Mechanical properties of electromagnetic waves: the Maxwell tensor'

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• Mechanical properties of electromagnetic waves
  the Maxwell tensor.
• Absorption cross section and Scattering cross
  section.
• Mie theory of light scattering.
• 4. Calculations of radiation forces exerted on
  small particles (Rayleigh regime) and
• big particle (geometrical optics
  regime).
• Stiffness of an optical trap.
• Radiation forces exerted by different types of
  optical beams Gaussian beam, Laguerre-
• Gaussian beam, Bessel beam, evanescent wave.
• Optical beam focused by an objective with high
  numerical aperture.
• Metal and dielectric particles, living cells.
• Calibration of optical trap.
• Photonic force microscope.
• Experimental realization of optical tweezers.
• Applications of optical tweezers in
• cell biology
• fluorescence spectroscopy of single cells
• Raman spectroscopy of single cells

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Represent incident field as a sum of vector
spherical functions M and N
Find Force and Torque
To find scattering field an and bn
n?
In investigation of the rainbow one needs to sum
about 12000 terms, assuming a water droplet
radius of 1 mm 100 nm sphere - 7 terms 1000
nm sphere 20 terms
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Arbitrary incident field
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• The Mie scattering theory can be applied to
  spheres of arbitrary radius and dielectric
  susceptibility. The main difficulty in using this
  theory is a low convergence of the terms
  describing partial waves.
• Different techniques were proposed to simplify
  algorithms
• However, great simplification of the problem can
  be achieved in the limit of a sphere of very
  small radius or very big radius

Mie theory
Rayleigh regime
Geometrical optics
sphere's radius
wavelength of light beam
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Rayleigh regime
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Consider the case when the sphere radius is much
smaller then the wavelength.
There are at least three techniques that can be
used to find the radiation pressure in this case.
1. The Mie theory
What are the scattering coefficients an and bn ?
The radiation force is determined by this
scattering coefficient
np refractive index of the sphere na refractive
index of the medium
xltlt1
Condition for Rayleigh regime
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Let us use the formula for the scattering
coefficients from the Mie theory
The scattering cross section of small
particles. This is a result of the Mie theory.
The Mie theory can described only the radiation
force along propagation direction because it
deals with the plane wave.
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2. The electrodynamics approach.
For particles much smaller than the wavelength we
can find the scattering cross section using
approximation of the homogeneous field.
Fields near a 50 nm dielectric sphere.
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Dipole moment induced by an homogeneous
electric field in a dielectric sphere (Jackson).
r
E0
a
z
E0
q
ep
ea
No free charges! The Laplace equation for the
electric potential in the spherical coordinates
with the boundary conditions at the interface ra
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with the boundary conditions at the interface ra
Tangential E at ra
Normal D at ra
The axial symmetry prompts one to take solutions
for potentials as
outside
Inside
(P are the Legandre polynomials)
Solutions are
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potential due to the external field
potetial due to the induced dipole
-

