Differential Gouy Phase Shifts

Updated 16 August 2025

Differential Gouy Phase Shifts are relative phase differences between wave components caused by differing modal orders, confinement, and propagation geometries.
They influence mode coupling and beam evolution in applications ranging from optical interference to quantum correlation measurements.
Understanding these shifts is key for optimizing nonlinear processes, state engineering, and precision quantum sensing techniques.

Differential Gouy phase shifts refer to the relative phase changes accumulated by distinct components—spatial, temporal, or polarization—of a propagating wave, owing to their different modal structure or propagation geometry. While the standard Gouy phase describes a universal shift for beams passing through a focus, the differential Gouy phase becomes crucial when multiple modes, orders, or physical paths with distinct confinement or evolution contribute to physical observables, affecting phenomena such as pulse reshaping, vector beam evolution, quantum correlations, and interferometric measurements across optical, electron, and matter-wave systems.

1. Fundamental Mechanisms of Differential Gouy Phase Shifts

The Gouy phase arises as an intrinsic phase anomaly in wave propagation, most commonly observed where a beam or wavefront is confined or focused. For a monochromatic scalar beam, the phase shift is generally given by

$\phi_N(z) = (N+1)\arctan\left(\frac{z}{z_R}\right)$

where $N$ is the modal order (e.g., for LG modes, $N=2p+|\ell|$ ) and $z_R$ is the Rayleigh range. For full-aperture spherical waves, the shift is $\pi$ upon passing through the focal point, and for cylindrical (2D) symmetry, the shift is $\pi/2$ (Tyc, 2012).

Differential Gouy phase shifts emerge when comparing phase accumulation between different spatial modes (e.g., LG or HG of varying $p$ or $\ell$ ), different propagation paths (classical vs. non-classical in interference), polarization components (as in polarization-selective phase engineering), or temporal counterparts (as in Hermite–Gauss temporal modes). For two modes of orders $N_1$ and $N_2$ , the relative phase evolution is

$\Delta\phi(z) = (N_2 - N_1)\arctan\left(\frac{z}{z_R}\right)$

as shown for vectorial beams (Zhong et al., 2021), establishing the differential Gouy phase as the effective phase shift governing beam or field evolution.

2. Mathematical Formulations and Physical Realizations

Differential Gouy phases are ubiquitous in systems involving multi-modal superpositions, vectorial structure, entanglement, or complex boundary conditions:

Full-Aperture Focusing: In 3D, the combination of converging and diverging spherical waves introduces a fixed $\pi$ phase difference; in 2D, the analogous combination using Bessel functions yields a fixed $\pi/2$ shift. This impacts the interference, notably reshaping sharply localized pulses (e.g., leading to single $\delta$ -like peaks before the focus and double peaks post-focus) (Tyc, 2012).
Modal Superpositions in Structured Light: In Hermite–Gauss or Laguerre–Gauss mode decompositions, each mode order acquires a Gouy phase proportional to its order, resulting in propagation-dependent mixing and periodic revivals of spatial/polarization structure—enabling design of beams with custom propagation properties (Zhong et al., 2021).
Temporal Gouy Phase: For temporal Hermite–Gauss modes (pulse shaping), the accumulated phase per mode is $(n+1/2)\gamma$ where $\gamma$ is a “fractional” angle set by the temporal ray (ABCD) matrix, critically important in interferometric temporal mode sorting (Horoshko et al., 2023).
Polarization-Selective Engineering: Devices employing polarization-selective astigmatic Gouy phase (e.g., using a pair of cylindrical lenses with engineered birefringence) achieve OAM sign reversal in one polarization, converting homogeneous scalar beams to cylindrical vector beams (CVBs) of arbitrary topological charge (Jia et al., 2021).
Gauge and Geometric Connection: The metaplectic group underpins the algebraic formalism for describing phase evolution under quadratic transformations. The Gouy phase is the geometric (Berry) phase accrued by the symplectic spinor under cyclic evolution in phase space (Fernandes et al., 2013).

3. Impact in Linear and Nonlinear Optical Processes

The differential Gouy phase is pivotal in dictating mode coupling and phase matching conditions:

Four-Wave Mixing (FWM): For LG modes, nonlinear interactions require not only conservation of OAM ( $\ell$ ) but also conservation of “mode order” $N=1+2p+|\ell|$ due to accumulated differential Gouy phases. In a thick medium ( $L \gg z_R$ ), only modes satisfying strict total mode order matching yield efficient conversion, driving unique radial-to-azimuthal mode coupling inaccessible by OAM selection alone (Offer et al., 2020).
Cavity QED and Self-Organization: In nearly-degenerate multimode cavities, the Gouy phase shift of each transverse mode controls atomic self-organization and the emergence or breaking of U(1) symmetry in the order parameter of the atomic density wave. The phase structure can be engineered via cavity design or multi-pump driving, enabling continuous, discrete, or locked symmetry-breaking (Guo et al., 2018).
Non-Interferometric Measurement: Geometric (Pancharatnam–Berry) and Gouy phases can be determined from simple centroid tracking after mode transformations and partial beam blockage, providing robust, high-resolution access to differential phase shifts (Malhotra et al., 2018).

