Understanding the static structure factor

The structure factor \(S(q)\) is an important quantity for characterizing disordered systems of particles, like colloids. Its significance comes from the fact that it can be directly measured in a light scattering experiment and is related to other quantities that characterize a system's microscopic arrangement and inter-particle interactions. However, it's difficult to learn about it the context of disorder because it's primarily used in crystallography. Crystals are far from being disordered.

In this post, I'll explore the nature of the static structure factor, which is something of an average structure factor over many microscopic configurations of a material. A dynamic structure factor describes the statistics of a material in time as well as space.

The structure factor of a disordered material can be measured by illuminating the material with a beam of some type of radiation (usually X-rays, neutrons, or light). The choice of radiation depends on the material. It is also important that the material should not scatter the incident beam too strongly because the structure function is typically found in singly-scattered radiation (see the first Born approximation for a discussion about a related concept). If the material is multiply scattering, the information about the material's structure, which is largely carried by the singly scattered light, is washed out.

In a measurement, the sample is usually placed at the center of rotation of a long rotating arm. A detector for the radiation is placed at the opposite end of the arm. The arm is rotated about this axis and the intensity of the scattered radiation as determined by the detector is recorded as a function of the angle. This data set essentially contains the structure factor, but must be transformed and corrected for, accordingly.

First, the structure factor is usefully represented as a function of the scattering wave number, \(q\), and not as a function of the scattering angle. In optics, \(q\) is usually given by

\[ q = \frac{4 \pi n}{\lambda} \sin( \theta / 2) \]

where \(n\) is the refractive index of the background material (usually a solvent like water) and \(\lambda\) is the wavelength of the light. \(\theta\) is the scattering angle.

Additionally, the structure factor must be corrected for a large number of confounding factors, such as scattering from the sample cell and radiation frequency-dependent detectors. A classic paper that details all these corrections to find \(S(q)\) in a neutron scattering experiment is given here.

Once the structure factor is found in an experiment, it may be Fourier transformed numerically to give the radial distribution function, \(g(r)\) (see Ziman for the proper conditions for which this applies) of particles. This function gives the probability of finding a particle at a radial distance from another particle in the system. Many important thermodynamic properties are related to \(g(r)\). Importantly, the pair-wise interaction potential between any two particles is related to \(g(r)\), and the pair-wise interaction determines many macroscopic system properties.

The structure factor as \(q\) (or equivalently the scattering angle) goes to zero is also an important quantity in itself. \(S(0)\) is equal to the macroscopic density fluctuations of particles in the medium (see Ziman, Section 4.4, p. 130). But density fluctuations can be calculated from thermodynamics and leads to the isothermal compressibility of a material.

In the language of optics, which I'll stick to for the rest of this post, the density fluctuations would correspond to large regions of refractive index variations across the sample.

This leads to an interesting problem, the resolution of which reminds me of the fallibility in taking some models too literally: for a homogeneous and non-scattering optical material, like a very nice piece of glass, the value of the density fluctuations in the refractive index are essentially zero (this is true because the disorder in a glass is at a length scale that is much smaller than the wavelength of light). This means that \(S(0) = 0\). At the same time, the scattered intensity in the type of experiment measured above is directly proportional to the structure factor:

\[I(q) \sim S(q).\]

So, if I illuminate a nice piece of glass with a laser beam, and I know that \(S(0)\) is equal to zero, the above expression means that there should be no scattered intensity in the forward direction. But this is a silly conclusion, because when I do this experiment in the lab I see the laser beam shining straight through the glass! In other words, \(I(0)\) is not zero.

The problem is that this expression is for the scattered intensity. In random media, we often talk about the scattered light and the ballistic light. The latter of these two is not scattered but directly transmitted through the material as if the material were not there. So, even though no light is scattered into the forward direction, there is still the ballistic, unscattered beam, that is passing straight through the sample.

Most small angle light scattering experiments measure as close as they can to \(q=0\) and extrapolate to the structure function's limiting value. \(S(0)\) can't actually be measured. But, it's determination is important for materials with significant long-range order, such as those near a phase transition, because the small angles correspond to large distances, due to their inverse Fourier relationship.

One can also engineer a material to not transmit any light into the forward direction. To do this, \(S(q)\) must be zero AND there must be no ballistic light passing through the material. This can be achieved with a crystal that diffracts all the light into directions other than the forward direction, such as a blazed grating.

