PT-symmetry and Waveguides / (2) The transverse case. Course 2 : PT symmetry basics : the transverse case

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1 PT-symmetry and Waveguides / () The transverse case Course : PT symmetry basics : the transverse case Imperfect PT symmetry [ plasmonics] Application example : switching below the exceptional point From here on : Credits to : Anatole Lupu and Aloyse Degiron (IEF, Orsay, France) Mondher Besbes and Jean-Paul Hugonin (my lab : LCF, IOGS, Palaiseau) My inspiration?

2 The observables of a quantum system are defined to be the self-adjoint operators A on H, a fixed complex Hilbert space. Φ Φ Φ Φ The Hamiltonian H is the operator associated to the total energy In a finite-dimensional space Φ }, the Hamiltonian matrix H = is Hermitian : This grants diagonal terms are real in any basis. Thus granting (eigen)energies are real, using the basis of H eigenstates }, The spectrum is real 3 Energy-conserving systems Linear systems Open systems Losses and/or Gain from an outside reservoir Hermiticity Real eigenvalues allowed Non-Hermiticity Real spectrum? Nature of this boundary 4

3 @ Hermitian operator H Real Eigenvalues Hermitian operator Real Eigenvalues C. M. Bender and S. Boettcher, Phys. Rev. Lett. 8, 543 (998) Real E But Hamiltonians with D degree of freedom are not that simple Harmonic osc 5 PT -Symmetric Non-Hermitian x Hamiltonian t s t P P P T Real Eigenvalues as long as r < s det ² ² t and T t det ² whose sign is or! or 6

4 Analogy : Time evolution in Quantum mechanics of Two-modes systems Spatial evolution of two guided modes in z-propagation PT -Symmetric Non-Hermitian Hamiltonian Coupled modes equations P P T Real Eigenvalues as long as r < s Real Eigenvalues as long as g < gain Gain [P sym] loss losses 7 Non conservative "atoms" (?!)(*) s A A t (*) Gain bosons t A Non-conservative Guides z A Evolution axis z Evanescent coupling 8

5 ω or Re(Energy) Split modes Too much damping Losses ω' or Im(Energy) Losses Change of behaviour in eigenvalues but in high loss regime The case of balanced gain/loss x GAIN T P P z LOSS L T ω Split modes coupling Gain/Loss Exceptional point (EP) : singularity of (eigenvalues)/ (gain) ω' Gain/Loss

6 The light s bet GAIN Betting that light given to other guide T Oops, losses.. Will come back re amplified??? T coupling LOSS Bet works? Below EP, oscillation, real ω. Bet lost? overall gain or loss? What are eigenstates Gain/Loss Eigenstates below/above EP c /c Gain/Loss

7 eigenstate behaviour vs. «gain loss» Symmetry breaking, pictorially A B Symmetry-breaking of eigenstates A B ("winner-takes-all") 3 Danger of unbalanced eigenstates with gain/loss Loss state Gain state c >+ > > c > -δc >+ δc > (c δc) >+ δc > δc/c) High growth of unbalanced part Can be misleading (Reciprocity based on exact eigenmodes) 4

8 Trajectory of eigenvalues in complex plane Eigenvalues in complex plane Im EP Re Note (tracking tool for real wg, hint from J P Hugonin): Product of eigenvalues Π=(ω ω ) is continuous at EP! Sum Σ=(ω +ω ) continuous as well PT symmetry in Optics «unnamed» (& very strong κ) Ctyroky & Nolting : Kulishov/Greenberg/Poladian/Agarwal: (tomorrow) Gratings with Δn= Δn r cos(kz)+ iδn i cos(kz+φ), «««nonreciprocity»»» "named" El Ganainy et al. (CREOL), «Theory of coupled optical PT-symmetric structures», Opt. Lett. 3, 63 (7) Klaiman et al. PRL 8; Guo et al. PRL 9; (topic starts to blow up) Observed with parametric gain/loss Rüter et al. (Clausthal u. CREOL, Technion) «Observation of parity time symmetry in optics», Nat. Phys. 6, 9 () 6

9 (Transmission matrix) distance L distance L (µm) map log(t ) of T T T 3 gain (cm - ) 7 PT-symmetry and Waveguides / () The transverse case Course : PT symmetry basics : the transverse case Imperfect PT symmetry [ plasmonics] Application example : switching below the exceptional point 8

