1. Penetration Depth & Reflectivity
For w<wp (“low” frequency regime)
The skin depths of some common metals at a frequency of 10 GHz (microwave region) are less than a micrometer:
Thus at microwave frequencies, most of the current flows in an extremely thin region near the surface. Ohmic losses of waveguides at microwave frequencies are therefore only dependent on the surface coating of the material. A layer of silver 3 mm thick evaporated on a piece of glass is thus an excellent conductor at such frequencies.
In copper, the skin depth can be seen to fall according to the square root of frequency:
Conductor
Skin depth (μm)
Aluminum
0.80
Copper
0.65
Gold
0.79
Silver
0.64
Frequency
Skin depth (μm)
60 Hz
8470
10 kHz
660
100 kHz
210
1 MHz 66
10 MHz 21
100 MHz
6.6
At low frequency, penetration decreases with frequency!

2. Penetration Depth & Reflectivity
For w>wp (“High” frequency regime)
Real dielectric function  no attenuation  transparency
No attenuation
independently
of the frequency
n is close to 1–  R=0 and T=1
Subtle point here!
It should also be noted that we are only specifying the decay of the field which may be due to absorption of the electromagnetic energy in a dissipative “lossy” medium or may simply describe the penetration of the field in a medium where no loss occurs (or a combination of the two).
For instance, a hypothetical substance may have a complex index of refraction 
A wave will enter that medium without significant reflection and will be totally absorbed in the medium with a penetration depth (in field strength) of ,
where  l  is the vacuum wavelength. A different hypothetical material with a complex index of refraction will also have a penetration depth of 16 wavelengths, however, since n=0, R=1 hence in this case the wave will be perfectly reflected from the material! No actual absorption of the radiation takes place, however the electric and magnetic fields extend well into the substance. In either case the penetration depth is found directly from the imaginary part of the material's refractive index as is detailed above.

3. Surface Reflectivity and surface plasmons
Novotny Ch. 12
Lüth Ch. 5
Metals look grey until intra-band transitions are considered. If not:
Inter-band transitions
Gold absorbs green/blue photons (2.75 eV),
And reflects yellow ones.
Copper absorbs green photons (2.1 eV),
and reflects orange/red ones.
Gold (Fig. 12.2): incident light at l=450 nm (blue) has a d.f.:
reflected blu light
transmitted (and absorbed) blu light
These are bulk properties…

4. Bulk phonon polariton:
ions oscillate in an
elastic regime under the
action of an external
electromagnetic field
“coupling” between
phonons & e.m.-field
oscillating dipole momentum
phonon susceptibility
reset of the dielectric constant
plotted in next slide
DC field dielectric constant
longitudinal oscillation

5. Bulk phonon polaritons
within the optical gap
e is real and negative!
Reflectivity

6. VACUUM-DIELECTRIC INTERFACE
Note:
The so called “evanescent fields”
are due to surface polaritons.
These fields are also “used” in
scanning near-field optical microscopy (“SNOM”) to scan and determine surface morphology
by analysis of the changes
induced by surface fields on
the “enhanced” fields of a tip.

7. Total internal reflection from inside dielectric
VACUUM-DIELECTRIC
INTERFACE
No solutions
These solutions
are partly reflected
waves
Evanescent waves
inside the dielectric
(<0, high reflectivity) w
Ibach Ch. 7
Total internal reflection from inside dielectric

