Scientific instruments
Optical tweezers (originally called single-beam gradient force trap) are scientific instruments that use a highly focused laser beam to hold and move microscopic and sub-microscopic objects like atoms, nanoparticles and droplets, in a manner similar to tweezers. If the object is held in air or vacuum without additional support, it can be called optical levitation.
The laser light provides an attractive or repulsive force (typically on the order of piconewtons), depending on the relative refractive index between particle and surrounding medium. Levitation is possible if the force of the light counters the force of gravity. The trapped particles are usually micron-sized, or even smaller. Dielectric and absorbing particles can be trapped, too.
Optical tweezers are used in biology and medicine (for example to grab and hold a single bacterium, a cell like a sperm cell or a blood cell, or a molecule like DNA), nanoengineering and nanochemistry (to study and build materials from single molecules), quantum optics and quantum optomechanics (to study the interaction of single particles with light). The development of optical tweezing by Arthur Ashkin was lauded with the 2018 Nobel Prize in Physics.
History and development.
The detection of optical scattering and the gradient forces on micron sized particles was first reported in 1970 by Arthur Ashkin, a scientist working at Bell Labs. Years later, Ashkin and colleagues reported the first observation of what is now commonly referred to as an optical tweezer: a tightly focused beam of light capable of holding microscopic particles stable in three dimensions. In 2018, Ashkin was awarded the Nobel Prize in Physics for this development.
One author of this seminal 1986 paper, Steven Chu, would go on to use optical tweezing in his work on cooling and trapping neutral atoms. This research earned Chu the 1997 Nobel Prize in Physics along with Claude Cohen-Tannoudji and William D. Phillips. In an interview, Steven Chu described how Ashkin had first envisioned optical tweezing as a method for trapping atoms. Ashkin was able to trap larger particles (10 to 10,000 nanometers in diameter) but it fell to Chu to extend these techniques to the trapping of neutral atoms (0.1 nanometers in diameter) using resonant laser light and a magnetic gradient trap (cf. Magneto-optical trap).
In the late 1980s, Arthur Ashkin and Joseph M. Dziedzic demonstrated the first application of the technology to the biological sciences, using it to trap an individual tobacco mosaic virus and "Escherichia coli" bacterium. Throughout the 1990s and afterwards, researchers like Carlos Bustamante, James Spudich, and Steven Block pioneered the use of optical trap force spectroscopy to characterize molecular-scale biological motors. These molecular motors are ubiquitous in biology, and are responsible for locomotion and mechanical action within the cell. Optical traps allowed these biophysicists to observe the forces and dynamics of nanoscale motors at the single-molecule level; optical trap force-spectroscopy has since led to greater understanding of the stochastic nature of these force-generating molecules.
Optical tweezers have proven useful in other areas of biology as well. They are used in synthetic biology to construct tissue-like networks of artificial cells, and to fuse synthetic membranes together to initiate biochemical reactions. They are also widely employed in genetic studies and research on chromosome structure and dynamics. In 2003 the techniques of optical tweezers were applied in the field of cell sorting; by creating a large optical intensity pattern over the sample area, cells can be sorted by their intrinsic optical characteristics. Optical tweezers have also been used to probe the cytoskeleton, measure the visco-elastic properties of biopolymers, and study cell motility. A bio-molecular assay in which clusters of ligand coated nano-particles are both optically trapped and optically detected after target molecule induced clustering was proposed in 2011 and experimentally demonstrated in 2013.
Optical tweezers are also used to trap laser-cooled atoms in vacuum, mainly for applications in quantum science. Some achievements in this area include trapping of a single atom in 2001, trapping of 2D arrays of atoms in 2002, trapping of strongly interacting entangled pairs in 2010, trapping precisely assembled 2-dimensional arrays of atoms in 2016 and 3-dimensional arrays in 2018. These techniques have been used in quantum simulators to obtain programmable arrays of 196 and 256 atoms in 2021 and represent a promising platform for quantum computing.
Researchers have worked to convert optical tweezers from large, complex instruments to smaller, simpler ones, for use by those with smaller research budgets.
Physics.
General description.
Optical tweezers are capable of manipulating nanometer and micron-sized dielectric particles, and even individual atoms, by exerting extremely small forces via a highly focused laser beam. The beam is typically focused by sending it through a microscope objective. Near the narrowest point of the focused beam, known as the beam waist, the amplitude of the oscillating electric field varies rapidly in space. Dielectric particles are attracted along the gradient to the region of strongest electric field, which is the center of the beam. The laser light also tends to apply a force on particles in the beam along the direction of beam propagation. This is due to conservation of momentum: photons that are absorbed or scattered by the tiny dielectric particle impart momentum to the dielectric particle. This is known as the scattering force and results in the particle being displaced slightly downstream from the exact position of the beam waist, as seen in the figure.
Optical traps are very sensitive instruments and are capable of the manipulation and detection of sub-nanometer displacements for sub-micron dielectric particles. For this reason, they are often used to manipulate and study single molecules by interacting with a bead that has been attached to that molecule. DNA and the proteins and enzymes that interact with it are commonly studied in this way.
