Fibre-optical analogue of the event horizon

Confinement to the Black Hole ... to be reserved for cases of Drunkeness, Riot, Violence, or Insolence to Superiors.

- British Army regulation (1844)

Simulated view of a black hole in front of the Milky Way.


Creating optical analogues of the event horizon has been an exciting adventure with many ups and downs, high hopes and deep disappointments, an adventure that seems destined to continue. So far, most of it is still theory, but we succeeded in the first small step of demonstrating in the laboratory the physics of horizons for light [1]. The theory [1] has been an unfinished intellectual adventure as well - quantum horizons have not lost their mystery nor lure. Working on artificial black holes seems seriously addictive. Maybe we are beyond the horizon, the point of no return - half in jest, half in earnest, of course.

In our experiment, we use ultrashort light pulses in microstructured optical fibers to demonstrate the formation of an artificial event horizon in optics. We create analogues of the horizon, not real black holes, they only act on light in the fibre, and our experiment is completely harmless. We observed a classical optical effect, the blue-shifting of light at a white-hole horizon. We also show by theoretical calculations that such a system is capable of probing the quantum physics of horizons, in particular Hawking radiation.

High-resolution images of our experiment and our team (Credit: Chris Kuklewicz)

Children's cartoons (Credit: Maria Leonhardt).

Video of a lecture on artificial black holes recorded about a week after U.L. got the idea for this project (Reisensburg, September 3, 2004).


Our experiment. The yellow tape marks our territory on the optical table of the Ultrafast Photonics Facility at St Andrews. The blue circuit is a spool that contains 1.5 meters of microstructured optical fibre. In this fibre we create 80 million analogues of the event horizon per second. The lines indicate the light paths (artificial lines not seen in reality). Pulses coming in from a master laser follow the red path; infrared probe light is produced in the small laser wrapped in tin foil and follows the green path. Both are coupled into the fibre where the pulses establish horizons for the probe light. The output probe, shown in magenta, is fed into the yellow-clad fibre that leads to the spectrum analyzer shown below. The spectrum reveals the blue-shifting of light at a white-hole horizon.


Most laboratory analogues of black holes [1-5] are inspired by a simple and intuitive idea of Bill Unruh [6] (and, independently, of Grigori Volovik [3], Matt Visser and U.L.): the space-time geometry of a black hole resembles a river [4], a moving medium flowing towards a waterfall, the singularity. Imagine that the river carries waves propagating against the current with speed c'. The waves play the role of light where c' represents c, the speed of light in vacuum. Suppose that the closer the river gets to the waterfall the faster it flows and that at some point the speed of the river exceeds c'. Clearly, beyond this point waves can no longer propagate upstream. The point of no return corresponds to the horizon of the black hole.

The "Point of no return" is the analogue of the event horizon for fish in a river. To the left of it, the water flows faster than a fish can swim. So, if a fish happens to drift beyond this line, it can never get back upstream; it is doomed to be crushed in Singularity Falls. Credit: SonntagsZeitung.

Imagine another situation: a fast river flowing out into the sea, getting slower. Waves cannot enter the river beyond the point where the flow speed exceeds the wave velocity; the river resembles an object that nothing can enter, a white hole. These analogies are more than analogies: the propagation of waves in moving media is mathematically equivalent to wave propagation in space-time geometries. Isn't it astonishing that rivers resemble objects as remote from our everyday experience as the horizon of the black hole?

You can make analogues of the event horizon in the kitchen. Just let water from the tap flow onto a flat surface, as shown in the picture by Piotr Pieranski below. You see a ring of water waves, but the interior of this ring is smooth. Here the water moves faster than the speed of the waves; no waves can enter this region. The water flows outwards and gets slower. Rings of waves form at the circle where the water slows down to the speed of the waves. This circle represents a white-hole horizon (more commonly known as the hydraulic jump).

White holes in the kitchen - hydraulic jump.

