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Abstract
In this paper, we present a novel form of a partially coherent beam characterized by classical entanglement in higher dimensions. We coin the term “twisted vector vortex (TVV) beam” to describe this phenomenon. Similar to multi-partite quantum entangled states in higher dimensions, the partially coherent twisted vector vortex beam possesses distinct properties such as non-uniform polarization, vortex phase, and twist phase. Through experiments, we offer empirical evidence for these three degrees-of-freedom in the light field. The results demonstrate that the state of the light is inseparable in terms of polarization and orbital angular momentum (OAM) modes. Additionally, the twist phase introduces an additional dimension in controlling the vector vortex beam. This research reveals the possibility of new controlling dimensions in classical entanglement through the chirality of coherence within partially coherent light. Consequently, this opens up new avenues for the utilization of partially coherent light in both classical and quantum domains.
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1. Introduction
Classical entanglement [1,2] has recently garnered significant advancements in various fields including communication [3,4], metrology [5–8], sensing [9,10], computation, and encryption [11–13]. Classical entanglement refers to a state of light that resembles quantum entanglement mathematically but exhibits greater robustness in noisy environments [14–17]. At the heart of classical entanglement lies the inseparability of the correlation function across different degrees-of-freedom (DoFs). For instance, a vector vortex light exhibits two-partite entanglement as its wavefunction cannot be algebraically separated in terms of polarization and orbital angular momentum (OAM) DoFs [18]. To expand information capacity and enable novel information protocols, proposals have been put forward for generating light with multiple DoFs and higher dimensions [2,19–22]. Two-partite classical entanglements have been observed within the DoFs of OAM, polarization, propagation direction, time, and ray path. Three-partite classical entanglements have also been realized in space-time-polarization and Ray-wave-polarization. However, to our current knowledge, all experimental realizations of multi-partite classical entanglement have been limited to fully coherent light beams. In complex environments, such as atmospheric turbulence, the degradation of coherence poses challenges for creating and sustaining multi-partite classical entanglement [23,24]. Coherence, as an inherent degree of freedom in light, has yet to be fully explored in the context of classical entanglement. Therefore, establishing high-dimensional entanglement in partially coherent light fields holds significant practical importance.
Extensive research has shown [25–27] that controlling the coherence of light can effectively enhance a beam's resistance to turbulence and its self-recovery capabilities. The emergence of twisted partially coherent light provides a new platform for manipulating classical entanglement within optical beams with low coherence. Twisted partially coherent light, an exemplary form of OAM beam, exudes a chiral structure within its phase of coherence. Unlike conventional vortex beams, the OAM within twisted partially coherent beams does not originate from the helical phase of the wave function; instead, it arises from the chiral nature of the spatial coherence function of the beam [28]. The distinctiveness of this chirality finds its roots in a special phase residing within the coherence function of the light field, known as the “twist phase”. Inextricably coupling the coordinates of two spatial points within the beam's cross-section plane, the twist phase engenders classical entanglement [29,30]. Noteworthy is the fact that this twist phase is particular to partially coherent light, as it disappears in the state of complete coherence and was first theoretically introduced in Ref. [28].
Under experimental conditions, Friberg et al. successfully generated a partially coherent beam containing the twist phase [31]. More recently, researchers have made significant headway by simplifying the experimental setup, enabling them to generate truly twisted partially coherent optical beams [32–34]. The twist phase engenders classical entanglement and ushers in new possibilities for leveraging high-dimensional light within noisy environments [35–37]. Given that high-dimensional entanglement constitutes a crucial resource for constructing communication networks, extending classical entanglement, derived from twisted partially coherent beams, to different degrees of freedom is of utmost practical significance. In a pioneering study, researchers reported a method for generating radially polarized twisted partially coherent beams, thereby expanding the DoFs of the light to encompass polarization, twist phase, and OAM [38]. However, they fell short of creating a non-separable correlation solely within the OAM DoF, thus constricting the dimensionality of classical entanglement in partially coherent light.
