Optical stacking of microparticles in a pyramidal structure created with a symmetric cubic phase Pedro A. Quinto-Su1,∗ and Roc´ıo J´auregui2 1 Instituto
de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico Apartado Postal 70-543, 04510, M´exico D.F., Mexico 2 Instituto de F´ısica, Universidad Nacional Aut´ onoma de M´exico Apartado Postal 20-364, 01000, M´exico D.F., Mexico ∗ [email protected]
Abstract: We show a simple way to generate three dimensional optical potentials consisting of tightly localized high intensity spots arranged in a structure with a pyramidal geometry. The three dimensional patterns are created by focusing a Gaussian beam with a symmetric cubic phase abs((ax)3 ) + abs((ay)3 ) imprinted by a spatial light modulator. We show that it is possible to trap and stack around a hundred dielectric microspheres (silica mean diameter 2.47 μ m) in pyramidal structures (characteristic dimensions H, W ∼ 15 − 20μ m) held together by optical binding with moderate laser power (P < 20 mW). Axial stability is mainly provided by balancing the light scattering force with the axial gradient and gravity. The microparticle structures are sufficiently stable to be easily displaced by moving the microscope stage. © 2014 Optical Society of America OCIS codes: (350.4855) Optical tweezers or optical manipulation; (090.1995) Digital holography.
References and links 1. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288–290 (1986). 2. K. Dholakia and P. Zem´anek, “Colloquium: gripped by light: optical binding,” Rev. Mod. Phys. 82, 1767–1791 (2010). 3. D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003). ˘ zm´ar, V. Koll´arov´a, X. Tsampoula, F. Gunn-Moore, W. Sibbett, Z. Bouchal, and K. Dholakia, “Generation 4. T. Ci˘ of multiple Bessel beams for a biophotonics workstation,” Opt. Express 18, 14024–14035 (2008). 5. E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets, and D. G. Grier, “Computer-generated holographic optical tweezer arrays,” Rev. Sci. Instrum. 72, 1810–1816 (2001). 6. H. Melville, G. F. Milne, G. C. Spalding, W. Sibbett, K. Dholakia, and D. McGloin, “Optical trapping of threedimensional structures using dynamic holograms,” Opt. Express 11, 3562–3567 (2003). 7. J. Leach, G. Sinclair, P. Jordan, J. Courtial, and M. J. Padgett, “3D manipulation of particles into crystal structures using holographic optical tweezers,” Opt. Express 12, 220–226 (2004). 8. V. Garc´es-Ch´avez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002). 9. M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science 296, 1101–1103 (2002). 10. M. V. Berry and N. L. Balazs, “Non-spreading wave packets,” Am. J. Phys. 47, 264–267 (1979). 11. ] J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev.Lett. 58, 1499–1501 (1987). 12. J. Durnin, J. J. Miceli, and J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. 13, 79–80 (1988).
#208549 - $15.00 USD Received 19 Mar 2014; revised 14 Apr 2014; accepted 6 May 2014; published 13 May 2014 (C) 2014 OSA 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.012283 | OPTICS EXPRESS 12283
13. D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005). 14. J. C. Guti´errez-Vega, M. D. Iturbe-Castillo, and S. Ch´avez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25(20), 1493–1495 (2000). 15. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007). 16. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007). 17. C. Alpmann, R. Bowman, M. Woerdemann, M. Padgett, and C. Denz, “Mathieu beams as versatile light moulds for 3D micro particle assemblies,” Opt. Express 18, 26084–26091 (2010). 18. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33, 207–209 (2008). ˘ z˘ m´ar, H. I. C. Dalgarno, R. F. Marchington, F. J. Gunn-Moore, and K. Dholakia, “Realization 19. J. E. Morris, T. Ci of curved Bessel beams: propagation around obstructions,” J. Opt. 12, 124002 (2010). 20. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Phot. 2, 675–678 (2008). 21. J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9, 1334–1336 (2009). 22. S. Vo, K. Fuerschbach, K. P. Thompson, M. A. Alonso, and J. P. Rolland, “Airy beams: a geometric optics perspective,” J. Opt. Soc. Am. 27, 2574–2582 (2010). 23. W. H. Wright, G. J. Sonek, and M. W. Berns, “Parametric study of the forces on microspheres held by optical tweezers,” Appl. Opt. 33, 1735–1748 (1994).
1.
