1726 J. Opt. Soc. Am. A/Vol. 18, No. 7 /July 2001 Serna et al.
Complete spatial characterization of a pulsed
doughnut-type beam by use of

spherical optics and a cylindrical lens

Julio Serna and Fernando Encinas-Sanz

Departamento de Óptica, Facultad de Ciencias Fı́sicas, Universidad Complutense, 28040 Madrid, Spain

George Nemeş

Astigmat, 1118 Starbird Circle, Suite 23, San Jose, California 95117

Received October 25, 2000; revised manuscript received January 22, 2001; accepted February 16, 2001

A complete spatial characterization (in second-order moments) of a doughnut-type beam from a pulsed trans-
versely excited atmospheric CO2 laser is described. It includes the measurement of the orbital angular mo-
mentum carried by the beam. The key element in the characterization is the use of a cylindrical lens in ad-
dition to the usual spherical optics. Internal features of the beam that would have remained hidden if only
spherical optics were employed were revealed by use of the cylindrical lens. The experimental results are
compared and agree with a theoretical Laguerre–Gauss mode beam. © 2001 Optical Society of America

OCIS codes: 140.3460, 140.3470.
1. INTRODUCTION
The spatial characterization of laser beams has become
an important topic in itself. The beams produced by in-
dustrial and research lasers need to be well known and
specified to permit useful and reproducible results to be
obtained from their use. Along this line characterization
based on the Wigner distribution function and on the ir-
radiance second-order moments of beams has proven to be
very useful.1–6 Such characterization remains the basis
of an International Organization for Standardization
(ISO) standard,7 although some limitations have been
proposed and introduced in the scope of the standard.7–9

Within this formalism the most general three-
dimensional beam is characterized completely if the ten
elements of its beam matrix are known.5 Nine of these
elements (those related to the beam size, the beam diver-
gence, and the radius of curvature) can be obtained from
irradiance measurements by use of spherical lenses and
free-propagation sections. But the tenth element, which
is related to the orbital angular momentum that is trans-
ported by the beam,6 remains hidden if only optical sys-
tems with rotational symmetry are used.8 Several set-
ups have been proposed for measuring all the elements of
the beam matrix.10,11 In these setups at least one cylin-
drical lens has to be used during measurement to reveal
the information that is related to the orbital angular mo-
mentum. Otherwise, the characterization of the beam is
only partial and could lead to errors.8 Some experimen-
tal results have been obtained for pseudostigmatic
beams8,9 that are generated from simple astigmatic
beams,10 and they have been compared with the theoret-
ical predictions.11

The purpose of this paper is to apply the theory of mea-
suring pseudosymmetrical, or pseudotype, beams8,9 to a
0740-3232/2001/071726-08$15.00 ©
beam that comes unmodified from a laser head and not to
a specially conditioned beam, as was done in previous
studies. To our knowledge, this study is among the first
to present measurements to reveal the pseudosymmetri-
cal behavior of a beam from a laser without special accom-
modations for inducing such behavior outside the laser
cavity. Note that, without the cylindrical lens in the
measurement, the pseudosymmetrical aspect of the beam
(which is related to the existence of the orbital angular
momentum) could not be proved. Therefore a beam char-
acterization that uses no cylindrical lenses is not enough,
as was shown theoretically in Ref. 8.

With the above result in mind, we show how to com-
plete a full ten second-order-moments characterization of
a laser beam (from these moments the physical param-
eters could also be determined, as was shown in Ref. 12).
For this purpose, we use a transversely excited atmo-
spheric (TEA) CO2 laser that delivers a doughnut beam.
We used this laser in a number of previous studies.13–16

Its doughnut-shaped profile suggests that the beam could
carry orbital angular momentum, although previously we
had no direct proof of that. To complete this study, we
start by summarizing the theory and the previous results
in Section 2. Section 3 is devoted to the experimental
setup and to a discussion of the results. Conclusions are
given in Section 4.

