Abstract

In simultaneous phase-shifting dual-wavelength interferometry, by matching both the phase-shifting period number and the fringe number in interferogram of two wavelengths to the integers, the phase with high accuracy can be retrieved through combining the principle component analysis (PCA) and least-squares iterative algorithm (LSIA). First, by using the approximate ratio of two wavelengths, we can match both the temporal phase-shifting period number and the spatial fringe number in interferogram of two wavelengths to the integers. Second, using above temporal and spatial hybrid matching condition, we can achieve accurate phase shifts of single-wavelength of phase-shifting interferograms through using PCA algorithm. Third, using above phase shifts to perform the iterative calculation with the LSIA method, the wrapped phases of single-wavelength can be determined. Both simulation calculation and experimental research demonstrate that by using the temporal and spatial hybrid matching condition, the PCA + LSIA based phase retrieval method possesses significant advantages in accuracy, stability and processing time.

© 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 (2)

2013 (3)

J. Vargas and C. O. S. Sorzano, “Quadrature component analysis for interferometry,” Opt. Lasers Eng. 51(5), 637–641 (2013).
[Crossref]

J. Vargas, J. M. Carazo, and C. O. S. Sorzano, “Error analysis of the principal component analysis demodulation algorithm,” Appl. Phys. B 115(3), 305–364 (2013).

J. Vargas, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Generalization of the principal component analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

2011 (5)

2007 (1)

2004 (1)

2003 (2)

1998 (1)

1985 (1)

1984 (2)

1973 (1)

1971 (1)

Abdelsalam, D. G.

Barada, D.

Belenguer, T.

Carazo, J. M.

J. Vargas, J. M. Carazo, and C. O. S. Sorzano, “Error analysis of the principal component analysis demodulation algorithm,” Appl. Phys. B 115(3), 305–364 (2013).

J. Vargas, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Generalization of the principal component analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

Charrière, F.

Cheng, Y.-Y.

Colomb, T.

Cuche, E.

Dakoff, A.

Depeursinge, C.

Emery, Y.

Estrada, J. C.

J. Vargas, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Generalization of the principal component analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

Fei, L.

Gaskill, J. D.

Gass, J.

Han, B.

Ishii, Y.

Kawata, S.

Kiire, T.

Kim, D.

Kim, M. K.

Kothiyal, M. P.

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

Kühn, J.

Kumar, U. P.

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

Lam, P. S.

Lu, X.

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]

Marquet, P.

Mohan, N. K.

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

Montfort, F.

Onodera, R.

Pförtner, A.

Polhemus, C.

Quiroga, J. A.

Schwider, J.

Sorzano, C. O. S.

J. Vargas and C. O. S. Sorzano, “Quadrature component analysis for interferometry,” Opt. Lasers Eng. 51(5), 637–641 (2013).
[Crossref]

J. Vargas, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Generalization of the principal component analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

J. Vargas, J. M. Carazo, and C. O. S. Sorzano, “Error analysis of the principal component analysis demodulation algorithm,” Appl. Phys. B 115(3), 305–364 (2013).

Sugisaka, J.

Tian, J.

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]

J. Vargas, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Generalization of the principal component analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

J. Vargas, J. M. Carazo, and C. O. S. Sorzano, “Error analysis of the principal component analysis demodulation algorithm,” Appl. Phys. B 115(3), 305–364 (2013).

J. Vargas and C. O. S. Sorzano, “Quadrature component analysis for interferometry,” Opt. Lasers Eng. 51(5), 637–641 (2013).
[Crossref]

J. Vargas, J. A. Quiroga, and T. Belenguer, “Analysis of the principal component algorithm in phase-shifting interferometry,” Opt. Lett. 36(12), 2215–2217 (2011).
[Crossref] [PubMed]

J. Vargas, J. A. Quiroga, and T. Belenguer, “Phase-shifting interferometry based on principal component analysis,” Opt. Lett. 36(8), 1326–1328 (2011).
[Crossref] [PubMed]

Wang, H.

Wang, Z.

Wyant, J. C.

Yatagai, T.

Zhang, W.

Zhao, H.

Zhong, L.

Appl. Opt. (9)

Appl. Phys. B (1)

J. Vargas, J. M. Carazo, and C. O. S. Sorzano, “Error analysis of the principal component analysis demodulation algorithm,” Appl. Phys. B 115(3), 305–364 (2013).

J. Opt. (1)

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

Opt. Commun. (2)

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]

J. Vargas, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Generalization of the principal component analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

Opt. Express (2)

Opt. Lasers Eng. (1)

J. Vargas and C. O. S. Sorzano, “Quadrature component analysis for interferometry,” Opt. Lasers Eng. 51(5), 637–641 (2013).
[Crossref]

Opt. Lett. (5)

Other (1)

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing (Chemical Rubber Company, 2005).

