1.A Periodicity (for electrons)

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1 PT-symmetry and Waveguides / (1) Waveguides & Bragg structures Henri BENISTY Laboratoire Charles Fabry Institut d Optique Graduate School Palaiseau, France [south of Paris].. Now part of a big «cluster» of universities : «Paris Saclay» Course 1 : Semiconductors Basics of coupling : two oscillators... two guides, two waves (à la Haus) (What if 3rd oscillator? Fano, EIT, ) Mode profiles and active behaviour of waveguide From periodic structures to lasers - Edge emitting DFB laser - Vertically emitting VCSEL Mach-Zehnder interferometer for switching. Course & 3 PT symmetry (transverse & longitudinal) Course 4 : PT symmetry (advanced) 1 1.A Periodicity (for electrons) Periodicity Floquet-Bloch theorem (plane wave generalization) Kronig-Peney eample ( multilayer in optics) Bloch modes with real k (in bands) never decay (in spite of infinitely many reflection s) Fourier space : 1st Brillouin Zone, Reciprocal Lattice - /a 0 + /a

2 One band Two bands + a gap a few ev Atomic potential B A N D 1D Electron Energy D /a E D G E S 3D... k m*~ 0.1m o Conduction band k U m* m*~ mo Valence band m*>0 U k m*<0 E(k ) E(k, k y ) Quantum wells Band edges potential, Conduction band edge Atomic potential Valence band edge 3 Quantum well Conduction band edge Atomic potential Valence band edge Levels of infinite well, 16 Fundamental transition (Gain or loss!) heavy holes Mode discretization Similar to dielectric waveguide 4

3 Epitay, MOCVD and MBE 3eV Egap GaN SiC AlAs gap ~0.37 ~0.5 refractive inde ,5eV 1,1eV 0,8eV Si 1.1 GaAs 0.88 InP 0.93 ~ InAs ~5 3. to 3.5 (Ge n~4.0)?6 %?6 % 8 % lattice parameter MetalOrganicVapourPhase Epitay Molecular Beam Epitay ~slow growth rate < 500 nm/h 5 Basics of waveguide y Field confined along, free along z E E o f () ep(i z t) e y ; z n clad n core n clad Helmholtz eqn. TE pol. here f () ² wave picture ray picture evanescent waves due to TIR (Tot. Int. Refl.) ² ² = Similar to Schrödinger eqn. with a swap ( ) E ( ) V( ) ( ) = 6

4 y 1 n 1 z Waveguide mode in k-space f() k o = (ω/c) k / k o E E o f () ep(ik z z) e y c k z n eff k z / k o f() Guided modes 1<n eff <n DEFINITION OF EFFECTIVE INDEX k z = n eff (ω/c) i.e. k z = n eff k o FUNDAMENTAL MODE highest eff & n eff (bottom of well) 7 Equivalence with potential (quantum) well Energy (& potential) 0 (bands...) y or z eff localized resonances (guided modes) ( ) ( n eff ) 8

5 Course 1 : Semiconductors Basics of coupling : two oscillators... two guides, two waves (à la Haus) (What if 3rd oscillator? Fano, EIT, ) Mode profiles and active behaviour of waveguide From periodic structures to lasers - Edge emitting DFB laser - Vertically emitting VCSEL Mach-Zehnder interferometer for switching. 9 Photonic structures Many of them can be considered as oscillators Guide Fabry-Perot L So what is the simple description? One mode 1 oscillator frequency, losses Two modes oscillators + phase matching phenomena Three modes 3 oscillators + interaction control (EIT, Fano) 10

6 Simplest (*) losses! Comple ω Eigenvalue Matri equation Oscillators treated with 1st order ODE (homog. 1st order ODE) Oscillator alone d a i o a dt d a a ioa dt d a i a dt o i/ o i d a i a dt (*) "square root of d a 0 a dt " Response to outside ecitation a ao ep( it) u uo ep( it) d a i a u dt Harmonic solutions Thus iao i ao uo ao (inhomog. 1st order ODE) uo i( ) Coupling Damped resonator response ao ( ) uo o 11 Oscillators Abs and Disp responses Lorentzian ao u ( ) o o Fabry-Perot (FP) response : locally a Lorentzian t 1 1 1msin ( /) 1 m( / p) ao uo i( ) ( o ) (Abs) Internal field has same denominator as Transmission... A FP is an oscillator! (Disp) ( o ) ( ) o 1

