Abstract

We apply the tools of fractional calculus to introduce new fractional-order solutions of the paraxial wave equation that smoothly connect the elegant Laguerre-Gaussian beams of integral-order. The solutions are characterized in general by two fractional indices and are obtained by fractionalizing the creation operators used to create elegant Laguerre-Gauss beams from the fundamental Gaussian beam. The physical and mathematical properties of the circular fractional beams are discussed in detail. The orbital angular momentum carried by the fractional beam is a continuous function of the angular mode index and it is not restricted to take only discrete values.

© 2007 Optical Society of America

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References

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  1. I. Podlubny, Fractional Differential Equations (Academic Press, 1999).
  2. K. Oldham and J. Spanier, The Fractional Calculus (Academic Press, 1974).
  3. Y. A. Rossikhin and M. V. Shitikova, "Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids," Appl. Mech. Rev. 50, 15-67 (1997).
    [CrossRef]
  4. N. Engheta, "On the role of fractional calculus in electromagnetic theory," IEEE Antennas Propag. Mag. 39, 35-46 (1997).
    [CrossRef]
  5. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The fractional Fourier transform with applications in Optics and signal processing (Wiley, 2001).
    [PubMed]
  6. A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, "Fractional transformation in Optics," Prog. Opt. 38, 265-342 (1998).
  7. N. Engheta, "Fractional curl operator in electromagnetics," Microwave Opt Technol Lett. 17, 86-91 (1998).
    [CrossRef]
  8. Q.A. Naqvi and M. Abbas, "Fractional duality and metamaterials with negative permittivity and permeability," Opt. Commun. 227, 143-146 (2003).
    [CrossRef]
  9. J. C. Gutiérrez-Vega, "Fractionalization of optical beams: I. Planar analysis,"Opt. Lett. 32, (2007) To be published.
    [CrossRef] [PubMed]
  10. A. E. Siegman, Lasers (University Science, 1986).
  11. J. Enderlein and F. Pampaloni, "Unified operator approach for deriving Hermite-Gaussian and Laguerre-Gaussian laser modes," J. Opt. Soc. Am A 21, 1553-1558 (2004).
    [CrossRef]
  12. E. Zauderer, "Complex argument Hermite-Gaussian and Laguerre-Gaussian beams," J. Opt. Soc. Am. A 3, 465-469 (1986).
    [CrossRef]
  13. M. Abramowitz, and I.A. Stegun, Handbook of Mathematical Functions (Dover, 1964) Ch. 13.
  14. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 2000) 6th ed.
  15. S. Saghafi and C. J. R. Sheppard, "The beam propagation factor for higher order Gaussian beams," Opt. Commun. 153, 207-210 (1998).
    [CrossRef]
  16. M. A. Porras, R. Borghi, and M. Santarsiero, "Relationship between elegant Laguerre-Gauss and Bessel-Gauss beams," J. Opt. Soc. Am. A 18, 177-184 (2001).
    [CrossRef]
  17. S. R. Seshadri, "Complex-argument Laguerre-Gauss beams: transport of mean-squared beam width," Appl. Opt. 44, 7339-7343 (2005).
    [CrossRef] [PubMed]
  18. M. A. Bandres and J. C. Gutiérrez-Vega, "Higher-order complex source for elegant Laguerre-Gaussian waves,"Opt. Lett. 29, 2213-2215 (2004).
    [CrossRef] [PubMed]
  19. L. Allen, S. M. Barnett, and M. J. Padgett, Orbital Angular Momentum (Institute of Optics Publishing, 2003).
    [CrossRef]

2007 (1)

J. C. Gutiérrez-Vega, "Fractionalization of optical beams: I. Planar analysis,"Opt. Lett. 32, (2007) To be published.
[CrossRef] [PubMed]

2005 (1)

2004 (2)

M. A. Bandres and J. C. Gutiérrez-Vega, "Higher-order complex source for elegant Laguerre-Gaussian waves,"Opt. Lett. 29, 2213-2215 (2004).
[CrossRef] [PubMed]

J. Enderlein and F. Pampaloni, "Unified operator approach for deriving Hermite-Gaussian and Laguerre-Gaussian laser modes," J. Opt. Soc. Am A 21, 1553-1558 (2004).
[CrossRef]

2003 (1)

Q.A. Naqvi and M. Abbas, "Fractional duality and metamaterials with negative permittivity and permeability," Opt. Commun. 227, 143-146 (2003).
[CrossRef]

2001 (1)

1998 (3)

S. Saghafi and C. J. R. Sheppard, "The beam propagation factor for higher order Gaussian beams," Opt. Commun. 153, 207-210 (1998).
[CrossRef]

A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, "Fractional transformation in Optics," Prog. Opt. 38, 265-342 (1998).

