Abstract

To reduce the environmental errors, a snapshot phase-shifting interference microscope (SPSIM) has been developed for surface roughness measurement. However, fringe-print-through (FPT) error widely exists in the phase-shifting interferometry (PSI). To ensure the measurement accuracy, we analyze the sources which introduce the FPT error in the SPSIM. We also develop a FPT error correction algorithm which can be used in the different intensity distribution conditions. The simulation and experiment verify the correctness and feasibility of the FPT error correction algorithm.

© 2017 Optical Society of America

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References

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2016 (2)

2015 (3)

2014 (1)

B. Kimbrough, “Correction of errors in polarization based dynamic phase shifting interferometers,” Int. J. Optomechatronics 8(4), 304–312 (2014).
[Crossref]

2013 (1)

2012 (1)

B. Kimbrough, N. Brock, and J. Millerd, “Dynamic surface roughness profiler,” Proc. SPIE 8501, 85010D (2012).
[Crossref]

2009 (2)

2008 (1)

2005 (2)

2004 (1)

1997 (2)

1995 (1)

1994 (1)

C. T. Farrell and M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5(6), 648–652 (1994).
[Crossref]

1992 (1)

C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992).
[Crossref]

1991 (2)

C. L. Koliopoulos, “Simultaneous phase-shift interferometer,” Proc. SPIE 1531, 119–127 (1991).
[Crossref]

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84 (3), 118–124 (1991).
[Crossref]

1988 (1)

1974 (1)

Brangaccio, D. J.

Brock, N.

Bruning, J. H.

Chai, L.

Chen, Y. C.

Chen, Z.

de Groot, P.

de Groot, P. J.

Deck, L. L.

Estrada, J. C.

Farrant, D. I.

Farrell, C. T.

C. T. Farrell and M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5(6), 648–652 (1994).
[Crossref]

C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992).
[Crossref]

Gallagher, J. E.

Han, B.

Hayes, J.

Herriott, D. R.

Hibino, K.

Kimbrough, B.

B. Kimbrough, “Correction of errors in polarization based dynamic phase shifting interferometers,” Int. J. Optomechatronics 8(4), 304–312 (2014).
[Crossref]

B. Kimbrough, N. Brock, and J. Millerd, “Dynamic surface roughness profiler,” Proc. SPIE 8501, 85010D (2012).
[Crossref]

Kinnstaetter, K.

Koliopoulos, C. L.

C. L. Koliopoulos, “Simultaneous phase-shift interferometer,” Proc. SPIE 1531, 119–127 (1991).
[Crossref]

Larkin, K. G.

Lee, C. M.

Liang, C. W.

Liang, R.

Lin, P. C.

Liu, F.

Liu, S.

Lohmann, A. W.

Millerd, J.

North-Morris, M.

Novak, M.

Okada, K.

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84 (3), 118–124 (1991).
[Crossref]

Oreb, B. F.

Player, M. A.

C. T. Farrell and M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5(6), 648–652 (1994).
[Crossref]

C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992).
[Crossref]

Quiroga, J. A.

Rosenfeld, D. P.

Sato, A.

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84 (3), 118–124 (1991).
[Crossref]

Schwider, J.

Servin, M.

Streibl, N.

Tian, C.

Tsujiuchi, J.

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84 (3), 118–124 (1991).
[Crossref]

Wang, D.

Wang, X.

Wang, Z.

White, A. D.

Wu, F.

Wu, Y.

Wyant, J.

Xu, J.

Xu, Q.

Appl. Opt. (7)

Int. J. Optomechatronics (1)

B. Kimbrough, “Correction of errors in polarization based dynamic phase shifting interferometers,” Int. J. Optomechatronics 8(4), 304–312 (2014).
[Crossref]

J. Opt. Soc. Am. A (3)

Meas. Sci. Technol. (2)

C. T. Farrell and M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5(6), 648–652 (1994).
[Crossref]

C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992).
[Crossref]

Opt. Commun. (1)

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84 (3), 118–124 (1991).
[Crossref]

Opt. Express (5)

Opt. Lett. (2)

Proc. SPIE (2)

C. L. Koliopoulos, “Simultaneous phase-shift interferometer,” Proc. SPIE 1531, 119–127 (1991).
[Crossref]

B. Kimbrough, N. Brock, and J. Millerd, “Dynamic surface roughness profiler,” Proc. SPIE 8501, 85010D (2012).
[Crossref]

Other (3)

D. Malacara, Optical Shop Testing, 3rd ed. (John Wiley & Sons, Inc., 2007), Chap. 1–7.

D. Malacara, Optical Shop Testing, 3rd ed. (John Wiley & Sons, Inc., 2007), Chap. 14.

https://www.4dtechnology.com/products/polarimeters/polarcam/ .

