Abstract

A small-displacement sensing system based on multiple total internal reflections in heterodyne interferometry is proposed. In this paper, a small displacement can be obtained only by measuring the variation in phase difference between s- and p-polarization states for the total internal reflection effect. In order to improve the sensitivity, we increase the number of total internal reflections by using a parallelogram prism. The theoretical resolution of the method is better than 0.417nm. The method has some merits, e.g., high resolution, high sensitivity, and real-time measurement. Also, its feasibility is demonstrated.

© 2009 Optical Society of America

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References

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  1. A. Nesci, R. Dändliker, and H.P. Herzig, “Quantitative amplitude and phase measurement by use of a heterodyne scanning near-field optical microscope,” Opt. Lett. 26, 208-210 (2001).
    [CrossRef]
  2. F. C. Demarest, “High-resolution, high-speed, low data age uncertainty, heterodyne displacement measuring interferometer electronics,” Meas. Sci. Technol. 9, 1024-1030 (1998).
    [CrossRef]
  3. X. Liu, W. Clegg, D. F. L. Jenkins, and B. Liu, “Polarization interferometer for measuring small displacement,” IEEE Trans. Instrum. Meas. 50, 868-871 (2001).
    [CrossRef]
  4. J.-Y. Lee, H.-Y. Chen, C.-C. Hsu, and C.-C. Wu, “Optical heterodyne grating interferometry for displacement measurement with subnanometric resolution,” Sens. Actuators A 137, 185-191 (2007).
    [CrossRef]
  5. M.-H. Chiu, B.-Y. Shih, C.-W. Lai, L.-H. Shyu, and T.-H. Wu, “Small absolute distance measurement with nanometer resolution using geometrical optics principles and a SPR angular sensor,” Sens. Actuators A 141, 217-223(2008).
    [CrossRef]
  6. M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), Chap. 1, pp. 45-50.
  7. M.-H. Chiu, S.-F. Wang, and R.-S. Chang, “Instrument for measuring small angles by use of multiple total internal reflections in heterodyne interferometry,” Appl. Opt. 43, 5438-5442 (2004).
    [CrossRef] [PubMed]
  8. S. Shen, T. Liu, and J. Guo, “Optical phase-shift detection of surface plasmon resonance,” Appl. Opt. 37, 1747-1751 (1998).
    [CrossRef]

2008 (1)

M.-H. Chiu, B.-Y. Shih, C.-W. Lai, L.-H. Shyu, and T.-H. Wu, “Small absolute distance measurement with nanometer resolution using geometrical optics principles and a SPR angular sensor,” Sens. Actuators A 141, 217-223(2008).
[CrossRef]

2007 (1)

J.-Y. Lee, H.-Y. Chen, C.-C. Hsu, and C.-C. Wu, “Optical heterodyne grating interferometry for displacement measurement with subnanometric resolution,” Sens. Actuators A 137, 185-191 (2007).
[CrossRef]

2004 (1)

2001 (2)

X. Liu, W. Clegg, D. F. L. Jenkins, and B. Liu, “Polarization interferometer for measuring small displacement,” IEEE Trans. Instrum. Meas. 50, 868-871 (2001).
[CrossRef]

A. Nesci, R. Dändliker, and H.P. Herzig, “Quantitative amplitude and phase measurement by use of a heterodyne scanning near-field optical microscope,” Opt. Lett. 26, 208-210 (2001).
[CrossRef]

1998 (2)

F. C. Demarest, “High-resolution, high-speed, low data age uncertainty, heterodyne displacement measuring interferometer electronics,” Meas. Sci. Technol. 9, 1024-1030 (1998).
[CrossRef]

S. Shen, T. Liu, and J. Guo, “Optical phase-shift detection of surface plasmon resonance,” Appl. Opt. 37, 1747-1751 (1998).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), Chap. 1, pp. 45-50.

Chang, R.-S.

Chen, H.-Y.

J.-Y. Lee, H.-Y. Chen, C.-C. Hsu, and C.-C. Wu, “Optical heterodyne grating interferometry for displacement measurement with subnanometric resolution,” Sens. Actuators A 137, 185-191 (2007).
[CrossRef]

Chiu, M.-H.

M.-H. Chiu, B.-Y. Shih, C.-W. Lai, L.-H. Shyu, and T.-H. Wu, “Small absolute distance measurement with nanometer resolution using geometrical optics principles and a SPR angular sensor,” Sens. Actuators A 141, 217-223(2008).
[CrossRef]

M.-H. Chiu, S.-F. Wang, and R.-S. Chang, “Instrument for measuring small angles by use of multiple total internal reflections in heterodyne interferometry,” Appl. Opt. 43, 5438-5442 (2004).
[CrossRef] [PubMed]

Clegg, W.

