Abstract

It is known that certain oscillating three-dimensional charge-current distributions do not radiate. In this paper we show that no monochromatic, well-behaved, two-dimensional scalar source distribution that is localized within a finite area may be nonradiating. We also generalize this result to fluctuating, stationary planar source distributions of any arbitrary state of coherence.

© 1978 Optical Society of America

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  1. See, for example, G. H. Geodecke, "Classical radiationless motion and possible implications for quantum theory," Phys. Rev. B 135, 281–288 (1964), and the references cited therein.
  2. The class of classical charge-current distributions in question consists of all the monochromatic charge-current distributions such that the space-dependent parts of the charge density and of the current density are continuous and differentiable functions of position, and that they vanish identically outside a sphere of finite radius.
  3. A. J. Devaney and E. Wolf, "Radiating and nonradiating classical current distributions and the fields they generate," Phys. Rev. D 8, 1044–1047 (1973).
  4. See, for example, E. Wolf, "Radiometry and coherence," J. Opt. Soc. Am. 68, 6–17 (1978), or H. P. Baltes, "Coherence and the radiation laws," Appl. Phys. 12, 221–244 (1977), Sec. 3.
  5. We wish to emphasize that the true scalar source distribution ρω(r) in the plane z = 0 [rather than the field distribution vω(r) in that plane] is considered here as the source of radiation into the half space z > 0. Conclusions that are analogous to the results obtained in this paper can be drawn also in the case when the field vω(r) in the plane z = 0 is regarded as the source, provided that vω(r) is a well-behaved function of position and vanishes identically outside some finite domain in the plane z = 0. Such conclusions must, however, be interpreted with caution since it seems unlikely that a field distribution can be realized in some plane such that it vanishes identically outside a finite area (cf. Ref. 7, Sec. 4), even though this type of a mathematical simplification is commonly used in optical scalar diffraction theory. In particular, the field vω(r) generated by a true planar-source distribution ρω(r) localized within some finite area in the plane z = 0 necessarily extends throughout the entire source plane z = 0.
  6. This result follows at once, for example, from Eq. (3.3) of the paper: A. J. Devaney and E. Wolf, "Multipole expansions and plane wave representations of the electromagnetic field," J. Math. Phys. 15, 234–244 (1974), if it is specialized to the case of a planar-source density.
  7. E. Wolf, "The radiant intensity from planar sources of any state of coherence," J. Opt. Soc. Am. (to be published), Appendix A.
  8. The cartesian components Sx and Sy of the unit vector S are related to the polar angles θ and φof the coordinate system by the expressions Sx = sinθ cosφ and Sy = sinθ sin φ. With the help of these relations the radiation patterns aω (Sx,Sy) can be converted into the more familiar form f(θ, φ), which is a function of the angles θ and φ.
  9. This result follows as a two-dimensional generalization of a wellknown theorem on analytic functions given, for example, by A. Rudin, Real and Complex Analysis (McGraw-Hill, New York, 1966), Sec. 19.1. The multi-dimensional extension of this theorem is discussed by B. A. Fuks, Introduction to the Theory of Analytic Functions of Several Complex Variables (American Mathematical Society, Providence, R. I., 1963), p. 352.
  10. This choice of ensemble does not imply any loss of generality, since different temporal frequency components of a stationary field are uncorrelated. This fact is a consequence of the general form of the Wiener-Khintchine theorem for stationary random processes [cf. R. L. Stratonovich, Topics in the Theory of Random Noise (Gordon and Breach, New York, 1963), Vol. I, p. 281.
  11. It has recently been shown that an incoherent planar source of finite energy density does not radiate [cf. E. Wolf and W. H. Carter, "On the Radiation Efficiency of Quasihomogeneous Sources of Different Degrees of Spatial Coherence", in Proc. Fourth Rochester Conference on Coherence and Quantum Optics, edited by L. Mandel and E. Wolf (Plenum, New York, in press)]. The incoherent source was described in terms of the field distribution vω(r) across the source plane. A similar result can be shown to be valid for a strictly incoherent primary source characterized by the true source distribution ρω(r), i.e., a source for which [Equation] However, such a source clearly does not belong to the class of well-behaved sources that we are considering in the present paper.

1978 (1)

1974 (1)

This result follows at once, for example, from Eq. (3.3) of the paper: A. J. Devaney and E. Wolf, "Multipole expansions and plane wave representations of the electromagnetic field," J. Math. Phys. 15, 234–244 (1974), if it is specialized to the case of a planar-source density.

1973 (1)

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

1964 (1)

See, for example, G. H. Geodecke, "Classical radiationless motion and possible implications for quantum theory," Phys. Rev. B 135, 281–288 (1964), and the references cited therein.

Carter, W. H.

It has recently been shown that an incoherent planar source of finite energy density does not radiate [cf. E. Wolf and W. H. Carter, "On the Radiation Efficiency of Quasihomogeneous Sources of Different Degrees of Spatial Coherence", in Proc. Fourth Rochester Conference on Coherence and Quantum Optics, edited by L. Mandel and E. Wolf (Plenum, New York, in press)]. The incoherent source was described in terms of the field distribution vω(r) across the source plane. A similar result can be shown to be valid for a strictly incoherent primary source characterized by the true source distribution ρω(r), i.e., a source for which [Equation] However, such a source clearly does not belong to the class of well-behaved sources that we are considering in the present paper.

