Abstract

Two sampling expansions are derived, from which the three-dimensional (3-D) field amplitude distribution in the vicinity of an axial image point can be recovered entirely from a 1-D set of amplitude samples. In the first of these expansions, the sampling points are regularly spaced along the system optical axis, whereas in the second these points lie along one radius in a plane perpendicular to this axis. The expansions are derived with a scalar diffraction theory by using the paraxial and classical approximations. Apart from leading to very efficient computational schemes, both expansions are shown to provide a good basis for analyzing the structure of this kind of diffraction pattern, and knowledge of this structure can be applied in turn to the solution of pupil synthesis problems.

© 1997 Optical Society of America

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  1. R. Barakat, “The calculation of integrals encountered in optical diffraction theory,” in The Computer in Optical Research, B. R. Frieden, ed. (Springer, Berlin, 1980), Chap. 2.
  2. P. Jacquinot, B. Roizen-Dossier, “Apodisation,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. III, Chap. II.
  3. By an apodized system we shall mean a system whose pupil function has been modified in some prescribed way, not necessarily to the effect of suppressing the secondary lobes of its PSF.
  4. Y. Li, E. Wolf, “Three-dimensional intensity distribution near the focus in systemsof different Fresnel numbers,” J. Opt. Soc. Am. A 1, 801–808 (1984).
    [CrossRef]
  5. C. J. R. Sheppard, Z. S. Hegedus, “Axial behavior of pupil-plane filters,” J. Opt. Soc. Am. A 5, 643–647 (1988).
    [CrossRef]
  6. J. Ojeda-Castañeda, E. Tepichin, A. Pons, “Apodization of annular apertures: Strehl ratio,” Appl. Opt. 28, 5140–5145 (1988).
    [CrossRef]
  7. E. H. Linfoot, Recent Advances in Optics (Clarendon, Oxford, 1955), Chap. I.
  8. A. Boivin, Théorie et Calcul des Figures de Diffraction de Révolution (Les Presses de L'Université Laval, Québec, 1968).
  9. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).
  10. By reference image-plane we shall mean the plane that is perpendicular to the optical axis at the center of the reference sphere that we use to establish the wave-front aberration of the system.
  11. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), p. 60.
  12. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), p. 21.
  13. A defocus term can, of course, also be included in the apodizing function F(ρ2). This term is very useful to produce an axial shift of the window that we use to display our graphical results (Fig. 5). This shift is commonly required with aberrated systems, where the diffraction patterns do not possess an image plane of symmetry.
  14. From Eqs. (1.1), (2.5), and (2.7), we have f/z=1-2δ/N. When N→∞ (the classical regime), 1/z∼1/f for all the relevant values of δ.
  15. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), pp. 435–449.
  16. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 475.
  17. A. Boivin, “On the theory of diffraction by concentric arrays of ring-shaped apertures,” J. Opt. Soc. Am. 42, 60–64 (1952).
    [CrossRef]
  18. J. C. Dainty, “The image of a point for an aberration free lens with a circular pupil,” Opt. Commun. 1, 176–178 (1969).
    [CrossRef]
  19. L. Beiser, “Perspective rendering of the field intensity diffracted at a circularaperture,” Appl. Opt. 5, 869–870 (1966).
    [CrossRef] [PubMed]
  20. This will be evident after Eq. (5.6).
  21. C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction aperture,” J. Opt. Soc. Am. 54, 240–244 (1964).
    [CrossRef]
  22. C. W. McCutchen, “Convolution relation within the three-dimensional diffraction image,” J. Opt. Soc. Am. A 8, 868–870 (1991).
    [CrossRef] [PubMed]
  23. A. Papoulis, Systems and Transforms with Applications in Optics (Krieger, Malabar, Fla., 1981), p. 130.
  24. A. W. Lohmann, Optical Information Processing (Lecture Notes), 2nd ed. (Erlangen, Germany, 1978), p. 21.
