Download Forces, light and waves

Forces, light and waves
Mechanical actions of radiation
Jacques Derouard
« Emeritus Professor », LIPhy
Example of comets
Cf Hale-Bopp comet (1997)
Exemple of comets
Successive positions of a comet
Sun
Tail is in the direction
opposite / Sun, as if
repelled by the Sun
radiation
Comet trajectory
A phenomenon known a long time ago...
the first evidence of radiative pressure predicted several centuries later
Peter Arpian « Astronomicum Caesareum » (1577)
• Maxwell 1873 electromagnetic waves
Energy flux associated with momentum flux (pressure):
a progressive wave exerts
Pressure = Energy flux (W/m2) / velocity of wave hence:
Pressure (Pa) = Intensity (W/m2) / 3.108 (m/s)
or:
Pressure (nanoPa) = 3,3 . I (Watt/m2)
• NB similar phenomenon with acoustic waves. Because the
velocity of sound is (much) smaller than the velocity of
light, acoustic radiative forces are potentially stronger.
• Instead of pressure, one can consider forces:
Force = Energy flux x surface / velocity
hence
Force = Intercepted power / wave velocity or:
Force (nanoNewton) = 3,3 . P(Watt)
Example of comet tail
composed of particles radius r
• I = 1 kWatt / m2 (cf Sun radiation at Earth)
– opaque particle diameter ~ 1µm, mass ~ 10-15 kg,
surface ~ 10-12 m2
– then P intercepted ~ 10-9 Watt
hence F ~ 3,3 10-18 Newton comparable to
gravitational attraction force of the sun at Earth-Sun
distance
Example of comet tail
Influence of the particles size r
• For opaque particles r >> 1µm
– Intercepted power increases like the cross section ~r2
thus less strongly than gravitational force that
increases like the mass ~r3
Example of comet tail
Influence of the particles size r
• For opaque particles r << 1µm
– Solar radiation wavelength λ ~0,5µm
– r << λ « Rayleigh regime»
– Intercepted power varies like the cross section ~r6
thus decreases much stronger than gravitational
force ~r3
Radiation pressure most effective for particles size ~ 1µm
Another example
Ashkin historical experiment (1970)
A. Ashkin, ‘Acceleration and trapping of particles by radiation pressure’, Physical
Review Letters, Vol. 24, No. 4, 156, 1970
Laser I ~ 19mW / 100µm2 thus ~ 2.108 W/m2
Ashkin experiment (1970)
A. Ashkin, ‘Acceleration and trapping of particles by radiation pressure’, Physical
Review Letters, Vol. 24, No. 4, 156, 1970
Polystyren beads suspended in water
Beads r=1,32µm
Plaser=19mW
λ=515nm
w0=6,2µm
Observes that
-the beads are pushed by the laser beam (and
slowed by water drag force) <V>=26µm/s
Ashkin experiment (1970)
• NB polystyren beads are transparent:
– no radiation absorption
– but deflection of light due to refraction
• Radiative force is the result of this deflection
Radiation pressure
• Absorption, reflexion or scattering of a light
beam by a particle
r
F
Absorption of light makes
the particle recoil
r
F
Deflexion (refraction or scattering)
of a uniform light beam yields to a
force directed along the light beam
Ashkin experiment (1970)
A. Ashkin, ‘Acceleration and trapping of particles by radiation pressure’, Physical
Review Letters, Vol. 24, No. 4, 156, 1970
Polystyren beads suspended in water
Observes that
-the beads are pushed by the laser beam
-the beads are attracted by the laser beam
« Gradient force »
• Deflection or scattering of a non uniform
intensity light beam by a particle
r
F
Deflection of light of non uniform
intensity across the particle yields to a
resulting force directed obliquely, that
tends (in this case) to push the particle
towards maximum intensity region
« Gradient force »
• Deflection or scattering of a non uniform
intensity light beam by a particle
r
F
When the particle index of
refraction is smaller than that of
the medium (bubble), the
deflection of light tends to expell
the particle from maximum
intensity region
(should also be observed with
reflective particles)
Also observed by Ashkin in 1970
In conclusion two types of forces
exerted by light on matter:
• Radiation pressure (or « scattering force »):
particles are pushed by a light beam
– effect proportional to absorption or scattering cross
section
• Gradient force: particles are (generally) attracted
towards high intensity regions (effect reversed with
refractive index contrast)
Radiatives forces
• Atomic particles: close to a resonant
absorption line σ is enormous, so are the
radiative forces (-> cold atoms physics)
• (NB for dielectric particles Ashkin has
observed scattering resonance through
radiative pressure resonance)
Expression of radiative forces
case of « small » particles (limit a<<λ)
Response of the particle to
radiation field:
complex polarisability
α = α '+iα "
Radiation field characterized by
Energy density
Poynting vector
= Intensity x propagation direction
r
U (r )
r r
< S (r ) >
Expression of radiative forces
case of « small » particles (limit a<<λ)
r
r
r
< F >= Fscat + Fgrad
Radiation pressure
Gradient force
Expression of radiative forces
case of « small » particles (limit a<<λ)
r
r
r
< F >= Fscat + Fgrad
rr
r
α" < S (r ) >
Fscat = k
nmed
ε0
c
Radiation pressure
- α’’ proportional to the sum of absorption and scattering cross
sections
r
Fscat
-α’’ > 0,
always towards the propagation of the wave,
maximum for absorption or scattering resonance frequencies
Expression of radiative forces
case of « small » particles (limit a<<λ)
r
r
r
< F >= Fscat + Fgrad
r
r α'
r 
Fgrad = −∇ −
U (r ) 
 2ε 0