So if you can easily make your own horizons, why are they interesting in science? They can demonstrate a possible solution to a puzzle of the classical wave dynamics at the event horizon and an even more fascinating piece of quantum physics. At the event horizon time stands still. Light would freeze. The wavelength of light would get infinitely short. But this is impossible! Something must stop the infinite freezing of light, something as yet unknown. Does space-time cease to be continuous for distances shorter than the Planck scale? Nobody knows yet. In laboratory analogues, however, the equivalent of trans-Planckian physics usually is known, at least in principle. The wave velocity changes for changing wavelength in a known way that depends on the medium. For example, the water waves of the kitchen white hole - capillary waves - get faster for shorter wavelengths. Short waves penetrate the horizon, as the picture also shows. Something similar ought to happen at the horizon of the astrophysical black hole.

What is the quantum physics at the horizon? According to quantum field theory, the vacuum (the quantum state of absolute nothing) is not empty, but teeming with possibilities - pairs of particles and anti-particles are appearing and disappearing all the time. If the partners of such a virtual pair are created on opposite sides of the horizon they are separated after birth; they can no longer annihilate each other and are forced to materialize. The black hole is not black, but emits particles, and so does the white-hole horizon.

Pair creation at the horizon of a black hole.

Stephen Hawking predicted this behavior of astrophysical black holes, known as Hawking radiation, in 1974 [7] and it has inspired many scientists ever since, but nobody has observed it yet. Hawking radiation is characterized by a temperature that depends on the size of the black hole (the radius of the event horizon that is proportional to its mass). The smaller the black hole, the hotter it is, for the following reason: smaller black holes have stronger tidal forces than larger ones, and it is the tidal forces that separate the unfortunate Hawking partners; the stronger they are the more particles materialize. Solar-mass black holes turn out to be about eight orders of magnitudes colder than the Cosmic Microwave Background, supermassive black holes like the one in the centre of our galaxy are even colder; so there probably is no chance of observing Hawking radiation in astrophysics.

Hawking's formula for the temperature of black holes gives a tantalizing glimpse into a mysterious connection between research areas that are not connected yet. You already see this from the natural constants that this formula contains - Planck's constant represent quantum mechanics, Newton's G and the speed of light c stand for gravity and Einstein's general relativity, and Boltzmann's constant associates the horizon with thermodynamics and statistical physics. Quantum mechanics usually is the physics of the incredibly small, general relativity the physics of the extremely large, and thermodynamics and statistics govern much of our daily lives. If we understand Hawking radiation better, we may better understand how these three worlds are connected.

Astrophysical black holes are too large for emitting noticeable Hawking radiation, but in the laboratory we can make small things. However, for the wave-horizons of fluids, like the kitchen white-holes, the Hawking temperatures are still very low. One would need superfluids such as liquid Helium-4, superfluid Helium-3 or Bose-Einstein condensates that do not solidify at extremely low temperatures. But even then the observation of Hawking radiation is not easy.

Grigori Volovik's book [3] shows many connections between the quantum physics of superfluid Helium-3 and the standard model of cosmology and particle physics. Volovik, one of the leading experts on superfluids and a pioneer of laboratory analogues of cosmology, calls this - perhaps more in earnest than in jest - the "Helium-centric world view": if something in our physical model for the universe cannot be reproduced in Helium-3, it is probably wrong ... .

We decided to use light for demonstrating the physics of the event horizon [1]. Optics offers unrivalled advantages: light is the purest and simplest quantum object imaginable. Our fibre-optical horizons are made of pure glass, light and air. The achievable Hawking temperatures are high, above 1000 Kelvin [1] (the temperature of fire), and the generated light quanta, photons, are directly detectable with high efficiency. One can see single photons with the naked eye as feeble flashes of light in complete darkness. Imagine what modern instruments can detect! The sophisticated tools of quantum optics have enabled researchers to perform tests and demonstrations of the fundamentals of quantum mechanics with unprecedented clarity and precision.

But making horizons for light sounds incredibly difficult. Remember Unruh's river. Imagine you replace the waves of the river by light. For establishing a horizon, the river needs to flow faster than the speed of light in the medium (say in water). This is possible in principle, because optical media such as water or glass typically reduce the speed of light in vacuum, c, by the refractive index n but it seems very difficult in practice; light is still fast in normal materials.

Fibre-optical analogue of the event horizon. A light pulse in an optical fibre slows down probe light attempting to overtake it.

