Abstract

We present the influence of alignment and the real properties of optical components on the performance of a two-detector homodyne displacement-measuring quadrature laser interferometer. An experimental method, based on the optimization of visibility and sensitivity, was established and theoretically described to assess the performance and stability of the interferometer. We show that the optimal performance of such interferometers is achieved with the iterative alignment procedure described.

© 2011 Optical Society of America

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  1. D. Y. Lee, D. M. Kim, D. G. Gweon, and J. Park, “A calibrated atomic force microscope using an orthogonal scanner and a calibrated laser interferometer,” Appl. Surf. Sci. 253, 3945–3951 (2007).
    [CrossRef]
  2. I. Misumi, S. Gonda, O. Sato, K. Sugawara, K. Yoshizaki, T. Kurosawa, and T. Takatsuji, “Nanometric lateral scale development using an atomic force microscope with directly traceable laser interferometers,” Meas. Sci. Technol. 17, 2041–2047 (2006).
    [CrossRef]
  3. T. Hausotte, B. Percle, and G. Jäger, “Advanced three-dimensional scan methods in the nanopositioning and nanomeasuring machine,” Meas. Sci. Technol. 20, 084004 (2009).
    [CrossRef]
  4. P. Gregorčič, T. Požar, and J. Možina, “Quadrature phase-shift error analysis using a homodyne laser interferometer,” Opt. Express 17, 16322–16331 (2009).
    [CrossRef] [PubMed]
  5. T. Požar and J. Možina, “Homodyne quadrature laser interferometer applied for the studies of optodynamic wave propagation in a rod,” Strojniški Vestnik—J. Mech. Eng. 55, 575–580(2009).
  6. F. Petrů and O. Číp, “Problems regarding linearity of data of a laser interferometer with a single-frequency laser,” Precis. Eng. 23, 39–50 (1999).
    [CrossRef]
  7. T. Keem, S. Gonda, I. Misumi, Q. X. Huang, and T. Kurosawa, “Simple, real-time method for removing the cyclic error of a homodyne interferometer with a quadrature detector system,” Appl. Opt. 44, 3492–3498 (2005).
    [CrossRef] [PubMed]
  8. J. J. Monzon and L. L. Sanchezsoto, “Absorbing beam splitter in a Michelson Interferometer,” Appl. Opt. 34, 7834–7839(1995).
    [CrossRef] [PubMed]
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    [CrossRef]
  10. J. Ahn, J. A. Kim, C. S. Kang, J. W. Kim, and S. Kim, “A passive method to compensate nonlinearity in a homodyne interferometer,” Opt. Express 17, 23299–23308 (2009).
    [CrossRef]
  11. P. Křen, “Linearisation of counting interferometers with 0.1 nm precision,” Int. J. Nanotechnology 4, 702–711 (2007).
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    [CrossRef] [PubMed]
  13. M. A. Zumberge, J. Berger, M. A. Dzieciuch, and R. L. Parker, “Resolving quadrature fringes in real time,” Appl. Opt. 43, 771–775 (2004).
    [CrossRef] [PubMed]
  14. M. Dobosz, T. Usuda, and T. Kurosawa, “The methods for the calibration of vibration pick-ups by laser interferometry: part I. Theoretical analysis,” Meas. Sci. Technol. 9, 232–239 (1998).
    [CrossRef]
  15. P. Gregorčič, T. Požar, and J. Možina, “Phase-shift error in quadrature-detection-based interferometers,” Proc. SPIE 7726, 77260X (2010).
    [CrossRef]
  16. Edmund Optics, Optics and Optical Instruments Catalog (2010), p. 141.
  17. Eksma Optics, Laser Components 2009/2010 (2009), vol.  18, p. 1.39.

