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October 15, 2000 / Vol. 25, No. 20 / OPTICS LETTERS

1493

Alternative formulation for invariant optical fields:

Mathieu beams

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda

Instituto Nacional de Astrofisica, Optica y Electrónica, Apartado Postal 51/216, Puebla, Puebla Mexico 72000

Received June 5, 2000

Based on the separability of the Helmholtz equation into elliptical cylindrical coordinates, we present another

class of invariant optical fields that may have a highly localized distribution along one of the transverse

directions and a sharply peaked quasi-periodic structure along the other.

radial and angular Mathieu functions.We identify the corresponding function in the McCutchen sphere that

produces this kind of beam and propose an experimental setup for the realization of an invariant optical

field.© 2000 Optical Society of America

OCIS codes:

260.1960, 350.5500, 110.6760, 200.4650, 140.3300.

These fields are described by the

The interest in invariant optical fields (IOF’s) is due

to the fact that, under ideal conditions, they propa-

gate indefinitely without change of their transverse

intensity distribution. Their potential applications

in wireless communications, optical interconnections,

laser machining, and surgery make them very rele-

vant.However, to create true IOF’s would require

sources with infinite extent.

real world it is possible to generate approximations

to such fields, and the distance over which they can

propagate without significant alteration is limited by

the aperture of the optical system.

A whole class of this kind of wave field was intro-

duced by Durnin et al.1

These fields were described

in terms of nonsingular Bessel function J0.

applications a ringed structure of Bessel beams can

be a disadvantage. For this reason it is important to

identify other three-dimensional propagating solutions

of the wave equation that have no ringed structure but

are still invariant, if they exist.

In group theory it is known that, whenever a

partial differential equation is invariant under a

continuous symmetry group, one can find a coordi-

nate system in which the equation is separable.2

particular, the Helmholtz equation is known to be

separable in 11 coordinate systems.

Cartesian (rectangular cylindrical), circular cylindri-

cal, parabolic cylindrical, and elliptical cylindrical

coordinates have translation symmetries such that the

equation is separable into a transverse and a longitu-

dinal part. The separability of the equation imposes

the condition that the solutions of the transverse

part not depend on the longitudinal coordinate.

condition can be trivially observed for plane waves

and Bessel beams,1and the same must be true for

solutions of the Helmholtz equation in any cylindrical

coordinate system.

Based on the separability of the Helmholtz equation

into elliptical cylindrical coordinates, we present a

new analytic formulation of IOF’s that may have a

highly localized distribution along one of the trans-

verse directions and a sharply peaked quasi-periodic

structure along the other.

by the Mathieu functions that are exact solutions of

the Helmholtz equation.We also present a general

Nevertheless, in the

For some

In

Of them, only

This

These fields are described

formalism for creating IOF’s in terms of the Mc-

Cutchen theorem3,4that, when it is applied to the

solutions found, provides a way to generate Mathieu

beams. A simple experimental setup is proposed for

the realization of these IOF’s.

For optical fields the Helmholtz equation is obtained

from the electromagnetic-wave equation, assuming a

temporal dependence of the form exp?2ivt?.

for the fundamental traveling-wave solutions in any

of the four cylindrical coordinate systems mentioned

above, we find that the simplest of the IOF’s is a plane

wave whose wave vector has a magnitude k0? v?v,

where v is the wave’s frequency and v is its phase

velocity.

In a Cartesian frame two interfering plane waves

can produce an IOF with a cos2transverse intensity

pattern that propagates without change in its struc-

ture.5

Further superposition of a finite or an infinite

number of plane waves may produce propagation-

invariant patterns.The wave vectors of the new set

of plane waves must fulfill the conditions imposed

by the McCutchen sphere (MS).3,4

condition let us write the three-dimensional ampli-

tude distribution of a scalar optical field in infinite

space as1,6

Looking

To formalize this

u?x,y,z? ? exp?ikzz?

Z 2p

0

A?w?

