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Abstract
Spin-orbit interactions (SOI) arising due to the propagation of a paraxial light beam, with spin-angular momentum and intrinsic- and extrinsic-orbital angular momentum degrees of freedom, along a curved trajectory and their mutual interactions is investigated due to several fundamental effects of interest. We demonstrate here all six direct and reciprocal SOI effects due to the propagation of light in an inhomogeneous-anisotropic medium, such as a graded-index rod. We show that each of light’s angular momentum component impacts the other in such a way as to have a unique effect, characterized using interferometry, polarimetry, and weak measurement methods. The results are expected to have significant impact on the basic understanding of light-matter interaction and its applications.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The direct and reciprocal influence between the spin- (S), intrinsic-orbital (IO) and the extrinsic-orbital (EO) angular momenta of light, broadly classified as spin-orbit interaction (SOI) of light, is already an exciting area of research of fundamental and applied interest, evidenced by growing number of original research articles [1–13] and authoritative reviews [14–18]. Many of this research have their roots in more fundamental observations [19–28] made even before the SOI terminology became popular.
The classification of the AM interactions and their manifestation in different locally isotropic media establishes a close link between the geometric Berry phase, AM dynamics and conservation and optical beam shifts [14–17]. More recently, the fundamental aspects of the SOI of light, divided into six categories, lists how one AM degree of freedom (DoF) affects the other and expands it further to list the joint effect of two types of AM on the third [12]. Alternately, the SOI effects, categorized into spin-to-orbital AM conversion (SOC) and spin-Hall effect (SHE) of light arising in inhomogeneous (but isotropic) and anisotropic (but homogeneous) systems gives more insight into the underlying interactions and their manifestation [11,17]. It is important to realize that the SOI effects are so fundamental to understanding light and its interaction with matter that other manifestations such as orbital Hall effect (OHE) of light due to orbit-orbit interaction (OOI) and the reciprocal effects of SOC, SHE and OHE should also be present, but have rarely, if any, been investigated. These fundamental effects have however been investigated in a large variety of optical systems including in optical fibers [1,12,24–27], reflection at dielectric interface [3,5,8,21,22], uniaxial crystals [11], focused beams [6,28] and scattering [10]; never more than one effect in a single system, except in metasurfaces [29,30].
As many of the SOI effects investigated are a manifestation of light-matter interaction, the medium’s inhomogeneity, which changes the direction of propagation of light, and its anisotropy, which induces a phase change between two polarization components of light, play a critical role in the overall behavior. Instead of limiting the investigation of the SOI effects independently to inhomogeneous (but isotropic) and anisotropic (but homogeneous) systems, we consider here a graded-index (GRIN) rod system wherein both inhomogeneity and anisotropy are present simultaneously and play a role together. Exploiting this combination, we carry out interferometry, polarimetry and weak measurements to observe all the three direct and three reciprocal SOI and OOI effects, arranged pair-wise. This investigation has also opened the possibility to observe higher-order effects such as joint effect of the one or two of the AM constituents on the other, which will be reported in subsequent publications.
The GRIN rod is an optical component in which light propagates along a curved path and is commonly used as a lens element for fiber coupling [31,32]. The rod has a graded refractive index profile, of parabolic type, typically in a direction perpendicular to its axis (Z). The refractive index gradient is achieved by doping process, wherein the dopant concentration changes from the outside towards the core of the rod, leading to the appearance of weak anisotropy in the GRIN rod, measured and confirmed using different methods [33–37]. Equally important is the evolution of polarization along curved trajectory and the resulting Berry phase, which have been reported in isotropic inhomogeneous medium as due to SOI [38]. More recently, it was shown that vector-vortex modes with entangled spin and orbital AM are the solutions of Maxwell equations in the GRIN medium [39]. Thus, the simultaneous presence of inhomogeneity and anisotropy in the GRIN rod offers itself as a model system that we exploit in the demonstration of joint propagation path-polarization dependent SOI effects of light beam.