  25. F. Bowman, Introduction to Bessel Functions (Dover, New York, 1958), p. 17.
  26. That is, a set of functions fn(r)(0, v).
  27. J. Ojeda-Castañeda, L. R. Berriel Valdos, “High focal depth by quasibifocus,” Appl. Opt. 27, 4163–4165 (1988).
    [CrossRef]
  28. J. Ojeda-Castañeda, L. R. Berriel Valdos, “Zone plate for arbitrarily high focal depth,” Appl. Opt. 29, 994–997 (1990).
    [CrossRef] [PubMed]
  29. Since, in general, we must have |F(ρ2)| ≤1(0≤ρ≤1), the expression for F(ρ2) given by Eq. (5.7b) must be normalized. The normalizing factor in this case is, evidently, F(1/2)=A(0)+2∑m=1∞A(m).
  30. J. Ojeda-Castañeda, P. Andrés, A. Dı ́az, “Annular apodizers for low sensitivity to defocus and to spherical aberration,” Opt. Lett. 11, 487–489 (1986).
    [CrossRef] [PubMed]
  31. J. Ojeda-Castañeda, E. Tepichin, A. Diaz, “Arbitrarily high focal depth with a quasioptimum real and positivetransmittance apodizer,” Appl. Opt. 28, 2666–2670 (1989).
    [CrossRef]
  32. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 416.
  33. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 365.
  34. Our aim in this example was to show the potential usefulness of the sampling expansions for tackling problems of pupil synthesis. We did not attempt to find another optimal solution to the problem of apodization; many of these have been given in the past, satisfying various criteria of optimization.2
  35. Ref. 9, Chap. 12.
  36. F. Bowman, Introduction to Bessel Functions (Dover, New York, 1958), p. 97.
  37. Notice that these linear transformations are defined in spaces of infinite dimension.
  38. The Fourier-transform relationship between these fields is a consequence of using Kirchhoff's (scalar) diffraction theory in the paraxial approximation.
  39. G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), p. 594.
  40. In fact, Lommel presented various integrals involving cylinder functions, the most important perhaps being those containing the product of two of these functions. What we now call Lommel's first and second integrals are special cases of these, which are very general in character.41
  41. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1966), p. 133.
  42. F. Bowman, Introduction to Bessel Functions (Dover, New York, 1958), p. 102.
  43. To derive Eqs. (A6a) and (A6b), we made use of the well-known recurrence relations (d/dx)[xmJm(x)]= xmJm-1(x) and (2m/x)Jm(x)=Jm-1(x)+Jm+1(x), with m=1. See F. Bowman, Introduction to Bessel Functions (Dover, New York, 1958), p. 93.
  44. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 397.
  45. A rigorous proof of the inequality |ϕn(λn,s)|<1 for λn,s<αn is not trivial. Some results that are due to Watson concerning the stationary values of cylinder functions might be needed. See G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1966), p. 488.
  46. G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), p. 620.
  47. Staff of the Bateman Manuscript Project, Higher Transcendental Functions, A. Erdély, ed. (Krieger, Malabar, Fla., 1970), Vol. II, p. 72.
  48. When giving his expansion, Bateman does not comment on the three possible cases that one has in every Dini expansion. For μ=0 these are H>0,H=0, and H<0 [Eq. (4.4)]. It is known that when H>0, the series expansion starts with n=1, whereas in the last two cases it starts with n=0(see Ref. 25). We have tacitly assumed H to be nonpositive when writing Eq. (4.1).
  49. Staff of the Bateman Manuscript Project, Higher Transcendental Functions (Krieger, Malabar, Fla, 1970), Vol. II, p. 70. Schlömilch's and Dini's expansions of Jμ(vρ) also appear in V. Mangulis, Handbook of Series for Scientist and Engineers (Academic, New York, 1966), p. 29.

1991 (1)

1990 (1)

1989 (1)

1988 (3)

1986 (1)

1984 (1)

1969 (1)

J. C. Dainty, “The image of a point for an aberration free lens with a circular pupil,” Opt. Commun. 1, 176–178 (1969).
[CrossRef]

1966 (1)

1964 (1)

1952 (1)

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 365.

Andrés, P.