2010 (1)

P. Gregorčič, T. Požar, and J. Možina, “Phase-shift error in quadrature-detection-based interferometers,” Proc. SPIE 7726, 77260X (2010).
[CrossRef]

2009 (4)

J. Ahn, J. A. Kim, C. S. Kang, J. W. Kim, and S. Kim, “A passive method to compensate nonlinearity in a homodyne interferometer,” Opt. Express 17, 23299–23308 (2009).
[CrossRef]

T. Hausotte, B. Percle, and G. Jäger, “Advanced three-dimensional scan methods in the nanopositioning and nanomeasuring machine,” Meas. Sci. Technol. 20, 084004 (2009).
[CrossRef]

P. Gregorčič, T. Požar, and J. Možina, “Quadrature phase-shift error analysis using a homodyne laser interferometer,” Opt. Express 17, 16322–16331 (2009).
[CrossRef] [PubMed]

T. Požar and J. Možina, “Homodyne quadrature laser interferometer applied for the studies of optodynamic wave propagation in a rod,” Strojniški Vestnik—J. Mech. Eng. 55, 575–580(2009).

2007 (2)

D. Y. Lee, D. M. Kim, D. G. Gweon, and J. Park, “A calibrated atomic force microscope using an orthogonal scanner and a calibrated laser interferometer,” Appl. Surf. Sci. 253, 3945–3951 (2007).
[CrossRef]

P. Křen, “Linearisation of counting interferometers with 0.1 nm precision,” Int. J. Nanotechnology 4, 702–711 (2007).

2006 (1)

I. Misumi, S. Gonda, O. Sato, K. Sugawara, K. Yoshizaki, T. Kurosawa, and T. Takatsuji, “Nanometric lateral scale development using an atomic force microscope with directly traceable laser interferometers,” Meas. Sci. Technol. 17, 2041–2047 (2006).
[CrossRef]

2005 (1)

2004 (1)

1999 (1)

F. Petrů and O. Číp, “Problems regarding linearity of data of a laser interferometer with a single-frequency laser,” Precis. Eng. 23, 39–50 (1999).
[CrossRef]

1998 (1)

M. Dobosz, T. Usuda, and T. Kurosawa, “The methods for the calibration of vibration pick-ups by laser interferometry: part I. Theoretical analysis,” Meas. Sci. Technol. 9, 232–239 (1998).
[CrossRef]

1996 (1)

C. M. Wu and C. S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. 7, 62–68(1996).
[CrossRef]

1995 (1)

1981 (1)

Ahn, J.

Berger, J.

Cíp, O.

F. Petrů and O. Číp, “Problems regarding linearity of data of a laser interferometer with a single-frequency laser,” Precis. Eng. 23, 39–50 (1999).
[CrossRef]

Dobosz, M.

M. Dobosz, T. Usuda, and T. Kurosawa, “The methods for the calibration of vibration pick-ups by laser interferometry: part I. Theoretical analysis,” Meas. Sci. Technol. 9, 232–239 (1998).
[CrossRef]

Dzieciuch, M. A.

Gonda, S.

I. Misumi, S. Gonda, O. Sato, K. Sugawara, K. Yoshizaki, T. Kurosawa, and T. Takatsuji, “Nanometric lateral scale development using an atomic force microscope with directly traceable laser interferometers,” Meas. Sci. Technol. 17, 2041–2047 (2006).
[CrossRef]

T. Keem, S. Gonda, I. Misumi, Q. X. Huang, and T. Kurosawa, “Simple, real-time method for removing the cyclic error of a homodyne interferometer with a quadrature detector system,” Appl. Opt. 44, 3492–3498 (2005).
[CrossRef] [PubMed]

Gregorcic, P.

P. Gregorčič, T. Požar, and J. Možina, “Phase-shift error in quadrature-detection-based interferometers,” Proc. SPIE 7726, 77260X (2010).
[CrossRef]

P. Gregorčič, T. Požar, and J. Možina, “Quadrature phase-shift error analysis using a homodyne laser interferometer,” Opt. Express 17, 16322–16331 (2009).
[CrossRef] [PubMed]

Gweon, D. G.

D. Y. Lee, D. M. Kim, D. G. Gweon, and J. Park, “A calibrated atomic force microscope using an orthogonal scanner and a calibrated laser interferometer,” Appl. Surf. Sci. 253, 3945–3951 (2007).
[CrossRef]

Hausotte, T.

