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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 18 — Aug. 27, 2012
  • pp: 19714–19725
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Wavefront reconstruction by modal decomposition

Christian Schulze, Darryl Naidoo, Daniel Flamm, Oliver A. Schmidt, Andrew Forbes, and Michael Duparré  »View Author Affiliations


Optics Express, Vol. 20, Issue 18, pp. 19714-19725 (2012)
http://dx.doi.org/10.1364/OE.20.019714


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Abstract

We propose a new method to determine the wavefront of a laser beam based on modal decomposition by computer-generated holograms. The hologram is encoded with a transmission function suitable for measuring the amplitudes and phases of the modes in real-time. This yields the complete information about the optical field, from which the Poynting vector and the wavefront are deduced. Two different wavefront reconstruction options are outlined: reconstruction from the phase for scalar beams, and reconstruction from the Poynting vector for inhomogeneously polarized beams. Results are compared to Shack-Hartmann measurements that serve as a reference and are shown to reproduce the wavefront and phase with very high fidelity.

© 2012 OSA

1. Introduction

Wavefront reconstruction of optical fields has become an important task in many domains of optics: astronomy, for example, is nowadays inconceivable without wavefront measurements. In combination with deformable mirrors for correction of distorted wavefronts, wavefront reconstruction has become a fundamental part of adaptive optics systems that enable high quality terrestrial observations [1

1. F. Roddier, M. Séchaud, G. Rousset, P.-Y. Madec, M. Northcott, J.-L. Beuzit, F. Rigaut, J. Beckers, D. Sandler, P. Léna, and O. Lai, Adaptive Optics in Astronomy (Cambridge, 1999). [CrossRef]

]. Correction and control of wavefronts is no less essential in the fields of microscopy, such as 2-photon microscopy [2

2. M. A. A. Neil, R. Jukaitis, M. J. Booth, T. Wilson, T. Tanaka, and S. Kawata, “Adaptive aberration correction in a two-photon microscope,” J. Microsc. 200, 105–108 (2000). [CrossRef] [PubMed]

, 3

3. M. Rueckel, J. A. Mack-Bucher, and W. Denk, “Adaptive wavefront correction in two-photon microscopy using coherence-gated wavefront sensing,” Proc. Natl. Acad. Sci. 103, 17137–17142 (2006). [CrossRef] [PubMed]

] and confocal microscopy [4

4. M. Booth, M. Neil, and T. Wilson, “Aberration correction for confocal imaging in refractive-index-mismatched media,” J. Microsc. 192, 90–98 (1998). [CrossRef]

], and ophthalmology, such as optical coherence tomography [5

5. B. Hermann, E. J. Fernández, A. Unterhuber, H. Sattmann, A. F. Fercher, W. Drexler, P. M. Prieto, and P. Artal, “Adaptive-optics ultrahigh-resolution optical coherence tomography,” Opt. Lett. 29, 2142–2144 (2004). [CrossRef] [PubMed]

] and scanning laser ophthalmoscopy [6

6. A. Roorda, F. Romero-Borja, I. William Donnelly, H. Queener, T. Hebert, and M. Campbell, “Adaptive optics scanning laser ophthalmoscopy,” Opt. Express 10, 405–412 (2002). [PubMed]

]. Another field of application for wavefront control is laser material processing, in particular processes that require a high beam quality, such as laser cutting and drilling, often performed by fiber lasers [7

7. R. Paschotta, Encyclopedia of Laser Physics and Technology (Wiley, 2008).

, 8

8. M. Paurisse, M. Hanna, F. Druon, and P. Georges, “Wavefront control of a multicore ytterbium-doped pulse fiber amplifier by digital holography,” Opt. Lett. 35, 1428–1430 (2010). [CrossRef] [PubMed]

]. Concerning the wavefront measurement itself there exists a variety of different sensor types, including laser ray tracing [9

9. R. Navarro and E. Moreno-Barriuso, “Laser ray-tracing method for optical testing,” Opt. Lett. 24, 951–953 (1999). [CrossRef]

