Micropheres, Photonic Atoms, and the Physics of Nothing

Stephen Arnold. American Scientist. Volume 89, Issue 5. Sep/Oct 2001.

The transfer of energy between neighboring molecules plans a pivotal role in nature. In photosynthesis, for example, a plant fuels its metabolism and growth with sunlight by taking advantage of a curious physical phenomenon that allows energy to hop from one chlorophyll molecule to another situated about a half a nanometer away. A couple of hundred chlorophyll molecules pass the energy they collect from the sun in this way to a single reaction center, the starting point for subsequent chemical reactions. Without this mechanism for transferring energy between molecules, photosynthesis would largely cease, and we would likely starve.

About 15 years ago I began to wonder whether similar forms of energy transfer could influence photochemistry within aerosol particles. In particular, I wanted to know whether there are subtleties in the way energy is conveyed between molecules in an isolated droplet about 10 micrometers in diameter. To most physicists, the idea must have seemed crazy. After all, the range of the longest substantial exchange of this sort, as Nobelist Jean Perrin had discovered and Theodore Forster had described in quantum-mechanical terms decades ago, is only about 5 nanometers. The vessels I was proposing to use would be 2,000 times larger. So there was no obvious reason to expect that their tiny size would have any influence at all. Still, research elsewhere with similar microscopic particles hinted of interesting physical effects, and I urged one of my graduate students, Lorcan Folan, to investigate. Little did I know that the results we and others were soon to obtain would distinguish the lowly aerosol particle as a high-tech item.

Such microscopic particles now stand poised to serve in a variety of ways, from lasers of exceptional efficiency to optical filters of unprecedented purity and chemical probes of tiny size-to name just a few obvious applications. But before delving into how tiny spheres can provide such valuable functions, it is worthwhile to review how this rapidly evolving creature of 21st-century technology first arose from a primordial soup of basic research.

Peering at Particles

Probing something as subtle as the transfer of energy between molecules may seem daunting, but in truth the procedure is straightforward. You first energize one type of molecule, known as a donor, by illuminating it with light from a properly tuned laser, which kicks ground-state electrons to a higher energy level. Then you look for a transfer of this energy to another type of molecule, known as an acceptor, by sensing the characteristic color of light it gives off when its excited electrons fall back to a lower-energy state (a familiar enough process called fluorescence). If no energy passes between donors and acceptors, only the donor molecules will fluoresce, giving off their own particular color. So the ratio of acceptor to donor fluorescence provides a convenient way to gauge the amount of energy transferred.

To perform the measurement on a microscopic droplet, one simply mixes in the appropriate donors and acceptors and observes the spectrum of laser-induced fluorescence. The not-so-minor complication is that it is difficult to hold a 10-micrometer sphere of liquid in place long enough to study it. Lorcan and I solved this problem by constructing an apparatus in which he could levitate and contain an electrically charged particle indefinitely using electrostatic force to balance gravity, just as Robert Millikan had done decades before in his famous oil-drop experiment. But Millikan’s scheme alone does not provide a “trap”-the particle can wander freely. To prevent such drift, Lorcan’s apparatus superimposed an oscillating electric field on the constant levitation field, following the example of Wolfgang Paul, who had shown with his Nobel Prize-winning work of the 1950s that a dynamic field can be used to trap an atomic ion. With a properly fashioned oscillating field, a microscopic droplet just hangs there in air, an easy target to hit with a laser and to view through a microscope.

The first spectrum Lorcan obtained from a levitated droplet sent a wave of surprise through our laboratory. He had planned a series of experiments at varying concentrations, starting on the dilute side where the average distance between donors and acceptors was 20 times the maximum range for Forster exchange. So we did not expect any energy transfer to be apparent. Yet the ratio of acceptor to donor fluorescence proved to be more than 10 percent.

As Lorcan increased the concentration of acceptor molecules in a series of droplets under study, we became increasingly puzzled. Had the wellknown Forster mechanism operated, the amount of energy transferred in such dilute solutions should have been proportional to the concentration of acceptors. But these experiments revealed a range of concentrations nearly two orders of magnitude wide for which we saw little change in the amount of energy conveyed between molecules. What is more, in the light spectra for both donors and acceptors, we observed spikes that were as obvious as fence pickets. Such distinctive features had never appeared in experiments that probed these very same molecules in a centimeterscale test tube. Although the explanation was not immediately obvious, these two pieces of evidence were ultimately to revamp our view of how energy was being passed between molecules.

