Full characterisation of Airy beams under physical principles

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by  J Rogel-Salazar

Full characterisation of Airy beams under physical principles
J Rogel-Salazar∗
Applied Mathematics and Quantitative Analysis Group,
Science and Technology Research Institute, School of Physics Astronomy and Mathematics,
University of Hertfordshire, Hatfield, AL10 9AB, U.K.†
H A Jim´enez-Romero
Escuela Superior de F´ısica y Matem´aticas, Instituto Polit´ecnico Nacional,
Edificio 9, U. P. Adolfo Lo´pez Mateos, M´exico, D. F. 07300, M´exico
S Ch´avez-Cerda‡
Group of Theoretical Optics, Instituto Nacional de Astrof´ısica,
4 O´ptica y Electro´nica, Apartado Postal 51/216, 72000, Puebla, Puebla, M´exico
1
0 ThepropagationcharacteristicsofAirybeamsisinvestigatedandfullydescribedunderthetrav-
2 elingwavesapproachanalogoustothatusedfornon-diffractingBesselbeams. Thisispossiblewhen
noticingthatAiryfunctionsareinfactBesselfunctionsoffractionalorder 1. Weshowhowphysical
n 3
principles impose restrictions such that the non-diffracting Airy beams cannot be of infinite extent
a
as has been argued, and introduce for the first time quantitative expressions for the maximum
J
transverse and longitudinal extent of Airy beams. We show that under the appropriate physical
1
conditions it is possible to obtain higher-order Airy beams.
2

]
s Bessel beams are known to belong to a class of optical travellingHankelwavedescription. Inthisworkweshow
c beams described by the (2+1)-dimensional Helmholtz thatAirybeamshavesimilarpropertiestothoseofBessel
i
t equation, and they have the property of being non- beams due to the fact that the former are the result of
p
diffracting [1, 2] and self-healing [3, 4] on propagation. the superposition of counter-propagating Hankel travel-
o
Originally, Bessel beam properties have been studied ling waves of fractional order. We also demonstrate that
.
s using the diffraction Rayleigh-Sommerfeld or Fresnel- theproposedwaveapproachimposestheconditionunder
c
i Kirchhoff integrals. Implicitly, this approach considers physical principles for these beams to be of finite extent
s
theBesselbeamtobeofinfinitetransverseextent[1]. Al- contrary to the “ideal” infinite Airy beam.
y
h ternatively, a more straightforward and physically sound The aim of this paper is to provide a better under-
p description of these beams can be done in terms of trav-
standingofthefundamentalnatureofthetravellingwave
[ elling Hankel waves [2]. This description based on the
methodology applied to Airy beams. In particular we
differential Helmholtz wave equation provides a better
1 show that all the known propagation characteristics of
understanding of the physical origin of these beams and
v Airy beams are straightforwardly and intuitively under-
5 their intriguing properties. Even further, this same ap- stoodusingtheHankeltravellingwaveapproachandthat
2 proachallowedtodemonstratetheexistenceofotherfam-
the focusing features of (1+1)-dimensional Airy beams
2 ilies of non-diffracting beams described by fundamental
can only be described in clear and simple terms with
5 solutions of the separable (2+1) Helmholtz wave equa-
. this methodology, which is similar to that for (2+1)-
1 tion. These families of beams show propagation charac-
dimensional Bessel beams. In Section I we provide a
0 teristics analogous to the Bessel beams [5, 6].
briefaccountofthetravellingwavedescriptionforBessel
4 In recent years, another kind of non-diffracting beam
beamsasawaytomotivateitsapplicationtoAirybeams.
