## Comprehensive model of 1550 nm MEMS-tunable high-contrast-grating VCSELs |

Optics Express, Vol. 22, Issue 7, pp. 8541-8555 (2014)

http://dx.doi.org/10.1364/OE.22.008541

Acrobat PDF (2359 KB)

### 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

1. K. Iga, “Surface-emitting laser-its birth and generation of new optoelectronics field,” IEEE J. Sel. Top. Quantum Electron. **6**, 1201–1215 (2000). [CrossRef]

4. D. I. Babic, K. Streubel, R. P. Mirin, N. M. Margalit, J. E. Bowers, E. L. Hu, D. E. Mars, L. Yang, and K. Carey, “Room-temperature continuous-wave operation of 1.54-μm vertical-cavity lasers,” IEEE Photon. Technol. Lett. **7**, 1225–1227 (1995). [CrossRef]

5. C. J. Chang-Hasnain, “Tunable VCSEL,” IEEE J. Sel. Top. Quantum Electron. **6**, 978–987 (2000). [CrossRef]

7. M. Y. Li, W. Yuen, G. S. Li, and C. J. Chang-Hasnain, “Top-emitting micromechanical VCSEL with a 31.6-nm tuning range,” IEEE Photon. Technol. Lett. **10**, 18–20 (1998). [CrossRef]

7. M. Y. Li, W. Yuen, G. S. Li, and C. J. Chang-Hasnain, “Top-emitting micromechanical VCSEL with a 31.6-nm tuning range,” IEEE Photon. Technol. Lett. **10**, 18–20 (1998). [CrossRef]

10. Y. Rao, W. J. Yang, C. Chase, M. C. Y. Huang, D. P. Worland, S. Khaleghi, M. R. Chitgarha, M. Ziyadi, A. E. Willner, and C. J. Chang-Hasnain, “Long-wavelength VCSEL using high-contrast grating,” IEEE J. Sel. Top. Quantum Electron. **19**, 1701311 (2013). [CrossRef]

9. C. Chase, Y. Rao, W. Hofmann, and C. J. Chang-Hasnain, “1550 nm high contrast grating VCSEL,” Opt. Express **18**, 15461–15466 (2010). [CrossRef] [PubMed]

11. D. B. Young, J. W. Scott, F. H. Peters, M. G. Peters, M. L. Majewski, B. J. Thibeault, S. W. Corzine, and L. A. Coldren, “Enhanced performance of offset-gain high-barrier vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. **29**, 2013–2022 (1993). [CrossRef]

## 2. Material gain of InGaAlAs quantum wells

**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.

*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

15. I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, “Band parameters for IIIV compound semiconductors and their alloys,” J. Appl. Phys. **89**, 5815 (2001). [CrossRef]

*α*and

*β*are the Varshni parameters [16

16. Y. P. Varshni, “Temperature dependence of the energy gap in semiconductors,” Physica **34**, 149–154 (1967). [CrossRef]

17. J. Minch, S. H. Park, T. Keating, and S. L. Chuang, “Theory and experiment of In_{1−x}Ga_{x}As_{y}P_{1−y} and In_{1−x−y}Ga_{x}Al_{y}As long-wavelength strained quantum-well lasers,” IEEE J. Quantum Electron. **35**, 771–782 (1999). [CrossRef]

^{−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], We see both the peak gain and the total spontaneous emission rate decrease with temperature.

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

18. V. Karagodsky, F. G. Sedgwick, and C. J. Chang-Hasnain, “Theoretical analysis of subwavelength high contrast grating reflectors,” Opt. Express **18**, 16973–16988 (2010). [CrossRef] [PubMed]

**R̃**

_{12}]

_{N1×N1}can be obtained as [19] where [

**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 where

*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.

*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.

*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.

