Forces, light and waves · Energy flux associated with momentum flux (pressure): a progressive wave exerts Pressure = Energy flux (W/m2) / velocity of wave hence: ... Numerical calculation

Forces, light and wavesMechanical actions of radiation

Jacques Derouard« Emeritus Professor », LIPhy

Example of comets

Cf Hale-Bopp comet (1997)

Exemple of comets

Sun

Comet trajectory

Successive positions of a comet

Tail is in the directionopposite / Sun, as ifrepelled by the Sunradiation

Peter Arpian « Astronomicum Caesareum » (1577)

A phenomenon known a long time ago...

the first evidence of radiative pressure predicted several centuries later

• 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 thevelocity of sound is (much) smaller than the velocity oflight, acoustic radiative forces are potentially stronger.

• Instead of pressure, one can consider forces:Force = Energy flux x surface / velocity

henceForce = Intercepted power / wave velocity or:

Force (nanoNewton) = 3,3 . P(Watt)

Example of comet tailcomposed 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-Sundistance

Example of comet tailInfluence of the particles size r

• For opaque particles r >> 1µm– Intercepted power increases like the cross section ~r2

thus less strongly than gravitational force thatincreases like the mass ~r3

Example of comet tailInfluence 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 gravitationalforce ~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’, PhysicalReview 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’, PhysicalReview Letters, Vol. 24, No. 4, 156, 1970

Observes that

-the beads are pushed by the laser beam (andslowed by water drag force) <V>=26µm/s

Polystyren beads suspended in water

Beads r=1,32µm

Plaser=19mW

λ=515nm

w0=6,2µm

• NB polystyren beads are transparent:– no radiation absorption

– but deflection of light due to refraction

• Radiative force is the result of this deflection

Ashkin experiment (1970)

Radiation pressure

• Absorption, reflexion or scattering of a lightbeam by a particle

Fr

Fr Deflexion (refraction or scattering)

of a uniform light beam yields to aforce directed along the light beam

Absorption of light makesthe particle recoil

Ashkin experiment (1970)A. Ashkin, ‘Acceleration and trapping of particles by radiation pressure’, PhysicalReview Letters, Vol. 24, No. 4, 156, 1970

Observes that

-the beads are pushed by the laser beam

-the beads are attracted by the laser beam

Polystyren beads suspended in water

« Gradient force »

• Deflection or scattering of a non uniformintensity light beam by a particle

Fr Deflection of light of non uniform

intensity across the particle yields to aresulting force directed obliquely, thattends (in this case) to push the particletowards maximum intensity region

« Gradient force »

• Deflection or scattering of a non uniformintensity light beam by a particle

Fr

When the particle index ofrefraction is smaller than that ofthe medium (bubble), thedeflection of light tends to expellthe particle from maximumintensity region

(should also be observed withreflective particles)

Also observed by Ashkin in 1970

In conclusion two types of forcesexerted 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) attractedtowards high intensity regions (effect reversed withrefractive index contrast)

Radiatives forces

• Atomic particles: close to a resonantabsorption line σ is enormous, so are theradiative forces (-> cold atoms physics)

• (NB for dielectric particles Ashkin hasobserved scattering resonance throughradiative pressure resonance)

Expression of radiative forcescase of « small » particles (limit a<<λ)

Response of the particle toradiation field: complex polarisability "' ααα i+=

Radiation field characterized by

Energy density

Poynting vector= Intensity x propagation direction

>< )(rSrr

)(rUr

gradscat FFFrrr

+>=<

Gradient force

Radiation pressure


medscat nc

rSkF

><= )("

0

rrr

εα

- α’’ proportional to the sum of absorption and scattering crosssections

-α’’ > 0, always towards the propagation of the wave,maximum for absorption or scattering resonance frequencies

Radiation pressure


gradscat FFFrrr

+>=<

scatFr

−∇−= )(

2

'

0

rUFgrad

rrr

εα

-If α’ > 0 attraction towards large U regions

-If α’ < 0 repulsion from large U regions

-large variation of α’ close to resonance frequencies, maychange of sign (« blue detuned optical atomic traps »)

Gradient force


gradscat FFFrrr

+>=<

Radiation pressure and gradient forcesboth exists also with acoustic waves:

• Radiation pressure: associated with momentumflux transported by acoustic wave = Energy flux / velocity in the simplest cases

• Gradient force: for small spherical particles itresults from « Gor’kov potential ». For largeparticules it can be estimated like in geometricaloptics, where the analogous of refractive index is1/ρc

• Gradient force: for spherical particles it resultsfrom « Gor’kov potential ».

Particles are trapped at thenodes of a 2D network ofmoveable stationnary 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’+i α’’) ∆p

– Acoustic resonance mode

20'

4

1pF ∇=

rrα

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)

FGRAD

FPrad

laser

FPrad

FGRADFGRAD

FGRAD

Scattered light

F

Optical trapping and propelling of particles overan optical wave guide

Light intensity profile

Particule

F

LIGHT

F

LIGHT

Numerical calculation of the electromagnetic field energydensity and forces applied on a glass microparticles ofdiameter 250nm immersed in water and lying over asilicon nitride optical waveguide.

