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A word u = u1 . . . un is a scattered factor of a word w if u can be obtained from w by deleting some of its letters: there exist the (potentially empty) words v0, v1, .., vn such that w = v0u1v1...unvn. The set of all scattered factors up to length k of a word is called its full k-spectrum. Firstly, we show an algorithm deciding whether the kspectra for given k of two words are equal or not, running in optimal time. Secondly, we consider a notion of scattered-factors universality: the word w, with alph(w) = Σ, is called k-universal if its k-spectrum includes all words of length k over the alphabet Σ; we extend this notion to kcircular universality. After a series of preliminary combinatorial results, we present an algorithm computing, for a given k -universal word w the minimal i such that w i is k-universal for some k > k . Several other connected problems are also considered.Supported by the DFG grant MA 5725/2-1. F.M. thanks Pawe l Gawrychowski for his comments and suggestions.

Photonic crystal slabs (PCS) are a promising platform for optical biosensing. Yet, flexible applications based on PCS for biosensing have been limited, as the mechanical properties influence the optical ones. Here, we show the suppression of the mechanochromism effect for flexible PCS. We obtained flexible photonic crystal slabs by sputtering of a dielectric 100 nm Nb2O5 high refractive index layer onto a flexible nanostructured polydimethylsiloxane (PDMS) substrate with 370 nm grating period. The PCS exhibit a guided mode resonance at around 650 nm. We demonstrate that these flexible photonic crystal slabs show less than 0.5 nm resonance shift for 4% strain and call them stabilized PCS (sPCS). We compare this to a resonance shift of ∼21 nm for ∼4% strain of a flexible photonic crystal with a flexible nanoparticle high index layer (mechanochromatic PCS, mPCS). This high resonance shift is expected from the Bragg equations, where 4% grating period change correspond to approximately 4% change of the resonance wavelength (i.e., ∼26 nm at a resonance wavelength of 650 nm), if changes in the mode effective refractive index are neglected. In a stretch series we obtain color-to-strain dependencies of 4.79 nm/% strain for mPCS and 0.11 nm/% strain for our stabilized sPCS. We analyze the suppression of the mechanochromism with detailed microscopy results. We observe that fissures and fractures form in the rigid waveguiding layer of the sPCS upon mechanical stress. An algorithm based on Holistically-Nested Edge Detection (HED) is used for automated counting of cracks. Rigid photonic crystal cells with sizes on the order of 10 µm to 100 µm are formed that explain the stable optical properties. Even more stable optical properties with less than 0.03 nm wavelength shift per 1% strain are demonstrated for sPCS with an additional dielectric 100 nm SiO2 low index layer beneath the Nb2O5 waveguide layer decoupling the waveguide further from the flexible PDMS substrate.

We present flexible photonic crystal slabs (PCS) obtained by sputtering of a dielectric 100 nm Nb2O5 high refractive index layer onto a flexible nanostructured polydimethylsiloxane (PDMS) substrate with 370 nm grating period. The PCS exhibit a guided mode resonance at around 650 nm. We demonstrate that these flexible photonic crystal slabs show less than 0.5 nm resonance shift for 4% strain and call them stabilized PCS (sPCS). We compare this to a resonance shift of ~21 nm for ~4% strain of a flexible photonic crystal with a flexible nanoparticle high index layer (mechanochromatic PCS, mPCS). This high resonance shift is expected from the Bragg equations, where 4 % grating period change correspond to approximately 4 % change of the resonance wavelength (i.e., ~26 nm at a resonance wavelength of 650 nm), if changes in the mode effective refractive index are neglected. In a stretch series we obtain color-to-strain dependencies of 4.79 nm/% strain for mPCS and 0.11 nm/% strain for our stabilized sPCS. We analyze the suppression of the mechanochromism with detailed microscopy results. We observe that fissures and fractures form in the rigid waveguiding layer of the sPCS upon mechanical stress. An algorithm based on Holistically-Nested Edge Detection (HED) is used for automated counting of cracks. Rigid photonic crystal cells with sizes on the order of 10 µm to 100 µm are formed that explain the stable optical properties. Even more stable optical properties with less than 0.03 nm wavelength shift per 1% strain are demonstrated for sPCS with an additional dielectric 100 nm SiO2 low index layer beneath the Nb2O5 waveguide layer decoupling the waveguide further from the flexible PDMS substrate.

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