Abstract

Synthesis of single-wavelength temporal phase-shifting algorithms (PSA) for interferometry is well-known and firmly based on the frequency transfer function (FTF) paradigm. Here we extend the single-wavelength FTF-theory to dual and multi-wavelength PSA-synthesis when several simultaneous laser-colors are present. The FTF-based synthesis for dual-wavelength (DW) PSA is optimized for high signal-to-noise ratio and minimum number of temporal phase-shifted interferograms. The DW-PSA synthesis herein presented may be used for interferometric contouring of discontinuous industrial objects. Also DW-PSA may be useful for DW shop-testing of deep free-form aspheres. As shown here, using the FTF-based synthesis one may easily find explicit DW-PSA formulae optimized for high signal-to-noise and high detuning robustness. To this date, no general synthesis and analysis for temporal DW-PSAs has been given; only ad hoc DW-PSAs formulas have been reported. Consequently, no explicit formulae for their spectra, their signal-to-noise, their detuning and harmonic robustness has been given. Here for the first time a fully general procedure for designing DW-PSAs (or triple-wavelengths PSAs) with desire spectrum, signal-to-noise ratio and detuning robustness is given. We finally generalize DW-PSA to higher number of wavelength temporal PSAs.

© 2016 Optical Society of America

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References

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2015 (1)

W. Zhang, X. Lu, C. Luo, L. Zhong, and J. Vargas, “Principal component analysis based simultaneous dual-wavelength phase-shifting interferometry,” Opt. Commun. 341, 276–283 (2015).
[Crossref]

2014 (1)

2013 (1)

R. Kulkarni and P. Rastogi, “Multiple phase estimation in digital holographic interferometry using product cubic phase function,” Opt. Lasers Eng. 51(10), 1168–1172 (2013).
[Crossref]

2011 (4)

2010 (1)

2009 (1)

2007 (1)

2005 (1)

2003 (1)

2000 (1)

C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 39(1), 79–85 (2000).
[Crossref]

1998 (1)

1985 (1)

1984 (1)

1973 (1)

1971 (1)

Abdelsalam, D. G.

Alfieri, D.

Awatsuji, Y.

Barada, D.

Charrière, F.

Cheng, Y. Y.

Colomb, T.

Cuche, E.

Dakoff, A.

De Nicola, S.

Depeursinge, C.

Emery, Y.

Falaggis, K.

Fei, L.

Ferraro, P.

Finizio, A.

Gass, J.

Grilli, S.

Ishii, Y.

Ito, K.

Kakue, T.

Kawata, S.

Kiire, T.

Kim, D.

Kim, M. K.

Kothiyal, M.

U. P. Kumar, N. K. Mohan, and M. Kothiyal, “Red–Green–Blue wavelength interferometry and TV holography for surface metrology,” J. Opt. 40(4), 176–183 (2011).
[Crossref]

Kubota, T.

Kühn, J.

Kulkarni, R.

R. Kulkarni and P. Rastogi, “Multiple phase estimation in digital holographic interferometry using product cubic phase function,” Opt. Lasers Eng. 51(10), 1168–1172 (2013).
[Crossref]

Kumar, U. P.

U. P. Kumar, N. K. Mohan, and M. Kothiyal, “Red–Green–Blue wavelength interferometry and TV holography for surface metrology,” J. Opt. 40(4), 176–183 (2011).
[Crossref]

Lu, X.

W. Zhang, X. Lu, C. Luo, L. Zhong, and J. Vargas, “Principal component analysis based simultaneous dual-wavelength phase-shifting interferometry,” Opt. Commun. 341, 276–283 (2015).
[Crossref]

W. Zhang, X. Lu, L. Fei, H. Zhao, H. Wang, and L. Zhong, “Simultaneous phase-shifting dual-wavelength interferometry based on two-step demodulation algorithm,” Opt. Lett. 39(18), 5375–5378 (2014).
[Crossref] [PubMed]

Luo, C.

W. Zhang, X. Lu, C. Luo, L. Zhong, and J. Vargas, “Principal component analysis based simultaneous dual-wavelength phase-shifting interferometry,” Opt. Commun. 341, 276–283 (2015).
[Crossref]

Magnusson, R.

Marquet, P.

Matoba, O.

Mohan, N. K.

U. P. Kumar, N. K. Mohan, and M. Kothiyal, “Red–Green–Blue wavelength interferometry and TV holography for surface metrology,” J. Opt. 40(4), 176–183 (2011).
[Crossref]

Montfort, F.

