Abstract

Expressions for describing Gaussian beams focused by a lens with spherical aberration have been derived. Numerical results show that, when the coefficient of the spherical aberration is negative, one can obtain flattened laser irradiance at two positions along the focused field. The larger the coefficient of negative spherical aberration, the larger the flatness is. The effect of the Fresnel number of the focusing lens on the flattened laser irradiance is also investigated.

© 1998 Optical Society of America

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References

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1998

J. Pu, “Focusing Gaussian beams by an annular lens with spherical aberration,” J. Mod. Opt. 45, 239–247 (1998).
[CrossRef]

1996

A. Yoshida, T. Asakura, “Propagation and focusing of Gaussian laser beams beyond conventional diffraction limit,” Opt. Commun. 123, 694–704 (1996).
[CrossRef]

S. N. Dixit, M. D. Feit, M. D. Perry, H. T. Powell, “Designing fully continuous phase screens for tailoring focal-plane irradiance profile,” Opt. Lett. 21, 1715–1717 (1996).
[CrossRef] [PubMed]

1995

R. Martinez-Herrero, P. M. Mejias, N. Hodgson, H. Weber, “Beam-quality changes generated by thermally induced spherical aberration in laser cavities,” IEEE J. Quantum Electron. 31, 2173–2176 (1995).
[CrossRef]

1994

R. M. Stevenson, M. J. Norman, T. H. Bett, D. A. Pepler, C. N. Danson, I. N. Ross, “Binary-phase zone plate arrays for generation of uniform focal profile,” Opt. Lett. 19, 363–365 (1994).
[PubMed]

J. A. Ruff, A. E. Siegman, “Measurement of beam quality degradation due to spherical aberration in a simple lens,” Opt. Quant. Electron. 26, 629–632 (1994).
[CrossRef]

1993

1992

1991

1986

1982

W. B. Veldkamp, “Technique for generating focal-plane flattop beam profiles,” Rev. Sci. Instrum. 53, 294–297 (1982).
[CrossRef]

1976

1972

Asakura, T.

A. Yoshida, T. Asakura, “Propagation and focusing of Gaussian laser beams beyond conventional diffraction limit,” Opt. Commun. 123, 694–704 (1996).
[CrossRef]

Bett, T. H.

Chen, Z.

Danson, C. N.

Deng, X.

Dew, S. K.

Dixit, S. N.

Feit, M. D.

Gupta, R.

Hodgson, N.

R. Martinez-Herrero, P. M. Mejias, N. Hodgson, H. Weber, “Beam-quality changes generated by thermally induced spherical aberration in laser cavities,” IEEE J. Quantum Electron. 31, 2173–2176 (1995).
[CrossRef]

Ih, C. S.

Kenney, C. S.

Klingsporn, P. E.

Liang, X.

Ma, R.

Martinez-Herrero, R.

R. Martinez-Herrero, P. M. Mejias, N. Hodgson, H. Weber, “Beam-quality changes generated by thermally induced spherical aberration in laser cavities,” IEEE J. Quantum Electron. 31, 2173–2176 (1995).
[CrossRef]

Mejias, P. M.

R. Martinez-Herrero, P. M. Mejias, N. Hodgson, H. Weber, “Beam-quality changes generated by thermally induced spherical aberration in laser cavities,” IEEE J. Quantum Electron. 31, 2173–2176 (1995).
[CrossRef]

Metcalf, H.

Norman, M. J.

Overfelt, P. L.

Parsons, R. R.

Pepler, D. A.

Perry, M. D.

Powell, H. T.

Pu, J.

J. Pu, “Focusing Gaussian beams by an annular lens with spherical aberration,” J. Mod. Opt. 45, 239–247 (1998).
[CrossRef]

Ross, I. N.

Ruff, J. A.

J. A. Ruff, A. E. Siegman, “Measurement of beam quality degradation due to spherical aberration in a simple lens,” Opt. Quant. Electron. 26, 629–632 (1994).
[CrossRef]

Siegman, A. E.

J. A. Ruff, A. E. Siegman, “Measurement of beam quality degradation due to spherical aberration in a simple lens,” Opt. Quant. Electron. 26, 629–632 (1994).
[CrossRef]

Stevenson, R. M.

Veldkamp, W. B.

W. B. Veldkamp, “Technique for generating focal-plane flattop beam profiles,” Rev. Sci. Instrum. 53, 294–297 (1982).
[CrossRef]

Weber, H.

R. Martinez-Herrero, P. M. Mejias, N. Hodgson, H. Weber, “Beam-quality changes generated by thermally induced spherical aberration in laser cavities,” IEEE J. Quantum Electron. 31, 2173–2176 (1995).
[CrossRef]

Xie, C.

Yoshida, A.

A. Yoshida, T. Asakura, “Propagation and focusing of Gaussian laser beams beyond conventional diffraction limit,” Opt. Commun. 123, 694–704 (1996).
[CrossRef]

Yu, W.

Appl. Opt.

IEEE J. Quantum Electron.

