Geometric (Pancharatnam–Berry) Phase in Polarization

Most of the phase a light wave accumulates is dynamical — it is the wave's frequency multiplied by the time it spends, or its wavevector times the distance it travels. But there is a second, subtler kind of phase that depends on nothing about speed or duration and everything about the path taken through the space of states. Take a beam of polarized light and carry its polarization slowly around a closed loop — say linear, to circular, to a different linear, back to the start — and the wave returns with an extra phase that no dynamical bookkeeping predicts. Its size is fixed purely by the geometry of the loop: it equals half the solid angle the loop encloses on the Poincaré sphere. This is the geometric phase of polarized light, and the Indian physicist S. Pancharatnam discovered it in 1956, a full twenty-eight years before Michael Berry gave the idea its general quantum-mechanical form. The geometric phase is one of physics' great unifying threads. The very same "the phase equals the enclosed solid angle" statement governs the precession of a Foucault pendulum, the Aharonov–Bohm effect, the quantized Hall conductance, and a spin carried adiabatically around a loop in parameter space. The general theory — the adiabatic geometric phase written as a loop integral of a connection, equal to the flux of a curvature — belongs to quantum mechanics and is developed in the companion article aq-qm-berry-phase; its classical counterpart, the Hannay angle exemplified by the Foucault pendulum, belongs to mechanics and is owned by aq-mech-hannays-angle. Optics owns the polarization instance, and it is the most tangible of all: the states live on a sphere you can draw, the loops are sequences of polarizers and wave plates you can put on a bench, and the phase — half the enclosed area — you can read out by interference. Modern "flat optics" turns this into hardware: metasurfaces and liquid-crystal q-plates that steer, focus, and shape light not by thickness and curvature but by imprinting a designed geometric phase across a wavefront.

Key takeaways

Beyond the ordinary dynamical phase ( $\omega t$ or $kz$ ), a wave carried around a closed loop in its space of states picks up a geometric phase set only by the loop's geometry.
For polarized light the states are points on the Poincaré sphere, and Pancharatnam (1956) showed the geometric phase equals half the solid angle the loop encloses: $\gamma = \tfrac12\Omega$ .
A concrete realization: a half-wave plate rotated by angle $\alpha$ imprints a geometric phase $2\alpha$ on a circularly polarized beam.
The general adiabatic version — Berry's connection and curvature, the phase as a loop integral equal to a solid angle — is owned by aq-qm-berry-phase; the classical Hannay angle (Foucault pendulum) by aq-mech-hannays-angle.
Geometric-phase optics (q-plates, Pancharatnam–Berry metasurfaces) shapes wavefronts by imprinting phase geometrically rather than through optical path length.
Key idea: carry a polarization state slowly around a closed loop on the Poincaré sphere and it returns with a purely geometric phase equal to half the enclosed solid angle, $\gamma = \tfrac12\Omega$ — independent of how fast or by what dynamics the loop was traversed.

The core insight

The geometric (Pancharatnam–Berry) phase is the phase a wave acquires from the shape of its path through state space, over and above the dynamical phase. For polarization the state space is the Poincaré sphere, and Pancharatnam's rule is that a closed loop of states imprints a phase $\gamma = \tfrac12\Omega$ , half the solid angle enclosed. It is anholonomy — a failure to return to the starting phase after a round trip — and it is the optical face of the same geometric phase that governs Berry's adiabatic phase in quantum mechanics and Hannay's angle in classical mechanics.

Where this lives today

The geometric phase has moved from a subtlety of crystal optics to a design principle for hardware. Pancharatnam–Berry metasurfaces — arrays of sub-wavelength antennas or nanopillars, each rotated to a designed local angle — imprint an arbitrary phase profile $2\alpha(x,y)$ on a wavefront and so act as ultrathin, flat lenses, holograms, and beam deflectors, a fast-growing branch of "flat optics." Q-plates, liquid-crystal devices with a spatially varying optic axis, convert spin angular momentum to orbital angular momentum and generate optical vortices for high-capacity communications and super-resolution microscopy. In photonic quantum information the geometric phase offers gates that are intrinsically robust to certain noise, because the phase depends on the loop's geometry rather than on dynamical details. And the same mathematics — anholonomy, a connection and its curvature — links optics to the Aharonov–Bohm effect, the Foucault pendulum, and the topological band theory behind the quantum Hall effect and topological insulators.

