Abstract

A synthesis method for arbitrary on-axis intensity distributions from axially symmetric fields is developed in the paraxial approximation. As an important consequence, a new pseudo-nondiffracting beam, the axially symmetric on-axis flat-top beam (AFTB), is given by an integral transform form. This AFTB is completely determined by three simple parameters: the central spatial frequency Sc, the on-axis flat-top length L, and the on-axis central position zc. When LSc1, this AFTB can give a nearly flat-top intensity distribution on the propagation axis. In particular, this AFTB approaches the nondiffracting zero-order Bessel J0 beam when L. It is revealed that the superposition of multiple AFTB fields can give multiple on-axis flattop intensity regions when some appropriate conditions are satisfied.

© 2000 Optical Society of America

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  1. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [CrossRef]
  2. J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  3. J. Turunen, A. Vasara, A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
    [CrossRef] [PubMed]
  4. A. Vasara, J. Turunen, A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989).
    [CrossRef] [PubMed]
  5. A. J. Cox, D. C. Dibble, “Holographic reproduction of a diffraction-free beam,” Appl. Opt. 30, 1330–1332 (1991).
    [CrossRef] [PubMed]
  6. A. J. Cox, J. D’Anna, “Constant-axial-intensity nondiffracting beam,” Opt. Lett. 17, 232–234 (1992).
    [CrossRef] [PubMed]
  7. Z. Jaroszewich, J. Sochacki, A. Kolodziejczyk, L. R. Staronski, “Apodized annular-aperture logarithmic axicon: smoothness and uniformity of intensity distributions,” Opt. Lett. 18, 1893–1895 (1993).
    [CrossRef]
  8. R. Piestun, J. Shamir, “Control of wave-front propagation with diffractive elements,” Opt. Lett. 19, 771–773 (1994).
    [CrossRef] [PubMed]
  9. B. Salik, J. Rosen, A. Yariv, “One-dimensional beam shaping,” J. Opt. Soc. Am. A 12, 1702–1706 (1995).
    [CrossRef]
  10. J. Rosen, B. Salik, A. Yariv, H. K. Liu, “Pseudonondiffracting slitlike beam and its analogy to the pseudonondispersing pulse,” Opt. Lett. 20, 423–425 (1995).
    [CrossRef] [PubMed]
  11. J. Rosen, B. Salik, A. Yariv, “Pseudo-nondiffracting beams generated by radial harmonic functions,” J. Opt. Soc. Am. A 12, 2446–2457 (1995).
    [CrossRef]
  12. Z. Jiang, Q. Lu, Z. Liu, “Propagation of apertured Bessel beams,” Appl. Opt. 34, 7183–7185 (1995).
    [CrossRef] [PubMed]
  13. B.-Z. Dong, G.-Z. Yang, B.-Y. Gu, O. K. Ersoy, “Iterative optimization approach for designing an axicon with long focal depth and high transverse resolution,” J. Opt. Soc. Am. A 13, 97–103 (1996).
    [CrossRef]
  14. A. T. Friberg, “Stationary-phase analysis of generalized axicons,” J. Opt. Soc. Am. A 13, 743–750 (1996).
    [CrossRef]
  15. L. Niggl, T. Lanzl, M. Maier, “Properties of Bessel beams generated by periodic grating of circular symmetry,” J. Opt. Soc. Am. A 14, 27–33 (1997).
    [CrossRef]
  16. S. Y. Popov, A. T. Friberg, “Design of diffractive axicons for partially coherent light,” Opt. Lett. 23, 1639–1641 (1998).
    [CrossRef]
  17. W.-X. Cong, N.-X. Chen, B.-Y. Gu, “Generation of nondiffracting beams by diffractive phase elements,” J. Opt. Soc. Am. A 15, 2362–2364 (1998).
    [CrossRef]
  18. M. Honkanen, J. Turunen, “Tandem systems for efficient generation of uniform-axial-intensity Bessel fields,” Opt. Commun. 154, 368–375 (1998).
    [CrossRef]
  19. S. Y. Popov, A. T. Friberg, “Apodization of generalized axicons to produce uniform axial line images,” Pure Appl. Opt. 7, 537–548 (1998).
    [CrossRef]
  20. A. G. Sedukhin, “Beam-preshaping axicon focusing,” J. Opt. Soc. Am. A 15, 3057–3066 (1998).
    [CrossRef]
  21. Z. Jaroszewicz, J. Morales, “Lens axicons: systems composed of a diverging aberrated lens and a perfect converging lens,” J. Opt. Soc. Am. A 15, 2383–2390 (1998).
    [CrossRef]
  22. Z. Jaroszewicz, J. Morales, “Lens axicons: systems composed of a diverging aberrated lens and a converging aberrated lens,” J. Opt. Soc. Am. A 16, 191–197 (1999).
    [CrossRef]
  23. N. Guérineau, J. Primot, “Nondiffracting array generation using an N-wave interferometer,” J. Opt. Soc. Am. A 16, 293–298 (1999).
    [CrossRef]
  24. J. Rosen, A. Yariv, “Synthesis of an arbitrary axial field profile by computer-generated holograms,” Opt. Lett. 19, 843–845 (1994).
    [CrossRef] [PubMed]
  25. R. Piestun, B. Spektor, J. Shamir, “Wave fields in three dimensions: analysis and synthesis,” J. Opt. Soc. Am. A 13, 1837–1848 (1996).
    [CrossRef]
  26. R. Liu, B.-Z. Dong, G.-Z. Yang, B.-Y. Gu, “Generation of pseudo-nondiffracting beams with use of diffractive phase elements designed by the conjugate-gradient method,” J. Opt. Soc. Am. A 15, 144–151 (1998).
    [CrossRef]
  27. R. Liu, B.-Y. Gu, B.-Z. Dong, G.-Z. Yang, “Design of diffractive phase elements that realize axial-intensity modulation based on the conjugate-gradient method,” J. Opt. Soc. Am. A 15, 689–694 (1998).
    [CrossRef]
  28. See, for examples, J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), pp. 55–58 and pp. 11–12; G. P. Agrawal, D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. 69, 575–578 (1979); Q. Cao, X. Deng, “Corrections to the paraxial approximation of an arbitrary free-propagation beam,” J. Opt. Soc. Am. A 15, 1144–1148 (1998).
    [CrossRef]

