Skip to main content
SpringerLink
Log in

A novel algorithm based on sub-fringe integration method for direct two-dimensional unwrapping phase reconstruction from the intensity of one-shot two-beam interference fringes

  • Published:
Applied Physics B Aims and scope Submit manuscript

Abstract

Phase reconstruction is extremely important in optical interferometry, where the quantitative and qualitative information is encoded in the form of two-dimensional interference interferograms. Most often, a majority of phase calculation methods return results of the wrapped phase limited in the range (− π, π) due to the fundamental properties of the mathematical arctangent function. It would be helpful if we could avoid the unwrapping phase step and calculate the phase directly. In this paper, we present an algorithm of phase reconstruction based on the sub-fringe integration method to avoid the ambiguities due to the properties of the arctangent function. The theoretical considerations of the proposed algorithm were presented. Also, the main steps required to implement the algorithm, in addition to providing the flowchart outlined for those steps, were presented. The algorithm is suitable to analyze the two-beam interference fringe. The proposed algorithm adds a power element to the sub-fringe method, which does not need the unwrapping process. Besides, it needs neither hardware to introduce phase shift, nor more than one interferogram. Both simulation and experimental results demonstrate the usefulness of the proposed algorithm to extract the phase from the two-beam interference interferogram, even if the fringe shifts are variable along the fiber and with the irregularity of the fiber radius across the interferogram. Appropriate and adequate interferograms were presented to clarify the basic idea of the article.

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

Similar content being viewed by others

References

  1. P. Hariharan, Optical Interferometry, 2nd edn. (Elsevier, Amsterdam, 2003)

    Google Scholar 

  2. T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley, New York, 2005)

    Google Scholar 

  3. D.P. Towers, T.R. Judge, P.J. Bryanston-Cross, Automatic interferogram analysis techniques applied to quasi-heterodyne holography and ESPI. Opt. Lasers Eng. 14, 239–281 (1991)

    Google Scholar 

  4. M. Jiang, W. Chen, Z. Zheng, M. Zhong, Fringe pattern analysis by S-transform. Opt. Commun. 285, 209–217 (2012)

    ADS  Google Scholar 

  5. A.A. Hamza, T.Z.N. Sokkar, M.A. Mabrouk, M.A. El-Morsy, Refractive index profile of polyethylene fiber using interactive multiple-beam Fizeau Fringe analysis. J. Appl. Polym. Sci. 77, 3099–3106 (2000)

    Google Scholar 

  6. M.A. El-Morsy, T. Yatagai, A.A. Hamza, M.A. Mabrouk, T.Z.N. Sokkar, Multiple-beam Fizeau Fringe-pattern analysis using Fourier transform method for accurate measurement of fiber refractive index profile of polymer fiber. J. Appl. Polym. Sci. 85(3), 475–484 (2002)

    Google Scholar 

  7. T.Z.N. Sokkar, M.A. El-Morsy, H.H. Wahba, Automatic fringe analysis of the induced anisotropy of bent optical fibres. Opt. Commun. 281, 1915–1923 (2008)

    ADS  Google Scholar 

  8. M.A. El-Morsy, On investigations of refractive indices and birefringence of fiber using duplicate and non-duplicate micro-interferograms of double refracting interferometer. Microsc. Res. Tech. 82, 596–614 (2018)

    Google Scholar 

  9. M.A. El-Morsy, T. Yatagai, A.A. Hamza, M.A. Mabrouk, T.Z.N. Sokkar, Automatic refractive index profiling of fibers by phase analysis method using Fourier transform. Opt. Lasers Eng. 38(6), 509–525 (2002)

    Google Scholar 

  10. H.H. Wahba, Reconstruction of 3D refractive index profiles of PM PANDA optical fiber using digital holographic method. Opt. Fiber Technol. 20, 520–526 (2014)

    ADS  Google Scholar 

  11. W.A. Ramadan, H.H. Wahba, M.A. Shams El-Din, Two-dimensional refractive index and birefringence profiles of a graded index bent optical fibre. Opt. Fiber Technol. 36, 115–124 (2017)

