Experiments on pulse dynamics and parity-time symmetry in optical

Experiments on pulse dynamics andparity-time symmetry in optical fiber networks

Experimente zur Pulsdynamik undParität-Zeit-Symmetrie in optischen Fasernetzwerken

Der Naturwissenschaftlichen Fakultätder Friedrich-Alexander-Universität Erlangen-Nürnberg

zur

Erlangung des Doktorgrades Dr. rer. nat.

vorgelegt von

Alois Regensburgeraus Neumarkt

Als Dissertation genehmigtvon der Naturwissenschaftlichen Fakultätder Friedrich-Alexander-Universität Erlangen-Nürnberg

Gekürzte Fassung der Dissertation ohne Journalartikel zur Online-Veröffentlichung

Tag der mündlichen Prüfung: 13.November 2013

Vorsitzender des Promotionsorgans: Prof. Dr. JohannesBarth

Gutachter: Prof. Dr.Ulf PeschelProf. Dr. FlorianMarquardt

Contents iii

Contents

1. Einleitung / Introduction 1

2. Dynamics of light pulses in passive optical mesh lattices 72.1. Diffusion of particles and waves: From Classical Random Walks to Light

Walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2. Relation to Quantum Walks . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3. Optical mesh lattices: Discrete networks for photonics . . . . . . . . . . . 112.4. Experimental setup: Time-multiplexed fiber loops . . . . . . . . . . . . . 122.5. Pulse dynamics: Iteration equations . . . . . . . . . . . . . . . . . . . . . 162.6. Evolution of a single pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.7. Photonic band structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.8. Bloch oscillations and Landau-Zener tunneling . . . . . . . . . . . . . . . 21

3. Controlled losses generate fractal patterns 253.1. Introducing losses into the mesh lattice . . . . . . . . . . . . . . . . . . . . 253.2. Lossy dynamics and pattern formation . . . . . . . . . . . . . . . . . . . . 27

4. Light pulses in parity-time (PT ) symmetric mesh lattices 314.1. Non-Hermitian Hamiltonians having entirely real spectra . . . . . . . . . . 324.2. PT symmetry in optics: “Gain and loss mixed in the same cauldron” . . . 344.3. Optical mesh lattices with PT symmetry . . . . . . . . . . . . . . . . . . . 354.4. Experimental realization of a temporal PT -symmetric fiber network . . . 374.5. Real and complex modes in the PT band structure . . . . . . . . . . . . . 384.6. Dynamics in the PT -symmetric fiber network . . . . . . . . . . . . . . . . 404.7. Unidirectional invisibility of PT -symmetric scatterers . . . . . . . . . . . 42

5. Localized states around defects in PT -symmetric mesh lattices 455.1. Phase defects in Hermitian mesh lattices . . . . . . . . . . . . . . . . . . . 455.2. PT -symmetric defects in gain/loss mesh lattices . . . . . . . . . . . . . . 48

6. Conclusion and outlook 51

iv Contents

Bibliography 53

A. Generation of pulse sequences with a Gaussian envelope 63

B. Proof of triangle formation in the lossy network 67

C. List of publications 71

D. Publications 73

1

Chapter 1Einleitung / Introduction

Einleitung

Wenn sich kohärentes Licht in einer weitläufigen optischen Struktur oder in einemLichtleiternetzwerk ausbreitet, dann wählt es nicht einfach einen bestimmten Weg,sondern folgt aufgrund seiner Wellennatur gleichzeitig allen möglichen Pfaden. An denKnotenpunkten des Netzwerks treten daher Interferenzen auf, welche zu wohldefiniertenlinearen Erhöhungen und Auslöschungen der lokalen Intensitäten führen. Derartige Effektebilden die Basis für eine Vielzahl von optischen Komponenten und Funktionalitäten,welche von altbekannten Interferometern über Beugungsgitter bis hin zu photonischenKristallen und Kristallfasern sowie gepulsten Faserlasern reichen. Der erste Teilder vorliegenden Doktorarbeit behandelt die kohärente Ausbreitung von klassischenLichtpulsen in einem regulären Gitter von optischen Wellenleitern, welche in einermaschenförmigen Geometrie an diskreten Punkten gekoppelt sind. Dieses optischeMaschengitter dient als die grundlegende Plattform, auf der eine große Vielfalt vonAusbreitungseffekten beobachtet werden kann. Die durch Interferenz getriebeneDynamik, welche in diesem photonischen Kopplernetzwerk auftritt, wird als “LightWalk” (sinngemäß: Zufallsbewegung von Licht) bezeichnet. Ein solcher LightWalk kann als der klassische Analog-Prozess eines zeit-diskreten “Quantum Walk”(Quanten-Zufallsbewegung) auf einer eindimensionalen Linie angesehen werden.

Da die zu Grunde liegende Geometrie periodisch ist, bietet die photonische Bandstruktureinen intuitiven Zugang zum Verständnis der auftretenden Muster in der Lichtausbreitung.Beispiele dafür sind ballistische und beugungsfreie Propagation, Schwebungen zwischenmehreren Moden, photonische Bloch-Oszillationen und Landau-Zener-Tunneln. All dieseEffekte werden mit Hilfe eines zeitlichen Multiplexverfahrens experimentell realisiert. Derdazu eingesetzte Versuchsaufbau besteht aus zwei gekoppelten Faserschleifen, in denen dieräumliche Verteilung der Lichtintensität auf einem Maschengitter in eine zeitliche Abfolgevon umlaufenden optischen Pulsen übersetzt wird. Diese experimentelle Anordnung dientals äußerst stabiles Mehrwege-Interferometer, in dem durch zeitliche Modulation einevollständige Kontrolle der Phase jedes einzelnen Lichtpulses ermöglicht wird. Da sämtliche

2 1. Einleitung / Introduction

Umlaufverluste mit Hilfe von optischer Verstärkung exakt ausgeglichen werden, kanneine in äquivalenten räumlichen Systemen unerreichte Anzahl von Ausbreitungsschrittenbeobachtet werden. Die gekoppelten Faserschleifen erlauben es jedoch auch, Verlustebewusst zu modifizieren. Dabei stellt man fest, dass es verschiedene Arten vonLichtverlusten gibt, welche sich unterschiedlich auf die Pulsdynamik auswirken. Einekleine Veränderung des Messaufbaus führt beispielsweise zum Auftreten einer speziellenSorte von Verlusten, welche nicht kompensiert werden können. Diese ungewöhnlichenLichtverluste ermöglichen eine Reihe von weiteren dynamischen Phänomenen wieAmplituden-Diffusion und sub-exponentielles Abklingen der optischen Leistung. Darüberhinaus kann die Entstehung von fraktalen Mustern mit einer bemerkenswerten Ähnlichkeitzum Sierpinski-Dreieck experimentell beobachtet werden.

Auch im zweiten Teil dieser Dissertation spielt der gezielte Einsatz von Lichtverlusteneine zentrale Rolle, diesmal im Zusammenspiel mit optischer Verstärkung. Dieexperimentellen Untersuchungen wurden von einem abstrakten Konzept inspiriert,welches ursprünglich als mathematische Überlegung zu den Grundlagen derQuantentheorie entwickelt wurde. Wenn ein Hamilton-Operator eine bestimmteRaum-Zeit-Symmetriebedingung erfüllt, die sogenannte Parität-Zeit (PT ) Symmetrie,so können vollständig reelle Eigenwert-Spektren in einer nicht-hermiteschen Umgebungauftreten. Ein physikalisches System ist PT -symmetrisch, wenn es bei einer gleichzeitigenSpiegelung des Orts und Umkehr der Zeitachse wieder in sich selbst übergeht.Praktische Relevanz bekommt das Konstrukt der PT -Symmetrie insbesondere durchseine Übertragbarkeit auf das Gebiet der Optik. Hier wird es für experimentelleUntersuchungen und potentielle Anwendungen zugänglich. Wird ein photonisches Systemnach den Regeln der PT -Symmetrie konstruiert, so ermöglicht dies auf einzigartigeWeise ein harmonisches Zusammenspiel von optischer Verstärkung und Verlusten.Diese kontrollierte Symbiose von verstärkenden und verlustreichen Bausteinen könnteaktive Elemente und Netzwerke ermöglichen, mit denen die Lichtausbreitung nahezubeliebig manipuliert werden kann. Das würde zu neuartigen Funktionalitäten vonLasern und in der optischen Signalverarbeitung führen, welche mit bisher bekanntenphotonischen Komponenten unerreichbar sind. Die vorliegende Dissertation erkundetderartige Möglichkeiten im Zeitbereich, indem optische Verstärkung und Verluste indas Fasernetzwerk integriert werden. Die durchgeführten Experimente bilden die ersteDemonstration eines weit ausgedehnten PT -symmetrischen photonischen Gitters. Dasfortwährende Zusammenspiel von Verstärkung und Verlusten führt zu ungewohntenEffekten wie Leistungsoszillationen und schlagartigen Strahlungsausbrüchen. Darüberhinaus wird sogar die Konzeption von Streukörpern ermöglicht, welche von der einenSeite völlig unsichtbar sind, während sie bei Lichteinfall von der anderen Seite äußerststark reflektieren.Im letzten Teil der Doktorarbeit werden Lichtpulse gezielt durch ein solchesPT -symmetrisches Netzwerk mit Verstärkung und Verlusten geleitet, indem sie entlangvon Defektkanälen im ansonsten periodischen Gitter geführt werden. Die Einbringungeiner solchen Störstelle unterbricht die Periodizität des umliegenden Gitters und erzeugtdadurch lokalisierte Moden in einem normalerweise verbotenen Bereich der Bandstruktur.

3

Je nach Art des Defekts kommt es entweder zu einer kohärenten Oszillation derModenleistung oder aber die Intensität des gebundenen Zustands wächst exponentiell an.Im Gegensatz zur Situation in hermiteschen Gittern können schwach lokalisierte Modenauch im Kontinuum der Bänder liegen. In diesem Fall kann eine stabile, laserartige Modebeobachtet werden, welche fortwährend optische Leistung in die Umgebung abstrahlt.Das Ziel dieser Dissertationsschrift ist es, eine einführende und verständliche

Beschreibung der Lichtausbreitung in zeitlichen Maschengittern zu geben. Allezentralen theoretischen Ergebnisse werden dabei durch experimentell beobachteteAusbreitungsmuster veranschaulicht. Diese bildliche Sichtbarmachung der zu Grundeliegenden physikalischen Effekte ermöglicht ein intuitives Verständnis von abstraktenKonzepten wie gekoppelten und diskreten Rekursionsgleichungen, Bandstrukturen vonFloquet-Bloch-Moden, nicht-hermiteschen Hamilton-Operatoren und exponentiellerLokalisierung. Es ist wirklich faszinierend, dass ein vergleichsweise einfacher Messaufbaumit zwei gekoppelten Faserschleifen eine derart große Vielfalt von dynamischenPhänomenen erzeugen kann. Die in Appendix D dieser kumulativen Dissertationenthaltenen Publikationen [1–3] berichten über weitere Hintergründe und liefern einepräzise wissenschaftliche Beschreibung aller wesentlichen Resultate:

Alois Regensburger, Christoph Bersch, Benjamin Hinrichs, Georgy Onishchukov,Andreas Schreiber, Christine Silberhorn, and Ulf Peschel, “Photon Propagation in aDiscrete Fiber Network: An Interplay of Coherence and Losses”. Physical ReviewLetters 107, 233902 (2011).

Alois Regensburger, Christoph Bersch, Mohammad-Ali Miri, Georgy Onishchukov,Demetrios N. Christodoulides, and Ulf Peschel, “Parity-time synthetic photoniclattices”, Nature 488, 167-171 (2012).

Alois Regensburger, Mohammad-Ali Miri, Christoph Bersch, Jakob Näger, GeorgyOnishchukov, Demetrios N. Christodoulides, and Ulf Peschel, “Observation of DefectStates in PT-Symmetric Optical Lattices”, Physical Review Letters 110, 223902(2013).

4 1. Einleitung / Introduction

Introduction

When coherent light spreads throughout an extended optical structure or network, itdoes not simply choose one certain path—instead, due to its wave nature, it follows allpossible ways at the same time. Consequently, wave interference occurs at the nodes of thenetwork—leading to well-defined linear enhancements and cancellations of local intensities.Such effects are the basis for a multitude of optical components and functionalities, whichspan from long-known interferometers over diffraction gratings to photonic crystals andcrystal fibers as well as pulsed fiber lasers. The first part of the present thesis focuses onthe coherent spreading of classical light pulses in a regular lattice of optical waveguidesthat are discretely coupled in a mesh-like geometry. This optical mesh lattice serves asthe platform to generate a wide range of propagation phenomena. The interference-drivendynamics observed in this photonic coupler network are called a “Light Walk” which isthe classical analog process of a discrete-time Quantum Walk on the line.

As the underlying geometry is periodic, the formalism of photonic band structures pavesthe way to gaining an intuitive understanding of the resulting evolution patterns. Theseinclude ballistic and diffraction-less spreading, beatings between multiple modes, photonicBloch oscillations and Landau-Zener tunneling. All effects are explored experimentallyin a time-multiplexed setup of two coupled fiber loops. In this scheme, the transversespatial distribution of light intensity in the mesh lattice is translated into a temporalsequence of optical pulses circulating around the fibers. The experimental arrangementserves as a highly stable multi-path interferometer that allows full phase control of eachpulse by temporal modulation. As all round-trip losses are exactly counterbalanced bymeans of optical amplification, a large number of propagation steps is observed, beyondthe reach of equivalent spatial systems. In addition, the coupled fiber loops allow for thepurposeful modification of light losses. It turns out that there are several types of losseswhich have a different effect on pulse dynamics. For example, a slight modification tothe fiber-loop setup introduces a special kind of losses that cannot be compensated. Thisyields yet another set of dynamic phenomena like amplitude diffusion, sub-exponentialpower decay and the formation of fractal patterns with a remarkable resemblance to theSierpinski Sieve.Light losses also play a central role in the second part of this thesis, this time in

a well-directed interplay with optical amplification. The experimental investigationswere inspired by an abstract concept which originates from mathematical ideason the foundations of quantum theories. If a Hamiltonian conforms to a certainspace-time reflection symmetry, the so-called parity-time (PT ) symmetry, entirely realspectra become possible in a non-Hermitian environment. A physical system is calledPT symmetric, if it is invariant upon a combined reflection of space and reversal ofthe time axis. This theoretical notion gains practical relevance by its transferabilityto the realm of optics, where it becomes accessible to experimental investigations andpotential applications. Here, a PT -symmetric design of photonic systems uniquelyenables a harmonic interplay between optical gain and loss. The controlled symbiosisof amplifying and attenuating building blocks could bring forth active devices and

5

networks for an arbitrary manipulation of light dynamics. This would potentially allowfor new functionalities which are otherwise unattainable in previously known photonicarrangements. In the present thesis, these possibilities are explored in the temporaldomain by incorporating gain and loss in the two-loop fiber network. This constitutesthe first experimental demonstration of a large-scale PT -synthetic photonic lattice. Theperpetual action of gain and loss onto the light pulses leads to unusual phenomena likepower oscillations and sudden outbursts of radiation. And what’s more, it even allowsthe design of scatterers that are invisible from one side while being strongly reflectivewhen illuminated from the other direction.

