Abstract

This work describes a procedure for an analytical calculation of the point spread function (PSF) of an optical system affected by defocus and spherical aberration. Explicit formulas are derived for the approximate calculation of the PSF of an optical system with spherical aberration up to the ninth order. Application of the derived formulas is performed on an example of optical systems with spherical aberration up to the third order.

© 2019 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
  2. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  3. A. Papoulis, System and Transforms with Applications in Optics (McGraw-Hill, 1968).
  4. E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, 1963).
  5. G. B. Airy, “On the diffraction of an object-glass with circular aperture,” Trans. Cambridge Philos. Soc. 5, 283–291 (1835).
  6. E. Lommel, “Diffraction by a slot and strips,” Abh. Bayer. Akad. 15, Abth. 2, 233 (1885).
  7. E. Lommel, “Theory and experimental investigations of diffraction phenomena at a circular aperture and obstacle,” Abh. Bayer. Akad. 15, Abth. 3, 531 (1886).
  8. B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. thesis (University of Groningen, 1942).
  9. F. Zernike and B. R. A. Nijboer, Contribution to La Théorie des Images Optiques (Revue d’Optique, 1949), p. 227.
  10. E. Wolf, “The diffraction theory of aberrations,” Rep. Prog. Phys. 14, 95–120 (1951).
    [Crossref]
  11. H. F. A. Tschunko, “Derivation of the point spread function,” Appl. Opt. 22, 1413–1414 (1983).
    [Crossref]
  12. J. J. Stamnes and H. Heier, “Scalar and electromagnetic diffraction point-spread functions,” Appl. Opt. 37, 3612–3622 (1998).
    [Crossref]
  13. A. B. Budgor, “Exact solutions in the scalar diffraction theory of aberrations,” Appl. Opt. 19, 1597–1600 (1980).
    [Crossref]
  14. J. J. M. Braat, P. Dirksen, and A. J. E. M. Janssen, “Assessment of an extended Nijboer-Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A 19, 858–870 (2002).
    [Crossref]
  15. A. J. E. M. Janssen, “Extended Nijboer-Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A 19, 849–857 (2002).
    [Crossref]
  16. J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread functions,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2008), Vol. 51, pp. 349–468.
  17. V. S. Ignatowsky, “Diffraction by a lens of arbitrary aperture,” Tr. Opt. Inst. Petrograd 1(4), 1–36 (1919).
  18. H. H. Hopkins, “The Airy disc formula for systems of high relative aperture,” Proc. Phys. Soc. 55, 116–128 (1943).
    [Crossref]
  19. E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London A 253, 349–357 (1959).
    [Crossref]
  20. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London A 253, 358–379 (1959).
    [Crossref]
  21. A. Miks, J. Novak, and P. Novak, “Calculation of point-spread function for optical systems with finite value of numerical aperture,” Optik 118, 537–543 (2007).
    [Crossref]
  22. B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens Principle, 2nd. ed. (Clarendon, 1950).
  23. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (World Scientific, 2006).
  24. C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–100 (1954).
    [Crossref]
  25. F. Kottler, “Zur Theorie der Beugung an Schwarzen Schirmen,” Ann. Phys. 375, 405–456 (1923).
    [Crossref]
  26. F. Kottler, “Diffraction at a black screen: Part I: Kirchoff’s theory,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1965), Vol. 4, Chap. VII.
  27. F. Kottler, “Diffraction at a black screen : Part II: Electromagnetic theory,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1967), Vol. 6, Chap. VIII.
  28. B. Karczewski, “Fraunhofer diffraction of an electromagnetic wave,” J. Opt. Soc. Am. 51, 1055–1057 (1961).
    [Crossref]
  29. B. Karczewski and E. Wolf, “Comparison of three theories of electromagnetic diffraction at an aperture. Part I: Coherence matrices,” J. Opt. Soc. Am. 56, 1207–1210 (1966).
    [Crossref]
  30. B. Karczewski and E. Wolf, “Comparison of three theories of electromagnetic diffraction at an aperture. Part II: The far field,” J. Opt. Soc. Am. 56, 1214–1218 (1966).
    [Crossref]
  31. E. W. Marchand and E. Wolf, “Consistent formulation of Kirchhoff’s diffraction theory,” J. Opt. Soc. Am. 56, 1712–1721 (1966).
    [Crossref]
  32. A. S. Marathay and J. F. McCalmont, “Vector diffraction theory for electromagnetic waves,” J. Opt. Soc. Am. A 18, 2585–2593 (2001).
    [Crossref]
  33. R. Burtin, “Two problems of optical diffraction at large relative apertures,” Opt. Acta 3, 104–109 (1956).
    [Crossref]
  34. A. S. Marathay and G. B. Parrent, “Use of scalar theory in optics,” J. Opt. Soc. Am. 60, 243–245 (1970).
    [Crossref]
  35. J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
    [Crossref]
  36. L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012).
  37. H. H. Hopkins, “Image formation by a general optical system. 1: General theory,” Appl. Opt. 24, 2491–2505 (1985).
    [Crossref]
  38. H. H. Hopkins, “Image formation by a general optical system. 2: Computing methods,” Appl. Opt. 24, 2506–2519 (1985).
    [Crossref]
  39. H. H. Hopkins and M. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157–182 (1970).
    [Crossref]
  40. A. Miks and J. Novak, “Method for numerical integration of rapidly oscillating functions in diffraction theory,” Int. J. Numer. Methods Eng. 82, 525–536 (2010).
    [Crossref]
  41. G. A. Korn and T. M. Korn, Mathematical Handbook (McGraw-Hill, 1968).
  42. V. Mahajan, Optical Imaging and Aberrations, Part II. Wave Diffraction Optics (SPIE, 2011).

