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Limits of the paraxial approximation in laser beams

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Limits of the paraxial approximationin laser beams
Pablo Vaveliuk
Departamento de Física, Universidade Estadual de Feira de Santana, BR 116, KM 03, Feira de Santana 44031-460,Bahia, Brazil and CEPPEV-Fundaçâo Visconde de Cairu, 40226-460, Salvador, Bahia, Brazil
Beatriz Ruiz and Alberto Lencina
Centro de Investigaciones Ópticas, cc 124 (1900), La Plata, Argentina
Received October 19, 2006; revised January 2, 2007; accepted January 8, 2007;posted January 16, 2007 (Doc. ID 77710); published March 19, 2007
The validity of the paraxial approximation for laser beams in free space is studied via an integral criterionbased on the propagation invariants of Helmholtz and paraxial wave equations. This approach allows one todetermine the paraxial limit for beams with nondeﬁned spot size and for beams described by more param-eters in addition to typical longitudinal wavelength and transverse waist. As examples, the paraxiality of higher-order Hermite, Laguerre, and Bessel–Gaussian beams was completely determined. This methodcould be extended to nonlinear optics and Bose condensates.
© 2007 Optical Society of America
OCIS codes:
140.3300, 350.5500, 260.2110
.
The paraxial approximation (PA) is a powerful tool tosimplify the time-independent wave equation, calledthe Helmholtz equation (HE). Looking at the propa-gation direction, the PA assumes that the ﬁeld is theproduct of a fast oscillatory part and a slowly varying complex amplitude whose second derivate is ne-glected within the HE. The simpler resultant equa-tion, called the paraxial equation (PE), can be ana-lytically solved for a great deal of linear as well asnonlinear phenomena such as free-space laser beampropagation,
1–3
second-harmonic generation,
1
andwave mixing.
1
The PA was also used in Bosecondensates.
4
The crucial point is to check the PA va-lidity since, otherwise, the PE could give nonphysicalsolutions leading to a misleading interpretation of the physical nature of the problem. The typical localmethod, which compares the HE derivates, is com-monly employed for beams with well-deﬁned spotsize and is characterized by two parameters, longitu-dinal wavelength and transverse beam waist.
2,5
Un-fortunately, this method could be imprecise in deter-mining the validity limits of beams with nondeﬁnedspot size as the higher-order Hermite–Gauss (HG)
2,3
and Laguerre-Gauss (LG)
3
beams. Besides, the localmethod could be deﬁcient for beams characterized byadditional parameters. For example, the Bessel–Gauss (BG) includes a transverse wave vector
6,7
andthe Ince–Gauss, an ellipticity parameter.
8
This Letter presents an integral criterion for deter-mining the paraxial approximation validity limits of free-space propagating beams by comparing thepropagation invariants of Helmholtz and paraxialequations. Then, a
paraxiality estimator
is intro-duced. This permits mapping of the paraxial–nonparaxial regions of any paraxial beam. As an ex-ample, the method was applied to higher-order HG,LG, and BG beams so that its paraxial and non-paraxial limits were completely identiﬁed.We start from the vector HE for the time-independent complex amplitude
E
in free space,
2
E
+
k
2
E
=0,
1
where
k
is the wave vector modulus. If the wavepropagation is assumed to be in the
z
direction, wedeﬁne
E
r
=
A
r
e
ikz
. Assuming that
A
varies slowlyin the
z
direction when compared with the fast oscil-lating part
e
ikz
, i.e.,
z
2
A
2
ik
z
A
, then Eq. (1) sim-pliﬁes into the PE
2
:
t
2
A
+2
ik
z
A
=0,
2
with the Laplacian operator broken up into trans- verse and longitudinal parts.When the PA is fulﬁlled, Eqs. (1) and (2) are physi-cally equivalent. However, at a mathematical level,they present different propagation invariants, whichwas noted earlier in the context of scalar nonlinearwave propagation.
9,10
We begin with the invariant of Eq. (1). Let
E
h
be a wave solution of this equation. After some algebraic manipulations and integrating over a transverse plane
z
=
cte
, assuming ﬁnite inten-sity distribution
lim
x
,
y
→
±
E
h
=0
, the result is
z
−
+
Im
E
h
*
·
z
E
h
d
x
d
y
=0,
3
so that
P
HE
=
−
+
Im
E
h
*
·
z
E
h
d
x
d
y
is an invariant of Eq. (1). Note that the parameter
0
P
HE
/
2
k
2
repre-sents the power crossing an inﬁnite plane
z
=
cte
.
