Optimization of bismuth-doped fiber amplifier gain spectrum based on genetic algorithm in 1200 nm-1600 nm

With the continuous evolution of communication technology, and the rapid rise of fiber optic communication in China However, the transmission distance of fiber optic communication is limited due to its fiber loss, so fiber optic amplifier has become one of the key research topics today. Currently on the market is a large-scale commercial bait-doped fiber amplifier, although it has the advantage of low noise and high gain, its effective amplification band is only concentrated in 1530 nm-1620 nm, however, there are many important transmission band amplification there are still research gaps. As a result, the focus of this paper’s research is on optimizing the gain spectrum of a bismuth-doped fiber amplifier at wavelengths between 1200 and 1600 nm using modelling and numerical simulation. The research method of this paper is as follows: first of all, after the literature survey, this topic is using the three-energy level structure, and then collecting the data of pump light wavelength and emission absorption cross-section, signal light emission, and absorption cross-section. fit the data to the graph line, and call the genetic optimization algorithm to derive the gain spectrum optimal value. It’s found that when the pump light wavelength is 1230 nm, fiber length is 2 m, bismuth ion doping concentration is 10.0×1024 ions/m3, the signal light wavelength is 1330 m, its amplifier gain spectrum reaches a peak of 57.17 dB, which is consistent with the results of the three-layer cycle (manual search).


Introduction
The optical fiber amplifier is an important optical device in the optical fiber communication system to amplify signals.When making optical fiber, a special method is used to add a very small amount of rare earth ions to the core layer of the optical fiber to obtain the corresponding rare earth ion-doped optical fiber amplifier.Nowadays, the most commercially used fiber amplifier is the erbium-doped one (EDFA) which has advantages such as high gain and low loss, but its bandwidth amplification range can only cover the C band and the L band (1520nm-1630nm).It will not satisfy the capacity need for the communication system with the rapid development of Cloud Computing and IoT technology [1,2].On this basis, fiber amplifiers doped with other rare earth ions came into being to improve the effective utilization of other bandwidths, such as O-band, S-band (total of about 400nm).Among them, the electrons outside the bismuth nucleus are not full, so it is easy to lose electrons and undergo transitions, which can be used for multi-band bandwidth amplification.In 1999, Murata et al. first observed broadband fluorescence radiation in bismuth-doped silicate glass [2].In the following two decades, researchers discovered the maximum magnification of bismuth-doped silicon glass [3,4], proved the laser action between 1150 nm and 1300nm of bismuth-doped aluminium silicate fiber [5], continuing to discover the amplifying features of bismuth-doped optical fibers, which vigorously promoted the research progress of bismuth-doped optical fibers.China is also conducting systematic research on bismuth-doped optical fibers.In 2022, the Shanghai Institute of Optics and Mechanics, based on a selfmade bismuth-doped optical fiber amplifier, used improved chemical vapour deposition technology combined with liquid phase doping technology (MCVD) to prepare the first domestic low lossy Bi/P co-doped silica-based fiber has realized E-band (1360nm-1460nm) broadband amplification for the first time [1].Therefore, bismuth-doped optical fiber occupies an important position in the field of optical fiber communication, and it is imminent to research its amplification gain.
Based on the results from prior studies, the purpose of this paper is to analyze the influences of fiber length and bismuth ion doping concentration on the gain spectrum of the amplifier for signal light with a wavelength between 1200nm and 1600nm.To this end, MATLAB is used to model, calculate, and optimize the parameters using cyclic and genetic optimization algorithms to get the fiber amplifier's maximum gain under specific conditions.

Theoretical models
At the specific wavelength, to research the change of gain spectrum of Bismuth-doped fiber amplifier with different fiber lengths and ion concentrations, it's necessary to find out the extra-nuclear electron level structure of the bismuth atom first.
The configuration of the extra-nuclear level is 4f 14 5d 10 6s 2 6p 3 , which can be simplified to a threelevel structure, where the ground level, metastable level, and the excited level are 3 p 0 , 3 p 1 , 3 p 2 , which are shown in Figure 1.And the absorption rates of the pump from the ground level to the excited level, the absorption and emission rates of signals between the ground level and the excited level, the spontaneous emission rates from the metastable level to the ground state, and the non-radiative transition rates of electrons from the excited level to the metastable level are all denoted by the symbols W pa , W sa , W se , A 21 , and A 32 [6].To prepare for subsequent optimization calculations, it is necessary to get the corresponding rate and power propagation equations.In the stable state, the transmission and transition of electrons in the threelevel structure can be generalized as the following three equations [6]: 2 () Where z is the wavelength of the signal, N is the total particle concentration, and N 1 , N 2 , and N 3 are the particle concentration of three levels respectively.Due to the higher rate ratio of A 21 and A 32 , the non-radiative transition of electrons from the metastable level to the ground level is disregarded [2].
Where h is Planck constant, P s and P p are signal power and pump power, σ 13 , σ 12 and σ 13 are the pump's absorption cross-section, the signal's absorption cross-section and the emission cross-section of the signal.Now the change of pump, signal and ASE with different propagation lengths, which is the power propagation equations, is given as following equations: () () Where α is the loss coefficient of fiber material (/m), Δυ is the full width at half maximum frequency (Hz), and γ is the overlap integration factor, which can be calculated by the following formula: Where r, w, V, n 1 , n 2 , λ p are the fiber core radius, the model field radius of the optical field, the fibre's normalised frequency, the fibre core's refractive index and cladding, and the wavelength of the pump respectively.Similarly, to find Γ s and Γ ase , only the pump wavelength in the formula is needed to be replaced with the signal wavelength and ASE wavelength.

