Topological Photonics
Topological photonics applies the mathematical concepts of topology from condensed matter physics to the control and manipulation of light. By engineering photonic structures with nontrivial topological properties, researchers have created systems that support robust optical modes immune to backscattering from defects, disorder, and sharp bends. These topologically protected states of light open remarkable possibilities for fault-tolerant optical devices that maintain their function despite fabrication imperfections and environmental perturbations.
The field emerged from the recognition that the band structure of photonic crystals can possess topological invariants analogous to those responsible for the quantum Hall effect in electronic systems. Just as electrons in quantum Hall materials flow along edges without scattering backward, light in topological photonic structures can propagate along interfaces in a unidirectional manner that is fundamentally protected by topology. This protection does not rely on material perfection but rather on the global properties of the band structure that cannot be changed by local perturbations.
This article provides comprehensive coverage of topological photonics, from the fundamental principles underlying topological protection to the practical implementation of topological photonic devices. Understanding these concepts is essential for researchers and engineers developing next-generation optical systems that require robustness against disorder and defects.
Fundamentals of Topology in Photonics
Topological Invariants
Topology is a branch of mathematics concerned with properties that remain unchanged under continuous deformations. A coffee cup and a donut are topologically equivalent because one can be smoothly transformed into the other without cutting or gluing, but neither can be transformed into a sphere. The number of holes, or genus, is a topological invariant that distinguishes these shapes. In photonic systems, analogous invariants characterize the band structure and determine the existence of protected edge states.
The Chern number is the most fundamental topological invariant in photonics, analogous to the TKNN invariant that characterizes the integer quantum Hall effect. For a two-dimensional photonic crystal, the Chern number is calculated by integrating the Berry curvature over the Brillouin zone. A nonzero Chern number indicates that the bulk bands have nontrivial topology, guaranteeing the existence of chiral edge states at interfaces with topologically distinct materials including vacuum or trivial photonic crystals.
Other topological invariants relevant to photonics include the Z2 invariant for time-reversal symmetric systems, the winding number for one-dimensional systems, and higher-order invariants that characterize corner and hinge states in three-dimensional structures. Each invariant is associated with specific symmetry requirements and protected boundary modes, providing a rich classification scheme for topological photonic phases.
Berry Phase and Berry Curvature
The Berry phase is a geometric phase acquired by a quantum state when it is adiabatically transported around a closed loop in parameter space. In photonic systems, the relevant parameter space is the Brillouin zone, and the Berry phase accumulated along a closed path encodes topological information about the band structure. The local density of Berry phase, called the Berry curvature, plays a role analogous to a magnetic field in momentum space.
Mathematically, the Berry connection is defined as the inner product of a Bloch eigenstate with its derivative with respect to the wavevector. The Berry curvature is the curl of the Berry connection, and its integral over the Brillouin zone gives the Chern number multiplied by 2pi. Regions of high Berry curvature often occur near band degeneracies and band crossings, where the eigenstates change rapidly as a function of wavevector.
In photonic systems, the Berry curvature can be engineered by controlling the spatial distribution of materials with different optical properties. Breaking certain symmetries, particularly time-reversal symmetry through magneto-optical effects, generates Berry curvature that gives rise to nontrivial topology. Understanding how structure and symmetry determine Berry curvature is essential for designing topological photonic devices.
Bulk-Boundary Correspondence
The bulk-boundary correspondence is the fundamental principle connecting the topological properties of the bulk material to the existence of states at its boundaries. For a system with a nonzero Chern number, the correspondence guarantees that the number of chiral edge modes equals the difference in Chern numbers across the interface. These edge modes exist within the bulk band gap and cannot be removed by any perturbation that does not close the gap.
The robustness of topological edge states arises from this bulk-boundary correspondence. Because the edge states are required by the bulk topology, they cannot be eliminated by edge disorder or defects as long as the bulk gap remains open. Backscattering is forbidden because there are no backward-propagating states available at the same energy into which the forward-propagating mode could scatter.
The correspondence extends to higher-order topological phases, where the bulk topology determines the existence of states on corners or hinges rather than edges. In a second-order topological insulator, for example, the bulk and edges may both be gapped, but topologically protected states appear at corners where edges meet. This hierarchy of bulk-boundary correspondences provides a rich taxonomy of topological phases in photonics.
Symmetry and Topological Classification
The topological classification of photonic systems depends critically on the symmetries present. Time-reversal symmetry, spatial inversion symmetry, and various crystallographic symmetries all constrain the possible topological phases and the types of protected edge states. Breaking specific symmetries is often necessary to realize nontrivial topology, while preserving others ensures the protection of boundary modes.
In systems with time-reversal symmetry, the Chern number must vanish, but a Z2 topological invariant can still be nontrivial. These quantum spin Hall analogs support pairs of counter-propagating edge states with opposite pseudospin, protected by time-reversal symmetry against backscattering between the two modes. Breaking time-reversal symmetry through magneto-optical effects enables nonzero Chern numbers and truly unidirectional edge propagation.
Crystallographic symmetries including rotation, mirror, and glide symmetries enable additional topological phases characterized by symmetry indicators. These crystalline topological phases support protected modes at boundaries that preserve the relevant symmetry. The interplay between different symmetries creates a rich landscape of topological phases that can be accessed by careful design of photonic structures.
Photonic Topological Insulators
Magneto-Optical Photonic Crystals
The first photonic analog of the quantum Hall effect was realized using magneto-optical materials that break time-reversal symmetry under an applied magnetic field. Ferrimagnetic materials such as yttrium iron garnet (YIG) exhibit strong magneto-optical effects at microwave frequencies, enabling photonic crystals with nonzero Chern numbers. The resulting chiral edge states propagate in only one direction, immune to backscattering from defects and disorder.
In a two-dimensional magneto-optical photonic crystal, the application of an out-of-plane magnetic field induces off-diagonal components in the permeability tensor. These gyrotropic components break time-reversal symmetry and generate Berry curvature in the photonic bands. When the band structure develops a gap with nontrivial Chern number, topological edge states appear at the boundaries of the crystal.
