Abstract

We present the reconstruction of a laser beam wavefront from its mode spectrum and investigate in detail the impact of distinct aberrations on the mode composition. The measurement principle is presented on a Gaussian beam that is intentionally distorted by displaying defined aberrations on a spatial light modulator. The comparison of reconstructed and programmed wavefront aberrations yields excellent agreement, proving the high measurement fidelity.

© 2013 Optical Society of America

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  1. L. W. Casperson, “Gaussian light beams in inhomogeneous media,” Appl. Opt. 12, 2434–2441 (1973).
    [CrossRef]
  2. R. F. Lutomirski and H. T. Yura, “Propagation of a finite optical beam in an inhomogeneous medium,” Appl. Opt. 10, 1652–1658 (1971).
    [CrossRef]
  3. 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 University, 1999).
  4. J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19, 1794–1802 (2002).
    [CrossRef]
  5. M. A. A. Neil, R. Juakaitis, 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]
  6. 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]
  7. T. Cizmar, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics 4, 388–394 (2010).
    [CrossRef]
  8. L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
    [CrossRef]
  9. 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]
  10. R. Navarro and E. Moreno-Barriuso, “Laser ray-tracing method for optical testing,” Opt. Lett. 24, 951–953 (1999).
    [CrossRef]
  11. 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]
  12. 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).
  13. S. Velghe, J. Primot, N. Guérineau, M. Cohen, and B. Wattellier, “Wave-front reconstruction from multidirectional phase derivatives generated by multilateral shearing interferometers,” Opt. Lett. 30, 245–247 (2005).
    [CrossRef]
  14. R. G. Lane and M. Tallon, “Wave-front reconstruction using a Shack-Hartmann sensor,” Appl. Opt. 31, 6902–6908 (1992).
    [CrossRef]
  15. 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]
  16. G. P. Andersen, L. Dussan, F. Ghebremichael, and K. Chen, “Holographic wavefront sensor,” Opt. Eng. 48, 085801 (2009).
    [CrossRef]
  17. C. Schulze, D. Naidoo, D. Flamm, O. A. Schmidt, A. Forbes, and M. Duparré, “Wavefront reconstruction by modal decomposition,” Opt. Express 20, 19714–19725 (2012).
    [CrossRef]
  18. N. Hodgson and H. Weber, Laser Resonators and Beam Propagation (Springer, 2005).
  19. 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]
  20. ISO, “Lasers and laser-related equipment—Test methods for determination of the shape of a laser beam wavefront—Part 1: Terminology and fundamental aspects,” , 2003.
  21. C. Schulze, S. Ngcobo, M. Duparré, and A. Forbes, “Modal decomposition without a priori scale information,” Opt. Express 20, 27866–27873 (2012).
    [CrossRef]
  22. 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]
  23. V. Arrizón, U. Ruiz, R. Carrada, and L. A. González, “Pixelated phase computer holograms for the accurate encoding of scalar complex fields,” J. Opt. Soc. Am. A 24, 3500–3507 (2007).
    [CrossRef]
  24. W.-H. Lee, “Binary computer-generated holograms,” Appl. Opt. 18, 3661–3669 (1979).
    [CrossRef]
  25. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1991).
  26. J. L. Rodgers and W. A. Nicewander, “Thirteen ways to look at the correlation coefficient,” Am. Statist. 42, 59–66 (1988).
    [CrossRef]

2012

2011

2010

T. Cizmar, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics 4, 388–394 (2010).
[CrossRef]

2009

2008

2007

2006

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[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]

2005

2004

2002

2000

M. A. A. Neil, R. Juakaitis, 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]

1999

1992

1988

J. L. Rodgers and W. A. Nicewander, “Thirteen ways to look at the correlation coefficient,” Am. Statist. 42, 59–66 (1988).
[CrossRef]

1979

1975

1973

1971

Andersen, G. P.

