Skulls in the Stars

As I’ve so far been restricting my ‘optics basics’ posts to discussions of fundamental concepts related to optics, it might seem strange at first glance to dedicate a post to a single optical experiment.  What will hopefully become clear, however, is that Young’s double slit experiment is connected to so many basic concepts in optical physics  (and still provides surprising new results to this day) that one post is hardly enough to describe all the interesting insights that can be gained by studying the experiment and its implications.

The experiment is named for its inventor, Thomas Young (1773-1829), pictured below:

Young demonstrated his genius early in life, having learned to read quite well by age 2, and by age thirteen he was well acquainted with Latin, Greek, French and Italian.  Later in life he would learn numerous other languages, and would have a hand in attempts to translate the Rosetta Stone (though I have read it argued that he actually left the translation problem worse than when he started).

By the early 1800s, scientists were still convinced of the so-called corpuscular theory of light.  Isaac Newton had, nearly 100 years earlier, offered seemingly solid evidence that the manifold observations of light were best interpreted by treating light as a stream of particles.  In 1802, however, Young presented a paper¹ in which laid the theoretical foundations for a view of light as a wave, and in 1804, in a paper entitled, “Experiments and Calculations Relative to Physical Optics,” he presented the results² of his famous double-slit experiment, which strongly illustrated the wave nature of light.

So what is the double-slit experiment, and how does it illustrate the wave properties of light?  A simple schematic of the experiment is shown below:

A light source is positioned some distance behind a screen A which possesses a pair of closely spaced narrow slits.  Light emerges from the pinholes and spreads as spherical waves; the waves overlap, and the interference pattern produced is projected on a measurement screen B.   For future reference, we label refer to the position on the measurement screen as ‘x‘.

Young’s own illustration of this wave interference effect is shown below:

From T. Young, “Course of Lectures on Natural Philosophy and the Mechanical Arts,” 1807.

A simulation of the pattern of light which would be observed on the screen B is shown below:

One should see bright lines of light, where the light from the two slits constructively interferes, alternating with dark lines where the light destructively interferes.

We can understand the general principle of Young’s experiment by a basic geometrical argument.  Let us assume that our source produces a monochromatic (single-frequency) wave which hits the two slits at the same time, i.e. so that the peaks of the wave arrive at the the two slits simultaneously.  The two waves are then said to be in phase with one another.  As a reminder, a snapshot of monochromatic wave traveling along a string appears as below:

We now consider what happens when light emerges from the pinholes and travels to the screen B:

Looking at light which travels to point 1 on the measurement screen, light coming from pinhole Q has to travel farther than light coming from pinhole P; therefore light from Q has a phase lag associated with it relative to the light at point P.  Light traveling to point 3 results in the opposite case: light from P has a phase lag associated with it relative to the light at point Q.  The distance between each pinhole and point 2 is the same, so the light arrives at that point with the same phase.

We can quantify this as follows: provided the distance L between the screens is large, the paths of light from pinholes P and Q to a point on the measurement screen are nearly parallel.  The difference in phase can then be calculated using the following figure:

Assuming the two paths meet at the observation screen, light emerging from pinhole Q must travel a distance $$d sin theta$$ farther than light from pinhole P.  Assuming monochromatic light of wavenumber $$k$$, this means that the light from the two pinholes arrives at the screen with a relative phase difference:

$$phi = kdsintheta$$.

Two waves are out of phase, and produce destructive interference, when this phase is an odd multiple of $$ pi$$, i.e.

$$ kdsintheta = (2n+1)pi, quad n = 0,1,2,ldots$$,

and produce constructive interference when the phase is an even multiple of $$pi$$, i.e.

$$ kdsintheta = 2n pi, quad n = 0,1,2,ldots$$.

A little more geometry demonstrates that $$sintheta approx x/L$$, so that bright fringes occur at positions

$$ x = 2n pi L/kd$$.

The distance between two successive bright fringes on the screen is given by:

$$Delta x = 2pi L/kd$$.

