Abstract

We show that the transmission of a periodic grating with circular symmetry can be composed of the sum of the transmissions of positive and negative axicons of different orders. The properties of the light beams generated by the gratings can be understood from the well-known properties of the light beam of an axicon, which is approximately a Bessel beam of zeroth order. The method is applied to a circular binary phase grating, which is investigated both theoretically and experimentally in the spatial domain and the Fourier domain. A simple and accurate method for determining the focal length of a lens with use of Bessel light beams is presented.

© 1997 Optical Society of America

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References

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  1. J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  2. A. Vasara, J. Turunen, and A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989).
    [CrossRef] [PubMed]
  3. J. K. Jabczynski, “A ‘diffraction-free’ resonator,” Opt. Commun. 77, 292–294 (1990).
    [CrossRef]
  4. A. J. Cox and D. C. Dibble, “Nondiffracting beam from a spatially filtered Fabry–Perot resonator,” J. Opt. Soc. Am. A 9, 282–286 (1992).
    [CrossRef]
  5. J. Rosen, B. Salik, and A. Yariv, “Pseudo-nondiffracting beams generated by radial harmonic functions,” J. Opt. Soc. Am. A 12, 2446–2457 (1995).
    [CrossRef]
  6. J. A. Davis, E. Carcole, and D. M. Cottrell, “Intensity and phase measurements of nondiffracting beams generated with a magneto-optic spatial light modulator,” Appl. Opt. 35, 593–598 (1996).
    [CrossRef] [PubMed]
  7. A. J. Cox and D. C. Dibble, “Holographic reproduction of a diffraction-free beam,” Appl. Opt. 30, 1330–1332 (1991).
    [CrossRef] [PubMed]
  8. R. Arimoto, C. Saloma, T. Tanaka, and S. Kawata, “Imaging properties of axicon in a scanning optical system,” Appl. Opt. 31, 6653–6657 (1992).
    [CrossRef] [PubMed]
  9. T. Wulle and S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 70, 1401–1404 (1993).
    [CrossRef] [PubMed]
  10. S. Ruschin, “Modified Bessel nondiffracting beams,” J. Opt. Soc. Am. A 11, 3224–3228 (1994).
    [CrossRef]
  11. R. P. MacDonald, J. Chrostowski, S. A. Boothroyd, and B. A. Syrett, “Holographic formation of a diode laser nondiffracting beam,” Appl. Opt. 32, 6470–6474 (1993).
    [CrossRef] [PubMed]
  12. J. H. McLeod, “The axicon: a new type of optical element,” J. Opt. Soc. Am. 44, 592–597 (1954).
    [CrossRef]
  13. Z. Jiang, Q. Lu, and Z. Liu, “Propagation of apertured Bessel beams,” Appl. Opt. 34, 7183–7185 (1995).
    [CrossRef] [PubMed]
  14. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 1.1.
  15. R. J. Collier, C. B. Burckhardt, and L. H. Lin, Optical Holography (Academic, New York, 1971), Chap. 6.2.

1996

1995

1994

1993

1992

1991

1990

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

1989

1987

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

1954

Arimoto, R.

Boothroyd, S. A.

Carcole, E.

Chrostowski, J.

Cottrell, D. M.

Cox, A. J.

Davis, J. A.

Dibble, D. C.

Durnin, J.

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

Eberly, J. H.

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

Friberg, A. T.

Herminghaus, S.

T. Wulle and S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 70, 1401–1404 (1993).
[CrossRef] [PubMed]

Jabczynski, J. K.

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

Jiang, Z.

Kawata, S.

Liu, Z.

Lu, Q.

MacDonald, R. P.

McLeod, J. H.

Miceli , Jr., and, J. J.

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

Rosen, J.

Ruschin, S.

Salik, B.

Saloma, C.

Syrett, B. A.

Tanaka, T.

Turunen, J.

Vasara, A.

Wulle, T.

T. Wulle and S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 70, 1401–1404 (1993).
[CrossRef] [PubMed]

Yariv, A.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

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

Phys. Rev. Lett.

T. Wulle and S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 70, 1401–1404 (1993).
[CrossRef] [PubMed]

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

Other

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 1.1.

R. J. Collier, C. B. Burckhardt, and L. H. Lin, Optical Holography (Academic, New York, 1971), Chap. 6.2.

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Figures (8)

Fig. 1
Fig. 1

Perspective and sectional view of a negative axicon with radius Rna and Hna.

Fig. 2
Fig. 2

Intensity distributions of a positive and a negative axicon (solid and dashed curves, respectively). The intensity of the negative axicon was multiplied by a factor of 500. (a) Radial intensity distribution I(r, z0). The horizontal scale is different for the right- and left-hand side. (b) Axial intensity distribution I(0, z).

Fig. 3
Fig. 3

Absolute value |Aax (u)| and imaginary part Im[Aax (u)], solid and dashed curves, respectively, of the Fourier transform of an axicon versus variable u in the Fourier plane.

Fig. 4
Fig. 4

Schematic diagram of the experimental setup for measuring the intensity distribution of Bessel beams. He–Ne laser; T, telescope; L1, L2, L3, L4, lenses; M1, M2, M3, mirrors; A1, A2, A3, apertures; PG, circular binary phase grating; CCD camera.

Fig. 5
Fig. 5

Axial intensity distribution I(0, z) versus distance z from the phase grating. The points and curves represent the measured values and calculated curves, respectively. (a) Circular binary phase grating, (b) selected first-order positive axicon.

Fig. 6
Fig. 6

Fourier transform of the circular binary phase grating in the back focal plane of a lens with a focal length f=25 cm. dm (m=1, 3, 5, ) are the diameters of the concentric rings.

Fig. 7
Fig. 7

Intensity distribution IF(rF) of the inner ring of the Fourier transform versus radial coordinate rF in the back focal plane of a lens with focal length f=387 cm. The points and curves represent the measured values and calculated Fourier transforms, respectively. (a) Circular binary phase grating, (b) selected first-order positive axicon.

Fig. 8
Fig. 8

Diameters dm of the rings of the Fourier transform of the binary phase grating versus order m of the rings. The line represents a least-squares fit through the experimental points.

Equations (21)

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Taxr=exp-2πiρaxr.
ρax=2πRaxkLHax(n-1).
Er, zE0=-ikLzexpikLz+r22z×0RTrexpikL2zr2J0kLrzrrdr.
Iax(r, z)I0=2πkLα2z J02(αr).
zmax=kLαRax
α=2πρax.
d0=2r04.81/α0.77ρax.
Tnar=exp+2πiρnar,
A(u)=2π0a(r)J0(2πru)r dr.
u=kLrF2πf.
dax=4πfkLρax.
Ana(u)=[Aax(u)]*
f(r)=m=-cm exp(-2πimρr)
cm=1ρ0ρf(r)exp(2πimρr)dr.
Tr=m=-cm exp-2πimρrfor0rR0elsewhere.
ρm=ρ/m,
Tr=c0+m=1cm exp-2πiρmr+m=1c-m exp2πiρmr.
f(r)=1forlρ<r<(l+1/2)ρ,lZe-iπ=-1elsewhere.
cm=2iπmforoddm0forevenm.
dm=m4πfkLρforoddm.
f=skLρ4π.

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