For instance, we have all seen the spectacular colors that can appear in soap bubbles:

Image from Microscopy-uk.org.uk, by Michael Reese Much. Borrowing his lovely images until I can produce my own!

These colors are produced by optical interference, as we will discuss below; the “thin film optics” that creates bright colors in soap films also results in the bright colors of oil slicks.

A rainbow of color produced by white light reflecting off of a thin layer of diesel fuel on water, via Wikipedia.

Most of us would look at a soap film image and marvel at the beautiful rainbow colors; others would investigate the optics underlying them.  But it took an exceptional physicist, Jean Baptiste Perrin (1870-1942), to realize that these colors concealed something more: direct evidence that matter consists of discrete atoms and molecules!

Today we take for granted that all material objects in the universe are comprised of discrete “bits” of matter, which we call atoms; however, even up until the early 20th century there were still proponents of the continuum hypothesis, in which all matter is assumed to be infinitely divisible.

This is quite surprising, because the first atomic hypothesis goes back to ancient Greece and the philosophical speculations of Democritus and his teacher Leucippus; the modern atomic theory dates to 1804, when John Dalton observed his law of multiple proportions.  This law is stated as follows:

If two elements form more than one compound between them, then the ratios of the masses of the second element which combine with a fixed mass of the first element will be ratios of small whole numbers.

That is, to use the example from Wikipedia, it is possible to react carbon with oxygen in two ways, one of which forms carbon monoxide and one of which forms carbon dioxide.  We can react 100 grams of carbon with 133 grams of oxygen to form CO, and that same mass of carbon can react with 266 grams of oxygen to form CO₂. The ratio of masses of oxygen in the two compounds is

CO₂ : CO = 2 : 1,

indicating that oxygen can only join with carbon in discrete amounts.  This strongly suggested to Dalton that matter such as oxygen exists in discrete quantities: the “atoms” of his theory.

However, other researchers of the time argued that this law could be explained by the continuum hypothesis as well, with a little theoretical gymnastics. Arguments between the atomists and the continuum proponents continued through the 19th century.  The fundamental difficulty for the atomic theory? Nobody could actually see atoms or molecules, or find any direct experimental evidence for their existence; without even an estimate of their size, the idea of atoms was purely academic.

The continuum hypothesis was finally put to rest when, in 1905, Albert Einstein proposed an explanation for the phenomenon of Brownian motion, which had mystified scientists for nearly one hundred years.  In 1827, the biologist Robert Brown noticed that minute particles ejected by pollen grains suspended in water were moving around in an irregular, jittery motion.  An example of this motion is shown in the video below.

Einstein explained this motion as due to the rapid and near continuous bombardment of the visible particles by the invisible molecules of the water surrounding them.  Einstein’s theory explained Brownian motion both physically and mathematically, and furthermore the theory gave for the first time a method to calculate the density and size of the molecules.

Jean Baptiste Perrin

Einstein’s theory still needed to be confirmed experimentally, however, and Jean Perrin was perfectly suited in both skills and interests for the task!  In the early 1890s he had been deeply involved in the investigations of mysterious cathode rays (now known to be streams of electrons) and the even more mysterious Röntgen rays (now called X-rays).  In 1897 he earned his doctorate in science and became a lecturer of physical chemistry at the Sorbonne in Paris.  With his new position, he became fascinated with the “molecular hypothesis” (more or less another term for the atomic theory) and began extensive investigations into the subject.  When Einstein’s theory of Brownian motion was put forth, Perrin devoted his efforts to confirming the theory and determining an estimate of molecular size; his results were published in 1910 and contributed to Perrin’s Nobel Prize in Physics for 1926.

Even though the molecular hypothesis was now on solid footing thanks to Einstein’s theory and Perrin’s experiments, Perrin was not completely satisfied.  The Brownian theory provided an indirect measurement of the size of molecules, in that the size could be deduced from Einstein’s formulas.  Perrin wanted to know if there was a way to directly measure the molecular size; as he recollected in his Nobel lecture,*

The objective reality of molecules and atoms which was doubted twenty years ago, can today be accepted as a principle the consequences of which can always be proved.

Nevertheless, however sure this new principle may be, it would still be a great step forward in our knowledge of matter, and for all that a certitude of a different order, if we could perceive directly these molecules the existence of which has been demonstrated.

To make this direct measurement, Perrin turned to a phenomenon that had fascinated him since childhood: the colors of soap bubbles!

The spectacular colors of soap films come from a wave interference effect, as illustrated below.

An example of destructive interference in thin films. The two reflected waves (shown horizontally offset for clarity) partially cancel out.

Let’s first imagine that a light wave comes in on a thin film which is two wavelengths λ thick — this means that two complete “ups and downs” of the wave exactly fit inside the film.  Part of the wave will be reflected from the exterior surface of the film (shown in the middle) and part of the wave will be reflected from the interior of the film (shown on the right).  Glossing over some subtlety in the nature of reflection from the inside and the outside, we can see that the two reflected waves are out of phase — where one is waving left, the other is waving right, and vice-versa.  The two waves destructively interfere, and the net result is that very little light reflects at that particular wavelength.

In a slightly different situation, let us imagine that a light wave comes in on a thin film which is two and one quarter wavelengths thick!

An example of constructive interference in thin films.  The two reflected waves are in phase and constructively interfere.

Now the two reflected waves are in phase — they both wave left and right at the same time.  These waves constructively interfere, and the reflection of the wave is increased.

It turns out that the reflection of light from such a thin film will be enhanced whenever the thickness d of the film is an odd number of quarter wavelengths:

This enhancement is the source of the bright colors of a thin film!  Because a thin film will only reflect light of certain wavelengths, and the reflected color depends on the film thickness, the reflected color is a direct measure of the film thickness.

