Circular-Beam Approximation

• Circular-beam approximation is a modeling approach that simplifies beam analysis by assuming a circular Gaussian profile and reducing multidimensional parameters to a single scalar beam-spot size.
• It is applied across paraxial optics for OAM-carrying modes, atmospheric quantum channels for beam-wandering models, and eigen-emittance tailoring for transverse beam dynamics.
• The method reduces computational complexity while providing explicit formulas that match experimental observations in structured light and quantum communication applications.

Searching arXiv for recent and relevant papers on circular-beam approximation and related usages. Calling arXiv search tool with targeted queries. In recent arXiv literature, the expression circular-beam approximation designates more than one construction. One use models atmospheric free-space channels by assuming that the instantaneous beam profile at the receiver remains a circular Gaussian with random beam-spot size and wandering but no ellipticity; another uses a circular-beam approximation in eigen-emittance tailoring to obtain a simple relation between transverse eigen-emittance variation and the beam phase integral; and a separate paraxial-optics literature studies Circular-Beams as normalized monochromatic solutions carrying Orbital Angular Momentum (Pechonkin et al., 17 Jul 2025, Groening et al., 2021, Vallone, 2015). A plausible implication is that the term should be interpreted from context rather than treated as a single standardized formalism.

1. Scope and terminological usage

The principal uses of the term in the cited literature can be organized as follows.

Context	Object modeled	Defining statement
Paraxial optics	Circular-Beams (CiBs)	“a very general solution of the paraxial wave equation carrying Orbital Angular Momentum”
Atmospheric quantum channels	PDT of transmittance	“a simplified alternative” to the elliptic-beam approximation
Eigen-emittance tailoring	Coupled transverse beam dynamics	a relation between eigen-emittance difference and beam phase integral

For CiBs, the object is not an approximation in the narrow sense but a beam family with explicit normalization, Laguerre–Gaussian decomposition, far-field divergence, and experimentally relevant special cases (Vallone, 2015). For turbulent channels, the approximation is explicitly a model reduction: the elliptic-beam description tracks two spot sizes plus orientation, whereas the circular-beam model tracks only a scalar beam-spot variable $S$, reducing integrals over beam-shape parameters from $4$D down to $1$D (Pechonkin et al., 17 Jul 2025). For eigen-emittance tailoring, the approximation applies under experimentally relevant conditions and yields the working formula $\Delta\varepsilon(s)=|\mathcal W(s)|$, with $\mathcal W$ the beam phase integral or vorticity (Groening et al., 2021).

2. Circular-Beams as paraxial OAM solutions

Circular-Beams are defined in cylindrical coordinates $(r,\phi,z)$, with time dependence $e^{-i\omega t}$ understood. Two complex beam parameters are introduced,

$q_0=i z_0-d_0,\qquad q_1=i z_1-d_1,$

with $z_0=\pi W_0^2/\lambda$, and then

$q(z)=z+q_0,\qquad \widetilde q(z)=z+q_1,$

$4$0

where $4$1. The normalized, monochromatic Circular–Beam carrying orbital index $4$2 and radial parameter $4$3 is (Vallone, 2015)

$4$4

with $4$5.

Normalization is imposed by

$4$6

and the closed form is

$4$7

Convergence, and hence square-integrability, holds under the conditions listed in eq. (8) of the paper, in particular whenever $4$8 or $4$9 if $1$0, etc. (Vallone, 2015).

Because $1$1 is an orthonormal basis for the paraxial wave equation, the CiB admits the expansion

$1$2

with

$1$3

The coefficients are independent of $1$4 except for the common Gouy-phase in the $1$5 modes. This establishes CiBs as a parametrized OAM-carrying family with an explicit modal representation in the standard Laguerre–Gaussian basis.

3. Asymptotics, divergence, and experimental generation of CiBs

Near the axis, $1$6, one has

$1$7

so that $1$8 and

$1$9

Hence $\Delta\varepsilon(s)=|\mathcal W(s)|$0 as $\Delta\varepsilon(s)=|\mathcal W(s)|$1 (Vallone, 2015).

In the far field, $\Delta\varepsilon(s)=|\mathcal W(s)|$2, the asymptotic forms $\Delta\varepsilon(s)=|\mathcal W(s)|$3, $\Delta\varepsilon(s)=|\mathcal W(s)|$4, $\Delta\varepsilon(s)=|\mathcal W(s)|$5, and $\Delta\varepsilon(s)=|\mathcal W(s)|$6 imply that the CiB acquires an effective Gaussian envelope in angle $\Delta\varepsilon(s)=|\mathcal W(s)|$7. Equivalently, from the LG expansion,

$\Delta\varepsilon(s)=|\mathcal W(s)|$8

where $\Delta\varepsilon(s)=|\mathcal W(s)|$9 is a slowly varying polynomial in $\mathcal W$0 determined by the coefficients $\mathcal W$1. The overall divergence half-angle (rms) is

$\mathcal W$2

with

$\mathcal W$3

For the special hypergeometric case $\mathcal W$4, a simpler closed form is available (Vallone, 2015).

