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 OJAppS  Vol.11 No.9 , September 2021
Mathematical Modelling of Einstein’s Rate Equations for Zinc Phosphate Glass with Er3+-Yb3+
Abstract: There is a constant growth in the demand of data information transmission capacity, that is, more and more people send data, voice, video signals, among others, through communications networks. Due to the above there is great interest in improving network devices, such as optical amplifiers, which must cover a large bandwidth and generate greater gain than those currently available. For this reason in this work a computational simulation for a Quasi-system was carried out three energy levels of Erbium and Ytterbium varying their concentrations and proving that they are optimal candidates in a zinc phosphate matrix as this type of glass contains properties such as, high transparency, low melting point, high thermal stability, high gain density due to high solubility, low refractive index and low dispersion, which makes them optimal as signal amplifiers. The results confirm that by increasing the doping of the Erbium ion the gain of the amplifier decreases, contrary to the Ytterbium ion that by increasing the doping the gain of the amplifier increases.

1. Introduction

Amorphous solids such as phosphates have been of great importance when doped with rare earth ions, because their excellent properties have applications in solid state lasers, optical amplifiers, three-dimensional screens, among others [1].

In fact, phosphate glasses allow a high concentration of rare earth ions (RE rare earth) (up to 1021 ions/cm3) to dissolve in the glass matrix without clumping, because of the presence of phosphorus, which introduces nonbridging oxygen (depending on its position), as shown in Figure 1, which will become a chain-like structure, compared to the random silicate glass network.

This allows the manufacture of various devices with high energy gain. In addition, this type of glass contains properties such as, high transparency, low melting point, high thermal stability, high gain density which is due to high solubility, low refractive index and low dispersion [2].

Specifically since 1995 zinc phosphate has been a case study because glass can be manufactured in a simpler way with ZnO contents, in addition the glass formation intervals of most other binary phosphates are generally more limited [3].

On the other hand, in optical communication networks, the information signals that travel in the optical fiber have to travel very long distances without presenting significant attenuation that prevents the recovery of the signal from the receiving side. However, when the distances covered are tens or even hundreds of kilometers, it is necessary to amplify the signal.

An optical amplifier is a device that amplifies an optical signal directly without converting it into an electrical signal. Amplified spontaneous fiber optic amplifiers provide in-line amplification of the optical input signals by the stimulated emission of photons by rare-earth ions that are implanted in the core of the fiber optic.

In recent years, the Er3+-Yb3+ co-doped glass waveguide (fiber optic) laser and amplifiers have been of great interest due to its advantages in cost, size, high gain, in addition to having a wide bandwidth, transparency at wavelengths and independence from polarization [4].

It is known that an Er-doped fiber optic amplifier (EDFA, Erbium-Doped Fiber Amplifier) requires a high concentration of Erbium ions (Er3+). However, a high concentration of Er3+ will reduce the spacing between the ions, the overlap between the electron clouds becomes severe, which causes the transfer of energy between the Er3+ ions, increasing the uptake of the excited state. Clustering greatly reduces the efficiency of external excitation (pump) and degrades performance to gain.

Fortunately, the Ytterbium element, as a sensitizer, has a better overlap between the emission spectrum of the Yb3+ (2F5/2-2F7/2) and the absorption spectrum of the Er3+ (4I13/2-4I15/2) also a wide and intense absorption in the wavelength range of 800 to 1080 nm, has a weak grouping effect and a large cross-sectional absorption compared to Erbium, therefore, a high concentration of dopant (Yb3+) can be performed in the waveguide, so the co-doping agent Erbium-Ytterbium (Er3+-Yb3+) can efficiently improve the gain characteristics in the waveguide amplifiers [5] [6] [7].