2. THEORY AND PREVIOUS RESULTS
Let us consider a laser beam that is propagating in the z
direction and is described by its Wigner distribution func-
tion h(x, y, u, v), where x and y are the transversal coor-
dinates and u and v represent angles in paraxial propa-
gation. In a second-order characterization the laser
2001 Optical Society of America



Serna et al. Vol. 18, No. 7 /July 2001 /J. Opt. Soc. Am. A 1727
beam is represented by its beam matrix P, which is de-
fined in terms of the second-order averages of
h(x, y, u, v).1,3,5,9 The beam matrix P is a 4 3 4 real,
symmetric, and positive definite matrix. It is composed
of three 2 3 2 submatrices, W, M, and U, and has a maxi-
mum of ten independent elements1,3,5,8–10,12:

P 5 F W M

Mt UG 5 F ^x2& ^xy& ^xu& ^xv&

^xy& ^ y2& ^ yu& ^ yv&

^xu& ^ yu& ^u2& ^uv&

^xv& ^ yv& ^uv& ^v2&

G . (1)

In matrix (1) W 5 Wt represents the (squared) beam
width, U 5 Ut is the (squared) beam divergence, and M is
a nonsymmetric mixed matrix. The symmetric part of
W21M is the beam curvature, and the antisymmetric part
of M is related to the orbital angular momentum that is
transported by the beam.6,12 Because we can do just ir-
radiance measurements, only the W submatrix can be
measured directly.10 But P is transformed by ABCD op-
tical systems (lenses and free spaces, among others) by
use of the ABCD law,1,5,8 allowing us to use different op-
tics to obtain all its elements.10,11 It is also important to
note that, when a beam is modified by (or is propagating
through) an ABCD optical system, there are two indepen-
dent combinations of its P beam-matrix elements that re-
main invariant. We can choose these invariant param-
eters (or the generalized beam-propagation parameters,
as we call them) to be the following normalized
quantities5,8–10,12:

Meff
4 5 4k2 ~det P!1/2 > 1, (2)

t 5 2k2T 5 2k2Tr~WU 2 M2! > 1. (3)

Here k 5 2p/l is the modulus of the wave vector and l is
the wavelength of the beam. Instead of t from expression
(3), it is better to define and use the normalized invariant
quantity a, called intrinsic astigmatism10

a 5 t 2 Meff
4 . (4)

Unlike t, a has a straightforward physical interpretation:
it allows one to define two disjoint classes of beams—the
intrinsic stigmatic beams (a 5 0) and the intrinsic astig-
matic beams (a . 0).8–10 The intrinsic astigmatism can
be used in combination with Meff

4 , as in Ref. 10, or with
another normalized invariant quantity, aM , called the
maximum intrinsic astigmatism, that is directly related
to Meff

4 through8,9,12

aM 5
1

2
~Meff

4 2 1 !2. (5)

We now have the following double inequality:

0 < a < aM . (6)

Meff
4 is related to the extension of the beam in phase

space, i.e., the closer Meff
4 is to one, the smaller is the

phase-space extent of the beam, approaching the phase-
space extent of a TEM00 beam of the same wavelength.
Because the beam matrix P changes when the beam
propagates through ABCD optical systems whereas a and
Meff

4 , or, alternatively, a and aM , remain invariant, we
consider the beam-propagation parameters to be even
more important than P to the characterization of a beam.
Four disjoint families of beams were defined on the basis
of the four possible combinations of pairs of beam-
propagation parameters: (a 5 0, aM 5 0), (a 5 0, aM
. 0), (a . 0; aM . 0, a , aM), and (a 5 aM . 0, aM
. 0). Their physical meaning was given in Ref. 9.

A physical parameter that is useful for our purposes is
the orbital angular momentum per photon lz . This
quantity can be obtained from the antisymmetric part of
M: m 5 (^xv& 2 ^ yu&)/2. The time-averaged orbital
angular-momentum flux J̄z

L of a partially coherent beam
through a z plane is given by6

J̄z
L 5

I

c
~^xv& 2 ^ yu&!, (7)

where I is the beam power and c is the light speed. For
such a beam the photon flux (photons per unit time) is
I/\v, with \ and v being the normalized Plank’s constant
and the angular frequency (v 5 kc), respectively. Fi-
nally, we obtain the orbital angular momentum per pho-
ton lz :

lz 5 J̄z
L

\v

I
5 k~^xv& 2 ^ yu&!\ 5 2km\. (8)

