Abstract

A method to obtain unambiguous surface height measurements using wavelength scanning interferometry with an improved repeatability, comparable to that obtainable using phase shifting interferometry, is reported. Rather than determining the conventional fringe frequency-derived z height directly, the method uses the frequency to resolve the fringe order ambiguity, and combine this information with the more accurate and repeatable fringe phase derived z height. A theoretical model to evaluate the method’s performance in the presence of additive noise is derived and shown to be in good agreement with experiments. The measurement repeatability is improved by a factor of ten over that achieved when using frequency information alone, reaching the sub-nanometre range. Moreover, the z-axis non-linearity (bleed-through or ripple error) is reduced by a factor of ten. These order of magnitude improvements in measurement performance are demonstrated through a number of practical measurement examples.

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References

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2015 (1)

P. de Groot, “Principles of interference microscopy for the measurement of surface topography,” Adv. Opt. Photonics 7(1), 1–65 (2015).
[Crossref]

2014 (3)

2012 (5)

C. L. Giusca, R. K. Leach, F. Helary, T. Gutauskas, and L. Nimishakavi, “Calibration of the scales of areal surface topography-measuring instruments: part 1. Measurement noise and residual flatness,” Meas. Sci. Technol. 23(3), 035008 (2012).
[Crossref]

Y.-S. Ghim and A. Davies, “Complete fringe order determination in scanning white-light interferometry using a Fourier-based technique,” Appl. Opt. 51(12), 1922–1928 (2012).
[Crossref] [PubMed]

H. Muhamedsalih, F. Gao, and X. Jiang, “Comparison study of algorithms and accuracy in the wavelength scanning interferometry,” Appl. Opt. 51(36), 8854–8862 (2012).
[Crossref] [PubMed]

F. Gao, H. Muhamedsalih, and X. Jiang, “Surface and thickness measurement of a transparent film using wavelength scanning interferometry,” Opt. Express 20(19), 21450–21456 (2012).
[Crossref] [PubMed]

A. Davila, J. M. Huntley, C. Pallikarakis, P. D. Ruiz, and J. M. Coupland, “Wavelength scanning interferometry using a Ti:Sapphire laser with wide tuning range,” Opt. Lasers Eng. 50(8), 1089–1096 (2012).
[Crossref]

2011 (2)

J. A. N. Buytaert and J. J. J. Dirckx, “Study of the performance of 84 phase-shifting algorithms for interferometry,” J. Opt. 40(3), 114–131 (2011).
[Crossref]

Y.-S. Ghim, A. Suratkar, A. Davies, and Y.-W. Lee, “Absolute thickness measurement of silicon wafer using wavelength scanning interferometer,” Proc. SPIE 8133, 813312 (2011).
[Crossref]

2010 (3)

2009 (1)

2004 (1)

P. D. Ruiz, Y. Zhou, J. M. Huntley, and R. D. Wildman, “Depth-resolved whole-field displacement measurement using wavelength scanning interferometry,” J. Opt. A, Pure Appl. Opt. 6(7), 679–683 (2004).
[Crossref]

2002 (2)

2000 (4)

P. de Groot, “Measurement of transparent plates with wavelength-tuned phase-shifting interferometry,” Appl. Opt. 39(16), 2658–2663 (2000).
[Crossref] [PubMed]

A. Harasaki, J. Schmit, and J. C. Wyant, “Improved vertical-scanning interferometry,” Appl. Opt. 39(13), 2107–2115 (2000).
[Crossref] [PubMed]

J. Kato and I. Yamaguchi, “Phase-Shifting Fringe Analysis for Laser Diode Wavelength-Scanning Interferometer,” Opt. Rev. 7(2), 158–163 (2000).
[Crossref]

I. Yamaguchi, A. Yamamoto, and M. Yano, “Surface topography by wavelength scanning interferometry,” Opt. Eng. 39(1), 40 (2000).
[Crossref]

1999 (1)

1998 (1)

X. Dai and K. Seta, “High-accuracy absolute distance measurement by means of wavelength scanning heterodyne interferometry,” Meas. Sc. Technol. 9(7), 1031 (1998).

