Abstract

A new derivation is presented for the analytical inversion of aerosol spectral extinction data to size distributions. It is based on the complex analytic extension of the anomalous diffraction approximation (ADA). We derive inverse formulas that are applicable to homogeneous nonabsorbing and absorbing spherical particles. Our method simplifies, generalizes, and unifies a number of results obtained previously in the literature. In particular, we clarify the connection between the ADA transform and the Fourier and Laplace transforms. Also, the effect of the particle refractive-index dispersion on the inversion is examined. It is shown that, when Lorentz’s model is used for this dispersion, the continuous ADA inverse transform is mathematically well posed, whereas with a constant refractive index it is ill posed. Further, a condition is given, in terms of Lorentz parameters, for which the continuous inverse operator does not amplify the error.

© 2000 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1987), pp. 633–664.
  2. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  3. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).
  4. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  5. P. T. Walters, “Practical applications of inverting spectral turbidity data to provide aerosol size distributions,” Appl. Opt. 19, 2353–2365 (1980).
    [CrossRef] [PubMed]
  6. R. A. Dobbins, G. S. Jizmagian, “Optical scattering cross sections for polydispersions of dielectric spheres,” J. Opt. Soc. Am. 56, 1345–1350 (1966).
    [CrossRef]
  7. A. Tarantola, B. Valette, “Generalized nonlinear inverse problems solved using the least squares criterion,” Rev. Geophys. Space Phys. 20, 219–232 (1982).
    [CrossRef]
  8. D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
    [CrossRef]
  9. S. Twomey, “Comparison of constrained linear inversion and an iterative non-linear algorithm applied to the indirect estimation of particle size distributions,” J. Comput. Phys. 18, 188–200 (1975).
    [CrossRef]
  10. K. S. Shifrin, A. Y. Perelman, “The determination of the spectrum of particles in a dispersed system from data on its transparency. I. The fundamental equation for the determination of the spectrum of the particles,” Opt. Spectrosc. (USSR) 15, 285–289 (1963).
  11. A. L. Fymat, “Analytical inversions in remote sensing of particle size distributions. 1. Multispectral extinctions in the anomalous diffraction approximation,” Appl. Opt. 17, 1675–1676 (1978).
  12. M. A. Box, B. H. McKellar, “Analytic inversion of multispectral extinction data in the anomalous diffraction approximation,” Opt. Lett. 3, 91–93 (1978).
    [CrossRef] [PubMed]
  13. A. L. Fymat, C. B. Smith, “Analytical inversions in remote sensing of particle size distributions. 4. Comparison of Fymat and Box–McKellar solutions in the anomalous diffraction approximation,” Appl. Opt. 18, 3595–3598 (1979).
    [CrossRef] [PubMed]
  14. M. A. Box, B. H. McKellar, “Relationship between two analytic inversion formulae for multispectral extinction data,” Appl. Opt. 18, 3599–3601 (1979).
    [CrossRef] [PubMed]
  15. M. A. Box, B. H. McKellar, “Further relations between analytic inversion formulas for multispectral extinction data,” Appl. Opt. 20, 3829–3831 (1981).
    [CrossRef] [PubMed]
  16. K. S. Shifrin, A. Y. Perelman, V. M. Volgin, “Calculations of particle-radius distribution density from the integral characteristics of the spectral attenuation coefficient,” Opt. Spectrosc. (USSR) 51, 534–538 (1981).
  17. J. Wang, F. R. Hallett, “Spherical particle size determination by analytical inversion of the UV–visible–NIR extinction spectrum,” Appl. Opt. 35, 193–197 (1996).
    [CrossRef] [PubMed]
  18. E. O. Brigham, The Fast Fourier Transform and Applications (Prentice-Hall, Englewood Cliffs, N.J., 1988).
  19. K. S. Shifrin, G. Tonna, “Inverse problems related to light scattering in the atmosphere and ocean,” Adv. Geophys. 34, 175–252 (1993).
    [CrossRef]
  20. C. B. Smith, “Inversion of the anomalous diffraction approximation for variable complex index of refraction near unity,” Appl. Opt. 21, 3363–3366 (1982).
    [CrossRef] [PubMed]
  21. J. D. Klett, “Anomalous diffraction model for inversion of multispectral extinction data including absorption effects,” Appl. Opt. 23, 4499–4508 (1984).
    [CrossRef] [PubMed]
  22. G. Viera, M. A. Box, “Information content analysis of aerosol remote-sensing experiments using an analytic eigenfunction theory: anomalous diffraction theory,” Appl. Opt. 24, 4525–4533 (1985).
    [CrossRef]
  23. M. Bertero, C. De Mol, E. R. Pike, “Particle size distributions from spectral turbidity: a singular-system analysis,” Inverse Probl. 2, 247–258 (1986).
    [CrossRef]
  24. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).
  25. R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1999).
  26. B. Davies, Integral Transforms and Their Applications (Springer-Verlag, Berlin, 1978).
    [CrossRef]
  27. M. A. Box, B. H. McKellar, “Determination of moments of the size distribution function in scattering by polydispersions,” Appl. Opt. 15, 2610 (1976).
    [CrossRef] [PubMed]
  28. A. L. Fymat, “Determination of moments of the size distribution function in scattering by polydispersions: a comment,” Appl. Opt. 17, 3516–3517 (1978).
    [CrossRef] [PubMed]
  29. K. Aki, P. Richards, Quantitative Seismology (Freeman, San Francisco, Calif., 1980), Vol. I, pp. 170–177.
  30. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  31. I. Gradshteyn, I. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).

