## Experimental detection of optical vortices with a Shack-Hartmann wavefront sensor |

Optics Express, Vol. 18, Issue 15, pp. 15448-15460 (2010)

http://dx.doi.org/10.1364/OE.18.015448

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### Abstract

Laboratory experiments are carried out to detect optical vortices in conditions typical of those experienced when a laser beam is propagated through the atmosphere. A Spatial Light Modulator (SLM) is used to mimic atmospheric turbulence and a Shack-Hartmann wavefront sensor is utilised to measure the slopes of the wavefront surface. A matched filter algorithm determines the positions of the Shack-Hartmann spot centroids more robustly than a centroiding algorithm. The slope discrepancy is then obtained by taking the slopes measured by the wavefront sensor away from the slopes calculated from a least squares reconstruction of the phase. The slope discrepancy field is used as an input to the branch point potential method to find if a vortex is present, and if so to give its position and sign. The use of the slope discrepancy technique greatly improves the detection rate of the branch point potential method. This work shows the first time the branch point potential method has been used to detect optical vortices in an experimental setup.

© 2010 Optical Society of America

## 1. Introduction

8. N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, and B. Y. Zel’dovich, “Wave-front dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. A **73**, 525–528 (1983). [CrossRef]

11. D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A **15**, 2759–2768 (1998). [CrossRef]

12. M. C. Roggemann and D. J. Lee, “Two-deformable-mirror concept for correcting scintillation effects in laser beam projection through the turbulent atmosphere,” Appl. Opt. **37**, 4577–4585 (1998). [CrossRef]

13. D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. **200**, 43–72 (2001). [CrossRef]

15. F. A. Starikov, V. P. Aksenova, V. V. Atuchind, I. V. Izmailova, F. Y. Kaneva, G. G. Kochemasov, A. V. Kudryashovb, S. M. Kulikov, Y. I. Malakhovc, A. N. Manachinsky, N. V. Maslov, A. V. Ogorodnikov, I. S. Soldatenkovd, and S. A. Sukharev, “Wave front sensing of an optical vortex and its correction in the close-loop adaptive system with bimorph mirror,” Proc. SPIE **6747**, 1–8 (2007).

16. F. A. Starikov, G. G. Kochemasov, M. O. Koltygin, S. M. Kulikov, A. N. Manachinsky, N. V. Maslov, S. A. Sukharev, V. P. Aksenov, I. V. Izmailov, F. Yu. Kanev, V. V. Atuchin, and I. S. Soldatenkov, “Correction of vortex laser beam in a closed-loop adaptive system with bimorph mirror,” Opt. Lett. **34**, 2264–2266 (2009). [CrossRef] [PubMed]

20. E. O. Le Bigot, W. J. Wild, and E. J. Kibblewhite, “Reconstruction of discontinuous light-phase functions,” Opt. Lett. **23**, 10–12 (1998). [CrossRef]

23. G. A. Tyler, “Reconstruction and assessment of the least-squares and slope discrepancy components of the phase,” J. Opt. Soc. Am. A **17**, 1828–1839 (2000). [CrossRef]

## 2. Shack-Hartmann wavefront sensing

### 2.1. Shack-Hartmann spot detection and centroiding using the Hotelling Observer

*H*

_{1}or the spot absent hypothesis, denoted

*H*

_{0}. The ideal observer for such a test is the likelihood ratio:

*λ*(

*g*|

*r⃗*)) =

_{spot}*ln*[Λ(

*g*|

*r⃗*)], where

_{spot}*pr*(

*g*|

*H*

_{1},

*r⃗*) is the probability density function (PDF) of the data

_{spot}*g*under the

*H*

_{1}hypothesis with a spot located at the position vector

*r⃗*and

_{spot}*pr*(

*g*|

*H*

_{0}) is the spot absent PDF. The Hotelling observer takes on the forum:

*A*is the intensity of the spot (a scalar quantity) and

*h*(

*r⃗*) is an estimate of the PSF at the position vector

*r⃗*. It can be shown [30

30. L. Caucci, H. H. Barrett, N. Devaney, and J. J. Rodríguez, “Application of the hotelling and ideal observers to detection and localization of exoplanets,” J. Opt. Soc. Am. A **24**, B13–B24 (2007). [CrossRef]

