Recent Optical and Photonic Technologies Part 15 ppt

b The damping coefficient filled square and the trap frequency filled circle as a function of the laser intensity.. Semiclassical theory of sub-Doppler forces in an asymmetric magneto-op

substituted by the averaged value (I s)av = 3.78 mW/cm2 As shown in Fig 13, the trap frequencies are in good agreement with the theoretical values The damping coefficients, on the other hand, are about twice larger than the simple theoretical predictions We provide a quantitative description of the theoretical model and explain the discrepancy found in the damping coefficient

The summary of the data of Fig 13 is presened in Fig 14 The damping coefficient and the

trap frequency are presented as a function of s0δ/(1+4δ2)2 and bs0δ /(1+4δ2), respectively

Fig 14 The damping coefficient versus s0δ/(1 + 4δ2)2 [filled circles, experimental data; dashed line, calculated results; dashed-dotted line, calculated results multiplied by 1.76] and the trap frequency versus bs0δ /(1 + 4δ2) [filled squares, experimental data; solid line, calculated results]

One can observe that the measured trap frequencies are in excellence agreement with the calculated results On the other hand, one has to multiply the simply calculated damping coefficients by a factor 1.76 to fit the experimental data We find that the discrepancy in the damping coefficients results from the existence of the sub-Doppler trap described in Sec 2.3

In order to show that the existence of the sub-Doppler force affects the Doppler-cooling parameters, we have performed Monte-Carlo simulation with 1000 atoms In the simulation,

we used sub-Doppler forces and momentum diffusions described in Sec 2.3 The results are presented in Fig 15 Here we averaged the trajectories for 1000 atoms by using the same

parameters as used in Fig 12 We have varied the intensity (I) associated with Fsub without affecting the intensity for the Doppler force, and obtained the averaged trajectory, where

Iexpt =0.17 mW/cm2 is the laser intensity used in the experiment [Fig 13] We then infer the damping coefficient and the trap frequency by fitting the averaged trajectory with Eq (17) The fitted results for the damping coefficient and the trap frequency are shown in Fig 15(b) While the trap frequency remains nearly constant, the damping coefficient increases with the intensity Note that to obtain an increase of factor 1.76 as shown in Fig 14, one should

use I/Iexpt = 1.6 The reason for the increase of the damping coefficient can be well explained qualitatively from the simulation

An Asymmetric Magneto-Optical Trap 405

(a) (b)

Fig 15 The Monte-Carlo simulation results (a) The averaged trajectories for 1000 atoms together with the fitted curves obtained from Eq (17) (b) The damping coefficient (filled square) and the trap frequency (filled circle) as a function of the laser intensity

4 Adjustable magneto-optical trap

When the detuning and intensity of the longitudinal (z-axis) lasers along the symmetry axis

of the anti-Helmholtz coil of the MOT are different from those of the transverse (x and y

axis) lasers, one can realize an array of several sub-Doppler traps (SDTs) with adjustable separations between traps (Heo et al., 2007; Noh & Jhe, 2007) As shown in Fig 16(a), it is similar to the conventional six-beam MOT, except that the detunings (δx and δy) and

intensities (I x and I y) of the transverse lasers can be different from those of the longitudinal ones (δz and I z) In the case of usual MOT, one obtains a usual Doppler trap superimposed with a tightly confined SDT at the MOT center, exhibiting bimodal velocity as well as spatial distributions (Dalibard, 1988; Townsend et al., 1995; Drewsen et al., 1994; Wallace et al.,

1994; Kim et al., 2004) Under equal detunings but unequal intensities (I x , I y  I z), which typically arise in the nonlinear dynamics study of nonadiabatically driven MOT (Kim et al., 2003; 2006), one still obtains the bimodal distribution However, as the transverse-laser detuning δt (≡ δx = δy) is different from the longitudinal one δz with the same configuration of laser intensity, the SDT at the center becomes suppressed with the usual Doppler trap still present The existence of the central SDT, available at equal detunings, contributes not only

to the lower atomic temperature but also to the larger damping coefficients than is expected

(a) (b)

