Figure 1

(Color online) (a) Example of PhC membrane $L_N$ cavity $\left(N=11\right)$. The PhC parameters used in the computation of the cavity modes $\left(a=198\text{nm},r/a=0.26\right)$ are indicated. The same parameters were used for the PhC W1 waveguide shown in (b). (c) Waveguide dispersion as obtained from 2DFD computations. The bands that are spatially confined in the waveguide are shown in gray (red) (odd modes) and black (blue) (even modes). (d) Real part of the $E_y$ component of four eigenmodes belonging to the odd waveguide band shown in (c). The modes were randomly selected around the minimum of the band, in $k_x=\pi /a$. The corresponding $k_x$ values are indicated.Reuse & Permissions

Figure 2

(Color online) (a) Absolute value of ${\phantom{|}\text{FT}\left\{{\mathcal{E}}_y^{m,N}\left(x\right)\right\}|}_{k_x}$ for the fundamental mode $\left({\text{M}}_0\right)$ of the $L_5$ cavity (the function is symmetric with respect to $k_x=0$). The spatial distribution of the $E_y$ component of this mode (computed by 2DFD) is shown in the inset. (b) Absolute value of ${\mathcal{W}}_G^{\text{w}{k}_x}$ (see definition in the text). The box delimits the range of values of ${\mathcal{W}}_G^{\text{w}{k}_x}$ $\left(G∊\left[-4\cdot \frac{2\pi }{a},4\cdot \frac{2\pi }{a}\right]\right)$ that were included in the sum in Eq. (19). (c) Absolute value of the $c_{\text{w}{k}_x}^{m,N}$ coefficients for the modes of four $L_N$ cavities $\left(N=5,9,15,35\right)$, superimposed to the odd waveguide band (displayed as a continuous line). The continuum bands of the PhC are shaded in gray. The color map on the right side of the figure is used in both panels (b) and (c). For each value of $m$ and $N$, the positioning of the $c_{\text{w}{k}_x}^{m,N}$ coefficients along the frequency axis was determined by the corresponding mode frequency.Reuse & Permissions

Figure 3

(Color online) (Top) Spatial distribution of the $E_y$ component of the fundamental mode $\left({\text{M}}_0\right)$ of four $L_N$ cavities $\left(N=3,7,11,21\right)$. The upper half shows the modes as obtained from 2D FD computations; in the lower half, the modes reconstructed using Eqs. (2, 19) are displayed. (Bottom) Integral along $y$ of the $E_y$ component of the same modes shown in the top panel. The black dots connected by a black curve refer to the modes obtained from 2DFD computations; the white squares connected by a gray (red) curve to the modes obtained using Eqs. (2, 19).Reuse & Permissions

Figure 4

(Color online) (a), (c), (e), and (g) Spatial distribution of the $E_y$ component of four modes $\left[\left(\text{a}\right){\text{-M}}_0,\left(\text{c}\right){\text{-M}}_4,\left(\text{e}\right){\text{-M}}_8,\left(\text{g}\right){\text{-M}}_{12}\right]$ of the $L_{35}$ cavity. The upper half shows the modes as obtained from 2DFD computations; in the lower half, the modes reconstructed using Eqs. (2, 19) are displayed. (b), (d), (f), and (h) Integral along $y$ of the $E_y$ component of the same modes shown in (a), (c), (e), and (g). The black dots connected by a black curve refer to the modes obtained from 2DFD computations; the white squares connected by a gray (red) curve to the modes obtained using Eqs. (2, 19).Reuse & Permissions

Figure 5

(Color online) (a) Spatial distribution of the $E_y$ component of a mode with Gaussian envelope function $\left(\sigma =3a\right)$, reconstructed as described in the main text. (b) Integral along $y$ of the $E_y$ component shown in (a). Two Gaussian curves corresponding to $\sigma =3a$ are plotted for comparison as continuous (red and blue) lines. (c) Comparison between the absolute value of the 1D Fourier Transform $\left({\phantom{|}\text{FT}\left\{{\mathcal{E}}_y\left(x\right)\right\}|}_{k_x}\right)$ of the ideal Gaussian mode defined by Eq. (29) (black dots connected by a black curve) and of the reconstructed mode shown in (a) and (b) [white squares connected by a gray (red) curve.] (d) 30× zoom of (c) in the light-cone region (shaded in light gray in the figure).Reuse & Permissions

Figure 6

(Color online) (a) False-color mapping of the radius $r$ (in units of $a$, see color bar on the right) of the holes forming the X cavity defined in the main text. The holes are also shown, as circles of varying radius (proportional to the actual $r$). (b) Spatial distribution of the $E_y$ component of the Gaussian mode supported by the X cavity presented in (a). The upper half shows the fundamental mode of the X cavity, as obtained from 2DFD computations. A mode with Gaussian envelope function $\left(\sigma =3a\right)$—reconstructed using the method described in the main text—is displayed in the lower half [same as in Fig. 5a]. (c) Integral along $y$ of the $E_y$ component of the same mode shown in (b). The black dots connected by a black curve refer to the mode obtained from 2DFD computations; the white squares connected by a gray (red) curve to the reconstructed mode with Gaussian envelope function [see main text and Fig. 5b]. (d) Comparison between the absolute value of the 1D Fourier Transform $\left({\phantom{|}\text{FT}\left\{{\mathcal{E}}_y\left(x\right)\right\}|}_{k_x}\right)$ of the ideal Gaussian mode defined by Eq. (29) (black dots connected by a black curve) and of the fundamental mode of the X cavity [white squares connected by a gray (red) curve]. (e) 30× zoom of (d) in the light-cone region (shaded in light gray in the figure). (f) Spectrum of the fundamental mode of the X cavity, obtained from 3D FDTD [gray (red) curve]. The $Q$ factor of the mode $\left(17.5\times {10}^6\right)$ is indicated. The spectrum of the fundamental mode of an $L_{11}$ cavity with nearly identical mode volume (see main text) is also shown for comparison [black (dark blue) curve, the mode wavelength was shifted to match that of the X-cavity mode; only a small portion of the mode peak fits in the displayed spectral range].Reuse & Permissions

Figure 7

(Color online) (a) Absolute value of ${\mathcal{W}}_G^{\left(e\right)\text{w}{k}_x}$ (see definition in the text). The box delimits the range of values of ${\mathcal{W}}_G^{\left(e\right)\text{w}{k}_x}$ $\left(G∊\left[-4\cdot \frac{2\pi }{a},4\cdot \frac{2\pi }{a}\right]\right)$ that were included in the sum in Eq. (A4). (b) Spatial distribution of the $E_y$ component of the lowest-frequency even mode $\left({\text{M}}_0^{\left(e\right)}\right)$ supported by the $L_{11}$ cavity. The upper half shows the mode as obtained from 2DFD computations; in the lower half, the mode reconstructed using Eqs. (2, A4) is displayed. (c) ${\mathcal{E}}_y^{\left(e\right)m,N}\left(x\right)$ (see definition in the text), calculated for the same mode shown in panel (b). The black dots connected by a black curve refers to the mode obtained from 2DFD computations; the white squares connected by a gray (red) curve to the mode obtained using Eqs. (2, A4).Reuse & Permissions