2010 (1)

A. Miks and J. Novak, “Method for numerical integration of rapidly oscillating functions in diffraction theory,” Int. J. Numer. Methods Eng. 82, 525–536 (2010).
[Crossref]

2007 (1)

A. Miks, J. Novak, and P. Novak, “Calculation of point-spread function for optical systems with finite value of numerical aperture,” Optik 118, 537–543 (2007).
[Crossref]

2002 (2)

2001 (1)

1998 (1)

1985 (2)

1983 (1)

1980 (1)

1970 (2)

H. H. Hopkins and M. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157–182 (1970).
[Crossref]

A. S. Marathay and G. B. Parrent, “Use of scalar theory in optics,” J. Opt. Soc. Am. 60, 243–245 (1970).
[Crossref]

1966 (3)

1961 (1)

1959 (2)

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London A 253, 349–357 (1959).
[Crossref]

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London A 253, 358–379 (1959).
[Crossref]

1956 (1)

R. Burtin, “Two problems of optical diffraction at large relative apertures,” Opt. Acta 3, 104–109 (1956).
[Crossref]

1954 (1)

C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–100 (1954).
[Crossref]

1951 (1)

E. Wolf, “The diffraction theory of aberrations,” Rep. Prog. Phys. 14, 95–120 (1951).
[Crossref]

1943 (1)

H. H. Hopkins, “The Airy disc formula for systems of high relative aperture,” Proc. Phys. Soc. 55, 116–128 (1943).
[Crossref]

1939 (1)

J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[Crossref]

1923 (1)

F. Kottler, “Zur Theorie der Beugung an Schwarzen Schirmen,” Ann. Phys. 375, 405–456 (1923).
[Crossref]

1919 (1)

V. S. Ignatowsky, “Diffraction by a lens of arbitrary aperture,” Tr. Opt. Inst. Petrograd 1(4), 1–36 (1919).

1886 (1)

E. Lommel, “Theory and experimental investigations of diffraction phenomena at a circular aperture and obstacle,” Abh. Bayer. Akad. 15, Abth. 3, 531 (1886).

1885 (1)

E. Lommel, “Diffraction by a slot and strips,” Abh. Bayer. Akad. 15, Abth. 2, 233 (1885).

1835 (1)

G. B. Airy, “On the diffraction of an object-glass with circular aperture,” Trans. Cambridge Philos. Soc. 5, 283–291 (1835).

Airy, G. B.

G. B. Airy, “On the diffraction of an object-glass with circular aperture,” Trans. Cambridge Philos. Soc. 5, 283–291 (1835).

Baker, B. B.

