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Abstract
In this work, a genetic algorithm based on differential evolution is proposed to identify sets of input driving signals used in the generation of optical frequency combs with Mach-Zehnder intensity modulators. Simulation results show that the algorithm is a powerful technique for solving a multiparameter optimization problem concerning the design of combs with a desired number of optical frequency lines under a given flatness. Results from experiments corroborate that combs with the desired number of lines can be achieved according to the driving voltages provided by the algorithm.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Since its proposal and development, Optical frequency Combs (OFC) have found use in several fields, ranging from spectroscopy measurements to optical communications [1–5]. OFCs are formed by equally spaced optical lines in the frequency spectrum domain. Techniques for generating combs are advancing rapidly and employ several nonlinear mechanisms and devices, such as mode-locked lasers that employ the Kerr-lens effect in the titanium-sapphire laser (KLM Ti:sapphire) [2,6] or the Brillouin and the Four-Wave-Mixing (FWM) mechanisms that are used in nonlinear fibers to generate combs that span several octaves [7,8]. Another interesting scheme employs microresonators of several shapes, in which the FWM effect is explored and the coupling of light to the structure is accomplished by means of evanescent waves using a tapered fiber or a prism [9]. Along these techniques the electro-optic modulation of a continuous-wave laser propagating in a guided structure has also been explored and whose principle of operation is based on the Pockels effect in materials with a high electro-optic coefficient [10]. Due to the availability of low-cost commercial devices, configurations using a cascade of Mach-Zehnder electro-optic modulators have become common [11–14]. For instance, in one such scheme an intensity and phase modulator are placed in tandem in order to obtain a comb output with the highest number of lines and flat spectrum. In this case, when properly biased, the intensity modulator produces a train of optical lines, whose shape is carved only where the chirping provided by the phase modulation is almost linear, thus equalizing the spectrum [15,16]. Other configurations employ a cascade of intensity electro-optic modulators, in which the electrical driving parameters must be set appropriately to obtain a comb output with a high number of lines and flat spectrum [17,18]. The problem is nontrivial as even in the simplest case the system is inherently nonlinear and the power distribution per comb line is not uniform.
In this work we look into a genetic algorithm that can be employed to obtain combs based on the cascade of Mach-Zehnder intensity electro-optic modulators (MZIM) that present a high number of lines, aiming a flat-topped spectrum and/or a high side-mode suppression ratio. This is a nontrivial problem as one is faced with the challenge of setting up correctly the driving input parameters of the devices. A Mach-Zehnder modulator generally presents two electrical input ports, one for the DC bias and the other for the RF signal, which control the electrical polarization and the modulation of the propagating optical wave, respectively. By adjusting the amplitudes and the frequencies of such signals, an optical comb, made of two or more MZIMs, can be designed to output a spectrum of regularly spaced optical lines within a certain bandwidth. However, determining the correct amplitudes of these signals for obtaining a comb with a predefined number of lines showing a desired flatness can be tricky, the difficulty increasing with the number of modulators used in the cascade of the optical frequency comb generator (OFCG).
Literature presents several examples of combs using Mach-Zhender intensity electo-optic modulators. In [12], Kun Qu et al. utilize two cascaded MZIMs to generate an OFC with 16 lines, but no information about the method used to identify the optimal input electrical parameters of the generator is provided. Xin Zhou et al. [13], on the other hand, use the least square method to identify possible solutions, but they state that adjustments have to be realized experimentally.
Other authors propose a graphical method to identify possible operational points for the system. For instance, Lei Shang et al. [17] utilize a method in which they use a constant value for the voltage of the continuous signals applied to the MZIMs and plot the flatness of the generated OFC as a function of a variable proportional to the amplitude of the sinusoidal signal. With the generated curve they identify the optimal operation point for the system as the one with the lower flatness. In a similar manner, Bo Li et al. [18] set the amplitude of the sinusoidal signals and plot the measured flatness for both sinusoidal and continuous electrical signals. In this way they identify the optimal operational point of the OFC. In both cases, such points were identified by using fixed parameters from the start, which limits the number of operational points available to use.
Given these previous experiments, the aim of this work is to employ the differential evolution algorithm to explore sets of input driving parameters to MZIMs that lead to combs with a specific number of lines, which are generated under the constraints given by the flatness and the side-mode suppression ratio (SMSR). We choose intensity electro-optic modulators due to their low cost and availability, although other modulators types and arrangements could also be employed for the demonstration.
