# Comparative Modeling of Infrared Fiber Lasers

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{0}, N

_{1}, and N

_{2}is equal to the total doping concentration N

_{Dy}, and the coefficients a

_{mn}are given by

_{xa}is the absorption cross section for signal s, idler i, and pump p, whereas σ

_{xe}gives the respective values for the emission cross section. The photon flux is ϕ

_{x}for signal s, idler i, and pump p. The branching ratio is β

_{21}for the 2-1 transition (Figure 2), and τ

_{1}and τ

_{2}are the radiative lifetimes for levels 1 and 2, respectively (Figure 2). The rate Equation (1) is complemented by the set of six ordinary differential equations that describe the spatial evolution of the pump, idler, and signal powers for both the forward- and backward-propagating waves along the z axis:

_{x}is the confinement factor for signal s, idler i, and pump p; α gives the loss coefficient; and P

_{p}, P

_{s}, and P

_{i}are the values of the power of the pump, signal, and idler, respectively. The numeric values of the parameters are given in Table 1. A more rigorous approach, which does not use the confinement factor approximation, involves the exact calculation of the overlapping integrals between the ion populations and the electromagnetic field by taking into account the spatial distribution of the optical propagation modes:

_{d}is the rare earth-doped region and i

_{p}, i

_{s}, and i

_{i}are the normalized intensities of the pump, signal, and idler optical modes, respectively.

^{3+}energy level diagram in Figure 3, using the rate equations approach, one obtained consistently the following set of algebraic equations that enabled calculation of the populations of the energy levels:

_{0}, N

_{1}, N

_{2}, N

_{3}, and N

_{4}(Figure 3) is equal to the total doping concentration N

_{Er}. Note that τ

_{1}, τ

_{2}, τ

_{3}, and τ

_{4}are the lifetimes of levels 1, 2, 3, and 4, respectively, whereas β

_{xy}gives the branching ratios from level x to y. W

_{11}and W

_{22}are the cooperative upconversion coefficients for levels 1 and 2, respectively. R

_{GSA}gives the ground state absorption rate, R

_{SE}gives the rate of stimulated emission between levels 1 and 2, and R

_{ESA}gives the rate of the excited state absorption from level 2 to level 4:

_{2}= g

_{1}= 2. The values of the relevant cross sections σ

_{se}, σ

_{ESA}, and σ

_{GSA}, confinement factors Γ

_{x}, wavelengths λ

_{x}, effective cross section A

_{eff}, and Boltzmann factors b

_{x}are given in Table 2. Aligning the fiber with the z axis of the coordinate system enabled the following four differential equations to be written in the following form:

_{s}and P

_{p}are the powers of the signal and pump, respectively, and the superscripts + and − denote the forward- and backward-propagating waves, respectively. In Equation (9), α

_{x}gives the value of loss.

- The fiber laser model developed at the Institute of Photonics and Electronics of the Czech Academy of Sciences (UFE) was implemented in C programming language (gcc 4.9.2) within the Windows 7 operating system, 64 bit Intel core i7-3930K CPU at 3.2 GHz. The UFE model is currently being developed for the study of longitudinal-mode instabilities and associated buildup of dynamic fiber Bragg gratings [40].
- The fiber laser model developed at the Politecnico di Bari (PB) was implemented in MATLAB within the Windows 10 operating system, 64-bit Intel Core i7-4790 CPU at 3.6 GHz. The numerical integration was carried out using a 4-5 Runge–Kutta algorithm, and the more rigorous overlap integrals approach was employed.
- The fiber laser model developed at the University of Nottingham and Wroclaw University of Science and Technology (NU–PWr) was implemented in MATLAB within the Windows 10 operating system, 64 bit Intel Core i5 7th Generation, CPU at 2.5 GHz. The numerical integration was carried out using a 4-5 Runge–Kutta algorithm.

## 3. Results

^{−34}J·s and the value of the speed of light in free space of 2.99792458 × 10

^{8}m/s were used.

