Spatiotemporal bessel beams:theory and experiments

Spatiotemporal bessel beams: theory and experiments

Michaël Dallaire, Nathalie McCarthy*

and Michel Piché

Centre d’Optique, Photonique et Laser (COPL)

Département de physique, de génie physique et d’optique, Université Laval,

Québec, Qc, Canada, G1V 0A6

*[email protected]

Abstract: We describe a family of dispersion-free and diffraction-free

optical beams consisting in two-dimensional wave packets with a

spatiotemporal Bessel (STB) profile propagating in media with anomalous

dispersion. We also describe quasi-invariant optical beams with a

spatiotemporal Bessel-Gauss (STBG) profile; these wave packets have finite

dimensions and energy, conditions to be representative of physical beams.

The paper provides a detailed account of the properties of STB and STBG

beams, including their spatially resolved frequency spectrum, their far-field

behaviour and a comparison of the propagation of STBG beams with that of

Gaussian wave packets. An experimental setup based on a folded pulse

shaper has allowed to generate STBG beams using the ultrashort pulses

from a Ti:sapphire laser. The analysis of the spatially resolved frequency

spectrum and of the spatial and temporal profiles obtained experimentally

shows good agreement with theory.

©2009 Optical Society of America

OCIS codes: (320.5540) Pulse shaping; (320.7090) Ultrafast lasers; (320.5550) Pulses.

References and links

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#114098 - $15.00 USD Received 9 Jul 2009; accepted 15 Sep 2009; published 24 Sep 2009

(C) 2009 OSA 28 September 2009 / Vol. 17, No. 20 / OPTICS EXPRESS 18148

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1. Introduction

Diffraction-free and dispersion-free optical beams or wave packets are special cases of optical

wave propagation that have caught general attention in the last decades. The generation of

beams with spatial or temporal invariance requires a thorough knowledge of the physical laws

governing their propagation in a given medium. Temporal solitons were the first class of

optical signals to be recognized as being invariant under propagation [1] and to be

experimentally demonstrated [2]. Solitons require a nonlinearity to preserve their shape: in the

case of temporal solitons, the effects of nonlinearity and dispersion cancel each other while,

for spatial solitons, nonlinearity compensates for diffraction. It was pointed out by Durnin that

nonlinearity was not a requirement for invariant propagation; indeed Bessel beams can

propagate in a purely linear material without deformation [3].

The diffraction-free character of ideal Bessel beams has triggered immense interest. It was

soon realized that, due to practical considerations, only quasi-Bessel beams could be produced

[3,4]. Gori et al. derived the complete solution describing the evolution of transverse Bessel-

Gauss beams during propagation [5]. These monochromatic solutions to the wave equation

have been generalized to polychromatic fields; for instance, pulsed Bessel and Bessel-Gauss

beams [6–9] and Bessel-X waves [10–12] have caught a lot of attention since they lead to

interesting spatiotemporal coupling phenomena. Other diffraction-free beams have also been

investigated, such as diffraction-free optical bottles [13], periodically focused propagation-

invariant beams [14] and helicon waves [15], to name a few.

In recent papers, Longhi [16] and Porras and Trapani [17] have introduced a family of

invariant three-dimensional wave packets propagating in media with anomalous dispersion

that they have called “envelope O waves” or “O-waves”, in reference to their spatiotemporal

concentric structure. The amplitude of these “O-waves” along their spatiotemporal radius is

modulated by a sinc function that induces a structure with multiple concentric shells.

In this paper, we explore in detail the properties of a family of two-dimensional (x and t)

optical wave packets that we call “spatiotemporal Bessel beams” (STB beams) and that we

have introduced in Ref [18]. These wave packets are characterized by a spatiotemporal profile

that corresponds to a Bessel function of first kind of order m. This set of solutions to the wave

equation is neither subject to dispersion nor diffraction and it is restricted to optical materials

with anomalous dispersion. However, unlike solitons, the propagation of STB beams takes

place in a linear optical medium.