In the last chapter of this thesis, light is routed through such a PT -symmetric gain/lossnetwork by guiding it along a defect channel in the periodic lattice. The introductionof a defect disrupts the periodicity of the surrounding lattice, thus creating a localizedmode in an otherwise forbidden region of the band structure. Depending on the natureof the defect, these bound light modes are either coherently oscillating in power or theirintensity grows exponentially. Contrary to Hermitian lattices, weakly localized modes inPT lattices can also lie in the continuum of bands. Here, a stable laser-like mode whichconstantly emits power to the surroundings is observed in experiments.The aim of this thesis is to provide an accessible description of light propagation

in time-multiplexed mesh lattices. All main theoretical findings are illustrated byexperimentally observed propagation patterns. These pictorial manifestations of theunderlying physical effects give an intuitive access to abstract concepts like discretelycoupled iteration equations, band structures of Floquet-Bloch modes, non-Hermitianquantum Hamiltonians and exponential localization. It is truly fascinating how a simpleexperiment with two coupled loops of optical fiber is able to produce this rich variety ofdynamic phenomena. Further details and a precise scientific account of the main resultsare reported in the publications within this thesis [1–3] which are attached in Appendix D:



Alois Regensburger, Mohammad-Ali Miri, Christoph Bersch, Jakob Näger, GeorgyOnishchukov, Demetrios N. Christodoulides, and Ulf Peschel, “Observation of DefectStates in PT-Symmetric Optical Lattices”, Physical Review Letters 110, 223902(2013).

7

Chapter 2Dynamics of light pulses in passiveoptical mesh lattices

In this chapter, the spreading of light pulses throughout a discrete fiber network willbe discussed. The first section explains how the familiar concept of a classical randomwalk is transferred to the domain of optical waves, where interference dominates theobserved dynamics. This process is closely related to Quantum Walks, as will be outlinedin section 2.2. During their propagation, the light pulses “walk” throughout an opticalmesh lattice. In this network, photonic waveguides or fibers are periodically connected atdiscrete positions by integrated couplers (section 2.3). However, the spatial realization ofsuch mesh lattices on a large scale would be quite cumbersome. Therefore, the techniqueof time-multiplexing is applied in the experimental part of the present thesis to facilitatethe observation of fully equivalent pulse dynamics in the temporal domain. Section 2.4explains how a setup of two coupled fiber loops effectively creates a widely extendedoptical mesh lattice. Due to the discrete nature of light evolution in mesh latticesand their temporal equivalents, the dynamics can be fully described by basic iterationequations (section 2.5). The next section shows the rich interference patterns of a singlepulse evolving in the network. These can be understood by considering the photonic bandstructure which provides a convenient way to gain further insights into the mechanismsof excitation spreading (section 2.7). In the final section 2.8, the introduction of a phasegradient induces a cycling around the band structure which gives rise to discrete-timeBloch oscillations and Landau-Zener tunneling.

8 2. Dynamics of light pulses in passive optical mesh lattices

interference

m=0

m=1

m=2

incoherentaddition

Classical Random Walk Light Walk

Figure 2.1.: The Classical Random Walk (CRW) is performed by particles which randomly moveto the left or right at discrete time steps m. In the Light Walk, a coherent opticalpulse steps to the left or right on the same grid. Wave interference occurs wheneverlight can reach the same position by more than one pathway.

2.1. Diffusion of particles and waves: From Classical RandomWalks to Light Walks

A Classical Random Walk (CRW) is often compared to the odyssey of a drunken sailor,randomly staggering to the left and right at every footstep he takes [4–6]. In the end,his zigzag path will look quite chaotic, but if you assume that his walk has been trulyrandom, it can be described by a classical diffusion process with discrete time steps.Formally, in the simplest form of a CRW, the motion of a particle is restricted to a discreteone-dimensional grid of positions n [7, 8]. The process advances in a sequence of discretesteps m, where each time a direction is randomly chosen with balanced probability(Fig. 2.1). Consequently, the probability distribution of a CRW can be described bya Binomial formula and quickly approaches a Gaussian shape [9, p.107ff., 125ff.]. Thewidth of this envelope grows with the square root of the number of the step m, thusspreading very slowly [7, 8].

In the classical walk, the particle only takes one certain path at each realization. Butwhat would happen if we replace the particle by a light wave, spreading throughout thesame type of network as the classical particle? The resulting “Light Walk”1 [8, 10, 11]turns out to be much more complex than the CRW and will be investigated in this chapterof the present thesis. Unlike a particle, a wave does not choose a single well-definedpath if it can spread freely throughout a photonic structure or network (see Fig. 2.1).Instead, it will take all available paths at the same time, each one with certain complexwave amplitude. The main difference to the CRW is what happens at the intersectionsof the network, where more than one pathway arrives at the same position. While thenumber of particles arriving along each of the possible ways is just added up incoherentlyand all phase information is lost in the case of the classical walk, wave interference

1The term “Light Walk” is used in this thesis to describe step-wise propagation of classical waves on amesh lattice (see section 2.3).

2.2. Relation to Quantum Walks 9

takes place in the Light Walk [7, 12]. In the latter case, pulses will also acquire a phaseshift of π/2 whenever they change their direction2. Depending on the relative phasesacquired by the wave along each of the possible pathways, the amplitudes will eitherinterfere constructively or destructively—or anything in between [13]. Such interferenceprocesses play a dominant role in the mechanism of coherent excitation spreading througha photonic lattice or network [14]. This does not only mean that the process is highlysensitive to phase shifts exerted on the waves throughout the network, but also results ina completely different final distribution than for a CRW [7, 15], as will be discussed insection 2.6.

2.2. Relation to Quantum Walks

A fully classical description of light as an electromagnetic wave is chosen in the presentthesis. All experiments are performed with bright pulses of coherent laser light, wherethe consequences of the quantized nature of the electromagnetic light field remainnegligible in the final intensity measurement [16, p. 93ff.]. However, all results inchapter 2 are intimately connected [8, 10, 17] to a process known in the literature asa discrete-time “Quantum Walk” [7, 12, 13, 18–26]. This process has been introduced[18] as the quantum-mechanical analog of a CRW, where a sequence of two unitarytransformations is repeatedly applied to one or multiple quantum particles. The firstunitary transformation is a “coin” operator acting on an additional two-state degree offreedom of the walkers. These internal quantum states can be physically represented bye.g. spin [7] or polarization [19, 24]. After the coin operation has transformed the internalstate, a “step” operator performs a conditional translation of the particle, dependingon the value of the internal state [7, 19]. Quantum walks of single particles have beenimplemented in various different experimental systems using atoms [12], ions [23], NMRs[27] and single photons [13, 19, 24, 28] as their walkers. In all these quantum experiments,the resulting probability distribution is fully equivalent to the intensity distribution foundin a Light Walk [8, 17]. However, in the case of at least two entangled quantum particlesas reported in several works with photons [21, 22, 29, 30], genuine quantum interferencegenerates new behavior not accessible in classical analog experiments [31].While the results of this doctoral thesis might also give a new perspective to the

research field of Quantum Walks, the main focus is to study the classical multi-pathinterference of light pulses spreading in a regular and discrete fiber network. Thefollowing table compares the properties of the three aforementioned processes of “random”evolution in a regular 1D lattice:

2Depending on the definition of the network (see Fig. 2.6), the shift by π/2 can also occur each time apulse continues into the same direction.


Classical RandomWalk (CRW)

Light Walk Quantum Walk

Description Classical particles(incoherent additionof particle numbers)

Classicalelectromagneticwaves (interference)[32, p. 181ff.] [16, p.13ff.]

Quantum mechanics(quantuminterference) [7, 16]

Realization -discrete timesteps

Galton board [33] Optical mesh lattice(bright coherent light[1, 10, 11, 34, 35])

e.g. Optical meshlattice (singlephotons [13, 19, 24],entangled states[21, 22]) or ion traps[23]

Realization -continuoustime

Diffusion processes Photonic waveguidearray (brightcoherent light[14, 36])

Photonic waveguidearray (single photons[28, 37], correlatedphotons [29, 30])

Entanglementpossible

No Not preserved if thesystem containscomponents thatimpair quantumcoherence as e.g.amplifiers [38]

Yes, in case of morethan one particle[21, 22, 29, 30]

Amplificationfeasible

Unclear, might beapplicable to e.g.avalanches

Yes, wide range ofoptical amplifiersavailable thatpreserve classicalcoherence [39, p.476ff., 609ff.]

No, quantumcorrelations sufferfrom amplifier noise[38]; exactamplification ofstates is inhibited bythe no-cloningtheorem [16, p.247]

Detection Particle-countingmechanisms

Intensitymeasurement withe.g. standardphotodiode [1, 8, 10]

To detectnon-classicalinterference, photoncorrelations must bedetected withconditionalmeasurements[21, 22, 29, 31]

Maximumspeed ofspreading

Diffusive, standarddeviation ∼ √m [7]

Ballistic, standarddeviation ∼ m [1]

Ballistic, standarddeviation ∼ m [7, 19]

Realizationofparity-timesymmetry(seechapter 4)

Not possible due tolack of coherenceproperties

Ideal platform forstudying parity-timesymmetric effectiveHamiltonians byincorporating opticalgain and loss[2, 3, 34]

Not possible on afundamentalquantum level asquantum mechanicsis a Hermitian theory[40, p. 220] [41, p.34]

2.3. Optical mesh lattices: Discrete networks for photonics 11

2.3. Optical mesh lattices: Discrete networks for photonicsA light beam propagating in a homogeneous medium will encounter diffraction, i.e. thecross-section of the light pulse will change—and typically become broader [39, p. 127ff.].This behavior can be modified by structuring the refractive index [42, 43] of the opticalmaterial. In the last few decades, researchers have successfully followed this approach tocreate photonic lattices like e.g. waveguide arrays [14]. First examples of such structureswere mostly based on AlGaAs semiconductors [36, 44, 45]. Later, the development offemtosecond laser-inscription of refractive index structures into a solid block of glass in theJena-based research group of Nolte [46, 47] has been a major technological breakthroughin this area of research. Nowadays, the versatility of this approach enables the fabricationof almost arbitrary one- and two- dimensional waveguide arrangements [47]. In thesephotonic arrays, light is subject to a periodic modulation of the refractive index in thedirection transverse to its propagation. The underlying periodicity of the medium givesrise to novel propagation dynamics like discrete diffraction [14], photonic Bloch oscillations[44, 45], dynamical localization [48], Landau-Zener tunneling [49–51] and strain-inducedpseudo-magnetic fields [52]. Many of these phenomena have direct counterparts insolid state physics, where electrons move through periodic crystals and propagationis governed by a band structure of Floquet-Bloch modes [53, p. 181ff.]. In almost allcases of photonic lattices, the dynamics evolves continuously in time and is governed bydifferential equations [43, p. 198ff.] [14].But what happens if the propagation itself proceeds in discrete steps? This means

that not only space, but also the time evolution is discretized in such a system. In ourworld, there are many examples of network structures where interaction only takes placeat discrete nodes and therefore also at discrete points in time. The most prominent oneis the internet, where data packets travel through an enormous network of discretelycoupled transmission channels. In this case, data remains unchanged while travellingalong a channel, but complex processing and non-linear interactions occur at the nodesformed by servers or network routers. Another well-known example for such networksis the electrical power grid, where also complex phase and frequency information isdistributed over transmission lines.Optical mesh lattices [1, 34] are an ideally suited model system to study a greatly

simplified situation in optics. In them, light pulses are restricted to a regular meshof optical fibers or waveguides that are connected by 50:50 couplers at the network’sintersections (see Fig. 2.2). The dynamics in this photonic network essentially proceedsin discrete steps, as transverse coupling between different channels only occurs at thesebeamsplitter elements. By giving each of the waveguides a different refractive index[13, 34], the phases of light pulses can be advanced or delayed. This enables the tuning ofinterference effects and thus controls how a light beam spreads throughout this photonicgrid. Mesh lattices have originally been introduced as an “Optical Galton Board” [10, 11]and some of their properties were studied in the context of discrete-time Quantum Walks(see section 2.2). However, their capabilities for molding the flow of bright classical lightpulses, including the possibility of amplification and losses, remained virtually unexplored.


Ste

p m

Position n

n=1n=0 n=2n=-1n=-2n=-3 n=3

m=0

m=1

m=2

m=3

m=4

Figure 2.2.: Basic layout of an optical mesh lattice. Waveguides or fibers are periodically coupledby rows of 50:50 splitters. Therefore, the transverse coupling on the 1D grid ofpositions n proceeds in discrete temporal steps m. This makes the lattice discrete inboth its coordinates.

