VCSEL with multi-transverse cavities with bandwidth beyond 100 GHz

1 Introduction

Light emitters utilizing directly modulated vertical-cavity surface-emitting lasers (VCSELs) are appealing for cost-effective photonic applications due to the unique properties of VCSELs, such as high efficiency, low power consumption, better temperature stability, and direct fabrication of dense arrays [1, 2]. However, the carrier-photon resonance (CPR), thermal effects, and parasitic resistance/capacitance limit the 3-dB bandwidth of VCSELs to 30 GHz [3, 4]. On a contrary, the VCSEL has been challenged to boost its bandwidth in order to meet the current demands of the Internet, supercomputers, and data centers, all of which require a bandwidth greater than 100 Gb/s [5]. Several ways to increase the transmission bitrate of semiconductor lasers have been proposed in the last decade. External optical feedback has been identified as a technique of improving the modulation bandwidth (MBW) of cost-effective directly modulated semiconductor lasers [6, 7]. Optical-feedback (OFB) has been shown to produce a variety of schemes in semiconductor laser dynamics, ranging from stable continuous wave (CW) to the most unstable chaotic dynamics, and passing through period-1 and period-doubling oscillations, quasi periodicity, and intermittency. The combined impacts of both the coupling strength of OFB and its phase relative to the phase of the lasing field at the laser facet determine the sort of path to chaos [8, 9]. The laser is driven to discrete chaos cycles separated by CW regimes when treated to strong OFB from a short-external cavity [10, 11]. The number of these chaotic cycles diminishes when the external cavity is shortened [10]. According to Alghamdi et al. [12], under the weak OFB regime, CPR frequency reduces as OFB increases, and is consequently linked with diminished MBW. The relaxation oscillations become undamped when OFB induces hopf-bifurcation, and the laser jumps to an external cavity mode with a frequency larger than the solitary laser’s CPR frequency [10, 11].

With intensifying feedback, the CPR frequency goes up, as does the MBW, which could be owing to an increase in photon lifetime [13]. When the OFB is adjusted further and the feedback allows CW operation again, the high-frequency component resurfaces. The interaction between the laser mode without OFB and an external mode produced by high OFB [14] is used to explain this photon–photon-resonance (PPR). This interaction occurs because the applied modulation signal produces carrier pulsation at the beating frequency of these two modes, causing a resonance peak in the intensity modulation (IM) response in addition to the CPR peak [15, 16]. When the linked feedback light is out of phase, the frequency gap between the CPR and PPR peaks becomes flat, and the 3 dB-bandwidth is increased, according to Dalir et al. [14].

It has been discovered that using external OFB can enable VCSELs to own a greater MBW [17, 18]. Dalir et al. [19], [20], [21], [22] demonstrated that enhancing MBW via adding a single transverse coupled cavity (TCC) to a primary VCSEL cavity. The stated design principle was to utilize the PPR effect to manipulate the slow-light delay time in the TCC and the generated slow-light feedback. The significant PPR effect creates peaky patterns in the IM response, which are referred to as “resonance modulation response” by Ahmed et al. [23]. The coupling strength variation affects the MBW enhancement effect [23]. Dr Dalir on the other hand observed that a shorter TCC is beneficial for boosting the CPR effect and bandwidth, but that it requires strong coupling to accomplish the same degree of MBW enrichment as a long TCC [23].

Dalir et al. recently demonstrated a new VCSEL architecture consisting of a hexagonal transverse-coupled-cavity adiabatically coupled through a central VCSEL cavity [24], as illustrated in Figure 1. The authors demonstrated a prototype with a 3-dB roll-off MBW of 45 GHz, which is five times larger than a conventional VCSEL fabricated on the same epiwafer structure [24]. The design offered more slow-light coupling into the VCSEL cavity than a VCSEL with a single TCC, which worked to enhance the bandwidth and increase the IM response beyond the CPR frequency [25]. The MBW of this MTCC VCSEL was expected to increase up to 100 GHz based on modeling [24]. These findings encourage the authors to further explore the modulation performance of the MTTC-VCSEL and optimize the device structure in hopes of improving MBW.

