Abstract

Tailoring of the transverse intensity profiles of propagation-invariant optical fields is considered. The design of diffractive elements capable of realizing such fields by Fourier synthesis is discussed. High-efficiency realization of finite-aperture approximations of the constructed fields is demonstrated in a system consisting of two multilevel diffractive elements. The first element is a diffractive toroidal lens, which focuses the incident field into a ring pattern. The second diffractive element, located at the focal plane of the first element, introduces the phase modulation necessary to realize the desired transverse intensity profile behind a separate collimating lens. The influence of the fabrication errors of the diffractive elements on the fidelity of the propagation-invariant spot array is simulated, and system-integration aspects based on substrate-mode planar-integrated optics are considered.

© 2000 Optical Society of America

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    [CrossRef]
  2. J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  3. G. Indebetouw, “Nondiffracting optical fields: some remarks on their analysis and synthesis,” J. Opt. Soc. Am. A 6, 150–152 (1989).
    [CrossRef]
  4. J. Durnin, J. H. Eberly, “Diffraction-free arrangement,” U.S. patent4,887,885 (December, 19, 1989).
  5. K. Uehara, H. Kikuchi, “Generation of nearly diffraction-free laser beams,” Appl. Phys. B: Photophys. Laser Chem. 48, 125–129 (1989).
    [CrossRef]
  6. J. K. Jabczynski, “A diffraction-free resonator,” Opt. Commun. 77, 292–294 (1990).
    [CrossRef]
  7. Y. Lin, W. Seka, J. H. Eberly, H. Huang, D. L. Brown, “Experimental investigation of Bessel beam characteristics,” Appl. Opt. 31, 2708–2713 (1992).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  12. J. Turunen, A. Vasara, A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
    [CrossRef] [PubMed]
  13. A. Vasara, J. Turunen, A. T. Friberg, “Realization of general nondiffractive beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989).
    [CrossRef] [PubMed]
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    [CrossRef]
  15. L. Niggl, T. Lanzl, M. Maier, “Properties of Bessel beams generated by periodic gratings of circular symmetry,” J. Opt. Soc. Am. A 14, 27–33 (1997).
    [CrossRef]
  16. P. Vahimaa, V. Kettunen, M. Kuittinen, J. Turunen, A. T. Friberg, “Electromagnetic analysis of nonparaxial Bessel beams generated by diffractive axicons,” J. Opt. Soc. Am. A 14, 1817–1824 (1997).
    [CrossRef]
  17. M. Honkanen, J. Turunen, “Tandem systems for efficient generation of uniform-axial-intensity Bessel fields,” Opt. Commun. 154, 368–375 (1998).
    [CrossRef]
  18. V. Kettunen, J. Turunen, “Propagation-invariant spot arrays,” Opt. Lett. 23, 1247–1249 (1998).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  21. P. Laakkonen, J. Lautanen, V. Kettunen, J. Turunen, M. Schirmer, “Multilevel diffractive elements in SiO2 by electron beam lithography and proportional etching with analogue negative resist,” J. Mod. Opt. 46, 1295–1307 (1999).
    [CrossRef]
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    [CrossRef]
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  25. J. Turunen, F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications (Wiley—VCH, Berlin, 1997), Sect. 1.2.1.
  26. Ref. 25, Sect. 1.3, Chap. 6.
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    [CrossRef]
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    [CrossRef] [PubMed]

1999 (2)

N. Guérineau, J. Primot, “Nondiffracting array generation using an N-wave interferometer,” J. Opt. Soc. Am. A 16, 293–298 (1999).
[CrossRef]

P. Laakkonen, J. Lautanen, V. Kettunen, J. Turunen, M. Schirmer, “Multilevel diffractive elements in SiO2 by electron beam lithography and proportional etching with analogue negative resist,” J. Mod. Opt. 46, 1295–1307 (1999).
[CrossRef]

1998 (2)

M. Honkanen, J. Turunen, “Tandem systems for efficient generation of uniform-axial-intensity Bessel fields,” Opt. Commun. 154, 368–375 (1998).
[CrossRef]

V. Kettunen, J. Turunen, “Propagation-invariant spot arrays,” Opt. Lett. 23, 1247–1249 (1998).
[CrossRef]

1997 (3)

1995 (1)

1994 (1)

J. Jahns, “Planar packaging of free-space optical interconnects,” Proc. IEEE 82, 1623–1631 (1994).
[CrossRef]

1992 (3)

1990 (1)

J. K. Jabczynski, “A diffraction-free resonator,” Opt. Commun. 77, 292–294 (1990).
[CrossRef]

1989 (6)

1988 (1)

1987 (2)

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

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

1980 (2)

G. Roy, R. Tremblay, “Influence of the divergence of a laser beam to the axial intensity distribution of an axicon,” Opt. Commun. 34, 1–3 (1980).
[CrossRef]

E. N. Leith, G. Collins, I. Khoo, T. Wynn, “Correlation image formation with an axicon,” J. Opt. Soc. Am. 70, 141–145 (1980).
[CrossRef]

1970 (1)

S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. A 60, 1168–1177 (1970).
[CrossRef]

1964 (1)

J. A. Nedler, R. Mead, “A simplex method for function minimization,” Comput. J. (UK) 7, 308–313 (1964).

1954 (1)

Bará, S.

