## Abstract

The micro–optical resonator of the microelectromechanical system(MEMS) tunable vertical cavity surface emitting laser(VCSEL), which is a gain–guided laser, consists of two mirrors fabricated on semiconductor chips of different materials. To understand the relationship between the curvature radius of the concave mirror and the diameter of the tunnel junction forming the active region, the diffraction loss map was obtained. The Fox–Li method was used with the integral kernel of the Rayleigh–Sommerfeld diffraction to simulate an optical resonator with a large Fresnel number. We derived the guideline for the actual bonding process of the different material chips by simulating with each parameter.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Vertical cavity surface emitting lasers(VCSELs) are commonly used in several areas, including optical communication, three–dimensional sensing, and gas analysis utilizing laser spectroscopy [1–3]. In this study, a widely–tunable VCSEL with a microelectromechanical system(MEMS) movable mirror, called a MEMS–VCSEL, was developed. MEMS–VCSEL exhibits rapid tuning of 500 kHz with a tunable range of 50 nm–60 nm in three bands of ${1.55}\;\mu \textrm {m}$, ${1.62}\;\mu \textrm {m}$, and ${1.69}\;\mu \textrm {m}$ [4–6].

Figure 1 shows the schematic of the MEMS–VCSEL. The MEMS–VCSEL consists of a VCSEL chip with an in–plane distributed Bragg reflector(DBR) on the upper side and a Si–MEMS chip with a concave DBR on the lower side. To form the optical resonator, the Au–electrode of each chip is bonded using a thermocompression method. To tune the wavelength, the length of the resonator is controlled by displacing the concave mirror on a membrane in the direction of the optical axis using an electrostatic force [7].

The MEMS–VCSEL is a gain–guided laser consisting of semiconductors made of different materials. This difference causes diffraction loss of the resonator, as the active layer and the optical axis of the mirror are inevitably inclined and/or misaligned during fabrication. As a result, the lasing threshold can be increased with an increase in the diffraction loss. Although the tolerance of the mirror inclination and the misalignment of the optical axis are important for the bonding process, fabricating several resonators to obtain the tolerance is time–consuming. Therefore, numerical simulation is preferable to conducting multiple experiments.

The beam propagation method(BPM), the conventional method of calculating the mode of the optical resonator, uses paraxial approximation [8–11]. Therefore, it is not applicable to micro–optical resonators such as MEMS–VCSEL, in which the diameter of the mirror is larger than the cavity length(the Fresnel number $N_{F}$ [12,13] is high). Although the finite difference time domain(FDTD) method [14,15], which discretizes Maxwell’s equations, does not require approximation, it has certain disadvantages, such as difficulty in calculating the mode and high consumption of computational resources.

On the other hand, the Fox–Li method can efficiently simulate an optical resonator with less approximation [16,17]. Therefore, in the present study, this method was adopted to calculate the mode profile and the dependence of the diffraction loss on the mirror’s concave curvature radius, the inclination of the mirror, and the misalignment of the optical axis [18,19]. Further, it allows the integral kernel to be derived without the paraxial approximation [20].

However, previous applications of the Fox–Li method did not consider the gain–guided structure, which squeezes the beam and plays an important role in the optical resonator. Thus, the calculated mode profiles were dependent on the mirror area, and the results differ significantly from the actual mode profile of the MEMS–VCSEL.

In this study, an optical aperture(AP) [21] is introduced in addition to the existing model using the Fox–Li method to simulate the gain–guided structure, that is, the tunnel junction(TJ) in the laser active layer. The dependence of the diffraction loss on the misalignment of the optical axis of the mirror and the TJ, as well as on the mirror inclination, was calculated using the modified model. Additionally, the diffraction loss map, which demonstrates the effect of mirror inclination and misalignment on the diffraction loss, is introduced to investigate the characteristics of the resonator. The guidelines for the actual bonding process of the different semiconductor chips were derived by investigating the dependence of the diffraction loss on the TJ diameter and curvature radius of the concave mirror from the diffraction loss map.

