1. No category

カー時空中での軌道角運動量
をもつ光の伝播
桝田 篤樹
大阪市立大学
共同研究者
石原秀樹（大阪市大） 木下俊一郎（大阪市大）
ブラックホール磁気圏勉強会
軌道角運動量をもつ光
＝光渦
＝Twisted wave
目次
•
Kerr時空でのTwisted
•
Twisted waveとその性質
waveの伝播
•
まとめ
Twisted waveとは
i(kz+m
t)
e
m=1
ˆ
Lz
z
=
i
=m
軌道角運動量の固有状態
Twisted wave
Max
0
m=1
amplitude
2π
0
phase
About Twisted wave
• 波数ベクトルがねじれている
Twisted wave 角運動量を運ぶ
•
et al.: Optical vortices with starlight
•
References
Twisted
wave は渦構造をもっている
G. Anz
Lee, J. H., Foo, G., Johnson, E. G., & Swartzland
Phys.
Rev. Lett., 97, 053901
Twisted
waveは実験室で実際に生成されている
Allen,Mawet,
L., Beijersbergen,
M.Absil,
W., Spreeuw,
R. J.J.C.,
& Woerd
D., Riaud, P.,
O.,
&
Surdej,
2005,
ApJ,
6
1995,Phys. Lett. 75, 826-829
J. P. 1992, Phys.
Rev.
A,
45,
8185
McAlister, H. A., Hartkopf, W. I., Sowell, J. R., Dombro
Arecchi, O.
F.G.T.,1989,
Giacomelli,
G., Ramazza, P. L., & Res
AJ, 97, 510
Rev.
Lett.,
67,
3749
S. 1991, Phys.
Molina-Terriza, G., Torres, J. P., & Torner, L. 2007, Natu
Barbieri,
C.,
Dravins,
D.,
Occhipinti,
T.,
et
al.
2007,
J.
Mod.
Op
近年、宇宙物理学の分野でも注目を集め始めている
Nye, J. F., & Berry, M. V. 1974, Proc. R. Soc. London A,
Bazhenov,
V.
Y.,
Vasnetsov,
M.
V.,
&
Soskin,
M.
S.
1990,
JETP
, 191
Oemrawsingh, S. S. R., Eliel, E. R., Nienhuis, G., & W
M. W.,
Coerwinkel, R. P. C., Kristensen, M., &
Rev. Lett. 101, 100801
(2008)
, 52, Phys.
429Beijersbergen,
J. Opt. Soc. Am. A, 21, 2089
wave front
m=-1
m=+2
m=0
m=+1
m=+3
具体的な解
Laguerre
Gaussian
beam
w = w0 1 +
2
Bessel
beam
=
2 /w
m
2
2
Lm
2
/w
exp(im )(w0 /w)
0
exp[
2 2
2z/kw0
,R = z 1 +
2
2
kw0 /2z
= Jm (q )exp[i(
分散関係
q =
2
2
1/w
,
= (m + 1)arctan 2z/kw02
2
ik/2R
i ]
t + kz)]exp(im )
2
k
2
Bessel beam
B
= Jm (q
iS S =
)e
q =
2
J3 ( )
0.4
0.3
0.2
0.1
-0.1
-0.2
-0.3
t + kz + m
2
4
6
8
10
2
k
2
Production of twisted wave 1
input
output
Circularly
liquid crystal
Twisted wave
Polarization
・Birefringence;refractive index depend on
the polarization
・Refractive depend on site
Propagation of twisted
waves in a Gravitational field
massless KG equation
g
µ
µ
=0
Propagation of hight
frequency wave
g
g
µ
µ
=0
µ
µ
(Ae
i
S
)
1
g
2
µ
(
µ S)(
S) = 0
k kµ = 0
µ
ハミルトン=ヤコビ
方程式
µ
wave vector
Orbit of Bessel beam in flat
:Bessel beam solution
(exact solution with wave equation in flat spacetime)
B
= Jm (q )e
iS
uµ =
=(
u
¯µ =
t + kz + m
µS
J. Opt. A: Pure Appl. Opt. 10 (2008) 035005
, 0, m, k)
uµ dS
′′
′ ′
r φ + 2r φ = 0,
u
¯µ
M V Berry and K T Mc
uniquely defined at each point. P (r ) depends nonlinearly on
ψ(r ), so the superposition of trajectories is different from the
trajectories of the superposition.
In vacuum or any homogeneous medium, the geometrical
rays are straight lines. This is an immediate consequence
of Snell’s law (or, more abstractly, Hamilton’s equations),
according to which rays bend only if there is a variation of
refractive index. The straightness is not immediately obvious
from the algebraic formulae describing the rays, but will be
confirmed by showing that they satisfy the following equations,
expressing the vanishing of the curvature |r ′ × r ′′ |/|r ′ |:
1
dS
=(
S=
′′
′2
r − r φ = 0.
(1.4)
By contrast, the trajectories of the exact Poynting vector are
usually curved.
