Skip to main content
SpringerLink
Log in

The phase retrieval problem: a spectral parameter power series approach

  • Published:
Journal of Engineering Mathematics Aims and scope Submit manuscript

Abstract

This paper presents the application of the spectral parameter power series method [Pauli, Math Method Appl Sci 33:459–468 (2010)] for constructing the Green’s function for the elliptic operator \(-\nabla \cdot I\nabla \) in a rectangular domain \(\varOmega \subset \mathbb R ^{2}\), where \(I\) admits separation of variables. This operator appears in the transport-of-intensity equation (TLE) for undulatory phenomena, which relates the phase of a coherent wave with the axial derivative of its intensity in the Fresnel regime. We present a method for solving the TIE with Dirichlet boundary conditions. In particular, we discuss the case of an inhomogeneous boundary condition, a problem that has not been addressed specifically in other works, under the restricted assumption that the intensity \(I\) admits separation of variables. Several simulations show the validity of the method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28
Fig. 29
Fig. 30
Fig. 31
Fig. 32
Fig. 33

Similar content being viewed by others

Notes

  1. In Feynman’s words [61, p. 30–31]: “No one has ever been able to define the difference between interference and diffraction satisfactorily.” Yet it seems more appropriate to classify this as an interference simulation.

References

  1. Pauli W (1933) Die allgemeinen Prinzipien der Wellenmechanik. In: Geiger H, Scheel K (eds) Handbuch der Physik, vol 24. Springer, Berlin, pp 83–272 (in German)

  2. Orlowski A, Paul H (1994) Phase retrieval in quantum mechanics. Phys Rev A 50:921–924

    Article  ADS  Google Scholar 

  3. Walther A (1963) The question of phase retrieval in optics. Opt Acta 10:41–49

    Article  ADS  MathSciNet  Google Scholar 

  4. Dialetis D, Wolf E (1967) The phase retrieval problem of coherence theory as a stability problem. Nuovo Cimento B 47:113–116

    Article  ADS  Google Scholar 

  5. Fienup JR (1982) Phase retrieval algorithms: a comparison. Appl Opt 21:2758–2769

    Article  ADS  Google Scholar 

  6. Teague MR (1983) Deterministic phase retrieval: a Green’s function solution. J Opt Soc Am 73:1434–1441

    Article  ADS  Google Scholar 

  7. Paganin DM (2006) Coherent X-ray optics. Oxford University Press, New York

    Book  Google Scholar 

  8. Spence JCH, Weierstall U, Howells M (2002) Phase recovery and lensless imaging by iterative methods in optical, X-ray and electron diffraction. Philos Trans R Soc Lond A 360:875–895

    Article  ADS  MATH  MathSciNet  Google Scholar 

  9. Reimer L, Kohl H (2008) Transmission electron microscopy. Physics of image formation, 5th edn. Springer, Berlin

  10. Ni W-l, Zhou T (2008) Algorithm for phase contrast X-ray tomography based on nonlinear phase retrieval. Appl Math Mech 29:101–112

    Article  MATH  MathSciNet  Google Scholar 

  11. Gerchberg RW, Saxton WO (1972) A practical algorithm for the determination of the phase from image and diffraction plane pictures. Optik 35:237–246

    Google Scholar 

  12. Fienup JR, Wackerman CC (1986) Phase-retrieval stagnation problems and solutions. J Opt Soc Am A 3:1897–1907

    Article  ADS  Google Scholar 

  13. Allen LJ, Oxley MP (2001) Phase retrieval from series of images obtained by defocus variation. Opt Commun 199:65–75

    Article  ADS  Google Scholar 

  14. Volkov VV, Zhu Y, De Graef M (2002) A new symmetrized solution for phase retrieval using the transport of intensity equation. Micron 33:411–416

    Article  Google Scholar 

  15. Gureyev TE (2003) Composite techniques for phase retrieval in the Fresnel region. Opt Commun 220:49–58

    Article  ADS  Google Scholar 

  16. Beleggia M, Schofield MA, Volkov VV, Zhu Y (2004) On the transport of intensity technique for phase retrieval. Ultramicroscopy 102:37–49

    Article  Google Scholar 

  17. Lewis A, Tikhonenkov I (2005) Noninterferometric phase calculation for paraxial beams using intensity distribution. Opt Commun 246:21–24

    Article  ADS  Google Scholar 

  18. Xue B, Zheng S (2011) Phase retrieval using the transport of intensity equation solved by the FMG-CG method. Optik 122:2101–2106