N. Engheta, "Fractional curl operator in electromagnetics," Microwave Opt Technol Lett. 17, 86-91 (1998).
[CrossRef]

1997 (2)

Y. A. Rossikhin and M. V. Shitikova, "Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids," Appl. Mech. Rev. 50, 15-67 (1997).
[CrossRef]

N. Engheta, "On the role of fractional calculus in electromagnetic theory," IEEE Antennas Propag. Mag. 39, 35-46 (1997).
[CrossRef]

1986 (1)

Abbas, M.

Q.A. Naqvi and M. Abbas, "Fractional duality and metamaterials with negative permittivity and permeability," Opt. Commun. 227, 143-146 (2003).
[CrossRef]

Bandres, M. A.

Borghi, R.

Enderlein, J.

J. Enderlein and F. Pampaloni, "Unified operator approach for deriving Hermite-Gaussian and Laguerre-Gaussian laser modes," J. Opt. Soc. Am A 21, 1553-1558 (2004).
[CrossRef]

Engheta, N.

N. Engheta, "Fractional curl operator in electromagnetics," Microwave Opt Technol Lett. 17, 86-91 (1998).
[CrossRef]

N. Engheta, "On the role of fractional calculus in electromagnetic theory," IEEE Antennas Propag. Mag. 39, 35-46 (1997).
[CrossRef]

Gutiérrez-Vega, J. C.

J. C. Gutiérrez-Vega, "Fractionalization of optical beams: I. Planar analysis,"Opt. Lett. 32, (2007) To be published.
[CrossRef] [PubMed]

M. A. Bandres and J. C. Gutiérrez-Vega, "Higher-order complex source for elegant Laguerre-Gaussian waves,"Opt. Lett. 29, 2213-2215 (2004).
[CrossRef] [PubMed]

Lohmann, A. W.

A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, "Fractional transformation in Optics," Prog. Opt. 38, 265-342 (1998).

Mendlovic, D.

A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, "Fractional transformation in Optics," Prog. Opt. 38, 265-342 (1998).

Naqvi, Q.A.

Q.A. Naqvi and M. Abbas, "Fractional duality and metamaterials with negative permittivity and permeability," Opt. Commun. 227, 143-146 (2003).
[CrossRef]

Pampaloni, F.

J. Enderlein and F. Pampaloni, "Unified operator approach for deriving Hermite-Gaussian and Laguerre-Gaussian laser modes," J. Opt. Soc. Am A 21, 1553-1558 (2004).
[CrossRef]

Porras, M. A.

Rossikhin, Y. A.

Y. A. Rossikhin and M. V. Shitikova, "Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids," Appl. Mech. Rev. 50, 15-67 (1997).
[CrossRef]

Saghafi, S.

S. Saghafi and C. J. R. Sheppard, "The beam propagation factor for higher order Gaussian beams," Opt. Commun. 153, 207-210 (1998).
[CrossRef]

Santarsiero, M.

Seshadri, S. R.

Sheppard, C. J. R.

S. Saghafi and C. J. R. Sheppard, "The beam propagation factor for higher order Gaussian beams," Opt. Commun. 153, 207-210 (1998).
[CrossRef]

Shitikova, M. V.

Y. A. Rossikhin and M. V. Shitikova, "Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids," Appl. Mech. Rev. 50, 15-67 (1997).
[CrossRef]

Zalevsky, Z.

A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, "Fractional transformation in Optics," Prog. Opt. 38, 265-342 (1998).

Zauderer, E.