Experimental demonstrations have confirmed unidirectional edge transport in magneto-optical photonic crystals at microwave frequencies, with edge modes that route around obstacles without reflection. The requirement for external magnetic fields and the limitation to microwave frequencies where magneto-optical effects are strong present challenges for practical applications, motivating the development of alternative approaches for higher frequencies.
Time-Reversal-Invariant Topological Photonics
Photonic analogs of the quantum spin Hall effect preserve time-reversal symmetry while supporting topological edge states. These systems use pseudospin degrees of freedom, typically constructed from degenerate modes of the photonic crystal, that behave analogously to electron spin. Counter-propagating edge states with opposite pseudospin are protected against backscattering by time-reversal symmetry, which forbids coupling between the two modes.
Various approaches have been developed to engineer the required pseudospin states. One method uses the transverse electric (TE) and transverse magnetic (TM) polarizations as the two pseudospin components, with appropriate coupling between them to open a topological gap. Another approach employs crystalline symmetries, such as the six-fold rotation symmetry of honeycomb lattices, to create degenerate orbital modes that serve as pseudospin.
The protection in time-reversal-invariant systems is more fragile than in systems with broken time-reversal symmetry because perturbations that couple the two pseudospin modes can cause backscattering. Nevertheless, careful design can minimize such perturbations, enabling robust edge transport in practical devices. These systems have the advantage of not requiring magnetic materials or external magnetic fields.
Valley Photonic Crystals
Valley photonic crystals exploit the valley degree of freedom in honeycomb lattices to create topological edge states without requiring magneto-optical materials. In these systems, breaking inversion symmetry while preserving time-reversal symmetry opens a gap at the Dirac points and generates opposite Berry curvature at the two inequivalent valleys (K and K'). Although the total Chern number vanishes, the valley Chern number is nonzero.
Edge states in valley photonic crystals appear at domain walls between regions with opposite inversion-symmetry breaking. These states are valley-polarized, with modes in different valleys propagating in opposite directions. While not as robustly protected as states in systems with broken time-reversal symmetry, valley edge states are resistant to many types of disorder that do not mix the valleys.
Valley photonic crystals have been demonstrated across a wide frequency range from microwaves to near-infrared, using platforms including photonic crystal slabs and coupled resonator arrays. The ability to realize topological edge transport without magnetic materials makes valley photonics particularly attractive for on-chip photonic applications where integration with standard fabrication processes is important.
Photonic Crystal Implementations
Photonic crystals provide a versatile platform for implementing topological phases through periodic modulation of the refractive index. Two-dimensional photonic crystal slabs with honeycomb or square lattice geometries have been extensively studied, with the band structure engineered through the size, shape, and arrangement of the constituent elements. Gap opening and topological transitions can be controlled by deforming the lattice or modifying the local geometry.
The photonic crystal platform enables precise control over the band structure through lithographic fabrication techniques. Silicon photonic crystals operating at telecommunications wavelengths have demonstrated topological edge states with high transmission through sharp corners and around defects. The compatibility with established silicon photonics fabrication makes these structures promising for integrated photonic circuits.
Three-dimensional photonic crystals offer additional possibilities for topological phases, including Weyl points and nodal lines in the bulk band structure. Fabrication of three-dimensional structures with the required precision is more challenging, but advances in two-photon polymerization and layer-by-layer assembly are enabling experimental studies of three-dimensional topological photonics.
Topological Edge States
Chiral Edge States
Chiral edge states propagate in only one direction along the boundary of a topological photonic insulator with broken time-reversal symmetry. This unidirectional propagation is absolute: there are no backward-propagating modes available at any energy within the gap, so backscattering is fundamentally impossible regardless of the nature of defects or disorder at the edge. This perfect protection is the hallmark of topological physics.
The group velocity of chiral edge states is determined by the slope of their dispersion relation, which crosses the bulk gap monotonically from one bulk band to the other. The number of chiral edge modes equals the Chern number of the lower bulk band, with positive Chern numbers giving clockwise propagation (for appropriate conventions) and negative giving counterclockwise. At interfaces between regions with different Chern numbers, the net number of chiral modes equals the Chern number difference.
Experimental signatures of chiral edge states include unidirectional transmission even around sharp corners, immunity to backscattering from deliberately introduced defects, and the absence of standing wave patterns that would indicate counter-propagating modes. These properties have been demonstrated in magneto-optical photonic crystals at microwave frequencies and in analogous systems using synthetic gauge fields at optical frequencies.
Helical Edge States
Helical edge states occur in time-reversal-invariant topological photonic insulators, where counter-propagating modes are protected by time-reversal symmetry rather than the absence of backward-propagating states. The two edge modes have opposite pseudospin, and scattering between them would require flipping the pseudospin, which is forbidden by symmetry for non-magnetic perturbations. This protection is robust against disorder that respects time-reversal symmetry.
The dispersion relation of helical edge states forms a Kramers pair, with the two branches related by time reversal. At time-reversal-invariant momenta, the modes are degenerate, while away from these points they split. The crossing at time-reversal-invariant points is protected by symmetry and cannot be removed without breaking time-reversal symmetry or closing the bulk gap.
Pseudospin-resolved measurements can distinguish the two helical edge modes by their different coupling to external probes with specific pseudospin polarization. By selectively exciting one pseudospin component, unidirectional excitation of the edge modes is possible. This pseudospin selectivity enables novel functionalities including spin-dependent routing of light.
Edge State Localization
Topological edge states are exponentially localized to the boundary, with a decay length determined by the bulk gap magnitude. Larger gaps produce more strongly confined edge modes, which is desirable for applications requiring tight confinement. However, larger gaps often require stronger perturbations that can introduce loss or fabrication challenges, creating tradeoffs in the design of practical devices.
The transverse profile of edge states depends on the specific structure and boundary conditions. In photonic crystal implementations, the edge geometry can be engineered to control the mode profile, enabling optimization for coupling to external waveguides or sources. The field distribution also determines the sensitivity of edge modes to perturbations located at different positions relative to the boundary.
