Abstract

It is well known that having 3 temporal phase shifting (PS) interferograms we do not have many possibilities of using an algorithm with a desired frequency spectrum, detuning, and harmonic robustness. This imposes severe restrictions on the possibilities to demodulate such set of temporal interferograms. It would be nice to apply for example a 7 step PS algorithm to these 3 images in order to have more possibilities to phase demodulate them; even further, it would be even better to apply a quadrature filter having a spatial spread given by a real number to these 3 interferograms. In this paper we propose to do just that; namely we show how to demodulate a set of M-steps phase shifting images with a quadrature filter having a real-number as spatial spread. The interesting thing in this paper is to use a higher than M spread quadrature filter to demodulate our interferograms; in traditional PS interferometry one is stuck to the use of M step phase shifting formula to obtain the searched phase. Using a less than M PS formula is not interesting at all given that we would not use all the available information. The main idea behind the “squeezing” phase shifting method is to re-arrange the information of the M phase shifted fringe patterns in such a way to obtain a single carrier frequency interferogram (a spatio-temporal fringe image) and use any two dimensional quadrature filter to demodulate it. In particular we propose the use of Gabor quadrature filters with a spread given by real-numbers along the spatial coordinates. The Gabor filter may be designed in such way that we may squeeze the frequency response of the filter along any desired spatio-temporal dimension, and obtain better signal to noise demodulation ratio, and better harmonic rejection on the estimated phase.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Servin and M. Kujawinska, "Modern fringe pattern analysis in Interferometry," in Handbook of Optical Engineering, D. Malacara and B. J. Thompson eds., (Marcel Dekker, 2001) Chap 14, 373-422.
  2. D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, (CRC Press, Taylor & Francis, second edition, 2005).
    [CrossRef]
  3. K. Freischland and C. L. Koliopoulos, "Fourier description of digital phase measuring interferometry," J. Opt. Soc. Am. A 7, 542-551 (1990).
    [CrossRef]
  4. M. Servin, D. Malacara, D. Marroquin, and F. J. Cuevas, "Complex linear filters for phase shifting with very low detuning sensitivity," J. Mod. Opt. 44, 1269-1278 (1997).
    [CrossRef]
  5. Y. Surrel, "Design of algorithms for phase measurements by the use of phase stepping," Appl. Opt. 35, 51-60 (1996).
    [CrossRef] [PubMed]
  6. M. Takeda, H. Ina, and S. Kobayashi, "Fourier transform methods of fringe-pattern analysis for computer based topography and interferometry," J. Opt. Soc. Am. A 72, 156-160 (1982).
    [CrossRef]
  7. K. H. Womack, "Interferometric phase measurement using spatial synchronous detection," Opt. Eng. 23, 391-395 (1984).
  8. J. H. Bruning, D. R. Herriot, J. E. Gallagher, D. P. Rosenfel, A. D. White, and D. J. Brangaccio, "Digital wavefront measuring interferometry for testing optical surfaces and lenses," Appl. Opt. 13, 2693-2703 (1974).
    [CrossRef] [PubMed]

1997

M. Servin, D. Malacara, D. Marroquin, and F. J. Cuevas, "Complex linear filters for phase shifting with very low detuning sensitivity," J. Mod. Opt. 44, 1269-1278 (1997).
[CrossRef]

1996

1990

1984

K. H. Womack, "Interferometric phase measurement using spatial synchronous detection," Opt. Eng. 23, 391-395 (1984).

1982

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

1974

Brangaccio, D. J.

Bruning, J. H.

Cuevas, F. J.

M. Servin, D. Malacara, D. Marroquin, and F. J. Cuevas, "Complex linear filters for phase shifting with very low detuning sensitivity," J. Mod. Opt. 44, 1269-1278 (1997).
[CrossRef]

Freischland, K.

Gallagher, J. E.

Herriot, D. R.

Ina, H.

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

Kobayashi, S.

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

Koliopoulos, C. L.

Malacara, D.

M. Servin, D. Malacara, D. Marroquin, and F. J. Cuevas, "Complex linear filters for phase shifting with very low detuning sensitivity," J. Mod. Opt. 44, 1269-1278 (1997).
[CrossRef]

Marroquin, D.

M. Servin, D. Malacara, D. Marroquin, and F. J. Cuevas, "Complex linear filters for phase shifting with very low detuning sensitivity," J. Mod. Opt. 44, 1269-1278 (1997).
[CrossRef]

Rosenfel, D. P.

Servin, M.

M. Servin, D. Malacara, D. Marroquin, and F. J. Cuevas, "Complex linear filters for phase shifting with very low detuning sensitivity," J. Mod. Opt. 44, 1269-1278 (1997).
[CrossRef]

Surrel, Y.

Takeda, M.

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

White, A. D.

Womack, K. H.

K. H. Womack, "Interferometric phase measurement using spatial synchronous detection," Opt. Eng. 23, 391-395 (1984).

Appl. Opt.

J. Mod. Opt.

M. Servin, D. Malacara, D. Marroquin, and F. J. Cuevas, "Complex linear filters for phase shifting with very low detuning sensitivity," J. Mod. Opt. 44, 1269-1278 (1997).
[CrossRef]

J. Opt. Soc. Am. A

K. Freischland and C. L. Koliopoulos, "Fourier description of digital phase measuring interferometry," J. Opt. Soc. Am. A 7, 542-551 (1990).
[CrossRef]

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

Opt. Eng.