The physical picture

Picture the Poincaré sphere, whose points are all the polarization states: the equator carries the linear states, the two poles the left- and right-circular states, and everything between the ellipses. Now transport a polarization slowly around a closed loop on this sphere — a triangle of three states, say, connected by the "shortest turns" a sequence of ideal polarizers or wave plates would make. When the state comes home, the light is back to its original polarization, but its phase is not: it has slipped by an amount equal to half the area (solid angle) the loop swept out. Enclose no area — a loop that backtracks on itself — and there is no geometric phase; enclose an octant of the sphere and the phase is a quarter turn. The phase is written on the geometry alone.

Figure

The punchline: a geodesic triangle of three mutually orthogonal-ish states on the Poincaré sphere encloses a solid angle Ω = π/2 sr (one octant); the geometric phase is γ = Ω/2 = 45°

Why half the solid angle, and not the full area? Because the phase Pancharatnam defined is a property of the amplitude — the wavefunction-like Jones field — while Poincaré-sphere points are built from that amplitude through quadratic, intensity-like combinations (the Stokes parameters). A state that travels once around the sphere therefore corresponds to the amplitude travelling only halfway — the same two-to-one, spinor relationship that flips a spin-½ by a sign under a $2\pi$ rotation, and the reason a loop enclosing a hemisphere ( $\Omega = 2\pi$ ) delivers a phase of $\pi$ . The octant triangle above, with its three right-angle corners, encloses one-eighth of the sphere, $\Omega = 4\pi/8 = \pi/2$ steradians, and delivers a geometric phase of $\pi/4$ , a clean $45^\circ$ .

Figure

A geometric-phase element: rotating a half-wave plate by angle α imprints phase 2α on a circular input (and flips its handedness) — the basis of q-plates and PB metasurfaces

The second figure is the working device. A half-wave plate flips a circular state to the opposite handedness; rotate the plate physically by an angle $\alpha$ and, on top of the handedness flip, the transmitted beam acquires a phase of exactly $2\alpha$ — twice the mechanical rotation. Crucially, nothing about the thickness or optical path of the plate changed; only its orientation did, so the phase is geometric. Tile a surface with tiny half-wave elements each rotated by a designed local angle $\alpha(x,y)$ and you imprint an arbitrary phase profile $2\alpha(x,y)$ across the wavefront — a lens, a hologram, a vortex generator — built entirely from geometry. That is the principle of the q-plate and the Pancharatnam–Berry metasurface.

The core equations

The central statement is Pancharatnam's: the geometric phase of a cyclic evolution of polarization is half the solid angle the state traces on the Poincaré sphere.

\gamma = \frac{1}{2}\,\Omega \qquad (1)

where: $\gamma$ — geometric (Pancharatnam–Berry) phase acquired over one closed circuit; $\Omega$ — solid angle (steradians) enclosed by the circuit of states on the Poincaré sphere. The phase is signed by the sense of circulation and is completely independent of how fast or by what physical elements the loop is traversed — it is geometry, not dynamics. The total phase a wave carries splits into two pieces, and disentangling them is the whole point.

\phi_{\text{total}} = \phi_{\text{dyn}} + \gamma, \qquad \phi_{\text{dyn}} = \frac{2\pi}{\lambda}\,nL \qquad (2)

where: $\phi_{\text{dyn}}$ — dynamical phase, from optical path length $nL$ ; $\gamma$ — geometric phase, from the loop's geometry. The dynamical part is what ordinary optics tracks; the geometric part is invisible to path-length accounting and shows up only when states are cycled and compared by interference. The solid angle of a spherical polygon is its spherical excess — the amount by which its interior angles overshoot the flat-geometry total.

\Omega = \left(\sum_i \theta_i\right) - (k-2)\pi \qquad (3)

where: $\theta_i$ — interior angles of a geodesic $k$ -gon on the unit sphere. For a triangle ( $k=3$ ), $\Omega = \theta_1+\theta_2+\theta_3 - \pi$ ; three right angles give $\Omega = 3(\pi/2) - \pi = \pi/2$ , one octant, and hence $\gamma = \pi/4 = 45^\circ$ . The concrete device relation ties a mechanical rotation to a geometric phase.

