Beam-Diverging Effect in Photonics

• Beam-Diverging Effect is the controlled increase in a beam’s transverse spread during propagation, measurable through rms and encircled-energy metrics in various fields.
• In OAM-carrying vortex beams, the divergence is bounded by the beam’s mean absolute orbital angular momentum, establishing a generalized uncertainty principle for focusing.
• Engineered divergence is exploited in near-field array beamforming and advanced optical, charged-particle, and plasma systems to achieve dynamic beam shaping and enhanced resolution.

The beam-diverging effect denotes the increase, control, or deliberate induction of a beam’s transverse spread during propagation or interaction with a medium or structure. In the cited literature, it appears as rms or encircled-energy divergence of optical vortex beams, angular centroid walk-off in reflected optical beams, beam broadening or narrowing at resonances, graded-index spreading of spin-wave beams, multiple-scattering-induced broadening of charged-particle beams, and broad spatial coverage generated intentionally in near-field array beamforming. A central result is that, for monochromatic paraxial beams carrying orbital angular momentum (OAM), the mean absolute value of the OAM imposes a strict lower bound on beam divergence, including arbitrary coherent superpositions in the Laguerre–Gaussian basis (Vallone et al., 2016).

1. Definitions and observables

The effect is not described by a single observable across all subfields. In paraxial optics, the standard definitions are the root-mean-square (rms) angular divergence,

$$\theta_{\mathrm{rms}}=\lim_{z\rightarrow\infty}\frac{\sigma_r(z)}{z},$$

and the encircled-energy (EE) divergence,

$$\theta_{\mathrm{EE}}=\lim_{z\to\infty}\frac{R_{\mathrm{EE}}(z)}{z},$$

where $R_{\mathrm{EE}}(z)$ is the radius that contains a fixed fraction $E_0$ of the total energy. The EE definition is used when the rms divergence is ill defined, for example when the intensity decays too slowly at large radii (Vallone et al., 2016).

In resonant reflection problems, the effect is quantified by the second-moment width change

$$\Delta=\frac{\int (x-\delta)^2 |E_r|^2 dx}{\int |E_r|^2 dx}-\frac{\int x^2 |E_i|^2 dx}{\int |E_i|^2 dx},$$

and by the relative change

$$\Xi=\sqrt{\frac{\int (x-\delta)^2 |E_r|^2 dx / \int |E_r|^2 dx}{\int x^2 |E_i|^2 dx / \int |E_i|^2 dx}},$$

so that broadening and narrowing are treated within the same formalism (Pollès et al., 2016). In beam-shift theory, divergence appears through angular Goos–Hänchen and Imbert–Fedorov shifts, obtained from the propagation dependence of the reflected-beam centroid (Aiello, 2011). In graded-index magnonics, the relevant observable is the beam width, reported through the full width at half maximum (FWHM), while in Doppler backscattering the filter function is inversely proportional to the beam width, making focusing and divergence directly measurable through the scattering response (Gruszecki et al., 2017, Ruiz et al., 2024).

Domain	Observable or manifestation	Representative result
Optical OAM beams	rms or EE divergence	$\sigma_k \sigma_r \geq 1+\langle|\ell|\rangle$
Resonant reflection	$\Delta$ or $\Xi$	a reflected beam can be $10\%$ narrower
Graded-index spin waves	FWHM evolution	gradual decrease in internal field leads to clear beam broadening
Near-field arrays	spatial coverage region	hierarchical search with $$\theta_{\mathrm{EE}}=\lim_{z\to\infty}\frac{R_{\mathrm{EE}}(z)}{z},$$0 pilots

This variety of observables indicates that “beam-diverging effect” is domain-specific. A plausible implication is that the common content is not a single metric but a common geometric event: the redistribution of energy or probability away from a nominally localized trajectory or focus.