2. Theoretical details
The spin-orbit interaction stems from the transverse nature of light, coupling the propagation vector $\vec{k}$ and the state-of-polarization (SoP) of the paraxial light beam thereby mutually affecting each other in a variety of ways. Thus, a beam of light propagating in an inhomogeneous-anisotropic GRIN medium couples the SAM, IOAM and EOAM degrees of freedom leading to the observation of SOC, SHE, OHE and their corresponding reciprocal effects. Important to note that geometric phase (GP) is the underlying mechanism for all the spin-orbit interactions of light [14,15,17] and the two types of GP, the Rytov-Vladimirskii-Berry (RVB) phase, resulting from the change in the direction of light propagation and the Pancharatnam-Berry (PB) phase due to manipulation of the SoP of light [14,15] together play critical role. In addition, depending on whether the interaction of a beam of light propagating through the medium is symmetric or with broken symmetry decides on how the intrinsic spin and orbital and the extrinsic orbital AM of the light couple with each other and affect output beam characteristics.
2.1 On-axis propagation
The GRIN rod considered here is an example inhomogeneous-anisotropic system with parabolic refractive index profile given by $n(r )= {n_0}\left( {1 - \frac{{{A_1}{r^2}}}{2}} \right)$, where, ${A_1}$ is the gradient parameter, transverse (XY) to the direction of light propagation (Z). The direction of linear polarization of light field propagating through the GRIN medium undergoes rotation due to the parabolic refractive index distribution of the GRIN rod. To understand this process, we consider non-commutative rotation matrices that leads to the demonstration of GP accumulation in the transformation of off-diagonal elements, leading to the generation of optical vortex beam of charge l = ± 2 due to conservation of total AM of light, for circularly polarized Gaussian input beam [10]. This transformation can be expressed as
Now consider the propagation of vertical (V-) polarized paraxial Gaussian beam of light through the GRIN rod. As the linear SoP can be expressed as a superposition of right- and left- circular polarization eigen modes, the above transformation for the input V- polarized light can be written as
Fig. 1. (a) Calculated output beam intensity pattern for orthogonal (H-) polarization projection for V- polarized input Gaussian beam into the GRIN rod, (b) residual phase gradient due to rotation of the propagation vector as a function of input position that results in parallel transport of SoP. Overall size of each image is (1.8 × 1.8) mm.
2.2 Off-axis propagation
It is by now well-established that systems with broken cylindrical symmetry due to angled propagation of light beam during reflection, refraction and transmission results in spin-Hall effect to preserve AM conservation [3,5,8,10,11]. Breaking the symmetry either by blocking a part of the input beam [42] or by shifting the input beam position with respect to the propagation axis is expected to lead to the observation of SHE of light. The manifestation of SHE is also anticipated for normal incidence due to refractive index gradient $n({\vec{r}} )$ across the beam or due to spread in the wave vector $({\Delta \vec{k}} )$ [17,18,29,30]. The SHE investigated here arises from the gradient in the refractive index profile of the medium and we exploit it to understand the SOI and OOI phenomenon in general and to realize both the spin- and orbital-Hall effect of light and their reciprocal processes.
The trajectory of the input beam changes symmetrically around the propagation axis in the radial plane due to gradient in the refractive index profile and the SoP of the output beam rotates around the Z-axis. Since the system is not rotationally invariant around Z-axis due to broken cylindrical symmetry, for off-axis illumination, the AM of light is not conserved, resulting in the appearance of transverse angular momentum dependent shift in the center-of-intensity (CoI) of the light beam. The CoI is calculated using $CoI = \frac{{\sum {I(x)x + I(y)y} }}{{\sum {I(x) + I(y)} }}$, where, $I(x),I(y)$ are the intensity at x- and y- positions respectively and is summed over the beam cross-section to find the weighted average position of the beam and to find the transverse shift (〈X〉) for different off-axis illumination of the GRIN rod with reference to the on-axis. The output optical field for off-axis propagation is written as [10]
Like the SAM case, a shift in the input position for the IOAM beam with respect to the propagation axis is expected to result in OHE owing to transverse orbital Berry phase arising due to index gradient as a function of the topological charge [42,43], as the beam propagates through the GRIN rod. The parallel transport law which governs the rotation of polarization vector of the beam field is also applicable equivalently to the beams with IOAM. The Berry curvature signifying the orbit-orbit (intrinsic and extrinsic) interaction results in different paths leading to l dependent shift of the OAM beam trajectory $\langle X \rangle $, given by replacing σ in Eq. (4) by the charge of the IOAM beam l [10].
