Abstract

Diffraction image of a point source due to an optical system with residual primary coma and nonuniform amplitude transmission is studied in order to obtain a suitable expression for the fractional encircled energy distribution in the Fraunhofer plane. Numerical results are obtained for a series of pupil filters and for various amounts of coma in the system. Intensity distribution, fractional encircled energy distribution, Strehl ratio, two-point resolution, and comatic elongation are the properties of the diffraction field that have been examined. A comparative study of the performance of these pupil filters under influence of primary coma is also presented.

© 1985 Optical Society of America

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References

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  1. J-E. Villeneuve, A. Boivin, S. C. Biswas, “L’image Tridimensionnelle du Point en Présence d’Aberration Sphérique Primaire et de Filtrage d’Amplitude: Unitaire ou Modal,” Can J. Phys. 63, 287 (1985).
    [CrossRef]
  2. J-E. Villeneuve, S. C. Biswas, A. Boivin, “Image Diffraction-nelle due à Une Pupille Aberrante Non-uniforme,” Can. J. Phys. 63, 275 (1985).
    [CrossRef]
  3. S. C. Biswas, J-E. Villeneuve, “Combined Effect of All Aberrations and a Pupil Filter on the Diffraction Image,” J. Opt. Soc. Am. A 1, 1316A (1984).
  4. Y. Li, “Establishment of the Maximum Encircled Energy in the Geometrical Focal Plane,” Opt. Acta 31, 1107 (1984).
    [CrossRef]
  5. V. N. Mahajan, “Line of Sight of an Aberrated Optical System,” J. Opt. Soc. Am. A 2, 833 (1985).
    [CrossRef]
  6. H. H. Hopkins, “Image Shift, Phase Distortion and Optical Transfer Function,” Opt. Acta 31, 345 (1984).
    [CrossRef]
  7. S. A. Self, “Focusing of Spherical Gaussian Beams,” Appl. Opt. 22, 658 (1983).
    [CrossRef] [PubMed]
  8. M. J. Yzuel, J. Calvo, “Point-Spread Function Calculation for Optical Systems with Residual Aberrations and a Non-Uniform Transmission Pupil,” Opt. Acta 30, 233 (1983).
    [CrossRef]
  9. G. I. Greisukh, S. A. Stepanor, “Aberrational Analysis of Optical Systems with Diffraction Elements,” Opt. Spectros. 54, 93 (1983).
  10. S. Szapiel, “Aberration Balancing Technique for Radially Symmetric Amplitude Distributions: A Generalization of the Maréchal Approach,” J. Opt. Soc. Am. 72, 947 (1982).
    [CrossRef]
  11. A. Dubik, “Analysis of The Effect of Diffraction and Apodization upon the Spatial-Energy Radiation—Characteristics in Nd-Glass Amplifiers,” J. Tech. Phys. 22, 3 (1981).
  12. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U.P., London, 1966).
  13. S. C. Biswas, A. Boivin, “Influence of Primary Astigmatism on the Performance of Optimum Apodizers,” J. Opt. (India) 4, 1 (1975).
  14. S. C. Biswas, A. Boivin, “Influence of Spherical Aberration on the Performance of Optimum Apodizers,” Opt. Acta 23, 569 (1976).
    [CrossRef]
  15. A. M. Clements, J. E. Wilkins, “Apodization for Maximum Encircled-Energy Ratio and Specified Rayleigh Limit,” J. Opt. Soc. Am. 64, 23 (1974).
    [CrossRef]
  16. R. Boivin, A. Boivin, “Optimized Amplitude Filtering for Superresolution over a Restricted Field,” Opt. Acta 27, 587, 1641 (1980); Opt. Acta 30, 681 (1983).
    [CrossRef]
  17. J-E. Villeneuve, “L’Image Tridimensionnelle du Point sous l’Influence Conjointe de l’Aberration Sphérique et du Filtrage d’Amplitude,” Doctoral Thesis, U. Laval, Québec (1981).
  18. H. Osterberg, J. E. Wilkins, “The Resolving Power of a Coated Objective,” J. Opt. Soc. Am. 39, 553 (1949).
    [CrossRef]
  19. H. Osterberg, F. C. Wissler, “The Resolution of Two Particles in a Bright Field by Coated Microscope Objectives,” J. Opt. Soc. Am. 39, 558 (1949).
    [CrossRef]
  20. J. E. Wilkins, “The Resolving Power of a Coated Objective II,” J. Opt. Soc. Am. 40, 222 (1950).
    [CrossRef]
  21. J. J. Stamnes, H. Heier, S. Ljunggren, “Encircled Energy for Systems with Centrally Obscured Circular Pupils,” Appl. Opt. 21, 1628 (1982).
    [CrossRef] [PubMed]
  22. S. C. Biswas, A. Boivin, “Performance of Optimum Apodizers in Presence of Primary Coma,” Can. J. Phys. 57, 1388 (1979).
    [CrossRef]
  23. G. Boyer, M. Séchaud, “Superresolution by Taylor Filters,” Appl. Opt. 12, 893 (1973).
    [CrossRef]
  24. B. R. Frieden, “The Extrapolation Pupil, Image Synthesis, and Some Thought Applications,” Appl. Opt. 9, 2489 (1970).
    [CrossRef] [PubMed]
  25. V. N. Mahajan, “Zernike Annular Polynomials for Imaging Systems with Annular Pupils,” J. Opt. Soc. Am. 71, 75 (1981).
    [CrossRef]

