Dark-hollow optical beams with a controllable shape for optical trapping in air A.P. Porfirev* and R.V. Skidanov Samara State Aerospace University, 34 Moskovskoye shosse, Samara 443086, Russia Image Processing Systems Institute of the Russian Academy of Sciences, 151 Molodogvardejskaya Street, Samara 443001, Russia * [email protected]
Abstract: A technique for generating dark-hollow optical beams (DHOBs) with a controllable cross-sectional intensity distribution is proposed and studied both theoretically and experimentally. Superimposed Bessel beams were used to generate such DHOBs. Variation of individual beam parameters enables the generation of Bessel-like non-diffracting beams. This technique allows the design of transmission functions for elements that shape both non-rotating and rotating DHOBs. We demonstrate photophoresis-based optical trapping and manipulation of absorbing airborne nanoclusters with such beams. ©2015 Optical Society of America OCIS codes: (140.3300) Laser beam shaping; (140.7010) Laser trapping; (350.4855) Optical tweezers or optical manipulation.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970). D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003). D. Cojoc, V. Garbin, E. Ferrari, L. Businaro, F. Romanato, and E. Di Fabrizio, “Laser trapping and micromanipulation using optical vortices,” Microelectron. Eng. 78–79, 125–131 (2005). N. Eckerskorn, L. Li, R. A. Kirian, J. Küpper, D. P. DePonte, W. Krolikowski, W. M. Lee, H. N. Chapman, and A. V. Rode, “Hollow Bessel-like beam as an optical guide for a stream of microscopic particles,” Opt. Express 21(25), 30492–30499 (2013). V. G. Shvedov, A. R. Davoyan, C. Hnatovsky, N. Engheta, and W. Krolikowski, “A long-range polarizationcontrolled optical tractor beam,” Nat. Photonics 8(11), 846–850 (2014). V. G. Shvedov, A. S. Desyatnikov, A. V. Rode, W. Krolikowski, and Y. S. Kivshar, “Optical guiding of absorbing nanoclusters in air,” Opt. Express 17(7), 5743–5757 (2009). V. G. Shvedov, A. V. Rode, Y. V. Izdebskaya, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Giant optical manipulation,” Phys. Rev. Lett. 105(11), 118103 (2010). R. Ling-Ling, G. Zhong-Yi, and Q. Shi-Liang, “Rotational motions of optically trapped microscopic particles by a vortex femtosecond laser,” Chin. Phys. B. 21(10), 104206 (2012). S. Chávez-Cerda, M. A. Meneses-Nava, and J. M. Hickmann, “Interference of traveling nondiffracting beams,” Opt. Lett. 23(24), 1871–1873 (1998). D. McGloin, V. Garcés-Chávez, and K. Dholakia, “Interfering Bessel beams for optical micromanipulation,” Opt. Lett. 28(8), 657–659 (2003). R. Vasilyeu, A. Dudley, N. Khilo, and A. Forbes, “Generating superpositions of higher-order Bessel beams,” Opt. Express 17(26), 23389–23395 (2009). R. Rop, A. Dudley, C. Lopez-Mariscal, and A. Forbes, “Measuring the rotation rates of superpositions of higherorder Bessel beams,” J. Mod. Opt. 59(3), 259–267 (2012). A. Dudley and A. Forbes, “From stationary annular rings to rotating Bessel beams,” J. Opt. Soc. Am. A 29(4), 567–573 (2012). A. P. Porfirev and R. V. Skidanov, “A simple method of the formation nondiffracting hollow optical beams with intensity distribution in form of a regular polygon contour,” Comput. Opt. 38(2), 243–248 (2014). V. V. Kotlyar, V. A. Soifer, and S. N. Khonina, “An algorithm for the generation of laser beams with longitudinal periodicity: rotating images,” J. Mod. Opt. 44(7), 1409–1416 (1997). A. Fedotowsky and K. Lehovec, “Optimal filter design for annular imaging,” Appl. Opt. 13(12), 2919–2923 (1974). A. P. Porfirev and R. V. Skidanov, “Generation of an array of optical bottle beams using a superposition of Bessel beams,” Appl. Opt. 52(25), 6230–6238 (2013).
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Received 5 Jan 2015; revised 5 Mar 2015; accepted 12 Mar 2015; published 24 Mar 2015 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.008373 | OPTICS EXPRESS 8373
18. H. Rohatschek, “Direction, magnitude and causes of photophoretic forces,” J. Aerosol Sci. 16(1), 29–42 (1985). 19. O. Jovanovic, “Photophoresis: light-induced motion of particles suspended in gas,” J. Quant. Spectrosc. Radiat. Transf. 110(11), 889–901 (2009). 20. H. Rohatschek, “Semi-empirical model of photophoretic forces for the entire range of pressures,” J. Aerosol Sci. 26(5), 717–734 (1995).
1. Introduction The optical manipulation of micro- and nanoscale objects, which was first demonstrated by A. Ashkin [1], has become a widespread technique. The development of holographic optical tweezers (HOTs) allows the generation of complex optical traps that can be used to manipulate various objects in various environments such as in liquids [2, 3] or air [4, 5]. The optical trapping and manipulation of different types of microscopic objects is possible due to the action of radiation pressure and thermal forces, e.g., photophoretic forces [6]. Using photophoretic forces, Shvedov et al. [7] implemented “giant optical manipulation” and performed the optical transport of microscopic objects over a meter-scale distance. The principle of action of the giant optical manipulator is based on the fact that the bright ring of light intensity acts as a repelling “pipe wall” on the particles trapped in the dark region near the axis, while the axial component of the thermal force pushes particles along the pipeline. In particular, Shvedov et al. [7] used a single fork-type amplitude diffractive hologram to form the dark-hollow optical beams (DHOBs). Unfortunately, there are many challenges with the annular cross-sectional shape of DHOBs for manipulating non-spherical objects. For example, controlled manipulation is difficult to achieve because of the rotation of the captured microobjects [8]. One possible solution of this problem requires matching the forms of the hollow beam and the captured microscopic object. In this article, we present a new method for shaping non-diffracting DHOBs, where the intensity distribution of the DHOB cross-section can form simple geometrical shapes. Our method allows controlled moving of non-spherical microscopic objects without rotation and minimizes the risk of thermal damage. This method is based on the variation of the parameters of interfering coaxial Bessel beams. The superposition of Bessel beams, which are a combination of beams of different orders, has been previously discussed by many authors [9–13]. We propose a new method for the formation of hollow beams with a controlled shape not only by varying the orders of the individual beams but also by varying the amplitude of the individual beams used in the superposition. 2. Description of technique for generating DHOBs 2.1 The interference of two coaxial Bessel beams An ideal n-th order Bessel beam is described by the complex amplitude E ( r , ϕ , z ) = A0 exp ( ik z z ) J n ( kr r ) exp ( inϕ ) ,
(1)
where J n is the n-th order Bessel function, k z and kr are the longitudinal and radial wave number, respectively, k = k z2 + kr2 = 2π λ , and ( r , ϕ , z ) are the cylindrical coordinates. Let us consider the interference of two coaxial Bessel beams with equal radial wave numbers and different topological charges n and m at z = 0 . The resultant complex amplitude is given by E ( r , ϕ , z ) = A0 n J n ( kr r ) exp ( inϕ ) + A0 m J m ( kr r ) exp ( imϕ ) .
