Efficient Focusing with an Ultra-Low Effective-Index Lens Based on Photonic Crystals
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Efficient Focusing with an Ultra-Low Effective-Index Lens Based on Photonic Crystals Eugen Foca1,2, Helmut Föll1, Frank Daschner3, Vladimir V. Sergentu2, Jürgen Carstensen1, Reinhard Knöchel3, Ion M. Tiginyanu2 1 Chair for General Materials Science, Faculty of Engineering, Christian-Albrechts-University of Kiel, Kiel, Germany; 2 Institute of Applied Physics, Technical University of Moldova, Chisinau, Moldova; 3 Microwave Laboratory, Faculty of Engineering, Christian-Albrechts-University of Kiel, Kiel, Germany. ABSTRACT This work focuses on photonic crystals (PC) that can be ascribed an effective index of refraction < 1 or even < 0. We investigate the possibility to design optical elements (in this case a lens) based on this type of PC. A new approach for determining the effective refractive index of PCs with unusual index of refraction is used, which is simpler than earlier methods based on analyzing equi-frequency surfaces in k-space. An ultra-low refractive index PC is given a form approximating a concave lens and is proven theoretically and experimentally that it efficiently focuses the electromagnetic radiation in the microwave range. Strong focusing effects are found for both polarizations (TE and TM mode). Intensity gains as large as 35 for TM polarizations and 29 for TE polarizations are found. Measurements are in a good accordance with simulations. INTRODUCTION Propagation of electromagnetic radiation in a Photonic Crystal (PC) is well understood in the long wavelength range, i.e. where the wavelength is large compared to the crystals’ periodicity and energetically below the band gap. In this case PCs can be treated as homogeneous materials and all their parameters like neff , the effective index of refraction, are (relatively) easily calculated. In this case devices based on a PC have an neff > 1, given by neff = ck ω , where ω is the radiation frequency and k is the wave vector of the Bloch wave propagating in the PC. However, more interesting phenomena are encountered in the short wavelength limit (energies above the band gap), where unusual dispersion functions and beam propagation are found. In this wavelength range, PCs may, for example behave as homogeneous material with an ultra-low index of refraction, meaning a neff < 1 or even a neff < 0 [1]. However, an index of refraction may also not be defined at all, or only in some approximation. We will therefore first discuss a method that allows to check rather easily if for a given wavelength and polarization state a given PC can be described, at least approximately, by an effective index of refraction, and what numerical value must be assigned to neff in this case. Using this method, suitable wavelength ranges are identified for which designing optical elements (here a concave lens) with ultra-low neff makes sense. The resulting lens is characterized exhaustively in two configurations: perfect order (perfect crystal) and strongly disturbed (“amorphous”). In what follows we report on extensions of the previous work [2] and present new results. “P
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