Optical manipulation of microobjects using binary diffractive elements ...

ISSN 1060992X, Optical Memory and Neural Networks (Information Optics), 2014, Vol. 23, No. 3, pp. 164–169. © Allerton Press, Inc., 2014.

Optical Manipulation of Microobjects Using Binary Diffractive Elements A. P. Porfiriev and R. G. Skidanov S.P. Korolev Samara State Aerospace University (National Research University), Samara, Russia Image Processing Systems Institute of the Russian Academy of Sciences, Samara, Russia email: [email protected] Received May 7, 2014; in final form, July 1, 2014

Abstract—The paper describes the computation of binary diffractive optical elements (DOE) produc ing specific complex light patterns. The approach is based on amplitude and phase coding methods. The experimental complex light distributions are shown to agree with the computer simulation results. Due to the phase gradient this kind of light beams can be used for moving microscopic particles along a particular path. The light fields like that have been used successfully to trap and move tiny transparent polystyrol spheres 5 micrometer in diameter. Keywords: Optical manipulation, optical traps, holographic optical tweezers, binary diffractive optical elements, amplitude coding, phase coding DOI: 10.3103/S1060992X14030102

1. INTRODUCTION The first experiments on trapping microscopic objects with laser light were carried out by A. Ashkin at AT&T Bell Lab in 1970 [1]. He devised a “levitation” trap which employed the pressure of an upward light beam to counterbalance gravitational forces acting on a particle. In 1986 A. Ashkin, S. Chu and colleagues from the Bell Telephone Lab developed a singlebeam threedimensional optical trap [2], which became a real breakthrough in the field of optical micromanipulations. Today this kind of singlebeam light trap is widely known as optical or laser tweezers. Simple optical tweezers allow one to handle either a single particle, or a set of microobjects [3]. Holo graphic optical tweezers (HOT) [4] make it possible to form a set of such optical traps and manipulate many tiny objects simultaneously [5–11]. In addition, HOT allow us to generate optical traps with partic ular amplitudephase shapes [12, 13]. In this case the trap area is determined by the intensity distribution, the phase gradient in the trap region defining the direction of movement of trapped particles [14]. Vortex light beams [15] and spatial light modulators (SLM) [16, 17] were used to generate such light beams. DOEs can also be used to form narrow optical traps to capture submicrometer objects [18, 19]. Use of vor tex light beams does not permit us to generate light fields of arbitrary amplitudephase distributions. Though SLMbased systems can theoretically be used to form light fields with nearly 100% efficiency, the light absorption in SLMs restricts their application [17]. The paper offers a method to calculate phase transmission functions of optical elements generating specific complex light distributions. This sort of elements can be made as DOEs engraved in optical sub strates. The approach allows us to overcome restrictions on laser light power, which, in turn, permits higher light energy densities in the optical trap area. The use of methods of amplitude and phase coding made it possible to compute binary transmission functions and reduce the time needed to fabricate DOEs. 2. COMPUTATION METHODS Let us need to generate a particular complex lightfield distribution:

B ( ξ, η) = I out ( ξ, η) exp (iΦ out ( ξ, η)) , where Iout(ξ, η) is the intensity distribution and Φout(ξ, η) is the phase distribution in the focal plane. 164

(1)

OPTICAL MANIPULATION OF MICROOBJECTS

(a)

(b)

(c)

165

(d)

(e)

Fig. 1. A crosslike optical trap. (a) the perfect intensity distribution (a negative); (b) the perfect phase distribution; (c) the computed phase of a DOE; simulated distribution of intensity (a negative) (d) and phase (e).

For the element to form this distribution in the focus of a lens with focal distance f, its transmission function must have the following form: ∞

T ( x, y ) = A ( x, y ) exp (iϕ ( x, y )) =

⎡ik

⎤

∫ ∫ B (ξ, η) exp ⎢⎣ f (ξx + ηy)⎥⎦ dξdη = ᑣ

−1

{B ( ξ, η)} ,

(2)

−∞ −1

where ᑣ { } is the inverse Fourier transform, k = 2π/λ is the wave number. Amplitude coding methods allow us to get the phase function of the element as a sum of original phase function ϕ(x, y) = arg(T(x, y)) and phase function ψ(x, y) that is the result of amplitude coding T(x, y) [20, 21]: ϕDOE(x, y) = ϕ(x, y) + ψ(x, y). (3) Let us use the local phase jump method [21] to compute ψ(x, y). In this case each sample of original amplitude An is replaced by a Δxwide extension of a transparent DOE with phase offset Δψn = π. The width of the offset is