\gamma = 2\alpha \quad\text{(rotated half-wave plate, circular input)} \qquad (4)

where: $\alpha$ — the physical rotation angle of a half-wave plate; the transmitted (handedness-flipped) circular beam gains geometric phase $2\alpha$ . This is Eq. (1) in disguise: rotating the plate carries the state around a loop on the Poincaré sphere enclosing solid angle $\Omega = 4\alpha$ , so $\gamma = \tfrac12\Omega = 2\alpha$ .

Derivation APancharatnam's phase as half the enclosed solid angle

The goal is to establish the central result — that transporting a polarization state around a closed geodesic loop on the Poincaré sphere imprints a geometric phase equal to half the enclosed solid angle — and to make clear why it is geometric (path-shape only) rather than dynamical. We build a loop from Pancharatnam's rule for comparing the phases of two non-orthogonal polarization states and sum around a spherical triangle.

Step 1. Adopt Pancharatnam's connection: two non-orthogonal polarization states are "in phase" when their superposition has maximum intensity, which fixes the relative phase of any two nearby states. Chaining this comparison along a path defines how phase is transported as the state moves on the sphere. Why: without a rule for comparing the phases of different polarizations there is no way to say the state "came back with an extra phase"; Pancharatnam's maximum-intensity criterion supplies exactly that connection.

Step 2. Carry the state around a closed geodesic triangle $A \to B \to C \to A$ (each leg the shortest turn between two states). Along each geodesic leg the Pancharatnam phase is stationary, so no phase accrues on the legs — all of it comes from the turning at the corners. Why: geodesics are the "parallel-transport" paths on the sphere, so the phase is stored entirely in the loop's global shape, not in the individual legs — the hallmark of a geometric quantity.

Step 3. Sum the corner contributions; the total equals half the spherical excess of the triangle, $\gamma = \tfrac12\big[(\theta_A+\theta_B+\theta_C)-\pi\big] = \tfrac12\Omega$ , the solid angle enclosed. Why: the spherical excess is exactly the solid angle a geodesic triangle subtends, so the accumulated phase is $\tfrac12$ the enclosed area — Pancharatnam's result, Eq. (1).

Step 4. Note the factor of one-half: Poincaré-sphere points are quadratic (Stokes-parameter) combinations of the amplitude, so one full circuit of the sphere is only half a circuit of the amplitude — the same $\tfrac12$ that gives a spin-½ its $\pi$ sign-flip under a $2\pi$ rotation. Why: this is why an octant ( $\Omega = \pi/2$ ) gives $45^\circ$ and a full hemisphere ( $\Omega = 2\pi$ ) gives $\pi$ , a sign flip — the geometric phase is a half-solid-angle, a genuinely optical fingerprint of the deeper Berry phase. *Verified: the spherical-excess area $\Omega = 3(\pi/2)-\pi = \pi/2$ for a right-angle triangle and the resulting $\gamma = \Omega/2 = 45^\circ$ are checked in verify_aq-optics-pancharatnam-berry-phase.py.*

Go deeper · Derivation B · 1 minA rotated half-wave plate imprints twice its angle

Derivation BA rotated half-wave plate imprints twice its angle

The goal is to prove, from Jones calculus alone, that rotating a half-wave plate by a mechanical angle $\alpha$ adds a geometric phase $2\alpha$ to a circularly polarized beam (while flipping its handedness) — the concrete engine of geometric-phase optics. We write the rotated-plate operator and act on a circular state.

Step 1. A half-wave plate with its fast axis at angle $\alpha$ is $J(\alpha) = R(\alpha)\,\mathrm{diag}(1,-1)\,R(-\alpha)$ , where $R$ is the $2\times2$ rotation by $\alpha$ . Why: a rotated element is its unrotated form conjugated by rotations into and out of the plate's frame — standard Jones bookkeeping.

Step 2. Act on a right-circular input $\hat{e}_+ = \tfrac1{\sqrt2}(1,i)^{\mathsf T}$ . Carrying out the matrix product gives an output proportional to $e^{-i2\alpha}\,\hat{e}_-$ — the opposite circular state, multiplied by a pure phase $e^{-i2\alpha}$ . Why: the half-wave plate must flip handedness (that is what a half-wave does to circular light), and the only freedom left is an overall phase, which the algebra fixes at $\mp2\alpha$ .

