Abstract

It is shown in an earlier paper dealing with flat-topped light beams [Opt. Lett. 27, 1007 (2002) ] that the profile of flat-topped beams can be expressed in the form 1[1exp(ξ2)]M, where ξ is a dimensionless parameter and M is a nonnegative number. The expansion of the proposed expression is a finite series containing only the lowest-order Gaussian modes. This situation provides the possibility of reformulating the scalar theory of diffraction at an aperture in an opaque screen if the Gaussian mode expansion is employed to describe the boundary values of the light incident on the screen. As an example of this effort, an asymptotic model is established for three-dimensional irradiance distributions near the focus in systems of different Fresnel numbers. The proposed expansions contain only elementary functions and permit all elementary operations; therefore no special functions or special algorithms are needed in the evaluation of either irradiance distributions or the integrated energy in a focused field.

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References

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  1. F. Gori, 'Flattened Gaussian beams,' Opt. Commun. 107, 335-341 (1994).
    [CrossRef]
  2. Y. Li, 'Light beams with flat-topped profiles,' Opt. Lett. 27, 1007-1009 (2002).
    [CrossRef]
  3. Y. Li, 'New expressions for flat-topped light beams,' Opt. Commun. 206, 225-334 (2002).
    [CrossRef]
  4. Y. Li, 'Flat-topped light beams with non-circular cross-sections,' J. Mod. Opt. 50, 1957-1966 (2003).
    [CrossRef]
  5. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), Sec. 8.2.2.
  6. G. N. Watson, A Treatise of the Theory of Bessel Functions (Cambridge U. Press, 1962), pp. 537-550.
  7. Y. Li and E. Wolf, 'Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,' J. Opt. Soc. Am. A 1, 801-808 (1984).
    [CrossRef]
  8. J. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980), Eq. (0.155) on p. 4.
  9. Y. Li, 'Degeneracy in the Fraunhofer diffraction of truncated Gaussian beams,' J. Opt. Soc. Am. A 4, 1237-1244 (1987).
    [CrossRef]
  10. Y. Li and E. Wolf, 'Focal shift in focused truncated Gaussian beams,' Opt. Commun. 42, 151-156 (1982).
    [CrossRef]
  11. E. H. Linfoot and E. Wolf, 'Phase distribution near focus in an aberration-free diffraction image,' Proc. Phys. Soc. London, Sect. B 69, 823-832 (1956).
    [CrossRef]
  12. E. Wolf, 'Light distribution near focus in an error-free diffraction image,' Proc. R. Soc. London, Ser. A 204, 533-548 (1951).
    [CrossRef]
  13. Y. Li, 'Encircled energy of diffracted converging spherical waves,' J. Opt. Soc. Am. 73, 1101-1104 (1983).
    [CrossRef]
  14. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), Sec. 4.5.1.
  15. F. W. J. Olver, Asymptotics and Special Functions (Academic, 1974), Chap. 1.
  16. Y. Li and F. T. S. Yu, 'Intensity distribution near the focus of an apertured focused Gaussian beam,' Opt. Commun. 70, 1-7 (1989).
    [CrossRef]

2003 (1)

Y. Li, 'Flat-topped light beams with non-circular cross-sections,' J. Mod. Opt. 50, 1957-1966 (2003).
[CrossRef]

2002 (2)

Y. Li, 'Light beams with flat-topped profiles,' Opt. Lett. 27, 1007-1009 (2002).
[CrossRef]

Y. Li, 'New expressions for flat-topped light beams,' Opt. Commun. 206, 225-334 (2002).
[CrossRef]

1994 (1)

F. Gori, 'Flattened Gaussian beams,' Opt. Commun. 107, 335-341 (1994).
[CrossRef]

1989 (1)

Y. Li and F. T. S. Yu, 'Intensity distribution near the focus of an apertured focused Gaussian beam,' Opt. Commun. 70, 1-7 (1989).
[CrossRef]

1987 (1)

1984 (1)

1983 (1)

1982 (1)

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

1956 (1)

E. H. Linfoot and E. Wolf, 'Phase distribution near focus in an aberration-free diffraction image,' Proc. Phys. Soc. London, Sect. B 69, 823-832 (1956).
[CrossRef]

1951 (1)

E. Wolf, 'Light distribution near focus in an error-free diffraction image,' Proc. R. Soc. London, Ser. A 204, 533-548 (1951).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), Sec. 8.2.2.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), Sec. 4.5.1.

