1. Introduction

The concept of twist phase, impressed on the cross-spectral density (CSD) was introduced by Simon and Mukunda more than two decades ago [1], which enables us to obtain the most general rotationally invariant Gaussian CSDs. As a matter of fact, the presence of twist phase has more far-reaching consequences. The CSD can not be factorized as a product of an $x$- and $y$- component, in a sense that it is genuinely two-dimensional. More precisely, it presents an intrinsic chiral nature responsible for the rotation of the beam about its axis upon propagation. Inspired by the pioneering work [1], imposing a twist to the Gaussian Schell-model (GSM) beams, namely, the TGSM beams, has received considerable attention. The first realization of a TGSM beam was demostrated experimentally by Friberg et al. in 1994 [2]. And also, a series of literature concerning exploration of striking properties and applications of these beams have witnessed the development of the theory of TGSM beams propagating in various media [3–7].

Despite most of attention over the years has been focused on the twisted beams, the examples are mainly restricted to the GSM family, little attention has been paid to the twisted non-GSM beams. The lack stems from the difficulty of attaining genuine twisted sources different from the usual TGSM model. It has been proved that simply multiplying a valid CSD by a twist does not necessarily work, since the resulting CSD must satisfy the definiteness property. The problem of under which conditions a twist phase can be impressed on a partially coherent beam remains unsettled for a comparativly long time [8]. Until recently, Borghi et al. addressed this issue [9,10]. They discussed the possibility, as well as the necessary and sufficient condition of generating genuine form of twisted Schell-model beams with axial symmetry, and, in particular, concluded that for certain correlation functions, the beam cannot be twisted.

The relationship between the spectral intensity and the spectral degree of coherence, also known as reciprocity theorem, was established long ago and is well understood [11]. Specifically, a significant step in this aera was taken in [12], where the CSD is expressed as a superposition integral, which enables us to access a set of mathematical models for sources producing random beams. Since then, shaping the statistical properties of beam at the far field by structuring source correlation has been extensively studied [13–17]. On this basis, several attempts have been made to generalize the twisted Schell-model beams, such as the elliptical TGSM beams, the twisted multi-GSM beams [18,19], the twisted GSM array beams [20,21], as well as the circular Laguerre Gaussian Schell-model (LGSM) beams with a twist [22]. Meanwhile, Gori suggested an alternative approach to devise genuine twisted sources without symmetry constraints [23]. Very recently, the experimental realizability of bona fide twisted CSD of the anisotropic GSM beam is acquired by a set of cylindrical lenses [24]. These publications have served as a useful tool for having in-depth insight into the nature of twist phase. Even so, it is highly desirable to provide non-trivial twisted beams with other geometry different from current sources and further investigate their intriguing features. In this paper, we begin with the theoretical foundations of producing bona fide CSD, aiming to introduce a new form of twisted beams, called the elliptical twisted LGSM (TLGSM) beam, to the best of our knowledge. The distinctive feature of this beam is that the dark hollow profile can be realized at the source plane, in contrast to the previous LGSM beam. Particularly, in the general case of elliptical TLGSM beam, it splits into two parts with rotational symmetry during propagation.

2. Elliptical TLGSM beam and its propagation in random media

Let a statistically stationary random source located in the plane $z=0$ generate a beam-like field that propagates into the positive half-space $z>0$. The second-order correlation properties, at a pair of points with transverse position vectors $\rho '_1=(x'_1,\;y'_1)$ and $\rho '_2=(x'_2,\;y'_2)$, at angular frequency $\omega$, are described in terms of the CSD function as

Now we consider the case that the weight function is expressed as

In order to introduce the twisted sources, the kernels are given by [18]

Upon substituting from Eqs. (3) and (4) into Eq. (2), and expanding $(v^2_x+v^2_y)^m$ in a binomial series, Eq. (2) can be written as

By applying these conditions, we arrive at a valid expression for the CSD function of the elliptical TLGSM beam at the source plane

