Generalized Gouy Phase in Wave Dynamics

• Generalized Gouy phase is a universal descriptor of phase accumulation in focused and mode-structured waves, unifying optical and accelerator beam dynamics.
• It governs phase evolution in diverse fields including relativistic matter waves, quantum optics, and electron vortex beams through a common mathematical framework.
• Its extensions facilitate practical applications in mode conversion, nonlinear selection rules, and quantum interference across various experimental geometries.

The generalized Gouy phase denotes a family of extensions of the phase anomaly acquired by confined, focused, or mode-structured waves beyond the standard paraxial Gaussian-beam setting. In its canonical optical form, the Gouy phase is $\varphi_G(z)=\arctan(z/z_R)$, but the literature generalizes it to accelerator optics, relativistic matter waves, phase-space interference, multi-photon and biphoton states, vector and spatiotemporal structured fields, and electron vortex beams in magnetic fields. A central unifying result is that the Gouy phase can be written as a phase advance $\int dz/\beta(z)$, placing it on the same footing as the Courant-Snyder phase of beam dynamics and making it a generic descriptor of focused-wave evolution rather than a phenomenon restricted to light beams (Floettmann, 2020).

1. Canonical formulation and unifying definitions

The standard paraxial Gouy phase for a focused beam is

$\varphi_G(z)=\arctan\!\left(\frac{z}{z_R}\right),$

with $z_R$ the Rayleigh length. In higher-order Hermite-Gaussian and Laguerre-Gaussian modes it appears multiplied by the modal order, so that phase accumulation is tied to transverse confinement and mode structure. The key generalization developed in accelerator-optical language is

$\varphi_G(z)=\phi(z)=\int \frac{dz}{\beta(z)},$

where $\beta(z)$ is the beta function of Courant-Snyder theory. In the adapted formalism, the Courant-Snyder phase and the Gouy phase are identical, and optical mode converters used for Laguerre-Gaussian/Hermite-Gaussian transformations are mathematically the same as converters developed for charged-particle beams (Floettmann, 2020).

Several later works extend this basic idea by changing the dynamical variables, the carrier system, or the notion of “mode order.” In relativistic quantum beams the phase depends jointly on longitudinal separation and relative time; in quantum optics it scales with photon number; in biphotons it depends on two correlation-controlled Rayleigh lengths; and in magnetic fields it acquires oscillatory and orbital-angular-momentum-dependent terms (Ducharme et al., 2015).

Context	Generalized phase expression	Distinctive feature
Accelerator-optical unification	$\varphi_G(z)=\phi(z)=\int dz/\beta(z)$	Equivalence of Gouy and Courant-Snyder phase (Floettmann, 2020)
Relativistic Hermite-Gaussian particle beam	$$\phi_{mn}=(1+m+n)\arctan\!\big((\xi_3+c\tau)/(2b)\big)$$	Explicit dependence on space and time (Ducharme et al., 2015)
$N$-photon Fock state	$$|N\rangle_{\ell p;0}\rightarrow e^{-iNkz-iN\Phi_G(z)}|N\rangle_{\ell p;z}$$	Phase scales linearly with excitation number (Hiekkamäki et al., 2022)
Type-I SPDC biphoton	$\int dz/\beta(z)$0	Controlled by two-photon correlations (Brito et al., 2020)
Electron vortex beam in uniform magnetic field	$\int dz/\beta(z)$1	Width oscillation and rotational dynamics (Meng et al., 2024)

This multiplicity of forms suggests that “generalized Gouy phase” is best understood not as a single alternative formula, but as a common phase-advance structure that survives changes of geometry, medium, statistics, and kinematics.

2. Beam dynamics, astigmatism, and nonparaxial optical reinterpretations

A major route to generalization comes from beam-dynamical reformulation. By rewriting Hermite-Gaussian fields using the beta function, the Gouy phase becomes the Courant-Snyder phase advance exactly, not only analogically. For a beam focused at $\int dz/\beta(z)$2, the optical field phase contains the usual $\int dz/\beta(z)$3 term, and the identity

$\int dz/\beta(z)$4

follows from $\int dz/\beta(z)$5 with $\int dz/\beta(z)$6. For astigmatic beams the phase splits into transverse-plane components, and the total mode phase becomes

$\int dz/\beta(z)$7

which is the anisotropic two-dimensional generalization of the same beam-dynamical structure (Floettmann, 2020).

