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Abstract
We introduce a class of twisted sinc-correlation partially coherent array sources, by applying the construction theory of correlation function. Spectral density of such novel focused beam propagating in free space is analyzed. It is shown that the intensity distribution presents a good twisted effect and splitting phenomenon upon propagation. The array dimension, the intensity distribution and spatial distribution of the lobes can be flexibly regulated by altering the source parameters. We also explore the spatial evolution of multiple correlation singularities of this light field, where the phase distribution appears as a rotational spiral windmill profile during propagation. Furthermore, the coherence orbital angular momentum of the twisted source beam is investigated. These findings could be useful in the particle manipulation and free-space optical communication.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Twisted partially coherent beams play a critical role in various technological applications, such as image, trapping, free-space optical communication and measurement of topological charge [1,2]. The twisted beams can be generated by introducing the twisted phase into the partially coherent beams. To generate a Gaussian Schell-model (GSM) beam with rotational symmetry, the original idea suggested introducing the cross product of points $\mathbf {\hat {r}}_{1}$ and $\mathbf {\hat {r}}_{2}$, as $iu_{0}\left (x_{1} y_{2}-y_{1} x_{2}\right )$ [3]. Including such a term in the beams gives it a definite handedness or "twist". And $u_{0}$ is defined as the twist strength. An incoherent superposition is performed for light beams, thus endowing partially coherent beams with twisted properties. For example, Gaussian beams with mutually inclined propagation axes, can be superimposed to generate twisted Gaussian Schell-model (TGSM) beams [4]. Partially coherent vortex beams, whose cross-spectral density (CSD) is expressed with modified Bessel functions, can also be used to construct TGSM beams [5]. Recent evidence suggests that the construction theory of correlation function [6] is an effective method to devise physical genuine CSD functions, as long as the mathematical forms of spatial correlation functions for optical fields satisfy the constraint of non-negative definiteness. Subsequently, theoretical exploration on twisted partially coherent light has made significant progress [7–18], and the related experiments have also been continuously verified by employing the pseudo-mode superposition method [19,20].
Researchers have studied the propagating statistical properties of random twisted scalar and vector light fields in atmospheric turbulence, non-Kolmogorov turbulence, oceanic turbulence, and weak turbulence, which has broad application value in optical communication [21,22]. The twisted phase also plays an crucial role in adjusting the orbital angular momentum (OAM) spectra of twisted Laguerre-Gaussian Schell-model (TLGSM) beams [23]. Besides, a recent study revealed the correlation between the twisted phase and classical entanglement in phase space, as well as the conditions for a TGSM source to generate a strongly classical entanglement in paraxial field [24,25].
Although researchers have paid considerable attention to twisted partially coherent beams and carried out lots of related work since their introduction, most twisted CSDs were generated by mapping the twisted phase to the un-twisted CSD. Considering the physical realizability problem, the selection of the un-twisted CSD is constrained. For example, $J_{0}$ - correlated Schell-model source would be "non-twistabe", even if it carries a twisted phase [26]. Therefore, it would be highly desirable to explore new methods to obtains a more general twisted sources. In this paper, we have introduced a new kind of twisted sinc-correlation Schell-model array (TSCSMA) beams without twisted phase, by employing the construction theory of correlation function. Spectral density of such a source beam would gradually evolve into a periodic array profile and rotate up to $90$ degrees during propagation. The spatial position and intensity distribution of each lobe can be flexibly controlled by changing the parameters, and the phase distribution of the propagating CSD function carries multiple spatially correlation singularities. At last, with the help of a spiral basis applied at both azimuthal arguments, the coherence orbital angular momentum (COAM) of such a twisted beam is studied.
2. Theory
According to the unified theory of coherence and polarization [27], the light field emitted from two arbitrary points in planar source can be denoted by the CSD function. With sufficient conditions met, the CSD can be expressed as the integral [6]
We shall choose the kernel function as follows [7]
Here, $w_{0}$ presents the beam width, $\mu$ denotes the twisted factor, and $\sigma$ is the coherence length of the source beam. Space factors $n_{0},m_{0}$ are arbitrary positive real numbers. $\chi _{m n_{0}}$ and $\zeta _{m m_{0}}$ are the modified parameters. On substituting from Eqs. (2)–(4) into the Eq. (1), we can get a CSD function of the new light field as follows
We assume this novel beam is focused by a lens, which is located at the incident plane, and propagates along the positive $z$ direction. The paraxial propagation of partially coherent beams through an ABCD optical system can be treated by the Collins formula [30]
After substituting Eqs. (2)–(4) into Eq. (5) and changing the order of integration, we obtain the CSD function at an arbitrary propagation distance
The twisted factor need satisfy the constraint $\mu ^{2}\leq 1$, to guarantee the validity of the integration of $\mathcal {L}_{\gamma _{i}}(\mathbf {\hat {r}}_{1},\mathbf {\hat {r}}_{2})$. Spectral density takes the form $S(\mathbf {\hat {r}},z)=W(\mathbf {\hat {r}}, \mathbf {\hat {r}}, z)$ with the formula $\mathbf {\hat {r}}=\mathbf {\hat {r}}_{1}=\mathbf {\hat {r}}_{2}$. The transfer matrix of the focused optical system is
3. Free-space propagation of focused isotropic TSCSMA beams
Figure 1 presents the transverse spectral density generated by isotropic TSCSMA beams in free space with parameters $M=N =3$. For a general SCSMA source (shown in row 1), its intensity distribution would evolve from a Gaussian distribution to a flat-topped distribution and finally splits into a 3*3 array as the propagation distance increases. According to the reciprocity theorem, the radiant intensity in the far plane is related to the 2D space Fourier transform of the spectral degree of coherence in the source plane [27], so that for a sinc-correlation array source beam, each lobe of the far-field spectral density appears as a square shape. Row 2 demonstrates the evolution of spectral density with a smaller positive value of the twisted factor. Differently from the analytical calculations performed in Ref. [12], where each lobe of the array rotates synchronously as an individual around its respective lobe center, our simulations have revealed that the spectral density would rotate to $90$ degrees clockwise as a whole upon propagation. Besides, the light intensity of each element of the array is not the same, and in particular one can observe a significant intensity attenuation of the element near the light-field center. When the value of the twisted factor is increased, as shown in row 3, the intensity distribution in the source plane no longer behaves as a Gaussian-like profile, but as a 2*2 array. Spectral density retains its rotational invariance during propagation, which means that parameters $M$ and $N$ have no effect on the control of array dimension.