G. P. Andersen, L. Dussan, F. Ghebremichael, and K. Chen, “Holographic wavefront sensor,” Opt. Eng. 48, 085801 (2009).
[CrossRef]

Arrizón, V.

Artal, P.

Beckers, J.

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 University, 1999).

Beuzit, J.-L.

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 University, 1999).

Booth, M. J.

M. A. A. Neil, R. Juakaitis, 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]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1991).

Borrego-Varillas, R.

Bueno, J. M.

Carrada, R.

Casperson, L. W.

Chamot, S. R.

Chen, K.

G. P. Andersen, L. Dussan, F. Ghebremichael, and K. Chen, “Holographic wavefront sensor,” Opt. Eng. 48, 085801 (2009).
[CrossRef]

Cizmar, T.

T. Cizmar, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics 4, 388–394 (2010).
[CrossRef]

Cohen, M.

Dainty, C.

Davidson, F. M.

de Aldana, J. R. V.

Dholakia, K.

T. Cizmar, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics 4, 388–394 (2010).
[CrossRef]

Drexler, W.

Duparré, M.

Dussan, L.

G. P. Andersen, L. Dussan, F. Ghebremichael, and K. Chen, “Holographic wavefront sensor,” Opt. Eng. 48, 085801 (2009).
[CrossRef]

Esposito, S.

Fercher, A. F.

Fernández, E. J.

Flamm, D.

Forbes, A.

Gbur, G.

Ghebremichael, F.

G. P. Andersen, L. Dussan, F. Ghebremichael, and K. Chen, “Holographic wavefront sensor,” Opt. Eng. 48, 085801 (2009).
[CrossRef]

González, L. A.

Guérineau, N.

Hermann, B.

Hodgson, N.

N. Hodgson and H. Weber, Laser Resonators and Beam Propagation (Springer, 2005).

Juakaitis, R.

M. A. A. Neil, R. Juakaitis, 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]

Kaiser, T.

Kawata, S.

M. A. A. Neil, R. Juakaitis, 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]

Lai, O.

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 University, 1999).

Lane, R. G.

Lee, W.-H.

Léna, P.

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 University, 1999).

Lutomirski, R. F.

Madec, P.-Y.

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 University, 1999).

Manzo, C.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef]

Marrucci, L.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef]

Mazilu, M.

T. Cizmar, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics 4, 388–394 (2010).
[CrossRef]

Moreno-Barriuso, E.

Naidoo, D.

Navarro, R.

Neil, M. A. A.

M. A. A. Neil, R. Juakaitis, 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]

Ngcobo, S.

Nicewander, W. A.

J. L. Rodgers and W. A. Nicewander, “Thirteen ways to look at the correlation coefficient,” Am. Statist. 42, 59–66 (1988).
[CrossRef]

Northcott, M.

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 University, 1999).

Paparo, D.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef]

Prieto, P. M.

Primot, J.

Ricklin, J. C.

Rigaut, F.

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 University, 1999).

Rimmer, M. P.

Roddier, F.

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 University, 1999).

Rodgers, J. L.

J. L. Rodgers and W. A. Nicewander, “Thirteen ways to look at the correlation coefficient,” Am. Statist. 42, 59–66 (1988).
[CrossRef]

Romero, C.

Roso, L.

Rousset, G.

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 University, 1999).

Ruiz, U.

Sandler, D.

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 University, 1999).

Sattmann, H.

Schmidt, O. A.

Schröter, S.

Schulze, C.

Séchaud, M.

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 University, 1999).

Tallon, M.

Tanaka, T.

M. A. A. Neil, R. Juakaitis, 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]

Tyson, R. K.

Unterhuber, A.

Velghe, S.

Wattellier, B.

Weber, H.

N. Hodgson and H. Weber, Laser Resonators and Beam Propagation (Springer, 2005).

Wilson, T.

M. A. A. Neil, R. Juakaitis, 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]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1991).