Moritani, Y.

Nishio, K.

Onodera, R.

Osten, W.

C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 39(1), 79–85 (2000).
[Crossref]

Pierattini, G.

Polhemus, C.

Rastogi, P.

R. Kulkarni and P. Rastogi, “Multiple phase estimation in digital holographic interferometry using product cubic phase function,” Opt. Lasers Eng. 51(10), 1168–1172 (2013).
[Crossref]

Sansone, L.

Seebacher, S.

C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 39(1), 79–85 (2000).
[Crossref]

Shimozato, Y.

Sugisaka, J.

Towers, C. E.

Towers, D. P.

Ura, S.

Vargas, J.

W. Zhang, X. Lu, C. Luo, L. Zhong, and J. Vargas, “Principal component analysis based simultaneous dual-wavelength phase-shifting interferometry,” Opt. Commun. 341, 276–283 (2015).
[Crossref]

Wagner, C.

C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 39(1), 79–85 (2000).
[Crossref]

Wang, H.

Wyant, J. C.

Yatagai, T.

Zhang, W.

W. Zhang, X. Lu, C. Luo, L. Zhong, and J. Vargas, “Principal component analysis based simultaneous dual-wavelength phase-shifting interferometry,” Opt. Commun. 341, 276–283 (2015).
[Crossref]

W. Zhang, X. Lu, L. Fei, H. Zhao, H. Wang, and L. Zhong, “Simultaneous phase-shifting dual-wavelength interferometry based on two-step demodulation algorithm,” Opt. Lett. 39(18), 5375–5378 (2014).
[Crossref] [PubMed]

Zhao, H.

Zhong, L.

W. Zhang, X. Lu, C. Luo, L. Zhong, and J. Vargas, “Principal component analysis based simultaneous dual-wavelength phase-shifting interferometry,” Opt. Commun. 341, 276–283 (2015).
[Crossref]

W. Zhang, X. Lu, L. Fei, H. Zhao, H. Wang, and L. Zhong, “Simultaneous phase-shifting dual-wavelength interferometry based on two-step demodulation algorithm,” Opt. Lett. 39(18), 5375–5378 (2014).
[Crossref] [PubMed]

Appl. Opt. (8)

J. Opt. (1)

U. P. Kumar, N. K. Mohan, and M. Kothiyal, “Red–Green–Blue wavelength interferometry and TV holography for surface metrology,” J. Opt. 40(4), 176–183 (2011).
[Crossref]

Opt. Commun. (1)

W. Zhang, X. Lu, C. Luo, L. Zhong, and J. Vargas, “Principal component analysis based simultaneous dual-wavelength phase-shifting interferometry,” Opt. Commun. 341, 276–283 (2015).
[Crossref]

Opt. Eng. (1)

C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 39(1), 79–85 (2000).
[Crossref]

Opt. Express (2)

Opt. Lasers Eng. (1)

R. Kulkarni and P. Rastogi, “Multiple phase estimation in digital holographic interferometry using product cubic phase function,” Opt. Lasers Eng. 51(10), 1168–1172 (2013).
[Crossref]

Opt. Lett. (4)

Other (1)

M. Servin, J. A. Quiroga, and M. Padilla, Interferogram Analysis for Optical Metrology, Theoretical Principles and Applications (Wiley-VCH, 2014).

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Figures (18)

Fig. 1
Fig. 1

Schematics for DW-DH using a single tilted reference mirror [6]. The orange-color light-path corresponds to the spatial superposition of the red and green lasers.

Fig. 2
Fig. 2

The hexagons are the spatial quadrature filters which demodulate φ 1 and φ 2 .

Fig. 4
Fig. 4

Fourier spectrum of the DW temporal-carrier interferograms.

Fig. 5
Fig. 5

Ideal spectra of two filters that passband the desired signals exp(i φ 1 ) and exp(i φ 2 ) from N temporal phase-shifted interferograms; all crossed Dirac deltas are filtered-out.

Fig. 6
Fig. 6

Spectral plots for the two DW-FTFs { H 1 (ω), H 2 (ω)} . The crossed Dirac deltas are filter-out signals. These FTFs can demodulate { φ 1 , φ 2 } with poor signal-to-noise ratio.

Fig. 7
Fig. 7

Graph of G SNR ( d ) . We kept the third (blue) local maximum at d=0.225 λ eq =751nm , for which G SNR ( d )=23.5 . Each DW-PSA thus have a signal-to-noise of 23.5 4.84 .

