Abstract

The scalar Huygens–Fresnel principle is reformulated to take into account the vector nature of light and its associated directed electric and magnetic fields. Based on Maxwell’s equations, a vector Huygens secondary source is developed in terms of the fundamental radiating units of electromagnetism: the electric and magnetic dipoles. The formulation is in terms of the vector potential from which the fields are derived uniquely. Vector wave propagation and diffraction formulated in this way are entirely consistent with Huygens’s principle. The theory is applicable to apertures larger than a wavelength situated in dark, perfectly absorbing screens and for points of observation in the right half-space at distances greater than a wavelength beyond the aperture. Alternatively, a formulation in terms of the fields is also developed; it is referred to as a vector Huygens–Fresnel theory. The proposed method permits the determination of the diffracted electromagnetic fields along with the detected irradiance.

© 2001 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1989), Sec. 8.3.
  2. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 30–76.
  3. E. Hecht, Optics (Addison-Wesley, Reading, Mass., 1987).
  4. B. B. Baker, E. T. Copson, The Mathematical Theory of Huygens’ Principle, 2nd ed. (Clarendon, Oxford, UK, 1949).
  5. H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. 66, 163–182 (1944).
    [CrossRef]
  6. K. Miyamoto, E. Wolf, “Generalization of Maggi–Rubinowicz theory of the boundary diffraction wave. I, II,” J. Opt. Soc. Am. 52, 615–625, 626–637 (1962).
    [CrossRef]
  7. E. W. Marchand, E. Wolf, “Boundary diffraction wave in the domain of the Rayleigh–Kirchhoff diffraction theory,” J. Opt. Soc. Am. 52, 761–767 (1962).
    [CrossRef]
  8. E. W. Marchand, E. Wolf, “Consistent formulation of Kirchhoff’s diffraction theory,” J. Opt. Soc. Am. 56, 1712–1722 (1966).
    [CrossRef]
  9. P. Debye, “Das Verhalten von Lichtwellen in der Nahe des Brennpunktes order einer Brennlinie,” Ann. Phys. (Leipzig) 30, 755–776 (1909).
    [CrossRef]
  10. F. Kottler, “Electromagnetische theorie der Beugung an schwarzen Schirmen,” Ann. Phys. (Leipzig) 71, 457–508 (1923).
    [CrossRef]
  11. J. A. Stratton, L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
    [CrossRef]
  12. E. Wolf, “Electromagnetic diffraction in optical systems. I,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
    [CrossRef]
  13. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
    [CrossRef]
  14. C.-T. TaiDyadic Green’s Functions in Electromagnetic Theory, 2nd ed. (IEEE Press, Piscataway, N.J., 1994).
  15. B. Karczewski, E. Wolf, “Comparison of three theories of electromagnetic diffraction at an aperture. Part I. Coherence matrices,” J. Opt. Soc. Am. 56, 1207–1214 (1966).
    [CrossRef]
  16. B. Karczewski, E. Wolf, “Comparison of three theories of electromagnetic diffraction at an aperture. Part II. The far field,” J. Opt. Soc. Am. 56, 1214–1219 (1966).
    [CrossRef]
  17. T. D. Visser, S. H. Wiersma, “An electromagnetic description of the image formation in confocal fluorescence microscopy,” J. Opt. Soc. Am. A 11, 599–608 (1994).
    [CrossRef]
  18. G. Bekefi, “Diffraction of electromagnetic waves by an aperture in a large screen,” J. Appl. Phys. 24, 1123–1130 (1953).
    [CrossRef]
  19. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999).
  20. J. R. Wait, Electromagnetic Wave Theory (Harper & Row, New York, 1985), pp. 76–195, 225–259.
  21. S. A. Schelkunoff, “Some equivalence theorems of electromagnetics and their application to radiation problems,” Bell Syst. Tech. J., January15, 1936, pp. 99–112 (1936).
  22. S. A. Schelkunoff, Electromagnetic Waves (Van Nostrand, Princeton, N.J., 1943), pp. 331–373.
  23. A. S. Marathay, J. F. McCalmont, “Diffraction of electromagnetic waves. Part I. Theory,” presented at the 1999 Annual Meeting, Optical Society of America, Santa Clara, Calif., September 28, 1999.
  24. J. F. McCalmont, A. S. Marathay, “Diffraction of electromagnetic waves. Part II. Practice,” presented at the 1999 Annual Meeting of the Optical Society of America, Santa Clara, Calif., September 28, 1999.
  25. P. Lorrain, D. R. Corson, F. Lorrain, Electromagnetic Fields and Waves (Freeman, New York, 1988).
  26. R. P. Bocker, B. R. Frieden, “Solution of the Maxwell field equations in vacuum for arbitrary charge and current distribution using matrix algebra,” IEEE Trans. Education 36, 350–356 (1993).
    [CrossRef]
  27. E. C. Jordan, K. G. Balmain, Electromagnetic Waves and Radiating Systems (Prentice-Hall, New Delhi, 1968).
  28. R. F. Harrington, Time-Harmonic Electromagnetic Fields and Waves (McGraw-Hill, New York, 1961), pp. 1–142.
  29. C. T. A. Johnk, Engineering Electromagnetic Fields and Waves (Wiley, New York, 1975), pp. 592–633.
  30. R. Loudon, The Quantum Theory of Light (Clarendon, Oxford, UK, 1973), Chap. 4, Secs. 4-1 and 4-2.
  31. E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
    [CrossRef]
  32. B. T. Draine, J. Goodman, “Beyond Clausius–Massotti: wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J. 405, 685–697 (1993).
    [CrossRef]
  33. J. F. McCalmont, “A vector Huygens–Fresnel model of the diffraction of electromagnetic waves,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1999).
  34. E. Yamashita, Analysis Methods for Electromagnetic Wave Problems (Artech House, Boston, Mass., 1990), pp. 177–211.
  35. E. Wolf, “Some recent research on the diffraction of light,” in Proceedings of the Symposium on Modern Optics, New York, March 22–24, Vol. XVII of Microwave Research Institute Symposia Series (Polytechnic Press, Brooklyn, N. Y., 1967), pp. 443–452.

