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References

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  1. L. D. Dickson, Appl. Opt. 9, 1854 (1970).
    [Crossref] [PubMed]
  2. D. D. Lowenthal, Appl. Opt. 13, 2126 (1974); Appl. Opt. 13, 2774 (1974).
    [Crossref] [PubMed]
  3. A. Maréchal, Rev. d’Optique 26, 257 (1947).
  4. P. Barrucand, C. R. Acad. Sci. Paris 264, A-792 (1967); C. R. Acad. Sci. Paris 265, A-807 (1967).
  5. A. Boivin, J. Opt. Soc. Am. 42, 60 (1952) and Théorie et Calcul des Figures de Diffraction de Révolution(Les Presses de l’Université Laval, Québec; Gauthier–Villars, Paris, 1964), p. 415.
    [Crossref]
  6. Reference 1, p. 1855.
  7. Reference 2, p. 2129.
  8. Reference 1, p. 1855.
  9. Reference 2, p. 2131.
  10. W. B. King, J. Opt. Soc. Am. 58, 655 (1968).
    [Crossref]

1974 (1)

1970 (1)

1968 (1)

1967 (1)

P. Barrucand, C. R. Acad. Sci. Paris 264, A-792 (1967); C. R. Acad. Sci. Paris 265, A-807 (1967).

1952 (1)

1947 (1)

A. Maréchal, Rev. d’Optique 26, 257 (1947).

Appl. Opt. (2)

C. R. Acad. Sci. Paris (1)

P. Barrucand, C. R. Acad. Sci. Paris 264, A-792 (1967); C. R. Acad. Sci. Paris 265, A-807 (1967).

J. Opt. Soc. Am. (2)

Rev. d’Optique (1)

A. Maréchal, Rev. d’Optique 26, 257 (1947).

Other (4)

Reference 1, p. 1855.

Reference 2, p. 2129.

Reference 1, p. 1855.

Reference 2, p. 2131.

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Figures (2)

FIG. 1
FIG. 1

Relative far-field irradiance versus W = kRX/f for a truncated gaussian beam (γ = 1) degraded by various amounts of third-order spherical aberration as measured by the polynomial coefficient a2 in wavelengths, calculated from Eq. (11).

FIG. 2
FIG. 2

Relative far-field irradiance versus W = kRX/f for a truncated gaussian beam (γ = 1) degraded by various amounts of third-order spherical aberration as measured by the polynomial coefficient a2 in wavelengths, calculated from Eq. (11).

Equations (14)

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I ( γ , W ) = ( 4 γ 4 ) 0 1 exp ( - ρ 2 / γ 2 ) J 0 ( W ρ ) [ 1 + i k Φ ( ρ ) - 1 2 k 2 Φ 2 ( ρ ) ] ρ d ρ 2 [ 1 - exp ( - 1 / γ 2 ) ] 2 ,
Δ j ( γ , W ) 2 γ 2 [ 1 - exp ( - 1 / γ 2 ) ] × 0 1 exp ( - ρ 2 / γ 2 ) J 0 ( W ρ ) Φ j ( ρ ) ρ d ρ ,
I ( γ , W ) = Δ 0 ( γ , W ) + i k Δ 1 ( γ , W ) - 1 2 k 2 Δ 2 ( γ , W ) 2 .
Φ ( ρ ) = n = 0 a n ρ 2 n ,             [ Φ ( ρ ) ] 2 = n = 0 b n ρ 2 n .
b 0 = a 0 2 , b 1 = 2 a 0 a 1 , b 2 = 2 a 0 a 2 + a 1 2 , b 3 = 2 a 0 a 3 + 2 a 1 a 2 , b 4 = 2 a 0 a 4 + 2 a 1 a 3 + a 2 2 , b 5 = 2 a 0 a 5 + 2 a 1 a 4 + 2 a 2 a 3 .
b n a 0 n = l = 1 n b n - l a l ( 3 l - n ) ,             b 0 = a 0 2 .
Δ 0 ( γ , W ) = 2 γ 2 [ 1 - exp ( - 1 / γ 2 ) ] m = 0 ( - 1 ) m ( 1 2 W ) 2 m ( m ! ) 2 × 0 1 ρ 2 m + 1 exp ( - ρ 2 / γ 2 ) ,
Δ 1 , 2 ( γ , W ) = 2 γ 2 [ 1 - exp ( - 1 / γ 2 ) ] n = 0 a n , b n × m = 0 ( - 1 ) m ( 1 2 W ) 2 m ( m ! ) 2 0 1 ρ 2 ( n + m ) + 1 exp ( - ρ 2 / γ 2 ) ρ d ρ .
I m , n ( γ ) = exp ( - 1 / γ 2 ) S m , n ( γ ) 2 ( m + n + 1 ) ,
S m , n ( γ ) = { 1 + 1 ( m + n + 2 ) · ( 1 γ 2 ) + 1 ( m + n + 2 ) ( m + n + 3 ) · ( 1 γ 2 ) 2 + } .
Γ n ( γ , W ) = exp ( - 1 / γ 2 ) γ 2 [ 1 - exp ( - 1 / γ 2 ) ] m = 0 ( - 1 ) m ( 1 2 W ) 2 m ( m ! ) 2 · S m , n ( γ ) ( m + n + 1 ) ,
Δ 0 ( γ , W ) = Γ 0 ( γ , W ) , Δ 1 ( γ , W ) = n = 0 a n Γ n ( γ , W ) , Δ 2 ( γ , W ) = n = 0 b n Γ n ( γ , W ) .
I ( γ , W ) = [ Γ 0 ( γ , W ) ] 2 { 1 - k 2 [ n = 0 b n ( Γ n ( γ , W ) Γ 0 ( γ , W ) ) - ( n = 0 a n Γ n ( γ , W ) Γ 0 ( γ , W ) ) 2 ] } .
I ( γ , W ) = [ Γ 0 ( γ , W ) ] 2 { 1 - ( k a n ) 2 [ Γ 2 n ( γ , W ) Γ 0 ( γ , W ) - ( Γ n ( γ , W ) Γ 0 ( γ , W ) ) 2 ] } ,