Abstract

Analytic properties of the angular spectrum amplitude of the field scattered from a finite object with a continuous scattering potential are established, and some of its consequences are discussed. In particular, it is shown that the scattered field must contain both homogeneous and evanescent waves, unless it vanishes everywhere outside the scattering volume.

© 1985 Optical Society of America

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References

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  1. H. G. Booker, J. A. Ratcliffe, D. H. Shinn, “Diffraction from an irregular screen with applications to ionospheric problems,” Phil. Trans. R. Soc. London Ser. A 242, 579–609 (1950).
    [CrossRef]
  2. D. R. Rhodes, “On a fundamental principle in the theory of planar antennas,” Proc. IEEE 52, 1013–1021 (1964).
    [CrossRef]
  3. D. M. Kerns, “Scattering matrix description and nearfield measurements of electroacoustic tranducers,” J. Acoust. Soc. Am. 57, 497–507 (1975).
    [CrossRef]
  4. A. J. Devaney, E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
    [CrossRef]
  5. E. Lalor, E. Wolf, “New model for the interaction between a moving charged particle and a dielectric, and the Čerenkov effect,” Phys. Rev. Lett. 26, 1274–1277 (1971).
    [CrossRef]
  6. A. Walther, “Gabor’s theorem and energy transfer through lenses,” J. Opt. Soc. Am. 57, 639–644 (1967);also “Systematic approach to the teaching of lens theory,” Am. J. Phys. 35, 808–816 (1967).
    [CrossRef]
  7. R. Mittra, P. L. Ransom, “Imaging with coherent fields” in Proceedings of the Symposium on Modern Optics, J. Fox, ed. (Wiley, New York, 1967), pp. 619–647.
  8. W. D. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. 57, 772–778 (1967).
    [CrossRef]
  9. B. Richards, E. Wolf, “Electromagnetics diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
    [CrossRef]
  10. D. Gabor, “Light and information” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1961), Vol. 1, pp. 109–153, App. I.
    [CrossRef]
  11. G. C. Sherman, “Integral-transform formulation of diffraction theory,” J. Opt. Soc. Am. 57, 1490–1498 (1967).
    [CrossRef]
  12. J. R. Shewell, E. Wolf, “Inverse diffraction and a new reciprocity theorem,” J. Opt. Soc. Am. 58, 1596–1603 (1968).
    [CrossRef]
  13. E. Wolf, “Three-dimensional structure determination of semitransparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
    [CrossRef]
  14. M. Nieto-Vesperinas, “Inverse scattering problems: A study in terms of the zeros of entire functions,” J. Math. Phys. 25, 2109–2115 (1984).
    [CrossRef]
  15. G. S. Agarwal, “Interaction of electromagnetic waves at rough dielectric surfaces,” Phys. Rev. B 15, 2371–2382 (1977).
    [CrossRef]
  16. E. Wolf, J. R. Shewell, “Diffraction theory of holography,” J. Math. Phys. 11, 2254–2267 (1970).
    [CrossRef]
  17. G. S. Agarwal, D. N. Pattanayak, E. Wolf, “Electromagnetic fields in spatially dispersive media,” Phys. Rev. B 10, 1447–1475 (1974).
    [CrossRef]
  18. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).
    [CrossRef]
  19. G. S. Agarwal, A. T. Friberg, E. Wolf, “Scattering theory of distortion correction by phase conjugation,” J. Opt. Soc. Am. 73, 529–538 (1983).
    [CrossRef]
  20. H. Weyl, “Ausbreitung elektromagnetischer Wellen über einem ebenen Leiter,” Ann. Phys. (Leipzig) 60, 481–500 (1919).Because the integrand of Eq. (2.8) is an entire function of α, the α integration contours C±may be deformed in accordance with the rules of integration in the complex plane. When these contours are chosen as shown in Fig. 2, Eq. (2.8) transforms to the following, frequently used, alternative form of the Weyl representation:(R1)exp(ik|r−r′|)|r−r′|=ik2π∫−∞∞1sz×exp{ik[sx(x−x′)+sy(y−y′)+sz|z−z′|]}dsxdsy.Here,(R2)sz=+(1−sx2−sy2)1/2when sx2+sy2≤1,=+i(sx2+sy2−1)1/2when sx2+sy2>1.The two forms of the Weyl expansion, and the transformation required to pass from one to the other, are discussed in A. Baños, Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon, New York, 1966), Sec. 2.13.
  21. If instead of Eq. (2.8) we make use in Eq. (2.6) of the alternative representation (R1) of G(r, r′), we readily obtain the following alternative form of the angular spectrum representation of the scattered fields:(R3)U(s)(r)=∫∫−∞∞a(±)(sx,sy)exp[ik(sxx+syy±szz)]dsxdsy.Here sz is given by Eq. (R2), and(R4)a(±)(sx,sy)=ik2πA(α,β).In Eq. (R3) the upper or the lower sign is taken according to whether the point r≡ (x, y, z) is situated in the half-space ℛ+or ℛ−, respectively. In Eq. (R4) the upper or lower sign is associated with polar angle α that is on the contour C+or C−, respectively. Angular spectrum representations of the form (R3) are frequently used in optics.
  22. E. T. Copson, Theory of Functions of a Complex Variable (Oxford U. Press, Oxford, 1970), pp. 107–108.
  23. W. F. Osgood, Topics in the Theory of Functions of Several Complex Variables (Dover, New York, 1966), p. 33.
  24. C. Müller, “Radiation patterns and radiation fields,” J. Rat. Mech. Anal. 4, 235–246 (1955), especially pp. 237–238.
  25. D. Colton, R. Kress, Integral Equation Methods in Scattering Theory (Wiley, New York, 1983), Theorem 3.5, p. 72.
  26. Whether there are finite range potentials that do not give rise to scattering [i.e., for which U(s)(r) ≡ 0 everywhere outside the scattering volume D] does not appear to be known. However, the answer to the corresponding question for the radiation problem is known: There are monochromatic source distributions of finite support D that do not generate any fields outside D[see, for example, A. J. Devaney, E. Wolf, “Radiating and non-radiating classical current distributions and the fields they generate,” Phys. Rev. D 8,1044–1047 (1973)].