1985 (3)

V. N. Mahajan, “Line of Sight of an Aberrated Optical System,” J. Opt. Soc. Am. A 2, 833 (1985).
[CrossRef]

J-E. Villeneuve, A. Boivin, S. C. Biswas, “L’image Tridimensionnelle du Point en Présence d’Aberration Sphérique Primaire et de Filtrage d’Amplitude: Unitaire ou Modal,” Can J. Phys. 63, 287 (1985).
[CrossRef]

J-E. Villeneuve, S. C. Biswas, A. Boivin, “Image Diffraction-nelle due à Une Pupille Aberrante Non-uniforme,” Can. J. Phys. 63, 275 (1985).
[CrossRef]

1984 (3)

S. C. Biswas, J-E. Villeneuve, “Combined Effect of All Aberrations and a Pupil Filter on the Diffraction Image,” J. Opt. Soc. Am. A 1, 1316A (1984).

Y. Li, “Establishment of the Maximum Encircled Energy in the Geometrical Focal Plane,” Opt. Acta 31, 1107 (1984).
[CrossRef]

H. H. Hopkins, “Image Shift, Phase Distortion and Optical Transfer Function,” Opt. Acta 31, 345 (1984).
[CrossRef]

1983 (3)

S. A. Self, “Focusing of Spherical Gaussian Beams,” Appl. Opt. 22, 658 (1983).
[CrossRef] [PubMed]

M. J. Yzuel, J. Calvo, “Point-Spread Function Calculation for Optical Systems with Residual Aberrations and a Non-Uniform Transmission Pupil,” Opt. Acta 30, 233 (1983).
[CrossRef]

G. I. Greisukh, S. A. Stepanor, “Aberrational Analysis of Optical Systems with Diffraction Elements,” Opt. Spectros. 54, 93 (1983).

1982 (2)

1981 (2)

V. N. Mahajan, “Zernike Annular Polynomials for Imaging Systems with Annular Pupils,” J. Opt. Soc. Am. 71, 75 (1981).
[CrossRef]

A. Dubik, “Analysis of The Effect of Diffraction and Apodization upon the Spatial-Energy Radiation—Characteristics in Nd-Glass Amplifiers,” J. Tech. Phys. 22, 3 (1981).

1980 (1)

R. Boivin, A. Boivin, “Optimized Amplitude Filtering for Superresolution over a Restricted Field,” Opt. Acta 27, 587, 1641 (1980); Opt. Acta 30, 681 (1983).
[CrossRef]

1979 (1)

S. C. Biswas, A. Boivin, “Performance of Optimum Apodizers in Presence of Primary Coma,” Can. J. Phys. 57, 1388 (1979).
[CrossRef]

1976 (1)

S. C. Biswas, A. Boivin, “Influence of Spherical Aberration on the Performance of Optimum Apodizers,” Opt. Acta 23, 569 (1976).
[CrossRef]

1975 (1)

S. C. Biswas, A. Boivin, “Influence of Primary Astigmatism on the Performance of Optimum Apodizers,” J. Opt. (India) 4, 1 (1975).

1974 (1)

1973 (1)

1970 (1)

1950 (1)

1949 (2)

Biswas, S. C.

