先端放射光源に関する研究会
岡崎コンファレンスセンター
２０１４年１１月２１日

佐々木 茂美広大放射光センター

On behalf of
佐々木茂美1, 宮本篤1, 加藤政博2, 許斐太郎2,保坂将人3, 山本尚人3，今園孝志4，小池雅人4
1広大HiSOR, 2分子研UVSOR, 3名大SR, 4原子力機構量子ビーム

Outline 光渦：波動方程式の解
Light vortices: solutions to the wave equation

 円偏光アンジュレータからの放射光Synchrotron radiation from helical undulators
 UVSORでの実験について

Experiments at UVSOR
 新奇な性質の利用可能性について
Possibility for using novel properties

Maxwell's equations
E  0 E
 B B  0cS EB4  source-free,in vacuum1 E
 0c t

1r  (EB) dr
4cJ Poynting vector

1 B
 0c tTotal angular momentum3
1(EA) Ei (r ) drJ 4 c i1

Spin S

Orbital LwhereB  A

Optical "donuts"
 Laguerre-Gaussian mode (azimuthal phase):

a p r, ,z  A p r,z exp i p r,z expi   The Poynting vector completely specifies the fieldr w

PhaseMagnitude/2

Phase singularities
Laguerre-Gaussian beams•On-axis singularity characterized by helical phase front•

Poynting vector follows wavefront as beam propagates•

Gives rise to orbital momentum component•

Intensity is zero on-axis (optical donut)

Orbital angular momentum

L•Distinct from spin AM associated with circular polarization•

Has magnitude of lh per photon, where topological "charge" l isnumber of (2πl) phase cycles around beam(  integer)

L. Allen, PRA 45, 8185 (1992)

Photon angular momentum
If every polarization vector rotates the light has spin.
If the phase structure rotates the light has orbital angular momentum- M. Padgett

SpinS0Orbital

linear/circularaxis-independent
intrinsic

azimuthalaxis-dependent
intrinsic or extrinsic(on-axis) (off-axis)S  1L0J LSL

Physics Today 57 May 2004 Miles Padgett

Can we produce an x-ray vortex?
Need:Intense, highly coherent X-ray or VUV beam
Means to generate the helical phase
Means to detect it
Will it be stable upon propagation, or fall apart due
to rapidly varying phases and imperfect optics?

Spiral phase platePolyimide phase plate, 34 µm thick, gives ~2π shift at 9 keVExperiment
Coherent xray beamTungsten wire

Spiral phaseplate

Helium flightpath

CdWO4scintillator

CCD camera

A. Peele, Opt. Lett. 27, 1752 (2002)

Spiral phase contrast x-ray microscopy

Images of 1 µm pinhole at = 2.73 nm

ordinary ZP

SZP, l=1

SZP, l=2Sakdinawat and Liu, Opt. Lett. 32, 2635 (2007)

"Charge 1" vortexLphasesingularity



Visualize singularity as split in Fresnel diffraction
fringes from a tungsten wire (9 keV)
A. Peele, JOSA A21, 1575 (2004)

Theory
For a variable polarizing undulator with N periods (e.g. APPLE undulator),
the radiated intensity of the nth harmonic is2d 2Ie2 2 n 2 2 2
 L( N  /  ),
A

Axy1

d d 
4 0 c 

where:A x  2 cos S0  K y (S1  S1 )Ay  2 sin S0  iK x (S1  S1 )

L(N/1) is the Laue function for
fundamantal 1 and -n1()Sq  J p (Y )Jn 2 p q (X)ei (n 2 p q )

q = –1, 0, 1p

X  2n K y2 cos2   K x2 sin 2 Y  n (K y2  K x2 ) /4 , tan  (K y /K y )tan   1/(1  2 2  K x2 /2  K y2 /2)

B. Kincaid, JAP 48, 2684 (1977)
R. Walker, CERN Acc. School, 1998

Theory
For the pure circular mode, Ky=Kx=K and X=2nξK, Y=0, =. Aftersimple manipulation, we obtainA  e 2 sin  J ( X )  iK J ( X )e  J ( X )e .
Ax  ein 2 cos  J n ( X )  K J n 1 ( X )ei  J n 1 ( X )e-i inyn 1n0.4i

n 1-i2ndIntensity [a.u.]