We now consider the measurement procedure for ob-
taining the beam-matrix elements. A simple way to ob-
tain the elements of P is to measure transversal profiles
of the beam irradiance at different z planes with a cam-
era. From those profiles ^x2&z , ^xy&z , and ^ y2&z can be
calculated. With a parabolic fit of those values, we can
obtain most of the elements of P at z 5 0 because in free-
space propagation we have

^x2&z 5 ^x2&0 1 2^xu&0z 1 ^u2&z2, (9)

^xy&z 5 ^xy&0 1 ~^xv&0 1 ^ yu&0!z 1 ^uv&z2, (10)

^ y2&z 5 ^ y2&0 1 2^ yv&0z 1 ^v2&z2. (11)

In Eqs. (9)–(11) the subindex 0 indicates values at z 5 0
(note that the divergence terms are z invariant; hence
they have no subindex). But with this procedure it is im-
possible to obtain all the terms. We can measure W, U,
and the symmetric part of M, but ^xv& and ^yu& always ap-
pear as (^xv& 1 ^ yu&), so we cannot recover the antisym-
metric part of M. It can be proved that this coupling can-
not be broken with spherical lenses either. That is why a
second set of measurements that includes the use of at
least one cylindrical lens is needed.8,10,11

The crossed term ^uv&L of the divergence submatrix UL
just after a cylindrical lens with a convergence 1/fx8 in the
x direction can be expressed with respect to the beam-
matrix elements just before the lens (with no subindex) by

^uv&L 5 ^uv& 2
^xv&

fx8
. (12)

In this way the antisymmetric part of M, m 5 (^xv&
2 ^ yu&)/2, can be recovered:



1728 J. Opt. Soc. Am. A/Vol. 18, No. 7 /July 2001 Serna et al.
m 5
1
2 ~^xv& 2 ^ yu&!

5 ~^uv& 2 ^uv&L!fx8 2
1
2 ~^xv& 1 ^ yu&!.

(13)

Using Eq. (13), one can complete the ten-parameter char-
acterization after the quantity ^uv&L is determined. This
value can be obtained from a parabolic fit for the beam
measured in free space after the cylindrical lens by use of
a formula analogous to Eq. (10).

To complete a full second-order characterization of a la-
ser beam according to the above theory, we used a pulsed
TEA CO2 laser. This laser was selected for two reasons:
First, we had obtained several previous results for that
laser (see Refs. 13–16), but they were only partial in
terms of second-order characterization, i.e., no cylindrical
lens was used in those previous studies for beam charac-
terization. Second, under some circumstances the output
laser beam irradiance profile has a doughnut shape.
Therefore it is considered to be a serious candidate as a
beam for which it is not enough that a characterization be
done with only rotationally symmetric optics [as is recom-
mended in ISO document 11146 (Ref. 7)], i.e., character-
ization without a cylindrical lens in which the antisym-
metric part of M would be ignored. The TEA CO2 laser
considered is a pulsed laser. It is important to note that
the second-order-characterization formalism can be ex-
tended to pulsed beams under very general conditions.16

3. MEASUREMENTS AND DISCUSSION
The CO2 laser is shown schematically in Fig. 1. It has a
stable, half-symmetric resonator with a spherical mirror
(radius of curvature of 10 m) and a flat output mirror.
The cavity length is 1120 mm. Inside the resonator there
is a Brewster plate for generating a linearly polarized
beam and a circular diaphragm close to the curved mir-
ror. The intracavity diaphragm controls the transverse
modes, and it is essential for our measurements (we used
diameters of 8, 10, and 11 mm for this circular dia-
phragm, as is explained below). Spherical lenses L1 ( f18
5 254 mm), L2 ( f28 5 127 mm), and L3 ( f38 5 508 mm)
were used to condition the beam. The laser beam at
l 5 10.6 mm is pulsed and has an energy per pulse of
.0.5 J. The pulse structure contains an .70-ns gain-
switch peak that is followed by an .2-ms tail. An electro-
optical chopper selects a temporal slice inside the laser
pulse. For reasons that are explained below, in our ex-
periments, we used a thick time slice (Dt0 5 100 ns) in
the tail region of the pulse (500 ns after the beginning of
the pulse, t0 . 500 ns). To measure the elements of the
beam matrix P, we took transversal profiles of the beam

Fig. 1. The cylindrical lens (cyl) is used to decouple the ^xv&
1 ^ yu& term.
irradiance by using a pyroelectric camera (Spiricon,
Model Pyrocam I). The cylindrical lens ( fx8 5 184 mm)
was used to put in evidence the antisymmetric part of the
submatrix M, as was explained in Section 2.