7 Coupled oscillators d a 1 i d 1 a 1 a i a dt dt Coupling without coupling a 1 a d a1 i 1 a1 1 i a1 i dt a i a i a Without losses, this is Hermitian evolution (eigenvalues = eigen- are real) with coupling 1 i 1 i i i Given by solving «secular equation» det 1 i i 0 13 Define detuning δ: Solution : Eigenfrequencies with simplest algebra 1 i i i det 0 i 0 1 has the same eigenvalues as ecept a shift of 4 0 i i (H+ 1) E («Energy has arbitrary origin in Hamiltonians») A simple plot shows core of anticrossing 4 / Eigenvalues 4 ( 0) 4 /

8 generality of splitting & anticrossing 4 / eigen / Abcissa : Any control parameter of detuning 15 Eciting the coupled system 1 Only δ=0 case tractable i.e. the two oscillators are degenerate, symmetric Any control parameter of detuning delta New eigenmodes = the famous "binding" and "antibinding" «orbitals» : means 1 and in opposition a a1 1 Also called «supermodes» 1 + means 1 and in phase a a1 16

9 Application to coupled waveguides in space Two Guides D k Adding space to time the control parameter is «k»,... it becomes «free» y as (antisym) A guide a time-dependent oscillator sym (symmetric) k k 17 Application to coupled waveguides in space Two Guides D k y A guide?=? a timedependent oscillator Subsequently, supermode components evolve «diagonally» as (antisym) k sym (symmetric) k 18

10 Application to coupled plane waves by periodicity Result = dispersion in a periodic system Step 1 : forward&backward modes Step : shift by /a Step 3: form anticrossing /a /a k /a k Many comments! Initial waves are plane waves Origin of : Fresnel reflections k modulo G cf. slide. waves : mi of and Appearance of band gap («photonic band gap») solutions = {sym} & {as} at /a (Brillouin zone edge) Stationary waves at this point «slow light» near these edges Use in bandgap : partial Bragg reflector E a 0 1 k= /a+ik Mode at ω Mode at ω 1 wave(s) with imaginary k (decay) Progressive wave + Stationary wave progressive wave E low Medium High 0

11 General treatment of weak periodicity Coupled Mode Theory We have seen one version : d a1 1 i a1 i dt a i a Time version (not so common : Hermann Haus microwave people)... swapping control parameter (k) and time s roles... d a1( z) 1( ) i a1( z) i dz a( z) i ( ) a( z) Spatial version (Kogelnik, BTSJ) (Yariv, Tamir) and in cm -1 (Obtained from Mawell equations at constant frequency, by taking out slowly varying «envelope» terms of the field. ) The spatial dephasing (spatial coherence) now plays a big role instead of temporal effects... but fundamentally same story 1 Distributed Bragg reflector Origin of coupling by periodicity : Fresnel reflections Strength ~ Δn/n depends on INDEX CONTRAST Very high Δn Air/Silicon: 1.0/3.5 Penetration length [# of periods] ~ [<n>/ Δn] 1 to Quite high Δn Medium Δn Low Δn Very low Δn SiO /TiO : 1.5/.5 to 3 AlAs/GaAs, SiO /Al O 3 3.0/ /1.7 4 to 6 InP/InGaAsP 3.30/ to 16 SiO /Ge:SiO 1.5/ to 00 Very very low Δn (Holograms) 1.5/ to 000 Very very very low n X-rays : 1.0 / > 0 000

12 Course 1 : Semiconductors Basics of coupling : two oscillators... two guides, two waves (à la Haus) (What if 3rd oscillator? Fano, EIT, ) Mode profiles and active behaviour of waveguide From periodic structures to lasers - Edge emitting DFB laser - Vertically emitting VCSEL Mach-Zehnder interferometer for switching. 3 L OVERLAP : CONFINEMENT FACTOR core ma E d min layer E d field profile width («mode volume») bottom_clad core top_clad 1 (vs. L) L / core n eff 1 n 0 Optimal confinement ~λ/ Δn n 1 (curve ~ valid for both & n scales) L / 4