1996 (1)

1995 (2)

P. de Groot and L. Deck, “Surface Profiling by Analysis of White-light Interferograms in the Spatial Frequency Domain,” J. Mod. Opt. 42(2), 389–401 (1995).
[Crossref]

P. de Groot and L. Deck, “Surface Profiling by Analysis of White-light Interferograms in the Spatial Frequency Domain,” J. Mod. Opt. 42(2), 389–401 (1995).
[Crossref]

1994 (1)

1993 (1)

G. Barwood, P. Gill, and W. Rowley, “Laser diodes for length determination using swept-frequency interferometry,” Meas. Sci. Technol. 4(9), 988–994 (1993).
[Crossref]

1982 (1)

1977 (1)

A. J. Jerri, “The Shannon Sampling Theorem-Its Various Extensions and Applications: A Tutorial Review,” Proc. IEEE 65(11), 1565–1596 (1977).
[Crossref]

1974 (1)

D. C. Rife and R. R. Boorstyn, “Single-Tone Parameter Estimation from Discrete-Time Observations,” IEEE Trans. Inf. Theory 20(5), 591–598 (1974).
[Crossref]

Barwood, G.

G. Barwood, P. Gill, and W. Rowley, “Laser diodes for length determination using swept-frequency interferometry,” Meas. Sci. Technol. 4(9), 988–994 (1993).
[Crossref]

Boorstyn, R. R.

D. C. Rife and R. R. Boorstyn, “Single-Tone Parameter Estimation from Discrete-Time Observations,” IEEE Trans. Inf. Theory 20(5), 591–598 (1974).
[Crossref]

Buytaert, J. A. N.

J. A. N. Buytaert and J. J. J. Dirckx, “Study of the performance of 84 phase-shifting algorithms for interferometry,” J. Opt. 40(3), 114–131 (2011).
[Crossref]

Colonna de Lega, X.

Coupland, J.

Coupland, J. M.

A. Davila, J. M. Huntley, C. Pallikarakis, P. D. Ruiz, and J. M. Coupland, “Wavelength scanning interferometry using a Ti:Sapphire laser with wide tuning range,” Opt. Lasers Eng. 50(8), 1089–1096 (2012).
[Crossref]

Dai, X.

X. Dai and K. Seta, “High-accuracy absolute distance measurement by means of wavelength scanning heterodyne interferometry,” Meas. Sc. Technol. 9(7), 1031 (1998).

Dale, J.

Davies, A.

Davila, A.

A. Davila, J. M. Huntley, C. Pallikarakis, P. D. Ruiz, and J. M. Coupland, “Wavelength scanning interferometry using a Ti:Sapphire laser with wide tuning range,” Opt. Lasers Eng. 50(8), 1089–1096 (2012).
[Crossref]

de Groot, P.

P. de Groot, “Principles of interference microscopy for the measurement of surface topography,” Adv. Opt. Photonics 7(1), 1–65 (2015).
[Crossref]

P. de Groot, X. Colonna de Lega, J. Kramer, and M. Turzhitsky, “Determination of Fringe Order in White-Light Interference Microscopy,” Appl. Opt. 41(22), 4571–4578 (2002).
[Crossref] [PubMed]

P. de Groot, “Measurement of transparent plates with wavelength-tuned phase-shifting interferometry,” Appl. Opt. 39(16), 2658–2663 (2000).
[Crossref] [PubMed]

P. de Groot and L. Deck, “Surface Profiling by Analysis of White-light Interferograms in the Spatial Frequency Domain,” J. Mod. Opt. 42(2), 389–401 (1995).
[Crossref]

P. de Groot and L. Deck, “Surface Profiling by Analysis of White-light Interferograms in the Spatial Frequency Domain,” J. Mod. Opt. 42(2), 389–401 (1995).
[Crossref]

L. Deck and P. de Groot, “High-speed noncontact profiler based on scanning white-light interferometry,” Appl. Opt. 33(31), 7334–7338 (1994).
[Crossref] [PubMed]

De Groot, P. J.

P. J. De Groot, “Progress in the specification of optical instruments for the measurement of surface form and texture,” Proc. SPIE 9110, Dimens. Opt. Metrol. Insp. Pract. Appl. III 9110, 1–12 (2014).

Deck, L.