1
hasbeenreportedbutitspropagationisgovernedbythe
: Section II explains the relationship between Airy and
v paraxial wave equation in (1 + 1) dimensions, the so-
Bessel functions and in Section III we show the charac-
i called Airy beams [7]. These beams do not belong to
X teriseofthefiniteAirybeaminadirectmanner. Finally
the same class as those of the Helmholtz equation and
in Section IV we show that there can exist higher order
r
thus it might be expected that their physical properties
a Airy beams by discussing the physical conditions that
are not actually described in the same terms. Nonethe-
enable this possibility.
less, an interesting fact that is hardly discussed in all
the published literature on Airy beams is that the Airy
functionsareBesselfunctionsoffractionalorderequalto
1/3. This allows the application of the aforementioned
I. TRAVELLING HANKEL WAVES AND
BESSEL BEAMS
∗ [email protected]
† Blackett Laboratory, Department of Physics, Imperial College
London,PrinceConsortRoad,London,SW72BZ,UK Light propagation in linear media is described with
‡ [email protected] the use of the scalar wave equation for the electric field
2
E(r,t) given by both of them interfere leaving only the Bessel function:
∇2E(r,t)= 1 ∂E(r,t). (1) Ein(r,ϕ,z,t)+Eout(r,ϕ,z,t)=
v2 ∂t2 2J (k r)exp(imϕ+ik z−iωt). (6)
m r z
In cylindrical coordinates r = (r,z), and v corresponds Recalling the Sommerfeld radiation condition, Equation
to the speed of light in the medium in question. It is (3), the incoming wave must be generated at a finite dis-
possible to solve Equation (1) by separation of variables tanceimplyingthatBesselbeamsmusthavefinitetrans-
[8] ending up with ordinary differential equations for the verse extent and thus a finite propagation distance. The
variables r, z and t. The separation constants can be interference of both Hankel waves only occurs within a
chosen such that ω2/v2 = k2 + k2 ≡ |k|2 so that we conic region, and it is only within this region that the
r z
can think of k as the wavevector. In this manner, the Bessel beam can be formed. This region is therefore
equation for the radial coordinate from Equation (1) is called the region of existence of the Bessel beam which
given by: canbeexploitedinapplicationssuchasthedesignoflaser
resonatorswhoseoutputisrelatedtoaBesselbeam[10].
d2H 1dH (cid:18) m2(cid:19)
+ + k2− H =0, (2) As we have mentioned before, Bessel beams are non-
dr2 r dr r r2
diffractive, i.e. they propagate without spreading or
changing their shape within their region of existence.
which is the Bessel differential equation of order m [9]
Similarly, they show the property of self-healing that oc-
and its solutions are given by the m−th order Bessel
curs when the beam is partially blocked, i.e. it recon-
function, J (k r) and the m−th order Neumann func-
m r structs itself after some distance [4]. These properties
tion, N (k r). In many cases the Neumann functions
m r can straightforwardly be understood in terms of travel-
mentioned above are discarded due to the singularity
ling Hankel waves that provides a clear frame for the
they present at the origin. However, it has been proved
physics behind of them and others like the evolution of
by one of the authors [2] that these functions do indeed
focused Bessel beams [11].
carry physical meaning. Also, the solutions J and N
m m
cannot be used separately to describe the propagation of
light, because they do not satisfy the Sommerfeld radia-
tion condition independently. For cylindrical waves this
condition reads,
II. TRAVELLING WAVES APPROACH TO
(cid:18) (cid:19) AIRY BEAMS
dH
lim r1/2 −ik H =0, (3)
r→∞ dr r
Berry and Balazs introduced the idea of the “non-
and it tells us that a wave equation cannot have waves spreading Airy wave-packet” by solving a force-free
coming from an infinite distance. Thus, the solution Schr¨odingerequation[12]. Thesolutionpropagateswith-
needed to describe propagating waves must be given by out change along a parabolic trajectory and thus show
the complex superposition of the Jm and Nm functions. acceleration. The relationship between the force-free
This leads to the so-called Hankel waves [2] Schr¨odinger equation and the paraxial wave equation in-
dicatesthatnon-diffractiveAirybeamsarethereforepos-
H(1)(k r)=J (k r)+iN (k r), (4)
m r m r m r sible and indeed they have been observed [7]. Further-
H(2)(k r)=J (k r)−iN (k r). (5) more, ithasbeenreportedthat, similartoBesselbeams,
m r m r m r
Airy beams also have the property of self-healing [13].
