Surface Plasmon?
 Propagating surface waves can exist between a metal and a dielectric.
 Although you won’t need to derive this, I will put the full derivation on Keats
for those interested - here, I will present the dispersion relationship as we’re
interested in the technology.
• First of all, what is the physical origin of these
surface waves?
Surface Plasmon?
 Well, we know that metals have free charges.
 Any wave propagating along the surface of a metal must therefore
exhibit strong interaction with these charges.
 Plasmons exist as longitudinal (along the surface) oscillations in
free charge density.
 This is best explained in a figure.
Surface Plasmons
Surface plasmons
 We normally consider the two polarisation
basis for light at an interface: the transverse
magnetic modes (TM-modes, or p-modes)
and the transverse electric modes (TEmodes, or s-modes).
𝜀1
𝜀2 y
x
z
 For TM modes, only the field components Ex, Ez, and
Hy are non zero.
 Whereas for TE modes, only Hx, Hz and Ey are non
zero
Surface plasmons
𝜀1
 Let’s remind ourselves of the different
between TM (p) and TE (s) polarisation.
ẑ
k
𝜀2 y
x
k
z
Surface plasmons
 Narrator: TE polarised modes are not supported.
 The dispersion relationship for a surface wave for TM
polarised surface wave is given as :
𝑘𝑠𝑝𝑝 = 𝑘0
𝜀1
𝜀2 y
𝜀𝑚 (𝜔)𝜀𝑑
𝜀𝑚 (𝜔) + 𝜀𝑑
 This is in the ‘classic’ form for a dispersion relationship
𝑘𝑠𝑝𝑝 =
𝑠𝑝𝑝
𝑛𝑒𝑓𝑓 =
𝜀𝑚 (𝜔)𝜀𝑑
𝜀𝑚 (𝜔) + 𝜀𝑑
𝑠𝑝𝑝
𝑘0 𝑛𝑒𝑓𝑓
x
z
For derivation,
see Keats
Surface plasmons
 So what does it look like?
𝝎𝒔𝒑
𝑘𝑠𝑝𝑝 = 𝑘0
𝜀𝑚 (𝜔)𝜀𝑑
𝜀𝑚 (𝜔) + 𝜀𝑑
Surface plasmons
 The SPP dispersion is ALWAYS on the “right-hand side” of the light
line (higher k): for all values of 𝜔 < 𝜔𝑠𝑝𝑝 , 𝑘𝑠𝑝𝑝 > 𝑐𝑘
 This means that (remember 𝒑 = ℏ𝒌) there
is a momentum mismatch between
incident light in the dielectric and the SPP
 An SPP CANNOT be excited by direct
illumination (without nanostructuring, as
we shall see)
Surface plasmons
 Visualising the momentum mismatch in the isofrequency surface in
the (𝑘𝑥 , 𝑘𝑦 ) plane
Surface plasmons
 We can have the complete dispersion by
adding the volume plasmons we obtained
before
 Within the model of an ideal free
electron plasma, the frequency range
between surface plasmon frequency
and plasma frequency represents a
gap where no propagating
electromagnetic mode exists.
We shall see how to find a simple expression
for ωspp in Problem sheet 2
𝜔𝑝
𝜔𝑠𝑝𝑝
Surface plasmons
 In real metals, damping modifies
the SPP dispersion curve and does
not allow for unlimited growth of
the wavenumber at 𝜔 → 𝜔𝑠𝑝
𝜔𝑝
𝜔𝑠𝑝𝑝
 However, 𝜔𝑠𝑝𝑝 remains the upper
frequency limit for surface
plasmons
S. A. Maier. Plasmonics:
Fundamentals and Applications
The transition between SPP and
VP is a region known as the
‘backbending’ region of the
dispersion.
Surface plasmons
 Let’s summarise SPPs?
 Surface waves, confined to the interface between a metal and a dielectric
material
 Commonly called Surface Plasmon Polaritons (SPP)


A plasmon is a quasi-particle that represents the quantum of the collective excitation of
free electrons in solids.
A polariton is a quasi-particle resulting from strong
coupling of electromagnetic waves with an electric or
magnetic dipole‐carrying excitation:
 phonon‐polaritons: coupling of photons with optic
phonons
 exciton‐polaritons: coupling of photons with an exciton
 Surface plasmon‐polaritons: coupling of surface
plasmons with light
Surface plasmons
 We can also define an SPP wavelength using the standard equation:
𝑘=
2𝜋
𝜆
 Therefore 𝜆𝑆𝑃𝑃 =
2𝜋
The EM field is confined to the interface, and
just like a surface wave on a dielectric, it
decays exponentially away from the interface.
