## Fractionalization of optical beams: II. Elegant Laguerre-Gaussian modes

Optics Express, Vol. 15, Issue 10, pp. 6300-6313 (2007)

http://dx.doi.org/10.1364/OE.15.006300

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### 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

## 1. Introduction

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]

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). [CrossRef] [PubMed]

9. J. C. Gutiérrez-Vega, “Fractionalization of optical beams:
I. Planar analysis,”Opt. Lett. **32**, ?–? (2007). [CrossRef] [PubMed]

*M*

^{2}factor of the circular fractional beam in closed form. We discuss also the adjoint beam solutions and establish the conditions for beam biorthogo-nality. 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. In this paper we focus our attention to the fractionalization of the elegant LG beams, and leave the fractionalization of the standard LG beams for a future article. We finally remark that this paper is fully self-contained as it introduces all relevant definitions as well as main motivations.

## 2. Fractional operators and fractionalization of elegant Laguerre–Gaussian beams

*iωt*) propagating in free-space along the positive

*z*axis of a coordinate system

**r**= (

*x*,

*y*,

*z*) = (

*r*cos

*θ*,

*r*sin

*θ*,

*z*):

*w*

_{0}is the beam width at the waist plane

*z*= 0, and

*z*=

_{R}*kw*

^{2}

_{0}/2 is the Rayleigh distance, and

*k*is the wave number. The field in Eq. (1) carries unit power (i.e.∫∫

^{∞}

_{-∞}|

*U*

_{0,0}|

^{2}d

*x*d

*y*= 1), and constitutes a fundamental solution of the PWE

_{2}is the transverse Laplacian operator.

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]

*n*= (0,1,2,…) and angular

*l*= (0,1,2,…) mode numbers

*U*

_{0,0}by the repeated application of the differential creation operators

*L*is the associated Laguerre polynomial [13], and (!) denotes the factorial operation. The basis for such a construction is the following theorem: let

_{n}^{l}*U*be a solution of the linear operator

*L*(i.e.

*LU*= 0), if other linear operator

*A*commutes with

*L*, then the function

*AU*is also a solution of

*L*. For the sake of simplicity, throughout the paper we restrict the analysis to positive values of the azimuthal index. Expressions for negative index can be straightforwardly derived from the positive ones by symmetry considerations.

*U*of the PWE resulting from the operation

_{η,λ}*η*and

*λ*are two positive arbitrary numbers denoting the fractional radial and angular indices of the beam, and

*c*is a normalization constant to be determined at a later stage. The operators (

_{η,λ}*A*

^{±})

^{α}(for arbitrary

*α*≥ 0) correspond to the fractionalization of the operators

*A*

^{±}and satisfy the following conditions: (a) for

*α*= 1, we get the original operators

*A*

^{±}; (b) for

*α*= 0, we obtain the identity operator, and (c) for two numbers

*α*and

*β*we have (

*A*

^{±})

^{α}(

*A*

^{±})

^{β}= (

*A*

^{±})

^{β}(

*A*

^{±})

^{α}= (

*A*

^{±})

^{α+β}. The fractional operators (

*A*

^{±})

^{α}commutes with the operator of the PWE, therefore we expect that the action of (

*A*

^{±})

^{α}on any solution of the PWE will give a new solution of the same equation.

*f*(

*r*,

*θ*) and its inverse FT are written in cylindrical coordinates as

*k*,

_{x}*k*) = (

_{y}*k*cos

_{t}*ϕ*,

*k*sin

_{t}*ϕ*) denotes the transverse position in the frequency space.

*∂/∂x*by

*ik*and

_{x}*∂/∂*by

_{y}*ik*. Therefore differential operators

_{y}*A*

^{±}are replaced according to

*U*

_{00}(

*r*,

*θ*) .

*U*̃

_{00}with a modulation function proportional to

*k*

_{t}^{2η+λ}exp (

*iλϕ*).

*U*(

_{η,λ}**r**) is now determined by inverse Fourier transforming Eq. (9). Inserting

*U*̃

_{00}into Eq. (7b) we obtain

*d*=

_{η,λ}*c*(-1)

_{η,λ}^{η+λ}(

*iw*

_{0}/√2)

^{2η+λ+1}/2

*πi*√

*π*is a constant.

## 3. Fractional beams with fractional radial index *η*
and integer angular index *l*

*λ*becomes integer. As we will see, this condition leads to fractional beams whose transverse intensity distribution is circularly symmetrical, or equivalently, whose transverse field is separable into radial and angular parts.

