1. Introduction

In addition to their low cost, VCSELs have the advantages of low power consumption, small footprint, wafer-scale testing, low-cost packaging, and ease of fabrication into arrays [1,2]. Therefore, VCSELs are attractive light sources for use in cost-effective radio over fiber (RoF) networks. Broadband RoF links require laser diodes with bandwidth in the mm-waveband to enable dense information transmission. The modulation bandwidth of the VCSEL, however, is restricted by the limited intrinsic carrier-photon resonance (CPR) of the VCSEL which is characterized by a relaxation frequency below ~20 GHz [3–7]. Various schemes have been demonstrated to increase the bandwidth frequency of VCSELs, including injection-locking [8,9], coupled cavity [10,11], and modulator-integration [12].

Dalir and Koyama proposed a lateral optical feedback (OFB) scheme for increasing the bandwidth of VCSELs and predicted 60% improvement in the small signal modulation bandwidth [13,14]. They experimentally demonstrated the bandwidth of a TCC-VCSEL beyond 29 GHz [15,16]. Also, the authors demonstrated the resonance modulation centered on a frequency of 23 to 35 GHz with large enhancement of over 30 dB in the modulation amplitude [15–18]. This resonance modulation was proved to be due to photon-photon resonance (PPR) between transverse oscillating modes [18]. An important question arises: How fast can the TCC-VCSEL be modulated?

The analysis of the modulation bandwidth enhancement due to lateral slow-light feedback in [13,14] was based on a small-signal approach applying the Lang-Kobayashi model [19]. Because the Lang-Kobayashi model is applicable to weak and moderate OFB, the predicated modulation bandwidth was limited to ~40 GHz. In addition, detailed information on the slow-light propagation in the feedback cavity was overseen, such as the cavity length and loss and the slowing factor. Boosting the bandwidth of the TCC-VCSEL to further mm-frequencies require generalization of the Lang-Kobayashi model to treat the regime of strong slow-light feedback and taking into account multiple round trips in the lateral active cavity.

In this paper, we present modeling on the modulation bandwidth enhancement to the mm-wave band of TCC-VCELs. We take into account the multiple round trips in the feedback cavity, and the optical loss and phase delay in each round trip, which enable us to discuss a strong coupling regime. We examine the influence the TCC length, feedback ratio and modulation index on the mm-wave modulation of the VCSEL. We show that the modulation bandwidth is boosted to frequencies reaching 70 GHz and resonance modulation is induced around frequencies as high as 90 GHz.

2. Theoretical and numerical modeling

The proposed structure of the TCC-VCSEL is schematically illustrated in Fig. 1(a), with Fig. 1(b) depicting an image of a developed device [17]. The VCSEL is laterally coupled with an external cavity through an oxide aperture, which forms lateral optical confinement and leads to a leaky traveling wave in the lateral direction. In the VCSEL cavity, the light is confined from top and bottom, so light travels zigzag with an angle Φ in the lateral direction. Near the cut-off condition of light propagation in the lateral direction, Φ will be close to 90° [14], therefore light travels perpendicular in a lateral movement which leads to slowing light in the lateral coupled waveguide. This slow light is totally reflected back at the far end of the lateral waveguide of length L_C and travels round trips in the cavity. The period of each round trip is $\tau =2n_g{L}_C/c$, where n_g = fn is the group index with n being the average material refractive index and f is the slow factor of light whose value ranges between 30 and 50 [14]. In each round trip in the coupled cavity, the slow light suffers loss of $\exp \left(-2{\alpha }_C{L}_C\right)$ and phase delay of $\exp \left(-2j{\beta }_C{L}_C\right)$, where α_C = fα_m and ${\beta }_C=2\pi n/(\lambda f)$ are the lateral optical loss and propagation constant with α_m being the material loss and λ the emission wavelength. After each round trip, the laser light is injected into the VCSEL cavity with a coupling ratio η.

 

Fig. 1 Scheme of (a) slow-light feedback in TCC-VCSEL, and (b) structure of a fabricated TCC-VCSEL.

