Abstract

We present a heterodyne Michelson interferometer for vibration measurement in which feedback is used to obviate the need to unwrap phase data. The Doppler shift of a vibrating target mirror is sensed interferometrically and compensated by means of a voltage-controlled oscillator driving an acousto-optic modulator. For frequencies within the servo bandwidth, the oscillator control voltage provides a direct measurement of the target velocity. Outside the servo bandwidth, phase-sensitive detection is used to evaluate high-frequency displacements. This approach is of great interest for the frequently-occurring situation where vibration amplitudes at low frequency exceed an optical wavelength, but knowledge of the vibration spectrum at high frequency is important as well.

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References

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  1. J. Kauffmann and H. J. Tiziani, “Time-resolved vibration measurement with temporal speckle pattern interferometry,” Appl. Opt. 45, 6682-6688 (2006).
    [CrossRef] [PubMed]
  2. D. J. Park, G. J. Park, T. S. Aum, J. H. Yi, and J. H. Kwon, “Measurement of displacement and vibration by using the oblique ray method,” Appl. Opt. 45, 3728-3732 (2006).
    [CrossRef] [PubMed]
  3. G. Pedrini, W. Osten, and M. E. Gusev, “High-speed digital holographic interferometry for vibration measurement,” Appl. Opt. 45, 3456-3462 (2006).
    [CrossRef] [PubMed]
  4. R. A. Bruce and G. L. Fitzpatrick, “Remote vibration measurement of rough surfaces by laser interferometry,” Appl. Opt. 14, 1621-1626 (1975).
    [CrossRef] [PubMed]
  5. S. C. Bartlett, F. Farahi, and D. A. Jackson, “A dual resolution noncontact vibration and displacement sensor based upon a two wavelength source,” Rev. Sci. Instrum. 61, 1014-1017(1990).
    [CrossRef]
  6. N. Mio and K. Tsubono, “Vibration transducer using an ultrashort Fabry-Perot cavity,” Appl. Opt. 34, 186-189 (1995).
    [CrossRef] [PubMed]
  7. H.-J. Yang, J. Deibel, S. Nyberg, and K. Riles, “High-precision absolute distance and vibration measurement with frequency scanned interferometry,” Appl. Opt. 44, 3937-3944 (2005).
    [CrossRef] [PubMed]
  8. O. B. Wright, “Stabilized dual-wavelength fiber-optic interferometer for vibration measurement,” Opt. Lett. 16, 56-58(1991).
    [CrossRef] [PubMed]
  9. L. M. Barker and R. E. Hollenbach, “Laser interferometer for measuring high velocities of any reflecting surface,” J. Appl. Phys. 43, 4669-4675 (1972).
    [CrossRef]
  10. J. Weng, H. Tan, S. Hu, Y. Ma, and X. Wang, “New all-fiber velocimeter,” Rev. Sci. Instrum. 76, 093301 (2005).
    [CrossRef]
  11. J. Shamir, “Compact interferometer for accurate determination of optical constants of thin films,” J. Phys. E 9, 499-503(1976).
    [CrossRef]
  12. F. P. Küpper and W. J. Mastop, “Stabilized and calibrated Michelson interferometer,” Rev. Sci. Instrum. 47, 434-436 (1976).
    [CrossRef]
  13. T. Kwaaitaal, “Contribution to the interferometric measurement of sub-angstrom vibrations,” Rev. Sci. Instrum. 45, 39-41 (1974).
    [CrossRef]
  14. P.-Y. Chien, “Two-frequency displacement measurement interferometer based on a double-heterodyne technique,” Rev. Sci. Instrum. 62, 254-255 (1991).
    [CrossRef]
  15. W. M. J. Haesen and T. Kwaaitaal, “Improvement in the interferometric measurement of subangstrom vibrations,” Rev. Sci. Instrum. 44, 954-955 (1973).
    [CrossRef]
  16. F. E. Terman, Electronic and Radio Engineering (McGraw-Hill, 1955), Chap. 17.
  17. J. Lawall, J. M. Pedulla, and Y. L. Coq, “Ultrastable laser array at 633 nm for real-time dimensional metrology,” Rev. Sci. Instrum. 72, 2879-2888 (2001).
    [CrossRef]
  18. Perkin Elmer Corporation, Model No. 7280, http://www.signalrecovery.com/. Certain commercial equipment, instruments, or materials are identified in this article in order to specify the experimental procedure adequately. Such identification is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor is it intended to imply that the materials or equipment identified are necessarily the best available for the purpose.
  19. E. O. Brigham, The Fast Fourier Transform and its Applications (Prentice-Hall, 1988).

