Electromagnetic plasma wave modes propagating along light-cone coordinates (2511.04554v1)

Abstract: We present new electromagnetic plasma modes that propagates in one time and one space coordinates. Differently to the usual plane wave solution, which is written in terms of separation of variables, all our solutions are along the light-cone coordinates. This allow us to find several new wavepacket solutions whose functionality properties rely on the conditions imposed on the choice for their light-cone coordinates dependence. The presented wavepacket solutions are constructed in terms of multiplications of Airy functions, Parabolic cylinder functions, Mathieu functions, or Bessel functions. We thoroughly analyze the case of a double Airy solution, which have new electromagnetic properties, as a defined wavefront, and velocity faster than the electromagnetic plane wave counterpart solution. It is also mentioned how more general structured wavepackets can be constructed from these new solutions.

• The paper presents exact electromagnetic plasma wave solutions by reformulating the traditional model using light-cone variables.
• It derives structured wavepacket solutions, including Double Airy modes, that exhibit tunable, shock-like propagation and acceleration.
• The study highlights potential for tailored plasma wave engineering and experimental realizability in plasma-based optical devices.

Electromagnetic Plasma Wave Modes Propagating along Light-Cone Coordinates

Introduction and Theoretical Framework

This paper introduces a family of exact, one-dimensional electromagnetic plasma wave solutions characterized by their propagation along light-cone coordinates, $\eta = x-t$ and $\xi = x+t$. Unlike conventional plane wave modes, these solutions emerge from a reformulation of the relativistic cold plasma wave equation in terms of light-cone variables. The authors provide a systematic derivation starting from the continuity, momentum, and Maxwell equations, leading to the central wave equation:

$$\left( \frac{\partial^2}{\partial t^2} - \frac{\partial^2}{\partial x^2} \right) A^z + \omega_p^2 A^z = 0,$$

where $A^z$ represents the electromagnetic potential component and $\omega_p$ is the plasma frequency. Traditionally, this equation is solved using plane wave ansätze with well-established subluminal group velocity constraints; however, the focus here is on alternative functional separability in terms of the light-cone variables, opening up a novel class of solutions not obtainable through standard separation-of-variable techniques.

Structured Wavepacket Solutions in Light-Cone Variables

The central technical advancement lies in seeking separable solutions $A^z(t,x) = f(\rho)g(\chi)$, with $\rho$ and $\chi$ being arbitrary functions of $\eta$ and $\xi$. The transformed wave equation admits analytic solutions for carefully selected forms of $\rho(\eta,\xi)$ and $\chi(\eta,\xi)$, leading to wavepackets expressible in terms of classical special functions.

The principal modes explored include:

• Double Airy Modes: These are constructed by taking $f$ and $g$ as Airy functions, producing explicit solutions with marked asymmetry in light-cone propagation directionality. The method generalizes to allow linear combinations of Airy functions of both kinds.

  Figure 1: Double Airy solution described in Sec.~\ref{douairy} evolving for three times ($\alpha=1$, $\alpha=0.05$, $\omega_p=1$); demonstrates localized, structured propagation as $\alpha \to 0$.

• Double Parabolic Cylinder, Mathieu, Modified Mathieu, and Modified Bessel Modes: These solutions exploit respective canonical forms (Weber, Mathieu, and Bessel equations) for $f$ and $g$, with ancillary parameters introducing arbitrary scaling and asymmetry. The general solution for each is provided in closed analytic form, dependent on arbitrary constants and the nature of the functional transform.

Of particular note, these wavepackets, despite being one-dimensional in spatial extent, do not exhibit constant propagation or energy velocities. The velocity profiles and localized wavefronts are tunable via parameters such as $\alpha$, the scale factor, influencing the dominance of the light-cone directionality.

Propagation Characteristics and Physical Implications

A detailed analysis is presented for the double Airy mode, including the temporal evolution and localization behavior as a function of $\alpha$. Explicit formulae are derived for the wavefront position and velocity:

$$v_{\text{WF}}(t,\alpha) = 1 + \frac{\alpha^6 \omega_p^2}{4}\left( \frac{\alpha^6 \omega_p^2}{2} t + \frac{\alpha^{12}\omega_p^4}{64} + \frac{\alpha^8 \beta\omega_p^2}{4} \right)^{-1/2},$$

demonstrating superluminal phase velocity behavior analogous to, but distinct from, classical plane wave phase velocities.

Figure 2: Double Airy solution described in Sec.~\ref{douairy} at $t=120$ with three $\alpha$ values; vertical lines indicate dynamically evolving wavefront positions and demonstrate parameter-dependent shock front formation.

The energy velocity, defined via the Poynting vector and energy density, is shown to be always subluminal and approaches the speed of light for small $\alpha$:

$$v_\epsilon(t,x,\alpha)\approx 1-\frac{\alpha^6\omega_p^2\Psi^2}{2\xi},$$

where $\Psi$ encapsulates the local wavepacket structure. This behavior underlines the accelerating nature of these plasma modes and positions them as candidates for engineered shock-like propagation.

Wavepacket Superposition and Generalization

A salient feature of the solutions is their linearity, permitting the construction of more complex wavepackets by averaging solutions over distributions of the parameter $\alpha$:

$${\mathfrak A}(t,x,\omega_p,\vec\kappa) = \int \zeta(\alpha)\, A^z(t,x,\omega_p,\alpha,\vec\kappa)\, d\alpha,$$

where $\zeta(\alpha)$ is an arbitrary weight function. This procedure allows the tailoring of wavepacket structure, intensity, and localization properties.

Figure 3: Wavepacket \eqref{wavepacketG} for the Double Airy solution with varying $\zeta(\alpha)=\exp(-\delta\, \alpha^2)$; increasing $\delta$ enhances shock front localization at $t=120$, $\omega_p=1$.

Moreover, the method admits generalization to three-dimensional wavepackets by incorporating transverse Bessel structures, showing that the effective plasma frequency becomes $\sqrt{\omega_p^2 + \lambda^2}$ for each mode with transverse mode index $\lambda$.

Practical and Theoretical Implications

The exact wavepacket solutions described have several implications:

• Experimental Realizability: The fundamentally new structure of these wavepackets suggests the possibility of laboratory realization, especially since previous classes of accelerating structured light have been observed in high-intensity laser-plasma experiments.
• Plasma Wave Engineering: The analytic tunability over wavefront shape, velocity profile, and localization presents new opportunities for tailored plasma wave manipulation, with potential applications in plasma-based optical elements and photonic devices.
• Generalizability: The approach is not restricted to one dimension or specific functional forms, and can be extended to more general plasma environments or more elaborate special function structures.

From a theoretical perspective, the existence of exact, non-eikonal, shock-like solutions to Maxwell’s equations in plasma—without requiring multidimensional extension—challenges conventional views of electromagnetic mode propagation and structure formation in dispersive media.

Conclusion

This paper provides a comprehensive construction of exact, analytic electromagnetic plasma wave modes propagating along light-cone coordinates, fundamentally distinct from the canonical plane wave solutions. Through judicious selection of functional forms and parameters, the authors deliver a toolbox of structured wavepacket solutions with tunable propagation and localization dynamics, including shock-like features and accelerating modes. The demonstrated methods of superposition, generalization, and three-dimensional extension highlight the versatility and relevance of this approach for both theoretical paper and practical plasma-wave engineering. Future directions include experimental validation and application of these modes in controlled plasma settings.