Spatiotemporal Gouy Phase

• Spatiotemporal Gouy phase is a propagation-induced phase shift where the phase evolution depends on both spatial mode order and time or frequency dynamics.
• It manifests in systems ranging from relativistic matter waves to structured optical fields, enabling effects such as mode conversion, resonance shifts, and Lorentz-covariant phase evolution.
• Understanding this phase aids advanced applications in interferometry, beam shaping, and quantum state manipulation by tailoring phase matching and intermodal interference.

Searching arXiv for recent and foundational papers on spatiotemporal Gouy phase across matter waves, structured light, and quantum optics. arXiv search: "spatiotemporal Gouy phase" The spatiotemporal Gouy phase denotes Gouy-phase phenomena in which propagation-induced phase evolution is not adequately described as a purely spatial waist-crossing anomaly, but instead depends explicitly on mixed space–time variables or couples transverse spatial structure to temporal, spectral, or frequency-domain dynamics. In relativistic matter-wave theory, this dependence appears through the 4-position of the beam waist; in dispersive and structured optical systems, it appears as an intermodal phase governing unitary evolution, resonance shifts, and mode conversion; and in quantum and nonlinear optics it organizes correlations between spatial and temporal degrees of freedom (Ducharme et al., 2015, Tang et al., 30 Aug 2025, Barros et al., 2020).

1. Foundational definitions and mathematical forms

For paraxial Gaussian and vortex beams, the Gouy phase is the extra phase shift acquired through focus relative to a plane wave. In vortex beams described by Laguerre-Gaussian modes, the standard expression is

$$\Phi_G(z)=-(2n+|m|+1)\arctan\!\left(\frac{z}{z_R}\right),$$

with dependence on the radial index $n$ and the absolute value of the OAM quantum number $m$. For Hermite-Gaussian modes of transverse order $S_j=m_j+n_j$, the corresponding phase is

$$\phi^j_{mn}(z)=-(S_j+1)\arctan\!\left(\frac{z}{z_0^{(j)}}\right).$$

These formulas already contain the key structural feature that persists in later spatiotemporal generalizations: the phase is mode-order dependent, so different modal components accumulate different propagation phases (Guzzinati et al., 2012, Barros et al., 2020).

A relativistic extension replaces purely longitudinal dependence by mixed space–time dependence. For exact Hermite-Gaussian Klein–Gordon beam solutions, the relativistic Gouy phase is

$$g_{mn}(\xi_3,\tau)=(1+m+n)\arctan\!\left(\frac{\xi_3+c\tau}{2b}\right),$$

where $\xi_3$ is the longitudinal coordinate relative to the focal point and $\tau$ is the time relative to the focal time. In Bateman–Hillion Dirac-beam solutions, the analogous expression is

$$g_{lp}=(1+|l|+2p)\arctan\!\left[2\kappa(\xi_3+\xi_0)\right].$$

In both cases, the Gouy phase depends jointly on longitudinal position and time, rather than on propagation distance alone (Ducharme et al., 2015, Ducharme et al., 2018).

Spatiotemporal optical modes admit a further decomposition into global and intermodal contributions. For spatiotemporal Hermite-Gaussian modes propagating in isotropic dispersive media, the total Gouy phase can be written as

$$\psi_{mn}(z)=\frac{N+1}{2}\sigma(Z)+\frac{l}{2}\delta(Z),$$

where the extermodal Gouy phase $n$0 depends only on total mode order, while the intermodal Gouy phase $n$1 governs observable reshaping:

$n$2

Here $n$3 is the ellipticity and $n$4 the group velocity dispersion. This suggests that “spatiotemporal Gouy phase” is best understood not as a single formula but as a family of mode-order-dependent propagation phases defined on mixed space–time or space–frequency manifolds (Tang et al., 30 Aug 2025).

2. Relativistic matter waves and covariant beam theory

A central development in the subject is the construction of exact relativistic beam solutions with explicit focal 4-position. In the Klein–Gordon treatment of relativistic quantum particles, exact Hermite-Gaussian solutions are Bateman–Hillion solutions with modified phase factors that make the focal point explicit in spacetime. Their envelopes depend on the combination $n$5, and the resulting Gouy phase is form invariant under Lorentz transformations. The same work shows that correspondence with Schrödinger solutions requires the constraint

$n$6

which ties spatial propagation to a single physical time variable (Ducharme et al., 2015).