13 Course 1 : Semiconductors Basics of coupling : two oscillators... two guides, two waves (à la Haus) (What if 3rd oscillator? Fano, EIT, ) Mode profiles and active behaviour of waveguide From periodic structures to lasers - Edge emitting DFB laser - Vertically emitting VCSEL Mach-Zehnder interferometer for switching. 5 INDEX CONTRAST size (by materials.) Very high n optimal µm Air/Silicon: 1.0/3.5 ~ 140 nm Quite high n SiO /TiO : 1.5/.5 ~ 150 nm Medium n Low n Very low n AlAs/GaAs, SiO /Al O 3 3.0/ /1.7 00, 350 nm InP/InGaAsP 3.30/3.17 ~ 450 nm Si0 /Ge:SiO 1.5/ µm Very very low n (Holograms) 1.5/ µm Very very very low n X-rays : 1.0 / µm? 6

14 DFB laser eample Distributed Bragg reflector Waveguide Gain QW = DFB SCH laser 7 DFB FP laser Ideally this but not feasible Rather this etra step : thin layer periodically etched away and replaced by another (epitaially grown) medium So Confinement factor of grating is very small ma grating E d min grating grating layer layer E d 8

15 DFB FP laser n eff model Reality Each section seen as a fraction of a waveguide effective indices n eff1 and n eff. Layers of n eff1 and n eff Model : laser considered as a multilayer with inde n eff1 and n eff. (and gain, abs) Because of small n eff1 and n eff only differ by ~ 0.1% So the actual stop band is ~1-3 nm, same order as FSR So lasing become much better on any of the two band edges (the slowest allowed modes) 9 DFB FP laser spectrum Fundamental transition (Gain or loss!) Gain g Two narrow modes ( in competition) heavy holes abs ω gap Still two modes left how to get one? With a modified grating : grating with a / defect among /4 steps Phase-shifted DFB laser 30

16 DFB with a defect ("phase-shifted") : It s a laser with two ("elongated") mirrors imaginary k (decay) Progressive wave + Stationary wave E E 31 DFB with phase-shift : spectrum Gain g abs h One frequency only is selected Need not be at peak gain 3

17 SCH ~ L Maimizing Light-Matter interaction: Separate Confinement Heterostructure (SCH) Use ~ half of "alloy span" to produce photon confinement Use remaining half to achieve electron confinement CB Electron energy 80%Al 40%Al recombination L g~l 1 L -1 0%Al 0%Al VB 40%Al 80%Al E Inde of refraction The VCSEL Need to parallelize optical links Vertical Surface Emitting Laser Mirrors Active Mirrors So business as usual? 1 1 g th Ln( ) L rr 1 scatt - Etreme R required > 99.5% - Quantum well location vs. (anti) node critical 34

18 VCSEL QW placement Having QW only where fmode field is large A new concept is naturally introduced : a periodic gain medium with ~/n periodicity (admittedly, in practice 850 nm VCSELs can «unfortunately» operate with 3QWs and no period, but lower gain ones or high power ones have to work so) 35 Course 1 : Semiconductors Basics of coupling : two oscillators... two guides, two waves (à la Haus) (What if 3rd oscillator? Fano, EIT, ) Mode profiles and active behaviour of waveguide From periodic structures to lasers - Edge emitting DFB laser - Vertically emitting VCSEL Mach-Zehnder interferometer for switching. 36

19 Integrated laser modulator(ilm) & Switching Mach-Zehnder Interferometer (MZI) in ILM Laser / E ² 1 cos ) MMI splitter / MMI splitter SWITCH with MZI / CTRL / Or with coupled wg s CTRL 37 E. Kogelnik s 1976 on couplers Electro optic modulation Kogelnik, IEEE JQE, 1976 Two parameters : Length L Propagation constant modulation Δβ by Re(n) + length Single section : no L tolerance of β=β o +Δβ/ «BAR STATE» β=β o Δβ/ Δβ=0, «CROSS STATE» NOK OK Δβ=0 Δβ>0 Re(n) modulation 38

20 E. Kogelnik s 1976 alternation idea for couplers Single section : no L tolerance of +! Kogelnik, IEEE JQE, 1976 Alternate sections : L tolerance of length NOK OK length Re(n) modulation OK OK OK more room for states Re(n) modulation 39 CONCLUSION : Still time for clever gain What a nice job in the 0th century! But «Much ado about real inde» (guide & DFB, SCH, couplers) Gain made better by QW Charles («Chuck») Henry VCSEL as first engineered periodic gain media but not the commonly shared description Kenichi Iga Switching (Telecom) optics developed with «functional» view point : gain and lasing are here, inde tuning is there, separately Still room & food for thought for a clever use of distributed gain 40