Bor, Zs.

Brown, D. L.

Cavallaro, J. R.

Collins, G.

Collins, S. A.

S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. A 60, 1168–1177 (1970).
[CrossRef]

Davidson, N. A.

N. A. Davidson, A. A. Friesem, E. Hasman, “Efficient formation of nondiffracting beams with uniform intensity along the propagation direction,” Opt. Commun. 88, 326–330 (1992).
[CrossRef]

Durnin, J.

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

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

J. Durnin, J. H. Eberly, “Diffraction-free arrangement,” U.S. patent4,887,885 (December, 19, 1989).

Eberly, J. H.

Y. Lin, W. Seka, J. H. Eberly, H. Huang, D. L. Brown, “Experimental investigation of Bessel beam characteristics,” Appl. Opt. 31, 2708–2713 (1992).
[CrossRef] [PubMed]

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

J. Durnin, J. H. Eberly, “Diffraction-free arrangement,” U.S. patent4,887,885 (December, 19, 1989).

Erdélyi, M.

Friberg, A. T.

Friesem, A. A.

N. A. Davidson, A. A. Friesem, E. Hasman, “Efficient formation of nondiffracting beams with uniform intensity along the propagation direction,” Opt. Commun. 88, 326–330 (1992).
[CrossRef]

Guérineau, N.

Hasman, E.

N. A. Davidson, A. A. Friesem, E. Hasman, “Efficient formation of nondiffracting beams with uniform intensity along the propagation direction,” Opt. Commun. 88, 326–330 (1992).
[CrossRef]

Hetherington, D.

Honkanen, M.

M. Honkanen, J. Turunen, “Tandem systems for efficient generation of uniform-axial-intensity Bessel fields,” Opt. Commun. 154, 368–375 (1998).
[CrossRef]

Horváth, Z. L.

Huang, A.

Huang, H.

Huang, Y.-T.

Indebetouw, G.

Jabczynski, J. K.

J. K. Jabczynski, “A diffraction-free resonator,” Opt. Commun. 77, 292–294 (1990).
[CrossRef]

Jahns, J.

J. Jahns, “Planar packaging of free-space optical interconnects,” Proc. IEEE 82, 1623–1631 (1994).
[CrossRef]

J. Jahns, A. Huang, “Planar integration of free-space optical components,” Appl. Opt. 28, 1602–1605 (1989).
[CrossRef] [PubMed]

Jaroszewicz, Z.

Kato, M.

Kettunen, V.

P. Laakkonen, J. Lautanen, V. Kettunen, J. Turunen, M. Schirmer, “Multilevel diffractive elements in SiO2 by electron beam lithography and proportional etching with analogue negative resist,” J. Mod. Opt. 46, 1295–1307 (1999).
[CrossRef]

V. Kettunen, J. Turunen, “Propagation-invariant spot arrays,” Opt. Lett. 23, 1247–1249 (1998).
[CrossRef]

P. Vahimaa, V. Kettunen, M. Kuittinen, J. Turunen, A. T. Friberg, “Electromagnetic analysis of nonparaxial Bessel beams generated by diffractive axicons,” J. Opt. Soc. Am. A 14, 1817–1824 (1997).
[CrossRef]

Khoo, I.

Kikuchi, H.

K. Uehara, H. Kikuchi, “Generation of nearly diffraction-free laser beams,” Appl. Phys. B: Photophys. Laser Chem. 48, 125–129 (1989).
[CrossRef]

Kolodziejczyk, A.

Kostuk, R. K.

Kuittinen, M.

Laakkonen, P.

P. Laakkonen, J. Lautanen, V. Kettunen, J. Turunen, M. Schirmer, “Multilevel diffractive elements in SiO2 by electron beam lithography and proportional etching with analogue negative resist,” J. Mod. Opt. 46, 1295–1307 (1999).
[CrossRef]

Lanzl, T.

Lautanen, J.