*λ*(

*g*|

*r⃗*)) =

_{spot}*t*(

_{Hot}*g*|

*r⃗*), as both are linear functions of the data and optimal. The Hotelling observer therefore only requires knowledge of the modeled spot signal and the data covariance matrix. Therefore the goal is to find a template which maximises the correlation between the template and the data. If

_{spot}*K*|

_{g}*H*for

_{i}*i*∊ {0,1} are the covariance matrices under the

*H*

_{0}and

*H*

_{1}hypotheses then the average covariance matrix is:

*b*, of each spot image can be estimated along with its variance,

*σ*

^{2}, and an estimate of the PSF is known it can be shown that

*K*is diagonal with it elements given by [31

_{g}31. L. Caucci, H. H. Barrett, N. Devaney, and J. J. Rodríguez, “Statistical decision theory and adaptive optics: A rigorous approach to exoplanet detection,” in “Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings,” (OSA, 2007), ATuA5.

*w*, and the data,

*g*, can be computed either in the image plane or in the Fourier plane. In the Fourier plane the position of the spot is estimated by using a parabolic fit to interpolate the peak of the cross-correlation [28

28. L. A. Poyneer, “Scene-based Shack-Hartmann wave-front sensing: analysis and simulation,” Appl. Opt. **42**, 5807–5815 (2003). [CrossRef] [PubMed]

29. C. Leroux and C. Dainty, “Estimation of centroid positions with a matched-filter algorithm: relevance for aberrometry of the eye,” Opt. Express **18**, 1197–1206 (2010). [CrossRef] [PubMed]

_{MF}will refer to using the Hotelling observer in this manner and

*MF*will refer to using a simple matched filter with this approach. In the image plane the peak of the cross-correlation is found, and hence the spot position, by shifting the position of the template spot until maximum is found. In practice this PSF fitting type approach is carried out by an unconstrained maximisation of the Hotelling observer where the value of the Hotelling observer is only dependent upon the position of the test spot in the template vector [25

25. D. Burke, S. Gladysz, L. Roberts, N. Devaney, and C. Dainty, “An Improved Technique for the Photometry and Astrometry of Faint Companions,” Pub. Astro. Soc. Pac. **121**, 767–777 (2009). [CrossRef]

_{ML}.

_{MF}showed the lowest mean error on the estimation of the position of the spot (in pixels) followed by

*MF*and Hot

_{ML}, see Table 1. The centroiding algorithm faired worst; however, it should be noted that this centroiding algorithm was not optimised as in [29

29. C. Leroux and C. Dainty, “Estimation of centroid positions with a matched-filter algorithm: relevance for aberrometry of the eye,” Opt. Express **18**, 1197–1206 (2010). [CrossRef] [PubMed]

30. L. Caucci, H. H. Barrett, N. Devaney, and J. J. Rodríguez, “Application of the hotelling and ideal observers to detection and localization of exoplanets,” J. Opt. Soc. Am. A **24**, B13–B24 (2007). [CrossRef]

*ε*= 4 × 10

^{−4}pixels, versus the false positive fraction of decisions made, i.e. the false alarm rate. The area under the LROC curve, denoted AUC [30

30. L. Caucci, H. H. Barrett, N. Devaney, and J. J. Rodríguez, “Application of the hotelling and ideal observers to detection and localization of exoplanets,” J. Opt. Soc. Am. A **24**, B13–B24 (2007). [CrossRef]

*Hot*has the highest area under the LROC curve, AUC = 0.82. Furthermore the

_{MF}*Hot*has a much higher SNR compared to the straight matched filter,

_{MF}*MF*, and centroiding algorithm. Hence this increased SNR coupled with a high performance in spot detection and localisation shows that the

*Hot*is capable of detecting spots with very low intensities which is critical when analysing spot images in close proximity to a vortex. It should be noted that computing the Hotelling observer in the image plane could be improved by interpolation of the PSF so that the resolution of the cross-correlation could be improved, as was done with the Fourier plane method.

_{MF}### 2.2. Vortex detection

*s*, to the wavefront phase,

*ϕ*.

*ϕ*

_{lmse}, as described by Fried [11

11. D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A **15**, 2759–2768 (1998). [CrossRef]

^{x}() are the horizontal phase differences and Δ

^{y}() are the vertical phase differences.