1994

1993

B. T. Draine, J. Goodman, “Beyond Clausius–Massotti: wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J. 405, 685–697 (1993).
[CrossRef]

R. P. Bocker, B. R. Frieden, “Solution of the Maxwell field equations in vacuum for arbitrary charge and current distribution using matrix algebra,” IEEE Trans. Education 36, 350–356 (1993).
[CrossRef]

1973

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

1966

1962

E. W. Marchand, E. Wolf, “Boundary diffraction wave in the domain of the Rayleigh–Kirchhoff diffraction theory,” J. Opt. Soc. Am. 52, 761–767 (1962).
[CrossRef]

K. Miyamoto, E. Wolf, “Generalization of Maggi–Rubinowicz theory of the boundary diffraction wave. I, II,” J. Opt. Soc. Am. 52, 615–625, 626–637 (1962).
[CrossRef]

1959

E. Wolf, “Electromagnetic diffraction in optical systems. I,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

1953

G. Bekefi, “Diffraction of electromagnetic waves by an aperture in a large screen,” J. Appl. Phys. 24, 1123–1130 (1953).
[CrossRef]

1944

H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. 66, 163–182 (1944).
[CrossRef]

1939

J. A. Stratton, L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[CrossRef]

1923

F. Kottler, “Electromagnetische theorie der Beugung an schwarzen Schirmen,” Ann. Phys. (Leipzig) 71, 457–508 (1923).
[CrossRef]

1909

P. Debye, “Das Verhalten von Lichtwellen in der Nahe des Brennpunktes order einer Brennlinie,” Ann. Phys. (Leipzig) 30, 755–776 (1909).
[CrossRef]

Baker, B. B.

B. B. Baker, E. T. Copson, The Mathematical Theory of Huygens’ Principle, 2nd ed. (Clarendon, Oxford, UK, 1949).

Balmain, K. G.

E. C. Jordan, K. G. Balmain, Electromagnetic Waves and Radiating Systems (Prentice-Hall, New Delhi, 1968).

Bekefi, G.

G. Bekefi, “Diffraction of electromagnetic waves by an aperture in a large screen,” J. Appl. Phys. 24, 1123–1130 (1953).
[CrossRef]

Bethe, H. A.

H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. 66, 163–182 (1944).
[CrossRef]

Bocker, R. P.