], pyramid sensors [10

10. S. R. Chamot, C. Dainty, and S. Esposito, “Adaptive optics for ophthalmic applications using a pyramid wavefront sensor,” Opt. Express 14, 518–526 (2006). [CrossRef] [PubMed]

], interferometric approaches [11

11. M. P. Rimmer and J. C. Wyant, “Evaluation of large aberrations using a lateral-shear interferometer having variable shear,” Appl. Opt. 14, 142–150 (1975). [PubMed]

13

13. S. Velghe, J. Primot, N. Guérineau, M. Cohen, and B. Wattellier, “Wave-front reconstruction from multidirectional phasederivatives generated by multilateral shearing interferometers,” Opt. Lett. 30, 245–247 (2005). [CrossRef] [PubMed]

] and the widely used Shack-Hartmann sensor (SHS) [14

14. R. G. Lane and M. Tallon, “Wave-front reconstruction using a Shack-Hartmann sensor,” Appl. Opt. 31, 6902–6908 (1992). [CrossRef] [PubMed]

]. Recent advances have seen the use of computer-generated holograms (CGHs) to encode certain aberrations to determine the Zernike coefficients [15

15. L. Changhai, X. Fengjie, H. Shengyang, and J. Zongfu, “Performance analysis of multiplexed phase computer-generated hologram for modal wavefront sensing,” Appl. Opt. 50, 1631–1639 (2011). [CrossRef] [PubMed]

], the use of ring-shaped phase masks for decomposition in azimuthal modes [16

16. I. A. Litvin, A. Dudley, F. S. Roux, and A. Forbes, “Azimuthal decomposition with digital holograms,” Opt. Express 20, 10996–11004 (2012). [CrossRef] [PubMed]

], and non-linear approaches for high intensity light pulses [17

17. R. Borrego-Varillas, C. Romero, J. R. V. de Aldana, J. M. Bueno, and L. Roso, “Wavefront retrieval of amplified femtosecond beams by second-harmonic generation,” Opt. Express 19, 22851–22862 (2011). [CrossRef] [PubMed]

].

2. Modal decomposition

3. Wavefront reconstruction

4. Measurement setup

Fig. 1 Scheme of the measurement setup: WF aberrated wavefront to be relay imaged onto the computer-generated hologram (CGH) and the Shack-Hartmann wavefront sensor (SHS), MO microscope objectives, QWP quarter-wave, P polarizer, L1,2,3 lenses, FL Fourier lens, BS beam splitter, CCD1,2 CCD cameras, M mirror.

5. Calibration of the setup

Fig. 2 Fundamental mode illumination of CGH and wavefront sensor for calibration. (a) Intensity measured with the wavefront sensor. (b) Intensity measured with the CCD camera. (c) Wavefront measured with the wavefront sensor (scale in μm). (d) Modal power spectrum (insets depict respective mode intensities).

6. Scalar beams

After aligning the hologram and the wavefront sensor with the fundamental mode, the position of the fiber seed beam with respect to the fiber front face was shifted to excite some higher order modes in the fiber. The experiments were done with a step-index fiber with core diameter of 7.7μm and a numerical aperture of 0.12. Hence, the fiber guides three modes at 1064 nm. As the CGH we used a solid amplitude-only diffractive device, fabricated via laser lithography, for all fiber experiments [18

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

]. To perform scalar experiments a polarizer was placed between the fiber end face and the first beam splitter that divides the beam for analysis with CGH and wavefront sensor. This procedure enables the reconstruction of the wavefront from the phase as described by Eq. 10, which is beneficial because no measurement of Stokes parameters is necessary. In this way, we reconstructed the wavefront with up to 30 Hz using the CGH, only limited by the maximum camera frame rate.

Fig. 3 Wavefront reconstruction for a higher order mode scalar beam. (a) Intensity measured with the wavefront sensor (SHS). (b) Reconstructed intensity (inset depicts directly measured intensity with CCD1). (c) Modal power spectrum (insets depict mode intensities). (d) Wavefront measured with the Shack-Hartmann sensor (scale in μm). (e) Wavefront determined from the phase reconstruction according to Eq. (10) (scale in μm). (f) Wavefront from the minimization according to Eq. (9) (scale in μm).

