Optimization of metallic biperiodic photonic crystals. Application to compact directive antennas

Transcription

1 Optimization of metallic biperiodic photonic crystals Application to compact directive antennas Nicolas Guérin Computational Optic Groups (COG) IFH, ETHZ, Keywords: 3D modeling, perfect electric conducting model, thin wire method of moments, biperiodic Green s function, optimization, compact directive antennas.

2 Is the perfect electric conducting model relevant at OF? Band diagrams for a photonic crystal with a square lattice, lattice constant a=75nm, radius r=.35a. Solid lines: H polarization, dotted: E polarization Perfect Electric Conductor (PEC) Silver For metallic photonic crystals with acceptable losses at OF, PEC models at RF are very useful for obtaining a first impression. 2

3 Outline 1. Three dimensional modeling 2. Compact directive antennas: Photonic crystal ~ ultra refractive metamaterial 3. Fast Optimization of the photonic crystal: Parameters of the structure Geometry of the grids 4. Conclusion 3

4 Modeled structures Three dimensional structures of wires (finites or infinites) d y d d d x Biperiodic plates + ground plane + source (monopole, patch) Crossed grids Monopole Ground plane 4

5 Three dimensional Modeling 1. Integral Formalism (perfect metallic conductor) 2. Method of Moments (matrix resolution) 3. Harrington s approximation (thin wires) 4. Biperiodic grating (biperiodic Green s function) 5. Ground plane and non periodic incident field 6. Numerical validation of the code 5

6 Biperiodic grating How to model a large structure with many cells? infinite periodic structure α β γ k z d y Advantage : the unknowns are reduced to the first cell d x y Drawback : compute the biperiodic Green s function x d y d x 6

7 Ground plane Image theory Biperiodic grating Source Ground Plane same field in the upper space z x y 7

8 Non periodic incident field Incident field FFT2D Solve a grating problem plane waves For each plane wave z x y 8

9 Validation of the code: source z Example: source =patch (342 wires) + e ϕ e r z(mm) mm -4-2 x (mm) mm y(mm) E Plane (x = ) x z ϕ θ + e θ y H Plane (y = ) y x 9

10 Validation of the code: source E plane (x=) with our code H plane (y=) with our code E plane (x=) with NEC E θ component 33 3 E θ component E θ component E ϕ component H plane (y=) with NEC E θ component E ϕ component

11 Validation of the code: FFT2D FFT2D : expansion of the field emitted by the patch into a packet of plane waves 2 Direct calculus of the emitted field 2 Reproduction of the field as a packet of plane waves z (mm) y (mm) y (mm) 11

12 Validation of the code: biperiodic grating Transmission infinite infinite finite 5 metallic grids of parallel wires Transmission x1-4 1x1-5 2D rigorous code (S matrix) incidence incidence Wavelength λ (cm) Transmission x1-4 1x1-5 3D code (grating) incidence incidence Wavelength λ (cm) 12

13 Summary: potential of the 3D code model realistic structures which have many cells along two directions of the space. up to 1 wires inside the first cell. simulation of more general structures with mesh of wires (e.g. patch, plates) non periodic incident field, ground plane 3D diagrams and polarisation of the field 3D directivity for the study of antennas 13

14 Metallic photonic crystal ~ metamaterial 2D metallic crystal Ultra refractive antenna d d vacuum ( n = 1) i ext 1 positive négative 2 2 ωc 2 2 c ω λ ε= 1 = 1 λ ε λ c d λ c = 2πd ln 2 r GAP λ 2 Ultra refraction i int source metamaterial with < n eff < 1-9 < i int < 9 et n eff << 1 i ext 14

15 Transmission of the photonic crystal Many properties of the antenna can be derived from the study of the transmission of the photonic crystal Transmission Numerical Experimental Homogeneous medium Effective index λ c Bandgap Real part of the Effective index Transmission of a stack of 6 grids (metallic strips) Free-space wavelentgh λ (mm) The code allows to find efficiently the parameters of the structure which will give high directivity for the required frequency area. 15

16 Directive antenna E Plane Numerical (x = ) Experimental Monopole z H Plane (y = ) y db E Plane ( x = ) H Plane ( y = ) db 33 3 x θ x θ y θ = 9,9 x θ = 7,8 y 16

17 Optimization of the parameters of the structure Parameters: Periods of the grids, cross section, Vertical distance between the grids, number of grids. Fast code for transmission: Just compute complex reflection and transmission for 1 grid (optimize periods and cross section) Layer by layer code (optimize vertical distance, number of grids) Application for directive antenna: Increase both directivity and bandwidth 17

18 Example : transmission / number of grids 1..8 Transmission λ c 6 grids 8 grids 1 grids 12 grids Re ( n eff ) Free-space wavelength λ (mm) Time for all curves (3pts/curve) = 1 hour 18

19 Optimization of the geometry L ω k y k x Capacitive effects (Jerusalem crosses) Ultra refractive phenomena at low frequencies 19

20 Optimization of the geometry Photonic crystal made of 6 biperiodic gratings (Jerusalem crosses) Transmission Wires : periods = 5,8 mm Crosses : L = 2,9 mm, =,4 mm Crosses : L = 2,9 mm, = 1 mm Crosses : L = 2,9 mm, = 2 mm L 5.8 mm Free space wavelength λ (mm) 2 frequency areas for directive antenna (bi-band antenna) Control of the width of the GAP and resonant frequencies 2

21 Conclusion Modeling of 3D PEC photonic crystal Fast optimization of such structures: - periods, vertical distance between grids - number of grids - geometry of grids (wires, strips) - geometry of the structure (Jerusalem crosses) Application to directive antennas: - Metamaterial antenna - Fabry-Perot antenna - bi-band antenna 21