*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 Figure 6(b) shows the round-trip phase spectra for different air-gap thicknesses, indicating the tunability of the cavity, where zero-crossing points correspond to cavity resonances. Due to the change of the resonance wavelength, the reflectivity at resonance also changes largely with the air-gap thickness, as indicated by the circles in Fig. 6(a), though the peak reflectivity remains nearly the same. The transfer matrix method can take into account complex effective indices in the layered medium as where

*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 where

*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 where Γ is the confinement factor calculated from the transfer matrix method at a given air-gap thickness. The photon lifetime can be found as where

*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.

*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).

*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

*g*(

*λ*,

*n*,

*T*) and spontaneous emission rate

*R*

_{sp}(

*n*,

*T*) from the

**k**·

**p**method, and the photon lifetime

*τ*, the confinement factor Γ and the mirror loss

_{p}*α*

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

21. S.-W. Chang, C.-Y. Lu, S. L. Chuang, T. D. Germann, U. W. Pohl, and D. Bimberg, “Theory of metal-cavity surface-emitting microlasers and comparison with experiment,” IEEE J. Sel. Top. Quantum Electron. **17**, 1681–1692 (2011). [CrossRef]

*n*and the photon density

*S*where

*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}*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}*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.

*β*

_{c1}and

*β*

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

22. J. W. Scott, D. B. Young, B. J. Thibeault, M. G. Peters, and L. A. Coldren, “Design of index-guided vertical-cavity lasers for low temperature-sensitivity, sub-milliamp thresholds, and single-mode operation,” IEEE J. Sel. Top. Quantum Electron. **1**, 638–648 (1995). [CrossRef]

23. P. V. Mena, J. J. Morikuni, S.-M. Kang, A. V. Harton, and K. W. Wyatt, “A comprehensive circuit-level model of vertical-cavity surface-emitting lasers,” J. Lightwave Technol. **17**, 2612–2632 (1999). [CrossRef]

*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*) to drop, as shown in Fig. 2(b). Therefore, compared to

_{a}*R*

_{Auger},

*R*

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

*B*coefficient as where

*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.

## 5. Electrostatic model for MEMS and tunable resonance

*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

*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.

*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

## Acknowledgments

## References and links

1. | K. Iga, “Surface-emitting laser-its birth and generation of new optoelectronics field,” IEEE J. Sel. Top. Quantum Electron. |

2. | M.-C. Amann and W. Hofmann, “InP-based long-wavelength VCSELs and VCSEL arrays,” IEEE J. Sel. Top. Quantum Electron. |

3. | P. A. Martin, “Near-infrared diode laser spectroscopy in chemical process and environmental air monitoring,” Chem. Soc. Rev. |

4. | D. I. Babic, K. Streubel, R. P. Mirin, N. M. Margalit, J. E. Bowers, E. L. Hu, D. E. Mars, L. Yang, and K. Carey, “Room-temperature continuous-wave operation of 1.54-μm vertical-cavity lasers,” IEEE Photon. Technol. Lett. |

5. | C. J. Chang-Hasnain, “Tunable VCSEL,” IEEE J. Sel. Top. Quantum Electron. |

6. | N. Satyan, A. Vasilyev, G. Rakuljic, V. Leyva, and A. Yariv, “Precise control of broadband frequency chirps using optoelectronic feedback,” Opt. Express |

7. | M. Y. Li, W. Yuen, G. S. Li, and C. J. Chang-Hasnain, “Top-emitting micromechanical VCSEL with a 31.6-nm tuning range,” IEEE Photon. Technol. Lett. |

8. | M. C. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A surface-emitting laser incorporating a high-index-contrast subwavelength grating,” Nat. Photonics |

9. | C. Chase, Y. Rao, W. Hofmann, and C. J. Chang-Hasnain, “1550 nm high contrast grating VCSEL,” Opt. Express |

10. | Y. Rao, W. J. Yang, C. Chase, M. C. Y. Huang, D. P. Worland, S. Khaleghi, M. R. Chitgarha, M. Ziyadi, A. E. Willner, and C. J. Chang-Hasnain, “Long-wavelength VCSEL using high-contrast grating,” IEEE J. Sel. Top. Quantum Electron. |

