## Rolling out the (optical) carpet: the Talbot effect

One of the wonderful things about having a career in science is that a deeper understanding of the science leads to a greater appreciation of its beauty.  In physics, this usually requires a nontrivial amount of mathematics, but there are some phenomena that are self-evidently beautiful; unfortunately, many of these are also not very well known!

In working on my textbook on optics, I delved rather deeply into one of these phenomena, known as the optical Talbot effect.  First observed in 1836 by Henry Fox Talbot, the effect went unnoticed for nearly fifty years before being rediscovered by the great Lord Rayleigh in 1881. The true subtlety of the phenomenon was still not understood, however, for another hundred years!

In short, the Talbot effect can be described as the self-imaging of a diffraction grating: at regular distances from the grating, the light diffracted through it forms a nearly perfect image of the grating itself. This simple statement does not do justice to the Talbot effect, however, which results in stunning images such as:

This is an example of what is known as a Talbot carpet,  presumably because it is reminiscent of an ornate Persian rug:

(Why isn’t it called a “Talbot rug”?  That I can’t answer.)

There’s a lot to explain in order to understand the significance of the Talbot carpet, starting with an explanation of what exactly a diffraction grating is!

A diffraction grating is a device that uses the wave properties of light to direct different wavelengths of light in different directions; the principle is illustrated below:

White light is split into its constituent colors on transmission through the grating; these constituent colors  travel in different directions.  This effect is very useful in optics applications, and can be used to make a monochromator of light and a spectrometer of light, as illustrated below:

A monochromator is used to select only a single color of light from a white light beam, while a spectrometer is used to separately measure the brightness of individual colors of light.

In general, a diffraction grating is a thin translucent material etched with a periodic pattern.  The simplest diffraction grating, however, is a thin opaque screen which has slits cut in it at regular intervals:

The slits will be taken to be separated by a distance $d$.  As the name implies, a diffraction grating operates on the optical principle of diffraction, which may be loosely defined as the tendency of a light wave to “spill out” in all directions after passing through a narrow aperture.  An old simulation I did that shows this effect is shown below; light that is incident vertically from below on an aperture in a opaque (blue) screen spreads in all directions:

The natural question to ask: how does diffraction through a large number of holes result in different colors of light traveling in different directions?

We can actually explain the effect by considering only two holes, using essentially the same calculation that was used in explaining Young’s double slit experiment.  Let us assume that a single-frequency (monochromatic) wave is illuminating a pair of holes separated by a distance $d$.  We consider the behavior of the light emanated from these holes at a very large distance away:

If the distance is large enough, the light passing from the two holes to the observation point travel in nearly parallel directions.  Let us “zoom in” on the immediate neighborhood of the holes:

Using a little trigonometry (we have to use a little math!), we find that the light from hole 2 has to travel a distance $d\sin\theta$ farther than light from hole 1.  If that distance is equal to a wavelength of light, then the light from the two holes remains in phase and it constructively interferes at the observation point:

If this condition holds for one pair of holes, it holds for all pairs; we may write the condition for constructive interference of light emanating from the grating as:

$d \sin \theta = n \lambda$,      (1)

where $\lambda$ is the wavelength of light and $n=1,2,3,\ldots$.  If this condition does not hold, then light emanating from more distant holes is progressively more out of phase, and the combination is negated; the result is that light of a given wavelength tends to appear only at observation angles given by the above equation.  Because the angle depends on the wavelength, different colors travel in different directions, producing the observed color spectrum.

The holes on a diffraction grating are separated by distances comparable to, though typically larger than, a wavelength.  Compact discs work as a crude grating because their data pits are separated by slightly more than a wavelength, producing nifty color patterns (picture from hyperphysics):

So this is how light behaves when it travels sufficiently far from a grating after diffraction.  In 1836, however, Henry Fox Talbot decided to study1 the behavior of light in the region immediately behind the grating, right after the light has been diffracted.  What he saw, in his own words:

I then viewed the light which had passed through this grating with a lens of considerable magnifying power.  The appearance was very curious, being a regular alternation of numerous lines or bands of red and green colour, having their direction parallel to the lines of the grating.  On removing the lens a little further from the grating, the bands gradually changed their colours, and became alternately blue and yellow.  When the lens was a little more removed, the bands again became red and green.  And this change continued to take place for an indefinite number of times, as the distance between the lens and grating increased. … It was very curious to observe that though the grating was greatly out of the focus of the lens, yet the appearance of the bands was perfectly distinct and well defined.