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Figures (5)

Fig. 1
Fig. 1 (a) One-frame simulated SPSDWIs; (b) the phase of synthetic wavelength achieved with the proposed method; (c) the theoretical phase; (d) the difference between (b) and (c).
Fig. 2
Fig. 2 The relationship between the RMSE of phase retrieval with the PCA + LSIA or PCA method and (a) the phase shifting period number at λ 1 ; (b) the fringe number in interferogram; (c) the amplitude of random phase-shifting error; (d) the number of SPSDWIs.
Fig. 3
Fig. 3 The relationship between the RMSE of phase retrieval or iteration number with PCA + LSIA or LSIA method and (a) the phase-shifting period number at λ 1 ; (b) the fringe number in interferogram; (c) the amplitude of random phase-shifting error; (d) the number of SPSDWIs.
Fig. 4
Fig. 4 (a) One-frame experimental SPSDWIs; (b) the phase of synthetic wavelength achieved with the PCA + LSIA method; (c) reference phase; (d) the difference between (b) and (c).
Fig. 5
Fig. 5 (a) One-frame experimental SPSDWIs; (b) the phase of synthetic wavelength achieved with the PCA + LSIA method; (c) reference phase; (d) the difference between (b) and (c).

Tables (2)

Tables Icon

Table 1 RMSE, PVE, and Processing Time Achieved with Different Methods (Simulation)

Tables Icon

Table 2 RMSE, PVE, and Processing Time of Phase Retrieval Achieved with Different Methods (Experiment)

Equations (27)

