2 R. El-Ganainy, and D.N.Christodoulides Collaborative groupsR. El-Ganainy, and D.N.ChristodoulidesCollege of Optics /CREOL, University of Central Florida, USAM. SegevTechnion, IsraelP. Ambichl, and S. RotterInstitute of Theoretical Physics, TU-Wien, Vienna, AustriaZ. MusslimaniMathematics department, Florida State University, USAG. Aqiang and G. Salamo – University of Arkansas, USAC. E. Rüter and D. Kip - Clausthal University, Germany
3 Overview Physical characteristics of PT-symmetric potentials Introduction to PT-symmetric OpticsPhysical characteristics of PT-symmetric potentialsGroup velocity in PT-symmetric latticesPT-symmetry breaking in Fabry-Perot cavitiesConclusions
5 PT-symmetry in Quantum Mechanics Should a Hamiltonian be Hermitian in order to have real eigenvalues?Parity and Time operatorsSchrödinger EquationPT-potentialPT symmetric Hamiltonian can exhibit entirely real eigenvalue spectrum!*C. M.Bender et al, Phys. Rev. Lett., 80, 5243 (1998); C. M.Bender et al, Phys. Rev. Lett., 89, (2002)C. M.Bender et al, Phys. Rev. Lett., 98, (2007); C. M.Bender, Contemporary Physics, 46, 277 (2005)
6 Quantum mechanics and Wave Optics Paraxial OpticsQuantum MechanicsParaxial equation of diffractionSchrödinger equationIn spite of this intensive theoretical studies, the concept of PT symmetry remain elusive until it was realized that PT symmetry can be implemented using optical geometries. This analogy can be understood by considering electromagnetic wave propagation through optically engineered media.We know that the propagation of wide beams respects an equation of diffraction analogous to Schrodinger equation, apart from replacing the time variable with z. We also know that potentials in SE can be realized using the so called optical potential which is nothing more than engineering the spatial distribution of refractive index. So complex potential would mean complex refractive index profile which can be achieved using gain and loss.Propagation constantsEnergy eigenvalues6
7 PT-symmetric potential PT symmetry in Optics*NonlinearSchrödingerEquationPT-symmetric potentialXTypical parametersGL*R. El-Ganainy, K.G.Makris, D. N. Christodoulides, and Z. H. Musslimani, Opt. Lett. 32, 2632 (2007).K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. Lett. 100, (2008).Z.H.Musslimani, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, Phys. Rev. Lett. 100, (2008).K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. A 81, (2010).*R. El-Ganainy, K.G.Makris, and D. N. Christodoulides, Phys. Rev. A 86, (2012).
8 realization of PT-lattice Observation ofPT-breakingin a passive couplerObservation ofPT-breakingin an active couplerExperimentalrealization of PT-latticeParity–time synthetic photonic lattices,A. Regensburger, C. Bersch,A. Miri, G. Onishchukov,D. N. Christodoulides , and U. PeschelNature, 488, 167–171 (09 August 2012)
9 Negative index materials Photonic crystalsNegative index materialsPT-symmetricOpticszPT-symmetricwaveguidesPT-symmetriccavities
11 Abrupt phase transition PT Phase transition in a single waveguide*Scarff potentialExceptional pointAbrupt phase transitionBiorthogonalitycondition*Z. Ahmed, Phys. Lett. A, 282, 343 (2001)W.D. Heiss, Eur. Phys. J. D 7, 1 (1999)
12 Orthonormality relation Floquet-Bloch modes in real lattices*Discrete DiffractionFloquet Bloch modek : Bloch wavenumber, n : number of bandD periodOrthonormality relationBandstructureSuperpositionprincipleProjectioncoefficientsParseval’sidentity*D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature, 424, 817 (2003).
13 Before phase transition After phase transition Bandstucture of a PT optical lattice*ExceptionalpointBefore phase transitionAfter phase transition*K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. Lett. 100, (2008).
15 Motion of Scattering matrix eigenvalues in the complex plane Scattering from Fabry-Perot PT cavities*brokenunbrokenGLg: gain/lossScatteringmatrixBasicrelationsExceptionalpointMotion of Scattering matrixeigenvalues in the complex plane*L. Ge, Y. D. Chong, and A.D. Stone, Phys. Rev. Lett. 106, (2011).
16 Relation between finite and open PT-cavities* Helmholtz equationin finite domain(Cavity length)/2General RobinBoundary ConditionsGain-loss amplitude*P. Amblich, K.G.Makris, L. Ge, Y. D. Chong, and S. Rotter, to be submitted (2013).