𝜀𝑚 𝜀𝑑
𝑘0
𝜀𝑚 +𝜀𝑑
Dielectric
Interface
Metal
Surface plasmons
 Excitations of the conduction electrons of real metals however suffer
from damping. Therefore, 𝜀𝑚 (𝜔) is complex, and with it also the
SPP propagation constant 𝑘𝑠𝑝𝑝
 The intensity decays with the square of the electric field, so at a
distance x, the intensity has decreased by a factor of:
 𝑒 −2𝐼𝑚 𝑘𝑠𝑝𝑝 𝑥
The factor of 2 is coming
from squaring the field
 The distance over which the SPP intensity has dropped to 𝑒 −1 of its
original value is called the propagation length:
 L = 2 𝐼𝑚 𝑘𝑠𝑝𝑝
−1
Surface plasmons
• Of course, in the same way as we did for TIR in a prism in
Problem sheet 1, we can determine the penetration depth in
either the metal, or the dielectric medium.
• The wavevector in the z-direction is given by:
𝜀1
𝜀2 y
 𝑘𝑧 = 𝑖
2
𝑘𝑠𝑝𝑝
x
2
− 𝑘0 𝜀𝑑(𝑚)
𝜀𝑚 𝜀𝑑
 With 𝑘𝑆𝑃𝑃 = 𝑘0
𝜀𝑚 +𝜀𝑑
• We use either the dielectric or
metal permittivity as needed.
 We get the penetration depth as δd(m)= 2 𝐼𝑚 𝑘𝑧
z
−1
The wavevector in z is completely imaginary (it lies to the right of the light line).
Surface plasmons
 For example, for 𝜆0 = 600 nm:
 For gold-air, we have 𝛿𝑔𝑜𝑙𝑑 = 31 nm and 𝛿𝑎𝑖𝑟 = 280 nm
 For silver-air, we have 𝛿𝑠𝑖𝑙𝑣𝑒𝑟 = 24 nm and 𝛿𝑎𝑖𝑟 = 390 nm
Surface plasmons
 Recall the SPP dispersion: for 𝜔 < 𝜔𝑠𝑝 , 𝑘𝑠𝑝𝑝 > 𝑐𝑘
𝝎𝒔𝒑
At higher frequencies
(lower wavelength),
the gap between the
light line and SPP
dispersion increases.
The magnitude of kz is proportional to
this gap, as these are what we subtracted
on the previous slide!
So higher frequencies, larger kz,
smaller penetration depth!
Surface plasmons
 Recall the SPP dispersion: for 𝜔 < 𝜔𝑠𝑝 , 𝑘𝑠𝑝𝑝 > 𝑐𝑘
𝝎𝒔𝒑
We cannot excite these
modes on the surface of
a metal directly without
some additional way to
increase the
momentum of the
photons.
 So how do we excite those modes?
It’s also key to remember that
the real part of the SPP
wavevector (or propagation
constant) is along the
interface – so for momentum
matching, we need a large ‘inplane’ wave-vector.
Surface plasmons
 Optically we need to access large in-plane wavevectors for momentum
matching:
 Recall 𝑘 =
2𝜋
𝜆
=
2𝜋 𝜀𝑟
𝜆0
 But, if we increase the
dielectric constant next to
the metal surface, we also
move the SPP dispersion!
Surface plasmons
 Optically we need to access larger wavevectors
 Recall 𝑘 =
2𝜋
𝜆
=
2𝜋 𝜀𝑟
𝜆0
 But, we can still use this if we
apply ourselves and think it
through a little.
 We know that if we had a
photon in glass, it has enough
momentum to couple to an
SPP at an air interface.
Key point to remember:
We need this momentum in the
direction of the SPP (along the
interface).
So the lines on this plot would
represent an angle of incidence of
90 degrees!
Surface plasmons
𝝎𝒔𝒑 ( 𝜀𝑟 = 1)
𝝎𝒔𝒑 ( 𝜀𝑟 = 1.5)
 Optically we need to access larger wavevectors
 Recall 𝑘 =
2𝜋
𝜆
=
Air
2𝜋 𝜀𝑟
𝜆0
Metal
 This is the situation before us
on the right.
 We have put a metal layer on a
glass substrate.
 We now have two SPP modes,
one at the metal/air side, and
one on the metals.
Glass
Surface plasmons
𝝎𝒔𝒑 ( 𝜀𝑟 = 1)
𝝎𝒔𝒑 ( 𝜀𝑟 = 1.5)
 Optically we need to access larger wavevectors
 Recall 𝑘 =
2𝜋
𝜆
=
Air
2𝜋 𝜀𝑟
𝜆0
Metal
 So, if we shine light from the
glass side, we should be able to
reach the necessary
momentum for the SPP on the
air side.
 However, if we use a planar
glass substrate, we still have a
problem.
Glass
Surface plasmons
Air
 Recall: we need the momentum in the plane of
the surface.
Metal
 But we know refraction conserves in-plane
momentum.