What Talbot had observed was the self-imaging of the diffraction grating, now known as the Talbot effect.  For a single wavelength of light, the Talbot effect manifests itself as illustrated in the following diagram:

At regular distances $z_T$, the light emanating from the grating forms perfect images of the grating itself, to be referred to as primary Talbot images.  Halfway between these images, the light forms perfect images which are shifted half a period from the original grating, to be referred to as secondary Talbot images (my terminology).  The Talbot distance $z_T$ depends on the wavelength of light, meaning that the Talbot images of different colors appear at different distances; this is the origin of the “alternation of… bands of… colour” that Talbot observed.

It is exceedingly tricky to try and explain the Talbot effect without introducing any mathematics, though I’ll do my best!  (If this gets too tedious to read, skip ahead to the pretty Talbot effect pictures.)  Let us forget about the grating for a moment and discuss plane waves.  A monochromatic plane wave is a wave of a single frequency whose wavefronts (surfaces of constant phase) form planes:

As this illustration (hopefully) demonstrates, a plane wave has parallel, equally-spaced planar surfaces on each of which the wave takes on its peak value; these surfaces we call the wavefronts.  If we just consider waves propagating in two-dimensions, the wavefronts are lines and the picture would look something like this:

Let us now suppose for the moment that a plane wave is propagating away from the plane of the grating, $z=0$, at an angle $\theta$.  The plane wave has an overall wavelength of $\lambda$, but if we travel in the x-direction or the z-direction, we see that the plane wave repeats itself over distances $\lambda_x$ and $\lambda_z$, respectively:

Using some trigonometry, we can show that the three quantities are related by the equation,

$\displaystyle \frac{1}{\lambda^2}=\frac{1}{\lambda_x^2}+\frac{1}{\lambda_z^2}$.

Now let us bring the grating back into the picture.  Because the grating has a period $d$ in the x-direction, it follows that the light emanating from the grating also must have a period $d$ in the x-direction.  This means that the only plane waves that emanate from the grating must satisfy $n\lambda_x=d$, where $n$ is an integer!  Several of these possibilities are illustrated below:

The picture here shows exaggerated angles, for clarity; typically, $d$ is appreciably larger than $\lambda$, which makes the angles of the contributing plane waves very small.  What we are describing here is, in essence, a mathematical concept known as a Fourier series, which states that a periodic signal — in our case, the wave in the grating plane — can be described entirely in terms of periodic functions of increasingly smaller wavelength that repeat over the distance $d$.

So the only plane waves which propagate away from a diffraction grating have “horizontal wavelengths” of $\lambda_x = d/n$; performing a little algebra, however, shows that this implies that the “vertical wavelengths” of these waves must satisfy

$\displaystyle \lambda_z = \frac{\lambda}{\sqrt{1-\lambda^2n^2/d^2}}$.              (2)

Because the vertical wavelengths can only take on special values, one can show that there are vertical distances at which all plane waves return to their original phase relationship.  This distance is the Talbot distance, and it is given by the formula,

$\displaystyle z_T = \frac{2d^2}{\lambda}$.                    (3)

Awright, I’m going to do a little more math to derive the Talbot distance, and then I’m done!  If we follow a plane wave traveling in the z-direction, its phase is given by $k_z z$, where $z$ is the propagation distance and $k_z$ is the wavenumber of light in the z-direction, defined as

$\displaystyle k_z = \frac{2\pi}{\lambda_z}$.                 (4)

From this formula, we see that the phase of a plane wave changes by $2\pi$ as it travels a distance $z = \lambda_z$.  We may write

$\displaystyle k_z z = \frac{2\pi z}{\lambda} \sqrt{1-\lambda^2n^2/d^2}$.         (5)

This is the phase of the $n$th order plane wave emanating from the grating.  Now we need to use an important theorem from calculus, the binomial theorem, which states that, for small $x$, we may write

$\displaystyle \sqrt{1-x^2}\approx 1-\frac{x^2}{2}$.   (6)

With this result, assuming that $\lambda^2n^2/d^2$ is a small quantity, we may write

$\displaystyle k_z z = \frac{2\pi z}{\lambda}z - \frac{2\pi \lambda n^2z }{2d^2}$.   (7)

The first term of this equation is independent of the order $n$ entirely!  If we consider the case that

$\displaystyle \frac{\lambda z }{2d^2}= 1$,   (8)

then our formula takes the form:

$\displaystyle k_z z = \frac{2\pi z}{\lambda}z - 2\pi n^2$.          (9)

Regardless of the value of $n$, the second term is a multiple of $2\pi$!  Every plane wave emanating from the grating will have the same phase at distances $z$ which satisfy equation (8)!  If we solve for $z$, we find that

$\displaystyle z = \frac{2d^2}{\lambda}$,    (10)

which is the Talbot distance!