3. Experimental details
The output from a stabilized He-Ne laser (λ = 632.8 nm) is passed through a 50-50 beam-splitter (BS) and is polarized using a Glan-Thompson (GT) polarizer. A half-wave plate (HWP) is introduced to change the SoP suitably to generate Gaussian or Laguerre-Gaussian beam using suitable phase mask projected on the spatial light modulator (SLM, Santec, Japan), as shown in Fig. 2. The reflected phase-encoded beam from the SLM is passed through a GT polarizer, HWP and QWP respectively to generate the required SoP for the input beam launched into the GRIN rod (Edmund optics, USA). The GRIN rod used is 3.93 mm long, 1.8 mm in diameter and of effective focal length 1.71 mm, having numerical aperture (NA) of 0.55. The on-axis refractive index of the GRIN rod is n0 = 1.629 and the gradient constant is 0.132 mm−2. For simplicity, the refractive index gradient and its effect are assumed to be present only in the transverse (XY) plane, neglecting its effect in other (XZ and YZ) planes. The refractive index profile of the GRIN rod is cylindrically symmetric with reference to the input beam axis and required asymmetry is realized by horizontally shifting the rod with reference to the beam axis. A polarized Gaussian beam with 2 mm beam waist is focused on to the input end of the GRIN rod, kept on a V-groove, using a 20X microscope objective lens. The GRIN rod position with respect to the beam axis can be adjusted precisely using a 5-axis precision alignment stage (Newport, USA). The focal point of the MO with respect to the input end of the GRIN rod is adjusted to achieve a nearly collimated output beam and is measured using the CCD camera (Thorlabs, USA) connected to a computer for image acquisition and processing. The vertical and horizontal light beam polarization at the input are oriented along X and Y axes respectively.
Fig. 2. Schematic of the experimental setup to investigate all the SOI effects due to propagation of polarized Gaussian and LG beams through the GRIN rod. BS: 50-50 Beam splitter, M: Mirror, SLM: Spatial light modulator, MO: Microscope objective lens, O/p: Output measurements, CCD: camera. Cross-section of the GRIN shows the location where the input beam is incident. Insets (i) and (ii) correspond to polarimetry and projection / weak measurement setup and interferometry setup.
The phase and polarization characteristics of the output beam are measured via interferometry, polarimetry, orthogonal projection and / or weak measurement method to study all the manifestation of SOI effect in the output beam. The interferometry measurements are carried out by arranging a Mach-Zehnder-like interferometer setup wherein the output beam from the GRIN rod is superposed with a reference beam to extract the phase structure of the output beam. Stokes polarimetry measurements are carried out on the output beam to calculate the S3 component and its behavior under different experimental conditions, as mentioned. The spin phase change arising due to the interaction of light with GRIN rod is extracted by measuring the residual intensity across the orthogonally (to the input SoP) oriented polarization element. For example, for circular polarized input beam the effect due to passing through the GRIN rod is extracted by projecting the output in orthogonal circular SoP to extract information on the residual effects due to the interaction of light with the inhomogeneous-anisotropic medium. Also, as many of the effects arise due to weak interaction between the light beam and the GRIN rod, we carry out standard weak measurement [8,11] protocol to extract the weak S3 component. By rotating the analyzer, we also measure the shift in the beam’s CoI to quantify the interaction strength. Any or all the above-mentioned measurements are carried out to extract the required SOI information for on-axis or off-axis launching of input beam into the GRIN rod. Though we observe small ellipticity in the SoP of the output beam as we move towards the periphery of the rod due to increase in birefringence related effects we ignore the effects and other higher-order effects here, to keep the treatment simple.
