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Robust light transport in non-Hermitian photonic lattices.

Longhi S, Gatti D, Della Valle G - Sci Rep (2015)

Bottom Line: Topological photonic structures, a new class of optical systems inspired by quantum Hall effect and topological insulators, can realize robust transport via topologically-protected unidirectional edge modes.While a forward propagating mode in the lattice is amplified, the corresponding backward propagating mode is damped, thus resulting in an asymmetric transport insensitive to disorder or imperfections in the structure.The possibility to observe non-Hermitian delocalization is suggested using an engineered coupled-resonator optical waveguide (CROW) structure.

View Article: PubMed Central - PubMed

Affiliation: Dipartimento di Fisica- Politecnico di Milano and Istituto di Fotonica e Nanotecnologie-Consiglio Nazionale delle Ricerche, Piazza Leonardo da Vinci, 32, I-20133 Milano, Italy.

ABSTRACT
Combating the effects of disorder on light transport in micro- and nano-integrated photonic devices is of major importance from both fundamental and applied viewpoints. In ordinary waveguides, imperfections and disorder cause unwanted back-reflections, which hinder large-scale optical integration. Topological photonic structures, a new class of optical systems inspired by quantum Hall effect and topological insulators, can realize robust transport via topologically-protected unidirectional edge modes. Such waveguides are realized by the introduction of synthetic gauge fields for photons in a two-dimensional structure, which break time reversal symmetry and enable one-way guiding at the edge of the medium. Here we suggest a different route toward robust transport of light in lower-dimensional (1D) photonic lattices, in which time reversal symmetry is broken because of the non-Hermitian nature of transport. While a forward propagating mode in the lattice is amplified, the corresponding backward propagating mode is damped, thus resulting in an asymmetric transport insensitive to disorder or imperfections in the structure. Non-Hermitian asymmetric transport can occur in tight-binding lattices with an imaginary gauge field via a non-Hermitian delocalization transition, and in periodically-driven superlattices. The possibility to observe non-Hermitian delocalization is suggested using an engineered coupled-resonator optical waveguide (CROW) structure.

No MeSH data available.


Related in: MedlinePlus

Numerically-computed wave packet evolution in (a) the Hatano-Nelson Hamiltonian Eq. (4), and (b) in the driven lattice Hamiltonian Eq. (7) in the absence (left panels) and in the presence (right panels) of disorder. The strength of disorder is as in Figs 2 and 4. The figures show the evolution of the normalized amplitude probabilities an(t) (modulus of an(t)) in a pseudo color map. Initial condition corresponds to a two-humped wave packet  with carrier Bloch wave number q0 = π/2 (upper plots) and q0 = 0 (lower plots).
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f7: Numerically-computed wave packet evolution in (a) the Hatano-Nelson Hamiltonian Eq. (4), and (b) in the driven lattice Hamiltonian Eq. (7) in the absence (left panels) and in the presence (right panels) of disorder. The strength of disorder is as in Figs 2 and 4. The figures show the evolution of the normalized amplitude probabilities an(t) (modulus of an(t)) in a pseudo color map. Initial condition corresponds to a two-humped wave packet with carrier Bloch wave number q0 = π/2 (upper plots) and q0 = 0 (lower plots).

Mentions: Let us consider an initially localized wave packet with carrier wave number q0 that propagates in the ordered lattice. The initial excitation can be written rather generally as with a slowly-varying amplitude F(n), where S(q) is the Bloch spectrum which is assumed to be a narrow function at around q = q0. For a static lattice with a single tight-binding band, like for the Hatano-Nelson model (2), the amplitude probabi-lities cn(t) evolve according to , where E(q) is the complex energy dispersion curve of the lattice band. A similar expression holds for a time-periodic Hamiltonian e.g. the model (7), provided that time t is taken at integer multiplies than the modulation period 2π/ω and E(q) is replaced by the quasi-energy band. Owing to the dependence of the complex energy E(q) on the Bloch wave number q, the wave packet is generally distorted during propagation. However, for a wave packet spectrally narrow at around the carrier wave number q = q0 such that and , at leading order one has , where , and . This means that the wave packet propagates nearly undistorted with a group velocity vg, apart from a uniform amplification (g > 0) or attenuation (g < 0). In the Hatano-Nelson Hamiltonian the condition of nearly-undistorted propagation is attained at q0 = π/2, whereas for the driven lattice model (7) nearly undistorted propagation is predicted for almost any wave number q0, owing to the flatness of ImE(q) and the linear dependence of ReE(q) on q inside the Brillouin zone Fig. 4(a). As an example, Fig. 7(a) shows the numerically-computed evolution of a two-humped wave packet for the Hatano-Nelson Hamiltonian for carrier wave number q0 = 0 and , in either an ordered and a disordered lattice. The figure clearly shows that, according to the previous analysis, even in the absence of disorder strong wave packet distortion is observed for q0 = 0, whereas wave packet distortion is much weaker for . For the modulated lattice Hamiltonian Eq. (7), wave packet distortion is almost absent, as shown in Fig. 7(b). The reason thereof is the special behavior of dispersion curves of the quasi energy minibands of the driven lattice, as discussed in the main text.


