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 (2)

1998 (8)

1997 (1)

1996 (3)

1995 (4)

1994 (2)

1993 (1)

1992 (1)

1991 (1)

1989 (1)

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]

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. (3)

J. Opt. Soc. Am. A (15)

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]

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]

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

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]

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

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

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

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

Opt. Commun. (1)

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

Opt. Lett. (6)

Phys. Rev. Lett. (1)

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

Pure Appl. Opt. (1)

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 (1)

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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