Plane Waves as Tractor Beams - KFKIarpi/talks/szeged15.pdf · Intro2chApplicationsConclusionsBackup slides Plane Waves as Tractor Beams arXiv:1303.3237, Phys. Rev. D88 125007 Árpád

Intro 2ch Applications Conclusions Backup slides

Plane Waves as Tractor BeamsarXiv:1303.3237, Phys. Rev. D88 125007

rpd LukcsCollaborators: Pter Forgcs, Tomasz Romanczukiewicz

Wigner RCP RMKI,Budapest, Hungary

SZTE Elmleti Fizikai Tanszk Szeminriuma2015. December 3.

Lukcs . Tactor beams 1

Intro 2ch Applications Conclusions Backup slides NRP NRP on kinks Other approaches

Outline

1 IntroductionWhat is negative radiation pressure?NRP on kinksOther approaches

2 NRP in two channel scattering1dExamplesForce in 2d scattering

3 ApplicationsNeutron scattering on XY model vortices

4 Conclusions



Radiation pressure

Scattering of light: the pressure acting on a platelet

P = (1 + R2 T 2)W ,

W = 0E20/2

R: reflection, T : transmission

F

R=0 : F=0

Can the pressure be negative?What objects can be pulled towards the radiation source?Unitarity (conservation of energy): R2 + T 2 = 1, P = 2WR2 > 0.How can P be negative?



NRP on kinks

Kinks in the 4 theorylinearized perturbations: scattering reflectionlessnonlinearities generate higher harmonics

(Video: Tomasz Romanczukiewicz)

More momentum in the forward direction NRP(Romanczukiewicz 2004, Forgcs, Lukcs and Romanczukiewicz 2007.)It should be possible with 2 channels, different momenta


tromneg.mpgMedia File (video/mpeg)


Other approaches

Gain media (Mizraki and Fainman, 2010)

F

Structured beams (Sukhov and Dogariu, 2011)

F

Negative , (Veselago 1967)



Other approaches

Gain media (Mizraki and Fainman, 2010)

F

Structured beams (Sukhov and Dogariu, 2011)

F

Negative , (Veselago 1967)Lukcs . Tactor beams 5

Intro 2ch Applications Conclusions Backup slides 1d Examples 2d

Two channel scattering in 1d

Plane waves as tractor beams in unitary scatteringTwo kinds of waves (channels) i = 1,2momentum Pichannel 1 incoming wave

p = |A1|2[P1(1 + |R11|2 |T11|2) + P2(|R21|2 |T21|2)

],

where Rij and Tij : reflection and transmission coefficient,channel i j .energy flux Si ; energy conservation

i

Si(|Rij |2 + |Tij |2) = Sj .

one channel: no NRP, p = 2|A|2P1|R|2 > 0.NRP expected: P2 > P1 and large T21.



Birefringent media

Anisotropic dielectric tensor

= diag(x , y , z)

Two types of wave: polarizationsWave propagation in the z direction

Two modes: the x and y polarizations, assume nx < nyMomentum flux of plane waves Pi = i/2Energy fluxes Si =

i//2

Pressurepz|Ex0|2

= Px(1 + |Rxx |2 |Txx |2) + Py (|Ryx |2 |Tyx |2) ,

where Tij transmission, Rij reflection, channel j i

Energy conservation:

i Si(|Rij |2 + |Tij |2) = Sj .



Force formula

ScattererPlane of thickness L, same material, rotated by

Neglect multiple surface interactions

pz = 0|Ex0|2(

P0 + P1 cosny nx

cL),

small angle

P0 = 2(nx ny )(4n2x + 3nxny + n2y )/4ny

P1 = 2(nx ny )(nx + ny )2/2/nyTractor beam: nx < nynx,y = n n: P0 = P1 = 2nn2

At 45: P0 = P1 = nn/2



Macroscopic example

Dielectric

nx = 3, ny = 6

Incoming wave: 1 kW/cm2 at = 1GHz

PressureP0 = 1.08, P1 = 1.99, avg. pressure Pz = 0.072 Pa

Radiation pressure at total reflection: 0.6 PaAccuracy: exact value 0.053 Pa

This is a macroscopic effect.