J-E. Villeneuve, A. Boivin, S. C. Biswas, “L’image Tridimensionnelle du Point en Présence d’Aberration Sphérique Primaire et de Filtrage d’Amplitude: Unitaire ou Modal,” Can J. Phys. 63, 287 (1985).
[CrossRef]

J-E. Villeneuve, S. C. Biswas, A. Boivin, “Image Diffraction-nelle due à Une Pupille Aberrante Non-uniforme,” Can. J. Phys. 63, 275 (1985).
[CrossRef]

S. C. Biswas, J-E. Villeneuve, “Combined Effect of All Aberrations and a Pupil Filter on the Diffraction Image,” J. Opt. Soc. Am. A 1, 1316A (1984).

S. C. Biswas, A. Boivin, “Performance of Optimum Apodizers in Presence of Primary Coma,” Can. J. Phys. 57, 1388 (1979).
[CrossRef]

S. C. Biswas, A. Boivin, “Influence of Spherical Aberration on the Performance of Optimum Apodizers,” Opt. Acta 23, 569 (1976).
[CrossRef]

S. C. Biswas, A. Boivin, “Influence of Primary Astigmatism on the Performance of Optimum Apodizers,” J. Opt. (India) 4, 1 (1975).

Boivin, A.

J-E. Villeneuve, S. C. Biswas, A. Boivin, “Image Diffraction-nelle due à Une Pupille Aberrante Non-uniforme,” Can. J. Phys. 63, 275 (1985).
[CrossRef]

J-E. Villeneuve, A. Boivin, S. C. Biswas, “L’image Tridimensionnelle du Point en Présence d’Aberration Sphérique Primaire et de Filtrage d’Amplitude: Unitaire ou Modal,” Can J. Phys. 63, 287 (1985).
[CrossRef]

R. Boivin, A. Boivin, “Optimized Amplitude Filtering for Superresolution over a Restricted Field,” Opt. Acta 27, 587, 1641 (1980); Opt. Acta 30, 681 (1983).
[CrossRef]

S. C. Biswas, A. Boivin, “Performance of Optimum Apodizers in Presence of Primary Coma,” Can. J. Phys. 57, 1388 (1979).
[CrossRef]

S. C. Biswas, A. Boivin, “Influence of Spherical Aberration on the Performance of Optimum Apodizers,” Opt. Acta 23, 569 (1976).
[CrossRef]

S. C. Biswas, A. Boivin, “Influence of Primary Astigmatism on the Performance of Optimum Apodizers,” J. Opt. (India) 4, 1 (1975).

Boivin, R.

R. Boivin, A. Boivin, “Optimized Amplitude Filtering for Superresolution over a Restricted Field,” Opt. Acta 27, 587, 1641 (1980); Opt. Acta 30, 681 (1983).
[CrossRef]

Boyer, G.

Calvo, J.

M. J. Yzuel, J. Calvo, “Point-Spread Function Calculation for Optical Systems with Residual Aberrations and a Non-Uniform Transmission Pupil,” Opt. Acta 30, 233 (1983).
[CrossRef]

Clements, A. M.

Dubik, A.

A. Dubik, “Analysis of The Effect of Diffraction and Apodization upon the Spatial-Energy Radiation—Characteristics in Nd-Glass Amplifiers,” J. Tech. Phys. 22, 3 (1981).

Frieden, B. R.

Greisukh, G. I.

G. I. Greisukh, S. A. Stepanor, “Aberrational Analysis of Optical Systems with Diffraction Elements,” Opt. Spectros. 54, 93 (1983).

Heier, H.

Hopkins, H. H.

H. H. Hopkins, “Image Shift, Phase Distortion and Optical Transfer Function,” Opt. Acta 31, 345 (1984).
[CrossRef]

Li, Y.

Y. Li, “Establishment of the Maximum Encircled Energy in the Geometrical Focal Plane,” Opt. Acta 31, 1107 (1984).
[CrossRef]

Ljunggren, S.

Mahajan, V. N.

Osterberg, H.

Séchaud, M.

Self, S. A.

Stamnes, J. J.

Stepanor, S. A.

G. I. Greisukh, S. A. Stepanor, “Aberrational Analysis of Optical Systems with Diffraction Elements,” Opt. Spectros. 54, 93 (1983).

Szapiel, S.

Villeneuve, J-E.

