Design and simulation of resonant-periodic-gain cavity high-speed VCSELs and carrier transport influence on their modulation characteristics

Jian Feng; Hui Li; Wei Miao; Shaochi Pan; Bin Wang; Zhao Chen; Xing Zhang; Chuyu Zhong; Shupeng Deng; Shihao Ding; Jinlong Lu; Nannan Li; Yanjing Wang; Cunzhu Tong; Cunzheng Ning

doi:10.3788/COL202624.012601

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Abstract

We demonstrate a high-speed vertical-cavity surface-emitting laser (VCSEL) design that has a resonant-periodic-gain (RPG) active region and 1λ optical length cavity. We have developed a simulation model for the high-frequency modulation of a VCSEL based on the rate equation theory. Both λ/2 cavity and 1λ RPG cavity VCSELs are in good agreement with the calculation results of our optimization model. The optical confinement factor, oxide aperture, escape and capture lifetime of carriers, p-doped distributed Bragg reflector (p-DBR) reflectivity and photon lifetime, and gain suppression coefficient are all taken into consideration. Their effects on the modulation frequency response characteristics of devices are analyzed. The 3 dB resonant frequency response bandwidth of this VCSEL can reach ∼60 GHz at an 11 mA bias current. Our advanced cavity design, in conjunction with the simulation model, offers valuable insights into enhancing the high-speed modulation performance of VCSELs.

Keywords

carrier transport high-speed modulation vertical-cavity surface-emitting laser

1. Introduction

Optical interconnects are extensively utilized in data centers and supercomputers for high-speed, short-distance communication. They have experienced a surge in demand for higher data rates and bandwidths, driven by advancements in artificial intelligence (AI), big data, blockchain, and cloud computing. The optical interconnect module primarily consists of directly electrically high-frequency-modulated vertical-cavity surface-emitting semiconductor lasers (VCSELs), multimode transmission fibers, and high-speed photodetectors, offering advantages such as miniaturization, low cost, and low energy consumption^[1,2]. Over the past two decades, the performance characteristics of VCSELs have facilitated the widespread adoption of multimode active optical cables (AOCs) and transceivers in short-reach optical communication channels^[3–5].

As a key light source, VCSELs have their modulation speed limited by relaxation oscillation frequency, typically less than 30 GHz. This restricts the data transmission rate to 100 Gb/s per lane using pulse-amplitude-modulation 4 (PAM4) modulation. In recent years, several teams have tried to use lithographically defined apertures^[6], transversely coupled cavities^[7], and bidirectional multimodal^[8] methods to achieve PAM4 results exceeding 100 Gb/s. Recently, Broadcom Inc. demonstrated a system with a transmission rate of 200 Gb/s per lane using 850 nm VCSELs^[9,10], incorporating pre-emphasis in the transmitter, adaptive equalization in the receiver, and forward error correction. The modulation bandwidth of a semiconductor laser depends on factors such as material gain properties, carrier and cavity lifetimes, optical mode confinement, electrical parasitic elements, photon lifetime, thermal effects, and other device parameters^[11,12]. Most of these factors are either optimized or constrained by fundamental physical processes. Therefore, it is essential to incorporate new ideas or approaches in cavity design and establish simulation models to optimize the high-speed signal modulation performance of VCSELs.

Furthermore, for maximum gain from the active region and to reduce the operational threshold of the device, quantum well (QW) layer groups are optimally positioned at anti-nodes of the optical standing wave by selecting specific separate confinement heterostructure (SCH) thicknesses. This design, known as a resonant-periodic-gain (RPG) structure^[13,14], is widely used in high-power surface-emitting devices, such as vertical-external-cavity surface-emitting lasers (VECSELs)^[15–17] and multi-junction VCSELs^[18–20]. In the design of high-speed VCSELs, the cavity is usually optimized to be as small as possible to pursue a higher modulation rate. With advancements in epitaxial growth technology, we can balance cavity volume and differential gain to further enhance modulation rates. Recently, a 1060 nm VCSEL with a multi-QW large active cavity and a dielectric top distributed Bragg reflector (DBR) was reported^[21], and its excellent high-speed modulation performance ( $f_{3 dB} \sim 45 GHz$ ) has garnered significant attention.