7. Vector beams

Fig. 4 Wavefront reconstruction for a higher order mode non-scalar beam. (a) Intensity measured with the wavefront sensor (SHS). (b) Reconstructed intensity (inset depicts directly measured intensity with CCD1). (c) Modal power spectrum. (d) Wavefront measured with the Shack-Hartmann sensor (scale in μm). (e) Phase distribution is not well defined. (f) Wavefront from the minimization according to Eq. (9).

8. Beams with phase singularities

It is pertinent to apply the technique to singular beams: beams containing optical vortices. An interesting example of a such a beam is a scalar donut beam, because of its exceptional phase distribution. Such a donut beam is formed by a coherent superposition of the two higher order fiber modes (cf. insets of Fig. 5(c)), each with nearly equal content and a relative phase difference of π/2. The fundamental mode is absent in this special case. The described superposition forms a phase singularity in the center of the beam with a corresponding zero intensity point.

Fig. 5 Wavefront reconstruction for a scalar donut beam. (a) Intensity measured with the wavefront sensor (SHS). (b) Reconstructed intensity (inset depicts directly measured intensity with CCD1). (c) Modal power spectrum. (d) Wavefront measured with the Shack-Hartmann sensor (scale in μm). (e) Wavefront determined from the phase reconstruction (scale in μm). (f) Wavefront from the minimization according to Eq. (9).

9. Decomposing beams including extrinsic aberrations

Fig. 6 Wavefront reconstruction of a fundamental Gaussian beam with extrinsically added wavefront curvature using a lens of focal length f = 1000 mm. (a) Modal power spectrum (insets depict Laguerre Gaussian modes LGp0 used for decomposition). (b) Inter-modal phase differences. (c) Comparison of reconstructed (CGH) and theoretically expected (Sim) wavefront (cross section through center). The inset in (c) depicts the measured two-dimensional wavefront (same scale as (c)). See Media 1 for the decay of higher order mode signal in the diffraction pattern of the hologram, and Media 2 for the corresponding hologram phase pattern, both for a lens of f = 500 mm.
Fig. 7 (a) Modal decomposition without physical lens and decomposition into modes without curvature. (b) Modal decomposition into modes that incorporate the wavefront curvature measured in Fig. 6 for a physical lens f = 1000 mm. The insets depict the corresponding mode intensities.

10. Discussion of techniques

11. Conclusion

Acknowledgments

The authors would like to thank Mr. B. Eppich for the productive discussions and valueable hints and ideas.

References and links

1.

F. Roddier, M. Séchaud, G. Rousset, P.-Y. Madec, M. Northcott, J.-L. Beuzit, F. Rigaut, J. Beckers, D. Sandler, P. Léna, and O. Lai, Adaptive Optics in Astronomy (Cambridge, 1999). [CrossRef]

2.

M. A. A. Neil, R. Jukaitis, M. J. Booth, T. Wilson, T. Tanaka, and S. Kawata, “Adaptive aberration correction in a two-photon microscope,” J. Microsc. 200, 105–108 (2000). [CrossRef] [PubMed]

3.

M. Rueckel, J. A. Mack-Bucher, and W. Denk, “Adaptive wavefront correction in two-photon microscopy using coherence-gated wavefront sensing,” Proc. Natl. Acad. Sci. 103, 17137–17142 (2006). [CrossRef] [PubMed]

4.

M. Booth, M. Neil, and T. Wilson, “Aberration correction for confocal imaging in refractive-index-mismatched media,” J. Microsc. 192, 90–98 (1998). [CrossRef]

5.

B. Hermann, E. J. Fernández, A. Unterhuber, H. Sattmann, A. F. Fercher, W. Drexler, P. M. Prieto, and P. Artal, “Adaptive-optics ultrahigh-resolution optical coherence tomography,” Opt. Lett. 29, 2142–2144 (2004). [CrossRef] [PubMed]

6.

A. Roorda, F. Romero-Borja, I. William Donnelly, H. Queener, T. Hebert, and M. Campbell, “Adaptive optics scanning laser ophthalmoscopy,” Opt. Express 10, 405–412 (2002). [PubMed]

7.

