Discrete, Nonlinear and Disordered Optics

Throughout the last years, topologically protected edge states have been measured in various branches of physics including solid state, BECs and optical systems like waveguide arrays and photonic crystals. Here, we experimentally demonstrate the implementation of the Su-Schrieffer-Heeger (SSH) model in a fiber network consisting of two loops of different length, where the variable mutual coupling determines the existence of topologically non-trivial states. Due to the length of the fibers of about 4 km, the hardly explored nonlinear regime is directly accessible. In our measurements, we excite both topologically trivial and non-trivial states with a nonlinear wave and provide a direct comparison of the coupling efficiencies.

In this paper we considered the mathermatical and geometrical construction of coherent states for fermion physical system in a quantum field theoretical context.

Properly modulated flat-band lattices have a divergent density of states at the flat-band energy. Quasiperiodic modulations are known to host a metal-insulator transition already in one space dimension. Their embedding into flat-band geometries consequently allows for a precise engineering and fine tuning of mobility edges. We obtain analytic expressions for singular mobility edges for two flat-band lattice examples. In particular, we engineer cases with arbitrarily small energy separations of mobility edge, zeroes, and divergencies.

Photonic metasurfaces can be used to locally control the phase of incoming electromagnetic waves by scattering the wave from the sub-wavelength building blocks of the metasurface. In combination with the Generalized Snell's law this allows to nearly arbitrarily control the refraction at such surfaces. Depending on the exact structure of the used metasurface one can construct various optical elements, e.g., lenses and holograms. Here, we want to demonstrate a metasurface that generates an optical vortex. For that purpose, we fabricate an antenna array using electron beam lithography. We employ gold antennas with a length of 200 nm to modify the properties of a circular polarized wave. More specific, we control the locally induced phase of light with the opposite helicity by rotating each antenna by a certain angle relatively to a reference axis. To achieve a radial phase gradient, necessary to generate an optical vortex, the orientation of the antennas rotates by $\pi$ per turn. Using an interferometer we demonstrate that our photonic metasurface generates an optical vortex exhibiting the characteristic spiral phase distribution. Our experimental results are in good agreement with the Fresnel approximation.

The ongoing development of light-based technologies requires a full control of the light field. Properties such as the frequency or the spatial mode have to be accurately controlled and converted, to fit the high demands. Applications of tailored light fields can be found, for instance, in optical tweezers or high-resolution microscopy. Furthermore, the propagation of the light field can be manipulated to achieve light guiding, localization or reflection, among others. An approach to set these properties is the design of a discrete system with a periodical or disordered refractive index structure. A powerful technique to create such structures is femtosecond laser lithography, also known as direct laser writing (DLW). It enables the nonlinear induction of three dimensional, permanent, and nanometer sized refractive index structures directly into the volume of transparent materials. We will present an integrated optical device with a periodical modulated potential in a nonlinear optical material, which allows performing frequency and mode conversion simultaneously. The concept is based on the induction of an appropriate computer-generated hologram into the volume of a nonlinear medium by DLW, adapting our previously reported nonlinear beam splitter approach to higher modes. The fundamental wave is spatially shaped and simultaneously split into several diffraction orders. The transmitted and diffracted waves are subsequently noncollinearly phase-matched, generating a spatially shaped second harmonic (SH). Applying this concept, we demonstrate SH optical vortices with desired topological charges and non-diffracting SH Bessel beams.