In this paper, we propose a method for generating high-dimensional non-separable states via the utilization of partially coherent twisted vector vortex (TVV) beams, possessing controllable radial and azimuthal polarization as well as high order OAM modes. By generating an optical beam encompassing three DoFs, namely twist phase, OAM, and polarization, we successfully observed a non-separable correlation among OAM and polarization through the polarization-projected light intensity. The twist phase establishes a connection between the OAM mode and the chirality of coherence, thus modulating the non-separable state of light in an individualized manner. Demonstrably, the non-separable light field manifests itself within a twelve-dimensional Hilbert space. The TVV beam, engendered within the confines of this study, thus provides a valuable resource in terms of dimensionality, spurring on the development of multi-partite high-dimensional classical entanglement. With far-reaching implications, this advancement holds promise for significant strides within both optical communication and optical manipulation.
2. Theoretical model
The TVV beams represent a novel variant of vector vortex beams, embodying three DoFs encompassing polarization, OAM, and the twist phase. Coherence-driven chirality serves to signify the orientation of the twist phase. Clockwise orientation of the twist phase indicates a positive chirality, while counterclockwise is for a negative chirality. This work would utilize the cross-spectral density (CSD) function to establish the theoretical model of the multi-DoFs state of the partially coherent light.
According to a “beam wander” model [39,40], the CSD function of a partially coherent beam is as follows
In the above expression, c is the radius of the distribution range of the wandering beam center. The parameter $\alpha $ is defined as $\alpha = F \cdot c$, with F a dimensionless coefficient and $F \ge 1$. w is a width parameter of the wandering optical beam ${\mu _0} = {{ \pm 1} / {({3{w^2}} )}}$. The analytical expression for the CSD function of a twisted partially coherent beam is [34]
In the expression, $\mu $ represents the twist factor and $\mu = F \cdot {\mu _0}$. By changing the sign of ${\mu _0}$, one can control the sign of the twist factor $\mu $. When $\mu > 0$ the synthesized beam possesses a positive twist; when $\mu < 0$ the beam has a negative twist. By adjusting the value of F, one can control the magnitude of the twist factor. In Eq. (5), R represents the radius of the Region of Interest (ROI), which also confines the distribution region of the wandering beam center. The beam parameters in Eq. (5) are defined as
By adjusting the parameters R, ${\mu _0}$, F, c, and w, the beam described by Eq. (5) can possess a tunable twist phase and a tunable hollow center in cross section plane [34]. To be concrete in simulation, we set $R = 4w$, and ${\mu _0} ={\pm} {1 / {({3{w^2}} )}}$.
The correlation function of the twisted partially coherent beam can be re-expressed in the degree-of-freedom of OAM. One can calculate the OAM spectrum of a partially coherent beam with the following formula [40]
Here, $S(l )$ represents the proportion of the OAM mode with the topological charge as l. In other words, $S(l )$ is the OAM spectrum of the beam. The integrals in Eq. (8) are performed in the polar coordinates, where $\textbf{r} \equiv ({\rho ,\theta ,z} )$. Here $\theta $ is the azimuthal angle, $\theta = \textrm{ta}{\textrm{n}^{ - 1}}({{y / x}} )$. $\rho $ is the radius, $\rho = \sqrt {{x^2} + {y^2}} $. The difference between the two spatial points ${\textbf{r}_1}$ and ${\textbf{r}_2}$ is restricted to the angular direction, which means that ${\rho _1} = {\rho _2} = \rho $ and ${z_1} = {z_2} = z$. The integral range of $\rho $ is limited by ROI. According to Ref. [42], the OAM spectrum is related to the angular CSD function according to the following formula
Here, $W(\Delta \theta)= 2\pi \int_0^{ROI} {W({{\textbf{r}_1},{\textbf{r}_2}} )} \rho \textrm{d}\rho $ is the angular CSD function with $\Delta \theta = {\theta _1} - \theta {}_2$. $W({\Delta \theta } )$ represents the coherence of the beam in the angular direction. It means that $S(l )$ is the angular Fourier transformation of $W({\Delta \theta } )$. The functions of $S(l )$ and $W({\Delta \theta } )$ are equivalent in information.
