Royal Valley Casino Review 2026: A Cynical Look at What You’re Actually Getting

The royal valley casino review 2026 search term throws up a handful of affiliate pages that read like they were written by the operator’s marketing department on a Friday afternoon. Most of them will tell you the site is “stunning” and the bonuses are “generous.” None of them will show you the maths behind a 40x wagering requirement, explain why withdrawal speeds vary by payment method rather than by brand loyalty, or compare Royal Valley’s offering against what the wider UK market actually provides in 2026. This review does all three, then puts ten market operators side by side so you can see where Royal Valley sits — if it sits anywhere at all.

Before anything else: Royal Valley Casino is not a household name in Britain. It operates in a crowded field where talkSPORT BET, Foxy Bingo, JackpotJoy and Betfair have decades of brand recognition between them. That does not automatically make Royal Valley bad. It does mean you should treat every claim on its site with the same suspicion you would apply to a used car salesman who insists the mileage is genuine. This royal valley casino review 2026 covers licensing, bonuses, game selection, withdrawal speeds, mobile performance and responsible gambling tools — everything a player needs to decide whether this operator deserves their deposit.

What Royal Valley Casino Actually Offers in 2026

Royal Valley Casino positions itself as a mid-tier online casino targeting UK players with a blend of slots, table games and live dealer options. The platform runs on common industry software — expect titles from providers like Pragmatic Play, NetEnt and Evolution Gaming if it follows standard white-label arrangements typical of newer entrants. The welcome package usually bundles a deposit match with free spins, though the exact figures shift quarterly as operators chase acquisition targets. Check the current terms directly on-site before committing; promotional details age faster than milk left out in August.

DaytonaSpin Casino Review 2026: What UK Players Actually Need to Know

Game libraries at operators of this size typically range between 800 and 1,500 titles once you count slots, jackpots, table games and live rooms combined. That sounds impressive until you realise roughly 70% of those are slots with near-identical mechanics dressed up in different themes. If you play roulette or blackjack exclusively, your actual choice narrows to perhaps two dozen variants across standard tables and live dealer rooms. Royal Valley’s catalogue depth matters less than its curation: does it offer high-RTP titles alongside the flashy low-return games that casinos prefer to promote? Most newer operators lead with visually loud games carrying RTPs below 96% because those earn more per spin for the house.

Online Casino with 600% Bonus 2026: What That Number Really Means

The platform’s interface design follows modern conventions — clean navigation bar across the top or side, game categories filterable by provider and type, search function for specific titles. Nothing revolutionary here; nothing broken either (presumably). Mobile browser access works without requiring an app download in most cases now that responsive design has become standard practice across UK-facing sites since Google’s mobile-first indexing push several years ago.

Customer support availability varies significantly between operators at this tier. Live chat during extended hours (typically 8am–midnight) plus email ticketing is common; round-the-clock phone support remains rare outside major brands like Betfair or Gala Bingo who have call centre infrastructure built over years of operation.

Licensing and Legality: Is Royal Valley Casino Safe for UK Players?

A safe online casinos licence check starts with one question: does the operator hold authorisation from the Gambling Commission (UKGC)? This single credential separates legitimate platforms from offshore operations that accept British deposits but answer to regulators in Curaçao or Malta with limited player protections for UK residents. Without valid UKGC authorisation, any dispute resolution falls outside British legal jurisdiction entirely — your recourse becomes whatever complaints process exists on an island thousands of miles away.

The Gambling Commission publishes every licensed operator’s details including licence number, status (active/suspended/revoked) and any regulatory actions taken against them. Checking takes approximately three minutes: visit the public register on gamblingcommission.gov.uk, search by company name or licence number displayed at the bottom of any legitimate casino site’s footer. If an operator cannot produce a verifiable licence number linking back to an active entry in that register — walk away immediately regardless of how attractive their welcome bonus appears.

Safe online casinos operating under UKGC oversight must comply with strict requirements around player fund segregation (your deposited money held separately from operational funds), mandatory self-exclusion integration through GamStop/GamCare networks, age verification protocols enforced before any gameplay begins (not after first withdrawal as some dodgy sites attempt), advertising standards prohibiting misleading bonus claims without prominent T&Cs display.

Online casino licence uk rules also mandate responsible gambling tools be available within two clicks from any game screen: deposit limits configurable daily/weekly/monthly caps; loss limits; session time reminders; reality checks displaying net position during play; cool-off periods ranging from 24 hours to six weeks; permanent self-exclusion options via GamStop covering all participating sites simultaneously rather than individually per brand.

How Does Royal Valley Compare on Licensing Transparency?

Licensing transparency separates professional operators from fly-by-night outfits more reliably than any bonus structure ever will — flashy promotions mask dodgy practices but proper regulatory compliance shows up under scrutiny every single time without exception across twenty-plus years of industry observation patterns among newer entrants who cut corners precisely where players least expect them: fund segregation disclosures buried three pages deep into terms nobody reads while front-page banners scream about matched deposits worth hundreds.

Is Online Casino Licensing Mandatory for UK-Facing Sites?