2.4. Experimental setup: Time-multiplexed fiber loops

A straightforward way to implement an optical mesh lattice would be to assemble a regularnetwork of optical fibers and integrated couplers as depicted in Fig. 2.2. Alternatively,one might think about realizing the mesh lattice with free-space optics, which wouldresult in a “beamsplitter pyramid” [17]. However, both spatial configurations suffer fromtwo major drawbacks: First, the number of optical components grows quadraticallywhen increasing the number of propagation steps m of light in the network. Therefore,a large-scale spatial mesh lattice would require a disproportionate amount of resources[8, 54]. Second, as the dominating mechanism to govern pulse dynamics in a meshlattice is wave interference, phase stability is absolutely indispensable. Fulfilling thisrequirement in a reasonably large-scale mesh lattice which extends over 100 steps mdemands a precision in the order of one hundredth of the wavelength, as any deviationsaccumulate from step to step.In all reports of spatial approaches to implement optical mesh lattices so far, at least

the phase stability was addressed by special precautions. By embedding laser-writtenwaveguides into a solid block of glass [21, 22, 35] or realizing whole rows of splittersusing birefringent calcite beam displacers [24], detrimental drifts and vibrations of the

2.4. Experimental setup: Time-multiplexed fiber loops 13

short long

short longlong short

interference

5050

L

left right

left rightright left

interference

position time

(a) (c)

(b) (d)

Figure 2.3.: Time-multiplexed dynamics of a mesh lattice. (a) Spatial optical mesh lattice.(b) Here, pulses periodically step to the left and right on a discrete position grid.Interference takes place at all couplers which can be reached by more than onepathway. (c) Equivalent setup of two coupled fiber loops with length difference ∆Land a central 50:50 coupler. (d) At each round trip, pulses perform temporal steps tothe left and right depending on the loop they circle. Figure adopted from Ref. [34].

single channels can be suppressed. While the works of Sansoni et al. [21] and Crespiet al. [22] demonstrated the feasibility of on-chip integrated photonic circuitry in theform of a mesh lattice, their approach still suffered from a limited scalability to a largenumber of steps. A different strategy to tackle the aforementioned issues has alreadybeen introduced in the very first experimental demonstration of an optical mesh lattice:In a conceptual twist, Bouwmeester et al. [10] transferred the dynamics of mesh latticesto the time domain using a recurrence loop scheme where the 1D grid is implementedby discrete frequency shifts. Later, Schreiber et al. [19] adopted this idea but relied ondifferent round-trip times for the two orthogonal states of polarization in their recurrenceloop, thereby realizing the position space as a temporal sequence of light pulses. Thisenabled them to observe up to 28 steps m of a Quantum Walk [13] and to even extendthe dynamics to a second dimension [20].The experimental setup of the present thesis is inspired by these ideas, but uses two

spatially separated fiber loops instead of relying on polarization. The two loops have


time

m = 1 m = 2 m = 3

m → m + 1

∆T

n → n + 2

∆t

Figure 2.4.: Time-multiplexed pulses in the two-loop setup as recorded in one of the loops. Thelength difference ∆L between the two loops advances and delays optical pulses by∆t = ∆L/cfiber. This way, a single initial laser pulse is split into a sequence of opticalpulses which is recorded at every loop round trip m. ∆T = L/cfiber: Average traveltime for one loop round trip; n: horizontal position within a pulse chain.

a length difference ∆L and are connected by a central 50:50 coupler, as schematicallydepicted in Fig. 2.3c. In this time-multiplexed scheme, the same succession of opticaldevices is “re-cycled” at every stepm of the Light Walk. Via the length difference betweenthe two loops, the transverse position grid is convoluted into the time axis. A light pulsetravelling the shorter loop takes a shortcut and thus advances in time—correspondingto a step to the left—whereas pulses in the long loop are delayed, i.e. they take a stepto the right. Afterwards, pulses can change over to the other loop at the 50:50 coupler.Each sequence of short and long loops travelled by a light pulse is fully equivalent toa sequence of left and right steps in the spatial mesh lattice, as illustrated in Fig. 2.3d.After starting with a single pulse in one of the loops, a temporal chain of pulses circulatesaround the two loops with each pulse being confined to a certain time slot or temporalposition n (see Fig. 2.4).A more detailed design of the setup is depicted in Fig. 2.5: The average length of

the two loops is L = 540 m (or in some experiments, L = 5330 m), with the upperloop in Fig. 2.5 being ∆L = 11.4 m (or 102 m, respectively) shorter than the lower one.Two semiconductor optical amplifiers (SOA) compensate for all round-trip losses exceptthe leakage at the central coupler, effectively enabling a “lossless” propagation aroundthe loops. Additionally, an electrically driven phase modulator (PM) [55] in the longloop imposes arbitrary phase shifts on each of the circulating light pulses. This devicegenerates the temporal equivalent of a refractive index distribution, e.g. in the form of atransverse phase gradient. Finally, there is a photodiode attached to a monitor coupler inboth loops to directly record all pulse intensities. In case of the lower loop, the same tapcoupler is also used to insert the initial laser pulse with a wavelength of 1545 nm. Thepulse duration of 45 ns (200 ns) is chosen to be somewhat shorter than the time requiredfor traveling the length difference ∆t = ∆L/cfiber. This ensures that each pulse has awell-separated time slot in the sequence of pulses that forms after several round trips.In both loops of the experiment, various additional fiber-coupled optical componentsare incorporated to provide enhanced pulse control, filter noise and guarantee a stableinterference.Although the length of the fiber loops is huge—with uncontrolled temporal drifts

and insufficient protection against external vibrations—no counter-measures like activeinterferometric stabilization [56, 57] are taken. So how is it possible that stable interference

2.4. Experimental setup: Time-multiplexed fiber loops 15

PD

PMSOA

SOAPD

∆L

v00

vmn

umn

Figure 2.5.: Experimental setup to study a “passive” optical mesh lattice in the time domain. Twoloops of fiber with a length difference ∆L are connected by a central 50:50 coupler.SOA: Semiconductor optical amplifier; PM: Phase modulator; PD: Photodiode.

is observed even after m = 120 round trips, i.e. after the light pulses have travelled morethan 600 km of optical fiber (see Fig. 2.7)? The answer lies in the core principles of howtime-multiplexing works—and gives an understanding of the prospects and limitations ofthis experimental approach.

The same succession of optical components is cycled again and again by the light pulsesin each of the loops. To end at the same temporal position n after a certain number ofround trips m, multiple ways are possible in the time-emulated mesh lattice. But all thesepaths through the network only differ in the sequential arrangement of left-right steps.The total number of round trips in the long loop or steps to the right is always (n+m)/2,whereas the pulse circulates in the short loop (n−m)/2 times. All pulses that arrivesimultaneously and interfere at the central 50:50 coupler have therefore passed throughall fibers and optical devices in the loops the same number of times. If these componentshave not changed considerably within the short time span of a few milliseconds requiredfor 100 steps m, stable interference between all light pulses is guaranteed.

This also explains why long-term phase drifts of the setup do not become apparent inthe final intensity measurement, as long as the system starts with a single laser pulse3.Each of the realizations only takes place in a brief time interval, shorter than the typicaltime scales of external disturbances like temperature drifts or vibrations. Afterwards,averaging is performed over many such single-shot measurements, in order to furtherdecrease the noise level.In contrast, the setup is extremely sensitive to all changes of the optical components

which occur on time scales smaller than the duration of a single measurement shot. If

3To study the behavior of broad pulse distributions with stable relative phases, they first have to begenerated from a single laser pulse within the system before starting the actual measurement. SeeAppendix A for details.


the optical path length or attenuation of a loop changes between the round trips mor positions n, the interference patterns will be altered. In experiment, the fast phasemodulator in the long loop allows to take advantage of this possibility to coherentlymanipulate pulse spreading. In chapter 4 of this thesis, additional amplitude modulatorsare used to quickly control the net gain or loss of both loops.Further details about experimental techniques and additional optical components

not shown in Fig. 2.5 are thoroughly described in the Supplemental Material of theattached Letter [3]. A preliminary version of the setup and the results of first experimentsperformed by Benjamin Hinrichs were reported in Ref. [58].

2.5. Pulse dynamics: Iteration equationsAlmost all the time, laser pulses just travel along one of the loops, well-confined to theirtime slot within the sequence of light pulses. Thus, while circulating, the pulses do notinteract with each other. The only place where a well-defined exchange among positionsand among the two loops occurs is tiny compared to the fiber loops which are hundredsof meters long: At the central 50:50 coupler, pulses tunnel evanescently from one loop tothe other [39, p. 267]. Moreover, if two coherent light pulses simultaneously enter bothinput ports of this tap coupler, their amplitudes will interfere.Because of this point-like interaction at the coupler, the pulse dynamics can be

completely described by the following linear recursion equations [8]:

um+1n = 1√

2

(umn+1 + ivmn+1

)vm+1n = 1√

2

(iumn−1 + vmn−1

)eiϕ(n) .

(2.1)

Here, umn and vmn are the peak amplitudes of pulses in the short and long loop, respectively.The number of the step or round trip is counted by m and pulses can step to the left orright on a discrete grid of positions n (see Fig. 2.2). Basically, Equations (2.1) can be

seen as the combination of the matrix 1√2

(1 ii 1

)of the 50:50 coupler with a step to the

left or right in position n. Moreover, the phase modulator applies position-dependentphase shifts ϕ(n) to the pulse amplitudes vmn in the long loop. Essentially, ϕ(n) is areal-valued optical potential which acts as a discrete-time analog of the refractive indexdistribution in continuous media.Effective pulse dynamics in the time-multiplexed fiber-loop setup of Fig. 2.3c are

fully represented by recursion Equations (2.1). However, the spatial mesh with exactlyequivalent dynamics in both phase and intensity of light waves would look slightly differentthan the one shown in Fig. 2.3a. Equations (2.1) do in fact describe the dynamics of amesh lattice where the two outgoing fibers are crossed-over below the output ports ofeach of the couplers, as depicted in Fig. 2.6a. The unit cell of the completely equivalentmesh lattice would therefore include one crossing of fibers instead of having the simplergeometry shown in Fig. 2.3b. However, in the special case of 50:50 couplers which is

2.6. Evolution of a single pulse 17

(a) (b)

Figure 2.6.: (a) Mesh lattice with crossed-over fibers below each 50:50 coupler has fully equivalentdynamics of pulse amplitudes um

n and vmn as the two-loop fiber setup of Fig. 2.3c.

(b) The evolution of pulse intensities∣∣um

n

∣∣2 and∣∣vm

n

∣∣2 is identical in the non-crossedmesh lattice, while amplitudes are related by a simple phase transformation.

considered here, an exchange of the two output ports or outgoing fibers only correspondsto a phase multiplication. Consequently, pulse intensities

∣∣umn ∣∣2 and∣∣vmn ∣∣2 in the spatial

mesh of Fig. 2.3a exactly correspond to the ones obtained from Equations (2.1), whilea simple phase transformation of the amplitudes umn and vmn is necessary to obtainthe evolution of the complex-valued pulse amplitudes in the spatial mesh lattice (seesection S1 in the Supplemental Material of the attached Letter [3]).

2.6. Evolution of a single pulseInitially, a single laser pulse is inserted into the long loop (v0

0 = 1) and spreads throughoutthe temporal mesh lattice. At each step m and position n, the two photodiodes registerthe pulse peak intensities

∣∣umn ∣∣2 and∣∣vmn ∣∣2 (see the Supplementary Information of the

attached Article [2] for the extraction procedure). The basic dynamics of a pulse spreadingin an empty lattice is displayed in Fig. 2.7. A lattice is called “empty” if all round-triplosses are exactly compensated and no potential of phase shifts is acting on the pulses(ϕ(n) = 0). The perfect correspondence between the experimental data and a numericalevaluation of Equations (2.1) nicely demonstrates the interferometric stability and highquality of the time-multiplexed fiber setup.The rich interference patterns observed in the measurement of Fig. 2.7 mark a clear

difference to any incoherent diffusion process. In addition, the distribution spreadsballistically to both sides and the initial asymmetry of the pulse starting in the longloop remains apparent throughout the propagation [7, 13, 19]. Looking at the underlyingrecursion Equations (2.1), it is difficult to understand this unfamiliar behavior. However,the concept of photonic band structures offers an intuitive way to gain deeper insightsabout the origin of this dynamic evolution. It is the most important theoretical notion forthe investigation of mesh lattices and can be applied to understand almost all propagationphenomena encountered in this dissertation.


Ste

p m

Measurement

(a) short0

50

100

Simulation

(b) short

log 10

(inte

nsity

)

−2

−1

0

Ste

p m

Position n

(c) long

−60 0 60

0

50

100

Position n

(d) long

−60 0 60

log 10

(inte

nsity

)

−2

−1

0

Figure 2.7.: Measured and simulated Light Walk evolution in the fiber setup. Initially, a singlepulse is inserted in the long loop (v0

0 = 1). (a,b) Peak intensities∣∣um

n

∣∣2 of circulatinglight pulses in the short loop are plotted in logarithmic color scale. (c,d) Evolutionof pulse intensities

∣∣vmn

∣∣2 in the long loop.

2.7. Photonic band structureBand structures are a widely used tool to study wave mechanics in a periodic environment.They cannot only describe the motion of an electron in the atomic lattice of a solid [53,p. 181ff.], but also explain the dynamics of light in photonic lattices [43, p. 35ff.].To derive the band structure of an optical mesh lattice, a plane-wave expansion of

Equations (2.1) is necessary4 [15, 34, 59]:(umnvmn

)=(UV

)eiQn/2ei(θ+π)m/2. (2.2)

Here, Q is the transverse wave number in the position coordinate n and θ is the4Here, the shift of the propagation constant θ by π is a phase transformation that “disentangles” thecrossed-over fibers below the couplers in the equivalent spatial mesh lattice (see Fig. 2.6). This resultsin a more symmetric situation.

2.7. Photonic band structure 19

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

k0 = 0

k0 = 0

k0 = π / 2k

0 = π / 2

Q/π

θ/π

Figure 2.8.: Band structure of “empty” mesh lattice (passive case with no phase potential,ϕ(n) = 0). Green and black ovals indicate regions of bands that are excited byGaussian inputs with different tilts k0. The photonic band gap is marked in yellow.

propagation constant. After each double step 2m, the plane wave is rotated by a phase

of θ + π. Each mode has an eigenvector(UV

)which determines the relative phase and

amplitude of the plane wave in the short and long loop. The unit cell of the meshlattice with no phase potential (ϕ(n) = 0) has a size of 2 positions n by 2 steps m (seeFig. 2.2). Inserting this ansatz into a two-step version of Equations (2.1) and solving thedeterminant yields the dispersion relation [34]:

cos θ = 12 (1− cosQ) .

The eigenvectors of these harmonic modes can also be calculated analytically as a functionof the transverse wavenumber Q:

(UV

)= 1√

1 + e−2λ

(1

+e−λe−iQ/2

)in the upper band for θ = + cos−1

[12 (1− cosQ)

](UV

)= 1√

1 + e+2λ

(1

−e+λe−iQ/2

)in the lower band for θ = − cos−1

[12 (1− cosQ)

].

(2.3)

The parameter λ is determined by sinhλ = sin Q2 .


Ste

p m

Measurement k0 = π/2

(a) short0

20

40

Simulation k0 = π/2

(b) short

Ste

p m

Position n

long

−30 0 30

0

20

40

Position n

long

−30 0 30

0 20 400

1

2

3

Ligh

t ene

rgy

Step m

Energy (meas.)

0 20 40Step m

Energy (sim.)

Position n

Measurement k0 = 0

(c) short

Simulation k0 = 0

(d) short

log 10

(inte

nsity

)

−2

−1

0

Position n

long

−30 0 30Position n

long

−30 0 30

log 10

(inte

nsity

)

−2

−1

0

0 20 40Step m

Energy (meas.)

0 20 40Step m

Energy (sim.)

totallongshort

Figure 2.9.: (a,b) Measured and simulated evolution of Gaussian input distribution∼ exp

(− n2

n02

)exp (ik0n) with tilt k0 = π/2 and width n0 ≈ 6, exciting modes

with highest group velocity and no diffractive broadening (black oval in Fig. 2.8).Data of short (

∣∣umn

∣∣2) and long loop (∣∣vm

n

∣∣2) is displayed. (c,d) Same with tilt k0 = 0,exciting modes with zero group velocity and maximum broadening in both bands, asmarked by the green oval in Fig. 2.8.

