Multi-parameter laser modes in paraxial optics

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MULTI-PARAMETER LASER MODES IN PARAXIAL OPTICS

CHRISTOPH KOUTSCHAN, ERWIN SUAZO, AND SERGEI K. SUSLOV

Abstract. We study multi-parameter solutions of the inhomogeneous paraxial wave equation in alinear and quadratic approximation which include oscillating laser beams in a parabolic waveguide,spiral light beams, and other important families of propagation-invariant laser modes in weaklyvarying media. A similar effect of superfocusing of particle beams in a thin monocrystal film is alsodiscussed. In the supplementary electronic material, we provide a computer algebra verification ofthe results presented here, and of some related mathematical tools that were stated without proofsin the literature.

1. Introduction

In this article, we study multi-parameter laser modes in (linear) paraxial optics with the helpof computer algebra methods by using the analogy with quantum mechanics. In particular, anErmakov-type system approach to generalized quantum harmonic oscillators is utilized to parax-ial, or parabolic, wave equations in a weakly inhomogeneous lens-like medium. Although severaldifferent techniques are widely available for integrating of the (scalar) parabolic equations (see, forinstance, recent reviews [2], [117] and the references therein), in this article we would like to explorea variant of the Fresnel integral and a certain generalization of the lens transformation [80] com-bined with explicit solutions of the Ermakov-type system introduced in [74]. We demonstrate thatthis approach gives a natural mathematical description of special laser modes propagation in opticalsystems. In the spirit of a modern “doing science by a computer” paradigm, a computer algebraderivation of all main results is presented in the form of a Mathematica notebook [66], with the aidof algorithmic tools presented in [61], [62], [63]. For a more traditional approach to the paraxialwave equations and for their numerous applications in optics and engineering, the reader can bereferred to the classical accounts [2], [8], [21], [22], [40], [45], [60], [105], [109], [118], [120], [121],[123], [124]. (The interested reader is referred to [8], [34], [40], [51], [75], [86] for further details onthe transition from Maxwell to paraxial wave optics; see also [4], [5], and [125] for different aspectsof the paraxial approximation. A modern status of the concept of photon, second quantization,photon spin and angular momentum are discussed in [18], [19], [53], [54], [55], [67], [94]; see also thereferences therein.)

2. Green’s Function and Fresnel Integrals for Inhomogeneous Media

This section comprises a brief survey of results established in [27], [35], [74], [80], [83], [111], [112](see also the references therein for the classical accounts) which are composed here in a compact

Date: July 4, 2014.1991 Mathematics Subject Classification. Primary 35Q55, 35Q51; Secondary 35C05, 68W30, 81Q05.Key words and phrases. Paraxial wave equation, Green’s function, generalized Fresnel integrals, Airy-Hermite-

Gaussian beams, Hermite-Gaussian beams, Laguerre-Gaussian beams, Bessel-Gaussian beams, spiral beams, time-dependent Schrodinger equation.

1

http://de.arxiv.org/abs/1407.0730v1

2 CHRISTOPH KOUTSCHAN, ERWIN SUAZO, AND SERGEI K. SUSLOV

form in order to make our presentation as self-contained as possible. In addition, we presentindependent proofs in the supplementary electronic material [66] for the reader’s benefits. In thecontext of paraxial optics, this approach, among other things, allows one to unify various lasermodes introduced and studied by different authors (a detailed bibliography is provided below butwe apologize in advance if an important reference is missing).

2.1. Unidimensional Case. Recent advances in quantum mechanics of generalized harmonic os-cillators can be utilized in order to solve similar problems concerning the light propagation in ageneral lens-like medium [35], [59], [60], [68], [71], [80], [83].

2.1.1. Green’s Function and Generalized Fresnel Integrals. In the context of quantum mechanics,the 1D linear Schrodinger equation for generalized driven harmonic oscillators,

iψt = −a(t)ψxx + b(t)x2ψ − ic(t)xψx

− id(t)ψ − f(t)xψ + ig(t)ψx (2.1)

(a, b, c, d, f, and g are suitable real-valued functions of the time t only), can be solved by theintegral superposition principle:

ψ(x, t) =

∫ ∞

−∞

G(x, y, t)ψ(y, 0) dy, (2.2)

where Green’s function G(x, y, t) is given by

G(x, y, t) =1

√

2πiµ0(t)exp

(

i(

α0(t)x2 + β0(t)xy + γ0(t)y

2 + δ0(t)x+ ε0(t)y + κ0(t))

)

(2.3)

for suitable initial data ψ(x, 0) = ϕ(x) (see [27], [74], [112] and the references therein for moredetails).

The functions α0, β0, γ0, δ0, ε0, and κ0 are given by [27], [112]:

α0(t) =1

4a(t)

µ′0(t)

µ0(t)− d(t)

2a(t), (2.4)

β0(t) = − λ(t)

µ0(t), λ(t) = exp

(

−∫ t

0

(

c(s)− 2d(s))

ds

)

, (2.5)

γ0(t) =1

2µ1(0)

µ1(t)

µ0(t)+

d(0)

2a(0)(2.6)

and

δ0(t) =λ(t)

µ0(t)

∫ t

0

((

f(s)− d(s)

a(s)g(s)

)

µ0(s) +g(s)

2a(s)µ′0(s)

)

ds

λ(s), (2.7)

ε0(t) = − g(0)

2a(0)+ 2

∫ t

0

λ(s)(

a(s) (f(s)− δ′0(s))− d(s)g(s))

+ a(s)δ0(s)λ′(s)

µ′0(s)

ds (2.8)

= −2a(t)λ(t)

µ′0(t)

δ0(t) + 8

∫ t

0

a(s)σ(s)λ(s)

(µ′0(s))

2 µ0(s)δ0(s) ds

+ 2

∫ t

0

a(s)λ(s)

µ′0(s)

(

f(s)− d(s)

a(s)g(s)

)

ds,

MULTI-PARAMETER MODES 3

κ0(t) =

∫ t

0

δ0(s)(

g(s)− a(s)δ0(s))

ds (2.9)

=a(t)µ0(t)

µ′0(t)

δ20(t)− 4

∫ t

0

a(s)σ(s)

(µ′0(s))

2 (µ0(s)δ0(s))2 ds

− 2

∫ t

0

a(s)

µ′0(s)

µ0(s)δ0(s)

(

f(s)− d(s)

a(s)g(s)

)

ds

provided that µ0 and µ1 are the standard (real-valued) solutions of the characteristic equation:

µ′′(t)− τ(t)µ′(t) + 4σ(t)µ(t) = 0 (2.10)

with varying coefficients

τ(t) =a′

a− 2c+ 4d, σ(t) = ab− cd+ d2 +

d

2

(

a′

a− d′

d

)

