Second-order moment equivalence of twisted Gaussian Schell model beams and orbital angular momentum eigenmodes

This paper demonstrates that cylindrically symmetric coherent orbital angular momentum eigenmodes and twisted Gaussian Schell-model beams share identical covariance matrices and second-order moment evolution under arbitrary ABCD transformations, enabling a direct parameter mapping and the application of established TGSM propagation tools to diverse coherent beam families like Laguerre-Gaussian, Bessel-Gaussian, and perfect vortex beams.

The Gist

Imagine you have two very different types of light beams. One is a perfectly organized, coherent beam (like a laser) that carries a specific amount of "twist" or spin, known as Orbital Angular Momentum (OAM). Think of this like a perfectly formed spiral staircase. The other is a partially chaotic, "twisted" beam that isn't perfectly ordered but has a statistical "twist" built into its structure. Think of this like a crowd of people moving in a swirling pattern, where no single person is perfectly aligned, but the group as a whole has a rotational flow.

For decades, physicists studied these two types of beams as if they lived in completely different worlds. One group studied the perfect spirals (coherent beams), and another group studied the swirling crowds (partially coherent beams).

The Big Discovery
This paper reveals a surprising secret: If you only look at the "average statistics" of these beams, they are indistinguishable.

The authors show that for any perfectly organized spiral beam, you can find a twisted, chaotic beam that has the exact same "fingerprint" regarding its size, spread, and how it moves through space. They call this fingerprint the covariance matrix.

The Analogy: The Shadow and the Object
Imagine holding a complex, 3D sculpture (the perfect spiral beam) and a fuzzy, blurry cloud (the twisted beam) in front of a light source.

• The sculpture has intricate details, sharp edges, and a specific shape.
• The cloud is amorphous and lacks those sharp details.

However, if you look at the shadow they cast on the wall (which represents the "second-order moments" or the covariance matrix), the shadows are identical. They have the same width, the same length, and the same angle of tilt.

The paper proves that for these specific types of light, the "shadow" (the statistical data) is enough to predict exactly how the beam will behave as it travels, even if the "object" casting the shadow looks totally different.

The Three Families of Beams
The authors tested this idea on three famous families of spiral beams:

1. Laguerre-Gaussian (LG): The standard "donut" shaped beams used in labs.
2. Perfect Vortex (PVB): Beams where the ring size stays the same even if you change the amount of twist.
3. Bessel-Gaussian (BG): Beams that can travel long distances without spreading out much.

For each of these, they calculated the "shadow" (the covariance matrix) and showed it matches perfectly with a specific type of twisted beam. They even provided a "translation guide" (a table in the paper) that tells you exactly how to convert the settings of a perfect spiral beam into the settings of a matching twisted beam.

Why Does This Matter?
The paper highlights a powerful practical benefit: The Toolbox Effect.

Physicists have spent years building a massive "toolbox" of mathematical formulas to predict how the twisted, chaotic beams will move through lenses and air. Because the "shadows" are identical, you can use that entire existing toolbox to predict how the perfect spiral beams will move, without doing any new, difficult calculations.

If you know how the twisted beam spreads out, you instantly know how the perfect spiral beam will spread out. If you know how the twisted beam behaves in a turbulent atmosphere, you know how the spiral beam will behave too.

The Catch (The "Fingerprint" Limit)
The paper also notes a limitation. While the "shadows" are identical, the beams themselves are not.

• If you look closely at the perfect spiral, you see its detailed structure.
• If you look at the twisted beam, you see a blur.

The "shadow" (covariance matrix) is enough to tell you the beam's size and how it travels, but it cannot tell you the full, detailed shape of the beam. However, for the specific families of beams studied (especially the Laguerre-Gaussian ones), the authors found that the shadow is actually so detailed that it can uniquely identify the beam's specific settings (like its size and twist level) with perfect accuracy.

In Summary
This paper connects two separate worlds of optics. It proves that a perfectly ordered, twisting light beam and a statistically twisted, partially ordered beam are "twins" when it comes to their basic movement and spread. This allows scientists to use the well-understood math of the chaotic beams to instantly understand the behavior of the perfect beams, simplifying the study of light for applications like high-speed data transmission and optical trapping.

Technical Summary

Technical Summary: Second-order moment equivalence of twisted Gaussian Schell model beams and orbital angular momentum eigenmodes

Problem Statement
The study of structured optical fields has historically developed along two largely independent tracks: the theory of fully coherent orbital angular momentum (OAM) eigenmodes (such as Laguerre-Gaussian, Bessel-Gaussian, and Perfect Vortex beams) and the theory of partially coherent twisted Gaussian Schell-model (TGSM) beams. While TGSM beams were recognized as the first class of partially coherent fields to encode nonzero OAM via a "twist" phase, and OAM eigenmodes are defined by their quantized topological charge $\ell$, a direct structural link between the second-order statistical properties of individual coherent OAM modes and TGSM beams had not been explicitly established. Specifically, it was unclear whether the covariance matrix (CM) of a single coherent OAM eigenstate could be mapped directly to the CM of a TGSM beam without decomposing the coherent state into a statistical mixture.