^{3+}-doped chalcogenide–selenide glass fiber laser, for the results calculated using the UFE and NU–PWr models, the relative difference, defined as the ratio between the absolute value of the difference and half of the sum of the results, was then less than 0.2% for the signal and below 0.22% for the idler wave at pump powers of 1 W and 5 W, respectively. In the case of the idler wave, the small values of the idler wave power for pump powers of 0.4 W and 0.2 W made it difficult to achieve small values of the relative difference. Nonetheless, these results consistently indicated that the idler was below the lasing threshold. In the case of the Er

^{3+}-doped fluoride glass fiber laser, both the NU–PWr and UFE models calculated results that agreed on all four digits. It is noted that the results shown in Table 4, Table 5 and Table 6 were rounded to the nearest decimal.

## 4. Conclusions

^{3+}-doped chalcogenide–selenide step-index glass fiber and in the case of the Er

^{3+}-doped fluoride glass fiber lasers, a very good agreement was achieved between the results calculated using the UFE and NU–PWr models.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**Dependence of the residual calculated using the Institute of Photonics and Electronics of the Czech Academy of Sciences (UFE) model at z = 0, and the central processing unit (CPU) time of the iteration number at a pump power of: (

**a**) 5 W, (

**b**) 1 W, (

**c**) 0.4 W, and (

**d**) 0.2 W.

**Figure 5.**Dependence of the residual calculated using the Politecnico di Bari (PB) model at z = 0, and CPU time of the iteration number at a pump power of (

**a**) 5 W, (

**b**) 1 W, (

**c**) 0.4 W, and (

**d**) 0.2 W.

**Figure 6.**Dependence of the residual, calculated using the University of Nottingham and Wroclaw University of Science and Technology (NU–PWr) model at z = 0, and CPU time on the iteration number at a pump power of (

**a**) 5 W, (

**b**) 1 W, (

**c**) 0.4 W, and (

**d**) 0.2 W.

**Table 1.**Numerical modeling parameters used in simulations, Dy

^{3+}-doped chalcogenide glass fiber laser.