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), p. 594.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), p. 620.

Barakat, R.

R. Barakat, “The calculation of integrals encountered in optical diffraction theory,” in The Computer in Optical Research, B. R. Frieden, ed. (Springer, Berlin, 1980), Chap. 2.

Beiser, L.

Berriel Valdos, L. R.

Boivin, A.

A. Boivin, “On the theory of diffraction by concentric arrays of ring-shaped apertures,” J. Opt. Soc. Am. 42, 60–64 (1952).
[CrossRef]

A. Boivin, Théorie et Calcul des Figures de Diffraction de Révolution (Les Presses de L'Université Laval, Québec, 1968).

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), pp. 435–449.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 475.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 416.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 397.

Bowman, F.

F. Bowman, Introduction to Bessel Functions (Dover, New York, 1958), p. 102.

To derive Eqs. (A6a) and (A6b), we made use of the well-known recurrence relations (d/dx)[xmJm(x)]= xmJm-1(x) and (2m/x)Jm(x)=Jm-1(x)+Jm+1(x), with m=1. See F. Bowman, Introduction to Bessel Functions (Dover, New York, 1958), p. 93.

F. Bowman, Introduction to Bessel Functions (Dover, New York, 1958), p. 97.

F. Bowman, Introduction to Bessel Functions (Dover, New York, 1958), p. 17.

Dainty, J. C.

J. C. Dainty, “The image of a point for an aberration free lens with a circular pupil,” Opt. Commun. 1, 176–178 (1969).
[CrossRef]

Di ´az, A.

Diaz, A.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), p. 60.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), p. 21.

Hegedus, Z. S.

Jacquinot, P.

P. Jacquinot, B. Roizen-Dossier, “Apodisation,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. III, Chap. II.

Li, Y.

Linfoot, E. H.

E. H. Linfoot, Recent Advances in Optics (Clarendon, Oxford, 1955), Chap. I.

Lohmann, A. W.

A. W. Lohmann, Optical Information Processing (Lecture Notes), 2nd ed. (Erlangen, Germany, 1978), p. 21.

McCutchen, C. W.

Ojeda-Castañeda, J.

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (Krieger, Malabar, Fla., 1981), p. 130.

Pons, A.

J. Ojeda-Castañeda, E. Tepichin, A. Pons, “Apodization of annular apertures: Strehl ratio,” Appl. Opt. 28, 5140–5145 (1988).
[CrossRef]

Roizen-Dossier, B.

P. Jacquinot, B. Roizen-Dossier, “Apodisation,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. III, Chap. II.

Sheppard, C. J. R.

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 365.

Tepichin, E.

J. Ojeda-Castañeda, E. Tepichin, A. Diaz, “Arbitrarily high focal depth with a quasioptimum real and positivetransmittance apodizer,” Appl. Opt. 28, 2666–2670 (1989).
[CrossRef]

J. Ojeda-Castañeda, E. Tepichin, A. Pons, “Apodization of annular apertures: Strehl ratio,” Appl. Opt. 28, 5140–5145 (1988).
[CrossRef]

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1966), p. 133.

A rigorous proof of the inequality |ϕn(λn,s)|<1 for λn,s<αn is not trivial. Some results that are due to Watson concerning the stationary values of cylinder functions might be needed. See G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1966), p. 488.

Wolf, E.

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

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 475.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), pp. 435–449.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 397.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 416.

Appl. Opt. (5)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

J. C. Dainty, “The image of a point for an aberration free lens with a circular pupil,” Opt. Commun. 1, 176–178 (1969).
[CrossRef]

Opt. Lett. (1)

Other (37)

Since, in general, we must have |F(ρ2)| ≤1(0≤ρ≤1), the expression for F(ρ2) given by Eq. (5.7b) must be normalized. The normalizing factor in this case is, evidently, F(1/2)=A(0)+2∑m=1∞A(m).