Data preparation
To calculate the optimized gain in different bismuth concentrations and fiber lengths, it's necessary to further improve the parameters in the above equations.
Figure 2 illustrates the spontaneous emission and absorption cross sections (σ 12 and σ 21 ) of the bismuth-doped fiber amplifier at the pump wavelength of 1230 nm [7].The equations can be obtained after fitting with the Gaussian function in MATLAB [8]: ) 2 (13) Figure 2. Referred to absorption cross-section and emission cross-section [7].The curve that pumps absorption cross-section against wavelength is shown in Figure 3.When the pump wavelength is still 1230nm, the absorption cross-section is about 2.1×10 -24 m 2 , which can be used to calculate the emission cross-section [7].The following table shows the used parameters in equations [9].This study is based on the simulation optimization of MATLAB software, which is an interactive development system and is currently one of the most commonly used and popular simulation and mathematical computing software in the world, which combines the functions of computing, programming, and graphing in one easy to develop an environment at the same time [10].

Two-layer cycle.
The rate equation is first solved in the course of the project using the symbols, then write the rate equation, power propagation equation, and function file expressing the rate equation, the main program for solving the rate and power propagation equations is then launched.In the initial stage of the experiment, a two-level cyclic deconstruction was used [11], whose outer layer function is the wavelength range (1200 nm-1600 nm) and whose inner layer functions are the fiber length (1 m-7 m) and the bismuth ion doping concentration (2.0×10 24 ions/m 3 -8.0×10 24ions/m 3 ) as loop variables.For the fluctuation of the gain spectrum with fiber length and bismuth ion doping concentration, two plots of peak gain maxima were obtained.

Three-layer cycle.
As shown in Figure 4, the first four steps are consistent with the two-layer loop, and this group applied the manual search for the gain maximum point in the middle stage of the experiment(the three-layer nested loop) [11].The signal light wavelength, fiber length, and doping concentration are cycled as three parameters in a certain range, and a large number of gain results are obtained by running MATLAB, in which a manual search is performed to obtain the peak gain maximum, and the flow chart is as follows:

Optimization of gain spectrum based on genetic algorithm.
First, write the adaptation function code with the fixed conditions: signal optical power P s _0=1350×10 -9 , pump optical power P p _0=950×10 - 9 , ASE power P ase _0=0, signal optical wavelength swl=1.33×10 - .Next, Visualization functions and functions expressing rate equations and power propagation equations are used to implement genetic algorithms.
The optimizing the gain spectrum based on the genetic algorithm is shown in Figure 6.The spontaneous emission cross-section and absorption cross-section spectra of the bismuth-doped fiber amplifier under pump light with a wavelength of 1230nm are shown in Figure 7 as the two curves [7].Using Gaussian function fitting, the functional expressions can be obtained as follows: Wherein, the first formula is the functional expression of the spontaneous emission cross-section spectrum, and the second is the functional expression of the absorption cross-section spectrum.

Parameters setting
Table 2. Values of different control variables.
Fiber length L(m) 1 3 5 7 Bismuth ion concentration N(10 24 ions/m 3 ) 2 4 6 8 By using the fiber length and bismuth ion concentration as variables, the gain spectra of a fiber amplifier doped with bismuth can be calculated.The corresponding images are drawn and the peak gains under different conditions are obtained at the same time.

