77 "2015/9/25 “ ”

2015/9/25"

77

"

""

"

mizutani@mech.eng.osaka;u.ac.jp"

06;6879;7319"

77 "2015/9/25 “ ”

•  "(JIS"Z"8103) "

!(measurement,"instrumentaIon)"

"

"

"

!(measurement)"

"

"

"

"(metrology)"

"

"

JIS

77 "2015/9/25 “ ”

NASA"homepage

77 "2015/9/25 “ ”

"

"

MEMS

1m

"

"

6000 " "

(1795 )1889 " 86 "

1960 "

"

1983 "

"

2009 "

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77 "2015/9/25 “ ”

" 1

!

!

y

x

z

"

" "

" "

"

" "

"

I = |E(z, t)|2 = |E0 sin(ωt− kz)|2 = |Ex(z, t) +Ey(z, t)|2k =

2π

λω = 2πft" " " " x " y "

"

"

!

!

!

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!xz yz

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!!

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x y !

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77 "2015/9/25 “ ”

!Schrödinger !

Maxwell !

"

!!

rot H = ε∂E

∂t

rot E = −µ∂H

∂tdiv H = 0

div E = 0

∆ε,∆µ << λ

E H "

n: c:

!

E = E 0 exp{−i(ωt− kz)}∇2u =n2

c2∂2u

∂t2

!n2

c2∂2E

∂t2−∇2E − grad logµ× rotE + grad(

ρ

εE log ε)− µ

∂J

∂t= 0

n2

c2∂2H

∂t2−∇2H − grad log ε× rotH − grad(Hgrad logµ)− rotJ + grad log ε× J = 0

∆ε,∆µ ≃ λE H

!

!

!

!

!

!

!

!E = !ω = hν = hc

λHΨ(r, t) = i! ∂

∂tΨ(r, t)

!

!

!( )! ∆x∆p ≥ !2

77 "2015/9/25 “ ”

00

77 "2015/9/25 “ ”

2015 InternaIonal"Year"of"the"Light"2015

Communica:ons!Health! Economy! Environment!! Social!

"

"

!

""

!"!

1015 "Ibn"Al"Haythem"Book!of!Op(cs!!!

1815 "Fresnel"and"the"wave"nature"of"light"

"!

1865 "Maxwell"and"electromagneIc"waves""

""

1915 "General"relaIvity"–"light"in"space"and"Ime"

"""""

1965" "Cosmic"microwave"background,"Charles"Kao"

" "and"opIcal"fibre"technology"

=

77 "2015/9/25 “ ”

•  1901"W."C."Röntogen"

•  1902"Hendrik"Antoon"Lorentz"and"Pieter"Zeeman"

•  1907"Albert"Abraham"Michelson"

•  1908"Gabriel"Lippmann"

•  1911"Wilhelm"Wien"

•  1919"Johannes"Stark"

•  1921"Albert"Einstein"

•  1922"Niels"Bohr"

•  1930"Chandrasekhara"Venkata"Raman"

•  1953"Frits"(Frederik)"Zernike"

•  1955"Willis"Eugene"Lamb"

•  1964"Charles"H."Townes,"Nicolay"G."Basov"and"Aleksandr"M"Prokhorav"

•  1966"Alfred"Kastler"

•  1966"Robert"S."Mulliken"

•  1967"Ronald"G.W."Norrish"and"George"Porter"

•  1971"Dennis"Gabor"

•  1981"Nicolaas"Bloembergen"and"Arthur"L."Schawlow"

•  1986"John"C."Polanyi"

•  1989"Norman"F."Ramsey"

•  1997"Steven"Chu,"Claude"Chohen;Tannoudji"and"William"D."Phillips"

•  1999"Ahmed"Zewail"

•  2001"Eric"A."Cornell,"Wolfgang"Kejerle"and"Carl"E."Wieman"

•  2005"Roy"J."Glauber"

•  2005"John"L."Hall"and"Theodor"W."Hänsch"