The fundamental solution (m = 0) of STB beams is similar to the localized wave packets

described by Malaguti et al. [19] for the case of anomalous dispersion. In practice, it would be

impossible to generate a pure STB beam because of its infinite dimensions and energy. The

introduction of a Gaussian envelope in the Bessel-type solution allows to circumvent this

difficulty. The resulting beam will be called a “spatiotemporal Bessel-Gauss” beam, or STBG

beam. The Gaussian envelope induces a z-dependent evolution of beam parameters which can

be described analytically. The spread of an STBG beam during propagation is found to be

much lower than that of a spatiotemporal Gaussian wave packet matching the central lobe of



the STBG beam. We have experimentally demonstrated how to synthesize STBG beams

through spatiotemporal reshaping of the ultrashort pulses emitted by a Ti:sapphire laser. The

folded pulse shaper developed for that purpose can be viewed as a spatiotemporal axicon.

Experimental measurements of the spatial and temporal profiles, along with the analysis of the

spatially resolved spectrum, show good agreement with theory.

This paper is organized as follow. In Section 2, we present the STB beam as a solution to

the two-dimensional wave equation in media with anomalous dispersion. In Section 3, we

introduce STBG beams which are physical solutions with finite energy due to their Gaussian

envelope. The propagation of STBG beams is described in Section 4. In Section 5 we examine

the spatially resolved spectrum of STB beams; propagation invariance is shown to result from

a proper selection of the plane wave spectrum constituting this type of optical wave packet.

The pulse shaper that we have designed to produce STBG beams is described in Section 6. In

Section 7, we present our experimental results showing that optical wave packets with a

spatiotemporal Bessel profile have been generated using the pulse shaper.

2. Spatiotemporal Bessel beams

We consider an optical field defined by the following expression

( ){ }( )( , ) Re , ,o oj t z

E r t u r t eω β−=

� �ɶ (1)

where ( ),u r t�ɶ is the complex envelope of the field,

oω is its central frequency,

oβ is the

corresponding wavenumber and ( , , )r x y z=�

. The envelope is assumed to be slowly varying

as compared to the carrier ( )o oj t z

eω β−

. The temporal envelope ( , )u r t�ɶ is related to its spectral

counterpart ( ), 'U r ω�

ɶ by a standard pair of Fourier transforms, with 'o

ω ω ω≡ − . We assume

that the propagation of such an optical field is governed by the paraxial wave equation. In the

frequency domain, each spectral component of the field then propagates in a lossless medium

according to [20]:

2( , ')

( , ') ( ) ( , ').2 ( )

t

U r jU r j U r

z

ωω β ω ω

β ω∂ −

= ∇ − ∆∂

�ɶ � �

ɶ ɶ (2)

The operator 2

t∇ is the transverse Laplacian, defined as 2 2 2 2 2

tx y∇ ≡ ∂ ∂ + ∂ ∂ . The

wavenumber ( )β ω is ( ) on cω ω with ( )n ω being the refractive index and

oc the speed of

light in vacuo. For a relatively narrow spectrum, ( )β ω can be expanded as a Taylor series

around o

ω with its frequency dependent component ( )β ω∆ defined as:

( )

( )

2

2

1

' ' ...

2

o

o

β ωβ ω β β ω

β β ω

= + + +

= + ∆ (3)

where ( )i i

i d dβ β ω≡ is evaluated at o

ω ω= . Moreover one can make the approximation

that the factor ( )2 2o

j jβ ω β− ≅ − in Eq. (2) since ( )β ω∆ << o

β , where

o o o on cβ ω= and

on is the index of refraction at

oω . The wave equation in a reference frame

moving at the pulse group velocity 1

1g

β=v is obtained by defining a new time coordinate

1 T t zβ= − . By limiting the expansion of ( )β ω to second order in 'ω and assuming that the

field ( ), 'U r ω�

ɶ has no dependency in y, an inverse Fourier transform of Eq. (2) leads to the



following wave equation in the reference frame travelling with the pulse at the group velocity

gv

( ) ( ) ( )2 2

22 2

, , , , , , ,

2 2o

v x z T v x z T v x z Tj j

z x Tβ

β

∂ ∂ ∂−= +

∂ ∂ ∂

ɶ ɶ ɶ (4)

where ( ), ,v x z Tɶ is the field ( ), ,u x z tɶ in which t is expressed in terms of T.

A class of solutions exhibiting propagation invariance can be obtained from Eq. (4) by

introducing the Laplacian operator in polar coordinates:

2 2 2 2

22 2 2 2 2

1 1 .

ox T

β βρ ρρ ρ θ

∂ ∂ ∂ ∂ ∂− ≡ + +

∂∂ ∂ ∂ ∂ (5)

This operator is expressed in terms of the spatiotemporal radial and angular coordinates

defined as 2 2

2ox Tρ β β≡ − and ( )1

2tan

oT xθ β β−= − , respectively. In order that ρ be

real for all combinations of x and T, parameter 2

β has to be negative; in other words, this

treatment is valid for materials with anomalous dispersion. In terms of coordinates ρ and θ ,

Eq. (4) becomes:

( ) ( )

2 2

2 2 2

, , 1 1 , , .