A different reinterpretation comes from the Madelung-Bohm formalism for paraxial propagation. There the field is written as $\int dz/\beta(z)$8, and the Gouy phase appears as the phase contribution required to preserve Gaussian propagation. In this treatment the Ermakov-Lewis invariant is explicitly time dependent even though the Hamiltonian is time independent, and the effective Bohm index of refraction

$\int dz/\beta(z)$9

produces a GRIN-like focusing mechanism. For the Gaussian solution the phase contains the term

$\varphi_G(z)=\arctan\!\left(\frac{z}{z_R}\right),$0

so the Gouy shift emerges as a consequence of envelope dynamics encoded in the invariant and in the effective refractive-index profile (Moya-Cessa et al., 2021).

The notion also extends beyond paraxial one-sided beams to full-aperture waves converging on a focal point from all directions. In three dimensions, the physically regular field is the superposition of converging and diverging spherical waves, yielding a $\varphi_G(z)=\arctan\!\left(\frac{z}{z_R}\right),$1 phase shift through the focus; in two dimensions, the corresponding cylindrical-wave analysis gives a $\varphi_G(z)=\arctan\!\left(\frac{z}{z_R}\right),$2 shift. For sharply localized pulses this distinction is consequential: in 3D the pulse preserves its sharpness up to a sign change, whereas in 2D the Gouy-phase-induced mode transformation reshapes the pulse into a broader, double-humped form (Tyc, 2012).

These formulations place the Gouy phase within a broader class of phase anomalies associated with envelope evolution, anisotropic focusing, and the regularization of fields at caustics or focal points.

3. Matter waves, relativistic particles, and semiclassical caustics

For relativistic quantum particles, exact Bateman-Hillion Hermite-Gaussian solutions of the Klein-Gordon equation yield a spacetime Gouy phase

$\varphi_G(z)=\arctan\!\left(\frac{z}{z_R}\right),$3

with $\varphi_G(z)=\arctan\!\left(\frac{z}{z_R}\right),$4. The phase depends on both the longitudinal separation $\varphi_G(z)=\arctan\!\left(\frac{z}{z_R}\right),$5 and the relative time $\varphi_G(z)=\arctan\!\left(\frac{z}{z_R}\right),$6, and the solutions are form invariant under Lorentz transformations. To recover the nonrelativistic Schrödinger result, the coordinates must satisfy the constraint $\varphi_G(z)=\arctan\!\left(\frac{z}{z_R}\right),$7, under which the relativistic solution reduces to the standard paraxial matter-wave form (Ducharme et al., 2015).

An exact Bateman-Hillion-Gaussian solution of the Dirac equation generalizes the Gouy phase further for relativistic electron beams beyond the paraxial limit. In that setting,

$\varphi_G(z)=\arctan\!\left(\frac{z}{z_R}\right),$8

contains coupled space-time dependence, and after eliminating time one obtains a purely spatial but nonparaxial expression with curved phase fronts. The work relates the total Gouy phase shift to the Berry phase through

$\varphi_G(z)=\arctan\!\left(\frac{z}{z_R}\right),$9

and connects this enhancement to intrinsic spin-orbit coupling and fractional conversion between spin and orbital angular momentum (Ducharme et al., 2018).

In semiclassical strong-field ionization, the generalized Gouy phase appears as a Maslov phase. Classical electron trajectories that cross focal points or caustics acquire the factor $z_R$0, with $z_R$1 the Maslov index. Because Coulomb focusing in three dimensions drives one branch of the holographic electron trajectories through a focal point as it crosses the symmetry axis in momentum space, each zero crossing of $z_R$2 adds a $z_R$3 phase jump. This correction shifts photoelectron interference fringes and is necessary for agreement with time-dependent Schrödinger-equation calculations (Brennecke et al., 2019).

The matter-wave literature also generalizes the Gouy phase to interference between states at different times. For a correlated Gaussian wave packet, the wave function acquires a global Gouy phase

$z_R$4

whereas the cross-Wigner function acquires a Gouy phase difference $z_R$5 because it compares the initial state and its later evolution. This establishes “temporal-like Gouy phases” as relevant to temporal interference and to phase-space reconstructions from double-slit intensity data (Marinho et al., 2023).

A path-dependent version appears in triple-slit interference with exotic trajectories. There the classical and non-classical paths carry different phases, $z_R$6 and $z_R$7, and the Sorkin parameter $z_R$8 depends directly on their difference through the crossed interference terms. For the electron parameters discussed in the study, $z_R$9 is of order $\varphi_G(z)=\phi(z)=\int \frac{dz}{\beta(z)},$0, and at $\varphi_G(z)=\phi(z)=\int \frac{dz}{\beta(z)},$1, $\varphi_G(z)=\phi(z)=\int \frac{dz}{\beta(z)},$2 ns, omitting the Gouy phase difference produces an error of approximately $\varphi_G(z)=\phi(z)=\int \frac{dz}{\beta(z)},$3 in $\varphi_G(z)=\phi(z)=\int \frac{dz}{\beta(z)},$4 (Paz et al., 2015).

