Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 4152895, 11 pages

http://dx.doi.org/10.1155/2016/4152895

## Numerical Estimation of Spectral Properties of Laser Based on Rate Equations

^{1}Faculty of Electrical Engineering, University of Zilina, Univerzitna 1, 010 26 Zilina, Slovakia^{2}Centre for Nanoscience and Nanotechnology, CNRS, Université Paris-Sud, Université Paris-Saclay, C2N-Orsay, 91405 Orsay Cedex, France

Received 27 July 2016; Accepted 26 October 2016

Academic Editor: Luciano Mescia

Copyright © 2016 Jan Litvik et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Laser spectral properties are essential to evaluate the performance of optical communication systems. In general, the power spectral density of the phase noise has a crucial impact on spectral properties of the unmodulated laser signal. Here the white Gaussian noise and -noise are taken into the consideration. By utilizing the time-dependent realizations of the instantaneous optical power and the phase simultaneously, it is possible to estimate the power spectral density or alternatively the power spectrum of an unmodulated laser signal shifted to the baseband and thus estimate the laser linewidth. In this work, we report on the theoretical approach to analyse unmodulated real-valued high-frequency stationary random passband signal of laser, followed by presenting the numerical model of the distributed feedback laser to emulate the time-dependent optical power and the instantaneous phase, as two important time domain laser attributes. The laser model is based on numerical solving the rate equations using fourth-order Runge-Kutta method. This way, we show the direct estimation of the power spectral density and the laser linewidth, when time-dependent laser characteristics are known.

#### 1. Introduction

Constant growth in transmission data and capacity leads to the deployment of new optical technologies that are used in the infrastructure of communication systems. Recently, the emphasis is devoted to the transmission systems with capacity in excess 100 Gb·s^{−1} per one channel. This yields to the overall capacity of the optical communication systems ranging from 10 Tb·s^{−1} to 100 Tb·s^{−1} [1]. These levels of the capacity can only be reached by utilizing and combining various optical technologies such as wavelength division multiplexing (WDM), polarization-division multiplexing (PDM), spatial division multiplexing (SDM), optical coherent systems with high-order modulation formats, and digital signal processing (DSP) [1–5].

However, the aforementioned progressive optical technologies desire high-quality sources with a spectral linewidth in order of kilohertz’s [1], with high degree of coherence, and low noise level. The optical systems that are able to use advanced modulation formats, for instance, 16-state quadrature amplitude modulation (16-QAM), require laser linewidth of around 1 MHz for symbol rate 10 GBaud [1, 6]. In case of quadrature phase shift keying (QPSK), laser linewidths of 10 MHz are usually sufficient [1, 7].

The laser phase noise is one of the critical parameters, having a substantial influence on the performance of optical coherent systems, utilizing WDM and high-order modulation formats [8]. Here, the prime goal is to estimate the laser characteristics. Lasers employed in the optical coherent systems are usually used two times in the transmission link: as signal sources at the side of optical transceiver and at the side of optical receiver as local oscillators [8–10].

In the laser active region placed in the cavity, there exists two dominant processes of light generation. First one is the spontaneous emission process, arising from the direct recombination. The second one is light generation caused by the stimulated emission. The recombination process requires the flow of the input optical power. From this reason the stimulated emission leads to the amplification of the incoming photon flux [11]. In general, the laser linewidth is considered as a consequence of phase fluctuation in the optical field. The main reason of these fluctuations is the effect of spontaneous emission. This randomly changes the phase and the amplitude of the lasing optical field. The second reason is due to the instantaneous changes the charge carrier density. These variations have an impact on the change of refractive index in the active region of the laser. During the relaxation period oscillations, the change of refractive index causes additional phase shift of the laser field and also leads to the broadening of the laser linewidth [12].

The effect of the laser spectral linewidth broadening can be described by the well-known Schawlow-Townes formula [13]where is the frequency of lasing light, is Planck’s constant, is the photon lifetime, is the spontaneous emission factor, is the linewidth enhancement factor, and is the total lasing power. From (1), it becomes apparent that three approaches can be used to reduce the laser spectral linewidth.

First is to increase the photon lifetime by using longer resonator cavity or by improving the reflectivity of the facet. Second approach is to reduce the linewidth enhancement factor by using the negative wavelength detuning of the distributed feedback (DFB) laser or by using the quantum well structures. The third option is to increase the laser output power [13, 14].

From (1), it is clear that the laser linewidth is inversely proportional to the output power . However, the laser linewidth cannot reach zero due to some residual noise of the source. The spectrum of the frequency modulated (FM) noise of the semiconductor lasers is characterized by white frequency noise [13, 15]. The white frequency noise is primarily caused by the spontaneous emission and the -noise. This quantity is also inversely proportional to the output power. The -noise is almost independent of the output power and it is independent of the time of signal observation [13]. If the white frequency noise is sufficiently reduced by increasing output optical power, the spectral linewidth becomes constant. Experimental results have shown that origin of this residual laser linewidth is caused by flicker () noise, which can be observed on the frequencies lower than 100 kHz [8, 11, 13].