Gradient force
-If α’ > 0 attraction towards large U regions
-If α’ < 0 repulsion from large U regions
-large variation of α’ close to resonance frequencies, may
change of sign (« blue detuned optical atomic traps »)
Radiation pressure and gradient forces
both exists also with acoustic waves:
• Radiation pressure: associated with momentum
flux transported by acoustic wave =
Energy flux / velocity in the simplest cases
• Gradient force: for small spherical particles it
results from « Gor’kov potential ». For large
particules it can be estimated like in geometrical
optics, where the analogous of refractive index is
1/ρc
Radiation pressure and gradient forces
both exists also with acoustic waves:
• Gradient force: for spherical particles it results
from « Gor’kov potential ».
Particles are trapped at the
nodes of a 2D network of
moveable stationnary waves
Radiation pressure and gradient forces
both exists also with acoustic waves:
• Gradient force on bubbles (P. Marmottant, P.
Thibault et al…)
– « Bjerknes force »: response of the bubble to
acoustic pressure is its variation of volume
∆V=(α’+iα’’)∆p
r 1 r 2
F = α ' ∇p 0
4
– Acoustic resonance mode
Change of sign of α’, hence F,
when crossing resonance frequency
Resonance for R~20µm:
Change of sign of radiative force
Optics
A variant of the first Ashkin’s experiment: propelling of
microparticles over optical waveguides.
• Gaugiran (CEA-LETI), Derouard et al
Opt. Express 13, 6956-6963 (2005);
Opt. Express 15, 8146-8156 (2007)
Optical trapping and propelling of particles over
an optical wave guide
FGRAD
Light intensity profile
FGRAD
FPrad
FGRAD
laser
FPrad
FGRAD
Particule
Scattered light
F
Numerical calculation of the electromagnetic field energy
density and forces applied on a glass microparticles of
diameter 250nm immersed in water and lying over a
silicon nitride optical waveguide.
LIGHT
F
Experimental set-up
CCD camera
Microscope
objective
Optical
waveguide
Microparticles
suspended in water
Silicon
substrate
Propelling of glass microparticles (diameter 1µm))
Gaugiran et al, (2005)
Propelling of biological cells (yeast and bacteria)
((Gaugiran et al. (2005)
Propelling of biological cells (red blood cells)
((Gaugiran et al. (2005)
Radiative forces and optical
trapping of particles
Radiative forces and optical
trapping of particles
• Need to balance the effects of radiation
pressure. Several possibilities:
–
–
–
–
gravity
substrate
2 counter propagating light beams
gradient force stronger than radiation pressure
(strongly focused beam : « optical tweezer »)
Radiative forces and optical
trapping of particles
• Need to balance the effects of radiation
pressure. Several possibilities:
–
–
–
–
gravity
substrate
2 counter propagating light beams
gradient force stronger than radiation pressure
(strongly focused beam : « optical tweezer »)
First trapping experiment: counter
propagating beams
Gradient forces attract beads towards beams axis
Opposite axial radiation pressure forces are balanced
A. Ashkin, ‘Acceleration and trapping of particles by radiation pressure’,
Physical Review Letters, Vol. 24, No. 4, 156, 1970
Recente version of this
configuration: «optical stretcher »
(Guck et al, 2000, 2005)
• Ytterbium fibered
laser injected in single
mode optical fibers
• Microfluidic channel
• Biological cells
suspended in water
100µm
Application: observation of the
deformation of a «fibroblast»
(Guck et al, 2005)
Trapped cell:
As a result of radiative
pressure the cell is
distorted
Monitoring of
laser beam
intensity
The cell is not squeezed,
it is streched!! ??
Radiation pressure in material media
• In vacuum radiation pressure = Intensity / c0
• In medium refractive index n, velocity of light = c0 / n
hence, we may guess that
radiation pressure = Intensity / (c0 / n )
thus
radiation pressure = (Intensity / c0 ) x n
•
• Actually it seems that in a number of cases, everything is as
if the photons transported by the wave had momentum equal
to nx hν /c0
Radiative forces on material media
Medium refractive
Medium refractive
index n1
index n2
Radiation
Radiation
Intensity I
Intensity I
Momentum flux
Momentum flux
I n1 /c
I n2 /c
Radiative forces on material media
Medium refractive
Medium refractive
index n1
index n2
If n2 > n1 then I/n2 > I/n1, hence a force F is
exerted at the interface that tends to pull the
medium 2
r
F
Momentum flux
Momentum flux
I n1 /c
I n2 /c
Radiative forces and optical
trapping of particles
• Need to balance the effects of radiation
pressure. Several possibilities:
–
–
–
–
gravity
substrate
2 counter propagating light beams
