Abstract

We present a new linear inversion formalism for the scalar inverse source problem in three-dimensional and one-dimensional (1D) spaces, from which a number of previously unknown results on minimum-energy (ME) sources and their fields readily follow. ME sources, of specified support, are shown to obey a homogeneous Helmholtz equation in the interior of that support. As a consequence of that result, the fields produced by ME sources are shown to obey an iterated homogeneous Helmholtz equation. By solving the latter equation, we arrive at a new Green-function representation of the field produced by a ME source. It is also shown that any square-integrable (L2), compactly supported source that possesses a continuous normal derivative on the boundary of its support must possess a nonradiating (NR) component. A procedure based on our results on the inverse source problem and ME sources is described to uniquely decompose an L2 source of specified support and its field into the sum of a radiating and a NR part. The general theory that is developed is illustrated for the special cases of a homogeneous source in 1D space and a spherically symmetric source.

© 2000 Optical Society of America

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References

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  1. A. J. Devaney, E. Wolf, “Radiating and nonradiating classical current distributions and the fields they generate,” Phys. Rev. D 8, 1044–1047 (1973).
    [CrossRef]
  2. N. Bleistein, J. K. Cohen, “Nonuniqueness in the inverse source problem in acoustics and electromagnetics,” J. Math. Phys. 18, 194–201 (1977).
    [CrossRef]
  3. A. J. Devaney, R. P. Porter, “Holography and the inverse source problem. Part II: inhomogeneous media,” J. Opt. Soc. Am. A 2, 2006–2011 (1985).
    [CrossRef]
  4. R. P. Porter, A. J. Devaney, “Holography and the inverse source problem,” J. Opt. Soc. Am. 72, 327–330 (1982).
    [CrossRef]
  5. E. A. Marengo, A. J. Devaney, “The inverse source problem of electromagnetics: linear inversion formulation and minimum energy solution,” IEEE Trans. Antennas Propag. 47, 410–412 (1999).
    [CrossRef]
  6. M. Berry, J. T. Foley, G. Gbur, E. Wolf, “Nonpropagating string excitations,” Am. J. Phys. 66, 121–123 (1998).
    [CrossRef]
  7. K. Kim, E. Wolf, “Non-radiating monochromatic sources and their fields,” Opt. Commun. 59, 1–6 (1986).
    [CrossRef]
  8. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
  9. M. Bertero, C. De Mol, E. R. Pike, “Linear inverse problems with discrete data. I: General formulation and singular system analysis,” Inverse Probl. 1, 301–330 (1985).
    [CrossRef]
  10. M. Bertero, C. De Mol, E. R. Pike, “Linear inverse problems with discrete data. II: Stability and regularisation,” Inverse Probl. 4, 573–594 (1988).
    [CrossRef]
  11. K. J. Langenberg, “Applied inverse problems for acoustic, electromagnetic, and elastic wave scattering,” in Basic Methods of Tomography and Inverse Problems, P. C. Sabatier, ed. (Institute of Physics, Bristol, UK, 1987), pp. 127–453.
  12. M. Bertero, “Linear inverse and ill-posed problems,” in Advances in Electronics and Electron Physics, P. W. Hawkes, ed. (Academic, San Diego, Calif.1989), Vol. 75, pp. 1–120.
  13. J. Weidmann, Linear Operators in Hilbert Spaces (Springer-Verlag, New York, 1980).
  14. C. Müller, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer-Verlag, Berlin, 1969).
  15. G. Arfken, Mathematical Methods for Physicists (Academic, San Diego, Calif.1985).
  16. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).
  17. J. B. Marion, Classical Electromagnetic Radiation (Academic, New York, 1968).
  18. G. S. Agarwal, A. J. Devaney, D. N. Pattanayak, “On the solution of the partial differential equation Πj=1n(∇2+kj2)ψ=0,” J. Math. Phys. 14, 906–908 (1973).
    [CrossRef]
  19. A. Gamliel, K. Kim, A. I. Nachman, E. Wolf, “A new method for specifying nonradiating monochromatic, scalar sources and their fields,” J. Opt. Soc. Am. A 6, 1388–1393 (1989).
    [CrossRef]

1999 (1)