4. Results and discussion
4.1 SAM-IOAM interaction
4.1.1 Spin-to-orbital AM conversion
Symmetric, on-axis propagation of right- or left- circularly polarized Gaussian beam through the GRIN rod and projected in the orthogonal SoP results in the generation of optical vortex beams with l = ±2 and is a classic example of spin-to-orbital AM conversion (SOC) of light. The propagation of RCP Gaussian beam through the inhomogeneous medium results in an output optical field that is a superposition of RCP Gaussian beam and LCP Laguerre-Gaussian (LG) beam of topological charge + 2. The situation reverses for LCP input Gaussian beam with the generation of -2 charge LG beam in the output with orthogonal SoP. The output beam-field projected in orthogonal SoP to the input are shown in Fig. 3 (a) and (f). On-axis interference of the projected output beams with a slightly diverging Gaussian beam in the same SoP shows counter clock-wise and clock-wise rotating helical interference patterns (Fig. 3 (b) and (g)), corresponding to l = ± 2 charge optical vortex beam. To extract the phase structure, we perform off-axis interference of the orthogonally projected output beam from the GRIN rod with a Gaussian beam of same SoP. The resulting forklet interference pattern is shown in Fig. 3 (c) and (h). Fourier transform analysis of the fringe pattern is carried out [40,45] to extract the phase information of the output optical field, which is shown in Fig. 3 (d, e) and (i, j), clearly showing counter- and clock-wise rotating helical phase structure of the beam with l = ±2 and the corresponding oppositely oriented forklet pattern respectively for RCP and LCP input beams. The results can be understood as due to parallel transport law and the resulting accumulation of geometric phase, attributed to the SOI of light.
Fig. 3. Orthogonal SoP projection of output beam field, after passing through the GRIN rod, for input (a) RCP and (f) LCP Gaussian beam. On-axis (b) and (g) and off-axis (c) and (h) interferograms of the orthogonally projected beams. Extracted phase from the interferograms for (d,e) RCP and (i,j) LCP input beams.
4.1.2 Orbital-to-spin AM conversion—reciprocal SOC
The reciprocal SOC effect, of converting IOAM (l) of the beam to SAM (σ) is realized by propagating linear vertically polarized IOAM beam with σ = 0 and l = ±1, ±2 and ± 3 through the GRIN rod. Due to inhomogeneity and anisotropy experienced by the paraxial beam propagating in the GRIN rod, the input vertical SoP of the beam, with σ = 0, accumulates l-dependent spin component characterized by measuring the accumulated spin component in the output beam using Stokes polarimetry from which one can see the effect of the conversion process.
On-axis propagation of vertically polarized Gaussian beam (σ = 0 and l = 0) when projected in orthogonal, horizontal SoP gives the 4-lobe intensity pattern as shown in Fig. 1 (a). This confirms that the experimental setup is well aligned. Introducing a QWP oriented at 90° after the GRIN rod, we can measure the S3 component in the output beam by rotating the analyzer. As the S3 component due to inhomogeneity and anisotropy of the GRIN rod are very weak, we carry out weak Stokes measurement by rotating the analyzer by δ = ±5° from the cross position [11]. Using the images captured on the CCD camera we calculate the right and left circular SoP in the output beam. Due to weak anisotropy in the GRIN rod we could measure very small right and left elliptical SoP in the beam cross section. This confirms that the eigen modes of the GRIN rod are not vertical and horizontal but elliptical SoP. We then propagate vertically-polarized IOAM modes with l = ±1, ±2 and ± 3 through the GRIN rod and measure the excess right and left spin components, corresponding to IOAM-to-SAM conversion. Figure 4 shows the experimentally measured S3 Stokes components for different l values of the beam propagating through the GRIN rod. We can clearly see that linearly polarized input beam with σ = 0 has resulted in spatially varying S3 components corresponding to right and left circular SoP indicated by red and blue colors in Fig. 4. The S3 components is independent of the l value due to conservation of total AM in the output beam. We also note that there is a small anti-clock wise rotation in the overall S3 pattern with increasing + l value, which changes direction when + l is changed to -l. We also notice that the S3 components spatially stretches in the transverse plane with increasing |l| value, possibly due to change in the wavefront curvature due to diffraction of the output beam from the GRIN rod.
Fig. 4. Experimentally measured S3 Stokes parameter for different IOAM values (l = ±1, ±2 and ± 3) of input beam with vertical SoP, corresponding to σ = 0. Red and blue color correspond to left and right circular SoP components in the output beam.
4.2 SAM-EOAM interaction
4.2.1 Spin-Hall effect of light
A linearly polarized light beam propagating in an inhomogeneous medium will split in to right- and left- circularly polarized light and deviate away from each other, which in the presence of anisotropy in addition will lead to ellipticity variations as well [46,47]. Known originally as the Magnus effect, this manifestation, in modern terminology is the spin-Hall effect of light [17,30].
