w w w .m o b ile b a rrie rs .c o m w w w .m o b ile b a rrie rs .c o m
LiNbO A ¥ w ? > « ¶ : r w yp x A ¥ w m Ý ¡ ) U A L t ) Q è ¹ t m M o \ } H6 p x O S ù ï ¬...
Transcript of LiNbO A ¥ w ? > « ¶ : r w yp x A ¥ w m Ý ¡ ) U A L t ) Q è ¹ t m M o \ } H6 p x O S ù ï ¬...
LiNbO3 r22
∗1 ∗2 ∗3
Improvement of Measurement Accuracy of the Multiple Reflection Interference Method
and Measurement of Electro-Optic Coefficient r22 of LiNbO3 Crystal
Kazuya YONEKURA∗1, Lianhua JIN∗2, Kuniharu TAKIZAWA∗3
ABSTRACT: Multiple reflection interferometry obtains electro-optic coefficient from the phase of interference sig-
nal between the light, which passes through the sample crystal without internal reflection and the lights which after
proceeding several internal reflection inside the crystal. To simplify the data process, conventional interferometry con-
sidered only two beam interference, which results in theoretical systematic error. We have analyze the error caused by
the theory and form tolerance of the EO crystal, furthermore, propose two measurement techniques to minimize these
errors. EO coefficient r22 of non-doped congruent LiNbO3 crystal was measured with multiple reflection interferome-
try, and the measurement results was 6.54 ± 0.02pm/V at 632.8nm by null method.
KEYWORDS: Elctrooptic coefficient, Dispersion, Electrooptic crystal, interference
(Received October 13, 2006)
1.
LiNbO3(LN) (1)
(Electro-Optic;EO) (2) (3)
(4)
(5)
LN [1–3]
[4]
[5,6]
LN
EO
∗1
(Doctor Candidate, Dept. of Applied Physics,
e-mail:[email protected])∗2
(Research Associate, Dept. of Materials and Life Science,∗3
(Professor, Dept. of Materials and Life Science
EO
EO
EO Mach-Zehnder [7–9]
Michelson [10] [11–13] Fabry
Perot [14] Senarmont [15]
1 Mach-Zehnder Michelson
EO
2
EO
EO
Mach-Zehnder
EO
2
EO [16]
1
─49─
成蹊大学理工学研究報告J. Fac. Sci. Tech., Seikei Univ.Vol.43 No.2 (2006) pp.49-61(研究装置設備紹介・報告)
Crystal
Mirror
Mirror
Half mirror
Half mirror
Photo Detector
disturbances
Crystal
Mirror
Halfmirror
Mirror
Photo Detector
disturbances
Crystal
Mirror
Mirror
Half mirror
Half mirror
Photo Detector
disturbances
Crystal
Mirror
Mirror
Half mirror
Half mirror
Photo Detector
disturbances
Crystal
Mirror
Halfmirror
Mirror
Photo Detector
disturbances
Crystal
Mirror
Halfmirror
Mirror
Photo Detector
disturbances
1 2 ( Mach-Zehnder
Michelson )
EO
EO
Senarmont
EO
EO
2
EO
LN LiTaO3
3m r22 Senarmont
LN EO r13 r33 r51
Fabry-Perot(FP) LN
FP EO
FP
FP
LN
V
Signal Generator
LightSource
Polarizer(45°)
Analyzer(-45°)
zy
x
PhotoDetectorWavelength
Plate(λ/4)
LN
V
Signal Generator
LightSource
Polarizer(45°)
Analyzer(-45°)
zy
x
PhotoDetectorWavelength
Plate(λ/4)
2 Senarmont
3
EO
EO [12,17]
EO
[18–20]
2 EO
[21,22]
FP
EO
EO
EO
EO
EO
LN EO
Chirakadze
LN 420-1150nm r22 [22]
Mendez
LN 458-632.8nm 1.04μm
r13 r33 [23]
2 FP
EO
[24] 3
EO
FP
FP
2
EO
EO
─50─
8
2 EO
3
null
4
5
6
null 7
LN EO r22
632.8nm 8
2.