X. Liu, W. Clegg, D. F. L. Jenkins, and B. Liu, “Polarization interferometer for measuring small displacement,” IEEE Trans. Instrum. Meas. 50, 868-871 (2001).
[CrossRef]

Dändliker, R.

Demarest, F. C.

F. C. Demarest, “High-resolution, high-speed, low data age uncertainty, heterodyne displacement measuring interferometer electronics,” Meas. Sci. Technol. 9, 1024-1030 (1998).
[CrossRef]

Guo, J.

Herzig, H. P.

Hsu, C.-C.

J.-Y. Lee, H.-Y. Chen, C.-C. Hsu, and C.-C. Wu, “Optical heterodyne grating interferometry for displacement measurement with subnanometric resolution,” Sens. Actuators A 137, 185-191 (2007).
[CrossRef]

Jenkins, D. F. L.

X. Liu, W. Clegg, D. F. L. Jenkins, and B. Liu, “Polarization interferometer for measuring small displacement,” IEEE Trans. Instrum. Meas. 50, 868-871 (2001).
[CrossRef]

Lai, C.-W.

M.-H. Chiu, B.-Y. Shih, C.-W. Lai, L.-H. Shyu, and T.-H. Wu, “Small absolute distance measurement with nanometer resolution using geometrical optics principles and a SPR angular sensor,” Sens. Actuators A 141, 217-223(2008).
[CrossRef]

Lee, J.-Y.

J.-Y. Lee, H.-Y. Chen, C.-C. Hsu, and C.-C. Wu, “Optical heterodyne grating interferometry for displacement measurement with subnanometric resolution,” Sens. Actuators A 137, 185-191 (2007).
[CrossRef]

Liu, B.

X. Liu, W. Clegg, D. F. L. Jenkins, and B. Liu, “Polarization interferometer for measuring small displacement,” IEEE Trans. Instrum. Meas. 50, 868-871 (2001).
[CrossRef]

Liu, T.

Liu, X.

X. Liu, W. Clegg, D. F. L. Jenkins, and B. Liu, “Polarization interferometer for measuring small displacement,” IEEE Trans. Instrum. Meas. 50, 868-871 (2001).
[CrossRef]

Nesci, A.

Shen, S.

Shih, B.-Y.

M.-H. Chiu, B.-Y. Shih, C.-W. Lai, L.-H. Shyu, and T.-H. Wu, “Small absolute distance measurement with nanometer resolution using geometrical optics principles and a SPR angular sensor,” Sens. Actuators A 141, 217-223(2008).
[CrossRef]

Shyu, L.-H.

M.-H. Chiu, B.-Y. Shih, C.-W. Lai, L.-H. Shyu, and T.-H. Wu, “Small absolute distance measurement with nanometer resolution using geometrical optics principles and a SPR angular sensor,” Sens. Actuators A 141, 217-223(2008).
[CrossRef]

Wang, S.-F.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), Chap. 1, pp. 45-50.

Wu, C.-C.

J.-Y. Lee, H.-Y. Chen, C.-C. Hsu, and C.-C. Wu, “Optical heterodyne grating interferometry for displacement measurement with subnanometric resolution,” Sens. Actuators A 137, 185-191 (2007).
[CrossRef]

Wu, T.-H.

M.-H. Chiu, B.-Y. Shih, C.-W. Lai, L.-H. Shyu, and T.-H. Wu, “Small absolute distance measurement with nanometer resolution using geometrical optics principles and a SPR angular sensor,” Sens. Actuators A 141, 217-223(2008).
[CrossRef]

Appl. Opt. (2)

IEEE Trans. Instrum. Meas. (1)

X. Liu, W. Clegg, D. F. L. Jenkins, and B. Liu, “Polarization interferometer for measuring small displacement,” IEEE Trans. Instrum. Meas. 50, 868-871 (2001).
[CrossRef]

Meas. Sci. Technol. (1)

F. C. Demarest, “High-resolution, high-speed, low data age uncertainty, heterodyne displacement measuring interferometer electronics,” Meas. Sci. Technol. 9, 1024-1030 (1998).
[CrossRef]

Opt. Lett. (1)

Sens. Actuators A (2)

J.-Y. Lee, H.-Y. Chen, C.-C. Hsu, and C.-C. Wu, “Optical heterodyne grating interferometry for displacement measurement with subnanometric resolution,” Sens. Actuators A 137, 185-191 (2007).
[CrossRef]

M.-H. Chiu, B.-Y. Shih, C.-W. Lai, L.-H. Shyu, and T.-H. Wu, “Small absolute distance measurement with nanometer resolution using geometrical optics principles and a SPR angular sensor,” Sens. Actuators A 141, 217-223(2008).
[CrossRef]

Other (1)

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), Chap. 1, pp. 45-50.