The x component of k is the same in the
air beneath the glass, as it is in the glass!
nglass koSinθglass=nair koSinθ
Glass
Surface plasmons
 Optically we need to access larger wavevectors
 Recall 𝑘 =
2𝜋
𝜆
=
2𝜋 𝜀𝑟
𝜆0
 So, if we use a planar substrate,
the refraction into the glass
from the air can only maintain
the in-plane momentum from
the air (blue dashed line).
 Back to square one!
Surface plasmons
 However, if we use a metal layer on a prism (as we showed for TIR)
then we solve the refraction problem at the air/glass interface.
 This idea gave rise to the Kretshmann geometry shown below!
A.V. Zayats et al., Physics Reports 408 (2005) 131–314
Surface plasmons
 Now, thanks to the prism, our incident light has access to the region
highlighted in peach below.
 Here we want to match the in-
plane component of
our incomping light, 𝒌:
 𝑘∥ =
2𝜋 𝜀𝑟
sin 𝜃
𝜆0
 We need to tune 𝜃 at a
particularly frequency or
wavelength, to overlap with 𝑘𝑠𝑝𝑝
𝑘∥
𝑘∥
Surface plasmons
 Matching the wavevectors for a given wavelength, we must satisfy the
following condition:
 𝑘𝑔𝑙𝑎𝑠𝑠 sin 𝜃= 𝑘𝑠𝑝𝑝
 𝑘0 𝜀𝑔𝑙𝑎𝑠𝑠 sin 𝜃 = 𝑘𝑠𝑝𝑝
 So
2𝜋 𝜀𝑔𝑙𝑎𝑠𝑠
𝜆0
sin 𝜃 = 𝑘𝑠𝑝𝑝
Solving the above equation we can
find the angle of incidence required
in the glass substrate!
Surface plasmons
 So let us look at an example: silver-air interface
𝑘𝑠𝑝𝑝 𝜆0
 So 𝜃𝑆𝑃𝑃 = arcsin
2𝜋 𝜀𝑟3
𝜀𝑚 𝜀𝑑
 with 𝑘𝑆𝑃𝑃 = 𝑘0
𝜀𝑚 +𝜀𝑑
 Here we only need to
consider the real part
of 𝑘𝑠𝑝𝑝
𝜀𝑟2
𝜀𝑟1
= 𝜀′ + 𝑖𝜀′′
𝜀𝑟3
Surface plasmons
 We consider the following:
 𝜆0 = 980 𝑛𝑚
 𝜀𝑑1 = 1 (air)
 𝜀𝑚 = −48.474 + i 0.55699
 𝜀𝑑2 = 1.52 (standard for glass)
Notice how
much of
the light
goes to the
SPP!
Heinz Raether, Surface plasmons on smooth and rough
surfaces and on grating, Springer
 This gives us:
 𝑘𝑠𝑝𝑝 = 6.4786 × 106 + 7.8392 × 102 𝑖
 Which corresponds to an internal incident
angle of 𝜃𝑆𝑃 = 42.3494°
Surface plasmons
 Keeping exactly the same parameters and
changing only 𝜀𝑟1 , we obtain:
 𝜀𝑑1 = 1, 𝜃𝑆𝑃 = 42.3494°
 𝜀𝑑1 = 1.1, 𝜃𝑆𝑃 = 45.0137°
Notice that this is a ‘resonant’ condition – and
this condition (via Kspp) varies as the
dielectric constant of the top interface.
Contrast this with a dielectric prism
evanescent wave, which is not resonant (exists
at all wavelengths at angles above TIR)
 𝜀𝑑1 = 1.01, 𝜃𝑆𝑃 = 42.6160° (this is n=1.005!)
 This is the principle behind SPR (Surface Plasmon
Resonance) sensing – which we shall meet later!
Surface plasmons - Nanostructuring
 Rather than using a prism, which is cumbersome, we can
use nanostructuring to couple light directly to SPPs.
 To see how we can do this, let’s consider a diffraction
grating and find the grating equation you learned in first
year!
 Just like refraction, this is derived by considering the in-
plane components of the wavevector, and the
wavevector of the grating!
Surface plasmons - Nanostructuring
We can describe the wavevector of a grating thus:
2𝜋
𝑘𝑔 =
𝑑𝑔𝑟𝑎𝑡𝑖𝑛𝑔
Where dgrating is the line spacing on the grating.