Fig. 1. Average intensity generated by focused isotropic TSCSMA sources at several propagation distances in free space with $w_{0}=\delta =2mm$.
Taking the far-field light intensity as an example, we presents the influence of parameters on spectral density in Fig. 2. Row 1 first shows the light intensity with different values of twisted factors. One can observe an attenuation of the spectral density near the light-field center induced by the twisted factor, where the larger the value of the twisted factor the stronger the intensity attenuation of the lobe. The far-field spectral density with various space factors is given in row 2. It is shown that space factors affect the spatial distribution of every lobe of the array. Specifically, the spectral density is a space uniformly distributed $4*4$ array when $n_{0}=m_{0}=1$; if $n_{0}$ ($m_{0}$) is less than 1, the lobes converge towards the center of the light field; on the contrary, the lobes would diverge in a direction away from the light-field center on condition that $n_{0}$ ($m_{0}$) is greater than 1. Furthermore, the light intensity can be modulated by space factors, thus our work may be of assistance for dynamic control multiple particles at specific locations in space. Row 3 illustrates the effect of parameter $M$ on the spectral density. To obtain a source beam whose intensity distribution splits, one can choose the weight function as $p(\mathbf {\hat {v}})=p_{1}(\mathbf {\hat {v}})-p_{2}(\mathbf {\hat {v}})$ , then the CSD takes on the form as $W^{0}=W_{1}^{0}-W_{2}^{0}$ [28]. The mathematical expression of the coherence structure in Eq. (5) can be sorted out as a difference of two spectral degrees of coherence (meeting the non-negative requirement), which leads to the split of intensity distribution.
Fig. 2. Spectral density of focused isotropic TSCSMA beams in far field: $w_{0}=\delta =2mm;$ row 1: $N=M=4, \enspace n_{0}=m_{0}=1,\enspace (a1) \enspace \mu =0, \enspace (a2) \enspace \mu =0.05, \enspace (a3) \enspace \mu =0.1, \enspace (a4) \enspace \mu =0.15;$ row 2: $\mu =0.05, \enspace N=M=4,\enspace n_{0}=m_{0},\enspace (b1) \enspace n_{0}=0.5,\enspace (b2)\enspace n_{0}=1,\enspace (b3)\enspace n_{0}=1.5, \enspace (b4) \enspace n_{0}=2;$ row 3: $\mu =0.05, \enspace N=M, \enspace n_{0}=m_{0}=1, \enspace (c1) \enspace N=1, \enspace (c2) \enspace N=2, \enspace (c3) \enspace N=3, \enspace (c4) \enspace N=4.$
Due to the adjustment of the twisted factor, spectral density in $z=0$ plane can split along $x$ and $y$ axis, and the far-field intensity distribution presents an attenuation. Thus the array dimension of the light intensity is also regulated by the twisted factor. To have an in-depth view of this phenomenon, we investigate the spectral density with various twisted factors (row 1) and propagating distances (row 2), respectively, in Fig. (3) with $M=N=4$. With the twisted factor increasing, the corresponding source-field spectral density would gradually evolved from Gaussian profile to flat-topped profile, and finally split into $2*2$ arrays. For the case of $\mu =0$ (seen in row 2), the light intensity in the central optical field is maximum at propagation, and an array of uniform intensity distributions also appears in the far field. That is to say, parameters $M$ and $N$ play a dominant role in regulating the splitting characteristics of the intensity distribution. Increasing the values of twisted factors ($\mu =0.1$ and $\mu =0.15$), the light intensity closed to the central light field would be attenuated during propagation. The far-field light intensity still appears as four lobes at $y=0$. However, intensity of these four arrays is not uniform. Here, the parameters $N$, $M$ and $\mu$ jointly control the splitting effect of the spectral density. If we choose a larger twisted factor, like $\mu =0.20$, the splitting of spectral density is completely controlled by the twisted factor.
