helical mathieu and parabolic localized pulses

1Lttwpt[olidtabfoMbtbtt

fcob[aIHfzpi

isaks

J. Davila-Rodriguez and J. C. Gutiérrez-Vega Vol. 24, No. 11 /November 2007 /J. Opt. Soc. Am. A 3449

Helical Mathieu and parabolic localized pulses

Josue Davila-Rodriguez and Julio C. Gutiérrez-Vega*

Photonics and Mathematical Optics Group, Tecnológico de Monterrey, Monterrey, México 64849*Corresponding author: [email protected]

Received August 16, 2007; accepted August 31, 2007;posted September 7, 2007 (Doc. ID 86457); published October 10, 2007

Using suitable polychromatic superpositions of helical Mathieu and parabolic nondiffracting beams, we studyfor the first time, to the best of our knowledge, the higher-order helical Mathieu X waves and the traveling andstationary parabolic X waves. The mathematical and physical properties of these new kinds of localized pulsesare discussed. © 2007 Optical Society of America

OCIS codes: 260.1960, 350.5500, 050.1960.

tvsnms

2NWt

wte

uvlt

w=pTasbvbc

f

. INTRODUCTIONocalized pulses are broadband wave-packet solutions ofhe wave equation that have been widely studied sincehe introduction by Lu and Greenleaf of the so-called Xaves [1,2]. Over the last 15 years, the theory of localizedulses has been developed and experimentally verified inhe fields of acoustics [1,2], microwaves [3], and optics4–7] for a variety of extended spectral functions. On thether hand, nondiffracting beams are monochromatic so-utions of the wave equation that propagate indefinitelyn free space without changing their transverse intensityistribution [8,9]. Optical generation and characteriza-ion of nondiffracting beams are now well established,nd new beam structures and applications are activelyeing reported [10]. Four fundamental families of nondif-racting beams are known: plane waves in Cartesian co-rdinates, Bessel beams in circular coordinates [8,9],athieu beams in elliptic coordinates [11,12], and para-

olic beams in parabolic coordinates [13,14]. Here we usehe term “fundamental” to refer to a complete family ofeams that constitutes a basis for expanding any arbi-rary nondiffracting beam with the same transverse spa-ial frequency.

The connection between localized pulses and nondif-racting beams arises from the fact that the former can beonstructed with a suitable polychromatic superpositionf the latter [5,6]. Until now, plane waves and Besseleams have served as the standard basis to build X waves2,5,6]; however, this construction can certainly also be re-lized using Mathieu and parabolic nondiffracting beams.n this direction, in a recent paper, Dartora andernandez-Figueroa [15] reported for the first time, as

ar as we know, an X wave based on the superposition oferoth-order, even Mathieu beams, obtaining a localizedulse that lacked the typical rotational symmetry foundn Bessel-based X waves.

In this paper, we show the general method for obtain-ng localized pulses based on the polychromatic superpo-ition of fundamental families of nondiffracting beamsnd then introduce for the first time, to the best of ournowledge, helical Mathieu X waves and traveling andtationary parabolic X waves. The physical properties of

1084-7529/07/113449-7/$15.00 © 2

hese localized pulses are discussed. This approach pro-ides alternative insight into the physics of X waves andheds light on the strong connection of these waves to theondiffracting solutions used in optics. Finally, we re-ark that this work consolidates and extends previous

tudies on zeroth-order Mathieu localized pulses [15].

. LOCALIZED PULSES IN TERMS OFONDIFFRACTING BEAMSe begin by writing the wave equation in free space for

he time-varying scalar field U�rt ,z , t�,

��t2 +

�2

�z2 −1

c2

�2

�t2�U�rt,z,t� = 0, �1�

here rt= �x ,y�= �r cos � ,r sin �� denotes the position athe transverse plane, �t

2 is the transverse Laplacian op-rator, and c is the light speed.