1984 (1)

M. Nieto-Vesperinas, “Inverse scattering problems: A study in terms of the zeros of entire functions,” J. Math. Phys. 25, 2109–2115 (1984).
[CrossRef]

1983 (1)

1977 (1)

G. S. Agarwal, “Interaction of electromagnetic waves at rough dielectric surfaces,” Phys. Rev. B 15, 2371–2382 (1977).
[CrossRef]

1975 (1)

D. M. Kerns, “Scattering matrix description and nearfield measurements of electroacoustic tranducers,” J. Acoust. Soc. Am. 57, 497–507 (1975).
[CrossRef]

1974 (2)

A. J. Devaney, E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
[CrossRef]

G. S. Agarwal, D. N. Pattanayak, E. Wolf, “Electromagnetic fields in spatially dispersive media,” Phys. Rev. B 10, 1447–1475 (1974).
[CrossRef]

1973 (1)

Whether there are finite range potentials that do not give rise to scattering [i.e., for which U(s)(r) ≡ 0 everywhere outside the scattering volume D] does not appear to be known. However, the answer to the corresponding question for the radiation problem is known: There are monochromatic source distributions of finite support D that do not generate any fields outside D[see, for example, A. J. Devaney, E. Wolf, “Radiating and non-radiating classical current distributions and the fields they generate,” Phys. Rev. D 8,1044–1047 (1973)].

1971 (1)

E. Lalor, E. Wolf, “New model for the interaction between a moving charged particle and a dielectric, and the Čerenkov effect,” Phys. Rev. Lett. 26, 1274–1277 (1971).
[CrossRef]

1970 (1)

E. Wolf, J. R. Shewell, “Diffraction theory of holography,” J. Math. Phys. 11, 2254–2267 (1970).
[CrossRef]

1969 (1)

E. Wolf, “Three-dimensional structure determination of semitransparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

1968 (2)

1967 (3)

1964 (1)

D. R. Rhodes, “On a fundamental principle in the theory of planar antennas,” Proc. IEEE 52, 1013–1021 (1964).
[CrossRef]

1959 (1)