Gori, F.

F. Gori, 'Flattened Gaussian beams,' Opt. Commun. 107, 335-341 (1994).
[CrossRef]

Gradshteyn, J. S.

J. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980), Eq. (0.155) on p. 4.

Li, Y.

Y. Li, 'Flat-topped light beams with non-circular cross-sections,' J. Mod. Opt. 50, 1957-1966 (2003).
[CrossRef]

Y. Li, 'New expressions for flat-topped light beams,' Opt. Commun. 206, 225-334 (2002).
[CrossRef]

Y. Li, 'Light beams with flat-topped profiles,' Opt. Lett. 27, 1007-1009 (2002).
[CrossRef]

Y. Li and F. T. S. Yu, 'Intensity distribution near the focus of an apertured focused Gaussian beam,' Opt. Commun. 70, 1-7 (1989).
[CrossRef]

Y. Li, 'Degeneracy in the Fraunhofer diffraction of truncated Gaussian beams,' J. Opt. Soc. Am. A 4, 1237-1244 (1987).
[CrossRef]

Y. Li and E. Wolf, 'Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,' J. Opt. Soc. Am. A 1, 801-808 (1984).
[CrossRef]

Y. Li, 'Encircled energy of diffracted converging spherical waves,' J. Opt. Soc. Am. 73, 1101-1104 (1983).
[CrossRef]

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

Linfoot, E. H.

E. H. Linfoot and E. Wolf, 'Phase distribution near focus in an aberration-free diffraction image,' Proc. Phys. Soc. London, Sect. B 69, 823-832 (1956).
[CrossRef]

Olver, F. W.

F. W. J. Olver, Asymptotics and Special Functions (Academic, 1974), Chap. 1.

Ryzhik, I. M.

J. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980), Eq. (0.155) on p. 4.

Watson, G. N.

G. N. Watson, A Treatise of the Theory of Bessel Functions (Cambridge U. Press, 1962), pp. 537-550.

Wolf, E.

Y. Li and E. Wolf, 'Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,' J. Opt. Soc. Am. A 1, 801-808 (1984).
[CrossRef]

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

E. H. Linfoot and E. Wolf, 'Phase distribution near focus in an aberration-free diffraction image,' Proc. Phys. Soc. London, Sect. B 69, 823-832 (1956).
[CrossRef]

E. Wolf, 'Light distribution near focus in an error-free diffraction image,' Proc. R. Soc. London, Ser. A 204, 533-548 (1951).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), Sec. 8.2.2.

Yu, F. T.

Y. Li and F. T. S. Yu, 'Intensity distribution near the focus of an apertured focused Gaussian beam,' Opt. Commun. 70, 1-7 (1989).
[CrossRef]

J. Mod. Opt. (1)

Y. Li, 'Flat-topped light beams with non-circular cross-sections,' J. Mod. Opt. 50, 1957-1966 (2003).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Commun. (4)

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

F. Gori, 'Flattened Gaussian beams,' Opt. Commun. 107, 335-341 (1994).
[CrossRef]

Y. Li, 'New expressions for flat-topped light beams,' Opt. Commun. 206, 225-334 (2002).
[CrossRef]

Y. Li and F. T. S. Yu, 'Intensity distribution near the focus of an apertured focused Gaussian beam,' Opt. Commun. 70, 1-7 (1989).
[CrossRef]

Opt. Lett. (1)

Proc. Phys. Soc. London, Sect. B (1)

E. H. Linfoot and E. Wolf, 'Phase distribution near focus in an aberration-free diffraction image,' Proc. Phys. Soc. London, Sect. B 69, 823-832 (1956).
[CrossRef]

Proc. R. Soc. London, Ser. A (1)

E. Wolf, 'Light distribution near focus in an error-free diffraction image,' Proc. R. Soc. London, Ser. A 204, 533-548 (1951).
[CrossRef]

Other (5)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), Sec. 8.2.2.

G. N. Watson, A Treatise of the Theory of Bessel Functions (Cambridge U. Press, 1962), pp. 537-550.

J. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980), Eq. (0.155) on p. 4.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), Sec. 4.5.1.