T. Hausotte, B. Percle, and G. Jäger, “Advanced three-dimensional scan methods in the nanopositioning and nanomeasuring machine,” Meas. Sci. Technol. 20, 084004 (2009).
[CrossRef]

Heydemann, P. L. M.

Huang, Q. X.

Jäger, G.

T. Hausotte, B. Percle, and G. Jäger, “Advanced three-dimensional scan methods in the nanopositioning and nanomeasuring machine,” Meas. Sci. Technol. 20, 084004 (2009).
[CrossRef]

Kang, C. S.

Keem, T.

Kim, D. M.

D. Y. Lee, D. M. Kim, D. G. Gweon, and J. Park, “A calibrated atomic force microscope using an orthogonal scanner and a calibrated laser interferometer,” Appl. Surf. Sci. 253, 3945–3951 (2007).
[CrossRef]

Kim, J. A.

Kim, J. W.

Kim, S.

Kren, P.

P. Křen, “Linearisation of counting interferometers with 0.1 nm precision,” Int. J. Nanotechnology 4, 702–711 (2007).

Kurosawa, T.

I. Misumi, S. Gonda, O. Sato, K. Sugawara, K. Yoshizaki, T. Kurosawa, and T. Takatsuji, “Nanometric lateral scale development using an atomic force microscope with directly traceable laser interferometers,” Meas. Sci. Technol. 17, 2041–2047 (2006).
[CrossRef]

T. Keem, S. Gonda, I. Misumi, Q. X. Huang, and T. Kurosawa, “Simple, real-time method for removing the cyclic error of a homodyne interferometer with a quadrature detector system,” Appl. Opt. 44, 3492–3498 (2005).
[CrossRef] [PubMed]

M. Dobosz, T. Usuda, and T. Kurosawa, “The methods for the calibration of vibration pick-ups by laser interferometry: part I. Theoretical analysis,” Meas. Sci. Technol. 9, 232–239 (1998).
[CrossRef]

Lee, D. Y.

D. Y. Lee, D. M. Kim, D. G. Gweon, and J. Park, “A calibrated atomic force microscope using an orthogonal scanner and a calibrated laser interferometer,” Appl. Surf. Sci. 253, 3945–3951 (2007).
[CrossRef]

Misumi, I.

I. Misumi, S. Gonda, O. Sato, K. Sugawara, K. Yoshizaki, T. Kurosawa, and T. Takatsuji, “Nanometric lateral scale development using an atomic force microscope with directly traceable laser interferometers,” Meas. Sci. Technol. 17, 2041–2047 (2006).
[CrossRef]

T. Keem, S. Gonda, I. Misumi, Q. X. Huang, and T. Kurosawa, “Simple, real-time method for removing the cyclic error of a homodyne interferometer with a quadrature detector system,” Appl. Opt. 44, 3492–3498 (2005).
[CrossRef] [PubMed]

Monzon, J. J.

Možina, J.

P. Gregorčič, T. Požar, and J. Možina, “Phase-shift error in quadrature-detection-based interferometers,” Proc. SPIE 7726, 77260X (2010).
[CrossRef]

P. Gregorčič, T. Požar, and J. Možina, “Quadrature phase-shift error analysis using a homodyne laser interferometer,” Opt. Express 17, 16322–16331 (2009).
[CrossRef] [PubMed]

T. Požar and J. Možina, “Homodyne quadrature laser interferometer applied for the studies of optodynamic wave propagation in a rod,” Strojniški Vestnik—J. Mech. Eng. 55, 575–580(2009).

Park, J.

D. Y. Lee, D. M. Kim, D. G. Gweon, and J. Park, “A calibrated atomic force microscope using an orthogonal scanner and a calibrated laser interferometer,” Appl. Surf. Sci. 253, 3945–3951 (2007).
[CrossRef]

Parker, R. L.

Percle, B.

T. Hausotte, B. Percle, and G. Jäger, “Advanced three-dimensional scan methods in the nanopositioning and nanomeasuring machine,” Meas. Sci. Technol. 20, 084004 (2009).
[CrossRef]

Petru, F.

