Abstract
As a classical or quantum system undergoes a cyclic evolution governed by slow change in its parameter space, it acquires a topological phase factor known as the geometric or Berry phase. One popular manifestation of this phenomenon is the Gouy phase which arises when the radius of curvature of the wavefront changes adiabatically in a cyclic manner, for e.g., when focused by a lens. Here, we report on a new manifestation of the Berry phase in 3D structured light which arises when its polarization state adiabatically evolves along the optical path. We show that such a peculiar evolution of angular momentum, which occurs under free space propagation, is accompanied by an accumulated phase shift that elegantly coincides with Berry’s prediction. Unlike the conventional dynamic phase, which accumulates monotonically with propagation, the Berry phase observed here can be engineered on demand, thereby enabling new possibilities; such as spindependent spatial frequency shifts, and modified phase matching in resonators and nonlinear interactions. Our findings expand the laws of wave propagation and can be applied in optics and beyond.
1 Introduction
One of the many wonders of the quantum world manifests when a charged particle passes around a long solenoid; although the magnetic field is negligible in the region through which the particle passes (outside the solenoid) and the particle’s wavefunction is negligible inside the solenoid, nevertheless the particle’s wavefunction still experiences a phase shift as a result of the enclosed magnetic field [1]. This mysterious interaction—confirmed by various experimental setups [2–10]—is known as the Aharonov–Bohm effect and highlights the role of electromagnetic potentials, φ and A, which were often debated as mere mathematical constructs, in enforcing the principle of locality [11]. Importantly, this phase shift is topological in nature; it does not depend on the shape of the path traversed by the particle but rather its topological invariants. As much as it is profound, however, this quantal phase accumulation descends from a more deeply rooted origin. Notably, in his 1984 seminal work [12], Sir Michael Berry showed that “A quantal system in one eigenstate, slowly transported around a circuit by varying the parameters in its Hamiltonian, will acquire a geometrical phase factor in addition to the familiar dynamical phase”. This additional phase factor is referredto today as the geometric or Berry phase.
The geometric phase is of a fundamental significance as it underpins many physical phenomena [13, 14]. For instance, its classical analog explains the angular displacement observed in the Foucault pendulum [15]—known as the Hannay angle [16], it underlies the Zak phase encountered by Bloch electrons in 1D periodic lattices [17], and manifests in spindependent deformations of optical fields like, for e.g., spin–orbit coupling [18]. In optics, the two main classes of geometric phase are [13, 19]: (a) the spinredirection geometric phase, and (b) the Pancharatnam–Berry phase [20]. The former arises when light with fixed polarization (or more generally; angular momentum) changes its direction in space—a situation encountered in helically wounded optical fibers [21–23] (Figure 1(a))—whereas the latter is typically observed when successively projecting light’s polarization in a cyclic manner using birefringent elements. For example, when light passes through a sequence of polarizing elements, causing its original state of polarization to traverse a cyclic trajectory on the Poincaré sphere, the output beam gains an additional phase shift governed by the topology of the path traversed in polarization space. The curvature of the Poincaré sphere, which visualizes all possible states of polarization, allows this phase factor to be geometrically evaluated as half the solid angle enclosed by the traversed topological path. Generalizations of this rule that apply to nonadiabatic and/or noncyclic topological deformations have also been reported [24, 25]. Notably, the wellknown Gouy phase which accompanies a Gaussian beam as it changes its waist size under focusing also has deep connection with the Berry phase (Figure 1(b)). This additional phase factor arises as the complex radius of curvature of the Gaussian beam is adiabatically cycled in its parameter space, introducing a spread in the transverse momentum and thus a perturbation to the axial propagation constant, which can also be reconciled from the position–momentum uncertainty principle [26, 27]. Other manifestations of the geometric phase in optics are evident in the spin Hall effect of light [28–30], spin–orbit conversion of circularly polarized Gaussian beams via strong focusing [31, 32] or using metasurfaces with locally varying anisotropy [33–35], as well as temporal beating of polychromatic polarized light [36], and more recently, in Young’s double slit experiment invoking polarized vector fields [37]. In addition, higher order manifestations of the geometric phase exist in beams carrying orbital angular momentum [38–41]. Besides its scientific significance, the geometric phase plays a key role in many applications: from precision metrology [42], to high resolution microscopy [43], optical micromanipulation [31, 44, 45], and polarimetry [46, 47].