1996 (1)

1993 (1)

K. S. Shifrin, G. Tonna, “Inverse problems related to light scattering in the atmosphere and ocean,” Adv. Geophys. 34, 175–252 (1993).
[CrossRef]

1986 (1)

M. Bertero, C. De Mol, E. R. Pike, “Particle size distributions from spectral turbidity: a singular-system analysis,” Inverse Probl. 2, 247–258 (1986).
[CrossRef]

1985 (1)

1984 (1)

1982 (2)

C. B. Smith, “Inversion of the anomalous diffraction approximation for variable complex index of refraction near unity,” Appl. Opt. 21, 3363–3366 (1982).
[CrossRef] [PubMed]

A. Tarantola, B. Valette, “Generalized nonlinear inverse problems solved using the least squares criterion,” Rev. Geophys. Space Phys. 20, 219–232 (1982).
[CrossRef]

1981 (2)

M. A. Box, B. H. McKellar, “Further relations between analytic inversion formulas for multispectral extinction data,” Appl. Opt. 20, 3829–3831 (1981).
[CrossRef] [PubMed]

K. S. Shifrin, A. Y. Perelman, V. M. Volgin, “Calculations of particle-radius distribution density from the integral characteristics of the spectral attenuation coefficient,” Opt. Spectrosc. (USSR) 51, 534–538 (1981).

1980 (1)

1979 (2)

1978 (3)

1976 (1)

1975 (1)

S. Twomey, “Comparison of constrained linear inversion and an iterative non-linear algorithm applied to the indirect estimation of particle size distributions,” J. Comput. Phys. 18, 188–200 (1975).
[CrossRef]

1966 (1)

1963 (1)

K. S. Shifrin, A. Y. Perelman, “The determination of the spectrum of particles in a dispersed system from data on its transparency. I. The fundamental equation for the determination of the spectrum of the particles,” Opt. Spectrosc. (USSR) 15, 285–289 (1963).

1962 (1)

D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
[CrossRef]

Aki, K.

K. Aki, P. Richards, Quantitative Seismology (Freeman, San Francisco, Calif., 1980), Vol. I, pp. 170–177.

Bertero, M.

M. Bertero, C. De Mol, E. R. Pike, “Particle size distributions from spectral turbidity: a singular-system analysis,” Inverse Probl. 2, 247–258 (1986).
[CrossRef]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1987), pp. 633–664.

Box, M. A.

Bracewell, R.

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1999).

Brigham, E. O.

E. O. Brigham, The Fast Fourier Transform and Applications (Prentice-Hall, Englewood Cliffs, N.J., 1988).

Davies, B.

B. Davies, Integral Transforms and Their Applications (Springer-Verlag, Berlin, 1978).
[CrossRef]

De Mol, C.

M. Bertero, C. De Mol, E. R. Pike, “Particle size distributions from spectral turbidity: a singular-system analysis,” Inverse Probl. 2, 247–258 (1986).
[CrossRef]

Deirmendjian, D.

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).

Dobbins, R. A.

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Fymat, A. L.

Gradshteyn, I.

I. Gradshteyn, I. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).

Hallett, F. R.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Jizmagian, G. S.

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

Klett, J. D.

McKellar, B. H.

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Perelman, A. Y.

K. S. Shifrin, A. Y. Perelman, V. M. Volgin, “Calculations of particle-radius distribution density from the integral characteristics of the spectral attenuation coefficient,” Opt. Spectrosc. (USSR) 51, 534–538 (1981).