At the outset, we were planning to probe the subtleties of Forster transfer, whereby the energy in an excited electron shifts to another molecule without ever generating a photon. How, one might reasonably ask, does that happen? Such transfer takes place because an excited molecule behaves something like a transmitting radio antenna. Close to this nanoscopic source, the oscillating electric field is especially intense (although it drops off extremely rapidly, with the cube of the distance). This field can, in fact, be sufficiently strong to induce oscillation in the electron cloud of a nearby molecule, and this coupling conveys energy if the acceptor sits close enough that the probability of transfer overwhelms the natural probability for the donor to fluoresce. Photons are not involved in such an exchange; that is to say, the donors do not have to give off electromagnetic radiation. The Forster process is thus akin to what happens to people who mysteriously hear radio programs because the electric field of a nearby radio transmitter is so powerful that it induces currents to flow in their fillings.

Of course, radio transmitters are not built to energize the mouths of people standing nearby: They depend on their antennas broadcasting electromagnetic waves to far-flung receivers. So too with excited molecules. If the donor has no near-field neighbors to capture this energy, the excited molecule emits a photon into the far field (at least one wavelength away). An acceptor looking back at the donor from this distance normally finds itself so far away that the probability of being struck by the photon and absorbing its energy is astronomically small. Yet this was the scale of separation between donors and acceptors in Lorcan’s first experiment.

What was going on? Given the low concentration of donor and acceptor molecules within the droplet, we knew that Forster transfer was not operating. Photons must have been leaving the donors and hitting the acceptors, but they were doing so with an unexpected efficiency The only reasonable explanation was that each of the emitted photons was returning many times to the same region, so that it had many chances to collide with an acceptor. The spikes we saw in the spectra gave us a good clue to the mechanism.

These spikes are resonance peaks, which correspond to special electromagnetic modes for the entire particle. The situation is akin to a violin string, which supports vibrational modes only at those frequencies that provide for an integral number of half wavelengths along its length. The electromagnetic modes of small particles– commonly known as Mie resonances or whispering-gallery modes-are analogous and are normally described by their wave character, too, in a way that takes full account of such complications as polarization and diffraction. However there is a more visceral description of these modes that clearly underscores the ability of a photon to revisit a region many times. H. M. Nussenzveig of the Universidade Federal do Rio de Janeiro realized that if the sphere is much larger than the wavelength of light involved, whispering-gallery modes can be represented as geometrical orbits. With diffraction stripped away, photons can be thought of as bouncing around inside the particle in well-defined trajectories, confined by total internal reflection-the same phenomenon that makes the surface of a swimming pool look like a mirror when you peer up from under water and look to the side.

The swimming-pool effect arises because rays hitting the surface from underneath at a shallow angle reflect completely back into the water. To a large extent this is also true inside a transparent particle, so long as it is much larger than the wavelength. When a ray of light strikes the spherical surface at a shallow angle, it just bounces back inside. A ricocheting photon thus remains within the particle for much longer than it would otherwise. Ultimately the photon lifetime is limited by diffraction, which causes photon trajectories to be less certain, allowing the energy eventually to “leak” out. The effects of diffraction grow as the particle shrinks, and when the radius approaches the wavelength of light passing through the interior, resonances are lost altogether.

With these considerations in mind, we began to make sense of the observations and started to appreciate the importance of the special electromagnetic modes of a tiny droplet. We called them photonic-atom modes, because the trajectories of the photons resemble the orbits of electrons in atoms. It turns out that the longest-lived modes have photons traveling along simple polygons, whereas for the shorter-lived modes the paths close after more than one revolution around the interior. Using our data and the photonic-atom picture, we estimated that before being absorbed or leaking out, the photons circulating within an acceptor-free droplet of 10-micrometer radius cover a total distance of about 0.6 meter 30,000 times the width of the sphere!

Just as with an electron in a Bohr atom, the photon in a microsphere, or “photonic atom,” has a quantized amount of orbital angular momentum. Photons have momentum? Indeed they do. Einstein had shown long ago that a photon has momentum proportional to its energy, which in turn is proportional to the optical frequency. The orbital angular momentum of a photon in a photonic atom is just the momentum Einstein had shown times the inner radius of the polygon (which for grazing photons is nearly the same as the particle radius). So having modes separated by a constant increment of angular momentum is equivalent to their being regularly spaced in frequency-the spectral fence pickets Lorcan had uncovered.

One can think of a given mode as a wave that circumnavigates the interior of the tiny sphere and returns in step with the oscillations at its starting point. The mode with the next higher value of angular momentum has a frequency increased by just the right amount to squeeze an additional wavelength into this circuit. Such properties of photonic atoms seemed neatly to explain what we were seeing in our experiments.

