Abstract

The optical forces of a (2 + 1)D Airy beam on a microsphere are studied in the ray optics regime. The ray model of a (2 + 1)D Airy beam is derived from its Fourier angular spectrum using a stable aggregate of the flexible elements theory. Numerical results demonstrate that the microsphere can be trapped by the transverse optical force and pulled towards the beam major lobe. Longitudinal optical forces further push the microsphere towards the positive z-direction. The trend for the movement of a microsphere in an Airy beam is clearly demonstrated as the stream line of optical forces, which is consistent with the observed phenomena in optical trapping experiments. In the meantime, both the transverse and the longitudinal optical forces increase when the relative refractive index of the trapped microsphere increases. Calculation of optical forces on microspheres with larger size reveals that the optical forces contributed by rays on each hemisphere are actually different due to the asymmetry of the Airy beam. The force difference could cause natural torque on the trapped objects if they are ovals or other asymmetric shapes.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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2017 (1)

2016 (1)

Z. Zhao, W. Zang, and J. Tian, “Optical trapping and manipulation of Mie particles with Airy beam,” J. Opt. 18(2), 025607 (2016).
[Crossref]

2013 (1)

M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photon. Rev. 7(6), 839–854 (2013).
[Crossref]

2011 (1)

K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics 5(6), 335–342 (2011).
[Crossref]

2010 (3)

2009 (1)

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(10), 1334–1336 (2009).
[Crossref]

2008 (3)

2007 (2)

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
[Crossref]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref]

2006 (1)

2002 (1)

2001 (2)

M. A. Alonso and G. W. Forbes, “Using rays better. II. Ray families to match prescribed wave fields,” J. Opt. Soc. Am. A 18(5), 1146–1159 (2001).
[Crossref]

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, “Numerical Modelling of Optical Trapping,” Comput. Phys. Commun. 142(1-3), 468–471 (2001).
[Crossref]

1997 (1)

1994 (1)

1992 (1)

A. Ashkin, “Forces of a Single-Beam Gradient Laser Trap on a Dielectric Sphere in the Ray Optics Regime,” Biophys. J. 61(2), 569–582 (1992).
[Crossref]

1989 (1)

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66(10), 4594–4602 (1989).
[Crossref]

1986 (1)

1980 (1)

M. V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980).
[Crossref]

1979 (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

1965 (1)

D. Ludwig, “Wave propagation near a smooth caustic,” Bull. Amer. Math. Soc. 71(5), 776–780 (1965).
[Crossref]

Alexander, D. R.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66(10), 4594–4602 (1989).
[Crossref]

Alonso, M. A.

Alpmann, C.

M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photon. Rev. 7(6), 839–854 (2013).
[Crossref]

Ambrosio, L. A.

Ashkin, A.

A. Ashkin, “Forces of a Single-Beam Gradient Laser Trap on a Dielectric Sphere in the Ray Optics Regime,” Biophys. J. 61(2), 569–582 (1992).
[Crossref]

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(5), 288–290 (1986).
[Crossref]

Balazs, N. L.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Barton, J. P.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66(10), 4594–4602 (1989).
[Crossref]

Baumgartl, J.

J. Baumgartl, T. Čižmár, M. Mazilu, V. C. Chan, A. E. Carruthers, B. A. Capron, W. McNeely, E. M. Wright, and K. Dholakia, “Optical path clearing and enhanced transmission through colloidal suspensions,” Opt. Express 18(16), 17130–17140 (2010).
[Crossref]

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(10), 1334–1336 (2009).
[Crossref]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

Berry, M. V.

M. V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980).
[Crossref]

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Bishop, A. I.

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, “Numerical Modelling of Optical Trapping,” Comput. Phys. Commun. 142(1-3), 468–471 (2001).
[Crossref]

Bjorkholm, J. E.

Block, S. M.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).

Broky, J.

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
[Crossref]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref]

Cai, J.

Capron, B. A.

Carruthers, A. E.

Chan, V. C.

Chen, H.

Cheng, H.

Christodoulides, D. N.

Chu, S.

Cižmár, T.

Collins, S. D.

Day, D.

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(10), 1334–1336 (2009).
[Crossref]

Denz, C.

M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photon. Rev. 7(6), 839–854 (2013).
[Crossref]

Dholakia, K.

K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics 5(6), 335–342 (2011).
[Crossref]

J. Baumgartl, T. Čižmár, M. Mazilu, V. C. Chan, A. E. Carruthers, B. A. Capron, W. McNeely, E. M. Wright, and K. Dholakia, “Optical path clearing and enhanced transmission through colloidal suspensions,” Opt. Express 18(16), 17130–17140 (2010).
[Crossref]

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(10), 1334–1336 (2009).
[Crossref]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

Dogariu, A.

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
[Crossref]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref]

Dziedzic, J. M.

Esseling, M.

M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photon. Rev. 7(6), 839–854 (2013).
[Crossref]

Fei, Z.

Forbes, G. W.

Fuxi, G.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Gu, M.

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(10), 1334–1336 (2009).
[Crossref]

Hannappel, G. M.