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In this work, we design a 1λ RPG cavity for high-speed VCSELs and develop a simulation model for the high-frequency direct current modulation of VCSELs based on the rate equation theory. The effects of the number, volume, and carrier parameters of QWs in the active region, the optical confinement factor increased by the standing wave matching, and the diameter of the oxide aperture on the device performance are analyzed. The carrier transport processes in the active region are discussed, and the impact of the carrier capture rate, escape rate, photon lifetime, and gain compression factor in the cavity on the small-signal 3 dB modulation bandwidth is investigated. Our results contribute to advancing the design of high-speed VCSELs.

2. VCSEL Cavity Design and Model

The device design framework for high-speed communication VCSELs primarily involves three critical components: 1) strained QW structures engineered to achieve high differential gain, improving lasing efficiency; 2) SCH active regions optimized to balance carrier confinement and transport loss through precise control of SCH layer thickness and bandgap offset; and 3) optical cavities requiring integrated optimization of optical field distribution, QW confinement factors, and oxidation-defined aperture structures to enable precise control of transverse mode profiles and threshold current. This study will primarily focus on the 850 nm VCSELs that are currently widely used for high-speed, low-power, and short-distance optical communications in data centers and AI supercomputers.

The conventional structure design of 850 nm wavelength VCSELs consists of a 110-nm-thick SCH containing five 4.5-nm-thick strained InGaAs/AlGaAs QWs. The structure includes a bottom n-doped DBR with 4 pairs of ${Al}_{0.12} GaAs / {Al}_{0.9} GaAs$ layers and 35 pairs of ${Al}_{0.12} GaAs / {Al}_{0.95} GaAs$ layers, as well as a top p-doped DBR with 23 pairs of ${Al}_{0.12} GaAs / {Al}_{0.9} GaAs$ layers. The optical thickness of the active region is precisely controlled to λ/2, as shown in Fig. 1.

Figure 1.Schematic cross-sectional view of the VCSEL with conventional λ/2 cavity and 1λ RPG cavity design.

The RPG cavity is designed based on the conventional structure, with specific details regarding its design and functionality. The active region of the RPG cavity is composed of five QWs per group, with two groups totaling 10 QWs. The SCH region employs ${Al}_{0.5} GaAs$ for compensation design, limiting injected carriers and controlling the optical thickness of the active region to 1λ, as shown in Fig. 1. Figure 2(a) illustrates the vertical profiles of the refractive index and vertical cavity mode for a conventional λ/2 cavity, while Fig. 2 (b) depicts those for a 1λ RPG cavity. The insets in Figs. 2(a) and 2(b) show the optical field standing wave matching in the active regions of the conventional and RPG structures, respectively. In the design of high-speed communication devices, it is usually necessary to design a large optical confinement factor ( $Γ$ ) of the QW layers to provide higher differential gain and obtain a higher modulation rate. In our RPG cavity design, compared to the conventional devices, the $Γ$ of the QW layers has an increase from 3% to 5.4%. To further investigate, we employ simulation models to study the effects of enhancing cavity and optical confinement factors on the modulation dynamic characteristics of VCSELs.

$Profiles of the refractive index and the optical field distribution for 850 nm VCSELs: (a) a conventional λ/2 cavity design; (b) a 1λ RPG cavity design. The x-axis is the vertical distance from the bottom of the n-DBR to the top of the p-DBR within the epitaxial structure. The insets depict the optical field distributions within the active regions.$

Figure 2.Profiles of the refractive index and the optical field distribution for 850 nm VCSELs: (a) a conventional λ/2 cavity design; (b) a 1λ RPG cavity design. The x-axis is the vertical distance from the bottom of the n-DBR to the top of the p-DBR within the epitaxial structure. The insets depict the optical field distributions within the active regions.