11,12
For a lossless medium, this power must be constant.Thus, it is licit to refer to
P
HE
as the HE-
power in-variant
.On the other hand, to derive the invariant of thePE, we start from
A
p
*
·
z
A
p
+
z
A
p
*
·
A
p
, assuming that
A
p
is a solution of Eq. (2). Thus, replacing
z
A
p
z
A
p
*
with the ﬁrst term of Eq. (2) [
c
.
c
. of Eq. (2)], integrat-ing with the conditions lim
x
,
y
→
±
E
p
=0, and noting that
A
p
·
A
p
*
=
E
p
·
E
p
*
results in
April 15, 2007 / Vol. 32, No. 8 / OPTICS LETTERS
927
0146-9592/07/080927-3/$15.00 © 2007 Optical Society of America
z
−
+
E
p
·
E
p
*
d
x
d
y
=0,
4
so that
P
PE
=
−
+
E
p
·
E
p
*
k
d
x
d
y
is an invariant of Eq.(2). The factor
k
was included so that
P
PE
agrees di-mensionally with
P
HE
. As
0
P
PE
/
2
k
2
is the powerwithin the paraxial approximation,
2
P
PE
can be calledthe PE-
power invariant
.Our method analyzes the ratio between invariants,separating
A
into real amplitude and phase:
A
r
=
r
e
i
r
, so that the ratio
P
HE
/
P
PE
reads as
−
+
Im
E
h
*
·
z
E
h
d
x
d
y
−
+
E
p
·
E
p
*
k
d
x
d
y
=
−
+
h
2
k
+
z
h
d
x
d
y
−
+
p
2
k
d
x
d
y
.
5
PA fulﬁls
⇔
k
z
h
⇔
h
p
(HE and PE solu-tions are physically equivalent). Under such condi-tions, the right-hand side of Eq. (5) shows that nec-essarily
P
HE
/
P
PE
1. Otherwise, if
z
h
is not muchless than
k
, then
h
”
p
so that
P
HE
/
P
PE
”
1. Thismeans that the fast oscillations are not completelyincluded in
e
ikz
but in
e
i
r
so that PA is not fulﬁlled.The above analysis suggests that Eq. (5) is the baseby which to deﬁne an efﬁcient
paraxiality estimator
P
˜
for any paraxial ﬁeld
E
p
r
,
z
as
P
˜
=
−
+
Im
E
p
*
·
z
E
p
d
x
d
y
−
+
E
p
·
E
p
*
k
d
x
d
y
.
6
P
˜
1 indicates that the paraxial approximationholds. The structure of
z
consists of a
z
-dependentpart (associated with the Guoy phase shift) and an-other part that is, in addition, dependent on
r
rep-resenting the radial phase factor, indicating how anequiphase surface curves from the planar phase front
z
=
cte
. The ﬁrst
r
-independent part is not affectedby the transverse integration, which does not allow aphase compensation, i.e., the conditions
z
k
and
−
+
2
k
+
z
d
x
d
y
−
+
2
k
d
x
d
y
cannot be satisﬁedsimultaneously so that
−
+
2
z
d
x
d
y
−
+
2
k
d
x
d
y
implies that
z
k
. The deviation from the unit in-dicates that
z
begins to be comparable with
k
.This integral method possesses some advantages: thePA limits can be determined for beams in which it ishard to deﬁne a
regular spot size
and also for beamsdescribed by additional parameters to the longitudi-nal wave vector and transverse beam waist.In the following, the power of Eq. (6) in determin-ing the validity range of HG, LG, and BG modes as-suming linearly polarized transverse beams will beseen. A certain care must be taken because a TEMmode with a nonuniform proﬁle propagating in freespace violates Maxwell’s equations. Then, a longitu-dinal component is mandatory.