It can help to put some numbers into this equation and see what it implies about the size of fringes.  Let us assume we have green light, $$lambda = 550 times 10^{-9}$$ meters, and an observation screen $$L = 1$$ meter from the slit screen.   This gives $$k = 1.1 times 10^{7}$$ 1/meters.  If we assume a slit separation of 1 mm, this results in a fringe separation of

$$Delta x = 0.5 $$ mm!

If we decrease the separation of the slits to 0.5 mm, we increase the fringe separation to 1 mm.  It is clear, though, that the slits must be very closely spaced together in order to observe sizable fringes; there is another coherence-related reason to have closely-spaced slits which we mention in a moment.

It is worth noting that a “double slit” experiment does not necessarily involve using slits at all!  The principle of the experiment is the division of a wave in space into two distinct parts which are then brought together to interfere.  Young himself performed his initial experiments by placing a piece of thin card into a beam of light and examining the shadows produced on a distant wall.  The card was approximately 1/30th of an inch in width.  Young also did not have the luxury of using monochromatic light: his interference experiments used sunlight, and the fringes produced were not light and dark bands but fringes of alternating color.

Young’s experiment could be considered the “tipping point” for acceptance of the wave theory of light.  However, it has proven useful in understanding a number of other phenomena, as well, as we will see.

One of the earliest extensions of the “double slit” idea was introduced by Augustin-Jean Fresnel and François Arago, who in 1819 published the results³ of experiments to determine the effect of polarization on the interference of light.

Their article (which I will discuss in more detail in a “classics” post in the near future) was clearly inspired by Young’s work, as they begin with a review of the relevant results of the double slit experiment! They then note that Young had only dealt with unpolarized light or light interfering with the same state of polarization, and they wondered what the result would be if a similar interference experiment was done where the light coming from the two pinholes was orthogonally polarized.

To investigate, they separated a beam of unpolarized light into two orthogonally polarized ones by the use of a rhomboid of Iceland spar (calcite) or an achromatic rock crystal prism.  The material is birefringent, which means that different polarizations of light travel at different speeds through the crystal, and are refracted differently.  The output of the crystal is two parallel beams.  A metal cylinder was placed in between these two beams, partly obstructing each, in the same manner as Young’s cardstock was used to divide sunlight.  Part of each beam gets diffracted into the shadow region behind the cylinder, and the wavefield in that region was examined for the presence of fringes:

The result of this initial experiment demonstrated that light beams that are polarized in perpendicular directions do not interfere.  Further observations about the relationship between interference and polarization are now referred to as the Fresnel-Arago laws.  This experiment was one of the major demonstrations that light must be a transverse wave, i.e. that the vibrations of light waves are perpendicular to the direction of wave propagation.

Returning momentarily to Young’s original experiment, it is to be noted that Young could only see fringes when he performed the experiment with sunlight passing through a very narrow hole:

I made a small hole in a window-shutter, and covered it with a piece of thick paper, which I perforated with a fine needle.

This point-like source of  light is necessary because sunlight has extremely low spatial coherence, and interference fringes can only be observed for light of high spatial coherence.  Turning this idea on its head, it turns out that Young’s double slit experiment can be used to directly measure the spatial coherence of an illuminating light field.

I have discussed coherence in another basics post, but we’ll review the relevant concepts in order to discuss their relationship to Young’s experiment.  Interference effects such as those seen in Young’s experiment require the interfering waves to have a definite phase relationship to one another — that is, if they are in phase at one point in time, they should be in phase for almost all times.  Realistic waves, however, are never perfectly monochromatic.  Ordinary thermal sources of radiation, such as a lightbulb or the Sun, consist of a large number of atoms radiating light almost independently of one another.  The field emanating from such a source has random fluctuations of amplitude and phase.  The net result of all these atoms radiating independently is that the wavefronts emanating from a light source are constantly changing their shape — and the more the shape fluctuates in time, the less spatially coherent the wavefield is:

In the left illustration above, the fields at points $$P_1$$ and $$P_2$$ are always oscillating in synch: there is a definite phase relationship between the fields at the two points.  In the right illustration, the fields at the two points may or may not be in synch – the fields are partially coherent.

