P7 – Scattering of Surface Plasmon Polaritons by Gold ... - repetit.dk

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CHAPTER 2. THEORY z x /c np /c kx θi Evanescent field Figure 2.3: Kretschmann-Raether geometry of the prism and the metal film. The evanescent field is indicated at the prism/metal interface. The projection of incoming light through a dielectric medium on the metal film yields kx = ω/c √ εp sin θ while the projection of light through air yields kx = ω/c sin θ. the angle of the incoming EM radiation is varied, kix changes and the phase of the evanescent wave changes. When a kix is chosen that satisfies the SPP dispersion relation at a given ω resonance occurs and an SPP can be excited. At this angle of incidence of the incoming EM radiation, the momentum of the radiation is equal to the momentum of the SPP. In Fig. 2.2(a) the dispersion relation for SPPs on a semi-infinite solid is plotted. At all kx the SPP dispersion is to the right of the light line in vacuum, i.e. light cannot be converted into SPPs because of the missing momentum ∆kx. When using the Kretschmann-Raether geometry, light is reflected at a metal surface covered with a dielectric medium, e.g. quartz. When light is travelling inside the dielectric medium its momentum is different from that of light travelling in vacuum (air). The momentum of light inside quartz is √ εpω/c, which is different from that of vacuum ω/c. kx is therefore different when light impinges the metal through a dielectric medium than if it impinges the metal through air, see Fig. 2.3. The dispersion relation of a medium characterized by ε is expressed through Eqn. 2.5 as k 2 z = k 2 ε cos 2 θ yielding k 2 x = k 2 ε − k 2 ε cos 2 θ k 2 x = k 2 ε sin 2 θ kx = √ ε ω c sin θ ⇒ ω = kx √ c ε sin θ (2.22) where θ is the angle at which SPPs are excited. The dispersions for a prism and for air are plotted on Fig. 2.4, where the blue line is the dispersion relation Eqn. 2.20, line (1) is the dispersion of the projection on the x-axis of light in vacuum, line (2) is the dispersion of light in vacuum, line (3) is the dispersion of the projection on the x-axis of light in a medium ε, line (4) is the dispersion of light in a medium ε. Notice that the dispersion of light in air is located left of the SPP dispersion relation for all kx, i.e. no SPPs can be excited. Since the dispersion of light through a dielectric medium lies at the right side of the SPP dispersion relation up to a certain kx, light can excite SPPs of frequencies below this point at the metal/air interface. 18
Angular Frequency, Ω Dispersion Relation 1 2 3 4 Wavenumber, kx 2.1. PHYSICS OF SPPS Figure 2.4: Dispersion relation of SPPs for a quartz/metal/air system. Eqn. 2.20 is compared to the dispersion of light in air and in quartz and the projections on the x-axis of light in air and in quartz. The black lines represent the dispersion of light in vacuum and the projection on the x-axis. The dashed lines are the dispersion of light in quartz and the projection on the x-axis. The dispersion of light in quartz intersects the SPP dispersion at the point marked by the red dotted line. 2.1.3 Quantitative Description of Minimum Reflected Intensity The resonance situation described above can be calculated for a given system by Fresnel reflection theory for a two interface system. The total reflectivity is calculated in appendix B and is given by R(θ) = Er Ein 2 = rpm + rmae2ikzmd 1 + rpmrmae2ikzmd 2 , (2.23) where rpm and rma is the single interface reflection coefficient of the prism/metal and the metal/air interface respectively. These are given by rpm = rma = kzp εp kzp εp kzm εm kzm εm − kzm εm + kzm εm kza − εa kza + εa where kzp, kzm and kza are expressed through Eqn. 2.5 k 2 x + k 2 zi = εi (2.24) , (2.25) ω 2 . (2.26) c Here kx is the x component of the wavevector of the incoming EM radiation in the prism and kzi is the z component of the EM radiation in medium i. kx can be written as kx = k sin θ 19
Page 3: Institute of Physics and Nanotechno
Page 7 and 8: Contents 1 Introduction 9 1.1 Surfa
Page 9 and 10: Chapter 1 Introduction Contents 1.1
Page 11 and 12: LR DR LR Objective Metal Glass Imme
Page 13 and 14: Chapter 2 Theory Contents 2.1 Physi
Page 15 and 16: E1x = E2x 2.1. PHYSICS OF SPPS (2.6
Page 17: Angular Frequency, Ω Dispersion R
Page 21 and 22: Intensity Arbitrary Units 1.0 0.8 0
Page 23 and 24: 2.2. SCALAR MODEL For λ = 650 nm,
Page 25 and 26: 2.3. SINGLE PARTICLE SCATTERING E0(
Page 27 and 28: Intensity AU 2.0 1.5 1.0 0.5 Intens
Page 29 and 30: 2.4. VECTORIAL MODEL and the field
Page 31 and 32: 2.4. VECTORIAL MODEL plane at z=20
Page 33 and 34: 2.4.4 Relation to Surface Plasmon P
Page 35 and 36: ˆGSPP (r, r ′ (1) iakSPPH 0 (kSP
Page 37 and 38: kSPP S in = 2 −2akSPPz 2 a −
Page 39 and 40: Chapter 3 Numerical Results Content
Page 41 and 42: (a) (b) 3.2. VECTORIAL MODEL RESULT
Page 43 and 44: 50 100 150 200 250 0 50 100 150 200
Page 45 and 46: 3.2. VECTORIAL MODEL RESULTS Figure
Page 47 and 48: Chapter 4 Experimental Setup Conten
Page 49 and 50: 4.2. LRM SETUP s-polarized light ca
Page 51 and 52: Chapter 5 Experimental Results and
Page 53 and 54: 5.2. LRM RESULTS the obtained thick
Page 55 and 56: • The spacing between the nanopar
Page 57 and 58: (a) Intensity [Arbitrary Units] 200
Page 59 and 60: Chapter 6 Conclusion In this projec
Page 61 and 62: Appendix A Surface Plasmon Excitati
Page 63 and 64: Appendix B Reflectivity of a two in
Page 65 and 66: Bibliography [Barnes et al., 2003]

P7 – Scattering of Surface Plasmon Polaritons by Gold ... - repetit.dk

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