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

1955 (1)

C. Müller, “Radiation patterns and radiation fields,” J. Rat. Mech. Anal. 4, 235–246 (1955), especially pp. 237–238.

1950 (1)

H. G. Booker, J. A. Ratcliffe, D. H. Shinn, “Diffraction from an irregular screen with applications to ionospheric problems,” Phil. Trans. R. Soc. London Ser. A 242, 579–609 (1950).
[CrossRef]

1919 (1)

H. Weyl, “Ausbreitung elektromagnetischer Wellen über einem ebenen Leiter,” Ann. Phys. (Leipzig) 60, 481–500 (1919).Because the integrand of Eq. (2.8) is an entire function of α, the α integration contours C±may be deformed in accordance with the rules of integration in the complex plane. When these contours are chosen as shown in Fig. 2, Eq. (2.8) transforms to the following, frequently used, alternative form of the Weyl representation:(R1)exp(ik|r−r′|)|r−r′|=ik2π∫−∞∞1sz×exp{ik[sx(x−x′)+sy(y−y′)+sz|z−z′|]}dsxdsy.Here,(R2)sz=+(1−sx2−sy2)1/2when sx2+sy2≤1,=+i(sx2+sy2−1)1/2when sx2+sy2>1.The two forms of the Weyl expansion, and the transformation required to pass from one to the other, are discussed in A. Baños, Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon, New York, 1966), Sec. 2.13.

Agarwal, G. S.

G. S. Agarwal, A. T. Friberg, E. Wolf, “Scattering theory of distortion correction by phase conjugation,” J. Opt. Soc. Am. 73, 529–538 (1983).
[CrossRef]

G. S. Agarwal, “Interaction of electromagnetic waves at rough dielectric surfaces,” Phys. Rev. B 15, 2371–2382 (1977).
[CrossRef]

G. S. Agarwal, D. N. Pattanayak, E. Wolf, “Electromagnetic fields in spatially dispersive media,” Phys. Rev. B 10, 1447–1475 (1974).
[CrossRef]

Booker, H. G.

H. G. Booker, J. A. Ratcliffe, D. H. Shinn, “Diffraction from an irregular screen with applications to ionospheric problems,” Phil. Trans. R. Soc. London Ser. A 242, 579–609 (1950).
[CrossRef]

Colton, D.

D. Colton, R. Kress, Integral Equation Methods in Scattering Theory (Wiley, New York, 1983), Theorem 3.5, p. 72.

Copson, E. T.

E. T. Copson, Theory of Functions of a Complex Variable (Oxford U. Press, Oxford, 1970), pp. 107–108.

Devaney, A. J.

A. J. Devaney, E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
[CrossRef]

Whether there are finite range potentials that do not give rise to scattering [i.e., for which U(s)(r) ≡ 0 everywhere outside the scattering volume D] does not appear to be known. However, the answer to the corresponding question for the radiation problem is known: There are monochromatic source distributions of finite support D that do not generate any fields outside D[see, for example, A. J. Devaney, E. Wolf, “Radiating and non-radiating classical current distributions and the fields they generate,” Phys. Rev. D 8,1044–1047 (1973)].

Friberg, A. T.

Gabor, D.

D. Gabor, “Light and information” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1961), Vol. 1, pp. 109–153, App. I.
[CrossRef]

Kerns, D. M.

D. M. Kerns, “Scattering matrix description and nearfield measurements of electroacoustic tranducers,” J. Acoust. Soc. Am. 57, 497–507 (1975).
[CrossRef]

Kress, R.

D. Colton, R. Kress, Integral Equation Methods in Scattering Theory (Wiley, New York, 1983), Theorem 3.5, p. 72.

Lalor, E.

E. Lalor, E. Wolf, “New model for the interaction between a moving charged particle and a dielectric, and the Čerenkov effect,” Phys. Rev. Lett. 26, 1274–1277 (1971).
[CrossRef]

Mittra, R.

R. Mittra, P. L. Ransom, “Imaging with coherent fields” in Proceedings of the Symposium on Modern Optics, J. Fox, ed. (Wiley, New York, 1967), pp. 619–647.