Wyant, J. C.

Yura, H. T.

Am. Statist.

J. L. Rodgers and W. A. Nicewander, “Thirteen ways to look at the correlation coefficient,” Am. Statist. 42, 59–66 (1988).
[CrossRef]

Appl. Opt.

J. Microsc.

M. A. A. Neil, R. Juakaitis, 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]

J. Opt. Soc. Am. A

Nat. Photonics

T. Cizmar, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics 4, 388–394 (2010).
[CrossRef]

Opt. Eng.

G. P. Andersen, L. Dussan, F. Ghebremichael, and K. Chen, “Holographic wavefront sensor,” Opt. Eng. 48, 085801 (2009).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef]

Other

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 University, 1999).

N. Hodgson and H. Weber, Laser Resonators and Beam Propagation (Springer, 2005).

ISO, “Lasers and laser-related equipment—Test methods for determination of the shape of a laser beam wavefront—Part 1: Terminology and fundamental aspects,” , 2003.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1991).

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Figures (8)

Fig. 1.
Fig. 1.

Decomposition (a)–(c) of an ideal fundamental Gaussian beam and (d)–(f) of the same beam aberrated with defocus. NF, near field intensity; W, wavefront; ρ2, modal power spectrum.

Fig. 2.
Fig. 2.

Simulated relative fundamental mode content ρ0,02 as a function of aberration strength b. Insets depict the shapes of the corresponding aberrations.

Fig. 3.
Fig. 3.

Schematic of the experimental setup to modally decompose differently aberrated Gaussian beams. He–Ne, helium–neon laser; L15, lenses; M, mirror; SLM1,2, spatial light modulator (actually reflective); A, aperture; CCD, camera.

Fig. 4.
Fig. 4.

Modal decomposition of a Gaussian beam aberrated with tilt. (a) Modal power spectrum, (b) modal phase spectrum, (c) theoretically expected wavefront, and (d) measured wavefront.

Fig. 5.
Fig. 5.

Modal decomposition of a Gaussian beam aberrated with astigmatism. (a) Modal power spectrum, (b) modal phase spectrum, (c) theoretically expected wavefront, and (d) measured wavefront.

Fig. 6.
Fig. 6.

Modal decomposition of a Gaussian beam aberrated with (a), (b) defocus and (c), (d) third order spherical. (a) Modal power spectrum, (b) reconstructed wavefront, (c) modal power spectrum, and (d) reconstructed wavefront.

Fig. 7.
Fig. 7.

Experimental setup for measuring the far field of differently aberrated Gaussian beams. He–Ne, helium–neon laser; L13, lenses; M, mirror; SLM1, spatial light modulator; CCD, camera.

Fig. 8.
Fig. 8.

Theoretical and measured far fields FFth and FFm of a fundamental Gaussian beam aberrated with (a), (b) tilt and (c), (d) trefoil.

Tables (1)

Tables Icon

Table 1. Summary of Results, Stating Aberration Strength b, Corresponding Relative Fundamental Mode Power ρ0,02, and the Correlation Coefficient between the Theoretically Expected and Measured Wavefronts (Zernike Polynomials Znm)

Equations (8)

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

U(r)=p,lρp,lup,l(r)exp(iϕp,l),
|P||Pt|P|tW|2dAmin,
P(r)=ε0ω4[i(UtU*U*tU)+2k|U|2ez],
Tp,l(r)=up,l*(r).
Tp,lcos(r)=[uref*(r)+up,l*(r)]/2,Tp,lsin(r)=[uref*(r)+iup,l*(r)]/2.
Ip,lsinρref2+ρp,l2+2ρrefρp,lsinϕp,l,Ip,lcosρref2+ρp,l2+2ρrefρp,lcosϕp,l.
ϕp,l=arctan[2Ip,lsinIp,lρIrefρ2Ip,lcosIp,lρIrefρ],
U=Uiexp(iπbZnm),

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