## Abstract

A comprehensive theoretical model for the long-wavelength micro-electro-mechanical-tunable high-contrast-grating vertical-cavity surface-emitting lasers is presented. Our band structure model calculates the optical gain and spontaneous emission of the InGaAlAs quantum well active region. The grating reflectivity and the cavity resonance condition are investigated through optical modeling. Correlating the results with the electrostatic model for the micro-electro-mechanical system, we accurately predict the measurements on the voltage-contolled lasing wavelength. Furthermore, our calculated temperature-dependent wavelength-tunable light output vs. current (L-I) curves show excellent agreement with experiment.

© 2014 Optical Society of America

## 1. Introduction

Long wavelength vertical-cavity surface-emitting lasers (VCSELs) emitting at around 1550 nm have drawn extensive research interest in the past few decades due to their fast-growing applications in fiber communication, optical interconnects, and laser spectroscopy [1–4]. VCSELs with tunable wavelengths are particularly important for wavelength-division-multiplexing and light ranging and detection [5–7]. The high contrast grating (HCG) controlled by a micro-electromechanical system (MEMS) has become a promising candidate for tunable VCSELs due to its high reflectivity, good mode selectivity, low power consumption, and low cost [7–10].

The InP-based HCG tunable VCSELs have been demonstrated with single mode operation, a wide wavelength tuning range, and fast MEMS tuning speed [9]. However, a theoretical model has yet to be developed for analyzing the laser performance. In this paper, we demonstrate a comprehensive model which covers the theories of quantum-well (QW) material gain, the reflectivity of the top and bottom mirrors composed of both the HCG and distributed Bragg reflectors (DBRs). The optical modeling of the resonant cavity is connected to the electrostatic modeling of the MEMS, and the results provide a deeper understanding of the tunable device properties and accurately predict the lasing wavelength and threshold current. This paper also establishes a rate-equation model that correlates the injected carrier density and output photon density, accurately predicting the laser light output vs. current (L-I) behavior. Thermal effects are especially important to consider for short cavity lasers such as VCSELs [11]. In this work we consider the temperature change of the active region as more current is injected. The degradation of the material gain as the active region temperature increases is considered, as well as the red-shift of both the gain peak and the cavity resonance due to thermal effects. The current that leaks through the quantum wells without undergoing recombination is also included. The unpinning effect of the carrier density above threshold is investigated. The temperature-dependent L-I curves calculated by our model agree very well with the experiment. This comprehensive model can be an important tool for designing high-speed, low-chirp, low-threshold tunable lasers with wide tuning ranges.

Figure 1(a) shows the schematic of the HCG tunable VCSEL to be modeled in this work. The active region consists of InGaAlAs multiple quantum wells. The top reflector consists of the HCG and DBR. The air gap between the HCG and top DBR is tunable by the MEMS control voltage. The device is electrically injected and the proton implantation serves to form the current confining aperture. Figure 1(b) shows the flow of the modeling procedure. The HCG optical properties are calculated by the mode matching method, and the air-gap thickness is correlated with the tuning voltage by the MEMS model. The VCSEL structure is modeled by the transfer matrix method, which can predict the MEMS-controlled lasing wavelength and provide the cavity parameters. The rate-equation model takes in the cavity parameters and the QW material gain calculated by the **k** · **p** method. Finally, the temperature-dependent MEMS-controlled L-I curves are calculated.

## 2. Material gain of InGaAlAs quantum wells

The active region of the long-wavelength HCG VCSEL consists of InGaAlAs multiple quantum wells. The valence band structure and eigenstates are solved with the 4-band **k** · **p** method using the Luttinger-Kohn model [12], which includes the heavy-hole and light-hole mixing. The conduction band structure and eigenstates are solved with the single-band effective-mass approximation. The strain effect is included through the Pikus-Bir strain terms in the Hamiltonian [13]. The optical transition matrix is calculated from the wavefunction overlap between subbands. The transition rate can be obtained from Fermi’s golden rule, which accounts for the Fermi-Dirac occupation of the conduction and valence subbands. Therefore we can write the material gain and spontaneous emission rate for the quantum wells as

