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Nanophotonics Class 2 Surface plasmon polaritons

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**Surface plasmon polariton: EM wave at metal-dielectric interface**

z x For propagating bound waves: - kx is real - kz is imaginary EM wave is coupled to the plasma oscillations of the surface charges

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**Derivation of surface plasmon dispersion relation: k()**

Wave equation: Substituting SP wave + boundary conditions leads to the Dispersion relation: x-direction: Note: in regular dielectric:

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Dispersion relation: x-direction: z-direction: Bound SP mode: kz imaginary: em + ed < 0, kx real: m < 0 so: m < -d

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**Dielectric constant of metals**

Drude model: conduction electrons with damping: equation of motion with collision frequency g and plasma frequency If g << wp, then: no restoring force

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**Measured data and model for Ag:**

Drude model: Modified Drude model: Contribution of bound electrons Ag:

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**Bound SP modes: m < -d**

bound SP mode: m < -d

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**Surface plasmon dispersion relation:**

w Radiative modes real kx real kz (e'm > 0) wp Quasi-bound modes imaginary kx real kz (-ed < e'm < 0) Dielectric: ed Metal: em = em' + em" x z real kx imaginary kz Bound modes (e'm < -ed) Re kx

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**Surface plasmons dispersion:**

w large k small wavelength 3.4 eV (360 nm) Ar laser: vac = 488 nm diel = 387 nm SP = 100 nm X-ray wavelengths at optical frequencies Ag/SiO2 Re kx

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**Surface plasmon dispersion for thin films**

Drude model ε1(ω)=1-(ωp/ω) 2 Two modes appear Thinner film: Shorter SP wavelength Propagation lengths: cm !!! (infrared) Example: HeNe = 633 nm SP = 60 nm L- L-(symm) L+(asymm)

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**Cylindrical metal waveguides**

k E z r E Fundamental SPP mode on cylinder: Can this adiabatic coupling scheme be realized in practice? Now when we let a small nanowire extend from this tip, the taper can serve as a coupling element to excite a wave propagating on the nanowire. This design is difficult to fabricate and integrate, however, so instead we will consider a structure that can be fabricated with standard planar lithography techniques, namely a laterally tapered metal stripe waveguide. To understand how to realize a similar adiabatic coupling in this geometry, we need to find the mode in the metal stripe that transforms to the nanowire mode as the stripe narrows. taper theory first demonstrated by Stockman, PRL 93, (2004)

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**Delivering light to the nanoscale**

+ 1 µm |E| nanoscale confinement Field symmetry at tip similar to SPP mode in conical waveguide k E x z Optics Express 16, 45 (2008) Ewold Verhagen, Kobus Kuipers

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**Concentration of light in a plasmon taper: experiment**

Au Er Al2O3 λ = 1.5 μm Ewold Verhagen, Kobus Kuipers

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**Concentration of light in a plasmon taper: experiment**

60 nm apex diam. (1490 nm) Er3+ energy levels transmission 1 µm 10 µm PL Intensity (counts/s) lexc = 1490 nm Nano Lett. 7, 334 (2007) Ewold Verhagen, Kobus Kuipers

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**Concentration of light in a plasmon taper: experiment**

Detecting upconversion luminescence from the air side of the film (excitation of SPPs at substrate side) 550 nm 660 nm k E x z Plasmonic hot-spot Theory: Stockman, PRL 93, (2004) Optics Express 16, 45 (2008) Ewold Verhagen, Kobus Kuipers

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**FDTD Simulation: nanofocussing to < 100 nm**

sym asym Et, H |E|2 z = -35 nm 1 µm + E n1 = 1 1 µm n2 = 1.74 start tip Nanofocusing predicted: 100 x |E|2 at 10 nm from tip 3D subwavelength confinement: 1.5 µm light focused to 92 nm (/16) limited by taper apex (r=30 nm) Saturating scale The structure is excited by a SP mode at the substrate side of the film, just like in the experiment. When we look at the resulting field intensity in the plane of the Er ions, we see that SPs are indeed concentrated at the tip, to a dimension much smaller than the wavelength The width of this spot is 92 nm, and this number is limited by the rounding radius that we used to reflect the finite sharpness of the tip in the experiment. At a depth of 10 nm below the tip, the intensity is enhanced by a factor 100, which is btw not visible in this saturated color scale. Optics Express 16, 45 (2008) Ewold Verhagen, Kobus Kuipers

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**Coaxial MIM plasmon waveguides**

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**FIB milling of coaxial waveguides**

<w>=100 nm, L=485 nm <w>=50 nm, L=485 nm 100 nm 100 nm Silica substrates with nm thick Ag Ring width: nm Two-step milling process ~7° taper angle Nano Lett. 9, in press (2009) René de Waele, Stanley Burgos 18

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**Narrow channels show negative index**

Excitation above resonance, w>wsp 25 nm-wide channel in Ag filled with GaP Simulation shows negative phase velocity with respect to power flow Negative refractive index of -2 René de Waele, Stanley Burgos

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**Positive and negative index modes**

René de Waele, Stanley Burgos

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**Plasmonic toolbox: , (), d - Engineer ()**

Plasmonic integrated circuits Plasmonic multiplexer Plasmonic concentrator Plasmonic lens thin section And much more …..

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**Conclusions: surface plasmon polariton**

Surface plasmon: bound EM wave at metal-dielectric interface Dispersion: (k) diverges near the plasma resonance: large k, small Control dispersion: control (k), losses, concentration Manipulate light at length scales below the diffraction limit

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