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Torqued Light

August 19, 2019

Like many other things that we are given in nature, mankind has used light to its advantage for many years before understanding anything about it. Its importance is underscored by the divinity bestowed upon the Sun by ancient people. As written in the Bible (Genesis, Book 1, verse 3), one of the first things that God created was light, and He "saw that the light was good."[1] The name of the chemical element, helium, derives from Helios, the Greek personification of the Sun, since it was first detected as an unknown spectral line at 587.49 nanometers in the Sun's chromosphere during the solar eclipse of 1868. Portion of the Genesis Vulgate

Two years of high school Latin are sufficient to translate the first few verses from the Vulgate, the Latin version of the Bible that was assembled in the 4th century from available Greek texts. "In the beginning God created heaven, and earth. And the earth was void and empty, and darkness was upon the face of the deep; and the spirit of God moved over the waters. And God said: Be light made. And light was made. And God saw the light that it was good; and he divided the light from the darkness. And he called the light Day, and the darkness Night; and there was evening and morning one day."[2] (Click for larger image.)

Slowly, scientists discovered the properties of light, the most interesting of which is the fact that light has the finite speed of 299,792,458 meters per second. This was discovered by Danish astronomer, Ole Rømer in observations of the orbit of Jupiter's moon Io in 1676. Rømer found that the times between eclipses of Io became shorter as the distance between Earth and Jupiter decreased, and longer as it increased. His observations give a speed of light that's nearly identical to today's value. Why light has such a particular speed is a mystery, but its finite speed makes life easier for some physicists, and harder for others.

Much slower than light speed was the progress in our understanding of light, since it took an additional two centuries for James Clerk Maxwell's 1865 theory that light is an electromagnetic wave. Einstein's 1905 theory of relativity underscored the unusual nature of light in the idea that the speed of light is the same in all inertial frames and is not dependent on the motion of the light source. That's why we observe the light analog of the Doppler effect in the redshifting of light from cosmic sources rapidly receding from us as a consequence of universal expansion.

Maxwell's electromagnetic equations

Maxwell's electromagnetic equations. These equations are a popular T-shirt image, at least among the geek elite. The equation at the upper left-hand corner is important, since it shows that there is no such object as a magnetic monopole. Lesser known are Maxwell's thermodynamic equations, usually called the Maxwell relations to distinguish them from his electromagnetic equations. I discussed these thermodynamic equations in an earlier article (Maxwell's Other Equations, April 6, 2011). (Created using Inkscape. Click for larger image.)

Max Planck's analysis of black body radiation in 1900 gave us Planck's law that demonstrated that energy was quantized. This was quickly followed by an explanation of the photoelectric effect by Albert Einstein Einstein showed that light energy is carried in discrete quantized units, now called photons, so that light has both particle and wave nature. It's interesting to note that Einstein's 1921 Nobel Prize in Physics was "for his services to theoretical physics, and especially for his discovery of the law of the photoelectric effect," worded that way since relativity was still controversial.

According to classical physics, photons should not have momentum, since they are massless and the momentum is the mass multiplied by velocity. In 1871, Maxwell calculated that electromagnetic radiation will exert a force on objects, a theory that was experimentally confirmed by Pyotr Lebedev in 1899. The photon momentum p in a vacuum is given as p = hk/2π, where h is Planck's Constant, and k is the wave vector. The wave vector has a value of 2π/λ, where λ is the wavelength, so shorter wavelengths have higher momentum.

Crookes radiometer

William Crookes invented a device called the Crookes radiometer in 1873 after observing that weight measurements in a partial vacuum changed slightly with illumination.

The paddle wheel on his radiometer moves when exposed to light, with the dark vanes retreating from a light source. Crookes thought that it was the pressure of light that caused the vanes to move, thereby proving that light had momentum.

It was subsequently discovered that the paddle wheel didn't move in a better vacuum, and a certain air pressure was needed to produce an optimal motion.

Careful thought would have indicated that the paddle wheel motion was in the opposite sense to what should be expected, since photons bouncing from the reflective surfaces would exert more force than those absorbed by the dark surfaces.

The motion is a thermal effect involving the motion of air molecules.[3]

(Wikimedia Commons image)

The radiation pressure that the Sun exerts on an object in a vacuum near Earth's orbit is quite feeble, just 9.2 micropascals on a perfectly reflecting mirror, but its affect will accumulate over time to cause an acceleration. Radiation pressure could be an enabler for interplanetary and interstellar travel by solar sailing; or, in a recently proposed version, the idea of using an Earth-based laser to propel small interstellar probes.[4]

As I wrote in an earlier article (The Yarkovsky Effect, June 12, 2012), radiation pressure can be used to steer asteroids away from the Earth. The Yarkovsky effect is the photon momentum effect when photons are emitted by an object, rather than impinging upon it. A temperature difference of one side of an object versus the other causes an imbalance in its thermal radiation, and this results in a small force. The anisotropic emission of thermal photons in the Yarkovsky effect has been proposed as a method of steering asteroids away from collision with the Earth.

