Bessel-modulated Gaussian beams with quadratic radial dependence

1 June 1999

Optics Communications 164 Ž1999. 83–93 www.elsevier.comrlocateroptcom

Full length article

Bessel-modulated Gaussian beams with quadratic radial dependence C.F.R. Caron 1, R.M. Potvliege

)

Physics Department, UniÕersity of Durham, Durham DH1 3LE, UK Received 4 February 1999; accepted 8 April 1999

Abstract We describe a class of closed-form solutions of the paraxial wave equation whose transverse profile is a Gaussian function multiplied by a Bessel function, the argument of which is quadratic in the distance from the axis. Their characteristics are compared to those of Bessel–Gauss beams. Unlike the latter, these Bessel-modulated Gaussian beams propagate colinearly. They have interesting non-Gaussian features for certain values of their parameters, such as concentric rings of phase dislocation outside the waist plane or a flat axial profile. q 1999 Elsevier Science B.V. All rights reserved. PACS: 42.25.Bs; 42.60.Jf Keywords: Paraxial wave equation; Nonparaxial beams; Bessel–Gauss beams; Phase dislocation

1. Introduction Various simple, closed-form paraxial solutions of the Helmholtz equation have been described over the years. A particularly interesting class of such solutions, known as Bessel–Gauss beams, has been introduced by Gori et al. in 1987 w1x. Their field amplitude, written in cylindrical coordinates Ž r , z, f ., takes on the form Ef JmŽ br .expŽyr 2rw 02 .expŽ im f . in the z s 0 plane, with r denoting the distance from the z-axis and Jm the Bessel function of the first kind of order m. Ef , b and w 0 are three arbitrary parameters. A Bessel–Gauss beam thus reduces to an ordinary Gaussian beam for b ™ 0 Žor to )

Corresponding author. E-mail: [email protected] Present address: Springer-Verlag, Tiergartenstrasse 17, D69121 Heidelberg, Germany. 1

an higher order Gauss–Laguerre mode for m / 0. and to a paraxial approximation of a ‘‘nondiffracting’’ Bessel beam w3–5x for w 0 ™ `. Bessel–Gauss beams can be generated by resonators constructed with aspheric phase-conjugating mirrors w2x. They also provide a convenient analytical model of the Bessel-like beams produced using other techniques. The purpose of this communication is to describe another class of simple closed-form solutions of the paraxial wave equation, which are similar to Bessel–Gauss beams in that their amplitude is also a product of a Bessel function and a Gaussian. In the solutions we are concerned with, however, the argument of the Bessel function is not linear in the transverse coordinate but quadratic. For example, the amplitude of the zeroth-order mode in the z s 0 plane is Ef J0 Ž b 2r 2 .expŽyr 2rw 02 ., written in terms of the three parameters Ef , b and w 0 . Because of

0030-4018r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 1 7 4 - 1

C.F.R. Caron, R.M. PotÕlieger Optics Communications 164 (1999) 83–93

84

this quadratic dependence, these Bessel-modulated Gaussian beams have very different propagation characteristics for non-zero b than Gori et al.’s Bessel–Gauss beams. Whereas the latter have an underlying conical geometry, being basically a continuous superposition of interfering Gaussian beams directed at an angle with respect to the z-axis w1,6x, the beams considered here have the familiar colinear geometry of pure Gaussian beams. Yet, they also have interesting non-Gaussian features for certain values of their parameters, such as concentric rings of phase dislocation, parallel to but at some distance of the z s 0 plane, or a flat axial profile. For the sake of simplicity, the Bessel-modulated Gaussian beams with quadratic radial dependence will be referred to as ‘QBG beams’ in the following. A brief review of Bessel–Gauss beams is given in Section 2, for clarity. The QBG beams are introduced and discussed in Section 3. The feasibility of realizing them with a simple amplitude transparency is considered in Section 4. Finally, QBG pulses, analogous to the Bessel–Gauss pulses studied by Overfelt w7x, are defined in Section 5.

This field is a continuous superposition of plane waves whose wave vectors are distributed on the surface of a cone of half-angle a . As a consequence of the oblique direction of these plane waves, its phase velocity, crcos a , is superluminal w10,11x. An approximate paraxial solution of the Helmholtz equation is obtained by replacing '1 y u 2 by 1 y u 2r2 in Eq. Ž1., which is meaningful if the integrand is negligible for u R 1 w9x. This paraxial solution is E Ž p. Ž r , z, f . s c Ž r , z, f .expŽ ikz ., where c Ž r , z, f . is an exact solution of the paraxial wave equation,

ž

1 E

1 E2

E r

r Er Er

q

2

r Ef

E 2

q 2 ik

/

Ez

c Ž r , z , f . s 0,

Ž 3. and can be represented by the following integral,

c Ž r , z ,f . s ei mf

`

H0

d u u f Ž u . J < m < Ž k r u . eyi k z u

2

r2

.

