Abstract

In the studies on optical diffraction in the presence of aberrations, it is normal practice to consider only one aberration at a time. The tolerance limits and all other important properties are determined on the basis of the presence of a single aberration despite the fact that most aberrations are normally present in an actual system. The present paper endeavors to look for the combined effect of a group of residual Seidel aberrations. A classical diffraction setup is used, in which the diffracting aperture (circular) is supposed to have primary spherical aberration, primary coma, and primary astigmatism and is illuminated with the fundamental mode (TEM00) of a laser beam. This illuminating laser beam is represented by the modal function ψ00 (c,ρ), which is the lowest-order solution of the Fredholm integral equation of the second kind. A relatively simple treatment is presented for this complicated problem. The diffraction of a uniform beam is also considered side by side. The isophote diagram of the diffracted field of a laser beam as well as that of a uniform beam under the joint influence of three important aberrations is presented for the first time known to us. Many interesting observations have been made from the various numerical results obtained. It is noted, for example, that the combined effect of all the residual aberrations of a system could be more severe than expected and that new restrictions should be imposed on all individual aberrations to compensate for their combined effect.

© 1986 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 Diffractionnelle due à une Pupille Aberrante Non-Uniforme,” Can. J. Phys. 63, 275 (1985).
    [CrossRef]
  3. S. C. Biswas, J.-E. Villeneuve, “Diffraction by an Aberrated Optical System with Nonuniform Amplitude Transmission: Results for Primary Coma,” Appl. Opt. 24, 4473 (1985).
    [CrossRef] [PubMed]
  4. V. N. Mahajan, “Line of Sight of an Aberrated Optical System,” J. Opt. Soc. Am. A 2, 833 (1985).
    [CrossRef]
  5. A. Arimoto, “Aberration Expansion and Evaluation of the Quasi-Gaussian Beam by a Set of Orthogonal Functions,” J. Opt. Soc. Am. 64, 850 (1974).
    [CrossRef]
  6. D. D. Lowenthal, “Marechal Intensity Criteria Modified for Gaussian Beams,” Appl. Opt. 13, 2126 (1974); J. Opt. Soc. Am. 65, 853 (1975).
    [CrossRef] [PubMed]
  7. A. Maréchal, “Etude de l’Eclairement au Centre de la Tâche de Diffraction pour les Différentes Aberrations Géométriques,” C. R. Acad. Sci. Paris 218, 395 (1944).
  8. A. Yoshida, T. Asakura, “Diffraction Pattern of Off-Axis Gaussian Beams in the Optical System with Astigmatism and Coma,” Opt. Commun. 25, 133 (1978).
    [CrossRef]
  9. A. Yoshida, “Spherical Aberration in Beam Optical Systems,” Appl. Opt. 21, 1812 (1982).
    [CrossRef] [PubMed]
  10. S. C. Biswas, A. Boivin, “Performance of Optimum Apodizers in Presence of Primary Coma,” Can. J. Phys. 57, 1388 (1979).
    [CrossRef]
  11. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1983).
  12. S. C. Biswas, A. Boivin, “Influence of Spherical Aberration on the Peformance of Optimum Apodizers,” Opt. Acta 23, 569 (1976).
    [CrossRef]
  13. J.-E. Villeneuve, “L’image Tridimensionnelle du Point sous I’Influence Conjointe de l’Aberration Sphérique et du Filtrage d’Amplitude,” Doctoral Thesis, U. Laval, Quebec (1981).
  14. D. Slepian, “Analysitic Solution of Two Apodization Problems,” J. Opt. Soc. Am. 55, 1110 (1965).
    [CrossRef]
  15. J. C. Heurtley, “Hyperspheroidal Functions: Optical Resonators with Circular Mirrors,” in Proceedings, Symposium on Quasi-Optics, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1964), p. 367.
  16. B. R. Frieden, “Evaluation, Design and Extrapolation Methods for Optical Signals, Based on Use of the Prolate Functions,” Prog. Opt. 9, 311 (1971).
    [CrossRef]
  17. R. Boivin, A. Boivin, “Laurent-Series Expansions for the Finite Hankel Self-Transforms,” Can. J. Phys. 63, 254 (1985).
    [CrossRef]
  18. S. C. Biswas, A. Boivin, “Electromagnetic Diffraction in the Focal Region of a Wide-Angle Spherical Mirror under Oblique Illumination II. Formal Expansion of the Diffraction Integrals,” Opt. Acta 26, 373 (1979).
    [CrossRef]
  19. F. B. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill, New York, 1956).
  20. P. J. Davis, P. Rabinowitz, Methods of Numerical Integration (Academic, New York, 1975).
  21. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).
  22. S. C. Biswas, A. Boivin, “Influence of Primary Astigmatism on the Performance of Optimum Apodizers,” J. Opt. India 4, 1 (1975).
  23. 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).
  24. K. Tanaka, N. Saga, H. Mizokami, “Field Spread of a Diffracted Gaussian Beam Through a Circular Aperture,” Appl. Opt. 24, 1102 (1985).
    [CrossRef] [PubMed]
  25. V. N. Mahajan, “Strehl Ratio for Primary Aberration: Some Analytical Results for Circular and Annular Pupils,” J. Opt. Soc. Am 72, 1258 (1982).
    [CrossRef]
  26. D. D. Lowenthal, “Far Field Pattern for Gaussian Beams in the Presence of Small Spherical Aberrations,” J. Opt. Soc. Am. 65, 853 (1975).
    [CrossRef]
  27. S. Szapiel, “Aberration Balancing Techniques for Radially Symmetric Amplitude Distribution: a Generalization of the Marechal Approach,” J. Opt. Soc. Am. 72, 947 (1982).
    [CrossRef]
  28. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. P., London, 1966).