11. | D. B. Young, J. W. Scott, F. H. Peters, M. G. Peters, M. L. Majewski, B. J. Thibeault, S. W. Corzine, and L. A. Coldren, “Enhanced performance of offset-gain high-barrier vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. |

12. | S. L. Chuang, |

13. | G. L. Bir and G. E. Pikus, |

14. | L. Condren and S. Corzine, |

15. | I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, “Band parameters for IIIV compound semiconductors and their alloys,” J. Appl. Phys. |

16. | Y. P. Varshni, “Temperature dependence of the energy gap in semiconductors,” Physica |

17. | J. Minch, S. H. Park, T. Keating, and S. L. Chuang, “Theory and experiment of In |

18. | V. Karagodsky, F. G. Sedgwick, and C. J. Chang-Hasnain, “Theoretical analysis of subwavelength high contrast grating reflectors,” Opt. Express |

19. | W. C. Chew, |

20. | M. Y. Li and C. J. Chang-Hasnain, “Tilt loss in wavelength tunable micromechanical vertical cavity lasers,” in CLEO: 1999, 457–458, May1999. |

21. | S.-W. Chang, C.-Y. Lu, S. L. Chuang, T. D. Germann, U. W. Pohl, and D. Bimberg, “Theory of metal-cavity surface-emitting microlasers and comparison with experiment,” IEEE J. Sel. Top. Quantum Electron. |

22. | J. W. Scott, D. B. Young, B. J. Thibeault, M. G. Peters, and L. A. Coldren, “Design of index-guided vertical-cavity lasers for low temperature-sensitivity, sub-milliamp thresholds, and single-mode operation,” IEEE J. Sel. Top. Quantum Electron. |

23. | P. V. Mena, J. J. Morikuni, S.-M. Kang, A. V. Harton, and K. W. Wyatt, “A comprehensive circuit-level model of vertical-cavity surface-emitting lasers,” J. Lightwave Technol. |

**OCIS Codes**

(050.2770) Diffraction and gratings : Gratings

(140.3600) Lasers and laser optics : Lasers, tunable

(230.4685) Optical devices : Optical microelectromechanical devices

(140.7260) Lasers and laser optics : Vertical cavity surface emitting lasers

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: February 25, 2014

Revised Manuscript: March 26, 2014

Manuscript Accepted: March 26, 2014

Published: April 2, 2014

**Citation**

Pengfei Qiao, Guan-Lin Su, Yi Rao, Ming C. Wu, Connie J. Chang-Hasnain, and Shun Lien Chuang, "Comprehensive model of 1550 nm MEMS-tunable high-contrast-grating VCSELs," Opt. Express **22**, 8541-8555 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-7-8541

