Abstract

Based on a far-field asymptotic solution of the Helmholtz equation a vector theory to describe the propagation of an off-axis Gaussian wave is developed, the accurate formulas represented in terms of elementary functions are derived, and the propagation properties such as wave spot size and divergence angle are discussed in detail. The applicable range of scalar theory is also presented. A relative error criterion of optical intensity is given by ∊ = sin2 α.

© 1999 Optical Society of America

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References

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  1. M. Lax, W. H. Louisell, W. B. Mcknight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [CrossRef]
  2. G. P. Agrawal, D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. 69, 575–578 (1979).
    [CrossRef]
  3. M. Couture, P. A. Belanger, “From Gaussian beam to complex source point spherical wave,” Phys. Rev. 24, 355–359 (1981).
    [CrossRef]
  4. G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
    [CrossRef]
  5. T. Takenaka, M. Yokota, O. Fukumitsu, “Propagation of light beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 2, 826–829 (1985).
    [CrossRef]
  6. S. Nemoto, “Nonparaxial Gaussian beams,” Appl. Opt. 29, 1940–1946 (1990).
    [CrossRef] [PubMed]
  7. M. A. Porras, “Non-paraxial vectorial moment theory of light beam propagation,” Opt. Commun. 127, 79–95 (1996).
    [CrossRef]
  8. X. Zeng, C. Liang, Y. An, “Far-field radiation of planar Gaussian sources and comparison with solutions based on the parabolic approximation,” Appl. Opt. 36, 4491–4496 (1997).
    [CrossRef]

1997

X. Zeng, C. Liang, Y. An, “Far-field radiation of planar Gaussian sources and comparison with solutions based on the parabolic approximation,” Appl. Opt. 36, 4491–4496 (1997).
[CrossRef]

1996

M. A. Porras, “Non-paraxial vectorial moment theory of light beam propagation,” Opt. Commun. 127, 79–95 (1996).
[CrossRef]

1990

1985

1983

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

1981

M. Couture, P. A. Belanger, “From Gaussian beam to complex source point spherical wave,” Phys. Rev. 24, 355–359 (1981).
[CrossRef]

1979

1975

M. Lax, W. H. Louisell, W. B. Mcknight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

G. P. Agrawal, D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. 69, 575–578 (1979).
[CrossRef]

An, Y.

X. Zeng, C. Liang, Y. An, “Far-field radiation of planar Gaussian sources and comparison with solutions based on the parabolic approximation,” Appl. Opt. 36, 4491–4496 (1997).
[CrossRef]

Belanger, P. A.

M. Couture, P. A. Belanger, “From Gaussian beam to complex source point spherical wave,” Phys. Rev. 24, 355–359 (1981).
[CrossRef]

Couture, M.

M. Couture, P. A. Belanger, “From Gaussian beam to complex source point spherical wave,” Phys. Rev. 24, 355–359 (1981).
[CrossRef]

Fukumitsu, O.

Lax, M.

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

M. Lax, W. H. Louisell, W. B. Mcknight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Liang, C.

X. Zeng, C. Liang, Y. An, “Far-field radiation of planar Gaussian sources and comparison with solutions based on the parabolic approximation,” Appl. Opt. 36, 4491–4496 (1997).
[CrossRef]

Louisell, W. H.

M. Lax, W. H. Louisell, W. B. Mcknight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Mcknight, W. B.

M. Lax, W. H. Louisell, W. B. Mcknight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Nemoto, S.

Pattanayak, D. N.

Porras, M. A.

M. A. Porras, “Non-paraxial vectorial moment theory of light beam propagation,” Opt. Commun. 127, 79–95 (1996).
[CrossRef]

Takenaka, T.

Yokota, M.

Zeng, X.

X. Zeng, C. Liang, Y. An, “Far-field radiation of planar Gaussian sources and comparison with solutions based on the parabolic approximation,” Appl. Opt. 36, 4491–4496 (1997).
[CrossRef]

Appl. Opt.

X. Zeng, C. Liang, Y. An, “Far-field radiation of planar Gaussian sources and comparison with solutions based on the parabolic approximation,” Appl. Opt. 36, 4491–4496 (1997).
[CrossRef]

S. Nemoto, “Nonparaxial Gaussian beams,” Appl. Opt. 29, 1940–1946 (1990).
[CrossRef] [PubMed]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

M. A. Porras, “Non-paraxial vectorial moment theory of light beam propagation,” Opt. Commun. 127, 79–95 (1996).
[CrossRef]

Phys. Rev.

M. Couture, P. A. Belanger, “From Gaussian beam to complex source point spherical wave,” Phys. Rev. 24, 355–359 (1981).
[CrossRef]

Phys. Rev. A

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

M. Lax, W. H. Louisell, W. B. Mcknight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Geometry of the far-field components.

Fig. 2
Fig. 2

Variance of the divergence angles with ω0/λ.

Equations (27)

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2Er+k2Er=0
Exx, y, 0=A exp-x2+y2ω02,
Eyx, y, 0=0,
Exx, y, z=B zr2expikrexp- 1C2x2+y2r2,
B=iAπω02λ,
C2=4k2ω02,
r2=x2+y2+z2.
Ezx, y, z=-xz Exx, y, z=-B xr2expikrexp-1C2x2+y2r2.
Ex, y, z=B expikrr2exp-1C2x2+y2r2 z0-x.
Ix, y, z=|Ex|2+|Ez|2=|B|2x2+z2r4exp-2C2x2+y2r2.
12z2=x02+z2r04exp-2C2x02+y02r02,
12z2=x02+z2r04exp-2C2x02r02.
12=1-x02r02exp-2C2x02r02.
x02x02+z2=s0
x0=s01-s0 z.
12=1-y02r022 exp-2C2y02r02.
y0=t01-t0 z.
θx=2 arctanx0z=2 arctans01-s0,
θy=2 arctany0z=2 arctant01-t0.
θx max=2 arctan 1=90°,
θy max=2 arctan2-11/265.5°.
Iθ, ρ, z=|B|2ρ2 cos2 θ+z2ρ2+z22exp-2C2ρ2ρ2+z2.
-+-+ Ix, y, zdxdy=|B|202πdθ 0+ρ2 cos2 θ+z2ρ2+z22×exp-2C2ρ2ρ2+z2ρdρ=|B|2π 0+ρ2+2z2ρ2+z22×exp-2C2ρ2ρ2+z2ρdρ.
-+-+ Ix, y, zdxdy=π|B|220+12-u1-u×exp-2C2 udu.
=|I-Is|I=x2x2+z2.
=sin2 α.
αarcsinδ.

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