•  2006"J."C."Mather,"G."F."Smoot"

•  008"Shimomura"

•  2009"C."Kao,"W."Boyle,"G."Smith"

•  2014"Akasaki,"Amano,"Nakamura"

"( )

"

"

Gabor "

CCD "(Science"Daily "

LED"(wikipedia "

77 "2015/9/25 “ ”

δ = δx − δy

Ex(z, t) = E0x cos(ωt− kz + δx)Ey(z, t) = E0y cos(ωt − kz + δy)

" "

+ +

y

x

z

77 "2015/9/25 “ ”

1800

CCD

hjp://www.ccs;inc.co.jp/s2_ps/s1/s_04/column/light_color/

( ) "(δ1;δ2 )

1 "

)

2

E1

E2

E1

E2

"

"

77 "2015/9/25 “ ”

"

"

"

1/4 "

I = a+ b cos(ωt− kz +∆)

"

"

"

"

1

fx

f y

"

"

"

" "M."Takeda,"et"al.,"J"OpIcal"Society"of"America"Vol."72,"(1982)"156.

77 "2015/9/25 “ ”

Zygo

LIGO (MIT,"Caltech)

4"km

LIGO

AMD"handbook

77 "2015/9/25 “ ”

•  ナノフォトニクス –  マイクロメートル以下のスケールで生じる特異的な光学現象 –  "

–  "

–  "

薄膜 ナノ粒子 ナノ構造 分子配向

関連する光学パラメータ 散乱，表面プラズモン共鳴

異方性物質 等方性物質

関連する光学パラメータ 偏光

77 "2015/9/25 “ ”

金属原子（粒子）

電子雲

電場(光)

・自由電子の集団振動 ・金属原子特有の現象 ・振動モードと光の波数が一致すると吸収が大きくなる ・金属薄膜，金属粒子 ・薄膜や粒子のサイズ：ナノオーダ

時間, 位置

400 500 600 700 800

20

40

60

80

100

粒径 [nm]

波長 [nm] 例：金ナノ粒子の吸収断面積

(吸収断面積)/(粒

径3) [×10

7 nm-1]

1.0

0.5

1.5

0 1

2.0

吸収率が極端に高い

77 "2015/9/25 “ ”

(4)

(3)

(1)

(2) 局在表面プラズモン共鳴

1.324

1.326

1.328

1.33

1.332

25 30 35 40 45 5025 30 35 40 45 50 温度 [℃]

1.330

1.332

1.328

1.326

屈折率 [-]

1.324

水の屈折率温度依存性

　　屈折率変化領域 　　　＝擬似的に拡大した粒子像

+� +�+�

-�-�-�

400 500 600 700 800

20

40

60

80

100

粒径 [nm]

波長 [nm] 金ナノ粒子の吸収断面積

(吸収断面積)/(粒

径3) [×10

7 nm-1]

1.0

0.5

1.5

0 1

2.0

(1) ナノ粒子分散溶液

励起光

(2) 励起光照射

(3) 粒子の温度上昇

(4) 熱拡散 ナノ粒子可視化プロセス

77 "2015/9/25 “ ”

E o

E r

I�

対物 レンズ

結像 レンズ

ビーム スプリッタ

コリメータ

He-Ne レーザ

Nd: YVO4 レーザ

サンプル

ダイクロイック ミラー

Nd: YVO4 レーザ （λ=532 [nm]）

He-Neレーザ （λ=632.8 [nm]）

オシレータ

ビデオ アンプ

ピンホール

PMT

ロックイン アンプ

EO振幅 変調器

ref.

Photothermal sig.

準共通光路型干渉計光熱変換顕微鏡

位相検出部：準共通光路型干渉計 物体光と参照光が共通光路なので環境のゆらぎに強い

繰り返し精度：1/300λ

InternaIonal"Journal"of"Optomechatronics,"Vol.7,"No.2,"pp.96;104,"2013.

