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Abstract
We generate optical fields whose polarization structures not only rotate about their propagation axis but also can be controlled to accelerate independently from their spatial profile. We show that by combining accelerated intensity transport with orthogonal polarization states, we can produce a vector beam that displays optical activity with periodical acceleration and deceleration of the Stokes vector during propagation. We achieve this with orthogonal, scalar fields, represented by weighted superpositions of oppositely charged Bessel beams. In addition to their creation, we show that the Stokes vector can be made to accelerate or decelerate at specific locations along the Poincaré sphere by tailoring the generating basis. We also witness an optical current, or intensity transport, between local positions in the field that corresponds with the occurrence of the state-of-polarization accelerating or decelerating.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Light travels on a straight line at a constant velocity regardless of the speed of the observer. Despite this universal law, light can be tailored to produce exotic effects by interference, for example, transversely accelerating [1], radially accelerating [2] and angularly accelerating [3,4] forms of structured light have been demonstrated, and even examples in curved space [5]. These effects have been extended to other waves too, including matter [6,7]. Such accelerating waves have found a myriad of applications, including optical sorting, plasma control, particle acceleration and many others (see Refs. [8,9] for good reviews).
The aforementioned work considered the intensity structure of the light (or its equivalent in other waves), whereas it is now topical to control other degrees of freedom in what is known as structured light [10], including polarization [11–13]. There has been significant advances in the latter, tailoring polarization not only in the transverse plane (2D) but also along the propagation direction for 3D polarization structures [14]. These include singularity lines [15], polarization knots [16,17], high-order Poincaré structures [18,19], longitudinally varying polarization structures [20–22] and rotating polarization structures in free-space [23–25]. Of particular relevance to our work is the advances in producing accelerating polarization states, thus so far achieved in free-space by varying the Gouy phase of scalar beams [22] and in non-linear media [26–28]. The interest in such beams is diverse, including selection rules in parametric processes [29,30], eletromagnetic induced transparency [31], and the chiral interactions of light with matter [32,33]. In this work, we demonstrate the first accelerating State-of-Polarization (SoP) with vectorial light propagating in free-space, where a non-homogeneous polarization structure can be made to periodically accelerate and decelerate, independently from the spatial profile. We derive the analytical expressions for the acceleration of the Stokes vector and intensity patterns for each polarization basis, which we interpret as a simulation of accelerated Optical Activity (OA) during free-space propagation. In addition to achieving accelerated OA, we demonstrate how the accelerated intensity transport associated with each polarization component, results in an accelerated trajectory in the Poincaré sphere, as well as the transfer of optical current between various local positions within the optical field.
2. Concept and theory
Consider the following two scalar fields, which represent weighted superpositions of oppositely charged Bessel beams (i.e. weighted petal beams)
Fig. 1. Diagrams illustrating: (a) the intensity of scalar components used to generate an angularly accelerating/decelerating polarization structure (phases included as insets) and (b) the angularly accelerating/decelerating trajectory with propagation of the right (red) and left (blue) circularly polarised components of the resulting vector mode [Visualization 1].
If these fields are in orthogonal polarization basis states, there are two other bases which will satisfy the following relationships, post path interference, $U_{3/4} \propto U_1 \pm U_2$ and $U_{5/6} \propto U_1 \pm iU_2$ (e.g. $1,2,3,4,5,6 = H,V,D,A,R,L=D,A,H,V,L,R=R,L,H,V,A,D$ - for purely linear and circular basis states). Therefore the Stokes intensity distributions, which can be used to fully describe the polarization structure of the resulting vector beam, can be determined by taking each field and multiplying by its complex conjugate (i.e. $I_i = U_i^{*}U_i$). These intensity distributions are given by
where: andHere the ’petal’ and ’ring’ factors, which contain the radial dependence, define central and outer regions of the transverse intensity profiles - while the $\Phi ^{\alpha /\beta }_{\pm }$ factors which contain the azimuthal and propagation dependence describe the angular rotation and intensity transport behaviour. It is clear from these expressions that the input component intensities $I_{1/2}$ are propagation invariant and, consequently, the total intensity profile. Other orthogonally polarized pairs of components exhibit a $\pi$ phase difference in their behavior during propagation, while these pairs exhibit a $\pi /2$ phase difference between each other. The propagation dependent behaviour of the polarization structure of the resulting vector mode is also shown in Fig. 1(a).