Once the azimuthal and longitudinal wave components
Airy beams can be obtained when considering the
are incorporated, exp(imϕ+ik z), the wavefronts of the
z
propagationofaplane-polarisedbeaminalinearmedium
solutions for the Helmholtz wave equation in cylindrical
described by the normalised paraxial wave equation
coordinatesareconichelicoids. H(1)andH(2)arerelated
m m
to outgoing (convex) and incoming (concave) conic solu- ∂U 1∂2U
tions respectively. The Hankel waves, in combination −i + =0, (7)
∂ξ 2 ∂s2
with the temporal part, describe cylindrical wavefronts
that collapse and are generated at the longitudinal z- where U(s,ξ) is the electric field envelope that depends
axis. This is the physical origin of the singularity of the on the normalised coordinates s = x/x and ξ = z/kx2,
0 0
Hankel functions, the longitudinal axis is simultaneously with x being a given transverse scale and k is the
0
sink and source for the incoming and outgoing cylindri- wavenumber. SinceweknowthattheAirybeamremains
cal wave components, respectively. In that sense, since invariantwhilepropagatingalongacurvedtrajectorywe
incomingwavesbecomeoutgoing,thereisaregionwhere candefineanacceleratingvariableΓ=s−aξ2+vξ with
4
aandv beingrealconstants. Wecannowwritetheelec-
tricfieldenvelopeasfollowsU(s,ξ)=w(Γ)exp(iθ(Γ,ξ)),
whichleadstothefollowingordinarydifferentialequation
3
for w(Γ) concentrate on the behaviour of Ai(s) in the rest of this
discussion. The Airy differential equation is defined in
A2 the entire real space and thus, the second term in Equa-
w(cid:48)(cid:48)−αΓw = , (8)
w3 tion (13) can have either a positive or a negative sign
depending on whether the value of s is positive or nega-
and θ(Γ,ξ) is given by
tive. Using either of the two signs yields the same Airy
solutionwiththeonlydifferencethatonewillbethemir-
(cid:90) Γ dΓ(cid:48) (cid:16)a (cid:17) a2 av v
θ(Γ,ξ)=A + ξ−v + ξ3− ξ2+ ξ, (9) rorreflectionoftheotherwithrespecttotheverticalaxis,
0 w2 2 24 4 2 s=0.
where A = A(ξ) is assumed to be a constant. We can
thinkoftheparametersaandvaboveastheacceleration Asimplecalculationshowsthatbymakingthechange
√
and velocity of the beam, respectively. Without loss of of variable w = sZ (cid:0)2s3/2(cid:1), Equation (13) can be
1/3 3
generality, let us consider the case where a = 1, v = 0 transformedintotheBesseldifferentialequationoforder
and A=0; Equation (8) becomes: 1/3, with Z being the cylindrical Bessel function of
1/3
order 1/3 [15, 16]. In a similar way, the Airy functions
(cid:18) (cid:19)
1
w(cid:48)(cid:48)− s− ξ2 w =0, (10) canbeexpressedintermsofmodifiedBesselfunctionsof
4 order 1/3, K , as follows [14]:
1/3
which is the well-known Airy differential equation [14] 1(cid:114)s (cid:18)2 (cid:19)
w(s)= K s3/2 , (14)
and therefore the solution can be expressed as: π 3 1/3 3
U(s,ξ)=Ai(cid:18)s− ξ2(cid:19)exp(cid:20)i(cid:18)sξ − ξ3(cid:19)(cid:21). (11) w(s)= 1(cid:114)se2πi/3H(1) (cid:18)2is3/2(cid:19). (15)
4 2 12 2 3 1/3 3
We can distinguish two important cases; let us consider
where Ai(·) is the Airy function. The expression above
the argument of these Bessel functions, K to be χ =
implies that the intensity of the beam has the profile of 1/3
2|s|3/2. On the one hand, when s ≥ 0, the profile has
the modulus squared of the Airy function, i.e. 3
a monotonic decreasing behaviour and is proportional to
(cid:12)(cid:12) (cid:18) ξ2(cid:19)(cid:12)(cid:12)2 the modified Bessel function K1/3:
I(s,ξ)=(cid:12)Ai s− (cid:12) . (12)
(cid:12) 4 (cid:12) (cid:114)
1 s
Ai(s)= K (χ). (16)
The argument of the Airy function in Equation (11) π 3 1/3
shows that the Airy beam follows a parabolic trajectory
On the other hand, when s < 0, the profile can be con-
that can be interpreted as having a transverse accelera-
sidered as the superposition of two waves which are es-
tion[12]. Itisalsoclearthatitdoesnotchangeneitherits
sentially Hankel functions of order 1/3, i.e.