With recent advances in wavefront shaping, enabled by digital holography [48] and metasurfaces [49, 50], it became possible to sculpt light into complex topologies by structuring all its degreesoffreedom, pointbypoint, at the subwavelength scale, thus enabling new behaviors. For instance, a new class of metaoptics can now perform successive polarization transformations along the optical path, after a single interaction with incident light, thereby modifying its polarization at each plane thereafter [51]. Light topologies of this nature should in principle incur new physical dynamics connected with the geometric phase, as their spin angular momentum traces some path on the Poincarè sphere with propagation. Hence, the propagation dynamics of such beams cannot be solely described by the dynamic phase. Here, we explore this further; using a metasurface with shape birefringent unit cells, we sculpt the amplitude, phase, and polarization of incident light, pointbypoint, transforming it into a quasi diffractionless pencillike beam. Importantly, we allow the spin angular momentum (polarization) of such a beam to evolve, in an adiabatic manner, as a function of the propagation distance (Figure 1(c)). We show that such a peculiar evolution in the angular momentum is accompanied by an additional phase shift which satisfies the criteria of the Pancharatnam–Berry phase and that is different from the familiar dynamic phase accumulated with propagation. Notably, the sign and accumulation rate of this geometric phase factor can be tailored on demand by judiciously designing the polarization transformation carried by the metaoptic. With this degreeoffreedom, one can design the phase gradient along the optical path, leading to new physical behaviors such as shifting the spatial frequency of the beam depending on its input polarization—a consequence of its energy–momentum conservation. In the following, we first revisit the design strategy of our recently polarization metaoptics then examine their underlying geometric phase factor and its physical consequences.
2 Longitudinally variable polarization metaoptics
2.1 Design strategy
Our goal is to construct vector beams with propagationdependent spin angular momentum and then examine the evolution of their geometric phase. Bound by angular momentum conservation laws [52], however, this behavior can be realized only locally; i.e., the global angular momentum across the transverse section of the beam should always be conserved. Here, we adopt the design strategy first introduced in Ref. [51]. Consider a discrete superposition of forward propagating modes with different polarization states and propagation constants (wave vectors). Due to the constructive and destructive interference among these copropagating modes, the polarization of the resulting waveform will be modulated with propagation in space. By properly selecting the weight (amplitude and phase) and polarization state of each mode, the polarization state of the envelope can be precisely controlled along the direction of propagation—see for e.g., Ref. [53] and review article [54]. We take a step further and assign 2by2 Jones matrices as the weighting coefficients for each forward propagating mode [51, 55]. Consequently, the resulting superposition will mathematically take the form of a 2by2 Jones matrix whose 4 elements undergo modulation in space. In essence, this propagationdependent Jones matrix describes a polarization optic whose eigenvalues (retardance) and eigen vectors (fast axis orientation) changes in space. Light incident on such a device will modify its polarization state along the direction of propagation. We chose the Bessel profile as our forward propagating modes; hence our device implements the superposition
The term
The matrixvalued coefficients
To realize the transverse profile
2.2 Results
We consider a polarization metaoptic whose response
As the beam propagates in space it acquires a monotonically increasing dynamic phase. Additionally, the evolution of the beam’s state of polarization with propagation gives rise to another phase factor, modifying the overall phase. To illustrate this, we examine the phase acquired by vector beams with spatially evolving polarization in comparison to a reference beam, with fixed polarization state, propagating for the same distance. In this case, the difference between the propagation phases accumulated by each beam will vanish, thus any relative phase shift will be attributed to the polarization transformation—i.e., a Pancharatnam–Berry phase arising only in one of the two beams. More specifically, we consider the output response of our metaoptic under the three input polarization states: RCP, LCP and xpolarized light. We compare this response to that of a reference device in which
Here, we adopted Pancharatnam’s operational definition [20] which implies that the relative phase shift between two beams of different polarizations is the phase retardation which allows the intensity resulting from their mutual interference to be maximized. Figure 3(d)–(f) show the predicted relative phase shift considering the three input polarizations above. Note that circularly polarized light incident on our metaoptic (i.e., zdependent HWP) would reverse its chirality at the output while gradually accumulating a phase that is negative (for RCP light) or positive (for LCP) as it propagates away from the device. At each zplane, the final polarization state is the same but the path taken on the Poincaré sphere from the initial to the final state is different. The red and blue trajectories on the Poincaré sphere depict the responses of the zdependent metaoptic and the reference device, respectively, where both trajectories coincide at the initial zplane. The geometric phase is equal to half the solid angle enclosed by the two paths. In contrast, linearly polarized light will adiabatically rotate, evolving along the equator of the Poincaré sphere, only encountering a constant (π) phase shift, relative to the reference beam, after crossing the diametrically opposite point on the Poincare’ sphere.
