Abstract

The principal maximum of axial irradiance of a focused beam with a low Fresnel number does not lie at its focal point; instead it lies at a point that is closer to the focusing pupil. It has been shown by the numerical example of a weakly truncated Gaussian beam that its value increases and its location moves closer to the pupil when spherical aberration is introduced into the beam. Such an increase has been referred to as “beyond the conventional diffraction limit.” Similarly, an increase in the value and a shift in the location of the principal maximum of axial irradiance of a uniform beam toward the pupil by the introduction of some spherical aberration has been characterized as an unexpected result. We explain why and how such a result comes about and that it neither invalidates any diffraction limit nor is it unexpected. We illustrate this for uniform as well as Gaussian beams of various truncation ratios. Both focused and collimated beams aberrated by spherical aberration or astigmatism are considered.

© 2005 Optical Society of America

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References

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    [CrossRef]
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2005

1997

D. Y. Jiang, J. J. Stamnes, “Focusing at low Fresnel numbers in the presence of cylindrical or spherical aberration,” Pure Appl. Opt. 6, 85–96 (1997).
[CrossRef]

1996

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

1994

1993

1986

1984

A. S. Dementev, D. P. Domarkene, “Diffraction of converging spherical waves by a circular aperture,” Opt. Spectrosc. 56, 532–534 (1984).

G. D. Sucha, W. H. Carter, “Focal shift for a Gaussian beam; an experimental study,” Appl. Opt. 23, 4345–4347 (1984).
[CrossRef] [PubMed]

1983

1982

1980

1881

Lord Rayleigh, Philos. Mag. 11, 214 (1881);also his Scientific Papers (Dover, 1964), Vol. 1, p. 513.
[CrossRef]

Asakura, T.

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

Born, M.

M. Born, E. Wolf, Principles of Optics (Oxford, 1999).
[CrossRef]

Carter, W. H.

Dementev, A. S.

A. S. Dementev, D. P. Domarkene, “Diffraction of converging spherical waves by a circular aperture,” Opt. Spectrosc. 56, 532–534 (1984).

Domarkene, D. P.

A. S. Dementev, D. P. Domarkene, “Diffraction of converging spherical waves by a circular aperture,” Opt. Spectrosc. 56, 532–534 (1984).

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978).

Jiang, D. Y.

D. Y. Jiang, J. J. Stamnes, “Focusing at low Fresnel numbers in the presence of cylindrical or spherical aberration,” Pure Appl. Opt. 6, 85–96 (1997).
[CrossRef]

Li, Y.

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

Mahajan, V. N.

Rayleigh, Lord

Lord Rayleigh, Philos. Mag. 11, 214 (1881);also his Scientific Papers (Dover, 1964), Vol. 1, p. 513.
[CrossRef]

Siegman, A. E.

A. E. Siegman, An Introduction to Lasers and Masers (McGraw Hill, 1971).

Stamnes, J. J.

D. Y. Jiang, J. J. Stamnes, “Focusing at low Fresnel numbers in the presence of cylindrical or spherical aberration,” Pure Appl. Opt. 6, 85–96 (1997).
[CrossRef]

Sucha, G. D.

Wolf, E.

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

M. Born, E. Wolf, Principles of Optics (Oxford, 1999).
[CrossRef]

Yoshida, A.

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

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

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

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

Opt. Lett.

Opt. Spectrosc.

A. S. Dementev, D. P. Domarkene, “Diffraction of converging spherical waves by a circular aperture,” Opt. Spectrosc. 56, 532–534 (1984).

Philos. Mag.

Lord Rayleigh, Philos. Mag. 11, 214 (1881);also his Scientific Papers (Dover, 1964), Vol. 1, p. 513.
[CrossRef]

Pure Appl. Opt.

D. Y. Jiang, J. J. Stamnes, “Focusing at low Fresnel numbers in the presence of cylindrical or spherical aberration,” Pure Appl. Opt. 6, 85–96 (1997).
[CrossRef]

Other

M. Born, E. Wolf, Principles of Optics (Oxford, 1999).
[CrossRef]

Ref. [6], Chap. 1.

