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We experimentally verified the interference resulting of the superposition of two Bessel beams propagating in free space and showed for first time the self imaging effect using nondiffracting beams. Our results are supported by numerical simulations and possible applications are discussed.
©1998 Optical Society of America
The Bessel beams are a topic of continuous interest since its proposal and generation by Durnin and collaborators [1, 2]. Many works have dealt with methods of generation, propagation, characterization [3, 4] and with its application to distinctive fields as medical 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 both coherent [10] and incoherent [11] propagation. The simultaneous use of nondiffracting beams 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. We analyzed the dynamical interplay between two interfering superimposed Bessel beams (SBB) supported by numerical simulations of the spatial beam propagation in 2+1 dimensions. Our discoveries showed that the stability of the resulting SBB is closely related to the associated Bessel beams and that the central width oscillates between the widths of the two separated Bessel beams. Our theoretical results also showed the existence of the self-imaging effect during the SBB’s propagation. In this work we experimentally obtain the SBB and confirm the predicted behavior, and demonstrate, for the first time, the phenomenon of self-imaging using nondiffractive beams.
Figure 1. The evolution of the SBB: density plot showing the simulated evolution of two Bessel beams in phase with same normalized amplitude and spatial frequencies k 0 = 4 and k 1 = k 0/5.
In order to provide a better understanding of our experimental results, initially we will do a brief theoretical discussion. Mathematically the SBB may be represented by the following equation obtained from a Hankel transform applied to the light produced by a double annular slit:
where is the radial coordinate, ki is the normalized radius of the annular slits and = k 2 - , 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 Bessel beams. From this equation we notice that along the longitudinal axis the intensity will oscillate 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 between the 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 the propagation distance z, where is possible to observe the periodical evolution of the SBB generated using Bessel beams with frequencies k 0 = 4 and k 1 = k 0/5. We observe that the maximum intensity initially is in the range between 3.6 - 4.05, what is expected because ideally the peak intensity should be four, since it is the result of the interference of two beams with maximum amplitude equal to one. This can be viewed directly from Eq. (1). At the fourth period the maximum intensity reachs the 4.05 - 4.5 intensity range as the result of the decaying of one of the SBB[15]. In the animated simulation of Fig. 2 we observe the behavior of the intensity during the propagation of the SBB along three and a half periods. The animation was constructed from the previous data which was produced by exactly solving numerically the nonparaxial wave equation (Helmholtz equation) in two transverse dimensions without any kind of approximation.
Figure 2. Animation showing the simulated evolution of two Bessel beams in phase with same normalized amplitude and spatial frequencies k 0 = 4 and k 1 = k 0/5. [Media 1]
In the experiment, the light of a 10 mW He-Ne laser at 6328 Å was spatially filtered focusing it with a 40X objective in a 5 μm pinhole. The resulting light was collimated placing the focus of a second lens exactly at the position of the pinhole. The light beam obtained was directed to a screen with two annular slits with the same 0.1 mm slit 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 a simplified experimental setup used to obtain the SBB. Using this setup we obtained a laser light transmission through the screen better than 15%. Higher transmission could be obtained with holographic techniques[16]. A sequence of photographic shots was taken along the propagation direction, always centering the single lens reflex camera in the optical axis. The camera was used without any objective, with a direct exposition of the film to the light generated. The shots were taken with a shutter speed of 1/15 seconds on a technical pan film. All the paper’s pictures shots correspond to a 35 mm film area of 24×36 mm. In Fig. 4 (a-g) we have a sequence of seven photographs taken at distances of 0, 6.8, 13.6, 20.4, 27.2, 34.0 and 40.8 cm, respectively. This distance covers a complete oscillation period of the SBB, and clearly is visible that the last photograph shown recovers the initial transversal shape. Fig. 4(d) corresponds to a half oscillation period and shows a situation where the interference of the propagating Bessel beams produces a minimum at the beam’s center. In the intermediary positions there is a complex pattern of evolution as was expected from the simulations. The propagation for 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 integer oscillation periods. Again we observe that the transverse patterns are reproduced along the propagation. After the distance corresponding to z max of the outer ring the SBB starts to decay as anticipated without recovering the previous transversal shape.
From the experimental results we confirm that the period of oscillation is a function of the longitudinal frequencies through , which produce a theoretical value for the period equal to 39.7 cm, an almost perfect match with the obtained experimental value of 40.8 cm, considering the intrinsical experimental errors of our measurements. This can be put in terms of the radial frequencies through . This relation allows to control the period by changing the radial frequency, which is related to the radii of the rings. The above analysis can be used to narrow or enlarge the region where the maxima or minima of intensity appear. In the case of the minimum it could be possible to use the SBB to generate a three dimensional dark spot in the space, with the light intensity increasing in all directions going out of the minimum. We believe that this could be useful to light trapping atoms in space using only a single beam of light. More work on this is under way.
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 density plot obtained through a numerical simulation of the evolution using the experimental parameters 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 shown in the experimental picture shots. We observe an almost perfect match between both results.
Figure 6. Simulated density plots to the positions corresponding to a central minimum (a) and maximum (b).
In conclusion, we have experimentally generated a superimposed Bessel beam and verified the existence of the interference of the two Bessel beams along the free propagation in the space. Our experimental results were supported by 2+1 dimensional numerical solution of the full wave equation with a perfect agreement between them. We have shown that the stability of the resulting SBB is closely related to the associated Bessel beams. Our experimental results clearly showed the existence of a self-imaging effect during the propagation of the SBB, which may be used successfully as an atom light trap [12, 14].
One of the authors (JMH) owe a debt to R. Pinheiro and H. Alencar. He also thanks the partial support by CNPq, FINEP, CAPES, FAPEAL, Brazilian agencies, and the TWAS (Third World Academy of Science) besides the INAOE (Instituto Nacional de Astrofísica Óptica y Electrónica) by the support during his stay in México. This work was partially supported by CONACYT (Consejo Nacional de Ciencia y Tecnología) under the grant number 3943P-E9607.
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