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44 OPTICS LETTERS / Vol. 29, No. 1 / January 1, 2004 Parabolic nondiffracting optical wave fields Miguel A. Bandres and Julio C. Gutiérrez-Vega Photonics and Mathematical Optics Group, Tecnológico de Monterrey, Monterrey, N.L. 64849, Mexico Sabino Chávez-Cerda Instituto Nacional de Astrofísica, Optica y Electrónica, Apartado Postal 51/216, Puebla 72000, Mexico Received June 5, 2003 We demonstrate the existence of parabolic beams that constitute the last member of the family of fundamental nondiffracting wave fields and determine their associated angular spectrum. Their transverse structure is described by parabolic cylinder functions, and contrary to Bessel or Mathieu beams their eigenvalue spectrum is continuous. Any nondiffracting beam can be constructed as a superposition of parabolic beams, since they form a complete orthogonal set of solutions of the Helmholtz equation. A novel class of traveling parabolic waves is also introduced for the first time. © 2004 Optical Society of America OCIS codes: 260.1960, 350.5500, 140.3300, 050.1960. The spatial evolution of propagation-invariant optical fields (PIOFs) has been a subject of interest, since un- der ideal conditions they propagate indefinitely with- out change of their transverse intensity distribution. PIOFs have an extensive range of applications in fields such as optical tweezers, metrology, microlithography, medical imaging and surgery, nonlinear optics, and wireless and optical communications, among others. 1,2 The three-dimensional Helmholtz equation (HE) is known to be separable in 11 orthogonal coordinate sys- tems. 3 Of them, only the rectangular (i.e., Cartesian), circular cylindrical, elliptic cylindrical, and parabolic cylindrical coordinates have translation symmetries and allow the HE to be separated into a transverse part and a longitudinal part. 3–5 This separability of the equation imposes the condition that the solutions of the transverse part are independent of the longitudinal coordinate. PIOFs are exact solutions of the HE and have been demonstrated theoretically and experimen- tally for the first three coordinate systems: They are plane waves in Cartesian coordinates, Bessel beams in circular cylindrical coordinates, 1 and Mathieu beams in elliptic coordinates. 6–8 However, the funda- mental family of PIOFs in parabolic coordinates had remained unexplored until now. We present the last class of PIOFs that are exact solutions of the HE. These solutions are expressed in terms of parabolic cylinder functions and have an inherent geometry imposed by the parabolic coordi- nates. By use of the group symmetries of the HE, a spectrum for these fields is found that, common to all PIOFs, lies on a ring. 9 The solutions describe satisfactorily transverse parabolic stationary waves, and with them a novel class of traveling waves is constructed. Fundamental PIOFs of the HE can be expressed in terms of a reduced Whittaker integral 10 U r exp2ik z z Z p 2p Aw 3 exp2ik t x cos w1 y sin wdw , (1) where Aw is the angular spectrum of field U r de- fined on a ring of radius k t in frequency space. The transverse and longitudinal wave-vector components satisfy the relation k 2 k 2 t 1 k 2 z , where k is the wave number. We assume a time dependence expivt for the optical fields. From Eq. (1) it is clear that the f ield intensity I jU j 2 is independent of propagation coor- dinate z. The known fundamental solutions of the HE that de- scribe PIOFs are fully determined after their angular spectrum is introduced in Eq. (1). For m plane waves we have Aw P m dw2w m , for mth-order Bessel beams Aw expimu, and for mth-order Mathieu beams Aw; q ce m w; q 1 i se m w; q, where ce and se are the angular Mathieu functions and q is the el- lipticity parameter. 6,8 To get the fundamental PIOF in parabolic cylinder coordinates the task is to find the corresponding spectrum. Finding PIOFs in parabolic cylinder coordinates re- quires one to solve the HE in this system. Parabolic cylinder coordinates are defined by the transforma- tion x 1 iy h1 ij 2 2 and z z with ranges in h [ 2`, `; j [ 0, `; z [ 2`, `. In these coordi- nates the three-dimensional HE separates into a lon- gitudinal part, which has a solution with dependence exp2ik z z, and a transverse part, which splits into d 2 Fh dh 2 1 k 2 t h 2 1 2k t aFh 0, (2) d 2 Rj dj 2 1 k 2 t j 2 2 2k t aRj 0, (3) where for convenience we have written the separation constant as 2k t a [ 2`, `. By a simple change of variables, sj ! v, where s 2k t 12 , Eqs. (2) and (3) can be transformed into the canonical form of the parabolic cylinder differential equation (PCDE), namely, d 2 P dv 2 1 v 2 4 2 aP 0. The solutions to the PCDE are found by standard methods (e.g., Frobenius), and its Taylor expansion with v 0 is 0146-9592/04/010044-03$15.00/0 © 2004 Optical Society of America

