## A comprehensive approach to deal with instrumental optical aberrations effects in high-accuracy photon's orbital angular momentum spectrum measurements |

Optics Express, Vol. 18, Issue 20, pp. 21111-21120 (2010)

http://dx.doi.org/10.1364/OE.18.021111

Acrobat PDF (1385 KB)

### Abstract

With the current and upcoming applications of beams carrying orbital angular momentum (OAM), there will be the need to generate beams and measure their OAM spectrum with high accuracy. The instrumental OAM spectrum distortion is connected to the effect of its optical aberrations on the OAM content of the beams that the instrument creates or measures. Until now, the effect of the well-known Zernike aberrations has been studied partially, assuming vortex beams with trivial radial phase components. However, the traditional Zernike polynomials are not best suitable when dealing with vortex beams, as their OAM spectrum is highly sensitive to some Zernike terms, and completely insensitive to others. We propose the use of a new basis, the OAM-Zernike basis, which consists of the radial aberrations as described by radial Zernike polynomials and of the azimuthal aberrations described in the OAM basis. The traditional tools for the characterization of aberrations of optical instruments can be used, and the results translated to the new basis. This permits the straightforward calculation of the effect of any optical system, such as an OAM detection stage, on the OAM spectrum of an incoming beam. This knowledge permits to correct, *a posteriori*, the effect of instrumental OAM spectrum distortion on the measured spectra. We also found that the knowledge of the radial aberrations is important, as they affect the efficiency of the detection, and in some cases its accuracy. In this new framework, we study the effect of aberrations in common OAM detection methods, and encourage the characterization of those systems using this approach.

© 2010 Optical Society of America

## 1. Introduction

1. G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: Preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. **88**, 013601 (2002). [CrossRef] [PubMed]

2. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature (London) **412**, 313–316 (2001). [CrossRef]

3. J. Lin, X.-C. Yuan, S. H. Tao, and R. E. Burge, “Multiplexing free-space optical signals using superimposed collinear orbital angular momentum states,” Appl. Opt. **46**, 4680–4685 (2007). [CrossRef] [PubMed]

4. G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas'ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express **12**, 5448–5456 (2004). [CrossRef] [PubMed]

5. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. **75**, 826–829 (1995). [CrossRef] [PubMed]

6. S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev. **2**, 299–313 (2008). [CrossRef]

7. G. A. Swartzlander Jr, “Peering into darkness with a vortex spatial filter,” Opt. Lett. **26**, 497–499 (2001). [CrossRef]

8. A. Jesacher, A. Schwaighofer, S. Fürhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, “Wavefront correction of spatial light modulators using an optical vortex image,” Opt. Express **15**, 5801–5808 (2007). [CrossRef] [PubMed]

9. G. A. Tyler and R. W. Boyd, “Influence of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett. **34**, 142–144 (2009). [CrossRef] [PubMed]

10. G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A **25**, 225–230 (2008). [CrossRef]

11. C. Paterson, “Atmospheric Turbulence and Orbital Angular Momentum of Single Photons for Optical Communication,” Phys. Rev. Lett. **94**, 153901 (2005). [CrossRef] [PubMed]

12. Z. Yi-Xin and C. Ji, “Effects of turbulent aberrations on probability distribution of orbital angular momentum for optical communication,” Chin. Phys. Lett. **26**, 074220 (4pp) (2009). [CrossRef]

13. C. Jenkins, “Optical vortex coronagraphs on ground-based telescopes,” Mon. Not. R. Astron. Soc. **384**, 515–524 (2008). [CrossRef]

14. B. R. Boruah and M. A. Neil, “Susceptibility to and correction of azimuthal aberrations in singular light beams,” Opt. Express **14**, 10377–10385 (2006). [CrossRef] [PubMed]

15. E. Compain, S. Poirier, and B. Drevillon, “General and self-consistent method for the calibration of polarization modulators, polarimeters, and mueller-matrix ellipsometers,” Appl. Opt. **38**, 3490–3502 (1999). [CrossRef]

16. H. G. Tompkins and E. A. Irene, ed., *Handbook of Ellipsometry* (Springer, Berlin, 2005). [CrossRef]

1. G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: Preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. **88**, 013601 (2002). [CrossRef] [PubMed]

2. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature (London) **412**, 313–316 (2001). [CrossRef]

3. J. Lin, X.-C. Yuan, S. H. Tao, and R. E. Burge, “Multiplexing free-space optical signals using superimposed collinear orbital angular momentum states,” Appl. Opt. **46**, 4680–4685 (2007). [CrossRef] [PubMed]

4. G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas'ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express **12**, 5448–5456 (2004). [CrossRef] [PubMed]

5. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. **75**, 826–829 (1995). [CrossRef] [PubMed]

7. G. A. Swartzlander Jr, “Peering into darkness with a vortex spatial filter,” Opt. Lett. **26**, 497–499 (2001). [CrossRef]

*a posteriori*, the experimental data. We already applied this work in a new technique for the measurement of the OAM superposition state of light [17

17. N. Uribe-Patarroyo, A. Alvarez-Herrero, and T. Belenguer, “Measurement of the quantum superposition state of an imaging ensemble of photons prepared in orbital angular momentum states using a phase-diversity method,” Phys. Rev. A **81**, 053822 (2010). [CrossRef]

## 2. Theoretical basis

*z*is

*c*

_{1,2}are constants and

*e*.

^{ilϕ}*lϕ*, with topological charge + 1 on the Zernike basis. Although in mathematical expressions we use the Zernikes

19. M. Born and E. Wolf, *Principles of optics: electromagnetic theory of propagation, interference and diffraction of light* (Pergamon Press, New York, 1959). [PubMed]

*ϕ*spans [0, 2

*π*]. If we had chosen [−

*π, π*], the only difference would be a zero-valued piston term. Analytically, the projection of the phase of a beam with topological charge

*l*on the Zernike basis is

*κ*= 2 for

*m*= 0, and

*κ*= 1 otherwise.

*I*

^{±}

*correspond to each one of the integrals on the left. Clearly, this projection (the value of*

^{m,l}*I*

^{±}

*) is not unique. We chose the one that reproduces the OAM phase modulo 2*

^{m,l}*π*(that is, wrapping the phase). If one chooses to unwrap the phase, then it follows trivially that

*I*

^{±}

*=*

^{m,l}*l*×

*I*

^{±}

^{m,}^{1}. This is an indication of the different nature of the OAM functions and Zernike polynomials: the first is a basis for the angular part of complex functions in a circular domain, while the second is a basis for real functions in the unit circle, which is usually used to describe the pure imaginary phase of the generalized pupil of optical systems. Now, using

19. M. Born and E. Wolf, *Principles of optics: electromagnetic theory of propagation, interference and diffraction of light* (Pergamon Press, New York, 1959). [PubMed]

*l*. It's clear that the + coefficients are zero due to the odd nature of vortex +1 phase. In Fig. 2 we can see the OAM +1 phase reconstructed from the first (Wyant ordering) 527 Zernike terms. However, one should not infer from this result that even (+) Zernike coefficients do not affect OAM: if one chooses to wrap the phase, the even coefficients will be non-zero in general, which means that they are not orthogonal to the OAM basis functions, and therefore affects the OAM spectrum, a result well-known for the case of astigmatism [14

14. B. R. Boruah and M. A. Neil, “Susceptibility to and correction of azimuthal aberrations in singular light beams,” Opt. Express **14**, 10377–10385 (2006). [CrossRef] [PubMed]

12. Z. Yi-Xin and C. Ji, “Effects of turbulent aberrations on probability distribution of orbital angular momentum for optical communication,” Chin. Phys. Lett. **26**, 074220 (4pp) (2009). [CrossRef]

*m*= 0. We have from the integral definition of

19. M. Born and E. Wolf, *Principles of optics: electromagnetic theory of propagation, interference and diffraction of light* (Pergamon Press, New York, 1959). [PubMed]

*a*and the orthogonality of

_{n}*independent from the wrapped or unwrapped phase convention.*This means that the radial Zernikes aberrations will never change the topological charge of a beam.

*b*, which should take into account the normalization factor of the radial polynomials. The complex weights

_{n}*c*define the azimuthal phase (the OAM spectrum); and

_{l,p}*r(ρ)*is a real-valued function, which modulates the amplitude of the beam in the radial direction (such as the Laguerre polynomials of Eq. (1)). As we see, this is a more general description than that of Eq. (1), as this can describe, for example, diverging Laguerre-Gaussian beams, or LG beams with radial aberrations. We want to point out that the fact that Eq. (9) applies only to cylindrically symmetric beams is a limitation of all basis, and is proper to the very nature of a basis. When a basis defined in some domain (here, unit circle) needs to be used in a different one, a Gram-Schmidt orthogonalization process is performed to adapt the basis to the new domain [20

20. W. Swantner and W. W. Chow, “Gram-schmidt orthonormalization of zernike polynomials for general aperture shapes,” Appl. Opt. **33**, 1832–1837 (1994). [CrossRef] [PubMed]

