## 2 Non-Linear VCSEL Behavior  
All directly modulated LASERs (and therefore also VCSELs) exhibit non-linear behavior in their frequency response when varying the operation current, which happens with every change of the data symbol i.e., changing the optical power level. Figure 2 shows the measured frequency response of a VCSEL [2] for different bias currents. It is evident that the VCSEL exhibits biasing-dependent resonance frequency shifts and non-linear bandwidth reduction that influences its frequency response and data transmission. At low currents, the resonance is very distinct, causing strong Inter Symbol Interference (ISI). To obtain a flatter frequency response and avoid strong resonances, VCSELs are usually biased above 3 mA. For higher currents, the resonances become increasingly less pronounced. However, larger VCSEL bias currents inherently cause self-heating effects  
Figure 1: VCSEL Transmitter System (**a**) data rates over publication dates (**b**) power efficiency. Figure 2: Frequency response of VCSEL for different bias currents. in turn limiting the VCSEL's bandwidth [3] or (when surpassing operating thresholds) damaging the device. It is important to note that the measurements performed for Figure 2 were executed at a constant bias current (so in a high self-heating state). In reality the VCSEL might exhibit such high currents only for very short duration, meaning that the VCSEL does not experience self-heating effects as severely due to its dynamic signaling. Because of this, the VCSEL can reach higher bandwidths than the Alternating Current (AC) measurements reveal. Because the AC-response measurement is somewhat limited for explaining VCSEL behavior, the so-called pulse response of a VCSEL is shown in Figure 3. The pulse response is very similar to an impulse response, the only difference is that 1 Unit Interval (UI) wide pulses are used. A PAM-4 modulated signal has 16 different possible data transition combinations for two consecutive data sets, of which four do not cause a modulation level change. Thus, these four Direct Current (DC) cases are omitted leaving 12 different data transitions. Please note that a different VCSEL was used for AC and pulse response measurement due to test setup and assembly. It becomes evident from Figure 3 that the VCSEL does not have sufficient bandwidth to transmit 56 Gbaud as the pulses are not reaching the DC values during the transition. Moreover, the data-dependent relaxation oscillations (different per output level) are clearly visible, contributing to the ISI and BER problem. The oscillations vary both in frequency and amplitude. This means that commonly used Finite Impulse Response (FIR)/Feed-Forward Equalizer (FFE) filtering is not effective for VCSEL equalization, since the filter operates in a linear fashion such that the coefficients can only be optimized for one of the responses, not all of them. Figure 3 also shows that the top eye will be more shifted to the left (so earlier in time) and the bottom eye more shifted to the right (so later in time) than the middle eye. This effect, called skewing, is due to the asymmetric rise-and-fall times of the VCSEL (originating from current-dependent bandwidth) and severely increases the BER of a transceiver system. It can only be prevented by non-linear filtering (like Volterra or the one proposed in this paper) or by having a specialized receiver with independent slicing times for all three PAM-4 eyes. To acquire a deeper understanding of VCSEL behavior, we introduce an accurate electro-optical VCSEL model [13]. The model used is shown in Figure 4, where the discrete components of the electronic interface are summarized in Table 1.  
The electronic interface consists of linear, passive components, mainly modelling parasitic (pad) capacitances, inductances, non-linear diode behavior as well as the non-linear active area impedance of the LASER. The equivalent current through the active area \(I(t)\) serves as an input for the rate equations.