K. S. Shifrin, A. Y. Perelman, “The determination of the spectrum of particles in a dispersed system from data on its transparency. I. The fundamental equation for the determination of the spectrum of the particles,” Opt. Spectrosc. (USSR) 15, 285–289 (1963).

Phillips, D. L.

D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
[CrossRef]

Pike, E. R.

M. Bertero, C. De Mol, E. R. Pike, “Particle size distributions from spectral turbidity: a singular-system analysis,” Inverse Probl. 2, 247–258 (1986).
[CrossRef]

Richards, P.

K. Aki, P. Richards, Quantitative Seismology (Freeman, San Francisco, Calif., 1980), Vol. I, pp. 170–177.

Ryzhik, I.

I. Gradshteyn, I. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).

Shifrin, K. S.

K. S. Shifrin, G. Tonna, “Inverse problems related to light scattering in the atmosphere and ocean,” Adv. Geophys. 34, 175–252 (1993).
[CrossRef]

K. S. Shifrin, A. Y. Perelman, V. M. Volgin, “Calculations of particle-radius distribution density from the integral characteristics of the spectral attenuation coefficient,” Opt. Spectrosc. (USSR) 51, 534–538 (1981).

K. S. Shifrin, A. Y. Perelman, “The determination of the spectrum of particles in a dispersed system from data on its transparency. I. The fundamental equation for the determination of the spectrum of the particles,” Opt. Spectrosc. (USSR) 15, 285–289 (1963).

Smith, C. B.

Tarantola, A.

A. Tarantola, B. Valette, “Generalized nonlinear inverse problems solved using the least squares criterion,” Rev. Geophys. Space Phys. 20, 219–232 (1982).
[CrossRef]

Tonna, G.

K. S. Shifrin, G. Tonna, “Inverse problems related to light scattering in the atmosphere and ocean,” Adv. Geophys. 34, 175–252 (1993).
[CrossRef]

Twomey, S.

S. Twomey, “Comparison of constrained linear inversion and an iterative non-linear algorithm applied to the indirect estimation of particle size distributions,” J. Comput. Phys. 18, 188–200 (1975).
[CrossRef]

Valette, B.

A. Tarantola, B. Valette, “Generalized nonlinear inverse problems solved using the least squares criterion,” Rev. Geophys. Space Phys. 20, 219–232 (1982).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

Viera, G.

Volgin, V. M.

K. S. Shifrin, A. Y. Perelman, V. M. Volgin, “Calculations of particle-radius distribution density from the integral characteristics of the spectral attenuation coefficient,” Opt. Spectrosc. (USSR) 51, 534–538 (1981).

Walters, P. T.

Wang, J.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1987), pp. 633–664.

Adv. Geophys. (1)

K. S. Shifrin, G. Tonna, “Inverse problems related to light scattering in the atmosphere and ocean,” Adv. Geophys. 34, 175–252 (1993).
[CrossRef]

Appl. Opt. (11)

M. A. Box, B. H. McKellar, “Determination of moments of the size distribution function in scattering by polydispersions,” Appl. Opt. 15, 2610 (1976).
[CrossRef] [PubMed]

A. L. Fymat, “Determination of moments of the size distribution function in scattering by polydispersions: a comment,” Appl. Opt. 17, 3516–3517 (1978).
[CrossRef] [PubMed]

A. L. Fymat, C. B. Smith, “Analytical inversions in remote sensing of particle size distributions. 4. Comparison of Fymat and Box–McKellar solutions in the anomalous diffraction approximation,” Appl. Opt. 18, 3595–3598 (1979).
[CrossRef] [PubMed]

M. A. Box, B. H. McKellar, “Relationship between two analytic inversion formulae for multispectral extinction data,” Appl. Opt. 18, 3599–3601 (1979).
[CrossRef] [PubMed]

P. T. Walters, “Practical applications of inverting spectral turbidity data to provide aerosol size distributions,” Appl. Opt. 19, 2353–2365 (1980).
[CrossRef] [PubMed]

M. A. Box, B. H. McKellar, “Further relations between analytic inversion formulas for multispectral extinction data,” Appl. Opt. 20, 3829–3831 (1981).
[CrossRef] [PubMed]

C. B. Smith, “Inversion of the anomalous diffraction approximation for variable complex index of refraction near unity,” Appl. Opt. 21, 3363–3366 (1982).
[CrossRef] [PubMed]

