern of vChGB in the initial plane as a function of the beam parameters. In Sect. 3, the propagation of the vChGB through a general paraxial ABCD optical system is investigated in the process of Huygens–Fresnel diffraction. Then, the propagation characteristics of the beam in free space and in FrFT plane are discussed with numerical examples in Sect. 4. The main results are summarized in the conclusion part.

2 Characteristics of vChGB in the z = 0 plane

In the Cartesian coordinates system, a vChGB propagating along the z-axis in the source plane z = 0 can be expressed as

$$E\left( {x_{0} ,y_{0} ,z = 0} \right) = \cosh \left( {b\frac{{x_{0} }}{{\omega_{0} }}} \right)\cosh \left( {b\frac{{y_{0} }}{{\omega_{0} }}} \right)e^{{ - \left( {\frac{{x_{0}^{2} + y_{0}^{2} }}{{\omega_{0}^{2} }}} \right)}} \left( {x_{0} + iy_{0} } \right)^{m} ,$$
(1)

where \(\left( {x_{0} {\kern 1pt} {\kern 1pt} ,{\kern 1pt} y_{0} {\kern 1pt} {\kern 1pt} } \right)\) are the Cartesian coordinates at arbitrary point in the source plane and \(\omega_{0}\) is the beam waist radius of the Gaussian part. b being the decentered parameter (or normalized modal factor) of the cosh-Gaussian beam and the integer parameter m denotes the topological charge of the vortex.

The field of Eq. (1) can be generated in practical simply by modulating the wave front of a cosh-Gaussian field at initial plane with a spiral phase plate. It is obvious that when b = 0, Eq. (1) reduces to the field of a vortex-Gaussian beam (Zhou et al. 2013a), while for m = 0, one obtains the simple cosh-Gaussian beam. In addition, when b is taken as pure imaginary complex, Eq. (1) may describe a vortex cosine-Gaussian beam.

A preliminarily graphical illustration of the vChGB at the initial plane is shown in Fig. 1, where we have presented the result of the intensity distribution of the beam for different values of b and m and for a fixed value of ω0 (ω0 = 1 mm). Through numerous calculations, we have observed that the field pattern depends strongly on the decentered parameter b. We have revealed two comportments with respect to the magnitude of b. From the plots of Fig. 1a, b, e, f, it is found that for small values of b (b < 1.5), the beam has a central dark spot surrounded by a bright ring spot whatever the value of m. Thus, the beam pattern resembles closely to the vortex-Gaussian beam, and its central dark spot width increases with larger m. While for high values of b (see the plots of Fig. 1), the beam presents four symmetrical bright lobes. When m is larger, the lobes separate and the inter-lobe distance increases. We note that in this case the beam profile resembles closely to the four-petal beam (Duan and Lü 2006). The phase distribution of the considered field in the source plane is presented in Fig. 2. As can be seen from the plots of this figure, the equiphase contours show a singularity at the point (0, 0) for an arbitrary value of b, because of the presence of the vortex charge. The phase increases counterclockwise and the distribution pattern takes on twofold or fourfold divergent rays for m = 2 or m = 4, respectively.

Fig. 1
figure 1

Intensity profile a vChGB at the source plane z = 0, with \(\omega_{0} = 1\,{\text{mm}}\) for different values of b and m. Top row (m = 1), bottom row (m = 2). a–e for b = 0, b–f for b = 1, c–g for b = 1.5 and d–h for b = 2.5

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Fig. 2
figure 2

Phase pattern of the vChGB at the source plane z = 0, with \(\omega_{0} = 1\,{\text{mm}}\) for different values of b. for different values of b and m. Top row (m = 1), bottom row (m = 2). a–e for b = 0, b–f for b = 1, c–g for b = 1.5 and d–h for b = 2.5

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3 Propagation of vChGB through a paraxial ABCD optical system

