1 Introduction

Semiconductor laser is an integral part of high-speed telecommunications. The next-generation optical networks require optical transmitters to operate in tens of GHz modulation speed [1]. A commonly used directly modulated laser has fundamental speed limits, due to the CPR effect and displays severe wavelength chirp [2,3,4]. The principal attraction of the direct modulation technique is its simplicity and low cost. With the laser biased above threshold and a modulation signal superimposed on the drive current, the output optical power of the laser is an analog of the modulation waveform. The intrinsic response is set mainly by the differential gain, and much effort has been put into increasing this value. Strained quantum well long-wavelength semiconductor lasers have achieved modulation bandwidths of 25 GHz for DFB structures [5, 6]. However, the frequency chirping of DFB laser is also directly related to the differential gain, with higher values producing lower chirp. One way to reduce the wavelength chirp is to use an external modulator based on a Mach–Zehnder interferometer in a material showing strong electro-optic effect, such as LiNbO [7,8,9]. However, such an external modulator is very expensive and cannot be integrated with III–V semiconductor lasers.

On the other hand, the vertical-cavity surface-emitting laser (VCSEL) is emerging as the light source of choice for modern high-speed and short-distance communications. Advantages of VCSELs over traditional edge-emitting lasers include inherent low cost [10], enhanced reliability [11], circularly symmetric optical output. The TCC-VCSEL has recently attracted much attention as a promising light source with wide bandwidth with low-frequency chirp and low relative intensity noise [12,13,14,15,16]. In this device, the VCSEL is coupled in the lateral direction with a passive feedback cavity, which re-radiates the VCSEL cavity with slow light from the transverse direction. This slow-light feedback happens to couple with the laser radiation in the VCSEL cavity and induces a type of photon–photon resonance (PPR) with the modulating signal. The PPR either boosts the modulation bandwidth of the VCSEL or generates enhanced peak in the modulation response between the CPR and PPR frequencies [14, 17]. Both types of improvement of the modulation response of the VCSEL support its applications in the next generation of ultra-high-speed photonics [17, 18]. Experiments on the TCC-VCSEL demonstrated bandwidth of 29 GHz by Dalir and Koyama [19], 30 GHz by Dalir et al. [20] and 44 GHz by Dalir et al. [21], compared with 10 GHz of the conventional VCSEL (C-VCSEL). Theoretical investigation of the slow-light feedback showed that the amount of bandwidth enhancement depends on how strong is the feedback [14].

In this paper, we propose a novel structure of a VCSEL coupled with a cascade of multiple TCCs which could provide transverse feedback stronger than the case of single TCC, and then enhances more the VCSEL bandwidth. The VCSEL cavity and TCCs are connected through oxide apertures through which the slow light is coupled among these cavities. We present the modeling of the device basing on a travelling-wave model of the electromagnetic field in the transverse direction. The feedback is treated as time delay of the slow-light due to round trips in the cascaded TCCs. In addition, we counted the multiple reflections and the corresponding multiple round trips of slow light in the closest TCC in which slow light might be strong. This consideration is reasonable to a high degree since the slow-light feedback is rather weak compared with the vertically oscillating light in the primary VCSEL cavity. While counting multiple reflections in the other TCCs makes the analysis awesome, it may not cause big changes in the simulated results. A formula is introduced to the effective reflectivity of the slow light at the oxide aperture of the VCSEL cavity, and the threshold gain of the structure is derived. This form of complex threshold gain is then used to modify the time-delay rate equations of the VCSEL under optical feedback to appropriately describe the time evolution of the field intensity and phase. The fast Fourier transform (FFT) is used to investigate the frequency content of the laser signal. Then, over a wide range of the modulation current, the Fourier spectrum of the signal is used to generate the modulation response of the device as well as to assess the signal distortion. The optimum parameters of the structure, including the coupling ratio and TCC length, for generating ultra-high-frequency signals are investigated. A parametric simulation is presented to the signal characteristics including the frequency of oscillation, frequency chirp and second-harmonic distortion (2HD). The obtained results show that the proposed structure could achieve 300% enhancement of the modulation bandwidth of the C-VCSEL due to extension of the CPR frequency or the PPR effect. Signals with frequencies as high as (40–49 GHz) with low distortion are presented.