profile nor its amplitude on propagation when observed
along this parabolic trajectory. In this sense the Airy (cid:114)
1 s (cid:104) (cid:105)
beam can be regarded as being a non-diffracting beam, Ai(−s)= ei6πH(1)(χ)+e−6iπH(2)(χ) . (17)
as it is the case for the Bessel beam discussed in Sec- 2 3 1/3 1/3
tion I. It may seem that although both Airy and Bessel
We can now define the following functions:
beams are non-diffracting, the physics are not described
in a similar way given that they do not belong to the 1(cid:114)s
same class of beams. However, in what follows we will AiH(1)(s)= ei6πH(1)(χ), (18)
2 3 1/3
show that Airy beams can be better understood when
(cid:114)
1 s
described by the travelling waves approach used in the AiH(2)(s)= e−6iπH(2)(χ). (19)
treatment of Bessel beams outlined in Section I. 2 3 1/3
In order to elucidate the properties and behaviour of With these definitions, we can now write the Airy func-
Airy beams, let us recall Airy’s differential equation, tion in a more compact form as follows:
namely
Ai(−s)=AiH(1)(s)+AiH(2)(s). (20)
d2w
∓sw =0, (13) WeidentifythesefunctionsasthetravellingHankelcom-
ds2
ponents of the Airy beam, in analogy to the way it was
whosesolutionsaregivenbytheAiryfunctionsAi(s)and done for Bessel beams in Section I. We note that it is
Bi(s) [14], where we are using normalised coordinates. possible to obtain an analytical form of the phase of
The Bi(s) function is defined as the solution with the these Hankel components by considering the asymptotic
same amplitude of oscillation as Ai(s) as s goes to minus expansion of Hankel functions, which approximate very
infinity and differs in phase by π/2. We will therefore welltotheoriginalfunctionfromthefirstmaximum[14].
4
Using this expansion we have that agation features of the AiH beams.
(cid:114) s
AiH(1)(s)(cid:39) eiφ, (21)
6πχ
(cid:114) s
AiH(2)(s)(cid:39) e−iφ, (22)
6πχ
withφ=χ−π,andtheirwavefrontbeingtheoppositeof
4
eachother. InFigure1,wecanseethewavefrontsofeach
of the components AiH(1)(s) and AiH(2)(s). Each phase
FIG. 2. (Color online) Behaviour of the independent propa-
gation of the AiH(1), a), and of the AiH(2), b). component
of the Airy beam in arbitrary dimensionless units.
We can now easily understand the self-healing prop-
erty of the Airy beam: since the Airy beam is formed by
the superposition of the fractional Hankel components
AiH(1)(s) and AiH(2)(s), the self-healing region arises
from the recombination of these two components after
FIG.1. (Coloronline)SchematicwavefrontsoftheAiH(1)(s)
the shadows of the obstructed waves as seen in Figure 3.
and AiH(2)(s) functions (arbitrary units).
determines a geometric wavefront, and thus according
to geometrical optics, the rays that make up each beam
will propagate perpendicularly to this front [17]. In fact,
when the Airy beam is cut, it will propagate along tra-
jectories determined by the rays of the AiH(2)(s) beam.