2 [um]

位相 [rad]

π

-π

標準偏差 [rad]

0.06

0 標準偏差の分布 平均

2 [um]

77 "2015/9/25 “ ”

サンプル 　　　PVA固定化金ナノ粒子: 10 [wt%] （スピーンコーティング: 3000 [rpm], 300 [s])

　　　

0

1.0

PT"sig

."[a."u.]

x�

y�

z�

2"[um]

x-y plane y-z plane x-z plane

CLEO,"2015.

""

20nm

77 "2015/9/25 “ ”

レンズ

反射率 [-]

1.0

0 カメラで観察した画像

SPR発生部

金属薄膜 基板

表面プラズモン エバネッセント光

空気層

反射光

Otto配置光学系 A. Otto: Z. Phys., 216, (1968), 398-410.

Optics Express, Vol.18, No.14, (2010), pp.14480-14487.

変形Otto配置光学系

・反射光強度はSPRを反映（高精度薄膜計測が可能） ・空気層のナノオーダの厚さ制御が必要

・反射光強度はSPRを反映（高精度薄膜計測が可能） ・空気層のナノオーダの厚さ制御が不要

金薄膜の観測結果

金薄膜の膜厚 54.0 ± 0.3nm

SPR発生部の反射光強度解析 77 "2015/9/25 “ ”

(1)　スピンコート

レジスト剤 Si 基盤

(7)　レンズ完成

(2)　電子線描画

電子線

照射パターン （格子状）

電磁レンズ

(4)　現像

現像液

プラズマ (SF6, C4F8, O2)

(5)　リアクティブイオンエッチング

PDMS ポリジメチルシロキサン

(6)　PDMS塗布

マイクロレンズアレイ作製プロセス

[µm]

0

0.7

φ5umレンズアレイ

金膜厚 t [nm

]

40

60

50

10 [um]

反射率 R [-]

0

1.0

10 [um]

反射光強度分布 薄膜分布

77 "2015/9/25 “ ”

x

y

z

x

y

z

77 "2015/9/25 “ ”

xy

z

y

xz

xy

z

y

xz

p – 偏光

s – 偏光

主軸方位 = θx

y

z

y

xz

θ

(主軸方位 = 0°)

( = 90)

77 "2015/9/25 “ ”

x

y

z

y

x

z

x

y

z

y

x

z

円偏光

楕円偏光

77 "2015/9/25 “ ”

x

y

z

y

x

z

y

x

z

y

xz

θ直線偏光

円・楕円偏光

自然偏光

水面での反射液晶

太陽ランプ

反射

77 "2015/9/25 “ ”

位相差

屈折率異方性 吸収率異方性 1. 複屈折性

2. 旋光性

3. 二色性

4. 円二色性

直交する振動成分の屈折率の差

左右円偏光の屈折率の差

直交する振動成分の吸収率の差

左右円偏光の吸収率の差

結晶

5. 偏光解消 散乱などで複数の偏光が混在した状態を作る

旋光角

進相軸

遅相軸

散乱

5

物性には固有の偏光作用があるため，5要素のうち特定の要素に着目した測定をすることで物性測定ができる 77 "2015/9/25 “ ”

PLZT

1mm

強誘電性結晶: PLZT

10um

SEMで観察した組成差像(参考)

半導体レーザ

サンプル

偏光子

対物レンズ

CMOS

20mm

λ/4板

検出光子

λ/4板

ミュラー行列偏光顕微鏡

λ=635nm

・電解を印加すると結晶構造が変化 ・偏光変調器 ・多結晶 ・散乱による偏光解消が問題 ・電解応答の詳細な解析が行われていない

・ミュラー行列：偏光５要素を一 　度に表現できる行列 ・λ/4波長板を回転させて検出し 　た信号を周波数解析する ・倍率は任意に変更可能

77 "2015/9/25 “ ”

�deg.]