Rate equations are fundamental tools for analyzing the intrinsic dynamic behavior of VCSELs. These equations are crucial for modeling the interaction between carriers and photons within the active region of VCSELs. The single-mode equations that describe the supply and loss of carriers and photons are analogous to those found in Refs. [22 –24]. The equations are given as $\frac{d N_{b}}{d t} = \frac{η_{i} I}{q V_{a}} - \frac{N_{b}}{τ_{cap}} + \frac{N_{w} V_{w}}{τ_{esc} V_{b}} - \frac{N_{b}}{τ_{b}},$ (1) $\frac{d N_{w}}{d t} = \frac{N_{b} V_{b}}{τ_{cap} V_{w}} - \frac{N_{w}}{τ_{esc}} - \frac{N_{w}}{τ_{w}} - v_{g} g N_{p},$ (2) $\frac{d N_{p}}{d t} = Γ v_{g} g N_{p} + Γ β_{sp} R_{sp} - \frac{N_{p}}{τ_{p}},$ (3)where $N_{w}$ and $N_{b}$ are the densities of carriers in the QWs and barriers, and $N_{p}$ is the density of photons in the cavity. $I$ is the injection current, and $q$ is the electronic charge. $V_{a}$ , $V_{b}$ , and $V_{w}$ are the volume of the active region, barrier region, and QW region, respectively. $η_{i}$ is the efficiency of the carrier injection from the SCH region to the barrier region, approximated as 1. $τ_{w}$ and $τ_{b}$ are the carrier recombination lifetimes in the QW and barrier states, respectively. $τ_{cap}$ is the capture time of carriers into the QWs, and $τ_{esc}$ is the escape time of carriers out of the QWs. $v_{g}$ is the group velocity of the lasing mode, $g$ is the gain coefficient, $β_{sp}$ is the spontaneous emission factor, $R_{sp}$ is the spontaneous recombination rate, and $τ_{p}$ is the photon lifetime.

The small-signal response in VCSELs can be analyzed by applying a small sinusoidal modulating current to the bias current. The modulation transfer function, denoted as $H_{i} (ω)$ , is defined as^[23,25] $H_{i} (ω) = \frac{ω_{r}}{ω_{r}^{2} - ω^{2} + j ω γ} .$ (4)Here, $ω_{r} = 2 π / f_{r}$ represents the relaxation resonance angular frequency, and $γ$ is the damping factor.

This transfer function exhibits the characteristics of a second-order low-pass filter with a damped resonance peak, where the relaxation resonance frequency represents the natural oscillation frequency between the carriers and photons within the laser cavity. The classical equations for the dependence of the small-signal resonance frequency $f_{r}$ and damping factor $γ$ on the bias current can be approximated as^[23,25] $f_{r} = \frac{ω_{r}}{2 π} = \frac{1}{2 π} \sqrt{\frac{η_{i} Γ v_{g}}{q V_{a}} \cdot \frac{\partial g / \partial N}{χ} (I - I_{th})},$ (5) $γ = K \cdot f_{r}^{2} + γ_{0},$ (6) $K = 4 π^{2} (τ_{p} + \frac{ε \cdot χ}{v_{g} (\partial g / \partial N)}),$ (7) $γ_{0} = \frac{1}{χ \cdot τ_{w}},$ (8) $χ = 1 + \frac{τ_{cap}}{τ_{esc}},$ (9)where $χ$ is the transport factor and $ε$ is the gain suppression factor. The damping represents the rate of energy loss in the cavity, which effectively reduces the strength of the resonance peak. The damping $γ$ increases linearly with the square of $f_{r}$ . The proportionality is the K-factor.