On a final note, the structure factor can sometimes be related to important material properties beyond the radial distribution function. Ziman says in section 4.1, pg. 126 that the direct correlation function (which measures interactions between pairs of particles) can be derived directly from the structure factor. This correlation function is related to the Percus-Yevick model for liquids.

Metamaterials for heat

There's a really cool experiment described in a recent PRL and summarized here about creating a metamaterial for cloaking objects from heat flow.

What connections to metamaterials for light can be drawn? The transport of heat is governed by a diffusion equation, which is very different from the wave equation and Maxwell's equations for governing light transport.

However, the diffusion equation can apply to light transport in disordered materials (see Ishimaru or van Rossum and Nieuwenhuizen). Is there some way, then to cloak objects in randomly scattering media from light?

The trouble with this thought is that one would have to add structure to a random material, and the only way I can think of doing this would be to create large-scale structures from a material with small-scale disorder.

Understanding the correlations between model parameters of speckle

Today I read "Structural correlations in Gaussian random wave fields," an old PRE by Freund and Shvartsman. The authors analytically found the existence of correlations between the amplitude and phase gradients in random electromagnetic fields commonly known as speckle. Notably, while the amplitude and phase are not correlated at certain points, the amplitude is correlated to the gradient of the phase. Higher amplitudes usually are found with smaller phase gradients and vice-versa.

What's not clear to me is if this treatment works for vector fields or only scalar fields. Notably, I'm not sure what phase means for a random vector field.

Perhaps the authors make the assumption that the components of the vector are independent and thus a scalar treatment is sufficient, but I'm not sure that this is so.

How do you teach what polarization is?

Today is Optics Day at CREOL, our annual public open house where we present demonstrations of various optical phenomena and technologies, speakers, and pizza. :)

During this year's Optics Day I am charged with explaining the phenomenon of polarization to visitors. Now, I find polarization incredibly difficult to explain to non-scientists, and here's why: the usual treatment of optical polarization in physics involves describing the direction of the electric field vector of an electromagnetic wave. If I were to start with this definition while speaking with somebody not trained in physics, I would then have to explain electromagnetic waves. This would be followed by an explanation of the equivalence of light and electromagnetic waves, wave phenomena in general, linear, circular and the more general elliptical polarization states, etc. etc. until the poor person who has come to see a cool demonstration and learn something new has completely been befuddled because it takes so much background understanding to comprehend what polarization even means.

This year, I am determined to find an explanation of polarization that is more intuitive to a non-scientist. A rough outline that I intend to give for polarization's foundation in observation goes as follows:

1) Our sense of sight is perhaps the most obvious sense we have. We see objects and from these objects we discern shape, size, color and other properties.

2) There are physical quantities that cannot be sensed by our eyes. For example, flowers have fragrance that our noses can detect. Wind is another example. We feel its effects or we see its effects on other things, but we don't directly see "wind." Therefore, there are physical quantities that cannot be seen but nevertheless may be sensed.

3) There are still more phenomena that exist but cannot be sensed by any of our sense organs. For example, a compass points north because the needle experiences a magnetic force. Additionally, small objects all fall towards the earth because of gravity. Magnetism and gravity require tools that sense things that we cannot: magnetic and gravitational fields. Where our senses fail us, we use tools to measure some quantity.

4) Polarization lies in this last classification of phenomena. It cannot be sensed by us (which isn't strictly true), but can be determined by appropriate tools. These tools are things that are found in nature, like quartz crystals, and man-made objects like polarizers and waveplates.

From this foundation, I will explain some of the consequences of the polarization of light, what it can be used for, and may even digress into the physicist's model if the visitors are interested enough. My hope is to build the concept of polarization up from a basis of observation, not to start with our model first, followed later by how we observe polarization.

What do biologists want? It's probably best to ask them.

Just a thought: more than likely, if a biologist wants to learn something, they'll need a tool that is specialized for measuring exactly the quantity that they're interested in. A general measurement tool will almost always be less-than-ideal for measuring some specific quantity. Therefore, we optical scientists should spend less effort in optimizing a mature imaging technology and instead work routinely with biologists to help solve their problems.

