Abstract

In this paper, we explore theoretically and experimentally the laser beam shaping ability resulting from the coaxial superposition of two coherent Gaussian beams (GBs). This technique is classified under interferometric laser beam shaping techniques contrasting with the usual ones based on diffraction. The experimental setup does not involve the use of some two-wave interferometer but uses a spatial light modulator for the generation of the necessary interference term. This allows one to avoid the thermal drift occurring in interferometers and gives a total flexibility of the key parameter setting the beam transformation. In particular, we demonstrate the reshaping of a GB into a bottle beam or top-hat beam in the focal plane of a focusing lens.

© 2013 Optical Society of America

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References

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  1. F. M. Dickey and S. G. Holswade, eds., Laser Beam Shaping (CRC-Taylor & Francis Group, 2006).
  2. V. A. Soifer, ed., Methods for Computer Design of Diffractive Optical Elements (Wiley, 2002).
  3. J. B. Steward, B. E. A. Saleh, M. C. Teich, and J. T. Fourkas, “Experimental demonstration of polarization-assisted transverse and axial optical superresolution,” Opt. Commun. 241, 315–319 (2004).
    [CrossRef]
  4. I. D. Maleev and G. A. Swartzlander, “Composite optical vortices,” J. Opt. Soc. Am. B 20, 1169–1176 (2003).
  5. V. Pyragaite and A. Stabinis, “Interference of intersecting singular beams,” Opt. Commun. 220, 247–255 (2003).
    [CrossRef]
  6. T. Ando, N. Matsumoto, Y. Ohtake, Y. Takiguchi, and T. Inoue, “Structure of optical singularities in coaxial superpositions of Laguerre–Gaussian modes,” J. Opt. Soc. Am. A 27, 2602–2612 (2010).
    [CrossRef]
  7. P. Xu, X. He, J. Wang, and M. Zhan, “Trapping a single atom in a blue detuned optical bottle beam trap,” Opt. Lett. 35, 2164–2166 (2010).
    [CrossRef]
  8. J. Arlt and M. J. Padgett, “Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical bottle beam,” Opt. Lett. 25, 191–193 (2000).
    [CrossRef]
  9. P. T. Tai, W. F. Hsieh, and C. H. Chen, “Direct generation of optical bottle beams from a tightly focused end-pumped solid-state laser,” Opt. Express 12, 5827–5833 (2004).
    [CrossRef]
  10. C. H. Chen, P. T. Tai, and W. F. Hsieh, “Bottle beam from a bare laser for single-beam trapping,” Appl. Opt. 43, 6001–6006 (2004).
    [CrossRef]
  11. B. P. S. Ahluwalia, X. C. Yuan, and S. H. Tao, “Generation of self-imaged optical bottle beams,” Opt. Commun. 238, 177–184 (2004).
    [CrossRef]
  12. L. Isenhower, W. Williams, A. Dally, and M. Saffman, “Atom trapping in an interferometrically generated bottle beam trap,” Opt. Lett. 34, 1159–1161 (2009).
    [CrossRef]
  13. 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]
  14. 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]

2012 (1)

2010 (2)

2009 (1)

2007 (1)

2004 (4)

P. T. Tai, W. F. Hsieh, and C. H. Chen, “Direct generation of optical bottle beams from a tightly focused end-pumped solid-state laser,” Opt. Express 12, 5827–5833 (2004).
[CrossRef]

C. H. Chen, P. T. Tai, and W. F. Hsieh, “Bottle beam from a bare laser for single-beam trapping,” Appl. Opt. 43, 6001–6006 (2004).
[CrossRef]

B. P. S. Ahluwalia, X. C. Yuan, and S. H. Tao, “Generation of self-imaged optical bottle beams,” Opt. Commun. 238, 177–184 (2004).
[CrossRef]

J. B. Steward, B. E. A. Saleh, M. C. Teich, and J. T. Fourkas, “Experimental demonstration of polarization-assisted transverse and axial optical superresolution,” Opt. Commun. 241, 315–319 (2004).
[CrossRef]

2003 (2)

I. D. Maleev and G. A. Swartzlander, “Composite optical vortices,” J. Opt. Soc. Am. B 20, 1169–1176 (2003).

V. Pyragaite and A. Stabinis, “Interference of intersecting singular beams,” Opt. Commun. 220, 247–255 (2003).
[CrossRef]

2000 (1)

Ahluwalia, B. P. S.

B. P. S. Ahluwalia, X. C. Yuan, and S. H. Tao, “Generation of self-imaged optical bottle beams,” Opt. Commun. 238, 177–184 (2004).
[CrossRef]

Ando, T.

Arlt, J.

Arrizón, V.

Carrada, R.

Chen, C. H.

Dally, A.

Duparré, M.

Flamm, D.

Forbes, A.

Fourkas, J. T.

J. B. Steward, B. E. A. Saleh, M. C. Teich, and J. T. Fourkas, “Experimental demonstration of polarization-assisted transverse and axial optical superresolution,” Opt. Commun. 241, 315–319 (2004).
[CrossRef]

González, L. A.

He, X.

Hsieh, W. F.

Inoue, T.

Isenhower, L.

Maleev, I. D.

Matsumoto, N.

Naidoo, D.

Ohtake, Y.

Padgett, M. J.

Pyragaite, V.

V. Pyragaite and A. Stabinis, “Interference of intersecting singular beams,” Opt. Commun. 220, 247–255 (2003).
[CrossRef]

Ruiz, U.