2. Generating frequency combs with MZIM
Integrated intensity modulators are constructed using a combination of phase modulators in the well known Mach-Zehnder interferometric arrangement [10]. The basic structure is formed by the input waveguide, which is then divided in two arms. At the exit both arms are reunited into a single output. The light of a coherent source is coupled to the input waveguide and is equally divided in the two arms. Electrical pads are strategically embedded close to the waveguides forming the arms so that the driving electrical signal can be applied, causing a change in the refractive index of the guiding structures by means of the electro-optic effect. In this way, light propagating through the arms suffers a phase change, which translates into a constructive or destructive interference at the output, resulting in a intensity modulation of the optical signal. The electric driving signal is composed by a DC bias voltage and a time-varying voltage (RF), which may be applied to a single arm or both arms of the device, corresponding to a single-drive or a dual-drive configuration, respectively. Most commercial Mach-Zehnder modulators are produced using lithium niobate (LiNbO3) as the optical material due to its high electro-optic coefficient.
Let’s assume that a continuous-wave coherent light (laser), represented as an electrical field given by $E_{IN} = E_i e^{j\omega _F t}$, where $E_i$ is the field amplitude, $\omega _F$ is the angular frequency of the laser and $t$ is the time [14], is coupled into the input waveguide of a dual-drive MZIM, which is the device used in this study, as shown in Fig. 1. Let’s also assume that the time-varying signal is described by a sinusoidal wave $V(t)=V_{RF}\sin (\omega _{RF}t)$, where $V_{RF}$ is the signal amplitude and $\omega _{RF}$ is the angular frequency of the signal. Besides contributing to the change of the refractive index this RF signal sets the spacing between the frequency lines in the optical comb spectrum. For the sake of simplification and visualization, these driving signals are represented separated and applied to just one arm in Fig. 1, however in a real modulator device such voltages are applied to both arms in the case of a dual-drive configuration. Moreover, the time-varying signal, $V_{RF}$, is also divided internally in half and applied to the pads of the modulator in a way that the amplitude of the signal in one arm is the inverse of the other in an arrangement know as push-pull [19]. Under such assumptions, the output field of the dual-drive MZIM can be written as [19],
Fig. 1. (a) Sketch of an OFCG using two cascade MZIMs and (b) Diagram of the comb spectrum with the terms of Eq. (3) responsible for the formation of some even and odd lines.
Using the Jacobi-Anger expansion [20], Eq. (2) can be presented as
When the RF signal is applied to the MZIM, a comb of lines is generated at the device output, whose spectrum is centered at the frequency of the light source ($\omega _F$) and its lines are separated by a multiple of the RF signal frequency ($\omega _{RF}$). The amplitude of the central line is related to the Bessel function of the first kind of zero order, $\mathfrak {J}_0$. In the same way, the side lines corresponding to the even harmonics have their amplitudes proportional to $\mathfrak {J}_{2n}$ and the ones related to the odd harmonics are proportional to $\mathfrak {J}_{2n-1}$.
In this way, the amplitude of all the lines depends on the argument of the Bessel functions, $\beta$, and therefore, $V_{RF}$. Through $\alpha$, the amplitude of groups of lines can be controlled since the central line and the even harmonics are multiplied by $\cos (\alpha )$ and the odd harmonics multiplied by $\sin (\alpha )$, respectively. As $\alpha$ depends on $V_{bias}$, when the driving signal is applied to the MZIM the amplitude of the OFC lines can be controlled. An example of a comb with the lines and some of the even and odd terms of Eq. (3) are sketched in Fig. 1(b).
From Eq. (3), it is seen that the comb is composed by the individual frequencies of the light source. In this way, in a cascade of several modulators, if the output of the first modulator is used as the input to the following device, each line of the first stage will contribute to generate new lines in the output of the second, generating a comb with multiple lines. As an example, when two cascaded MZIMs are used to generate an OFCG, as shown in Fig. 1, a multiplication takes place. In this case, if the first device generates three lines from its single light frequency and the second generates five, the output of the cascaded devices gives a spectrum with 15 lines.
The challenge of the problem is, however, to identify the amplitudes of the input driving signals so that it results in the generation of an OFC with the desired parameters, generally, with a number of lines of ideally the same amplitude and side lines of much lower amplitude. This challenge becomes even more difficult as more than one MZIM device is employed in the cascade, as is the case of the present study, where two cascaded dual-drive MZIMs are employed. One possible solution for solving this problem is discussed in the next section.