Fig 16 (a) Schematic of the asymmetric magneto-optical trap (b) Measured damping coefficients versus normalized laser-detuning differences

by the Doppler theory In order to confirm the enhanced damping, we have measured the

damping coefficients of MOT versus the laser detuning differences, δt – δz, by using the

transient oscillation method described in Sec 3.2 (Kim et al., 2005) As is shown in Fig 16(b),

one can observe a ‘resonance’ behaviour; the damping coefficient is suppressed by more

than a factor of 2 and approaches the usual Doppler value at unequal detunings, which is

directly associated with the disappearance of the central SDT

When the transverse laser intensity is increased above a certain value at unequal detunings,

we now observe the appearance of novel SDTs In Fig 17, the fluorescence images of the

trapped atoms, obtained with I t ≡ I x + I y = 11.4I z fixed, are presented for various values of

δt – δz The central peak, corresponding to the usual SDT, becomes weak when the detunings

are different, as discussed in Fig 16(b) However, the two side peaks, associated with the

novel SDTs, are displaced symmetrically with respect to the MOT center, in proportion to δt

– δz In addition to these two adjustable side SDTs, there also exist another two weak SDTs

located midway between each side SDT and the central one, which will be discussed later

Fig 17 (a) Fluorescence images that show two adjustable side SDTs for several values of δt –

δz (b) SDT pictures plotted in series with the increasing detuning differences

In Fig 18(a), we plot the positions of the two side SDTs for various values of δt – δz,

represented by filled squares, which are also shown in Fig 17(b) Attributed to the

coherences between the ground-state magnetic sublevels with Δm = ±1 transitions (see Fig

18(b)), the two side SDTs appear at the positions z S=± (δ δt− z)/(g gμB b) and thus their

where Δν = (δt – δz)/(2π) and μB is the Bohr magneton Since the ground-state g-factor is g g =

1/3 for 85Rb atoms and the magnetic field gradient is b = 0.17 T/m, the calculated value

An Asymmetric Magneto-Optical Trap 407

(solid line) is Δz/Δν = 1.26 mm/MHz, which agrees well with the experimental result of 1.25 (±0.12) mm/MHz, considering 10% error of position measurements On the other hand, the

two weak SDTs, resulting from the coherences due to Δm = ±2 transitions (refer to Fig 18(b)), are located midway at z M = z S/2, as shown in Fig 18(a) (open circles) The fitted result

is 0.61 mm/MHz, which is almost half the value given by Eq (18), in good agreement with

the ‘doubled’ energy differences of the Δm = ±2 transitions with respect to the Δm = ±1 ones,

responsible for the side SDTs

(a) (b)

Fig 18 Measured positions of available SDTs versus negative detuning differences

In order to have a qualitative understanding of the detuning-difference dependence, we have calculated the cooling and trapping forces in two dimension by using the optical Bloch equation approach (Dalibard, 1988; Chang & Minogin, 2002; Noh & Jhe, 2007) In Fig 19(a),

we present the calculated forces F(z,v = 0) for F g = 3→ F e = 4 atomic transition In the

presence of the transverse lasers, the ground-state sublevels with Δm = ±1 transitions can be

coupled by a π photon from the transverse lasers in combination with a σ ± photon from the longitudinal lasers (see Fig 18(b)) As a result, for unequal detunings, there exists a position where the Zeeman shift compensates the laser-frequency difference, such that

(a) (b)

Fig 19 (a) Calculated forces F(z,v =0) for various detuning differences The maximum forces

at 0.3 Γ corresponds to 5 × 10–3 kΓ Here δz = –2.7Γ, I z = 0.11 mW/cm2, and I t = 5.6I z (b) Five SDTs, including two weak SDTs midway between the two side SDTs and the central one, for δt – δz = –0.24Γ

=

At this position, atoms can feel the sub-Doppler forces associated with the Δm = ±1

coherences and thus the novel SDT is obtained at two positions of ± (δx – δz )/(g gμB b), as

confirmed in Fig 18(a) As shown in Fig 19(b), the two weak midway SDTs arise because

the weak σ ± photons, in addition to the dominant π ones, from the transverse lasers can

contribute to the atomic coherences in the z-direction Therefore, besides the Δm = ±1

transitions responsible for the side SDTs, the two-photon-assisted Δm = ±2 coherences (here,

each σ ± photon comes from the longitudinal and the transverse laser, as shown in Fig 18(b))

can be generated, and atoms at the position z M, satisfying the relation ωt – ωz =

±2g gμB bz M, feel this additional coherence As a result, the midway SDTs can be obtained at

z M = z S/2 (see Fig 18(a)) The typically observed image and the calculated force are

presented in Fig 19(b)