The Dirac-beam extension adds spin and nonparaxial structure. The Bateman–Hillion solution to the Dirac equation yields a scalar Laguerre-Gaussian factor whose Gouy phase depends on $n$7, so the beam evolution is explicitly spacetime dependent. In the paraxial regime, the phase reduces to

$n$8

and the Gouy phase fronts are flat. Beyond the paraxial regime, tightly focused beams acquire curved Gouy phase fronts, and the total Gouy phase shift exceeds the usual $n$9 by an amount related to the Berry phase,

$m$0

The same analysis connects this increase to intrinsic spin-orbit coupling and fractional angular momenta in relativistic electron beams (Ducharme et al., 2018).

Matter-wave Gouy phase was also proposed for direct interferometric observation before the electron-vortex measurements. In the Ramsey-interferometry proposal using Rydberg atoms, a collimated atomic beam focused cylindrically in one transverse direction acquires a Gouy phase shift of $m$1, detected as a Ramsey-fringe shift,

$m$2

The proposal also states that Gouy-phase differences between transverse modes can implement rotations in a mode basis, motivating the use of atomic spatial modes as q-dits (Paz et al., 2010).

3. Electron vortex beams, magnetic lenses, and generalized Gouy rotation

Electron vortex beams provided the first experimental observation of Gouy phase for quantum matter waves. In transmission electron microscopy, truncation of a vortex beam breaks cylindrical symmetry and converts Gouy-phase differences into a directly visible image rotation. For a truncated beam of topological charge $m$3, the rotation is

$m$4

so the sense of rotation depends on the sign of $m$5. OAM-balanced superpositions such as $m$6 and $m$7 do not exhibit Gouy-induced rotation because each component acquires the same magnitude Gouy phase. The same experiment also observed an OAM-independent Larmor rotation,

$m$8

and showed that Gouy and Larmor rotations can add or subtract depending on the OAM sign. This behavior was identified as unique to electron vortex beams and without an optical counterpart, because Larmor rotation occurs only for charged particles (Guzzinati et al., 2012).

A later generalization addressed electron vortex beams in uniform magnetic fields beyond the fixed-frequency picture of conventional Landau states. In this framework, the relevant modes are extended Landau states with a periodically oscillating beam width. Their generalized Gouy phase is

$m$9

with $S_j=m_j+n_j$0, and the observable rotation angle becomes

$S_j=m_j+n_j$1

This broader spectrum of angular frequencies captures reversal of rotation direction for negative topological charge and unifies Gouy, Landau, and Larmor regimes. Numerical simulations using the Chebyshev method were reported to validate the analytic prediction (Meng et al., 2024).

These charged-particle results establish an important distinction. In free-space optics, the Gouy phase primarily describes the evolution of a focused neutral-wave mode. In electron optics, the Gouy phase coexists with Zeeman and magnetic-confinement dynamics, so the experimentally observed angular evolution is often a composite of Gouy and magnetic effects rather than a pure optical analogue (Guzzinati et al., 2012, Meng et al., 2024).

4. Temporal Gouy phase and phase-space formulations

A temporal analogue of the Gouy phase appears when Hermite-Gauss temporal modes undergo temporal focusing. In an interferometric sorter built from $S_j=m_j+n_j$2 Mach–Zehnder interferometers, the first $S_j=m_j+n_j$3 temporal Hermite-Gauss modes can be sorted by adjusting the accumulated temporal Gouy phase in one arm using a fractional Fourier transform implemented by a time lens. The mode-order-dependent phase is

$S_j=m_j+n_j$4

with

$S_j=m_j+n_j$5

For two interferometers, the work reports a theoretical lower bond on the cross-talk probability of 5.5% in sorting Schmidt modes of a photon pair generated in spontaneous parametric downconversion (Horoshko et al., 2023).

A phase-space formulation based on the cross-Wigner transform sharpens the distinction between global and relative Gouy phases. For a correlated Gaussian matter-wave packet under free evolution, the wavefunction acquires a global Gouy phase

$S_j=m_j+n_j$6

whereas the cross-Wigner function acquires a Gouy phase difference

$S_j=m_j+n_j$7

The same framework extends to double-slit evolution, where the cross-Wigner phase difference becomes a spatiotemporal Gouy phase difference depending on the evolution times and initial correlations. The results suggest that temporal like-Gouy phases are important for an accurate description of temporal interference (Marinho et al., 2023).