The behavior of the spatial structure in the polarization indicates the presence of an angularly accelerating trajectory of the Stokes vectors for both the central and outer regions on the surface of the Poincaré sphere with propagation. This is clear when one considers the Stokes parameters (where generation in the H-V basis will be used throughout the rest of this article, i.e. $U_{1/2} = U_{H/V}$)
Fig. 2. Diagram showing the change in total azimuthal angle of the beam profile with right-circular polarization ($S_3>0$) changing with propagation [Visualization 2].
The optical current, which describes the intensity transport, for the case of a coherent vector beam is best described by the Poynting vector. This vector current contains both a term associated with the phase structure (orbital component) as well as a term associated with the polarization structure (spin component) according to [34–37]
Fig. 3. Simulations of the Poynting vector (top) [Visualization 3] and total intensity (bottom) of a H-V superposition of weighted petal beams, the total intensity $S_0$ remains invariant with propagation.
3. Experiment
3.1 Generation setup
In this work, vector petal beams with radially self-accelerating polarization structures were generated experimentally using the set-up shown in Fig. 4. A 633 nm HeNe laser was used to produce a horizontally polarized Gaussian beam, which was expanded and collimated using lenses EL and CL, respectively. The beam was then used to illuminate a PLUTO-2-NIR-015 liquid crystal on Silicon spatial light modulator (LCoS-SLM). The SLM was addressed with holograms encoded via a commonly employed algorithm which facilitates manipulation of the entire complex field illuminating the hologram [38]. The desired modulation $H(x,y)$ was determined by computing the, pixel-wise, inverse Fourier transform of the weighted petal beams using the discrete inverse Fourier transform matrix $\mathcal {F}^{-1}$, according to:
Fig. 4. Diagram showing the experimental set-up used to generate and characterize petal beams with azimuthally accelerating polarization structures.
The two petal beams were given: $l=-1$, $k_{r2}=0.75k_{r1}=0.75(0.0025k)$ (where $k$ is the total wavevector magnitude) and $w_0=2.25\times 10^{-3}$ m; while the $x$-component of the holographic grating frequency for the two beams were given opposite directions. The, now spatially separated, first order diffraction components associated with the two petal beams were then directed into a 50:50 BS through a series of mirrors (M1 - M3), D-shaped mirror (MD) and Fourier lenses (FL - with focal lengths $f=750\times 10^{-3}$ m forming a $2f$ system between the SLM screen and the coinciding focal planes of these lenses). Prior to entering the BS a half-wave plate (HWP) was used to convert the polarization of one of the beams to vertical. The intensity of the six polarization components of the resulting vector beam was then imaged using a FLIR Grasshopper3 CCD camera, after passing through a conventional Stokes polarimetry arrangement (quarter-wave plate (QWP) and linear polarizer (LP) [39]).
3.2 Propagation techniques
The Stokes intensity measurements were taken within $z\in [-z_{max},z_{max}]$, where $z=0$ corresponds to the focal plane of the Fourier lenses (as the beam paths were made equal) and $z_{max}=\frac {2\pi w_0}{\lambda k_r}$ is the maximum distance in which the BG beams are valid. The measurements over this range were acquired in two manners: through mounting of the CCD on a rail then measuring physical propagation effects and through encoding a lens phase ($e^{i(k-kr^{2}/f)z}$)) into the holograms to digitally move the position of the Fourier plane as shown in Fig. 5. The latter propagation technique facilitated more polarimetry measurements, over a larger range with greater stability and therefore the results presented in this article were acquired as such; the observed effects agree with physical propagation measurements which can be seen in Visualization 4 (taken in the range $z\in [-0.25,0.1]$ m with $k_2=0.5k_1$). From the expression for $z_{max}$ it is also clear that beams with differing $k_r$ will maintain their structure over different ranges. The results presented in the following sections ensure that $z\leq (z_{max}\approx 0.9$ m$)$ of $U_1$ which has a shorter valid range.
Fig. 5. Diagram showing the equivalence of focusing and positive propagation of a beam generated in the focal plane of lens FL.
4. Results
4.1 Structural acceleration
In Fig. 6 we can see the SoP of a beam generated in the manner described above, at equally spaced values for $z$. From visual inspection, it is clear that the structured SoPs in the petals change in a non-linear manner, i.e. a transition from right-circular [R] to anti-diagonal [A] to left circular [L] is visible in the range shown. This is further highlighted by inspecting the orientation of the polarization components, the R component begins to accelerate while the L component begins to decelerate at the same time the D component is accelerating and the A component is decelerating - the energy flow away from (in to) accelerating (decelerating) components observed in previous work is also verified here [3]. The local SoP progression of a point in the beam with propagation is representative of propagation through a birefringent material (i.e. QWP) which is experiencing periodically accelerating/decelerating rotation of its fast axis. Visualization 5 and Visualization 6 illustrates, more clearly, the propagation of the SoP and intensities, respectively. Note that in Visualization 6 the frames have been independently normalized to increase visibility of the outer region.