2. OAM-carrying beams and lower bounds on divergence

The most explicit theorem in the supplied literature concerns vortex beams. For a monochromatic paraxial beam with complex amplitude $$\theta_{\mathrm{EE}}=\lim_{z\to\infty}\frac{R_{\mathrm{EE}}(z)}{z},$$1, mean absolute OAM

$$\theta_{\mathrm{EE}}=\lim_{z\to\infty}\frac{R_{\mathrm{EE}}(z)}{z},$$2

spatial standard deviation $$\theta_{\mathrm{EE}}=\lim_{z\to\infty}\frac{R_{\mathrm{EE}}(z)}{z},$$3, and transverse-wavevector standard deviation $$\theta_{\mathrm{EE}}=\lim_{z\to\infty}\frac{R_{\mathrm{EE}}(z)}{z},$$4, the bound

$$\theta_{\mathrm{EE}}=\lim_{z\to\infty}\frac{R_{\mathrm{EE}}(z)}{z},$$5

holds for arbitrary coherent superpositions of Laguerre–Gaussian modes and, more generally, for any beam that can be decomposed in the Laguerre–Gaussian basis, including Circular Beams as special cases. In terms of the beam quality factor and rms divergence,

$$\theta_{\mathrm{EE}}=\lim_{z\to\infty}\frac{R_{\mathrm{EE}}(z)}{z},$$6

For beams with no OAM, the familiar bound $$\theta_{\mathrm{EE}}=\lim_{z\to\infty}\frac{R_{\mathrm{EE}}(z)}{z},$$7 is recovered; for beams carrying OAM, the lower bound is increased proportionally to the mean absolute OAM (Vallone et al., 2016).

The same work treats rms and encircled-energy divergence separately. For fixed $$\theta_{\mathrm{EE}}=\lim_{z\to\infty}\frac{R_{\mathrm{EE}}(z)}{z},$$8, the EE bound is

$$\theta_{\mathrm{EE}}=\lim_{z\to\infty}\frac{R_{\mathrm{EE}}(z)}{z},$$9

where $R_{\mathrm{EE}}(z)$0 is the minimal encircled-energy radius and $R_{\mathrm{EE}}(z)$1 depends on the chosen energy fraction $R_{\mathrm{EE}}(z)$2. The results are described as tight, and the bounds can be saturated, for example, by pure Laguerre–Gaussian modes with $R_{\mathrm{EE}}(z)$3 and the desired OAM. The theorem is also written as a generalized uncertainty principle for a two-dimensional free particle,

$R_{\mathrm{EE}}(z)$4

and yields an OAM-dependent focusing limit,

$R_{\mathrm{EE}}(z)$5

with applications in long-range communication, microscopy, and $R_{\mathrm{EE}}(z)$6D quantum systems (Vallone et al., 2016).

A related question concerns how the divergence scales with OAM. One analysis resolves the apparent conflict between linear and square-root scaling by distinguishing the launch constraint. If the Gaussian beam waist $R_{\mathrm{EE}}(z)$7 is held constant, then

$R_{\mathrm{EE}}(z)$8

so $R_{\mathrm{EE}}(z)$9. If instead the rms beam radius $E_0$0 is held constant, then

$E_0$1

so $E_0$2. The paper identifies the first regime with mode converter systems and the second with spatial-light-modulator or forked-grating systems (Padgett et al., 2014).

OAM also modifies angular beam shifts after reflection. For a Laguerre–Gauss beam with OAM index $E_0$3, the angular evolution factor becomes

$E_0$4

and the centroid obeys

$E_0$5

The factor $E_0$6 amplifies the angular shift, while the terms proportional to $E_0$7 mix spatial and angular shifts (Aiello, 2011).

3. Resonant, diffractive, and optofluidic manifestations

The beam-diverging effect is not always monotone broadening. Near a resonance in a multilayered structure, the reflected beam can become narrower than the incident beam. For a wide beam, the asymptotic deformation is

$E_0$8

At resonance, $E_0$9 and $$\Delta=\frac{\int (x-\delta)^2 |E_r|^2 dx}{\int |E_r|^2 dx}-\frac{\int x^2 |E_i|^2 dx}{\int |E_i|^2 dx},$$0, so $$\Delta=\frac{\int (x-\delta)^2 |E_r|^2 dx}{\int |E_r|^2 dx}-\frac{\int x^2 |E_i|^2 dx}{\int |E_i|^2 dx},$$1, implying narrowing. Off resonance, $$\Delta=\frac{\int (x-\delta)^2 |E_r|^2 dx}{\int |E_r|^2 dx}-\frac{\int x^2 |E_i|^2 dx}{\int |E_i|^2 dx},$$2 and $$\Delta=\frac{\int (x-\delta)^2 |E_r|^2 dx}{\int |E_r|^2 dx}-\frac{\int x^2 |E_i|^2 dx}{\int |E_i|^2 dx},$$3, so $$\Delta=\frac{\int (x-\delta)^2 |E_r|^2 dx}{\int |E_r|^2 dx}-\frac{\int x^2 |E_i|^2 dx}{\int |E_i|^2 dx},$$4, implying the usual widening. The reported narrowing can reach $$\Delta=\frac{\int (x-\delta)^2 |E_r|^2 dx}{\int |E_r|^2 dx}-\frac{\int x^2 |E_i|^2 dx}{\int |E_i|^2 dx},$$5, occurs on a very narrow angular range close to a resonance, and improves the resolution of sensors based on the detection of surface plasmon resonances by a factor three (Pollès et al., 2016).