A complementary interpretation comes from the Madelung–Bohm formalism for paraxial propagation. Writing the propagated Gaussian field as $S_j=m_j+n_j$8 yields a phase

$S_j=m_j+n_j$9

so the Gouy phase is

$$\phi^j_{mn}(z)=-(S_j+1)\arctan\!\left(\frac{z}{z_0^{(j)}}\right).$$0

Within this treatment, the effective Bohm index of refraction generates a GRIN medium that produces the focusing needed for the Gouy phase (Moya-Cessa et al., 2021).

5. Nonlinear optics, biphotons, and quantum-state evolution

In nonlinear cavities, the Gouy phase couples transverse mode order to spectral structure. For a type-I optical parametric oscillator below threshold, the accumulated round-trip Gouy phase for transverse order $$\phi^j_{mn}(z)=-(S_j+1)\arctan\!\left(\frac{z}{z_0^{(j)}}\right).$$1 is

$$\phi^j_{mn}(z)=-(S_j+1)\arctan\!\left(\frac{z}{z_0^{(j)}}\right).$$2

and the total round-trip phase is

$$\phi^j_{mn}(z)=-(S_j+1)\arctan\!\left(\frac{z}{z_0^{(j)}}\right).$$3

As a result, beat notes and comb lines are shifted by the difference in transverse mode order, and multiple interleaved combs corresponding to different spatial orders are generated simultaneously. The work identifies the Gouy phase as the parameter linking spatial and frequency degrees of freedom in realistic OPO cavities and relates this to hybrid multipartite entanglement in spatiotemporal modes (Barros et al., 2020).

In four-wave mixing, Gouy phase matching governs conversion between orthogonal modal dimensions. For Laguerre-Gauss modes with order

$$\phi^j_{mn}(z)=-(S_j+1)\arctan\!\left(\frac{z}{z_0^{(j)}}\right).$$4

the thick-medium regime enforces both OAM conservation and total mode-order conservation,

$$\phi^j_{mn}(z)=-(S_j+1)\arctan\!\left(\frac{z}{z_0^{(j)}}\right).$$5

This explains radial-to-azimuthal and azimuthal-to-radial conversion that would otherwise appear counterintuitive. In the thin-medium regime, Gouy phase matching is relaxed and many radial modes contribute coherently (Offer et al., 2020).

For type-I SPDC biphotons in the double-Gaussian approximation, the propagating two-photon wavefunction develops a Gouy phase

$$\phi^j_{mn}(z)=-(S_j+1)\arctan\!\left(\frac{z}{z_0^{(j)}}\right).$$6

with Rayleigh lengths $$\phi^j_{mn}(z)=-(S_j+1)\arctan\!\left(\frac{z}{z_0^{(j)}}\right).$$7 and $$\phi^j_{mn}(z)=-(S_j+1)\arctan\!\left(\frac{z}{z_0^{(j)}}\right).$$8. The logarithmic negativity is independent of propagation distance but depends on the same Rayleigh-length ratio, so the biphoton Gouy phase and entanglement are both Rayleigh-length-related. The work further reports reasonable agreement with the experimental data of D. Kawase et al. for a focused biphoton (Brito et al., 2020).

Spatiotemporally structured pumps extend Gouy-phase control from state diagnosis to state synthesis. In SPDC pumped by spatiotemporal structured light, the Gouy phase difference

$$\phi^j_{mn}(z)=-(S_j+1)\arctan\!\left(\frac{z}{z_0^{(j)}}\right).$$9

enters the effective phase-matching term, allowing the pump spatial structure to define the phase-matching function when signal and idler are projected onto high-order spatial modes. The method is presented as an all-optical Gouy phase matching scheme capable of generating both spectrally uncorrelated and high-dimensional spectrally entangled photon pairs in a standard periodically-poled crystal (Montenegro et al., 17 Jun 2026).