In Sect. 2, we present the proposed structure of the cascaded TCCs-VCSEL and the theoretical model of the time-delay rate equations. In Sect. 3, we introduce the procedure of numerical simulation. The simulation results on the modulation bandwidth enhancement and the generated ultra-high oscillations are presented. Finally, conclusions of the present work appear in Sect. 4.

2 Theoretical model

A scheme of the proposed designed structure of the VCSEL with cascaded TCCs is given in Fig. 1. The VCSEL is laterally coupled with a series of N-external lossy cavities through an oxide aperture. The TCCs, each with length LC, are also coupled through oxide apertures. This structure of coupled waveguides introduces lateral optical confinement and a leaky traveling wave in the lateral direction.

Fig. 1
figure 1

A scheme of slow-light feedback in VCSEL with a cascade of multiple TCCs

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In the VCSEL cavity, light is confined from top and bottom, and travels in zigzag with an angle close to 90° near the cut-off condition of light propagation in the lateral direction [22]. That is, light travels perpendicularly and is slowed in the laterally coupled waveguides. Within each TCC, the slow light propagates with group velocity of vg = c/ng, where ng = fn is the group index, n is the average material refractive index and f is the slow factor of light. At the aperture of an arbitrary TCC (mth TTC), the slow light is partially transmitted to the following (m + 1)th TCC, and is partially reflected back and coupled to the preceding (m − 1)th TCC with coupling ratio ηm. The coupling ratio of the slow-light feedback to the primary VCSEL cavity is η. At the last TCC (m = M), the slow light is totally reflected back and propagates back toward the VCSEL cavity. Within each TCC, the back-and-forth propagating slow light suffers loss of \(\exp \left( { - 2\alpha_{Cm} L_{Cm} } \right)\) and phase delay of \(\exp \left( { - 2j\beta_{Cm} L_{Cm} } \right)\), where αCm = fmκm and \(\beta_{Cm} = {{2\pi n} \mathord{\left/ {\vphantom {{2\pi n} {\left( {\lambda f_{m} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\lambda f_{m} } \right)}}\) are the lateral optical loss and propagation constant with κm being the material loss and λ the emission wavelength. The period of the round trip between the VCSEL cavity and the far end of the mth TCC is \(\tau_{m} = {{2n_{mg} L_{Cm} } \mathord{\left/ {\vphantom {{2n_{mg} L_{Cm} } c}} \right. \kern-\nulldelimiterspace} c}\).

In this structure, there a number of M external cavities in addition to the VCSEL cavity related to the lasing process, which contains M + 2 mirrors in the transverse directions. The end mirrors of these cavities are perfectly reflecting. In the present model of optical feedback, the external mirrors and the reflecting portion at the oxide aperture of the VCSEL are assumed to form an equivalent virtual mirror to the slow light propagating in the transverse direction. The effective reflectivity of the forward propagating field E(t) of slow light at instant t and at the oxide aperture of the VCSEL cavity is given by

$$R_{eff} = \sqrt \eta \left\{ {1 + \frac{\eta }{1 - \eta }\left[ \begin{gathered} \sum\limits_{m = 1}^{M - 1} {\sqrt {\eta \left( {1 - \eta_{m} } \right)} \left( {\prod\limits_{n = 1}^{m} {\eta_{n - 1}^{n - 1} } } \right)\frac{{E\left( {t - \tau_{m} } \right)}}{E\left( t \right)}e^{{ - j2\beta_{Cm} L_{Cm} }} e^{{ - 2m\alpha_{cm} L_{cm} }} } \hfill \\ + \sqrt \eta \left( {\prod\limits_{n = 1}^{M} {\eta_{n - 1}^{n - 1} } } \right)\frac{{E\left( {t - \tau_{M} } \right)}}{E\left( t \right)}e^{{ - j2\beta_{CM} L_{CM} }} e^{{ - 2m\alpha_{cM} L_{cM} }} \hfill \\ \end{gathered} \right]} \right\}$$
(1)