J. D. Klett, “Anomalous diffraction model for inversion of multispectral extinction data including absorption effects,” Appl. Opt. 23, 4499–4508 (1984).
[CrossRef] [PubMed]

G. Viera, M. A. Box, “Information content analysis of aerosol remote-sensing experiments using an analytic eigenfunction theory: anomalous diffraction theory,” Appl. Opt. 24, 4525–4533 (1985).
[CrossRef]

J. Wang, F. R. Hallett, “Spherical particle size determination by analytical inversion of the UV–visible–NIR extinction spectrum,” Appl. Opt. 35, 193–197 (1996).
[CrossRef] [PubMed]

A. L. Fymat, “Analytical inversions in remote sensing of particle size distributions. 1. Multispectral extinctions in the anomalous diffraction approximation,” Appl. Opt. 17, 1675–1676 (1978).

Inverse Probl. (1)

M. Bertero, C. De Mol, E. R. Pike, “Particle size distributions from spectral turbidity: a singular-system analysis,” Inverse Probl. 2, 247–258 (1986).
[CrossRef]

J. Assoc. Comput. Mach. (1)

D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
[CrossRef]

J. Comput. Phys. (1)

S. Twomey, “Comparison of constrained linear inversion and an iterative non-linear algorithm applied to the indirect estimation of particle size distributions,” J. Comput. Phys. 18, 188–200 (1975).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Lett. (1)

Opt. Spectrosc. (USSR) (2)

K. S. Shifrin, A. Y. Perelman, “The determination of the spectrum of particles in a dispersed system from data on its transparency. I. The fundamental equation for the determination of the spectrum of the particles,” Opt. Spectrosc. (USSR) 15, 285–289 (1963).

K. S. Shifrin, A. Y. Perelman, V. M. Volgin, “Calculations of particle-radius distribution density from the integral characteristics of the spectral attenuation coefficient,” Opt. Spectrosc. (USSR) 51, 534–538 (1981).

Rev. Geophys. Space Phys. (1)

A. Tarantola, B. Valette, “Generalized nonlinear inverse problems solved using the least squares criterion,” Rev. Geophys. Space Phys. 20, 219–232 (1982).
[CrossRef]

Other (11)

K. Aki, P. Richards, Quantitative Seismology (Freeman, San Francisco, Calif., 1980), Vol. I, pp. 170–177.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

I. Gradshteyn, I. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1999).

B. Davies, Integral Transforms and Their Applications (Springer-Verlag, Berlin, 1978).
[CrossRef]

E. O. Brigham, The Fast Fourier Transform and Applications (Prentice-Hall, Englewood Cliffs, N.J., 1988).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1987), pp. 633–664.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

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Equations (134)