Fig. 8
Fig. 8

Spectral plots for the FTFs H 1 (ω) and H 2 (ω) for the SNR-optimized DW-PSA. Note that ω 1 =W[(2π/ λ 1 )d]=1.2 and ω 2 =W[(2π/ λ 2 )d]=2.6 ; with W(x)=arg[exp(ix)] .

Fig. 9
Fig. 9

The upper row shows 5 simulated overlapped interferograms without noise. The lower panel shows the same interferograms corrupted with phase-noise uniformly distributed in [0,π]. The noisy fringes were low-pass filtered by a 3x3 averaging window.

Fig. 10
Fig. 10

The demodulated phases φ1(x,y) and φ2(x,y) corresponding to the noiseless (panel (a)) and noisy (panel (b)) 5-steps interferograms in Fig. 9. Please note that there is absolutely no cross-talking between the two demodulated phases φ1(x,y) and φ2(x,y).

Fig. 11
Fig. 11

The effect of detuning (Δ) greatly exaggerated for clarity. The amount of detuning is Δ (radians/sample). The well-tuned frequencies are { ω 1 , ω 2 , ω 1 , ω 2 } , while the detuned frequencies are {( ω 1 Δ),( ω 2 Δ),( ω 1 +Δ),( ω 2 +Δ)} .

Fig. 12
Fig. 12

Joint signal-to-noise product G SNR ( d ) of the two detuning-robust FTF-filters { H 1 (ω), H 2 (ω)} in Eq. (22). The second maximum has a PZT-displacement of d = 381nm.

Fig. 13
Fig. 13

Spectra of detuning-robust DW-PSA tuned at ω 1 =2.5rad and ω 2 =1.05rad . The second-order zeroes tolerate a fair amount of frequency detuning Δ.

Fig. 14
Fig. 14

Harmonic amplitudes for | H 1 (n ω 1 )| in red, and | H 2 (n ω 2 )| in green. The ideal would be to bandpass just the Dirac-deltas at ω= ω 1 and ω= ω 2 ; but this is not possible.

Fig. 15
Fig. 15

Five DW phase-shifted temporal interferograms with high amplitude saturation.

Fig. 16
Fig. 16

The demodulated distorted-phases { φ 1 , φ 2 } from the 5 saturated fringe patterns. Please note that there is a slight harmonics cross-talking between the distorted phases.

Fig. 17
Fig. 17

Simplified schematics for a temporal 3-wavelenght phase-shifting interferometer.

Fig. 18
Fig. 18

Fourier spectrum I(ω) for a 3-wavelength temporal phase-shifted interferograms.

Equations (31)