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44 OPTICS LETTERS / Vol. 29, No. 1 / January 1, 2004

Parabolic nondiffracting optical wave fields

Miguel A. Bandres and Julio C. Gutiérrez-Vega

Photonics and Mathematical Optics Group, Tecnológico de Monterrey, Monterrey, N.L. 64849, Mexico

Sabino Chávez-Cerda

Instituto Nacional de Astrofísica, Optica y Electrónica, Apartado Postal 51/216, Puebla 72000, Mexico

Received June 5, 2003

We demonstrate the existence of parabolic beams that constitute the last member of the family of fundamentalnondiffracting wave fields and determine their associated angular spectrum. Their transverse structure isdescribed by parabolic cylinder functions, and contrary to Bessel or Mathieu beams their eigenvalue spectrumis continuous. Any nondiffracting beam can be constructed as a superposition of parabolic beams, since theyform a complete orthogonal set of solutions of the Helmholtz equation. A novel class of traveling parabolicwaves is also introduced for the first time. © 2004 Optical Society of America

OCIS codes: 260.1960, 350.5500, 140.3300, 050.1960.

The spatial evolution of propagation-invariant opticalfields (PIOFs) has been a subject of interest, since un-der ideal conditions they propagate indef initely with-out change of their transverse intensity distribution.PIOFs have an extensive range of applications in f ieldssuch as optical tweezers, metrology, microlithography,medical imaging and surgery, nonlinear optics, andwireless and optical communications, among others.1,2

The three-dimensional Helmholtz equation (HE) isknown to be separable in 11 orthogonal coordinate sys-tems.3 Of them, only the rectangular (i.e., Cartesian),circular cylindrical, elliptic cylindrical, and paraboliccylindrical coordinates have translation symmetriesand allow the HE to be separated into a transverse partand a longitudinal part.3 – 5 This separability of theequation imposes the condition that the solutions of thetransverse part are independent of the longitudinalcoordinate. PIOFs are exact solutions of the HE andhave been demonstrated theoretically and experimen-tally for the first three coordinate systems: They areplane waves in Cartesian coordinates, Bessel beamsin circular cylindrical coordinates,1 and Mathieubeams in elliptic coordinates.6 – 8 However, the funda-mental family of PIOFs in parabolic coordinates hadremained unexplored until now.

We present the last class of PIOFs that are exactsolutions of the HE. These solutions are expressedin terms of parabolic cylinder functions and have aninherent geometry imposed by the parabolic coordi-nates. By use of the group symmetries of the HE,a spectrum for these fields is found that, common toall PIOFs, lies on a ring.9 The solutions describesatisfactorily transverse parabolic stationary waves,and with them a novel class of traveling waves isconstructed.

Fundamental PIOFs of the HE can be expressed interms of a reduced Whittaker integral10

U �r� � exp�2ikzz�Z p

2p

A�w�

3 exp�2ikt�x cos w 1 y sin w��dw , (1)

0146-9592/04/010044-03$15.00/0

where A�w� is the angular spectrum of field U �r� de-fined on a ring of radius kt in frequency space. Thetransverse and longitudinal wave-vector componentssatisfy the relation k2 � k2

t 1 k2z , where k is the wave

number. We assume a time dependence exp�ivt� forthe optical fields. From Eq. (1) it is clear that the f ieldintensity I � jU j2 is independent of propagation coor-dinate z.