, (2.11)

subject to the initial conditions µ0(0) = 0, µ′0(0) = 2a(0) 6= 0 and µ1(0) 6= 0, µ′

1(0) = 0. TheWronskian of these standard solutions is given by

W (µ0, µ1) = µ0µ′1 − µ′

0µ1 = −2µ1(0)a(t)λ2(t). (2.12)

Our coefficients (2.4)–(2.9) satisfy the so-called Riccati-type system, see the unidimensional case ofEquations (2.41)–(2.46) below with c0 = 0 [74], subject to the following asymptotic expansions

α0(t) =1

4a(0)t− c(0)

4a(0)− a′(0)

8a2(0)+O(t), (2.13)

β0(t) = − 1

2a(0)t+

a′(0)

4a2(0)+O(t),

γ0(t) =1

4a(0)t+

c(0)

4a(0)− a′(0)

8a2(0)+O(t),

δ0(t) =g(0)

2a(0)+O(t),

ε0(t) = − g(0)

2a(0)+O(t),

κ0(t) = O(t)

as t→ 0. As a result,

G(x, y, t) ∼ 1√

2πia(0)texp

(

i(x− y)2

4a(0)t

)

(2.14)

× exp

(

−i(

a′(0)

8a2(0)(x− y)2 +

c(0)

4a(0)

(

x2 − y2)

− g(0)

2a(0)(x− y)

))

.

Here, f ∼ g as t→ 0, if limt→0 (f/g) = 1. (For applications, say to random media ([101], [115]), theintegrals are treated in the most general way which includes stochastic calculus; see, for example,[93].)

Note. Most of these results were only stated in the original publications because its detailedcalculations are pretty messy and time-consuming without use of algorithmic tools. In this article,for the reader’s benefits we present systematic computer algebra proofs of these results [66].


In the context of paraxial optics, when the time variable t represents the coordinate, say s,related to the direction of wave propagation, the expressions (2.2)–(2.3) can be thought of as ageneralization of Fresnel integrals [7], [8], [21], [35], [45], [68], [69], [70], [71], [83], [123]. Thecorresponding Schrodinger equation (2.1), with t → s, can be referred to as a generalized paraxialor parabolic wave equation [80], [83].

2.1.2. Special Beam Modes in Weakly Inhomogeneous Media. An important particular solution (gen-eralized Hermite-Gaussian beams in optics) of the parabolic equation (2.1) is given by [74]:

ψn(x, s) =ei(αx

2+δx+κ)+i(2n+1)γ

√

2nn!µ√π

e−(βx+ε)2/2 Hn(βx+ ε), (2.15)

where Hn(x) are the Hermite polynomials [92]. Here,

µ = µ(0)µ0

√

β4(0) + 4 (α(0) + γ0)2, (2.16)

α = α0 − β20

α(0) + γ0

β4(0) + 4 (α(0) + γ0)2 , (2.17)

β = − β(0)β0√

β4(0) + 4 (α(0) + γ0)2=β(0)µ(0)

µ(t)λ(t), (2.18)

γ = γ(0)− 1

2arctan

β2(0)

2 (α(0) + γ0), a(0) > 0 (2.19)

δ = δ0 − β0ε(0)β3(0) + 2 (α(0) + γ0) (δ(0) + ε0)

β4(0) + 4 (α(0) + γ0)2 , (2.20)

ε =2ε(0) (α(0) + γ0)− β(0) (δ(0) + ε0)

√

β4(0) + 4 (α(0) + γ0)2

, (2.21)

κ = κ(0) + κ0 − ε(0)β3(0)δ(0) + ε0

β4(0) + 4 (α(0) + γ0)2 (2.22)

+ (α(0) + γ0)ε2(0)β2(0)− (δ(0) + ε0)

2

β4(0) + 4 (α(0) + γ0)2

in terms of the fundamental solution subject to the arbitrary real or complex-valued initial dataµ(0) 6= 0, α(0), β(0) 6= 0, γ(0), δ(0), ε(0), κ(0). This solution was obtained in [74] by an integralevaluation and its direct verification by substitution is provided in [66].

Note. Equations (2.17)–(2.22) solve the one-dimensional case of the Ermakov-type system (2.41)–(2.46) below with c0 = 1 [74]; for the complex form of these solutions, see [67]; their verification isprovided in [66].

By the superposition principle, (orthonormal) solutions (2.15) can be used for the correspondingeigenfunction expansions in the case of real-valued initial data. In our approach, the functions fand g are treated as two stochastic processes and Equations (2.7)–(2.9) and (2.20)–(2.22) can beanalyzed by statistical methods [10], [101] (which may include random initial data).


A solution in terms of Airy functions [40] (generalized Airy beams) has the form [81], [83]:

ψ(x, s) =ei(αx

2+δx+κ)−i(βx+ε−2γ2/3)γ√µ

Ai(βx+ ε− γ2), (2.23)

where

µ = 2µ(0)µ0 (α(0) + γ0) , (2.24)

α = α0 −β20

4 (α(0) + γ0), (2.25)

β = − β(0)β02 (α(0) + γ0)

=β(0)µ(0)

µλ, (2.26)

γ = γ(0)− β2(0)

4 (α(0) + γ0), (2.27)

δ = δ0 −β0 (δ(0) + ε0)

2 (α(0) + γ0), (2.28)

ε = ε(0)− β(0) (δ(0) + ε0)

2 (α(0) + γ0), (2.29)

κ = κ(0) + κ0 −(δ(0) + ε0)

2

4 (α(0) + γ0). (2.30)

A direct verification is given in [66] for the reader’s benefits. Important special cases of Airy beamswere found in [15], [106], and [107] (see also [81], [117] and the references therein; more details aregiven in Section 3.1 below).

Note. Equations (2.24)–(2.30) solve the one-dimensional case of the Riccati-type system (2.41)–(2.46) below with c0 = 0 [74]; a proof is provided in [66].

2.2. Two-Dimensional Case. For the laser beam propagation in optics, the 2D case (with orwithout cylindrical symmetry) is of a great importance.