Looking again at the soap bubble picture from the beginning of the post, we can now interpret what is going on.

This film is oriented vertically, so gravity is pulling down on it, and it is therefore on average thickest at the bottom and thinnest at the top.  Perrin himself describes the colors to be seen as follows:

You know the properties of thin laminae: each ray reflected from such a lamina is formed by the superposition of a ray reflected from the front side of the lamina on a ray reflected from the rear side. For each elementary colour these rays add together or subtract from one another according to a classical formula, depending on whether they are in phase or out of phase; in particular, there is extinction when the thickness of the lamina is an even multiple of one quarter of the wavelength, and there is maximum reflection when it is an odd multiple.

If, therefore, white light strikes a lamina which has a thickness increasing continuously from zero, the reflected light is at first non-existent (black lamina), then weak (grey lamina), then lively and still almost white, becoming successively straw yellow, orange yellow, red, violet, blue (tints of the first order), then again (but with different tints) yellow, red, violet, blue, green (second order); and so on, the reflected colour becoming continuously more complex and more off-white up to the “white of a higher order” (the spectrum is furrowed with black grooves the number of which increases with the thickness of the lamina). All these tints will be present at the same time on a lamina which has not a uniform thickness and which will be black or grey in its thinnest region, straw yellow in a thicker region, red in an even thicker region, and so forth.

At the top of the film where it appears black, it is in fact almost perfectly transparent: the film is so thin that it reflects almost no light at all, and we see the black screen behind it.  It is to be noted, in the figure above, that near the top of the film, there are regions of bright and nearly constant color, bounded by lines where the colors change dramatically.  Perrin realized that something interesting was happening at these transition lines.

Having examined a large number of stratified laminae, it occurred to me, before I made any measurement, that the difference in thickness between two adjacent bands cannot fall below a certain value and that this elementary minimum difference, a kind of “step of a staircase”, is included a whole number of times in each band. Similarly, if we throw playing-cards on the table, the thickness at each point is that of a whole number of cards, without all possible thicknesses being necessarily present, since two or three cards may remain stuck together. The stratified liquid strips would, therefore, be formed by the piling up of identical sheets, more or less overlapping each other, their liquid state imposing on the free contours the form of arcs of a circle (which are fixed at their extremities on globules or on the non-stratified periphery, according to conditions so far unknown).

In short: where the soap bubble is at its thinnest, it can only be a finite number of molecules thick.  At any point where it gets thicker or thinner, the thickness must change by a discrete number of molecular thicknesses.  Perrin likens this to a random pile of playing cards, whose thickness on the table can only change by the thickness of one card; I crudely illustrate the idea for layers of soap molecules below.

By measuring the reflectivity of the thin film for the different regions of uniform colors, Perrin was able to deduce how much the thickness of the film changed between regions.  He collected a large number of these differences in thickness; assuming each must be a multiple of the molecular thickness, he was able to determine the size of the soap molecules!

The measurements confirmed this impression. From 1913 onwards I found a value ranging between 4.2 and 5.5 mµ. And since then, precise photometric determinations made under my direction in 1921 by P.V. Wells, who otherwise had to overcome serious experimental difficulties, have fully established what we can call a law of multiple thicknesses.

We first of all applied simply the classical relationship between the thickness of the lamina and the intensity of the reflected light, using monochromatic lighting.

On the first-order band 120 measurements were made, giving thicknesses grouped according to the law of chances around 4.4 mµ. It is certainly the best measurement made so far of the thickness of the “black spot ” for which Johonnott gave 6 mµ. The extreme thinness of this band, the faintness of the reflected light, and the difficulties due to parasitic lights make this determination particularly interesting.

The set of the measurements for the first fifteen bands give similarly thicknesses which are, within several hundredths, of the successive multiples of 4.5 mµ.

Perrin therefore found a value of 4.5 nanometers (4.5 billionths of a meter) for the size of soap molecules.  This number may seem much too large to those who know a little atomic physics: atomic sizes are usually measured in angstroms (0.1 nanometers), and the soap molecule is over a factor of ten larger.  Perrin’s result is not surprising, however, because soap molecules are fashioned as large chains of other molecules, as shown below.

Two chemical representations of sodium stearate, a typical soap molecule. Via Wikipedia.

Soap derives its effective cleaning properties due to the fact that one end of this chain is repelled by water (hydrophobic) and the other end is attracted to water (hydrophilic).  The right side of the molecule shown above, the polar end, is attracted to water; a film of soap therefore consists of a collection of soap molecules standing at attention, side by side.

Monolayer of soap molecules encasing a thin film of water. The red spheres represent the polar ends of the soap molecules.

It is interesting to note that Perrin’s experiment is similar in philosophy to the Millikan oil drop experiment, performed a few years earlier in 1909.  In the Millikan experiment, microscopic droplets of oil were imbued with a small amount of electric charge, with presumably only a small and countable number of electrons in each droplet.  Though it was not possible to reliably imbue a droplet with a single electron charge, Millikan measured the charge for a large number of drops and demonstrated that the difference between droplets was always an integer multiple of a fundamental value: the charge of the electron. Similarly, Perrin demonstrated that the difference in soap film thicknesses was always an integer multiple of the size of the soap molecule.  Perrin was aware of Millikan’s work, and it seems not unreasonable to assume that he was inspired by it, whether he realized it or not.

