Abstract

Refractive optical trapping forces can be nonconservative in the vicinity of a stable equilibrium point even in the absence of radiation pressure. We discuss how nonconservative 3D force fields reduce, near an equilibrium point, to circular forcing in a plane; a simple model of such forcing is the refractive trapping of a sphere by four rays. We discuss in general the diffusion of an anisotropically trapped, circularly forced particle and obtain its spectrum of motion. Equipartition of potential energy holds, even though the nonconservative flow does not follow equipotentials of the trap. We find that the dissipated nonconservative power is proportional to temperature, providing a mechanism for runaway heating instability in traps.

© 2011 Optical Society of America

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References

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  1. A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992).
    [Crossref] [PubMed]
  2. W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, 2nd ed. (Addison-Wesley, 1962).
  3. Y. Roichman, B. Sun, A. Stolarski, and D. G. Grier, “Influence of nonconservative optical forces on the dynamics of optically trapped colloidal spheres: the fountain of probability,” Phys. Rev. Lett. 101, 128301 (2008).
    [Crossref] [PubMed]
  4. B. Sun, J. Lin, E. Darby, A. Y. Grosberg, and D. G. Grier, “Brownian vortexes,” Phys. Rev. E 80, 010401 (2009).
    [Crossref]
  5. S. H. Simpson and S. Hanna, “First-order nonconservative motion of optically trapped nonspherical particles,” Phys. Rev. E 82, 031141 (2010).
    [Crossref]
  6. D. G. B. Edelen, Applied Exterior Calculus (Wiley, 1985).
  7. W. L. Burke, Applied Differential Geometry (Cambridge University Press, 1985).
  8. T. Frankel, The Geometry of Physics, 2nd ed. (Cambridge University Press, 2004).
  9. D. W. Jordan and P. Smith, Nonlinear Ordinary Differential Equations, 2nd ed. (Oxford University Press, 1987).
  10. G. B. Burch, Matrix Methods in Optics (Wiley, 1975).
  11. P. Nelson, Biological Physics: Energy, Information, Life(Freeman, 2008).
  12. F. Gittes and C. F. Schmidt, “Thermal noise limitations on micromechanical experiments,” Euro. Biophys. J. 27, 75–81(1998).
    [Crossref]
  13. L. D. Landau and E. M. Lifshitz, Statistical Physics, 3rd ed., Part 1 (Pergamon, 1980).
  14. K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594–612 (2004).
    [Crossref]
  15. F. Gittes, B. Schnurr, P. D. Olmsted, F. C. MacKintosh, and C. F. Schmidt, “Microscopic viscoelasticity: shear moduli of soft materials determined from thermal fluctuations,” Phys. Rev. Lett. 79, 3286–3289 (1997).
    [Crossref]
  16. E. J. G. Peterman, F. Gittes, and C. F. Schmidt, “Laser-induced heating in optical traps,” Biophys. J. 84, 1308–1316 (2003).
    [Crossref] [PubMed]

2010 (1)

S. H. Simpson and S. Hanna, “First-order nonconservative motion of optically trapped nonspherical particles,” Phys. Rev. E 82, 031141 (2010).
[Crossref]

2009 (1)

B. Sun, J. Lin, E. Darby, A. Y. Grosberg, and D. G. Grier, “Brownian vortexes,” Phys. Rev. E 80, 010401 (2009).
[Crossref]

2008 (1)

Y. Roichman, B. Sun, A. Stolarski, and D. G. Grier, “Influence of nonconservative optical forces on the dynamics of optically trapped colloidal spheres: the fountain of probability,” Phys. Rev. Lett. 101, 128301 (2008).
[Crossref] [PubMed]

2004 (1)

K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594–612 (2004).
[Crossref]

2003 (1)

E. J. G. Peterman, F. Gittes, and C. F. Schmidt, “Laser-induced heating in optical traps,” Biophys. J. 84, 1308–1316 (2003).
[Crossref] [PubMed]