Ž 4. In the paraxial approximation, the Bessel beam Ž2. can thus be written as E BŽ p., m Ž r , z , f . s Ef J < m < Ž k r sin a .

2. Bessel–Gauss beams

=exp ikz Ž 1 y 12 sin2a . exp Ž im f . .

As is well known, exact solutions of the Helmholtz equation may be constructed by superposing plane waves. This approach leads, in particular, to solutions of the form w8,9x E Ž r , z ,f . s ei mf

`

H0

d u u f Ž u . J < m < Ž k r u . e i k z'1yu . 2

Ž 1. With the branch of the square root function chosen so that '1 y u 2 s i'u 2 y 1 for u ) 1, the solution defined by Eq. Ž1. is valid in the half space z ) 0. The spectral function f Ž u. is arbitrary, provided the integral and its second order derivatives with respect to z and r exist. For instance, taking f Ž u. s Ef d Ž u y sin a .ru yields the Žnon square-integrable. Bessel beam w3–5x

Ž p. Ž . Clearly, E B, m r , z, f is a satisfactory approximation Ž of the exact EB, m r , z, f . only to the extent that a is small enough for 1 y Žsin2a .r2 to be sufficiently close to cos a . Bessel–Gauss beams w1,6x are paraxial solutions of the Helmholtz equation of the form Ž p. E BG , m Ž r , z ,f . s

Ef 1 q izrz R

ž

=exp y

J< m<

ž

krrw 0 1 q izrz R

r 2rw 02 1 q izrz R

ž

=exp ikz 1 y

1 2

Ž 2.

/

/ sin2a

1 q izrz R

=exp Ž im f . ,

E B , m Ž r , z , f . s Ef J < m < Ž k r sin a . exp Ž ikzcos a . =exp Ž im f . .

Ž 5.

z R s kw 02r2.

/ Ž 6.

with k s kw 0 sin a and The parameters Ef , z R and w 0 are, respectively, the amplitude at the

C.F.R. Caron, R.M. PotÕlieger Optics Communications 164 (1999) 83–93

origin, the Rayleigh range and the spot size of the Ž p. fundamental Gaussian mode to which EBG ,0 reduces for a ™ 0. We will suppose that Ef G 0. The field amplitude Ž6. can be obtained from the Huygens– Fresnel diffraction integral w1x or from the integral representation Ž4., the corresponding spectral function being f Ž u. s

Ef k

2

w 02

2

=I < m <

ž

ž

exp y

k 2 w 02

k

2

w 02

4

sin2a

/ ž

usin a exp y

2

/ k 2 w 02 4

u

2

/

. Ž 7.

Ž p. E BG , m Ž r , z s 0, f . s Ef J < m < Ž krrw 0 .

=exp Ž yr 2rw 02 . exp Ž im f . , Ž 8. and on axis, Ef

ž

=exp ikz 1 y

1 2

1 q izrz R

/

Ž 9.

It is clear that Bessel–Gauss beams reduce to Bessel beams near the origin, i.e., for r < w 0 and < z < < z R , but have very different large-r and large-z beŽ p. Ž . is haviours. The far-field form of EBG , m r , z, f simply obtained by writing r s r sin u , z s rcos u , with 0 F u - pr2, and letting r becoming very large. Neglecting terms of higher order in 1rr, one then has

rcos u

I< m <

=exp y

ž

k 2 w 02 2

k 2 w 02 4

sin a tan u

Ef z R Ž p k 2 w 02 sin a tan u .

y1 r2

rcos u 2

=exp y

k w 02

2 Ž sin a y tan u . .

4

Ž 11 .

Ž p. < < E BG , m Ž r s 0, z , f .

f

/

Ž sin2a q tan2u . .

Ž 10 .

Recall that for large arguments Ž x 4 0., I < m < Ž x . f Ž2p x .y1 r2 expŽ x . w12x. Replacing the modified

Ef z R z

ž

exp y

k 2 w 02 4

/

sin2a dm0 .

Ž 12 .

Eqs. Ž11. and Ž12. show that paraxial Bessel–Gauss beams propagate in the far-field region with the shape of a cone of half-aperture arctanŽsin a . when kw 0 is very large w1x. It is interesting to compare this result to the asymptotic form of the exact Žnonparaxial. solution of the Helmholtz equation correŽ p. Ž . sponding to EBG , m r , z, f . This asymptotic form is easily derived from the angular spectrum representation of the beam; one has indeed, under fairly general conditions and to leading order in 1rr, < E
sin a

=exp Ž im f . dm0 .

