Resonances in high-contrast gratings with complex unit cell topology

Transcription

1 Resonances in high-contrast gratings with comle unit cell toology Milan Maksimovic Focal-Vision & Otics, Oldenzaal, The Netherlands The XXI International Worksho on Otical Wave & Waveguide Theory and Numerical Modelling Enschede, The Netherlands, Aril 19-, 13

2 Outline Introduction Modeling Eamles Concluding remarks / questions

3 Introduction High-Contrast Grating: high inde (bars) medium grating fully immersed in low inde medium Sub-wavelength structure: eriod smaller than wavelength of the incident wave Suerstrate and substrate are simle low inde media (air) Refractive inde is real and frequency indeendent Only zero-th diffraction order is roagating Alications: Ultrathin broadband reflectors for VCSEL (relacement for DBR reflectors) High-Q resonator mirrors Resonant olarization sensitive filters and beam slitters Planar flat focusing mirrors (wavefront shaing) Hollow core waveguides (ultracomact otical couler and filters). Recent review ref.: C. Chang-Hasnain and W. Yang, "High-contrast gratings for integrated otoelectronics," Adv. Ot. Photon. 4, (1). 3

4 High-Contrast Gratings: modeling Decomosition: TE (with E y,h,h z ) & TM (with H y,e,e z ) non-null comonents Φ(,z) -either E y or H y L 1 Φ = zφ k (, z) (, z) TE( TM ) 1 TE n k L = + ( ) L ( ) 1 1 n n k n TM = + ( ) ( ) Eigenvalue roblem, seudo-eriodicity, ± ikz Φ (, z) = ϕ( ) e L ± ik ϕ( ) ϕ( ) e α Λ TE( TM ) ϕ( ) = r ϕ( ) +Λ = α = sinθ Grating Parameters: Λ -eriod η -the duty t g -grating height n() - refractive inde Plane Wave Ecitation: Θ-incidence angle k =π/λ wave number,, m Outside the grating filed is eanded in finite number of roagating and evanescent waves! RCWA/FMM All quantities eanded in Fourier series,.. Other rigorous numerical methods FEM FDTD 4

5 High-Contrast Gratings: modeling Couled (Bloch) Mode Theory: Periodic waveguide-array modes as an eansion basis Several modes are ecited in the wavelength range of interest Couling of waveguide-array modes at interfaces Dual-mode regime resonsible for most of novel henomena: (destructive/constructive ) interference between modes P. Lalanne, et. al., J. Lightwave Technol. 4, 44 (6). V. Karagodsky, et al. Ot. Eress, (1). Effective Medium Theory: Assumes single roagating mode: does not cature resonant henomena in the intermediate regime HCG as an Photonic Crystal hotonic bands originate from the resonance of the HCG suermodes, Adv. Ot. Photon. 4, (1) HCG resonse in terms of resonant ecitation of leaky modes Scattering matri ω, Resonance reresentation of the scattering matri (comle oles /zeros) decomosition ω ω ωω Sectral features = Non-resonant background + Pole/zero resonance contribution 5

6 Sectral resonse tailoring: how to generate unit cell toology? Sectral resonse tailoring : resonances ositions, number and shae, etc. IniUal design : one that already ossesses desired (sectral ) features reducuon of comutational burden in design and otimization tasks! Fundamental grating arameters for otimization of sectral resonse (simle unit cell) eriod, thickness, duty cycle, refractive inde Etended feature/arameter set (comle unit cell) transition oints in the unit cell symmetry of the unit cell Possible aroach to (restricted) toology otimization using the local change in toology of the unit cell? How to generate initial unit cell toology? Perturbations ( defects ) in the simle unit cell Deterministic aeriodic sequence for multile transition oints Randomly generated transition oints 6

7 7 HCG with the unit cell toology derived from erturbations Refractive inde rofile for: a) basic unit cell; b)- c) symmetrically erturbed unit cell; d) asymmetrically erturbed unit cell 1) Start with the simle unit cell (defined transition oints) ) Generate new transition oints by the cut and slit rocedure 3) Transition oints shift: Δ=ηΛ=L H 4) Parameter controls the strength of erturbation c c + = = + = = ,,, ± + = ± = = + = /, /,, 1 5, ( ) ( ) H c L = = + = 1 1, / 1 1 c 3 4 ηλ=l H Λ

8 HCG ehibiting broadband high reflectivity : symmetric erturbation of the unit cell HCG with the: eriod Λ=.6 μm, duty cycle η=.3548, height t g =.14 µm, n g =3.,n =1; TE-olarization Eamle HCG otimized for broadband reflectivity and with the small thickness comared to the wavelength. Symmetric erturbations : Sectral shae does not change (no resonances)! Bandwidth and central wavelength shift 8

