Abstract

In this paper, we use the irradiance transport equation and the Fourier transform-based experimental solution given by Ichikawa–Lohmann–Takeda. We analyze experimental factors such as the digital filter, the introduced error for the rotation and period of the Ronchi ruling, and a new method is demonstrated for the measurement of 3D wavefront information.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
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  2. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [Crossref]
  3. G. Ade, “On the validity of the transport equation for the intensity in optics,” Opt. Commun. 52, 307–310 (1985).
    [Crossref]
  4. T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
    [Crossref]
  5. M. Beleggia, M. Schofield, V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
    [Crossref]
  6. K. Ichikawa, A. W. Lohmann, and M. Takeda, “Phase retrieval based on the irradiance transport equation and the Fourier transform method: experiments,” Appl. Opt. 27, 3433–3436 (1988).
    [Crossref]
  7. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [Crossref]
  8. J. A. Quiroga, J. A. Gomez-Pedrero, and J. C. Martinez-Anton, “Wave front measurement by solving the irradiance transport equation for multifocal systems,” Opt. Eng. 40, 2885–2891 (2001).
    [Crossref]
  9. C. Dorrer and J. D. Zuegel, “Optical testing using the transport-of-intensity equation,” Opt. Express 15, 7165–7175 (2007).
    [Crossref]
  10. R. Shomali, A. Darudi, and S. Nasiri, “Application of irradiance transport equation in aspheric surface testing,” Optik 123, 1282–1286 (2012).
    [Crossref]
  11. J. R. Fienup, J. C. Marron, T. J. Schulz, and J. H. Seldin, “Hubble Space Telescope characterized by using phase-retrieval algorithms,” Appl. Opt. 32, 1747–1767 (1993).
    [Crossref]
  12. C. Roddier and F. Roddier, “Wave-front reconstruction from defocused images and the testing of ground-based optical telescopes,” J. Opt. Soc. Am. 10, 2277–2287 (1993).
    [Crossref]
  13. M. Soto, Sensores de curvatura: optimización de su rendimeinto, Ph.D. dissertation (Universidad de Santiago de Compostela, 2006).
  14. M. Soto, E. Acosta, and S. Ríos, “Performance analysis of curvature sensors: optimum positioning of the measurement planes,” Opt. Express 11, 2577–2588 (2003).
    [Crossref]
  15. D. Malacara, Optical Shop Testing (Wiley, 2007).
  16. P. A. Magaña, A. Granados-Agustín, and F. Cornejo-Rodríguez, “Medición de la fase o frente de onda con un banco nodal,” Rev. Mex. Fis. 46, 54–58 (2000).

2012 (1)

R. Shomali, A. Darudi, and S. Nasiri, “Application of irradiance transport equation in aspheric surface testing,” Optik 123, 1282–1286 (2012).
[Crossref]

2007 (1)

2004 (1)

M. Beleggia, M. Schofield, V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[Crossref]

2003 (1)

2001 (1)

J. A. Quiroga, J. A. Gomez-Pedrero, and J. C. Martinez-Anton, “Wave front measurement by solving the irradiance transport equation for multifocal systems,” Opt. Eng. 40, 2885–2891 (2001).
[Crossref]

2000 (1)

P. A. Magaña, A. Granados-Agustín, and F. Cornejo-Rodríguez, “Medición de la fase o frente de onda con un banco nodal,” Rev. Mex. Fis. 46, 54–58 (2000).

1997 (1)

T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
[Crossref]

1993 (2)

C. Roddier and F. Roddier, “Wave-front reconstruction from defocused images and the testing of ground-based optical telescopes,” J. Opt. Soc. Am. 10, 2277–2287 (1993).
[Crossref]

J. R. Fienup, J. C. Marron, T. J. Schulz, and J. H. Seldin, “Hubble Space Telescope characterized by using phase-retrieval algorithms,” Appl. Opt. 32, 1747–1767 (1993).
[Crossref]

1988 (1)

1985 (1)

G. Ade, “On the validity of the transport equation for the intensity in optics,” Opt. Commun. 52, 307–310 (1985).
[Crossref]

1983 (1)

1982 (2)

Acosta, E.