Ef z R

Ž p. < < E BG ,m f

1 q izrz R 2

Ž p. < < EBG ,m f

Bessel function by its asymptotic form, in Eq. Ž10., gives

Eq. Ž11. is not valid for u s 0; instead, from Eq. Ž9.,

In this last equation, I < m < Ž x . s Žyi . < m < J < m < Ž ix . is the modified Bessel function of first kind of order < m <. Ž p. Ž . EBG , m r , z, f takes on a simple form in the waist plane,

Ž p. E BG , m Ž r s 0, z , f . s

85

cos u kr

< f Ž sin u . < ,

Ž 13 .

if E is the field defined by Eq. Ž1. and 0 F u - pr2 w9,13x. In view of Eq. Ž7., Eq. Ž13. reads, in the present case, < E BG , m < f

Ef z R cos u r =exp y

I< m <

ž

k 2 w 02 4

k 2 w 02 2

sin a sin u

/

Ž sin2a q sin2u . .

Ž 14 .

The upshot is that when kw 0 is very large, nonparaxial Bessel–Gauss beams propagate in the far-field region with the shape of a cone of half-aperture a , not arctanŽsin a . as found in the paraxial approximation. The difference between these two angles is only of third order in a , and may be smaller, when a < 1, than the contribution of terms that are not negligible when kw 0 is not asymptotically large. As Ž p. < 2 an illustration, < EBG and < EBG,0 < 2 , calculated ac,0 cording to Eqs. Ž10. and Ž14., are compared in

C.F.R. Caron, R.M. PotÕlieger Optics Communications 164 (1999) 83–93

86

Fig. 1. Far-field profile, vs the angle u s arctanŽ rrz ., Ža. for a zeroth-order Bessel–Gauss beam with k s 3 or 12; Žb. for a zeroth-order Ž p. < 2 < <2 Ž QBG beam with m s 0, 3 or 12. Solid curves: paraxial results, < E BG ,0 ; dotted curves: nonparaxial results, E BG ,0 . These two sets of results can be distinguished on the scale of the figure only for k s m s 12.. All curves are normalized to unity at their maximum, and are drawn for w 0 s 12prk.

Fig. 1Ža. for kw 0 s 12p and k s 3 and 12. These two cases correspond, respectively, to a s 4.6 deg and a s 18.6 deg. An excellent agreement between the paraxial and the nonparaxial results is observed for k s 3, but not for k s 12. ŽNote how little the width of the angular distribution about a changes when k increases from 3 to 12, although this increase in k reduces by 75% the diameter of the central spot at z s 0.. 3. QBG beams A simple, paraxial solution of the wave equation can also be written down in closed form for Gaussian fields modulated by a Bessel function quadratic in r , namely

w0 WŽ z.

=exp y

Ef k 2 w 02

(4m q 4 2

ž

=exp y

2

`

H0

1 q i Ž m 2 q 1 . zrz R W 2Ž z.

r2

Ž 15 .

where Ef , z R and w 0 have the same meanings than in Section 2, and 2

W Ž z . s w 0 1 y Ž m2 q 1 . Ž zrz R . q 2 izrz R .

Ž 16 .

ž

k 2 w 02 4m 2 q 4

m k 2 w 02 4m 2 q 4 u2

/

d x x Jn Ž Ax . Jn r2 Ž Bx 2 . eyC x 1r2

'B 2 q C 2 =exp y

W Ž z.

J < m < r2

u2

/ Ž 17 .

in the angular spectrum representation Ž4., or, with more effort, by working out the corresponding Huygens–Fresnel diffraction integral. In either case, the calculation is made possible by the relation w12x

mr 2 J < m < r2

=exp Ž ikz . exp Ž im f . ,

(

f Ž u. s

s

Ž p. EBM G , m Ž r , z ,f .

s Ef

Eq. Ž15. can be obtained directly by letting

Jn r2

2

A2 B 4Ž B 2 q C 2 .

A2 C 4Ž B 2 q C 2 .

,

Ž 18 .

which applies when A ) 0, Re C )
C.F.R. Caron, R.M. PotÕlieger Optics Communications 164 (1999) 83–93 Ž p. Ž r , z, f . to a pure to w 0 Ž1 q izrz R . and E BMG,0 Gaussian beam,

EGŽ p. Ž r , z . s

Ef 1 q izrz R

ž

exp y

r 2rw 02 1 q izrz R

=exp Ž ikz . .

/ Ž 19 .

Fig. 2 shows the transverse profile of the axisymmetric QBG beam in the z s 0 plane, i.e., Ž p. 2 2 EBM G ,0 Ž r , z s 0, f . s Ef J 0 Ž mr rw 0 .