Montgomery, W. D.

Müller, C.

C. Müller, “Radiation patterns and radiation fields,” J. Rat. Mech. Anal. 4, 235–246 (1955), especially pp. 237–238.

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, “Inverse scattering problems: A study in terms of the zeros of entire functions,” J. Math. Phys. 25, 2109–2115 (1984).
[CrossRef]

Osgood, W. F.

W. F. Osgood, Topics in the Theory of Functions of Several Complex Variables (Dover, New York, 1966), p. 33.

Pattanayak, D. N.

G. S. Agarwal, D. N. Pattanayak, E. Wolf, “Electromagnetic fields in spatially dispersive media,” Phys. Rev. B 10, 1447–1475 (1974).
[CrossRef]

Ransom, P. L.

R. Mittra, P. L. Ransom, “Imaging with coherent fields” in Proceedings of the Symposium on Modern Optics, J. Fox, ed. (Wiley, New York, 1967), pp. 619–647.

Ratcliffe, J. A.

H. G. Booker, J. A. Ratcliffe, D. H. Shinn, “Diffraction from an irregular screen with applications to ionospheric problems,” Phil. Trans. R. Soc. London Ser. A 242, 579–609 (1950).
[CrossRef]

Rhodes, D. R.

D. R. Rhodes, “On a fundamental principle in the theory of planar antennas,” Proc. IEEE 52, 1013–1021 (1964).
[CrossRef]

Richards, B.

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

Sherman, G. C.

Shewell, J. R.

E. Wolf, J. R. Shewell, “Diffraction theory of holography,” J. Math. Phys. 11, 2254–2267 (1970).
[CrossRef]

J. R. Shewell, E. Wolf, “Inverse diffraction and a new reciprocity theorem,” J. Opt. Soc. Am. 58, 1596–1603 (1968).
[CrossRef]

Shinn, D. H.

H. G. Booker, J. A. Ratcliffe, D. H. Shinn, “Diffraction from an irregular screen with applications to ionospheric problems,” Phil. Trans. R. Soc. London Ser. A 242, 579–609 (1950).
[CrossRef]

Walther, A.

Weyl, H.

H. Weyl, “Ausbreitung elektromagnetischer Wellen über einem ebenen Leiter,” Ann. Phys. (Leipzig) 60, 481–500 (1919).Because the integrand of Eq. (2.8) is an entire function of α, the α integration contours C±may be deformed in accordance with the rules of integration in the complex plane. When these contours are chosen as shown in Fig. 2, Eq. (2.8) transforms to the following, frequently used, alternative form of the Weyl representation:(R1)exp(ik|r−r′|)|r−r′|=ik2π∫−∞∞1sz×exp{ik[sx(x−x′)+sy(y−y′)+sz|z−z′|]}dsxdsy.Here,(R2)sz=+(1−sx2−sy2)1/2when sx2+sy2≤1,=+i(sx2+sy2−1)1/2when sx2+sy2>1.The two forms of the Weyl expansion, and the transformation required to pass from one to the other, are discussed in A. Baños, Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon, New York, 1966), Sec. 2.13.

Wolf, E.

G. S. Agarwal, A. T. Friberg, E. Wolf, “Scattering theory of distortion correction by phase conjugation,” J. Opt. Soc. Am. 73, 529–538 (1983).
[CrossRef]

G. S. Agarwal, D. N. Pattanayak, E. Wolf, “Electromagnetic fields in spatially dispersive media,” Phys. Rev. B 10, 1447–1475 (1974).
[CrossRef]

A. J. Devaney, E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
[CrossRef]

Whether there are finite range potentials that do not give rise to scattering [i.e., for which U(s)(r) ≡ 0 everywhere outside the scattering volume D] does not appear to be known. However, the answer to the corresponding question for the radiation problem is known: There are monochromatic source distributions of finite support D that do not generate any fields outside D[see, for example, A. J. Devaney, E. Wolf, “Radiating and non-radiating classical current distributions and the fields they generate,” Phys. Rev. D 8,1044–1047 (1973)].

