# Cylindrical Resonator Utilizing a Curved Resonant Grating as a Cavity Wall

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## Abstract

**:**

## 1. Introduction

**Figure 1.**Schematic view of part of a flat/curved resonant grating and a circular cavity. (

**a**) Conventional thin-film grating with a sinusoidal profile on a flat substrate; (

**b**) Curved structure; (

**c**) Cylindrical cavity with a curved grating wall.

## 2. Characteristics of a Curved RG

#### 2.1. Structure and Method of Calculation

_{z}, H

_{θ}, and H

_{ρ}).

**Figure 2.**Detail of the sample structures. (

**a**) Curved resonant grating (proposed structure). ρ

_{0}, 2a and d

_{0}are the average radial position, depth and base thickness of the grating, respectively. The region surrounded by a solid line indicates the computational domain for FDTD. PBC and RBC stand for Periodic and Radiation Boundary Conditions, respectively; (

**b**) Reference curved slab structure. The grating region is replaced by an average refractive index layer.

_{0}. The depth and average thickness of the grating layer are 2a and d

_{0}, respectively. The index of the grating layer, n

_{2}= 2.63, is set close to the effective index of the fundamental TM mode of the InP/air disk structure [24]. The refractive indexes of the inner and outer spaces are n

_{1}= 1.45 and n

_{3}= 1.0, respectively. The local thickness of the grating layer, d(θ), is defined as:

_{0}= 0.68Λ and a = 0.25Λ.

_{0}denotes the azimuthal extent of a unit grating, and is expressed as θ

_{0}= 2π/m [rad] where m is the total number of gratings on a wall edge. The pitch of the grating, Λ, is defined as the length of an arc at the average grating height, .

**Figure 3.**Reflection spectra viewed from inside the curved grating. “m” is the number of gratings on the outer edge of the cavity wall. The broken line shows the spectrum of the average index structure of Figure 2(b) for m = 50.

_{4}, is defined as:

_{0}− d

_{0}) is calculated by a cylindrical coordinate version of the two-dimensional FDTD method [25]. The computational domain is surrounded by a solid line in Figure 2(a). The angular direction is limited by one grating period. Both sides are connected by periodic boundary conditions (PBCs). The radial ends are terminated by radiation boundary conditions (RBCs) [26] to absorb inward converging and outward diverging waves.

#### 2.2. Reflection Spectra

_{0}= 2Λ with a temporal Gaussian envelope was excited near the inner RBC. Because there is insufficient space between the excitation position and the inner surface of the RG, it is difficult to temporally separate the excited and reflected waves. Therefore we tried to estimate the reflectivity using the waveform monitored outside the RG. The recorded time-domain waveform of E

_{z}and H

_{θ}are Fourier transformed and then multiplied to obtain the radial component of the Poynting vector, . A similar calculation is carried out for a uniform space of n = 1.45 to obtain a reference Poynting vector . The reflectivity of RG is then evaluated by . The results for various curvature radii are plotted in Figure 3. It is clearly seen that the spectra are the superposition of average structure reflectivity (Figure 2(b)) and a sharp resonance peak.

#### 2.3. Effective High-Index Mirror Position

_{c}and A

_{d}are their complex amplitudes. k

_{0}and ρ are the free space wave number and the radial coordinate, respectively.

_{eff}). First, we calculated the field distribution at a wavelength of interest using the FDTD method. An example for the m = 50 structure is shown by the solid line in Figure 5. Then the data of ρ/Λ < 6 was least-squares fitted to Equation (3) to obtain A

_{c}and A

_{d}. The fitted field is shown by the dotted line in the figure. Finally, |r| and ρ

_{eff}were calculated using Equation (6). The inset of Figure 5 is a magnified view of the fitted field near the grating layer. The calculated relationships between wavelength and effective mirror position for RGs and average index structures are summarized in Figure 6. This result clarifies that the mirror positions for both structures behave similarly far from the GMR wavelength. For the average layer structure, as the wavelength moves across the minimum reflection point, the mirror position slowly shifts by about a quarter wavelength; this phenomenon can be commonly seen in a flat dielectric film. On the other hand, for a curved RG this shift is more abrupt: the mirror position appears almost to jump at the GMR wavelength. This can be interpreted to mean that the complex reflectivity of the RG follows a sharp Lorentzian function near the resonance wavelength as described by Fan et al. [28].

**Figure 5.**Example of a calculated field intensity profile upon reflection (solid line) and its extrapolated curve (dotted line), for the m = 50 structure. The latter is a linear summation of and .

**Figure 6.**Estimated effective high-index mirror positions, for the m = 50 structure. Solid and dashed lines are curved RG and average index slab, respectively.

## 3. Characteristics of RGC

#### 3.1. Resonance Wavelength of the Cavity Mode

_{z}at the center, and monitored the electromagnetic fields at two pitches outside the grating (ρ = ρ

_{0}+ 2Λ). Peak wavelengths contained in the monitored waveform were extracted using Fast Fourier Transformation (FFT). These wavelengths correspond to the resonance wavelengths of the cavity system.

**Figure 7.**Calculated resonance wavelengths for various cavity sizes. The dashed line indicates the GMR wavelength of the curved RG wall. (

**a**) Determined from the peaks of radiation spectra monitored outside the cavity; (

**b**) Estimated by , where ρ

_{eff}is the radial position of the effective high-index mirror.

#### 3.2. Quality Factor

_{0}and τ

_{0}are the angular frequency and the decay time, respectively. The results are plotted as filled circles in Figure 8. Practically, the total number of gratings on the cavity wall (m) must be an integer number. However, according to our FDTD method, non-integer m can be handled without difficulty, because we need to specify only the azimuthal extent (θ

_{0}) of the unit grating. In Figure 8, the results for non-integer m structures are also plotted as a dotted line.

**Figure 8.**Quality factor of the resonance series indicated by A-A’ in Figure 7(a).

## 4. Conclusions

## Acknowledgment

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**MDPI and ACS Style**

Ohtera, Y.; Iijima, S.; Yamada, H.
Cylindrical Resonator Utilizing a Curved Resonant Grating as a Cavity Wall. *Micromachines* **2012**, *3*, 101-113.
https://doi.org/10.3390/mi3010101

**AMA Style**

Ohtera Y, Iijima S, Yamada H.
Cylindrical Resonator Utilizing a Curved Resonant Grating as a Cavity Wall. *Micromachines*. 2012; 3(1):101-113.
https://doi.org/10.3390/mi3010101

**Chicago/Turabian Style**

Ohtera, Yasuo, Shohei Iijima, and Hirohito Yamada.
2012. "Cylindrical Resonator Utilizing a Curved Resonant Grating as a Cavity Wall" *Micromachines* 3, no. 1: 101-113.
https://doi.org/10.3390/mi3010101