Propagation of vortex beam around a Kerr black hole Atsuki Masuda - - PowerPoint PPT Presentation

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Propagation of vortex beam around a Kerr black hole Atsuki Masuda - - PowerPoint PPT Presentation

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Propagation of vortex beam around a Kerr black hole Atsuki Masuda Osaka City University collaborator: Hideki Ishihara(Osaka City University) Shunichiro Kinoshita (Osaka City University) Spin angular momentum Orbital angular momentum

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Propagation of vortex beam around a Kerr black hole

Atsuki Masuda Osaka City University

collaborator: Hideki Ishihara(Osaka City University) Shunichiro Kinoshita (Osaka City University)

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Angular momentum of light

  • Spin angular momentum
  • Orbital angular momentum

circular polarization

helical wavefront

Vortex beam

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Contents

  • What is a vortex beam


Property, Production, Observation

  • Propagation of plane wave
  • Propagation of a vortex beam
  • Results
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What is

Vortex beam?

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:integer :azimuthal angule

ψ ∝ ei(kz+mφ−ωt)

m=1

m

φ

z

What is vortex beam?

ˆ Lzψ = −i ∂ ∂φψ

= mψ

eigenstate of orbital angular momentum

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vortex beam

phase

2π

Transverse plane

m=1

amplitude

Max

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Recently, some application of vortex beam to astrophysics have been considered

Application information science, the vortex beam has more information than the plane wave

・Allen et. al , Phvs. Rev. A, 45, 8185 (1992)

・ ・

  • J. Wangi et. al, Nature Photonics 6, 488-496(2012)
  • F. Tamburini et. al, Nature Physics 7, 195(2011)

Why vortex beam?

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Vortex beam carries Orbital Angular Momentum

the vortex beam carries the orbital angular momentum about the propagation axis.

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wave fronts

m=-1 m=+3 m=+2 m=+1 m=0

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Production and Observation

  • f the vortex beam
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The pattern of interference with plane wave

m=1 m=3

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Production of vortex beam

vortex beam Plane wave

input Output

This plate is made of glass ,called spiral phase plate

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w2 = w0

  • 1 +
  • 2z/kw2

2 R = z

  • 1 +
  • kw2

0/2z

2

Φ = (m + 1)arctan

  • 2z/kw2
  • ,

,

Laguerre Gaussian beam

Solutions of vortex beam

ψ =

  • 2ρ/w

m Lm

  • −2ρ2/w2

exp(imφ)(w0/w)

Laguerre function

Bessel beam

ψ = Jm(qρ)exp[i(−ωt + kz)]exp(imφ)

q2 = ω2 − k2

dispersion relation

Bessel function

exp[−ρ2(1/w2 − ik/2R) − iΦ)]

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Propagation of plane waves in a Gravitational field

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Eikonal approximation

wave vector

1 2 gµν(µS)(νS)Aei S

= 0

gµν µ νψ =0

ψ ≡ Aei S

  • Hamilton equation of massless particle

˙ xα = ∂H ∂kα ˙ kα = − ∂H ∂xα

D ˙ xµ Dτ = 0

gµν ˙ xµ ˙ xν = 0

,

,

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plane wave

vortex beam

?

Geodesic equation

Propagation of wave

Eikonal approximation Eikonal approximation

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Propagation of vortex beam ~flat spacetime~

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Orbit of Bessel beam in flat spacetime

uµ =

= (−ω, 0, m, k)

µS

satisfying geodesic equation

¯ uµ = 1

  • dS
  • uµdS

= (−ω, 0, 0, k)

:Bessel beam solution

(exact solution of wave equation in flat spacetime)

ψB = Jm(qρ)eiS

S = −ωt + kz + mφ

q,k,ω:constant Jm:Bessel function

  • 1
1
  • 1
1 2π π

z

y x

¯ uµ

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vµ

¯ uµ

¯ uµ∂µ¯ uν = 0

transverse plane

Decomposition of wave vectors

uµ = ¯ uµ + vµ

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~curved spacetime~ Propagation of vortex beam

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Scale of beam radius

L

curvature scale

d

beam radius

L d >>

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Orbit of Bessel beam in a curved spacetime

+hµν

gµν = ηµν

metric perturbation

correction term

Orbit

¯ kµ := ¯ uµ + δ¯ uµ

ψ = Jm(qρ)ei S+S

  • δuµ ≡ ∂µδS
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Perturbed eikonal equation

Averaging

Ansatz

ψ = ψBei S

  • kµ = uµ + δuµ

H := 1 2gµνkµkν − kµhµνvν + 1 2q2 = 0

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Perturbed eikonal equation

the extra force between angular momentum

  • f the vortex beam and curved space-time.

H := 1 2gµνkµkν − kµhµνvν + 1 2q2 = 0

D ˙ xµ Dτ = ˙ xνgναµhαβvβ

˙ xα = ∂H ∂kα

˙ kα = − ∂H ∂xα

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Riemann normal coordinate

hµν = −1 3Rµανβ(xα − xα

B)(xβ − xβ B)

xB

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Sνβ = 1 2(Xν

Bvβ − Xβ Bvν)

Xµ

B = xµ − xµ B

where

D ˙ xµ Dτ = ˙ xαgνανhαβvβ

D ˙ xµ Dτ = − 1 2q RµναβuνSαβ

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How does vortex beam propagate around Kerr B.H?

D ˙ xµ Dτ = − 1 2q RµναβuνSαβ

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Orbit of vortex beam

  • n the equatorial plane
  • f a Kerr Black hole
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a：Kerr parameter M:mass of black hole

perturbative form

  • f Kerr metric

Φ = M r

hti = 2Ma r3 (−y, x, 0)

ds2 = −(1 − 2Φ)dt2 + 2htidxidt + (1 + 2Φ)δijdxidxj

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Expanding metric around Beam

Bij = 1 4 ∂hti ∂xj − ∂htj ∂xi

  • ,

where

= 1 2µ( Bg · l)

− 1 2q RµναβuνSαβ = 1

2q ∂µ(∂lhtk − ∂khtl)utSkl

li = ut 2q ijkSjk

D ˙ xµ D = 1 2µ( Bg · l)

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  • Bg

Configuration of Bg

  • l

D ˙ xµ D = 1 2µ( Bg · l)

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y x

Propagation of parallel to axis

  • f black hole

attracting force！ D ˙ xµ D = 1 2µ( Bg · l)

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y x z

Toward black hole

  • n equatorial plane

the force acts in the z direction

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Propagation to the azimuth direction

x z y z

not acting extra force

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Summary

  • We obtained the equation for orbit of

the vortex beam in the Kerr spacetime.

  • Extra force depend on .
  • Bg =

h D ˙ xµ D = 1 2µ( Bg · l)

li = ut 2q ijkSjk

m q

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Future Work

  • By using vortex beam, 


we determine spin parameter

  • f Black hole
  • observing distribution of m of light

emitted by a same source in the Kerr space-time

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Observation of vortex photon

photons with even values of l into Port A1 photons with odd values of l into Port B1

eimφ

eimφ+imα

→

  • Phys. Rev. Lett. 88, 257901(2002)