Gain maximization of thulium-doped fiber amplifiers based on genetic algorithm with 1900 nm signal

With the development of fiber communication technology, the existing operating wavelength band of fiber amplifiers, such as 1530-1560 nm and 1480-1510 nm, may not be satisfied with the growing requirement in many fields. Therefore, this research emphasizes expanding the available band (1800-2000 nm) and maximizing the gain of the thulium doped fiber amplifier at the central wavelength (1900 nm). The study is based on the two-level energy system of thulium ion, which can cover the objective wavelength. According to the related rate equations solved by the symbol function method and power propagation equations solved by the Fourth order Runge-Kutta method, use MATLAB to simulate a fiber amplifier and study the coeffect amplification effect of thulium doped fiber amplifier to 1900 nm signal light. Use the Genetic algorithm to find the best parameters match (fiber length 1m and doping concentration 3.5) when the gain reaches the maximum (27.5 dB). Through optimizing the collocation of parameters, the gain can be maximized.


Introduction
With the fifth-generation mobile communication gradually becoming universal and commercial, the realization of its advanced application, such as the extremely high transmission rate with 10-20Gbit/s (Enhanced Mobile Broadband: eMBB), The connection latency should reach the 1ms level, and the connection should support high reliability (99.999%) at high-speed movement (500Km/h)(Ultra-high reliable and low delay communication: uRLLC), and the robust connectivity can quickly promote the deep integration of various vertical industries (smart city, smart home, environmental monitoring, Etc.)(Massive Machine Type Communication: mMTC), is inseparable from the rapid development of optical fiber communication.
Thus, how to satisfy the growing demand for communication indicators in all fields (Larger capacity, longer distance, higher speed) becomes the advanced research direction of optical fiber communication development.One of the ways to improve the performance of optical fiber communication systems is to use high-performance optical fiber communication devices, such as fiber amplifiers.However, rareearth doped optical fiber amplifiers do not need the optical signal to go through the complex process of photoelectric conversion, electro-optical conversion and regeneration and can directly amplify the optical signal, which significantly simplifies the system structure of the optical fiber communication system and reduces the overall cost of the system [1].Among them, TDFA has the advantages of high gain, broad band width, low noise and gain sensitivity to polarization [2], so it is generally used.Nevertheless, how to set up the collocation of thulium doped fiber amplifier device parameters to optimize its operating performance is a research frontier in the fields, having existed for many years.Hence, this research has practical significance in propelling the development of fiber communication.The study will use MATLAB to simulate a thulium doped fiber amplifier and find its gain spectrum.Using Genetic algorithm, optimize its parameter match to improve the performance of the fiber amplifier.

Research methods
The principle of the research is simulating a TDFA by MATLAB and expressing its gain spectrum function, then using Genetic algorithm to find the best parameters collocation that can maximize the gain of thulium-doped fiber amplifier to 1900 nm signal light.

Get the data of emission cross-section and absorption cross-section
Before simulating a fiber amplifier, searching for basic parameters is the first step of the research.The curve figure about the absorption and emission cross-section of thulium ion, which is in the wavelength period of 1500-2100 nm, can be gained by seeking literature (Figure 1) [3].However, because of looking for the specific data of 1900 nm, complete some preprocessing to filter the redundant parts of this plot and focusing on the 1900 nm, as the central wavelength of 1800-2000 nm, crop the crucial section about 1800-2000 nm in the image (Figure 2, Figure 3).In virtue of MATLAB codes, operate the image identification by scanning each pixel from the figure.Set up the top and bottom bounds of the X and Y coordinate axes.After that, get the objective emission and absorption cross-section plots (Figure 4, Figure 5) and call the cftool tool cabinet in MATLAB to fit the curves.The cftool will read each data of x and y and fit them with each fitting model.In order to find the best fitting means, all the models will be examined by comparing their root-mean-square errors: RMSE (RMSE: An index measures the difference between the observed value and the actual value.)because there is a negative correlation between RMSE and fitting results, the smallest RMSE, the best fitting results.After testing all the fitting methods, it is easy to get the best model to fit the curves is four-term Fourier (Figure6, Figure 7).The order of magnitudes of root-mean-square error is -27, which means this method can provide a good effect of fitting curves.The following equation ( 1) is the fitting formula of the four-term Fourier.Table (1) shows the values of parameters in that fitting formula.