2006

2005

2001

J. Lawall, J. M. Pedulla, and Y. L. Coq, “Ultrastable laser array at 633 nm for real-time dimensional metrology,” Rev. Sci. Instrum. 72, 2879-2888 (2001).
[CrossRef]

1995

1991

O. B. Wright, “Stabilized dual-wavelength fiber-optic interferometer for vibration measurement,” Opt. Lett. 16, 56-58(1991).
[CrossRef] [PubMed]

P.-Y. Chien, “Two-frequency displacement measurement interferometer based on a double-heterodyne technique,” Rev. Sci. Instrum. 62, 254-255 (1991).
[CrossRef]

1990

S. C. Bartlett, F. Farahi, and D. A. Jackson, “A dual resolution noncontact vibration and displacement sensor based upon a two wavelength source,” Rev. Sci. Instrum. 61, 1014-1017(1990).
[CrossRef]

1976

J. Shamir, “Compact interferometer for accurate determination of optical constants of thin films,” J. Phys. E 9, 499-503(1976).
[CrossRef]

F. P. Küpper and W. J. Mastop, “Stabilized and calibrated Michelson interferometer,” Rev. Sci. Instrum. 47, 434-436 (1976).
[CrossRef]

1975

1974

T. Kwaaitaal, “Contribution to the interferometric measurement of sub-angstrom vibrations,” Rev. Sci. Instrum. 45, 39-41 (1974).
[CrossRef]

1973

W. M. J. Haesen and T. Kwaaitaal, “Improvement in the interferometric measurement of subangstrom vibrations,” Rev. Sci. Instrum. 44, 954-955 (1973).
[CrossRef]

1972

L. M. Barker and R. E. Hollenbach, “Laser interferometer for measuring high velocities of any reflecting surface,” J. Appl. Phys. 43, 4669-4675 (1972).
[CrossRef]

Aum, T. S.

Barker, L. M.

L. M. Barker and R. E. Hollenbach, “Laser interferometer for measuring high velocities of any reflecting surface,” J. Appl. Phys. 43, 4669-4675 (1972).
[CrossRef]

Bartlett, S. C.

S. C. Bartlett, F. Farahi, and D. A. Jackson, “A dual resolution noncontact vibration and displacement sensor based upon a two wavelength source,” Rev. Sci. Instrum. 61, 1014-1017(1990).
[CrossRef]

Brigham, E. O.

E. O. Brigham, The Fast Fourier Transform and its Applications (Prentice-Hall, 1988).

Bruce, R. A.

Chien, P.-Y.

P.-Y. Chien, “Two-frequency displacement measurement interferometer based on a double-heterodyne technique,” Rev. Sci. Instrum. 62, 254-255 (1991).
[CrossRef]

Coq, Y. L.

J. Lawall, J. M. Pedulla, and Y. L. Coq, “Ultrastable laser array at 633 nm for real-time dimensional metrology,” Rev. Sci. Instrum. 72, 2879-2888 (2001).
[CrossRef]

Deibel, J.

Farahi, F.

S. C. Bartlett, F. Farahi, and D. A. Jackson, “A dual resolution noncontact vibration and displacement sensor based upon a two wavelength source,” Rev. Sci. Instrum. 61, 1014-1017(1990).
[CrossRef]

Fitzpatrick, G. L.

Gusev, M. E.

Haesen, W. M. J.

W. M. J. Haesen and T. Kwaaitaal, “Improvement in the interferometric measurement of subangstrom vibrations,” Rev. Sci. Instrum. 44, 954-955 (1973).
[CrossRef]

Hollenbach, R. E.

L. M. Barker and R. E. Hollenbach, “Laser interferometer for measuring high velocities of any reflecting surface,” J. Appl. Phys. 43, 4669-4675 (1972).
[CrossRef]

Hu, S.

J. Weng, H. Tan, S. Hu, Y. Ma, and X. Wang, “New all-fiber velocimeter,” Rev. Sci. Instrum. 76, 093301 (2005).
[CrossRef]

Jackson, D. A.

S. C. Bartlett, F. Farahi, and D. A. Jackson, “A dual resolution noncontact vibration and displacement sensor based upon a two wavelength source,” Rev. Sci. Instrum. 61, 1014-1017(1990).
[CrossRef]

Kauffmann, J.

Küpper, F. P.

F. P. Küpper and W. J. Mastop, “Stabilized and calibrated Michelson interferometer,” Rev. Sci. Instrum. 47, 434-436 (1976).
[CrossRef]

Kwaaitaal, T.

T. Kwaaitaal, “Contribution to the interferometric measurement of sub-angstrom vibrations,” Rev. Sci. Instrum. 45, 39-41 (1974).
[CrossRef]

W. M. J. Haesen and T. Kwaaitaal, “Improvement in the interferometric measurement of subangstrom vibrations,” Rev. Sci. Instrum. 44, 954-955 (1973).
[CrossRef]

Kwon, J. H.

Lawall, J.