A. Papoulis, Systems and Transforms with Applications in Optics (Krieger, Malabar, Fla., 1981), p. 130.

A. W. Lohmann, Optical Information Processing (Lecture Notes), 2nd ed. (Erlangen, Germany, 1978), p. 21.

F. Bowman, Introduction to Bessel Functions (Dover, New York, 1958), p. 17.

That is, a set of functions fn(r)(0, v).

This will be evident after Eq. (5.6).

R. Barakat, “The calculation of integrals encountered in optical diffraction theory,” in The Computer in Optical Research, B. R. Frieden, ed. (Springer, Berlin, 1980), Chap. 2.

P. Jacquinot, B. Roizen-Dossier, “Apodisation,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. III, Chap. II.

By an apodized system we shall mean a system whose pupil function has been modified in some prescribed way, not necessarily to the effect of suppressing the secondary lobes of its PSF.

E. H. Linfoot, Recent Advances in Optics (Clarendon, Oxford, 1955), Chap. I.

A. Boivin, Théorie et Calcul des Figures de Diffraction de Révolution (Les Presses de L'Université Laval, Québec, 1968).

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).

By reference image-plane we shall mean the plane that is perpendicular to the optical axis at the center of the reference sphere that we use to establish the wave-front aberration of the system.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), p. 60.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), p. 21.

A defocus term can, of course, also be included in the apodizing function F(ρ2). This term is very useful to produce an axial shift of the window that we use to display our graphical results (Fig. 5). This shift is commonly required with aberrated systems, where the diffraction patterns do not possess an image plane of symmetry.

From Eqs. (1.1), (2.5), and (2.7), we have f/z=1-2δ/N. When N→∞ (the classical regime), 1/z∼1/f for all the relevant values of δ.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), pp. 435–449.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 475.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 416.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 365.

Our aim in this example was to show the potential usefulness of the sampling expansions for tackling problems of pupil synthesis. We did not attempt to find another optimal solution to the problem of apodization; many of these have been given in the past, satisfying various criteria of optimization.2

Ref. 9, Chap. 12.

F. Bowman, Introduction to Bessel Functions (Dover, New York, 1958), p. 97.

Notice that these linear transformations are defined in spaces of infinite dimension.

The Fourier-transform relationship between these fields is a consequence of using Kirchhoff's (scalar) diffraction theory in the paraxial approximation.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), p. 594.

In fact, Lommel presented various integrals involving cylinder functions, the most important perhaps being those containing the product of two of these functions. What we now call Lommel's first and second integrals are special cases of these, which are very general in character.41

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1966), p. 133.

F. Bowman, Introduction to Bessel Functions (Dover, New York, 1958), p. 102.

To derive Eqs. (A6a) and (A6b), we made use of the well-known recurrence relations (d/dx)[xmJm(x)]= xmJm-1(x) and (2m/x)Jm(x)=Jm-1(x)+Jm+1(x), with m=1. See F. Bowman, Introduction to Bessel Functions (Dover, New York, 1958), p. 93.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 397.

A rigorous proof of the inequality |ϕn(λn,s)|<1 for λn,s<αn is not trivial. Some results that are due to Watson concerning the stationary values of cylinder functions might be needed. See G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1966), p. 488.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), p. 620.

Staff of the Bateman Manuscript Project, Higher Transcendental Functions, A. Erdély, ed. (Krieger, Malabar, Fla., 1970), Vol. II, p. 72.

When giving his expansion, Bateman does not comment on the three possible cases that one has in every Dini expansion. For μ=0 these are H>0,H=0, and H<0 [Eq. (4.4)]. It is known that when H>0, the series expansion starts with n=1, whereas in the last two cases it starts with n=0(see Ref. 25). We have tacitly assumed H to be nonpositive when writing Eq. (4.1).

Staff of the Bateman Manuscript Project, Higher Transcendental Functions (Krieger, Malabar, Fla, 1970), Vol. II, p. 70. Schlömilch's and Dini's expansions of Jμ(vρ) also appear in V. Mangulis, Handbook of Series for Scientist and Engineers (Academic, New York, 1966), p. 29.

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Figures (5)

Fig. 1
Fig. 1

Geometry of the problem.

