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Beam Dynamics in PT -waveguides and cavities

Konstantinos Makris Electrical Engineering Department, Princeton University, USA. Superoscillatory diffractionless beams. Beam Dynamics in PT -waveguides and cavities. Collaborative groups. R. El-Ganainy, and D.N.Christodoulides.

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Beam Dynamics in PT -waveguides and cavities

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

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

  3. 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

  4. Introduction to PT-symmetric Optics

  5. PT-potential PT-symmetry in Quantum Mechanics Should a Hamiltonian be Hermitian in order to have real eigenvalues? Schrödinger Equation Parity and Time operators PT symmetric Hamiltonian can exhibitentirelyreal eigenvalue spectrum! *C. M.Bender et al, Phys. Rev. Lett., 80, 5243 (1998); C. M.Bender et al, Phys. Rev. Lett., 89, 270401 (2002) C. M.Bender et al, Phys. Rev. Lett., 98, 040403 (2007); C. M.Bender, Contemporary Physics, 46, 277 (2005)

  6. Quantum mechanics and Wave Optics Paraxial Optics Quantum Mechanics Schrödinger equation Paraxial equation of diffraction Propagation constants Energy eigenvalues

  7. PT symmetry in Optics* X G L Nonlinear Schrödinger Equation PT-symmetric potential 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, 103904 (2008). Z.H.Musslimani, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008). K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. A 81, 063807 (2010). *R. El-Ganainy, K.G.Makris, and D. N. Christodoulides, Phys. Rev. A 86, 033813 (2012).

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

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

  10. Physical characteristics of PT–potentials

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

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

  13. Bandstucture of a PT optical lattice* Exceptional point Before phase transition After phase transition *K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. Lett. 100, 103904 (2008).

  14. PT–symmetric optical cavities

  15. Scattering from Fabry-Perot PT cavities* L G broken unbroken g: gain/loss Scattering matrix Basic relations Exceptional point Motion of Scattering matrix eigenvalues in the complex plane *L. Ge, Y. D. Chong,and A.D. Stone, Phys. Rev. Lett. 106, 093902(2011).

  16. Relation between finite and open PT-cavities* Helmholtz equation in finite domain (Cavity length)/2 General Robin Boundary Conditions Gain-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 Open scattering PT-system Finite PT-system Each eigenstate of S is also an eigenstate of an effective Hamiltonian Heff with the appropriate Robin boundary conditions The effective Hamiltonian Heff is PT-symmetric when The 2D union of all the eigenvalue curves of Heff for is identical to the unbroken phase of the open scattering problem

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

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

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

  21. Scattering coefficients in 2 layer PT-cavities* Reflectance from left to right Reflectance from left to right Transmittance For both transmission resonance points are below the EP Normal incidence L. Ge, Y. Chong, D. Stone, PRA 85, 023802 (2012)

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

  24. Superoscillatory diffractionless beams K.G. Makris Electrical Engineering Department, Princeton University, USA E. Greenfield, and M. Segev Physics Department, Solid State Institute, Technion, Israel 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 D. Psaltis School 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-resolution with no evanescent waves 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, 022003 (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 field with no evanescent waves 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) Fabrication of Superoscillatory lens E. T. F. Rogers, et al. Nature Materials 11, 432 (2012).

  26. Diffraction-free beams in Optics* m=0 m=3 Helmholtz equation Bessel beam of mth 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 where The lobes of a Bessel beam are always of the order of l Question:Can we have diffractionless beams with sub-l features? Intensity profiles of Bessel beams Answer:YES, by using the concept of superoscillations m=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 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 Pm points are subwavelength, the coefficients cm will give us a superoscillatory superposition K. 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 Jn Polar coordinates Superposition of Bessel beams Specific example Superoscillatory diffractionless beam K. G. Makris and D. Psaltis, Opt. Lett. 36, 4335 (2011).

  29. Different 1D and 2D patterns

  30. Example 1: Superposition of J0,J1,J2 beams zoom subwavelength 3-point pattern

  31. Example 2: Superposition of J2,J6,J10 beams Phase singularities on sub-wavelength scale subwavelength subwavelength 12-point pattern

  32. Experimental set-up* Superposition of two spatially translated J2 Bessel beams Superpostion and not an interference effect Diffraction limit: Wavelength: *E. Greenfield, R. Schley, H. Hurwitz, J. Nemirovsky, K.G. Makris, and M. Segev, Optics Express, accepted (2013)

  33. Observation of superoscillatory beams

  34. w w=2.5 mm~4l

  35. 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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