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

Fractional calculus is the theory of differentiation and integration to arbitrary fractional or even complex order [1, 2]. Although the theory dates back to 1695 when Leibniz presented his analysis of the derivative of order one half, for three centuries it developed mainly as a pure theoretical field of mathematics. In the last few decades there has been a revival of interest in fractional calculus because its theory has had applications in several areas, such as fractals, viscoelasticity, transmission lines, and electrodynamics, to name a few [1, 2, 3, 4]. Optics is a field in which the use of conventional calculus plays a major role, and it is of interest to see the potential application of fractional calculus in this field. In this direction, the fractionalization of the Fourier transform and its optical applications have been already studied for the last 15 years [5, 6], and more recently, the fractionalization of the curl operator was applied also in electromagnetism [7] and metamaterials [8].

The present work is part of our efforts in exploring the applications of fractional calculus and fractional operators to optics and beam theory. In particular, in a previous paper [9], we introduced fractional-order beam solutions to the paraxial wave equation (PWE) that constitute continuous transition modes between the Hermite–Gaussian beams of integral-order [10]. These new solutions were obtained formally by applying the fractional derivative operator on the fundamental Gaussian beam.

In this paper, we extend our previous work to the case of cylindrical geometry, and present new fractional-order solutions that smoothly connect the Laguerre–Gaussian (LG) beams of integral-order [10]. The new circular fractional solutions are characterized in general by two fractional indices and are obtained by fractionalizing the creation operators used to create integer-order LG beams from the fundamental Gaussian beam. Remarkably, the mathematical form of the fractional solutions in circular coordinates turned out to be much simpler than the expression for the fractional beams in Cartesian coordinates [9]. This fact allowed us to determine the width, the moments, and the 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

We begin the analysis by writing the complex field amplitude of the normalized Gaussian beam with time dependence exp(-iωt) propagating in free-space along the positive z axis of a coordinate system r = (x,y, z) = (rcosθ, rsinθ, z):

$U0,0(r)=2π1μw0exp(−r2μw02),μ≡μ(z)=1+izzR$

where w 0 is the beam width at the waist plane z = 0, and zR = 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|2dxdy = 1), and constitutes a fundamental solution of the PWE

$(Δ2+i2k∂∂z)U=(∂2∂r2+1r∂∂r+1r2∂2∂θ2+i2k∂∂z)U=0,$

where Δ2 is the transverse Laplacian operator.

It is known [11,12] that normalized elegant LG beams with radial n= (0,1,2,…) and angular l = (0,1,2,…) mode numbers

$Un,l(r)=2(n!)2μ(2n+l)!U0,0(r)(rμw0)lLnl(r2μw02)exp(ilθ),$

can be derived from the fundamental Gaussian beam U 0,0 by the repeated application of the differential creation operators

$A±≡∓w02(∂∂x±i∂∂y)=∓w02exp(±iθ)(∂∂r±i1r∂∂θ),$

applying the following operator prescription

$Un,l(r)=1(2n+l)!(A+)n+l(A−)nU00l≥0,$

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

To explore the application of fractional operators in beam theory, we look for new fractional-order solutions Uη,λ of the PWE resulting from the operation

$Uη,λ(r)=cη,λ(A+)η+λ(A−)ηU00,$

where η and λ are two positive arbitrary numbers denoting the fractional radial and angular indices of the beam, and cη,λ is a normalization constant to be determined at a later stage. The operators (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.

Evaluation of the fractional operators in Eq. (6) can be performed analytically in the frequency space. The two-dimensional Fourier transform (FT) of a function f(r,θ) and its inverse FT are written in cylindrical coordinates as

$f˜ktϕ=12π∫0∞∫02πfrθexp[−iktcos(ϕ−θ)]rdrdθ,,$
$frθ=12π∫0∞∫02πf˜ktϕexp[iktrcos(ϕ−θ)]ktdktdϕ,$

where (kx,ky) = (kt cos ϕ,ktsinϕ) denotes the transverse position in the frequency space.

To get the FT of Eq. (6), we replace ∂/∂x by ikx and ∂/∂y by iky. Therefore differential operators A ± are replaced according to

$A±⇒∓w02i(kx±iky)=∓w02iktexp(+iϕ).$

The FT of Eq. (6) is then given by

$U˜η,λkxky=cη,λ(−1)η+λ(iktw02)2η+λexp(iλϕ)U˜00ktϕ,$

where

$U˜00ktϕ=w02πexp(−μkt2w024),$

is the FT of the fundamental Gaussian beam U 00 (r, θ) .

