# Twisted Soft Photon Hair Implants on Black Holes

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## Abstract

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## 1. Introduction

**J**of a particle. This quantity can be decomposed in spin angular momentum (SAM),

**S**, related to polarization, and orbital angular momentum (OAM),

**L**.

**L**and

**S**from the classical field formulation [21,22,23] down to the single photon level, where intensity corresponds to the probability of generating a photon in a specific region of spacetime [24,25,26]. In contrast to plane waves (which carry linear momentum with no azimuthal component), those fields can have nonzero total azimuthal momentum ${p}_{\varphi}$ but an identically zero azimuthal component of linear momentum [27]. EM-OAM is currently applied in many research fields [28] and technologies: quantum and classical communications [29,30,31], astrophysics [32], and nanotechnology. Prototype examples of fields carrying specific SAM and OAM eigenvalues are Laguerre–Gaussian (LG) modes that provide an orthogonal fundamental basis to expand any OAM field and present a well-known spatial structure in intensity and phase distributions. They represent cylindrically symmetric structured EM beams that carry $l\hslash $ OAM per photon relative to their symmetry axis with amplitude, in cylindrical coordinates $(r,\phi ,z)$,

## 2. Results

## 3. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Normalized electric field (E) probability of photon emission (intensity) and phase distribution of an $l=1$ vortex generated by four dipoles oriented across the y-axis (upper panel). The four dipoles—much smaller than $\lambda $—are distributed on a circle with radius $\lambda /10$. In the lower panel we find the probability of associating a soft photon hair in the soft photon implant and the spatial phase information generated by these currents. Units are in $\lambda $, and the field properties also remain valid for any frequency when $\nu \to 0$.

**Figure 2.**(

**Left**): spiral spectrum of the radiation field emitted by the dipolar currents shows a dominant contribution of the $l=1$ vorticity of the source; (

**Right**): spiral spectrum of the soft photon field is peaked on a dominant value ($l=2$), which is one unit larger that the orbital angular momentum (OAM) peak of the real photon field.

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**MDPI and ACS Style**

Tamburini, F.; Laurentis, M.D.; Licata, I.; Thidé, B. Twisted Soft Photon Hair Implants on Black Holes. *Entropy* **2017**, *19*, 458.
https://doi.org/10.3390/e19090458

**AMA Style**

Tamburini F, Laurentis MD, Licata I, Thidé B. Twisted Soft Photon Hair Implants on Black Holes. *Entropy*. 2017; 19(9):458.
https://doi.org/10.3390/e19090458

**Chicago/Turabian Style**

Tamburini, Fabrizio, Mariafelicia De Laurentis, Ignazio Licata, and Bo Thidé. 2017. "Twisted Soft Photon Hair Implants on Black Holes" *Entropy* 19, no. 9: 458.
https://doi.org/10.3390/e19090458