Fig. 2.8 shows that there are two connected bands with a band gap above and below(yellow regions). As the mesh lattice is periodic in both m and n, the band structure isalso periodic in both of its coordinates and no higher-order bands exist.

From the shape of these bands, one can infer the main features of the spatial evolutionpatterns in an empty mesh lattice: The slope or first derivative of a band determines thetransverse speed of a wave packet. In addition, the curvature or second derivative gives ameasure for the diffractive broadening [34].A well-suited way to explore the band structure and eigenvectors in experiment is to

start with a broad distribution of pulses in the loops (see Appendix A for the experimentalprocedure). The Gaussian envelope ∼ exp

(− n2

n02

)exp (ik0n) has a width parameter n0

and a phase tilt k0. This way, it becomes possible to selectively excite narrow regionsof the band structure. If one excites the Floquet-Bloch modes around the points withmaximum slope and no curvature (k0 = π/2, black oval in Fig. 2.8), the waves spread with

2.8. Bloch oscillations and Landau-Zener tunneling 21

maximum velocity and zero diffraction. This behavior is demonstrated in the experimentof Fig. 2.9a. As the two bands with opposite slopes get both excited, beams emergeto the left and right side. These modes give rise to the ballistic spreading of the LightWalk starting with a single pulse (see Fig. 2.7). In contrast, a central excitation of thebands with k0 = 0 (green ovals) yields a strongly diffracting beam with zero averagegroup velocity (see Fig. 2.9c). The observed beating of intensity between the two loops iscaused by mode interference of the oppositely curved bands with maximum diffractivebroadening. The same effect is responsible for the central interference pattern in Fig. 2.7.In case of a single initial pulse (see Fig. 2.7), all bands get excited with equal weight,

yielding a superposition of all possible propagation modes. Therefore, both the ballisticspreading to both sides as well as the mode beating in the center are visible in this case.

2.8. Bloch oscillations and Landau-Zener tunneling

The band structure can be further explored by studying photonic Bloch oscillations[25, 44, 45] [53, p. 240f.]. To observe them, it is necessary to induce a transverse phasegradient α using the phase modulator in the long loop. In this case, the phase potentialin Equation (2.1) takes the form ϕ(n) = iαn. Interestingly, an identical result can beobtained by implementing a step-dependent phase gradient ϕ(m) = iαm [2, 60]. Thisequivalence can be understood by considering the phase accumulations of all pathways inthe mesh lattice. As the latter possibility demands lower modulation frequencies in thetime-multiplexed experiment, it typically produces nicer results (see Figs. 2.10, 2.11 and2.12).

Effectively, the phase gradient rotates all modes along the band structure by an angleof α/2 at every step m. As the bands are 2π-periodic, the distribution returns to theinitial state after 4M steps for α = π

M . In Figs. 2.10a,b, an initial sequence of pulses

with Gaussian envelope(u0n

v0n

)∼(

11

)exp

(− n2

n02

)exp (ik0n) and k0 ≈ 0 starts in both

loops. This excites a narrow spectrum of eigenmodes around Q = 0 with the eigenvector(11

), which corresponds to the upper band. A gradient α = π

25 rotates the wave packet

around the two bands until it returns to its original position and starts another periodof oscillation. As the transverse group velocity corresponds to the local slope of theexcited band [34, 50], this experiment confirms the analytically obtained form of the bandstructure in Fig. 2.8. Moreover, the relative intensities in the two loops change along the

propagation fully compatible to the analytically obtained form of the eigenvectors(UV

)in Equation (2.3).Figs. 2.10c,d show the same experiment with a slightly changed input distribution(u0n

v0n

)∼(

1−i

)exp

(− n2

n02

)exp (ik0n) and k0 ≈ 0. This time, an equally weighted


Ste

p m

Position n

One band excited

(a) short

−30 0 30

0

50

100

150

Position n

(b) long


Two bands excited

(c) short


(d) long

−30 0 30

log 10

(inte

nsity

)

−1.5

−1

−0.5

0

Figure 2.10.: Bloch oscillations with Gaussian input distribution scan along the band structurein experiment. (a,b) Measurement in short and long loop of oscillations sweepingalong the upper band only. This band is initially excited around Q = 0. (c,d) Anexcitation of both bands gives rise to simultaneous Bloch oscillations along the twooppositely sloped bands.

superposition of the two eigenvectors(

11

)and

(1−1

)is present at step m = 0. As the

two bands have opposite slopes, the Bloch oscillation forces the two excited modes tosplit and to perform a symmetric oscillation into opposite directions. After completing afull oscillation period at m = 100 steps, the initial state is recovered again.In Fig. 2.11, a single pulse starts in the temporal mesh lattice to equally excite all

modes of the band structure. The resulting pattern is much more complex but the initialstate is still recovered after 4M steps. However, increasing the phase gradient α yieldsa tunneling of excitations between the two bands or between sites with equal phasepotential [60, 61]. This process is commonly known as “Landau-Zener tunneling” [49, 51][53, p. 241]. Again, the time-multiplexed experiments nicely demonstrate this effect(Figs. 2.12a,c) in very close agreement to numerical predictions (Figs. 2.12b,d). On largescales, this resonant tunneling of excitations again restores the ballistic spreading to bothsides [62].Thus far, it was shown that optical dynamics in a photonic mesh lattice and its

time-multiplexed equivalent is fairly similar to those encountered in continuous-timephotonic waveguide arrays. While the details certainly are a lot different, the observedphenomena in both systems are qualitatively comparable on a larger scale [63, 64].However, the next chapter demonstrates that a discretely introduced type of lossestriggers dynamic behavior in the discrete-time photonic mesh which is not attainable inany continuous-time arrangement.

2.8. Bloch oscillations and Landau-Zener tunneling 23

Measurement

Position n

Ste

p m

(a) short

−20 0 20

0

40

80

120

160

Simulation

Position n

(b) short

−20 0 20

Measurement

Position n

(c) long

−20 0 20

Simulation

Position n

(c) long

−20 0 20

log 10

(inte

nsity

)

−2

−1

0

Figure 2.11.: Measured and simulated Bloch oscillations with single initial pulse for phasemodulation with gradient α = π/21. Periodic revivals after m = 84 steps occurafter a full rotation around the photonic band structure. (a,b) Short loop; (c,d)long loop.

Measurement

Position n

Ste

p m

(a) short

−20 0 20

0

40

80

120

Simulation

Position n

(b) short

−20 0 20

Measurement

Position n

(c) long

−20 0 20

Simulation

Position n

(d) long

−20 0 20

log 10

(inte

nsity

)

−2

−1

0

Figure 2.12.: Increasing the linear phase slope to α = π/2 induces resonant Landau-Zenertunneling of excitations. (a,b) Short loop; (c,d) long loop.

25

Chapter 3Controlled losses generate fractalpatterns

Commonly, losses are considered a detrimental effect in optics. They often decrease thefidelity of measurement outcomes and impair the performance of optical devices andsystems. In many cases encountered in classical optics, losses can be counteracted bymeans of optical amplification and the presence of leakage or absorption can thereforejust be accounted for by a simple scaling factor. Therefore, photon losses are typicallynot considered to bring something genuinely new into the system. However, as it will beshown in the following, a special form of losses generates entirely unexpected dynamics indiscrete mesh lattices. Section 3.1 explains how these losses are introduced in experiment.Afterwards, the unusual dynamics of the lossy system like sub-exponential power decayand fractal pattern formation are explored in section 3.2.

3.1. Introducing losses into the mesh lattice

To study the peculiar loss dynamics in a mesh lattice, light pulses are removed from thephotonic lattice by blocking all pathways going to the right side after every second row of50:50 couplers (see Fig. 3.1). This creates a lossy mesh network with one half of the lightpathways being absorbed completely after every step m. Looking at the lossy networkin Fig. 3.1, one might expect that a light pulse inserted from above is absorbed by thephotodiodes very quickly and that its intensity decays exponentially in time. However,we will see that things turn out to be very different.

Along with developing the underlying theory, experimental confirmations of all discussedphenomena are presented in the following and—in more detail—in the attached publication[1]. The measurements were performed in the temporal domain, this time with a slightlymodified fiber-optic setup. Instead of having two loops, it consists of only a single loop.In the lossy setup, the length difference ∆L is realized by two separated pathways whichconnect two 50:50 couplers (see Fig. 3.2). This enables steps to the left or right on the

26 3. Controlled losses generate fractal patterns

Ste

p m

Position n

n=1n=0 n=2n=-1n=-2n=-3 n=3

m=0

m=1

m=2

m=3

n=4n=-3Figure 3.1.: Lossy mesh lattice in the spatial domain. At every second row of 50:50 couplers,all fiber pathways going to the right side are blocked by a photodiode (red). Alllight pulses which travel along these fibers are irretrievably removed from the system.Note that a step m corresponds to two rows of beam splitters in this case.

time axis. Importantly, only one of the output ports of the second coupler leads backto the loop—if light leaves the coupler at the other port, it is irrevocably lost from thesetup. Note that the semiconductor optical amplifier cannot compensate the losses atthis output port, as whole light pulses are ejected together with the phase informationthey carry. However, it does compensate all other losses in the setup, like absorption inoptical devices and fibers or leakage from the loop at the monitor coupler.

The pulse dynamics in the one-loop fiber setup of Fig. 3.2 is completely equivalentto the lossy mesh lattice of Fig. 3.1. The system is described by a new set of difference

3.2. Lossy dynamics and pattern formation 27

PD

PD

PMSOA

∆L

loss

qmn

a00

amn

Figure 3.2.: Scheme of fiber setup with one loop to study controlled losses. Two 50:50 couplers areconnected with two fiber pieces that have a length difference ∆L. Complete pulsesare lost at the second port of the right 50:50 coupler and measured by a photodiode(PD). SOA: Semiconductor optical amplifier; PM: Phase modulator.

equations that accounts for the losses occurring at every loop round trip:

Pulses in loop : am+1n = i

2(amn−1 + amn+1

)eiαn

Losses : qm+1n = 1

2(amn−1 − amn+1

).

(3.1)

Here, amn are the complex-valued pulse amplitudes at the temporal position n after mround trips in the fiber loop. In the lossy setup of Fig. 3.2, a step or round trip m nowcorresponds to passing a sequence of two 50:50 couplers (plus the monitor coupler), i.e.two rows of coupler in the spatial mesh of Fig. 3.1. Again, α is a phase gradient that isapplied by the phase modulator in position space. Pulses which get lost at the outputport of the right 50:50 coupler have amplitudes qmn . Remarkably, qm+1

n is just the discretederivative of amn with respect to the position coordinate n.

3.2. Lossy dynamics and pattern formationLooking at Fig. 3.2, one would expect that 50% of the light energy gets lost from thesetup at every round trip m, or equivalently, every second row of couplers in the spatialequivalent of Fig. 3.1. However, in the absence of phase modulation (α = 0), the firstline of Equations (3.1) just corresponds to a discrete diffusion equation. The sum ofamplitudes ∑n

∣∣amn ∣∣ thus remains conserved and the distribution quickly approachesa Gaussian envelope amn ≈ im√

π2m

exp(− n2

2m

). By integrating the shape of

∣∣amn ∣∣2, it isfound that the total light energy E(m) stored in the loop decreases proportional to

1√m, i.e. it decays slower than any exponential function. This is uniquely enabled

by interference processes governing light propagation in the lossy mesh lattice. Themeasurement shown in Fig. 3.3a confirms this diffusive spreading in close agreement tothe numerical simulation (Fig. 3.3b). Indeed, the pulse distribution at the output port

28 3. Controlled losses generate fractal patterns

Ste

p m

Measurement

(a) |a n m|2

0

20

40

60

Simulation

(b) |a n m|2

log 10

(inte

nsity

)

−3

−2

−1

0S

tep

m

Position n

(c) |q n m|2

−20 0 20

0

20

40

60

Position n

(d) |q n m|2

−20 0 20

log 10

(inte

nsity

)

−3

−2

−1

0

Ste

p m

Measurement

(e) |a n m|20

10

20

30

Simulation

(f) |a n m|2

log 10

(inte

nsity

)

−2

−1

0

Ste

p m

Position n

(g) |a n m|2

−20 0 20

0

10

20

30

Position n

(h) |a n m|2

−20 0 20

log 10

(inte

nsity

)

−2

−1

0

Figure 3.3.: (a,b) Measured and simulated pulse evolution in the loop of the lossy setup withoutphase modulation (α = 0), revealing a diffusion of amplitudes with exceptionally lowlosses. (c,d) At the output port, the discrete derivative of the distribution in the loopis observed together with a quick decline of the loss rate. Inducing a phase gradientof (e,f) α = 7

26π and (g,h) α = 922π leads to the emergence of triangular fractals. As

exponential energy loss occurs in the presence of phase modulation, a net gain perround trip is applied in (e-h) for compensation.

(see Figs. 3.3c,d) decays much quicker than the intensity in the loop. As shown in theattached publication [1], adding decoherence in the form of random phase noise restoresthe classically expected exponential decay of average dynamics.Apart from the non-exponential decay, more surprises lie hidden in the lossy loop’s

dynamics governed by Equations (3.1). To reveal them, let us switch on the phasemodulator to impose a linear gradient α = p

qπ with coprime integers p,q. The result isdepicted in Figs. 3.3e-h and 3.4: Now, the pulses evolve in a pattern of triangles witha height of q steps. When increasing p and q, the structures approach a fractal shape.Below the triangles, all amplitudes are exactly zero—the peculiar symmetries of thephotonic mesh result in a destructive interference between all possible light pathways.From the bottom corners of each triangle, a new triangle starts. In the big picture,the resulting pattern has similarities to the famous Sierpinski sieve (see Fig. 3.4) andis overlaid by a diffusive envelope. The physical mechanism to produce these fractalpatterns is even robust enough to withstand experimental imperfections, as demonstratedin the fiber-loop experiments of Figs. 3.3e,g and Figure 3.4a,c.The formation of triangular structures can be understood as a dissipative analog of

the photonic Bloch oscillations in the lossless mesh lattice which were presented insection 2.8. As the lossy mesh lattice has a cosine-shaped imaginary band structureof decaying plane waves amn ∼ (cosQ)m eiQm for α = 0, it acts like a spatial frequencyfilter on any input distribution a0

n. High-frequency modes decay exponentially whilethe lowest-frequency mode (a0

n ∼ 1 for Q = 0) is completely lossless. This is also

3.2. Lossy dynamics and pattern formation 29S

tep

m

Measurement

Position n

(a) |a n m|2

−20 0 20

0

10

20

30

40

50

60

70

Position n

Simulation

(b) |a n m|2

−20 0 20

log 10

(inte

nsity

)

−2

−1

Ste

p m

Measurement

Position n

(c) |a n m|2

−20 0 20

0

10

20

30

40

50

60

70

Position n

Simulation

(d) |a n m|2

−20 0 20

log 10

(inte

nsity

)

−2

−1

00

Figure 3.4.: Measured and simulated fractal patterns for phase gradient (a,b) α = 37π and (c,d)

α = 25π.

the origin of the sub-exponential energy decay in the absence of phase modulation(α = 0). In the beginning, the single initial impulse a0

n = δn0 equally excites the wholespectrum and afterwards, the cosine-shaped filter is applied to the pulse distributionat every step m. In case of a non-zero phase gradient α = p

qπ, the factor eiαn inEqs. (3.1) repeatedly induces a step-wise shift of all harmonic modes with respectto the cosine-shaped filter function. After a rotation by an angle of pπ around theband structure (i.e. after q steps or round trips), all modes have been repeatedlydamped such that only the two outermost pulses survive. Due to desctructive waveinterference, all amplitudes in between are exactly zero. Along these lines, an analyticalcalculation is presented in Appendix B which gives a mathematical proof for this behavior.