Steve Holler, an undergraduate student I was advising at the time, set about to shore up the hypothesis by taking pictures of a glowing microdroplet through discriminating color filters that captured the light of donor and acceptor emissions separately. His acceptor image boosted our confidence enormously, because light was concentrated close to the surface, where the photons in a photonic atom circulate. Holler’s donor image was less interesting, with almost all of the interior aglow, but it displayed a distinctive asymmetry, which arose because the spherical droplet focused much of the incident laser light onto its back side. Surprisingly, the acceptor image showed no difference between back and front. Rather, it revealed two curious bright spots, diametrically opposed.

The cloud of mystery eventually dissipated when we realized that donor molecules excited preferentially at the back of the droplet launch photons at every “azimuth.” Imagine a busy airport sending jets in every direction, with each of the pilots instructed to fly a great-circle route. These aircraft will eventually converge at the antipodal point on the other side of the globe. Unless they collide in midair, they will then return to their starting point, and so forth. Similarly, the concentration of the donor emission at the back of a microsphere sends photons on great-circle trajectories, which intersect at two antipodal points, seen as bright points in the image of acceptor fluorescence. Because our image averaged many revolutions of the photons from one side of the sphere to the other, we saw a completely symmetrical picture-just as if we had taken a time exposure of a swinging pendulum over an interval much longer than one oscillation.

The overall efficiency of energy transfer we measured for a sphere was about 10 percent. But the acceptor image showed that this transfer only takes place in a thin shell near the surface; most donor molecules do not participate. Indeed, only one out of five are involved, indicating that for these the transfer efficiency is about 50 percent. Noel Goddard, a master’s student working in my lab, was later able to obtain such an enhanced efficiency by embedding all donor molecules just under the surface of the droplet. But even with all donors in the right place, a key question remains: Why should they emit photons so efficiently into the special directions needed for photonic-atom modes? The answer requires a brief consideration of why an excited atom or molecule emits light in the first place.

Much Ado about Nothing

When I took quantum mechanics as a graduate student, I recall being told by my professor that the excited states of hydrogen are mathematically stationary. I realized that the atom should emit a photon and so asked, “Does this mean that if I put an excited hydrogen atom in my pocket, go home and come back tomorrow, it will still be excited?” The instructor confirmed that it would not, showing some discomfort about where the discussion was headed. Still, I pushed on: “How long, then?” He said it was on the order of nanoseconds, prompting me to reply, “That’s stationary?” My professor had little more to offer except to remind me that we had already done calculations on stimulated emission.

In stimulated emission, a passing photon induces an excited electron to return to the ground state. The energy of the triggering photon must match the difference between these two levels, and the emission produces a second, identical photon. Although this process makes lasers possible, it is not nearly as interesting to me as spontaneous emission, which provides almost all of the light we see. The sun, incandescent bulbs, fluorescent lamps, even fireflies shine without requiring external photons to trigger their emissions. Einstein used thermodynamic arguments to show the need for both stimulated and spontaneous emission, but he was not able to offer an explanation for the latter. Nor could the mechanism be obtained from the quantum mechanics of atoms. The answer appeared only after 1948, when physicists began to appreciate that the electromagnetic field of empty space is quantized.

As high school students, we were taught to fl-think that if we removed all the atoms, molecules and photons from a vessel, it would then contain nothing. But this notion is incorrect. Even in a cold enclosure held near absolute zero, an excited atom is bathed in electromagnetic fluctuations, which are particularly intense when the atom’s emission wavelength matches a resonant mode. They are the result of an electromagnetic Heisenberg uncertainty principle: The product of the electric and magnetic fields of a mode has a fixed minimum–both fields cannot simultaneously be zero. Even in a completely dark, empty enclosure, each electromagnetic mode will have a residual “zero-point energy,” equal to a constant times its frequency.

The time-varying fields associated with zero-point energy trigger spontaneous emission, which in this sense is quite similar to stimulated emission, although the source of stimulation is rather subtle. Such quantum fluctuations also apply pressure, as the Dutch physicist H. B. G. Casimir realized decades ago. Two neutral metallic plates, for example, attract each other with a force that increases as their separation diminishes. A simple explanation is that the limited space between the plates supports fewer electromagnetic modes than the surrounding regions, giving rise to a net inward push.

The universe is thought to have an infinite number of electromagnetic modes and a zero-point energy density that increases continuously with frequency. Within an enclosure, the spectrum of the energy density becomes bunched around discrete frequencies, each one associated with a particular mode. As the container is reduced in size, a mode of a given frequency occupies a smaller volume and consequently has a higher zero-point energy density. If the physical enclosure measures just a few wavelengths on a side, the zero-point energy densities within a mode can exceed the energy densities of free space by orders of magnitude. As a result, an excited molecule, which would otherwise radiate over a broad band of frequencies, is easily induced to emit photons into such a mode-so long as the emission band contains the frequency required for that mode.