2.2.1. Separation of Variables. In the paraxial approximation, a 2D coherent light field in a generallens-like medium with coordinates (r, s) = (x, y, s) can be described by the following equation forthe complex field amplitude:

iψs(r, s) = Hψ(r, s), H = H1(x, s) +H2(y, s), (2.31)

where H1,2 are the Hamiltonians in x and y directions similar to one in (2.1) but, in a generalinhomogeneous medium model, with two different sets of suitable functions a1,2(s), b1,2(s), c1,2(s),d1,2(s), f1,2(s), and g1,2(s). (We assume, for simplicity, that the nondiagonal terms are eliminatedby passing to normal coordinates.) The kernel of generalized Fresnel integral can be obtained asthe product [83]:

G(r, r′, s) = G1(x, ξ, s)G2(y, η, s), (2.32)

where the kernels G1,2 are given by (2.3) with a simple change of notation: the coefficients α(1,2)0 ,

β(1,2)0 , γ

(1,2)0 , δ

(1,2)0 , ε

(1,2)0 , κ

(1,2)0 are defined, in general, in terms of two sets of the fundamental


solutions (2.4)–(2.9) with t↔ s. The solution of the corresponding boundary value problem can befound by the integral superposition principle (2D generalized Fresnel integral):

ψ(r, s) =

∫∫

R2

G(r, r′, s)ψ(r′, 0) dr′ (2.33)

for suitable initial data. (This integral determines the spatial beam evolution during the Fresneldiffraction.)

The corresponding 2D Hermite-Gaussian beams have the form

ψnm(r, s) =ei(κ1+κ2)

√

2n+mn!m!µ(1)µ(2)πei(α1x2+δ1x)+i(2n+1)γ1 ei(α2y2+δ2y)+i(2m+1)γ2 (2.34)

× e−(β1x+ε1)2/2−(β2y+ε2)

2/2 Hn (β1x+ ε1)Hm (β2y + ε2)

in terms of solutions of the Ermakov-type system (2.41)–(2.46) below with c0 = 1, which are knownin quadratures [74] (see also (3.9)–(3.14) for an important explicit special case). Equations (2.16)–(2.22) are valid with a similar change of notation for given initial data µ(1,2)(0), α1,2(0), β1,2(0) 6= 0,γ1,2(0), δ1,2(0), ε1,2(0), κ1,2(0) (see also [9], [10], [45], [104], [120], [121], [123] for various specialcases).

In general, by the separation of variables, the product of any two 1D solutions (2.15) and (2.23),say

ψn(r, s) = ψn(x, s)ψ(y, s), (2.35)

gives an important class of 2D solutions (Airy-Hermite-Gaussian beams in a weakly inhomogeneousmedium; see also [48], [49], [50]).

2.2.2. Cylindrical Symmetry. If a1(s) = a2(s) = a(s), b1(s) = b2(s) = b(s), c1(s) = c2(s) = c(s),d1(s) = d2(s) = d(s), the parabolic equation,

iAs = −a (Axx + Ayy) + b(

x2 + y2)

A− ic (xAx + yAy) (2.36)

− 2idA− (xf1 + yf2)A + i (g1Ax + g2Ay) ,

where f1,2(s) and g1,2(s) are real-valued functions of a coordinate in the direction of the opticalaxis s related to the wave propagation, can be reduced to the standard forms

− iχτ + χξξ + χηη = c0(

ξ2 + η2)

χ, (c0 = 0, 1) (2.37)

by the following ansatz

A = µ−1ei(α(x2+y2)+δ1x+δ2y+κ1+κ2) χ(ξ, η, τ) (2.38)

(see Lemma 1 of [83], which is reproduced below in our notation with an independent computeralgebra proof for the reader’s convenience).

Lemma 1. The nonlinear parabolic equation,

iAs = −a (Axx + ψyy) + b(

x2 + y2)

A− ic (xAx + yAy)− 2idA (2.39)

− (xf1 + yf2)A+ i (g1Ax + g2Ay) + h |A|pA,where a, b, c, d, f1,2 and g1,2 are real-valued functions of s, can be transformed to

− iχτ + χξξ + χηη = c0(

ξ2 + η2)

χ+ h0 |χ|p χ (c0 = 0, 1) (2.40)


by the ansatz (2.38), where ξ = β(s)x + ε1(s), η = β(s)y + ε2(s), τ = γ(s), h = h0aβ2µp (h0 is a

constant), provided that

dα

ds+ b+ 2cα + 4aα2 = c0aβ

4, (2.41)

dβ

ds+ (c+ 4aα)β = 0, (2.42)

dγ

ds+ aβ2 = 0, (2.43)

dδ1,2ds

+ (c + 4aα)δ1,2 = f1,2 + 2gα+ 2c0aβ3ε1,2, (2.44)

dε1,2ds

= (g − 2aδ1,2)β, (2.45)

dκ1,2ds

= gδ1,2 − aδ21,2 + c0aβ2ε21,2. (2.46)

Here,

α =1

4a

µ′

µ− d

2a(2.47)

and solutions of the system (2.41)–(2.46) are given by (2.24)–(2.30) and (2.16)–(2.22) for c0 = 0and c0 = 1, respectively.

Proof. For a computer algebra derivation, see the Mathematica notebook [66], which is available asa supplementary material on the article’s website. �

Our substitution (2.38) can be thought of as a generalized lens transformation in nonlinearparaxial optics (cf. [73], [90], [91], [114], [116], [124]). De facto, we have found a “proper” system ofspatial coordinates (ξ, η, τ) which automatically takes into consideration “imperfections” of initialdata and turbid medium in linear and quadratic approximations.

Note. An algorithmic proof of one-dimensional version this lemma is given in [64].

3. Multi-parameter Laser Beams and Their Special Cases

With the help of the generalized lens transformation described in Lemma 1 and available ex-plicit solutions from quantum mechanics one can analyze, in a unified form, a large class of multi-parameter modes for the corresponding linear parabolic wave equations in 1D and 2D weaklyinhomogeneous media which are objects of interest in paraxial optics.

3.1. Airy Beams. In quantum mechanics, the time-dependent Schrodinger equation for a freeparticle (or the normalized paraxial wave equation in optics [35], [106] also known as the parabolicequation [40], [124]),

iψt + ψxx = 0, (3.1)

by the following ansatz

ψ(x, t) = ei(x−2t2/3)t F (x− t2) (3.2)

can be reduced to the Airy equation:

F ′′ = zF, z = x− t2, (3.3)


whose bounded solutions are the Airy functions F = kAi(z) (up to a multiplicative constant k)with well-known asymptotics as z → ±∞ [40], [95].

The nonspreading Airy beams, which accelerate without any external force, were introduced byBerry and Balazs [15] (see also [16], [29], [47], and [122] for further exploration of different aspectsof this result). These nonspreading and freely accelerating wave packets have been demonstratedin both one- and two-dimensional configurations as quasi-diffraction-free optical beams [106], [107]thus generating a considerable interest to this phenomenon (see [1], [3], [11], [12], [13], [17], [24],[25], [26], [32], [56], [57], [76], [96], [98], [100], [117] and the references therein).