Perrin’s result was the first direct measurement of molecular size.  It seems to have been largely forgotten today for a number of reasons:

• It was not the first strong evidence of molecular existence; Einstein’s Brownian motion explanation was the deciding factor.
• It was a technique of limited usefulness: the method works pretty much only for thin films of soap.**
• With modern technology such as atomic force microscopy, we can produce images of objects, including molecules, with sub-nanometer resolution!

Though Perrin’s work with soap films did not fundamentally alter our understanding of the natural world, it is an imaginative and beautiful result that demonstrates that profound insights can come from even the most mundane observations.  Who knows what other scientific discoveries are hidden in plain sight, right before our eyes?

***************************

* Quotes on the soap bubble experiments will be taken from Perrin’s Nobel lecture, as his original 1913 paper is brief and less detailed.  The original paper is J. Perrin, “Observations sur les lames minces,” Archives des Sciences Physiques et Naturelles 35 (1913), 384-385.

** Perrin himself notes, however, that a similar technique was used to determine the molecular size of mica, a solid which can be divided into thin planar sheets.

This pair of iron hemispheres, with handles attached and a valve on one side, are a small scale model of one of the earliest and most dramatic displays of the power of atmospheric pressure.  They are now known as the Magdeburg hemispheres, and they still work as a great demo to this day.

Otto von Guericke

Curiously, the hemispheres are not named for a person, but a place: Magdeburg, Germany.  The spheres were designed by the German scientist (and mayor of Magdeburg) Otto von Guericke (1602-1686) in 1654 to show off his newly-invented vacuum pump and the concept of atmospheric pressure.

The premise is simple: with the hemispheres pressed together, air is pumped out of the interior, creating at least a partial vacuum.  This seals together the hemispheres with a remarkable force.

Guericke first demonstrated this force in 1654 for the Emperor Ferdinand III.  Thirty horses, in two teams of 15, were unable to pull apart the evacuated hemispheres!  He performed a smaller scale performance in 1656 in his hometown of Magdeburg, using two teams of eight horses.  This latter performance was immortalized in a famous sketch by the scientist Gaspar Schott in a 1657 book, Mechanicahydraulica-pneumatica.

So how do the spheres stay together?  Simply put: atmospheric pressure!  All objects within the atmosphere are under constant bombardment from air molecules traveling every which way; this atmospheric pressure is not noticeable to us because our bodies have an internal pressure that matches and balances it.  (Though you won’t explode if exposed to a vacuum, movie tropes notwithstanding.)

When the hemispheres are first placed together, the air pressure within them balances the air pressure outside, and they are easily pulled apart.  When air is removed from the interior of the hemispheres, however, there is no longer any force pushing outward: the atmospheric pressure outside dominates, pushing the hemispheres together and keeping them from being separated.

An attentive reader may have already noticed a similarity between the Magdeburg hemispheres and a very common, everyday object: a suction cup!

A suction cup gets its sticking force by the same action as the Magdeburg hemispheres: when one pushes the cup against a smooth wall, one forces the air out of the cup, allowing atmospheric pressure to hold it to the wall.  We could say that the Magdeburg hemispheres are an early primitive form of suction cup (though suction cups apparently have a history that goes back much further)!

A good set of Magdeburg hemispheres, however, can provide suction forces significantly stronger, as Guericke’s original horsepower demonstration shows.  In mathematical terms, the force holding together a pair of hemispheres of radius can be shown to be

,

where is the pressure difference between the interior and exterior of the hemispheres.  My pair of hemispheres have a radius of 5 cm; assuming conservatively that my hand vacuum pump can reduce the interior pressure to half of atmospheric pressure (atmospheric pressure being ), I find that the hemispheres will stick together with a force of 392 Newtons, or 88 lbs!  Guericke’s hemispheres had a radius of about 25 cm; if he had almost completely evacuated them of air, they would provide a force of about 4400 lbs!

For those not willing to spend some $60 for a metal set of hemispheres, it is possible to perform a simple version of Guericke’s demonstration at home with readily available supplies.  You will need:

• two identical glasses
• a candle
• a damp piece of paper that covers the opening of a glass

Put the candle in one of the glasses, and light it.  Place the other glass on the top of the first, with the wet piece of paper between them; this paper will help form an airtight seal between the glasses.  The candle will slowly consume all of the available oxygen in the glasses, and be extinguished.  If you have done things correctly, you will find that the glasses are strongly stuck together due to the drop of atmospheric pressure inside!  If you have set up a really good seal, it might be impossible to pull the glasses apart by hand — they can be twisted apart in this case.

Obviously, the seal between the two glasses is key: I’ve only been able to get a weak seal on my preliminary home experiments; a heavier piece of paper should work better than a lighter one.

How does the home experiment work?  It looks like several factors can play a role in reducing the pressure within the glasses.  For one, a candle burns by combining the oxygen in the air with hydrogen present in the candle to produce water vapor and carbon dioxide.  Less CO₂ gas is produced than the amount of O₂ consumed (thanks to the creation of water, as well); when the water vapor condenses, this results in a pressure drop inside the glasses, serving as a crude form of vacuum pump. Another aspect: the candle heats up the air within the glass, causing it to expand.  When the candle goes out, the temperature drops, causing the pressure to drop within the glass as well.  The more I look at this simple setup, the less simple it seems to be!

The Magdeburg experiment is a delightful glimpse of the hidden and powerful atmospheric forces that are with us at all times.  It is just as fascinating today as it was some 350 years ago!

Update: revised my calculations for the force of attraction of the hemispheres!  Also, related to comments below, I revised my explanation of the “candle effect”; it looks like it is another experiment in which a number of factors can play a role!