So, considering the conservation of in-plane momentum:
𝑘𝑆𝑖𝑛𝜃𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡 ± 𝑚𝑘𝑔 = 𝑘𝑆𝑖𝑛𝜃𝑑𝑖𝑓𝑓𝑟𝑎𝑐𝑡𝑒𝑑
Where m is an integer describing diffraction order (m=0, 1,2 𝑒𝑡𝑐. )
𝑛𝑘0 𝑆𝑖𝑛𝜃𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡 ± 𝑚𝑘𝑔 = 𝑛𝑘0 𝑆𝑖𝑛𝜃𝑑𝑖𝑓𝑓𝑟𝑎𝑐𝑡𝑒𝑑
(𝑛𝑆𝑖𝑛𝜃𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡 + n𝑆𝑖𝑛𝜃𝑑𝑖𝑓𝑓𝑟𝑎𝑐𝑡𝑒𝑑 )𝑑𝑔𝑟𝑎𝑡𝑖𝑛𝑔 = ±𝑚𝜆0
At normal incidence and with n=1 (air) we have the familiar equation:
𝑑𝑔𝑟𝑎𝑡𝑖𝑛𝑔 𝑆𝑖𝑛𝜃𝑑𝑖𝑓𝑓𝑟𝑎𝑐𝑡𝑒𝑑 = ±𝑚𝜆0
Surface plasmons - Nanostructuring
 So we can use a grating to couple to SPPs, i.e. we need to periodically
nanostructure the metal surface.
2𝜋
𝑑
𝑛1
 𝑛1 𝑘0 sin 𝜃 ± 𝑚
𝑧
= ±𝑘𝑆𝑃𝑃
𝜃
𝑥
𝑘𝑖𝑛 𝑥
𝑘𝑔𝑟𝑎𝑡𝑖𝑛𝑔
𝑘𝑆𝑃𝑃
 kgrating adds (and substracts) from kspp
 This is best illustrated by considering a dispersion plot.
SPPs on periodically nanostructured surfaces
Original dispersion
𝜔
Bragg scattering of the dispersion
𝜔
We’ve drawn the negative
side too, as an SPP can go
either direction along the
interface
k
-2p/d
-p/d
0
p/d
• The grating shifts the SPP dispersion by multiples of kgrating (dashed lines)
• Remembering that the wavevector is in-plane for SPPs, so normal incidence
corresponds to k=zero.
• We can clearly see in the plot (above right) that the curves cross zero. This means
that we can excite SPPs at normal incidence, with the grating providing all of the
momentum in the plane of the surface.
2p/d
k
Periodic nanostructures in nature
 Even periodic dielectric structures in nature show amazing
visual effects via diffraction.
So do periodic
nanostructures
on metallic
films!
We shall see some
methods for making
these nanostructures
later!
Periodic nanostructures in nature
We shall see some
methods for making
these nanostructures
later!
So do periodic
nanostructures
on metallic
films!
10 µm
SPPs – Gold Air Interface
 An example, a grating with a 600 nm period on gold.
Au
SiO2
Here, the bright
colours represent
more
transmission, the
dark colours less.
We can also
couple to the
modes on the
glass/Au side –
impossible using
a prism!
Here are some experimental results for a gold film on glass.
The transmission is measured through the nanostructured area (right figure)
The dark bands correspond to ‘missing light’ which are SPPs!
SPPs Summary
 Light can couple to longitudinal resonant electron oscillations at the interface between a
metal and a dielectric, these are termed Surface Plasmon Polaritons.
 As they are confined to the interface, the lie to the right of the light line in the dispersion plots.
 This also means that we need to either use a prism or a periodic nanostructure to provide incident
light with the necessary momentum to couple to SPPs.
 For a given metal (and therefore permittivity) higher frequencies have larger wavevectors,
this means that the SPP is more confined to the interface.
 The penetration depth into the dielectric can easily be λ0/4 and higher.
 However, this will also come at the expense of more losses in the propagation direction
(travels shorter distances)!
• The SPP will also have higher wavevectors if the dielectric
has a higher dielectric constant (slide 29) so light is
confined even more tightly to the interface.
SPPs Summary
 SPPs can provide nanoscale electromagnetic fields on metal/dielectric interfaces.
 As they allow light to be squeezed, they have potential to become optical chip interconnects with
high integration density.
 The challenge is minimising the attenuation of the signal. (prop length)
 SPPs are resonant, and therefore very sensitive to the dielectric constant close to the surface
of the metal film – any variation close to the surface will change the SPP coupling condition
(angle required at some wavelength, or the wavelength required at some angle).
 This means the surface is an incredibly sensitive sensor.
 Any process which modifies the refractive index close to the surface can be detected.
• The incident light can couple very efficiently to the SPP, and
as it is squeezed (high wavevector) the local EM field at the
surface can be many times higher than the incident field.
SPPs Summary
 This can be incredibly useful for any process that depends on intensity.
 Fluorescence
 Photocatalysis
 Raman Scattering – Inelastic optical scattering which provides a spectra fingerprint
of molecules.
 Optical forces e.g. optical tweezers for manipulating nano-objects, (force
proportional to intensity gradients)
 Nonlinear optics – materials where the refractive index has a contribution
proportional to intensity (can make active optical active devices).