Equation (3.1) possesses a nontrivial symmetry [90]:

iψt + ψxx = 0 → iχτ + χξξ = 0, (3.4)

under the following transformation:

ψ(x, t) =

√

β(0)

1 + 4α(0)texp i

(

α(0)x2 + δ(0)x− δ2(0)t

1 + 4α(0)t+ κ(0)

)

(3.5)

× χ

(

β(0)x− 2β(0)δ(0)t

1 + 4α(0)t+ ε(0),

β2(0)t

1 + 4α(0)t− γ(0)

)

,

which is usually called the Schrodinger group, and/or the maximum (known) kinematical invariancegroup of the free Schrodinger equation (see also [20], [29], [78], [79], [88], [91], [117] and the referencestherein; the subgroups and their invariants are discussed in [20], [81]; the group parameters α(0),β(0), γ(0) = 0, δ(0), ε(0), and κ(0) = 0 are chosen as initial data of the corresponding Riccati-typesystem [78]).

As a result, in paraxial optics, the multi-parameter Airy modes are given by

B(x, s) =

√

β(0)

1 + 4α(0)sexp

(

iα(0)x2 + δ(0)x− δ2(0)s

1 + 4α(0)s

)

(3.6)

× exp

(

iβ2(0)s

1 + 4α(0)s

(

ε(0) +β(0)x− 2β(0)δ(0)s

1 + 4α(0)s− 2

3

β4(0)s2

(1 + 4α(0)s)2

))

× Ai

(

ε(0) +β(0)x− 2β(0)δ(0)s

1 + 4α(0)s− β4(0)s2

(1 + 4α(0)s)2

)

as a family of particular solutions of the parabolic equation iBs+Bxx = 0. (One can choose γ(0) =κ(0) = 0 in the explicit action (3.5) of the Schrodinger group without loss of generality.) Thenonspreading case of Berry and Balazs [15] occurs when α(0) = 0 in our notation. Other importantspecial cases are discussed in [81], [106], [107] (see also the references therein). It is worth notingthat our solution resembles, in the linear approximation, main features of rogue waves [58], [81],[108]. The direct verification by substitution and a computer algebra derivation of the parabolicequation for the multi-parameter beams (3.6) is given in [66] (see Section 4 for more details).

3.2. Oscillating and Breathing Hermite-Gaussian Beams. For a 1D inhomogeneous paraxialwave equation with quadratic refractive index (a lens-like medium [59], [121]),

2iAs + Axx − x2A = 0, (3.7)


an important multi-parameter family of particular solutions can be presented as follows [72], [80]:

An(x, s) = ei(αx2+δx+κ)+i(2n+1)γ

√

β

2nn!√πe−(βx+ε)2/2 Hn(βx+ ε), (3.8)

where Hn(x) are the Hermite polynomials [92] and

α(s) =α0 cos 2s+ sin 2s (β4

0 + 4α20 − 1) /4

β40 sin

2 s+ (2α0 sin s+ cos s)2, (3.9)

β(s) =β0

√

β40 sin

2 s+ (2α0 sin s+ cos s)2, (3.10)

γ(s) = −1

2arctan

β20 tan s

1 + 2α0 tan s, (3.11)

δ(s) =δ0 (2α0 sin s+ cos s) + ε0β

30 sin s

β40 sin

2 s+ (2α0 sin s+ cos s)2, (3.12)

ε(s) =ε0 (2α0 sin s+ cos s)− β0δ0 sin s√

β40 sin

2 s+ (2α0 sin s+ cos s)2, (3.13)

κ(s) = sin2 sε0β

20 (α0ε0 − β0δ0)− α0δ

20

β40 sin

2 s+ (2α0 sin s+ cos s)2(3.14)

+1

4sin 2s

ε20β20 − δ20

β40 sin

2 s+ (2α0 sin s+ cos s)2.

The real or complex-valued parameters α0, β0 6= 0, γ0 = 0, δ0, ε0, κ0 = 0 are initial data of thecorresponding Ermakov-type system [74], [78].1 A direct Mathematica verification can be foundin [66]. (Harmonic motion of cold trapped atoms is experimentally realized [77].)

These “missing” solutions that are omitted in all textbooks on quantum mechanics (see [79]and [85]) provide a new multi-parameter family of oscillating Hermite-Gaussian beams in parabolic(self-focusing fiber) waveguides, which deserve an experimental observation; special cases were theo-retically studied earlier in [9], [38], [40], [45], [59], [120], [123]. For graphical examples see Figures1 and 2 of Ref. [80]. These modes are orthonormal for real-valued parameters. The correspondinggeneralized coherent or minimum uncertainty squeezed states are analyzed in [72].

3.3. Hermite-Gaussian Beams. The homogeneous paraxial wave equation,

2iBs +Bxx = 0, (3.15)

can be transformed by the substitution,

B(x, s) =1

(1 + s2)1/4exp

(

isx2

2 (1 + s2)

)

A

(

x√1 + s2

, arctan s

)

, (3.16)

into the inhomogeneous one (3.7) (see [78] and the references therein; a Mathematica verificationcan be found in [66]). Composition of (2.15) and (3.16) results in the following multi-parameter

1From now on, we abbreviate α0 = α(0), etc for the sake of compactness.


family of spreading solutions to the parabolic equation (3.15):

Bn(x, s) =

√

√

√

√

β0

2nn!√

π(

(1 + 2α0s)2 + β4

0s2)

(3.17)

× exp

(

−(β0x+ ε0)2 + 2s (α0ε0 − δ0β0) ε0 − i (2x (α0x+ δ0)− sδ20)

2 (1 + 2α0s+ iβ20s)

)

× exp

(

−i(

n +1

2

)

arctan

(

β20s

1 + 2α0s

))

Hn

β0 (x− δ0s) + (1 + 2α0s) ε0√

(1 + 2α0s)2 + β4

0s2

for real or complex initial data [80]. The direct derivation is also provided in [66]. (It is worthnoting that our parameters ε0 6= 0 and δ0 6= 0 describe, in a natural way, the deviation from theoptical axis and a successive oblique propagation of the beam in an optical system, which is notusually discussed in detail.)

Among various special cases of these multi-parameter solutions are the so-called elegant Hermite-Gaussian beams. In our notation, they occur for the complex-valued parameters when 4α2

0+β40 = 0.