Now we briefly explain the measurements that were
done previously and the results that were obtained by use
of the three values of the circular diaphragm before we
describe the full beam characterization. For the 8-mm
diaphragm, a rounded Gaussian-like beam is obtained in
free space (M2 close to 1), and its behavior is nearly con-
stant along the pulse.14 With the 10-mm diaphragm the
output beam has a significant doughnut-shaped contribu-
tion and remains rotationally symmetric in free-space
propagation. For that beam, we measured the output
profiles, performed mode-beating measurements, and car-
ried out a second-order characterization by using rota-
tionally symmetric optics.15 From those measurements,
we know that, at the leading edge of the pulse (from the
beginning of the pulse to t0 . 100 ns), there is only one
mode. After that, there is a transition region, and,
finally, the beam reaches a doughnutlike spatial structure
with two modes (t0 . 500 ns), although the central mini-
mum is not zero.

For the 11-mm diaphragm preliminary results showed
highly similar behavior for the t0 . 500-ns region. From
the mode-beating measurements, we also found two
transversal modes, but the irradiance profile was closer to
a purely doughnut-shaped spatial structure with a nearly
zero irradiance at the central minimum. Because of
these characteristics, the 11-mm diaphragm was used in
the measurements considered in this paper, and the
second-order-moments characterization was done for a
thick time slice (Dt0 5 100 ns) in the tail region of the
pulse (t0 . 500 ns) that was produced with the 11-mm in-
tracavity diaphragm.

We now describe the actual measurements done in this
experiment. To check the setup, we took a first set of
measurements by using the 8-mm diaphragm. We do not
expect the Gaussian-like beam with M2 close to one14 to
have any general astigmatism or angular momentum.
This beam was used to check the orientation of the cylin-
drical lens. Measurements of this beam were done with-
out and with the cylindrical lens.

Typical beam profiles obtained with the 8-mm dia-
phragm are shown in Figs. 2(a) and 2(b). In both cases
the laser output profile was very reproducible from shot to
shot. Figure 2(c) shows the second-order moments that
characterized the (squared) beam size. With no cylindri-
cal lens, beam profiles were taken from z 5 350 mm to
z 5 1350 mm, but those taken after z 5 600 mm were
hidden to simplify the figure. One can see that, before
the cylindrical lens, the beam is rotationally symmetric.
After the lens, the beam remains Gaussian like, but it is
deformed along the x axis, as is expected. The cylindrical
lens acts on ^x2&, leaving ^ y2& unaffected. At the same
time the ^xy& parabola after the lens is close to zero and is
nearly flat, so we find that the output beam does not ro-
tate (or that its total rotation is negligible), as is shown in
the lower part of Fig. 2(c). As expected, from a stigmatic
input beam, before the cylindrical lens an aligned simple
astigmatic beam is obtained after a cylindrical lens with
power along the x axis.



Serna et al. Vol. 18, No. 7 /July 2001 /J. Opt. Soc. Am. A 1729
Fig. 2. Typical beam profiles obtained with the 8-mm diaphragm (a) with no cylindrical lens and (b) after the cylindrical lens, both
taken at z 5 650 mm. (c) Experimental values of ^x2&, ^xy&, and ^ y2& obtained before and after the cylindrical lens with the 8-mm
diaphragm. Before the lens the beam is rotationally symmetric, and after the lens the beam is a simple astigmatic beam aligned along
the lens axes. Note that the rotation angle is negligible.
After checking the setup with the Gaussian-like beam
that was produced by the 8-mm diaphragm, we started
the characterization of the doughnut-shaped beam that
was produced with the 11-mm diaphragm. A first group
of measurements was taken without the cylindrical lens.
It is important to note that, with no cylindrical lens, the
laser output profile is quite reproducible from shot to
shot. A rotationally symmetric beam with a profile simi-
lar to the one shown in Fig. 3(a) is always obtained.
From those irradiance profiles, we measured the following
coefficients for Eqs. (9)–(11):

^x2& 5 ~5.08 6 0.02! 2 @~12.04 6 0.04!