δx n = Δx (1 − An ) , 2

(4)

where 0 ≤ An ≤ 1. By changing the amplitude coding inapplicability threshold, we can vary the efficiency and error of the light field generation within a wide range. To reduce phase ϕDOE(x, y) to a binary form, let us make the phase encoding by superimposing carrier frequency ν: ϕbin(x, y) = sgn{cos[ϕDOE(x, y) + 2πνx]}. (5) An element with transmission function ϕbin(x, y) generates light field pattern B(ξ, η) in the +1 diffrac tion order and its complex conjugate B*(ξ, η) in the –1 diffraction order. According to (4) when the original amplitude at a point is 0, the largest width of the phase step should be Δx/2, its height being constant and equal to π rad. The energy efficiency of a phase step like that is about 80% [22]. This kind of steps introduces much distortion into the resultant light field pattern. When low frequencies predominate in the spectrum computed with (2), the generation of light field patterns suffers from significant errors no matter what amplitude coding inapplicability threshold we choose. To overcome this problem, we can use the trick of supplementing distribution T(x, y) with zeros. From the physical point of view, by doing this we decrease the radius of the incident beam to make it light only the DOE cen ter whose contribution to the generation of the light pattern is greatest in this case. Figures 1–3 show the examples of phase transmission functions of DOEs computed with the help of formulae (1)–(5) and the simulation results of complex light distributions generated by these DOEs. The amplitude coding methods give good results when the elements are designed to generate simple light field patterns. If the resulting patterns are not simple (e.g. the sample distributions are sets of separate geometric figures), the generation of patterns begins to suffer from errors and its efficiency drops. The errors grow because of noticeable changes of intensity between different portions of generated patterns. An approach that requires another amplitude coding iteration given a changed sample lightfield distribu tion can be used to cope with the problem. In this case the algorithm can be as follows: 1) formulae (1)–(4) are used to calculate transmission functions ϕDOE(x, y); OPTICAL MEMORY AND NEURAL NETWORKS (INFORMATION OPTICS)

Vol. 23

No. 3

2014

166

PORFIRIEV, SKIDANOV

(a)

(b)

(c)

(d)

(e)

Fig. 2. An optical trap in the form of two lines. (a) the perfect intensity distribution (a negative); (b) the perfect phase dis tribution; (c) the computed phase of a DOE; simulated distribution of intensity (a negative) (d) and phase (e).

(a)

(b)

(c)

(d)

(e)

Fig. 3. An optical trap in the shape of an equilateral triangle. The perfect distributions of intensity (a negative) (a) and phase (b); (c) the computed phase of a DOE; simulated distribution of intensity (a negative) (d) and phase (e).

2) the direct Fourier transform is utilized to find the output light field distribution ∞

F ( ξ, η) =

∫ ∫ exp [iϕ

−∞

DOE

( x, y )] exp ⎡⎢ik ( ξx + ηy )⎤⎥ d ξd η = ᑣ{ϕDOE ( x, y )} , ⎣f

(6)

⎦

3) if the pattern generation error is too high, original amplitude |B(ξ, η)| is replaced by |B'(ξ, η)| con sidering the resultant distribution F(ξ, η): |B'(ξ, η)| = | |B(ξ, η)| + μ(|B(ξ, η)| – |F(ξ, η)|)|,

(7)

where 0 ≤ μ ≤ 1; 4) the amplitude coding procedure is repeated for the new distribution B'(ξ, η) = |B'(ξ, η)|exp(iΦout(ξ, )) The algorithm was used to compute the phase transmission functions of elements generating light fields that can be represented as a set of simple patterns. For example, Figure 4 gives the simulation results of generation of light grids in the focal plane.

(a)

(b)

(c)

(d)

(e)

Fig. 4. An optical trap in the form of eightnod grid. The perfect distributions of intensity (a negative) (a) and phase (b); (c) the computed phase of a DOE; simulated distribution of intensity (a negative) (d) and phase (e). OPTICAL MEMORY AND NEURAL NETWORKS (INFORMATION OPTICS)

Vol. 23

No. 3

2014

OPTICAL MANIPULATION OF MICROOBJECTS

167

M1 I

L3

L M3 DOE L1

500 µm

500 µm (a)

V (b)

CCD

L2

PC M2

500 µm

500 µm (c)

(d)

Fig. 6. The optical arrangement used in experiments on manipulation of microobjects. L—laser, M1, M2 are turning mirrors, I—an illuminator, M3—a semitransparent mirror, L1—a focusing microobjec tive, V—a substrate with water suspension of micro particles, L2—an imaging microobjective, L3—a converging lens (spatial light modulator), F—light filters, TV—a CCD camera, PC—a computer.

