Abstract

The monochromatic scalar fields that achieve the maximum focal irradiance for a given input power and directional spread are found through a variational approach. In two dimensions, the maximum focal irradiance is found to be proportional to the sine of the directional spread as well as the input power. In three dimensions, the corresponding relation is found in parametric form.

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References

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  1. I. M. Bassett, "Limit to concentration by focusing," Opt. Acta 33, 279-286 (1986).
    [Crossref]
  2. C. J. R. Sheppard and K. G. Larkin, "Optimal concentration of electromagnetic radiation," J. Mod. Opt. 41, 1495-1505 (1994).
    [Crossref]
  3. R. K. Luneburg, Mathematical Theory of Optics (U. of California Press, 1964), pp. 386-395.
  4. R. Barakat, "Solution of the Luneberg apodization problems," J. Opt. Soc. Am. 52, 264-275 (1962).
    [Crossref]
  5. J. E. Wilkins, Jr., "Luneberg apodization problems," J. Opt. Soc. Am. 53, 420-424 (1963).
    [Crossref]
  6. M. W. Kowarz, "Energy constraints in optimum apodization problems," Opt. Commun. 110, 274-278 (1994).
    [Crossref]
  7. See, for example, N. Bokor and N. Davidson, "Toward a spherical spot distribution with 4π focusing of radially polarized light," Opt. Lett. 29, 1968-1970 (2004).
    [Crossref] [PubMed]
  8. N. J. Moore, M. A. Alonso, and C. J. R. Sheppard, "Monochromatic electromagnetic fields with maximum focal energy density" (submitted to J. Opt. Soc. Am. A ).
  9. M. A. Alonso and G. W. Forbes, "Uncertainty products for nonparaxial wave fields," J. Opt. Soc. Am. A 17, 2391-2402 (2000).
    [Crossref]
  10. M. A. Alonso, R. Borghi, and M. Santarsiero, "Nonparaxial fields with maximum joint spatial-directional localization. I. Scalar case," J. Opt. Soc. Am. A 23, 691-700 (2006).
    [Crossref]
  11. See, for example, L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), pp. 109-127.
  12. It is sufficient to consider variations due to the complex conjugate of the angular spectrum as shown in Ref. .
  13. Notice that Eqs. are consistent with this statement. Incidentally, from combining these equations, one can see that the Lagrange multiplier μ2 equals Δp/DR(2) in this case.
  14. The fact that 2πα2/(β22−1)1/2 equals unity is consistent with Eqs. . In establishing this, one can find that μ2=tanΔθ/DR(2) by combining these equations.
  15. This equality is consistent with Eqs. .
  16. C. J. R. Sheppard and S. Saghafi, "Beam modes beyond the paraxial approximation: a scalar treatment," Phys. Rev. A 57, 2971-2979 (1998).
    [Crossref]
  17. C. J. R. Sheppard, "High-aperture beams," J. Opt. Soc. Am. A 18, 1579-1587 (2001).
    [Crossref]
  18. M. A. Alonso, R. Borghi, and M. Santarsiero, "Joint spatial-directional localization features of wave fields focused at a complex point," J. Opt. Soc. Am. A 23, 933-939 (2006).
    [Crossref]
  19. V. Bargmann, "Irreducible unitary representations of the Lorentz group," Ann. Math. 48, 568-640 (1947).
    [Crossref]
  20. M. A. Alonso, R. Borghi, and M. Santarsiero, "New basis for rotationally symmetric nonparaxial fields in terms of spherical waves with complex foci," Opt. Express 14, 6894-6905 (2006).
    [Crossref] [PubMed]

2006 (3)

2004 (1)

2001 (1)

2000 (1)

1998 (1)

C. J. R. Sheppard and S. Saghafi, "Beam modes beyond the paraxial approximation: a scalar treatment," Phys. Rev. A 57, 2971-2979 (1998).
[Crossref]

1995 (1)

See, for example, L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), pp. 109-127.

1994 (2)

C. J. R. Sheppard and K. G. Larkin, "Optimal concentration of electromagnetic radiation," J. Mod. Opt. 41, 1495-1505 (1994).
[Crossref]

M. W. Kowarz, "Energy constraints in optimum apodization problems," Opt. Commun. 110, 274-278 (1994).
[Crossref]

1986 (1)

I. M. Bassett, "Limit to concentration by focusing," Opt. Acta 33, 279-286 (1986).
[Crossref]

1964 (1)

R. K. Luneburg, Mathematical Theory of Optics (U. of California Press, 1964), pp. 386-395.

1963 (1)

1962 (1)

1947 (1)

V. Bargmann, "Irreducible unitary representations of the Lorentz group," Ann. Math. 48, 568-640 (1947).
[Crossref]

Alonso, M. A.

Barakat, R.

Bargmann, V.