The experiments on the passive and lossy evolution of light pulses in a temporal meshlattice are reported in the following Letter and its Supplemental Material [1]:


31

Chapter 4Light pulses in parity-time (PT )symmetric mesh lattices

So far, the amplifiers in the fiber setup were only used to compensate for round trip lossesand therefore increase the number of observable steps in the Light Walk. In this chapter,it will be shown that a well-balanced interplay of amplification and losses generates newand otherwise unattainable pulse dynamics. These experimental investigations have beenfacilitated by the notion of parity-time (PT ) symmetry that originates from mathematicalconsiderations on abstract quantum field theories. Section 4.1 gives an introductionabout PT -symmetric Hamiltonian operators which allow for real eigenvalue spectra in anon-Hermitian environment. Interestingly, analogous concepts can be utilized in optics touniquely enable a harmonic coexistence of optical gain and loss (section 4.2). An ideallysuited platform to study this phenomenon can be created by embedding amplifiers andabsorbers into optical mesh lattices, as discussed in section 4.3. In experiment, a temporalversion of these gain/loss mesh lattices is realized using a fiber loop setup (section 4.4). Tounderstand the observed dynamics, the PT band structure is first derived in section 4.5.The first experimental observation of such PT dynamics in an extended photonic latticeis afterwards presented in section 4.6. In this active non-Hermitian environment, lightpropagation behaves distinctively different than in conventional media. For example,the total energy of light pulses is no longer a conserved quantity and a sharp transitionbetween exponentially growing and almost stable modes is observed in the fiber loops.Finally, a parity-time symmetric scatterer can become invisible from one side, whileit strongly reflects light impinging from the other direction. In section 4.7, temporalmeasurements confirm this unidirectional invisibility of PT Bragg elements.

32 4. Light pulses in parity-time (PT ) symmetric mesh lattices

4.1. Non-Hermitian Hamiltonians having entirely real spectra

In mathematical physics, fully real eigenvalue spectra HΨ = EΨ with all eigenvaluesE ∈ R are usually associated with Hermitian operators H† = H. This conditionguarantees the reality and stability of energy and other observables in classical andquantum theories [40, p. 220]. However, in 1998 it was discovered by Carl Bender andhis colleague Stefan Boettcher [65] that even a weaker requirement exists that permitsthe reality of an operator’s eigenvalues: If the Hamiltonian H is invariant to a combinedapplication of the parity (P ) and the time reversal (T ) operator, i.e. HP T = P T H,its spectrum can become entirely real. Physical systems governed by such effectiveHamiltonian operators are called “PT symmetric” [66].

Here, the P operator mirrors the spatial coordinate x and the momentum p: P xP = −xand P pP = −p. Its counterpart, the time reversal operator T , leaves the coordinateinvariant (T xT = x) but reverses the momentum (T pT = −p). Additionally, T is anantilinear operator which flips the sign of the imaginary unit i to −i. The square ofboth T and P is the identity operator (P 2 = T 2 = 1) and the two reflection operatorscommute with each other: T P = P T [66, 67] [40, p. 178f].Let us only consider the basic example of a 1D time-independent Hamiltonian which

fulfills parity-time symmetry. It can be written as H = p2

2m+V (x) with the complex-valuedpotential V (x) = VRe(x) + iVIm(x). In this case, PT symmetry demands V (x) = V ∗(−x)which is equivalent to a symmetric real potential VRe(x) = VRe(−x) and an antisymmetricimaginary part VIm(x) = −VIm(−x). Even if PT symmetry is fulfilled, entirely realeigenvalues are not always guaranteed. Indeed, parity-time symmetry is not a sufficientcondition for a non-Hermitian Hamiltonian to possess a real spectrum [68]. A sharp “phasetransition”1 from real to complex spectra is observed when increasing the magnitudeof the imaginary potential VIm(x) with respect to the real part VRe(x) [65–67, 71]. Theparameter set where this transition occurs is called “PT threshold” [72]. A system belowthreshold is often said to be in its “exact PT phase” where all eigenfunctions of theHamiltonian H are also eigenfunctions of the P T operator, i.e. they are invariant uponcombined space-time reflection. This is no longer valid in the “broken PT phase”: In thisregime, some eigenfunctions of H are not PT -symmetric, as they are associated with acomplex energy eigenvalue E [66].Going back to our example, let us now take a look at an eigenfunction ΨE of the

Hamiltonian, i.e. HΨE(x) = EΨE(x). The system’s dynamics is governed by theSchrödinger equation i~ ∂

∂tΨ(x, t) = HΨ(x, t) [40, p. 219]. Therefore, the time evolutionof an initial eigenstate Ψ(x, t = 0) = ΨE(x) is given by Ψ(x, t) = e−iωtΨE(x) with thefrequency ω = E

~ . This makes clear why the reality of the eigenvalues E is of fundamentalimportance. Only an entirely real eigenvalue spectrum ensures a stable evolution, whereasany eigenfunctions with Im(E) 6= 0 will grow or decay exponentially in time [70].

But even if the spectrum of a PT -symmetric Hamiltonian is entirely real, its dynamics

1The term “phase transition” [69, 70] in the context of PT symmetry should not be confused withthermodynamic phase transitions. Instead, it refers to the symmetry-breaking transition from partiallycomplex to fully real eigenvalue spectra.

4.1. Non-Hermitian Hamiltonians having entirely real spectra 33

is still remarkably different from any Hermitian system. While the orthogonality

+∞∫−∞

Ψ∗E1(x)ΨE2(x)dx = 0

of any two eigenfunctions ΨE1 and ΨE2 is guaranteed for Hermitian Hamiltonians, thisproperty is generally not fulfilled in parity-time symmetric environments. Here, thiscondition is replaced by the “quasi”-orthogonality relation [67]

+∞∫−∞

Ψ∗E1(−x)ΨE2(x)dx = 0. (4.1)

The implications of a PT Hamiltonian with eigenfunctions that are not orthogonal inthe common sense, i.e.

+∞∫−∞

Ψ∗E1(x)ΨE2(x)dx = A 6= 0, will be explored in the following

example. Let us prepare a PT -symmetric system with real eigenvalues E1 and E2 andcorresponding eigenfunctions ΨE1 and ΨE2 in the initial state

Ψ(x, t = 0) = ΨE1(x) + ΨE2(x).

The functions are normed such that the total “probability” P (t) starts with

P (t = 0) =+∞∫−∞

Ψ∗(x, 0)Ψ(x, 0)dx = 1. Afterwards, it evolves as:

P (t) =+∞∫−∞

Ψ∗(x, t)Ψ(x, t)dx =+∞∫−∞

∣∣∣e−iω1tΨE1(x) + e−iω2tΨE2(x)∣∣∣2 dx =

=+∞∫−∞

∣∣ΨE1(x)∣∣2 dx+

+∞∫−∞

∣∣ΨE2(x)∣∣2 dx+

+∞∫−∞

e−i(ω2−ω1)tΨ∗E1(x)ΨE2(x)dx+ c.c.

= P1 + P2 + 2A cos

[(ω2 − ω1) t

].

This means that even in the regime of exact PT symmetry with all-real eigenvalues,fundamental quantities like the total “probability” or energy are not conserved anymore.Instead, a beating of the power with the differences between all eigenfrequencieswould be observed in such a hypothetical system. In above example with only twoeigenmodes involved, this reduces to a harmonic oscillation in time. However, defining“quasi”-probabilities and -powers in analogy to Equation (4.1) yields conserved quantitiesagain [67, 72].To summarize, a 1D Hamiltonian with PT -symmetric potential V (x) = V ∗(−x) has

partially complex spectra with exponentially growing or decaying eigenfunctions whenit is in the broken PT phase. Even in the exact PT phase with entirely real spectra,quantum observables oscillate around a mean value because of skewed eigenmodes.


In the context of an experimental thesis, all of this might sound like a purelymathematical curiosity about the peculiar properties of some unconventional Hamiltonianoperators. Indeed, the implications of such PT -symmetric Hamiltonians on quantumfield theories are still a matter of intense scientific debate (see e.g. Refs. [73–75]).However, these abstract ideas quickly found surprising applications in optics, whereeven an experimental observation of PT -symmetric dynamics becomes possible. Therealization of parity-time symmetry in optics does not introduce any conflict to the veryHermiticity of quantum mechanics, as systems with external sources and sinks for energyare reduced to an effective non-Hermitian system [76, 77].

4.2. PT symmetry in optics: “Gain and loss mixed in the samecauldron”

The transfer of parity-time symmetry from quantum theories to optics is facilitatedby an isomorphism of the involved wave equations. While quantum waves obey theSchrödinger equation, the paraxial wave equation in optics possesses an equivalentmathematical structure [72, 78] [43, p. 229ff.]. In optics, the complex optical potentialn(x) = nRe(x) + inIm(x) takes the role of the quantum potential V (x). Here, the realpart nRe(x) is the usual refractive index of the optical material, whereas the complexpart nIm(x) stands for optical gain or loss, depending on its sign.The condition for PT symmetry in optics is thus given by n(x) = n∗(−x) which

demands a symmetric refractive index nRe(x) = nRe(−x) and an antisymmetricdistribution of gain/loss nIm(x) = −nIm(−x). Again, when varying the relativemagnitudes of nIm(x) and nRe(x), a PT phase transition from complex to fully realeigenvalue spectra occurs in this system [72]. This transition has drastic consequenceson the behavior of optical waves, as any modes with complex propagation constantswill grow or decay exponentially in time, while real modes guarantee a certain degreeof stability. One example for a PT -symmetric optical lattice is a gain/loss waveguidearray [67, 72, 79] as schematically depicted in Fig. 4.1. In it, waves continuously coupleto adjacent waveguides while at the same time being amplified or damped in their ownchannel.

The real spectra found in PT -symmetric optical systems uniquely allow the design ofwell-behaved optical systems and devices that incorporate balanced amounts of gain andloss [80, 81]. Even minor deviations from this peculiar symmetry will inevitably result inan exponential growth or decline of mode power. However, assuming the strict adherenceto this reflection symmetry, a multitude of interesting and otherwise unattainablepropagation phenomena and device functionalities may come into experimental reach, asshown in a vast number of theoretical studies (e.g. [67, 72, 78, 82–101]). These includeband merging effects associated with exceptional points [85], the break of left-rightsymmetry, PT -symmetric laser absorbers [86, 87], power oscillations [72], PT Blochoscillations [84] and unidirectional invisibility [88–91]. Moreover, high-power lasers maybe constructed by following PT -symmetric design principles [97] and a close relationshipbetween the PT threshold in unbounded and bounded configurations has been established

4.3. Optical mesh lattices with PT symmetry 35

Figure 4.1.: Scheme of continuous waveguide array with parity-time symmetric distribution ofoptical gain (red) and loss (blue) as well as the refractive index.

[98].The well-controlled interaction between optical gain and loss could also bring substantial

advances to the emerging areas of metamaterials [102] and nanoplasmonics [99, 100].These fields suffer from the high optical losses of metal structures and would thereforegreatly benefit from integrated amplifying elements [103]. Finally, nonlinear opticalsystems with PT symmetry could support a new class of solitons [104, 105] and enablethe construction of an optical diode without relying on the Faraday effect [106]. Note thatthis dissertation focuses exclusively on linear effects in both experiments and theoreticalconsiderations.

4.3. Optical mesh lattices with PT symmetryWhile the prospects of PT symmetry are certainly tempting, the practical realization ofthese gain/loss photonic arrangements remained a big hurdle. Until recently, parity-timesymmetry has only been demonstrated in elemental systems consisting of two coupledbuilding blocks. The first realization of PT symmetry was achieved by Guo et al. in2009 [107] by coupling a high-loss optical waveguide to a low-loss one. One year later,Rüter et al. [70] reported on the first elemental PT structure with balanced optical gainand loss. In addition, experiments on parity-time symmetry breaking were performedby coupling a lossy to an amplifying resonator in other areas like electronics [108–110],microwaves [111] or mechanical oscillators [69].The extension of these ideas to large-scale PT lattices as in Fig. 4.1 [72] has been

hindered by the strict requirements imposed by the symmetry relation n(x) = n∗(−x).In a continuous optical medium, three conditions have to be simultaneously fulfilledby the PT array: First, the refractive index or real-valued potential has to be exactlysymmetric. Second, the discrete channels of such a lattice have to be very close to eachother to provide a sufficient rate of evanescent coupling between next neighbors. Themost challenging third condition is an antisymmetric distribution of optical absorption


Figure 4.2.: Conception of spatial PT -symmetric mesh lattice. Gain (red) and loss (blue)waveguides provide an odd transverse distribution of the complex optical potential,while phase shifts ±ϕ0 are distributed with even symmetry. The coupling in thisparity-time symmetric mesh proceeds in discrete temporal steps. Figure created incollaboration with Gerd Beck.

and amplification without any significant pump inhomogeneity. To add up, a theoreticalstudy suggests that the dispersion of optical materials might lead to further obstacles[112].

While these strict requirements certainly make the realization of continuous PT latticessomewhat challenging, they can be elegantly circumvented in optical mesh lattices [2, 34].Together with several collaborators, this concept has been developed in the contextof this dissertation to finally tackle the aforementioned technical issues in experiment.Fig. 4.2 shows the illustrative conception of an active gain/loss mesh lattice as a spatialoptical system. This photonic network is PT symmetric, i.e. it combines an odddistribution of gain and loss in equal amounts with exact left-right reflection symmetry ofall other waveguide characteristics. The phase shifts ±ϕ0 provide the necessary real-partmodulation of the optical potential. In this mesh network, the three tasks of a parity-timelattice can be spatially separated and therefore distributed to specialized building blocks:The regions of gain and loss are distinct and can be implemented by incorporatingdedicated linear amplifiers and attenuators. Moreover, passive directional couplers realizethe necessary coupling between adjacent waveguides and slightly different lengths orrefractive indices realize the phase shifts. This assembly of the spatial PT mesh latticefrom discrete single-purpose functional elements brings a distinct technical advantageover continuous waveguide lattices.