E. Lalor, E. Wolf, “New model for the interaction between a moving charged particle and a dielectric, and the Čerenkov effect,” Phys. Rev. Lett. 26, 1274–1277 (1971).
[CrossRef]

E. Wolf, J. R. Shewell, “Diffraction theory of holography,” J. Math. Phys. 11, 2254–2267 (1970).
[CrossRef]

E. Wolf, “Three-dimensional structure determination of semitransparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

J. R. Shewell, E. Wolf, “Inverse diffraction and a new reciprocity theorem,” J. Opt. Soc. Am. 58, 1596–1603 (1968).
[CrossRef]

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

Ann. Phys. (Leipzig) (1)

H. Weyl, “Ausbreitung elektromagnetischer Wellen über einem ebenen Leiter,” Ann. Phys. (Leipzig) 60, 481–500 (1919).Because the integrand of Eq. (2.8) is an entire function of α, the α integration contours C±may be deformed in accordance with the rules of integration in the complex plane. When these contours are chosen as shown in Fig. 2, Eq. (2.8) transforms to the following, frequently used, alternative form of the Weyl representation:(R1)exp(ik|r−r′|)|r−r′|=ik2π∫−∞∞1sz×exp{ik[sx(x−x′)+sy(y−y′)+sz|z−z′|]}dsxdsy.Here,(R2)sz=+(1−sx2−sy2)1/2when sx2+sy2≤1,=+i(sx2+sy2−1)1/2when sx2+sy2>1.The two forms of the Weyl expansion, and the transformation required to pass from one to the other, are discussed in A. Baños, Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon, New York, 1966), Sec. 2.13.

J. Acoust. Soc. Am. (1)

D. M. Kerns, “Scattering matrix description and nearfield measurements of electroacoustic tranducers,” J. Acoust. Soc. Am. 57, 497–507 (1975).
[CrossRef]

J. Math. Phys. (3)

A. J. Devaney, E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
[CrossRef]

M. Nieto-Vesperinas, “Inverse scattering problems: A study in terms of the zeros of entire functions,” J. Math. Phys. 25, 2109–2115 (1984).
[CrossRef]

E. Wolf, J. R. Shewell, “Diffraction theory of holography,” J. Math. Phys. 11, 2254–2267 (1970).
[CrossRef]

J. Opt. Soc. Am. (6)

J. Rat. Mech. Anal. (1)

C. Müller, “Radiation patterns and radiation fields,” J. Rat. Mech. Anal. 4, 235–246 (1955), especially pp. 237–238.

Opt. Commun. (1)

E. Wolf, “Three-dimensional structure determination of semitransparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

Phil. Trans. R. Soc. London Ser. A (1)

H. G. Booker, J. A. Ratcliffe, D. H. Shinn, “Diffraction from an irregular screen with applications to ionospheric problems,” Phil. Trans. R. Soc. London Ser. A 242, 579–609 (1950).
[CrossRef]

Phys. Rev. (1)

Whether there are finite range potentials that do not give rise to scattering [i.e., for which U(s)(r) ≡ 0 everywhere outside the scattering volume D] does not appear to be known. However, the answer to the corresponding question for the radiation problem is known: There are monochromatic source distributions of finite support D that do not generate any fields outside D[see, for example, A. J. Devaney, E. Wolf, “Radiating and non-radiating classical current distributions and the fields they generate,” Phys. Rev. D 8,1044–1047 (1973)].

Phys. Rev. B (2)

G. S. Agarwal, “Interaction of electromagnetic waves at rough dielectric surfaces,” Phys. Rev. B 15, 2371–2382 (1977).
[CrossRef]

G. S. Agarwal, D. N. Pattanayak, E. Wolf, “Electromagnetic fields in spatially dispersive media,” Phys. Rev. B 10, 1447–1475 (1974).
[CrossRef]

Phys. Rev. Lett. (1)

E. Lalor, E. Wolf, “New model for the interaction between a moving charged particle and a dielectric, and the Čerenkov effect,” Phys. Rev. Lett. 26, 1274–1277 (1971).
[CrossRef]

Proc. IEEE (1)

D. R. Rhodes, “On a fundamental principle in the theory of planar antennas,” Proc. IEEE 52, 1013–1021 (1964).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

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

Other (6)

D. Gabor, “Light and information” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1961), Vol. 1, pp. 109–153, App. I.
[CrossRef]

R. Mittra, P. L. Ransom, “Imaging with coherent fields” in Proceedings of the Symposium on Modern Optics, J. Fox, ed. (Wiley, New York, 1967), pp. 619–647.