R. P. Bocker, B. R. Frieden, “Solution of the Maxwell field equations in vacuum for arbitrary charge and current distribution using matrix algebra,” IEEE Trans. Education 36, 350–356 (1993).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1989), Sec. 8.3.

Chu, L. J.

J. A. Stratton, L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[CrossRef]

Copson, E. T.

B. B. Baker, E. T. Copson, The Mathematical Theory of Huygens’ Principle, 2nd ed. (Clarendon, Oxford, UK, 1949).

Corson, D. R.

P. Lorrain, D. R. Corson, F. Lorrain, Electromagnetic Fields and Waves (Freeman, New York, 1988).

Debye, P.

P. Debye, “Das Verhalten von Lichtwellen in der Nahe des Brennpunktes order einer Brennlinie,” Ann. Phys. (Leipzig) 30, 755–776 (1909).
[CrossRef]

Draine, B. T.

B. T. Draine, J. Goodman, “Beyond Clausius–Massotti: wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J. 405, 685–697 (1993).
[CrossRef]

Frieden, B. R.

R. P. Bocker, B. R. Frieden, “Solution of the Maxwell field equations in vacuum for arbitrary charge and current distribution using matrix algebra,” IEEE Trans. Education 36, 350–356 (1993).
[CrossRef]

Goodman, J.

B. T. Draine, J. Goodman, “Beyond Clausius–Massotti: wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J. 405, 685–697 (1993).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 30–76.

Harrington, R. F.

R. F. Harrington, Time-Harmonic Electromagnetic Fields and Waves (McGraw-Hill, New York, 1961), pp. 1–142.

Hecht, E.

E. Hecht, Optics (Addison-Wesley, Reading, Mass., 1987).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999).

Johnk, C. T. A.

C. T. A. Johnk, Engineering Electromagnetic Fields and Waves (Wiley, New York, 1975), pp. 592–633.

Jordan, E. C.

E. C. Jordan, K. G. Balmain, Electromagnetic Waves and Radiating Systems (Prentice-Hall, New Delhi, 1968).

Karczewski, B.

Kottler, F.

F. Kottler, “Electromagnetische theorie der Beugung an schwarzen Schirmen,” Ann. Phys. (Leipzig) 71, 457–508 (1923).
[CrossRef]

Lorrain, F.

P. Lorrain, D. R. Corson, F. Lorrain, Electromagnetic Fields and Waves (Freeman, New York, 1988).

Lorrain, P.

P. Lorrain, D. R. Corson, F. Lorrain, Electromagnetic Fields and Waves (Freeman, New York, 1988).

Loudon, R.

R. Loudon, The Quantum Theory of Light (Clarendon, Oxford, UK, 1973), Chap. 4, Secs. 4-1 and 4-2.

Marathay, A. S.

A. S. Marathay, J. F. McCalmont, “Diffraction of electromagnetic waves. Part I. Theory,” presented at the 1999 Annual Meeting, Optical Society of America, Santa Clara, Calif., September 28, 1999.

J. F. McCalmont, A. S. Marathay, “Diffraction of electromagnetic waves. Part II. Practice,” presented at the 1999 Annual Meeting of the Optical Society of America, Santa Clara, Calif., September 28, 1999.

Marchand, E. W.

McCalmont, J. F.

J. F. McCalmont, “A vector Huygens–Fresnel model of the diffraction of electromagnetic waves,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1999).

J. F. McCalmont, A. S. Marathay, “Diffraction of electromagnetic waves. Part II. Practice,” presented at the 1999 Annual Meeting of the Optical Society of America, Santa Clara, Calif., September 28, 1999.

A. S. Marathay, J. F. McCalmont, “Diffraction of electromagnetic waves. Part I. Theory,” presented at the 1999 Annual Meeting, Optical Society of America, Santa Clara, Calif., September 28, 1999.

Miyamoto, K.

K. Miyamoto, E. Wolf, “Generalization of Maggi–Rubinowicz theory of the boundary diffraction wave. I, II,” J. Opt. Soc. Am. 52, 615–625, 626–637 (1962).
[CrossRef]

Pennypacker, C. R.

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

Purcell, E. M.

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

Richards, B.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Schelkunoff, S. A.

S. A. Schelkunoff, “Some equivalence theorems of electromagnetics and their application to radiation problems,” Bell Syst. Tech. J., January15, 1936, pp. 99–112 (1936).