1985 (6)

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 Diffractionnelle due à une Pupille Aberrante Non-Uniforme,” Can. J. Phys. 63, 275 (1985).
[CrossRef]

R. Boivin, A. Boivin, “Laurent-Series Expansions for the Finite Hankel Self-Transforms,” Can. J. Phys. 63, 254 (1985).
[CrossRef]

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

K. Tanaka, N. Saga, H. Mizokami, “Field Spread of a Diffracted Gaussian Beam Through a Circular Aperture,” Appl. Opt. 24, 1102 (1985).
[CrossRef] [PubMed]

S. C. Biswas, J.-E. Villeneuve, “Diffraction by an Aberrated Optical System with Nonuniform Amplitude Transmission: Results for Primary Coma,” Appl. Opt. 24, 4473 (1985).
[CrossRef] [PubMed]

1984 (1)

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).

1982 (3)

1979 (2)

S. C. Biswas, A. Boivin, “Electromagnetic Diffraction in the Focal Region of a Wide-Angle Spherical Mirror under Oblique Illumination II. Formal Expansion of the Diffraction Integrals,” Opt. Acta 26, 373 (1979).
[CrossRef]

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

1978 (1)

A. Yoshida, T. Asakura, “Diffraction Pattern of Off-Axis Gaussian Beams in the Optical System with Astigmatism and Coma,” Opt. Commun. 25, 133 (1978).
[CrossRef]

1976 (1)

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

1975 (2)

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

D. D. Lowenthal, “Far Field Pattern for Gaussian Beams in the Presence of Small Spherical Aberrations,” J. Opt. Soc. Am. 65, 853 (1975).
[CrossRef]

1974 (2)

1971 (1)

B. R. Frieden, “Evaluation, Design and Extrapolation Methods for Optical Signals, Based on Use of the Prolate Functions,” Prog. Opt. 9, 311 (1971).
[CrossRef]

1965 (1)

1944 (1)

A. Maréchal, “Etude de l’Eclairement au Centre de la Tâche de Diffraction pour les Différentes Aberrations Géométriques,” C. R. Acad. Sci. Paris 218, 395 (1944).

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

Arimoto, A.

Asakura, T.

A. Yoshida, T. Asakura, “Diffraction Pattern of Off-Axis Gaussian Beams in the Optical System with Astigmatism and Coma,” Opt. Commun. 25, 133 (1978).
[CrossRef]

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 Diffractionnelle due à une Pupille Aberrante Non-Uniforme,” Can. J. Phys. 63, 275 (1985).
[CrossRef]

S. C. Biswas, J.-E. Villeneuve, “Diffraction by an Aberrated Optical System with Nonuniform Amplitude Transmission: Results for Primary Coma,” Appl. Opt. 24, 4473 (1985).
[CrossRef] [PubMed]

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, “Electromagnetic Diffraction in the Focal Region of a Wide-Angle Spherical Mirror under Oblique Illumination II. Formal Expansion of the Diffraction Integrals,” Opt. Acta 26, 373 (1979).
[CrossRef]

S. C. Biswas, A. Boivin, “Influence of Spherical Aberration on the Peformance 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.