*σ*accounts for the valence band spin degeneracy, and the lineshape function

*L*(

*k*,

_{t}*h̄ω*) accounts for the finite transition linewidth due to various scattering mechanisms. ${M}_{nm}^{\sigma}$ is the momentum matrix element, and

*f*and

_{c}*f*are the Fermi distribution functions for electrons in the

_{v}*n*-th conduction subband and

*m*-th valence subband, respectively. The large densities of electrons and holes in the laser active region bring in the many-body effects, which cause the band gap renormalization. Thus, we need to account for the red-shift of the band edge with the increasing injection level. The band gap shrinkage is modeled with a cubic-root dependence on the carrier density [14] as where Δ

*E*

_{BR}is the band gap renormalization constant for quantum wells, and

*n*

_{2D}is the surface carrier density in each quantum well normalized by 10

^{12}cm

^{−2}. Furthermore, we include the temperature dependence of the material band gap [15] as where

*α*and

*β*are the Varshni parameters [16]. Our InGaAlAs gain model has been verified by experimental data [17]. Figure 2(a) and Fig. 2(b) show the transverse electric (TE) polarized (electric field parallel to QWs) material gain and the TE spontaneous emission rate per unit volume per unit energy interval (s

^{−1}cm

^{−3}eV

^{−1}), respectively, for the InGaAlAs QWs at different temperatures and different carrier densities. Increasing temperature results in the red-shift of the gain and spontaneous emission spectra. Increasing carrier density results in the blue-shift of both spectra under low injection due to band-filling, but red-shift under high injection due to band gap renormalization. The total spontaneous emission rate per unit volume (s

^{−1}cm

^{−3}) is the integration over the emission spectrum averaged among the TE and TM polarizations [12],

## 3. Optical modeling of high contrast gratings and tunable VCSELs

The complex reflection coefficient of the HCG is an important parameter for the design and modeling of HCG VCSELs. For normal incidence on a subwavelength HCG, i.e. the wavelength is smaller than the grating period Λ, only the zeroth-order Floquet mode (normal reflection) is propagating, while all higher-order Floquet modes are evanescent. In this case, a complex reflection coefficient instead of a reflection matrix can be defined. We calculate the reflection spectrum using three different methods: analytical mode matching [18], mode matching [19] using numerically solved eigenmodes, and parameter extraction from the finite element simulation. As shown in Fig. 3(a), we define Region I and Region III to be the incidence and transmission air regions, respectively, and Region II to be the HCG layer.

For the analytical mode matching method, the modes inside Region II are expanded in terms of sinusoidal functions in both the transverse and longitudinal directions. In Region I and III, the field is expanded into a summation of discrete propagating waves, the wavenumbers and propagation directions of which are governed by the Floquet theorem. The boundary conditions for the electric and magnetic fields are matched at the air-HCG interfaces, and we can obtain a reflection matrix, out of which the zeroth-order reflection coefficient can be extracted.

For the numerical mode matching method, we first solved the eigenmodes in the HCG layer using the 1D finite-difference frequency-domain method, and using Floquet modes in the air regions. Applying the boundary conditions, the reflection and transmission matrices are obtained at each interface. The generalized reflection matrix [**R̃**_{12}]_{N1×N1} can be obtained as [19]

**R**

_{12}]

_{N1×N1}, [

**R**

_{23}]

_{N2×N2}, [

**R**

_{21}]

_{N2×N2}, [

**T**

_{12}]

_{N2×N1}, and [

**T**

_{21}]

_{N1×N2}are the reflection and transmission matrices at the interfaces. [

**I**]

_{N2×N2}is the identity matrix. The subscript

*ij*indicates wave incidence from Region

*i*to Region

*j*.

*N*

_{1}and

*N*

_{2}are the number of modes in Region I and Region II, respectively. The propagation matrix [

**K**

_{2}]

_{N2×N2}can be written as

*t*is the thickness of the grating, and

_{g}*k*is the propagation constant for the

_{iz}*i*-th eigenmode in Region II. The complex reflection coefficient can then be obtained from the

**R̃**

_{12}matrix. As shown in Fig. 3(b), the idea of generalized reflection for layered medium is still applicable when there are multiple modes in each region, except that the reflection and transmission at each interface are characterized by matrices solved from mode matching. The dimensions of the matrices also match with the number of modes in each region.

