Hollow Gaussian Beam (HGB) Overview

• Hollow Gaussian Beam (HGB) is a class of annular optical beams with a Gaussian-apodized intensity ring and ideally dark core, differing by vortex phase and amplitude design.
• HGBs are generated through methods such as refractive shaping, digital amplitude cleaning, and nonlinear OAM cancellation, enabling applications in atom guiding and quantum gas traps.
• Their unique propagation, radial confinement, and diagnostic techniques support practical use in nonlinear frequency conversion, attosecond optics, and metamaterial studies.

Hollow Gaussian beam (HGB) denotes a class of annular optical beams whose transverse intensity is concentrated in a bright ring surrounding a dark core. In the contemporary literature, the term is not fully uniform: in some contexts it refers to the $p=0$ Laguerre–Gaussian donut modes whose on-axis null is enforced by an azimuthal phase singularity, whereas in other contexts it denotes non-vortex hollow beams with the same Gaussian-type radial hollowness but an azimuthally uniform phase. What remains common across these usages is a Gaussian-apodized annular profile with ideally zero on-axis intensity, a structure exploited in blue-detuned atom guiding, optical box potentials, nonlinear wavelength conversion, attosecond high-harmonic generation, and propagation studies in nontrivial media and geometries (Poulin et al., 2011, Chaitanya et al., 2016, Ren et al., 2024, Martín-Hernández et al., 6 Jul 2025).

1. Definition, classification, and terminological scope

The literature uses HGB in at least three technically distinct senses. In cold-atom funneling through hollow-core fiber, the beam is identified with the $p=0$ family of Laguerre–Gaussian modes with azimuthal index $l\ge 1$, whose factor $e^{i l\varphi}$ forces the field amplitude to vanish at $r=0$. In nonlinear-optics work, HGB is instead defined as a dark-hollow paraxial beam that retains the radial structure of a vortex mode but omits the azimuthal phase term, and therefore carries zero orbital angular momentum (OAM). In optical-box implementations for quantum gases, the term is often used operationally for a non-vortex annulus with zero or near-zero on-axis intensity produced by refractive reshaping and amplitude masking rather than by phase topology (Poulin et al., 2011, Chaitanya et al., 2016, Ren et al., 2024).

This nonuniformity is not merely lexical. A Laguerre–Gaussian donut, a non-vortex hollow Gaussian beam, and a hard-edged annular aperture can share a ring-shaped intensity profile while differing in OAM content, phase singularity, diffraction sidelobes, and propagation behavior. In particular, the attosecond-HHG work distinguishes an HGB from an annular aperture by emphasizing the smooth Gaussian apodization and the absence of azimuthal spatial phase variation, while the generalized paraxial analysis distinguishes vortex-free HGBs from vortex HGBs and notes that a persistent on-axis null under scalar free-space propagation requires nonzero azimuthal index; a vortex-free HGB can be exactly dark on axis at a chosen plane, but the core generally fills for $z\neq 0$ (Martín-Hernández et al., 6 Jul 2025, Radożycki, 2021).

A common misconception is that all donut beams are OAM-carrying vortices. The opposite misconception also appears: that an HGB, by definition, must be OAM-free. The record is mixed. Some authors explicitly equate the practical HGB used in guiding with an LG donut mode, whereas others reserve HGB for the zero-OAM case and treat the LG donut as a related but distinct beam family (Poulin et al., 2011, Chaitanya et al., 2016).

2. Canonical field models and diagnostic structure

For the $p=0$, $l\ge 1$ Laguerre–Gaussian realization used in blue-detuned guiding, the paraxial field envelope may be written as

$$E(r,\varphi,z)=E_0 \frac{w_0}{w(z)}\left(\frac{\sqrt{2}\,r}{w(z)}\right)^{|l|}\exp\!\left(-\frac{r^2}{w(z)^2}\right)e^{i l\varphi}e^{-i\zeta(z)},$$

with

$$w(z)=w_0\sqrt{1+\left(\frac{z}{z_R}\right)^2},\qquad z_R=\frac{\pi w_0^2}{\lambda}.$$

The corresponding intensity scales as

$p=0$0

which is zero on axis for $p=0$1. For $p=0$2, the ring peaks at $p=0$3, so an $p=0$4 mode has $p=0$5 (Poulin et al., 2011).