5 Conclusions

In this article we have presented experimental and theoretical works on the asymmetric

magneto-optical trap In Sec 2, we have studied parametric resonance in a magneto-optical

trap We have described a theoretical aspect of parametric resonance by the analytic and

numerical methods We also have measured the amplitude and phase of the limit cycle

motions by changing the modulation frequency or the amplitude We find that the results

are in good agreement with the calculation results, which are based on simple Doppler

cooling theory In the final subsection we described direct observation of the sub-Doppler

part of the MOT without the Doppler part by using the parametric resonance which We

compared the spatial profile of sub-Doppler trap with the Monte-Carlo simulation, and

observed they are in good agreements

In Sec 3, we have presented two methods to measure the trap frequency: one is using

parametric resonance and the other transient oscillation method In the case of parametric

resonance method, we could measure the trap frequency accurately by decreasing the

modulation amplitude of the parametric excitation down to its threshold value While only

the trap frequency were able to be obtained by the parametric resonance method, we could

obtain both the trap frequency and the damping coefficient by the transient oscillation

method We have made a quantitative study of the Doppler cooling theory in the MOT by

measuring the trap parameters We have found that the simple rate-equation model can

accurately describe the experimental data of trap frequencies

In Sec 4, we have demonstrated the adjustable multiple traps in the MOT When the laser

detunings are different, the usual sub-Doppler force and the corresponding damping

coefficient at the MOT center is greatly suppressed, whereas the novel sub-Doppler traps are

generated and exist within a finite range of detuning differences We have found that π and

σ ±atomic transitions excited by the transverse lasers in the longitudinal direction are

responsible for the strong side and the weak middle sub-Doppler traps, respectively The

adjustable array of sub-Doppler traps may be useful for controllable

atom-interferometer-type experiments in atom optics or quantum optics

The AMOT described in this article can be used for study of nonlinear dynamics using cold

atoms such as critical phenomena far from equilibrium (Kim et al., 2006) or a nonlinear

Duffing oscillation (Nayfeh & Moore, 1979; Strogatz, 2001)

An Asymmetric Magneto-Optical Trap 409

6 Acknowledgement

This work was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00355)

7 References

Bagnato, V S., Marcassa, L G., Oria, M., Surdutovich, G I., Vitlina, R & Zilio, S C (1993) Spatial

distribution of atoms in a magneto-optical trap, Phys Rev A Vol 48(No 5): 3771–3775

Chang, S & Minogin, V (2002) Density-matrix approach to dynamics of multilevel atoms in

laser fields, Phys Rep Vol 365(No 2): 65–143

Dalibard, J (1998) Laser cooling of an optically thick gas: The simplest radiation pressure

trap?, Opt Commun Vol 68(No 3): 203–208

di Stefano, A., Fauquembergue, M., Verkerk, P & Hennequin, D (2003) Giant oscillations in

a magneto-optical trap, Phys Rev A Vol 67(No 3): 033404-1–033404-4

Dias Nunes, F., Silva, J F., Zilio, S C & Bagnato, V S (1996) Influence of laser fluctuations

and spontaneous emission on the ring-shaped atomic distribution in a

magneto-optical trap, Phys Rev A Vol 54(No 3): 2271–2274

Drewsen, M., Laurent, Ph., Nadir, A., Santarelli, G., Clairon, A., Castin, Y., Grinson, D &

Salomon, C (1994) Investigation of sub-Doppler cooling effects in a cesium

magnetooptical trap, Appl Phys B Vol 59(No 3): 283–298

Friebel, S., D’Andrea, C., Walz, J., Weitz, M & H¨ansch, T W (1998) CO2-laser optical

lattice with cold rubidium atoms, Phys Rev A Vol 57(No 1): R20–R23

Heo, M S., Kim, K., Lee, K H., Yum, D., Shin, S., Kim, Y., Noh, H R & Jhe, W (2007)

Adjustable multiple sub-Doppler traps in an asymmetric magneto-optical trap,

Phys Rev A Vol 75(No 2): 023409-1–023409-4

Hope, A., Haubrich, D., Muller, G., Kaenders, W G & Meschede, D (1993) Neutral cesium

atoms in strong magnetic-quadrupole fields at sub-doppler temperatures, Europhys Lett Vol 22(No 9): 669–674