Anderson localization [1] is the halt of diffusion of waves in random lattices above a certain critical strength of disorder. In photonics, the quest to observe the Anderson localization of light has fueled much of the modern experimental research [2]. The localization occurs because the wave process is scattered by inhomogeneities of the medium, causing the different components of the wave field to interfere with itself. In a basic theory of wave processes, the renewed interest in Anderson localization is due to the suggestions (mostly motivated by the numerical simulation) that a weak nonlinearity might destroy the localized state and that the ensuing loss of localization is itself a critical phenomenon [3]. That means that above a certain critical strength of nonlinear interaction the nonlinear field can slowly diffuse across the lattice (in spite of the underlying disorder) and is Anderson localized despite those nonlinearities otherwise. The possible destruction of Anderson localization by a weak nonlinearity has since been studied in the fashion of nonlinear Schrodinger models with disorder. A modified perturbation theory with regard to the strength of the nonlinear term has been developed [4] and extensive numerical simulations have been carried out [5]. A subdiffusive scaling for the onset spreading has been introduced and numerically measured [3,5]. More recently, a non-perturbative approach to the nonlinear Anderson dynamics has been proposed based on topological approximations, using random walks and the formalism of fractional Fokker-Planck equation [6]. We consider the problem of dynamical localization of waves in a discrete nonlinear Schrodinger equation (NLSE) with random potential, which is characterized by a random potential and a nolinear term with an amplitude $\beta$. For $\beta\rightarrow 0$, the NLSE reduces to the original Anderson model [1]. All eigenstates are localized in that limit. With the aim of understanding the asymptotic ($t\rightarrow+\infty$) spreading of initially localized wave packet under the action of the nonlinear term, we obtained that the four-wave interaction always destroys Anderson localization, giving rise to unlimited spreading of the wave field to large distances. In the quantum domain of the energy leveles $n$, the process is not thresholded, contrary to its classical counterpart and leads to a faster spreading of the subdiffusive type, with the dispersion $\langle(\Delta n)\rangle \sim t^{1/2}$ for $t\rightarrow+\infty$, while in the classical dispersion is $\langle(\Delta n)\rangle \sim t^{1/3}$ . [1] P. W. Anderson, Phys. Rev. {\bf 109}, 1492 (1958). [2] C. M. Aegerter, M. Storzer, and G. Maret, Europhys. Lett. {\bf 75}, 562 (2006); M. Storzer et al., Phys. Rev. Lett. {\bf 96}, 063904 (2006); T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature \textbf{446}, 52 (2007); C. Conti, Nature Photonics {\bf 7}, 5 (2013). G. M. Akselrod et al., Nature Comm. {\bf 5}, 3646 (2014). [3] A. S. Pikovsky and D. L. Shepelyansky, Phys. Rev. Lett. {\bf 100}, 094101 (2008). [4] Y. Krivolapov, S. Fishman, and A. Soffer, New J. Phys. {\bf 12}, 063035 (2010). [5] S. Flach, D. O. Krimer, and Ch. Skokos, Phys. Rev. Lett. {\bf 102}, 024101 (2009); Ch. Skokos et al., Phys. Rev. E {\bf 79}, 056211 (2009). [6] A. Iomin, Phys. Rev. E {\bf 81}, 017601 (2010); A. V. Milovanov and A. Iomin, Europhys. Lett. {\bf 100}, 10006 (2012); A. V. Milovanov and A. Iomin, Phys. Rev. E {\bf 89}, 062921 (2014).

Edge modes in topological insulators are said to be robust against defects. In a Floquet system of helically curved waveguides (as in [1]) we study the influence of defects with time-dependent coupling on the robustness of the transport along the edge. To this end, the inverse of the waveguide array is fabricated via 3D-lithography (direct laser writing) in negative tone photoresist. Subsequently the sample is infiltrated with a material of higher refractive index, creating low-loss 3D waveguides. We find that dynamic defects do not destroy the chiral edge current, even when waveguides temporally overlap with the defect. [1] M. C. Rechtsman et al., Nature 496, 196-200 (2013).

Snowflakes, flash of lightning and Romanesco are just a few examples for self-similar structure, which we encounter in our everyday life. While in mathematics, Julia sets and Mandelbrot fractals are the most famous fractal patterns, in physics it is undoubtedly the butterfly spectrum, which was first investigated by Douglas Hofstadter about 40 years ago. As it arises from the interplay between a discrete lattice and an applied magnetic field, it is not only limited to solid states as originally proposed but also observable in Bose-Einstein condensates and microwave systems. Here, Hofstadter’s idea is extended to the optical regime, where we demonstrate high resolution measurements of the butterfly spectrum in a mesh lattice of optical fibers. The lattice is implemented by a time multiplexing technique based on two coupled fiber loops of different length. The magnetic field is emulated by phase modulator in the shorter loop. Within this system the influence of the fractal spectrum on the transport properties is investigated and a cyclic exchange of ballistic and diffusive mobility is observed.

Random lasing is a phenomena which has continuously received attention since its experimental discovery in 2000. We study the propagation of light in an optically active, nonlinear and dielectrically disordered medium using a functional integral formalism, which enables the derivation of an effective low-energy action, known as the Keldysh nonlinear sigma model. In addition, we were able to calculate explicit transport quantities in the form of counting statistics which we could show, in the case of nonactive media, to be quantitatively the same as previous diagrammatical calculations. Finally we study random lasing by including the full matter-light interacting picture and integrating out the matter fields, thus exposing the nonlinear nature of effective photonic interactions in our theory. By obtaining the full interacting saddlepoint of the nonlinear sigma model and systematically taking into account fluctuations around the saddlepoint we obtain the diffusive modes.