3rdTo explore whether the higher harmonics carry
OAM in general, we consider the time-independent
complex amplitude. For the circular mode this canbe written asNEXT SLIDE

4th

0.3
0.2
0.1
0.0
-1.5

-1-0.5

0



0.5

1

1.5

S. Sasaki, PAC07 Proc. TUPMN097, 2007
S. Sasaki, Nucl. Instrum . Methods A582 (2007) 43.

Helical caseIn this mode, Kx = Ky = K, X = 2nξK, Y = 0, and  = A

Ax  iAy2

 2ei ( n 1)Radiated amplitude has
the characteristic centralminimum and exp(il)signature of LG modes

Sasaki and McNulty, PRL 100, 124801 (2008)

J n ( X )  J n1 ( X )Comparisons between two equations
In the circular mode; Kx = Ky = K, X = 2nξK, Y = 0, and  = (Ax  iAy ) /2 2 ei(n1) Jn (X)  Jn1(X)

In Laguerre-Gaussian beam;a p r, ,z  A p r,z exp i p r,z expi  
Sasaki and McNulty, PRL 100, 124801 (2008)

Intensity, magnitude, and phase for the first few harmonics

First Observation of OAM in Undulator Radiation

J. Bahrdt, et. al. Phys. Rev. Lett. 111, 034801 (2013).

First Observation of Light Vortices
in Undulator Radiation19

2010年時点の日本の光源リング

挿入光源が設置されている放射光源リングに限った

Fac ilit y

En e r gy [Ge V] Em it t an c eSpring-8
PFSagaLS
NewSubaru
UVSOR-IIHiSOR
NagoyaLS
HiSOR-II82.5
1.4
1-1.5
0.75
0.7
1.20.73.436253727400
53
14MAX III

0.7

13

回折限界とエミッタンス

x   4Experiments at UVSOR-III
x   4 : 10 nmrad ~ 10 eV
0 = 17.5 nmrad @ 750 MeV
Diffraction Limited LS below Ep =6 eV

750 MeV ：Emittance = 17.5 nmrad, DL wavelength=220 nm, 6eV
600 MeV：Emittance = 10.9 nmrad, DL wavelength=138 nm, 9eV
500 MeV：Emittance = 7.6 nmrad, DL wavelength=100 nm, 12eV
400 MeV：Emittance = 4.8 nmrad, DL wavelength=63 nm, 20eV

2台のアンジュレータ光の干渉• 2次光と1次光の干渉を見ると– 1本の渦巻きが見えるはず
上流のアンジュレータ光(1次光)
下流のアンジュレータ光
(2次光)

23

ファイバー分光器による観測• ステージを使ってファイバー分光器をXY平面内にスキャン
ファイバー分光器

1.6 m

7m

APPLE-IIアンジュレータ
XYステージ(20 x 20 mmを自動スキャン)24

Results of Experiments
E = 600 MeV, 0 = 11 nmrad

Sasaki, Miyamoto (Hiroshima U), Katoh, Konomi (UVSOR), Hosaka, Yamamoto (Nagoya U)

CCDカメラによる観測• VUVを撮影できるCCDによる直接撮影を試みた
Ee = 500 MeV

 アンジュレータ
 上流：円偏光, 2次光 下流：直線偏光, 1次光

バンドパスフィルタ 波長(バンドパスフィルターによる)
 355 nm

CCDカメラUndulators

Polarizer
Band-pass filter
CCD

Interferences between both circular polarizations• 同符号のSを持つ円偏光

光が強め合う位置は時間的に変化しない

強め合う位置偏光子を通して方向の電場のみ通過するようにすると？

• 逆符号のSを持つ円偏光

時間的に積分するとドーナツ状に光る！

28

Circular 1st (upstream: s=+1, l =0) & 2nd harmonic (downstream: s=-1, l =-1) interference.
Measurement were done with a polarimeter. The spiral pattern rotates with the rotation ofpolarimeter. E = 600 MeV, e0 = 11 nmrad, l = 240 nm. Rotation angles of porarimeter are: Top
Left: 0°, Top Right: 30°, Bottom Left: 60°, Bottom Right: 90°OAMを持つアンジュレータ光• アンジュレータ光(APPLE-II)– 任意の波長– 偏光可変(直線/円-左右)– 整数倍の高調波
電子ビーム

