Abstract

Huygens's principle that each point on a wave front represents a source of spherical waves is conceptually useful but is incomplete; the backward parts of the wavelets have to be neglected ad hoc, otherwise backward waves are generated. The problem is solved mathematically by Kirchhoff's rigorous integration of the wave equation, but the intuitive appeal of Huygens's simple principle is lost. I show that, by using spatiotemporal dipoles instead of spherical point sources, one can recover a simple principle of scalar wave propagation that is correct whenever the concept of a wave front is meaningful.

© 1991 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. C. Huygens, Traité de la Lumière (Leyden, 1690) [English translation by S. P. Thompson, Treatise on Light (Macmillan, London, 1912)].
  2. A. Fresnel, Ann. Chem. Phys. 1, 239 (1816);Mem. Acad. 5, 339 (1826).
  3. H. von Helmholtz, J. Math. 57, 7 (1859).
  4. G. Kirchhoff, Berl. Sitzungsber. 641 (1882);Ann. Phys. 18, 663 (1883);Vorlesungen über Math. Phys. 2 (1891).
  5. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).
  6. B. B. Baker, E. T. Copson, Mathematical Theory of Huygens' Principle, 2nd ed. (Oxford U. Press, London, 1950).
  7. Part of this research was presented by D. A. B. Miller, in Digest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1990), paper PD16.
  8. Note that such scalar solutions, while valid, e.g., for acoustic waves, are not complete solutions for electromagnetic waves because Maxwell's equations impose additional constraints (see Ref. 6).
  9. See, for example, J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  10. The spatial dipole, consisting of equal and opposite infinitesimally separated point sources, should not be confused with a radiating electric dipole, which is a source of vector electromagnetic waves.
  11. Some waves (e.g., the net wave from two discrete point sources) cannot be adequately described using wave fronts; despite this, the wave front is, however, undoubtedly useful conceptually in understanding waves.
  12. Other features of the spatiotemporal dipoles include the following: (i) Although only considered here for the monochromatic case, the spatiotemporal dipoles also work for the general time-dependent case. (ii) There are two different kinds of spatiotemporal dipoles, the second kind having the opposite relative delay; this second kind corresponds to waves propagating inward rather than outward. (iii) Kirchhoff's surface integral can be rewritten exactly in terms of the two kinds of spatiotemporal dipoles instead of point and doublet sources; thus, instead of having surface sources to set the overall wave amplitude (doublets) and its normal derivative (point sources), we can use the two kinds of spatiotemporal dipoles to set the outward and inward wave amplitudes, respectively.

1882 (1)

G. Kirchhoff, Berl. Sitzungsber. 641 (1882);Ann. Phys. 18, 663 (1883);Vorlesungen über Math. Phys. 2 (1891).

1859 (1)

H. von Helmholtz, J. Math. 57, 7 (1859).

1816 (1)

A. Fresnel, Ann. Chem. Phys. 1, 239 (1816);Mem. Acad. 5, 339 (1826).

Baker, B. B.

B. B. Baker, E. T. Copson, Mathematical Theory of Huygens' Principle, 2nd ed. (Oxford U. Press, London, 1950).

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Copson, E. T.

B. B. Baker, E. T. Copson, Mathematical Theory of Huygens' Principle, 2nd ed. (Oxford U. Press, London, 1950).

Fresnel, A.

A. Fresnel, Ann. Chem. Phys. 1, 239 (1816);Mem. Acad. 5, 339 (1826).

Huygens, C.

C. Huygens, Traité de la Lumière (Leyden, 1690) [English translation by S. P. Thompson, Treatise on Light (Macmillan, London, 1912)].

Kirchhoff, G.

G. Kirchhoff, Berl. Sitzungsber. 641 (1882);Ann. Phys. 18, 663 (1883);Vorlesungen über Math. Phys. 2 (1891).

Miller, D. A. B.

Part of this research was presented by D. A. B. Miller, in Digest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1990), paper PD16.

Stratton, J. A.

See, for example, J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

von Helmholtz, H.