On our first full day in Seville, we spent the morning visiting the Seville Cathedral, as described in the previous Spain post.  That was only the beginning of the day, however, as we went directly from the cathedral to the Alcázar, a beautiful royal palace with a long  and storied history.  It is still used by the Spanish Royal family as a residence when staying in Seville.

Entrance to the Real Alcázar of Seville, the Puerta del León.

This palace-fortress has such a long history, with such extensive renovations and changes, that even its date of origin is unclear!  It seems that many of the surviving structures date from the 14th century, though some remains could date back as early at the 10th century.

Appearances can be deceiving, however!  The tile lion greeting visitors to the Alcázar is one of the most recent additions, and was created in 1894 by artist José Gestoso.  The Gothic script Ad Utrumque means “Prepared for All”.

Detail of the lion at the Lion’s Gate.

The gate itself is believed to have been constructed in the 14th century by Peter of Castile, a king with a notoriously bloody reputation.  Through the gate one comes across the Patio del León, which incorporates a section of an older Muslim wall on the far side.

The Patio del León.

This Patio has a bit of a secret history: in the 1600s, it was the location of El Corral de Comedies de la Montería, a theater of a style popular in that era that were located in the inner courtyards of homes.  The theater burned down in 1695, though theatrical productions had been banned in Seville years earlier.  Last year, researchers designed a virtual reconstruction of what the theater would have looked like.

Off to the left of the Patio del León one can enter the Sala de Justica (“Hall of Justice”).

The Sala de Justica.

This room, which definitely exhibits Muslim artistic influence (including a fountain in the center of the room, not seen), was likely used as a residence of Peter I during the construction of his palace.  Also, it is said that Peter’s step-brother, Don Fadrique, was murdered in this room by order of Peter himself.

Passing through the rear arch of the Patio del León, one enters the heart of the complex, the Patio de la Montería.

The Patio de la Montería.

This open area connects three major buildings in the Alcázar.  In front is the Palacio del Ray Pedro I de Castilla (“Palace of King Peter I of Castille”), built in 1364. On the left is the Palacio Gótico (“Gothic Palace”), built in the mid 13th century under Alfonso X.  On the right is the Casa de la Contratación (“House of Trade”), which was established in that location in 1503 but whose facade dates to 1755, rebuilt in the wake of the devastating Lisbon earthquake.

At this point, everybody needed a restroom break (remember, we had already spent much of the morning in the Seville Cathedral).  While I waited for everyone else to come back, I stepped into the area of the Gothic Palace and photographed what I would later learn to be the Patio del Crucero (“Court of the Crossing”).

The Patio del Crucero.

The courtyard is one of the few things that remains from the original 12th century Moorish palace, though the Gothic Palace was essentially built around it.  Directly ahead in the photograph is the entrance to the Gothic Palace.

Entrance to the Gothic Palace.

Once we had all found each other again, we moved on to the Palacio del Ray Pedro I, also known as the Mudéjar Palace because it incorporates both Muslim and Christian styles in its construction.  What this means is immediately evident when one enters the Patio de las Doncellas (“The Courtyard of the Damsels”).

The Patio de las Doncellas.

The Courtyard of the Damsels is named for the legend that the Moorish rulers of Andalusia would demand 100 virgins from the Christian kingdoms each year.  This story was used to spur the Christian reconquest of the Moorish territories that took place during the Middle Ages.

Directly behind the Courtyard, one enters the magnificent Salón de Embajadores, the throne room of Peter I.

The Salón de Embajadores.

I wish I had gotten more photographs for a more complete panorama!  This ornately-decorated room is a perfect example of the phrase “horror vacui” — fear of the void — in which no space is left blank.

Another ornate, and subtly creepy, chamber is the Patio de las Muñecas (“Court of the Dolls”).

The Patio de las Muñecas.

The room is subtly creepy because the busy decor includes numerous small stucco heads.  I didn’t even notice them while passing through the chamber, and have no good pictures of them myself!

From the Mudéjar Palace we passed without realizing it into the Gothic Palace, and entered the Gothic Palace Chapel.

The Gothic Palace Chapel.

The picture on the altarpiece is Our Lady of La Antigua, a beloved saint of Seville.  The image here is an 18th century reproduction of a 13th century original in the Seville Cathedral that was created soon after the mosque was converted into a church.

From the palace we passed out into the massive Gardens of the Alcázar.  The first thing that caught our eye was the lovely Pond of Mercury.

The Pond of Mercury.

This pond was originally an irrigation reservoir that was fed by a Roman aqueduct!  In 1575, it was converted into a more decorative pool with a theme based on the Roman god Mercury.

The remainder of the gardens are divided into too many sections to describe in detail!  Let me simply provide a few photographs of the various sections, to give a feel for its style.

A view from above of the Gardens of the Alcazár.

Another view of the Alcazár Gardens.

The above ground view of the gardens were taken from the Galería de los Grutescos, an old Muslim wall that was given a facelift in the 17th century by the Italian artist Vermondo Resta.

The Galería de los Grutescos.

The Galería de los Grutescos was a great place to take a picture of my lovely wife Beth!

Beth in the gardens of the Alcázar.

There was so much more to be seen at the Alcázar, but we had so much more to see that day in Seville!  Our next stop would be the Plaza de España… but I’ll leave that for my next post.

For those not familiar with optics, there’s a lot to unpack in even the title of the paper: What is “coherence”?  What is a “plasmon”?  Why do we care about “converting” coherence?  Let’s take a look at each of these ideas in turn as we build an explanation of what my collaborators and I have accomplished!

Let’s start with a short review of what we mean by optical coherence.  Light has long been known to possess wave properties, and like all waves it can generate interference patterns: alternating regions of high and low intensity.  The archetypical example is produced in Young’s double slit experiment, in which light is transmitted through a pair of slits in an opaque screen.  The light emanating from the two slits interferes and produces a pattern on a secondary screen further “downstream”.