One can show (though this involves mathematics outside the scope of a basics post) that the visibility of the interference fringes in Young’s experiment is a direct measure of the spatial coherence of the field at the two slits.  The visibility $$mathcal{V}$$ of interference fringes is defined as

$$displaystyle mathcal{V} = frac{I_{max}-I_{min}}{I_{max}+I_{min}}$$,

where $$I_{max}$$ and $$I_{min}$$ are the brightest and dimmest intensity the wavefield achieves in the neighborhood of a single bright interference fringe.  If there are no fringes, i.e. no spatial coherence, the light intensity is uniform, $$I_{max}=I_{min}$$, and $$mathcal{V}=0$$.  If there is complete destructive interference, $$I_{min}=0$$ and $$mathcal{V}=1$$.  The visibility of the interference fringes are a direct measure of the spatial coherence of the wavefield: when $$mathcal{V}=0$$, the field is incoherent at the two slits, and when $$mathcal{V}=1$$, the field is fully coherent.  In essence, Young’s experiment is the archetypical technique for measuring the spatial coherence of a light wave.⁴

We have seen how Young’s experiment was originally used to demonstrate the wave properties of light.  In an ironic twist, it is now used quite regularly to show not only the wave-like properties of matter, but also the particle-like properties of waves!

Historically, the latter of these concepts was the first to appear.  With this in mind, a little historical background is in order.  By 1900, although Newton’s corpuscular theory of light had been discarded in favor of a wave theory of light, “cracks” in the wave theory began to appear.  Perhaps the most significant of these was the photoelectric effect, in which light illuminating a metal surface ejects electrons from that surface.  Wave theories of light could not seemingly account for the observed properties of the photoelectric effect, and in 1905 Einstein suggested that the experiment could be readily and easily explained if one imagined that light in fact consists of a stream of particles.  These particles, later named photons, of course also possessed wave-like properties, and this was the first illustration of the idea of wave-particle duality, i.e. that all matter and energy must possess both wave-like and particle-like properties.

Einstein’s interpretation raised interesting questions concerning Young’s interference experiment.  If light consists of particles, what, exactly, is producing the interference pattern?  If one photon at a time is sent through Young’s interferometer, what will be the result?

The first attempt to answer this second question was undertaken by G.I. Taylor in 1909.  Taylor performed the experiment with increasingly dim beams of light, the attenuation being undertaken with a collection of smoked glass screens.  Taylor’s “double slit”, incidentally, was produced by looking at the shadow cast by a needle.  Taylor recorded his interference patterns with photographic film, and was careful to make certain that the total amount of light falling on the film was the same in each case.  For the weakest beam of light, this involved running the experiment continuously for an astounding three months!  In such a case, typically one photon at a time would have been passing through the double slits.  Taylor found, however, that the interference pattern looked essentially the same in each of his images.  Evidently each photon “carries” information about the interference pattern with it, though the exact nature of this relationship would require further discoveries, both experimental and theoretical.

In 1924, Louis de Broglie successfully defended his doctoral thesis, in which he proposed that, in analogy with light, matter must also possess a dual wave/particle behavior.  Only three years later, in 1927, Clinton Davisson and Lester Germer proved⁶, essentially by accident, de Broglie’s hypothesis.  Davisson and Germer were studying the behavior of electrons when scattered from a crystalline nickel surface.  To their surprise, they found that the electrons were preferentially being scattered in certain special direcctions, a result that was already known from X-ray experiments as Bragg diffraction, a wave phenomena. Davisson won the Nobel prize for his work in 1937.

With the wave properties of electrons demonstrated, it was natural to expect that Young’s experiment could be done with electrons, as well.  This was finally done in 1964 by Claus Jönsson⁷, in an experiment which is now referred to as one of the most beautiful physics experiments of all time⁸.  Jönsson etched slits into a thin silver film; the slits were of a thickness on the order of 0.3 μm (μm = 1 millionth of a meter) and were separated by roughly 1 μm.  The experiment was done with 1-5 slits; for two slits, he produced the following photograph:

Jönsson, Am. J. Phys. 42 (1974), 4- 11; Figure 8.

This can be seen to be directly analogous to the interference pattern for light shown previously.

