We report the generation of a zero-order Bessel beam of continuously variable spot size using a simple optical setup. We have used a pair of metal axicon mirrors to generate a hollow beam of variable dark diameter. This beam was subsequently focused by a convex lens to get a Bessel beam of variable spot size. We also studied the effect of a hollow-beam ring width on nondiffracting propagation range of the generated beam.
© 2012 Optical Society of America
Bessel beams have received significant attention since their conception and realization by Durnin et al. . The size of the central spot as well as the peak intensity of a Bessel beam does not change significantly over a propagation distance. This unique and remarkable property, called nondiffracting property, of Bessel beams has made them useful for various technological applications such as atom guiding [2,3], optical tweezers , laser ablation, and laser machining [5–7]. Besides nondiffracting property, Bessel beams also show self-healing property which can be exploited for some purposes . While generation of high-order Bessel beams is possible , we consider, in the present work, generation of only zero-order Bessel beams. The cross-section of a zero-order Bessel beam has a bright spot in the center surrounded by concentric rings, with overall intensity distribution given as square of the zero-order cylindrical Bessel function (). The intensity of side-rings decreases rapidly as one goes away from the beam center, however power in any ring is equal to that in the central lobe in intensity profile of the beam. These beams also show interesting electromagnetic field structure when spot size of the central lobe is comparable to the wavelength .
The electromagnetic field, having transverse variation as -Bessel function, results from the superposition of plane waves with wave vectors distributed over the surface of a cone. The spot size, which is referred to here as full width at half-maximum (FWHM), of the central peak in the transverse intensity profile, is equal to for an ideal -Bessel beam; where , is wave-vector magnitude and is half of the vertex angle of the cone formed by the wave vectors in the superposition. It is well known that generation of an ideal -Bessel beam is not possible as it requires plane waves having infinite width as its constituent waves. However, beams having an intensity profile close to -Bessel beam intensity profile have been experimentally generated over finite transverse extent as well as with finite nondiffracting propagation range. Such a beam is referred to here as a generated -Bessel beam (or Bessel beam).
Various optical elements and setup have been used to generate Bessel beams. These include a ring aperture placed in the focal plane of a convex lens , binary-coded holograms , axicon lenses , a converging lens with spherical aberrations , and reflective axicon mirrors [13–18]. The reflective axicon mirrors are easy to fabricate and have no aberrations (as compared to refractive elements such as axicon lenses) due to propagation in bulk of refractive medium. The inhomogeneity in bulk medium can be a source of such aberrations.
Generally, the spot size of a Bessel beam is fixed for a given experimental setup. To change the Bessel-beam spot size, one needs to replace the components of the setup with those having modified specifications. However, some active optical devices such as spatial light modulators (SLMs)  and tunable acoustic gradient lens [20,21] have been used to generate Bessel beams with variable spot parameters. But, SLMs are more expensive whereas acoustic gradient lenses have a complex design as compared to optical axicons. In an interesting paper, Milne et al.  have reported use of a fluidic axicon lens to obtain a Bessel beam with variable spot size. This method required refilling of fluid having a different value of refractive index to obtain the changed spot size of the Bessel beam. While changing fluid each time is inconvenient, this approach may also suffer from limitations on continuous variation of spot size due to discrete values of the refractive index of various fluids. Recently, Brousseau et al.  and Brunne et al.  have used axicon mirrors whose conical surface could be manipulated by applied magnetic field and applied voltage, respectively, to obtain variable spot parameters of the generated Bessel beam. Earlier in a work by Vaičaitis and Paulikas , the generation of a Bessel beam of variable spot size has also been studied by using passive optical elements. In this paper we report the generation of a -Bessel beam of continuously variable spot size using a simple and convenient method with a low-cost setup. We used a setup consisting of a pair of axicon mirrors and a few other standard optical elements. The generated hollow beam (HB) was focused using a convex lens to obtain the -Bessel beam. The Bessel-beam spot size could be varied just by varying the separation between axicon mirrors in the setup. The range over which spot size remains variable can be altered by changing the focal length of the lens. A -Bessel beam with such a continuous variation possible in its spot size can be useful in several applications such as atom guiding, optical tweezers, and laser machining.