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens Principle, 2nd. ed. (Clarendon, 1950).

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Bouwkamp, C. J.

C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–100 (1954).
[Crossref]

Braat, J. J. M.

J. J. M. Braat, P. Dirksen, and A. J. E. M. Janssen, “Assessment of an extended Nijboer-Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A 19, 858–870 (2002).
[Crossref]

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread functions,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2008), Vol. 51, pp. 349–468.

Budgor, A. B.

Burtin, R.

R. Burtin, “Two problems of optical diffraction at large relative apertures,” Opt. Acta 3, 104–109 (1956).
[Crossref]

Chu, L. J.

J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[Crossref]

Copson, E. T.

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens Principle, 2nd. ed. (Clarendon, 1950).

Dirksen, P.

J. J. M. Braat, P. Dirksen, and A. J. E. M. Janssen, “Assessment of an extended Nijboer-Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A 19, 858–870 (2002).
[Crossref]

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread functions,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2008), Vol. 51, pp. 349–468.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012).

Heier, H.

Hopkins, H. H.

H. H. Hopkins, “Image formation by a general optical system. 1: General theory,” Appl. Opt. 24, 2491–2505 (1985).
[Crossref]

H. H. Hopkins, “Image formation by a general optical system. 2: Computing methods,” Appl. Opt. 24, 2506–2519 (1985).
[Crossref]

H. H. Hopkins and M. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157–182 (1970).
[Crossref]

H. H. Hopkins, “The Airy disc formula for systems of high relative aperture,” Proc. Phys. Soc. 55, 116–128 (1943).
[Crossref]

Ignatowsky, V. S.

V. S. Ignatowsky, “Diffraction by a lens of arbitrary aperture,” Tr. Opt. Inst. Petrograd 1(4), 1–36 (1919).

Janssen, A. J. E. M.

Karczewski, B.

Korn, G. A.

G. A. Korn and T. M. Korn, Mathematical Handbook (McGraw-Hill, 1968).

Korn, T. M.

G. A. Korn and T. M. Korn, Mathematical Handbook (McGraw-Hill, 1968).

Kottler, F.

F. Kottler, “Zur Theorie der Beugung an Schwarzen Schirmen,” Ann. Phys. 375, 405–456 (1923).
[Crossref]

F. Kottler, “Diffraction at a black screen: Part I: Kirchoff’s theory,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1965), Vol. 4, Chap. VII.

F. Kottler, “Diffraction at a black screen : Part II: Electromagnetic theory,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1967), Vol. 6, Chap. VIII.

Lommel, E.

E. Lommel, “Theory and experimental investigations of diffraction phenomena at a circular aperture and obstacle,” Abh. Bayer. Akad. 15, Abth. 3, 531 (1886).

E. Lommel, “Diffraction by a slot and strips,” Abh. Bayer. Akad. 15, Abth. 2, 233 (1885).

Mahajan, V.

V. Mahajan, Optical Imaging and Aberrations, Part II. Wave Diffraction Optics (SPIE, 2011).

Marathay, A. S.

Marchand, E. W.

McCalmont, J. F.

Miks, A.

A. Miks and J. Novak, “Method for numerical integration of rapidly oscillating functions in diffraction theory,” Int. J. Numer. Methods Eng. 82, 525–536 (2010).
[Crossref]

A. Miks, J. Novak, and P. Novak, “Calculation of point-spread function for optical systems with finite value of numerical aperture,” Optik 118, 537–543 (2007).
[Crossref]

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (World Scientific, 2006).

Nijboer, B. R. A.

B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. thesis (University of Groningen, 1942).

F. Zernike and B. R. A. Nijboer, Contribution to La Théorie des Images Optiques (Revue d’Optique, 1949), p. 227.

Novak, J.

A. Miks and J. Novak, “Method for numerical integration of rapidly oscillating functions in diffraction theory,” Int. J. Numer. Methods Eng. 82, 525–536 (2010).
[Crossref]

A. Miks, J. Novak, and P. Novak, “Calculation of point-spread function for optical systems with finite value of numerical aperture,” Optik 118, 537–543 (2007).
[Crossref]

Novak, P.

