Abstract

We present a real-time complex amplitude reconstruction method for determining the beam propagation ratio M2 of laser beams based on the transport of intensity equation (TIE). In this work, a synchronous acquisition system consisting of two identical CCDs is established. Once two beam intensity images at different cross-section positions along the optical axis are captured simultaneously by the system, the complex amplitude of the laser beam can be rapidly reconstructed using TIE algorithm. Then the beam intensity distribution at any section position along its propagation direction can be obtained by using angular spectrum (AS) theory. The beam quality M2 factor is therefore calculated utilizing the second-order moments and hyperbola fitting methods, which conform to the ISO standard. The suitability of this method is verified by the numerical analysis and experiments with the He-Ne and high-power fiber laser sources, respectively. The experimental technique is simple and fast, which allows to investigate laser beams under conditions inaccessible to other methods.

© 2017 Optical Society of America

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References

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  1. K. C. Jorge, R. Riva, N. A. S. Rodrigues, J. M. S. Sakamoto, and M. G. Destro, “Scattered light imaging method (SLIM) for characterization of arbitrary laser beam intensity profiles,” Appl. Opt. 53(20), 4555–4564 (2014).
    [Crossref] [PubMed]
  2. F. Chen, J. Ma, R. Zhu, Q. Yuan, W. Zhou, J. Su, J. Xu, and S. Pan, “Coupling efficiency model for spectral beam combining of high-power fiber lasers calculated from spectrum,” Appl. Opt. 56(10), 2574–2579 (2017).
    [Crossref] [PubMed]
  3. E. Perevezentsev, A. Poteomkin, and E. Khazanov, “Comparison of phase-aberrated laser beam quality criteria,” Appl. Opt. 46(5), 774–784 (2007).
    [Crossref] [PubMed]
  4. A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).
    [Crossref]
  5. A. E. Siegman, “How to (maybe) measure laser beam quality,” in DPSS (Diode Pumped Solid State) Lasers: Applications and Issues, M. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics (Optical Society of America, 1998), paper MQ1.
  6. C. Borgentun, J. Bengtsson, and A. Larsson, “Full characterization of a high-power semiconductor disk laser beam with simultaneous capture of optimally sized focus and farfield,” Appl. Opt. 50(12), 1640–1649 (2011).
    [Crossref] [PubMed]
  7. R. D. Niederriter, J. T. Gopinath, and M. E. Siemens, “Measurement of the M2 beam propagation factor using a focus-tunable liquid lens,” Appl. Opt. 52(8), 1591–1598 (2013).
    [Crossref] [PubMed]
  8. International Organization for Standardization, ISO 11146–1/2/3 Test methods for laser beam widths, divergence angles and beam propagation ratios – Part 1: Stigmatic and simple astigmatic beams / Part 2: General astigmatic beams / Part 3: Intrinsic and geometrical laser beam classification, propagation and details of test methods (ISO, Geneva, 2005).
  9. Ophir-Spiricon, Inc., “M-200 Operator’s Manual” (2007).
  10. J. Strohaber, G. Kaya, N. Kaya, N. Hart, A. A. Kolomenskii, G. G. Paulus, and H. A. Schuessler, “In situ tomography of femtosecond optical beams with a holographic knife-edge,” Opt. Express 19(15), 14321–14334 (2011).
    [Crossref] [PubMed]
  11. O. A. Schmidt, C. Schulze, D. Flamm, R. Brüning, T. Kaiser, S. Schröter, and M. Duparré, “Real-time determination of laser beam quality by modal decomposition,” Opt. Express 19(7), 6741–6748 (2011).
    [Crossref] [PubMed]
  12. B. Schäfer, M. Lübbecke, and K. Mann, “Hartmann-Shack wave front measurements for real time determination of laser beam propagation parameters,” Rev. Sci. Instrum. 77(5), 053103 (2006).
    [Crossref]
  13. J. V. Sheldakova, A. V. Kudryashov, V. Y. Zavalova, and T. Y. Cherezova, “Beam quality measurements with Shack-Hartmann wavefront sensor and M2-sensor: comparison of two methods,” Proc. SPIE 6452, 645207 (2007).