2. Experimental Setup
Figure 1 shows the schematic of our experimental setup. We used two homemade metal axicon mirrors [ (convex) and (concave)] of equal angle for conical surface. These gold-coated diamond-turned copper axicon mirrors have high reflectivity () at 633 nm laser wavelength which we used in the experiments. A linearly polarized (vertical-direction) laser beam having a nearly Gaussian spatial profile with FWHM and power from an H–Ne laser system was used as an input laser beam. Appropriate polarization-selective optical elements such as a polarizing-cube beam splitter (PBS) and quarter-wave () plates (Fig. 1) were used for coupling the input laser beam to axicon mirrors. The collimated HB generated from the setup was focused using a convex spherical lens () of focal length to obtain the -Bessel beam. A digital CCD camera (Pixelfly, PCO, Germany) having pixels (pixel-size ), in conjunction with peripheral component interconnect interface-board on a computer, was used to record the beam images at different positions. We have recorded an image of a HB by taking an image of the screen on which the HB was incident. Direct imaging of the HB was not possible because its size was larger than the size of the CCD element. Therefore surface quality of the screen used may also affect the intensity profiles of the recorded images. On the other hand, for a Bessel beam, images were recorded keeping CCD directly in the beam path. The transverse intensity profiles of the beams were obtained from the recorded images. The overall conversion efficiency from input laser-beam power to -Bessel-beam power in our experiments was . This efficiency can be further increased by using high-reflectivity dielectric coating on the metal axicon mirrors.
After the lens (L), the generated HB gets converged such that wave vectors of different rays form a conical surface with cone axis the same as optical axis (-axis) of the lens (Fig. 1). The region of this conical superposition of waves [shown by shaded part in Fig. 1] is the region of formation of the Bessel beam with average direction of propagation along the -axis. The transverse intensity variation in this region is approximately a square of -Bessel function. Here we note that angle , which governs the Bessel-beam spot size, is dependent on the dark diameter of the HB at lens (L). The dark diameter can be varied by varying the separation () between axicon mirrors in the setup. This provided the opportunity to vary continuously the spot size of the generated Bessel beam just by varying the mirror separation ().
3. Results and Discussion
A. Variation of FWHM with Axicon Mirror Separation ()
We recorded images of generated Bessel beam at different values of position and for different values of mirror separation . Corresponding to different values of , the images of the HB were also recorded at lens plane (). Figures 2 and 3 show the CCD images and intensity profiles of the generated HB (at the lens plane ) and Bessel beam (at ) respectively when we kept mirror separation of in our setup. The graphs shown in Figs. 2 and 3 are the plot of intensity profiles of hollow and generated Bessel beams across the horizontal diameter in the beam cross-section. Nearly identical profiles were observed across the vertical diameter of the beam, indicating a good circular symmetry in the generated beam. In Fig. 3(b), the curve having variation as square of -Bessel function is also shown for comparison with experimentally observed profile.
From the recorded CCD images, we measured the dark diameter of the generated HB for various values of separation . In the measurements, we have taken dark diameter as separation between nearest half-maximum intensity points (on the intensity profile) which are diametrically opposite about the beam center. We found that measured dark diameter was nearly equal to the geometrically estimated dark diameter () given by , and its variation with mirror separation was linear as shown by hollow circles in Fig. 4. Similar results have been observed by us earlier also . However, this striking dependence of hollow-beam diameter on mirror separation has been exploited in the present work to generate the Bessel beam of variable spot size after focusing the HB using the lens. Owing to dependence of Bessel-beam spot size on hollow-beam diameter in our geometry (Fig. 1), we could obtain a continuous variation in the spot size of the generated Bessel beam by varying the mirror separation , as shown in Fig. 4. In this figure, data show the measured variation in hollow-beam diameter (hollow circles) at the lens plane (position ) and corresponding variation in Bessel beam spot size (filled circles) measured at with separation . It is clearly evident from the data in this figure that a larger mirror separation resulted in a larger hollow-beam diameter and a smaller Bessel-beam spot size. Though data shown in Fig. 4 are for some selected values of , using our setup we could vary the Bessel-beam spot size continuously over a range by varying the mirror separation . We note that for our geometry, Bessel-beam spot size is given as , where (, i.e. focal length of lens) is the distance of the Bessel-beam formation region from the lens and is the geometrically estimated dark diameter of the HB. Hence FWHM of a Bessel beam is expected to vary inversely with mirror separation as well as with diameter . Our data presented in Fig. 5 show similar behavior. A small variation in can result in variation in Bessel-beam spot size which is given approximately as , where refers the variation. Thus a variation (at ) is expected to change FWHM of Bessel beam by value . However, such a small change is difficult to sense in our setup as resolution of our CCD is (size of a pixel). Nevertheless, such a small variation in Bessel-beam spot size can be measured by other more accurate methods .