アンジュレータ光APPLE-IIアンジュレータ
 円偏光の高調波が軌道角運動量を運ぶ
 1次光(基本波)：平面波 2次光：ℓ=±1
 3次光：ℓ=±2, …

1次光と高次光の干渉による渦巻きパターン• 上流アンジュレータからの放射光𝒂(𝒓)𝝅𝒅𝝅𝟐𝝅𝑳𝟐𝑨 𝒓, 𝝋 =𝐜𝐨𝐬 𝟐 +𝒓 ± 𝒏−𝟏 𝝋+− 𝝎𝒕
𝑳+𝒅𝜸 𝝀 (𝑳 + 𝒅)𝝀𝝀

• 下流アンジュレータ

10𝒃(𝒓)
𝝅 𝟐 𝟐𝝅𝑳𝑩 𝒓, 𝝋 =𝐜𝐨𝐬𝒓 +− 𝝎𝒕
𝑳𝑳𝝀𝝀5• 𝑨 + 𝑩を0~/2の間tで積分して 0cos関数の中身が0になるjは、𝝅𝒅 𝝅𝒅 𝟐𝝋 = ± − 𝟐 + 𝟐 𝒓 /(𝒏 − 𝟏) -5𝜸 𝝀 𝑳 𝝀-10-10-505103次の光で見ると…
• 3次光と1次光の干渉を見ると– 2本の渦巻きが見えるはず
上流のアンジュレータ光(3次光)
下流のアンジュレータ光(1次光)

Double spiral by linear 1st & circular 3rd harmonic (Top row) and that by the circular 1st & 3rd
harmonic with the same helicity interference (Bottom row). E = 500 MeV, e0 = 8 nmrad, l =248nm, downstream undulator setting: left column; s=-1, l =-2, rightcolumn; s=+1, l =+2,33
without a polarimeter.

Double spiral by 2nd (l =±1) & 2nd (l =∓1) harmonic interference.
E = 500 MeV, 0 = 8 nmrad, l = 355 nm, with a polarimeter.

Triple spiral by the circular 2nd (s=+1, l =+1) and 3rd (s=-1, l =-2) harmonic interference.
E = 400 MeV, 0 = 5 nmrad,  = 355 nm, with a polarimeter (band width  = 1.3 nm).





X-ray orbital dichroism•

Dichroic effects are expected with by x-ray beams carrying OAM.

•Sensitive to DOS occupancy difference between orbitals with j ± l and j = 0.

•Enables study of unoccupied states in absence of strong core-hole effects.

•Subtracting spectra measured with ± OAM states allows separation ofquadrupolar from much stronger dipolar transitions.

 A r  eˆ A   expi   ikzw fep  A R  r  gme.g.,

Probe j = ±2 transitions(e.g. 3d  1s ) in:––––cuprates
manganites
ruthenates
rare-earths

But: cross-section scales as (/w)|l| !
A. Alexandrescu, PRL 96, 243001 (2006)

Conventional dichroism
subtract spectra with left and right polarized lightm=-2e1-10123d

q=1e-1

Transition metal
1s"Orbital dichroism"
subtract spectra with positive and negative orbital angular momentum of the beam
m=-2-1012

3dn=1

n=-1

q=1
Use left (q=1) circularly polarized light

Transition metal
1sPredicted spectra(a) OAM dichroic signal using circularly (e1,e-1)
and linearly (ex/y) polarized x-rays for a Cu2+ ionin spherical symmetry and z-axis B-field. (b) Cu2+
ion in crystal field with D4h symmetry.

OAM dichroism spectrum of Mn3+ ion.

van Veenendaal and McNulty, PRL (2007)

まとめ• 軌道角運動量を持つアンジュレータ放射光
回折限界光条件での検証実験＠UVSOR-III– 2次光と1次光、3次光と1次光の干渉による渦巻の観測を行った。
• ℓ の符号による回転方向の違いも確認– 1次光のヘリシティーを逆転した観測も行った。
• 逆ヘリシティーの場合、偏光子を回転すると干渉パターンも回転した– 逆ヘリシティーの2次光どうし、2次光と3次光の干渉も観測した。
– 新しい放射光利用研究のプローブとなる可能性を秘めている。