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(10), 1334–1336 (2009).
[Crossref]

Heckenberg, N. R.

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, “Numerical Modelling of Optical Trapping,” Comput. Phys. Commun. 142(1-3), 468–471 (2001).
[Crossref]

Hernández-Figueroa, H. E.

Knoesen, A.

Li, Y. M.

Lin, Z.

Liu, S.

Lu, W.

Ludwig, D.

D. Ludwig, “Wave propagation near a smooth caustic,” Bull. Amer. Math. Soc. 71(5), 776–780 (1965).
[Crossref]

Mazilu, M.

McNeely, W.

Nieminen, T. A.

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, “Numerical Modelling of Optical Trapping,” Comput. Phys. Commun. 142(1-3), 468–471 (2001).
[Crossref]

Ren, H. L.

Rubinsztein-Dunlop, H.

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, “Numerical Modelling of Optical Trapping,” Comput. Phys. Commun. 142(1-3), 468–471 (2001).
[Crossref]

Schaub, S. A.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66(10), 4594–4602 (1989).
[Crossref]

Sidick, E.

Siviloglou, G. A.

Stevenson, D. J.

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(10), 1334–1336 (2009).
[Crossref]

Svoboda, K.

Tian, J.

Z. Zhao, W. Zang, and J. Tian, “Optical trapping and manipulation of Mie particles with Airy beam,” J. Opt. 18(2), 025607 (2016).
[Crossref]

H. Cheng, W. Zang, W. Zhou, and J. Tian, “Analysis of optical trapping and propulsion of Rayleigh particles using Airy beam,” Opt. Express 18(19), 20384–20394 (2010).
[Crossref]

Upstill, C.

M. V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980).
[Crossref]

Wendong, X.

Woerdemann, M.

M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photon. Rev. 7(6), 839–854 (2013).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).

Wright, E. M.

Xiaosong, G.

Zang, W.

Z. Zhao, W. Zang, and J. Tian, “Optical trapping and manipulation of Mie particles with Airy beam,” J. Opt. 18(2), 025607 (2016).
[Crossref]

H. Cheng, W. Zang, W. Zhou, and J. Tian, “Analysis of optical trapping and propulsion of Rayleigh particles using Airy beam,” Opt. Express 18(19), 20384–20394 (2010).
[Crossref]

Zhao, Z.

Z. Zhao, W. Zang, and J. Tian, “Optical trapping and manipulation of Mie particles with Airy beam,” J. Opt. 18(2), 025607 (2016).
[Crossref]

Zhou, J. H.

Zhou, W.

Am. J. Phys. (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Appl. Opt. (2)

Biophys. J. (1)

A. Ashkin, “Forces of a Single-Beam Gradient Laser Trap on a Dielectric Sphere in the Ray Optics Regime,” Biophys. J. 61(2), 569–582 (1992).
[Crossref]

Bull. Amer. Math. Soc. (1)

D. Ludwig, “Wave propagation near a smooth caustic,” Bull. Amer. Math. Soc. 71(5), 776–780 (1965).
[Crossref]

Comput. Phys. Commun. (1)

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, “Numerical Modelling of Optical Trapping,” Comput. Phys. Commun. 142(1-3), 468–471 (2001).
[Crossref]

J. Appl. Phys. (1)

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66(10), 4594–4602 (1989).
[Crossref]

J. Opt. (1)

Z. Zhao, W. Zang, and J. Tian, “Optical trapping and manipulation of Mie particles with Airy beam,” J. Opt. 18(2), 025607 (2016).
[Crossref]

J. Opt. Soc. Am. A (1)

Lab Chip (1)

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(10), 1334–1336 (2009).
[Crossref]

Laser Photon. Rev. (1)

M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photon. Rev. 7(6), 839–854 (2013).
[Crossref]

Nat. Photonics (2)

K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics 5(6), 335–342 (2011).
[Crossref]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

Opt. Express (7)

Opt. Lett. (3)

Phys. Rev. Lett. (1)

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref]

Prog. Opt. (1)

M. V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980).
[Crossref]

Other (2)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).

Supplementary Material (1)

NameDescription
» Visualization 1       Evolution of rays for 3D Airy beam.

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Figures (7)