Step 3. Read off the phase: rotating the plate by $\alpha$ multiplied the (flipped) output by $e^{\mp i2\alpha}$ , a geometric phase $\gamma = 2\alpha$ set by the plate's orientation, not its thickness. Why: the plate's retardance ( $\pi$ ) never changed, so this phase cannot be dynamical — it is purely the geometric phase of Eq. (4), the loop on the sphere having area $4\alpha$ .

Step 4. Generalize to a surface: let the local fast-axis angle vary as $\alpha(x,y)$ and the imprinted phase is $2\alpha(x,y)$ — a designable wavefront (lens, grating, vortex) built from orientation alone. Why: this is why q-plates and Pancharatnam–Berry metasurfaces can be thin and flat yet do the work of curved, thick optics — the phase is geometric. *Verified: the rotated-HWP Jones product mapping $\hat{e}_+ \to e^{-i2\alpha}\hat{e}_-$ (handedness flip + phase $2\alpha$ ) is checked symbolically in verify_aq-optics-pancharatnam-berry-phase.py.*

Limiting cases

Case	Result	Meaning
Loop encloses no area	$\Omega = 0 \Rightarrow \gamma = 0$	a path that backtracks stores no geometric phase
Octant triangle (three right angles)	$\Omega = \pi/2 \Rightarrow \gamma = 45^\circ$	the standard worked circuit
Hemisphere ( $\Omega = 2\pi$ )	$\gamma = \pi$	a sign flip of the amplitude — the spin-½ analogue
Rotated half-wave plate	$\gamma = 2\alpha$	mechanical rotation ↦ geometric phase, the q-plate
Adiabatic quantum spin in a field loop	$\gamma = -\tfrac12\Omega$ (spin-½)	the general Berry phase — aq-qm-berry-phase

Common trap“The geometric phase is just some extra optical path length”

It is not — and conflating the two erases what makes it useful. Dynamical phase is $\tfrac{2\pi}{\lambda}nL$ : it grows with how far and how slowly the wave travels, and it is what every thickness-and-index calculation already tracks. The geometric phase depends on none of that — not on the wavelength, not on the plate thickness, not on the traversal speed — only on the shape of the loop the state traces in its space of states. You can prove it is separate by traversing the loop faster or slower, or with thicker elements: the dynamical phase changes, the geometric phase does not. This is exactly why Pancharatnam–Berry optics is achromatic-by-design in a way ordinary lenses are not, and why a metasurface can imprint a phase profile by rotating sub-wavelength elements rather than by piling up path length. Treat the geometric phase as "hidden path length" and you will predict the wrong wavelength dependence and miss the whole point of flat optics.

Worked examples

1 · The octant circuit: 45° from pure geometry (confidence builder)

A polarization state is carried slowly around a closed geodesic triangle on the Poincaré sphere whose three corners are mutually orthogonal directions — a horizontal linear state, a right-circular state, and a diagonal linear state — so that each corner is a right angle. Find the solid angle enclosed and the geometric phase acquired.

Step 1. The triangle is a geodesic (great-circle) triangle with three right-angle corners, $\theta_A = \theta_B = \theta_C = 90^\circ = \pi/2$ .

Step 2. Solid angle from the spherical excess, Eq. (3): $\Omega = (\pi/2 + \pi/2 + \pi/2) - \pi = 3\pi/2 - \pi = \mathbf{\pi/2\ \text{sr}}$ — exactly one octant of the sphere ( $4\pi/8$ ).

Step 3. Geometric phase from Eq. (1): $\gamma = \tfrac12\Omega = \tfrac12(\pi/2) = \pi/4 = \mathbf{45^\circ}$ .

Step 4. Appreciate what did not enter: no wavelength, no plate thickness, no traversal time. The $45^\circ$ is fixed by the triangle's area alone, and reversing the sense of circulation would give $-45^\circ$ . An octant circuit gives $\Omega = \pi/2$ sr and a geometric phase of exactly $45^\circ$ .

2 · A rotating half-wave plate as a phase shifter (exam level)

A right-circularly polarized beam passes through a half-wave plate. The plate is rotated in its own plane by $\alpha = 90^\circ$ . What geometric phase does the beam acquire, what happens to its handedness, and how large a Poincaré-sphere loop does this correspond to?

Step 1. Geometric phase from Eq. (4): $\gamma = 2\alpha = 2 \times 90^\circ = \mathbf{180^\circ}$ . A quarter-turn of the plate has imprinted a half-wave of phase — geometrically, not by any change in path length.