Next, we recast the angular coherence function $W({\Delta \theta } )$ and the OAM mode function to Dirac notations, i.e., $W({\Delta \theta } )\to |\mu \rangle $, ${e^{ - il \times ({\Delta \theta } )}} \to |l \rangle $. Equation (9) can be re-expressed as:
The sign of the twist factor signifies the orientation of the twist phase. Positive $\mu $ means that the twist phase is in the clockwise orientation, while negative $\mu $ indicates that the twist phase is in the counterclockwise direction.
Figure 1 shows the theoretical results of the normalized OAM spectrum of a twisted partially coherent beam with opposite twist factors. The simulations are performed using Eq. (5) and Eq. (8), with $R = 2\textrm{mm}$, ${\mu _0} ={\pm} 1.333\textrm{m}{\textrm{m}^{ - 2}}$, $F = 8$, $c = 2\textrm{m}{\textrm{m}^{ - 1}}$, $w = 0.5\textrm{mm}$. It is worth to note that the values for the parameters c, w, R, etc. are selected based on a previously theoretical simulation for the purpose of finding out an obvious twist phenomenon [34] where a numerical simulation method called Efficient Matrix Approach (EMA) [43,44] was used. As shown in Fig. 1(a), $S(l )$ does not distributes symmetrically around $l = 0$. The OAM modes with negative topological charges have a higher proportion than those with positive topological charges. With an opposite sign of the twist factor, in Fig. 1(b) the distribution of the OAM modes is in an opposite direction. The sign of the twist factor $\mu $ determines the asymmetric direction of the OAM spectrum. Since the twist phase is a correlation phase existing in the DoF of coherence and the spatial structure of the twist phase reveals the spatial property of coherence, in this work the chirality of coherence is actually presented by the chirality of twisted phase.
Fig. 1. Theoretical simulation of the normalized OAM spectrum of a twisted partially coherent beam. (a) OAM spectrum for $\mu > 0$; (b) OAM spectrum for $\mu < 0$. The horizontal axis represents the topological charge, denoted as l.
One can construct the polarization-OAM classical entanglement in a vector vortex beam utilizing the non-separable states as follows [18]:
$|R \rangle $ and $|L \rangle $ represent right- and left-circular polarization states, respectively. $|{{l_0}} \rangle $ represents the OAM mode with the topological charge as ${l_0}$. The vector vortex states, namely $|{TM} \rangle $, $|{TE} \rangle $, $|{H{E^e}} \rangle $, and $|{H{E^o}} \rangle $, are orthogonal to each other and inseparable in DoFs of polarization and OAM.
By considering the twist factor as an independent dimension within the realm of coherence degree-of-freedom, one can construct a multidimensional state encompassing three DoFs: polarization, OAM, and twist phase. By coupling the non-separable states described in Eqs. (11) (12) with the twist phase, one can derive the TVV states in the following manner:
These TVV states, denoted as ${|{RP} \rangle _{\textrm{twist}}}$, ${|{AP} \rangle _{\textrm{twist}}}$ possess orthogonality and inseparability within the polarization and OAM DoFs, simultaneously being embedded within the state of light's coherence. It is worth noting that the states of Eqs. (13) and (14) are actually related to those of Eqs. (11) (12). In realization, the states of Eqs. (13) and (14) can be obtained according to the strategy of Eqs. (11) and (12) by replacing the OAM mode with an opposite. Equation (15) represents a radially polarized TVV state, while Eq. (16) embodies an azimuthally polarized TVV state. Equations (15) (16) epitomize the principal findings of the theoretical framework developed in this study.
3. Experiment
The experimental strategy to synthesized the TVV beam is as follows. First, we utilize a spatial light modulator (SLM) to produce a twisted partially coherent beam; then we input the beam to a Sagnac interferometer, where the DoFs of OAM and polarization are coupled with each other and embedded into the DoF of coherence. At the output of the interferometer, a radially polarized TVV beam can be obtained, i.e., ${|{RP} \rangle _{\textrm{twist}}}$. To obtain another twisted two-partite classical entanglement state, i.e., the azimuthally polarized TVV state ${|{AP} \rangle _{\textrm{twist}}}$, we add two more half-wave plates to modulate the polarization of the output beam.
