Zero Group Velocity Modes of Insulator Metal Insulator and Insulator Insulator Metal Waveguides

Transcription

1 Zero Group Velocity Modes of Insulator Metal Insulator and Insulator Insulator Metal Waveguides Dmitry Fedyanin, Aleksey Arsenin, Vladimir Leiman and Anantoly Gladun Department of General Physics, Moscow Institute of Physics and Technology (State University) Dolgoprudny, Russia e mail: feddu@mail.ru Acknowledgement: Vladimir Tarakanov, the athor of PIC code "KARAT"

2 OUTLINE Lossless case: Dispersion relations and dispersion curves of SPPs in IMI and IIM waveguide structures Existence conditions of waves with zero group velocity Lossy case: Effect of losses on dispersion curves Numerical calculations of energy velocity An analysis of points of zero energy velocity Excitation of zero group velocity modes Results of numerical simulation Applications

3 History Backward waves and waves with zero group velocity are typically associated with periodic structures. Nevertheless a periodicity is not the only way to obtain such waves - particularly investigations of slow wave propagation in plasma-dielectric structures showed that plasma waveguides with a specially designed circular cross section allow the existence of backward waves, so are some kinds of planar plasma waveguides: Trivelpiece A W, Gould R W 1959 Space charge waves in cylindrical plasma columns J. Appl. Phys. 30, Paik S F 1962 A backward wave in plasma waveguide Proc. IRE Allis W P, Buchsbaum S J, Bers A 1963 Waves in Anisotropic Plasma (Cambridge, MA: M.I.T. Press) Oliner A A, Tamir T 1962 Backward waves on isotropic plasma slabs J. Appl. Phys Schumann W O 1960 Z. angew Phys. 12, 4, 145

4 Lossless case Metal is described by Drude Zener model:

5 Dispersion Relation, where and a is a half thickness of the film For IMI waveguide structures with ε2=ε3 dispersion relation can be easily simplified and rearranged as two branches: κ 2 ε1 th κ 1 a = κ1 ε2 anti symmetric mode κ1 ε2 th κ 1 a = κ2 ε1 symmetric mode

6 Dispersion Curves (Lossless Case) IMI AS S AS anti symmetric mode, S symmetric mode AS S IIM ε2>ε3 The decay constant κ2 may be complex while κ1 and κ3 are real.

7 Existence Conditions of Waves with Zero Group Velocity Equation for Poynting vector in a complex form SPP's group velocity is negative if and only if net energy flux is opposite to the phase velocity direction that could be written as [ Allis W P, Buchsbaum S J, Bers A 1963 Waves in anisotropic plasma ] and the group velocity is zero if It will be recall that the only non radiative modes interest us, thus the projection of net power flow on z axis is equal to zero., where k = ω /c and

8 Existence Conditions of Waves with Zero Group Velocity Re(Sx) x IMI? IIM Single metal insulator interface

9 Existence Conditions of Waves with Zero Group Velocity IMI IIM small kx large kx small kx large kx

10 Existence Conditions of Waves with Zero Group Velocity (IMI structures) In order to satisfy the existence condition of waves with zero group velocity the power flow inside the metal must be equal (in absolute value) to the power flow in dielectric media. Solving Maxwell equations we obtain that this requirement is equivalent to the following equality This lengthy expression can be significantly simplified in case of anti symmetric mode and ε2=ε3: [ Allis W P, Buchsbaum S J, Bers A 1963 Waves in anisotropic plasma, chapter 7 ]

11 Existence Conditions of Waves with Zero Group Velocity (IMI structures) silver Dispersion curves of SPP in the insulator silver insulator structures for various permittivities. Non radiative modes are presented. The value of ε2 is fixed and is equal to 2.5, film thickness d=15 nm. Permittivity ε3 takes values from 1 till 12 (1; 2; 2.5; 3; 4; 5; 7; 12). The upper curve corresponds to ε3=1. According to [Palik E D 1998 Handbook of Optical Constants of Solids I (San Diego, CA: Academic Press) ] and Drude Zener model in a wavelength range nm [ Johnson P B, Christy R W 1972 Phys Rev B 6, 4370 ]