The relevance of multiphoton microscopy to physicists and biologists

There is a comprehensive review article in this month's Nature Photonics concerning many of the technological capabilities and recent advancements of multiphoton microscopy (MPM). The article details many recent advances in MPM engineering for achieving faster image acquistion, increased signal-to-noise ratios, and deeper imaging capabilities.

These advances are impressive and lead me to believe that MPM has become a rather mature technology. I'm curious to know to what extent biologists have used MPM to solve problems in their research since the review article is somewhat lacking in references that come from journals outside of physics and optics.

This is the same problem I encounter again and again in optics. It's very difficult to identify worthwhile work in a research field that primarily develops tools for researchers from other fields to use. I do not blame the optical scientists for this difficulty, though, and here's why. An optical sensing technique is usually not suited for publication in pure biology journals, so they must publish in optics and applied science journals. In this arena, they must argue for their technique relative to other related techniques, not to the suitability of their work for solving biological problems. A sentence in the introduction and conclusion of an article is usually sufficient for reviewers to acknowledge the technique's worth towards a biological problem of interest.

Does improving microscopy mean improving biology?

In my last article I began exploring the relationship between optics and biology to better determine to what extent optics is capable of solving problems in biology, particularly molecular and microbiology. I posed a set of questions, one of them asking whether "...the current trends in improving microscopies [will] lead to answers of the fundamental questions of molecular and microbiology."

Let me start this brief essay by stating my own current opinion, which is based primarily on speaking with biologists and perusing the internet. I believe that the fundamental problems in biology lay at the molecular level and at the systems level. The molecular-level problems include how certain proteins fold and are transported through organelles like the Golgi bodies [1].  The systems-level problems deal with the coherent interaction of the many elements within an organism. To illustrate this, consider how the coordinated actions of various cells (such as Schwann cells, astrocytes, and neurons) lead to an effective functionality of the nervous system.

Microscopy unfortunately is ill-suited to exploring either of these two levels. It is true that fluorescence microscopy has allowed us to specifically target some structures of interest inside a cell and that superresolving microscopies for beating the diffraction limit exist, like PALM and STORM. However, fluorescent markers--which are also used in PALM and STORM--are known to adversely affect the behaviors of live cells. PALM and STORM are furthermore very complex to implement and limited to some degree by their data acquisition times [2].

One popular line of microscopy research is label-free microscopy, whereby images are acquired without introducing any artificial contrast-generating mechanism into the sample. One example is based on stimulated Raman spectroscopy (SRS). This approach usually is a spectroscopic technique that involves inferring what collection of known substances contributed to a measured spectrum from an image. Achieving a good resolution with SRS or any other label-free technique usually means allowing for a severe increase in the measurement time. At the time of this writing, I see neither the spatial nor temporal resolution of label-free microscopies as good enough for addressing the current open-ended biological problems.

At the systems level, microscopy is simply not the tool to use. I think that computer modeling and experiments on live animals are the norm here, though I am not saying that optics cannot play any role.

Overall, I think that we optical scientists are placing too much emphasis on improving light microscopy [3]. It seems to me that the information that biologists require is not found in images but rather in some other form. This is not to say that optics is of no use to biology. Take the technique known as dual polarization interferometry, for example, which uses light to probe protein crystal and lipid bilayer growth on waveguides. As another example, consider that optical tweezers have been influential in measuring the mechanics of biopolymers like DNA.

So what should we focus our attention on? I think label-free sensing mechanisms are in the right direction because they risk minimal alteration to cell and biomolecule functionality. I also think that techniques for sensing dynamic phenomena will trump anything that looks at the structure of fixed (dead) cells. Structure at this point seems well-known to biologists, but how structure evolves in time is not. Finally, controlling biological systems with light seems incredibly promising (e.g. optogenetics), though I think it is too early to tell whether it'll be valuable in deepening our knowledge.

I will hopefully address whether optics is the best tool for fulfilling these characteristics in the future.

Some references
The Wikipedia articles on Molecular biology, the Central dogma of molecular biology, and Biophysics are worth reading.

Stanford Encyclopedia of Philosophy entry on Molecular Biology

Seven fundamental, unsolved questions in molecular biology: Cooperative storage and bi-directional transfer of biological information by nucleic acids and proteins: an alternative to “central dogma”

Notes
[1] I have personally been exposed to a problem of the mechanics of certain biopolymers in regulating the structure of mitochondria. Biopolymers may arguably lie outside the realm of molecular biology since they are made of many, many molecules and not just a few, but I believe that my experience with this problem gave me some good insight.

