1 Scattering and Absorption Cross-sections of a Metal Nano-particle
Consider a spherical metal nanoparticle (NP) of radius a and dielectric constant NP in air ( 1 = 1) as an oscillating dipole. Using the expression of the effective polarizability with radiative corrections for the NP, i.e.,
α= α0
1−ık3 6πα0
, (1)
whereα0 is the quasistatic polarizability, i.e.,
α0= 4πa3 NP−1
NP+ 21
. (2)
• Calculate the scattering and the absorption cross sections of the NP. The cross section (σ) is defined as the scattered and absorbed power (Psca,abs) divided by the incident intensity Iinc, i.e.,σsca,abs =Psca,abs/Iinc.
Consider a spherical metal nanoparticle (NP) of radiusa and dielectric constantNP in air (1 = 1) as an oscillating dipole. Using the expression of the effective polarizability with radiative corrections for the NP, i.e.,
α= α0
1−ik6π3α0, (3)
Whereα0 is the quasi-static polarizability, i.e., α0 = 4πa3 NP−1
NP+ 21, (4)
derive the optical properties of the nanoparticle are based on the polarizability model.
If a plane wave is incident on the nanoparticle, the response can be written as an oscillating dipole~p given by the expression
~
p=oα ~Einc. (5)
According to the optical theorem, the extinguished powerPextis the sum of scatteredPsca and the absorbedPabs powers, i.e, Pext=Psca+Pabs. Using the well-known expressions for the radiation and absorption of electromagnetic energy by an oscillating dipole, i.e.,
Psca = p2ω4
6π0c3, (6)
Pabs = ωIm{~p·E~inc}, (7) The correspondingscattering cross sectionscan be written as
σsca = Psca Iinc
=⇒ k4
6π|α|2×(0cZ) (8)
Where0√
0µ0 ×
0 = 1,hence we can write :
σsca = k4
6π|α|2, (9)
Thetotal power dissipatedcan be written as:
Ptot =ω Im [P . ~~ Einc
∗] =ω Im [0 α E~inc .E~inc
∗] (10)
Hence,
Ptot =ω 0 |E~inc|2 Im{α} (11) Now,Iinc =0c|Einc|2is the incident intensity, hence thetotal scattering cross-section will be as follows:
σtot = Ptot
Iinc =⇒ ω 0|Einc|2 Im{α}Z
|Einc|2 (12)
0.Z=0. rµ0
0 = 1
c (13)
σtot =kIm{α}, (14)
Whereσtot=σsca−σabs.
kIm{α}= k4
6π|α|2−σabs (15)
α= α0
1−ik6π3α0, (16)
Where,Z = 1−ik6π3α0
α= α0
1−ik6π3α0
, =⇒ |α|2= |α0|2
1 +(6π)k62|α0|2 (17) We get:
Im{α}= Im{α0 Z ×Z∗
Z∗} (18)
=⇒ Im{α0.Z∗} 1
|Z|2 (19)
=⇒ 1
1 +(6π)k62|α0|2 ×Im{α0.(1 + 1k3
6π )α0∗} (20) By solving this equation we get:
kIm{α}= k4
6π|α|2+ kIm{α0}
1 +(6π)k62|α0|2 (21) Here:
kIm{α}=σtot (22)
k4
6π|α|2=σsca (23)
and,
kIm{α0}
1 +(6π)k62|α0|2 =σabs (24)
• Show that the sum of the scattered and absorbed power is equal to the extinguished power, i.e, that σext = σsca+σabs, where σext is the extinction cross section and the extinguished powerPext is the power removed by the NP in the propagation direction of the incident plane wave.
According to thescattering theory we have,
S~tot=S~inc+S~sca+S~int (25) Where,
S~int= 1
2 Re{E~sca×Hinc∗ }+1
2 Re{E~inc×Hsca∗ } (26) ˆ
4π
S~tot ~n.r2dΩ =−Pabs (27) ˆ
4π
S~inc ~n.r2dΩ = 0 (28) ˆ
4π
S~sca ~n.r2dΩ =Psac (29)
−Pabs=Psac+ ˆ
4π
S~int ~n.r2dΩ (30) Hence we get:
Psac+Pabs=− ˆ
4π
S~int~n.r2dΩ (31)
=⇒ (σabs+σsca)Iinc=σtot.Iinc (32) We know that,
σtot =kIm{α} (33)
Whereα fulfills the optical theorem.
− ˆ
4π
S~int~n.r2dΩ =Pext (34) Here Pext is the extinguished power. The extinguished scattering cross-section can be written as: σext =σsca+σabs.
2 Surface Plasmon-Polariton Waves
• Briefly describe the conditions for the excitation and observation of surface plasmon- polariton (SPP) waves.
A surface plasmon-polariton (SPP) corresponds to a coherent oscillation of electrons at the interface between a metal and a dielectric medium. These are essentially electro- magnetic waves that are trapped on the surface, because of their interaction with the free electrons of the conductor. The first observation of SPPs was made by A. Otto in 1968 and later the experimental setup was improved in 1971 by E. Kretschmann. To couple light to the SPP mode, energy and momentum conservation must be maintained.