On the other hand, it is important to know whether the glass is stable or tends to crystallize, as this phenomenon could take place during the manufacturing process of an optical fibre. In particular, the crystallisation temperature of the glass must be sufficiently above the temperature of this process to avoid the possible recrystallization phenomenon. In addition, the thermal expansion coefficient of the central glass should be slightly higher than that of the coating glass and the viscosity behaviors of the two crystals should be compatible. Reason that the viability of zinc phosphate glass as a host matrix in doping is proposed Er3+-Yb3+ [8].

Therefore, in this project, Einstein’s equations of ratio for Zn3(PO4)2 glasses co-doped with Er3+-Yb3+, at different doping concentrations are mathematically modeled, confirming that they are optimal candidates as signal amplifiers.

2. Methodology

Einsteins rate equations

Einstein’s ratio equations were implemented for a Quasi-three energy level model for the Er3+-Yb3+ co-doping.

In the Er3+-Yb3+ co-doping model of Figure 1, both Er3+ and Yb3+ are excited at 980 nm, Yb3+ acts as a sensitizer for the Er3+ molecules contained in the matrix. The Yb3+ ions at their excited level 2F5/2 allow the transfer of ions into the excited state of the Er3+ 4I11/2, by energy cooperation, to then have a non-radiative decay of the ions to the lower level 4I13/2 of the Er3+, and finally radiatively dropping to base level 4I15/2.

This gives the system two forms of pumping, through external excitation and through energy cooperation due to the sensitization of both rare earths, thus improving the emission power. The stimulated optical absorption and emission transitions are due to the pumping beam and the signal beam. For this study only the transitions between the three lowest levels are considered, assuming that the excited state to higher levels and the up conversion processes are weak. the system of equations obtained is as follows:

d N 1 d t = W 12 N 1 W 13 N 1 + A 21 N 2 + W 21 N 2 (1)

d N 2 d t = W 12 N 1 A 21 N 2 W 21 N 2 + A 32 N 3 (2)

Figure 1. Energy levels and transitions in a gain medium co-doped with Er3+-Yb3+.

d N 3 d t = W 13 N 1 A 32 N 3 + A 43 N 4 (3)

d N 4 d t = W 45 N 4 + A 54 N 5 + C c r N 1 N 5 (4)

where:

W ( i j ) = σ i j ( ν S ) / h ν S

W ( i j ) = σ i j ( ν p ) / h ν p

Nj indicates the fractional population level of level j.

The Ajk parameters indicate the spontaneous transition rates from level j to k, with units of s−1.

In addition, the equations contain absorption and stimulated emission rates, which are determined by the cross-sections of each σjk transition (whose values depend on the wavelengths involved), the optical intensities Ip, Is in the pump, the wavelength of the signal, and the photon energy.

Analytical technique for solving Einsteins rate equations

In this analytical proposal it is important to consider the following:

Assuming that N1 and N2 are the concentrations of Er3+ ions at levels 4I15/2 and 4I13/2, respectively; NEr is the total concentration of Er3+ ions; N4 and N5 are the concentrations of the Yb3+ ions at level 2F7/2 and 2F5/2, respectively; NYb is the Yb3+ total ion concentration. Under the conditions of the uniform dopant and the steady state, the ion Er3+ and the ion Yb3+ at the corresponding levels depend on the wavelength guide z, i.e., Ni = Ni(z). Therefore, the multilevel rate equations for a system co-doped with Er3+-Yb3+ is given by Yu-Hai Wang et al. [5]

σ 12 ( ν S ) P S ( z ) Γ s A c h ν S N 1 ( z ) + σ 13 ( ν p ) P p ( z ) Γ p ( A c h ν p ) N 1 ( z ) σ 21 ( ν S ) P S ( z ) Γ s A c h ν S N 2 ( z ) N 2 ( z ) τ 21 + σ 45 ( ν p ) P p ( z ) Γ p A c h ν p N 4 ( z ) σ 54 ( ν p ) P p ( z ) Γ p A c h ν p N 5 ( z ) N 5 ( z ) τ 54 = 0 (5)

With

N 1 ( z ) + N 2 ( z ) = N E r (6)