We note that for our geometry, Bessel-beam spot size varies with focal length and dark diameter which is given as . For a given value of focal length , the range over which FWHM can be varied is governed by the cross-section area of optical components (i.e. size of optics) to accommodate the generated HB and the minimum achievable separation in the setup. In our setup using , we could vary spot size from to by varying mirror separation from 130 to 70 mm. As FWHM is varying linearly with focal length , the spot size of the generated Bessel beam can be further reduced if a lens of smaller focal length is used. We have earlier demonstrated the generation of a Bessel beam of FWHM smaller than the size of a CCD pixel by using a shorter focal length lens . Thus the range over which the spot size can be varied by varying separation can be further extended by replacing the lens in the setup.
B. Variation in Nondiffracting Propagation Range with Axicon Mirror Separation ()
The variation in peak intensity of a Bessel beam with propagation distance is another important issue. The range over which peak intensity as well as spot size remain unchanged is called nondiffracting propagation range (). For its experimental determination, we define as a range along -axis over which variation in both the parameters (i.e., intensity and spot size) is within 10% from their values at the peak intensity position. For our geometry, is expected to scale as the length of Bessel beam formation region (and is given as , for ). Here is ring-width parameter defined as times the FWHM of the Gaussian intensity profile of the ring of the HB [expressed mathematically later in Eq. (3)]. Figure 6 shows the measured variation in peak intensity of the Bessel beam (hollow circles) generated in our experiments with distance , for some selected different values of separation . A faster variation in peak intensity with for the larger value indicates the smaller value of for larger . The experimentally measured ( for ) for the generated Bessel beam was much larger (approximately five times) than the Rayleigh range for a Gaussian beam of equal FWHM. It can be pointed out here that since the nondiffracting propagation range is large, for a sample having a length much smaller than value, a careful positioning of the sample in the beam path would not be required for unchanged spot-size and intensity values on a sample during continuous variation in spot size over a certain range. This may be useful in several applications.
Further, the nondiffracting propagation range can be connected to FWHM as [as ]. Thus the propagation-invariant range is larger for a larger FWHM spot size of the Bessel beam. This is reflected from our data in Fig. 6 where the nondiffracting range is reduced with a decrease in FWHM [see graphs in Figs. 6(a)–6(d)].
As can be noted from Fig. 6, the FWHM of the generated beam (filled circles) also varies with distance . These observations are in agreement with our simulation results presented afterwards in this paper. Since the variation in FWHM spot size, as well as peak intensity, is slow with , the beam seems to evidently preserve its nondiffracting property.
C. Effect of HB Ring-Width on Bessel-Beam Generation
It is known that a Fourier transform of a ring function is a -Bessel function, where the radius of the ring determines the transverse frequency of the -Bessel function variation. Hence, focusing of an annular intensity profile with a very small ring width should result in a Bessel-beam field, and Bessel-beam FWHM spot size should vary with hollow-beam diameter as we experimentally observed (Fig. 5). These conditions were satisfied in our experiments described above (with ), and a good quality Bessel beam was obtained as shown in Fig. 3. In order to investigate the effect of on the quality of a beam generated in the focal region of the lens, we measured the transverse intensity profile of a generated beam for different values of ring-width parameter of the HB. In these measurements, the input beam was expanded to increase the ring width of a HB at the lens plane for a fixed value of mirror separation (). When input beam spot size was changed, we noted that, in addition to change in , the peak intensity radius of HB was also changed slightly. This may be due to diffraction effects involved in the propagation of a ring from source position to the lens plane. The measured profiles of a generated Bessel beam are shown in Fig. 7 for different values of ring-width parameter of a HB. As can be noted from this figure, the profile of a generated beam gets modified in a qualitative way with an increase in a ring-width parameter . The higher central-peak intensity (with relatively weaker side peaks) for larger values of shows that a generated beam was degraded and became much different from a -Bessel beam at ring-width parameter [Fig. 7(c)]. We have also measured the variation in peak intensity and FWHM of the beam with axial distance , for different values of ring-width parameter . Results of these measurements are shown in Fig. 8. The results show reduction in nondiffracting propagation range with increase in . This is in agreement with the nature of spatial profiles shown in Fig. 7. These experimental results are also qualitatively in agreement with the results of simulations presented in next subsection.