J-E. Villeneuve, A. Boivin, S. C. Biswas, “L’image Tridimensionnelle du Point en Présence d’Aberration Sphérique Primaire et de Filtrage d’Amplitude: Unitaire ou Modal,” Can J. Phys. 63, 287 (1985).
[CrossRef]

J-E. Villeneuve, S. C. Biswas, A. Boivin, “Image Diffraction-nelle due à Une Pupille Aberrante Non-uniforme,” Can. J. Phys. 63, 275 (1985).
[CrossRef]

S. C. Biswas, J-E. Villeneuve, “Combined Effect of All Aberrations and a Pupil Filter on the Diffraction Image,” J. Opt. Soc. Am. A 1, 1316A (1984).

J-E. Villeneuve, “L’Image Tridimensionnelle du Point sous l’Influence Conjointe de l’Aberration Sphérique et du Filtrage d’Amplitude,” Doctoral Thesis, U. Laval, Québec (1981).

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U.P., London, 1966).

Wilkins, J. E.

Wissler, F. C.

Yzuel, M. J.

M. J. Yzuel, J. Calvo, “Point-Spread Function Calculation for Optical Systems with Residual Aberrations and a Non-Uniform Transmission Pupil,” Opt. Acta 30, 233 (1983).
[CrossRef]

Appl. Opt. (4)

Can J. Phys. (1)

J-E. Villeneuve, A. Boivin, S. C. Biswas, “L’image Tridimensionnelle du Point en Présence d’Aberration Sphérique Primaire et de Filtrage d’Amplitude: Unitaire ou Modal,” Can J. Phys. 63, 287 (1985).
[CrossRef]

Can. J. Phys. (2)

J-E. Villeneuve, S. C. Biswas, A. Boivin, “Image Diffraction-nelle due à Une Pupille Aberrante Non-uniforme,” Can. J. Phys. 63, 275 (1985).
[CrossRef]

S. C. Biswas, A. Boivin, “Performance of Optimum Apodizers in Presence of Primary Coma,” Can. J. Phys. 57, 1388 (1979).
[CrossRef]

J. Opt. (India) (1)

S. C. Biswas, A. Boivin, “Influence of Primary Astigmatism on the Performance of Optimum Apodizers,” J. Opt. (India) 4, 1 (1975).

J. Opt. Soc. Am. (6)

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

S. C. Biswas, J-E. Villeneuve, “Combined Effect of All Aberrations and a Pupil Filter on the Diffraction Image,” J. Opt. Soc. Am. A 1, 1316A (1984).

V. N. Mahajan, “Line of Sight of an Aberrated Optical System,” J. Opt. Soc. Am. A 2, 833 (1985).
[CrossRef]

J. Tech. Phys. (1)

A. Dubik, “Analysis of The Effect of Diffraction and Apodization upon the Spatial-Energy Radiation—Characteristics in Nd-Glass Amplifiers,” J. Tech. Phys. 22, 3 (1981).

Opt. Acta (5)

S. C. Biswas, A. Boivin, “Influence of Spherical Aberration on the Performance of Optimum Apodizers,” Opt. Acta 23, 569 (1976).
[CrossRef]

R. Boivin, A. Boivin, “Optimized Amplitude Filtering for Superresolution over a Restricted Field,” Opt. Acta 27, 587, 1641 (1980); Opt. Acta 30, 681 (1983).
[CrossRef]

H. H. Hopkins, “Image Shift, Phase Distortion and Optical Transfer Function,” Opt. Acta 31, 345 (1984).
[CrossRef]

M. J. Yzuel, J. Calvo, “Point-Spread Function Calculation for Optical Systems with Residual Aberrations and a Non-Uniform Transmission Pupil,” Opt. Acta 30, 233 (1983).
[CrossRef]

Y. Li, “Establishment of the Maximum Encircled Energy in the Geometrical Focal Plane,” Opt. Acta 31, 1107 (1984).
[CrossRef]

Opt. Spectros. (1)

G. I. Greisukh, S. A. Stepanor, “Aberrational Analysis of Optical Systems with Diffraction Elements,” Opt. Spectros. 54, 93 (1983).

Other (2)

J-E. Villeneuve, “L’Image Tridimensionnelle du Point sous l’Influence Conjointe de l’Aberration Sphérique et du Filtrage d’Amplitude,” Doctoral Thesis, U. Laval, Québec (1981).

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U.P., London, 1966).

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Figures (20)

Fig. 1
Fig. 1

Optical setup and coordinate system.

Fig. 2
Fig. 2

Intensity distribution along the 0–180° axis for different values of β031, Airy pupil.

