The light that pulls rather than pushes

Light has a pressure because it carries momentum, and momentum arriving at a surface pushes it. That force points along the beam, is proportional to the intensity, and is the one this collection has met before.

There is a second force, it points across the beam, and no amount of momentum arriving can produce it.

Two forces from one beam #

A particle small compared with the wavelength responds to a field by polarising: an induced dipole $\mathbf{p} = \alpha\mathbf{E}$, with $\alpha$ fixed by the particle’s size and by how much more polarisable it is than its surroundings.

A dipole in a uniform field feels no net force. The pull on one end matches the push on the other, and the only effect is a torque.

A dipole in a gradient does feel one. The two ends are in different fields, they do not match, and the net force is $(\mathbf{p}\cdot\nabla)\mathbf{E}$ — which for an induced dipole becomes $\tfrac12\alpha\nabla E^2$, proportional to the gradient of the intensity and pointing up it.

That is the gradient force. It is not a pressure, it does not point along the beam, and it does not care about the sign of the field — which is why an oscillating optical field, whose $E$ averages to zero, produces a steady force.

The induced dipole is what the gradient acts on, and its size is set by how much more polarisable the particle is than the medium around it — not by its polarisability alone. That relative quantity is the one worth holding, because it is what makes the same beam behave oppositely on two objects in the same water: a glass bead is drawn in and an air bubble is driven out, with nothing about the light altered between the two experiments.

The scattering force is the other one and is the ordinary radiation pressure: the particle scatters light out of the beam, the removed momentum goes somewhere else, and the particle recoils forward. For a small sphere the scattering cross-section goes as the sixth power of the radius and the inverse fourth power of the wavelength — the same dependence that makes the sky blue — while the gradient force goes only as the third power of the radius.

That difference in scaling is the whole design problem. For a large particle the scattering force wins and nothing can be trapped; for a small one the gradient force wins, and the crossover is where the trap becomes possible.

Where the energy for the pull comes from #

A force that pulls a particle toward a light source looks as though something is being got for nothing, so it is worth saying where the bookkeeping balances.

The particle in the field has an interaction energy $-\tfrac12\alpha E^2$: negative, because the induced dipole aligns with the field it is induced by, and larger in magnitude where the field is larger. Moving up the gradient therefore lowers the energy, and the force is the gradient of that energy exactly as in any potential problem.

What supplies it is the light, and the transaction is visible if the beam is followed. A particle sitting in a focus scatters light, and it also phase-shifts the light passing it — a small lens in the beam. The redistribution of the beam’s own momentum by that scattering is what the particle recoils against, and integrating the momentum flux over a surface enclosing the whole arrangement gives the same force as the dipole calculation.

So there is no free force. There is a beam whose momentum distribution is altered by the particle’s presence, and the alteration is not confined to the forward direction. Calling it a “gradient force” is a description in terms of the particle’s polarisability; calling it a momentum flux is a description in terms of the field, and the two agree because both are Maxwell’s equations.

Where the particle actually sits #

Along the axis of a focused beam the intensity rises to the waist and falls after it, so the gradient force pulls forward before the waist and backward after it — restoring, in both cases, toward the brightest point.

The scattering force pushes forward everywhere and does not vanish at the waist. So the equilibrium is not at the focus: it is downstream of it, at the point where the backward pull has grown enough to match the forward push. For the beam computed here — a hundred milliwatts at $1064$ nm focused to a half-micrometre waist, acting on a $60$ nm polystyrene sphere in water — that is $0.066$ micrometres past the focus.

Whether there is an equilibrium at all is the design question. The gradient force at its largest must exceed the scattering force, and since the axial gradient scales as the intensity over the Rayleigh range while the scattering force scales as the intensity, the ratio improves as the beam is focused more tightly and does not improve at all with power.

That is why optical tweezers required high-numerical-aperture microscope objectives. Ashkin’s first experiments in 1970 used two counter-propagating beams pointed at each other, so that the two scattering forces cancelled and only the gradient forces remained — a workable arrangement, and one that needs an alignment. The single-beam trap, in 1986, waited for objectives that could focus to a waist comparable with the wavelength.

A spring made of light #

Near the equilibrium the force is linear in displacement, so the trap is a harmonic potential and the particle in it is a mass on a spring.

An optical trap is a potential well tens of $kT$ deep, with a stiffness set by the beam’s intensity and its waist. Calling it a spring is exact rather than figurative: the restoring force is linear in displacement near the bottom, the particle oscillates in it, and the stiffness is measured in newtons per metre like any other.

For the beam above, the depth is $47$ times $k_BT$ and the transverse stiffness is about $3.1\times10^{-6}$ newtons per metre — which is $0.0031$ piconewtons per nanometre, and which is an extraordinarily soft spring — every minimum is a parabola and this one is a very shallow parabola indeed by any standard except the one that matters. A protein motor pulls with a few piconewtons; a strand of DNA resists being stretched with a few; a bacterium swims with a fraction of one. The trap’s stiffness is chosen to sit in that range deliberately.