Ade, G.

G. Ade, “On the validity of the transport equation for the intensity in optics,” Opt. Commun. 52, 307–310 (1985).
[Crossref]

Beleggia, M.

M. Beleggia, M. Schofield, V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[Crossref]

Cornejo-Rodríguez, F.

P. A. Magaña, A. Granados-Agustín, and F. Cornejo-Rodríguez, “Medición de la fase o frente de onda con un banco nodal,” Rev. Mex. Fis. 46, 54–58 (2000).

Darudi, A.

R. Shomali, A. Darudi, and S. Nasiri, “Application of irradiance transport equation in aspheric surface testing,” Optik 123, 1282–1286 (2012).
[Crossref]

Dorrer, C.

Fienup, J. R.

Gomez-Pedrero, J. A.

J. A. Quiroga, J. A. Gomez-Pedrero, and J. C. Martinez-Anton, “Wave front measurement by solving the irradiance transport equation for multifocal systems,” Opt. Eng. 40, 2885–2891 (2001).
[Crossref]

Granados-Agustín, A.

P. A. Magaña, A. Granados-Agustín, and F. Cornejo-Rodríguez, “Medición de la fase o frente de onda con un banco nodal,” Rev. Mex. Fis. 46, 54–58 (2000).

Gureyev, T. E.

T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
[Crossref]

Ichikawa, K.

Ina, H.

Kobayashi, S.

Lohmann, A. W.

Magaña, P. A.

P. A. Magaña, A. Granados-Agustín, and F. Cornejo-Rodríguez, “Medición de la fase o frente de onda con un banco nodal,” Rev. Mex. Fis. 46, 54–58 (2000).

Malacara, D.

D. Malacara, Optical Shop Testing (Wiley, 2007).

Marron, J. C.

Martinez-Anton, J. C.

J. A. Quiroga, J. A. Gomez-Pedrero, and J. C. Martinez-Anton, “Wave front measurement by solving the irradiance transport equation for multifocal systems,” Opt. Eng. 40, 2885–2891 (2001).
[Crossref]

Nasiri, S.

R. Shomali, A. Darudi, and S. Nasiri, “Application of irradiance transport equation in aspheric surface testing,” Optik 123, 1282–1286 (2012).
[Crossref]

Nugent, K. A.

T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
[Crossref]

Quiroga, J. A.

J. A. Quiroga, J. A. Gomez-Pedrero, and J. C. Martinez-Anton, “Wave front measurement by solving the irradiance transport equation for multifocal systems,” Opt. Eng. 40, 2885–2891 (2001).
[Crossref]

Ríos, S.

Roddier, C.

C. Roddier and F. Roddier, “Wave-front reconstruction from defocused images and the testing of ground-based optical telescopes,” J. Opt. Soc. Am. 10, 2277–2287 (1993).
[Crossref]

Roddier, F.

C. Roddier and F. Roddier, “Wave-front reconstruction from defocused images and the testing of ground-based optical telescopes,” J. Opt. Soc. Am. 10, 2277–2287 (1993).
[Crossref]

Schofield, M.

M. Beleggia, M. Schofield, V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[Crossref]

Schulz, T. J.

Seldin, J. H.

Shomali, R.

R. Shomali, A. Darudi, and S. Nasiri, “Application of irradiance transport equation in aspheric surface testing,” Optik 123, 1282–1286 (2012).
[Crossref]

Soto, M.

M. Soto, E. Acosta, and S. Ríos, “Performance analysis of curvature sensors: optimum positioning of the measurement planes,” Opt. Express 11, 2577–2588 (2003).
[Crossref]

M. Soto, Sensores de curvatura: optimización de su rendimeinto, Ph.D. dissertation (Universidad de Santiago de Compostela, 2006).

Takeda, M.

Teague, M. R.

Volkov, V.

M. Beleggia, M. Schofield, V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[Crossref]

Zhu, Y.