According to Eqs. (1), (2), (5), (8), and (9), the carrier capture and escape rates in QWs depend on the thickness and ratio of the QW and barrier layers. Based on our previous work^[24], $τ_{esc}$ is mainly affected by temperature, with an order of magnitude difference within the range of 100 K (approximately 10 to 100 ps). As temperature increases, $τ_{esc}$ approaches $τ_{cap}$ , reducing the effective differential gain through the transport factor. This results in a lower resonance frequency and a higher damping rate. In the design of the RPG active region of a 1λ short cavity, it is essential to increase the number of QWs and optical confinement factors while ensuring the barrier layer has a specific thickness and bandgap height. Additionally, an extra carrier confinement layer can be incorporated at the boundary of the active region.

In addition to the parameters determined by the active region design, which influence the resonance frequency, $τ_{p}$ is a critical factor affecting the modulation response^[26]. This parameter is closely related to the internal cavity loss rate ( $α_{i}$ ) and the coupling mirror loss rate ( $α_{m}$ ), as expressed in the following equation^[26,27]: $τ_{p} \approx \frac{1}{α_{i} + α_{m}^{T} + α_{m}^{B}} .$ (10)Here, $α_{m}^{T}$ and $α_{m}^{B}$ are the mirror loss rates of the top p-DBR and bottom n-DBR, respectively. As the gain increases in the design of the RPG cavity, the optical cavity absorption loss also rises. The reflectivity of the p-DBR not only determines the optical output coupling rate but also significantly impacts the photon lifetime. In addition to adjustments in the epitaxial structure design, $α_{m}^{T}$ can be modified by coating the p-DBR surface, thereby altering the photon lifetime and modulation response characteristics. To achieve the highest possible modulation frequency, a shorter $τ_{cap}$ is required, and $τ_{p}$ should be set to balance the effects of $τ_{cap}$ and $τ_{p}$ , ensuring a high-bandwidth, critically damped response.

To demonstrate the modulation characteristics, we developed our model based on the VISTAS model^[28] and applied it to an oxide-confined 850 nm wavelength RPG cavity VCSEL. The intrinsic VCSEL model parameters are listed in Table 1. The values for the 1λ RPG cavity model parameters were determined based on the λ/2 cavity model.

Parameter	Value (λ/2 cavity)	Value (1λ RPG cavity)	Unit
Wavelength	850	850	nm
Number of QWs	5	10	—
Thickness of QW	4.5	4.5	nm
Thickness of barrier	50.2	99.2	nm
Oxidation-defined aperture	6	6	µm
Optical confinement factor Γ	0.03	0.054	—
Capture time of carrier τ_cap	7	5	ps
Escape time of carrier τ_esc	100	150	ps
Logarithmic gain coefficient g	7.5 × 10³	10⁴	cm⁻¹
Group velocity v_g	7.14 × 10⁷	7.14 × 10⁷	m/s
Top p-DBR reflectivity	99.8%	99.6%
Average internal loss α_i	30	35	cm⁻¹
Spontaneous recombination coefficient β_sp	5 × 10⁻⁴	5 × 10⁻⁴	—

Table 1. Numerical Values of Typical Parameters in the VCSEL Model

View all Tables

3. Results and Discussion

In the simulations, we first fixed and verified the simulation parameters based on the conventional λ/2 cavity to ensure they were within a reasonable range. As shown in Fig. 3(a), we calculated the small-signal frequency responses of the VCSEL under different bias currents. The results demonstrate that the observed effect of bias current shifts both the peak frequency of the frequency response and the associated bandwidth, $f_{3 dB}$ , to larger values. $f_{3 dB}$ can reach a maximum of 44 GHz with a 12 mA bias current. Since our research model does not calculate the parasitic cut-off frequency ( $f_{p}$ ) of the device, the typical value of $f_{p}$ for common device structures would result in a reduction of approximately 10–15 GHz in $f_{3 dB}$ ^[29,30]. Consequently, the 3 dB bandwidth of the corresponding device ranges from 30 to 35 GHz, which aligns with previously reported experimental results^[31,32].

Figure 3.Plot of the frequency response spectra at different values of the bias current. The value of parameters is the same as in Table 1. (a) Conventional λ/2 cavity design; (b) 1λ RPG cavity design.