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We treated the present case of lateral OFB by adopting our time-delay rate equation model of edge-emitting laser diodes under strong OFB in [20] to the present structure. OFB is treated as time delay of the electric field E(t) of the slow light in the TTC due round trips, each of period τ in the TCC. The dynamic behavior and modulation characteristics of the TCC-VCSEL are then described by the following rate equations of the injected electron number N(t), photon number S(t) contained in the lasing mode and the optical phase θ(t),

The intensity modulation is included by exciting the laser with a sinusoidal component with amplitude I_m and frequency f_m in addition to a bias component I_b.

The rate equations are solved by the 4th order Runge-Kutta method using an integration step as short as 0.2ps. We considered 6 round trips (p = 1→6 in Eq. (7)) beyond which there are no significant changes in the modulation characteristics. 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 = 2mA. The numerical values of the laser parameters listed in Table 1 are used.

Table 1. Definition and numerical values of the VCSEL parameters^a

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3. Results and discussion

Figure 2 plots several examples of the IM response of the TCC-VCSEL with improved modulation characteristics over those of the VCSEL without feedback, η = 0, when the length of the TCC is L_C = 20 μm and the modulation index is m = I_m / I_b = 0.01. The IM response of the VCSEL without feedback is plotted for comparison, which exhibits a peak at the frequency of f_m = 12 GHz due to the CPR and a modulation bandwidth frequency of f_3dB0 = 23.5 GHz. These values are given for an ideal small oxide aperture (8μm²) VCSEL at a low bias of 2mA while larger oxide aperture VCSELs have been practically used in order to realize a modulation bandwidth of over 20 GHz [6]. We omitted the thermal effect and parasitics in the present model. But the modulation response is dominated by the injection current density, differential gain and photon lifetime, thus the result could be valid by increasing the oxide aperture area and increasing the bias current. The IM responses given in Fig. 2(a) are characterized by improved modulation bandwidth frequency f_3dB over f_3dB0. When the coupling ratio is η = 0.5, f_3dB increases to 33.5 GHz and the CPR peak increases to ~5 dB and shifts to the modulation frequency of f_m = 16 GHz. When OFB becomes stronger with η = 0.7, f_3dB increases further to 42 GHz, i.e., f_3dB increases by an amount of 79% beyond that of the VCSEL without feedback f_3dB0, which is higher than the percentage of 60% predicted by Dalir and Koyama [14]. On the other hand, the IM response does not drops below the –3dB level and reveals another peak with a value of 1.4 dB at the mm-wave frequency of f_m = 39 GHz. The figure shows also an interesting IM response under very strong OFB of η = 0.96, where the mm-wave peak is more enhanced than the CPR peak and occurs at a higher mm-wave frequency of 49.6 GHz. This IM response can be considered as a type of extended CPR resonance to the mm-wave region [22].

 

Fig. 2 IM responses when L_C = 20 μm under different values of η: (a) with CPR, and (b) with PPR. The IM response of the solitary VCSEL (η = 0) is also plotted for comparison.

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In Fig. 2(b), the IM response drops below the –3dB level at bandwidth frequencies of f_3dB = 24 and 18 GHz when η is 0.76 and 0.78, respectively. However, the TCC-VCSEL is characterized by resonance modulation where the IM responses are enhanced over frequency passbands of f_m = 33 ~43 GHz and f_m = 36.5 ~43.5 GHz centered on the frequencies of 40 and 41 GHz when η = 0.76 and 0.78, respectively. The increase in the feedback is associated with an increase in the peak value of the resonance modulation response from 4.8 dB to 12 dB. Resonance modulation exceeding 29 GHz and approaching 40 GHz was demonstrated in experiments of the fabricated TCC-VCSEL by our group [15–17], and the shown shift of the resonance modulation was also observed. This mm-wave resonance modulation is attributed to a PPR effect [16–18,23–26]. Although coupling among the cavity modes are not treated in the present time-delay model, the resonance in the TCC appears by counting the multiple roundtrips, which makes the PPR effect. The figure shows also that when OFB becomes very strong with η = 0.98, the PPR resonance occurs at higher mm-wave band of f_m = 38 ~54.5 GHz centered on the frequency 50 GHz. The peak value of this PPR is 18 dB higher than the corresponding response of the VCSEL without feedback.