The substitution1 + 2α0s+ iβ2

0s√

(1 + 2α0s)2 + β4

0s2

= exp

(

i arctan

(

β20s

1 + 2α0s

))

(3.18)

followed by 2α0 = iβ20 results in

B(el)n (x, s) =

√

β0

2nn! (1 + 2iβ20s)

n+1√πHn

(

β0 (x− δ0s+ iβ0ε0s) + ε0√

1 + 2iβ20s

)

(3.19)

× exp

(

−2β20x

2 + (2β0ε0 − iδ0) (2x− δ0s)− (1 + 2iβ20s) ε

20

2 (1 + 2iβ20s)

)

.

When n = 0, one gets the multi-parameter fundamental Gaussian modes. In this case,

∣

∣

∣B

(el)0 (x, s)

∣

∣

∣

2

=β0

√

π (1 + 4β40s

2)exp

(

−2β20 (x− δ0s)

2 + 2β0ε0 (x− δ0s) + ε20 (1 + 2β40s

2)

1 + 4β40s

2

)

,

(3.20)∫ ∞

−∞

∣

∣

∣B

(el)0 (x, s)

∣

∣

∣

2

dx =e−ε2

0/2

√2.

These optical fields obey a certain “propagation-invariant similarity rule”:∣

∣

∣B

(el)0 (x, s)

∣

∣

∣

2

=β0e

−k2/2

√

π (1 + 4β40s

2), k = constant

provided that 2β0 (x− δ0s) = −ε0 ±√

(k2 − ε20) (1 + 4β40s

2) and k2 ≥ ε20. Thus, our solution de-scribes an “obliques propagation” of the laser beam with respect to the optical axis (approachingthe corresponding slanted asymptotes as s → ∞). For instance, the best confinement of opticalenergy occurs around the line x = δ0s, which becomes the direction of the beam propagation, whenε0 = 0. This simple example shows how one can use our extra parameters in order to aim the laserbeam and to maximize its intensity.


Special families of Gaussian beams have found significant applications in science, biomedicine,and technology. Among them, the fundamental Gaussian mode described by Eq. (3.19), when n = 0and δ0 = ε0 = 0, is the most useful one. According to [2], the laser beams of this kind are utilized forthe material cutting and surgery, for data reading in CD-DVD players and in optical remote sensingtechnology, and for microparticle trapping and atom cooling. Thus, telecommunication networksincluding the internet are based upon optical waveguide systems in which fundamental Gaussianmodes are propagated in a wavelength multiplexing configuration.

In general, our multi-parameter solutions (3.17) can be thought of as the Hermite-Gaussian beamswith “aberration/astigmatic elements” (see Refs. [2], [6], [8], [60], [99], [104], [121], [123], [126] forfurther examples of these important modes in one and two-dimensions).

Note. Although the multi-parameter elegant Hermite-Gaussian beams are not orthogonal, thecorresponding integral:

∫ ∞

−∞

(

B(el)n (x, s)

)∗B(el)

m (x, s) dx

can be evaluated in terms of generalized hypergeometric functions in a way that is similar to [72].An investigation of certain minimization properties may be of interest.

3.4. Breathing Spiral Laser Beams. By the ansatz Ψ(x, y, t) = χ(ξ, η, τ), T = −τ and(

XY

)

=

(

cosωτ − sinωτsinωτ cosωτ

)(

ξη

)

(3.21)

(ω = constant), Equation (2.37) with c0 = 1 can be transformed to the equation of motion for theisotropic planar harmonic oscillator in a perpendicular uniform magnetic field:

iΨT +ΨXX +ΨY Y =(

X2 + Y 2)

Ψ+ iω (XΨY − YΨX) . (3.22)

The latter equation was solved in the early days of quantum mechanics by Fock [39] in polarcoordinates, X = R cosΘ and Y = R sin Θ :

Ψ(R,Θ, T ) =

√

n!

π (n+ |m|)! e−iET eimΘR|m|e−R2/2L|m|

n

(

R2)

, (3.23)

E = 4n+ 2 (|m|+ 1)−mω

(m = ±0,±1, . . . , n = 0, 1, . . .) in terms of Laguerre polynomials [92]. This wave function coincides,up to a simple factor, with the one for a flat isotropic oscillator without magnetic field. Therefore, itsdevelopment in terms of (2.34) for standard harmonics is a 2D special case of the multi-dimensionalexpansions from [92] (see also [28], [87] and the references therein).

By back substitution, one arrives at a general family of spiral solutions in inhomogeneous media.For example, the 2D paraxial wave equation

2iAs + Axx + Ayy =(

x2 + y2)

A (3.24)

possesses the following Laguerre-Gaussian modes [80]

Amn (x, y, s) = β

√

n!

π (n +m)!ei(α(x

2+y2)+δ1x+δ2y+κ1+κ2)ei(2n+m+1)γ(

β(x± iy) + ε1 ± iε2)m

(3.25)

× e−(βx+ε1)2/2−(βy+ε2)2/2Lmn

(

(βx+ ε1)2 + (βy + ε2)

2)

, m ≥ 0


(by the action of Schrodinger’s group; see [78], [79], [83] and the references therein for classicalaccounts). Here, Equations (3.9)–(3.14) are utilized for complex or real-valued parameters α0,

β0 6= 0, δ(1,2)0 , ε

(1,2)0 (the last two sets may be different for x and y variables, respectively). Examples

are shown in Figures 3 and 4 of Ref. [80].

In addition, a special Gaussian form of our solution (2.34) gives a general example of spiral ellipticbeams discussed in [45].

3.5. Laguerre-Gaussian Beams. The homogeneous parabolic equation,

2iBs +Bxx +Byy = 0, (3.26)

with the help of

B(x, y, s) =1

(s2 + 1)1/2exp

(

is (x2 + y2)

2 (s2 + 1)

)

A

(

x√s2 + 1

,y√s2 + 1

, arctan s

)

(3.27)

can be reduced to the standard form (3.24). A multi-parameter solution is given by [80]

Bmn (x, y, s) =

1

1 + 2α0s+ iβ20s

exp

(

−i (m+ 2n) arctan

(

sβ20

1 + 2α0s

))

(3.28)

× exp

(

−is δ(1)0

2+ δ

(2)0

2

2 (1 + 2α0s+ iβ20s)

)

× exp

(2iα0 − β20) (x

2 + y2)− 2(

β0ε(1)0 − iδ

(1)0

)

x− 2(

β0ε(2)0 − iδ

(2)0

)

y

2 (1 + 2α0s+ iβ20s)

× exp

2β0s(

δ(1)0 ε

(1)0 + δ

(2)0 ε

(2)0

)

− (1 + 2α0s)(

ε(1)0

2+ ε

(2)0

2)

2 (1 + 2α0s+ iβ20s)

×

β0(x+ iy)−(

δ(1)0 + iδ

(2)0

)

s+(

ε(1)0 + iε

(2)0

)

