Modified genetic algorithm for high-efficiency dispersive waves emission at 3 &#x00B5;m

Zimiao Wang; Feng Ye; Qian Li

doi:10.1364/OE.444411

1. Introduction

In recent years, the integrated photonic platforms have made significant progress, which are vital for the high-density integrated optical network. Silicon (Si) and silicon nitride (Si₃N₄) are excellent candidate materials in integrated photonics [1–3], while lithium niobite (LN) on insulator has proved to be a powerful platform for various photonic applications including electro-optic modulation, microcomb, supercontinuum generation, second-harmonic generation [4–6], etc. The LN integrated waveguide has low propagation loss (0.03 dB/cm), strong ${\chi ^{(2 )}}$ nonlinearity, large transparency window ranging from ultraviolet to mid-infrared and various optical properties. It is widely used in second-order nonlinear frequency conversion process such as optical parametric amplification (OPA) [7], optical parametric oscillation (OPO) [8] and difference frequency generation (DFG) [9] to generate spectrum in specific spectral regions such as ultraviolet and mid-infrared, which are generally difficult to obtain. However, those processes require both energy conservation and momentum conservation (so called phase-matching) conditions [10–12], which can be achieved by either exploiting inherent birefringence of materials or by periodically poled process to realize quasi-phase matching (QPM) [13]. These methods generally need strict experimental requirements or complex fabricating techniques. In contrast, there is another novel method to obtain those spectrum components called Cherenkov radiation [14–16].

The Cherenkov radiation, also known as dispersive wave (DW) generation, is a well-known nonlinear mechanism in supercontinuum (SC) generation, which emits DW according to the DW phase-matching condition [17]. The DW could acquire a fraction of energy from the pump and generate different ultraviolet or mid-infrared spectrum by designing the structure of waveguides. The DW has been widely used in diverse fields such as imaging [18,19], sensing [20,21] and spectroscopy [22] applications. SC generation is a process affected by multiple parameters, which means that different input parameters will lead to DW with different intensity and central wavelength. In order to obtain the DW that meets the application requirements, a numerical strategy is proposed to facilitate the process of optimization.

As a fully developed method in machine learning, GA has been widely used in ultrafast and nonlinear optics such as coherent control of ultrafast dynamics [23–26] and pulse characterization [27,28]. GA is a computational model that simulates the evolution process of Darwin’s biological evolution theory which includes natural selection and genetic mechanism [29]. GA starts from an initial Population which consists of many coded Individuals. Then, Parents are selected according to the Fitness to generate better Children by crossover and mutation operators. After sufficient iterations, the best Individual in the final Population can be recognized as the approximate optimal solution of the problem. Besides, GA in this work has been modified by using two strategies, which is called MGA. Firstly, we use the simulated binary crossover (SBX) method to improve the accuracy of GA. Next, a non-uniform mutation process is applied to enhance the convergence capabilities of GA. Therefore, MGA has more accurate searching capability and more stable convergence capability than conventional GA.

Considering that the conversion efficiency (CE) of mid-infrared DW is degraded when the DW generates at long wavelength [17], we are committed to maximize the CE of mid-infrared DW by tuning the input parameters with machine learning method. In this paper, we present a numerical strategy based on MGA to optimize central wavelength ($\lambda$), peak power ($P$) and time duration (${T_{\textrm{FWHM}}}$) to maximize the CE of mid-infrared DW in a 5-mm-long LN ridge waveguide. This method only adjusts the input parameters without redesigning the structure of waveguide, which is helpful to find the most suitable input parameters for the experimental conditions. The combination between MGA and DW generation provides a novel idea for obtaining mid-infrared spectral components with high-CE near 3 µm, which can be applied in detecting acetylene (C₂H₂) gas that could bring safety hazards to industrial production [21].

