Abstract

Two approaches for generating flat-top beams (uniform intensity profile) with extended depth of focus are presented. One involves two diffractive optical elements (DOEs) and the other only a single DOE. The results indicate that the depth of focus of such beams strongly depends on the phase distribution at the output of the DOEs. By having uniform phase distribution, it is possible to generate flat-top beams with extended depth of focus.

© 2018 Optical Society of America

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References

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2016 (2)

2013 (1)

2009 (1)

2008 (2)

2007 (4)

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M. Takamoto, F.-L. Hong, R. Higashi, and H. Katori, “An optical lattice clock,” Nature 435, 321–324 (2005).
[Crossref]

2004 (1)

2003 (2)

E. Jane, G. Vidal, W. Dur, P. Zoller, and J. I. Cirac, “Simulation of quantum dynamics with quantum optical systems,” Quantum Inf. Comput. 3, 15–37 (2003).

J. A. Hoffnagle and C. M. Jefferson, “Beam shaping with a plano-aspheric lens pair,” Opt. Eng. 42, 3090–3099 (2003).
[Crossref]

2002 (2)

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1992 (2)

P. W. Malyak, “Two-mirror unobscured optical system for reshaping the irradiance distribution of a laser system,” Appl. Opt. 31, 4377–4383 (1992).
[Crossref]

A. Parent, M. Morin, and P. Lavigne, “Propagation of super-Gaussian field distributions,” Opt. Quantum Electron. 24, S1071–S1079 (1992).
[Crossref]

1982 (3)

1981 (1)

1974 (1)

1972 (1)

R. W. Grechberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

1965 (1)

Aagedal, H.

Agresti, J.

Auerbach, J. M.

Ballüder, K.

Barton, I. M.

Becker, M. F.

Beth, J.

Bryngdahl, O.

Burch, J.

Caley, A. J.

Chen, D.

Cheng, W.

Cirac, J. I.

E. Jane, G. Vidal, W. Dur, P. Zoller, and J. I. Cirac, “Simulation of quantum dynamics with quantum optical systems,” Quantum Inf. Comput. 3, 15–37 (2003).

D’Ambrosio, E.

Dearden, G.

DeSalvo, R.

Dickey, F. M.

L. A. Romero and F. M. Dickey, “Lossless laser beam shaping,” J. Opt. Soc. Am. A 13, 751–760 (1996).
[Crossref]

F. M. Dickey and S. C. Holswade, Laser Beam Shaping—Theory and Techniques (Marcel Dekker, 2000).

Dong, B. Z.

Dorrer, C.

Dur, W.

E. Jane, G. Vidal, W. Dur, P. Zoller, and J. I. Cirac, “Simulation of quantum dynamics with quantum optical systems,” Quantum Inf. Comput. 3, 15–37 (2003).

Edwardson, S.

Egner, S.

Ersoy, O. K.

Fienup, J. R.

Forest, D.

Frieden, B. R.

Grechberg, R. W.

R. W. Grechberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Gu, B. Y.

Han, W.

Hao, B.

Heinzen, D. J.

Higashi, R.

M. Takamoto, F.-L. Hong, R. Higashi, and H. Katori, “An optical lattice clock,” Nature 435, 321–324 (2005).
[Crossref]

Hoffnagle, J. A.

Holswade, S. C.

F. M. Dickey and S. C. Holswade, Laser Beam Shaping—Theory and Techniques (Marcel Dekker, 2000).

Hong, F.-L.

M. Takamoto, F.-L. Hong, R. Higashi, and H. Katori, “An optical lattice clock,” Nature 435, 321–324 (2005).
[Crossref]

Jane, E.

E. Jane, G. Vidal, W. Dur, P. Zoller, and J. I. Cirac, “Simulation of quantum dynamics with quantum optical systems,” Quantum Inf. Comput. 3, 15–37 (2003).

Jefferson, C. M.

Karpenko, V. P.

Kastner, C. J.

Katori, H.

