## Abstract

It is known that a twisted Gaussian Schell-model (TGSM) beam with elliptical Gaussian amplitude will rotate its beam spot upon propagation because of the vortex structure of the transverse energy flux. In this paper, we study a special kind of twisted partially coherent beams named twisted Hermite-Gaussian correlated Schell model (HGCSM) beam whose degree of coherence (DOC) is non-circularly symmetric but the source amplitude is of the circular Gaussian profile. Our results reveal that the beam spot (average intensity distribution) does not rotate during propagation even if the circular symmetry of the beam spot is broken. However, the DOC pattern shows the rotation under propagation. From the investigation of the transverse energy flux and OAM density flux, we attribute the nontrivial rotation phenomenon to the redistribution of the transverse energy flux by non-circular DOC. Furthermore, based on Hyde‘s approach [J. Opt. Soc. Am. A **37**, 257 (2020) [CrossRef] ], we introduce a method for the generation of this class of twisted partially coherent sources. The non-rotation of the beam spot and rotation of the DOC are demonstrated in experiment.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

## 1. Introduction

In 1992, Allen et al. found that laser beams with vortex phase carries well-defined orbital angular momenta (OAM) [1]. Each photon in such vortex beams has an OAM of $l\hbar$, where $l$ is the topological charge and $\hbar$ is the reduced Plank constant. This finding opens a new chapter in optics, i.e., singular optics. Since then, vortex beams have received a great deal of attention and have found important application in optical processing [2], optical communication [3], optical manipulation [4], quantum information [5], microscopy and imaging [6,7] and so on. When a vortex beam interacts with dielectric particles such as silica particles, it accompanies the exchange of the OAM between the photons and particles, leading the rotation of particles with respect to the phase singularity of the vortex beam [8]. In general, the transverse energy flux of the vortex beam in beam’s cross section is also of vortex structure, which implies photons take spiral trajectories along the beam propagation axis. However, one can’t directly observe these trajectories because the beam spot of the vortex beam is usually circular symmetry. When the rotational symmetry of the vortex beam is broken, the beam spot upon propagation displays clockwise or anticlockwise rotation with respect to its optical axis, depending on the sign of topological charge. The best examples are the asymmetric Bessel-Gauss beam [9] and the asymmetric Laguerre-Gaussian (LG) beam [10]. Nevertheless, Kotlyar et al. revealed from both theory and experiment that the OAM is not the necessary condition responsible for the rotation of beam spot during propagation [11]. It was shown that for the appropriate coherent superposition of LG modes, Bessel modes or hypergeometric modes, they will display rotation of beam spot with zero OAM or no rotation even having non-zero OAM. The reason behind this phenomenon is closely related to differences of Gouy phase among superposition modes under propagation. The Poynting vector, OAM and rotation rates of multi-mode Bessel beams were studied in [12,13]. The aforementioned laser beams are fully coherent. Strictly speaking, optical fields are partially coherent owing to that the source may emit several modes or is mixed with thermal noise, such as the wide class of partially coherent Gaussian Schell-model source [14] J0-correlated Schell-model sources [15], the partially coherent quasi-Airy sources [16] and so on [17,18]. Therefore, the incoherent superposition of similar partially coherent source will produce partially coherent source [19,20], too. In addition, if the field propagates through a random medium or reflects by a rough surface, it will become the partially coherent field.