An optical example

Dielectric

Liquid crystal 5CB, at 25 C, = 5893 : nx = 1.53, ny = 1.72

Scatterer: rotated birefringent plateletE.g., L = 0.1 mm, = /4

Pressure

x polarization pz = 5.52 1013Pa (m/V)2|E0|2

y polarization pz = 7.48 1013Pa (m/V)2|E0|2

Accuracy: 4.95 1013Pa (m/V)2|E0|2 and8.14 1013Pa (m/V)2|E0|2A few percents of radiation pressure on a totally reflecting mirror!



Force in 2d scattering

Scattering in 2d, rotational invariancePartial wave expansion (Fourier trf. in )S-matrix elements S`: n n matrix (n: channels)S` unitary (conservation of energy)

Momentum balance: force master formula

F = Fx + iFy = 4`

{AS`+1K S`A A

KA},

A = (A1, . . . ,An)T amplitude, K = diag(k1, . . . , kn) wave numbers

A consequence of unitarity:

Re |Aa|2ka(1 Saa,`+1Saa`) > 0

one channel radiation pressure positive

NRP: kb > ka necessary


Intro 2ch Applications Conclusions Backup slides XY model

Scattering on vortices

Two types of wave (particle species/spin/etc.)Neglect vortex corer asymptotic form of A a two channel Aharonov-Bohm scattering problem(

+ iA22

)2 K 2 = 0 , 2 =

(i

i

),

whereA = e/r = (u, d), u: heavy and d light modefermionic bdry cond. (r , + 2) = (r , )

Solution: partial waves, radial eq. numericallyNRP for large rane of parametersCross section geometric, 1/k1/sin/2 in scattering amplitudeLarge cross section from one channel to another



Scattering off cosmic strings

Inside a cosmic string: GUT Higgs zero, flux of broken gauge fieldCosmic string catalyzed baryon number violation:

B + string `+ string

A simplified description:Neglect spin degrees of freedom2 channel AharonovBohm scattering1 heavy baryon, 1 light lepton, mass ratio 1.5 : 2.Large cross section:cosmic string catalyzed baryon number violationForce: string friction (moving in a plasma)

Decoupled approximation

Fi = 4niv(1 exp(2ii ))

ni density, i coupling to the X-bosononly valid for light modes, heavy modes give negative contributionat v = 0.65

F ux = 6.09|A|2 , F dx = 7.44|A|2 .scattering energy mi/

1 v2





B + string `+ stringA simplified description:

Neglect spin degrees of freedom2 channel AharonovBohm scattering1 heavy baryon, 1 light lepton, mass ratio 1.5 : 2.

Large cross section:cosmic string catalyzed baryon number violationForce: string friction (moving in a plasma)





1 v2





B + string `+ stringA simplified description:

Neglect spin degrees of freedom2 channel AharonovBohm scattering1 heavy baryon, 1 light lepton, mass ratio 1.5 : 2.Large cross section:cosmic string catalyzed baryon number violationForce: string friction (moving in a plasma)





1 v2



Force: x

x component

-10

-5

0

5

10

15

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

Fx/

(Am

plitu

de)2

Fx,u

Fx,d

Fx,ddc



Force: y

y component

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

Fy/

(Am

plitu

de)2

Fy,u

Fy,d



Neutron scattering: XY model vortices I

XY modelrotators (spins) in a plane, with nearest neighbor interaction

Magnetic vortex: singularity of magnetization M

gM energy difference between parallel and antiparallel spin neutrons

Diagonalize Hamiltonian locally2 modes, ~2/2/m(k2d k2u ) = gMsmall momentum transfer

Measurable manifestation of the same phenomenon as NRPLarge cross sections, 1/k (A-B)Large spin-flip cross sectionE = 4.1 105 eV (45 neutrons), du = 1.19 104 mCan be calculated perturbatively


Intro 2ch Applications Conclusions Backup slides

Conclusions

Many approaches to tractor beams (e.g., structured beams)Multi-channel scattering, ki 6= kj fairly generalOne-dimensional examples:

polarizations of EM waveshigher harmonics (kink)

Two dimensionsCosmic strings: baryon decayMagnetic XY-vortex

THANK YOUFOR

YOURATTENTION!


