1. Introduction

The state of polarization of light is related to its vectorial nature of electromagnetic filed. It has became one of the most salient features of the light field [1]. The polarization state in the scalar regime, such as the state containing homogeneous linear, circular, or elliptic polarization, is associated with the spin angular momentum and therefore it can be mapped as a point on the conventional spin-based Poincaré sphere [2]. Such a geometric description relies on orthogonal basis of two pure spin states (i.e., the left and the right circular polarizations) mapped on the north and south poles of the sphere, respectively [3]. When the orthogonal spin basis is endowed with the orbital angular momentum of light, it constructs a spin-orbit based Poincaré sphere, also called as the higher-order Poincaré sphere [4,5]. A linear combination of these spin-orbit bases results in vector vortex state, which exhibits inhomogeneous polarization distributions in transverse plane and are featured by helical wave front characterized by a topological charge [4,5]. It is worth mentioning that the state of polarization of the higher-order vector vortex beam can also be geometrically described as a point on the surface of the higher-order sphere, in a similar way to that shown by the conventional Poincaré sphere. Among them, the two pure states mapped on the poles of the sphere degrade into homogeneous circular polarizations having vortex phase profiles, while the equator characterizes vector beam with inhomogeneous linear polarizations. Other points between the poles and equator exhibit vector states of elliptical polarizations. It is emphasized that these higher-order vector vortex states are solutions to the vectorial Maxwell wave equation [1,6], and can be experimentally realized by many techniques implemented both inside [7,8] or outside [9,10] the laser cavities. This geometric representation provides not only deep insight into the light field, but also greatly simplifies problems of description for the complex polarizations.

In recent years, the vector vortex beams have been widely studied in various aspects owing to their intriguing spin-orbit coupling properties of the light fields [11–14]. For example, a specific type of vector vortex state undergoes an adiabatic transformation through an anisotropic medium where the intrinsic spin and orbital components evolve in a particular way, resulting in the spin-orbit Hall effect of light [15]. Manifestations of spin-orbit interactions of the vector vortex beams have been reported as spin state splitting on the Poincaré sphere, accompanied with vectorial spin Hall effect of light [16]. When tightly focusing the vector vortex light beam, a strong longitudinal polarization of light field can be achieved [17], accompanied by a phenomenon of spin-to-orbital angular momentum conversion [14]. In addition to these intriguing classical properties, the vector vortex light beam has also been applied to quantum regime [18,19], where the vector vortex beam can be possible to encode unlimited quantum states with discrete topological charges.

Despite great potential shown by the vector vortex light beams, to date, on-demand controlled conversion between such complex states is not achieved. We note that several attempts have been made to structure design that allows vector vortex state conversions between specific points on the higher-order sphere [11,15,20]. Nevertheless, each conversion requires its own design, and no theory so far has been developed to account for the spin-orbit coupling dynamic of the vector vortex state in the course of state conversion.

In this article, we come up with and demonstrate a novel theoretical platform to realize geometrical control of the vector vortex state of light fields via a linear coupling system. We take advantage of analogies between the Poincaré sphere and the Bloch sphere that is used for describing quantum state dynamics in two-level system [21,22], as well as the three-wave mixing process in nonlinear optics [23]. In this regard, we built the connection between these different settings, and present, for the first time, a geometric theory that clearly reveals how a tailored phase mismatching quantity, together with a suitable coupling strength, supports robust state conversion between these complex vector vortex light beams.

2. Principle for the geometric control

A very powerful method to represent polarization state is the Poincaré sphere. This sphere maps the complex Jones vectors into points on this sphere, where the north and south poles represent two orthotropic states of polarizations. This concept of geometric representation was later adopted for representing other types of dynamic two-level systems, including spins in magnetic field [24], as well as atomic two-level system coupled by electromagnetic waves [21,25], and is known in quantum mechanics as the Bloch sphere. As shown in Fig. 1(a), the quantum state in a two-level quantum system can be described by a Bloch sphere, expressed as $\vec \rho =(U, V, W)$ [21]. On the sphere, the south and north poles represent the two orthogonal eigenstates of the two-level system, i.e., the ground state $|w_g>$ and the excited state $|w_e>$, as marked in the Fig. 1(a). The quantum wave packet evolves according to an action of an external electromagnetic field. The mechanism for driving the evolution process of the quantum state is that the two pure states become interactive with each other under the driving of external field. The coupling effect can be mapped onto a form of the torque vector $\vec g$, as is illustrated in the figure. Therefore, the state evolution on the Bloch sphere can be geometrically controlled by the torque vector $\vec g$, which is associated with the external field and their detuning. In this case, the dynamics of the quantum system satisfies the classical gyro equation [22], represented by the precession of the state around the torque vector $\vec g$, as is illustrated in Fig. 1(a).

