Abstract

The shift in the plane of best focus away from the geometrical focal plane is calculated for a Laguerre–Gaussian beam. It is found that the plane of best focus is closer to the focusing lens than predicted by geometrical optics by an amount dependent not only on the Fresnel number of the field at the exit plane of the lens but also on the mode order.

© 1987 Optical Society of America

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References

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  1. E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
    [Crossref]
  2. Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–21.5 (1981).
    [Crossref]
  3. W. H. Carter, “Focal shift and concept of effective Fresnel number for a Gaussian laser beam,” Appl. Opt. 21, 1989–1994 (1982).
    [Crossref] [PubMed]
  4. G. Goubau, in Millimetre and Submillimetre Waves, F. A. Benson, ed. (Iliffe, London, 1969), pp. 337–367.
  5. R. L. Phillips, L. C. Andrews, “Spot size and divergence for Laguerre Gaussian beams at any order,” Appl. Opt. 22, 643–644 (1983).
    [Crossref] [PubMed]
  6. W. H. Carter, “Electromagnetic beam fields,” Opt. Acta 21, 871–892 (1974).
    [Crossref]
  7. W. H. Carter, “Anomalies in the field of a Gaussian beam near focus,” Opt. Commun. 7, 211–218 (1973).
    [Crossref]

1983 (1)

1982 (1)

1981 (2)

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[Crossref]

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–21.5 (1981).
[Crossref]

1974 (1)

W. H. Carter, “Electromagnetic beam fields,” Opt. Acta 21, 871–892 (1974).
[Crossref]

1973 (1)

W. H. Carter, “Anomalies in the field of a Gaussian beam near focus,” Opt. Commun. 7, 211–218 (1973).
[Crossref]

Andrews, L. C.

Carter, W. H.

W. H. Carter, “Focal shift and concept of effective Fresnel number for a Gaussian laser beam,” Appl. Opt. 21, 1989–1994 (1982).
[Crossref] [PubMed]

W. H. Carter, “Electromagnetic beam fields,” Opt. Acta 21, 871–892 (1974).
[Crossref]

W. H. Carter, “Anomalies in the field of a Gaussian beam near focus,” Opt. Commun. 7, 211–218 (1973).
[Crossref]

Goubau, G.

G. Goubau, in Millimetre and Submillimetre Waves, F. A. Benson, ed. (Iliffe, London, 1969), pp. 337–367.

Li, Y.

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[Crossref]

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–21.5 (1981).
[Crossref]

Phillips, R. L.

Wolf, E.

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–21.5 (1981).
[Crossref]

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[Crossref]

Appl. Opt. (2)

Opt. Acta (1)

W. H. Carter, “Electromagnetic beam fields,” Opt. Acta 21, 871–892 (1974).
[Crossref]

Opt. Commun. (3)

W. H. Carter, “Anomalies in the field of a Gaussian beam near focus,” Opt. Commun. 7, 211–218 (1973).
[Crossref]

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[Crossref]

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–21.5 (1981).
[Crossref]

Other (1)

G. Goubau, in Millimetre and Submillimetre Waves, F. A. Benson, ed. (Iliffe, London, 1969), pp. 337–367.

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Figures (4)

Fig. 1
Fig. 1

Illustrating the notation.

Fig. 2
Fig. 2

Magnitude of the field amplitude for a Laguerre–Gaussian beam with σ = 1λ and N = 3. The contour lines indicate loci of constant magnitude as labeled: 1, 1.0; 2, 0.78; 3, 0.606; 4, 0.472; 5, 0.368; 6, 0.286; 7, 0.223; 8, 0.174; 9,0.135; A, 0.105; B, 0.082; C, 0.064; D, 0.050; E, 0.039; F, 0.030; G, 0.023; H, 0.018; and I, 0.014. These data are normalized with a maximum of unity at the focal point in the lower left-hand corner. The contour levels are also given by the formula exp[−0.25(n − 1)] for integer values of n (where A = 10, B = 11, etc.).

Fig. 3
Fig. 3

Phase of the field amplitude for a Laguerre–Gaussian beam with σ = 1λ and N = 3, corresponding to the amplitude in Fig. 2. The contour lines indicate loci of constant phase 30 deg apart.

Fig. 4
Fig. 4

Plot of the fractional defocus as a function of effective exposure and mode order obtained from field data without the use of the paraxial approximation.

Tables (2)

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Table 1 Computer Data for a(z)a

Tables Icon

Table 2 Computer Data for C(z)

Equations (23)

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Δ f ( z ) = C ( z ) - R ( z ) R ( z ) = 1 π 2 F 2 ( z ) .
F ( z ) = a 2 ( z ) λ C ( z )
C ( z ) = z + k 2 σ 4 / z ,
a 2 ( z ) 2 - ( x 2 + y 2 ) ψ ( x ) 2 d x d y - ψ ( x ) 2 d x d y ,
a ( z ) = 2 σ ( 1 + z 2 k 2 σ 4 ) 1 / 4 ( 2 N + 1 ) 1 / 2 ,
F ( z ) = 2 z 2 π k σ 2 ( 2 N + 1 ) ,
C ( z ) z = 1 + ( 2 N + 1 ) 2 π 2 F 2 ( z ) ,
Δ f ( z ) = ( 2 N + 1 ) 2 π 2 F 2 ( z ) .
ψ ( x ) = - A ( p , q ) exp ( i k p · x ) d p d q ,
ψ ( ) ( x ) k r - i λ z r A ( ± x r , ± y r ) e ± i k i r r .
A ( p , q ) = k 2 ψ 0 2 π N ! exp [ - k 2 σ 2 ( p 2 + q 2 ) 2 ] × L N ( 0 ) [ k 2 σ 2 ( p 2 + q 2 ) ]             i f f i p 2 + q 2 1 = 0             i f f i p 2 + q 2 > 1 ,
ψ ( x ) = k 2 ψ 0 2 π N ! p 2 + q 2 = 1 exp [ - k 2 σ 2 ( p 2 + q 2 ) 2 ] × L N ( 0 ) [ k 2 σ 2 ( p 2 + q 2 ) ] exp ( i k p · x ) d p d q
p = ρ cos θ , x = r cos θ , q = ρ sin θ , y = r sin θ
ψ ( x ) = k 2 ψ 0 N ! 0 1 exp ( - k 2 σ 2 ρ 2 2 ) L N ( 0 ) ( k 2 σ 2 ρ 2 ) × exp ( i k z 1 - ρ 2 ) J 0 ( k r ρ ) ρ d ρ .
ρ = sin ϕ
ψ ( x ) = k 2 ψ 0 N ! 0 π / 2 exp ( - k 2 σ 2 sin 2 ϕ 2 ) L N ( 0 ) ( k 2 σ 2 sin 2 ϕ ) × exp ( i k z cos ϕ ) J 0 ( k r sin ϕ ) sin ϕ cos ϕ d ϕ .
d d C r 0 r 1 [ ϕ ( r ) - 2 π λ r 2 2 C ] 2 d r = 0 ,
C ( z ) = π λ r 1 5 - r 0 5 5 / r 0 r 1 r 2 ϕ ( r ) d r .
Δ f ( z ) = K ( N ) / F 2 ( z ) ,
F ( z ) ( 2 N + 1 ) .
K ( N ) = K 0 ( 2 N + 1 ) 2 ,
Δ f ( z ) = K 0 ( 2 N + 1 ) 2 F 2 ( z ) .
Δ f ( z ) = ( 2 N + 1 ) 2 π 2 F 2 ( z ) .

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