    ADS  Google Scholar 

  12. Z. Pan, Z. Liang, S. Li, J. Weng, J. Zhong, Microtomography of the polarization-maintaining fiber by digital Holography. Opt. Fiber Technol. 22, 46–51 (2015)

    ADS  Google Scholar 

  13. J. Zhong, J. Weng, Generalized Fourier analysis for phase retrieval of fringe pattern. Opt. Exp. 18(26), 26806–26820 (2010)

    ADS  Google Scholar 

  14. R. Porras-Aguilar, K. Falaggis, Absolute phase recovery in structured light illumination systems: sinusoidal vs. intensity discrete patterns. Opt. Lasers Eng. 84, 111–119 (2016)

    Google Scholar 

  15. G.A. Ayubi, I. Duarte, Ferrari JA (2019) Optimal phase-shifting algorithm for interferograms with arbitrary steps and phase noise. Opt. Lasers Eng. 114, 129–135 (2019)

    Google Scholar 

  16. S. Zhang, Absolute phase retrieval methods for digital fringe projection profilometry: a review. Opt. Lasers Eng. 107, 28–37 (2018)

    Google Scholar 

  17. Y. Wei, D. Yan, S.S. Cha, Comparison of results of interferometric phase extraction algorithms for three-dimensional flow-field tomography. Opt. Lasers Eng. 32, 147–155 (1999)

    Google Scholar 

  18. R. Zhao, P. Sun, Deformation-phase measurement by optical flow method. Opt. Commun. 371, 144–149 (2016)

    ADS  Google Scholar 

  19. C.K. Yee, K.S. Yen, Single frame profilometry with rapid phase demodulation on colour-coded fringes. Opt. Commun. 397, 44–50 (2017)

    ADS  Google Scholar 

  20. J. Cheng, Q. Yuan, Y. Dou, Y. Yao, J. Huang, C. Zuo, Z. Gao, Phase extraction for dual-wavelength phase-shift Fizeau interferometry in the presence of multi-beam interference. Opt. Commun. 402, 489–497 (2017)

    ADS  Google Scholar 

  21. Z. Malacara, M. Servín, Interferogram Analysis for Optical Testing, 2nd edn. (Taylor & Francis Group, London, 2005)

    Google Scholar 

  22. P. Rastogi, E. Hack, Phase Estimation in Optical Interferometry (Taylor & Francis Group, London, 2015)

    Google Scholar 

  23. C. Zuo, S. Feng, L. Huang, T. Tao, W. Yin, Q. Chen, Phase shifting algorithms for fringe projection profilometry: a review. Opt. Lasers Eng. 109, 23–59 (2018)

    Google Scholar 

  24. W. Osten (Ed.), Fringe 2005, in The 5th International Workshop on Automatic Processing of Fringe Patterns. Springer. ISBN-13 9783540260370

  25. A. Patil, P. Rastogi, Approaches in generalized phase shifting interferometry. Opt. Lasers Eng. 43, 475–490 (2005)

    MATH  Google Scholar 

  26. H. Lei, X. Chang, F. Wang, H. Xiao-Tang, H. Xiao-Dong, A novel algorithm based on histogram processing of reliability for two-dimensional phase unwrapping. Optik 126, 1640–1644 (2015)

    ADS  Google Scholar 

  27. Q. Kemao, S.H. Soon, A. Asundi, A simple phase unwrapping approach based on filtering by windowed Fourier transform. Opt. Laser Technol. 37, 458–462 (2005)

    ADS  Google Scholar 

  28. K. Chen, J. Xi, Y. Yu, Quality-guided spatial phase unwrapping algorithm for fast three-dimensional measurement. Opt. Commun. 294, 139–147 (2013)

    ADS  Google Scholar 

  29. Y. Guo, X. Chen, T. Zhang, Robust phase unwrapping algorithm based on least squares. Opt. Lasers Eng. 63, 25–29 (2014)

    Google Scholar 

  30. F. Da, H. Huang, A fast, accurate phase unwrapping method for wavelet-transform profilometry. Opt. Commun. 285, 421–432 (2012)

    ADS  Google Scholar 

  31. S. Fang, L. Meng, L. Wang, P. Yang, M. Komori, Quality-guided phase unwrapping algorithm based on reliability evaluation. Appl. Opt. 50(28), 5446–5452 (2011)