N 4 ( z ) + N 5 ( z ) = N Y b (7)

where Γp and Γs are the overlapping factors of the pump and the signal, respectively; AC is the cross-sectional area of the glass; σ 12 ( ν S ) and σ 21 ( ν S ) are the cross-sectional absorption and emission signals respectively; σ 13 ( ν p ) the pump absorption cross-section; σ 45 ( ν p ) and σ 54 ( ν p ) are the pump absorption and emission cross-sections, respectively; h is the Planck constant. Letting Pp and Ps be the bomb powers and steady state signal, respectively, which satisfy the following transmission equations:

d P p ( z ) d z = Γ p [ σ 13 ( ν p ) N 1 ( z ) + σ 45 ( ν p ) N 4 ( z ) σ 54 ( ν p ) N 5 ( z ) ] P p ( z ) (8)

d P s ( z ) d z = Γ s [ σ 21 ( ν s ) N 2 ( z ) σ 12 ( ν s ) N 1 ( z ) ] P s ( z ) (9)

with

G ( z ) = P s ( z ) P s ( 0 )

σ = σ 12 + σ 21 σ 13 + ( σ 45 + σ 54 ) 1 η 0 η 0 ( σ 13 + σ 45 N Y b N E r ) σ 12 (10)

where G(z) will be the amplifier gain.

Computer Simulator

The mathematical model based on Einstein’s rate equations was implemented in the Matlab numerical computation system.

The parameters used in the first simulation were [5].

3. Results and Discussion

Below are the gain graphs in (dB) versus glass length in (m), varying the concentration of Er by 1.0%, 2.0%, 3.0%, 4.0% and 5.0% and leaving Yb fixed at 1.0%.

And likewise varying the concentrations of Yb in 1.0%, 2.0%, 3.0%, 4.0% and 5.0%, leaving Er fixed in 1.0%.

In Figure 2 the values shown in Tables 1-5 are used, we can observe that in a

Figure 2. Representation of gain (dB) vs Length (m) at different concentrations of Yb (1.0%, 2.0%, 3.0% 4.0% and 5.0%,), Er at 1.0%.

Table 1. Parameters used for pump length and signal.

Table 2. Er3+-Yb3+ emission lifetime parameters.

Table 3. Area parameters and overlapping factors.

Table 4. Parameters used for signal power and ion concentration Er3+-Yb3+.

Table 5. Parameters used in the absorption and emission cross section.

ratio of Yb 1.0% - Er 1.0% you can obtain a gain of approximately 15 dB, when now the concentration of Yb increases to 2.0% the gain can reach approximately 28 dB, By further increasing the concentration of the Yterbium ion the maximum gain is maintained at approximately 28 dB, with a length of glass up to 0.02 m.

In Figure 3 we can observe in a ratio of Er 1.0% - Yb 1.0% which is equally 15 dB as in Figure 2, by increasing the concentration of the Erbium ion to 2.0% the gain decreases by about 6 dB and each time the concentration of Erbium is increased this phenomenon is repeated, which confirms what is established in the theoretical part where a high concentration of Er3+ will reduce the spacing between the ions, the overlap between the electron clouds becomes severe, which will cause the transfer of energy between the Er3+ ions, increasing the uptake of the excited state. Clustering greatly reduces the efficiency of external excitation (pump) and degrades performance to gain.

4. Conclusions

The results of the simulation confirm that by increasing the doping of the

Figure 3. Representation of gain (dB) vs Length (m) at different concentrations of Er (1.0%, 2.0%, 3.0% 4.0% and 5.0%), Yb at 1.0%.

Erbium ion the gain of the amplifier decreases, contrary to the Ytterbium ion that by increasing the doping the gain of the amplifier increases.

With a glass length of 0.02 m and a concentration ratio of 1% Er - 2% Yb the maximum gain obtained is 28 dB. The highest gain was 28 dB with a ratio of 1% Er-2% Yb. By increasing the concentration of Erbium by 1% - 2% and leaving the concentration of Ytterbium fixed at 1%, there is a decay of the gain of approximately 8 dB.