We note that these waves must satisfy the Sommerfeld
radiation condition that imposes the restriction of Airy
beamsbeingoffiniteextentand,similartoBesselbeams,
they can only exist within a finite region of space.
In order to show that it is in fact the AiH(2) com-
ponent the one that bears the property of the parabolic
caustics we carried out the propagation of each compo-
nent independently. In Figure 2a we see the propagation
of the AiH(1) component that simply travels away from
the propagation axis diminishing its amplitude. In Fig-
ure 2b the parabolic caustic associated to the normal
rays to its concave wavefront can indeed be appreciated.
We note that the caustic determined by the rays defines FIG. 3. (Color online) Propagation of an obstructed Airy
the parabolic motion of the main maximum of the beam beam (in arbitrary dimensionless units) showing self-healing.
under consideration [12]. This behaviour is similar to Notice the presence of two shadows. The red dot at the edge
that produced by a third order aberration (coma). The ofthefigureindicatesthepositionwheretheobstructionhas
numerical simulations shown in this paper use a compu- been located.
tational window such that the dimensionless parameters
ares∈[−30,30]andξ ∈[0,5],andwehaveusedasuper- In regards to focusing, we note that the Airy beam
Gaussian window t(x) = exp−(s/t )50, t the width of shows the non-common behaviour of presenting two dif-
0 0
the window, to reduce diffraction effects introduced if a ferent focusing regions, one for AiH(2)(s) in which focus-
hard aperture was used instead and emphasize the prop- ing is observed and a second for AiH(1)(s), farther away,
5
where a defocusing beam appears. This behaviour can can indeed go further and assume that it is possible to
be seen in Figure 4, where we have marked the two re- make use of a packet that is either only the real or only
gions as I and II. This behaviour is easily explained by the imaginary part of one of these two components, say
noting that the composing Hankel waves, besides having AiH(1) for example.
opposite travelling directions, have opposite wave front Finally, we can now take a more general view of the
curvatures: one is positive focusing and the other is neg- behaviour of an Airy-Gauss beam when it is off-Axis.
ativedefocusing. Whenpassingthroughthepositivelens In Figure 6 we show the propagation of the Airy-Gauss
these add or subtract accordingly. beam, where it is clear that each of the Hankel compo-
nentsprovidesthetwocontrastingbehavioursoffocusing
and defocusing mentioned before.
FIG.4. (Coloronline)BehaviourofafocussedAirybeam(in
arbitrarydimensionlessunits),observethetworegionsdueto
the focusing of the two Hankel travelling waves. In the edge
FIG. 6. (Color online) Propagation of off-axis Airy-Gauss
of the figure we have indicated the profile of the Airy beam
beam using arbitrary dimensionless units
as well as the two wavefronts of the beam.
Aswehaveseen,theapplicationofthetravellingHan-
Asimplegeometricalanalysisunderthisconsideration
kelwaveapproachhasprovideduswithastraightforward
easily explains the two observed regions. In Figure 5 we
descriptionofthepropagationcharacteristicsoftheAiry
show a schematic of the optical arrangements including
beam under different circumstances. We will now show
a lens; the wavefronts have been marked in two differ-
how to determine the propagation distance of finite en-
ent colours. Notice how each region is generated by the
ergy Airy beams.
focusing of the rays coming from each of the two wave-
fronts, i.e. from each of the Hankel waves that compose
theAirybeam. Wecanthinkofeachregiontobeformed
III. FINITE AIRY BEAMS
Itisusuallyarguedthatnon-diffractingbeams,suchas
the Airy beam, require an infinite amount of energy for
their generation. This might be to give an explanation
for the non-diffraction feature or because the range of
the mathematical function that describes the profile is
infinite. However, in this Section we show that from a
physical perspective this cannot be the case as the beam
must be constrained by physical principles. One of them
wasmentionedabovefortheBesselbeamsthatalsoapply
for the Airy beams and this is the Sommerfeld radiation
FIG. 5. Schematic ray tracing of the double focusing of an condition. Within the travelling wave approach, to have
Airy beam related to that shown in Figure 4. an infinite Airy beam would require having sources at
infinity.
