Abstract

This study presents an analysis of the nonlinearity resulting from polarization crosstalk at a polarizing beam splitter (PBS) and a wave plate (WP) in a homodyne interferometer. From a theoretical approach, a new compensation method involving a realignment of the axes of WPs to some specific angles according to the characteristics of the PBS is introduced. This method suppresses the nonlinearity in a homodyne interferometer to 0.36 nm, which would be 3.75 nm with conventional alignment methods of WPs.

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References

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  1. I. Misumi, S. Gonda, T. Kurosawa, and K. Takamasu, “Uncertainty in pitch measurements of one-dimensional grating standards using a nanometrological atomic force microscope,” Meas. Sci. Technol. 14(4), 463–471 (2003).
    [CrossRef]
  2. P. L. M. Heydemann, “Determination and correction of quadrature fringe measurement errors in interferometers,” Appl. Opt. 20(19), 3382–3384 (1981).
    [CrossRef] [PubMed]
  3. C.-M. Wu, C.-S. Su, and G.-S. Peng, “Correction of nonlinearity in one-frequency optical interferometry,” Meas. Sci. Technol. 7(4), 520–524 (1996).
    [CrossRef]
  4. T. B. Eom, J. Y. Kim, and K. Jeong, “The dynamic compensation of nonlinearity in a homodyne laser interferometer,” Meas. Sci. Technol. 12(10), 1734–1738 (2001).
    [CrossRef]
  5. J.-A. Kim, J. W. Kim, C.-S. Kang, T. B. Eom and J. Ahn, “A digital signal processing module for real-time compensation of nonlinearity in a homodyne interferometer using a field-programmable gate array,” Meas. Sci. Technol 20(1), 017003.1–017003.5 (2009).
  6. T. Keem, S. Gonda, I. Misumi, Q. 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(17), 3492–3498 (2005).
    [CrossRef] [PubMed]
  7. Z. Li, K. Herrmann, and F. Pohlenz, “A neural network approach to correcting nonlinearity in optical interferometers,” Meas. Sci. Technol. 14(3), 376–381 (2003).
    [CrossRef]
  8. O. P. Lay and S. Dubovitsky, “Polarization compensation: a passive approach to a reducing heterodyne interferometer nonlinearity,” Opt. Lett. 27(10), 797–799 (2002).
    [CrossRef] [PubMed]
  9. G. Dai, F. Pohlenz, H.-U. Danzebrink, K. Hasche, and G. Wilkening, “Improving the erformance of interferometers in metrological scanning probe microscopes,” Meas. Sci. Technol. 15(2), 444–450 (2004).
    [CrossRef] [PubMed]

2005 (1)

2004 (1)

G. Dai, F. Pohlenz, H.-U. Danzebrink, K. Hasche, and G. Wilkening, “Improving the erformance of interferometers in metrological scanning probe microscopes,” Meas. Sci. Technol. 15(2), 444–450 (2004).
[CrossRef] [PubMed]

2003 (2)

I. Misumi, S. Gonda, T. Kurosawa, and K. Takamasu, “Uncertainty in pitch measurements of one-dimensional grating standards using a nanometrological atomic force microscope,” Meas. Sci. Technol. 14(4), 463–471 (2003).
[CrossRef]

Z. Li, K. Herrmann, and F. Pohlenz, “A neural network approach to correcting nonlinearity in optical interferometers,” Meas. Sci. Technol. 14(3), 376–381 (2003).
[CrossRef]

2002 (1)

2001 (1)

T. B. Eom, J. Y. Kim, and K. Jeong, “The dynamic compensation of nonlinearity in a homodyne laser interferometer,” Meas. Sci. Technol. 12(10), 1734–1738 (2001).
[CrossRef]

1996 (1)

C.-M. Wu, C.-S. Su, and G.-S. Peng, “Correction of nonlinearity in one-frequency optical interferometry,” Meas. Sci. Technol. 7(4), 520–524 (1996).
[CrossRef]

1981 (1)

Dai, G.

G. Dai, F. Pohlenz, H.-U. Danzebrink, K. Hasche, and G. Wilkening, “Improving the erformance of interferometers in metrological scanning probe microscopes,” Meas. Sci. Technol. 15(2), 444–450 (2004).
[CrossRef] [PubMed]

Danzebrink, H.-U.

G. Dai, F. Pohlenz, H.-U. Danzebrink, K. Hasche, and G. Wilkening, “Improving the erformance of interferometers in metrological scanning probe microscopes,” Meas. Sci. Technol. 15(2), 444–450 (2004).
[CrossRef] [PubMed]

Dubovitsky, S.

Eom, T. B.

T. B. Eom, J. Y. Kim, and K. Jeong, “The dynamic compensation of nonlinearity in a homodyne laser interferometer,” Meas. Sci. Technol. 12(10), 1734–1738 (2001).
[CrossRef]

Gonda, S.