If the experiment is prepared correctly, one sees alternating bands of light and darkness on the second screen, as shown in a simulation below.

However, if one does this experiment with an ordinary light source, like a light bulb, the interference pattern will almost certainly not be seen.  The reason for this is that a light bulb produces light that has different wave vibrations at different points in space — the light reaching the two slits in Young’s experiment are uncorrelated, or spatially incoherent.  A crude illustration of the difference between correlated and uncorrelated waves is shown below; basically, correlated waves are doing the same thing, and uncorrelated waves are “doing their own thing”.

We can only get interference patterns by the interference of light from highly spatially coherent sources, such as a laser or a conventional source which has been appropriately collimated and filtered.

The propagation characteristics of light strongly depend on its spatial coherence.  In addition to interference effects, the spatial coherence affects the directionality of light.  A coherent laser beam, for instance, creates a beam of light that is essentially in a single direction, whereas a light bulb radiates in almost every direction. There are applications where it is useful to have light which is partially coherent: highly directional, but not able to produce strong interference patterns.  For instance, it has been shown that partially coherent light will propagate through atmospheric turbulence with less distortion than a comparable fully coherent beam, making partially coherent beams potentially useful for free-space optical communications systems.  Any technique which allows us to modify the spatial coherence of light would therefore be an excellent tool.

This is where surface plasmons come in!  Speaking technically, a surface plasmon is an electron density wave that can be excited along the surface of certain metals by light.  This is illustrated schematically below.

The “+” and “-” represent lower and higher densities of electrons along the surface, respectively, and the “E” and “H” represent the directions of the electric and magnetic fields.  (The right-most figure crudely illustrates how the intensity of the surface plasmon is maximum at the surface and decays away rapidly away from it.)

A plasmon is an electron density wave, which means the thing that oscillates and travels along the surface of the metal is the highs and lows of the density of electrons.  This is quite analogous to the behavior of sound waves, in which the density of the air molecules is “waving”.

There are a number of important aspects of surface plasmons (to be referred to simply as “plasmons” from now on) that make them very useful for optics applications.  First, plasmons have a shorter wavelength, and consequently higher momentum, than the light that excites them.  This means that plasmons cannot be excited on a smooth metal surface, but only at bumps or holes on that surface.  Also, once excited, those plasmons will carry energy along the surface until they are either absorbed or hit another hole and get converted back into light.  Second, plasmons are coherently excited at one of these holes or bumps, which means that there is a perfect correlation between the light illuminating the hole and the plasmon wave that gets generated.

Around 2007, my postdoc advisor started to think that these surface plasmons could change the state of coherence of a light wave.  How would this work?  Let’s look at a cartoon version of Young’s double slit experiment, and imagine that each slit is illuminated by an independent light beam.  Therefore one hole is illuminated by “apples”, and the other by “oranges”.

The “apple light” and the “orange light” cannot interfere, and after passing through the holes, that hasn’t changed — no interference pattern will be created.

But now let us suppose that the screen with the slits can support surface plasmons.  When the apple light hits the slit, part of it will be directly transmitted through the hole, but some of it will convert into a surface plasmon.  This “apple plasmon” will travel to the other hole, where it will convert back into apple light!  A similar argument applies for the orange light, and the result is illustrated below.

The light going into the pair of slits was completely uncorrelated (apples & oranges), but the light coming out of the slits is now correlated — apple/orange & apple/orange! By essentially swapping a part of the light between the two slits, the spatial coherence of the light has been increased.

My collaborators and I published a theoretical paper¹ on these results in 2007, titled, “Surface Plasmons Modulate the Spatial Coherence of Light in Young’s Interference Experiment.”  In this work, we demonstrated that, depending on the separation of the slits and the wavelength of light, the spatial coherence of light could be increased or even decreased on passing through the pair of slits.  The increase in spatial coherence was demonstrated experimentally and published later that same year.²

This work got me thinking: is it possible to change the spatial coherence of an entire beam of light using surface plasmons?  In the plasmonic double slit experiment, we are only changing the spatial coherence of light at two points of a beam of light — the rest of the light is being blocked by the metal screen.

The natural thought was to imagine transmitting light through an array of holes in a metal screen.

When this array is illuminated by a beam of light, plasmons will propagate back and forth between multiple holes, and the light emanating from the collection of illuminated holes will presumably form a beam of light that has a modified state of coherence.  But will it work?

Studying the interactions of plasmons bouncing around between multiple holes on a metal surface requires rather complicated numerical simulations, and things cannot be done with simple pen and paper calculations.  As an intermediate step, my student Choon How and I studied³ the behavior of surface plasmons in Young’s three slit experiment!

The big question in the three slit study was whether the middle slit would hinder the propagation of plasmons to the outer holes.  That is: can plasmons from the left hole “jump” the middle hole and influence the right hole?

It was found via simulation that plasmons do indeed jump the middle hole, and that the presence of the middle hole could help enhance the coherence between the outer holes.  This opened the door to studying how the overall coherence of a beam of light could be enhanced by an array of holes in a plasmonic screen.

Every researcher eventually runs into an annoying project that just takes forever to get completed and written up.  This study of a “coherence-converting plasmonic array” was such a project for me, and it took roughly 3 years to get it submitted for publication.  It turned out to be difficult in large part for two reasons: 1. the complexity of the simulations, and 2. the challenge in defining the “overall” coherence of a beam of light.