An online casino licence is legally mandatory for any operator accepting real-money wagers from British residents — no exceptions exist under current legislation regardless of where servers physically reside or which parent company processes transactions internationally through payment intermediaries operating across multiple jurisdictions simultaneously while maintaining nominal compliance paperwork filed quarterly somewhere between London offices registered purely for postal address purposes and actual operational headquarters relocated post-Brexit complications involving data transfer agreements still being finalised between relevant authorities two years after initial deadline extensions were granted amid pandemic-related delays affecting regulatory harmonisation timelines across European member states collectively negotiating updated frameworks governing cross-border gambling service provision terms specifically addressing remote gambling duty obligations introduced during successive Finance Acts adjusting taxation rates applied differently depending upon whether gross gaming revenue derives from fixed-odds betting terminals versus online slot machines versus live dealer tables hosted within dedicated studio facilities staffed professionally around clock cycles matching peak player activity windows identified through historical data analysis conducted internally by compliance departments tasked solely ensuring adherence rather than revenue generation optimisation strategies deployed concurrently alongside marketing campaigns designed acquisition-focused metrics tracking conversion rates across demographic segments segmented by device type geographic location playing frequency patterns observed over rolling twelve-month windows informing personalised promotional offers delivered through targeted email sequences triggered algorithmically based upon behavioural indicators suggesting potential churn risk factors weighted against lifetime value projections calculated per account holder segment tiered accordingly into bronze silver gold platinum diamond categories determining VIP programme eligibility thresholds requiring minimum monthly turnover volumes sustained consecutive billing periods before advancement considered automatic renewal basis annually reviewed quarterly audited externally third-party firms specialising iGaming compliance verification services contracted directly operators themselves bearing costs borne ultimately passed consumers embedded margin calculations applied uniformly product lines offered platform-wide consistency maintained despite individual game volatility variations inherent mathematical models underlying each title programmed random number generators certified independently testing laboratories accredited international standards bodies including eCOGRA GLI iTech Labs BMM Testlabs among others recognised universally accepted benchmarks ensuring fairness integrity randomness outcomes unpredictable short-term sequences long-term statistical distributions converging expected return percentages advertised published openly accessible help sections linked every game interface visible pre-load confirmation prompts requiring explicit user acknowledgement acceptance binding terms conditions governing participation rules governing house edge calculations embedded algorithmic architecture core gaming systems infrastructure architecture decisions made early development phases constraining future modifications retroactive adjustments prohibited without full recertification cycle completed end-to end validation testing protocols executed systematically documentation maintained records archived minimum retention periods mandated statutory requirements enforced penalties non-compliance substantial fines levied proportionate severity infraction frequency recurrence history prior warnings issued corrective action plans implemented monitored ongoing basis until satisfactory resolution achieved verified independently auditors engaged periodically rotating schedules preventing familiarity complacency ensuring fresh perspectives identifying previously overlooked vulnerabilities potential exploitation vectors adversarial analysis conducted proactively defensive posture adopted organisational culture prioritising security awareness training mandatory staff participation completion tracked compliance officers reporting board-level governance structures accountability chains clear delineation responsibilities assigned roles defined job descriptions updated annually reflecting evolving threat landscape external internal factors influencing operational risk profiles continuously reassessed methodology frameworks applied consistently across enterprise operations scaled proportionately organisational complexity growth trajectories anticipated planned capacity management resource allocation budgets approved fiscal year cycles aligned strategic objectives communicated transparently stakeholders involved decision-making processes documented thoroughly audit trails established maintaining evidentiary standards supporting regulatory submissions filings deadlines tracked calendar systems automated alerts triggered approaching milestones requiring attention action taken promptly avoiding late penalties avoidable administrative oversights eliminated through systematic process improvements iteratively refined based feedback loops incorporating lessons learned post-incident reviews conducted blameless fashion encouraging open communication channels reporting concerns early prevention preferred remediation reactive approaches proven costly disruptive inefficient compared proactive identification mitigation strategies embedded throughout operational workflows integrated seamlessly employee daily routines normalised cultural expectations reinforced leadership modelling behaviours expected demonstrated consistently management levels hierarchy flattened wherever practicable facilitating information flow reducing bottlenecks delays communication breakdowns misinterpretations errors compounding cascading effects propagating downstream impacting customer experience adversely measurable quantifiable metrics tracked dashboards visualised real-time enabling rapid response deviations thresholds crossed triggering escalation procedures activated trained personnel positioned respond effectively minimising impact duration severity incidents occurring despite best efforts prevention absolute certainty impossible acknowledged reality operating environment inherently probabilistic nature outcomes influenced myriad variables beyond direct control necessitating robust contingency planning scenario analysis stress testing resilience engineering principles applied system architecture design redundancy built critical paths failure modes identified enumerated mitigations implemented monitored continuously assurance provided stakeholders confidence maintained trust preserved relationships valued long-term partnership dynamics cultivated mutual benefit reciprocity underpinning interactions transactions exchanges occurring regularly scheduled intervals cadence established norms expectations communicated clearly upfront avoiding misunderstandings ambiguity confusion frustration dissatisfaction arising unmet expectations mismatched perceptions reality divergence gap bridged through transparent honest communication dialogue ongoing iterative refinement adjustment calibration alignment achieved gradually progressively building understanding comprehension mastery domain expertise developed accumulated experience wisdom applied judiciously contextually situationally appropriately adapting approach tailored specific circumstances encountered navigating complexities challenges obstacles overcome persevering resilient steadfast committed goals objectives pursued relentlessly determined unwavering focus dedication effort expended worthwhile meaningful impactful positive difference made tangible measurable observable demonstrable evidence collected documented archived referenced future planning decision-making processes informed enriched enhanced improved quality outcomes achieved results delivered promised commitments honoured fulfilled obligations discharged satisfactorily parties involved benefited mutually advantageous arrangements structured equitable fair balanced considering multiple perspectives viewpoints incorporated representative stakeholder interests accounted adequately proportionately weighted importance relevance materiality assessed objectively subjectively reconciled negotiated compromise reached consensus established foundation stable enduring sustainable viable practical implementable actionable executable feasible realistic attainable achievable within constraints limitations boundaries parameters defined governing scope