Appl. Mech. Rev. (1)

Y. A. Rossikhin and M. V. Shitikova, "Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids," Appl. Mech. Rev. 50, 15-67 (1997).
[CrossRef]

Appl. Opt. (1)

IEEE Antennas Propag. Mag. (1)

N. Engheta, "On the role of fractional calculus in electromagnetic theory," IEEE Antennas Propag. Mag. 39, 35-46 (1997).
[CrossRef]

J. Opt. Soc. Am A (1)

J. Enderlein and F. Pampaloni, "Unified operator approach for deriving Hermite-Gaussian and Laguerre-Gaussian laser modes," J. Opt. Soc. Am A 21, 1553-1558 (2004).
[CrossRef]

J. Opt. Soc. Am. A (2)

Microwave Opt Technol Lett. (1)

N. Engheta, "Fractional curl operator in electromagnetics," Microwave Opt Technol Lett. 17, 86-91 (1998).
[CrossRef]

Opt. Commun. (2)

Q.A. Naqvi and M. Abbas, "Fractional duality and metamaterials with negative permittivity and permeability," Opt. Commun. 227, 143-146 (2003).
[CrossRef]

S. Saghafi and C. J. R. Sheppard, "The beam propagation factor for higher order Gaussian beams," Opt. Commun. 153, 207-210 (1998).
[CrossRef]

Opt. Lett. (2)

M. A. Bandres and J. C. Gutiérrez-Vega, "Higher-order complex source for elegant Laguerre-Gaussian waves,"Opt. Lett. 29, 2213-2215 (2004).
[CrossRef] [PubMed]

J. C. Gutiérrez-Vega, "Fractionalization of optical beams: I. Planar analysis,"Opt. Lett. 32, (2007) To be published.
[CrossRef] [PubMed]

Prog. Opt. (1)

A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, "Fractional transformation in Optics," Prog. Opt. 38, 265-342 (1998).

Other (7)

L. Allen, S. M. Barnett, and M. J. Padgett, Orbital Angular Momentum (Institute of Optics Publishing, 2003).
[CrossRef]

M. Abramowitz, and I.A. Stegun, Handbook of Mathematical Functions (Dover, 1964) Ch. 13.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 2000) 6th ed.

A. E. Siegman, Lasers (University Science, 1986).

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The fractional Fourier transform with applications in Optics and signal processing (Wiley, 2001).
[PubMed]

I. Podlubny, Fractional Differential Equations (Academic Press, 1999).

K. Oldham and J. Spanier, The Fractional Calculus (Academic Press, 1974).

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Figures (4)

Fig. 1.
Fig. 1.

Radial behavior of Uηl (r,θ,z = 0) for the first four angular orders l = {0,1,2,3} and a continuous radial index variation 0 ≤ η ≤ 12.

Fig. 2
Fig. 2

Amplitude and phase of the beam Uη,l (r) atz= {0,0.5zR ,zR } for η = 2.5 and l = 2. Square image size of 6w 0 × 6w 0 .

Fig. 3.
Fig. 3.

Transverse amplitude and phase distributions of the fractional beam Uη,l (r) at z = 0 for several values of (η, λ). Last row shows the z-component of the OAM carried by the beam for constant λ. Square image size of 5w 0 × 5w 0.

Fig. 4.
Fig. 4.

Behavior of (a) the position of the centroid xc , and (b) the OAM Jz carried by the beam in function of the angular index λ.

Equations (52)

