Abstract

The signal-to-noise ratios for the path-stabilized, the active-heterodyne, the Sag, and the Fabry-Perot interferometers are calculated for cases in which dynamic surface displacements are detected. Expressions for the minimum-detectable displacements are given for the sensitivity limit of each interferometer. It is found that the ultimate sensitivities of the interferometers considered are nearly equal for similar conditions of use. Sensitivity values are shown to be about 10−15 m Hz−1/2 for systems that use typical laboratory instruments.

© 1987 Optical Society of America

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References

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  1. A. Ambrozy, Electronic Noise (McGraw-Hill, New York, 1982).
  2. Th. Kwaaitaal, B. J. Luymes, and G. A. van der Pijll, “Noise limitations of Michelson laser interferometers,” J. Phys. D 13, 1005 (1980).
    [CrossRef]
  3. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).
  4. R. M. De La Rue, R. F. Humphryes, I. M. Mason, and E. A. Ash, “Acoustic-surface-wave amplitude and phase measurements using laser probes,” Proc. IEE 119, 117 (1972).
  5. J. E. Bowers, R. L. Jungerman, B. T. Khuri-Yakub, and G. S. Kino, “An all fiber-optic sensor for surface acoustic wave measurements,” IEEE J. Lightwave Technol. LT-1, 429 (1983).
    [CrossRef]
  6. D. A. Jackson, A. Dandridge, and S. K. Sheem, “Measurement of small phase shifts using a single-mode optical-fiber interferometer,” Opt. Lett. 5, 139 (1980).
    [CrossRef] [PubMed]
  7. R. Ulrich, S. C. Rashleigh, and W. Eickhoff, “Bending-induced birefringence in single-mode fibers,” Opt. Lett. 5, 273 (1980).
    [CrossRef] [PubMed]
  8. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1984), pp. 323–327.
  9. J. P. Monchalin, “Optical detection of ultrasound,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control UFFC-33, 485 (1986).
    [CrossRef]
  10. J. P. Monchalin, “Optical detection of ultrasound using a confocal Fabry–Perot interferometer,” Appl. Phys. Lett. 47, 14 (1985).
    [CrossRef]

1986 (1)

J. P. Monchalin, “Optical detection of ultrasound,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control UFFC-33, 485 (1986).
[CrossRef]

1985 (1)

J. P. Monchalin, “Optical detection of ultrasound using a confocal Fabry–Perot interferometer,” Appl. Phys. Lett. 47, 14 (1985).
[CrossRef]

1983 (1)

J. E. Bowers, R. L. Jungerman, B. T. Khuri-Yakub, and G. S. Kino, “An all fiber-optic sensor for surface acoustic wave measurements,” IEEE J. Lightwave Technol. LT-1, 429 (1983).
[CrossRef]

1980 (3)

1972 (1)

R. M. De La Rue, R. F. Humphryes, I. M. Mason, and E. A. Ash, “Acoustic-surface-wave amplitude and phase measurements using laser probes,” Proc. IEE 119, 117 (1972).

Ambrozy, A.

A. Ambrozy, Electronic Noise (McGraw-Hill, New York, 1982).

Ash, E. A.

R. M. De La Rue, R. F. Humphryes, I. M. Mason, and E. A. Ash, “Acoustic-surface-wave amplitude and phase measurements using laser probes,” Proc. IEE 119, 117 (1972).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1984), pp. 323–327.

Bowers, J. E.

J. E. Bowers, R. L. Jungerman, B. T. Khuri-Yakub, and G. S. Kino, “An all fiber-optic sensor for surface acoustic wave measurements,” IEEE J. Lightwave Technol. LT-1, 429 (1983).
[CrossRef]

Dandridge, A.

De La Rue, R. M.

R. M. De La Rue, R. F. Humphryes, I. M. Mason, and E. A. Ash, “Acoustic-surface-wave amplitude and phase measurements using laser probes,” Proc. IEE 119, 117 (1972).

Eickhoff, W.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

Humphryes, R. F.

R. M. De La Rue, R. F. Humphryes, I. M. Mason, and E. A. Ash, “Acoustic-surface-wave amplitude and phase measurements using laser probes,” Proc. IEE 119, 117 (1972).

Jackson, D. A.

Jungerman, R. L.