The propagation of the vChGB through a paraxial ABCD optical system is governed by the Huygens–Fresnel integral transformation (Collins 1970)

$$E\left( {x,y,z} \right) = \frac{ik}{2\pi B}e^{ - ikz} \int\limits_{ - \infty }^{ + \infty } {\int\limits_{ - \infty }^{ + \infty } {E_{0} \left( {x_{0} ,{\kern 1pt} y_{0} ,{\kern 1pt} {\kern 1pt} 0} \right)\exp \left\{ { - \frac{ik}{2B}\left[ {A\left( {x_{0}^{2} {\kern 1pt} {\kern 1pt} + {\kern 1pt} y_{0}^{2} {\kern 1pt} } \right) - 2\left( {xx_{0} {\kern 1pt} {\kern 1pt} + {\kern 1pt} yy_{0} } \right) + D\left( {x^{2} + y^{2} {\kern 1pt} } \right)} \right]} \right\}dx_{0} dy_{0} } ,}$$
(2)

where \(E_{0} \left( {x_{0} ,{\kern 1pt} y_{0} ,{\kern 1pt} {\kern 1pt} 0} \right)\) and \(E\left( {x,y,z} \right)\) are the fields at the source plane z = 0 and in the receiver plane z, respectively. z is the distance from the initial plane. A, B and D are the matrix elements of the paraxial ABCD optical system, \(k = \frac{2\pi }{\lambda }\) is the wave number and λ is the wavelength of laser light in vacuum.

On substituting from Eq. (1) into Eq. (2) and recalling the well-known binomial formula (Gradshteyn and Ryzhik 1994)

$$\left( {x_{0} + iy_{0} } \right)^{m} = \sum\limits_{l = 0}^{m} {C_{l}^{m} } x_{0}^{l} \left( {iy_{0} } \right)^{{_{m - l} }} ,$$
(3a)

where

$$C_{l}^{m} = \frac{m!}{{l!\left( {m - l} \right)!}},$$
(3b)

then by making some algebraic operations, Eq. (2) can be expressed as

$$E\left( {x,y,z} \right) = \frac{ik}{2\pi B}e^{ - ikz} {\kern 1pt} e^{{ - \frac{ikD}{2B}\left( {x^{2} + y^{2} } \right)}} \sum\limits_{l = 0}^{m} {C_{l}^{m} } \left( i \right)^{{_{m - l} }} I_{l} \left( x \right)I_{m - l} \left( y \right),$$
(4a)

where

$$I_{n} \left( u \right){\kern 1pt} {\kern 1pt} = \int\limits_{ - \infty }^{ + \infty } {u_{0}^{n} \cosh \left( {b\frac{{u_{0} }}{{\omega_{0} }}} \right)e^{{ - \left( {\frac{ikA}{2B} + \frac{1}{{\omega_{0}^{2} }}} \right)u_{0}^{2} + \frac{ik}{B}uu_{0} }} } du_{0} ,$$
(4b)

with \(u = x{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} or\,\,y\) and n is integer.

Using the definition of cosh function, the integral of Eq. (4b) can be rewritten as

$$I_{n} \left( u \right){\kern 1pt} = \frac{1}{2}\left( {J_{n}^{ + } \left( u \right) + J_{n}^{ - } \left( u \right)} \right){\kern 1pt} ,$$
(5a)

where

$$J_{n}^{ \pm } \left( u \right){\kern 1pt} = \int\limits_{ - \infty }^{ + \infty } {x_{0}^{n} e^{{ - \alpha x_{0}^{2} + \left( {\frac{ik}{B}u \pm \frac{b}{{\omega_{0} }}} \right)x_{0} }} } dx_{0} ,$$
(5b)

and the auxialiary parameter α is defined as

$$\alpha {\kern 1pt} = {\kern 1pt} \frac{ikA}{2B} + \frac{1}{{\omega_{0}^{2} }}.$$
(5c)

Now, by recalling the integrale formula (Prudnikov et al. 1986)

$$\int\limits_{ - \infty }^{ + \infty } {x^{n} e^{{ - px^{2} + 2qx}} } dx = e^{{\frac{{q^{2} }}{p}}} \sqrt {\frac{\pi }{p}} \left( {\frac{1}{2i\sqrt p }} \right)^{n} H_{n} \left( {\frac{iq}{\sqrt p }} \right),$$
(6)

where \(H_{n} \left( {.{\kern 1pt} {\kern 1pt} } \right)\) is the Hermite polynomial of nth-order, the integral of Eq. (5b) yields

$$J_{n}^{ \pm } \left( u \right){\kern 1pt} = f^{ \pm } \left( u \right)\sqrt {\frac{\pi }{\alpha }{\kern 1pt} {\kern 1pt} } \left( {\frac{1}{2i\sqrt \alpha }} \right)^{n} H_{n} \left( {\frac{i}{2\sqrt \alpha }\left( {\frac{ik}{2B}u \pm \frac{b}{{\omega_{0} }}} \right)} \right),$$
(7a)

where

$$f^{ \pm } \left( u \right) = e^{{\frac{1}{4\alpha }\left( {\frac{ik}{B}u \pm \frac{b}{{\omega_{0} }}} \right)^{2} }} .$$
(7b)