Fig. 2
Fig. 2

Discrete spectrum of bounded spherical waves that gives rise to the diffracted field beyond the exit pupil of the system. The points O m ( m = 0 ,   ± 1 ,   ± 2 ,   ) are the foci of the waves of this spectrum and the centers of the Lommel patterns in the ASE. The amplitude of the mth wave of the spectrum is the amplitude of the diffracted field at O m . The phase difference between two consecutive waves is 2π at the rim of the pupil.

Fig. 3
Fig. 3

Plots of the first three reference image-plane interpolation functions and their corresponding envelopes.

Fig. 4
Fig. 4

Plots of Sonine apodizing functions F k ( ρ 2 ) = ( 1 - ρ 2 ) k for k = 0 , 1, 2, 3 ( 0 ρ 1 ) . The dashed curve corresponds to the apodizing function F ( ρ 2 ) = [ J 0 ( α 1 ρ ) - J 0 ( α 1 ) ] / [ 1 - J 0 ( α 1 ) ] [Eq. (5.24)], where α 1 is the first (nonzero) root of J 1 ( x ) .

Fig. 5
Fig. 5

Meridional sections of the 3-D diffraction patterns that surround an axial image point in a system apodized with Sonine functions [Eq. (5.25)]: (a) k = 0 (free pupil), (b) k = 1 , (c) k = 2 , (d) k = 3 . The intensity distributions are normalized and shown with isophotes (intensity contour lines). In all cases the axial image point is at the origin, and the patterns are symmetric with respect to the δ and τ axes. The dashed lines represent the boundary of the geometrical shadow.

Equations (115)