due to the simultaneous incidence of each Hankel com- For Airy beams, whose profile exhibits reduction of
ponent of the Airy beam. We would like to emphasise the separation distance between intensity peaks, there
thatthisisthecaseregardlessofanyinterferencepattern existsanothermorebasicphysicalconstraintthatwedis-
causedbythefunctionsAiH(1) andAiH(2). However,we cuss now. The distance between two consecutive peaks
6
in the transverse Airy intensity profile should not be- wherelisanon-negativeintegerthatprovidesuswithin-
come smaller than the wavelength of the light used. If formationaboutthenumberofcyclesthathaveoccurred
that were the case we would end up with an unphysical for a particular value of l.
situation. This situation is analogous to the treatment,
In this way, the distance between two consecutive ex-
for instance, of wave excitations with a fractal boundary
tremes is thus given by:
[18, 19]; although mathematically the fractal structure
continues to infinitely small scales, in reality there are (cid:18) (cid:19)2 (cid:34)(cid:18) (cid:19)2 (cid:18) (cid:19)2(cid:35)
1 3π 3 3 3 1 3
physical constraints that avoid this situation. ∆s= 2l− − 2l+ . (26)
α 2 4 4
We know that the distance between two consecutive
peaks in the beam intensity cannot be smaller than a
transverse wavelength λ , i.e. ∆s ≥ λ /2. This argu-
c c
mentprovidesaphysicalcriterionforthecut-offpointof
a finite Airy beam after which it must dampen.
Todefinethecriticaltransversewavelengthλ wemust
c
also take into consideration that Airy beams propagate
within the paraxial regime as governed by Equation (7).
For this purpose we require to provide the maximum
angle allowed for a ray in the wavefront to be consid-
ered paraxial. Curiously enough, to date in the litera-
ture there is not an established quantitative criterion for
such angle and, as noticed by Beck several decades ago:
“The term paraxial rays is a relative one and to some
extent a matter of arbitrary choice.” It will depend on
the tolerance error that is planned to be accepted [20].
We will introduce the quantitative criterion of paraxi-
FIG. 7. (Color online) Schematic of the behavior of an Airy
−→ alityputforwardbyAgrawal,Siegmanandothers[21–23]
beam: When the wavevector k of the Airy beam, which is
where paraxial optical beams can be focused or diverge
normal to the wavefront φ, reaches the maximum paraxial
at angles up to a maximum critical value of θ = π/6.
value θ = π the oscillations in the profile must dampen in c
c 6 With this in mind, we consider a ray perpendicular to
ordertoremainphysical. Atthatpoint,thedistancebetween
thewavefrontofAiH(2)atthecut-offposition. Whenthe
twoextremepointsintheprofilemustbeoftheorderofhalf →−
the critical wavelength λ . wavevector k to this ray reaches the maximum paraxial
c →−
angle θ =π/6, recalling that |k|=2π/λ, a simple cal-
c
culation gives λ = 2λ. Thus, substituting ∆s = 2λ in
Next, we proceed to find a critical value of the beam c
Equation (26) and solving for l we can get the position
extentafterwhichitsprofilemustdampen,givingriseto c
s (l ) where the oscillations of the Airy profile must
a finite energy Airy beam. In order to tackle the issue, max c
dampen.
letustaketheasymptoticexpansionforlargearguments
of the Airy function given by the corresponding Bessel
functions of order 1/3:
(cid:18) (cid:19)
Ai(αs)∝cos 2(αs)32 − π , (23) A. Maximum propagation distance of finite Airy
3 4 beams
where we have taken the normalised coordinates and α
We now propose a method for determining the prop-
is a parameter that allows us to change the frequency in
agation distance of the finite Airy beam based on the
the Airy function. We need to find the distance between
observations made above. Consider the parabolic tra-
two consecutive extreme points of the equation above,
jectoryfollowedbythemainmaximumoftheAirybeam
seeFigure7. Thesepointsoccurwhenthefollowingcon-
andtakearaycomingfromthewavefrontattheopposite
ditions are met: For minima we have
extremeofthewindowandfindthepointofintersection.