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I(x,y)=a(x,y)+ b 1 (x,y)cos[ φ 1 (x,y)+ u 1 x ]+ b 2 (x,y)cos[ φ 2 (x,y)+ u 2 x ].
I(x,y,t)=a(x,y)+ b 1 (x,y)cos[ φ 1 (x,y)+( 2π λ 1 d )t ]+ b 2 (x,y)cos[ φ 2 (x,y)+( 2π λ 2 d )t ].
λ eq = λ 1 λ 2 | λ 1 λ 2 | ; λ eq >>( λ 1 or λ 2 ).
ω 1 = 2π λ 1 d,and ω 2 = 2π λ 2 d.
I(x,y,t)=a(x,y)+ b 1 (x,y)cos[ φ 1 (x,y)+ ω 1 t ]+ b 2 (x,y)cos[ φ 2 (x,y)+ ω 2 t ],
I 0 (x,y)=a+ b 1 cos[ φ 1 ]+ b 2 cos[ φ 2 ], I 1 (x,y)=a+ b 1 cos[ φ 1 + ω 1 ]+ b 2 cos[ φ 2 + ω 2 ], I 2 (x,y)=a+ b 1 cos[ φ 1 +2 ω 1 ]+ b 2 cos[ φ 2 +2 ω 2 ], I 3 (x,y)=a+ b 1 cos[ φ 1 +3 ω 1 ]+ b 2 cos[ φ 2 +3 ω 2 ], I 4 (x,y)=a+ b 1 cos[ φ 1 +4 ω 1 ]+ b 2 cos[ φ 2 +4 ω 2 ].
I(ω)=aδ(ω)+ b 1 2 [ e i φ 1 δ(ω ω 1 )+ e i φ 1 δ(ω+ ω 1 ) ]+ b 2 2 [ e i φ 2 δ(ω ω 2 )+ e i φ 2 δ(ω+ ω 2 ) ].
H 1 (ω)=( 1 e iω )[ 1 e i(ω+ ω 2 ) ][ 1 e i(ω ω 2 ) ][ 1 e i(ω+ ω 1 ) ], H 2 (ω)=( 1 e iω )[ 1 e i(ω ω 1 ) ][ 1 e i(ω+ ω 1 ) ][ 1 e i(ω+ ω 2 ) ].
h 1 (t)= F 1 { H 1 (ω) }= n=0 4 c 1, n ( ω 1 , ω 2 )δ(tn) , h 2 (t)= F 1 { H 2 (ω) }= n=0 4 c 2, n ( ω 1 , ω 2 )δ(tn) .
1 2 H 1 ( ω 1 ) b 1 (x,y) e i φ 1 (x,y) = n=0 4 c 1, n ( ω 1 , ω 2 ) I n (x,y) , 1 2 H 2 ( ω 2 ) b 2 (x,y) e i φ 2 (x,y) = n=0 4 c 2, n ( ω 1 , ω 2 ) I n (x,y) .
A 1 e i φ 1 = e i ω 2 I 0 + c 1, 1 ( ω 1 , ω 2 ) I 1 c 1, 2 ( ω 1 , ω 2 ) I 2 + c 1, 3 ( ω 1 , ω 2 ) I 3 e i( ω 2 ω 1 ) I 4 , c 1, 1 ( ω 1 , ω 2 )=1+ e i ω 2 + e 2i ω 2 + e i( ω 2 ω 1 ) , c 1, 2 ( ω 1 , ω 2 )=1+ e i ω 2 + e 2i ω 2 + e i( ω 2 ω 1 ) + e i ω 1 + e i(2 ω 2 ω 1 ) , c 1, 3 ( ω 1 , ω 2 )=[ 1+ e i ω 1 + e i( ω 2 + ω 1 ) + e i( ω 2 ω 1 ) ] e i ω 2 .
A 2 e i φ 2 = e i ω 1 I 0 + c 2, 1 ( ω 1 , ω 2 ) I 1 c 2,2 ( ω 1 , ω 2 ) I 2 + c 2,3 ( ω 1 , ω 2 ) I 3 e i( ω 1 ω 2 ) I 4 , c 2,1 ( ω 1 , ω 2 )=1+ e i ω 1 + e 2i ω 1 + e i( ω 1 ω 2 ) , c 2,2 ( ω 1 , ω 2 )=1+ e i ω 1 + e 2i ω 1 + e i( ω 1 ω 2 ) + e i ω 2 + e i(2 ω 1 ω 2 ) , c 2,3 ( ω 1 , ω 2 )=[ 1+ e i ω 2 + e i( ω 1 + ω 2 ) + e i( ω 1 ω 2 ) ] e i ω 1 .
SNR 1 = | H 1 ( ω 1 ) | 2 1 2π π π | H 1 (ω) | 2 dω , SNR 2 = | H 2 ( ω 2 ) | 2 1 2π π π | H 2 (ω) | 2 dω .
d= ω 1 ( λ 1 2π )= ω 2 ( λ 2 2π ) ω 2 = ω 1 ( λ 1 λ 2 ) ω 2 =1.49 radians sample .
H 1 (ω)=( 1 e iω )[ 1 e i[ω+1.49] ][ 1 e i[ω1.49] ][ 1 e i(ω+1.26) ], H 2 (ω)=( 1 e iω )[ 1 e i(ω1.26) ][ 1 e i(ω+1.26) ][ 1 e i[ω+1.49] ].
