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:dd053501@cc.seikei.ac.jp)∗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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voltaic effects in doped linbo3”, Journal of Modern Op-

tics, 24, 4, pp. 475–482 (1977).

[2] S. Kawada: “The trend of the high-density optical

─59─

recording technology. the multilayer optic recording by

the applcation of the photo-refractive material”, O plus

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d speckle-shift hologram and its storage capacity”, Opt

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[6] M. Doi, M. Sugiyama, K. Tanaka and M. Kawai: “Ad-

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[8] Y. Fujii and T. Sakudo: “Interferometric determination

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man: “Electrooptic coefficients and elastic-wave propa-

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[13] E. H. Turner: “High-frequency electro-optic coefficients

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[14] S. K. Mathew: “Experimentally determined r13 electro-

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N. Theofanous and G. Alexakis: “Influence of the

temperature-dependent spontaneous birefringence in the

electro-optic measurements of LiNbO3”, J. Appl. Phys.,

65, 65, pp. 2406–2408 (1989).

[16] J. A. de Toro, M. D. Serrano, A. G. Cabanes and

J. M. Cabrera: “Accurate interferometric measure-

ment of electro-optic coefficients: application to quasi-

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(1998).

[17] R. Johannes and W. Haas: “Linear electrooptic effect in

potassium tantalate niobate crystals”, Appl. Opt., 6, 6, p.

1059 (1967).

[18] A. R. Johanson: “The strain-free electro-optic effect in

single-crystal barium titanate”, Appl. Phys. Lett., 7, 7,

pp. 195–198 (1965).

[19] D. F. Nelson and E. H. Turner: “Electro-optic and piezo-

electric coefficients and refractive index of gallium phos-

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