Abstract

Recent technological innovations have enabled the development of a new class of dynamic (vibration-insensitive) interferometer based on a CCD pixel-level phase-shifting approach. We present theoretical and experimental results for an interferometer based on this pixelated phase-shifting technique. Analyses of component errors and instrument functionality are presented. We show that the majority of error sources cause relatively small magnitude peak-to-valley errors in measurement of the order of 0.002–0.005λ. These errors are largely mitigated by high-rate data acquisition and consequent data averaging.

© 2005 Optical Society of America

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References

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  1. K. Creath, “Phase measurement interferometry techniques,” Prog. Opt. 26, 349–393 (1988).
    [CrossRef]
  2. K. Creath, J. C. Wyant, “Direct phase measurement of aspheric surface contours,” in Proc. SPIE 645, 101–106 (1986).
    [CrossRef]
  3. D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).
  4. J. E. Greivenkamp, J. H. Bruning, “Phase-shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 1992), pp. 501–598.
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  6. J. Schwider, T. Dresel, B. Manzke, “Some considerations of reduction of reference phase error in phase-stepping interferometry,” Appl. Opt. 38, 655–659 (1999).
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  7. R. Smythe, R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361–364 (1984).
    [CrossRef]
  8. P. J. De-Groot, L. L. Deck, “Numerical simulations of vibration in phase-shifting interferometry,” Appl. Opt. 35, 2172–2178 (1996).
    [CrossRef] [PubMed]
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    [CrossRef]
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  14. B. K. A. Ngoi, K. Venkatakrishnan, N. R. Sivakumar, T. Bo, “Instantaneous phase shifting arrangement for microsur-face profiling of flat surfaces,” Opt. Commun. 190, 109–116 (2001).
    [CrossRef]
  15. B. K. A. Ngoi, K. Venkatakrishnan, N. R. Sivakumar, “Phase-shifting interferometry immune to vibration,” Appl. Opt. 40, 3211–3214 (2001).
    [CrossRef]
  16. J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” in Proc. SPIE 5531, 304–314 (2004).
    [CrossRef]
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2004 (1)

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” in Proc. SPIE 5531, 304–314 (2004).
[CrossRef]

2001 (2)

B. K. A. Ngoi, K. Venkatakrishnan, N. R. Sivakumar, T. Bo, “Instantaneous phase shifting arrangement for microsur-face profiling of flat surfaces,” Opt. Commun. 190, 109–116 (2001).
[CrossRef]

B. K. A. Ngoi, K. Venkatakrishnan, N. R. Sivakumar, “Phase-shifting interferometry immune to vibration,” Appl. Opt. 40, 3211–3214 (2001).
[CrossRef]

1999 (1)

1996 (1)

1995 (1)

1992 (1)

C. Koliopoulis, “Simultaneous phase-shift interferometer,” in Proc. SPIE 1531, 119–127 (1992).
[CrossRef]

1990 (1)

M. Kuechel, “The new Zeiss interferometer,” in Proc. SPIE 1332, 655–663 (1990).
[CrossRef]

1989 (1)

1988 (1)

K. Creath, “Phase measurement interferometry techniques,” Prog. Opt. 26, 349–393 (1988).
[CrossRef]

1986 (2)

K. Creath, J. C. Wyant, “Direct phase measurement of aspheric surface contours,” in Proc. SPIE 645, 101–106 (1986).
[CrossRef]

J. McLaughlin, B. Horwitz, “Real-time snapshot interferometer,” in Proc. SPIE 680, 35–43 (1986).
[CrossRef]

1985 (1)

1984 (2)

R. Smythe, R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361–364 (1984).
[CrossRef]

O. Kwon, “Multichannel phase-shifted interferometer,” Opt. Lett. 9, 59–61 (1984).
[CrossRef] [PubMed]

Bo, T.

B. K. A. Ngoi, K. Venkatakrishnan, N. R. Sivakumar, T. Bo, “Instantaneous phase shifting arrangement for microsur-face profiling of flat surfaces,” Opt. Commun. 190, 109–116 (2001).
[CrossRef]

Brock, N.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” in Proc. SPIE 5531, 304–314 (2004).
[CrossRef]

Bruning, J. H.

J. E. Greivenkamp, J. H. Bruning, “Phase-shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 1992), pp. 501–598.

Creath, K.