Jun, J W., Chang, S., Kwon, T Y., Lee, H S & Minogin, V G (1999) Kinetic theory of the

magneto-optical trap for multilevel atoms, Phys Rev A Vol 60(No 5): 3960–3972

Kim, K., Noh, H R., Yeon, Y H & Jhe,W (2003) Observation of the Hopf bifurcation in

parametrically driven trapped atoms, Phys Rev A Vol 68(No 3): 031403(R)-1–

031403(R)-4

Kim, K., Noh, H R & Jhe, W (2004) Parametric resonance in an intensity-modulated

magneto-optical trap, Opt Commun Vol 236(No 4-6): 349–361

Kim, K., Noh, H R., Ha, H J & Jhe, W (2004) Direct observation of the sub-Doppler trap in

a parametrically driven magneto-optical trap, Phys Rev A Vol 69(No 3): 033406-1–

033406-5

Kim, K., Noh, H R & Jhe, W (2005) Measurements of trap parameters of a magneto-optical

trap by parametric resonance, Phys Rev A Vol 71(No 3): 033413-1–033413-5

Kim, K., Lee, K H., Heo, M., Noh, H R & Jhe, W (2005) Measurement of the trap

properties of a magneto-optical trap by a transient oscillation method, Phys Rev A

Vol 71(No 5): 053406-1–053406-5

Kim, K., Heo, M S., Lee, K H., Jang, K., Noh, H R., Kim, D & Jhe, W (2006) Spontaneous

Symmetry Breaking of Population in a Nonadiabatically Driven Atomic Trap: An

Ising-Class Phase Transition, Phys Rev Lett Vol 96(No 15): 150601-1–150601-4

Kohns, P., Buch, P., Suptitz, W., Csambal, C & Ertmer, W (1993) On-Line Measurement of

Sub-Doppler Temperatures in a Rb Magneto-optical Trap-by-Trap Centre

Oscillations, Europhys Lett Vol 22(No 7): 517–522

Kulin, S., Killian, T C., Bergeson, S D & Rolston, S L (2000) Plasma Oscillations and

Expansion of an Ultracold Neutral Plasma, Phys Rev Lett Vol 85(No 2): 318–321

Labeyrie, G., Michaud, F & Kaiser, R (2006) Self-Sustained Oscillations in a Large

Magneto-Optical Trap, Phys Rev Lett Vol 96(No 2): 023003-1–023003-4

Landau, L D & Lifshitz, E M (1976) Mechanics, Pergamon, London

Lapidus, L J., Enzer, D & Gabrielse, G (1999) Stochastic Phase Switching of a

Parametrically Driven Electron in a Penning Trap, Phys Rev Lett Vol 83(No 5):

899–902

Metcalf, H J & van der Straten, P (1999) Laser Cooling and Trapping, Springer, New York Nayfeh, N H & Moore, D T (1979) Nonlinear Oscillations, Wiley, New York

Noh, H R & Jhe, W (2007) Semiclassical theory of sub-Doppler forces in an asymmetric

magneto-optical trap with unequal laser detunings, Phys Rev A Vol 75(No 5):

053411-1–053411-9

Raab, E L., Prentiss, M., Cable, A., Chu, S & Pritchard, D E (1987) Trapping of Neutral

Sodium Atoms with Radiation Pressure, Phys Rev Lett Vol 59(No 23): 2631–2634

Razvi, M A N., Chu, X Z., Alheit, R., Werth, G & Blümel, R (1998) Fractional frequency

collective parametric resonances of an ion cloud in a Paul trap, Phys Rev A Vol

58(No 1): R34–R37

Sesko, D., Walker, T & Wieman, C (1991) Behavior of neutral atoms in a spontaneous force

trap, J Opt Soc Am B Vol 8(No 5): 946–958

Steane, A M., Chowdhury, M & Foot, C J (1992) Radiation force in the magneto-optical

trap, J Opt Soc Am B Vol 9(No 12): 2142–2158

Strogatz, S H (2001) Nonlinear Dynamics and Chaos, Perseus, New York

Tabosa, J W R., Chen, G., Hu, Z., Lee, R B & Kimble, H J (1991) Nonlinear spectroscopy

of cold atoms in a spontaneous-force optical trap, Phys Rev Lett Vol 66(No 25):

3245–3248

Tan, J & Gabrielse, G (1991) Synchronization of parametrically pumped electron oscillators

with phase bistability, Phys Rev Lett Vol 67(No 22): 3090–3093

Tan, J & Gabrielse, G (1993) Parametrically pumped electron oscillators, Phys Rev A Vol

48(No 4): 3105–3121

Townsend, C G., Edwards, N H., Cooper, C J., Zetie, K P., Foot, C J., Steane, A M.,