E. A. Marengo, A. J. Devaney, “The inverse source problem of electromagnetics: linear inversion formulation and minimum energy solution,” IEEE Trans. Antennas Propag. 47, 410–412 (1999).
[CrossRef]

1998 (1)

M. Berry, J. T. Foley, G. Gbur, E. Wolf, “Nonpropagating string excitations,” Am. J. Phys. 66, 121–123 (1998).
[CrossRef]

1989 (1)

1988 (1)

M. Bertero, C. De Mol, E. R. Pike, “Linear inverse problems with discrete data. II: Stability and regularisation,” Inverse Probl. 4, 573–594 (1988).
[CrossRef]

1986 (1)

K. Kim, E. Wolf, “Non-radiating monochromatic sources and their fields,” Opt. Commun. 59, 1–6 (1986).
[CrossRef]

1985 (2)

M. Bertero, C. De Mol, E. R. Pike, “Linear inverse problems with discrete data. I: General formulation and singular system analysis,” Inverse Probl. 1, 301–330 (1985).
[CrossRef]

A. J. Devaney, R. P. Porter, “Holography and the inverse source problem. Part II: inhomogeneous media,” J. Opt. Soc. Am. A 2, 2006–2011 (1985).
[CrossRef]

1982 (1)

1977 (1)

N. Bleistein, J. K. Cohen, “Nonuniqueness in the inverse source problem in acoustics and electromagnetics,” J. Math. Phys. 18, 194–201 (1977).
[CrossRef]

1973 (2)

A. J. Devaney, E. Wolf, “Radiating and nonradiating classical current distributions and the fields they generate,” Phys. Rev. D 8, 1044–1047 (1973).
[CrossRef]

G. S. Agarwal, A. J. Devaney, D. N. Pattanayak, “On the solution of the partial differential equation Πj=1n(∇2+kj2)ψ=0,” J. Math. Phys. 14, 906–908 (1973).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal, A. J. Devaney, D. N. Pattanayak, “On the solution of the partial differential equation Πj=1n(∇2+kj2)ψ=0,” J. Math. Phys. 14, 906–908 (1973).
[CrossRef]

Arfken, G.

G. Arfken, Mathematical Methods for Physicists (Academic, San Diego, Calif.1985).

Berry, M.

M. Berry, J. T. Foley, G. Gbur, E. Wolf, “Nonpropagating string excitations,” Am. J. Phys. 66, 121–123 (1998).
[CrossRef]

Bertero, M.

M. Bertero, C. De Mol, E. R. Pike, “Linear inverse problems with discrete data. II: Stability and regularisation,” Inverse Probl. 4, 573–594 (1988).
[CrossRef]

M. Bertero, C. De Mol, E. R. Pike, “Linear inverse problems with discrete data. I: General formulation and singular system analysis,” Inverse Probl. 1, 301–330 (1985).
[CrossRef]

M. Bertero, “Linear inverse and ill-posed problems,” in Advances in Electronics and Electron Physics, P. W. Hawkes, ed. (Academic, San Diego, Calif.1989), Vol. 75, pp. 1–120.

Bleistein, N.

N. Bleistein, J. K. Cohen, “Nonuniqueness in the inverse source problem in acoustics and electromagnetics,” J. Math. Phys. 18, 194–201 (1977).
[CrossRef]

Cohen, J. K.

N. Bleistein, J. K. Cohen, “Nonuniqueness in the inverse source problem in acoustics and electromagnetics,” J. Math. Phys. 18, 194–201 (1977).
[CrossRef]

De Mol, C.

M. Bertero, C. De Mol, E. R. Pike, “Linear inverse problems with discrete data. II: Stability and regularisation,” Inverse Probl. 4, 573–594 (1988).
[CrossRef]

M. Bertero, C. De Mol, E. R. Pike, “Linear inverse problems with discrete data. I: General formulation and singular system analysis,” Inverse Probl. 1, 301–330 (1985).
[CrossRef]

Devaney, A. J.