F. W. J. Olver, Asymptotics and Special Functions (Academic, 1974), Chap. 1.

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Figures (8)

Fig. 1
Fig. 1

Diffraction of a uniform, converging spherical wave at an aperture (circular or rectangular) in an opaque screen. The point Q is located on the wavefront W of radius f, centered on the geometrical focal point F and passing the center O of the aperture.

Fig. 2
Fig. 2

(a) Flat-topped profiles F M ( ρ ) as a function of dimensionless variable ρ for different values of M given on each curve. (b) Rescaled profiles P M ( ρ ) as a function of ρ for the values of M and β M given on each curve.

Fig. 3
Fig. 3

(a) Normalized focal-plane irradiance patterns formed by the diffraction of a focused Gaussian beam truncated by a hard-edged aperture ( M ) and the flat-topped Gaussian apertures of M = 10 and 50, assuming the truncation parameter σ = 1 . (b) The same diffraction pattern enlarged to show the difference in sidelobe irradiance.

Fig. 4
Fig. 4

(a) Normalized axial irradiance patterns formed by the diffraction of a focused Gaussian beam truncated by a hard-edged aperture ( M ) and by the flat-topped Gaussian apertures of M = 10 and 50, assuming Fresnel number N a = 1 and the truncation parameter σ = 1 . (b) The same pattern enlarged to show the difference in sidelobe irradiance. (c) Adjustment of the Fresnel number from N a = 1 to 1.08 to obtain the same amount of the focal shift predicted by different theories.

Fig. 5
Fig. 5

(a) Isophotes (i.e., contour lines of equal irradiance) in a meridional plane near the focus of a focused Gaussian beam truncated by a hard-edged aperture ( M ) , assuming Fresnel number N a = 1 and the truncation parameter σ = 1 . Irradiance is normalized to unity at geometrical focal point F. The dashed lines represent the boundary of geometrical shadow. (b) Diffraction at a flat-topped Gaussian aperture of M = 50 . The complete distribution would be found by rotating (a) and (b) about the z axis.

Fig. 6
Fig. 6

Isophotes (i.e., contour lines of equal irradiance) in a meridional plane of a focused Gaussian beam truncated by an aperture that is much larger than the beam waist. Irradiance is normalized to unity at the geometrical focal point F.

Fig. 7
Fig. 7

(a) Contour lines of encircled energy, i.e., the fraction of the total energy that falls within circles centered on the axis in receiving planes z f = constant , predicted by the classical theory for diffraction of a focused Gaussian beam truncated by a hard-edged aperture of N a = 1 and σ = 1 . (b) Pattern predicted by asymptotic expansion (30) for a flat-topped Gaussian aperture of M = 20 . Dark dots show variation of the locations of optimum focusing of a Gaussian laser beam onto a finite-sized target.

Fig. 8
Fig. 8

(a) Diffraction of an elliptical Gaussian beam at a rectangular aperture. (b) Focal-plane diffraction pattern predicted by the classical theory for the case of an elliptical Gaussian beam of ellipticity ϵ = 2 diffracted at a rectangular hard-edged aperture ( M ) with an aspect ratio a y a x = 2 : 1 . (c) Pattern predicted by asymptotic expansion (38) for a flat-topped Gaussian aperture of M = 50 .

Equations (68)