2o

v z jv z

z

ρ θρ θ

β ρ ρρ ρ θ

∂ − ∂ ∂ ∂= + + ∂ ∂∂ ∂

ɶɶ (6)

The last equation can be solved by separation of variables with

( ) ( ) ( ) ( ), ,v z f g h zρ θ ρ θ=ɶ , as outlined at page 1259 of ref [21]. To obtain a general

solution to that equation, we assume the angular dependency to have the form

( ) jm

mg A e

θθ = , m being the integer associated to a particular azimuthal order and m

A the

corresponding amplitude. One can also obtain ( ) 1ej z

h zφ+= with

1/ 2

oaφ β≡ , where a is

a constant introduced for the separation of variables. The radial function ( )f ρ is ( )mJ aρ ,

where ( )m

J ξ is the well-known Bessel function of first kind of order m . A solution to Eq. (4)

is then given by:

( )2

( / )2

2

, , ,o pj t zjm

m m m

o

TE x z t A J a x e e

ωθ

β β−

= −

vɶ (7)

where a, of units mm−2

, will be called the beam parameter as it allows scaling the dimensions

of the solution; p

v is the phase velocity of the wave packet given by 1

( )p o o

ω β φ= −v .

Equation (7) is the general expression of a spatiotemporal Bessel (STB) beam of order m.

Except for the factor representing the carrier, this expression is clearly independent of z in the

reference frame moving at group velocity g

v . Therefore, during propagation, the

spatiotemporal profile of an STB beam remains unchanged.

Figure 1 shows the intensity pattern of an STB beam of order m = 0 along normalized axes

x a and 2o

T a β β− . This beam consists in a series of concentric annuli with a high

intensity central lobe that propagates without deformation along the z-axis, as shown in the

video of Fig. 2. The total width of the central lobe along the x-axis and its duration along the

T-axis, both taken at the first zero of Bessel function ( )m

J ξ (as shown in Fig. 3), are

respectively given by



4.8

,Xa

∆ = (8)

24.8 .oTa

β β−∆ = (9)

In the case of a medium with positive GVD (i.e. normal dispersion), ρ would become

imaginary for certain values of x and T, leading to a solution involving a modified Bessel

function of first kind ( )m

I ξ with an asymptotic divergence.

Fig. 1. Relative intensity distribution of a spatiotemporal Bessel beam of azimuthal order m = 0

along normalized space and time axes.

Fig. 2. Schematic representation of the propagation of an ideal STB beam (see Media1).

The spatiotemporal frequency content of the STB beam in the polar coordinate system is

given by the Hankel transform of ( )mJ aρ . Using [22], one finds

( ) ( )F k k a kρ ρ ρδ= − , where ( )δ ξ is the Dirac delta function. This result means that the

STB beam is formed by a set of plane waves whose spectrum forms a ring of infinitesimal

thickness in the polar coordinate system. From the definition of the spatiotemporal coordinate



ρ one finds that ( )22 2

2x o ok k aρ β β ω ω= − − = , where kρ is the spatiotemporal frequency in

the polar coordinate system and x

k is the projection of kρ along the x-axis.

3. Spatiotemporal Bessel-Gauss beams

The spatiotemporal Bessel beam defined by Eq. (7) has infinite extent along x and t (or z) and,

as such, it would carry infinite power [3]. To circumvent this problem, one can limit the extent

of the beam by introducing a spatiotemporal Gaussian envelope in the expression that defines

the field of an STB beam [5]. The resulting wave packet, named spatiotemporal Bessel-Gauss

beam (or STBG beam), can be expressed as follows in the plane of beam waist:

( )2

2( , , 0) exp ,jm

m m m

o

u z A J a ew

θρρ θ ρ

= = −

ɶ (10)

with o

w the size of the Gaussian envelope at the beam waist. A comparison between STB and

STBG beams as a function of the normalized space-time position is presented in Fig. 3.