Szriftgiser, P., Perrin, H & Dalibard J (1995) Phase-space density in the

magneto-optical trap, Phys Rev A Vol 52(No 2): 1423–1440

Walhout, M., Dalibard, J., Rolston, S L & Phillips, W D (1992) σ+–σ– Optical molasses in a

longitudinal magnetic field, J Opt Soc Am B Vol 9(No 11): 1997–2007

Tseng, C H., Enzer, D., Gabrielse, G & Walls, F L (1999) 1-bit memory using one electron:

Parametric oscillations in a Penning trap, Phys Rev A Vol 59(No 3): 2094–2104

Walker, T., Sesko, D & Wieman, C (1990) Collective behavior of optically trapped neutral

atomstitle, Phys Rev Lett Vol 64(No 4): 408–411

Walker, T & Feng, P (1994) Measurements of Collisions Between Laser-Cooled Atoms,

Adv At Mol Opt Phys Vol 34: 125–170

Wallace, C D., Dinneen, T P., Tan, K Y N., Kumarakrishnan, A., Gould, P L & Javanainen,

J (1994) Measurements of temperature and spring constant in a magneto-optical

trap, J Opt Soc Am B Vol 11(No 5): 703–711

Wilkowski, D., Ringot, J., Hennequin, D & Garreau, J C (2000) Instabilities in a

Magnetooptical Trap: Noise-Induced Dynamics in an Atomic System, Phys Rev Lett Vol 85(No 9): 1839–1842

Xu, X., Loftus, T H., Smith, M J., Hall, J L., Gallagher, A & Ye, J (2002) Dynamics in a

two-level atom magneto-optical trap, Phys Rev A Vol 66(No 1): 011401-1–011401-4

20

The Photonic Torque Microscope: Measuring Non-conservative Force-fields

1ICFO – The Institute of Photonic Sciences, Castelldefels (Barcelona),

3Universität Stuttgart, Stuttgart,

4Università di Napoli “Federico II”, Napoli,

1Spain

2,3Germany

4Italy

1 Introduction

Over the last 20 years the advances of laser technology have permitted the development of

an entire new field in optics: the field of optical trapping and manipulation The focal spot of a

highly focused laser beam can be used to confine and manipulate microscopic particles ranging from few tens of nanometres to few microns (Ashkin, 2000; Neuman & Block, 2004)

Fig 1 PFM setups with detection using forward (a) and backward (b) scattered light

Such an optical trap can detect and measure forces and torques in microscopic systems – a technique now known as photonic force microscope (PFM) This is a fundamental task in many areas, such as biophysics, colloidal physics and hydrodynamics of small systems

The PFM was devised in 1993 (Ghislain & Webb, 1993) A typical PFM comprises an optical trap that holds a probe – a dielectric or metallic particle of micrometre size, which randomly moves due to Brownian motion in the potential well formed by the optical trap – and a position sensing system The analysis of the thermal motion provides information about the local forces acting on the particle (Berg-Sørensen & Flyvbjerg, 2004) The PFM can measure forces in the range of femtonewtons to piconewtons This range is well below the limits of techniques based on micro-fabricated mechanical cantilevers, such as the atomic force microscope (AFM)

However, an intrinsic limit of the PFM is that it can only deal with conservative force-fields, while it cannot measure the presence of a torque, which is typically associated with the presence of a non-conservative (or rotational) force-field

In this Chapter, after taking a glance at the history of optical manipulation, we will briefly review the PFM and its applications Then, we will discuss how the PFM can be enhanced to deal with non-conservative force-fields, leading to the photonic torque microscope (PTM) (Volpe & Petrov, 2006; Volpe et al., 2007a) We will also present a concrete analysis workflow to reconstruct the force-field from the experimental time-series of the probe position Finally, we will present three experiments in which the PTM technique has been successfully applied:

1. Characterization of singular points in microfluidic flows We applied the PTM to

microrheology to characterize fluid fluxes around singular points of the fluid flow (Volpe et al., 2008)

2 Detection of the torque carried by an optical beam with orbital angular momentum We used the

PTM to measure the torque transferred to an optically trapped particle by a Gaussian beam (Volpe & Petrov, 2006)

Laguerre-3 Quantitative measurement of non-conservative forces generated by an optical trap We used the

PTM to quantify the contribution of non-conservative optical forces to the optical trapping (Pesce et al., 2009)