1998 (1)

F. Gittes and C. F. Schmidt, “Thermal noise limitations on micromechanical experiments,” Euro. Biophys. J. 27, 75–81(1998).
[Crossref]

1997 (1)

F. Gittes, B. Schnurr, P. D. Olmsted, F. C. MacKintosh, and C. F. Schmidt, “Microscopic viscoelasticity: shear moduli of soft materials determined from thermal fluctuations,” Phys. Rev. Lett. 79, 3286–3289 (1997).
[Crossref]

1992 (1)

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992).
[Crossref] [PubMed]

Ashkin, A.

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992).
[Crossref] [PubMed]

Berg-Sørensen, K.

K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594–612 (2004).
[Crossref]

Burch, G. B.

G. B. Burch, Matrix Methods in Optics (Wiley, 1975).

Burke, W. L.

W. L. Burke, Applied Differential Geometry (Cambridge University Press, 1985).

Darby, E.

B. Sun, J. Lin, E. Darby, A. Y. Grosberg, and D. G. Grier, “Brownian vortexes,” Phys. Rev. E 80, 010401 (2009).
[Crossref]

Edelen, D. G. B.

D. G. B. Edelen, Applied Exterior Calculus (Wiley, 1985).

Flyvbjerg, H.

K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594–612 (2004).
[Crossref]

Frankel, T.

T. Frankel, The Geometry of Physics, 2nd ed. (Cambridge University Press, 2004).

Gittes, F.

E. J. G. Peterman, F. Gittes, and C. F. Schmidt, “Laser-induced heating in optical traps,” Biophys. J. 84, 1308–1316 (2003).
[Crossref] [PubMed]

F. Gittes and C. F. Schmidt, “Thermal noise limitations on micromechanical experiments,” Euro. Biophys. J. 27, 75–81(1998).
[Crossref]

F. Gittes, B. Schnurr, P. D. Olmsted, F. C. MacKintosh, and C. F. Schmidt, “Microscopic viscoelasticity: shear moduli of soft materials determined from thermal fluctuations,” Phys. Rev. Lett. 79, 3286–3289 (1997).
[Crossref]

Grier, D. G.

B. Sun, J. Lin, E. Darby, A. Y. Grosberg, and D. G. Grier, “Brownian vortexes,” Phys. Rev. E 80, 010401 (2009).
[Crossref]

Y. Roichman, B. Sun, A. Stolarski, and D. G. Grier, “Influence of nonconservative optical forces on the dynamics of optically trapped colloidal spheres: the fountain of probability,” Phys. Rev. Lett. 101, 128301 (2008).
[Crossref] [PubMed]

Grosberg, A. Y.

B. Sun, J. Lin, E. Darby, A. Y. Grosberg, and D. G. Grier, “Brownian vortexes,” Phys. Rev. E 80, 010401 (2009).
[Crossref]

Hanna, S.

S. H. Simpson and S. Hanna, “First-order nonconservative motion of optically trapped nonspherical particles,” Phys. Rev. E 82, 031141 (2010).
[Crossref]

Jordan, D. W.

D. W. Jordan and P. Smith, Nonlinear Ordinary Differential Equations, 2nd ed. (Oxford University Press, 1987).

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Statistical Physics, 3rd ed., Part 1 (Pergamon, 1980).

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Statistical Physics, 3rd ed., Part 1 (Pergamon, 1980).

Lin, J.

B. Sun, J. Lin, E. Darby, A. Y. Grosberg, and D. G. Grier, “Brownian vortexes,” Phys. Rev. E 80, 010401 (2009).
[Crossref]

MacKintosh, F. C.

F. Gittes, B. Schnurr, P. D. Olmsted, F. C. MacKintosh, and C. F. Schmidt, “Microscopic viscoelasticity: shear moduli of soft materials determined from thermal fluctuations,” Phys. Rev. Lett. 79, 3286–3289 (1997).
[Crossref]

Nelson, P.