    [Crossref]
  14. V. Akondi, A. R. Jewel, and B. Vohnsen, “Digital phase-shifting point diffraction interferometer,” Opt. Lett. 39(6), 1641–1644 (2014).
    [Crossref] [PubMed]
  15. V. Akondi and B. Vohnsen, “Myopic aberrations: impact of centroiding noise in Hartmann Shack wavefront sensing,” Ophthalmic Physiol. Opt. 33(4), 434–443 (2013).
    [Crossref] [PubMed]
  16. R. Cortés, R. Villag’omez, V. Coello, and R. L’opez, “Laser beam quality factor (M2) measured by distorted fresnel zone plates,” Rev. Mex. Fis. 54(4), 279–283 (2008).
  17. M. Scaggs and G. Haas, “Real time laser beam analysis system for high power lasers,” Proc. SPIE 7913, 791306 (2011).
    [Crossref]
  18. T. Kaiser, D. Flamm, S. Schröter, and M. Duparré, “Complete modal decomposition for optical fibers using CGH-based correlation filters,” Opt. Express 17(11), 9347–9356 (2009).
    [Crossref] [PubMed]
  19. D. Flamm, C. Schulze, R. Brüning, O. A. Schmidt, T. Kaiser, S. Schröter, and M. Duparré, “Fast M2 measurement for fiber beams based on modal analysis,” Appl. Opt. 51(7), 987–993 (2012).
    [Crossref] [PubMed]
  20. C. Schulze, D. Flamm, M. Duparré, and A. Forbes, “Beam-quality measurements using a spatial light modulator,” Opt. Lett. 37(22), 4687–4689 (2012).
    [Crossref] [PubMed]
  21. W. Shi, Z. Zhang, X. He, Q. Liu, Z. Zhang, Y. Ma, and Y. Jin, “Measuring laser beam quality by use of phase retrieval and Fraunhofer diffraction,” Proc. SPIE 9255, 92552Q (2015).
    [Crossref]
  22. Y. Du, Y. Fu, and L. Zheng, “Complex amplitude reconstruction for dynamic beam quality M2 factor measurement with self-referencing interferometer wavefront sensor,” Appl. Opt. 55(36), 10180–10186 (2016).
    [Crossref] [PubMed]
  23. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73(11), 1434–1441 (1983).
    [Crossref]
  24. M. Silva-López, J. M. Rico-García, and J. Alda, “Measurement limitations in knife-edge tomographic phase retrieval of focused IR laser beams,” Opt. Express 20(21), 23875–23886 (2012).
    [Crossref] [PubMed]
  25. Goodman J. W., Introduction to Fourier Optics (McGraW-Hill Companies, 2005).
  26. C. Zuo, Q. Chen, and A. Asundi, “Boundary-artifact-free phase retrieval with the transport of intensity equation: fast solution with use of discrete cosine transform,” Opt. Express 22(8), 9220–9244 (2014).
    [Crossref] [PubMed]
  27. X. Tian, W. Yu, X. Meng, A. Sun, L. Xue, C. Liu, and S. Wang, “Real-time quantitative phase imaging based on transport of intensity equation with dual simultaneously recorded field of view,” Opt. Lett. 41(7), 1427–1430 (2016).
    [Crossref] [PubMed]
  28. 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] [PubMed]
  29. Z. Jingshan, R. A. Claus, J. Dauwels, L. Tian, and L. Waller, “Transport of Intensity phase imaging by intensity spectrum fitting of exponentially spaced defocus planes,” Opt. Express 22(9), 10661–10674 (2014).
    [Crossref] [PubMed]
  30. K. Komuro and T. Nomura, “Quantitative phase imaging using transport of intensity equation with multiple bandpass filters,” Appl. Opt. 55(19), 5180–5186 (2016).
    [Crossref] [PubMed]
  31. M. Reed Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73(11), 1434–1441 (1983).
    [Crossref]
  32. D. A. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer, 2001), 224.
  33. 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] [PubMed]

2017 (1)

2016 (3)

2015 (1)

W. Shi, Z. Zhang, X. He, Q. Liu, Z. Zhang, Y. Ma, and Y. Jin, “Measuring laser beam quality by use of phase retrieval and Fraunhofer diffraction,” Proc. SPIE 9255, 92552Q (2015).
[Crossref]

2014 (5)

2013 (3)

2012 (3)

2011 (4)

2009 (1)

2008 (1)

R. Cortés, R. Villag’omez, V. Coello, and R. L’opez, “Laser beam quality factor (M2) measured by distorted fresnel zone plates,” Rev. Mex. Fis. 54(4), 279–283 (2008).