Equation (9) describes the angular spectrum of plane waves for the desired fractional beam. From a physical point of view, the circular fractional beam can be derived by modulating the plane-wave spectrum of the fundamental Gaussian beam Ũ00 with a modulation function proportional to kt 2η+λexp (iλϕ).

The field Uη,λ (r) is now determined by inverse Fourier transforming Eq. (9). Inserting Ũ00 into Eq. (7b) we obtain

$Uη,λ(r)=dη,λ∫0∞dktkt2η+λ+1exp(−μkt2w024)∫02πdϕexp(iλϕ)exp[iktrcos(ϕ−θ)],$

where dη,λ = cη,λ(-1)η+λ (iw 0/√2)2η+λ+1/2πiπ is a constant.

Equation (11) is the integral representation for the fractional beam Uη,λ (r) and effectively corresponds to the evaluation of the fractional operators given in Eq. (6). In the following sections we will study in detail the solutions of Eq. (11) for both integral and fractional angular index λ.

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

In this section we consider the solutions Eq. (11) when the angular index 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.

Let 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 lth-order Bessel functions Jl , we have [14]

$∫02πexp(ilϕ)exp[iRcos(ϕ−θ)]dϕ=il2πexp(ilθ)Jl(R)$

Substitution into Eq. (11) yields

$Uη,l(r)=il2πdη,lexp(ilθ)∫0∞dktkt2η+l+1exp(−μkt2w024)Jl(ktr)$

Physically speaking, Eq. (13) corresponds to the continuous expansion of the field Uη,l in terms of l-th order Bessel beams of the form Jl(ktr)exp(ilθ).

The evaluation of the integral in Eq. (13) [using Eq. (6.631.1) of Ref. [14]] leads to the desired expression for the circular fractional beam, namely

$Uη,l(r)=Nη,lU0,0(r)(rμw0)lΦ(−η,l+1;r2μw02)exp(ilθ),$

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.

where Φ(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

$Nη,l=Γ(η+l+1)Γ(l+1)Γ(2η+l+1)(2μ)(2η+l)2,$

has been determined to ensure unit beam power, i.e. ∫∫ -∞|U ηl|2 dxdy = 1 at any z plane. Here Γ(x) denotes the Gamma function.

Equation (14) is an exact solution of the PWE and is the main result of this work. It constitutes the radial fractionalization of the fundamental Gaussian beam in circular coordinates and indeed corresponds to the analytical evaluation of the fractional operator relation

$Uη,l=1Γ(2η+l+1)(A+)η+l(A−)ηU00,$

for arbitrary positive values of the radial index 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.

In Fig. 1 we show the radial behavior of Uη,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 Uη,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, Uηl (r, θ,z = 0) has n radial zeros when η falls in the interval (n - 1) < ηn.

While the radial part of the fractional beam Uη,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 = {0,0.5zR,zR} . Because of its azimuthal dependence of the form exp(ilθ), the fractional beam Uη,l (r) is azimuthally symmetric, and carries an intrinsic orbital angular momentum of lh̄ per photon which is independent of the fractional radial index η.

From the theory of the confluent hypergeometric functions [see Eq. 8.972.1 of Ref. [14]], when η becomes a positive integer n = 0,1,2,…, then Φ(-n,l+ 1;x) = n!l!Lln(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.

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 .

3.1. Moments, width, and M2 factor of the circular fractional beams

The significant parameters to characterize the circular fractional beam Uη,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 Uη,l (r).

The second-order moments σ0 and σ associated with the intensity distributions at the waist and at the far field, respectively, provide the irradiance spot size w = 2σ0 and the quality factor M 2 = 2πσ0σ of the beam. For circularly symmetric beams, the moments σ0 and σ are given by

$σ02=∫∫r2∣Uη,l∣2rdrdθ∫∫∣Uη,l∣2rdrdθ;σ∞2=∫∫kt2∣U˜η,l∣2ktdktdϕ∫∫∣U˜η,l∣2ktdktdϕ.$

where Uη,l and Ũη,l are given by Eqs. (14) and (9) evaluated at 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 σ0 and σ

$σ02=Γ2(η+l+1)22η+l+1Γ2(l+1)Γ(2η+l+1)∫0∞r2(r2w02)lexp(−2r2w02)Φ2(−η,l+1;r2w02)2rdrw02,$
$σ∞2=2ππΓ(2η+l+1)(w022)2η+l+1∫0∞kt2(2η+l+1)exp(−kt2w022)ktdkt.$