P. de Groot and L. Deck, “Surface Profiling by Analysis of White-light Interferograms in the Spatial Frequency Domain,” J. Mod. Opt. 42(2), 389–401 (1995).
[Crossref]

P. de Groot and L. Deck, “Surface Profiling by Analysis of White-light Interferograms in the Spatial Frequency Domain,” J. Mod. Opt. 42(2), 389–401 (1995).
[Crossref]

L. Deck and P. de Groot, “High-speed noncontact profiler based on scanning white-light interferometry,” Appl. Opt. 33(31), 7334–7338 (1994).
[Crossref] [PubMed]

Deck, L. L.

L. L. Deck, “Absolute distance measurements using FTPSI with a widely tunable IR laser,” Proc. SPIE 4778, 218–226 (2002).
[Crossref]

Dirckx, J. J. J.

J. A. N. Buytaert and J. J. J. Dirckx, “Study of the performance of 84 phase-shifting algorithms for interferometry,” J. Opt. 40(3), 114–131 (2011).
[Crossref]

Estrada, J. C.

Gao, F.

Ghim, Y.-S.

Gill, P.

G. Barwood, P. Gill, and W. Rowley, “Laser diodes for length determination using swept-frequency interferometry,” Meas. Sci. Technol. 4(9), 988–994 (1993).
[Crossref]

Giusca, C. L.

C. L. Giusca, R. K. Leach, F. Helary, T. Gutauskas, and L. Nimishakavi, “Calibration of the scales of areal surface topography-measuring instruments: part 1. Measurement noise and residual flatness,” Meas. Sci. Technol. 23(3), 035008 (2012).
[Crossref]

Gutauskas, T.

C. L. Giusca, R. K. Leach, F. Helary, T. Gutauskas, and L. Nimishakavi, “Calibration of the scales of areal surface topography-measuring instruments: part 1. Measurement noise and residual flatness,” Meas. Sci. Technol. 23(3), 035008 (2012).
[Crossref]

Harasaki, A.

Helary, F.

C. L. Giusca, R. K. Leach, F. Helary, T. Gutauskas, and L. Nimishakavi, “Calibration of the scales of areal surface topography-measuring instruments: part 1. Measurement noise and residual flatness,” Meas. Sci. Technol. 23(3), 035008 (2012).
[Crossref]

Howard, L.

Hughes, B.

Huntley, J. M.

A. Davila, J. M. Huntley, C. Pallikarakis, P. D. Ruiz, and J. M. Coupland, “Wavelength scanning interferometry using a Ti:Sapphire laser with wide tuning range,” Opt. Lasers Eng. 50(8), 1089–1096 (2012).
[Crossref]

P. D. Ruiz, Y. Zhou, J. M. Huntley, and R. D. Wildman, “Depth-resolved whole-field displacement measurement using wavelength scanning interferometry,” J. Opt. A, Pure Appl. Opt. 6(7), 679–683 (2004).
[Crossref]

Ina, H.

Jerri, A. J.

A. J. Jerri, “The Shannon Sampling Theorem-Its Various Extensions and Applications: A Tutorial Review,” Proc. IEEE 65(11), 1565–1596 (1977).
[Crossref]

Jiang, X.

Kato, J.

J. Kato and I. Yamaguchi, “Phase-Shifting Fringe Analysis for Laser Diode Wavelength-Scanning Interferometer,” Opt. Rev. 7(2), 158–163 (2000).
[Crossref]

Kobayashi, S.

Kramer, J.

Lancaster, A. J.

Larkin, K. G.

Leach, R.

Leach, R. K.

C. L. Giusca, R. K. Leach, F. Helary, T. Gutauskas, and L. Nimishakavi, “Calibration of the scales of areal surface topography-measuring instruments: part 1. Measurement noise and residual flatness,” Meas. Sci. Technol. 23(3), 035008 (2012).
[Crossref]

G. Moschetti, H. Muhamedsalih, D. O’Connor, X. Jiang, and R. K. Leach, “Vertical axis non-linearities in wavelength scanning interferometry,” in 11th International Conference and Exhibition on Laser Metrology,Machine Tool, CMM & Robotic Performance (2015), pp. 31–39.

R. K. Leach, “Is one step height enough,” in Proceedings of ASPE (2015).

Lee, Y.-W.

