. INTRODUCTIONhe generation of nondiffracting beams has attracteduch interest in the past two decades [1–5]. Different
echniques have been proposed, and the zeroth-orderessel beam has been particularly stressed. In theory,essel beams are of infinite extent, so they cannot be pro-uced exactly. Various finite-extent approximations ofessel beams have, however, been demonstrated. Mostommon techniques involve an annular aperture [1], aomputer-generated hologram [2], a conical prism (axi-on) [3,4], and a spatial light modulator [6]. Recently,coustic waves in fluids have been shown to create similareams with concentric ring patterns [7]. A wide number ofpplications exist, from metrology and classical optics8,9] to nonlinear optics [10–13].
In the past few years, new studies have been devoted tohe generation of dark hollow beams for use in opticalweezers and in atom traps or atom guiding [14]. Higher-rder Bessel beams (that present a zero-field amplitude inheir center) could be produced with computer-generatedolograms [2] or phase screw components [15].In this context, the search of techniques for nondiffract-
ng beam generation deserves much attention. The pur-ose of this paper is to present a new method of generat-ng nondiffracting beams. Our setup involves only aocused Gaussian beam and an opaque disk. The diameterf the central peak of the nondiffracting beam can be cho-en by adding a suitable collimating lens. We demon-trate the actual generation of a beam with a central peakhose diameter is 800 �m and that is diffraction-free
ver more than 5 m. The generation of a beam of diameter00 �m is similarly possible.Section 2 presents our experimental setup and the re-
ults obtained. Section 3 presents the theoretical analysishat we have developed to model our configuration. Our
xperimental results are well predicted by the theoreticalevelopment.
. EXPERIMENTAL RESULTShe experimental setup is shown in Fig. 1. Our laserource is a CW He-Ne laser emitting at 632.8 nm. Epura-ion is ensured by focusing the laser beam with a micro-cope objective through a pinhole. The filtered divergingeam, which is a quasi-Gaussian beam, is then focusedith a 2 cm diameter lens L1 whose focal length is 10 cm.ocusing occurs 15 cm after L1. The diameter of the focalpot is 70±6 �m at 1/e.
An opaque disk fabricated by e-beam lithography of di-meter 300 �m is positioned just before the focal spot. Its well centered on the optical axis of the experiment, but.5±0.1 mm before the focal spot. Note that this defocus-ng is not a very sensitive parameter to achieve nondif-racting beam generation. We would obtain quite similaresults if this distance were 5 mm. In this configuration,he opaque disk stops the center of the laser beam whilehe peripheral part of the Gaussian beam is diffracted byhe disk. This diffraction of the Gaussian beam leads tohe generation of a thin intense peak centered on the op-ical axis and accompanied by a wide number of concen-ric rings.
The diffracted pattern observed with a CCD camerapositioned 3.3±0.2 cm after the opaque disk) is presentedn Fig. 2. [Lens L2 is not present in these first experi-
ents (see Fig. 1)]. We can observe the thin central peaknd concentric beams. The diameter of the first dark ringround the central intense peak is 110±11 �m. Precisions limited by the pixel size in this case. This pattern isery similar to a Bessel beam generated by an annularperture as shown for example in [1].
007 Optical Society of America
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3754 J. Opt. Soc. Am. A/Vol. 24, No. 12 /December 2007 M. Brunel and S. Coetmellec
The number of concentric rings that can be observed inig. 2 is �21. The total number of concentric rings is,owever, much higher. By displacing the CCD cameraransversely out of the optical axis and increasing the in-ensity of the incident beam more than 250 rings can bebserved. Figure 3 shows some portion of the rings num-ered 100–150 approximately. These results raise theuestion whether the beam generated is a nondiffractingeam.Figure 4 shows the pattern observed when the CCD is
ositioned on the optical axis of the experiment, but farrom the opaque disk (11.5±0.2 cm after the disk). We
Fig. 1. Ex
Fig. 2. Diffracted bea
till observe an intense peak with concentric rings. It isowever clear that the beam that has been generated isiverging. This is confirmed by precise measurementsade with the CCD placed at different positions after
he disk, i.e., 3.3±0.2 cm, 11.5±0.2 cm, 28.5±0.3 cm,03±1 cm, and 157±2 cm. The diameter of the first darking around the central peak is then 110±11 �m,80±11 �m, 935±22 �m, 3.6±0.2 mm, and 5.4±0.3 mm,espectively.