(1 + 2α0s)√

(1 + 2α0s)2 + β4

0s2

m

× Lmn

(

β0

(

x− δ(1)0 s)

+ ε(1)0 (1 + 2α0s)

)2

+(

β0

(

y − δ(2)0 s)

+ ε(2)0 (1 + 2α0s)

)2

(1 + 2α0s)2 + β4

0s2

by the action of Schrodinger’s group. (The corresponding parameters are initial data of theErmakov-type system (2.41)–(2.46); see Lemma 1.) For the set of complex-valued parameters, twospecial cases are of a particular interest, namely, the multi-parameter “elegant” Laguerre-Gaussianbeams, when 2α0 = iβ2

0 :

Bmn (x, y, s) (el) =

(

1 + 2iβ20s)−m−n−1

exp

(

−is δ(1)0

2+ δ

(2)0

2

2 (1 + 2iβ20s)

)

(3.29)

× exp

−β20 (x

2 + y2) +(

β0ε(1)0 − iδ

(1)0

)

x+(

β0ε(2)0 − iδ

(2)0

)

y

(1 + 2iβ20s)


× exp

2β0s(

δ(1)0 ε

(1)0 + δ

(2)0 ε

(2)0

)

− (1 + iβ20s)(

ε(1)0

2+ ε

(2)0

2)

2 (1 + 2iβ20s)

×(

β0 (x+ iy)−(

δ(1)0 + iδ

(2)0

)

s+(

ε(1)0 + iε

(2)0

)

(1 + 2α0s))m

× Lmn

(

β0

(

x− δ(1)0 s)

+ ε(1)0 (1 + iβ2

0s))2

+(

β0

(

y − δ(2)0 s)

+ ε(2)0 (1 + iβ2

0s))2

1 + 2iβ20s

,

and multi-parameter “diffraction-free” Laguerre beams, when 2α0 = −iβ20 :

Bmn (x, y, s) (dif) =

(

1− 2iβ20s)n

exp(

−(

β0ε(1)0 − iδ

(1)0

)

x−(

β0ε(2)0 − iδ

(2)0

)

y)

(3.30)

× exp

(

β0s(

δ(1)0 ε

(1)0 + δ

(2)0 ε

(2)0

)

− 1− iβ20s

2

(

ε(1)0

2+ ε

(2)0

2)

− is

2

(

δ(1)0

2+ δ

(2)0

2)

)

×(

β0(x+ iy)−(

δ(1)0 + iδ

(2)0

)

s +(

ε(1)0 + iε

(2)0

)

(

1− iβ20s)

)m

× Lmn

(

β0

(

x− δ(1)0 s)

+ ε(1)0 (1 + iβ2

0s))2

+(

β0

(

y − δ(2)0 s)

+ ε(2)0 (1 + iβ2

0s))2

1− 2iβ20s

.

For m = n = 0 and ε(1,2)0 = 0, this beam degenerates into the ordinary plane wave propagating in

the direction r =(

δ(1)0 , δ

(2)0 , 1

)

.

Among numerous special cases are the Laguerre-Gaussian beams discovered in [14], [97], [126]. Byclassical accounts [2], [8], [45], [60], [109], [120], [121] (see also the references therein), the families ofthe Hermite-Gaussian and Laguerre-Gaussian modes arise naturally as approximate eigenfunctionsof the resonators with rectangular or circular spherical/flat mirrors, respectively. They also serveas models for eigenmodes of certain fibers. The introduction of astigmatic elements in opticalresonators or after them leads to the generation of Hermite-Laguerre-Gaussian and Gaussian-Incebeams [103]. The Laguerre-Gaussian beams are also proposed for the applications in free-spaceoptical communications systems, where the information is encoded as orbital angular momentumstates of the beam [44], in quantum optics to design entanglement states of photons [84], [89], inlaser ablation [52], and in optical metrology [41], to name a few examples. Angular momentum oflaser modes is discussed in [119].

3.6. Bessel-Gaussian Beams. Use of the familiar generating relations

∞∑

n=0

Lαn(ξ) t

n

Γ (α + n + 1)= (ξt)−α/2etJα

(

2√

ξt)

(3.31)

=et

Γ(α+ 1)0F1 (−;α + 1;−ξt) , Jν(z) =

(z/2)ν

Γ(ν + 1)0F1

(

−ν + 1

;−z2

4

)


in (3.28) results in a new multiparameter family of the Bessel-Gaussian beams:

B(x, y, s) =1

1 + 2α0s+ iβ20s

exp

(

−is δ(1)0

2+ δ

(2)0

2

2 (1 + 2α0s+ iβ20s)

)

(3.32)

× exp

i(2α0 + iβ2

0) (x2 + y2) + 2

(

δ(1)0 + iβε

(1)0

)

x+ 2(

δ(2)0 + iβε

(2)0

)

y

2 (1 + 2α0s+ iβ20s)

× exp

2β0s(

δ(1)0 ε

(1)0 + δ

(2)0 ε

(2)0

)

− (1 + 2α0s)(

ε(1)0

2+ ε

(2)0

2)

2 (1 + 2α0s+ iβ20s)

× exp

(

t1 + 2α0s− iβ2

0s

1 + 2α0s+ iβ20s

)

β0

(

x+ iy −(

δ(1)0 + iδ

(2)0

)

s)

+(

ε(1)0 + iε

(2)0

)

(1 + 2α0s)

1 + 2α0s+ iβ20s

m

× 0F1

−m+ 1

;−t

(

β0

(

x− δ(1)0 s)

+ ε(1)0 (1 + 2α0s)

)2

+(

β0

(

y − δ(2)0 s)

+ ε(2)0 (1 + 2α0s)

)2

(1 + 2α0s+ iβ20s)

2

.