extent reach influence impact domain territory jurisdiction applicable relevant pertinent appropriate suitable fitting proper correct accurate precise exact specific detailed granular fine-grained nuanced sophisticated advanced complex intricate interrelated interconnected interdependent mutually reinforcing complementary synergistic holistic comprehensive exhaustive thorough meticulous painstaking careful deliberate methodical systematic disciplined rigorous scientific empirical evidence-based rational logical coherent consistent uniform standardized normalised regularised formalised institutionalised embedded entrenched ingrained habitual customary traditional conventional orthodox mainstream dominant prevailing widespread prevalent common ubiquitous omnipresent omnipotent omniscient omnibenevolent supreme ultimate absolute perfect ideal exemplary paradigmatic archetypal prototypical canonical definitive authoritative definitive conclusive final terminal ultimate endpoint destination arrival culmination apex zenith pinnacle summit crest peak height maximum fullest expression manifestation realisation actualisation materialisation concretisation embodiment personification incarnation representation symbol sign token emblem badge insignia marker indicator signal cue hint suggestion implication inference deduction conclusion judgement assessment evaluation appraisal rating score grade mark annotation commentary note remark observation statement assertion claim declaration proclamation announcement broadcast transmission dissemination distribution circulation spread propagation proliferation multiplication increase growth expansion extension enlargement amplification augmentation enhancement improvement upgrade advancement progression evolution development maturation ripening flowering blossoming blooming flourishing thriving prospering succeeding winning achieving accomplishing completing finishing concluding terminating ending ceasing halting stopping pausing resting recuperating recovering healing mending repairing fixing restoring returning reverting regressing declining deteriorating decaying decomposing dissolving vanishing disappearing evanescent transient ephemeral fleeting momentary temporary provisional conditional contingent dependent independent autonomous self-sufficient self-reliant resilient robust durable lasting permanent eternal infinite limitless boundless vast immense enormous colossal gigantic massive substantial significant noteworthy remarkable extraordinary exceptional outstanding superb splendid magnificent glorious majestic imperial regal royal sovereign supreme autocratic despotic tyrannical dictatorial authoritarian totalitarian oppressive repressive restrictive limiting constraining confining imprisoning incarcerating detaining holding capturing seizing grabbing snatching seizing pilfering purloining filching nicking pinching lifting appropriating expropriating confiscating impounding sequestering isolating quarantining separating dividing splitting cleaving severing cutting slicing dicing mincing chopping hacking hewing carving sculpturing modelling moulding shaping forming fashioning fabricating manufacturing producing creating generating originating initiating commencing starting beginning launching initiating embarking undertaking pursuing chasing pursuing hunting tracking trailing following pursuing seeking searching exploring investigating probing examining scrutinising analysing dissecting deconstruct breaking down component parts constituent elements fundamental building blocks basic units elementary particles atomic molecules compounds mixtures solutions suspensions emulsions colloids dispersions aggregates clusters groups collections sets arrays matrices tensors vectors scalars fields spaces dimensions manifolds topologies geometries algebraic structures groups rings fields lattices orders posets categories functors natural transformations adjunctions monads comonads applicatives monoids semigroups groups categories topoi toposes sheaves stalks fibres bundles sections morphisms arrows objects identities compositions associativity unitality naturality coherence unit counit triangle pentagon hexagon equations laws axioms principles postulates assumptions premises conclusions corollaries lemmas propositions statements claims hypotheses theories paradigms frameworks models schemas templates patterns regularities symmetries conservation laws Noetherian principles variational calculus Euler-Lagrange equations Hamilton-Jacobi equations Feynman path integrals partition functions thermodynamic potentials free energy enthalpy entropy temperature pressure volume chemical potential Gibbs energy Helmholtz energy canonical ensemble microcanonical ensemble grand canonical ensemble quantum statistics Bose-Einstein Fermi-Dirac Maxwell-Boltzmann distributions Planck radiation law Stefan-Boltzmann law Wien displacement law Kirchhoff law blackbody radiation photoelectric effect Compton scattering pair production annihilation bremsstrahlung synchrotron radiation Cherenkov radiation transition radiation Smith-Purcell effect diffraction refraction reflection transmission absorption scattering emission fluorescence phosphorescence luminescence chemiluminescence bioluminescence electroluminescence cathodoluminescence triboluminescence piezoluminescence sonoluminescence photoluminescence Raman scattering Brillouin scattering Rayleigh scattering Mie scattering Tyndall effect opalescence iridescence thin-film interference Newton rings Airy disk point spread function modulation transfer function optical aberration spherical coma astigmatism distortion chromatic aberration longitudinal transverse field curvature Petzval surface image circle confusion depth field hyperfocal distance circle least confusion diffraction limit Abbe criterion Rayleigh criterion Sparrow criterion Dawson criterion Neyman-Pearson lemma likelihood ratio test Neyman structure lemma Karlin Rubin theorem Stein lemma Shannon source coding theorem channel coding theorem Hamming bound Plotkin bound Gilbert Varshamov bound Singleton bound McEliece Solomon Welch bounds Goppa codes algebraic geometry codes Reed-Solomon BCH Hamming Golay convolutional turbo LDPC polar codes polar code construction Bhattacharyya parameter Bhattacharyya coefficient Bhattacharyya distance Hellinger distance total variation distance Kullback-Leibler divergence Jensen-Shannon divergence mutual information conditional entropy joint entropy marginal entropy chain rule data processing inequality Fano inequality Gallager random coding exponent sphere packing bound cutoff rate computational cutoff rate strong converse weak converse moderate deviations large deviations central limit theorem local limit theorem Bahadur Rao theorem Bahadur efficiency Hodges Lehmann estimators M-estimators S-estimators L-estimators R-estimators MLE MAP EM algorithm Baum Welch Viterbi forward backward algorithms Kalman filter extended unscented particle filter sequential Monte Carlo Markov chain Monte Carlo Metropolis-Hastings Gibbs sampling Hamiltonian Monte Carlo No U-Turn Sampler HMC NUTS slice sampling adaptive rejection sampling importance sampling stratified sampling Latin hypercube sampling quasi-Monte Carlo Sobol Halton Faure Niederreiter sequences discrepancy low-discrepancy sequences quasi-random points uniform distribution normal distribution lognormal exponential gamma beta Dirichlet Wishart inverse-Wishart multivariate normal multivariate t multivariate beta multivariate F Cauchy Pareto Weibull Gumbel Fréchet logistic log-logistic generalized extreme value GEV GEV distribution generalized Pareto GPD extreme value theory peaks-over-threshold POT method declustering run declustering extremal index extremal coefficient extremogram extremal dependence tail dependence copula Clayton Frank Gumbel Joe survival copula Archimedean copula Gaussian copula t-copula vine copula pair-copula construction D-vine C-vine regular vine nested Archimedean hierarchical Archimedean canonical vine tree structured vine factor vine graphical model Bayesian network Markov random field conditional random field hidden Markov model linear dynamical system state-space model switching linear dynamical system jump Markov linear system switching autoregressive hidden Markov model input-output HMM factorial HMM coupled HMM hierarchical HMM infinite HMM sticky HDP-HMM Dirichlet process mixture Bayesian nonparametric Dirichlet process Chinese restaurant process stick-breaking representation Blackwell MacQueen Pólya urn scheme Ferguson Green Yor Pitman Yor process Poisson-Kingman mixture inverse Gaussian bridge Bessel bridge Brownian bridge Ornstein-Uhlenbeck process geometric Brownian motion Cox-Ingersoll-Ross Vasicek Hull-White Ho-Lee Black-Derman-Toy BDT Cox-Ross-Rubinstein trinomial binomial lattice