These results are plotted in Fig. 5. In this figure, it ap-ears clearly that the central peak (and actually thehole pattern with all concentric rings) exhibits a linear
ntal setup.
0.2 cm after the disk.
perime
m 3.3±
M. Brunel and S. Coetmellec Vol. 24, No. 12 /December 2007 /J. Opt. Soc. Am. A 3755
Fig. 3. Out-of-axis rings numbered 100 to 150.
Fig. 4. Diffracted beam 11.5±0.2 cm after the disk.
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3756 J. Opt. Soc. Am. A/Vol. 24, No. 12 /December 2007 M. Brunel and S. Coetmellec
preading versus the distance from the opaque disk. Thisroperty provides the way to realize a nondiffractingeam: In a recent paper, Khonina et al. [16] have indeedemonstrated that a nondiffracting Bessel beam could beransformed into a diverging Bessel beam by introducingspherical lens. We can thus reasonably expect that ouressel-like diverging beam can be transformed into aondiffracting Bessel-like beam by introducing a collimat-
ng lens.To confirm this and observe a nondiffracting beam with
ero divergence, lens L2 is positioned after the opaqueisk (see Fig. 1). Typically, the opaque disk is at the focaloint of this lens L2. In a first case, we use a lens whoseocal length is 25 cm. In the plane of the lens, the diam-ter of the central peak is 800±22 �m. According to Fig., the lens should then collimate a beam whose centraleak has a dimension of 800 �m approximately. We canhus expect the generation of a nondiffracting beam thatxhibits such a dimension.
Figure 6 shows the pattern observed 5±0.2 cm afterens L2 [6(a)] and 1.3±0.02 m after L2 [6(b)]. As we areimited by the length of our table, Fig. 6(c) shows finallyhe pattern observed 5.2±0.05 m after L2 and after twoeflections from plane mirrors. We can see that the cen-ral peak conserves its size during propagation: The di-meter of the central peak is effectively 800±22 �m in allhree cases. We have further verified along the z axis thathis dimension remains unchanged during the wholeropagation.In conclusion a beam is generated whose central peak
s surrounded by a wide number of bright rings ��250�.fter collimation, the beam propagates without changing
he rings’ radii, similar to a diffraction-free beam. Theentral peak can conserve its dimension over more thanm.It is important to note that the diameter of the central
eak can be easily adjusted by the choice of a collimating
Fig. 5. Divergenc
ens. To reduce the dimension of the central peak, we haveeplaced the collimating lens L2 by another lens whose fo-al length is shorter, i.e., 6 cm. This lens has been posi-ioned in such a way that the opaque disk is at its focaloint, i.e., 6 cm before L2. The beam could then be colli-ated. We observed that the size of the central peak
200 �m in this case) remained unchanged over morehan 1 m.
In summary, our setup allows the generation of beamsith diffraction-free characteristics and of variable di-ension.
. THEORY. Paraxial Diffraction
t is particularly important to predict these results theo-etically. It is actually possible to determine the exact ex-ression of the diffraction pattern under such conditions.he beam that illuminates the particle is a Gaussianeam. The expression of its electric field can be written
E��,�,z� = E0�z�exp�−�2 + �2
��z�2 �exp�−i�
�
�2 + �2
R�z� � , �1�
here � and � represent the transverse coordinates of theeam. The waist and the radius of curvature along theropagation axis z are given by
��z� =��02�1 + � z
z0�2� , �2�
R�z� = − z�1 + � z0
z �2� , �3�
here z0=��02 /� represents the Rayleigh length of the
eam and � its width at the waist. With these notations,
e diffracted beam.
