October 15, 2000 / Vol. 25, No. 20 / OPTICS LETTERS
Alternative formulation for invariant optical fields:
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
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
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
Of them, only
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
In a Cartesian frame two interfering plane waves
can produce an IOF with a cos2transverse intensity
pattern that propagates without change in its struc-
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
To formalize this
u?x,y,z? ? exp?ikzz?
3 exp?ikt?x cos w 1 y sin w??dw ,
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
0146-9592/00/201493-03$15.00/0© 2000 Optical Society of America
OPTICS LETTERS / Vol. 25, No. 20 / October 15, 2000
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
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
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
IOF’s are characterized by the ring formed by the
The case in which A?w?
When A?w? is
When A?w? is a ran-
n?0d?w 2 2pn?N?, which produces kaleido-
In these conditions
?cosh 2j 2 cos 2h?ut?j,h? ? 0,
which can be split into the radial and angular Mathieu
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
u?2??j,h,z;q? ? ?Ce0?j;q?
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
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?,
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
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
After applying the
Assuming that illumina-
October 15, 2000 / Vol. 25, No. 20 / OPTICS LETTERS
zero-order Mathieu beam.
zero-order Mathieu beam.
(a) Transverse intensity pattern of a truncated
(b) Angular spectrum of the
(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 effects of diffraction
annular slit with transmittance modulated by A?w? ?
ce0?w; q? and a lens, as in the experiment of Durnin
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 firstname.lastname@example.org.
S. Chavez-Cerda’s e-mail
1. J. E. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev.
Lett. 58, 1499 (1987).
2. E. G. Kalnins and W. Miller, Jr., J. Math. Phys. 17,
3. C. W. McCutchen, J. Opt. Soc. Am. 54, 240 (1964).
4. G. Indebetouw, J. Opt. Soc. Am. A 6, 150 (1989).
5. Y. Y. Ananev, Opt. Spectrosc. (USSR) 64, 722 (1988).
6. J. A. Stratton, Electromagnetic Theory (McGraw-Hill,
New York, 1941).
7. A. Vasara, J. Turunen, and A. T. Friberg, J. Opt. Soc.
Am. A 6, 1748 (1989).
8. R. Piestun and J. Shamir, J. Opt. Soc. Am. A 15, 3039
9. Z. Bouchal and J. Wagner, Opt. Commun. 176, 299
10. J. Turunen, A. Vasara, and A. T. Friberg, J. Opt. Soc.
Am. A 8, 282 (1991).
11. N. W. McLachlan, Theory and Applications of Mathieu
Functions (Oxford University, London, 1951).
12. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals,
Series and Products, 5th ed. (Academic, London, 1994).
13. J. C. Gutiérrez-Vega, “Formal analysis of the propa-
gation of invariant optical fields with elliptical sym-
metries,” Ph.D. dissertation (Instituto Nacional de
Astrofísica, Optica y Electroníca, Puebla, Mexico,
14. M. D. Feit and J. A. Fleck, Jr., J. Opt. Soc. Am. B 5,