[2] I have heard that Nikon microscopes are now offering STORM capabilities.

[3] I should point out that I think that the work in improving microscopies IS worth doing, I'm just not so certain that so much attention should be given to it.

An optical scientist considers the question: what do biologists want from a microscope?

Optics and biology have been intertwined for hundreds of years. Robert Hooke and Antonie van Leeuwenhoek both contributed greatly to the fields of microscopy and microbiology in their infancy, advancing each field by establishing a greater understanding in the other. As optics evolved and technologies derived from it became more refined, the number of discoveries in the realm of microbiology witnessed a concomitant increase. This fact was perhaps recognized in part with the award of the Nobel Prize in Physics in 1953 to Fritz Zernike for the phase contrast microscope, a tool which rendered otherwise invisible cells visible with relatively modest modifications to an existing microscope. Much work in microbiology followed as a result of this along with other developments in optics.

The relationship seemed to change, though, starting in the mid-twentieth century with the advent of molecular biology. During this time, molecular biological technologies evolved and matured to the point where discoveries were facilitated primarily by non-optical means, with microscopes serving as more of a tool for routine lab work than as significant drivers for learning something new. After all, a traditional light microscope is limited to observing structures no smaller than about one wavelength of light across, or about half a micron (one millionth of a meter). DNA, proteins, and all the other biomolecules are just too small to see, even for the most powerful microscope objectives.

Of course one could argue that the development of the targeted fluorescent proteins that reveal the location of a molecule's existence within a cell helped to advance the field of optics, but in this case the role of enabler switched sides; molecular biology led to an increase in the number of optical technologies for imaging fluorescent markers, such as fluorescence correlation spectroscopy. From the viewpoint of a scientist, this reversal is a bit distasteful. We would like for technology to enable new discoveries about the fundamentals of life, not for new discoveries to lead to technology that tells us what we already know.

Now we are well into the twenty first century and are rooted firmly within the scientific age of molecular biology and biotechnology. (The age of physics is now past and now concerns itself primarily with the ultimate limits of space: the infinitesimal quark and the awesomely large cosmos.) Given the history between optics and biology and the recent change in their relationship, I think it's necessary to make an assessment, so to speak, of this relationship.

In the near future I will write posts that explore this topic. I hope to answer questions like

1. What do biologists want out of a measurement technique?
2. Will the current trends in improving microscopies lead to answers of the fundamental questions of molecular and microbiology, or are we moving in the wrong direction?
3. Are optical scientists misguided in the search for improved images? Are there other forms of information carried by light that are more useful than images?
4. Will it be possible to better control biological processes using light?

Will light from two independent lasers give a beat note?

In my line of research I usually assume that light from two independent lasers will not produce an interference pattern when combined. This is because the light from one is not coherent with the other. For this reason I was surprised when a colleague of mine who works with frequency combs told me that light from two independent lasers will produce a beat note when combined interfered. This means, for a short time, there will be an interference pattern, though it may change too rapidly for our eyes to see. One can understand this by assuming an ever-shrinking line width for each laser until each one is perfectly monochromatic. [It also helps that the center wavelengths be slightly different.]

Of course, in retrospect, I realize now the error in my thinking. Supposedly "general" rules when applied to the topic of coherence are almost always wrong because one must specify the relevant timescales involved to determine whether light is coherent or not. These time scales include the integration time of the detector, the width of a wave packet, the period of the carrier wave, etc. Because there are so many different parameters, optical coherence problems do not lend themselves to an easy generalization, at least when one is first learning the topic.

And even when one is experienced with it, he or she will likely continuously be surprised like I was this morning.

Hyper-ballistic transport of waves

In this month's Nature Physics there is a paper entitled "Hyper-transport of light and stochastic acceleration by evolving disorder" by Levi, et al. The work is an experimental and numerical study of the propagation of light in a disordered medium that has been carefully constructed to serve as a model for the transport of a 2D quantum wavepacket in a spatio-temporal random potential, i.e. a potential energy landscape that changes randomly in space and time. The authors demonstrate that a beam's spot-size and angular spectrum spreads faster as it propagates through this particular medium than it would if the beam propagated in free space or in a random distribution of parallel waveguides (the Anderson localized regime).