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M. Brunel and S. Coetmellec Vol. 24, No. 12 /December 2007 /J. Opt. Soc. Am. A 3757
he origin of the z axis is in the plane of the waist of theeam.The opaque disk is now positioned at distance zq before
he waist (see Fig. 1). Replacing z by −zq in previous ex-ressions, it is possible to define the beam size and its ra-ius of curvature on the particle �q and Rq, respectively.he amplitude of the electric field that is diffracted at dis-ance zc after the opaque disk can then be written
E�x,y,zc − zq� =exp�i2�zc/��
i�zc
�−�
+�−�
+�
E��,�,− zq�1 − T��,���
�exp� i�
�zc�� − x�2 + �� − y�2� d�d�,
�4�
ig. 6. Diffracted beam pattern after a 25 cm focal length collimens.
Fig. 7. Transverse intensity profile
T��,�� = �1 if ��2 + �2 D/2,
0 otherwise. . �5�
is the diameter of the opaque disk.This calculus has been done in the case of an incident
lliptic and astigmatic beam [17]. This is here a very spe-ific case. After some developments, some simplificationsre found. Developing expressions of [17] in our case of aircular Gaussian beam, the diffracted field recorded atistance zc of the particle in the plane of the camera is
here the amplitude is E�x ,y ,zc−zq�=1/ i�zc�A1−A2�,ith
A1 = K2 expr2�iM − N��, �6�
lens at a distance of (a) 5 cm, (b) 1.3 m, and (c) 5.2 m after this
lated with truncated developments.
ating
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3758 J. Opt. Soc. Am. A/Vol. 24, No. 12 /December 2007 M. Brunel and S. Coetmellec
A2 =�D2
2exp�ir2�T0�r�. �7�
The system exhibits a symmetry of revolution aroundhe z axis, such that we have introduced the radial coor-inate r=�x2+y2. The function T0�r� is given by
T0�r� = exp�− iu/4��2�
u �1/2
�s=0
�
Ks�− 1�sJ2s+1�Dr�
Dr�8�
ith
Ks = �− i�s�2s + 1�Js+1/2�u/4�. �9�
he different parameters of these expressions are giveny
= �/�zc, �10�
u = D2/2�− 1 + �zcb1� − ia1�D2/2�, �11�
K = ���q
2
1 + i�q2� zc
Rq− 1��
1/2
, �12�
Fig. 8. Theoretic
N =�q
2
1 + �2�q4/�2� 1
Rq−
1
zc�2 , �13�
M = 1 + N��q
2
�� 1
Rq−
1
zc� , �14�
a1 =1
�q2 , �15�
b1 =1
�Rq. �16�
Let us mention that we do not take into account theens of collimation L2 here. It must be noted that the dif-racted pattern appears as a combination of Bessel func-ions of odd orders J2s+1. The Ks coefficients have a z de-endent complex argument. Each term changes with itswn “phase velocity.” A priori, the resulting interferencef all terms will thus be different at different distances zc.ith these relations, it is possible to simulate the beam
ulation of Fig. 2.
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M. Brunel and S. Coetmellec Vol. 24, No. 12 /December 2007 /J. Opt. Soc. Am. A 3759
hat is diffracted by the opaque disk. Function T0�r� isowever a development over an infinity of functions. Themplitude of these functions decreases rapidly when pa-ameter s increases, and the sum can be truncated into aew functions 0�s�N.
Typically the central peak is well described consideringnly a few functions �N=5�, but the other rings are nothen well predicted. The first two concentric rings aroundhe central peak are well described considering a tenth ofhe functions �N=8�. For N=14, we can describe the sevenrst concentric rings while other rings are not predicted.inally, when a twentieth of the functions is considered
N�16�, the theoretical development describes well theattern over more than 55 concentric rings. In this case,he use of other higher-order functions does not modifyhe pattern calculated (for the transverse dimension thatould be considered here and that is limited to 57 rings).
To illustrate this, Fig. 7 shows the transverse intensityrofile obtained using the truncated developments N=14dotted curve), 15 (dashed curve), 20 (solid curve) and N25 (solid curve). Differences in the profiles using 20 or5 functions are not discernible. Parameters are thosef our experimental conditions, i.e., �=632.8 nm, �035 �m, D=300 �m, zq=3.5 mm, zc=3.3 cm.Figure 8 shows the corresponding diffraction pattern
btained with a development of 21 functions �N=20�. Inig. 2, the CCD camera was saturated in the domain ofhe central peak; this effect allowed better visualization ofhe concentric rings. To obtain a similar visual impres-ion, the theoretical intensity pattern of Fig. 8 is pre-ented in gray levels on a logarithmic scale.