*λ*=

*l*be an integer number, for this case the angular integral in Eq. (11) can be evaluated in closed form in terms of the

*l*th-order Bessel functions

*J*, we have [14]

_{l}*U*in terms of

_{η,l}*l*-th order Bessel beams of the form

*J*(

_{l}*k*)exp(

_{t}r*ilθ*).

*a*,

*b*;

*z*) is the Kummer confluent hypergeometric function [often denoted also as

_{1}

*F*

_{1}(

*a*,

*b*;

*z*)],

*U*

_{0,0}(

**r**) is the Gaussian beam [Eq. (1)], and the normalization constant

^{∞}

_{-∞}|

*U*

*|*

_{ηl}^{2}d

*x*d

*y*= 1 at any

*z*plane. Here Γ(

*x*) denotes the Gamma function.

*η*, and integers values of the azimuthal index

*l*. The presence of the Gaussian beam in Eq. (14) ensures the physical requirement that the field amplitude vanishes for

*r*arbitrarily large, and that the beam is square integrable.

*U*(

_{η,l}*r*,

*θ*,

*z*= 0) at the waist plane for the first four angular orders

*l*= {0,1,2,3} and a continuous radial index variation 0 ≤ η≤ 12. At this plane

*μ*= 1, thus the radial part of

*U*(

_{η,l}**r**) becomes a purely real function. The function Φ creates maxima, minima, and beam nulls in the amplitude distribution as

*η*increases. In particular,

*U*(

_{ηl}*r*,

*θ*,

*z*= 0) has

*n*radial zeros when

*η*falls in the interval (

*n*- 1) <

*η*≤

*n*.

*U*(

_{η,l}**r**) at the waist plane

*z*= 0 is purely real, outside this plane it becomes complex leading to a continuous variation of the transverse pattern. This effect is illustrated in Fig. 2, where we show the transverse amplitude and phase of the fractional beam

*U*

_{2.5, 2}(

**r**) at

*z*= {0,0.5

*z*,

_{R}*z*} . Because of its azimuthal dependence of the form exp(

_{R}*ilθ*), the fractional beam

*U*(

_{η,l}**r**) is azimuthally symmetric, and carries an intrinsic orbital angular momentum of

*lh*̄ per photon which is independent of the fractional radial index

*η*.

*η*becomes a positive integer

*n*= 0,1,2,…, then Φ(-

*n,l*+ 1;

*x*) =

*n*!

*l*!

*L*(

^{l}_{n}*x*) /(

*n*+

*l*)! and thus Eq. (14) reduces to the elegant LG beams given by Eq. (3) and shown with solid lines in Fig. 1. So, effectively, the integer-order solutions have been “smoothly connected” by varying the order of fractional differentiation of the Gaussian beam.

### 3.1. Moments, width, and M^{2} factor of the circular fractional
beams

*U*(

_{η,l}**r**) are determined in terms of the moments of the intensity distribution and its Fourier transform. The first-order moment provides the position of the centroid of the beam on the transverse plane, and vanishes by virtue of the circular symmetry of

*U*(

_{η,l}**r**).

_{0}and σ

_{∞}associated with the intensity distributions at the waist and at the far field, respectively, provide the irradiance spot size

*w*= 2σ

_{0}and the quality factor

*M*

^{2}= 2

*π*σ

_{0}σ

_{∞}of the beam. For circularly symmetric beams, the moments σ

_{0}and σ

_{∞}are given by

*U*and

_{η,l}*Ũ*are given by Eqs. (14) and (9) evaluated at

_{η,l}*z*= 0, respectively. Taking into account that Eqs. (14) and (9) are already normalized [i.e. both denominators in Eqs. (17) are unity] we have for σ

_{0}and σ

_{∞}

*M*

_{2}factor of the fractional beam

*U*(

_{η,l}**r**) turns out to be

*M*

^{2}factor increases with orders (

*η,l*). For integer values of

*η*, Eq. (22) reduces to the

*M*

^{2}factor of the elegant LG beams [15

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]

### 3.2. The fractional radial creation operator

*U*, and derive a useful fractional creation operator for this radial part which allows to rewrite the two-variable operator prescription [Eq. (16)] using operators depending exclusively on derivatives with respect to

_{η,l}*r*.

*U*(

_{η,l}**r**) can take the separated form

*U*(

_{ηl}**r**) =

*f*(

_{ηl}*R*)exp(

*ilθ*), with

*f*(

_{η,l}*R*) being the radial part given by

*R*≡

*r*/ √

*μw*

_{0}is the normalized radius, and unnecessary constants have been omitted. Taking advantage of the fact that Φ fulfills the confluent hypergeometric equation [13] it can be demonstrated (see Appendix B) that

*f*(

_{ηl}*R*) is solution of the ordinary differential equation

*Q*denotes the linear operator of the equation.