We now demonstrate the SOI effect due to off-axis propagation of V-polarized Gaussian beam through the GRIN rod. The out-of-plane (XZ) separation of the input beam into two-lobe Hermite-Gaussian (HG) like beams in the two orthogonal circular SoP is the spin-Hall effect of light [11,17,30]. This effect is due to the position-dependent transverse phase gradient experienced by the right- and left- circular polarization components in a direction perpendicular to refractive index gradient (XY plane) induced by shifting the input beam launch position. To realize this effect in the GRIN rod, the input beam position is moved by 0.7 mm from the on-axis position along the Y-axis in the setup shown in Fig. 2. The output beam projected in the orthogonal (horizontal, H-) SoP shows two-lobe pattern (Fig. 5 (b)). The small magnitude of the spin-Hall shift is amplified and measured by rotating the analyzer (Fig. 2) employing the weak measurement scheme [8,11]. The output beam for the maximum shift positions, shown in Fig. 5 (a) and (c), correspond respectively to the left- and right- circular polarization projection of the output beam-field. The S3 Stokes parameter polarization map of the output beam is shown in Fig. 5 (d) demonstrates the presence of orthogonal CP components (split along X) for linearly polarized input beam. For the weak measurement, the QWP is kept fixed at 90° and the analyzer is rotated from -30° to + 30° through 0° crossed position and a computer-controlled CCD camera kept at 37 cm from the GRIN rod captures the images as a function of the analyzer rotation angle. From the measurements the CoI of the beam is calculated and plotted as a function of analyzer angle (Fig. 5 (e)). From the measurements we calculate a maximum shift of $\approx 341.5\mu m$ for RCP projection and $\approx 344.7\mu m$ for LCP projection.
Fig. 5. (a) Experimentally measured output beam intensity due to spin-Hall effect in the GRIN rod, for vertically polarized input Gaussian beam, shifted by 0.7 mm from on-axis. The transverse shift of the beam centroid and projected at angles, ɛ = (a) - 9°, (b) 0° and (c) +9° of the analyzer, (d) S3 Stokes parameter of the output beam and (e) calculated beam shift as a function of analyzer rotation angle.
4.2.2 Effect of EOAM on SAM—reciprocal SHE
In the SHE of light, the initial plane path deviates and becomes curved depending on the SAM of light. The additional bending of the plane due to curvature of the path is the reciprocal SHE of light, also referred to as Rytov effect [48,49]. In the reciprocal SHE of light the SAM in addition depends on the EOAM of light. Thus, a linearly polarized light beam propagating in an inhomogeneous medium split into right- and left-hand circularly polarized beam and in addition will show an increase in the polarization component and shift position as a function of increasing the curvature of the path. This trajectory dependent variation in the beam polarization results in the accumulation of Berry phase and in this situation with broken symmetry, the effect will also be measured in the plane transverse to the plane of incidence, as with the SHE of light.
The input linearly polarized Gaussian light beam is focused at different points along the y-axis from close to on-axis (y = 0.07 mm) to far-off position of y = 0.7 mm. For each of these input beam positions we measure the S3 Stokes parameters and the maximum beam shift corresponding to weak measurement for crossed polarizer orientation at ± ε, as discussed in the previous section. As can be seen from Fig. 6 (a) to (h) the S3 content in the beam increases with off-axis position, corresponding to the increase in the torsion experienced by the linearly polarized input beam propagating through the GRIN rod. Shown in Fig. 6 (i) is the quadratic increase in the S3 behavior, as can be expected [38]. The corresponding increase in the maximum beam shift position (Fig. 6 (j)), measured using the weak measurement method also shows a similar behavior, confirming that the entire beam structure experiences rotation and diffraction induced elongation of the beam in the transverse direction. We also noticed an increase in polarization ellipticity in the beam cross-section along with the quadratic increase in the SoP of the beam due to increase in the anisotropy in the GRIN rod for off-axis position.
Fig. 6. Experimental measurement to demonstrate the effect of trajectory curvature on SHE of light. (a) – (h) shows the S3 parameter as a function of the input beam position (indicated in the bottom left side of each, in mm) from close to on-axis (0.07 mm) to off-axis (0.7 mm). Graphs (i) and (j) are the data extracted from the S3 Stokes measurement and beam shift from weak measurement, as a function of input beam position.
