Abstract

We investigate segmented Bessel beams that are created by placing different ring apertures behind an axicon that is illuminated with a plane wave. We find an analytical estimate to determine the shortest possible beam segment by deriving a scale-invariant analytical model using appropriate dimensionless parameters such as the wavelength and the axicon angle. This is verified using simulations and measurements, which are in good agreement. The size of the ring apertures was varied from small aperture sizes in the Frauhofer diffraction limit to larger aperture sizes in the classical limit.

© 2017 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
  6. Z. Ding, H. Ren, Y. Zhao, J.S. Nelson, and Z. Chen, “High-resolution optical coherence tomography over a large depth range with an axicon lens,” Opt. Commun. 27, 243–245 (2002).
  7. F. Courvoisier, P. Lacourt, M. Jacquot, M.K. Bhuyan, L. Furfaro, and J.M. Dudley, “Surface nanoprocessing with nondiffracting femtosecond Bessel beams,” Opt. Lett. 3, 3163–3165 (2009).
    [Crossref]
  8. J. Dudutis, P. Gecys, and R. Raciukaitis, “Non-ideal axicon-generated Bessel beam application for intra-volume glass modification,” Opt. Express 24, 28433–28443 (2016).
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  12. J. W Goodman, Introduction to Fourier Optics (Roberts and Company publisher, 2005)
  13. A. Müller, M.C. Wapler, U.T. Schwarz, M. Reisacher, K. Holc, O. Ambacher, and U. Wallrabe, “Quasi-Bessel beams from asymmetric and astigmatic illumination sources,” Opt. Express 24, 17433–17452 (2016).
    [Crossref] [PubMed]

2016 (2)

2010 (1)

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nature Photonics 4, 780–785 (2010).
[Crossref]

2009 (1)

F. Courvoisier, P. Lacourt, M. Jacquot, M.K. Bhuyan, L. Furfaro, and J.M. Dudley, “Surface nanoprocessing with nondiffracting femtosecond Bessel beams,” Opt. Lett. 3, 3163–3165 (2009).
[Crossref]

2008 (1)

2002 (1)

Z. Ding, H. Ren, Y. Zhao, J.S. Nelson, and Z. Chen, “High-resolution optical coherence tomography over a large depth range with an axicon lens,” Opt. Commun. 27, 243–245 (2002).

1995 (1)

1991 (1)

1987 (2)

J. Durnin, J. J. Miceli, and J. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[Crossref]

1960 (1)

Ambacher, O.

Bhuyan, M.K.

F. Courvoisier, P. Lacourt, M. Jacquot, M.K. Bhuyan, L. Furfaro, and J.M. Dudley, “Surface nanoprocessing with nondiffracting femtosecond Bessel beams,” Opt. Lett. 3, 3163–3165 (2009).
[Crossref]

Brzobohatý, O.

Chen, Z.

Z. Ding, H. Ren, Y. Zhao, J.S. Nelson, and Z. Chen, “High-resolution optical coherence tomography over a large depth range with an axicon lens,” Opt. Commun. 27, 243–245 (2002).

Cižmár, T.

Courvoisier, F.

F. Courvoisier, P. Lacourt, M. Jacquot, M.K. Bhuyan, L. Furfaro, and J.M. Dudley, “Surface nanoprocessing with nondiffracting femtosecond Bessel beams,” Opt. Lett. 3, 3163–3165 (2009).
[Crossref]

Ding, Z.

Z. Ding, H. Ren, Y. Zhao, J.S. Nelson, and Z. Chen, “High-resolution optical coherence tomography over a large depth range with an axicon lens,” Opt. Commun. 27, 243–245 (2002).

Dudley, J.M.

F. Courvoisier, P. Lacourt, M. Jacquot, M.K. Bhuyan, L. Furfaro, and J.M. Dudley, “Surface nanoprocessing with nondiffracting femtosecond Bessel beams,” Opt. Lett. 3, 3163–3165 (2009).
[Crossref]

Dudutis, J.

Durnin, J.

J. Durnin, J. J. Miceli, and J. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[Crossref]

Eberly, J.

J. Durnin, J. J. Miceli, and J. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Fahrbach, F. O.

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nature Photonics 4, 780–785 (2010).
[Crossref]

Furfaro, L.