(2)
The intensity of these superimposed beams is I ( r , ϕ ) = Aon J n ( kr r ) exp ( inϕ ) + A0 m J m ( kr r ) exp ( imϕ ) = 2
A02n J n2 ( k r r ) + A02m J m2 ( kr r ) + 2 A0 n A0 m J n ( kr r ) J m ( kr r ) cos ( n − m ) ϕ .
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(3)
Received 5 Jan 2015; revised 5 Mar 2015; accepted 12 Mar 2015; published 24 Mar 2015 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.008373 | OPTICS EXPRESS 8374
The phase of these superimposed beams is P ( r , ϕ ) = arg { A0 n J n ( kr r ) exp ( inϕ ) + A0 m J m ( kr r ) exp ( imϕ )} = A J ( k r ) sin ( nϕ ) + A0 m J m ( kr r ) sin ( mϕ ) arc tan 0 n n r . A0 n J n ( kr r ) cos ( nϕ ) + A0 m J m ( kr r ) cos ( mϕ )
(4)
where arg{} is an argument of a complex number, arctan{} is an arctangent. From Eq. (3), we see that the generated intensity profile is determined not only by the difference of beam orders but also by the real factors A0n and A0 m = α A0 n ( 0 ≤ α < 1 ), which determine the weight contribution of each beam amplitude to the overall distribution. The factors A0m and A0n provide an additional degree of freedom in the calculation of the transmission function of the elements forming non-diffracting DHOBs with a predetermined shape. 2.2 Generation of DHOBs with an intensity distribution in the form of a regular polygon In case of superimposed different order Bessel beams ( n > m ), we can generate DHOBs with a cross-sectional intensity distribution that has a predetermined shape in the form of a regular polygon contour [14]. For example, this polygon contour can be generated when α = 0.2 . In 2 this case, the additive with a factor A0m in Eq. (3) can be neglected, and we can write I ( r , ϕ ) = A02n J n2 ( kr r ) + 2 A0 n A0 m J n ( kr r ) J m ( kr r ) cos ( n − m ) ϕ .
(5)
When the angle ϕ varies from 0 to 2π , cosine changes its sign 2(n-m) times, and, accordingly, it has 2(n-m) extreme points. Depending on the sign of the product J n ( kr r ) J m ( kr r ) , the value of cosine leads to an increase or a decrease in the light field
2 J n2 ( kr r ) in Eq. (5), and changes the profile intensity, which is defined by the first additive A0n
of the annular beam. Thus, periodic modulation of the beam profile with a period 2π ( n − m )
will depend on ϕ . The value (n-m) determines the type of the regular polygon. Figure 1 shows the results of the calculation for the parameters of n = 8, m = 3, kr = 51996 m−1, and α = 0.2 .
Fig. 1. The intensity (top row) and phase (bottom row) distributions of two superimposed Bessel beams (n = 8, m = 3, kr = 51996 m−1, and α = 0.2): (a), (e) z = 100 mm; (b), (f) z = 300 mm; (c), (g) z = 500 mm; and (d), (h) z = 700 mm. Mesh step is 200 μm.
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2.3 Generation of DHOBs with an intensity distribution in form of a “cogwheel” Let us consider the case where the factor α ranges from 0.3 to 0.6. In this case, we cannot 2 neglect the additive terms with the factor A0m . Therefore, each additive term in Eq. (3) makes a significant contribution to the shape of the generated light field. Different generated intensity distributions will depend on the difference between the radii of the central rings that are defined by the orders of corresponding Bessel beams. For example, when the difference between the orders is m − n = 2 , the intensity profile given by Eq. (3) has the shape of a “cogwheel.” The number of “teeth” on the cogwheel is determined by the difference of the orders m − n . Note, that in order to form a “cogwheel,” the orders of n and m must have opposite signs. As mentioned above, the third additive term in Eq. (3) determines the number and type of modulation of the intensity profile determined by sum A02n J n2 ( kr r ) + A02m J m2 ( kr r ) . In the case of two superimposed Bessel beams with orders m = -n, the cross-sectional intensity distribution is a set of 2m light peaks placed in a circle, which has a radius determined by the value m. If the following conditions hold true:
m ≠ n , m > 0, n < 0, m − n = 2, and 0.3 ≤ α ≤ 0.6 the intensity profile defined by the sum A02n J n2 ( kr r ) + A02m J m2 ( kr r ) undergoes
(6) m−n
modulations. These modulations lead to an increase or a decrease in the intensity of various 2 2 sections, which correspond the additive A0n J n2 ( kr r ) or A0m J m2 ( kr r ) , and to the generation of the cross-sectional intensity distribution in the shape of a “cogwheel.” Figure 2 shows the results of the calculation for the parameters of n = 4, m = −2, kr = 51996 m−1, and α = 0.4 .
