## Abstract

The inverse method of proving the twistability of cross-spectral density (CSD) inevitably falls into spontaneous difficulties. Based on a nonnegative self-consistent design guideline for generating genuine CSDs introduced by Gori and Santarsiero, we demonstrate a feasible way for twisting partially coherent sources by sticking a Schell-model function to CSDs, which also determines the upper bound of the twisting strength. Analysis shows that the degree of coherence of a new class of twisted pseudo-Gaussian Schell-model beam is neither shift invariant nor shift-circular symmetric. In the presence of a vortex phase, the two different types of chiral phases affect each other and together control the propagation behavior. We further carry out an experiment to generate this non-uniformly correlated twisted beam using weighted superposition of mutually uncorrelated pseudo modes. The result is beneficial for devising nontrivial twisted beams and offers new opportunities.

© 2021 Optical Society of America

Recent years have seen exponential interest in design and synthesis of physically realizable twisted partially coherent sources, fueled with exciting opportunities for fundamental science and applications. The twist phase is an intrinsically two-dimensional inseparable quadratic phase factor first proposed by Simon and Mukunda [1]. Interestingly, the twist phase appears to be biased towards only certain partially coherent sources. An elementary but striking consequence of this prejudice is that the twist phase cannot be present in a coherent source. This noticeable bound on the strength of the twist phase also brings out the subtle nature of this phase and distinguishes it from the more familiar phase curvature [2–4]. Such a spatial coupled phase and the connections with an optical vortex reveal a subtle connection between macroscopic statistical optics and quantum optics. It has been shown that light beams with twist phase carry orbital angular momentum, which offers new opportunities for applications such as optical tweezers, optical imaging, and wireless optical communications [5–8].

Twisted Gaussian Schell-model (TGSM) beams were introduced by Simon and Mukunda as typical partially coherent fields [1]. Owing to their nontrivial properties, TGSM beams have attracted considerable interest. The propagation of TGSM beams and their interaction with matter have been extensively studied in detail [9–13]. The first experimental demonstration of TGSM beams was carried out by Friberg *et al*. using a six-cylinder lens system [2]. Recently, Wang *et al*. further developed an optimized three-cylinder lens system for converting an anisotropic GSM beam into a TGSM beam [8]. More recently, we reported a universal experimental method for accurately customizing general partially coherent beams by decomposition of such beams into weighted superpositions of overlapping mutually uncorrelated pseudo modes [13,14].

Despite tremendous progress, a regrettable fact is that the twist phase studied so far has been restricted to uniformly correlated Schell-model (SM) sources, and the spectral degree of coherence (DoC) is shift invariant. Whether the twist phase can successfully be mapped to more general partially coherent sources has long been an open and interesting question. A mutually incoherent superposition realization and an analysis of coherent-mode decomposition have promoted much insight into the nature of twistable sources [2–4]. Recently, this issue has re-aroused great interest, driven by the new opportunities offered by the complex design of the correlation function and phase twist [15–19]. Importantly, Gori and Santarsiero introduced an elegant method to conceive twisted cross-spectral densities (CSDs), for which no symmetry constraints are required and the nonnegative constraint is automatically satisfied [16]. Immediately after, Borghi discussed the necessary and sufficient condition for SM sources endowed with axial symmetry [17]. Subsequently, several classes of twisted partially coherent sources have been theoretically designed [10–12,19,20].

In this Letter, we introduce a feasible kernel for generating a new class of non-uniformly correlated twisted pseudo-GSM (TPGSM) sources, which can be further extended to design general twisted SM sources. It has been shown that the DoC has neither shift invariance nor shift-circular symmetry. Moreover, we carry out an experiment to generate TPGSM beams using weighted superpositions of overlapping mutually uncorrelated pseudo modes. The dependence of the spectral intensity on twist phase and vortex phase with different chirality has been studied. The results demonstrate the subtle nature of the coherence, twist phase, and vortex phase, and promise new opportunities.

Consider a random light source propagating close to the $z$ direction. The second-order correlation function at two typical points ${{\textbf{r}}_1}$ and ${{\textbf{r}}_2}$ across the source plane may be described by the CSD. The CSD satisfies the nonnegative definiteness condition, which can be written in the form [15,21]

Let us assume that the kernel takes on the following form:

In particular, consider that $p({\boldsymbol u})$ has the following form:

On substituting from Eqs. (2) and (3) into Eq. (1), the expression for the CSD is obtained as

The DoC can be alternatively evaluated using the definition

Figure 1 shows contour plots of the absolute value of the DoC as a function of ${{\textbf{r}}_1}$ for different values of ${{\textbf{r}}_2}$. It is clearly seen that the DoC reduces to Gaussian distribution when ${{\textbf{r}}_2}$ is placed at the source center. In this case, the effective coherence width ${\delta _0}{\delta _1}/\sqrt {\delta _0^2 + \delta _1^2}$ coincides with the deviation of the Gaussian. When ${{\textbf{r}}_2}$ moves away from the center, it can be observed that the DoC evolves into an eccentric doughnut shape, which is significantly different from the shift invariance of GSM sources. Meanwhile, one also finds that not only does the maximum value of DoC shift from the center as the position of the relative point shifts, but also the circular symmetric structure no longer exists, which differs from PSM sources with circular symmetry [14,22]. As a consequence, the Van Cittert–Zernike theorem is not suitable for experimentally generating such a non-uniformly correlated source.

Let us now consider the paraxial propagation of this source through a rotationally symmetric system. The CSD across a plane at a distance $z$ is given by the formula

It is obvious that the rms beam width of a TPGSM beam is larger than that of a TGSM beam due to the pseudo-Gaussian correlation function. In addition, the beam width when the twisted phase and the vortex phase have the same chirality is also always larger than the width when the chirality is different.