Localized waves are defined through the condition ofniform propagation U�rt ,z , t�=U�rt ,z−vt�, where v is theelocity of propagation of the wave field. This conditioneads to the known Fourier representation U�kt ,kz ,�� ofhe X waves given by [5,6]

U�kt,kz,�� = G��,��kz −�

ccos ��kt −

�

csin �� ,

�2�

here � is the angular frequency, kt= �kx ,ky��kt cos � ,kt sin �� denotes the position at the transverselane in the k space, and � is the Dirac delta function.he wave vectors k of the X wave lie on a cone of topngle 2� that opens from the kz axis in k space. The con-tituent plane waves propagate at speed c along the coneut their interference pattern has a superluminal velocity=c / cos �. The complex amplitude function G�� ,�� is ar-itrary and determines the transverse shape and the lo-alization properties of the beam.

X waves in space are obtained by inverse Fourier trans-orming Eq. (2):

007 Optical Society of America

wftf

s

T(pattffwboGX

bt

w

i[

ascoGfi

abgasfX

3Mfwctmacma

ATs

wM�tdc

fi

wrttlt

wotidwbi

io

Eo

3450 J. Opt. Soc. Am. A/Vol. 24, No. 11 /November 2007 J. Davila-Rodriguez and J. C. Gutiérrez-Vega

U�rt,�� =�0

�

d� exp�i��

ccos ��

��−

d�G��,��exp�i��

csin ��r cos�� − �� ,

�3�

here ��z−vt is the longitudinal coordinate of a movingrame traveling at speed v along the z axis. Throughouthe paper, the fields U�rt ,z , t� are restricted to positiverequencies � only.

It is convenient to write the angular spectrum in theeparable form

G��,�� = A��f��. �4�

his separation is reasonable for the following reasons:a) writing G�� ,�� as a product only assumes a decou-ling between the spectral composition of the light sourcend the spatial structure of the macroscopic field; (b) inhe case when these functions are not decoupled, arbi-rary G�� ,�� can be built with a proper superposition ofunctions with the separable form, and (c) writing theunction as a product allows for a representation of Xaves in terms of fundamental families of nondiffractingeams, whose properties have been explored in several co-rdinate systems [8,11,13]. For example, the choice�� ,��=exp�im��f�� leads to the well-known mth-orderwaves based on Bessel beams Jm�tr�exp�im��.By defining the transverse and longitudinal wave num-

ers t�� /c�sin � and z�� /c�cos � and using Eq. (4),hen U�rt ,�� [Eq. (3)] takes the form

U�rt,�� =�0

�

f��rt,��exp�iz��d�, �5�

here

��rt,�� −

d�A��expit��r cos�� − ��, �6�

s a solution of the two-dimensional Helmholtz equation10]

�t2 + t��2� = 0, �7�

nd physically describes the transverse field of an idealcalar nondiffracting beam with transverse wavenumbert�� /c�sin �. The expressions in Eqs. (3) and (6) areompletely general in the sense that they do not dependn a particular coordinate system. Since the spectrum�� ,�� is arbitrary, an infinite number of transverse pro-les can be obtained.It is clear from Eq. (5) that an X wave is obtained frompolychromatic superposition of pure nondiffracting

eams with transverse and longitudinal wave numbersiven by t and z, respectively. The wave is stationary inmoving frame which travels along the z axis with con-

tant speed v=c / cos �. In Sections 3 and 4 we apply theormalism developed in this section to introduce localized

waves based on Mathieu and parabolic beams.

. HELICAL MATHIEU PULSESathieu beams constitute a complete and orthogonal

amily of nondiffracting beams that are solutions of theave equation in elliptic coordinates [11,12,16,17]. Re-

ently, these beams have been applied for example to pho-onic lattices [18] and the transfer of orbital angular mo-entum using optical tweezers [19,20]. Gaussian

podized Mathieu beams, which carry a finite power andan be generated experimentally to a very good approxi-ation, have been already reported in free space [21,22]

nd through ABCD optical systems [23,24].

. Spectral Integral Representationo construct X waves based on helical Mathieu beams weet the spatial part of the spectrum to be

Am± ��,�� = cem��,�� ± isem��,��, �8�

here cem�·� and sem�·� are the even and odd angularathieu functions of mth-order and ellipticity parameter

0. The positive and negative signs correspond to posi-ive and negative helicity of the spectrum and eventuallyetermine the direction of the orbital angular momentumarried by the wave field.