 

Fig. 1. Principle for geometric control of wave packets in different physical settings: (a) The Bloch sphere describing quantum states in the two-level quantum systems. The pure states denoted by $|\omega _g>$ and $|\omega _e>$ are mapped on the south and north poles of the sphere. The state evolves according to the rotation of the torque vector $\vec g$. (b) The Bloch sphere representing three-wave mixing process in $\chi ^{(2)}$ nonlinear optics. The idler $|A_1>$ and signal $|A_3>$ waves represent the two orthotropic pure states, while the pump wave $|A_2>$ drives the nonlinear process, as a result, the nonlinear state $\vec \rho =(U_{SFG}, V_{SFG}, W_{SFG})$ rotates according to the torque vector $\vec g$. Here SFG denotes the process of sum-frequency generation. (c) Higher-order Poincaré sphere ($\ell =1$) used to map the vector vortex states. The two pure sates are represented by the left $|L>$ and $|R>$ circularly polarized vortex beams. In an analogous manner, the state $\vec S=(S_1, S_2, S_3)$ evolves according the the torque vector $\vec g$. (d)-(f) Representations of typical vector vortex states mapped on the equator of the sphere, see the points marked as 1, 2, 3 in (c), respectively.

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An equivalent of a spinor has been recently shown in nonlinear optics. The optical structure with $\chi ^{(2)}$ nonlinearity couple the idler $|A_1>$, signal $|A_3>$ and pump waves $|A_2>$. In this three-wave mixing process, the idler and the signal waves represent the spin down and spin up equivalents [26], as depicted, respectively, on the south and north poles of the Bloch sphere, see Fig. 1(b); while the pump wave acts as an external field that drives the nonlinear process. Particularly note that, a geometric representation for the nonlinear three-wave mixing process was already developed [23,26]. In that scenario, all possible nonlinear light states can be shown on the surface of a Bloch sphere constructed by the three components of a state vector $\vec {\rho }_{SFG}=(U_{SFG},V_{SFG},W_{SFG})$, which are related to the idler and the signal waves. Note that here SFG denotes the sum-frequency generation. This representation is closely connected to the description of the vector vortex states on the higher-order Poincaré sphere, and it generates fully optical manipulations for the idler and signal waves, hence leading to novel nonlinear phenomena. For instance, the light-matter interactions in a modulated $\chi ^{(2)}$ nonlinear crystal have led to notable outcomes including the recent finding of a new form of geometric phase [27–29], the all-optical Stern-Gerlach effect [26], and the optical skyminons (a topological texture of spin distribution) [30].

We explore the underlying SU(2) structure of the higher-order Poincaré sphere, compared to that discussed in the two-level quantum settings [21,25] as well as nonlinear optics [23,26]. We define a higher-order poincaré sphere constructed by three Stokes parameters, represented by $S_1$, $S_2$, and $S_3$, respectively, as illustrated in Fig. 1(c). On the Poincaré sphere, the north and south poles of the sphere represent two orthogonal eigenmodes, corresponding to the left ($|L>$) and right ($|R>$) circularly-polarized vortex light fields. Note that these two eigenmodes have opposite topological charges $\pm \ell$. As mentioned, the vector vortex state mapped at other point on the sphere is a linear combination of the two eigenstates. For instance, we present three typical mixed states mapped on the equator of the sphere, see the points marked as 1, 2, 3 on the equator, corresponding to an azimulthal angle of $0$, $\pi /2$, and $\pi$. Their corresponding polarization distributions are illustrated in Fig. 1(d), 1(e) and 1(f), respectively. In order to geometrically control the vector vortex states on the sphere, it is required to couple the two orthogonal polarized light modes in an optical system, in an analogous way to the two-level system as well as the three-wave mixing process. In this work, the coupling effect between two orthotropic polarization components could take place in a single-mode anisotropic fiber [31], where these two modes exhibit different refringence index, let’s say $n_x$ and $n_y$, respectively, while having the same transverse profile during propagation. The birefringence effect of the fiber structure gives rise to different group velocities of the two components. As a result, phase mismatching appears in the course of beam propagation, which in turn affects the dynamics of the vector vortex state. This effect plays a role of ’external field’ that drives the motion of the vector vortex state on the sphere. It can be predicted that the coupling effect and the phase mismatching quantity can be mapped into an analogous torque vector $\vec g$, which determines the evolution process of the vector vortex state.