3. Using the differential evolution algorithm for OFC generation
Due to its flexibility, genetic algorithms are one of the main choices when solving complex problems with multi-objectives and non linear characteristics, such as the one reported in this work. The capability of adapting these algorithms to a particular scenario is very important. For example, the problem tackled in this paper requires a change from the time to the frequency domain during the simulation, a change that could not be easily done with other types of algorithms.
The differential evolution algorithm (DEA) is an example of genetic algorithm that provides accurate results concerning the described task. It was proposed by Storn and Price in 1995 and it has been utilized to solve problems related to water piping in big cities [21,22]. In summary, the algorithm is composed by a population of individuals. Each population is submitted to certain operations in order to create new and better versions of themselves. One of these operations is executed through a function known as fitness. It is within this function that the main adaptations occur in order to allow the algorithm to solve the proposed problem.
The fitness function is used to evaluate an individual, retrieved from a population calculated by the genetic algorithm. In our case, the individual is formed by a group of values that is utilized in the comb generation. For example, if the comb configuration presents only one MZIM, the individuals will require two values: $V_{bias}$ and $V_{RF}$. Then, the fitness function is used to analyze one individual at a time by simulating the OFC with both parameters and calculating the associated features, such as the flatness ($M_{flat}$) and the Side-Mode Suppression Ratio ($M_{SMSR}$). Once such features are obtained, the function calculates a value ($V_{Ind}$) using Eq. (4) and returns it to the algorithm. The problem proposed is a minimization one, so that the smallest such a value is, the best solution it represents for the individual. The returned value in Eq. (4) consists of a weighted summation of three factors: the first includes the flatness in the calculation of the individual’s value and it should be as low as possible. The second term makes the value of the individual dependent on the inverse of SMSR: the higher the SMSR value, the lower is its contribution to the weighted sum. The third term is the validity factor ($Val$), that equals $0$ if the individual generates a valid OFC and $1$ if not. It is used to prevent individuals with bad results to propagate through future populations by adding a high value to the sum. The weights used in the simulations are given by $P_{flat} = 5$, $P_{SMSR} = 7$ and $P_{Val} = 1000$, respectively.
For instance, $P_{Val}$ must be high enough to prevent individuals that generate invalid OFCs passing through future generations, once only individuals with low evaluation values should have "descendants". $P_{flat}$ was tested with $0, 1, 3, 5, 7$ and $10$ units, respectively, with $P_{SMSR}$ set to $0$. The obtained results, except for $P_{flat}$ set to zero, were very similar in a way that the magnitude of such parameters does not matter, but the proportion between the values of $P_{flat}$ and $P_{SMSR}$ should be taken into consideration. In order to test the influence of such parameters, $P_{flat}$ was set to $5$ and $P_{SMSR}$ was tested with $1, 3, 5, 7$ and $10$, respectively. Results showed that the optimal proportion between both parameters depends on the number of desired lines for the OFC, but not on the difference between them. In this way, the following values were chosen as default in the simulations: $P_{flat} = 5$ and $P_{SMSR} = 7$.
The fitness function in this work was set to simulate combs produced by two cascaded MZIMs, so its parameters have the electrical signals for two MZIMs, including $V_\pi$, $V_{bias}$ and $V_{RF}$. In a similar work the DEA has been employed with its fitness function adjusted to two MZIMs working in a Dual-Parallel configuration [23].
As stated before, the objective of this work is to generate an OFC with a fixed number of lines of ideally the same amplitude and with the lowest possible amplitudes for the side lines. Considering only the lines that form the main comb, the difference in amplitude between the highest and lowest values is measured as the flatness of the comb. The measure of the SMSR is given by the difference between the line with the lowest amplitude, among the ones forming the comb, and the line with the highest amplitude among those located in the sides of the main lines. So the objective of the algorithm is to find the values of the electrical signals (bias and RF) that produce an OFC with the lowest flatness and the highest SMSR.