The particle does not sit still in it. Thermal agitation kicks it about, and the size of the wander is what a spring in a thermal bath always has:

$$\tfrac12 k\langle x^2\rangle = \tfrac12 k_BT,$$

which gives $36$ nanometres of root-mean-square displacement for the numbers above.

Thermal wandering in a trap does not grow without limit. Free Brownian motion has a mean square displacement rising linearly with time; in a trap it saturates at $kT$ over the stiffness — and that saturation is how the stiffness is measured. Watch the particle jiggle, take the variance, and the trap has calibrated itself.

That relation is usually a nuisance and here it is the calibration. Record the particle’s position with a photodetector for a few seconds, compute the variance, and the stiffness follows as $k_BT/\langle x^2\rangle$ — with no knowledge whatever of the laser power, the beam profile, the particle’s size or its refractive index.

A force measurement calibrated by a thermometer is an unusual and rather beautiful thing. It works because equipartition is a statement about any quadratic degree of freedom in contact with a bath, whatever the degree of freedom is made of, and an optical trap makes one out of light.

The three ways to calibrate, and why two are worse #

Equipartition is one of three standard calibrations, and comparing them shows what makes it good.

Drag. Move the fluid past the trapped bead at a known speed and measure the displacement; the force is Stokes’ drag, $6\pi\eta a v$. This requires knowing the bead’s radius and the fluid’s viscosity, and the viscosity near a surface is not the bulk value, so the method carries a systematic that depends on where in the chamber the measurement was made.

The power spectrum. The bead’s position fluctuates with a Lorentzian spectrum whose corner frequency is the stiffness divided by the drag coefficient. Fitting it gives both, and it is the most informative method — but it needs the drag coefficient again, and it needs the detector’s own frequency response to be known and flat well past the corner.

Equipartition. Take the variance of the position and divide $k_BT$ by it. No hydrodynamics, no beam parameters, no bead radius.

Its weakness is the one thing it does depend on: the detector’s length calibration, since a variance in nanometres squared requires knowing what a nanometre is at the detector. It is also biased by any noise added to the measurement, which inflates the variance and therefore underestimates the stiffness — the one direction of error a careful experimenter has to watch, and the reason the power-spectrum method, which separates real motion from white detector noise by their different spectra, is preferred when precision matters.

The same force elsewhere #

The expression $\tfrac12\alpha\nabla E^2$ is not about light.

The force pulling a dielectric slab into a capacitor is the same gradient force at zero frequency, and it lives entirely in the fringing field at the edge. That is worth having beside the optical case because it shows the mechanism is not about light: a polarisable object in any non-uniform field is pulled toward the strong region, and the only thing an optical trap adds is that the field oscillates at $10^{14}$ Hz.

At zero frequency it is what pulls a dielectric slab into a capacitor, and it lives there in the fringing field for the same reason it lives here in the focus: a uniform field exerts no such force and the entire effect is where the field is not uniform.

At radio frequency it is dielectrophoresis, used to trap and sort cells between microfabricated electrodes. At optical frequency it is the trap. And at the frequency of a detuned laser acting on an atom’s resonance it is the dipole force that holds atoms in an optical lattice, where the polarisability is enormous near resonance and the trap can be made from milliwatts.

The sign is worth a sentence in all of them. The force is up the gradient for a particle more polarisable than its surroundings and down the gradient for one less polarisable. A bubble in water is pushed out of a bright region rather than into it; an atom illuminated above its resonance is repelled from bright regions rather than attracted; and a low-index particle can be trapped only by a beam with a dark centre.

What it was built for #

The instrument’s importance is that it made single-molecule mechanics possible.

Attach one end of a DNA molecule to a trapped bead and the other to a surface, move the surface, and read the bead’s displacement: the force–extension curve of one molecule, measured to a fraction of a piconewton. Attach a bead to a kinesin motor walking along a microtubule and the trap resists it: the motor’s steps appear as $8$-nanometre displacements, one per ATP molecule consumed, and its stall force comes out at about $6$ piconewtons — a discreteness as unmistakable in a mechanical trace as it is in a photocurrent.

Those are measurements on one molecule rather than on an ensemble average, and they revealed behaviour no bulk experiment could have shown — that motors step discretely, that some of the steps fail, that the distribution of dwell times says how many rate-limiting transitions there are per step. Ashkin’s share of the 2018 Nobel prize was for the technique that made them possible, forty-eight years after the first demonstration.

The numbers, collected #

It is worth having the scales in one place, because the instrument works in a corner of parameter space that is not obvious.

The beam: $100$ mW at $1064$ nm, focused to a waist of $0.5$ µm. The peak intensity is $2.5\times10^{11}$ W/m², which is a hundred million times the intensity of sunlight at the Earth and is nevertheless harmless to a cell for the reasons in the next section.

The particle: $60$ nm radius polystyrene, index $1.59$, in water at $1.33$. The ratio $(m^2-1)/(m^2+2)$ that carries all the material dependence is $0.125$ — a number under a fifth, which is why trapping works better in air than in water and better still for a metal particle, whose polarisability is much larger.