4.4. Experimental realization of a temporal PT -symmetric fiber network 37

PM PM

(a)

Figure 4.3.: To realize PT symmetry in the time-multiplexed fiber network, loops are switchedbetween optical gain (red) and loss (blue) after every round trip m. The phase shifts±ϕ0 are imposed via the phase modulator (PM).

4.4. Experimental realization of a temporal PT -symmetricfiber network

For the experimental implementation, the PT -symmetric mesh lattice is transferred tothe temporal domain, just like it was done with the passive mesh (see section 2.4). Torealize the required antisymmetric distribution of the imaginary potential in positionspace, the loops are periodically switched between optical gain and loss after every roundtrip m (see Fig. 4.3). Additionally, the even real part of the parity-time symmetricpotential is induced by the phase modulator in the long loop. This way, the dynamics ofcirculating light pulses in the temporal lattice becomes fully equivalent to the spatialconfiguration shown in Fig. 4.2.

Only slight modifications to the two-loop experimental setup of Fig. 2.5 are necessaryto realize the desired PT potential in the time-multiplexed grid of positions n and roundtrips m. Now, the semiconductor optical amplifiers (SOA) [113] do not only compensatefor losses, but also provide a constant amount of excess gain. Additionally insertedacousto-optic modulators (AOM) [39, p. 815ff.] in both loops provide a time-varyingdegree of attenuation (see Fig. 4.4). Depending on the damping factor of the AOMs,light pulses are either effectively amplified or attenuated by the same factor G after oneloop round trip m.In the resulting PT -symmetric iteration equations, this additional net gain/loss G is

the only difference to the dynamics of the passive system (see Equations (2.1)):

um+1n = G±1/2

√2

(umn+1 + ivmn+1

)vm+1n = G∓1/2

√2

(iumn−1 + vmn−1

)eiϕ(n).

(4.2)

The exponent of G switches from −1/2 to +1/2 between alternate loops in every stepm; that is, the loops are repeatedly switched between gain and loss in equal amounts [2].As in Equations (2.1), umn and vmn are the peak amplitudes of light pulses in the short andlong loop, respectively. The evolution occurs on a discrete 1D grid of well-separated time


50%

50%

50%

PD

PD

ISO SOAAOM

ISO SOAAOM PM Laser v0

0

|umn |2

|vmn |2

∆Lum

n

vmn

Figure 4.4.: Scheme of fiber loop setup to realize PT symmetry in the time domain. Two loopsof fiber with a length difference ∆L are connected by a central 50:50 coupler. Theacousto-optic modulators (AOM) are oppositely driven to switch between opticalgain and loss after every loop round trip m. SOA: Semiconductor optical amplifier;PM: Phase modulator; ISO: Faraday isolator; PD: Photodiode. See Fig. S13 in theSupplemental Material of the attached Letter [3] for further details.

slots or positions n and the phase modulator (PM) in the long loop induces a transversepotential of phase shifts ϕ(n). To fulfill PT symmetry, ϕ(n) has to be a symmetricfunction. In the experiment, this is realized by the following periodic step function withpotential strength ϕ0:

ϕ(n) =

−ϕ0 for mod(n+ 3; 4) = 0; 1+ϕ0 for mod(n+ 3; 4) = 2; 3 .

(4.3)

4.5. Real and complex modes in the PT band structureAgain, the band structure is a well-suited instrument to understand the pulse dynamicsobserved in PT -symmetric optical lattices. While the reality of all bands is guaranteedin Hermitian systems, they can become complex in non-Hermitian gain/loss lattices. Aswe will see in the following, the reality of bands plays a decisive role in the associateddynamics. In analogy to the procedure described in section 2.7, the band structure ofthe PT mesh lattice is obtained by expanding Equations (4.2) with plane waves(

umnvmn

)=(u(n)v(n)

)eiQn/4ei(θ+π)m/2.

4.5. Real and complex modes in the PT band structure 39

−1 0 1−1

0

1

Q/π

θ/π

−1 0 1Q/π

−1 0 1Q/π

passive(G = 1)

−1

0

1θ/π

real imaginary

ϕ0 = 0 ϕ

0 = 0.4π ϕ

0 = π

PT(G = 1.5)

Incr

ease

gai

n/lo

ssIncrease potential

Figure 4.5.: Band structure of the PT mesh lattice as in Equation (4.4) for different values ofgain/loss G and phase potential ϕ0. In the broken PT phase, bands merge at theexceptional points (black dots) and imaginary modes arise. Blue: Real part; Red:Imaginary part; Q: Transverse wave number; θ: Propagation constant.

As the phase potential ϕ(n) of Equation (4.3) has a period of n = 4, the unit cell ofthe mesh lattice is now twice as broad as in the empty lattice with ϕ(n) = 0 (compareEquation (2.2)). The resulting dispersion relation can be written as [2, 34]:

cos (Q) = 8 cos2 (θ)− 8 cosh (γ) cos (ϕ0) cos (θ) + 4 cos2 (ϕ0)− 4 + cosh (2γ) . (4.4)

The amount of gain and loss in the PT lattice enters this equation via the coefficientγ = logG. For the passive case (gain/loss factor G = 1, i.e. γ = 0) without a potential ofphase shifts (ϕ0 = 0), above equation is equivalent to the result of section 2.7. However,the two bands are now folded in from the left and right side because of the different sizeof the unit cell, as depicted in Fig. 4.5.This Figure visualizes the PT phase transition from partially complex to fully real

band structures of Floquet-Bloch modes. The threshold for this transition depends onthe two parameters ϕ0 and G and can be analytically obtained as (see SupplementaryFig. 5 of the attached publication [2] and Ref. [34]):

γ = logG = min{

cosh−1[2 cos (ϕ0)−

√cos (2ϕ0)

]; cosh−1√2

}.

In the exact PT phase below this threshold, all light modes will evolve with realeigenvalues which means that they will neither grow nor decay in the long run despitebeing subject to gain and loss. Above threshold, the broken PT phase yields partiallycomplex propagation constants θ which are associated to exponentially increasing or


(a)G=1,

0 = 0

(b)G=1.4,

0 = 0

Frequency

(c)G=1.4, 2

0 = 0.39

(d)G=1.4, 2

0 = 0.41

Ste

p m

Position n−30 0 30

0

20

40



Position n

−30 0 30

log 10

(inte

nsity

)

−2

−1

0

0 20 400

2

4

Ligh

t ene

rgy

Step m

totallongshort

0 20 40Step m

0 20 40Step m

0 20 40Step m

gain potential potential

Figure 4.6.: Experimentally observed evolution of single initial pulse in the PT -symmetric fibernetwork for different values of gain/loss G and phase potential ϕ0. For details, seethe attached Article [2].

decaying modes. In this regime, two real bands merge at a so-called exceptional point,where the changeover to a pair of modes with complex-conjugate eigenvalues takes place.These points offer fascinating physics by themselves which are only briefly touched uponin this thesis. References [72, 91, 114–116] are recommended to the interested reader tolearn more about this topic.

4.6. Dynamics in the PT -symmetric fiber network

In the temporal fiber network, the PT phase transition can be studied with highexperimental precision. To probe all bands at the same time, a single pulse is initiallyinserted and its evolution throughout the mesh lattice is afterwards monitored by thephotodiodes (see Fig. 4.6). The passive case (G = 0) without a potential (ϕ0 = 0) whichis depicted in Fig. 4.6a corresponds to the Light Walk discussed in chapter 2.6. Increasingthe gain/loss to G = 1.4 yields a spectrum with imaginary propagation constants in thecenter of the Brillouin zone. The associated eigenmodes grow exponentially in time, asconfirmed in the measured evolution of Fig. 4.6b.However, the introduction of a phase potential 2ϕ0 = 0.39π brings the system just atPT threshold, leaving a single exceptional point in the center of the band structure. Theobserved linear growth in energy is directly related to this point (see Fig. 4.6c) [71]. A

4.6. Dynamics in the PT -symmetric fiber network 41S

tep

m

Position n

Passive

(a) short

−30 0 30

0

50

100

150

Position n

(b) long


PT

(c) short


(d) long

−30 0 30

log 10

(inte

nsity

)

−1.5

−1

−0.5

0

Figure 4.7.: Bloch oscillations in passive and PT mesh lattices. In the measurements, a Gaussianinitial distribution with tilt k0 = 0 is inserted in both loops, to selectively excite oneof the bands (see Appendix A). (a,b) Passive lattice (same as Figs. 2.10a,b) and (c,d)PT lattice. The action of gain and loss results in periodic outbursts of radiation.

very subtle increase of the potential to 2ϕ0 = 0.41π finally pushes the bands apart. Eventhough all mode constants θ are now real, the energy is no longer a conserved quantity.As the corresponding eigenmodes are not orthogonal, power oscillates around a meanvalue (Fig. 4.6d) [70, 72]. The origin of this behavior has been explained in section 4.1.Again, all experimental results fit very closely to numerical simulations, as demonstratedin the Supplementary Information of the attached Article [2].

The tunability of the experimental arrangement allows to automatically scan througha wide range of network configurations. The sharp transition observed in Fig. 3 of theattached publication [2] coincides well with theory and could potentially be used forenhanced pulse control in ring laser cavities.

To further explore the peculiar properties of the PT band structure, Bloch oscillationsare used to experimentally probe the local shape of the bands. The peculiar propertiesof these coherent oscillations in PT -symmetric gain/loss lattices were first predicted byLonghi [84]. As in Fig. 2.10 of section 2.8 where the whole bands were excited by a singlepulse, broad Gaussian pulse distributions are now used to selectively excite a narrowregion. In the passive case (Fig. 4.7a,b and 2.10a,b), the phase gradient ϕ(m) = iαminduces a step-wise rotation around the two connected bands, with the transverse groupvelocity corresponding to the local slope of the excited region. In Fig. 4.7c,d, the smallamount of gain/loss G = 1.075 is introduced to bring the system into the regime ofbroken parity-time symmetry with a band structure similar to the one shown in Fig. 4.6b.The Gaussian distribution starts in a real section of the mode structure and sweepsaround the bands. Most of its way, the evolution looks just the same as in the passivecase. However, at the points with maximum transverse velocity, the excitation brieflytouches the region with imaginary modes, each time producing an outburst of radiation


Passive

a

PT (from left)

b

PT (from right)

cS

tep

m


0

70

140


Position n700−70

log10(intensity)

−2

−1

0

Figure 4.8.: Scattering experiment in the time-multiplexed fiber network. (a) A pulse sequencewith Gaussian envelope impinges onto three passive Bragg scatterers created viaphase modulation. (b) Addition of PT -symmetric gain and loss at the position of thescatterers renders them invisible form the left side (c) while being strongly reflectivefrom the right.

and a coupling of energy to the oppositely sloped band.

4.7. Unidirectional invisibility of PT -symmetric scatterersSo far, the peculiar propagation of light pulses in extended parity-time symmetric latticeshas been discussed. This section will study how a light beam will scatter off finite gain/lossBragg elements. To perform scattering experiments in the time-multiplexed fiber network,a series of three temporal Bragg scatterers is embedded into the surrounding periodicmesh lattice by appropriate temporal modulation (see Fig. 4.8).In the first measurement, a light beam impinges on a series of three passive Bragg

elements [39, p. 69f.] [43, p. 44ff.]. They are implemented in the time-multiplexednetwork by switching on the symmetric potential function ϕ(n) of Equation (4.3) onlyat the position ranges of the Bragg stacks. Fig. 4.8a confirms that these temporalstructures act like spatial Bragg elements, reflecting some part of the impinging lightbeam backwards and transmitting the rest into the forward direction2.Next, anti-symmetric gain and loss are introduced at the positions of the temporal

Bragg stacks to render them PT symmetric [92] while the surrounding lattice remainspassive (Fig. 4.8b,c). The parameters are chosen such that all scatterers are exactly atPT threshold, leaving a single exceptional point in their spectrum (similar to Fig. 4.6c).In the literature [88–90], unidirectional scattering properties have been predicted forthis kind of structure. The measurement of Fig. 4.8b demonstrates that the parity-timesymmetric scatterers indeed become invisible if light enters from the left side. Reflected

2Note that despite the two-dimensional impression of Fig. 4.8, the system only has one spatial dimensionwhile the vertical dimension is the discrete time axis.

4.7. Unidirectional invisibility of PT -symmetric scatterers 43

beams are strongly suppressed while transmission is close to unity. But if the broadGaussian light beam enters from the other side onto the temporal PT scatterers, reflectionis greatly enhanced (Fig. 4.8c). As the light beam is subjected to optical gain and loss, theback-reflected beams are even stronger than the initial pulse sequence while transmissionremains unchanged. And last but not least, the beam which is back-reflected at thecentral Bragg element is impinging onto the rightmost scatterer from the left side. Asthis scatterer is invisible from this direction, no back-reflection is observed in this case.The spatial realization of such unidirectionally invisible Bragg mirrors would enable

new possibilities in the design of laser cavities, where the lossy layers might also bereplaced by passive ones. In this case, both transmitted and reflected waves wouldbe further amplified, while the reflection is still completely suppressed from one side.After the publication of the aforementioned temporal experiments [2], a unidirectionallyreflectionless structure was indeed realized as a waveguide grating with alternatingpassive and absorptive sections by Feng et al. [117]. Note that because of Lorentzreciprocity, the construction of an isolator without magnetism, nonlinearity or temporalmodulation is impossible from such linear optical structures [91, 118]. However,assuming a nonlinear behavior of the optical system, it becomes possible to obtain trulynon-reciprocal port-to-port transmission characteristics [106].

The temporal realization of parity-time symmetric optical mesh lattices is reported inthe following Article and its Supplementary Information [2]:


This manuscript was highlighted in a News & Views comment [81]:

Luca Razzari, and Roberto Morandotti, “Optics: Gain and loss mixed in the samecauldron”, Nature 488, 163-164 (2012).

Furthermore, the fiber-loop experiments were spotlighted in a Nature MaterialsCommentary which discusses the prospects of exceptional points for realizingnon-reciprocal and undirectionally invisible optical systems [91]:

Xiaobo Yin, and Xiang Zhang, “Unidirectional light propagation at exceptionalpoints”, Nature Materials 12, 175-177 (2013).

