# A New Type of Shape-Invariant Beams with Structured Coherence: Laguerre-Christoffel-Darboux Beams

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## Abstract

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## 1. Introduction

## 2. Preliminaries

#### 2.1. Shape Invariant Partially Coherent Fields: LG Modes

#### 2.2. Some Closed-Form CSDs with LG Modes

## 3. Laguerre-Christoffel-Darboux Sources

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CSD | Cross-spectral density |

GSM | Gaussian Schell-model |

LG | Laguerre–Gaussian |

LCD | Laguerre–Christoffel–Darboux |

OAM | Orbital angular momentum |

TGSM | Twisted Gaussian Schell-model |

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**Figure 1.**Intensity profiles ${I}_{Nm}$ given by Equation (35) for several values of N and m (as indicated in Figure labels).

**Figure 2.**Maps of the intensity ${I}_{Nm}$ (normalized to the maximum) across the source plane for three different pairs $(N,m)$ (as indicated in the figure labels).

**Figure 3.**Intensity profiles as functions of the radial coordinate and intensity maps (normalized to its maximum) for $N=100$ and $m=25,50,100$.

**Figure 4.**Absolute value of the complex degree of coherence and funtion in the ${r}_{1}$ ${r}_{2}$ plane for $N=1,2,5,10$ (from left to right) and $m=0$ (upper row) and $m=5$ (lower row).

**Figure 5.**Absolute degree of coherence (Equation (40)) relative to three different points, indicated by red dots, for $N=2$, $m=4$.

**Figure 6.**Upper row: intensity profile given by Equation (35) for $N=5$ and several values of m. Second, third and fourth rows: absolute value of the degree of coherence for relative to a point located at a distance ${r}_{2}$ from the source center (indicated with an arrow on the intensity profile and a dashed line in the degree of coherence profile).

**Figure 7.**Upper row: intensity profile given by Equation (35) for $N=20$ and several values of m. Second, third and fourth rows: absolute value of the degree of coherence for relative to a point located at a distance ${r}_{2}$ from the source center (indicated with an arrow on the intensity profile and a dashed line in the degree of coherence profile).

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**MDPI and ACS Style**

Martínez-Herrero, R.; Santarsiero, M.; Piquero, G.; González de Sande, J.C.
A New Type of Shape-Invariant Beams with Structured Coherence: Laguerre-Christoffel-Darboux Beams. *Photonics* **2021**, *8*, 134.
https://doi.org/10.3390/photonics8040134

**AMA Style**

Martínez-Herrero R, Santarsiero M, Piquero G, González de Sande JC.
A New Type of Shape-Invariant Beams with Structured Coherence: Laguerre-Christoffel-Darboux Beams. *Photonics*. 2021; 8(4):134.
https://doi.org/10.3390/photonics8040134

**Chicago/Turabian Style**

Martínez-Herrero, Rosario, Massimo Santarsiero, Gemma Piquero, and Juan Carlos González de Sande.
2021. "A New Type of Shape-Invariant Beams with Structured Coherence: Laguerre-Christoffel-Darboux Beams" *Photonics* 8, no. 4: 134.
https://doi.org/10.3390/photonics8040134