Propagation of surface Propagation of surface plasmons through planar plasmons through planar interfaceinterface

Tomáš VáryPeter MarkošDept. Phys. FEI STU, Bratislava

Introduction

•Properties of SPP

•Surface waves on interface of two dielectrics – method of solution

•Normal incidence

•Oblique angle incidence

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

Properties of SPP Planar wave bound to metal-

dielectric interface

1, 1,, ~ exp( ) exp( )z m z mE H A ik z B ik z

( , , , ) ( , , ) exp[ ( ) ],

( , , , ) ( , , ) exp[ ( ) ],

x y z x y z

x y z x y z

H x y z t H H H i k x k y k z i t

E x y z t E E E i k x k y k z i t

1 1zk iK

zm mk iK

for z>0, B=0

for z<0, A=0

Exponential decay of electromagnetic field

with increasing distance from surface

1, 0mK K

Field components for p-polarization

1 1 10, ,0 exp( )xH ik x K z i t H

0, ,0 exp( )m m x mH ik x K z i t H

1 1 1 11

1,0, exp( )x xH iK k ik x K z i t

E

1,0, exp( )m m m x x m

m

H iK k ik x K z i t

E

z>0

z<0

TM polarized SPP

Metal dielectric function – Drude formula: εm(ω) = 1 - (ωp2 / ω2)

Bulk plasma frquency: ωp2 = ne2/ ε0m

Dielectric : ε1 = const.

Dispersion relationFrom conservation of tangential components of field on interface z = 0:

H1y = Hmy

E1x = Emx 1

1

( ) ( )0

( )x m x

m

K k K k

we get dispersion relation in form:

Dispersion dependence of SPP for metal – vacuum interface (ε1 = 1)

Wave vectorDispersion for plane waves:

22 2 2

1 1 12 xk k Kc

22 2 2

2 m m x mk k Kc

Components of wave vector:

Prerequisite for existence of SPP: 1 0m 11

p

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

1 2m m

Continuity of fields on interface

On interface x = 0 are equations of continuity of Ez and Hy of form:

Electric field:

Magnetic field:

Infinite number of propagating plane waves has to be taken into count

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: ( ), ( ) ( ), ( ) ( )i i i i i i i ie k h q E k H q k q

Continuity of fields on interface

Indicies α = β = 0 belong to amplitudes of SPP

Using orthogonality integral we can form system of linear equations:

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:

111 (1 ) (1 )T TS CC CC

112 (1 ) 2TS CC C

121 (1 ) 2T TS C C C

122 (1 ) (1 )T TS C C C C

Scattering matrix symmetries:

Dependance of transmission, reflection and scattering of SPP on dielectric permitivity ratio –

normal incidence

ω = 0,23 ωp

Frequency dependence of transmission, reflection and scattering – normal incidence

ε1 = 1, ε2 = 5 ε1 = 5, ε2 = 1

Relative dependence of density of scattered energy for normal incidence SPP

ε1 = 1, ε2 = 3

ω = 0,23 ωp

Area 1 Area 2

Relative dependence of density of scattered energy for normal incidence SPP

ε1 = 1, ε2 = 5

ω = 0,23 ωp

ε1 = 1, ε2 = 10

ω = 0,39 ωp

Snell's law for SPP

Change of an angle of refraction compared to planar waves due to the metal-

dependent character of SPP:

ε1 = 1; ε2 = 5

Dependence of refraction angle on angle of incidence for SPP

Dependence of transmission, reflection and scattering of SPP on angle of incidence

ε1 = 1, ε2 = 2 ε1 = 1, ε2 = 6ω = 0,23 ωp

Dependence of transmission, reflection and scattering of SPP on angle of incidence

ε1 = 2, ε2 = 1 ε1 = 6, ε2 = 1

ω = 0,23 ωp

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]

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

Thank you for your attention