Arising from: Y. Akahane, T. Asano, B.-S. Song & S. Noda Nature 425, 944–947 (2003); Akahane et al. reply One challenge in photonics is strongly to confine light in small volumes in order to increase light–matter interaction. Akahane et al.1 propose a new concept for increasing the lifetime of this interaction, based on tailoring of the Fourier spectrum of cavity modes, which they believe is demonstrated by the surprising enhancement (roughly tenfold) of the quality factor Q of the cavity as a result of fine-tuning the mirror-hole geometry in a photonic-crystal nanocavity. Here we question the validity of their concept and argue that the improvement in Q is due to an increase in the impedance wave matching at the cavity edges and to a slow-wave effect. This alternative interpretation opens the way to new cavity designs.
Like Akahane et al.1, we first consider two cavities formed in a silicon slab with air claddings (Fig. 1a). The light is confined in the x-direction by perfect-metal mirrors (left cavity) or by Bragg mirrors (right cavity). For the cavity parameters given in the figure legend and for y-polarization, the resonant wavelength is λ = 1.5 µm. Using the Fourier modal method2, we calculate the cavity mode profiles at the centre plane of the slab (Fig. 1b) and their Fourier spectra (Fig. 1c).
These curves closely resemble those in Fig. 2 of Akahane et al., which would suggest1 that the metallic cavity, with large Fourier components in the leaky region, should show a smaller Q factor than the dielectric one. However, the metallic-cavity Q is 105, a value that is 200 times larger than that of the dielectric cavity. In addition, for a null metal height of e, the mode profile of the metallic cavity remains completely unchanged but its Q is only 350.
We infer that the mode profile inside the cavity conveys nothing about the mode lifetime. The Fourier spectrum method is valid for calculating mode profiles across a surface positioned above the cavity in the air clad3. For such x-invariant surfaces, the cavity-mode profile and its Fourier spectrum contain all the information on the far-field radiation losses.
But this is not true for a surface inside the cavity. The reason is that the parallel kx momentum of the cavity mode is not strictly matched at the slab–cladding interface because neither Snell's law nor Fresnel's law apply at a corrugated interface. Design rules that overlook this parallel-momentum mismatch ignore the essence of the longitudinal confinement and, in turn, the associated impedance-mismatch problem4 that determines radiation losses at the cavity edges.
To understand the physical reasons for the roughly tenfold enhancement in Q, we consider a Fabry–Perot model. We assume that the nanocavity mode is formed by the recirculation of the fundamental Bloch wave of the line-defect waveguide between the mirrors. The model, which relies on a three-dimensional computation2 for the Bloch-wave modal reflectivity |rm|exp(iΦ), reproduces the experimental trends well: a red-shift of λ as the hole shift d increases, and a peak value of Q for d ≈ 0.17a , where a is the lattice constant of the photonic crystal. For Fabry–Perot cavities5,
where L = 3a is the defect length. From equation (1), three effects explain the Q improvement between d = 0 and 0.17a: an increase of |rm|2 from 98.3% to 99.7%, a doubling of the group-index ng of the Bloch wave, and a doubling of the penetration depth δΦ/δλ into the mirrors.
The first effect has been analysed previously for a one-dimensional slab6 and two-dimensional air-bridge cavities7, and the second and third effects result from the highly dispersive nature8,9 of the Bloch wave in the spectral window. Our analysis highlights the importance of wave impedance matching in nanocavities and the key role of slow waves in further improvements.
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Sauvan, C., Lalanne, P. & Hugonin, JP. Tuning holes in photonic-crystal nanocavities. Nature 429, 1 (2004). https://doi.org/10.1038/nature02602
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