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N = a 2 / λ f ,
U ( x 0 ,   y 0 ,   z ) = ( - i / λ z ) exp ( ikz ) exp [ ik ( x 0 2 + y 0 2 ) / 2 z ] × - - U ( x ,   y ,   0 ) × exp [ ik ( x 2 + y 2 ) / 2 z ] × exp [ - ik ( x 0 x + y 0 y ) / z ] d x d y ,
U ( x ,   y ,   0 ) = circ ( ρ ) F ( ρ 2 ) exp ( - ika 2 ρ 2 / 2   f   ) ,
ρ = x 2 + y 2 / a
G ( δ ,   v ) = 2 π ( a 2 / λ z ) 0 1 F ( ρ 2 ) exp ( - i 2 π δ ρ 2 ) J 0 ( v ρ ) ρ   d ρ ,
δ = ( a 2 / 2   fz ) ζ / λ ,
v = 2 π τ ,
ζ = z - f ,
τ = a ( x 0 2 + y 0 2 / z ) / λ .
δ = 1 2   ( a / f   ) 2 ζ / λ ,
τ = ( a / f   ) x 0 2 + y 0 2 / λ ,
G ( δ ,   v ) = 2 0 1 F ( ρ 2 ) exp ( - i 2 π δ ρ 2 ) J 0 ( v ρ ) ρ   d ρ .
L ( δ ,   v ) = 2 0 1   exp ( - i 2 π δ ρ 2 ) J 0 ( v ρ ) ρ   d ρ
= C ( δ ,   v ) - iS ( δ ,   v ) .
K ( ρ ;   δ ,   v ) = exp ( - i 2 π δ ρ 2 ) J 0 ( v ρ ) .
K ( ρ ;   δ ,   v ) = m = - f m ( a ) ( δ ,   v ) K ( ρ ;   m ,   0 )
( 0 ρ 1 ) ,
K ( ρ ;   m ,   0 ) = exp ( - i 2 π m ρ 2 )
G ( δ ,   v ) = 2 0 1 F ( ρ 2 ) m = - f m ( a ) ( δ ,   v ) K ( ρ ;   m ,   0 ) ρ   d ρ
= m = - 2 0 1 F ( ρ 2 ) K ( ρ ;   m ,   0 ) ρ   d ρ f m ( a ) ( δ ,   v )
= m = - G ( m ,   0 ) f m ( a ) ( δ ,   v ) .
f m ( a ) ( δ ,   v ) = 0 2 π 0 1 K ( ρ ;   δ ,   v ) K * ( ρ ;   m ,   0 ) ρ   d ρ d θ 0 2 π 0 1 K ( ρ ;   m ,   0 ) K * ( ρ ;   m ,   0 ) ρ   d ρ d θ
= 2 0 1   exp [ - i 2 π ( δ - m ) ρ 2 ] J 0 ( v ρ ) ρ   d ρ .
f m ( a ) ( δ ,   v ) = L ( δ - m ,   v ) .
G ( δ ,   v ) = m = - G ( m ,   0 ) L ( δ - m ,   v ) .
L ( δ ,   0 ) = 2 0 1   exp ( - i 2 π δ ρ 2 ) ρ   d ρ
= exp ( - i π δ ) sinc ( δ ) ,
G ( m ,   0 ) = 2 0 1 F ( ρ 2 ) exp ( - i 2 π m ρ 2 ) ρ   d ρ .
G ( m ,   0 ) = -   rect ( ξ - 1 / 2 ) F ( ξ ) exp ( - i 2 π m ξ ) d ξ ,
G ( δ ,   v ) = -   rect ( ξ - 1 / 2 ) F ( ξ ) J 0 ( v ξ ) × exp ( - i 2 π δ ξ ) d ξ ,
G ( δ ,   0 ) L ( δ ,   v ) - G ( δ ,   0 ) L ( δ - δ ,   v ) d δ
= m = - G ( m ,   0 ) L ( δ - m ,   v ) ,
G ( δ ,   0 ) L ( δ ,   v ) = - [ rect ( ξ - 1 / 2 ) F ( ξ ) ] × [ rect ( ξ - 1 / 2 ) J 0 ( v ξ ) ] × exp ( - i 2 π δ ξ ) d ξ
= G ( δ ,   v ) ,
K ( ρ ;   δ ,   v ) = n = 0 f n ( r ) ( δ ,   v ) k ( ρ ;   0 ,   α n ) ( 0 < ρ < 1 ) ,
K ( ρ ;   0 ,   α n ) = J 0 ( α n ρ )
G ( δ ,   v ) = n = 0 G ( 0 ,   α n ) f n ( r ) ( δ ,   v ) .
xJ 0 ( x ) + HJ 0 ( x ) = 0 ,
xJ 0 ( x ) = 0 ,