2. Model

The nonlinear propagation along the LN waveguide is simulated by using generalized nonlinear Schrӧdinger equation (GNLSE) [30]:

(1)$$\frac{{\partial A(z,T)}}{{\partial z}} + \frac{\alpha }{2}A(z,T) - \sum\limits_{k \ge 2} {\frac{{{i^{k + 1}}}}{{k!}}} {\beta _k}\frac{{{\partial ^k}A(z,T)}}{{\partial {T^k}}} = i\gamma (1 + i{\tau _{\textrm{shock}}}\frac{\partial }{{\partial T}})A(z,T){|{A(z,T)} |^2}, $$

where $A(z,T)$ is the time-domain envelope, $z$ is the propagation distance along the waveguide, $T$ is the time and linear loss $\alpha$ of this 5-mm-long LN waveguide is 0.03 dB/cm [31]. The ${\beta _\textrm{k}}$ represents the linear dispersion of this LN waveguide at the pump frequency ${\omega _0} = 2\pi c/{\lambda _0}$. The Kerr nonlinear coefficient $\gamma$ is defined as $\gamma = {{{n_2}{\omega _0}} / {c{A_{\textrm{eff}}}}}$, the nonlinear Kerr parameter ${n_2}$ and effective mode area ${A_{\textrm{eff}}}$ of this LN waveguide can be simulated by COMSOL. The effects of self-steepening and optical shock formation are characterized by a time scale ${\tau _{\textrm{shock}}} = {\tau _0} = {1 / {{\omega _0}}}$. The contribution from ${\chi ^{(2 )}}$ nonlinearity, Raman self-frequency shift and Raman-induced soliton fission to the DW generation is neglected for this waveguide [32].

In Fig. 1(a), we present a typical structure of LN ridge waveguide which is able to be fabricated and further used in experiments for generating mid-infrared DW near 3 µm. Besides, the waveguide structure can be designed to be more complicated according to different application scenarios. Figure 1(b) shows that the propagation of pump will lead to the SC generation. In addition, we consider the DW phase-matching condition that leads to the DW generation which enables octave-spanning SC generation. And the DW phase-matching condition can be written to express the phase mismatch between DW and the soliton pulse [17]:

(2)$$\Delta \beta (\omega ) \approx \beta (\omega ) - \beta ({\omega _\textrm{s}}) - (\omega - {\omega _\textrm{s}}){\beta _1}({\omega _\textrm{s}}) = \sum\limits_{k = 2} {\frac{{{{(\omega - {\omega _\textrm{s}})}^k}}}{{k!}}} \frac{{{d^k}}}{{d{\omega ^k}}}\beta ({\omega _\textrm{s}}) = {\beta _{{\mathop{\rm int}} }}(\omega ), $$

where $\beta (\omega )$ is the propagation constant, ${\omega _\textrm{s}}$ and ${\beta _1}({\omega _\textrm{s}})$ are respectively the term of central frequency and first-order dispersion of the incident soliton pulse, ${\beta _{{\mathop{\rm int}} }}$ is defined as the integrated dispersion. It should be noted that the curve of ${\beta _{{\mathop{\rm int}} }}$ and the DW phase matching wavelength (${\beta _{{\mathop{\rm int}} }}(\omega ) = 0$) will be changed as the ${\omega _\textrm{s}}$ changes, which means that tuning the central wavelength of incident pulse leads to DW generation at different wavelength. Therefore, we associate the change of the dispersion component with the change of pump wavelength and describe it with Eq. (2). Figure 2(a) shows that the dispersive waves are excited near the phase matching wavelength (${\beta _{{\mathop{\rm int}} }}(\omega ) = 0$), while DW can be divided into short-wavelength DW (SWDW) and long-wavelength DW (LWDW). Figure 2(b) shows that C-band pump in the anomalous group velocity dispersion (GVD) region leads to mid-infrared DW generation near 3 µm in this LN ridge waveguide. Since the characteristic absorption band of C₂H₂ is distributed between 3 − 3.1 µm as shown in Fig. 2(b), the mid-infrared DW is useful for C₂H₂ gas sensing [21]. Considering that the structure shown in Fig. 1(a) is able to be obtained and C-band pump is widely used, this method has great potential for practical applications. The input pulses used in our simulations are in the form of Gaussian $\sqrt {{P_0}} \exp ({ - {T^2}/2/T_0^2} )$.