The execution steps of the algorithm is presented in Fig. 2. The initialization consists in generating a base population formed by $n$ individuals. The individuals of this first population are created with random values, between given limits of each input parameter. So, if the OFCG has only one MZIM, each individual has two only parameters: ($V_{bias}$ and $V_{RF}$). If the comb is constructed with two cascaded modulators, as in the case of this work, then the individual has six parameters ($V_{bias 1,2}$, $V_{RF 1,2}$ and $\omega _{RF 1,2}$). The initial population has to be evaluated by the fitness function, generating a vector with the corresponding value returned for each individual. With this first population and its evaluation vector, the recursive part of the algorithm takes place in the execution procedure.
The recursive step is executed until a stop condition is satisfied. These conditions can be adapted according to the situation. In this paper, the main condition is the execution time, fixed in 15 minutes. Another condition, such as the value of the current best individual, could not be used because each comb with its particular number of lines has a specific optimal value for the current best individual, therefore being different from each other. Using this as the stop condition would cause the algorithm not to converge in several cases.
Each iteration of the recursive part of the process starts with a base population ($X$) with its respective evaluation vector. This base population is processed by a mutation operation generating a second population, the mutated one ($V$). The mutation operator is standard and it is given by
where $i$ is the current new individual under generation, $r1$, $r2$ and $r3$, corresponds to the index of the individuals of the population used in the operation, which are randomly chosen, $G$ is the current generation and $F_{mut}$ is the mutation factor, a value that is randomly calculated, between $0.5$ and $1.0$, each time it is executed.The next step is the cross-over operation. In this operation, a new population is formed ($U$) using the base and mutated populations. First, a number is randomly generated between $0$ and $1$ and it is compared to the cross-over factor ($CR$). If the number is lower than $CR$, the current individual under generation is copied from $V$, with the corresponding index, if not, the copy is taken from the base population. This operation results in a population with individuals from both $X$ and $V$ populations. The value of $CR$ lies also between $0$ and $1$, with lower values being more appropriate for optimization of parameters that are independent, which is not the case of this work. Simulations were performed to identify the effects of the $CR$ parameter in the results of the algorithm. The tested values were $0$, $0.3$, $0.6$, $0.9$ and $1$. Tests have shown that the $CR$ value has only a small influence in the final value of the individual and the resulting simulated spectrum, with flatness values presenting a slight increase in the second decimal digit, as the value comes closer to $1$. Since the results for different values of $CR$ are similar, the value $0.9$ was adopted as default [24]. It is important to note that the effects of the variation of this parameter are small in this particular case, but its behaviour may be different in other applications.
After the cross-over operation, the resulting population must be evaluated by the fitness function. At the end of the process, there is a base population ($X$) in a vector with the values attributed to its respective individuals and a new population ($U$), also with its own vector. The creation of the base population for the next generation, or iteration of the recursive step of the algorithm, takes place by selecting the $n$ best individuals from $X$ and $U$. These best individuals form the base population for the next generation, continuing the iteration until the stop condition is satisfied.
When the recursive step of the algorithm is finalized, the best individual of the current population is selected and presented as the result of the execution. The use of the individual’s best characteristics generates the best possible output for the comb, considering the specified number of lines and the corresponding flatness and SMSR values.
4. Simulation results
The results presented here were obtained using the weights described in section 3., populations of 1000 individuals and a configuration using two cascaded MZIMs. The simulations went on for 15 minutes. After this time, simulations halted and the best individual of the current population was presented as the result. Table 1 shows some of the results achieved, where $f_{RF_n}$ is the RF frequency applied to the first and the second MZIM in the cascade, with $n = {1,2}$.
Table 1. Values of the electrical voltages calculated by the DEA using the configuration with two cascaded MZIMs.
Figure 3 shows results of the comb spectra with $3$, $7$, $9$ and $15$ lines using the voltages calculated by the DEA, whose flatness is $0.36$ dB, $1.56$ dB, $1.76$ dB and $12.17$ dB, respectively, and SMSR values ranging from $13.19$ to $57.96$ dB. We note that as the number of lines increases, the flatness value also increases, the worst result observed with the 15-line comb in the simulation.
Fig. 3. Simulations using two cascaded Mach-Zehnder modulators and the operational points calculated by the DEA for (a) 3 lines, (b) 7 lines, (c) 9 lines and (d) 15 lines.
Within the 15 minutes stop condition used in the simulations, it was noticed, however, that the values attributed to the individuals decreased by $99\%$ after $38$ seconds, in average. But in some cases the execution time took as long as $3$ minutes, approximately, to reach the best values.