K. H. Womack, "Interferometric phase measurement using spatial synchronous detection," Opt. Eng. 23, 391-395 (1984).

Other

M. Servin and M. Kujawinska, "Modern fringe pattern analysis in Interferometry," in Handbook of Optical Engineering, D. Malacara and B. J. Thompson eds., (Marcel Dekker, 2001) Chap 14, 373-422.

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, (CRC Press, Taylor & Francis, second edition, 2005).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1.
Fig. 1.

Schematic of the proposed technique to combine M phase shifted interferograms of size L×L into a single-image, spatio-temporal, carrier-frequency image of size LM×L.

Fig. 2.
Fig. 2.

Panels (a) (b) and (c) represents three phase shifted interferograms of a defocused wavefront. Panel 2(d) is the single spatial interferogram obtained using Eq. (3). Panel (d) is the carrier frequency interferogram with a spatial frequency of 2π/3 radians/pixel.

Fig. 3.
Fig. 3.

Panels (a) (b) and (c) on this figure represent the shifted phase of the interferograms in Fig. 2. Panel (d) shows the blended modulating phase of interest ϕ(x,y). We can see that three consecutive pixels have the same value of modulating phase.

Fig. 4.
Fig. 4.

Panel (a) shows a computer generated interferogram of a noisy phase, we have generated 2 additional phase shifted (α=2π/3) interferograms (not shown) with the same noisy phase. Panel (b) represents the estimated phase ϕ ^ (x,y) obtained from demodulating the carrier frequency interferogram shown in Panel (d) and extracting only the “middle” pixel from the three consecutive pixels that have the same phase ϕ(x,y). Panel (c) shows the resulting estimated phase using the three step phase shifting algorithm in Eq. (2). We can see that the estimated phase in panel (c) has substantially more noise than the one shown in panel (b).

Fig. 5.
Fig. 5.

In panels (a), (b), and (c) we show three phase shifted (with α=2π/3) interferograms which have been fully saturated to show the harmonic rejection capability of the squeezing interferometry technique. Panel (d) shows the blended carrier frequency fringe image obtained using the images in panels (a), (b) and (c).

Fig. 6.
Fig. 6.

Panel (a) shows the staircase phase obtained from the three saturated phase shifted interferograms shown in Fig. 5 and demodulated according to Eq. (2). Panel (b) shows the phase obtained (taking the “middle” phase value) from the carrier interferogram of Panel (d) in Fig. 5. We can see that the estimated phase using our proposed squeezing interferometric technique shown in panel (b) has less harmonic components that the one obtained using the standard 3 step phase shifting algorithm (Eq. (2)) shown in panel (a). Panel (c) show gray level “cuts” of panels (a) and (b) to clearly show the difference between the two recovered wrapped phases. Panel (d) shows the same phase information than in panel (c) but now the phase “cuts” are shown unwrapped.

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

I ( x , y , α ) = a ( x , y ) + b ( x , y ) cos [ ϕ ( x , y ) α ]
I ( x , y , 0 ) = a ( x , y ) + b ( x , y ) cos [ ϕ ( x , y ) ]
I ( x , y , + α ) = a ( x , y ) + b ( x , y ) cos [ ϕ ( x , y ) + α ]
ϕ w ( x , y ) = tan 1 [ ( 1 cos ( α ) ) [ I ( x , y , α ) I ( x , y , + α ) ] sin ( α ) [ 2 I ( x , y , 0 ) I ( x , y , α ) I ( x , y , + α ) ] ]
I ( 3 x , y ) = I ( x , y , α ) I ( 3 x + 1 , y ) = I ( x , y , 0 ) I ( 3 x + 2 , y ) = I ( x , y , + α ) } ( 0 , 0 ) ( x , y ) ( L , L )
I ( x , 3 y ) = I ( x , y , α ) I ( x , 3 y + 1 ) = I ( x , y , 0 ) I ( x , 3 y + 2 ) = I ( x , y , + α ) } ( 0 , 0 ) ( x , y ) ( L , L ) .
I ( M x + m , y ) = I ( x , y , m α ) , α = 2 π M , ( 0 , 0 ) < ( x , y ) < ( L , L ) , m = 0 , . . . , M 1
I ( x , y ) = a ( x , y ) + b ( x , y ) cos [ α x + ϕ ( x , y ) ] , ( 0 , 0 ) < ( x , y ) < ( M L , L ) , α = 2 π M
G α ( x , y ) = [ cos ( α x ) + i sin ( α x ) ] e [ x 2 ( M σ ) 2 + y 2 σ 2 ] , α = 2 π M .
ϕ ( x , y ) = arctan { Im [ I ( x , y ) * * G α ( x , y ) ] Re [ I ( x , y ) * * G α ( x , y ) ] } , ( 0 , 0 ) < ( x , y ) < ( L , L ) ,
G α ( x , y ) = [ cos ( α x x ) + i sin ( α x x ) ] e [ x 2 ( M σ x ) 2 + y 2 σ y 2 ] , α = 2 π M ,
I ( x , M y + m ) = I ( x , y , m α ) , α = 2 π M , ( 0 , 0 ) < ( x , y ) < ( L , L ) , m = 0 , . . . , M 1

Metrics