Fig. 3. Relative intensity profile at z = 0 of an STB beam and an STBG beam of order m = 0 as

a function of the normalized space-time position. The size o

w of the Gaussian envelope has

been set to 14.4 in normalized units (a value equivalent to 6∆X or 6∆T).

The solution describing an STBG beam of arbitrary order m at any position z can be

obtained by the following Huygens-Fresnel integral in cylindrical coordinates:

1 2

2

0

2( , , ) exp

2exp ( , , 0) ' .

m

m

o o

m m

o o

j ju z

z z

jJ u z d

z z

π πρρ θ

λ λ

πρρ πρρ θ ρ ρ

λ λ

+

∞

= −

′ ′′× − =

∫

ɶ

ɶ

(11)

Using Eq. (6).633.2) of ref [23], the complete expression of the STBG beam takes the

following form, analogous to that of standard Bessel-Gauss beams [5]:

( )

( )( )

2( )

2 2

2

e( , , ) exp

4

exp I e e ,o o

m j z

oR

m m

RR

j t zjmR

m

o R R

a w zj zE z A

z jzz z

a zj

z jz z jz

ω βθ

ρ θ

ρπ ρλ

Φ

−

−=

+ +

× − + +

ɶ

(12)

with R

z being the Rayleigh range of the Gaussian envelope defined as 2

R o oz wπ λ= . The

Gouy phase shift is given by ( ) arctan( / )R

z z zΦ = , and ( ) ( )m m

I J jξ ξ= is the modified

Bessel function of the first kind of order m.



The temporal size of the Gaussian envelope ot

w can be obtained from o

w as:

( )2 2

2.

ot o ow w β β= × − (13)

Even for beam sizes o

w containing few annuli of the Bessel function ( )o

J aρ , the Gaussian

envelope has virtually no effect on the central lobe of the Bessel profile at the waist but it

brings gradually the outer annuli to zero. The finite size of the STBG beam has important

consequences on its properties during propagation, as shown in the next section.

4. Propagation of STBG beams

Equation (12) describes the evolution of STBG beams as they propagate in media with

anomalous dispersion. A typical example of STBG beam propagation is shown in Fig. 4,

where the movie reveals two different beam structures: 1) co-propagating concentric annuli

near the beam waist (near field) and 2) a single diverging ring far from the waist (far field).

Fig. 4. (Media 2) illustrates the evolution of an STBG beam during propagation.

4.1 Near-field distribution

Figure 5(a) shows the evolution of the relative intensity distribution of an STBG beam (order

m = 0) near its waist during propagation in fused silica. The envelope of the signal is centered

at 1550 nm and its spectral width is 40 nm. For the purpose of comparison, the evolution of a

spatiotemporal Gaussian wave packet with a beam size c

w matched to the central lobe of the

STBG beam is shown in Fig. 5(b). The size c

w is obtained by taking the width of the central

lobe at 1/e2 of its normalized intensity as it is defined for a Gaussian beam. The relation

between the size c

w of the central lobe of the STBG beam and beam parameter a is

1.752c

w a≈ . The Rayleigh distance zc associated to the beam size c

w is defined as

2

c c oz wπ λ= . For the example shown in Fig. 5(a), the size wo of the Gaussian envelope has

been set to 10c

w .



Fig. 5. Evolution of the relative intensity distribution during propagation in fused silica of a) an

STBG beam and b) a spatiotemporal Gaussian wave packet matched to central lobe of the

STBG beam. All curves are normalized such that their maximum is unity. The local intensity at

center of both beams is compared in c) as a function of propagation distance; the gray lines

indicate the distance where the intensity has fallen to 50% of its value at the waist.

Near the beam waist, the Bessel profile is the dominant contribution of the STBG beam

distribution. Figure 5(a) shows that the basic profile of the STBG beam is maintained over a

long distance (15 zc), even if the intensity of the central lobe is in fact reduced by 98% (see

Fig. 5 (c)). For the same distance, the residual intensity at center of the Gaussian wave packet

is even lower (by a factor of 6). Energy redistribution from the inner structure to an outer ring

gradually takes place during propagation, as seen in Fig. 5(a). The Gaussian envelope has a

direct impact on the profile of the propagated beam; a broader envelope will produce quasi-

invariant propagation over a longer distance than a narrower envelope. Figure 5(c) shows the

intensity at center as a function of propagation distance; the intensity at center of the STBG

beam decreases by 50% at a distance BG

z exceeding 6 zc, e.g. 6 times longer than for the

Gaussian wave packet. The ratio BG c

z z can be expressed as

0

0.38 ,BG c

z z a w≈ (14)

pointing out that the enhancement of the collimation length of the STBG beam scales as the

size of its Gaussian envelope and the square root of its beam parameter.