2.3 Significance
Unlike conventional dynamic phase accumulated with propagation, the geometric phase shift observed here can be controlled along the direction of propagation, on demand, by choosing the polarization response,
3 Direct observation of spatially evolving geometric phase
The propagationdependent modification in the spin angular momentum gives rise to an additional spatially evolving Berry phase, as discussed in the previous section. The accumulated Berry phase can be directly measured using a wavefront sensor or by performing an interferometric measurement with a reference Gaussian beam at different propagation distances. Measurements of this nature, however, have their own challenges; the former is limited to low resolution whereas the latter is extremely sensitive to misalignment. Instead, here we perform an alternative interferometric measurement using a single metasurface without the need to include a reference arm beam or a wavefront sensor. To achieve this, we fabricated a metasurface that generates a superposition of two vortex beams of opposite helicity (ℓ = 1 and ℓ = −1). Vortex beams are a class of structured light that carries orbital angular momentum owing to their helical wavefront and onaxis phase singularity, where ℏℓ signifies the OAM per photon [60–65]. When two coherent OAM modes with opposite helicity are superimposed they interfere to produce a petallike structure which in turn rotates clockwise (or counter clock wise) depending on the relative phase shift between the two OAM modes [66], see for e.g. Figure 4(a). The angular orientation of these petal structures provides a direct measure of the relative phase shift between its individual modes; simply by detecting the intensity profile.
More specifically, our metasurface implements superposition of two waveforms (ψ ^{ ℓ = −1} + ψ ^{ ℓ = 1}). The polarization response F ^{ ℓ } of ψ ^{ ℓ = −1} is chosen to mimic an HWP which rotates its fast axis along the zdirection (i.e., akin to the response in Figure 3), whereas the polarization response of ψ ^{ ℓ = 1} is set as an HWP whose fast axis is fixed. When illuminated by a plane wave, the metasurface will produce two copropagating vortex modes of opposite helicity, ℓ = −1 and ℓ = 1, creating petallike interference patterns like the ones in Figure 4(a). Since only one of the two waveforms (ψ ^{ ℓ = −1}) changes its polarization state with propagation, a relative Pancharatnam–Berry phase shift will arise between ψ ^{ ℓ = −1} and ψ ^{ ℓ = 1} which can be inferred from intensity measurements on a CCD. To demonstrate full control, we designed the polarization behavior of ψ ^{ ℓ = −1} to mimic an HWP whose axis slowly rotates in one direction (counter clockwise, CCW) over one space region and then rotates back (clockwise, CW), as a function of propagation distance, as illustrated by the blue arrows in Figure 4(b). By design, we chose the reversal in this adiabatic rotation to occur at the plane z = 16 mm. Figure 4(c) shows how a conventional HWP responds to circularly polarized light; reversing its chirality at the output while imparting a geometric phase that is twice the angle of the fastaxis. This behavior is spindependent. Similarly, when illuminated by right handed circularly polarized (RCP) light, our metasurface produces copropagating vortices with left hand circular polarization (LCP), reversing the input chirality as expected from an HWP, while rotating its polarization adiabatically with propagation. This transformation allows ψ ^{ ℓ = −1} to accumulate a Berry phase factor besides its propagation phase. In contrast, ψ ^{ ℓ = 1} only accumulates the usual propagation phase with no Berry phase. Therefore, any rotation in the resulting petal structure will serve as a direct observation of the Berry phase factor accumulated by ψ ^{ ℓ = −1}, given that the dynamic phase difference accumulated with propagation is cancelled out. A device that can generate this petallike profile is shown in Figure 4(d) which exhibits optical micrographs and SEM images of a metasurface (924 μm in diameter).