3 exp?ikt?x cos w 1 y sin w??dw ,

(1)

where A?w? is an arbitrary modulation complex func-

tion, w is an angular variable, and kt? k0sin u0and

kz? k0cos u0 are the magnitudes of the transverse

and the longitudinal components of the wave vector

k0, respectively.The integral in Eq. (1) represents

the superposition of all the plane waves in the MS

whose wave vectors lie on a cone of angle tan u0?

kt?kz. Observe that the wave vectors delineate the

circumference of radius kt on the MS; see Fig. 1.

is important to remark that Eq. (1) is the solution of

the Helmholtz equation in any cylindrical coordinate

system by performance of the corresponding coordinate

transformation.

It

0146-9592/00/201493-03$15.00/0© 2000 Optical Society of America

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OPTICS LETTERS / Vol. 25, No. 20 / October 15, 2000

Fig. 1.

intersection between the cone u ? u0and the McCutchen

sphere jkj ? k0.

Because of the form of Eq. (1), it is clear that it

represents the propagation of invariant transverse

patterns.4,5

Function A?w? defines the structure of

the transverse pattern.7

is constant was analyzed in Ref. 1, and the IOF

is described by function J0?ktr?.

of the form exp?inw?, the field can be written in

terms of the higher Bessel functions Jn?ktr?exp?inw?,

anda linear superposition

kind is also invariant.7 –9

dom function it produces IOF’s with an irregular

transverseintensity distribution.10

A?w? ?PN21

scopic patterns, corresponds to the case of the super-

position of a discrete number of N plane waves

regularly distributed on the circumference.

Now, to find the description of IOF’s in elliptical

cylindrical coordinates requires that we solve the

Helmholtz equation. Elliptical cylindrical coordinates

are defined by x ? h cosh j cos h, y ? h sinh j sin h,

and z ? z, where j [ ?0, `? and h [ ?0, 2p? are the

radial and the angular variables, respectively, and 2h

is the interfocal separation.11,12

the Helmholtz equation separates into a longitudinal

part, which has a solution with dependence exp?ikzz?,

and a transverse part, whose solution ut?j, h? must

satisfy

≠2ut?j,h?

≠j2

≠h2

IOF’s are characterized by the ring formed by the

The case in which A?w?

When A?w? is

of solutions

When A?w? is a ran-

of this

Notethat

n?0d?w 2 2pn?N?, which produces kaleido-

In these conditions

1≠2ut?j,h?

1h2kt2

2

?cosh 2j 2 cos 2h?ut?j,h? ? 0,

(2)

which can be split into the radial and angular Mathieu

differential equations.6,11

relation k02? kt21 kz2. From Eq. (2), the zero-order

fundamental traveling-wave solutions are

u?1??j,h,z;q? ? ?Ce0?j;q?

kt satisfies the dispersion

1 iFey0?j;q??ce0?h;q?exp?ikzz?

u?2??j,h,z;q? ? ?Ce0?j;q?

2 iFey0?j;q??ce0?h;q?exp?ikzz?,

(3)

where Ce0 and Fey0 are the even radial Mathieu

functions of the first and second kinds, respectively,

ce0is the angular Mathieu function, and q ? h2kt2?4

is a parameter related to the ellipticity of the coor-

dinate system. The expressions within the brackets

are the first and second Mathieu–Hankel functions

of zero order.6,11

We remark that higher-order mode

solutions of Eq. (2) also exist with rotating phase

features.11–13

Equations (3) represent traveling coni-

cal waves, modulated azimuthally, as will be shown

below, slanting outward, u?1?, and inward, u?2?, whose

wave vectors form an angle u0 ? tan21?kt?kz? with

respect to the z axis.

In infinite space, where both traveling conical wave

solutions coexist and overlap, we have

u?j,h,z;q? ? Ce0?j;q?ce0?h;q?exp?ikzz?,

(4)

which is an IOF that will be referred to as the zero-

order Mathieu beam. Comparing Eq. (1) with thislast

result [Eq. (4)], we can deduce that for cylindrical el-

liptical coordinates the integral must be proportional

to the product of the Mathieu functions in Eq. (4).

fact,when we set A?w? ? ce0?w;q? within the integral of

Eq. (1), transform ?x, y? to polar coordinates ?r, f? by

use of the relation x cos w 1 y sin w ? r cos?w 2 f?,

and use the Euler formula for the complex exponen-

tial, it is possible to evaluate the integral,11–13and

the resulting field is proportional to Ce0?j; q?ce0?h; q?.

