Abstract

The oblate spheroidal wavefunctions proposed by Rodríguez-Morales and Chávez-Cerda are shown to be possible representations of physical beams only when the angular function Smn(β,η) has odd nm. This condition makes Smn odd in η, which ensures the convergence of integrals of physical quantities over a cross section of the beam. The odd nm condition also makes Smn(β,η) zero in the focal plane z=0 outside the circle ρ=b, and thus allows for the physically necessary discontinuity in phase at z=0 on the ellipsoidal surfaces of otherwise constant phase. Only a subset of the oblate spheroidal functions can be exact representations of nonparaxial scalar beams.

© 2010 Optical Society of America

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References

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  1. G. Rodríguez-Morales and S. Chávez-Cerda, Opt. Lett. 29, 430 (2004).
    [CrossRef] [PubMed]
  2. C. Flammer, Spheroidal Wavefunctions (Stanford U. Press, 1957).
  3. A. N. Lowan, Handbook of Mathematical Functions, M.Abramowitz and I.A.Stegun, eds. (Dover, 1972), Chap. 21.
  4. C. J. R. Sheppard and S. Saghafi, Phys. Rev. A 57, 2971(1998).
    [CrossRef]
  5. Z. Ulanowski and I. K. Ludlow, Opt. Lett. 25, 1792 (2000).
    [CrossRef]
  6. J. Lekner, J. Opt. A 3, 407 (2001).
    [CrossRef]
  7. J. Lekner, J. Opt. A 4, 491 (2002).
    [CrossRef]
  8. J. Lekner, J. Phys. B 37, 1725 (2004).
    [CrossRef]
  9. A. April, Opt. Lett. 33, 1392 (2008).
    [CrossRef] [PubMed]
  10. A. April, Opt. Lett. 33, 1563 (2008).
    [CrossRef] [PubMed]
  11. A. April, J. Opt. Soc. Am. A 27, 76 (2010).
    [CrossRef]
  12. N. Bokor and N. Davidson, Opt. Commun. 281, 5499 (2008).
    [CrossRef]

2010

2008

2004

2002

J. Lekner, J. Opt. A 4, 491 (2002).
[CrossRef]

2001

J. Lekner, J. Opt. A 3, 407 (2001).
[CrossRef]

2000

1998

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

April, A.

Bokor, N.

N. Bokor and N. Davidson, Opt. Commun. 281, 5499 (2008).
[CrossRef]

Chávez-Cerda, S.

Davidson, N.

N. Bokor and N. Davidson, Opt. Commun. 281, 5499 (2008).
[CrossRef]

Flammer, C.

C. Flammer, Spheroidal Wavefunctions (Stanford U. Press, 1957).

Lekner, J.

J. Lekner, J. Phys. B 37, 1725 (2004).
[CrossRef]

J. Lekner, J. Opt. A 4, 491 (2002).
[CrossRef]

J. Lekner, J. Opt. A 3, 407 (2001).
[CrossRef]

Lowan, A. N.

A. N. Lowan, Handbook of Mathematical Functions, M.Abramowitz and I.A.Stegun, eds. (Dover, 1972), Chap. 21.

Ludlow, I. K.

Rodríguez-Morales, G.

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]

Ulanowski, Z.

J. Opt. A

J. Lekner, J. Opt. A 3, 407 (2001).
[CrossRef]

J. Lekner, J. Opt. A 4, 491 (2002).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. B

J. Lekner, J. Phys. B 37, 1725 (2004).
[CrossRef]

Opt. Commun.

N. Bokor and N. Davidson, Opt. Commun. 281, 5499 (2008).
[CrossRef]

Opt. Lett.

Phys. Rev. A

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

Other

C. Flammer, Spheroidal Wavefunctions (Stanford U. Press, 1957).

A. N. Lowan, Handbook of Mathematical Functions, M.Abramowitz and I.A.Stegun, eds. (Dover, 1972), Chap. 21.

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Figures (3)

Fig. 1
Fig. 1

The oblate spheroidal coordinate system. The confocal ellipsoids (solid curves) are surfaces of constant ξ. The central ellipsoid (the disk ρ b , z = 0 ) is at ξ = 0 , with ξ increasing or decreasing by 0.2 in each of the outer half-ellipsoids. The confocal hyperboloids (dashed curves) are surfaces of constant η. In the focal plane z = 0 , the region outside the disk ρ > b is shown by the solid line η = 0 . The z axis is given by η = 1 . The hyperboloids increase in η by 0.2 from the η = 0 central hyperboloid.