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### References

- K. Iga, “Surface-emitting laser-its birth and generation of new optoelectronics field,” IEEE J. Sel. Top. Quantum Electron. 6, 1201–1215 (2000). [CrossRef]
- M.-C. Amann, W. Hofmann, “InP-based long-wavelength VCSELs and VCSEL arrays,” IEEE J. Sel. Top. Quantum Electron. 15, 861–868 (2009). [CrossRef]
- P. A. Martin, “Near-infrared diode laser spectroscopy in chemical process and environmental air monitoring,” Chem. Soc. Rev. 31, 201–210 (2002). [CrossRef] [PubMed]
- D. I. Babic, K. Streubel, R. P. Mirin, N. M. Margalit, J. E. Bowers, E. L. Hu, D. E. Mars, L. Yang, K. Carey, “Room-temperature continuous-wave operation of 1.54-μm vertical-cavity lasers,” IEEE Photon. Technol. Lett. 7, 1225–1227 (1995). [CrossRef]
- C. J. Chang-Hasnain, “Tunable VCSEL,” IEEE J. Sel. Top. Quantum Electron. 6, 978–987 (2000). [CrossRef]
- N. Satyan, A. Vasilyev, G. Rakuljic, V. Leyva, A. Yariv, “Precise control of broadband frequency chirps using optoelectronic feedback,” Opt. Express 17, 15991–15999 (2009). [CrossRef] [PubMed]
- M. Y. Li, W. Yuen, G. S. Li, C. J. Chang-Hasnain, “Top-emitting micromechanical VCSEL with a 31.6-nm tuning range,” IEEE Photon. Technol. Lett. 10, 18–20 (1998). [CrossRef]
- M. C. Huang, Y. Zhou, C. J. Chang-Hasnain, “A surface-emitting laser incorporating a high-index-contrast subwavelength grating,” Nat. Photonics 1, 119–122 (2007). [CrossRef]
- C. Chase, Y. Rao, W. Hofmann, C. J. Chang-Hasnain, “1550 nm high contrast grating VCSEL,” Opt. Express 18, 15461–15466 (2010). [CrossRef] [PubMed]
- Y. Rao, W. J. Yang, C. Chase, M. C. Y. Huang, D. P. Worland, S. Khaleghi, M. R. Chitgarha, M. Ziyadi, A. E. Willner, C. J. Chang-Hasnain, “Long-wavelength VCSEL using high-contrast grating,” IEEE J. Sel. Top. Quantum Electron. 19, 1701311 (2013). [CrossRef]
- D. B. Young, J. W. Scott, F. H. Peters, M. G. Peters, M. L. Majewski, B. J. Thibeault, S. W. Corzine, L. A. Coldren, “Enhanced performance of offset-gain high-barrier vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 29, 2013–2022 (1993). [CrossRef]
- S. L. Chuang, Physics of Photonic Devices, 2 (Wiley, 2009), Chap. 4 and 9.
- G. L. Bir, G. E. Pikus, Symmetry and Strain-Induced Effects in Semiconductors (Wiley, 1974), Chap. 5.
- L. Condren, S. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, 1995), Chap. 4.
- I. Vurgaftman, J. R. Meyer, L. R. Ram-Mohan, “Band parameters for IIIV compound semiconductors and their alloys,” J. Appl. Phys. 89, 5815 (2001). [CrossRef]
- Y. P. Varshni, “Temperature dependence of the energy gap in semiconductors,” Physica 34, 149–154 (1967). [CrossRef]
- J. Minch, S. H. Park, T. Keating, S. L. Chuang, “Theory and experiment of In1−xGaxAsyP1−y and In1−x−yGaxAlyAs long-wavelength strained quantum-well lasers,” IEEE J. Quantum Electron. 35, 771–782 (1999). [CrossRef]
- V. Karagodsky, F. G. Sedgwick, C. J. Chang-Hasnain, “Theoretical analysis of subwavelength high contrast grating reflectors,” Opt. Express 18, 16973–16988 (2010). [CrossRef] [PubMed]
- W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE, 1995), Chap. 2.
- M. Y. Li, C. J. Chang-Hasnain, “Tilt loss in wavelength tunable micromechanical vertical cavity lasers,” in CLEO: 1999, 457–458, May1999.
- S.-W. Chang, C.-Y. Lu, S. L. Chuang, T. D. Germann, U. W. Pohl, D. Bimberg, “Theory of metal-cavity surface-emitting microlasers and comparison with experiment,” IEEE J. Sel. Top. Quantum Electron. 17, 1681–1692 (2011). [CrossRef]
- J. W. Scott, D. B. Young, B. J. Thibeault, M. G. Peters, L. A. Coldren, “Design of index-guided vertical-cavity lasers for low temperature-sensitivity, sub-milliamp thresholds, and single-mode operation,” IEEE J. Sel. Top. Quantum Electron. 1, 638–648 (1995). [CrossRef]
- P. V. Mena, J. J. Morikuni, S.-M. Kang, A. V. Harton, K. W. Wyatt, “A comprehensive circuit-level model of vertical-cavity surface-emitting lasers,” J. Lightwave Technol. 17, 2612–2632 (1999). [CrossRef]

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