Set up the initial parameters to simulate a fiber amplifier
Thulium-doped optical fiber amplifier (TDFA) is a relatively novel type of Rare-earth doped fiber amplifier.The essential feature is thulium ion doped.The thulium ion is a multilevel system (Figure 6) [4].The trivalent thulium ion has a great energy level structure to absorb many kinds of pump light with different wavelengths.Meanwhile, after absorbing the pump light, the particles are stimulated and transition to other energy levels, releasing laser with other wavelengths [5].However, this study focuses on the effect of coherent amplification of signal light with a wavelength of 1900 nm by thulium-doped fiber amplifiers.Therefore, according to Figure 6, the multilevel energy system can be simplified to a two-level energy system.The rate equations are shown by equations ( 2), ( 3) and ( 4), with their answers equations ( 5) and ( 6).The power propagation equations are demonstrated by equations ( 7), ( 8) and ( 9).
In the MATLAB simulating codes, set up fiber length and the Thulium ion doping concentration as the two independent variables to control the gain of fiber amplifier (It also means to observe how fiber length and the ion doping concentration affect the gain.).Use the symbol function to solve the rate equations to get the bulk density of ions with electrons at each energy level.Then, substitute into power propagation equations.Use the fourth order Runge-Kutta method to solve the power propagation equations to get signal power.Finally, get the gain function of this fiber amplifier: equations ( 10) and (11).Table (2) illustrates some critical parameters for this research.Especially in the wavelength period 1800-2000 nm, the emission cross-section curve shows a peak at 1865 nm as the emission central, so use the data of 1865 nm to approximately represent the one of 1900 nm.However, there is no absorption central in 1800-2000 nm.According to Figure 6, it appears at approximately 1600-1700 nm.Hence, use the data of proximate wavelength 1800 nm to similarly equal the 1900 nm one.

Rate equations:
A 21 +W 12 +W 21 +W P (5) A 21 +W 12 +W 21 +W P (6) Power propagation equations: Gain function: (P p (z),P s (z),N 1 ,N 2 ,z)=10log 10 P s (z) After inputting the initial parameters, set up the two independent variables: fiber length L is limited by a lower limit of 1 m and upper limit 20 m, with 1 m step size (L= [1:1:20]); Thulium ion doping concentration N is limited by lower limit2 × 10 25  −3 and upper limit 5 × 10 25  −3 , with step size 1 × 10 24  −3 (N=[2× 10 25 : 1 × 10 24 : 5 × 10 25 ]).Use the previous two independent variables as the circulation volumes to build up a double circulation structure.Write the two nested loop structures for circulation statements: the matrix of ion doping concentration is set as external circulation; the matrix of fiber length is set as internal circulation.In virtue of the double loop structure, substitute each element in the two matrixes into the gain spectrum calculation function, which will generate this fiber amplifier's gain spectrum controlled by the range of fiber length and doping concentration.In the end, using the maximizing function in MATLAB can obtain the maximum gain.

Genetic algorithm.
Genetic Algorithm, a search algorithm, is a computational model simulating Darwinian genetic selection and natural selection.It uses specific coding techniques to act on binary strings of chromosomes, and its basic idea is to simulate the evolutionary process of the population composed of these strings [8].Moreover, GA is a global optimization algorithm that will not fall into the trap of a fast decline of an optimal local solution.Compared with traditional optimization algorithms, the genetic algorithm can achieve rapid convergence and has less computation time and high robustness when calculating accuracy requirements [9].Use the gain spectrum function as the fitness function (objective function) and substitute it into the Genetic algorithm built-in code of MATLAB.Set up the range of independent variables (fiber length(m) L [0 20]; Thulium ion doping concentration N [2× 10 25 5 × 10 25 ]).Then, encode the two independent variables according to the range, aiming to digitalize the relationship of Genotype and phenotype by binary, floating-point or symbol coding method.To randomly generate a population, the individuals in this population are chromosomes that represent different values of gain determined by different fiber lengths and doping concentrations.Calculate the fitness of each individual.The classical genetic algorithm has problems such as precocity and poor local optimization ability [10].Use the Roulette Wheel Selection method to select some individuals with relatively high fitness, which will continue to involve the evolution and eliminate the ones with lower fitness through competition.By operating the cross exchange (cross exchange: Simultaneous cross-switching of multiple gene segments between two parental chromosomes) and variation (variation: During the generation of progeny chromosomes, a random change occurs in one of the progeny's gene segments, such as the inverse code) of chromosomes, the new individuals (progeny individual) are generated.The previous two operations can add new individuals to the population, increase data diversity, and better perform optimization.The algorithm will repeat the operations described above until the circulation reaches maximum iterations.When the process ends, the optimization codes return the best solutions (the best match of fiber length and thulium ion doping concentration to maximize the fiber amplifier gain) and corresponding fitness (the maximum gain) value in the command line window.
The following figure illustrates the flow chart of the Genetic algorithm in this research (Figure 7).