The measured intensity profile at the output of the metasurface is shown in Figure 5(a) in response to input RCP light. The arrows depict the orientation of the rotating petals and θ denotes the polarization response of ψ ^{ ℓ = −1}; namely the fast axis orientation of its HWP with respect to the horizontal axis (blue arrows in Figure 4(b)). Note how the petal structure rotates in the CCW direction then stops and reverses its sense of rotation to the CW direction, suggesting a variable phase shift between ψ ^{ ℓ = −1} and ψ ^{ ℓ = 1}. When the same metasurface is illuminated by the orthogonal (LCP) polarization, the petal structure still rotates but the sense of rotation at each location is reversed, as shown in Figure 5(c). We attribute this rotation to the Pancharatnam–Berry phase accompanying the polarization transformation of ψ ^{ ℓ = −1}. To reconcile this, note that at each zplane the polarization state of input RCP light becomes LCP at the output. The two copropagating modes ψ ^{ ℓ = −1} and ψ ^{ ℓ = 1} undergo this polarization transformation via two different paths on the Poincaré sphere, as illustrated in Figure 5(b): (i) the blue paths (signified by different longitudes on the Poincaré sphere) are the trajectories traversed by ψ ^{ ℓ = −1} and (ii) the red path is the fixed trajectory of ψ ^{ ℓ = 1}. The z = 0 plane lies at the focus of a 4f optical system after the metasurface (see Figure 2(e)) where both the red and blue trajectories coincide. As the waveform propagates, the polarization of ψ ^{ ℓ = −1} and ψ ^{ ℓ = 1} remain the same but the solid angle between the red and blue trajectories (given by 4θ) increases adiabatically. In this case, a variable Berry phase factor is accumulated, only by ψ ^{ ℓ = −1}, and hence the petal structure is rotated. After a longer propagation distance, precisely at z = 16 mm (as per our design), these dynamics are reversed, the solid angle between the blue and red trajectories progressively decreases, and the petal structure eventually retains its initial orientation. Evidently, this response is spindependent; it reverses its topology depending on the chirality of incident light. In both cases, the additional Berry phase factor can be geometrically evaluated as half the solid angle between the red path and the blue path on the Poincaré spheres of Figure 5(b) and (d), which illustrate both the case of input RCP (top) and LCP (bottom).
To quantify the accumulated phase, we measured the angular orientation of the rotating petal structure at each zplane under the two input polarizations RCP and LCP. We achieved this by tracking the petal orientation using a robust image processing algorithm which tracks the center of mass of each lobe and estimates their tilt angle. Figure 5(e) depicts the result of this analysis. Under RCP illumination, the petal structure rotates in the CCW direction, acquiring a negative Berry phase factor which is accumulated at a linear rate along the optical path. At z = 16 mm, the rotating petal stops then reverses its sense of rotation as well as the slope by which the Berry phase is accumulated. This picture is mirrored for the case of input LCP light, as shown in Figure 5(f). For a petal structure composed of vortex modes ±ℓ, the acquired Berry phase value is equal to −2σℓθ as depicted on the right axis for the plots, where θ denotes the fast axis orientation of HWP1 (which also coincides with the magnitude of the petal’s angular orientation) and σ is the polarization handedness. As the beam reverses its sense of rotation, at z = 16 mm, it experiences angular acceleration/deceleration—a behavior that is captured by the flat valley and peak in the measured and simulated results of Figure 5(e) and (f), respectively, but not seen in their target response (which neglects these effects). Note that the measured orientation deviates from the simulation towards the edges of the region of interest (i.e., at z = 9 and 22 mm). One possible reason for this discrepancy, besides fabrication tolerances, is that we are dealing with apertured Bessel beams in which a small contribution of the beam’s angular momentum (stored in its outer most rings) is truncated and thus does not contribute to the propagation dynamics. This perturbs the angular rotation, especially at the edges of the propagation range, where the contributions from the beam’s peripherals become more significant. One can mitigate this discrepancy by extending the aperture size (metasurface diameter) and/or by including more Bessel terms in Eq. (1) to better approximate the target behavior.
The phase gradient along z translates to an effective momentum which perturbs the k _{ z } component of the wavevector, thus modifying the beam’s size (see also Figure 3(g)–(i) and associated discussion). Owing to the energymomentum conservation, such perturbation manifests as a shift in the transverse spatial frequencies of the output waveform. Figure 5(a) confirms this behavior; under RCP illumination the transverse beam size is slightly larger over the space region z > 16 mm (where the Berry phase gradient is positive). This effect is reversed under LCP illumination, where the beam’s dimensions are larger over the region z < 16 mm. Therefore, judicious design of the polarization transformation provides a new degreeoffreedom for tailoring the phase response along the optical path. It is worth noting that an analogous effect has been previously observed in timedomain; a coherent Gaussian beam experiencing an adiabatic evolution in its polarization state as a function of time also accumulates a timedependent linear Berry phase which translates to a temporal frequency shift [67].
