Abstract

We derive reciprocity relations for the generalized reflection and transmission coefficients of vector wave fields containing evanescent components. This is done by using Lorentz’s reciprocity theorem with sources at finite distance from the scatterer.

© 1998 Optical Society of America

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  1. H. Helmholtz, J. Reine Angew. Math. 67, 1 (1859); see also Rayleigh, The Theory of Sound (Dover, New York, 1945), Sect. 294.
  2. P. Roman, Advanced Quantum Theory (Addison-Wesley, Reading, Mass., 1965), pp. 282, 294.
  3. R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966), p. 46.
  4. E. Gerjuoy, D. S. Saxon, “Variational principles for the acoustic field,” Phys. Rev. 94, 1445–1458 (1954).
    [CrossRef]
  5. D. S. Saxon, “Tensor scattering matrix for the electromagnetic field,” Phys. Rev. 100, 1771–1775 (1955).
    [CrossRef]
  6. A. T. de Hoop, “A reciprocity theorem for the electromagnetic field scattered by an obstacle,” Appl. Sci. Res. Sect. B 8, 135–140 (1959).
    [CrossRef]
  7. Ph. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part I, Sect. 7.5.
  8. M. Nieto-Vesperinas, E. Wolf, “Generalized Stokes reciprocity relations for scattering from dielectric objects of arbitrary shape,” J. Opt. Soc. Am. A 3, 2038–2046 (1986).
    [CrossRef]
  9. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), Chap. 5.
  10. M. Nieto-Vesperinas, “Reciprocity of the impulse response for scattering from inhomogeneous media and arbitrary dielectric bodies,” J. Opt. Soc. Am. A 5, 360–365 (1988).
    [CrossRef]
  11. H. A. Lorentz, Versl. Gewone Vergad. Afd. Natuurkd. K. Ned. Akad. Wet. 4, 176–188 (1896); H. A. Lorentz, Collected Papers (Nijhoff, Den Haag, The Netherlands, 1936), Vol. III; see also M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), Chap. 1, p. 8.
  12. G. C. Sherman, “Diffracted fields expressible by plane-wave expansions containing only homogeneous waves,” J. Opt. Soc. Am. 59, 697–711 (1969).
    [CrossRef]
  13. A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
    [CrossRef]
  14. D. W. Pohl, “Scanning near-field optical microscopy,” in Advances in Optical and Electron Microscopy, C. J. R. Sheppard, T. Mulvey, eds. (Academic, New York, 1991), pp. 243–311.
  15. L. D. Landau, E. M. Lifshitz, L. P. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, Oxford, 1984), Sect. 89.
  16. M. Born, Natural Philosophy of Cause and Chance (Dover, New York, 1964), pp. 22–26.
  17. The incident field is here the field created by the dipole p1, where p1 has to be understood as the value of the dipole in the presence of the scatterer. This corresponds to the usual definition of the incident field in scattering problems. Usually, the source is assumed to be located at infinity and also to be unaffected by the presence of the scatterer.
  18. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), Chap. 3, pp. 83–84.
  19. R. W. P. King, Electromagnetic Engineering (McGraw-Hill, New York, 1945), Vol. I, p. 311.
  20. D. S. Jones, The Theory of Electromagnetism (Pergamon, Oxford, 1964), pp. 59–65.
  21. E. Wolf, G. S. Agarwal, A. T. Friberg, “Wavefront correction and scattering with phase-conjugated waves,” in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), pp. 107–116.
  22. M. Nieto-Vesperinas, E. Wolf, “Phase conjugation and symmetries with wave fields in free space containing evanescent components,” J. Opt. Soc. Am. A 2, 1429–1434 (1985).
    [CrossRef]
  23. R. J. Glauber, in Lectures in Theoretical Physics, W. E. Brittin, L. G. Dunham, eds. (Wiley-Interscience, New York, 1959), Vol. I, p. 315.
  24. P. Grivet, in Proceedings of the Symposium on Modern Optics, Vol. XVII of Microwave Research Institute Symposia Series (Wiley-Interscience, New York, 1967), pp. 467–479.
  25. R. Cadilhac, in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, New York, 1980), pp. 58–59.
  26. L. Onsager, “Reciprocal relations in irreversible processes,” Phys. Rev. 37, 405–426 (1931).
    [CrossRef]
  27. H. B. G. Casimir, “On Onsager’s principle of microscopic reversibility,” Rev. Mod. Phys. 17, 343–350 (1945); “Reciprocity theorems and irreversible processes,” Proc. IEEE 51, 1570–1573 (1963).
    [CrossRef]

1990 (1)

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

1988 (1)

1986 (1)

1985 (1)

1969 (1)

1959 (1)