Fig. 2. The intensity (top row) and phase (bottom row) distributions of two superimposed Bessel beams (n = 4, m = −2, kr = 51996 m−1, and α = 0.4): (a), (e) z = 100 mm; (b), (f) z = 300 mm; (c), (g) z = 500 mm; and (d), (h) z = 700 mm. Mesh step is 100 μm.
2.4 Generation of rotating DHOBs with a predetermined intensity distribution Now, let us consider the case when the superimposed beams have different values of kr . In this case, the transverse profile of the intensity will be rotated as defined by Eq. (3) [15]. The period of rotation is determined by the kr values of each beam. Figure 3 shows the results of the calculation for the parameters of n = 4, m = −2, kr1 = 51996 m−1, kr2 = 48296 m−1, and α = 0.4 .
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Fig. 3. The intensity (the top row) and phase (the bottom row) distributions of two superimposed Bessel beams (n = 4, m = −2, kr1 = 51996 m−1, kr2 = 48296 m−1, and α = 0.4):
(a), (e) z = 100 mm; (b), (f) z = 300 mm; (c), (g) z = 500 mm; and (d), (h) z = 700 mm. Mesh step is 100 μm.
2.5 Calculation of the transmission function for the elements forming the predetermined hollow light beams For the formation of the Bessel beam, we can use a diffractive optical element (DOE), which has a phase transmission function of the form [16]:
τ ( r , ϕ ) = sgn ( J n ( kr r ) ) × exp ( inϕ ) .
(7)
This element shapes a light field with an amplitude proportional to the Bessel function J n ( kr r ) exp ( inϕ ) . To estimate the distance along which a generated single diffraction-free beam retains its properties, we can use the following expression: zmax = Rk z kr (8) The diffractive optical elements for the generation of N superimposed Bessel beams have previously been considered [17]. The transmission function of these elements is given by the expression: N
( ( ))
τ ( r , ϕ ) = C p sgn J n kr r × exp ( in pϕ ) p =1
p
p
(9)
where C p are complex factors. We used the following equation to calculate the complex transmission function of the elements generating two coaxial superimposed Bessel beams:
τ ( r , ϕ ) = C1 sgn ( J n ( kr r ) ) × exp ( inϕ ) + C2 sgn ( J m ( kr r ) ) × exp ( imϕ )
(10)
Neglecting the amplitude component in Eq. (9) and only considering the phase of the transmission function, we achieved an efficient and high-quality generation of the light fields described by Eq. (2). Generally, the factors C1 and C2 in Eq. (10) are not equal to the factors A0n and A0m in Eq. (3) because ignoring the amplitude component in Eq. (10) leads to a redistribution of the energy of the light field generated by each beam in Eq. (3). Figure 4 shows the calculated according to Eq. (10) for the DOE phase function with the following parameters: n1 = 10 ; n2 = 10 , kr1 = kr2 = 51996 m−1; C1 = 1 ; C2 = 0.6 ; and R = 3 mm.
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Fig. 4. The phase function of the DOE forming an intensity distribution with a regular pentagon contour.
3. Experimental results
3.1 Experimental generation of two superimposed Bessel beams by spatial light modulation To generate the superimposed Bessel beams, we utilized a spatial light modulator (SLM) PLUTO-VIS (1920 × 1080 pixel resolution, 8-µm pixel size). Figure 5 shows the experimental optical setup. The output beam from a solid-state laser ( λ = 532 nm) was attenuated using a neutral density filter. A system composed of a microobjective MO (40 × , NA = 0.6), lens L1 ( f1 = 350 mm ), and pinhole PH (40-µm aperture) was utilized to generate a homogeneous Gaussian intensity profile of the laser beam incident on the SLM. Then, lenses L2 and L3 with focal lengths 350 and 150 mm, respectively, formed an image of the plane conjugated to the SLM display in the plane z = 0 . A CMOS-camera MDCE-5A (1/2 in, 1280 × 1024-pixel resolution) was used to record the formed intensity distributions. In the experiment, with the phase pattern of 1000 × 1000 pixels used, the size of the phase element at the SLM output was approximately equal to 8 × 8 mm and 6 × 6 mm, and the illuminating Gaussian beam diameter was about 7 mm. The zero and first diffraction orders were spatially separated by the superposition of the original phase function and a linear phase mask.
Fig. 5. Experimental optical setup: L is a solid-state laser, F is a neutral density filter, MO is a microobjective (40x, NA = 0.6), PH is a pinhole (40-µm aperture),L1, L2, and L3 are the lenses with focal lengths f1 = 350 mm, f 2 = 350 mm, and f 3 = 150 mm, respectively, BS is a beam splitter, SLM is a spatial light modulator (PLUTO Spatial Light Modulator, 1920x1080), RP is a rectangular prism, D is a diaphragm, CMOS is a video camera (VSTT-252), and Rail is an optical rail.
To analyze the intensity distributions of DHOBs formed at different distances, the CMOS camera along the optical axis was relocated. Figure 6 shows the experimentally obtained intensity distribution at different distances for two superimposed Bessel beams (n = 8, m = 3, kr = 51996 m−1, and α = 0.2 ; the size of the phase element was equal to 8 × 8 mm). Figure 7 shows the experimentally obtained intensity distribution at different distances for two
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superimposed Bessel beams (n = 4, m = −2, kr1 = 51996 m−1, kr2 = 48296 m−1, and α = 0.4 ; the size of the phase element was equal to 6 × 6 mm). The longitudinal intensity distribution was reconstructed on the cross-sectional transverse distributions images recorded by the CMOS camera.
Fig. 6. Experimental intensity distribution for two superimposed Bessel beams (n = 8, m = 3, kr = 51996 m−1, and α = 0.2).
Fig. 7. Experimental intensity distribution for two superimposed (n = 4, m = −2, kr1 = 51996 m−1, kr2 = 48296 m−1, and α = 0.4 ).