## 3. Usage of the new basis in an optical system

17. N. Uribe-Patarroyo, A. Alvarez-Herrero, and T. Belenguer, “Measurement of the quantum superposition state of an imaging ensemble of photons prepared in orbital angular momentum states using a phase-diversity method,” Phys. Rev. A **81**, 053822 (2010). [CrossRef]

21. N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, “Laser beams with phase singularities,” Opt. Quant. Electron. **24**, S951–S962 (1992). [CrossRef]

21. N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, “Laser beams with phase singularities,” Opt. Quant. Electron. **24**, S951–S962 (1992). [CrossRef]

2. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature (London) **412**, 313–316 (2001). [CrossRef]

4. G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas'ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express **12**, 5448–5456 (2004). [CrossRef] [PubMed]

22. S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, “Uncertainty principle for angular position and angular momentum,” New J. Phys. **6**, 103 (2004). [CrossRef]

*≠ Ψ*

_{i}*in general, due to the non-ideal nature of the OSA. The weights of each vortex mode can be different*

_{o}*d*≠

_{l,p}*c*due to azimuthal aberrations, such as astigmatism or coma, or non-symmetrical amplitude changes, such as non-circular apertures or misalignment. The functions

_{l,p}*r*are in general different from

_{o}*r*, and represent radial changes in intensity, such as diffraction at a circular aperture. Finally,

_{i}*d*coefficients, and radial aberrations via the

_{l}*b*coefficients. We will concentrate now on the effect of the

_{n}*d*coefficients, and postpone to the next section how the radial aberrations affect the accuracy of the detection.

_{l}*m*beam coming from the OSG will be modified by the OSA into

*d*coefficients, correspond to the OAM spectrum that an incident 0 OAM beam will have after passage through the system. For a perfect system with no aberrations, only

_{l}*d*≠ 0 and therefore

_{0}*GP*is a constant.

17. N. Uribe-Patarroyo, A. Alvarez-Herrero, and T. Belenguer, “Measurement of the quantum superposition state of an imaging ensemble of photons prepared in orbital angular momentum states using a phase-diversity method,” Phys. Rev. A **81**, 053822 (2010). [CrossRef]

## 4. Effect of aberrations in measurements

14. B. R. Boruah and M. A. Neil, “Susceptibility to and correction of azimuthal aberrations in singular light beams,” Opt. Express **14**, 10377–10385 (2006). [CrossRef] [PubMed]

*ρ*= 0), we will always have unwanted non-zero intensity entering the pinhole if we do not have a pure OAM state, regardless of how small the pinhole is. This is a limitation of this particular measurement technique, and presents a trade-off between the amount of light arriving at the detector and the maximum SNR of the system. To measure the

*m*OAM state, one uses a −

*m*forked hologram, and therefore the original OAM spectrum […

*,m*− 1

*,m,m*+1,…] is transformed into […,−1,0,+1,…]. When focusing onto the pinhole, some unwanted light from the [

*m*±1

*,m*±2,…] states, now [±1, ±2,…], enters the pinhole. As the intensity of vortex beams goes as

*ρ*

^{|}^{2}

*for*

^{l|}*ρ*→ 0, the ratio of the intended signal intensity to the unwanted intensity inside a circular pinhole of radius

*r*is

_{p}*A*is the weight of the

_{m}*m*OAM state in the beam, and only contributions from neighboring states where considered. We see that the reduced SNR comes from a crosstalk between neighboring states, and therefore depend directly on the amplitude of

*A*

_{m}_{±1}. We stress now the importance of this kind of calculations to asses the accuracy of OAM spectrum measurements. Although this technique has been used in several works, a discussion about its accuracy, to our knowledge, has never been made.

### 4.1. Azimuthal aberrations

**14**, 10377–10385 (2006). [CrossRef] [PubMed]

### 4.2. Radial aberrations

*l*vortex with a lower

*l*′ <

*l*vortex with some radial aberrations,

*but without changing the OAM content of the beam.*This is the reason why, if only intensity measurements of OAM spectra are carried out, compensation of the radial aberrations is very important for a correct measurement. In Fig. 5 we have the case of PSF profiles with 0,

*larger*intensity inside both pinhole sizes than the plane wave. This implies that in this case not only

*A*

_{m}_{±1}of Eq. (14) affects SNR, but now the radial dependence of intensity is changed completely, so that

*ρ*

^{2}

*for*

^{l}*ρ*→ 0 is no longer satisfied. Therefore, an optical characterization of this type of system is mandatory, to asses the effect of the system in the measurement of the OAM spectrum of the incoming light.

24. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the Orbital Angular Momentum of a Single Photon,” Phys. Rev. Lett. **88**, 257901 (2002). [CrossRef] [PubMed]

*ρ*from the center of the beam, in the same way that fringes would appear in a normal interferometer. Although this would be evident in applications with high light throughput, not so in low-light applications, and this aberrations can become critical. Therefore, it is important also to characterize this type of measurement devices.

7. G. A. Swartzlander Jr, “Peering into darkness with a vortex spatial filter,” Opt. Lett. **26**, 497–499 (2001). [CrossRef]

25. G. C. G. Berkhout and M. W. Beijersbergen, “Using a multipoint interferometer to measure the orbital angular momentum of light in astrophysics,” J. Opt. A: Pure Appl. Opt. **11**, 094021 (2009). [CrossRef]

## 5. Conclusions

## References and links

1. | G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: Preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. |

2. | A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature (London) |

3. | J. Lin, X.-C. Yuan, S. H. Tao, and R. E. Burge, “Multiplexing free-space optical signals using superimposed collinear orbital angular momentum states,” Appl. Opt. |

4. | G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas'ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express |

5. | H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. |

6. | S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev. |

7. | G. A. Swartzlander Jr, “Peering into darkness with a vortex spatial filter,” Opt. Lett. |

8. | A. Jesacher, A. Schwaighofer, S. Fürhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, “Wavefront correction of spatial light modulators using an optical vortex image,” Opt. Express |

9. | G. A. Tyler and R. W. Boyd, “Influence of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett. |

10. | G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A |

11. | C. Paterson, “Atmospheric Turbulence and Orbital Angular Momentum of Single Photons for Optical Communication,” Phys. Rev. Lett. |

12. | Z. Yi-Xin and C. Ji, “Effects of turbulent aberrations on probability distribution of orbital angular momentum for optical communication,” Chin. Phys. Lett. |

13. | C. Jenkins, “Optical vortex coronagraphs on ground-based telescopes,” Mon. Not. R. Astron. Soc. |

14. | B. R. Boruah and M. A. Neil, “Susceptibility to and correction of azimuthal aberrations in singular light beams,” Opt. Express |

15. | E. Compain, S. Poirier, and B. Drevillon, “General and self-consistent method for the calibration of polarization modulators, polarimeters, and mueller-matrix ellipsometers,” Appl. Opt. |

16. | H. G. Tompkins and E. A. Irene, ed., |

17. | N. Uribe-Patarroyo, A. Alvarez-Herrero, and T. Belenguer, “Measurement of the quantum superposition state of an imaging ensemble of photons prepared in orbital angular momentum states using a phase-diversity method,” Phys. Rev. A |

18. | J. C. Wyant and K. Creath, “Basic Wavefront Aberration Theory for Optical Metrology,” in “ |

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

20. | W. Swantner and W. W. Chow, “Gram-schmidt orthonormalization of zernike polynomials for general aperture shapes,” Appl. Opt. |

21. | N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, “Laser beams with phase singularities,” Opt. Quant. Electron. |

22. | S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, “Uncertainty principle for angular position and angular momentum,” New J. Phys. |

23. | J. W. Goodman, |

24. | J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the Orbital Angular Momentum of a Single Photon,” Phys. Rev. Lett. |

25. | G. C. G. Berkhout and M. W. Beijersbergen, “Using a multipoint interferometer to measure the orbital angular momentum of light in astrophysics,” J. Opt. A: Pure Appl. Opt. |

26. | N. Uribe-Patarroyo, A. Alvarez-Herrero, A. López Ariste, A. Asensio Ramos, T. Belenguer, R. Manso Sainz, C. LeMen, and B. Gelly, “Detecting photons with orbital angular momentum in extended astronomical objects: application to solar observations,” (unpublished). |

**OCIS Codes**

(350.5030) Other areas of optics : Phase

(080.1005) Geometric optics : Aberration expansions

(080.4865) Geometric optics : Optical vortices

(260.6042) Physical optics : Singular optics

**ToC Category:**

Physical Optics

**History**

Original Manuscript: April 26, 2010

Revised Manuscript: June 9, 2010

Manuscript Accepted: June 17, 2010

Published: September 22, 2010

**Citation**

Néstor Uribe-Patarroyo, Alberto Alvarez-Herrero, and Tomás Belenguer, "A comprehensive approach to deal with instrumental optical aberrations effects in high-accuracy photon’s orbital angular momentum spectrum measurements," Opt. Express **18**, 21111-21120 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-20-21111