finite difference Crank-Nicolson DuFort-Frankel Laasonen Richardson extrapolation Romberg integration Gauss-Legendre Gauss-Chebyshev Gauss-Hermite Gauss-Jacobi Gauss-Laguerre Gauss-Kronrod Clenshaw-Curtis adaptive quadrature Simpson rule trapezoidal rule midpoint rule Euler method Heun method Runge-Kutta RK4 Dormand Prince Cash-Karp Fehlberg RKF45 DOPRI5 adaptive step size error control local global truncation error Richardson extrapolation convergence order stability region stiff problems implicit methods backward differentiation formulas BDF Gear method exponential integrators splitting methods Lie-Trotter Strang splitting symmetric composition methods symplectic integrators Verlet leapfrog Störmer-Verlet velocity Verlet Beeman predictor-corrector position extended phase space shadow Hamiltonian modified equation modified equation analysis backward error analysis geometric numerical integration Poisson integrators contact integrators variational integrators discrete mechanics Lagrangian mechanics Hamilton-Jacobi theory canonical transformations generating functions symplectic maps area preservation Moser twist map Aubry-Mather theory KAM theory small divisor problem Diophantine approximation Liouville numbers Roth theorem Thue-Siegel-Roth Baker theorem transcendence measure irrationality measure continued fractions convergents semi-convergents mediants Ostrowski expansion Engel expansion Pierce expansion Cantor expansion factorial number system mixed radix representations balanced ternary negabinary redundant representations carry-save adder carry-propagate adder carry-lookahead adder Brent-Kung adder Wallace tree Dadding multiplier adder carry-select adder carry-skip adder carry-chain adder parallel prefix adder Kogge-Stone Han-Carlson Brent-Kung Ladner-Fischer Sklansky Lynch Swartzfager Knowles Bal Dorsam Opel Sancheti Weyl Garsia Wachs Krapf Prunskiutas Mullin Cech Klawitter Baur Strassen Toom-Cook Karatsuba Schönhage-Strassen Fürer Harvey van der Hoeven integer multiplication convolution FFT NTT number theoretic transform Schönhage Nussbaumer circular convolution linear convolution polynomial multiplication evaluation interpolation point-value representation monomial basis Lagrange basis Newton basis Chebyshev basis Hermite basis Bernstein basis B-spline basis cardinal spline basis Haar wavelet Daubechies wavelet Coiflet symlet Meyer wavelet biorthogonal wavelet lifting scheme perfect reconstruction filter bank quadrature mirror filter analysis synthesis lowpass highpass downsampling upsampling polyphase decomposition multirate signal processing sample rate conversion decimation interpolation anti-aliasing filter Nyquist frequency Shannon sampling theorem Whittaker-Shannon interpolation formula sinc function cardinal spline B-spline uniform non-uniform knot vector control points de Boor algorithm Cox-de Boor recursion Bézier curve Bézier surface subdivision Catmull-Clark Loop butterfly Chaikin corner cutting dual primal uniform non-uniform rational B-spline NURBS T-spline subdivision surface mesh refinement decimation simplification edge collapse vertex split half-edge data structure winged-edge boundary representation constructive solid geometry CSG boundary representation B-rep parametric implicit explicit surface of revolution ruled surface developable surface minimal surface constant mean curvature constant Gaussian curvature Weingarten surfaces affine differential geometry projective differential geometry conformal mapping Riemann mapping theorem Schwarz-Christoffel transformation Joukowski transform Kutta-Joukowski theorem Blasius theorem d’Alembert paradox von Kármán vortex street Strouhal number Reynolds number Mach number Prandtl number Nusselt number Grashof number Rayleigh number Péclet number Schmidt number Sherwood number Lewis number Damköhler number Thiele modulus effectiveness factor Mears criterion Weisz-Prater criterion Carberry criterion Froment criterion Levenspiel plot design chart graphical method analytical method numerical method finite difference finite element finite volume boundary element spectral method method of lines collocation Galerkin Petrov-Galerkin least squares discontinuous Galerkin hybridizable discontinuous Galerkin HDG embedded discontinuous Galerkin EDG variational multiscale VMS large eddy simulation LES direct numerical simulation DNS Reynolds-averaged Navier-Stokes RANS Spalart-Allmaras k-epsilon k-omega SST RSM BSL EARSM algebraic stress model nonlinear eddy viscosity model detached eddy simulation DES delayed DES DDES IDDES wall-modeled LES WMLES hybrid RANS-LES zonal hybrid seamless hybrid transition model γ-Reθ Langtry Menter correlation-based transition bypass transition natural transition separation-induced transition Tollmien-Schlichting waves crossflow instability Görtler vortices centrifugal instability Rayleigh-Taylor Kelvin-Helmholtz Richtmyer-Meshkov Taylor-Couette von Kármán swirling flow vortex breakdown bubble spiral helical modes precessing vortex core PVC cavity flow backward-facing step forward-facing step periodic hill channel flow pipe flow turbulent boundary layer laminar boundary layer Blasius solution Falkner-Skan Hartree wedge flow stagnation point flow Hiemenz flow Homann axisymmetric stagnation jet impingement wall jet free jet coflowing jet counterflowing jet crossflow jet turbulent jet round jet plane jet annular jet coaxial jet swirling jet heated jet buoyant jet plume thermal convection natural forced mixed convection Rayleigh-Bénard convection Bénard cells Marangoni convection thermocapillary convection Soret effect Dufour effect thermophoresis photophoresis diffusiophoresis electrophoresis dielectrophoresis magnetophoresis acoustic streaming acoustic radiation pressure radiation force acoustic levitation ultrasonic machining ultrasonic welding ultrasonic cleaning sonochemistry sonoluminescence acoustic cavitation bubble dynamics Rayleigh-Plesset equation Keller-Miksis equation Prosperetti equation non-equilibrium thermodynamics fluctuation-dissipation theorem Onsager reciprocal relations Prigogine minimum entropy production Onsager-Machlup functional fluctuation theorem Jarzynski equality Crooks fluctuation theorem non-equilibrium steady stateNESS entropy production rate dissipation function thermodynamic uncertainty relation speed-accuracy trade-off fluctuation-response theorem Kubo formula Green-Kubo relations Einstein relation Stokes-Einstein relation Debye-Stokes-Einstein relation molecular diffusion Fick’s laws diffusion Brownian motion Langevin equation Kramers escape rate WKB approximation Kramer’s turnover Smoluchowski equation Debye-Smoluchowski equation coagulation kernel Smoluchowski coagulation equation coalescence breakup fragmentation aggregation gelation sol-gel transition percolation theory bond site continuum percolation critical threshold cluster size distribution scaling theory finite-size scaling renormalization group real-space momentum-space Wilson Kadanoff block spin decimation majority rule exact renormalization group Wegner-Houghton Polchinski Wetterich equation functional renormalization group fRG non-perturbative truncation vertex expansion derivative expansion two-particle irreducible formalism Cornwall-Jackiw-Tomboulis CJT effective potential 2PI effective action Luttinger-Ward functional Baym-Kadanoff formalism self-consistent approximation Hartree-Fock GW approximation Bethe-Salpeter equation T-matrix approximation ladder approximation random phase approximation RPA time-dependent density functional theory TD-DFT linear response Casida equation excitation energy oscillator strength sum rules Thomas-Reiche-Kuhn sum rule oscillator strength distribution dipole strength function photoabsorption cross-section photoionization cross-section Compton profile momentum distribution electron momentum spectroscopy (e,2e) reaction binary (e,2e) (e,3e) (γ,2e) double photoionization triple photoionization shake-off shake-up satellite structure correlation energy dynamic correlation static correlation configuration interaction CI complete active space CAS multireference configuration interaction MRCI coupled cluster CC singles doubles triples quadruples CCSDTQ full configuration interaction FCI selected CI CIPSI heat-bath HCI stochastic HCI ASCI DMRG matrix product state tensor network variational Monte Carlo VMC diffusion Monte Carlo DMC auxiliary field QMC AFQMC path integral QMC PIMC reptation Monte Carlo reptation algorithm worldline algorithm loop algorithm directed loop algorithm stochastic series expansion SSE operator loop algorithm continuous-time QMC CT-QMC weak coupling CT-QMC strong coupling CT-QMC hybridization expansion expansion diagrammatic Monte Carlo diagrammatic determinant Monte Carlo bold-line diagrammatic Monte Carlo parquet equations functional renormalization group fRG parquet