In general, it can be tricky to perform exact simulations of the interaction of light with matter.  The multi-hole plasmonic array we were interested in turned out to be a particularly difficult system to model, and in fact exact simulations were beyond the computing capabilities I had available at the time.  However, it turned out that the exact results of the double slit and the triple slit cases could be reproduced quite accurately with a “toy model” of the system.  For the double slit case, this toy model really involved treating the light propagation as consisting of the two parts described above: direct transmission through the slit and slit-plasmon-slit coupling.  We implemented a similar  model to study the interaction of light and plasmons for an array of holes; we were partially justified by the fact that others had done essentially the same thing to study different plasmonic systems.

Defining the “overall” or “global” state of coherence of a light beam also turned out to be somewhat of a challenge.  Spatial coherence is usually characterized between two points in a light wave, as for instance the coherence between the light emanating from the two slits in Young’s experiment.  Though it is clear that some light beams are more coherent than others (a laser produces light more coherent than a light bulb), quantifying this overall coherence has never been done satisfactorily.  In the end, we opted to simply look at the coherence of light emanating from multiple pairs of holes in the array; if the coherence of most or all pairs increased/decreased dramatically, it was a good sign that the overall coherence had been modified.

We studied 3 by 3 hole arrays, 4 by 4 hole arrays, and a 7-hole hexagonal array.  We looked at how the degree of coherence of light emanating from the holes, called , changed as a function of hole separation .  A sketch of the hexagonal array, and the coherence results for that hexagonal array, are shown below.

Hexagonal hole array with hole separation d.

The left figure above shows the degree of coherence — 0 for completely incoherent, 1 for fully coherent — between the holes A and C as a function of the array spacing , and compares it to what the degree of coherence would look like in the absence of plasmons (dashed line).  It can be seen that for certain hole separations, the spatial coherence increases dramatically.  In the right figure, we see the degree of coherence compared for holes A and B and for holes A and C as a function of array spacing.  It can be seen that, for certain values of , the coherence increases dramatically for both hole pairs; this is an indication that the transmitted beam as a whole has had its coherence increased!  For instance, at hole separation 1.1 μm, 1.6 μm and 2.2 μm, both coherence values spike sharply, almost to full coherence!

We learned a few things from the simple simulations done in this research project.  First and foremost, we found that we could in fact change the global state of coherence of a light beam by the use of a plasmonic hole array.  This puts me one step further to my goal of designing “coherence converting plasmonic devices”, which could be attached to the front of a light source to change its statistical properties for various applications.  We also noted that, unlike the two slit and three slit cases, there does not seem to be a hole separation that results in a global decrease in coherence.  For some hole separations, the coherence between particular pairs of holes can be lower than the illuminating light, but we found no unique values of for which all holes exhibited a decrease in coherence.  This may change, however, if we use more complicated hole arrangements.

Future research will involve the development of exact numerical simulations of the plasmonic systems in question, as well as investigations of more complicated hole arrays.

This work is also a good example of why it is important to diversify one’s research interests.  I had developed a strong independent backgrounds in the theories of optical coherence and surface plasmons; this made it very easy to combine my knowledge to study the coherence of surface plasmons, a topic that nobody else had investigated before!

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¹ C.H. Gan, G. Gbur and T.D. Visser, “Surface plasmons modulate the spatial coherence in Young’s interference experiment,” Phys. Rev. Lett. 98 (2007), 043908.

² N. Kuzmin, G.W. ‘t Hooft, E.R. Eliel, G. Gbur, H.F. Schouten and T.D. Visser, “Enhancement of spatial coherence by surface plasmons,” Opt. Lett. 32 (2007), 445.

³ C.H. Gan and G. Gbur, “Spatial coherence conversion with surface plasmons using a three-slit interferometer,” Plasmonics 3 (2008), 111.

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Gan, C., Gu, Y., Visser, T., & Gbur, G. (2012). Coherence Converting Plasmonic Hole Arrays Plasmonics DOI: 10.1007/s11468-011-9309-1

Unfortunately, the article is only available to subscribers; however, if you’re interested in reading about Arago’s exploits, you can look at a blog post I did on the subject a few years ago at this link.

The myth of Pandora came to my mind while pondering the novel Exogene, by T.C. McCarthy.

In this second book of the Subterrene War trilogy, one gets the ominous feeling that, like Pandora, humanity is being enticed by the allure of short-term gain into making decisions that will lead to doom.  As happened with the first book in the trilogy, Germline, I found Exogene pretty much impossible to put down.

The novel is set at an unspecified point in the future, when the United States has become locked in a brutal war against Russia in Kazakhstan for control of rare earth metals vital for technology.  To back up their regular forces, the U.S. has bred genetically-engineered super-soldiers — Germline units — in the form of nearly identical teenage girls.  Raised, trained and indoctrinated in vats, the girls are released into combat at the (effective) age of 15, and are terminated at age 18.  They are designed to break down both physically and mentally at that age, in a process known as the “spoiling”.  The termination, and the spoiling, are necessary because, as they mature, the Germline units develop thoughts counter to their programming.

In Exogene, we follow one of these solders — simply named Catherine — as she nears the end of her service.  She may very well be the most skilled killer ever produced by the program, but like others before her she begins to question her predetermined fate and the “religion” that the military has raised her into.  Soon, she has fled from the U.S. forces and, fighting the spoiling and anyone who comes across her path, seeks to stay alive at any cost.  Traveling away from the front lines, Catherine still must deal with the far-reaching consequences of such a massive conflict.