Replacement of Eq. (8) into Eq. (6) yields the transverseeld of the helical Mathieu beams, namely

�m± �rt,�� = CmJem��,��cem��,�� ± iSmJom��,��sem��,��,

�9�

here Jem�·� and Jom�·� are the mth-order even and oddadial Mathieu functions, and Cm and Sm are normaliza-ion factors to ensure that the even and odd parts carryhe same power. In Eq. (9), the radial �� 0, � � and angu-ar �� 0,2� elliptic coordinates are related to the Car-esian coordinates according to

x = h cosh � cos �, �10a�

y = h sinh � sin �, �10b�

here h=2�� /t=2��c /�� csc � is the semifocal distancef the elliptic coordinate system. The transverse distribu-ion of the helical Mathieu beams [Eq. (9)] is character-zed by a set of confocal elliptic rings whose eccentricity isetermined by the parameter �. [19] The special casehen �=0 corresponds to the known mth-order Besseleams Jm�tr�exp�±im�� for which the spectrum [Eq. (8)]s Am

± �� ,0�=cos m�± i sin m�.Substitution of Eq. (9) into Eq. (5) leads to the general

ntegral representation for helical Mathieu based X wavef mth-order

Um± �rt,�� =�

0

�

fm��

�CmJem��,��cem��,�� ± iSmJom��,��sem��,��

�exp�iz��d�. �11�

very nondiffracting X wave can be obtained from wavesf this form by suitably weighing and summing over m.

BWhu

S(rh

Ssmr

whntRbs

CXTssctweo

M

wei

wi

w

w

iB

DMIoo

Tt�rtEt�t[

iiss

Fpssf


. Orthogonality of Helical Mathieu X Wavese now turn to consider the orthogonality condition for

elical Mathieu X waves. At a fixed time, the scalar prod-ct of two waves is defined by

�Um Un� =�−�

�

Um* �rt,z,t�Un�rt,z,t�dxdydz. �12�

ubstituting the Fourier representation of the wave [Eq.3)] into the scalar product, integrating over a cylinder ofadius R, and substituting the angular spectrum of theelical Mathieu X waves [Eq. (8)], we obtain

�Um Un� =R tan �

22 �0

�

d��fm* ��fn��

��−

cem�� isem��cen�� ± isen��d�.

�13�

ince the angular Mathieu functions cem�� and sem��atisfy the same orthogonality relations as their trigono-etric counterparts cos � and sin � [25,26], then Eq. (13)

educes to

�Um Un� =R tan �

2�m,n�

0

�

d��fm* ��fn��, �14�

here �m,n is the Kronecker delta symbol. Note that twoelical Mathieu X waves with the same order are orthogo-al inside the volume enclosed by the cylinder R if the in-egral in Eq. (14) vanishes. The inner product diverges as→�, which accounts for the fact that ideal Mathieu

eams and X waves are not square integrable over allpace.

. Helical Mathieu X Waves in Terms of BesselWaves

he two-dimensional Helmholtz equation [Eq. (7)] can beolved in several orthogonal coordinate systems using theeparation of variables method [26]. This fact leads toomplete and orthogonal families of eigenfunctions of thewo-dimensional Helmholtz equation and eventually of Xaves. In this sense, helical Mathieu X waves can also bexpressed as a summation of the standard X waves basedn Bessel beams of the form

BBn±�r,�� = Jn�tr�exp�±in��. �15�

From [27] we know that monochromatic even and oddathieu beams admit of the Bessel expansions

Jem��,��cem��,�� = �n=0

�

An�m�Jn�tr�cos�n��, �16a�

Jom��,��sem��,�� = �n=0

�

Bn�m�Jn�tr�sin�n��, �16b�

here An�m� and Bn

�m� are the expansion coefficients whosexplicit expressions can be found in [27,28]. Replacingnto Eq. (9) and using the intermediate results

Jn�tr�cos�n�� = 12 �BBn

+ + BBn−�, �17�

Jn�tr�sin�n�� = 12i �BBn

+ − BBn−�, �18�

e obtain the transverse field of the helical Mathieu beamn terms of a summation of Bessel beams, namely

�m+ �rt,�� = �

n=0

�

Dn+BBn

+�r,�� + Dn−BBn

−�r,��, �19�

hereDn

± = 12 �CmAn

�m� ± SmBn�m��. �20�

Finally, substitution into Eq. (11) produces

Um+ �rt,�� = �

n=0

�

Dn+Xn

+�r,�,�� + Dn−Xn

−�r,�,��,

here

Xn±�r,�,�� =�

0

�

d�fm��BBn±�r,��exp�iz��, �21�

s the well-known integral expression for the nth-orderessel X wave [5,6].