In partially coherent domain, in addition to vortex phase and astigmatic phase, there exists another nontrivial phase, named twist phase, which will induce the beam carrying OAM [21]. Different from other conventional phases, the twist phase $\exp [{\rm i}\mu (x_1y_2-x_2y_1)]$, is non-separable with two spatial points, where $\mu$ is the twist factor and $(x_i, y_j) (i,j=1,2)$ are two spatial points. It was coined by Friberg that the twist phase opens up a new dimension, in the theory of partially coherent beams, providing one a new freedom to manipulate partially coherent beams [22]. Similar with vortex phase, the twist phase also has an intrinsic handedness property and is responsible for the rotation of beam spot with respect to its propagation axis during propagation [21]. The direct observation of such rotation both in experiment and in theory can be achieved if the circular symmetry of beam spot is broken [23–25]. Owing to these intriguing characteristic, the twisted beam has potential applications in optical imaging [26], optical trapping [27], stimulated down-conversion [28] and so on. Over past two decades, the most studied twisted beam models is the twisted Gaussian Schell-model beam (TGSM) whose source profile and degree of coherence (DOC) are all circular Gaussian function. In recent years, researches paid attention to whether the twist phase can be imposed on partially coherent beams with arbitrary DOC even non-circular symmetry [19,29,30]. What is the rotation behavior of beam spot and DOC of the twisted beam during propagation if the circular symmetry of the DOC is broken? In this paper, we construct a new twisted model whose DOC function violates the circular symmetry, named twisted Hermite-Gaussian correlated Schell-model (HGCSM) beam, to study the rotation behavior of the beam on propagation.

The paper is organized as follows: In section 2, we first theoretically devised a twisted HGCSM source based on the Gori‘s critical condition. In section 3, the average intensity, DOC, transverse Poynting vector and OAM density flux are examined through numerical examples. The relation between the rotation of beam profile or DOC and their symmetry is revealed. In section 4, we carry out the experiment to observe the rotation behavior of beam profile and DOC during propagation and verify our theoretical prediction. The conclusion is summarized in section 5.

## 2. Twisted Hermite-Gaussian correlated source

In space frequency domain, the second-order statistics of a partially coherent source ($z=0$) propagating along $z$-axis can be described by the cross-spectral density (CSD) function, defined as [31]

By making use of the variable change $\textbf{r}_s=( \textbf{r}_1+ \textbf{r}_2)/2$ and $\textbf{r}_0= \textbf{r}_s- \textbf{r}$, Eq. (2) becomes

At the right side of Eq. (3), the phase term is just the twist phase, i.e.,

Therefore, Eq. (3) provides an elegant way to devise genuine twisted beam if the twist phase is not compensated by the function $R( \textbf{r}_1, \textbf{r}_2)$. Here we construct a new type of twisted beam, name twisted HGCSM beam. Suppose that $p_0( \textbf{r}_0)=1/2\pi \delta _0^2$ and the partially coherent source is the HGCSM beam [32]

## 3. Average intensity, energy flow, OAM and DOC for twisted HGCSM beams during free space propagation

Within the accuracy of paraxial approximation, the CSD function of a partially coherent beam from the source plane ($z = 0$) to the plane $z$ in free space can be connected by the following Huygens-Fresnel integral

From Eq. (16), one finds that in contrast to the TGSM beam whose spectral density is of a Gaussian profile [the first Gaussian function at the right side of Eq. (16)], the spectral density of the twisted HGCSM beam is the product of the Gaussian function and the parabolic function. Figure 1 illustrates the spectral density of the twisted HGCSM beam propagation in free space with different propagation distances. Near the source plane ($z$ is small), the effect of the parabolic function can be neglected. Thus, the beam profile is similar to the Gaussian profile. With the increase of the propagation distance, the parabolic function gradually plays the important role. As a result, the beam pattern evolves into two beamlets in the far field. An interesting phenomenon is that the beam pattern upon propagation does not rotate, although the circular symmetry of the beam pattern is broken.

To insight the state of beam profile evolution, we now examine the average transverse energy flux (Poynting vector) during the propagation. According to [1], the quantity is defined as

In order to see the variation of total energy in the radial and azimuthal components, we integrate the average energy flux across the transverse plane

Figure 2 shows the energy of radial component (blue dashed line) and azimuthal component (red solid line) as a function of propagation distance. The azimuthal component gradually decreases upon propagation, implying “rotation” component during propagation become less and less and nearly zero after 2 m propagation distance. Therefore, one can not observe the rotation of beam spot upon propagation even if the beam spot is not circular symmetry.