gradient force stronger than radiation pressure
(strongly focused beam : « optical tweezer »)
Ashkin 1986: First experiment of trapping a
particle using a single focused light beam:
“optical tweezer”
A. Ashkin et al ‘Observation of single-beam gradient force optical trap for
dielectric particles’, Optics Letters, Vol. 11, No. 5, 288, 1986
Radiatives forces
• These forces are due to the momentum flux
tranported by the radiation.
• But light transports angular momentum as
well
« radiative torques »
Radiative torques
• Light transports angular momentum
– Photon spin and polarization of light
• photon of circularly polarized wave has spin h/2π
along the direction of propagation
• Beth’s experiment (1936): mechanical action of
circularly polarized wave on a birefringent plate
Radiative torques
• Beth’s experiment (1936): mechanical action of
circularly polarized wave on a birefringent plate
r
r
– Birefringent medium: P not parallel to E
r r r
– then torque per unit volume Γ = P × E
– in the same time change of polarization of the
transmitted light (the total angular momentum of
light+material is conserved)
Beth (1936)
Radiative torques
• Recent version: micro viscosimetry with
« vaterite » (sort of calcite) particles (cf
Rubinsztein-Dunlop et al 2007)
Another configuration:
« form birefringence »
r
• Non spherical object:
induced polarization P
r
not parallel
to
E
r
r
r
– torque Γ = P × E
r
• Conservation of J implies that the angular
momentum of the scattered wave is
affected/ incident wave
Radiative torques
• Light transports angular momentum
– « orbital » angular momentum of radiation related to
spatial modes of electromagnetic field
Radiative torques
• Light transports angular momentum
– cf « transverse modes » of laser cavities, LaguerreGauss modes
Propagation / z
Gauss
E p ,l (r , z , ϕ ) = coeff . exp(ikz ) ⋅ exp[ −( r / w) ].
2
.L (2r / w ). exp(−ilϕ )
l
p
2
Laguerre
polynomia
2
non axisymetric
mode
Laguerre-Gauss E0 l modes
Wave Surfaces (Padgett, Courtial et Allen, 2004)
l=0
« Mode TEM00 »
l = +1
l = +3
« Doughnut modes »
Intensity distribution (Beijersbergen et al, 1992)
Radiative torques
• Light transports angular momentum
– Allen et al (1992):
• E0,l corresponds to photons having angular momentum
of projection l.h/2π along z axis
• N photons/second correspond to an energy flux of
I = N.hω/2π
• E0,l with N photons/second corresponds to an angular
momentum flux (torque!) of J = N.l.h/2π = Ι.l/ω
The larger the smaller ω !
Generation of modes Ep,l
One possibility: transmission of a TEM00 wave
through a helicoidal phase plate
Grier, 2003
Generation of modes Ep,l
Other possibility: diffraction of a mode TEM00 by a
« fork » hologram
Binary, ( not « blazed »)
Or phase hologram
(« blazed »)
Application to trapping and rotation of
microbeads
Grier, 2003
Acoustics
Pionnier in the
study of « optical
vortices»
Acoustics
Conclusions and Résumé
• Radiative forces and torques: linear and angular
momenta transport by the waves
– Light wave
– Also sound waves (strong analogies but some more or
less subtle differences). Potentially larger effects thanks
to sound wave velocity and frequency much smaller
than light’s
– Other waves: water surface waves (= «gravity waves »)
and Stokes drift …
• Exotic wave modes carrying angular momentum
Conclusions and Résumé
• Applications
– Cold atoms
– Measurement of molecular motor forces,
characterization of mechanical properties of
microparticles, microviscosimetry...
– Manipulation of microparticles, Lab on chips
Thank you!
Expression of real and imaginary parts of the polarizability of a
spherical particle compex refraction index n, radius a << λ/ n
(Rayleigh regime)
 n2 −1  3
α ' = 4πε 0ℜe  2
a
 n + 2 
2
2
2

 n −1  3 2
n −1 3 6 
 a + 4πε 0
α " = 4πε 0 ℑm 2
k a 
2

n
+
2
3
n +2






Absorption
Scattering
Crookes radiometer (1873)
• Initially improperly taken as
evidence for the existence
of radiative forces
• Actually a thermal
(« radiometric ») effect
Crookes radiometer
Crookes radiometer
hν
c
Mirror
Black
Recoil of black surface
following the absorption
of light
Crookes radiometer
hν
c
But recoil of reflecting surface
is twice that of black surface!
Black
Mirror
Crookes radiometer puts in evidence the heating
of black surface that induces a mechanical
reaction of the residual gas in the glass cell
« Radiometric effects »
• Thermodynamic forces induced by the
heating of ambiant medium
• For Crookes radiometer a simplistic model
yields to :
Fradiometric ~ Fradiation
c
Vthermal −velocity
Fradiometric~Fradiation x 3.108/300m/s = Fradiation x 106