In Fig. 3(b), the frequency responses of a 1λ RPG cavity VCSEL under varying bias currents are presented. The observed increase in $f_{3 dB}$ is attributed to the enhanced optical confinement factor inherent in the RPG design, which incorporates multiple QWs. Under consistent 6 µm oxidation-defined aperture settings, the elevated peak frequency and overall frequency response are attributed to the extended optical path facilitated by the RPG cavity design. This design significantly elevates the device’s performance ceiling, pushing $f_{3 dB}$ from 30–35 GHz to over 40 GHz.

The potential drawbacks of the extended longitudinal optical length are mitigated by reducing the oxidation-defined aperture diameter and regulating the active region volume. As illustrated in Figs. 4(a)–4(c), varying the aperture diameter (5, 4, and 3 µm) results in distinct frequency response characteristics. With the decrease of oxide aperture and active region volume $V_{a}$ , the peak resonance frequency increases, the peak strength decreases, and the frequency response curve of the device decreases and tends to be smooth as a whole. In the design of a 4 µm oxide aperture, $f_{3 dB}$ of the RPG cavity VCSEL has a maximum value of $\sim 60 GHz$ and has a relatively wide linear operating range of high-speed modulation, as shown in Fig. 4(b). This design is expected to enable high-speed VCSEL devices with $f_{3 dB}$ reaching approximately 45 GHz, meeting the demands of 200 GB/s per lane applications.

Figure 4.Plot of the frequency response spectra of the 1λ RPG cavity VCSEL with different advice region volumes, which has different oxidation-defined aperture diameters of (a) 5, (b) 4, and (c) 3 µm.

The RPG cavity design significantly alters the active region, necessitating careful consideration of QW parameters and their associated error ranges to ensure optimal frequency response characteristics. Referring to previous simulation results, we maintain the oxide aperture and bias current at 4 µm and 7 mA, respectively. Figure 5(a) illustrates the frequency response spectra of the 1λ RPG cavity VCSEL under varying $Γ$ . Under the same $V_{a}$ design and the same bias current, with the increase of $Γ$ , the peak of the resonance frequency increases significantly, and the strength of the peak frequency decreases. The frequency response curve becomes smooth, which enhances the modulation bandwidth $f_{3 dB}$ and improves the device’s capability for high-frequency modulation. In the design of the RPG epitaxial structure, the active region can be optimized by adjusting the QW width, as well as the number and position of QWs, and by modifying the composition of QWs and barriers at the same time to obtain higher $Γ$ .

Figure 5.Plot of the frequency response spectra of the 1λ RPG cavity VCSEL with a 4 µm oxidation-defined aperture and a 7 mA bias current, and the advice region has different (a) optical confinement factors, (b) escapes, and (c) capture lifetimes of carriers.

Figures 5(b) and 5(c) illustrate the frequency response spectra of the 1λ RPG cavity VCSEL under varying carrier escape and capture lifetimes in the QW. An increase in the carrier escape rate within the QW leads to a reduction in the peak strength of the frequency response, consequently decreasing $f_{3 dB}$ . In VCSELs, the carrier escape process is primarily governed by thermal escape, and the application of these devices typically requires maintaining acceptable frequency response characteristics over a broad temperature range (25°C to 85°C). In the design of RPG cavities, increasing the thickness of the active region can lead to heat accumulation within the QWs. The high-temperature operation can be achieved through a strong detuning design of QW gain and cavity mode^[33,34]. Additionally, incorporating an extra carrier confinement layer at the boundary of the active region can help suppress the carrier escape rate.

On the other hand, an increase in the carrier capture rate within the QW enhances the peak strength of the frequency response. Typically, the carrier capture rate is more critical for λ/2 cavity VCSELs, where the compact design often results in insufficient thickness for barrier and SCH layers. However, in the case of 1λ RPG cavity VCSELs, increasing the thickness of the barrier and SCH layers is necessary to compensate for the extended optical path. Furthermore, enhancing the bandgap height of the barrier layer allows simultaneous control of carrier escape and capture rates, directly influencing the damping factor.