Near corners or other geometric features, the edge state profile may differ from its form along straight edges. Understanding this dependence is important for predicting transmission through corners and around obstacles. In many topological systems, edge states smoothly navigate corners without reflection, but the detailed behavior depends on the corner geometry and how it affects the local band structure.
Coupling and Excitation of Edge States
Efficient coupling between external sources and topological edge states is essential for practical applications. The coupling depends on matching the mode profile, polarization, and momentum of the external light to those of the edge state. In photonic crystal implementations, grating couplers or tapered waveguides can bridge between conventional waveguides and the topological edge modes.
Selective excitation of specific edge modes in systems with multiple edges or degenerate modes requires control over the spatial and momentum distributions of the input light. Near-field probes can locally excite edge states with high spatial resolution, while far-field excitation typically requires momentum matching through appropriate coupling structures. The chiral or helical nature of edge states also enables directional excitation based on the handedness or pseudospin of the input.
The emission from sources placed near topological edges couples preferentially to edge modes rather than bulk modes, enabling enhanced light-matter interaction. Quantum emitters coupled to topological edge states can exhibit preferential emission into one propagation direction, useful for integrated quantum photonic circuits. This chiral coupling is a direct consequence of the locked relationship between propagation direction and mode properties in topological systems.
Higher-Order Topological Phases
Topological Corner States
Higher-order topological insulators support protected states at corners or hinges rather than edges. In a two-dimensional second-order topological insulator, both the bulk and edges are gapped, but zero-dimensional corner states appear at the intersections of edges. These corner states are protected by the bulk topology and crystallographic symmetries that stabilize the higher-order phase.
The corner states in higher-order topological photonic systems have been realized using breathing kagome lattices, where the ratio of inter-cell to intra-cell coupling determines the topological phase. When this ratio crosses a critical value, the system transitions between trivial and topological phases, with corner states appearing in the topological regime. These states are localized to specific corners determined by the geometry and symmetry of the structure.
The existence of corner states is governed by a generalized bulk-boundary correspondence involving higher-order topological invariants. These invariants can be calculated from the bulk band structure and predict the presence and properties of corner modes. The classification of higher-order phases depends on both the spatial dimension and the protecting symmetries, leading to a rich variety of possible phases.
Higher-Order Topological Insulators
A d-dimensional nth-order topological insulator has gapped bulk and gapped boundaries of dimensions greater than d-n, with topologically protected states only appearing at (d-n)-dimensional boundaries. Thus, a three-dimensional second-order topological insulator has gapped bulk and surfaces, but supports one-dimensional hinge states at the intersections of surfaces. A three-dimensional third-order topological insulator has protected zero-dimensional corner states.
Photonic implementations of higher-order topological phases have been demonstrated in both microwave and optical regimes. Coupled resonator arrays provide a particularly flexible platform, where the coupling between resonators can be engineered to realize different topological phases. By controlling the coupling pattern, researchers have demonstrated transitions between different higher-order phases and observed the associated corner and hinge states.
The protection of higher-order topological states typically requires crystallographic symmetries that constrain the structure of the boundaries. Disorder that respects these symmetries does not destroy the corner states, but symmetry-breaking perturbations can gap them out. Understanding which symmetries protect specific higher-order phases is essential for designing robust structures.
Applications of Corner States
Topological corner states offer unique properties for photonic applications. Their strong localization in all dimensions makes them natural candidates for cavity modes with small mode volumes. Unlike conventional cavity modes that require perfect fabrication to maintain high quality factors, topological corner states are protected against certain types of disorder, potentially offering more robust performance.
Corner states in topological photonic crystals have been proposed for single-mode lasing, where the topological protection helps select a specific lasing mode and stabilize its frequency. The localization of corner states also enhances light-matter interaction for applications in sensing and nonlinear optics. Quantum emitters coupled to corner states experience modified emission properties due to the altered local density of optical states.
Networks of topological corner states connected by topological edge channels enable novel architectures for photonic circuits. The robustness of both the localized corner modes and the connecting edge channels promises reliable operation despite fabrication imperfections. These hybrid architectures combining different orders of topological protection are an active area of research.
Quadrupole and Multipole Moments
Higher-order topology can be characterized by multipole moments that generalize the electric polarization used to describe first-order topological insulators. A quantized quadrupole moment indicates a second-order topological phase, while higher multipoles characterize still-higher orders. These multipole moments are bulk quantities that determine the existence of boundary modes through a generalized bulk-boundary correspondence.
The quadrupole moment is defined through a nested Wilson loop calculation that captures the topological structure of the Wannier bands. When this moment is quantized to a nontrivial value, corner charges appear at the intersections of edges, manifesting as the corner-localized modes in photonic systems. The mathematical framework connecting multipole moments to corner states provides predictive power for designing higher-order topological phases.
Experimental detection of multipole moments requires measuring the spatial distribution of charges or field amplitudes at boundaries. In photonic systems, this can be accomplished through near-field optical microscopy or by observing the spectral signatures of corner-localized modes. The quantization of multipole moments to specific values provides a robust signature of higher-order topology.
Synthetic Dimensions
Frequency as a Synthetic Dimension
Synthetic dimensions exploit internal degrees of freedom as additional spatial dimensions, enabling the study of higher-dimensional physics in lower-dimensional structures. In photonics, the frequency modes of a resonator provide a natural synthetic dimension, with electro-optic or acousto-optic modulation coupling adjacent frequency modes to create an effective tight-binding lattice in the frequency domain.
The coupling between frequency modes induced by modulation creates a synthetic lattice where the frequency index plays the role of a spatial coordinate. By controlling the modulation phase and amplitude, arbitrary coupling patterns can be engineered, including the complex hoppings needed to realize synthetic gauge fields. This approach has enabled the observation of topological edge states in the frequency domain using just a single physical resonator.
The synthetic dimension approach offers several advantages over spatial implementations. The frequency modes of a high-finesse resonator can provide many synthetic lattice sites with uniform properties, avoiding the fabrication variations that plague spatial photonic crystals. Reconfigurability is inherent because the modulation parameters can be changed dynamically, allowing exploration of different topological phases in the same physical structure.