    ADS  Google Scholar 

  32. Y.T. Zhang, M.J. Huang, H.R. Liang, F.Y. Lao, Branch cutting algorithm for unwrapping photo elastic phase map with isotropic point. Opt. Lasers Eng. 50, 619–631 (2012)

    Google Scholar 

  33. Z. Dongliang, D. Feipeng, A novel algorithm for branch cut phase unwrapping. Opt. Lasers Eng. 49(5), 609–617 (2011)

    Google Scholar 

  34. Y. Lia, J. Zhua, W. Shen, Phase unwrapping algorithms, respectively, based on path-following and discrete cosine transform. Optik 119, 545–547 (2008)

    ADS  Google Scholar 

  35. Q. Liu, Y. Wang, F. Ji, J. He, A three-step least-squares iterative method for tilt phase-shift interferometry. Opt. Exp. 21(24), 29505–29515 (2013)

    ADS  Google Scholar 

  36. L.A. Romero, Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods. J. Opt. Soc. Am. A Opt. Image Sci. Vis. 11(1), 107–117 (1994)

    ADS  MathSciNet  Google Scholar 

  37. Z.Q. Wei, F. Xu, Y.Q. Jin, Phase unwrapping for SAR interferometry based on an ant colony optimization algorithm. Int. J. Remote Sens. 29(3), 711–725 (2008)

    ADS  Google Scholar 

  38. M. Yan, L. Wang, Weighted Kalman filter phase unwrapping algorithm based on in SAR image. Eng. Rev. 33, 227–231 (2013)

    Google Scholar 

  39. K. Zhong, Z. Li, Y. Shi, C. Wang, Y. Lei, Fast phase measurement profilometry for arbitrary shape objects without phase unwrapping. Opt. Lasers Eng. 51, 1213–1222 (2013)

    Google Scholar 

  40. F.W.Y. Chan, A novel optical method without phase unwrapping for subsurface flaw detection. Opt. Lasers Eng. 47, 186–193 (2009)

    Google Scholar 

  41. W. Lohry, V. Chen, S. Zhang, Absolute three-dimensional shape measurement using coded fringe patterns without phase unwrapping or projector calibration. Opt. Exp. 22(2), 1287–1301 (2014)

    ADS  Google Scholar 

  42. E.-H. Kim, J. Hahn, H. Kim, B. Lee, Profilometry without phase unwrapping using multi-frequency and four-step phase-shift sinusoidal fringe projection. Opt. Exp. 17(10), 7818–7830 (2009)

    ADS  Google Scholar 

  43. D. Guangliang, M. Wang, C. Zhou, S. Si, H. Li, Z. Lei, Y. Li, A simple spatial domain algorithm to increase the residues of wrapped phase maps. J. Mod. Opt. 64(3), 231–237 (2017)

    ADS  Google Scholar 

  44. G. Paez, M. Strojnik, Fringe analysis and phase reconstruction from modulated intensity patterns. Opt. Lett. 22(22), 1669–1671 (1997)

    ADS  Google Scholar 

  45. G. Paez, M. Strojnik, Phase-shifted interferometry without phase unwrapping: reconstruction of a decentered wave front. J. Opt. Soc. Am. A 16(3), 475–480 (1999)

    ADS  Google Scholar 

  46. J. Muroz, G. Paez, M. Strojnik, Two-dimensional phase unwrapping of subsampled phase-shifted interferograms. J. Mod. Opt. 51(1), 49–63 (2004)

    ADS  Google Scholar 

  47. P. Gao, B. Yao, J. Han, L. Chen, Y. Wang, M. Lei, Phase reconstruction from three interferograms based on integral of phase gradient. J. Mod. Opt. 55(14), 2233–2242 (2008)

    ADS  MATH  Google Scholar 

  48. F.W.Y. Chan, A novel optical method without phase unwrapping for subsurface flaw detection. Opt. Lasers Eng. 47, 186–193 (2009)

    Google Scholar 

  49. K. Song, S. Hu, X. Wen, Y. Yan, Fast 3D shape measurement using Fourier transform profilometry without phase unwrapping. Opt. Lasers Eng. 84, 74–81 (2016)

