We present a unique method that utilizes high intensity core of the zero-order nondiffracting beam (NDB) to fabricate a homogeneous polymer fiber as narrow as 2 µm and as long as centimeters. The constant diameter of the fiber along all its length is done by the propagation invariant properties of the NDB. The length of the fiber is determined by the maximum propagation distance of the NDB which is much longer than the classical Gaussian beam of comparable width. Moreover, we also proved that the self-writing waveguide mechanism prolongs the length of the developed fibers. Circular movement of the NDB creates hollow fiber, several co-axial, or overlapping fibers.
©2006 Optical Society of America
Photopolymerization induced by a single focused laser beam or by an interference of several beams forming an optical lattice found recently extremely interesting applications in the field of microfabrication [1–3] and nanofabrication , light driven micromachines [5–7], and fabrication of photonic crystals [8–12]. Special type of applications represents fabrication of polymer fibers or microstructured polymer fibers [13–17]. When a polymer fiber grew at the end of an optical fiber an interesting phenomena of self-writing has been described [18,19] and extensively studied theoretically and experimentally [20,21]. Combination of the above mentioned methods led to fiber tip modifications , coupling of diode lasers to optical fibers  or even fabrication of artificial insects’ compound eyes .
A group of nondiffracting beams (NDBs) represents one of many possible propagation invariant solutions of wave equation . The unique property of these beams is that they keep their radial intensity profile unchanged while they propagate. The simplest example of these beams is so called Bessel beam because its lateral intensity profile is described by the first class Bessel function of the zero order. This beam is formed as a result of interference of plane waves with wavevectors covering the surface of the cone. Due to this type of generation the beam has another useful property – self reconstruction. If an obstacle is placed to the narrow nondiffracting beam, it reconstructs itself on a short distance . Unfortunately these beams are only a theoretical solution and in reality only approximations to them can be generated. Such beams are generally called pseudo-nondiffraction beams and can be obtained by several ways . For simplicity let us call in the rest of the paper this pseudo-nondiffractiong beam formed behind an axicon illuminated by a Gaussiam beam (GB) as the Bessel beam (BB). BBs have been used in many unique applications  like microparticle guiding and delivery [29,30], microparticle and living cells sorting , and atom guiding .
In this paper we present for the first time how the properties of non-diffracting beams can be effectively used to create long and uniform polymer fibers of radius as small as micrometers and length as long as centimeters. We even demonstrate fabrication of hollow fiber or rotational symmetric hollow objects even mutually penetrated by off-axis gyration of BB. Such formations can be used as single-mode fibers or hollow fibers, they can interconnect optical components in opto-fluidic systems or lab-on-a-chip systems, link several surfaces or microobjects.
2. Properties of a Bessel beam
In real circumstances it is not possible to obtain a beam that does not change its lateral properties over infinity range of propagation. In the case of the GB incident on an axicon the following scalar form of the spatial light intensity distribution in the BB can be used :
where θ is the polar angle between propagation axis of the BB and the plane waves wave-vectors forming the BB behind the axicon (see Fig. 1):
where Č is the apex angle of the axicon, na is the refractive index of the axicon and nm is the refractive index of the surroundings medium, k is the wavenumber of the beam in the medium, P is the power of the GB incident on the axicon, w is the half-width of the GB at the axicon, and zmax is the maximum propagation distance expressing the axial range where the BB exists:
In the lateral plane the highest light intensity is at the centre of the BB. Let us define the radius of the BB core (RBBC) as the radial distance from the beam centre to the first intensity minimum done by:
Therefore, narrower BB is generated for bigger θ but at the same time the maximum BB propagation distance zmax will be shorter using the same width of the incident GB.
Using the commercially available axicons the resulting RBBC is too wide for our purposes and therefore a telescope made from lenses L1 and L2 (see Fig. 1) is used to decrease it. The intensity of the BB passing through the telescope can be rewritten in the new coordinate system z′, r′ (see Fig. 1):
where the parameters of the new BB are transformed as:
where M=f2/f1 is the magnification of the telescope and f1 or f2 is the focal length of the lens L1 or L2, respectively, T is the transmissivity of the telescope. In the majority of our experiments the angle θ’ satisfies cos θ’≅1and consequently 2max≅M2zmax . Therefore Eq. (5) can be approximated by:
Hence we see that the telescope scales the RBBC rB by a factor M and the distance of BB existence zmax by a factor M2 . The wider the incident GB is used, the longer BB is obtained but the lower BB intensity is on the optical axis.
