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Beam Dynamics in PT -waveguides and cavities Konstantinos Makris Electrical Engineering Department, Princeton University, USA Superoscillatory diffractionless beams

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G. Aqiang and G. Salamo – University of Arkansas, USA C. E. Rüter and D. Kip - Clausthal University, Germany Collaborative groups Mathematics department, Florida State University, USA Z. Musslimani College of Optics /CREOL, University of Central Florida, USA R. El-Ganainy, and D.N.Christodoulides P. Ambichl, and S. Rotter Institute of Theoretical Physics, TU-Wien, Vienna, Austria M. Segev Technion, Israel

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Overview Introduction to PT -symmetric Optics Physical characteristics of PT -symmetric potentials Group velocity in PT -symmetric lattices PT -symmetry breaking in Fabry-Perot cavities Conclusions

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Introduction to PT -symmetric Optics

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PT -symmetry in Quantum Mechanics Parity and Time operators PT symmetric Hamiltonian can exhibit entirely real eigenvalue spectrum! Should a Hamiltonian be Hermitian in order to have real eigenvalues? Schrödinger Equation PT -potential * 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)

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Quantum mechanics and Wave Optics Paraxial equation of diffraction Schrödinger equation Quantum Mechanics Energy eigenvaluesPropagation constants Paraxial Optics

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PT symmetry in Optics* Nonlinear Schrödinger Equation X G L Typical parameters *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). PT -symmetric potential

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Observation of PT -breaking in an active coupler Experimental realization of PT -lattice Observation of PT -breaking in a passive coupler Parity – time synthetic photonic lattices, A. RegensburgerA. Regensburger, C. Bersch,C. Bersch A. Miri, G. Onishchukov,A. MiriG. Onishchukov D. N. ChristodoulidesD. N. Christodoulides, and U. PeschelU. Peschel Nature, 488, 167–171 (09 August 2012)

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PT -symmetric Optics Photonic crystals Negative index materials PT -symmetric waveguides z PT -symmetric cavities

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Physical characteristics of PT –potentials

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Scarff potential Abrupt phase transition PT Phase transition in a single waveguide * 0 * Z. Ahmed, Phys. Lett. A, 282, 343 (2001) W.D. Heiss, Eur. Phys. J. D 7, 1 (1999) Exceptional point Biorthogonality condition

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Floquet Bloch mode Bandstructure k : Bloch wavenumber, n : number of band Orthonormality relation Floquet-Bloch modes in real lattices * Superposition principle Projection coefficients Parseval’s identity * D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature, 424, 817 (2003). Discrete Diffraction D period

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After phase transition Before phase transition * K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. Lett. 100, (2008). Bandstucture of a PT optical lattice * Exceptional point

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PT –symmetric optical cavities

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Scattering from Fabry-Perot PT cavities* G L * L. Ge, Y. D. Chong, and A.D. Stone, Phys. Rev. Lett. 106, (2011). g: gain/loss Scattering matrix Basic relations Motion of Scattering matrix eigenvalues in the complex plane Exceptional point broken unbroken

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Relation between finite and open PT -cavities* General Robin Boundary Conditions Helmholtz equation in finite domain Gain-loss amplitude (Cavity length)/2 *P. Amblich, K.G.Makris, L. Ge, Y. D. Chong, and S. Rotter, to be submitted (2013).