V. Bargmann, "Irreducible unitary representations of the Lorentz group," Ann. Math. 48, 568-640 (1947).
[Crossref]

Bassett, I. M.

I. M. Bassett, "Limit to concentration by focusing," Opt. Acta 33, 279-286 (1986).
[Crossref]

Bokor, N.

Borghi, R.

Davidson, N.

Forbes, G. W.

Kowarz, M. W.

M. W. Kowarz, "Energy constraints in optimum apodization problems," Opt. Commun. 110, 274-278 (1994).
[Crossref]

Larkin, K. G.

C. J. R. Sheppard and K. G. Larkin, "Optimal concentration of electromagnetic radiation," J. Mod. Opt. 41, 1495-1505 (1994).
[Crossref]

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. of California Press, 1964), pp. 386-395.

Mandel, L.

See, for example, L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), pp. 109-127.

Moore, N. J.

N. J. Moore, M. A. Alonso, and C. J. R. Sheppard, "Monochromatic electromagnetic fields with maximum focal energy density" (submitted to J. Opt. Soc. Am. A ).

Saghafi, S.

C. J. R. Sheppard and S. Saghafi, "Beam modes beyond the paraxial approximation: a scalar treatment," Phys. Rev. A 57, 2971-2979 (1998).
[Crossref]

Santarsiero, M.

Sheppard, C. J. R.

C. J. R. Sheppard, "High-aperture beams," J. Opt. Soc. Am. A 18, 1579-1587 (2001).
[Crossref]

C. J. R. Sheppard and S. Saghafi, "Beam modes beyond the paraxial approximation: a scalar treatment," Phys. Rev. A 57, 2971-2979 (1998).
[Crossref]

C. J. R. Sheppard and K. G. Larkin, "Optimal concentration of electromagnetic radiation," J. Mod. Opt. 41, 1495-1505 (1994).
[Crossref]

N. J. Moore, M. A. Alonso, and C. J. R. Sheppard, "Monochromatic electromagnetic fields with maximum focal energy density" (submitted to J. Opt. Soc. Am. A ).

Wilkins, J. E.

Wolf, E.

See, for example, L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), pp. 109-127.

Ann. Math. (1)

V. Bargmann, "Irreducible unitary representations of the Lorentz group," Ann. Math. 48, 568-640 (1947).
[Crossref]

J. Mod. Opt. (1)

C. J. R. Sheppard and K. G. Larkin, "Optimal concentration of electromagnetic radiation," J. Mod. Opt. 41, 1495-1505 (1994).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (5)

Opt. Acta (1)

I. M. Bassett, "Limit to concentration by focusing," Opt. Acta 33, 279-286 (1986).
[Crossref]

Opt. Commun. (1)

M. W. Kowarz, "Energy constraints in optimum apodization problems," Opt. Commun. 110, 274-278 (1994).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (1)

C. J. R. Sheppard and S. Saghafi, "Beam modes beyond the paraxial approximation: a scalar treatment," Phys. Rev. A 57, 2971-2979 (1998).
[Crossref]

Other (6)

R. K. Luneburg, Mathematical Theory of Optics (U. of California Press, 1964), pp. 386-395.

See, for example, L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), pp. 109-127.

It is sufficient to consider variations due to the complex conjugate of the angular spectrum as shown in Ref. .

Notice that Eqs. are consistent with this statement. Incidentally, from combining these equations, one can see that the Lagrange multiplier μ2 equals Δp/DR(2) in this case.

The fact that 2πα2/(β22−1)1/2 equals unity is consistent with Eqs. . In establishing this, one can find that μ2=tanΔθ/DR(2) by combining these equations.

This equality is consistent with Eqs. .

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Figures (7)

Fig. 1
Fig. 1

Geometric interpretation of the measure of angular spread Δ θ . The unit circle for 2D (left) or the unit sphere for 3D (right) is weighted by A ( u ) 2 . The measure Δ θ is the half-angle subtended at the center of the circle (sphere) by the perpendicular chord (plane) through the centroid. The distance from centroid to center is u z = cos Δ θ .

Fig. 2
Fig. 2

(a) Geometric interpretation of the measure of spatial spread D R ( 2 ) for 2D paraxial fields as the half-width of a rectangular distribution with the area equal to the field’s total power and height equal to the field’s irradiance at the focal plane. (b) Spatial spread D R ( 3 ) for 3D paraxial fields is the radius of a cylindrical distribution with the volume equal to the field’s total power and height equal to the field’s irradiance at the focal plane.

Fig. 3
Fig. 3

Irradiance U ( r ) 2 of the optimal 2D paraxial field, normalized at the focus, presented using the scaled coordinates ϵ 2 x and ϵ 2 2 z .

