Mie theory explains light scattering of spherical particles of size approximately equal to or larger

1. Mie theory of light scattering
R.Gandhimathi

2. During light matter interaction, charge excitation and re-radiation of the
electromagnetic energy happens
Accelerated charges radiate electromagnetic energy in all directions
Redirection of radiation from the actual propagation direction is called
scattering
Also, the excited elementary charges absorb and transform some part of the
incident energy into other forms, i.e. heat energy. This is called absorption.
Both scattering and absorption remove energy from a beam of light and the
beam is attenuated.
Scattering and Absorption

3. Mie theory
.
:

4. Mie theory
▪ Gustav Mie provided the mathematical description for the spectral
dependence of the scattering by a spherical nanoparticle
▪ Describes the scattering of an electromagnetic wave by a homogeneous
spherical medium having refractive index different from that of the
surrounding medium through which the wave is traversing
▪ A definite solution to the Maxwell's equations for the multipoles
(dipole, quadrupole, octupole) radiation due to the electric polarization
of the scattering particles
▪ Mie solution to Maxwell’s equations provides the basis for measuring the
size of particles through the scattering of electromagnetic radiation

5. ▪ Mie theory of the scattering and absorption of electromagnetic
radiation by a sphere is developed in order to understand the colors of
colloidal gold particles in solution
▪ The approach is used to expand the internal and scattered fields into a
set of normal modes described by vector harmonics
▪ Actually, it expands the incident and the scattered wave into spherical
waves and matches the boundary conditions at the particle interface
(Fresnel) to solve wave equation

6. Mie theory
.
:

7. ▪ The field pattern of propagating electromagnetic waves is knows mode of
electromagnetic radiation
▪ Transverse mode: Electromagnetic field pattern of the radiation in the plane
perpendicular to the propagation direction
▪ Longitudinal mode: Electromagnetic field pattern of the radiation in the plane
parallel to the propagation direction
Electromagnetic Modes

8. Let us consider a homogeneous, isotropic sphere of radius R located at the origin
in a uniform, static electric field E = E0 ˆz
The surrounding medium is isotropic and non-absorbing with dielectric constant
εm, and the field lines are parallel to the z-direction at sufficient distance from
the sphere
With the electrostatic or Quasistatic approach, the calculated electric field is
E = −∇.
Wave vector K=0
2=0Laplace equation
Modes of Subwavelength Metal Particles
-Electric potential

9. ( 1)
0
( , ) [ ] (cos )l l
l l l
l
r Ar B r P  

− +
=
= +
Pl (cos) -Legendre Polynomials of order l
θ -angle between the position vector r at point P and the z-axis
Solution for the potentials in and out the sphere can be written as
B1 is determined at the boundary
condition r →∞
out → −E0z = −E0r cos θ
B1 = −E0 and for0lB = 1l 
The electric potential in terms of polar coordinates
0
( , ) (cos )l
in l l
l
r Ar P  

=
= 
( 1)
0
( , ) [ ] (cos )l l
out l l l
l
r B r C r P  

− +
=
= +

10. Al and Cl are defined by the boundary conditions at r = a
1 1in out
r a r aa a
 
 = =
 
− = −
 
equality of the normal components of the displacement field
Equality of the tangential components of the electric field
0 0
in out
m
r a r ar
 
   
= =
 
− = −
 
Application of these boundary conditions leads to Al = Cl = 0 for l  1,

11. 0
3
cos
2
m
in
m
E r

 
 
−
=
+
out is given by Applied field + dipole located at the particle center
0 3
0
.
cos
4
out
m
P r
E r
r
 
 
= − +
3
0 0 2
cos
cos
2
m
out
m
E r E a
r
  
 
 
−
= − +
+
3
0 04
2
m
m
m
P a E
 
 
 
−
=
+
Dipole moment
3
4
2
m
m
a
 
 
 
−
=
+
Polarizability
p = ε0εmE0,

12. 0
0 3
0
3
2
3 ( . ) 1
4
m
in
m
out
m
E E
n n p p
E E
r

 
 
=
+
−
= +
2
4
2 4 68
6 3 2
m
sca
m
k
C k a
 

  
−
= =
+
  3
Im 4 Im
2
m
abs
m
C k ka
 
 
 
 −
= =  
+ 
There is a resonant
enhancement in both the
internal and dipolar fields and
the nanoparticle acts as an
electric dipole, resonantly
absorbing and scattering
electromagnetic fields
For smaller particles, the efficiency of absorption, scaling with a3,dominates
over the scattering efficiency, which scales with a6
Scattering and absorption cross sections