=exp Ž yr 2rw 02 . ,

Ž 20 .

compared to that of the axisymmetric Bessel–Gauss beam, Eq. Ž8. with m s 0. When m is non-zero but imaginary, the Bessel function J0 Ž mr 2rw 02 . increases like expŽ< m < r 2rw 02 .rr for r ™ ` and tends to compensate the decreasing Gaussian factor: the result is a nearly-Gaussian profile broader than for m s 0. The Bessel function oscillates when m is real, but this manifests essentially by a narrowing of the Ž p. Ž . when m Q 2. For waist of EBM G,0 r , z s 0, f smaller values of this parameter, the z s 0 profile of the QBG beam is qualitatively similar to that of a Bessel–Gauss beam with k s m. The latter has a still narrower waist, though, and consequently carries less

87

power for a same intensity at r s 0 Žsee Appendix A.. There is, however, an important difference between the two: In Eq. Ž20., the Bessel function oscillates faster and faster for increasing values of r , while in Eq. Ž8. it oscillates quasi-periodically. At a distance rl f kw 02r4m from the origin, the separaŽ p. Ž tion between consecutive zeros of EBM G,0 r , z s 0, f . becomes less than a wavelength. This happens within the range of the Gaussian, i.e., rl is less than w 0 , when m R kw 0r4. On axis, the QBG amplitude simply is Ž p. E BM G , m Ž r s 0, z , f .

Ef exp Ž ikz . exp Ž im f .

s

(1 y Ž m q 1. Ž zrz 2

R

2 . q 2 izrz R

dm0 . Ž 21 .

Ž p. Ž .< 2 for r s 0 The variation with z of < E BM G,0 r , z, f is illustrated by Fig. 3Ža. for m s 0 Žthe Gaussian limit., 1 and 2. Its variation in the x y z plane is displayed in Fig. 4, for m s 0 and 2. The most striking feature of these results is the development Ž p. Ž .< 2 , symon axis of two maxima of < EBM G,0 r , z, f metric with respect to the z s 0 plane, when m increases beyond 1. These maxima occur at z s "z max , with

(m y 1 2

z max s

m2 q 1

zR ,

Ž 22 .

and their height is Ž p. < E BM <2 G ,0 Ž r s 0, z s "z max , f . s

m2 q 1 2m

Ef2 .

Ž 23 . They move towards the z s 0 plane and become sharper and higher when m increases beyond 2. However, increasing m also reduces the depth of Ž p. Ž r s 0, the field: the distance z 1r2 at which < EBMG,0 2 Ž p. z s z 1r2 , f .< s Ž1r2.< EBMG,0 Ž r s 0, z s 0, f .< 2 is

(m y 1 q 2(m q m q 1 s 2

Fig. 2. Transverse profile of QBG beams in the z s 0 plane. The Ž p. Ž . Ž . results plotted are EBM G ,0 r , z s 0, f for m s 0 solid curve , m s 2 Ždashed curve., and m s i r'2 Ždot-dashed curve.. The dotted curve represents the transverse profile of the Bessel–Gauss Ž p. Ž . beam, E BG ,0 r , z s 0, f , for k s 2. All results are normalized to unity at r s 0.

z 1r2

4

m2 q 1

2

zR ,

Ž 24 .

and thus z 1r2 f '3 z R rm for m 4 1. The only maximum for m F 1 or m imaginary occurs at the origin. The case m s 1 is remarkable in that the correspond-

88

C.F.R. Caron, R.M. PotÕlieger Optics Communications 164 (1999) 83–93

Ž p. Ž . Ž . Ž . Ž . Ž . Fig. 3. The variation of E BM G ,0 r , z, f on the z-axis, for m s 0 solid curve , m s 1 dot-dashed curve and m s 2 dashed curve . a Ž p. Ž p. Ž .x w Ž .x Normalized intensity, < EBM G ,0 Ž r s 0, z, f .rEf < 2 ; Žb. relative phase, d Ž z . s argw E BMG ,0 r s 0, z, f y arg exp ikz . The dotted curves correspond to the polynomial model of Section 4, Eq. Ž31., for m s 1.

ing axial intensity profile is very flat near z s 0. Indeed, for m s 1, Ž p. < E BM <2 G ,0 Ž r s 0, z , f . s

Ef2

(1 q 4Ž zrz

R

.