*π*then there is a phase singularity enclosed in the loop. By convention the (i,j)th pixel is the top left of the four pixels and the vortex is placed there, a more exact position of the branch point could be established by further numerical methods but we know of no standard technique. The overall robustness of the contour sum method has been questioned under heavy noise conditions [18, 21

21. E. O. Le Bigot and W. J. Wild, “Theory of branch-point detection and its implementation,” J. Opt. Soc. Am. A **16**, 1724–1729 (1999). [CrossRef]

32. M. Chen, F. S. Roux, and J. C. Olivier, “Detection of phase singularities with a Shack-Hartmann wavefront sensor,” J. Opt. Soc. Am. A **24**, 1994–2002 (2007). [CrossRef]

21. E. O. Le Bigot and W. J. Wild, “Theory of branch-point detection and its implementation,” J. Opt. Soc. Am. A **16**, 1724–1729 (1999). [CrossRef]

22. W. J. Wild and E. O. Le Bigot, “Rapid and robust detection of branch points from wave-front gradients,” Opt. Lett. **24**, 190–192 (1999). [CrossRef]

*R*

_{π/2}) is a 90 degree rotation,

*s*are the slope values measured by the wavefront senor with

*s*and

_{x}*s*the slopes x-values and y-values respectively.

_{y}*A*

^{†}is the adjoint of the geometry matrix and the

*m*

^{2}value is inserted to prevent the matrix from becoming singular. The pseudo-inverse could also be calculated using a singular value decomposition.

*ϕ*

_{slpdis}are the phase gradients of the slope discrepancy, ∇

*ϕ*are the phase gradients of the least-squares phase and

_{lmse}*s*are the phase gradients calculated from the wavefront sensor.

## 3. Laboratory experiments

### 3.1. Experimental method

33. J. M. Martin and S. M. Flatte, “Simulation of point-source scintillation through three-dimensional random media,” J. Opt. Soc. Am. A **7**, 838–847 (1990). [CrossRef]

34. R. A. Johnston and R. G. Lane, “Modeling scintillation from an aperiodic kolmogorov phase screen,” Appl. Opt. **39**, 4761–4769 (2000). [CrossRef]

*X*,

_{k}*Y*) with charge

_{k}*m*can be generated using the following product wave function, where

_{k}*K*is the index of the vortex. In this experiment we seeded the initial plane wave with a single vortex, (

*X*

_{1},

*Y*

_{1}), at the centre of the field:

*X*and

_{k}*Y*defining the positions of the zero crossings. These simulations and all of the other analysis code is being implemented using MATLAB.

_{k}*π*radians and applied to the SLM [35

35. M. A. A. Neil, M. J. Booth, and T. Wilson, “Dynamic wave-front generation for the characterization and testing of optical systems,” Opt. Lett. **23**, 1849–1851 (1998). [CrossRef]

36. M. A. A. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” J. Microsc. **197**, 219–223 (2000). [CrossRef] [PubMed]

### 3.2. Experimental results

*σ*

^{2}

_{I}= 0.154 and

*σ*

^{2}

_{I}= 0.652, where

*σ*

^{2}

_{I}is the scintillation index given by

*I*is the irradiance of the beam at the receiver.

*σ*

^{2}

_{I}> 1 is described as strong turbulence [37] and therefore the scintillation encountered here can only be described as being in either the weak or moderate turbulence regimes. This means that few vortices will be created naturally by the numerical simulation of propagation through the atmosphere. Those that have been created are very close together (ie. within one Shack-Hartmann lenslet) and do not effect the Shack-Hartmann slopes. The reason that the simulations are initially seeded with one vortex is to ensure that there is a detectable vortex in the final optical field, post propagation. At higher turbulence levels (ie. in the strong regime) one would expect to see more naturally occurring vortices and with the pairs spaced further apart, this has not been investigated in this paper.

*σ*

^{2}

_{I}= 0.154 the detection rate goes from 0.23 for the raw Shack-Hartmann slopes to 0.63 for the slope discrepancy technique. For

*σ*

^{2}

_{I}= 0.652 the detection rate goes from 0.07 for the raw Shack-Hartmann slopes to 0.21 for the slope discrepancy technique. This is a result for a comparison of 30 fields, using a numerical simulation of 15 phase screens employing Kolmogorov statistics with a

*λ*of 633 nm and a 2.8 km propagation path, for both

*σ*

^{2}

_{I}= 0.154 and

*σ*

^{2}

_{I}= 0.652. A failure is defined as either a non-detection or as a detection along with false positives, in the case of

*σ*

^{2}

_{I}= 0.154 most of the failures are due to false detection. The reason for the drop in detection rates when going into a higher turbulence regime can be explained by the existence of more phase aberrations and a higher degree of scintillation.