4 Discussion and outlook
To the best of our knowledge, we reported the first direct observation of a longitudinally evolving Pancharatnam–Berry phase under freespace propagation. We examined a new class of polarization metaoptics which implements a superposition of Bessel beams with different cone angles, each weighted by a different Jones matrix, allowing the spin angular momentum (polarization state) of the ensemble to be tailored atwill along the optical path. These polarization transformations are accompanied by a propagationdependent geometric phase factor which, unlike the monotonically accumulated propagation phase, can be designed on demand along the direction of propagation. For e.g., in Figure 5(e) we presented a scenario in which the phase gradient (with respect to z) can be negative over one space region and positive over another.
Here, we paid particular attention to Pancharatnam–Berry phase accompanying the polarization transformation of Bessel beams. This has been done by designing the response of our metasurface to mimic an HWP whose fast axis rotates by an angle of θ along the optical path, giving rise to a phase factor of −2σθ. Besides the HWP profile, our approach can realize other polarization responses and trajectories on the Poincaré sphere. To demonstrate this, we considered another device that mimics the function of a longitudinally variable quarterwave plate thus modifying the chirality—i.e., the spin angular momentum—of incident light along the optical path. More details on this device can be found in Supplementary Figure 2. More generally, one can, in principle, implement a metasurface that performs a propagationdependent rotation of the entire spatial coordinate system (not only the polarization state) by designing the longitudinal response F ^{ ℓ } to mimic the 2by2 rotation matrix R(θ(z)). In this case, the accumulated Berry phase becomes a function of the total angular momentum and is given by −(σ + ℓ)θ, as predicted by Bliokh for the 2D case [38]. A metasurface profile of this nature, composed of an asymmetric 2by2 matrix, however, cannot be implemented using the single layer metasurface deployed in this work as it requires elliptical form birefringence for the metasurface unit cells. This requirement can be achieved by using cascaded or bilayer metasurfaces. We reserve the demonstration of these higher order Berry phases [39, 40] and their analysis to other future work.
Controlling the spin angular momentum and Berry phase, as demonstrated in this work, can inspire new directions in science and technology. It redefines basic rules of wave propagation and points towards a new route to tailoring the phase gradient along the optical path. This can have significant impact on nonlinear interactions and can enable more compact cavity designs and photonic devices [68]. Given our choice of Bessel functions as the OAM modes, our devices generate pencillike beams characterized by a nondiffracting and selfhealing behavior [56] which are also desirable in micromanipulation and free space optical communications. Furthermore, our work enables topologically complex states of light which in turn can lead to many new phenomena in quantum and classical optics [69]. Besides their potential application in light–matter interaction and freespace communications, the compact form of our devices enables their integration in laser cavities, defining new rules for the phase matching condition, and generating new topologically complex combinations of SAM and OAM states of light at the source [70–72]. Lastly, the multidisciplinary nature of angular momentum across different fields may inspire related research efforts in the areas of microfluidics, acoustics, and electron beams, to name a few.
Funding source: Natural Sciences and Engineering Research Council of Canada (NSERC) http://dx.doi.org/10.13039/501100000038
Award Identifier / Grant number: PDF5330132019
Funding source: Office of Naval Research (ONR) http://dx.doi.org/10.13039/100000006
Award Identifier / Grant number: N000142012450
Funding source: National Science Foundation (NSF)
Award Identifier / Grant number: ECCS2025158

Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

Research funding: A. H. D. acknowledges financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC) under grant number PDF5330132019. This work was performed in part at the Harvard University Center for Nanoscale Systems (CNS); a member of the National Nanotechnology Coordinated Infrastructure Network (NNCI), which is supported by the National Science Foundation under NSF award no. ECCS2025158. Additionally, financial support from the Office of Naval Research (ONR), under the MURI program, grant no. N000142012450, and from the Air Force Office of Scientific Research (AFOSR), under grant no. FA955501910135, is acknowledged.

Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
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