3. EXPERIMENTS ON MOVEMENT OF TINY TRANSPARENT POLYSTYROL BEADS DOEs with designed phase transmission func tions were made on a glass substrate by photoli thography with a 2micrometer accuracy. The 500 µm 500 µm radius of the DOEs was 2 mm. Figure 3.2 shows threedimensional photos of the microreliefs of (e) (f) the DOEs whose phase functions are given in paragraph 1 in Figures 1.3 and 1.24, correspond Fig. 5. Real intensity distributions (negatives) of optical traps. Crosslike pattern (a), twoline pattern (c), equilat ingly. Interference microscope ZYGO NewView eral triangle contour (e) and corresponding interfero 5000 (50× objective, NA = 0.55) was used to take grams (b, d, f). Only one of the generated diffraction the photos. orders is shown. Figure 5a shows the work of a binary DOE (the transmission function is given in the figure): a light cross with a phase gradient. The corresponding interferogram is given in Fig. 5b. It is seen that the curvature of interference fringes changes along the rays forming the cross, which indicates the phase shift. Similar images related to generation of an optical trap in the form of two lines and equilateral triangle con tour are shown in Figs. 5c–5d. Additionally, 5micrometer polystyrol beads were used in experiments on optical manipulation with the DOEmodified light beams. The optical arrangement used in the experiments is shown in Fig. 6. A 532nm rareearthelementsdoped neodymium laser was used as a monochromatic light source. The output power was about 500 mW. To minimize the reflection losses of laser power on lens surfaces, the necessary diameter of the laser beam falling on a DOE was achieved by increasing the distance between the laser output aperture and first turning mirror M1 rather than using a collimator. This distance was as long as 1500 mm. After reflecting from mirror M1, the laser beam goes to the DOE. A microobjective L1 (20×, NA = 0.4) focuses the DOEmodified beam onto a glass substrate holding a water suspension of tiny particles. A combination of light I, lens L3 and semitransparent mirror M3 was used to illuminate the work ing area. A few light filters (F) were chosen from a set of colored optical glasses to secure concurrent obser vation of the beam and microparticles. The filters absorbed light in the wavelength range from 380 to OPTICAL MEMORY AND NEURAL NETWORKS (INFORMATION OPTICS)

Vol. 23

No. 3

2014

168

PORFIRIEV, SKIDANOV

5 µm

5 µm

(a)

5 µm

(b)

5 µm

(c)

(d)

Fig. 7. Movement of a microscopic polystyrol bead in the crosslike light beam (the arrow points to the particle being moved). The interframe time interval is 0.75 s. The grid spacing is 10 µm.

5 µm

5 µm (a)

5 µm

5 µm (b)

5 µm (e)

5 µm (c)

5 µm (f)

(d)

5 µm (g)

(h)

Fig. 8. Movement of a microscopic polystyrol bead captured by one of the light line (the arrow points to the particle being moved). The time interval between the frames is 1.25 s. The grid spacing is 10 µm.

540 nm to a sufficient degree and transmitted the illuminating light almost fully. Microobjective L2 (16×, NA = 0.4) was used to image the manipulation plane onto the CCD array of a VSCTT252 camera. The movement of microbeads captured within the crosslike light area is shown in Fig. 7. The time interval between the frames is 0.75 s. It is seen that the captured microsphere (marked by the black arrow) moves from one end of the cross towards its center. The reason is that the phase gradient is directed along the lines towards the cross center. The speed of the particle was measured to be 3.84 ± 0.60 μm/s. Figure 8 shows the movement of microparticles caught within the light lines. The interframe interval is 1.25 s. As seen from Fig. 8, captured in the area around a light line, the polystyrol bead (marked by the white arrow) begins to move in the phasegradient direction perpendicular to the main flow. So particles depart away from the mainstream. The velocity of the microparticle was 0.82 ± 0.60 μm/s. CONCLUSIONS We have developed a method for computing binary phase focusing elements that can produce specific amplitudephase distributions of light field which have predefined phase distributions. The method is based on the amplitude coding technique using a local phase jump. Optical traps formed by light lines with a particular phase gradient were used in experiments to capture and move tiny polystyrol beads 5 μm in diameter along specific paths. The experiments showed that opti cal traps make particles turn aside from the mainstream and can be used for sorting out microobjects. OPTICAL MEMORY AND NEURAL NETWORKS (INFORMATION OPTICS)