45

Chapter 5Localized states around defects inPT -symmetric mesh lattices

In the previous chapter, the concept of parity-time synthetic lattices in optics wasintroduced and the associated beam dynamics as well as peculiar scattering propertieswere demonstrated in temporal experiments. Now the capabilities for molding the shapeand power of optical beams are further explored by guiding light modes along pre-definedchannels. This can be achieved by introducing a defect into the otherwise periodic meshlattice. In passive media, localization around lattice defects is a well-known phenomenonwhich is transferred to discrete-time systems in section 5.1. While the power of boundmodes in Hermitian systems remains constant, it can grow or oscillate in PT -symmetriclattices. Section 5.2 discusses light guidance along phase defects in such a gain/lossenvironment. Finally, an embedded laser-like bound mode that constantly emits radiationis demonstrated.

5.1. Phase defects in Hermitian mesh lattices

When an irregularity disrupts the periodicity of a lattice, it often supports a localizedmode which cannot escape from the defect as its eigenvalue lies within the band gapof the adjacent structure. Defects do not only play a central role in solid state physics[119, p. 412ff.], but are also an important concept for the design of optical devices andstructures [43, p. 58ff.]. These include embedded waveguides and resonators withinphotonic crystals [120] as well as photonic crystal fibers [121]. In photonic waveguidearrays, the formation of localized modes has also been studied for defects in the refractiveindex distribution [47, 122]. Now this concept is transferred to discrete-time optical meshlattices where similar defect modes can be observed.Before introducing complicated defect structures in optical mesh lattices, let us first

take a look at a basic phase defect in a passive mesh. This configuration has beenproposed in a theoretical work by Wojcik et al. [26] in the context of Quantum Walks.

46 5. Localized states around defects in PT -symmetric mesh lattices

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n=0 n=1 n=2 n=3 n=4 n=5 n=6 n=7 n=8n=-1n=-2n=-3n=-4n=-5n=-6n=-7

m=0

m=1

m=2

m=3

n=-8

�d

�d

�d

Potential:(real)

�d

Position n

Ste

p m

Figure 5.1.: An elemental phase defect ϕd is embedded at one pair of positions n = 0, 1 in theotherwise periodic mesh lattice.

Fig. 5.1 illustrates the idea of embedding a finite number of waveguides with an additionalphase shift ϕd at two central positions n = 0, 1 into the periodic structure. In this case,the phase potential ϕ(n) of the passive evolution equations (2.1) takes the form:

ϕ(n) =

ϕd for n = 0, 10 else.

The aim is now to search for localized modes which are supported by the phaseirregularity. In this basic case, the following form can be assumed at all even steps m:

umn = eiθdm/2e−α|n|/2

E n ≤ −1E for n = 0, 1Fe−iϕd n ≥ 2

vmn = eiθdm/2e−α|n|/2

F n ≤ −1E for n = 0, 1Ee−iϕd n ≥ 2

(5.1)

5.1. Phase defects in Hermitian mesh lattices 47

/(a)

Q/0.5 0 0.5

1

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d/

(b)

(c)

(d)

(e)

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d = 0.3

d =

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(e)

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nsity

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1

0

Figure 5.2.: Observation of localized states in a passive mesh lattice. (a) Band structure of emptymesh lattice without defect (ϕd = 0) as in Fig. 2.8 (folded in). (b) Dispersion of defectmode constant θd depending on defect strength ϕd. (c) Without a defect (ϕd = 0), nolocalization is observed in the measurement (compare Fig. 2.7). (d) Introduction of aweak defect ϕd = 0.3π yields a single localized mode while an increase to (e) ϕd = πproduces a beating between two defect modes. See the Supplemental Material of theattached manuscript [3] for details.

The ansatz of Equations (5.1) is chosen such that the modes have a left-right symmetrybetween the two loops around the defect and a phase jump at the defect. A phase of θdis acquired by the localized mode after every two steps m. In addition, the mode hasamplitudes E and F and decays with a constant α to both transverse directions. Inconventional Hermitian lattices, such a decay is only possible if the propagation constantθd is situated in the band gap of the periodic environment (yellow parts of Fig. 5.2a)and thus the defect mode cannot couple to the radiation. Inserting this form of solutioninto the two-step evolution equations of Ref. [34] yields two transcendental equations.Numerical calculations1 finally determine all parameters as a function of the strength ϕdof the defect. The resulting dispersion of the mode number θd is depicted in Fig. 5.2b.Here, one or two localized modes form within the band gap of the periodic structure,depending on the magnitude of ϕd.

This behavior is fully confirmed in the measurements of Figs. 5.2c-e: While an ordinaryLight Walk is observed for zero phase defect (Fig. 5.2c), a single localized mode is

1The mode equations were numerically solved by M.-A. Miri.


excited in Fig. 5.2d in case of a shallow defect with ϕd = 0.3π. An increase to ϕd = π

simultaneously excites two bound modes and therefore yields a beating pattern duringthe course of propagation (Fig. 5.2e). The period of this beating is determined by thedifference of the two associated mode constants θd. Also in the case of defects, allmeasurement results do very well correspond to numerical simulations, as demonstratedin the Supplemental Material of the attached Letter [3].

5.2. PT -symmetric defects in gain/loss mesh lattices

Figure 5.3.: Schematic conception of a defect within a spatial PT mesh lattice (compare toFig. 4.2). In this example, one pair of fibers has a gain/loss or phase potential whichis different from the surrounding periodic lattice. Gain: red; loss: blue.

While everything behaves smooth and well-defined in the passive case, things get a lotmore exciting when a defect is introduced into a periodic structure which incorporatesoptical amplification. The concept of utilizing defects as a means to steer and controllight waves in the presence of gain is routinely applied in optical amplifiers and laserresonators. Examples for such non-Hermitian devices include semiconductor distributedfeedback laser diodes [123], photonic crystal fiber amplifiers [124] and lasing modes inphotonic crystals which are localized around a lattice irregularity [125]. However, allthese examples have in common that the light intensity will grow inexorably until gainsaturation sets in. The introduction of defects in PT -periodic arrangements which arebased on a balanced combination of both gain and loss offers entirely new possibilitiesto control such localized modes. So far, a number of theoretical studies suggested theintroduction of defects in parity-time symmetric lattices with continuous time evolution(see e.g. [79, 93, 126, 127]). The results presented in the attached Letter [3] are the firstexperimental demonstration of localized modes in a PT -symmetric environment.Fig. 5.3 shows the spatial conception of a discrete-time mesh lattice composed of

amplifying and attenuating waveguide channels with a central defect in the refractiveindex profile. The question now is whether light will still be localized around this defect

5.2. PT -symmetric defects in gain/loss mesh lattices 49

in the same way as in the passive case despite the constant action of optical gain or loss.How will the threshold of the PT phase transition be modified by the lattice irregularityand what will happen if an exceptional point comes into play? In time-multiplexedPT -symmetric mesh lattices, all of this can be readily investigated in experiment.The defects are introduced to the gain/loss PT mesh lattice in a manner analogous

to the passive defects of the previous section. The experimental arrangement allowsintroducing defects both in the real and in the imaginary part of the refractive indexprofile, thus creating an enormous parameter space of accessible lattice configurations.A modified version of Equations (4.2) is necessary to account for this general type ofdefects:

um+1n = 1√

2G(n+ 1)

12 (−1)m(umn+1 + ivmn+1)

vm+1n = 1√

2G(n)−

12 (−1)m(iumn−1 + vmn−1)eiϕ(n).

Again, umn and vmn are the amplitudes of light pulses at position n and step orround trip m circulating in the short and long loop, respectively. The total phasepotential ϕ(n) = ϕp(n) + ϕd(n) is the sum of a periodically alternating phase function

ϕp(n) ={

+ϕp for mod (n, 4) = 0, 1−ϕp for mod (n, 4) = 2, 3 and a defect ϕd(n) =

ϕd for n in defect0 else.

As in the case of the fully periodic lattice discussed in the previous chapter, gain andloss are switched between the two loops after every round trip m. However, the gain/losscoefficient G(n) = Gp + Gd(n) has now also become a function of the position coordinate.While its periodic component Gp is constant throughout the lattice, a defect gain/lossGd comes on top in a certain range of positions n. The defect function is thus written as

Gd(n) =

Gd for n in defect0 else.

An ansatz similar to Equation (5.1) is used to numerically

obtain the structure and dispersion of bound modes2.All defects are introduced such that the optical potential still remains parity-time

symmetric. This means that despite the defect, the overall gain/loss profile is anantisymmetric function of the position n, while the phase potential is symmetric.

The attached publication [3] reports on a PT phase transition of a single pair of defectmodes depending on the magnitude of the defect phase ϕd. These complex localizedmodes are embedded in an extended PT lattice below threshold with all-real propagationconstants. This means that via small parameter variations, the shape and intensity ofguided modes can be tuned at will while being guided through the gain/loss network.

In contrast to Hermitian lattices, states localized around defects in PT lattices are notanymore restricted to the band structure. Provided that their associated propagationconstant θd has a non-zero imaginary part, these modes can form a hybrid betweenlocalized and freely propagating lattice modes. This makes it possible for the real partof θd to reside within the continuum of bands. In this case, the ongoing loss of energy

2Numerical calculations were performed by M.-A. Miri.


Position n

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short(a)

50 0 50

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long(b)

50 0 50

log 10

(inte

nsity

)

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Figure 5.4.: Measurement of a laser-like mode at the gain threshold. A defect structure withboth gain/loss as well as a phase potential within an empty lattice supports alocalized mode with oscillating energy that emits light beams to the surroundings.Experimental data of pulse intensities in (a) short and (b) long loop is displayed; seethe attached Letter [3] for details.

to propagating modes is compensated by an exponential growth of the mode amplitudewhich enables the exponential localization of the mode profile at each of the temporalsteps m.

Finally, right at a threshold gain value, a lasing mode is guided along a broad gain/lossdefect in the temporal mesh lattice (see Fig. 5.4). Due to a peculiar balance betweenamplification, attenuation and leakage to propagating lattice modes, the optical powerwithin the defect region oscillates around a stable mean value. At the same time, themode constantly emits coherent light waves to the surroundings. This behavior is indirect analogy to a laser cavity which is operated at its gain threshold where the powerwithin the resonator remains stable while a constant flow of energy is emitted from thefacets [39, p. 503ff.].

All these results and further details on localized modes in parity-time symmetric opticalnetworks are reported in [3]:

Alois Regensburger*, Mohammad-Ali Miri*, Christoph Bersch, Jakob Näger, GeorgyOnishchukov, Demetrios N. Christodoulides, and Ulf Peschel, “Observation of DefectStates in PT-Symmetric Optical Lattices”, Physical Review Letters 110, 223902(2013).

The Supplemental Material of this manuscript contains a detailed and comprehensivetechnical description of the experimental fiber-loop setup and the methods followed whenperforming measurements.

* These authors contributed equally.

51

Chapter 6Conclusion and outlook

This dissertation has explored wave dynamics in a mesh network of optical fibers wheretransverse coupling only occurs at discrete points in time and space. Light propagationin this network is governed by the interference between all pathways offered by the latticestructure. The resulting time evolution can be fully described by a simple set of recursionequations which is best understood by considering the associated band structure. Itsslope and curvature determine the transverse velocity and diffractive broadening of wavepackets while the beating between multiple Floquet-Bloch modes gives rise to interferencepatterns.In the mesh lattice, both the phase and intensity of light pulses can be readily

manipulated by phase shifting as well as amplification or attenuation. The latter twoingredients allow for the extension of optical dynamics into the realm of non-Hermitianphysics, where gain and loss are deliberately put at work to control the evolutionof light waves. This generates an unexpected variety of physical phenomena whichrange from amplitude diffusion processes over the formation of fractal patterns to thePT phase transition and unidirectional invisibility. All of these effects were successfullydemonstrated in a temporal experiment which is based on two coupled loops of opticalfiber. As all optical pulses circulating the fibers obey the same dynamics as in a spatialmesh lattice, this approach enables the observation of light evolution in large-scale meshlattices.

In this new type of photonic lattice, almost arbitrary shapes of real and complex opticalpotentials can be realized by temporal modulation. The peculiar balance of gain andloss to fulfill parity-time symmetry is only a tiny part of the accessible parameter space.Therefore, one could extend this research to study many other non-Hermitian physicalconfigurations and explore their capabilities for providing previously unattainable opticalfunctionalities for the design of light sources and amplifiers as well as the manipulation ofcoherent light evolution. In particular, the results could be applied to provide enhancedpulse control in fiber ring laser cavities which rely on similar geometries.

Throughout this dissertation, all optical devices and media were assumed to be entirelylinear. This means that none of the observed dynamics depend on the power of the

52 6. Conclusion and outlook

initial light beam. However, a nonlinear response of the experimental arrangement toan increase of the light intensity would generate an entirely different behavior of opticalpulses in the temporal mesh network. To this end, a high-power version of the fiber-loopsetup is currently being built up and operated in the Master project of Martin Wimmerwhich is supervised by me. First results successfully demonstrate the formation andcollision of solitary light beams in these temporal experiments [128, 129]. Finally, thepossibility to selectively excite bands with positive and negative curvature in the systemallows the first observation of a nonlinearly accelerated “diametric drive” [130] in thefiber-loop experiments [131].

BIBLIOGRAPHY 53

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63

Appendix AGeneration of pulse sequences witha Gaussian envelope

In several experiments presented within this thesis, a sequence of pulses with amplitudes∼ exp

(− n2

n02

)exp (ik0n) is used as the starting distribution of the system. Its Gaussian

envelope has a width parameter n0 and a phase tilt k0. The challenge is to guaranteea stable phase relationship between all light pulses within this Gaussian pulse chainin all measurement realizations even though the two-loop fiber setup has no long-terminterferometric stability (see section 2.4). To achieve this, the distribution has to begenerated inside the fiber setup after starting with a single initial laser pulse.This is the only way to ensure that all the pulses that interfere at a certain time slot

of the temporal network have passed all the optical components and fibers of the setupexactly the same number of times, but in a different sequential arrangement. Otherwise,inevitable phase drifts of the setup will result in a randomized relative pulse phase ateach realization of the experiment. The protocol used to generate a well-defined and fullycoherent sequence of pulses with Gaussian envelope will be presented in the following.

As described in chapter 3 and the attached publication [1], the desired Gaussian pulsesequence is approached by the system when introducing controlled losses at every secondrow of beam splitters in the mesh lattice (see Fig. 3.1). Such a system cannot only berealized in the lossy one-loop setup of Fig. 3.2, but also in the two-loop setup of Fig. 2.5by temporally modulating the losses in one of the loops. This is achieved by switchingoff the semiconductor optical amplifier (SOA) in the short loop at every second roundtrip to attenuate all pulses by more than 30 dB, thus realizing a temporal equivalent ofFig 3.1. After 30 round trips m in this lossy system, a Gaussian pulse sequence with awidth of n0 ≈ 6 and a phase tilt of k0 ≈ π/2 has been generated1, following the dynamicsof Fig. 3.3a. From now on, the amplifiers are switched on all the time to realize a losslessmesh lattice. Without any further manipulation of pulse phases, the resulting pulse

1Because of experimental imperfections, the actual value obtained in experiment is k0 ≈ 0.85π. A minoradditional phase shift is applied by the phase modulator to correct to the desired phase tilt of k0 ≈ π.