From Figure 2 and Figure 3, it can be verified that the Ytterbium element, as a sensitizer, has a better overlap between the emission spectrum of Yb3+ and the absorption spectrum of Er3+, by increasing the concentration of dopant.

It will be important to consider in the future the change in amplifier gain with parameters such as pumping power in order to analyze and obtain the ideal design characteristics of the glass.

Acknowledgements

The authors are grateful for the support of CONACyT through the project through the projects 254280 and 285600, Frida Lissete Flores Rivera thanks to the National Council of Science and Technology (CONACyT-Mexico) for the scholarship received during her Postgraduate studies.

Cite this paper: Rivera, F. , Pérez-Sánchez, G. , Barron-Meza, M. , Miranda, J. and Velázquez, D. (2021) Mathematical Modelling of Einstein’s Rate Equations for Zinc Phosphate Glass with Er3+-Yb3+. Open Journal of Applied Sciences, 11, 1038-1045. doi: 10.4236/ojapps.2021.119076.
References

[1]   Hraiech, S., Ferid, M., Guyot, Y. and Boulon, G. (2018) Spectroscopic Characterization and Temperature-Dependent Upconversion Behavior of Er3+ and Yb3+ Co-Doped Zinc Phosphate Glass. Journal of Luminescence, 197, 153-158.
https://doi.org/10.1016/j.jlumin.2018.01.029

[2]   Boetti, N.G., Scarpignato, G.C., Lousteau, J., Pugliese, D., Bastard, L., Broquin, J.-E. and Milanese, D. (2015) High Concentration Yb-Er Co-Doped Phosphate Glass for Optical Fiber Amplification. Journal of Optics, 17, Article ID: 065705.
https://doi.org/10.1088/2040-8978/17/6/065705

[3]   Brow, R.K., Tallant, D.R., Myers, S.T. and Phifer, C.C. (1995) The Short-Range Structure of Zinc Polyphosphate Glass. Journal of Non-Crystalline Solids, 191, 45-55.
https://doi.org/10.1016/0022-3093(95)00289-8

[4]   Vasudevan, B., Sivasubramanian, A. and Ramesh Babu, M. (2017) Optical Study on Er-Yb Co-Doped Boro-Tellurite Glasses for Optical Amplifiers. Journal of Optoelectronics and Advanced Materials, 19, 11-15.

[5]   Wang, Y.-H., Ma, C.-C., Li, D.-L. and Zhang, D.-M. (2008) Formulized Analytical Technique for Gain Characteristics of Phosphate Glass Er3+/Yb3+ Co-Doped Waveguide Amplifiers. Optica Applicata, 38, 329-339.

[6]   Xu, S.-H., Yang, Z.-M., Feng, Z.-M., Zhang, Q.-Y., Jiang, Z.-H., Xu, W.-C. (2009) Efficient Fibre Amplifiers Based on a Highly Er3+/Yb3+ Codoped Phosphate Glass-Fibre. Chinese Physics Letters, 26.

[7]   Langar, A., Bouzidi, C., Elhouichet, H. and Férid, M. (2013) Er-Yb Codoped Phosphate Glasses with Improved Gain Characteristics for an Efficient 1.55 mm Broadband Optical Amplifiers. Journal of Luminescence, 148, 249-255.
https://doi.org/10.1016/j.jlumin.2013.12.008

[8]   Seneschal, K., Smektala, F., Bureau, B., Le Floch, M., Jiang, S., Luo, T., Lucas, J. and Peyghambarian, N. (2005) Properties and Structure of High Erbium Doped Phosphate Glass for Short Optical Fibers Amplifiers. Materials Research Bulletin, 40, 1433-1442.
https://doi.org/10.1016/j.materresbull.2005.05.004

 
 
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