Abstract

A new method for the characterization of coherent laser beams is proposed. It is based on the non-iterative solution of the transport-of-intensity-equation. The phase to recover is decomposed into paraxial properties of laser beams and a set of lateral shifted radial basis functions, which allows for the derivation of a direct solution of the phase by a least-squares fit without the need of an initial guess. The method is tested with synthetic data to deduce an accuracy metric. Additionally, two real laser beams are characterized. Including the real light source in terms of the reconstructed field allows for a more holistic simulation of optical systems.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
Phase retrieval on annular and annular sector pupils by using the eigenfunction method to solve the transport of intensity equation

Shengyang Huang, Fengjie Xi, Changhai Liu, and Zongfu Jiang
J. Opt. Soc. Am. A 29(4) 513-520 (2012)

Boundary-artifact-free phase retrieval with the transport of intensity equation II: applications to microlens characterization

Chao Zuo, Qian Chen, Hongru Li, Weijuan Qu, and Anand Asundi
Opt. Express 22(15) 18310-18324 (2014)

References

  • View by:
  • |
  • |
  • |

  1. T. S. Ross, Laser beam quality metrics (Society of Photo-Optical Instrumentation Engineers, 2013).
  2. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21(15), 2758–2769 (1982).
    [Crossref]
  3. J. R. Fienup, “Phase retrieval algorithms: a personal tour [invited],” Appl. Opt. 52(1), 45–56 (2013).
    [Crossref]
  4. B. Boehme and H. Gross, “Characterization of complex optical systems based on wavefront retrieval from point spread function,” in Optical Systems Design2005, vol. 5965 (SPIE).
  5. P. Dirksen, J. Braat, A. Janssen, and C. Juffermans, “Aberration retrieval using the extended nijboer-zernike approach,” J. Micro/Nanolithogr., MEMS, MOEMS 2(1), 61 (2003).
    [Crossref]
  6. M. R. Teague, “Deterministic phase retrieval: a green’s function solution,” J. Opt. Soc. Am. 73(11), 1434–1441 (1983).
    [Crossref]
  7. S. Pan, J. Ma, R. Zhu, T. Ba, C. Zuo, F. Chen, J. Dou, C. Wei, and W. Zhou, “Real-time complex amplitude reconstruction method for beam quality m2 factor measurement,” Opt. Express 25(17), 20142–20155 (2017).
    [Crossref]
  8. Y. Zhu, Z. Zhang, and G. Barbastathis, “Phase imaging for absorptive phase objects using hybrid uniform and structured illumination transport of intensity equation,” Opt. Express 22(23), 28966–28976 (2014).
    [Crossref]
  9. (2013).
  10. L. Waller, L. Tian, and G. Barbastathis, “Transport of intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express 18(12), 12552–12561 (2010).
    [Crossref]
  11. T. E. Gureyev, A. Roberts, and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of zernike polynomials,” J. Opt. Soc. Am. A 12(9), 1932–1941 (1995).
    [Crossref]
  12. T. E. Gureyev and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. ii. orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13(8), 1670–1682 (1996).
    [Crossref]
  13. J. Stock, A. Broemel, J. Hartung, D. Ochse, and H. Gross, “Description and reimplementation of real freeform surfaces,” Appl. Opt. 56(3), 391–396 (2017).
    [Crossref]
  14. N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49(1), 6–10 (1984).
    [Crossref]
  15. E. Reinhard, G. Ward, S. Pattanaik, and P. Debevec, High Dynamic Range Imaging: Acquisition, Display, and Image-Based Lighting (Elsevier Science, 2005).
  16. P. E. Debevec and J. Malik, Recovering high dynamic range radiance maps from photographs, Proceedings of the 24th annual conference on Computer graphics and interactive techniques (ACM Press/Addison-Wesley Publishing Co., 1997).
  17. D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. iii. the effects of noise,” J. Microsc. 214(1), 51–61 (2004).
    [Crossref]
  18. C. Zuo, Q. Chen, Y. Yu, and A. Asundi, “Transport-of-intensity phase imaging using savitzky-golay differentiation filter - theory and applications,” Opt. Express 21(5), 5346–5362 (2013).
    [Crossref]
  19. A. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. 36(8), 1627–1639 (1964).
    [Crossref]
  20. A. Siegman, Lasers (University Science Books, 1986).
  21. M. Born, E. Wolf, and A. Bhatia, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 2000).
  22. N. Hodgson and H. Weber, Laser Resonators and Beam Propagation: Fundamentals, Advanced Concepts, Applications (Springer Berlin Heidelberg, 2005).
  23. W. Singer, M. Totzeck, and H. Gross, Handbook of Optical Systems, Volume 2: Physical Image Formation (Wiley, 2006).
  24. A. E. Siegman, “Analysis of laser beam quality degradation caused by quartic phase aberrations,” Appl. Opt. 32(30), 5893–5901 (1993).
    [Crossref]