K. Creath, “Phase measurement interferometry techniques,” Prog. Opt. 26, 349–393 (1988).
[CrossRef]

K. Creath, J. C. Wyant, “Direct phase measurement of aspheric surface contours,” in Proc. SPIE 645, 101–106 (1986).
[CrossRef]

Deck, L. L.

De-Groot, P. J.

Delisle, R.

Dresel, T.

Greivenkamp, J. E.

J. E. Greivenkamp, J. H. Bruning, “Phase-shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 1992), pp. 501–598.

Hayes, J.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” in Proc. SPIE 5531, 304–314 (2004).
[CrossRef]

Horwitz, B.

J. McLaughlin, B. Horwitz, “Real-time snapshot interferometer,” in Proc. SPIE 680, 35–43 (1986).
[CrossRef]

Koliopoulis, C.

C. Koliopoulis, “Simultaneous phase-shift interferometer,” in Proc. SPIE 1531, 119–127 (1992).
[CrossRef]

Kothiyal, M. P.

Kuechel, M.

M. Kuechel, “The new Zeiss interferometer,” in Proc. SPIE 1332, 655–663 (1990).
[CrossRef]

Kwon, O.

Malacara, D.

D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).

Malacara, Z.

D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).

Manzke, B.

Mazzoni, A.

McLaughlin, J.

J. McLaughlin, B. Horwitz, “Real-time snapshot interferometer,” in Proc. SPIE 680, 35–43 (1986).
[CrossRef]

Melozzi, L. P. M.

Millerd, J.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” in Proc. SPIE 5531, 304–314 (2004).
[CrossRef]

Moore, R.

R. Smythe, R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361–364 (1984).
[CrossRef]

Ngoi, B. K. A.

B. K. A. Ngoi, K. Venkatakrishnan, N. R. Sivakumar, T. Bo, “Instantaneous phase shifting arrangement for microsur-face profiling of flat surfaces,” Opt. Commun. 190, 109–116 (2001).
[CrossRef]

B. K. A. Ngoi, K. Venkatakrishnan, N. R. Sivakumar, “Phase-shifting interferometry immune to vibration,” Appl. Opt. 40, 3211–3214 (2001).
[CrossRef]

North-Morris, M.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” in Proc. SPIE 5531, 304–314 (2004).
[CrossRef]

Novak, M.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” in Proc. SPIE 5531, 304–314 (2004).
[CrossRef]

Schwider, J.

Servin, M.

D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).

Sivakumar, N. R.

B. K. A. Ngoi, K. Venkatakrishnan, N. R. Sivakumar, T. Bo, “Instantaneous phase shifting arrangement for microsur-face profiling of flat surfaces,” Opt. Commun. 190, 109–116 (2001).
[CrossRef]

B. K. A. Ngoi, K. Venkatakrishnan, N. R. Sivakumar, “Phase-shifting interferometry immune to vibration,” Appl. Opt. 40, 3211–3214 (2001).
[CrossRef]

Smythe, R.

R. Smythe, R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361–364 (1984).
[CrossRef]

Venkatakrishnan, K.

B. K. A. Ngoi, K. Venkatakrishnan, N. R. Sivakumar, T. Bo, “Instantaneous phase shifting arrangement for microsur-face profiling of flat surfaces,” Opt. Commun. 190, 109–116 (2001).
[CrossRef]

B. K. A. Ngoi, K. Venkatakrishnan, N. R. Sivakumar, “Phase-shifting interferometry immune to vibration,” Appl. Opt. 40, 3211–3214 (2001).
[CrossRef]

Wyant, J. C.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” in Proc. SPIE 5531, 304–314 (2004).
[CrossRef]

K. Creath, J. C. Wyant, “Direct phase measurement of aspheric surface contours,” in Proc. SPIE 645, 101–106 (1986).
[CrossRef]

Appl. Opt. (6)

Opt. Commun. (1)

B. K. A. Ngoi, K. Venkatakrishnan, N. R. Sivakumar, T. Bo, “Instantaneous phase shifting arrangement for microsur-face profiling of flat surfaces,” Opt. Commun. 190, 109–116 (2001).
[CrossRef]

Opt. Eng. (1)

R. Smythe, R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361–364 (1984).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (5)