The crux of their demonstration is provided by the comparison of their measured transport regime to the two aforementioned regimes: ballistic and localized transport. Ballistic transport is characterized by a beam spot size that grows with propagation distance and a constant angular spectrum with many longitudinal plane wave components. Localized transport is characterized by a beam spot size that does not grow in size with propagation and takes place in a disordered medium with refractive index fluctuations that possess an angular spectrum of plane waves with all the same longitudinal components. These characteristics are illustrated in Figure 2 of the article.

In contrast, hyper-transport is defined by a spot size that grows faster with propagation than in the ballistic case and by an angular spectrum of the beam (not of the disorder!) that widens with propagation as well (see Figure 3C).

The authors do not provide a comparison with the case of diffusive propagation of the waves. I think that this may be because diffusive transport (increasing spot size and constant but uniform angular spectrum over all propagation directions) is a limiting case of the transport regime that they studied. In other words, they looked at the transient process of the waves becoming diffusive, but not the limiting case. I think that similar work has already been done in the area of beam propagation through atmospheric turbulence, though I can't provide any references.

To be fair, the authors do state that:

"Strictly within the domain of optics, the results described below are intuitive. However, this direct analogy to transport in quantum systems makes our findings relevant for very many wave systems containing disorder."

I would have liked to have seen a comparison to or discussion about the diffusive regime since it would reveal that any multiply scattering medium displays this hyper-transport for short propagation distances.

Finally, I like their technique for controlling the disorder's correlation distance in the z-direction. This seems to be a very good tool for studying transport in disordered systems.

Our latest paper on optically-controlled active media

My colleagues and I just published a paper in Nature Photonics entitled "Superdiffusion in Optically Controlled Active Media." You may access the paper via this link to Nature Photonics or read about it in UCF Today.

I should point out that the claims made in the article in UCF Today is a bit over-reaching. What we have done is demonstrated that the coupling between light and particles in suspension models nonequilibrium processes that share some characteristics with similar processes inside cells. This is because the colloidal particles exchange energy randomly with their thermal bath (the water) and with the laser's radiation.

Importantly, the nature of the light-matter coupling is random due to multiple scattering by the particles, which establishes a three dimensional speckle inside the suspension.  This speckle exerts random forces on the particles which adds an additional component to their motion, besides that of Brownian motion. The resulting effect is that the particles move superdiffusively for times that are shorter than the decorrelation time of the speckle (which was about 1 millisecond).

Besides serving as a model system, I think it may be interesting to explore its ability to control some types of reaction kinetics. If reactants in solutions are driven apart from one another before they have a chance to react, then this would present a mechanical way of slowing the reaction of the bulk solution.

If you can access the article, then I hope you enjoy it!

Come see my talk at FiO

I'm giving a talk on optically controlled active media today at Frontiers in Optics. The talk number is FTh3D.7 and it's in Highland E.

The talk is about our work concerning the optical forces on colloidal particles in 3D, space and time dependent speckle. I'm pushing it as a model system for testing ideas from nonequilibrium thermodynamics, but I think that our method of treating the field-particle coupling is equally interesting.

Come check it out.

Tommorow at Frontiers in Optics

Right now I'm in Rochester, New York for this year's Frontiers in Optics conference hosted by the OSA. So far I've attended the plenary talks, which dealt with quantum optics and thermodynamics, 2D IR spectroscopy, retinal imaging, and, of course, the Higgs boson. In addition, I visited the Omega laser facility, which was incredibly fascinating. If you're in the area and you have any interest in incredibly powerful lasers or inertial confinement nuclear fusion, then I recommend making a visit.

Tomorrow I plan to visit some talks and work at the CREOL exhibition booth from 1:00 PM to 3:00 PM. Additionally, I'm giving a talk for a group mate who couldn't make it to the conference on Wednesday morning at 11:30 AM and my own talk on optically-controlled active media on Thursday at 3:00 PM. Briefly, the talk deals with solutions pumped by light to expand their free energy so that they may do carry out additional work. I hope the project will eventually be applied to controlling reaction kinetics in cells.

If  you're there, let me know and we can talk over coffee or a beer!