17 Finite and open PT-cavities Finite PT-systemOpen scattering PT-systemEach eigenstate of S is also an eigenstate of an effectiveHamiltonian Heff with the appropriate Robin boundary conditionsThe effective Hamiltonian Heff is PT-symmetric whenThe 2D union of all the eigenvalue curves of Heff foris identical to the unbroken phase of the open scattering problem
18 Practical considerations for observing PT-scattering in cavities Symmetric output powerbelow EPEigenvectorof S-matrixEigenvectorof S-matrixbrokenAsymmetric output power above EP
19 Physical value of gain at the exceptional point brokenTypical physical valuesWe need longcavities to observePT-phase transitionunbrokeng-mismatchtolerance
20 Effect of incidence angle in scattering in PT-cavities It is experimentally easier if the angle of incidence is non-zerounbrokenunbrokenTE polarizationTM polarization
21 Scattering coefficients in 2 layer PT-cavities* Reflectancefromleft to rightReflectancefromleft to rightTransmittanceFor bothtransmission resonancepoints are below the EPNormalincidenceL. Ge, Y. Chong, D. Stone, PRA 85, (2012)
22 Multilayer Fabry-Perot PT-cavities* 12 layers, TE, normal incidencebrokenbrokenzoomMultiple phase transitions*K.G.Makris, P. Amblich, L. Ge, S. Rotter, and D. N. Christodoulides to be submitted (2013).
23 Multilayer Fabry-Perot PT-cavities TE-polarizationTM-polarization12 layersbrokenbrokenEP1Closed pathsof scatteringeigenvalues incomplex planeExperimentally,we do not need toscan the length of cavity, but the angleEP2
24 Superoscillatory diffractionless beams K.G. MakrisElectrical Engineering Department, Princeton University, USAE. Greenfield, and M. SegevPhysics Department, Solid State Institute, Technion, IsraelD. Papazoglou, and S. TzortzakisMaterials Science and Technology Department, University of Crete, Heraklion, GreeceInstitute of Electronic Structures and Laser, Foundation for Research and Technology Hellas, Heraklion, GreeceD. PsaltisSchool of Engineering, Swiss Federal Institute of Technology Lausanne(EPFL), Switzerland
25 Optical Superoscillations Superoscillatory field: A field that locally has subwavelength features but no evanescent waves.Theoretical suggestion:Optical super-resolutionwith no evanescent wavesM. V. Berry, and S. Popescu, J. Phys. A: Math. Gen. 39, 6965 (2006)M. V. Berry, and M. R. Dennis, J. Phys. A: Math. Theor. 42, (2009)P. J. S. G. Ferreira and A. Kempf, IEEE Trans. Signal Process., 54, 3732, (2006)M. R. Dennis, A. C. Hamilton, and J. Courtial, Opt. Lett. 33, 2976 (2008)Optical Experiment:Subwavelength focus in the far fieldwith no evanescent wavesN. I. Zheludev, Nature 7, 420 (2008); F. M. Huang, et al.,J. Opt. A: Pure Appl. Opt. 9, S285 (2007); F. M. Huang,and N. I. Zheludev, Nano Lett. 9, 1249 (2009)Fabrication of Superoscillatory lensE. T. F. Rogers, et al.Nature Materials 11, 432 (2012).
26 Diffraction-free beams in Optics* Helmholtzequationm=0Bessel beamof mth orderThey are stationary solutions of Helmholtz equationThey have no-evanescent wave components (band-limited)They carry infinite power, thus they do not diffractm=3whereThe lobes of a Bessel beam are always of the order of lQuestion: Can we have diffractionless beams with sub-l features?Intensity profilesof Bessel beamsAnswer: YES, by using the concept of superoscillationsm=3*J. Durnin, J. Opt. Soc. Am. A 4, 651 (1987), J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
27 Stationary Superoscillatory beams* Stationary solution of Helmholtz equationWe force the field to pass through Npredetermined points in the x-y planeThe field as superposition ofsolutions of Helmholtz equationSolution of the problemIf the distances between the Pm pointsare subwavelength, the coefficients cmwill give us a superoscillatorysuperpositionK. G. Makris and D. Psaltis, Opt. Lett. 36, 4335 (2011).
28 Analytical form of a superoscillatory beam* We choose to write our field as superposition of Bessel beams JnPolarcoordinatesSuperposition ofBessel beamsSpecific exampleSuperoscillatorydiffractionless beamK. G. Makris and D. Psaltis, Opt. Lett. 36, 4335 (2011).
30 Example 1: Superposition of J0,J1,J2 beams zoomsubwavelength3-point pattern
31 Example 2: Superposition of J2,J6,J10 beams Phase singularities on sub-wavelength scalesubwavelengthsubwavelength12-point pattern
32 Experimental set-up* Superposition of two spatially translated J2 Bessel beamsSuperpostion and notan interference effectDiffraction limit:Wavelength:*E. Greenfield, R. Schley, H. Hurwitz, J. Nemirovsky, K.G. Makris, and M. Segev, Optics Express, accepted (2013)
35 Conclusions PT-symmetry in optical periodic potentials Group velocity in PT-latticesPT- symmetric scattering in cavitiesRelation between PT open and finite systemsDiffractionless superoscillatory beamsObservation of stationary superoscillatory beams