12 Existence Conditions of Waves with Zero Group Velocity (IMI structures) silver Zero group velocity points of the dispersion curves of SPPs as parametric functions of permittivity ε3. Permittivity ε3 takes values from 1 till 20. The value of ε2 is fixed (three materials are considered ZrO 2, SiO2, Al2O3 with permittivities 5.5, 2.2 and 2.84 correspondibly), film thickness d=15 nm.

13 Existence Conditions of Waves with Zero Group Velocity (IMI structures) silver Zero group velocity points of the dispersion curves of SPPs as parametric functions of permittivity ε3. Permittivity ε3 takes values from 1 till 20. The value of ε2 is fixed (three materials are considered ZrO2, SiO2, Al2O3 with permittivities 5.5, 2.2 and 2.84 correspondibly), film thickness d=15 nm.

14 Existence Conditions of Waves with Zero Group Velocity (IMI structures) Zero group velocity points of the dispersion curves of SPPs as parametric functions of permittivity ε3. Permittivity ε3 takes values from 1 till 20. The value of ε2 is fixed (three materials are considered ZrO2, SiO2, Al2O3 with permittivities 5.5, 2.2 and 2.84 correspondibly), film thickness d=15 nm.

15 Existence Conditions of Waves with Zero Group Velocity (IMI structures) Zero group velocity points of the dispersion curves of SPPs as parametric functions of film thickness d which takes values from 3 nm till 50 nm. The values of ε2 and ε3 are fixed.

16 Existence Conditions of Waves with Zero Group Velocity (IMI structures) Zero group velocity points of the dispersion curves of SPPs as parametric functions of film thickness d which takes values from 3 nm till 50 nm. The values of ε2 and ε3 are fixed.

17 Existence Conditions of Waves with Zero Group Velocity (IMI structures) Resonance circular frequency as function of permittivites of surrounding insulator media for fixed value of film thickness d. d=10 nm d=15 nm

18 Existence Conditions of Waves with Zero Group Velocity (IIM structures) vacuum silver Dispersion curves of SPP in the vacuum insulator silver structures for various permittivities ε2. Film thickness d=8 nm.

19 Existence Conditions of Waves with Zero Group Velocity (IIM structures) vacuum silver Zero group velocity points of the dispersion curves of SPPs as parametric functions of permittivity ε2. Permittivity ε2 takes values from 1 till 15. The value of ε3 is fixed (vacuum). Film thickness is fixed too, and three different values are considered.

20 Existence Conditions of Waves with Zero Group Velocity (IIM structures) Zero group velocity points of the dispersion curves of SPPs as parametric functions of permittivity ε2. Permittivity ε2 takes values from 1 till 15. The value of ε3 is fixed (vacuum). Film thickness is fixed too, and three different values are considered.

21 Existence Conditions of Waves with Zero Group Velocity (IIM structures) vacuum silver Dispersion curves of SPP in the vacuum insulator silver structures for various film thicknesses. Permittivity ε2=5.5 (ZrO2)

22 Existence Conditions of Waves with Zero Group Velocity (IIM structures) vacuum λ=350 nm silver Zero group velocity points of the dispersion curves of SPPs as parametric functions of film thickness d which takes values from 2nm till 20nm. The value of ε3 is fixed (vacuum). Four different materials as medium 2 are considered. dcoff is a cutoff thickness, i.e. the maximum thickness that allow the existence of zero group velocity mode. λ=450 nm

23 Existence Conditions of Waves with Zero Group Velocity (IIM structures) Zero group velocity points of the dispersion curves of SPPs as parametric functions of film thickness d which takes values from 2nm till 20nm. The value of ε3 is fixed (vacuum). Four different materials as medium 2 are considered.