Introduction
Microscopic objects can be trapped by laser light with large intensity gradients as in the case of highly focused beams in optical tweezers [1]. The gradient force is attractive or repulsive depending on the relative refractive index between the particles and the surrounding medium. Furthermore, dielectric particles trapped at an intensity maximum get polarized, so it is possible that objects in the vicinity might attach creating microparticle chains; this has been called optical binding [2]. In the last two decades great progress in optical manipulation has been achieved with structured and non Gaussian laser beams created with phase masks, spatial light modulators (SLM’s) [3], axicons and other optical elements [4]. These advances have allowed to simultaneously trap, manipulate and sort three dimensional (3d) microparticle arrays [5–9]. Non diffractive beams [10, 11] or propagation invariant beams have also been used to trap multiple particles in 3d configurations. These beams propagate with little change in their transverse intensity profiles for longer distances than standard Gaussian beams [12]. Furthermore, some non-diffracting beams (for example: Bessel [13], Mathieu [14], Airy [10,15,16]) are ”selfhealing”, that is, when part of the beam is obstructed, the original intensity profile is recovered further down the optical path (depending on the size of the obstruction and beam angle). In this way, a single propagation invariant beam can simultaneously trap particles (at maxima or in between maxima) in different planes along the optical axis [8]. Also, 3d microparticle structures have been produced by stacking microparticles on top of each other with optical potentials or ”optical moulds” generated by Mathieu beams [17]. Non diffracting beams can also propagate in curved trajectories like in the case of Airy beams (parabolic trajectory) [18] and curved Bessel beams [19]. Some applications of Airy beams in micromanipulation involve removal of microparticles in what has been called ”optical path clearing” [20] and sorting microparticles in microfluidic microwells [21]. Experimentally, Airy beams (3d) are implemented by focusing a Gaussian beam with a 2d cubic phase profile (φ (x, y) ∝ x3 + y3 ), since the Fourier transform of an Airy wavepacket modulated by an exponential aperture function is proportional to a cubic frequency modulated by a Gaussian amplitude [15]. Airy beams also exhibit interesting properties from the geometric optics perspective [22]. In this work we focus a Gaussian beam with a symmetric cubic phase φ (x, y) = abs((ax)3 ) + #208549 - $15.00 USD Received 19 Mar 2014; revised 14 Apr 2014; accepted 6 May 2014; published 13 May 2014 (C) 2014 OSA 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.012283 | OPTICS EXPRESS 12284
abs((ay)3 ) to create three dimensional optical potentials formed by tightly localized high intensity spots arranged in a structure with a geometry similar to a square pyramid with the apex along the optical axis (close to the geometric focus of the lens). We show that these beams can be used to stack around one hundred microscopic particles (silica, mean diameter 2.47 μ m) in pyramidal structures with moderate laser power P in the range of a few milli Watts, and that the structures are sufficiently stable to allow displacements. 2.
Experimental setup
The experimental setup is depicted in Fig. 1. The laser source is a CW butterfly laser diode with a wavelength of 975 nm coupled to a single mode fiber (Thorlabs, PL980P330J). The beam is collimated and then expanded so that it overfills the screen of a reflective SLM (Holoeye, Pluto NIR2). The reflected beam is focused by a 40 cm lens (L1) where a spatial filter blocks the undiffracted order. The beam is relayed and coupled to the microscope objective (Nikon E Plan, 100x, oil, 1.25 NA) of our custom inverted microscope by three identical lenses (L2, L3, L4) with a focal length of 20 cm. The microscope sample holder is a 3d piezo stage (Thorlabs, Max301) that also has differential micrometer drives for manual displacement. A short pass filter is placed in front of the CCD (Thorlabs, DCU223M) camera to block the retroreflected laser light from the bottom coverslip of the container. constant phase
symmetric cubic phase
z LED and condenser 1080 px
y Sample x 100X/1.25
f1
f1
f2
3d Piezo stage
Dichroic L1 SLM
Spaal filter Expanded laser beam
L2
L3
L4 filter
Tube lens CCD
Fig. 1. Experimental setup and holograms projected in the SLM (upper left corner), constant phase and symmetric cubic with a = 0.001 μ m−1 .