Perhaps one of the most intriguing characteristics induced by the twist phase is that the beam carries the OAM along the $z$ direction [33]. According to [34,35], the $z$-component average OAM flux density of a partially coherent beam can be connected with the transverse energy flux by the following formula

It follows from Eq. (22) that such OAM flux only depends on the azimuthal direction of the average energy flux. On substituting Eq. (19) into Eq. (22), the $z$-component OAM flux density of the twisted HGCSM beam becomes

In the source plane, the OAM flux density reduces to $M_z( \textbf{r},z)=-\epsilon _0r^2\mu k^{-1}\exp (-r^2/2\sigma _s^2)$, independence of the DOC. When the beam leaves from the source plane, the DOC and the coherence length will affect the OAM flux density distribution. Figure 3 shows the OAM flux density at different propagation distances in free-space propagation. The circular symmetry (only in the source plane) of the OAM flux pattern is broken during the propagation owing to the special coherence of the source. When the propagation distance is larger than 0.5 m, the pattern become four beamlets. The two is positive OAM flux, and another two is negative. In our region of interest, the OAM flux density decreases as the propagation distance increases. One reason is that the beam expands due to diffraction, and another is that the azimuthal energy flows decreases with the increase of the propagation distance.

Although the OAM flux density varies with propagation distance, the total average OAM is conserved on propagation, defined as

where $n_p$ is the photon flux along the propagation distance. The result is consistent with that reported in [33].Let us now pay attention to the behavior of the DOC of the twisted HGCSM beam, defined by the normalized CSD function [31]

On substituting Eq. (12) into Eq. (25), we can obtain the DOC of a twisted HGCSM beam at pair of points $\textbf{r}$ and $- \textbf{r}$ in plane $z$ as

From Eq. (26), one can find the third term i.e., $(kx+\mu zy)^2$ in the first parentheses will result the DOC pattern rotating during propagation. The rotation angle is $\phi =\arctan (\mu z/k)$. Thus, at the propagation distance $z=k/\mu$, the rotation angle is $\pi /4$, while for the sufficiently large distance, the rotation angle tends to $\pi /2$. In Fig. 4, we plot the DOC distribution of the twisted HGCSM beam at several propagation distance in free space. Obviously, the DOC pattern rotates counter clockwise with the propagation distance. The variation of the rotation angle with propagating distance (red solid curve) is shown in Fig. 5. The DOC pattern rotates relatively fast when the beam is near the source. When the propagation distance is larger than 20 m, the beam pattern rotates very slowly and tends to $\pi /2$. The angular velocity can be derived as $v_{\phi }=d\phi /dz=k\mu /(k^2+\mu ^2z^2)$ which is plotted in Fig. 5 (blue solid curve). The angular velocity decreases monotonously as the propagation distance increases, tending to zero for sufficiently large propagation distance.

## 4. Experimental generation of a twisted HGCSM beam and its propagation characteristics

In this section, we carry out the experiment to generate the twisted HGCSM beam and measure the average intensity distribution and the DOC of the generated beam upon propagation. Before discussion about the experimental results, let us first briefly discuss the method for the generation of such kind of twisted beam using the incoherent superposition of random modes. It refers from Eq. (7) that the DOC can be represented as the product of the DOC of the HGCSM beam and the DOC of the TGSM beam, i.e.,

For $\mu _{HG}( \textbf{r}_1, \textbf{r}_2)$, it can be written as the following alternative integral form

For the next step, we also could write the $\mu _{TG}( \textbf{r}_1, \textbf{r}_2)$ as the following integral form

Equation (33) provides a way to synthesize the twisted HGCSM beam in experiment.