D. Colton, R. Kress, Integral Equation Methods in Scattering Theory (Wiley, New York, 1983), Theorem 3.5, p. 72.

If instead of Eq. (2.8) we make use in Eq. (2.6) of the alternative representation (R1) of G(r, r′), we readily obtain the following alternative form of the angular spectrum representation of the scattered fields:(R3)U(s)(r)=∫∫−∞∞a(±)(sx,sy)exp[ik(sxx+syy±szz)]dsxdsy.Here sz is given by Eq. (R2), and(R4)a(±)(sx,sy)=ik2πA(α,β).In Eq. (R3) the upper or the lower sign is taken according to whether the point r≡ (x, y, z) is situated in the half-space ℛ+or ℛ−, respectively. In Eq. (R4) the upper or lower sign is associated with polar angle α that is on the contour C+or C−, respectively. Angular spectrum representations of the form (R3) are frequently used in optics.

E. T. Copson, Theory of Functions of a Complex Variable (Oxford U. Press, Oxford, 1970), pp. 107–108.

W. F. Osgood, Topics in the Theory of Functions of Several Complex Variables (Dover, New York, 1966), p. 33.

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Figures (3)

Fig. 1
Fig. 1

Illustrating the notation.

Fig. 2
Fig. 2

The α contours C+ and C.

Fig. 3
Fig. 3

Notation relating to the derivation of the asymptotic approximation (2.14) to the free-space Green’s function G(r, r′).

Equations (25)

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( 2 + k 2 ) U ( i ) ( r ) = 0 ,
[ 2 + k 2 n 2 ( r ) ] U ( r ) = 0 ,
U ( r ) = U ( i ) ( r ) + U ( s ) ( r ) ,
( 2 + k 2 ) U ( s ) ( r ) = F ( r ) U ( r ) ,
F ( r ) = k 2 [ n 2 ( r ) 1 ]
U ( s ) ( r ) = 1 4 π D U ( r ) F ( r ) G ( r , r ) d 3 r ,
G ( r , r ) = exp ( ik | r r | ) | r r | .
G ( r , r ) = ik 2 π π π d β C ± d α sin α exp [ i k s · ( r r ) ] ,
s x = sin α cos β , s y = sin α sin β , s z = cos α .
U ( s ) ( r ) = i k 2 π π π d β C ± d α sin α A ( α , β ) exp ( i k s · r ) ,
A ( α , β ) = 1 4 π D U ( r ) F ( r ) exp ( i k s · r ) d 3 r .
u = r / r
u ( sin θ cos φ , sin θ sin φ cos θ ) .
G ( r , r ) exp ( ik r ) r exp ( i k u · r ) ( k r ) .
U ( s ) ( r u ) 1 4 π exp ( i k r ) r D U ( r ) F ( r ) exp ( i u · r ) d 3 r ( k r ) .
U ( s ) ( r u ) A ( θ , φ ) exp ( ik r ) r ( k r ) .
A ( α , β ) = 1 4 π D U ( r ) F ( r ) exp [ i k ( x sin α cos β + y sin α sin β + z cos α ) ] d 3 r
0 α π / 2 , 0 β < 2 π
π 2 < α π , 0 β < 2 π
α = π 2 i γ , 0 γ < , 0 β < 2 π
α = π 2 + i γ , > γ 0 , 0 β < 2 π
exp(ik|rr|)|rr|=ik2π1sz×exp{ik[sx(xx)+sy(yy)+sz|zz|]}dsxdsy.
sz=+(1sx2sy2)1/2whensx2+sy21,=+i(sx2+sy21)1/2whensx2+sy2>1.
U(s)(r)=a(±)(sx,sy)exp[ik(sxx+syy±szz)]dsxdsy.
a(±)(sx,sy)=ik2πA(α,β).

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