On substituting from Eqs. (7a) and (5a) into (4a), we obtain

$$\begin{aligned} E\left( {x,y,z} \right) & = \frac{ik}{2B}e^{ - ikz} \frac{1}{4\alpha }\left( {\frac{1}{2\sqrt \alpha }} \right)^{m} e^{{ - \frac{ikD}{2B}\left( {x^{2} + y^{2} } \right)}} \\ & \quad \times \sum\limits_{l = 0}^{m} {C_{l}^{m} } \left( i \right)^{{_{ - l} }} {\kern 1pt} {\kern 1pt} \left[ {f^{ + } \left( x \right)H_{l} \left( {\frac{i}{2\sqrt \alpha }\left( {\frac{ik}{B}x + \frac{b}{{\omega_{0} }}} \right)} \right) + f^{ - } \left( x \right)H_{l} \left( {\frac{i}{2\sqrt \alpha }\left( {\frac{ik}{B}x - \frac{b}{{\omega_{0} }}} \right)} \right)} \right] \\ & \quad \times \left[ {f^{ + } \left( y \right)H_{m - l} \left( {\frac{i}{2\sqrt \alpha }\left( {\frac{ik}{B}y + \frac{b}{{\omega_{0} }}} \right)} \right) + f^{ - } \left( y \right)H_{m - l} \left( {\frac{i}{2\sqrt \alpha }\left( {\frac{ik}{B}y - \frac{b}{{\omega_{0} }}} \right)} \right)} \right]. \\ \end{aligned}$$
(8a)

This last formula can be also written in terms of confluents hypergeometric functions \({}_{1}F_{1} \left( {{\kern 1pt} {\kern 1pt} .{\kern 1pt} {\kern 1pt} } \right)\) as

$$\begin{aligned} E\left( {x,y,z} \right) & = \frac{ik}{2B\alpha }e^{ - ikz} {\kern 1pt} \left( {\frac{1}{2\sqrt \alpha }} \right)^{m} \exp \left( {\frac{b}{{\omega_{0} \sqrt {2\alpha } }}} \right)^{2} \exp \left[ { - \frac{ikD}{2B}\left( {1 + \frac{k}{2i\alpha BD}} \right)\left( {x^{2} + y^{2} } \right)} \right] \\ & \quad \times \left\{ \begin{aligned} \cosh \left( {Sx} \right)\cosh \left( {Sy} \right)\sum\limits_{p = 0}^{{\left[ {m/2} \right]}} {C_{2p}^{m} \left( { - 1} \right)^{(m - 2p)/2} \frac{{\left( {2p} \right)!}}{p!}\,\frac{{\left( {m - 2p} \right)!}}{{\left[ {(m - 2p)/2} \right]!}}} \left[ \begin{aligned} \hfill \\ {}_{1}F_{1} \left( { - p;\,\frac{1}{2};\,X^{2} } \right){}_{1}F_{1} \left( { - \frac{{\left( {m - 2p} \right)}}{2};\,\frac{1}{2};\,Y^{2} } \right) \hfill \\ \end{aligned} \right] \hfill \\ + 4\sinh \left( {Sx} \right)\sinh \left( {Sy} \right)\sum\limits_{q = 0}^{{\left[ {{\raise0.7ex\hbox{$m$} \!\mathord{\left/ {\vphantom {m 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right] - 1}} {C_{2q + 1}^{m} } \left( i \right)^{m - 2q - 1} \frac{{\left( {2q + 1} \right)!}}{q!}\frac{{\left( {m - 2q - 1} \right)!}}{{\left( {\frac{m - 2q}{2}} \right)!}}\,\left[ {X\,Y\,{}_{1}F_{1} \left( { - q\,;\,\frac{3}{2}\,;X^{2} } \right)\,{}_{1}F_{1} \left( { - \frac{m - 2q}{2}\,;\,\frac{3}{2}\,;Y^{2} } \right)} \right] \hfill \\ \end{aligned} \right\} \\ \end{aligned}$$
(8b)

where

$$R = \frac{ - k}{2B\sqrt \alpha }r + \frac{ib}{{2\omega_{0} \sqrt \alpha }},\,{\text{R}} = {\text{X}}\,{\text{or}}\,{\text{Y}}\,{\text{and}}\,r = {\text{x}}\,{\text{or}}\,{\text{y}},$$
(8c)

and

$$S = \frac{ikb}{{2\alpha B\omega_{0} }}.$$
(8d)

