# High-Q Defect-Free 2D Photonic Crystal Cavity from Random Localised Disorder

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## Abstract

**:**

## 1. Introduction

## 2. Theory and Method

_{Si}= 3.4) PhC slab with a hexagonal array of cylindrical air holes which extends for 36 periods both in the x- and y-directions (in-plane). The structure has a periodicity of a = 0.449 μm, slab thickness of h = 0.6a, and the radius of the cylindrical airholes is R = 0.25a. The plane positions of the 19 innermost airholes are all randomly perturbed. A schematic of the structure is shown in Figure 1a.

**Figure 1.**(

**a**) Schematic of the hexagonal array 2D silicon (grey) photonic crystal (PhC) slab with the dashed circle showing the boundary inside which the position of the cylindrical airholes are randomly perturbed. The origin, i.e., (x,y) = (0,0), is measured from the centre of the unperturbed blue airhole. Note: the centre airhole is perturbed in the figure; (

**b**) Normalised frequency u of high-Q modes found for various disorder amplitudes, δ, (solid lines) relative to the air-band edge of the cavity. An arbitrarily shaped local potential well U

_{l}(δ) created from the local refractive index distribution governed by its dependence on δ and a random function (see Equation (1)). The positions of the airholes are indicated by the colours blue (C—centre), green (F—first ring) and red (S—second ring) as a function of the x-position of the airholes. In some regions of the 2D PhC slab, the local refractive index either increases (below the band gap), Δn

_{+}, or decreases (above the band gap), Δn

_{−}, depending on the random positions of the airholes. In cases where the refractive index is increased, it is possible to obtain localised states inside the bandgap as indicated by the modes with disorder amplitudes of δ = 0.1, 0.2 and 0.3.

_{i}is the unperturbed position of the ith airhole in either of the in-plane coordinates (see Figure 1a) of the random region, δ is the amplitude of randomness which controls the minimum and maximum value of the shift, and f

_{ran}returns a random uniformly distributed number that lies within the range [−0.5,0.5]. Note, results shown below refer to a single realisation of f

_{ran}only. We have also considered different f

_{ran}functions, i.e., ensemble behaviour, and the same trends in frequencies, total Qs, modal volume and mode profiles were also observed for each instance f

_{ran}.

_{l}(δ) for Figure 1b, where U

_{l}is the local potential well induced that traps the cavity mode induced by the disorder. The variation is not uniform; it is dependent on the position of the airholes which can be either negative or positive. The modes shown in Figure 1b exist in regions with the overall increased refractive index, which redistributes modes locally deeper into the bandgap.

_{||}and out-of-plane Q

_{⊥}with respect to the structure’s orientation (in-plane coordinates: xy-plane; out-of-plane direction: z-axis) and are related via 1/Q = 1/Q

_{||}+ 1/Q

_{⊥}. The in-plane term is further related to the partial Qs in each respective Cartesian direction: 1/Q

_{||}= 1/Q

_{x}+ 1/Q

_{y}[30].

## 3. Results and Discussion

_{δ=0.1}= 0.295, u

_{δ=0.2}= 0.293 and u

_{δ=0.3}= 0.290. Increasing δ causes the modes to shift down in frequency away from the air band-edge and towards the centre of the PBG, which in turn corresponds to a mode which is more strongly confined in-plane. These modes are localised due to a local potential well being created and its functional form depends on δ and on the realization of the random process. Our calculations recover the frequency, the total and partial Qs and V of the disordered cavity modes. These results are given in the next section.

#### 3.1. Total, Partial Qs and Purcell Effect

_{||}

_{.}

_{||}(crosses) and out-of-plane Q

_{⊥}(squares) as a function of δ. It can be seen that both partial Qs behave very differently, Q

_{||}increases with random disorder, and conversely, Q

_{⊥}decreases. The modal volumes were also calculated for the range of δ = 0.1 and 0.3 cavities, where they were found to vary between 2.03(λ/n)

^{3}and 0.93(λ/n)

^{3}respectively (see Table 1). As mentioned in Section 2 the same trends in total and partial Qs were also obtained for different f

_{ran}. We have tested four different random functions.

**Table 1.**Total, partial Qs (Q

_{⊥},Q

_{||}), V and F of the cavity modes as a function of disorder amplitude δ. The modal volume for δ = 0.05 could not be determined because the computational domain was too large.

δ | Q_{⊥} (×10^{4}) | Q_{||} (×10^{4}) | Q (×10^{4}) | V (λ/n)^{3} | F |
---|---|---|---|---|---|

0.05 | 20.2 | 0.45 | 0.44 | – | – |

0.1 | 12 | 1.9 | 1.65 | 2.03 | 618 |

0.15 | 3.77 | 2.5 | 1.5 | 1.6 | 712 |

0.2 | 1.5 | 10.3 | 1.21 | 1.20 | 766 |

0.25 | 1.1 | 23.8 | 1.1 | 1.02 | 820 |

0.3 | 0.66 | 38.2 | 0.65 | 0.93 | 531 |

**Figure 2.**In-plane (crosses) and out-of-plane (square) partial Qs of the cavity as a function of disorder amplitude δ.

#### 3.2. Mode Profiles

_{x}and E

_{y}, and their Fourier transforms, of the in-plane coordinates for the disorder amplitudes δ = 0.1, 0.2 and 0.3 are shown in Figure 3. Figure 3a,c show the changes in-plane confinement for each respective mode given their associated disorder amplitude and it can be seen that the confinement around the central region of cavity improves as the disorder increases, subsequently V decreases. This is confirmed quantitatively since Q

_{||}from Figure 2 varies from 1.90 × 10

^{4}to 3.82 × 10

^{5}for increasing disorder. Another indicator is the k-space distribution of Figure 3b,d which shows that the fraction of the field that falls within the leaky light-cone is higher for larger δ, confirming the behaviour seen in Figure 2 [30].

**Figure 3.**The major electric field components in the middle of the slab for (

**a**) E

_{x}and its associated; (

**b**) Fourier transform. Similarly; the (

**c**) E

_{y}major field component; and its (

**d**) Fourier transform (circle represents the light). The top, middle and bottom rows correspond to disorder amplitudes δ = 0.1, 0.2 and 0.3 respectively.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Chung, K.; Karle, T.J.; Rajasekharan, R.; De Sterke, C.M.; Tomljenovic-Hanic, S.
High-*Q* Defect-Free 2D Photonic Crystal Cavity from Random Localised Disorder. *Crystals* **2014**, *4*, 342-350.
https://doi.org/10.3390/cryst4030342

**AMA Style**

Chung K, Karle TJ, Rajasekharan R, De Sterke CM, Tomljenovic-Hanic S.
High-*Q* Defect-Free 2D Photonic Crystal Cavity from Random Localised Disorder. *Crystals*. 2014; 4(3):342-350.
https://doi.org/10.3390/cryst4030342

**Chicago/Turabian Style**

Chung, Kelvin, Timothy J. Karle, Ranjith Rajasekharan, C. Martijn De Sterke, and Snjezana Tomljenovic-Hanic.
2014. "High-*Q* Defect-Free 2D Photonic Crystal Cavity from Random Localised Disorder" *Crystals* 4, no. 3: 342-350.
https://doi.org/10.3390/cryst4030342