2.1.

GA Theory

The GA technique is inspired by Darwinian evolution and follows the law of survival of the fittest.¹⁹ This algorithm is popular in many fields for its self-organization, self-learning, and self-adaptation properties. Using a binary tree data structure, this algorithm selects the optimal individual in the child generations through natural selection and genetic mechanisms, including selection, crossover, and mutation.²⁰ The optimal individual selection criterion in the algorithm is defined using a fitness function. Initial individuals are generated randomly. Other individuals with high fitness are saved by the program automatically and enters the next generation through the genetic mechanism.

In the selection process, models for laser power and wavelength established by GA are chosen by evaluating the fitness function at each iteration. During selection, the program picks from the current existing models rather than generating a new combination of operators and constants. The structure of the model, like the various chromosomes in natural selection, is constantly changing via the crossover and mutation operations. Crossover and mutation change the model’s binary tree composition and diversify the structure, thus supporting selection of the fittest model.

It should be noted that the elements in the function set are chosen after studying well-known theories,²¹ since these determine the model complicity. Generally, the elements in the function set include basic algebraic operators, Boolean algebraic operators, and some other user defined operators.²² The terminal set in GA contains input variables, numerical constants, logical constants, and so on. The tree-structured model is established by randomly selecting the elements from the function and terminal sets as the root node. The binary tree is extended by selection, crossover, and mutation process until termination criteria are reached. Usually, the termination condition is defined based on the number of generations.

2.2.

Modeling $P-(I,T)$

To ensure the accuracy of the model, we analyze the classic model as a benchmark for selecting the function and termination sets. The output power of a semiconductor laser is studied by analyzing the carrier rate equation in semiconductor physics. As a result, the mechanistic model for laser power output can be expressed as²³

The impact of junction temperature on the threshold current and external differential quantum efficiency are studied from its physical mechanism defined by the following equations:²⁴

Referring to Eq. (4), we choose the function set and the terminal set of the $P-(I,T)$ model as $\{+,-,\times ,÷,\exp ,\ln \}$ and $\{x_1,x_2,x_3,A\}$, respectively. The terminal set is the set including all the independent variables, which can be written as $\{x_1,x_2,x_3\}$. $\{A\}$ is a random constant set containing the model coefficients. Since GA is applied to establish a model for the output power, current, and temperature, we let $\{x_1=I,x_2=T,x_3=P\}$. The composition of elements determines that the model constructed by GA will be a double-input and single-output model, and the expression will involve one or more relations including simple arithmetic, exponential, and logic operations.

The fitness function plays an important role in GA modeling since it directly determines the efficiency of program execution and the accuracy of the constructed model. Usually, there are four kinds of fitness functions: raw fitness, standardized fitness, adjusted fitness, and normalized fitness.²⁰ Given the complexity of the laser power model and our demands, a raw fitness function is adopted due to its simple implementation. In this paper, a fitness function aiming to evaluate the fitness level of the model is designed based on the minimum variance principle, like most regression procedures. We take a segmented assessment methodology in order to more precisely describe the trend of power with varying current and temperature. That is to say, all the test data in GA modeling are divided equally into several groups, and the minimum variance principle is applied to each part. The adaptive evaluation function used in GA modeling can be expressed as²²

All measurement data are divided into $n$ groups, with $m$ data in each part. $D[i]$ is the $i$’th laser output power measurement in each group, $\mathrm{avg}[j]$ is the $j$’th average of the measurements of the $j$’th part, and $\mathrm{GP}[i]$ is the $i$’th solution of the GA model. For a specific set of measurements, $\sum _{i=1}^m{(D[i]-\mathrm{avg}[i])}^2$ is a constant. Thus $\sum _{i=1}^m{(D[i]-\mathrm{GP}[i])}^2$, which is related to the GA model, characterizes the accuracy of the fitting results.

2.3.

Modeling $\lambda -(I,T)$

The refractive index of the semiconductor material and the laser wavelength corresponding to the band gap will change as temperature, carrier concentration, and electric-field intensity fluctuate. The DBR laser is a multielectrode structure with an internal grating reflector. Laser tuning is primarily achieved by changing the grating period and the effective refractive index, which is accurately controlled by current and temperature. To be more specific, once the injected current fluctuates, the carrier concentration will shift, which will lead to changes in the active area refractive index and material gain factor. A simultaneous temperature rise can change the refractive index and band gap of the material. The tuning mechanism is demonstrated in Fig. 1.²⁵

Fig. 1

Current and temperature are tuned by affecting the gain factor, carrier concentration, and band gap.

Although the output frequency from DBR laser has been discussed in many monographs, there is a lack of quantitative models for lasers with different output characteristics or in various operating conditions. Referring to the qualitative analysis above, we choose the function set and terminal set as $\{+,-,\times ,÷,\exp ,\ln \}$ and $\{x_4,x_5,x_6,B\}$, respectively, where $x_4=I$, $x_5=T$, $x_6=\lambda$, and $\lambda$ is the wavelength measured with a wavelength meter. As for modeling $\lambda -(I,T)$, the fitness function is shown as Eq. (5) as well, where measurements are wavelength in place of power.

3.1.