+V E

�deg.] 180

0

45

90

135 複屈折位相差

-V

800V 0V 0V

100µm

Sensors"and"Actuators"A:"Physical,"Vol.200,"pp.37;;43,"2013. ""

InternaIonal"Journal"of"Optomechatronics,"Vol.7,"No.4,"p.253,"2013. ""

PLZT

結晶粒界 結晶粒界の移動

分極方向 偏光軸

電圧印加前 電圧印加 電圧除加後

・電圧印加前後で偏光軸の角度が変化 結晶流界の移動が原因

・電圧除加後も結晶流界の移動は保持される 　　偏光メモリデバイスとして応用

印加電圧 [kV/cm]

メモリ量[ deg.]

77 "2015/9/25 “ ”

2 μmd ≈ λ

p ≈ λ

ε < 1/10λ

h ≈ λ

n1n2

incident ligh(polarized light)

reflected light(depolarized light)

θ

1

0

-11

0

-11

0

-11

0

-1

valu

e of

Mue

ller m

atrix

incident angle θ [deg]30 50 7030 50 7030 50 7030 50 70ナノインプリンティングにより

作製されたサブ波長構造 ・SiO2 ・反射防止構造

“Ellipsometry"at"the"Nanostructure,”"Chapter"8,"Springer,"2013.

""

i n c i d e n t p l a n e

nanostructure

azimuth angle α

Εip

Εis

ΕrpΕrs

エリプソメトリ（偏光解析）によるナノ構造評価 ナノ構造

反射前後での偏光の変化を解析

driver for stepping motor

1/4 wave plate 1/4 wave plate

polarizer analyzer

halogen lamp

CCD camera

sample stage

spectroscope field lens

collimator lens

driver for stepping motor

motor

fiber

PC

偏光５特性（ミュラー行列）偏光測定装置

測定結果（偏光解消成分） ・理想的な構造では偏光解消は生じない

計算結果と実験結果 の不一致要素

偏光解消の解析モデル 波長の10分の1以下の構造 の不均一性が影響

77 "2015/9/25 “ ”

2 μm

77 "2015/9/25 “ ”

各種光学現象を利用して 目的シグナルを顕在化

悪　←　計測精度　→　良

光応用計測

小 ← 対象物・現象の大きさ →　大

微弱シグナル中の 目的シグナルを顕在化 統計的アプローチ

・相関計測の適用 アクティブにノイズを与えることで

　　　シグナルを顕在化 　　　（ただし，多数のNが必要）　 ・デバイスの進展により可能 　 DMDプロジェクター：30kHz

積層型画像センサ：空間相関 空間光変調器：空間相関 時間相関カメラ：時間相関 偏光カメラ：偏光相関

・装置が簡易になる

組み合わせが容易 種々の測定装置を拡張できる　

統計光学応用計測 （Photon metrology）

77 "2015/9/25 “ ”

Photon"metrology( )

"

( )

"

"

"

=

+ "

( )"

=" "+" "

="

E = !ω = hν = hc

λ

∆x∆p ≥ !2

(x:" ,"p:" )

77 "2015/9/25 “ ”

"

10nm "

"

"

mentum resulting from the rotational symmetry.However, only in certain circumstances will theresult be exact (16). [For a general review oflight’s angular momentum, see (17).]

We can describe the SHEL as a consequenceof a geometric phase (Berry’s phase) (18), whichcorresponds to the spin-orbit interaction. It is al-ready known that photons, when guided by anoptical fiber with torsion, acquire a geometricphase whose sign is determined by the spin state(19, 20). When a photonic wave packet changesdirection because of a spatial variation in the re-fractive index, the plane-wave components withdifferent wave vectors experience different geo-metric phases, affecting the spatial profile andresulting in the SHEL.

For a paraxial beam, the transverse beam state(for the relevant direction y and its associated wavevector ky) at the air side of the interface (Fig.1A), including the spin state js⟩, can be written asjYa⟩ ¼ ∫dyYð yÞj y⟩js⟩ ¼ ∫dkyFðkyÞjky⟩js⟩, withF(ky) being the Fourier transform of Y( y). Atthe glass side of the interface, under the action ofthe geometric phases, the state becomes |Yg⟩ =∫dkyF(ky)exp(−ikys%3d)|ky|s⟩ = ∫dyY(y − sd)|y⟩|s⟩,with %s3js⟩ ¼ sjs⟩, indicating +d and −d shifts forthe wave packets of the parallel and antiparallelspin states. Here, the term expð−ikys%3dÞ repre-sents a coupling between the spin and the trans-verse momentum of the photons.