3. Theoretical model

Any state of polarization of the complex vector vortex light fields can be represented as a form of superposition of two antipodal circular polarization basis [4,5]

A linearly coupled system for the vector vortex states is exploited, in which the two components of the states become coupled during state evolution. As mentioned above, this could be implemented possibly through optical birefringent fiber where the state components $\Psi _x$ and $\Psi _y$ undergo different refractive index, leading to different group velocities [31]. Besides, one can also explore possibilities with optical cavities [32], coupled waveguides [33,34], or space-varying birefringent wave plates [20] for coupling the two polarization components. To demonstrate geometric control of the complex light field, we consider a vector vortex state carried by an evolving envelope field $E(x, y, z)$, where $z$ denotes the evolution axis in the setting. In this case, the full vector of the vector vortex light field is expressed as $\textbf {E}=E(x,y,z)\Psi$. Evolution of state of polarization on the Poincaré sphere can be realized by utilizing a linearly coupled wave equation under the paraxial condition. In the normalized form, it can be written as [31]

We further note that the second and third terms of the coupled wave equations stand for spatial diffraction of the light field, which does not alter the state of polarization during evolution in the system. In general, this terms can be neglected owing to the fact that the light wave propagates in a specific spatial mode which does not change significantly in the course of propagation. Therefore, to look for a geometric representation for the state evolution on the higher-order Poincaré sphere, it is convenient to remove these diffraction terms. As a result, we simplify the coupled wave equation as

We define a state vector, represented by three Stokes parameters, $\vec S(z) = ({S_1(z)},{S_2(z)},{S_3(z)})$ whose z dependence can be obtained from the above coupled wave equation. For a given $z$, the state vector $\vec S(z)$ can be represented as a point on the surface of the sphere. For example, taking into account the north and south poles of a normalized sphere, the corresponding state vectors are mapped as $\vec S=(0, 0, 1)$ and $\vec S=(0, 0, -1)$, respectively. We further deduce the motion equation of the state vector $\vec S(z)$, expressed in a form of $d\vec S/dz = \vec g \times \vec S$, where $\vec g = ({g_1},{g_2},{g_3})$ is a torque vector. Three components of $\vec g$ are obtained as $g_1=0$, $g_2=-2C\cos (\Delta \beta \cdot z)$, and $g_3=2C\sin (\Delta \beta \cdot z)$, respectively. Clearly, the torque vector $\vec g$ represents the coupling between the two fields. Note that the symbol $\times$ denotes the vector product. The picture resembles the geometric descriptions for other settings, as illustrated in Fig. 1(c), showing the precession of the vector vortex state around the torque vector $\vec g$. The explicit form of the underlying three-wave coupling equation for the vector $\vec S(z)$ can be written as

4. Adiabatic conversions between vector vortex states

Firstly, we examine the situation of $\Delta \beta = 0$, also called as a perfect phase matching condition. This resembles the resonant condition of the two-level quantum states. In this case, full conversion of the initial vector vortex state to another state is achievable. The perfect-phase-matched solution for the vector vortex state exhibits similar dynamical features as those reported in resonant condition of the two-level physical settings. This results in periodic evolution of the vector vortex state, as visualized in the Poincaré sphere as a periodic rotation of the state vector $\vec S$ started form the point: $\vec S=(1, 0, 0)$, see the red circle on the sphere in Fig. 2(a). This phenomenon is known as the Rabi oscillation, which was already investigated extensively in the quantum physics [35–40], but it is rarely studied with the vector vortex light beams. Here we demonstrate the Rabi oscillation of light waves in the framework of higher-order vector vortex regime. This new oscillating states of light are characterized by a topologically shaped wavefront and coupled by spatially varying polarization. Figure 2(b) shows the corresponding variation of the three stokes components along with the propagation distance. It is clearly seen that the components of $S_1$ and $S_3$ exhibit harmonic oscillation following a $\sin$ or $\cos$ function but with different phase offsets, a phenomenon of Rabi oscillation. Contrary to $S_1$ and $S_3$, the value of $S_2$ shows no variation during state evolution on the Poincaré sphere. This is because the derivative of the second wave equation in Eq. (7) becomes zero and hence it is independent on the value of $S_1$ and $S_3$. To further observe this intriguing ’Rabi oscillation’, we simulate propagation of the vector vortex beams based on Eq. (4), by means of the numerical method of the 4th Runge-Kutta. To this end, we take into account the light field which contains vortex phase with topological charge of $|\ell |=1$. The vector vortex state is carried by an envelope having the Laguerrre-Gauss profile. In this scenario, the initial condition for the Eq. (4) takes the following form