These integrals can be evaluated analytically (see Appendix A), and the results are given by the following surprisingly simple formulas

$σ02=w022(2η+l+l22η+l);σ∞2=2w02(2η+l+1).$

Thus the irradiance spot size and the M 2 factor of the fractional beam Uη,l (r) turns out to be

$w=2σ0=w02(2η+l+l2)2η+l$
$M2=2πσ0σ∞=2π(2η+l+l2)(2η+l+1)2η+l.$

As expected, the 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, 16, 17]. It is important to note that the parameters defined in this section must be taken as global parameters of the circular fractional beams. They do not describe local variations of the intensity distribution.

3.2. The fractional radial creation operator

In this section we discuss the differential equation satisfied by the radial part of the fractional beams Uη,l, 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 r.

We commence by noting from Eq. (14) that Uη,l (r) can take the separated form Uηl (r) = fηl (R)exp(ilθ), with fη,l(R) being the radial part given by

$fη,l(R)=Rlexp(−R2)Φ(−η,l+1;R2).$

where Rr/ √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 fηl (R) is solution of the ordinary differential equation

$Qfη,l(R)=[d2dR2+1+2R2RddR+(4+4η+2l−l2R2)]fη,l(R)=0,$

where Q denotes the linear operator of the equation.

From Eq. (16) and the commutability of the operators A ± , it is clear that if Uη,l is a solution of the PWE with azimuthal index l, then A ± Uη,l is a solution with index l ± 1, that is

$A+Uη,l=Uη,l+1,A−Uη,l=Uη+1,l−1.$

The special case of the repeated application of A + on a function g(r) which depends exclusively on r leads to the operator relation

$(A+)mg(r)=[exp(iθ)(∂∂r+i1r∂∂θ)]mg(r)=exp(imθ)rm(1r∂∂r)mg(r).$

where m is an integer.

For arbitrarily positive β, the individual application of the fractional operators (A +)β or (A -)β on Uη,l leads to azimuthally asymmetric solutions of the PWE (see Sect. 3 for details). By applying the operator (A +)β (A -)β on Uη,l it is possible to cancel out the opposite angular effects, and consequently to modify the radial index η of Uη,l by keeping constant its azimuthal index l. From the definition of A ± in Eq. (4), the operator A + A - is recognized to be the same as the transverse Laplacian operator Δ2 acting on a field with azimuthal dependence exp(ilθ) (i.e. the angular derivative /∂θ is then replaced by il). We have explicitly

$Uη+β,l=(A+)β(A−)βUη,l=Δ2βUη,l=(∂2∂r2+1r∂∂r−l2r2)βUη,l.$

We then conclude that the fractional operator Δβ 2 is indeed a radial creation operator for the field Uη,l.

The effect of acting the fractional operator Δβ 2 on the radial function fη,l (r) is determined also by noting that the commutator of Δβ 2 and the operator Q in Eq. (24) is given by [Δβ 2, Q] = 4β Δβ 2. It follows that the new function Δβ 2 fη,l satisfies the same differential equation (24) with the parameter η changed to η+β. We then conclude that the effect of Δβ 2 on fη,l is simply to change the order, while remaining the original functional form, i.e. Δβ 2 fη,l = fη+β,l.

We complete this section mentioning that results discussed above allow us to rewrite the two-variable operator prescription in Eq. (16) in terms of operators depending on radial derivatives exclusively. Omitting unnecessary constants we write

$Uη,l=(A+)η+l(A−)ηU00=(A+)l(A+)η(A−)ηU00.$

Now, by virtue of Eq. (27) with 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

$Uη,l=exp(ilθ)rl(1r∂∂r)l(∂2∂t2+1r∂∂r)ηU00,$

for arbitrarily positives values of η. Equation (29) allows to generate a fractional beam Uη,l starting from the fundamental Gaussian beam using only radial operators. It is worth mentioning that this operator formula (with η being an integer number) has been applied recently in Ref. [18] to determine the higher-order complex source for the elegant LG non paraxial waves.

