# Analysis of the Emission Features in CdSe/ZnS Quantum Dot-Doped Polymer Fibers

^{1}

^{2}

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## Abstract

**:**

^{12}.

## 1. Introduction

## 2. Theoretical Model

_{P}), emitting light (P), and the molecule population density in the excited state (N

_{2}) with time (t) and position along the fiber (z) can be analyzed based on a set of rate equations similar to those of rare-earth-doped fiber. It is important to note that the emission spectra of CdSe/ZnS QDs-doped POFs can be turned by changing the fiber length, taking advantage of the significant overlap between the absorption and emission cross-sections of CdSe/ZnS QDs. Therefore, the independent variables have to take into account the wavelength λ, which allows us to carry out computational simulations of spectra and its evolution with fiber parameters, such as QDs-doped concentration and doped fiber length. To introduce this dependence, we divided the absorption and PL spectra into discrete subintervals centered at wavelengths λ

_{k}. Therefore, either the absorption cross-section or the emission cross-section should be λ-dependent, as ${\sigma}_{a}({\lambda}_{k})$, ${\sigma}_{e}({\lambda}_{k})$ or ${\sigma}_{a}({\lambda}_{P})$, ${\sigma}_{e}({\lambda}_{P})$.

_{2}(z,t) is the density of CdSe/ZnS QDs in the excited states, N

_{1}(z,t) is the density of CdSe/ZnS QDs in the non-excited state, and N

_{1}= N − N

_{2}, N is the total density. P(z,t,λ) is the resulting light power at the wavelength λ

_{p}, P

_{p}(z,t) is the pump power. h is the Planck’s constant, c is the speed of light, and τ is the spontaneous lifetime of CdSe/ZnS QDs. A

_{core}is the diameter of doped fiber core, v

_{z}is the speed of light in the fiber core. σ

_{a}(λ

_{k}) and σ

_{e}(λ

_{k}) represent the absorption and stimulated emission cross-section at wavelength λ

_{k}, respectively. Similarly, σ

_{a}(λ

_{p}) and σ

_{e}(λ

_{p}) represent the absorption and stimulated emission cross-section at wavelength λ

_{p}. ∑

_{e-sp}(λ

_{k}) represents the spontaneous emission cross-section at wavelength λ

_{k}.

_{p}(z,t) are considered in the second term. The last term is the absorption and stimulated radiation caused by P(z,t,λ), correspond to the re-absorption effect.

_{p}(z,t)in the fiber core.

_{p}(0) = P

_{0}; P

_{p}(0,t,λ

_{k}) = 0 (k = 1, 2, …, K), which means that the CW pump power is P

_{0}at t = 0 and will be launched into the fiber at z = 0. The aforementioned rate equations under steady-state conditions can be numerically solved through the finite-difference method, and expressed as

## 3. Model Parameters Derived Experimentally

_{clad}and n

_{core}were 1.493 and 1.458, respectively.

_{q}, and can be calculated from an empirical equation [22]: σ

_{a}(λ

_{peak}) = 1600∆ED

^{3}cn

_{q}. The absorption emission cross-section at any arbitrary wavelength could be deduced from the absorption spectra of the QDs. The stimulated emission cross-section can be obtained by solving the Mc-Cumber equation [23]:

_{0}is the intersection of the absorption and stimulated emission spectra, which is 571.3 nm in this paper.

## 4. Results and Discussion

#### 4.1. Analysis of the Spontaneous Emission

_{1}to λ

_{1}+ dλ might be re-absorbed and generated a new light with relative longer wavelength at λ

_{2}to λ

_{2}+ dλ, which would lead to the symmetry of the whole spectrum being broken and the average wavelength will be larger than the peak wavelength. The curves of peak wavelength and average wavelength of two kinds of fiber with doping concentrations of 2 ppm and 5 ppm as a function of doped fiber length are shown in Figure 6.

_{e}and σ

_{overlap}were the emission cross-section and the emission cross-section that can be reabsorbed, respectively. We calculated the overlap coefficients of ten kinds of SE spectra with the peak wavelength from 581.9 to 626.9 nm and intervals of 5 nm, under the assumption that the SE spectra shapes were consistent. The overlapping coefficients were 0.34, 0.27, 0.21, 0.16, 0.12, 0.17, 0.078, 0.064, 0.05, and 0.05, respectively, which gradually decrease with the increase of peak wavelength. The average wavelength red shift of the output spectra at the end of the 17-cm-long fibers with the overlap coefficient for four doping concentrations of 2 ppm, 3 ppm, 4 ppm, and 5 ppm is shown in Figure 9. Under the same doping concentration, the larger the overlap coefficient, the larger the red shift. The maximum red shift under 2 ppm and 5 ppm doping concentration were 8.6 and 29.6 nm. It should be noted that as the SE wavelength becomes larger, the overlap coefficient decreases, so the maximum red shift in the experiment should be somewhat smaller than the value shown in Figure 10. The calculated red shift was smaller than that of 12.0 and 33.1 nm obtained in the experiment shown in Figure 6 because the effect of output SE intensity was not considered. The final red shift of the output spectra was the result of the combined effect of SE intensity and overlap coefficient.

#### 4.2. Analysis of the Amplified Spontaneous Emission

_{av}of the output spectra generated under different pump power were calculated and shown in Figure 11d,e. Special attention was paid to clarify the re-absorption effect, especially during the conversion of SE to ASE.