Figure 6 shows our experimental setup for the synthesis of the twisted HGCSM beam. A linearly polarized laser beam ($\lambda =532$ nm) emitted from a diod-pump Nd:YAG laser is first expanded by a beam expander (BE), reflected by a mirror, then split by a 50:50 beam splitter. The reflection part goes toward a phase-only spatial light modulator (SLM) to modulate the phase and amplitude of the incident beam. The complex field to be modulated first encodes into a phase-only computer generated hologram (CGH), and then is loaded on the SLM screen. The method for the synthesis of the CGHs can be found in [37]. We first generate N=3000 CGHs, each of which is recorded the amplitude and phase information of a realization of the random field. The SLM works in such a way that at each time step, the chronologically earliest CGH is removed from the SLM‘s screen and replaced by a new CGH. The SLM‘s screen plays 3000 CGHs in cycle with each CGH being equal displaying time, about 18 ms. The modulated light beam from the SLM passes through a 4f optical system consisting of L1 and L2 to image the beam with unit magnification. A circular aperture is place the rear focal plane of L1 to block unwanted diffraction orders and background noise. We could regard the rear focal plane of L2 as the source plane of the generated twisted HGCSM beam. A focal lens L3 is located in the source plane and a CCD is used to measure the average intensity distributions and the DOC at different distances after the L3.

Figure 7 illustrates the experimental results of the normalized average intensity distribution of the generated twisted HGCSM beams at several propagation distances after the L3. The displayed intensity distribution is the average of N=3000 intensity realizations (instantaneous intensity) captured by the CCD. The beam pattern in the source is Gaussian profile and evolves into two beamlets at the focal plane. In the evolution process, the beam spot indeed does not rotates, which agrees with the theoretical prediction.

The experimental results of the square of the modulus of the DOC at two point $\textbf{r}$ and $- \textbf{r}$ are shown in Fig. 8. For each picture, the result is obtained from the intensity correlation of the 3000 intensity realizations. In our experiment, the random process obeys Gaussian statistics. Thus, the square of the modulus of the DOC is proportional to the intensity correlation. The DOC pattern rotates upon propagation, and reaches 90 degree of rotation angle in the focal plane.

## 5. Conclusions

We have investigated the propagation characteristics of the twisted HGCSM beams, a special kind of twisted partially coherent beams whose DOC is of non-circular symmetry and the initial amplitude is circular Gaussian. It finds that the average intensity distribution (mode order: m=1 and n =0) breaks into two beamlets under propagation whereas the beam spot does not rotates, which is quite different from the TGSM beam with initial elliptical Gaussian amplitude. The reason is that the transverse energy flux is no more vortex structure during propagation owing to its non-circular DOC, resulting the non-rotation of beam spot. The azimuthal component of the transverse energy flux, which is the main contribution of the beam spot rotation and z-component OAM, gradually decreases and tends to zero on propagation. Nevertheless, the DOC profile exhibit the rotation behavior with respect to the beam propagation axis. It shows that the rotation switch to the DOC pattern induced by the non-circular DOC of the source. The rotation angle from 0 to $\pi /2$ when the beam leaves from source plane to sufficiently larger distance. The rotation angle of the DOC of order m = 1 and n=0 only depends on the twist factor and wave number. We also develop a method proposed by Hyde [36] to experimentally generate the twisted HGCSM beams with the use of a phase-only SLM. This method is the combination of two independent random processes where the product of the autocorrelation functions of two statistics is just the prescribed DOC function. The evolution of the average intensity and square of modulus of the DOC of the generated beam are measured in experiment. The experimental results show that the average intensity indeed does not rotates whereas the DOC rotates upon propagation, reaching to $\pi /2$ angle in the focal plane, consistent with the theoretical predictions.

## Funding

National Key Research and Development Program of China (2019YFA0705000); National Natural Science Foundation of China (11974218, 12074310, 12192254, 91750201); Natural Science Basic Research Program of Shaanxi Province (2021JM-315); Innovation Group of Jinan (2018GXRC010); Local Science and Technology Development Project of the Central Government (YDZX20203700001766).

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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