24 Lossy case Metal is described by Drude Zener model:

25 Drude Zener dielectric function with dumping factor Silver: Γ 7 10 s 13 1

26 Effect of Losses on Dispersion Curves silver IMI Al2O3 Dispersion curves of SPP in the Al2O3 silver insulator structures for various permittivities ε2. Film thickness d=10 nm. Dashed lines show dispersion curves in a lossless system. Only the region, where Im(kx) < Re(kx), is considered.

27 Effect of Losses on Dispersion Curves IMI

28 Effect of Losses on Dispersion Curves IMI A B C D Re( ) Film thickness d=10 nm. Dashed lines show dependecies for lossless systems. Only the region, where Re(kx) >(ω/c) max(ε2,ε3), is considered. Points A, B, C, D correspond to cross points of dispersion curves with light lines.

29 Effect of Losses on Dispersion Curves vacuum IIM silver Dispersion curves of SPP in the vacuum insulator silver structures for various permittivities ε2. Film thickness d=8 nm. Dashed lines show dispersion curves in a lossless system. Only the region, where Im(kx) < Re(kx), Re( ) is considered.

30 Effect of Losses on Dispersion Curves B A B A C C Re ( ) Dispersion curves of SPP in the vacuum insulator silver. Film thickness d=8 nm. Negative values of Im(kx) correspond to backward waves. IIM Im(kx) < Re(kx)

31 Modified Drude Zener model Γ=Γ0+β ω 2 Γ0= s 18 β= s 13 [ Chen L, Lynch D W 1987 Phys Rev B 36, 1425 ] (red curve) 1

32 Effect of Losses on Dispersion Curves (cm 1) Dispersion curves of SPP in the vacuum insulator silver structures for various permittivities ε2. Modified Drude Zener model is used. Film thickness d=8 nm. Im(kx) < Re(kx)

33 Effect of Losses on Dispersion Curves vacuum Al2O3 silver Comparison of dispersion curves with different dumping factors Γ. Silver dielectric function obeys Drude Zener model. Silver is covered by ultra thin insulator film of Al2O3. Film thickness d=8 nm. Im(kx) < Re(kx) IIM

34 Effect of Losses on Dispersion Curves IIM vacuum Al2O3 silver Dispersion curves of SPP in the vacuum insulator silver structures for various dumping factors Γ. Film thickness d=8 nm.

35 Excitation of Zero Group Velocity Modes

36 Excitation of Zero Group Velocity Modes Three-layer Kretschmann geometry IMI εpr > ε2, ε3 Three-layer Kretschmann geometry or Otto configuration εpr > ε2> ε3 IIM Simple Kretschmann geometry εpr > ε2> ε3

37 Excitation of Zero Group Velocity Modes

38 Results of Numerical Simulation and Applications

39 Stored Light in Nano Scale Plasmonic Cavity s=50 nm Γ= s 1 d=10 nm L=180 nm and Γ= s 1 IMI εpr=7.4 ϴ=45⁰ ω= s 1

40 Stored Light in Nano Scale Plasmonic Cavity IMI

41 Stored Light in Nano Scale Plasmonic Cavity Γ=7 10 s 12 1 E=E0 sin(ω t) point A point B IMI

42 Stored Light in Nano Scale Plasmonic Cavity Γ=7 10 s 12 1 E=E0 sin(ω t) IMI point A point B V=V0 sin(ω t)

43 Stored Light in Nano Scale Plasmonic Cavity Γ=7 10 s 12 1 IMI

44 Stored Light in Nano Scale Plasmonic Cavity IMI Γ= s 1

45 Future

46 Thank you for your attention! Dmitry Fedyanin, Aleksey Arsenin, Vladimir Leiman and Anantoly Gladun Moscow Institute of Physics and Technology (State University), Russia e mail: feddu@mail.ru