The origin of this “spin-orbit” interactionterm lies in the transverse nature of the photonpolarization: The polarizations associated withthe plane-wave components experience differentrotations in order to satisfy the transversalityafter refraction. This is depicted pictorially inFig. 1B with incoming horizontal polarization(|H ⟩) (along xI). In the spin basis, this state

corresponds to jH⟩ ¼ 1ffiffiffi

2p ðjþ⟩þ j−⟩Þ. In the

lowest-order approximation, the change in thestate after refraction is jky⟩jH⟩→jky⟩ðjH⟩ þkydjV ⟩Þ ¼ jky⟩jϕ⟩, withϕ ¼ kyd << 1 and |V⟩being vertical polarization. In the spin basis,

jϕ⟩ ¼ 1ffiffiffi

2p ðexpð−ikydÞjþ⟩þ expðikydÞj−⟩Þ, indi-

cating the coupling expð−ikys%3dÞ.As a result of the polarization-dependent

Fresnel reflections at the interface, the oppositedisplacements of the two spin components ac-tually depend on the input polarization state (seeSOM for details). For |H⟩ and |V ⟩ input po-larizations, the displacements dH and dV are givenby (Fig. 1C)

dHjT⟩ ¼ Tl2p

cosðqTÞ − ðts=tpÞcosðqIÞsinðqIÞ

,

dVjT⟩ ¼ Tl2p

cosðqTÞ − ðtp=tsÞcosðqIÞsinðqIÞ

ð1Þ

Here, qI and qT are, respectively, the central in-cident and transmitted angles related by Snell’slaw; ts and tp are the Fresnel transmission coef-ficients at qΙ; and l is the wavelength of the light

in the incident medium. In a continuously varyingrefractive index, the input polarization dependencedisappears, and the motion can be formulated interms of a particle moving in a vector potential inmomentum space (4, 14, 21, 22), along the samelines as with electronic systems (23, 24).

For optical wavelengths, precise characteriza-tion of the displacements requires measurementsensitivities at the angstrom level. To achieve thissensitivity, we use a signal enhancement tech-nique known from quantum weak measurements(25). In a quantum measurement, a property (ob-servable A% ) of a system is first coupled to a sep-arate degree of freedom (the “meter”), and thenthe information about the state of the observableis read out from the meter. At the single-photonlevel, the SHEL is actually equivalent to a quan-tum measurement of the spin projection alongthe central propagation direction (observable s% 3,with eigenstates |+⟩ and |–⟩), with the transversespatial distribution serving as the meter [similarto a Stern-Gerlach spin-projection measurement(26)]. However, the displacements generated bythe SHEL here are much smaller than the widthof the transverse distribution, resulting in a weakmeasurement: The meter states associated withdifferent spin eigenstates overlap to a large ex-

tent. Therefore, the meter carries very little infor-mation about the state of the observable, leavingthe initial state almost undisturbed. Although ourexperiment is at a classical level with a largenumber of photons in a quantum-mechanical co-herent state, the results remain the same, witheach photon behaving independently. Furthermore,in the paraxial regime, the dynamics of the trans-verse distribution are given by the Schrödingerequation with time replaced by path length, mak-ing the analysis identical to nonrelativistic quan-tum mechanics with an impulsive measurementinteraction Hamiltonian HI ¼ ky %Ad.