Devaney, A. J.

This result follows at once, for example, from Eq. (3.3) of the paper: A. J. Devaney and E. Wolf, "Multipole expansions and plane wave representations of the electromagnetic field," J. Math. Phys. 15, 234–244 (1974), if it is specialized to the case of a planar-source density.

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

Geodecke, G. H.

See, for example, G. H. Geodecke, "Classical radiationless motion and possible implications for quantum theory," Phys. Rev. B 135, 281–288 (1964), and the references cited therein.

Rudin, A.

This result follows as a two-dimensional generalization of a wellknown theorem on analytic functions given, for example, by A. Rudin, Real and Complex Analysis (McGraw-Hill, New York, 1966), Sec. 19.1. The multi-dimensional extension of this theorem is discussed by B. A. Fuks, Introduction to the Theory of Analytic Functions of Several Complex Variables (American Mathematical Society, Providence, R. I., 1963), p. 352.

Wolf, E.

See, for example, E. Wolf, "Radiometry and coherence," J. Opt. Soc. Am. 68, 6–17 (1978), or H. P. Baltes, "Coherence and the radiation laws," Appl. Phys. 12, 221–244 (1977), Sec. 3.

This result follows at once, for example, from Eq. (3.3) of the paper: A. J. Devaney and E. Wolf, "Multipole expansions and plane wave representations of the electromagnetic field," J. Math. Phys. 15, 234–244 (1974), if it is specialized to the case of a planar-source density.

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

E. Wolf, "The radiant intensity from planar sources of any state of coherence," J. Opt. Soc. Am. (to be published), Appendix A.

J. Math. Phys. (1)

This result follows at once, for example, from Eq. (3.3) of the paper: A. J. Devaney and E. Wolf, "Multipole expansions and plane wave representations of the electromagnetic field," J. Math. Phys. 15, 234–244 (1974), if it is specialized to the case of a planar-source density.

J. Opt. Soc. Am. (1)

Phys. Rev. B (1)

See, for example, G. H. Geodecke, "Classical radiationless motion and possible implications for quantum theory," Phys. Rev. B 135, 281–288 (1964), and the references cited therein.

Phys. Rev. D (1)

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

Other (7)

The class of classical charge-current distributions in question consists of all the monochromatic charge-current distributions such that the space-dependent parts of the charge density and of the current density are continuous and differentiable functions of position, and that they vanish identically outside a sphere of finite radius.

We wish to emphasize that the true scalar source distribution ρω(r) in the plane z = 0 [rather than the field distribution vω(r) in that plane] is considered here as the source of radiation into the half space z > 0. Conclusions that are analogous to the results obtained in this paper can be drawn also in the case when the field vω(r) in the plane z = 0 is regarded as the source, provided that vω(r) is a well-behaved function of position and vanishes identically outside some finite domain in the plane z = 0. Such conclusions must, however, be interpreted with caution since it seems unlikely that a field distribution can be realized in some plane such that it vanishes identically outside a finite area (cf. Ref. 7, Sec. 4), even though this type of a mathematical simplification is commonly used in optical scalar diffraction theory. In particular, the field vω(r) generated by a true planar-source distribution ρω(r) localized within some finite area in the plane z = 0 necessarily extends throughout the entire source plane z = 0.

E. Wolf, "The radiant intensity from planar sources of any state of coherence," J. Opt. Soc. Am. (to be published), Appendix A.

The cartesian components Sx and Sy of the unit vector S are related to the polar angles θ and φof the coordinate system by the expressions Sx = sinθ cosφ and Sy = sinθ sin φ. With the help of these relations the radiation patterns aω (Sx,Sy) can be converted into the more familiar form f(θ, φ), which is a function of the angles θ and φ.

This result follows as a two-dimensional generalization of a wellknown theorem on analytic functions given, for example, by A. Rudin, Real and Complex Analysis (McGraw-Hill, New York, 1966), Sec. 19.1. The multi-dimensional extension of this theorem is discussed by B. A. Fuks, Introduction to the Theory of Analytic Functions of Several Complex Variables (American Mathematical Society, Providence, R. I., 1963), p. 352.

This choice of ensemble does not imply any loss of generality, since different temporal frequency components of a stationary field are uncorrelated. This fact is a consequence of the general form of the Wiener-Khintchine theorem for stationary random processes [cf. R. L. Stratonovich, Topics in the Theory of Random Noise (Gordon and Breach, New York, 1963), Vol. I, p. 281.

It has recently been shown that an incoherent planar source of finite energy density does not radiate [cf. E. Wolf and W. H. Carter, "On the Radiation Efficiency of Quasihomogeneous Sources of Different Degrees of Spatial Coherence", in Proc. Fourth Rochester Conference on Coherence and Quantum Optics, edited by L. Mandel and E. Wolf (Plenum, New York, in press)]. The incoherent source was described in terms of the field distribution vω(r) across the source plane. A similar result can be shown to be valid for a strictly incoherent primary source characterized by the true source distribution ρω(r), i.e., a source for which [Equation] However, such a source clearly does not belong to the class of well-behaved sources that we are considering in the present paper.

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