S. A. Schelkunoff, Electromagnetic Waves (Van Nostrand, Princeton, N.J., 1943), pp. 331–373.

Stratton, J. A.

J. A. Stratton, L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[CrossRef]

Tai, C.-T.

C.-T. TaiDyadic Green’s Functions in Electromagnetic Theory, 2nd ed. (IEEE Press, Piscataway, N.J., 1994).

Visser, T. D.

Wait, J. R.

J. R. Wait, Electromagnetic Wave Theory (Harper & Row, New York, 1985), pp. 76–195, 225–259.

Wiersma, S. H.

Wolf, E.

B. Karczewski, E. Wolf, “Comparison of three theories of electromagnetic diffraction at an aperture. Part II. The far field,” J. Opt. Soc. Am. 56, 1214–1219 (1966).
[CrossRef]

E. W. Marchand, E. Wolf, “Consistent formulation of Kirchhoff’s diffraction theory,” J. Opt. Soc. Am. 56, 1712–1722 (1966).
[CrossRef]

B. Karczewski, E. Wolf, “Comparison of three theories of electromagnetic diffraction at an aperture. Part I. Coherence matrices,” J. Opt. Soc. Am. 56, 1207–1214 (1966).
[CrossRef]

E. W. Marchand, E. Wolf, “Boundary diffraction wave in the domain of the Rayleigh–Kirchhoff diffraction theory,” J. Opt. Soc. Am. 52, 761–767 (1962).
[CrossRef]

K. Miyamoto, E. Wolf, “Generalization of Maggi–Rubinowicz theory of the boundary diffraction wave. I, II,” J. Opt. Soc. Am. 52, 615–625, 626–637 (1962).
[CrossRef]

E. Wolf, “Electromagnetic diffraction in optical systems. I,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1989), Sec. 8.3.

E. Wolf, “Some recent research on the diffraction of light,” in Proceedings of the Symposium on Modern Optics, New York, March 22–24, Vol. XVII of Microwave Research Institute Symposia Series (Polytechnic Press, Brooklyn, N. Y., 1967), pp. 443–452.

Yamashita, E.

E. Yamashita, Analysis Methods for Electromagnetic Wave Problems (Artech House, Boston, Mass., 1990), pp. 177–211.

Ann. Phys. (Leipzig)

P. Debye, “Das Verhalten von Lichtwellen in der Nahe des Brennpunktes order einer Brennlinie,” Ann. Phys. (Leipzig) 30, 755–776 (1909).
[CrossRef]

F. Kottler, “Electromagnetische theorie der Beugung an schwarzen Schirmen,” Ann. Phys. (Leipzig) 71, 457–508 (1923).
[CrossRef]

Astrophys. J.

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

B. T. Draine, J. Goodman, “Beyond Clausius–Massotti: wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J. 405, 685–697 (1993).
[CrossRef]

Bell Syst. Tech. J.

S. A. Schelkunoff, “Some equivalence theorems of electromagnetics and their application to radiation problems,” Bell Syst. Tech. J., January15, 1936, pp. 99–112 (1936).

IEEE Trans. Education

R. P. Bocker, B. R. Frieden, “Solution of the Maxwell field equations in vacuum for arbitrary charge and current distribution using matrix algebra,” IEEE Trans. Education 36, 350–356 (1993).
[CrossRef]

J. Appl. Phys.

G. Bekefi, “Diffraction of electromagnetic waves by an aperture in a large screen,” J. Appl. Phys. 24, 1123–1130 (1953).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Phys. Rev.

H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. 66, 163–182 (1944).
[CrossRef]

J. A. Stratton, L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[CrossRef]

Proc. R. Soc. London Ser. A

E. Wolf, “Electromagnetic diffraction in optical systems. I,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Other

C.-T. TaiDyadic Green’s Functions in Electromagnetic Theory, 2nd ed. (IEEE Press, Piscataway, N.J., 1994).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1989), Sec. 8.3.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 30–76.