Figure 1:

Schematic structure of our hexagonal transverse-coupled cavity vertical-cavity surface-emitting lasers (VCSEL) (a) top view and (b) cross-sectional view [24].

In this article, we use the theoretical model in [24] to look at VCSELs in combination with various multi-lateral and short TCC methods for increasing MBW above 100 GHz. The TCCs are planned to encircle the VCSEL, providing direct slow-light input into the main cavity from each TCC. As a result, even if each TCC’s direct OFB is weak or intermediate, it adds up to a couple more slow light into the VCSEL cavity and induces OFB strong enough to accomplish additional MBW augmentation.

We compare the IM responses of the VCSEL design utilizing one, two, four, and six TCCs based on numerical integration of the time-delay rate equations of the studied MTCCs VCSEL. We show that coupling the VCSEL with multiple TCCs has benefits over coupling it with single or double TCCs in terms of not only producing higher MBW enhancement but also utilizing considerably lower optical coupling into the VCSEL cavity. Thanks to the MTCC structure and PPR effect, we obtain data on ultra-high bandwidth surpassing 170 GHz employing VCSEL integrated with four and six TCCs, which is, to the best of our knowledge, the highest predicted value. In addition, we analyze the 4TCCs-noise VCSEL’s characteristics.

The design and modeling of the MTCCs VCSEL are discussed in the next section, and the numerical calculation methods are provided in Section 3. Section 4 discusses the result of the IM response with MBW augmentation and noise characteristics. Finally, in Section 5, we then provide the presented work’s closing conclusions.

2 Model of slow-light feedback in VCSEL due multi-surrounding TCCs

Figure 2(a) depicts the TCC-VCSEL structure, whereas Figure 2(b) shows light propagation in both the VCSEL cavity and the TCC [23]. Through an oxide aperture in which light is laterally restricted, the VCSEL is laterally linked with the TCC. Light is limited from the top and bottom of the VCSEL cavities, and it travels in a zigzag pattern with an angle close to 90° at the cut-off condition of lateral light propagation [23]. Slow light is partially transferred to the TCC through the oxide aperture, resulting in a leaky traveling wave and light slowing in the TCC.

Figure 2:

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

The slow light then propagates with the group velocity of v _g = c/n _g , where n _g  = fn is the group index, n is the average material refractive index, and f is the slow factor of light. Thus slow light is totally reflected back at the far end of the lateral waveguide and travels round trips in the cavity. After each round trip, the slow light is coupled to the primary VCSEL cavity through the oxide aperture. In each round trip in the TCC, the slow light suffers the loss of exp ( − 2 α C L C ) and phase delay of exp ( − 2 j β C L C ) , where α _C  = fα _m and β C = 2 π n / ( λ f ) are the lateral optical loss and propagation constant with α _m being the material loss and λ the emission wavelength.

Figure 3 depicts the proposed design and model of the MTTC-VCSEL, which shows a top view of the VCSEL surrounded in the lateral direction by multiple lossy cavities (arms) connected by oxide apertures. In the direction of each TCC, this coupled waveguide structure introduces a leaky traveling wave. In the laterally coupled waveguides, light travels perpendicularly and is slowing down. Within each TCC, the slow light propagates between the end of the TCC (arm) to the VCSEL cavity (at the device center) for a number of round trips with a group velocity of v _g. The slow light is totally reflected back at the corresponding aperture and is coupled into the VCSEL cavity with a coupling ratio η. The period of the round trip between the VCSEL cavity and the far end of the mth TCC is τ m = 2 n g m L C m / c . In this case, the threshold gain level of the VCSEL cavity G _thc is modified to the following form:

which represents a generalized form of the gain derived in references [23, 26]. The OFB function U _m (t − τ _m ) is a time-delay function that describes the slow-light feedback from the mth TCC.

where the summation is over the multiple round trips in the TCC. In this case, θ(t − pτm) − θ(t) represents the deviation in the optical phase due to chirping in the mth cavity.

