Physicists Derive Exact Gouy Phase Law for Higher-Order Laser Modes

More than a century ago, the French physicist Louis Georges Gouy noticed something strange about focused light. When a spherical wave passes through its focal point, it picks up an extra bit of phase beyond what the distance traveled alone can explain, a shift he observed as a fringe displacement between the two sides of a focus. That extra phase, now called the Gouy phase, has become one of the quiet workhorses of modern optics, quietly shaping everything from ultrashort laser pulses to quantum information experiments. Yet for all its ubiquity, a fully explicit, analytic account of how the Gouy phase behaves for higher-order laser modes passing through general optical systems has remained surprisingly incomplete. A new paper in the journal Results in Optics by Nazarin Nizam, Dakshin Tillo and J. Solomon Ivan closes that gap with a rigorous derivation that is as elegant as it is practical.

The story begins with the laser itself. In their landmark 1966 analysis, Kogelnik and Li showed that a paraxial light field emerging from a laser cavity, the fundamental Gaussian mode being the canonical example, accumulates an additional phase on free propagation beyond the trivial plane-wave factor that simply tracks the distance traveled. This Gouy phase arises because a focused beam cannot be a perfect plane wave; its transverse profile must spread and reconverge, and that spreading carries a phase cost. Later work by Hariharan and Robinson, and by Feng and Winful, probed the physical origin of the effect, while Simon and Mukunda, and Subbarao, revealed its deeper identity as a geometric phase, the same family of phases that underlies the Berry phase in quantum mechanics. The concept has since been traced in matter waves, surface plasmon polaritons, phonon polaritons and even acoustic waves, and it has found applications in pulse shaping, atom optics and quantum information processing.

What the new work tackles is the higher-order case. Laser beams are not limited to the simple bell-shaped fundamental mode; they can be sculpted into Hermite-Gaussian modes, the characteristic ladder of patterns with nodal lines that emerges whenever a paraxial beam is described in rectangular coordinates. Each such mode is labeled by two mode numbers, nx and ny, counting the nodes along the transverse axes. While earlier researchers had derived analytic Gouy phase expressions for the fundamental Gaussian mode passing through rotationally symmetric first-order optical systems, and had extended the treatment to Bessel-Gauss beams, higher-order modes had only been handled with empirical expressions, notably in the design of a spatial mode multiplexer by Linares and colleagues in 2017. The new paper replaces that empirical shortcut with a complete analytic proof.

The mathematical machinery at the heart of the derivation is the Iwasawa decomposition, a result from the theory of symplectic groups that was brought into optics by Arvind, Dutta, Mukunda and Simon in the mid-1990s. Every rotationally symmetric first-order optical system, meaning any combination of free propagation and thin spherical lenses, is described by a two-by-two ray transfer matrix with unit determinant, whose entries a, b, c and d encode how the system transforms ray positions and angles. The Iwasawa decomposition guarantees that any such matrix can be factored into a sequence of three elementary operations: a lens-like element, a scaling element, and a rotation-like element. In the wave picture, each of these corresponds to a unitary transformation generated by a quadratic Hamiltonian, the same kind of Hamiltonians that govern harmonic oscillators in quantum mechanics.

The key insight is that the Hermite-Gaussian modes are eigenstates of the rotation-like element. When the scale parameter of the decomposition is chosen to match the beam waist width, the rotation-like unitary acts on a Hermite-Gaussian mode simply by multiplying it by a phase factor proportional to nx plus ny plus one. Because the lens-like and scaling elements act on these modes in a straightforward, universal manner, the entire evolution through an arbitrary rotationally symmetric system can be carried out explicitly. The result is a closed-form expression for the Gouy phase picked up by any Hermite-Gaussian mode through any such system: it equals minus nx plus ny plus one times the arctangent of a specific ratio involving the matrix entries b and a and the beam width. For the special case of free propagation, the formula reproduces exactly the familiar Kogelnik-Li result, a reassuring consistency check.

The authors also worked through the complications that real experiments introduce. If the incoming mode already carries a quadratic phase, as happens when it is injected away from its waist plane, the effective ray transfer matrix is modified by a lens-like factor, and the Gouy phase is computed from the updated matrix entries. If the mode is displaced or tilted, which in the formalism corresponds to a unitary displacement operator acting on the state, the analysis shows something remarkable: the Gouy phase itself is untouched. Misalignments and small tilts of the optical components impart a transverse shift, a tilt phase and an additional overall phase to the mode, but the Gouy phase accumulated through the system remains exactly what the ideal, perfectly aligned calculation predicts. The Gouy phase, in other words, is intrinsic to the mode and the system, robust against displacement noise.

Several further structural results fall out of the formalism. All Hermite-Gaussian modes sharing the same value of nx plus ny acquire the same Gouy phase in a rotationally symmetric system, which immediately implies that Laguerre-Gaussian modes, the circular cousins of the Hermite-Gaussian family, are similarly degenerate whenever their indices satisfy the corresponding relation. Composite systems are handled simply by multiplying their ray transfer matrices before applying the phase formula. The phase varies continuously with the ratio b over a, vanishing when that ratio is zero and saturating at plus or minus pi over two in the extreme limits. Two different optical systems described by the same ray transfer matrix impart the same Gouy phase, differing only in propagation phases and an overall pi ambiguity tied to the sign convention of the unitary representation.

Perhaps the most immediately useful outcome is a recipe for engineering relative phases between modes. The authors illustrate with a simple arrangement: a single convex lens of fixed focal length that can slide within a fixed total system distance. As the lens moves, the ratio b over a of the composite system changes, and with it the Gouy phase of each mode. Crucially, the width change, the quadratic phase and the propagation phase acquired are identical across all modes, so the modes retain their identity, while the Gouy phases differ according to their mode numbers. Numerical exploration, using a wavelength of 632.8 nanometers and a 10-centimeter focal length lens, shows that by tuning the input beam width and its quadratic phase, the relative phase between modes can be swept in a controlled fashion as a quantity proportional to nx plus ny plus one, with the lens needing to travel only a few centimeters. This provides a compact, tunable knob for sculpting structured light fields, where superpositions of spatial modes with controlled relative phases are the raw material for optical vortices, mode-division multiplexing and high-dimensional quantum protocols.

The authors are careful about the limits of their result. Rotationally asymmetric systems, built from astigmatic lenses, do not preserve Hermite-Gaussian modes and involve a far richer decomposition with ten parameters instead of three, so extending the Gouy phase concept there demands new care. Even so, the framework reaches beyond spatial beams: the same mathematics applies in the time domain to pulsed laser sources, where space-time duality maps temporal lenses onto spatial ones, and to quantum states of a single mode of radiation, with the reduced wavelength replaced by Planck’s constant. What began as a small anomaly in nineteenth-century fringe patterns thus emerges, in twenty-first-century form, as a precisely controllable resource, one that a single sliding lens can now dial at will.

Subject of Research: Analytical derivation of the Gouy phase acquired by higher-order Hermite-Gaussian modes in rotationally symmetric first-order optical systems

Article Title: Gouy phase for higher-order Gaussian modes

Article References: Nizam, N., Tillo, D., & Ivan, J. S. (2026). Gouy phase for higher-order Gaussian modes. Results in Optics, 25, Article 101176. https://doi.org/10.1016/j.rio.2026.101176

Image Credits: AI Generated

DOI: 10.1016/j.rio.2026.101176

Keywords: Gouy phase, Hermite-Gaussian modes, paraxial optics, ray transfer matrix, Iwasawa decomposition, structured light, laser beams, geometric phase, mode multiplexing, first-order optical systems, quantum optics, spatial light modes