J. E. Bowers, R. L. Jungerman, B. T. Khuri-Yakub, and G. S. Kino, “An all fiber-optic sensor for surface acoustic wave measurements,” IEEE J. Lightwave Technol. LT-1, 429 (1983).
[CrossRef]

Khuri-Yakub, B. T.

J. E. Bowers, R. L. Jungerman, B. T. Khuri-Yakub, and G. S. Kino, “An all fiber-optic sensor for surface acoustic wave measurements,” IEEE J. Lightwave Technol. LT-1, 429 (1983).
[CrossRef]

Kino, G. S.

J. E. Bowers, R. L. Jungerman, B. T. Khuri-Yakub, and G. S. Kino, “An all fiber-optic sensor for surface acoustic wave measurements,” IEEE J. Lightwave Technol. LT-1, 429 (1983).
[CrossRef]

Kwaaitaal, Th.

Th. Kwaaitaal, B. J. Luymes, and G. A. van der Pijll, “Noise limitations of Michelson laser interferometers,” J. Phys. D 13, 1005 (1980).
[CrossRef]

Luymes, B. J.

Th. Kwaaitaal, B. J. Luymes, and G. A. van der Pijll, “Noise limitations of Michelson laser interferometers,” J. Phys. D 13, 1005 (1980).
[CrossRef]

Mason, I. M.

R. M. De La Rue, R. F. Humphryes, I. M. Mason, and E. A. Ash, “Acoustic-surface-wave amplitude and phase measurements using laser probes,” Proc. IEE 119, 117 (1972).

Monchalin, J. P.

J. P. Monchalin, “Optical detection of ultrasound,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control UFFC-33, 485 (1986).
[CrossRef]

J. P. Monchalin, “Optical detection of ultrasound using a confocal Fabry–Perot interferometer,” Appl. Phys. Lett. 47, 14 (1985).
[CrossRef]

Rashleigh, S. C.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

Sheem, S. K.

Ulrich, R.

van der Pijll, G. A.

Th. Kwaaitaal, B. J. Luymes, and G. A. van der Pijll, “Noise limitations of Michelson laser interferometers,” J. Phys. D 13, 1005 (1980).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1984), pp. 323–327.

Appl. Phys. Lett. (1)

J. P. Monchalin, “Optical detection of ultrasound using a confocal Fabry–Perot interferometer,” Appl. Phys. Lett. 47, 14 (1985).
[CrossRef]

IEEE J. Lightwave Technol. (1)

J. E. Bowers, R. L. Jungerman, B. T. Khuri-Yakub, and G. S. Kino, “An all fiber-optic sensor for surface acoustic wave measurements,” IEEE J. Lightwave Technol. LT-1, 429 (1983).
[CrossRef]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

J. P. Monchalin, “Optical detection of ultrasound,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control UFFC-33, 485 (1986).
[CrossRef]

J. Phys. D (1)

Th. Kwaaitaal, B. J. Luymes, and G. A. van der Pijll, “Noise limitations of Michelson laser interferometers,” J. Phys. D 13, 1005 (1980).
[CrossRef]

Opt. Lett. (2)

Proc. IEE (1)

R. M. De La Rue, R. F. Humphryes, I. M. Mason, and E. A. Ash, “Acoustic-surface-wave amplitude and phase measurements using laser probes,” Proc. IEE 119, 117 (1972).

Other (3)

A. Ambrozy, Electronic Noise (McGraw-Hill, New York, 1982).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1984), pp. 323–327.

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Figures (8)

Fig. 1
Fig. 1

Path-stabilized Michelson interferometer.

Fig. 2
Fig. 2

Normalized SNR versus reference mirror displacement for stabilized Michelson interferometer.

Fig. 3
Fig. 3

Heterodyne Michelson interferometer.

Fig. 4
Fig. 4

Normalized SNR versus fraction of laser power directed to the reference path for the heterodyne interferometer.

Fig. 5
Fig. 5

Sagnac interferometer adapted for surface-displacement detection.

Fig. 6
Fig. 6

Normalized SNR versus modulation parameter group for the Sagnac interferometer.

Fig. 7
Fig. 7

Fabry–Perot interferometer used in surface-disturbance detection.

Fig. 8
Fig. 8

Normalized SNR versus nominal effective cavity length for the Fabry–Perot interferometer.