Next, we focus on the experimental generation of such a non-uniformly correlated twisted source. In principle, any partially coherent light can be thought of as the weighted superpositions of a set of mutually uncorrelated, but perfectly coherent components [15]. For a TPGSM source, the CSD can be represented in the following superposition form:

Since the weights of the pseudo modes are closely related to the DoC, it can be used to determine the number of samples. According to the weight function, we consider ${{\boldsymbol u}_{{mn}}}$ uniformly distributed in the confidence interval $[{- 2\sqrt {1/a} ,\;2\sqrt {1/a)}}],$ and ${{\boldsymbol u}_t}$ is equally spaced in the interval $[{- 2\sqrt 2 /({k{\delta _1}}),\;2\sqrt 2 /({k{\delta _1}})}],$ where $\sqrt {1/a}$ and $\sqrt 2 /({k{\delta _1}})$ are the equivalent waist widths of the corresponding components of the weighting function. Also, Eq. (11) indicates that the pseudo modes are statistically independent. Based on this idea, a feasible experimental protocol is to use a dynamic spatial light modulator (SLM) holographic technique to implement the mutually independent superpositions [14].

According to Eqs. (1 )–(4), (10), and (11), the complex holograms can be designed as

The experimental sketch used for the generation of the TPGSM beam is illustrated in Fig. 2. A He–Ne laser beam of wavelength 633 nm is polarized, expanded, and collimated to illuminate the reflective phase-only SLM1 with a pixel pitch of 8 µm and refresh rate of 60 Hz. The reflected beam from SLM1 passes through an amplitude filtering system to select the first-order diffracted beam. SLM1 displays phase holograms calculated according to Eq. (12). The focal lengths of Fourier lenses L1 and L2 are 15 cm. A TPGSM source is generated after L2 and then tailored by an amplitude-type SLM2. Finally, the generated TPGSM beam is focused by a lens L3 (${f_3} = {{40}}\;{\rm{cm}}$) and recorded by a CCD.

According to the weight function and values of parameters, ${{1}}{{{5}}^3}$ pseudo modes have been sampled for calculating the holograms. The twist parameter is set as $\mu = - 0.001k$ . Since the number of sampled modes is independent of the hologram number and the recording time, a sufficient number of modes can be sampled to accurately synthesize the predicted CSD. In fact, due to the limitation of phase modulation precision, excessive samples will not improve the synthesis quality. Also, 300 holograms have been calculated for a dynamic loop display, and the actual recording time of CCD statistics is about 2 s. Figure 3 shows the normalized intensity distribution of the generated TPGSM beam behind SLM2. The corresponding theoretical fits are plotted, and the beam widths along $x$ and $y$ directions are ${\sigma _{0x}} \approx 0.39\,\,{\rm{mm}}$ and ${\sigma _{0y}} \approx 0.18\,\,{\rm{mm}}$, respectively. Figure 4 illustrates the experiment results of the square of the absolute value of the DoC of the synthesized TPGSM beam before SLM2. The corresponding theoretical fits (solid lines) show that the coherence parameters are ${\delta _0} \approx 0.28\,\,{\rm{mm}}$ and ${\delta _1} \approx 0.11\,\,{\rm{mm}}$

Figures 5 and 6 illustrate the normalized intensity distribution at different propagation distances for different values of topological charge $l$. We let the beam propagate a distance ${{{z}}_0} = {{10}}\;{\rm{cm}}$ and then focus by a thin lens L3. The elements of the transfer matrix of the whole optical system can be described as $A = 1 - {{z}}/{f_3}$, $B = z + {z_0} - {{z}}{z_0}/{f_3}$. In the absence of the vortex phase ($l = 0$), it can be seen from Fig. 5 that not only does the beam undergo rotation during propagation, but also a strong self-focusing bright spot appears in the center of the beam. According to our previous study, this bright spot is induced by circular pseudo-Gaussian coherence [14,22]. In addition, a careful observation also found that the rotation speed of the central bright spot and the whole beam is not the same. Since the beam spot on propagation is not a standard ellipse, it is difficult to give a strict definition of the rotation angle. Under the case of $l = - 1$, Fig. 6 clearly shows that there is always an intensity null at the center induced by phase singularity, regardless of propagation distance. This is similar to vortex PGSM beams but different from GSM vortex beams [14]. However, in the case of $l = 1$, it is found that the center of the beam is no longer null intensity. The physical interpretation for this phenomenon is that both the twist factor and the optical vortex are intrinsically related and both have chirality. When they have the same chirality, the vortex phase is stable, and the twist phase leads to a rotation of beam spot and a larger divergence angle. Conversely, the optical vortex is weakened, and the null intensity in the center disappears. A more in-depth analysis can be performed through orthogonal pseudo-modal decomposition [4,22].

In conclusion, we suggest a feasible way to twist partially coherent sources by sticking an SM coherence function to CSDs. The nonnegative requirement of the CSD shows that the upper bound of the twist phase of the TGSM source is valid for all twisted SM family sources. A new class of non-uniformly correlated TPGSM beams have been theoretically designed and experimentally generated using a dynamic SLM holographic technique. The nontrivial propagation characteristics indicate that the correlation function plays a key role in the propagation of twisted beams. Due to the inherent correlation between the twist factor and the vortex phase, the intensity distribution exhibits a close dependence on the phase chirality. The results help in devising nontrivial twisted partially coherent lights and promote further insight into the twisting effect.

## Funding

National Key Research and Development Program of China (2019YFA0705000); National Natural Science Foundation of China (11504172, 11974218, 61875088, 91750201); National Postdoctoral Program for Innovative Talents (7131701018); Fundamental Research Funds for the Central Universities (30919011293); Innovation Group of Jinan (2018GXRC010).

## 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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