1. Propagation of surface plasmons through planar interface Tomáš Váry Peter Markoš Dept. Phys. FEI STU, Bratislava

2. Introduction‏ Properties of SPP Surface waves on interface of two dielectrics – method of solution Normal incidence Oblique angle incidence

3. Surface Plasmon Polaritons (SPP)‏ Electromagnetic oscillations propagating along metal–dielectric or left-handed – dielectric interface in wave-like fashion High sensitivity to surface features Two dimensional character of propagation – planar optics

4. Properties of SPP Planar wave bound to metal- dielectric interface for z>0, B=0 for z<0, A=0 Exponential decay of electromagnetic field with increasing distance from surface

5. Field components for p-polarization z>0 z<0 TM polarized SPP Metal dielectric function – Drude formula: ε m (ω) = 1 - (ω p 2 / ω 2 ) Bulk plasma frquency: ω p 2 = ne 2 / ε 0 m Dielectric :ε 1 = const.

6. Dispersion relation From conservation of tangential components of field on interface z = 0: H 1y = H my E 1x = E mx we get dispersion relation in form: Dispersion dependence of SPP for metal – vacuum interface (ε 1 = 1)‏

7. Wave vector Dispersion for plane waves: Components of wave vector: Prerequisite for existence of SPP:

8. SPP on interface of two different dielectrics Surface of metal is divided by x = 0 plane on two half-spaces, waves aproaches from the left side Problem with conservation of tangential field components on both sides of interface

9. Continuity of fields on interface On interface x = 0 are equations of continuity of E z and H y of form: Electric field: Magnetic field: Infinite number of propagating plane waves has to be taken into count

10. SPP transmitting through interface between two media Total field is superposition of all possible surface and planar waves. Relations between individual waves are described by coupling coefficients determined by ortogonality integral [Oulton et al]: Norm condition for i = j:

11. Continuity of fields on interface Indicies α = β = 0 belong to amplitudes of SPP Using orthogonality integral we can form system of linear equations:

12. Scattering matrix Whole system can be expressed in matrix form: Here components of C matrix are coupling coefficients from ortogonality integral. This form allows us to write scattering matrix for this system: where components of scattering matrix are: Scattering matrix symmetries:

13. Dependance of transmission, reflection and scattering of SPP on dielectric permitivity ratio – normal incidence ω = 0,23 ω p

14. Frequency dependence of transmission, reflection and scattering – normal incidence ε 1 = 1, ε 2 = 5ε 1 = 5, ε 2 = 1

15. Relative dependence of density of scattered energy for normal incidence SPP ε 1 = 1, ε 2 = 3 ω = 0,23 ω p Area 1Area 2

16. Relative dependence of density of scattered energy for normal incidence SPP ε 1 = 1, ε 2 = 5 ω = 0,23 ω p ε 1 = 1, ε 2 = 10 ω = 0,39 ω p

17. Snell's law for SPP Change of an angle of refraction compared to planar waves due to the metal- dependent character of SPP:

18. ε 1 = 1; ε 2 = 5 Dependence of refraction angle on angle of incidence for SPP

19. Dependence of transmission, reflection and scattering of SPP on angle of incidence ε 1 = 1, ε 2 = 2ε 1 = 1, ε 2 = 6 ω = 0,23 ω p

20. Dependence of transmission, reflection and scattering of SPP on angle of incidence ε 1 = 2, ε 2 = 1ε 1 = 6, ε 2 = 1 ω = 0,23 ω p

21. Conclusion - Energy losses due to radiation of planar waves up to 50% - Strong angular dependence of radiated waves - Negative permeability materials enable excitations of TE SPP with much smaller scattering losses [TV: in preparation]

22. References - Zayats, A. V., Smolyaninov, I. I., Maradudin, A. A.: Nano-optics of surface plasmon polaritons. In: Phys. Reports 408, 131 -314, (2005)‏ - Oulton, R. F et al.: Scattering of surface plasmon polariton on abrupt surface interfaces: Implications for nanoscale cavities. In: Phys. Rev B 76, 035408, (2007)‏ - Stegeman, G. I. Et al: Refraction of surface plasmon polariton by an interface. In: Phys. Rev. B, vol. 12, 1981, no. 6 - Vary, T.: Surface plasmon polaritons, Diploma thesis - Acknowledgment: This work was supported by Slovak Grant Agency APVV

23. Thank you for your attention