4.2 Far-field distribution

Far away from its waist, the STBG beam transforms into a spatiotemporal ring whose radius

grows linearly as a function of distance z. This annulus can be interpreted as being due to the

spatial and temporal separation of multiple pulsed beams travelling at different angles with

respect to the z-axis. As these pulsed beams propagate toward the waist, they spatially and

temporally converge to a central point moving at group velocity g

v ; when they overlap,



interference occurs, creating a circular pattern with the shape of a Bessel function. Once the

circular structure appears near the waist, the dimension of the concentric annuli remains

invariant. The visibility of the concentric annuli forming the Bessel profile increases closer to

the waist until a maximum is reached at z = 0. Beyond the waist, the combined effect of

divergence and dispersion has the consequence that the original pulsed beams separate

spatially and temporally, resulting in the gradual fading of the circular beam structure which

evolves into a ring distribution in the far field. In the far field the beam pattern reduces to a

spreading and fading spatiotemporal donut, whose intensity distribution is asymptotically

given by:

( ) ( )( )( )

( )

2

2

2

2, exp ,

o

f

zE z P z

w z

ρ σρ

− − ≈

ɶ (15)

with ( )fw z representing the size of the envelope in the far field, ( )zσ its offset and ( )o

P z

its peak intensity. These quantities can be expressed as:

( ) ,o

f

o

zw z

w

λπ

= (16)

( ) ,2

oa zz

λσ

π= (17)

and

( )( )

22

.2

o

o o

o

wP z A

a z zλ σ= (18)

Equations (15-18) are valid only for z >> zc. Equation (15) indicates that the Bessel

structure has vanished in the far field where the only remaining feature is an annular Gaussian

profile having an increasing radial offset and a decreasing intensity maximum as it propagates

along z. A typical far-field profile is shown in Fig. 6. For a Gaussian envelope of very large

size, the far-field envelope becomes extremely thin; this is consistent with the fact that the

Hankel transform of an STB beam is an annular Dirac delta function, as discussed in Section

2.

The donut beam profile in the far field produces a sequence of two pulses along the z-axis.

The two pulses are separated by a maximum delay at transverse position x = 0 ; that delay

decreases for increasing values of |x| to produce a single pulse at the edges of the donut.

Equation (17) gives the spatiotemporal radial position of maximum intensity in the far field;

one then finds directly the delay between the front and rear short pulses:

22 .z

o

at z

ββ

−∆ = (19)

This delay should easily be observed experimentally.

It should be noted that the offset is independent of the size o

w of the Gaussian envelope at

the beam waist. We can see from Eq. (17) that the transverse offset ( )zσ increases as

/o

z a β , leading to the following relation defining the divergence angle δ along the

propagation axis:

( )arctan / / .o o

a aδ β β= ≈ (20)



This relationship, that reflects the conical nature of the beam, has been obtained by Gori et al,

for the purely spatial Bessel-Gauss beam [5]. It points out that a can be viewed as the

spatiotemporal frequency of the STB beam in the coordinate system ( , , zρ θ ), as discussed in

Section 2.

Fig. 6. Intensity distribution at distance z = 30 zc for an STBG beam having the following

parameters: ∆λ = 40 nm with a central wavelength λo = 1550 nm (which gives a = 40 mm−2),

index of refraction n0 = 1.444, β2 = −27.95 ps2/km, Gaussian envelope sizes wo = 2.95 mm and

wot = 1.19 ps.

5. Spatially resolved spectrum of STB beams

In order to generate propagation invariant two-dimensional wave packets, it is important to

know their spectrum of optical frequencies as a function of lateral position x. For the STB

beam this information is obtained by a temporal Fourier transform of expression (7). Putting

m = 0 and using Eq. (6).677.3) of ref [23], one finds the following analytical expressions

defining the spatially resolved spectrum of plane waves constituting the STB beam:

( ) ( ) ( ), , cos ,zj k z

xF x z A k x eω ω −=ɶ (21)

with

( )2

2 ,x o ok a β β ω ω= + − (22)

( )12

z o o ok aβ β β ω ω= − + − (23)

and

( ) 2,

o

o

x

A Ak

β βω

−= (24)

where x

k and z

k are the wave vector components at optical frequency ω . As shown in Fig.