## 2. Calculation model and method

#### 2.1 Calculation model

Figure 2 shows a calculation model composed of two circular mirrors, $\mathrm {m}_1$ and $\mathrm {m}_2$, and a circular AP. $\mathrm {m}_1$ and $\mathrm {m}_2$ are equivalent to a concave mirror on the Si–MEMS chip and a plane mirror on the VCSEL chip. A concave mirror was fabricated onto the MEMS membrane by chemical mechanical polishing(CMP) [4]. The CMP process condition determines the curvature radius $R$ [22], which is also a design parameter for the MEMS–VCSEL. The AP with diameter $a_{0}$ is the approximate structure of the active region formed by the TJ, as explained below. $\theta$ is an inclination angle of $\mathrm {m}_2$. $d$ is the position shift of the optical axis of $\mathrm {m}_2$ and AP, which shows the difference in the distances between the centers of $\mathrm {m}_1$ and $\mathrm {m}_2$ in the $xy$–plane. Here, $a$ is the diameter of the two mirrors, and $b$ is the cavity length, which indicates the distance between $\mathrm {m}_1$ and $\mathrm {m}_2$. $b^{\prime }$ and $b^{\prime \prime }$ are the distances between $\mathrm {m}_1$ and AP and AP and $\mathrm {m}_2$, respectively. The reflectivities of $\mathrm {m}_1$ and $\mathrm {m}_2$ are $R_{1}$ and $R_{2}$, respectively, which allows consideration of the reflection loss. However, $R_{1}=R_{2}=100\%$ were set in this study.

The MEMS–VCSEL has a gain–guiding mechanism owing to the active region formed by a TJ in the VCSEL. The transverse mode is strongly confined by the active region because the light is gained when passing through the region. In fact, because of the sufficiently short distance between TJ and Si–MEMS(several wavelengths), the beam diameter on the Si–MEMS mirror is almost equal to the TJ diameter. From this experimental result, the active region was simply approximated using the AP.

Therefore, the design parameters of a micro–optical resonator are the mirror diameter $a$, cavity length $b$, TJ diameter $a_{0}$, and curvature radius $R$. When the TJ diameter is larger than the mirror diameter($a_{0} \gg a$), the former can be disregarded in the design parameters. In this study, the parameters are assumed to be $a={24}\lambda$, $b^{\prime }={7}\lambda$, $b^{\prime \prime }={3}\lambda$, and $b=b^{\prime } + b^{\prime \prime }={10}\lambda$, as shown in Fig. 2. The other design parameters are the TJ diameter and curvature radius.

When the beam diameter is sufficiently confined by the TJ, the Fresnel number, which expresses the characteristics of the optical resonator, is approximated as below:

#### 2.2 Fox–Li method

The Fox–Li method enables the calculation of the characteristics of Fabry–Pérot resonators, that is, the diffraction loss and mode profile. This method is governed by an integral equation that can be solved by a computer by discretizing and diagonalizing the equation. In this section, the applied governing equation, integral kernel, and integral method are introduced.

The electrical fields propagated from one mirror to another in the resonator are expressed using diffraction equations. Now, $f^q_{\mathrm {m}_1}$ is the electrical field on $\mathrm {m}_1$ after being reflected $q$ times. $f^{q+1}_{\mathrm {m}_2}$ on $\mathrm {m}_2$ and $f^{q+2}_{\mathrm {m}_1}$ on $\mathrm {m}_1$ are also the fields after being reflected $q+1$ times and $q+2$ times, respectively. These electrical fields are expressed using the following equations:

In this study, the integral kernel $K{(s_{i^{\prime }})},{(s_{i})}$ is expressed as follows:

The integral equation was discretized to calculate Eq. (4) and Eq. (6) as follows:

## 3. Results

This section first presents the calculation of the dependency of the beam diameter on the curvature radius of the concave mirror and the TJ diameter in the ideal case, where both the inclination angle and the position shift are zero($\theta =0, d=0$). Then, the effects of the curvature radius and TJ diameter on the beam diameter are discussed. Next, a diffraction loss map is introduced to visualize the dependence of the diffraction loss of the optical resonator on the position shift and inclination angle. Lastly, the dependency of the diffraction loss map on the curvature radius and TJ diameter is discussed.

