Abstract

When a Gaussian beam is apertured, it undergoes a focal shift as well as a phase shift. The focal shift has been investigated extensively although the phase shift has seldom been discussed. We analyze the phase shift of the apertured Gaussian beam. Furthermore we point out that the phase aperture may be used to transform the intensity distribution of the Gaussian beam to obtain a more concentrated beam, to realize uniformity of the Gaussian beam, and to obtain a ring beam.

© 1997 Optical Society of America

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References

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  1. A. Yariv, Quantum Electronics (Wiley, New York, 1989), Chap. 6.
  2. Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
    [CrossRef]
  3. Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
    [CrossRef]
  4. Y. Li, “Propagation of focal shift through axisymmetrical optical systems,” Opt. Commun. 95, 13–17 (1993).
    [CrossRef]
  5. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1980), Chap. 8.
  6. B. R. Frieden, “Lossless conversion of a plane laser wave to a plane wave of uniform irradiance,” Appl. Opt. 4, 1400–1403 (1965).
    [CrossRef]
  7. W. B. Veldkamp, C. J. Kastner, “Beam profile shaping for laser radars that use detectors arrays,” Appl. Opt. 21, 345–355 (1982).
    [CrossRef] [PubMed]
  8. D. Shafer, “Gaussian to flat-top intensity distributing lens,” Opt. Laser Technol. 14, 159–160 (1982).
    [CrossRef]
  9. C.-Y. Han, Y. Ishii, K. Murata, “Reshaping collimated laser beams with Gaussian profile to uniform profiles,” Appl. Opt. 22, 3644–3647 (1983).
    [CrossRef] [PubMed]
  10. M. A. Karim, A. M. Hanafi, F. Hussain, S. Mustafa, Z. Samberid, N. M. Zain, “Realization of a uniform circular source using a two-dimensional binary filter,” Opt. Lett. 10, 470–472 (1985).
    [CrossRef] [PubMed]
  11. S. R. Jahan, M. A. Karim, “Refracting systems for Gaussian-to-uniform beam transformations,” Opt. Laser Technol. 21, 27–30 (1989).
    [CrossRef]
  12. E. Tervonen, A. T. Friberg, J. Turunen, “Acousto-optic conversion of laser beams into flat-top beams,” J. Mod. Opt. 40, 625–635 (1993).
    [CrossRef]
  13. R. M. Stevenson, M. J. Norman, T. H. Bett, D. A. Pepler, C. N. Danson, I. N. Ross, “Binary-phase zone plate arrays for the generation of uniform focal profiles,” Opt. Lett. 19, 363–365 (1994).
    [PubMed]
  14. M. D. McNeill, T. Poon, “Gaussian-beam profile shaping by acousto-optic Bragg diffraction,” Appl. Opt. 33, 4508–4515 (1994).
    [CrossRef] [PubMed]
  15. M. Duparre, M. A. Golub, B. Ludge, V. S. Pavelyev, V. A. Soifer, G. V. Uspleniev, S. G. Volotovskii, “Investigation of computer-generated diffractive beam shapers for flattening of single-modal CO2 laser beams,” Appl. Opt. 34, 2489–2497 (1995).
    [CrossRef]
  16. K. Tyson, Principles of Adaptive Optics (Academic, San Diego, Calif., 1991), Chap. 6.
  17. S. Wang, “New beam of CO2 laser,” Appl. Laser (China) 14, 160–161 (1994).

1995 (1)

1994 (3)

1993 (2)

E. Tervonen, A. T. Friberg, J. Turunen, “Acousto-optic conversion of laser beams into flat-top beams,” J. Mod. Opt. 40, 625–635 (1993).
[CrossRef]

Y. Li, “Propagation of focal shift through axisymmetrical optical systems,” Opt. Commun. 95, 13–17 (1993).
[CrossRef]

1989 (1)

S. R. Jahan, M. A. Karim, “Refracting systems for Gaussian-to-uniform beam transformations,” Opt. Laser Technol. 21, 27–30 (1989).
[CrossRef]

1985 (1)

1983 (1)

1982 (3)

W. B. Veldkamp, C. J. Kastner, “Beam profile shaping for laser radars that use detectors arrays,” Appl. Opt. 21, 345–355 (1982).
[CrossRef] [PubMed]

D. Shafer, “Gaussian to flat-top intensity distributing lens,” Opt. Laser Technol. 14, 159–160 (1982).
[CrossRef]

Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
[CrossRef]

1981 (1)

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

1965 (1)

Bett, T. H.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1980), Chap. 8.