-

-

The potential of an ideal dipole
r
q
p
Comparing we find that the induced moment is
Polarizability (electrostatic value paE)
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This dipole oscillates and hence, emits
(scatters) radiation in all directions (Rayleigh
scattering). The electric field of the scattered
wave at a distance rgtgtl from the dipole is
The scattered radiation intensity (time average
of the Poynting vector)
Integrating over a big sphere we find the total
energy scattered in all directions per units
time and dividing this on the incident intensity
we obtain the scattering cross section
This is the same formula that the Mie theory
gives.
blue sky
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If the optical field is inhomogeneous?
The dipole approximation how to include
gradient (dipole) forces
Non-homogeneous electric field
The electrical charges pf the dipole experience
a Lorentz force
Force acting on dipole in nonuniform electric
filed
At the optical frequencies only the time average
of the electromagnetic force is observed
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The motion of the charges particles in external
fields necessarily involves the emission of
radiation whenever the charges are accelerated.
The emitted radiation carries off energy,
momentum and angular momentum and so must
influence the subsequent motion of the charged
particles. Consequently the correct treatment
must include the reaction of the radiation on the
motion of the source.
modified equation of motion a charged particle in
external fields the Abraham-Lorentz equation
Due to this effect the polarizabilty a for the
electrodynamics differs from its static value a0
(that we found early)
electrodynamics correction
For a wave propagating along k the electric field
is
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GRADIENT FORCE
SCATTERING FORCE
Now we use the electrostatic polarizability a0
For a plane wave (Mie scattering) only the
scattering force exists.
scattering cross section
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The gradient and the scattering forces in the
Rayleign approximation.
the time average Poynting vector (intensity)
Example, a Gaussian beam
waist (m)
total power (W)
confocal parameter (m)
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intensity near the focus
Z
X
Axial force lt Radial force Radial force gt short
range
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Optical trap made of a Gaussian beam repeals a
low-refractive index particle.
Z
X
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intensity near the focus
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Stiffness of the optical trap
Whenever a bead leaves the center of the optical
trap the restoring force pulls it back to the
center.
stiffness
F optical -k Dx k is the stiffness of the OT
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radial force RA
axial force RA
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Criteria for stability of a single-beam laser trap
1. Z-Gradient ForcegtgtScattering Force
First condition of stable trapping
max of gradient force
Polysterene in water m1.2 w01.5 mm l1 mm
a(max)65 nm
Gravitation force and other forces (Brownian
motion)
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Geometrical optics limit
the sphere size is much bigger then the wavelength
phase shift gained after transmission through the
sphere is very big
With these are two conditions the geometrical
optics limits can be used.
phase shiftgtgt1
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In geometrical optics one describes an optical
wave as a ray with direction along the
propagation direction.
How a ray passes through the sphere
angle
a2q-2r bp-2r
an angle - clockwise
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The total force in the Z direction is the net
change in momentum per second in the Z direction
due to the scattered rays
The total force in the Y direction is the net
change in momentum per second in the Y direction
due to the scattered rays
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Using a mathematical trick
The net force that a ray exerts on the sphere is
given by
this is a definition the scattering force is
directed along the propagation direction of the
incident ray, and the gradient force is directed
perpendicular to the incident ray
Here R and T are reflectance and
transmittance (that depends on polarization of
the incident ray
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optical ray
The radiation force depends on the incident angle
FGRAD
FSCAT
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The radiation forces produced by a focused beam
input beam with a given distribution of
intensity, (for example, a gaussian beam)
lens
Fgrad
Each ray creates the force in the axial and in
the radial direction.
qmax
Fscat
RAY
BEAM AXIS
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Radiation forces for trap produced by a focused
beam (Gaussian)
Axial radiation force depends on the shift of the
sphere from the focus and NA of the lens
qmax
qmax800 NA1.2
s
qmax400 NA0.4
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Geometrical optics limit
The equlibrium position moves to bigger z when
the spheres radius grows.
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bk a
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Metal Spheres
1. Rayleigh particles
volume of the sphere
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Metal Spheres
If we define
refractive index
extinction
An electromagnetic field attenuates to e-times
inside a skin layer
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dielectric
metal
Electric dipoles are "active" only inside this
volume
A metal sphere interacts mechanically less with
optical waves!
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Geometrical optics regime
Dielectric sphere
R P
P
q
z
Metal sphere No contributions of refractive
waves
r
z
q
y
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The net force that a ray exerts on the metal
sphere is given by
Gradient and scattering forces exerted by a ray
versus the incident angle
FORCE
scattering force
COMPARE WITH A DIELECTRIC SPHERE
gradient force
FORCE
ANGLE
ANGLE
at normal incidence the scattering
force increases two times
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a 20 nm
pN
absorption force
total force
Z-gradient force
scattering force
pN
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a50 nm
AXIAL FORCE pN
SHIFT (m)
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a10 mm, a focused Gaussian beam
total force
scattering force
For big metal particles a 2D trapping is possible
only.
gradiend force
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Optical potential energy
A particle in the optical trap experiences forces
hence one can indroduce the acquired trapping
energy which is at position R is given by the
integral over the scalar product of force and
path
For small displacement around the position of
force equilibrium the bottom of the
potential well can be approximated by a harmonic
potential
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Brownian motion in optical traps
Brownian motion
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2. Optical potential gtgtthe kinetic energy of the
Brownian motion.
second condition of stable trapping
Archimede force
Archimedes' principle is the fundamental natural
law of buoyancy, first identified by the Greek
mathematician and inventor Archimedes in the 3d
century BC. It states that any object floating
upon or submerged in a fluid is buoyed upward by
a force equal to the weight of the displaced
fluid.
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• Mechanical properties of electromagnetic waves
  the Maxwell tensor.
• Absorption cross section and Scattering cross
  section.
• Mie theory of light scattering.
• 4. Calculations of radiation forces exerted on
  small particles (Rayleigh regime) and
• big particle (geometrical optics
  regime).
• Stiffness of an optical trap.
• Radiation forces exerted by different types of
  optical beams Gaussian beam, Laguerre-
• Gaussian beam, Bessel beam, evanescent wave.
• Optical beam focused by an objective with high
  numerical aperture.
• Metal and dielectric particles, living cells.
• Calibration of optical trap.
• Photonic force microscope.
• Experimental realization of optical tweezers.
• Applications of optical tweezers in
• cell biology
• fluorescence spectroscopy of single cells
• Raman spectroscopy of single cells