13. ▪ In this context, the phase of the resonant oscillating electromagnetic
field is constant over the particle volume and the spatial field
distribution can be calculated by assuming a particle in an electrostatic
field
▪ If the field distributions are known, the harmonic time dependence can
be added to the solution
By solving the Laplace equation with the use of an expansion in
spherical harmonics of order l, the polar modes of a spherical
metallic particle is given by
2 1
l p
l
l
 =
+

14. ▪ It describes the optical properties of nanoparticles of dimensions
below 100 nm adequately for many applications
▪ Thus in the QSA, when the particle size is much smaller than the
wavelength, the size of particle doesn’t play any role and the metallic
nanosphere exhibits just a single LSPR which is dipolar, irrespective
of the size
▪ However, for particles of larger dimensions, the QSA is no longer
valid due to the changes in phase of the driving field over the particle
volume and the LSPR of the bigger particles shifts toward infrared
and higher multipolar resonances emerge in the spectrum
▪ In such a situation, a rigorous electrodynamic approach is required i.e.
Helmholtz equation

15. In larger nanoparticles, the field distribution inside the nanoparticle is
inhomogeneous and the only possible solution is Helmholtz equation
To solve Helmholtz equation the best approach is to consider the field E1
inside the nanoparticle and the field outside the sphere E2
The field outside the nanoparticle is decomposed into external field E0, and
scattered part Escat E2=E0+Escat
As per boundary conditions, to satisfy the tangential component of electric
and magnetic field at the boundary of the metal nanoparticle/dielectric
interface, the surface of the metal sphere should be continuous

16. 1
R


R-Radius of the particle
2 0 scE E E= +
E1
medium 2
1 0
1 0
( ) | ( ) |
( ) | ( ) |
SC
SC
E E E
H H H
 
 
 
 
= +

= +
Boundary condition
2=0
Quasi-static approximation
Laplace equation
2
2
2 2
1
0
c t



 − =

Wave equation in terms of a scalar quantity ψ
- sphere surface
Have solutions
only for discrete
values of k,
eigenvalues
Helmholtz equation
2 2
( ) ( ) 0r k r  + =
2
k


=
Wave vector
c


= k
c

=

17. The above one can be written with three independent scalar equations, Ex, Ey,
and Ez.
2
2
2 2
1
0x
x
c t



 − =

𝛥2 𝜓 + 𝜀𝑘2 𝜓 = 0
If  is a scalar function and 2 will be the scalar Laplacian
If  is a vector function and 2 will be the vector Laplacian
Representation of the Laplacian in spherical polar coordinates
1 2 3
2
2
2 2 2 2 2
1 1 1
( ) (sin )
sin sin
r
r r r r r
  
  

    
    
 = + +
    
Radial part Angular part

18. 1 2 3( ) ( ) ( )r     =
By spherical symmetry (nano particles) we use spherical coordinates
ሻ𝜓(𝑟, 𝜃, 𝜑ψ is a function of the polar coordinates
i.e., Radical functions and spherical harmonics
2 2
0
0 ( , , ) ( , ) ( )
n
nm nm n
n m n
k r A z         
+ +
= =−
 + =  =  
▪ Coordinate r-radius of the particle
▪ -the angle the scattering direction makes with the direction of
propagation of the incident wave
▪ Azimuthal angle φ- the angle of the xy -plane projection of r from the x
axis to the y axis

19. 1 2 3
(cos( ))cos( )
(cos( ))sin( )
m
n
m
nm
n
P m even
P m odd
  
 
  
 −

=  −


spherical functions
Using separation of variables technique involving separation constants m and n
1 2 3
2
2 2
2 2 2 2 2
1 1 1
( ) (sin ) 0
sin sin
r k
r r r r r
  
  
 
    
    
+ + + =
    
To solve the Helmholtz Differential Equation in Spherical Coordinates, we
need to use Separation of Variables method

20. 2 2
0
0 ( , ) ( )
n
nm nm n
n m n
k A z      
+ +
= =−
 + =   
electric potential =angle variables and Z which depends on radius𝜓
M= 0, and there was no dependence on  and we had only this dependence
on theta angle.
Scalar Helmholtz equation
To solve Scalar Helmholtz equation
the scalar analogues of the spherical harmonic functions
Angle dependent function is a multiplication of Legendre polynomials over
sines or cosines. The Legendre polynomial is responsible for theta angle
dependence, and the cosines and sines have this phi dependence,
dependence on phi angle