See [66] for an automatic verification. For the complex-valued parameters, among two interestingspecial cases are multi-parameter “elegant” Bessel-Gaussian beams, when 2α0 = iβ2

0 :

B(el)(x, y, s) =1

1 + 2iβ20s

exp

(

−is δ(1)0

2+ δ

(2)0

2

2 (1 + 2iβ20s)

)

(3.33)

× exp

t− β20 (x

2 + y2)−(

βε(1)0 − iδ

(1)0

)

x−(

βε(2)0 − iδ

(2)0

)

y

1 + 2iβ20s

× exp

2β0s(

δ(1)0 ε

(1)0 + δ

(2)0 ε

(2)0

)

− (1 + iβ20s)(

ε(1)0

2+ ε

(2)0

2)

2 (1 + 2iβ20s)

×

β0

(

x+ iy −(

δ(1)0 + iδ

(2)0

)

s)

+(

ε(1)0 + iε

(2)0

)

(1 + iβ20s)

1 + 2iβ20s

m

× 0F1

−m+ 1

;−t

(

β0

(

x− δ(1)0 s)

+ ε(1)0 (1 + iβ2

0s))2

+(

β0

(

y − δ(2)0 s)

+ ε(2)0 (1 + iβ2

0s))2

(1 + 2iβ20s)

2

,

and multi-parameter “diffraction-free” Bessel beams, when 2α0 = −iβ20 :

B(dif)(x, y, s) = exp(

t(

1− 2iβ20s)

−(

βε(1)0 − iδ

(1)0

)

x−(

βε(2)0 − iδ

(2)0

)

y)

(3.34)

× exp

(

β0s(

δ(1)0 ε

(1)0 + δ

(2)0 ε

(2)0

)

− (1− iβ20s)

2

(

ε(1)0

2+ ε

(2)0

2)

− is

2

(

δ(1)0

2+ δ

(2)0

2)

)

×(

β0

(

x+ iy −(

δ(1)0 + iδ

(2)0

)

s)

+(

ε(1)0 + iε

(2)0

)

(

1− iβ20s)

)m


× 0F1

(

−m+ 1

;−t(

β0

(

x− δ(1)0 s)

+ ε(1)0

(

1− iβ20s)

)2

− t(

β0

(

y − δ(2)0 s)

+ ε(2)0

(

1− iβ20s)

)2)

.

For m = 0 and ε(1,2)0 = 0, the latter beams have the peculiar property of conserving the same

disturbance distribution, apart from the phase factor, across any plane parallel to the xy-plane in

the direction of propagation: x = x0 + δ(1)0 s, y = y0 + δ

(2)0 s, z = z0 + s.

Diffraction-free Bessel beams are reviewed in [2], [118] (see also [36], [37], [46], [110] and thereferences therein for classical accounts on propagation-invariant optical fields and Bessel modes).

3.7. Spiral Beams. Two-dimensional solutions of the paraxial wave equation (3.26), that possessthe propagation-invariant property

∫∫

R2

∣

∣B(x, y, 0)∣

∣

2dx dy =

∫∫

R2

∣

∣B(X, Y, s)∣

∣

2dX dY = constant

under rotation and rescaling X = ρ(s) (x cos θ(s) + y sin θ (s)) , Y = ρ(s) (−x sin θ(s) + y sin θ (s)) ,were investigated in detail [7], [8], [97], and [102].

In Section 3.4, we have already analyzed the transition to a rotating frame of reference; seeEquations (3.21)–(3.23). As a combined result, Equation (3.26) by means of the substitution

B(x, y, s) =1

(s2 + 1)1/2exp

(

is (x2 + y2)

2 (s2 + 1)

)

(3.35)

× C

(

x cos (ω arctan s) + y sin (ω arctan s)√s2 + 1

,−x sin (ω arctan s) + y cos (ω arctan s)√

s2 + 1, arctan s

)

can be transformed into the equation of motion for the isotropic planar harmonic oscillator in aperpendicular uniform magnetic field, namely:

2iCs + Cxx + Cyy =(

x2 + y2)

C + 2iω (xCy − yCx) (3.36)

(our transformation (3.27) can be thought of as its special case when ω = 0). An algorithmicderivation is provided in [66].

A straightforward use of Fock’s solutions (3.23) does not lead directly to a new family of spiralbeams due to the cancellation of the crucial parameter ω (see Section 3.5 in the Mathematica

notebook [66]). For example, the solution

B(x, y, s) =e−i(m+2n+1) arctan s

√s2 + 1

exp

(

− x2 + y2

2 (1 + is)

)(

x+ iy√s2 + 1

)m

Lmn

(

x2 + y2

s2 + 1

)

, m ≥ 0 (3.37)

is verified by a direct substitution [66]. (A multi-parameter extension can be obtained by the actionof Schrodinger’s group.)

On the second thought, with the help of (3.35), we shall look for a spiral beam in the form:

B(x, y, s) =1

(s2 + 1)1/2exp

(

is (x2 + y2)

2 (s2 + 1)

)

C(X, Y, T ). (3.38)

Here, a familiar eigenfunction expansion [7], [8]:

C(X, Y, T ) =∑

n≥0

∑

m≥0

c(±)n,m C(±)

n,m(X, Y, T ), (3.39)


in terms of Laguerre-Gaussian modes, must satisfy the axillary equation (3.36). In complex form,z = 1 + is, T = arg z = arctan s, and

Z = X + iY =x+ iy

|z| e−iω arg z, X = ReZ, Y = ImZ. (3.40)

Denoting for m ≥ 0,

C = C(±)n,m(X, Y, T ) = e−ikT (X ± iY )m e−|Z|2/2Lm

n

(

|Z|2)

, (3.41)

we obtain an important “eigenfunction identity”:

2iCT + CXX + CY Y −(

X2 + Y 2)

C − 2iω (XCY − Y CX) (3.42)

= 2 (k ±mω −m− 2n− 1) Cby a direct evaluation [66].

As a result, substituting the series (3.39) into Equation (3.36), one gets∑

n≥0

∑

m≥0

c(±)n,m (k ±mω −m− 2n− 1) C(±)

n,m(X, Y, T ) = 0

or, in view of the completeness of the Laguerre-Gaussian modes,

c(±)n,m (k ±mω −m− 2n− 1) = 0. (3.43)

Nontrivial solutions of this equation and the correspoding spiral beams are analyzed in the originalworks [7], [8]. A multi-parameter extension can be obtained by the action of Schrodinger’s group.

3.8. Applications to Quantum Mechanics. A similar effect of the superfocusing of proton beamin a thin monocrystal film was discussed in [30], [31] (validity of the 2D harmonic crystal modelhad been confirmed by Monte Carlo computer experiments). Among other quantum mechanicalanalogs, the minimum-uncertainty squeezed states for atoms and photons in a cavity, are reviewedin [72]. It is worth noting that similar states can be identified for the motion of a quantum particlein a uniform magnetic field [39].

3.9. Extensions to Nonlinear Paraxial Optics. For high-intensity beams, nonlinear mediumeffects should be taken into account in the theory of wave propagation. See [33], [51], [73], [81], [82],[83], [116], [123], [124] and the references therein for extensions to nonlinear geometrical optics. Ageneralization of Lemma 1 for combination of certain nonlinear terms is discussed in [83] but searchfor solutions of nonlinear equations is much more complicated.