The experimental setup for generating the state ${|{RP} \rangle _{\textrm{twist}}}$ is shown in Fig. 2(a) [excluding parts (b)-(d)]. A laser beam (He-Ne laser, linearly polarized beams, wavelength 632 nm) is incident on a spatial light modulator (SLM1, phase modulation type, 1280 × 720, pixel size 6.3µm). A half-wave plate (HWP1) and a polarization beam splitter (PBS1) have been used to modulate the polarization of the input laser beam to match with the SLM1. The SLM1 loads continuously number of computer-generated holograms (CGHs) which are designed to generate the twisted partially coherent beam. A sequence of 500 holograms is dynamically loaded onto the liquid crystal screen of SLM1 and played at a frame rate of 50 fps. The details for the generation of the twisted partially coherent beam are in Ref. [34]. The reflected beam from SLM1 passes through a half-wave plate (HWP2) and a beam splitter (BS2), and enters the Sagnac interferometer loop. At the input port of the Sagnac loop, the polarization beam splitter (PBS2) splits the beam into two parts. The transmitted part is in horizontal polarization (p state, $|p \rangle $), while the reflected part is in vertical polarization (s state, $|s \rangle $). The horizontal polarization part passes through a half-wave plate (HWP3) and a Dove prism (DP). The transmitted beam is turned to vertically polarized ($|s \rangle $), and impinges into the second spatial light modulator (SLM2). SLM2 loads a stationary CGH of the OAM state. The aim is to load the vortex phase to the twisted partially coherent beam. Reflected from SLM2, the beam is in a twisted polarized vortex state, i.e., $|\mu \rangle |{{l_0}} \rangle |s \rangle $. This beam passes through a circular aperture (CA1) and enters PBS2 again, leaving the Sagnac loop through the reflection path of PBS2. The output beam which propagates counterclockwise in the Sagnac loop is in the state $|{ - \mu } \rangle |{ - {l_0}} \rangle |s \rangle $. Meanwhile, at the input port of the Sagnac loop, the vertical polarization part of PBS2 goes through CA1 to interact with SLM2. The reflected beam from SLM2 is in the state $|{ - \mu } \rangle |{{l_0}} \rangle |s \rangle $. The beam passes through the CA2, DP, and HWP3 to arrive at the p-polarized twisted vortex state $|\mu \rangle |{ - {l_0}} \rangle |p \rangle $. After reflected by a mirror, the beam enters PBS2 again and directed out of the Sagnac loop through the transmission path of PBS2. The output beam which propagates clockwise in the Sagnac loop is in the state $|{ - \mu } \rangle |{{l_0}} \rangle |p \rangle $. As a result, the outgoing light of the Sagnac loop is in a superposed state: $|{ - \mu } \rangle |{ - {l_0}} \rangle |s \rangle + |{ - \mu } \rangle |{{l_0}} \rangle |p \rangle $. After reflected by BS2, the outgoing light goes through a quarter-wave plate (QWP), and turns to be a radially polarized TVV beam which is in a polarization-OAM non-separable state: ${|{RP} \rangle _{\textrm{twist}}}$. The intensity distribution of the radially polarized TVV beam is observed by a camera (CCD2).
Fig. 2. (a) Experimental setup for generating a TVV beam. (b) Settings to the radially polarized beam to an azimuthally polarized beam. (c) Settings to project the light intensity to the horizontal polarization direction. (d) Settings to verify the twist property of the optical beam. (e) Experimental results of the average intensity distribution of the beam captured by CCD1 at different propagation distance Z. (f)-(g) Experimental results of the normalized OAM spectrum of the twisted partially coherent beam produced in SLM1 with opposite twist factors. HWP: half wave plate; PBS: polarization beam splitter; SLM: spatial light modulator; BS: beam splitter; CA: circular aperture; DP: dove prism; M: mirror; QWP: quarter-wave plate; CL: cylindrical lens; L: plano-convex lens; CCD: camera.