Bessel
beams
3.2 Optical trapping experiments with airborne particles In optical trapping experiments with DHOBs, we used a solid-state laser ( λ = 532 nm, with a maximum output power of 500 mW) [Fig. 8]. The laser beam was expanded with a telescope (L1 with f1 = 15 mm and L2 with f 2 = 35 mm) to illuminate the DOE, which has the transmission function shown in Fig. 4. A generated beam was focused by a microobjective MO1 (20 × , NA = 0.4). An airborne absorbing particle in a cuvette is trapped in the area of minimum intensity of the generated DHOB. Observation of the particle trapping was possible due to the scattered light recorded by the video camera Cam1 (MDCE-5, 1280 × 1024 pixels). The particles were imaged through a microobjective MO2 (10 × , NA = 0.3).
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Fig. 8. Experimental setup for optical trapping experiments: L is a solid-state laser (λ = 532 nm), L1, L2 are lenses with f1 = 15 mm and f2 = 35 mm, respectively, MO1 is a microobjective (20 × , NA = 0.4), MO2 is a microobjective (10 × , NA = 0.3), DOE is a diffractive optical element with the phase function shown in Fig. 4, C is a cuvette, and Cam1 is a video camera (MDCE-5, 1280 × 1024 pixels).
To demonstrate the optical trapping and holding of absorbing particles, we used carbon nanoparticle agglomeration. The typical size of the agglomerations ranged up to tens of micrometers [Fig. 9]. The trapping of such particles could be observed with the naked eye due to scattered light from their surfaces.
Fig. 9. Carbon nanoparticle agglomerations used in the experiments.
Figure 10 shows the typical results of experiments on the trapping and holding of absorbing particles. In order to put carbon nanoparticle agglomerations inside the minimum intensity region of the DHOB, the particles were sprayed with a syringe pump. Thus, the particles initially had a significant acceleration directed to the bottom of the cuvette. An experiment shows that using DHOBs we have been able carry out a multiple simultaneous particles trapping. It is clearly seen that some of the particles are trapped stable, and some of the particles keep on moving inside volume defined by DHOB structure.
Fig. 10. Experimental optical trapping and holding of absorbing microparticles with the DHOB generated by DOE (Media 1). Three particles are trapped stable (in the bottom right of the images). One particle (denoted by an arrow) keeps on moving inside volume defined by DHOB structure.
To explain the mechanism behind such particle trapping, it is necessary to review the combined effect of all possible forces acting on a particle in our experimental system: the gravity G, radiation force FR, and photophoretic force Fp = FΔT + FΔα [Fig. 11(a)]. The gravity is always directed downward, regardless of the position of the particle. The radiation
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force is directed along the direction of beam propagation. There are two types of photophoretic forces: FΔT resulting from a temperature gradient and FΔα resulting from a different thermal accommodation coefficient α [18, 19]. The photophoretic force FΔT can be directed from the hotter to the colder side of the body (positive photophoresis) or from the colder side to the hotter side of the body (negative photophoresis) [20]. The direction of FΔα is particle body-fixed, meaning that it is always directed from α1 to α 2 (suppose α1 > α 2 ), as shown in Fig. 11(a) [20]. The ratio of magnitude of the FΔα to that of the FΔT is inversely proportional to the pressure. At atmospheric pressure or below the FΔα is practically alone present [20]. The photophoretic force is orders of magnitude larger than the radiation force [7]. Therefore, magnitude and direction of the resultant force is defined mainly by gravity and the photophoretic force. Figures 11(b)-11(e) show directions of the forces depending on the position of the particles inside the minimum intensity region of the DHOB. If the direction of the photophoretic force is not equal to the gravity and its magnitude can exceed that of gravity, the particle moves upwards (from the maximum intensity of the DHOB to the minimum intensity) [Figs. 11(b) and 11(c)]. When the gravity becomes more than the photophoretic force, the particle moves downward [Figs. 11(d) and 11(e)]. For massive particles trapped in the maximum intensity of the DHOB, the gravity is balanced with the photophoretic force. Such particles can maintain their position along the beam or move over relatively short distances (for instance, three particles trapped in the bottom right of the images in Fig. 10).
Fig. 11. Illustration of trapping mechanism involved in DHOB trapping. (a) shows possible forces acting on a particle in our experimental system; (b-e) show directions of the forces depending on the position of the particles inside the minimum intensity region of the DHOB.
Figure 12 shows the motion stages of a carbon nanoparticle agglomeration trapped by the DHOB with an intensity distribution in the form of a regular pentagon contour. The particle moves up and downward (along the axis Y) and along the beam (along the axis Z). The particle image size changes because of defocusing that occurs when agglomeration moves in a plane YX perpendicular to image plane. Figure 13 shows all the points in which trapped particle was located during observation. It can be seen that the particle is located in an area of about 200x200 μm. Thus, using DHOBs, we can keep the absorbing particles in a bounded volume with a predetermined shape. In the case of non-rotating beams, it is optical tubes with different cross-sectional shape; in the case of rotating beams, it is more complex optical threedimensional structure.
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Received 5 Jan 2015; revised 5 Mar 2015; accepted 12 Mar 2015; published 24 Mar 2015 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.008373 | OPTICS EXPRESS 8381
Fig. 12. Motion stages of a carbon nanoparticle agglomeration trapped by the DHOB with an intensity distribution in the form of a regular pentagon contour.
Fig. 13. The points in which trapped particle was located during observation.
4. Conclusions
We proposed a method that allows the generation of DHOBs with a controllable crosssectional intensity distribution. We used this technique to shape various rotating and nonrotating DHOBs. Such beams can be used for optical trapping particles in air because the hollow shape of the beams allows us to hold the particles in the region of minimum intensity. Furthermore, we have generated DHOBs by SLM and demonstrated that these beams have diffraction-free properties. In experiments with airborne particles using DHOBs with an intensity distribution in form of a regular polygon, we trapped and held absorbing particles in area of minimum intensity. The ability to generate DHOBs with a controllable cross-sectional intensity distribution can be useful in experimental studies of various airborne particles and microbiology. Acknowledgments
The research was financially supported by RSF grant No. 14-19-00114. We thank D.P. Porfirev for valuable discussions and assistance.