Equations (63)

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P D = P 0 { I [ m [ α m f m ] 1 / 2 E m ] I 2 } ,
E s = A s exp j [ ω 0 t - k 0 L s + 2 k 0 δ sin ( ω s t ) ] ,
E r = A r exp j ( ω 0 t - k 0 L r - 2 k 0 Δ ) ,
E Resultant = A r ( exp j [ ω 0 t - k 0 ( L r + 2 Δ ) ] + K exp j { ω 0 t - k 0 [ L s - 2 δ sin ( ω s t ) ] } ) ,
E Resultant = A r { 1 + K 2 + 2 K cos [ 2 k 0 ( Δ + δ sin ω s t ) ] } 1 / 2 ,
E Resultant = A r { 1 + K 2 + 2 K [ cos ( 2 k 0 Δ ) cos ( 2 k 0 δ sin ω s t ) - sin ( 2 k 0 Δ ) sin ( 2 k 0 δ sin ω s t ) ] } 1 / 2 ,
cos ( 2 k 0 δ sin ω s t ) = J 0 ( 2 k 0 δ ) + 2 I = 1 J 2 I ( 2 k 0 δ ) cos ( 2 I ω s t ) , sin ( 2 k 0 δ sin ω s t ) = 2 I = 0 J 2 I + 1 ( 2 k 0 δ ) sin [ ( 2 I + 1 ) ω s t ] ,
E Resultant = A r [ 1 + K 2 + 2 K ( [ J 0 ( 2 k 0 δ ) + 2 I = 1 J 2 I ( 2 k 0 δ ) cos ( 2 I ω s t ) ] cos ( 2 k 0 Δ ) - { 2 I = 0 J 2 I + 1 ( 2 k 0 δ ) sin [ ( 2 I + 1 ) ω s t ] } sin ( 2 k 0 Δ ) ) ] 1 / 2 .
E Resultant = A r { 1 + K 2 + 2 K [ cos ( 2 k 0 Δ ) - 2 k 0 δ sin ( 2 k 0 Δ ) sin ( ω s t ) ] } 1 / 2 .
I D = P D η q h ν ,
i q 2 = 2 q Δ ν I D ,
SNR = ( I s 2 i q 2 ) 1 / 2 .
SNR = ( P 0 η 4 h ν Δ ν ) 1 / 2 2 K sin ( 2 k 0 Δ ) [ 1 + K 2 + 2 K cos ( 2 k 0 Δ ) ] 1 / 2 k 0 δ .
SNR = ( P 0 η h ν Δ ν ) 1 / 2 0.707 sin ( 2 k 0 Δ ) [ 1 + cos ( 2 k 0 Δ ) ] 1 / 2 k 0 δ .
δ min = ( k 0 ) - 1 ( h ν Δ ν P 0 η ) 1 / 2 { 2 1 / 2 [ 1 + cos ( 2 k 0 Δ ) ] 1 / 2 sin ( 2 k 0 Δ ) } ,
δ min = 5.64 × 10 - 15 mHz - 1 / 2 .
E s = A s exp j [ ω 0 t - k 0 L s + 2 k 0 δ sin ( ω s t ) ] ,
E r = A r exp j [ ( ω 0 + 2 ω B ) t - k 0 L r ] ,
E Resultant = A r ( exp j [ ( ω 0 + 2 ω B ) t - k 0 L r ] + K exp j { ω 0 t - k 0 [ L s - 2 δ sin ( ω s t ) ] } ) ,
E Resultant = A r { 1 + K 2 + 2 K cos [ 2 ω B t + 2 k 0 δ sin ω s t - k 0 ( L s - L r ) ] } 1 / 2 ,
E Resultant = A r ( 1 + K 2 + 2 K { cos [ 2 ω B t - k 0 ( L s - L r ) ] × cos ( 2 k 0 δ sin ω s t ) - sin [ 2 ω B t - k 0 ( L s - L r ) ] × sin ( 2 k 0 δ sin ω s t ) } ) 1 / 2 ,
E Resultant = A r [ 1 + K 2 + 2 K ( cos [ 2 ω B t - k 0 ( L s - L r ) ] × [ J 0 ( 2 k 0 δ ) + 2 I = 1 J 2 I ( 2 k 0 δ ) cos ( 2 I ω s t ) ] - sin [ 2 ω B t - k 0 ( L s - L r ) ] × { 2 I = 0 J 2 I + 1 ( 2 k 0 δ ) sin [ ( 2 I + 1 ) ω s t ] } ) ] 1 / 2 .