We further simulate the total field distribution using the finite element method (COMSOL Multiphysics). By fitting the field distribution using the complex reflection coefficient as the fitting parameter, we are able to extract the reflection spectrum.

Figure 4(a) shows the total field distribution when a normal incident wave (*λ* = 1550 nm) is reflected by a TE-HCG (electric field parallel to HCG bars), where the cross-sections of HCG bars are indicated by the white boxes. Figure 4(b) and Fig. 4(c) show excellent agreement among the three methods for both the magnitude and the phase of the complex reflection coefficient. The green dashed line indicates *λ* = Λ. When *λ* > Λ, we no longer have a single reflected mode since higher order Floquet modes become propagating. The power is not conserved for the zeroth-order mode, and incident power will be carried away by higher order Floquet modes.

The top mirror of the tunable HCG VCSEL consists of 2–4 pairs of *p*-doped DBR and a TE-HCG (electric field parallel to HCG bars) with an air gap in between the two regions. The bottom mirror consists of 40–55 pairs of *n*-doped DBR composed of alternating InGaAlAs and InP layers. The air-gap thickness and, consequently, the lasing wavelength can be tuned by the MEMS control voltage. Since the device diameter is large (between 10–25 *μ*m) compared to the emission wavelength, the fundamental transverse mode profile approaches a plane wave, and the effective index approaches the material refractive index. In this case, the transfer matrix method [12] can reduce the 3D problem to 1D, and provide an accurate prediction of the top and bottom mirror reflectivity, cavity resonance wavelength, confinement factor, quality factor, and threshold material gain.

The complex reflection coefficient of the TE-HCG is used as the boundary condition for the transfer matrix method. Figure 5 shows the reflectivity of the top mirror (including the DBR, air gap and HCG), bottom mirror, and HCG alone. The top DBR increases the reflection bandwidth, and the round-trip high reflection window is determined by the bottom DBR.

By adjusting the air-gap thickness, we can investigate the tunability of both the magnitude and phase of the top mirror reflection. The peak reflectivity of the top mirror changes little with the air-gap thickness. However, there is significant variation in the shape and asymmetry of the reflection spectrum, as shown in Fig. 6(a). The air-gap thickness largely perturbs the phase of the wave reflected by HCG, which ultimately determines the resonance wavelength. Similar to the Fabry-Perot model, we define the total round-trip phase as

*ϕ*

_{top}is the top reflection phase and

*ϕ*

_{cavity+bottom}is the bottom reflection phase that includes the cavity region. The resonance condition is determined by

*g*is the QW material gain and

*α*

_{i}is the material intrinsic loss. The effective index real part

*n′*is assumed constant since the change induced by the gain in QWs is negligible. Further, the small change in the thin QWs has little effect on optical modes. In order to calculate the threshold material gain

*g*

_{th}and mirror loss

*α*

_{m}, we define the round-trip gain at resonance to be

*r*

_{top}and

*r*

_{cavity+bottom}are the complex reflection coefficients of the top mirror, and the bottom region (including cavity region and bottom DBR), respectively. Then the threshold material gain can be found by setting the round-trip gain to be zero Since the mirror loss is equal to the threshold modal gain

*G*

_{th}when the intrinsic loss is zero, we can find the mirror loss as

*α*

_{d}accounts for the diffraction loss due to the finite-size effect, and the tilting and bending of the HCG caused by the MEMS tuning [20].

*Q*

_{rad},

*Q*

_{mat}, and

*Q*

_{d}refer to the quality factors associated with the radiation loss, material loss, and diffraction loss, respectively.