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I n (x,y)=A(x,y)+ B λ 1 (x,y)cos[ φ λ 1 (x,y)+ θ λ 1 ,n ]+ B λ 2 (x,y)cos[ φ λ 2 (x,y)+ θ λ 2 ,n ]
I n,k = A k + B λ 1 ,k cos( φ λ 1 ,k + θ λ 1 ,n )+ B λ 2 ,k cos( φ λ 2 ,k + θ λ 2 ,n )
I ˜ n,k = I n,k A k = u λ 1 ,k a λ 1 ,n + v λ 1 ,k b λ 1 ,n + u λ 2 ,k a λ 2 ,n + v λ 2 ,k b λ 2 ,n
A k 1 N n=1 N I n,k
I ˜ = [ I ˜ 1 , I ˜ 2 , , I ˜ N ] T
C= I ˜ I ˜ T
D=UC U T
C ij = k=1 K [ u λ 1 ,k 2 a λ 1 ,i a λ 1 ,j + v λ 1 ,k 2 b λ 1 ,i b λ 1 ,j + u λ 2 ,k 2 a λ 2 ,i a λ 2 ,j + v λ 2 ,k 2 b λ 2 ,i b λ 2 ,j ] +S
S= k=1 K [ u λ 1 ,k v λ 1 ,k ( a λ 1 ,i b λ 1 ,j + a λ 1 ,j b λ 1 ,i )+ u λ 2 ,k v λ 2 ,k ( a λ 2 ,i b λ 2 ,j + a λ 2 ,j b λ 2 ,i ) + u λ 1 ,k u λ 2 ,k ( a λ 1 ,i a λ 2 ,j + a λ 1 ,j a λ 2 ,i )+ u λ 1 ,k v λ 2 ,k ( a λ 1 ,i b λ 2 ,j + a λ 1 ,j b λ 2 ,i ) + u λ 2 ,k v λ 1 ,k ( a λ 2 ,i b λ 1 ,j + a λ 2 ,j b λ 1 ,i )+ v λ 1 ,k v λ 2 ,k ( b λ 1 ,i b λ 2 ,j + b λ 1 ,j b λ 2 ,i )]
k=1 K cos φ λ 1 ,k cos φ λ 2 ,k 0, k=1 K cos φ λ 1 ,k sin φ λ 2 ,k 0, k=1 K cos φ λ 2 ,k sin φ λ 1 ,k 0, k=1 K sin φ λ 1 ,k sin φ λ 2 ,k 0, k=1 K cos φ λ 1 ,k sin φ λ 1 ,k 0, k=1 K cos φ λ 2 ,k sin φ λ 2 ,k 0
S<< k=1 K u λ 1 ,k 2 a λ 1 ,i a λ 1 ,j , S<< k=1 K v λ 1 ,k 2 b λ 1 ,i b λ 1 ,j , S<< k=1 K u λ 2 ,k 2 a λ 2 ,i a λ 2 ,j , S<< k=1 K v λ 2 ,k 2 b λ 2 ,i b λ 2 ,j
C ij k=1 K ( u λ 1 ,k 2 a λ 1 ,i a λ 1 ,j + v λ 1 ,k 2 b λ 1 ,i b λ 1 ,j + u λ 2 ,k 2 a λ 2 ,i a λ 2 ,j + v λ 2 ,k 2 b λ 2 ,i b λ 2 ,j )
C k=1 K u λ 1 ,k 2 G 1 + k=1 K v λ 1 ,k 2 F 1 + k=1 K u λ 2 ,k 2 G 2 + k=1 K v λ 2 ,k 2 F 2
G 1 = [cos θ λ 1 ,1 ,,cos θ λ 1 ,N ] T [cos θ λ 1 ,1 ,,cos θ λ 1 ,N ], F 1 = [sin θ λ 1 ,1 ,,sin θ λ 1 ,N ] T [sin θ λ 1 ,1 ,,sin θ λ 1 ,N ], G 2 = [cos θ λ 2 ,1 ,,cos θ λ 2 ,N ] T [cos θ λ 2 ,1 ,,cos θ λ 2 ,N ], F 2 = [sin θ λ 2 ,1 ,,sin θ λ 2 ,N ] T [sin θ λ 2 ,1 ,,sin θ λ 2 ,N ]
λ G 1 = n=1 N cos 2 θ λ 1 ,n , λ F 1 = n=1 N sin 2 θ λ 1 ,n , λ G 2 = n=1 N cos 2 θ λ 2 ,n , λ F 2 = n=1 N sin 2 θ λ 2 ,n
w G 1 = [cos θ λ 1 ,1 ,,cos θ λ 1 ,N ] T , w F 1 = [sin θ λ 1 ,1 ,,sin θ λ 1 ,N ] T , w G 2 = [cos θ λ 2 ,1 ,,cos θ λ 2 ,N ] T , w F 2 = [sin θ λ 2 ,1 ,,sin θ λ 2 ,N ] T
n=1 N cos θ λ 1 ,n cos θ λ 2 ,n 0, n=1 N cos θ λ 1 ,n sin θ λ 2 ,n 0, n=1 N sin θ λ 1 ,n cos θ λ 2 ,n 0, n=1 N sin θ λ 1 ,n sin θ λ 2 ,n 0, n=1 N cos θ λ 1 ,n sin θ λ 1 ,n 0, n=1 N cos θ λ 2 ,n sin θ λ 2 ,n 0
F 1 w G 1 =0, G 2 w G 1 =0, F 2 w G 1 =0, G 1 w F 1 =0, G 2 w F 1 =0, F 2 w F 1 =0, G 1 w G 2 =0, F 1 w G 2 =0, F 2 w G 2 =0, G 1 w F 2 =0, F 1 w F 2 =0, G 2 w F 2 =0
θ ˜ λ 1 =±arctan( U 2 '/ U 1 '), θ ˜ λ 2 =±arctan( U 4 '/ U 3 ')
E k = n=1 N ( I n,k I n,k r ) 2 = n=1 N ( A k + u λ 1 ,k a λ 1 ,n + v λ 1 ,k b λ 1 ,n + u λ 2 ,k a λ 2 ,n + v λ 2 ,k b λ 2 ,n I n,k r ) 2
E k A k =0, E k u λ 1 ,k =0, E k v λ 1 ,k =0, E k u λ 2 ,k =0, E k v λ 2 ,k =0
φ λ 1 ,k =arctan( v λ 1 ,k / u λ 1 ,k ), φ λ 2 ,k =arctan( v λ 2 ,k / u λ 2 ,k )
E n = k=1 K ( I n,k I n,k r ) 2 = k=1 K ( A k +u ' λ 1 ,n a ' λ 1 ,k +v ' λ 1 ,n b ' λ 1 ,k +u ' λ 2 ,n a ' λ 2 ,k +v ' λ 2 ,n b ' λ 2 ,k I n,k r ) 2
E n A k =0, E n u ' λ 1 ,n =0, E n v ' λ 1 ,n =0, E n u ' λ 2 ,n =0, E n v ' λ 2 ,n =0
θ λ 1 ,n =arctan(v ' λ 1 ,n /u ' λ 1 ,n ), θ λ 2 ,n =arctan(v ' λ 2 ,n /u ' λ 2 ,n )
max{ | θ λ 1 (m) θ λ 1 (m1) | }+max{ | θ λ 2 (m) θ λ 2 (m1) | }<ε
φ Λ = φ λ 2 φ λ 1 =2πh( 1 λ 2 1 λ 1 )= 2πh Λ

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