The first calculation of the Talbot distance was performed by Lord Rayleigh in an 1881 paper entitled, “On copying diffraction-gratings, and on some phenomena connected therewith.”2 He seems to have rediscovered Talbot’s long-forgotten work, and he notes:

In the course of last summer, however, I found accidentally that Fox Talbot had made, many years ago, some kindred observations; and the perusal of his account of them induced me to alter somewhat my proposed line of attack.

Rayleigh’s “line of attack” was in developing an optical method for creating new diffraction gratings.  By placing a photographic material in the position of one of the Talbot planes, one can record a near-perfect image of the original grating, which can in turn be used as a new grating.

A near-complete theory of the Talbot effect took another 80 years to be developed, by Cowley and Moodie3 in the 1950s and Winthrop and Worthington4 in the 1960’s. They were evidently the first to demonstrate that there is in fact an infinite family of Talbot images between the primary and secondary images. We may now return to the tantalizing image we presented at the beginning of this post, with some additional context:

This figure shows the intensity of light after being diffracted by a grating.  Light is incident from the left, passes through the grating, and produces an intricate interference pattern.  The horizontal axis shows one Talbot distance; it can be seen that the light wave converges to form an image of the light field in the grating on the right of the picture.  Halfway through the picture, we can see a secondary Talbot image, shifted vertically from the primary image by half a period.  Halfway between the primary and secondary image, however, we can see another image of the grating, though one that is of half the size or, equivalently, double the frequency.  At 1/3 distances between primary and secondary images, we can see a triple-frequency image.  These shrunken images are referred to as fractional Talbot images.

The Talbot carpet has a very self-similar structure, which immediately brings to mind images of fractals.  It comes as no surprise, then, that in 1996 — now 160 years after Talbot’s original observation — Berry and Klein demonstrated5 that the Talbot carpet possesses a fractal structure.

It is this very intricate fractal structure that makes Talbot carpets such a visual treat; last week, I wrote myself a little piece of software to generate my own Talbot carpets of differing styles.  Without further ado, I present: my gallery of Talbot carpets!  (All colors were chosen for artistic purposes and do not represent the true wavelength of light.)

This one I call “ice palace.”  It was drawn by assuming that the aperture had a two-tiered transmission function of the form:

I call this one “Audrey II”; see if you can guess why!  I also used a two-tiered transmission function to draw this one, except the central part of the aperture was taken to be out of phase:

I call this one “Spring Garden”.  The unusual aspect of this image is that the apertures were taken to have asymmetric transmission functions, resulting in a Talbot image which looks slightly different if turned upside down.  The aperture function looks like:

I call this one “Lava”.  It was also designed with an asymmetric transmission function for the aperture.

I call this masterpiece “fractal chessboard”.  The difference here is that the grating was taken to be a pure phase grating, i.e. no light is blocked by the grating, only delayed.  Nevertheless, this phase delay results in a modulation of the intensity of light beyond the grating.  The behavior of the images is quite different, however, and they are now referred to as Lohmann images, after the researcher who introduced them6,7.

Effects like the Talbot effect are not limited to optics.  It has been demonstrated8 that similar effects occur in quantum-mechanical systems, in particular the famous “particle in a box” problem in quantum mechanics, in which a single quantum particle is trapped in a “well” of infinite sides.  As time passes, the wavefunction of a particle in a box can undergo a so-called “quantum revival” in which it somewhat miraculously returns to its original state.

It can be shown that the particle in a box satisfies almost exactly the same mathematical equation as light does in the solution of the Talbot effect, with the exception that the Talbot effect arises when light evolves through space, while quantum revivals of a wavefunction arise from the evolution of the wavefunction in time.

Research on the Talbot effect may lead to more useful applications.  In a generalization of the idea of using the effect to write diffraction gratings, it has been suggested that one could use the effect to photographically construct three-dimensional periodic structures, that could be used as photonic crystals.  The Talbot effect is periodic in three-dimensions, and in principle one could etch a three-dimensional pattern into a block of photoresist by placing the photoresist into the Talbot region.  There are some subtle and not-so-subtle complications in such a strategy; however, some authors have started to tailor Talbot gratings with specific optical patterns with an eye towards such applications9.

