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We demonstrate that photonic and phononic crystals consisting of closelyspaced inclusions constitute a versatile class of subwavelength metamaterials.Intuitively, the voids and narrow gaps that characterise the crystal form aninterconnected network of Helmholtz-like resonators. We use this intuition toargue that these continuous photonic (phononic) crystals are in factasymptotically equivalent, at low frequencies, to discrete capacitor-inductor(mass-spring) networks whose lumped parameters we derive explicitly. Thecrystals are tantamount to metamaterials as their entire acoustic branch, orbranches when the discrete analogue is polyatomic, is squeezed into asubwavelength regime where the ratio of wavelength to period scales like theratio of period to gap width raised to the power 1/4; at yet larger wavelengthswe accordingly find a comparably large effective refractive index. The fullyanalytical dispersion relations predicted by the discrete models yielddispersion curves that agree with those from finite-element simulations of thecontinuous crystals. The insight gained from the network approach is used toshow that, surprisingly, the continuum created by a closely packed hexagonallattice of cylinders is represented by a discrete honeycomb lattice. Theanalogy is utilised to show that the hexagonal continuum lattice has aDirac-point degeneracy that is lifted in a controlled manner by specifying thearea of a symmetry-breaking defect.

Asymptotic network models of subwavelength metamaterials formed by closely packed photonic and phononic crystals