PHYSICAL REVIEW B 88, 125138 (2013) Homogenization limit in a graded photonic crystal Eric Cassan, 1,* Jean Dellinger, 2 Xavier Le Roux, 1 K. Van Do, 1 Fr´ ed´ erique de Fornel, 2 and Benoˆıt Cluzel 2,* 1 Institut d’Electronique Fondamentale, Universit´ e Paris-Sud, Centre National de la Recherche Scientiﬁque, 91405 Orsay, France 2 Groupe d’Optique de Champ Proche–Laboratoire Interdisciplinaire Carnot de Bourgogne, Unit´ e Mixte de Recherche Centre National de la Recherche Scientiﬁque No. 6303, Universit´ e de Bourgogne, Dijon, France (Received 23 July 2013; revised manuscript received 11 September 2013; published 27 September 2013) The transition between the long-wavelength and the short-wavelength regimes of light propagation in two- dimensional graded photonic crystal is investigated using a hyperspectral near-ﬁeld scanning microscope. The experiments show an invariant quantity of only 1.78 times the lattice period as the criterion for the possible application of homogenization theories. These results are discussed in light of Fourier decomposition of the electromagnetic Bloch waves, and a physical interpretation of the observed transition between the two light propagation regimes is proposed. These results indicate the robustness of the homogenization approaches and suggest that the sharp transition between the two light propagation regimes could be proﬁtably combined in graded optical artiﬁcial materials. DOI: 10.1103/PhysRevB.88.125138 PACS number(s): 42.70.Qs, 78.67.Pt, 81.05.Xj I. INTRODUCTION Photonic crystals (PhC) are artiﬁcial optical materials made of a periodic arrangement of dielectric elementary constituents. The optical properties which are often desired for PhCs, such as photonic band gap, 1 slow light, 2 or unusual dispersion 3,4 such as negative and ultrarefraction or self-collimation, are obtained in the short-wavelength regime (SWR) or the diffractive regime. These effects are obtained for a frequency range close to the photonic band gap where the light behavior is governed by the interferences between for- ward and backward electromagnetic waves. From the classical Bragg relationship, this corresponds to normalized frequencies ω = a/λ equal to ω B = (2n) −1 , where a is the lattice period, λ the vacuum wavelength, and n the optical effective index to be considered. For the semiconductors and dielectrics usually employed for planar PhC, the n value ranges from 1.7 to 3, which corresponds to ω B values ranging from 0.2 to 0.3. Below the photonic band gap (ω  ω B ), the wavelength be- comes much larger than the lattice period, and light then prop- agates in a homogeneous-like material. This long-wavelength regime (LWR) is described by the theories of homogenization of periodic or composite materials 5–8 (and references therein). In the most general case, planar PhCs behave as anisotropic biaxial optical media, which give rise to unusual physical properties such as in-plane anisotropy in comparison with natural crystals. 9 Using a rotationally symmetric lattice cell unit deﬁned both by the Bravais lattice and the geometry of the inclusions, media with long-wavelength in-plane isotropic behaviors can be also deﬁned. This leads to a homogenized index tensor reduced to a single scalar value that can be evaluated in the LWR, i.e., for ω → 0, as the square root of the dispersion diagram curve slope. TE (in-plane electric ﬁeld) and TM (in-plane magnetic ﬁeld) modes have different slopes, 8 but only the effective long-range index, n eff , for TM modes has an analytical expression given by 8,9 n eff =  f × n 2 hole + (1 − f ) × n 2 diel , (1) with f as the lattice ﬁlling factor, and n hole and n diel as the optical index of the hole and the surrounding dielectric media, respectively. Several recent works relying on the formalism of spatial coordinate transforms for the design of artiﬁcial optical materials have directly used this homogenized index approach. This regime has attracted an increasing interest with the emergence of metamaterials and their fascinating optical properties. Experimental demonstration of electromagnetic invisibility devices or Luneburg lensing structures, 10,11 for instance, have been reported. While LWR and SWR have been separately explored, the gap between them has not received much attention. We provide here a direct experimental observation of an abrupt transition between the two regimes of light propagation in a graded PhC. Thanks to a hyperspectral scanning near-ﬁeld optical microscope (HSNOM), 12,13 this transition is observed over a broad spectral range at near-infrared wavelengths for an invariant ratio between the propagating wavelength and the PhC periodicity. This allows us to deﬁne a homogenization limit close to λ/2.2, which is a signiﬁcant much larger value than the λ/10 limit usually considered for homogenization theory. II. METHODS We have considered here a square lattice made of cylindrical air holes etched in a silicon on insulator (SOI) wafer. As shown in Fig. 1(a), the fabricated artiﬁcial material is obtained by breaking the lattice symmetry through the introduction of a gradual in-plane modiﬁcation of the air hole ﬁlling factor (f ). Lower and higher values of the normalized hole radius range from r/a = 0.20 to r/a = 0.35. Light is injected inside the gradual PC with a silicon waveguide, and a HSNOM is used to provide a direct visualization of the light path over a broad spectral range. The near-ﬁeld measurements are restricted here to propagating wavelengths ranging from λ = 1300 nm (ω = 0.296) to 1550 nm (ω = 0.248), which correspond to a normal- ized frequency range where the transition between the SWR and LWR is observed for the TM mode considered throughout this work. A multimedia ﬁle showing the continuous variation of the near-ﬁeld images across the whole spectral range is provided online (see Supplemental Material in Ref. 14). 125138-1 1098-0121/2013/88(12)/125138(5) ©2013 American Physical Society