4. Quantum optical, multi-particle, and correlated-state generalizations

In quantum optics, the Gouy phase is not restricted to classical field envelopes. For an $\varphi_G(z)=\phi(z)=\int \frac{dz}{\beta(z)},$5-photon Fock state in a transverse Laguerre-Gaussian mode, propagation yields

$\varphi_G(z)=\phi(z)=\int \frac{dz}{\beta(z)},$6

with

$\varphi_G(z)=\phi(z)=\int \frac{dz}{\beta(z)},$7

The phase anomaly is therefore multiplied by the total excitation number. In the reported experiment, two-photon N00N states exhibit a doubled oscillation frequency relative to the classical or one-photon case, and the work shows that this behavior cannot be reproduced by replacing the wavelength with an “effective de Broglie wavelength” $\varphi_G(z)=\phi(z)=\int \frac{dz}{\beta(z)},$8 because the Gouy phase depends nonlinearly on the wavenumber (Hiekkamäki et al., 2022).

The same study analyzes quantum metrological consequences using the quantum Fisher information

$\varphi_G(z)=\phi(z)=\int \frac{dz}{\beta(z)},$9

and for radial-mode N00N states finds an $\beta(z)$0 term in the displacement sensitivity, reflecting Heisenberg scaling. In this setting the generalized Gouy phase becomes both a propagation phase and a metrological resource (Hiekkamäki et al., 2022).

For type-I SPDC biphotons in the double-Gaussian approximation, the two-photon wavefunction develops a Gouy phase upon propagation that depends on the correlation-controlled Rayleigh lengths $\beta(z)$1 and $\beta(z)$2: $\beta(z)$3 The covariance-matrix analysis shows that the logarithmic negativity depends on the Rayleigh lengths but not on the propagation distance $\beta(z)$4. The Gouy phase and the entanglement are thus related through the same geometric parameters. After a thin lens, the focused biphoton Gouy phase remains calculable in closed form and is reported to be in reasonable agreement with the experiment of Kawase et al. (Brito et al., 2020).

These quantum versions replace the single-mode “extra phase through focus” by number-dependent, correlation-dependent, and interference-dependent phase structures. A plausible implication is that the generalized Gouy phase serves as a bridge between transverse-mode geometry and nonclassical resource quantifiers such as entanglement and Fisher information.

5. Structured, astigmatic, vectorial, and spatiotemporal fields

In hybrid structured Gaussian beams, the generalized Gouy phase is explicitly parameterized. The SHEN family is written as

$\beta(z)$5

where $\beta(z)$6 encodes the generalized Gouy phase difference induced by astigmatism and $\beta(z)$7 encodes ellipticity or intrinsic coordinate aberration. For a perfect astigmatic mode converter, $\beta(z)$8; for a general astigmatic system, $\beta(z)$9 can take any value in $\varphi_G(z)=\phi(z)=\int dz/\beta(z)$0. The associated SHEN sphere uses longitude $\varphi_G(z)=\phi(z)=\int dz/\beta(z)$1 and latitude $\varphi_G(z)=\phi(z)=\int dz/\beta(z)$2 to represent the topological evolution of multi-singularity vortex beams, including Hermite-Laguerre-Gaussian and helical-Ince-Gaussian subfamilies (Shen et al., 2018).

Vectorially structured light introduces a relative-order version of the phase. For a superposition of modes of different orders, each component acquires its own Gouy phase $\varphi_G(z)=\phi(z)=\int dz/\beta(z)$3, so the physically relevant quantity is the intramodal phase difference

$\varphi_G(z)=\phi(z)=\int dz/\beta(z)$4

This fractional Gouy phase, interpreted in the work as a geometric-phase difference between spatial modes with different orders under the same unitary transformation, governs propagation variations and revivals of both spatial and polarization structures in non-eigen vector modes (Zhong et al., 2021).

Higher-order spatiotemporal wave packets extend the same logic to nonseparable space-time pulses. There the modal order is a continuous real parameter $\varphi_G(z)=\phi(z)=\int dz/\beta(z)$5, and the Gouy phase becomes

$\varphi_G(z)=\phi(z)=\int dz/\beta(z)$6

with $\varphi_G(z)=\phi(z)=\int dz/\beta(z)$7 for flying-pancake pulses and $\varphi_G(z)=\phi(z)=\int dz/\beta(z)$8 for flying-doughnut pulses. The modal order is strongly coupled to the Gouy phase, producing ultrafast cycle-switching evolution, spatiotemporal self-healing, and sub- or superluminal propagation effects. Temporal revivals occur where

$\varphi_G(z)=\phi(z)=\int dz/\beta(z)$9

and the introduced stretch parameter keeps the Gouy-phase coefficient unchanged while stretching the temporal envelope, thereby decoupling pulse duration from modal order (Yu et al., 3 Nov 2025).