5
This could be signiﬁ-cant for spot diameters of the order of or less than1/
k
. In this case, the paraxiality estimator should beanalyzed by considering the longitudinal corrections.Within the scalar context, HG, LG, and BG modesare solutions of Eq. (2). HG and LG are expressed inrectangular and cylindrical coordinates, respectively,and are characterized by two basic parameters:wavenumber
k
(longitudinal) and waist size
w
0
(transverse). Other well-known parameters such asthe Rayleigh length
z
0
=
kw
02
/2, the spot size
w
z
, theradial phase curvature
R
z
, and the Gouy shift areconstructed from those basics.
2
By using dimension-less variables, the HG and LG eigenmodes can becompactly written as
A
m
,
n
HG
E
0
=11+
iz
˜
H
m
2
x
˜
1+
z
˜
2
H
n
2
y
˜
1+
z
˜
2
1−
iz
˜
1+
z
˜
2
m
+
n
exp
−
r
˜
2
1+
iz
˜
,
7a
A
,
p
LG
E
0
=
2
r
˜
1−
iz
˜
p
1+
iz
˜
p
+
+1
L
p
2
r
˜
2
1+
z
˜
2
e
i
−
r
˜
2
/1+
iz
˜
,
7b
where
E
0
is maximum ﬁeld value,
r
˜
=
r
/
w
0
=
x
˜
2
+
y
˜
2
isthe transverse dimensionless radial coordinate,
z
˜
=
z
/
z
0
is the dimensionless longitudinal coordinate,
is the azimuthal coordinate,
k
˜
=
kw
0
is the dimension-less characteristic beam parameter, and
H
j
,
L
p
arethe Hermite and the generalized Laguerre polynomi-als, respectively. The order mode for both families is
N
, where
N
=
m
+
n
for HG and
N
=2
p
+
for LG. Themode
N
=0 is the Gaussian beam. Note that the HGand LG eigenmodes are characterized by a single di-mensionless length parameter
k
˜
, which greatly sim-pliﬁes the calculus. The paraxiality estimator,
P
˜
, wascalculated for all HG and LG
N
orders with help of Ref. 13, giving
P
˜
N
HG
k
˜
=
P
˜
N
LG
k
˜
=1−
N
+1
/
k
˜
2
.
8
Figure 1 depicts the curves
P
˜
versus
k
˜
for several HG
Fig. 1. (Color online) Curves
P
˜
versus
k
˜
for the ﬁrst ﬁveHG and LG modes. The shaded region indicates where theparaxial approximation is fulﬁlled. The inset depicts theproﬁles of
E
·
E
*
(thick curve) and Im
2
E
*
·
z
˜
E
/
k
˜
2
(thincurve) for
N
=0 at
z
˜
=0, 0.25, 0.75 and 1.
r
˜
−2,2
.
928
OPTICS LETTERS / Vol. 32, No. 8 / April 15, 2007
and LG modes. The shaded region where the paraxialapproximation is valid,
P
1, is clearly visualized byany order mode. The tonality change (to a lightershade) for lower
P
˜
values shows that the PA remainsunfulﬁlled. Note that when the order mode increases,the PA validity range shifts toward greater
k
˜
values,despite the fact that all modes continue to be charac-terized by only
k
and
w
0
. For example, a paraxialbeam with
P
˜
0.88 is assumed. Then, a typical vis-ible wavelength,
=0.5
m implies a waist
w
0
0.23
m for the lowest-order
N
=0. In turn, for
N
=4, it must be
w
0
0.51
m to reach the same
P
˜
value. The HG and LG estimators are invariant un-der propagation, their values being the same in thewhole space (the area under the curves
E
·
E
*
andIm
2
E
*
·
z
˜
E
/
k
˜
2
). The inset of Fig. 1 shows that thesecurves tends to overlap at central region (around
r
=0). This possibly would suggest that a nonparaxialfar ﬁeld of the beam waist behaves like a paraxialfrom a given distance
z
=
z
p
(the beginning of the over-lap) such that
z
p
increases when
P
˜
→
0.Now we analyze the PA validity limits of a BGbeam formed by a Bessel function modulated by aGaussian proﬁle. Their propagation properties werewidely studied
6,7
because the function represents a variant of nondiffractive Bessel beams
14
but does notrequire inﬁnite energy for its formation. The