EO
[24]
EO
2
EO
(1)
(2)
(3)
EO
EO
(1)
(2) EO EO
1 LN LiTiO3
EO r13T r33
T [24] Ti
LN
Γ EO r22T
EO Γr22T Ti Γr22
T
[25] NH4H2PO4 KH2PO4 EO
r41T d14
[26]
EO 2
EO
null
3
EO
3. null
3.1 null
4
VAC sin(ωθt)
VAC ωθ
1.5
ωθ
sinωθt
EO
3
R
A
VAC sinωθt θ sinωθt
VDC VDC
4 EO
─51─
φ
A sin[ωt + (θ sinωθt + φ)]
θ sinωθt φ
1
1.5 2
3
3(θ sinωθt + φ)
A√R 2 AR sin[ωt + 3(θ sinωθt + φ)]
2.5
AR sin[ωt + 3(θ sinωθt + φ)]
TEM00 (m+ 0.5)
(m + 1)
I (1)
I = A2
⎧⎪⎪⎨⎪⎪⎩m∑k=0
Rk sin[ωt + (2k + 1) (θ sinωθt + φ)
]⎫⎪⎪⎬⎪⎪⎭2
(1)
T 2π/ωθ(� 2π/ω)
(1) P
P=A2
T
T∫0
{ m∑k=0
Rk sin[ωt + (2k + 1)(θ sinωθt + φ)]}2dx (2)
(2) P
P =A2
2
m∑k=0
R2k+ A2m−1∑l=0
m−l∑k=1
Rk+2l cos[2k (θ sinωθt+φ)
]
=A2
2
m∑k=0
R2k + A2m−1∑l=0
m−l∑k=1
Rk+2l
⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩cos(2kφ)
[J0 (2θ) + 2
∞∑h=1
J2h (2kθ) cos (2hωθt)
]
−2 sin (2kφ)∞∑h=0
J2h+1 (2kθ) sin [(2h + 1)ωθt]
⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭(3)
Jh h
sinωθt PFm
PFm = −2A2m−1∑l=0
m−l∑k=1
Rk+2l sin (2kφ) J1 (2kθ) sinωθt (4)
1.5
m = 1
PF1 = −2A2R sin (2φ) J1 (2θ) sinωθt (5)
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0 0.5 1 1.5 2 2.5 3
θ
J 1(2
θ) θ =1.9159
5 J1(2θ) = 0 θ
φ θ
J1(2θ) = 0 θ 5
θ VAC
LN EO r22
θ =πn3
or22LVACλD
(6)
no LN λ L
D 0
PF1 = 0
φ R (6)
EO
2 m = 1
m
3.2 m
m→ ∞ (3) Fabry-Perot
Fabry-Perot 3
m
LN
─52─
[11]
DL= S 2 4λ
nπ(7)
L D n S
S 3 ∼ 6
(2m+ 1)
(2m + 1)L
(7)
m
m =12
(noπD2
4λLS 2− 1
)(8)
λ = 632.8nm
no = 2.2868 [27]
D = 2mm L = 20mm (8) m = 7.383
6 m D
m = 7 m ≥ 8
0
5
10
15
20
25
30
35
0 0.5 1 1.5 2 2.5 3 3.5 4
D (mm)
m
6 m D L = 20mm λ = 632.8nm
n = 2.2868
3.3 null
[24]
3
m = 3 PF3
PF3 =−2q∑i=1
A2i R sinωθt
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
(1+R2+R4
)sin (2φ) J1 (2θ)
+(R + R3
)sin (4φ) J1 (4θ)
+R2 sin (6φ) J1 (6θ)
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (9)
φ = π/4 2
3 m > 3 φ = π/4
0.00
0.01
0.10
1.00
10.00
1.0 1.5 2.0 2.5 3.0 3.5 4.0R
Syst
emat
ic E
rror
(%
)
7 R
1
MaterialRefractive
indices
Power
Reflectivity
Wavelength
(nm)
Systematic
Error(%)
KH2PO4 1.5079 0.0410 632.8 0.03
LiIO3 1.8830 0.0930 632.8 0.17
LiNbO3 2.3870 0.1677 441 0.58
LiNbO3 2.2864 0.1532 632.8 0.48
LiNbO3 2.2364 0.1459 1000 0.43
BaTiO3 2.4037 0.1701 632.8 0.60
InGaAsP 3.4 0.2975 1550 1.62
GaAs 3.4550 0.3037 1150 2.17
i
φ = π/4
2
3.2
m = 7
m = 7 EO
7
EO EO
1 KH2PO4
KH2PO4
LN
2.2
0.5% EO
GaAs InGaAsP
3
1%
3.4 null
(4) PFm = 0 θ
(4) m R φ θ 4