R. Boivin, A. Boivin, “Laurent-Series Expansions for the Finite Hankel Self-Transforms,” Can. J. Phys. 63, 254 (1985).
[CrossRef]

J.-E. Villeneuve, S. C. Biswas, A. Boivin, “Image Diffractionnelle 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]

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, “Electromagnetic Diffraction in the Focal Region of a Wide-Angle Spherical Mirror under Oblique Illumination II. Formal Expansion of the Diffraction Integrals,” Opt. Acta 26, 373 (1979).
[CrossRef]

S. C. Biswas, A. Boivin, “Influence of Spherical Aberration on the Peformance 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, “Laurent-Series Expansions for the Finite Hankel Self-Transforms,” Can. J. Phys. 63, 254 (1985).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1983).

Davis, P. J.

P. J. Davis, P. Rabinowitz, Methods of Numerical Integration (Academic, New York, 1975).

Frieden, B. R.

B. R. Frieden, “Evaluation, Design and Extrapolation Methods for Optical Signals, Based on Use of the Prolate Functions,” Prog. Opt. 9, 311 (1971).
[CrossRef]

Heurtley, J. C.

J. C. Heurtley, “Hyperspheroidal Functions: Optical Resonators with Circular Mirrors,” in Proceedings, Symposium on Quasi-Optics, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1964), p. 367.

Hildebrand, F. B.

F. B. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill, New York, 1956).

Lowenthal, D. D.

Mahajan, V. N.

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

V. N. Mahajan, “Strehl Ratio for Primary Aberration: Some Analytical Results for Circular and Annular Pupils,” J. Opt. Soc. Am 72, 1258 (1982).
[CrossRef]

Maréchal, A.

A. Maréchal, “Etude de l’Eclairement au Centre de la Tâche de Diffraction pour les Différentes Aberrations Géométriques,” C. R. Acad. Sci. Paris 218, 395 (1944).

Mizokami, H.

Rabinowitz, P.

P. J. Davis, P. Rabinowitz, Methods of Numerical Integration (Academic, New York, 1975).

Saga, N.

Slepian, D.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

Szapiel, S.

Tanaka, K.

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 Diffractionnelle due à une Pupille Aberrante Non-Uniforme,” Can. J. Phys. 63, 275 (1985).
[CrossRef]

S. C. Biswas, J.-E. Villeneuve, “Diffraction by an Aberrated Optical System with Nonuniform Amplitude Transmission: Results for Primary Coma,” Appl. Opt. 24, 4473 (1985).
[CrossRef] [PubMed]

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 I’Influence Conjointe de l’Aberration Sphérique et du Filtrage d’Amplitude,” Doctoral Thesis, U. Laval, Quebec (1981).

Watson, G. N.

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

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1983).

Yoshida, A.

A. Yoshida, “Spherical Aberration in Beam Optical Systems,” Appl. Opt. 21, 1812 (1982).
[CrossRef] [PubMed]

A. Yoshida, T. Asakura, “Diffraction Pattern of Off-Axis Gaussian Beams in the Optical System with Astigmatism and Coma,” Opt. Commun. 25, 133 (1978).
[CrossRef]

Appl. Opt. (4)

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

A. Maréchal, “Etude de l’Eclairement au Centre de la Tâche de Diffraction pour les Différentes Aberrations Géométriques,” C. R. Acad. Sci. Paris 218, 395 (1944).