Although intensity of the slow light is rather weak compared with that of the vertically oscillating light in the primary VCSEL cavity, we count the multiple reflections and corresponding multiple round trips in the first TCC (m = 1). In this case, the effective reflectivity is modified to

$$R_{eff} = \sqrt \eta \left\{ \begin{gathered} 1 + \frac{\eta }{1 - \eta }\left[ \begin{gathered} \sum\limits_{m = 1}^{M - 1} {\sqrt {\eta \left( {1 - \eta_{m} } \right)} \left( {\prod\limits_{n = 1}^{m} {\eta_{n - 1}^{n - 1} } } \right)\frac{{E\left( {t - \tau_{m} } \right)}}{E\left( t \right)}e^{{ - j2\beta_{Cm} L_{Cm} }} e^{{ - 2m\alpha_{cm} L_{cm} }} } \hfill \\ + \sqrt \eta \left( {\prod\limits_{n = 1}^{M} {\eta_{n - 1}^{n - 1} } } \right)\frac{{E\left( {t - \tau_{M} } \right)}}{E\left( t \right)}e^{{ - j2\beta_{CM} L_{CM} }} e^{{ - 2m\alpha_{cM} L_{cM} }} \hfill \\ \end{gathered} \right] \hfill \\ {\text{ X}}\sum\limits_{n = 1}^{N} {\sqrt {\left( {1 - \eta } \right)\left( {1 - \eta_{1} } \right)}^{n - 1} e^{{j(n - 1)\omega \tau_{1} }} \frac{{E\left( {t - \tau_{1} } \right)}}{E\left( t \right)}e^{{ - j2\beta_{CM} L_{C1} }} e^{{ - 2m\alpha_{c1} L_{c1} }} } \hfill \\ \end{gathered} \right\}$$
(2)

where N is the total number of multiple reflections. While counting multiple reflections in the other TCCs makes the analysis awesome, it may not cause big changes in the simulated results.

In this case, the time-delay rate equations of the VCSEL with single TCC [14] are modified to the following forms of the number of photons \(S(t) \propto \left| {E(t)} \right|\) contained in the lasing mode and the optical phase θ(t) in addition to the carrier number N(t):

$$\frac{dS}{{dt}} = \left[ {G - \frac{1}{{\tau_{p} }} + \frac{{v_{g} }}{W}\ln \left| U \right|} \right] S + \Gamma R_{sp}$$
(3)
$$\frac{d\theta }{{dt}} = 2\pi \Delta f(t) = \frac{1}{2}\left( {\alpha \Gamma \frac{a}{V}\left( {N - N_{th} } \right) - \frac{v_{g}}{{ W}}\phi } \right)$$
(4)
$$\frac{dN}{{dt}} = \frac{{\eta_{i} }}{e}I(t) - \frac{N}{{\tau_{s} }} - GS$$
(5)

where Δf(t) is the associated frequency shift. In the above equations, G is the optical gain defined as [23]

$$G = \frac{\Gamma a}{V}\frac{{N - N_{T} }}{1 + \varepsilon S}$$
(6)

where a as the differential gain of the active region whose volume is V, NT the electron numbers at transparency, and ε the gain suppression coefficient. Γ is the confinement factor, τp is the photon lifetime, ηi is the injection efficiency, τs is the electron lifetime due to the spontaneous emission, Rsp is the spontaneous emission rate, and Nth is the electron number at threshold. The power P(t) emitted from the primary VCSEL cavity is calculated from the photon number S(t) as

$$P\left( t \right) = h\nu \alpha_{m} S\left( t \right)$$
(7)

where h is the Planck’s constant, f is the frequency of the laser output and αm is the power loss through the top mirrors.

The lateral OFB is included in the time-delay function U(t − τ) as

$$\begin{gathered} U = \left| U \right|e^{ - j\phi } = 1 + \hfill \\ \, \frac{\eta }{1 - \eta }\left[ \begin{gathered} \sum\limits_{m = 1}^{M - 1} {\sqrt {\eta \left( {1 - \eta_{m} } \right)} \left( {\prod\limits_{n = 1}^{m} {\eta_{n - 1}^{n - 1} } } \right)\sqrt {\frac{{S\left( {t - \tau_{m} } \right)}}{S\left( t \right)}} e^{{j\left[ {\theta \left( {t - \tau_{m} } \right) - \theta \left( t \right)} \right]}} e^{{ - j2\beta_{Cm} L_{Cm} }} e^{{ - 2m\alpha_{cm} L_{cm} }} } \hfill \\ + \sqrt \eta \left( {\prod\limits_{n = 1}^{M} {\eta_{n - 1}^{n - 1} } } \right)\sqrt {\frac{{S\left( {t - \tau_{M} } \right)}}{S\left( t \right)}} e^{{j\left[ {\theta \left( {t - \tau_{M} } \right) - \theta \left( t \right)} \right]}} e^{{ - j2\beta_{CM} L_{CM} }} e^{{ - 2\alpha_{cM} L_{cM} }} \hfill \\ \end{gathered} \right] \hfill \\ {\text{ X}}\sum\limits_{n = 1}^{N} {\sqrt {\left( {1 - \eta } \right)\left( {1 - \eta_{1} } \right)}^{n - 1} \sqrt[{}]{{\frac{{S\left[ {t - \left( {n - 1} \right)\tau_{1} } \right]}}{S\left( t \right)}}}e^{{j\left\{ {\theta \left[ {t - \left( {n - 1} \right)\tau_{1} } \right] - \theta \left( t \right)} \right\}}} e^{{ - j2(n - 1)\beta_{C1} L_{C1} }} e^{{ - 2(n - 1)\alpha_{c1} L_{c1} }} } \hfill \\ \end{gathered}$$
(8)