4

,

Ž 25 .

which differs from Ef2 only by terms of order Ž zrz R . 4 . Increasing m from 0 to a few units thus changes Ž p. Ž .< 2 f the shape of the region where < EBM G,0 r , z, f 2 Ef : this region becomes more elongated along the

z-axis Žunless m gets to large. and more squeezed transversally. This results, up to m f 2, in an increase of the volume of space where the intensity reaches values close to that at Ž r s 0, z s 0. ŽFig. 5.. Ž p. Ž . The phase variation of E BM G,0 r , z, f along the Ž . z-axis is shown in Fig. 3 b . The quantity plotted is Ž p. Ž .x the phase difference d Ž z . s argw E BM G,0 r s 0, z, f w Ž .x y arg exp ikz . As in the case of pure Gaussian beams, and whatever the value of m , d Ž z . changes by p across the origin. However, its slope depends on m outside the immediate vicinity of z s 0: for

Fig. 4. The amplitude distribution in the x y z plane of a zeroth-order QBG beam with m s 2 Žpositive values of x . compared to that of a Ž p. Ž .< 2 , normalized to unity at pure Gaussian beam of same w 0 and z R Žnegative values of x .. The figure represents < E BMG ,0 r s 0, z, f Ž x s 0, z s 0..

C.F.R. Caron, R.M. PotÕlieger Optics Communications 164 (1999) 83–93

Ž p. Ž .< 2 is at least 90% Žsolid Fig. 5. Volume wherein < E BM G ,0 r , z, f . Ž . Ž curve , 80% dashed curve , or 70% dotted curve. of its value at Ž r s 0, z s 0., relative to the corresponding volume for m s 0 Ži.e., for a pure Gaussian beam of same w 0 and z R ..

m 4 0, d Ž z . varies more rapidly with z than in the Gaussian case Ž m s 0., when < z < - z R , and more slowly when < z < R z R . Wave front patterns of QBG beams and Bessel– Gauss beams are compared in Fig. 6. The curves Ž p. Ž .x represent the surfaces on which argw EBM G,0 r , z, f , Ž p. Ž w .x or arg EBG ,0 r , z, f in the lower right quadrant of the figure, is an integral multiple of p . The variations of the phase depicted in Fig. 3Žb. and the higher phase velocity of Bessel–Gauss beams are hardly noticeable on the scale of Fig. 6, but some of the wave front singularities associated with the zeros of the Bessel function are clearly visible. These zeros

89

are located in the z s 0 plane when m Žor k for Bessel–Gauss beams. is purely real. The phase behaves in their vicinity in the same way than in the vicinity of the dark Airy rings observed in the focal plane of a converging spherical wave diffracted by a circular aperture w14x. The wave fronts of QBG beams are more severely perturbed by these singularities, since they swerve towards the zeros of the Bessel functions and those succeed each other more rapidly than in the case of Bessel–Gauss beams. The accuracy of the paraxial approximation is questionable in the regions where the ensuing deviation of the gradient of the phase from the z-direction is significant. There are no wave front singularities in the z s 0 plane when m is imaginary. Instead, concentric rings of wave front dislocation form in the planes z s "z R rŽ1 y < m < 2 .1r2 ; Rew W 2 Ž z .x vanishes on these planes, and the dislocations originate at the zeros of the oscillating Bessel function J < m < r2 w mr 2rW 2 Ž z .x. The one nearest to the axis is clearly visible in the lower left quadrant of Fig. 6. In essence, these singularities are the same as those occurring in the z s 0 plane when m is real. However, they have the distinction of appearing in untruncated beams whose amplitude is quasi-Gaussian and everywhere positive in the waist plane. We have not examined in detail whether such singularities exist at finite distance for the nonparaxial QBG beams corresponding to the

Fig. 6. Intersections of the x y z plane with the surfaces on which the phase of the amplitude is an integral multiple of p , for w 0 s 4prk. Upper left quadrant: pure Gaussian beam; upper right quadrant: QBG beam with m s 2; lower left quadrant: QBG beam with m s ir '2 ; lower right quadrant: Bessel–Gauss beam with k s 2.

C.F.R. Caron, R.M. PotÕlieger Optics Communications 164 (1999) 83–93

90

spectral function Ž17.. They are not related to the similar phase dislocations recently described for nonparaxial Gaussian beams with discontinuous spectral function w15x. The far-field behaviour of the QBG beams can be examined by proceeding as for the Bessel–Gauss Ž p. beams, i.e., by expressing E BM G, m in terms of the angle u s arctanŽ rrz . and of the radius r s Ž r 2 q z 2 .1r2 , taking r very large, and retaining only the leading order term in the result. This yields, for 0 F u - pr2, Ž p. < E BM < G ,m f

Ef z R

(

rcos u m2 q 1

ž

=exp y

J < m < r2

k 2 w 02 4m 2 q 4

ž

m k 2 w 02 4m 2 q 4

tan2u

/

tan2u .

/

Ž 26 .