Can. J. Phys. (4)

S. C. Biswas, A. Boivin, “Performance of Optimum Apodizers in Presence of Primary Coma,” Can. J. Phys. 57, 1388 (1979).
[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 Diffractionnelle due à une Pupille Aberrante Non-Uniforme,” Can. J. Phys. 63, 275 (1985).
[CrossRef]

R. Boivin, A. Boivin, “Laurent-Series Expansions for the Finite Hankel Self-Transforms,” Can. J. Phys. 63, 254 (1985).
[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 (1)

V. N. Mahajan, “Strehl Ratio for Primary Aberration: Some Analytical Results for Circular and Annular Pupils,” J. Opt. Soc. Am 72, 1258 (1982).
[CrossRef]

J. Opt. Soc. Am. (4)

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

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

Opt. Acta (2)

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

S. C. Biswas, A. Boivin, “Electromagnetic Diffraction in the Focal Region of a Wide-Angle Spherical Mirror under Oblique Illumination II. Formal Expansion of the Diffraction Integrals,” Opt. Acta 26, 373 (1979).
[CrossRef]

Opt. Commun. (1)

A. Yoshida, T. Asakura, “Diffraction Pattern of Off-Axis Gaussian Beams in the Optical System with Astigmatism and Coma,” Opt. Commun. 25, 133 (1978).
[CrossRef]

Prog. Opt. (1)

B. R. Frieden, “Evaluation, Design and Extrapolation Methods for Optical Signals, Based on Use of the Prolate Functions,” Prog. Opt. 9, 311 (1971).
[CrossRef]

Other (7)

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

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1983).

F. B. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill, New York, 1956).

P. J. Davis, P. Rabinowitz, Methods of Numerical Integration (Academic, New York, 1975).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

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

J. C. Heurtley, “Hyperspheroidal Functions: Optical Resonators with Circular Mirrors,” in Proceedings, Symposium on Quasi-Optics, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1964), p. 367.

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Figures (15)

Fig. 1
Fig. 1

Geometry of the diffraction setup.

Fig. 2
Fig. 2

Modal function ψ0,0(5,ρ).

Fig. 3
Fig. 3

Axial intensity and amplitude distribution for the uniform aperture with spherical aberration and coma.

Fig. 4
Fig. 4

Axial intensity and amplitude distribution for the uniform aperture with spherical aberration and astigmatism.

Fig. 5
Fig. 5

Axial intensity and amplitude distribution for the uniform aperture with spherical aberration, coma, and astigmatism.

Fig. 6
Fig. 6

Axial intensity and amplitude distribution for the modal aperture with coma.

Fig. 7
Fig. 7

Axial intensity and amplitude distribution for the modal aperture with astigmatism.

Fig. 8
Fig. 8

Axial intensity and amplitude distribution for the modal aperture with spherical aberration and coma.

Fig. 9
Fig. 9

Axial intensity and amplitude distribution for the modal aperture with spherical aberration and astigmatism.

Fig. 10
Fig. 10

Axial intensity and amplitude distribution for the modal aperture with spherical aberration, coma, and astigmatism.

Fig. 11
Fig. 11

Isophote diagram on the best focal plane for the modal aperture with coma.

Fig. 12
Fig. 12

Isophote diagram on the best focal plane for the modal aperture with astigmatism.

Fig. 13
Fig. 13

Isophote diagram on the best focal plane for the uniform aperture with spherical aberration, coma, and astigmatism.

Fig. 14
Fig. 14

Isophote diagram on the best focal plane for the modal aperture (i.e., the fundamental laser mode) with spherical aberration, coma, and astigmatism.

Fig. 15
Fig. 15

Fractional encircled energy (FEE) distribution for different pupil conditions.

Tables (2)

Tables Icon

Table I Focal Shifts and Maximum Axial/Off-Axial Intensities for Different Pupil Conditions

Tables Icon

Table II Spot Size of the Diffraction Image Based on the 66.5% Encircled Energy Criterion

Equations (65)

Equations on this page are rendered with MathJax. Learn more.