Harnessing Light 2

The US National Academy of Sciences has released the second iteration of Harnessing Light, which analyzes and recommends action for maintaining or increasing US competitiveness in global photonics markets.

I just watched the OSA webinar of the roundtable discussion on this document that occurred today at Stanford. Of interest was the committee's strong recommendation to increase US manufacturing capabilities in both optics and other areas that utilize optics for their manufacturing processes. They also addressed the stigma of manufacturing being a blue-collar field and made note that manufacturing engineers address very challenging and technical problems. One member also mentioned that the personal satisfaction from manufacturing jobs is often very great because of the tangible reward of seeing a product that one has designed come to market.

Tom Baer said that industry is better-suited for multidisciplinary research because the historical barriers across fields do not exist there.

Also of interest was the notion that the US has been a good innovator for ideas and technologies but has increasingly lost its ability to capitalize on these ideas to other countries.

I love when new concepts are explained by old ones

I just now learned that photon bunching has an entirely classical analog which corresponds to the fluctuations of the intensity of light with a finite bandwidth. I am well-versed with concepts dealing with the latter as they are tied to optical coherence theory, an important group of concepts in my research. However, I never properly learned what causes photon bunching and always regarded it as somewhat mysterious. Placing it in terms of something I know well is deeply satisfying to me.

I think this type of reaction is true for many scientifically-minded people. When a mysterious phenomenon can be explained by something familiar, we feel satisfied in placing the phenomenon within the logical, mental framework we've built for ourselves. It also explains why creating new concepts and shattering old ones can be so difficult.

An optical method for finding exoplanets

This morning I read an Optics Letter from 2005 entitled "Optical Vortex Coronograph" that described an optical system for detecting exoplanets orbiting a star that could be up to 1e8 times brighter than the planet's reflected light.

The system is detailed below. In a traditional coronograph (i.e. one not employing a vortex phase mask), the mask in focal plane FP1 is a zero light-transmitting block of very small angular extent. Because the image of a star that the system is pointed at is formed in plane FP1, its light is filtered out of the final image by this mask. The Lyot stop in plane PP2 then blocks the light from the star that is diffracted by the mask. The resulting intensity collected in plane FP3 is largely contributed to by any point source near the star, e.g. an exoplanet.

 

What is not clear to me is why replacing the block in FP1 by a vortex phase mask improves the performance of the coronograph. Mathematical arguments are presented, but I find an intuitive explanation lacking.

Understanding the generalized Stokes-Einstein equation

Mason and Weitz published a paper in 1995 about a technique for extracting bulk material parameters from dynamic light scattering measurements on complex fluids. That is, they established a mathematical relationship between the fluctuations of scattered light intensity from a colloidal suspension and the shear moduli of the complex fluid as a whole.

One primary assumption in this derivation is the equivalence of the frequency-dependent viscosity to a so-called memory function:

 

where η(s) is the Laplace frequency-dependent viscosity and ς(s) is the memory function. As a special case example, ς(s) is a delta-function at s=0 for purely viscous fluids since they do not store energy (i.e. they do not possess any elasticity). Substituting this into the well-known Stokes-Einstein equation leads to a relation between the colloidal particles' mean-squared-displacement (measured by dynamic light scattering) and the complex shear modulus of the fluid, G*(ω) (after conversion to the Fourier frequency domain):

The authors note in the end of the paper that it's unknown why light scattering techniques should produce the shear modulus of the fluid since they measure elements along the diagonal of the system's linear response tensor, whereas the shear moduli are contained in the off-diagonal elements.

They also note (with explanations I don't quite understand) that "...the light scattering may not provide a quantitatively exact measure of the elastic moduli; nevertheless, as our results show, the overall trends are correctly captured, and the agreement is very good." (emphasis mine)

Question everything

As you may know, my major field is optics, which concerns the study and application of light. Throughout my studies I've been constantly amazed that Maxwell's electromagnetic theory of light, which has been around since the late 1800's, still contains features that have not been settled or have been simply overlooked by scientists. One such artifact is the dissimilarity between Minkowski's and Abraham's descriptions of the momentum carried by an electromagnetic wave.

