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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

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} [

However, the aforementioned progressive optical technologies desire high-quality sources with a spectral linewidth in order of kilohertz’s [

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 [

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 [

The effect of the laser spectral linewidth broadening can be described by the well-known Schawlow-Townes formula [

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 [

From (

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 [

We consider the unmodulated signal of the laser as a real-valued random passband signal

In order to determine the laser spectrum, first we need to know the autocorrelation function

Based on (^{2}; then the expression

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,

In the first case, where

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 [

The time-dependent optical power

Averaged one-sided power spectral density of

It is worth noting that, at the beginning of each individual realization, the laser transient state comes into play. This artificial effect has to be suppressed in order to correctly estimate the power spectrum. The suppression of the undesired laser transient states can be performed in the frequency domain as follows:

By using (

To determine the power spectrum

Autocorrelation function of the complex envelope of the unmodulated laser signal.

From the reason of higher numerical accuracy, only those samples of power spectrum

In Figure

Calculated samples and approximation of power spectrum

In Figure

Dependence of average output laser power at wavelengths 1550 nm and 1310 nm for different values of input current.

In Figures

Approximated values of semiconductor DFB laser linewidth obtained by numerical approach for wavelength 1310 nm.

Approximated values of semiconductor DFB laser linewidth obtained by numerical approach for wavelength 1550 nm.

On the other hand, the laser with the smallest output power

In summary, we reported on the detailed mathematical methodology, supported by numerical calculations based on laser rate equations, to accurately estimate the laser spectral properties. In particular, we developed a straightforward theoretical approach to numerically evaluate the spectral characteristics of optical sources, with the focus on distributed feedback semiconductor lasers, typically used by fiber-optic communications systems. This description and presented numerical model are appropriate in situations, when time-dependent laser parameters are readily attainable, specifically output laser power and its instantaneous optical phase. Here, the mathematical analysis provided in the work directly begins with the complex envelope of the laser signal in the baseband and this approach is advantageously used to characterize desired parameters of optical sources, especially laser linewidth, as numerically demonstrated and evaluated in this work. Furthermore, this work provides explicit characterization of the laser linewidth in the baseband, being a parameter of practical importance, because it can be shown by the optical spectral analysers. Specifically, the laser power spectral density was simulated using two important near-infrared wavelengths of 1310 nm and 1550 nm. This was done for various injection currents and different levels of the Langevin noise sources. The narrowest laser linewidth obtained by simulation was 252.6 kHz. This value was achieved for the highest average laser output power of 2.9492 mW and for the lowest parameter

In this part, the shifted power spectrum to the zero frequency area, which is expressed by (

In this part of the appendix the statistical averaging of squared difference of instantaneous phase values which are expressed in (

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The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the Slovak Research and Development Agency (APVV-0025-12). The research is supported by the European Regional Development Fund and the Slovak state budget for the project “Research Centre of University of Zilina,” ITMS 26220220183.