10 3 FLAVOURS OF IMPERFECT PT SYMMETRY gain = Losses = or : fixed, variable / / 9 ixed losses : Survival of abrupt transition matched losses g= g distance (µm) (a) T 4 6 abrupt behaviour 8 GAIN g EP log(t) 3 gain (cm-) g LOSS fixed (metal) losses distance (µm) (b) GAIN FIXED LOSS +g g g and should be "matched" T «active- PT-structures» EP remains abrupt behaviour 3 gain (cm-) «passive- PT-structures» e.g. Guo et al. PRL 9 H. Benisty et al. Opt. Express, 9, 84,

11 PT Kogelnik 97 s + _ PT-sym, s Gain / no gain Device length L Commutation loci Electro-optic tuning ΔRe(n) Very few tunable ΔRe(n) proposals in plasmonics We d better tune ΔIm(n)! Fixed losses (Au)?? What are the new diagrams? ~ cm ~4 μm~ nm Au BCB Substrate Wideband mode (VIS-IR range) Courtesy of A. Degiron

12 Co directional coupling between plasmonic guide and dielectric guide Two Eigenmodes Air BCB SiO t t Degiron et al, New J. Phys. 9 (Duke U) Detuning control by BCB thickness 3 Experimental waveguides A combination of negative and positive lithography steps are used to fabricate plasmonic stripes coupled to SU8 waveguides, embedded in BCB polymer. CL = coupling length 4

13 Measurements vs. coupling length Arbitrary BCB thickness Simulations Experiments BCB t=6.6 μm Optimized BCB thickness BCB t=5.4 μm 5 Dephasing of beating is a first sign Gain g Losses -g T T Coupled Mode Theory d i dz M ig / M ig / Transmission.5 g= g < g crit g >> g crit Propagation distance 5 6

14 PT-symmetry and Waveguides / () The transverse case Course : PT symmetry basics : the transverse case Imperfect PT symmetry [ plasmonics] Application example : switching below the exceptional point 7 The exact transfer matrix of the arbitrarily detuned case Im detuning δ complex detuning Ω δ ² detuning & coupling =M(z) Transfer matrix cos Ω Ω sin Ω sin Ω Ω sin Ω Ω cos Ω sin Ω Ω exp (Kogelnik 976) 8

15 The exact transfer matrix of Im only detuning case Im detuning δ complex detuning Ω ² =M(z) detuning & coupling Transfer matrix cos Ω Ω sin Ω sin Ω Ω sin Ω Ω cos Ω sin Ω Ω exp 9 The exact transfer matrix of: Im only detuning & perfect PT case Im detuning if PT-sym!! δ complex detuning Ω detuning & coupling =M(z) Transfer matrix cos Ω sin Ω sin Ω sin Ω cos Ω sin Ω exp 3

16 The exact transfer matrix of no detuning perfect case Im detuning δ complex detuning Ω =M(z) Transfer matrix cos sin sin cos Simply rotation 3 Switching requirements cross Ω sin Ω cos Ω sin Ω? sin Ω cos Ω sin Ω etc. exp bar cos sin sin cos Change of beat length cos & sin cancellations 3

17 Switching and all the transfer matrix exp ) Ω Δ π sin π 33 Bar and Cross perfect switch states in ideal (gain=loss) PT symmetric coupler (PTSC) Bar Cross perfect switch Cross Bar perfect switch T T T T db db T T Smallest length Switch Not good Switch 34

18 What is the new concept replacing V π? Real index modulation Imaginary index modulation Δ(Im) replaces Δ(Re) Δ(Im)= g + χ (the «sum» of gain and loss) CMT solution tan iml iml iml (usual electro optic switch s product) So 33% shorter couplers are possible (insofar as measured by phase) 35 What about less perfect systems? = or alwaysexist levelishigher or lower than unity But if ±5 db are tolerable Then large margin exist! Extra db in T 36

19 References on switching and coupled guides/plasmonics [] H. Benisty, A. Degiron, A. Lupu, A. De Lustrac, S. Chénais, S. Forget, M. Besbes, G. Barbillon, A. Bruyant, S. Blaize, and G. Lérondel, "Implementation of PT symmetric devices using plasmonics: principle and applications," Optics Express, vol. 9, pp ,. [] H. Benisty, C. Yan, A. T. Lupu, and A. Degiron, "Healing Near-PT-Symmetric Structures to Restore Their Characteristic Singularities: Analysis and Examples," IEEE J. Lightwave Technol., vol. 3, pp ,. [3] A. Lupu, H. Benisty, and A. Degiron, "Switching using PT symmetry in plasmonic systems: positive role of the losses," Opt. Express, vol., pp , 6 3. [4] A. Lupu, H. Benisty, and A. Degiron, "Using optical PT-symmetry for switching applications," Photonics and Nanostructures-Fundamentals and Applications, vol., pp. 35-3, 4. [5] H. Benisty, A. Lupu, and A. Degiron, "Transverse periodic PT symmetry for modal demultiplexing in optical waveguides," Phys. Rev. A, vol. 9, p. 5385, 5. [6] H. Benisty and M. Besbes, "Plasmonic inverse rib waveguiding for tight confinement and smooth interface definition " J. Appl. Phys., vol. 8, pp. 638 (-8),. [7] H. Benisty and M. Besbes, "Confinement and optical properties of the plasmonic inverserib waveguide," J. Opt. Soc. Am. B, vol. 9, pp , March CONTEXT : Gain with plasmons SPASER (Stockman, Oulton with nanorods,...) Optical Amplifiers with LRSPP (Berini) 38