8. Quasi-particle (phonon/polariton) creation/annihilation
Figure 4.27, curve 1 illustrates the micro-Raman spectrum from a bulk n-GaP crystal. Curves 2 and 3 were obtained from the samples anodized at current densities of 5 and 15 mA/cm2. The as-grown crystal exhibits only one Raman band centered at 404 cm-1, which corresponds to the Brillouin-zone-center LO phonon. The q = 0 TO phonon is forbidden in (100) geometry and therefore it is absent in the bulk GaP spectrum. The anodization process leads to a small downward frequency shift and broadening by 15 to 20 % of the LO-phonon band accompanied by a significant Raman signal intensification (up to 5–6 times). A complete breakdown of the polarization selection rules is reflected in Figure 4.27 by the appearance of a strong TO-phonon band. Like the LO phonon, the TO phonon also exhibits a downward frequency shift with increasing the anodization current.
wq q
Incident light
(ki, wi)
Scattered light
(ks, ws)
Quasi-particle (phonon/polariton)
creation/annihilation
wq≡Dw=wi-ws, q=ki-ks
In technical context, usually frequency is expressed in
m-1 or more often cm-1, i.e. through the corresponding
reduced wavevector k=k/2p unit of measurement:
Raman: inelastic scattering
of photons on surface: the reflected photons have different energy and k, and the shift is due to creation/annihilation of
excitations with energy the energy shift, and wavevector
the wavevector shift

10. (SEMI)METAL-DIELECTRIC
INTERFACE
Plasmonic dielectric function
Metal/Dielectric interface (surface) dispersion relation
Note: w+, in both limits, would
give a dispersion of the kind:
w=vk//, with v>c (physical nonsense)
wp is at 1.0 in ordinate!
(silver)
Lüth: case of metal/vacuum interface (e1=1)
Free electron

11. (SEMI)METAL-DIELECTRIC
INTERFACE
Dissipation included
Surface Plasmon Polariton electric field

12. Surface polariton dispersion for free electrons + inter-band transitions

13. Electron energy loss spectra
see below for basics on the technique.
Inelastic scattering of low energy e-
(20÷200 eV) on crystal surface:
the scattered e- energy loss is due to
creation of a quantum of excitation ω
either phonon or plasmon polaritons
Scattering cross section (event probability):
i.e. large peaks (“structure”) in the spectrum
Fuchs-Kliewer SPhP
Red: surface-plasmon-polariton (DE=40 meV)
extremely sensitive to doping concentration and exposure
Blue: surface-phonon-polariton (DE=37 meV)
not sensitve to doping concentration and exposure

14. Loss peak position Phonon-like With varying doping
with the GaAs dielectric function this relation
becomes a quadratic equation, which provides
two “hybrid” solutions plotted in the graph
dashed line in the graph is:
Plasmon-like
With varying doping
or adsorbate density
we have that:
nphonon is constant
nelectron varies
Phonon-like
ideal phonon-like curve, independent of
electron concentration n
Phonon-like
Plasmon-like
(semi-Log scale!)

15. EELS and HREELS
Lüth Panel IX (Ch. 4)
(Ch. 4 Dielectric Theory
EELS: primary energy of the electron beam: 20÷500 eV (but <200 eV for surf. analysis)
HREELS: (High Resolution EELS): energy below 20 eV

17. Scattering on adsorbed atoms
In order to have momentum conservation K=Dk = kmode
The activated/destroyed mode of the adsorbate must possess a wavevector parallel to Dk.
If e.g. bending oscillations are involved, the scattering is off-specular:

18. Surface plasmon polariton excitation
Novotny Ch. 12

19. Sensitivity to adsorbates
SPP reflectivity curves
Sensitivity to adsorbates

21. Confinement effects in nanogold particles (size dlA)
Localized plasmons
Confinement effects in nanogold particles (size dlA)
Novotny Ch. 12
Electric field distribution
for a
nanogold/dielectric system
The e.m. field can penetrate the nanogold particle and make it “resonating” as an antenna (at w=wSPP) re-irradiating a scattered field only thanks to free electron “oscillations” i.e. plasmon polaritons, (even if these oscillations are quasi-static, and the field is uniform inside the particle).
Uniform (!)
internal field
Scattered field

22. Estinction(a)=Absorption-a3+Scattering-a6
“a” being the particle radius
Lycurgus’cup

23. DIATHRETE CUP