Lecture 4: Electromagnetic Waves III - Diffraction and Scattering in Nanophotonics | Study notes Electrical and Electronics Engineering

Lecture 4 - Electromagnetic Waves III 15

EECS 598-002 Nanophotonics and Nanoscale Fabrication Winter 2006, P.C.Ku

4. Electromagnetic Waves III

4.1. Diffraction and Scattering

In the previous sections, we have discussed the generation of EM waves and their propagation through a

homogeneous medium. We have also discussed how the interface between two semi-infinite media

interacts with the waves. In this section, we will continue our discussions to the interaction of EM waves with

a nanoscale structure. We consider two general cases, the diffraction and the scattering. Diffraction is one

type of scattering in which the EM wave is scattered off from a small aperture. From our previous

discussions, the details of the inter-atomic and atomic-EM wave interactions can be lumped into a

polarization vector (after ignoring higher order terms) as in (2.7). This process can be thought of as an

ensemble of dipole moments being induced by the incoming EM wave and a new EM wave is generated

from those dipole antennas. This picture, rigorous speaking, is not quite correct as the induced EM wave

can in turn interact with the medium itself and generate new dipole moments. The process will iterate over

and over for an infinite amount of times. But in most cases we can still use the above picture (that is to

truncate the iteration process) to get a good approximation.

4.1.1. Scattering by a Small Sphere (Rayleigh scattering)

We consider the scattering of an EM wave by a small sphere. If the sphere is small enough (still > 10 nm but

much smaller than the wavelength), we can approximate the induced polarization vector as an infinitesimal

dipole moment. The scattered wave is the radiation generated by this dipole antenna as given by (3.26). To

determine the dipole moment, we match the boundary conditions for the fields at the surface of the sphere

(i.e. the interface between two macroscopic media and in this case the air and the sphere). Assume the

incident wave is a plane wave given by:

ˆˆ

as 1

ikx

ii i

EzEe zE kx=≈

 (4.1)

The fields (including the field inside the sphere and the scattered field) near the origin of the sphere (i.e. the

location of the induced dipole moment) should have the near field nature, i.e. quasi-static. Because there

are no boundaries inside the sphere and the field inside the sphere satisfies the Poisson equation, the

solution is a plane wave polarized in the same direction as the incoming wave:

(

)

ˆcos sin

tt t

EzE r E

θθ θ

== −

(4.2)

The scattered field is given by (3.29):

1ˆˆ

ˆˆ

() 2cos sin 2cos sin

iIl a

Er r r E

θθ θ θθ θ

πωε

⎛⎞ ⎛⎞

⎡⎤⎡⎤

=+≡+

⎜⎟ ⎜⎟

⎣⎦⎣⎦

⎝⎠ ⎝⎠

KK (4.3)

To match the boundary condition at the surface of the sphere, we demand:

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4. Electromagnetic Waves III

4.1. Diffraction and Scattering

Et = zE ˆ^ t = ( ˆ r cos θ −θˆ sinθ) Et

K (4.2)

πωε r^ θ^ θ^ θ θ^ θ^ θ r

= ⎛⎜^ ⎞⎟^ ⎡^ + ⎤^ ≡ ⎡^ + ⎤⎛⎜^ ⎞⎟

⎝ ⎠ ⎣^ ⎦^ ⎣^ ⎦⎝ ⎠

K K (4.3)

E E

D D

E E E

ε E ε E ε E

E = ε^ ε +− εε E (4.6)

Il = − i π ka με ⎛⎜^ ε^ ε +−εε⎞⎟ E

θ^ ε^ ε θ θ

= − ⎛^ − ⎞ ≡

K K (4.8)

H^ K^ ( ) r K^ = ϕˆ^ με E θ≡ϕˆ H ϕ (4.9)

S = E × H ⋅ ds = π^ με ⎛⎜^ k a E ε^ ε +−εε⎞⎟

K K K (4.10)

σ= π⎛⎜^ k a ε^ ε +−εε⎞⎟ ∝ω a

In a special case when ε s = − 2 ε, the scattering cross section goes to infinity. Rigorously speaking, if the

nonetheless, it tells us a there is a resonance when ε s = − 2 ε. This again corresponds to the surface

= ∆^ − (4.17)

propagation constant kz via (4.13). The plot of ω vs k z is called the dispersion relation.

RHS LHS