Making these substitutions of Eq. (15), both $w_i$ and $\delta$ have significant physical interpretations. Similar to $\sigma$, $w_i$ is a measure of the beam size, and thus directly quantified as the rms width in $i$- direction illuminated by the beam source. $\delta$ is the rms coherence length. Alternatively, the spatial correlation of the beam source can be measured by the ratio $w_i/\delta$. For very small $\delta$, it is equivalent to saying that the beam is partially coherent. Therefore, we may say that Eq. (15) gives the rms width and rms coherence length relationships with $a$ and $g$, from which the spatial coherence can be determined. From the realizations of the CSD function some quantities must obey constraints. Apart from these conditions well established in order to ensure the nonnegative difiniteness of $W^{(0)}(\boldsymbol{\rho}'_1,\boldsymbol{\rho}'_2)$[11], it is easily seen that the inequality $u\delta ^2\leq 1$, which for TGSM sources must be satisfied, holds regardless of the value of the twist parameter $u$[1]. In particular, we note that Eq. (14) could turn back to the basic model. When the model index $m$ equals to zero, it corresponds to the conventional TGSM beam. Also, it is reduced to the LGSM beam if the twisted parameter $a$ is set to be zero. Equation (14) therefore represents a general model for the elliptical TLGSM beam source.

Based on the extended Huygens-Fresnel principle [26], the spectral density at a point, in any transverse plane of the far field filled with random media, is obtained by setting $\boldsymbol{\rho}_1=\boldsymbol{\rho}_2=\boldsymbol{\rho}$, leading to

3. Numerical results and discussions

3.1 Circular TLGSM beams

We first begin with computed examples of the spectral density generated by the circular TLGSM beam. The initial parameters are chosen to be: $\lambda =632\textrm{n}m$, $l_0=10^{-3}\textrm{m}$, $L_0=1\textrm{m}$, $\alpha =3.667$, and other parameters are specified in figure captions. The intensity distribution at the source plane is plotted in Fig. 1 to show the modulations of the parameter $g$ and the mode index $m$. The results of $m=0$ [Fig. 1(a), 1(d)], in which it is reduced to the basic TGSM beam, are well in agreement with the theoretical expectation. However, the TLGSM beam with $m\neq 0$ shows a simgificantly different picture, compared to the LGSM beam, that the ring-shaped beam is available at the source plane. Whereas for the LGSM beam, it can only be formed at distances far away from the source [28,29]. Thus, we can say that, by setting specific values of the source quantities, the TLGSM beam has the richer content of intensity profiles. In addition, for the source with smaller $g$ and larger $m$, its dark core expands wider, which is beneficial to set up a dark optical trapping. Physically this is understandable that the beam structure is dominated by the spatial coherence, as evident already from Eq. (15), that indicates the coherence length is proportional to the term $g$. Therefore, the first conclusion can be drawn that the TLGSM beam with lower coherence has the advantage of generating ring-shape profiles.

 

Fig. 1. Distribution of the spectral density of the circular TLGSM beam at the source plane with parameters of $\sigma =1.3\textrm{cm}$, $a=0.3$, $g=2\times 10^{-5}\textrm{m}^2$ (upper row), $g=9\times 10^{-6}\textrm{m}^2$ (lower row), and $m=0$ (first column), $m=1$ (second column), $m=2$ (third column).

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Distribution of the spectral density, along the $z$-axis, is illustrated in Fig. 2, where a comparison between propagating in free space and atmospheric turbulence is made. The impacts of $a$ and $g$ are taken into account. For quantitative comparison, we assume $a=0.03$ for [Figs. 2(a)–(c)] and $a=0.3$ for [Figs. 2(d)–(i)], with $g$ being $4\times 10^{-5}{\textrm{m}}^{2}$, $2\times 10^{-5}{\textrm{m}}^{2}$, $9\times 10^{-6}{\textrm{m}}^{2}$ from the left column to the right column. In free space [Figs. 2(a)–(f)], our results for larger $g$ and smaller $a$ that lead to smaller twisted factor $u$, which, as noted above, is defined as $u=a/g$, are analogous to that of the LGSM beams, developing the dark hollow shape at appreciable distances from the source. In particular, it was found that the larger $u$ enables the beam center to reach the minimum value sooner on propagation. Whereas for large enough value of $u$, there is an important distinction that the beam produces, and maintains the hollow spectral density during propagation [Fig. 2(f)]. Comparisons based on equal $g$ values imply that the spectral density at the center of beam decreases as $a$ increases. The profound difference from the LGSM beams comes from the fact that the presence of twist pahse $a$. The analysis therefore substantiates the intuitive conclusion that, introducing a twisted phase to the LGSM beam may induce more intriguing propagation properties.