2007 (2)

J. V. Sheldakova, A. V. Kudryashov, V. Y. Zavalova, and T. Y. Cherezova, “Beam quality measurements with Shack-Hartmann wavefront sensor and M2-sensor: comparison of two methods,” Proc. SPIE 6452, 645207 (2007).
[Crossref]

E. Perevezentsev, A. Poteomkin, and E. Khazanov, “Comparison of phase-aberrated laser beam quality criteria,” Appl. Opt. 46(5), 774–784 (2007).
[Crossref] [PubMed]

2006 (1)

B. Schäfer, M. Lübbecke, and K. Mann, “Hartmann-Shack wave front measurements for real time determination of laser beam propagation parameters,” Rev. Sci. Instrum. 77(5), 053103 (2006).
[Crossref]

1990 (1)

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).
[Crossref]

1983 (2)

Akondi, V.

V. Akondi, A. R. Jewel, and B. Vohnsen, “Digital phase-shifting point diffraction interferometer,” Opt. Lett. 39(6), 1641–1644 (2014).
[Crossref] [PubMed]

V. Akondi and B. Vohnsen, “Myopic aberrations: impact of centroiding noise in Hartmann Shack wavefront sensing,” Ophthalmic Physiol. Opt. 33(4), 434–443 (2013).
[Crossref] [PubMed]

Alda, J.

Asundi, A.

Barbastathis, G.

Bengtsson, J.

Borgentun, C.

Brüning, R.

Chen, F.

Chen, Q.

Cherezova, T. Y.

J. V. Sheldakova, A. V. Kudryashov, V. Y. Zavalova, and T. Y. Cherezova, “Beam quality measurements with Shack-Hartmann wavefront sensor and M2-sensor: comparison of two methods,” Proc. SPIE 6452, 645207 (2007).
[Crossref]

Claus, R. A.

Coello, V.

R. Cortés, R. Villag’omez, V. Coello, and R. L’opez, “Laser beam quality factor (M2) measured by distorted fresnel zone plates,” Rev. Mex. Fis. 54(4), 279–283 (2008).

Cortés, R.

R. Cortés, R. Villag’omez, V. Coello, and R. L’opez, “Laser beam quality factor (M2) measured by distorted fresnel zone plates,” Rev. Mex. Fis. 54(4), 279–283 (2008).

Dauwels, J.

Destro, M. G.

Du, Y.

Duparré, M.

Flamm, D.

Forbes, A.

Fu, Y.

Gopinath, J. T.

Haas, G.

M. Scaggs and G. Haas, “Real time laser beam analysis system for high power lasers,” Proc. SPIE 7913, 791306 (2011).
[Crossref]

Hart, N.

He, X.

W. Shi, Z. Zhang, X. He, Q. Liu, Z. Zhang, Y. Ma, and Y. Jin, “Measuring laser beam quality by use of phase retrieval and Fraunhofer diffraction,” Proc. SPIE 9255, 92552Q (2015).
[Crossref]

Jewel, A. R.

Jin, Y.

W. Shi, Z. Zhang, X. He, Q. Liu, Z. Zhang, Y. Ma, and Y. Jin, “Measuring laser beam quality by use of phase retrieval and Fraunhofer diffraction,” Proc. SPIE 9255, 92552Q (2015).
[Crossref]

Jingshan, Z.

Jorge, K. C.

Kaiser, T.

Kaya, G.

Kaya, N.

Khazanov, E.

Kolomenskii, A. A.

Komuro, K.

Kudryashov, A. V.

J. V. Sheldakova, A. V. Kudryashov, V. Y. Zavalova, and T. Y. Cherezova, “Beam quality measurements with Shack-Hartmann wavefront sensor and M2-sensor: comparison of two methods,” Proc. SPIE 6452, 645207 (2007).
[Crossref]

L’opez, R.

R. Cortés, R. Villag’omez, V. Coello, and R. L’opez, “Laser beam quality factor (M2) measured by distorted fresnel zone plates,” Rev. Mex. Fis. 54(4), 279–283 (2008).

Larsson, A.

Liu, C.

Liu, Q.

W. Shi, Z. Zhang, X. He, Q. Liu, Z. Zhang, Y. Ma, and Y. Jin, “Measuring laser beam quality by use of phase retrieval and Fraunhofer diffraction,” Proc. SPIE 9255, 92552Q (2015).
[Crossref]

Lübbecke, M.

B. Schäfer, M. Lübbecke, and K. Mann, “Hartmann-Shack wave front measurements for real time determination of laser beam propagation parameters,” Rev. Sci. Instrum. 77(5), 053103 (2006).
[Crossref]

Ma, J.