_{av}and a slow growth of the FWHM. The FWHM of the three kinds of doped fiber with concentrations of 1 ppm, 1.5 ppm, and 2 ppm increased from 26.62, 25.27, and 24.25 nm to their respective maximum values of 27.36 nm, 26.38 nm, and 25.21 nm. The pump powers corresponding to the three maximum FWHM were 43.2, 52.7, and 61.6 mW, and were labeled with vertical lines in Figure 11a–c. All of them were in the places where the output light intensity would increase significantly later. As we continued to increase the pump power, the output light was in a state of transition from SE to ASE, the slope of λ

_{av}became significantly larger, leading to the FWHM of the output spectra narrowing significantly, and a blue shift of the λ

_{av}toward the SE peak wavelength, where the position of the maximum emission cross-section was observed. When the pump power was larger than PTs, the number of modes decreases due to energy transfer from lower power modes to those situated near the emission peak and the ASE generates, the output intensity increases sharply and and the λ

_{av}will blue shift slightly toward the SE peak wavelength, indicating that re-absorption effect was significantly suppressed. However, the slope of FWHM was still high, and it will continue to decrease as ASE intensity increases.

_{av}were greatly affected by the doping concentration. Under the same pump power, the larger the doping concentration, the narrower the FWHM and relatively larger λ

_{av}will be. The output ASE intensity and FWHM of many different fibers with concentrations from 0.5 ppm to 5 ppm under the pump power of 150 mW were calculated and shown in Figure 12. The output light grows rapidly with the increase of doping concentration, the ASE intensity of the 5 ppm POF was about 150 times larger than that of the 0.5 ppm POF. Correspondingly, the FWHM decreased from 20.58 nm to 7.85 nm.

_{av}of these POFs with concentrations from 0.5 ppm to 5 ppm under the pump power of 50 mW, 100 mW, 150 mW, and 200 mW are shown in Figure 13. The output λ

_{av}was much larger when the pump power was 50 mW, which might be attributed to the fact that the pump power was not large enough to reach the ASE PTs, so the output light was mainly generated by the SE. When the pump power up to 100 mW, 150 mW, and 200 mW, the main output light was generated by ASE, and the λ

_{av}increased almost linearly, which means that although the re-absorption effect will decrease, it will still exist and affect the λ

_{av}slightly.

^{12}. This means that if the total number of QDs in POFs is insufficient, ASE will not be generated, and all the output light is produced by SE.

## 5. Conclusions

^{12}. The analyses shown in this work might be helpful to design amplifiers and lasers based on QDs and POFs.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**TEM image of the 3.8 nm CdSe/ZnS QDs, and the corresponding emission and absorption spectra.

**Figure 3.**The comparison of spontaneous emission spectra between calculation and experiment under different fiber lengths (1–15 cm) in which the directions of the arrows represent the increase of the fiber length from 1 to 2, 3, 4, 5, 7, 10, 12, and 15 cm.

**Figure 4.**Peak wavelength of the CdSe/ZnS QD-doped POFs as a function of the doped fiber length (the solid lines were the simulation results, and the dots were the experimental results).

**Figure 5.**Peak wavelength and output SE intensity of a 15-cm-long POFs with concentration of 2 ppm as a function of time.

**Figure 6.**Peak wavelength and average wavelength of two kinds of fiber with doping concentrations of 2 ppm and 5 ppm as a function of doped fiber length (the solid lines were the simulation results, and the dots were the experimental results).

**Figure 7.**Output SE intensity of four kinds of POFs as a function of doped fiber length (the solid lines were the simulation results, and the dots were the experimental results).

**Figure 8.**FWHM of the output SE spectra as a function of doped fiber length (the solid lines were the simulation results and the dots were the experimental results).

**Figure 9.**The overlap coefficients between the absorption and emission cross-sections with different peak wavelengths.

**Figure 11.**The output light intensity at the end of the 17-cm-long POFs with concentrations of (

**a**) 1 ppm, (

**b**) 1.5 ppm and (

**c**) 2 ppm. The corresponding λ

_{av}(

**d**) and FWHM (

**e**) of the output light as a function of the pump power.

**Figure 14.**The ASE intensity of 20 cm-long QDs-doped POFs as function of pump powers with concentrations of 0.1 ppm–5 ppm.

Parameter | Notation | Value |
---|---|---|

Pump wavelength | λ_{peak} | 473 nm |

Core radius | r | 66 μm |

Number density of QDs | n_{q} | 1.8 6× 10^{15} cm^{−3} |

Absorption cross-section (558.2 nm) | σ_{a}(λ_{peak}) | 6.22 × 10^{−22} m^{2} |

Emission cross-section (581.9 nm) | σ_{e}(λ_{peak}) | 6.55 × 10^{−22} m^{2} |

Absorption cross-section (473 nm) | σ_{a}(λ_{p}) | 7.85 × 10^{−22} m^{2} |

Emission cross-section (473 nm) | σ_{e}(λ_{p}) | 0 |

Spontaneous lifetime | τ | 20 ns |

_{q}corresponding to l ppm concentration of CdSe/ZnS QDs.

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**MDPI and ACS Style**

Peng, X.; Wu, Z.; Ye, C.; Ding, Y.; Liu, W. Analysis of the Emission Features in CdSe/ZnS Quantum Dot-Doped Polymer Fibers. *Photonics* **2023**, *10*, 327.
https://doi.org/10.3390/photonics10030327

**AMA Style**

Peng X, Wu Z, Ye C, Ding Y, Liu W. Analysis of the Emission Features in CdSe/ZnS Quantum Dot-Doped Polymer Fibers. *Photonics*. 2023; 10(3):327.
https://doi.org/10.3390/photonics10030327

**Chicago/Turabian Style**

Peng, Xuefeng, Zhijian Wu, Chen Ye, Yang Ding, and Wei Liu. 2023. "Analysis of the Emission Features in CdSe/ZnS Quantum Dot-Doped Polymer Fibers" *Photonics* 10, no. 3: 327.
https://doi.org/10.3390/photonics10030327