The known fundamental solutions of the HE that de-scribe PIOFs are fully determined after their angularspectrum is introduced in Eq. (1). For m plane waveswe have A�w� �

Pm d�w 2 wm�, for mth-order Bessel

beams A�w� � exp�imu�, and for mth-order Mathieubeams A�w;q� � cem�w; q� 1 i sem�w; q�, where ce andse are the angular Mathieu functions and q is the el-lipticity parameter.6,8 To get the fundamental PIOFin parabolic cylinder coordinates the task is to find thecorresponding spectrum.

Finding PIOFs in parabolic cylinder coordinates re-quires one to solve the HE in this system. Paraboliccylinder coordinates are defined by the transforma-tion x 1 iy � �h 1 ij�2�2 and z � z with ranges inh [ �2`,`�; j [ �0,`�; z [ �2`,`�. In these coordi-nates the three-dimensional HE separates into a lon-gitudinal part, which has a solution with dependenceexp�2ikzz�, and a transverse part, which splits into

d2F�h�dh2 1 �k2

t h2 1 2kta�F�h� � 0 , (2)

d2R�j�dj2 1 �k2

t j2 2 2kta�R�j� � 0 , (3)

where for convenience we have written the separationconstant as 2kta [ �2`,`�. By a simple change ofvariables, sj ! v, where s � �2kt�1�2, Eqs. (2) and(3) can be transformed into the canonical form ofthe parabolic cylinder differential equation (PCDE),namely, d2P�dv2 1 �v2�4 2 a�P � 0. The solutionsto the PCDE are found by standard methods (e.g.,Frobenius), and its Taylor expansion with v � 0 is

© 2004 Optical Society of America

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January 1, 2004 / Vol. 29, No. 1 / OPTICS LETTERS 45

given by11

P �v,a� �X̀n�0

cnvn

n!, cn12 � acn 2

n�n 2 1�cn22

4.

(4)

We denote the even and odd solutions of the PCDEas Pe and Po, respectively. The first two coefficientsfor the Pe function are c0 � 1 and c1 � 0, whereasfor the Po solution they are c0 � 0 and c1 � 1. Forlarge values of argument v these solutions have anoscillatory behavior and an envelope that decaysas v21�2.

We emphasize that our approach to expressing thesolutions of the PCDE differs from the common meth-ods found in the literature.3,10,12 In the typical ap-proach the PCDE is transformed into a Weber differen-tial equation whose solutions are the parabolic cylinderfunctions Dn�m�, where n and m are in general complexparameters. Although mathematical connections be-tween the Pe and Po functions with the Dn functionscan be established, we believe that the Weber approachmakes the physics confusing and difficult to extract.Instead we prefer to deal with the real and physicallyachievable Pe and Po solutions.

In searching for solutions in free space, we see thatthe discontinuity of the h coordinate at the positive xaxis imposes the condition that only products of func-tions of the same parity in h and j are continuousin the whole space. Then the f irst (even) and second(odd) transverse stationary solutions are

Ue�h, j;a� �1

pp2jG1j

2Pe�sj;a�Pe�sh;2a� , (5)

Uo�h, j; a� �2

pp2jG3j

2Po�sj;a�Po�sh;2a� , (6)

where G1 � G��1�4� 1 �1�2� ia�, G3 � G��3�4� 1

�1�2� ia�, and the coefficients in front of both equa-tions are for normalization purposes. There aresome mathematical properties of the transversesolutions to be discussed here. The solutionsare orthonormal with respect to eigenvalue a;this means that

RRUe, o�h, j; a�U�

e,o�h,j;a0�dS �d�a 2 a0�, where d is the Dirac delta function andthe integration is carried out over the whole space.Therefore, analogous to plane waves, Bessel beams,and Mathieu beams, parabolic beams form a completeset of functions in the sense that any PIOF can berepresented as a linear superposition of fundamentalparabolic beams. Note that parabolic beams, such asplane waves, have a continuous eigenvalue spectrumof multiplicity 2.

Transverse patterns of parabolic beams for a � 0,1.5, 24 are shown in Fig. 1. The fields exhibit well-defined parabolic nodal lines. Note that the patternsare symmetrical about the x axis, i.e., Ue, o�x,2y;a� �6Ue,o�x, y;a�, where the plus and minus signs corre-spond to the even and odd fields, respectively. Evi-dently the odd solutions vanish along the whole x axisfor any value of a. For the fundamental mode a � 0

the y axis is also an axis of symmetry. For a . 0a dark parabolic region around the positive x axis ispresent. As a increases, this region grows, its ver-tex recedes from the origin, and the field inside itdecreases asymptotically to zero. The solutions fora , 0 have a similar behavior, since fields satisfy thesymmetry relation Ue, o�x, y;2a� � Ue, o�2x, y;a�.