For the beams we study here, it turns out, unexpectedly, to
be easier to calculate the exact Poynting trajectories than the
geometrical rays, which require knowledge of the asymptotics
of Bessel and Laguerre functions.
, 0, 0, k)
y
0
-1
2π
z
π
0
1 -1
0
x
1
wave vector of Bessel beam in flat
spacetime
uµ = u
¯µ + vµ
u
¯µ
u
¯
µ
¯
µu
vµ
transverse plane
=0
Bessel beam in a curved spacetime
gµ =
+hµ
µ
iS +i
Jm (q )e
S
Orbit
uµ
u
¯µ+
u
¯=
1
dS
1
uµ dS + dS
µ
S
uµ dS
Perturbation equation
Ansatz
(
µ
¯µ )(¯
¯ )=
h )(¯
u+ u
u + u
µ
1 µ ¯ ¯
H = g kµ k
2
q + hµ u
¯µ v
2
1 µ
h u
¯µ v =
2
1 2
q
2
Perturbation equation
1 µ ¯ ¯
1
2
µ
H = g kµ k
h u
¯µ v =
q
2
2
Differentiation
µ
¯
k
¯
k
=
µ
µ
h
µ
u
¯µ v
How does twisted wave
propagate around Kerr B.H?
µ
¯
k
¯
µk =
µ
h
13
u
¯µ v
perturbative metric of Kerr
ds =
2
(1
M
=
r
2
)dt + 2 Ai dx dt +
2
i
ij dx
i
dx
j
2M a
Ai = 2 ( y, x, 0)
r
M:mass of black hole
a：Kerr parameter
Scaleの比較
時空の曲率半径
>>
ビームの局在スケール
>>
ビームの波長
Beam周りでの展開
hµ
u
¯µ v
hµ |XB
hµ
x
=
1
=
4
hµ
+
x
xB ) u
¯µ v
xB
XB u
¯µ v
(XB = x
xB
Ai
xj
1
= Bg · l
4
µ
¯
k
(x
Aj
xi
i vj
(XB
xB )
j i
XB v )
XB
1
¯
µk =
4
(Bg · l)
Bgの配位
l
Bg
重力場中の運動方程式
u
µ
u
=
0
µ
1
+
4
(Bg · l)
弱重力
dv
=
dt
+v
1
Bg + (Bg · l)
4
Orbit of twisted wave
on the equatorial plane
of Black hole
propagation of parallel to
axis of black hole
z
y
x
y
x
¯µ
k
1
¯
µk =
2
(Bg · l)
Propagation of parallel to
axis of black hole
z
y
x
y
x
attracting force！
Torward black hole
on equator
z
y
z
x
y
Tangential on the equatorial plane
z
z
y
x
y
x acting interaction term
not
SUMMARY
• Kerrブラックホールの周りのTwisted wave
の軌道が満た方程式を得た
¯µ
k
1
¯
µk =
2
(Bg · l)
• 相互作用項は外部磁場と磁気モーメント相互作用
• 求まった方程式はa=0では測地線方程式になる
• 外力は
1
4
r
Future Work
• 測地線とのずれからKerr時空の情報を得る
• 偏光情報を含めての光の軌道を考える
• Twisted waveのB.Hシャドー
Propagation of wave
Plane wave
Eikonal
approximation
Geodesic equation
Twisted wave
?
Eikonal
approximation
Production of twisted wave 2
ref NaturePhysics 7, 195-197
input
Circularly
Polarization
output
Twisted wave
It is known that there is analogy between Maxwell
equation in medium and curved spacetime
Perturbative approach
他の角運動量-角運動量相互
作用との比較
バイナリー連星の相互作用
ポストニュートン
円偏光平面波とKerr
アイコナール近似の高精度
外力の働く向
きは一致
すべて遠方では
1
r4 で効く
Perturbation equation
Ansatz
Wave equation
in flat spacetime
source term
Perturbation equation
u
¯
u
¯µ +
µ
µ
(
u
¯µ ) u +
µ
u
¯µ
1
u =
2
(h u
¯µ u
¯ )+
µ
(h u
¯µ v )
µ
averaging around axis of
beam
µ
(
u
¯µ ) u
¯ +
=
µ
u
¯µ
u
1 µ
1
h u
¯µ u
¯ +
2
S
h u
¯µ v dS
µ
Perturbation equation
µ
(
u
¯µ ) u
¯ +
µ
u
1 µ
1
h u
¯µ u
¯ +
2
S
=
u
¯
u
¯µ
¯
µu
µ
+
=0
h u
¯µ v dS
µ
equation of orbit
of zero order
motion equation of the beam
k
µ
1
µk =
2
1
(h kµ k ) +
S
µ
kµ = u
¯µ + u
¯µ
h u
¯µ v dS
µ
interaction term of Kerr
metric and twisted wave
u
¯µ
vµ
uµ
µ(
t + kz)
µ (m
µ
S
)
u
¯µ
vµ
uµ
µ(
t + kz)
µ (m
µ
S
)