 

Fig. 2. Changes in the spectral density generated by the circular TLGSM beam on propagation in free space (top two rows) and atmospheric turbulence (third row). The related quantities are: $\sigma =1.3\textrm{cm}$, $m=2$, $\tilde {C}_n^2=5\times 10^{-13}\textrm{m}^{2/3}$, $a=0.03$ for Figs. 2(a)–2(c), and $a=0.3$ for Figs. 2(d)–2(i), $g=4\times 10^{-5}\textrm{m}^2$ (left column), $g=2\times 10^{-5}\textrm{m}^2$ (middle column), and $g=9\times 10^{-6}\textrm{m}^2$ (right column).

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For ease of comparison with the above results, following, we discuss the additional effects incurred in the presence of turbulence [Figs. 2(g)–2(i)], where the optical path is assumed to have moderate-strength ($\tilde {C}_n^2=5\times 10^{-13}\textrm{m}^{2/3}$) turbulence. Under this circumstance, the profiles with larger $g$ approach to the Gaussian-like or flatten-top shape, the evolution is to be expected since the effect of turbulence, as mentioned in the preceding paragraph, is the Gaussian process. However, in case of smaller $g$, the turbulence influence can be quite slight [Fig. 2(i)], in keeping with the dark hollow profile, with the only difference being expansion due to the diffraction. Physically this is reasonable, in fact, the source coherence dominates the evolution of the spectral density in the primary propagation process, where the strength of the turbulence is negligible. With the propagation distance increases, the beam’s performance is gradually governed by the turbulence.

3.2 Elliptical TLGSM beams

We now proceed with the analysis concerning the general case of elliptical TLGSM beams. The transverse intensity distribution at various values of the propagation distance $z$ is given in Fig. 3, where the impact of the quantity $a$ is considered. The value of $a$ is set to be 0.1 for the left column and 0.3 for the right column, respectively. It is firstly noticed that the intensity profile with small value of $a$ is identical to that generated by the elliptical TGSM beams [see Fig. 3(a)]. And the intensity ellipse undergoes a spatial rotation around the $x$ axis during propagation, thereby achieving the same behaviour as is realized with the TGSM beams. A further important difference is, in comparison with the circular TLGSM beam [22], the elliptical beam splits into two parts with rotational symmetry for large enough $a$. Ultimately, we estimate the performance of such beams when propagating in turbulent environment over the transverse plane at $z=600\textrm{m}$ [Figs. 3(d) and 3(h)], where the turbulence strength is assumed to be the same as in Fig. 2. We note that the presence of turbulence does not change the angular velocity of rotation, instead, it stimulates the two isolate parts getting closer, thus exhibiting the trend of being Gaussian.

 

Fig. 3. Evolution of the spectral density of the elliptical TLGSM beam on propagation in free space (top three rows) and atmospheric turbulence (last row). The parameters are set as: $\sigma _x=1.3\textrm{cm}$, $\sigma _y=0.8\textrm{cm}$, $m=2$, $g=9\times 10^{-6}\textrm{m}^2$, $\tilde {C}_n^2=5\times 10^{-13}\textrm{m}^{2/3}$, $a=0.1$ [Figs. 3(a)–3(d)], and $a=0.3$ [Figs. 3(e)–3(h)], the propagation distance $z=0$ (first row), $z=300\textrm{m}$ (second row), and $z=600\textrm{m}$ (last two rows)