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U 0 ( Q ) = A exp [ ( a ρ ) 2 w 2 ] exp ( i k f ) f ( k = 2 π λ ) ,
U ( P ) = i λ W U 0 ( Q ) exp ( i k s ) s d S .
s = ( x ξ ) 2 + ( y η ) 2 + ( z ς + f ) 2 .
s ( f + z ) 1 2 ( f + z ) [ 2 a r ρ cos ( θ + ψ ) + z f ( a ρ ) 2 r 2 ] ,
ρ = 1 a ξ 2 + η 2 ,
r = x 2 + y 2
U ( P ) = i π a 2 A λ f 2 exp [ i Φ ] ( 1 u N 2 π N a ) K ( u N , v N ) ,
K ( u N , v N ) = 2 0 1 J 0 ( v N ρ ) exp [ i 2 ( u N 2 i σ ) ρ 2 ] ρ d ρ = 2 0 1 J 0 ( v N ρ ) exp ( i 2 u N ρ 2 ) ρ d ρ
u N = 2 π N a z f 1 + z f ,
v N = 2 π N a r a 1 + z f ,
u N = u N 2 i σ .
N a = a 2 λ f ,
σ = ( a w ) 2 .
U n ( w , z ) = p = 0 ( 1 ) p ( w z ) n + 2 p J n + 2 p ( z ) ,
V n ( w , z ) = p = 0 ( 1 ) p ( w z ) n 2 p J n 2 p ( z ) ,
F M ( ρ ) = 1 [ 1 exp ( ρ 2 ) ] M ,
P M ( ρ ) = 1 [ 1 exp ( β M ρ 2 ) ] M .
0 P M ( ρ ) ρ d ρ = 0 P 1 ( ρ ) ρ d ρ = 1 2 .
P M ( ρ ) = m = 1 M α m exp ( m β M ρ 2 ) ,
α m = ( 1 ) m 1 ( M m )
β M = m = 1 M α m m .
β M = m = 1 M 1 m C + 1 M + ln ( M ) ,
K ( u N , v N ) = 2 0 P M ( ρ ) J 0 ( v N ρ ) exp ( i 2 u N ρ 2 ) ρ d ρ = m = 1 M α m { 2 0 J 0 ( v N ρ ) exp [ ( m β M + i u N 2 ) ρ 2 ] ρ d ρ } = m = 1 M α m m β M + i u N 2 exp ( 1 4 v N 2 m β M + i u N 2 ) .
I ( P ) = U ( P ) 2 = I F ( 1 u N 2 π N a ) 2 m = 1 M α m m β M + i u N 2 exp ( 1 4 v N 2 m β M + i u N 2 ) 2 ,
I ( 0 , v ) = I F [ s = 1 ( 2 σ v ) s J s ( v ) ] 2 ( for v > 2 σ ) ,
I ( 0 , v ) = I F [ exp ( σ v 2 4 σ ) s = 0 ( v 2 σ ) s J s ( v ) ] 2 ( for v < 2 σ ) .
I ( 0 , v ) = I F 4 { 1 ( 1 2 σ ) J 0 ( 2 σ ) + π σ [ J 1 ( 2 σ ) H 0 ( 2 σ ) J 0 ( 2 σ ) H 1 ( 2 σ ) ] } 2 ( for v = 2 σ ) ,
I ( P ) = I F m = 1 M α m m β M + σ exp ( v 2 4 1 m β M + σ ) 2 .
I ( u N , 0 ) = I F 4 σ 2 + u N 2 ( 1 u N 2 π N a ) 2 cosh σ cos ( u N 2 ) cosh σ 1 .
I ( P ) = I F ( 1 u N 2 π N a ) 2 m = 1 M α m m β M + i u N 2 2 .
m = 1 M α m m β M + i u N 2 exp ( 1 4 v N 2 m β M + i u N 2 ) m = 1 M α m i u N 2 exp ( 1 4 v N 2 i u N 2 ) = 1 i u N 2 exp ( 1 4 v N 2 i u N 2 ) m = 1 M α m .
m = 1 M α m 1 ,
I N ( P ) = I F 1 + i u G 2 ( 1 u G 2 π N w ) 2 exp ( v G 2 4 1 + i u G 2 ) ,
u G = 2 π N w z f 1 + z f ,
v G = 2 π N w r a 1 + z f
N w = w 2 λ f .
E = 0 0 2 π I N ( 0 , v ) r d r d θ
L ( z , r 0 ) = 1 E 0 r 0 0 2 π I N ( u N , v N ) r d r d θ .
L ( z , r 0 ) = 1 E 0 [ m = 1 M ( α m ) 2 m β M + σ ( 1 exp ( v 0 2 2 m β M + σ ( m β M + σ ) 2 + ( u N 2 ) 2 ) ) + m = 1 M n = m + 1 M α m α n [ H m , n ( u N , v 0 ) H m , n ( u N , 0 ) ] ( m β M + σ ) 2 + ( u N 2 ) 2 ( n β M + σ ) 2 + ( u N 2 ) 2 ] ,
