Laguerre GaussianEdit

Laguerre-Gaussian beams are a fundamental family of light modes that arise when solving the paraxial wave equation in cylindrical coordinates. Named for their mathematical underpinning in Laguerre polynomials and their Gaussian envelope, these beams carry orbital angular momentum and exhibit rich, tunable intensity patterns. They have become essential tools in optics and photonics, with wide-ranging applications from high-capacity communications to precise micromanipulation.

Laguerre-Gaussian beams, often abbreviated as LG modes, form a complete set of solutions to the paraxial approximation of the Helmholtz equation. They are characterized by two integer indices: a radial index p (nonnegative) and an azimuthal index l (which can be any integer). The azimuthal index l is directly tied to the beam’s orbital angular momentum, giving each photon an angular momentum of lħ. The radial index p determines the number of radial nodes in the intensity pattern. This dual-index structure makes LG modes especially versatile for encoding information and shaping light in three dimensions.

Mathematical formulation

The electric field amplitude of a Laguerre-Gaussian mode E_p^l(r, φ, z) is typically written in its paraxial form as a product of a Gaussian envelope with a Laguerre polynomial modulation and phase factors. In standard notation:

E_p^l(r, φ, z) ∝ (w0 / w(z)) ⋅ (√2 r / w(z))^{|l|} ⋅ L_p^{|l|}(2 r^2 / w^2(z)) ⋅ exp(- r^2 / w^2(z)) ⋅ exp(- i k r^2 / (2 R(z))) ⋅ exp(- i (2p + |l| + 1) ζ(z)) ⋅ exp(i l φ)

where: - r, φ are cylindrical coordinates in the transverse plane, - z is the propagation direction, - w(z) is the beam radius, w0 is the waist, and z_R = π w0^2 / λ is the Rayleigh range, - R(z) is the radius of curvature of the wavefronts, - ζ(z) is the Gouy phase, given by arctan(z/z_R), - L_p^{|l|}(x) is the generalized Laguerre polynomial.

The term exp(i l φ) imparts an azimuthal phase dependence that gives the beam its orbital angular momentum. The Laguerre polynomials L_p^{|l|} control the radial structure, while the Gaussian envelope ensures the beam remains well-behaved under propagation. For reference, LG modes reduce to simpler Gaussian-like forms in special cases, and they are connected to other mode families such as Hermite–Gaussian beams through mode-conversion techniques.

Mode indices and physical meaning

l (azimuthal index): an integer that sets the helical phase structure exp(i l φ) and determines the orbital angular momentum per photon, lħ. Nonzero l values produce a phase singularity at the beam center and a characteristic doughnut-shaped intensity profile when p ≥ 0.
p (radial index): a nonnegative integer that specifies the number of radial nodes in the intensity distribution. Increasing p adds more rings to the beam’s cross-section.

LG modes form a complete, orthogonal set in the paraxial regime, enabling mode-division multiplexing and the faithful representation of arbitrary field distributions. The ability to encode information in the OAM degree of freedom has been explored extensively in optical communication and quantum information contexts.

Optical properties

Intensity patterns: For l ≠ 0, the central intensity is suppressed due to a phase singularity, producing doughnut-like profiles with p determining the number of bright rings.
Phase structure: The azimuthal phase term exp(i l φ) yields a helical wavefront, a hallmark of orbital angular momentum in light.
Gouy phase: The accumulated phase shift ζ(z) influences the propagation and interference behavior of LG modes, particularly in systems with focusing elements.
Paraxial propagation: LG beams are solutions within the paraxial approximation and are most accurate when the beam is near-axis and not strongly divergent. Nonparaxial corrections can be incorporated for tightly focused or highly divergent fields.
Degeneracy and superposition: Modes with the same total order (2p + |l|) share Gouy-phase behavior and can be superposed to form more complex field patterns.

Generation and detection

Generation methods: LG modes can be produced by imparting the appropriate azimuthal phase to a beam, using devices such as spiral phase plates, spatial light modulators, or reflective diffraction gratings. Polarization-to-OAM conversion with devices like q-plates also enables efficient generation from other beam forms.
Mode converters: Cylindrical lens systems can transform between LG and Hermite–Gaussian beams to access different mode bases.
Detection and analysis: Interferometric methods, forked holograms, and mode sorter devices can separate LG modes by l and p. Interference with a reference beam reveals the characteristic fork dislocations associated with different OAM states, while mode sorters map OAM states to distinct position channels for direct measurement.

Applications and impact

Optical communications: The ability to multiplex data on multiple OAM channels offers a path to higher-capacity links, especially when combined with other degrees of freedom such as polarization and wavelength. See discussions of optical communication and related mode-multiplexing concepts.
Quantum information science: Entangled or high-dimensional OAM states enable more information per photon and robust protocols for certain tasks, linking to broader quantum information research.
Optical manipulation: The angular momentum carried by LG beams can exert torque on microscopic particles, enabling precise rotation and control in optical tweezers setups and related micromanipulation techniques.
Imaging and metrology: LG modes contribute to specialized illumination schemes and high-resolution imaging modalities, where their unique phase and intensity structures provide advantages in certain contexts.
Fundamental optics: The study of LG modes intersects with broader topics in beam theory, mode decomposition, and the interplay between phase, amplitude, and polarization in complex fields.

Relationship to other beam families

LG beams complement other families such as the Hermite–Gaussian beams; through mode converters and appropriate transformations, researchers can relate LG and HG solutions within the broader framework of the paraxial wave equation. The Laguerre polynomial structure and radial dependence offer a natural way to describe circularly symmetric systems, in contrast to the Cartesian symmetry of HG modes. The full landscape of paraxial beams also interacts with more general solutions that consider nonparaxial regimes and vector fields, expanding the toolbox for light shaping and control.

History and development

The mathematical underpinnings trace back to the Laguerre polynomials, a classical family of orthogonal polynomials. The practical use of Laguerre-Gaussian modes in optics came to prominence after experiments demonstrating orbital angular momentum in light and the ability to manipulate it in experiments such as optical trapping and multiplexed communications. The work connecting the mathematical structure to a physically realizable beam profile bridged abstract mathematical physics with experimental photonics, and it continues to influence contemporary research in structured light and photonic technologies.