a ( v 0 , θ 0 ) = R 2 cos θ ¯ λ d 0 d 0 1 0 2 π τ ( ρ , θ ) exp ( i u ρ 2 ) / 2 × exp [ - i v 0 ρ cos ( θ - θ 0 ) ] ρ d ρ d θ ,
ρ = r / R , v 0 = 2 π R λ d r 0 , u = 2 π λ ( R d ) 2 z , }
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 ,
k W ( ρ , θ ) = W P S + W P C + W P A .
W P S = 2 π λ δ ρ 4 ,
W P C = β 031 R 3 1 ( ρ ) cos θ , W P A = β 022 R 2 2 ( ρ ) cos 2 θ .
β N , n ψ N , n ( c , ρ 2 ) = 0 1 ψ N , n ( c , ρ 1 ) J N ( c ρ 1 ρ 2 ) ρ 1 d ρ 1 ,             0 ρ 2 1 and N , n = 0 , 1 , 2 , ,
ψ o , n ( c , ρ ) = j = 0 ψ j ( 0 ) ρ 2 j j ! , 0 ρ 1 ( MacLaurin ) ,
ψ o , n ( c , ρ ) = j = 0 ( - 1 ) j ψ j ( 1 ) ( 1 - ρ 2 ) j j ! , 0 ρ 1 ( Taylor ) .
a ( v 0 , θ 0 ) = K 0 0 1 0 2 π F ( ρ ) exp [ - i v 0 ρ cos ( θ - θ 0 ) ] + i β cos θ + i γ cos 2 θ ] ρ d ρ d θ ,
K 0 = R 2 cos θ ¯ λ d 0 d , F ( ρ ) = f ( ρ ) exp [ i ϕ ( ρ ) ] , ϕ ( ρ ) = exp ( ½ u ρ 2 + i α ρ 4 ) , α = 2 π δ λ β = β 031 R 3 1 ( ρ ) , γ = β 022 R 2 2 ( ρ ) .
a ( v 0 θ 0 ) = K 0 0 1 F ( ρ ) { 0 2 π exp [ i w cos ( θ - θ 0 ) + i γ cos 2 θ ] d θ } ρ d ρ ,
w = ( v 0 2 ρ 2 + β 2 - 2 v 0 ρ β cos θ 0 ) 1 / 2 , θ 0 = arctang ( - v 0 ρ sin θ 0 β - v 0 ρ cos θ 0 ) .
0 2 π exp ( i w cos ( θ - θ 0 ) + i γ cos 2 θ ) d θ = 2 π J 0 ( w ) J 0 ( γ ) + 4 π l = 1 ( - 1 ) l J 2 l ( γ ) J 4 l ( w ) cos 4 l θ 0 1 - 4 π l = 0 ( - 1 ) l J 2 l + 1 ( γ ) J 2 l + 2 ( w ) cos ( 4 l + 2 ) θ 0 .
a R ( v 0 , θ 0 ) = K 0 0 1 f ( ρ [ C cos ϕ ( ρ ) - D sin ϕ ( ρ ) ] ρ d ρ , a I ( v 0 , θ 0 ) = K 0 0 1 f ( ρ [ C sin ϕ ( ρ ) + D cos ϕ ( ρ ) ] ρ d ρ ,
C = 2 π J 0 ( w ) J 0 ( γ ) + 4 π l = 1 ( - 1 ) l J 21 ( γ ) J 4 l ( w ) cos 4 l θ 0 , D = - 4 π l = 0 ( - 1 ) l J 2 l + 1 ( γ ) J 4 l + 2 ( w ) cos ( 4 l + 2 ) θ 0 .
I ( v 0 , θ 0 ) = a R 2 + a I 2 .
E ( V 0 ) = 0 V 0 0 2 π ( I v 0 , θ 0 ) v 0 d v 0 d θ 0 ,
I ( v 0 , θ 0 ) = a ( v 0 , θ 0 ) a * ( v 0 , θ 0 ) .
I ( v 0 ) = 0 2 π { 0 2 π exp [ - i ν 1 cos ( θ 1 - θ 0 ) + i β 1 cos θ 1 + i γ 1 cos 2 θ 1 ] d θ 1 } × { 0 2 π exp [ - i ν 2 cos ( θ 2 - θ 0 ) + i β 2 cos θ 2 + i γ 2 cos 2 θ 2 ] d θ 2 } d θ 0 ,
I ( v 0 ) = 2 π m = 0 m J m ( ν 1 ) J m ( ν 2 ) × 0 2 π 0 2 π exp [ i β 1 cos θ 1 - i β 2 cos θ 2 + i γ 1 cos 2 θ 1 - i γ 2 cos 2 θ 2 ) × cos m ( θ 1 - θ 2 ) d θ 1 d θ 2 ,