R
─53─
3
PFm = 0 m φ θ
PFm = 0
EO
m φ
(4) θ
3.2 m = 7
m = 7 φ
m = 7 φ θ PF7 (θ, φ)
R(no − 1no + 1
)2= 1.5328 φ
0.3 0.5 0.9 1.2mrad PF7 (θ, φ)
φ θ φ = 0.76567
θ = 0.90437 PF7 φ
φ = 0.76567 8 PF7 (θ, φ)
0 < φ <π
4φ 9
θ φ PF7
0 ≤ θ ≤ 2 0 ≤ φ ≤ π4
PF7
PF7 φ 0.76567 (4)
PF7 = 0 θ0 10 8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 0.5 1 1.5 2 2.5 3θ
PF7
( θ ,
φ)
φ = 0.3φ = 0.5φ = 0.76421φ = 0.9φ = 1.2
8 φ θ PF7 (θ, φ)
0
1.0
0.5
0.5 1.0 1.5 2.0
1.5
�(r
ad)
�(rad)
PF7(
θ,φ)
high
low
0
1.0
0.5
0.5 1.0 1.5 2.0
1.5
�(r
ad)
�(rad)
PF7(
θ,φ)
high
low
9 PF7 (θ, φ)
PF7 = 0 θ0 = 1.9251
PF7 = 0
m = 1
PF1 = 0 θ0 = 1.9159
0.5%
m θ0
λ = 632.8nm m = 1 ∼ 10
θ0 11 m ≥ 7
θ0 = 1.9251
m = 7
-0.2
-0.1
-0.1
0.0
0.1
0.1
0.2
1.86 1.88 1.9 1.92 1.94 1.96 1.98 2θ
PF7
( θ ,
φ)
φ = 0.3φ = 0.5φ = 0.76421φ = 0.9φ = 1.2
θ=1.9746
θ=1.9571
θ=1.9251
θ=1.9074θ=1.8719
10 φ θ PFm(θ, φ) θ0
(m = 7)
1.914
1.916
1.918
1.920
1.922
1.924
1.926
1 2 3 4 5 6 7 8 9 10m
θ 0 (
rad)
11 m θ0
4.
4.1
null
EO
EO
[28]
─54─
4 EO
3
1 Takizawa
MgO LN
632.8nm r13 r33
rc [28]
Takizawa LN
[28]
EO
(3)
(4)
3ωθ
3 sin 3ωθt
3 PTm
PTm=−2A2m−1∑l=0
m−l∑k=1
Rk+2l sin (2kφ) J3 (2kθ) sin 3ωθt (10)
(m = 1 ) (10)
(11)
PT1 = −2A R sin (2φ) J3 (2θ) sin 3ωθt (11)
(5) (5) (11)
3
PF1PT1
∣∣∣∣∣∣rms
=−√2A2R sin (2φ) J1 (2θ)
−√2A2R sin (2φ) J3 (2θ)=J1 (2θ)J3 (2θ)
(12)
(12) 3
J1/J3
12 α
β
EO
3.3
2
4.2 II( )
13 4 3
0
20
40
60
80
100
0 0.5 1 1.5 2θ
J1(
2 θ)/J
3(2 θ
)
β
α
12 J1(θ)/J3(θ) θ EO
α
4 3
φ
13
EO
null
(4) (10) θ
θ
PFmPTm
∣∣∣∣∣∣rms
=
m−1∑l=0
m−l∑k=1
Rk+2l sin (2kφ) J1 (2kθ)
m−1∑l=0
m−l∑k=1
Rk+2l sin (2kφ) J3 (2kθ)(13)
3.2 m = 7 null PF7
φPF7PT7
∣∣∣∣∣∣rms
3
θ VAC (6) EO
LN
25 632.8nm φ = 0.76567
φ (4) (10) PF7 PT7 θ
Laser BeamCrystal
����
����
Multi Meter
Photo Detector
Signal Generator + High Power Amp.Lock-in Amp.
sinAC DCV t Vθω +Lock-in Amp.Laser Beam
Crystal
����
����
Multi Meter
Photo Detector
Signal Generator + High Power Amp.Lock-in Amp.
sinAC DCV t Vθω +Lock-in Amp.
13
─55─
14PF7PT7
∣∣∣∣∣∣rms
J1(2θ)J3(2θ)
θ 15
3
α β
15 Δβ m = 1 m = 7
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
0 0.5 1 1.5 2 2.5θ (rad)
PF
7(2 θ
), P
T7(
2 θ)
Fundamental wave amplitude
Third harmonic wave amplitude (P T7)
(P F7)
14 θ 3
0
20
40
60
80
100
0 0.5 1 1.5 2θ
( )( )
7
7
2
2
F
T
rms
P
P
θθ
( )( )
1
3
2
2
J
J
θθ
βΔ
α
0
20
40
60
80
100
0 0.5 1 1.5 2θ
( )( )
7
7
2
2
F
T
rms
P
P
θθ
( )( )
1
3
2
2
J
J
θθ
βΔ
α
15
5.