Danson, C. N.

Duparre, M.

Friberg, A. T.

E. Tervonen, A. T. Friberg, J. Turunen, “Acousto-optic conversion of laser beams into flat-top beams,” J. Mod. Opt. 40, 625–635 (1993).
[CrossRef]

Frieden, B. R.

Golub, M. A.

Han, C.-Y.

Hanafi, A. M.

Hussain, F.

Ishii, Y.

Jahan, S. R.

S. R. Jahan, M. A. Karim, “Refracting systems for Gaussian-to-uniform beam transformations,” Opt. Laser Technol. 21, 27–30 (1989).
[CrossRef]

Karim, M. A.

Kastner, C. J.

Li, Y.

Y. Li, “Propagation of focal shift through axisymmetrical optical systems,” Opt. Commun. 95, 13–17 (1993).
[CrossRef]

Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
[CrossRef]

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

Ludge, B.

McNeill, M. D.

Murata, K.

Mustafa, S.

Norman, M. J.

Pavelyev, V. S.

Pepler, D. A.

Poon, T.

Ross, I. N.

Samberid, Z.

Shafer, D.

D. Shafer, “Gaussian to flat-top intensity distributing lens,” Opt. Laser Technol. 14, 159–160 (1982).
[CrossRef]

Soifer, V. A.

Stevenson, R. M.

Tervonen, E.

E. Tervonen, A. T. Friberg, J. Turunen, “Acousto-optic conversion of laser beams into flat-top beams,” J. Mod. Opt. 40, 625–635 (1993).
[CrossRef]

Turunen, J.

E. Tervonen, A. T. Friberg, J. Turunen, “Acousto-optic conversion of laser beams into flat-top beams,” J. Mod. Opt. 40, 625–635 (1993).
[CrossRef]

Tyson, K.

K. Tyson, Principles of Adaptive Optics (Academic, San Diego, Calif., 1991), Chap. 6.

Uspleniev, G. V.

Veldkamp, W. B.

Volotovskii, S. G.

Wang, S.

S. Wang, “New beam of CO2 laser,” Appl. Laser (China) 14, 160–161 (1994).

Wolf, E.

Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
[CrossRef]

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1980), Chap. 8.

Yariv, A.

A. Yariv, Quantum Electronics (Wiley, New York, 1989), Chap. 6.

Zain, N. M.

Appl. Laser (China) (1)

S. Wang, “New beam of CO2 laser,” Appl. Laser (China) 14, 160–161 (1994).

Appl. Opt. (5)

J. Mod. Opt. (1)

E. Tervonen, A. T. Friberg, J. Turunen, “Acousto-optic conversion of laser beams into flat-top beams,” J. Mod. Opt. 40, 625–635 (1993).
[CrossRef]

Opt. Commun. (3)

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
[CrossRef]

Y. Li, “Propagation of focal shift through axisymmetrical optical systems,” Opt. Commun. 95, 13–17 (1993).
[CrossRef]

Opt. Laser Technol. (2)

D. Shafer, “Gaussian to flat-top intensity distributing lens,” Opt. Laser Technol. 14, 159–160 (1982).
[CrossRef]

S. R. Jahan, M. A. Karim, “Refracting systems for Gaussian-to-uniform beam transformations,” Opt. Laser Technol. 21, 27–30 (1989).
[CrossRef]

Opt. Lett. (2)

Other (3)

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1980), Chap. 8.

A. Yariv, Quantum Electronics (Wiley, New York, 1989), Chap. 6.

K. Tyson, Principles of Adaptive Optics (Academic, San Diego, Calif., 1991), Chap. 6.

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Figures (9)

Fig. 1
Fig. 1

Dependencies of relative phase shifts ΔΨ of the apertured Gaussian beam on aperture radius a and distance L when z 1 = z 0; ω0 = 0.5 mm, λ = 632.8 nm are used throughout.