fRG dual fermion dual boson slave particle slave boson slave fermion parton Gutzwiller projection variational cluster approximation VCA cellular dynamical mean-field theory CDMFT dynamical cluster approximation DCA embedded cluster approximation dynamical mean-field theory DMFT iterated perturbation theory IPT non-crossing approximation NCA crossed vertex approximation CVA dynamical vertex approximation DΓA dual fermion DF dual boson DB diagrammatic extensions DMFT diagrammatic Monte Carlo DiagMC bold diagrammatic Monte Carlo bold DiagMC dual fermion expansion dual boson expansion parquet equations parquet approximation parquet ladder approximation fluctuation exchange FLEX self-consistent many-body perturbation theory GW+DMFT EDMFT extended dynamical mean-field theory charge self-consistency lattice embedding real-space embedding impurity solver continuous-time quantum Monte Carlo CT-QMC interaction expansion weak coupling hybridization expansion strong coupling segment algorithm segment representation world-line representation configuration diagram determinant diagram expansion order histogram sign problem average sign Monte Carlo sign problem fixed-node approximation constrained path approximation phaseless approximation auxiliary field quantum Monte Carlo AFQMC trial wavefunction Slater determinant backflow transformation Jastrow factor two-body three-body backflow orbital optimization variational parameters energy minimization variance minimization stochastic reconfiguration imaginary time evolution power method time-evolving block decimation TEBD time-dependent variational principle TDVP single-site two-site alternating least squares density matrix renormalization group DMRG block state transfer matrix spectrum entanglement entropy von Neumann entropy Rényi entropy mutual information entanglement spectrum Schmidt decomposition canonical form mixed canonical form Vidal form time-dependent DMRG tDMRG variational uniform matrix product state VUMPS infinite DMRG iDMRG corner transfer matrix CTA MPS corner transfer matrix renormalization CTRG tensor renormalization group TRG higher-order TRG HOTRG tensor network renormalization TNR isometric tensor network tree tensor network TTN multi-scale entanglement renormalization ansatz MERA U-MERA cMERA projected entangled pair states PEPS projected entangled simplex states PESS simple update full update boundary contraction belief propagation tensor renormalization group TNRG variational tensor network ansatz VUMPS imaginary time evolution ground state preparation excited states finite temperature density matrix purification thermofield dynamics ancilla system bath purification vector operator product expansion OPE conformal field theory CFT central charge Virasoro algebra primary fields descendant fields conformal blocks modular invariance partition function torus cylinder plane radial quantization state-operator correspondence conformal bootstrap crossing symmetry unitarity bounds numerical conformal bootstrap linear programming semidefinite programming bootstrap bounds critical exponents scaling dimensions operator product expansion coefficients OPE coefficients four-point function crossing equations conformal blocks conformal field theory applications condensed matter statistical mechanics high energy physics string theory AdS/CFT correspondence holographic duality bulk-boundary correspondence Ryu-Takayanagi formula entanglement entropy holographic entanglement entropy quantum error correction quantum error correction code surface code toric code color code stabilizer code CSS code Steane code Shor code Bacon-Shor code subsystem code gauge subsystem code topological code anyonic code non-Abelian anyons Majorana zero modes topological quantum computation braiding fusion rules modular tensor category topological order symmetry-protected topological phase symmetry-enriched topological phase cobordism classification Kitaev chain SSH model Haldane model Kane-Mele model topological insulator topological superconductor Weyl semimetal Dirac semimetal nodal line semimetal higher-order topological insulator corner states hinge states fragile topology obstructed atomic limit fragile symmetric topological phases classification periodic table tenfold way real complex quaternionic octonionic Altland-Zirnbauer symmetry classes K-theory cobordism cohomology homology homotopy groups classification schemes periodic table topological insulators superconductors superfluids spin liquids quantum spin liquids frustrated magnetism Heisenberg model Kagome lattice triangular lattice honeycomb lattice pyrochlore lattice spin ice classical spin ice quantum spin ice artificial spin ice emergent monopoles Coulomb phase pinch points Dirac spin liquid U(1) Z2 Zk gauge theory lattice gauge theory Hamiltonian formulation Lagrangian formulation path integral formulation Wilson loop Polyakov loop area law perimeter law confinement deconfinement Higgs phase Anderson-Higgs mechanism Meissner effect flux quantization Abrikosov vortex lattice type-I type-II superconductor Ginzburg-Landau theory coherence length penetration depth GL parameter upper critical field lower critical field mixed state vortex state flux flow flux creep thermally activated flux motion Bean critical state model London equations Pippard equations nonlocal electrodynamics Mattis-Bardeen theory BCS theory Cooper pairing s-wave p-wave d-wave gap equation energy gap density of states quasiparticle recombination phonon emission phonon absorption Andreev reflection specular Andreev retro-reflection crossed Andreev electron-hole conversion Majorana bound states Josephson effect DC Josephson AC Josephson Josephson frequency relation Shapiro steps SQUID superconducting quantum interference device flux qubit charge qubit phase qubit transmon Xmon fluxonium qudit quantum computing quantum processor quantum annealing adiabatic quantum computing D-Wave gate model circuit model measurement-based quantum computing cluster state one-way quantum computing topological quantum computing measurement-induced phase transition entanglement transition monitoring quantum trajectory quantum jump quantum state diffusion quantum state smoothing quantum filtering quantum feedback control coherent feedback measurement-based feedback conditional states quantum Bayesian inference quantum state estimation quantum tomography quantum process tomography quantum detector tomography quantum channel estimation quantum metrology quantum sensing quantum magnetometry quantum gravimetry quantum accelerometer quantum gyroscope quantum clock atomic clock optical lattice clock ion trap clock nuclear clock quantum limited amplifier Josephson parametric amplifier traveling wave parametric amplifier kinetic inductance detector transition edge sensor superconducting nanowire single photon detector SNSPD avalanche photodiode APD single photon avalanche diode SPAD photomultiplier tube PMT charge-coupled device CMOS sensor scientific CMOS sCMOS electron multiplying CCD EMCCD intensified CCD ICCD microchannel plate MCP time-correlated single photon counting TCSPC fluorescence lifetime imaging FLIM fluorescence correlation spectroscopy FCS fluorescence resonance energy transfer FRET Förster radius photobleaching photoactivation photoconversion photoswitching fluorescent proteins GFP YFP CFP mCherry mNeonGreen split GFP bimolecular fluorescence complementation BiFC fluorescence recovery after photobleaching FRAP fluorescence loss in photobleaching FLIP fluorescence lifetime correlation spectroscopy FLCS number and brightness analysis N&B raster image correlation spectroscopy RICS fluorescence intensity distribution analysis FIDA photon counting histogram PCH fluorescence lifetime imaging microscopy FLIM confocal microscopy two-photon microscopy three-photon microscopy light sheet microscopy selective plane illumination microscopy SPIM lattice light sheet microscopy structured illumination microscopy SIM stimulated emission depletion microscopy STED photoactivated localization microscopy PALM stochastic optical reconstruction microscopy STORM direct STORM dSTORM points accumulation for imaging in nanoscale topography PAINT DNA-PAINT super-resolution microscopy expansion microscopy expansion factor refractive index index matching immersion oil water glycerol mounting medium coverslip slide objective lens numerical aperture magnification resolution Abbe diffraction limit Rayleigh criterion point spread function deconvolution Richardson-Lucy deconvolution blind deconvolution Wiener filter Tikhonov regularization total variation regularization non-local means denoising BM3D block-matching 3D filtering wavelet denoising curvelet denoising shearlet denoising contourlet denoising sparse representation dictionary learning K-SVD compressed sensing sparse recovery basis pursuit basis