Fig. 6. Experimental results showing the evolution of the SoP (top) [Visualization 5] and individual right-circular (R), left-circular (L), diagonal (D) and anti-diagonal (A) polarized component intensities (bottom) [Visualization 6] with propagation distance.
4.2 Intensity transport
Figure 7(a) shows how radial functions $R_{petal/ring}$ characterize the inner and outer regions as well as a transverse profile showing the circular cropping region which was used to approximate measurements of these components experimentally. It should be noted that although the ring components appear to contain a greater overall fraction of the total power, that this power distributed over a large area and the ring region therefore has a general low intensity - for this reason it is more convenient to consider the petal region when making experimental observations. Figure 7(b) shows the normalized power of the petal and ring components of the four propagation dependent polarization components. These agree with previous observations of energy exchange in scalar superposed petal beams with different radial and axial wavevector components [3]. The phase difference of the exchange behaviour between the polarized components also agrees with the intensity expressions in Eqs. (4a)–(4f).
Fig. 7. Plots showing: (a) the contribution of radial functions $R_{petal/ring}$ to the total intensity and (b) the normalized power of the ring and petal structures of four propagation dependent polarization components.
4.3 Accelerating optical activity
Considering only the petal region and observing the non-linear changes to the Stokes parameters (Cartesian coordinates of the Stokes vector on the Poincaré sphere) within this region, one can observe the accelerating polarization behaviour in a more general manner. In Fig. 8 we can see the spatially averaged SoP of the inner region at different points along the propagation distance plotted on the Poincaré sphere. From the density of points, it is clear that the Stokes vector is experiencing periodic angular acceleration/deceleration, while executing rotation about the input axis (i.e. accelerated OA). Theoretical predictions using the intensity expressions reveal that deceleration occurs at the equator and poles (meridians) for H-V (R-L) generation. This observation is reasonably verified in the experimental plot (highlighted by the dashed ovals), furthermore the measured magnitude of the Stokes vector change with propagation $|\Delta \vec {S}|$ is included alongside, displaying periodic acceleration/deceleration. Visualization 7 displays simulated trajectories of the petal Stokes vector with propagation for generation in the three bases $1,2=(H,V)/(D,A)/(R,L)$.
Fig. 8. Diagrams showing the trajectory of the average ’petal’ Stokes vector along the Poincaré sphere (left) as well as views along the input axis (right) for generation in the H-V (top) and R-L (bottom) bases, simulated comparisons are included as insets [Visualization 7].
5. Discussion and conclusion
Optical Activity is the effect of the rotation of the state of polarization of linearly polarized light, first observed more than 200 years ago, and is associated with the transmission of light through a birefringent medium [40]. Unlike normal birefringence, where the refractive indices differ for the $x$ and $y$ axes, in order to demonstrate OA, the medium needs to emulate circular (or chiral) birefrigence. Optical Activity has been obserevd in liquid crystals [41], Raman scattering [42,43] and non-chiral metasurfaces [44,45]. Recently OA has been simulated by inducing the Gouy phase in free space, resulting in a longitudinally changing local polarization vector [22], e.g., from radial to azimuthal. Here we have shown how to employ non-diffracting vectorial fields to produce long-range, periodic acceleration in the rotation of the global polarization structure, i.e., the entire polarization structure experiences angular acceleration, while maintaining a constant intensity profile. By judicious choice of superposition states it will be possible to accelerate arbitrary polarization structures by this approach. This is analogous to OA with periodical acceleration and deceleration of the Stokes vector during propagation, an effect not observed previously in free-space.
During propagation, the SoP of the petal structure rotates around a given axis in the Poincaré sphere, which can be rotated by changing the generating basis. When the rotation occurs along the equator of the Poincaré sphere, the behaviour exhibited is known traditionally as OA.
The structure of angular sectors of right and left handedness is maintained at any given plane, rotating with periodic acceleration and deceleration.
Even though this kind of evolution is typical of propagation in birefringent media, either linear or chiral, we have shown that it is achievable in free space. The acceleration and deceleration can be associated with propagation in a non-uniform birefrigent medium.
Adding this to the degrees of control presented in the structure, can make a powerful tool to characterize materials regarding their interactions with polarized light and chiral structures.
This can motivate new studies of light with changing features in free-space propagation, as well as interaction of vectorial light with different types of media.
Funding
Department of Science and Technology, Republic of South Africa.
Disclosures
The authors declare no conflicts of interest.
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