2 Brief history of optical manipulation

Optical trapping and manipulation did not exist before the invention of the laser in 1960 (Townes, 1999) It was already known from astronomy and from early experiments in optics that light had linear and angular momentum and, therefore, that it could exert radiation pressure and torques on physical objects Indeed, light’s ability to exert forces has been

recognized at least since 1619, when Kepler’s De Cometis described the deflection of comet

tails by sunrays

In the late XIX century Maxwell’s theory of electromagnetism predicted that the light momentum flux was proportional to its intensity and could be transferred to illuminated objects, resulting in a radiation pressure pushing objects along the propagation direction of light

Early exciting experiments were performed in order to verify Maxwell’s predictions Nichols and Hull (Nichols & Hull, 1901) and Lebedev (Lebedev, 1901) succeeded in detecting radiation pressure on macroscopic objects and absorbing gases A few decades later, in 1936, Beth reported the experimental observation of the torque on a macroscopic object resulting from interaction with light (Beth, 1936): he observed the deflection of a quartz wave plate suspended from a thin quartz fibre when circularly polarized light passed through it These effects were so small, however, that they were not easily detected Quoting J H Poynting’s

The Photonic Torque Microscope: Measuring Non-conservative Force-fields 413 presidential address to the British Physical Society in 1905, “a very short experience in attempting to measure these forces is sufficient to make one realize their extreme minuteness – a minuteness which appears to put them beyond consideration in terrestrial affairs.” (Cited in Ref (Ashkin, 2000))

Things changed with the invention of the laser in the 1960s (Townes, 1999) In 1970 Ashkin showed that it was possible to use the forces of radiation pressure to significantly affect the dynamics of transparent micrometre sized particles (Ashkin, 1970) He identified two basic

light pressure forces: a scattering force in the direction of the incident beam and a gradient force in the direction of the intensity gradient of the beam He showed experimentally that,

using just these forces, a focused laser beam could accelerate, decelerate and even stably trap small micrometre sized particles

Ashkin considered a beam of power P reflecting on a plane mirror: P h/ ν photons per second strike the mirror, each carrying a momentum hν/c , where h is the Planck constant,

ν is the light frequency and c the speed of light If they are all reflected straight back, the

total change in light momentum per second is 2⋅(P h/ ν) (⋅ hν/c)=2 /P c, which, by conservation of momentum, implies that the mirror experiences an equal and opposite force

in the direction of the light This is the maximum force that one can extract from the light Quoting Ashkin (Ashkin, 2000), “Suppose we have a laser and we focus our one watt to a

small spot size of about a wavelength 1 m≅ μ , and let it hit a particle of diameter also of

1 mμ Treating the particle as a 100% reflecting mirror of density ≅1gm cm/ 3, we get an acceleration of the small particle=A F m= / =10− 3dynes/10− 12gm=109cm sec/ 2 Thus,

6

10

A≅ g, where g≅103cm sec/ 2, the acceleration of gravity This is quite large and should give readily observable effects, so I tried a simple experiment [ ] It is surprising that this simple first experiment [ ], intended only to show forward motion due to laser radiation pressure, ended up demonstrating not only this force but the existence of the transverse force component, particle guiding, particle separation, and stable 3D particle trapping.”

In 1986, Ashkin and colleagues reported the first observation of what is now commonly

referred to as an optical trap (Ashkin et al., 1986): a tightly focused beam of light capable of

holding microscopic particles in three dimensions One of Ashkin’s co-authors, Steven Chu, would go on to use optical tweezing in his work on cooling and trapping atoms This research earned Chu, together with Claude Cohen-Tannoudji and William Daniel Phillips, the 1997 Nobel Prize in Physics

In the late 1980s, the new technology was applied to the biological sciences, starting by

trapping tobacco mosaic viruses and Escherichia coli bacteria In the early 1990s, Block, Bustamante and Spudich pioneered the use of optical trap force spectroscopy, an alternate

name for PFM, to characterize the mechanical properties of biomolecules and biological motors (Block et al., 1990; Finer et al., 1994; Bustamante et al., 1994) Optical traps allowed these biophysicists to observe the forces and dynamics of nanoscale motors at the single-molecule level Optical trap force spectroscopy has led to a deeper understanding of the nature of these force-generating molecules, which are ubiquitous in nature

Optical tweezers have also proven useful in many other areas of physics, such as atom trapping (Metcalf & van der Straten, 1999) and statistical physics (Babic et al., 2005)