. Three-Dimensional Distribution of the Helicalathieu X Waves

n this paper we study the three-dimensional distributionf the helical Mathieu X waves using a spectral functionf the general form

f�� = �p�� − �min��max − ��exp�− ��. �22�

he spectrum contains four parameters: (i) p and � con-rol the skewness of the distribution (see Fig. 1), and (ii)min and �max adjust the bandwidth of the function to theange �min,�max since the form of the polynomial forceshe end points to be zero. The spectral distribution inq. (22) can be visualized as a superposition of two spec-

ral functions of the standard form �polynomial in ��exp�−�� which is the usual form adopted for the spec-

ral dependency of the X waves based on Bessel beams5–7].

The amplitude of a fifth-order helical Mathieu X waven the planes y=0 and �= −2,−1,0,1,2 �m is illustratedn Fig. 2(a). The conical angle � is 15° and the frequencypectrum is given by Eq. (22) with the parameters corre-ponding to the function fI�� included in the caption of

ig. 1. (Color online) Spectral functions given by Eq. (22) with=1, �min=1.71�1015 rad/s and �max=5.54�1015 rad/s. For thepectrum fI �=2.13�10−15 m−1, and for fII �=−8.53�10−16 s. Thepectra are normalized such that the maximum of the spectralunction is unity.

Ftsi�pie[

dttpltXm

caflT[bMmfsed

sc

FMiv�qt

FMiv�qt


ig. 1. The three-dimensional field distribution was ob-ained by solving Eq. (11) at 201 transverse planes evenlypaced through the range � �2 �m. The wave is local-zed and remains invariant in a coordinate systemx ,y ,�=z−vt� that propagates in the z direction at a su-erluminal velocity v=c / cos �. The Mathieu X wave is ofnfinite transverse extent and it has a divergent total en-rgy, which are typical properties of the ideal X wave5–7].

In Fig. 2(b) we show the transverse intensity and phaseistributions of the wave at �=0. The pattern is charac-erized by a well-defined bright elliptic ring and a phasehat rotates following an elliptic trajectory. For m�1, thehase exhibits m in-line vortices, each with unitary topo-ogical charge such that the total charge (along a closedrajectory enclosing all the vortices) is m. Unlike Bessel

waves, the Mathieu X waves are not rotationally sym-etric with respect to the propagation axis.

ig. 2. (Color online) (a) Amplitude of a fifth-order helicalathieu X wave in the planes y=0 and �= −2,−1,0,1,2 �m us-

ng the spectral function fI�� shown in Fig. 1. (b) and (c) Trans-erse intensity and phase distributions of the wave at �=0 and=1 �m. (d) Intensity and phase of a Mathieu beam whose fre-uency � corresponds to the maximum value of the spectral func-ion fI.

The intensity and phase of the wave at �=1 �m are in-luded in Fig. 2(c). The spirallike structure of the phase isn indication that the X wave acquires a spherical wave-ront as the longitudinal coordinate increases. The topo-ogical charge remains constant at any transverse plane.o compare the transverse pattern of the Mathieu X waveFig. 2(b)] with respect to the monochromatic Mathieueam, in Fig. 2(d) we show the intensity and phase of aathieu beam whose frequency � corresponds to theaximum value of the spectral function fI in Fig. 1. While

or the monochromatic wave the energy is shared betweeneveral elliptic rings, for the polychromatic wave the en-rgy is much more concentrated in a single and well-efined elliptic ring.The spectral function fI�� used for the Mathieu X wave

hown in Fig. 2 is skewed to the lower frequencies. Foromparison purposes, in Fig. 3 we show the same fifth-