Equation (8a) (or equivalently Eq. (8b)) is the main result of this paper; it expresses analytically the field of a vChGB propagating through a paraxial optical system versus the beam parameters ω0, b, and m, and the matrix elements A, B, D of the paraxial media.

The case m = 0 corresponds the cosh-Gaussian beam, then Eq. (8a) reduces to

$$E\left( {x,y,z} \right) = \left( {\frac{ik}{2\alpha B}} \right)e^{ - ikz} {\kern 1pt} e^{{ - \frac{ikD}{2B}\left( {x^{2} + y^{2} } \right)}} \exp \left[ { - \frac{{k^{2} \left( {x^{2} + y^{2} } \right)}}{{4\alpha B^{2} }} + \frac{{b^{2} }}{{2\alpha \omega_{0}^{2} }}} \right]\cosh \left( {\frac{ikbx}{{2\alpha B\omega_{0} }}} \right)\cosh \left( {\frac{ikby}{{2\alpha B\omega_{0} }}} \right),$$
(9a)

or equivalently

$$E\left( {x,y,z} \right) = \left( {\frac{ik}{2\alpha B}} \right)e^{ - ikz} {\kern 1pt} e^{{ - \frac{ikD}{2B}\left( {x^{2} + y^{2} } \right)}} \exp \left[ { - \frac{{k^{2} \left( {x^{2} + y^{2} } \right)}}{{4\alpha B^{2} }} + \frac{{b^{2} }}{{2\alpha \omega_{0}^{2} }}} \right]\cos \left( {\frac{kbx}{{2\alpha B\omega_{0} }}} \right)\cos \left( {\frac{kby}{{2\alpha B\omega_{0} }}} \right).$$
(9b)

This last equation is consistent with the result obtained directly for chG beam in Lu et al. (1999) and Hricha and Belafhal (2005b) in one-dimensional space consideration.

The limiting case b = 0 corresponds to the vortex-Gaussian beam, so Eq. (8) is simplified to

$$\begin{aligned} E\left( {x,y,z} \right) & = \left( {\frac{ik}{2B}} \right)e^{ - ikz} {\kern 1pt} \frac{1}{4\alpha }\left( {\frac{1}{2\sqrt \alpha }} \right)^{m} e^{{ - \left( {\frac{ikD}{2B} + \frac{{k^{2} }}{{4\alpha B^{2} }}} \right)\left( {x^{2} + y^{2} } \right)}} \\ & \quad \times \sum\limits_{l = 0}^{m} {C_{l}^{m} } \left( i \right)^{{_{ - l} }} \left( { - 1} \right)^{m} H_{l} \left( {\frac{kx}{2B\sqrt \alpha }} \right)H_{m - l} \left( {\frac{ky}{2B\sqrt \alpha }} \right). \\ \end{aligned}$$
(10)

We note that we have numerically checked that Eq. (10) is equivalent to Eq. (8) of Zhou et al. (2013a) even if they are different in form.

3.1 Free space propagation of vChGB

The free space propagation of vChGB is directly obtained by substituting B = z, and A = D=1 into Eq. (8a), so this yields