Methodology
The authors employ a second-order moment analysis based on the covariance matrix formalism. They define the CM for a cylindrically symmetric coherent OAM eigenmode $\psi(\vec{r}) = U(r)e^{i\ell\phi}$ in terms of the transverse position and wave vector coordinates $(x, k_x, y, k_y)$. By calculating the expectation values of the relevant operators (e.g., $\langle x^2 \rangle$, $\langle k_x^2 \rangle$, $\langle x k_y \rangle$), they derive a universal form for the CM of any such eigenmode.

This derived matrix is then compared term-by-term with the known covariance matrix of a TGSM beam. The authors establish a mapping between the parameters of the coherent beams (radial profile moments $\langle r^2 \rangle$, $\langle k_r^2 \rangle$, and topological charge $\ell$) and the TGSM parameters (beam waist $\sigma$, coherence length $\delta$, and twist phase $u$).

To validate this equivalence, the authors apply the formalism to three specific families of coherent beams:

1. Laguerre-Gaussian (LG) modes: Calculating exact moments for arbitrary radial ($p$) and azimuthal ($\ell$) indices.
2. Perfect Vortex Beams (PVBs): Analyzing the regime where the ring radius $R$ is much larger than the width $w$ ($R \gg w$).
3. Bessel-Gaussian (BG) beams: Analyzing the regime where the Gaussian envelope is wide relative to the Bessel oscillations.

For each family, the authors derive the specific conditions under which the coherent beam's CM corresponds to a physically realizable TGSM beam (satisfying the inequality $|u| \leq 1/(k\delta^2)$). They also investigate the invertibility of this mapping: determining if the CM uniquely identifies the beam parameters within each family.

Key Contributions and Results

1. Universal Structural Equivalence: The primary result is that the covariance matrix of any cylindrically symmetric coherent OAM eigenmode takes a universal form identical to that of a TGSM beam. Both matrices share the same pattern of zero and non-zero entries. Specifically, the off-diagonal blocks representing angular momentum correlations ($\langle x k_y \rangle$ and $\langle y k_x \rangle$) are proportional to $\ell$ for coherent modes and to the twist parameter $u$ for TGSM beams.
2. Parameter Identification: The authors provide explicit term-by-term identification formulas. For a coherent mode, the TGSM beam waist $\sigma^2$ corresponds to $\langle r^2 \rangle/2$, the wave vector variance $\tau^2$ to $\langle k_r^2 \rangle/2$, the twist phase $u$ to $\ell/(k\langle r^2 \rangle)$, and the inverse coherence length squared $1/\delta^2$ to a specific combination of $\langle k_r^2 \rangle$, $\langle r^2 \rangle$, and $\ell$.
3. Second-Order Equivalence Classes: The paper defines a second-order equivalence relation ($\psi_1 \sim \psi_2$) where two beams are equivalent if they share the same CM. Consequently, any two beams in the same equivalence class (e.g., a specific LG mode and its matched TGSM) are indistinguishable regarding all second-order observables. This includes beam-width evolution, far-field divergence, and the $M^2$ beam-quality factor under arbitrary symplectic (ABCD) transformations.
4. Family-Specific Realizability:
   • LG Modes: The CM of any LG mode corresponds to a realizable TGSM beam. However, the mapping is not surjective; not every TGSM beam corresponds to an LG mode because the product of second moments for LG modes is quantized (dependent on integer mode order $N$), whereas TGSM parameters can vary continuously.
   • PVBs and BG Beams: In the regimes where these beams are "perfect" or exhibit quasi-non-diffracting behavior, their CMs correspond to TGSM beams in the low-coherence regime (where $\sigma^2/\delta^2 \gg 1$). In this limit, the spatial structure and OAM content effectively decouple.
5. Uniqueness of Parameter Determination:
   • For LG modes, the CM uniquely and exactly determines the beam parameters ($w_0$ and $p$) given the fixed $\ell$.
   • For PVBs and BG beams, the CM determines the parameters to leading order within the relevant approximations, though exact uniqueness from the full moment equations is not rigorously proven.
6. Completeness of PVBs: The authors prove in the appendix that PVB modes form a complete orthonormal basis in the limit of vanishing ring width ($w \to 0$). For finite $w$, the expansion represents a convolution of the field with a Gaussian kernel, faithfully preserving the OAM spectrum while smoothing radial features smaller than $w$.

Significance and Claims
The paper claims to provide a "direct bridge" between the coherent structured-beam community and the partially coherent community. By demonstrating that individual coherent OAM eigenstates possess a covariance matrix structure identical to TGSM beams, the work allows the extensive "TGSM propagation toolbox" (developed for partially coherent fields) to be applied directly to the second-order moment evolution of coherent beams.

The authors emphasize that this equivalence allows for the design of partially coherent TGSM beams that mimic the propagation dynamics (beam width, divergence, $M^2$) and OAM content of specific coherent eigenstates, without needing to reproduce the full field structure. Conversely, it implies that in experimental scenarios where only second-order statistics are measured, a coherent OAM beam and its matched TGSM beam are indistinguishable. The work is presented as a fundamental theoretical connection rather than a proposal for new experimental setups, though it suggests utility in structured-light applications where second-order properties are the primary concern.