## Wavefront reconstruction by modal decomposition |

Optics Express, Vol. 20, Issue 18, pp. 19714-19725 (2012)

http://dx.doi.org/10.1364/OE.20.019714

Acrobat PDF (1264 KB)

### 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

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]

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]

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]

## 2. Modal decomposition

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]

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]

**r**= (

*x*,

*y*) the spatial coordinates,

*c*=

_{l}*ρ*

_{l}e^{iΔφl}the complex expansion coefficient, Ψ

*(*

_{l}**r**) =

*ψ*(

_{l}**r**)

**e**

*the*

_{l}*l*mode with amplitude

^{th}*ρ*, intermodal phase differences Δ

_{l}*φ*(phase difference between two modes, with one mode acting as a reference), polarization

_{l}**e**

*, and*

_{l}*N*the number of modes (which may count to infinity). Since the sum of the modes represents the entire optical field, the intensity and phase distributions of this field are easily computed via

*I*(

**r**) = |

**U**(

**r**)|

^{2}and Φ

*(*

_{j}**r**) = arg

*U*(

_{j}**r**) for each vector component

*U*. To perform the modal decomposition into a given set of modes Ψ

_{j}*(for example modes of a fiber or Laguerre-Gaussian modes of free space) and to determine the set of coefficients*

_{l}*c*, as described by Eq. (1), is the main task of the correlation filter method. Note that the modes Ψ

_{l}*are calculated prior to any experiment and are thus known (an overview of the description of modes can be found in [20*

_{l}20. B. E. A. Saleh and M. C. Teich, *Fundamentals of Photonics* (Wiley, 1991) [CrossRef]

**U**by a one-dimensional set of coefficients

*c*, since the spatial information is stored in the modal fields

_{l}*ψ*.

_{l}**17**, 9347–9356 (2009). [CrossRef] [PubMed]

**17**, 9347–9356 (2009). [CrossRef] [PubMed]

*ψ*. To measure the intermodal phase difference Δ

_{l}*φ*of a mode

_{l}*ψ*to a chosen reference mode

_{l}*ψ*

_{0}(mostly the fundamental mode, but in general arbitrary), two transmission functions are necessary, each representing an interferometric superposition of the two mode fields [18

**17**, 9347–9356 (2009). [CrossRef] [PubMed]

*φ*is calculated unambiguously according to [18

_{l}**17**, 9347–9356 (2009). [CrossRef] [PubMed]

*ρ*and phases Δ

_{l}*φ*simultaneously from one diffraction pattern we use angular multiplexing. Accordingly, the final transmission function

_{l}*T*(

**r**) of the hologram is a superposition of all single transmissions functions

*T*(

_{n}**r**) (for each modal amplitude and sine and cosine of the phase), each modulated with a particular carrier frequency

**K**

*to achieve spatial seperation of the information in the Fourier plane [18*

_{n}**17**, 9347–9356 (2009). [CrossRef] [PubMed]

**U**is provided by the analysis of the cartesian field components

*U*and

_{x}*U*, including the proper phase difference

_{y}*δ*between them. This can be done by measuring the Stokes parameters

*S*

_{0}...

*S*

_{3}of the beam, which necessitates six (assuming completely polarized light) modal decompositions with a quarter-wave plate and a polarizer in appropriate orientations in front of the hologram [22, 23

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]

*I*(

*α*) is the measured (respectively reconstructed) intensity behind the polarizer at angular orientations

*α*= 0°, 45°, 90°, 135° and

*I*

_{λ}_{/4}(

*α*) denotes two additional measurements with the polarizer placed at

*α*= 45°, 135° and a preceding quarter-wave plate. Having performed the depicted six modal decompositions, the optical field, including the polarization properties, is completely known [24

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]

## 3. Wavefront reconstruction

**P**is computable from [22]: where

*ω*is the angular frequency,

*ε*

_{0}the vacuum permittivity,

*ε*the permittivity distribution, and

**E**and

**H**are the electric and magnetic field respectively, that both obey the decomposition of Eq. (1). On the right-hand side of Eq. (7),

**H**was assumed to be the vector field

**U**, and

**E**was calculated according to Maxwell’s Equations. Note that alternatively the vector field could have been chosen to be the electric field, which yields the same Poynting vector. The wavefront is then usually defined as the continuous surface that is normal to the time average direction of energy propagation, i.e., normal to the time average Poynting vector

**P**[25

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

*z*denotes the position of the measurement plane. Since there might be no continuous surface that fulfills this condition, the wavefront can be more generally defined as the continuous surface that minimizes the power density weighted deviations of the direction of its normal vectors to the direction of energy flow in the measurement plane [25

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

**P**

*= [*

_{t}*P*,

_{x}*P*, 0]′. In the simple case of scalar, i.e. linearly polarized, beams the wavefront is equal to the phase distribution Φ(

_{y}**r**) of the beam, except for a proportionality factor [26]: It is important to note that this expression is only valid as long as there are no phase jumps or phase singularities, because the wavefront is always considered to be a continuous surface.