Spectral properties of a laser are determined via spectral properties of the phase noise. It follows that spectral properties of the unmodulated laser signal are naturally determined by power spectral density (PSD) of phase noise [13]. In the literature, these spectral characteristics of laser are typically described by using linewidth. The spectral linewidth of the laser is the only parameter of the PSD and it is characterized as full width at half maximum (FWHM). This parameter is used to comparing different sources of radiation. On the other hand, the linewidth parameter does not provide in-depth information about laser spectral properties. The approach to describe laser characteristics by using the power spectral density of phase noise provides complex information about the laser spectral properties [16–18]. The laser spectrum approximation by Voigt probability distribution allows inclusion of the essential contributions of the white phase noise and -phase noise simultaneously. The white phase noise has a substantial influence on a spectrum shape according to the Cauchy-Lorentz probability density function. The -noise causes spectrum shaping according to the Gaussian probability density function [13]. The Rice probability density function is additional option to approximate the laser spectrum shape, assuming that the laser phase noise has white Gaussian noise character. However, the methods are limited by the attainable bandwidth and the number of samples used in numerical evaluation. By taking this into consideration, the laser spectral PSD can be directly modelled by Rice distribution [18, 19]. In Section 2, we describe the derivation of the laser PSD and linewidth formulas. The derivation of theoretical laser PSD formulas has been stated in [13, 20]. In this work, we have performed a revision of this theoretical analysis. The theoretical analysis has been extended in order to obtain relationships in more compact form. Subsequently, it is possible to use the approximation of PSD by Voigt profile for particular laser type, which is given through the numerical model in the form of laser rate equations. The analysis begins from the description of the laser signal as a real-valued random passband signal. The phase fluctuations of the laser signal are considered as a stationary Gaussian random process. By using the autocorrelation function, the laser linewidth is expressed for two theoretical cases. In the first case, the shape of linewidth is approximated by Gaussian shape, while in the second case the Lorentzian shape is considered. In Section 3, we show the application of the presented theoretical approach and its implementation for particular PSD calculation. At the end, we show that the resulting shape of the laser PSD is given by combination of the Gaussian and Lorentzian shape, yielding the Voigt profile. Finally, conclusions are drawn in Section 4.

#### 2. Derivation of the PSD and Linewidth Formulas

We consider the unmodulated signal of the laser as a real-valued random passband signal , which can be mathematically expressed aswhere is the arbitrary realization of the random laser signal, is its complex envelope, that is, complex-valued baseband signal, is the time, and is the reference frequency of the signal . The overall spectrum of the signal has a symmetrical distribution around the frequency . Furthermore, has a constant envelope and its random character is determined by the random phase variations as an argument of its complex envelope [12, 13, 15, 16]. Thus, the complex envelope of the signal can be expressed aswhere is the constant average power of the laser and is the instantaneous phase of the signal . We expect that the instantaneous phase is stationary Gaussian random process [14, 21, 22]. For the instantaneous phase, it is valid thatwhere is the function of instantaneous frequency fluctuations of signal from mean frequency (hereafter abbreviated as phase noise). The instantaneous frequency of the signal is then . Since is Gaussian random process and according to (4), the quantity must also be a Gaussian random process.

In order to determine the laser spectrum, first we need to know the autocorrelation function of the signal , which can be defined aswhereis the autocorrelation function of the complex envelope of signal , is the operator of ensemble averaging (expected value), and represents complex conjugation. The power spectrum of unmodulated laser signal can be determined by using the autocorrelation theorem. From this theorem, it is clear that double-sided power spectral density of signal is represented by the Fourier transform of the autocorrelation function of the signal [23] and thenwhere denotes Fourier transform. From a physical point of view, we take into account only those spectral components that are situated on the positive frequencies . Instead of double-sided PSD, we use only single-sided PSD of unmodulated signal, here denoted as . Single-sided PSD is defined as follows:It is suitable to operate with shifted version of single-sided PSD of unmodulated laser signal . The frequency shift toward zero frequency by frequency value is performed. Therefore, we obtain the power spectrum centred around zero frequency. It can be shown that this operation is valid and it is described in more detail in Appendix A. As a result of this approach, we getIn the following, we will determine the unmodulated signal spectrum only by using (9). Therefore as a first step, it is suitable to determine the autocorrelation function of complex envelope of unmodulated laser signal. By substituting (3) into (6), we getFor Gaussian random variable, the following relation is valid [23, 24]:If we utilize (11), (10) becomes of the form