Orbital Angular Momentum Dimensions
Orbital angular momentum (OAM) states of light provide another synthetic dimension characterized by the topological charge of the helical wavefront. Coupling between OAM modes can be induced by appropriately designed optical elements or nonlinear interactions, creating effective lattices in the OAM space. Combined with other degrees of freedom, OAM synthetic dimensions enable the study of multidimensional topological physics.
The unbounded nature of OAM, which can in principle take any integer value, provides a semi-infinite synthetic dimension that can host edge states at the OAM equals zero boundary. This natural edge enables the study of topological phenomena without needing to engineer a boundary between topologically distinct regions. Edge states localized near OAM equals zero have been observed in coupled OAM mode systems.
The combination of OAM and frequency synthetic dimensions creates effective two-dimensional lattices where each dimension has distinct physical character. This hybrid approach has been used to realize synthetic magnetic fields and observe Landau levels, the hallmark of quantum Hall physics, in photonic systems without requiring real magnetic fields.
Higher-Dimensional Topology
Synthetic dimensions enable the study of topological phases in four and higher dimensions that cannot be directly realized in three-dimensional space. Four-dimensional quantum Hall effects and other exotic phases have been observed using synthetic dimensions combined with spatial dimensions, revealing physics inaccessible through other means. These experiments demonstrate topological invariants and boundary modes unique to higher dimensions.
The second Chern number characterizes four-dimensional topological phases and determines the existence of three-dimensional boundary states. While four-dimensional bulk materials cannot exist in our three-dimensional space, synthetic dimensions provide access to the physics of four-dimensional systems, including the observation of second Chern number effects and four-dimensional quantum Hall responses.
Higher-dimensional topological phases exhibit response functions and boundary modes not present in lower dimensions. The theoretical richness of these phases provides motivation for synthetic dimension experiments, while the experimental observations validate theoretical predictions and reveal unexpected phenomena. This interplay between theory and experiment in higher-dimensional topology is a frontier of topological photonics research.
Synthetic Gauge Fields
Synthetic gauge fields in photonic systems mimic the effect of magnetic fields on charged particles, enabling neutral photons to exhibit quantum Hall-like physics. The Peierls substitution relates complex hopping phases to the gauge field, with the accumulated phase around a closed loop giving the synthetic magnetic flux through the loop. Engineering these complex phases in photonic structures creates effective magnetic fields that affect photon dynamics.
Various approaches create synthetic gauge fields in photonic systems. Dynamical modulation with appropriate phase relationships induces complex couplings in coupled resonator arrays. Carefully designed waveguide structures with helical or periodic bending create geometric phases equivalent to gauge fields. Floquet driving provides another route to synthetic gauge fields through time-periodic modulation.
Synthetic gauge fields enable topological phases in systems that would otherwise be topologically trivial. The tunability of synthetic fields, achieved by adjusting modulation parameters or structural design, allows exploration of phase diagrams and transitions between different topological phases. This control is difficult to achieve with real magnetic fields, making synthetic approaches particularly valuable for studying topological physics.
Non-Hermitian Topology
Non-Hermitian Band Theory
Optical systems with gain and loss are fundamentally non-Hermitian, requiring extensions of the standard Hermitian band theory used in condensed matter physics. Non-Hermitian band structures have complex eigenvalues representing both the frequency and gain or loss rate of each mode. The topological classification of non-Hermitian systems differs from the Hermitian case, with new invariants and phenomena that have no Hermitian analog.
The non-Hermitian extension of Bloch theory reveals that the Brillouin zone can develop nontrivial topology even in one-dimensional systems, which are always topologically trivial in the Hermitian case. Complex energy bands can wind around the origin in the complex plane, characterized by a winding number that determines the existence of edge states. This non-Hermitian skin effect causes all bulk modes to localize at boundaries, dramatically different from Hermitian systems.
The biorthogonal formalism with separate left and right eigenvectors is necessary for properly defining inner products and observables in non-Hermitian systems. The Berry phase and topological invariants must be reformulated using this biorthogonal structure. When left and right eigenvectors become parallel at exceptional points, the standard formalism breaks down, requiring careful mathematical treatment.
Exceptional Points
Exceptional points are degeneracies unique to non-Hermitian systems where not only the eigenvalues but also the eigenvectors coalesce. Unlike Hermitian degeneracies where eigenvectors remain orthogonal, at exceptional points the system becomes defective with fewer linearly independent eigenvectors than its dimension. This coalescence leads to extreme sensitivity to perturbations and novel dynamical behavior.
The order of an exceptional point refers to how many eigenvectors coalesce. Second-order exceptional points, where two eigenvectors merge, are most commonly studied, but higher-order exceptional points with three or more coalescing eigenvectors exhibit even more dramatic physics. The response to perturbations near an nth-order exceptional point scales as the nth root of the perturbation strength, enabling enhanced sensitivity for sensing applications.
Encircling exceptional points in parameter space produces topological mode switching that depends on the direction of encirclement. This chiral behavior, where clockwise and counterclockwise loops produce different final states, has been demonstrated in microwave and optical systems. The state evolution during encirclement can be engineered for mode conversion, switching, and other functionalities.
Parity-Time Symmetry
Parity-time (PT) symmetric systems have balanced gain and loss arranged with specific spatial symmetry. Despite being non-Hermitian, PT-symmetric Hamiltonians can have entirely real eigenvalues below a threshold called the exceptional point. Above this threshold, PT symmetry is spontaneously broken and eigenvalues become complex conjugate pairs. This phase transition between PT-symmetric and PT-broken phases has important consequences for laser physics and optical device design.
In the PT-symmetric phase, modes are delocalized across both gain and loss regions, enabling power oscillation between them. Above the exceptional point, modes localize preferentially in either the gain or loss region, breaking the balance. Lasers operating near the exceptional point exhibit enhanced mode discrimination because only modes in the gain region experience amplification, improving single-mode operation.