The phase pattern encoded into a reflecting SLM is a symmetric cubic phase φ (x, y) = abs((ax)3 ) + abs((ay)3 ). Where xy are the coordinates of the pixels in the SLM (dimensions of microns, pixel size of the SLM 8 μ m/px, 1080 pixels), and a is a scale factor with dimensions of inverse microns. This parameter controls the size of the projected pattern, such that decreasing the value of a results in a smaller scale. For all the experiments shown the scaling parameter in the phase is a = 0.001 μ m−1 . Linear prism phases are used to displace the beam in order to spatially filter the undiffracted order. Also, negative lens phases displace the 3d pattern along the optical axis. Laser powers are measured after the beam is transmitted by the microscope objective. For the images of the experiments that are shown in this article, the microparticle solution is an aqueous suspension of silica microspheres (Bangs Laboratories) with a mean diameter d of 2.47 μ m and density ρ of 2 gr/cm3 , refractive index n p ∼ 1.4 at 975 nm. The container is prepared by sandwiching spacers (height ∼ 100 μ m) between two glass coverslips (no 1, 0.13-0.16 mm thick). Then, the chamber is filled with the microparticle solution and it is sealed with nail polish.
#208549 - $15.00 USD Received 19 Mar 2014; revised 14 Apr 2014; accepted 6 May 2014; published 13 May 2014 (C) 2014 OSA 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.012283 | OPTICS EXPRESS 12285
3.
Results and analysis
Beam structure. In order to image the beam emerging from the microscope objective at different planes along the optical axis, a single microscope coverslip (No. 1, 0.13-0.16 mm thick) is placed on the microscope stage and the short pass filter is removed. In this way, we can image the retroreflected part of the beam with the CCD. Initially, the glass slide is positioned below the geometrical focus of the microscope objective. The slide is moved up (z direction, along optical axis) in steps of 0.5 μ m using the piezo stage (closed loop mode) until it reaches the plane of the geometrical focus of the objective. The position of the geometrical focus of the lens is labeled as zero, while the positions before that point are negative. We image the cross sections of the beam for two cases: symmetric cubic phase and for a constant phase for comparison (holograms in Fig. 1 with an added linear phase for spatial filtering). The images are shown in Fig. 3, top row: constant phase, bottom row: symmetric cubic phase. The cross sections for the symmetric cubic phase resemble the classic intensity pattern of a beam diffracted by a square aperture. However, with the constant phase (square aperture at the SLM) we don’t observe these patterns (Fig. 2, top row). Another interesting feature of the structured beam (symmetric cubic phase) is that it vanishes beyond the focus, in contrast to the focusing of a beam apodised by a square aperture, where the localized intensity patterns would repeat after the geometrical focus of the lens. The beam cross sections in Fig. 2 (bottom row) in the range of -15 to 0 μ m are used -6.5 μm
-5.5μm
-4.5 μm
-3.5 μm
-14 μm
-10.5 μm
-9.5μm
-7 μm
-6 μm
0 μm
Constant
-7.5 μm
Symmetric cubic
9 μm 0 μm
9 μm
Fig. 2. Cross sections. Top row: Constant phase. Bottom row: Symmetric cubic phase. Digital holograms shown in Fig. 1 (upper left corner).
to reconstruct the 3d structure of the beam during propagation. The isosurfaces are shown in Fig. 3. The left side (Fig. 3(a)) shows the 3d structure which resembles a square pyramid with localized spots. A view from the top along the optical axis (Fig. 3(b)) reveals that most of the high intensity spots are located along the four apex edges of the pyramidal structure. The side view (xz plane) is shown in Fig. 3(c). The axial length of the central lobe (∼ 8μ m) is larger than the Rayleigh length (2z0 = 1.46 μ m, diffraction limited) for a Gaussian beam focused with the same microscope objective, suggesting a weaker axial intensity gradient. C
y (μm)
z (μm)
B
z (μm)
A
x(
μm
)
m)
y (μ
x (μm)
x (μm)
Fig. 3. Isosurfaces from the cross sections in Fig. 2 (bottom row).
Microparticle stacking and manipulation. The container with the aqueous microparticle solution is placed on the piezo stage of our custom inverted microscope. The imaging plane is #208549 - $15.00 USD Received 19 Mar 2014; revised 14 Apr 2014; accepted 6 May 2014; published 13 May 2014 (C) 2014 OSA 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.012283 | OPTICS EXPRESS 12286
located at the focal plane of the microscope objective. The sample is moved with the microscope stage to a position (z direction along optical axis) where we image the bottom of the container where most of the particles are located due to their higher density. This means that the apex of the pyramidal structure is also located at the bottom. In order to move the apex up (z direction, away from the bottom) we add a negative lens phase to the digital hologram, shifting the beam in the axial direction controlling the effective volume of the structured beam that interacts with the particles. Figures 4(a), 4(b), 4(c), and 4(d) show loading of the microparticles A
B
C
D
E
10 μm
F
20 μm
Fig. 4. A. Apex at bottom. Raising gradually the apex: frames B.,C.,D. E., (Media 1, frame width 30 μ m). F. Microparticle pyramids with different sizes. P=12.5 mW.
