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 Figure 2(a). One of our future targets is to design three-dimensional structures like a micro-disk resonator on a planar integrated light circuit. However, in this study, we simplified the structure as a two-dimensional one using the effective index method [23], to identify the main resonance characteristics of the curved RG-type resonator. Refractive index and electromagnetic fields were assumed to be uniform in the z (normal to the paper) direction of the figure. Throughout this paper we consider only TM modes (E_z, H_θ, and H_ρ).

Let the distance between the center of curvature and the average height of the grating be ρ₀. The depth and average thickness of the grating layer are 2a and d₀, respectively. The index of the grating layer, n₂ = 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 n₁ = 1.45 and n₃ = 1.0, respectively. The local thickness of the grating layer, d(θ), is defined as:

   (1)

where d₀ = 0.68Λ and a = 0.25Λ.

These parameters were determined so that the lowest-order GMR wavelength is in the valley region of the reflection spectrum (spectra are shown in Figure 3).

The angle θ₀ denotes the azimuthal extent of a unit grating, and is expressed as θ₀ = 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 2(b) shows a simple cylindrical structure where the grating region is replaced by a mean refractive index layer. This structure was also analyzed to see the resonance characteristics where the grating does not exhibit GMR. The average index of the replaced region, n₄, is defined as:

   (2)

The reflectivity for a cylindrical wave launched from inside the curvature (ρ < ρ₀ − d₀) 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.

A concentric cylindrical wave of a center wavelength of λ₀ = 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.

The relationships between GMR wavelength, bandwidth and curvature are summarized in Figure 4, which shows that the resonance wavelength depends almost linearly on the curvature. This can be explained as follows: the modal field of a curved waveguide shifts to the outer side as the curvature increases [27]. In the structure of Figure 2(a), the average refractive index outside the grating layer is low, and so the mean optical path length for the guided mode becomes small as the curvature increases. This tends to shorten the resonance wavelength.

We also confirmed that the peak reflectivity at the GMR wavelength remains at almost 100% for structures with m > 20.

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:

   (3)

where and denote the 0th-order Hankel function of the first and second kind, and correspond to converging and diverging waves, respectively. A_c and A_d are their complex amplitudes. k₀ and ρ are the free space wave number and the radial coordinate, respectively.

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 should therefore be a negative real number. The electric field at such a plane can thus be expressed as:

   (4)

where A and |r| are another complex amplitude and the norm of the amplitude reflectance, respectively. Considering Equation (3), the field inside the curvature can be expressed as follows.

   (5)

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

,    (6)

We tried to calculate the relationship between the wavelength and the effective mirror position (ρ_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].

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 E_z at the center, and monitored the electromagnetic fields at two pitches outside the grating (ρ = ρ₀ + 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 m in Figure 7(a). This figure shows that there are roughly two kinds of resonance series: (a) an almost linear series extending from the top right to bottom left; and (b) A wavy series located around 2.1 < λ/Λ < 2.2. The former is the resonance modes where RG almost serves as a mean dielectric structure as in Figure 2(b). The resonance wavelength and Q are determined by the Fresnel reflection constant of the multilayer, so we call these modes “Fresnel modes” hereafter. The latter series are dependent on the GMR spectrum of the RG, and we call them “Grating modes” for convenience.

The resonance condition for TM modes of a hollow circular metallic cavity is given as zeros of a Bessel function as follows [29]:

   (7)

where is the radius of the cavity. We tried to check whether or not this condition also applies to our RG-type resonator. We replaced the physical radius ( ) by the effective one () and plotted the wavelengths which satisfy the above relationship for various cavity sizes (m). The effective position, , for various curvatures was calculated by the procedure described in Section 2.3. The result is shown in Figure 7(b). These results look similar for both Fresnel and Grating modes. It is clear that the deformation of the resonance curve near the GMR wavelength corresponds to the behavior of seen in Figure 6. This indicates that the resonance wavelength of the proposed cavity is also governed by the conventional resonance condition of Equation (7), regardless of the confinement mechanism (Fresnel reflection or resonant reflection) of the modes.

Next, Q-factors of the resonance series marked by A-A’ in Figure 7(a) are evaluated using the following equation [30]:

   (8)

where ω₀ and τ₀ 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 (θ₀) of the unit grating. In Figure 8, the results for non-integer m structures are also plotted as a dotted line.

Figure 8 shows that Q of the grating mode (52 < m < 53) becomes extremely high, corresponding to the high reflectivity of the RG wall. Q of other Fresnel modes is less than 100. In other words, only one high-Q resonant mode exists in a wide range of wavelength, which means that this RG type cavity effectively behaves as a single-mode resonator.

Electric field distributions of a Fresnel mode (point “F” in Figure 7) and high-Q grating mode (point “G” in Figure 7 and Figure 8) are displayed in Figure 9(a,b), respectively. In Figure 9(a), the standing wave ratio (SWR) inside the cavity looks small due to the low reflectivity of the cavity wall. The wavelength is far from GMR, and therefore the field intensity in the grating layer is of the same order as that inside the cavity. In addition, substantial field leakage is seen outside. On the contrary, in Figure 9(b) the field intensity in the grating layer is enhanced due to the guided mode resonance. This characteristic is well known for RGs [31]. Also, the high reflectivity of the wall keeps the SWR almost at infinity. There is almost no field leakage outside the cavity.

As we showed in Figure 3, the basic characteristics of our curved RG mirror are similar to those of the conventional flat RG, and so the tolerance to changes in various structural or material parameters will be approximately the same. The cavity characteristics (tolerance for achieving high-Q) will also have a similar level of tolerance.