1999

1998

1997

1996

1995

1994

1993

1992

1991

1989

1988

1987

J. Durnin, J. J. Miceli, J. H. 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]

Chen, N.-X.

Cong, W.-X.

Cox, A. J.

D’Anna, J.

Dibble, D. C.

Dong, B.-Z.

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]

Eberly, J. H.

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

Ersoy, O. K.

Friberg, A. T.

Goodman, J. W.

See, for examples, J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), pp. 55–58 and pp. 11–12; G. P. Agrawal, D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. 69, 575–578 (1979); Q. Cao, X. Deng, “Corrections to the paraxial approximation of an arbitrary free-propagation beam,” J. Opt. Soc. Am. A 15, 1144–1148 (1998).
[CrossRef]

Gu, B.-Y.

Guérineau, N.

Honkanen, M.

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

Jaroszewich, Z.

Jaroszewicz, Z.

Jiang, Z.

Kolodziejczyk, A.

Lanzl, T.

Liu, H. K.

Liu, R.

Liu, Z.

Lu, Q.

Maier, M.

Miceli, J. J.

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

Morales, J.

Niggl, L.

Piestun, R.

Popov, S. Y.

S. Y. Popov, A. T. Friberg, “Design of diffractive axicons for partially coherent light,” Opt. Lett. 23, 1639–1641 (1998).
[CrossRef]

S. Y. Popov, A. T. Friberg, “Apodization of generalized axicons to produce uniform axial line images,” Pure Appl. Opt. 7, 537–548 (1998).
[CrossRef]

Primot, J.

Rosen, J.

Salik, B.

Sedukhin, A. G.

Shamir, J.

Sochacki, J.

Spektor, B.

Staronski, L. R.

Turunen, J.

Vasara, A.

Yang, G.-Z.

Yariv, A.

Appl. Opt.