Results presentation 3.2.1. Variation of the Gain Spectrum with a Single Variable.
When the concentration of bismuth ions is 8×10 24 ions/m 3 , the gain spectrum of the fiber amplifier is plotted between 1m and 7 m in Figure 8 [13].In the figure, the peak gain first increases from a minimum of 45 dB to a maximum of 57 dB, and then decreases to 53 dB when the fiber length is between 1m and 5 m.It shows that the peak gain can reach the maximum when the fiber length is between 1 m and 5 m. Figure 9 shows the gain spectra of the optical fiber amplifier between the concentration of bismuth ions of 2×10 24 ions/m 3 and 8×10 24 ions/m 3 when the optical fiber length is 6 m.When the bismuth ion concentration is from 2×10 24 ions/m 3 to 8×10 24 ions/m 3 , the peak gain of the gain spectrum starts from the minimum of 52 dB to the maximum of 55 dB.Then it decreased to 54 dB.It shows that the peak gain can reach the maximum when the bismuth ion concentration is between 2×10 24 ions/m 3 and 6×10 24 ions/m 3 .Gain spectrum of fiber amplifier with bismuth ion concentration.Analyzing Figure 8 and Figure 9 shows that when the concentration of bismuth ions is 8×10 24 ions/m 3 and the fiber length is between 1 m and 5 m, the peak gain of the gain spectrum can reach the maximum value.Or when the fiber length is 6m and the bismuth ion concentration is between 2×10 24 ions/m 3 and 6×10 24 ions/m 3 , the peak gain can reach the maximum.

The Variation of the gain spectrum with two variables.
Therefore, the gain spectrum of the bismuth-doped fiber amplifier can be calculated within the above-mentioned variation range of fiber length and bismuth ion concentration, and the three-dimensional waveform diagrams as shown in Figure 10 and Figure 11 can be obtained [14].
When the concentration of bismuth ions is 8×10 24 ions/m 3 , the three-dimensional gain spectrum can be plotted between 1 m and 5 m in Figure 10.When the signal light wavelength is 1330 nm and the fiber length is 2.37 m, the peak gain can reach the maximum value of 56.9718 dB.

Figure 10.
Gain spectrum of fiber amplifier with fiber lengths.When the fiber length is 6 m, the three-dimensional gain spectrum can be plotted between 2×10 24 ions/m 3 and 6×10 24 ions/m 3 in Figure 11.When the signal light wavelength is 1330 nm and the bismuth ion concentration is 3.05×10 24 ions/m 3 , the peak gain can reach the maximum value of 55.237 dB.Gain spectrum of fiber amplifier with bismuth ion concentration.After the calculation and analysis, although the maximum gain of the gain spectrum of the fiber amplifier is obtained, it just can be obtained under only one variable condition.So it is impossible to obtain the maximum gain when the fiber length and the bismuth ion concentration change at the same time.
Therefore, taking the fiber length and bismuth ion concentration as variables at the same time, a large number of three-dimensional vector results can be obtained between the fiber length of 1m and 5 m and the bismuth ion concentration of 2×10 24 ions/m 3 and 6×10 24 ions/m 3 .Because the three-dimensional vector can not make the corresponding graph, the peak gain only can be found manually.So it can reach the maximum value of 57.17 dB at the central wavelength of the signal light of 1330 nm when the fiber length is 2m and the concentration of bismuth ions is 10×10 24 ions/m 3 .

Optimization using genetic algorithms.
Next, the genetic optimization algorithm [12] is used to optimize the results and the image is shown in Figure 12.  12, after optimizing the results by genetic algorithm, it can be obtained that the maximum gain of the bismuth-doped fiber amplifier is 57.17 dB, which is the same as that obtained by manually searching.Therefore, it can be considered that the peak gain of the gain spectrum of the 1200 nm-1600 nm bismuth-doped fiber amplifier can reach the maximum value of 57.17 dB at the central wavelength of the signal light of 1330 nm when the fiber length is 2 m and the concentration of bismuth ions is 10×10 24 ions/m 3 .

Discussion
First, the results do find the maximum gain between 1200 nm and 1600 nm for the bismuth-doped fiber amplifier, but the method is poor.In the early stage of the study, the genetic algorithm was not correctly used, which wasted a lot of time.As a result, there were still some problems in the process of using a genetic optimization algorithm to solve the maximum gain in the later stage, but the correct result was finally obtained.
In the research process, the ant colony algorithm, as a more advanced optimization algorithm, is also considered in the research scope.However, this algorithm is relatively difficult and has not been successfully combined with the simulation program at present, so we temporarily give up using the ant colony algorithm and choose a genetic algorithm to solve it.In future studies, we will also continue to try to combine the ant colony algorithm with a simulation program to optimize the solution to obtain the maximum gain of the fiber amplifier gain spectrum peak.
The general steps are as follows: Firstly, the bottom function of the ant colony optimization algorithm needs to be set and can run successfully.Secondly, it is consistent with the above genetic algorithm process.The fitness function (fit.m) and the visualization function (plot.m)should be set.Finally, the upper and lower bounds of the definition domain are set in the main function and the function is called to run.