T. Keem, S. Gonda, I. Misumi, Q. 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(17), 3492–3498 (2005).
[CrossRef] [PubMed]

I. Misumi, S. Gonda, T. Kurosawa, and K. Takamasu, “Uncertainty in pitch measurements of one-dimensional grating standards using a nanometrological atomic force microscope,” Meas. Sci. Technol. 14(4), 463–471 (2003).
[CrossRef]

Hasche, K.

G. Dai, F. Pohlenz, H.-U. Danzebrink, K. Hasche, and G. Wilkening, “Improving the erformance of interferometers in metrological scanning probe microscopes,” Meas. Sci. Technol. 15(2), 444–450 (2004).
[CrossRef] [PubMed]

Herrmann, K.

Z. Li, K. Herrmann, and F. Pohlenz, “A neural network approach to correcting nonlinearity in optical interferometers,” Meas. Sci. Technol. 14(3), 376–381 (2003).
[CrossRef]

Heydemann, P. L. M.

Huang, Q.

Jeong, K.

T. B. Eom, J. Y. Kim, and K. Jeong, “The dynamic compensation of nonlinearity in a homodyne laser interferometer,” Meas. Sci. Technol. 12(10), 1734–1738 (2001).
[CrossRef]

Keem, T.

Kim, J. Y.

T. B. Eom, J. Y. Kim, and K. Jeong, “The dynamic compensation of nonlinearity in a homodyne laser interferometer,” Meas. Sci. Technol. 12(10), 1734–1738 (2001).
[CrossRef]

Kurosawa, T.

T. Keem, S. Gonda, I. Misumi, Q. 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(17), 3492–3498 (2005).
[CrossRef] [PubMed]

I. Misumi, S. Gonda, T. Kurosawa, and K. Takamasu, “Uncertainty in pitch measurements of one-dimensional grating standards using a nanometrological atomic force microscope,” Meas. Sci. Technol. 14(4), 463–471 (2003).
[CrossRef]

Lay, O. P.

Li, Z.

Z. Li, K. Herrmann, and F. Pohlenz, “A neural network approach to correcting nonlinearity in optical interferometers,” Meas. Sci. Technol. 14(3), 376–381 (2003).
[CrossRef]

Misumi, I.

T. Keem, S. Gonda, I. Misumi, Q. 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(17), 3492–3498 (2005).
[CrossRef] [PubMed]

I. Misumi, S. Gonda, T. Kurosawa, and K. Takamasu, “Uncertainty in pitch measurements of one-dimensional grating standards using a nanometrological atomic force microscope,” Meas. Sci. Technol. 14(4), 463–471 (2003).
[CrossRef]

Peng, G.-S.

C.-M. Wu, C.-S. Su, and G.-S. Peng, “Correction of nonlinearity in one-frequency optical interferometry,” Meas. Sci. Technol. 7(4), 520–524 (1996).
[CrossRef]

Pohlenz, F.

G. Dai, F. Pohlenz, H.-U. Danzebrink, K. Hasche, and G. Wilkening, “Improving the erformance of interferometers in metrological scanning probe microscopes,” Meas. Sci. Technol. 15(2), 444–450 (2004).
[CrossRef] [PubMed]

Z. Li, K. Herrmann, and F. Pohlenz, “A neural network approach to correcting nonlinearity in optical interferometers,” Meas. Sci. Technol. 14(3), 376–381 (2003).
[CrossRef]

Su, C.-S.

C.-M. Wu, C.-S. Su, and G.-S. Peng, “Correction of nonlinearity in one-frequency optical interferometry,” Meas. Sci. Technol. 7(4), 520–524 (1996).
[CrossRef]

Takamasu, K.

I. Misumi, S. Gonda, T. Kurosawa, and K. Takamasu, “Uncertainty in pitch measurements of one-dimensional grating standards using a nanometrological atomic force microscope,” Meas. Sci. Technol. 14(4), 463–471 (2003).
[CrossRef]

Wilkening, G.

G. Dai, F. Pohlenz, H.-U. Danzebrink, K. Hasche, and G. Wilkening, “Improving the erformance of interferometers in metrological scanning probe microscopes,” Meas. Sci. Technol. 15(2), 444–450 (2004).
[CrossRef] [PubMed]

Wu, C.-M.

C.-M. Wu, C.-S. Su, and G.-S. Peng, “Correction of nonlinearity in one-frequency optical interferometry,” Meas. Sci. Technol. 7(4), 520–524 (1996).
[CrossRef]

Appl. Opt. (2)

Meas. Sci. Technol. (5)

C.-M. Wu, C.-S. Su, and G.-S. Peng, “Correction of nonlinearity in one-frequency optical interferometry,” Meas. Sci. Technol. 7(4), 520–524 (1996).
[CrossRef]

T. B. Eom, J. Y. Kim, and K. Jeong, “The dynamic compensation of nonlinearity in a homodyne laser interferometer,” Meas. Sci. Technol. 12(10), 1734–1738 (2001).
[CrossRef]

I. Misumi, S. Gonda, T. Kurosawa, and K. Takamasu, “Uncertainty in pitch measurements of one-dimensional grating standards using a nanometrological atomic force microscope,” Meas. Sci. Technol. 14(4), 463–471 (2003).
[CrossRef]