Fig. 4
Fig. 4

Irradiance U ( r ) 2 of an optimal 2D field for a given total power U 2 for (a) β 2 = 1.01 , (b) β 2 = 1.1 , and (c) β 2 = 2 . Note that the focal irradiance increases significantly as β 2 (and consequently the directional spread of the field) increases.

Fig. 5
Fig. 5

U ( x , 0 ) U ( 0 ) with β 2 = 1 + ϵ 2 2 2 for ϵ 2 = 0.01 (solid curve), ϵ 2 = 0.1 (dashed curve), and ϵ 2 = 1 (dashed–dotted curve), as a function of ϵ 2 x . As the paraxial limit is approached ( β 2 1 ) , the normalized field tends toward exp ( ϵ 2 x ) .

Fig. 6
Fig. 6

Irradiance U ( r ) 2 of an optimal 3D filed for a given total power U 2 for (a) β 3 = 1.01 , (b) β 3 = 1.1 , and (c) β 3 = 2 . Note that the focal irradiance increases significantly as β 3 (and consequently the directional spread of the field) increases. For optimal 3D fields, the irradiance is more concentrated at the focus than for optimal 2D fields.

Fig. 7
Fig. 7

Forbidden regions (dark) over the plane d R ( N ) versus Δ θ for (a) 2D fields and (b) 3D fields. Also displayed are the spreads of CF fields (dashed curve) and of fields with truncated angular spectra (dashed–dotted curve) in both two and three dimensions.

Equations (70)