R. Paschotta, Encyclopedia of Laser Physics and Technology (Wiley, 2008).

8.

M. Paurisse, M. Hanna, F. Druon, and P. Georges, “Wavefront control of a multicore ytterbium-doped pulse fiber amplifier by digital holography,” Opt. Lett. 35, 1428–1430 (2010). [CrossRef] [PubMed]

9.

R. Navarro and E. Moreno-Barriuso, “Laser ray-tracing method for optical testing,” Opt. Lett. 24, 951–953 (1999). [CrossRef]

10.

S. R. Chamot, C. Dainty, and S. Esposito, “Adaptive optics for ophthalmic applications using a pyramid wavefront sensor,” Opt. Express 14, 518–526 (2006). [CrossRef] [PubMed]

11.

M. P. Rimmer and J. C. Wyant, “Evaluation of large aberrations using a lateral-shear interferometer having variable shear,” Appl. Opt. 14, 142–150 (1975). [PubMed]

12.

J.-C. Chanteloup, F. Druon, M. Nantel, A. Maksimchuk, and G. Mourou, “Single-shot wave-front measurements of high-intensity ultrashort laser pulses with a three-wave interferometer,” Opt. Lett. 23, 621–623 (1998). [CrossRef]

13.

S. Velghe, J. Primot, N. Guérineau, M. Cohen, and B. Wattellier, “Wave-front reconstruction from multidirectional phasederivatives generated by multilateral shearing interferometers,” Opt. Lett. 30, 245–247 (2005). [CrossRef] [PubMed]

14.

R. G. Lane and M. Tallon, “Wave-front reconstruction using a Shack-Hartmann sensor,” Appl. Opt. 31, 6902–6908 (1992). [CrossRef] [PubMed]

15.

L. Changhai, X. Fengjie, H. Shengyang, and J. Zongfu, “Performance analysis of multiplexed phase computer-generated hologram for modal wavefront sensing,” Appl. Opt. 50, 1631–1639 (2011). [CrossRef] [PubMed]

16.

I. A. Litvin, A. Dudley, F. S. Roux, and A. Forbes, “Azimuthal decomposition with digital holograms,” Opt. Express 20, 10996–11004 (2012). [CrossRef] [PubMed]

17.

R. Borrego-Varillas, C. Romero, J. R. V. de Aldana, J. M. Bueno, and L. Roso, “Wavefront retrieval of amplified femtosecond beams by second-harmonic generation,” Opt. Express 19, 22851–22862 (2011). [CrossRef] [PubMed]

18.

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

19.

D. Flamm, D. Naidoo, C. Schulze, A. Forbes, and M. Duparré, “Mode analysis with a spatial light modulator as a correlation filter,” Opt. Lett. 37, 2478–2480 (2012). [CrossRef] [PubMed]

20.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991) [CrossRef]

21.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill Publishing Company, 1968).

22.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1991).

23.

H. G. Berry, G. Gabrielse, and A. E. Livingston, “Measurement of the Stokes parameters of light,” Appl. Opt. 16, 3200–3205 (1977). [CrossRef] [PubMed]

24.

D. Flamm, O. A. Schmidt, C. Schulze, J. Borchardt, T. Kaiser, S. Schröter, and M. Duparré, “Measuring the spatial polarization distribution of multimode beams emerging from passive step-index large-mode-area fibers,” Opt. Lett. 35, 3429–3431 (2010). [CrossRef] [PubMed]

25.

B. Neubert and B. Eppich, “Influences on the beam propagation ratio M2,” Opt. Commun. 250, 241 – 251 (2005). [CrossRef]

26.

ISO, “ISO 15367-1:2003 lasers and laser-related equipment – test methods for determination of the shape of a laser beam wavefront – Part 1: Terminology and fundamental aspects,” (2003).

27.

R. T. Schermer, “Mode scalability in bent optical fibers,” Opt. Express 15, 15674–15701 (2007). [CrossRef] [PubMed]

28.

