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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Mechanical properties of electromagnetic waves
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Mechanical values

Electric fields exert forces on charges and
magnetic fields exert forces on currents
E,H
How to connect E,H with mechanical forces?
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Poyntings theorem and conservation of Energy and
Momentum for a System of Charged Particles and
Electromagnetic Fields
1. An electromagnetic field E,H exerts a force
Fq(EVxH) at a single charge q moving with
velocity V.
2. The rate of doing work by these fields
The magnetic field does not work since it
perpendicular to the velocity.
3. For a system with continuous distribution of
charges and currents the total rate of doing work
by the fields in a finite volume
This power represents a conservation of
electromagnetic energy into mechanical or thermal
energy. It must be balanced by a corresponding
rate of decrease of energy in the electromagnetic
field within the volume. Using the Ampère and
Faraday laws one can obtain
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Force (newton) Nm kg/s2 Energy (joule) JN
m Nm2 kg/s2 Power (watt) WJ/sm2
kg/s3 Electrical potential (volt)
VW/Am2 kg/s3A Capacitance
(farad) FC/V s4 A2/m2kg
the total energy density
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Since the volume V is arbitrary
conservation of energy
with the vector S representing energy flux (the
Poynting vector) (energy/area.sec)
The statement of conservation of energy the time
rate of change of electromagnetic energy within a
certain volume plus the energy flowing out
through the boundary surfaces of the volume per
unit time is equal to the negative of the total
work done by the fields on the sources within
the volume.
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The Poynting vector can be expressed via
experimental measured values
(Intensity of the beam)
For a harmonic field
Knowing S we can easily find the electric field
in the optical beam. Let us consider a 10 mW
beam of 2 mm diameter the usual values for
a He-Ne laser. If the beam is homogeneous in its
section
From Maxwell equations for a plane wave
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The conservation of the linear momentum The
Maxwell stress tensor
The total electromagnetic force on a charge
particle
If the sum of all the momenta of all the
particles in the volume V is denoted by Pmech
following the Newton ley
We use again the Maxwell equations to obtain
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We tentatively identify the volume integral
as the total electromagnetic momentum of the
field Pfield in the volume V
then
is the density of the electromagnetic momentum
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Our example ( 10 mW , 2 mm diameter-beam) has the
momentum density
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Back to the comservation of momentum
The expression
is the Maxwell stress tensor
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is the component of the flow per unit area of
momentum across the surface S. where n is the
outward normal to the closed surface A.
This is a statement of conservation of
momentum the time rate of change of total
momentum within a certain volume is equal to the
force acting on the surface A of the volume V
containing the combined system of
particles. This expression can be used to
calculate the forces acting on material objects
in electromagnetic fields.
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With the same manner we may introduce the
conservation of angular momentum.
is the flux of angular momentum with T the
Maxwell stress tensor and
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L
mechanical angular momentum
E
g
x
S
L(field)
z
Angular momentum of an electromagnetic beam
H
A Gaussian beam with circular polarisation has
both the angular momentum and momentum
A Gaussian beam with linear polarization has no
angular momentum but has a momentum
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A Laguerre-Gaussian beam has both the angular
momentum and momentum
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Physical meaning of the Maxwell tensor
The Maxwell tensor is just a different way of
writing the fact that electric fields exert
forces on charges and magnetic fields exert
forces on current.
Units
the units for pressure
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Example a linearly polarised plane wave
propagating in the z-direction
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The pressure is along the propagating direction.
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Example. The Maxwell tensor for some simple
cases.
Reflection of a plane wave on the interface
between two medium with refractive indices n1 and
n2
The force acting on the surface for the incident
wave
S
z
i.e. the wave presses the surface along the
positive direction
For the reflected wave
Also along the positive direction
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The summary force acting on 1 m2 of the
reflecting surface from the wave in the left
side of the interface is
Let us consider a force acting on a particle of
surface D1 mm2 from a beam I10 mW focussed on
this surface
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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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Where we are? An optical wave carries
ENERGY
flux of energy (Poynting vector)
MOMENTUM
momentum density
ANGULAR MOMENTUM
density of angular momentum
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The transfer of momentum from the optical wave to
an object follows this equation
with the Maxwell stress tensor
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Mechanical pressure exerted on a sphere by a
plane wave the Mie theory
Gustav Adolf Feodor Wilhelm Ludwig Mie
Boundary conditions at the surface Internal
field TANG. COMPON External field TANG COMPON
External field incident wave scattering wave
Internal wave
We need to solve each scattering problem twice
(for a given direction of propagation) in order
to determine the scattering of an arbitrarily
polarized plane wave.