f n ( r ) ( δ ,   v ) = 0 2 π 0 1 K ( ρ ;   δ ,   v ) K * ( ρ ;   0 ,   α n ) ρ   d ρ d θ 0 2 π 0 1 K ( ρ ;   0 ,   α n ) K * ( ρ ;   0 ,   α n ) ρ   d ρ d θ
= 0 1 J 0 ( α n ρ ) exp ( - i 2 π δ ρ 2 ) J 0 ( v ρ ) ρ   d ρ 0 1 J 0 2 ( α n ρ ) ρ   d ρ ,
f n ( r ) ( δ ,   v ) = 1 J 0 2 ( α n )   2 0 1 J 0 ( α n ρ ) × exp ( - i 2 π δ ρ 2 ) J 0 ( v ρ ) ρ   d ρ .
f n ( r ) ( δ ,   v ) = 1 J 0 2 ( α n )   m = - × 2 0 1 J 0 ( α n ρ ) exp ( - i 2 π m ρ 2 ) ρ   d ρ × L ( δ - m ,   v ) ,
f n ( r ) ( δ ,   v ) = 1 J 0 2 ( α n )   m = - L ( m ,   α n ) L ( δ - m ,   v ) ,
G ( δ ,   v ) = n = 0   G ( 0 ,   α n ) J 0 2 ( α n )   m = - L ( m ,   α n ) L ( δ - m ,   v ) .
G ( δ ,   0 ) = m = - G ( m ,   0 ) f m ( a ) ( δ ,   0 ) .
f m ( a ) ( δ ,   0 ) = exp [ - i π ( δ - m ) ] sinc ( δ - m ) .
G ( δ ,   0 ) = exp ( - i π δ ) m = - ( - 1 ) m G ( m ,   0 ) sinc ( δ - m ) .
G ( m ,   0 ) = ( - 1 ) m A ( m ) .
| G ( δ ,   0 ) | 2 = m = - A ( m ) sinc ( δ - m ) 2 .
F ( ρ 2 ) = m = - G ( m ,   0 ) exp ( i 2 π m ρ 2 ) ,
F ( ρ 2 ) = m = - A ( m ) exp [ i 2 π m ( ρ 2 - 1 / 2 ) ]
= A ( 0 ) + 2 m = 1 A ( m ) cos [ 2 π m ( ρ 2 - 1 / 2 ) ] ,
A ( m ) = R | m | | m |   M 0 otherwise   ,
G ( 0 ,   v ) = n = 0 G ( 0 ,   α n ) f n ( r ) ( 0 ,   v ) .
f n ( r ) ( 0 ,   v ) = 1 J 0 2 ( α n ) × 2 0 1 J 0 ( α n ρ ) J 0 ( v ρ ) ρ   d ρ .
f n ( r ) ( 0 ,   v ) = 1 J 0 ( α n )   1 1 - ( α n / v ) 2   2 J 1 ( v ) v .
G ( 0 ,   v ) = 2 J 1 ( v ) v   n = 0   G ( 0 ,   α n ) J 0 ( α n )   1 1 - ( α n / v ) 2
= L ( 0 ,   v ) n = 0   G ( 0 ,   α n ) J 0 ( α n )   1 1 - ( α n / v ) 2 .
| f n ( r ) ( 0 ,   v ) |   1 ,
f n ( r ) ( 0 ,   α q ) = δ q - n ,
J 1 ( v ) = [ J 1 2 ( v ) + Y 1 2 ( v ) ] 1 / 2   cos [ θ 1 ( v ) ] ,
θ 1 ( v ) = tan - 1 [ Y 1 ( v ) / J 1 ( v ) ]
e n ( v ) = ± 1 J 0 ( α n )   1 1 - ( α n / v ) 2   2 [ J 1 2 ( v ) + Y 1 2 ( v ) ] 1 / 2 v .
f n ( r ) ( 0 ,   v ) 1 J 0 ( α n )   2 J 1 ( v ) v
= f 0 ( r ) ( 0 ,   v ) / J 0 ( α n ) .
G ( 0 ,   α 0 ) f 0 ( r ) ( 0 ,   v ) + G ( 0 ,   α 1 ) f 1 ( r ) ( 0 ,   v )
[ G ( 0 ,   α 0 ) + G ( 0 ,   α 1 ) / J 0 ( α 1 ) ] f 0 ( r ) ( 0 ,   v )
= 0 ;
G ( 0 ,   α 1 ) = - J 0 ( α 1 ) G ( 0 ,   0 ) .
G ( 0 ,   α n ) = 2 0 1 F ( ρ 2 ) J 0 ( α n ρ ) ρ   d ρ
F ( ρ 2 ) = n = 0   G ( 0 ,   α n ) J 0 2 ( α n )   J 0 ( α n ρ ) .
F ( ρ 2 ) = G ( 0 ,   0 ) [ 1 - J 0 ( α 1 ρ ) / J 0 ( α 1 ) ] .
F ( ρ 2 ) = J 0 ( α 1 ρ ) - J 0 ( α 1 ) 1 - J 0 ( α 1 ) .
F k ( ρ 2 ) = ( 1 - ρ 2 ) k .