7, the spectrum of optical frequencies needed to generate an STB beam is limited by two

cutoffs. This is due to the condition that x

k must be real for propagating wave packets. From

Eq. (22) one finds that the boundaries of the spectral distribution are given by

( ) ( ) ,o HW o HWω ω ω ω ω− ∆ < < + ∆ (25)



with

2

,HW

o

aω

β β−

∆ = (26)

from which we obtain

( ) 2

0 2.

HWa β β ω= − ×∆ (27)

In practice, the half-width of the spectrum HW

ω∆ will be fixed by the laser source used to

generate the invariant wave packets. It can be seen that the size and duration of the central

lobe (and the annuli) of an STB beam are set by parameter a (see Fig. 8). Equation (27) can be

used to rewrite Eqs. (8-9) in terms of physical parameters only, showing that the scale of the

temporal variation of the Bessel function at the beam waist depends only upon the spectral

half-width HW

ω∆ according to

4.8

.HW

Tω

∆ =∆

(28)

As an example, the last relation would give a wave packet propagating in fused silica with a

central lobe of duration T∆ ≈ 600 fs and, according to Eq. (8), a spatial size X∆ ≈ 1500 µm

for a spectrum of 10-nm half-width centered at 1550 nm.

From Eqs. (22-23), it can easily be shown that, within the slowly-varying envelope and

paraxial approximations, the plane waves constituting the STB beam verify the following

dispersion relation

( )2 2 2.

x zk k β ω+ = (29)

Since x

k is real and 2

β < 0, Eq. (22) establishes that the range of values of x

k is limited by

.x

a k a− < < (30)

Wave vector components outside these boundaries will induce distortions to the ideal

Bessel profile. It is easily observed from (22) that x

k a= ± if o

ω ω= ; this means that, in

order to verify the dispersion relation of the STB beam, its frequency components at center of

the spectrum are propagating at the largest angle with respect to the z-axis. Conversely, its

frequency components at the boundaries of the spectrum o HW

ω ω± ∆ are propagating parallel

to the z-axis. In order to generate an STB beam, one has to use a set of plane waves such that

their frequency ω and the component x

k of their wave vector along the x-axis verify Eq.

(22).

Finally, it should be observed that Eq. (23) predicts that the group velocity at all

frequencies ω is constant (1

1 β≡ ) along the z-axis. Propagation invariance can be viewed as

the consequence of the fact that all waves interfering to produce the STB beam have the same

group velocity along the z-axis.



Fig. 7. Theoretical spatially resolved spectrum of a spatiotemporal Bessel beam. The vertical

(spatial) scale goes to infinity.

Fig. 8. Beam parameter a and pulse duration ∆T as a function of the half-width

of the spectrum of an STB beam (λ0 = 1550 nm, propagation in fused silica).

6. Experimental setup

The annular intensity distribution presented in Fig. 6 is the spatiotemporal Fourier transform

of an STBG beam. Diffraction has the effect of separating the spatial frequencies of the beam,

and dispersion does the same with its optical frequencies. The resulting far-field distribution

can be viewed as a mapping of optical frequencies as a function of transverse position x and of

spatial frequencies as a function of time T. A strategy to produce an STBG beam would

consist of generating its far field by positioning a reflective mask in a pulse shaper designed to

split the optical and spatial frequencies of a short laser pulse; if the mask has an annular

shape, it is then possible to obtain an STBG beam at its waist by doing the inverse Fourier

transform in the space and time domains of the signal reflected by the mask.

We have designed a pulse shaper that performs the required Fourier transforms. The

experimental setup is composed of a Ti:sapphire laser emitting femtosecond pulses with a

spectrum of 10- to 30-nm width centered at 800 nm, a spatiotemporal pulse shaper (shown in

Fig. 9), and beam/pulse diagnostics (autocorrelator, CCD camera, etc.). The pulse shaper is

folded by the use of a reflective annular mask; temporal and spatial Fourier transforms are



made sequentially. The primary goal of our experiments is to demonstrate a pulse shaping

technique allowing to generate the proper field distribution at the waist of an STBG beam.

Fig. 9. Experimental setup based on a pulse shaper with a folded diffraction grating. The figure

follows the path of a short laser pulse through the pulse shaper. The reflective mask selects the

optical frequencies that are recombined on the diffraction grating. Cylindrical lens #2 produces

the spatial Fourier transform of the mask. See also (Media 3).