Fig. 1. (a) The cross-sectional view of the LN waveguide used in this work. (b) Schematic of the supercontinuum generation in this 5-mm-long LN waveguide. The femtosecond pulse located in C-band is input into the waveguide (left), expanded into supercontinuum and output (right). DWs are formed in the visible and mid-infrared region.

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Fig. 2. (a) Dispersive waves are generated at the DW phase matching points where ${\beta _{{\mathop{\rm int}} }}(\omega ) = 0$. (b) The integrated dispersion corresponding to the input which is pumped in the anomalous GVD regime, the grey area marked the C-H fundamental vibrational transition bands of C₂H₂, and red points represent the phase matching points (${\beta _{{\mathop{\rm int}} }}(\omega ) = 0$).

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Fig. 3. The flow chart of this 3-stage MGA process (top). Sketch of the three MGA stages (bottom), where m represents the number of individuals contained in the current Population and is the indicator to determine which stage should be implemented.

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Each Individual generated by MGA is consist of a set of parameters $i = {[{i_1},{i_2},{i_3}]^T} = {[\lambda ,P,{T_{\textrm{FWHM}}}]^T}$, which is used as the input of SC generation. The simulation output is used to evaluate the CE defined by the following equation [33], which is also used as the Fitness function,

(3)$$\eta = {{\int_{{\upsilon _\textrm{c}} - \delta }^{{\upsilon _\textrm{c}} + \delta } {d\upsilon |\tilde{A}(\upsilon ){|^2}} } / {\int_{ - \infty }^{ + \infty } {d\upsilon |{{\tilde{A}}_0}(\upsilon ){|^2}} }}, $$

where ${\tilde{A}_0}(\upsilon )$ and $\tilde{A}(\upsilon )$ are respectively the input and output envelope in frequency-domain, $2\delta$ is the defined spectral channel width and ${\upsilon _\textrm{c}}$ is the central frequency of the mid-infrared DW. We are committed to improve the CE of mid-infrared DW in the range of ${\lambda _{\textrm{DW}}} \in$ [3000, 3100] nm, which exactly corresponds to the absorption range of C₂H₂, i.e., the central wavelength of the mid-infrared DW ${{{\lambda _\textrm{c}} = c} / {{\upsilon _\textrm{c}}}}$, the minimum and maximum wavelength of the defined spectral channel are ${c / {({{\upsilon_\textrm{c}} + \delta } )}} = 3000$ nm and ${c / {({{\upsilon_\textrm{c}} - \delta } )}} = 3100$ nm. Since the width and position of the DW waveform will change with different input parameters, the 100 nm bandwidth helps us to directly and efficiently evaluate the CE of DW in the desired wavelength regime, therefore we do not enlarge the bandwidth to cover the entire waveform of DW.

Then, we should discuss how these three input parameters $[{\lambda _\textrm{s}},{P_0},{T_{\textrm{FWHM}}}]$ affect the CE of DW before discussing the concept of GA. As for the central wavelength ${\lambda _\textrm{s}}$, its corresponding central frequency is of great significance in Eq. (2), which means the integrated dispersion ${\beta _{{\mathop{\rm int}} }}$ and the DW phase-matching wavelength will vary by changing the central wavelength ${\lambda _\textrm{s}}$. If the DW phase-matching wavelength deviates far from the ${\lambda _{\textrm{DW}}} \in$ [3000, 3100] nm, the CE defined by Eq. (3) will drop significantly, which means that the generated DW deviates from our expectations. As for the peak power ${P_0}$, its impact on CE needs to be discussed in different situations. When providing a low ${P_0}$, mid-infrared DW is difficult to form and the CE of mid-infrared DW is low. As the peak power increases, the mid-infrared DW is gradually formed by the action of Cherenkov radiation, and the CE will gradually increase. When ${P_0}$ is relatively high, the mid-infrared DW can be formed stably. The increase in peak power will not significantly increase the intensity of DW in the defined channel (${\lambda _{\textrm{DW}}} \in$ [3000, 3100] nm), but will increase the intensity of the secondary peaks on both sides of the defined channel, which decrease the CE of DW in ${\lambda _{\textrm{DW}}} \in$ [3000, 3100] nm. As for the time duration ${T_{\textrm{FWHM}}}$, its impact on the CE of DW is similar to peak power ${P_0}$. The different spectrum components of DW are generated by acquiring a fraction of energy from the pump. The increase in ${T_{\textrm{FWHM}}}$ leads to the increase of the incident energy, which will increase the intensity of both the SWDW and the LWDW. Therefore, when the intensity of SWDW increase more than the intensity of LWDW, the CE of DW will be decreased. To comprehensively evaluate their impact on the CE of mid-infrared DW, the MGA described below is used to find the optimal input parameters