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Received 5 Jan 2015; revised 5 Mar 2015; accepted 12 Mar 2015; published 24 Mar 2015 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.008373 | OPTICS EXPRESS 8382
Abstract: A technique for generating dark-hollow optical beams (DHOBs) with a controllable cross-sectional intensity distribution is proposed and studied both theoretically and experimentally. Superimposed Bessel beams were used to generate such DHOBs. Variation of individual beam parameters enables the generation of Bessel-like non-diffracting beams. This technique allows the design of transmission functions for elements that shape both non-rotating and rotating DHOBs. We demonstrate photophoresis-based optical trapping and manipulation of absorbing airborne nanoclusters with such beams. ©2015 Optical Society of America OCIS codes: (140.3300) Laser beam shaping; (140.7010) Laser trapping; (350.4855) Optical tweezers or optical manipulation.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970). D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003). D. Cojoc, V. Garbin, E. Ferrari, L. Businaro, F. Romanato, and E. Di Fabrizio, “Laser trapping and micromanipulation using optical vortices,” Microelectron. Eng. 78–79, 125–131 (2005). N. Eckerskorn, L. Li, R. A. Kirian, J. Küpper, D. P. DePonte, W. Krolikowski, W. M. Lee, H. N. Chapman, and A. V. Rode, “Hollow Bessel-like beam as an optical guide for a stream of microscopic particles,” Opt. Express 21(25), 30492–30499 (2013). V. G. Shvedov, A. R. Davoyan, C. Hnatovsky, N. Engheta, and W. Krolikowski, “A long-range polarizationcontrolled optical tractor beam,” Nat. Photonics 8(11), 846–850 (2014). V. G. Shvedov, A. S. Desyatnikov, A. V. Rode, W. Krolikowski, and Y. S. Kivshar, “Optical guiding of absorbing nanoclusters in air,” Opt. Express 17(7), 5743–5757 (2009). V. G. Shvedov, A. V. Rode, Y. V. Izdebskaya, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Giant optical manipulation,” Phys. Rev. Lett. 105(11), 118103 (2010). R. Ling-Ling, G. Zhong-Yi, and Q. Shi-Liang, “Rotational motions of optically trapped microscopic particles by a vortex femtosecond laser,” Chin. Phys. B. 21(10), 104206 (2012). S. Chávez-Cerda, M. A. Meneses-Nava, and J. M. Hickmann, “Interference of traveling nondiffracting beams,” Opt. Lett. 23(24), 1871–1873 (1998). D. McGloin, V. Garcés-Chávez, and K. Dholakia, “Interfering Bessel beams for optical micromanipulation,” Opt. Lett. 28(8), 657–659 (2003). R. Vasilyeu, A. Dudley, N. Khilo, and A. Forbes, “Generating superpositions of higher-order Bessel beams,” Opt. Express 17(26), 23389–23395 (2009). R. Rop, A. Dudley, C. Lopez-Mariscal, and A. Forbes, “Measuring the rotation rates of superpositions of higherorder Bessel beams,” J. Mod. Opt. 59(3), 259–267 (2012). A. Dudley and A. Forbes, “From stationary annular rings to rotating Bessel beams,” J. Opt. Soc. Am. A 29(4), 567–573 (2012). A. P. Porfirev and R. V. Skidanov, “A simple method of the formation nondiffracting hollow optical beams with intensity distribution in form of a regular polygon contour,” Comput. Opt. 38(2), 243–248 (2014). V. V. Kotlyar, V. A. Soifer, and S. N. Khonina, “An algorithm for the generation of laser beams with longitudinal periodicity: rotating images,” J. Mod. Opt. 44(7), 1409–1416 (1997). A. Fedotowsky and K. Lehovec, “Optimal filter design for annular imaging,” Appl. Opt. 13(12), 2919–2923 (1974). A. P. Porfirev and R. V. Skidanov, “Generation of an array of optical bottle beams using a superposition of Bessel beams,” Appl. Opt. 52(25), 6230–6238 (2013).
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18. H. Rohatschek, “Direction, magnitude and causes of photophoretic forces,” J. Aerosol Sci. 16(1), 29–42 (1985). 19. O. Jovanovic, “Photophoresis: light-induced motion of particles suspended in gas,” J. Quant. Spectrosc. Radiat. Transf. 110(11), 889–901 (2009). 20. H. Rohatschek, “Semi-empirical model of photophoretic forces for the entire range of pressures,” J. Aerosol Sci. 26(5), 717–734 (1995).
1. Introduction The optical manipulation of micro- and nanoscale objects, which was first demonstrated by A. Ashkin [1], has become a widespread technique. The development of holographic optical tweezers (HOTs) allows the generation of complex optical traps that can be used to manipulate various objects in various environments such as in liquids [2, 3] or air [4, 5]. The optical trapping and manipulation of different types of microscopic objects is possible due to the action of radiation pressure and thermal forces, e.g., photophoretic forces [6]. Using photophoretic forces, Shvedov et al. [7] implemented “giant optical manipulation” and performed the optical transport of microscopic objects over a meter-scale distance. The principle of action of the giant optical manipulator is based on the fact that the bright ring of light intensity acts as a repelling “pipe wall” on the particles trapped in the dark region near the axis, while the axial component of the thermal force pushes particles along the pipeline. In particular, Shvedov et al. [7] used a single fork-type amplitude diffractive hologram to form the dark-hollow optical beams (DHOBs). Unfortunately, there are many challenges with the annular cross-sectional shape of DHOBs for manipulating non-spherical objects. For example, controlled manipulation is difficult to achieve because of the rotation of the captured microobjects [8]. One possible solution of this problem requires matching the forms of the hollow beam and the captured microscopic object. In this article, we present a new method for shaping non-diffracting DHOBs, where the intensity distribution of the DHOB cross-section can form simple geometrical shapes. Our method allows controlled moving of non-spherical microscopic objects without rotation and minimizes the risk of thermal damage. This method is based on the variation of the parameters of interfering coaxial Bessel beams. The superposition of Bessel beams, which are a combination of beams of different orders, has been previously discussed by many authors [9–13]. We propose a new method for the formation of hollow beams with a controlled shape not only by varying the orders of the individual beams but also by varying the amplitude of the individual beams used in the superposition. 2. Description of technique for generating DHOBs 2.1 The interference of two coaxial Bessel beams An ideal n-th order Bessel beam is described by the complex amplitude E ( r , ϕ , z ) = A0 exp ( ik z z ) J n ( kr r ) exp ( inϕ ) ,
(1)
where J n is the n-th order Bessel function, k z and kr are the longitudinal and radial wave number, respectively, k = k z2 + kr2 = 2π λ , and ( r , ϕ , z ) are the cylindrical coordinates. Let us consider the interference of two coaxial Bessel beams with equal radial wave numbers and different topological charges n and m at z = 0 . The resultant complex amplitude is given by E ( r , ϕ , z ) = A0 n J n ( kr r ) exp ( inϕ ) + A0 m J m ( kr r ) exp ( imϕ ) .