Ref. [6], p. 354.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978).

A. E. Siegman, An Introduction to Lasers and Masers (McGraw Hill, 1971).

Ref. [6], p. 343.

Ref. [6], p. 349.

V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffraction Optics (2nd printing) (SPIE Press, 2004), Chap. 4.

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Figures (7)

Fig. 1
Fig. 1

Axial irradiance of a truncated Gaussian beam focused at a distance R with a Fresnel number N = 1 . The truncation parameter γ = a ω , where a is the radius of the pupil truncating a beam of radius ω representing the beam radius at which the amplitude reduces to 1 e of its value at the center. γ = 0 corresponds to a uniform beam and γ = 3 represents a weakly truncated beam. The irradiance is normalized by the focal-point irradiance P S p λ 2 R 2 for a uniform beam, where P is the power transmitted by a pupil of area S p .

Fig. 2
Fig. 2

Axial irradiance of a focused beam with a Fresnel number N = 1 aberrated by spherical aberration A s . The axial irradiance when A s = 0 is shown for comparison. The defocus aberration B d in units of wavelength is also shown. (a) Uniform beam ( γ = 0 ) . (b) Gaussian beam with γ = 1 . (c) Weakly truncated Gaussian beam with γ = 3 .

Fig. 3
Fig. 3

Axial irradiance of a focused beam with a Fresnel number N = 1 aberrated by astigmatism A a . The axial irradiance when A a = 0 is shown for comparison. The defocus aberration B d in units of wavelength is also shown. (a) Uniform beam ( γ = 0 ) . (b) Gaussian beam with γ = 1 . (c) Weakly truncated Gaussian beam with γ = 3 .

Fig. 4
Fig. 4

Strehl ratio of a beam for a given value of defocus aberration B d as a function of spherical aberration A s . Both B d and A s are in units of wavelength λ. (a) Uniform beam ( γ = 0 ) ; the Strehl ratio in this case is zero when B d is an integral number of wavelengths, as may be seen from Eq. (14). (b) Gaussian beam with γ = 1 . (c) Weakly truncated Gaussian beam with γ = 3 .

Fig. 5
Fig. 5

Axial irradiance of a collimated Gaussian beam, i.e., one with a Fresnel number N = 0 . Uniform ( γ = 0 ) and Gaussian beams with γ = 1 , 2 , 3 are considered. The irradiance is in units of the pupil irradiance P S p for a uniform beam and the distance z is in units of the far-field distance D 2 λ .

Fig. 6
Fig. 6

Axial irradiance of a collimated beam, i.e., one with Fresnel number N = 0 , aberrated by spherical aberration A s . The axial irradiance when A s = 0 is shown for comparison. The units of irradiance and z are the same as in Fig. 5. The defocus aberration B d in units of wavelength is also shown. (a) Uniform beam ( γ = 0 ) . (b) Gaussian beam with γ = 1 . (c) Weakly truncated Gaussian beam with γ = 3 . In (a) and (b) the right-hand scale is for A s = 0 and B d .

Fig. 7
Fig. 7

Axial irradiance of a collimated beam, i.e., one with Fresnel number N = 0 , aberrated by astigmatism A a . The axial irradiance when A a = 0 is shown for comparison. The units of irradiance and z are the same as in Fig. 5. The defocus aberration B d in units of wavelength is also shown. (a) Uniform beam ( γ = 0 ) . (b) Gaussian beam with γ = 1 . (c) Weakly truncated Gaussian beam with γ = 3 .