E. A. Marengo, A. J. Devaney, “The inverse source problem of electromagnetics: linear inversion formulation and minimum energy solution,” IEEE Trans. Antennas Propag. 47, 410–412 (1999).
[CrossRef]

A. J. Devaney, R. P. Porter, “Holography and the inverse source problem. Part II: inhomogeneous media,” J. Opt. Soc. Am. A 2, 2006–2011 (1985).
[CrossRef]

R. P. Porter, A. J. Devaney, “Holography and the inverse source problem,” J. Opt. Soc. Am. 72, 327–330 (1982).
[CrossRef]

A. J. Devaney, E. Wolf, “Radiating and nonradiating classical current distributions and the fields they generate,” Phys. Rev. D 8, 1044–1047 (1973).
[CrossRef]

G. S. Agarwal, A. J. Devaney, D. N. Pattanayak, “On the solution of the partial differential equation Πj=1n(∇2+kj2)ψ=0,” J. Math. Phys. 14, 906–908 (1973).
[CrossRef]

Feshbach, H.

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

Foley, J. T.

M. Berry, J. T. Foley, G. Gbur, E. Wolf, “Nonpropagating string excitations,” Am. J. Phys. 66, 121–123 (1998).
[CrossRef]

Gamliel, A.

Gbur, G.

M. Berry, J. T. Foley, G. Gbur, E. Wolf, “Nonpropagating string excitations,” Am. J. Phys. 66, 121–123 (1998).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

Kim, K.

Langenberg, K. J.

K. J. Langenberg, “Applied inverse problems for acoustic, electromagnetic, and elastic wave scattering,” in Basic Methods of Tomography and Inverse Problems, P. C. Sabatier, ed. (Institute of Physics, Bristol, UK, 1987), pp. 127–453.

Marengo, E. A.

E. A. Marengo, A. J. Devaney, “The inverse source problem of electromagnetics: linear inversion formulation and minimum energy solution,” IEEE Trans. Antennas Propag. 47, 410–412 (1999).
[CrossRef]

Marion, J. B.

J. B. Marion, Classical Electromagnetic Radiation (Academic, New York, 1968).

Morse, P. M.

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

Müller, C.

C. Müller, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer-Verlag, Berlin, 1969).

Nachman, A. I.

Pattanayak, D. N.

G. S. Agarwal, A. J. Devaney, D. N. Pattanayak, “On the solution of the partial differential equation Πj=1n(∇2+kj2)ψ=0,” J. Math. Phys. 14, 906–908 (1973).
[CrossRef]

Pike, E. R.

M. Bertero, C. De Mol, E. R. Pike, “Linear inverse problems with discrete data. II: Stability and regularisation,” Inverse Probl. 4, 573–594 (1988).
[CrossRef]

M. Bertero, C. De Mol, E. R. Pike, “Linear inverse problems with discrete data. I: General formulation and singular system analysis,” Inverse Probl. 1, 301–330 (1985).
[CrossRef]

Porter, R. P.

Weidmann, J.

J. Weidmann, Linear Operators in Hilbert Spaces (Springer-Verlag, New York, 1980).

Wolf, E.

M. Berry, J. T. Foley, G. Gbur, E. Wolf, “Nonpropagating string excitations,” Am. J. Phys. 66, 121–123 (1998).
[CrossRef]

A. Gamliel, K. Kim, A. I. Nachman, E. Wolf, “A new method for specifying nonradiating monochromatic, scalar sources and their fields,” J. Opt. Soc. Am. A 6, 1388–1393 (1989).
[CrossRef]

K. Kim, E. Wolf, “Non-radiating monochromatic sources and their fields,” Opt. Commun. 59, 1–6 (1986).
[CrossRef]

A. J. Devaney, E. Wolf, “Radiating and nonradiating classical current distributions and the fields they generate,” Phys. Rev. D 8, 1044–1047 (1973).
[CrossRef]

Am. J. Phys. (1)

M. Berry, J. T. Foley, G. Gbur, E. Wolf, “Nonpropagating string excitations,” Am. J. Phys. 66, 121–123 (1998).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

E. A. Marengo, A. J. Devaney, “The inverse source problem of electromagnetics: linear inversion formulation and minimum energy solution,” IEEE Trans. Antennas Propag. 47, 410–412 (1999).
[CrossRef]

Inverse Probl. (2)