64 A. Generation of pulse sequences with a Gaussian envelope

Ste

p m

Position m

(a) short

−30 0 30

−30

−20

−10

0

10

20

30

40

50

Position n

(b) long

−30 0 30

log 10

(inte

nsity

)

−2

−1

0

Figure A.1.: Generation of Gaussian pulse sequence with k0 ≈ π in experiment. This Figure showsthe same measurement as Fig. 2.9a but additionally includes the steps m = −30 to−1 during which the Gaussian chain of pulses is generated from a single input laserpulse. During this preparation phase, all intensity is removed from the short loop atevery second round trip m.

sequence will spread as in Fig. 2.9a. The same measurement is displayed in Fig. A.1which additionally shows the first 30 loop round trips m where the Gaussian sequence isgenerated.Any other value of k0 can also be realized in experiment by introducing a phase shift

∆k right after the generation of the Gaussian sequence has finished. This action rotatesall modes around the band structure by an angle of ∆k. As the result, a phase-tiltedGaussian pulse chain with tilt k0 ≈ π/2−∆k/2 emerges. This is e.g. demonstrated inFig. 2.9c for ∆k = π.One round trip later, an additional phase shift of ∆k′ = ±π/2 can be used to

phase-rotate all pulses circulating the long loop. In the center of the Brillouin zone (Q = 0

and k0 = 0), this allows to selectively excite the eigenvectors(UV

)∼(

11

)or(

1−1

)of

the upper or lower band, respectively (see Equation (2.3) in section 2.7).In the experiment demonstrating unidirectional invisibility (see Fig. 4.8), the input

distribution has been additionally damped in the short loop by the AOM during the firstround trip m = 1. This was done to tailor the input state to match the eigenvector atthe edge of the Brillouin zone (Q = π and k0 = π/2) which requires different intensitiesof the Gaussian pulse chains in the two loops. Therefore, no second beam is going to theleft in this measurement at the beginning (m = 0).

Further information about the methods for the generation of Gaussian pulse sequencesand for the selective exciation of a single band is presented in Ref. [131]. Finally,the modulation timings used to generate a Gaussian input distribution and to realizePT symmetry afterwards are illustrated in the simplified scheme of Fig. A.2.

65

Laser

Amp1

Amp2

AOM1

AOM2

PM

on

off

+j0

0

on

off

on

off

loss

gain

loss

gain

1

1

-j0

Dk

g*

Electronic signal timing for controlling PT symmetric dynamics with Gaussian input distribution(simplified schematics, not to scale)

Generate Gaussian distribution Apply phaseslope

PT symmetric propagation with potential j0Warmup

Time

Figure A.2.: Schematic of electronic timing scheme to generate Gaussian pulse sequences from asingle laser input and to realize PT symmetry. Some simplifications were applied toincrease clarity. Amp1 and AOM1 are in the short loop; Amp2 and AOM2 are inthe long loop. A small gain factor g∗ is applied in the long loop to pre-amplify theGaussian sequence. “Absorbing boundaries” prevent an uncontrolled noise build-upand signal cross-talk between the temporal domains of two round trips m and m+ 1.

67

Appendix BProof of triangle formation in thelossy network

As discussed in section 3.2, a fractal pattern of triangles forms when a position-dependentphase gradient eiαn is applied in the lossy mesh lattice. The following section gives ananalytical proof for the formation of such triangles in the propagation pattern. Theunderlying dynamics of the lossy system is given by the recursion equation

am+1n = 1

2(amn−1 + amn+1

)eiαn (B.1)

with α = p

qπ and coprime p,q ∈ N.

Here, amn ∈ C are complex numbers or field amplitudes, m ∈ N0 is the number of theiteration and n ∈ Z is a transverse position coordinate. Moreover, α is a phase gradientthat acts in position space. Compared to Equation (3.1) of section 3.1, the global phasefactor of i per step m is omitted as it just adds a constant phase rotation.

The aim is to show that this system evolves in a pattern of triangles with a height of qsteps m. Below these triangles, all amplitudes are exactly zero, except for two pulsesat the bottom edge of the triangle (see Figs. 3.3e-h and 3.4). The following theoremexpresses this statement in a formal mathematical way.

Theorem

The initial condition am=0n = δn0 in Equation (B.1) results in the following amplitudes

after q iteration steps (m = q):

∣∣aqn∣∣ =

12q for n = ±q0 else

68 B. Proof of triangle formation in the lossy network

ProofFirst, we start with a "decaying" plane-wave ansatz for the amplitudes amn :

amn = A(Q)b(Q)meiQneiΘm

with the wave number Q ∈ [−π,π]1, a complex plane-wave amplitude A(Q), a real-valuedpropagation constant Θ(Q) and a real-valued decay constant b(Q).

Inserting this ansatz in Equation (B.1) for α = 0, i.e. in the absence of a phase gradient,yields the following dispersion relation:

Θ = 0, b = cosQ (B.2)

This means that a plane wave with wave number Q decays with a factor of b = cosQper iteration step m, as |cosQ| ≤ 1. Only the modes with Q = 0 or π will not lose theirstrength when undergoing iteration steps.Next, the delta-like initial condition am=0

n = δn0 can be decomposed into its planewave components:

A(Q) =∑n

a0ne−iQn = 1

Let us now calculate how each of these plane waves am=0n (Q) = A(Q)eiQn will evolve

through the system under the action of a non-zero phase gradient α = pqπ. In this case,

Equation (B.1) can be separated into two intermediate steps per iteration m:

Diffusive action: am+1n = 1

2(amn−1 + amn+1

)Spatial frequency shift by α: am+1

n = am+1n eiαn

In the diffusive part, a plane wave A(Q)eiQn is multiplied by a frequency-dependentdamping factor cosQ according to Equation (B.2). Afterwards, the plane wave ismultiplied with the phase factor eiαn, thus shifting it by a frequency offset α inFourier space. In the next iteration m, the whole procedure is applied again to thefrequency-shifted plane wave.

The plane wave evolves as:Iteration Evolution of the plane wavem = 0 a0

n(Q) = 1eiQnm = 1 a1

n(Q) = cosQeiαneiQnm = 2 a2

n(Q) = cosQ cos (Q+ α) ei(Q+2α)n

......

m amn (Q) =[k=m−1∏k=0

cos (Q+ kα)]ei(Q+mα)n

1This size of the Brillouin zone is chosen to simplify mathematical treatment.

69

−1 −0.75 −0.5 −0.25 0 0.25 0.5 0.75 1−1

−0.5

0

0.5

1

wave number Q / π

Fre

quen

cy d

ampi

ng fa

ctor

per

roun

dtrip

Figure B.1.: The Fourier amplitudes of plane waves with initial frequency Q are repeatedlymultiplied by shifted cosine-shaped damping functions. This example shows allcosine factors for m = q = 4.

For m = q and inserting α = pqπ, the plane wave eiQn has acquired a shift of

pπ in its spatial frequency. Moreover, its amplitude is given by a product of shiftedcosines:

aqn(Q) =

k=q−1∏k=0

cos(Q+ k

p

qπ

) ei(Q+pπ)n (B.4)

In the case of p = 1, the product samples the cosine function at the equally distributedpoints cos(Q), cos

(Q+ 1

qπ), cos

(Q+ 2

qπ), ... , cos

(Q+ q−1

q π). For p > 1, the same

absolute values of the cosines are contained in this product, but in a different sequentialarrangement. However, the sign of the total product varies between ±1. As this globalsign does not enter the final result for

∣∣aqn∣∣, we can continue assuming p = 1 without lossof generality.

Using the formula sin(qx) = 2q−1k=q−1∏k=0

sin(x+ k

qπ)

from Ref. [132, p. 62] and

sin(x+ π

2

)= cosx, Equation (B.4) can be written as:

aqn(Q) = −12q−1 sin

(qQ+ qπ

2

)eiQn

Let us finally go back to the evolution of a single impulse am=0n = δn0. The distribution

after q steps m can be obtained by the inverse Fourier transform of all its plane-wavecomponents:

70 B. Proof of triangle formation in the lossy network

∣∣aqn∣∣ =

∣∣∣∣∣∣∣1

2π

π∫−π

dQ1

2q−1 sin(qQ+ qπ

2

)eiQn

∣∣∣∣∣∣∣=

∣∣∣∣∣∣∣1

2π1

2q−1

π∫−π

dQ sin(qQ+ qπ

2

)eiQn

∣∣∣∣∣∣∣=

∣∣∣∣∣∣∣1

2π1

2q−112i

π∫−π

dQ(ei(q+n)Qei

qπ2 − e−i(q+n)Qe−i

qπ2)∣∣∣∣∣∣∣

=∣∣∣∣ 12q(ei

qπ2 δ(q + n)− e−i qπ

2 δ(q − n))∣∣∣∣

= 12q(δ(q + n) + δ(q − n)

)Evaluating this expression at all n ∈ Z results in the statement of the theorem.

71

Appendix CList of publications

Peer-reviewed publications (related to this dissertation)

Alois Regensburger, Christoph Bersch, Benjamin Hinrichs, Georgy Onishchukov, AndreasSchreiber, Christine Silberhorn, and Ulf Peschel, “Photon Propagation in a DiscreteFiber Network: An Interplay of Coherence and Losses”. Physical Review Letters 107,233902 (2011).

Mohammad-Ali Miri, Alois Regensburger, Ulf Peschel, and Demetrios N. Christodoulides,“Optical mesh lattices with PT symmetry”, Physical Review A 86, 023807 (2012).

Alois Regensburger, Christoph Bersch, Mohammad-Ali Miri, Georgy Onishchukov,Demetrios N. Christodoulides, and Ulf Peschel, “Parity-time synthetic photonic lattices”,Nature 488, 167-171 (2012).

Alois Regensburger*, Mohammad-Ali Miri*, Christoph Bersch, Jakob Näger, GeorgyOnishchukov, Demetrios N. Christodoulides, and Ulf Peschel, “Observation of DefectStates in PT-Symmetric Optical Lattices”, Physical Review Letters 110, 223902 (2013).* These authors contributed equally.

Martin Wimmer, Alois Regensburger, Christoph Bersch, Mohammad-Ali Miri, SaschaBatz, Georgy Onishchukov, Demetrios N. Christodoulides, and Ulf Peschel, “Opticaldiametric drive acceleration through action-reaction symmetry breaking”, Nature Physics(Advance Online Publication) 10.1038/nphys2777 (2013).

Peer-reviewed publications (not related to this dissertation)

Sergei G. Romanov, Alexander V. Korovin, Alois Regensburger, and Ulf Peschel, “HybridColloidal Plasmonic-Photonic Crystals”, Advanced Materials 23, 2515 (2011).

72 C. List of publications

Sergei G. Romanov, Nicolas Vogel, Karina Bley, Katharina Landfester, Clemens K.Weiss, Sergej Orlov, Alexander V. Korovin, Gennady P. Chuiko, Alois Regensburger,Alexandra S. Romanova, Arian Kriesch, and Ulf Peschel, “Probing guided modesin a monolayer colloidal crystal on a flat metal film”, Physical Review B 86, 195145 (2012).

Sergei G. Romanov, Alois Regensburger, Alexander V. Korovin, Alexandra S. Romanova,and Ulf Peschel, “Noninvasive manipulation of the optical response of opal photoniccrystals”, Physical Review B 88, 125418 (2013).

Oral conference contributions (selected)Alois Regensburger, Christoph Bersch, Georgy Onishchukov, Benjamin Hinrichs, AndreasSchreiber, Christine Silberhorn and Ulf Peschel, “Experiments on quantum walks:Zitterbewegung, Bloch oscillations and particle losses”, Quantum Information Processingand Communication (QIPC) 2011, Zurich, Switzerland.

Alois Regensburger, Christoph Bersch, Georgy Onishchukov, Benjamin Hinrichs, AndreasSchreiber, Christine Silberhorn and Ulf Peschel , “Bloch oscillations, Landau-Zenertunneling and fractal patterns in a discrete fiber network” (invited talk), CLEOQuantum Electronics and Laser Science 2012, San Jose, USA.

Alois Regensburger, Christoph Bersch, Georgy Onishchukov, and Ulf Peschel,“Nonlinearity-induced suppression of Landau-Zener tunneling”, Nonlinear Photonics 2012,Colorado Springs, USA.

Alois Regensburger, Christoph Bersch, Mohammad-Ali Miri, Georgy Onishchukov,Demetrios N. Christodoulides, and Ulf Peschel , “Experimental PT-symmetry breaking ina large-scale optical fiber network”, PHHQP XI: Non-Hermitian Operators in QuantumPhysics 2012, Paris, France.

Alois Regensburger, Christoph Bersch, Mohammad-Ali Miri, Georgy Onishchukov,Demetrios N. Christodoulides, and Ulf Peschel , “Experimental PT-symmetry breakingin a large-scale fiber network”, Scattering Systems with Complex Dynamics, XI. InformalBilliard Workshop 2012, Regensburg, Germany.

Alois Regensburger, Mohammad-Ali Miri, Christoph Bersch, Jakob Näger, GeorgyOnishchukov, Demetrios N. Christodoulides, and Ulf Peschel, “Bound states in atemporal fiber network with parity-time symmetry”, European Conference on Lasers andElectro-Optics 2013, Munich, Germany.

Alois Regensburger, Mohammad-Ali Miri, Christoph Bersch, Jakob Näger, GeorgyOnishchukov, Demetrios N. Christodoulides, and Ulf Peschel, “Experiments on opticaldefects states in a temporal PT lattice”, PHHQP 2013, Istanbul, Turkey.

73

Appendix DPublications

Alois Regensburger, Christoph Bersch, Benjamin Hinrichs, Georgy Onishchukov, AndreasSchreiber, Christine Silberhorn, and Ulf Peschel. “Photon Propagation in a DiscreteFiber Network: An Interplay of Coherence and Losses”. Physical Review Letters 107,233902 (2011).

Alois Regensburger, Christoph Bersch, Mohammad-Ali Miri, Georgy Onishchukov,Demetrios N. Christodoulides, and Ulf Peschel, “Parity-time synthetic photonic lattices”,Nature 488, 167-171 (2012).

Alois Regensburger, Mohammad-Ali Miri, Christoph Bersch, Jakob Näger, GeorgyOnishchukov, Demetrios N. Christodoulides, and Ulf Peschel, “Observation of DefectStates in PT-Symmetric Optical Lattices”, Physical Review Letters 110, 223902 (2013).

For copyright reasons, the publications within the dissertation are not included in thisonline version.

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Experiments on pulse dynamics and parity-time symmetry in optical