J. Opt. Soc. Am. A

A. G. Sedukhin, “Beam-preshaping axicon focusing,” J. Opt. Soc. Am. A 15, 3057–3066 (1998).
[CrossRef]

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

B.-Z. Dong, G.-Z. Yang, B.-Y. Gu, O. K. Ersoy, “Iterative optimization approach for designing an axicon with long focal depth and high transverse resolution,” J. Opt. Soc. Am. A 13, 97–103 (1996).
[CrossRef]

A. T. Friberg, “Stationary-phase analysis of generalized axicons,” J. Opt. Soc. Am. A 13, 743–750 (1996).
[CrossRef]

R. Piestun, B. Spektor, J. Shamir, “Wave fields in three dimensions: analysis and synthesis,” J. Opt. Soc. Am. A 13, 1837–1848 (1996).
[CrossRef]

B. Salik, J. Rosen, A. Yariv, “One-dimensional beam shaping,” J. Opt. Soc. Am. A 12, 1702–1706 (1995).
[CrossRef]

J. Rosen, B. Salik, A. Yariv, “Pseudo-nondiffracting beams generated by radial harmonic functions,” J. Opt. Soc. Am. A 12, 2446–2457 (1995).
[CrossRef]

Z. Jaroszewicz, J. Morales, “Lens axicons: systems composed of a diverging aberrated lens and a converging aberrated lens,” J. Opt. Soc. Am. A 16, 191–197 (1999).
[CrossRef]

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

R. Liu, B.-Z. Dong, G.-Z. Yang, B.-Y. Gu, “Generation of pseudo-nondiffracting beams with use of diffractive phase elements designed by the conjugate-gradient method,” J. Opt. Soc. Am. A 15, 144–151 (1998).
[CrossRef]

R. Liu, B.-Y. Gu, B.-Z. Dong, G.-Z. Yang, “Design of diffractive phase elements that realize axial-intensity modulation based on the conjugate-gradient method,” J. Opt. Soc. Am. A 15, 689–694 (1998).
[CrossRef]

W.-X. Cong, N.-X. Chen, B.-Y. Gu, “Generation of nondiffracting beams by diffractive phase elements,” J. Opt. Soc. Am. A 15, 2362–2364 (1998).
[CrossRef]

Z. Jaroszewicz, J. Morales, “Lens axicons: systems composed of a diverging aberrated lens and a perfect converging lens,” J. Opt. Soc. Am. A 15, 2383–2390 (1998).
[CrossRef]

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

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

Opt. Commun.

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

Opt. Lett.

Phys. Rev. Lett.

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

Pure Appl. Opt.

S. Y. Popov, A. T. Friberg, “Apodization of generalized axicons to produce uniform axial line images,” Pure Appl. Opt. 7, 537–548 (1998).
[CrossRef]

Other

See, for examples, J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), pp. 55–58 and pp. 11–12; G. P. Agrawal, D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. 69, 575–578 (1979); Q. Cao, X. Deng, “Corrections to the paraxial approximation of an arbitrary free-propagation beam,” J. Opt. Soc. Am. A 15, 1144–1148 (1998).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Normalized on-axis intensity distributions I(0, η) of the AFTB’s for (a) LSc=10, (b) LSc=100, and (c) LSc=200.

Fig. 2
Fig. 2

Normalized on-axis intensity distribution I(0, η) of the AFTB corresponding to LSc=10 in different choices of the high limit of the integral. (a) H=20, (b) H=50, and (c) H=100.

Fig. 3
Fig. 3

On-axis intensity distribution I(0, z) with different central position zc. (a) zc=0.0 m, (b) zc=5.0 m, and (c) zc=10.0 m. The parameters L and Sc are chosen such that L=10 m and Sc=10 m-1.

Fig. 4
Fig. 4

Radial intensity distributions I(r) of the AFTB at the (a) z=0.0-m, (b) z=9.0-m, (c) z=9.5-m, (d) z=9.9-m, (e) z=10.0-m, and (f) z=10.1-m planes. The parameters are chosen such that λ=0.6328 µm,L=20 m,zc=0 m,Sc=10 m-1, and LSc=200.