However, it turns out [1] that moving an optical medium at the speed of light is easy. In fact, it happens all the time in optical telecommunication. In fibre-optic communication, the information carriers are light pulses confined to the core of optical fibres. Each pulse adds a slight contribution to the refractive index of the fibre [8] (a contribution proportional to the intensity profile of the pulse), as if the pulse were adding an extra piece of glass. This is called the Kerr effect (see Agrawal's book [8] on Nonlinear Fibre Optics). The contribution to the refractive index moves with the pulse. The pulse thus establishes a moving medium [1], although nothing material is moving, only light. This medium naturally moves at the speed of light, because it is made by light itself. In materials, the velocity of light depends on the colour (the frequency) and sometimes on the polarization as well; different colours move at different speeds. This works to our advantage, because we can launch probe light that follows the pulse at a faster velocity, but is slowed down due to the additional contribution to the refractive index, experiencing a horizon.

But there is an important subtlety: are we talking about the group velocity or the phase velocity of light? The group velocity is the speed at which a wave packet moves, whereas the phase velocity is the velocity at which the oscillations of light propagate. For example, for a light pulse in the fibre, the intensity profile moves with the group velocity, while oscillations of light scintillate across it at the phase velocity. Which speed does the pulse need to match for establishing a horizon, the group or the phase velocity? It turns out [1] that the classical freezing of light happens when the group velocity of probe light matches the speed of the pulse. In our experiment [1] we measured the wavelength shifting caused by such a group-velocity horizon for infrared light of about 1500 nm wavelength. Hawking radiation should be created when the pulse matches the phase velocity of light [1], which happens in the near ultraviolet [1].

Schematic figures that summarize our experimental programme. (a) Classical horizons. The probe is slowed down by the pulse until its group velocity matches the pulse speed at the points indicated in the figure, establishing a white-hole horizon at the back and a black-hole horizon at the front of the pulse. The probe light is blue-shifted at the white hole until the optical dispersion releases it from the horizon. (b) Quantum pairs. Even if no probe light is incident, the horizon emits photon pairs corresponding to waves of positive frequencies from the outside of the horizon paired with waves at negative frequencies from beyond the horizon. An optical shock has steepened the pulse edge, increasing the luminosity of the white hole.

Each pulse establishes two artificial event horizons for ultraviolet light, a black-hole horizon at the front where the pulse enhances the refractive index, reducing the phase velocity below the pulse speed, and a white-hole horizon at the trailing end of the pulse. These horizons would also exist and have a physical effect if no ultraviolet light is present, which is normally the case. They should still spontaneously emit Hawking radiation consisting of ultraviolet light quanta. So, whenever people communicate via fibre optics, using the internet or making long-distance phone calls, they create numerous artificial event horizons as a side effect without noticing it. However, the Hawking temperature of such telecommunication horizons is astronomically low and therefore such effects have remained unnoticed and are completely negligible in practice.

The few-cycle laser (Rainbow from Femtolasers) in our new dedicated laboratory that we plan to use for demonstrating Hawking radiation with fibre optics. This "Ferrari laser" fires micrometer-size bullets of red light with a peak intensity of half a megawatt.

In order to achieve substantial Hawking temperatures with fibre optics we need extremely short, intense light pulses where the intensity changes more rapidly than the wavelength. It takes both the latest advances in the generation of few-cycle optical pulses [9] and some of the most advanced optical fibres, photonic-crystal fibres [10] also called microstructured fibres, in order to enhance an effect that in-principle exists in ordinary fibre-optic communication to the level where it becomes observable in the laboratory.

Cross-sections of various types of photonic-crystal fibres. Reproduced from the review article [10].

The optical Kerr effect is at the heart of ultrashort-pulse Ti:sapphire lasers. Here the Kerr effect acts not only in the external optics, but in the lasing material itself (Titanium Sapphire). The laser uses a mechanism called Kerr-lens modelocking that was discovered by Wilson Sibbett in St Andrews. Pulses as short as a few optical cycles are made in Kerr-lens-modelocked lasers with sophisticated high-tech mirrors that enable the laser to operate over the entire spectral range of the lasing material. Such lasers may also synchronize the oscillations of light with the pulse intensity profile (carrier-envelope stabilization) using the frequency-comb technique invented by Nobel laureate Ted Hansch. These "Ferrari lasers" create extremely short, intense and precisely controlled pulses of light.