After the form of the parabolic PIOFs was obtained,we proceeded to find their spectrum. The approachwe used is based on group theory. It can be shownthat applying second-order symmetry operators tothe solution of the HE and transforming to frequencyspace allows the spectrum of the parabolic PIOFs tobe obtained.4,5 The spectra for the even and odd solu-tions are

Ae�w;a� �1

2�pjsin wj�1�2 expµia ln

Çtan

w

2

Ç∂, (7)

Ao�w;a� �1i

(2Ae�w;a� , w [ �2p, 0�Ae�w;a� , w [ �0,p�

. (8)

An interesting feature of these spectra is that theyare expressed in terms of elementary functions only.

Fig. 1. Transverse patterns of the stationary even andodd parabolic beams for a � 0, 1.5, 24.

Fig. 2. (a) Amplitude and (b) phase of the angular spec-tra of even (solid curve) and odd (dashed curves) parabolicbeams with a � 1.

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46 OPTICS LETTERS / Vol. 29, No. 1 / January 1, 2004

Fig. 3. Evolution of an apertured parabolic beam for a �1.5 along the plane �x,z�, which can be observed in thewell-defined conical region where the f ield preserves in-variance.

Fig. 4. Transverse structure of (a) the amplitude jU j and(b) the phase of a traveling solution TU1�h,j,a � 101�2�.

The symmetry properties of the amplitude and phaseof the angular spectra are shown in Fig. 2. Both am-plitude and phase are singular at w � 0,6p for anyvalue of a, with the exception of a � 0, for which thephase is zero although the amplitude is still singular.The rapid growth caused by the tangent function atthe singular points is slowed down by the log function,resulting in a rather smooth phase.

Until now we have assumed that the f ields are ofinfinite extent; however, in real physical situations thefields have finite transverse extension, introducingdiffraction effects into the evolution of the otherwisePIOF.13 To take this into account, we simulated thepropagation of an even parabolic beam, using as theinitial condition the field depicted in Fig. 1(b) fora � 1.5, through a circular aperture of radius 6 mm.We chose the parameters of the simulation to yield ageometric maximum propagation distance of �6.0 m,assuming an illumination at l � 632.8 nm and aspatial frequency of kt � 10, 000 m21. The evolutionalong the plane �x, z� is shown in Fig. 3. Note thecharacteristic cone-shaped region where constituentplane waves superpose to build up the parabolic beam.As part of the family of PIOFs the parabolic beamsalso reconstruct themselves after they have beenpartially blocked, as is known to occur with Besselbeams and Mathieu beams.

From the stationary beam solutions described byEqs. (5) and (6) it is possible to construct travelingsolutions of the form TU6�h,j; a� � Ue�h,j; a� 6

iUo�h, j; a� whose associated spectra are A6�w;a� �Ae�w;a� 6 iAo�w; a�, in which the sign defines thetraveling direction. Plots for a . 0 are shown inFig. 4. The traveling-wave feature can be observedin the gradient of their phase structure.

In conclusion, we have shown that parabolic beamscomplete the family of fundamental PIOFs that areexact solutions of the HE. Analogous to plane waves,Bessel beams, and Mathieu beams, parabolic beamsalso form an orthonormal and complete set in thesense that any generalized PIOF can be expanded interms of parabolic beams. We have demonstratedthat parabolic beams have a continuous eigenvaluespectrum. Using group theory, we have deducedtheir angular spectrum in frequency space. We alsoconstructed traveling solutions as a superpositionof parabolic beams. Finally, simulations of thepropagation of the parabolic beams verified theirnondiffracting behavior.

We are grateful to W. Miller, Jr., who provideduseful references. This research was supported bythe Consejo Nacional de Ciencia y Tecnología and theTecnológico de Monterrey Research Chair in Optics.J. Gutierrez-Vega’s e-mail address is [email protected].

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