(cid:18) (cid:19)2 (cid:18) (cid:19)2 Figure 8 shows this situation; in order to guide the eye
1 3π 3 3 3
smin = α 2 2l− 4 , (24) wehavemarkedtheregionofexistenceoftheAirybeam
given by the ray as well as the parabolic trajectory fol-
lowed by the beam (white dotted lines).
and for maxima:
We know that for an Airy beam propagating in a ho-
(cid:18) (cid:19)2 (cid:18) (cid:19)2
1 3π 3 1 3 mogeneous medium, the normalised solution is given by
s = 2l+ , (25)
max α 2 4 Equation (11) and thus it is clear that the trajectory of
7
IV. HIGHER-ORDER AIRY MODES IN GRIN
MEDIA
Up to now we have concentrated on the treatment of
the Airy beam propagating in homogeneous media as a
result of solving the normalised paraxial wave equation
given by expression (7) and ended up with a solution
given by the Airy function. However, the Airy functions
also appear in the description of the propagating modes
ininhomogeneousmedia,inparticularinonedimensional
linear gradient index media [24–26].
Itisknownthatinsol-gelmaterialsthegradient-index
profilecanbecontrolledbydiffusionandprecipitationof
dopants, including quadratic, quartic and higher gradi-
ent profiles [27–29]. This materials are used to construct
Wood lenses that are thin radial gradient index lenses
with plane parallel surfaces whose index of refraction is
FIG.8. (Coloronline)Themethodtodeterminethepropaga- of the form N = N +N r2 +N r4 +... With this
00 10 20
tiondistanceξ ofanAirybeamisgivenbytheintersection
max antecedent, it can be possible to create a material whose
of the parabolic trajectory followed by the main maximum
index of refraction be the one dimensional equivalent of
and a ray coming from the edge of the profile (in arbitrary
a Wood lens.
dimensionless units). We also show a diagram of the phases
of the Hankel components of the Airy beam. In this Section we investigate the propagating modes
for power law gradient index media and show that they
are what we have come to call higher-order Airy beams.
Let us start with the Helmholtz equation in (1+1)D
the main maximum satisfies the following equation:
∇2E+k2E =0, (32)
(cid:18)ξ(cid:19)2
s− =a (27)
2 1 and let the medium have a dispersion relation such
that the wave number can be expressed as k2(x) =
(cid:104) (cid:105)
where a1 =−1.0187297... is the first zero of the deriva- k02 + β −k0−xn+ β4 , where β is a parameter that
tive of the Airy function, that is to say, the position of
depends on the medium. We can now propose an
the first intensity maximum at the onset of propagation.
Ansatz such that the electric field is given by E(x,z) =
Furthermore, we know that light rays are normal to the (cid:104) (cid:16) (cid:17) (cid:105)
wavefrontandthereforeinthiscase, theequationforthe U(x,z)exp i k0− β2 z . In the paraxial approxima-
ray we are interested in can be expressed as tion we have that
ξ =s−01/2(s−s0), (28) 2i(cid:18)k0− β2(cid:19)∂∂Uz + ∂∂2xU2 −βxnU =0. (33)
where s is the point where the beam is truncated, in
0
other words, where the edge of the window is located. If we require that β = 2k0 the wave number is given by
Using Equation (27) and Equation (28) for the ray we k2(x)=−2k0xn and Equation (33) is simplified into
have that
∂2U
(cid:16)(cid:112) (cid:112) (cid:17) −βxnU =0, (34)
ξ =2 |s |± 2|s |−|a | , (29) ∂x2
± 0 0 1
(cid:112) (cid:112)
s =3s ±2 |s | 2|s |−|a |. (30) which corresponds to a generalised form of the Airy dif-
± 0 0 0 1
ferential equation. Making the transformation [16]
The solution we seek is the one with the negative sign
√
because it is the first intersection between the parabola √ (cid:18) 2 β (cid:19)
n+2
and the ray, i.e. the point where the propagation ends. U(x)= xZn+12 n+2x 2 , (35)
In this way, the propagation distance is given by:
substituting into Equation (34) and after some algebra,
(cid:16)(cid:112) (cid:112) (cid:17)