#### 3.1 Dependency of the beam diameter on curvature radius and TJ diameter

Figure 3 shows the dependency of the beam diameter on curvature radius $R$ and TJ diameter $a_{0}$ when the inclination angle of the plane mirror and the position shift are zero($\theta =0, d=0$). $R$ and $a_{0}$ are changed from ${50}\lambda$ to ${1200}\lambda$ and from ${4}\lambda$ to ${24}\lambda$, respectively. The beam diameter was calculated from the TEM$_{00}$ mode on the mirror $\mathrm {m}_1$ and defined as the diameter at which the electric fields are $1/e$ from the peak. The beam diameter without TJ is represented by the black line in Fig. 3. Therefore, the beam diameter depends exclusively on $R$, and not on the TJ diameter, even in the case where a TJ with $a_{0}={24}\lambda$ is present. This is indicated by the overlap between the black line(without TJ) and the green line($a_{0}={24}\lambda$) in Fig. 3. When the TJ diameter is smaller than the green line, the beam diameter remains almost constant even if $R$ changes(orange and blue lines in Fig. 3). This means that the beam is confined by the TJ rather than the concave mirror. When the TJ diameter is $a_{0}={4}\lambda$, the beam diameter is constant for $R > {50}\lambda$, indicating that the beam confinement by TJ is always dominant in that region. In the case of $a_{0}={12}\lambda$, the beam diameter begins to decrease at approximately $R={200}\lambda$, which clearly shows the change in the main cause of beam confinement from the TJ diameter to the curvature radius.

#### 3.2 Diffraction loss maps

The diffraction loss map is a contour plot of the dependence of the diffraction loss $\alpha _\mathrm {D}$ of the optical resonator on the position shift $d$ and the inclination angle $\theta$. Specifically, $d$ is taken as the horizontal axis, $\theta$ is the vertical axis, and the diffraction loss $\alpha _\mathrm {D}$ is a contour. In this study, the case of the position shift $d$ and inclination angle $\theta$ in the $y$–axis were considered. $d$ varies over a range of $\pm {2}\lambda$ in increments of ${0.2}\lambda$, and $\theta$ varies over a range of $\pm 1^{\circ }$ in increments of $0.1^{\circ }$.

#### 3.3 Dependency of the diffraction loss map on the TJ diameter

Figure 4 shows the dependency of the diffraction loss map on the TJ diameter $a_{0}$ for a curvature radius $R={500}\lambda$, calculated by changing $a_{0}$ from ${4}\lambda$ to ${14}\lambda$. The white line shows the contour line of diffraction loss in $5\%$ increments. The diffraction loss map is symmetrical at the origin $(d, \theta )=(0, 0)$ owing to the same optical resonator structure between the resonator with position shift $d$ and inclination angle $\theta$ and that with $-d$ and $-\theta$. The diffraction loss is always lowest when there is no position shift or inclination angle, that is, $(d, \theta )=(0, 0)$. The contour line of the diffraction loss becomes elliptical because the beam transmission is higher at the AP when the beam shift by tilt is in the same direction as the AP shift. On the other hand, the diffraction loss increases at a higher rate when the beam shift direction by tilt differs from the AP shift direction. The cusps of the contour line are observed in some diffraction loss maps because of the rough mesh size, where a position shift of ${0.2}\lambda$ or less is negligible.

When $a_{0}={4}\lambda$, the diffraction loss $\alpha _\mathrm {D}$ primarily depends on the position shift $d$. However, as $a_{0}$ increases, the effect of the inclination angle $\theta$ is observed. When $a_{0}={14}\lambda$, the diffraction loss primarily depends on $\theta$. As the TJ diameter decreases, the diffraction loss increase. This can be interpreted as a decrease of the light in the resonator not pass through the active region.

#### 3.4 Dependency of the diffraction loss map on the curvature radius

Figure 5 shows the obtained results of the dependency of the diffraction loss map on the curvature radius $R$ for the TJ diameter $a_{0}={8}\lambda$. The curvature radius $R$ changed from ${20}\lambda$ to $\infty$. The white line shows the contour line of diffraction loss in $5\%$ increments basically and the numbers on the line show the value of the diffraction loss. The basic characteristics of the diffraction loss map are the same as shown in the Section3.3. The diffraction loss is always lowest when there is no position shift or inclination angle. The symmetry at the origin $(d, \theta )=(0, 0)$ is present and the contour line of the diffraction loss becomes elliptical. As the curvature radius decreases, the diffraction loss decreases, and the dependency on $\theta$ reduces owing to the focusing effect of the concave mirror on the optical axis.