A. Miks, J. Novak, and P. Novak, “Calculation of point-spread function for optical systems with finite value of numerical aperture,” Optik 118, 537–543 (2007).
[Crossref]

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012).

O’Neill, E. L.

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, 1963).

Papoulis, A.

A. Papoulis, System and Transforms with Applications in Optics (McGraw-Hill, 1968).

Parrent, G. B.

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London A 253, 358–379 (1959).
[Crossref]

Stamnes, J. J.

Stratton, J. A.

J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[Crossref]

Tschunko, H. F. A.

van Haver, S.

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread functions,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2008), Vol. 51, pp. 349–468.

Wolf, E.

B. Karczewski and E. Wolf, “Comparison of three theories of electromagnetic diffraction at an aperture. Part I: Coherence matrices,” J. Opt. Soc. Am. 56, 1207–1210 (1966).
[Crossref]

E. W. Marchand and E. Wolf, “Consistent formulation of Kirchhoff’s diffraction theory,” J. Opt. Soc. Am. 56, 1712–1721 (1966).
[Crossref]

B. Karczewski and E. Wolf, “Comparison of three theories of electromagnetic diffraction at an aperture. Part II: The far field,” J. Opt. Soc. Am. 56, 1214–1218 (1966).
[Crossref]

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London A 253, 358–379 (1959).
[Crossref]

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London A 253, 349–357 (1959).
[Crossref]

E. Wolf, “The diffraction theory of aberrations,” Rep. Prog. Phys. 14, 95–120 (1951).
[Crossref]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Yzuel, M.

H. H. Hopkins and M. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157–182 (1970).
[Crossref]

Zernike, F.

F. Zernike and B. R. A. Nijboer, Contribution to La Théorie des Images Optiques (Revue d’Optique, 1949), p. 227.

Abh. Bayer. Akad. (2)

E. Lommel, “Diffraction by a slot and strips,” Abh. Bayer. Akad. 15, Abth. 2, 233 (1885).

E. Lommel, “Theory and experimental investigations of diffraction phenomena at a circular aperture and obstacle,” Abh. Bayer. Akad. 15, Abth. 3, 531 (1886).

Ann. Phys. (1)

F. Kottler, “Zur Theorie der Beugung an Schwarzen Schirmen,” Ann. Phys. 375, 405–456 (1923).
[Crossref]

Appl. Opt. (5)

Int. J. Numer. Methods Eng. (1)

A. Miks and J. Novak, “Method for numerical integration of rapidly oscillating functions in diffraction theory,” Int. J. Numer. Methods Eng. 82, 525–536 (2010).
[Crossref]

J. Opt. Soc. Am. (5)

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

Opt. Acta (2)

R. Burtin, “Two problems of optical diffraction at large relative apertures,” Opt. Acta 3, 104–109 (1956).
[Crossref]

H. H. Hopkins and M. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157–182 (1970).
[Crossref]

Optik (1)

A. Miks, J. Novak, and P. Novak, “Calculation of point-spread function for optical systems with finite value of numerical aperture,” Optik 118, 537–543 (2007).
[Crossref]

Phys. Rev. (1)

J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[Crossref]

Proc. Phys. Soc. (1)

H. H. Hopkins, “The Airy disc formula for systems of high relative aperture,” Proc. Phys. Soc. 55, 116–128 (1943).
[Crossref]

Proc. R. Soc. London A (2)

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London A 253, 349–357 (1959).
[Crossref]

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London A 253, 358–379 (1959).
[Crossref]

Rep. Prog. Phys. (2)

E. Wolf, “The diffraction theory of aberrations,” Rep. Prog. Phys. 14, 95–120 (1951).
[Crossref]

C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–100 (1954).
[Crossref]

Tr. Opt. Inst. Petrograd (1)

V. S. Ignatowsky, “Diffraction by a lens of arbitrary aperture,” Tr. Opt. Inst. Petrograd 1(4), 1–36 (1919).

Trans. Cambridge Philos. Soc. (1)

G. B. Airy, “On the diffraction of an object-glass with circular aperture,” Trans. Cambridge Philos. Soc. 5, 283–291 (1835).

Other (14)

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread functions,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2008), Vol. 51, pp. 349–468.

