Experimental observation of interfering Bessel beams

Experimental observation of interfering

Bessel beams

S. Chavez-Cerda, E. Tepichin, M. A. Meneses-Navaand G. Ramirez

Instituto Nacional de Astrofisica, Optica y ElectronicaApartado Postal 51 y 216, 72000, Puebla, Puebla, Mexico

J. Miguel Hickmann

Departamento de Fısica, Universidade Federal de Alagoas57072-970, Maceio, AL, Brazil

[email protected]

Abstract: We experimentally verified the interference resulting of thesuperposition of two Bessel beams propagating in free space and showedfor first time the self imaging effect using nondiffracting beams. Our re-sults are supported by numerical simulations and possible applicationsare discussed.c©1998 Optical Society of AmericaOCIS codes: (260.1960) Diffraction theory; (140.3300) Laser beam shaping

References

1. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A4, 651-654 (1987).

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

3. G. Indebetouw, “Nondiffracting optical fields: some remarks on their analysis and synthesis,” J.Opt. Soc. Am. A 6, 150-152 (1989).

4. R. Borghi and M. Santarsiero, “M2 factor of Bessel-Gauss beams,” Opt. Lett. 22, 262-264 (1997).5. S. Ruschin and A. Leizer, “Evanescent Bessel beams,” J. Opt. Soc. Am. A 15, 1139-1143 (1998).6. D. Ding and Z. Lu, “The second harmonic component in the Bessel beams,” Appl. Phys. Lett.68, 608-610 (1996).

7. X. Liu, “Comment on ‘The second harmonic component in the Bessel beams,’ ” Appl. Phys.Lett., 71, 722 (1997) .

8. V. E. Peet and R. V. Tsubin, “Third harmonic generation and multiphoton ionization in Besselbeams,” Phys. Rev. A 56, 1613-1620 (1997).

9. S. Klewitz, P. Leiderer, S. Herminghaus and S. Sogomonian, “Tunable stimulated Raman scat-tering by pumping with Bessel beams,” Opt. Lett. 21, 248-250 (1996).

10. K. Patorsky “The self-imaging phenomenon and its applications,” in Progress in Optics XXVII,E. Wolf, ed., p. 3 -108 (Elsevier, Amsterdan, 1990) and references there in.

11. E. Tepichin, P. Andres and J. Ibarra, “2-D Lau patterns: in-register incoherent joint superposi-tionof Montgomery patterns,” Opt. Commun. 125, 27-35 (1996).

12. Yu. B. Ovchinnikov, I. Manek, R. Grimm, “Surface trap for Cs atoms based on evanescent-wavecooling,” Phys. Rev. Lett. 79, 2225-2228 (1997).

13. I. Manek, Yu. B. Ovchinnikov, R. Grimm, “Generation of a hollow laser beam for atom trappingusing an axicon,” Opt. Comm. 147, 67-70 (1998).

14. J. Yin, Y. Zhu, “Dark-hollow-beam gravito-optical atom trap above an apex of a hollow opticalfiber,” Opt. Comm. 152, 421-428 (1998).

15. S. Chavez-Cerda, M. A. Meneses-Nava and J. M. Hickmann, “Interference of travelling non-diffracting beams,” Opt. Lett. 23, 1871-1873 (1998).

16. W-H Lee, “Computer generated holograms: techniques and applications,” in Progress in Optics

XVI, p. 121 (1978).

The Bessel beams are a topic of continuous interest since its proposal and generationby Durnin and collaborators [1, 2]. Many works have dealt with methods of generation,

(C) 1998 OSA 21 December 1998 / Vol.3, No. 13 / OPTICS EXPRESS 524#7100 - $15.00 US Received November 12, 1998; Revised December 07, 1998

propagation, characterization [3, 4] and with its application to distinctive fields as med-ical physics, microscopy[5], second/third harmonic[6, 7, 8] and Raman generation [9].The self-imaging phenomenon has been used in many powerful applications under bothcoherent [10] and incoherent [11] propagation. The simultaneous use of nondiffractingbeams and self-imaging effect may lead to exploit new three-dimensional phenomena as,for example, a new kind of atom trapping with light [12, 13, 14]. In an early work[15]we have studied the interference of Bessel beams from a theoretical point of view. Weanalyzed the dynamical interplay between two interfering superimposed Bessel beams(SBB) supported by numerical simulations of the spatial beam propagation in 2+1dimensions. Our discoveries showed that the stability of the resulting SBB is closelyrelated to the associated Bessel beams and that the central width oscillates betweenthe widths of the two separated Bessel beams. Our theoretical results also showed theexistence of the self-imaging effect during the SBB’s propagation. In this work we ex-perimentally obtain the SBB and confirm the predicted behavior, and demonstrate, forthe first time, the phenomenon of self-imaging using nondiffractive beams.

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Figure 1. The evolution of the SBB: density plot showing the simulated evo-lution of two Bessel beams in phase with same normalized amplitude and spatialfrequencies k0 = 4 and k1 = k0/5.

In order to provide a better understanding of our experimental results, initiallywe will do a brief theoretical discussion. Mathematically the SBB may be represented bythe following equation obtained from a Hankel transform applied to the light producedby a double annular slit:

I(r, z) = J20 (k0r) + a2J20 (k1r) + 2a J

20 (k0r) J

20 (k1r) cos((kz0 − kz1)z + θ) (1)

where r =√x2 + y2 is the radial coordinate, ki is the normalized radius of the annular

slits and k2iz = k2−k2ir, with i = 0, 1 referring to the two annular slits. The symbols a and

θ are, respectively, an amplitude factor and a phase difference relating the two Besselbeams. From this equation we notice that along the longitudinal axis the intensity willoscillate due to the presence of the last term. In order to simplify from now on we will


deal with the particular case where there is no phase difference nor amplitude betweenthe two Bessel beams, i.e. θ = 0 and a = 1 without loss of generality.