In Fig. 3(a) we examine the stability of the induced resonance modulation in the regime of large-signal modulation. The IM responses corresponds to η = 0.78 at different values of the modulation index m. The figure indicates that the IM responses have almost similar spectral characteristics including the response enhancement over the mm-frequency band of f_m = 32.5 – 39 GHz. The peak frequency of this resonance modulation little shifts to a lower frequency of 40 GHz when m = 0.6. The noticeable influence of increasing m is to suppress the response enhancement within these enhanced bands, which is manifestation of harmonic distortion in the modulated signal [20]. In Fig. 3(b), we clarify this relationship by plotting the 2nd and 3rd-order harmonic distortions, 2HD and 3HD, respectively, versus m when the TCC-VCSEL is modulated at the peak frequency f_m = 41 GHz. The figure indicates an increase in the harmonic distortions with the increase in m, with 2HD being larger than 3HD. The increase in both 2HD and 3HD is noticeable up to m < 0.15 and then becomes slight.

 

Fig. 3 (a) IM responses as a function of m, and (b) variation of 2HD and 3HD with m, when η = 0.78.

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We examine the modulation bandwidth enhancement to the mm-wave using shorter feedback cavities. Figure 4(a) plots examples of the induced resonance modulation by strong OFB of η = 0.61 and 0.96 when L_C = 10 μm. When η = 0.61, the PPR-induced resonant modulation induces enhanced response over the mm-wave passband of f_m = 58 ~67 GHz and centered on f_m = 62 GHz. The peak value of this passband is 7 dB. This resonance modulation is associated with deterioration of f_3dB to 15 GHz. When OFB is very strong of η = 0.96, the central frequency of the mm-wave passband becomes as high as f_m = 91 GHz, which is comparable to the resonance frequency f_comp = 85 GHz of the compound cavity formed between the far ends of the VCSEL and feedback cavities. That is, the PPR is due to coupling between transverse modes separated by a frequency ~f_comp [2].

 

Fig. 4 IM responses of the TCC-VCSEL when (a) L_C = 10 μm and (b) L_C = 3 μm.

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It is worth noting that the peaks of the resonance modulation are lower than the case when L_C = 20 μm in Fig. 2(b). That is, the IM-response becomes less enhanced with shortening the length of the TCC. When L_C is decreased further, resonance modulation with response above the −3dB level was not noticed. However, the modulation bandwidth is more improved. An example is given in Fig. 4(b) when L_C = 3 μm, which indicates improvement of the modulation bandwidth to f_3dB = 61 GHz when η = 0.88. That is, f_3dB increases by an amount of 160% beyond f_3dB0, which is higher than the percentage of 60% predicted in [14].

In Figs. 5(a) and 5(b), we examine the variation of the modulation bandwidth f_3dB with η when L_C = 3 and 10 mm, respectively. Figure 5(a) shows that the increase in OFB strength up to η = 0.45 causes deterioration in f_3dB, which reaches 0.87 f_3dB0. The further increase in η then causes continuous improvement in f_3dB indicating that the strong coupling plays an important role for high-modulation bandwidths. The increase in η improves f_3dB up to 3f_3dB0 when η = 0.94. Figure 5(b) shows that up to η = 0.56, the TCC-VCSEL exhibits improvement of f_3dB up 2.51 f_3dB0. When η = 0.56 – 0.61, the VCSEL exhibits the resonance modulation shown in Fig. 4(a) associated with deterioration of f_3dB. The resonant modulation is attained again under very strong OFB.

 

Fig. 5 Variation of the modulation bandwidth f_3dB with the coupling ratio η when (a) L_C = 3 μm and (b) L_C = 10 μm.

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7. Conclusions

The transfer function of IM responses in VCSELs can be tailored by transverse coupled cavities, which exhibit a large enhancement in the modulation bandwidth by a factor of 3, reaching at 70 GHz or higher. Strong slow-light feedback can also induce resonant passband modulation centered on frequencies reaching 4.8 times the bandwidth of the VCSEL without feedback by controlling the length of the feedback cavity. Further optimizations could be expected for ultrahigh speed VCSELs. The bandwidth enhancement is strongly dependent on the OFB phase in the Lang-Kobayashi equation model. The multiple roundtrip model including frequency chirping presented here makes the modulation response more insensitive to the OFB phase. Detailed investigation on the influence of the OFB phase on the modulation characteristics of the TCC-VCSEL would be published elsewhere.