B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. thesis (University of Groningen, 1942).

F. Zernike and B. R. A. Nijboer, Contribution to La Théorie des Images Optiques (Revue d’Optique, 1949), p. 227.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

A. Papoulis, System and Transforms with Applications in Optics (McGraw-Hill, 1968).

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, 1963).

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens Principle, 2nd. ed. (Clarendon, 1950).

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (World Scientific, 2006).

L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012).

F. Kottler, “Diffraction at a black screen: Part I: Kirchoff’s theory,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1965), Vol. 4, Chap. VII.

F. Kottler, “Diffraction at a black screen : Part II: Electromagnetic theory,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1967), Vol. 6, Chap. VIII.

G. A. Korn and T. M. Korn, Mathematical Handbook (McGraw-Hill, 1968).

V. Mahajan, Optical Imaging and Aberrations, Part II. Wave Diffraction Optics (SPIE, 2011).

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Figures (5)

Fig. 1.
Fig. 1. Imaging using an optical system.
Fig. 2.
Fig. 2. Wave aberration (δsK=16λF2, s0=δsK/2).
Fig. 3.
Fig. 3. PSF (δsK=16λF2, s0=δsK/2).
Fig. 4.
Fig. 4. Wave aberration (δsK=24λF2, s0=δsK/2).
Fig. 5.
Fig. 5. PSF (δsK=24λF2, s0=δsK/2).

Equations (46)