Figure 9 shows the path followed by an short laser pulse in the pulse shaper. First the

optical frequencies of the laser pulse are spatially dispersed by the grating. After being

collimated by cylindrical lens #1 of focal length f1 = 100 mm, the spectrally dispersed beam is

reflected by the circular mask that selects the proper optical frequencies along the horizontal

axis. The beam is then focused by lens #1 on the grating; this set of operations generates the

inverse Fourier transform in the time domain of the signal reflected by the mask. The beam

propagates up to cylindrical lens #2 (f2 = 500 mm) which does the inverse Fourier transform

in the spatial domain (i.e. along the vertical axis) of the signal reflected by the mask. The

resulting STBG beam is obtained at the Fourier plane of lens #2. The focal length 2

f of the

lens that generates the spatial Fourier transform must be matched to the half-width HW

ω∆ of

the spectrum using Eq. (27):

2

0 2

,mask mask

HW

d df

a

π π

λ λ ω β β= =

∆ − (31)

where dmask is the diameter of the reflective mask.

The pulse shaper, based on a folded geometry, allows to control independently the

dispersion and the diffraction acting on the propagated wave packet. Its global dispersion is

set to zero by properly adjusting the distance between lens # 1 and the grating. Parameters

have been chosen such that diffraction from the grating up to the observation plane has little

effect on the structure of the packet along the T-axis.

7. Experimental results

We have characterized the STBG beam profile by monitoring it independently along the space

and time axes. A homemade autocorrelator has been used to acquire its temporal profile,

integrated along the transverse axis; its temporally integrated spatial profile has been acquired

with a CCD camera placed at the beam waist. Furthermore, the spatiotemporal intensity



distribution of the STBG beam has been analyzed in terms of its spatially resolved spectrum

of optical frequencies, in view of the fact that its temporal (spectral) profile is different at

every position x. By extracting this information, it is possible to compute its spatiotemporal

intensity distribution by means of an inverse Fourier transform.

Figure 10 shows an autocorrelation trace obtained with an STBG beam produced with

pulses having an 18-nm bandwidth. One important element to mention is that the two-photon

photodiode employed in the autocorrelator integrates over the entire lateral x-axis, meaning

that all the annuli of the spatiotemporal Bessel-Gauss profile are spatially integrated. The

main consequence of this operation is that the autocorrelation signal has no zero. A very good

agreement is found between the theoretical trace and the experimental measurements, as

shown in Fig. 10. The theoretical trace was calculated from the exact expression of the

spatially and temporally dependent autocorrelation of an STBG beam, followed by an

integration along the spatial axis for all values of the temporal coordinate.

Fig. 10. Autocorrelation trace of the central part of an STBG beam generated with pulses

having an 18-nm spectral bandwidth.

We have also recorded the spatial profile of the STBG beam by placing a CCD camera at

its waist. This measurement produces a time-integrated spatial profile along the x and y axes

(see Fig. 11(a)). The two-dimensional picture shows that the beam profile has no structure

along the y-axis (except for a Gaussian envelope), verifying our assumption of a planar

geometry along the x-axis. Here again, the measurement does not lead to a Bessel-Gauss

intensity distribution due to the temporal integration over a sequence of short pulses whose

structure varies with position x; furthermore the theoretical intensity distribution does not fall

to zero at any point along the x-axis, as shown in Fig. 11b. The experimental spatial profile

shown in Fig. 11a is in very good agreement with the theoretical prediction. The intensity

distribution at the center of the beam, shown in Fig. 11c, reveals that the minima and the

peaks of the intensity distribution are located very closely to the corresponding positions

expected from theory.

We have also investigated the spatially resolved spectrum of the STBG beam by placing a

diffraction grating in the plane of its waist. The spatially resolved spectrum was recorded on a

CCD camera. A typical result is shown in Fig. 12. According to theory, the spectrum of

optical frequencies of the STBG beam evolves along the x-axis; this is a consequence of the

nature of the STBG beam which, at each position x, is made of a specific sequence of pulses

in the time domain (see Figs. 3 and 4). The presence of the Gaussian envelope modifies the

spatially resolved spectrum found for STB beams; the comparison of Figs. 7 and 12(a) shows

that the Gaussian envelope softens the sharp discontinuities at the spectral boundaries and

limits the distribution along the x-axis. The experimental spatially resolved spectrum,

presented in Fig. 12(b), is in satisfactory agreement with the theoretical prediction shown in

Fig. 12(a).