Figure 3:

Schematic top-view of VCSEL with each of the multiple surrounding cavities providing direct slow-light feedback.

The rate equations of the MTCC-VCSEL are given for the injected electron number N(t), photon number S(t) contained in the lasing mode, and the optical phase θ ( t ) = arg [ E ( t ) ] as:

where a is the differential gain of the active region whose volume is V, N _T is the electron number at transparency, and ε is the gain suppression coefficient. Γ is the confinement factor, τ _p = 1/G_thD is the photon lifetime, η _i is the injection efficiency, τ _s is the electron lifetime due to the spontaneous emission, R _sp is the spontaneous emission rate, and N _th is the electron number at the threshold. In Eq. (3), the injection current is assumed to have sinusoidal current modulation with bias component I _b, modulation component I _m, and modulation frequency f _m.

The Langevin noise sources in Eqs. (3)–(5) are given in respective by [27]

where x _s , x _N , and x _θ are random numbers having normal distributions with zero mean and variance of unity. The frequency content of intensity fluctuations is measured in terms of relative intensity noise (RIN), which is calculated from the fluctuations δ S ( t ) = S ( t ) − S b in S(t), where S _b is the bias value of S(t). Over a finite time T, RIN is given as [28]

where f is the Fourier frequency.

3 Numerical calculations

In the present calculations and for simplicity, we assume that the TCCs are identical, each TCC has length L _C , group index n _g, propagation constant β _C , optical loss α _C , and hence round trip τ. Also, the lateral slow-light is coupled to the primary VCSEL cavity with an equal coupling ratio η. Therefore, the threshold gain in Eq. (1) is reduced to

with the feedback function U(t − τ) id then given by

We integrate rate equation (3)–(5) by means of the fourth-order Runge–Kutta algorithm using a time step as short as Δt = 0.2 ps. The calculations of the time-delayed values of the photon number and phase are achieved as follows. In the time interval 0 ≤ t ≤ τ , the integration is done for VCSEL without feedback. The calculated values of S and θ are then stored for use as time-delayed values S ( t − τ ) and θ ( t − τ ) for integration of the rate equations over the interval t = τ → 2 τ including the OFB terms. Then the calculated values S ( t − τ ) , S ( t − 2 τ ) , θ ( t − τ ) and θ ( t − 2 τ ) are used as time-delayed values for integration over the interval t = 2 τ → 3 τ . This process continues considering further roundtrip and the corresponding terms of S ( t − p τ ) and θ ( t − p τ ) as time-delayed values until the laser output is unchanged. This accumulation of slow-light feedback due to the round trips in the MTCCs guarantees larger feedback required for MBW enhancement. This effect is supported by the fact that the present model is a generalization of the famous Lang–Kobayashi model [26], which is obtained by reducing the model to the limit of weak feedback, η 2 / ( 1 − η ) ≪ 1 , and counting one round trip [23].

The data samples for characterization of laser dynamics and noise are collected after the laser operation is stabilized. We apply the numerical values of the VCSEL parameters given in Table 1 [3]. The slow factor and material absorption loss are set to be f = 40 and α _m = 10 cm⁻¹, respectively, and the bias current is I _b = 2 mA. For simulation of the IM response of the MTCC-VCSEL, we apply the fast Fourier transform (FFT) to the modulated laser signal as,

where a ₁(f _m) is the fundamental peak of the FFT spectrum of the laser intensity at the modulation frequency f _m. In this calculation, we drop the noise sources in rate equations (3)–(5).

4 Results and discussions

4.1 Modulation response of VCSEL with single TCC

Examples of the IM response spectra with improved MBW (f _3dB) of a VCSEL integrated with a single TCC are plotted in Figure 4. These spectra correspond to TCC length of L _C  = 5 μm and coupling ratio of η = 0.75, 0.8, and 0.9. The figure indicates an increase of MBW beyond that of the conventional VCSEL (C-VCSEL), f _3dB0 = 21.5 GHz, to f _3dB = 27.5, 36, and 40 GHz when η = 0.75, 0.8, and 0.9, respectively. This range of η corresponds to very strong transverse optical feedback. The increase of MBW can be attributed to the strong anti-phase coupling between the transverse coupled radiation and the vertically lasing radiation in the VCSEL cavity [20].

