January1,1998/Vol.23,No.1/OPTICS LETTERS7 Interference model for back-focal-plane displacement
detection in optical tweezers
Frederick Gittes and Christoph F.Schmidt
Department of Physics and Biophysics Rearch Division,University of Michigan,Ann Arbor,Michigan48109-1120
Received August22,1997
The lateral position of an optically trapped object in a microscope can be monitored with a quadrant photodiode
to within nanometers or better by measurement of intensity shifts in the back focal plane of the lens that
is collimating the outgoing lar light.This detection is largely independent of the position of the trap in
the field of view.We provide a model for the esntial mechanism of this type of detection,giving a simple,
clod-form analytic solution with simplifying assumptions.We identify intensity shifts as first-order far-field
interference between the outgoing lar beam and scattered light from the trapped particle,where the latter
is pha advanced owing to the Gouy pha anomaly.This interference also ref lects momentum transfer to
the particle,giving the spring constant of the trap.Our respon formula is compared with the results of experiments.©1998Optical Society of America
OCIS codes:120.3180,180.3170,140.7010,170.0180,230.0230,330.4150.
Optical trapping of micrometer-scale particles in a mi-croscope was introduced by Ashkin and co-workers1and
has found wide u in biology,physics,and materials
science.2,3One can achieve trapping in an ordinary light microscope by focusing a lar in the specimen
plane with a high-N.A.lens(in our experiments,a mi-
croscope objective;e Fig.1).Changes in the beam emerging from the trap(in our ca,collimated by a
denr)can then be ud for nonimag-
ing detection of nanometer-scale displacements4,5of the trapped particle.In biological samples,for example,
nanometer resolution at bandwidths up to100kHz can
reveal details of macromolecular motion.
One convenient way to detect particle motion us
a quadrant diode placed in the back focal plane(BFP)
of the collimating lens or at a conjugate optical plane (Fig.1).Similar geometries were ud to generate
pha contrast in scanning transmission electron
microscopy6,7and lar scanning microscopy8and to measure displacements in optical traps.9–12Our own
experimental tup will be described in the future.
The intensity pattern in the BFP does not depend on position of the focus,and one can reposition a
trapped bead in the specimen plane without changing
the photodiode signal(at least for ideal optics).The pattern in the BFP reprents the angular-intensity
distribution of light that has pasd through the focus,人畜相交
and thus BFP detection is equivalent to performing an angular-scattering experiment.
Recent extensive investigations of trapping and
scattering have ud generalized Lorenz–Mie scat-tering13–18to describe spheres in focud Gaus
sian
beams.That theory,although it is rather compli-
cated,is quite accurate up to s1͞kw0ഠ0.15on axis16and sഠ0.1off axis17(where k2p͞l and w0
is the1͞e radius of the waist).Harada and Asakura19
also considered a Rayleigh scatterer in a zero-order Gaussian beam,restricting their analysis to s#0.08 to compare the results with generalized Lorenz–Mie results.All the preceding studies sought to predict local fields and forces as accurately as possible.By contrast,our goal is to explain how scattered light in the far field can be ud for motion and force detection. We provide a model highlighting the esntial physi-cal mechanism that creates the light-intensity distri-bution in the BFP.We u first-order interference to explain far-field intensity shifts in a simple way, even for more highly focud beams.This interfer-ence occurs throughout the angular range of the
focus, Fig.1.Optical trapping tup in a microscope with dis-placement detection.A dichroic mirror diverts the lar from the imaging path,and a quadrant diode(10-mm di-ameter)is placed in a plane conjugate to the BFP of the 1.4-N.A.oil immersion condenr.In practice,light is not exactly collimated by the condenr so that an intermediate image is formed(this angle is exaggerated in the figure). Analog and digital electronics convert current from the diode quadrants into normalized X and Y signals,related as explained in the text to the displacement of a trapped particle from the focus in the specimen plane.
0146-9592/98/010007-03$10.00/0©1998Optical Society of America
8OPTICS LETTERS/Vol.23,No.1/January1,1998 unlike in more-familiar plane-wave scattering prob-lems.13The momentum transfer implicit in the an-gular shift of intensity also allows us to estimate the lateral constant for a trapped particle.To emphasize the interference mechanism,we simplify the actual ex-perimental situation by considering Rayleigh scatter-ing within a zero-order Gaussian beam.The latter is only approximate for our focus,which has sഠ0.19, whereas a zero-order Gaussian beam has been shown even at s0.1to contain an average error of4.4%,14 and the Rayleigh approximation is not highly accurate for spheres as large as ours͑d0.5m m͒.Our cal-culated detector respon profile is nevertheless very similar to the experimental respon.