    Google Scholar 

  50. K. Zhong, Z. Li, Y. Shi, C. Wang, Y. Lei, Fast phase measurement profilometry for arbitrary shape objects without phase unwrapping. Opt. Lasers Eng. 51, 1213–1222 (2013)

    Google Scholar 

  51. Y. Park, C. Depeursinge, G. Popescu, Quantitative phase imaging in biomedicine. Nat. Photonics 12, 578–589 (2018)

    ADS  Google Scholar 

  52. K. Lee, K. Kim, J. Jung, J. Heo, S. Cho, S. Lee, G. Chang, Y. Jo, H. Park, Y. Park, Quantitative phase imaging techniques for the study of cell pathophysiology: from principles to applications. Sensors 13, 4170–4191 (2013)

    Google Scholar 

  53. M.K. Kim, Principles and techniques of digital holographic microscopy. SPIE Rev. 1, 018005:1–50 (2010)

    Google Scholar 

  54. B. Kemper, G. von Bally, Digital holographic microscopy for live cell applications and technical inspection. Appl. Opt. 47(4), A52–A61 (2008)

    Google Scholar 

  55. M.A. El-Morsy, K. Harada, M. Itoh, T. Yatagai, A subfringe integration method for multiple-beam Fizeau fringe analysis. Opt. Laser Technol. 35, 223–232 (2003)

    ADS  Google Scholar 

  56. G. Lai, T. Yatagai, Generalized phase-shifting interferometry. J. Opt. Soc. Am. A 8(5), 822–827 (1991)

    ADS  Google Scholar 

  57. L.M. Guerrero-Sánchez, A. Aguilar, Generalized phase-shifting interferometry by parameter estimation with the least squares method. Opt. Lasers Eng. 51, 626–632 (2013)

    Google Scholar 

  58. F. Liu, Y. Wu, F. Wu, W. Song, Generalized phase shifting interferometry based on Lissajous calibration technology. Opt. Lasers Eng. 83, 106–115 (2016)

    Google Scholar 

  59. T.Z.N. Sokkar, K.A. El-Farahaty, M.A. El-Bakarya, E.Z. Omar, M. Agour, Characterization of axially tilted fibres utilizing a single-shot interference pattern. Opt. Lasers Eng. 91, 144–150 (2017)

    Google Scholar 

  60. M. Agour, K. El-Farahaty, E. Seisa, E. Omar, T. Sokkar, Single-shot digital holography for fast measuring optical properties of fibers. Appl. Opt. 54(28), 188–195 (2015)

    ADS  Google Scholar 

  61. M. Wang, L. Ma, D. Li, J. Zhong, Subfringe integration method for automatic analysis of moire deflection tomography. Opt. Eng. 39(10), 2726–2733 (2000)

    ADS  Google Scholar 

  62. N. Barakat, A.A. Hamza, Interferometry of fibrous materials (Adam Hilger, Bristol, 1990)

    Google Scholar 

  63. H.M. El-Dessouky, Birefringent characterization of necking phenomenon along cold-drawn polypropylene fibers. I. Offline drawing. J. Appl. Polym. Sci. 105, 757–764 (2007)

    Google Scholar 

Download references

Acknowledgment

This work was supported by the Deanship of Scientific Research at Prince Sattam bin Abdulaziz University, Saudi Arabia, under Grant No. 7353/01/2017. The author extends his thanks and appreciation to Prof. A.A. Hamza and Prof. T.Z.N. Sokkar for their continuous encouragement and valuable effort.

Author information

Authors and Affiliations

  1. Plasma Technology and Material Science Unit, Physics Department, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj, 11942, Saudi Arabia

    M. A. El-Morsy

  2. Physics Department, Faculty of Science, University of Damietta, New Damietta, 34517, Egypt

    M. A. El-Morsy

Authors
  1. M. A. El-Morsy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. A. El-Morsy.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

El-Morsy, M.A. A novel algorithm based on sub-fringe integration method for direct two-dimensional unwrapping phase reconstruction from the intensity of one-shot two-beam interference fringes. Appl. Phys. B 125, 216 (2019). https://doi.org/10.1007/s00340-019-7330-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00340-019-7330-9

Search

Navigation