## 4. Conclusion

## Acknowledgments

## References and links

1. | J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A |

2. | D. G. Grier, “A revolution in optical manipulation,” Nature (London) |

3. | G. Gibson, J. Courtial, and M. J. Padgett, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express |

4. | F. S. Roux, “Dynamical behavior of optical vortices,” J. Opt. Soc. Am. B |

5. | D. L. Fried and J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. |

6. | M. C. Roggemann and B. M. Welsh, “Imaging through Turbulence”, Laser and Optical Science and Technology (CRC Press, 1996). |

7. | R. Mackey and C. Dainty, “Adaptive optics correction over a 3km near horizontal path,” Proc. SPIE |

8. | N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, and B. Y. Zel’dovich, “Wave-front dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. A |

9. | C. A. Primmerman, T. R. Price, R. A. Humphreys, B. G. Zollars, H. T. Barclay, and J. Herrmann, “Atmospheric-compensation experiments in strong-scintillation conditions,” Appl. Opt. |

10. | J. D. Barchers, D. L. Fried, and D. J. Link, “Evaluation of the performance of Hartmann sensors in strong scintillation,” Appl. Opt. |

11. | D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A |

12. | M. C. Roggemann and D. J. Lee, “Two-deformable-mirror concept for correcting scintillation effects in laser beam projection through the turbulent atmosphere,” Appl. Opt. |

13. | D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. |

14. | D. C. Ghiglia and M. D. Pritt, |

15. | F. A. Starikov, V. P. Aksenova, V. V. Atuchind, I. V. Izmailova, F. Y. Kaneva, G. G. Kochemasov, A. V. Kudryashovb, S. M. Kulikov, Y. I. Malakhovc, A. N. Manachinsky, N. V. Maslov, A. V. Ogorodnikov, I. S. Soldatenkovd, and S. A. Sukharev, “Wave front sensing of an optical vortex and its correction in the close-loop adaptive system with bimorph mirror,” Proc. SPIE |

16. | F. A. Starikov, G. G. Kochemasov, M. O. Koltygin, S. M. Kulikov, A. N. Manachinsky, N. V. Maslov, S. A. Sukharev, V. P. Aksenov, I. V. Izmailov, F. Yu. Kanev, V. V. Atuchin, and I. S. Soldatenkov, “Correction of vortex laser beam in a closed-loop adaptive system with bimorph mirror,” Opt. Lett. |

17. | V. E. Zetterlind III, “Distributed beacon requirements for branch point tolerant laser beam compensation in extended atmospheric turbulence,” Master’s thesis, Air Force Institute of Technology (2002). |

18. | K. Murphy, R. Mackey, and C. Dainty, “Branch point detection and correction using the branch point potential method,” Proc. SPIE |

19. | K. Murphy, R. Mackey, and C. Dainty, “Experimental detection of phase singularities using a Shack-Hartmann wavefront sensor,” Proc. SPIE |

20. | E. O. Le Bigot, W. J. Wild, and E. J. Kibblewhite, “Reconstruction of discontinuous light-phase functions,” Opt. Lett. |

21. | E. O. Le Bigot and W. J. Wild, “Theory of branch-point detection and its implementation,” J. Opt. Soc. Am. A |

22. | W. J. Wild and E. O. Le Bigot, “Rapid and robust detection of branch points from wave-front gradients,” Opt. Lett. |

23. | G. A. Tyler, “Reconstruction and assessment of the least-squares and slope discrepancy components of the phase,” J. Opt. Soc. Am. A |

24. | H. H. Barrett, K. J. Myers, M. N. Devaney, and J. C. Dainty, “Objective assessment of image quality. IV. Application to adaptive optics,” J. Opt. Soc. Am. A |

25. | D. Burke, S. Gladysz, L. Roberts, N. Devaney, and C. Dainty, “An Improved Technique for the Photometry and Astrometry of Faint Companions,” Pub. Astro. Soc. Pac. |

26. | H. H. Barrett and K. Myres, “Foundations of Image Science” (Weily Series in Pure and Applied Optics, 2004). |

27. | L. Caucci, H. H. Barrett, and J. J. Rodriguez, “Spatio-temporal hotelling observerfor signal detection from imagesequences,” Opt. Express |

28. | L. A. Poyneer, “Scene-based Shack-Hartmann wave-front sensing: analysis and simulation,” Appl. Opt. |