Conclusion
To sum up, the energy level, the numerical model of BDFA and the transition structure are provided in this study.The model's associated rate and power propagating equations are solved using the bismuth ion three-level system, and the effects of the length of the fiber and bismuth ion level on the peak gain of the gain spectrum are investigated.Finally, with the genetic algorithm, the changes in the gain spectrum with signal light wavelength, fiber length and doping concentration are calculated by MATLAB programming, and the fiber length and doping concentration are optimized to maximize the peak gain.
The results show that the signal light wavelength, fiber length and bismuth ion concentration influence the amplifier gain and each other.When the signal light wavelength is from 1200nm to 1600nm, the gain spectrum first increases and then decreases.Using a genetic algorithm, it shows that when the wavelength of the signal light is 1330nm, the fiber length is 2m, and the bismuth ion concentration is 10.0×10 24 ions/m 3 , a bismuth-doped optical fiber amplifier gets a maximum gain of 57.17 dB, and broadband amplification in O-band is realized.
Although the correct results are obtained, there are still some problems.In the beginning, because the genetic algorithm was not used correctly, it took a lot of time.Instead, using the method of a threelayer cyclic manual search with huge calculations, the problem is solved.Then, trying to use the genetic algorithm in the simulation program, the wavelength of the signal light is set between 1200 nm and 1600 nm.However, the results did not match the expectations.Therefore, in the later period of research, the wavelength of the signal light is set as a constant value,1330 nm (the central wavelength of signal light).And we will still actively try to change the value of signal light wavelength from 1200 nm to 1600 nm to find the maximum value of the gain spectrum.

Figure 4 .
Figure 4. Flowchart of circular method.2.4.Optimization algorithm -genetic algorithm2.4.1.Fundamentals.A genetic algorithm simulates the natural evolutionary process to find the optimal answer.And it is a class of randomized search algorithms that draw on natural selection and natural genetic mechanisms in the biological world.During the operation of the algorithm in Figure5, a set of candidate individuals is retained in each iteration through mechanisms such as selection, crossover, and variation, and the process is repeated, and its fitness reaches a near-optimal state[12].

Figure 5 .
Figure 5. Flow chart of genetic algorithm steps.

Figure 7 .
Figure 7. Cross-section showing the emission and absorption of a bismuth-doped fiber amplifier with a 1230 nm pump light.

Figure 8 .
Figure 8. Gain spectrum of fiber amplifiers with fiber lengths.Figure9shows the gain spectra of the optical fiber amplifier between the concentration of bismuth ions of 2×10 24 ions/m 3 and 8×10 24 ions/m 3 when the optical fiber length is 6 m.When the bismuth ion concentration is from 2×10 24 ions/m 3 to 8×10 24 ions/m 3 , the peak gain of the gain spectrum starts from the minimum of 52 dB to the maximum of 55 dB.Then it decreased to 54 dB.It shows that the peak gain can reach the maximum when the bismuth ion concentration is between 2×10 24 ions/m 3 and 6×10 24 ions/m 3 .

Figure 9 .
Figure 9. Gain spectrum of fiber amplifier with bismuth ion concentration.Analyzing Figure8and Figure9shows that when the concentration of bismuth ions is 8×10 24 ions/m 3 and the fiber length is between 1 m and 5 m, the peak gain of the gain spectrum can reach the maximum value.Or when the fiber length is 6m and the bismuth ion concentration is between 2×10 24 ions/m 3 and 6×10 24 ions/m 3 , the peak gain can reach the maximum.

Figure 11 .
Figure 11.Gain spectrum of fiber amplifier with bismuth ion concentration.After the calculation and analysis, although the maximum gain of the gain spectrum of the fiber amplifier is obtained, it just can be obtained under only one variable condition.So it is impossible to obtain the maximum gain when the fiber length and the bismuth ion concentration change at the same time.Therefore, taking the fiber length and bismuth ion concentration as variables at the same time, a large number of three-dimensional vector results can be obtained between the fiber length of 1m and 5 m and the bismuth ion concentration of 2×10 24 ions/m 3 and 6×10 24 ions/m 3 .Because the three-dimensional vector can not make the corresponding graph, the peak gain only can be found manually.So it can reach the maximum value of 57.17 dB at the central wavelength of the signal light of 1330 nm when the fiber length is 2m and the concentration of bismuth ions is 10×10 24 ions/m 3 .

Figure 12 .
Figure 12.Optimization of peak gain maximum of gain spectrum of fiber amplifier by genetic algorithm.As shown in Figure12, after optimizing the results by genetic algorithm, it can be obtained that the maximum gain of the bismuth-doped fiber amplifier is 57.17 dB, which is the same as that obtained by manually searching.Therefore, it can be considered that the peak gain of the gain spectrum of the 1200 nm-1600 nm bismuth-doped fiber amplifier can reach the maximum value of 57.17 dB at the central () =