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( 2 + k 2 ) U ( r ) = 0 .
U ( r ) = S N A ( u ) exp ( i k r u ) d Ω ,
U ̃ 0 ( p ) = ( 1 2 π ) N 1 R N 1 U 0 ( x ) exp ( i x p ) d N 1 x ,
U 0 ( x ) = A ( u ) exp [ i ( u x x + u y y ) ] d u x d u y u z = U ̃ 0 ( p ) exp [ i ( p x x + p y y ) ] d p x d p y .
U ̃ 0 ( p ) = A ( u ) u z = A ( u ) cos θ ,
U 2 = S N A ( u ) 2 d Ω .
Δ θ = arccos u z ,
u j = 1 U 2 S N u j A ( u ) 2 d Ω ,
U ̃ ( p , z ) = U ̃ 0 ( p ) exp ( i z 1 p 2 ) U ̃ 0 ( p ) exp [ i z ( 1 p 2 2 ) ] .
U ( x , z ) U ̃ 0 ( p ) exp ( i x p ) exp [ i z ( 1 p 2 2 ) ] d N 1 p .
cos Δ θ = cos θ A ( u ) 2 d θ A ( u ) 2 d θ ( 1 p 2 2 ) U ̃ 0 ( p ) 2 d N 1 p U ̃ 0 ( p ) 2 d N 1 p .
Δ θ Δ p = [ p 2 U ̃ 0 ( p ) 2 d N 1 p U 2 ] 1 2 ,
U 2 U ̃ 0 ( p ) 2 d N 1 p = ( 1 2 π ) N 1 U 0 ( x ) 2 d N 1 x .
D R ( 2 ) = π U 2 U 0 ( 0 ) 2 .
D R ( 3 ) = 4 π U 2 U 0 ( 0 ) 2 .
δ Δ p δ U ̃ 0 * + μ N δ D R ( N ) δ U ̃ 0 * = 0 ,
δ Δ p δ U ̃ 0 * = δ δ U ̃ 0 * [ U ̃ 0 * ( p ) p 2 U ̃ 0 ( p ) d N 1 p U ̃ 0 * ( p ) U ̃ 0 ( p ) d N 1 p ] 1 2 = p 2 Δ p 2 2 Δ p U 2 U ̃ 0 ( p ) .
δ D R ( 2 ) δ U 0 * = D R ( 2 ) U 2 U ̃ 0 ( p ) [ D R ( 2 ) ] 2 U 0 ( 0 ) π U 2 ,
δ D R ( 3 ) δ U 0 * = D R ( 3 ) 2 U 2 U ̃ 0 ( p ) [ D R ( 3 ) ] 3 U 0 ( 0 ) 8 π U 2 .
U ̃ 0 ( p ) = γ N p 2 + ϵ N 2 U 0 ( 0 ) ,
γ 2 = 2 μ 2 Δ p [ D R ( 2 ) ] 2 π ,
ϵ 2 = Δ p ( 2 μ 2 D R ( 2 ) Δ p ) ,
γ 3 = μ 3 Δ p [ D R ( 3 ) ] 3 4 π ,
ϵ 3 = Δ p ( μ 3 D R ( 3 ) Δ p ) .
U 0 ( x ) = U 0 ( 0 ) γ 2 p 2 + ϵ 2 2 exp ( i p x ) d p = U 0 ( 0 ) π γ 2 ϵ 2 exp ( ϵ 2 x ) .
U ( x , z ) = 2 π 2 i U 0 ( 0 ) γ 2 ϵ 2 exp [ i z ( 1 + ϵ 2 2 2 ) ] { exp ( ϵ 2 x ) [ 1 + i 2 F ( x + i ϵ 2 z π z ) ] + exp ( ϵ 2 x ) [ 1 + i 2 F ( x + i ϵ 2 z π z ) ] } ,
D R ( 2 ) = 1 2 ϵ 2 , Δ p = ϵ 2 .
D R ( 2 ) Δ p 1 2 .
U 0 ( 0 ) 2 2 π Δ p U 2 .
U 2 = U 0 ( 0 ) 2 π γ 3 2 ϵ 3 2 .
U ( ρ cos ψ , ρ sin ψ , 0 ) K 0 ( ϵ 3 ρ ) ,
U ( 0 , 0 , z ) exp [ i z ( 1 + ϵ 3 2 2 ) ] E 1 ( i z ϵ 3 2 2 ) ,
δ Δ θ δ A * + μ N δ D R ( N ) δ A * = 0 .
δ Δ θ δ A * = cos Δ θ u z U 2 sin Δ θ A ( u ) ,
δ D R ( 2 ) δ A * = D R ( 2 ) U 2 A ( u ) [ D R ( 2 ) ] 2 U ( 0 ) π U 2 ,
δ D R ( 3 ) δ A * = D R ( 3 ) 2 U 2 A ( u ) [ D R ( 3 ) ] 3 U ( 0 ) 8 π U 2 .
A ( u ) = α N β N u z U ( 0 ) ,
α 2 = μ 2 sin Δ θ [ D R ( 2 ) ] 2 π ,
β 2 = cos Δ θ + μ 2 sin Δ θ D R ( 2 ) ,
α 3 = μ 3 sin Δ θ [ D R ( 3 ) ] 3 8 π ,
β 3 = cos Δ θ + 1 2 μ 3 sin Δ θ D R ( 3 ) .
U 2 = U ( 0 ) 2 2 π α 2 2 β 2 ( β 2 2 1 ) 3 2 ,
U ( 0 ) = U ( 0 ) 2 π α 2 ( β 2 2 1 ) 1 2 ,
D R ( 2 ) = β 2 2 β 2 2 1 .
u z = cos θ A ( u ) 2 d θ U 2 = 1 β 2 ,
sin Δ θ = β 2 2 1 β 2 .
D R ( 2 ) sin Δ θ 1 2 .
U ( 0 ) 2 2 π sin Δ θ U 2 .
U 2 = U ( 0 ) 2 4 π α 3 2 β 3 2 1 ,
U ( 0 ) = U ( 0 ) 2 π α 3 ln β 3 + 1 β 3 1 .
D R ( 3 ) = 2 β 3 2 1 ( ln β 3 + 1 β 3 1 ) 1 .
u z = β 3 2 1 4 π α 3 2 0 2 π 0 π α 3 2 ( β 3 cos θ ) 2 cos θ sin θ d θ = β 3 β 3 2 1 2 ln β 3 + 1 β 3 1 .
a 1 2 ln β 3 + 1 β 3 1 ,
D R ( 3 ) = sinh a a ,
Δ θ = arccos ( coth a a csch 2 a ) .
D R ( 3 ) sinh a a ,
U ( 0 ) 2 4 π a 2 U 2 sinh 2 a ,
coth a a csch 2 a = cos Δ θ .
U ( 0 , 0 , z ) = 2 π α 3 0 π exp ( i z cos θ ) β 3 cos θ sin θ d θ = 2 π α 3 exp ( i z β 3 ) { E 1 [ i z ( β 3 1 ) ] E 1 [ i z ( β 3 + 1 ) ] } ,
d R ( N ) arccos D ¯ R ( N ) D R ( N ) ,
d R ( N ) + Δ θ π 2 .
A CF ( u ; q ) = U 0 exp ( q u z ) .
Δ θ = arccos I 1 ( 2 q ) I 0 ( 2 q ) , D R ( 2 ) = I 0 ( 2 q ) 2 I 0 2 ( q ) ,
Δ θ = arccos [ coth ( 2 q ) 1 2 q ] , D R ( 3 ) = q coth q ,
A EP ( u ; θ M ) = { U 0 , u z > cos θ M 0 , u z < cos θ M .
Δ θ = arccos sin θ M θ M . D R ( 2 ) = π 2 θ M ,
Δ θ = arccos [ cos 2 ( θ M 2 ) ] , D R ( 3 ) = csc ( θ M 2 ) ,
A 0 ( θ ) 2 sin θ d θ = A ( θ ) 2 sin θ d θ ,
sin θ d θ = β 3 2 1 ( β 3 cos θ ) 2 sin θ d θ .
cos θ = cos θ β 3 1 β 3 cos θ = cos θ coth a 1 coth a cos θ ,

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