# Parameter-tolerant design of high contrast gratings

## Summary (3 min read)

### 1. INTRODUCTION

- High contrast gratings (HCG) are usually 1D photonic structures made of two materials with a large index step (∼ 2).
- The coupling between the propagating modes and the outside waves as well as the resonances of the modes between the two slab interfaces can be tailored by adjusting the grating dimensions and the index contrast to give a wide range of optical properties.
- 13 HCG mirrors have originally been developed for VCSEL devices : with only two subwavelength layers, they can surpass Bragg reflectors performances in term of high reflectivity.
- The tolerance with respect to the errors of fabrication is an important aspect and has to be taken into account during the design of the structure to achieve a robust and efficient mirror.

### 2. HCG-VCSEL FABRICATION

- The device is composed of five GaInAsSb quantum wells in a cavity composed by a Bragg mirror made from 24 pairs of AlAsSb/GaSb20 and high contrast grating on top .
- To enhance the device performances, an electro-optic confinement solution has been introduce with the use of an oxide aperture.
- Since oxidation of the antimonide alloy (AlSb) has not returned encouraging results for its application as a confinement layer, a metamorphic approach with oxidation of AlGaAs layer is used.
- 21 A GaAs layer is thus grown on top of the Sb-based cavity of the VCSEL and the HCG can be obtained thanks to the high contrast between AlOx low index material (n ∼1.7) with low absorption22 and high index of GaAs (n ∼ 3.3).
- This technological approach presents the advantage of using mature industrial process such as reactive ion etching (RIE) for the grating fabrication and wet oxidation for the AlOx layer.

### 3. VCSEL MIRROR REQUIREMENTS

- Compared to other semiconductor lasers, VCSEL devices have a very thin gain region and require high Q cavities made from high reflectivity mirrors higher than 99.5% with typical bandwidth of 150 nm in the mid infrared.
- Besides the mirror effect, a polarization selectivity is wanted to prevent polarization instabilities of the emitted beam which is an well known problem in VCSEL structures due to their circular symmetry.
- The high reflectivity of 99.5% is chosen for the transverse magnetic (TM) polarization which is perpendicular to the grating slabs.
- The transverse electric polarization (TE) is kept lower than 90% for the whole high reflectivity bandwidth to prevent any lasing for this polarization mode.

### 4. TECHNOLOGICAL CONSTRAINTS

- The fabrication of the HCG structure is constrained by technological considerations.
- The aluminum oxide layer thickness TA is limited between 300 and 400 nm around the optimum 23 (2k−1)λ/4 to prevent optical losses and delamination during the wet oxidation process.
- The fill factor FF of the grating is limited to values between 35% and 55% by the e-beam lithography process and the RIE process used to ensure a well defined square shape pattern of the grating slabs.
- Besides boundaries on the HCG dimensions, tolerances on the structure parameters have to be taken into account to ensure that the design is feasible and keep high performances despite the pitfalls of the manufacturing process.
- The etching of the grating slab is the most critical step in the process flow since it is done without the use of a stop layer.

### 5. ROBUST OPTIMIZATION ALGORITHM

- The HCG design can be done with a theoretical analysis of the Bloch mode resonances inside the grating,2 but the authors chose here to solve an inverse problem by computing the reflectivity of the mirror with a numerical analysis and adjust the grating parameters to increase its performances.
- This method allows to design easily more complex structures which are not ideal HCG gratings and include the previously defined VCSEL requirements during the design process.

### 5.1 The optimization algorithm

- An optimization algorithm has been used to automate the search of the most efficient mirror design in a same way that it can be done manually with a study of the evolution of the reflectivity of the mirror with respect to its dimensions.
- The normalized bandwidth is multiplied by a Gaussian weighted average of the transverse magnetic reflection coefficients RTM of the bandwidth to ensure the centering around λ0.
- Several optimization algorithm are available to search for the maximum of a function and the choice of the best algorithm is done with respect to the function characteristics.
- PSO algorithm is based on a set of particles which are potential candidates sharing their knowledge of optimum positions when exploring the search space.
- Secondly, a local velocity term moves the particle toward the local best position xlp known by the particle p.

### 5.2 The anti-optimization algorithm

- Even if the optimization algorithm succeed in finding efficient HCG mirrors, a detailed study of the optimum found by the algorithm can show that this point is not realistic from a fabrication point of view due to its very low tolerance.
- The vicinity of the point is defined in their case as the space ∆X = {∆Tg,∆FF,∆TA,∆Λ,∆TL} which represents the estimation of the errors that can be made during the fabrication process.
- Thus, the minimum value of the MF function in the space X ±∆X corresponds to the worst case scenario of fabrication.
- First, a memoization method has been introduced to store in memory the results of the MF (X) function for each point X met during the optimization.
- By evaluating the robustness of only the best solutions during the optimization, the PSO algorithm has to compare solutions which have been tested in tolerance and solutions which have not yet been.

### 6. RESULTING DESIGNS

- The execution of the robust optimization algorithm has been done to adjust the parameters of the GaAs/AlOx grating structure defined previously.
- The boundary of the search space and tolerance requirements on the structure dimensions are summarized in Table 1 and come either from technological limitations as described previously for AlOx thickness TA and fill factor FF or chosen arbitrarily from empirical rules.
- 24 The resulting mirror which dimensions are given in Table 2 exhibits a very high reflectivity and polarization selective bandwidth of 425 nm .
- The tolerance evaluation of the parameters shows a good robustness with an optimum value well centered within large variation ranges.

### 7. CONCLUSION

- An original robust optimization algorithm for high contrast grating design has been presented in this work which searches not only for the best solution but takes also into account the accuracy of the manufacturing technology.
- The algorithm has shown its ability to design HCG mirrors for a VCSEL application at 2.3 µm while taking into account user-defined technological constraints.
- The execution of the robust optimization algorithm returns not only efficient mirrors with 99.5% high reflectivity for 425nm large bandwidth but also fulfills the robustness requirements with large tolerance values of more than 10% on the most critical parameters.
- This design method can be quickly adapted to design other high contrast grating structures for other manufacturing conditions by only adapting the figure of merit and the technological constraints.

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