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) / 3.10 8 (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 ~ 2.10 8 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)
• 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 Absorption of light makes F the particle recoil
r Deflexion (refraction or scattering) F 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 Deflection of light of non uniform F 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
When the particle index of refraction is smaller than that of the medium (bubble), the deflection of light tends to expell r the particle from maximum F 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 r Energy density U (r ) r r Poynting vector < S(r ) > = Intensity x propagation direction 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 r r r α" < S(r) > F = k n Radiation pressure scat ε med 0 c
- α’’ proportional to the sum of absorption and scattering cross sections r α - ’’ > 0 , F scat 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 F = −∇− U (r) Gradient force grad ε 2 0
-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 F = α'∇p 2 4 0 – 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 F GRAD Light intensity profile FGRAD
FPrad FPrad F GRAD FGRAD
laser 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
Optical waveguide Microscope Microparticles objective 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 x radiation pressure = (Intensity / c 0 ) n
• • Actually it seems that in a number of cases, everything is as if the photons transported by the wave had momentum equal x ν to n h /c0 Radiative forces on material media Medium refractive Medium refractive index n index n1 2 Radiation Radiation
Intensity I Intensity I
Momentum flux Momentum flux
I n 1 /c I n 2 /c Radiative forces on material media Medium refractive Medium refractive index n index n1 2
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 n 1 /c I n 2 /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/2 π 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 rparallelr rto E – 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:r induced polarization P not parallelr to rE 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, Laguerre- Gauss modes Propagation / z Gauss ϕ = ⋅ − 2 E p,l (r, z, ) coeff .exp( ikz ) exp[ (r / w) ]. l 2 2 − ϕ .Lp (2r / w ). exp( il )
Laguerre non axisymetric polynomia mode Laguerre-Gauss E0 l modes Wave Surfaces (Padgett, Courtial et Allen, 2004)
l = 0 l = +1 l = +3
« Mode TEM00 » « 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 /2 π along z axis
• N photons/second correspond to an energy flux of I = N.hω/2 π
• E0,l with N photons /second corresponds to an angular π = Ι. /ω momentum flux ( torque!) of J = N.l.h /2 l 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 α = πε ℜ 3 ' 4 0 e 2 a n + 2
2 2 2 n −1 2 n −1 α"= 4πε ℑm a3 + 4πε k 3a6 0 n2 + 2 3 0 n2 + 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
Recoil of black surface Mirror Black following the absorption of light Crookes radiometer hν c
But recoil of reflecting surface is twice that of black surface! Mirror Black 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 : c Fradiometri c ~ Fradiation Vthermal −velocity
8 6 Fradiometric ~Fradiation x 3.10 /300m/s = Fradiation x 10