With the weak measurement taking place inbetween, the signal enhancement technique usesan appropriate preselection and postselection ofthe state of the observable to achieve an en-hanced displacement in the meter distribution(25, 27) (Fig. 2A). Given the preselected andpostselected states |y1⟩ and |y2⟩, for sufficient-ly weak measurement strengths, the final po-sition of the meter is proportional to the real partof the so-called “weak value” of the measuredobservable A%

Aw ¼ ⟨y2jA% jy1⟩⟨y2jy1⟩

ð2Þ

604020

–60–40–20

0

6040200 80

zT

zI

xI

y

y

xI

y

xT

zI

zT

z

x

y

xT

zT

y zI

y

xI

AirGlass

θΙ

+

θΤ

θI (degrees)

y-di

spla

cem

ent (

nm)

A B

C

λ = 633 nm

δ H

δ H

δ V

δ V

Fig. 1. The SHEL at an air-glass interface. (A) jþ⟩ and j−⟩ spin components of a wave packet incident atangle qI experience opposite transverse displacements (not deflections) upon refraction at an angle qT.(B) Different plane-wave components acquire different polarization rotations upon refraction to satisfytransversality. The input polarization is in the xI direction (equivalent to horizontal according to Fig. 3)for all constituent plane waves. Arrows indicate the polarization vectors associated with each plane wavebefore and after refraction. The insets clarify the orientation of the vectors. (C) Theoreticaldisplacements of the spin components (Eq. 1) for horizontally and vertically polarized incident photonswith wavelength l = 633 nm.

8 FEBRUARY 2008 VOL 319 SCIENCE www.sciencemag.org788

REPORTS

on

Febr

uary

16,

200

8 w

ww

.sci

ence

mag

.org

Dow

nloa

ded

from

O.Hosten"and"P.Kwiat,"science"319"(2008)"787.

~Ei6 ¼ ðeix þ ireiyÞw0ffiffiffiffiffiffi2p

p exp %w20ðk2ix þ k2iyÞ

4

" #

; (1)

where w0 is the beam waist. The polarization operator r ¼ 61corresponds to left- and right-circularly polarized light,respectively. In this work, we only consider the incident lightbeam with horizontal polarization and vertical polarizationcan be analyzed in the similar way. Using the reflectionmatrix,16,17 we can obtain the expressions of the reflectedangular spectrum

~Er ¼rpffiffiffi2

p ½expðþikrydrÞ~Erþ þ expð%ikrydrÞ~Er%': (2)

Here, dr ¼ ð1þ rs=rpÞ cothi=k0; rp and rs denote Fresnelreflection coefficients for parallel and perpendicular polar-izations, respectively. k0 is the wave number in free space.And the ~Er6 can be written as

~Er6 ¼ ðerx þ ireryÞw0ffiffiffiffiffiffi2p

p exp %w20ðk2rx þ k2ryÞ

4

" #

: (3)

At any given plane zr ¼ const:, the transverse displacementof field centroid compared to the geometrical-optics predic-tion is given by

d6 ¼

ðð~n(r6i@kry

~nr6dkrxdkryðð

~n(r6~nr6dkrxdkry

; (4)

where ~nr6 ¼ rp expð6ikrydrÞ~Er6. Calculating the reflecteddisplacements of the SHE of light requires the explicit solu-tion of the boundary conditions at the interfaces. Thus, weneed to know the generalized Fresnel reflection of the gra-phene film

rA ¼ RA þ R0

A expð2ik0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2 % sin2hi

pdÞ

1þ RAR0A expð2ik0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2 % sin2hi

pdÞ

: (5)

Here, A 2 fp; sg; RA and R0

A is the Fresnel reflection coeffi-cients at the first interface and second interface, respectively.n and d represent the refractive index and thickness of thegraphene film, respectively. The thickness of the graphenefilm is about d ¼ mDd in which m denotes the layer numbersand Dd represents the thickness of single layer graphene (Ddis about 0.34 nm). So far, we have established the relation-ship between the transverse shifts and the graphene layers.After obtaining the displacements of SHE of light, we candetermine the layer numbers of graphene.