In Fig. 1 is depicted a contour plot for the evolution of the intensity along thepropagation distance z, where is possible to observe the periodical evolution of the SBBgenerated using Bessel beams with frequencies k0 = 4 and k1 = k0/5. We observe thatthe maximum intensity initially is in the range between 3.6 − 4.05, what is expectedbecause ideally the peak intensity should be four, since it is the result of the interferenceof two beams with maximum amplitude equal to one. This can be viewed directly fromEq. (1). At the fourth period the maximum intensity reachs the 4.05 − 4.5 intensityrange as the result of the decaying of one of the SBB[15]. In the animated simulation ofFig. 2 we observe the behavior of the intensity during the propagation of the SBB alongthree and a half periods. The animation was constructed from the previous data whichwas produced by exactly solving numerically the nonparaxial wave equation (Helmholtzequation) in two transverse dimensions without any kind of approximation.

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Figure 2. Animation showing the simulated evolution of two Bessel beamsin phase with same normalized amplitude and spatial frequencies k0 = 4 andk1 = k0/5.

In the experiment, the light of a 10 mW He-Ne laser at 6328 A was spatiallyfiltered focusing it with a 40X objective in a 5 µm pinhole. The resulting light wascollimated placing the focus of a second lens exactly at the position of the pinhole. Thelight beam obtained was directed to a screen with two annular slits with the same 0.1 mmslit thickness and 1.63 mm and 2.11 mm radius for inner and outer rings respectively.Behind the screen there is another lens with a 75 cm focal length. Fig. 3 shows asimplified experimental setup used to obtain the SBB. Using this setup we obtained alaser light transmission through the screen better than 15%. Higher transmission couldbe obtained with holographic techniques[16]. A sequence of photographic shots wastaken along the propagation direction, always centering the single lens reflex camera inthe optical axis. The camera was used without any objective, with a direct expositionof the film to the light generated. The shots were taken with a shutter speed of 1/15seconds on a technical pan film. All the paper’s pictures shots correspond to a 35 mmfilm area of 24x36 mm. In Fig. 4 (a-g) we have a sequence of seven photographs taken atdistances of 0, 6.8, 13.6, 20.4, 27.2, 34.0 and 40.8 cm, respectively. This distance coversa complete oscillation period of the SBB, and clearly is visible that the last photographshown recovers the initial transversal shape. Fig. 4(d) corresponds to a half oscillationperiod and shows a situation where the interference of the propagating Bessel beamsproduces a minimum at the beam’s center. In the intermediary positions there is a


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complex pattern of evolution as was expected from the simulations. The propagationfor longer distances is presented in Fig. 5 (a-e), which corresponds to distances of 61.2,83.2, 102.0, 122.4 and 142.8 cm, respectively. There are pictures for only half and integeroscillation periods. Again we observe that the transverse patterns are reproduced alongthe propagation. After the distance corresponding to zmax of the outer ring the SBBstarts to decay as anticipated without recovering the previous transversal shape.

Figure 3. Experimental setup.

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Figure 4. A sequence of photografic shots showing the SBB evolution in the firstperiod.

From the experimental results we confirm that the period of oscillation is a functionof the longitudinal frequencies through 2π

k0z−k1z, which produce a theoretical value for

the period equal to 39.7 cm, an almost perfect match with the obtained experimentalvalue of 40.8 cm, considering the intrinsical experimental errors of our measurements.


This can be put in terms of the radial frequencies through kz =√k2 − k2r . This relation

allows to control the period by changing the radial frequency, which is related to theradii of the rings. The above analysis can be used to narrow or enlarge the region wherethe maxima or minima of intensity appear. In the case of the minimum it could bepossible to use the SBB to generate a three dimensional dark spot in the space, withthe light intensity increasing in all directions going out of the minimum. We believe thatthis could be useful to light trapping atoms in space using only a single beam of light.More work on this is under way.

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Figure 5. A sequence of photografic shots showing the subsequent central mini-mums and maximums along the evolution of the SBB.

In order to compare the theory with our experimental results in Fig. 6 we show a densityplot obtained through a numerical simulation of the evolution using the experimentalparameters for the distances corresponding to a minimum (a) and to a maximum (b)at the beam’s center. The area depicted in the picture is equivalent to the area shownin the experimental picture shots. We observe an almost perfect match between bothresults.


Figure 6. Simulated density plots to the positions corresponding to a centralminimum (a) and maximum (b).

In conclusion, we have experimentally generated a superimposed Bessel beam and veri-fied the existence of the interference of the two Bessel beams along the free propagationin the space. Our experimental results were supported by 2+1 dimensional numericalsolution of the full wave equation with a perfect agreement between them. We haveshown that the stability of the resulting SBB is closely related to the associated Besselbeams. Our experimental results clearly showed the existence of a self-imaging effectduring the propagation of the SBB, which may be used successfully as an atom lighttrap [12, 14].

One of the authors (JMH) owe a debt to R. Pinheiro and H. Alencar. He alsothanks the partial support by CNPq, FINEP, CAPES, FAPEAL, Brazilian agencies, andthe TWAS (Third World Academy of Science) besides the INAOE (Instituto Nacional deAstrofisica Optica y Electronica) by the support during his stay in Mexico. This workwas partially supported by CONACYT (Consejo Nacional de Ciencia y Tecnologia)under the grant number 3943P-E9607.


Experimental observation of interfering Bessel beams

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