Figure 7(a) shows the tuning of the cavity resonance wavelength by controlling the air-gap thickness through the MEMS. The linear tuning range can be as wide as 20 nm, with a tuning efficiency around 0.04 nm/nm. Figure 7(b) shows the cavity mirror loss and the radiation *Q* at different air-gap thicknesses. We can see when air-gap thickness is 1.83 *μ*m, the reflection spectrum in Fig. 6(a) is most symmetric. It also corresponds to the center of the tuning range in Fig. 7(a), and the lowest mirror loss and the highest radiation *Q* in Fig. 7(b).

Moving away from the linear tuning range, the lasing wavelength begins to change very quickly with the air-gap thickness, which is mainly caused by the bottom DBR. The phase delay from the air gap changes linearly with the gap thickness, while the HCG phase dispersion is also nearly linear as in Fig. 4(c), yet the phase dispersion from the bottom DBR is highly nonlinear for off-resonance. When the tuning goes beyond a certain point, there is a sudden jump in the resonance wavelength, indicated by the discontinuities in Fig. 7(a). The discontinuity is due to the switching between two longitudinal cavity modes. Within the linear tuning range, the mirror loss and radiation *Q* change very little, while outside of this range, the mirror loss sharply increases and the radiation *Q* sharply decreases. This is due to the significant decrease in reflectivity from the bottom DBR as the resonance wavelength shifts away from the center of the reflection bandwidth spectrum.

## 4. Rate equations for HCG tunable VCSELs

After obtaining the gain *g*(*λ*, *n*, *T*) and spontaneous emission rate *R*_{sp}(*n*, *T*) from the **k** · **p** method, and the photon lifetime *τ _{p}*, the confinement factor Γ and the mirror loss

*α*

_{m}from the transfer matrix method, the output power of the HCG tunable VCSELs is modeled using the rate equations [12, 14, 21] for the carrier density

*n*and the photon density

*S*

*β*

_{sp}is the spontaneous emission coupling factor, and

*η*is the current injection efficiency. The active region temperature

_{i}*T*can be obtained from the substrate temperature

_{a}*T*

_{sub}, input electric power (

*VI*), and output light power

*P*as where

*R*

_{th}is the thermal resistance in K/mW. The cavity resonance wavelength also has a red-shift with increasing temperature due to the change of the material refractive index and the thermal expansion of the cavity. The change of lasing wavelength due to thermal effects is where

*dλ/dT*, obtained from experiments, is around 0.102 nm/K, and Δ

*T*is known once the active region temperature is obtained in Eq. (15). The non-radiative recombination rate and the stimulated emission rate can be calculated as where

*v*is the surface recombination velocity,

_{s}*C*is the Auger recombination coefficient,

*A*and

_{a}*V*are the surface area and volume of the active region, respectively, and

_{a}*v*is the group velocity in the active region.

_{g}In order to account for injected carriers that pass through the quantum wells without undergoing recombination, we consider the series leakage current as

*F*and

_{c}*F*are the quasi-Fermi levels in conduction band and valence band, respectively,

_{v}*E*

_{g,barrier}is the band gap of the QW barrier, and

*I*

_{l0}is a leakage current parameter. As the quasi-Fermi level separation (

*F*−

_{c}*F*) becomes closer to the QW barrier band gap, the leakage current significantly increases, which indicates large leakage currents at high injection levels.

_{v}Due to the incomplete electrical isolation of the proton implantation, we include *I*_{sh}(*I*) in the rate equations as the shunt leakage current. The shunt leakage is dependent on the injection current rather than the carrier density, and the carrier pinning effect does not clamp the shunt leakage. The shunt leakage path can be considered as a leakage diode in parallel with the laser diode. When the laser diode has a small turn-on voltage compared to the shunt diode, the laser diode path behaves like a small resistance, and the voltage is almost linear with the total current. The shunt diode current depends on the voltage exponentially. Thus, in this case, it is a good approximation to model the shunt leakage current as an exponential function of the total current.

If the shunt diode turns on earlier than the laser diode, the shunt leakage current increases with total current linearly at first, and the laser diode is nearly an open circuit. As the voltage increases, the laser diode turns on, and the circuit becomes two parallel diodes. Since the current through each diode depends on the voltage exponentially, the two currents are polynomial functions of each other. Thus, in this case, we can relate the total current to the shunt leakage current as a linear function at first and a polynomial function after the laser diode turns on.