E 0 = m = 1 M ( α m ) 2 m β M + σ + 2 m = 1 M n = m + 1 M α m α n 0.5 ( m + n ) β M + σ ,
H m , n ( u N , v 0 ) = ( q m + q n ) cos [ ( l m l n ) v 0 2 ( ϕ m ϕ n ) ] + ( l m l n ) sin [ ( l m l n ) v 0 2 ( ϕ m ϕ n ) ] ( q m + q n ) 2 + ( l m l n ) 2 exp [ ( q m + q n ) v 0 2 ] ,
v 0 = 2 π N a r 0 a 1 + z f ,
q κ + i l κ = 1 4 1 κ β M + σ + i u N 2 ,
ϕ κ = arctan ( u N 2 κ β M + σ ) ( κ = m , n ) .
d L ( z , r 0 ) d z = 0 .
U 0 ( Q ) = A exp ( σ x ξ a 2 ) exp ( σ y η a 2 ) exp ( i k f ) f ,
ξ a = ξ a x ,
η a = η a y .
P M ( ξ a , η a ) = ( m = 1 M α m exp ( m β M ξ a 2 ) ) ( m = 1 M α m exp ( m β M η a 2 ) ) ,
I N ( u x , v x ; u y , v y ) = I F 1 ( 1 u x 2 π N x ) ( 1 u y 2 π N y ) I x ( u x , v x ) I y ( u y , v y ) ,
I p ( u p , v p ) = m = 1 M α m m β M + σ p + i u p 2 exp ( v p 2 4 m β M + σ p + i u N 2 ) 2 ( p = x , y ) ,
u p = 2 π N p z f 1 + z f ,
v p = 2 π N p p a p 1 + z f ,
N p = a p 2 λ f .
K ( u N , v N ) = 2 0 1 J 0 ( v N ρ ) exp ( i 2 u N ρ 2 ) ρ d ρ ,
u N = u N 2 i σ = u N 2 + 4 σ 2 exp ( i ϑ ) ,
K ( u N , v N ) = exp ( i u N 2 ) u N 2 { U 1 ( u N , v N ) + i U 2 ( u N , v N ) }
= exp ( i u N 2 ) u N 2 { i exp ( i u N + v N 2 u N 2 ) + i V 0 ( u N , v N ) + V 1 ( u N , v N ) } ,
I ( u N , v N ) = U ( u N , v N ) 2 = ( π a 2 A λ f 2 ) ( 1 u N 2 π N a ) 2 K ( u N , v N ) 2 = I 0 [ exp ( σ ) 1 ] 2 4 σ 2 u N 2 + 4 σ 2 ( 1 u N 2 π N a ) 2 [ ( U 1 ( c ) U 2 ( s ) ) 2 + ( U 1 ( s ) + U 2 ( c ) ) 2 ]
= I 0 [ exp ( σ ) 1 ] 2 4 σ 2 u N 2 + 4 σ 2 ( 1 u N 2 π N a ) 2 { [ V 0 ( s ) + V 1 ( c ) + exp [ σ ( 1 v N 2 u N 2 ) ] sin [ u N 2 ( 1 v N 2 u N 2 ) ] ] 2 + [ V 0 ( c ) + V 1 ( s ) exp [ σ ( 1 v N 2 u N 2 ) ] cos ( u N 2 ( 1 v N 2 u N 2 ) ) ] 2 } ,
U n ( u N , v N ) = U n ( c ) ( u N , v N ) + i U n ( s ) ( u N , v N ) = p = 0 ( 1 ) p ( u N v N ) n + 2 p exp [ i ( n + 2 p ) ϑ ] J n + 2 p ( v N ) ,
V n ( u N , v N ) = V n ( c ) ( u N , v N ) + i V n ( s ) ( u N , v N ) = p = 0 ( 1 ) p ( u N v N ) n 2 p exp [ i ( n + 2 p ) ϑ ] J n 2 p ( v N ) .
U n ( c ) ( v N , v N ) = p = 0 ( 1 ) p J n + 2 p ( v N ) cos [ ( n + 2 p ) ϑ ] ,
U n ( s ) ( v N , v N ) = p = 0 ( 1 ) p J n + 2 p ( v N ) sin [ ( n + 2 p ) ϑ ] .
U 1 ( c ) ( v N , v N ) = 1 2 sin ( v N 2 4 σ 2 ) ,
U 2 ( c ) ( v N , v N ) = 1 2 [ J 0 ( v N ) cos ( v N 2 4 σ 2 ) ] .
U 1 ( s ) ( v N , v N ) = 1 2 cos ( v N 2 4 σ 2 ) ,
U 2 ( s ) ( v N , v N ) = 1 2 [ J 0 ( v N ) sin ( v N 2 4 σ 2 ) ] .

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