9 HCG resonances with the symmetric and asymmetric erturbations of the unit cell HCG with the: eriod Λ=.6 μm, duty cycle η=.3548, height t g =.14 µm, n g =3.,n =1; TE-olarization a) b) c) d) Transition oints shift: =ηλ a) =.5 b) P=.5 c) =.75 d) =.1 Symmetry breaking is the origin of sectral resonances! Ecitation of reviously non-ecited modes 9

10 HCG ehibiting broadband high reflectivity : resonances under the oblique incidence HCG with the: eriod Λ=.6 μm, duty cycle η=.3548, height t g =.14 µm, n g =3.,n =1; TE-olarization Resonances (sectral reflectivity anomalies ) aear under the oblique incidence! Higher diffraction order(s) become(s) roagating under the oblique incidence! 1

11 HCG reflectivity: the symmetric and asymmetric erturbations of the unit cell Sectral reflectivity : eriodic structure (left), symmetrically erturbed structure with arameter =.1 (middle) and asymmetrically erturbed structure with the arameter =.1 (right) 11

12 HCG reflectivity: randomly generated (asymmetric) transition oints within unit cell Transition oints in the unit cell generated from uniform distribution in the interval (,1) Total thickness constrained to be within 1% difference from the initial unit cell 1

13 HCG ehibiting high-q resonances: simle unit cell HCG with the: eriod Λ=.716μm, duty cycle η=.7, height t g =1.494µm, n g =3.48,n =1; TE-olarization Dee subwavelength regime Near-wavelength regime Diffraction regime 13

14 HCG ehibiting high-q resonances: symmetrically erturbed unit cell Reflectivity for a symmetrically erturbed structure with =.1 (solid) and unerturbed structure (dashed) Resonance line shae ( switch ): Lorentzian Fano tye! Unit cell symmetry reserved under erturbation! HCG with the: eriod Λ=.716μm, duty cycle η=.7, height t g =1.494µm, n g =3.48,n =1; TEolarization Ey in simle unit cell structure close to resonance.16354; Ey for the erturbed structure at resonance

15 HCG ehibiting high-q resonances: asymmetrically erturbed unit cell Sectral reflectivity for eriodic (dashed) and asymmetrically erturbed structure (solid) Angular deendence at resonance! Original structure oerates in the dee subwavelength regime! Breaking the symmetry in the unit cell enables ecitation of resonances! Ey close to resonances.6966,.9337, 3.17 resectively. 15

16 HCG unit cell transition oints deterministically generated from aeriodic sequence Photonic aeriodic suerlattices( hotonic quasi-crystals) offer rich variety of comle atterns Aeriodic sequence of letters over finite alhabet: Associate letters of the sequence with the high-or low-inde material (H) and (L) Eamle: Thue-Morse (model aeriodic) sequence formed by substitutions L HL and H LH Generate transition oints by re-scaling initial structure: 1. Start with initial simle unit cell: η-duty cycle, Λ eriod. Select generation (n) with the N H =N L = n (number of letters in the sequence) 3. Scaled (normalized) thicknesses: l H =η/n H and l L =(1-η)/N H 4. Generate transition oints according to chosen sequence (reuted letter interreted as double thicknesses) a) basic eriodic structure b) scaled structure according to 3rd generation of Thue-Morse sequence c) scaled structure according to 4th generation of Thue-Morse sequence 16

17 HCG reflectivity: 3 rd generation Thue-Morse sequence based unit cell HCG with the: eriod Λ=.716μm, duty cycle η=.7, height t g =1.494µm, n g =3.48,n =1; TEolarization Additional resonances not resent in the simle unit cell HCG Localization of field ( hot sots ) outside the grating bars Resonances moved into the (reviously) dee subwavelength domain 17

18 HCG reflectivity for 3 rd & 4 th generation Thue-Morse sequence based unit cell Hierarchical fragmentation of the sectral resonse Increasing the number of resonances in the given sectral range Unit cell symmetry switches between subsequent generations What is the largest generation number accessible subject to fabrication resolution and tolerances? 18

19 Concluding remarks / Questions Summary; HCG unit cell toology influences strongly the sectral resonse and field distributions Symmetry of the unit cell (erturbations) lays major role in shaing sectral reflectivity resonse under normal and oblique incidence(number and osition of resonances) Sectral resonance line shae and field localization may be controlled with comle unit cells: Lorentzian shae transforms into the Fano-tye resonance shae and vice versa under erturbations Resonances introduced into the (reviously) dee subwavelength region Formation of the hot-sots in the free sace Unit cell designed using deterministic aeriodic sequences gives rise to a highly fragmented sectral resonse dislaying hierarchical structure Designed comle unit cell toology as a the efficient initial toology rior to global otimization rocedure Questions: Is there an otimal choice of initial unit cell toology? Choice of modeling method? Stability of resonances under fabrication resolution and tolerances? Conditions and constraints for robust toology otimization?.. 19

20 Resonances in high-contrast gratings with comle unit cell toology Thank you for the attention! Milan Maksimovic The XXI International Worksho on Otical Wave & Waveguide Theory and Numerical Modelling Enschede, The Netherlands, Aril 19-, 13