The phase ϕ of the function U is determined as

$$\phi = - \tan^{ - 1} \left[ {\frac{{{\text{Im}} \left( U \right)}}{{{\text{Re}} \left( U \right)}}} \right] + p\pi$$
(9)

where p is an integer. The injection current I(t) in Eq. (3) is assumed to include a bias component Ib and a modulation signal with amplitude Im and frequency fm.

3 Procedures of numerical calculations

In the present study, we consider the case of identical transverse cavities with length LC, group velocity ng, and round-trip time \(\tau = {{2n_{g} L_{C} } \mathord{\left/ {\vphantom {{2n_{g} L_{C} } c}} \right. \kern-\nulldelimiterspace} c}\). The feedback function U in Eq. 7 is reduced to the form

$$\begin{gathered} U = 1 + \left( {\sum\limits_{m}^{M} {\frac{{\eta^{m} }}{{\sqrt {1 - k\eta } }}\sqrt {\frac{{S\left( {t - m\tau } \right)}}{S\left( t \right)}} e^{{j\left[ {\theta \left( {t - m\tau } \right) - \theta \left( t \right)} \right]}} e^{{ - j2m\beta_{C} L_{C} }} e^{{ - 2m\alpha_{C} L_{C} }} } } \right) \hfill \\ \, \left( {\sum\limits_{n = 1}^{N} {\left( {1 - \eta } \right)^{n - 1} \sqrt {\frac{{S\left[ {t - \left( {n - 1} \right)\tau } \right]}}{S\left( t \right)}} e^{{j\left\{ {\theta \left[ {t - \left( {n - 1} \right)\tau } \right] - \theta \left( t \right)} \right\}}} e^{{ - j2(n - 1)\beta_{C} L_{C1} }} e^{{ - 2(m - 1)\alpha_{c} L_{C} }} } } \right) \hfill \\ \end{gathered}$$
(10)

with k = 1 for the last TCC (m = M) and k = 0 for other TCC. Therefore, U(t − τ) is a time-delay function that counts the slow-light feedback as time delay of the electric field in the lateral cavities including N-reflections in the closest feedback cavity with round-trip \(\tau = {{2n_{g} L_{C} } \mathord{\left/ {\vphantom {{2n_{g} L_{C} } c}} \right. \kern-\nulldelimiterspace} c}\).

Equation (10) can be reduced to analyze the slow feedback by the famous Lang–Kobayashi model by ignoring the multiple reflections and considering the limit of weak OFB with \(\sqrt {{{\eta^{2} } \mathord{\left/ {\vphantom {{\eta^{2} } {(1 - \eta )}}} \right. \kern-\nulldelimiterspace} {(1 - \eta )}}} < < 1\). In this case, the feedback function U becomes

$$U = \sum\limits_{m}^{M} {\frac{{\eta^{m} }}{{\sqrt {1 - k\eta } }}\sqrt {\frac{{S\left( {t - m\tau } \right)}}{S\left( t \right)}} e^{{j\left[ {\theta \left( {t - m\tau } \right) - \theta \left( t \right)} \right]}} e^{{ - j2m\beta_{C} L_{C} }} e^{{ - 2m\alpha_{C} L_{C} }} }$$
(11)

The time-delay rates Eqs. (3)–(5) are solved by means of the 4th order Runge–Kutta method using an integration step as short as Δt = 0.4 ps. The laser dynamics are considered to characterize the VCSEL modulation when the operation under optical feedback is stabilized [24]. Then, over a time of integration T, the power spectrum Pf of the laser power P(t) is calculated using the FFT as [25]

$$P_{f} = \sqrt {\frac{\Delta t}{T}} \left| {FFT\left[ {P\left( t \right)} \right]} \right|$$
(12)

We use the numerical values of the parameters of 850-nm VCSEL as listed in Table 1 [23].