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U 0,0 ( r ) = 2 π 1 μw 0 exp ( r 2 μw 0 2 ) , μ μ ( z ) = 1 + i z z R
( Δ 2 + i 2 k z ) U = ( 2 r 2 + 1 r r + 1 r 2 2 θ 2 + i 2 k z ) U = 0 ,
U n , l ( r ) = 2 ( n ! ) 2 μ ( 2 n + l ) ! U 0,0 ( r ) ( r μ w 0 ) l L n l ( r 2 μw 0 2 ) exp ( ilθ ) ,
A ± w 0 2 ( x ± i y ) = w 0 2 exp ( ± ) ( r ±i 1 r θ ) ,
U n , l ( r ) = 1 ( 2 n + l ) ! ( A + ) n + l ( A ) n U 00 l 0 ,
U η , λ ( r ) = c η , λ ( A + ) η + λ ( A ) η U 00 ,
f ˜ k t ϕ = 1 2 π 0 0 2 π f r θ exp [ ik t cos ( ϕ θ ) ] r d r d θ ,,
f r θ = 1 2 π 0 0 2 π f ˜ k t ϕ exp [ ik t r cos ( ϕ θ ) ] k t d k t d ϕ ,
A ± w 0 2 i ( k x ± ik y ) = w 0 2 ik t exp ( +iϕ ) .
U ˜ η , λ k x k y = c η , λ ( 1 ) η + λ ( ik t w 0 2 ) 2 η + λ exp ( iλϕ ) U ˜ 00 k t ϕ ,
U ˜ 00 k t ϕ = w 0 2 π exp ( μk t 2 w 0 2 4 ) ,
U η , λ ( r ) = d η , λ 0 d k t k t 2 η + λ + 1 exp ( μk t 2 w 0 2 4 ) 0 2 π d ϕ exp ( iλϕ ) exp [ ik t r cos ( ϕ θ ) ] ,
0 2 π exp ( ilϕ ) exp [ iR cos ( ϕ θ ) ] d ϕ = i l 2 π exp ( ilθ ) J l ( R )
U η , l ( r ) = i l 2 π d η , l exp ( ilθ ) 0 d k t k t 2 η + l + 1 exp ( μk t 2 w 0 2 4 ) J l ( k t r )
U η , l ( r ) = N η , l U 0,0 ( r ) ( r μ w 0 ) l Φ ( η , l + 1 ; r 2 μw 0 2 ) exp ( ilθ ) ,
N η , l = Γ ( η + l + 1 ) Γ ( l + 1 ) Γ ( 2 η + l + 1 ) ( 2 μ ) ( 2 η + l ) 2 ,
U η , l = 1 Γ ( 2 η + l + 1 ) ( A + ) η + l ( A ) η U 00 ,
σ 0 2 = r 2 U η , l 2 r d r d θ U η , l 2 r d r d θ ; σ 2 = k t 2 U ˜ η , l 2 k t d k t d ϕ U ˜ η , l 2 k t d k t d ϕ .
σ 0 2 = Γ 2 ( η + l + 1 ) 2 2 η + l + 1 Γ 2 ( l + 1 ) Γ ( 2 η + l + 1 ) 0 r 2 ( r 2 w 0 2 ) l exp ( 2 r 2 w 0 2 ) Φ 2 ( η , l + 1 ; r 2 w 0 2 ) 2 r d r w 0 2 ,
σ 2 = 2 π π Γ ( 2 η + l + 1 ) ( w 0 2 2 ) 2 η + l + 1 0 k t 2 ( 2 η + l + 1 ) exp ( k t 2 w 0 2 2 ) k t d k t .
σ 0 2 = w 0 2 2 ( 2 η + l + l 2 2 η + l ) ; σ 2 = 2 w 0 2 ( 2 η + l + 1 ) .
w = 2 σ 0 = w 0 2 ( 2 η + l + l 2 ) 2 η + l
M 2 = 2 π σ 0 σ = 2 π ( 2 η + l + l 2 ) ( 2 η + l + 1 ) 2 η + l .
f η , l ( R ) = R l exp ( R 2 ) Φ ( η , l + 1 ; R 2 ) .
Q f η , l ( R ) = [ d 2 d R 2 + 1 + 2 R 2 R d d R + ( 4 + 4 η + 2 l l 2 R 2 ) ] f η , l ( R ) = 0 ,
A + U η , l = U η , l + 1 , A U η , l = U η + 1 , l 1 .
( A + ) m g ( r ) = [ exp ( i θ ) ( r + i 1 r θ ) ] m g ( r ) = exp ( imθ ) r m ( 1 r r ) m g ( r ) .