ig. 3. (Color online) (a) Amplitude of a fifth-order helicalathieu X wave in the planes y=0 and �= −2,−1,0,1,2 �m us-

ng the spectral function fII�� shown in Fig. 1. (b) and (c) Trans-erse intensity and phase distributions of the wave at �=0 and=1 �m. (d) Intensity and phase of a Mathieu beam whose fre-uency � corresponds to the maximum value of the spectral func-ion f .
II

oscrsia

4Wwfctft

ATs

w

atib

w+oec−tp

twtdsoippr

i

Ebg

BTtpcEf�Xv

boecs�grsqs

i�aTa pt

Ft


rder helical Mathieu X wave but now we are using thepectral function fII�� which is skewed to higher frequen-ies (see Fig. 1). The separation between consecutiveings is now smaller than in Fig. 2 because the transversepatial frequency t of each spectral Mathieu components proportional to �, and we now have more contributionst larger frequencies.

. PARABOLIC X WAVESe now turn to consider parabolic nondiffracting beams,hich constitute the fourth family of fundamental nondif-

racting beams [13,14] and can also serve as a basis toonstruct localized X waves. The transverse structure ofhe parabolic beams is described by the parabolic cylinderunctions, and, contrary to Bessel or Mathieu beams,heir eigenvalues are continuous instead of discrete.

. Spectral Integral Representationo construct X waves based on parabolic beams we set thepatial part of the spectrum to be

A±�� = Ae��;a� ± iAo��;a�, �23�

here

Ae��;a� =�

2 sin � exp�ia ln�tan

�

2�� , �24�

Ao��;a� =1

i �− Ae��;a�, � � �− ,0�

Ae��;a�, � � �0,�, �25�

nd a� �−� , � � is the continuous parabolicity parameterhat plays the role of an order. Replacement of Eq. (23)nto Eq. (6) yields the transverse field of the paraboliceams [13,14], namely

�m± �rt,�� = �m

e �rt,�� ± i�mo �rt,��, �26�

�m± �rt,�� = �1 2Pe��;a�Pe��;− a�

± i2 �3 2Po��;a�Po��;− a�, �27�

here ��2t�1/2, �1��1/4+ ia /2�, and �3��3/4ia /2�. In Eq. (27), Pe�v ;a� and Po�v ;a� are the even anddd real solutions of the parabolic cylinder differentialquation �d2 /dx2+x2 /4−a�P�x ;a�=0, and the parabolicylindrical coordinates rt= �� ,�� are defined as x= ��2

�2� /2, y=��, where �� 0, � � and �� −� , � �. The sta-ionary solutions �m

e and �mo are called the even and odd

arabolic beams, respectively.Figure 4 shows the transverse intensity and phase dis-

ributions for an even and a traveling parabolic beamith a=−1.618. For a�0, the transverse intensity pat-

ern of the parabolic beams consists of well-defined non-iffracting parabolic fringes with a dark parabolic regionurrounding the negative x axis [13]. For the even anddd parabolic beams the intensity and phase patterns arenvariant under propagation; however, for the travelingarabolic beams when observed at a fixed transverselane the phase travels along confocal parabolic trajecto-ies around the semiplane �x�0,z� for a�0.

Substitution of Eq. (27) into Eq. (5) leads to the generalntegral representation for parabolic X waves

Ua±�rt,�� = Ua

e �rt,�� ± iUao�rt,��, �28�

Ua±�rt,�� =�

0

�

fa�� 1 2Pe��;a�Pe��;− a�

± i2 �3 2Po��;a�Po��;− a�exp�iz��d�.

�29�

very nondiffracting X wave can be obtained from para-olic X waves of this form by suitably weighing and inte-rating over the continuous order a.

