S-matrix approach for calculations of the optical properties of metallic-dielectric photonic crystal slabs

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1 S-matrix approach for calculations of the optical properties of metallic-dielectric photonic crystal slabs N. I. Komarevskiy1,2, T. Weiss3, and S. G. Tikhodeev2 1 Faculty of Physics, Lomonosov Moscow State University, Russia A. M. Prokhorov General Physics Institute, RAS, Moscow, Russia 2 3 LASMEA, Universite Blaise Pascal, France 1

2 Contents 1) Introduction a) Photonic crystal slabs (PCSs) b) Optical peculiarities of the PCSs c) Transfer and scattering matrices 2) Additions to the S-matrix approach a) Comparison of the S-matrix and improved S-matrix calculation schemes 3) S-matrix calculations of the optical properties a) Dispersion relation b) Spectra c) Near-field distribution 2

$..... Layers may be composed of dielectrics, metals, semiconductors Diffractive (opening of diffractive channels as λ decreases)$

3 Photonic crystal slabs (PCS) 1D 2D Layers may be composed of dielectrics, metals, semiconductors Diffractive (opening of diffractive channels as λ decreases) Peculiarities in optical spectra Waveguide Plasmonic Resonances (localized and delocalized) 3

4 Transfer-matrix method Transfer matrix combines the amplitudes at different planes a and b + Ab Ab = Ta,b + Aa Aa Scheme of calculations 1) 2) 3) Splitting of the structure into layers along Oz Floquet-Fourier expansion of the Maxwell's equations in each layer Calculation of transfer, interface and total transfer matrices through the structure 4) Calculation of transmission, reflection, absorption T-matrix is numerically instable for PCSs (mixing of exponentially increasing and decreasing modes) 4

Scattering-matrix method S-matrix combines the input amplitudes with the output ones + Ab Aa = S a,b out + Aa Ab in Scheme of calculations 1) 2) 3) 4) 5) Splitting of the structure into layers along

5 Scattering-matrix method S-matrix combines the input amplitudes with the output ones + Ab Aa = S a,b out + Aa Ab in Scheme of calculations 1) 2) 3) 4) 5) Splitting of the structure into layers along Oz Floquet-Fourier expansion of the Maxwell's equations in each layer Calculation of the transfer matrix for field amplitudes Iterative calculation of the scattering matrix, Ko&Inkson (1988) Calculation of transmission, reflection, absorption and diffraction Whittaker&Culshaw, PRB 60, 2610 (1999) S.G.Tikhodeev et al., Phys. Rev. B 66, (2002) 5

6 Scattering matrix approach Advantages of the method: 1) No additional fitting parameters, only structure's geometry and epsilon data 2) Calculation of the near-field distribution 3) All calculation can be performed on PC Limitations of the method: 1) Slow convergence for metallic-dielectric layers 2) Time of calculations ~Ng3, Ng - number of used harmonics Additions to the method1: adaptive spatial resolution2 Improved convergence1!!! Accuracy of the solution (eigenvalues)??? 1 2 T. Weiss, N. A. Gippius, S. G. Tikhodeev, G. Granet, and H. Giessen, J. Opt. A (2009) G. Granet, J. Opt. Soc. Am. A 16, 2510 (1999). 6

Accuracy of the eigenvalue calculations S-matrix (factorization rules) Compare two calculation schemes S-matrix improved (with adaptive spatial resolution and factorization rules) eigenvalues in the

7 Accuracy of the eigenvalue calculations S-matrix (factorization rules) Compare two calculation schemes S-matrix improved (with adaptive spatial resolution and factorization rules) eigenvalues in the periodic layer K(n)=K(n)(kx0,ω) n=1,,2ng, Ng - number of used harmonics gold nanowires on top of quartz substrate It is not possible to check the accuracy of K directly! Using T-matrix over the period and some math Td = Td 2 T2 1Td1 T1 2 = > k (xn ) = k (xn ) (K ( n ), ω ) ω k x0 = cos θ c - compare two values 7

8 Accuracy of the eigenvalue calculation k = k (K, ω ) ω k x = cos θ c (n) x (n) x (n) lim ln Ng 0 TE - polarization k (xn ) k x0 k x0 = n=const (fixed harmonic) TM - polarization - improved S-matrix (with adaptive spatial resolution) - S-matrix 8

9 S-matrix calculations of the optical properties dispersion relation of the waveguide modes air waveguide substrate det(s 1 (ω, k )) = 0 9

10 S-matrix calculations of the optical properties dispersion relation of the surface plasmon polariton air silver film substrate det(s 1 (ω, k )) = 0 εag calculated using Lorentz-Drude model1 A. D. Rakić et al., Applied Optics, Vol. 37, No. 22 (1998) 1 10

S-matrix calculations of the optical properties Structure parameters: wau=100 nm, hwire=20 nm, Lz=40 nm, hfilm=10 nm, dx=200-350 nm bare localized plasmon (no silver film) 1) Localized

11 S-matrix calculations of the optical properties Structure parameters: wau=100 nm, hwire=20 nm, Lz=40 nm, hfilm=10 nm, dx= nm bare localized plasmon (no silver film) 1) Localized plasmon in gold nanowires (Energy~1.8 ev) Extinction spectra, TM - polarization 2) Delocalized plasmon in the silver film (excitation due to periodic corrugation) 3) Interaction of the resonances 11

12 S-matrix calculations of the optical properties Structure parameters: wau=100 nm, hwire=20 nm, Lz=40 nm, hfilm=10 nm, dx= nm Period=200 nm Dispersion relation of bare plasmon modes Calculated extinction spectrum Normal light incidence, TM - polarization 12

13 S-matrix calculations of the optical properties Structure parameters: wau=100 nm, hwire=20 nm, Lz=40 nm, hfilm=10 nm, dx= nm Period=350 nm Dispersion relation of bare plasmon modes Calculated extinction spectrum Normal light incidence, TM - polarization 13

14 S-matrix calculations of the optical properties Near-field distribution at resonances Electric field Field Energy 1220 mev Magnetic field 14

15 S-matrix calculations of the optical properties Near-field distribution at resonances Electric field Field Energy 2040 mev Magnetic field 15

16 Conclusions 1) Scattering matrix approach is a powerful tool for calculations of the PCS 1) Accuracy of the two calculation schemes were compared. Improved scheme shows better results for TM polarization and not large number of harmonics<121 1) Extinction spectra and near-field distribution of the specific structure have been modeled theoretically 16

17 Acknowledgements Georgy Kichin 1, Anton Akimov1, and Nikolay Gippius1,2 1 A. M. Prokhorov General Physics Institute, RAS, Moscow, Russia 2 LASMEA, Universite Blaise Pascal, France 17

18 Thank you for your attention! 18

arxiv: v1 [physics.optics] 3 Nov 2017

arxiv: v1 [physics.optics] 3 Nov 2017 arxiv:1711.01195v1 [physics.optics] 3 Nov 2017 This is a translation of the following paper: D. A. Bykov and L. L. Doskolovich, On the use of the Fourier modal method for calculation of localized eigenmodes