Fig. 5
Fig. 5

On-axis intensity distribution I(0, z) of the superposition field of two AFTB’s. The parameters are chosen such that c1=c2=1, λ=0.6328 µm, L1=L2=10 m, zc,1=10 m, zc,2=25 m, Sc,1=Sc,2=10 m-1, and L1Sc,1=L2Sc,2=100.

Equations (37)

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i2kϕz+2ϕx2+2ϕy2=0,
ψ(f, z)=ψ(f, 0)exp(-iπλ f2z)
ϕ(r, z)=2π0ψ(f, z)J0(2πrf)f d f,
ϕ(r, z)=0kψ(2S/λ, 0)exp(-i2πSz)×J0(2πr2S/λ)dS,
ϕ(0, z)=0kψ(2S/λ, 0)exp(-i2πSz)dS,
φ(S)=-Ig(0, z) exp(i2πSz)dz.
--Slφ(S)exp(-i2πSz)d S
ϕ(r, z)=0φ(S-Sc)exp(-i2πSz)×J0(2πr2S/λ)dS.
Ig(0, z)=-φ(S1)exp(-i2πS1z)dS1,
ϕ(0, z)=exp(-i2πScz)[Ig(0, z)-g(z)],
g(z)=--Scφ(S1)exp(-i2πS1z)dS1,
I(0, z)=|Ig(0, z)-g(z)|2.
ϕ(r, z)=0φ(S)exp(-i2πSz)J0(2πr2S/λ)dS.
φ(S)=L sinc(LπS)exp(i2πSzc),
ϕ(r, z)=0L sinc[Lπ(S-Sc)]×exp[i2π(Szc-Sczc-Sz)]×J02πr2Sλ1/2dS.
I(0, z)=|ϕ(0, z)|2=|rectz-zcL-g(z)|2,
g(z)=--ScL sinc(LπS1)×exp[-i2πS1(z-zc)]dS1,
g(η)=--LScsinc(πξ)exp(-i2πξη)dξ.
I(0, η)=-LScH sinc(πξ)exp(-i2πξη)dξ2.
ϕ(r, z)exp(i2πSczc)=0L sinc[Lπ(S-Sc)]×exp[i2πS(z-zc)]×J02πr2Sλ1/2dS.
I(r, zc)=|ϕ(r, zc)|2=0L sinc[Lπ(S-Sc)]J02πr2Sλ1/2dS2.
L sincLπr122λF2-Scexpi2πzcr122λF2-Sc×exp-ikr122F
ϕ(r, z)=n=1mcnϕn(r, z),
ϕn(r, z)=0Ln sinc[Lnπ(S-Sc,n)]×exp[i2π(Szc,n-Sc,nzc,n-Sz)]×J0(2πr2S/λ)dS,
ϕ(0, z)=n=1mcn exp(-i2πSc,nz)×rectz-zc,nLn-gn(z),
gn(z)=--Sc,nLn sinc(LnπS1)×exp[-i2πS1(z-zc,n)]dS1.
φ(S)=zc-L/2zc+L/2exp(i2πSz)dz.
φ(S)=sin(πSL)πSexp(i2πSzc).
φ(S-Sc)=Lsinc[Lπ(S-Sc)]exp[i2π zc(S-Sc)].
ϕ(0, z)=0L sinc[Lπ(S-Sc)]×exp[i2π(Szc-Sczc-Sz)]dS=exp(-i2πScz)0L sinc[Lπ(S-Sc)]×exp[-i2π(S-Sc)(z-zc)]dS.
ϕ(0, z)=exp(-i2πScz)-ScL sinc(LπS1)×exp[-i2πS1(z-zc)]dS1.
rectz-zcL=-L sinc(LπS1)exp(i2πS1zc)×exp(-i2πS1z)dS1,
ϕ(0, z)exp(i2πScz)=rectz-zcL-g(z),
g(z)=--ScL sinc(LπS1)×exp[-i2πS1(z-zc)]dS1,
g(z)=--LScsinc[π(LS1)]×exp-i2π(LS1)z-zcLd(LS1).
g(η)=--LScsinc(πξ)exp(-i2πξη)dξ,
I(0, η)=|rect(η)-g(η)|2,

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