Experimental set-up

Siliconsubstrate

Opticalwaveguide

CCD camera

Microscopeobjective

Microparticlessuspended in water

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 opticaltrapping of particles


• Need to balance the effects of radiationpressure. Several possibilities:– gravity

– substrate

– 2 counter propagating light beams

– gradient force stronger than radiation pressure(strongly focused beam : « optical tweezer »)



– substrate



First trapping experiment: counterpropagating beams

A. Ashkin, ‘Acceleration and trapping of particles by radiation pressure’,Physical Review Letters, Vol. 24, No. 4, 156, 1970

Gradient forces attract beads towards beams axis

Opposite axial radiation pressure forces are balanced

Recente version of thisconfiguration: «optical stretcher »

(Guck et al, 2000, 2005)

• Ytterbium fiberedlaser injected in singlemode optical fibers

• Microfluidic channel

• Biological cellssuspended in water

100µm

Application: observation of thedeformation of a «fibroblast»

(Guck et al, 2005)

Monitoring oflaser beamintensity

Trapped cell:

As a result of radiativepressure the cell isdistorted

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 equalto nx hν /c0

Medium refractive

index n2

Medium refractive

index n1

Radiation

Intensity I

Radiative forces on material media

Momentum flux

I n1 /c

Radiation

Intensity I

Momentum flux

I n2 /c

If n2 > n1 then I/n2 > I/n1, hence a force F isexerted at the interface that tends to pull themedium 2

Fr

Radiative forces on material media

Medium refractive

index n1

Medium refractive

index n2

Momentum flux

I n1 /cMomentum flux

I n2 /c



– substrate



Ashkin 1986: First experiment of trapping aparticle using a single focused light beam:

“optical tweezer”

A. Ashkin et al ‘Observation of single-beam gradient force optical trap fordielectric particles’, Optics Letters, Vol. 11, No. 5, 288, 1986

Radiatives forces

• These forces are due to the momentum fluxtranported by the radiation.

• But light transports angular momentum aswell

« 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 ofcircularly polarized wave on a birefringent plate

Radiative torques

• Beth’s experiment (1936): mechanical action ofcircularly polarized wave on a birefringent plate

– Birefringent medium: not parallel to

– then torque per unit volume

– in the same time change of polarization of thetransmitted light (the total angular momentum oflight+material is conserved)

Pr

Er

EPrrr

×=Γ

Beth (1936)

Radiative torques

• Recent version: micro viscosimetry with« vaterite » (sort of calcite) particles (cfRubinsztein-Dunlop et al 2007)

Another configuration:« form birefringence »

• Non spherical object: induced polarizationnot parallel to– torque

• Conservation of implies that the angularmomentum of the scattered wave isaffected/ incident wave

EPrrr

×=Γ

Pr

Er

Jr

• Light transports angular momentum

– « orbital » angular momentum of radiation related tospatial modes of electromagnetic field

Radiative torques

• Light transports angular momentum– cf « transverse modes » of laser cavities, Laguerre-

Gauss modes

Radiative torques

)exp()./2(.

].)/(exp[)exp(.),,(22

2,

ϕ

ϕ

ilwrL

wrikzcoeffzrElp

lp

−

−⋅=Gauss

Laguerrepolynomia

non axisymetricmode

Propagation / z

l = 0 l = +1 l = +3

Laguerre-Gauss E0 l modes

Intensity distribution (Beijersbergen et al, 1992)

« Doughnut modes »« Mode TEM00 »

Wave Surfaces (Padgett, Courtial et Allen, 2004)

• Light transports angular momentum– Allen et al (1992):

• E0,l corresponds to photons having angular momentumof projection l.h/2π along z axis

• N photons/second correspond to an energy flux ofI = N.hω/2π

• E0,l with N photons/second corresponds to an angularmomentum flux (torque!) of J = N.l.h/2π = Ι.l/ω The larger the smaller ω !

Radiative torques

Generation of modes Ep,l

Grier, 2003

One possibility: transmission of a TEM00 wavethrough a helicoidal phase plate

Other possibility: diffraction of a mode TEM00 by a« fork » hologram

Binary, ( not « blazed »)

Or phase hologram(« blazed »)

Generation of modes Ep,l

Grier, 2003

Application to trapping and rotation ofmicrobeads

Acoustics

Pionnier in thestudy of « opticalvortices»

Acoustics

Conclusions and Résumé

• Radiative forces and torques: linear and angularmomenta transport by the waves– Light wave

– Also sound waves (strong analogies but some more orless subtle differences). Potentially larger effects thanksto sound wave velocity and frequency much smallerthan 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 ofmicroparticles, microviscosimetry...

– Manipulation of microparticles, Lab on chips

Thank you!

32

2

0 2

14' a

n

ne

+−

ℜ= πεα

+−

+

+−

ℑ= 63

2

2

2

03

2

2

0 2

14

3

2

2

14" ak

n

na

n

nm πεπεα

Absorption Scattering

Expression of real and imaginary parts of the polarizability of aspherical particle compex refraction index n, radius a << λ/ n(Rayleigh regime)

Crookes radiometer (1873)

• Initially improperly taken asevidence for the existenceof radiative forces

• Actually a thermal(« radiometric ») effect

Crookes radiometer

Crookes radiometer

Recoil of black surfacefollowing the absorptionof light

BlackMirror

c

hν

Crookes radiometer

BlackMirror

c

hν

But recoil of reflecting surfaceis twice that of black surface!

Crookes radiometer puts in evidence the heatingof black surface that induces a mechanicalreaction of the residual gas in the glass cell

« Radiometric effects »

• Thermodynamic forces induced by theheating of ambiant medium

• For Crookes radiometer a simplistic modelyields to :

velocitythermalradiationcradiometri V

cFF

−

~

Fradiometric~Fradiation x 3.108/300m/s = Fradiation x 106

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Forces, light and waves · Energy flux associated with momentum flux (pressure): a progressive wave exerts Pressure = Energy flux (W/m2) / velocity of wave hence: ... Numerical calculation