For the non-vortex convention, the field of an HGB of order $p=0$6 is written as

$p=0$7

with intensity

$p=0$8

The on-axis null follows from $p=0$9 for $l\ge 1$0, and the ring radius obeys $l\ge 1$1. A different but related order convention writes

$l\ge 1$2

for which the intensity ring peaks at $l\ge 1$3. The two peak-radius formulas therefore reflect different order parametrizations rather than contradictory radial physics (Chaitanya et al., 2016, Mishra et al., 24 Dec 2025).

Several diagnostic constructions recur across the literature. The nonlinear-generation work shows that the Fourier-transform intensity of an HGB of order $l\ge 1$4 is

$l\ge 1$5

which has a bright center followed by $l\ge 1$6 concentric ripples; plotting the negative logarithm reveals the order clearly. The generalized paraxial solution likewise shows that hollowness can be synthesized by superposing shifted Gaussian beams and that the condition for an on-axis null is the vanishing of the DC angular component, $l\ge 1$7, or, in the paper’s specific parameterization, $l\ge 1$8 (Chaitanya et al., 2016, Radozycki, 2022).

3. Generation strategies

A major linear strategy combines fixed refractive optics with digital amplitude cleaning. In the 2024 quantum-gas beam-shaping work, a 532 nm CW laser is first pre-shaped by axicons or prism assemblies into a ring or polygonal annulus, then corrected by a TI DLPLCR67EVM with DLP670S chip and 5.4 $l\ge 1$9m pixel pitch operating in binary ON/OFF amplitude mode. At the atom plane, the ring diameter is $e^{i l\varphi}$0–$e^{i l\varphi}$1m, the total ring width is $e^{i l\varphi}$2–$e^{i l\varphi}$3m, the inner boundary width is $e^{i l\varphi}$4m, and the residual interior light after dark-count subtraction is $e^{i l\varphi}$5 of the ring peak. The inner-wall steepness, fitted by $e^{i l\varphi}$6, reaches $e^{i l\varphi}$7 for a ring and $e^{i l\varphi}$8 for a square. Using the efficiency metric $e^{i l\varphi}$9, the combined method is higher by a factor of $r=0$0 at $r=0$1 compared to mask or DMD-only shaping, whereas DMD-only is lower than the mask method by a factor $r=0$2 because the measured DMD diffraction efficiency is $r=0$3 (Ren et al., 2024).

A distinct nonlinear route generates HGBs by OAM cancellation in $r=0$4 three-wave mixing. In that scheme, vortex pumps at $r=0$5 nm and $r=0$6 nm with equal OAM magnitude and opposite helicity, $r=0$7 and $r=0$8, drive sum-frequency generation in BIBO so that the azimuthal phase factors cancel while the hollow radial structure is retained. The generated field at $r=0$9 nm then has the HGB form with order equal to the pump-vortex order. Experimentally, ultrafast HGBs of orders $z\neq 0$0, $z\neq 0$1, and $z\neq 0$2 were produced with power $z\neq 0$3 mW at $z\neq 0$4 nm and single-pass IR$z\neq 0$5UV conversion efficiency $z\neq 0$6 for $z\neq 0$7 (Chaitanya et al., 2016).

A third route uses higher-order cylindrical vector beams and a binary multi-zone diffractive optical element. The proposed 2025 method employs $z\neq 0$8-$z\neq 0$9 modes, a central opaque disk, and three concentric annular zones with alternating transparency. Within the reported simulations, the dark-core diameter remains $p=0$0 across the tested $p=0$1 and NA values, while the bright-ring FWHM is strongly tunable by the focusing NA. For $p=0$2, the FWHM decreases from $p=0$3 at $p=0$4 to $p=0$5 at $p=0$6 and $p=0$7 at $p=0$8; corresponding values are also reported for $p=0$9 and $l\ge 1$0. The study states that radial and azimuthal input polarization perform identically under this scheme (Mishra et al., 24 Dec 2025).