$$\begin{aligned} E\left( {x,y,z} \right) & = \frac{ik}{2z}e^{ - ikz} {\kern 1pt} \frac{1}{4\beta }\left( {\frac{1}{2\sqrt \beta }} \right)^{m} e^{{ - \frac{ik}{2z}\left( {x^{2} + y^{2} } \right)}} \\ & \quad \times \sum\limits_{l = 0}^{m} {C_{l}^{m} } \left( i \right)^{{_{ - l} }} {\kern 1pt} {\kern 1pt} \left[ {f_{\beta }^{ + } \left( x \right)H_{l} \left( {\frac{i}{2\sqrt \beta }\left( {\frac{ik}{2z}x + \frac{b}{{\omega_{0} }}} \right)} \right) + f_{\beta }^{ - } \left( x \right)H_{l} \left( {\frac{i}{2\sqrt \beta }\left( {\frac{ik}{2z}x - \frac{b}{{\omega_{0} }}} \right)} \right)} \right] \\ & \quad \times \left[ {f_{\beta }^{ + } \left( y \right)H_{m - l} \left( {\frac{i}{2\sqrt \beta }\left( {\frac{ik}{2z}y + \frac{b}{{\omega_{0} }}} \right)} \right) + f_{\beta }^{ - } \left( y \right)H_{m - l} \left( {\frac{i}{2\sqrt \beta }\left( {\frac{ik}{2z}y - \frac{b}{{\omega_{0} }}} \right)} \right)} \right]{\kern 1pt} {\kern 1pt} , \\ \end{aligned}$$
(11)

where

$$f_{\beta }^{ \pm } \left( u \right){\kern 1pt} = e^{{\frac{1}{4\beta }\left( {\frac{ik}{z}u \pm \frac{b}{{\omega_{0} }}} \right)^{2} }} ,$$
(12)

with

$$\beta {\kern 1pt} = {\kern 1pt} \frac{ik}{2z} + \frac{1}{{\omega_{0}^{2} }}.$$
(13)

3.2 Propagation of vChGB through a FrFT system

As it is well-known, the FrFT system is an extension of the Fourier transformation (Namias 1993). Generally speaking, this transformation can be implemented in optics either by a combination of lenses and spaces or by using of a quadratic graded-index medium (Mendolvic and Ozaktas 1993; Ozaktas and Mendolvic 1993; Lohmann et al. 1998; Lin and Cai 2002; Zhou 2009; Zhou et al. 2013b; Tang et al. 2014). To save the paper space, we present only the schematic of so called as Lohmann I setup for the fractional Fourier transform, where f denotes a standard focal length, and the lens focal length is \(\frac{f}{\sin \left( \varphi \right)}\) (see Fig. 3).

Fig. 3
figure 3

Schematic of fractional Fourier transformation setup

Full size image

The transfer matrix of p-order fractional Fourier transformation system is given by

$$\left( {\begin{array}{*{20}c} A & B \\ C & D \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {\cos \left( \varphi \right)} & {f\sin \left( \varphi \right)} \\ { - \sin (\varphi )/f} & {\cos \left( \varphi \right)} \\ \end{array} } \right),$$
(14)

where \(\varphi = p\frac{\pi }{2}\) and p is the fractional order of FrFT system.

By substituting A, B, D from Eq. (14) into Eq. (8a), one directly obtains

$$\begin{aligned} E\left( {x,y,z} \right) = & \frac{ik}{2f\sin \left( \varphi \right)}e^{ - ikz} {\kern 1pt} \frac{1}{4\gamma }\left( {\frac{1}{2\sqrt \gamma }} \right)^{m} e^{{ - \frac{ik}{2ftg\left( \varphi \right)}\left( {x^{2} + y^{2} } \right)}} \\ & \quad \times \sum\limits_{l = 0}^{m} {C_{l}^{m} } \left( i \right)^{{_{ - l} }} {\kern 1pt} {\kern 1pt} \left[ {f^{ + } \left( x \right)H_{l} \left( {\frac{i}{2\sqrt \gamma }\left( {\frac{ik}{{{\kern 1pt} {\kern 1pt} f\sin \left( \varphi \right)}}x + \frac{b}{{\omega_{0} }}} \right)} \right) + f^{ - } \left( x \right)H_{l} \left( {\frac{i}{2\sqrt \gamma }\left( {\frac{ik}{{{\kern 1pt} {\kern 1pt} f\sin \left( \varphi \right)}}x - \frac{b}{{\omega_{0} }}} \right)} \right)} \right] \\ & \quad \times \left[ {f^{ + } \left( y \right)H_{m - l} \left( {\frac{i}{2\sqrt \gamma }\left( {\frac{ik}{{{\kern 1pt} f{\kern 1pt} \sin \left( \varphi \right)}}y + \frac{b}{{\omega_{0} }}} \right)} \right) + f^{ - } \left( y \right)H_{m - l} \left( {\frac{i}{2\sqrt \gamma }\left( {\frac{ik}{{{\kern 1pt} {\kern 1pt} f\sin \left( \varphi \right)}}y - \frac{b}{{\omega_{0} }}} \right)} \right)} \right]{\kern 1pt} {\kern 1pt} , \\ \end{aligned}$$
(15)