Across these structured-field settings, the generalized Gouy phase is not merely a longitudinal scalar phase. It becomes a tunable parameter of topology, astigmatic aberration, polarization evolution, revival structure, and space-time nonseparability.

6. Mode control, nonlinear conversion, magnetic rotation, cavity QED, and measurement

The Gouy phase can be engineered as an operational degree of freedom. A radial-mode sorter based on accumulated Gouy phases implements the mode-dependent phase shift

$$\phi_{mn}=(1+m+n)\arctan\!\big((\xi_3+c\tau)/(2b)\big)$$0

with $$\phi_{mn}=(1+m+n)\arctan\!\big((\xi_3+c\tau)/(2b)\big)$$1 for $$\phi_{mn}=(1+m+n)\arctan\!\big((\xi_3+c\tau)/(2b)\big)$$2. Using a Mach-Zehnder geometry with lens systems designed through ABCD-matrix analysis, the device sorts Laguerre-Gaussian radial modes; experimentally, visibilities greater than $$\phi_{mn}=(1+m+n)\arctan\!\big((\xi_3+c\tau)/(2b)\big)$$3 were obtained, with theoretical values greater than $$\phi_{mn}=(1+m+n)\arctan\!\big((\xi_3+c\tau)/(2b)\big)$$4, and tests were reported up to $$\phi_{mn}=(1+m+n)\arctan\!\big((\xi_3+c\tau)/(2b)\big)$$5 and $$\phi_{mn}=(1+m+n)\arctan\!\big((\xi_3+c\tau)/(2b)\big)$$6 (Gu et al., 2017).

In nonlinear optics, four-wave mixing in heated rubidium vapor shows that Gouy phase matching acts as a selection rule for total mode order in thick media: $$\phi_{mn}=(1+m+n)\arctan\!\big((\xi_3+c\tau)/(2b)\big)$$7 When the medium is thick, this permits efficient azimuthal-to-radial conversion and radial-to-radial conversion; in the thin-medium regime the selection is relaxed and the output becomes a coherent superposition of radial modes (Offer et al., 2020).

For electron vortex beams in uniform magnetic fields, the generalized Gouy phase of an extended Landau state is

$$\phi_{mn}=(1+m+n)\arctan\!\big((\xi_3+c\tau)/(2b)\big)$$8

and the corresponding mean rotation angle is

$$\phi_{mn}=(1+m+n)\arctan\!\big((\xi_3+c\tau)/(2b)\big)$$9

This framework predicts a broader spectrum of angular frequencies than conventional Landau-state analysis and captures the reversal of rotation direction for negative topological charge, in agreement with experimental parameters and Chebyshev-method simulations (Meng et al., 2024).

In multimode cavity QED, accounting for Gouy phase shifts of nearly degenerate cavity modes changes the form of cavity-mediated atom-atom interactions. The longitudinal phase of a cavity mode is

$N$0

and the resulting interaction kernel is naturally expressed in terms of a complex order parameter. In the confocal limit, the theory yields an emergent $N$1 symmetry corresponding to the phase of the atomic density wave, while finite mode spacing, loss, cloud size, or displacement from the midplane break and lock that phase. The same work proposes restoring the $N$2 symmetry by using two pump fields (Guo et al., 2018).

Free-electron modulation with two counter-propagating Gaussian beams adds another operational generalization. The relevant total Gouy phase is

$N$3

and it enters the ponderomotive interaction seen by the electron. In the reported theory, this phase is crucial for generating comb-shape spectra with similar amplitudes and states with high degree of coherence, and the same logic extends to chirped femtosecond pulses through a generalized focal phase (Zhao et al., 28 Oct 2025).

Direct measurement proposals for matter waves also rely on Gouy-phase engineering. Ramsey interferometry with Rydberg atoms and cavity-induced matter-wave “lenses” is predicted to produce a $N$4 fringe shift when one internal-state component is focused in one transverse direction while the other is not, giving a direct determination of the matter-wave Gouy phase (Paz et al., 2010).

Taken together, these developments establish the generalized Gouy phase as a control parameter for mode conversion, quantum interference, nonlinear selection rules, angular rotation, emergent many-body order, and free-electron wavefunction shaping. The common thread is that focusing, confinement, or modal nonseparability produce a phase advance whose detailed form depends on the underlying dynamical system, but whose physical role remains the same: it governs how structured waves accumulate relative phase under propagation.