    Article  ADS  Google Scholar 

  19. Rubinstein J, Wolansky G (2004) A variational principle in optics. J Opt Soc Am A 21:2164–2172

    Article  ADS  MathSciNet  Google Scholar 

  20. Rosenblatt J (1984) Phase retrieval. Commun Math Phys 95:317–343

    Article  ADS  MATH  MathSciNet  Google Scholar 

  21. Ersoy OK (2007) Diffraction Fourier optics and imaging. Wiley, Hoboken

    Book  MATH  Google Scholar 

  22. Goodman JW (2004) Introduction to Fourier optics. Roberts & Company Publishers, Englewood

    Google Scholar 

  23. Feiock FD (1978) Wave propagation in optical systems with large apertures. J Opt Soc Am 68:485–489

    Article  ADS  Google Scholar 

  24. Tay CJ, Quan C, Yang FJ, He XY (2004) A new method for phase extraction from a single fringe pattern. Opt Commun 239:251–258

    Article  ADS  Google Scholar 

  25. Creath K (1988) Phase-measurement interferometry techniques. Prog Opt 26:349–393

    Article  Google Scholar 

  26. Stahl HP (1990) Review of phase-measuring interferometry. In: Proceedings of SPIE, vol 1332, optical testing and metrology III: recent advances in industrial optical inspection, San Diego, July 1990, pp 704–719

  27. Vladimirov VS (1971) Equations of mathematical physics. Marcel Dekker, Inc., New York

    Google Scholar 

  28. Courant R, Hilbert D (1989) Methods of mathematical physics, vol II. Wiley, New York

    Book  Google Scholar 

  29. Sweers G (1997) Hopf’s lemma and two dimensional domains with corners. Rend Istit Mat Univ Trieste Suppl 28:383–419

    Google Scholar 

  30. Nye JF (1999) Natural focusing and fine structure of light: caustics and wave dislocations. Institute of Physics Publication, Bristol

    MATH  Google Scholar 

  31. Freund I (1994) Optical vortices in Gaussian random wave fields: statistical probability densities. J Opt Soc Am A 11:1644–1652

    Article  ADS  Google Scholar 

  32. Coullet P, Gil L, Rocca F (1989) Optical vortices. Opt Commun 73:403–408

    Article  ADS  Google Scholar 

  33. Foley JT, Butts RR (1981) Uniqueness of phase retrieval from intensity measurements. J Opt Soc Am 71:1008–1014

    Article  ADS  Google Scholar 

  34. Gureyev TE, Roberts A, Nugent KA (1995) Partially coherent fields, the transport-of-intensity equation, and phase uniqueness. J Opt Soc Am A 12:1942–1946

    Article  ADS  MathSciNet  Google Scholar 

  35. Gureyev TE, Nugent KA (1996) Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination. J Opt Soc Am A 13:1670–1682

    Article  ADS  Google Scholar 

  36. Lee C-m, Rubinstein J (2006) Elliptic equations with diffusion coefficient vanishing at the boundary: theoretical and computational aspects. Q Appl Math 64:735–747

    MATH  MathSciNet  Google Scholar 

  37. Fienup JR (1986) Phase retrieval using boundary conditions. J Opt Soc Am A 3:284–288

    Article  ADS  Google Scholar 

  38. Soto M, Ríos S, Acosta E (2000) Role of boundary conditions in phase estimation by transport of intensity equation. Opt Commun 184:19–24

    Article  ADS  Google Scholar 

  39. Shengyang H, Fengjie X, Changhai L, Zongfu J (2011) Eigenfunctions of Laplacian for phase estimation from wavefront gradient or curvature sensing. Opt Commun 284:2781–2783

    Article  ADS  Google Scholar 

  40. Nugent KA, Gureyev TE, Cookson DF, Paganin D, Barnea Z (1996) Quantitative phase imaging using hard X rays. Phys Rev Lett 77:2961–2964

    Article  ADS  Google Scholar 

  41. Born M, Wolf E (2001) Principles of optics, 7th edn. Cambridge University Press, Cambridge

    Google Scholar 

  42. Zernike F (1934) Beugungstheorie des schneidenverfahrens und seiner verbesserten form, der phasenkontrastmethode. Physica 1:689–704 (in German)

    Article  ADS  MATH  Google Scholar 

  43. Castillo-Pérez R, Kravchenko VV, Reséndiz-Vázquez R (2011) Solution of boundary and eigenvalue problems for second-order elliptic operators in the plane using pseudoanalytic formal powers. Math Method Appl Sci 34:455–468