4. Atom-optical and quantum-gas applications

In atom optics, the principal attraction of an HGB is the coexistence of strong radial confinement and low on-axis intensity. In the far-detuned limit,

$l\ge 1$1

so blue detuning makes $l\ge 1$2 and pushes atoms toward the dark core while large detuning suppresses scattering. In the fiber-coupling study, a blue-detuned $l\ge 1$3 beam diffracting from a hollow-core photonic-crystal fiber acts as a single funnel and guide for cold Rb atoms released from a MOT above a vertically oriented fiber. For a $l\ge 1$4m radius core, $l\ge 1$5 mW, $l\ge 1$6 GHz, and MOT-to-fiber distance $l\ge 1$7 mm, the simulations predict $l\ge 1$8 coupling efficiency. The same study identifies a gravito-optical bottle trap: atoms with substantial orbital angular momentum can be indefinitely recycled in the conservative potential instead of entering the core. The resulting optimization rule is to increase the light-force parameter $l\ge 1$9 only up to the onset of trapping, not beyond. Larger cores and longer $$E(r,\varphi,z)=E_0 \frac{w_0}{w(z)}\left(\frac{\sqrt{2}\,r}{w(z)}\right)^{|l|}\exp\!\left(-\frac{r^2}{w(z)^2}\right)e^{i l\varphi}e^{-i\zeta(z)},$$0, with corresponding scaling of $$E(r,\varphi,z)=E_0 \frac{w_0}{w(z)}\left(\frac{\sqrt{2}\,r}{w(z)}\right)^{|l|}\exp\!\left(-\frac{r^2}{w(z)^2}\right)e^{i l\varphi}e^{-i\zeta(z)},$$1, yield predicted efficiencies $$E(r,\varphi,z)=E_0 \frac{w_0}{w(z)}\left(\frac{\sqrt{2}\,r}{w(z)}\right)^{|l|}\exp\!\left(-\frac{r^2}{w(z)^2}\right)e^{i l\varphi}e^{-i\zeta(z)},$$2. In the same geometry and trap depth, the paper reports $$E(r,\varphi,z)=E_0 \frac{w_0}{w(z)}\left(\frac{\sqrt{2}\,r}{w(z)}\right)^{|l|}\exp\!\left(-\frac{r^2}{w(z)^2}\right)e^{i l\varphi}e^{-i\zeta(z)},$$3 optimal coupling with a red Gaussian versus $$E(r,\varphi,z)=E_0 \frac{w_0}{w(z)}\left(\frac{\sqrt{2}\,r}{w(z)}\right)^{|l|}\exp\!\left(-\frac{r^2}{w(z)^2}\right)e^{i l\varphi}e^{-i\zeta(z)},$$4 for a blue $$E(r,\varphi,z)=E_0 \frac{w_0}{w(z)}\left(\frac{\sqrt{2}\,r}{w(z)}\right)^{|l|}\exp\!\left(-\frac{r^2}{w(z)^2}\right)e^{i l\varphi}e^{-i\zeta(z)},$$5 in a $$E(r,\varphi,z)=E_0 \frac{w_0}{w(z)}\left(\frac{\sqrt{2}\,r}{w(z)}\right)^{|l|}\exp\!\left(-\frac{r^2}{w(z)^2}\right)e^{i l\varphi}e^{-i\zeta(z)},$$6m diameter core, but the blue guide strongly suppresses internal-state shifts and heating because atoms reside in the dark region (Poulin et al., 2011).

For homogeneous quantum gases, HGBs are used as blue-detuned optical box walls. The 2024 optical-generation work characterizes the inner wall by a power-law fit $$E(r,\varphi,z)=E_0 \frac{w_0}{w(z)}\left(\frac{\sqrt{2}\,r}{w(z)}\right)^{|l|}\exp\!\left(-\frac{r^2}{w(z)^2}\right)e^{i l\varphi}e^{-i\zeta(z)},$$7, with $$E(r,\varphi,z)=E_0 \frac{w_0}{w(z)}\left(\frac{\sqrt{2}\,r}{w(z)}\right)^{|l|}\exp\!\left(-\frac{r^2}{w(z)^2}\right)e^{i l\varphi}e^{-i\zeta(z)},$$8 values over $$E(r,\varphi,z)=E_0 \frac{w_0}{w(z)}\left(\frac{\sqrt{2}\,r}{w(z)}\right)^{|l|}\exp\!\left(-\frac{r^2}{w(z)^2}\right)e^{i l\varphi}e^{-i\zeta(z)},$$9, and notes that the optical dipole potential follows the same wall exponent because $$w(z)=w_0\sqrt{1+\left(\frac{z}{z_R}\right)^2},\qquad z_R=\frac{\pi w_0^2}{\lambda}.$$0. The reported combination of high $$w(z)=w_0\sqrt{1+\left(\frac{z}{z_R}\right)^2},\qquad z_R=\frac{\pi w_0^2}{\lambda}.$$1, high $$w(z)=w_0\sqrt{1+\left(\frac{z}{z_R}\right)^2},\qquad z_R=\frac{\pi w_0^2}{\lambda}.$$2, and near-diffraction-limited edge sharpness is used to realize nearly ideal box boundaries. Combined with a one-dimensional optical lattice, the method is stated to prepare a nearly ideal two-dimensional uniform quantum gas with different geometrical boundaries (Ren et al., 2024).

