Abstract

Recently, pixelated spatial carrier interferograms have been used in optical metrology and are an industry standard nowadays. The main feature of these interferometers is that each pixel over the video camera may be phase-modulated by any (however fixed) desired angle within [0,2π] radians. The phase at each pixel is shifted without cross-talking from their immediate neighborhoods. This has opened new possibilities for experimental spatial wavefront modulation not dreamed before, because we are no longer constrained to introduce a spatial-carrier using a tilted plane. Any useful mathematical model to phase-modulate the testing wavefront in a pixel-wise basis can be used. However we are nowadays faced with the problem that these pixelated interferograms have not been correctly demodulated to obtain an error-free (exact) wavefront estimation. The purpose of this paper is to offer the general theory that allows one to demodulate, in an exact way, pixelated spatial-carrier interferograms modulated by any thinkable two-dimensional phase carrier.

© 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. R. Smythe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361–364 (1984).
  4. O. Y. Kwon, “Multichannel phase-shifted interferometer,” Opt. Lett. 9(2), 59–61 (1984).
    [CrossRef] [PubMed]
  5. C. L. Koliopoulos, “Simultaneous phase-shift interferometer,” Proc. SPIE 1531, 119–127 (1992).
    [CrossRef]
  6. B. K. A. Ngoi, K. Venkatakrishnan, and N. R. Sivakumar, “Phase-shifting interferometry immune to vibration,” Appl. Opt. 40(19), 3211–3214 (2001).
    [CrossRef]
  7. 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]
  8. 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]
  9. 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]
  10. B. T. Kimbrough, “Pixelated mask spatial carrier phase shifting interferometry algorithms and associated errors,” Appl. Opt. 45(19), 4554–4562 (2006).
    [CrossRef] [PubMed]

2009 (1)

2006 (1)

2005 (1)

2004 (1)

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 (1)

1992 (1)

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

1984 (2)

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

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

1982 (1)

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]

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.

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. Smythe 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.

Smythe, R.

R. Smythe 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. (3)

J. Opt. Soc. Am. A (1)

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. (1)

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

Opt. Express (1)

Opt. Lett. (1)

Proc. SPIE (2)

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 (1)

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

The basic building block (or superpixel) of the phase-mask proposed in [7,8]. The superpixel is periodically repeated over the entire CCD, giving the spatially homogeneous two dimensional carrier pm(x,y) required by Eq. (2).

Fig. 2
Fig. 2

Four phase-shifted interferograms obtained by sorting the phase-masked CCD pixels according to their phase-shift. The CCD has 2Nx2N pixels, and the four smaller interferograms have NxN pixels.

Fig. 3
Fig. 3

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

Fig. 4
Fig. 4

Here we show the real (in panel (a)) and the imaginary (in panel (b)) signals of the product I(x,y)R(x,y). The function I(x,y) is the measured interferogram in Eq. (2), and the signal R(x,y) is the reference wavefront exp[i pm(x,y)].

Fig. 5
Fig. 5

Phase demodulation of the interferogram I = a + bcos[ϕ + pm], phase modulated by the periodic phase-mask pm(x,y). Panel (a) shows the pixelated interferogram. Panel (b) is the spectrum of the interferogram. Panel (c) shows the spectra of the product I(x,y)exp[i pm(x,y)]. Note that the conjugate spectra are separated π radians Finally Panel (d) shows the wrapped (error-free) demodulated phase.

Fig. 6
Fig. 6

Another possible phase-mask pm2(x,y) that may be used to modulate the wavefront under measurement.

Fig. 7
Fig. 7

Phase estimation of the interferogram modulated by pm2(x,y); I = a + bcos[ϕ + pm2]. Panel (a) shows the pixelated carrier interferogram. Panel (b) shows the spectrum of the interferogram. Note that the conjugate spectra are separated (√2)π radians. Panel (c) shows the spectra of the product I(x,y)exp[i pm2(x,y)]. And finally Panel (d) shows the wrapped (error-free) demodulated phase.

Equations (9)

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I ( x , y ) = a ( x , y ) + b ( x , y ) cos [ φ ( x , y ) + ω x x + ω y y ] .
I ( x , y ) = a ( x , y ) + b ( x , y ) cos [ φ ( x , y ) + p m ( x , y ) ] .
φ ^ ( x , y ) = tan 1 [ I ( 0 ) I ( π ) I ( π / 2 ) I ( 3 π / 2 ) ] = 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 ) ] .
R ( x , y ) = exp [ i p m ( x , y ) ] .
I ( x , y ) R ( x , y ) = { a ( x , y ) + b ( x , y ) cos [ φ ( x , y ) + p m ( x , y ) ] } exp [ i p m ( x , y ) ] .
Re [ I ( x , y ) R ( x , y ) ] = a cos ( p m ) + ( b / 2 ) cos ( φ + 2 p m ) + ( b / 2 ) cos ( φ ) Im [ I ( x , y ) R ( x , y ) ]     = a sin ( p m ) + ( b / 2 ) sin ( φ + 2 p m )     + ( b / 2 ) sin ( φ ) .
| p m ( x , y ) x | > | φ ( x , y ) x | max , a n d     | p m ( x , y ) y | > | φ ( x , y ) y | max .
L P F { Re [ I exp ( i p m ) ] } = ( b / 2 ) cos ( φ ) L P F { Im [ I exp ( i p m ) ] }     = ( b / 2 ) sin ( φ ) .
φ ^ ( x , y ) = tan 1 { [ b ( x , y ) / 2 ] sin [ φ ( x , y ) ] [ b ( x , y ) / 2 ] cos [ φ ( x , y ) ] } .

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