Figure 2(b) is the setting to transform the radially polarized beam to an azimuthally polarized beam. HWP4 has its fast axis in the horizontal direction, while the fast axis of HWP5 is at a 45-degree angle to that of HWP4. The beam passing through HWP4 and HWP5 becomes an azimuthally polarized twisted vortex beam, i.e., ${|{AP} \rangle _{\textrm{twist}}}$. Figure 2(c) is the setting to project the TVV beam to the horizontal polarization.
Figure 2(d) is the setting to verify the twisting behavior of the twisted partially coherent beam. The output beam of SLM1 passes through a cylindrical lens (CL) and a convex lens (L), and then captured by CCD1 at the propagation distance Z.
Figure 2(e) is the experimental results of the average intensity profile of the beam captured by CCD1 at different propagation distances. One can see that after the interaction with CL and L the light spot of the beam generated by SLM1 rotates during propagation. When $\mu > 0$, the intensity profile of the beam rotates counterclockwise during propagation; when $\mu < 0$ the intensity profile rotates clockwise. This phenomenon indicates that the beam generated by SLM1 is a twisted partially coherent beam.
Figure 2(f)-(g) are the experimental results of the OAM spectrum of the beam produced by SLM1. When the twist factor is positive, the OAM modes on the negative half of the spectrum are higher than those on the positive half. When the twist factor is negative, the OAM modes on the positive half of the spectrum are higher than those on the negative half. The results agree with the theoretical predictions in Fig. 1. According to the results of Figs. 2(e)-(g), one can verify that the beam injected into the Sagnac loop is indeed a twisted partially coherent beam.
Figure 3 illustrates the evolution of the state of the twisted light in the Sagnac loop given that $\mu > 0$ and ${l_0} > 0$. At the entrance of PBS2, the input state is $|\mu \rangle $ which means the OAM modes on the negative half-axis of the spectrum are higher than those on the positive half. In Fig. 3(a), the input twisted light propagates clockwise in the triangular loop. Reflected by PBS2, the state becomes $|{ - \mu } \rangle |s \rangle $ which means the positive half is higher in the OAM spectrum. When the light is reflected by SLM2, it interacts with a vortex phase. SLM2 acts as a mode filter which subtracts an OAM mode with topological charge ${l_0}$ out from the incident light. Reflected by SLM2, the light is in the state $|{ - \mu } \rangle |{{l_0}} \rangle |s \rangle $, which means the negative twisted light lost an OAM mode with the topological charge ${l_0}$. After two reflections (DP, M) and two transmissions (HWP3, PBS2), the output beam of the clockwise path is in the state of $|{ - \mu } \rangle |{{l_0}} \rangle |p \rangle $. Meanwhile, in Fig. 3(b) the input twisted light propagates counterclockwise in the triangular loop. After two reflections (M, DP) and one transmission (HWP3), the light is in the state $|\mu \rangle |s \rangle $. Reflected by SLM2, the state of the beam changes to $|\mu \rangle |{{l_0}} \rangle |s \rangle $, which means the positive twisted light lost an OAM mode with the topological charge ${l_0}$. After a reflection (PBS2), the output beam of the counterclockwise path is in the state $|{ - \mu } \rangle |{ - {l_0}} \rangle |s \rangle $.
Fig. 3. Evolution of the state of the twisted partially coherent light in the Sagnac loop when propagates (a) clockwise and (b) counterclockwise. It is given that $\mu > 0$ and ${l_0} > 0$.