Step 2. Handedness: a half-wave plate always flips circular light, so the output is left-circular regardless of $\alpha$ ; the rotation only sets the phase $2\alpha$ on that flipped state.

Step 3. Poincaré-sphere picture: as the plate rotates by $\alpha$ , the state's image sweeps a loop enclosing solid angle $\Omega = 4\alpha = 4(90^\circ) = 360^\circ = 2\pi$ sr — a hemisphere — and $\gamma = \tfrac12\Omega = \pi = 180^\circ$ , consistent with Step 1.

Step 4. This is exactly how a Pancharatnam–Berry deflector works: array many such rotated plates with $\alpha$ increasing linearly across the aperture, and the imprinted phase ramp $2\alpha(x)$ steers the beam — a grating with no grooves. A $90^\circ$ plate rotation gives a $180^\circ$ geometric phase, flips the handedness, and corresponds to a hemisphere ( $\Omega = 2\pi$ ) on the sphere.

Physics of the portrait

The phase is named for the young physicist who found it, working alone in India two decades before the idea swept through quantum mechanics.

Origins · 1 minPancharatnam and how this was discovered

Pancharatnam

1934–1969 · Bangalore & Oxford

Shivaramakrishnan Pancharatnam was an Indian physicist and a nephew of C. V. Raman, in whose Bangalore institute he began working on crystal optics as a teenager. In a remarkable 1956 paper, "Generalized Theory of Interference and Its Applications," he asked how to compare the phases of two beams of different polarization — a question with no obvious answer, since orthogonal states never interfere — and gave the maximum-intensity rule that now bears his name. From it he found that carrying a polarization around a closed cycle of states leaves behind an extra phase equal to half the solid angle enclosed on the Poincaré sphere: the geometric phase, discovered twenty-eight years before Michael Berry's 1984 general derivation of the adiabatic phase in quantum mechanics. Berry himself, on learning of the earlier work, named the effect the Pancharatnam–Berry phase and wrote appreciatively of his predecessor. Pancharatnam went to Oxford in the 1960s to work at the Clarendon Laboratory, but died suddenly in 1969 at just thirty-five, his deepest result unrecognized in his lifetime. It is now the foundation of geometric-phase optics, from q-plates to the metasurfaces reshaping how lenses and holograms are built.

History

1892

— Poincaré introduces the sphere that represents all polarization states

1956

— Pancharatnam finds the geometric phase of polarized light, equal to half the solid angle on the Poincaré sphere

1984

— Berry derives the general adiabatic geometric phase in quantum mechanics

1986

— Tomita and Chiao observe the geometric phase of light in a helically wound optical fibre

1987

— Berry identifies Pancharatnam's 1956 phase as an early instance of the adiabatic geometric phase

Going further

Texts. Berry, "Quantal phase factors accompanying adiabatic changes," Proc. R. Soc. Lond. A 392 (1984) · Pancharatnam, "Generalized theory of interference," Proc. Indian Acad. Sci. 44 (1956) · Hecht, Optics, 5th ed. (Poincaré sphere and polarization) · Shapere & Wilczek, Geometric Phases in Physics. Cross-track: aq-qm-berry-phase (adiabatic Berry connection and curvature) and aq-mech-hannays-angle (the classical Hannay angle) develop the general theory.

Provenance. A verifier should confirm: (1) $\gamma = \tfrac12\Omega$ , the geometric phase as half the enclosed solid angle, with the octant circuit $\Omega = 3(\pi/2)-\pi = \pi/2 \Rightarrow \gamma = 45^\circ$ (Pancharatnam 1956; Berry 1984, 1987); (2) the rotated-half-wave-plate result $\gamma = 2\alpha$ with handedness flip, from Jones calculus. The general adiabatic phase — the loop integral of Berry's connection equal to the flux of the Berry curvature, and its solid-angle form for a spin — is developed in aq-qm-berry-phase (recalled here by name, not re-derived); its classical counterpart, the Hannay angle (Foucault pendulum), in aq-mech-hannays-angle. Historical dates (Poincaré 1892, Pancharatnam 1956, Berry 1984, Tomita–Chiao 1986) are standard.

Prerequisites. Stokes Parameters and the Poincaré Sphere · Circular Polarization · Wave Plates and Retarders · Jones Calculus. Leads to. Polarization Aberrations and Vector Beams · geometric-phase (flat) optics.