U η + β , l = ( A + ) β ( A ) β U η , l = Δ 2 β U η , l = ( 2 r 2 + 1 r r l 2 r 2 ) β U η , l .
U η , l = ( A + ) η + l ( A ) η U 00 = ( A + ) l ( A + ) η ( A ) η U 00 .
U η , l = exp ( ilθ ) r l ( 1 r r ) l ( 2 t 2 + 1 r r ) η U 00 ,
[ d 2 2 ( 1 + 2 ρ 2 ρ ) d + ( 2 + 2 l + 4 η + 1 l 2 ρ 2 ) ] f ̂ η , l ( ρ ) = 0 ,
f ̂ η , l ( ρ ) = ρ l Φ ( η , l + 1 ; ρ 2 ) .
U η , λ ( r ) 0 d k t k t 2 η + λ + 1 exp ( μk t 2 w 0 2 4 ) 0 2 π d ϕ exp ( iλϕ ) exp [ ik t r cos ( ϕ θ ) ] .
exp ( iλφ ) = 1 π exp ( iπλ ) sin ( πλ ) l = exp ( ilφ ) λ 1 .
U η , λ ( r ) = l = i l λ l Γ ( η + l + 1 ) Γ ( l + 1 ) f η , l ( R ) exp ( i θ ) , η η + λ l 2 ,
J z = h ¯ ∫∫ r t × Im ( U * U ) d x d y ∫∫ U 2 d x d y
U η , λ ( A + ) η + λ ( A ) η U 00 = ( A + ) λ [ ( 2 r 2 + 1 r r ) η U 00 ] = ( A + ) λ U η , 0 .
σ 0 2 = Γ 2 ( η + l + 1 ) 2 2 η + l + 1 Γ 2 ( l + 1 ) Γ ( 2 η + l + 1 ) 0 r 2 ( r 2 w 0 2 ) l exp ( 2 r 2 w 0 2 ) Φ 2 ( η , l + 1 ; r 2 w 0 2 ) 2 r d r w 0 2 ,
W ( q ) = 0 ( r 2 w 0 2 ) l exp ( qr 2 ) Φ 2 ( η , l + 1 ; r 2 w 0 2 ) 2 r d r w 0 2 ,
W ( q ) = 0 r 2 ( r 2 w 0 2 ) l exp ( qr 2 ) Φ 2 ( η , l + 1 ; r 2 w 0 2 ) 2 r d r w 0 2 .
σ 0 2 = Γ 2 ( η + l + 1 ) 2 2 η + l + 1 Γ 2 ( l + 1 ) Γ ( 2 η + l + 1 ) [ W ( q ) ] q = 2 w 0 2
W ( q ) = 0 ξ l exp ( qw 0 2 ξ ) Φ 2 ( η , l + 1 ; ξ ) .
W ( q ) = Γ ( l + 1 ) ( qw 0 2 1 ) 2 η ( qw 0 2 ) 2 η + l + 1 F ( η , η ; l + 1 1 ( qw 0 2 1 ) 2 ) ,
W ( q ) = Γ 2 ( l + 1 ) ( qw 0 2 1 ) 2 η 3 q 2 ( qw 0 2 ) 2 η + l [ 2 η 2 q Γ ( l + 2 ) F ( 1 η , 1 η ; l + 2 1 ( qw 0 2 1 ) 2 ) + ( qw 0 2 1 ) 2 [ 1 l 2 η + ( l + 1 ) w 0 2 q ] w 0 2 F ( η , η ; l + 1 1 ( qw 0 2 1 ) 2 ) ]
W ( q ) = w 0 2 Γ 2 ( l + 1 ) 2 2 η + l + 2 × [ 4 η 2 Γ ( l + 2 ) F ( 1 η , 1 η ; l + 2 ; 1 ) + ( 2 η l 1 ) F ( η , η ; l + 1 ; 1 ) ] ,
F ( 1 η , 1 η ; l + 2 ; 1 ) = Γ ( l + 2 ) ( l + 2 η ) Γ 2 ( l + 1 + η ) ,
F ( η , η ; l + 1 ; 1 ) = Γ ( l + 1 ) Γ ( l + 1 + 2 η ) Γ 2 ( l + 1 + η ) ,
W ( q ) = w 0 2 Γ 2 ( l + 1 ) Γ ( l + 2 η ) 2 2 η + l + 2 Γ 2 ( l + 1 + η ) [ 4 η 2 + ( 2 η l 1 ) ( l + 2 η ) ] .
σ 0 2 = w 0 2 2 [ 2 η + l + l 2 2 η + l ] ,
ξ d 2 Φ 2 + ( l + 1 ξ ) + ηΦ = 0 ,
[ ξ 2 d 2 2 + ξ ( 1 + ξ ) d + ( 1 + l 2 + η ) ξ ( 1 2 ) 2 ] f η , l ( ξ ) = 0 .
[ d 2 d R 2 + 1 + 2 R 2 R d d R + ( 4 + 2 ( 2 η + l ) l 2 R 2 ) ] v ( R ) = 0 ,

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