| H 1 ( ω 1 ) | 2 1 2π π π | H 1 (ω) | 2 dω =0.94; | H 2 ( ω 2 ) | 2 1 2π π π | H 2 (ω) | 2 dω =1.04; ω 1 =1.26; ω 2 =1.49.
G SNR ( d )=( | H 1 ( ω 1 ) | 2 1 2π π π | H 1 (ω) | 2 dω )( | H 2 ( ω 2 ) | 2 1 2π π π | H 2 (ω) | 2 dω );d[0, λ eq ].
ω 1 =arg[ exp( id2π/ λ 1 ) ]=1.2, ω 2 =arg[ exp( id2π/ λ 2 ) ]=2.6.
A 1 ( ω 1 ) e i φ 1 = e 2.6i I 0 +(0.78+0.62i) I 1 (0.5i) I 2 (1+0.19i) I 3 e 1.4i I 4
A 2 ( ω 2 ) e i φ 2 = e 1.2i I 0 +(0.8+0.6i) I 1 (0.920.1i) I 2 +(0.650.77i) I 3 e 1.4i I 4 .
A 2 e i φ ^ 2 = H 2 ( ω 1 Δ) e i φ 1 + H 2 ( ω 2 Δ) e i φ 2 + H 2 ( ω 1 +Δ) e i φ 1 + H 2 ( ω 2 +Δ) e i φ 2 .
H 1 (ω)=( 1 e iω ) [ 1 e i(ω+ ω 2 ) ] 2 [ 1 e i(ω ω 2 ) ] 2 [ 1 e i(ω+ ω 1 ) ] 2 , H 2 (ω)=( 1 e iω ) [ 1 e i(ω ω 1 ) ] 2 [ 1 e i(ω+ ω 1 ) ] 2 [ 1 e i(ω+ ω 2 ) ] 2 .
H R 1 = | H 1 ( ω 1 ) | 2 | n |2 { ( 1 n 2 ) 2 [ | H 1 (n ω 1 ) | 2 + | H 2 (n ω 2 ) | 2 ] } =11.83, H R 2 = | H 2 ( ω 2 ) | 2 | n |2 { ( 1 n 2 ) 2 [ | H 1 (n ω 1 ) | 2 + | H 2 (n ω 2 ) | 2 ] } =12.2
I(x,y,t)=a+ b 1 cos[ φ 1 + ω 1 t ]+ b 2 cos[ φ 2 + ω 2 t ]+ b 3 cos[ φ 3 + ω 3 t ].
H 1 (ω)=( 1 e iω )[ 1 e i(ω+ ω 2 ) ][ 1 e i(ω ω 2 ) ][ 1 e i(ω+ ω 3 ) ][ 1 e i(ω ω 3 ) ][ 1 e i(ω+ ω 1 ) ], H 2 (ω)=( 1 e iω )[ 1 e i(ω ω 1 ) ][ 1 e i(ω+ ω 1 ) ][ 1 e i(ω+ ω 3 ) ][ 1 e i(ω ω 3 ) ][ 1 e i(ω+ ω 2 ) ], H 3 (ω)=( 1 e iω )[ 1 e i(ω ω 1 ) ][ 1 e i(ω+ ω 1 ) ][ 1 e i(ω+ ω 2 ) ][ 1 e i(ω ω 2 ) ][ 1 e i(ω+ ω 3 ) ].
G SNR (d)=( | H 1 ( ω 1 ) | 2 1 2π π π | H 1 (ω) | 2 dω )( | H 2 ( ω 1 ) | 2 1 2π π π | H 2 (ω) | 2 dω )( | H 3 ( ω 3 ) | 2 1 2π π π | H 3 (ω) | 2 dω ).
ω 1 =W( 2π λ 1 d ), ω 2 =W( 2π λ 2 d ), ω 3 =W( 2π λ 3 d );W(x)=arg[ exp(ix) ].
h 1 (t)= F 1 { H 1 (ω) }= n=0 6 c 1, n ( ω 1 , ω 2 , ω 3 )δ(tn) , h 2 (t)= F 1 { H 2 (ω) }= n=0 6 c 2, n ( ω 1 , ω 2 , ω 3 )δ(tn) , h 3 (t)= F 1 { H 3 (ω) }= n=0 6 c 3, n ( ω 1 , ω 2 , ω 3 )δ(tn) ,
I n =a+ b 1 cos[ φ 1 +n ω 1 ]+ b 2 cos[ φ 2 +n ω 2 ]+ b 3 cos[ φ 3 +n ω 3 ];n=0,...,6.
A 1 e i φ 1 (x,y) = n=0 6 c 1,n ( ω 1 , ω 2 , ω 3 ) I n (x,y) , A 2 e i φ 2 (x,y) = n=0 6 c 2,n ( ω 1 , ω 2 , ω 3 ) I n (x,y) , A 3 e i φ 3 (x,y) = n=0 6 c 3,n ( ω 1 , ω 2 , ω 3 ) I n (x,y) ,
H 1 (ω)=( 1 e iω )[ 1 e i(ω+ ω 2 ) ][ 1 e i(ω ω 2 ) ][ 1 e i(ω+ ω 1 ) ], H 2 (ω)=( 1 e iω )[ 1 e i(ω ω 1 ) ][ 1 e i(ω+ ω 1 ) ][ 1 e i(ω+ ω 2 ) ].

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