Table 1 Definition and numerical values of the VCSEL parameters [23]
Full size table

4 Results and discussion

4.1 Modulation characteristics of C-VCSEL

The small-signal modulation of the C-VCSEL is characterized in Fig. 2. Figure 2a plots the IM response of the C-VCSEL at three bias currents of Ib = 0.5, 1.5, and 2.5 mA. The IM response is calculated by generating the FFT spectrum of the signal power via Eq. 12, picking up the spectrum amplitude a1 at the corresponding modulation frequency fm and applying the following equation

$${\text{IM } - \text{ response}} = 10\log_{10} \left[ {\frac{{a_{1} \left( {f_{m} } \right)}}{{a_{1} \left( {f_{m} \to 0} \right)}}} \right]$$
(13)
Fig. 2
figure 2

Modulation characteristics of the C-VCSEL (η = 0): a IM response when Ib = 0.5, 1.5 and 2.5 mA, b variation of carrier-photon resonance fCR0 and bandwidth f3dB0 as well as frequency chirp Δν with Ib, and c variations of distortion 2HD with Ib

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The figure shows the typical modulation character of semiconductor laser that the IM response peaks at a frequency comparable to the CPR frequency fCP0 and then drops to lower response values. The modulation bandwidth f3dB0 is the frequency at which the M responses drop to one-half (− 3 dB). As seen in the figure, the response at the peak frequency fCP0 is suppressed with the increase of the bias current Ib, and the drop to lower values beyond this frequency becomes smoother. This effect is known to be due to increase of the damping rate of the relaxation oscillations of the laser as found in [26]. The shown increase in the values of the modulation 3 dB-bandwidth f3dB0 and fCP0 with Ib is investigated in Fig. 2b. The figure indicates that the ranges are f3dB0 = 12 ~ 29 GHz and fCP0 = 5 ~ 16 GHz when the current ranges as Ib = 0.5 ~ 4 mA. Since the regime of CPR is richest with laser non-linearity, we characterize this regime in Fig. 2c, which plots the frequency chirp Δν at the peak frequency fCP0 as well as the associated second-harmonic distortion (2HD). 2HD is calculated from the FFT spectrum of the laser signal and then picking up the spectral lines a1 and a2 at the fundamental frequency and the second-harmonic frequency, respectively. Then,

$$2HD = 10\log_{10} \left( {\frac{{a_{2} }}{{a_{1} }}} \right)$$
(14)

The figure indicates that the frequency chirp of the C-VCSEL is as low as ranging between 3.5 and 5.5 GHz. Also, 2HD is lower than − 20 dB over the relevant current range. These low values of the harmonic distortion indicate high degree of the signal periodicity.

4.2 Modulation characteristics of cascaded TCCs-VCSEL

We examine the modulation characteristics of the cascaded TCCs-VCSEL in two cases of rather weak coupling (η = 0.5) and strong coupling (η = 0.8) of slow-light feedback into the primary VCSEL cavity. Figure 3a and b characterize the IM response of these two cases, respectively, considering 6TCCs-VCSEL with TCC length of LC = 4 µm under different multiple reflections of N = 1, 2, 4 and 5. The bias current is set to be Ib = 2.5 mA. Figure 3a shows that the IM response has almost similar spectral characteristics to that of the C-VCSEL in Fig. 2, displaying the CPR peak. That is, the intensity the feedback light and the induced phase change are not strong enough to change the modulated signal characteristics. The CPR peak frequency decreases little from fCR = 17 GHz when N = 1 to fCR = 15 GHz when N = 5, and the 3 dB-bandwidth decreases also a little from f3dB = 30 to 28 GHz. That is, the multiple reflections have minor influence on spectral characteristics of the cascaded TTCs-VCSEL when the slow-light coupling is weak. These shown IM-response characteristics may correspond to the Lang–Kobayashi model following Eq. (11). Dalir and Koyama reported in [22] that the highest bandwidth predicted by applying the Lang–Kobayashi model to VCSEL with single TCC is ~ 40 GHz.