These atom-optical uses rely on the same basic operational principle: for blue detuning, intensity minima define the low-perturbation region. In the fiber case the dark channel is used for transport through a narrow core; in the box-potential case the dark interior is used as a flat trap volume bounded by steep repulsive walls. This suggests that the central dark region, rather than OAM content per se, is the application-defining property in many cold-atom implementations.

5. Nonlinear conversion and strong-field attosecond optics

The nonlinear $$w(z)=w_0\sqrt{1+\left(\frac{z}{z_R}\right)^2},\qquad z_R=\frac{\pi w_0^2}{\lambda}.$$3 generation of HGBs makes explicit a point that is often obscured in beam taxonomy: removing the azimuthal phase of a vortex does not produce a Gaussian beam. In the sum-frequency experiment, opposite-helicity input vortices obey $$w(z)=w_0\sqrt{1+\left(\frac{z}{z_R}\right)^2},\qquad z_R=\frac{\pi w_0^2}{\lambda}.$$4, so the generated beam has zero OAM, yet its radial dependence remains hollow. Zero-OAM output is verified interferometrically by the absence of the $$w(z)=w_0\sqrt{1+\left(\frac{z}{z_R}\right)^2},\qquad z_R=\frac{\pi w_0^2}{\lambda}.$$5-petal ring lattice that is observed for the pump vortices, and the order is read from the Fourier-plane ripple count after negative-log scaling. The work presents this as a direct demonstration that phase removal leaves a hollow beam rather than collapsing the mode to $$w(z)=w_0\sqrt{1+\left(\frac{z}{z_R}\right)^2},\qquad z_R=\frac{\pi w_0^2}{\lambda}.$$6 (Chaitanya et al., 2016).

In high-order harmonic generation, the 2025 attosecond-source study defines the HGB as a ring-shaped driving beam with no azimuthal phase variation and uses it to improve both harmonic generation and refocusing. The far-field angular spectrum of a radially symmetric HGB is a Hankel transform,

$$w(z)=w_0\sqrt{1+\left(\frac{z}{z_R}\right)^2},\qquad z_R=\frac{\pi w_0^2}{\lambda}.$$7

so a ring-like near-field profile yields a narrow central lobe with weak secondary rings. The study reports that HGB-driven harmonics are generated on a ring with low divergence, and that the divergence decreases with harmonic order. For attosecond synthesis, harmonics $$w(z)=w_0\sqrt{1+\left(\frac{z}{z_R}\right)^2},\qquad z_R=\frac{\pi w_0^2}{\lambda}.$$8th–$$w(z)=w_0\sqrt{1+\left(\frac{z}{z_R}\right)^2},\qquad z_R=\frac{\pi w_0^2}{\lambda}.$$9th are propagated $p=0$00 cm and refocused with a reflective element of focal length $p=0$01 cm. When the intrinsic dipole phase is artificially removed, the spatial extent of the refocused train becomes about five times larger than with the dipole phase included, which the paper interprets as evidence that the dipole phase makes the EUV emission appear as if coming from a smaller virtual source. The HGB geometry reduces the chromatic focal spread relative to a Gaussian driver, although it does not eliminate dipole-phase-induced spatiotemporal coupling (Martín-Hernández et al., 6 Jul 2025).

The same work also shows that subtle azimuthal amplitude or phase inhomogeneities are imprinted onto the HHG far field. A perfect HGB produces a bright central lobe with well-defined outer rings, whereas the experimentally retrieved HGB, containing minor azimuthal aberrations, suppresses the secondary rings and leaves a single dominant central spot. The result elevates azimuthal phase flatness from a secondary quality factor to a central design constraint in strong-field HGB applications (Martín-Hernández et al., 6 Jul 2025).