Both experimental and theoretical patterns appearualitatively very similar (Figs. 2 and 8). Quantitatively,
Fig. 9. Transverse beam in
he diameter of the first dark ring around the centraleak is 110 �m in both cases. Concentric rings are ob-erved both theoretically and experimentally. We coulderify that their diameters are identical (verification haseen done for the different rings of Figs. 2 and 8; the pre-ision of the verification is limited by the precision of thexperiment). Note that all simulations are now carriedut considering developments of 21 functions �N=20�. Theivergence of the central peak has then been calculatednd compared with experimental results: i.e., the diam-ter of the first dark ring around the central peak haseen calculated for different positions zc after the opaqueisk. The theoretical results are shown in Fig. 5 for com-arison with the experimental results. As can be seen, thexperimental data are well predicted by theory. The beamhat is generated is diverging, and the experimental andheoretical values of the divergence conform. We have fur-her verified that the divergence of the fifth dark ring isdentical experimentally and theoretically.
Finally, Fig. 9 shows a transverse intensity profile ofhe pattern that is observed 11.5 cm after the opaque diskthe corresponding pattern has been presented in Fig. 4).
theoretical fit of the profile is presented for comparison.e have added a constant background level to simulate
he background noise observed with the camera. We canee that the theoretical profile fits well the experimentalrofile. In summary, our experimental observations areell predicted by our theoretical analysis.
. Beam Collimationxperimentation has demonstrated that the beam can beollimated by lens L2. It is now necessary to model thisehavior. Subsection 3.A gives the exact field amplitude
y profile and theoretical fit.
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3760 J. Opt. Soc. Am. A/Vol. 24, No. 12 /December 2007 M. Brunel and S. Coetmellec
t the entrance of the collimating lens. Lens L2 and free-pace propagation must now be taken into account. Lens2 is easily modeled through a complex transmission
unction of the form
t�x,y� = exp�−i��x2 + y2�
�fL2� , �17�
here fL2is the focal length of L2. Free-space propagation
s expressed by the propagation operator (Fresnel trans-orm). The amplitude of the electric field that is diffractedt distance z after L can thus be written
ig. 10. Propagation through the lens L2: pattern calculated (a)ens.
� 2
E�x�,y�,z�� =exp�i2�z�/��
i�z�
L2
E��,�,zc − zq�
�exp�−i�
�fL2
��2 + �2���exp� i�
�z��� − x��2 + �� − y��2� d�d�.
�18�
The incident electric field E�� ,� ,z −z � has the expres-
front of the lens, (b) 5 cm after the lens, and (c) 1.3 m after the
just in
c q
ig. 11. Normalized transverse intensity profiles just in front of the lens, 5 cm after the lens, and 1.3 m after the lens. The three curvesre superposed.
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M. Brunel and S. Coetmellec Vol. 24, No. 12 /December 2007 /J. Opt. Soc. Am. A 3761
ion that has been developed in Subsection 3.A. Unfortu-ately, the integral of Eq. (18) does not lead to an analyti-al expression as in Subsection 3.A. We have thusvaluated this integral numerically. Experimentally, theollimation of the beam is observed when the opaque disks at the focal point of L2, i.e., when fL2
=zc. Transmissionhrough L2 then leads numerically to a simplification ofhe expression of A2 [suppression of the term exp�ir2� inq. (7)]. We have simulated the patterns that should bebserved in the case of L2 of focal length 25 cm (as in ourxperiment).
Figure 10(a) shows the theoretical pattern calculatedust in front of L2 using the analytical expressions of Sub-ection 3.A. For comparison, Figs. 10(b) and 10(c) showhe patterns that are predicted 5 cm after L2 [Fig. 10(b)]nd 1.3 m after [Fig. 10(c)]. These two patterns are ob-ained using our numerical integration of expression (18).