The narrow BB is formed inside a cuvette filled with a fluid of refractive index np . The previous results assumed the beam propagates in the air, therefore using the Snell’s law for refraction into the fluid sinθ′p =sinθ′/np the following form of BB maximal propagation distance should be considered in the fluid:
The experimental set-up is shown in Fig. 1. In all experiments we used a GB coming from the CW laser (Verdi V5, Coherent, wavelength 532 nm, maximal power 5.5 W, w=1.125 mm). This beam was transformed by an axicon into a BB and further decreased by a telescope formed from lenses L1 and L2. The apex angle α of the axicon and the focal lengths f1, f2 of the lenses were chosen according to the actual experimental demands and their values are specified below for each experiment. The transformed BB propagated vertically up through a cuvette filled with a solution of UV light indurate optical glue (Norland NOA 63, np =1.52). The created polymer structures were observed by the long-working-distance microscope objective (Mitutoyo M Plan Apo SL 80X) and a CCD camera (Kampro KC-381CG or IDT X-StreamVISION XS-3).
3. Formation of polymer fibers
Polymer fiber grows in the central core of the BB if the monomer is exposed by a BB. Changing the RBBC (by the selection of the appropriate axicon apex angle α and the telescope magnification M) the polymer fibers of different diameter dfiber and lengths Lfiber were created. Figure 2 presents some of our experimental results for three different RBBC (columns A-C) obtained with the same axicon (Eksma 130–0270) having cone angle α=170° and refractive index na =1.52. Top row shows the lateral intensity profiles of three generated BB together with parameters of the set-up obtained experimentally and theoretically. The lateral intensity profile of the BB was obtained from a calibrated CCD camera placed instead of the cuvette and facing the incident beam. Since the RBBC is the same regardless of the medium where the beam propagates [see Eq. (4)], we fitted a simplified form of Eq. (7) (Br’) to this CCD intensity image. The RBBC r’B is then determined from r’B =2.4048/B. The theoretical values of r’BT, z’maxT , and z’maxTp=np z’maxT were obtained from Eqs. (2)–(6) and Eq. (8) using related values of f1, f2 , axicon parameters and refractive index of the monomer. The length of the created fiber (Lfiber ) was determined from the side-view of the CCD camera but for wider BBs (B and C columns in Fig. 2) the length of the fiber was limited by the dimensions of the used cuvette.
We developed a technique how to extract the fibers out from the solution and carry them to a different medium or place. The fiber was attached by one its end to an optical fiber and mechanically pulled out of the solution, washed by acetone and transferred. This procedure was used to observe the fibers and to measure the fiber diameter dfiber by the environmental scanning electron microscope Aquasem-Vega (ESEM) under the high pressure conditions and without metallic coating (bottom row in Fig. 2).
Figure 2 summarizes the results obtained for BBs of different widths. In the case of the narrowest fiber (column A) the experimental and related theoretical estimates of the BB parameters coincide very well but the fiber dimensions differ from the expectations. The radius of the fiber is larger than the related RBBC. In this narrow BB the intensity in its center is high enough to induce photopolymerization in the whole area of the BB core. Therefore, when the fiber is growing, it scatters more light and changes the light distribution around this structure. But its intensity is still strong enough to polymerize the wider fiber. The length of the fiber is about 4× longer than the corresponding length of the BB z’maxTp . This discrepancy is caused by the self-guiding mechanism of the light in the created fiber which significantly prolongs the fiber length as it will be shown below. In wider BBs (columns B, C) the coincidence between the experimental and theoretical values of RBBC is slightly worse. Since here the incident laser power is not sufficient to initiate polymerization in the whole area of the central core, the created fiber radius is narrower than the corresponding RBBC. The BB z’maxTp is longer than the dimensions of the cuvette and therefore the fiber fills the cuvette over all its length. The homogeneity of the fiber wide 2 µm and long 15 mm (column B in Fig. 2) is demonstrated in Fig. 3. With the available laser power we have not observed formation of any polymer structure in the outer BB intensity rings because the field intensity has not been high enough to initiate the photopolymerization here.
