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Here, we report the formation of Bessel-like beam array from periodic patterns fabricated by the four-beam interference lithography. Characteristics of the generated Bessel-like beams depend on geometrical parameters of the fabricated microaxicon-like structures, which can be easily controlled via the laser processing parameters. The output beam characteristics disclose the attributes of Bessel beams. The demonstrated method enables an easy fabrication of angular-tolerant wavefront detectors, optical tweezers, optical imaging systems or materials processing tools, having a broad range of applications.
© 2015 Optical Society of America
1. Introduction
An ideal Bessel beam [1] is a non-diffracting beam characterized by the field distribution that is proportional to the zero-order Bessel function J0. It contains equal energy in all fringes, infinite total energy and propagates with a stable radial intensity pattern [2]. Of course, the ideal Bessel beam cannot be experimentally realized, but over a limited distance its perfect approximation known as Bessel-like beam can be generated [3]. Bessel beams due to their unusual properties (propagation invariance and self-reconstruction [4, 5]) are widely used in many fields: optical manipulation [6, 7], fabrication of polymer microfibers [8], optical lithography [9], microscopy [10], cell transfection [11], as virtual tips for near-field optics [12], optical pumping [13], data distribution [14], laser drilling [15, 16], imaging [17], atomic transport [18] and trapping [19, 20], etc. Bessel beams are mainly generated using circular slits [1, 21], axicons [22, 23] or holographic gratings [24–26]. The use of axicons [27] due to their high efficiency is the most popular method for Bessel beam formation. Microaxicons are very desirable elements for Bessel beam applications in microscopic scale (e. g. trapping of particles or cells) due to integrability and low weight. There are various ways to fabricate microaxicons or axicon-like elements: vapor phase deposition of dielectric layers through shadow masks [28], reactive-ion etching [29], electron beam lithography [30], laser lithography [31, 32], electric-field induced self-organization [33], or optical lithography combined with chemical etching [34]. In this work, we demonstrate an easy, fast and flexible method to fabricate microaxicon-like elements over a large area based on the four-beam interference lithography and investigate the influence of geometrical parameters of microaxicon-like structures to the Bessel-like beam formation. Characteristics of the output beams show that all types of analyzed structures generate Bessel-like beams, having many potential applications at microscopic scale, such as material treatment, manipulation of biological or colloidal material, optical imaging or sensing.
2. Materials and methods
Microaxicon-like elements were fabricated using the four-beam interference lithography method from hybrid organic-inorganic Zr-containing negative photoresist SZ2080 [35] (FORTH, Greece) enriched with the photoinitiator 4,4‘-bis(dimethylamino)-benzophenone (concentration by weight equal to 0.5%). This material perfectly fits to production of optical components as exhibits sufficiently high optical resistance to the laser beam radiation, comparable with the damage threshold of conventional optical coatings used for optical glasses and nonlinear crystals [36], has a refractive index similar to the glass (about 1.5) [35], exhibits low shrinkage and high mechanical stability. Moreover, it is transparent in a near infrared region [36].
The principle of microaxicons-like fabrication process via the four-beam interference lithography is shown in Fig. 1. In the fabrication process, a Yb:KGW femtosecond laser (Pharos, Light Conversion) emitting pulses with a duration of 250 fs at a 100 kHz repetition rate and 515 nm wavelength of irradiation was used. Four identical laser beams were obtained by splitting the laser beam with a diffractive optical element (DOE) (Holo-Or Ltd.). Afterward, these four beams were collected by the two-lens imaging system in order to generate a four-beam interference pattern (Fig. 1(a) (right)). The period of the obtained interference pattern can be adjusted by varying magnification of the two-lens imaging system [37]. The four-beam interference pattern with 60 μm period was recorded into SZ2080 photopolymer sample (Fig. 1(a) (left)), which was prepared by using a spin-coating method. Before the laser processing, the samples were heated for ~20 min at 95°C to evaporate the solvent and solidify the samples. The photopolymerization process was initiated in a solid state of the material. After the photomodification, samples were immersed into 4-methyl-2-pentanone for 20 min (Fig. 1(b)) to dissolve unmodified polymer parts and develop the periodically arranged polymeric round-tip microaxicons (Fig. 1(c)).