M. Takamoto, F.-L. Hong, R. Higashi, and H. Katori, “An optical lattice clock,” Nature 435, 321–324 (2005).
[Crossref]

Kohn, N.

Kreuzer, J.

J. Kreuzer, Laser Light Redistribution in Illuminating Optical Signal Processing Systems (MIT, 1965).

Kuang, Z.

Kunkel, W. M.

Lagrange, B.

Lavigne, P.

A. Parent, M. Morin, and P. Lavigne, “Propagation of super-Gaussian field distributions,” Opt. Quantum Electron. 24, S1071–S1079 (1992).
[Crossref]

Leger, J.

Leger, J. R.

Legger, J. R.

Li, J.

Li, Y.

Liang, J.

Liu, D.

Liu, J.

Mackowsky, J. M.

Malyak, P. W.

Michel, C.

Miller, J.

Montorio, J. L.

Morgado, N.

Morin, M.

A. Parent, M. Morin, and P. Lavigne, “Propagation of super-Gaussian field distributions,” Opt. Quantum Electron. 24, S1071–S1079 (1992).
[Crossref]

Muller-Quade, J.

Oak, S. M.

Parent, A.

A. Parent, M. Morin, and P. Lavigne, “Propagation of super-Gaussian field distributions,” Opt. Quantum Electron. 24, S1071–S1079 (1992).
[Crossref]

Perrie, W.

Pinard, L.

Ranganathan, K.

Remilleux, A.

Romero, L. A.

Rudolph, J.

Saxton, W. O.

R. W. Grechberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Schmid, M.

Scott, P. W.

Shealy, D. L.

Simoni, B.

Southwell, W. H.

Sundar, R.

Taghizadeh, M. R.

Takamoto, M.

M. Takamoto, F.-L. Hong, R. Higashi, and H. Katori, “An optical lattice clock,” Nature 435, 321–324 (2005).
[Crossref]

Tarallo, M. G.

Thomson, M. J.

Veldkamp, W. B.

Vidal, G.

E. Jane, G. Vidal, W. Dur, P. Zoller, and J. I. Cirac, “Simulation of quantum dynamics with quantum optical systems,” Quantum Inf. Comput. 3, 15–37 (2003).

Waddie, A. J.

Wang, Z.

Willems, P.

Wyrowski, F.

Yang, G. Z.

Yang, Z. H.

Zhan, Q.

Zhuang, J. Y.

Zoller, P.

E. Jane, G. Vidal, W. Dur, P. Zoller, and J. I. Cirac, “Simulation of quantum dynamics with quantum optical systems,” Quantum Inf. Comput. 3, 15–37 (2003).

Zuegel, J. D.

Appl. Opt. (19)

P. W. Scott and W. H. Southwell, “Reflective optics for irradiance redistribution of laser beams: design,” Appl. Opt. 20, 1606–1610 (1981).
[Crossref]

W. B. Veldkamp and C. J. Kastner, “Beam profile shaping for laser radars that use detector arrays,” Appl. Opt. 21, 345–356 (1982).
[Crossref]

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[Crossref]

W. B. Veldkamp, “Laser beam profile shaping with interlaced binary diffraction gratings,” Appl. Opt. 21, 3209–3212 (1982).
[Crossref]

P. W. Malyak, “Two-mirror unobscured optical system for reshaping the irradiance distribution of a laser system,” Appl. Opt. 31, 4377–4383 (1992).
[Crossref]

J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. 32, 1737–1746 (1993).
[Crossref]

G. Z. Yang, B. Z. Dong, B. Y. Gu, J. Y. Zhuang, and O. K. Ersoy, “Gerchberg–Saxton and Yang–Gu algorithms for phase retrieval in a nonunitary transform system: a comparison,” Appl. Opt. 33, 209–218 (1994).
[Crossref]

J. M. Auerbach and V. P. Karpenko, “Serrated-aperture apodizers for high-energy laser systems,” Appl. Opt. 33, 3179–3183 (1994).
[Crossref]