16
ξ W
1/e2
2 [100]
[010] 16
[010] ξmrad
noW tan ξ
1
Δφ (14)
Δφ =2πλ
(no − 1)W tan ξ (14)
W = 1mm λ = 632.8nm no = 2.2868
ξ = 0.1mrad Δφ 1.277rad
null
17
M
w
(15)
Δφ =2πλ
(no − 1)wM
tan ξ (15)
M = 20
w = 0.5mm Δφ 0.0319rad
φ
0.21%
6. null
(1)
ξ
Laser Beam
LN
Wtanξ
W
xz
y
ξ
Laser Beam
LN
Wtanξ
W
xz
y
xz
y
16
Beam expanderLN crystal
WMW
Pinhole
w
xzy Beam expander
LN crystal
WMW
Pinhole
w
xzy
17 M
─56─
(2) Mach-Zehnder Michelson
(3)
(4)
(5)
(6) EO
EO
(7) EO
null
(1) null EO
8
10 φ
θ
EO
(2) null
EO
(3) EO
3
null
1 10 1 EO
EO
(4) 3
(5)
null
7.
7.1 LN
Y Z LN
EO r22
18 NEL
( ) EO
NEXIV VM-150M
2
19
L = 20.01mm d = 2.00mm LN
(100) [100]
90 ± 0.1 (010) [010] 90 ± 0.01
(001) [001] 90 ± 0.1
EO r22
0.05mrad
2
LN Crystal length (mm) Axial accuracy (deg)
L d X Axis Y Axis Z Axis
20.01 2.00 ±0.1 ±0.01 ±0.1
7.2
LN EO r22
20
x
z
y
L=20.01mm
D=2.00mm
x
z
y
x
z
y
L=20.01mm
D=2.00mm
18 non-doped congruent LN
─57─
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
1 2 3 4 5 6 7 8 9
���X����(mm)
Wed
ge A
ngle
(m
rad)
L
E
CW LaserBeam
x [100]z [001]
y [010]
LN Crystal (010)
(100)
(001)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
1 2 3 4 5 6 7 8 9
���X����(mm)
Wed
ge A
ngle
(m
rad)
L
E
CW LaserBeam
x [100]z [001]
y [010]
LN Crystal (010)
(100)
(001)
19 2 (001) [010]
ξ
• He-Ne MELLES GRIOT 05-LHP-171
( λ=632.8nm 15mW)
• Si-PIN NEOARK RD-252
• Agilent 33220A
• Agilent 34401A 6 1/2
• NF 3628
1 48dB
• NF LI5640
• HEOPS-3B10
• CS-4125
He-Ne
ND 100μW
LN VAC sin(ωθt)
VDC
P ωθ
PFm
P
P PFm
sin(ωθt)
3 PTm
sin(3ωθt)
25
25
0.05mrad
M = 20
w = 1mm
High PowerAmplifier
SignalGenerator
Lock-in Amp.(FW)
Band passFilter
OSC.
LN CrystalND FilterBeam
expander
PinholeLaser Source
Signal(P)
Signal(Pm)
Ref. Sig.
Multimeter
PhotoDetector
xz
y
Lock-in Amp.(THW)
High PowerAmplifier
SignalGenerator
Lock-in Amp.(FW)
Band passFilter
OSC.OSC.
LN CrystalND FilterBeam
expander
PinholeLaser Source
Signal(P)
Signal(Pm)
Ref. Sig.
Multimeter
PhotoDetector
xz
y
Lock-in Amp.(THW)
20
7.3 null
null EO r22
PFm VDC VAC
VDC VAC A
VAC (6) EO r22
no 2.2868 [27]
EO r22 = 6.54±0.02pm/V 0.02pm/V
20
VAC
EO r22
6.51pm/V
0.5 %
7.4
null EO r22
null PF7 VDC VAC
3 PT7
VACPF7PT7
∣∣∣∣∣∣rms
4.2 15
VAC = 168.0V PF7 β = 5.83
(6) θ = β r22 = 6.50 ± 0.05pm/V
0.05pm/V 20
─58─
7.5
null
3 null
2 null
3
3 r22
r22(pm/V)
6.54±0.02 Our group (null )
6.50±0.05 Our group ( )
6.3 Iwasaki [29]
6.4
Chirakadze [22]
Abdi [30]
Abarkan [31]
6.7Lenzo [12]
Smakula [32]
6.8 Smakula [32]
8.
EO
(1) EO
EO
•
•••
•
(2) L = 20mm D = 2mm
LN
(λ = 632.8nm)
m
m ≤ 7
(3) θ0 m m ≥ 8 θ = 1.9251
7.5
8
(4) null
θ EO
• EO φ
θ
• EO
(5) 3
EO
• null 1 ∼ 10 1
EO
• 3
• EO
(6)
(7) LiNbO3
632.8nm EO r22
• 6.54 ± 0.02pm/V (null )
• 6.50 ± 0.05pm/V ( )
(8)
LN 5% M O LN
1.8% MgO LN
(440nm) (3,39μ m) EO r22
r13 r33
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