Comparing the output beams in Figs. 3(a) and (b), one can observe a difference in the darkness of the hollow center of the average intensity distribution. The state of $|{ - \mu } \rangle |{ - {l_0}} \rangle |s \rangle $ in Fig. 3(b) has a deeper center than that of $|{ - \mu } \rangle |{{l_0}} \rangle |p \rangle $ in Fig. 3(a). As a carrier wave, the twisted light which is in the state $|{ - \mu } \rangle $ is low on the negative side of the OAM spectrum. Subtraction of a negative mode $- {l_0}$ on the negative side of the OAM spectrum has less influence on the total OAM than the subtraction of a positive mode ${l_0}$ when the interacting light is in $|{ - \mu } \rangle $. As a result, the absolute value of the total OAM in $|{ - \mu } \rangle |{ - {l_0}} \rangle |s \rangle $ is higher than that in $|{ - \mu } \rangle |{{l_0}} \rangle |p \rangle $. This explains the difference in the depth of the hollow center in the states of $|{ - \mu } \rangle |{ - {l_0}} \rangle |s \rangle $ and $|{ - \mu } \rangle |{{l_0}} \rangle |p \rangle $. As shown in the inserted graphs of Fig. 3(a) and (b), the interaction between the twist chirality and the OAM mode ${l_0}$ can also be observed by changing the sign of the twist factor. When the input twisted light is in the state $|{ - \mu } \rangle $, the state of the output beam changes to $|\mu \rangle |{{l_0}} \rangle |p \rangle $ in Fig. 3(a) and $|\mu \rangle |{ - {l_0}} \rangle |s \rangle $ in Fig. 3(b) correspondingly. Since the OAM spectrum of $|\mu \rangle $ is high on the negative side, the state $|\mu \rangle |{{l_0}} \rangle |p \rangle $ has a deeper hollow center than that of $|\mu \rangle |{ - {l_0}} \rangle |s \rangle $.
4. Results and discussions
After the generation of the radially polarized TVV beam, we would verify the influence from the coherence.
First, we examine the intensity distribution of the TVV beam under different topological charges and twist factors. Here, we took the topological charge as ${l_0} = 1$ and ${l_0} = 2$, and the twist factor as $\mu ={\pm} 8$ to investigate the intensity distribution and the differences under different twist conditions.
Figures 4 and 5 show respectively the experimentally obtained intensity distribution of the focused TVV beam with radial and azimuthal polarizations. The horizontal axis in the one-dimensional graph is in unit of pixel, with each pixel 5.2${\mathrm{\mu} \mathrm{m}}$. The vertical axis represents the normalized average intensity. From the experimental results, it can be observed that under the same topological charge, the sign of twist factor influences the darkness of the hollow center. This is because when the beams from clockwise and counterclockwise are superposed, contributions from the two paths are not equal. The light intensities in the clockwise path and counterclockwise path are slightly different caused by an unequal beam splitting at PBS2. In other words, when the influence from the shallow centered beam is not the same as that from the deep centered beam, the combined beam from the two paths at output would change with the direction of the OAM spectrum of the input beam. The difference between the cases of positive and negative twist factors indicates that the DoF of twist phase has been coupled with the DoF of OAM.
It can also be observed from Figs. 4 and 5 that increasing the topological charge ${l_0}$ reduces the difference caused by the sign of the twist factor. Comparisons between Fig. 4 and 5 show that changing the polarization from radial polarization to azimuthal polarization decreases the intensity at the beam center. The explanation for the phenomena about the sign of $\mu $ can be draw out from the perspective of OAM. When the chirality of the twist phase and that of the vortex phase are opposite (e.g., positive ${l_0}$ and negative $\mu $), the twist phase in coherence increases the OAM flux of the beam. Accordingly, the hollow center in the optical beam becomes more obvious. On the other hand, if the chirality of the twist phase is the same as that of the vortex phase, the twist phase in coherence decreases the OAM value of the beam, leading to a weaker hollow center in light intensity. Our experimental observations are consistent with the findings in Ref. [38]. The difference caused by the opposite signs of $\mu $ is not such obvious for case of high ${l_0}$. This is because the OAM contribution from the twist phase decreases as the topological charge of the vortex phase increases.
In order to investigate the influence from the sign of the twist factor on TVV beams, we measured the average intensity distribution in the horizontal polarization direction utilizing the settings in Fig. 2(c). We studied the effects of coherence chirality and topological charge on the polarization distribution by examining the segmented patterns of the projected beam.