Fig. 2
Fig. 2

Surfaces of constant phase for ψ G (upper figure), j 0 ( k R ) (middle figure), and ψ 01 (lower figure), drawn for β = 2 in increments of π / 4 (darker curves are at multiples of π). The equiphase surfaces for ψ G all go off to infinite ρ in the z = 0 plane, while those of j 0 ( k R ) converge on to the zeros of j 0 ( k R ) , which lie on the circles ρ ν = b [ ( ν π / β ) 2 + 1 ] 1 2 . The surfaces with phase equal to an integer multiple of π converge onto the circles ρ = b [ ( X / β ) 2 + 1 ] 1 2 , where tan ( X ) = X . For the spheroidal wavefunction ψ 01 , the equiphase surfaces are half-ellipsoids ξ = ξ 0 , ξ = ξ 0 . These half-ellipsoids tend to hemispheres for large ξ 0 2 .

Fig. 3
Fig. 3

The modulus squared of the spheroidal wavefunction ψ 01 ( β , ξ , η ) = i R 01 ( 3 ) ( β , ξ ) S 01 ( β , η ) in the y = 0 plane, plotted for a tightly focused beam with β = k b = 2 . The z axis defines the propagation direction of the beam, and ρ = x 2 + y 2 is the radial distance from the beam axis. Note that the solution is identically zero outside the circle ρ = b in the focal plane z = 0 .

Equations (16)

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ψ m n ( β , ξ , η ) = R m n ( 3 ) ( β , ξ ) S m n ( β , η ) e i m ϕ ,
ρ = b [ ( ξ 2 + 1 ) ( 1 η 2 ) ] 1 2 , z = b ξ η , ϕ = ϕ .
{ ξ ( ξ 2 + 1 ) ξ + η ( 1 η 2 ) η + ξ 2 + η 2 ( ξ 2 + 1 ) ( 1 η 2 ) ϕ 2 + β 2 ( ξ 2 + η 2 ) } ψ = 0.
( ξ 2 + 1 ) d 2 R d ξ 2 + 2 ξ d R d ξ + ( β 2 ξ 2 + m 2 ξ 2 + 1 α ) R = 0 ,
( 1 η 2 ) d 2 S d η 2 2 η d S d η + ( β 2 η 2 m 2 1 η 2 + α ) S = 0.
R m n ( 1 ) ( β , ξ ) 1 β ξ cos [ β ξ π 2 ( n + 1 ) ] ,
R m n ( 2 ) ( β , ξ ) 1 β ξ sin [ β ξ π 2 ( n + 1 ) ] .
R m n ( 3 ) ( β , ξ ) 1 β ξ exp { i [ β ξ π 2 ( n + 1 ) ] } .
ψ m n ( β , ξ , η ) = i R m n ( 3 ) ( β , ξ ) S m n ( β , η ) e i m ϕ .
P m n ( β , ξ , ϕ ) = m ϕ + arctan [ R m n ( 1 ) ( β , ξ ) / R m n ( 2 ) ( β , ξ ) ] .
ρ 2 b 2 ( ξ 0 2 + 1 ) + z 2 b 2 ξ 0 2 = 1.
ρ 2 b 2 ( 1 η 0 2 ) = z 2 b 2 η 0 2 + 1.
ψ G ( ρ , z ) = b b + i z exp [ i k z k ρ 2 2 ( b + i z ) ] ,
j 0 ( k R ) = sin k R k R , R = b ( ξ i η ) = ( z i b ) [ 1 + ρ 2 ( z i b ) 2 ] 1 2 .
0 d ρ ρ | ψ | 2 = b 2 0 1 d η η 1 [ η 2 + ( z 0 b η ) 2 ] | ψ | 2 ,
| ψ m n | 2 ( β ξ ) 2 [ S m n ( β , η ) ] 2 = η 2 ( k z 0 ) 2 [ S m n ( β , η ) ] 2 .

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