M. Beleggia, M. Schofield, V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[Crossref]

Zuegel, J. D.

Appl. Opt. (3)

J. Opt. Soc. Am. (3)

Opt. Commun. (2)

G. Ade, “On the validity of the transport equation for the intensity in optics,” Opt. Commun. 52, 307–310 (1985).
[Crossref]

T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
[Crossref]

Opt. Eng. (1)

J. A. Quiroga, J. A. Gomez-Pedrero, and J. C. Martinez-Anton, “Wave front measurement by solving the irradiance transport equation for multifocal systems,” Opt. Eng. 40, 2885–2891 (2001).
[Crossref]

Opt. Express (2)

Optik (1)

R. Shomali, A. Darudi, and S. Nasiri, “Application of irradiance transport equation in aspheric surface testing,” Optik 123, 1282–1286 (2012).
[Crossref]

Rev. Mex. Fis. (1)

P. A. Magaña, A. Granados-Agustín, and F. Cornejo-Rodríguez, “Medición de la fase o frente de onda con un banco nodal,” Rev. Mex. Fis. 46, 54–58 (2000).

Ultramicroscopy (1)

M. Beleggia, M. Schofield, V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[Crossref]

Other (2)

D. Malacara, Optical Shop Testing (Wiley, 2007).

M. Soto, Sensores de curvatura: optimización de su rendimeinto, Ph.D. dissertation (Universidad de Santiago de Compostela, 2006).

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Figures (10)

Fig. 1.
Fig. 1. Experimental setup: LS is the He–Ne laser source, P is the spatial filter system, L2 is the lens under test, and G is a Ronchi ruling. The CCD camera registers irradiance in two planes separated by 0.7 mm.
Fig. 2.
Fig. 2. Experimentally measured irradiance I ( x , 0 ; 0 ) .
Fig. 3.
Fig. 3. Periods of the rulings from top to bottom are: 20, 60, and 120 line pairs/cm. (a), (d), and (g) are the frequency spectra; (b), (e), and (h) show the filter positions for order n = 1 in the spectrum; and (c), (f), and (i) show the experimental irradiance distributions.
Fig. 4.
Fig. 4. Images for the tilt correction of the Ronchi ruling. (a) Distribution of irradiance I ( x , y ; 0 ) ; (b) selection of a central region; (c) comparison of the vertical angle of the stripes relative to the y axis; and (d) the differences determining the tilt angle of the Ronchi ruling.
Fig. 5.
Fig. 5. ZYGO interferograms for lenses with focal lengths (a)  f = 250    mm , (b)  f = 200    mm .
Fig. 6.
Fig. 6. Experimental results using a grid of 20 line pairs/cm. (a) Positive lens under test with 50 mm diameter and 250 mm focal distance; (b) registered irradiance distributions; and (c) profile wavefront W ( x , 0 ; 0 ) along the x axis.
Fig. 7.
Fig. 7. Experimental results using a grid of 20 line pairs/cm. (a) Positive lens under test with 50 mm diameter and 200 mm focal distance; (b) registered irradiance distributions; and (c) profile of wavefront W ( x , 0 ; 0 ) along the x axis.
Fig. 8.
Fig. 8. Ronchi ruling rotated to four angles. (a) 0°, (b) 90°, (c) 45°, (d)  45 ° .
Fig. 9.
Fig. 9. For lens one: (a) 1D wavefronts for each of the ruling positions; (b) 3D wavefront obtained using the MATLAB fitting procedure; (c) 3D wavefront cutoff over the diameter of the lens.
Fig. 10.
Fig. 10. For lens two: (a) 1D wavefronts for each ruling position; (b) 3D wavefront obtained using the MATLAB fitting procedure; (c) 3D wavefront cutoff over the diameter of the lens.

Equations (2)

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2 π λ I ( x , y ; z ) z = T [ I ( x , y ; z ) T ϕ ( x , y ; z ) ] ,
I ( x , y ; z ) z = T I ( x , y ; z ) T W ( x , y ; z ) + I ( x , y ; z ) T 2 W ( x , y ; z ) .

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