J. Lawall, J. M. Pedulla, and Y. L. Coq, “Ultrastable laser array at 633 nm for real-time dimensional metrology,” Rev. Sci. Instrum. 72, 2879-2888 (2001).
[CrossRef]

Ma, Y.

J. Weng, H. Tan, S. Hu, Y. Ma, and X. Wang, “New all-fiber velocimeter,” Rev. Sci. Instrum. 76, 093301 (2005).
[CrossRef]

Mastop, W. J.

F. P. Küpper and W. J. Mastop, “Stabilized and calibrated Michelson interferometer,” Rev. Sci. Instrum. 47, 434-436 (1976).
[CrossRef]

Mio, N.

Nyberg, S.

Osten, W.

Park, D. J.

Park, G. J.

Pedrini, G.

Pedulla, J. M.

J. Lawall, J. M. Pedulla, and Y. L. Coq, “Ultrastable laser array at 633 nm for real-time dimensional metrology,” Rev. Sci. Instrum. 72, 2879-2888 (2001).
[CrossRef]

Riles, K.

Shamir, J.

J. Shamir, “Compact interferometer for accurate determination of optical constants of thin films,” J. Phys. E 9, 499-503(1976).
[CrossRef]

Tan, H.

J. Weng, H. Tan, S. Hu, Y. Ma, and X. Wang, “New all-fiber velocimeter,” Rev. Sci. Instrum. 76, 093301 (2005).
[CrossRef]

Terman, F. E.

F. E. Terman, Electronic and Radio Engineering (McGraw-Hill, 1955), Chap. 17.

Tiziani, H. J.

Tsubono, K.

Wang, X.

J. Weng, H. Tan, S. Hu, Y. Ma, and X. Wang, “New all-fiber velocimeter,” Rev. Sci. Instrum. 76, 093301 (2005).
[CrossRef]

Weng, J.

J. Weng, H. Tan, S. Hu, Y. Ma, and X. Wang, “New all-fiber velocimeter,” Rev. Sci. Instrum. 76, 093301 (2005).
[CrossRef]

Wright, O. B.

Yang, H.-J.

Yi, J. H.

Appl. Opt.

J. Appl. Phys.

L. M. Barker and R. E. Hollenbach, “Laser interferometer for measuring high velocities of any reflecting surface,” J. Appl. Phys. 43, 4669-4675 (1972).
[CrossRef]

J. Phys. E

J. Shamir, “Compact interferometer for accurate determination of optical constants of thin films,” J. Phys. E 9, 499-503(1976).
[CrossRef]

Opt. Lett.

Rev. Sci. Instrum.

S. C. Bartlett, F. Farahi, and D. A. Jackson, “A dual resolution noncontact vibration and displacement sensor based upon a two wavelength source,” Rev. Sci. Instrum. 61, 1014-1017(1990).
[CrossRef]

F. P. Küpper and W. J. Mastop, “Stabilized and calibrated Michelson interferometer,” Rev. Sci. Instrum. 47, 434-436 (1976).
[CrossRef]

T. Kwaaitaal, “Contribution to the interferometric measurement of sub-angstrom vibrations,” Rev. Sci. Instrum. 45, 39-41 (1974).
[CrossRef]

P.-Y. Chien, “Two-frequency displacement measurement interferometer based on a double-heterodyne technique,” Rev. Sci. Instrum. 62, 254-255 (1991).
[CrossRef]

W. M. J. Haesen and T. Kwaaitaal, “Improvement in the interferometric measurement of subangstrom vibrations,” Rev. Sci. Instrum. 44, 954-955 (1973).
[CrossRef]

J. Weng, H. Tan, S. Hu, Y. Ma, and X. Wang, “New all-fiber velocimeter,” Rev. Sci. Instrum. 76, 093301 (2005).
[CrossRef]

J. Lawall, J. M. Pedulla, and Y. L. Coq, “Ultrastable laser array at 633 nm for real-time dimensional metrology,” Rev. Sci. Instrum. 72, 2879-2888 (2001).
[CrossRef]

Other

Perkin Elmer Corporation, Model No. 7280, http://www.signalrecovery.com/. Certain commercial equipment, instruments, or materials are identified in this article in order to specify the experimental procedure adequately. Such identification is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor is it intended to imply that the materials or equipment identified are necessarily the best available for the purpose.

E. O. Brigham, The Fast Fourier Transform and its Applications (Prentice-Hall, 1988).

F. E. Terman, Electronic and Radio Engineering (McGraw-Hill, 1955), Chap. 17.

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Figures (7)

Fig. 1
Fig. 1

Optical and electronic setups. OI, optical isolator. Other acronyms defined in text. Light from a stabilized laser is split into two beams with frequencies ν 1 (fixed) and ν 2 (variable). The light is passed through a fiber and sent into a Michelson interferometer. The “reference” arm of the interferometer is driven by light of frequency ν 1 , and the “measurement” arm is driven by light of frequency ν 2 . A feedback loop varies frequency ν 2 so as to maintain a constant frequency difference between the light returned from the two arms of the interferometer in the presence of a Doppler shift arising from the motion of the mirror reflecting light of frequency ν 2 . Dashed box, lock-in amplifier; solid box, setup used to evaluate the noise floor.