Fig. 3
figure 3

IM response of cascaded 6TCCs-VCSEL under (a) weak feedback (η = 0.5) and (b) strong feedback (η = 0.8), under different number of multiple reflections N = 1 ~ 5 with LC = 4 µm when Ib = 2.5 mA

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The case of strong coupling illustrated in Fig. 3b shows that the 3 dB-bandwidth increases with and without multiple reflections. When N = 1, the bandwidth is f3dB = 32 GHz = 1.23f3dB0. This bandwidth is much more enhanced to f3dB = 64 ~ 65 GHz ≈ 2.5f3dB by considering more round trips (N > 1) due to the multiple reflections. In the cases of N = 1 and 2, the enhancement of the bandwidth is seen due to extension of the CPR frequency to higher values of fCP = 19 GHz when N = 1 and 45 GHz when N = 2. Since the CPR frequency is proportional to the photon lifetime, this extension of the CPR frequency indicates reduction in the photon lifetime of the 6TCC-VCSEL structure below that of the VCSEL itself [23]. Although this resonance frequency increases also to fCP = 20 GHz for larger numbers of the multiple reflections (N > 2), a PPR effect appears in the form of resonant modulation around frequencies fPP = 39 GHz when N = 3, and 41 GHz when N = 5. In the latter cases of larger numbers of multiple reflections, other response peaks appear in the 3/2th and 2nd-order harmonics of fPPR. That is, the bandwidth enhancement in these cases is attributed to the PPR effect, which is explained in terms of the interaction between the transverse laser mode without slow-light feedback and an external mode induced by strong feedback [20]. This interaction occurs due to the carrier pulsation introduced by the applied modulation signal at the beating frequency of these two modes and introduces a resonance in the IM response at this frequency. The flatness of the frequency gap between the CPR and PPR peaks could be attributed to the out-of-phase relationship between the coupled transverse slow-light and the verical light in the primary cavity of the VCSEL as indicated by Dalir and Koyama [20].

It is important to examine influence of the parameters of the cascaded TCCs on the modulation characteristics of the proposed TCC-VCSEL. In Fig. 4, we investigate influence of the coupling ratio η on the IM-response of the 6TCC-VCSEL. The figure plots the response spectrum when η = 0.3, 0.5, 0.7, 0.75 and 0.8. In this case, we assume LC = 4 µm. The figure shows that the bandwidth is f3dB = 29 GHz under weak coupling of η = 0.3, whereas it decreases by the increase of η to 0.5. When η = 0.7, the PPR effect works to enhance the bandwidth to f3dB = 37 GHz preceded by an enhanced resonant modulation peak around the PPR frequency fPP = 31 GHz. By further increase of η to 0.75, f3dB increase to 42 GHz and the PPR peak is shifted to fPP = 36 GHz. As discussed before, the case of stronger coupling of η = 0.8 is associated with enhancing the 3 dB-dnadwidth to f3dB = 66 GHz. This behavior of variation of the modulation bandwidth was also predicted in theory by Ahmed et al. [14] for the VCSEL with single TCC. It was attributed to the combined effect of the coupling strength of the slow-light feedback and its phase compared with the phase of the lasing field at the oxide aperture [14].

Fig. 4
figure 4

IM response of cascaded 6TCCs-VCSEL under different lengths LC with η = 0.8 when N = 5 and Ib = 2.5 mA