In the conventional design of top-emitting VCSEL structures, the multi-period bottom DBR typically has reflectivity exceeding 99.99%. The top DBR’s reflectivity significantly influences the device’s differential efficiency, threshold, and $τ_{p}$ within the cavity^[35]. Transitioning to the RPG cavity design, incorporating additional QWs enhances the gain coefficient. The RPG cavity design for a communication application, however, necessitates a delicate balance between $τ_{p}$ and laser threshold, achievable through strategic p-DBR design adjustments. Equation (10) facilitates the calculation of $τ_{p}$ as a function of p-DBR reflectivity, with results illustrated in Fig. 6(a). Utilizing these computed $τ_{p}$ values, frequency response spectra for various $τ_{p}$ are plotted in Fig. 6(b), demonstrating a decrease in the frequency resonance peak strength as $τ_{p}$ increases from 1 to 3.4 ps. In the preliminary design of the RPG cavity, we selected a moderate value with a p-DBR reflectivity of 99.6% and $τ_{p}$ of about 2.5 ps, as listed in Table 1. This choice balances gain, threshold, and modulation bandwidth optimization. In fabricated devices, potential deviations in DBR reflectivity may result in degraded high-speed modulation characteristics. To mitigate this, precise reflectivity and $τ_{p}$ control can be achieved through the deposition of an additional dielectric layer on the p-DBR’s surface^[26,27].

Figure 6.(a) Variation of photon lifetime τ_p with different top p-DBR reflectivity. Plot of the frequency response spectra of the 1λ RPG cavity VCSEL with a 4 µm oxidation-defined aperture and a 7 mA bias current, and with different (b) photon lifetimes τ_p and (c) gain suppression factors ε (cm³).

The gain suppression coefficient $ε$ is a critical parameter in the semiconductor theory, influencing the scaling characteristics of laser performance in the nonlinear gain of QWs. In addition to its role in scaling, $ε$ contributes to resonance damping, as indicated by Eqs. (6) and (7). Typically, $ε$ exhibits a nonlinear relationship with carrier density, photon density, and temperature^[36]. This parameter cannot be directly measured experimentally but is often set within an experimental range during simulations. As shown in Fig. 6(c), the frequency response spectra of the 1λ RPG cavity VCSEL with varying $ε$ reveal that introducing compact, high-gain multiple QWs may lead to faster deterioration of modulation characteristics under extreme conditions. Consequently, systematic optimization of the device is necessary to mitigate the impact of nonlinear gain effects.

4. Conclusion

In summary, we have demonstrated a high-speed VCSEL design featuring an RPG active region and 1λ optical length cavity. A simulation model for the high-frequency direct current modulation of VCSELs has been developed. Both the λ/2 cavity conventional VCSEL and 1λ RPG cavity VCSEL exhibit good agreement with the optimization model’s calculation results. The bias current, optical confinement factor, oxidation-defined aperture, carrier escape and capture lifetimes, p-DBR reflectivity and photon lifetime, and gain suppression coefficient have been considered, and their effects on the devices’ modulation frequency response characteristics have been analyzed.

The optimized VCSEL employs a 1λ RPG cavity design, incorporating: 1) an active region with two periods and a total of 10 QWs; 2) an optical confinement factor of 5.4%; 3) a bottom n-DBR reflectivity of 99.99% and a top p-DBR reflectivity of 99.6%; and 4) an oxidation-defined aperture diameter of 4 µm. This VCSEL achieves a 3 dB resonant response bandwidth of nearly 60 GHz at an 11 mA bias current. When accounting for parasitic responses, the expected modulation bandwidth is approximately 45 GHz, which meets the requirements of next-generation optical interconnects. For datacom VCSELs, the large signal modulation characteristics are also important for fabricated devices, and equivalent circuit parameters need to be taken into consideration, especially under higher-order-level modulation. This is left for future work.

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