The MGA is implemented in the range of input: $\lambda \in$ [1530, 1565] nm, $P \in$ [1000, 2500] W, ${T_{\textrm{FWHM}}} \in$ [60, 200] fs. C-band is the central wavelength range of the input soliton pulse. The range of peak power ensures the sufficient intensity of mid-infrared DW while the peak of defined spectral channel (${\lambda _{\textrm{DW}}} \in$ [3000, 3100] nm) is higher than the secondary peaks on both sides. Pulse duration ${T_{\textrm{FWHM}}}$ is in the range of 60 fs−200 fs which is reasonable and achievable in the experiments. The optimization process is shown in Fig. 3, using a three-stage MGA which is similar to Ref. [33]. Firstly, the algorithms randomly generate an initial Population that includes 100 individuals (stage 1). Then, the crossover operator $\hat{X}$ and mutation operator $\hat{M}$ are applied on the selected Parents to generate new Children which are added to the Population until the maximize size (stage 2). The maximum iteration of MGA is 300 and the selection process applies the method of roulette. Once the maximize Population size of 300 is reached, the new Child will replace the worst Individual if ${\eta _{\textrm{child}}} > {\eta _{\textrm{worst}}}$ or be disregarded otherwise (stage 3).

It’s also necessary to describe $\hat{X}$ and $\hat{M}$, which are designed to sufficiently achieve the accurate optimization result. The crossover operator $\hat{X}$ combines two randomly selected Parents $i_{\textrm{ P}}^{\textrm{ }1,2}$ to generate two new Children $i_{\textrm{ C}}^{\textrm{ }1,2}$:

(4)$${[i_{\textrm{ C}}^{\textrm{ }1},i_{\textrm{ C}}^{\textrm{ }2}]^T} = \hat{X} \cdot {[i_{\textrm{ P}}^{\textrm{ }1},i_{\textrm{ P}}^{\textrm{ }2}]^T} = \hat{X} \cdot \left[ {\begin{array}{lll} {\lambda_{\textrm{ P}}^{\textrm{ }1}}&{P_{\textrm{ P}}^{\textrm{ }1}}&{T_{\textrm{ P}}^{\textrm{ }1}}\\ {\lambda_{\textrm{ P}}^{\textrm{ }2}}&{P_{\textrm{ P}}^{\textrm{ }2}}&{T_{\textrm{ P}}^{\textrm{ }2}} \end{array}} \right] = \left[ {\begin{array}{lll} {\lambda_{\textrm{ C}}^{\textrm{ }1}}&{P_{\textrm{ C}}^{\textrm{ }1}}&{T_{\textrm{ C}}^{\textrm{ }1}}\\ {\lambda_{\textrm{ C}}^{\textrm{ }2}}&{P_{\textrm{ C}}^{\textrm{ }2}}&{T_{\textrm{ C}}^{\textrm{ }2}} \end{array}} \right]. $$

To search the parameter space accurately and reduce the coding complexity, we choose the method of SBX [34], and the crossover operator:

(5)$$\hat{X} = \left[ {\begin{array}{ll} {0.5(1 + \kappa )}&{0.5(1 - \kappa )}\\ {0.5({1 - \kappa } )}&{0.5(1 + \kappa )} \end{array}} \right];\textrm{ }\kappa = \left\{ {\begin{array}{*{20}{l}} {{{(2u)}^{1/(n + 1)}}\textrm{ , }u \le 0.5}\\ {{{(2 - 2u)}^{ - 1/(n + 1)}}\textrm{ , }u > 0.5} \end{array}} \right.\textrm{ },\textrm{ }u \in [0,1), $$

where transmission factor $\kappa$ is defined as the ratio of the difference between the Children and Parents. The right part of Eq. (5) is used to fit the binary code process by a uniform distribution random number $u \in [0,1)$. The similarities between Children and Parents are determined by the spread factor n, i.e., a large value of n gives higher probability for generating Children near the Parents while a small n generates Children far from Parents.

After the crossover process, the mutation term $\hat{M}$ is applied to the Individual to generate mutated Individual: $i \to i^{\prime}$,

(6)$${i_\textrm{k}}\mathrm{^{\prime}} = {i_\textrm{k}} + \Delta (t,\textrm{ }i_\textrm{k}^{\max } - {i_\textrm{k}})\textrm{ or }{i_\textrm{k}}\mathrm{^{\prime}} = {i_\textrm{k}} - \Delta (t,\textrm{ }{i_\textrm{k}} - i_\textrm{k}^{\min }), {\kern 3pt}\textrm{k} = 1,2,3,$$

where ${i_\textrm{k}}$ respectively represents the $\lambda$, $P$, ${T_{\textrm{FWHM}}}$ when k = 1, 2, 3, and ${i_\textrm{k}}\mathrm{^{\prime}}$ represents the corresponding mutated parameter, while the terms of $i_\textrm{k}^{\max }$ and $i_\textrm{k}^{\min }$ are the maximum and minimum value of ${i_\textrm{k}}$, respectively. Hence, only one parameter is randomly mutated to its upper limit or decrease to its lower limit when the mutation is applied. And $\Delta (t,y)$ is the amount of mutation, which can be written as:

(7)$$\Delta (t,\sigma ) = \sigma \cdot (1 - {u^{{{(1 - t/T)}^{\textrm{ }h}}}}), $$

where $\sigma = i_\textrm{k}^{\max } - {i_\textrm{k}}$ or $\sigma = {i_\textrm{k}} - i_\textrm{k}^{\min }$, which depends on whether ${i_\textrm{k}}$ becomes larger or smaller, $u \in [0,1)$ is uniform distribution random number, t is the current iteration while $T$ is the total iterations of MGA. The mutation equation shown above describes a non-uniform mutation process [35] that allows the algorithm to possess a large mutation range in the early stage of MGA, and the mutation range will be gradually shrunk to the approximate solution along with the continuous iteration. And h is the term that determines the property of non-uniform, which is generally distributed between 2 − 5.

In a word, the accuracy of MGA has been enhanced by using SBX and non-uniform mutation processes, which also means that the closer these three parameters are to the optimal solution, the smaller their tuning step size are. The MGA uses the method of roulette to pick the approximate solutions as the Parents for crossover, which consists of $\lambda$, $P$ and ${T_{\textrm{FWHM}}}$. Then, the crossover operator $\hat{X}$ is applied to explore the parameter space around the selected Parents and generates new Children, while the mutation operator $\hat{M}$ provides the diversity of Children to avoid premature convergence. Three parameters of Children are used as the input for SC generation to obtain the Fitness values corresponding to them. This cycle repeats until the optimal solution is found. Due to DW generation typically involves Cherenkov radiation and soliton fission, it is generally hard to achieve precise analytical estimates. Therefore, the combination between MGA and numerical simulation is of great usefulness. Each Individual requires ∼180 s for GNLSE simulation using MATLAB v2019b running on a 2.9 GHz dual-core Intel Core i5 processor.