Light emerging from the trapping focus at angles
u and f(Fig.2)will be detected in the BFP at a ra-dius f sin u(by the sine condition20),where f is the focal length of the lens.We consider angular-intensity changes caud by first-order interference for a given lateral displacement of a bead away from the focus in the specimen plane.Our approximations are as fol-lows:(1)We are concerned with small beads,and
so we assume a Rayleigh scatterer(a delta-function polarizability),but we u the uniform-field polariz-ability(at optical frequencies)that corresponds to our actual sphere volume[Eq.(3),below].(2)The focus is described in a paraxial scalar Gaussian beam approxi-mation.21In our experiment the focus is roughly30±
in angular radius;the input has w inഠ1.0mm,with the objective f2.1mm.A small-angle approxima-tion will not correctly predict axial trapping forces,but we are concerned with lateral displacement detection and trapping force.We also assume that the trapped particles lie in the focal plane;in reality there is some axial displacement.
Defining r0to be at the focus,we obrve diverging light at an angle u from the optical axis. The outgoing unscattered electric field21is
E͑r͒ഠ2ikw0I1/2
tot
r͑pe s c s1/2
exp͑ikr2k2w20u2͞4͒.(1)
A factor exp͑2i v t͒is implicit.Here k2p n s͞l0(in our experiment l01.064m m);e s,n s,and c s are the permittivity,the refractive index,and the speed of light in the solvent,respectively;w0is the1͞e radius of the focus,and I tot is the total power in the beam.We u the SI system.The factor2i in Eq.(1)reprents the pha anomaly of the focus(the Gouy pha jump)20 and introduces an important90±pha shift between scattered and unscattered light in the far field.
A small spherical particle of diameter d2a is located at r s in the focal plane,taken for simplicity to be a lateral displacement by x.The unscattered field at this point is
E͑r s͒E͑x͒
2I1/2
tot
w0͑pe s c s͒1/2
exp͑2x2͞w20͒.(2)
The particle will have an induced dipole moment p4pe s a E.For an index of refraction n rn͞n s relative to the solvent,the uniform-field susceptibility is22
aa3n2r21
n2r12
0.0074d3,(3)
assuming that n1.45for silica beads and n s1.33in aqueous solution(at l01.064m m).In a Rayleigh approximation the scattered field at large r is
E0͑r͒ഠ
k2a
r
E͑x͒exp͑ik j r2r s j͒,
ഠk
2a
r
E͑x͒exp͓ik͑r2x sin u cos f͔͒(4)
at obrvation angle u and f(f0along the x axis).We consider only first-order interference,which is justified since intensity modulation owing to the bead is small throughout the BFP.Discarding an j E0j2yields the average change in intensity I:
d I
e s c s
2股票止损
͑j E1E0j22j E j2͒ഠe s c s Re͑EE0ء͒.(5)
Substituting expressions(1),(2),and(4)into Eq.(5) gives us
d I͑x͒
I tot
2k
3a
p r2
exp͑2x2͞w20͒
3sin͑kx sin u cos f͒exp͑2k2w20u2͞4͒.(6) Equation(6)describes the angular-interference pat-tern caud by a particle displaced by x from the optical axis in the focal plane,obrved in a direction͑u,f͒.
A split diode monitoring the BFP of the collimating lens(Fig.1)is oriented for detection along the6x axis; intensity changes on the͑1͒and͑2͒halves are equal and opposite.Integrating Eq.(6)over angles u a
nd f with2p͞2,f,p͞2and with sin uഠ0gives
I12I2
I11I2
ഠ16
p
p
k a
w0
G͑x͞w0͒,
支气管炎食疗
G͑u͒exp͑22u2͒
Z u
雪中送碳
exp͑y2͒d y.(7)
Expression(7)predicts absolute detector respon. For x,,w0,the respon is proportional to d3͞w30, showing a nsitive dependence on particle and focus size.G͑u͒is calculable with Dawson’s inte-gral,23and in Fig.3expression(7)is compared with the experimental data.In Fig.3we apply expres-sion(7)(dashed curve),using the
nominal sphere size
Fig.2.Focus geometry as discusd in the text.A refrac-tive particle,located in the focal plane of a Gaussian beam, is displaced laterally by x.Interference of scattered and unscattered light is considered at a large distance j r j and is obrved in the BFP at a radius f sin u(the collimating lens,of focal length f,obeys the sine condition).The an-gular radius of the Gaussian focus(ϳ30±)is exaggerated.
January1,1998/Vol.23,No.1/OPTICS LETTERS
9
Fig.3.Experimental calibration data compared with model respon.Solid curve,a silica bead fixed upon a cover slip was moved by a piezoelectric stage through the nsitive range of the detector.Dashed curve,expres-sion(7)with no adjustable parameters besides the nominal sphere size͑0.5m m͒and a focus size͑w0ഠ0.53m m͒estimated from the input beam width.The function G͑x͞w0͒in expression(7)has a slope of w021at x0and
extrema G͑x͞w0͒60.334at x60.552w0.