We have then extracted three intensity profiles fromigs. 10(a)–10(c). Figure 11 shows the superposition of
he three profiles. Each profile is normalized with respecto its maximum value at the center of the central peak.he three profiles are similar. These results show clearlyhat the beam conserves its distribution after the lensver more than 1 m; i.e., a nondiverging central peak (ofiameter 800 �m in this case, as in our experiment) andoncentric rings, as with a diffraction-free beam. In con-lusion, our experimental results are well predicted byhe theoretical analysis.
. CONCLUSIONnew method of generating nondiffracting beams is pre-
ented. It consists of focusing a Gaussian beam in the vi-inity of an opaque disk. A beam is generated whose cen-ral peak is surrounded by a wide number of bright rings�250�. After collimation, the beam propagates withouthanging the rings’ radii, similar to a diffraction-freeeam. The central peak can conserve its dimension overore than 5 m. The diameter of the central peak is ad-
usted simply by choosing the focal length of a collimatingens. We could, for example, generate a beam of 800 �miameter that is nondiffracting over more than 5.2 m. Ourxperimental observations are well predicted by our the-retical developments that simulate exactly the paraxial
iffraction.
EFERENCES1. J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, “Diffraction-
free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).2. A. Vasara, J. Turunen, and A. T. Friberg, “Realization of
general nondiffracting beams with computer-generatedholograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989).
3. G. Indebetouw, “Nondiffracting optical fields: someremarks on their analysis and synthesis,” J. Opt. Soc. Am.A 6, 150–153 (1989).
4. V. Jarutis, R. Paskauskas, and A. Stabinis, “Focusing ofLaguerre Gaussian beams by axicon,” Opt. Commun. 184,105–112 (2000).
5. V. Kotlyar, S. Khonina, V. Soifer, M. Shinkaryev, and G.Uspleniev, “Trochoson,” Opt. Commun. 91, 158–162 (1992).
6. J. A. Davis, J. Guertin, and D. M. Cottrell, “Diffraction-freebeams generated with programmable spatial lightmodulators,” Appl. Opt. 32, 6368–6370 (1993).
7. K. A. Higginson, M. A. Costolo, and E. A. Rietman,“Adaptive geometric optics derived from nonlinear acousticeffects,” Appl. Phys. Lett. 84, 843–845 (2004).
8. T. Aruga, S. W. Li, S. Yoshikado, M. Takabe, and R. Li,“Nondiffracting narrow light beam with small atmosphericturbulence-influenced propagation,” Appl. Opt. 38,3152–3156 (1999).
9. K. Wang, L. Zeng, and C. Yin, “Influence of the incidentwave-front on intensity distribution of the nondiffractingbeam used in large-scale measurement,” Opt. Commun.216, 99–103 (2003).
0. T. Wulle and S. Herminghaus, “Nonlinear optics of Besselbeams,” Phys. Rev. Lett. 70, 1401–1404 (1993).
1. S. P. Tewari, H. Huang, and R. W. Boyd, “Theory of third-harmonic generation using Bessel beams, and self-phase-matching,” Phys. Rev. A 54, 2314–2325 (1996).
2. L. Niggl and M. Maier, “Efficient conical emission ofstimulated Raman Stokes light generated by a Besselpump beam,” Opt. Lett. 22, 910–912 (1997).
3. J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “Efficiencyof second-harmonic generation with Bessel beams,” Phys.Rev. A 60, 2438–2441 (1999).
4. J. Arlt and K. Dholakia, “Generation of high-order Besselbeams by use of an axicon,” Opt. Commun. 177, 297–301(2000).
5. N. E. Andreev, S. S. Bychkov, V. V. Kotlyar, L. Ya.Margolin, L. N. Pyatnitskii, and P. G. Serafimovich,“Formation of high-power hollow Bessel light beams,”Quantum Electron. 26, 126–130 (1996).
6. S. Khonina, V. Kotlyar, R. Skidanov, V. Soifer, K. Jefimovs,J. Simonen, and J. Turunen, “Rotation of microparticleswith Bessel beams generated by diffractive elements,” J.Mol. Spectrosc. 51, 2167–2184 (2004).
7. F. Nicolas, S. Coetmellec, M. Brunel, D. Allano, D. Lebrun,and A. J. E. M. Janssen, “Application of the fractionalFourier transformation to digital holography recorded byan elliptical and astigmatic Gaussian beam,” J. Opt. Soc.