Thus, contrary to the geometrically estimated behavior as discussed before that nondiffracting propagation range () should scale linearly with ring width , we observed reduction in the nondiffracting propagation range with increase in . This was due to deviation of generated beam profile from a -Bessel beam profile for larger values of .
D. Results of Simulations
Earlier in a work by Pu et al. , it was shown that a focusing of a Gaussian beam (with its central part blocked) by using an aberrated converging lens can result in zero-order Bessel function type transverse field distribution. To show theoretically for our setup that focusing a HB using a converging spherical lens can result in a field distribution close to that of a -Bessel beam, we numerically solved the Fresnel integral for our geometry to find field distribution after the lens. We used formulism similar to that used in earlier works [12,26]. When a lens is illuminated by a HB (Fig. 9), we can write the field distribution at point (, ) after the lens as
In the above definition of , is the function describing the variation of the HB field with at lens plane. Here is the lens aberration coefficient and is paraxial focal length of the lens. For a uniform intensity HB, . For a HB having a spatial intensity profile of its ring, is a function of position . To model our observed hollow-beam intensity profile, we have assumed to be of form1), we have set parameters and by comparing Eq. (3) with the experimentally measured intensity profile of the HB at the lens plane.
Figure 10 shows the transverse intensity profiles, obtained after evaluating Eq. (1), for different values of axial position for two different values of ring-width parameter (). These profiles were evaluated for different hollow-beam parameters which were set after measuring the hollow-beam profiles for a fixed but varying the expansion factor in the input beam (as done for taking data shown in Figs. 7 and 8 for different values of ). While changing the spot size of the input beam, we noted that, in addition to change in , there was a small change in the hollow-beam peak radius also. This may be because of diffraction effects. We estimated the lens parameters and by measuring the focal positions of HBs of different radii and applying the formulism similar to given in .
As can be noted from Fig. 10, for larger value of [Fig. 10(b)] the calculated intensity profile shows more deviation from an ideal -Bessel beam profile. These calculated profiles show the trend similar to the experimentally observed profiles shown in Fig. 7. Similarly, it can be seen that the peak intensity (Fig. 11) and FWHM spot size (Fig. 12) obtained from numerical simulations follow the similar trend in variation with axial distance as observed experimentally for different values of (Fig. 8). Thus our simulations show a qualitative agreement with the experimentally observed results. From these findings, it is clear that smaller ring width of the HB is required for generation of a good-quality Bessel beam.
Using a simple setup we have demonstrated the generation of a zero-order Bessel beam of continuously variable central spot size. This was achieved by focusing a variable-diameter collimated HB which was generated after passing a laser beam through a setup consisting of a pair of convex and concave metal axicon mirrors. In the experimental setup, by varying the mirror separation, we could obtain the variation in the spot size of the generated Bessel beam. Such a variable spot-size -Bessel beam may find various potential applications in atom guiding, optical tweezers, and laser machining. We note that range over which Bessel-beam spot size can be varied in our setup remains limited due to limits on separation between mirrors and size of the optics used. This range can be extended by using a lens of different focal length.
We have also investigated the effect of hollow-beam ring width on spatial profile and non-diffracting propagation range of Bessel beams generated in our setup. Our results show that smaller ring width of the HB gives the better quality of generated Bessel beams.
We thank R. Balasubramaniam (Machine Dynamics Division, BARC, Mumbai), C. Mukharjee (MOSS, RRCAT, Indore), and L. Sudarsanam (Laser Workshop, RRCAT, Indore) for helping in fabrication of axicon mirrors; and H. S. Vora (LESD, RRCAT, Indore) for providing image processing software. We acknowledge the contribution of Optical Workshop, RRCAT, Indore for lapping the copper disks used in making axicon mirrors.
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