*A*

^{±}, it is clear that if

*U*is a solution of the PWE with azimuthal index

_{η,l}*l*, then

*A*

^{±}

*U*is a solution with index

_{η,l}*l*± 1, that is

*A*

^{+}on a function

*g*(

*r*) which depends exclusively on

*r*leads to the operator relation

*m*is an integer.

*β*, the individual application of the fractional operators (

*A*

^{+})

^{β}or (

*A*

^{-})

^{β}on

*U*leads to azimuthally asymmetric solutions of the PWE (see Sect. 3 for details). By applying the operator (

_{η,l}*A*

^{+})

_{β}(

*A*

^{-})

^{β}on

*U*it is possible to cancel out the opposite angular effects, and consequently to modify the radial index

_{η,l}*η*of

*U*by keeping constant its azimuthal index

_{η,l}*l*. From the definition of

*A*

^{±}in Eq. (4), the operator

*A*

^{+}

*A*

^{-}is recognized to be the same as the transverse Laplacian operator Δ

_{2}acting on a field with azimuthal dependence exp(

*ilθ*) (i.e. the angular derivative

*∂*/

*∂θ*is then replaced by

*il*). We have explicitly

*β*

_{2}is indeed a radial creation operator for the field

*U*.

_{η,l}^{β}

_{2}on the radial function

*f*(

_{η,l}*r*) is determined also by noting that the commutator of Δ

^{β}

_{2}and the operator

*Q*in Eq. (24) is given by [Δ

^{β}

_{2},

*Q*] = 4

*β*Δ

^{β}_{2}. It follows that the new function Δ

^{β}_{2}

*f*satisfies the same differential equation (24) with the parameter

_{η,l}*η*changed to

*η*+

*β*. We then conclude that the effect of Δ

^{β}_{2}on

*f*is simply to change the order, while remaining the original functional form, i.e. Δ

_{η,l}

^{β}_{2}

*f*=

_{η,l}*f*.

_{η+β,l}*l*= 0, the operation (

*A*

^{+})

^{η}(

*A*

^{-})

^{η}

*U*

_{00}is fully equivalent to [

*∂*

_{r}^{2}+ (1/

*r*)

*∂*]

_{r}^{η}

*U*

_{00}, which yields a function depending on

*r*only. Since

*l*is an integer number, then from Eq. (26) we finally have

*η*. Equation (29) allows to generate a fractional beam

*U*starting from the fundamental Gaussian beam using only radial operators. It is worth mentioning that this operator formula (with

_{η,l}*η*being an integer number) has been applied recently in Ref. [18

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]

### 3.3. On the adjoint radial equation and adjoint fractional beams

*ρ*=

*R*

^{*}=

*r*/

*w*

_{0}√

*μ*

^{*}and (

^{*}) denotes complex conjugate. The solutions to the adjoint equation are the adjoint fractional beams given by

*f*(

_{η,l}*R*), note that there is no Gaussian factor associated with the adjoint functions. It is now clear that the radial equation (24) is not a self-adjoint equation, then its solutions

*f*do not form an orthonormal set. Applying the theory of the confluent hypergeometric functions [13] it is possible to show that the biorthogonality integral ∫

_{η,l}^{∞}

_{0}

*f*(

_{η,l}*r*)

*f*̂

*(*

_{γ,l}*r*)

*r*d

*r*between

*f*and its adjoint functions

_{η,l}*f*̂

*diverges for arbitrary values of*

_{γl}*η*and

*γ*unless both

*η*and

*γ*become integer numbers (that indeed is the case of the known biorthogonal relation for the elegant LG beams [10]). We then conclude that it is not possible to formulate an orthogonality relation for functions

*f*(

_{η,l}*r*) with arbitrary

*η*in the semi-infinite domain 0 ≤

*r*< ∞. Nevertheless, it may be established in a finite domain by converting Eq. (24) into a self-adjoint form and applying the Sturm-Liouville theory.

## 4. Discussion of the case when the angular index *λ* is
not integer

*λ*of the fractional beam

*U*(

_{η,λ}**r**) is not an integer number. As we will see, this case leads the transverse field to be non-circularly symmetrical.