Figure 4:

IM response with bandwidth improvement of the design of TCC VCSEL with L _C  = 5 μm when η = 0.75, 0.8, and 0.9. The IM response of the C-VCSEL is plotted for comparison with the dashed-dotted line.

The predicted values of MBW f _3dB are plotted as a function of the coupling ratio η in Figure 5 for TCC lengths of L _C  = 5 and 6 μm. The figure indicates the increase of f _3dB in the regime of very strong feedback with coupling ratios of η > 0.71 when L _C  = 5 μm and η > 0.78 when L _C  = 6 μm. The values of f _3dB are smaller when L _C  = 6 μm than those when L _C  = 5 μm since the strength of OFB decreases with the increase of the length of the feedback waveguide.

Figure 5:

Variation of the bandwidth f _3dB of the TTC VCSEL with the coupling ratio η when L _C  = 5 and 6 μm.

4.2 Modulation response of MTCC-VCSEL

We show how increasing the number of lateral TCCs not only reduces the range of coupling that corresponds to an increase in MBW but also increases the bandwidth to much higher and more interesting levels. Figure 6(a)–(c) plot examples of the IM responses with MBW enhancement when the number of lateral TCCs increases to M = 2, 4, and 6, respectively. All cases correspond to the TCC length of L _C  = 5 μm. Figure 6(a) of M = 2 shows that the MBW values of f _3dB = 40 and 45 GHz are obtained when η = 0.4 and 0.5, respectively, which are almost one half the values in Figure 4 that results in the same MBW in the TCC-VCSEL. The bandwidth increases further to 60 and 61 GHz when the coupling ratio increases to η = 0.7 and 0.9, respectively. It is worth noting that a PPR effect is induced with the increase of η; a PPR peak is seen around the modulation frequency f _PP = 180 GHz when η = 0.9. This PPR is a result of modulation at frequencies close to the beating frequencies of external cavity oscillating modes [14, 17, 18, 29, 30]. When the VCSEL is surrounded by 4TCCs (M = 4), Figure 6(b) shows that the low values of the coupling ratio of η = 0.2 and 0.3 accumulate more slow-light feedback in the VCSEL cavity so that the MBW is increased to f _3dB = 48 and 64, respectively. The PPR effect is remarkable when η = 0.5 which is seen as resonant modulation with a PPR peak of 5.8 dB around a very high frequency of f _PP = 176 GHz. In this case, MBW is enhanced to f _3dB = 80 GHz. The further increase of the light coupling into the VCSEL cavity to η = 0.6 compensates the loss in the drop of the IM response under the −3 dB level and results in an ultra-high MBW of f _3dB = 170 GHz and a high PPR peak around f _PP = 145 GHz. These values, to the best of our knowledge, are the highest predicted values for the semiconductor laser MBW. Figure 6(c) indicates more improvement of the modulation performance with higher values of MBW at lower values of the coupling ratio η when the number of TCCs increases to M = 6. At the low coupling of η = 0.15, MBW of the 6TTC-VCSEL is f _3dB = 56 GHz. The PPR effect is initiated at lower values of the coupling ratio; when η = 0.3 a PPR peak is seen around frequency f _PP = 175 GHz. The associated MBW is f _3dB = 76 GHz. When η increases to 0.45, the PPR is too enhanced to reveal resonant modulation of 6.2 dB around frequency f _PP = 165 GHz and the associated bandwidth of f _3dB = 85 GHz. In this case of 6TCC-VCSEL, the ultra-high MBW of f _3dB = 165 GHz is obtained at a lower coupling of η = 0.5, which is associated also with a higher PPR peak around f _PP = 147 GHz.

Figure 6:

IM response of the design of (a) 2TCC-VCSEL, (b) 4TCC-VCSEL, and (c) 6TCC-VCSEL with L _C  = 5 μm at different values of η that results in MBW enhancement.