In a 2010 PRA Rapid Communication, Chaumet et al. expand on earlier work by Hinds and Barnett that examines the force on a dipole in a time-varying (i.e. pulsed) plane wave. This force is written completely as

 

where Pj is the dipole moment, E is the electric field, B is the magnetic induction, and ε is the Levi-Civita tensor. The first term in the sum relates to both the radiation pressure and the gradient force. The second term, according to Hinds and Barnett, is usually absent in laser trapping and cooling texts because it is proportional to the time-derivative of the Poynting vector, which is zero in common cooling setups. This term is responsible for repulsion of systems such as a two-level atom from the leading edge of the wave when the first term alone predicts an attraction.

Works like this make me wary of blindly using formulas when performing calculations since it reminds me that a theory may not be complete or its assumptions explicit when presented to a niche audience.

Back in action... and optomechanical backaction

I returned to Orlando on the red eye from LA Monday morning and am back in the swing of things (paper writing, data analysis, fixing broken equipment, etc.). I unfortunately had the flu for part of the FiO conference, so I did not attend many talks. A few that I did see and were interesting included FMD1: Near Threshold Optomechanical Backaction Amplifier, FTuZ1: Extracting information from optical fields through spatial
and temporal modulation, and FTuS7: Optically Induced and Directed Manipulation on Surfaces. The abstracts and submissions should be up at http://www.opticsinfobase.org/ within the next month.

FTuS7 was especially interesting. This group out of Oxford used an optically heated metallic substrate to form colloidal crystals from thermophoretically and convectively trapped silica microspheres. They employed standard video microscopy to observe the grain boundaries between two crystals and recorded the annealing time—the amount of time it took for the grain boundary to disappear due to large scale reorientation of the two crystals. The position of the nucleation sites for the crystals were controlled by splitting and directing the laser beam through the microscope objective with a spatial light modulator.

Pretty cool stuff. Fortunately, I got better in time to do some climbing in Yosemite. This time we hit the Five Open Books and I followed on my first Yosemite 5.9, Commitment. The crux on Commitment is ridiculous.

Amplitude and phase: what are they really?

For sinusoidal signals, the idea of amplitude and phase are pretty clear. The amplitude is one half of the peak to valley distance of the signal and phase coincides with the position of the signal at a point in time or space. By signal, I mean any physical quantity that is undergoing simple harmonic motion, e.g. a mass on a spring. In optics, the amplitude and phase of plane waves are well-defined as well: the amplitude is related to the irradiance and the phase typically refers to the position of the wavefronts with respect to some reference.

What's often taken for granted by new physics students (and even experienced ones) is that plane waves and sinusoidal signals are not real. Any real signal must be of finite duration; a sinusoid extends to infinity both forwards and backwards in time. As a result, the amplitude and phase of real signals become somewhat ambiguous.

The two concepts are retained in practice, however, because they simplify the interpretation of many physical situations, such as the effects of a filter on a signal or the output of an optical interferometer or two-slit experiment. A real signal s(t) has an analytic representation z(t) that consists of s(t) plus the signal's Hilbert transform times the imaginary number, j (I was trained as an engineer, but consider it the same as i used by physicists and mathematicians). Two things should be noted here: 1) the analytic signal is unique, and 2) the analytic signal is complex; it has a real and imaginary part. Like all complex numbers, it can also be represented by a magnitude and phase angle. The magnitude is often denoted the amplitude and the phase angle is thought of as the phase of the signal.

Unfortunately, the amplitude and phase of the analytic signal can not always be directly correlated to physically meaningful quantities. For frequency modulated signals, the time-varying amplitude and phase are independent only so long as their individual spectra are non-overlapping. Furthermore, even if the time-varying amplitude and phase are spectrally separate, it is very counter-intuitive (at least to me) to be able to represent one function of time (the real signal) with two functions of time. This last point I believe is related to the Kramers-Kronig relationship between the real and imaginary parts of a signal. So, while a time-varying amplitude and phase can represent a real modulated signal and simplify interpretation, the only thing about the signal with any physical significance is what can be measured, such as current or voltage.

Most of this post comes from thoughts I've had recently concerning how casually we in optical sciences use the terms amplitude and phase. Once an optical signal becomes incoherent, either in space or time, its amplitude and phase are said to vary randomly. This statement only really makes since once one considers that amplitude and phase are purely theoretical constructs for describing real data and that one must not place too much physical significance in their meaning.

I also learned a lot and reproduced some information from this very good article by Boashash in the Proceedings of the IEEE: http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=135376&tag=1.