3 1023#z 1 @~7.69 6 0.03! 3 1026#z2, (14)

^xy& 5 ~51 6 5 ! 3 1023 2 @~90 6 20!

3 1026#z 1 @~47 6 9 ! 3 1029#z2, (15)

^ y2& 5 ~5.37 6 0.01! 2 @~12.64 6 0.04!

3 1023#z 1 @~7.97 6 0.02! 3 1026#z2. (16)
In Eqs. (14)–(16), the units for z are millimeters, and
those for ^x2&, ^xy&, and ^ y2& are millimeters squared.
The error factors included in those formulas are due to
the parabolic fitting of the experimental data. Crossed
terms in the errors of each parabola are not written but
were considered in the calculations that follow. The ex-
perimental results and the parabolic fits can be seen on
the left-hand side of Fig. 3(c) in which measurements
with no cylindrical lens after z 5 600 mm were sup-
pressed for the sake of clarity. The results were remark-
ably reproducible.

For the second set of measurements a cylindrical lens
was used. For this type of beam at least two kinds of be-
havior are possible after the cylindrical lens: A
doughnut-shaped beam can be obtained by the addition of
the irradiances of a TEM01 and a TEM10 Hermite–Gauss
mode. We can call that type of beam an irradiance
doughnut. Another way to obtain a doughnut-shaped
beam is to add those two modes but in a coherent way
(i.e., to add their electric fields) with 1p or 2p phases.
In such a case, we would have a Laguerre–Gauss beam:



1730 J. Opt. Soc. Am. A/Vol. 18, No. 7 /July 2001 Serna et al.
LG0
11 or LG0

21. We can call them amplitude doughnuts.
The difference between these two types of doughnut-
shaped beams can be seen by the propagation of the
beams through a cylindrical lens. An irradiance dough-
nut indicates a stigmatic beam and therefore an intrinsic
stigmatic beam,9 and its propagation through a cylindri-
cal lens merely makes the doughnut elliptical (it flattens
the irradiance profile), imposing on it the symmetry axes
of the cylindrical lens. The beam is still intrinsic stig-
matic after the lens, but it has an aligned simple astig-
matic symmetry. On the other hand, an amplitude
doughnut indicates an intrinsic astigmatic beam of the
pseudostigmatic type,8,9 and propagation through a cylin-
drical lens maintains its intrinsic astigmatism but de-
stroys the pseudosymmetry of the beam; the beam be-
comes a rotating general astigmatic beam.9 In the case
of a LG0

61 beam the beam is divided into two lobes.
For these measurements the cylindrical lens was

placed at zL 5 565 mm. A typical beam profile after the
cylindrical lens is shown in Fig. 3(b). But, in contrast
with the high reproducibility of the output beam profile
and the corresponding measurements made without the
cylindrical lens, in this case, we found that the laser pat-
tern could change from shot to shot. This fact is reflected
in Fig. 4. For most pulses [approximately 75% of them,
Fig. 4(a)] the beam profiles are rotated clockwise, and in
only a few cases [approximately 5%, Fig. 4(c)] they are ro-
tated counterclockwise. These are original intrinsic as-
tigmatic beams from the laser that had a pseudostigmatic
(rotational) symmetry in free space [the round irradiance
profile of Fig. 3(a)] before the cylindrical lens. In some
other cases [approximately 20%, Fig. 4(b)] the beam pro-
file is elliptical (flattened), indicating a true stigmatic
beam before the cylindrical lens and an aligned simple as-
tigmatic one after the cylindrical lens.9 Note that pulsed
laser beams that seem to be fully reproducible when we
consider only rotationally symmetric optics (suggesting
only truly stigmatic beams) are not as simple as they
appear at first sight. Also note that, for most pulses, the
beam is of the intrinsic astigmatic, pseudostigmatic type.
Fig. 3. Typical beam profiles obtained with the 11-mm diaphragm (a) with no cylindrical lens and (b) after the cylindrical lens, both
taken at z 5 725 mm. (c) Experimental values of ^x2&, ^xy&, and ^ y2& obtained before and after the cylindrical lens with the 11-mm
diaphragm. Only pulses that rotate clockwise were considered after the lens. Before the lens the beam is rotationally symmetric, but
after the lens the beam profiles rotate as shown.



