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

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 Angular momentum of light circular polarization helical wavefront Vortex beam

Contents • What is a vortex beam 
 Property, Production, Observation • Propagation of plane wave • Propagation of a vortex beam • Results

What is Vortex beam?

:integer :azimuthal angule m=1 z What is vortex beam? eigenstate of orbital angular momentum ψ ∝ e i ( kz + m φ − ω t ) φ m L z ψ = − i ∂ ˆ ∂φψ = m ψ

vortex beam phase 0 2π Transverse plane m=1 0 amplitude Max

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?

Vortex beam carries Orbital Angular Momentum the vortex beam carries the orbital angular momentum about the propagation axis.

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

Production and Observation of the vortex beam

The pattern of interference with plane wave m=1 m=3

Production of vortex beam vortex beam Plane wave This plate is made of glass ,called spiral phase plate input Output

Laguerre Bessel Solutions of vortex beam Gaussian Laguerre function beam beam dispersion relation Bessel function ψ = J m ( q ρ )exp[ i ( − ω t + kz )]exp( im φ ) q 2 = ω 2 − k 2 � m � � − 2 ρ 2 /w 2 � � ψ = 2 ρ /w L m exp( im φ )( w 0 /w ) 0 exp[ − ρ 2 (1 /w 2 − ik/ 2 R ) − i Φ )] � � 2 � � � 2 � w 2 = w 0 2 z/kw 2 kw 2 2 z/kw 2 � � � � R = z 0 / 2 z Φ = ( m + 1)arctan 1 + 1 + 0 0 , ,

Propagation of plane waves in a Gravitational field

Hamilton equation of massless particle Eikonal approximation wave vector ψ ≡ Ae i S g µ ν � µ � ν ψ = 0 � � 1 � 2 g µ ν ( � µ S )( � ν S ) Ae i S � = 0 x α = ∂ H k α = − ∂ H ˙ ˙ ∂ k α ∂ x α , D ˙ x µ x µ ˙ x ν = 0 g µ ν ˙ D τ = 0 ,

plane wave vortex beam ? Geodesic equation Propagation of wave Eikonal approximation Eikonal approximation

Propagation of vortex beam ~flat spacetime~

Orbit of Bessel beam in flat spacetime :Bessel beam solution Jm:Bessel function q,k,ω:constant satisfying geodesic equation (exact solution of wave equation in flat spacetime) ψ B = J m ( q ρ ) e iS S = − ω t + kz + m φ � µ S -1 1 u µ = y 0 x 1 0 = ( − ω , 0 , m, k ) ¯ -1 u µ 2 π � 1 z u µ = u µ dS ¯ � dS π = ( − ω , 0 , 0 , k ) 0

transverse plane Decomposition of wave vectors u µ = ¯ u µ + v µ v µ u µ ∂ µ ¯ ¯ u ν = 0 ¯ u µ

~curved spacetime~ Propagation of vortex beam

Scale of beam radius d beam radius L L >> d curvature scale

Orbit of Bessel beam in a curved spacetime metric perturbation correction term Orbit + h µ ν g µ ν = η µ ν ψ = J m ( q ρ ) e i S + � S � δ u µ ≡ ∂ µ δ S ¯ u µ + δ ¯ k µ := ¯ u µ

Perturbed eikonal equation Averaging Ansatz ψ = ψ B e i � S � k µ = u µ + δ u µ H := 1 2 g µ ν k µ k ν − k µ h µ ν v ν + 1 2 q 2 = 0

Perturbed eikonal equation H := 1 2 g µ ν k µ k ν − k µ h µ ν v ν + 1 2 q 2 = 0 x α = ∂ H k α = − ∂ H ˙ ˙ ∂ k α ∂ x α D ˙ x µ x ν g να � µ h αβ v β D τ = ˙ the extra force between angular momentum of the vortex beam and curved space-time.

Riemann normal coordinate x B h µ ν = − 1 B )( x β − x β 3 R µ ανβ ( x α − x α B )

D ˙ x µ x α g να � ν h αβ v β D τ = ˙ D ˙ x µ D τ = − 1 2 q R µ ναβ u ν S αβ S νβ = 1 B v β − X β 2( X ν B v ν ) where X µ B = x µ − x µ B

How does vortex beam propagate around Kerr B.H? D ˙ x µ D τ = − 1 2 q R µ ναβ u ν S αβ

Orbit of vortex beam on the equatorial plane of a Kerr Black hole

a：Kerr parameter M:mass of black hole perturbative form of Kerr metric ds 2 = − (1 − 2 Φ ) dt 2 + 2 h ti dx i dt + (1 + 2 Φ ) δ ij dx i dx j h ti = 2 Ma Φ = M r 3 ( − y, x, 0) r

Expanding metric around Beam 2 q R µ ναβ u ν S αβ = 1 − 1 2 q ∂ µ ( ∂ l h tk − ∂ k h tl ) u t S kl = 1 B g · � 2 � µ ( � l ) l i = u t where � ∂ h ti � ∂ x j − ∂ h tj B ij = 1 2 q � ijk S jk ∂ x i 4 , D ˙ D � = 1 x µ B g · � 2 � µ ( � l )

Configuration of Bg D ˙ D � = 1 x µ B g · � 2 � µ ( � l ) � l � B g

y x Propagation of parallel to axis of black hole attracting force！ D ˙ D � = 1 x µ B g · � 2 � µ ( � l )

y x z Toward black hole on equatorial plane the force acts in the z direction

Propagation to the azimuth direction x z y z not acting extra force

Summary the vortex beam in the Kerr spacetime. • We obtained the equation for orbit of D ˙ D � = 1 x µ B g · � 2 � µ ( � l ) l i = u t � � � B g = � � 2 q � ijk S jk h m • Extra force depend on . q

Future Work • By using vortex beam, 
 we determine spin parameter of Black hole emitted by a same source in the Kerr space-time • observing distribution of m of light

Observation of vortex photon Phys. Rev. Lett. 88, 257901(2002) e im φ e im φ + im α photons with even → values of l into Port A1 photons with odd values of l into Port B1