Fig. 3
Fig. 3

Intensity distribution along the 0–180° axis for different values of β031, Clements-Wilkins pupil.

Fig. 4
Fig. 4

Intensity distribution along the 0–180° axis for different values of β031, Boivin-Boivin pupil.

Fig. 5
Fig. 5

Intensity distribution along the 0–180° axis for different values of β031, Gaussian pupil.

Fig. 6
Fig. 6

Intensity distribution along the 0–180° axis for different values of β031, Sonine pupil.

Fig. 7
Fig. 7

Intensity distribution along the 0–180° axis for different values of β031, phase-step pupil.

Fig. 8
Fig. 8

Intensity distribution along the 0–180° axis for different values of β031, annular pupil (ɛ = 0.3).

Fig. 9
Fig. 9

Intensity distribution along the 0–180° axis for different values of β031, annular pupil (ɛ = 0.5).

Fig. 10
Fig. 10

Fractional encircled-energy distribution for different values of β031, Airy pupil.

Fig. 11
Fig. 11

Fractional encircled-energy distribution for different values of β031, Clements-Wilkins pupil.

Fig. 12
Fig. 12

Fractional encircled-energy distribution for different values of β031, Boivin-Boivin pupil.

Fig. 13
Fig. 13

Fractional encircled-energy distribution for different values of β031, Gaussian pupil.

Fig. 14
Fig. 14

Fractional encircled-energy distribution for different values of β031, Sonine pupil.

Fig. 15
Fig. 15

Fractional encircled-energy distribution for different values of β031, phase-step pupil.

Fig. 16
Fig. 16

Fractional encircled-energy distribution for different values of β031, annular pupil (ɛ = 0.3).

Fig. 17
Fig. 17

Fractional encircled-energy distribution for different values of β031, annular pupil (ɛ = 0.5).

Fig. 18
Fig. 18

Variation of the Strehl ratio with respect to β031.

Fig. 19
Fig. 19

Elongation of the central lobe along the 0–180° axis for different values of β031.

Fig. 20
Fig. 20

Variation of angular resolution according to the Huber-Hopkins criterion.

Equations (46)