Serna et al. Vol. 18, No. 7 /July 2001 /J. Opt. Soc. Am. A 1731
Fig. 4. Beam profiles taken at z 5 775 mm after the cylindrical lens: (a) clockwise rotated (75% of the pulses), (b) aligned elliptical
(flattened) profile (20% of the pulses), (c) counterclockwise rotated (5% of the pulses).
To explain the existence of the three types of beams
emitted by our laser, we have to consider that in TEA CO2
lasers usually there is more than one longitudinal mode.
In our case the laser emits approximately 20 longitudinal
modes. The intracavity diaphragm selects the transver-
sal modes, but the longitudinal modes are mutually inde-
pendent; thus some of them could rotate clockwise,
whereas others rotate counterclockwise. If nearly all the
modes rotate clockwise, we obtain a beam such as that
shown in Fig. 3(b) or Fig. 4(a), whereas if most of them
rotate counterclockwise something similar to that shown
in Fig. 4(c) is measured. Intermediate behaviors appear
between those two pure cases. The predominance of
clockwise-rotating beams could be explained by a small
asymmetry inside the laser cavity.

We completed the second-order-moments characteriza-
tion for the clockwise-rotating beams [those shown in
Figs. 3(b) and 4(a)]. The ^x2&, ^xy&, and ^ y2& values for
those beams together with the parabolic fits are shown on
the right-hand side of Fig. 3(c). The rotation angle for
those profiles is also shown in the figure. From the ex-
perimental values, we obtained the following free-
propagation parabolae:

^x2&L 5 ~23.3 6 0.1! 2 @~67.8 6 0.3!

3 1023#z 1 @~49.5 6 0.2! 3 1026#z2, (17)

^xy&L 5 ~1.8 6 0.3! 2 @~5.7 6 0.7!

3 1023#z 1 @~4.5 6 0.5! 3 1026#z2, (18)

^ y2&L 5 ~4.51 6 0.07! 2 @~10.2 6 0.2!

3 1023#z 1 @~6.2 6 0.1! 3 1026#z2, (19)

with z in millimeters and ^x2&L , ^xy&L , and ^ y2&L in units
of millimeters squared.

Equations (14)–(16) and (17)–(19) contain all the infor-
mation that we need to complete the characterization
(only the crossed terms in the error arising from the para-
bolic fit were not explicitly written). From Eqs. (14)–
(16), we have for the initial beam at z 5 zL
WzL 5 F ~730 6 10! 3 1023 ~14.6 6 0.7! 3 1023

~14.6 6 0.7! 3 1023 ~770 6 10! 3 1023 G ,
(20)

MzL 5 F 2~1.66 6 0.02! 3 1023 2~19 6 2 !

3 1026 1 m

2~19 6 2 ! 3 1026 2 m 2~1.80 3 0.03!

3 1023

G ,

(21)

UzL 5 F ~7.69 6 0.03! 3 1026 ~47 6 9 ! 3 1029

~47 6 9 ! 3 1029 ~7.97 6 0.02!

3 1026
G ,

(22)

where m is the antisymmetric part of the MzL submatrix.
In a similar way, from Eqs. (17)–(19) at z 5 zL , we have

WL
zL 5 F ~760 6 40! 3 1023 ~20 6 20! 3 1023

~20 6 20! 3 1023 ~770 6 10! 3 1023G ,
(23)

ML
zL

5 F 2~5.83 6 0.2! 3 1023 2~300 6 100!

3 1026 1 mL

2~300 6 100! 3 1026 2 mL 2~1.53 6 0.03!