(2)
The intensity of these superimposed beams is I ( r , ϕ ) = Aon J n ( kr r ) exp ( inϕ ) + A0 m J m ( kr r ) exp ( imϕ ) = 2
A02n J n2 ( k r r ) + A02m J m2 ( kr r ) + 2 A0 n A0 m J n ( kr r ) J m ( kr r ) cos ( n − m ) ϕ .
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(3)
Received 5 Jan 2015; revised 5 Mar 2015; accepted 12 Mar 2015; published 24 Mar 2015 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.008373 | OPTICS EXPRESS 8374
The phase of these superimposed beams is P ( r , ϕ ) = arg { A0 n J n ( kr r ) exp ( inϕ ) + A0 m J m ( kr r ) exp ( imϕ )} = A J ( k r ) sin ( nϕ ) + A0 m J m ( kr r ) sin ( mϕ ) arc tan 0 n n r . A0 n J n ( kr r ) cos ( nϕ ) + A0 m J m ( kr r ) cos ( mϕ )
(4)
where arg{} is an argument of a complex number, arctan{} is an arctangent. From Eq. (3), we see that the generated intensity profile is determined not only by the difference of beam orders but also by the real factors A0n and A0 m = α A0 n ( 0 ≤ α < 1 ), which determine the weight contribution of each beam amplitude to the overall distribution. The factors A0m and A0n provide an additional degree of freedom in the calculation of the transmission function of the elements forming non-diffracting DHOBs with a predetermined shape. 2.2 Generation of DHOBs with an intensity distribution in the form of a regular polygon In case of superimposed different order Bessel beams ( n > m ), we can generate DHOBs with a cross-sectional intensity distribution that has a predetermined shape in the form of a regular polygon contour [14]. For example, this polygon contour can be generated when α = 0.2 . In 2 this case, the additive with a factor A0m in Eq. (3) can be neglected, and we can write I ( r , ϕ ) = A02n J n2 ( kr r ) + 2 A0 n A0 m J n ( kr r ) J m ( kr r ) cos ( n − m ) ϕ .
(5)
When the angle ϕ varies from 0 to 2π , cosine changes its sign 2(n-m) times, and, accordingly, it has 2(n-m) extreme points. Depending on the sign of the product J n ( kr r ) J m ( kr r ) , the value of cosine leads to an increase or a decrease in the light field
2 J n2 ( kr r ) in Eq. (5), and changes the profile intensity, which is defined by the first additive A0n
of the annular beam. Thus, periodic modulation of the beam profile with a period 2π ( n − m )
will depend on ϕ . The value (n-m) determines the type of the regular polygon. Figure 1 shows the results of the calculation for the parameters of n = 8, m = 3, kr = 51996 m−1, and α = 0.2 .
Fig. 1. The intensity (top row) and phase (bottom row) distributions of two superimposed Bessel beams (n = 8, m = 3, kr = 51996 m−1, and α = 0.2): (a), (e) z = 100 mm; (b), (f) z = 300 mm; (c), (g) z = 500 mm; and (d), (h) z = 700 mm. Mesh step is 200 μm.
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2.3 Generation of DHOBs with an intensity distribution in form of a “cogwheel” Let us consider the case where the factor α ranges from 0.3 to 0.6. In this case, we cannot 2 neglect the additive terms with the factor A0m . Therefore, each additive term in Eq. (3) makes a significant contribution to the shape of the generated light field. Different generated intensity distributions will depend on the difference between the radii of the central rings that are defined by the orders of corresponding Bessel beams. For example, when the difference between the orders is m − n = 2 , the intensity profile given by Eq. (3) has the shape of a “cogwheel.” The number of “teeth” on the cogwheel is determined by the difference of the orders m − n . Note, that in order to form a “cogwheel,” the orders of n and m must have opposite signs. As mentioned above, the third additive term in Eq. (3) determines the number and type of modulation of the intensity profile determined by sum A02n J n2 ( kr r ) + A02m J m2 ( kr r ) . In the case of two superimposed Bessel beams with orders m = -n, the cross-sectional intensity distribution is a set of 2m light peaks placed in a circle, which has a radius determined by the value m. If the following conditions hold true:
m ≠ n , m > 0, n < 0, m − n = 2, and 0.3 ≤ α ≤ 0.6 the intensity profile defined by the sum A02n J n2 ( kr r ) + A02m J m2 ( kr r ) undergoes
(6) m−n
modulations. These modulations lead to an increase or a decrease in the intensity of various 2 2 sections, which correspond the additive A0n J n2 ( kr r ) or A0m J m2 ( kr r ) , and to the generation of the cross-sectional intensity distribution in the shape of a “cogwheel.” Figure 2 shows the results of the calculation for the parameters of n = 4, m = −2, kr = 51996 m−1, and α = 0.4 .
Fig. 2. The intensity (top row) and phase (bottom row) distributions of two superimposed Bessel beams (n = 4, m = −2, kr = 51996 m−1, and α = 0.4): (a), (e) z = 100 mm; (b), (f) z = 300 mm; (c), (g) z = 500 mm; and (d), (h) z = 700 mm. Mesh step is 100 μm.
2.4 Generation of rotating DHOBs with a predetermined intensity distribution Now, let us consider the case when the superimposed beams have different values of kr . In this case, the transverse profile of the intensity will be rotated as defined by Eq. (3) [15]. The period of rotation is determined by the kr values of each beam. Figure 3 shows the results of the calculation for the parameters of n = 4, m = −2, kr1 = 51996 m−1, kr2 = 48296 m−1, and α = 0.4 .
