Abstract

A family of closed-form expressions for the scalar field of strongly focused Gaussian beams in oblate spheroidal coordinates is given. The solutions satisfy the wave equation and are free from singularities. The lowest-order solution in the far field closely matches the energy density produced by a sine-condition, high-aperture lens illuminated by a paraxial Gaussian beam. At the large waist limit the solution reduces to the paraxial Gaussian beam form. The solution is equivalent to the spherical wave of a combined complex point source and sink but has the advantage of being more directly interpretatable.

© 2000 Optical Society of America

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References

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    [CrossRef]
  2. C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957).
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    [CrossRef]
  4. G. A. Deschamps, Electron. Lett. 7, 684 (1971).
    [CrossRef]
  5. B. Richards and E. Wolf, Proc. R. Soc. London Ser. A 253, 358 (1959).
    [CrossRef]
  6. M. Couture and P. A. Bélanger, Phys. Rev. A 24, 355 (1981).
    [CrossRef]

1998

C. J. R. Sheppard and S. Saghafi, Phys. Rev. A 57, 2971 (1998).
[CrossRef]

1988

1981

M. Couture and P. A. Bélanger, Phys. Rev. A 24, 355 (1981).
[CrossRef]

1971

G. A. Deschamps, Electron. Lett. 7, 684 (1971).
[CrossRef]

1959

B. Richards and E. Wolf, Proc. R. Soc. London Ser. A 253, 358 (1959).
[CrossRef]

Barrett, H. H.

Bélanger, P. A.

M. Couture and P. A. Bélanger, Phys. Rev. A 24, 355 (1981).
[CrossRef]

Couture, M.

M. Couture and P. A. Bélanger, Phys. Rev. A 24, 355 (1981).
[CrossRef]

Deschamps, G. A.

G. A. Deschamps, Electron. Lett. 7, 684 (1971).
[CrossRef]

Flammer, C.

C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957).

Landesman, T. B.

Richards, B.

B. Richards and E. Wolf, Proc. R. Soc. London Ser. A 253, 358 (1959).
[CrossRef]

Saghafi, S.

C. J. R. Sheppard and S. Saghafi, Phys. Rev. A 57, 2971 (1998).
[CrossRef]

Sheppard, C. J. R.

C. J. R. Sheppard and S. Saghafi, Phys. Rev. A 57, 2971 (1998).
[CrossRef]

Wolf, E.

B. Richards and E. Wolf, Proc. R. Soc. London Ser. A 253, 358 (1959).
[CrossRef]

Electron. Lett.

G. A. Deschamps, Electron. Lett. 7, 684 (1971).
[CrossRef]

J. Opt. Soc. Am. A

Phys. Rev. A

C. J. R. Sheppard and S. Saghafi, Phys. Rev. A 57, 2971 (1998).
[CrossRef]

M. Couture and P. A. Bélanger, Phys. Rev. A 24, 355 (1981).
[CrossRef]

Proc. R. Soc. London Ser. A

B. Richards and E. Wolf, Proc. R. Soc. London Ser. A 253, 358 (1959).
[CrossRef]

Other

C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957).

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Figures (4)

Fig. 1
Fig. 1

Real part of the wave, Eq.  (1), plotted in the vicinity of the geometric focus with kd=2.3, showing the singularity and the discontinuity.

Fig. 2
Fig. 2

Field contours at t=0 for the combined wave, Eq.  (3), with kd=2.3, shown in the plane of the beam axis.

Fig. 3
Fig. 3

Time-averaged energy density of the wave, Eq.  (3), with kd=1, shown in the focal plane z=0 and normalized to 1 at x=0.

Fig. 4
Fig. 4

Far-field irradiance of the Gaussian wave, expression  (6), for beams with (a) kd=5 and (b) kd=2.3 (solid curves) compared with irradiance at a sine-condition lens, expression  (7) (dotted curves), plotted versus polar angle. The divergence half-angles are (a) 39° and (b) 69°.

Equations (9)

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

ψ001=expkdη+iξexpkdkdη+iξ,
x=d1+ξ21-η2 cos φ,  y=d1+ξ21-η2 sin φ,  z=dξη.
ψ00=ψ001-exp-kdη+iξexpkdkdη+iξ=2sinkdξ-iηexpkdkdξ-iη.
exp-kx2+y2/2d/kd,
-exp-kd1-cos αexp-ikr/ikr,
exp-2kd1-cos α.
cos α exp-2f sin αw2,
ψmn=exp-kdjnpPnmsexp±imφ,
ψmn=exp-kdjnkRPnmz-idRexp±imφ,

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