3 1023

G ,

(24)

UL
zL 5 F ~49.5 6 0.2! 3 1026 ~4.5 6 0.5! 3 1026

~4.5 6 0.5! 3 1026 ~6.2 6 0.1! 3 1026G ,
(25)

for the beam after the cylindrical lens with mL being the
antisymmetric part of ML

zL. Finally, by using Eq. (13), we
find

m 5 2~810 6 90! 3 1026 mm. (26)

With this value, we have completed the measurement of
all the elements in PzL for the clockwise-rotating beams.



1732 J. Opt. Soc. Am. A/Vol. 18, No. 7 /July 2001 Serna et al.
We return now to calculate the beam-propagation pa-
rameters and the orbital angular momentum that is car-
ried by some of the above-measured beams. If we had ig-
nored the intrinsic astigmatism of the beam by not using
the cylindrical lens in the measurement and thus setting
m 5 0, we would have had

Meff
4 5 4.03 6 0.01, (27)

t 5 4.03 6 0.01, (28)

a 5 ~2.0 6 0.2! 3 1023, (29)

aM 5 4.58 6 0.03 (30)

and no orbital angular momentum

lz 5 0. (31)

With the numbers from Eqs. (27)–(31), we would have
considered that we had almost an intrinsic stigmatic
beam (a . 0). But with the full characterization the re-
sult is quite different:

Meff
4 5 3.1 6 0.2, (32)

t 5 5.0 6 0.2, (33)

a 5 1.8 6 0.4, (34)

aM 5 2.2 6 0.4, (35)

lz 5 2~1.0 6 0.1!\. (36)

Now we know that we have an intrinsic astigmatic beam
belonging to the third of the four disjoint families of
beams mentioned in Section 2, a . 0 and a , aM , and in

Fig. 5. Beam irradiance profiles at z 5 920 mm (after the cylin-
drical lens): (a) an actual experimental profile, (b) a pure (the-
oretical) LG0

21 beam.
its initial form the beam transports orbital angular mo-
mentum, lz Þ 0. Note that, in Eqs. (27)–(36), we con-
sider only the errors derived from the fitting of the experi-
mental data to Eqs. (9)–(11).

From the irradiance profiles, we also know the general
shape of the beam. It is a doughnut-shaped beam, so it
would be interesting to compare our results with those of
a LG beam. We could compare the experimental irradi-
ance profiles with those obtained from a numerical propa-
gation of a LG0

21 beam. As one can see from Fig. 5, for a
z plane situated after the lens the results are very simi-
lar. It could also be interesting to calculate the invariant
parameters and lz for a LG0

21 beam. For such a beam the
result is

Meff
4 5 3, (37)

t 5 5, (38)

a 5 2, (39)

aM 5 2, (40)

lz 5 2\. (41)

These values are close to the experimental ones [Eqs.
(32)–(36)]. We could say that the clockwise-rotating
pulses contain nearly pure LG0

21 modes.

4. SUMMARY AND CONCLUSIONS
From the theory and the experiments presented in this
paper, one can see that a cylindrical lens is needed to
complete a full second-order-moments characterization of
a laser beam, including the measurement of the angular
momentum transported by the beam. A characterization
without such a lens would lead us to incomplete or even
erroneous conclusions, as was explained in Ref. 8. But
even if we were not interested in a full second-order char-
acterization, a cylindrical lens could show us features of
the beam that remain hidden if only rotationally symmet-
ric optical systems are considered. We have seen that
our laser had this type of behavior. Its beam profile
seemed to be very reproducible in free space without the
cylindrical lens; only after the cylindrical lens, as is
shown in Fig. 4, was the fact revealed that the laser emit-
ted different kinds of beam profiles. The cylindrical lens
also allowed us to complete a second-order characteriza-
tion of our beam in the case of pulses that display a
clockwise-rotating transverse irradiance pattern. We
have seen that those pulses consist of nearly pure LG0

21

modes and transport an orbital angular momentum of lz
. 2\ per photon.

ACKNOWLEDGMENTS
This research was supported by the Comisión Interminis-
terial de Ciencia y Tecnologı́a of Spain under project
PB97-0295.

J. Serna, the corresponding author, can be reached at
the address on the title page or by e-mail,
fiopt01@sis.ucm.es. G. Nemeş’s e-mail address is
gnemes98@hotmail.com.



Serna et al. Vol. 18, No. 7 /July 2001 /J. Opt. Soc. Am. A 1733
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