 

Fig. 2. Periodic evolution of the vector vortex states represented by the first-order Poincaré sphere, under the setting: $\Delta \beta = 0$, and $C = 1$. (a) Geometrical visualization of the state evolution on a sphere. Two periodic trajectories are plotted for different initial states, expressed as: $(\theta ,\phi ) = (\pi /2,0)$ (see the red curve), and $(\theta ,\phi ) = (\pi /2,\pi /4)$ (see the blue curve). (b) and (c) present their corresponding variations of the Stokes parameters along with the evolution. (d) Direct observation of the periodic conversion of the vector vortex light beams, based on the model in Eq. (4). The panels denote intensity distributions of $x$-components of the light fields at different points of the sphere, see the red curve in (a). In the color bars, L: Low; H: High.

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Next, we consider the case where the phase of the two orthotropic components of the complex light field are mismatched in the course of propagation. In this ’non-resonant’ condition, the setting does not support complete cycle conversion between two target states, since all components of the state vector $\vec S$ become coupled, as indicated by the theoretical model in Eq. (7). As a result, the value of each Stokes component is a function of distance $z$ and is related to $\Delta \beta$. To see this detuned effect, we take the value of $\Delta \beta =2.5$ into account, and set the initial state located at the equator of the sphere, expressed as $(\theta ,\phi ) = (\pi /2,0)$. With these settings, we obtain the evolution trajectory mapped on the Poincaré sphere, as illustrated by the blue curve in Fig. 3(a). Distinctive features can be indicated: first, the evolution trajectory becomes noncyclic and flopping around the initial point of the sphere; second, the detuning effect leads to a relatively low oscillating amplitude of $S_1$ around a specific value. This phenomenon is further confirmed by the real-time value of the Stokes parameters, as illustrated in Fig. 3(b). On the other hand, we consider another process where the state starts from another point on the sphere, expressed as $(\theta ,\phi ) = (\pi /2, -\pi )$. This vector vortex state exhibits $\pi$-phase shift as compared to the initial state depicted by the process (see the blue curves). Interestingly, it is shown from the Stokes parameters [see Fig. 3(c)] that the value of the Stokes parameter is also $\pi$-phase shifted for the same detuning, as compared to the results in Fig. 3(b). The outcome for this process is presented in Fig. 3(a), see the red trajectory on the sphere. It is shown that the two initial $\pi$-symmetric states (z=0) exhibit similar behaviors in the course of their evolutions, as suggested from the resultant evolution trajectories both having flower patterns on the sphere.

 

Fig. 3. Non-periodic evolution of the vector vortex states represented by the first-order Poincaré sphere, under the setting: $\Delta \beta = 2.5$, and $C = 1$. (a) Geometric evolution trajectories of the vector vortex states starting from different initial points: $(\theta ,\phi ) = (\pi /2,0)$ (see the blue curve), and $(\theta ,\phi ) = (\pi /2,\pi )$ (see the red curve). (b) and (c) present their corresponding variations of the Stokes parameters along with the evolutions.

