^{1}, H. T. Johnson

^{1,a)}and Kent D. Choquette

^{2}

### Abstract

Using nonlinear programing and the geometry projection method, the quality factor of the monopole mode of a single defectphotonic crystal lasercavity is improved from 38 000 to 87 000. Beginning with a design that considers only round air holes shifted away from the cavity, the radius of the nearest neighbor and of the surrounding air holes are optimized while satisfying a constraint on the resonant frequency. The total reflectivity of the photonic crystal laserstructure is then defined, and it is shown that this quantity correlates strongly to the total quality factor. The reflectivity of the structure is improved by altering the shape of the holes immediately surrounding the cavity, thus leading to an improvement in quality factor. The geometry projection method is used to define the shape of the holes and the finite element and adjoint methods are used to compute the objective function and sensitivities required by the optimizer. This work demonstrates one way to optimize the factor of a photonic crystal laser by altering the hole shape.

The support of NSF Grant No. ECS-05-08473 is gratefully acknowledged.

INTRODUCTION

COMPUTING THE FACTOR

FINDING AN INITIAL CONDITION FOR THE OPTIMIZATION

CORRELATING FACTOR WITH REFLECTIVITY

OPTIMIZING FACTOR VIA THE GPM

SUMMARY

### Key Topics

- Reflectivity
- 37.0
- Laser resonators
- 14.0
- Finite element methods
- 12.0
- Crystal defects
- 8.0
- Photonic crystal lasers
- 6.0

## Figures

The finite element model used to calculate the factor. Only four rows of holes are shown here for clarity; six rows of holes are used in all calculations.

The finite element model used to calculate the factor. Only four rows of holes are shown here for clarity; six rows of holes are used in all calculations.

The factor as a function of nearest neighbor hole radius for various choices of surrounding hole radius . The parameter search is refined via the interval bisection near the peak. The thickness of the slab is adjusted such that the computed resonant frequency satisfies the constraint .

The factor as a function of nearest neighbor hole radius for various choices of surrounding hole radius . The parameter search is refined via the interval bisection near the peak. The thickness of the slab is adjusted such that the computed resonant frequency satisfies the constraint .

The maximum factor as a function of surrounding hole radius . The nearest neighbor hole radius at each point is shown in Fig. 2. The function does not have a sharp peak; a peak factor of over 40 000 is found for .

The maximum factor as a function of surrounding hole radius . The nearest neighbor hole radius at each point is shown in Fig. 2. The function does not have a sharp peak; a peak factor of over 40 000 is found for .

The finite element model used to calculate the total reflectivity. The model has the same boundary conditions and material properties as the model used to calculate the factor, but it has a PML region in the center, a line of current around the PML.

The finite element model used to calculate the total reflectivity. The model has the same boundary conditions and material properties as the model used to calculate the factor, but it has a PML region in the center, a line of current around the PML.

The factor and total structural reflectivity as a function of nearest neighbor holes for a structure with surrounding hole radius of . The structural reflectivity and factor exhibit the same trends.

The factor and total structural reflectivity as a function of nearest neighbor holes for a structure with surrounding hole radius of . The structural reflectivity and factor exhibit the same trends.

The finite element model used for reflectivity optimization (left) and the mesh used (right). Some elements are removed for clarity.

The finite element model used for reflectivity optimization (left) and the mesh used (right). Some elements are removed for clarity.

The magnitude of the fields of the (left) unoptimized and (right) optimized designs.

The magnitude of the fields of the (left) unoptimized and (right) optimized designs.

The improvement in , , and during the optimization. The improvement in drives the improvement in .

The improvement in , , and during the optimization. The improvement in drives the improvement in .

The intermediate designs during the optimization. The initial nearest neighbor hole shape is shown in gray.

The intermediate designs during the optimization. The initial nearest neighbor hole shape is shown in gray.

The improvement in factor and reflectivity per optimization iteration. The two curves approximately follow each other.

The improvement in factor and reflectivity per optimization iteration. The two curves approximately follow each other.

Effective mode volume and Purcell factor of the lasing cavity during the optimization.

Effective mode volume and Purcell factor of the lasing cavity during the optimization.

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