Evolution of Geometric Phase of light since 1956: A Catalog Review

The geometric phase of light is a fascinating phenomenon in optics and arises whenever there is a change in the polarization state of light. It is a fundamentally well-established concept and has recently found extensive applications, particularly in the development of geometric phase elements that enable efficient manipulation of light. In this tutorial review, we discuss the evolution of the geometric phase of polarization on the Poincaré sphere, from its inception by Shivaramakrishnan Pancharatnam in 1956 to its recent advances and applications. This review article aims to focus on core papers related to the geometric phase of polarization rather than providing an exhaustive literature survey. In this review, first, we introduced the basic parameters and corresponding parameter spheres involved in the geometric phase of light. Then, we provide an in-depth analysis of geometric phase in polarization modes, spatial modes, vector modes, and electromagnetic fields. A brief discussion of applications of the geometric phase is also provided. The intriguing explanation given in this review can awaken new ideas related to the geometric phase of light and can open new directions in fundamental and applied optics. Finally, the tutorial is structured as a comprehensive catalog of the geometric phase of light.

💡 Research Summary

The paper presents a comprehensive tutorial review of the geometric (Pancharatnam‑Berry, PB) phase of light, tracing its development from Pancharatnam’s 1956 discovery to the most recent applications. It begins by classifying optical phases into dynamical, Gouy, and geometric components, emphasizing that the geometric phase is a topological quantity accumulated when a light beam undergoes a closed evolution in a parameter space such as polarization, propagation direction, spatial mode, or the full electromagnetic field.

The authors first revisit the classic Poincaré sphere representation of polarization. By mapping the complex electric‑field components onto an Argand plane and stereographically projecting onto a unit sphere, any polarization state is described by latitude 2χ (ellipticity) and longitude 2ψ (orientation). Optical elements that modify polarization—wave plates, liquid‑crystal retarders, metasurfaces—are treated as “geometric gears” that trace specific paths on the sphere; the solid angle subtended by the closed path directly yields the PB phase. Experimental verification methods such as Mach‑Zehnder interferometry, polarization state tomography, and digital holographic phase retrieval are discussed.

The review then extends the concept to the orbital degree of freedom. The authors introduce the orbital Poincaré sphere, where Laguerre‑Gaussian (LG) and Hermite‑Gaussian (HG) modes occupy distinct points defined by mode order and orientation. Devices such as q‑plates, spatial light modulators, and mode converters generate closed loops in this modal sphere, producing a spin‑orbit coupled geometric phase that can be interpreted as a solid angle on the orbital sphere.

For vector beams, the polarization and orbital spheres are combined into a higher‑dimensional “composite Poincaré sphere” (tensor product of the two). The total geometric phase of a vector beam is shown to decompose into the sum of its polarization‑derived PB phase and its modal phase, providing a clear framework for understanding spin‑orbit interactions in complex beams.

A recent frontier covered is the geometric phase of the full electromagnetic field. By treating the electric‑magnetic field tensor as a point on a four‑dimensional complex sphere, the authors describe how inhomogeneous, anisotropic, or nonlinear media (e.g., plasmonic interfaces, multilayer structures, nonlinear frequency conversion) generate additional geometric contributions. Experimental approaches include interferometric measurements of phase shifts in nonlinear crystals and near‑field scanning of plasmonic polaritons.

The paper catalogs a wide variety of interferometric configurations used to measure geometric phases, comparing their sensitivity, alignment tolerance, and suitability for different beam types. It then surveys practical applications: metasurface‑based PB elements (geometric lenses, vortex generators, spin‑dependent beam splitters), high‑efficiency polarization‑controlled holography, quantum information encoding via geometric phase, fast optical scanning, optical tweezers with spin‑controlled torque, and phase‑encoded LiDAR/communication systems.

In the concluding section, the authors organize the core literature into a “catalog” and outline future research directions: (1) extending geometric‑phase concepts to nonlinear and quantum regimes, (2) exploiting higher‑dimensional parameter spaces for multi‑degree‑of‑freedom phase control, and (3) developing reconfigurable, real‑time metasurfaces for dynamic phase programming. By systematically mapping parameter spaces to optical devices, the review unifies disparate studies, clarifies underlying mechanisms, and opens new avenues for both fundamental investigations and applied photonic technologies.