The combination of PT symmetry and topology creates non-Hermitian topological phases with unique properties. PT-symmetric topological insulators can support edge states with real energies despite the non-Hermitian bulk. The interplay between PT symmetry breaking and topological transitions provides rich physics and opportunities for device engineering that are actively being explored.
Non-Hermitian Skin Effect
The non-Hermitian skin effect refers to the dramatic localization of all bulk eigenstates at system boundaries under open boundary conditions. Unlike Hermitian systems where bulk states extend throughout the material, non-Hermitian systems can exhibit skin modes that pile up at one edge, determined by the non-Hermitian topology of the bulk. This effect has no Hermitian analog and fundamentally changes the bulk-boundary correspondence.
The skin effect is characterized by a nonzero winding number of the complex energy bands in the Brillouin zone. When the bands wind around the origin, the corresponding modes localize at the boundary in the direction determined by the winding sense. This winding topology is distinct from and can coexist with conventional band topology characterized by Chern numbers or other invariants.
Experimental observations of the non-Hermitian skin effect have been reported in various photonic platforms including coupled resonator arrays and acoustic metamaterials. The extreme mode localization has implications for light harvesting, sensing, and novel device functionalities. Understanding and controlling the skin effect is essential for designing non-Hermitian topological photonic devices.
Floquet Topological Systems
Periodically Driven Photonics
Floquet systems are periodically driven in time, with the driving creating effective band structures that can have topological properties absent in the static system. The Floquet theorem states that eigenstates of periodically driven systems are products of a periodic function and a phase factor, analogous to Bloch states in spatially periodic systems. The quasienergy, defined modulo the driving frequency, plays the role of energy in the effective band structure.
Periodic driving can induce topological phase transitions by opening gaps at quasienergy values determined by the driving frequency and amplitude. Even systems with trivial static topology can become topologically nontrivial under appropriate driving, enabling dynamical control of topological properties. This tunability is a major advantage of Floquet approaches for reconfigurable topological devices.
Photonic implementations of Floquet systems use various forms of temporal modulation including electro-optic modulation, acoustic waves, and mechanical oscillation. Waveguide arrays with helical or periodically modulated geometry create effective time-periodic evolution for propagating light. These realizations have demonstrated Floquet topological phases and edge states in both microwave and optical regimes.
Floquet Topological Phases
Floquet topological phases are classified differently from static phases because the quasienergy is periodic, with the Brillouin zone boundary at quasienergy equal to half the driving frequency equivalent to negative half the driving frequency. This periodicity allows for topological phases unique to Floquet systems, including anomalous Floquet topological insulators where edge states exist without corresponding bulk topological invariants in the conventional sense.
The classification of Floquet phases requires new topological invariants that account for the time-periodic structure. The winding of the time-evolution operator over a driving period encodes topological information not captured by the effective static Hamiltonian. Gaps can open at both quasienergy zero and quasienergy equal to half the driving frequency, each potentially hosting different types of edge states.
Anomalous Floquet edge states appear at quasienergy gaps where the Chern number of the bands below the gap equals the Chern number of the bands above. Despite the apparently trivial bulk topology, edge states exist due to the winding structure of the time evolution. These states have been observed in photonic lattices and demonstrate physics unique to driven systems.
Waveguide Array Implementations
Waveguide arrays with periodic modulation along the propagation direction provide a platform for Floquet topological photonics where the spatial propagation coordinate plays the role of time. Light propagating through the array experiences periodic evolution due to the modulation, mapping directly to Floquet physics. The effective driving frequency and amplitude are controlled by the modulation period and depth.
Helically modulated waveguide arrays create circular motion of the waveguide centers, inducing effective magnetic fields through the resulting geometric phase. This approach has been used to demonstrate photonic Floquet topological insulators with chiral edge states. The helical modulation can be fabricated using femtosecond laser writing, enabling rapid prototyping of different Floquet phases.
The propagation direction serving as time means that input and output facets of the waveguide array correspond to initial and final times. Edge states appear at the spatial boundaries of the array and can be excited by injecting light at the input facet near the edge. Observations of robust edge transport through defects and around corners confirm the topological protection of Floquet edge states.
Dynamic Topological Transitions
Floquet driving enables dynamic transitions between different topological phases by varying the driving parameters. Unlike static systems where topology changes require material modification, Floquet topology can be switched by changing the driving frequency, amplitude, or phase. This dynamic control enables reconfigurable topological devices and the study of topological phase transitions in real time.
Quenches across topological transitions, where driving parameters are suddenly changed, produce transient dynamics that reveal information about the topology of both initial and final phases. The evolution of edge state populations during quenches and the formation of edge states after quenching from trivial to topological phases have been studied theoretically and observed experimentally.
Adiabatic transitions between Floquet phases, where parameters are slowly varied, can pump edge states or transfer population between edge and bulk modes. This pumping is a dynamical manifestation of the topological character and can be used for controlled transport of light in topological photonic circuits. The interplay between driving-induced topology and adiabatic evolution creates rich dynamics for exploration and application.
Topological Lasers
Concept and Advantages
Topological lasers exploit topological edge states as the lasing mode, inheriting the robustness of topological modes for improved laser performance. The topological protection of edge states against scattering from defects and disorder translates to reduced sensitivity of the laser to fabrication imperfections and environmental perturbations. This robustness addresses a key challenge in conventional laser arrays where disorder disrupts the desired mode pattern.
In a topological laser, gain is provided to the edge states, which experience amplification while bulk states remain below threshold. The unidirectional or helical nature of topological edge states can improve mode selection by suppressing standing wave patterns that would otherwise compete for gain. Single-mode operation is facilitated by the topological constraint that only specific edge configurations are allowed.
The concept of topological lasers extends beyond improved robustness to enable new functionalities. Unidirectional output from chiral edge state lasers provides intrinsic isolation without requiring separate optical isolators. The distributed nature of edge states enables laser arrays that maintain coherence despite disorder, useful for high-power applications requiring many coupled laser elements.
Implementations
The first topological laser demonstrations used magneto-optical photonic crystals at microwave frequencies, where the strong magneto-optical effects of ferrite materials enable large bandgaps and robust chiral edge states. These proof-of-concept devices demonstrated unidirectional emission and immunity to defects, establishing the basic principles of topological lasing.
