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⁺ (²F_5/2-²F_7/2) and the absorption spectrum of the Er³⁺ (⁴I_13/2-⁴I_15/2) 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.

Einstein’s 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 ²F_5/2 allow the transfer of ions into the excited state of the Er^{3+ 4}I_11/2, by energy cooperation, to then have a non-radiative decay of the ions to the lower level ⁴I_13/2 of the Er³⁺, and finally radiatively dropping to base level ⁴I_15/2.

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)

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

N_j indicates the fractional population level of level j.

The A_jk 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 σ_jk transition (whose values depend on the wavelengths involved), the optical intensities I_p, I_s in the pump, the wavelength of the signal, and the photon energyhν.

Analytical technique for solving Einstein’s 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 ⁴I_15/2 and ⁴I_13/2, respectively; N_Er is the total concentration of Er³⁺ ions; N₄ and N₅ are the concentrations of the Yb³⁺ ions at level ²F_7/2 and ²F_5/2, respectively; N_Yb 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., N_i = N_i(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 Γ_p and Γ_s are the overlapping factors of the pump and the signal, respectively; A_C 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 P_p and P_s 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].

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

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.

The results of the simulation 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.

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.

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.

The authors declare no conflicts of interest regarding the publication of this paper.

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