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Received 5 Jan 2015; revised 5 Mar 2015; accepted 12 Mar 2015; published 24 Mar 2015 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.008373 | OPTICS EXPRESS 8376
Fig. 3. The intensity (the top row) and phase (the bottom row) distributions of two superimposed Bessel beams (n = 4, m = −2, kr1 = 51996 m−1, kr2 = 48296 m−1, and α = 0.4):
(a), (e) z = 100 mm; (b), (f) z = 300 mm; (c), (g) z = 500 mm; and (d), (h) z = 700 mm. Mesh step is 100 μm.
2.5 Calculation of the transmission function for the elements forming the predetermined hollow light beams For the formation of the Bessel beam, we can use a diffractive optical element (DOE), which has a phase transmission function of the form [16]:
τ ( r , ϕ ) = sgn ( J n ( kr r ) ) × exp ( inϕ ) .
(7)
This element shapes a light field with an amplitude proportional to the Bessel function J n ( kr r ) exp ( inϕ ) . To estimate the distance along which a generated single diffraction-free beam retains its properties, we can use the following expression: zmax = Rk z kr (8) The diffractive optical elements for the generation of N superimposed Bessel beams have previously been considered [17]. The transmission function of these elements is given by the expression: N
( ( ))
τ ( r , ϕ ) = C p sgn J n kr r × exp ( in pϕ ) p =1
p
p
(9)
where C p are complex factors. We used the following equation to calculate the complex transmission function of the elements generating two coaxial superimposed Bessel beams:
τ ( r , ϕ ) = C1 sgn ( J n ( kr r ) ) × exp ( inϕ ) + C2 sgn ( J m ( kr r ) ) × exp ( imϕ )
(10)
Neglecting the amplitude component in Eq. (9) and only considering the phase of the transmission function, we achieved an efficient and high-quality generation of the light fields described by Eq. (2). Generally, the factors C1 and C2 in Eq. (10) are not equal to the factors A0n and A0m in Eq. (3) because ignoring the amplitude component in Eq. (10) leads to a redistribution of the energy of the light field generated by each beam in Eq. (3). Figure 4 shows the calculated according to Eq. (10) for the DOE phase function with the following parameters: n1 = 10 ; n2 = 10 , kr1 = kr2 = 51996 m−1; C1 = 1 ; C2 = 0.6 ; and R = 3 mm.
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Received 5 Jan 2015; revised 5 Mar 2015; accepted 12 Mar 2015; published 24 Mar 2015 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.008373 | OPTICS EXPRESS 8377
Fig. 4. The phase function of the DOE forming an intensity distribution with a regular pentagon contour.
3. Experimental results
3.1 Experimental generation of two superimposed Bessel beams by spatial light modulation To generate the superimposed Bessel beams, we utilized a spatial light modulator (SLM) PLUTO-VIS (1920 × 1080 pixel resolution, 8-µm pixel size). Figure 5 shows the experimental optical setup. The output beam from a solid-state laser ( λ = 532 nm) was attenuated using a neutral density filter. A system composed of a microobjective MO (40 × , NA = 0.6), lens L1 ( f1 = 350 mm ), and pinhole PH (40-µm aperture) was utilized to generate a homogeneous Gaussian intensity profile of the laser beam incident on the SLM. Then, lenses L2 and L3 with focal lengths 350 and 150 mm, respectively, formed an image of the plane conjugated to the SLM display in the plane z = 0 . A CMOS-camera MDCE-5A (1/2 in, 1280 × 1024-pixel resolution) was used to record the formed intensity distributions. In the experiment, with the phase pattern of 1000 × 1000 pixels used, the size of the phase element at the SLM output was approximately equal to 8 × 8 mm and 6 × 6 mm, and the illuminating Gaussian beam diameter was about 7 mm. The zero and first diffraction orders were spatially separated by the superposition of the original phase function and a linear phase mask.
Fig. 5. Experimental optical setup: L is a solid-state laser, F is a neutral density filter, MO is a microobjective (40x, NA = 0.6), PH is a pinhole (40-µm aperture),L1, L2, and L3 are the lenses with focal lengths f1 = 350 mm, f 2 = 350 mm, and f 3 = 150 mm, respectively, BS is a beam splitter, SLM is a spatial light modulator (PLUTO Spatial Light Modulator, 1920x1080), RP is a rectangular prism, D is a diaphragm, CMOS is a video camera (VSTT-252), and Rail is an optical rail.
To analyze the intensity distributions of DHOBs formed at different distances, the CMOS camera along the optical axis was relocated. Figure 6 shows the experimentally obtained intensity distribution at different distances for two superimposed Bessel beams (n = 8, m = 3, kr = 51996 m−1, and α = 0.2 ; the size of the phase element was equal to 8 × 8 mm). Figure 7 shows the experimentally obtained intensity distribution at different distances for two
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Received 5 Jan 2015; revised 5 Mar 2015; accepted 12 Mar 2015; published 24 Mar 2015 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.008373 | OPTICS EXPRESS 8378
superimposed Bessel beams (n = 4, m = −2, kr1 = 51996 m−1, kr2 = 48296 m−1, and α = 0.4 ; the size of the phase element was equal to 6 × 6 mm). The longitudinal intensity distribution was reconstructed on the cross-sectional transverse distributions images recorded by the CMOS camera.
Fig. 6. Experimental intensity distribution for two superimposed Bessel beams (n = 8, m = 3, kr = 51996 m−1, and α = 0.2).
Fig. 7. Experimental intensity distribution for two superimposed (n = 4, m = −2, kr1 = 51996 m−1, kr2 = 48296 m−1, and α = 0.4 ).
Bessel
beams
3.2 Optical trapping experiments with airborne particles In optical trapping experiments with DHOBs, we used a solid-state laser ( λ = 532 nm, with a maximum output power of 500 mW) [Fig. 8]. The laser beam was expanded with a telescope (L1 with f1 = 15 mm and L2 with f 2 = 35 mm) to illuminate the DOE, which has the transmission function shown in Fig. 4. A generated beam was focused by a microobjective MO1 (20 × , NA = 0.4). An airborne absorbing particle in a cuvette is trapped in the area of minimum intensity of the generated DHOB. Observation of the particle trapping was possible due to the scattered light recorded by the video camera Cam1 (MDCE-5, 1280 × 1024 pixels). The particles were imaged through a microobjective MO2 (10 × , NA = 0.3).