In Figure 7(a), we plot variation of MBW, f _3dB, with the coupling ratio η for the three cases of MTCC-VCSEL (M = 2, 4, and 6) when L _C  = 5 μm. The figure shows that the MBW improvement when M = 2 is initiated when η > 0.35 and f _3dB reaches 62 GHz when η > 0.75. In the case of the 4TCC VCSEL, MBW is much more enhanced when η > 0.15. When η > 0.4 the PPR effect is remarkable and MBW reaches f _3dB = 84 GHz when η = 0.64. In the range of 0.65 ≤ η ≤ 0.77, the strong light feedback works to recover the gap between the CPR and PPR peaks above the 3 dB level, and MBW is much more enhanced to values between f _3dB = 88 and f _3dB = 178 GHz. The BW improvement is achieved at weaker optical feedback coupling of η > 0.10 by the 6TTC-VCSEL. When η > 0.30, the boosted PPR effect is associated with enhanced MBW to values reaching f _3dB = 84 GHz. The further increase of η between 0.45 and 0.65 raises the IM response above the 3 dB level, which is smaller than the range achieved by the 4TTC-VCSEL. In this case, the predicted values of MBW are in the ultra-high frequency range of f _3dB = 88–180 GHz. The corresponding variations of f _3dB with η when L _C  = 6 μm are plotted in Figure 7(b). The figure shows that the behaviors of f _3dB with the variation of η for the three cases of M = 2, 4, and 6 are similar in general to those when L _C  = 5 μm. However, the increase of MBW of the MTCC-VCSEL above that of the C-VCSEL occurs at smaller values of the coupling ratio η, and the predicted bandwidth is lower. The highest MBW ranges between f _3dB = 80 and 155 GHz. It is worth reporting that when the TCC is shorter than 5 μm the laser operates under CW, and the slow-light feedback is coupled in such a way to make the frequency gap between the CPR and PPR peak lower than the 3 dB level, which then does not support MBW enhancement.

Figure 7:

Variation of the bandwidth f _3dB of the TTC VCSEL with the coupling ratio η when (a) L _C  = 5 μm and (b) L _C  = 6 μm for 2TCC-VCSEL, 4TCC-VCSEL, and 6TCC-VCSEL.

4.3 Noise properties of MTTC VCSEL

In this subsection, we investigate the noise properties of the MTCC-VCSEL under current modulation. We focus on the regime of ultra-high bandwidth, f _3dB > 100 GHz, shown in Figure 7 for cases of VCSEL integrated with four TCCs (M = 4) and six TCCs (M = 6). The noise is evaluated in terms of the frequency spectrum of RIN when the MTCC-VCSEL is modulated with modulation frequency f _m equal to one of the PPR frequencies in the regimes of ultra-high bandwidth. As explored in Figure 7, these regimes are 0.68 ≤ η ≤ 0.78 and 0.66 ≤ η ≤ 0.76 for 4TCC-VCSEL with L _C  = 5 and 6 μm, respectively, while they are 0.475 ≤ η ≤ 0.645 and 0.0.475 ≤ η ≤ 0.52 for 4TCC-VCSEL, respectively. Figure 8(a)–(d) plot four examples of the simulated RIN spectra of MTCC-VCSEL with L _C  = 5 μm when (M = 4, η = 0.7, f _m  = 145 GHz) and when (M = 6, η = 0.64, f _m  = 149 GHz), and of MTCC-VCSEL with L _C  = 6 μm when (M = 4, η = 0.76, f _m  = 129 GHz) and when (M = 6, η = 0.52, f _m  = 127 GHz), respectively. The figures show that the RIN spectra exhibit pronounced peaks around the modulation frequency f _m due to the high degree of periodicity. At frequencies lower than the regime of the resonance peak, the RIN level increases in general with the decrease of frequency and then exhibits flat (white) noise in the regime of low frequencies f _m  < 1 GHz. The figures indicate also that the level of the low-frequency RIN (LF-RIN) of the 4TTC-VCSEL is almost one-order of magnitude lower than that of the 6TCC-VCSEL. This difference may indicate that the modulated signal of the 6TCC-VCSEL at this ultra-high frequency is a little more irregular than that of the 4TCC-VCSEL.