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a 0 ( υ 0 , θ 0 ) = R 2 cos θ ¯ λ d 0 d 0 1 0 2 π τ ( ρ , θ ) exp [ i υ 0 ρ cos ( θ θ 0 ) ] × ρ d ρ d θ ,
ρ = r R , υ 0 = 2 π R λ d r 0 .
τ ( ρ , θ ) = f ( ρ ) exp [ i k W ( ρ , θ ) ] .
k W ( ρ , θ ) = β l n m R n m ( ρ ) cos m θ ,
I ( υ 0 , θ 0 ) = a 0 ( υ 0 , θ 0 ) a 0 * ( υ 0 , θ 0 ) ,
E ( V 0 ) = 0 V 0 0 2 π I ( υ 0 , θ 0 ) υ 0 d υ 0 d θ 0 .
F ( V 0 ) = E ( V 0 ) / E ( ) ,
k W ( ρ , θ ) = β 031 R 3 1 ( ρ , θ ) cos θ .
a 0 ( υ 0 , θ 0 ) = K 0 0 1 0 2 π f ( ρ ) exp [ i υ 0 ρ cos ( θ θ 0 ) + i β 031 R 3 1 ( ρ ) cos θ ] ρ d ρ d θ ,
K 0 = R 2 cos θ ¯ λ d 0 d .
a 0 ( υ 0 , θ 0 ) = 2 π K 0 0 1 f ( ρ ) J 0 [ υ 0 ( ρ ) ρ ] ρ d ρ ,
υ 0 ( ρ ) ρ = { ( υ 0 ρ ) 2 2 υ 0 ρ β 031 R 3 1 ( ρ ) cos θ 0 + [ β 031 R 3 1 ( ρ ) ] 2 } 1 / 2 ,
E ( V 0 ) = 0 V 0 0 2 π a 0 ( υ 0 , θ 0 ) α 0 * ( υ 0 , θ 0 ) υ 0 d υ 0 d θ 0 .
J 0 { Z 2 + z 2 2 Z z cos ϕ } = m = 0 ε m J m ( Z ) J m ( z ) cos m ϕ ,
E ( V 0 ) = 4 ( π K 0 ) 2 0 1 0 2 π [ m = 0 ε m 0 1 f ( ρ 1 ) J m ( υ ) J m ( w ) cos m θ 0 ρ 1 d ρ 1 × n = 0 ε n 0 1 f ( ρ 2 ) J n ( υ ) J n ( w ) cos n θ 0 ρ 2 d ρ 2 ] υ 0 d υ 0 d θ 0 ,
υ = υ 0 ρ , w = β 031 R 3 1 ( ρ ) .
0 2 π cos m θ 0 cos n θ 0 d θ 0 = 0 , = 2 π , = π , m n , m = n = 0 , m = n 0 . }
E ( V 0 ) = 4 m = 0 ε ˜ m 0 V 0 [ 0 1 f ( ρ 1 ) J m ( υ ) J m ( w ) ρ 1 d ρ 1 ] × [ 0 1 f ( ρ 2 ) J m ( υ ) J m ( w ) ρ 2 d ρ 2 ] υ 0 d υ 0 ,
ε ˜ 0 = 2 π , ε ˜ m = 4 π , m 0 .
E ( V 0 ) = 4 m = 0 ε ˜ m 0 V 0 [ 0 1 f ( ρ 1 ) J m ( w ) k = 0 × ( 1 ) k ( υ 0 ρ 1 ) 2 k + m 2 2 k + m k ! ( m + k ) ! ρ 1 d ρ 1 ] × [ 0 1 f ( ρ 2 ) J m ( w ) n = 0 × ( 1 ) n ( υ 0 ρ 2 ) 2 n + m 2 2 n + m n ! ( m + n ) ! ] ρ 2 d ρ 2 υ 0 d υ 0 .
E ( V 0 ) = 8 m = 0 ε ˜ m 0 V 0 ( V 0 2 4 ) m [ k = 0 ( V 0 2 4 ) k a k ( m ) ] × [ n = 0 ( V 0 2 4 ) n a n ( m ) ] d ( υ 0 2 4 ) ,
a k ( m ) = 0 1 f ( ρ ) J m ( w ) ρ 2 k + m ( 1 ) k k ! ( m + k ) ! ρ d ρ .
E ( V 0 ) = 8 ( V 0 2 4 ) m = 0 ε ˜ m n = 0 C n ( m ) m + n + 1 ( V 0 2 4 ) m + n ,
C n ( m ) = k = 0 n a k ( m ) a n k ( m ) .
E ( V 0 ) = 8 ( V 0 2 4 ) m = 0 n = 0 m ε ˜ m n C n ( m n ) m + 1 ( V 0 2 4 ) m .
f ( ρ ) = m = 0 N 1 α m ( 1 ρ 2 ) m ,
E ( ) = 4 π m = 0 N 1 n = 0 N 1 α m α n m + n + 1 ,
f ( ρ ) = n = 0 N ( 4 n + 2 ) 1 / 2 x n P n ( 1 2 ρ 2 ) ,
λ = 0.71329 , μ = 0.224 ,
x 0 = 0.602296 , x 1 = 0.106937 , x 2 = 0.0610535 , x 3 = 0.00693752 .
f ( ρ ) = n = 0 N α n ( 1 ρ 2 ) n ,
α 0 = 0.946606 , α 1 = 0.323033 , α 2 = 0.379674 , α 3 = 0.519157 .
f ( ρ ) = 1 C [ 1 i = 1 M b i J 0 ( z i ρ ) ] ,
b 1 = 2.1857049 , b 3 = 1.9391646 , z 1 = 3.193080 , z 3 = 8.420 , C = 1.3495944 . b 2 = 2.3702677 , b 4 = 1.6214321 , z 2 = 5.550 , z 4 = 11.1031 ,
E ( ) = 2 C 2 { 1 2 i = 1 M b i J 1 ( z i ) z i } .
f ( ρ ) = exp ( k ρ 2 ) ,
E ( ) = 4 π m = 0 n = 0 m b n m + n + 1 exp ( 2 k ) ,
b n = k n n ! .
f ( ρ ) = 1 V 0 2 ( 1 ρ 2 ) J 1 ( V 0 ) J 2 ( V 0 ) .
f ( ρ ) = 1 1.046 ( 1 ρ 2 ) .
f ( ρ ) = 1 + t [ exp ( ± i π ) 1 ] ,
t = 1 , ρ ε , = 0 , ρ > ε ,
E ( ) = 4 π .
f ( ρ ) = 1 t ,
E ( ) = 4 π ( 1 ε 2 ) .
β 031 = 2 π 3 λ W 31 .

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