While the momentum of light in a vacuum is well established, there's the further question of the momentum of light in a dielectric. Since the speed of light is reduced in a dielectric by a factor equal to the refractive index, our classical mechanical sensibilities would conclude that the momentum of light is reduced by the same factor; that is, p = hk/2πn, where n is the refractive index. This is the value calculated by Max Abraham in 1909.[5] However, Hermann Minkowski calculated a momentum value p = nhk/2π in 1908.[6]

These two theoretical predictions differ by the square of the dielectric constant; so, you would think that the matter would have been experimentally resolved after more than a century. We still don't know, and it appears that the values obtained might even depend on the type of experiment used to make the measurement.[7-9] That's because the calculations done by Minkowski and Abraham assume an idealized case that combines the real and imaginary components of the dielectric constant, while experiments partition the transfer of photon momentum differently for the real and imaginary parts of the refractive index.

Light continues to offer surprises. A 1992 study, published in Physical Review A by physicists from Leiden University (Leiden, The Netherlands), showed that light beams can carry an orbital angular momentum (OAM).[10] These OAM beams carry a rotated spatial shape, and they have been named vortex beams. Such OAM beams have the potential to enable new methods of optical communications. All these vortex beams have been static; that is, the OAM does not vary in time.

Now, physicists from the University of Salamanca (Salamanca, Spain), the University of Colorado (Boulder, Colorado), the National Institute of Standards and Technology (NIST, Boulder, Colorado), the Barcelona Institute of Science and Technology (Barcelona, Spain), ICREA (Barcelona, Spain), and the Kapteyn-Murnane Laboratories Inc. (Boulder, Colorado) have observed a time-varying OAM on a light pulse, a property they call self-torque of light.[11-13]

Vortex beams present a donut-like shape when imaged on a surface, and torqued light beams show a croissant shape.[13] While the self-torque phenomenon can occur when electromagnetic fields interact with matter, there has been no optical analog until now.[11-12] The self-torque of light is an inherent property of the light, itself.[12] The research team demonstrated self-torque using beams in the extreme-ultraviolet that were driven by time-delayed pulses of different OAM.[12]

Spatial profile of a torqued light beam

Spatial profile of a torqued light beam.

While OAM light beams project donuts onto a surface, torqued light beams produce croissants.

This is an experimental image of harmonics 13-23 in the experiment, selected by an aluminum filter.

(arXiv image from ref. 12.)[12]

The torqued light beams were generated by the interference of two vortex beams of different orbital angular momentum.[11] A time delay between the OAM pulses produced a time-dependent angular momentum, the self-torque.[11] The self-torqued beams are created by high harmonic generation in which an ultrafast laser pulse is coherently upconverted into extreme-ultraviolet and X-ray spectra.[11] The self torque is a function of the OAM amplitude of the generating beams, their relative time delay, and the harmonic order.[11]

Says Laura Rego of the University of Salamanca, first author of the paper, "This is the first time that anyone has predicted or even observed this new property of light... For example, we think we can modulate the orbital angular momentum of light in the same way frequency is modulated in communications."[13] Torqued light could be used for manipulation of nanostructures and atoms on attosecond time scales and nanometer spatial scales.[11]


  1. The Book of Genesis, Vatican Website.
  2. The Latin Vulgate Bible With Douay-Rheims English Translation, Vulgate.org.
  3. Explanation of the motion of a Crookes radiometer, Wikipedia.
  4. Philip Lubin, "A Roadmap to Interstellar Flight," arXiv, October 31, 2016
  5. M. Abraham, Rend. Circ. Matem. Palermo vol. 28 (1909), p. 1.
  6. H. Minkowski, Nachr. Ges. Wiss. Göttn. Math.-Phys. Kl.(1908), p. 53.
  7. Mark Buchanan, "Minkowski, Abraham and the photon momentum," Nature Physics, vol. 3, no. 2 (February, 2007), p. 73.
  8. Berian James, Letter to Nature (Friday, Jan 5 2007).
  9. Bradley W. Carroll, Farhang Amiri, and J. Ronald Galli, "An Effective Photon Momentum in a Dielectric Medium: A Relativistic Approach," arXiv, January 15, 2018.
  10. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A, vol. 45, no. 11 (June 1992), pp. 8185-8189 (1992), DOI:https://doi.org/10.1103/PhysRevA.45.8185.
  11. Laura Rego, Kevin M. Dorney, Nathan J. Brooks, Quynh L. Nguyen, Chen-Ting Liao, Julio San Román, David E. Couch, Allison Liu, Emilio Pisanty, Maciej Lewenstein, Luis Plaja, Henry C. Kapteyn, Margaret M. Murnane, and Carlos Hernández-García, "Generation of extreme-ultraviolet beams with time-varying orbital angular momentum," Science, vol. 364, no. 6447 (June 28 2019), Article eaaw9486, DOI: 10.1126/science.aaw9486.
  12. Laura Rego, Kevin M. Dorney, Nathan J. Brooks, Quynh L. Nguyen, Chen-Ting Liao, Julio San Román, David E. Couch, Allison Liu, Emilio Pisanty, Maciej Lewenstein, Luis Plaja, Henry C. Kapteyn, Margaret M. Murnane, and Carlos Hernández-García, "Light with a self-torque: extreme-ultraviolet beams with time-varying orbital angular momentum," arXiv, January 30, 2019.
  13. Discovered: A new property of light, American Association for the Advancement of Science Press Release, June 27, 2019.

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