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Qcκr=2-4i exp-iκrκr+4 1-exp-iκrκr2.
2igκ=0+πr2QcκrNrdr.
Fκ=0+ Kκ, rPrdr,
Fκ  κEκ,
Eκ  gκ+iA
=aκ-ibκ-A,
Kκ, r  κri42-Qcκr=-exp-iκr+1-exp-iκriκr
=iκrddκr1-exp-iκrκr
=i 1κr0κr τ exp-iτdτ
=- m=1+mm+1-iκrmm!
lim|κ|+0φπ Fκ=0.
limk+ κkβk-A=0.
lim|κ|+0φπ Eκ=0.
β  limk+ βk=A=0+πr2Nrdr.
limκ0 Eκ=iA.
β0  limk0 βk=0.
limκ0 Fκ=0.
Eκ=aκ-ibκ-A=2πi m=1+ m-iκm-10+rm+1m+1! Nrdr,
aκ=κ 23 π 0+ r3Nrdr+Oκ3,
bκ=κ214 π 0+ r4Nrdr+Oκ4.
Kκ, r  Krκ, r-iKiκ, r=-cosκr+sinκrκr-i-sinκr+1-cosκrκr.
0+ Krκ, rKrκ, rdκ=π2 δr-r,
0+ Kiκ, rKiκ, rdκ=π2 δr-r.
0+Kκ, rK*κ, r+K*κ, rKκ, r×dκ=2πδr-r.
Pr=12π0+Kκ, rF*κ+K*κ, rFκdκ.
Pr=2π0+ Krκ, rκaκdκ,
Pr=2π0+ Kiκ, rκbκ-Adκ.
Pr=2π0+ Kiκ, rκbκ-Adκ, =2π0+ Kiκk, rκkβk-Adκkdkdk, =-1π0+ κkiκ ddκ1-exp-iκrκrdκkdk×2βk-2βdk, =-1π0+ κ2kddk1-cosκkrκkr×2βk-2βdk
Nr=-12π21r0+ κ2kddk1-cosκkrκkr×τk-τdk,
Frκ=2π0+0+ Krκ, rKiκ, rdrFiκdκ.
0+ Krκ, rKiκ, rdr=121κ+κ-1κ-κ,
Frκ=1π0+1κ+κ-1κ-κFiκdκ
Frκ=12π-+1κ+κ-1κ-κFiκdκ.
aκ=2π1κ0+κ2κ2-κ2bκ-Adκ, =2π1κ0+bκ-Adκ+2π κ 0+1κ2-κ2×bκ-Adκ, =1π0+κκ2-κ22bκ-2Adκ.
Tκhκ  κ-1ddκκhκ,
Tr-1fr=r-10r τfτdτ.
-iTκKκ, r=exp-iκrr.
-iTκFκ=0+exp-iκrrPrdr.
rPr=12πiσ-iσ+i TpF-ipexp+prdp, σ0.
12π -i2TκFκ=12π-+exp-iκrrPrdr
12π i -i2TκFκ=12π-+exp-iκrrPrdr.
rPr=12π-+12π -i2TκFκ×exp+iκrdκ,  P-r=-Pr,
rPr=12π-+12π i-i2TκFκ×exp+iκrdκ,  P-r=Pr.
rPr=12πi-+ TκFκexp+iκrdκ,  r0,
Fˆp  F-ip-2πNtot/p.
rPr=12πiσ-iσ+i TpFˆpexp+prdp,  σ0.
drPrdr=12πiσ-iσ+idpFˆpdpexp+prdp=12πi pFˆpexp+prp=σ-ip=σ+i-r 12πiσ-iσ+i pFˆpexp+prdp.
-TrPr=12πiσ-iσ+i pFˆpexp+prdp,  σ0.
Pr=-12πiσ-iσ+i pFˆpTr-1exp+prdp,  σ0=-12πiσ-iσ+i pFˆpr-10r τ exp+pτdτdp=12πiσ-iσ+i Fˆp1pr0ipr τ exp-iτdτdp=12πiσ-iσ+i FˆpKip, rdp.
Pr=12πiσ-iσ+iF-ip-2πNtot/pKip, rdp=12πiσ-iσ+i Kip, rF-ipdp-2πNtot12πiσ-iσ+i p-1Kip, rdp=12πiσ-iσ+i Kip, rF-ipdp-2πNtot12πiσ-iσ+iddp1-exp+prprdp=12πiσ-iσ+i Kip, rF-ipdp-2πNtot12πi1-exp+prprp=σ-ip=σ+i=12πiσ-iσ+i Kip, rF-ipdp,
Pr=12π--iσ+-iσ K-κ, rFκdκ,  σ0, r0.
Δn-k=Δn*k.
κ-k=-κ*k.
Kκ, r=K*-κ*, r.
Kκ-k, r=K*-κ*-k, r=K*κk, r.
Fκ-k=F*-κ*-k=F*κk,
Eκ-k=-E*-κ*-k=-E*κk.