29. | C. Leroux and C. Dainty, “Estimation of centroid positions with a matched-filter algorithm: relevance for aberrometry of the eye,” Opt. Express |

30. | L. Caucci, H. H. Barrett, N. Devaney, and J. J. Rodríguez, “Application of the hotelling and ideal observers to detection and localization of exoplanets,” J. Opt. Soc. Am. A |

31. | L. Caucci, H. H. Barrett, N. Devaney, and J. J. Rodríguez, “Statistical decision theory and adaptive optics: A rigorous approach to exoplanet detection,” in “Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings,” (OSA, 2007), ATuA5. |

32. | M. Chen, F. S. Roux, and J. C. Olivier, “Detection of phase singularities with a Shack-Hartmann wavefront sensor,” J. Opt. Soc. Am. A |

33. | J. M. Martin and S. M. Flatte, “Simulation of point-source scintillation through three-dimensional random media,” J. Opt. Soc. Am. A |

34. | R. A. Johnston and R. G. Lane, “Modeling scintillation from an aperiodic kolmogorov phase screen,” Appl. Opt. |

35. | M. A. A. Neil, M. J. Booth, and T. Wilson, “Dynamic wave-front generation for the characterization and testing of optical systems,” Opt. Lett. |

36. | M. A. A. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” J. Microsc. |

37. | L. C. Andrews and R. L. Phillips, “Laser Beam Propagation through Random Media” (SPIE Publications, 2005). |

**OCIS Codes**

(010.0010) Atmospheric and oceanic optics : Atmospheric and oceanic optics

(010.1080) Atmospheric and oceanic optics : Active or adaptive optics

(010.7350) Atmospheric and oceanic optics : Wave-front sensing

(080.4865) Geometric optics : Optical vortices

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: May 17, 2010

Revised Manuscript: June 15, 2010

Manuscript Accepted: June 23, 2010

Published: July 6, 2010

**Citation**

Kevin Murphy, Daniel Burke, Nicholas Devaney, and Chris Dainty, "Experimental detection of optical
vortices with a Shack-Hartmann
wavefront sensor," Opt. Express **18**, 15448-15460 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-15-15448