P. Laakkonen, J. Lautanen, V. Kettunen, J. Turunen, M. Schirmer, “Multilevel diffractive elements in SiO2 by electron beam lithography and proportional etching with analogue negative resist,” J. Mod. Opt. 46, 1295–1307 (1999).
[CrossRef]

Leith, E. N.

Lin, Y.

Maier, M.

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Sect. 3.2.

McLeod, J. H.

Mead, R.

J. A. Nedler, R. Mead, “A simplex method for function minimization,” Comput. J. (UK) 7, 308–313 (1964).

Miceli, J. J.

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

Nedler, J. A.

J. A. Nedler, R. Mead, “A simplex method for function minimization,” Comput. J. (UK) 7, 308–313 (1964).

Niggl, L.

Primot, J.

Roy, G.

G. Roy, R. Tremblay, “Influence of the divergence of a laser beam to the axial intensity distribution of an axicon,” Opt. Commun. 34, 1–3 (1980).
[CrossRef]

Sauer, F.

Schirmer, M.

P. Laakkonen, J. Lautanen, V. Kettunen, J. Turunen, M. Schirmer, “Multilevel diffractive elements in SiO2 by electron beam lithography and proportional etching with analogue negative resist,” J. Mod. Opt. 46, 1295–1307 (1999).
[CrossRef]

Seka, W.

Sochacki, J.

Sogno, L.

Szabò, G.

Tittel, F. K.

Tremblay, R.

G. Roy, R. Tremblay, “Influence of the divergence of a laser beam to the axial intensity distribution of an axicon,” Opt. Commun. 34, 1–3 (1980).
[CrossRef]

Turunen, J.

P. Laakkonen, J. Lautanen, V. Kettunen, J. Turunen, M. Schirmer, “Multilevel diffractive elements in SiO2 by electron beam lithography and proportional etching with analogue negative resist,” J. Mod. Opt. 46, 1295–1307 (1999).
[CrossRef]

V. Kettunen, J. Turunen, “Propagation-invariant spot arrays,” Opt. Lett. 23, 1247–1249 (1998).
[CrossRef]

M. Honkanen, J. Turunen, “Tandem systems for efficient generation of uniform-axial-intensity Bessel fields,” Opt. Commun. 154, 368–375 (1998).
[CrossRef]

P. Vahimaa, V. Kettunen, M. Kuittinen, J. Turunen, A. T. Friberg, “Electromagnetic analysis of nonparaxial Bessel beams generated by diffractive axicons,” J. Opt. Soc. Am. A 14, 1817–1824 (1997).
[CrossRef]

A. Vasara, J. Turunen, A. T. Friberg, “Realization of general nondiffractive beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989).
[CrossRef] [PubMed]

J. Turunen, A. Vasara, A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
[CrossRef] [PubMed]

J. Turunen, F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications (Wiley—VCH, Berlin, 1997), Sect. 1.2.1.

Uehara, K.

K. Uehara, H. Kikuchi, “Generation of nearly diffraction-free laser beams,” Appl. Phys. B: Photophys. Laser Chem. 48, 125–129 (1989).
[CrossRef]

Vahimaa, P.

Vasara, A.

Wolf, E.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Sect. 3.2.

Wynn, T.

Wyrowski, F.

J. Turunen, F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications (Wiley—VCH, Berlin, 1997), Sect. 1.2.1.

Appl. Opt. (5)

Appl. Phys. B: Photophys. Laser Chem. (1)

K. Uehara, H. Kikuchi, “Generation of nearly diffraction-free laser beams,” Appl. Phys. B: Photophys. Laser Chem. 48, 125–129 (1989).
[CrossRef]

Comput. J. (UK) (1)

J. A. Nedler, R. Mead, “A simplex method for function minimization,” Comput. J. (UK) 7, 308–313 (1964).

J. Mod. Opt. (1)

P. Laakkonen, J. Lautanen, V. Kettunen, J. Turunen, M. Schirmer, “Multilevel diffractive elements in SiO2 by electron beam lithography and proportional etching with analogue negative resist,” J. Mod. Opt. 46, 1295–1307 (1999).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (9)

A. Vasara, J. Turunen, A. T. Friberg, “Realization of general nondiffractive beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989).
[CrossRef] [PubMed]

L. Niggl, T. Lanzl, M. Maier, “Properties of Bessel beams generated by periodic gratings of circular symmetry,” J. Opt. Soc. Am. A 14, 27–33 (1997).
[CrossRef]

P. Vahimaa, V. Kettunen, M. Kuittinen, J. Turunen, A. T. Friberg, “Electromagnetic analysis of nonparaxial Bessel beams generated by diffractive axicons,” J. Opt. Soc. Am. A 14, 1817–1824 (1997).
[CrossRef]