Tables (1)

Tables Icon

Table 1 Standard Deviation of Defocus Aberration and Defocus Aberration Balanced a with Spherical Aberration or Astigmatism for Minimum Variance

Equations (30)

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A ( ρ ) = A 0 exp ( γ ρ 2 ) , 0 ρ 1 ,
γ = ( a ω ) 2
A 0 2 = 2 γ 1 exp ( 2 γ ) ( P S p ) .
I ( z ; γ ) = ( R z ) 2 ( 2 γ B d 2 + γ 2 ) 1 sinh γ ( cosh γ cos B d ) ,
B d ( z ) = π N ( R z 1 )
I ( R ; γ ) = [ tanh ( γ 2 ) ] ( γ 2 ) .
2 ( λ z S p B d B d 2 + γ 2 ) ( cosh γ cos B d ) = sin B d .
I ( z ; γ ) = ( R z ) 2 2 γ B d 2 + γ 2 ,
I ( R ) = 2 γ ,
I ( 0 ; γ ) = 2 γ π 2 N 2 .
I ( z p ; γ ) = ( 2 γ ) + ( 2 γ π 2 N 2 ) ,
z p R = [ 1 + ( γ π N ) 2 ] 1 .
S = [ 1 + ( B d γ ) 2 ] 1 .
I ( z ) = ( R z ) 2 { [ sin ( B d 2 ) ] ( B d 2 ) } 2 ,
tan ( B d 2 ) = ( R z ) B d 2 , z R ,
I ( z ; γ ) = 2 γ 1 exp ( 2 γ ) ( R π z ) 2 0 1 0 2 π exp ( γ ρ 2 ) exp { i [ Φ ( ρ , θ ) + B d ρ 2 ] } ρ d ρ d θ 2 .
I ( z ; γ ) = 2 γ 1 exp ( 2 γ ) ( R z ) 2 0 1 exp ( γ x ) exp [ i ( A s x 2 + B d x ) ] d x 2 .
I ( z ; γ ) = 2 γ 1 exp ( 2 γ ) ( R 2 ) 2 0 1 exp ( γ x ) exp [ i ( 0.5 A a + B d ) x ] J 0 ( 0.5 A a x ) d x 2 ,
0 2 π exp ( i A a ρ 2 cos 2 θ ) d θ
= exp ( 0.5 i A a ρ 2 ) 0 2 π exp ( 0.5 i A a ρ 2 cos 2 θ ) d θ = 2 π exp ( 0.5 i A a ρ 2 ) J 0 ( 0.5 A a ρ 2 ) .
σ Φ 2 = Φ 2 Φ 2 ,
Φ n = 0 1 0 2 π A ( ρ ) [ Φ ( ρ , θ ) + B d ρ 2 ] n ρ d ρ d θ 0 1 0 2 π A ( ρ ) ρ d ρ d θ = { γ π [ 1 exp ( γ ) ] } 0 1 0 2 π exp ( γ ρ 2 ) [ Φ ( ρ , θ ) + B d ρ 2 ] n ρ d ρ d θ ,
S = [ γ 1 exp ( γ ) ] 2 0 1 exp ( γ x ) exp [ i ( B d x + A s x 2 ) ] d x 2 ,
I ( z ; γ ) = 2 γ ( B d π ) 2 1 exp ( 2 γ ) 0 1 0 2 π exp ( γ ρ 2 ) exp { i [ Φ ( ρ , θ ) + B d ρ 2 ] } ρ d ρ d θ 2 ,
B d = π 4 z
I ( z ; γ ) = { 2 γ [ 1 + ( 4 γ z π ) 2 ] } [ coth γ cos ( π 4 z ) sinh γ ] .
I ( z ; 0 ) = 4 sin 2 ( π 8 z )
I ( z ; γ ) = 2 γ [ 1 + ( 4 γ z π ) 2 ] ,
I ( z ; γ ) = 2 γ B d 2 1 exp ( 2 γ ) 0 1 exp ( γ x ) exp [ i ( A s x 2 + B d x ) ] d x 2 ,
I ( z ; γ ) = 2 γ B d 2 1 exp ( 2 γ ) 0 1 exp ( γ x ) exp [ i ( 0.5 A a + B d ) x ] J 0 ( 0.5 A a x ) d x 2 ,

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