P. Nelson, Biological Physics: Energy, Information, Life(Freeman, 2008).

Olmsted, P. D.

F. Gittes, B. Schnurr, P. D. Olmsted, F. C. MacKintosh, and C. F. Schmidt, “Microscopic viscoelasticity: shear moduli of soft materials determined from thermal fluctuations,” Phys. Rev. Lett. 79, 3286–3289 (1997).
[Crossref]

Panofsky, W. K. H.

W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, 2nd ed. (Addison-Wesley, 1962).

Peterman, E. J. G.

E. J. G. Peterman, F. Gittes, and C. F. Schmidt, “Laser-induced heating in optical traps,” Biophys. J. 84, 1308–1316 (2003).
[Crossref] [PubMed]

Phillips, M.

W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, 2nd ed. (Addison-Wesley, 1962).

Roichman, Y.

Y. Roichman, B. Sun, A. Stolarski, and D. G. Grier, “Influence of nonconservative optical forces on the dynamics of optically trapped colloidal spheres: the fountain of probability,” Phys. Rev. Lett. 101, 128301 (2008).
[Crossref] [PubMed]

Schmidt, C. F.

E. J. G. Peterman, F. Gittes, and C. F. Schmidt, “Laser-induced heating in optical traps,” Biophys. J. 84, 1308–1316 (2003).
[Crossref] [PubMed]

F. Gittes and C. F. Schmidt, “Thermal noise limitations on micromechanical experiments,” Euro. Biophys. J. 27, 75–81(1998).
[Crossref]

F. Gittes, B. Schnurr, P. D. Olmsted, F. C. MacKintosh, and C. F. Schmidt, “Microscopic viscoelasticity: shear moduli of soft materials determined from thermal fluctuations,” Phys. Rev. Lett. 79, 3286–3289 (1997).
[Crossref]

Schnurr, B.

F. Gittes, B. Schnurr, P. D. Olmsted, F. C. MacKintosh, and C. F. Schmidt, “Microscopic viscoelasticity: shear moduli of soft materials determined from thermal fluctuations,” Phys. Rev. Lett. 79, 3286–3289 (1997).
[Crossref]

Simpson, S. H.

S. H. Simpson and S. Hanna, “First-order nonconservative motion of optically trapped nonspherical particles,” Phys. Rev. E 82, 031141 (2010).
[Crossref]

Smith, P.

D. W. Jordan and P. Smith, Nonlinear Ordinary Differential Equations, 2nd ed. (Oxford University Press, 1987).

Stolarski, A.

Y. Roichman, B. Sun, A. Stolarski, and D. G. Grier, “Influence of nonconservative optical forces on the dynamics of optically trapped colloidal spheres: the fountain of probability,” Phys. Rev. Lett. 101, 128301 (2008).
[Crossref] [PubMed]

Sun, B.

B. Sun, J. Lin, E. Darby, A. Y. Grosberg, and D. G. Grier, “Brownian vortexes,” Phys. Rev. E 80, 010401 (2009).
[Crossref]

Y. Roichman, B. Sun, A. Stolarski, and D. G. Grier, “Influence of nonconservative optical forces on the dynamics of optically trapped colloidal spheres: the fountain of probability,” Phys. Rev. Lett. 101, 128301 (2008).
[Crossref] [PubMed]

Biophys. J. (2)

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992).
[Crossref] [PubMed]

E. J. G. Peterman, F. Gittes, and C. F. Schmidt, “Laser-induced heating in optical traps,” Biophys. J. 84, 1308–1316 (2003).
[Crossref] [PubMed]

Euro. Biophys. J. (1)

F. Gittes and C. F. Schmidt, “Thermal noise limitations on micromechanical experiments,” Euro. Biophys. J. 27, 75–81(1998).
[Crossref]

Phys. Rev. E (2)