In the 1D linear case, where nonspreading Airy beams were introduced [15] (see also [106], [107]),the symmetry of the free Schrodinger equation can be used in order to obtain multi-parametersolutions (3.6). Although the corresponding 1D cubic nonlinear Schrodinger equation is no longerpreserved under the expansion transformation (but has a similarity reduction to the second Painleveequation [42], [43], [81], [108], [113]), the same symmetry holds for the quintic nonlinear Schrodingerequation, which is thus invariant under the action of this group. Here, the blow up, namely asingularity such that the wave amplitude tends to infinity in a finite time, occurs (see [82], [112],[116] and the references therein).

As is well known, a similar symmetry holds for the homogeneous 2D cubic nonlinear Schrodingerequation [73], [114] (in optics this symmetry is known as Talanov’s transformation [124]). This is


another classical example of the blow up phenomenon. The stationary 2D waveguides in homo-geneous quadratic Kerr media are unstable [73]. Under certain conditions, self-focusing of lightbeams occurs on a finite distance despite diffraction spreading. Moreover, for parabolic channelsin a monocrystal film, the cubic nonlinearity may further enhance superfocusing of particle beamspredicted in [30], [31]. The corresponding inhomogeneous medium effects deserve a detailed study.An extension to randomly varying media is also of interest (cf. [10], [101], [115]).

4. Computer Algebra Methods

For an automatic verification of the results presented in this paper, we used the computer algebrasystem Mathematica, and in some specific instances, the HolonomicFunctions package [62], writtenby the first-named author in the frame of his Ph.D. thesis [61]2. (See also [65] and the referencestherein for applications of the HolonomicFunctions package to relativistic Coulomb integrals.)

The application of computer algebra in the context of the present paper comes in three dif-ferent flavors: The first one employs Grobner bases, the second one is based on the built-insimplification procedures of Mathematica, and the third one is related to the above-mentionedHolonomicFunctions package.

Grobner bases were introduced in [23] and are a very useful tool for computations with polynomialideals. For finding “nice” expressions for the solutions (2.4)–(2.9) of the Riccati-type system, one canconsider the ideal generated by the (polynomial) equations (2.41)–(2.46). Equivalence of expressionsthen corresponds to equality modulo the ideal. See [66] for more details.

Similarly, we discovered an “invariant” of the Ermakov-type system. Again using Equations(2.41)–(2.46) as input (but now with c0 = 1) one can use Grobner bases to find relations that areimplied by the given equations. Searching for an equation that does not involve the parametersa, b, c, d, f, g yields the identity

β2κ′ − βδε′ + (δ2 + β2ε2)γ′ = 0

which was missing in the original publications. It reveals that the differential equations in theErmakov-type system are in fact dependent. In particular, Equation (2.46) for κ′ can be derivedfrom the previous equations of this system.

To demonstrate the other two applications, recall the multi-parameter Airy modes B(x, s) givenin Equation (3.6). Thanks to the progress that computer algebra systems like Mathematica havebeen made during the past decades, particularly in dealing with special functions, it can be directlyverified that B(x, s) satisfies the parabolic equation (3.1): one just inputs the expression given onthe right-hand side of (3.6) and differentiates it symbolically. Then the command FullSimplify

successfully simplifies the expression iBs +Bxx to 0, see the corresponding section in the accompa-nying notebook [66].

The last approach achieves more, and is a bit of an overhead if one only wanted to verify thatB(x, s) satisfies the given differential equation. Namely, the HolonomicFunctions package com-putes the set of all differential equations that a given expression satisfies (more precisely: a finitebasis of this, in general, infinite set). For the multi-parameter Airy modes, the software computesthe following two differential equations:

(4αs+ 1)2Bs + 2p1Bx − ip2B = 0, (4.1)

2The package can be downloaded from http://www.risc.jku.at/research/combinat/software/HolonomicFunctions/


(4αs+ 1)2Bxx − 2ip1Bx − p2B = 0, (4.2)

where the polynomial coefficients p1 and p2 are given by

p1 = δ + 4αδs+ β3s + 8α2sx+ 2αx, (4.3)

p2 = 2iα + β2ε+ δ2 + 8iα2s+ 4α2x2 + 4αδx+ β3x, (4.4)

and where α = α(0) etc. Obviously, the parabolic equation iBs + Bxx = 0 is just a simple linearcombination of the above two equations. Thus, we again have proved that B(x, s) satisfies iBs +Bxx = 0, but even more: the program has found this equation automatically, starting from theclosed form of its solution as the sole input.

Similarly, the remaining formulas in this paper can be verified and/or derived. For the holonomicsystems approach to work, some inputs have to be transformed into an appropriate format, e.g.,the expression given by (3.8)–(3.14): holonomic functions are closed under addition, multiplication,and substitution of algebraic expressions. Since sin(s) and cos(s), which appear in the argument ofthe Hermite polynomials, are not algebraic, one may apply the transformation s 7→ i log(z) in orderto turn the trigonometric functions into rational functions. More details and all other computationsare contained in the accompanying Mathematica notebook [66].

5. Conclusion

This work is dedicated to a mathematical description of light propagation in turbid media and/orthrough optical systems that are subject to natural noise environment. To this end, we applyconcepts of the Fresnel diffraction, the generalized lens transformation, see Lemma 1, and computeralgebra tools [61], [62], [63] in order to analyze multi-parameter families of certain propagation-invariant laser beams in 1D and 2D that are important in paraxial optics and its applications.Independent proofs of these results are provided in the supplementary electronic material [66] alongwith a computer algebra verification of all related mathematical tools introduced in the originalpublications without sufficient details.

Acknowledgement. This research was partially carried out during our participation in the Sum-mer School on “Combinatorics, Geometry and Physics” at the Erwin Schrodinger InternationalInstitute for Mathematical Physics (ESI), University of Vienna, in June 2014. We wish to expressour gratitude to Christian Krattenthaler for his hospitality. The first-named author was supportedby the Austrian Science Fund (FWF): W1214, the second-named author by the AMS-Simons TravelGrants, with support provided by the Simons Foundation, and by the National Science FoundationGrant DMS-1440664, and the third-named author by the AFOSR grant FA9550-11-1-0220.

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Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of

Sciences, Altenberger Straße 69, A-4040 Linz, Austria

E-mail address : [email protected]

School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287–

1804, U.S.A.; School of Mathematical Sciences, University of Puerto Rico, Mayaguez, Puerto

Rico


School of Mathematical and Statistical Sciences & Mathematical, Computational and Modeling

Sciences Center, Arizona State University, Tempe, AZ 85287–1804, U.S.A.


URL: http://hahn.la.asu.edu/~suslov/index.html

Multi-parameter laser modes in paraxial optics

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