Although the tiny droplets my students and I were studying are not evacuated cavities per se, they confine photons within modes in a small region and thus act in essentially the same way. So it is not surprising that the physics of nothingness applies to them as well. Indeed, detailed calculations confirm that enhanced quantum fluctuations within the photonic-atom modes account for the efficiency of the energy transfer we observed. Consistent with this, we find experimentally that the energy-transfer efficiency is increased even further as the particle size is reduced to a diameter of 5 micrometers. This is a result of the so-called cavity quantum electrodynamic effect.

An important application of this effect involves miniature lasers. Everyone is familiar with these semiconductor devices, which are found in laser pointers, laser printers and compact-disc players. Indeed examples are now so commonplace that many people forget what the word laser means: light amplification by stimulated emission of radiation. A fair fraction of the power fed into semiconductor lasers is wasted because there is a threshold current needed to start the lasing process. But power losses associated with this threshold can be much diminished by taking advantage of the cavity quantum electrodynamic effect. How? By making small lasers even smaller.

Laser action always begins with spontaneous emission. But in traditional devices, only a tiny fraction of the spontaneously generated photons–fewer than one in a thousand-go into a lasing mode. Thus considerable power is needed to ensure that an adequate number of these photons will be available to start the laser going. Here is where shrinking dimensions and the cavity quantum electrodynamic effect help out. Reducing the size of the laser cavity limits the number of available modes. In principle, the cavity can be made small enough that the emission spectrum of the laser material will overlap just one mode. The tiny size of the cavity then provides enhanced quantum fluctuations and rapid emission into this mode. Thus very little power is needed to start the laser working.

Recently investigators at several laboratories have used the confinement of photons in small spherical particles to create such low-threshold lasers. This elegant approach does, however, have its limits, because mode confinement is lost as the size of the sphere approaches the wavelength of the light. So it may not be possible to eliminate the threshold current completely Still, these microspheres make quite efficient lasers.

Microspheres also provide optical filters of exceptional spectral purity, because they hold the energy of a photon over a long time in comparison with the period of one oscillation in the corresponding light wave. A tiny sphere is thus something like a fine wineglass, which tapped with a spoon will sound an extended note, although each acoustic oscillation lasts just a millisecond or so.

A correlate of this property is that such a wineglass responds to external stimulation over a narrow range of frequency-so narrow that it takes an Ella Fitzgerald to sing just the right pitch and shatter the glass. Microspheres, too, respond to excitation over a slim range of frequencies.

For quantum optical systems, this basic physical notion is contained within another Heisenberg uncertainty principle. In a refined form, this relationship says that for a given mode, the product of the frequency width and the lifetime of a photon has a minimum value, Y2. We determined from our energy-transfer experiments that a photon typically bounces around the interior of a 10-micrometer microsphere for about 3 nanoseconds before it is absorbed by the material or flies out. Consequently, the corresponding minimum frequency width is around 50 megahertz. Although this may seem like a lot of bandwidth to a radio engineer, for optical communication it is really quite small. To understand just how narrow 50 megahertz is in this context, consider using such spheres to sort transmissions carried by light over an optical fiber.

How would one do something like this? The ideal way would be to use multiple microspheres of slightly different sizes, so that each one would serve to select a particular band of information. The number of separate channels available depends on distinguishing one resonance on a given sphere from the adjacent modes (those having the next higher or lower allowed value of angular momentum). For a sphere that is about 10 micrometers in radius, these modes are 3,000 gigahertz apart. So in principle one could distinguish (3 x 10^sup 12^)/(5 x 10^sup 7^) or 60,000 different channels. In practice, distortion of the sphere would probably reduce the number of channels to something closer to 10,000, which would still constitute a technical tour de force (although I suspect that if this scheme were ever employed to select from among television transmissions, there might still be nothing to watch).

In 1995, two members of my research group-Ali Serpenguzel and Giora Griffel-and I reported a means for sorting optical signals in just this way. The technique involves another quantum-mechanical principle called tunneling. When a microsphere-made, say, of glass or plastic-is placed within a wavelength of the core of an optical fiber, a photon traveling through the fiber has a fair probability of exciting a photonic-atom mode in the sphere, so long as the frequency of this photon corresponds to that of the mode. Physicists say the photon tunnels resonantly across this gap.