Photonic-crystal fibres were invented by Philip Russell. They represent the latest in the series of ingenious inventions that turned fibre-optic communication from a Victorian amusement to the dominant communication technology of the early 21st century [11]. Fibre glass - ultrapure fused silica - is the most transparent material ever made; it is more transparent than air. A pane of window glass looks dark green from the side, but fibres transmit light through dozens of kilometers of solid glass. If the oceans were filled with fibre glass it would be bright daylight at the bottom of the Mariana trench. Photonic-crystal fibres contain structures of microscopic longitudinal air holes (in fibres as thin as human hair). The arrangement of air holes gives the fibre tailor-made optical properties. Photonic-crystal fibres are examples of metamaterials, because in a metamaterial man-made structures smaller than the wavelength of light govern the optical properties of the material.

We hope to exploit the fascinating properties of these advanced fibres to, literally, shed light on various aspects of quantum horizons. For example, one could detect the photons created on either side of the horizon and see how they are correlated, a feat that is utterly impossible in astrophysics, because there the partner particles are lost beyond the horizon of the black hole. Or one could design the optical properties of the fibre such the black-hole and white-hole horizons act like the mirrors and the amplifying medium of a laser - a black-hole laser [12].

Another aspect of the Kerr effect should work to our advantage: optical wave breaking - the formation of optical shocks. The Kerr effect [8] influences the pulse as well, shaping the pulse while it propagates in the fibre. Regions of high intensity lag behind, because for them the refractive index is increased. The black-hole horizon at the front is stretched, but the trailing edge becomes extremely steep: the pulse develops an optical shock [8]. The pulse breaks like a wave at the ocean shore. The steep white-hole horizon will dominate the Hawking effect of the pulse. Our idea [1] thus solves two problems at once in a natural way: how to let an effective medium move at superluminal speed and how to generate a steep intensity profile at the horizon; for the first time the various aspects of the physics of artificial black holes conspire together.

Tahitian wave breaking.

This scientific adventure has been financed in unusual ways. The "start-up capital" for our experiment came from a private donation by Leonhardt Group AG, the corporation of Ulf Leonhardt's cousins Uwe and Helge, businessmen from former East-Germany. The theoretical groundwork was financed by a research project of the Leverhulme Trust, a charity of Unilever that supports innovative research in the sciences and arts, and that also financed Ulf Leonhardt's work on invisibility. The personnel costs for the initial experimental work were supported by a generous loan from the European project COntinuous VAriable Quantum Information with Atoms and Light. After the foundations had been laid, the Engineering and Physical Sciences Research Council UK financed most of the equipment and the personnel needed at this stage of our project.

References

  1. T.G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. Konig, and U. Leonhardt, Fiber-Optical Analog of the Event Horizon, Science 319, 1367-1370 (2008); Fiber-optical analogue of the event horizon: Appendices, Science (supporting online material).
  2. Artificial Black Holes edited by M. Novello, M. Visser, and G.E. Volovik (World Scientific, Singapore, 2002).
  3. G.E. Volovik, The Universe in a Helium Droplet (Clarendon Press, Oxford, 2003).
  4. Quantum Analogues: From Phase Transitions to Black Holes and Cosmology edited by W.G. Unruh and R. Schutzhold.
  5. G. Rousseaux, C. Mathis, P. Maissa, T.G. Philbin, and U. Leonhardt, Observation of negative-frequency waves in a water tank: A classical analogue to the Hawking effect? preprint arxiv:/0711.4767.
  6. W.G. Unruh, Experimental Black-Hole Evaporation? Physical Review Letters 21, 1351 (1981).
  7. S.W. Hawking, Black-hole explosions? Nature 248, 30 (1974).
  8. G. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2001).
  9. Few-Cycle Laser Pulse Generation and Its Applications edited by F.X. Kartner (Springer, Berlin, 2004).
  10. P. Russell, Photonic Crystal Fibers, Science 299, 358 (2003).
  11. J. Hecht, City of Light: The Story of Fiber Optics (Oxford University Press, Oxford, 1999).
  12. S. Corley and T. Jacobson, Black hole lasers, Physical Review D 59, 124011 (1999); U. Leonhardt and T.G. Philbin, Black-hole lasers revisited in Ref. [4]; preprint arxiv:/0803.0669.