F. Petrů and O. Číp, “Problems regarding linearity of data of a laser interferometer with a single-frequency laser,” Precis. Eng. 23, 39–50 (1999).
[CrossRef]

Požar, T.

P. Gregorčič, T. Požar, and J. Možina, “Phase-shift error in quadrature-detection-based interferometers,” Proc. SPIE 7726, 77260X (2010).
[CrossRef]

T. Požar and J. Možina, “Homodyne quadrature laser interferometer applied for the studies of optodynamic wave propagation in a rod,” Strojniški Vestnik—J. Mech. Eng. 55, 575–580(2009).

P. Gregorčič, T. Požar, and J. Možina, “Quadrature phase-shift error analysis using a homodyne laser interferometer,” Opt. Express 17, 16322–16331 (2009).
[CrossRef] [PubMed]

Sanchezsoto, L. L.

Sato, O.

I. Misumi, S. Gonda, O. Sato, K. Sugawara, K. Yoshizaki, T. Kurosawa, and T. Takatsuji, “Nanometric lateral scale development using an atomic force microscope with directly traceable laser interferometers,” Meas. Sci. Technol. 17, 2041–2047 (2006).
[CrossRef]

Su, C. S.

C. M. Wu and C. S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. 7, 62–68(1996).
[CrossRef]

Sugawara, K.

I. Misumi, S. Gonda, O. Sato, K. Sugawara, K. Yoshizaki, T. Kurosawa, and T. Takatsuji, “Nanometric lateral scale development using an atomic force microscope with directly traceable laser interferometers,” Meas. Sci. Technol. 17, 2041–2047 (2006).
[CrossRef]

Takatsuji, T.

I. Misumi, S. Gonda, O. Sato, K. Sugawara, K. Yoshizaki, T. Kurosawa, and T. Takatsuji, “Nanometric lateral scale development using an atomic force microscope with directly traceable laser interferometers,” Meas. Sci. Technol. 17, 2041–2047 (2006).
[CrossRef]

Usuda, T.

M. Dobosz, T. Usuda, and T. Kurosawa, “The methods for the calibration of vibration pick-ups by laser interferometry: part I. Theoretical analysis,” Meas. Sci. Technol. 9, 232–239 (1998).
[CrossRef]

Wu, C. M.

C. M. Wu and C. S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. 7, 62–68(1996).
[CrossRef]

Yoshizaki, K.

I. Misumi, S. Gonda, O. Sato, K. Sugawara, K. Yoshizaki, T. Kurosawa, and T. Takatsuji, “Nanometric lateral scale development using an atomic force microscope with directly traceable laser interferometers,” Meas. Sci. Technol. 17, 2041–2047 (2006).
[CrossRef]

Zumberge, M. A.

Appl. Opt. (4)

Appl. Surf. Sci. (1)

D. Y. Lee, D. M. Kim, D. G. Gweon, and J. Park, “A calibrated atomic force microscope using an orthogonal scanner and a calibrated laser interferometer,” Appl. Surf. Sci. 253, 3945–3951 (2007).
[CrossRef]

Int. J. Nanotechnology (1)

P. Křen, “Linearisation of counting interferometers with 0.1 nm precision,” Int. J. Nanotechnology 4, 702–711 (2007).

Meas. Sci. Technol. (4)

M. Dobosz, T. Usuda, and T. Kurosawa, “The methods for the calibration of vibration pick-ups by laser interferometry: part I. Theoretical analysis,” Meas. Sci. Technol. 9, 232–239 (1998).
[CrossRef]

I. Misumi, S. Gonda, O. Sato, K. Sugawara, K. Yoshizaki, T. Kurosawa, and T. Takatsuji, “Nanometric lateral scale development using an atomic force microscope with directly traceable laser interferometers,” Meas. Sci. Technol. 17, 2041–2047 (2006).
[CrossRef]

T. Hausotte, B. Percle, and G. Jäger, “Advanced three-dimensional scan methods in the nanopositioning and nanomeasuring machine,” Meas. Sci. Technol. 20, 084004 (2009).
[CrossRef]