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Received 5 Jan 2015; revised 5 Mar 2015; accepted 12 Mar 2015; published 24 Mar 2015 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.008373 | OPTICS EXPRESS 8379
Fig. 8. Experimental setup for optical trapping experiments: L is a solid-state laser (λ = 532 nm), L1, L2 are lenses with f1 = 15 mm and f2 = 35 mm, respectively, MO1 is a microobjective (20 × , NA = 0.4), MO2 is a microobjective (10 × , NA = 0.3), DOE is a diffractive optical element with the phase function shown in Fig. 4, C is a cuvette, and Cam1 is a video camera (MDCE-5, 1280 × 1024 pixels).
To demonstrate the optical trapping and holding of absorbing particles, we used carbon nanoparticle agglomeration. The typical size of the agglomerations ranged up to tens of micrometers [Fig. 9]. The trapping of such particles could be observed with the naked eye due to scattered light from their surfaces.
Fig. 9. Carbon nanoparticle agglomerations used in the experiments.
Figure 10 shows the typical results of experiments on the trapping and holding of absorbing particles. In order to put carbon nanoparticle agglomerations inside the minimum intensity region of the DHOB, the particles were sprayed with a syringe pump. Thus, the particles initially had a significant acceleration directed to the bottom of the cuvette. An experiment shows that using DHOBs we have been able carry out a multiple simultaneous particles trapping. It is clearly seen that some of the particles are trapped stable, and some of the particles keep on moving inside volume defined by DHOB structure.
Fig. 10. Experimental optical trapping and holding of absorbing microparticles with the DHOB generated by DOE (Media 1). Three particles are trapped stable (in the bottom right of the images). One particle (denoted by an arrow) keeps on moving inside volume defined by DHOB structure.
To explain the mechanism behind such particle trapping, it is necessary to review the combined effect of all possible forces acting on a particle in our experimental system: the gravity G, radiation force FR, and photophoretic force Fp = FΔT + FΔα [Fig. 11(a)]. The gravity is always directed downward, regardless of the position of the particle. The radiation
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force is directed along the direction of beam propagation. There are two types of photophoretic forces: FΔT resulting from a temperature gradient and FΔα resulting from a different thermal accommodation coefficient α [18, 19]. The photophoretic force FΔT can be directed from the hotter to the colder side of the body (positive photophoresis) or from the colder side to the hotter side of the body (negative photophoresis) [20]. The direction of FΔα is particle body-fixed, meaning that it is always directed from α1 to α 2 (suppose α1 > α 2 ), as shown in Fig. 11(a) [20]. The ratio of magnitude of the FΔα to that of the FΔT is inversely proportional to the pressure. At atmospheric pressure or below the FΔα is practically alone present [20]. The photophoretic force is orders of magnitude larger than the radiation force [7]. Therefore, magnitude and direction of the resultant force is defined mainly by gravity and the photophoretic force. Figures 11(b)-11(e) show directions of the forces depending on the position of the particles inside the minimum intensity region of the DHOB. If the direction of the photophoretic force is not equal to the gravity and its magnitude can exceed that of gravity, the particle moves upwards (from the maximum intensity of the DHOB to the minimum intensity) [Figs. 11(b) and 11(c)]. When the gravity becomes more than the photophoretic force, the particle moves downward [Figs. 11(d) and 11(e)]. For massive particles trapped in the maximum intensity of the DHOB, the gravity is balanced with the photophoretic force. Such particles can maintain their position along the beam or move over relatively short distances (for instance, three particles trapped in the bottom right of the images in Fig. 10).
Fig. 11. Illustration of trapping mechanism involved in DHOB trapping. (a) shows possible forces acting on a particle in our experimental system; (b-e) show directions of the forces depending on the position of the particles inside the minimum intensity region of the DHOB.
Figure 12 shows the motion stages of a carbon nanoparticle agglomeration trapped by the DHOB with an intensity distribution in the form of a regular pentagon contour. The particle moves up and downward (along the axis Y) and along the beam (along the axis Z). The particle image size changes because of defocusing that occurs when agglomeration moves in a plane YX perpendicular to image plane. Figure 13 shows all the points in which trapped particle was located during observation. It can be seen that the particle is located in an area of about 200x200 μm. Thus, using DHOBs, we can keep the absorbing particles in a bounded volume with a predetermined shape. In the case of non-rotating beams, it is optical tubes with different cross-sectional shape; in the case of rotating beams, it is more complex optical threedimensional structure.
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Received 5 Jan 2015; revised 5 Mar 2015; accepted 12 Mar 2015; published 24 Mar 2015 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.008373 | OPTICS EXPRESS 8381
Fig. 12. Motion stages of a carbon nanoparticle agglomeration trapped by the DHOB with an intensity distribution in the form of a regular pentagon contour.
Fig. 13. The points in which trapped particle was located during observation.
4. Conclusions
We proposed a method that allows the generation of DHOBs with a controllable crosssectional intensity distribution. We used this technique to shape various rotating and nonrotating DHOBs. Such beams can be used for optical trapping particles in air because the hollow shape of the beams allows us to hold the particles in the region of minimum intensity. Furthermore, we have generated DHOBs by SLM and demonstrated that these beams have diffraction-free properties. In experiments with airborne particles using DHOBs with an intensity distribution in form of a regular polygon, we trapped and held absorbing particles in area of minimum intensity. The ability to generate DHOBs with a controllable cross-sectional intensity distribution can be useful in experimental studies of various airborne particles and microbiology. Acknowledgments
The research was financially supported by RSF grant No. 14-19-00114. We thank D.P. Porfirev for valuable discussions and assistance.
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Received 5 Jan 2015; revised 5 Mar 2015; accepted 12 Mar 2015; published 24 Mar 2015 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.008373 | OPTICS EXPRESS 8382
