Science
Whispering gallery modes
Whispering Gallery Modes ("WGMs") are a wave interference effect in (typically) circular geometries. The name was coined by Lord Rayleigh to describe the acoustic effects in St. Paul's Cathedral, London. Essentially, the sound waves can reflect along the interior wall of the dome, traveling around a great circle and coming back to the starting position. This causes the sound waves to interfere, building up a standing wave that represents an acoustic resonance of the chamber.
Photo by Ross Lockwood
About 30 meters from the cathedral floor, there is a gallery with a 32-m diameter at which the effect is especially strong. There, a whisper at one side of the gallery can easily be heard by a listener at the other side, exactly as if someone was speaking into your ear. Over 100 years ago, Lord Rayleigh used the Bessel equation to show that the acoustic energy can get confined very close to the wall of the gallery (with virtually none in the center). They actually used candles to measure that, back in 1910. Not only that, but each resonance corresponds to a particular sound frequency, which becomes amplified - while non-resonant frequencies are lost to destructive interference.
This was when the term "whispering gallery mode" was coined.
These effects can be useful in many ways in optics as well. In optics, we use light instead of sound waves to create the whispering gallery modes. The "galleries" are also a lot smaller, proportionally to the ratio of wavelengths. Since visible light has wavelengths about 7 orders of magnitude smaller than sound, we use micro-scale structures to trap light. These "optical microcavities" are useful for a whole range of things, from the creation of microscale lasers and light sources, to quantum optics and quantum computation, to the development microfluidic "Lab-on-a-Chip" sensors.
One of the fun things is that the electric field associated with optical resonances can be calculated pretty easily. The derivation of the field comes straightforwardly from the Helmholtz equation. It take some messing around with Bessel functions, as one would expect for cylindrical or spherical resonances, but good math software can really make this job easy. We like Mathematica especially for many reasons, and students in our group all get trained up in how to do basic coding. Once you have that down, it's easy to make plots of the electric field going around the microcavity, as shown below for a layered capillary structure.
Q-factors
The quality (Q) factor is one of the two most important properties of optical resonances in any kind of cavity (the other is the optical mode volume, which we will also discuss). Q-factors are a direct measure of the ability of the cavity to confine and store light. Higher Q-factors mean the cavity confines the energy inside more effectively, and will produce stronger effects on light trapped inside.
Mathematically, the Q-factor has a number of simple definitions. It is technically proportional to the ratio of the optical energy stored in the cavity to the energy lost per oscillation cycle. So, a high Q-factor means a lot of optical energy stored in the cavity with very little loss. The same thing is, of course, true for any type of resonating cavity - even a standing wave on a fixed string.
Unless you do certain specialized techniques called "ringdown spectroscopy", it can be very hard to figure out how much energy is lost in your cavity per cycle. For example, in our work all we normally see is a fluorescence spectrum with a bunch of peaks in it where the resonant modes are. Luckily, there is a simpler, equivalent definition that lets us work out the Q-factors. In this definition, the Q-factor is simply the ratio of resonance frequency divided by the resonance linewidth. Although it may not seem like it, this is a direct consequence of the first definition and can be proven mathematically. For example, in the image below, we see a photograph of one of our microspheres and the corresponding emission spectrum. The red spectrum is fluorescence coming from silicon quantum dots that coat the sphere, using a method we've patented. The physics behind this emission is associated with the quantum mechanics of small semiconductor nanocrystals, but that is a large topic that perhaps I will cover later.
The important thing is that the oscillations in the corresponding luminescence spectrum show Q-factors of several thousand. This is actually very high for a fluorescent microstructure. It is interesting to think that the red light we see emitted from these microspheres is really composed of all those very sharp peaks.
The bare cavities show Q-factors up to almost one billion. This is exciting because of the quantum optical effects that can occur at such high Q. Coating the cavity with the quantum dots lowers the Q-factor though - mainly because of scatting and absorption. Any "energy loss mechansisms" like these in the cavity cause a lowering of the Q-factor and a broadening of the spectral peaks. Still, Q-factors of several thousand are excellent for fluorescent microcavities.
One of our microspheres, in the image below, is the optical analogue of St. Paul's cathedral. This structure is about 50 micrometers in diameter.