M. Bertero, C. De Mol, E. R. Pike, “Linear inverse problems with discrete data. I: General formulation and singular system analysis,” Inverse Probl. 1, 301–330 (1985).
[CrossRef]

M. Bertero, C. De Mol, E. R. Pike, “Linear inverse problems with discrete data. II: Stability and regularisation,” Inverse Probl. 4, 573–594 (1988).
[CrossRef]

J. Math. Phys. (2)

N. Bleistein, J. K. Cohen, “Nonuniqueness in the inverse source problem in acoustics and electromagnetics,” J. Math. Phys. 18, 194–201 (1977).
[CrossRef]

G. S. Agarwal, A. J. Devaney, D. N. Pattanayak, “On the solution of the partial differential equation Πj=1n(∇2+kj2)ψ=0,” J. Math. Phys. 14, 906–908 (1973).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (1)

K. Kim, E. Wolf, “Non-radiating monochromatic sources and their fields,” Opt. Commun. 59, 1–6 (1986).
[CrossRef]

Phys. Rev. D (1)

A. J. Devaney, E. Wolf, “Radiating and nonradiating classical current distributions and the fields they generate,” Phys. Rev. D 8, 1044–1047 (1973).
[CrossRef]

Other (8)

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

K. J. Langenberg, “Applied inverse problems for acoustic, electromagnetic, and elastic wave scattering,” in Basic Methods of Tomography and Inverse Problems, P. C. Sabatier, ed. (Institute of Physics, Bristol, UK, 1987), pp. 127–453.

M. Bertero, “Linear inverse and ill-posed problems,” in Advances in Electronics and Electron Physics, P. W. Hawkes, ed. (Academic, San Diego, Calif.1989), Vol. 75, pp. 1–120.

J. Weidmann, Linear Operators in Hilbert Spaces (Springer-Verlag, New York, 1980).

C. Müller, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer-Verlag, Berlin, 1969).

G. Arfken, Mathematical Methods for Physicists (Academic, San Diego, Calif.1985).

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

J. B. Marion, Classical Electromagnetic Radiation (Academic, New York, 1968).

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Figures (4)

Fig. 1
Fig. 1

Radiating part (thick curves) and NR part (thin curves) of the homogeneous source ρ(x)=M(x) versus x/a for (a) ka=π/6, (b) ka=π/2, (c) ka=π (NR case), (d) ka=1.5π, (e) ka=2.5π, (f) ka=5.5π.

Fig. 2
Fig. 2

Field magnitude for |x|a produced by the homogeneous source ρ(x)=M(x) versus x/a (solid curves). Also shown are the magnitudes of the radiating part (dashed curves) and the NR part (dotted–dashed curves) of the total field for |x|a. (a) ka=π/6, (b) ka=π/2, (c) ka=π (NR case), (d) ka=1.5π, (e) ka=2.5π, (f) ka=5.5π.

Fig. 3
Fig. 3

Radiating part (dotted curves) and NR part (solid curves) of the homogeneous spherical source ρ(r)=U(a-r) versus r/a for (a) ka=π/2, (b) ka=π, (c) ka=4.493 (NR case), (d) ka=2π, (e) ka=12, (f) ka=24.

Fig. 4
Fig. 4

Field magnitude for ra produced by the homogeneous spherical source ρ(r)=U(a-r) versus r/a (solid curves). Also shown are the magnitudes of the radiating part (dashed curves) and the NR part (dotted–dashed curves) of the total field for ra. (a) ka=π/2, (b) ka=π, (c) ka=4.493 (NR case), (d) ka=2π, (e) ka=12, (f) ka=24.