Figure 8:

RIN spectra of MTCC-VCSEL with L _C  = 5 μm when (a) M = 4, η = 0.7, f _m  = 145 GH, (b) M = 6, η = 0.64, f _m  = 149 GHz, and of MTCC VCSEL with L _C  = 6 μm when (c) M = 4, η = 0.76, f _m  = 129 GHz, and (d) M = 6, η = 0.52, f _m  = 127 GHz.

In Figure 9(a) and (b), we compare variations of the LF-RIN level of the 4TCC-VCSEL and 6TCC-VCSEL, respectively, when the TCC length is L _C  = 5 and 6 μm. The figure indicates an increase of LF-RIN with the increase of the coupling ratio η in general. The noise levels when L _C  = 5 μm are comparable to those of the nonmodulated TCC VCSEL as investigated by Ibrahim et al. [27]. The figure shows also that the noise levels when L _C  = 6 μm are higher than those when L _C  = 5 μm. Finally, the noise levels in the 6TCC-VCSEL are almost one-order of magnitude higher than those in the 4TCC-VCSEL over the same range of slow-light feedback. However, the investigated ranges of LF-RIN of the MTTC-VCSEL are still much lower than the level >10⁻⁸ 1/Hz that characterizes the unstable dynamics of the VCSEL [27].

Figure 9:

Variation of the LF-RIN level of (a) the 4TCC-VCSEL and (b) 6TCC-VCSELwhen the TCC length is L _C  = 5 and 6 μm.

Before concluding, we will briefly discuss this paper’s results in the wider context of advancements in nanophotonic emitters and lasers [31]. The interplay between the EM field, gain feedback, and laser performance is indeed forming an intricate system composed of the optical mode (or few-mode) cavities [32], [33], [34] and laser physics [35], [36], [37], [38], [39], [40], [41], [42], both at the nanoscale and (sub) diffraction-limited optical modes. Our work demonstrating enhanced laser modulating speed performance can also be seen as an extension of the ongoing discussion in the field of miniaturized laser devices. Here the debate around whether the Purcell factor, which captures the light–matter-interaction strength of such a laser cavity being proportional to the cold-cavities’ quality factor (Q) divided by the cavities’ mode volume, has (or has not) an influence on both the PPR and gain relaxation frequencies, i.e. speed of the laser [35], is an open debate to date. The Purcell factor is especially high in light emitters and lasers featuring a sub diffraction-limited optical mode, primarily due to the nonlinear scaling of volume and the introduced loss due to the cavities’ inability to provide OFB. This impacts the PPR in such small-volume cavities at (or below) the diffraction-limit of light [35], [36], [37], [38] because the laser design is also capable of increasing the temporal relaxations oscillations of the laser cavity. This in turn expands the “speed” of the laser under direct modulation. An example of this is the transverse-coupled cavity laser design that provides coherent feedback from a plurality of cavities and thus enhances the light emission from a central lasing cavity [24]. Looking ahead, future research should also investigate the effects of PPR in cavities with high longitudinal modes, for example in fiber optic-based laser systems for Brillouin amplification [43]. As it stands, the coherent feedback design of opportunely engineering a multiple TCC-based laser, as shown here, offers a new degree of freedom in laser and VCSEL design explorations. Given the predicted performance, these emerging VCSELs are poised to show a significant impact on the next-generation 5G and 6G network systems, data centers, and high-end sensors systems.

5 Conclusions

Finally, for the enhancement of MBW, potential designs and modeling of VCSEL encircled in the lateral direction by numerous TCCs were given. We demonstrated that the linked slow-light from each of the surrounding TCCs is collected in such a way that strong OFB is produced in the VCSEL cavity, outperforming VCSELs coupled with a single TCC or two TCCs. In VCSELs with four and six transverse cavities, this high feedback increases the PPR and MBW to about 150 GHz. The RIN spectrum is prominent around the modulation frequency, while the low-frequency portion is flat, with levels equivalent to nonmodulated TCC VCSELs.