2017 (2)

2014 (1)

2013 (2)

2010 (1)

2004 (1)

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. iii. the effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[Crossref]

2003 (1)

P. Dirksen, J. Braat, A. Janssen, and C. Juffermans, “Aberration retrieval using the extended nijboer-zernike approach,” J. Micro/Nanolithogr., MEMS, MOEMS 2(1), 61 (2003).
[Crossref]

1996 (1)

1995 (1)

1993 (1)

1984 (1)

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49(1), 6–10 (1984).
[Crossref]

1983 (1)

1982 (1)

1964 (1)

A. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. 36(8), 1627–1639 (1964).
[Crossref]

Asundi, A.

Ba, T.

Barbastathis, G.

Barty, A.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. iii. the effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[Crossref]

Bhatia, A.

M. Born, E. Wolf, and A. Bhatia, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 2000).

Boehme, B.

B. Boehme and H. Gross, “Characterization of complex optical systems based on wavefront retrieval from point spread function,” in Optical Systems Design2005, vol. 5965 (SPIE).

Born, M.

M. Born, E. Wolf, and A. Bhatia, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 2000).

Braat, J.

P. Dirksen, J. Braat, A. Janssen, and C. Juffermans, “Aberration retrieval using the extended nijboer-zernike approach,” J. Micro/Nanolithogr., MEMS, MOEMS 2(1), 61 (2003).
[Crossref]

Broemel, A.

Chen, F.

Chen, Q.

Debevec, P.

E. Reinhard, G. Ward, S. Pattanaik, and P. Debevec, High Dynamic Range Imaging: Acquisition, Display, and Image-Based Lighting (Elsevier Science, 2005).

Debevec, P. E.

P. E. Debevec and J. Malik, Recovering high dynamic range radiance maps from photographs, Proceedings of the 24th annual conference on Computer graphics and interactive techniques (ACM Press/Addison-Wesley Publishing Co., 1997).

Dirksen, P.

P. Dirksen, J. Braat, A. Janssen, and C. Juffermans, “Aberration retrieval using the extended nijboer-zernike approach,” J. Micro/Nanolithogr., MEMS, MOEMS 2(1), 61 (2003).
[Crossref]

Dou, J.

Fienup, J. R.

Golay, M. J. E.

A. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. 36(8), 1627–1639 (1964).
[Crossref]

Gross, H.

J. Stock, A. Broemel, J. Hartung, D. Ochse, and H. Gross, “Description and reimplementation of real freeform surfaces,” Appl. Opt. 56(3), 391–396 (2017).
[Crossref]

B. Boehme and H. Gross, “Characterization of complex optical systems based on wavefront retrieval from point spread function,” in Optical Systems Design2005, vol. 5965 (SPIE).

W. Singer, M. Totzeck, and H. Gross, Handbook of Optical Systems, Volume 2: Physical Image Formation (Wiley, 2006).

Gureyev, T. E.