The experimentally relevant special cases are explicit. For phase-only OAM modes, $\mathcal W$5 and $\mathcal W$6, one merely needs a Gaussian TEM$\mathcal W$7 input passed through a phase-only element such as a q-plate or spiral phase plate $\mathcal W$8; no amplitude mask is needed. For higher-order radial shapes, $\mathcal W$9 and $(r,\phi,z)$0, an amplitude mask with transmittance $(r,\phi,z)$1 plus the same $(r,\phi,z)$2 yields the full CiB of given $(r,\phi,z)$3. Since a CiB is specified by $(r,\phi,z)$4, standard Gaussian-beam ABCD laws apply separately to $(r,\phi,z)$5 and $(r,\phi,z)$6, and $(r,\phi,z)$7 is invariant. One may thus design lenses or telescopes to adjust divergence or focus (Vallone, 2015).

4. Circular-beam approximation for turbulent atmospheric quantum channels

In a free-space atmospheric quantum channel, the input–output relation for the Glauber–Sudarshan $(r,\phi,z)$8-function is

$(r,\phi,z)$9

where $e^{-i\omega t}$0 is the random channel transmittance and $e^{-i\omega t}$1 is the PDT. All statistics of the fluctuating channel enter via $e^{-i\omega t}$2; once $e^{-i\omega t}$3 is known, one can compute output photon-number distributions, quadrature variances, and nonclassicality witnesses such as Mandel $e^{-i\omega t}$4 via averages over $e^{-i\omega t}$5 (Pechonkin et al., 17 Jul 2025).

The circular-beam approximation assumes that the instantaneous beam profile at the receiver remains a circular Gaussian with random beam-spot size and wandering but no ellipticity. Three fluctuating random variables characterize the beam: the centroid position $e^{-i\omega t}$6 with $e^{-i\omega t}$7; the instantaneous squared beam-spot radius

$e^{-i\omega t}$8

modeled by a log-normal distribution; and the transmittance

$e^{-i\omega t}$9

where $q_0=i z_0-d_0,\qquad q_1=i z_1-d_1,$0 is the circular receiver aperture of radius $q_0=i z_0-d_0,\qquad q_1=i z_1-d_1,$1. The key simplification versus the elliptic-beam model is that one need only track scalar $q_0=i z_0-d_0,\qquad q_1=i z_1-d_1,$2, instead of two principal axes and orientation, reducing integrals over beam-shape parameters from $q_0=i z_0-d_0,\qquad q_1=i z_1-d_1,$3D down to $q_0=i z_0-d_0,\qquad q_1=i z_1-d_1,$4D (Pechonkin et al., 17 Jul 2025).

For fixed $q_0=i z_0-d_0,\qquad q_1=i z_1-d_1,$5, the conditional PDT is obtained from the beam-wandering model. For a circular Gaussian of spot size $q_0=i z_0-d_0,\qquad q_1=i z_1-d_1,$6 centered at random $q_0=i z_0-d_0,\qquad q_1=i z_1-d_1,$7,

$q_0=i z_0-d_0,\qquad q_1=i z_1-d_1,$8

for $q_0=i z_0-d_0,\qquad q_1=i z_1-d_1,$9, with

$z_0=\pi W_0^2/\lambda$0

and explicit functions $z_0=\pi W_0^2/\lambda$1 and $z_0=\pi W_0^2/\lambda$2 given in terms of modified Bessel functions $z_0=\pi W_0^2/\lambda$3 and $z_0=\pi W_0^2/\lambda$4. Marginalizing over the log-normal $z_0=\pi W_0^2/\lambda$5 yields the full PDT through a one-dimensional integration over $z_0=\pi W_0^2/\lambda$6 (Pechonkin et al., 17 Jul 2025).

5. Moment matching, validity range, and quantum observables

The first two moments of transmittance are

$z_0=\pi W_0^2/\lambda$7

$z_0=\pi W_0^2/\lambda$8

Two procedures are given to determine the log-normal parameters $z_0=\pi W_0^2/\lambda$9 for $q(z)=z+q_0,\qquad \widetilde q(z)=z+q_1,$0. One matches beam-spot moments $q(z)=z+q_0,\qquad \widetilde q(z)=z+q_1,$1 and $q(z)=z+q_0,\qquad \widetilde q(z)=z+q_1,$2, then uses

$q(z)=z+q_0,\qquad \widetilde q(z)=z+q_1,$3

The other matches transmittance moments by solving numerically for $q(z)=z+q_0,\qquad \widetilde q(z)=z+q_1,$4 from the system

$q(z)=z+q_0,\qquad \widetilde q(z)=z+q_1,$5

where explicit closed forms for $q(z)=z+q_0,\qquad \widetilde q(z)=z+q_1,$6 and $q(z)=z+q_0,\qquad \widetilde q(z)=z+q_1,$7 exist in terms of Marcum–$q(z)=z+q_0,\qquad \widetilde q(z)=z+q_1,$8 functions (Pechonkin et al., 17 Jul 2025).