͑d0.5m m͒and the focus size estimated roughly21 from the input Gaussian beam width͑w inഠ1.0mm͒and obtain kw0ഠ2f͞w in,implying that w0ഠ0.53m m. This ab initio model respon,with no adjustable parameters besides w0and d,is quite similar to the measured respon.The cloness of the agreement may perhaps be partly fortuitous,given the strong dependence of expression(7)on w20.
The d3dependence of the linear respon in ex-pression(7)is intrinsic for detection bad on small-particle scattering.Our model works for spheres with d0.5m mഠl͞2.For beads much larger than l,ray optics applies and l drops out of the problem.Then d is the only length scale,and the respon can d
epend only on x͞d;conquently the large-sphere linear re-spon must decrea as1͞d.
劳动价值理论
Our model provides a simple picture not only of detection but also of lateral trapping.The rate of light momentum transfer implied by the interference modulation,Eq.(6),can be shown to equal,as it must,the lateral trapping force of the lar focus. The Minkowski form of the stress tensor22for the momentum f lux of outgoing light has T rrI͞c s. Using I from Eq.(6)and integrating the x projection ͑T rr sin u cos f͒over u and f in the paraxial ca gives the spring constant for the trapped particle:
F x͞xഠ16a I tot
c s w40
,(8)
which agrees with the transver gradient force F gpe s a=j E j2evaluated at the waist.Thus one can also u BFP detection to track directly the time-dependent force being exerted on the particle(which was done in a counterpropagating-beam trap11).
In the past,lar trapping of spheres was mod-eled quantitatively for particles large enough for a geometric-optics treatment.24For smaller particles, for which ray optics is inappropriate,intuitive un-d
erstanding of trapping and detection has been rela-tively inaccessible,although rigorous electromagnetic expansions describing spheres near foci have been pre-nted and ud.14–18Here we have found that a simple model bad on pure interference throughout the numerical aperture of the focus evidently captures the physical mechanism of lateral trapping and detec-tion for smaller particles.
This study was supported in part by the Whitaker Foundation,the National Science Foundation(grant BIR95-12699),and donors to the Petroleum Rearch Fund,which is administered by the American Chemical Society.We thank Winfield Hill and the Rowland Institute for Science for generous technical support. References
1.A.Ashkin,J.M.Dziedzic,J.C.Bjorkholm,and S.Chu,
补肾喝什么Opt.Lett.11,288(1986).
2.A.Ashkin,Proc.Nat.Acad.Sci.USA94,4853(1997).
3.K.Svoboda and S.M.Block,Annu.Rev.Biophys.
Biomol.Struct.23,247(1994).
4.W.Denk and W.W.Webb,Appl.Opt.29,2382(1990).
5.K.Svoboda,C.F.Schmidt,B.J.Schnapp,and S.M.
Block,Nature(London)365,721(1993).
6.N.H.Dekkers and H.de Lang,Philips Tech.Rev.37,1
(1977).
7.N.H.Dekkers and H.de Lang,Optik41,452(1974).
8.T.Wilson,Optik80,167(1988).
9.L.P.Ghislain,N.A.Switz,and W.W.Webb,Rev.Sci.
Instrum.65,2762(1994).
10.L.P.Ghislain and W.W.Webb,Opt.Lett.18,1678
(1993).
11.S.B.Smith,Y.J.Cui,and C.Bustamante,Science271,
795(1996).
12.K.Visscher,S.P.Gross,and S.M.Block,IEEE J.
Select.Topics Quantum Electron.2,1066(1996).
13.H.C.van de Hulst,Light Scattering by Small Particles
(Wiley,New Y ork,1957).
14.J.P.Barton and D.R.Alexander,J.Appl.Phys.66,
2800(1989).有爱
15.J.P.Barton,D.R.Alexander,and S.A.Schaub,J.
Appl.Phys.66,4594(1989).
16.J.A.Lock and G.Gouesbet,J.Opt.Soc.Am.A11,2503
(1994).
17.G.Gouesbet and J.A.Lock,J.Opt.Soc.Am.A11,2516
(1994).
18.K.F.Ren,G.Gr´e han,and G.Gouesbet,Appl.Opt.35,
2702(1996).
19.Y.Harada and T.Asakura,Opt.Commun.124,529
(1996).
20.M.Born and E.Wolf,Principles of Optics,6th ed.
(Pergamon,London,1989).
21.A.E.Siegman,Lars(University Science,Mill Valley,
Calif.,1986).
22.J. D.Jackson,Classical Electrodynamics,2nd ed.
(Wiley,New Y ork,1975).
23.W.H.Press,B.P.Flannery,S.A.Teukolsky,and W.T.
采药去Vetterling,Numerical Recipes in C,2nd ed.(Cambridge U.Press,Cambridge,1992).
24.R.C.Gauthier and S.W allace,J.Opt.Soc.Am.B12,
1680(1995).