Fig. 11. - a) Experimental and b) theoretical spatial profiles of an STBG beam. c) Comparison

between the theoretical and experimental intensity distributions along X (at Y = 0).

Fig. 12. a) Theoretical and b) experimental spatially resolved spectra of an STBG beam.

Figure 13 shows the spatiotemporal profile of the STBG beam retrieved from the

measured spatially resolved spectrum and the assumption that it is unchirped. This assumption

is justified by the fact that we used transform-limited pulses and a zero-dispersion pulse

shaper. The obtained spatiotemporal distribution exhibits the characteristic circular profile

expected from an STBG beam. Some distortion along the concentric annuli can be observed,

caused by excess intensity along the space and time axes, and possibly some spatiotemporal



astigmatism. Based on the spectral bandwidth found in Fig. 12(b), the central lobe of the

STBG beam would last approximately 70 fs (at full width at half maximum).

Fig. 13. Experimental spatiotemporal profile of an STBG beam, retrieved from the spatially

resolved spectrum of Fig. 12(b).

8. Conclusion

We have described the properties of quasi-invariant, two-dimensional (x and t) optical wave

packets propagating along the z-axis in materials with anomalous dispersion. In contrast to

solitons, these wave packets propagate in linear optical media. The functional expression of

these wave packets involves a Bessel function of first kind of order m and a Gaussian

envelope that limits their size. In the limit case of a Gaussian envelope with infinite size, full

invariance during propagation would take place; however such beams would have infinite

extent and, in practice, it would be impossible to generate them. Quasi-invariant propagation

results from the mutual compensation of diffraction and anomalous dispersion when beam

parameters are properly selected. We have analyzed the spatially resolved frequency spectrum

of spatiotemporal Bessel beams. This analysis has revealed that these beams are produced by

a superposition of plane waves of different optical frequencies and directions of propagation

traveling at the same group velocity along the z-axis; the plane waves produce an interference

pattern that takes the shape of a space-time Bessel function. The Gaussian envelope used to

limit the size of a spatiotemporal Bessel beam induces a variation of the beam profile during

propagation; however this variation is much slower than that of a Gaussian wave packet

matched to the central lobe of the STBG beam. The intensity of the central lobe of an STBG

beam fades gradually to zero during propagation; the beam energy is redistributed from the

Gaussian envelope to an outer ring whose radius grows linearly with propagation distance. In

the far field, the beam profile takes the form of a diverging spatiotemporal donut.

The equations describing the propagation of STB and STBG beams involve only the x-

and z-axes, thus assuming a constant field along the y-axis. In the case of propagation in a

bulk dielectric with anomalous dispersion, these wave packets would then have a constant

field along the y-axis; this would produce concentric cylinders of light propagating along the

z-axis. STB and STBG beams could also be produced in guided-wave optics; they represent

field distributions suitable for quasi-invariant propagation in planar waveguides with

anomalous dispersion. In both cases, it would be interesting to investigate the effect on optical

nonlinearities on their propagation and, in particular, their resistance to self-focusing and

filamentation.



STBG beams have been synthesized experimentally with a femtosecond laser and a pulse

shaper designed to provide their spatiotemporal frequency spectrum. The pulse shaper is

effectively emulating a spatiotemporal axicon. It allows to match the spatial and temporal

sizes of an STBG beam to the bandwidth of the pulse spectrum. The experimental

measurement of the spatially-resolved spectrum of the so-produced wave packets has led to

the spatiotemporal reconstruction of their intensity distribution. Future avenues being

considered experimentally include the generation of STBG beams with an orbital angular

momentum (i.e. 0m ≠ ) and the shaping of three-dimensional O-waves. To verify the

propagation invariance of STBG beams, we are considering two strategies: propagation at 800

nm in a dispersion line made of diffraction gratings providing anomalous dispersion, or direct

propagation in a dielectric with anomalous dispersion, such as silica glass at 1550 nm.

Acknowledgements

This work was supported by grants from Natural Sciences and Engineering Research Council

of Canada (NSERC), Fonds québécois de la recherche sur la nature et les technologies

(FQRNT) and Institut canadien pour les innovations en photonique (ICIP/CIPI).



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Spatiotemporal bessel beams:theory and experiments