E Resultant = A r { 1 + K 2 + 2 K [ J 0 ( 2 k 0 δ ) cos [ 2 ω B t - k 0 ( L s - L r ) ] + I = 1 J 2 I ( 2 k 0 δ ) { cos [ ( 2 ω B + 2 I ω s ) t - k 0 ( L s - L r ) ] + cos [ ( 2 ω B - 2 I ω s ) t - k 0 ( L s - L r ) ] } + I = 0 J 2 I + 1 ( 2 k 0 δ ) ( cos { [ 2 ω B + ( 2 I + 1 ) ω s ] t - k 0 ( L s - L r ) } - cos { [ 2 ω B - ( 2 I + 1 ) ω s ] t - k 0 ( L s - L r ) } ) ] } 1 / 2 .
E Resultant = A r [ 1 + K 2 + 2 K ( cos [ 2 ω B t - k 0 ( L s - L r ) ] + k 0 δ { cos [ ( 2 ω B + ω s ) t - k 0 ( L s - L r ) ] - cos [ ( 2 ω B - ω s ) t - k 0 ( L s - L r ) ] } ) ] 1 / 2 .
SNR = ( P 0 η 4 h ν Δ ν ) 1 / 2 2 K ( 1 + K 2 ) 1 / 2 k 0 δ ,
δ min = k 0 - 1 ( 2 h ν Δ ν P 0 η ) 1 / 2 ,
δ min = 7.94 × 10 - 15 mHz - 1 / 2 .
E cw = A cw exp j [ ω 0 t - k L cw - 2 k 0 δ sin ( ω s t ) - M k sin ( ω M t + ϕ ) ] ,
E acw = A acw exp j { ω 0 t - k L acw - 2 k 0 δ sin [ ω s ( t + T ) ] - M k sin [ ω M ( t + T ) + ϕ ] } ,
E Resultant = A ( exp j [ ω 0 t - k L c w - 2 k 0 δ sin ( ω s t ) - M k sin ( ω M t + ϕ ) ] + exp j { ω 0 t - k L acw - 2 k 0 δ sin [ ω s ( t + T ) ] - M k sin [ ω M ( t + T ) + ϕ ] } ) ,
E Resultant = 2 1 / 2 A [ 1 + cos ( 2 k 0 δ [ sin ω s t - sin ω s ( t + T ) ] + M k { sin ( ω m t + ϕ ) - sin [ ω m ( t + T ) + ϕ ] } ) ] 1 / 2 ,
β i = tan - 1 ( sin ω i T cos ω i T - 1 ) ,             γ i = 2 sin ( ω i T 2 ) ,
E Resultant = 2 1 / 2 A { 1 + cos [ 2 k 0 δ γ s sin ( ω s t + β s ) + M k γ M sin ( ω M t + β M + ϕ ) ] } 1 / 2 .
E Resultant = 2 1 / 2 A { 1 + cos [ 2 k 0 δ γ s sin ( ω s t + β s ) ] × cos [ M k γ M sin ( ω M t + β M + ϕ ) ] - sin [ 2 k 0 δ γ s sin ( ω s t + β s ) ] × sin [ M k γ M sin ( ω M t + β M + ϕ ) ] } 1 / 2 .
E Resultant = 2 1 / 2 A ( 1 + [ J 0 ( 2 k 0 δ γ s ) + 2 i = 1 J 2 i ( 2 k 0 δ γ s ) × cos 2 i ( ω s t + β s ) ] [ J 0 ( M k γ M ) + 2 I = 1 J 2 I ( M k γ M ) cos 2 I ( ω M t + β M + ϕ ) ] - { 2 m = 0 J 2 m + 1 ( 2 k 0 δ γ s ) sin [ ( 2 m + 1 ) ( ω s t + β s ) ] } × { 2 n = 0 J 2 n + 1 ( M k γ M ) × sin [ ( 2 n + 1 ) ( ω M t + β M + ϕ ) ] } ) 1 / 2 .
E Resultant = 2 1 / 2 A ( 1 + { J 0 ( M k γ M ) + 2 I = 1 J 2 I ( M k γ M ) cos [ 2 I ( ω M t + β M + ϕ ) ] } - 4 k 0 δ γ s sin ( ω s t + β s ) { n = 0 J 2 n + 1 ( M k γ M ) × sin [ ( 2 n + 1 ) ( ω M t + β M + ϕ ) ] } ) 1 / 2 .
