Abstract

The performance of intracavity birefringence sensing by use of a standing-wave laser is theoretically analyzed when the cavity involves internal reflection. On the three-mirror compound cavity model, the condition for converting an optical path length into a laser frequency or a retardation into an optical beat frequency with good linearity and little uncertainty is derived as a function of the cavity parameters and is numerically analyzed.

© 2008 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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2006 (1)

1999 (2)

1998 (1)

1997 (1)

1993 (3)

1979 (1)

T. Yoshino, “Reflection anisotropy of 6328A laser mirrors,” Jpn. J. Appl. Phys. l8, 1503-1507 (1979).
[CrossRef]

1965 (1)

Ball, G.

Battiato, J.

Born, M.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergomon, 1964), p. 62.

Cordero, J. H.

Doyle, W. M.

Finnenmann, N.

Holzapfel, W.

Kawata, M.

Kim, B.

Kim, H.

Kim, S.

Kobayashi, Y.

Li, N.

Luo, F.

Meltz, G.

Morey, W.

Morse, T. F.

Park, H. G.

Quide, B.

Takahashi, Y.

Unlu, S.

Wang, D.

White, M. B.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergomon, 1964), p. 62.

Yoshino, T.

Appl. Opt. (2)

J. Lightwave Technol. (2)

J. Opt. Soc. Am. (1)

Jpn. J. Appl. Phys. (1)

T. Yoshino, “Reflection anisotropy of 6328A laser mirrors,” Jpn. J. Appl. Phys. l8, 1503-1507 (1979).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Other (1)

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergomon, 1964), p. 62.

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Figures (2)

Fig. 1
Fig. 1

Schematic diagram of the three-mirror compound cavity; the symbols are defined in the text.

Fig. 2
Fig. 2

Calculated uncertainty e in intracavity birefringence sensing as a function of internal surface reflectance r 2 2 and sensing-to-amplifying cavity ratio ρ with r 3 = 1 in the range of 10 6 r 2 2 0.5 .

Equations (48)

