Abstract

Recently a new type of spatial phase shifting interferometer was proposed that uses a phase-mask over the camera’s pixels. This new interferometer allows one to phase modulate each pixel independently by setting the angle of a linear polarizer built in contact over the camera’s CCD. In this way neighbor pixels may have any desired (however fixed) phase shift without cross taking. The standard manufacturing of these interferometers uses a 2x2 array with phase-shifts of 0, π/2, π, and 3π/2 radians. This 2x2 array is tiled all over the video camera’s CCD. In this paper we propose a new way to phase demodulate these phase-masked interferograms using the squeezing phase-shifting technique. A notable advantage of this squeezing technique is that it allows one the use of Fourier interferometry wiping out the detuning error that most phase shifting algorithms suffers. Finally we suggest the use of an alternative phase-mask to phase modulate the camera’s pixels using a linear spatial carrier along a given axis.

© 2010 OSA

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References

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  1. D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, 2 ed., (Taylor & Francis Group, CRC Press, 2005).
  2. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72(1), 156–160 (1982).
    [CrossRef]
  3. M. Servin, M. Cywiak, D. Malacara-Hernandez, J. C. Estrada, and J. A. Quiroga, “Spatial carrier interferometry from M temporal phase shifted interferograms: Squeezing Interferometry,” Opt. Express 16(13), 9276–9283 (2008).
    [CrossRef] [PubMed]
  4. R. Smithe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361–364 (1984).
  5. O. Y. Kwon, “Multichannel phase-shifted interferometer,” Opt. Lett. 9(2), 59–61 (1984).
    [CrossRef] [PubMed]
  6. C. L. Koliopoulos, “Simultaneous phase-shift interferometer,” Proc. SPIE 1531, 119–127 (1992).
    [CrossRef]
  7. B. K. A. Ngoi, K. Venkatakrishnan, and N. R. Sivakumar, “Phase-shifting interferometry immune to vibration,” Appl. Opt. 40(19), 3211–3214 (2001).
    [CrossRef]
  8. J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
    [CrossRef]
  9. M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. 44(32), 6861–6868 (2005).
    [CrossRef] [PubMed]
  10. J. F. Mosiño, M. Servin, J. C. Estrada, and J. A. Quiroga, “Phasorial analysis of detuning error in temporal phase shifting algorithms,” Opt. Express 17(7), 5618–5623 (2009).
    [CrossRef] [PubMed]
  11. B. T. Kimbrough, “Pixelated mask spatial carrier phase shifting interferometry algorithms and associated errors,” Appl. Opt. 45(19), 4554–4562 (2006).
    [CrossRef] [PubMed]

2009

2008

2006

2005

2004

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

2001

1992

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

1984

O. Y. Kwon, “Multichannel phase-shifted interferometer,” Opt. Lett. 9(2), 59–61 (1984).
[CrossRef] [PubMed]

R. Smithe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361–364 (1984).

1982

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72(1), 156–160 (1982).
[CrossRef]

Brock, N.

M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. 44(32), 6861–6868 (2005).
[CrossRef] [PubMed]

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

Cywiak, M.

Estrada, J. C.

Hayes, J.

M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. 44(32), 6861–6868 (2005).
[CrossRef] [PubMed]

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

Ina, H.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72(1), 156–160 (1982).
[CrossRef]

Kimbrough, B. T.

Kobayashi, S.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72(1), 156–160 (1982).
[CrossRef]

Koliopoulos, C. L.

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

Kwon, O. Y.

Malacara-Hernandez, D.

Millerd, J.

M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. 44(32), 6861–6868 (2005).
[CrossRef] [PubMed]

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

Moore, R.

R. Smithe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361–364 (1984).

Mosiño, J. F.

Ngoi, B. K. A.

North-Morris, M.

M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. 44(32), 6861–6868 (2005).
[CrossRef] [PubMed]

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

Novak, M.

M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. 44(32), 6861–6868 (2005).
[CrossRef] [PubMed]

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

Quiroga, J. A.

Servin, M.

Sivakumar, N. R.

Smithe, R.

R. Smithe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361–364 (1984).

Takeda, M.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72(1), 156–160 (1982).
[CrossRef]

Venkatakrishnan, K.

Wyant, J.

Wyant, J. C.

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

Appl. Opt.

J. Opt. Soc. Am. A

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72(1), 156–160 (1982).
[CrossRef]

Opt. Eng.

R. Smithe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361–364 (1984).

Opt. Express

Opt. Lett.

Proc. SPIE

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

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

Other

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, 2 ed., (Taylor & Francis Group, CRC Press, 2005).

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Figures (7)

Fig. 1
Fig. 1

This figure shows the CCD superpixel’s 2x2 square array at (x,y) with a zoomed-in portion of this building block. The amount of phase shifting (in radians) introduced to each pixel is also shown.

Fig. 2
Fig. 2

Four phase-shifted interferograms obtained from the phase-masked CCD. The CCD has 128x128 pixels, and the four smaller interferograms 64x64 pixels.

Fig. 3
Fig. 3

In panel (a) we show the estimated phase according to Eq. (3), and in panel (b) the demodulated detuning-error ϕerror (x,y) due to the use of this 4-steps algorithm.

Fig. 4
Fig. 4

Geometrical mapping (Eq. (5)) to obtain a single LCFI of size 4LxL from the four smaller phase-shifted interferograms of size LxL pixels above.

Fig. 5
Fig. 5

Demodulation of the spatial LCFI by the FTD technique. The passing from panel (b) to panel (c) is the “backward” mapping to obtain all the camera’s CCD pixels demodulated.

Fig. 6
Fig. 6

Forward-backward mapping between the phase-masked CCD pixels and the LCFI. The forward mapping is the direction from the CCD to the linear carrier interferogram.

Fig. 7
Fig. 7

Phase-mask to obtain in a direct way a standard LCFI. This mask introduces a linear carrier along the x axis of π/2 radians per pixel. As a consequence one may demodulate the LCFI obtained by the (detuning-free) FTD technique.

Equations (7)

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I ( x , y , t ) = a ( x , y ) + b ( x , y ) cos [ φ ( x , y ) + ω 0 t ] .
I ( x , y ) = a ( x , y ) + b ( x , y ) cos [ φ ( x , y ) + ω 0 x ] .
φ ^ ( x , y ) = tan 1 [ I ( x , y , 0 ) I ( x , y + 1 , π ) I ( x + 1 , y , π / 2 ) I ( x + 1 , y + 1 , 3 π / 2 ) ] .
h ( x , y ) = δ ( x + 1 , y ) e i π / 2 + δ ( x + 1 , y + 1 ) e i 3 π / 2 + δ ( x , y ) + δ ( x , y + 1 ) e i π .
I c ( 4 x , y )           = I ( x , y , 0 ) I c ( 4 x + 1 , y )     = I ( x , y , π / 2 ) I c ( 4 x + 2 , y ) = I ( x , y , π ) I c ( 4 x + 3 , y ) = I ( x , y , 3 π / 2 )     } ( 0 , 0 ) ( x , y ) ( L , L ) .
I ( x , y ) = a ( x , y ) + b ( x , y ) cos [ φ ( x , y ) + ( π / 2 ) x ] .
H ( ω x , ω y ) = { 1 f o r ω x < ε 0 f o r ω x ε .

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