Pr=12π--iσ+-iσ K-κ, rFκdκ=12πΓ  κ=κk, -<k<+ K-κ, rFκdκ=12π-+ κkK-κk, rdκkdk Eκkdk.
αk=-1π P -+βk-βk-kdk.
Hk, r  κkKκk, rdκkdk.
H-k, r=-H*k, r.
Pr=-12π-+1π P -+1k-k Hk, -rdk×βk-βdk-i 12π-+ Hk, -r×βk-βdk.
1π P -+1k-k Hk, -rdk=iHk, -r.
Pr=-iπ-+ Hk, -rβk-βdk=2π0+ Hk, -rβk-βdk.
Nr=12π21r0+ K-κk, rdκ2kdkτk-τdk.
2πNr=12π--iσ+-iσ κ2K-κ, rκr Eκdκ.
2π 0r Nrdr=12πi--iσ+-iσ 1-exp+iκrr+iκ×Eκdκ.
2πr 0r Nrdr-12πi--iσ+-iσ Eκdκ-r2π--iσ+-iσ×Fκdκ=-12πi--iσ+-iσexp+iκrEκdκ.
2πr0r Nrdr-Ntot/2=12πiσ-iσ+iexp+priE-ipdp.
Eκ=2πi 0+ddκ1-exp-iκrκNrdr.
0κ Eκdκ=2πi 0+1-exp-iκrκ-irNrdr.
p2πi0p E-ipdp-iPtot+Ntot=0+exp-prNrdr.
|Δλ|λ14dΔnλdλ-Δnλλrmax.
|Δλ|λλ4|Δn|maxrmax.
12π K-κk, rdκ2kdk12π K-κk, rdκ2kdk<dκ2kdk.
κk=2nk-1kn2k-1k.
n2k-1=K2p=1Papkp2-k2+i2bpkpk,
κk=K2p=1P apkkp2-k2+i2bpkpk,
dκkdk=K2p=1P apkp2+k2kp2-k2+i2bpkpk2.
|nk-1|12 |n2k-1|12K2k12a1b1  1.
dκ2kdk12 K4a12b1k13.
dκ2kdk212K2k12a1b12k1b1  2 k1b1.
212K2k12a1b12k1b11.
dκ2kdk8|Δn|2k
Sr=-2π0+ κ2kddk1-cosκkrκkτk-τdk.
Sr=-2π0+ κ2ddκ1-cosκrκ×2bκ-limκ+ 2bκdκ.
Sr=-4π Δn 0+cos2Δnkr+2Δnkr sin2Δnkr×τk-τdk+4π Δn 0+τk-τdk,
Sr=-2π0+dκ22bκκdκ-limκ+dκ22bκκdκcosκrdκ.
Sr=-2π0+ κ2ddκ1-cosκrκ×2bκ-limκ+ 2bκdκ=-2π1-cosκrκ2bκ-limκ+ 2bκκ=0κ=++2π0+1-cosκrdκdκκ22bκ-limκ+ 2bκdκ.
Sr=+2π0+dκdκκ22bκ-limκ+ 2bκdκ-2π0+cosκrdκdκκ22bκ-limκ+ 2bκdκ.
0+dκdκκ22bκ-limκ+ 2bκdκ=κ2bκ-limκ+2bκ|κ=0κ=++0+2bκ-limκ+ 2bκdκ=0.
dκ2limκ+ 2bκκdκ=limκ+dκ22bκκdκ.
Sr=-2π0+dκ22bκκdκ-limκ+dκ22bκκdκcosκrdκ,
Sr=2π0+κr2-sinκrκr+1-cosκrκr2×2bκ-limκ+ 2bκdκ
Nr=1π2r2 Δn 0+1-cos2Δnkr-2Δnkr sin2Δnkrτk-τdk,
Pr=1π0+ κkK-κk, rdκkdkτk-τdk,
Pr=1π1r0+ iiκkr-1×exp+iκkrdκdkτk-τdk.
Pr=1π0+ iiκkiκkr-1iκkr×exp+iκkrdκdkτk-τdk=-1π0+ κkexp+iκkr+1-exp+iκkriκkr-1iκkrdκdkτk-τdk=-1π0+ κkexp+iκkr+1-exp+iκkriκkrdκdkτk-τdk-1π1r0+ i dκdkτk-τdk.
Pr=1π0+ κkK-κk, rdκkdkτk-τdk-1π1r0+ i dκdkτk-τdk.
-1π1r0+ i dκdkτk-τdk=-1π1r i 0+τk-τdκkdkdk,
limκ0 Eκ=12limκ0 0+Kκ, rκr Srdr=120+limκ0Kκ, rκr Srdr=i40+ Srdr=iStot/4=iA.
lim|κ|+0φπ Fκ=lim|κ|+0φπ 0+ Kκ, rPrdr=lim|κ|+0φπ 0+-exp-iκr+1-exp-iκriκrPrdr=lim|κ|+0φπ 0+-exp-i|κ|cos φ rexp-|κ|sin φ rPrdr+lim|κ|+0φπ 0+expiφ1-exp-i|κ|cos φ rexp-|κ|sin φ ri|κ|rPrdr=-lim|κ|+0φπ 0+exp-i|κ|cos φ rexp-|κ|sin φ rPrdr-iπ expiφ lim|κ|+0φπ1|κ|0+ Nrdr+iπ expiφ lim|κ|+0φπ1|κ|0+exp-i|κ|cos φ rexp-|κ|sin φ rNrdr=0,