In Fresnel diffraction beams, divergence is coupled to self-acceleration and deceleration. The main lobe follows a parabolic law $$\Delta=\frac{\int (x-\delta)^2 |E_r|^2 dx}{\int |E_r|^2 dx}-\frac{\int x^2 |E_i|^2 dx}{\int |E_i|^2 dx},$$6, but the propagation contains both a deceleration region and an acceleration region separated by a critical propagation distance. Before the critical distance, the main lobe approaches the edge, the oscillatory fringes compress, and the beam undergoes self-smoothing; after the critical distance, the main lobe diverges away and the fringes reappear. This is explicitly contrasted with Airy beams, which only accelerate and do not exhibit a deceleration phase or a critical propagation distance (Zhang et al., 2013).

A cavity with net roundtrip gain presents a different divergence problem. A naive partial-wave sum diverges when $$\Delta=\frac{\int (x-\delta)^2 |E_r|^2 dx}{\int |E_r|^2 dx}-\frac{\int x^2 |E_i|^2 dx}{\int |E_i|^2 dx},$$7, yet the boundary-condition solution of Maxwell’s equations remains finite. The proposed mechanism is that the side-tail of a Gaussian beam leaks into the cavity before the main lobe arrives, is amplified by roundtrips, and produces a pre-excited field that interferes with the main portion of the beam. The consequence is convergence rather than runaway field amplitude, including in the discussion of amplified total internal reflection (Mansuripur et al., 2013).

Optofluidic systems furnish a geometric lensing analogue. An air microbubble in water acts as a diverging lens, spreading and defocusing a Gaussian laser beam inside the bubble. The internal intensity is reduced, especially at the rear interface, the factor $$\Delta=\frac{\int (x-\delta)^2 |E_r|^2 dx}{\int |E_r|^2 dx}-\frac{\int x^2 |E_i|^2 dx}{\int |E_i|^2 dx},$$8 changes sign, and the resulting radiation pressure leads to much smaller deformations than in droplets under otherwise identical conditions. The reported qualitative bubble shapes are “Acorn” for a wide beam and “Sea urchin” for a narrow beam (Ellingsen, 2012).

A defocused optical tweezer can also create a diverging Gaussian beam in the sample plane. In that setting, a Janus particle experiences self-thermophoresis with

$$\Delta=\frac{\int (x-\delta)^2 |E_r|^2 dx}{\int |E_r|^2 dx}-\frac{\int x^2 |E_i|^2 dx}{\int |E_i|^2 dx},$$9

a small outward radiation-pressure contribution,

$$\Xi=\sqrt{\frac{\int (x-\delta)^2 |E_r|^2 dx / \int |E_r|^2 dx}{\int x^2 |E_i|^2 dx / \int |E_i|^2 dx}},$$0

and an optically induced torque. In the deterministic limit, the coupled dynamics yield rosette-like looping trajectories described as hypotrochoids (Moyses et al., 2016).

4. Graded and inhomogeneous media

In magnonics, the beam-diverging effect is realized through spatial modulation of the spin-wave refractive index in a thin ferromagnetic film. In the out-of-plane configuration, the iso-frequency contours are circular, and the propagation direction follows

$$\Xi=\sqrt{\frac{\int (x-\delta)^2 |E_r|^2 dx / \int |E_r|^2 dx}{\int x^2 |E_i|^2 dx / \int |E_i|^2 dx}},$$1