by gradually shifting the apex above the imaging plane, resulting in a larger square base (10-20 μ m, Figs. 4(e) and 4(f)) as the apex is raised. We can see that the particles that are closer to the apex appear blurred since the imaging plane is at the bottom. Also, it is interesting that the bottom is not completely filled, since many of the microspheres arrange closer to the apex and along the four apex edges forming microparticle chains (as expected from the isosurfaces, Fig. 3). Loading is achieved by a careful balancing of the forces: the optical gradient forces ∝ ∇I (I is the intensity), the scattering force ∝ I, gravity and optical binding due to the light induced dipoles [2]. First we look at the gradient force. The number of localized spots in the beam decreases for planes that are closer to the apex, while the intensity per spot increases (Fig. 3). The size of the particles is larger than the axial separation between spots, so that the gradient force attracts the particles toward the apex, where the axial gradient of the main lobe is much smaller than the transverse one (Fig. 3). This is confirmed by experiments (not shown) with polystyrene microspheres (d = 2 μ m, n p ∼ 1.6 at 975 nm, ρ = 1 gr/cm3 neutrally buoyant) which typically result in more efficient optical traps (larger Q [23]) than with silica. The polystyrene beads quickly migrate towards the apex and are blown away (for P > 1 mW), since the axial gradient is not large enough to overcome the scattering force. In the case of silica microparticles (higher density), axial stability is achieved by the apparent weight and the axial gradient which balance the scattering force. Considering gradient, scattering and gravity the main limit in the stability of the structures is the overall laser power P, since beyond a certain threshold Pth , gravity and the axial gradient are not sufficient to balance the scattering force. This is observed in the videos of Media 1 that show stable loading (2.47 μ m silica) for P=12 and 18 mW, unstable at 29 mW, and for P=34 mW the particles are blown away. Experiments with 1.5 μ m silica particles (not shown, hologram with a = 0.0009 μ m−1 ) revealed that the maximum threshold Pth decreases to about 8 mW, consistent with smaller apparent weight per particle. The binding force between the light induced dipoles in the silica microspheres produces the chain-like structures along the four apex edges (Fig. 4) and holds the particles together. The dynamics of the particle chains (perturbed by Brownian motion) are similar to that of a flexible rope (Media 1–3). Preliminary experiments with different states of linear polarization suggest subtle changes in the structure and the distribution of particles. The number of stacked particles depends on the size of the 3d structure, to count them,
#208549 - $15.00 USD Received 19 Mar 2014; revised 14 Apr 2014; accepted 6 May 2014; published 13 May 2014 (C) 2014 OSA 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.012283 | OPTICS EXPRESS 12287
0s
0.46 s
0.69 s
1.39 s
29.96 s
20 μm
Fig. 5. Releasing the trap to estimate the number of trapped particles with P=12.5 mW (Media 2, frame width 42 μ m).
Vertical
Horizontal
the laser light is blocked and the microspheres fall back to the bottom. Hence, subtracting the number of particles that initially (before release) were in the background (not in the structure) from the number of particles after the release yields a good estimate of the number of stacked objects (a few might go out of the frame when the particles fall to the bottom ). Selected video frames showing the release of a microparticle pyramid are displayed in Fig. 5 (compressed video in Media 2). To count the objects, we wait for the particles to form a single layer at the bottom, since initially the particles fall on top of each other making a couple of layers. The estimated number of particles in the structure is around one hundred.
20 μm
Fig. 6. Translation of the pattern by displacing the microscope stage P=12.5 mW (Media 3, frame width 42 μ m).
Finally, we show that it is possible to displace the structures by moving the microscope stage in the transverse directions xy, thus dragging the microparticle patterns. Selected frames (compressed video in Media 3) are shown in Fig. 6. We have drawn white circles around some microparticles in the background so that it is easier to observe the displacements. It is interesting that the pattern can move above other objects that are at the bottom, since the bottom of the pyramid is mainly empty. 4.
Conclusion
We have shown a simple method to create 3d optical potentials with a pyramidal geometry using a symmetric cubic phase. These potentials can be used to stack large numbers of microparticles (∼ 100) with moderate laser powers. Some applications could include sorting of microparticles by trapping those with higher density while lifting the others. These beams could also be used in the context of optical clearing with lower power than what is needed for the equivalent Airy beam [20] but with the disadvantage that the particles are lifted vertically. Acknowledgments We thank V. Carmona and R. Guti´errez for exploratory experiments. This work was partially funded by the following grants: CONACYT (153821 and 166961) and DGAPA, UNAM (PAPIIT TB100312 and IN101511).