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Figures (11)

Fig. 1
Fig. 1 Snapshot phase-shifting interference microscope. NDF - neutral density filter, P - polarizer, PBS - polarization beam splitter, QWP - quarter-wave plate, L - lens, M - mirror.
Fig. 2
Fig. 2 Schematic diagram of the mismatching between the PA and the camera sensor.
Fig. 3
Fig. 3 Simulated tested surface, interferogram and intensity distribution for double frequency FPT error. (a) the tested surface, (b) the first frame interferogram, (b) the intensity distribution of a line in four phase-shifted interferograms.
Fig. 4
Fig. 4 Simulation results of the double frequency FPT error correction with irregular non-uniform intensity distribution. (a) the 2D map of the calculated surface with the FPT error, (b) the phase error before error correction (PV = 42.1 nm, RMS = 9.3 nm), (c) the 2D map of calculated surface after the FPT error correction, (d) the phase error after error correction (PV = 14.1 nm, RMS = 2.2 nm).
Fig. 5
Fig. 5 Simulated Lissajous figures by plotting N against D for double frequency FPT error. (a) an ellipse: ax = 2.05, ay = 1.67 (with errors); (b) a circle (without errors).
Fig. 6
Fig. 6 Simulation results of the single frequency FPT error correction with irregular non-uniform intensity distribution. (a) the 2D map of the calculated surface with the FPT error, (b) the phase error before error correction (PV = 33.4 nm, RMS = 8.1 nm), (c) the 2D map of calculated surface after the FPT error correction, (d) the phase error after error correction (PV = 16.7 nm, RMS = 2.9 nm).
Fig. 7
Fig. 7 Simulated Lissajous figures by plotting N against D for single frequency FPT error. (a) an ellipse: ax = 1.81, ay = 1.72 (with errors); (b) a circle (without errors).
Fig. 8
Fig. 8 Interferogram and intensity distribution before and after the response non-uniformity calibration of the polarization camera. (a) the first frame interferogram before the calibration, (b) the intensity distribution of a line in four phase-shifted interferograms before the calibration, (c) the first frame interferogram after the calibration, (d) the intensity distribution of a line in four phase-shifted interferograms after the calibration.
Fig. 9
Fig. 9 Experimental results before the FPT error correction with irregular non-uniform intensity distribution. (a), (b) and (c) the 3D, 2D map and line profile of the calculated phase with the FPT error (PV = 39.7 nm, RMS = 4.7 nm).
Fig. 10
Fig. 10 Experimental results after the FPT error correction with irregular non-uniform intensity distribution. (a), (b) and (c) the 3D, 2D map and line profile of the calculated phase after FPT error correction (PV = 39.7 nm, RMS = 3.9 nm).
Fig. 11
Fig. 11 Lissajous figures by plotting N against D. (a) an ellipse (with errors): ax = 1.66, ay = 1.60; (b) a circle (without errors).

Equations (30)