The mode structure of the field in a sphere can be very neat looking. Here is an example, simulated with Mathematica, for the electric field intensity of three different resonances. These are regions where the electric field trapped inside the sphere oscillates most strongly. This image corresponds to the resonance with quantum numbers l = 33 and m = 33, 31, and 29.
If the fluorescence from a set of quantum dots can be coupled to these resonances, the result can be very striking. Check out the emission spectrum below. Usually it is broad and featureless, but when deposited on the microsphere (in two coatings, in the case below) we can see an amazing set of strong, narrow resonances. Each of those peaks corresponds to a resonance like the one on the above left, but with a different number of intensity maxima fitting around the equator and therefore a different wavelength.
The mode volume is a slightly more difficult concept than the Q-factor. Essentially, the mode volume is a measure of the spatial confinement of electromagnetic radiation inside the cavity. Just like a higher Q factor will lead to stronger cavity effects, so does a smaller mode volume.
Mathematically, the mode volume is defined as the volume integral of the electric field intensity in and around the cavity, normalized to the maximum field intensity (the latter means that the mode volume does not depend on the magnitude of the field itself). This implies that we are integrating the electric field intensity over all space. The problem is, these are "open resonators" - that is, energy leaks out into space. This is different from the conventional solutions for closed resonator cases such as a vibrating drumhead, and it makes the physics a bit more complicated. For an open resonator we can set an integration limit at some reasonable value, such as 100 times the cavity radius. Obviously, this is an approximation that improves as the integral extends further out, but unfortunately it diverges at infinity. This is the "traditional" problem of an open resonator, and it means the mode volume should be though of as approximate.
So the ways that the fluorescent quantum dots interact with the microcavity are controlled largely by the mode volume and Q-factor of the cavity, and by the emission properties and geometry of the dot inside the cavity. In our case, we typically deal with cavities having a circular symmetry, such as rings, disks, or spheres. The QDs are generally at or very close to the WGM electric field maximum, which is the optimal location. Thus, we can observe very strong, intense resonances in the emission spectra.
In the image below, you can see how the field intensity looks for a WGM of a sphere (in cross section). The color represents field amplitude. If you know something about modes already, these are three different TE-polarized radial modes. The mode volume in this case is very large and extends well outside the cavity, but the Q-factors of these structures can be extraordinarily high. We've measured Q-factors approaching one billion in spheres without QDs, and about one million in quantum-dot-coated spheres.
Importantly for sensing applications, you can see that a portion of the field (the evanescent part) extends outside the resonator, sampling the nearby medium. How much that occurs depends on the radial mode order. In the simulations below for a layered sphere, you can see that the third-order modes have the largest evanescent component extending outside the sphere.
Sensors
First, let's talk microcavities again. The exact wavelength at which these resonances occur can be sensitive to the nature of the medium surrounding the microcavity (both its refractive index and absorption). In the simplest cases, the sensing action is simply a refractometric effect: as you change the refractive index outside the cavity, the mode resonant positions shift. We can measure these shifts to determine the change in refractive index of a fluid medium.
In our work, we have a thin layer of fluorescent quantum dots coating the surface of a capillary channel. The way we get the quantum dots into the capillary is the subject of a patent. Ultimately, the WGMs are confined by the QD layer but they extend slightly into the interior channel, thus enabling fluid sensing. Analytes are then pumped through the capillary core using a micro-syringe pump and the fluorescence WGM positions measured with a spectrometer. This combines microfluidics with quantum-dot detection methods.
We've used this method for detecting refractive changes due to the addition of a variety of solutes in water; for example, sucrose, glucose, alcohols, heavy oils, and other chemicals. We are now turning to mixtures of oil and water, and also true biosensing using surface functionalization chemistry. All this really means is that by treating the channel surface (the QD film) with certain solvents, we can make certain biomolecules preferentially stick to the sensing surface. Microfluidic biosensing of this type is likely to be important in a wide variety of technologies, from medical diagnositcs to enviornmental monitoring and control.
So, that's a short summary of the science. Obviously, one could write a book to describe everything. Some microcavity books have been written but they unfortunately tend to be very technical and quite hard to follow unless you've already got a fair bit of expertise in the field. Someday, someone will have to write a good 400- or 500-level book on microcavities, as they are becoming more and more important in optics, photonics, sensing, microlasers, and other applications. Meanwhile, feel free to send me any comments you may have!