Ma, Y.

W. Shi, Z. Zhang, X. He, Q. Liu, Z. Zhang, Y. Ma, and Y. Jin, “Measuring laser beam quality by use of phase retrieval and Fraunhofer diffraction,” Proc. SPIE 9255, 92552Q (2015).
[Crossref]

Mann, K.

B. Schäfer, M. Lübbecke, and K. Mann, “Hartmann-Shack wave front measurements for real time determination of laser beam propagation parameters,” Rev. Sci. Instrum. 77(5), 053103 (2006).
[Crossref]

Meng, X.

Niederriter, R. D.

Nomura, T.

Pan, S.

Paulus, G. G.

Perevezentsev, E.

Poteomkin, A.

Reed Teague, M.

Rico-García, J. M.

Riva, R.

Rodrigues, N. A. S.

Sakamoto, J. M. S.

Scaggs, M.

M. Scaggs and G. Haas, “Real time laser beam analysis system for high power lasers,” Proc. SPIE 7913, 791306 (2011).
[Crossref]

Schäfer, B.

B. Schäfer, M. Lübbecke, and K. Mann, “Hartmann-Shack wave front measurements for real time determination of laser beam propagation parameters,” Rev. Sci. Instrum. 77(5), 053103 (2006).
[Crossref]

Schmidt, O. A.

Schröter, S.

Schuessler, H. A.

Schulze, C.

Sheldakova, J. V.

J. V. Sheldakova, A. V. Kudryashov, V. Y. Zavalova, and T. Y. Cherezova, “Beam quality measurements with Shack-Hartmann wavefront sensor and M2-sensor: comparison of two methods,” Proc. SPIE 6452, 645207 (2007).
[Crossref]

Shi, W.

W. Shi, Z. Zhang, X. He, Q. Liu, Z. Zhang, Y. Ma, and Y. Jin, “Measuring laser beam quality by use of phase retrieval and Fraunhofer diffraction,” Proc. SPIE 9255, 92552Q (2015).
[Crossref]

Siegman, A. E.

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).
[Crossref]

Siemens, M. E.

Silva-López, M.

Strohaber, J.

Su, J.

Sun, A.

Teague, M. R.

Tian, L.

Tian, X.

Villag’omez, R.

R. Cortés, R. Villag’omez, V. Coello, and R. L’opez, “Laser beam quality factor (M2) measured by distorted fresnel zone plates,” Rev. Mex. Fis. 54(4), 279–283 (2008).

Vohnsen, B.

V. Akondi, A. R. Jewel, and B. Vohnsen, “Digital phase-shifting point diffraction interferometer,” Opt. Lett. 39(6), 1641–1644 (2014).
[Crossref] [PubMed]

V. Akondi and B. Vohnsen, “Myopic aberrations: impact of centroiding noise in Hartmann Shack wavefront sensing,” Ophthalmic Physiol. Opt. 33(4), 434–443 (2013).
[Crossref] [PubMed]

Waller, L.

Wang, S.

Xu, J.

Xue, L.

Yu, W.

Yu, Y.

Yuan, Q.

Zavalova, V. Y.

J. V. Sheldakova, A. V. Kudryashov, V. Y. Zavalova, and T. Y. Cherezova, “Beam quality measurements with Shack-Hartmann wavefront sensor and M2-sensor: comparison of two methods,” Proc. SPIE 6452, 645207 (2007).
[Crossref]

Zhang, Z.

W. Shi, Z. Zhang, X. He, Q. Liu, Z. Zhang, Y. Ma, and Y. Jin, “Measuring laser beam quality by use of phase retrieval and Fraunhofer diffraction,” Proc. SPIE 9255, 92552Q (2015).
[Crossref]

W. Shi, Z. Zhang, X. He, Q. Liu, Z. Zhang, Y. Ma, and Y. Jin, “Measuring laser beam quality by use of phase retrieval and Fraunhofer diffraction,” Proc. SPIE 9255, 92552Q (2015).
[Crossref]

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] [PubMed]

Zheng, L.

Zhou, W.

Zhu, R.

Zhu, Y.

Zuo, C.