Optical frequency topological lasers have been realized using valley photonic crystals in III-V semiconductor platforms. Ring resonator arrays with alternating coupling strengths create topological edge modes at domain walls, with gain provided by quantum wells embedded in the resonators. These devices have demonstrated robust single-mode lasing with emission wavelengths in the near-infrared telecommunications bands.
Higher-order topological lasers exploit corner states rather than edge states for lasing. The strong localization of corner modes creates high-quality-factor cavities with small mode volumes, desirable for low-threshold lasing. Corner state lasers have been demonstrated in two-dimensional photonic crystals and show promise for dense integration of nanoscale laser arrays.
Mode Competition and Stability
Understanding mode competition in topological lasers requires considering how the topological protection affects gain competition between modes. In conventional lasers, modes compete for gain through spatial hole burning and other saturation effects. Topological edge modes, being spatially separated from bulk modes, experience different saturation dynamics that can favor edge mode lasing.
The spectral properties of topological lasers depend on the dispersion of the edge states and how gain is distributed. Broad gain bandwidth can lead to multimode operation along the edge dispersion, while narrow gain selects a specific wavevector. The relationship between cavity length and edge state dispersion determines the mode spacing and single-mode stability.
Noise properties of topological lasers have been studied theoretically and experimentally. The reduced scattering from defects translates to improved frequency stability and reduced linewidth compared to conventional lasers with similar disorder. The inherent mode selection by topology also reduces partition noise that arises from competition between multiple modes.
Future Directions
Research in topological lasers continues toward practical devices with improved performance and new functionalities. Electrically injected topological lasers would enable integration with electronic circuits and eliminate the need for optical pumping. Progress requires developing topological photonic structures compatible with semiconductor growth and processing while maintaining sufficient bandgap and edge state quality.
High-power topological laser arrays could leverage the coherence of topological edge states for beam combining. Unlike conventional laser arrays that lose coherence due to disorder, topological protection may maintain phase relationships across large arrays. This application requires scaling to larger structures while preserving the topological gap and edge state properties.
Integration of topological lasers with other photonic components would enable on-chip systems with topologically protected light sources. Coupling topological laser output to topological waveguides creates fully robust optical interconnects. The combination of active and passive topological elements is an exciting direction for photonic circuit design.
Topological Quantum Optics
Quantum Emitters in Topological Environments
Quantum emitters such as atoms, quantum dots, and color centers coupled to topological photonic structures experience modified emission properties due to the unique characteristics of topological modes. The chiral nature of edge states leads to directional emission, where photons are preferentially emitted into one propagation direction. This chiral quantum optics enables deterministic routing of single photons and entanglement generation between distant emitters.
The local density of states at topological edges can enhance emission rates through the Purcell effect, while the protection of edge modes ensures that emitted photons propagate without backscattering. These properties are valuable for quantum information applications where photons must be efficiently generated and routed to specific destinations. The combination of enhanced emission and robust transport is difficult to achieve in conventional photonic structures.
Coupling strength between emitters and topological modes depends on the spatial and spectral overlap. Emitters positioned at edge state maxima and tuned to the edge band experience optimal coupling. The finite penetration depth of edge states into the bulk creates a region where coupling is possible while still benefiting from topological protection during subsequent propagation.
Chiral Quantum Optics
Chiral quantum optics studies light-matter interaction where emission is directionally dependent, with photons coupling preferentially to modes propagating in a specific direction. Topological photonic systems provide a natural platform for chiral quantum optics because the edge states are inherently unidirectional or have spin-momentum locking. This chirality enables functionalities impossible in conventional waveguide quantum electrodynamics.
When a quantum emitter couples to a chiral waveguide, the emission probability into the forward and backward directions can be dramatically asymmetric. In the limit of perfect chirality, photons are emitted exclusively in one direction, enabling deterministic single-photon routing. This determinism simplifies quantum network architectures that would otherwise require probabilistic operations and heralding.
Multiple emitters coupled to a common topological waveguide interact through the propagating photons in a cascaded manner determined by the chirality. The unidirectional nature of the interaction enables non-reciprocal entanglement generation and quantum state transfer. These cascaded quantum systems exhibit dynamics distinct from their bidirectional counterparts and enable new quantum information protocols.
Topological Protection of Quantum States
Topological protection may extend to quantum states carried by photons in topological waveguides. Single-photon wave packets propagating in topological edge modes experience suppressed backscattering that would otherwise cause decoherence and state degradation. Entangled photon pairs transmitted through topological channels may maintain higher fidelity than through conventional waveguides with similar disorder.
The extent to which topology protects quantum information is an active research question. While classical transmission is clearly protected against linear scattering, nonlinear effects and coupling to external baths can still cause quantum decoherence. Understanding the limits of topological quantum protection requires careful theoretical analysis and experimental characterization of quantum state fidelity in topological systems.
Topological error correction is a distinct concept from topological protection of photon transport, referring to encoding quantum information in topological degrees of freedom of many-body states. Photonic implementations of topological quantum error correction use entangled photon states with intrinsic protection against local errors. The connection between topological photonic systems and topological quantum computation is an area of fundamental and practical interest.
Many-Body Topological Photonics
When photon-photon interactions become significant, many-body effects emerge that can create new topological phases. Fractional quantum Hall analogs in photonic systems require strong effective interactions between photons, which can be mediated by atoms in electromagnetic induced transparency configurations or by nonlinear optical materials. These interacting topological phases support anyonic excitations with exotic exchange statistics.
The realization of strongly correlated topological photonic states requires interaction strengths comparable to the bandwidth, a challenging regime to reach in optical systems. Rydberg atoms coupled to photonic structures provide one promising platform where the strong atom-atom interactions translate to effective photon-photon interactions. Circuit quantum electrodynamics systems at microwave frequencies offer another route through the strong nonlinearity of superconducting qubits.