20 No électro-optic modulation in metals PT -symmetry offers a plausible alternative to elaborate active devices. Application example : Plasmonic modulator 3 Power (db) - SU8 WG SP WG Normalized Material Gain Lupu et al., Opt. Express Model Realisation : Gain with organics S. Chénais & S. Forget team, LPL, Paris 3 fvin Variable Stripe Length (VSL) technique: + Molecular film Thermally evaporated fvin layer on glass substrate Pump LASER Characteristics Frequency Doubled Q-switched Nd:YVO4 λ = 53 nm Hz, pulse duration < 5 ps Spectro (Ocean Optics) Cylindrical Lens Diverging Lens H. Rabbani-Haghighi, S. Forget, S. Chénais et al. Appl. Phys. Lett. 95, 3335 (9). Adjustable razor blades Sample Stripe quality (a) Without imaging (b) With imaging CCD (stripe length measurement) (a) (b) Pump Stripe Width : 3 µm 4

21 Gain with organics can be high & fast S. Chénais & S. Forget team, LPL, Paris 3 E=,4 mj/cm Gain= 5 cm Comparison : in a 5% doped DCM:PMMA of same thickness : Gain= 4 cm nanosec. pump probe gain 4 More miniature plasmonics? Example of Plasmonic Inverse Rib Optical WG n~ n~.4 JAP, H. Benisty and M. Besbes E-field in 3-5 nm tip... (+JOSA ) Like Oulton s nanorod/spaser, but deterministic Single WG? Gain brings just loss compensation Then yeah gain! 4

22 Two coupled «PIROWs» : good? A gain one and a normal one GAIN FIXED LOSSES METAL Can we have a good EP? H. Benisty and M. Besbes, "Confinement and optical properties of the plasmonic inverse-rib waveguide," JOSA. B, vol. 9, pp ,. Something wrong? 43 «Danger, LASER!» Fabry-Perot formula with gain Pole at = ² Well-known Threshold for lasing (no amplification limit within a linear saturation-free ansatz) But?? ω Gain Loss waveguide No mention of «lasing» or «threshold»?? Actually a sign that PT-symmetry needs our work to be a melting-pot! 44

23 CONCLUSION Basic (transverse) PT-symmetry New use of gain beyond plain loss compensation New entry for open systems (new inner products ) thanks to new quantum theory input OK for Adaptation to plasmonics (constant loss media) 45 References on switching and coupled guides/plasmonics [] H. Benisty, A. Degiron, A. Lupu, A. De Lustrac, S. Chénais, S. Forget, M. Besbes, G. Barbillon, A. Bruyant, S. Blaize, and G. Lérondel, "Implementation of PT symmetric devices using plasmonics: principle and applications," Optics Express, vol. 9, pp ,. [] H. Benisty, C. Yan, A. T. Lupu, and A. Degiron, "Healing Near-PT-Symmetric Structures to Restore Their Characteristic Singularities: Analysis and Examples," IEEE J. Lightwave Technol., vol. 3, pp ,. [3] A. Lupu, H. Benisty, and A. Degiron, "Switching using PT symmetry in plasmonic systems: positive role of the losses," Opt. Express, vol., pp , 6 3. [4] A. Lupu, H. Benisty, and A. Degiron, "Using optical PT-symmetry for switching applications," Photonics and Nanostructures-Fundamentals and Applications, vol., pp. 35-3, 4. [5] H. Benisty, A. Lupu, and A. Degiron, "Transverse periodic PT symmetry for modal demultiplexing in optical waveguides," Phys. Rev. A, vol. 9, p. 5385, 5. [6] H. Benisty and M. Besbes, "Plasmonic inverse rib waveguiding for tight confinement and smooth interface definition " J. Appl. Phys., vol. 8, pp. 638 (-8),. [7] H. Benisty and M. Besbes, "Confinement and optical properties of the plasmonic inverserib waveguide," J. Opt. Soc. Am. B, vol. 9, pp , March 9. 46