Equations (97)

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(2+k2)ψ(r)=-4πρ(r)(k0),
ψ(r)=Dd3r ρ(r)exp(ik|r-r|)|r-r|
Dd3r|ρˆ(r)|21/2
(2+k2)ρME(r)=0
(2+k2)2ψ(r)=0
(ρ, ρ)X=Dd3r ρ*(r)ρ(r),
ψ(r)=ikl=0m=-llgl,mhl(1)(kr)Yl,m(rˆ),
gl,m=(ψl,m, ρ)X,
ψl,m(r)=4πM(r)jl(kr)Yl,m(rˆ),l=0, 1,;m=-l,-l+1,,l,
M(r)=1ifrD0else.
(g, g)Y=l=0m=-llgl,m* gl,m.
Lρ=g,
(Lρ)l,m=(ψl,m, ρ)X.
(Lρ, g)Y=(ρ, Lg)X,
(Lg)(r)=l=0m=-llgl,mψl,m(r).
ρME=Lg˜,
g˜=(LL)-1g
ρME(r)=l=0m=-llgl,mψl,m(r)/σl2=4πM(r)l=0m=-llgl,mjl(kr)Yl,m(rˆ)/σl2,
σl2=(4π)20adr r2jl2(kr)=8π2a3[jl2(ka)-jl-1(ka)jl+1(ka)].
[N(L)]=R(L)¯,
X=N(L)[N(L)].
(2+k2)ψl,m(r)=0
l=0m=-ll|gl,m|2/σl2<,
(2+k2)G(r, r)=-4πδ(r-r).
-14πDdS ρME(r)nGD(r, r)=ρME(r)ifrD0ifrD,
14πDdSG0(r, r)nρME(r)-ρME(r)nG0(r, r)=ρME(r)ifrD0ifrD.
Dd3r ρME(r)G0(r, r)+14πDdSG0(r, r)nψ(r)-ψ(r)nG0(r, r)=ψ(r)ifrD0ifrD.
14πDdSG0(r, r)nψ(r)-ψ(r)nG0(r, r)=-ψ(r)ifrD0ifrD.
ψ(r)=18πkDdS(2+k2)ψ(r)2knG0(r, r)-n(2+k2)ψ(r)kG0(r, r)+14πDdSG0(r, r)nψ(r)-ψ(r)nG0(r, r).
ψ(r)=12kDdSnρME(r)kG0(r, r)-ρME(r)2knG0(r, r),
18πkDdS(2+k2)ψ(r)2knG0(r, r)-n(2+k2)ψ(r)kG0(r, r)=14πDdSψ(r)nG0(r, r)-G0(r, r)nψ(r),
ψ(r)=14πDdSnρME(r)×d3r G0(r, r)G0(r, r)-ρME(r)nd3r G0(r, r)G0(r, r).
2πkkG0(r, r)=d3r G0(r, r)G0(r, r).
G0(r, r)=12π2d3Kexp[iK·(r-r)]K2-k2,
2πkkG0(r, r)=2πd3Kexp[iK·(r-r)](K2-k2)2=d3r G0(r, r)G0(r, r),
(d2/dx2+k2)Ψ(x)=-ρ(x),
Ψ(x)=i2k-aadxρ(x)exp(ik|x-x|),
Ψ(x)=F+ exp(ikx)ifx>aF- exp(-ikx)ifx<-a,
F+=i2k-aadx ρ(x)exp(-ikx),
F-=i2k-aadx ρ(x)exp(ikx).
(ρ, ρ)U=-aadx ρ*(x)ρ(x).
(F, F)V=(F+)*F++(F-)*F-.
(Tρ)±=i2kdx M(x)ρ(x)exp(ikx),
(TF)(x)=-i2kM(x)[F+ exp(ikx)+F- exp(-ikx)],
ρME=TF˜,
F˜=(TT)-1F.
ρME(x)=ikM(x)a[sinc2(2ka)-1]×{[F+-F- sinc(2ka)]exp(ikx)+[F--F+ sinc(2ka)]exp(-ikx)},
ρME(x)ρ¯M(x)ask0,
ρ¯=12a-aadx ρ(x).
(d2/dx2+k2)2ΨME(x)=0if|x|<a,
ΨME(x)=A cos(kx)+B sin(kx)+Cx sin(kx)+Dx cos(kx)if|x|<a
ΨME(x)=A exp(ikx)+B exp(-ikx)+Cx exp(ikx)+Dx exp(-ikx)if|x|<a,
C=F- sinc(2ka)-F+2a[sinc2(2ka)-1],
