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

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2012 (2)

2011 (2)

2010 (2)

2009 (1)

2007 (2)

2006 (3)

2005 (2)

2004 (1)

2002 (1)

2000 (1)

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]

1999 (1)

1998 (2)

1992 (1)

1977 (1)

1975 (1)

Aksenov, V. P.

Artal, P.

Atuchin, V. V.

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, 1999).
[CrossRef]

Berry, H. G.

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, 1999).
[CrossRef]

Booth, M.

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

Booth, M. J.

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]

Borchardt, J.

Born, M.

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

Borrego-Varillas, R.

Bueno, J. M.

Campbell, M.

Chamot, S. R.

Changhai, L.

Chanteloup, J.-C.

Cohen, M.

Dainty, C.

de Aldana, J. R. V.

Denk, W.

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]

Drexler, W.

Druon, F.

Dudley, A.

Duparré, M.

Eppich, B.

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

Esposito, S.

Fengjie, X.

Fercher, A. F.

Fernández, E. J.

Flamm, D.

Forbes, A.

Gabrielse, G.

Georges, P.

Goodman, J. W.

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

Guérineau, N.

Hanna, M.

Hebert, T.

Hermann, B.

Izmailov, I. V.

Jukaitis, R.

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]

Kaiser, T.

Kanev, F. Y.

Kawata, S.

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]

Keen, S.

Kochemasov, G. G.

Kulikov, S. M.

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, 1999).
[CrossRef]

Lane, R. G.

Leach, J.

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, 1999).
[CrossRef]

Litvin, I. A.

Livingston, A. E.

Love, G. D.

Mack-Bucher, J. A.

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]

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, 1999).
[CrossRef]

Maksimchuk, A.

Manachinsky, A. N.

Maslov, N. V.

Moreno-Barriuso, E.

Mourou, G.

Naidoo, D.

Nantel, M.

Navarro, R.

Neil, M.

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

Neil, M. A. A.

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]

Neubert, B.

B. Neubert and B. Eppich, “Influences on the beam propagation ratio M2,” Opt. Commun. 250, 241 – 251 (2005).
[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, 1999).
[CrossRef]

Ogorodnikov, A. V.

Padgett, M. J.

Paschotta, R.

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

Paurisse, M.

Prieto, P. M.

Primot, J.

Queener, H.

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, 1999).
[CrossRef]

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, 1999).
[CrossRef]

Romero, C.

Romero-Borja, F.

Roorda, A.

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, 1999).
[CrossRef]

Roux, F. S.

Rueckel, M.

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]

Saleh, B. E. A.

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

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, 1999).
[CrossRef]

Sattmann, H.

Saunter, C.

Schermer, R. T.

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, 1999).
[CrossRef]

Shengyang, H.

Soldatenkov, I. S.

Starikov, F. A.

Sukharev, S. A.

Tallon, M.

Tanaka, T.

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]

Teich, M. C.

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

Unterhuber, A.

Velghe, S.

Wattellier, B.

William Donnelly, I.

Wilson, T.

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

Wolf, E.

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

Wyant, J. C.

Zongfu, J.

Appl. Opt. (4)

J. Microsc. (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]

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

Opt. Commun. (1)

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

Opt. Express (7)

Opt. Lett. (8)

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]

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]

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]

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]

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]

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]

Proc. Natl. Acad. Sci. (1)

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]

Other (6)

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

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).

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).

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]

Supplementary Material (2)

» Media 1: MPEG (224 KB)     
» Media 2: MPEG (324 KB)     

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Figures (7)

Fig. 1
Fig. 1

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

Fig. 2
Fig. 2

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

Fig. 3
Fig. 3

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

Fig. 4
Fig. 4

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

Fig. 5
Fig. 5

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

Fig. 6
Fig. 6

Wavefront reconstruction of a fundamental Gaussian beam with extrinsically added wavefront curvature using a lens of focal length f = 1000 mm. (a) Modal power spectrum (insets depict Laguerre Gaussian modes LGp0 used for decomposition). (b) Inter-modal phase differences. (c) Comparison of reconstructed (CGH) and theoretically expected (Sim) wavefront (cross section through center). The inset in (c) depicts the measured two-dimensional wavefront (same scale as (c)). See Media 1 for the decay of higher order mode signal in the diffraction pattern of the hologram, and Media 2 for the corresponding hologram phase pattern, both for a lens of f = 500 mm.

Fig. 7
Fig. 7

(a) Modal decomposition without physical lens and decomposition into modes without curvature. (b) Modal decomposition into modes that incorporate the wavefront curvature measured in Fig. 6 for a physical lens f = 1000 mm. The insets depict the corresponding mode intensities.

Equations (10)

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

U ( r ) = l = 1 N c l Ψ l ( r ) ,
T l ( r ) = ψ l * ( r ) ,
T l cos ( r ) = [ ψ 0 * ( r ) + ψ l * ( r ) ] / 2 , T l sin ( r ) = [ ψ 0 * ( r ) + i ψ l * ( r ) ] / 2
Δ φ l = arctan [ 2 I l sin ρ l 2 ρ 0 2 2 I l cos ρ l 2 ρ 0 2 ] .
T ( r ) = n = 1 3 N 2 T n ( r ) e i K n r .
S = [ S 0 S 1 S 2 S 3 ] = [ | U x | 2 + | U y | 2 | U x | 2 | U y | 2 2 | U x | | U y | cos δ 2 | U x | | U y | sin δ ] = [ I ( 0 ° ) + I ( 90 ° ) I ( 0 ° ) I ( 90 ° ) I ( 45 ° ) I ( 135 ° ) I λ / 4 ( 45 ° ) I λ / 4 ( 135 ° ) ] ,
P ( r ) = 1 2 [ E ( r ) × H * ( r ) ] = 1 2 [ i ω ε 0 ε 1 ( r ) [ × U ( r ) ] × U * ( r ) ] ,
w ( r , z ) P ( r , z ) ,
| P | | P t | P | t w | 2 d A min ,
w ( r ) = λ 2 π Φ ( r ) .

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