Finite Orbital Angular momentum Bessel beams propagating along light-cone coordinates

This paper investigates new solutions for Bessel electromagnetic beams propagating along light-cone coordinates, specifically focusing on an asymmetric structure involving a product of Airy functions and analyzing the conditions under which these beams carry finite orbital angular momentum density.

The Gist

Imagine you are shining a flashlight. In the classic physics textbook, the light beam is treated like a perfect, straight arrow flying through the air at the speed of light. It's simple, predictable, and boring. But what if light could be more like a swirling tornado or a spiraling galaxy? What if it could twist as it moves, carrying a hidden "spin" with it?

This paper is about discovering a new, exotic type of light beam that does exactly that. The authors, Felipe Asenjo and Swadesh Mahajan, have found a mathematical recipe for creating light beams that are far more complex and interesting than the standard "straight arrow" we are used to.

Here is the breakdown of their discovery using everyday analogies:

1. The "Light Cone" Highway

Usually, we think of light moving forward in a straight line (like a car on a highway). The authors decided to look at light moving along a "light cone."

• The Analogy: Imagine a highway that splits into two lanes: one lane going slightly into the future and one slightly into the past, but both converging on the same destination. These are the "light-cone coordinates."
• The Discovery: Most light beams just drive straight down the middle. These new beams, however, can weave between these two lanes in a very specific, twisted way.

2. The "Bessel" Twist (The Donut Shape)

The paper focuses on "Bessel beams."

• The Analogy: Think of a standard laser pointer as a solid cylinder of light. A Bessel beam is more like a donut or a ring. It has a bright center, a dark hole, and then rings of light around it.
• The Magic: These donut-shaped beams have a special property called Orbital Angular Momentum (OAM). Imagine a figure skater spinning. A standard laser is just sliding forward; a Bessel beam is spinning as it slides. It carries a "twist" or a "vortex" of energy.

3. The "Airy" Recipe (The Secret Sauce)

This is the most exciting part of the paper. The authors found a new way to build these spinning donut beams. They used a mathematical ingredient called the Airy function.

• The Analogy: If a standard light beam is a simple loaf of bread, this new beam is a marble cake or a swirled pastry.
• The "Double Airy" Structure: The authors mixed two Airy functions together (like mixing two different flavors of batter). This created a beam that is asymmetric.
  • What does that mean? Imagine a spinning top. Usually, it spins perfectly symmetrically. This new beam is like a top that is slightly lopsided. It leans more heavily toward one side of its path than the other.
  • Because of this lopsidedness, the beam doesn't just move forward; it has a complex internal dance. It's not just a simple wave; it's a structured, twisting packet of energy.

4. Why It Doesn't Go "Full Speed"

You might think light always travels at the speed of light ($c$).

• The Analogy: Imagine a runner on a track. A standard plane wave is a sprinter running in a perfectly straight line at top speed.
• The Reality: This new Bessel beam is like a runner who is also dancing or spinning while running. Because all that energy is going into the "twist" and the "dance" (the transverse structure), the runner moves forward slightly slower than the speed limit. The beam travels just a tiny bit slower than light because it's so busy spinning and twisting.

5. Why Should We Care?

The authors show that these beams carry Orbital Angular Momentum (OAM).

• The Analogy: Think of light as a delivery truck.
  • Old Light: Delivers a package (energy) from Point A to Point B.
  • New Light: Delivers a package and a spinning top.
• The Application: This "spinning top" (the OAM) can be used to grab and spin microscopic particles, like tiny gears in a machine made of atoms. It opens the door to new ways of manipulating matter at the microscopic level, potentially leading to better optical tweezers, faster data transmission (since you can encode data in the "twist"), and new ways to study the universe.

Summary

The paper is essentially saying: "We found a new way to make light that isn't just a straight line. It's a twisting, donut-shaped, slightly lopsided wave that spins as it travels. It's mathematically complex (using Airy functions), but it creates a beam of light that carries a unique 'spin' that could be incredibly useful for future technology."

They didn't just find a new wave; they found a new shape for light that behaves more like a fluid vortex than a rigid beam, and they proved it works perfectly within the laws of physics.

Technical Summary

1. Problem Statement

Standard electromagnetic wave solutions in a vacuum are typically plane waves propagating at the speed of light. While Bessel beams are a known class of non-diffracting solutions that possess intrinsic orbital angular momentum (OAM) and propagate along a specific axis (e.g., $z$) with a transverse Bessel function profile, they are often treated within the paraxial approximation or as simple plane wave extensions.

The authors investigate whether the vacuum Maxwell equations support more complex, exact solutions that:

1. Propagate along light-cone coordinates ($\eta = z - t$ and $\xi = z + t$) rather than a single linear direction.
2. Exhibit structural asymmetry between these light-cone coordinates.
3. Carry finite orbital angular momentum (OAM) density without relying on the paraxial approximation.
4. Propagate at sub-luminal speeds due to effective inertia induced by transverse structure.