Fig. 1.
Fig. 1. Ray model of (2 + 1)D Airy beam. Each point X is crossed by four rays at the same time. The transverse directions of rays are represented by arrows, the length of arrow is proportional to the sine of the angle between the ray and the z axis. (a) $z = 0$ ; (b) $z = 90$ μm and (c) $z = 180$ μm. Along the diagonal $y = x$ , the magnitude of arrows becomes stronger when the beam propagates, which implies the self-acceleration of Airy beam (see Visualization 1).
Fig. 2.
Fig. 2. Sketch of single ray tracing. (a) Ray validation. L1 is valid (green tick) and L2 is invalid (red cross). (b) Physical vectors associated with force calculation. Incident point, X, the outward normal unit vector, N1. Unit vectors E and N2 indicate directions of beam polarization, and the incident plane normal direction, respectively.
Fig. 3.
Fig. 3. Trapping efficiency of microbead along specific paths. Transverse intensity profiles of the Airy beam at propagation distance of (a) $z = 0$ , (b) $z = 90$ μm and (c) $z = 130$ μm. White straight lines l1 and l2 indicate the positions under study. Trapping efficiencies along respective lines [(d), (e) and (f) along l1, and (g), (h) and (i) along l2] for Qx [(d) and (g)], Qy [(e) and (h)], and Qz [(f) and (i)].
Fig. 4.
Fig. 4. Stream lines of optical force in (a) xoy plane, (b) vertical plane along line y = x (l2), and (c) (2 + 1)D. The caustic surface is demonstrated by a translucent surface in the 3D stream line plot (c). Optical forces with $|{\textbf Q} |< |{{{\textbf Q}_{{\textrm {max}}}}} |/50$ are not shown due to negligible magnitude.
Fig. 5.
Fig. 5. Trapping efficiency on microsphere with $\rho = 2\; \mu \mbox{m}$ in three cases of refractive index. The microsphere is placed at $z = 0$ plane along line l1 [Fig. 3(a)]. The (a) Qx, (b) Qy and (c) Qz components on microsphere with refractive indexes of 1.46 (black), 1.57 (red) and 1.65 (blue), respectively.
Fig. 6.
Fig. 6. Splitting trapping efficiency along line l1. The microsphere with ${n_b} = 1.46$ is placed at $z = 0$ plane. The Qx (a) and Qz (b) components of the trapping efficiency are summation of trapping efficiency on left and right hemispheres (c) in three cases of radius ρ with 2 μm, 4 μm and 6 μm. (d), (e) and (f) are decomposed trapping efficiency Qx on left and right sphere for ρ=2 μm, 4 μm and 6 μm. (g), (h) and (i) are decomposed trapping efficiency Qz on left and right sphere for ρ=2 μm, 4 μm and 6 μm.
Fig. 7.
Fig. 7. Optical torque on ellipsoid object mediated by asymmetric force in Airy beam. The axial optical force on right part of the object is stronger than on the left part, which will induce an optical torque on the object. (a) initially, the object lays horizontal in the transverse plane. (b) the object gradually rotates under the asymmetric axial optical forces. (c) The object reaches the final vertical configuration.

Equations (15)

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A ( u , v , 0 ) exp { 1 3 [ ( a i k x 0 u ) 3 + ( b i k y 0 v ) 3 ] } ,
A ( u , v , z ) exp { 1 3 [ ( a i k x 0 u ) 3 + ( b i k y 0 v ) 3 ] } exp ( i k z ) exp [ i k z 2 ( u 2 + v 2 ) ] .
x = 1 k u arg [ A ( u , v , z ) ] = k 2 x 0 3 u 2 + z u + a 2 x 0 ,
y = 1 k v arg [ A ( u , v , z ) ] = k 2 y 0 3 v 2 + z v + b 2 x 0 .
S 1 = ( υ Δ x 2 k 2 x 0 3 , υ Δ y 2 k 2 y 0 3 , 1 ) ,   S 2 = ( υ Δ x 2 k 2 x 0 3 , υ + Δ y 2 k 2 y 0 3 , 1 ) , S 3 = ( υ + Δ x 2 k 2 x 0 3 , υ Δ y 2 k 2 y 0 3 , 1 ) ,   S 4 = ( υ + Δ x 2 k 2 x 0 3 , υ + Δ y 2 k 2 y 0 3 , 1 ) .
I ( x , y , z )  =  C | Airy ( x x 0 z 2 4 k 2 x 0 4 + i a z k x 0 2 ) Airy ( y y 0 z 2 4 k 2 y 0 4 + i b z k y 0 2 ) | 2   × exp ( 2 a x x 0 + 2 b y y 0 a z 2 k 2 x 0 4 b z 2 k 2 y 0 4 ) .
C = 8 π a b x 0 y 0 exp [ 2 3 ( a 3 + b 3 ) ] .
t 2 = 2 ( X c ) S > 0.
d P = 1 4 P 0 I | cos θ | d σ ,
Q s = 1 + R cos ( 2 α ) T 2 [ cos ( 2 α 2 β ) + R cos ( 2 α ) 1 + R 2 + 2 R cos ( 2 β ) ] ,
Q g = R sin ( 2 α ) + T 2 [ sin ( 2 α 2 β ) + R sin ( 2 α ) 1 + R 2 + 2 R cos ( 2 β ) ] .
R = tan 2 ( α β ) tan 2 ( α + β ) sin 2 γ + sin 2 ( α β ) sin 2 ( α + β ) cos 2 γ ,
T = sin ( 2 α ) sin ( 2 β ) sin 2 ( α + β ) cos 2 ( α β ) sin 2 γ + sin ( 2 α ) sin ( 2 β ) sin 2 ( α + β ) cos 2 γ .
d Q χ = c n m P 0 d F χ = 1 4 Q χ I | cos θ | ρ 2 sin θ d φ d θ F ^ χ ,   χ = s ,  g,
Q χ = 1 4 Σ Q χ I | cos θ | ρ 2 sin θ F ^ χ d φ d θ ,   χ = s, g,

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