A. T. de Hoop, “A reciprocity theorem for the electromagnetic field scattered by an obstacle,” Appl. Sci. Res. Sect. B 8, 135–140 (1959).
[CrossRef]

1955 (1)

D. S. Saxon, “Tensor scattering matrix for the electromagnetic field,” Phys. Rev. 100, 1771–1775 (1955).
[CrossRef]

1954 (1)

E. Gerjuoy, D. S. Saxon, “Variational principles for the acoustic field,” Phys. Rev. 94, 1445–1458 (1954).
[CrossRef]

1945 (1)

H. B. G. Casimir, “On Onsager’s principle of microscopic reversibility,” Rev. Mod. Phys. 17, 343–350 (1945); “Reciprocity theorems and irreversible processes,” Proc. IEEE 51, 1570–1573 (1963).
[CrossRef]

1931 (1)

L. Onsager, “Reciprocal relations in irreversible processes,” Phys. Rev. 37, 405–426 (1931).
[CrossRef]

1896 (1)

H. A. Lorentz, Versl. Gewone Vergad. Afd. Natuurkd. K. Ned. Akad. Wet. 4, 176–188 (1896); H. A. Lorentz, Collected Papers (Nijhoff, Den Haag, The Netherlands, 1936), Vol. III; see also M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), Chap. 1, p. 8.

1859 (1)

H. Helmholtz, J. Reine Angew. Math. 67, 1 (1859); see also Rayleigh, The Theory of Sound (Dover, New York, 1945), Sect. 294.

Agarwal, G. S.

E. Wolf, G. S. Agarwal, A. T. Friberg, “Wavefront correction and scattering with phase-conjugated waves,” in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), pp. 107–116.

Born, M.

M. Born, Natural Philosophy of Cause and Chance (Dover, New York, 1964), pp. 22–26.

Cadilhac, R.

R. Cadilhac, in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, New York, 1980), pp. 58–59.

Casimir, H. B. G.

H. B. G. Casimir, “On Onsager’s principle of microscopic reversibility,” Rev. Mod. Phys. 17, 343–350 (1945); “Reciprocity theorems and irreversible processes,” Proc. IEEE 51, 1570–1573 (1963).
[CrossRef]

de Hoop, A. T.

A. T. de Hoop, “A reciprocity theorem for the electromagnetic field scattered by an obstacle,” Appl. Sci. Res. Sect. B 8, 135–140 (1959).
[CrossRef]

Feshbach, H.

Ph. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part I, Sect. 7.5.

Friberg, A. T.

E. Wolf, G. S. Agarwal, A. T. Friberg, “Wavefront correction and scattering with phase-conjugated waves,” in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), pp. 107–116.

Gerjuoy, E.

E. Gerjuoy, D. S. Saxon, “Variational principles for the acoustic field,” Phys. Rev. 94, 1445–1458 (1954).
[CrossRef]

Glauber, R. J.

R. J. Glauber, in Lectures in Theoretical Physics, W. E. Brittin, L. G. Dunham, eds. (Wiley-Interscience, New York, 1959), Vol. I, p. 315.

Grivet, P.

P. Grivet, in Proceedings of the Symposium on Modern Optics, Vol. XVII of Microwave Research Institute Symposia Series (Wiley-Interscience, New York, 1967), pp. 467–479.

Helmholtz, H.

H. Helmholtz, J. Reine Angew. Math. 67, 1 (1859); see also Rayleigh, The Theory of Sound (Dover, New York, 1945), Sect. 294.

Jones, D. S.

D. S. Jones, The Theory of Electromagnetism (Pergamon, Oxford, 1964), pp. 59–65.

King, R. W. P.

R. W. P. King, Electromagnetic Engineering (McGraw-Hill, New York, 1945), Vol. I, p. 311.

Landau, L. D.

L. D. Landau, E. M. Lifshitz, L. P. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, Oxford, 1984), Sect. 89.

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, L. P. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, Oxford, 1984), Sect. 89.

Lorentz, H. A.

H. A. Lorentz, Versl. Gewone Vergad. Afd. Natuurkd. K. Ned. Akad. Wet. 4, 176–188 (1896); H. A. Lorentz, Collected Papers (Nijhoff, Den Haag, The Netherlands, 1936), Vol. III; see also M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), Chap. 1, p. 8.

Maradudin, A. A.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

McGurn, A. R.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

Méndez, E. R.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

Michel, T.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

Morse, Ph. M.

Ph. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part I, Sect. 7.5.

Newton, R. G.

R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966), p. 46.

Nieto-Vesperinas, M.

Onsager, L.

L. Onsager, “Reciprocal relations in irreversible processes,” Phys. Rev. 37, 405–426 (1931).
[CrossRef]

Pitaevskii, L. P.