Fig. 1 The principle of microaxicons-like formation process via the four-beam interference lithography: a) irradiation of a photopolymer by the four-beam interference intensity distribution (right) and enlarged photomodified area (left); b) development process; c) fabricated periodically arranged round-tip microstructures.
3. Results and discussion
Three kinds of microaxicons-like arrays have been fabricated using the four-beam interference lithography technique, which we call them type I, II and III (shown in Fig. 2). The detail investigation of the structures formation using the four-beam interference lithography was studied in our recent paper [38]. The investigation results show that the geometry of the structures correlates with the laser irradiation dose. The higher the laser irradiation dose, the higher and wider structures can be obtained (profiles in Fig. 2 and data in Table 1). The maximum height of microaxicon-like assembly is limited by the layer thickness of the photopolymer. The shape of structures is determined by the rates of photopolymerization reaction and monomer diffusion during the fabrication process of the structures [39]. The standard deviation of any geometrical parameter (height, diameter and radius of curvature) does not exceed 7%. By using the identical laser processing parameters the fabricated structures are reproducible.
Fig. 2 SEM images and profiles of microaxicon-like structures fabricated via the four-beam interference lithography using a different laser irradiation dose: a) ~3.7 J (the average laser power ~0.37 W and laser exposure time 10 s); b) ~4.7 J (the average laser power ~0.47 W and laser exposure time 10 s); c) ~27.9 J (the average laser power ~0.93 W and laser exposure time 30 s). The period is ~60 μm. SEM micrographs of the structures are tilted by 40 deg. The scale bars represent 20 µm.
Table 1. Geometrical parameters of the structures.
Optical properties of the fabricated structures were investigated by the optical performance test system (Fig. 3(a)). This system consists of a laser (wavelength 532 nm), a 3D positioning system, an objective and a CCD camera. The magnification of the system was 19. In order to characterize beams formed by manufactured structures, the sample was placed before the objective in the distance larger than the focal length of the objective (11 mm). The propagation of beam behind the sample was registered by moving the sample in the z direction. The example of registered images of beam arrays, exiting from microstructures of different type at the distance of 200 μm or 350 μm from the sample is shown in Fig. 3. The 3D intensity distribution of the beams marked in the white dashed squares (Figs. 3(b)-3(d)) are depicted in Figs. 3(e)-3(g).
Fig. 3 The scheme of the optical performance test system (a) and 2D and 3D intensity distribution of beams exiting from different types of the structures: b,e – I type; c,f – II type and d,g – III type, at distance 200 μm (b,c) and 350 μm (d) from the sample. The scale bars represent 20 µm.
The results of optical performance test show that all fabricated microstructures generate the beams with concentric rings. The intensity distribution with the series of concentric rings alludes to the Bessel beam intensity distribution that is proportional to the square of zero-order Bessel function of the first kind:
where r is the radial coordinate; k⊥ is the perpendicular wave vector component which is related to the wavelength and the half apex angle β of the cone byComparison of the average of the transverse intensity profiles of the beams (black lines in Figs. 4(b), 4(d) and 4(f)), generated by different types of structures (Fig. 4(a) – I type, Fig. 4(c) – II type, Fig. 4(e) – III type) at the distance of 200 μm (Figs. 4(a) and 4(c)) and 350 μm (Fig. 4(e)) from the sample with the theoretical Bessel beam transverse intensity distribution calculated by using Eq. (1) when the perpendicular wave vector (k⊥) is 0.9 (red line in Fig. 4(b)), 1.15 (red line in Fig. 4(d)) and 0.65 (red line in Fig. 4(f)) indicates that the generated beams have a Bessel beam characteristics. This result can be caused by significant spherical aberrations of the used microstructures [41]. Bessel-like beam propagation behind the spherical structure with significant aberration when β1 > β2 > β3 is schematically shown in Fig. 5(a).Fig. 4 Intensity distributions of the Bessel-like beams formed using different types of the microstructures: a) I; c) II; e) III at the distance of 200 μm (a,c) and 350 μm (e) from the sample and comparison of the average of the transverse intensity profiles taken along the white lines (black curves) with the numerical fit of the Bessel beam intensity distribution when k⊥ = 0.9 (b), k⊥ = 1.15 (d) and k⊥ = 0.65 (f) (red lines). The scale bars represent 20 µm.
