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A thin-film grating on a curved substrate functions as a highly reflective and wavelength sensitive mirror for a diverging wave that has the same curvature as the substrate. In this paper we propose a cylindrical cavity surrounded by a curved resonant grating wall, and describe its resonance characteristics. Through finite-difference time-domain (FDTD) simulation we have clarified that this type of cavity supports two resonance modes: one is confined by Fresnel reflection and the other by resonance reflection of the wall. We have also demonstrated that the latter mode exhibits a Q factor several orders of magnitude higher than that of the former mode.

The cylindrical cavity is one of the most fundamental types of resonant cavities, and has been used to construct, for example, wavelength filters for microwaves [

Similar to other types of resonators, the quality factor (Q) of a cylindrical cavity, which is a basic characteristic, depends on the performance of the cavity walls. External Q increases with wall reflectivity, provided that the wall is lossless. The full width at half maximum (FWHM) of the resonance peak becomes narrow, while the free spectral range (FSR) widens as the cavity radius increases. This dependence is the same as conventional Fabry-Perot etalons which utilize a pair of flat mirrors. Various trials have been conducted in recent years aiming to control the resonance characteristics of circular cavities. One of the most straightforward approaches is to add sub-wavelength scale index perturbation to the walls. For example, the application of periodic index modulation along the radial [

Guided-mode resonance (GMR) is another useful phenomenon for controlling the reflection spectrum at material boundaries [

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

Recently, attempts have been made to extend the GMR concept to curved substrate structures [

Before calculating the characteristics of the cavity, we first analyzed the reflection property of the wall part. A schematic diagram of the structure is shown in _{z}_{θ}_{ρ}

Detail of the sample structures. (_{0}, 2_{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

Let the distance between the center of curvature and the average height of the grating be _{0}. The depth and average thickness of the grating layer are 2_{0}, respectively. The index of the grating layer, _{2} = 2.63, is set close to the effective index of the fundamental TM mode of the InP/air disk structure [_{1} = 1.45 and _{3} = 1.0, respectively. The local thickness of the grating layer,

where _{0} = 0.68Λ and

These parameters were determined so that the lowest-order GMR wavelength is in the valley region of the reflection spectrum (spectra are shown in

The angle _{0} denotes the azimuthal extent of a unit grating, and is expressed as _{0} = 2π/

Reflection spectra viewed from inside the curved grating. “

_{4}, is defined as:

The reflectivity for a cylindrical wave launched from inside the curvature (_{0} − _{0}) is calculated by a cylindrical coordinate version of the two-dimensional FDTD method [

A concentric cylindrical wave of a center wavelength of λ_{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 _{z}_{θ}

The relationships between GMR wavelength, bandwidth and curvature are summarized in

We also confirmed that the peak reflectivity at the GMR wavelength remains at almost 100% for structures with

Relationships between the curvature of the grating, GMR wavelength and 1-dB bandwidth.

As we will show in the next section, resonance wavelengths of a RG-type cylindrical cavity exhibit strange behavior near the GMR wavelength of the RG wall. We thought that the key physics behind such behavior would be a rapid phase shift of the RG upon reflection. Because the phase of the reflection is directly connected with the effective reflection plane where the field amplitude becomes either minimum or maximum, it is convenient to regard RGs as a uniform medium with an effective reflection position. By substituting an effective cavity radius, which is a summation of the nominal cavity radius and the penetration depth of the effective reflection plane, into the equation for the conventional resonance condition, we will be able to obtain resonance wavelengths for the curved RG-type cavity. Also, through such calculation we will be able to explain the strange behavior of the resonance wavelength of the cavity. Note that outside GMR, the RG shows finite transmission. Therefore it is more reasonable to regard the RG as a high-index or low-index medium rather than a metallic mirror. The principle of calculating the effective reflection positions of curved RGs as a high-index medium is as follows.

The electric field inside the curvature can be well approximated as a superposition of diverging and converging waves as follows, except for the region very close to the grating layer:

where _{c}_{d}_{0} and

We define the effective mirror plane as the radial position beyond which the space can be regarded as an infinitely extending high-index medium. The amplitude reflectivity of the electric field on such a plane

where A and |

Amplitude factors can be expressed using Equations (3) to (5) as follows:

We tried to calculate the relationship between the wavelength and the effective mirror position (_{eff}_{c}_{d}_{eff}

Example of a calculated field intensity profile upon reflection (solid line) and its extrapolated curve (dotted line), for the

Estimated effective high-index mirror positions, for the

The relationships between the structural parameters of the cavity (such as radius and refractive index) and the resonance wavelength are fundamental information for the resonator. We calculated the resonance wavelengths of cylindrical cavities surrounded by the RG walls with various curvature radii. The domain for FDTD simulation was extended to the center of the cavity.

We placed a point excitation source _{z}_{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.

The above calculation was carried out for various cavity radii, and the results are plotted as a function of

Calculated resonance wavelengths for various cavity sizes. The dashed line indicates the GMR wavelength of the curved RG wall. (_{eff}

The resonance condition for TM modes of a hollow circular metallic cavity is given as zeros of a Bessel function as follows [

where

Next, Q-factors of the resonance series marked by A-A’ in

where _{0} and _{0} are the angular frequency and the decay time, respectively. The results are plotted as filled circles in _{0}) of the unit grating. In

Quality factor of the resonance series indicated by A-A’ in

Electric field distributions of a Fresnel mode (point “F” in

(

As we showed in

We clarified the resonance characteristics of cylindrical resonant cavities surrounded by a curved grating wall through numerical simulation. The cavity was found to support two kinds of resonant modes, whose Q are determined by the Fresnel and GMR reflection constant at the cavity/wall boundary, respectively. We also showed that, in the example structure, the quality factor of the latter modes can be increased by several orders of magnitude compared with the former modes. The proposed cavity was found to function as a large-area, effectively single-mode cavity. This type of structure will be a useful platform for realizing micro-photonic devices such as single-mode lasers, high-finesse wavelength filters and biosensors. Having clarified the fundamental characteristics, we will investigate as our next study the optimum structural design which is more robust against fabrication errors of size, shape, and refractive index.

This research was in part supported by KAKENHI, Grant-in-Aid for Scientific Research (B) 23360145 of Japan Society for the Promotion of Science (JSPS).