I ( v 0 ) = 2 π m = 0 m J m ( ν 1 ) J m ( ν 2 ) ( A c A c * + A s A s * ) ,
A c = 0 2 π exp ( i β cos θ + i γ cos 2 θ ) cos m θ d θ , A s = 0 2 π exp ( i β cos θ + i γ cos 2 θ ) sin m θ d θ .
A c = l = 0 i l γ l l ! 0 2 π exp ( i β cos θ ) cos l 2 θ cos m θ d θ .
A c = 2 l = 0 ¯ l i l J l ( γ ) 0 2 π exp [ i β cos θ ] cos 2 l θ cos m θ d θ ,
˜ 0 = ½ ,             ˜ l = 1 for l > 0.
A s = 2 l = 0 ¯ l i l J l ( γ ) 0 2 π exp [ i β cos θ ] cos 2 l θ sin m θ d θ .
0 2 π cos n ϕ exp [ i S cos ( p ϕ - q ) ] d ϕ = 2 π ( i ) n / p J n / p ( S ) cos [ ( n / p ) q ] ,
0 2 π sin n ϕ exp [ i S cos ( p ϕ - q ) ] d ϕ = 2 π ( i ) n / p J n / p ( S ) sin [ ( n / p ) q ] .
cos 2 l θ cos m θ = ½ [ cos ( 2 l + m ) θ + cos ( 2 l - m ) θ ] ,
I 1 = ½ 0 2 π exp ( i β cos 2 θ ) cos ( 2 l + m ) θ d θ + ½ 0 2 π exp ( i β cos θ ) cos ( 2 l - m ) θ d θ .
I 1 = π ( i ) 2 l + m [ J m + 2 l ( β ) + J m - 2 l ( β ) ] ,
A c = 2 l = 0 ˜ l i l J l ( γ ) I 1 .
A s = 0.
I ( v 0 ) = 8 π 3 m = 0 m J m ( ν 1 ) { l = 0 ˜ l ( - i ) l J l ( γ 1 ) × [ J m + 2 l ( β 1 ) + J m - 2 l ( β 1 ) ] } × J m ( ν 2 ) { l = 0 ˜ l ( - i ) l J l ( γ 2 ) × [ J m + 2 l ( β 2 ) + J m - 2 l ( β 2 ) ] } ,
I ( v 0 ) = 8 π 3 m = 0 m F ¯ 1 ( m ) F ¯ 2 * ( m ) ,
F ¯ j ( m ) = J m ( v j ) l = 0 ˜ l ( - 1 ) l J l ( γ j ) × [ J m + 2 l ( β j ) + J m - 2 l ( β j ) ] .
E ( V 0 ) = 8 π m = 0 m 0 V 0 [ 0 1 F ( ρ ) J ˜ m ( β 1 , γ 1 ) J m ( v 0 ρ 1 ) ρ 1 d ρ 1 ] × [ 0 1 F ( ρ ) J ˜ m * ( β 2 , γ 2 ) J m ( v 0 ρ 2 ) ρ 2 d ρ 2 ] v 0 d v 0 ,
J ˜ m ( β j , γ j ) = l = 0 ˜ l ( - i ) l J l ( γ j ) × [ J m + 2 l ( β j ) + J m - 2 l ( β j ) ] .
E ( V 0 ) = 16 π m = 0 m 0 V 0 ( v 0 2 4 ) m [ n = 0 ( v 0 2 4 ) n a ˜ n ( m ) ] [ l = 0 ( v 0 2 4 ) l a ˜ l ( m ) ] * d v 0 2 4 ,
a ˜ n ( m ) = ( - 1 ) n n ! ( m + n ) ! 0 1 F ( ρ ) J ˜ m ( β j , γ j ) ρ m + 2 n + 1 d ρ .
E ( V 0 ) = 16 π V 0 2 4 m = 0 m n = 0 C ˜ n ( m - n ) m + 1 ( V 0 2 4 ) m ,
C ˜ n ( m ) = l = 0 n a ˜ l ( m ) a ˜ n - l * ( m ) .
C ˜ n ( m ) = l = 0 n [ a ˜ l R ( m ) a ˜ ( n - l ) R ( m ) + a ˜ l I ( m ) a ˜ ( n - l ) I ( m ) ] ,