Fig. 5 Bessel beam formation by spherically-shaped (a) and cone-shaped (b) refractive axicons (schematically) [41].
Images (Fig. 6) of the beam propagation behind of all types (I type - Figs. 6(a)-6(d); II type - Figs. 6(e)-6(h); III type - Figs. 6(i)-6(l)) of the structures demonstrate that the intensity distributions of the generated Bessel-like beams are changing with the propagation distance in all cases. It is seen that the transversal spatial frequency in the Bessel zone is slightly varying with the axial position z. The variation of the spatial frequency of the Bessel-like fringe pattern means that the conical beam angle β is a function of the axial position z (β(z)) (Fig. 5(a)), contrary to an “ideal” axicon (conical-shaped) where β is an invariant along the axial position z (Fig. 5(b)).
Fig. 6 Beams intensity distributions exiting from the different type of the structures (type I (a,b,c,d), type II (e,f,g,h) and type III (i,j,k,l)) in the transverse plane, at different distances from the apex of the microstructures: 50 μm (a,e,i); 100 μm (b,f,j); 200 μm (c,g,k); 500 μm(d,h,l). The scale bars represent 20 µm.
The experimental results present that the spatial frequency is decreasing along the axial position z (Fig. 6). The relation between the spatial frequency of the Bessel-like fringes in the transverse direction and the conical beam angle β can be expressed as
where is the fringe distance.From Eq. (3) it follows that the higher spatial frequency corresponds to the larger incident angle β. The measured fringe distance and the estimated (by Eq. (3)) angular dependence along the axial position z for the different type of the structures are shown in Figs. 7(a), 7(c) and 7(e). As seen from Figs. 7(a), 7(c) and 7(e) the conical angle dependence along z is decreasing for all structures and is varying in the range from ~11 deg to ~2 deg (brown circles), contrary to the dependence of the fringe distance (olive circles). The standard deviation of the conical angle is decreasing along the axial position z for all types of structures and does not exceed 10%.
Fig. 7 Alteration of the generated beams parameters (the conical angle β, fringe distance, central spot diameter and intensity) along the axial position z from different type of the structures: I type (a,b); II type (c,d); III type (e,f). Brown circles show the estimated conical angle along the propagation direction (a,c,e); olive circles – the measured fringe distance along z (a,c,e); red circles – the measured intensity values at different axial distance (b,d,f); black squares – the measured central spot diameter along z (b,d,f). Blue curves illustrate theoretically estimated Gaussian beam divergence when the beam waist diameter is 3 μm (b), 2.4 μm (d) and 5.4 μm (f); green curves – the estimated Gaussian beam intensity distribution along z when the initial spot size is equal to the spot size at the measured highest intensity value (4.2 μm (b), 3.9 μm (d) and 7.1 μm (f)). Arrows show the ordinate axis. DOF means depth of focus and belongs to black curves.