    MATH  Google Scholar 

  44. Paganin DM, Nugent KA (1998) Noninterferometric phase imaging with partially coherent light. Phys Rev Lett 80:2586–2589

    Article  ADS  Google Scholar 

  45. Schmalz JA, Gureyev TE, Paganin DM, Pavlov KM (2011) Phase retrieval using radiation and matter-wave fields: validity of Teague’s method for solution of the transport-of-intensity equation. Phys Rev A 84:023808

    Google Scholar 

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

    Article  ADS  Google Scholar 

  47. Soto M, Acosta E (2007) Improved phase imaging from intensity measurements in multiple planes. Appl Opt 46:7978–7981

    Article  ADS  Google Scholar 

  48. Waller L, Tian L, Barbastathis G (2010) Transport of intensity phase-amplitude imaging with higher order intensity derivatives. Opt Express 18:12552–12561

    Article  ADS  Google Scholar 

  49. Kravchenko VV, Porter RM (2010) Spectral parameter power series for Sturm-Liouville problems. Math Method Appl Sci 33:459–468

    MATH  MathSciNet  Google Scholar 

  50. Kravchenko VV (2008) A representation for solutions of the Sturm-Liouville equation. Complex Var Elliptic Equ 53:775–789

    Article  MATH  MathSciNet  Google Scholar 

  51. Kravchenko VV (2009) Applied pseudoanalytic function theory series: frontiers in mathematics. Birkhäuser, Basel

    Book  Google Scholar 

  52. Demidenko Eu (2006) Separable Laplace equation, magic Toeplitz matrix, and generalized Ohm’s law. Appl Math Comput 181:1313–1327

    Article  MATH  MathSciNet  Google Scholar 

  53. Cain GL, Meyer GH (2006) Separation of variables for partial differential equations: an eigenfunction approach. Chapman & Hall/CRC, New York

    Google Scholar 

  54. Roach GF (1970) Green’s functions: introductory theory with applications. VNR Company, London

    MATH  Google Scholar 

  55. Coleman MP (2005) An introduction to partial differential equations with MATLAB. Chapman & Hall/CRC, New York

    MATH  Google Scholar 

  56. Il’in VA (1950) On the convergence of bilinear series of eigenfunctions. Uspehi Matem Nauk (N. S.) 54(38):135–138 (in Russian)

    Google Scholar 

  57. Melnikov YA (2011) Green’s functions and infinite products: bridging the divide. Springer, New York

    Book  Google Scholar 

  58. Jackson JD (1998) Classical electrodynamics, 3rd edn. Wiley, New York

    Google Scholar 

  59. Lanczos C (1997) Linear differential operators. Dover Publications, Inc., New York

    Google Scholar 

  60. Peterson TE (1998) Eliminating Gibb’s effect from separation of variables solutions. SIAM Rev 40:324–326

    Article  ADS  MATH  MathSciNet  Google Scholar 

  61. Feynman R, Leighton R, Sands M (1964) The Feynman lectures on physics, vol 1. Addison Wesley, Boston

    Google Scholar 

Download references

Acknowledgments

V.B.F. acknowledges the support by COTEBAL-IPN for all the facilities given in the realization of this work. V.K. acknowledges the support by CONACyT via Project 166141.

Author information

Authors and Affiliations

  1. Departamento de Telemática, UPIITA-IPN. Av. Inst. Politécnico Nal., 2580, Col. Barrio la Laguna Ticomán, C.P. 07340, Mexico, DF, Mexico

    Víctor Barrera-Figueroa

  2. Facultad de Ingeniería, Universidad Autónoma de Querétaro, Av. 5 de Febrero esq. Hidalgo s/n, C.U. Cerro de las Campanas, C.P. 76010, Santiago de Querétaro, QRO, Mexico

    Herminio Blancarte

  3. Departamento de Matemáticas, CINVESTAV del IPN, Unidad Queretaro, Libramiento Norponiente 2000, Fracc. Real de Juriquilla, C.P. 76230, Santiago de Querétaro, QRO, Mexico

    Vladislav V. Kravchenko

Authors
  1. Víctor Barrera-Figueroa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Herminio Blancarte
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Vladislav V. Kravchenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Víctor Barrera-Figueroa.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Barrera-Figueroa, V., Blancarte, H. & Kravchenko, V.V. The phase retrieval problem: a spectral parameter power series approach. J Eng Math 85, 179–209 (2014). https://doi.org/10.1007/s10665-013-9644-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10665-013-9644-7

Keywords

Search

Navigation