*U*∝ (

_{η,λ}*A*

^{+})

*(*

^{η+λ}*A*

^{-})

^{η}*U*

_{00}, and its integral representation is given by Eq. (11), namely

^{2π}

_{0}exp(

*iλϕ*)exp[

*ik*cos(

_{t}r*ϕ*-

*θ*)]d

*ϕ*cannot be evaluated in closed form for non integer values of

*λ*. Although there is always the possibility of using numerical methods to evaluate directly Eq. (32), an alternative expression for

*U*(

_{ηλ}**r**) may be derived by expanding exp (

*iλϕ*) in its Fourier series as follows:

*l*. The result is

*f*

_{η′,|l|}(

*R*) is given by Eq. (23), and an overall amplitude factor has been omitted for simplicity.

*η,λ*) can be constructed with a suitable superposition of fractional beams with integer indices

*l*. Figure 3 shows the transverse amplitude and phase distributions of the fractional beam

*U*at

_{η,λ}*z*= 0 for several values of (

*η, λ*) in the ranges

*η*∈ [5

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

*λ*∈ [1,2]. The patterns were obtained by adding 101 terms of the series in Eq. (34) from

*l*= -50 to

*l*= 50. As the radial and the angular indices increase, the irradiance and phase patterns vary continuously exhibiting an azimuthally asymmetric shape which becomes circularly symmetrical only when

*λ*is integer.

*λ*the centroid of the beam

*U*is slightly displaced from the origin in the horizontal direction. Figure 4(a) shows the position of the beam centroid

_{η,λ}*x*in normalized units of

_{c}*w*

_{0}as a function of the angular index

*λ*. The curve was determined by evaluating numerically the definition of the first-order moment

*x*= ∫∫

_{c}^{∞}

_{-∞}

*x*|

*U*|

^{2}d

*x*d

*y*/∫∫

^{∞}

_{-∞}|

*U*|

^{2}d

*x*d

*y*for the range

*λ*∈ [0,6] and

*η*= 0.. As

*λ*increases

*x*exhibits a decreasing oscillatory behavior around the origin and vanishes for integer

_{c}*λ*.

*z*component of the OAM per photon in unit length about the origin of a transverse slice of a beam

*U*(

**r**) is given by [19

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

**r**

*=*

_{t}*x*

**x**̂ +

*y*

**y**̂ is the transverse radius vector. From the last paragraph of Sect. 3.2, we know that

*U*is proportional to

_{η,λ}*U*

_{η,0}depends on

*r*only and thus it does not carry OAM. We then conclude that the existence of OAM in

*U*comes from the application of (

_{η,λ}*A*

^{+})

^{λ}on

*U*

_{η,0}, and thus it follows that

*J*depends on

_{z}*λ*only, while it is independent on

*η*. To determine

*J*, we evaluated numerically Eq. (35) using a two-dimensional Gauss-Legendre quadrature for a large number of combinations of radial and angular indices (

_{z}*η*,

*λ*). The numerical analysis corroborated that the OAM carried by the beam is independent of the radial index

*η*, and a continuous function of the angular index

*λ*. For the beam shapes shown in Fig. 3, the values of

*J*are included in the bottom line. Figure 4(b) depicts also the behavior of

_{z}*J*as a function of the angular index within the range

_{z}*λ*∈ [0,6]. It is clear that by adjusting the value of

*λ*it is possible to tune OAM carried by the fractional beam.

## 5. Conclusions

- The fractional beams are characterized by two fractional radial
*η*and angular*λ*indices. When both indices become integers, the fractional solutions reduce to the known elegant LG beams of integer-order. - For integer values of
*λ*: (a) the mathematical description of the fractional beam acquires a particular simple form expressed in terms of a confluent hypergeometric function [Eq. (14)], (b) the transverse pattern is azimuthally symmetric, (c) the relevant beam parameters (e.g. the moments,*M*^{2}factor, and the carried OAM) can be also determined in closed form, and (d)taking advantage of the separability of the radial and angular parts of the fractional beam with*l*integer, it was possible to reformulate the two-variable operator prescription [Eq. (16)] in terms of fractional operators depending exclusively on derivatives with respect to*r*[Eq. (29)]. - For fractional values of
*λ*: (a) the field is not expressible in closed form but it can be written as a superposition of fractional beams with integer angular indices [Eq. (34)], (b) the transverse pattern is not circularly symmetric, (c) numerical evaluation is needed to compute the beam parameters, and (d) the position of the beam centroid slightly oscillates around the origin as*λ*increases. The excursion is smaller for larger values of*λ*. - The mode obtained for every particular index
*λ*. is stable and possesses fractional OAM. The OAM carried by the beam is independent of the radial index*η*, and a continuous function of the angular index*λ*. This property may be useful in applications where tuning of the OAM carried by the beam is important such as in optical trapping, and optical tweezers. - The differential equation satisfied by the radial part of the beam is not indeed a self-adjoint equation. The analysis of the adjoint equation and the corresponding adjoint beams revealed that it is not possible to formulate an orthogonality relation for the radial functions
*f*(_{η,l}*r*) with arbitrary*η*in the semi-infinite domain 0 ≤*r*< ∞. Nevertheless, it may be established in a finite domain by converting Eq. (24) into a self-adjoint form and applying the Sturm-Liouville theory.