21. M -A curl of a radius vector multiplied by scalar potential function 
N - taking the curl of M
2
( , , )
0
( , , )
nm
nm
mn nm
M r
E k E E C
N r
 

 

 + =  = 


( )nm nmM rot r= ( )nm mnN rot M
Vectorial Helmholtz equation
i.e. the vector analogues of the
spherical harmonic functions
The electric and magnetic fields in a source less region satisfy the vector
Helmholtz equation. Expand the Electric field into series over different
functions
M and N Vectorial functions

22. Solution for the scattered field
Scattered Electric field is decomposed into infinite sum of electric and
magnetic modes
(used to expand the magnetic and electric field in spherical waves)

23.  
2
0
2
2 2
0
(2 1)Re
2
(2 1)
2
ext n n
n
scat n n
n
n a b
n a b







=

=
= + +
= + +


The rate of the total amount of incident energy abstracted from the
incident beam due to interactions with a single particle is calculated directly
from the extinction cross section (cm2)
▪ For nanoparticles with sizes down to the optical wavelength, the surface
plasmon resonances are clearly dipolar
▪ However for larger particles, retardation effects (the phase difference
between the fields propagating from two different regions of the
nanoparticle and it is related to the effects of re-radiation) inside the
plasmonic particles lead to the appearance of higher order multipolar plasmon
resonances that show off as new features in the optical spectra

24. Mie Theory-Modes Structure
Magnetic modes
▪ Formed by loops. The electric field which is
going over a loop generates magnetic field
perpendicular to this loop i.e. magnetic dipole
moment looking out of the plane of the screen
▪ magnetic dipole is about six orders of magnitude
weaker than the electric dipole radiation
Electric modes
▪ Field lines go from one pole to another which
resembles the field lines of a dipole placed
inside the sphere
▪ The functions have nulls at certain  values,
thus creating directive lobes in between the
nulls
Magnetic modes MElectric modes N
E field= 0
dipole moment
N=1
• Mie scattering cross section is expressed as a combination of different orthogonal modes
• Dipole, quadrupole and octupole higher modes form a full set of solutions of Helmholtz equation
Dipole mode (N=1)

25. When the retardation of light across the
nanostructure is significant, by far-field
radiation, quadrupolar modes are excited and
these modes are higher in energy and have
much narrower absorption peaks than the
corresponding dipolar modes.
Quadrupole mode (N=2)
Octupole mode (N=3)
These multipole plasmonic resonance modes
are found potential applications in bio-sensing,
fluorescence, nano lasers or nonlinear nano-
photonics.

26. ▪ The multipolar modes are identified through the size dependence of
their spectral position or the angular distribution of the scattered light
▪ When the particle size further increase to the size regime beyond the
quasi static limit the overall spectral line shape becomes more
complicated.
▪ And the higher order multipolar resonances such as quadrupole and
octupole become increasingly significant in the extinction spectra in
addition to the dipolar plasmon resonances due to the phase retardation
effects, resulting in further redshifted and broadened dipolar plasmons
bands.

27. Surface charge distribution
▪ For a dipole -one hemisphere, charged positively on one side and
on the other side charged negatively and the color shows the
sign and amplitude
▪ In quadrupolar the charge changes the sign alternatively as
positive & negative (four times across the sphere)
▪ For octupolar mode the signs changes six times
The charge density distribution are calculated at each corresponding
resonance wavelength which shows distinct two, four, and six charge lobes,
which evidently demonstrates the plasmonic resonance characteristics of
dipole, quadrupole, and octupole

28. .
➢ Radiation pattern in the far field, in which direction and with which
intensity the modes radiate the energy
Radiation Pattern
Quadrupolar mode
➢ Quadrupolar mode has four lobes, so it emits energy in four
different directions
Dipolar mode
➢ The dipolar mode looks like a torus.
➢ The arrow represents the direction of a dipole moment
➢ So, perpendicular to the axis of a dipole it emits energy equally while it
doesn't emit energy at all in the direction of a dipole moment.
Octupolar mode
➢ Octupolar mode has six directions of energy radiation
➢ The higher is the order of the mode, the more lobes it will have

29. References
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Device Applications, Nanoelectronics. 2019
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and hybridization in nanoparticle dimers Mariano Pascale1, Scientific Reports | (2019) 9:14524
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30. Thk ou