The adjoint equation to Eq. (24) is found to be

$[d2dρ2−(1+2ρ2ρ)ddρ+(2+2l+4η+1−l2ρ2)]f̂η,l(ρ)=0,$

where ρ = R * = r/w 0μ * and (*) denotes complex conjugate. The solutions to the adjoint equation are the adjoint fractional beams given by

$f̂η,l(ρ)=ρlΦ(−η,l+1;ρ2).$

Comparing with respect to functions fη,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 fη,l do not form an orthonormal set. Applying the theory of the confluent hypergeometric functions [13] it is possible to show that the biorthogonality integral ∫ 0 fη,l(r)f̂γ,l(r) rdr between fη,l and its adjoint functions f̂γl diverges for arbitrary values of 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 fη,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

Let us now discuss the general case when the azimuthal index λ of the fractional beam Uη,λ (r) is not an integer number. As we will see, this case leads the transverse field to be non-circularly symmetrical.

As discussed in Sect. 2, the fractional beam results from the operation Uη,λ∝ (A +)η+λ (A -)η U 00, and its integral representation is given by Eq. (11), namely

$Uη,λ(r)∝∫0∞dktkt2η+λ+1exp(−μkt2w024)∫02πdϕexp(iλϕ)exp[iktrcos(ϕ−θ)].$

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.

The point here is that it appears that the angular integral ∫ 0 exp(iλϕ)exp[iktrcos(ϕ - θ)]dϕ 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 Uηλ (r) may be derived by expanding exp (iλϕ) in its Fourier series as follows:

$exp(iλφ)=1πexp(iπλ)sin(πλ)∑l=−∞∞exp(ilφ)λ−1.$

After inserting this expansion into Eq. (11), we can integrate each term of the summation following the same procedure described in Sect. 3 for integer values of l. The result is

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.

$Uη,λ(r)=∑l=−∞∞i∣l∣λ−lΓ(η′+∣l∣+1)Γ(∣l∣+1)fη′,∣l∣(R)exp(iθ),η′≡η+λ−∣l∣2,$

where f η′,|l| (R) is given by Eq. (23), and an overall amplitude factor has been omitted for simplicity.

Equation (34) says that a fractional beam with arbitrarily fractional indices (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 Uη,λ at z = 0 for several values of (η, λ) in the ranges η∈ [5,6] and 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.

For fractional values of λ the centroid of the beam Uη,λ is slightly displaced from the origin in the horizontal direction. Figure 4(a) shows the position of the beam centroid xc in normalized units of w 0 as a function of the angular index λ. The curve was determined by evaluating numerically the definition of the first-order moment xc = ∫∫ -∞ x|U|2dxdy/∫∫ -∞|U|2 dxdy for the range λ ∈ [0,6] and η = 0.. As λ increases xc exhibits a decreasing oscillatory behavior around the origin and vanishes for integer λ.

It is important to see what happens with the orbital angular momentum (OAM) carried by the beam when both indices vary. Within the paraxial regime, the 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]

$Jz=h¯∫∫−∞∞rt×Im(U*∇U)dxdy∫∫−∞∞∣U∣2dxdy$

where r t = x x̂ +y ŷ is the transverse radius vector. From the last paragraph of Sect. 3.2, we know that Uη,λ is proportional to

$Uη,λ∝(A+)η+λ(A−)ηU00=(A+)λ[(∂2∂r2+1r∂∂r)ηU00]=(A+)λUη,0.$

Now, it is clear that the function U η,0 depends on r only and thus it does not carry OAM. We then conclude that the existence of OAM in Uη,λ comes from the application of (A +)λ on U η,0, and thus it follows that Jz depends on λ only, while it is independent on η. To determine Jz, we evaluated numerically Eq. (35) using a two-dimensional Gauss-Legendre quadrature for a large number of combinations of radial and angular indices (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 Jz are included in the bottom line. Figure 4(b) depicts also the behavior of Jz as a function of the angular index within the range 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

We have introduced new fractional-order solutions of the PWE in cylindrical coordinates that constitute continuous transition modes between the integer-order elegant LG beams. The new solutions [Eqs. (14) and (34)] were obtained from the fundamental Gaussian beam by defining appropriate fractional creation operators which commute with the operator of the PWE. In the course of obtaining the new solutions, we also determined in closed form its Fourier transform and its expansion in terms of Bessel beams. The most important results of this work are summarized as follows:

• 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 fη,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.

Some of the interesting issues to pursue is whether the concept of fractional beams can be extended to multipole complex-source point solutions of the Helmholtz equation [18]. Of interest are also the effect of making the fractional parameters (η,λ) complex, and the behavior of the vortex structure exhibited by the fractional beam under a continuous variation of the fractional indices. These are currently under study by the author.