Photons circumnavigating the sphere can also tunnel into the fiber, if they do not just leak out first. Although such leakage might seem a bad thing, in fact it is very convenient, because it provides information about what is going on inside the microsphere. Indeed, we first investigated the phenomenon of tunneling by observing light leaking out of a polystyrene microsphere. We had placed the tiny plastic sphere on an optical fiber that was polished so as to eliminate all but the slimmest sliver of cladding. We then shined a tunable laser into the fiber and varied the frequency of light while viewing the microsphere at right angles. At most frequencies the scene was completely dark, except for some faint glints of light scattering from the polished surface of the fiber, but when the frequency of the laser matched a photonic-atom mode, the sphere lit up, thanks to leaking photons.

Within a month of our publishing an article about this work, physicists at the Ecole normale superieure in Paris saw a different sort of evidence for the tunneling of photons between an optical fiber and a microsphere. They measured a diminution in the light transmitted through the fiber when the frequency was set equal to a photonicatom mode. Such dips in transmission arise in part because leakage and absorption in the sphere reduce the probability of the photon returning to the fiber. But this simple picture of photons jumping between sphere and fiber neglects the wave character of light and thus misses an important physical effect: the destructive interference that can take place when light reenters the fiber after having made a relatively long visit to the sphere. Griffel and I first described this phenomenon in 1996. More recently, another group of investigators led by Kerry Vahala at the California Institute of Technology obtained nearly 100 percent efficiency for the coupling of light from an optical fiber to a microsphere. With their experimental arrangement, there is hardly any transmission at all through the fiber at resonance.

Many Points of Light

Our experiments with tiny plastic spheres and optical fibers helped launch a field that is best described by the phrase “microsphere photonics.” The centers of work in this area are now at Caltech and MIT, and many applications have emerged. One of the most intriguing came from Vahala’s laboratory at Caltech, where workers induced photons of a specific frequency to tunnel resonantly into a sphere and then into a second fiber. This photonic device constitutes what is called an add-drop filter, because it can be used either to add or to remove the signal carried on a given optical channel. With it, one can route information between optical fibers at select frequencies, without having to employ electronic circuitry at all.

Plastic microspheres can also serve as sensors, because their photonic-atom modes change in frequency when the temperature varies or when they come in contact with material of similar optical properties-for example, DNA molecules attached to their surfaces-a possibility I am now investigating with my colleague Iwao Teraoka and with Frank Vollmer of Rockefeller University. The idea here is to affix to a microsphere many strands of DNA carrying one particular base sequence. Genetic material having the complementary sequence can readily bind to the surface of such a sphere. When it does, the added coating (like the expansion that accompanies heating) alters the effective radius of the particle, which forces the resonant frequency of a given mode to shift. Other kinds of oscillators are sensitive to changes in dimension as well. For instance, the frequency of a pendulum changes when the rod connecting the bob to the pivot expands thermally. But pendula do not make very good thermometers, so it might come as something of a surprise that microspheres would have high sensitivity. They do, at least in principle.

The constancy of angular momentum for a particular mode dictates that the fractional decrease in frequency must be the same as the fractional increase in dimension, and vice versa. The minimum size change that can be detected is the smallest measurable fractional change in frequency times the radius of the sphere. One can easily observe a full line-width shift, some 50 megahertz, which at a typical optical frequency corresponds to a fractional change in size of one part in 10 million. So one can potentially discern a change in radius for a 10-micrometer sphere of 10-12 meters-one-hundredth of an atomic diameter. This exquisitely high sensitivity opens the door for a range of applications, from thermometry to biochemical sensing, for which multiple probes specific enough to detect the activity of particular genes could be interrogated over a single optical fiber using spheres of different sizes.

Thinking of tiny spheres as photonic atoms indeed appears to be a fruitful endeavor, but one can go a step further and push the analogy into the molecular realm. Just recently, scientists working with Makoto Kuwata-Gonokami at the University of Tokyo built the photonic equivalent of a hydrogen molecule. In HZ, two protons share two electrons in a covalent bond, which gives rise to a splitting of the atomic electronic states. Gonokami and his research collaborators touched two nearly identical fluorescent microspheres together and recorded a similar splitting in photonic-atom modes. Appropriately, the researchers dubbed their creation a photonic molecule.

Physicists continue to search for clever ways to benefit from the optical properties of tiny spheres. One can, for example, imagine that grouping more than two spheres together will offer yet more interesting or useful optical modes. Photonic “polymers” might even be in the future. Whatever further advances grow out of this research, I take great pleasure in remembering the experiments of 1980s that led me into the field of microsphere photonics in the first place-attempts to probe how energy is transferred between molecules without photons. In that quest I failed, which I realize now was the best thing I could have hoped for.