H. von Helmholtz, J. Math. 57, 7 (1859).

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Ann. Chem. Phys. (1)

A. Fresnel, Ann. Chem. Phys. 1, 239 (1816);Mem. Acad. 5, 339 (1826).

Berl. Sitzungsber. (1)

G. Kirchhoff, Berl. Sitzungsber. 641 (1882);Ann. Phys. 18, 663 (1883);Vorlesungen über Math. Phys. 2 (1891).

J. Math. (1)

H. von Helmholtz, J. Math. 57, 7 (1859).

Other (9)

C. Huygens, Traité de la Lumière (Leyden, 1690) [English translation by S. P. Thompson, Treatise on Light (Macmillan, London, 1912)].

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

B. B. Baker, E. T. Copson, Mathematical Theory of Huygens' Principle, 2nd ed. (Oxford U. Press, London, 1950).

Part of this research was presented by D. A. B. Miller, in Digest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1990), paper PD16.

Note that such scalar solutions, while valid, e.g., for acoustic waves, are not complete solutions for electromagnetic waves because Maxwell's equations impose additional constraints (see Ref. 6).

See, for example, J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

The spatial dipole, consisting of equal and opposite infinitesimally separated point sources, should not be confused with a radiating electric dipole, which is a source of vector electromagnetic waves.

Some waves (e.g., the net wave from two discrete point sources) cannot be adequately described using wave fronts; despite this, the wave front is, however, undoubtedly useful conceptually in understanding waves.

Other features of the spatiotemporal dipoles include the following: (i) Although only considered here for the monochromatic case, the spatiotemporal dipoles also work for the general time-dependent case. (ii) There are two different kinds of spatiotemporal dipoles, the second kind having the opposite relative delay; this second kind corresponds to waves propagating inward rather than outward. (iii) Kirchhoff's surface integral can be rewritten exactly in terms of the two kinds of spatiotemporal dipoles instead of point and doublet sources; thus, instead of having surface sources to set the overall wave amplitude (doublets) and its normal derivative (point sources), we can use the two kinds of spatiotemporal dipoles to set the outward and inward wave amplitudes, respectively.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

Wave from a spatiotemporal dipole for plane wave propagation for the approximate case of finite separation of the dipole. The position of the two sources is indicated by the vertical lines. The sources are separated by a distance d. (a) Waves propagating out from the right source. (b) Waves propagating out from the left source, which is delayed by a time τ = d/c compared with the right source. (c) Net wave from the two sources. Note that there is no resulting wave in the left direction, and the waves do not cancel on the right-hand side.

Fig. 2
Fig. 2

(a) Exact wave from a (spatial) dipole radiator. (b) Wave calculated using the proposed wave propagation principle, using spatiotemporal dipoles on a spherical surface of radius 2.25 wavelengths. The amplitude of the spatiotemporal dipoles per unit area is chosen equal to the wave amplitude on the spherical wave front of radius 2.25 wavelengths in (a). The dipoles are oriented perpendicular to the surface. For visual clarity, the wave amplitude is multiplied by the distance from the center to remove the underlying 1/r dependence. A seven-level gray scale is used. In the region outside the chosen wave front, both waves are seen to be similar.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

2 ϕ 1 c 2 2 ϕ t 2 = g ( r , t ) ,
ϕ ( r 1 , t ) = 1 4 π s { 1 r [ ϕ n ] + [ ϕ ] n ( 1 r ) 1 c r × r n [ ϕ t ] } d a .
2 ϕ n 2 2 ϕ q 2 2 , 2 ϕ q 3 2 .
ϕ n = 1 c ϕ t .
2 ϕ q 2 2 , 2 ϕ q 3 2 k 2 ϕ ,
1 4 π { 1 r [ ϕ n ] + [ ϕ ] n ( 1 r ) 1 c r r n [ ϕ t ] } = [ ϕ ] 4 π r { i k ( 1 + cos θ ) + cos θ r } ,
ϕ T D = a d e i ( ω t k r ) 4 π r { i k ( 1 + cos θ ) + cos θ r } .

Metrics