ϕ ( V 0 ) = E ( V 0 ) / E ( ) .
E ( ) = 4 π m = 0 n = 0 ( - 1 ) m + n ψ 0 , 0 m ( 1 ) ψ 0 , 0 n ( 1 ) m ! n ! ( m + n + 1 ) .
0 1 ψ 0 , 0 ( c , ρ ) 2 d ρ = 1 ,
0 1 [ m = 0 ( - 1 ) m ψ 0 , 0 m ( 1 ) ( 1 - ρ 2 ) m m ! ] [ n = 0 ( - 1 ) n ψ 0 , 0 n ( 1 ) ( 1 - ρ 2 ) n n ! ] ρ d ρ = 1 ,
½ m = 0 n = 0 ( - 1 ) m + n ψ 0 , 0 m ( 1 ) ψ 0 , 0 n ( 1 ) m ! n ! ( m + n + 1 ) = 1.
E ( ) = 8 π .
I θ = 0 2 π exp [ i υ 1 cos ( θ 0 - θ 1 ) + i υ 2 cos ( θ 0 - θ 2 ) ] d θ 0 .
I θ = 0 2 π exp [ i υ 1 cos ( θ 0 - θ 1 - θ 2 ¯ ) + i υ 2 cos θ 0 ] d θ 0 .
I θ = 0 2 π exp [ i x cos ( θ 0 - ϕ ) ] d θ 0 ,
x = [ υ 1 2 + υ 2 2 - 2 υ 1 υ 2 cos ( θ 1 - θ 2 ) ] 1 / 2 ,
ϕ = arctan υ 1 cos ( θ 1 - θ 2 ) - υ 2 υ 1 sin ( θ 1 - θ 2 ) .
I θ = 2 π J 0 ( x ) .
I θ = 2 π m = 0 m J m ( υ 1 ) J m ( υ 2 ) cos m ( θ 1 - θ 2 ) ,
m = 1 for m = 0 , = 2 for m 0.
A c = l = 0 i l γ l l ! 0 2 π exp ( i β cos θ ) cos l 2 θ cos m θ d θ .
A c = 0 2 π exp ( i β cos θ ) cos m θ + l = 0 ( - 1 ) l γ 2 l ( 2 l ) ! 0 2 π exp ( i β cos θ ) cos m θ d 0 θ cos 2 l 2 u d u + i l = 0 ( - 1 ) l γ 2 l + 1 ( 2 l + 1 ) ! 0 2 π × exp ( i β cos θ ) cos m θ d 0 θ cos 2 l + 1 2 u d u .
A c = 0 2 π exp ( i β cos θ ) cos m θ d θ + exp ( i β ) l = 1 ( - 1 ) l γ 2 l ( 2 l ) ! 0 2 π cos 2 l 2 u d u - l = 1 ( - 1 ) l ( γ / 2 ) 2 l l ! l ! [ 2 π exp ( i β ) - 0 2 π exp ( i β cos θ ) cos m θ d θ ] + l = 0 ( - 1 ) l ( γ / 2 ) 2 l + 2 ( 2 l + 2 ) ! 0 2 π n = 0 l ( 2 l + 2 n ) sin ( 2 l - 2 n + 2 ) 2 θ ( 2 l + 2 n + 2 ) d [ exp ( i β cos θ ) cos m θ ] - i l = 0 ( - 1 ) l ( γ / 2 ) 2 l + 1 ( 2 l + 1 ) ! 0 2 π n = 0 l ( 2 l + 1 n ) sin ( 2 l + 2 n + 1 ) 2 θ ( 2 l - 2 n + 1 ) d [ exp ( i β cos θ ) cos m θ ] .
A c = J 0 ( γ ) 0 2 π exp ( i β cos θ ) cos m θ d θ - 2 l = 0 n = 0 ( - 1 ) l + n ( γ 2 ) 2 l + 2 n + 2 ( 2 l + n + 2 ) ! n ! 0 2 π exp ( i β cos θ ) × cos ( 2 l + 1 ) cos m d θ + 2 i l = 0 n = 0 ( - 1 ) l + n ( γ / 2 ) 2 l + 2 n + 1 ( 2 l + n + i ) ! l ! 0 2 π exp ( i β cos θ ) × cos ( 2 l + 1 ) 2 θ cos m θ d θ .
A c = 2 l = 0 ˜ l i l J l ( γ ) 0 2 π exp ( i β cos θ ) cos 2 l θ cos m θ d θ ,
˜ l = 1 / 2 for l = 0 , = 1 for l 0.

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