Robust light transport in non-Hermitian photonic lattices.

Longhi S, Gatti D, Della Valle G - Sci Rep (2015)

Numerically-computed wave packet evolution in (a) the Hatano-Nelson Hamiltonian Eq. (4), and (b) in the driven lattice Hamiltonian Eq. (7) in the absence (left panels) and in the presence (right panels) of disorder. The strength of disorder is as in Figs 2 and 4. The figures show the evolution of the normalized amplitude probabilities an(t) (modulus of an(t)) in a pseudo color map. Initial condition corresponds to a two-humped wave packet  with carrier Bloch wave number q0 = π/2 (upper plots) and q0 = 0 (lower plots).
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4551998&req=5

f7: Numerically-computed wave packet evolution in (a) the Hatano-Nelson Hamiltonian Eq. (4), and (b) in the driven lattice Hamiltonian Eq. (7) in the absence (left panels) and in the presence (right panels) of disorder. The strength of disorder is as in Figs 2 and 4. The figures show the evolution of the normalized amplitude probabilities an(t) (modulus of an(t)) in a pseudo color map. Initial condition corresponds to a two-humped wave packet with carrier Bloch wave number q0 = π/2 (upper plots) and q0 = 0 (lower plots).
Mentions: Let us consider an initially localized wave packet with carrier wave number q0 that propagates in the ordered lattice. The initial excitation can be written rather generally as with a slowly-varying amplitude F(n), where S(q) is the Bloch spectrum which is assumed to be a narrow function at around q = q0. For a static lattice with a single tight-binding band, like for the Hatano-Nelson model (2), the amplitude probabi-lities cn(t) evolve according to , where E(q) is the complex energy dispersion curve of the lattice band. A similar expression holds for a time-periodic Hamiltonian e.g. the model (7), provided that time t is taken at integer multiplies than the modulation period 2π/ω and E(q) is replaced by the quasi-energy band. Owing to the dependence of the complex energy E(q) on the Bloch wave number q, the wave packet is generally distorted during propagation. However, for a wave packet spectrally narrow at around the carrier wave number q = q0 such that and , at leading order one has , where , and . This means that the wave packet propagates nearly undistorted with a group velocity vg, apart from a uniform amplification (g > 0) or attenuation (g < 0). In the Hatano-Nelson Hamiltonian the condition of nearly-undistorted propagation is attained at q0 = π/2, whereas for the driven lattice model (7) nearly undistorted propagation is predicted for almost any wave number q0, owing to the flatness of ImE(q) and the linear dependence of ReE(q) on q inside the Brillouin zone Fig. 4(a). As an example, Fig. 7(a) shows the numerically-computed evolution of a two-humped wave packet for the Hatano-Nelson Hamiltonian for carrier wave number q0 = 0 and , in either an ordered and a disordered lattice. The figure clearly shows that, according to the previous analysis, even in the absence of disorder strong wave packet distortion is observed for q0 = 0, whereas wave packet distortion is much weaker for . For the modulated lattice Hamiltonian Eq. (7), wave packet distortion is almost absent, as shown in Fig. 7(b). The reason thereof is the special behavior of dispersion curves of the quasi energy minibands of the driven lattice, as discussed in the main text.

Bottom Line: Topological photonic structures, a new class of optical systems inspired by quantum Hall effect and topological insulators, can realize robust transport via topologically-protected unidirectional edge modes.While a forward propagating mode in the lattice is amplified, the corresponding backward propagating mode is damped, thus resulting in an asymmetric transport insensitive to disorder or imperfections in the structure.The possibility to observe non-Hermitian delocalization is suggested using an engineered coupled-resonator optical waveguide (CROW) structure.

View Article: PubMed Central - PubMed

Affiliation: Dipartimento di Fisica- Politecnico di Milano and Istituto di Fotonica e Nanotecnologie-Consiglio Nazionale delle Ricerche, Piazza Leonardo da Vinci, 32, I-20133 Milano, Italy.

ABSTRACT
Combating the effects of disorder on light transport in micro- and nano-integrated photonic devices is of major importance from both fundamental and applied viewpoints. In ordinary waveguides, imperfections and disorder cause unwanted back-reflections, which hinder large-scale optical integration. Topological photonic structures, a new class of optical systems inspired by quantum Hall effect and topological insulators, can realize robust transport via topologically-protected unidirectional edge modes. Such waveguides are realized by the introduction of synthetic gauge fields for photons in a two-dimensional structure, which break time reversal symmetry and enable one-way guiding at the edge of the medium. Here we suggest a different route toward robust transport of light in lower-dimensional (1D) photonic lattices, in which time reversal symmetry is broken because of the non-Hermitian nature of transport. While a forward propagating mode in the lattice is amplified, the corresponding backward propagating mode is damped, thus resulting in an asymmetric transport insensitive to disorder or imperfections in the structure. Non-Hermitian asymmetric transport can occur in tight-binding lattices with an imaginary gauge field via a non-Hermitian delocalization transition, and in periodically-driven superlattices. The possibility to observe non-Hermitian delocalization is suggested using an engineered coupled-resonator optical waveguide (CROW) structure.

No MeSH data available.


Related in: MedlinePlus