E Resultant = 2 1 / 2 A [ 1 + [ J 0 ( M k γ M ) + 2 I = 1 J 2 I ( M k γ M ) × cos 2 I ( ω M t + β m + ϕ ) ] + 2 k 0 δ γ s ( n = 0 J 2 n + 1 ( M k γ M ) { cos [ ( 2 n + 1 ) × ( ω M t + β M + ϕ ) + ω s t + β s ] - cos [ ( 2 n + 1 ) × ( ω M t + β M + ϕ ) - ω s t - β s ] } ) ] 1 / 2 .
E Resultant = 2 1 / 2 A ( 1 + J 0 ( M k γ M ) + 2 k 0 δ γ s J 1 ( M k γ M ) × { cos [ ( ω M + ω s ) t + β M + β s + ϕ ] - cos [ ( ω M - ω s ) t + β M - β s + ϕ ] } ) 1 / 2 .
SNR = ( P 0 η h ν Δ ν ) 1 / 2 ζ J 1 ( M k γ M ) 2 1 / 2 [ 1 + ζ 2 + 2 ζ [ 1 + J 0 ( M k γ M ) ] } 1 / 2 k 0 δ γ s ,
M k γ M = M k 2 sin ( ω M T 2 ) ,
δ min = k 0 - 1 ( h ν Δ ν P 0 η ) 1 / 2 [ 2 + J 0 ( M k γ M ) ] 1 / 2 J 1 ( M k γ M ) ,
δ min = 1.46 × 10 - 14 mHz - 1 / 2 .
I T = ( 1 - R ) 2 ( 1 - R ) 2 + 4 R sin 2 ( ξ ) I I ,
ξ = 2 π n λ L cos θ ,
β = L / λ 0 ,
ξ = 2 π β ,
λ = { [ 1 - ( v c ) 2 ] 1 / 2 ( 1 - v c ) } 2 λ 0 ,
λ = ( 1 - 2 v c ) - 1 λ 0 ,
ξ = 2 π L λ 0 ( 1 - 2 v c ) .
ξ = 2 π β ( 1 - 2 v c ) .
Δ = δ sin ( ω s t ) ,
v = δ ω s cos ( ω s t ) .
ξ = 2 π β [ 1 - 2 δ ω s c cos ( ω s t ) ] .
I T Signal = ( 1 - R ) 2 I I Signal ( 1 - R ) 2 + 4 R sin 2 { 2 π β [ 1 - 2 δ ω s c cos ( ω s t ) ] } .
I T Signal = ( 1 + R 2 - 2 R ) I I Signal 1 + R 2 - 2 R { cos ( 4 π β ) cos [ 8 π β δ ω s c cos ( ω s t ) ] + sin ( 4 π β ) sin [ 8 π β δ ω s c cos ( ω s t ) ] } ,
I T Signal = ( 1 + R 2 - 2 R ) I I Signal 1 + R 2 - 2 R [ cos ( 4 π β ) + 8 π β δ ω s c sin ( 4 π β ) cos ( ω s t ) ] .
I T Signal = ( 1 + R 2 - 2 R ) I I Signal 1 + R 2 - 2 R cos ( 4 π β ) ,
( I T Signal ) 2 = 1 2 { 16 π R β δ ω s c sin ( 4 π β ) ( 1 + R 2 - 2 R ) I I Signal [ 1 + R 2 - 2 R cos ( 4 π β ) ] 2 } 2 ,
I q 2 1 / 2 = ( P I 2 q 2 η Δ ν h ν ) 1 / 2 [ 1 + R 2 - 2 R 1 + R 2 - 2 R cos ( 4 π β ) ] 1 / 2 ,
i Signal 2 1 / 2 = 2 - 1 / 2 η q h ν P I { 16 π R β δ ω s c sin ( 4 π β ) ( 1 + R 2 - 2 R ) [ 1 + R 2 - 2 R cos ( 4 π β ) ] 2 } .
SNR = ( η P I h ν Δ ν ) 1 / 2 ( 1 - R ) [ 8 π R β sin ( 4 π β ) ] [ 1 + R 2 - 2 R cos ( 4 π β ) ] 3 / 2 δ ω s c ,
δ min = ( h ν Δ ν η P I ) 1 / 2 [ 1 + R 2 - 2 R cos ( 4 π β ) ] 3 / 2 ( 1 - R ) [ 8 π R β sin ( 4 π β ) ] c ω s .
δ min = 9.2 × 10 - 16 mHz - 1 / 2 ,

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