Intro 2ch Applications Conclusions Backup slides Physics of cosmic strings

Cosmic strings

Classical field theoretical solutionThin, elongated objectString core: a zero of a Higgs fieldEnergy density localized in the core

Important parameter: string tension

= E/L

Electroweak string: G 1032 (: 10 mg/Solar diam)GUT string: G 106 (: Solar mass/Solar diam)

Cosmic strings are high energy localized objects that provide a linkbetween astrophysics and particle physics.



Physics of cosmic strings

Formation: during phase transitionsEvolution of a string network

Friction dominated era: scattering of particlesScaling v 0.65collisions, interlinkingradiation (e.g. at cusps formed in collisions)string tension contracts loopsexpansion: the network becomes more diluted

Signatures of cosmic stringsScattering of material off strings: structure formation: galaxies,voids, filaments (fractal dimension)Contribution to CMB anisotropy: best fit with GUT stringsG = (2.04 0.13) 106, Contribution to multipole` = 10:f10 = 0.11 0.05 (Hindmarsh et al., 2007, 2008)Gravitational lensingGravitational radiation



Artificial gauge potential

Diagonalize locally the Hamiltonian

U(V )U = K

K = diag(k1, . . . , kn)kinetic term:

UiU = + U(iU) = iAi

A : artificial gauge potentialU = U(): Aharonov-Bohm form

A =Ar

e ,

where A = UU/



Scalar perturbations of the global vortex

-5

0

5

10

15

20

25

6 8 10 12 14 16 18 20

Fx(

2)=

Fx/

(am

plitu

de)2

2

Fx,a(2)

Fx,g(2)

One massive, one Goldstone (massless) modeLukcs . Tactor beams 21


Perturbations of the superfluid vortex

-3

-2

-1

0

1

2

3

4

5

0 0.5 1 1.5 2 2.5 3 3.5

F(2

) =F

/(A

mpl

itude

)2

x

y

x PT

y PT

Asymptotics: H(r/2), Ki(2r), one channelLukcs . Tactor beams 22


AharonovBohm scattering I

The AharonovBohm effect:Motion of a charged particle in a region with B = 0

Double slit expreriment: Scattering:

Both experiments show flux dependenceHolonomy is also physical not just field strength Pei

Adr

Reaction force (deflected beam): Force acting on the scatterer



AharonovBohm scattering II

Schrdinger-equation

i = ( iA)2 ,

with electromagnetic vector potential

A(r , , z) =A0r

e .

2A0 flux; outside B = 0.Fixed energy: (r, t) = eit(r).Scattering asymptotics (?)

eikx + f ()r

eikr

Cross sections: d/d = |f ()|2Scattering amplitude: f () sinA02

1sin(/2) .



AharonovBohm scattering III

Partial waves: s` J|`A0| (note index shift wrt plane wave)Effect of scattering in the outgoing wave: phase shift `,S` = exp(2i`)(J(z) cos(z /2 /4)/

2z)

` 0 : ` =A0

2, ` 1 : ` = `

A02

.

thus

Fx = |0|24k

`=

{cos [2(` `1)] 1}

= |0|24k (cos(2A0) 1) 16|0|22A20k ,

Fy = |0|24k

`=

sin [2(` `1)] = |0|24k sin(2A0) 8|0|2A0k .

Analog problem: force acting on a superfluid vortex (GPe)



AharonovBohm scattering IV

Iordanskii force controversy C. Wexler, D.J. Thouless, Phys. Rev. B58R8897R8900 (1998):

f () = f () Fy = 0

A.L. Shelankov Europhys. Lett. 43 (1998) 623M.V. Berry J. Phys. A: Math. Gen. 32 (1999) 5627:scattering asymptotics does not hold in forward direction

(r, t) = ei(tkx)(r)

and for y

x

(x > 0, y) cos(A0)2i1/2

sin(A0)

k2

yx

transversal force Fy from this region (although not Fx )



References I

Forgcs,P., Lukcs,., Romanczukiewicz,T., Plane waves astractor beams Phys. Rev. D88, 125007.