K. Ballüder and M. R. Taghizadeh, “Regenerative ring-laser design by use of an intracavity diffractive mode-selecting element,” Appl. Opt. 38, 5768–5774 (1999).
[Crossref]

B. R. Frieden, “Lossless conversion of a plane laser wave to a plane wave of uniform irradiance,” Appl. Opt. 4, 1400–1403 (1965).
[Crossref]

J. A. Hoffnagle and C. M. Jefferson, “Design and performance of a refractive optical system that converts a Gaussian to a flat-top beam,” Appl. Opt. 39, 5488–5499 (2000).
[Crossref]

D. L. Shealy and J. A. Hoffnagle, “Laser beam shaping profiles and propagation,” Appl. Opt. 45, 5118–5131 (2006).
[Crossref]

M. G. Tarallo, J. Miller, J. Agresti, E. D’Ambrosio, R. DeSalvo, D. Forest, B. Lagrange, J. M. Mackowsky, C. Michel, J. L. Montorio, N. Morgado, L. Pinard, A. Remilleux, B. Simoni, and P. Willems, “Generation of a flat-top laser beam for gravitational wave detectors by means of a nonspherical Fabry–Perot resonator,” Appl. Opt. 46, 6648–6654 (2007).
[Crossref]

B. Hao and J. R. Legger, “Polarization beam shaping,” Appl. Opt. 46, 8211–8217 (2007).
[Crossref]

R. Sundar, K. Ranganathan, and S. M. Oak, “Generation of flattened Gaussian beam profiles in a Nd:YAG laser with a Gaussian mirror resonator,” Appl. Opt. 47, 147–152 (2008).
[Crossref]

B. Hao, J. Burch, and J. Leger, “Smallest flattop focus by polarization engineering,” Appl. Opt. 47, 2931–2940 (2008).
[Crossref]

J. Liang, J. Rudolph, N. Kohn, M. F. Becker, and D. J. Heinzen, “1.5% root-mean-square flat-intensity laser beam formed using a binary-amplitude spatial light modulator,” Appl. Opt. 48, 1955–1962 (2009).
[Crossref]

W. Cheng, W. Han, and Q. Zhan, “Compact flattop laser beam shaper using vectorial vortex,” Appl. Opt. 52, 4608–4612 (2013).
[Crossref]

J. Li, Z. Kuang, S. Edwardson, W. Perrie, D. Liu, and G. Dearden, “Imaging-based amplitude laser beam shaping for material processing by 2D reflectivity tuning of a spatial light modulator,” Appl. Opt. 55, 1095–1100 (2016).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

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

Nature (1)

M. Takamoto, F.-L. Hong, R. Higashi, and H. Katori, “An optical lattice clock,” Nature 435, 321–324 (2005).
[Crossref]

Opt. Eng. (1)

J. A. Hoffnagle and C. M. Jefferson, “Beam shaping with a plano-aspheric lens pair,” Opt. Eng. 42, 3090–3099 (2003).
[Crossref]

Opt. Express (3)

Opt. Lett. (3)

Opt. Quantum Electron. (1)

A. Parent, M. Morin, and P. Lavigne, “Propagation of super-Gaussian field distributions,” Opt. Quantum Electron. 24, S1071–S1079 (1992).
[Crossref]

Optik (1)

R. W. Grechberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Quantum Inf. Comput. (1)

E. Jane, G. Vidal, W. Dur, P. Zoller, and J. I. Cirac, “Simulation of quantum dynamics with quantum optical systems,” Quantum Inf. Comput. 3, 15–37 (2003).