The driving voltages obtained from the DEA were used in an experimental arrangement using the setup with two cascaded devices shown in Fig. 4. Both MZIMs, Thorlabs model LN81S-FC, presents a high $V_{\pi }$ (approx. $4.5$ V) and were connected with a short optical cord. The laser source employed as the input optical signal to the MZIMs is a DFB laser from ILX Lightwave, model 79800D/315. The output of the generator is measured using an optical heterodyne detection technique. In this technique a convolution takes place between the output spectrum of the OFCG and the frequency line of a local oscillator (tunable DFB laser, Santec model TSL-510) operating in a frequency ($F_{osc}$) very close to the frequency in which the OFC is centered ($F_{OFC}$). The convolution is performed through an optical splitter and results in a spectrum with two copies of the comb spectrum centered at the frequencies $F_{OFC} - F_{osc}$ and $F_{OFC} + F_{osc}$, respectively. The copy produced at the $F_{OFC} - F_{osc}$ side lies within the photodetector bandwidth ($10$ GHz), which is easily seen with an oscilloscope (Keysight, model DSO91204A). The utilized photodetector is a Discovery Semiconductor, model DSC-R402AC-39-SC/APC-K-2 [25]. The radio frequency sources were the SGS100A model of Rohde & Schwarz, with $1$ mHz resolution and frequency range from $1$ MHz to $12.75$ GHz and the Arbitrary Waveform Generator, model M8195A of Keysight with 65 GSa/s, 25 GHz bandwidth and 8 bit vertical resolution. Both radio frequency sources had their signal amplified by a Tektronix RF driver amplifier, model PSPL5865, with 12.5 GHz bandwidth and up to 26 dB gain. The DC source is a generic model with output in the range from $0$ V to $30$ V.
Fig. 4. Layout of the experimental arrangement, where PD means photodetector, Pol. Cont. is the polarization controller, RF Amp. is the radio frequency driver amplifier and CW Laser is the Continuous Wave laser.
Figure 5 shows the experimental spectra of combs generated using the $V_{bias}$ and $V_{RF}$ voltages provided by the DEA. Combs with $3$ and $9$ lines present a flatness of $2.71$ and $4.89$ dB, respectively, with the 15-line comb presenting a much worse flatness, but in qualitative agreement with the simulated spectrum seen in Fig. 3(d). Although the resulting combs shows the correct number of lines, the flatness and SMSR values deviated from those provided by the algorithm (see Table 1), which demanded an adjustment of the bias voltages applied to the modulators. Such voltages are presented in Table 2 for the spectra shown in Fig. 5. We believe that this deviation has origin on two factors. The first is the fact that the true response curves (Power x $V_{bias}$) of the modulators are not perfectly sinusoidal, which causes a different response of the device when the simulated bias voltage is applied to its input. The second factor lies on the voltage of the employed bias sources and the driver amplifier gain, which present low adjustment resolution, therefore limiting the accurate setting of the input values. As measurements were performed in a short time, the known drift behavior of the modulator’s operating point due to thermal effects did not contribute to change the comb response, playing no role on the observed results.
Table 2. Driving voltages related to the experimental spectra seen in Fig. 5.
5. Conclusions
The challenging task of identifying an operational point for the electrical signals that drive the cascade of MZIMs forming the optical comb is tackled with the DEA algorithm, which delivers values for $V_{bias}$ and $V_{RF}$ under the established stop condition. Experimental results show that the technique is functional for obtaining combs with the desired number of lines, although an adjustment was required in the bias and RF amplifier driver voltages to obtain flatness and SMSR that approximated those provided by the DEA. Further improvement may be implemented in the technique in order to take into account the effective response curves (Power x $V_{bias}$) of the modulators, so that simulated and experimental results match. Yet, we believe that the algorithm has shown its potential as a powerful technique for establishing the driving parameters for optical combs based upon cascaded Mach-Zehnder intensity modulators. Further improvement in the experimental arrangement can also be implemented, such as a dynamic control loop that contributes to reduce the drift of the modulator’s operational point due to thermal effects, when the comb is used over longer times. Such control loop can be implemented using a high-resolution spectrum analyser at the cascade output, whose signal may feed a micro-controller that drives the bias and RF voltage sources, establishing a stable operational point for the OFCG.
Funding
Conselho Nacional de Desenvolvimento Científico e Tecnológico (481581/2013-8).
Disclosures
On behalf of all authors, the corresponding author states that there is no conflict of interest.
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