## 1 Introduction  
Vertical-cavity surface-emitting lasers (VCSELs) [1; 2] are nowadays substituting the traditional edge emitting semiconductor lasers (EEL) in many applications, such as Fiber to the Home links, computer networks, optical interconnects, optical sensing, laser printing, etc. The reason lies in their significant advantages, such as low power consumption, small footprint, high reliability, easy packaging, single longitudinal mode emission, high modulation speed, low cost, circular output beam and easy fabrication in two-dimensional arrays. The last couple of years brought a huge deployment of VCSEL for proximity and 3D sensing and autofocus functions in smart phones with overall VCSEL revenue of \(\sim\)1.78 billion USD in 2018 and a trend to reach 3.8 billion USD by 2023 [3]. Short and medium range optical communication links, the very first wide deployment of VCSELs, demand for continuous increase of VCSEL modulation speed [4]. Consequently, many efforts have been paid to boost the VCSEL direct modulation speed [5; 6; 7; 8; 9; 10] with continuously increasing the modulation speed up to 71 Gb/s [11]. Further increase of the VCSEL modulation speed would require abandoning the direct modulation approach and, eventually, integrating VCSEL with an electro-optic modulator [12; 13]. Recently, an impressive small-signal modulation cut-off frequency of about 30 GHz for such an integrated VCSEL with electro-absorption modulator (EAM), i.e., EAM-VCSEL has been reported [14]. Separating the functions of the VCSEL as a light source and the EAM as a modulator allows for optimizing each of the two parts of the integrated device with respect to its functions. Electrical optimization of the integrated EAM has been reported in [15; 16] for a lumped element and traveling electrode design, respectively. Comparison of optimized and experimental characteristics of a stand alone EAM and integrated EAM-VCSEL has been reported in [13; 14], respectively.  
In an EAM-VCSEL, the EAM layer is very thin compared to the EAM section in an edge emitter laser. To obtain enough extinction ratio a resonant structure is needed where the light passes the EAM layer several times. A problem is that a resonant modulator normally gives strong and time varying back reflections that will perturb the carrier density in the VCSEL and hence lead to spurious VCSEL modulation. In this paper we show that it is possible to modulate a resonant EAM without disturbing the amplitude lasing condition of the VCSEL. By careful design, utilizing detuned loading and the chirp factor of the EAM material, one can obtain VCSEL-EAMs with a resonantly enhanced, flat modulation response with ultrahigh bandwidth. We carry out a thorough analysis of the small- and large-signal modulation response of such an integrated EAM-VCSEL based on the coupled-cavity approach as reported in the recent experimental studies [13; 14]. The paper is organized as follows. In Section 2, we provide the single-mode rate equations for the EAM-VCSEL and their stead-state solutions. On the base of these equations, we first derive in Section 3 expressions for the modulation transfer function (MTF) of EAM-VCSEL for small-signal modulation of either the EAM losses or the VCSEL injection current. We then present a number of MTF characteristics of coupled-cavity EAM-VCSEL for three typical cases of strong, intermediate and small coupling strength. These three cases are also considered in Section 6 where we carry out the large-signal modulation response of EAM-VCSEL. Finally, in Section 7 we conclude.