L. D. Landau, E. M. Lifshitz, L. P. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, Oxford, 1984), Sect. 89.

Pohl, D. W.

D. W. Pohl, “Scanning near-field optical microscopy,” in Advances in Optical and Electron Microscopy, C. J. R. Sheppard, T. Mulvey, eds. (Academic, New York, 1991), pp. 243–311.

Roman, P.

P. Roman, Advanced Quantum Theory (Addison-Wesley, Reading, Mass., 1965), pp. 282, 294.

Saxon, D. S.

D. S. Saxon, “Tensor scattering matrix for the electromagnetic field,” Phys. Rev. 100, 1771–1775 (1955).
[CrossRef]

E. Gerjuoy, D. S. Saxon, “Variational principles for the acoustic field,” Phys. Rev. 94, 1445–1458 (1954).
[CrossRef]

Sherman, G. C.

Wolf, E.

Ann. Phys. (N.Y.) (1)

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

Appl. Sci. Res. Sect. B (1)

A. T. de Hoop, “A reciprocity theorem for the electromagnetic field scattered by an obstacle,” Appl. Sci. Res. Sect. B 8, 135–140 (1959).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Reine Angew. Math. (1)

H. Helmholtz, J. Reine Angew. Math. 67, 1 (1859); see also Rayleigh, The Theory of Sound (Dover, New York, 1945), Sect. 294.

Phys. Rev. (3)

E. Gerjuoy, D. S. Saxon, “Variational principles for the acoustic field,” Phys. Rev. 94, 1445–1458 (1954).
[CrossRef]

D. S. Saxon, “Tensor scattering matrix for the electromagnetic field,” Phys. Rev. 100, 1771–1775 (1955).
[CrossRef]

L. Onsager, “Reciprocal relations in irreversible processes,” Phys. Rev. 37, 405–426 (1931).
[CrossRef]

Rev. Mod. Phys. (1)

H. B. G. Casimir, “On Onsager’s principle of microscopic reversibility,” Rev. Mod. Phys. 17, 343–350 (1945); “Reciprocity theorems and irreversible processes,” Proc. IEEE 51, 1570–1573 (1963).
[CrossRef]

Versl. Gewone Vergad. Afd. Natuurkd. K. Ned. Akad. Wet. (1)

H. A. Lorentz, Versl. Gewone Vergad. Afd. Natuurkd. K. Ned. Akad. Wet. 4, 176–188 (1896); H. A. Lorentz, Collected Papers (Nijhoff, Den Haag, The Netherlands, 1936), Vol. III; see also M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), Chap. 1, p. 8.

Other (15)

R. J. Glauber, in Lectures in Theoretical Physics, W. E. Brittin, L. G. Dunham, eds. (Wiley-Interscience, New York, 1959), Vol. I, p. 315.

P. Grivet, in Proceedings of the Symposium on Modern Optics, Vol. XVII of Microwave Research Institute Symposia Series (Wiley-Interscience, New York, 1967), pp. 467–479.

R. Cadilhac, in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, New York, 1980), pp. 58–59.

P. Roman, Advanced Quantum Theory (Addison-Wesley, Reading, Mass., 1965), pp. 282, 294.

R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966), p. 46.

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), Chap. 5.

Ph. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part I, Sect. 7.5.

D. W. Pohl, “Scanning near-field optical microscopy,” in Advances in Optical and Electron Microscopy, C. J. R. Sheppard, T. Mulvey, eds. (Academic, New York, 1991), pp. 243–311.

L. D. Landau, E. M. Lifshitz, L. P. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, Oxford, 1984), Sect. 89.

M. Born, Natural Philosophy of Cause and Chance (Dover, New York, 1964), pp. 22–26.

The incident field is here the field created by the dipole p1, where p1 has to be understood as the value of the dipole in the presence of the scatterer. This corresponds to the usual definition of the incident field in scattering problems. Usually, the source is assumed to be located at infinity and also to be unaffected by the presence of the scatterer.

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), Chap. 3, pp. 83–84.

R. W. P. King, Electromagnetic Engineering (McGraw-Hill, New York, 1945), Vol. I, p. 311.

D. S. Jones, The Theory of Electromagnetism (Pergamon, Oxford, 1964), pp. 59–65.

E. Wolf, G. S. Agarwal, A. T. Friberg, “Wavefront correction and scattering with phase-conjugated waves,” in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), pp. 107–116.

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Figures (4)

Fig. 1
Fig. 1

Scattering geometry and definition of the half-spaces R- and R+.

Fig. 2
Fig. 2

Geometry considered in the demonstration of the reciprocity of the generalized transmission coefficient. Situation 1: The dipole source p1 is in R-, and the scattered (transmitted) field is evaluated in R+. Situation 2: The dipole source p2 is in R+, and the scattered (transmitted) field is evaluated in R-.