The output light power can be obtained as

*β*

_{c1}and

*β*

_{c2}account for the coupling efficiencies for the stimulated emission and spontaneous emission power.

Figure 8 shows the theoretical temperature-dependent L-I curves of a fixed-gap (fixed-wavelength) TE-HCG VCSEL and the excellent agreement with experimental data. Our theory can accurately explain the temperature dependence of the threshold current and the rollover of the output power due to thermal effects. The threshold current increases with temperature because the material gain is reduced at higher temperature for a given carrier density. Therefore more carriers are required to increase gain to a high enough level to overcome the loss. Thus the threshold current is larger at higher temperature. The rollover can be caused by a combination of many mechanisms. Firstly, as larger current is injected, the active region temperature increases, which causes the thermal expansion of the cavity and the change of the material refractive indices. Thus, the lasing wavelength has a red-shift with temperature. Meanwhile, the material gain also has a red-shift with increasing temperature, as shown in Fig. 2. However, the lasing wavelength red-shift is slower than that of the material gain, causing the detuning of the gain peak and the lasing wavelength and the reduction of the stimulated emission rate and output power. Secondly, the material gain itself decreases with temperature even without considering the detuning, as also shown in Fig. 2. Thirdly, the series leakage current increases at higher carrier densities and high temperatures, as indicated in Eq. (19). Higher carrier density also gives rise to a larger non-radiative recombination current and larger spontaneous emission current. Therefore, the current contributing to the lasing mode is reduced, resulting in less lasing power. Furthermore, the shunt leakage increases with the injection current, and directly contributes to the rollover.

Figure 9(a) and Fig. 9(b) show the gain and carrier density solved from the rate equations as functions of the injection current. The kinks in the curves correspond to the lasing threshold, with higher substrate temperatures resulting in larger threshold currents, as described above. Both the gain and the carrier density should be pinned at their threshold values if thermal effects are not considered. However, the unpinning effect is observed in our theoretical results. The red-shift of the lasing wavelength at higher injection currents causes the decrease of the HCG and DBR reflectivity and the increase of mirror loss. Therefore, the gain is pinned at slightly higher values to overcome the loss when current increases, as shown in Fig. 9(a).

The carrier unpinning [22, 23] shown in Fig. 9(b) is mainly caused by the degradation of the material gain at elevated temperatures as injection current increases, and the detuning between the gain peak and the cavity resonance. To compensate the reduction of gain at cavity resonance, more carriers are required as current increases. At the same injection current above threshold, higher substrate temperature also requires a larger carrier density to maintain enough material gain, thus larger non-radiative, spontaneous emission, and leakage currents. This results in a smaller portion of the injected current contributing to the lasing mode at a higher temperature, and both the output power and the wall-plug efficiency are reduced, as shown in Fig. 8.

Figure 10(a) shows the spontaneous emission rate *R*_{sp} and the Auger recombination rate *R*_{Auger} calculated as functions of the injection current. Due to the unpinning of the carrier density, both *R*_{sp} and *R*_{Auger} keep increasing above threshold, and they are both larger for higher substrate temperatures. However, *R*_{sp} is less temperature-sensitive than *R*_{Auger}. Even though the carrier density *n* is larger with higher substrate temperature at a given injection current, as shown in Fig. 9(b), the increase of temperature also causes *R*_{sp}(*n*, *T _{a}*) to drop, as shown in Fig. 2(b). Therefore, compared to

*R*

_{Auger},

*R*

_{sp}increases with substrate temperature much slower at a fixed injection current.

To see the temperature-dependent spontaneous emission, we obtain the *B* coefficient as

*R*

_{sp},

*n*, and

*T*are all solved from the rate equations at a given injection current. The relationship among

_{a}*R*

_{sp},

*n*, and

*T*at different substrate temperatures are shown in Fig. 10(b). The

_{a}*B*coefficient decreases with carrier density due to the increasing active region temperature. At the same carrier density, the

*B*coefficient with lower substrate temperature is indeed larger. The four curves are pinned to the same curve due to stimulated emission, where the kinks indicate the thresholds. At the same carrier density,

*R*

_{sp}is also larger with lower substrate temperature. Below threshold,

*R*

_{sp}increases with

*n*almost quadratically, yet the curvature is reduced by the increase of the active region temperature.

