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**Electrical Engineering Department, Princeton University, USA**

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

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**R. El-Ganainy, and D.N.Christodoulides**

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

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**Overview Physical characteristics of PT-symmetric potentials**

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**

Should a Hamiltonian be Hermitian in order to have real eigenvalues? Parity and Time operators Schrödinger Equation PT-potential PT 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)

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**Quantum mechanics and Wave Optics**

Paraxial Optics Quantum Mechanics Paraxial equation of diffraction Schrödinger equation In 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 constants Energy eigenvalues 6

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**PT-symmetric potential**

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

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**realization of PT-lattice**

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)

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**Negative index materials**

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

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**Physical characteristics**

of PT–potentials

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**Abrupt phase transition**

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

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**Orthonormality relation**

Floquet-Bloch modes in real lattices* 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).

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**Before phase transition After phase transition**

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, (2008).

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PT–symmetric optical cavities

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**Motion of Scattering matrix eigenvalues in the complex plane**

Scattering from Fabry-Perot PT cavities* broken unbroken G L 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, (2011).

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**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).

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**Finite and open PT-cavities**

Finite PT-system Open scattering 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

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**Practical considerations for observing PT-scattering in cavities**

Symmetric output power below EP Eigenvector of S-matrix Eigenvector of S-matrix broken Asymmetric output power above EP

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**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

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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 unbroken unbroken TE polarization TM polarization

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**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, (2012)

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**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).

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**Multilayer Fabry-Perot PT-cavities**

TE-polarization TM-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

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**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

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**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, (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).

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**Diffraction-free beams in Optics***

Helmholtz equation m=0 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 m=3 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).

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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 Pm points are subwavelength, the coefficients cm 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 Jn Polar coordinates Superposition of Bessel beams Specific example Superoscillatory diffractionless beam K. G. Makris and D. Psaltis, Opt. Lett. 36, 4335 (2011).

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**Different 1D and 2D patterns**

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**Example 1: Superposition of J0,J1,J2 beams**

zoom subwavelength 3-point pattern

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**Example 2: Superposition of J2,J6,J10 beams**

Phase singularities on sub-wavelength scale subwavelength subwavelength 12-point pattern

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**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)

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**Observation of superoscillatory beams**

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w w=2.5 mm~4l

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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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