Anderson localization of light in a random dielectric system is still a controversial issue due to the vector nature of light. We investigate the effects of the vector nature of light on propagation properties in photonic crystals with binary disorder. Due to the transverse nature of light $(\nabla\cdot\vec{D} = 0)$, the three vector components reduce to the well-known, two-fold polarization degrees of freedom, that is, in two orthogonal polarization modes on each lattice site in a three-dimensional lattice. It can be described as a pseudospin degree of freedom. Hopping in the random lattice induces flipping of the pseudospin (mixing of the polarization modes) in analogy to random spin-orbit scattering in electronic systems. We generalize the photonic Coherent Potential Approximation (CPA) to this pseudospin system in order to calculate single-photon properties like the self-energy and density of states (DOS). To calculate transport properties, we generalize the Vollhardt-Wölfle theory of Anderson localization to the case of vector waves, using the pseudospin representation. We find localizing and antilocalizing contributions to the diffusion coefficient $D(\Omega)$ in the pseudospin singlet and triplet channels, analogous to random spin orbit scattering. We calculate the corresponding phase diagram of Anderson localization. Our results may provide a systematic way of analyzing the difficulties in achieving light localization in experiments.

Nondiffracting waves are highly relevant in optics and atom physics, particularly because their transverse intensity distributions propagate unchanged for hundreds of diffraction lengths. Thus, they feature potential applications in free-space wireless communications, optical interconnections, laser machining, and surgery. One prominent class of nondiffracting waves is given by Mathieu beams, which appear as translationally invariant solution of the Helmholtz equation in elliptic cylindrical coordinates. They can be classified according to their symmetry properties as even and odd and according to their order. So called elliptical Mathieu beams are obtained as a linear superposition of even and odd Mathieu beams of the same order. Whereas the even and odd Mathieu beams exhibit spatial phase distributions with discrete phase values, the elliptic Mathieu beams show outstanding continuously modulated spatial phase distributions that act as orbital angular momenta. During linear propagation, energy is continuously redistributed along elliptical paths of the invariant transverse intensity distribution. However, the energy flow of elliptic Mathieu beams withstands a direct observation. In our contribution, we visualize this energy flow of elliptic Mathieu beams by adding a nonlinearity in form of an anisotropic photorefractive crystal that breaks the sensitive equilibrium of continuous intensity redistribution. Consequently, a new type of rotating beam formation arises with high intensity filaments corresponding to the energy flow in an enforced preferential direction. This process is beneficially applied to realize tunable helically twisted photonic lattices, capable to study and utilize the light-matter interaction with chiral light, as well as for advanced waveguiding purposes.

States with exponentially decaying observables are not vectors belonging to a conventional Hilbert space, but they belong to a rigged Hilbert space. We study states with complex energy, the so-called Gamow vectors, to model temporal irreversibility in quantum mechanics. We analyze the reversed harmonic oscillator Hamiltonian and get a time asymmetric evolution dominated by a summation of decaying states with quantized decay rates. Finally, we test this result in nonlinear optics and report both simulations and experimental evidences

We propose a recursive scheme for the design of scatterers acting simultaneously as emitters and absorbers, such as lasers and coherent perfect absorbers in optics, at multiple prescribed frequencies. The approach is based on the assembly of non-Hermitian emitter and absorber units into self-dual emitter-absorber trimers at different composition levels, exploiting the simple structure of the corresponding transfer matrices. In particular, lifting the restriction to parity-time-symmetric setups enables the realization of emitter and absorber action at distinct frequencies and provides flexibility in the choice of realistic parameters. We further show how the same assembled scatterers can be rearranged to produce unidirectional and bidirectional transparency at the selected frequencies. With the design procedure being generically applicable to wave scattering in single-channel settings, we demonstrate it with concrete examples of photonic multilayer setups. [arXiv:1609.04211]

We predict novel localized modes supported by surface magnetic currents at interfaces between lattices with different synthetic gauge fields yet identical photonic band-gaps, and formulate their implementation in fiber loop mesh lattices with phase modulators.

