Localized States Emerging from Singular and Nonsingular Flat Bands in a Frustrated Fractal‐Like Photonic Lattice

Hanafi, Haissam; Menz, Philip; Denz, Cornélia

doi:10.1002/adom.202102523

Cited by 14 publications

(18 citation statements)

References 48 publications

(67 reference statements)

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“…The distribution is preserved even if the finite size of the gasket is arbitrarily large. Interestingly, almost exact replicas of the amplitude patterns discussed above, have been experimentally observed recently in photonic lattices 26,27 where, a variant of the SGF structure reported here has been studied in two dimensions. In spite of the variation in construction, the ring-like CLS and the armchair patterns described here in Fig.…”

Section: The Sierpinski Gasket: Engineering Compact Localization and ...supporting

confidence: 61%

“…This works easily when the unit cell has simple geometry, but turns out to be a tedious job in cases where the unit cell has an intricate internal structure. This issue has become pertinent and needs to be addressed, in particular, after a series of fascinating experiments reported laboratory realizations of the classic Sierpinski gasket fractal (SGF) [23][24][25] and observation of CLS [26][27][28][29][30] and photonic Floquet topological phase 31 on engineered SGF substrates. The latter experiments were done on SGF structures grafted either mainly by the femtosecond laser writing techniques, or designed with evanescently coupled helical waveguides, for example.…”

Section: Introductionmentioning

confidence: 99%

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Designer quantum states on a fractal substrate: compact localization, flat bands and the edge modes

Biswas¹,

Chakrabarti²

2023

Preprint

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Compact localized single particle eigenstates on a deterministic fractal substrate, modelled by a triangular Sierpinski gasket of arbitrarily large size, are unravelled and examined analytically. We prescribe an exact real space renormalization group (RSRG) decimation scheme within a tight binding formalism to discern these states, and argue that the number of such states can be infinite if the fractal substrate is enlarged to its thermodynamic limit. Interestingly, these localized states turn out to populate the non-dispersive, flat bands in a periodic array of Sierpinski gasket motifs, however large they may be. Our results match and corroborate the recently observed compact localized, flat band states engineered on two dimensional photonic waveguide networks with a fractal geometry, and provide a whole subset of them, which, in principle, should be observable in fractal photonic lattice experiments.

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Section: The Sierpinski Gasket: Engineering Compact Localization and ...supporting

confidence: 61%

Section: Introductionmentioning

confidence: 99%

Designer quantum states on a fractal substrate: compact localization, flat bands and the edge modes

Biswas¹,

Chakrabarti²

2023

Preprint

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“…Since the CLS‐2 and CLS‐3 share the same eigenvalue (propagation constant), their superimposed states (CLS‐5 and CLS‐6) are also eigenstates of the plane‐touching flat bands. [ 18,19 ] Thus, for the doubly‐degenerate flat bands β 5, 6 , there exist two different sets of basic CLSs, one set consists of {CLS‐2} and {CLS‐3}, the other one consists of {CLS‐5} and {CLS‐6}, where the notation “

\{\cdot \}

” indicates N translated copies of the selected basic CLS. When we choose one of the sets, {CLS‐2, CLS‐3} for example, {CLS‐2} corresponds to the flat band β 5 , then {CLS‐3} automatically corresponds to flat band β 6 , and vice versa.…”

Section: Resultsmentioning

confidence: 99%

Topological Flatband Loop States in Fractal‐Like Photonic Lattices

Song

Xie

Xia

et al. 2023

Laser & Photonics Reviews

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Noncontractible loop states (NLSs) are a recently realized topological entity in flatband lattices, arising typically from the band touching at a point where a flat band intersects one or more dispersive bands. There exists also band touching across a plane, where one flat band overlaps another all over the Brillouin zone without crossing a dispersive band. Such isolated plane‐touching flat bands remain largely unexplored. For example, what are the topological features associated with such flatband degeneracy? Here, nontrivial NLSs and robust boundary modes in a system with such degeneracy are demonstrated. Based on a tailored photonic lattice constructed from the well‐known fractal Sierpinski gasket, the wavefunction singularities and the conditions for the existence of the NLSs are theoretically analyzed. It is shown that the NLSs can exist in both singular and nonsingular flat bands, as a direct reflection of the real‐space topology. Experimentally, directly such flatband NLSs in a laser‐written Corbino‐shaped fractal‐like lattice are observed. This work not only leads to a deep understanding of the mechanism behind the nontrivial flatband states, but also opens up new avenues to explore fundamental phenomena arising from the interplay of flatband degeneracy, fractal structures, and band topology.

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“…In order to verify our results experimentally, we fabricated photonic lattices in fused silica (SiO 2 ) using the direct laser writing technique [24,40]. Table II shows the main differences in parameters compared to the simulations presented so far.…”

Section: Experiments In Fused Silicamentioning

confidence: 87%

Tilted Dirac cones and asymmetric conical diffraction in photonic Lieb-kagome lattices

Lang,

Hanafi,

Imbrock

et al. 2022

Preprint

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The Lieb lattice and the kagome lattice, which are both well known for their Dirac cones and flat bands, can be continuously converted into each other by a shearing transformation. During this transformation, the flat band is destroyed, but the Dirac cones remain and become tilted, with types I, II, and III occurring for different parameters. In this work, we first study these tilted Dirac cones using a tight-binding model, revealing how they can be engineered into the different types. We then demonstrate conical diffraction in a photonic lattice realization of the Lieb-kagome lattice using split-step beam propagation simulations, obtaining evidence of the presence of Dirac cones tilted in different directions. Finally, we performed experiments with photonic lattices laser-written in fused silica (SiO2) to validate the results of the simulations. These studies advance the understanding of the Lieb-kagome lattice and tilted Dirac cones in general and provide a basis for further research into this interesting tunable lattice system.

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Localized States Emerging from Singular and Nonsingular Flat Bands in a Frustrated Fractal‐Like Photonic Lattice

Cited by 14 publications

References 48 publications

Designer quantum states on a fractal substrate: compact localization, flat bands and the edge modes

Designer quantum states on a fractal substrate: compact localization, flat bands and the edge modes

Topological Flatband Loop States in Fractal‐Like Photonic Lattices

Tilted Dirac cones and asymmetric conical diffraction in photonic Lieb-kagome lattices

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