How is the optical geometric phase related to Berry's quantum phase?

They are the same phenomenon in two settings. Berry's 1984 result, developed in aq-qm-berry-phase, says that a quantum state carried adiabatically around a closed loop in parameter space acquires a geometric phase equal to a loop integral of the Berry connection, which by Stokes' theorem is the flux of the Berry curvature through the loop — and for a spin in a magnetic field that flux is half the solid angle the field traces. Pancharatnam's 1956 optical result is exactly this for polarization: the "parameter space" is the Poincaré sphere, and the geometric phase is half the enclosed solid angle. Pancharatnam found the polarization instance 28 years earlier; Berry generalized it and named the combined effect. The classical mechanical cousin — the Hannay angle, illustrated by the Foucault pendulum's precession — is owned by aq-mech-hannays-angle. Optics owns the most concrete version: the states are points on a sphere you can draw and the loops are wave plates on a bench.

Why is it half the solid angle and not the whole thing?

Because the Poincaré sphere is built from quadratic combinations of the field amplitude (the Stokes parameters are intensities and intensity differences), so one full trip around the sphere corresponds to the amplitude going only halfway. The geometric phase is a property of the amplitude, so it accumulates half the sphere's solid angle. This is the very same factor of two that gives a spin-½ its famous behaviour under rotation: a loop enclosing a hemisphere ( $\Omega = 2\pi$ ) produces a phase of $\pi$ , a sign flip of the amplitude. The half is not a convention; it is the signature that polarization states are spin-½-like objects represented on a sphere.

Can I actually measure a geometric phase, since a global phase is usually unobservable?

Yes — you measure it by interference, comparing a beam that went around the loop with a reference that did not, or by splitting a beam so the two arms traverse different loops. Pancharatnam's original argument was itself interferometric: his rule for when two polarizations are "in phase" is the maximum-intensity condition of their superposition. In practice, geometric phases are read out with interferometers, with polarization gratings that convert the phase into a measurable deflection, or through the beam steering and focusing that a metasurface's phase profile produces. The phase is global for a single beam in isolation, but the moment you can compare two paths — which any interferometer or any spatially varying element does — it becomes fully observable.

What is the difference between the geometric phase and the ordinary phase from a wave plate?

An ordinary wave plate inserts a dynamical retardance between two polarization components — a phase from the different optical path lengths the fast and slow axes present, which depends on the plate thickness and the wavelength. The geometric phase is different: it is the overall phase a state acquires from being carried around a closed loop of states, and it depends only on the loop's geometry, not on thickness or wavelength. A rotated half-wave plate shows both at once: its half-wave retardance is dynamical (it flips the handedness), but the $2\alpha$ phase from its orientation is geometric. The clean test is to change the thickness or wavelength: the dynamical part shifts, the geometric part does not.

What makes q-plates and Pancharatnam–Berry metasurfaces so useful?

They imprint phase by geometry rather than by path length, which unlocks two things ordinary optics cannot easily do. First, a flat, thin element can carry an arbitrary phase profile: rotate sub-wavelength half-wave elements to local angles $\alpha(x,y)$ and the wavefront gains $2\alpha(x,y)$ , so a nanometre-thin metasurface acts as a lens, hologram, or vortex generator. Second, because the phase is tied to the orientation pattern, the same element couples polarization to spatial structure — a q-plate converts spin angular momentum into orbital angular momentum, spinning a plane wave into an optical vortex. This spin–orbit conversion and the freedom to draw any phase map are why geometric-phase optics underlies flat lenses, structured-light generation, and high-capacity optical communication.

Does the geometric phase require the loop to be traversed slowly (adiabatically)?

In Berry's original quantum formulation, yes — adiabaticity keeps the system in a single instantaneous eigenstate so that the geometric phase is cleanly defined. Pancharatnam's optical version is more forgiving: because a polarization state passing through ideal polarizers and wave plates is projected exactly onto the next state, the "loop" can be traversed by discrete optical elements without any slowness requirement, and the half-solid-angle phase still emerges. Later work (Aharonov–Anandan) removed the adiabatic condition from the quantum case too, showing the geometric phase is a property of the closed path in state space itself, not of how slowly it is walked. So for polarization optics you can build the loop from a handful of elements on a bench and still read out the full geometric phase.