. Three-Dimensional Distribution of Parabolic X Waveso gain insight into parabolic X waves, in Fig. 5 we showhe amplitude of an even parabolic X wave Ua

e �rt ,�� in thelanes y=0 and �= −2,−1,0 �m for a=−1.618 and aonical angle �=15�. The frequency spectrum is given byq. (22) with the parameters corresponding to the

unction fII�� included in the caption of Fig. 1 with=−9.26�10−16 s. Like Mathieu X waves, the parabolicwaves are of infinite transverse extent and have a di-

ergent total energy.For the particular case shown in Fig. 5 the even para-

olic X wave comes from the polychromatic superpositionf real parabolic nondiffracting beams Ua

e �rt ,�� with anven parity about the x axis; therefore the beam does notarry orbital angular momentum and the pattern remainsymmetrical about the x axis for any longitudinal position. The phase structure of the wave shown in Fig. 5(d) sug-ests that the field at the plane �=0 reduces to a purelyeal function resembling the original even parabolic beamhown in Fig. 4(a). Outside the plane �=0, the wave ac-uires a spherical phase front leading to an intrincatetructure of vortices.

The three-dimensional distribution of a complex travel-ng parabolic X wave is depicted in Fig. 6 at the planes= −2,−1,0 �m. All the beam parameters are the sames for the corresponding even component shown in Fig. 5.he spatial distribution was obtained by solving Eq. (29)t 201 transverse planes evenly spaced through the range

� �2 �m. The propagating behavior of the travelingarabolic X waves is expected to be different from that ofhe stationary even and odd parabolic X waves. Because

ig. 4. (Color online) Intensity and phase transverse distribu-ion of an even and a traveling parabolic nondiffracting beam.

omtawat

bvXBtXm

5Wife

aptwcccc

ATntg

R

FXt

Fbs


f its complex phase distribution, transverse energy flowust occur along parabolic trajectories. Such behavior for

he transverse energy flow is clearly observed in the im-ge sequence shown in Fig. 6. Note that the energy flowsithin the parabolic nodal lines around the negative xxis, and that the field is symmetrical about the x axis athe plane �=0.

The results shown in Fig. 6 are of particular interestecause they clearly illustrate the behavior of the trans-erse energy flow occurring in the traveling parabolic

waves. Even though this effect also takes place inessel X waves and helical Mathieu X waves for which

heir nodal lines are closed, in the case of the parabolicwaves the nodal lines never close making this effect

ore significant.

. CONCLUSIONSe have shown the general method for constructing local-

zed pulses based on the polychromatic superposition ofundamental families of nondiffracting beams. We putmphasis on the helical Mathieu X waves of higher order

ig. 5. (Color online) Amplitude and phase of an even parabolicwave in the planes y=0 and �= −2,−1,0 �m using the spec-

ral function fII�� shown in Fig. 1.

nd the traveling and stationary parabolic X waves. Thearticular mathematical and physical properties of thesewo new localized waves were discussed and exemplifiedith some illustrative examples and plots. Several appli-

ations for the new Mathieu and parabolic X wave pulsesould be explored in high-speed optical processing andommunications, medical real-time imaging, optical mi-rolithography, and acoustic waves in crystals [29].

CKNOWLEDGMENTShis research was partially supported by Consejo Nacio-al de Ciencia y Tecnología of México grant 42808, and byhe Tecnológico de Monterrey Research Chair in Opticsrant CAT–007.

EFERENCES1. J. Lu and J. F. Greenleaf, “Ultrasonic nondiffracting

transducer for medical imaging,” IEEE Trans. Ultrason.Ferroelectr. Freq. Control 37, 438–447 (1990).

2. J. Lu and J. F. Greenleaf, “Nondiffracting X waves: exactsolutions to free-space scalar wave equation and their finite

ig. 6. (Color online) Amplitude and phase of a traveling para-olic X wave in the planes y=0 and �= −2,−1,0 �m using thepectral function fII�� shown in Fig. 1.

1

1

1

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

2

2


aperture realizations,” IEEE Trans. Ultrason. Ferroelectr.Freq. Control 39, 19–31 (1992).

3. E. Recami, M. Zamboni-Rached, K. Z. Nóbrega, C. A.Dartora, and H. E. Hernández-Figueroa, “On the localizedsuperluminal solutions to the Maxwell equations,” IEEE J.Sel. Top. Quantum Electron. 9, 59–73 (2003).

4. P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79,4135–4138 (1997).

5. J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan,and M. M. Salomaa, “Angular-spectrum representation ofnondiffracting X waves,” Phys. Rev. E 54, 4347–4352(1996).