C. M. Wu and C. S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. 7, 62–68(1996).
[CrossRef]

Opt. Express (2)

Precis. Eng. (1)

F. Petrů and O. Číp, “Problems regarding linearity of data of a laser interferometer with a single-frequency laser,” Precis. Eng. 23, 39–50 (1999).
[CrossRef]

Proc. SPIE (1)

P. Gregorčič, T. Požar, and J. Možina, “Phase-shift error in quadrature-detection-based interferometers,” Proc. SPIE 7726, 77260X (2010).
[CrossRef]

Strojniški Vestnik—J. Mech. Eng. (1)

T. Požar and J. Možina, “Homodyne quadrature laser interferometer applied for the studies of optodynamic wave propagation in a rod,” Strojniški Vestnik—J. Mech. Eng. 55, 575–580(2009).

Other (2)

Edmund Optics, Optics and Optical Instruments Catalog (2010), p. 141.

Eksma Optics, Laser Components 2009/2010 (2009), vol.  18, p. 1.39.

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Figures (9)

Fig. 1
Fig. 1

Schematic top view of the ideal HQLI. The polarization states are designated as arrows on the beam. The optical components used are: an optical Faraday isolator (OFI), a nonpolarizing beam splitter (NBS) or beam splitter (BS), an octadic wave plate (OWP) mounted on the motorized rotation stage, a reference mirror (RM), a target mirror (TM) attached to the piezoelectric transducer (PZT), and a polarizing beam splitter (PBS). The two signals are detected with photodiodes (PDx and PDy) and acquired by the oscilloscope (OSC). The signal from the harmonic-signal generator (HSG) is current-amplified and used to drive the PZT. A personal computer (PC) runs the automated measurements and turns the motorized rotation stage.

Fig. 2
Fig. 2

Comparison between the measured photodiode signals obtained with (a) a prealigned interferometer without OWP and (b) a postaligned interferometer in quadrature. (c) The parametric Lissajous representation of the measured signals. The ellipse is obtained from (a) and the circle from (b).

Fig. 3
Fig. 3

Dependence of the phase shift α on the OWP angle φ. (a) The solid curve shows the theoretical curve [Eq. (6)] for the HQLI with ideal components. The crosses and filled circles show the measurements of α ( φ ) for the NBS with an additional phase shift α 0 = 57.5 ° in the case of a post- and prealigned HQLI, respectively. The open circles show the same dependence, when the same NBS was rotated by 180 ° around the y axis in the prealigned HQLI. Filled squares show the measurement of α ( φ ) for the BS with α 0 = 15 ° placed in the prealigned interferometer. (b) The theoretical sensitivity of the phase shift on the OWP rotation [Eq. (7)].

Fig. 4
Fig. 4

Dependence of the inherent phase shift α 0 originating from the NBS as a function of the incidence angle ϑ NBS . Each point is an average of 10 measurements.

Fig. 5
Fig. 5

Theoretical results for the ideal HQLI. (a) The maximum and minimum, and (b) visibility of both signals as a function of the OWP rotation angle. The curves are obtained from Eqs. (5, 10) for OWP ( n = π / 4 ). The value of I 0 is replaced with 2.6 V .

Fig. 6
Fig. 6

Measured results for the prealigned HQLI with the NBS. (a) The maximum and minimum, and (b) the visibility of both signals as a function of the OWP rotation angle. The vertical dotted lines indicate the values of φ where the quadrature is achieved.

Fig. 7
Fig. 7

Measured results for the prealigned HQLI with the BS. (a) The maximum and minimum, and (b) the visibility of both signals as a function of the OWP rotation angle. The vertical dotted lines indicate the values of φ where the quadrature is achieved.

Fig. 8
Fig. 8

Measured results for the postaligned HQLI with the NBS. (a) The maximum and minimum, and (b) the visibility of both signals as a function of the OWP rotation angle. The vertical dotted lines indicate the values of φ where the quadrature is achieved.