0+ Eκdκ=0+0+1κ Kκ, rPrdrdκ=i 0+0+1κκrddκr1-exp-iκrκr×Prdrdκ=i 0+ Pr0+ddκ1-exp-iκrκrdκdr=i 0+ Pr1-exp-iκrκrκ=0κ=+dr=0+ Prdr=Ptot.
0+ αkdκdkdk=Ptot,
0+βk-βdκdkdk=0.
lim|κ|+0φπ κ dEκdκ=lim|κ|+0φπ κ ddκ0+1κ Kκ, rPrdr=lim|κ|+0φπ 0+-1κ Kκ, r+dKκ, rdκ×Prdr=lim|κ|+0φπ 0+ir exp-iκr+2 exp-κrκ-2 1-exp-iκriκ2rPrdr=0+lim|κ|+0φπ ir exp-iκr+2 exp-iκrκ-2 1-exp-iκriκ2rPrdr=i 0+lim|κ|+0φπexp-iκrrPrdr=0,
limpσ±i pFˆp=limpσ±i pF-ip-2πNtot=limpσ±i p 0+ K-ip, rPrdr-2πNtot=limpσ±i -p 0+exp-prPrdr+p 0+1-exp-prpr Prdr-2πNtot=limpσ±i -p 0+exp-prPrdr-2π 0+exp-prNrdr+2π 0+ Nrdr-2πNtot
limpσ±i pFˆp=limpσ±i 0+d exp-prdr Prdr-2π×0+exp-prNrdr=limpσ±i exp-prPrr=0r=+-0+exp-prdPrdrdr-2π 0+exp-prNrdr=limτ±exp-iτrexp-σrPrr=0r=+-limτ± 0+exp-iτrexp-σrdPrdr×dr-2π limτ± 0+exp-iτr×exp-σrNrdr=0.
12limr0- Nr+limr0+ Nr=-14π12πiσ-iσ+i pFˆpdp.
12 N0+=-14π12πiσ-iσ+ipF-ip-2πNtotdp=-18π2--iσ+-iσiκFκ-2πNtotdκ=-18π2Γ  κ=κk, -<k<+×iκFκ-2πNtotdκ=14π12πi-+κFκ+2πiNtotdκkdkdk.
--iσ+-iσ Eκdκ=--iσ+-iσ0+1κ Kκ, rPrdrdκ=--iσ+-iσ0+1κ iκrddκr×1-exp-iκrκrPrdrdκ=i --iσ+-iσ0+ddκ1-exp-iκrκr×Prdrdκ=i 0+--iσ+-iσddκ1-exp-iκrκr×dκPrdr=i 0+ Pr1-exp-iκrκrκ=--iσκ=+-iσdr=0,
--iσ+-iσ Fκdκ=--iσ+-iσ0+ Kκ, rPrdrdκ=i --iσ+-iσ0+ κ ddκ1-exp-iκrκr×Prdrdκ=i 0+--iσ+-iσ κ ddκ1-exp-iκrκr×dκPrdr=i 0+1-exp-iκrrκ=--iσκ=+-iσ Prdr-2πi 0+--iσ+-iσ1-exp-iκrκ×dκNrdr=0-2πi 0+-+1-cosκrκ+i sinκrκdκNrdr=2π 0+-+sinκrκdκNrdr=2π20+ Nrdr=2π2Ntot.
0+ Krκ, rKrκ, rdκ=0+cosκr-sinκrκr×cosκr-sinκrκrdκ=0+cosrκcosrκdκ+1rr0+sinrκsinrκκ2dκ-1r0+cosrκsinrκκdκ-1r0+cosrκsinrκκdκ.
0+cosrκcosrκdκ=π2 δr-r,  r>0, r>0, 0+sinrκsinrκκ2dκ=π2r0rrr0rr, 0+cosrκsinrκκdκ=ddr0+sinrκsinrκκ2dκ=π210r<r00r<r, 0+cosrκsinrκκdκ=ddr0+sinrκsinrκκ2dκ=π200r<r10r<r
0+ Krκ, rKrκ, rdκ=π2 δr-r+π21rrrr-π21r10-π21r01, 0r<r0r<r=π2 δr-r,  r>0,  r>0.
0+ Kiκ, rKiκ, rdκ=0+sinκr-1-cosκrκr×sinκr-1-cosκrκrdκ=2 0+sin2κr-sin2κrκr×sin2κr-sin2κrκrdκ=0+sin2rκsin2rκd2κ+2 1rr0+sin2rκsin2rκκ2×dκ-2r×0+sin2rκsin2rκκdκ-2r0+sin2rκsin2rκκdκ.