#208549 - $15.00 USD Received 19 Mar 2014; revised 14 Apr 2014; accepted 6 May 2014; published 13 May 2014 (C) 2014 OSA 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.012283 | OPTICS EXPRESS 12288
de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico Apartado Postal 70-543, 04510, M´exico D.F., Mexico 2 Instituto de F´ısica, Universidad Nacional Aut´ onoma de M´exico Apartado Postal 20-364, 01000, M´exico D.F., Mexico ∗ [email protected]
Abstract: We show a simple way to generate three dimensional optical potentials consisting of tightly localized high intensity spots arranged in a structure with a pyramidal geometry. The three dimensional patterns are created by focusing a Gaussian beam with a symmetric cubic phase abs((ax)3 ) + abs((ay)3 ) imprinted by a spatial light modulator. We show that it is possible to trap and stack around a hundred dielectric microspheres (silica mean diameter 2.47 μ m) in pyramidal structures (characteristic dimensions H, W ∼ 15 − 20μ m) held together by optical binding with moderate laser power (P < 20 mW). Axial stability is mainly provided by balancing the light scattering force with the axial gradient and gravity. The microparticle structures are sufficiently stable to be easily displaced by moving the microscope stage. © 2014 Optical Society of America OCIS codes: (350.4855) Optical tweezers or optical manipulation; (090.1995) Digital holography.
References and links 1. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288–290 (1986). 2. K. Dholakia and P. Zem´anek, “Colloquium: gripped by light: optical binding,” Rev. Mod. Phys. 82, 1767–1791 (2010). 3. D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003). ˘ zm´ar, V. Koll´arov´a, X. Tsampoula, F. Gunn-Moore, W. Sibbett, Z. Bouchal, and K. Dholakia, “Generation 4. T. Ci˘ of multiple Bessel beams for a biophotonics workstation,” Opt. Express 18, 14024–14035 (2008). 5. E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets, and D. G. Grier, “Computer-generated holographic optical tweezer arrays,” Rev. Sci. Instrum. 72, 1810–1816 (2001). 6. H. Melville, G. F. Milne, G. C. Spalding, W. Sibbett, K. Dholakia, and D. McGloin, “Optical trapping of threedimensional structures using dynamic holograms,” Opt. Express 11, 3562–3567 (2003). 7. J. Leach, G. Sinclair, P. Jordan, J. Courtial, and M. J. Padgett, “3D manipulation of particles into crystal structures using holographic optical tweezers,” Opt. Express 12, 220–226 (2004). 8. V. Garc´es-Ch´avez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002). 9. M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science 296, 1101–1103 (2002). 10. M. V. Berry and N. L. Balazs, “Non-spreading wave packets,” Am. J. Phys. 47, 264–267 (1979). 11. ] J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev.Lett. 58, 1499–1501 (1987). 12. J. Durnin, J. J. Miceli, and J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. 13, 79–80 (1988).
#208549 - $15.00 USD Received 19 Mar 2014; revised 14 Apr 2014; accepted 6 May 2014; published 13 May 2014 (C) 2014 OSA 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.012283 | OPTICS EXPRESS 12283
13. D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005). 14. J. C. Guti´errez-Vega, M. D. Iturbe-Castillo, and S. Ch´avez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25(20), 1493–1495 (2000). 15. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007). 16. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007). 17. C. Alpmann, R. Bowman, M. Woerdemann, M. Padgett, and C. Denz, “Mathieu beams as versatile light moulds for 3D micro particle assemblies,” Opt. Express 18, 26084–26091 (2010). 18. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33, 207–209 (2008). ˘ z˘ m´ar, H. I. C. Dalgarno, R. F. Marchington, F. J. Gunn-Moore, and K. Dholakia, “Realization 19. J. E. Morris, T. Ci of curved Bessel beams: propagation around obstructions,” J. Opt. 12, 124002 (2010). 20. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Phot. 2, 675–678 (2008). 21. J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9, 1334–1336 (2009). 22. S. Vo, K. Fuerschbach, K. P. Thompson, M. A. Alonso, and J. P. Rolland, “Airy beams: a geometric optics perspective,” J. Opt. Soc. Am. 27, 2574–2582 (2010). 23. W. H. Wright, G. J. Sonek, and M. W. Berns, “Parametric study of the forces on microspheres held by optical tweezers,” Appl. Opt. 33, 1735–1748 (1994).
1.