The comparison with the elliptic-beam approximation is specific. The elliptic-beam model matches experiments best near $q(z)=z+q_0,\qquad \widetilde q(z)=z+q_1,$9 for moderate turbulence, but circular-beam with transmittance-moment matching extends good agreement to a wider range of $4$00. By matching $4$01 and $4$02 directly, the circular-beam PDT reproduces the true first two moments for all $4$03, even outside the narrow elliptic-beam regime. The limitation is equally explicit: the method requires accurate inputs for $4$04 and $4$05, and errors in analytical moment formulas propagate into PDT biases, especially for bright inputs (Pechonkin et al., 17 Jul 2025).

The model is used for several quantum observables. For sub-Poissonian light,

$4$06

For click-detector nonclassicality,

$4$07

For quadrature squeezing,

$4$08

Using the circular-beam PDT with moment matching, $4$09 is found in excellent agreement with phase-screen simulation; the model reproduces simulated $4$10 within a few percent; and accurate modeling of $4$11 allows prediction of loss-induced anti-squeezing and verification of operating regimes for squeezed-state QKD over free-space links. The summary given in the source attributes to the method a simple $4$12D-integral PDT model, extended range of validity across beam-aperture ratios and moderate turbulence, sufficient accuracy for predicting nonclassicality transfer in quantum communication, and orders-of-magnitude reduction in computational effort versus the full elliptic-beam model (Pechonkin et al., 17 Jul 2025).

6. Circular-beam approximation in eigen-emittance tailoring

In beam dynamics, the circular-beam approximation concerns transverse eigen-emittances rather than optical field envelopes. Let $4$13 be the normalized $4$14D beam distribution, with second moments $4$15, and define the rms area

$4$16

The beam phase integral, also called the beam vorticity, is

$4$17

with centroid-slope field

$4$18

Its algebraic form is

$4$19

Under experimentally relevant conditions, the paper introduces an approximation providing a very simple and powerful relation between transverse eigen-emittance variation and the beam phase integral (Groening et al., 2021).

The derivation starts from the extended Busch theorem. For short coupling elements, such as solenoid fringes, one may assume: the beam rms area is effectively constant, $4$20; behind the element all inter-plane correlations vanish, so $4$21; and the change of $4$22 is dominated by the change of field, $4$23. Under these approximations,

$4$24

If the beam enters the coupling region uncorrelated, $4$25, then

$4$26

becomes the working formula. Within these conditions—short coupling sections, fully decoupled output, no substantial evolution of $4$27 in subsequent drifts, and constant beam rigidity—one may use $4$28 without recourse to the full $4$29 moment-matrix diagonalization (Groening et al., 2021).

The interpretation is that any beam can be viewed as the superposition of two circular modes rotating with opposite angular velocities. Following Brinkmann et al. (1999), the inverse round-to-flat projector maps the beam into two rigidly rotating $4$30D objects at frequencies $4$31, and the two $4$32D eigen-emittances coincide with the areas of these two oppositely rotating circular modes. The paper states that the difference of eigen-emittances is given by the beam phase integral or vorticity rather than by angular momentum. For perfectly round beams, $4$33 reduces to twice the rms angular momentum and one recovers the Kim formula $4$34; the result thus extends the angular-momentum picture to arbitrary non-cylindrical beams by replacing angular momentum with the more general phase integral (Groening et al., 2021).

The practical examples given are a flat-beam photoinjector, where $4$35, and the EMTEX emittance-transfer experiment, where $4$36, in excellent agreement with measured $4$37 mm mrad. Simple model objects include rigid rotation at constant $4$38, giving $4$39, and pure shear $4$40, giving $4$41 (Groening et al., 2021).

7. Related circular-symmetry approximations and common distinctions

A separate paraxial approximation concerns beams passing apertures with sharp boundaries rather than the two circular-beam approximations just described. For a uniform circular aperture of radius $4$42, with initial field $4$43 for $4$44 and zero outside, the sharp-boundary method gives

$4$45

No other fitting parameters appear; the edge-thickness parameter is taken $4$46 for a truly sharp boundary (Luz et al., 2016).

Its regime of validity is also explicit: paraxial propagation, $4$47, with the physical transition from inside to outside occurring over a transverse scale $4$48. The approximation is most accurate in the region $4$49 near the edge. It captures the onset of Fresnel diffraction as edge-blurring, with the initially sharp jump at $4$50 smoothed over a radial width $4$51, but it does not generate the full oscillatory ring structure of the Airy pattern (Luz et al., 2016).

This distinction matters conceptually. In the atmospheric-channel model, “circular” refers to the assumed instantaneous beam shape at the receiver; in eigen-emittance tailoring, it refers to the decomposition into two circular modes rotating with opposite angular velocities; and in the sharp-boundary diffraction problem, it refers to the geometry of the aperture. This suggests that the phrase circular-beam approximation is best read as a family resemblance across circular symmetry reductions, rather than as a unique formalism.