F. Courvoisier, P. Lacourt, M. Jacquot, M.K. Bhuyan, L. Furfaro, and J.M. Dudley, “Surface nanoprocessing with nondiffracting femtosecond Bessel beams,” Opt. Lett. 3, 3163–3165 (2009).
[Crossref]

Gecys, P.

Goodman, J. W

J. W Goodman, Introduction to Fourier Optics (Roberts and Company publisher, 2005)

Holc, K.

Jacquot, M.

F. Courvoisier, P. Lacourt, M. Jacquot, M.K. Bhuyan, L. Furfaro, and J.M. Dudley, “Surface nanoprocessing with nondiffracting femtosecond Bessel beams,” Opt. Lett. 3, 3163–3165 (2009).
[Crossref]

Jiang, Z.

Kenney, C. S.

Lacourt, P.

F. Courvoisier, P. Lacourt, M. Jacquot, M.K. Bhuyan, L. Furfaro, and J.M. Dudley, “Surface nanoprocessing with nondiffracting femtosecond Bessel beams,” Opt. Lett. 3, 3163–3165 (2009).
[Crossref]

Liu, Z.

Lu, Q.

McLeod, J.

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Müller, A.

A. Müller, M.C. Wapler, U.T. Schwarz, M. Reisacher, K. Holc, O. Ambacher, and U. Wallrabe, “Quasi-Bessel beams from asymmetric and astigmatic illumination sources,” Opt. Express 24, 17433–17452 (2016).
[Crossref] [PubMed]

A. Müller, M.C. Wapler, and U. Wallrabe, “Depth-controlled Bessel beams,” 2016 International Conference on Optical MEMS and Nanophotonics (OMN), (2016).

Nelson, J.S.

Z. Ding, H. Ren, Y. Zhao, J.S. Nelson, and Z. Chen, “High-resolution optical coherence tomography over a large depth range with an axicon lens,” Opt. Commun. 27, 243–245 (2002).

Overfelt, P. L.

Raciukaitis, R.

Reisacher, M.

Ren, H.

Z. Ding, H. Ren, Y. Zhao, J.S. Nelson, and Z. Chen, “High-resolution optical coherence tomography over a large depth range with an axicon lens,” Opt. Commun. 27, 243–245 (2002).

Rohrbach, A.

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nature Photonics 4, 780–785 (2010).
[Crossref]

Schwarz, U.T.

Simon, P.

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nature Photonics 4, 780–785 (2010).
[Crossref]

Wallrabe, U.

A. Müller, M.C. Wapler, U.T. Schwarz, M. Reisacher, K. Holc, O. Ambacher, and U. Wallrabe, “Quasi-Bessel beams from asymmetric and astigmatic illumination sources,” Opt. Express 24, 17433–17452 (2016).
[Crossref] [PubMed]

A. Müller, M.C. Wapler, and U. Wallrabe, “Depth-controlled Bessel beams,” 2016 International Conference on Optical MEMS and Nanophotonics (OMN), (2016).

Wapler, M.C.

A. Müller, M.C. Wapler, U.T. Schwarz, M. Reisacher, K. Holc, O. Ambacher, and U. Wallrabe, “Quasi-Bessel beams from asymmetric and astigmatic illumination sources,” Opt. Express 24, 17433–17452 (2016).
[Crossref] [PubMed]

A. Müller, M.C. Wapler, and U. Wallrabe, “Depth-controlled Bessel beams,” 2016 International Conference on Optical MEMS and Nanophotonics (OMN), (2016).

Zemánek, P.

Zhao, Y.

Z. Ding, H. Ren, Y. Zhao, J.S. Nelson, and Z. Chen, “High-resolution optical coherence tomography over a large depth range with an axicon lens,” Opt. Commun. 27, 243–245 (2002).

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

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

Nature Photonics (1)

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nature Photonics 4, 780–785 (2010).
[Crossref]

Opt. Commun. (1)

Z. Ding, H. Ren, Y. Zhao, J.S. Nelson, and Z. Chen, “High-resolution optical coherence tomography over a large depth range with an axicon lens,” Opt. Commun. 27, 243–245 (2002).