where

$$\gamma {\kern 1pt} = \frac{{{\kern 1pt} ik}}{2}\frac{1}{f\sin \left( \varphi \right)} + \frac{1}{{\omega_{0}^{2} }}.$$
(16)

and

$$f^{ \pm } \left( u \right) = e^{{\frac{1}{4\gamma }\left( {\frac{ik}{{{\kern 1pt} {\kern 1pt} f\sin \varphi }}u \pm \frac{b}{{\omega_{0} }}} \right)^{2} }} .$$
(17)

Equation (15) expresses the vChGB at the FrFT plane as a function of the fractional order p of the transform system and the parameters of the initial beam. When \(p = \left( {2n + 1} \right),n = 1,2,3 \ldots\), Eq. (15) may describe the standard Fourier Transformation of the initial vChGB field.

4 Numerical analysis and discussions

For illustrating the evolution of vChGB upon propagating in free space, we depicted numerically the intensity and the phase distributions based on Eq. (11). The calculation parameters are taken to be λ = 632.8 nm, ω0 = 1 mm, and \(z_{R} = \frac{{k\omega_{0}^{2} }}{2}\) denotes the Gaussian Rayleigh length, which is used here as the unit scale for the distance z.

Figure 4 represents the output transverse intensity of the beam for small values of b and for different topological charge m (m = 1, 2 and 4). As can be seen from the plots of Fig. 4, the beam spot widens upon the propagation; both the central dark and bright ring radius increase, while the beam profile is retained even in the far-field (z = 15zR).

Fig. 4
figure 4

Intensity profile of the outgoing beam in free space for different values of the topological charge m with \(\omega_{0} = 1\,{\text{mm}}\), \(\lambda = 632.8\,{\text{nm}}\) and b = 0.1. Top row (\(z = z_{R}\)), bottom row (\(z = 15z_{R}\)). a, bm = 1, c, dm = 2 and e, fm = 4

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Figure 5 presents the phase distributions corresponding to the propagating field of Fig. 4. From the plots of Fig. 5, it is found that when b is small, the equiphase contours change into the anticlockwise spiral distributions. The number of branches on the spiral distribution is exactly equal to the topological charge number m. Furthermore, it is seen that the spiral phase structure keeps the topological singularity, and becomes more and more divergent during the beam propagation in free space.

Fig. 5
figure 5

Phase pattern of the outgoing beam in free space for different values of the topological charge m with \(\omega_{0} = 1\,{\text{mm}}\), \(\lambda = 632.8\,{\text{nm}}\) and b = 0.1. Top row (\(z = z_{R}\)), bottom row (\(z = 15z_{R}\)). a, bm = 1, c, dm = 2 and e–f) m = 4

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Figure 6 shows the output beam pattern for large b (b = 4) for different values of m, as a function of the propagation distance z. It is observed from the plots of this figure that the spot size of the beam increases upon the propagation. One can note an angle rotation of the beam spot around the center, moreover a drastic change of the beam pattern is observed in the far-field compared to the initial beam profile. In fact, many outside lobes with weak intensity appear upon the propagation. The multi-lobe structure and the number of the lobes depend strongly on the value of m: For m = 1, four central lobes are dominant and all the lobes are linked. The central lobes seem to have been rotated a π/4 angle rotation around the center. For m = 2, the four central lobes observed in the far-field z = (15zR) resemble to the initial ones, and all the petals are separated (Fig. 4h). For m = 4, a bright core appears in the central region surrounded by many side-lobes upon the propagation. It is to be noted that for all values of m and b, the beam profile keeps its mirror symmetry. The phase distributions of the considered vChGb are illustrated in Fig. 7. The plots of this figure show that the equiphase contours obtained for large values of b are different from those corresponding to small values of b; the phase structure is not the same on the four lobes location. The phase profile changes drastically versus the propagation distance, in addition one can note the absence of the singularity at the phase pattern center in far field, for the case m = 4. This is due to the existence of the bright core in the center of the field pattern (see Fig. 6f–j).