When the spin-wave beam traverses a region with a slowly decreasing magnetic field, corresponding to an increasing refractive index, the group velocity angle decreases and the beam spreads, mimicking a diverging lens. When the field slowly increases, the beam narrows, yielding a mirage effect. Graded-index waveguides preserve the width of the spin-wave beam for a long distance, and the ratio of maximum to minimum FWHM during propagation remained $$\Xi=\sqrt{\frac{\int (x-\delta)^2 |E_r|^2 dx / \int |E_r|^2 dx}{\int x^2 |E_i|^2 dx / \int |E_i|^2 dx}},$$2, with no systematic broadening even over tens of $$\Xi=\sqrt{\frac{\int (x-\delta)^2 |E_r|^2 dx / \int |E_r|^2 dx}{\int x^2 |E_i|^2 dx / \int |E_i|^2 dx}},$$3m, limited only by intrinsic damping (Gruszecki et al., 2017).

The contrast between gradual and step-like variation is central. Gradual variation minimizes scattering and reflection, produces no interference fringes, and can yield repeated focusing and defocusing without cumulative spreading. Step-index interfaces generate pronounced reflection, fringe patterns, abrupt lateral shifts, and rapid beam degradation after several passes (Gruszecki et al., 2017). This suggests that, in beam engineering, a graded transition can suppress interface-induced divergence even when it deliberately redistributes the trajectory.

In plasma microwave propagation near a turning-point caustic, the beam width $$\Xi=\sqrt{\frac{\int (x-\delta)^2 |E_r|^2 dx / \int |E_r|^2 dx}{\int x^2 |E_i|^2 dx / \int |E_i|^2 dx}},$$4 is determined by the imaginary part of the Gaussian-beam matrix element,

$$\Xi=\sqrt{\frac{\int (x-\delta)^2 |E_r|^2 dx / \int |E_r|^2 dx}{\int x^2 |E_i|^2 dx / \int |E_i|^2 dx}},$$5

The filter function governing Doppler backscattering is inversely proportional to the beam width,

$$\Xi=\sqrt{\frac{\int (x-\delta)^2 |E_r|^2 dx / \int |E_r|^2 dx}{\int x^2 |E_i|^2 dx / \int |E_i|^2 dx}},$$6

For small incident angles, beam focusing near the caustic produces enhanced scattering contributions from the focusing region; for large incident angles, focusing is negligible or absent and the response becomes delocalized. The reported conclusion is that the Doppler backscattering signal enhancement for small incident angles is due to beam focusing and not due to forward scattering (Ruiz et al., 2024).

5. Charged-particle and accelerator beams

In high-power proton transport, the beam-diverging effect is produced by multiple Coulomb scattering in proton beam windows. The scattering changes proton direction randomly, increases angular and spatial spread, and degrades the beam distribution at the target. For the CSNS-I example, as the window thickness $$\Xi=\sqrt{\frac{\int (x-\delta)^2 |E_r|^2 dx / \int |E_r|^2 dx}{\int x^2 |E_i|^2 dx / \int |E_i|^2 dx}},$$7 increases from $$\Xi=\sqrt{\frac{\int (x-\delta)^2 |E_r|^2 dx / \int |E_r|^2 dx}{\int x^2 |E_i|^2 dx / \int |E_i|^2 dx}},$$8 to $$\Xi=\sqrt{\frac{\int (x-\delta)^2 |E_r|^2 dx / \int |E_r|^2 dx}{\int x^2 |E_i|^2 dx / \int |E_i|^2 dx}},$$9 mm, the horizontal rms emittance grows from $\sigma_k \sigma_r \geq 1+\langle|\ell|\rangle$0 to $\sigma_k \sigma_r \geq 1+\langle|\ell|\rangle$1 $\sigma_k \sigma_r \geq 1+\langle|\ell|\rangle$2. The beam loss outside the target rises from $\sigma_k \sigma_r \geq 1+\langle|\ell|\rangle$3 W for $\sigma_k \sigma_r \geq 1+\langle|\ell|\rangle$4 mm to $\sigma_k \sigma_r \geq 1+\langle|\ell|\rangle$5 W for $\sigma_k \sigma_r \geq 1+\langle|\ell|\rangle$6 mm. Increasing the distance $\sigma_k \sigma_r \geq 1+\langle|\ell|\rangle$7 from the proton beam window to the target also increases spread: for $\sigma_k \sigma_r \geq 1+\langle|\ell|\rangle$8 m, the beam loss is $\sigma_k \sigma_r \geq 1+\langle|\ell|\rangle$9 W, while for $\Delta$0 m it is $\Delta$1 W. The design consequence is a preference for thin, low-$\Delta$2, single-layer aluminum windows placed as close as practical to the target (Meng et al., 2010).