The asymptotic behaviour of the corresponding exact Žnonparaxial. QBG beam is obtained by substituting the spectral function Ž17. for f in Eq. Ž13.: < E BM G , m < f

Ef z R cos u

(

r m2 q 1

ž

=exp y

J < m < r2 k 2 w 02

4m 2 q 4

ž

m k 2 w 02 4m 2 q 4

/

sin2u .

sin2u

/ Ž 27 .

Ž p. < EBM < 2 and < EBMG,0 < 2 are plotted in Fig. 1Žb., as G,0 given by Eqs. Ž26. and Ž27. for kw 0 s 12p and m s 0, 3 and 12. For kw 0 s 12p , the spot size of the m s 0 beam, w 0 , is six times the wavelength. This radius is sufficiently large for the corresponding paraxial scalar Gaussian beam to be an excellent representation of its nonparaxial counterpart w8,16x, although it may already be too small for the scalar approximation itself to be entirely reliable w17x. Discrepancies between the paraxial and the exact results gradually appear when m increases, kw 0 fixed, due to the concomitant narrowing of the beam waist. This narrowing is indeed accompanied by a broadening of the far-field profile into the angular region where sin u and tan u differ significantly. The discrepancies become prominent in this angular region when m R kw 0r4, as may be expected from the previous discussion of the transverse profile of Ž p. Ž . Ž Ž . EBM G,0 r , z s 0, f . For example, note in Fig. 1 b the breakdown of the paraxial approximation at large

Fig. 7. The divergence angle of a zeroth-order QBG beam relative to the divergence angle of a pure Gaussian beam having, for each value of m , the same wave number and the same spot size in the z s 0 plane Žhalf-radius at 1r e of the central peak of the amplitude profile. than the QBG beam. The divergence angle is defined as the angle at which, asymptotically far from the origin, the modulus of the nonparaxial amplitude is 1r e its value at u s 0. ŽThis quantity is not a good measure of the beam divergence for m much larger than shown in the figure, as the amplitude then remains significant beyond the first zero of the Bessel function.. From top to bottom: w 0 s12p r k, 10p r k, 8p r k, 6p r k, 4p r k, and 2p r k.

u , for m s 12, and the excellent agreement between paraxial and nonparaxial results for m s 3.. The non-Gaussian character of QBG beams manifests in the far-field profile both by an oscillatory dependence in u and by a much faster broadening for increasing values of m than if these beams were purely Gaussian with the same decreasing spot size. The oscillations are not always significant, but, as seen from Fig. 7, the abnormally large divergence is noticeable even when m is small. The far-field behaviour of QBG beams contrasts strikingly with that of Bessel–Gauss beams. It implies that these two types of beams have fundamentally different propagation characteristics: while the latter have an essentially conical geometry, the former propagate colinearly. 4. Realization of QBG beams Axisymmetric QBG beams with small values of m can be realized as described in Ref. w18x for

C.F.R. Caron, R.M. PotÕlieger Optics Communications 164 (1999) 83–93

flattened Gaussian beams, i.e., by illuminating a transparency so as to reproduce the Žreal. field amplitude in the waist plane, Ž20.. However, the success of this method depends on whether this amplitude may be represented accurately enough by a non-negative function. While this is not an issue for Ž p. Ž . imaginary values of m , for which E BM G,0 r , z s 0, f is positive everywhere Žand is essentially a Gaussian function., the Bessel function becomes negative within the range of the Gaussian for real values of m of the order of unity or larger. We explored this approach by comparing the axisymmetric QBG field to the field which reduces to the function 2

F Ž r . s Ef

Ý Ž y1.

m Ž rrw 0 .

p

p

4 Ž p! .

ps0

=exp y Ž rrw 0 .

Ž p. EAB MG Ž r , z . s yi

z

`

H0

=e i kŽ2 z

d r X r X F Ž r X . J0 Ž k rr Xrz . X2

.

Ž 29 .

With the help of Eq. Ž6.631.10. of Ref. w12x, we find, after some algebra,

s

1 q izrz R 2

=

Ž y1.

Ý ps0

=L2 p

ž

iz R

ž

exp y p

r 2rw 02 1 q izrz R

m2 p Ž 2 p . ! 4 p Ž p! .

r 2rw 02

z 1 q izrz R

/

2

,

ž

1 q izrz R

ž

m2 1y

izrz R 1 q izrz R

2

ž

izrz R 1 q izrz R

2

/

4

/

.

Ž 31 .

Ž p. Ž . As should be the case, EAB MG r s 0, z differs from Ž p. E BM G,0 Ž r s 0, z, f . only by terms of order m6 and Ž p. Ž r s 0, z . for m s 1 higher. The variation of EABMG is represented in Fig. 3. The flat axial profile and steeper phase variation of the QBG beam is well reproduced by the model. This is also the case for the intensity maxima appearing on each side of the z s 0 plane for m ) 1; however, the agreement deteriorates rapidly when m increases beyond 2.