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This behavior can be understood by routing to influence the slow-light feedback strength on the dynamics of the non-modulated version of the device. Figure 5 plots the bifurcation diagram of the non-modulated laser in terms of the coupling ratio η. The diagram is constructed by picking up the peak Speak of the photon number S(t), normalized to its average value \(\overline{S}\), when the feedback induces time variation of S(t) and plotting Speak/\(\overline{S}\) versus η. As shown in the figure, the VCSEL operates under continuous wave (CW) when η ≤ 0.66, self-oscillation when 0.66 < η < 0.86, and chaos at larger values of η. In the regime of CW operation, the VCSEL exhibits damped oscillations before it relaxes to the steady state. The frequency of these damped oscillations is 17.5 GHz when η = 0.3 ns decreases to 16 GHz when η = 0.5. These frequencies correspond to the peak (CPR) frequencies of the IM responses of Fig. 5 when η = 0.3 and 0.5. The corresponding decrease in the bandwidth frequency f3dB is associated with reduction of the CPR frequency, as predicted by Ahmed et al. [18]. In the border with the regime of self-oscillation (η = 0.665), the oscillation has another high-frequency component with 33 GHz. This regime with self-pulsation was reported by Ahmed et al. [29] for edge-emitting laser under strong feedback. In the self-pulsing regime, this high-frequency oscillation becomes undamped and the oscillation frequency increases with the increase of the feedback strength (η). This frequency is ranging between 34 and 46 GHz when η increases from 0.7 to 0.85. This high-frequency oscillation indicates that the 6TTC-VCSEL jumps to one of the external cavity modes whose frequency is much higher than the solitary laser’s CPR frequency. These feedback-induced high-frequency oscillations represent a main cause of the PPR effect shown in the IM-responses of Fig. 5 of the 6TTC-VCSEL.

Fig. 5
figure 5

Variation of the IM response of cascaded 6TCCs-VCSEL with the coupling ratio η using N = 5 when LC = 4 µm, η = 0.8 and Ib = 2.5 mA

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On the other hand, influence of the TCC length LC on the modulation characteristics of the VCSEL is given in Fig. 6. We consider the case of five multiple reflections (N = 5) along with bias current of Ib = 2.5 mA. In this case, LC changes between LC = 4 µm and 7 µm, while the coupling ratio is set to be η = 0.8. The figure indicates that the bandwidth of the cascaded 6TCCs-VCSEL is lowered below that of the C-VCSEL as long as the TCC length is less than LC = 7 µm at which f3dB = 30 GHz. When LC = 4 and 5 µm, the PPR-induced bandwidth enhancement is remarkable revealing f3dB = 66 and 58 GHz, respectively. That is, as long as LC < 7 µm, the shorter is the feedback cavity, the higher will be the modulation bandwidth, which is due to increase in the frequency of the feedback-induced external cavity mode and the corresponding increase in the beating frequency of the PPR.

Fig. 6
figure 6

Bifurcation diagram of the 6TCC-VCSEL output in terms of the coupling ratio η when N = 5, LC = 4 µm and Ib = 2.5 mA

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Influence of the bias current Ib on the investigated IM responses that are characterized with bandwidth enhancement due to both extended CRP and PPR is illustrated in Fig. 7. Figure 7a and b plot the IM responses when N = 2 and 5, respectively, under current values of Ib = 1.0, 1.5 and 4.0 mA. In Fig. 7a, the increase of Ib is shown to increase both the CPR peak frequency and the 3 dB-bandwidth. This is because the increase of the bias current results in an increase in the photon density and a corresponding increase in the CPR frequency [23]. The bandwidth enhancement is remarkable at higher currents of Ib = 2.5 and 4.0 mA and the highest predicted bandwidth is f3dB = 74 GHz = 2.84f3dB0. Figure 7b displays similar changes on the IM response; the PPR peak frequency and 3 dB-bandwidth increase with the increase of Ib, reaching fPPR = 43 GHz and f3dB = 68 GHz when Ib = 4 mA.

Fig. 7
figure 7

Variation of the IM response of cascaded 6TCCs-VCSEL with bias current when (a) N = 2 and (b) N = 5, using η = 0.8, M = 6, LC = 4µm and Ib = 2.5 mA

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4.3 Modulated signal characteristics