In Exogene, I got the feeling that I was descending further into the madness of war — no small feat, considering Germline was unrelentingly horrifying.  At this point, I’m getting a strong “Apocalypse Now” vibe from the two books of the trilogy I’ve read: each book is taking us further from sanity and deeper into monstrosity.  In Germline, we saw the war from the point of view of a reporter who becomes an active participant, and the Germline units were an integral part of the story but somewhat aloof.  In Exogene, we get to see the war from a Germline unit’s perspective, and Catherine in turn gives us our first close up encounters with the Russian forces and the even more extreme things they are willing to do for victory.

It is the events in the Russian camp that gave me the strange feeling about Pandora’s Box.  A super-solider is still essentially a human at heart, and the Russians have plans to do even more extreme genetic modifications to make an even more powerful and indestructible warrior.  It all seems coldly reasonable, much like Pandora’s decision to open the box, but also suggests that the nations are so focused on the short-term goal of winning the war that they fail to see how their efforts might lead to disaster.  To me, it sounds like the same cold reason that led to the creation of the atomic bomb.

One of the great strengths about T.C. McCarthy’s writing is that he can make insanity sound reasonable.  The cult-like religious indoctrination of the Germline units might come across as utterly ridiculous in the hands of a lesser author, but McCarthy makes it plausible, even inevitable.  One can almost picture a gathering of scientists and soldiers, brainstorming the best way to keep their new genetic tools under control.

For me, Exogene did everything I hoped it would do.  Though it is set in the same war as Germline, it is a completely different story that shows us entirely different actors and aspects of the conflict.  I read the book within a couple of days (including reading non-stop on the plane to Spain) and thought about it for many days afterward.  It both stands on its own and builds upon its predecessor.  Highly recommended.

At first glance, it seems like a very ordinary, if ornate, drinking cup:

There is an odd bump in the center of the cup, but otherwise, it seems quite normal, and if it is filled to a certain level can be used without incident.

However, if the cup is filled higher than the bump, the drink starts to drain out of the bottom — in fact, the cup will completely empty itself!

This is the trick of the Pythagorus cup, also known as the Pythagorean cup or the Tantalus cup!  It can be used as a nice prank to play on someone (don’t use red wine, unless you need to replace your carpeting anyway), but also serves as a nice demonstration of some physics of fluids.

The cup is known as a “Pythagoras cup” because its invention is credited to the great Greek mathematician and philosopher Pythagoras of Samos (570 B.C.E. — 495 B.C.E.), best known for the Pythagorean theorem, which states that:

For a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two lesser sides, i.e.

.

Tradition holds that Pythagoras was supervising workers at the water supply works on Samos and invented the “fair cup” (or “greedy cup”) to moderate the workers’ wine intake.  Provided they filled their cups only to a moderate level, they could drink in peace.  It is unknown if there is any truth to this, but it certainly makes for a fun story!  Ornate Pythagoras cups are sold as souvenirs on Samos, and can be purchased (relatively) cheap on eBay.

So how does it work?  Here is a see-through version of the cup, purchased from Acme Klein Bottles.

There is a tube (or other channel) that runs from an opening near the bottom of cup upwards to a certain height, and then down through the base of the glass.  Provided the water level does not go above the upper part of the loop, it cannot escape out through the base.

But if the cup is filled above the loop, it spills over on the inside and drains out of the bottom!

The surprising thing here, and what makes this a reasonable physics demonstration as well as a cute trick, is that the cup drains completely once the process is started.  The unlucky drinker has created a siphon in their glass, the same process by which one can use a hose to drain a fuel tank, or (ick) a septic tank.

It is a little more tricky to explain how a siphon works!  In short, the weight of the water falling through the lower portion of the tube at the base of the glass reduces the pressure at the upper portion of the tube, allowing water to be “pushed” into the tube by the weight of the water remaining in the glass.  This explanation has been surprisingly controversial, however, as it has been demonstrated that some siphons can work in a vacuum, suggesting that intermolecular forces can play a significant role in “dragging” the liquid through the tube.  As far as I know, however, the atmospheric pressure explanation sufficiently matches the observational evidence under most circumstances.

Do you want your own Pythagoras cup?  As I have noted, you can purchase decorative ones on eBay directly from a Greek seller.  You can purchase a clear one, showing clearly the inner workings of the device, from Acme Klein Bottles, which sells a lot of entertaining mathematical glassware (including, obviously, Klein bottles).  The packaging of the cup includes a lot of great mathematical humor and is almost worth the price of purchase all by itself!

• How Jewish leaders reacted to Darwin’s theory of evolution,
• An astronomer who prepared years to record an event but fainted as it was happening,
• A medieval cookbook that shares recipes on things as exotic as unicorns,
• Emile du Chatelet, a brilliant female mathematician, and much more!

Thanks to Romeo Vitelli for hosting an excellent carnival!

The next edition of The Giant’s Shoulders will be hosted by The Medical Heritage Library and entries are due by May 15.  Entries can be submitted directly to the host blog or throughblogcarnival.com, as usual.

Updated some descriptions thanks to commenter VA!

When we woke up on Monday morning, the massive crowds of Palm Sunday had to a large extent dispersed, though the city of Seville was still quite lively.  We had a lot to do, so we wasted little time in making our way to the Seville Cathedral to do some sightseeing.  There was already a lengthy queue waiting to tour the inside of the massive building.

Entrance to the Seville Cathedral.

It is hard to do justice to the size of the Seville Cathedral (Catedral de Santa María de la Sede).  It is the third-largest church in the world, with some 11,500 square feet of interior space, and the largest Gothic cathedral in the world.  Like much of the early architecture in southern Spain (Andalusia), its architectural style is a curious mix of Christian and Muslim tastes.

In fact, the Seville Cathedral, whose construction began in 1402, incorporates a number of elements from an earlier mosque built on the site, most notably the Giralda minaret that was converted into a bell tower.