Vol. 23

No. 3

2014

OPTICAL MANIPULATION OF MICROOBJECTS

169

ACKNOWLEDGMENTS The research was financially supported by the Ministry of Education and Science of the Russian Fed eration within the scope of the state program on Fabrication of diffractive optical elements on curved axi symmetric surfaces (code number 02vR073035). REFERENCES 1. Ashkin, A., Acceleration and trapping of particles by radiation pressure, Phys. Rev. Lett., 1970, vol. 24, no. 4, pp. 156–159. 2. Ashkin, A., Dziedzic, J.M., Bjorkholm, J.E., and Chu, S., Observation of a singlebeam gradient force optical trap for dielectric particles, Opt. Lett., 1986, vol. 11, no. 5, pp. 288–290. 3. Skidanov, R.V., Morozov, A.A., and Porfirev, A.P., Composite light beam and microexplosion for optical micro manipulation, Comput. Opt., 2012, vol. 36, no. 3, pp. 371–376. 4. Grier, D.G. and Roichman, Y., Holographic optical trapping, Appl. Opt., 2006, vol. 45, no. 5, pp. 880–887. 5. Belloni, F. and Monneret, S., Quadrant kinoform: an approach to multiplane dynamic threedimensional holo graphic trapping, Appl. Opt., 2007, vol. 46, no. 21, pp. 4587–4593. 6. van der Horst, A. and Forde, N.R., Calibration of dynamic holographic optical tweezers for force measurements on biomaterials, Opt. Express, 2008, vol. 16, no. 25, pp. 20987–21003. 7. Padgett, M. and Di Leonardo, R., Holographic optical tweezers and their relevance to lab on chip devices, Lab. Chip, 2011, vol. 11, pp. 1196–1205. 8. Hesseling, C., Woerdemann, M., Hermerschmidt, A., and Denz C., Controlling ghost traps in holographic optical tweezers, Opt. Lett., 2011, vol. 36, no. 18, pp. 3657–3659. 9. Conkey, D.B., Trivedi, R.P., Pavani, S.R.P., Smalyukh, I.I., and Piestun, R., Threedimensional parallel parti cle manipulation and tracking by integrating holographic optical tweezers and engineered point spread func tions, Opt. Express, 2011, vol. 19, no. 5, pp. 3835–3842. 10. Porfirev, A.P. and Skidanov, R.V., Generation of an array of optical bottle beams using a superposition of Bessel beams, Appl. Opt., 2013, vol. 52, no. 25, pp. 6230–6238. 11. Woerdemann, M., Alpmann, C., Esseling, M., and Denz, C., Advanced optical trapping by complex beam shaping, Laser Photon. Rev., 2013, vol. 7, no. 6, pp. 839–854. 12. Skidanov, R.V., Kotlyar, V.V., Khonina, S.N., Volkov, A.V., and Soifer, V.A., Micromanipulation in higherorder bessel beams, Opt. Mem. Neural Networks, Ser. Inform. Opt., 2007, vol. 16, no. 2, pp. 91–98. 13. Skidanov, R.V. and Porfirev, A.P., Formation of the massif of hollow beams for sedimentation and positioning of microparticles, Comput. Opt., 2012, vol. 36, no. 3, pp. 387–394. 14. Skidanov, R.V., Calculation of force of interaction of a light beam with microparticles of any form, Comput. Opt., 2005, vol. 29, pp. 18–21. 15. Abramochkin, E.G., Kotova, S.P., Korobtsov, A.V., Losevsky, N.N., Mayorova, A.M., Rakhmatulin, M.A., and Volostnikov, V.G., Microobject manipulations using laser beams with nonzero orbital angular momentum, Laser Phys., 2006, vol. 16, no. 5, pp. 842–848. 16. Roichman, Y. and Grier, D.G., Projecting extended optical traps with shapephase holography, Opt. Lett., 2006, vol. 31, no. 11, pp. 1675–1677. 17. Jesacher, A., Maurer, C., Schwaighofer, A., Bernet, S., and RitschMarte, M., Fully phase and amplitude con trol of holographic optical tweezers with high efficiency, Opt. Express, 2008, vol. 16, no. 7, pp. 4479–4486. 18. Khonina, S.N. and Pelevina, E.A., Reduction of the focal spot size in highaperture focusing systems at insert ing of aberrations, Opt. Mem. Neural Networks, Ser. Inform. Opt., 2011, vol. 20, no. 3, pp. 155–167. 19. Khonina, S.N., Nesterenko, D.V., Morozov, A.A., Skidanov, R.V., and Soifer, V.A., Narrowing of a light spot at diffraction of linearlypolarized beam on binary asymmetric axicons, Opt. Mem. Neural Networks, Ser. Inform. Opt., 2012, vol. 21, no. 1, pp. 17–26. 20. MontesUsategui, M., Mas, J., Roth, M.S., and MartinBadosa, E., Adding functionalities to precomputed holograms with random mask multiplexing in holographic optical tweezers, Appl. Opt., 2011, vol. 50, no. 10, pp. 1417–1424. 21. Khonina, S.N., Balalayev, S.A., Skidanov, R.V., Kotlyar, V.V., Palvanranta, B., and Turunen, J., Encoded binary diffractive element to form hypergeometric laser beams, J. Opt., Ser. A, 2009, vol. 11, no. 6, pp. 065702– 065708. 22. Soifer, V.A., Methods of Computer Optics, Moscow: Fizmatlit, 2003.

OPTICAL MEMORY AND NEURAL NETWORKS (INFORMATION OPTICS)

Vol. 23

No. 3

2014