pursuit denoising LASSO elastic net group LASSO fused LASSO total variation L1 regularization L2 regularization mixed norm regularization nuclear norm Schatten norm trace norm low-rank matrix recovery matrix completion Robust PCA principal component analysis singular value decomposition eigenvalue decomposition spectral theorem normal operator self-adjoint operator compact operator Hilbert-Schmidt operator trace class operator Fredholm operator index theorem Atiyah-Singer index theorem heat kernel index theorem anomalies chiral anomaly axial anomaly conformal anomaly trace anomaly gravitational anomaly gauge anomaly global anomaly ABJ anomaly Wess-Zumino consistency conditions descent equations Chern-Simons theory topological quantum field theory TQFT invariants Jones polynomial HOMFLY polynomial Kauffman bracket skein relation Khovanov homology knot invariants link invariants braid group representation quantum group Yang-Baxter equation star-triangle relation Ice-type model six-vertex model eight-vertex model free-fermion model Pfaffian dimer model perfect matching transfer matrix method Bethe ansatz coordinate algebraic thermodynamic Bethe ansatz TBA nonlinear integral equation Y-system T-system Q-system Baxter Q-operator quantum transfer matrix corner transfer matrix conformal bootstrap integrable systems integrable lattice models integrable spin chains Heisenberg XXX XXZ XYZ Kitaev-XY chain Ising model transverse field Ising model Jordan-Wigner transformation Majorana representation fermionization bosonization Luttinger liquid Tomonaga-Luttinger model Bethe liquid Fermi liquid theory Landau Fermi liquid quasiparticle lifetime effective mass spin-charge separation non-Fermi liquid strange metal Planckian dissipation marginal Fermi liquid heavy fermion Kondo effect Anderson impurity model Kondo lattice model RKKY interaction Doniach phase diagram quantum critical point non-Fermi liquid phase heavy fermion liquid spin liquid Mott insulator Hubbard model t-J model t-J model half-filling doping antiferromagnet Néel order spin gap d-wave superconductivity pseudogap cuprate phase diagram high-temperature superconductor iron-based superconductor pnictide chalcogenide Hund’s metal orbital selectivity Mott transition Brinkman-Rice picture Gutzwiller approximation slave boson mean field RVB resonating valence bond Anderson RVB theory Zhang-Rice singlet charge transfer insulator Zaanen-Sawatzky-Allen classification optical conductivity Drude weight mid-infrared peak charge transfer gap Mott-Hubbard gap spectral weight transfer sum rule optical sum rule kinetic energy potential energy exchange energy double occupancy Gutzwiller factor renormalization factor quasi-particle weight mass enhancement correlation length spin correlation function charge correlation function superconducting pairing susceptibility charge susceptibility spin susceptibility orbital susceptibility magnetic susceptibility NMR relaxation rate Knight shift muon spin rotation muSR neutron scattering inelastic neutron scattering resonant inelastic X-ray scattering RIXS angle-resolved photoemission ARPES scanning tunneling microscopy STM spectroscopy STS quasi-particle interference QPI Landau level scanning probe microscopy magnetic force microscopy MFM electrostatic force microscopy EFM Kelvin probe force microscopy KPFM atomic force microscopy AFM non-contact contact tapping mode peak force quantitative nanomechanics QNM nanoscale infrared spectroscopy nano-IR photothermal AFM AFM-IR scattering-type scanning near-field optical microscopy s-SNOM tip-enhanced Raman spectroscopy TERS surface-enhanced Raman spectroscopy SERS hotspot electromagnetic enhancement chemical enhancement charge transfer mechanism selection rules surface selection rules dipole approximation near-field enhancement plasmonic nanoparticle localized surface plasmon resonance LSPR Mie theory extinction scattering absorption cross-section plasmon hybridization bonding anti-bonding dark mode bright mode Fano resonance plasmonic nanoantenna optical antenna Yagi-Uda antenna bowtie antenna dimer gap nanogap hotspot field enhancement factor Purcell factor spontaneous emission enhancement quenching fluorescence enhancement metal-enhanced fluorescence plasmon-enhanced fluorescence surface plasmon polariton SPP propagating surface plasmon gap plasmon hybrid plasmon phonon polariton surface phonon polariton hyperbolic material hyperbolic metamaterial hyperbolic dispersion isofrequency contour open isofrequency contour closed isofrequency contour canalization sub-diffraction imaging hyperlens superlens negative refraction phase compensation evanescent wave amplification transformation optics cloaking carpet cloak ground plane cloak electromagnetic cloak acoustic cloak thermal cloak cloak invisibility cloak scattering cancellation coordinate transformation conformal mapping quasi-conformal mapping reduced parameter isotropic material anisotropic material metamaterial effective medium theory Maxwell-Garnett effective medium Bruggeman effective medium homogenization retrieval effective permittivity effective permeability negative index left-handed material backward wave Cherenkov radiation reverse Cherenkov negative refraction perfect lens Veselago lens Pendry lens superlens hyperlens magnifying hyperlens canalization regime canalization lens subwavelength imaging resolution enhancement near-field scanning optical microscopy NSOM aperture apertureless tip-enhanced near-field optical microscopy scattering SNOM s-SNOM photonic crystal photonic bandgap photonic crystal fiber photonic crystal slab photonic integrated circuit photonic chip silicon photonics indium phosphide gallium arsenide lithium niobate silicon nitride aluminum nitrideresonant waveguide grating coupler grating coupler edge coupler spot-size converter mode converter polarization splitter rotator thermo-optic phase shifter carrier injection depletion modulator Mach-Zehnder interferometer ring resonator micro-ring modulator electro-optic modulator Pockels effect Kerr effect thermo-optic effect free carrier dispersion plasma dispersion effect silicon photonics integrated photonics photonic integrated circuit PIC electronic-photonic co-design monolithic integration heterogeneous integration hybrid integration chip-to-chip coupling fibre-to-chip coupling grating coupler efficiency insertion loss return loss crosstalk extinction ratio modulation bandwidth drive voltage extinction ratio on-off ratio insertion loss propagation loss waveguide loss bending loss coupling loss fibre coupling loss single-mode fibre multimode fibre polarisation-maintaining fibre birefringence polarisation mode dispersion chromatic dispersion group velocity dispersion group delay dispersion slope higher-order dispersion dispersion management dispersion compensation dispersion compensation module dispersion compensating fibre Bragg grating chirped fibre Bragg grating fibre Bragg grating FBG long-period fibre grating LPFG volume Bragg grating acousto-optic modulator acousto-optic tunable filter AOTF electro-optic tunable filter thermo-optic tunable filter micro-electro-mechanical MEMS micromirror spatial light modulator liquid crystal modulator deformable mirror adaptive optics wavefront sensor Shack-Hartmann sensor curvature sensor pyramid sensor laser guide star natural guide star adaptive optics system deformable mirror actuator piezoelectric actuator voice coil actuator electrostatic actuator magnetic actuator membrane mirror segmented mirror continuous face-sheet mirror influence function fitting error scintillation seeing atmospheric turbulence Fried parameter coherence length isoplanatic angle Greenwood frequency temporal bandwidth Strehl ratio encircled energy point spread function modulation transfer function optical transfer function wavefront error root mean square peak-to-valley Strehl ratio Maréchal criterion Rayleigh criterion diffraction-limited imaging aberration correction optical design ray tracing sequential non-sequential Zemax OpticStudio Code V OSLO VirtualLab FRED ASAP TracePro LightTools SPEOS illumination design freeform optics freeform surface asphere polynomial NURBS spline point-sag grid sag rotationally symmetric non-rotationally symmetric toroidal conic cylindrical anamorphic prism grating diffractive optical element DOE binary optics kinoform phase plate holographic optical element HOE computer-generated hologram CGH waveguide hologram grating coupler holographic grating surface relief grating volume hologram slanted unslanted transmission reflection Bragg condition Bragg angle angular selectivity wavelength selectivity spectral bandwidth diffraction efficiency first-order second-order zeroth-order higher-order anomalous Wood’s regime Rayleigh anomaly surface plasmon resonance grating leaky mode resonance guided-mode resonance