Therefore, the incoming beam has to match its momentum to that of the SPP. In the case of p-polarized light (polarization occurs parallel to the plane of incidence), there are three main techniques by which the missing momentum can be provided. The first makes use of a prism to enhance the momentum of the incident light. The second involves scattering from a topological defect on the surface, such as a subwavelength protrusion or hole, which provides a convenient way to generate SPPs locally. The third makes use of a periodic corrugation in the metal’s surface. S-polarized light (polarization occurs perpendicular to the plane of incidence) cannot excite SPPs.
Figure 1: (a) Otto configuration. (b) Kretschmann configuration.
Otto and Kretschmann configurations are the two well-known simple methods to achieve coupling using a prism. In the Otto setup, light illuminates the wall of a prism and it undergoes total internal reflection. An air gap (or a spacer of low index), less than a few optical wavelengths thick, provides the tunnel barrier across which radiation couples, via the evanescent component of the totally reflected wave, to SPPs at the air (dielectric) metal interface (e.g. gold), as shown in Fig??(a). By varying the angle of incidence of p-polarized radiation at the prism/dielectric interface, one can vary the momentum in the propagation direction and this allows for simple tuning through the coupling condition, called surface plasmon-polariton resonance (SPR). The other alternative and simpler ge- ometry was realized by Kretschmann. Rather than using a dielectric spacer the metal itself could be used as the evanescence tunnel barrier provided it is thin enough to allow radiation to penetrate to the other side. All that is now needed is a prism with a thin coating of some suitable metal, as shown in Fig. ??(b).
Figure 2: Schematics of a SPR-based sensor.
The sensing activity can be performed once efficient coupling is obtained. A schematic of a SPR-based sensor is depicted in Fig. ??. A glass prism is coated with a thin layer of a noble metal (or brought near to using Otto configuration) to create a biosensor sur- face. Antibody or another type of biomolecules (chemicals) immobilized on the sensor surface is also shown. At a particular angle (θo) the incoming wave couples to SPP. This special incidence angle is called SPR angle. At the SPR angle reflectivity hence drops to a minimum as shown in the figure (right). However, the binding of biomolecules onto the sensor’s surface affects the SPR condition: the refractive index of the dielectric com- ponent increases and shifts the SPR angle to another value (θ1). The measurement of the SPR-angle shift offers an opportunity to measure molecular binding to the sensor’s surface or other physical quantities (e.g. index of refraction).
• For a medium characterized by a complex frequency-dependent dielectric function(ω) = 1− ωp2
ω(ω+iγ), whereωp = 2π×2.068×1015rad/s andγ = 2π×4.449×1012rad/s derive an approximate expression for the propagation length of the SPP wave as a function of ω.
Propagation length of SPP waves. For a medium characterized by a complex frequency- dependent dielectric function,m(ω) = 1− ω2p
ω(ω−iγ), whereωp = 2π∗2.068∗1015rad/s and γ = 2π ∗4.449∗1012 rad/s derive an approximate expression for the propagation length of the SPP wave as a function ofω.
Becausem is complex there must e absorption, i.e. the intensity drops as a function of distance according toI(z) = I0e−αz, where α= L1 is the absorption coefficient andL is the propagation length. Assuming for simplicity d = 1, the dispersion relation of the SPP wave reads
β=βr+iβi= ω c
s
m(ω)
m(ω) + 1. (35)
Moreover, ifγ ω, we can write the expression for m(ω)as m(ω) = 1− ω2p
ω(ω+iγ) = 1−ωp2 ω2
1
1 +iωγ '1−ω2p ω2
1−iγ
ω
, (36)
where the imaginary part γ/ω in the equation corresponds to the absorption losses. By writingr(ω) = 1−ω2p/ω2, the dispersion relation of the SPP wave becomes
βr+iβi= ω c
s
r(ω)(1−iγ/ω) r(ω)(1−iγ/ω) + 1 = ω
c s
r(ω)(1−iγ/ω)
r(ω) + 1−ir(ω)γ/ω. (37) By rearranging the parameters, we can write,
βr+iβi=βr v u u u t
1−iγ/ω 1−i r(ω)
r(ω) + 1γ/ω 'βr
1−i γ
2ω 1 r(ω) + 1
, (38)
whereβr= ω c
s
r(ω)
r(ω) + 1. Considering thatr(ω) + 1<0, we get expression βi=− γ
2ω βr
r(ω) + 1 = 1
2L. (39)
Hence the propagation length as a function ofω can be written as L=−ω
γ
r(ω) + 1 βr
=−c γ
(r(ω) + 1)32
pr(ω) . (40) Replace r(ω) by 1−ω2p/ω2 and we get the final expression for the propagation length
Figure 3: Propagation length of SPP waves as a function ofω as follows,
L= c γ
(2−ωωp22)32 q
1− ωωp22
. (41)
3 References
• Principles of Nano-Optics (Second edition) by Lukas Novotny