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### References

- J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A 336, 165–190 (1974). [CrossRef]
- D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003). [CrossRef] [PubMed]
- G. Gibson, J. Courtial, and M. J. Padgett, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004). [CrossRef] [PubMed]
- F. S. Roux, “Dynamical behavior of optical vortices,” J. Opt. Soc. Am. B 12, 1215–1221 (1995). [CrossRef]
- D. L. Fried and J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. 31, 2865–2882 (1992). [CrossRef] [PubMed]
- M. C. Roggemann and B. M. Welsh, Imaging through Turbulence, Laser and Optical Science and Technology (CRC Press, 1996).
- R. Mackey and C. Dainty, “Adaptive optics correction over a 3km near horizontal path,” Proc. SPIE 7108, 1–9 (2008).
- N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, and B. Y. Zel’dovich, “Wave-front dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. A 73, 525–528 (1983). [CrossRef]
- C. A. Primmerman, T. R. Price, R. A. Humphreys, B. G. Zollars, H. T. Barclay, and J. Herrmann, “Atmospheric compensation experiments in strong-scintillation conditions,” Appl. Opt. 34, 2081–2088 (1995). [CrossRef] [PubMed]
- J. D. Barchers, D. L. Fried, and D. J. Link, “Evaluation of the performance of Hartmann sensors in strong scintillation,” Appl. Opt. 41, 1012–1021 (2002). [CrossRef] [PubMed]
- D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A 15, 2759–2768 (1998). [CrossRef]
- M. C. Roggemann and D. J. Lee, “Two-deformable-mirror concept for correcting scintillation effects in laser beam projection through the turbulent atmosphere,” Appl. Opt. 37, 4577–4585 (1998). [CrossRef]
- D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. 200, 43–72 (2001). [CrossRef]
- D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley-InterScience, 1998).
- F. A. Starikov, V. P. Aksenova, V. V. Atuchind, I. V. Izmailova, F. Y. Kaneva, G. G. Kochemasov, A. V. Kudryashovb, S. M. Kulikov, Y. I. Malakhovc, A. N. Manachinsky, N. V. Maslov, A. V. Ogorodnikov, I. S. Soldatenkovd, and S. A. Sukharev, “Wave front sensing of an optical vortex and its correction in the close-loop adaptive system with bimorph mirror,” Proc. SPIE 6747, 1–8 (2007).
- F. A. Starikov, G. G. Kochemasov, M. O. Koltygin, S. M. Kulikov, A. N. Manachinsky, N. V. Maslov, S. A. Sukharev, V. P. Aksenov, I. V. Izmailov, F. Yu. Kanev, V. V. Atuchin, and I. S. Soldatenkov, “Correction of vortex laser beam in a closed-loop adaptive system with bimorph mirror,” Opt. Lett. 34, 2264–2266 (2009). [CrossRef] [PubMed]
- V. E. ZetterlindIII, “Distributed beacon requirements for branch point tolerant laser beam compensation in extended atmospheric turbulence,” Master’s thesis, Air Force Institute of Technology (2002).
- K. Murphy, R. Mackey, and C. Dainty, “Branch point detection and correction using the branch point potential method,” Proc. SPIE 695105, 1–9 (2008).
- K. Murphy, R. Mackey, and C. Dainty, “Experimental detection of phase singularities using a Shack-Hartmann wavefront sensor,” Proc. SPIE 74760O, 1–9 (2009).
- E. O. Le Bigot, W. J. Wild, and E. J. Kibblewhite, “Reconstruction of discontinuous light-phase functions,” Opt. Lett. 23, 10–12 (1998). [CrossRef]
- E. O. Le Bigot and W. J. Wild, “Theory of branch-point detection and its implementation,” J. Opt. Soc. Am. A 16, 1724–1729 (1999). [CrossRef]
- W. J. Wild and E. O. Le Bigot, “Rapid and robust detection of branch points from wave-front gradients,” Opt. Lett. 24, 190–192 (1999). [CrossRef]
- G. A. Tyler, “Reconstruction and assessment of the least-squares and slope discrepancy components of the phase,” J. Opt. Soc. Am. A 17, 1828–1839 (2000). [CrossRef]
- H. H. Barrett, K. J. Myers, M. N. Devaney, and J. C. Dainty, “Objective assessment of image quality. IV. Application to adaptive optics,” J. Opt. Soc. Am. A 23, 3080–3105 (2006). [CrossRef]
- D. Burke, S. Gladysz, L. Roberts, N. Devaney, and C. Dainty, “An Improved Technique for the Photometry and Astrometry of Faint Companions,” Publ. Astron. Soc. Pac. 121, 767–777 (2009). [CrossRef]
- H. H. Barrett and K. Myres, Foundations of Image Science (Weily Series in Pure and Applied Optics, 2004).
- L. Caucci, H. H. Barrett, and J. J. Rodriguez, “Spatio-temporal hotelling observer for signal detection from image sequences,” Opt. Express 17, 10946–10958 (2009). [CrossRef] [PubMed]
- L. A. Poyneer, “Scene-based Shack-Hartmann wave-front sensing: analysis and simulation,” Appl. Opt. 42, 5807–5815 (2003). [CrossRef] [PubMed]
- C. Leroux and C. Dainty, “Estimation of centroid positions with a matched-filter algorithm: relevance for aberrometry of the eye,” Opt. Express 18, 1197–1206 (2010). [CrossRef] [PubMed]
- L. Caucci, H. H. Barrett, N. Devaney, and J. J. Rodríguez, “Application of the hotelling and ideal observers to detection and localization of exoplanets,” J. Opt. Soc. Am. A 24, B13–B24 (2007). [CrossRef]
- L. Caucci, H. H. Barrett, N. Devaney, and J. J. Rodríguez, “Statistical decision theory and adaptive optics: A rigorous approach to exoplanet detection,” in “Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings,” (OSA, 2007), ATuA5.
- M. Chen, F. S. Roux, and J. C. Olivier, “Detection of phase singularities with a Shack-Hartmann wavefront sensor,” J. Opt. Soc. Am. A 24, 1994–2002 (2007). [CrossRef]
- J. M. Martin and S. M. Flatte, “Simulation of point-source scintillation through three-dimensional random media,” J. Opt. Soc. Am. A 7, 838–847 (1990). [CrossRef]
- R. A. Johnston;/[p and R. G. Lane, “Modeling scintillation from an aperiodic kolmogorov phase screen,” Appl. Opt. 39, 4761–4769 (2000). [CrossRef]
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- M. A. A. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” J. Microsc. 197, 219–223 (2000). [CrossRef] [PubMed]
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