Experimental Setup

A DBR laser is an instrument that converts electrical energy to optical energy. Generally, drive current is injected to the diode after the temperature of the heat sink settles, and the laser output can be affirmatory. Changes in the injection current and junction temperature can directly alter the output power and wavelength. The experimental setup is shown in Fig. 2. There are mainly data acquisition and GA modeling steps in our experiment. For data acquisition, a current controller and a temperature controller were used to drive the DBR laser diode. The laser is divided into two beams by a glass plate, and both beams are used for frequency and power measurements. The laser diode we used as the light source was PH852DBR (Photodigm) with 40- to 50-mA threshold current. A high-performance current controller (Thorlabs LDC205C) with a large range of 0 to $\pm 500\mathrm{mA}$ was used for driving the laser diode, and its ripple was no more than $2\mu \mathrm{A}$ at the full driving range. The temperature controller (Thorlabs TED200C) was used to measure the temperature in real time, and its output voltage, through internal calculation, was used to drive a thermoelectric cooler. The temperature control range and stability are $-25°\mathrm{C}$ to 105°C and less than 2 mK, respectively. The power meter we used is PM100D (Thorlabs), the accuracy of which is only $\pm 3\%$. The wavelength of laser was measured with a wavelength meter (High Finesse WS7-60) with range from 192 to 2250 nm, with 60-MHz resolution.

Fig. 2

Experimental setup.

During data acquisition, we measured the power and wavelength values by changing the current from 40 to 80 mA (0.5-mA increments), while temperature remains constant. The measurement process was then repeated by adjusting the temperature at constant current. Both processes were repeated three times, and the measured data were filtered by the $3\sigma$ principle. The 284 data points we collected were used to analyze the effect of injection current and junction temperature on the DBR laser output power and wavelength with GA. Among the experimental data, we select the first 200 data points (70.42%) as a training set and the remaining 84 points (29.58%), which were measured at 21.5°C, as the test data.

The GA modeling flow is mentioned in Sec. 2, as shown in Fig. 2. In order to obtain accurate models with the GA program, the GA parameters should be carefully selected to obtain the proper complexity and correlation coefficient of the fitting results, which indicate the quality of the established models. The GA parameters we used are shown in Table 1. Tournament selection is chosen for GA, since it has several advantages over other alternative selection methods: it is efficient to code, runs in parallel, and is easy to adjust.²⁶

Table 1

GA parameters.

3.2.

Establishing the Model for $P-(I,T)$

The program repeats the calculations for the fitness function. Low-fitness individuals are eliminated until the optimal GA model is obtained. After simplifying the GA model and keeping the same valid figures as the original data, the GA model can be described as

If we notice that the testing data are not used during the modeling process, they are irrelevant to the models we obtained. Therefore, we can use them to evaluate the performance of the models. The GA model is much better since the root-mean-square error (RMSE) of the GA model is 0.0134 mW, nearly one third of that for the classic model (0.0415 mW). By comparing the GA model for the laser output power with the classic model [described as Eq. (4)], we find that the two models mostly overlap from 47 to 75 mA. We also find that the classic model overestimates the power output at low- or high-injection current, particularly when the current slightly exceeds the threshold ($\sim 40\mathrm{mA}$) or goes beyond 75 mA, as shown in Fig. 3. For the low-power probe light, below 5 mW usually, using GA method can reduce deviation by 0.1 mW or so. As a result, the uncertainty of optical rotation angle will decrease by over 5 deg, which will improve the accuracy of the atomic gyros and magnetometers, whose value is 8.11 deg per s.

Fig. 3

(a) Comparison between classic model and GA model for current ranging from 40 to 47 mA and (b) comparison between the classic model and GA model for current ranging from 75 to 80 mA.

The influence of current and temperature on the laser output power can be predicted easily with the GA model for injection currents ranging from 45 to 200 mA and for temperatures ranging from 21.5°C to 26°C, given the common controller parameters of the pump and probe laser. As the model describes, the output power of the DBR grows higher as the current increases or temperature decreases. Obviously, the output power exhibits a curvilinear rise as the current increases and as temperature stays constant over the range shown in Fig. 4, while the nonlinear relationship is not significant for power and temperature.

Fig. 4

Output power prediction with the $P-(I,T)$ model with current and temperature ranging from 45 to 200 mA and 21.5°C to 26°C, respectively.

The forms and parameters of GA model depend on the characteristics of the laser. According to the classic model, the power of laser varies linearly with current at a constant temperature, whereas the linearity of the measurements is 0.9996 on average, experimental results shown. This will lead to an uncertainty of prediction for power, which is nearly 0.05 mW. The bias of classic model is more remarkable at the low power (less than 5 mW) and the high power (about or above 20 mW), and the error exceeds 0.1 mW. For atomic gyros and magnetometers, the probe beam is usually 3 mW or so, and the pump beam is usually beyond 20 mW. GA model provides a more accurate prediction, which will help to improve the performance of quantum sensors.

3.3.

Establishing the Model for $\lambda -(I,T)$

The GA model we obtained can be expressed as

Fig. 5

Test results of the GA model.

Based on this model, we can predict wavelength fluctuations over a wide range of current and temperature changes, which is conducive to laser frequency stabilization and fast tuning. We use this model to predict the output wavelength under normal operating conditions as the control input changes, as shown in Fig. 6. Figure 6 describes a smooth surface with the wavelength changing linearly as temperature fluctuates and quadratically as current fluctuates.

Fig. 6

Wavelength prediction with the $\lambda -(I,T)$ model with current and temperature ranging from 45 to 200 mA and 21.5°C to 26°C, respectively.