Hartung, J.

Hodgson, N.

N. Hodgson and H. Weber, Laser Resonators and Beam Propagation: Fundamentals, Advanced Concepts, Applications (Springer Berlin Heidelberg, 2005).

Janssen, A.

P. Dirksen, J. Braat, A. Janssen, and C. Juffermans, “Aberration retrieval using the extended nijboer-zernike approach,” J. Micro/Nanolithogr., MEMS, MOEMS 2(1), 61 (2003).
[Crossref]

Juffermans, C.

P. Dirksen, J. Braat, A. Janssen, and C. Juffermans, “Aberration retrieval using the extended nijboer-zernike approach,” J. Micro/Nanolithogr., MEMS, MOEMS 2(1), 61 (2003).
[Crossref]

Ma, J.

Malik, J.

P. E. Debevec and J. Malik, Recovering high dynamic range radiance maps from photographs, Proceedings of the 24th annual conference on Computer graphics and interactive techniques (ACM Press/Addison-Wesley Publishing Co., 1997).

McMahon, P. J.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. iii. the effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[Crossref]

Nugent, K. A.

Ochse, D.

Paganin, D.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. iii. the effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[Crossref]

Pan, S.

Pattanaik, S.

E. Reinhard, G. Ward, S. Pattanaik, and P. Debevec, High Dynamic Range Imaging: Acquisition, Display, and Image-Based Lighting (Elsevier Science, 2005).

Reinhard, E.

E. Reinhard, G. Ward, S. Pattanaik, and P. Debevec, High Dynamic Range Imaging: Acquisition, Display, and Image-Based Lighting (Elsevier Science, 2005).

Roberts, A.

Ross, T. S.

T. S. Ross, Laser beam quality metrics (Society of Photo-Optical Instrumentation Engineers, 2013).

Savitzky, A.

A. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. 36(8), 1627–1639 (1964).
[Crossref]

Siegman, A.

A. Siegman, Lasers (University Science Books, 1986).

Siegman, A. E.

Singer, W.

W. Singer, M. Totzeck, and H. Gross, Handbook of Optical Systems, Volume 2: Physical Image Formation (Wiley, 2006).

Stock, J.

Streibl, N.

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49(1), 6–10 (1984).
[Crossref]

Teague, M. R.

Tian, L.

Totzeck, M.

W. Singer, M. Totzeck, and H. Gross, Handbook of Optical Systems, Volume 2: Physical Image Formation (Wiley, 2006).

Waller, L.

Ward, G.

E. Reinhard, G. Ward, S. Pattanaik, and P. Debevec, High Dynamic Range Imaging: Acquisition, Display, and Image-Based Lighting (Elsevier Science, 2005).

Weber, H.

N. Hodgson and H. Weber, Laser Resonators and Beam Propagation: Fundamentals, Advanced Concepts, Applications (Springer Berlin Heidelberg, 2005).

Wei, C.

Wolf, E.

M. Born, E. Wolf, and A. Bhatia, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 2000).

Yu, Y.

Zhang, Z.

Zhou, W.

Zhu, R.

Zhu, Y.

Zuo, C.

Anal. Chem. (1)

A. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. 36(8), 1627–1639 (1964).
[Crossref]

Appl. Opt. (4)

J. Micro/Nanolithogr., MEMS, MOEMS (1)

P. Dirksen, J. Braat, A. Janssen, and C. Juffermans, “Aberration retrieval using the extended nijboer-zernike approach,” J. Micro/Nanolithogr., MEMS, MOEMS 2(1), 61 (2003).
[Crossref]

J. Microsc. (1)

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. iii. the effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49(1), 6–10 (1984).
[Crossref]

Opt. Express (4)

Other (9)

T. S. Ross, Laser beam quality metrics (Society of Photo-Optical Instrumentation Engineers, 2013).

B. Boehme and H. Gross, “Characterization of complex optical systems based on wavefront retrieval from point spread function,” in Optical Systems Design2005, vol. 5965 (SPIE).