Z. L. Horváth, M. Erdélyi, G. Szabò, Zs. Bor, F. K. Tittel, J. R. Cavallaro, “Generation of nearly nondiffracting Bessel beams with Fabry–Perot interferometer,” J. Opt. Soc. Am. A 14, 3009–3013 (1997).
[CrossRef]

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

G. Indebetouw, “Nondiffracting optical fields: some remarks on their analysis and synthesis,” J. Opt. Soc. Am. A 6, 150–152 (1989).
[CrossRef]

N. Guérineau, J. Primot, “Nondiffracting array generation using an N-wave interferometer,” J. Opt. Soc. Am. A 16, 293–298 (1999).
[CrossRef]

J. Primot, L. Sogno, “Achromatic three-wave (or more) lateral shearing interferometer,” J. Opt. Soc. Am. A 12, 2679–2685 (1995).
[CrossRef]

S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. A 60, 1168–1177 (1970).
[CrossRef]

Opt. Commun. (4)

J. K. Jabczynski, “A diffraction-free resonator,” Opt. Commun. 77, 292–294 (1990).
[CrossRef]

M. Honkanen, J. Turunen, “Tandem systems for efficient generation of uniform-axial-intensity Bessel fields,” Opt. Commun. 154, 368–375 (1998).
[CrossRef]

N. A. Davidson, A. A. Friesem, E. Hasman, “Efficient formation of nondiffracting beams with uniform intensity along the propagation direction,” Opt. Commun. 88, 326–330 (1992).
[CrossRef]

G. Roy, R. Tremblay, “Influence of the divergence of a laser beam to the axial intensity distribution of an axicon,” Opt. Commun. 34, 1–3 (1980).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. Lett. (1)

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

Proc. IEEE (1)

J. Jahns, “Planar packaging of free-space optical interconnects,” Proc. IEEE 82, 1623–1631 (1994).
[CrossRef]

Other (4)

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Sect. 3.2.

J. Turunen, F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications (Wiley—VCH, Berlin, 1997), Sect. 1.2.1.

Ref. 25, Sect. 1.3, Chap. 6.

J. Durnin, J. H. Eberly, “Diffraction-free arrangement,” U.S. patent4,887,885 (December, 19, 1989).

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Figures (8)

Fig. 1
Fig. 1

Optical setup for the production of approximately propagation-invariant fields of arbitrary forms: TL, toroidal lens; DE, phase-modulation plate; L, collimating lens.

Fig. 2
Fig. 2

Required phase function for an element generating J0 and J1 beams centered at (αx, αy)=(-50, 0) and (αx, αy)=(50, 0), respectively.

Fig. 3
Fig. 3

Signal calculated with Eq. (5) and phase from Fig. 2.

Fig. 4
Fig. 4

(a) Cross section of the intensity distributions from Fig. 3 plotted at αy=10 and (b) corresponding cross section of an intensity distribution without optimization of the parameters an.

Fig. 5
Fig. 5

(a) Simulated signal with a real experimental system at a distance of z=200 mm behind lens L; (b)–(d) cross sections of the simulated signal at z=100 mm, z=200 mm, and z=400 mm, respectively.

Fig. 6
Fig. 6

(a) Intensity distribution of the toroidal lens at the focal plane (the source image grabbed is by a CCD camera) and (b) a radial cross section along the line x=y.

Fig. 7
Fig. 7

Experimental intensity distributions across different planes behind the collimating lens: (a) z=100 mm, (b) z=250 mm, and (c) z=400 mm.

Fig. 8
Fig. 8

Planar integration of the diffractive optical system for the generation of propagation invariant beam arrays on the fused silica substrate: ICG, input coupling grating; TL, toroidal lens; BG, Bessel beam generator; CL, collimating lens; OCG, output coupling grating.

Equations (8)

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

U(x, y, z)=002πfT(f, ϕ)×exp[if(x cos ϕ+y sin ϕ)+iwz]dfdϕ,
T(f, ϕ)=1(2π)2-U(x, y, 0)×exp[-if(x cos ϕ+y sin ϕ)]dxdy
k=(f cos ϕ, f sin ϕ, w),
T(f, ϕ)=A(ϕ)δ(f-α),
U(x, y, z)=exp(iβz)02πA(ϕ)×exp[iα(x cos ϕ+y sin ϕ)]dϕ
A(ϕ)A(ϕ)exp[-iα(Δx cos ϕ+Δy sin ϕ)].
A(ϕ)=n=1NanAn(ϕ)exp[-iα(xn cos ϕ+yn sin ϕ)],
ϕ(r)=k{fTL-[fTL2-(r-R)2]1/2},

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