## A. Appendix: Evaluation of the second-order moment Eq. (18)

*z*= 0 is given by the integral

*a*,

*b*;

*x*) is the Kummer Confluent Hypergeometric function. To evaluate Eq. (37) we introduce the auxiliary function

*q*= 2/

*w*

^{2}

_{0}. The derivative of

*W*(

*q*) with respect to

*q*reads as

_{0}

^{2}can be expressed in terms of the derivative of

*W*(

*q*) as follows

*F*(

*a*,

*b*;

*c*;

*x*) is the hypergeometric function [13]. From the latter equation, the derivative of

*W*(

*q*) with respect to

*q*is evaluated explicitly to be

*q*= 2/

*w*

^{2}

_{0}we get

*W*′ (

*q*) becomes

## B. Appendix: Derivation of Eq. (24)

*f*(

_{η,l}*R*) =

*R*exp (-

^{l}*R*

^{2})Φ(-

*η,l*,+ 1;

*R*

^{2}), we first note that Φ (

*η,l*+ 1;ξ) satisfies the Confluent Hypergeometric differential equation (CHDEq)

*R*

^{2}. By replacing Φ(-

*η,l*+ 1;ξ) = ξ

^{-l/2}exp(ξ)

*v*(ξ) into the CHDEq we determine that

*f*(ξ) satisfies the differential equation

_{η,l}*R*, we find that

*f*(

_{η,l}*R*) satisfies the equation

## Acknowledgments

## References and links

1. | I. Podlubny, |

2. | K. Oldham and J. Spanier, |

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. |

4. | N. Engheta, “On the role of fractional calculus
in electromagnetic theory,” IEEE Antennas
Propag. Mag. |

5. | H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, |

6. | A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, “Fractional transformation in
Optics,” Prog. Opt. |

7. | N. Engheta, “Fractional curl operator in
electromagnetics,” Microwave Opt Technol
Lett. |

8. | Q.A. Naqvi and M. Abbas, “Fractional duality and metamaterials
with negative permittivity and permeability,”
Opt. Commun. |

9. | J. C. Gutiérrez-Vega, “Fractionalization of optical beams:
I. Planar analysis,”Opt. Lett. |

10. | A. E. Siegman, |

11. | J. Enderlein and F. Pampaloni, “Unified operator approach for
deriving Hermite-Gaussian and Laguerre-Gaussian laser
modes,” J. Opt. Soc. Am A |

12. | E. Zauderer, “Complex argument Hermite-Gaussian
and Laguerre-Gaussian beams,” J. Opt.
Soc. Am. A |

13. | M. Abramowitz and I.A. Stegun, |

14. | I. S. Gradshteyn and I. M. Ryzhik, |

15. | S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for
higher order Gaussian beams,” Opt.
Commun. |

16. | M. A. Porras, R. Borghi, and M. Santarsiero, “Relationship between elegant
Laguerre-Gauss and Bessel-Gauss beams,”
J. Opt. Soc. Am. A |

17. | S. R. Seshadri, “Complex-argument Laguerre-Gauss
beams: transport of mean-squared beam width,”
Appl. Opt. |

18. | M. A. Bandres and J. C. Gutiérrez-Vega, “Higher-order complex source for
elegant Laguerre-Gaussian waves,”Opt.
Lett. |

19. | L. Allen, S. M. Barnett, and M. J. Padgett, |

**OCIS Codes**

(050.1970) Diffraction and gratings : Diffractive optics

(140.3300) Lasers and laser optics : Laser beam shaping

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Physical Optics

**History**

Original Manuscript: March 19, 2007

Revised Manuscript: May 3, 2007

Manuscript Accepted: May 3, 2007

Published: May 7, 2007

**Citation**

Julio C. Gutiérrez-Vega, "Fractionalization of optical beams: II. Elegant Laguerre–Gaussian modes," Opt. Express **15**, 6300-6313 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-10-6300

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### References

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