Z. Li, K. Herrmann, and F. Pohlenz, “A neural network approach to correcting nonlinearity in optical interferometers,” Meas. Sci. Technol. 14(3), 376–381 (2003).
[CrossRef]

G. Dai, F. Pohlenz, H.-U. Danzebrink, K. Hasche, and G. Wilkening, “Improving the erformance of interferometers in metrological scanning probe microscopes,” Meas. Sci. Technol. 15(2), 444–450 (2004).
[CrossRef] [PubMed]

Opt. Lett. (1)

Other (1)

J.-A. Kim, J. W. Kim, C.-S. Kang, T. B. Eom and J. Ahn, “A digital signal processing module for real-time compensation of nonlinearity in a homodyne interferometer using a field-programmable gate array,” Meas. Sci. Technol 20(1), 017003.1–017003.5 (2009).

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Figures (7)

Fig. 1
Fig. 1

Simple diagram of a homodyne interferometer with quadrature detection system: Polarizing Beam Splitter (PBS), Quarter-Wave Plate (QWP), Reference Mirror (RM), Target Mirror (TM), Non-Polarizing Beam Splitter (NPBS), Half-Wave Plate (HWP), Photo Diode (PD)

Fig. 2
Fig. 2

The principle of passive compensation method

Fig. 3
Fig. 3

Experiment setup for measuring the optical properties of PBS: Half-Wave Plate (HWP), Polarizing Beam Splitter (PBS), Polarizer (P), Photo Detector (PD)

Fig. 4
Fig. 4

Nonlinear error according to the alignment angle of wave plates: When the axis of the alignment angle of the QWP1 rotates, the alignment axes of other WPs are fixed. The QWP2, QWP3, and HWP graphs are obtained in the same manner.

Fig. 5
Fig. 5

Theoretically predicted nonlinear error in a homodyne interferometer: The passive compensation method suppresses the innate nonlinearity of the homodyne interferometer effectively.

Fig. 6
Fig. 6

Nonlinear error according to the alignment angle of wave plates: (a) QWP1, (b) QWP2, (c) QWP3, (d) HWP

Fig. 7
Fig. 7

Experimentally acquired nonlinearity: The passive compensation method suppresses the nonlinearity of the homodyne interferometer to one tenth in terms of the original peak-to-valley value.

Tables (2)

Tables Icon

Table 1 Calculation of Jones matrix parameters that represent the transmission and reflection properties of the polarizing beam splitter

Tables Icon

Table 2 Step for applying the passive compensation method practically without theoretical approach.

Equations (20)

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I x = A cos θ     ,     I y = A sin θ
θ = arctan ( I y / I x )
I x = A cos θ + B     ,     I y = C sin ( θ + δ ) + D
I x ' = ( I x B ) / A = cos θ I y ' = ( I y D ) / C = sin ( θ + δ )
Δ x λ δ 4 π
T = [ t p t s c t p c t s ]         R = [ r p r s c r p c r s ]
[ E x t E y t ] = [ t p t s c t p c t s ] [ E x E y ]
[ E x r E y r ] = [ r p r s c r p c r s ] [ E x E y ]
I x t = E x t E x t * = ( t p ) 2 | E x | 2 + ( E x E y * + E y E x * ) t p t s c + ( t s c ) 2 | E y | 2
I y t = E y t E y t * = ( t p c ) 2 | E x | 2 + ( E x E y * + E y E x * ) t p c t s + ( t s ) 2 | E y | 2
I x r = E x r E x r * = ( r p ) 2 | E x | 2 + ( E x E y * + E y E x * ) r p r s c + ( r s c ) 2 | E y | 2
I y r = E y r E y r * = ( r p c ) 2 | E x | 2 + ( E x E y * + E y E x * ) r p c r s + ( r s ) 2 | E y | 2
W = R ( ( ψ + Δ θ ) ) W o R ( ψ + Δ θ ) = [ w 11 w 12 w 21 w 22 ]
w 11 = e i ( Γ / 2 + ε / 2 ) cos 2 ( ψ + Δ θ ) + e i ( Γ / 2 + ε / 2 ) sin 2 ( ψ + Δ θ ) w 21 = i sin ( Γ / 2 + ε / 2 ) sin ( 2 ( ψ + Δ θ ) ) w 12 = i sin ( Γ / 2 + ε / 2 ) sin ( 2 ( ψ + Δ θ ) ) w 22 = e i ( Γ / 2 + ε / 2 ) sin 2 ( ψ + Δ θ ) + e i ( Γ / 2 + ε / 2 ) cos 2 ( ψ + Δ θ )
E t a r = R 1 Q ψ 1 e i φ R m e i φ Q ψ 1 T 1 E i n
E r e f = T 1 Q ψ 2 R m Q ψ 2 R 1 E i n
D 1 = R 2 T Q ψ 3 E i n t
D 2 = T 2 T Q ψ 3 E i n t
D 3 = R 3 H ψ R Q ψ 3 E i n t
D 4 = T 3 H ψ R Q ψ 3 E i n t

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