5. QBG pulses It has been known for some time that to each solution c Ž r , z, f . of the paraxial wave equation corresponds an exact Žnonparaxial. solution of the scalar wave equation, describing a pulse-like nonstationary wave 2 , and given by w21x F Ž r , z , f ,t . s exp ik Ž z q ct . r2 c Ž r , z y ct , f . . Ž 32 .

Ž p. EAB MG Ž r , z .

Ef exp Ž ikz .

Ef exp Ž ikz .

8

Ž 28 .

qr 2qr .r2 z

s

3 m4

2

2

Ž p. EAB MG Ž r s 0, z .

q

in the z s 0 plane. F Ž r . is nothing else than the Ž p. Ž . Taylor expansion of EBM G,0 r , z s 0, f in powers of m , truncated to the first three terms. It is easy to show that F Ž r . has a double zero at r s Ž8rm2 .1r4 w 0 and is otherwise positive. This function can be taken, therefore, as representing the amplitude distribution in the waist plane of a suitably attenuated Gaussian beam. The Huygens–Fresnel diffraction integral yields the corresponding approximate zeroth-order QBG beam, k

where L2 p denotes the Laguerre polynomial of deŽ p. Ž . gree 2 p. Note that by construction, EAB MG r , z is the product of expŽ ikz . and of an exact solution of the paraxial wave equation w19x. Eq. Ž30. is simpler Ž p. Ž . than the equivalent expansion of E BM G,0 r , z, f in orthogonal Gauss–Laguerre modes, which is given in Appendix B. Ž p. Ž . On axis, EAB MG r , z is

2 2p

2

91

/

izrz R 1 q izrz R

2p

/ Ž 30 .

2 Here we mean by ‘stationary field’ one ‘‘such that, when averages over any macroscopic time interval are considered rather than the instantaneous values, the properties of the field are found to be independent of the instant of time at which the average is taken’’ w20x.

C.F.R. Caron, R.M. PotÕlieger Optics Communications 164 (1999) 83–93

92

where If s c e 0 Ef2r2 is the intensity at Ž r s 0, z s 0.. P BM G,0 can be expressed in terms of a complete elliptic integral of the first kind w12x: with PG s p If w 02r2 denoting the power carried by the pure Gaussian beam Ž19. and in the notation of Ref. w12x, the result reads 2 1 m P BM G ,0 s K PG . Ž A.2 . p 1 q m2 1 q m2

Hence, to Ž15. correspond QBG pulses, FBM G , m Ž r , z , f ,t . w0 s Ef exp ik Ž z q ct . r2 W Ž z y ct .

mr 2 = J < m < r2

W 2 Ž z y ct .

=exp y

1 q i Ž m 2 q 1 . Ž z y ct . rz R 2

W Ž z y ct .

ž(

(

r2

=exp Ž im f . . Ž 33 . These pulses are the QBG analogues of Overfelt’s Bessel–Gauss pulses w7x. They propagate at velocity c in the z-direction, without deformation apart for phase variations caused by the dependence in z q ct of the first exponential factor: for an observer moving with the pulse, < FBM G, m < 2 is stationary and identical to the modulus squared of the paraxial wave Ž15.. Nonparaxial, vectorial electromagnetic fields are easily associated to FBM G, mŽ r , z, f ,t . by identifying it to a particular component of the Hertz vector potential w22x. Fields constructed in that way are exact solutions of the Maxwell’s equations, in spite of the increasingly fast oscillations of the Bessel function. As noted above, and by contrast, these oscillations compromise, too far from the propagation axis, the relevance of the QBG beams Ž15. as approximate representations of stationary nonparaxial fields.

/

The corresponding result for the zeroth-order Bessel–Gauss beam is P BG ,0 s exp Ž yk 2r4 . I0 Ž k 2r4 . PG ,

Ž A.3 .

where k s kw 0 sin a . Both P BMG,0 and PBG,0 reduce to PG when m s k s 0 and decrease monotonically for increasing values of these parameters. For m s k ) 0, the power of the QBG beam always exceeds that of the Bessel–Gauss beam. Appendix B. Expansion in Gauss–Laguerre modes Any f-independent solution of the paraxial wave equation may be represented by a superposition of orthogonal Gauss–Laguerre modes of order 0, i.e., of particular solutions of the form w19x

(2rp w c Ž r,z. s p

2 0

1 q izrz R

=L p

ž

1 y izrz R

ž

1 q izrz R

2 r 2rw 02 1 q z 2rz R2

p

/

/ ž

exp y

r 2rw 02 1 q izrz R

CFRC has been supported during the course of this work by the UK Engineering and Physical Sciences Research Council.