It is shown how in the Complex Ginzburg-Landau equation a periodic modulation of spectral losses in alternating (zig-zag) way for different spectral modes leads to modulation instability of the homogeneous field solution [1]. The instability induces pattern formation with practical applications for the generation of pulses in lasers [2]. [1] A.M.Perego, N.Tarasov, D.V.Churkin, S.K.Turitsyn,and K.Staliunas, Phys. Rev. Lett. 116, 028701 (2016). [2] N.Tarasov, A.M.Perego, D.V.Churkin, K.Staliunas, and S.K.Turitsyn, Nat. Commun. 7, 22441 (2016).

Local symmetries are spatial symmetries present in a subdomain of a complex system. We use the occurrence of such symmetries to derive nonlocal currents in one- and two-dimensional photonic waveguide arrays effectively described by Schrödinger's equation with a tight-binding Hamiltonian. These currents obey a nonlocal continuity equation where broken symmetries appear as sources or sinks, thereby expressing the effects of local symmetries in the spatial structure of evolving states [1]. For the case of one-dimensional systems, the nonlocal currents of stationary states are translationally invariant within local symmetry domains for arbitrary wavefunction profiles. Two distinct versions of the nonlocal currents enable a mapping between wave amplitudes of symmetry-related sites, thereby generalizing the global Bloch and parity mapping to local translation and inversion symmetry in discrete systems [1,2]. Additionally, the invariant currents can be used to classify perfect transmitting states with respect to their local symmetry in scattering setups [1,3]. Similar invariants also exist for Floquet states in periodically driven systems [1,4]. For two-dimensional systems, the nonlocal currents are in general no longer constant, but can still be used as a tool to analyze the eigenstates of a given system [5]. Encoding local potential and coupling symmetries into arbitrary stationary states, the theory of symmetry-induced continuity and local invariants may contribute to the understanding of wave structure and response in systems with localized spatial order. [1] arXiv:1607.06577 [2] P.A. Kalozoumis et. al., Phys. Rev. Lett. 113, 050403 [3] P.A. Kalozoumis et. al., Phys. Rev. A 88, 033857 [4] T. Wulf et. al., Phys. Rev. E 93, 052215 [5] arXiv:1612.07924

In nonlinear gap transmission process, the knowledge of the amplitude threshold is very important. When the system is strongly discrete, the continuous approximation generally used to find supratransmission threshold up to now is no longer appropriate. Following Togueu Motcheyo et al. [Commun Nonlinear Sci Numer Simulat 50 (2017) 29–34], we find supratransmission threshold in strongly discrete optical waveguide arrays using two dimensional map approach. This threshold is in agreement with the numerical one even for the large frequencies.

We show a possibility of realization of PT-symmetry breaking in a pair of coupled fiber loops of equal lengths with alternating coupling coefficient, and discuss an experimental implementation of the system.

Alessandro Zannotti, Carsten Mamsch, Matthias Rüschenbaum, and Cornelia Denz Nonlinear dynamical systems can perform sudden shifts due to perturbations of external control parameters. In optics, modelling the distribution of light according to their nonlinear potentials leads to the formation of caustics as geometrically stable structures whose characteristics are predicted by an analysis in terms of catastrophe theory. Catastrophes are classified by the number of state and control parameters, and can be found in nature as well as created artificially. For one control parameter, the only structurally stable catastrophe is the fold catastrophe, which exhibits the Airy function as a diffraction pattern. Whereas in fold catastrophe, two trajectories coalesce in a singular point, higher order catastrophes may emerge, as the cusp catastrophe with three merging trajectories leading to the caustic Pearcey beam with unique propagation properties. In recent years, caustic beams have experienced a renaissance in optics due to computer-aided spatial light modulator techniques. With these, both the fold catastrophe-based Airy beam and the cusp catastrophe-based Pearcey beam have been realized in paraxial approximation. Striking properties of these beams include acceleration in the transverse direction, auto-focusing, and form-invariant propagation. In our contribution, we utilize these higher-order cusp and also swallowtail catastrophes in paraxial light to transform them into photonic refractive index caustic lattices in nonlinear media. We demonstrate waveguiding with a rich diversity of light guiding paths along 2D quasi cubic and 3D curved caustic structures, and present their functionality as e.g. optical splitters. Finally, taking advantage of the strong auto-focusing of Pearcey beams, we demonstrate the formation of a Pearcey solitary wave.