Figure 11 shows the five current mechanisms that comprise the injection current, including the current contributing to stimulated emission, spontaneous emission, non-radiative recombination, the current leaking through quantum wells, and the shunt leakage current, at four different substrate temperatures. We can see indeed smaller percentage of the injection current goes into the lasing mode when the substrate temperature is higher.

## 5. Electrostatic model for MEMS and tunable resonance

In order to correlate the lasing wavelength and the L-I curve with the MEMS tuning voltage, we use an electrostatic model for the MEMS that controls the air-gap thickness, as shown in Fig. 12(a). The gravity of the MEMS top plate and the attractive force due to the opposite charges on the two MEMS plates are balanced by the MEMS elastic force. The force equations are

*k*is the spring constant for the elastic force

*F*,

_{k}*F*is the electrostatic force,

_{E}*h*

_{0}is the air-gap thickness when no charge is on the plate and gravity is not considered, i.e., the MEMS has no elastic deformation.

*x*

_{0}is the air-gap thickness when the control voltage is zero (no charge), and

*x*is the air-gap thickness when the control voltage is

*V*. From Eq. (22) we can obtain the mapping between the control voltage

*V*and the air-gap thickness

*x*as

Since the mapping between the air-gap thickness and resonance wavelength is obtained from Fig. 7(a), we can correlate the tuning voltage and resonance wavelength using Eq. (23). By taking the spring constant *k* as the only fitting parameter, our theoretical results match very well with the experimental data, as shown in Fig. 12(b). The fitted spring constant *k* is 0.16 N/m.

At a given MEMS control voltage, we can calculate the air-gap thickness, which is used as the input to the transfer matrix model, outputting the mirror loss *α*_{m}, confinement factor Γ, and the quality factor *Q*. Our rate-equation model further produces the L-I curves at different tuning voltages, as shown in Fig. 13(a). We can also see that as we increase the tuning voltage, the air-gap thickness is tuned away from the center of the linear tuning range in Fig. 7(a), and the threshold current increases due to the increase of the mirror loss. As shown in Fig. 13(b), the change of the threshold current and the peak power is small below *V* = 4 V because the shift of the lasing wavelength is small, as indicated in Fig. 12(b). Yet above *V* = 4 V we see a fast increase of the threshold current. Besides the increase of the mirror loss, the increase of diffraction loss also has a contribution to the large increase of the threshold current. The increase of diffraction loss can be caused by the bending of the HCG reflector due to MEMS tuning. From our model, we estimate the additional diffraction loss Δ*α*_{d} (relative to 0 V) at 5 V, 7 V, and 8 V to be 12, 20, and 26 cm^{−1}, respectively, which equates to a 0.1%, 0.17%, and 0.23% reduction in the reflectivity, respectively. Both the peak power and the slope of the L-I curve increase slightly with tuning voltage due to the increase of mirror loss. The parameters used in our theoretical model are listed in Table. 1.

## 6. Conclusion

We have successfully demonstrated a comprehensive model for the MEMS-tunable HCG VC-SELs. The model calculates the temperature-dependent material gain and spontaneous emission spectra of the quantum-well active region. The optical properties of the HCG reflector are modeled with both analytical and numerical methods, showing good convergence. The HCG VC-SEL cavity is modeled with the transfer matrix method, which produces important parameters for device-level simulation. The rate-equation model takes into account the thermal effects and our calculated temperature-dependent L-I curves show excellence agreement with experiment. Our MEMS model further correlates the tuning voltage with the resonance wavelength, threshold current, and peak power. The measurements can be accurately explained by our model.

## Acknowledgments

This work is supported by the DARPA under the E-PHI program (Grant number HR0011-11-2-0021). The authors would like to thank Professor Weng Cho Chew at University of Illinois at Urbana-Champaign, Dr. Chien-Yao Lu, and Dr. Chi-Yu Adrian Ni for the insightful discussion.

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