## 4. Discussion

The design guidline for MEMS–VCSEL was discussed in this Section. From the diffraction loss maps in the Section3., when the beam diameter becomes small, which means the curvature radius and/or the TJ diameter is small, the inclination angle of the mirror becomes insensitive to the diffraction loss, while the deviation of the optical axis is sensitive to it. Therefore, by designing the small beam diameter, the focus is on decreasing the deviation of the optical axis in the process of bonding the Si–MEMS chip and the VCSEL chip because misalignment of the optical axis is dominant in the diffraction loss. In the actual device, the small TJ diameter is usually designed to decrease the threshold current in laser operation by increasing the current density at the active region. However, the large diffraction losses were obtained in the small TJ diameter less than $a_{0}={10}\lambda$ in Fig. 4. The results indicate that most of the electrical fields in the resonator do not pass through the active region. The small curvature radius can improve the diffraction loss, for example, the diffraction loss was $2.9\%$ when the TJ diameter is $a_{0}={6}\lambda$ and the curvature radius is $R={20}\lambda$. Therefore the beam confinement by the concave mirror is preferred to that by the TJ because of the low diffraction loss. Especially for MEMS–VCSEL, the low diffraction loss leads to the laser characteristics such as low threshold current, high power, and wide tunable range.

## 5. Conclusion

We have calculated the micro–optical resonator to study the influence of the inclination angle of the mirror and the deviation of the optical axis, which can be problematic when bonding the Si–MEMS chip and the VCSEL chip. In the numerical calculation, the Fox–Li method was used, and its integral kernel was adopted to the Rayleigh–Sommerfeld integral kernel changing from an existing one so that the calculation could be performed in cases involving a very large Fresnel number. Moreover, the AP, which approximates the laser active layer(the TJ), was adopted, and the gain–guided mechanism of the active region formed by TJ was replicated as an approximate design model for bonding the Si–MEMS chip and the VCSEL chip in the MEMS–VCSEL.

In the resonator configuration of the concave mirror – the TJ – the plane mirror, we demonstrate the dependence of the curvature radius of the concave mirror and the TJ diameter on the beam diameter. The smaller the curvature radius of the concave mirror and the TJ diameter, the smaller the beam diameter. In addition, when comparing the curvature radius and the TJ diameter, the beam diameter is determined by the parameter with a larger confinement effect of the beam.

We also denoted the dependence of the curvature radius of the concave mirror and the TJ diameter on the diffraction loss caused by the mirror inclination and misalignment by using the diffraction loss map. The diffraction loss map, which shows the effect of the mirror inclination and misalignment on the diffraction loss, is introduced. The diffraction loss map facilitates understanding the dependence of the curvature radius of the concave mirror and the TJ diameter.

The dependencies of the diffraction loss map on the TJ diameter and the curvature radius were investigated. The smaller the TJ diameter, the greater the dependence on the optical misalignment, and the larger the TJ diameter, the greater the dependence on the inclination angle of the mirror. The smaller the curvature radius, the less the dependence on inclination angle of the mirror owing to the focusing effect of the concave mirror on the optical axis.

Finally, the design guideline for bonding different materials was obtained from this study. Designing the small beam diameter enables the focus on decreasing the misalignment of the optical axis in the process of bonding the Si–MEMS chip and the VCSEL chip because the inclination angle is not sensitive to the diffraction loss.

In the actual design, the curvature radius was designed to have a smaller impact on the diffraction loss, and the diameter of the TJ was made larger to reduce the impact on the diffraction loss. This magnitude relation was designed using diffraction loss maps. In this manner, the dependency of the inclination angle of the mirror on the diffraction loss was decreased, and focus was placed on joining the two chips while considering the deviation of the optical axis.

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

## References

**1. **T. Anan, N. Suzuki, K. Yashiki, K. Fukatsu, H. Hatakeyama, T. Akagawa, K. Tokutome, and M. Tsuji, “High-speed 1.1-*μ*m-range InGaAs VCSELs,” in Optical Fiber Communication Conference/National Fiber Optic Engineers Conference, (OSA, 2008), pp. 1–3.

**2. **A. Larsson, “Advances in VCSELs for Communication and Sensing,” IEEE J. Sel. Top. Quantum Electron. **17**(6), 1552–1567 (2011). [CrossRef]

**3. **H. Moench, M. Carpaij, P. Gerlach, S. Gronenborn, R. Gudde, J. Hellmig, J. Kolb, and A. van der Lee, “VCSEL-based sensors for distance and velocity,” in * Vertical-Cavity Surface-Emitting Lasers XX*, vol. 9766K. D. Choquette and J. K. Guenter, eds., International Society for Optics and Photonics (SPIE, 2016), pp. 40–50.