2. Methodology

The study begins with the vacuum Maxwell equations expressed via the vector potential $\mathbf{A}$ and scalar potential $\Phi$ in the Lorentz gauge. The dynamics are governed by the wave equation:
$$\left( \frac{\partial^2}{\partial t^2} - \nabla^2 \right) \mathbf{A} = 0$$

Key Steps in the Derivation:

• Separation of Variables: The authors assume a separable solution for the vector potential $\mathbf{A}(t, \mathbf{x}) = a(t, z)\mathbf{u}(x, y)$, where $a(t, z)$ describes propagation in the $z-t$ plane and $\mathbf{u}(x, y)$ describes the transverse structure.
• Transverse Structure: The transverse part $\mathbf{u}$ is solved to yield Bessel functions ($J_l$), establishing the "Bessel beam" character.
• Light-Cone Transformation: The propagation function $a(t, z)$ is transformed into light-cone coordinates $\eta = z - t$ and $\xi = z + t$. The authors introduce a separable ansatz $a = f(\rho)g(\chi)$, where $\rho$ and $\chi$ are specific combinations of $\eta$ and $\xi$.
• Simplification via Ansatz: By choosing specific functional forms for $\rho(\eta, \xi)$ and $\chi(\eta, \xi)$ (specifically $\rho = \Theta(\eta) + \Phi(\xi)$ and $\chi = \Theta(\eta) - \Phi(\xi)$), the wave equation is reduced to a separable form involving arbitrary functions $f$ and $g$.
• Specific Solution Selection: The authors focus on the case where $f$ and $g$ are Airy functions ($\text{Ai}$), satisfying $\partial^2 f / \partial \rho^2 = \rho f$ and $\partial^2 g / \partial \chi^2 = \chi g$. This leads to a "Double Airy-Bessel" solution.
• Field Calculation: The electric ($\mathbf{E}$) and magnetic ($\mathbf{B}$) fields are derived from the potentials. The Poynting vector ($\mathbf{S}$) and OAM density ($\mathbf{L} = \mathbf{r} \times \mathbf{S}$) are calculated exactly, without neglecting time-derivative terms usually dropped in paraxial approximations.

3. Key Contributions

• New Exact Solutions: The paper presents a novel class of exact electromagnetic wavepackets that propagate along light-cone coordinates, distinct from standard plane waves or paraxial Bessel beams.
• Double Airy-Bessel Beam: The primary contribution is the derivation of a solution where the longitudinal propagation profile is the product of two Airy functions with arguments dependent on $\sqrt{z+t}$ and linear combinations of $z-t$.
• Asymmetry and Tunability: The solution exhibits a fundamental asymmetry between the two light-cone coordinates. By tuning an arbitrary parameter ($\alpha$), the beam's propagation direction can be biased to travel almost purely along one light-cone coordinate ($\eta$ or $\xi$) at a velocity arbitrarily close to, but strictly less than, the speed of light.
• Finite OAM in Exact Regime: The authors demonstrate that these beams carry finite OAM density. Crucially, they show that unlike plane waves where OAM depends only on transverse structure, the OAM of these beams has a fully four-dimensional structure (dependent on both space and time) due to the non-trivial time dependence of the Airy product.
• Generalization: The methodology is shown to be applicable to other special functions, including Parabolic Cylinder, Mathieu, and Modified Bessel functions (detailed in the appendices).

4. Results

• Propagation Characteristics: The derived "Double Airy-Bessel" beam does not propagate at the speed of light ($c$). The effective inertia induced by the non-zero transverse dependence ($\lambda \neq 0$) results in sub-luminal propagation.
• Poynting Vector: The energy flow (Poynting vector) is not confined to the propagation axis; it possesses non-zero components in the transverse ($r-\phi$) plane. This transverse flow is responsible for the sub-luminal speed.
• OAM Density Formula: The paper derives an exact expression for the longitudinal OAM density ($L_z$):
  $$L_z = \frac{\lambda r}{2} \sin(2l\phi) \left[ \frac{ab}{r} J_l^2(\lambda r) - \lambda ab J_l(\lambda r)J_{l-1}(\lambda r) + \left( a \frac{\partial a}{\partial t} - \frac{\partial a}{\partial z} \frac{\partial b}{\partial z} \right) J_l(\lambda r)J_{l+1}(\lambda r) \right]$$
  This expression confirms that $L_z$ is non-zero for all angular momentum modes $l \neq 0$ and includes terms involving time derivatives of the propagation functions, which are typically neglected in standard treatments.
• Complex Variants: The authors also construct complex-valued solutions (using combinations of $\text{Ai}$ and $\text{Bi}$ or imaginary parameters) that maintain OAM properties, expanding the solution space.

5. Significance

• Theoretical Advancement: This work extends the theoretical understanding of vacuum electromagnetic waves beyond the plane wave and paraxial Bessel beam approximations. It proves that the vacuum wave equation supports highly structured, asymmetric, and exact solutions with rich longitudinal dynamics.
• Experimental Potential: The authors suggest that these complex wavepackets, while mathematically intricate, could be experimentally realizable in laboratory settings, similar to how standard Bessel beams are generated.
• OAM Applications: The discovery of exact solutions with finite, time-dependent OAM density opens new avenues for applications in optical manipulation, quantum information, and high-intensity laser physics where precise control of angular momentum and propagation speed is required.
• Foundation for Future Work: The paper establishes a framework for generating a wide variety of "Light-Cone Bessel Beams" using different special functions, suggesting a vast, unexplored space of electromagnetic wavepackets.

In summary, Asenjo and Mahajan have successfully identified and characterized a new family of exact electromagnetic solutions that combine Bessel transverse profiles with Airy-function-based longitudinal propagation along light-cone coordinates, revealing unique properties regarding asymmetry, sub-luminal speed, and complex orbital angular momentum.