4.3 IOAM-EOAM interaction
4.3.1 Orbital-Hall effect
It was unified and shown that a spinning paraxial beam of light with IOAM, propagating in a graded-index media also experiences transverse AM dependent deflection akin to the SHE and is known as orbital-Hall effect (OHE) of light [43,44]. The OHE, due to the bending of the light beam trajectory in an inhomogeneous medium is also attributed to the parallel transport law, orbital Berry phase, conservation of total AM and the OOI. This order-of-wavelength shift, independent of the SoP of the light beam, can be amplified and measured using weak measurement method [50] or by using higher-order IOAM beams or by interacting the beam with higher index gradient.
The off-axis propagation of IOAM beams through GRIN rod, due to the refractive index gradient results in l-dependent transverse deflection of the beam due to the interaction of intrinsic and extrinsic OAM of the light beam. This leads to OAM-dependent deformation in the beam structure and the OHE of light. Unlike the SHE, the propagation of IOAM beams through the GRIN rod is susceptible to diffraction induced beam distortions in the medium [43], making the measurement process involved and more susceptible to errors. Ignoring these effects and following closely the SHE of light derivation, the beam shift due to OHE can be calculated by simply changing σ to l in Eq. (4) [10].
The OHE of light is also measured using the setup shown in Fig. 2. The different IOAM beams are launched into the GRIN rod for a fixed input position at 0.6 mm from on-axis position and the output beam intensity is captured on the CCD camera, kept at 38 cm. As the OHE is independent of SAM, we do not use any polarization components in these measurements. From the measured intensity the OHE is calculated from the CoI position for + l and -l (l = 1–5) values projected on the SLM. The experimentally measured output beam intensity profiles for different ± l’s are shown in Fig. 7 (a) – (e) for + l and (f) – (j) for -l respectively. The solid (white) lines are the CoI of reference Gaussian beam and the dashed (white) lines are the CoI for + l and -l values with reference to the Gaussian input beam. As the input l of the IOAM beam is increased, the transverse l-dependent CoI shift increases, seen from the difference between the solid and dashed lines in the figures. The total shift is calculated from the CoI for + l and -l input beams. The measured CoI shift varies approximately linearly with l values (Fig. 7 (k)). Important to note that the symmetric input beam with different IOAM breaks up into tilted HG-like beams in the output, wherein the dark phase-discontinuity lines correspond to the |l| value [51,52] and the orientation of the modes gives information on the +/- l value. More importantly, we note that the beam intensity pattern is clearly asymmetric, indicated by green color dashed ellipses in Fig. 7 (a) – (j) indicating the high-intensity portion in the beam cross-section that arises due to OOI and indicating l-dependent transverse shift of the beam. In addition, we also noticed an in-plane shift in the beam position and an additional l dependent angular shift in the output beam, due to higher-order effects, which will be analyzed further and reported elsewhere.
Fig. 7. Experimentally measured transverse IOAM-dependent CoI shift due to OHE for beams with intrinsic OAM with different (+) and (-) topological charges (a) to (e) and (f) to (j) respectively. The solid (white color) horizontal lines in the figures represent the CoI of reference Gaussian beam and dashed lines are the CoI of corresponding to + l and -l IOAM input beams. The green ellipse indicates the high intensity areas of shifted output OAM beams. (k) transverse OHE beam shift increases linearly with l.
4.3.2 Effect of EOAM on IOAM—reciprocal OHE
Arising due to SOI, the mutual influence of path and polarization of paraxial light beam propagating along a curved trajectory lead to the demonstration of direct and reciprocal spin-Hall effect in the GRIN rod. The orbital-Hall effect of light arises due to shift in the CoI of paraxial beam propagating along a curved trajectory but due to transverse AM-dependent deflection as demonstrated in Section 4.3.1. Equivalent to the reciprocal SHE, working now in the modal (OAM) domain, changing the path and hence the trajectory curvature (EOAM) of the paraxial beam with l = 0, propagating in the inhomogeneous medium should lead to the accumulation of orbital Berry phase and change in the orientation of the output beam field, and appearance of IOAM (l ≠ 0) in the output beam. Additionally, magnitude of the OHE also varies to conserve the total AM of the paraxial beam of light [43,44]. This path dependent appearance of non-zero IOAM (l) beam is termed reciprocal OHE of light and establishes the equivalence between IOAM and SAM of a paraxial beam in the medium, validity of parallel transport law and appearance of geometric Berry phase [43,44]. Thus, propagating a paraxial HG beam with l = 0, we demonstrate that the output beam acquires non-zero l value due to beam rotation and accumulation of orbital Berry phase, by measuring the rotation angle of the output beam and its phase structure.
