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Two Dimensional Photonic Crystal Modes and Resonances in Three-dimensional Structures Shanhui Fan1 and J. D. Joannopoulos2 1 Department of Electrical Engineering, Stanford University, Stanford, CA 94305, U. S. A 2 Department of Physics and Center for Material Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, U. S. A ABSTRACT We present three-dimensional analysis of two-dimensional guided resonances in photonic crystal slab structures. This analysis leads to a new understanding of the complex spectral properties of such systems. Specifically, we calculate the dispersion diagrams, the modal patterns, and transmission and reflection spectra of these resonances. From these calculations, a key observation emerges involving the presence of two temporal pathways for transmission and reflection processes. Using this insight, we introduce a general physical model that explains the essential features of complex spectral properties. Finally, we show that the quality factors of these resonances are strongly influenced by the symmetry of the modes, and the strength of the index modulation. INTRODUCTION Photonic crystal slabs are a particularly important class of photonic crystal structures. A photonic crystal slab consists of a two-dimensionally periodic index contrast introduced into a high-index guiding layer (inset in Figure 1a). These structures support in-plane guided modes that are completely confined by the slab without any coupling to external radiations. These guided modes allow the control of light within the layer at the wavelength scale. Therefore, the slab structure may provide the basic substrate for large-scale on-chip integration of photonic components and circuits. [1-8] even modes

0.4 0.3 0.2 0.1 0 Γ

X M wavevector

odd modes

(b)

Frequency (2πc/a)

Frequency (2πc/a)

(a)

0.4 0.3 M

0.2 Γ

0.1

Γ

X

0 Γ

X

M wavevector

Γ

Figure 1. The band structure for (a) even and (b) odd modes in a photonic crystal slab. The structure of the slab is shown in the inset of (a), and consists of a square lattice of air holes with a radius of 0.2a introduced into a high-index dielectric slab with a dielectric constant of 12 and a thickness of 0.5a. Even and odd modes are defined with respect to the mirror parallel to the slab. The gray regions are the continuum of radiation modes. Solid lines outside the gray region are guided modes. Solid lines within the gray region are guided resonances. K8.5.1

In addition to in-plane wave guiding, photonic crystal slabs can also interact with external radiations in complex and interesting ways. Of particular importance here is the presence of guided resonances in the structures. [9-15]. Similar to the guided mode, a guided resonance also has its electromagnetic power strongly confined within the slab. Unlike the guided mode, however, the resonance can couple to external radiations. Therefore, guided resonances can provide an efficient way to channel light from within the slab to the external environment. This property has been exploited in the desi