For quantitative scientific measurements, most optical traps are operated in such a way that the dielectric particle rarely moves far from the trap center. The reason for this is that the force applied to the particle is linear with respect to its displacement from the center of the trap as long as the displacement is small. In this way, an optical trap can be compared to a simple spring, which follows Hooke's law.
Detailed view.
Proper explanation of optical trapping behavior depends upon the size of the trapped particle relative to the wavelength of light used to trap it. In cases where the dimensions of the particle are much greater than the wavelength, a simple ray optics treatment is sufficient. If the wavelength of light far exceeds the particle dimensions, the particles can be treated as electric dipoles in an electric field. For optical trapping of dielectric objects of dimensions within an order of magnitude of the trapping beam wavelength, the only accurate models involve the treatment of either time dependent or time harmonic Maxwell equations using appropriate boundary conditions.
Ray optics.
In cases where the diameter of a trapped particle is significantly greater than the wavelength of light, the trapping phenomenon can be explained using ray optics. As shown in the figure, individual rays of light emitted from the laser will be refracted as it enters and exits the dielectric bead. As a result, the ray will exit in a direction different from which it originated. Since light has a momentum associated with it, this change in direction indicates that its momentum has changed. Due to Newton's third law, there should be an equal and opposite momentum change on the particle.
Most optical traps operate with a Gaussian beam (TEM00 mode) profile intensity. In this case, if the particle is displaced from the center of the beam, as in the right part of the figure, the particle has a net force returning it to the center of the trap because more intense beams impart a larger momentum change towards the center of the trap than less intense beams, which impart a smaller momentum change away from the trap center. The net momentum change, or force, returns the particle to the trap center.
If the particle is located at the center of the beam, then individual rays of light are refracting through the particle symmetrically, resulting in no net lateral force. The net force in this case is along the axial direction of the trap, which cancels out the scattering force of the laser light. The cancellation of this axial gradient force with the scattering force is what causes the bead to be stably trapped slightly downstream of the beam waist.
The standard tweezers works with the trapping laser propagated in the
direction of gravity and the inverted tweezers works against gravity.
Electric dipole approximation.
In cases where the diameter of a trapped particle is significantly smaller than the wavelength of light, the conditions for Rayleigh scattering are satisfied and the particle can be treated as a point dipole in an inhomogeneous electromagnetic field. The force applied on a single charge in an electromagnetic field is known as the Lorentz force,
formula_1
The force on the dipole can be calculated by substituting two terms for the electric field in the equation above, one for each charge. The polarization of a dipole is formula_2 where formula_3 is the distance between the two charges. For a point dipole, the distance is infinitesimal, formula_4 Taking into account that the two charges have opposite signs, the force takes the form
formula_5
Notice that the formula_6 cancel out. Multiplying through by the charge, formula_7, converts position, formula_8, into polarization, formula_9,
formula_10
where in the second equality, it has been assumed that the dielectric particle is linear (i.e. formula_11).
In the final steps, two equalities will be used: (1) a vector analysis equality, (2) Faraday's law of induction.
# formula_12
#formula_13
First, the vector equality will be inserted for the first term in the force equation above. Maxwell's equation will be substituted in for the second term in the vector equality. Then the two terms which contain time derivatives can be combined into a single term.
formula_14
The second term in the last equality is the time derivative of a quantity that is related through a multiplicative constant to the Poynting vector, which describes the power per unit area passing through a surface. Since the power of the laser is constant when sampling over frequencies much longer than the frequency of the laser's light ~1014 Hz, the derivative of this term averages to zero and the force can be written as
formula_15
where in the second part we have included the induced dipole moment (in MKS units) of a spherical dielectric particle: formula_16, where formula_17 is the particle radius, formula_18 is the index of refraction of the particle and formula_19 is the relative refractive index between the particle and the medium. The square of the magnitude of the electric field is equal to the intensity of the beam as a function of position. Therefore, the result indicates that the force on the dielectric particle, when treated as a point dipole, is proportional to the gradient along the intensity of the beam. In other words, the gradient force described here tends to attract the particle to the region of highest intensity. In reality, the scattering force of the light works against the gradient force in the axial direction of the trap, resulting in an equilibrium position that is displaced slightly downstream of the intensity maximum. Under the Rayleigh approximation, we can also write the scattering force as
formula_20
Since the scattering is isotropic, the net momentum is transferred in the forward direction. On the quantum level, we picture the gradient force as forward Rayleigh scattering in which identical photons are created and annihilated concurrently, while in the scattering (radiation) force the incident photons travel in the same direction and ‘scatter’ isotropically. By conservation of momentum, the particle must accumulate the photons' original momenta, causing a forward force in the latter.
Harmonic potential approximation.
A useful way to study the interaction of an atom in a Gaussian beam is to look at the harmonic potential approximation of the intensity profile the atom experiences. In the case of the two-level atom, the potential experienced is related to its AC Stark Shift,
formula_21
where formula_22 is the natural line width of the excited state, formula_23 is the electric dipole coupling, formula_24 is the frequency of the transition, and formula_25 is the detuning or difference between the laser frequency and the transition frequency.
The intensity of a gaussian beam profile is characterized by the wavelength formula_26, minimum waist formula_27, and power of the beam formula_28. The following formulas define the beam profile:
formula_29
formula_30
formula_31
formula_32
To approximate this Gaussian potential in both the radial and axial