Results and discussion
This study uses two optimization methods to find the maximum value of the gain spectrum function and related solutions (the values of two independent variables).By comparing the results between them, seek a better method to optimize the match of thulium ion doped fiber amplifier's parameters, maximizing the gain of 1900 nm signal light.

Double circulation
Set up the range of independent variables: �  ℎ (): [0 20]   ( −3 ): [2 × 10 25 : 5 × 10 25 ] Substitute the fitness function (the gain function) into visualization codes.Use the surf function to create a 3-dimension surface diagram: X axis represents the doping concentration; Y axis represents the fiber length; Z axis represents the gain of this thulium doped fiber amplifier.According to the plot, easily to find that when the doping concentration N is 3.5× 10 25  −3 and the fiber length is 1 m, the fiber amplifier will provide the maximum gain 27.5353 dB to the signal with 1900 nm wavelength (Figure 8).

Genetic algorithm
Set up the range of independent variables: �  ℎ (): [0 ∶ 20]   ( −3 ): [2 × 10 25 : 5 × 10 25 ] In the codes of Genetic algorithm, call the gain spectrum function as the fitness function.According to the figures of results, the following comments can be easily observed.

Gain.
Through continuous iterative optimization of genetic algorithm, the best match of parameters is �  ℎ (): 0.972   ( −3 ): 35.4851 × 10 24 With this match, the gain for the 1900 nm signal can be the maximum value of 27.57dB (Figure 9).Under the premise of the doping concentration, the fiber length L< 0.972 m, and the gain is positively correlated with the fiber length.In optical fiber length L> 0.972 m, the gain is negatively correlated with the fiber length.

Signal power.
Under the premise that doping concentration N=35.4851× 10 24  −3 , the fiber length L< 0.972 m, the optical signal power is positively correlated with the fiber length.In optical fiber length L> 0.972 m, the optical signal power is negatively correlated with the fiber length.The optical signal power also reaches a maximum value of about 5.7×10 −4 dB under this condition (Figure 10).

ASE power.
Same gain and signal optical power at fiber length L< 0.972 m, the spontaneous radiation power is positively correlated with the fiber length.In optical fiber length L> 0.972 m, spontaneous radiation negatively correlates with fiber length.Under this condition, the maximum spontaneous radiated power reaches about 5.27× 10 −6 dB (Figure 11).

Pump power.
Unlike the approximate Gaussian distribution of the previous three graphs, the fourth figure is like an exponent function.The pump power is negatively correlated with the fiber length in the whole definition domain and gradually approaches 0 when the fiber length exceeds 1 m (Figure 12).

Comparation
This research uses the two optimizing methods to find the best parameter match, which makes the fiber amplifier work with the max gain.Although their results are similar, there are some differences between the two methods.The following table-Table (3) displays the distinctions.

Conclusion
Using the two optimization methods, the max gain of the thulium-doped fiber amplifier is similar.The maximum Genetic algorithm's gain is just 0.04 dB bigger than the one of double circulation.In this research, set up the range of 0-20 m of fiber length and 2-5 of ion concentration.After optimizing the gain spectrum function, the conclusion can be got that when the fiber length selects approximately 1m, and the doping concentration selects approximately 3.5, the max gain is 27.5 dB.However, there are also some deficiencies in the research.Firstly, the independent variables are not concerned comprehensively.In the study, only two independent variables: fiber length and thulium ion doping concentration, are set to control the fiber amplifier gain.Therefore, adding other independent variables, such as main pump power and input signal power, is necessary for further research to supplement the incomplete section.Secondly, the genetic algorithm highly depends on the initial population.The initial population (the initial range of independent variables) will significantly impact the optimization results, leading to deviation from the expected results.In future studies, multiple populations with different initial conditions will be introduced for parallel optimization, or simulated annealing genetic algorithm will be used to combine the two effectively, relieve the selection pressure of the genetic algorithm, enhance the global convergence of the genetic algorithm, and avoid falling into local optimal in the search process by utilizing the solid local search ability of the simulated annealing algorithm.

Figure 7 .
Figure 7.The flow chart of this research using Genetic algorithm.

Figure 8 .
Figure 8.The surface plot of double circulation's result.

Table 1 .
The parameters of absorption and emission cross-section curve expressions.

Table 2 .
The crucial parameters.

Table 3 .
The comparation between the two optimizing methods.