In electron-beam-driven plasma wakefield acceleration, finite transverse beam size introduces focusing and defocusing fields. When the transverse size is greater than the longitudinal extension, the wake is purely electrostatic; when the transverse dimensions are equal or smaller than the longitudinal extension, the wake is electromagnetic in nature. The driver beam undergoes transverse pinching much earlier than longitudinal modification, and the $\Delta$3D rigidity limit is modified in $\Delta$4D: a beam with $\Delta$5 exhibits significant profile modification within a hundred plasma periods, whereas truly rigid behavior is reported only for $\Delta$6. Narrower beams relative to their longitudinal length also yield a higher transformer ratio (Bera et al., 2019).

In circular accelerators, the supplied literature distinguishes apparent and real beam splitting. An AC dipole can create $\Delta$7 separated spots on diagnostics, but these are not true split beams; the whole beam successively occupies each position in time. The divergence or emittance can still change because off-axis orbits produce feed-down in quadrupoles and sextupoles. By contrast, stable resonance islands in nonlinear optics generate true split beams in transverse phase space, and each island can have different optics and equilibrium emittance (Franchi et al., 2022).

6. Engineered divergence in near-field communications and vortex-wave RF systems

In near-field beamforming, the beam-diverging effect is not a parasitic consequence of diffraction but a designed codebook feature. For a uniform linear array, a diverging codeword is defined as the complex conjugate of a classical near-field focusing codeword,

$\Delta$8

with the virtual focal point placed behind the array. The transmitted power is then high over a wide angular sector rather than concentrated at a single point. On that basis, a diverging polar-domain codebook is constructed hierarchically, and the coarse angular search requires only $\Delta$9 pilots. For a $\Xi$0-antenna array, the reported total pilot count is $\Xi$1–$\Xi$2, versus $\Xi$3 for exhaustive search, with accuracy $\Xi$4 and a single RF chain. Two further techniques are reported: DPC angular range reduction and pilot set expansion (Li et al., 19 Sep 2025).

For uniform planar arrays, the same idea is extended to three dimensions. A diverging codeword associated with a virtual focal point on the opposite side of the array produces a broad, adjustable beam that covers a rectangular planar region, or, as $\Xi$5 is stacked, an unbounded pyramidal frustum. The proposed hierarchical codebook performs coarse localization with approximately $\Xi$6 beams per tier, so that localization within one out of $\Xi$7 regions after $\Xi$8 tiers requires only $\Xi$9 pilots. For a $10\%$0 UPA and the stated coverage region, the reported overhead is approximately $10\%$1 pilots with $10\%$2 RF chain, and the identification accuracy is consistently $10\%$3 in simulation (Li et al., 19 Sep 2025).

Near-field divergence engineering is conceptually related to RF mitigation of OAM-wave divergence. For vortex waves generated by a uniform circular patch antenna array at $10\%$4 GHz, a tailored lens and a tailored reflector are introduced specifically to reduce the large beam divergence inherent to OAM waves. The tailored reflector is reported to outperform the conventional reflector when the reflector height $10\%$5 is less than around $10\%$6 and the opening angle $10\%$7 is less than $10\%$8 for a UCA with $10\%$9. Measured gains for the lens case are $$\theta_{\mathrm{EE}}=\lim_{z\to\infty}\frac{R_{\mathrm{EE}}(z)}{z},$$00 without lens, $$\theta_{\mathrm{EE}}=\lim_{z\to\infty}\frac{R_{\mathrm{EE}}(z)}{z},$$01 with conventional lens, and $$\theta_{\mathrm{EE}}=\lim_{z\to\infty}\frac{R_{\mathrm{EE}}(z)}{z},$$02 with tailored lens. The general conclusion is that vortex waves need a special lens or a special reflector to reduce effectively the beam divergence, especially when the radius of the UCA is very large (Hassan et al., 2020).

Taken together, these works show that the beam-diverging effect is both a limitation and a resource. In OAM optics and charged-particle transport it imposes lower bounds or loss mechanisms; in resonant optics it can sharpen sensing through controlled narrowing; in graded media it can be traded against reflection and interference; and in near-field communications it becomes a deliberate wide-beam mechanism for low-overhead localization and training.