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Figures (14)

Fig. 1
Fig. 1

Ray of light in air incident at θ on one side surface of a right-angle prism with refractive index n.

Fig. 2
Fig. 2

Rays returning from the displacement probe: (a) parallel ( Δ z = 0 ), (b) converging ( Δ z > 0 ). f, focal length of the objective lens; D, diameter of the beam; Δ θ , angular deviation from the optic axis; β 0 , slope angle of the incident ray; Δ z , displacement of the mirror; Δ y , distance off the optical axis if Δ z 0 .

Fig. 3
Fig. 3

Incident angles of the two marginal rays that lie on the plane perpendicular to one side surface of the right-angle prism are θ 0 + Δ θ and θ 0 Δ θ , respectively, where θ 0 is the initial incident angle when Δ z = 0 .

Fig. 4
Fig. 4

Parallelogram prism with a base angle of 45 ° .

Fig. 5
Fig. 5

Experimental configuration. AN, analyzer; PZT, piezo electric transducer; LIA, lock-in amplifier; BS, beam splitters; PC, personal computer.

Fig. 6
Fig. 6

A heterodyne light source. PBS, polarization beam splitters; AOMs, acousto-optic modulators.

Fig. 7
Fig. 7

Phase difference variation versus the incident angle θ on one side surface of the right-angle prism as shown in Fig. 1.

Fig. 8
Fig. 8

Plot of the phase difference variation versus time for (a)  Δ z = 0 nm , (b)  Δ z = 10 nm .

Fig. 9
Fig. 9

Experimental and theoretical curves of the total phase difference variation δ ϕ 1 versus the displacement Δ z .

Fig. 10
Fig. 10

Real-time measurement ( Δ z increases 10 nm per each 5   s ).

Fig. 11
Fig. 11

Theoretical resolution of the system.

Fig. 12
Fig. 12

Experimental and theoretical curves of the total phase difference variation δ ϕ 1 versus the displacement Δ z in the measurement range 10 nm Δ z + 10 nm (in steps of 1 nm displacement).

Fig. 13
Fig. 13

Theoretical sensitivity of the system.

Fig. 14
Fig. 14

Relative phase error (precent) versus displacement Δ z .

Equations (21)

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θ 1 = 45 ° + sin - 1 ( sin θ n ) .
Δ ϕ = 2 tan - 1 { sin 2 [ 45 ° + sin - 1 ( sin θ n ) ] 1 / n 2 tan [ 45 ° + sin - 1 ( sin θ n ) ] sin [ 45 ° + sin - 1 ( sin θ n ) ] } .
Δ θ ( n 2 tan 2 θ 1 1 ) ( n 2 sin 2 θ 1 1 ) 1 / 2 2 n sin θ 1 [ 2 ( n 2 1 ) tan 2 θ 1 ] ( n 2 sin 2 θ ) 1 / 2 cos θ δ ϕ = A ( θ ) δ ϕ ,
A ( θ ) = ( n 2 tan 2 θ 1 - 1 ) ( n 2 sin 2 θ 1 - 1 ) 1 / 2 2 n sin θ 1 [ 2 - ( n 2 - 1 ) tan 2 θ 1 ] ( n 2 - sin 2 θ ) 1 / 2 cos θ .
Δ z f 2 D Δ θ .
Δ z f 2 D A ( θ ) δ ϕ .
δ ϕ t 2 D f 2 1 A ( θ ) Δ z .
1 × sin ( Δ θ ) = n sin θ ,
l = h tan ( 45 ° + θ ) ,
m L / l ,
I 1 ( t ) = lim N 1 i = 1 N 1 I i ( 1 + V i cos ( 2 π f t + φ i ) )
= I 1 t [ 1 + V 1 cos ( 2 π f t + ϕ 1 ) ] ,
ϕ 1 = lim N 1 tan 1 ( i = 1 N 1 I i V i sin ϕ i i = 1 N 1 I i V i cos ϕ i ) ,
I 1 t = lim N 1 i = 1 N 1 I i ,
V 1 2 = lim N 1 i = 1 N 1 ( I i V i ) 2 + 2 j > i N 1 i = 1 N 1 ( I i V i ) ( I j V j ) cos ( ϕ i ϕ j ) I 1 2 ,
ϕ i = m i δ i .
I 2 ( t ) = I 2 t [ 1 + V 2 cos ( 2 π f t + φ 2 ) ] ,
δ ϕ 1 = ϕ 2 ϕ 1 2 m D f 2 1 A ( θ ) Δ z .
δ ϕ 1 2 m D f 2 Δ z .
R = d z d ( δ ϕ ) Δ Φ .
S = d ( δ ϕ ) d z ,

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