In this work, a signal enhancement technique known asthe weak measurements18,19 is used to measure the tinytransverse displacements. The experimental setup shown inFig. 2 is similar to that in Refs. 8 and 11. A Gaussian beamgenerated by a He-Ne laser passes through a short focallength lens (L1) and a polarizer (P1) to produce an initiallyhorizontal polarization beam. Here, the half-wave plate(HWP) is used to control the light intensity. When the beamimpinges onto the graphene-prism interface, the SHE of light

takes place, manifesting itself as the opposite displacementsof the two spin components. Then the two components inter-fere destructively after the second polarizer (P2) which isoblique to P1 with an angle of 90)6D. We note that the inci-dent light beam is preselected in the H polarization state byP1 (along to the xi-axis) and then postselected by P2 in thepolarization state with V ¼ sinDerx þ cosDery. In this condi-tion, we choose the angle D ¼ 2). Then we use L2 to colli-mate the beam and make the beam shifts insensitive to thedistance between L2 and the CCD. The reflected field at theplane of zr can be obtained with V * Er . The amplified dis-placement dw at the CCD is much larger than the initial shiftjd6j. Calculating the distribution of V * Er yields the ampli-fied factor Aw ¼ dw=dþ. Hence the amplified displacementsat the CCD are Awd6. It should be mentioned that the ampli-fied factor Aw is not a constant, which verifies the similarresult of our previous work.8

Now we focus our attention on identifying graphenelayers. However, there exists two unknown parameters (re-fractive index and layer numbers of graphene) to be identi-fied. Before identifying the graphene layers, we need tochoose the suitable refractive index parameter of graphene.There has several measured values of the refractive index ofgraphene reported recently.4,20–22 Here, we choose one suita-ble refractive index according from the work of Bruna andBorini.22 They concluded that the refractive index of gra-phene in the visible range consists of real refractive index(constant) and complex refractive index (depending on thewavelength). Here, the refractive index of graphene is about3.0þ 1.149i at 633 nm. We first need to prove that this re-fractive index is suitable for our graphene film. Our sampleconsists of graphene films with two different layers: onelayer, two layers. The graphene films (made from ACS Ma-terial company) were first grown on 25 lm thick copper foilin a quartz tube furnace system using a CVD method andthen were transferred to the prism. The Raman spectra ofthese two samples are shown in Fig. 3(a). We measure thedisplacements of the SHE of light on the graphene film every2) from 40) to 70) in the case of horizontal polarization andthe results are shown in Figs. 3(b) and 3(c). It should be

FIG. 2. Experimental setup: The sample is a BK7 glass transferred with thegraphene film. L1 and L2, lenses with effective focal length 50mm and250mm, respectively. HWP, half-wave plate (for adjusting the intensity). P1and P2, Glan laser polarizers. CCD, charge-coupled device (Coherent Laser-Cam HR). The light source is a 17 mW linearly polarized He-Ne laser at633 nm (Thorlabs HRP170). The inset shows that the angle between P1 andP2 is 90)6D.

251602-2 Zhou et al. Appl. Phys. Lett. 101, 251602 (2012)

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noted that the quality of the material (graphene film) and theexperimental environment will affect the measurement. Agroup of experiment for measuring the SHE of light at a pureair-prism interface was also carried out for making a refer-ence. We can find that the experimental results fit well withthe transverse displacement curve calculated from the litera-ture of Bruna.22 We can obtain that the refractive index ofgraphene is really close to 3.0þ 1.149i at 633 nm. Therefore,the SHE of light provides us an alternative way for choosingthe refractive index of graphene.

Using the suitable refractive index n¼ 3.0þ 1.149i at633 nm, we can identify the layer numbers of an unknowngraphene film with the weak measurements. The experimen-tal sample is also prepared with the CVD method. It should

be noted that, in our experimental condition, we cannot fabri-cate the sample with the precise layer numbers when the gra-phene film has more than two layers. Because it wouldunavoidably involve large technical errors. We just know theapproximate layer numbers ranges. Therefore, we prepare asample with the possible layer numbers ranging from threeto five layers. Our aim is to determine the actual layer num-bers of this graphene film. Figure 4 shows the theoretical andexperimental results of the graphene layer numbers determi-nation. From Fig. 4(a), we find that it is hard to distinguishthe transverse shifts of the different graphene layer numbersfrom 40# to 70#. Thence we measure the transverse displace-ments in a small range of incident angle (from 56# to 62#) to

Raman shift (cm-1)

Intensity(a.u.)