3 The photonic force microscope

One of the most prominent uses of optical tweezers is to measure tiny forces, in the order of 100s of femtonewtons to 10s of piconewtons A typical PFM setup comprises an optical trap

to hold a probe - a dielectric or metallic particle of micrometer size - and a position sensing

system In the case of biophysical applications the probe is usually a small dielectric bead

tethered to the cell or molecule under study The probe randomly moves due to Brownian

motion in the potential well formed by the optical trap Near the centre of the trap, the

restoring force is linear in the displacement The stiffness of such harmonic potential can be

calibrated using the three-dimensional position fluctuations To measure an external force

acting on the probe it suffices to measure the probe average position displacement under the

action of such force and multiply it by the stiffness

In order to understand the PFM it is necessary to discuss these three aspects:

1 the optical forces that act on the probe and produce the optical trap;

2 the position detection, which permits one to track the probe position with nanometre

resolution and at kilohertz sampling rate;

3 the statistics of the Brownian motion of the probe in the trap, which are used in the

calibration procedure

3.1 Optical forces

It is well known from quantum mechanics that light carries a momentum: for a photon at

wavelength λ the associated momentum is p h= /λ For this reason, whenever an atom

emits or absorbs a photon, its momentum changes according to Newton’s laws Similarly, an

object will experience a force whenever a propagating light beam is refracted or reflected by

its surface However, in most situations this force is much smaller than other forces acting

on macroscopic objects so that there is no noticeable effect and, therefore, can be neglected

The objects, for which this radiation pressure exerted by light starts to be significant, weigh

less than 1 gμ and their size is below 10s of microns

A focused laser beam acts as an attractive potential well for a particle The equilibrium

position lies near – but not exactly at – the focus When the object is displaced from this

equilibrium position, it experiences an attractive force towards it In first approximation this

restoring force is proportional to the displacement; in other words, optical tweezers force

can generally be described by Hooke’s law:

where x is the particle’s position, x0 is the focus position and k x is the spring constant of the

optical trap along the x -direction, usually referred to as trap stiffness In fact, an optical

tweezers creates a three-dimensional potential well, which can be approximated by three

independent harmonic oscillators, one for each of the x -, y- and z-directions If the optics are

well aligned, the x and y spring constants are roughly the same, while the z spring constant

is typically smaller by a factor of 5 to 10

Considering the ratio between the characteristic dimension L of the trapped object and the

wavelength λ of the trapping light, three different trapping regimes can be defined:

1 the Rayleigh regime, when L<< ; λ

2 an intermediate regime, when L is comparable to λ;

3 the geometrical optics regime, when L>> λ

In Fig 2 an overview of the kind of objects belonging to each of these regimes is presented,

considering that the trapping wavelength is usually in the visible or near-infrared spectral

region In any of these regimes, the electromagnetic equations can be solved to evaluate the

The Photonic Torque Microscope: Measuring Non-conservative Force-fields 415 force acting on the object However, this can be a cumbersome task For the Rayleigh regime and geometrical optics regime approximate models have been developed However, most of the objects that are normally trapped in optical manipulation experiments fall in the intermediate regime, where such approximations cannot be used In particular, this is true for the probes usually used for the PFM: typically particles with diameter between 0.1 and

10 micrometres

Fig 2 Trapping regimes and objects that are typically optically manipulated: from cells to viruses in biophysical experiments, and from atoms to colloidal particles in experimental statistical physics The wavelength of the trapping light is usually in the visible or near-infrared

3.2 Position detection

The three-dimensional position of the probe is typically measured through the scattering of

a light beam illuminating it This can be the same beam used for trapping or an auxiliary beam

Typically, position detection is achieved through the analysis of the interference of the forward-scattered (FS) light and unscattered (incident) light A typical setup is shown in Fig

1(a) The PFM with FS detection was extensively studied, for example, in Ref (Rohrbach &

Stelzer, 2002)

In a number of experiments, however, geometrical constraints may prevent access to the FS

light, forcing one to make use of the backward-scattered (BS) light instead This occurs, for

example, in biophysical applications where one of the two faces of a sample holder needs to

be coated with some specific material or in plasmonics applications where a plasmon wave

needs to be coupled to one of the faces of the holder (Volpe et al., 2006) A typical setup that

uses the BS light is presented in Fig 1(b) The PFM with BS detection has been studied

theoretically in Ref (Volpe et al., 2007b) and experimentally in Ref (Huisstede et al., 2005)