B. Sun, J. Lin, E. Darby, A. Y. Grosberg, and D. G. Grier, “Brownian vortexes,” Phys. Rev. E 80, 010401 (2009).
[Crossref]

S. H. Simpson and S. Hanna, “First-order nonconservative motion of optically trapped nonspherical particles,” Phys. Rev. E 82, 031141 (2010).
[Crossref]

Phys. Rev. Lett. (2)

F. Gittes, B. Schnurr, P. D. Olmsted, F. C. MacKintosh, and C. F. Schmidt, “Microscopic viscoelasticity: shear moduli of soft materials determined from thermal fluctuations,” Phys. Rev. Lett. 79, 3286–3289 (1997).
[Crossref]

Y. Roichman, B. Sun, A. Stolarski, and D. G. Grier, “Influence of nonconservative optical forces on the dynamics of optically trapped colloidal spheres: the fountain of probability,” Phys. Rev. Lett. 101, 128301 (2008).
[Crossref] [PubMed]

Rev. Sci. Instrum. (1)

K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594–612 (2004).
[Crossref]

Other (8)

L. D. Landau and E. M. Lifshitz, Statistical Physics, 3rd ed., Part 1 (Pergamon, 1980).

W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, 2nd ed. (Addison-Wesley, 1962).

D. G. B. Edelen, Applied Exterior Calculus (Wiley, 1985).

W. L. Burke, Applied Differential Geometry (Cambridge University Press, 1985).

T. Frankel, The Geometry of Physics, 2nd ed. (Cambridge University Press, 2004).

D. W. Jordan and P. Smith, Nonlinear Ordinary Differential Equations, 2nd ed. (Oxford University Press, 1987).

G. B. Burch, Matrix Methods in Optics (Wiley, 1975).

P. Nelson, Biological Physics: Energy, Information, Life(Freeman, 2008).

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Figures (3)

Fig. 1
Fig. 1 Nonconservative 3D force field near a stable equilibrium point. The circulation of force may be viewed as purely circular in some plane. (a) Vector flow-field point of view. (b) Combed-hair-on-a-sphere point of view.

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Fig. 2
Fig. 2 (a) Refraction of a single ray by a sphere. In the paraxial limit, r ⊥ is much smaller than the radius of the sphere. The force f acts to return the sphere center to the line of k ^ , but it also has a component along k ^ . (b) System of four rays for nonconservative trapping. The limit b ≪ a allows a paraxial approximation for each ray.

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Fig. 3
Fig. 3 Rms radius 〈 r 2 〉 1 / 2 (solid curves) of the position distribution P ( x , y ) of a diffusing particle in a trap with stiffness anisotropy κ y / κ x = 5 , for various values ξ of counterclockwise circular forcing in the x – y plane. The solid curves are lines of flow, which, for ξ ≠ 0 , do not follow the ξ = 0 equipotential contour (dotted curves). Circular flow at ξ = ∞ (final panel, solid curve) has the rms radius [ 2 k B T / ( κ x + κ y ) ] 1 / 2 .

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Equations (44)