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We turn our attention now to the evaluation of the angular velocity of rotation, an essential quantity for assessing the behaviour of any twisted beams, and examine the impact of quantity that should be accounted for the rotation, which is plotted in Fig. 4. It is worth noting that, here, we have adopted the Orientation function in MATLAB in order to characterize the rotation angles of the intensity ellipse, which refers to the angle between the major axis of the ellipse and the $x$-axis. The behaviours of rotation angle as a function of $z$ for various value of beam width, coefficient $a$, model index $m$, as well as $g$ are illustrated, respectively, in Figs. 4(a)–4(d), from which one can see that there is a monotonic rotation of the beam in space by an amount $\theta \approx \pi /2$ degree about the $x$ axis. Figure 4(a) shows that the bigger the ratio $\sigma _x/\sigma _y$ is, the shorter the distance over which the rotation angle approaches to $\pi /2$, arising from the stronger anisotropy. From Fig. 4(b), we observe that, within a certain range, the elliptical TLGSM beam with smaller $a$ is a worse rotating performer, nevertheless, beyond which whose rotation angle greatly exceeds that of the beam with larger $a$. We noticed that, for different values of $a$, the rotational angle reaches $\pi /4$ at the same propagation distance. The beam undergoes a rotation of $\pi /4$ within the special distance and another $\pi /4$ beyond the range. The result agrees well with the Fig. 2 in [1] and the Fig. 3 in [20]. When the beam propagates for a Rayleigh range, the denominator of the term $\textrm{tan}(2\theta )$ ($\theta$ denotes the rotation angle), which is derived from the equation of the intensity ellipse, equals zero, independent on the choice of the twist parameter $a$. We note also that for the larger value of $m$, the beam rotates more quickly than the TGSM beam, and the three curves are nearly identical at appreciate distance [Fig. 4(c)]. This result, in a sense, might be ascribed to the fact that TLGSM beams with non-zero model index carry orbital angular momentum (OAM). Indeed, it is in good agreement with another paper [30], in which it was concluded that OAM can induce anisotropic diffraction. The property of the OAM of elliptical TLGSM beams and the correlation between OAM and rotation of intensity ellipse will be further exploited in our next work. In general, smaller value of $g$, hence the shorter coherence length, allows rotation angle to reach its limit well before the larger $g$ [Fig. 4(d)]. To gain some simple physical insight about the influence of atmospheric turbulence on the rotation of the beam, we compared the change of rotation angle with different turbulence strength in Fig. 4(e). It was found that the presence of turbulence has no effect on the angular velocity. This is well understood since the model for the power spectrum of atmospheric turbulence we chosen is homogeneous and isotropic, which generates the same diffraction effect along with the $x$ and $y$ directions of the beam.

 

Fig. 4. Rotation angles of the intensity ellipse of the TLGSM beam on propagation with other parameter $\sigma _x=1.3\textrm{cm}$, $\sigma _y=0.8\textrm{cm}$, $m=1$, $a=0.3$ and $g=2\times 10^{-5}\textrm{m}^2$.

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4. Conclusion

By starting with the theoretical foundations of devising bona fide spatial correlation functions, insofar as a new type of partially coherent beams, termed elliptical TLGSM beams, has been introduced. Its primary characteristic is its twist in both the Laguerre polynomial and the exponential term. This kind of circular beams are capable of generating dark hollow or Gaussian-shape intensity profiles, depending on the exact values of the characterized parameters of the source. Particularly notable for the elliptical case is its splitting into two parts with rotational symmetry. We would like to make it clear that, the introduced mathematical expression of the TLGSM beam, as is evidenced, can be reduced to some special models under certain assumptions. In view of this, the TLGSM beam can be viewed as the generalized, complete formulation of the LGSM beam. Then, analytical expression and computed examples have been given to elucidate the spectral density of such a beam during propagation in random media, for example, the atmospheric turbulence. From the obtained results it is shown that the distribution of the spectral density is dominated by the quantities both of the beam source and the turbulent medium, i.e., twisted phase $a$, $g$, model index $m$ and turbulence strength $\tilde {C}_n^2$. Further, the changes of rotation angle of the intensity ellipse with different quantities are studied in detail by numerical examples.

Albeit in slightly different name, the principal distinction between the TLGSM beam and the LGSM beam is that an additional freedom, the twisted phase, is impressed, which enables the TLGSM beam to possess much richer intensity properties than that of the latter case. Apart from ring-shape profile being formed at appreciable distance, it can also be realized at the source plane, identical to the hollow beams. Of particular interest is that the elliptical beam will split into two isolated parts rotationally symmetry during propagation. This new type of TLGSM beams may find its potential applications in optical trapping, optical communication, as well as in the interaction between beam and material, due to the advantages of adjustable intensity profiles, low sensitivity to turbulence and the physical nature of carrying OAM.