The Giralda of the Seville Cathedral.

The cathedral construction was completed in 1506, but only five years later the central dome collapsed and needed to be rebuilt.  Another collapse occurred in 1888, and reconstruction continued until 1903.

One is welcomed into the cathedral with a bronze statue that represents “The Triumph of Faith”.

"The Triumph of Faith" at entrance to Seville Cathedral.

Once inside, it is difficult not to be somewhat awestruck at the massive size of the cathedral; the ceilings are some 13 stories high.  The legend has it that the planners of the cathedral said, “Let a church so beautiful and so great that those who see it built will think we were mad“.

Panorama of the interior of the Seville Cathedral.

Another view of the interior of the Seville Cathedral.

As could be expected from such an opulent and grandiose building, the Seville Cathedral contains many old works of art, some quite ghastly.  In a small entrance museum one is treated to a lifelike sculpture of the severed head of John the Baptist from 1625 (image behind link, since it is quite gory).

Pretty much everything in the cathedral is a piece of artwork, regardless of its role or overall importance.  For instance, the entrance to the “retro-choir” (space behind the high altar) is very old and quite compelling.

Entrance to the retro-choir (retroquire).

We had purchased audioguides to learn more about the cathedral’s history and the roles of the various chambers within, but I was so overwhelmed by everything that I hardly took note of the tour — I suppose this is what the architects would have hoped for!

There are many, many chapels spread throughout the cathedral, each a work of art in itself.  The one pictured below is, I believe, the chapel depicting the Virgin Mary.  (I’m looking at a map of the cathedral right now, trying to retrace my steps and determine what I was likely to be passing as I toured!)

Chapel of the Virgin Mary, in the Seville Cathedral.

There are also multiple altars within the Seville Cathedral; it appeared that the altar of the main chapel was under renovation, so much attention was being paid to the still magnificent “Altare dell’Argento” (Silver Altar).

The Altare dell' Argento, in the Seville Cathedral.

Many of the church’s sights appear far above ground level; the stained glass window in the following picture was very near the ceiling, probably 10 floors high.

Stained glass rose in the Seville Cathedral.

Everything is huge in the cathedral, and designed to impress; check out, for instance, the massive facade of the cathedral organ.

Organ of the Seville Cathedral.

Even in all this splendor, perhaps the most interesting part of the cathedral is a non-religious one: the burial place of Christopher Columbus!

Tomb of Christopher Colombus.

The sculpture of the tomb matches well with the cathedral; at first glance, one doesn’t realize that the remains were brought to Seville only in 1898, and brought with them some controversy.  Columbus died in Valladolid, Spain, in 1506, and his remains were originally interred there.  At the request of his son, however, they were transferred in 1542 to the Dominican Republic, and later transferred again to Cuba in 1795.  They were finally brought to Seville in 1898, and the tomb was constructed for him.  Even with such a large amount of travel it was assumed that the correct body was in fact buried in Seville, but another coffin had been discovered in the Dominican Republic in 1877 with Columbus’ name upon it.  In 2003, DNA tests confirmed that the remains in Seville are likely Columbus’, but it is possible that part of his remains are buried in the other location, as well!

From Columbus’ tomb, we moved on to the main sacristy of the cathedral, where sacred artifacts are stored.

The main sacristy of the Seville Cathedral.

From the sacristy, we moved on to the “ante-chapter house”, the antechamber to the cathedral’s chapter house, where meetings would traditionally be held.

The ante-chapterhouse.

The walls of the ante-chapter house were lined with images of Biblical scenes; I’ll just post one for folks to try and identify!

Biblical scene in the ante-chapter house.

The chapter house itself is a magnificent chamber; my panorama can only capture a portion of its elegance.

The chapter house.

If you aren’t convinced yet of the opulence of the Seville Cathedral, you should know that the next room we visited is called the treasure room!  It contains many priceless artifacts, and a number of quite bizarre ones; the next photo I took is of the cathedral’s collections of reliquaries, containing the supposed remains of saints.

Reliquaries of the Seville Cathedral. Note the bone in the left-most container.

Our next stop was the Giralda tower.  At 343 feet, it is a magnificent place to see the city from above; as a former minaret, one ascends it by ramp — the imam would ride on horseback to the peak to sound the call for prayer.  One would think that a ramp would make the climb easier, but the last 50 feet is still a pretty painful journey!  Nevertheless, the view of the city from the top is well worth it, as the next two panoramas hopefully show.

Panorama of Seville, from the Giralda.

Southern-facing panorama of Seville, from the Giralda.

In the second panorama, one can see in the distance a building that at first glance I thought was another cathedral.  In fact, I was looking at the Plaza de España, which we would visit later that day (and talk about in the next post).

The Plaza de Espana, from the Giralda.

At the top of the Giralda, Beth and I ran into the rest of our group (we had lost track of them soon after entering the cathedral).  We agreed to meet at the orange tree courtyard when we were finished wandering.

The orange tree courtyard of the Seville Cathedral.

This courtyard is the other significant structure remaining from the mosque that had previously been on the site.  It was a lovely place to relax after an extremely exhausting tour of the cathedral.

But our tour of Seville had hardly begun at this point!  On that same day we would also visit the Alcazar Palace, the Plaza de España, and take in a flamenco show.  But I’ll leave a discussion of those activities for the next post.

If my calculations are correct, £38.50 = $62, so it’s quite inexpensive right now — grab a copy while you can!

I should also note that Cambridge has put a large number of books in their optics catalog at 30% off until the end of the month.  Some great titles looks to be available for really cheap (including mine).