4. Dynamics of the fiber formation
As the previous results shown, it is worth studying how the dynamics of the fiber growth depends on the incident laser power. We used a different setup where the BB z’maxTp was short enough to observe both - the whole BB and the manufactured fiber by the CCD from a side. The setup consisted of the axicon with the apex angle α=179.065° and a combination of lenses with focal lengths f1=500 mm and f2=8 mm. The generated BB had the following parameters: 2r’B ~1.5 µm and z’maxTp ~100 µm. The top plot in Fig. 4 shows the calculated on-axis intensity profile in the BB using Eq. (5) and the images bellow show the side view of the created fiber if the image background at zero time was subtracted. Obviously the fiber grew asymmetrically with respect to the high intensity part of the BB. This also proves that the self-writing mechanism prolongs the fiber in the direction of the BB propagation. Under the studied circumstances the fiber growth stopped after 3 s due to the power losses in the process. The long fiber began to bend especially at the high intensity part of the BB. We speculate that this effect could be caused by the laser radiation pressure pushing this part forward.
Figure 5 summarizes how the length of the fiber develops in time for different illumination powers of the same BB as above. The formation has two regimes. In the first the fiber grows fast and almost linearly in time. Afterwards the growth is much slower till it is almost stopped. The higher the laser power was, the longer fiber was created. However, the growth stopped almost at the same time of 3.2 s of the BB illumination. These observations indicate that the fiber growth stopped when the power losses in the self-writing mechanism are so high that the laser power is not sufficient to initiate photopolymerization.
5. Self-written waveguide mechanism
As it was mentioned above, the fiber length exceeds several times the BB z’maxTp . We are persuaded that this phenomenon is related to the effect known as spatial soliton or self-writing waveguide [18–21]. Behind the region of the BB existence the polymer fiber continues in the growth and at the same time it serves itself as an optical waveguide. Till the intensity leaving the fiber is high enough to polymerize, the fiber grows but with decreasing speed.
To confirm that self-writing mechanism takes place in the fiber formation we carried out the experiment presented in Fig. 6. We used the same BB and image processing as in the previous Section. The process described in the caption to Fig. 6 proved that the formed fiber guides the light efficiently enough to induced photopolymerization at its end and proved the presence of the self-writing mechanism.
Figure 7 demonstrates chaotic formation of polymer fibers if high laser power was used. This phenomenon resembles creation of more fibers or chaotic fiber growth described in Refs. [34,35]. Moreover, here we observed also a reversed growth of the fibers as a result of combination of the light backscattered from the fiber structures and self-writing waveguide mechanism.
6. Fabrication of hollow fibers
We created hollow fibers by revolving the single BB around an axis parallel to the BB propagation. In this case the BB passed through rotating slightly tilted plan-parallel plate. The width of the polymer fiber wall was determined by the diameter of the BB core and the inner diameter of the hollow fiber was defined by the tilt of the plan-parallel plate. The BB parameters correspond to those in Fig. 2(A). Instead of a cuvette a pair of coverglasses separated by 150 µm was used and filled with the optical glue. Figure 8 shows an example of a single hollow fiber that was not attached to the coverglasses. Figure 9 visualizes more complicated structures attached by both ends to the coverglasses and so they are easily observable by an optical microscope.
This method provides easy manufacturing of symmetric hollow structures. Moreover, if two monomers sensitive to different wavelength were used, a narrow fiber core can be created by one narrow BB and wider cladding of manufacturing optical fiber would be cured by the revolving second BB at different wavelength  and proper width.
We demonstrate a new method how to employ the nondiffracting laser beam to form photopolymerized fibers, hollow fibers or rotationally symmetric structures. We manufactured polymer fibers of diameter close to 2 µm and of length exceeding 1.5 cm. Visualization of the fibers by environmental scanning electron microscope reveals that their surface is smooth and width is uniform along the fiber length. We proved that the manufactured fiber exceed the length of the illuminating region of the nondiffracting beam due to the self-writing mechanism. We concluded from fiber growth analyses that higher illumination laser power creates longer fibers but fiber growth stopped about after the same illumination time regardless of the illumination power. At high illumination powers we observed chaotic fiber growth not only in the direction of the light propagation but also against it. We also demonstrated formation of hollow fibers, intersected or concentric hollow fibers by circular movement of the illuminating nondiffracting beam. Our results could lead to manufacturing of cheap optical fibers, interconnecting elements in opto-fluidic lab-on-a-chip systems.
This work was supported by the GA AS CR (project No. KJB2065404), ISI IRP (AV0Z20650511), and MEYS CR (LC06007). The authors are obliged to Dr. V. Garcés-Chávez, Dr. Francisco Renero, and Taller de Óptica INAOE for manufacturing and providing the axicon.
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