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

U(M)=F(M)exp(ikR)R,
U(P)=iλSU(M)exp(ikrPM)rPMcosαdS,
rPM=(ξX)2+(ηY)2+(ζZ)2
F(M)=U0(M)exp[ikW(M)]U0(M)[1+ikW(M)k22W2(M)],
U(P)iλR2SF(M)exp[ik(rPMR)]dS,
I(P)=KU(P)U*(P)=K|U(P)|2,
U(P)iλR2Sexp[ik(rPMR)]dS,
rPMR(Xa)aR(ξξ0)(Ya)aR(ηη0)+β,
β=ξ2+η22Rξ02+η022R.
rPM2R2=(rPMR)(rPM+R)2R(rPMR).
U(P)=C02π01exp(ikaρrcos(φψ))rdrdφ=2πC01J0(kaρr)rdr=Cπ[2J1(kaρ)kaρ],
C=iλ(aR)2exp(ikβ),
I(τ)=I0[2J1(τ)τ]2,
τ=kaρ/R=2πλ(aR)(ξξ0)2+(ηη0)2=πρλF,
dA=2.44λF.
U(P)C02π01F(r,φ)exp[irτcos(φψ)]rdrdφ,
F(r,φ)=U0(r,φ)[1+ikW(r,φ)k22W2(r,φ)],C=iλ(aR)2exp(ikβ).
U(τ)2πC01[1+ikW(r)k22W2(r)]J0(τr)rdr.
U(τ)πC(V0+ikV1k22V2),
V0=201J0(τr)rdr=2J1(τ)τ,V1=201W(r)J0(τr)rdr,V2=201W2(r)J0(τr)rdr.
W=n=1MW2nr2n,
V1=201(n=1MW2nr2n+1)J0(τr)dr.
V1=2n=1MW2n01r2n+1J0(τr)dr=2n=1MW2nIn=n=1MCnIn,
In=01r2n+1J0(τr)dr,Cn=2W2n.
In=01r2n+1J0(τr)dr=p=0nBnpJp+1(τ)τp+1,
Bnp=(2)pn!(np)!.
I1=1τ40τJ0(z)z3dz=J1(τ)τ2J2(τ)τ2,I2=1τ60τJ0(z)z5dz=J1(τ)τ4J2(τ)τ2+8J3(τ)τ3,I3=1τ80τJ0(z)z7dz=J1(τ)τ6J2(τ)τ2+24J3(τ)τ348J4(τ)τ4,I4=1τ100τJ0(z)z9dz=J1(τ)τ8J2(τ)τ2+48J3(τ)τ3192J4(τ)τ4+384J5(τ)τ5,I5=1τ120τJ0(z)z11dz=J1(τ)τ10J2(τ)τ2+80J3(τ)τ3480J4(τ)τ4+1920J5(τ)τ53840J6(τ)τ6.
V1(τ)=k=1M+1(1)k+1UkJk(τ)τk,
U1=C1+C2+C3+C4+C5,U2=2C1+4C2+6C3+8C4+10C5,U3=8C2+24C3+48C4+80C5,U4=48C3+192C4+480C5,U5=384C4+1920C5,U6=3840C5.
V1(τ)=J0(τ)(U2τ24U3+U4τ4+24U4+12U5+U6τ6192U5+144U6τ8+1920U6τ10)+J1(τ)(U1τ2U2+U3τ3+8U3+8U4+U5τ548U4+72U5+18U6τ7+384U5+768U6τ93840U6τ11).
W2=(n=1MW2nr2n)2=n=12M1j=1nW2jW2(nj+1)r2n+2,
V2=201(n=12M1j=1nW2jW2(nj+1)r2n+3)J0(τr)dr.
V2=2n=12M1j=1nW2jW2(nj+1)In+1=n=12M1KnIn+1,
Kn=2j=1nW2jW2(nj+1),
V2(τ)=k=12M+1(1)k+1QkJk(τ)τk,
Q1=K1+K2+K3+K4+K5+K6+K7+K8+K9,Q2=4K1+6K2+8K3+10K4+12K5+14K6+16K7+18K8+20K9,Q3=8K1+24K2+48K3+80K4+120K5+168K6+224K7+288K8,Q4=48K2+192K3+480K4+960K5+1680K6+2688K7+4032K8+5760K9,Q5=384K3+1920K4+5760K5+13440K6+26880K7+48384K8+80640K9,Q6=3840K4+23040K5+80640K6+215040K7+483840K8+967680K9,Q7=46080K5+322560K6+1290240K7+3870720K8+9676800K9,Q8=645120K6+5160960K7+23224320K8+77414400K9,Q9=10321920K7+92897280K8+464486400K9,Q10=185794560K8+1857945600K9,Q11=3715891200K9.
V2(τ)=J0(τ)(Q2τ24Q3+Q4τ4+24Q4+12Q5+Q6τ6192Q5+144Q6+22Q7+Q8τ8+1920Q6+1632Q7+364Q8+32Q9+Q10τ1019200Q7+18240Q8+5272Q9+684Q10+42Q11τ12+192000Q8+201600Q9+70960Q10+12112Q11τ141920000Q9+2208000Q10+911200Q11τ16+19200000Q10+24000000Q11τ18+192000000Q11τ20)+J1(τ)(Q1τ2Q2+Q3τ3+8Q3+8Q4Q5τ548Q4+72Q5+18Q6+Q7τ7+384Q5+768Q6+252Q7+28Q8+Q9τ93840Q6+8064Q7+3288Q8+532Q9+38Q10+Q11τ11+38400Q7+84480Q8+40944Q9+8608Q10+912Q11τ13384000Q8+883200Q9+493920Q10+127024Q11τ15+3840000Q9+9216000Q10+5822400Q11τ1738400000Q10+96000000Q11τ19+3840000000Q11τ21).
Jn(τ)τn=12ns=0(1)ss!(n+s)!(τ2)2s,
W=W2r2+W4r4.
V1(τ)=[U1J1(τ)τU2J2(τ)τ2+U3J3(τ)τ3],
U1=C1+C2=2(W2+W4),U2=2C1+4C2=4(W2+2W4),U3=8C2=16W4,C1=2W2,C2=2W4
V2(τ)=[Q1J1(τ)τQ2J2(τ)τ2+Q3J3(τ)τ3Q4J4(τ)τ4+Q5J5(τ)τ5],
Q1=K1+K2+K3=2W22+4W2W4+2W42,Q2=4K1+6K2+8K3=8W22+24W2W4+16W42,Q3=8K1+24K2+48K3=16W22+96W2W4+96W42,Q4=48K2+192K3=192W2W4+384W42,Q5=384K3=768W42,K1=2W22,K2=4W2W4,K3=2W42.
PSF(τ)V02k2(V0V2V12),
SD=1k2(W2212+W2W46+4W4245).
W2=s08F2,W4=δsK16F2.

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