(2013).

E. Reinhard, G. Ward, S. Pattanaik, and P. Debevec, High Dynamic Range Imaging: Acquisition, Display, and Image-Based Lighting (Elsevier Science, 2005).

P. E. Debevec and J. Malik, Recovering high dynamic range radiance maps from photographs, Proceedings of the 24th annual conference on Computer graphics and interactive techniques (ACM Press/Addison-Wesley Publishing Co., 1997).

A. Siegman, Lasers (University Science Books, 1986).

M. Born, E. Wolf, and A. Bhatia, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 2000).

N. Hodgson and H. Weber, Laser Resonators and Beam Propagation: Fundamentals, Advanced Concepts, Applications (Springer Berlin Heidelberg, 2005).

W. Singer, M. Totzeck, and H. Gross, Handbook of Optical Systems, Volume 2: Physical Image Formation (Wiley, 2006).

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 (4)

Fig. 1.
Fig. 1. Accuracy of the reconstructed wavefront in relation to the error of the reconstructed intensity stack within $\pm 1RU$ . The data-points contain the results of different strengths of spherical aberration and astigmatism, as well as different far field divergence angles.
Fig. 2.
Fig. 2. Sketch of the experimental setup.
Fig. 3.
Fig. 3. Two-dimensional cross-sections $I(x=0,y,z)$ of the measured and reconstructed intensities of the beam with $\Theta =$ 3 mrad. The intensity values are normalized to the global maximum. Below, the corresponding on-axis intensities $I(x=0,y=0,z)$ are plotted for comparison.
Fig. 4.
Fig. 4. Two-dimensional cross-sections $I(x=0,y,z)$ of the measured and reconstructed intensities of the beam with $\Theta =$ 7 mrad. The intensity values are normalized to the global maximum. Below, the corresponding on-axis intensities $I(x=0,y=0,z)$ are plotted for comparison.

Equations (19)

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

2 U + k 2 U = 0 ,
U ( r ) = V ( r ) e i k z z .
δ 2 V δ z 2 k | δ V δ z | .
2 V + 2 i k δ V δ z = 0 ,
V ( r ) = I ( r ) e i ϕ ( r ) ,
k δ I δ z = I ϕ + I 2 ϕ .
I = ( I x I y I z ) , I = ( I x I y ) , ϕ = ( ϕ x ϕ y ) .
ϕ = ϕ t + ϕ c + ϕ R B F .
ϕ t = k ( t x x + t y y )
ϕ c = k c 2 ( x 2 + y 2 ) ,
ϕ R B F = m = 1 M c m e ϵ 2 ( ( x x m ) 2 + ( y y m ) 2 ) .
k I z = k ( I z , t + I z , c + I z , R B F ) ,
k I z , t = k ( I x t x + I y t y ) ,
k I z , c = k c ( I x x + I y y + 2 I ) ,
k I z , R B F = m = 1 M c m [ I x 2 ϵ 2 ( x m x ) e ϵ 2 ( ( x x m ) 2 + ( y y m ) 2 ) + I y 2 ϵ 2 ( y m y ) e ϵ 2 ( ( x x m ) 2 + ( y y m ) 2 ) + I ( 4 ϵ 4 ( x m x ) 2 2 ϵ 2 ) e ϵ 2 ( ( x x m ) 2 + ( y y m ) 2 ) + I ( 4 ϵ 4 ( y m y ) 2 2 ϵ 2 ) e ϵ 2 ( ( x x m ) 2 + ( y y m ) 2 ) ] .
| I z + c ( I x x + I y y + 2 I ) | 2 = min .
k ( I z + I z , t + I z , c ) = k I z , R B F .
rms w ( I z ( z ) ) = I z ( x , y , z ) 2 I ( x , y , z ) d x d y I ( x , y , z ) d x d y .
R U = λ π Θ 2 .

Metrics