Generalized solution of the paraxial equation Tomasz Rado_ zycki Faculty of Mathematics and Natural Sciences College of Sciences Institute of Physical Sciences

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Generalized solution of the paraxial equation

Tomasz Rado˙zycki∗

Faculty of Mathematics and Natural Sciences, College of Sciences, Institute of Physical Sciences,

Cardinal Stefan Wyszy´nski University, W´oycickiego 1/3, 01-938 Warsaw, Poland

A fairly general expression for a light beam is found as a solution of the paraxial Helmholtz

equation. It is achieved by exploiting appropriately chosen complex variables which entail the

separability of the equation. Next, the expression for the beam is obtained independently by super-

imposing shifted Gaussian beams, whereby the shift can be made either by a real vector (in which

case the foci of the Gaussian beams are located on a circle) or by a complex one. The solutions

found depend on several parameters, the speciﬁc choice of which allows to obtain beams with quite

diﬀerent properties. For several selected parameter values ﬁgures are drawn, demonstrating the

spatial distribution of the energy density and phase. In special cases, the eﬀect of a shift of the

intensity peak from one branch to another and phase singularities are observed.

I. INTRODUCTION

Within the scalar optics approximation a number of

monochromatic paraxial light beams of intriguing prop-

erties have been found. From the mathematical perspec-

tive, assuming the propagation along the z-axis, they are

described as solutions of the so-called paraxial equation:

4⊥ψ(r, z)+2ik∂zψ(r, z) = 0,(1)

where the stationary function ψ(r, z), called the envelope,

is related to the electric ﬁeld via

E(r, z, t) = E0eik(z−ct)ψ(r, z),(2)

with E0representing a constant vector. The Laplace op-

erator 4⊥in (1) is the two-dimensional one acting in

the transverse plane only (here r= [x, y]). The symbol

∂zstands for the partial derivative ∂

∂z . The approxima-

tions leading to the form (1) have been discussed in detail

elsewhere [1, 2].

A commonly accepted fundamental solution of the

paraxial equation is the Gaussian beam, which in the

cylindrical coordinates has the form (apart from the nor-

malization constant):

ψ(r, ϕ, z) = w0

w(z)n+1rneinϕ (3)

×exp h−r2

w(z)2+ikr2

2R(z)−i(n+ 1)ψG(z)i

endowed (n6= 0) or not (n= 0) with orbital angular

momentum with respect to the propagation axis [2–12].

The basic parameters that characterize this beam (w0

– the waist radius, w(z) = w0p1+(z/zR)2– radius at

a distance z,R(z) = z(1 + (zR/z)2) – the wavefront-

curvature radius, zR=kw2

0/2 – the Rayleigh length

and ψG(z) = arctan(z/zR) – the Gouy’s phase) are well

known and need no further explanation. This beam, writ-

ten in a slightly modiﬁed although equivalent form, will

be of use in the present work (see (18)).

∗t.radozycki@uksw.edu.pl

Other beams of similar nature include Bessel-Gaussian

(BG) [12–15], modiﬁed BG [16–18], Laguerre-Gaussian

(LG) [2, 12, 15, 19–23], Kummer-Gaussian (KG) (i.e.,

Hypergeometric-Gaussian) [24, 25], or γbeams [26], the

latter containing no Gaussian fall-oﬀ factor. All of them

are cylindrical in the sense that the wave intensity dis-

plays axial symmetry. Equation (1) has also solutions of

another kind, which do not manifest cylindrical symme-

try. As examples, one can mention Hermite-Gaussian

(HG) beams [2, 27] of rectangular symmetry, or Airy

beams [28–30].

The literature on this subject is extremely vast due

to the important and broad applications of light beams,

especially those with nontrivial structure, ranging from

trapping and guiding of particles, through image pro-

cessing, optical communication, harmonics generation in

nonlinear optics, quantum cryptography, up to biology

and medicine.

Some attempts to obtain a more general description

and/or derivation of various paraxial beams have been

undertaken in the past [18, 31–35]. In this paper, we

wish to present, along these lines, somewhat more general

solution to the equation (1), which depends on several

parameters remaining at our disposal. These general so-

lution does not exhibit cylindrical symmetry (in the sense

spoken of above), except for some special cases. A partic-

ular choice of the parameters values enables on one hand

to recover some of the above-mentioned modes, and on

the other to obtain other modes with equally interesting

properties.

The solution in question will be obtained below in two

ways. First, in Section II, some specially deﬁned substi-

tutions are used, which leads in two steps to the com-

plete separation of the paraxial equation. It is known

that the Helmholtz equation in 3 dimensions, owing to

the Robertson-Eisenhart condition [36, 37], turns out to

be separable in 11 orthogonal coordinate systems [38].

However, in order to obtain new solutions, in this work

complex variables will be employed.