Y.-S. Ghim, A. Suratkar, A. Davies, and Y.-W. Lee, “Absolute thickness measurement of silicon wafer using wavelength scanning interferometer,” Proc. SPIE 8133, 813312 (2011).
[Crossref]

Lewis, A. J.

Mandal, R.

Mansfield, D.

Moschetti, G.

G. Moschetti, H. Muhamedsalih, D. O’Connor, X. Jiang, and R. K. Leach, “Vertical axis non-linearities in wavelength scanning interferometry,” in 11th International Conference and Exhibition on Laser Metrology,Machine Tool, CMM & Robotic Performance (2015), pp. 31–39.

Muhamedsalih, H.

Nimishakavi, L.

C. L. Giusca, R. K. Leach, F. Helary, T. Gutauskas, and L. Nimishakavi, “Calibration of the scales of areal surface topography-measuring instruments: part 1. Measurement noise and residual flatness,” Meas. Sci. Technol. 23(3), 035008 (2012).
[Crossref]

O’Connor, D.

G. Moschetti, H. Muhamedsalih, D. O’Connor, X. Jiang, and R. K. Leach, “Vertical axis non-linearities in wavelength scanning interferometry,” in 11th International Conference and Exhibition on Laser Metrology,Machine Tool, CMM & Robotic Performance (2015), pp. 31–39.

Pallikarakis, C.

A. Davila, J. M. Huntley, C. Pallikarakis, P. D. Ruiz, and J. M. Coupland, “Wavelength scanning interferometry using a Ti:Sapphire laser with wide tuning range,” Opt. Lasers Eng. 50(8), 1089–1096 (2012).
[Crossref]

Quiroga, J. A.

Reichold, A. J. H.

Rife, D. C.

D. C. Rife and R. R. Boorstyn, “Single-Tone Parameter Estimation from Discrete-Time Observations,” IEEE Trans. Inf. Theory 20(5), 591–598 (1974).
[Crossref]

Rowley, W.

G. Barwood, P. Gill, and W. Rowley, “Laser diodes for length determination using swept-frequency interferometry,” Meas. Sci. Technol. 4(9), 988–994 (1993).
[Crossref]

Ruiz, P. D.

A. Davila, J. M. Huntley, C. Pallikarakis, P. D. Ruiz, and J. M. Coupland, “Wavelength scanning interferometry using a Ti:Sapphire laser with wide tuning range,” Opt. Lasers Eng. 50(8), 1089–1096 (2012).
[Crossref]

P. D. Ruiz, Y. Zhou, J. M. Huntley, and R. D. Wildman, “Depth-resolved whole-field displacement measurement using wavelength scanning interferometry,” J. Opt. A, Pure Appl. Opt. 6(7), 679–683 (2004).
[Crossref]

Schmit, J.

Servin, M.

Seta, K.

X. Dai and K. Seta, “High-accuracy absolute distance measurement by means of wavelength scanning heterodyne interferometry,” Meas. Sc. Technol. 9(7), 1031 (1998).

Stejskal, A.

Stone, J. A.

Suratkar, A.

Y.-S. Ghim, A. Suratkar, A. Davies, and Y.-W. Lee, “Absolute thickness measurement of silicon wafer using wavelength scanning interferometer,” Proc. SPIE 8133, 813312 (2011).
[Crossref]

Y.-S. Ghim, A. Suratkar, and A. Davies, “Reflectometry-based wavelength scanning interferometry for thickness measurements of very thin wafers,” Opt. Express 18(7), 6522–6529 (2010).
[Crossref] [PubMed]

Takeda, M.

Turzhitsky, M.

Wang, K.

Warden, M. S.

Wildman, R. D.

P. D. Ruiz, Y. Zhou, J. M. Huntley, and R. D. Wildman, “Depth-resolved whole-field displacement measurement using wavelength scanning interferometry,” J. Opt. A, Pure Appl. Opt. 6(7), 679–683 (2004).
[Crossref]

Wyant, J. C.

Yamaguchi, I.

I. Yamaguchi, A. Yamamoto, and M. Yano, “Surface topography by wavelength scanning interferometry,” Opt. Eng. 39(1), 40 (2000).
[Crossref]

J. Kato and I. Yamaguchi, “Phase-Shifting Fringe Analysis for Laser Diode Wavelength-Scanning Interferometer,” Opt. Rev. 7(2), 158–163 (2000).
[Crossref]

Yamamoto, A.