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It is worth noting that both the phase matching and mismatching conditions give rise to oscillations of the vector vortex states when given an input state. These instable state conversion of the vector vortex light field hinders its potential applications. For example, to achieve conversion between two target states, one should carefully design an anisotropic structure with delicate detuning and interaction length. Slight difference would lead to a failure conversion of the vector vortex states. In the following, we investigate possibilities for robust conversion between two arbitrary vector vortex states, by means of an adiabatic technique. In this case, the phase mismatching quantity $\Delta \beta$ is no longer a constant. It should vary adiabatically along the propagation distance. It means that the refractive index of $n_x$ and $n_y$ is no longer independent of propagation distance, but it is subject to a corresponding modulation function of $z$. The adiabatic process can be realized by sweeping the rate of the phase mismatch along the $z$ axis, so that it changes slowly with respect to the adiabatic condition [41]

 

Fig. 4. Robust conversion between two arbitrary vector vortex light states by means of the adiabatic technique. Without loss of generality, four different types of conversion processes are presented based on the first-order Poincaré sphere: (a) mixed state$\rightarrow$mixed state, i.e., ($\pi /2, 0$)$\rightarrow$($\pi /2, \pi$); (b) pure state$\rightarrow$pure state, i.e., it starts from the north pole and ends by the south pole; (c) pure state$\rightarrow$mixed state, i.e., it starts from the north pole, and ends at a point at ($\pi /2, 0$); and (d) mixed state$\rightarrow$pure state, i.e., it starts from a point ($\pi /2, 0$) and ends at the south pole. (a1)-(d1) The corresponding variations of the Stokes parameters along with state evolutions, obtained from the model in Eq. (7); whereas (a2)-(d2) show the intensity distributions of $x$-components of the initial and targets states, corresponding to the processes in (a)-(d). Note that the simulations were performed based on Eq. (4), under the settings: $C=1$, $\beta _0=115.2$, and (a, a1, a2) $\gamma _1=-2.00\times 10^{-4}$, $\gamma _2=2.04\times 10^{-4}$, and $\gamma _3=-6.10\times 10^{-4}$; (b, b1, b2) $\gamma _1=-55.10\times 10^{-4}$, $\gamma _2=42.86\times 10^{-4}$, and $\gamma _3=-10.20\times 10^{-4}$; (c, c1, c2) $\gamma _1=59.18\times 10^{-4}$, $\gamma _2=-38.80\times 10^{-4}$, and $\gamma _3=6.12\times 10^{-4}$; (d, d1, d2) $\gamma _1=59.18\times 10^{-4}$, $\gamma _2=38.78\times 10^{-4}$, and $\gamma _3=6.12\times 10^{-4}$.

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To further demonstrate the availability of the technique, we consider other different processes. We classify them into three types of conversion processes: the pure state to pure state, the pure state to mixed state, and the mixed state to pure state, corresponding to the results as illustrated in Fig. 4(b, b1, b2), Fig. 4(c, c1, c2), and Fig. 4(d, d1, d2), respectively. We should note that these different conversion processes are implemented under the same settings, but with different values of $\gamma _j$, as specified in the caption of Fig. 4. As seem from their evolution trajectories on the sphere, all the considered states exhibit slightly evolutions on the sphere at their early stages, owing to the large mismatch. Afterwards, they evolve into their target states adiabatically and finally become stable. Such an evolution feature is further indicated by their corresponding value of $\vec S$. As direct evidence, we also present intensity distributions of the horizontal components of the fields at the beginning and final stages of state evolution, see the corresponding results in Fig. 4(b2)–4(d2), respectively. These results suggest that using the adiabatic technique, we can realize robust conversion between two arbitrary vector vortex states.

5. Conclusion

In conclusion, we have demonstrated a novel geometric technique for controlling the state of polarization of light in the framework of a spin-orbit based Poincaré sphere. Coupled wave theory was adopted to account for the spin-orbit coupling dynamics of the complex light field in the setting, which plays a role in a manner similar to the two-level quantum system [21,25], as well as three-wave mixing process in nonlinear optics [23,26]. We have shown the intriguing Rabi oscillating properties of the vector vortex state in the phase matching condition. Such an effect of Rabi oscillation takes place in the framework of the higher-order optical regime, hence it contrasts to previously demonstrated Rabi oscillations [35–40]. On the other hand, we have considered the behaviors of vector vortex state in the phase mismatching condition, featuring by the nonclyclic evolution process. This detuned effect leads to inefficient state conversion of the light field. Importantly, in order to obtain a robust and high efficient conversion between two target complex states, adiabatic technique was utilized for tailoring the phase mismatch profile. In this regards, we have demonstrated four different types of adiabatic conversion processes, including conversion from any mixed state to mixed state of the light field. We believe our demonstrations open new possibilities for control and manipulation of the vector vortex light fields, and might find intriguing applications both in the quantum and the classical optics [42,43]. This works would also trigger new studies of the spin-orbit properties of light, accompanied by the evolution processes.