Figure 6 is the experimental results of the horizontal component of the light intensity of the radially polarized TVV beam $({{{|{RP} \rangle }_{\textrm{twist}}}} )$ and azimuthally polarized TVV beam $({{{|{AP} \rangle }_{\textrm{twist}}}} )$. Comparing Fig. 6(a1) with (d1), one can see that the direction for segmentation of a radially polarized beam is perpendicular to that of the azimuthally polarized beam. For both TVV beams, the number of the segmented intensity patterns in horizontal component is twice the absolute value of the topological charge, i.e., $2|{{l_0}} |$. It means that the DoFs of polarization and OAM are inseparable in the TVV beams produced in this work. It is worth to note that, in the work of Ref. [38], the number of segmentations of the radially polarized twisted partially coherent vortex beam does not change with the topological charge.
Fig. 6. The experimental figures show the average intensity distribution of radially and azimuthally polarized twisted vortex beams after PBS separation. (a)-(c) represent the average optical intensity of radially polarized twisted vortex beams under different topological charges and twists. (d)-(f) represent the average optical intensity of azimuthally twisted vector vortex beams under different topological charges and twists.
Through the comparison between Fig. 6(a1) and (a2), it can be observed that under the same topological charge, when the twist factor is positive, the upper right pattern is higher in intensity, while when the twist factor is negative, the lower left pattern is higher in intensity. The chirality of the twist phase alters the horizontal intensity distribution of the radially polarized TVV beam. Similar results can be observed in the comparisons in Figs. 6(b1)-(c2). It means that the DoFs of spatial modes (OAM modes) and coherence chirality are coupled together in the TVV beams.
The change in sign of the twist factor causes the horizontally projected light intensity distribution rotating 180° at the cross section. As the topological charge increases, the number of segmented patterns also increases. The results of Fig. 6 demonstrate that in a TVV beam the polarization-OAM inseparable state has coupled with the DoF of coherence.
This work provides a way to control three DoFs in a partially coherent light beam: controlling the twist chirality with SLM1, controlling the OAM mode with SLM2, and controlling the polarization with the Sagnac loop. In the DoF of polarization, there exist two dimensions: $|R \rangle $ and $|L \rangle $; in OAM, there are three dimensions, i.e., ${l_0} = 1,2,3$; in twist chirality, there exist two dimensions: ${\pm} \mu $. As a result, the experiment system allows a control in $2 \times 3 \times 2 = 12$ dimensional space, i.e.,
In this paper, we proposed a light field with three degrees of freedom (DoFs) namely polarization, OAM and twisted phase. Only two of them which are polarization and OAM are correlated inseparably. It means that the light is in a polarization-OAM two-partite classical entanglement. Although the third DoF, i.e., the twisted phase does not participate in the classical entanglement, as an individual dimension the chirality of the twist phase provides a modulation approach to the two-partite classical entanglement. It is worth noting that the TVV beam still has the potential to extend the dimensions by increasing ${l_0}$. Moreover, one can also use the DoF of coherence to form non-separable states, realizing a true three-partite classically entangled states. When the polarization, OAM and twisted phase are correlated with each other in a non-separable way, the light field is in a true three-partite classical entanglement. For example, one state of the polarization-OAM-twist entangled light can be
5. Conclusion
In this study, we have introduced a novel approach to generate the TVV beam with both radial and azimuthal polarizations. Furthermore, we have successfully achieved classical entanglement controllable in 12-dimensions within the realm of partial coherence. By harnessing a twisted partially coherent beam and a Sagnac interferometric loop, we have established an inseparable correlation between polarization and OAM while simultaneously coupling the non-separable states with the degree of freedom of coherence.
The experimental verification of polarization-OAM inseparability can be observed through the linear-polarization projected intensity distribution of the TVV beam. This work marks the initial exploration of high-dimensional classical entanglement in partially coherent light beams, boasting three DoFs. The proposed high-dimensional light beams hold tremendous potential in a myriad of applications, including optical information transmission, optical information processing, and optical field control.
Funding
Major Scientific Project of Zhejiang Laboratory (2019DE0KF01); Zhejiang Province Excellent Graduate Course Construction Project (2022YJSKC14).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at the time but may be obtained from the author upon reasonable request.
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