G ( 0 ,   v ) = 1 1 - ( v / α 1 ) 2   2 J 1 ( v ) v .
I ( 0 ,   v ) ( α 1 / v ) 4 I L ( 0 ,   v ) ,
G ( 0 ,   α n ) = 2 0 1 ( 1 - ρ 2 ) s J 0 ( α n ρ ) ρ   d ρ
= 2 s + 1 Γ ( s + 1 ) α n s + 1   J s + 1 ( α n ) .
G ( 0 ,   α n ) = m = - G ( m ,   0 ) f m ( a ) ( 0 ,   α n )
= m = - L * ( m ,   α n ) G ( m ,   0 ) ,
G ( m ,   0 ) = n = 0 G ( 0 ,   α n ) f n ( r ) ( m ,   0 )
= n = 0   L ( m ,   α n ) J 0 2 ( α n )   G ( 0 ,   α n ) ,
m = - L ( m ,   α n ) L * ( m ,   α n ) = δ n - n J 0 2 ( α n ) ,
n = 0   1 J 0 2 ( α n )   L ( m ,   α n ) L * ( m ,   α n ) = δ m - m .
m = - I L ( m ,   α n ) = J 0 2 ( α n ) ,
n = 0 I L ( m ,   α n ) / J 0 2 ( α n ) = 1 ,
( β 2 - α 2 ) 0 1 J μ ( α x ) J μ ( β x ) x   d x
= α J μ ( α ) J μ ( β ) - β J μ ( β ) J μ ( α )
( β 2 - α 2 0 ,   μ > - 1 ) ,
0 1 [ J μ ( α x ) ] 2 x   d x
= 1 2   { [ J μ ( α ) ] 2 - J μ - 1 ( α ) J μ + 1 ( α ) } .
( v 2 - α n 2 ) 0 1 J 0 ( α n ρ ) J 0 ( v ρ ) ρ   d ρ = vJ 0 ( α n ) J 1 ( v ) ,
0 1 [ J 0 ( α n ρ ) ] 2 ρ   d ρ = 1 2   [ J 0 ( α n ) ] 2 .
ϕ n ( v ) = 1 J 0 ( α n )   2 vJ 1 ( v ) v 2 - α n 2 .
ϕ n ( v ) = 1 J 0 ( α n )   2 v ( v 2 - α n 2 ) 2   [ ( v 2 - α n 2 ) J 0 ( v ) - 2 vJ 1 ( v ) ]
= - 1 J 0 ( α n )   2 v ( v 2 - α n 2 ) 2   [ α n 2 J 0 ( v ) + v 2 J 2 ( v ) ]
= - 1 J 0 ( α n )   2 v v 2 - α n 2   [ J 0 ( v ) - J 0 ( α n ) ϕ n ( v ) ] .
α n 2 J 0 ( v ) + v 2 J 2 ( v ) = 0 .
ϕ n ( λ n , s ) = J 0 ( λ n , s ) / J 0 ( α n ) .
| ϕ n ( λ n , s ) |   =   | J 0 ( λ n , s ) | / | J 0 ( α n ) |
= λ n , s α n 2   | J 2 ( λ n , s ) | | J 0 ( α n ) | .
J m ( x ) 2 π 1 / 2 x - 1 / 2   cos x - ( 2 m + 1 )   π 4 ,
| J 0 ( α n ) |   ( 2 / π ) 1 / 2 α n - 1 / 2 .
| ϕ n ( λ n , s ) | ( λ n , s / α n ) 3 / 2 | cos ( λ n , s - 5 π / 4 ) | .
F ( ρ 2 ) = n = 0 c n J 0 ( α n ρ ) ,
G n ( m ,   0 ) = 2 0 1 J 0 ( α n ρ ) exp ( - i 2 π m ρ 2 ) ρ   d ρ
= L ( m ,   α n ) .
G ( m ,   0 ) = n = 0 c n G n ( m ,   0 )
= n = 0 c n L ( m ,   α n ) .
G ( δ ,   v ) = m = - n = 0 c n L ( m ,   α n ) L ( δ - m ,   v ) .
c n = 0 1 F ( ρ 2 ) J 0 ( α n ρ ) ρ   d ρ 0 1 J 0 2 ( α n ρ ) ρ   d ρ
= G ( 0 ,   α n ) / J 0 2 ( α n ) .
G ( δ ,   v ) = m = - n = 0   G ( 0 ,   α n ) J 0 2 ( α n )   L ( m ,   α n ) L ( δ - m ,   v )
= n = 0   G ( 0 ,   α n ) J 0 2 ( α n )   m = - L ( m ,   α n ) L ( δ - m ,   v ) .

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