Two types of photodetectors are typically used The quadrant photodetector (QPD) works

by measuring the intensity difference between the left-right and top-bottom sides of the

detection plane The position sensing detector (PSD) measures the position of the centroid of

the collected intensity distribution, giving a more adequate response for non-Gaussian

profiles Note that high-speed video systems are also in use, but they do not achieve the

acquisition rate available with photodetectors

3.3 Brownian motion of an optically trapped particle

Assuming a very low Reynolds number regime (Happel & Brenner, 1983), the Brownian

motion of the probe in the optical trap is described by a set of Langevin equations:

( )t ( )t 2D ( ),t

where r( )t =[x t y t z t( ), ( ), ( )]T is the probe position, γ=6 Rπ η its friction coefficient, R its

radius, η the medium viscosity, K the stiffness matrix, 2Dγ⎡⎣h t h t h t x( ), ( ), ( )y z ⎤⎦ a vector of T

independent white Gaussian random processes describing the Brownian forces, D k T= B /γ

the diffusion coefficient, T the absolute temperature and k B the Boltzmann constant The

orientation of the coordinate system can be chosen in such a way that the restoring forces

are independent in the three directions, i.e K=diag(k k k x, ,y z) In such reference frame the

stochastic differential Eqs (2) are separated and, without loss of generality, the treatment

can be restricted to the x-projection of the system

When a constant and homogeneous external force f ext x, acting on the probe produces a shift

in its equilibrium position in the trap, its value can be obtained as:

where ( )x t is the probe mean displacement from the equilibrium position

There are several straightforward methods to experimentally measure the trap parameters –

trap stiffness and conversion factor between voltage and length – and, therefore, the force

exerted by the optical tweezers on an object, without the need for a theoretical reference

model of the electromagnetic interaction between the particle and the laser beam The most

commonly employed ones are the drag force method, the equipartition method, the potential

analysis method and the power spectrum or correlation method (Visscher et al., 1996;

Berg-Sørensen & Flyvbjerg, 2004) The latter, in particular, is usually considered the most reliable

one Experimentally the trap stiffness can be found by fitting the autocorrelation function

(ACF) of the Brownian motion in the trap obtained from the measurements to the theoretical

one, which reads

The Photonic Torque Microscope: Measuring Non-conservative Force-fields 417

*

k B xx

4 The photonic torque microscope

The PFM measures a constant force acting on the probe This implies that the force-field to

be measured has to be invariable (homogeneous) on the scale of the Brownian motion of the

trapped probe, i.e in a range of 10s to 100s of nanometres depending on the trapping

stiffness In particular, as we will see, this condition implicates that the force-field must be

conservative, excluding the possibility of a rotational component

Fig 3 Examples of physical systems that produce force-fields that cannot be correctly

probed with a classical PFM, because they vary on the scale of the Brownian motion of the

trapped probe (a possible range is indicated by the red bars): (a) forces produced by a

surface plasmon polariton in the presence of a patterned surface on a 50nm radius dielectric

particle (adapted from Ref (Quidant et al., 2005)); (b) trapping potential for 10nm diameter

dielectric particle near a 10nm wide gold tip in water illuminated by a 810nm

monochromatic light beam (adapted from Ref (Novotny et al., 1997)); and (c) force-field

acting on a 500nm radius dielectric particle in the focal plane of a highly focused

Laguerre-Gaussian beam (adapted from Ref (Volpe & Petrov, 2006))

However, there are cases where these assumptions are not fulfilled The force-field can vary

in the nanometre scale, for example, considering the radiation forces exerted on a dielectric

particle by a patterned optical near-field landscape at an interface decorated with resonant

gold nanostructures (Quidant et al., 2005) (Fig 3 (a)), the nanoscale trapping that can be

achieved near a laser-illuminated tip (Novotny et al., 1997) (Fig 3(b)), the optical forces

produced by a beam which carries orbital angular momentum (Volpe & Petrov, 2006) (Fig

3(c)), or in the presence of fluid flows (Volpe et al., 2008) In order to deal with these cases,

we need a deeper understanding of the Brownian motion of the optically trapped probe in

the trapping potential

In the following we will discuss the Brownian motion near an equilibrium point in a

force-field and we will see how this permits us to develop a more powerful theory of the PFM: the

Photonic Torque Microscope (PTM) Full details can be found in Ref (Volpe et al., 2007a)