3. Simulation and results

As mentioned above, the desired DW spectral channel in this work intend to cover the C-H fundamental vibrational transition bands of C₂H₂, which lies in the spectral range between 3 and 3.1 µm. The 5-mm-long LN waveguide used in this work was designed to obtain the optimal mid-infrared DW by using the MGA to vary the pulse parameters $\lambda \in$ [1530, 1565] nm, $P \in$ [1000, 2500] W, ${T_{\textrm{FWHM}}} \in$ [60, 200] fs.

Figure 4(a) shows the evolution of CE values as a function of the iteration times. In the first stage, MGA randomly generates Individuals with different CE values, and these Individuals determine the initial values of CE. In the subsequent stages, CE value of each Individual is optimized according to the crossover operator $\hat{X}$ and mutation operator $\hat{M}$, while the best CE and average CE values of Population are also correspondingly increased. And the optimal CE appears for the first time after the 215 iterations, which is marked with red dot. Figure 4(b) shows the evolution of the optimal Individual in the entire parameter space in 3-dimensional view. Each point denotes the optimal individual of each generation, where the red points represent high CE and the blue points represent low CE. The best Individual will maintain for several iterations before being promoted due to the nondirectional property of MGA searching, which leads the number of points shown in Fig. 4(b) to be less than the 300 iterations. Optimal CE (2.29%) of DW is reached after using MGA, which is 53% higher than the initial result (1.5%) of the randomly generated Individuals. Figure 4(c) shows that the MGA promotes the DW to be more concentrated in the desired spectral channel and further improves the CE. Besides, the value of CE depends on how far the DW phase-matching wavelength is from the incident soliton pulse [16], which means the optimal CE (2.29%) can be further improved by dispersion engineering. In order to reduce the difficulty of fabrication, here we only consider the typical structure of LN ridge waveguide.

Fig. 4. (a) The best CE and average CE develop with the iteration. Each blue dot represents the CE value of an Individual in the Population and the optimal CE appears after 215 iterations. (b) The evolution of best Individuals. Optimal result is marked with dashed line in the zoom-in graph ($\lambda$= 1563 nm, $P$= 2357 W and ${T_{\textrm{FWHM}}}$= 100 fs). (c) The output waveform before (blue) and after (red) using MGA. Desired spectral width is marked with pink dashed lines.

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Figure 5(a) is the time-frequency diagram obtained by using optimal Individual as the input parameters, while Fig. 5(b) is obtained by using Individual corresponding to the largest CE in the first stage. The mid-infrared spectral components in Fig. 5(a) are more localized to the predefined spectral channel than the components in Fig. 5(b). The predefined spectral channel of the mid-infrared DW has been marked with pink dashed lines (${\lambda _{\textrm{DW}}} \in$ [3000, 3100] nm). Figure 5(c) shows the curve of integrated dispersion, input spectra, output spectra (top), and the spectral evolutions (bottom) corresponding to the Fig. 5(a). The location of DW is related to the zero points of ${\beta _{{\mathop{\rm int}} }}$ curve, though the position may be shifted due to the soliton fission and nonlinear effects. In addition, we repeat the MGA simulation three times to ensure that the local optimum solution does not affect the convergence of MGA. The obtained results all converged in the vicinity of $\lambda = 1563\textrm{ nm}$, $P = 2357\textrm{ W}$ and ${T_{\textrm{FWHM}}} = 100\textrm{ fs}$, due to the good design of crossover and mutation operators. Perhaps some exhaustive algorithms such as searching the entire parameter space can finally find this set of optimal parameters, but it will take more time compared to MGA. As suggested in Ref. [36], the efficiency obtained in the optimization using GA is better than the optimization using exhaustive search. Therefore, we apply the MGA to further improve the efficiency.

Fig. 5. The time-frequency diagrams of the output by using the optimal Individual searched by MGA (a) and the Individual before using MGA (b) as input parameters. (c) The input spectra (grey), GNLSE-based simulation (blue) and integrated dispersion (red) for this LN waveguide by using the optimal Individual (top). Evolution of the spectrum in 5-mm-long LN waveguide. The spectral width (${\lambda _{\textrm{DW}}} \in [3000,3100]$ nm) with the optimal CE (2.29%) is marked by pink dashed lines (bottom).