## 4. Measurement setup

*μ*m) in the first instance, excited with a Nd:YAG laser (

*λ*=1064 nm) and guiding three fiber modes. Hence, the aberrated wavefront was generated at the fiber end face by multimode interference. The reliability of the modal decomposition approach was proved by comparing the results to those of a Shack-Hartmann wavefront sensor. A beam splitter (non-polarizing) was used to analyze the beam with the wavefront sensor and with the modal decomposition setup at the same time. Two different 4f-setups with magnifications of

*f*

_{L3}/

*f*

_{L1}= 750 mm/4 mm = 188 and

*f*

_{L2}/

*f*

_{L1}= 375 mm/4 mm = 94, respectively, relay imaged the plane of the aberrated wavefront (here end face of the respective fiber) to the wavefront sensor and to the CGH (cf. Fig. 1), with the focal lengths chosen that first, the beam matched the design radius of the hologram, and second, the beam was well sampled by the microlens array of the wavefront sensor. The wavefront sensor consisted of 69×69 microlenses with an array pitch of 0.108 mm. Via a second beam splitter (non-polarizing), the fiber end face was simultaneously imaged onto the CGH and a CCD camera for direct recording of the near field intensity (CCD

_{1}, cf. Fig. 1). Thereby, a polarizer and an optional quarter-wave plate were used to perform the Stokes measurements, whereas the CGH performed the modal decomposition for each polarization component. The hologram diffraction pattern of first order was observed with a second CCD camera (CCD

_{2}) in the far field of the hologram using a single lens in a 2f-setup (

*f*

_{FL}= 180 mm), and was used to determine the modal powers and phases of the beam for the scalar projection provided by the pair of quarter-wave plate and polarizer. Note that for a linearly polarized beam, quarter-wave plate and polarizer would not have been necessary. The hologram itself was written to either a commercial pixelated phase-only spatial light modulator (SLM, Holoeye Pluto) [19

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]

**17**, 9347–9356 (2009). [CrossRef] [PubMed]

## 5. Calibration of the setup

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

*μ*m, which is less than

*λ*/10. Additionally, the intensities measured with the wavefront sensor (Fig. 2(a)) and with the CCD camera (Fig. 2(b)) are in good agreement, which is, alongside with the flat wavefront, another proof that the optical setup itself adds no severe aberrations. The modal decomposition results in a modal spectrum (Fig. 2(d)) with a fundamental mode power of 99% of total power, which demonstrates the pureness of the reference beam.

## 6. Scalar beams

*μ*m 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

**17**, 9347–9356 (2009). [CrossRef] [PubMed]

## 7. Vector beams

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]

*U*and

_{x}*U*as well as their relative phase difference

_{y}*δ*and hence, the entire optical field, including the Poynting vector

**P**. The wavefront

*w*is infered according to Eq. (9) by numerically minimizing the direction deviations of Poynting vector and wavefront normal vectors, weighted with the power density |

**P**|. Results of this procedure are depicted in Fig. 4. In this example, the beam consists of 63% fundamental mode, 23% mode 2 and 14% mode 3 as shown in Fig. 4(c) (modal powers of x- and y-component added up). As seen from the intensity plots Fig. 4(a) and (b), the beam is shifted slightly off-axis due to modal interference. Measured intensities from both the Shack-Hartmann (Fig. 4(a)) and the CGH reconstruction (Fig. 4(b)) are in very good agreement. Regarding the wavefronts, results from direct Shack-Hartmann measurement (Fig. 4(d)) and from minimization based on Eq. (9) (Fig. 4(f)) correlate very well. However, since there is no phase of the entire vector field (each component has its own phase) in this case, there is no way to determine the wavefront according to Eq. (10).

## 8. Beams with phase singularities

*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.