m
-orderBessel function of the ﬁrst kind,
J
m
, is related withthe BG
m
mode. This function can be compactly writ-ten in dimensionless cylindrical coordinates as
A
m
BG
E
0
=
e
im
1+
iz
˜
exp
−
i
˜
2
z
˜
+4
r
˜
2
4
1+
iz
˜
J
m
˜
r
˜
1+
iz
˜
,
9
where
˜
=
w
0
is the dimensionless transverse wave-number. Note that the BG modes depend on two di-mensionless parameters,
k
˜
,
˜
, making the determi-nation of the PA validity range harder. Closed-formexpressions were obtained for
P
˜
m
BG
with the aid of Ref. 13. Here, the ﬁrst two lowest modes are given:
P
˜
0
BG
k
˜
,
˜
=1−1
k
˜
2
−
˜
2
+84
k
˜
2
I
1
˜
2
/4
I
0
˜
2
/4
,
10a
P
˜
1
BG
k
˜
,
˜
=1−
˜
2
k
˜
2
1+
I
0
˜
2
/4
I
1
˜
2
/4
,
10b
where
I
m
is the
m
-order modiﬁed Bessel function of the ﬁrst kind. Figure 2 shows a density plot of
P
˜
ver-sus
k
˜
,
˜
for the ﬁrst two BG orders: (a)
m
=0 and (b)
m
=1. We distinguish two different behavior regionsin terms of
˜
. For lesser values, the paraxial approxi-mation does not depend on this parameter, being de-termined only by
k
˜
as in Gaussian beams. However,for
˜
1, the paraxial region limit goes linearly in theplane
k
˜
,
˜
with slope
45° and below the straight
˜
=
k
˜
. Therefore, the PA validity must be carefullyevaluated in beams with
k
. Also note that theparaxial region area is smaller as the mode order in-creases. For example, for
k
˜
=100, PA is no longer valid from
˜
57 for
m
=0
P
˜
0
0.92
. In turn, for
k
˜
=100,
P
˜
1
0.92 is produced by
˜
40.In summary, we derive an integral method to de-termine the PA validity limits of any paraxial beambased on the comparison of the HE and PE power in- variants. This allows us to deﬁne a paraxiality esti-mator that becomes close to unity when the PA is valid. The method was applied in determining theparaxial limits of higher-order HG, LG, and BGbeams. Closed-form expressions were derived for
P
˜
sothat the paraxial and nonparaxial regions of thesebeams were mapped. The method could be extensibleto nonsolitonic nonlinear beams and correlated areasas Bose condensates.P. Vaveliuk’s e-mail address is pablov@ ﬁsica.ufpb.br.
References
1. A. Yariv,
Quantum Elecronics
(Wiley, 1989).2. J. T. Verdeyen,
Laser Electronics
(Prentice-Hall, 1981).3. A. E. Siegman,
Lasers
(University Science, 1986).4. E. V. Goldstein, K. Plättner, and P. Meystre, J. Res.Natl. Inst. Stand. Technol.
101
, 583 (1996).5. M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A
11
, 1365 (1975).6. F. Gori, G. Guattari, and C. Padovani, Opt. Commun.
64
, 491 (1987).7. R. Borghi, M. Santarsiero, and M. A. Porras, J. Opt.Soc. Am. A
18
, 1618 (2001).8. M. A. Bandres and J. C. Gutiérrez-Vega, Opt. Lett.
29
,144 (2004).9. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo,Opt. Lett.
18
, 411 (1993).10. A. P. Sheppard and M. Haelterman, Opt. Lett.
23
, 1820(1998).11. A. Lencina and P. Vaveliuk, Phys. Rev. E
71
, 056614(2005).12. A. Lencina, P. Vaveliuk, B. Ruiz, M. Tebaldi, and N.Bolognini, Phys. Rev. E
74
, 056614 (2006).13. I. S. Gradshteyn and I. M. Ryzhik,
Table of Integrals, Series and Products
(Academic, 1980).14. J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev.Lett.
58
, 1499 (1987).Fig. 2. (Color online) Density plot of
P
˜
m
BG
versus
k
˜
,
˜
for(a)
m
=0 and (b)
m
=1 with
k
˜
,
˜
10
−1
,10
3
. The paraxialregion is indicated by PR. April 15, 2007 / Vol. 32, No. 8 / OPTICS LETTERS
929

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