Experimentally, we demonstrate reciprocal OHE by propagating paraxial HG beam (l = 0) tilted + 45° and -45° with reference to vertical direction along different curved paths through the GRIN rod. Additionally, we also have the freedom to change the phase structure (0-π) of the HG beam using the SLM in the experimental setup shown in Fig. 2. The input HG beam position is changed from on-axis (y = 0 mm) to different off-axis positions and the output beam intensity is measured on the CCD camera, kept at a fixed distance from the GRIN rod. Figure 8 shows the measured output beam intensity, without using polarization component, from which we can see that the output mode rotates as a function of y. The rotation angle, with reference to the input beam, is measured by connecting the maximum intensity points (white line) of the beam as shown in Fig. 8 (a). As can be seen, the output beam also gets distorted due to transverse stretching and curvature experienced due to propagation through the GRIN rod. The beam rotation due to orbital Berry phase accumulation leads to the output beam acquiring non-zero IOAM. This is measured by collimating the output beam and performing single slit diffraction measurement [53] to reveal the hidden phase structure, shown in Fig. 8 (b). The diffraction pattern clearly shows the accumulation of non-zero IOAM due to off-axis propagation of HG beam, a reciprocal OHE. Changing the input HG mode orientation from + 45° to -45° flips the output diffraction pattern hinting at l’s sign reversal. This input HG mode mediated reversal of output beam phase structure is equivalent to σ -dependent generation of IOAM beams for input RCP / LCP Gaussian beam (Fig. 3). The position (y-) dependent beam rotation angle measured for different input HG beam intensity and phase structure plotted in Fig. 8 (c) clearly shows a linear change close to on-axis, which saturates as we move towards the edge of the GRIN rod, due to increase in anisotropy and diffraction-induced beam distortion [7].
Fig. 8. Experimentally measured output beam intensity for + 45° and -45° oriented input HG beam, after propagating through the GRIN rod, as a function of input beam position y. (a) Rotation of output beam intensity as a function of input position, (b) collimated output beam for fixed input beam position of y = -0.56 mm and the corresponding single slit diffraction pattern confirming the presence of phase dislocation and (c) plot of rotation angle as a function of input beam position along ± y direction for two different intensity distribution and phase structure.
5. Summary
We have experimentally demonstrated all the direct and reciprocal spin-orbit interaction effects of light in an inhomogeneous medium like GRIN rod. The parabolic refractive index profile is responsible for the curvature in the propagating beam path through the medium leading to the appearance of anisotropy as well. The complementarity between the Berry phase and the spin-Hall effect due to the mutual influence of the beam trajectory and the SoP of electromagnetic waves in a medium with smooth trajectory is the underlying mechanism for the observations reported. The advantage of our scheme relies on the demonstration of all the direct and reciprocal effects in a single optical system, the GRIN rod with potential to observe higher-order effects. Interferometry, polarimetry and weak measurement methods allowed us to characterize these weak effects accurately. Important to note that the anisotropy arising due to inhomogeneity in the path plays an important role in the SOI effects leading to the observation of reciprocal or anomalous effects as well [54–56]. In addition to addressing several fundamental aspects of SOI, our investigation offers an opportunity to further investigate the propagation of EM waves along non-planar trajectory, a potential platform to manifest the gauge structure of EM waves and consequences [57]. Also, important to note that despite more than two decades of expanding interest in the measurement and applications of direct SOI of light, limited to SOC and SHE, the OHE and the reciprocal effects and their applications are just beginning to be unraveled [58].
Funding
Science and Engineering Research Board (SERB); Department of Science and Technology, Ministry of Science and Technology (DST).
Acknowledgments
The authors thank Science and Engineering Research Board (SERB), Department of Science and Technology (DST), India for financial support to this area of research. PC acknowledges DST-INSPIRE scheme for research fellowship. The authors also thank the reviewers for pointing out issues with reciprocal SOI effects, which lead to further improvement in the data presented here.
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