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### References

- G. Molina-Terriza, J. P. Torres, and L. Torner, "Management of the angular momentum of light: Preparation of photons in multidimensional vector states of angular momentum," Phys. Rev. Lett. 88, 013601 (2002). [CrossRef] [PubMed]
- A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, "Entanglement of the orbital angular momentum states of photons," Nature 412, 313-316 (2001). [CrossRef]
- J. Lin, X.-C. Yuan, S. H. Tao, and R. E. Burge, "Multiplexing free-space optical signals using superimposed collinear orbital angular momentum states," Appl. Opt. 46, 4680-4685 (2007). [CrossRef] [PubMed]
- G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, "Free-space information transfer using light beams carrying orbital angular momentum," Opt. Express 12, 5448-5456 (2004). [CrossRef] [PubMed]
- H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995). [CrossRef] [PubMed]
- S. Franke-Arnold, L. Allen, and M. Padgett, "Advances in optical angular momentum," Laser Photonics Rev. 2, 299-313 (2008). [CrossRef]
- G. A. Swartzlander, Jr., "Peering into darkness with a vortex spatial filter," Opt. Lett. 26, 497-499 (2001). [CrossRef]
- A. Jesacher, A. Schwaighofer, S. Fürhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, "Wavefront correction of spatial light modulators using an optical vortex image," Opt. Express 15, 5801-5808 (2007). [CrossRef] [PubMed]
- G. A. Tyler, and R. W. Boyd, "Influence of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum," Opt. Lett. 34, 142-144 (2009). [CrossRef] [PubMed]
- G. Gbur, and R. K. Tyson, "Vortex beam propagation through atmospheric turbulence and topological charge conservation," J. Opt. Soc. Am. A 25, 225-230 (2008). [CrossRef]
- C. Paterson, "Atmospheric Turbulence and Orbital Angular Momentum of Single Photons for Optical Communication," Phys. Rev. Lett. 94, 153901 (2005). [CrossRef] [PubMed]
- Z. Yi-Xin and C. Ji, "Effects of turbulent aberrations on probability distribution of orbital angular momentum for optical communication," Chin. Phys. Lett. 26, 074220 (4pp) (2009). [CrossRef]
- C. Jenkins, "Optical vortex coronagraphs on ground-based telescopes," Mon. Not. R. Astron. Soc. 384, 515-524 (2008). [CrossRef]
- B. R. Boruah, and M. A. Neil, "Susceptibility to and correction of azimuthal aberrations in singular light beams," Opt. Express 14, 10377-10385 (2006). [CrossRef] [PubMed]
- E. Compain, S. Poirier, and B. Drevillon, "General and self-consistent method for the calibration of polarization modulators, polarimeters, and Mueller-matrix ellipsometers," Appl. Opt. 38, 3490-3502 (1999). [CrossRef]
- H. G. Tompkins, and E. A. Irene, eds., Handbook of Ellipsometry (Springer, Berlin, 2005). [CrossRef]
- N. Uribe-Patarroyo, A. Alvarez-Herrero, and T. Belenguer, "Measurement of the quantum superposition state of an imaging ensemble of photons prepared in orbital angular momentum states using a phase-diversity method," Phys. Rev. A 81, 053822 (2010). [CrossRef]
- J. C. Wyant, and K. Creath, "Basic Wavefront Aberration Theory for Optical Metrology," in "Applied Optics and Optical Engineering, Volume XI,", vol. 11, R. R. Shannon & J. C. Wyant, ed. (1992), vol. 11, pp. 27-39.
- M. Born, and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (Pergamon Press, New York, 1959). [PubMed]
- W. Swantner, and W. W. Chow, "Gram-schmidt orthonormalization of Zernike polynomials for general aperture shapes," Appl. Opt. 33, 1832-1837 (1994). [CrossRef] [PubMed]
- N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, "Laser beams with phase singularities," Opt. Quantum Electron. 24, S951-S962 (1992). [CrossRef]
- S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, "Uncertainty principle for angular position and angular momentum," N. J. Phys. 6, 103 (2004). [CrossRef]
- J. W. Goodman, Introduction to Fourier optics (Roberts and Co. Publishers, Englewood, CO, 2005), 3rd ed.
- J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, "Measuring the Orbital Angular Momentum of a Single Photon," Phys. Rev. Lett. 88, 257901 (2002). [CrossRef] [PubMed]
- G. C. G. Berkhout, and M. W. Beijersbergen, "Using a multipoint interferometer to measure the orbital angular momentum of light in astrophysics," J. Opt. A, Pure Appl. Opt. 11, 094021 (2009). [CrossRef]
- N. Uribe-Patarroyo, A. Alvarez-Herrero, A. López Ariste, A. Asensio Ramos, T. Belenguer, R. Manso Sainz, C. LeMen, and B. Gelly, "Detecting photons with orbital angular momentum in extended astronomical objects: application to solar observations," (unpublished).

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