Fig. 6
figure 6

Intensity profile of the outgoing beam in free space for different values of the topological charge m with \(\omega_{0} = 1\,{\text{mm}}\), \(\lambda = 632.8\,{\text{nm}}\) and b = 4. Top row (\(z = z_{R}\)), middle row (\(z = 5z_{R}\)). and bottom row (\(z = 15z_{R}\)). a–d–gm = 1, b–e–hm = 2 and c–f–jm = 4

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Fig. 7
figure 7

Phase pattern of the outgoing beam in free space for different values of the topological charge m with \(\omega_{0} = 1\,{\text{mm}}\), \(\lambda = 632.8\,{\text{nm}}\) and b = 4. Top row (\(z = z_{R}\)), middle row (\(z = 5z_{R}\)). and bottom row (\(z = 15z_{R}\)). a–d–gm = 1, b–e–hm = 2 and c–f–jm = 4

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In order to illustrate the properties of a vChGB in FrFT plane, we investigated numerically Eq. (15) with paying attention to the effect of the fractional order p on the considered beam. The calculations parameters are chosen as λ = 632.8 nm, ω0 = 1 mm and f = 1 m. Figure 8, 9, 10 and 11 represent the intensity and the phase distributions of the vChGB for small and large values of b, respectively, versus different fractional order p in the FrFT plane. As can be seen, the evolution of the intensity is periodical, and the period is equal 2.

Fig. 8
figure 8

Intensity distribution of a vChGB at FrFT plane with m = 1, b = 0.1, and f = 1 m versus the fractional order p

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For a small b, i.e., when 0 < p<1, the initial beam spot is gradually focused, until the beam spot reaches minimum size for p = 1. From Fig. 9, one can find that the phase contours spiral rotate clockwise and gradually converge with increasing p. In the second half of the period, that means 1 < p<2, the beam is defocused progressively and retrieves its initial shape for p = 2. The phase distributions change to anticlockwise rotation, and evolve in opposite process of the former one.

Fig. 9
figure 9

Phase pattern distribution of a vChGB at FrFT plane with m = 1, b = 0.1, and f = 1 m versus the fractional order p

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For large b (b = 4) when 0 < p<1, the initial four petals are gradually focused towards the center until the overlapping, then the beam spot reaches minimum size for p = 1.

Figure 10 shows that the phase contours contain two spiral branches with clockwise rotation, the number of branches is characteristic of vortex charge m = 2. The phase pattern distributions converge gradually with the increase of the order p.

Fig. 10
figure 10

Intensity distribution at the output FrFT plane for an incident vChGB with m = 2, b = 4 and f = 1 m for different fractional order p

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For 1 < p<2, the beam is defocused progressively and retrieves its initial shape for p = 2. The equiphase distributions (see Fig. 11) change to anticlockwise rotation, and evolve in inverse process compared to the one of the former half period. One can note from the above plots that the change situation of the intensity of vChGB through FrFT system is independent on the vortex charge value m. In all cases, the evolution is periodical and the profile has minimums and maximums for p = 2n + 1 and 2n, respectively. Hence, the beam spot can be controlled by adequately choice of the FrFT order p and the parameters of the beam.

Fig. 11
figure 11

Phase pattern distribution at the output FrFT plane for an incident vChGB with m = 2, b = 4 and f = 1 m for different fractional order p

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5 Conclusion

In summary, a vortex cosh-Gaussian beam is introduced and its analytical formula through a paraxial ABCD optical system is obtained. The evolution of the characteristics of the beam upon propagating in free space and through a FrFT system are illustrated graphically with numerical examples. It is demonstrated that the initial profile with small value of b is stable under propagation whereas for large value of b the original four-lobe profile is transformed into a multi-petals structure. The beam pattern in the far-field depends strongly on the topological vortex charge m and the decentered beam parameter b. In addition, the evolution of the intensity and the phase distributions of the considered beam in FrFT system shows a periodical dependence on the fractional order p. It is demonstrated that the beam profile in FrFT plane depends on the parameters of the beam, and the spot has a minimum width as p = 2n and a maximum width when p = 2n + 1. This work may contribute to research dealing with vortex beams, and it can be beneficial to applications in optical trapping, optical micromanipulation, and beam shaping, etc.