I. Yamaguchi, A. Yamamoto, and M. Yano, “Surface topography by wavelength scanning interferometry,” Opt. Eng. 39(1), 40 (2000).
[Crossref]

Yano, M.

I. Yamaguchi, A. Yamamoto, and M. Yano, “Surface topography by wavelength scanning interferometry,” Opt. Eng. 39(1), 40 (2000).
[Crossref]

Zhou, Y.

P. D. Ruiz, Y. Zhou, J. M. Huntley, and R. D. Wildman, “Depth-resolved whole-field displacement measurement using wavelength scanning interferometry,” J. Opt. A, Pure Appl. Opt. 6(7), 679–683 (2004).
[Crossref]

Adv. Opt. Photonics (1)

P. de Groot, “Principles of interference microscopy for the measurement of surface topography,” Adv. Opt. Photonics 7(1), 1–65 (2015).
[Crossref]

Appl. Opt. (9)

A. Harasaki, J. Schmit, and J. C. Wyant, “Improved vertical-scanning interferometry,” Appl. Opt. 39(13), 2107–2115 (2000).
[Crossref] [PubMed]

P. de Groot, X. Colonna de Lega, J. Kramer, and M. Turzhitsky, “Determination of Fringe Order in White-Light Interference Microscopy,” Appl. Opt. 41(22), 4571–4578 (2002).
[Crossref] [PubMed]

J. A. Stone, A. Stejskal, and L. Howard, “Absolute interferometry with a 670-nm external cavity diode laser,” Appl. Opt. 38(28), 5981–5994 (1999).
[Crossref] [PubMed]

Y.-S. Ghim and A. Davies, “Complete fringe order determination in scanning white-light interferometry using a Fourier-based technique,” Appl. Opt. 51(12), 1922–1928 (2012).
[Crossref] [PubMed]

L. Deck and P. de Groot, “High-speed noncontact profiler based on scanning white-light interferometry,” Appl. Opt. 33(31), 7334–7338 (1994).
[Crossref] [PubMed]

X. Jiang, K. Wang, F. Gao, and H. Muhamedsalih, “Fast surface measurement using wavelength scanning interferometry with compensation of environmental noise,” Appl. Opt. 49(15), 2903–2909 (2010).
[Crossref] [PubMed]

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Figures (13)

Fig. 1
Fig. 1 Example interferograms from a) PSI, b) CSI and c) WSI.
Fig. 2
Fig. 2 WSI setup. Top left block: control and computing electronics. Bottom block: wavelength sweeping light source by means of an acousto-optic tuneable filter (AOTF). Top right: Linnik type interferometer with optional reference mirror control for vibration compensation [9].
Fig. 3
Fig. 3 Explanatory plot of terms in the demodulated phase, Eq. (4). The measured phase differs from the ideal value due to dispersion (τ) and phase change upon reflection (γ0).
Fig. 4
Fig. 4 Example of tilted flat profile measurement. zf is the unambiguous measurement via estimation of the frequency of the fringe pattern. zamb is the ambiguous profile measurement via estimation of the phase of the fringe pattern.
Fig. 5
Fig. 5 Fringe order determination. The profile h’ can be employed to determine the fringe order m.
Fig. 6
Fig. 6 a) Tilted flat profile measured via estimation of the frequency and via the phase. An offset of 1 µm has been added for clarity. b) Difference between the two profiles.
Fig. 7
Fig. 7 System model response and its linear approximation. The ideal fringe pattern intensities observed at N points are a function of the model parameters α to the vector of observed data S(α) . Noise causes the observations to not be exactly at the ideal point along the system response curve, but in a point cloud around the ideal. The statistical property of the noise can be propagated to obtain the uncertainty of the model parameters.
Fig. 8
Fig. 8 Comparison of RMS error of z-height estimation through the fringe pattern frequency (a) and through the phase (b) as a function of the SNR and the number of samples (N).
Fig. 9
Fig. 9 Measurements of a 12.5 µm step height.a) surface and ISO 5436 analysis of step height via frequency estimation. b) surface and ISO 5436 step height analysis via phase estimation c) Measurement noise via frequency estimation and d) measurement noise via phase estimation.
Fig. 10
Fig. 10 Noise as a function of samples acquired for measurement via frequency estimation, and phase estimation. In both cases the noise is compared with the square root of the number of samples acquired curve. Note the different scales for the two curves.
Fig. 11
Fig. 11 Areal surface topography measurements and extracted profiles of a 15 nm nominal type ACG surface using z-height estimation via the frequency (a) and via the phase (b).
Fig. 12
Fig. 12 Areal surface topography measurements and extracted profiles of a calibrated tilted flat (maximum surface height Sz of 17.5 nm at coverage probability of 95%) using z-height estimation via the frequency (a) and via the phase (b).
Fig. 13
Fig. 13 Profile of steel sphere measurements via estimation of the frequency (a)and the phase (b).