Appl. Opt. (8)

E. Perevezentsev, A. Poteomkin, and E. Khazanov, “Comparison of phase-aberrated laser beam quality criteria,” Appl. Opt. 46(5), 774–784 (2007).
[Crossref] [PubMed]

C. Borgentun, J. Bengtsson, and A. Larsson, “Full characterization of a high-power semiconductor disk laser beam with simultaneous capture of optimally sized focus and farfield,” Appl. Opt. 50(12), 1640–1649 (2011).
[Crossref] [PubMed]

D. Flamm, C. Schulze, R. Brüning, O. A. Schmidt, T. Kaiser, S. Schröter, and M. Duparré, “Fast M2 measurement for fiber beams based on modal analysis,” Appl. Opt. 51(7), 987–993 (2012).
[Crossref] [PubMed]

R. D. Niederriter, J. T. Gopinath, and M. E. Siemens, “Measurement of the M2 beam propagation factor using a focus-tunable liquid lens,” Appl. Opt. 52(8), 1591–1598 (2013).
[Crossref] [PubMed]

K. C. Jorge, R. Riva, N. A. S. Rodrigues, J. M. S. Sakamoto, and M. G. Destro, “Scattered light imaging method (SLIM) for characterization of arbitrary laser beam intensity profiles,” Appl. Opt. 53(20), 4555–4564 (2014).
[Crossref] [PubMed]

K. Komuro and T. Nomura, “Quantitative phase imaging using transport of intensity equation with multiple bandpass filters,” Appl. Opt. 55(19), 5180–5186 (2016).
[Crossref] [PubMed]

Y. Du, Y. Fu, and L. Zheng, “Complex amplitude reconstruction for dynamic beam quality M2 factor measurement with self-referencing interferometer wavefront sensor,” Appl. Opt. 55(36), 10180–10186 (2016).
[Crossref] [PubMed]

F. Chen, J. Ma, R. Zhu, Q. Yuan, W. Zhou, J. Su, J. Xu, and S. Pan, “Coupling efficiency model for spectral beam combining of high-power fiber lasers calculated from spectrum,” Appl. Opt. 56(10), 2574–2579 (2017).
[Crossref] [PubMed]

J. Opt. Soc. Am. (2)

Ophthalmic Physiol. Opt. (1)

V. Akondi and B. Vohnsen, “Myopic aberrations: impact of centroiding noise in Hartmann Shack wavefront sensing,” Ophthalmic Physiol. Opt. 33(4), 434–443 (2013).
[Crossref] [PubMed]

Opt. Express (8)

T. Kaiser, D. Flamm, S. Schröter, and M. Duparré, “Complete modal decomposition for optical fibers using CGH-based correlation filters,” Opt. Express 17(11), 9347–9356 (2009).
[Crossref] [PubMed]

O. A. Schmidt, C. Schulze, D. Flamm, R. Brüning, T. Kaiser, S. Schröter, and M. Duparré, “Real-time determination of laser beam quality by modal decomposition,” Opt. Express 19(7), 6741–6748 (2011).
[Crossref] [PubMed]

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] [PubMed]

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] [PubMed]

C. Zuo, Q. Chen, and A. Asundi, “Boundary-artifact-free phase retrieval with the transport of intensity equation: fast solution with use of discrete cosine transform,” Opt. Express 22(8), 9220–9244 (2014).
[Crossref] [PubMed]

Z. Jingshan, R. A. Claus, J. Dauwels, L. Tian, and L. Waller, “Transport of Intensity phase imaging by intensity spectrum fitting of exponentially spaced defocus planes,” Opt. Express 22(9), 10661–10674 (2014).
[Crossref] [PubMed]

M. Silva-López, J. M. Rico-García, and J. Alda, “Measurement limitations in knife-edge tomographic phase retrieval of focused IR laser beams,” Opt. Express 20(21), 23875–23886 (2012).
[Crossref] [PubMed]

J. Strohaber, G. Kaya, N. Kaya, N. Hart, A. A. Kolomenskii, G. G. Paulus, and H. A. Schuessler, “In situ tomography of femtosecond optical beams with a holographic knife-edge,” Opt. Express 19(15), 14321–14334 (2011).
[Crossref] [PubMed]

Opt. Lett. (3)

Proc. SPIE (4)

M. Scaggs and G. Haas, “Real time laser beam analysis system for high power lasers,” Proc. SPIE 7913, 791306 (2011).
[Crossref]

W. Shi, Z. Zhang, X. He, Q. Liu, Z. Zhang, Y. Ma, and Y. Jin, “Measuring laser beam quality by use of phase retrieval and Fraunhofer diffraction,” Proc. SPIE 9255, 92552Q (2015).
[Crossref]

J. V. Sheldakova, A. V. Kudryashov, V. Y. Zavalova, and T. Y. Cherezova, “Beam quality measurements with Shack-Hartmann wavefront sensor and M2-sensor: comparison of two methods,” Proc. SPIE 6452, 645207 (2007).
[Crossref]

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).
[Crossref]

Rev. Mex. Fis. (1)

R. Cortés, R. Villag’omez, V. Coello, and R. L’opez, “Laser beam quality factor (M2) measured by distorted fresnel zone plates,” Rev. Mex. Fis. 54(4), 279–283 (2008).