1200 1500 1800 2100 2400 2700

(a)

2D

1 layer

2 layers

G

θi (degrees)

A wδ +(µm)

40 45 50 55 60 65 70-1200

-800

-400

0

400

800

1200

Glass3.0+1.149iData 1Data 2

(b)

Graphene: 1 layer

θi (degrees)

A wδ +(µm)

40 45 50 55 60 65 70-1200

-800

-400

0

400

800

1200

Glass3.0+1.149iData 1Data 2

(c)

Graphene: 2 layers

FIG. 3. Raman spectra of the samples and the graphene refractive indexselection in the case of horizontal polarization. (a) Raman spectra of one,two graphene layers. (b) represents the transverse displacements under thecondition of single layer graphene. We choose the thickness of one layergraphene film as 0.34 nm. The transverse shifts in the case of two layers gra-phene film are shown in (c). Here, the lines represent the theoretical results.The circle and triangle show the experimental results obtained from the air-prism and different graphene-prism conditions via weak measurements. Therefractive index of the BK7 substrate is chosen as n¼ 1.515 at 633 nm.

θi (degrees)

A wδ +(µm)

56 57 58 59 60 61 62-1200

-800

-400

0

400

800

12001 layer2 layers3 layers4 layers5 layersData 1Data 2Data 3

(b)

Raman shift (cm-1)

Intensity(a.u.)

1200 1500 1800 2100 2400 2700

(c)

2D

3 layers

G

θi (degrees)

A wδ +(µm)

40 45 50 55 60 65 70-1200

-800

-400

0

400

800

1200

1 layer2 layers3 layers4 layers5 layers

(a)

FIG. 4. The theoretical and experimental results of determining the layernumbers of graphene. (a) represents the theoretical transverse displacementsunder the condition of graphene layer numbers changing from one to five.Here, the refractive index of graphene is 3.0þ 1.149i at 633 nm. (b)describes the transverse shifts in the case of different incident angles rangingfrom 56# to 62#. The lines represent the theoretical results. The circle,square, and triangle show the experimental data obtained from three differ-ent areas of the graphene sample. (c) Raman reference data of the sample.

251602-3 Zhou et al. Appl. Phys. Lett. 101, 251602 (2012)

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X.Zio,"et,"al.,"APL"101"(2012)"251602.

Y."Aharonov,"D."Z."Albert,"and"L."Vaidman,"PRL,"60"(1988)"1351.

Aw ≡ ⟨Ψf |A|Ψi⟩⟨Ψf |Ψi⟩

|Ψi⟩ |Ψf ⟩A

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CCD

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相関計測 相関計測

弱 ← 光強度(SN比) → 強 弱 ← 光強度(SN比) → 強

低 ← 像の鮮明度 → 良

撮影対象物体 （デジタルカメラにて撮影）

G(x, y) =< ∆B∆I(x, y) >=< BI(x, y) > − < B >< I(x, y) >

規格化相関係数

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相関計測 （4096回積算）

相関計測（100000回積算）

パターン最適化 （4096回）

相関係数分布 検出光強度 照明光強度分布

シグナルのゆらぎ

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n=1

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T (x, y)

In (x, y)

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10um 10um

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光検出部 蛍光ゴーストイメージング顕微鏡

100um25um

骨芽細胞の観察像

標準試料（蛍光ビーズ）を用いた感度確認

200分の１以下の強度の励起光で検出可能 ・長時間観察が可能

z=0um

三次元計測にも適用可能 Z=10um z=20um Z=-10um z=-20um

深さ方向の感度

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