*w*

_{max}< 0.1

*μ*m). This deviance is actually based on the problem of defining the wavefront as a physical quantity itself. Other than the phase, the wavefront is connected to the Poynting vector [26]. It is known that the Poynting vector spirals for a vortex beam [28

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]

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

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]

## 9. Decomposing beams including extrinsic aberrations

**37**, 2478–2480 (2012). [CrossRef] [PubMed]

_{p0}modes respond in case of pure defocus. The good agreement between the expected wavefront for a lens with focal length

*f*= 1000 mm and the measured one is emphasized in Fig. 6(c), depicting a wavefront slice through the center. Another proof of accuracy for the measured wavefront is to incorporate the conjugated curvature into the modal basis set and running the decomposition again, easily achieved by changing the hologram on the SLM. This time, as shown in Fig. 7(b), the modal spectrum consists to 99% of fundamental mode. Since the conjugate of the measured curvature is already included in each mode on the hologram, this procedure yields a perfectly adapted mode set and consequently a response of the fundamental mode only. Or in other words, the curvature of the physical lens is balanced by the conjugated curvature programmed on the hologram. Hence, the modal spectrum resembles that of a modal decomposition performed without a physical lens in place and into modes with a flat wavefront, as illustrated in Fig. 7(a).

## 10. Discussion of techniques

## 11. Conclusion

## Acknowledgments

## 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, |

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. |

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. |

4. | M. Booth, M. Neil, and T. Wilson, “Aberration correction for confocal imaging in refractive-index-mismatched media,” J. Microsc. |

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. |

6. | A. Roorda, F. Romero-Borja, I. William Donnelly, H. Queener, T. Hebert, and M. Campbell, “Adaptive optics scanning laser ophthalmoscopy,” Opt. Express |

7. | R. Paschotta, |

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. |

9. | R. Navarro and E. Moreno-Barriuso, “Laser ray-tracing method for optical testing,” Opt. Lett. |

10. | S. R. Chamot, C. Dainty, and S. Esposito, “Adaptive optics for ophthalmic applications using a pyramid wavefront sensor,” Opt. Express |

11. | M. P. Rimmer and J. C. Wyant, “Evaluation of large aberrations using a lateral-shear interferometer having variable shear,” Appl. Opt. |

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. |

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. |

14. | R. G. Lane and M. Tallon, “Wave-front reconstruction using a Shack-Hartmann sensor,” Appl. Opt. |

15. | L. Changhai, X. Fengjie, H. Shengyang, and J. Zongfu, “Performance analysis of multiplexed phase computer-generated hologram for modal wavefront sensing,” Appl. Opt. |

16. | I. A. Litvin, A. Dudley, F. S. Roux, and A. Forbes, “Azimuthal decomposition with digital holograms,” Opt. Express |

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 |

18. | T. Kaiser, D. Flamm, S. Schröter, and M. Duparré, “Complete modal decomposition for optical fibers using CGH-based correlation filters,” Opt. Express |

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. |

20. | B. E. A. Saleh and M. C. Teich, |

21. | J. W. Goodman, |

22. | M. Born and E. Wolf, |

23. | H. G. Berry, G. Gabrielse, and A. E. Livingston, “Measurement of the Stokes parameters of light,” Appl. Opt. |

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. |

25. | B. Neubert and B. Eppich, “Influences on the beam propagation ratio M2,” Opt. Commun. |

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 |

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 |

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. |

**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

Sort: Year | Journal | Reset

### References

- 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]
- 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]
- 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]
- M. Booth, M. Neil, and T. Wilson, “Aberration correction for confocal imaging in refractive-index-mismatched media,” J. Microsc. 192, 90–98 (1998). [CrossRef]
- 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]
- 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]
- R. Paschotta, Encyclopedia of Laser Physics and Technology (Wiley, 2008).
- 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]
- R. Navarro and E. Moreno-Barriuso, “Laser ray-tracing method for optical testing,” Opt. Lett. 24, 951–953 (1999). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- R. G. Lane and M. Tallon, “Wave-front reconstruction using a Shack-Hartmann sensor,” Appl. Opt. 31, 6902–6908 (1992). [CrossRef] [PubMed]
- 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]
- I. A. Litvin, A. Dudley, F. S. Roux, and A. Forbes, “Azimuthal decomposition with digital holograms,” Opt. Express 20, 10996–11004 (2012). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991) [CrossRef]
- J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill Publishing Company, 1968).
- M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1991).
- H. G. Berry, G. Gabrielse, and A. E. Livingston, “Measurement of the Stokes parameters of light,” Appl. Opt. 16, 3200–3205 (1977). [CrossRef] [PubMed]
- 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]
- B. Neubert and B. Eppich, “Influences on the beam propagation ratio M2,” Opt. Commun. 250, 241 – 251 (2005). [CrossRef]
- 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).
- R. T. Schermer, “Mode scalability in bent optical fibers,” Opt. Express 15, 15674–15701 (2007). [CrossRef] [PubMed]
- 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]
- 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]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.