E. Hecht, Optics (Addison-Wesley, Reading, Mass., 1987).

B. B. Baker, E. T. Copson, The Mathematical Theory of Huygens’ Principle, 2nd ed. (Clarendon, Oxford, UK, 1949).

E. C. Jordan, K. G. Balmain, Electromagnetic Waves and Radiating Systems (Prentice-Hall, New Delhi, 1968).

R. F. Harrington, Time-Harmonic Electromagnetic Fields and Waves (McGraw-Hill, New York, 1961), pp. 1–142.

C. T. A. Johnk, Engineering Electromagnetic Fields and Waves (Wiley, New York, 1975), pp. 592–633.

R. Loudon, The Quantum Theory of Light (Clarendon, Oxford, UK, 1973), Chap. 4, Secs. 4-1 and 4-2.

S. A. Schelkunoff, Electromagnetic Waves (Van Nostrand, Princeton, N.J., 1943), pp. 331–373.

A. S. Marathay, J. F. McCalmont, “Diffraction of electromagnetic waves. Part I. Theory,” presented at the 1999 Annual Meeting, Optical Society of America, Santa Clara, Calif., September 28, 1999.

J. F. McCalmont, A. S. Marathay, “Diffraction of electromagnetic waves. Part II. Practice,” presented at the 1999 Annual Meeting of the Optical Society of America, Santa Clara, Calif., September 28, 1999.

P. Lorrain, D. R. Corson, F. Lorrain, Electromagnetic Fields and Waves (Freeman, New York, 1988).

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999).

J. R. Wait, Electromagnetic Wave Theory (Harper & Row, New York, 1985), pp. 76–195, 225–259.

J. F. McCalmont, “A vector Huygens–Fresnel model of the diffraction of electromagnetic waves,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1999).

E. Yamashita, Analysis Methods for Electromagnetic Wave Problems (Artech House, Boston, Mass., 1990), pp. 177–211.

E. Wolf, “Some recent research on the diffraction of light,” in Proceedings of the Symposium on Modern Optics, New York, March 22–24, Vol. XVII of Microwave Research Institute Symposia Series (Polytechnic Press, Brooklyn, N. Y., 1967), pp. 443–452.

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Figures (5)

Fig. 1
Fig. 1

Orientation of the EM dipole at rs on the primary wave front (solid circle). A source of a monochromatic spherical wave is located at S0; point of observation P is shown with position vector r. Difference vector Rs=r-rs makes an angle θs with nˆ(rs), the unit normal to the wave front. pe, pm, electric and magnetic dipole moments, respectively.

Fig. 2
Fig. 2

Mapping of the EM dipole’s normalized irradiance on a spherical wave front in the far zone (represented by the outermost ring) as a function of the angle between the direction of observation and nˆ, the outward unit normal to the primary wave front. (Here nˆ lies along the θ=0 direction.) The electric dipole is located at the origin, and its axis is perpendicular to the page. The magnetic dipole is also located at the origin, oriented in the vertical direction, and lies in the plane of the page. The Poynting vector points radially outward. The electric field vector is tangent to the spherical wave front and is perpendicular to the plane of the page. The magnetic field vector is also tangent to the wave front but lies in the plane of the page. The normalized magnitude of the Poynting vector is proportional to the length of the chord from the origin to the cardioid along the radius to the point of tangency. The plot is rotationally symmetric about nˆ.

Fig. 3
Fig. 3

(a) Diffraction geometry. The primary source of a monochromatic spherical wave is located at S0. The primary wave front is W. A dark screen, S, contains a clear aperture A. The point of observation is located at P. (b) EM dipoles are represented as small open circles along the arc of the wave front in the geometry of (a). Only the EM dipoles exposed by the aperture are shown; they contribute to the diffracted field at P. Primary source S0, primary wave front W, the open aperture A, and dark screen S have been removed; they are shown as dotted lines.

Fig. 4
Fig. 4

Axial distribution of the normalized field amplitude beyond circular apertures of diameters D calculated by the vector Huygens–Fresnel model as a function of distance z from the aperture. (a) D=1λ, (b) D=3λ, (c) D=6λ, (d) D=10λ. The incident field is a plane wave of unit amplitude falling normally onto the plane of the aperture.

Fig. 5
Fig. 5

Irradiance profiles along the horizontal y axis on an observation screen at distances (a) 5λ, (b) 100λ, (c) 500λ, (d) 15,000λ that are due to diffraction by a rectangular slit and produced by the vector Huygens–Fresnel model. Slit width, W=20λ. The first zero of the Fraunhofer pattern is at 2.81°. The incident field is a plane wave of unit amplitude falling normally onto the plane of the aperture and polarized in the vertical x direction. The irradiance profile is given as a function of wavelengths from the origin of the observation plane along the horizontal y axis.