Quantity | Unit | Value |
---|---|---|

Dy^{3+} ion concentration N_{Dy} | cm^{−3} | 7 × 10^{19} |

A_{eff} | m^{2} | 95 × 10^{−12} |

Fiber length L | m | 2.1 |

Fiber loss at all wavelengths α | dB/m | 1 |

Lifetime of level 2 (Figure 2) | ms | 2 |

Lifetime of level 1 (Figure 2) | ms | 5.2 |

Branching ratio for 2-1 transitions | 0.15 | |

Reflectivity for idler, signal, and pump at z = 0 | 0.2 | |

Reflectivity for idler, signal, and pump at z = L | 0.2 | |

Confinement factor for signal | 0.8 | |

Confinement factor for idler | 0.9 | |

Confinement factor for pump | 0.034 | |

Pump wavelength | μm | 1.71 |

Signal wavelength (λ_{1}) | μm | 4.6 |

Idler wavelength (λ_{2}) | μm | 3.35 |

Pump emission cross section | m^{2} | 0.318 × 10^{−24} |

Pump absorption cross section | m^{2} | 0.501 × 10^{−24} |

Signal emission cross section | m^{2} | 0.912 × 10^{−24} |

Signal absorption cross section | m^{2} | 0.485 × 10^{−24} |

Idler emission cross section | m^{2} | 0.097 × 10^{−24} |

Idler absorption cross section | m^{2} | 0.016 × 10^{−24} |

Quantity | Unit | Value |
---|---|---|

b_{1}/b_{2} | 0.1/0.16 | |

W_{11} | m^{3}/s | 1 × 10^{−24} |

W_{22} | m^{3}/s | 0.3 × 10^{−24} |

σ_{GSA} | m^{2} | 2.1 × 10^{−25} |

σ_{SE} | m^{2} | 4.5 × 10^{−25} |

σ_{ESA} | m^{2} | 1.1 × 10^{−25} |

Γ_{p} | 0.009 | |

Γ_{s} | 1.0 | |

Er^{3+} ion concentration N_{Er} | m^{−3} | 9.6 × 10^{26} |

Pump wavelength λ_{p} | Nm | 976 |

Pump wavelength λ_{s} | Nm | 2800 |

Fiber length L | m | 2.5 |

A_{eff} | m^{2} | 314 × 10^{−12} |

α_{p} | 1/m | 3 × 10^{−3} |

α_{s} | 1/m | 23 × 10^{−3} |

R_{p} (z = 0) | 0 | |

R_{p} (z = L) | 0.04 | |

R_{s} (z = 0) | 0.96 | |

R_{s} (z = L) | 0.04 |

**Table 3.**Branching ratios and level lifetimes for erbium trivalent ions doped into a fluoride glass.

Quantity | Unit | Value |
---|---|---|

τ_{1} | ms | 9 |

τ_{2} | ms | 6.9 |

τ_{3} | ms | 0.12 |

τ_{4} | ms | 0.57 |

β_{21}, β_{20} | 0.37, 0.63 | |

β_{32}, β_{31}, β_{30} | 0.856, 0.004, 0.14 | |

β_{43}, β_{42}, β_{41}, β_{40} | 0.34, 0.04, 0.18, 0.44 |

**Table 4.**Calculated values of signal output power values for the Dy

^{3+}-doped chalcogenide–selenide glass fiber laser.

Pump Power/W | Signal Power (NU–PWr)/W | Signal Power (UFE)/W | Relative Difference |
---|---|---|---|

0.2 | 4.733 × 10^{−3} | 4.731 × 10^{−3} | 0.422 × 10^{−3} |

0.4 | 8.744 × 10^{−3} | 8.736 × 10^{−3} | 0.915 × 10^{−3} |

1 | 49.13 × 10^{−3} | 49.04 × 10^{−3} | 1.833 × 10^{−3} |

5 | 319.1 × 10^{−3} | 318.6 × 10^{−3} | 1.568 × 10^{−3} |

**Table 5.**Calculated values of the idler output power values for the Dy

^{3+}-doped chalcogenide–selenide glass fiber laser.

Pump Power/W | Idler Power (NU–PWr)/W | Idler Power (UFE)/W | Relative Difference |
---|---|---|---|

0.2 W | 0 W | 4.140 × 10^{−6} | NA |

0.4 W | 0 W | 9.591 × 10^{−4} | NA |

1 W | 55.38 × 10^{−3} W | 55.26 × 10^{−3} | 2.169 × 10^{−3} |

5 W | 426.0 × 10^{−3} W | 425.4 × 10^{−3} | 1.409 × 10^{−3} |

**Table 6.**Calculated values of signal output power values for the Er

^{3+}ion-doped fluoride glass fiber laser.

Pump Power | Signal Power (NU–PWr)/W | Signal Power (UFE)/W | Relative Difference |
---|---|---|---|

5 W | 1.432 | 1.432 | 0 |

10 W | 3.171 | 3.171 | 0 |

15 W | 4.868 | 4.868 | 0 |

20 W | 6.458 | 6.458 | 0 |

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**MDPI and ACS Style**

Sujecki, S.; Sojka, L.; Seddon, A.B.; Benson, T.M.; Barney, E.; Falconi, M.C.; Prudenzano, F.; Marciniak, M.; Baghdasaryan, H.; Peterka, P.;
et al. Comparative Modeling of Infrared Fiber Lasers. *Photonics* **2018**, *5*, 48.
https://doi.org/10.3390/photonics5040048

**AMA Style**

Sujecki S, Sojka L, Seddon AB, Benson TM, Barney E, Falconi MC, Prudenzano F, Marciniak M, Baghdasaryan H, Peterka P,
et al. Comparative Modeling of Infrared Fiber Lasers. *Photonics*. 2018; 5(4):48.
https://doi.org/10.3390/photonics5040048

**Chicago/Turabian Style**

Sujecki, Slawomir, Lukasz Sojka, Angela B. Seddon, Trevor M. Benson, Emma Barney, Mario C. Falconi, Francesco Prudenzano, Marian Marciniak, Hovik Baghdasaryan, Pavel Peterka,
and et al. 2018. "Comparative Modeling of Infrared Fiber Lasers" *Photonics* 5, no. 4: 48.
https://doi.org/10.3390/photonics5040048