Saffman, M.

Saleh, B. E. A.

J. B. Steward, B. E. A. Saleh, M. C. Teich, and J. T. Fourkas, “Experimental demonstration of polarization-assisted transverse and axial optical superresolution,” Opt. Commun. 241, 315–319 (2004).
[CrossRef]

Schulze, C.

Stabinis, A.

V. Pyragaite and A. Stabinis, “Interference of intersecting singular beams,” Opt. Commun. 220, 247–255 (2003).
[CrossRef]

Steward, J. B.

J. B. Steward, B. E. A. Saleh, M. C. Teich, and J. T. Fourkas, “Experimental demonstration of polarization-assisted transverse and axial optical superresolution,” Opt. Commun. 241, 315–319 (2004).
[CrossRef]

Swartzlander, G. A.

Tai, P. T.

Takiguchi, Y.

Tao, S. H.

B. P. S. Ahluwalia, X. C. Yuan, and S. H. Tao, “Generation of self-imaged optical bottle beams,” Opt. Commun. 238, 177–184 (2004).
[CrossRef]

Teich, M. C.

J. B. Steward, B. E. A. Saleh, M. C. Teich, and J. T. Fourkas, “Experimental demonstration of polarization-assisted transverse and axial optical superresolution,” Opt. Commun. 241, 315–319 (2004).
[CrossRef]

Wang, J.

Williams, W.

Xu, P.

Yuan, X. C.

B. P. S. Ahluwalia, X. C. Yuan, and S. H. Tao, “Generation of self-imaged optical bottle beams,” Opt. Commun. 238, 177–184 (2004).
[CrossRef]

Zhan, M.

Appl. Opt. (1)

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (1)

Opt. Commun. (3)

V. Pyragaite and A. Stabinis, “Interference of intersecting singular beams,” Opt. Commun. 220, 247–255 (2003).
[CrossRef]

J. B. Steward, B. E. A. Saleh, M. C. Teich, and J. T. Fourkas, “Experimental demonstration of polarization-assisted transverse and axial optical superresolution,” Opt. Commun. 241, 315–319 (2004).
[CrossRef]

B. P. S. Ahluwalia, X. C. Yuan, and S. H. Tao, “Generation of self-imaged optical bottle beams,” Opt. Commun. 238, 177–184 (2004).
[CrossRef]

Opt. Express (1)

Opt. Lett. (4)

Other (2)

F. M. Dickey and S. G. Holswade, eds., Laser Beam Shaping (CRC-Taylor & Francis Group, 2006).

V. A. Soifer, ed., Methods for Computer Design of Diffractive Optical Elements (Wiley, 2002).

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Figures (11)

Fig. 1.
Fig. 1.

Transverse intensity distribution I(ρ)=|u(ρ)|2 of the CGB u(ρ) defined by Eqs. (1) and (2) for K=2 and W=1mm.

Fig. 2.
Fig. 2.

Three possible scenarios for the behavior of the two GBs superimposed in the field u.

Fig. 3.
Fig. 3.

On-axis intensity distribution of the focused mixed two GBs for K=3 and W=2.5mm. The focusing lens has a focal length f=200mm. As parameter K is reduced, one observes that the two intensity peaks get closer until overlapping.

Fig. 4.
Fig. 4.

Transverse intensity distribution, for K=3 and W=2.5mm, in three planes: (a) Z=166mm, i.e., the first focusing plane in Fig. 4, (b) Z=250mm, i.e., the second focusing plane in Fig. 4, and (c) Z=f=200mm, i.e., the focal plane of the lens.

Fig. 5.
Fig. 5.

Transverse intensity distribution, for W=2.5mm, in the focal plane of the focusing lens of focal length f=125mm. For K=0.17, the incident GB is transformed into a top-hat profile.

Fig. 6.
Fig. 6.

For K=0.4 the coaxial superposition of two coherent GBs is able to produce a bottle beam straddling the focal plane Z=f of the focusing lens.

Fig. 7.
Fig. 7.

Experimental setup for investigating the behavior of the superposition of two GBs.

Fig. 8.
Fig. 8.

Experimental beam shaping in the focal plane of the focusing lens with K=0.17, K=0.2, and K=0.32.

Fig. 9.
Fig. 9.

Experimental beam shaping for K=0.4 before the focal plane, i.e., at (zf)/f=0.08.

Fig. 10.
Fig. 10.

Experimental beam shaping for K=0.4 at the focal plane, i.e., z=f.

Fig. 11.
Fig. 11.

Experimental beam shaping for K=0.4 after the focal plane, i.e., at (zf)/f=+0.08.

Equations (11)

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u(r)=E0cos(βr2)exp[r2W2],
β=2πKW2,
u(r)=E02{exp[ikr22R]exp[r2W2]+exp[+ikr22R]exp[r2W2]}.
KW2=12λR.
Kc=W22λR.
R1,2=fR1,2fR1,2
W01,2=W1+(πW2λR1,2)2.
Z01,2=R1,21+(λR1,2πW2)2.
u1,2=E01,2·W01,2W1,2exp[r2W1,22]exp[i(kZ1,2+kr22R1,2arctg(Z1,2ZR1,2))],
ZR1,2=πW01,22λandW1,2=W01,21+(Z1,2ZR1,2)2andR1,2=Z1,2[1+(ZR1,2Z1,2)2].
E02=E01·W01W02.

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