4. Quantum and Matter-Wave Manifestations

Quantum systems inherit the classical Gouy phase but with notably enhanced or modified behavior:

Photon Number and Quantum Gouy Phase: For $N$ -photon Fock states, the phase shift is multiplied by $N$ : the entire state acquires a phase $e^{-iN\Phi_G}$ on propagation, leading to $N$ -fold enhancement of phase sensitivity in N00N-state interferometry. The quantum Gouy phase cannot be mimicked by simply assigning a reduced de Broglie wavelength (Hiekkamäki et al., 2022).
Phase-Space and Nonclassical Correlations: For biphotons generated by SPDC, the Gouy phase acquired upon propagation (or post-diffraction by spatial apertures) governs entanglement as quantified by the logarithmic negativity, since both phase and entanglement depend on the Rayleigh lengths linked to the spatial correlation parameters (Brito et al., 2020, Brito et al., 2021). Measuring the differential Gouy phase (e.g., between two slits of different widths) provides experimental access to phase-space entanglement, with empirical mapping between Gouy shift and entanglement content.
Matter Waves and Squeezing: Confined quantum wavepackets in harmonic traps accumulate Gouy phase both by spreading and (quantum) squeezing. Over a full oscillation, this produces a total Gouy phase of $\pi/2$ due to alternating spreading (each contributing $\pi/4$ ), linking phase evolution directly to genuinely quantum phenomena like squeezing (Oliveira et al., 21 Apr 2025).
Cross-Wigner Function Analysis: For matter waves, cross-Wigner transforms between differently evolved states encode differential Gouy phases as interference terms. Their magnitude and variation govern both spatial and temporal interference, and can be extracted by reconstructive tomographic techniques (Marinho et al., 2023).

5. Electron Vortex Beams and Magnetic Field Effects

Electron vortex beams (EVBs) present additional differential Gouy phase phenomena due to charge and OAM coupling to magnetic fields:

Larmor and Gouy Rotation: In TEM, the rotation of EVB images is governed both by an OAM-independent Larmor rotation (from Zeeman coupling) and an OAM-dependent Gouy rotation, the latter producing a rotation angle proportional to the sign of the OAM (i.e., the $m$ quantum number). The interplay enables unique control and measurement schemes, including field mapping and OAM manipulation not possible in optics (Guzzinati et al., 2012).
Generalized Gouy Rotation in Landau States: For EVBs in uniform magnetic fields, the paraxial Landau mode solutions yield a generalized Gouy phase,

$\Phi_G(z) = N \arctan\left(\frac{z_m}{z_R}\tan\frac{z}{z_m}\right) + \ell \frac{z}{z_m}$

where the arctan term encodes the free-space-like Gouy phase, while the linear term encodes the Larmor rotation. The observed direction reversal for negative $\ell$ demonstrates the fundamental quantum origin of the phase (Meng et al., 3 Jul 2024).

Berry Phase and SOI in Relativistic Beams: Strongly confined or relativistic Bateman-Hillion–Gaussian beams mix spin and orbital angular momentum, producing fractional angular momenta whose magnitude is set by Berry phase contributions. Here, the differential Gouy phase quantifies geometric phase accumulation due to SOI (Ducharme et al., 2018).

6. Interference, Path Integrals, and Nonclassical Trajectories

Differential Gouy phase shifts are critical in interference scenarios:

Nonclassical Paths in Interference: In triple-slit experiments, the Gouy phase acquired along non-classical (looped or exotic) paths differs from that of classical (straight) paths due to different propagation times and spatial confinement. This phase difference directly modulates the Sorkin parameter, which quantifies deviations from standard quantum interference, and neglecting it can lead to errors exceeding 50% in the estimation of non-classical path contributions (Paz et al., 2015).
Double-Slit and Cross-Wigner Approaches: For matter and biphoton waves, measuring intensity and visibility in double-slit or asymmetric double-slit configurations provides access to the differential Gouy phase, enabling indirect but quantitative measurements of underlying quantum correlations (Brito et al., 2021, Marinho et al., 2023).

7. Implications and Applications

The control and understanding of differential Gouy phase shifts have broad consequences:

Quantum Sensing and Metrology: Manipulation of the Gouy phase via state preparation (e.g., squeezing, higher-order superpositions) underpins quantum-enhanced sensing protocols, allowing Heisenberg-scaling sensitivity in displacement and phase measurements (Hiekkamäki et al., 2022, Oliveira et al., 21 Apr 2025).
State Engineering and Quantum Information: Devices exploiting polarization-selective or temporal Gouy phase engineering offer robust tools for creating structured quantum and classical light fields (CVBs, temporal mode sorting) essential for high-dimensional quantum communication, on-chip integration, and metrological applications (Jia et al., 2021, Horoshko et al., 2023).
Beam Shaping, Mode Conversion, and Advanced Imaging: Via FWM or multimode cavity design, careful management of Gouy phase underlies efficient mode conversion between radial and azimuthal components and enables the emulation of quantum liquid crystalline states and spin glasses in many-body cavity QED (Offer et al., 2020, Guo et al., 2018).
Fundamental Physics: The Gouy phase serves as a model system for Berry phase accumulation, geometric holonomy in symplectic Clifford systems, and quantum–classical correspondence through phase-space techniques such as the cross-Wigner transformation (Fernandes et al., 2013, Marinho et al., 2023).

In summary, differential Gouy phase shifts represent a unifying principle underlying phase evolution in multi-modal, multi-path, and multi-degree-of-freedom wave systems. Their quantitative role is essential for understanding and controlling both classical and quantum interference, state manipulation, entanglement evolution, and modal dynamics in optical, electron, and matter-wave platforms. Their exploitation enables new capabilities in precision measurement, quantum communication, and the engineering of exotic wave and many-body states.