Quantum Gouy phase also appears at the level of photon-number states. When an $$g_{mn}(\xi_3,\tau)=(1+m+n)\arctan\!\left(\frac{\xi_3+c\tau}{2b}\right),$$0-photon Fock state occupies a single mode, propagation through focus yields

$$g_{mn}(\xi_3,\tau)=(1+m+n)\arctan\!\left(\frac{\xi_3+c\tau}{2b}\right),$$1

so the Gouy phase is accrued $$g_{mn}(\xi_3,\tau)=(1+m+n)\arctan\!\left(\frac{\xi_3+c\tau}{2b}\right),$$2 times faster. Two-photon N00N states were observed to accrue the phase twice as fast as classical or single-photon counterparts. The same work explicitly rejects a common misconception: the enhanced phase sensitivity cannot be reproduced by replacing the field by a classical beam with effective de Broglie wavelength $$g_{mn}(\xi_3,\tau)=(1+m+n)\arctan\!\left(\frac{\xi_3+c\tau}{2b}\right),$$3 or by simply increasing mode order (Hiekkamäki et al., 2022).

6. Structured-light dynamics, SU(2) geometry, and higher-order spatiotemporal wave packets

Spatiotemporal Gaussian modes in dispersive media admit an SU(2) description in which propagation is a unitary rotation within a fixed-order subspace. For spatiotemporal Laguerre-Gaussian modes, the propagation dynamics are generated by a conserved quantity, and the rotation angle is exactly the intermodal Gouy phase $$g_{mn}(\xi_3,\tau)=(1+m+n)\arctan\!\left(\frac{\xi_3+c\tau}{2b}\right),$$4. The phase depends on the ellipticity of the wave packets and the group velocity dispersion, and its propagation law falls into three regimes: zero dispersion, normal dispersion, and anomalous dispersion. In anomalous dispersion, the non-monotonic behavior induces both distortion and revival of the intensity distribution, establishing a phase-locked mechanism analogous to the Talbot effect (Tang et al., 30 Aug 2025).

Higher-order spatiotemporal wave packets generalize this picture by introducing a continuous modal order parameter strongly coupled to the Gouy phase. For these modes, the Gouy-phase coefficient is $$g_{mn}(\xi_3,\tau)=(1+m+n)\arctan\!\left(\frac{\xi_3+c\tau}{2b}\right),$$5 for FP pulses and $$g_{mn}(\xi_3,\tau)=(1+m+n)\arctan\!\left(\frac{\xi_3+c\tau}{2b}\right),$$6 for FD pulses, so increased modal order produces more rapid ultrafast cycle-switching evolution. The same framework introduces a stretch parameter that stretches the temporal envelope while keeping the Gouy-phase coefficient unchanged, yielding stretch invariance: pulse duration can be tuned without shifting temporal-revival positions or altering the phase or group-velocity laws. The reported dynamics include ST self-healing and sub- or super-luminal propagation (Yu et al., 3 Nov 2025).

Related structured-light systems make the phase directly visible. In SU(2) structured beams whose spatial wave packets follow caustic trajectories, the evolution of caustic-linked wave packets directly visualizes both geometric phase and Gouy phase without interferometers or beam truncation. In vectorially structured light, propagation variations and revivals of spatial and polarization structure are mediated by fractional Gouy phases, described as geometric-phase differences between spatial modes with different orders under a same unitary transformation (Li et al., 17 Nov 2025, Zhong et al., 2021).

Quantum correlations can also be engineered through Gouy-phase-designed structured propagation. In spontaneous parametric down conversion driven by a pump superposition

$$g_{mn}(\xi_3,\tau)=(1+m+n)\arctan\!\left(\frac{\xi_3+c\tau}{2b}\right),$$7

the relative Gouy phase causes a self-splitting and recombining pump profile, and this structure is transferred to the joint two-photon probability distribution. The resulting propagation was described as implementing a Mach–Zehnder-like interferometer and was used to observe heralded single-photon interference and two-photon NOON state interference (Junior et al., 29 Apr 2026).

A recurrent misconception is that the Gouy phase is only a spatial correction associated with a monochromatic beam waist. The literature instead supports several nonexclusive meanings: a Lorentz-covariant spacetime phase in relativistic matter waves, a temporal focusing phase for Hermite-Gauss temporal modes, an intermodal SU(2) rotation angle in spatiotemporal Gaussian-mode spaces, and a phase-matching resource linking spatial structure to spectral or correlation dynamics in nonlinear and quantum optical systems (Ducharme et al., 2015, Horoshko et al., 2023, Tang et al., 30 Aug 2025, Montenegro et al., 17 Jun 2026).