6. J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa,“Unified description of nondiffracting X and Y waves,”Phys. Rev. E 62, 4261–4275 (2000).

7. J. Salo, A. T. Friberg, and M. M. Salomaa, “Orthogonal Xwaves,” J. Phys. A 34, 9319–9327 (2001).

8. J. Durnin, “Exact solutions for nondiffracting beams. I. Thescalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).

9. J. Durnin, J. J. Micely Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).

0. Z. Bouchal, “Nondiffracting optical beams: physicalproperties, experiments, and applications,” Czech. J. Phys.53, 537–578 (2003).

1. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields:Mathieu beams,” Opt. Lett. 25, 1493–1495 (2000).

2. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramírez,E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda,and G. H. C. New, “Experimental demonstration of opticalMathieu beams,” Opt. Commun. 195, 35–40 (2001).

3. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda,“Parabolic nondiffracting optical wave fields,” Opt. Lett. 29,44–46 (2004).

4. C. López-Mariscal, M. A. Bandres, S. Chávez-Cerda, and J.C. Gutiérrez-Vega, “Observation of parabolic nondiffractingoptical fields,” Opt. Express 13, 2364–2369 (2005).

5. C. A. Dartora and H. E. Hernández-Figueroa, “Propertiesof a localized Mathieu pulse,” J. Opt. Soc. Am. A 21,662–667 (2004).

6. C. A. Dartora, M. Zamboni-Racheda, K. Z. Nobrega, E.
Recami, and H. E. Hernández-Figueroa, “General
formulation for the analysis of scalar diffraction-free beamsusing angular modulation: Mathieu and Bessel beams,”Opt. Commun. 222, 75–80 (2003).

7. A. Chafiq, Z. Hricha, and A. Belafhal, “Paraxialpropagation of Mathieu beams through an aperturedABCD optical system,” Opt. Commun. 253, 223–230 (2005).

8. Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L.Torner, “Shaping soliton properties in Mathieu lattices,”Opt. Lett. 31, 238–240 (2006).

9. S. Chávez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J.C. Gutiérrez-Vega, A. T. O’Neil, I. MacVicar, and J.Courtial, “Holographic generation and orbital angularmomentum of high-order Mathieu beams,” J. Opt. B:Quantum Semiclassical Opt. 4, S52–S57 (2002).

0. C. López-Mariscal, J. C. Gutiérrez-Vega, G. Milne, and K.Dholakia, “Orbital angular momentum transfer in helicalMathieu beams,” Opt. Express 14, 4182–4187 (2006).

1. J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz–Gausswaves,” J. Opt. Soc. Am. A 22, 289–298 (2005).

2. C. López-Mariscal, M. A. Bandrés, and J. C. Gutiérrez-Vega, “Observation of the experimental propagationproperties of Helmholtz-Gauss beams,” Opt. Eng.(Bellingham) 45, 068001 (2006).

3. A. Chafiq, Z. Hricha, and A. Belafhal, “A detailed study ofMathieu–Gauss beams propagation through an aperturedABCD optical system,” Opt. Commun. 265, 594–602 (2006).

4. M. Guizar-Sicairos and J. C. Gutiérrez-Vega, “GeneralizedHelmholtz-Gauss beam and its transformation by paraxialoptical systems,” Opt. Lett. 31, 2912–2914 (2006).

5. J. C. Gutiérrez-Vega, R. M. Rodríguez-Dagnino, M. A.Meneses-Nava, and S. Chávez-Cerda, “Mathieu functions,a visual approach,” Am. J. Phys. 71, 233–242 (2003).

6. M. Abramowitz and I. Stegun, Handbook of MathematicalFunctions (Dover, 1964).

7. J. C. Gutiérrez-Vega and M. A. Bandres, “Normalization ofthe Mathieu–Gauss optical beams,” J. Opt. Soc. Am. A 24,215–220 (2007).

8. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals,Series, and Products (Academic, 2000).

9. J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa,“Nondiffracting bulk-acoustic X waves in crystals,” Phys.Rev. Lett. 83, 1171–1174 (1999).

helical mathieu and parabolic localized pulses

Documents