J. Leach, S. Keen, M. J. Padgett, C. Saunter, and G. D. Love, “Direct measurement of the skew angle of the poynting vector in a helically phased beam,” Opt. Express 14, 11919–11924 (2006). [CrossRef] [PubMed]

29.

F. A. Starikov, G. G. Kochemasov, S. M. Kulikov, A. N. Manachinsky, N. V. Maslov, A. V. Ogorodnikov, S. A. Sukharev, V. P. Aksenov, I. V. Izmailov, F. Y. Kanev, V. V. Atuchin, and I. S. Soldatenkov, “Wavefront reconstruction of an optical vortex by a Hartmann-Shack sensor,” Opt. Lett. 32, 2291–2293 (2007). [CrossRef] [PubMed]

OCIS Codes
(010.7350) Atmospheric and oceanic optics : Wave-front sensing
(030.4070) Coherence and statistical optics : Modes
(120.5050) Instrumentation, measurement, and metrology : Phase measurement
(090.1995) Holography : Digital holography
(140.3295) Lasers and laser optics : Laser beam characterization
(050.4865) Diffraction and gratings : Optical vortices

ToC Category:
Image Processing

History
Original Manuscript: June 29, 2012
Revised Manuscript: August 2, 2012
Manuscript Accepted: August 3, 2012
Published: August 13, 2012

Citation
Christian Schulze, Darryl Naidoo, Daniel Flamm, Oliver A. Schmidt, Andrew Forbes, and Michael Duparré, "Wavefront reconstruction by modal decomposition," Opt. Express 20, 19714-19725 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-18-19714