Opt. Express (3)

Opt. Lett. (1)

F. Courvoisier, P. Lacourt, M. Jacquot, M.K. Bhuyan, L. Furfaro, and J.M. Dudley, “Surface nanoprocessing with nondiffracting femtosecond Bessel beams,” Opt. Lett. 3, 3163–3165 (2009).
[Crossref]

Phys. Rev. Lett. (1)

J. Durnin, J. J. Miceli, and J. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Other (2)

A. Müller, M.C. Wapler, and U. Wallrabe, “Depth-controlled Bessel beams,” 2016 International Conference on Optical MEMS and Nanophotonics (OMN), (2016).

J. W Goodman, Introduction to Fourier Optics (Roberts and Company publisher, 2005)

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Figures (5)

Fig. 1
Fig. 1 A: Schematic drawing of the segmented Bessel beam region Δz using a ring aperture with aperture size d and an axicon with apex angle γ. The ring aperture directly in front of the axicon limits the Bessel beam region such that we get a smaller, segmented Bessel beam region Δz at position zstart until zend. B: Simulated transverse intensity profiles of segmented Bessel beams with axicon angle γ = 170° and wavelength λ = 632.8 nm. The intensity is scaled logarithmically to illustrated the side lobes. The optimal apertures dmin that we used at different radial positions r are: (i) r = 1 mm, dmin = 197.85 μm, (ii) r = 2 mm, dmin = 281.58 μm and (iii) r = 3 mm, dmin = 344.90 μm.
Fig. 2
Fig. 2 Left: Normalized segmented Bessel beam width Δ as a function of the aperture sizes . The intersection of the Fraunhofer and classical regime is marked with min, where the smallest segmented Bessel beam width Δmin is expected. Right: Comparison of the simulation to the analytical estimate as a function of the apertures sizes , at a fixed scale invariant radius ≈ 79200, different axicon apex angles (γ = 178° and γ = 170°), wavelength (632.8 nm and 473 nm).
Fig. 3
Fig. 3 Comparison of the normalized core intensity around the segmented Bessel beam region at a radius r = 2mm in (i) the Fraunhofer limit d = 100 μm, (ii) near the minimal aperture d = 200 μm and (iii) in the classical limit d = 800 μm for simulations and measurements with wavelength λ = 632.8 nm and an axicon apex angle of γ = 170°. The shaded areas indicate the analytical estimate for the segmented Bessel beam region. The corresponding transverse simulated intensity profile along the optical axis is shown at the bottom of each graph. To illustrate the side lobes the intensities were scaled logarithmically. The actual aperture width in the measurement are: 101.9 ± 0.7 μm, 199.6 ± 1.1 μm and 801.0 ± 0.6 μm.
Fig. 4
Fig. 4 Comparison of the analytical approximation and simulation of the beam length Δ as a function of different aperture size with measurements for two different configurations of the axicon apex angles, radius and wavelength.
Fig. 5
Fig. 5 Comparison of the analytical considerations and simulation of min (red) and Δmin (green) as a function of different radii .

Equations (17)

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

z = r tan ( α ) .
Δ z C = d tan ( α ) .
I ( θ ) = I 0 sinc 2 ( d 2 k sin ( θ ) ) ,
θ = α α = arctan ( r z + Δ z ) arctan ( r z ) r Δ z z 2 1 1 + r z 2 .
θ α .
θ = Δ z tan 2 ( α ) r 2 1 1 + tan 2 ( α ) .
Δ z F = 2 λ r d tan 2 ( α ) ( 1 + tan 2 ( α ) ) .
sinc 2 ( π ζ ) = e 2 , Δ z F 2 λ r ζ d tan 2 ( α ) .
d tan ( α ) 2 λ r ζ d tan 2 ( α ) , i . e . , d 2 2 λ r ζ tan ( α ) .
d ˜ : = d λ and r ˜ : = r λ tan ( α ) .
z ˜ : = z tan ( α ) λ .
Δ z ˜ C = d ˜ and Δ z ˜ F = 2 r ˜ ζ d ˜ .
2 d ˜ α .
d ˜ min = 2 r ˜ ζ = Δ z ˜ min .
Ψ initial = Ψ 0 e i k α r .
P ( r , z ) = e i z k 2 κ r 2 .
Ψ ( z image ) = 1 ( P ( z image ) ( B Ψ initial ) ) .

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