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GEOMETRY OF THE PROBLEM
na
np
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Incident field
Outside the sphere
Inside the sphere
At the spherical surface
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Wave equations
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Solutions of the vector wave equations
Suppose that, given a scalar function y and an
arbitrary constant vector C, we construct a
vector function M
(For example, ME)
The divergence of the curl of any vector
functions vanishes
Using vector identities we arrive to
Therefore, M satisfies the vector wave equation
if y is a solution to the scalar wave equation
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We construct from M another vector function N
which also satisfies the vector wave equation
if
and we also can find that
Therefore, M and N have all the required
properties of an electromagnetic field 1. they
satisfy the vector wave equation, 2. they are
divergence-free 3. the curl of M is proportional
to N and the curl of N is proportional to M
We reduce the problem of solving the vector wave
equations to solving a scalar wave equation
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The spherical coordinates correspond to the
spherical symmetry of the problems
We choose the radius vector r as the vector c
X
E
r
The scalar wave equotion
Z
Y
in the spherical polar coordinates is
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The way to solve this equation is well known.
where m and n are integration constants and we
find it from boundary conditions
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General solutions are
Pn(m) are the associated Legendre functions
zn are any of the four spherical Bessel functions
We use those functions that satisfy physical
conditions.
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With this function y given one can find the
vector functions M and N. Any solution of the
wave equation can now be expanded in an
infinite series of the functions M and N because
these functions are mutually orthogonal sets of
functions.
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The field lines are shown on the surface of an
imaginary sphere concentric with, but at a
distance from, the particle. The are transverse
magnetic modes (N) no radial magnetic field
component
transverse electric modes (M) no radial
electric field component
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These eigen modes of a sphere remind us that each
time when we are dealing with a system of finite
dimensions a solution contains a set of modes
with structure that depends on a symmetry of the
system. For example, for a planar optical
waveguide we have TE- and TM-modes of different
structure E(r) planar symmetry
An optical fiber ( cylindrical symmetry)
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Let now apply this tool to the Mie-problem
scattering a plane wave on a sphere of the
arbitrary refractive index and radius.
X
E
R
Z
Y
Step 1 Expansion of a plane wave in vector
spherical harmonic.
(This is simply a plane wave in the spherical
coordinates)
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Applying the orthogonality conditions
and similar expressions for Bomn, Aemn,, and Aomn.
These expressions are valid for ARBITRARY
incident wave,
however, for the incident plane wave
and after calculations B and A
"Expanding a plane wave in spherical wave
functions is somewhat like trying to force a
square peg into a round hole."
and Hi can be found from Maxwells equotions
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Step 2 The internal and scattered fields
The boundary conditions and the expansion of the
incident field dictate the form of the expansion
coefficients for the scattered and internal
fields
The coefficients an, bn, cn, dn we find from the
boundary conditions at the spheres surface
at ra
For purpose of calculations of the radiation
forces we need expressions for an and bn
np refractive index of the sphere na refractive
index of the medium
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Illustrations electrical field distribution near
a sphere
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dielectric sphere n1.55 l1 mm
sphere radius
50 nm
100 nm
500 nm
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gold sphere e-54i 5.9 l1 mm
sphere radius
50 nm
100 nm
500 nm
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Arbitrary incident field
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Hence, all fields are known now and we can
calculate Maxwell tensor Tij and mechanical
forces
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Torque
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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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General solutionhow light energy is
transferedto mechanical pressure
Both scattering and absorption remove energy from
a beam of light traversing the medium
Extinction Absorption Scattering
Only inhomogeneities causes scattering.
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The incident field gives rise to a field inside
the particle and a scattered field in the medium
surrounding the particle.
The main properties of the waves are its
intensity energy flux per unit
area phase polarization
Far from the particle the scattered wave is a
spherical wave in which energy flows outward from
the particle and its intensity must be
proportional to the incident intensity I0 S and
r-2 and for a given direction (q,f) is given by
Let us define a scattering cross section Csca
An absorption cross section Cabs
The cross section of extinction Cext Cscat Cabs
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The mechanical momentum removed from the incident
wave is proportional to Cext. but the part Csca
is partially replaced by the forward component of
the momentum of the scattered wave.
The total forward momentum carried by the
scattered radiation is proportional
q
The cross section of pressure is given by
This momentum is transfered to the particle and
hence the radiation force exerted by the incident
wave on the area Cpres

For a plane wave illumination and a spherical
particle there is only the force along the
propagation direction, however an assymetric
particle will suffer the force in the
perpendicular direction
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The cross section of radiation pressure
For non-absorbing particles
component of the Poynting vector along the
radius-vector
Using the expressions for the scatterind fields
when the incident wave is a plane wave
In general case of the particles with absorption
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Cpres?