J. McLaughlin, B. Horwitz, “Real-time snapshot interferometer,” in Proc. SPIE 680, 35–43 (1986).
[CrossRef]

M. Kuechel, “The new Zeiss interferometer,” in Proc. SPIE 1332, 655–663 (1990).
[CrossRef]

C. Koliopoulis, “Simultaneous phase-shift interferometer,” in Proc. SPIE 1531, 119–127 (1992).
[CrossRef]

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” in Proc. SPIE 5531, 304–314 (2004).
[CrossRef]

K. Creath, J. C. Wyant, “Direct phase measurement of aspheric surface contours,” in Proc. SPIE 645, 101–106 (1986).
[CrossRef]

Prog. Opt. (1)

K. Creath, “Phase measurement interferometry techniques,” Prog. Opt. 26, 349–393 (1988).
[CrossRef]

Other (2)

D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).

J. E. Greivenkamp, J. H. Bruning, “Phase-shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 1992), pp. 501–598.

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Figures (12)

Fig. 1
Fig. 1

Polarization-based Twyman–Green interferometer used for pixelated sensor experimentation. HWP, half-wave plate.

Fig. 2
Fig. 2

Superpixel layout showing the phase-shift variation with pixel position.

Fig. 3
Fig. 3

Phase grid for 3 × 3 weighted average phase calculation.

Fig. 4
Fig. 4

(a) 3 × 3 circular pixel grid algorithm showing the corresponding expected error. (b) 3 × 3 stacked pixel grid algorithm showing the corresponding expected error. w/p, waves per pixel.

Fig. 5
Fig. 5

Error in calculated phase for QWP retardance errors of 0.05λ and 0.1λ.

Fig. 6
Fig. 6

Phase error due to 45° polarizer misalignment.

Fig. 7
Fig. 7

Phase-dependent error plot for α = 0.01 linear diattenuator.

Fig. 8
Fig. 8

Calculated phase-step transition showing a two-pixel transition.

Fig. 9
Fig. 9

Cross section of the phase-step measurement showing a two-pixel response as seen in the theoretical calculation.

Fig. 10
Fig. 10

Tilt fringes obtained from a flat mirror under test.

Fig. 11
Fig. 11

Analysis of a single measurement of a flat mirror showing a double-frequency fringe print-through. WV, waves.

Fig. 12
Fig. 12

Sixteen measurements of average data for a flat mirror. The double-frequency errors apparent in the unaveraged case have been greatly reduced.

Equations (6)

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I = 1 2 [ I r + I l + 2 I r I l cos ( Δ ϕ + 2 θ ) ] .
tan ( ϕ ) = I 0 ° I 180 ° I 90 ° I 270 ° .
tan ( ϕ ) = 1 / 4 ( 4 + sin [ ϕ 1 , 1 ] + sin [ ϕ 1 , 3 ] + sin [ ϕ 3 , 1 ] + sin [ ϕ 3 , 3 ] ) 1 / 2 ( 2 sin [ ϕ 2 , 1 ] sin [ ϕ 2 , 3 ] ) 1 / 2 ( 2 + cos [ ϕ 1 , 2 ] + cos [ ϕ 3 , 2 ] ) ( 1 cos [ ϕ 2 , 2 ] ) .
M = [ exp [ 1 / 2 i ( δ + ϕ ) ] { cos 2 [ π ɛ 180 + θ ] + exp [ 1 / 2 i ( δ + ϕ ) ] sin 2 [ π ɛ 180 + θ ] } i sin [ π ɛ 90 + 2 θ ] sin [ δ + ϕ 2 ] i sin [ π ɛ 90 + 2 θ ] sin [ δ + ϕ 2 ] exp [ 1 / 2 i ( δ + ϕ ) ] { exp [ i ( δ + ϕ ) ] cos 2 [ π ɛ 180 + θ ] + sin 2 [ π ɛ 180 + θ ] } ] .
D = [ ( 1 α ) cos 2 [ θ ] + α sin 2 [ θ ] ( 1 α ) cos [ θ ] sin [ θ ] α sin [ θ ] cos [ θ ] ( 1 α ) cos [ θ ] sin [ θ ] α sin [ θ ] cos [ θ ] α cos 2 [ θ ] + ( 1 α ) sin 2 [ θ ] ] .
I 1 = 0.82 + 0.8 sin [ ϕ t ] , I 2 = 0.82 + 0.8 cos [ ϕ t ] , I 3 = 0.82 0.8 sin [ ϕ t ] , I 4 = 0.82 0.8 cos [ ϕ t ] .

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