**4. **N. Kanbara, S. Tezuka, and T. Watanabe, “MEMS Tunable VCSEL with Concave Mirror using the Selective Polishing Method,” in IEEE/LEOS International Conference on Optical MEMS and Their Applications Conference, 2006., (IEEE, 2006), pp. 9–10.

**5. **T. Yano, H. Saitou, N. Kanbara, R. Noda, S. Tezuka, N. Fujimura, M. Ooyama, T. Watanabe, T. Hirata, and N. Nishiyama, “Wavelength Modulation Over 500 kHz of Micromechanically Tunable InP-Based VCSELs With Si-MEMS Technology,” IEEE J. Sel. Top. Quantum Electron. **15**(3), 528–534 (2009). [CrossRef]

**6. **Y. Kitagawa, Y. Suzuki, T. Saruya, K. Tezuka, N. Kanbara, and N. Nishiyama, “1.6um to 1.7um micromachined tunable VCSELs,” (2016), pp. 1–5.

**7. **T. Watanabe, T. Hirata, N. Kanbara, N. Fujimura, T. Yano, S. Tezuka, H. Saitou, M. Ooyama, and R. Noda, “Tunable Laser Using Silicon MEMS Mirror,” IEEJ Trans. SM **130**(5), 176–181 (2010). [CrossRef]

**8. **J. V. Roey, J. van der Donk, and P. E. Lagasse, “Beam-propagation method: analysis and assessment,” J. Opt. Soc. Am. **71**(7), 803–810 (1981). [CrossRef]

**9. **L. Thylen and D. Yevick, “Beam propagation method in anisotropic media,” Appl. Opt. **21**(15), 2751–2754 (1982). [CrossRef]

**10. **D. Yevick and L. Thylén, “Analysis of gratings by the beam-propagation method,” J. Opt. Soc. Am. **72**(8), 1084–1089 (1982). [CrossRef]

**11. **D. Yevick, P. Meissner, and E. Patzak, “Modal analysis of inhomogeneous optical resonators,” Appl. Opt. **23**(13), 2127–2133 (1984). [CrossRef]

**12. **F. A. Jenkins and H. E. White, * Fundamentals of optics* (McGraw-Hill, 1957).

**13. **M. Born and E. Wolf, * Principles of optics* (Cambridge University, 2000).

**14. **K. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. **14**(3), 302–307 (1966). [CrossRef]

**15. **A. Taflove, “Application of the Finite-Difference Time-Domain Method to Sinusoidal Steady-State Electromagnetic-Penetration Problems,” IEEE Trans. Electromagn. Compat. **EMC-22**(3), 191–202 (1980). [CrossRef]

**16. **A. G. Fox and T. Li, “Resonant Modes in a Maser Interferometer,” Bell Syst. Tech. J. **40**(2), 453–488 (1961). [CrossRef]

**17. **A. Fox and T. Li, “Modes in a maser interferometer with curved and tilted mirrors,” Proc. IEEE **51**(1), 80–89 (1963). [CrossRef]

**18. **S. Tezuka and N. Kanbara, “Stability diagram of higher modes of the Fabry-Perot resonator,” Opt. Rev. **15**(1), 1–5 (2008). [CrossRef]

**19. **S. Tezuka, “The diagonalization of the 3D Fox-Li integral equation with the gaussian quadrature formula of very high order,” (2010), pp. 20PSa–11.

**20. **Y. Suzuki and S. Tezuka, “Numerical simulation of 3D Fox-Li integral equation described by Rayleigh-Sommerfeld diffraction for MEMS-VCSEL,” Opt. Rev. **26**(5), 430–435 (2019). [CrossRef]

**21. **W. W. Rigrod, “Diffraction loss of stable optical resonators with internal limiting apertures,” IEEE J. Quantum Electron. **19**(11), 1679–1685 (1983). [CrossRef]

**22. **Y. Kitagawa, Y. Suzuki, and S. Tezuka, “Mathematical Shape Evaluation of a Concave MEMS Mirror,” IEEJ Trans. SM **140**(5), 109–112 (2020). [CrossRef]