Fig. 2
Fig. 2

(a) Displacement versus time as measured from phase evolution of interferometer fringe. The total excursion corresponds to about 38 optical fringes, and the fundamental frequency is 1.4 Hz . (b) Corresponding frequency spectrum. Harmonics of the fundamental frequency are clearly visible up to 20 Hz .

Fig. 3
Fig. 3

(a) Demodulated X (upper trace) and Y (lower trace) quadrature signals coming from the lock-in amplifier under condition of weak feedback. (b) Corresponding polar plot, showing that the phase angle remains within a range of approximately ± 1 rad .

Fig. 4
Fig. 4

Dark trace: position spectrum inferred from quadrature output of lock-in amplifier under condition of weak feedback forcing it to zero. The spectrum is useful only above several hundred hertz, where the feedback is not effective. Gray (red online) trace: spectrum obtained without feedback, as shown in Fig. 2b.

Fig. 5
Fig. 5

(a) Velocity spectrum inferred from variations in VCO frequency under conditions of strong feedback. (b) Black line, position spectrum obtained by dividing velocity spectrum by 2 π ν ; gray (red online) line, measurement made without feedback shown in Fig. 2b.

Fig. 6
Fig. 6

(a) Solid (green online) line: Position spectrum obtained with “high-gain” signal. Dotted line: Position spectrum obtained with “low-gain” signal. Note the broad range of overlap, from about 256 Hz to 6 kHz . (b) Typical complete spectrum, consisting of two high gain spectra and one low gain spectrum, all concatenated. The upper trace is the cryostat measurement, and the lower trace is the noise floor.

Fig. 7
Fig. 7

Signal for three different oscillator configurations. Solid (blue online) line, AOM2 driven by a synthesizer employing direct digital frequency synthesis; dotted (black) line, AOM2 driven by a synthesizer employing frequency multiplication of a high-quality fixed reference oscillator; dashed (red online) line, AOM1 and AOM2, each driven by a different synthesizer employing frequency multiplication of a high-quality fixed reference oscillator.

Equations (25)

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E 1 ( t ) cos [ 2 π ( ν L + ν 1 ) t ] ,
E 2 ( t ) cos [ 2 π ( ν L t + 2 x ( t ) / λ + t ν 2 ( t ) d t ) ] ,
P ( t ) = P 0 { 1 + cos [ 2 π ( ν 1 t t ν 2 ( t ) d t 2 x ( t ) / λ ) ] } ,
P ( t ) = P 0 { 1 + cos [ 2 π ( Δ ν t 2 x ( t ) / λ ) ] } ,
Δ ν = ν 1 ν 2
X ( t ) = GK P ( t ) cos ( 2 π ν ref t ) ,
= GK P 0 2 cos { 2 π [ ( ν ref Δ ν ) t + 2 x ( t ) / λ ] } ,
Y ( t ) = GK P ( t ) sin ( 2 π ν ref t )
= GK P 0 2 sin { 2 π [ ( ν ref Δ ν ) t + 2 x ( t ) / λ ] } ,
Φ ( t ) = tan 1 Y ( t ) X ( t ) ,
= 2 π [ ( ν ref Δ ν ) t + 2 x ( t ) / λ ] .
Y ( t ) = GK P ( t ) sin ( 2 π ν ref t ) ,
Y ( t ) = GK P 0 2 sin [ 2 π ( t ν 2 ( t ) d t ν 1 t + ν ref t + 2 x ( t ) / λ ) ] .
t ν 2 ( t ) d t ν 1 t + ν ref t + 2 x ( t ) / λ = 0 ;
ν 2 ( t ) = ν 1 ν ref 2 x ˙ ( t ) / λ .
Y ( t ) = GK P 0 2 [ 2 π ( t ν 2 ( t ) d t ν 1 t + ν ref t + 2 x ( t ) / λ ) ] .
x ( t ) = x slow ( t ) + x fast ( t ) ,
Y ( t ) = 2 π GK P 0 x fast ( t ) / λ .
R = GK P 0 2 ,
x fast ( t ) = λ 4 π Y ( t ) R .
ν 2 ( t ) = ν 1 ν ref 2 x ˙ ( t ) / λ ,
δ ν 2 = 2 x ˙ ( t ) / λ
δ ν 2 = K VCXO V .
x ˙ ( t ) = λ 2 K VCXO V ( t ) ,
| x ( ν ) | = | x ˙ ( ν ) | 2 π ν .

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