Introduction
Microscopic objects can be trapped by laser light with large intensity gradients as in the case of highly focused beams in optical tweezers [1]. The gradient force is attractive or repulsive depending on the relative refractive index between the particles and the surrounding medium. Furthermore, dielectric particles trapped at an intensity maximum get polarized, so it is possible that objects in the vicinity might attach creating microparticle chains; this has been called optical binding [2]. In the last two decades great progress in optical manipulation has been achieved with structured and non Gaussian laser beams created with phase masks, spatial light modulators (SLM’s) [3], axicons and other optical elements [4]. These advances have allowed to simultaneously trap, manipulate and sort three dimensional (3d) microparticle arrays [5–9]. Non diffractive beams [10, 11] or propagation invariant beams have also been used to trap multiple particles in 3d configurations. These beams propagate with little change in their transverse intensity profiles for longer distances than standard Gaussian beams [12]. Furthermore, some non-diffracting beams (for example: Bessel [13], Mathieu [14], Airy [10,15,16]) are ”selfhealing”, that is, when part of the beam is obstructed, the original intensity profile is recovered further down the optical path (depending on the size of the obstruction and beam angle). In this way, a single propagation invariant beam can simultaneously trap particles (at maxima or in between maxima) in different planes along the optical axis [8]. Also, 3d microparticle structures have been produced by stacking microparticles on top of each other with optical potentials or ”optical moulds” generated by Mathieu beams [17]. Non diffracting beams can also propagate in curved trajectories like in the case of Airy beams (parabolic trajectory) [18] and curved Bessel beams [19]. Some applications of Airy beams in micromanipulation involve removal of microparticles in what has been called ”optical path clearing” [20] and sorting microparticles in microfluidic microwells [21]. Experimentally, Airy beams (3d) are implemented by focusing a Gaussian beam with a 2d cubic phase profile (φ (x, y) ∝ x3 + y3 ), since the Fourier transform of an Airy wavepacket modulated by an exponential aperture function is proportional to a cubic frequency modulated by a Gaussian amplitude [15]. Airy beams also exhibit interesting properties from the geometric optics perspective [22]. In this work we focus a Gaussian beam with a symmetric cubic phase φ (x, y) = abs((ax)3 ) + #208549 - $15.00 USD Received 19 Mar 2014; revised 14 Apr 2014; accepted 6 May 2014; published 13 May 2014 (C) 2014 OSA 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.012283 | OPTICS EXPRESS 12284
abs((ay)3 ) to create three dimensional optical potentials formed by tightly localized high intensity spots arranged in a structure with a geometry similar to a square pyramid with the apex along the optical axis (close to the geometric focus of the lens). We show that these beams can be used to stack around one hundred microscopic particles (silica, mean diameter 2.47 μ m) in pyramidal structures with moderate laser power P in the range of a few milli Watts, and that the structures are sufficiently stable to allow displacements. 2.
Experimental setup
The experimental setup is depicted in Fig. 1. The laser source is a CW butterfly laser diode with a wavelength of 975 nm coupled to a single mode fiber (Thorlabs, PL980P330J). The beam is collimated and then expanded so that it overfills the screen of a reflective SLM (Holoeye, Pluto NIR2). The reflected beam is focused by a 40 cm lens (L1) where a spatial filter blocks the undiffracted order. The beam is relayed and coupled to the microscope objective (Nikon E Plan, 100x, oil, 1.25 NA) of our custom inverted microscope by three identical lenses (L2, L3, L4) with a focal length of 20 cm. The microscope sample holder is a 3d piezo stage (Thorlabs, Max301) that also has differential micrometer drives for manual displacement. A short pass filter is placed in front of the CCD (Thorlabs, DCU223M) camera to block the retroreflected laser light from the bottom coverslip of the container. constant phase
symmetric cubic phase
z LED and condenser 1080 px
y Sample x 100X/1.25
f1
f1
f2
3d Piezo stage
Dichroic L1 SLM
Spaal filter Expanded laser beam
L2
L3
L4 filter
Tube lens CCD
Fig. 1. Experimental setup and holograms projected in the SLM (upper left corner), constant phase and symmetric cubic with a = 0.001 μ m−1 .