This implies that if the cone of wave vectors in Eq. (1)

is modulated by ce0?w; q? the resulting IOF will be a

Mathieu beam.

Until now we have assumed that the fields are of

infinite extent; however, in real physical situations the

fields are of finite transverse extension, introducing

diffraction effects into the evolution of the otherwise

invariant optical field.We have simulated the propa-

gation of a truncated Mathieu beam, using as the

initial condition Eq. (4) at z ? 0, through a circular

aperture.As we know of no available numerical

libraries that one can use to compute the whole family

of Mathieu functions, in particular the radial ones,

we developed a library to compute them based on the

theory of Mathieu functions.11–13

coordinate transformations defined above, we used

an algorithm to solve the Helmholtz equation.14

have chosen the parameters of the simulation to

give a geometric maximum propagation distance of

?15.5 m.An aperture of radius 10 mm gives an

angle u0 ? 0.00065 rad.

tion at l ? 632.8 nm produces kt ? k0sin u0 ?

6324.64 m21. We also set the interfocal distance of

the elliptical coordinates such that h ? 1 mm.

In Fig. 2(a) we show the transverse pattern of the

truncated zero-order Mathieu beam for the given

parameters, and in Fig. (b) its corresponding angular

spectrum is shown.The evolution of the transverse

intensity profiles along the planes x z and y z is

shown in Fig. 3.One may observe the very peculiar

evolution of the intensity in the x z plane [Figs. 3(a)

and 3(c)]; the beam propagates as a one-dimensional

quasi-invariant beam.The sudden vanishing of the

intensity is due to the fact that the superposition of

In

After applying the

We

Assuming that illumina-

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October 15, 2000 / Vol. 25, No. 20 / OPTICS LETTERS

1495

Fig. 2.

zero-order Mathieu beam.

zero-order Mathieu beam.

(a) Transverse intensity pattern of a truncated

(b) Angular spectrum of the

Fig. 3.

(a) x z and (b) y z.

in plane x z.

quasi-invariant beam within the conic overlapping region.

Evolution of an apertured Mathieu beam on planes

(c) Another view of the evolution

One may observe the very well-defined

conical waves that form the finite Mathieu beam can

occur only within a finite conical region.

teristic cone-shaped region is clearly observed in the

perpendicular plane as shown in Fig. 3(b).

well-defined pattern in one direction contrasts with

the resulting pattern in the other direction, which has

a transverse oscillating behavior, and the central peak

does not differ much from the lateral structure.

is contrary to what is found for the J0–Bessel beam,

in which the central peak is much more intense than

the surrounding structure.

can clearly be observed in the behavior of the axial

peak intensity and the variations along the edges of

the region of invariance.

To create Mathieu beams in the laboratory we refer

to the experimental setup suggested by Fig. 2(b), an

The charac-

The very

This

The effects of diffraction

annular slit with transmittance modulated by A?w? ?

ce0?w; q? and a lens, as in the experiment of Durnin

et al.1

This kind of transmittance can be difficult to

achieve.However, the required transmittance func-

tion can be approximated by an annular slit of radius

r0 illuminated with a one-dimensional strip pattern

with a Gaussian profile produced, for instance, with a

cylindrical lens.Mathematically this function is rep-

resented by d?r 2 r0?exp?2x2?w2?, where width w is

related to parameter q and can be adjusted according

to the desired pattern.This feature is of great impor-

tance since it provides a very simple way to create a

quasi-Mathieu beam in the laboratory.

In conclusion, we have shown that the Mathieu–

Hankel functions, which form a whole set of exact trav-

eling-wave solutions, can be used to describe a class

of IOF’s, Mathieu beams. Using the McCutchen theo-

rem, we demonstrated the relation between the general

class of IOF’s and these beams.

waves and Bessel beams, Mathieu beams also form an

orthogonal and complete set, in the sense that any in-

variant field can be represented as the superposition

of Mathieu beams.Finally, we proposed a simple ex-

perimental setup for creating an approximation to this

kind of beam.

Analogously to plane

J. C. Gutiérrez-Vega acknowledges support by the

Consejo Nacional de Ciencia y Tecnología (CONA-

CyT). This work was partially supported by CONA-

CyT grant 3943P-E9607.

address is sabino@inaoep.mx.

S. Chavez-Cerda’s e-mail

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