Tables (1)

Tables Icon

Table 1 Dual Replacements of Electric and Magnetic Sources

Equations (45)

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

·E=ρeo,·H=ρmμo,
×E=-Jm-μo Ht,×H=Je+o Et.
·Jm+ρmt=0
·Je+ρet=0.
E=Ee+Em,H=He+Hm.
A=Ae+Am,ϕ=ϕe+ϕm.
Je=nˆ×(Htt-Htr)=nˆ×Hti,
Jm=-nˆ×(Ett-Etr)=-nˆ×Eti,
Je=nˆ×Htt=nˆ×Hti,
Jm=-nˆ×Ett=-nˆ×Eti.
nˆ·E=ρseo, nˆ·H=ρsmμo,
nˆ×E=-Jsm,nˆ×H=Jse,
nˆ×E=-Jsm,nˆ×H=Jse.
Je(r, t)=Je(r)exp(-iωt),
Je(r)=-iωpeδ(r).
Jm(r)=-iωpmδ(r).
nˆ×H=+Jse=-i2πcpeδ(r-rs),
nˆ×E=-Jsm=i2πcpmδ(r-rs),
pe(rs)=+i2πcnˆ(rs)×H(rs),
pm(rs)=-i2πcnˆ(rs)×E(rs),
pepm=|H||E|=oμo.
2Ae(r)+k2Ae(r)=-μoJe(r).
·Ae(r)=0
2Am(r)+k2Am(r)=-oJm(r).
·Am(r)=0.
E(r)=Ee(r)+Em(r)=iωAe(r)-1o×Am(r),
H(r)=He(r)+Hm(r)=iωAm(r)+1μo×Ae(r).
A(r)=Ae(r)+Am(r),
2A(r)+k2A(r)=-μoJe(r)-oJm(r).
A(r)=14π allspace [μoJe(r)+oJm(r)] exp(ik|r-r|)|r-r|d(3)r,
A(r, t)=A(r)exp(-iωt),
Je(Rs)=-iωpeδ(Rs),
Jm(Rs)=-iωpmδ(Rs),
AEM(r, rs)=Ae+Am=-iω4π(μope+opm)×exp(ikRs)Rs,
EEM(r, rs)=k24πo exp(ikRs)Rs (Rˆs×pe)×Rˆs-oμo (Rˆs×pm),
HEM(r, rs)=k24πμo exp(ikRs)Rs (Rˆs×pm)×Rˆs+μoo (Rˆs×pe),
SEM=k48π2oμo |pe||pm|Rs2 1+cos θs22Rˆs
A(r)=ESD[Ae(r, rs)+Am(r, rs)]d(3)rs,
E(r)=ESDEEM(r, rs)d(3)rs=-i2λ2 EPWμoo {Rˆs×[H(rs)×nˆ(rs)]}×Rˆs-{Rˆs×[nˆ(rs)×E(rs)]}×exp(ikRs)Rs d(3)rs,
H(r)=ESDHEM(r, rs)d(3)rs=-i2λ2 EPWoμo{Rˆs×[nˆ(rs)×E(rs)]}×Rˆs+{Rˆs×[H(rs)×nˆ(rs)]} exp(ikRs)Rs d(3)rs,
H(rs)=oμo nˆ(rs)×E(rs),
E(r)=-i2λ2 EPW[Rˆs×E(rs)]×Rˆs-μoo [Rˆs×H(rs)] exp(ikRs)Rs d(3)rs
=-i2λ2 EPW(E(rs){1+[Rˆs·nˆ(rs)]}-[Rˆs·E(rs)][Rˆs+nˆ(rs)]) exp(ikRs)Rs d(3)rs,
H(r)=-i2λ2 EPW[Rˆs×H(rs)]×Rˆs+oμo [Rˆs×E(rs)] exp(ikRs)Rs d(3)rs
=-i2λ2 EPW{H(rs)[1+Rˆs·nˆ(rs)]-[Rˆs·H(rs)][Rˆs+nˆ(rs)]} exp(ikRs)Rs d(3)rs,

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