In “ideal” case, when an axicon has a perfect conical shape (Fig. 5(b)), the Bessel beam is a non-diffracting beam, maintaining the unchanged transversal distribution as it propagates. However, in our case, the fabricated structures have a spherical shape, and the central core diameter varies along z, but this variation disobeys the alteration of the Gaussian beam diameter along the z axis, given by the expression
where d0 is the diameter of the beam waist; z0 is the coordinate of the beam waist; z is the axial coordinate; zR is the Rayleigh length, which can be estimated from the waist diameter and the wavelength:The alteration of the central spot diameter along the z direction of the beam exiting from different type of structures (I type - Fig. 7(b), II type - Fig. 7(d), III type - Fig. 7(f)) are shown in Figs. 7(b), 7(d) and 7(f) (black solid lines). The smallest central spot of the beam exiting from I and II type structures is at ~100 μm distance from the sample, and for the III type structures is at the ~300 μm distance. The smallest core spot for the I type structures is ~3 μm, II type is ~2.4 μm and III type is ~5.4 μm, and is determined by the geometrical parameters of the spherical structure (proportional to the ratio of the radius of curvature and the diameter (~R/D)). The results (Figs. 7(b), 7(d) and 7(f)) show that the generated beam has a smaller divergence than the Gaussian beam if the Gaussian beam waist is equal to the initial spot size of the generated beam (I type: ~3 μm; II type: ~2.4 μm; III type: ~5.4 μm) (blue lines). A smaller diffraction effect means that the spreading of the generated beam is much lower than the divergence of the respective Gaussian beam. Divergence of the beam can be estimated from the depth of focus. For Gaussian beam, the depth of focus is equal to 2zR and it follows from Eq. (5) that the depth of focus is equal to ~26.6 μm, ~17.0 μm and ~86.1 μm when the wavelength (λ) is 532 nm and the beam waist (d0) is ~3 μm, ~2.4 μm and ~5.4 μm, respectively.
The depth of focus of the generated beams can be estimated from experimental results. As the depth of focus is the distance either side of the beam waist (d0), over which the beam diameter grows by . The estimated depth of focus of the generated beams from the I, II and III type structures (magenta arrows in Figs. 7(b), 7(d) and 7(f)) is ~134 μm, ~157 μm and ~488 μm, respectively. It is about 5, 9 and 6 times larger than the depth of focus of the Gaussian beam estimated above.
One more feature indicating that the generated beams are the Bessel-like beams is the axial variation of the intensity. A comparison of the axial variation of the intensity of the formed Bessel-like beam (red lines) and Gaussian beam (green lines) with a similar initial spot size is illustrated in Figs. 7(b), 7(d) and 7(f). As seen from Figs. 7(b), 7(d) and 7(f), the largest measured intensity for the I and II type structures is close to the focal length distance (~50 μm), and for the III type structures, it is closer than the measured focal length of the structure (~150 μm). This result can be determined from the shape of the structure as the III type structure differs from the spherical shape, and only the apex of the structure has the spherical part (Fig. 2(c)). Figures 7(b), 7(d) and 7(f), show the modulation of the axial intensity due to the round-tip of the structures in all cases [42]. Furthermore, Figs. 7(b), 7(d) and 7(f), exhibit that the measured axial intensity distribution of the generated beams is decreasing not so fast as the Gaussian beam intensity distribution. The decrease of intensity from the maximum value to 40% of the maximum value for the generated beams is ~3.5 (I type), ~4.5 (II type) and ~1.3 (III type) times lower than for the respective Gaussian beams, and this is one more evidence that the generated beams has the Bessel-like characteristics.
4. Conclusions
A novel and versatile method to generate the Bessel-like beam array by using the four-beam laser interference technique was developed, and an easy manipulation of the Bessel-like beam characteristics was demonstrated. The generated beams exhibit all the fundamental features of a Bessel-like beam: 1) the transverse intensity profile matches the Bessel beam transverse intensity distribution; 2) the axial divergence of the central beam is significantly lower than that of the Gaussian beam with the same initial waist size; 3) the decrease of the intensity behind the focus is either lower than for the respective Gaussian beam. The statistical deviation of the geometrical parameters of the microaxicon-like structures in the array does not exceed 7%. The variation of structure geometry leads to the maximum standard deviation of the conical angle of 10%, which is appropriate for applications where high precision is not needed. The possibility to fabricate the periodically arranged polymeric microaxicon-like arrays by using a simple and reproducible four-beam interference lithography technique paves the way for the development of a tilt-tolerant wavefront sensors, an optical tweezers, a new optical imaging system or a novel materials processing tool.
Acknowledgments
The authors acknowledge the financial support of this work to National Program “An improvement of the skills of researchers” under the supervision of Lithuanian Ministry of Education and Science, the project No. VP1-3.1-ŠMM-08-K-01-009.
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