Note that the Laguerre polynomials have not the same argument in Eq. Ž30. than in Eq. ŽB.1.. Owing to the orthonormality relation `

Appendix A. Power carried by the zeroth-order beam For small divergence angles, the power carried across the z s 0 plane by the electromagnetic field represented by the paraxial scalar electric field E Ž r , z, f , t . s Re w E BŽ pM. G ,0 Ž r , z, f . exp Ž yictrk .x takes the form w23x `

H0

.

Ž B.1 .

Acknowledgements

P BM G ,0 s 2p If

/

Ž p. <2 d r r < EBMG ,0 Ž r , z s 0, f . rEf ,

Ž A.1 .

2p

H0

d r r cp Ž r , z .

w

c pX Ž r , z . s d p pX ,

Ž B.2 .

the coefficients of the expansion ` Ž p. E BM G ,0 Ž r , z , f . s

Ý

a p cp Ž r , z . exp Ž ikz .

ps0

Ž B.3 . can be found for each p in turn by multiplying Ž p. wŽ Ž . EBM r , z s 0. and integrating G,0 r , z s 0, f by c p the product over r . The integral can be expressed as sums of Legendre polynomials, by writing the Laguerre polynomial appearing in cpw Ž r , z s 0. as a

C.F.R. Caron, R.M. PotÕlieger Optics Communications 164 (1999) 83–93

sum of powers and making use of Eq. Ž6.621. of Ref. w12x to integrate each term separately. The result is a p s Ef

(

=

p

p w 02 2

p!

n Ý Ž y1. n! Ž p y n . !

ns0

nq 1

1

1

Pn

(1 q m r4 2

. Ž B.4 .

(1 q m r4 2

Incidentally, the expansion of the zeroth-order Bessel–Gauss beam in Gauss–Laguerre modes is much simpler: E BG ,0 Ž r , z , f . s Ef

(

p w 02

`

=

Ý ps0

2 1 p!

ž

exp y

ž

k 2 w 02 sin2a

k 2 w 02 sin2a 8

8

/

p

/

=cp Ž r , z . exp Ž ikz . . Ž B.5 . This result follows from Eq. Ž8.975.3. of Ref. w12x. References w1x F. Gori, G. Guattari, C. Padovani, Optics Comm. 64 Ž1987. 491. w2x P. Paakkonen, J. Turunen, Optics Comm. 156 Ž1998. 359. ¨¨ ¨

93

w3x J. Durnin, J. Opt. Soc. Am. A 4 Ž1987. 651. w4x J. Durnin, J.J. Micelli Jr., J.H. Eberly, Phys. Rev. Lett. 58 Ž1987. 1499. w5x P.W. Milonni, J.H. Eberly, Lasers, Wiley, New York, 1988. w6x V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, J. Mod. Optics 43 Ž1996. 1155. w7x P.L. Overfelt, Phys. Rev. A 44 Ž1991. 3941. w8x G.P. Agrawal, D.N. Pattanayak, J. Opt. Soc. A 69 Ž1979. 575. w9x L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, Cambridge, 1995. w10x P. Saari, K. Reivelt, Phys. Rev. Lett. 79 Ž1997. 4135. w11x S. Klewitz, S. Sogomonian, M. Woerner, S. Herminghaus, Optics Comm. 154 Ž1998. 186. w12x I.S. Gradshteyn, I.M. Ryzhik, Tables of Integrals, Series, and Products, 5th Ed., Academic Press, New York, 1994. w13x J.T. Foley, E. Wolf, J. Opt. Soc. A 69 Ž1979. 761. w14x E.H. Linfoot, E. Wolf, Proc. Phys. Soc. B 69 Ž1956. 823. w15x M.V. Berry, J. Mod. Optics 45 Ž1998. 1845. w16x X. Zeng, C. Liang, Y. An, Appl. Optics 36 Ž1997. 2042. w17x P. Varga, P. Torok, ¨ ¨ Optics Comm. 152 Ž1998. 108. w18x V. Bagini, R. Borghi, F. Gori, A.M. Pacileo, M. Santarsiero, D. Ambrosini, G. Schirripa Spagnolo, J. Opt. Soc. Am. A 13 Ž1996. 1385. w19x A.E. Siegman, Lasers, University Science, Mill Valley, 1986. w20x M. Born, E. Wolf, Principles of Optics, 6th Ed., Cambridge University Press, Cambridge, 1980. w21x P.A. Belanger, J. Opt. Soc. Am. A 1 Ž1984. 723. ´ w22x S. Feng, H.G. Winful, R.W. Hellwarth, Optics Lett. 23 Ž1998. 385. w23x Q. Cao, X. Deng, Optics Comm. 151 Ž1998. 212.