Rev. Sci. Instrum. (1)

B. Schäfer, M. Lübbecke, and K. Mann, “Hartmann-Shack wave front measurements for real time determination of laser beam propagation parameters,” Rev. Sci. Instrum. 77(5), 053103 (2006).
[Crossref]

Other (5)

A. E. Siegman, “How to (maybe) measure laser beam quality,” in DPSS (Diode Pumped Solid State) Lasers: Applications and Issues, M. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics (Optical Society of America, 1998), paper MQ1.

International Organization for Standardization, ISO 11146–1/2/3 Test methods for laser beam widths, divergence angles and beam propagation ratios – Part 1: Stigmatic and simple astigmatic beams / Part 2: General astigmatic beams / Part 3: Intrinsic and geometrical laser beam classification, propagation and details of test methods (ISO, Geneva, 2005).

Ophir-Spiricon, Inc., “M-200 Operator’s Manual” (2007).

Goodman J. W., Introduction to Fourier Optics (McGraW-Hill Companies, 2005).

D. A. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer, 2001), 224.

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Figures (13)

Fig. 1
Fig. 1

The schematic diagram of phase retrieval and M2 factor calculation. (a)(b) the axial intensity images at two different longitudinal positions, (c) the phase retrieval by TIE, (d) the reconstructed intensity distribution at any selected plane, (e) performing a hyperbolic fit to the beam widths and calculating the M2x,y.

Fig. 2
Fig. 2

The schematic diagram of the field tracing process of Gaussian beam.

Fig. 3
Fig. 3

The reconstructed phase distributions and phase errors. (a) the true phase, (b)-(e) the reconstructed phase distributions when Δz=0.1mm,Δz=10mm,Δz=100mm,Δz=500mm, (f)-(i) the phase errors when Δz=0.1mm,Δz=10mm,Δz=100mm,Δz=500mm.

Fig. 4
Fig. 4

The RMSE values of phase errors with different axial acquisition distances when adding white Gaussian noise with different SNRs to the intensity images.

Fig. 5
Fig. 5

The experimental setup for He-Ne laser.

Fig. 6
Fig. 6

The phase retrieval process of the He-Ne laser with different axial acquisition distances.

Fig. 7
Fig. 7

The calculated M2 values of He-Ne laser with different axial acquisition distances.

Fig. 8
Fig. 8

The measurement errors of M2 values of the He-Ne laser.

Fig. 9
Fig. 9

The experimental setup for fiber laser.

Fig. 10
Fig. 10

The phase retrieval process of the fiber laser under the different laser powers.

Fig. 11
Fig. 11

The comparison results of M2 values of the fiber laser determined by proposed method and beam propagation analyzer, respectively. (a) M2x, (b) M2y.

Fig. 12
Fig. 12

The measurement errors of M2 values of the fiber laser.

Fig. 13
Fig. 13

The comparison results of M2 values of the fiber laser determined by proposed method before and after moving CCDs, respectively. (a) M2x, (b) M2y.

Tables (3)

Tables Icon

Table 1 The RMSE values of the phase errors

Tables Icon

Table 2 Specification of He-Ne laser

Tables Icon

Table 3 Specification of the CCD

Equations (10)

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

k I( r ) z =[ I( r )ϕ( r ) ]
k I( r ) z =I( r )ϕ( r )+I( r ) 2 ϕ( r )
I ϕ n | Ω =g
ψ=Iϕ
ϕ=k DCT 2 { DCT [ I 1 DCT DCT 2 ( I/ z ) ] }
U( x,y,z )= F 1 { G 0 ( f x , f y )exp[ j 2π λ z 1 ( λ f x ) 2 ( λ f y ) 2 ] }
ω x 2 ( z )= A x + B x z+ C x z 2
ω y 2 ( z )= A y + B y z+ C y z 2
M x 2 = π λ A x C x B x 2 4
M y 2 = π λ A y C y B y 2 4

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