Abstract

A representation of the general solution of the Maxwell equations is proposed in terms of the plane-wave spectrum of the electromagnetic field. In this representation the electric field solution is written as a sum of two terms that are orthogonal to each other at the far field: One is transverse to the propagation axis, and the magnetic field associated with the other is also transverse. The concept of the so-called closest field to a given beam is introduced and applied to the well-known linearly polarized Gaussian beam.

© 2001 Optical Society of America

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References

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  1. D. G. Hall, “Vector-beam solutions of Maxwell’s wave equation,” Opt. Lett. 21, 9–11 (1996).
    [Crossref] [PubMed]
  2. S. R. Seshadri, “Electromagnetic Gaussian beam,” J. Opt. Soc. Am. A 15, 2712–2719 (1998).
    [Crossref]
  3. S. R. Seshadri, “Partially coherent Gaussian Schell-model electromagnetic beams,” J. Opt. Soc. Am. A 16, 1373–1380 (1999).
    [Crossref]
  4. S. R. Seshadri, “Average characteristics of partially coherent electromagnetic beams,” J. Opt. Soc. Am. A 17, 780–709 (2000).
    [Crossref]
  5. P. Varga, P. Török, “Exact and approximate solutions of Maxwell’s equations for a confocal cavity,” Opt. Lett. 21, 1523–1525 (1996).
    [Crossref] [PubMed]
  6. P. Varga, P. Török, “The Gaussian wave solution of Maxwell’s equations and the validity of the scalar wave approximation,” Opt. Commun. 152, 108–118 (1998).
    [Crossref]
  7. C. J. R. Sheppard, “Polarization of almost-plane waves,” J. Opt. Soc. Am. A 17, 335–341 (2000).
    [Crossref]
  8. C. J. R. Sheppard, S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 16, 1381–1386 (1999).
    [Crossref]

2000 (2)

1999 (2)

1998 (2)

P. Varga, P. Török, “The Gaussian wave solution of Maxwell’s equations and the validity of the scalar wave approximation,” Opt. Commun. 152, 108–118 (1998).
[Crossref]

S. R. Seshadri, “Electromagnetic Gaussian beam,” J. Opt. Soc. Am. A 15, 2712–2719 (1998).
[Crossref]

1996 (2)

Hall, D. G.

Saghafi, S.

Seshadri, S. R.

Sheppard, C. J. R.

Török, P.

P. Varga, P. Török, “The Gaussian wave solution of Maxwell’s equations and the validity of the scalar wave approximation,” Opt. Commun. 152, 108–118 (1998).
[Crossref]

P. Varga, P. Török, “Exact and approximate solutions of Maxwell’s equations for a confocal cavity,” Opt. Lett. 21, 1523–1525 (1996).
[Crossref] [PubMed]

Varga, P.

P. Varga, P. Török, “The Gaussian wave solution of Maxwell’s equations and the validity of the scalar wave approximation,” Opt. Commun. 152, 108–118 (1998).
[Crossref]

P. Varga, P. Török, “Exact and approximate solutions of Maxwell’s equations for a confocal cavity,” Opt. Lett. 21, 1523–1525 (1996).
[Crossref] [PubMed]

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Figures (1)

Fig. 1
Fig. 1

Spatial structure of the transverse term (ETE)G that is associated with a linearly polarized Gaussian beam: (a) The intensity profile I normalized so that the integrated intensity equals 1. (b) The modulus of the x component |Ex|. (c) The modulus of the y component |Ey| (the z component is zero). At each point, I=|Ex|2+|Ey|2, as expected. For comparison, note that (|Ex|)max/(|Ey|)max=3.23, where the subscript max refers to the respective maximum values.

Equations (23)

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×H+ikE=0,
×E-ikH=0,
  E=  H=0,
E(x, y, z)=E˜(u, v, z)exp[ik(xu+yv)]dudv,
H(x, y, z)=H˜(u, v, z)exp[ik(xu+yv)]dudv,
L×H˜+ikE˜=0,
L×E˜-ikH˜=0,
L  E˜=L  H˜=0,
E˜(ρ, ϕ, z)=E˜0(ρ, ϕ)exp[ikz(1-ρ2)1/2],
E˜0(ρ, ϕ)  s(ρ, ϕ)=0,
s(ρ, ϕ)=[ρ cos ϕ, ρ sin ϕ, (1-ρ2)1/2],
H˜(ρ, ϕ, z)=s(ρ, ϕ)×E˜0(ρ, ϕ)exp[ikz(1-ρ2)1/2].
E˜0(ρ, ϕ)=a(ρ, ϕ)e1(ρ, ϕ)+b(ρ, ϕ)e2(ρ, ϕ),
e1=(sin ϕ, -cos ϕ, 0),
e2=[(1-ρ2)1/2cos ϕ, (1-ρ2)1/2sin ϕ, -ρ].
ETE(r)=0102πa(ρ, ϕ)e1exp(ikr  s)ρdρdϕ,
ETM(r)=0102πb(ρ, ϕ)e2exp(ikr  s)ρdρdϕ,
f(x, y)=0102πf˜(ρ, ϕ)×exp[ik(xρ cos ϕ+yρ sin ϕ)]ρdρdϕ.
Ef(r)=0102π( f˜  e1)e1exp(ikr  s)ρdρdϕ+0102π( f˜  e2)e2exp(ikr  s)ρdρdϕ.
fG(x, y)=iC0102πexp[-(ρ2/D2)]×exp[ik(xρ cos ϕ+yρ sin ϕ)]ρdρdϕ,
EG=(ETE)G+(ETM)G,
(ETE)G(r)=C0102πsin ϕ exp[-(ρ2/D2)]e1×exp(ikr  s)ρdρdϕ,
(ETM)G(r)=C0102πcos ϕ(1-ρ2)1/2×exp[-(ρ2/D2)]e2exp(ikr  s)ρdρdϕ.

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