0+sinrκsinrκdκ=π2 δr-r,  r>0, r>0, 0+sin2rκsin2rκκ2dκ=π4r0rrr0rr, 0+sin2rκsin2rκκdκ=ddr0+sin2rκsin2rκκ2dκ=π410r<r00r<r, 0+sin2rκsin2rκκdκ=ddr0+sin2rκsin2rκκ2dκ=π400r<r10r<r,
0+ Kiκ, rKiκ, rdκ=π2 δr-r+π21rrrr-π21rr0-π21r01, 0r<r0r<r=π2 δr-r,  r>0, r>0.
0+ Krκ, rKiκ, rdr=0+cosκr-sinκrκrsinκr-1-cosκrκrdr=0+cosrκsinrκdr+1κκ0+sinrκ1-cosrκr2dr-1κ×0+cosrκ1-cosrκrdr-1κ×0+sinrκsinrκrdr.
0+cosrκsinrκdr=1212i0+expirκ+exp-irκ×expirκ-exp-irκdr=14i0+expiκ+κr-exp-iκ+κrdr+14i0+-expiκ-κr+exp-iκ-κrdr=140+ expiκ+κr+exp-iκ+κrdr+140+ -expiκ-κr+exp-iκ-κrdr=14  -+ urexpiκ+κr-exp-iκ+κrdr+14  -+ ur-expiκ-κr+exp-iκ-κrdr=14  1-iκ+κ-1iκ+κ-1-iκ-κ+1iκ-κ=12 1κ+κ-1κ-κ,
0+sinrκ1-cosrκr2dr=κ2ln|κ2-κ2|κ2+κ2lnκ+κ|κ-κ|, 0+cosrκ1-cosrκrdr=12ln|κ2-κ2|κ2, 0+sinrκsinrκrdr=12lnκ+κ|κ-κ|
0+ Krκ, rKiκ, rdr=121κ+κ-1κ-κ+1κκκ2ln|κ2-κ2|κ2+κ2lnκ+κ|κ-κ|-1κ12ln|κ2-κ2|κ2-1κ12lnκ+κ|κ-κ|=121κ+κ-1κ-κ.
12π--iσ+-iσ K-κ, rKκ, rdκ=12π--iσ+-iσ-exp+iκr+i 1-exp+iκrκr×-exp-iκr-i 1-exp-iκrκrdκ=12π--iσ+-iσexp+iκr-rdκ+12π--iσ+-iσexp-iκr1-exp+iκriκrdκ-12π--iσ+-iσ1-exp-iκriκrexp+iκrdκ-12π--iσ+-iσ1-exp-iκriκr1-exp+iκriκrdκ=12πiσ-iσ+iexp+pr-rdp+12πiσ-iσ+iexp-pr1-exp+prprdp-12πiσ-iσ+i1-exp-prprexp+prdp-12πiσ-iσ+i1-exp-prpr1-exp+prprdp=δr-r+1ru-r-ur-r-1rur-ur-r-1rr0-rur+ru-r+r-rur-r=δr-r+1ru-r-ur-r-u-r+ur-r-1rur-ur-r-ur+ur-r=δr-r,
0+ Kκ, rKκ, rdr=0+-exp-iκr+1-exp-iκriκr×-exp-iκr+1-exp-iκriκrdr=0+exp-p+prdr-1p0+exp-pr×1-exp-prrdr-1p0+1-exp-prr×exp-prdr+1p1p×0+1-exp-prr1-exp-prrdr,
0+1-exp-prr1-exp-prrexp-μrdr=p+p+μlnp+p+μ-p+μlnp+μ-p+μlnp+μ+μ ln μ.
0+1-exp-prr1-exp-prrdr=p+plnp+p-p ln p-p ln p.
0+1-exp-prexp-μrdrr2=- p+μlnp+μ+μ ln μ.
0+1-exp-prrexp-μrdr=lnp+μ-ln μ.
0+exp-prdr=1/p.
0+exp-p+prdr=1p+p, -1p0+exp-pr1-exp-prrdr=-1p×lnp+p-ln p, -1p0+1-exp-prrexp-prdr=-1p×lnp+p-ln p, 1p1p0+1-exp-prr1-exp-prrdr=1p1p× p+plnp+p-p ln p-p ln p.
0+ Kκ, rKκ, rdr=1i1κ+κ, κ<0, κ<0.
12π0+ Kκ, rK-κ, rdr=12πi1κ-κ, κ<0<κ.
12π0+ Kκ, rK-κ, rdr=12πi1κ-κ,
12π--iσ+-iσ F-κFκdκ=12π--iσ+-iσ0+ K-κ, r×Prdr×0+ Kκ, rPrdrdκ=0+ Prdr 0+ Prdr 12π--iσ+-iσ×K-κ, rKκ, rdκ=0+0+ PrPrδr-rdrdr=0+Pr2dr.

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