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Finite PT -system Finite and open PT -cavities Open scattering PT -system Each eigenstate of S is also an eigenstate of an effective Hamiltonian H eff with the appropriate Robin boundary conditions The effective Hamiltonian H eff is PT -symmetric when The 2D union of all the eigenvalue curves of H eff for is identical to the unbroken phase of the open scattering problem

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Practical considerations for observing PT -scattering in cavities GL Eigenvector of S-matrix Eigenvector of S-matrix Asymmetric output power above EP broken Symmetric output power below EP

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Physical value of gain at the exceptional point broken unbroken Typical physical values We need long cavities to observe PT -phase transition g-mismatch tolerance

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Effect of incidence angle in scattering in PT -cavities It is experimentally easier if the angle of incidence is non-zero TE polarizationTM polarization unbroken

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Scattering coefficients in 2 layer PT -cavities* Transmittance Reflectance from left to right L. Ge, Y. Chong, D. Stone, PRA 85, (2012) Normal incidence Reflectance from left to right For both transmission resonance points are below the EP

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Multilayer Fabry-Perot PT -cavities* Multiple phase transitions zoom 12 layers, TE, normal incidence *K.G.Makris, P. Amblich, L. Ge, S. Rotter, and D. N. Christodoulides to be submitted (2013). broken

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Multilayer Fabry-Perot PT -cavities TE-polarization TM-polarization Experimentally, we do not need to scan the length of cavity, but the angle Closed paths of scattering eigenvalues in complex plane EP1 EP2 12 layers broken

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Superoscillatory diffractionless beams School of Engineering, Swiss Federal Institute of Technology Lausanne (EPFL), Switzerland D. Psaltis K.G. Makris Electrical Engineering Department, Princeton University, USA D. Papazoglou, and S. Tzortzakis Materials Science and Technology Department, University of Crete, Heraklion, Greece Institute of Electronic Structures and Laser, Foundation for Research and Technology Hellas, Heraklion, Greece E. Greenfield, and M. Segev Physics Department, Solid State Institute, Technion, Israel

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Optical Superoscillations M. 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) N. 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) Superoscillatory field: A field that locally has subwavelength features but no evanescent waves. Theoretical suggestion: Optical super-resolution with no evanescent waves Optical Experiment: Subwavelength focus in the far field with no evanescent waves M. R. Dennis, A. C. Hamilton, and J. Courtial, Opt. Lett. 33, 2976 (2008) E. T. F. Rogers, et al. Nature Materials 11, 432 (2012). Fabrication of Superoscillatory lens

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Helmholtz equation Bessel beam of m th order They are stationary solutions of Helmholtz equation They have no-evanescent wave components (band-limited) They carry infinite power, thus they do not diffract Diffraction-free beams in Optics* *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). m=0 m=3 The lobes of a Bessel beam are always of the order of Question: Can we have diffractionless beams with sub- features? Answer: YES, by using the concept of superoscillations where Intensity profiles of Bessel beams

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Stationary Superoscillatory beams* Stationary solution of Helmholtz equation We force the field to pass through N predetermined points in the x-y plane The field as superposition of solutions of Helmholtz equation Solution of the problem If the distances between the P m points are subwavelength, the coefficients c m will give us a superoscillatory superposition K. G. Makris and D. Psaltis, Opt. Lett. 36, 4335 (2011).

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Analytical form of a superoscillatory beam* We choose to write our field as superposition of Bessel beams J n Superoscillatory diffractionless beam Superposition of Bessel beams Specific example Polar coordinates K. G. Makris and D. Psaltis, Opt. Lett. 36, 4335 (2011).

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Different 1D and 2D patterns

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subwavelength Example 1: Superposition of J 0,J 1,J 2 beams zoom 3-point pattern

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subwavelength Phase singularities on sub-wavelength scale Example 2: Superposition of J 2,J 6,J 10 beams 12-point pattern

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Experimental set-up* Diffraction limit: *E. Greenfield, R. Schley, H. Hurwitz, J. Nemirovsky, K.G. Makris, and M. Segev, Optics Express, accepted (2013) Superposition of two spatially translated J 2 Bessel beams Wavelength: Superpostion and not an interference effect

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Observation of superoscillatory beams

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w=2.5 m~4 w

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Conclusions PT -symmetry in optical periodic potentials Group velocity in PT -lattices PT - symmetric scattering in cavities Relation between PT open and finite systems Diffractionless superoscillatory beams Observation of stationary superoscillatory beams

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