6. Propagation in metamaterials, curved spaces, and generalized paraxial formalisms

In metamaterials, HGB dynamics have been studied with the normalized spatio-temporal equation

$p=0$02

Here $p=0$03 measures dispersion relative to diffraction. For first-order HGBs in the negative-index regime with normal GVD and defocusing nonlinearity, the reported critical powers for self-trapped dynamics are $p=0$04, $p=0$05, and $p=0$06, with an approximately linear increase of $p=0$07 with $p=0$08. Low $p=0$09 preserves a single-ring intensity distribution over longer distances, whereas higher $p=0$10 accelerates the disappearance of the ring and the appearance of a tightly focused central bright spot with higher peak intensity and smaller HWHM. The paper points to trapping of nanosized particles as a consequence of these tighter focusing events (Ali et al., 2020).

On surfaces of constant Gaussian curvature, HGB propagation acquires a matrix-optics structure. For $p=0$11, the ABCD matrix along a geodesic is

$p=0$12

which is exactly of the fractional Fourier transform form with order $p=0$13 and scale $p=0$14. Within this framework, propagation on a positively curved surface is periodic with period $p=0$15, second moments have period $p=0$16, and integer-order HGBs evolve into multi-peak structures with a maximum of $p=0$17 transverse peaks. The paper further distinguishes two longitudinal-axis regimes using the divergence coefficient $p=0$18: type A occurs for $p=0$19, and type B for $p=0$20. For fractional order, half-integer $p=0$21 gives a dark axis for all $p=0$22, whereas other fractional orders develop asymmetric profiles in adjacent half periods (Ding et al., 2021).

Generalized paraxial constructions by shifted Gaussian superposition extend these results beyond standard modal bases. In that formalism, hollow beams can be synthesized either as vortex solutions with $p=0$23, $p=0$24, which produce modified Bessel–Gauss beams with a robust on-axis null, or as non-vortex solutions created by enforcing a zero-mean angular weight. This provides a unified language in which LG-like, Bessel–Gauss, modified Bessel–Gauss, Kummer–Gaussian, and other Gaussian-type cylindrical beams occupy the same analytic family (Radozycki, 2022, Radożycki, 2021).

7. Design rules, limitations, and recurring misconceptions

Several design rules recur across the application literature. For HGB-based loading of hollow-core fiber, the practical recommendation is to use $p=0$25, $p=0$26, with $p=0$27 as a robust choice, to match $p=0$28 to the fiber core, and to ensure that the ring radius at the MOT plane satisfies $p=0$29 so that the funnel encompasses the atom cloud. In the exemplar geometry, $p=0$30m gives $p=0$31m at the fiber end, and coupling is maximized by tuning $p=0$32 to the largest value that does not yet produce gravito-optical trapping (Poulin et al., 2011).

For fixed-optics plus DMD generation, the dominant limitations are alignment sensitivity, residual aberrations from axicon or prism imperfections, binary-amplitude diffraction orders, and wavelength dependence. The ON region on the DMD must overlap the pre-shaped annulus; misalignment reintroduces interior light or clips the ring. The same work notes chromatic retuning for other wavelengths and the need to remain below DMD damage thresholds at high power. In the cylindrical-vector DOE approach, performance depends on accurate placement of the zone edges $p=0$33, since fabrication error leaks energy into the core and compromises the claimed uniform dark-core diameter (Ren et al., 2024, Mishra et al., 24 Dec 2025).

In high-field applications, hollowness by itself is an insufficient performance descriptor. The attosecond-HHG study shows that slight azimuthal aberrations materially modify far-field structure, while the generalized paraxial analyses show that vortex-free hollowness at one plane does not guarantee a permanently dark core under propagation. Another recurring misconception is that removal of the vortex phase should recover a Gaussian beam; the nonlinear frequency-conversion study explicitly shows that the radial hollow structure survives azimuthal phase cancellation (Martín-Hernández et al., 6 Jul 2025, Radożycki, 2021, Chaitanya et al., 2016).

Taken together, these results locate HGBs not as a single uniquely defined mode, but as a family of Gaussian-apodized hollow beams whose classification depends on whether the annulus is produced by vortex phase, radial amplitude design, or geometric reshaping. The common utility of the family lies in the same optical invariant: strong radial gradients surrounding a dark center. The phase content, OAM status, and propagation law then determine which member of the family is appropriate for cold-atom guiding, homogeneous box trapping, wavelength conversion, attosecond driving, or propagation engineering in structured media.