Probability of radiation of twisted photons by classical currents

Abstract: The general formula for the probability of radiation of a twisted photon by a classical current is derived. The general theory of generation of twisted photons by undulators is developed. It is proved that the probability to record a twisted photon produced by a classical current is equal to the average number of twisted photons in a given state. The general formula for the projection of the total angular momentum of twisted photons with given the energy, the longitudinal projection of momentum, and the helicity is obtained. The symmetry property of the average number of twisted photons produced by a charged particle moving along a planar trajectory is found. The explicit formulas for the average number of twisted photons generated by undulators both in the dipole and wiggler regimes are obtained. It is established that, for the forward radiation of an ideal right-handed helical undulator, the harmonic number $n$ of the twisted photon coincides with its projection of the total angular momentum $m$. As for the ideal left-handed helical undulator, we obtain that $m=-n$. It is found that the forward radiation of twisted photons by a planar undulator obeys the selection rule that $n+m$ is an even number. It turns out that the average number of twisted photons produced by the undulator and detected off the undulator axis is a periodic function of $m$ in a certain spectral band of the quantum numbers $m$.

• The paper derives a general formula for the probability of twisted photon radiation from classical currents using the first Born approximation.
• It employs Bessel functions to compute angular momentum distributions and reveals symmetric emission properties for specific current trajectories.
• Applications to undulators in both dipole and wiggler regimes demonstrate potential impacts on quantum communication and astrophysical observations.

Probability of Radiation of Twisted Photons by Classical Currents

Introduction

The study presented in the paper deals with the derivation of a general formula for the probability of radiation of twisted photons from classical currents. This exploration is crucial as twisted photons—quanta of light with orbital angular momentum (OAM)—hold the potential for numerous applications ranging from quantum communication to astrophysical observations. The paper not only derives this general formula but also applies it to specific scenarios like radiation from undulators, both in dipole and wiggler regimes, to comprehend the angular momentum features of the emitted photons.

Derivation of the General Formula

The paper systematically derives the transition amplitude for the radiation of twisted photons by classical currents. The first Born approximation is used to obtain the transition amplitude for the process:

0 → ℓ

where a twisted photon with quantum numbers (s, m, k₃, k⊥) is emitted by a classical current. Here, m denotes the projection of total angular momentum onto the propagation axis, k₃ and k⊥ are momentum components, and s represents helicity. The probability, and hence the average number of photons with specific attributes, is calculated in terms of an integral over the classical current's trajectory, showing proportionality to the Bessel function and exponential terms derived from the space-time coordinates of the current.

Analytical Treatment and Symmetry Properties

The symmetry properties of the emitted twisted photons are identified, specifically for planar trajectories where the expectation is that:

dP(s, m, k₃, k₀) = dP(-s, -m, k₃, k₀)

denoting that for certain trajectories, the distribution over angular momentum projection is symmetric. The paper contributes a detailed formal explanation and pictorial representation of these properties.

Radiation by Undulators

The core application described is the radiation of twisted photons by undulators:

1. Dipole Regime:
   • The analysis in the dipole approximation considers weak undulator strength (K).
   • The radiation spectrum is dominated by lower harmonics, and the forward radiation consists of twisted photons where m=n for right-handed helices, and m=-n for left-handed helices.
   • A selection rule emerges for planar undulators, with only even n+m terms contributing to the radiation.
2. Wiggler Regime:
   • For strong fields (K ≫ 1), the analysis shows increased harmonic generation.
   • Forward radiation primarily emits twisted photons with m aligning with harmonic numbers of the undulator.
   • Angular dependence of photon number indicates potential for applications like photon-mode multiplexing in communications.

Computational Considerations

The computational aspect is tackled through integration over phase space, utilizing properties of Bessel functions to accommodate the complexity introduced by the twisted nature of photons. The mathematical framework accomplishes reliable predictive power of angular momentum distribution and photon energy distributions, applicable to the design of experiments involving twisted photons.

Conclusion

The work done in this paper establishes a comprehensive theoretical framework for understanding and predicting the behavior of twisted photon radiation in diverse electromagnetic settings beyond undulators, potentially benefiting the design of future photonics and communication technologies leveraging structured light.