Even without reaching the strongly correlated regime, photon interactions in topological systems lead to interesting nonlinear effects including soliton formation in topological edge modes and intensity-dependent shifts of edge state frequencies. These weakly nonlinear effects are more accessible experimentally and may have applications in optical switching and signal processing.
Robust Waveguides and Unidirectional Propagation
Defect-Immune Waveguiding
Topological edge states propagate around defects without backscattering, enabling waveguides with unprecedented robustness to disorder. This immunity arises from the absence of backward-propagating states at the edge rather than from specific symmetry of the defect. Any perturbation that does not close the bulk gap cannot cause backscattering, providing protection against a broad class of imperfections.
Demonstrations of defect-immune waveguiding have shown edge states navigating around deliberately introduced obstacles with minimal loss. Sharp corners that would cause significant reflection in conventional waveguides are traversed with high transmission in topological systems. This robustness relaxes fabrication tolerances and enables photonic circuits with complex routing geometries.
The degree of protection depends on the size of the bandgap relative to perturbation strength and the spectral width of the signal. Perturbations that close the gap locally can create scattering, while signals with bandwidth extending outside the gap experience reduced protection. Understanding these limits is essential for designing practical topological waveguides.
Unidirectional Transmission
Unidirectional transmission in topological waveguides arises from the chirality of edge states in systems with broken time-reversal symmetry. Unlike conventional optical isolators that rely on magneto-optical effects in bulk materials, topological isolators achieve nonreciprocity through the structure of the edge modes. This structural approach can provide isolation without requiring strong magneto-optical materials.
The isolation ratio achievable in topological waveguides depends on the purity of the chirality and the coupling between forward and backward directions due to disorder or design imperfections. In ideal systems, the isolation can be very high because backward-propagating modes simply do not exist. Practical devices show finite isolation determined by how well the ideal topology is realized.
Applications of topological unidirectional waveguides include optical isolation for laser protection, signal routing in photonic circuits, and elimination of spurious reflections in interferometric systems. The compatibility with standard photonic fabrication processes makes topological isolators attractive alternatives to traditional magneto-optical isolators for integrated photonics.
Delay Lines and Slow Light
Topological edge states can have low group velocity near band edges, enabling slow light effects with topological protection. Slow light delay lines suffer in conventional systems from disorder-induced scattering that scales with the group index, but topological protection may mitigate this scattering. The combination of delay functionality with robustness is valuable for optical buffering and signal processing.
The dispersion of topological edge states determines the achievable group delay and bandwidth. Engineering flat-band edge states maximizes delay but limits bandwidth, while dispersive edges provide broader bandwidth at the cost of reduced delay. Understanding this tradeoff and designing edge state dispersion for specific applications is an active area of research.
Coupled resonator optical waveguides (CROWs) in topological configurations provide another approach to slow light with protection. The resonator modes create narrow transmission bands with high group delay, while topological protection ensures robust transmission despite resonator-to-resonator variations. Topological CROWs have been demonstrated with enhanced tolerance to disorder compared to conventional implementations.
Integrated Topological Circuits
Practical applications of topological photonics require integration of topological elements with conventional photonic components. Couplers between topological edge waveguides and standard waveguides must efficiently transfer power while transitioning between different mode structures. Design of these interfaces is critical for system-level performance.
Splitters, combiners, and other passive components can be designed using topological principles to maintain protection throughout the circuit. Topological beam splitters that route light based on pseudospin or valley degree of freedom enable novel circuit functionalities. Active components including modulators and switches integrated with topological waveguides would complete the toolkit for topological photonic systems.
Demonstrations of integrated topological circuits have shown multiple interconnected components functioning together with robust performance. As the complexity of these circuits increases, maintaining topological protection throughout becomes more challenging but also more valuable. The development of design tools and fabrication processes for large-scale topological photonic circuits is ongoing.
Nonlinear Topological Effects
Nonlinear Edge Solitons
Topological edge states in nonlinear media can support solitons that combine the localization of solitons with the robustness of topological modes. Edge solitons balance dispersive spreading with nonlinear self-focusing while propagating along the topological edge. The topological protection may extend to these nonlinear excitations, enabling robust soliton propagation around defects.
The formation of edge solitons requires sufficient nonlinearity to overcome dispersion at the power levels achievable in the edge mode. The competition between nonlinearity, dispersion, and topological confinement creates rich dynamics including soliton bifurcation and instabilities. Understanding these dynamics is important for applications requiring high-power operation of topological waveguides.
Experimental observations of edge solitons have been reported in photonic lattices with Kerr nonlinearity. The solitons exhibit robust propagation characteristic of topological edge states while maintaining the localized profile associated with nonlinear self-trapping. The power threshold for soliton formation and the stability range provide information about the interplay between topology and nonlinearity.
Intensity-Dependent Topology
Strong nonlinearity can modify the effective band structure of photonic systems, potentially inducing topological phase transitions at high intensity. A system that is topologically trivial at low power may become nontrivial when nonlinear index changes alter the band structure sufficiently to close and reopen gaps with different topology. This intensity-dependent topology enables all-optical control of topological properties.
The reverse process, where a topological system becomes trivial at high intensity, is also possible and can be used for power-dependent switching. Edge states that exist at low power may be destroyed by intensity-dependent gap closing, providing a mechanism for power limiting or optical switching. The dynamics of this transition and the associated edge state behavior are areas of active research.
Self-induced topological transitions occur when light modifies the medium to create the conditions for its own topological protection. This bootstrap process requires careful design of the system parameters and initial conditions. Experimental realization remains challenging but would demonstrate a fundamentally new type of optical self-organization.
Parametric Processes in Topological Systems
Parametric processes including frequency conversion and parametric amplification take on new character in topological photonic systems. The phase matching conditions for parametric processes are modified by the dispersion of topological edge states, and the generated light inherits the topological protection of the edge modes. Topological parametric amplifiers could provide gain with enhanced stability against disorder.
Frequency conversion between topological edge modes enables routing of different frequency channels along protected pathways. The conversion efficiency depends on overlap integrals involving the spatial profiles of edge states at different frequencies, which may be optimized through design of the photonic structure. Wavelength-selective topological circuits become possible with integrated frequency conversion elements.