Fig. 3
Fig. 3

Geometry considered in the demonstration of the reciprocity of the generalized reflection coefficient. Situation 1: The dipole source p1 is in R-, and the scattered (reflected) field is evaluated in R-. Situation 2: The dipole source p2 is in R-, and the scattered (reflected) field is evaluated in R-.

Fig. 4
Fig. 4

Geometry considered in the derivation of Lorentz’s reciprocity theorem with sources.

Equations (57)

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E1i(r)=e1i(K)exp(iK·R+iγz)d2K
forz1<z<0,
γ(K)=k2-|K|2
for|K|k(homogeneouscomponents),
γ(K)=i|K|2-k2
for|K|>k(evanescentcomponents),
E1t(r)=e1t(K)exp(iK·R+iγz)d2Kforz>L,
E1r(r)=e1r(K)exp(iK·R-iγz)d2K
forz1<z<0.
e1t(K)=t(K, K)·e1i(K)d2K,
e1r(K)=r(K, K)·e1i(K)d2K,
E2i(r)=e2i(K)exp(iK·R-iγz)d2K
forL<z<z2.
E2t(r)=e2t(K)exp(iK·R-iγz)d2Kforz<0,
E2r(r)=e2r(K)exp(iK·R+iγz)d2K
forL<z<z2.
e2t(K)=τ(K, K)·e2i(K)d2K,
e2r(K)=ρ(K, K)·e2i(K)d2K,
e1i(K)=iμ0ω28π2γI(K)·p1 exp(-iK·R1-iγz1),
I(K)=I-kkk2,
E1t(r)=iμ0ω28π2 d2K exp(iK·R+iγz)×d2K t(K, K)·I(K)·p1γ×exp(-iK·R1-iγz1),
e2i(K)=iμ0ω28π2γI(K)·p2 exp(-iK·R2+iγz2).
E2t(r)=iμ0ω28π2d2K exp(iK·R-iγz)
×d2Kτ(K, K)·I(K)·p2γ×exp(-iK·R2+iγz2).
p1·E2t(r1)=p2·E1t(r2).
γ[τ(-K,-K)·I(-K)]T=γt(K, K)·I(K),
γ[τ(-K,-K)]T=γt(K, K),
E1i+(r)=e1i+(K)exp(iK·R+iγz)d2K
forz1<z<0,
e1i+(K)=iμ0ω28π2γT(K)·p1 exp(-iK·R1-iγz1),
E1i-(r)=e1i-(K)exp(iK·R-iγz)d2Kforz<z1,
e1i-(K)=iμ0ω28π2γI(K)·p1 exp(-iK·R1+iγz1).
E1r(r)=iμ0ω28π2d2K exp(iK·R-iγz)×d2K r(K, K)·I(K)·p1γ×exp(-iK·R1-iγz1).
E2i+(r)=e2i+(K)exp(iK·R+iγz)d2K
forz2<z<0,
e2i+(K)=iμ0ω28π2γI(K)·p2 exp(-iK·R2-iγz2),
E2i-(r)=e2i-(K)exp(iK·R-iγz)d2Kforz<z2,
e2i-(K)=iμ0ω28π2γI(K)·p2 exp(-iK·R2+iγz2).
E2r(r)=iμ0ω28π2d2K exp(iK·R-iγz)×d2K r(K, K)·I(K)·p2γ×exp(-iK·R2-iγz2).
p1·[E2i-(r1)+E2r(r1)]=p2·[E1i+(r2)+E1r(r2)].
p1·E2i-(r1)=p2·E1i+(r2),
p1·E2r(r1)=p2·E1r(r2).
γ[r(-K,-K)·I(-K)]T=γr(K, K)·I(K).
γ[r(-K,-K)]T=γr(K, K).
γ[ρ(-K,-K)]T=γρ(K, K).
×Ek=iωBk,×Hk=Jk-iωDk,
Dk(r)=0(r, ω)·Ek(r),
Bk(r)=μ0μ(r, ω)·Hk(r).
(H2·×E1-E1·×H2)+(E2·×H1-H1·×E2)=iω(B1·H2-H1·B2)-iω(D1·E2-E1·D2)+J1·E2-J2·E1.
(r)=[(r)]T,μ(r)=[μ(r)]T.
·(E1×H2-E2×H1)=J1·E2-J2·E1.
Ek(kr)=2iπγ(K)ek(K) exp(ikr)r,
Hk(kr)=2iπγ(K)hk(K) exp(ikr)r,
ωμ0hk(K)=k×ek(K).
V1J1(r)·E2(r)d3r=V2J2(r)·E1(r)d3r,
Jk(r)=-iωpkδ(r-rk),
p1·E2(r1)=p2·E1(r2).

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