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r ^ 23 = [ r 3 exp ( i 2 δ 2 ) + r 2 ] / [ 1 + r 2 r 3 exp ( i 2 δ 2 ) ] ,
δ 2 = 2 π f L 2 / c ,
r 1 r ^ 23 g 2 exp ( i 2 δ 1 ) = exp ( i 2 m π ) ( i 2 = 1 ; m : integer ) ,
δ 1 = 2 π f L 1 / c ,
r ^ 23 = | r ^ 23 | exp ( i 2 ϕ )
2 ϕ = tan 1 { r 3 ( 1 r 2 2 ) sin ( 2 δ 2 ) / [ r 3 ( 1 + r 2 2 ) cos ( 2 δ 2 ) + r 2 ( 1 + r 3 2 ) ] } ,
R | r ^ 23 | 2 = 1 ( 1 r 2 2 ) ( 1 r 3 2 ) / [ 1 + r 2 2 r 3 2 + 2 r 2 r 3 cos ( 2 δ 2 ) ] 2 .
2 δ + 1 2 ϕ = 2 m π , ( m : positive   integer )
2 δ 1 + tan - 1 { r 3 ( 1 - r 2 2 ) sin ( 2 δ 2 ) / [ r 3 ( 1 + r 2 2 ) cos ( 2 δ 2 ) + r 2 ( 1 + r 3 2 ) ] } = 2 m π
4 π f L 1 / c + tan - 1 { r 3 ( 1 r 2 2 ) sin ( 4 π f L 2 / c ) / [ r 3 ( 1 + r 2 2 ) cos ( 4 π f L 2 / c ) + r 2 ( 1 + r 3 2 ) ] } = 2 m π .
δ 1 , 0 = ( 2 π / c ) f 0 L 1 , 0 ,
δ 2 , 0 = ( 2 π / c ) f 0 L 2 , 0 ,
Δ δ 1 = ( 2 π / c ) Δ f L 1 , 0 ,
Δ δ 2 = ( 2 π / c ) ( Δ f L 2 , 0 + f 0 Δ L 2 ) .
Δ δ + 1 Δ ϕ = 0 ,
Δ δ + 1 ( ϕ / δ 2 ) 0 Δ δ 2 = 0 ,
q ( δ 2 , 0 ) = ( ϕ / δ 2 ) 0
Δ f = { f 0 / [ q ( δ 2 , 0 ) L 1 , 0 + L 2 , 0 ] } Δ L 2 .
L * = L 1 , 0 q ( δ 2 , 0 ) + L 2 , 0 ,
Δ f = ( f / 0 L * ) Δ L 2 ,
S Δ f / Δ L 2 = f / 0 L * .
q ( δ 2 , 0 ) = [ ( 1 a 2 ) cos 2 ( 2 δ 2 , 0 ) + 2 b cos ( 2 δ 2 , 0 ) + a 2 + b 2 ] / { a [ 1 + b cos ( 2 δ 2 , 0 ) ] } ,
= { [ b + cos ( 2 δ 2 , 0 ) ] 2 + a 2 sin 2 2 δ 2 , 0 } / { a [ 1 + b cos ( 2 δ 2 , 0 ) ] } ,
a = ( 1 r 2 2 ) / ( 1 + r 2 2 ) , ( 0 a 1 )
b = [ r 2 / ( 1 + r 2 2 ) ] / [ r 3 / ( 1 + r 3 2 ) , ( 0 b < )
b = ( 1 a 2 ) 1 / 2 / [ 2 r 3 / ( 1 + r 3 2 ) ] ,
a 2 + b 2 1 .
d q ( δ 2 , 0 ) / d δ 2 , 0 q ' ( δ 2 , 0 ) = 2 a sin ( 2 δ 2 , 0 ) g ( δ 2 , 0 ) / { a [ 1 + b cos ( 2 δ 2 , 0 ) ] 2 } ,
g ( δ 2 , 0 ) = b ( 1 a 2 ) cos 2 ( 2 δ 2 , 0 ) + 2 ( 1 a 2 ) × cos ( 2 δ 2 , 0 ) b ( a 2 + b 2 2 ) ,
= b ( 1 a 2 ) [ cos ( 2 δ 2 , 0 ) + 1 / b ] 2 ( b 2 1 ) ( a 2 + b 2 1 ) / b .
q ( n π + π / 2 ) q ( δ 2 , 0 ) q ( n π ) for   b < 1 , ( n : positive integer )
( 1 b ) / a q ( δ 2 , 0 ) ( 1 + b ) / a for  b < 1     . [ Eq . ( 19 b ) ]
[ ( r 3 r 2 ) ( 1 r 2 r 3 ) ] / [ ( 1 r 2 2 ) r 3 ] q ( δ 2 , 0 ) [ ( r 2 + r 3 ) ( 1 + r 2 r 3 ) ] / [ ( 1 r 2 2 ) r 3 ] ( r 2 < r 3 ) .
q ( δ 2 , 0 ) 1 ,
L * ( ideal ) = L 1 , 0 + L 2 , 0 .
L * = L 1 , 0 + L 2 , 0 + [ q ( δ 2 , 0 ) 1 ] L 1 , 0 .
e = Max | L * L * ( ideal ) | / L * ( ideal ) ,
= Max | q ( δ 2 , 0 ) 1 | L 1 , 0 / ( L 1 , 0 + L 2 , 0 ) [ Eqs . ( 27 ) and ( 28 ) ] ,
e = ( b / a ) L / 1 , 0 ( L 1 , 0 + L ) , ( b < 1 ) 2 , 0
e = { [ r 2 ( 1 + r 3 2 ) ] / [ r 3 ( 1 r 2 2 ) ] } / ( 1 + ρ ) , ( r 2 < r 3 ) ,
ρ = L 2 , 0 / L 1 , 0 .
e = [ 2 r 2 / ( 1 r 2 2 ) ] / ( 1 + ρ ) . ( r 3 1 )
e 2 r 2 / ( 1 + ρ ) . ( r 2 2 1 , r 3 1 )
L = L 1 , 0 + L + 2 , 0 Δ L 2 / 2 ,
L = L 1 , 0 + L 2 , 0 Δ L 2 / 2 ,
| Δ L 2 | = ( Γ / 2 π ) λ 0 = ( Γ / 2 π ) ( c / f 0 ) , ( λ 0 : optical wavelength )
Δ f f f = f 0 | Δ L 2 | / ( L 1 , 0 + L 2 , 0 ) ,
f b = ( c / 2 π ) Γ / ( L 1 , 0 + L 2 , 0 ) .

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