It is very striking that an effect that started as an almost off-hand observation 170 years ago could result in such a wealth of subtle optics and still be the focus of research today.  It is unclear how much more there is to uncover about the Talbot effect; in any case, it deserves more attention at the very least as an example of the beauty of physics.

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1 H.F. Talbot, “Facts relating to optical science. No. IV,” Phil. Mag. 9 (1836), 401-407.

2 Lord Rayleigh, “On copying diffraction-gratings, and on some phenomena connected therewith,” Phil. Mag. 11 (1881), 196-205.

3 J.M. Cowley and A.F. Moodie, “Fourier Images: I — The point source,” Proc. Phys. Soc. 70 (1957), 486-496.

4 J.T. Winthrop and Worthington, “Theory of Fresnel images. I. Plane periodic objects in monochromatic light,” J. Opt. Soc. Am. 55 (1965).

5 M.V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43 (1996), 2139-2164.

6 A.W. Lohmann, “An array illuminator based on the Talbot-effect,” Optik 79 (1988), 41-45.

7 T.J. Suleski, “Generation of Lohmann images from binary-phase Talbot array illuminators,” Appl. Opt. 36 (1997), 4686-4691.

8 M. Berry, I. Marzoli and W. Schleich, “Quantum carpets, carpets of light,” Physics World, June 2001, p. 39-44. This is a relatively math-free introduction to the relationship between the Talbot effect and quantum revivals.

9 M. Testorf, T.J. Suleski and Y-C. Chuang, “Design of Talbot array illuminators for three-dimensional intensity distributions,” Opt. Exp. 14 (2006), 7623-7629.

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H.F. Talbot (1836). Facts relating to optical science. No. IV Philosophical Magazine, 9, 401-407

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### 29 Responses to Rolling out the (optical) carpet: the Talbot effect

1. ColonelFazackerley says:

Great post, with some very pretty images. I guess the name “Audrey II” is due to the luscious lips!

• Thanks! Regarding the name, you’re right — somehow I look at that image and immediately see this instead.

2. stuwat says:

I see a small business opportunity here. How soon before I can place an order for a fractal chessboard carpet in blue and white?

Thanks for the detailed explanation of this phenomenon.

• I pity the people who actually have to weave the fractal carpet…

3. T says:

nice article! see also the optical realization of these beautiful carpets! cheers

http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-23-20966

4. Excelent post. Quite clear the explanation. I’d like comment that the Talbot effect is also used as a measurement tool in optical metrology.

5. Tanmayee Pathre says:

Nice article. Informative. In one of the sentences its written that self imaging is observed at regular distances from the grating. My question is do we need a screen to observe this effect?

• Thanks for the comment! It would probably be difficult to observe the self imaging on a screen, as the grating that produces it has a spacing on the order of a wavelength. I believe Talbot himself used a microscope to look at the image, “blowing it up” so that he could see it.

6. Tanmayee Pathre says:

Thanks for information. I have one more question if the effect is observed immediately after grating, where is it exactly observed?

7. Tanmayee Pathre says:

Nice article. I have a question. How far distance of the source from grating is important to observe talbot effect?

• Hi — as Eq. (10) shows above, the “Talbot distance” — the distance at which the image of the grating appears — is given by 2d^2/lambda, where d is the period of the grating and lambda is the wavelength of light. If we were to choose a wavelength in the visible region of light — say 0.5 micrometers — and a grating with a period of 0.5 micrometers, then the Talbot distance would be one micrometer, a very small distance! The use of a larger grating period will make the Talbot distance larger.

8. Mike Kim says:

i have a question that… can i get s program that you said in your writting, make talbot carpet images??

• Well, I wrote the code in Mathematica, I believe, but had to do the integrals for the different aperture shapes. I didn’t make a program that allows one to easily transition between different aperture types.

9. Holger says:

Hello,
could you comment on the numerical realization of the calculations?
I would like to create the Talbot “carpet” for a limited number of grating slits (say 10 or 50) and see the edge effects. Do you need to solve the Kirchhoff integrals or is there a simpler way to create the images?
Thanks,

10. Pingback: Open Lab Update | Child's Play

11. Zepei Li says:

Nice article. Sir, Your Talbot carpets is most beautiful! I’am a student and I also study talbot effect. Can I get one of your Mathematica program, learn to talbot carpet images?
Thanks!