The modulated signals of interest in applications are those with frequencies extending up to the 3 dB-bandwidth of the IM-response. That is, the proposed cascaded TCCs-VCSEL is predicted to generate ultra-high-frequency radio waves reaching 70 GHz. As we characterized the C-VCSEL signal at the CPR peak of the IM-response in Fig. 2, we present in Fig. 8 numerical characterization of the modulated signal at the extended CPR frequency fCP (N = 2 of Fig. 7a) and the PPR peak frequency fPP (N = 5 of Fig. 7b). Figure 8a and b plot examples of the variations of the signal power P(t) and frequency shift Δf(t), respectively, at the characteristic frequencies fCP = 40 GHz and fPP = 40 GHz considering current Ib = 2.5 mA. The corresponding signal of the C-VCSEL at the CPR frequency fCR0 = 16 GHz is plotted for comparison. Both figures show the periodic oscillations of the cascaded TCCs-VCSEL at these high frequencies. The figure displays much larger amplitude of the case of resonant modulation around fPP compared with those of the signal around the extended CPR frequency fCP and the C-VCSEL signal. However, the signal distortion is lower in case of response with the extended CPR frequency. Numerically, the distortions are 2HD = − 32 dB and − 30.5 dB when N = 2 and 5, respectively. This distortion is lower than that of the C-VCSEL that gives 2HD = − 30 dB around the CPR frequency fCP0. On the other hand, Fig. 8b indicates blue-shift of the frequency variation Δf(t) of the three cases. Although the signal at the extended CPR frequency corresponds to largest frequency shift of 388 GHz, the frequency amplitude (chirp) is only Δν = 5.5 GHz, which is lower than that of the C-VCSEL of 5.6 GHz. Although the average frequency chirp of resonant modulation is lower (120 GHz), the associated frequency chirp is highest of Δν = 118 GHz. This difference in the frequency shift Δf between the two cases of N = 2 and N = 5 is attributed to the difference in the phase φ of the feedback function U as given in Eq. 4.

Fig. 8
figure 8

Time domain characteristics of high-frequency signals of the cascaded 6-TCCs-VCSEL (a) signal power P(t) and (b) signal frequency Δf(t), when N = 2 with modulation frequency fm = 40 GHz and when N = 5 with fm = 40 GHz

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In Fig. 9a–c, we investigate the signal characteristics over the ranges of the extended CPR and PPR frequencies investigated with variation of the bias current Ib. Figure 9a and b plot the characteristics frequencies fCP and f3dB for the case of extended CPR frequencies (N = 2) and fPP and f3dB for the case of PPR frequencies (N = 5), respectively. The right-hand axis plots the corresponding frequency chirp Δν. Figure 9a indicates that the extended CPR frequency ranges between fCP = 43 and 49 GHz, while Fig. 9b indicates that the PPR frequency ranges between fPP = 40 and 43 GHz. These characteristic frequencies correspond to enhanced 3 dB-bandwidth of f3dB = 56 – 64 GHz under the extended CPR frequency and f3dB = 65 – 74 GHz under the PPR effect. The figures indicate also that the frequency chirp Δν in case of the extended CPR frequency is much lower than that under the PPR effect. Figure 9c plots variation of the associated 2HD with Ib, showing an increase of 2HD with the increases of the peak frequencies fCP and fPP. The range of 2HD in case of the extended CPR frequency is almost similar to that of the C-VCSEL while it is much larger in case of the PPR effect. However, the predicted range of signal distortion is less than − 20 dB. These results reflect the high degree of periodicity of the generated high-frequency signals.

Fig. 9
figure 9

Modulation characteristics of the cascaded 6TCCs-VCSEL: (a) CPR frequency fCP and PPR frequency fPP as well as the frequency chirp Δν, and (b) 2HD, as functions of current Ib

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5 Conclusion

In summary, we demonstrated simulation results on a novel VCSEL with a cascaded TCCs structure toward achieving modulation bandwidth of several tens GHz and generating ultra-high-speed oscillations. We developed a travelling-wave model to describe the slow-light propagation down TCCs and formulate the induced effective reflectivity of the TCCs structure as well as the threshold gain. The device was then simulated by numerical integration of the developed time-delay rate equations. The proposed device was predicted to exhibit modulation bandwidth approaching ~ 300% of that of the C-VCSEL by proper design of the cascaded TCCs. When counting a few numbers of multiple reflections in the TCC, the bandwidth is enhanced up to 63 GHz due to extension of the CPR towards frequencies as high as 45 GHz. When the number of multiple reflections is large, the PPR effect enhances the bandwidth, which could reach 74 GHz. In principle, the 6TTC-VCSEL could emit appreciated signals with frequencies reaching the bandwidth frequency, which is as high as 63 GHz under extended CPR and 70 GHz under PPR. We presented results of strongest oscillations with frequency range of 40 ~ 49 GHz. These peaks correspond to either the CPR or PPR frequencies. For the 6TCC-VCSEL with structure parameters listed in Table 1, the ranges of parameters required to generate such high-frequency pulsations are cavity length LC < 7 mm, coupling ratio η = 0.75 ~ 0.85 and biasing current exceeding 1.5 mA.