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References

  1. F. Roddier, M. Séchaud, G. Rousset, P.-Y. Madec, M. Northcott, J.-L. Beuzit, F. Rigaut, J. Beckers, D. Sandler, P. Léna, and O. Lai, Adaptive Optics in Astronomy (Cambridge, 1999). [CrossRef]
  2. M. A. A. Neil, R. Jukaitis, M. J. Booth, T. Wilson, T. Tanaka, and S. Kawata, “Adaptive aberration correction in a two-photon microscope,” J. Microsc. 200, 105–108 (2000). [CrossRef] [PubMed]
  3. M. Rueckel, J. A. Mack-Bucher, and W. Denk, “Adaptive wavefront correction in two-photon microscopy using coherence-gated wavefront sensing,” Proc. Natl. Acad. Sci. 103, 17137–17142 (2006). [CrossRef] [PubMed]
  4. M. Booth, M. Neil, and T. Wilson, “Aberration correction for confocal imaging in refractive-index-mismatched media,” J. Microsc. 192, 90–98 (1998). [CrossRef]
  5. B. Hermann, E. J. Fernández, A. Unterhuber, H. Sattmann, A. F. Fercher, W. Drexler, P. M. Prieto, and P. Artal, “Adaptive-optics ultrahigh-resolution optical coherence tomography,” Opt. Lett. 29, 2142–2144 (2004). [CrossRef] [PubMed]
  6. A. Roorda, F. Romero-Borja, I. William Donnelly, H. Queener, T. Hebert, and M. Campbell, “Adaptive optics scanning laser ophthalmoscopy,” Opt. Express 10, 405–412 (2002). [PubMed]
  7. R. Paschotta, Encyclopedia of Laser Physics and Technology (Wiley, 2008).
  8. M. Paurisse, M. Hanna, F. Druon, and P. Georges, “Wavefront control of a multicore ytterbium-doped pulse fiber amplifier by digital holography,” Opt. Lett. 35, 1428–1430 (2010). [CrossRef] [PubMed]
  9. R. Navarro and E. Moreno-Barriuso, “Laser ray-tracing method for optical testing,” Opt. Lett. 24, 951–953 (1999). [CrossRef]
  10. S. R. Chamot, C. Dainty, and S. Esposito, “Adaptive optics for ophthalmic applications using a pyramid wavefront sensor,” Opt. Express 14, 518–526 (2006). [CrossRef] [PubMed]
  11. M. P. Rimmer and J. C. Wyant, “Evaluation of large aberrations using a lateral-shear interferometer having variable shear,” Appl. Opt. 14, 142–150 (1975). [PubMed]
  12. J.-C. Chanteloup, F. Druon, M. Nantel, A. Maksimchuk, and G. Mourou, “Single-shot wave-front measurements of high-intensity ultrashort laser pulses with a three-wave interferometer,” Opt. Lett. 23, 621–623 (1998). [CrossRef]
  13. S. Velghe, J. Primot, N. Guérineau, M. Cohen, and B. Wattellier, “Wave-front reconstruction from multidirectional phasederivatives generated by multilateral shearing interferometers,” Opt. Lett. 30, 245–247 (2005). [CrossRef] [PubMed]
  14. R. G. Lane and M. Tallon, “Wave-front reconstruction using a Shack-Hartmann sensor,” Appl. Opt. 31, 6902–6908 (1992). [CrossRef] [PubMed]
  15. L. Changhai, X. Fengjie, H. Shengyang, and J. Zongfu, “Performance analysis of multiplexed phase computer-generated hologram for modal wavefront sensing,” Appl. Opt. 50, 1631–1639 (2011). [CrossRef] [PubMed]
  16. I. A. Litvin, A. Dudley, F. S. Roux, and A. Forbes, “Azimuthal decomposition with digital holograms,” Opt. Express 20, 10996–11004 (2012). [CrossRef] [PubMed]
  17. R. Borrego-Varillas, C. Romero, J. R. V. de Aldana, J. M. Bueno, and L. Roso, “Wavefront retrieval of amplified femtosecond beams by second-harmonic generation,” Opt. Express 19, 22851–22862 (2011). [CrossRef] [PubMed]
  18. T. Kaiser, D. Flamm, S. Schröter, and M. Duparré, “Complete modal decomposition for optical fibers using CGH-based correlation filters,” Opt. Express 17, 9347–9356 (2009). [CrossRef] [PubMed]
  19. D. Flamm, D. Naidoo, C. Schulze, A. Forbes, and M. Duparré, “Mode analysis with a spatial light modulator as a correlation filter,” Opt. Lett. 37, 2478–2480 (2012). [CrossRef] [PubMed]
  20. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991) [CrossRef]
  21. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill Publishing Company, 1968).
  22. M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1991).
  23. H. G. Berry, G. Gabrielse, and A. E. Livingston, “Measurement of the Stokes parameters of light,” Appl. Opt. 16, 3200–3205 (1977). [CrossRef] [PubMed]
  24. D. Flamm, O. A. Schmidt, C. Schulze, J. Borchardt, T. Kaiser, S. Schröter, and M. Duparré, “Measuring the spatial polarization distribution of multimode beams emerging from passive step-index large-mode-area fibers,” Opt. Lett. 35, 3429–3431 (2010). [CrossRef] [PubMed]
  25. B. Neubert and B. Eppich, “Influences on the beam propagation ratio M2,” Opt. Commun. 250, 241 – 251 (2005). [CrossRef]
  26. ISO, “ISO 15367-1:2003 lasers and laser-related equipment – test methods for determination of the shape of a laser beam wavefront – Part 1: Terminology and fundamental aspects,” (2003).
  27. R. T. Schermer, “Mode scalability in bent optical fibers,” Opt. Express 15, 15674–15701 (2007). [CrossRef] [PubMed]
  28. J. Leach, S. Keen, M. J. Padgett, C. Saunter, and G. D. Love, “Direct measurement of the skew angle of the poynting vector in a helically phased beam,” Opt. Express 14, 11919–11924 (2006). [CrossRef] [PubMed]
  29. F. A. Starikov, G. G. Kochemasov, S. M. Kulikov, A. N. Manachinsky, N. V. Maslov, A. V. Ogorodnikov, S. A. Sukharev, V. P. Aksenov, I. V. Izmailov, F. Y. Kanev, V. V. Atuchin, and I. S. Soldatenkov, “Wavefront reconstruction of an optical vortex by a Hartmann-Shack sensor,” Opt. Lett. 32, 2291–2293 (2007). [CrossRef] [PubMed]

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