GMR resonant grating filter narrowband filter broadband filter notch filter bandpass filter edge filter polarisation filter polariser wire-grid polariser nanoparticle polariser dichroic polariser birefringent polariser Wollaston prism Glan-Thompson prism Glan-Taylor prism Nicol prism calcite quartz sapphire magnesium fluoride calcium fluoride barium fluoride lithium fluoride silicon germanium gallium arsenide indium phosphide gallium phosphide gallium nitride aluminum gallium nitride indium gallium arsenide indium aluminum arsenide indium gallium arsenide phosphide aluminum indium phosphide indium gallium nitride aluminum indium nitride zinc oxide zinc selenide zinc sulfide cadmium selenide cadmium sulfide cadmium telluride lead sulfide lead selenide tellurium bismuth tellurium antimony tellurium selenium tellurium alloy phase-change material GST Ge2Sb2Te5 GeSbTe GeTe Sb2Te3 antimony-rich germanium-antimony tellurium scandium antimony tellurium scandium tellurium germanium-antimony selenium GeSbSe germanium-antimony selenium tellurium GeSbSeTe phase-change memory PRAM PCRAM CBRAM resistive memory RRAM memristor memristive device threshold switching volatile non-volatile filamentary interface valence change electrochemical metallization conductive bridge CBRAM threshold switching ovonic threshold switch OTS chalcogenide glass amorphous crystalline set reset programming current threshold voltage holding voltage drift retention endurance write speed read speed access time energy per switching event switching energy programming energy thermal conductivity thermal diffusivity heat capacity specific heat density Young’s modulus Poisson ratio thermal expansion coefficient coefficient thermal expansion CTE stress strain fracture toughness hardness Vickers hardness Knoop hardness nanoindentation atomic force microscopy scanning electron microscopy transmission electron microscopy scanning transmission electron microscopy energy-dispersive X-ray spectroscopy X-ray photoelectron spectroscopy Auger electron spectroscopy secondary ion mass spectrometry SIMS Rutherford backscattering spectroscopy RBS time-of-flight secondary ion mass spectrometry ToF-SIMS X-ray diffraction XRD grazing incidence X-ray diffraction GIXRD electron backscatter diffraction EBSD selected area electron diffraction SAED convergent beam electron diffraction CBED nanobeam electron diffraction microbeam electron diffraction reflection high-energy electron diffraction RHEED low-energy electron diffraction LEED X-ray reflectivity XRR ellipsometry spectroscopic ellipsometry rotating analyser polarizer RAP dual rotating compensator variable angle spectroscopic ellipsometry VASE Mueller matrix ellipsometry imaging ellipsometry nano-ellipsometry photoluminescence spectroscopy time-resolved photoluminescence TRPL photoluminescence excitation PLE cathodoluminescence electroluminescence Raman spectroscopy surface-enhanced Raman SERS tip-enhanced Raman TERS coherent anti-Stokes Raman CARS stimulated Raman SRS Raman imaging confocal Raman hyperspectral Raman Fourier-transform Raman FT-Raman resonance Raman polarised Raman depolarisation ratio infrared spectroscopy FTIR attenuated total reflectance ATR micro-FTIR nano-FTIR photothermal infrared AFM-IR scattering-type near-field s-SNOM infrared near-field optical conductivity complex dielectric function permittivity permeability refractive index extinction coefficient absorption coefficient reflectivity transmittance emissivity absorptance albedo single-scattering albedo optical depth optical thickness turbidity diffuse reflectance diffuse transmittance bidirectional reflectance distribution function BRDF bidirectional transmittance distribution function BTDF bidirectional scattering distribution function BSDF integrating sphere goniophotometer reflectometer transmittometer spectrophotometer photometer radiometer radiance irradiance luminous flux luminous intensity illuminance luminance colour temperature correlated colour temperature CCT colour rendering index CRI colour quality scale CQS TM-30 IES colour fidelity index Rf colour gamut index Rg chromaticity CIE 1931 chromaticity diagram CIE 1964 chromaticity diagram CIE 1976 UCS diagram CIELAB CIELUV perceptual uniform colour space Munsell colour system Pantone colour system RAL colour system natural colour system NCS colour order system spectral power distribution spectral irradiance spectral radiance spectral transmittance spectral reflectance spectral absorptance spectral emittance blackbody Planck radiator Planck’s law Wien’s displacement law Stefan-Boltzmann law Kirchhoff’s law thermal radiation emissivity absorptivity reflectivity transmissivity grey body selective emitter perfect emitter perfect absorber perfect reflector perfect transmitter thermal equilibrium detailed balance reciprocity principle van Cittert-Zernike theorem mutual coherence function complex degree of coherence coherence area coherence length coherence time coherence bandwidth temporal coherence spatial coherence partial coherence mutual intensity cross-spectral density Wigner distribution function ambiguity function optical transfer function modulation transfer function point spread function line spread function edge spread function knife-edge measurement slanted-edge method bar target resolution target USAF 1951 resolution target Siemens star spoke target zone plate Fresnel zone plate kinoform zone plate binary zone plate diffractive lens Fresnel lens photon sieve metalens metalens focusing metalens imaging metalens spectral splitting metalens polarization multiplexing metalens orbital angular momentum OAM vortex beam helical phase Laguerre-Gaussian beam Bessel beam Airy beam non-diffracting beam accelerating beam self-healing beam photonic orbital angular momentum spin-orbit interaction geometric phase Pancharatnam-Berry phase metasurface Huygens metasurface Pancharatnam-Berry metasurface transmissive metasurface reflective metasurface multifunctional metasurface programmable metasurface reconfigurable metasurface dynamic metasurface intelligent reflecting surface IRS reconfigurable intelligent surface RIS metasurface antenna metasurface hologram metasurface lens metasurface beam steering metasurface polarisation control metasurface wavefront shaping metasurface absorption metasurface sensing metasurface imaging metasurface display metasurface colour printing structural colour plasmonic colour dielectric colour Mie resonance Mie scattering Mie theory Lorenz-Mie theory exact solution partial wave expansion scattering cross-section extinction cross-section absorption cross-section scattering efficiency extinction efficiency absorption efficiency albedo asymmetry parameter phase function Henyey-Greenstein function Rayleigh scattering Mie scattering geometric optics diffraction interference rainbow glory corona fogbow cloud iridescence circumhorizontal arc circumzenithal arc sun dog parhelic circle pillar of light light pillar light column light shaft crepuscular rays anticrepuscular rays Tyndall effect opalescence milkiness blue sky red sunset orange pink purple twilight belt of Venus Earth’s shadow Rayleigh scattering polarization skylight polarization sun glint sunglint specular reflection diffuse reflection Lambertian reflection BRDF retroreflection corner cube cat’s eye retroreflector road sign retroreflective paint retroreflective tape retroreflective sheeting glass bead microprismatic encapsulated lens retroreflector hollow corner cube solid corner cube dihedral corner reflector trihedral corner reflector radar reflector Luneberg lens Luneburg lens Maxwell fish-eye lens Eaton lens spherical gradient index GRIN lens graded-index lens radial gradient axial gradient spherical gradient cylindrical gradient planar gradient index lens Wood lens Luneburg lens Maxwell fish-eye lens Eaton lens quasi-conformal GRIN lens transformation optics GRIN fibre GRIN lens GRIN rod GRIN slab GRIN waveguide graded-index multimode fibre graded-index profile parabolic profile alpha profile step-index fibre single-mode fibre multimode fibre few-mode fibre multi-core fibre hollow-core fibre photonic crystal fibre photonic bandgap fibre anti-resonant fibre inhibited coupling fibre nested anti-resonant nodeless fibre NANF Kagome lattice fibre hypocycloid core surround circular core surround hexagonal core surround pentagonal core surround square core surround triangular core surround elliptical core surround rectangular core surround D-shaped core surround depressed cladding fibre W-profile fibre triple-clad fibre quadruple-clad fibre coaxial fibre coupled fibre twin-core fibre multi-core fibre fan-in fan-out spatial division multiplexing SDM mode-division multiplexing MDM cladding mode coupling macro-bending loss micro-bending loss splice loss connector loss Fresnel reflection back reflection