ξ =2 |s |− 2|s |−|a | . (31) yields to
max 0 0 1
(cid:18) (cid:19)
1 1
This expression defines the maximum propagation dis- Z(cid:48)(cid:48) (ζ)+ Z(cid:48) (ζ)+ 1− Z (ζ)=0
tance for a finite Airy beam. This is, to our knowl- n+12 ζ n+12 (n+2)2ζ2 n+12
edge,thefirsttimethatthepropagationdistanceforAiry (36)
√
beams is formally defined. where ζ = 2 βxn+22. Equation (36) is the Bessel differ-
n+2
8
ential equation whose solutions are the family of Bessel travellingHankelwavesoriginallyintroducedtodescribe
functions of order 1 . Besselbeams. ThisispossibleduetothefactthatBessel
n+2
Equation (34) has been studied in detail by Swanson and Airy functions are intimately related to each other,
and Headley [30] who defined its solutions as A (x) and with the latter being the Bessel functions of fractional
n
B (x). It is clear that when n = 1 the functions A (x) order equal to 1. We introduced the two Hankel compo-
n n 3
andB (x)becomethestandardAiryfunctionsAi(x)and nents of the Airy beam, namely AiH(1)(·) and AiH(2)(·),
n
Bi(x), respectively. An important remark is that they and showed that the later bears the parabolic caustic
found that the general behaviour of the solutions is dif- property of the beam. It was shown that the superposi-
ferentdependingontheparityofn. Thisissomethingto tion of these Hankel components fully explain in simple
be expected since the branches of the power law parabo- and straightforward terms propagation characteristics of
lae can have different signs depending on which side of the Airy beam, such as self-healing and double-focusing.
the origin they extend to. Also,thisapproachallowedtoestablishforthefirsttime
We can now refer to Equation (34) as the Airy differ- the needed expression to compute the maximum prop-
ential equation of order n. In other words, its solutions agation distance of finte energy Airy beams. We ad-
can be seen as higher order Airy functions, and similarly dressedphysicalconstraintsofwhyan“ideal”Airybeam
to the Airy functions, they can also be cast in terms of of infinite extent cannot exist and provided a quantita-
Bessel functions of fractional order. It should therefore tive method to obtain the maximum extent of an Airy
be clear that given the adequate conditions in a power beam of a given wavelength. And finally, by studying
law gradient index medium it is then possible to obtain the solution of the paraxial wave equation in power law
higher-order Airy beams. GRIN media we demonstrated the possibility of creating
We want to remark that the study presented in this higher-order Airy beams.
section can also be applied to investigate the homolo-
gousprobleminQuantumMechanicsfortheSchr¨odinger
equationofaparticleconfinedwithinaninfinitewalland
a power law potential V(x)=βxn, with β constant [31– ACKNOWLEDGMENTS
35]. For the particular case of the linear potential [36],
the standing Airy wave packet arises from considering a The authors would like to acknowledge to Prof. Roc´ıo
particle subject to a constant force, e.g. gravitational Jauregui Renaud for fruitful and motivating discussions
force, that when it takes the value of zero the particle andbybringingtoourattentionRef. [31]. Wewouldalso
wave packet is still described by the Airy function, but like to thank Prof. Demetrios Christodoulides his valu-
acceleratingawayfromtheinfinitewallasaconsequence able comments during the development of the present
of removing the stabilising force [26]. work, and to Prof. Duncan Moore for enlightening dis-
cussions on GRIN materials. Finally, we also acknowl-
edge support of INAOE, Mexico and the STRI, UK dur-
V. CONCLUSIONS ing the development of this project.
Inthispaperweintroducedthephysicalprinciplesthat
govern the existence and propagation of Airy beams.
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