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However, the actual input parameters are not always totally equal to the designed values, we further investigate the DW performance degradation caused by the parameter deviation. Figures 6(a), 6(b) and 6(c) represent the degradation of CE by the deviation of $P \in$ [2307, 2407] W, $\lambda \in$ [1553, 1573] nm and ${T_{\textrm{FWHM}}} \in$ [90, 110] fs, respectively. The optimum CE is 2.29% and the corresponding input parameters are marked with red dots. In Fig. 6(a), the CE values are maintained above 2.1% (90% of the optimum CE) when the peak power deviate from 2307 W to 2407 W. In Fig. 6(b), CE values degrade to 2.1% (90% of the maximum CE) when the wavelength is 1557 nm. And in Fig. 6(c), the values of CE are less than 2.1% when the deviation of time duration is greater than 10 fs. Even though the deviation of $\lambda$ and ${T_{\textrm{FWHM}}}$ leads to the prominent degradation of CE, the value of CE is still above 2.1% (90% of the maximum CE) in the range of $\lambda \in$ [1557, 1573] nm and ${T_{\textrm{FWHM}}} \in$ [91, 110] fs. It shows that the CE of DW will not be affected significantly even if the input parameters fluctuate near the optimal solution found by MGA, we can use MGA to lock the position of the optimal parameters by setting various conditions corresponding to the experimental configuration, which can assist the experiment generating mid-infrared DW with the maximum CE. This method provides the flexibility and robustness for the mid-infrared emission with high CE and further applying to the gas detection.

Fig. 6. The degradation of CE by the deviation of (a)$\lambda$, (b)$P$ and (c) ${T_{\textrm{FWHM}}}$. Only one parameter changes while the other two parameters remaining. The values marked in red correspond to the optimal solutions given by MGA.

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In order to meet the application requirements of C₂H₂ gas detection, we design the LN ridge waveguide as shown in Fig. 1(a), and optimize the CE of mid-infrared DW by optimizing three parameters with MGA. It is worth noting that optimizing the waveguide structure is another potential method to further improve the CE of mid-infrared DW. X. Liu et al. [37] and H. Guo et al. [38] have reported that optimization on the waveguide structure do promote the optical nonlinear interactions, which adds a new degree of freedom to the optimization space of this MGA. In addition, this MGA-based method has potential applications in other fields, such as the octave-spanning self-referenced Kerr frequency combs. It is critical to achieve Kerr frequency combs with octave spectra via dispersive waves. H. Guo et al. have achieved self-referencing Kerr frequency combs with bandwidths exceeding one octave for pump residing in the telecom C-band by the dispersion engineering of Si₃N₄ microresonators [39]. The optical nonlinear interactions of integrated photonic devices can be promoted by combining the MGA and structure optimization, which is of great significance for complex optical applications.

4. Conclusions

In summary, we have presented an efficient optimization procedure to obtain the maximum CE of mid-infrared DW in a 5-mm-long LN waveguide. This procedure automatically adjusts three input parameters $\lambda$, $P$ and ${T_{\textrm{FWHM}}}$ by using the MGA, which has more accurate searching capability and more stable convergence capability than conventional GA. The CE of mid-infrared DW at around 3 µm is increased from 1.5% to 2.29% by using MGA which also provides the corresponding optimal input parameters. The combination of DW generation and MGA provides a novel idea for obtaining mid-infrared spectral components with high-CE, which offers a low power consumption approach to trace gas sensing in industrial safety and environmental monitoring. In addition, GA will provide new insights for the optimization of complex optical system when the structure optimization is regarded as a new degree of freedom.

Funding

Shenzhen Fundamental Research Program (No. GXWD20201231165807007-20200827130534001); Youth Science and Technology Innovation Talent of Guangdong Province (2019TQ05X227); Peking University Shenzhen Graduate School Faculty Start-up Fund.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Modified genetic algorithm for high-efficiency dispersive waves emission at 3 µm

Abstract

1. Introduction

2. Model

3. Simulation and results

4. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (7)

Optics Express