Pter Forgcs, rpd Lukcs, and Tomasz Romanczukiewicz,Negative radiation pressure exerted on kinks Phys. Rev. D77,125012 (2008).

T. Romanczukiewicz, Negative radiation pressure in case of twointeracting fieldsActa. Phys. Polonica B39 (2008) 3449-3462.

A. Mizrahi, and Y. Fainman, Negative radiation pressure on gainmedium structures Opt. Lett. 35 3405 (2010); K.J. Webb andShivanand, Negative electromagnetic plane-wave force in gainmedia Phys. Rev. E84, 057602 (2011).

A. Dogariu, S. Sukhov, and J.J. Senz Optically inducednegative forces Nat. Pho. 7, 24 (2012);



References II

S. Sukhov and A. Dogariu, On the concept ot tractor beamsOpt. Lett. 35 3847 (2010); S. Sukhov and A. Dogariu, NegativeNonconservative Forces: Optical Tractor Beams for ArbitraryObjects Phys. Rev. Lett. 107, 203602 (2011); O. Brzobohat,V. Karsek, M. iler, L. Chvtal, T. Cimr, and P. Zemnek,Experimental demonstration of optical transport, sorting andself-arrangement using a tractor beam Nat. Pho. 7, 123 (2013).

F. Capolino, Theory and phenomena of metamaterials, (CRCPress, Boca Raton London New York, 2009).

V.G. Veselago, The electrodynamics of substances withsimultaneously negative values of and Sov. Phys. Usp. 10509 (1968).

P.P. Karat and N.V. Madhusudana, Elastic and optical propertiesof some 4-n-Alkyl-CyanobiphenylsMol. Cryst. Liq. Cryst.3651-64 (1976).



References III

A.L. Shelankov, Magnetic force exerted by the AharonovBohmlineEurophys. Lett. 43 (6) pp. 623-628 (1998).

A. Vilenkin and E.P.S. Shellard, Cosmic strings and othertopological defects, (Cambridge University Press, Cambridge,1994).

M.G. Ahlford, F. Wilczek, AharonovBohm Interaction of CosmicStrings with Matter Phys. Rev. Letters 62, 1071 (1989);M.G. Ahlford, J. March-Russell, and F. Wilczek, Enhanced baryonnumber violation due to cosmic strings Nucl. Phys. B328 (1989)140-158; A.C. Davis and A.P. Martin, Global strings and theAharonovBohm effect Nucl. Phys. B419 (1994) 341-351;L. Perivolaropoulos, A. Matheson, A.-C. Davis, andR.H. Brandenberger, Nonabelian AharonovBohm baryondecayPhys. Lett. B245 (1990) 556-560. W.B. Perkins,L. Perivolaropoulos, A.-C. Davis, R.H. Brandenberger and



References IV

A. Matheson, Scattering of fermions from a cosmic stringNucl.Phys. B353 (1990) 237-270;

T.-P. Cheng and L.-F. Li, Gauge theory of elementary particlephysics, (Clarendon Press, Oxford, 1988); R.N. Mohapatra,Unification and Supersymmetry, (Springer-Verlag, New York,2003).

J. March-Russell, J. Preskill, and F. Wilczek, Internal framedragging and a global analog of the AharonovBohm effect Phys.Rev. Letters 68, 2567 (1992).


IntroductionWhat is negative radiation pressure?NRP on kinksOther approaches

NRP in two channel scattering1dExamplesForce in 2d scattering

ApplicationsNeutron scattering on XY model vortices

Conclusions

Documents

Plane Waves as Tractor Beams - KFKIarpi/talks/szeged15.pdf · Intro2chApplicationsConclusionsBackup slides Plane Waves as Tractor Beams arXiv:1303.3237, Phys. Rev. D88 125007 Árpád