Then, in Section III, it is demonstrated that, at least

for integer values of the parameter l(see (14)), these so-

lutions are superpositions of shifted zero-order Gaussian

beams [39] with some appropriately tailored weight func-

arXiv:2210.01088v1 [physics.optics] 3 Oct 2022

tion. It should be stressed that the ﬁrst approach does

not require lto be an integer, thus being more general.

Up to our knowledge such a solution of the paraxial equa-

tion has not been published before.

Section IV is devoted to certain special properties of

so derived beams. In particular, it is shown how a con-

crete choice of parameter values (l, µ, χ) leads to known

solutions previously obtained in the literature. Then the

properties of the waves of Section II in their general form

are analyzed and their spatial distributions are plotted,

for several selected parameter values. Necessarily, we had

to limit ourselves here to merely few cases, since there

are many possible options, especially when considering

that the parameters can also assume complex values. In

some cases, these spatial distributions exhibit quite spe-

cial properties, which deserve some attention and are dis-

cussed in following sections in detail. Here let us only

mention a transfer of the energy-density peak between

two beam-forming branches.

II. DERIVATION OF THE GENERAL

FORMULA FOR PARAXIAL BEAMS

In place of Cartesian coordinates x, y, z let us intro-

duce in (1) three complex variables ξ, η, α deﬁned in the

following way

ξ(x, y, z) = µ+2χ

α(z)(x+iy)1/2

,(4a)

η(x, y, z) = µ+2χ

α(z)(x−iy)1/2

,(4b)

α(z) = w2

0+2iz

k,(4c)

where quantities µand χare certain parameters. Their

values stay at our disposal for the moment and their role

in the structure of the beam will be determined later.

The choice of the particular branches of the complex

roots in (4a) and (4b) is inessential for the energy density

represented, up to a constant, by |ψ(r, z)|2. As regards

the third variable, i.e. α, it is in fact the quantity known

as the “complex beam parameter”, apart from some con-

stant coeﬃcient. Consequently w0stands for the beam’s

waist radius.

Now let us try to rewrite the paraxial equation in

terms of these new variables isolating, however, from the

very beginning the standard Gaussian factor. In order to

achieve this we set

ψ(r, z) = e−r2

α˜

ψ(ξ, η, α) (5)

and derive the diﬀerential equation satisﬁed by ˜

ψ. As

can be veriﬁed in the straightforward way, this equation

takes the form

ξη

∂2˜

∂ξ∂η −α2

χ2

∂˜

∂α −α

χ2˜

ψ= 0.(6)

The solution of (6) can be looked for in the form of the

product

ψ(ξ, η, α) = A(ξ, η)B(α),(7)

which allows for the full separation of the variables ξ, η

from α:

ξηA(ξ, η)

∂2A

∂ξ∂η =α

χ2α

B(α)

∂B

∂α + 1.(8)

Both sides of this equation have to be equal to the same

constant, which, by virtue of χbeing arbitrary at this

point (real or complex), can be set equal to 1. The solu-

tion of the ﬁrst equation

∂B

∂α =1

αχ2

α−1B(α),(9)

can be obtained in the standard way in the form

B(α) = B0

αe−χ2

α,(10)

with B0standing for a certain constant. The second

equation, i.e.,

ξη

∂2A

∂ξ∂η =A(ξ, η),(11)

can be solved as well upon ﬁrst introducing two new vari-

ables

u=ξ

η, v =ξ η, (12)

and then assuming

A(ξ, η) = Au(u)Av(v).(13)

Standard variables-separation procedure leads to two

equations

A00

u+1

uA0

u−l2

u2Au= 0,(14a)

A00

v+1

vA0

v−1 + l2

v2Av= 0, , (14b)

with l2standing for a separation constant. It should be

pointed out here that l, despite the symbol used, need

not be an integer. It can represent a fractional, real or

even complex number, and the separation of the variables

in the equation (11) proceeds in the same manner. This

fact implies that the resultant expression (17) will de-

scribe a whole wide variety of beams of diﬀerent nature,

depending on the choice made for the parameter’s value.

Several interesting examples will be provided in Sect. IV.

Equation (14a) has the two obvious solutions

Au(u) = ul, Au(u) = u−l,(15)

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GeneralizedsolutionoftheparaxialequationTomaszRado_zyckiFacultyofMathematicsandNaturalSciences,CollegeofSciences,InstituteofPhysicalSciences,CardinalStefanWyszynskiUniversity,Woycickiego1/3,01-938Warsaw,PolandAfairlygeneralexpressionforalightbeamisfoundasasolutionoftheparaxialHelmholtzequation.It...

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Generalized solution of the paraxial equation Tomasz Rado_ zycki Faculty of Mathematics and Natural Sciences College of Sciences Institute of Physical Sciences

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