The phase pattern encoded into a reflecting SLM is a symmetric cubic phase φ (x, y) = abs((ax)3 ) + abs((ay)3 ). Where xy are the coordinates of the pixels in the SLM (dimensions of microns, pixel size of the SLM 8 μ m/px, 1080 pixels), and a is a scale factor with dimensions of inverse microns. This parameter controls the size of the projected pattern, such that decreasing the value of a results in a smaller scale. For all the experiments shown the scaling parameter in the phase is a = 0.001 μ m−1 . Linear prism phases are used to displace the beam in order to spatially filter the undiffracted order. Also, negative lens phases displace the 3d pattern along the optical axis. Laser powers are measured after the beam is transmitted by the microscope objective. For the images of the experiments that are shown in this article, the microparticle solution is an aqueous suspension of silica microspheres (Bangs Laboratories) with a mean diameter d of 2.47 μ m and density ρ of 2 gr/cm3 , refractive index n p ∼ 1.4 at 975 nm. The container is prepared by sandwiching spacers (height ∼ 100 μ m) between two glass coverslips (no 1, 0.13-0.16 mm thick). Then, the chamber is filled with the microparticle solution and it is sealed with nail polish.
#208549 - $15.00 USD Received 19 Mar 2014; revised 14 Apr 2014; accepted 6 May 2014; published 13 May 2014 (C) 2014 OSA 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.012283 | OPTICS EXPRESS 12285
3.
Results and analysis
Beam structure. In order to image the beam emerging from the microscope objective at different planes along the optical axis, a single microscope coverslip (No. 1, 0.13-0.16 mm thick) is placed on the microscope stage and the short pass filter is removed. In this way, we can image the retroreflected part of the beam with the CCD. Initially, the glass slide is positioned below the geometrical focus of the microscope objective. The slide is moved up (z direction, along optical axis) in steps of 0.5 μ m using the piezo stage (closed loop mode) until it reaches the plane of the geometrical focus of the objective. The position of the geometrical focus of the lens is labeled as zero, while the positions before that point are negative. We image the cross sections of the beam for two cases: symmetric cubic phase and for a constant phase for comparison (holograms in Fig. 1 with an added linear phase for spatial filtering). The images are shown in Fig. 3, top row: constant phase, bottom row: symmetric cubic phase. The cross sections for the symmetric cubic phase resemble the classic intensity pattern of a beam diffracted by a square aperture. However, with the constant phase (square aperture at the SLM) we don’t observe these patterns (Fig. 2, top row). Another interesting feature of the structured beam (symmetric cubic phase) is that it vanishes beyond the focus, in contrast to the focusing of a beam apodised by a square aperture, where the localized intensity patterns would repeat after the geometrical focus of the lens. The beam cross sections in Fig. 2 (bottom row) in the range of -15 to 0 μ m are used -6.5 μm
-5.5μm
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-14 μm
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-7.5 μm
Symmetric cubic
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Fig. 2. Cross sections. Top row: Constant phase. Bottom row: Symmetric cubic phase. Digital holograms shown in Fig. 1 (upper left corner).
to reconstruct the 3d structure of the beam during propagation. The isosurfaces are shown in Fig. 3. The left side (Fig. 3(a)) shows the 3d structure which resembles a square pyramid with localized spots. A view from the top along the optical axis (Fig. 3(b)) reveals that most of the high intensity spots are located along the four apex edges of the pyramidal structure. The side view (xz plane) is shown in Fig. 3(c). The axial length of the central lobe (∼ 8μ m) is larger than the Rayleigh length (2z0 = 1.46 μ m, diffraction limited) for a Gaussian beam focused with the same microscope objective, suggesting a weaker axial intensity gradient. C
y (μm)
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Fig. 3. Isosurfaces from the cross sections in Fig. 2 (bottom row).
Microparticle stacking and manipulation. The container with the aqueous microparticle solution is placed on the piezo stage of our custom inverted microscope. The imaging plane is #208549 - $15.00 USD Received 19 Mar 2014; revised 14 Apr 2014; accepted 6 May 2014; published 13 May 2014 (C) 2014 OSA 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.012283 | OPTICS EXPRESS 12286
located at the focal plane of the microscope objective. The sample is moved with the microscope stage to a position (z direction along optical axis) where we image the bottom of the container where most of the particles are located due to their higher density. This means that the apex of the pyramidal structure is also located at the bottom. In order to move the apex up (z direction, away from the bottom) we add a negative lens phase to the digital hologram, shifting the beam in the axial direction controlling the effective volume of the structured beam that interacts with the particles. Figures 4(a), 4(b), 4(c), and 4(d) show loading of the microparticles A
B
C
D
E
10 μm
F
20 μm
Fig. 4. A. Apex at bottom. Raising gradually the apex: frames B.,C.,D. E., (Media 1, frame width 30 μ m). F. Microparticle pyramids with different sizes. P=12.5 mW.