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

From Eq. (18), the second-order moment at z = 0 is given by the integral

$σ02=Γ2(η+l+1)22η+l+1Γ2(l+1)Γ(2η+l+1)∫0∞r2(r2w02)lexp(−2r2w02)Φ2(−η,l+1;r2w02)2rdrw02,$

where Φ(a,b;x) is the Kummer Confluent Hypergeometric function. To evaluate Eq. (37) we introduce the auxiliary function

$W(q)=∫0∞(r2w02)lexp(−qr2)Φ2(−η,l+1;r2w02)2rdrw02,$

where q = 2/w 2 0. The derivative of W (q) with respect to q reads as

$W′(q)=−∫0∞r2(r2w02)lexp(−qr2)Φ2(−η,l+1;r2w02)2rdrw02.$

Thus σ0 2 can be expressed in terms of the derivative of W (q) as follows

$σ02=Γ2(η+l+1)22η+l+1Γ2(l+1)Γ(2η+l+1)[W′(q)]q=2w02$

Making the change of variable ξ = r 2 /w 2 0 the integral for W (q) in Eq. (38) becomes

$W(q)=∫0∞ξlexp(−qw02ξ)Φ2(−η,l+1;ξ)dξ.$

This integral can be evaluated in closed form using the formula 7.622.1 of Ref. [14], we obtain

$W(q)=Γ(l+1)(qw02−1)2η(qw02)2η+l+1F(−η,−η;l+11(qw02−1)2),$

where 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

$W′(q)=−Γ2(l+1)(qw02−1)2η−3q2(qw02)2η+l[2η2qΓ(l+2)F(1−η,1−η;l+21(qw02−1)2)+(qw02−1)2[−1−l−2η+(l+1)w02q]w02F(−η,−η;l+11(qw02−1)2)]$

Evaluating at q = 2/w 2 0 we get

$W′(q)=w02Γ2(l+1)22η+l+2×[−4η2Γ(l+2)F(1−η,1−η;l+2;1)+(2η−l−1)F(−η,−η;l+1;1)],$

Now taking into account the following special cases of the hypergeometric function [13]

$F(1−η,1−η;l+2;1)=Γ(l+2)(l+2η)Γ2(l+1+η),$
$F(−η,−η;l+1;1)=Γ(l+1)Γ(l+1+2η)Γ2(l+1+η),$

W′ (q) becomes

$W′(q)=w02Γ2(l+1)Γ(l+2η)22η+l+2Γ2(l+1+η)[−4η2+(2η−l−1)(l+2η)].$

Replacing into Eq. (40) and after some simple algebraic simplifications we finally obtain

$σ02=w022[2η+l+l22η+l],$

which is the desired expression.

B. Appendix: Derivation of Eq. (24)

To find the governing equation for fη,l (R) = Rl exp (- R 2)Φ(-η,l,+ 1;R 2), we first note that Φ (η,l + 1;ξ) satisfies the Confluent Hypergeometric differential equation (CHDEq)

$ξd2Φdξ2+(l+1−ξ)dΦdξ+ηΦ=0,$

where ξ = R 2. By replacing Φ(-η,l+ 1;ξ) = ξ-l/2exp(ξ)v(ξ) into the CHDEq we determine that fη,l(ξ) satisfies the differential equation

$[ξ2d2dξ2+ξ(1+ξ)ddξ+(1+l2+η)ξ−(12)2]fη,l(ξ)=0.$

By re-expressing this differential equation in terms of R, we find that fη,l(R) satisfies the equation

$[d2dR2+1+2R2RddR+(4+2(2η+l)−l2R2)]v(R)=0,$

which is the same as Eq. (24).

Acknowledgments

The author acknowledges the financial support from Consejo Nacional de Ciencia y Tecnología of México (grant 42808), and from Tecnológico de Monterrey (grant CAT007).

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). [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]

References

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• |
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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).
[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).
[Crossref] [PubMed]

2004 (2)

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]

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]

Abbas, M.

Q.A. Naqvi and M. Abbas, “Fractional duality and metamaterials with negative permittivity and permeability,” Opt. Commun. 227, 143–146 (2003).
[Crossref]

Abramowitz, M.

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

Allen, L.

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

Barnett, S. M.

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

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]

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

Gutiérrez-Vega, J. C.

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

Kutay, M. A.

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

Oldham, K.

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

Ozaktas, H. M.

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

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

Podlubny, I.

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

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]

Ryzhik, I. M.

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

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]

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]

Siegman, A. E.

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

Spanier, J.

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

Stegun, I.A.

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

Zalevsky, Z.

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

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

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]

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]

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)

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

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Figures (4)

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

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.

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.

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)

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