R. Martinez-Herrero, P. M. Mejias, N. Hodgson, H. Weber, “Beam-quality changes generated by thermally induced spherical aberration in laser cavities,” IEEE J. Quantum Electron. 31, 2173–2176 (1995).
[CrossRef]

J. Mod. Opt.

J. Pu, “Focusing Gaussian beams by an annular lens with spherical aberration,” J. Mod. Opt. 45, 239–247 (1998).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

A. Yoshida, T. Asakura, “Propagation and focusing of Gaussian laser beams beyond conventional diffraction limit,” Opt. Commun. 123, 694–704 (1996).
[CrossRef]

Opt. Lett.

Opt. Quant. Electron.

J. A. Ruff, A. E. Siegman, “Measurement of beam quality degradation due to spherical aberration in a simple lens,” Opt. Quant. Electron. 26, 629–632 (1994).
[CrossRef]

Rev. Sci. Instrum.

W. B. Veldkamp, “Technique for generating focal-plane flattop beam profiles,” Rev. Sci. Instrum. 53, 294–297 (1982).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Geometry for focusing Gaussian beams by a lens with spherical aberration. ∑ is the wave front behind the lens; ∑* is the spherical reference sphere centered at z = f.

Fig. 2
Fig. 2

Transverse intensity profiles at two positions of the focused field: (a) z/ f = 0.499, (b) z/ f = 0.562. Other parameters are N w = 1, N a = 10, kS 1 = -0.35 (δ1 = -5.57).

Fig. 3
Fig. 3

Transverse intensity profiles at two positions of the focused field: (a) z/ f = 0.518, (b) z/ f = 0.577. Other parameters are N w = 1, N a = 10, kS 1 = -0.3 (δ1 = -4.77).

Fig. 4
Fig. 4

Transverse intensity profiles at two positions of the focused field: (a) z/ f = 0.576, (b) z/ f = 0.605. Other parameters are N w = 1, N a = 10, kS 1 = -0.2 (δ1 = -3.18).

Fig. 5
Fig. 5

Transverse intensity profile evolution along the focused field. The positions of the focused field are indicated on the right side: N w = 1, N a = 10. (a) kS 1 = -0.3 (δ1= -4.77); (b) kS 1 = -0.2 (δ1 = -3.18).

Fig. 6
Fig. 6

Positions of the flattened laser beam irradiation related to the coefficient of the spherical aberration: N w = 1, N a = 10.

Fig. 7
Fig. 7

Transverse intensity profiles at two positions of the focused field: (a) z/ f = 0.504, (b) z/ f = 0.57. Other parameters are N w = 1, N a = 4, kS 1 = -0.3 (δ1 = -0.76).

Fig. 8
Fig. 8

Transverse intensity profiles at two positions of the focused field: (a) z/ f = 0.418, (b) z/ f = 0.505. Other parameters are N w = 1, N a = 2, kS 1 = -0.3 (δ1 = -0.19).

Fig. 9
Fig. 9

Transverse intensity profiles at two positions of the focused field: (a) z/ f = 0.367, (b) z/ f = 0.458. Other parameters are N w = 1, N a = 1.5, kS 1 = -0.3 (δ1 = -0.107).

Fig. 10
Fig. 10

Transverse intensity profiles at two positions of the focused field: (a) z/ f = 0.28, (b) z/ f = 0.38. Other parameters are N w = 1, Na = 1, kS1 = -0.3 (δ1 = -0.048).

Tables (1)

Tables Icon

Table 1 Wave-Front Deformation at Aperture Edge δ1, Two Positions at which Flattened Laser Irradiance Occurs, and the Difference between Them Related to Na a

Equations (16)

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U 0 Q = U 0 ρ ,   0 = A 0 exp - ρ 2 w 2 exp ik Φ R - ρ 2 2 f ,
U P = - ik 2 π s   U 0 Q exp iks s d s ,
s z ,
s z - ρ ρ   cos ϕ - ϕ z + ρ 2 2 z .
U ρ ,   ϕ ,   z = - ik 2 π z 0 2 π 0 a   U 0 ρ ,   ϕ ,   z = 0 × exp ik ρ 2 2 z - ρ ρ   cos ϕ - ϕ z ρ d ρ d ϕ .
2 π J 0 k ρ ρ z ,
U ρ ,   ϕ ,   z = - ik z 0 a   U 0 ρ ,   ϕ ,   z = 0 × J 0 k ρ ρ z exp ik   ρ 2 2 z ρ d ρ .
U ρ ,   ϕ ,   z = - ikA 0 z 0 a exp - ρ 2 w 2 exp ik Φ R - ρ 2 2 f × J 0 k ρ ρ z exp ik   ρ 2 2 z ρ d ρ .
Φ R = S 1 / w 4 ρ 4 .
U ρ ,   ϕ ,   z = - i π A 0 z / f 0 N a exp - X N w exp ikS 1 X 2 N w 2 × exp i π 1 z / f - 1 X J 0 2 π r X z / f d X ,
X = ρ 2 λ f ,     r = ρ λ f .
N w = w 2 λ f
N a = a 2 λ f
I r ,   z = U r ,   z U * r ,   z .
Φ R = δ 1 λ ρ a 4 ,
δ 1 = kS 1 2 π N a N w 2 .

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