Transcript Highligh in Physics 2005

Congresso del Dipartimento di Fisica
Highlights in Physics 2005
11–14 October 2005, Dipartimento di Fisica, Università di Milano
An application of the Optical Theorem to
the sizing of sub-wavelength particles
*
Potenza ,
M.A.C.
* Dipartimento
di Fisica, Università di Milano
Light scattering from a single particle
ABSTRACT
We present a novel technique based on the Optical Theorem
(OT) for the determination of the size of scattering particles in
the sub-wavelength range. When a plane wave impinges on a
particle, a scattered spherical wave is generated. The
scattered amplitude is in general made up by both in phase
and in quadrature components. The OT states that the
scattering cross section that describes the integral of the
scattered intensity at any angle is proportional to the
amplitude of the quadrature component at zero scattering
angle.
This remarkable result is the consequence of conservation
arguments that require that the interference between the
scattered wave and the incoming wave should exactly account
for the power loss of the incoming beam due to scattering of
power in different directions. We show an application of the
method in the realistic case where an assembly of particles is
present under the incoming beam. The scattered radiation in
the far field is then a speckle field, with the intensity that
fluctuates as a consequence of the stochastic interference
between individual scattered waves. If the scattered intensity
is collected fairly close to the scattering sample, and the
incoming beam is wide enough, then low visibility speckle field
is generated due to the interference between the weak
scattered fields and the powerful transmitted beam. The
method relies on the statistical analysis of these low visibility
fringes due to the heterodyning between scattered field and
the transmitted beam that acts as a self referencing local
oscillator. The statistical processing of the instantaneous
speckle distribution consists of the evaluation of the two
dimensional power spectrum of the intensity distribution. It
will be shown that the spectrum exhibits a multiple zero
structure, from which an accurate estimate of the amplitude
of the quadrature term can be performed. Spherical,
calibrated polystyrene colloidal particles have been used. The
results have been compared with the theoretical predictions
based on Mie scattering functions and the OT. The data show a
continuous variation of the phase of the scattered amplitude
from zero to p/2 as the particle diameter is changed from well
subwavelength values to many microns sizes, as expected from
the OT as one moves from the Rayleigh scattering regime to
the diffraction regime. Excellent accuracy for the
subwavelength diameters determination is reported.
Extinction factor
Qext = σ/πa2
diameter
0.1-4 um
4
3
M.
*
Giglio
Incident plane wave
Scattered wave
u  u0 s( ,  )e
ikzit
u0  E0e
i ( , )
Interference pattern
(paraxial approximation)
ikrit
e
 ikr
Small particle (with respect to the wavelength)
The forward scattered and transmitted waves are in phase
The particle cross section is:
300 nm
Cext
1000 nm
The intensity profile
is the sum of the
elementary patterns
diameter
0.1-2 um
1
Despite the speckled appearance, the power spectrum of the
intensity distribution provides the information about the phase
100 nm
5 10
Re[S(0)]
1
0.2
0.4
0.6
0.8
-7
1 10
-6
-6
1.5 10
2 10
-0.5
1
-1
-0.2
-1.5
x = ka
Relative refractive index: m = 1.55 (typical for airborne particles)
Im[S(0)]
3
2.5
2000 nm
Single
interference
pattern
-6
15
diameter
0.1-4 um
600 nm
0.5
0.2
3.5

Light scattering from a collection of particles
Phase lag of the forward
scattered wavefront
1.5

Due to the interference, the
dephasing of the scattered
wave reduces the field
amplitude in the forward
direction. This accounts
for the power scattered away:
conservation of the energy !
Large particle (the size of the wavelength or larger)
The forward scattered and transmitted waves are in quadrature
“An experiment by which this spherical
wave can be observed is impossible, for a
From the general theory of
telescope (…) sees the primary and
scattering, the phase of the
secondary source in the same direction:
scattered wave depends on
their images coincide”
the particle dimension
(Van de Hulst)
0.6
4
i ( 0 )
 2 Re is(0)e
k
OPTICAL THEOREM
The removed power
is appreciable by
the intensity at the
centre of the pattern.
2
10
The removed power is negligible.
Nevertheless the interference
pattern is visible, and the
spherical wave is measured !!
300 nm
Complex scattered field
(normalized by πa2)
Im[S(0)]
diameter
0.1-2 um

transmitted power = integral of the interference pattern
100 nm
0.4
5

2u0
i ( 0 ) ik ( x 2  y 2 ) / 2 z
u0  u  u0 
Re is(0)e e
kz
2
0.4
A few particles
1.5
diameter
0.1-4 um
0.5
Power
spectrum of
the speckle
field
diameter
0.1-4 um
1
0.5
0.3
2
0.2
1 10
1.5
0.1
1
0.5
0.4
0.6
2 10
-6
3 10
-6
4 10
Radial profile
of the power
spectum
-6
-0.5
Re[S(0)]
0.2
-6
-1
0.8
-0.1
5
10
15
20
25
1
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
1
0
0
-1.5
0
Many particles (speckles)
-0.2
x = ka
Relative refractive index: m = 1.18 (polystyrene in a water suspension)
10000
20000
30000
40000
0
0
10000
20000
30000
40000
0
10000
20000
30000
40000
The extrema positions depend on the dephasing value.
EXPERIMENTAL APPARATUS AND DATA ANALYSIS
The measurements have been performed by the self referencing
optical system scketched in the figure. A plane wave transilluminates a
monodisperse, dilute particle suspension and the transmitted beam
falls over the plane of a CCD camera, acting as a local oscillator for
the scattered wavefornts. The intensity distribution is the stochastic
sum of the intensities of all the elementary interference patterns
generated by each particle in the sample.
EXPERIMENTAL RESULTS
Comparison of the positions of the extrema in the radial profiles of the measured power spectra to the expected values.
Water suspensions of 400 nm, 600 nm, 1000 nm, 2000 nm in diameter have been used in a cell 2 mm thick.
Different sample-sensor distances have been used.
2
2
1.75
1.75
1.5
1.5
1.25
1
1
0.5
0.5
20000
Collimating
lens
scattered
Light is
If E0 >> ES :
ES
transmitted E0
I = | E0
|2
2
by the sample
+ E0 ES +
E0* ES
+ ES
2
The signal ES is heterodyned by the transmitted field E0
ES being stochastic, by averaging over a number of
realizations of the speckle field Ii (x,y), we have:
< Ii (x,y)> = | E0 |2 = I0
that is the transmitted field plus the static stray light !
40000 60000
20000 40000
80000 100000 120000 140000
Diameter 1000 nm
z = 29 mm
1.75
1.75
1.5
1.5
1.5
1.25
1.25
1.25
1
0.75
0.75
0.5
0.5
0.5
0.25
0.25
0.25
60000
80000
100000
Diameter 2000 nm
z = 62 mm
The dephasing of the scattered
waves depends on the particle size
accordingly to the Mie Theory.
1
0.75
40000
80000 100000 120000 140000
Diameter 600 nm
z = 29 mm
1.75
20000
60000
2
2
1
*
20000
40000 60000 80000 100000 120000 140000
Diameter 2000 nm
z = 29 mm
CCD
sensor
0.5
0.25
0.25
Sample
cell
1
0.75
0.75
Laser
source
1.5
1.25
20000
40000
60000
80000
100000
Diameter 600 nm
z = 62 mm
10000 20000 30000 40000 50000 60000 70000
Diameter 400 nm
z = 147 mm
Good accordance to the expected positions of the maxima and minima.
No free parameters are present to fit the dephasing values.
The method has been filed for PCT
patent in 2005 (University of Milan)