# Impact of Chirp in High-Capacity Optical Metro Networks Employing Directly-Modulated VCSELs
## 2 Chirp Measurements  
CD leads to a different phase accumulation for each spectral component of a modulated optical signal after fiber propagation. Thanks to coherent detection, the CD cumulated on the whole transmission path can be totally compensated for by DSP, removing the distortion on the signal and avoiding significant limitations to the transported capacity, as otherwise appears in case of DD [9]. Anyway, the distortion of the received electrical spectrum due to cumulated CD can be exploited to characterize the employed optical source in terms of chirp parameters. Owing to the double-sideband spectrum achieved by DM of the laser, destructive beating may occur at specific frequencies after the DD process, resulting in a frequency-selective channel that could induce significant distortions on the signal spectrum. The frequency of the fading dips depends on the cumulated CD, moving towards lower frequencies for higher dispersion. Moreover, laser chirp parameters have a strong impact on the shape of the frequency-selective channel, modifying the position and the extent of the spectrum dips [10]. Therefore, a complete characterization of the optical source in terms of chirp properties can be achieved by fitting the uncompensated channel transfer function. The interaction between phase and intensity modulation in a directly-modulated laser is described using the relation between the instantaneous frequency deviation \(\Delta\
u(t)\) and the optical modulated power \(P(t)\)[11]:  
\[\Delta\
u(t)=\frac{\alpha}{4\pi}\bigg{(}\frac{1}{P(t)}\frac{\partial P(t)}{ \partial t}+\kappa P(t)\bigg{)}, \tag{1}\]  
where \(\alpha\) is the linewidth enhancement factor and \(\kappa\) is the laser-specific adiabatic constant related to thermal effects, which depends on several laser parameters such as its gain saturation, carrier lifetime and confinement factor. The first term of Equation (1), which mainly depends on the \(\alpha\) factor, is the transient component, while the second one is the adiabatic chirp, which depends on the \(\kappa\) factor. A small-signal frequency response analysis can be used for modeling the optical source and obtain its specific chirp coefficients [12]. For a directly-modulated laser (DML), the general channel transfer function can be expressed as [10]:  
\[H(\omega)=\Big{|}\text{cos}(\theta)-\text{sin}(\theta)\alpha\Big{(}1-j\frac{ \omega_{c}}{\omega}\Big{)}\Big{|}, \tag{2}\]  
where  
\[\theta=\frac{LD\lambda_{0}^{2}\omega^{2}}{4\pi c},\ \omega_{c}=\kappa P_{0}, \tag{3}\]  
and \(L\) is the fiber length, \(D\) the dispersion coefficient, \(\lambda_{0}\) the wavelength of the CW source, \(c\) the speed of light and \(P_{0}\) the mean optical power. The position of the frequency dips and the height of the secondary lobes are determined by an interplay of \(\alpha\) and \(D\), while the extent of the dips (in particular of the first one) is primarily due to \(\omega_{c}\), i.e., to the adiabatic component of the chirp. This analysis technique has been used for measuring the chirp parameters of two different devices: the first is a 20-GHz long-wavelength VCSEL emitting at 1533.4 nm [13], while the second is an 8-GHz VCSEL tunable over about 70 nm in the third fiber window [14]. The former device is based on a short-cavity design, while the latter exploits a micro electro-mechanical systems (MEMS) top-mirror to achieve its very wide tunability. The two devices show different chirp performance owing to their specific technology. The experimental setup exploited for VCSEL chirp parameter estimation is shown in Figure 1. In particular, the long-wavelength VCSEL is directly modulated by a vector network analyzer with 40-GHz bandwidth, which generates the RF signal to perform the small-signal modulation analysis. The VCSEL bias is 9 mA and 16 mA for the fixed and tunable devices, emitting around 2.5 mW and 0.9 mW, respectively. In case of the tunable VCSEL, the MEMS bias is fixed to 20 mA, while the temperature is maintained at 25 \({}^{\circ}\)C by a standard temperature controller.