Spontaneous parametric down-conversion in topological systems generates photon pairs in edge modes, potentially with correlations reflecting the topological character of the modes. The entangled photons would propagate robustly along topological edges, combining quantum optical functionality with topological protection. This intersection of quantum optics and topological photonics offers opportunities for robust quantum light sources.
Four-Wave Mixing and Optical Switching
Four-wave mixing in topological edge states enables all-optical switching and signal processing with topological protection. The confined spatial extent of edge modes enhances nonlinear interaction strengths, potentially reducing the power required for switching. The wavelength conversion capability of four-wave mixing combined with wavelength-selective topological routing enables complex signal processing architectures.
Cross-phase modulation between edge modes and bulk or external light provides a mechanism for controlling edge state propagation without destroying topological protection. A control beam can shift the edge state frequency or modify the local gap, affecting transmission of a signal beam in the edge mode. This controlled interaction enables functionality beyond purely passive topological circuits.
The combination of nonlinear effects with topological protection is still being understood theoretically and explored experimentally. Challenges include reaching the required nonlinear strength while maintaining edge state quality and understanding how nonlinearity affects the fundamental topological properties. Progress in these areas will determine the viability of nonlinear topological photonic devices.
Topological Photonic Crystals
Design Principles
Designing topological photonic crystals requires engineering band structures with gaps possessing nontrivial topology. The process typically begins with identifying a target topological phase and the symmetries that protect it, then designing a unit cell geometry that realizes the required band structure. Perturbations to high-symmetry lattices can open topological gaps when they break specific symmetries while preserving others.
Common approaches include expanding and contracting elements of honeycomb lattices to create valley-selective gaps, introducing asymmetric elements to break inversion symmetry, and using combinations of materials to engineer band inversions. The lattice geometry, element shapes, and material contrasts all contribute to the band structure and must be optimized together for the desired topological properties.
Numerical methods including plane wave expansion and finite element analysis compute band structures and topological invariants for candidate designs. Topology optimization algorithms can search design spaces for structures meeting specified topological criteria. These computational tools enable exploration of complex geometries beyond intuitive designs based on known topological phases.
Material Platforms
Silicon photonics provides a mature platform for topological photonic crystals at telecommunications wavelengths. The high refractive index contrast between silicon and air enables wide bandgaps, while established fabrication processes allow precise control of feature sizes. Silicon-based topological photonic crystals have demonstrated robust edge transport and are compatible with integrated photonic circuits.
III-V semiconductors including gallium arsenide and indium phosphide offer gain functionality for topological lasers and amplifiers. The direct bandgap of these materials enables light emission, while quantum well structures provide optical gain. Fabrication requires different processes than silicon but is well established for conventional semiconductor lasers and photonic devices.
Polymer and other organic materials enable rapid prototyping and three-dimensional fabrication through techniques including two-photon polymerization. The lower refractive index contrast compared to semiconductors results in narrower bandgaps, but the fabrication flexibility allows exploration of complex three-dimensional topological structures. These platforms are valuable for proof-of-concept demonstrations and fundamental studies.
Fabrication Considerations
Fabrication of topological photonic crystals requires precision comparable to the feature sizes, typically hundreds of nanometers for optical wavelengths. Electron beam lithography and deep ultraviolet lithography provide the necessary resolution for planar structures. Pattern transfer through dry etching must maintain vertical sidewalls and smooth surfaces to minimize scattering loss.
Fabrication imperfections including size variations, positioning errors, and surface roughness can affect topological properties differently than conventional photonic devices. The robustness of topological edge states provides some tolerance to disorder, but perturbations that close the bandgap destroy topological protection. Understanding which imperfections are most detrimental guides fabrication process development.
Three-dimensional topological photonic crystals require fabrication methods that build structures layer by layer or create features throughout a volume. Layer-by-layer assembly, direct laser writing, and self-assembly approaches have produced three-dimensional photonic crystals, but achieving the precision needed for topological phases remains challenging. Advances in these techniques continue to expand the range of realizable topological structures.
Characterization Techniques
Characterization of topological photonic crystals involves measuring both the band structure and the properties of edge states. Angle-resolved transmission and reflection spectroscopy reveals the dispersion of bulk bands, while near-field scanning optical microscopy maps the spatial distribution of edge modes. These complementary techniques provide comprehensive information about topological properties.
Testing topological protection requires introducing controlled disorder and measuring the effect on edge state transmission. Deliberately fabricated defects including removed elements, displaced features, or sharp corners probe the robustness of edge transport. Comparison between topological and non-topological structures with similar disorder quantifies the benefit of topological protection.
Time-resolved measurements track the propagation of pulses through topological waveguides, revealing group velocity dispersion and any temporal distortion due to scattering. Phase-resolved techniques including holography provide additional information about mode structure and can distinguish different edge state components in systems with multiple edge modes.
Conclusion
Topological photonics has emerged as a vibrant field at the intersection of fundamental physics and photonic technology. The application of topological concepts from condensed matter physics to optical systems has revealed a wealth of new phenomena and enabled devices with unprecedented robustness to disorder and defects. From chiral edge states immune to backscattering, to corner states in higher-order topological phases, to the exotic physics of non-Hermitian topology, the field continues to expand in both fundamental understanding and practical applications.
The development of topological lasers, topological quantum optical systems, and robust topological waveguides demonstrates the practical value of topological protection for photonic devices. These advances address key challenges in integrated photonics including sensitivity to fabrication variations and performance degradation from defects. As fabrication techniques improve and design methods mature, topological photonic devices are likely to find increasing application in communications, computing, and sensing systems.
Looking forward, the field of topological photonics continues to evolve with discoveries of new topological phases, development of new material platforms, and exploration of nonlinear and quantum effects in topological systems. The fundamental insights gained from topological photonics extend beyond photonics itself, contributing to broader understanding of topological phases in physical systems. For researchers and engineers working in photonics, familiarity with topological concepts and their implementation is increasingly essential as these powerful ideas reshape the landscape of optical device design.