Based on (4), for the difference of the instantaneous phase values at the two different time instants, which are seconds apart from each other on the time scale , we can write down the following expression:The statistical averaging of square of (13) is valid (see Appendix B)where is the autocorrelation function of the phase noise and it is valid thatEquation (14) can be adjusted to the formwhere the triangular function is expressed as follows:From the autocorrelation theorem, we can obtain the autocorrelation function of the phase noise by using the inverse Fourier transform of double-sided PSD of the phase noiseBy substituting for from (18) to (16), we getAfter the change of the integration order and adjusting the previous equation, we getFor the expression in the square brackets on the right side in the last equation, the following is valid:where the function is defined asBy substituting (21) into (19), the mean square value of difference of the instantaneous phase at the two different time instants which are seconds apart from each other on the time scale iswhere is single-sided PSD of the phase noise , which is defined as follows:It is suitable to introduce another parameter which is indirectly describing the spectral properties of the unmodulated signal of the laser . This parameter is average power of the phase noise which is given by following relation:At the beginning of this section, we assumed that the phase noise is the stationary Gaussian random process and in the baseband of bandwidth , the phase noise has a white noise character. From this reason, it is possible to express the single-sided PSD as follows:After substitution of from (26) to (23) and after further adjustment, we getThe unmodulated laser signal can be viewed as one spectral line located at the frequency with the zero bandwidth that is frequency modulated (FM) by phase noise and causing spectral line spreading. Thus, the real lasers have nonzero bandwidth, which is caused exactly by this frequency modulation. Usually the bandwidth is characterized by full width at half maximum, which is denoted as . The frequency deviation of that FM is equal to the average value of the instantaneous frequency deviation of the signal from frequency [14, 17, 18, 20]. Instantaneous value of this deviation is denoted as and its average value is equal to . Afterwards, the quantity is considered as a frequency deviation of the aforementioned frequency modulation. Because the bandwidth of modulating signal is , then the index of FM modulation is . The quantity of phase noise has dimensions in Hz, and its average power is the quantity expressed in Hz^{2}; then the expression is dimensionless. As we will show below, the value will determine the character of the unmodulated laser spectrum [17, 20].

The value of the FM modulation index determines two characteristics of power spectra of the unmodulated laser signal. For a case, where the index of FM modulation is small, that is, is large enough compared to , the resulting spectrum shape is Lorentzian. For a case, when the FM index is high, that is, is small enough compared to , the final spectrum follows the Gaussian shape.

In the first case, where is large compared to , that is, increases to the infinity, for the integral on the right side of (27), we can write this approximationand (27) becomesBy substituting the last expression to (12), then for the autocorrelation function of the complex envelope of the unmodulated laser signal, the following is valid:By using (9), we can obtain the power spectrum of the unmodulated laser signal shifted to the zero frequency region. This step is performed by using Fourier transform of the autocorrelation function of complex envelope. Then the power spectrum can be expressed as follows:The shape of the power spectral density follows Lorentzian shape and the FWHM is given byAssuming that the root mean square (RMS) value of the phase noise is much lower than the bandwidth , then the linewidth of the unmodulated laser signal definitely depends on the power spectral density of the phase noise. In the second case, where is small compared to , the integral on the right side of (27) is rewritten as follows:and (27) becomes toBy substituting the last expression to (12), the autocorrelation function of the complex envelope of the unmodulated laser signal can be expressed asBy using (9), we can apply the Fourier transform on the last equation. This way, we obtain the power spectrum of the unmodulated laser signal shifted to the zero frequency asFrom the last expression it is obvious that the power spectral density has a Gaussian shape and the FWHM can be expressed as follows:By considering the condition that the RMS of the phase noise is much larger than the bandwidth , then the linewidth of unmodulated laser signal definitely depends on the RMS of phase noise .

#### 3. Numerical Simulation and Results

In this section, we present the application of the afore-described theoretical analysis to estimate the linewidth of distributed feedback semiconductor laser (DFB) by using the numerical approach. The DFB laser model is based on the standard equivalent circuit model [21, 22]. Here, the model is built up by using a set of coupled differential equations, where the noise is modelled via Langevin noise sources. The light emitting sources analysed by this approach have output characteristics that are comparable to the demonstrated experiments [21, 22]. The main output parameters of the numerical model are time-dependent optical power and its instantaneous phase . The signal characteristics of the numerical model are described in [9, 21].

The time-dependent optical power and the frequency fluctuations were generated for following parameters. Specifically, we considered two wavelengths of 1310 nm and 1550 nm, respectively, different input powers of the laser, ranging from 12.5 mA to 25 mA, and various levels of Langevin noise sources. In the case that Langevin noise sources satisfy Markovian assumptions [22], the noise sources are appropriately simulated by three random Gaussian variables with zero mean value and variance . We used the parameter to modify the impact of noise in the numerical model. Particular time-dependent variables were generated in the time interval ns with integration (time) step ps. This integration step is sufficient to achieve the numerical stability of fourth-order Runge-Kutta method that was used to solve the laser rate equations [10, 21, 22, 25]. From the statistical point of view, we performed realizations for the time-dependent optical powers series and fluctuations of the instantaneous frequency that deviates from the reference frequency . From the generated time-dependent instantaneous powers , we have determined the estimation of aswhere is the number of time-dependent realizations, index is the number of the particular realisation of and , and is the duration of these realizations. Subsequently, we have determined the one-sided power spectral density (see Figure 1) of the phase noise as follows: is determined only for positive frequencies from each realization of . Then, the values of the individual power spectral densities (particular statistical realisation) were averaged. As a result, we obtained a one mean single-sided power spectral density of the phase noise.