The forces: a trap depth of $47\,k_BT$, a stiffness of $0.0031$ pN/nm, and a scattering force smaller than the peak gradient force by a factor of $4.8$. The equilibrium sits $0.066$ µm past the waist.

And the motion: $36$ nm root-mean-square, on a timescale set by the drag over the stiffness, which for this bead in water is about a hundred microseconds. That last figure is why the position detector has to be fast — sampling more slowly than the corner frequency averages the fluctuation away and the equipartition calibration then reports a stiffness that is too large.

Where it stops #

The Rayleigh expressions used here need a particle much smaller than the wavelength. At $60$ nm in a medium of index $1.33$ against a wavelength of $1064$ nm that holds comfortably; at $100$ nm the ratio of the two forces has already fallen to $1.05$ and the trap barely exists. It does not hold for the micrometre-scale beads most experiments actually use, where the particle is comparable with the wavelength and neither the Rayleigh formulae nor the geometrical-optics ones apply; that intermediate regime requires the full Mie calculation, and the numbers change by factors of order one rather than qualitatively.

The trap formulae used here belong to the small-particle end of the scattering range; most real traps operate in the middle, where neither the small limit nor the large one applies and the force has to be computed numerically. That is where the model stops — not at a boundary of principle, but at the point where the closed forms run out and the answer becomes a computation.

The light heats things. A hundred milliwatts in a focal volume of a cubic micrometre is an enormous intensity, and although water absorbs very little at $1064$ nm — which is why that wavelength is used — a trapped bacterium is nonetheless being cooked slowly. Wavelength selection in biological trapping is entirely a compromise between absorption by water at longer wavelengths and by biological material at shorter ones.

And a trap is not a clamp. The particle explores a region tens of nanometres across, always, and no amount of stiffening removes the thermal motion — it only shrinks it as the inverse square root of the stiffness. Every measurement made with such an instrument is a measurement of a mean over that wandering, and the wandering is the noise floor.

A trap tens of $kT$ deep is escaped from at a rate the Boltzmann factor sets, so “trapped” is a statement about a timescale rather than about a permanent condition. At 20 $kT$ the escape time is long compared with any experiment; at 5 it is not, and the particle wanders off while being watched. The depth is chosen against the patience of the experimenter.

The last point deserves its own sentence, because it is where the language misleads. A trap of finite depth is escaped from eventually: the rate is set by the Boltzmann factor of the depth, and $47\,k_BT$ makes the expected escape time longer than any experiment. “Trapped” means a rate rather than a prohibition, exactly as it does for a nucleus behind a barrier.

What the figures leave out #

Two things are missing from the drawn forces and both matter for anybody comparing the numbers with an experiment.

The first is that the transverse and axial stiffnesses are not equal. A focused Gaussian beam varies over a waist across and over a Rayleigh range along, and the Rayleigh range is larger than the waist by roughly the waist divided by the wavelength — so the axial gradient is gentler and the axial stiffness is smaller, typically by a factor of three to ten. A trap is an anisotropic spring, and quoting one number for it is a simplification.

The second is that the calculation treats the beam as an undisturbed Gaussian. The particle is in the beam and alters it: it scatters, it acts as a small lens, and near a surface it interacts with its own reflection. The force it feels is the force in the actual field, not the field that would be there in its absence, and the difference grows with the particle’s size. That is precisely why the Rayleigh treatment is a small-particle treatment: the approximation being made is that the particle does not matter to the field it is sitting in.

Both corrections are computed in practice by solving the full scattering problem, which is a numerical exercise rather than a formula. The value of the expressions here is that they say what depends on what — power, waist, wavelength, index contrast, radius — and those dependences survive the corrections even where the numbers do not.

Why it is called a tweezer rather than a tractor beam #

The distinction is worth making, because the two are often conflated and only one of them exists.

What has been described holds a particle at a point and moves it by moving the point. The pull toward the focus is real and it acts over a distance of about a wavelength — the scale over which the intensity varies — so nothing is being drawn in from far away. Move the microscope stage and the trapped bead follows, which is a tweezer.

A tractor beam would pull an object toward the source along the beam over an extended distance, and arrangements that do something like it exist: structured beams whose intensity increases along the propagation direction can pull a particle backward over the region where that holds, and beams with particular angular spectra can produce a net backward scattering force on a particle that scatters preferentially forward. Both are real, both are laboratory demonstrations at micrometre scales, and neither scales to anything larger, because the forces are proportional to the polarisability and inversely proportional to the scale over which the field varies.

The ladder from here #

Later rungs on this anchor: the Mie regime treated properly, with the force computed from the full scattering solution and the resonances that appear when the particle is comparable with the wavelength; optical binding, where two trapped particles interact through the light they scatter at one another; the photonic force microscope, which uses the trapped bead as a probe rather than as a handle; and laser cooling, where the same beam is detuned so that the scattering force depends on velocity and removes energy rather than merely pushing.

The neighbouring ladders are light has a pressure, which is the other force; the force that lives where the model is not, which is the same gradient force at zero frequency; and the jiggle that proved atoms, which is what the calibration measures against.