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E T = Q 3 * T pbs * Q 1 * M T * Q 1 * R pbs * E i *exp(iφ).
E R = Q 3 * R pbs * Q 2 * M R * Q 2 * T pbs * E i .
E=[ E x E y ]=P* E T +P* E R .
I= E 2 = E 2 x + E 2 y .
φ=arctan( I 4 I 2 I 1 I 3 ).
T pbs =[ 1 0 0 0 ], R pbs =[ 0 0 0 1 ].
E T =[ cos 2 ( β 3 )+i sin 2 ( β 3 ) cos( β 3 )sin( β 3 )( 1i ) cos( β 3 )sin( β 3 )( 1i ) i cos 2 ( β 3 )+ sin 2 ( β 3 ) ]*[ T p 0 0 T s ] *[ cos 2 ( β 1 )+i sin 2 ( β 1 ) cos( β 1 )sin( β 1 )( 1i ) cos( β 1 )sin( β 1 )( 1i ) i cos 2 ( β 1 )+ sin 2 ( β 1 ) ]*[ 1 0 0 1 ] . *[ cos 2 ( β 1 )+i sin 2 ( β 1 ) cos( β 1 )sin( β 1 )( 1i ) cos( β 1 )sin( β 1 )( 1i ) i cos 2 ( β 1 )+ sin 2 ( β 1 ) ]*[ R p 0 0 R s ]* 1 2 [ 1 1 ]*exp(iφ)
E R =[ cos 2 ( β 3 )+i sin 2 ( β 3 ) cos( β 3 )sin( β 3 )( 1i ) cos( β 3 )sin( β 3 )( 1i ) i cos 2 ( β 3 )+ sin 2 ( β 3 ) ]*[ R p 0 0 R s ] *[ cos 2 ( β 2 )+i sin 2 ( β 2 ) cos( β 2 )sin( β 2 )( 1i ) cos( β 2 )sin( β 2 )( 1i ) i cos 2 ( β 2 )+ sin 2 ( β 2 ) ]*[ 1 0 0 1 ] . *[ cos 2 ( β 2 )+i sin 2 ( β 2 ) cos( β 2 )sin( β 2 )( 1i ) cos( β 2 )sin( β 2 )( 1i ) i cos 2 ( β 2 )+ sin 2 ( β 2 ) ]*[ T p 0 0 T s ]* 1 2 [ 1 1 ]
E 1 =m* P 1 *( E T + E R )+n* P 2 *( E T + E R )+p* P 3 *( E T + E R )+q* P 4 *( E T + E R ) E 2 =m* P 2 *( E T + E R )+n* P 1 *( E T + E R )+p* P 4 *( E T + E R )+q* P 3 *( E T + E R ) E 3 =m* P 3 *( E T + E R )+n* P 4 *( E T + E R )+p* P 1 *( E T + E R )+q* P 2 *( E T + E R ). E 4 =m* P 4 *( E T + E R )+n* P 3 *( E T + E R )+p* P 2 *( E T + E R )+q* P 1 *( E T + E R )
I 1 =A+Bcos( φ+ ε 1 ) I 2 =A+Bcos( φ+ π 2 + ε 2 )=ABsin( φ+ ε 2 ) I 3 =A+Bcos( φ+π+ ε 3 )=ABcos( φ+ ε 3 ). I 4 =A+Bcos( φ+ 3 2 π+ ε 4 )=A+Bsin( φ+ ε 4 )
N= I 4 I 2 =Bsin( φ+ ε 4 )+Bsin( φ+ ε 2 )=2B[ sin( φ+ ε 4 + ε 2 2 )cos( ε 4 ε 2 2 ) ].
D= I 1 I 3 =Bcos( φ+ ε 1 )+Bcos( φ+ ε 3 )=2B[ cos( φ+ ε 1 + ε 3 2 )cos( ε 1 ε 3 2 ) ].
N= a x sin( Φ ).
D= a y cos( Φ+ε ).
N= a x sinφ.
D= a y cos( φ+ε ).
ϕ=arctan( N D )=arctan( a x sinφ a y cos( φ+ε ) ).
N a 2 x + D a 2 y + 2ND a x a y sinε= cos 2 ε.
F=a x 2 +bxy+c y 2 +dx+fy+g.
a x = 2g ( ac ) 2 + b 2 ( a+c ) , a y = 2g ( ( ac ) 2 + b 2 ( a+c ) ) .
θ= 1 2 arctan b ac fora<c. θ= π 2 + 1 2 arctan b ac fora>c
[ N C D C ]=T*[ N D ].
φ=arctan( N c D c ).
I i,j = A i,j + B i,j cos( φ j + Δ i ).
I i,j = a j + b j cos Δ i + c j sin Δ i .
S j = i=1 M ( I i,j I i,j ) 2 = i=1 M ( a j + b j cos Δ i + c j sin Δ i I i,j ) 2 .
X j = S -1 R j .
S=[ M i=1 M cos Δ i i=1 M sin Δ i i=1 M cos Δ i i=1 M cos 2 Δ i i=1 M sin Δ i cos Δ i i=1 M sin Δ i i=1 M sin Δ i cos Δ i i=1 M sin 2 Δ i ].
X j = [ a j b j c j ] T .
R j = [ i=1 M I i,j i=1 M I i,j cos Δ i i=1 M I i,j sin Δ i ] T .

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