Fig. 3. Spectral density in source plane with various twisted factors (row 1) and spectral density in $y=0$ at several propagation distances. The relevant parameters set here are the same as in Fig.1.
Prior research indicated that the spot size and coherent feature acted vitally in the regulation on partially coherent beams. It’s natural to wonder whether these parameters can affect the splitting effect of spectral density. Figure 4 plots the spectral density at $y=0$ in the source field and far field, respectively. It turns out that the greater the value of beam width, the stronger the intensity attenuation in the central area of the light field. And spectral density in the central area declines rapidly with smaller value of coherence length. Therefore, source beams with a larger value of coherent length and smaller value of beam width can be more resistant to the splitting tendency caused by the twisted factor.
Fig. 4. Spectral density in source plane (row 1) and in far-field plane (row 2) with (a) and (c) $w_{0}=2\,mm;\,(b)\,{\textrm{and}}\,(d)\,\delta =2\,mm; \,\mu =0.15, \,N=4,\, n_{0}=1$.
Fig. 5. Phase (row a) and amplitude (row b) distribution of propagating CSD function with $w_{0}=2\,mm;\,\delta =2\,mm; \,\mu =0.2,\,N=M=3$.
The coherent structure of the source CSD function in Eq. (5) contains the correlation vortices, which makes the propagating light fields exhibit various novel properties. A number potential applications of the correlation singularities have been suggested, like probing random media [31] and measuring the DOC of any partially coherent light wave [32]. Therefore, we finally look at the behavior of the transverse phase and amplitude distributions of the CSD function, at a fixed point $\mathbf {\hat {\rho }}_{2}=0$. As shown in Fig. 5, the phase distribution appears as a spiral windmill structure, and one can observe a twist in the phase and amplitude during propagation. Due to the periodicity of the sinc-correlation structure, multiple correlation singularities can be observed in the evolution of the phase distribution. The amplitude at the corresponding phase singularity is 0. Optical vortices are the typical features of a monochromatic wave. For the coherent vortex light field, the spectral density at the phase singularity is zero. For the TSCSMA beams, fixing the spatial point $\hat {\rho }_{2}$ , the CSD can propagate like a monochromatic wave with respect to the variable $\hat {\rho }_{1}$. The value of the amplitude of the CSD is zero at the correlation singularity, which means the fields at the point $\hat {\rho }_{1}$ and $\hat {\rho }_{2}$ are uncorrelated at the correlation singularity.
4. COAM matrix of isotropic TSCSMA beams
To better study the interplay of OAM and coherence of this partially coherent beam, we then perform an in-depth analysis on the COAM of the source beam. For simplicity, we take the case of $M=N=1$ as an example. For a stationary light field with the CSD function $W^{0}(\mathbf {\hat {\rho }}_{1},\mathbf {\hat {\rho }}_{2})$, it’s COAM matrix elements can be defined as [33]
Figure 6(b) plots $S_{m^{'}n^{'}}$ as a function of $n^{'}$ for fixed $m^{'}$. One can find that the diagonal elements play an important role, while the off-diagonal modes are trivial there.
Fig. 6. (a) Plot of $S_{m^{'}n^{'}}$ and $S$ as a function of $\rho$. (b) Plot of $S_{m^{'}n^{'}}$ as a function of $n^{'}$ for fixed $m^{'}$ at $\rho =1\,mm$. $\mu =0.1, \,w_{0}=1\,mm;\,\delta =2\,mm$.
5. Conclusion
In summary, we introduced a new typical twisted partially coherent array source, named TSCSMA beams, and analyzed the spectral density of this novel focused beam propagating in free space. Such sources carry multiple coherent singularities and can generate beams whose lattice-like non-average intensities rotate at propagation. To illustrate the possibility in the control of the propagating beam, we considered the intensity distribution with different parameters setting. It has demonstrated that for the far-field transverse spectral density, the lobes of the array in the central optical field would be attenuated caused by the twisted factor. Space-factor can adjust the spatial distribution of every lobe of the array, and parameters M(N) and twisted factor jointly regulate the array dimensions. For the source-field transverse spectral density, a larger choice of twisted factor would induce the spectral density to perform a tendency to split. The splitting of the intensity distribution would be more obvious when source beam is endowed with smaller value of coherent length and larger value of beam width. In the end, by applying the COAM matrix theory, we expanded the CSD function into a 2D array and analyzed the connection between the modes $S_{m^{'}n^{'}}$ and spectral density $S(\mathbf {\hat {\rho }})$. These results might be beneficial to some potential applications, such as optical tweezers, beam shaping and free-space optical communication [1,2,21,22].
Funding
National Natural Science Foundation of China (11974101, 11974102, 12174089, 12274116); Natural Science Foundation of Henan Province (222300420042, 232300421019); the Key Scientific Research Projects of Institutions of Higher Learning of the Henan Province Education Department (No. 21zx002); State Key Laboratory of Transient Optics and Photonics (No. SKLST202216).
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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