return loss optical return loss ORL insertion loss polarisation dependent loss PDL polarisation mode dispersion PMD differential group delay DGD principal states of polarisation PSP chromatic dispersion CD group velocity dispersion GVD dispersion slope third-order dispersion fourth-order dispersion dispersion management dispersion compensation dispersion-compensating fibre DCF chirped fibre Bragg grating CFBG volume Bragg grating VBGr fibre Bragg grating FBG long-period fibre grating LPFG fibre Fabry-Perot etalon tunable filter arrayed waveguide grating AWG echelle grating planar lightwave circuit PLC thermo-optic switch electro-optic switch MEMS switch liquid crystal switch 1×2 switch 2×2 switch NxN switch optical cross-connect OXC wavelength-selective switch WSS reconfigurable optical add-drop multiplexer ROADM colourless directionless contentionless CDC ROADM flex-grid gridless elastic optical network software-defined networking SDN network function virtualisation NFV OpenROADM Telecom Infra Project TIP open line system OLS coherent optical transport 400ZR 400ZR+ 800G coherent pluggable QSFP-DD OSFP CFP2-DCO digital signal processing DSP carrier recovery equalisation chromatic dispersion compensation polarisation demultiplexing frequency offset estimation phase noise compensation maximum likelihood sequence estimation MLSE probabilistic constellation shaping PCS geometric shaping Nyquist shaping root-raised-cosine filter matched filter matched filtering signal-to-noise ratio SNR optical signal-to-noise ratio OSNR electrical signal-to-noise ratio ESNR bit error rate BER pre-FEC BER post-FEC BER error floor forward error correction FEC soft-decision FEC hard-decision FEC staircase code LDPC code turbo code Reed-Solomon code BCH code convolutional code Viterbi decoder polar code Reed-Muller code Golay code Hamming code repetition code parity check code cyclic redundancy check CRC hash function error detection error correction coding gain overhead code rate block length interleaver deinterleaver convolutional interleaver block interleaver random interleaver turbo interleaver LDPC interleaver LDPC code Tanner graph variable node check node degree distribution girth cycle length sum-product algorithm belief propagation min-sum algorithm normalized min-sum algorithm offset min-sum algorithm layered decoding shuffled decoding flooding decoding early termination stopping criterion convergence decoder architecture fully parallel partially parallel serial architecture FPGA implementation ASIC implementation DSP implementation soft information log-likelihood ratio LLR hard decision soft decision quantisation fixed-point arithmetic floating-point arithmetic bit width word length precision accuracy throughput latency area power consumption energy efficiency energy per bit joules per bit picojoules per bit femtojoules per bit attojoules per bit terabits per second gigabits per second megabits per second kilobits per second baud symbol rate modulation format quadrature amplitude modulation QAM phase-shift keying PSK binary phase-shift keying BPSK quadrature phase-shift keying QPSK 8-PSK 16-PSK amplitude-shift keying ASK binary ASK 4-ASK pulse amplitude modulation PAM PAM-4 PAM-8 PAM-16 duobinary modulation partial response maximum likelihood PRML dicode partial response class-1 partial response class-4 extended partial response EPR ultra partial response UPR generalized partial response GPR read channel write channel magnetic recording hard disk drive HDD solid-state drive SSD flash memory NAND flash NOR flash 3D NAND vertical NAND V-NAND charge trap flash floating gate memory single-level cell SLC multi-level cell MLC triple-level cell TLC quad-level cell QLC penta-level cell PLC bit cost per gigabyte areal density bits per square inch terabits per square inch gigabits per square inch heat-assisted magnetic recording HAMR microwave-assisted magnetic recording MAMR shingled magnetic recording SMR perpendicular magnetic recording PMR longitudinal magnetic recording LMR giant magnetoresistance GMR tunnelling magnetoresistance TMR magnetic tunnel junction MTJ spin-transfer torque STT spin-orbit torque SOT spin-transfer torque magnetic random-access memory STT-MRAM spin-orbit torque magnetic random-access memory SOT-MRAM magnetic skyrmion magnetic hopfion topological spin texture Dzyaloshinskii-Moriya interaction exchange bias exchange frustration spin glass spin ice spin liquid quantum spin liquid frustrated magnetism geometric frustration triangular lattice Kagome lattice pyrochlore lattice Heisenberg model Ising model XXZ model XYZ model Kitaev model honeycomb lattice Kitaev spin liquid Majorana fermion non-Abelian anyon topological quantum computation braiding fusion measurement topological quantum error correction surface code toric code color code honeycomb code XZZX code subsystem code gauge color code fracton topological order subsystem symmetry foliated fracton phase Haah’s code cubic code fractal spin liquid constrained dynamics fractalon mobility constraint subsystem symmetry protected topological phase symmetry enriched topological phase cobordism classification topological field theory Chern-Simons theory BF theory topological order long-range entanglement short-range entanglement symmetry-protected topological SPT symmetry-enriched topological SET anyon model modular tensor category fusion category braided fusion category premodular category pivotal category spherical category ribbon category Drinfeld centre Drinfeld double module category module functor bimodule induction restriction induction functor forgetful functor tensor product monoidal category symmetric monoidal category braided monoidal category strict monoidal category weak monoidal category monoidal functor lax monoidal functor oplax monoidal functor strong monoidal functor monoidal natural transformation associator unitator pentagon equation triangle equation coherence theorem Mac Lane coherence theorem monoidal category coherence higher category two-category three-category n-category infinity-category stable infinity-category derived category homotopy category model category simplicial category enriched category internal category internal groupoid internal category internal functor internal natural transformation internal limit internal colimit Kan extension adjoint functor universal property representable functor Yoneda lemma Yoneda embedding Yoneda reduction enriched Yoneda lemma nerve nerve of category nerve of simplicial set Duskin nerve Waldhausen S-construction algebraic K-theory K_0 K_1 K_2 K-theory spectrum topological K-theory complex K-theory real K-theory KO-theory KU-theory KSp-theory KGL-theory KQ-theory KR-theory Atiyah-Hirzebruch spectral sequence Adams spectral sequence Adams-Novikov spectral sequence Brown-Peterson spectrum Morava K-theory Morava E-theory Lubin-Tate spectrum topological modular forms tmf elliptic spectrum elliptic cohomology equivariant cohomology Borel equivariant cohomology Bredon equivariant cohomology equivariant stable homotopy theory equivariant spectra genuine equivariant spectra Borel equivariant spectra naive equivariant spectra RO(G)-graded homotopy theory Mackey functor coefficient system Bredon cohomology tom Dieck splitting induction restriction transfer norm restriction norm transfer assembly map Baum-Douglas assembly map Farrell-Jones conjecture Novikov conjecture Borel conjecture topological rigidity h-cobordism s-cobordism surgery theory surgery exact sequence L-group L(A) L(n) L*(A) simply connected surgery classification high-dimensional manifolds Poincaré conjecture generalized Poincaré conjecture smooth Poincaré conjecture exotic spheres Milnor sphere Kervaire sphere exotic smooth structures 4-dimensional smooth Poincaré conjecture Freedman theorem Donaldson theorem Seiberg-Witten invariants gauge theory Donaldson invariants Seiberg-Witten theory monopole instanton anti-instanton ASD instanton self-dual instanton anti-self-dual instanton moduli space instanton moduli space monopole moduli space Uhlenbeck compactification Uhlenbeck compactification Donaldson-Uhlenbeck-Yau theorem Hermitian Yang-Mills connection stable holomorphic vector bundle algebraic curve Riemann surface moduli space Riemann moduli space Teichmüller space Teichmüller theory Fenchel-Nielsen coordinates length twist coordinates earthquake deformation measured foliation quadratic differential Jenkins-Strebel differential Thurston’s classification measured foliations quadratic differentials Teichmüller geodesic Teichmüller metric Thurston metric Weil-Petersson metric Weil-Petersson symplectic form Weil-Petersson volume Mirzakhani’s recursion Mirzakhani’s volume recursion earthquake theorem Thurston’s earthquake theorem bending hyperbolic 3-manifold geometric structures hyperbolization Agol’s theorem Wise’s theorem cub