D=F--F+ sinc(2ka)2a[sinc2(2ka)-1].
A=12i sin(2ka)[F+ exp(2ika)-F--2aC cos(2ka)-2aD],
B=12i sin(2ka)[F- exp(2ika)-F++2aC+2aD cos(2ka)].
ρME(x)=-2ika[sinc(2ka)+1]F+M(x)cos(kx).
F+=ik2sin(ka)
ρME(x)=ηM(x)cos(kx),
η=2 sinc(ka)sinc(2ka)+1.
ρNR(x)=M(x)[1-η cos(kx)].
Ψ(x)=i2k-axdx exp[ik(x-x)]+xadx exp[-ik(x-x)]if|x|a
=1k2[exp(ika)cos(kx)-1]if|x|a.
ΨME(x)=η2k{ia[exp(ika)sinc(ka)+1]cos(kx)-x sin(kx)}.
ΨME(x)=iη2k-axdx cos(kx)exp[ik(x-x)]+xadx cos(kx)exp[-ik(x-x)]=η4k2[2ika+exp(2ika)-1]cos(kx)-η2kx sin(kx).
(ρME, ρ)U=2aη sinc(ka)=4a sinc2(ka)sinc(2ka)+1,
ΨNR(x)=anπ2[(-1)n cos(nπx/a)-1]if|x|a.
ρ(x)=M(x)+M(x),
P=0if2a<λ1ifλ2a<2λ2if2λ2a<3λ,
(d2/dr2+k2)[rψ(r)]=-4πrρ(r).
G0(r, r)=exp(ik|r-r|)|r-r|=4πikl=0m=-lljl(kr<)hl(1)(kr>)×Yl,m(rˆ)Yl,m*(rˆ),
ψ(r)=4πik0adr r2ρ(r)j0(kr<)h0(1)(kr>),
ψ(r)=Q(r)exp(ikr)/r+R(r)sinc(kr),
Q(r)=4πk-10rdr rρ(r)sin(kr),
R(r)=4πradr rρ(r)exp(ikr).
ψ(r)=Q(a)exp(ikr)/r,
ψ(r)=ik4πg0,0h0(1)(kr)=g0,04πexp(ikr)/r(r>a),
g0,0=(4π)3/20adr r2ρ(r)j0(kr)=(4π)3/2k-10adr rρ(r)sin(kr).
ρME(r)=4πg0,0U(a-r)j0(kr)/σ02,
σ02=8π2a3[j02(ka)-j-1(ka)j1(ka)]=8π2ak2[1-sinc(2ka)].
(d2/dr2+k2)2[rψME(r)]=0ifra,
ψME(r)=A sinc(kr)+Bcos(kr)r+C sin(kr)+D cos(kr)ifra,
ψME(a)=g0,04πexp(ika)/a.
ψME(r)=g0,014πaexp(ika)-4π3/2k-2σ0-2 cos(ka)×sinc(kr)sinc(ka)+4π3/2g0,0k2σ02cos(kr).
g0,0=(4π)3/2k-10adr rj0(kr)=(4π)3/2k2-a cos(ka)+sin(ka)k.
ρME(r)=2sinc(ka)-cos(ka)1-sinc(2ka)U(a-r)sinc(kr),
ψ(r)=4πk2[(1-ika)exp(ika)sinc(kr)-1].
sinc(ka)=cos(ka),
ψNR(r)=4πk2[sinc(kr)/cos(ka)-1],
(2+k12)(2+k22)ψ(r)=0
ψ1(r)=(2+k22)ψ(r),ψ2(r)=(2+k12)ψ(r),
ψ(r)=[(2+k12)-(2+k22)]ψ(r)k12-k22=ψ1(r)-ψ2(r)k22-k12(k12k22).
(2+k12)ψ1(r)=0(2+k22)ψ2(r)=0.
(2+ki2)G0(i)(r, r)=-4πδ(r-r),i=1, 2,
ψ1(r)=14πDdSG0(1)(r, r)n(2+k22)ψ(r)-(2+k22)ψ(r)nG0(1)(r, r),
ψ2(r)=14πDdSG0(2)(r, r)n(2+k12)ψ(r)-(2+k12)ψ(r)nG0(2)(r, r).
ψ(r)=limk2k1=kψ2(r)-ψ1(r)k12-k22=12klimk2k1=kk2ψ2(r)-k2ψ1(r).

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