Fig. 9
Fig. 9

(a) Displacement of the target mirror. (b) The measured sensitivities, corresponding to the given displacement, for: the prealigned HQLI without OWP, i.e., the signals have equal amplitudes, but are lacking the quadrature (triangles); the prealigned HQLI, i.e., the signals are in quadrature, but have different amplitudes (circles); and the postaligned HQLI, i.e., the signals in quadrature have equal amplitudes (crosses). The corresponding theoretical sensitivities are shown with the solid curves.

Equations (13)

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I x ( δ , 2 φ , n ) = I 0 8 [ ( cos δ + cos n ) 2 + ( sin δ sin n ( cos 2 φ + sin 2 φ ) ) 2 ] = c x + a x cos ( δ δ I x max ) , I y ( δ , 2 φ , n ) = I 0 8 [ ( cos δ + cos n ) 2 + ( sin δ + sin n ( cos 2 φ sin 2 φ ) ) 2 ] = c y + a y cos ( δ δ I y max ) .
I x ( δ 45 ° , 0 ° , π / 4 ) = I 0 4 ( 1 + cos δ ) , I y ( δ 45 ° , 0 ° , π / 4 ) = I 0 4 ( 1 + sin δ ) .
u ( t ) = λ 4 π ( arctan I y ( t ) I 0 / 4 I x ( t ) I 0 / 4 + m π ) , m = 0 , ± 1 , ± 2 , .
δ I x max ( 2 φ , n ) = arctan [ tan n ( sin 2 φ + cos 2 φ ) ] , δ I x min ( 2 φ , n ) = π + δ I x max , δ I y max ( 2 φ , n ) = arctan [ tan n ( sin 2 φ cos 2 φ ) ] , δ I y min ( 2 φ , n ) = π + δ I y max .
I x max ( 4 φ , n ) = I x ( δ I x max , φ , n ) = I 0 8 ( 1 + k + 1 ) 2 , I x min ( 4 φ , n ) = I x ( δ I x min , φ , n ) = I 0 8 ( 1 + k 1 ) 2 , I y max ( 4 φ , n ) = I y ( δ I y max , φ , n ) = I 0 8 ( 1 k + 1 ) 2 , I y min ( 4 φ , n ) = I y ( δ I y min , φ , n ) = I 0 8 ( 1 k 1 ) 2 ,
α ( 2 φ , n ) = arctan [ tan n ( cos 2 φ sin 2 φ ) ] + arctan [ tan n ( cos 2 φ + sin 2 φ ) ] ,
d α d φ = 4 tan n sin 2 φ [ 1 + cos 4 φ cos 2 n ( 2 + cos 4 φ ) ] 1 + 2 tan 2 n + tan 4 n cos 2 4 φ .
c x ( 4 φ , n ) = I x max + I x min 2 = I 0 2 + k 8 , c y ( 4 φ , n ) = I y max + I y min 2 = I 0 2 k 8 ,
a x ( 4 φ , n ) = I x max I x min 2 = I 0 1 + k 4 , a y ( 4 φ , n ) = I y max I y min 2 = I 0 1 k 4 .
vis x ( 4 φ , n ) = a x c x = I x max I x min I x max + I x min = 2 1 + k 2 + k , vis y ( 4 φ , n ) = a y c y = I y max I y min I y max + I y min = 2 1 k 2 k .
S = S x 2 + S y 2 = [ ( d V x d u ) 2 + ( d V y d u ) 2 ] 1 / 2 .
S = 4 π G λ [ ( d d δ ( c x + a x cos ( δ δ I x max ) ) ) 2 + ( d d δ ( c y + a y cos ( δ δ I y max ) ) ) 2 ] 1 / 2 = 4 π G λ [ a x 2 sin 2 ( δ δ I x max ) + a y 2 sin 2 ( δ δ I y max ) ] 1 / 2 .
S = 4 π G λ [ a 2 sin 2 ( δ δ I x max ) + a 2 sin 2 ( δ ( δ I x max + 90 ° ) ) ] 1 / 2 = 4 π G a λ [ sin 2 ( δ δ I x max ) + cos 2 ( δ δ I x max ) ] 1 / 2 = 4 π G a λ .

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