by gradually shifting the apex above the imaging plane, resulting in a larger square base (10-20 μ m, Figs. 4(e) and 4(f)) as the apex is raised. We can see that the particles that are closer to the apex appear blurred since the imaging plane is at the bottom. Also, it is interesting that the bottom is not completely filled, since many of the microspheres arrange closer to the apex and along the four apex edges forming microparticle chains (as expected from the isosurfaces, Fig. 3). Loading is achieved by a careful balancing of the forces: the optical gradient forces ∝ ∇I (I is the intensity), the scattering force ∝ I, gravity and optical binding due to the light induced dipoles [2]. First we look at the gradient force. The number of localized spots in the beam decreases for planes that are closer to the apex, while the intensity per spot increases (Fig. 3). The size of the particles is larger than the axial separation between spots, so that the gradient force attracts the particles toward the apex, where the axial gradient of the main lobe is much smaller than the transverse one (Fig. 3). This is confirmed by experiments (not shown) with polystyrene microspheres (d = 2 μ m, n p ∼ 1.6 at 975 nm, ρ = 1 gr/cm3 neutrally buoyant) which typically result in more efficient optical traps (larger Q [23]) than with silica. The polystyrene beads quickly migrate towards the apex and are blown away (for P > 1 mW), since the axial gradient is not large enough to overcome the scattering force. In the case of silica microparticles (higher density), axial stability is achieved by the apparent weight and the axial gradient which balance the scattering force. Considering gradient, scattering and gravity the main limit in the stability of the structures is the overall laser power P, since beyond a certain threshold Pth , gravity and the axial gradient are not sufficient to balance the scattering force. This is observed in the videos of Media 1 that show stable loading (2.47 μ m silica) for P=12 and 18 mW, unstable at 29 mW, and for P=34 mW the particles are blown away. Experiments with 1.5 μ m silica particles (not shown, hologram with a = 0.0009 μ m−1 ) revealed that the maximum threshold Pth decreases to about 8 mW, consistent with smaller apparent weight per particle. The binding force between the light induced dipoles in the silica microspheres produces the chain-like structures along the four apex edges (Fig. 4) and holds the particles together. The dynamics of the particle chains (perturbed by Brownian motion) are similar to that of a flexible rope (Media 1–3). Preliminary experiments with different states of linear polarization suggest subtle changes in the structure and the distribution of particles. The number of stacked particles depends on the size of the 3d structure, to count them,
#208549 - $15.00 USD Received 19 Mar 2014; revised 14 Apr 2014; accepted 6 May 2014; published 13 May 2014 (C) 2014 OSA 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.012283 | OPTICS EXPRESS 12287
0s
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Fig. 5. Releasing the trap to estimate the number of trapped particles with P=12.5 mW (Media 2, frame width 42 μ m).
Vertical
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the laser light is blocked and the microspheres fall back to the bottom. Hence, subtracting the number of particles that initially (before release) were in the background (not in the structure) from the number of particles after the release yields a good estimate of the number of stacked objects (a few might go out of the frame when the particles fall to the bottom ). Selected video frames showing the release of a microparticle pyramid are displayed in Fig. 5 (compressed video in Media 2). To count the objects, we wait for the particles to form a single layer at the bottom, since initially the particles fall on top of each other making a couple of layers. The estimated number of particles in the structure is around one hundred.
20 μm
Fig. 6. Translation of the pattern by displacing the microscope stage P=12.5 mW (Media 3, frame width 42 μ m).
Finally, we show that it is possible to displace the structures by moving the microscope stage in the transverse directions xy, thus dragging the microparticle patterns. Selected frames (compressed video in Media 3) are shown in Fig. 6. We have drawn white circles around some microparticles in the background so that it is easier to observe the displacements. It is interesting that the pattern can move above other objects that are at the bottom, since the bottom of the pyramid is mainly empty. 4.
Conclusion
We have shown a simple method to create 3d optical potentials with a pyramidal geometry using a symmetric cubic phase. These potentials can be used to stack large numbers of microparticles (∼ 100) with moderate laser powers. Some applications could include sorting of microparticles by trapping those with higher density while lifting the others. These beams could also be used in the context of optical clearing with lower power than what is needed for the equivalent Airy beam [20] but with the disadvantage that the particles are lifted vertically. Acknowledgments We thank V. Carmona and R. Guti´errez for exploratory experiments. This work was partially funded by the following grants: CONACYT (153821 and 166961) and DGAPA, UNAM (PAPIIT TB100312 and IN101511).
#208549 - $15.00 USD Received 19 Mar 2014; revised 14 Apr 2014; accepted 6 May 2014; published 13 May 2014 (C) 2014 OSA 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.012283 | OPTICS EXPRESS 12288