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ω = f x d x + f y d y + f z d z ,
ω ( 1 ) = d q 1 ,
ω ( 2 ) = q 1 d q 2 ,
ω ( 3 ) = d q 1 + q 2 d q 3 .
ω = f d θ − d Φ ,
− a y d x + b x d y = 1 2 ( a + b ) ρ 2 d θ − 1 2 ( a − b ) d ( x y ) ,
f = − I 0 c ( k ^ ′ − k ^ ) ,
f = − I 0 c [ ( cos ⁡ Δ θ − 1 ) k ^ + ( sin ⁡ Δ θ ) r ^ ⊥ ] ,
F out P 2 a F in = 2 n 1 − n / 2 − ( n − 1 ) / a a 1 − n / 2 ,
Δ θ = 2 ( n − 1 ) n a r ⊥
f ≈ I 0 c [ r ⊥ 2 2 a 2 k ^ − r ⊥ a ] .
k ^ = x ^ , r ⊥ = ( y − b ) y ^ + z z ^
f = 2 I 0 c a [ b a ( x y ^ − y x ^ ) − ( x x ^ + y y ^ + 2 z z ^ ) ] .
ω = 2 b I 0 c a 2 ρ d θ − d [ I 0 c a ( x 2 + y 2 + 2 z 2 ) ] ,
f = ξ z ^ × r = ξ ρ θ ^ ,
ω = ξ ρ d θ .
ν ± ∼ k B T γ ϵ 2 exp ⁡ ( ± Δ W k B T ) ,
Δ ν Δ W ∼ ξ 2 ϵ 2 γ .
〈 f ω x ω ′ * 〉 = 2 π ( f x ) ω δ ( ω − ω ′ ) ,
〈 d W d t 〉 = 1 2 π ∫ ( f v ) ω d ω = 1 2 π ∫ i ω ( f x ) ω d ω .
N ω = − i γ ( ω + i α x ) x ω ,
〈 N ω N ω * 〉 = γ 2 ( α x 2 + ω 2 ) 〈 x ω x ω * 〉 .
( x 2 ) ω = 2 k B T / γ α x 2 + ω 2 .
〈 x 2 〉 = 1 2 π ∫ ( x 2 ) ω d ω = k B T κ x ,
N ω = ( α − i ω − η z ^ × ) r ω ,
N ω = [ N x , ω N y , ω ] , r ω = [ x ω y ω ] ,
N ω = − i γ M r ω , M = [ ω + i α x i η − i η ω + i α y ]
〈 N ω N ω † 〉 = γ 2 M 〈 r ω r ω † 〉 M † ,
2 γ k B T 1 = γ 2 M ( r r † ) ω M † ,
( r r † ) ω = [ ( x 2 ) ω ( x y ) ω ( y x ) ω ( y 2 ) ω ] .
( x 2 ) ω = 2 k B T γ ω 2 + α y 2 + η 2 [ ω 4 + ( α x 2 + α y 2 − 2 η 2 ) ω 2 + ( η 2 + α x α y ) 2 ] ,
〈 x 2 〉 = k B T κ x + κ y [ 1 + ξ 2 + κ y 2 ξ 2 + κ x κ y ] .
〈 x 2 〉 = 2 k B T κ x + κ y , ξ ≫ κ x , κ y ,
〈 1 2 κ x x 2 + 1 2 κ y y 2 〉 = k B T ,
( x y ) ω = 2 k B T γ η ( α y − α x ) − 2 i η ω [ ω 4 + ( α x 2 + α y 2 − 2 η 2 ) ω 2 + ( η 2 + α x α y ) 2 ] ,
〈 x y 〉 = k B T κ x + κ y ξ ( κ y − κ x ) ξ 2 + κ x κ y .
P ( x , y ) ∝ exp ⁡ [ − 1 2 r ⊤ Gr ]
r ⊤ Gr = r ⊤ 〈 r r ⊤ 〉 − 1 r = 1.
r 2 = 〈 x 2 〉 〈 y 2 〉 − 〈 x y 〉 2 〈 y 2 〉 cos 2 θ − 2 〈 x y 〉 cos ⁡ θ sin ⁡ θ + 〈 x 2 〉 sin 2 θ ,
tan ⁡ 2 θ ξ = 2 ξ ( κ x + κ y ) .
〈 d W d t 〉 = 1 2 π ∫ ( f · v ) ω d ω ,
f ω · v ω * = i ω ξ [ x ω y ω ] ⊤ [ 0 1 − 1 0 ] [ x ω * y ω * ] = i ω ξ ( x ω y ω * − y ω x ω * ) = − 2 ξ ω Im ⁡ x ω y ω * ,
( f · v ) ω = − 2 ξ ω Im ⁡ ( x y ) ω .
〈 d W d t 〉 = 4 k B T ξ 2 κ x + κ y .

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