Fig. 2
Fig. 2

Dependencies of ΔΨ on a and z 1 when L → ∞ [Eq. (11)].

Fig. 3
Fig. 3

Dependencies of |S| 2 on δ and a when z 1 = z 0.

Fig. 4
Fig. 4

Plot of |S| 2 versus δ and z 1 with a = 0.8 mm.

Fig. 5
Fig. 5

Transverse intensity distribution in the far field (L = 1 km) with δ = 1.57 rad, z 1 = z 0, and a = 0.8 mm (b). (a) Original Gaussian beam.

Fig. 6
Fig. 6

Transverse intensity distribution with L = 5 m, δ = 4.6 rad, z 1 = z 0, and a = 0.5 mm (b). (a) Original Gaussian beam.

Fig. 7
Fig. 7

Phase-aperture-generated flat-top beam: (a) original Gaussian beam; (b) a = 0.6 mm, δ = 5.61 rad; (c) a = 0.5 mm, δ = 5.48 rad. z 1 = z 0 and L = 5 m.

Fig. 8
Fig. 8

Evolution of the flat-top beam of curve (c) in Fig. 7.

Fig. 9
Fig. 9

Dependencies of the transverse intensity distribution on phase delay δ of the phase aperture: z 1 = z 0, a = 0.5 mm, and L = 5 m.

Equations (23)

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Er2, θ2, z2=kiL002π Ar1, θ1, z1×exp-ikDr1dr1dθ1,
D=L+r12+r222L-2r1r2 cosθ1-θ22L,
Ar, z=A0ω0ωz exp-r2ω2z-ikr22Rz+kz-Ψz,
ωz=ω01+z/z021/2,  Rz=z0z/z0+z0/z,  Ψz=arctanz/z0,
Er2, z2=exp-ikL-ikr222LkiL×0r1Ar1,z1J0kr1r2Lexp-ikr122Ldr1,
Ir2, z2=kL20 r1Ar1, z1×J0kr1r2Lexp-ikr122Ldr12,
Er2, z2=exp-ikL-ikr222LkiL×0ar1Ar1,z1J0kr1r2Lexp-ikr122Ldr1,
E0, z2=exp-ikLkiL0ar1Ar1,z1exp-ikr122Ldr1=A0ω0ωz2 exp-ikz2-Ψz2S1,
S1=1-exp-a21ω2z1+ik2Rz1+ik2L.
Ψz2=arctanz2/z0=Ψz1+arctanz0Lz02+z12+Lz1.
S1=S1expiΔΨ,
S1=2 exp-a22ω2z1cosha2ω2z1-coska221Rz1+1L1/2,
ΔΨ=arctan×exp-a2/ω2z1sina2k1/2Rz1+1/2L1-exp-a2/ω2z1cosa2k1/2Rz1+1/2L,
ΔΨ|L=arctan×exp-a2/ω2z1sina2k/2Rz11-exp-a2/ω2z1cosa2k/2Rz1.
t=exp-iδr  a1r > a,
Ar, z=exp-iδA0ω0ωz exp-r2ω2z-ikr22Rz+kz-Ψzr < aA0ω0ωz exp-r2ω2z-ikr22Rz+kz-Ψzr > a.
Ar z=A0ω0ωz exp-r2ω2z-ikr22Rz+kz-Ψz+A0ω0ωz exp-r2ω2z-ikr22Rz+kz-Ψzexp-iδ-1Wr,
Wr=1,r < a0,r < a.
Er2 z2=A0ω0ωz2 exp-r22ω2z2-ikr222Rz2+kz2-Ψz2+exp-iδ-1×exp-ikz2-ikr222L-Ψz1A0ω0ωz1kiL×0a exp-r12ω2z1-i kr122Rz1-ikr122L×J0kr1r2Lr1dr1.
E0, z2=A0ω0ωz2 exp-ikz2-Ψz2×1+exp-iδ-1S1.
S=1+exp-iδ-1S1
=1+exp-iδ-1expiΔΨS1,
S2=1+2S12-2S12 cos δ+2S1×cosΔΨ-δ-cosΔΨ,

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