Equations (31)

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I(k, z m )=q(k)[1+V(k)cos(2πk z m )]
φ(k)=4πk z m =4π k 0 z m +4π(k k 0 ) z m
z m = 1 4π Δφ Δk
φ(k)=4π k 0 z+4π(k k 0 )z+τ(k k 0 )+ γ 0
z f = 1 4π dφ dk =z+ τ 4π + δ(z) 4π
z ^ f = 1 4π ( dφ dk τδ(z) ).
φ( k 0 )=4π k 0 z amb + γ 0
z p = 1 4π φ( k 0 ) k 0 + m 2 k 0 = z amb + 1 4π γ 0 k 0 + m 2 k 0 =z+ 1 4π γ 0 k 0
z ^ = 1 4π k 0 ( φ( k 0 )+2πm γ 0 )
m=Round[ h ' ]=Round[ 1 4π ( Δφ Δk δ(z)τ φ( k 0 ) γ 0 k 0 )2 k 0 ]
| 1 4π ( δ+τ γ 0 k 0 ) |< 1 4 k 0
Δz= z f z p = 1 4π ( τ+δ(z) γ 0 k 0 )
I n (α)= S n (α)+ W n , n[0,...,N1]
I=S(α)+W
U NL σ 2 [ J T J] 1
J nj = S n (α) α j | α= α NL n[0,...,N1];j[0,...,p1]
S n ([b, z f , z p ])=b e i[4π k 0 z p +4π( k n k 0 ) z f ] n[0,...,N1]
U NL σ 2 [ c b 0 0 0 c f c fp 0 c fp c p ] 1
c b =N
c f = (4πb) 2 n=0 N1 ( k n k 0 ) 2 = (4πb) 2 N(N1)(2N1) δ k 2 6
c p = (4π k 0 b) 2 N
c fp = (4πb) 2 k 0 n=0 N1 ( k n k 0 )= (4πb) 2 k 0 N(N1) δk 2
U NL σ 2 [ J T J] 1 = σ 2 adj[ J T J] det[ J T J]
U NL σ 2 [ J T J] 1 = σ 2 c b ( c p c f c fp 2 ) [ c p c f c fp 2 0 0 0 c b c p c b c fp 0 c b c fp c b c f ] = σ 2 [ d b 0 0 0 d f d fp 0 d fp d p ]= σ 2 D
c f c p c fp 2 = (4πb) 2 N(N1)(2N1) δ k 2 6 (4π k 0 b) 2 N [ (4πb) 2 k 0 N(N1) δk 2 ] 2 = (4πb) 4 12 N 2 (N1)(N+1) k 0 2 δ k 2 .
d b = 1 c b = 1 N
d f = c p c f c p c fp 2 = 1 (4πb) 2 12 N(N1)(N+1)δ k 2 1 (4πb) 2 12 N 3 δ k 2 = 1 (4πb) 2 12 NΔ k 2
d p = c f c f c p c fp 2 = 1 (4πb) 2 2(2N1) N(N+1) k 0 2 1 (4πb) 2 4 N k 0 2
d fp = c fp c f c p c fp 2 = 6 (4πb) 2 N(N+1) k 0 δk 1 (4πb) 2 6 N 2 k 0 δk
d f d p = 12 NΔ k 2 N k 0 2 4 = 3 k 0 Δk
d f d p = 2 k 0 Δk

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