Other (3)

F. M. Dickey and S. C. Holswade, Laser Beam Shaping—Theory and Techniques (Marcel Dekker, 2000).

J. Kreuzer, Laser Light Redistribution in Illuminating Optical Signal Processing Systems (MIT, 1965).

D. L. Shealy, History of Beam Shaping (CRC Press, 2005).

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Figures (10)

Fig. 1.
Fig. 1. Representative intensity distribution of super-Gaussian flat-top beams before and after propagation. (a) and (b), the uniform intensity distribution of a nearly ideal flat-top beam (super-Gaussian, n=50) distorts very fast upon propagation; (c) and (d), a nearly flat-top beam (super-Gaussian, n=10) with round corners maintains the uniform intensity distribution upon propagation. Beam waist ω0=2  mm, and wavelength is λ=1  μm.
Fig. 2.
Fig. 2. Conversion of a Gaussian beam to a super-Gaussian flat-top beam of order n=10 by a diffractive optical element. (a) Basic arrangement; (b) and (c), the evolution of intensity and phase distribution of the beam as a function of the working distance z, where phases are wrapped in between [π to π]. Here DOE1 is designed such that the flat-top beam is formed at z=0.4  m.
Fig. 3.
Fig. 3. Simulated results of converting a Gaussian beam to a super-Gaussian flat-top beam with order n=10, for a two-DOE-based design. Input/output beam waist ω0=2  mm, and working distance z=0.4  m. The green dashed lines mark the region of 90% of the total energy of the beams.
Fig. 4.
Fig. 4. Evolution of intensity cross section of Gaussian and super-Gaussian flat-top beams with different orders n as a function of propagation length. (a) n=2; (b) n=6; (c) n=10; and (d) n=50. The depth of focus decreases when n increases. The size of beam waist is ω0=2  mm, which corresponds to a Rayleigh range of ZR=11.8  m (ZR is calculated for a Gaussian beam). The working distance z=0.4  m.
Fig. 5.
Fig. 5. Calculated flatness as a function of propagation length for Gaussian and super-Gaussian beams with different orders n. The flatness of the beam increases with n, but at the same time also leads to decrease in the depth of focus. Blue curve, Gaussian beam (n=2); green curve, super-Gaussian flat-top beam with n=6; red curve, super-Gaussian flat-top beam with n=10; black curve, super-Gaussian flat-top beam with n=50. The size of beam waist is ω0=2  mm, which corresponds to a Rayleigh range of ZR=11.8  m (ZR is calculated for a Gaussian beam). The working distance z=0.4  m.
Fig. 6.
Fig. 6. Required phase range (blue) and sum of squares error (red) as a function of the working distance between the two DOEs for super-Gaussian flat-top beam with order n=6 and beam waist ω0=2  mm. The error is small for 0.3  m<z<2  m, and the phase range decreases for longer working distance.
Fig. 7.
Fig. 7. Evolution of intensity cross section of Gaussian and super-Gaussian flat-top beams with different orders n as a function of propagation length. (a) n=2; (b) n=6; (c) n=10; and (d) n=50. The beams with large n show shorter depth of focus than those with small n. The size of beam waist is ω0=2  mm, which corresponds to a Rayleigh range of ZR=11.8  m (ZR is calculated for a Gaussian beam). The working distance z=0.4  m.
Fig. 8.
Fig. 8. Calculated flatness as a function of propagation length for Gaussian and super-Gaussian beams with different orders n. The flatness of the beam increases with n, but at the same time also leads to decrease in the depth of focus. Blue curve, Gaussian beam (n=2); green curve, super-Gaussian flat-top beam with n=6; red curve, super-Gaussian flat-top beam with n=10; black curve, super-Gaussian flat-top beam with n=50. The size of beam waist is ω0=2  mm, which corresponds to a Rayleigh range of ZR=11.8  m (ZR is calculated for a Gaussian beam). The working distance z=0.4  m.
Fig. 9.
Fig. 9. Evolution of intensity cross section of super-Gaussian flat-top beam with n=6 as a function of propagation length for various working distances (a) z=10  cm, (b) z=20  cm, (c) z=50  cm, and (d) z=150  cm. The depth of focus increases with the working distance z. Beam waist ω0=2  mm.
Fig. 10.
Fig. 10. Calculated flatness as a function of propagation length for super-Gaussian flat-top beam with order n=6, for various working distances z. The results with various working distances are also compared with two-DOE-based design. The depth of focus increases with the working distance z.

Equations (1)

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E(x,y)=E0e(x2+y2ω0)n,