By the 1930s, the basic foundations of quantum mechanical theory had been laid.  One of the more perplexing aspects of the theory, however, was the measurement problem.  In short, the “wave” of a quantum-mechanical particle such as an electron or photon should be interpreted as relating to the probability of finding the particle at that point in space.  The wavefunction of the particle therefore follows all possible paths of the particle simultaneously, including going through both slits in the double slit experiment.  When the position of the particle is measured, however, it can only be detected at one place: the wavefunction “collapses” on measurement and the particle is detected as a point particle.  The likelihood of finding the particle at a particular point can only be described probabilistically.

(If this doesn’t really make sense to you, you’re not alone; the measurement problem is one of the still-unresolved questions of quantum mechanics.  We can say, however, that even though it doesn’t completely make sense, it works perfectly well in interpreting experiments.)

What does this mean?  If we were to perform an experiment with a single electron/photon, and I really mean just one electron/photon, we would not see any wave behavior at all.  It is fair to say that one cannot detect any wave properties of a single particle in isolation.  When we repeat the experiment many times, we will notice that the particles tend to appear more often at some places (regions of constructive interference) and don’t appear very often at other places (regions of destructive interference).  If we can fire a single electron/photon at a time through a double slit experiment, and keep track of where we detect them, we will see the classical wave interference pattern ‘build up’ over time.

In 1976, three Italian researchers Merli Missiroli and Pozzi performed this exact experiment⁹, and recorded the creation of an electron interference pattern on a television monitor.  The results of their experiment are shown below:

This picture was actually taken by looking at the image on a television monitor for differing intensities of electron beams, but the researchers also developed a similar picture by looking at the cumulative effect over time of a large number of electron detections.  In essence, individual electrons appear as points on the detector, but over time one sees that they distribute themselves according to the classical double slit interference pattern.

This experiment was repeated in 1989 by a Japanese research group¹⁰, with more advanced equipment and even more spectacular results:

Perhaps even more fascinating is the less-often reported comparable experiment with photons.  Just like electrons, photons can only be detected at a single point, and an interference pattern with light is built up by a large collection of discrete photon detections.  A lovely example of this experiment is depicted in the paper by Dimitrova and Weis from 2008¹¹.

Young’s experiment continues to produce interesting physical insights¹².  We have stated that the interference pattern produced by Young’s experiment can be used to deduce the state of coherence of the light field at the two slits.  In recent years, researchers have looked at the spatial coherence of the light in the interference pattern and found that, surprisingly, there exist regions where the light is fully coherent¹³ and fully incoherent¹⁴, regardless of the state of coherence of the light at the two slits.

Young’s experiment has been used to provide insights into nano-optical research, as well.  I have previously discussed the relationship between “extraordinary optical transmission” and electric charge surface waves on metal plates known as surface plasmons.  Early experiments in the field, and their interpretations, were somewhat contradictory, and this led to a heated debate about the exact role of surface plasmons in extraordinary transmission.  The confusion was cleared up somewhat when, instead of studying transmission through a large array of holes, researchers studied the transmission through only a pair of holes, essentially Young’s experiment with surface plasmons¹⁵. More recently, it has been shown that the presence of plasmons can affect the coherence properties of light in Young’s experiment as well¹⁶.

Thomas Young’s experiment has proven persistently useful because it boils down interference phenomena into their most fundamental form: the interference of two spatially separated waves.  I would not be surprised to see yet more new physical insights arise from Young’s simple but powerful idea.  (And I’m sure I’m forgetting at least a few interesting results which can be derived from it!)

1. T. Young, “On the Theory of Light and Colours,” Proc. Roy. Soc. Lond. A 92 (1802), 12-48.

2.  T. Young, “Experiments and Calculations Relative to Physical Optics,” Proc. Roy. Soc. Long. A 94 (1804), 1-16.

3.  F. Arago and A. Fresnel, “Sur l’Action que les rayons de lumière polarisés exercent les uns sur les autres,” Ann. Chim. et Phys. 10 (1819), 288-305.

4. See, for instance, the relatively recent book by E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, Cambridge, 2007).

5.  G.I. Taylor, “Interference fringes with feeble light,” Proc. Camb. Phil. Soc. 15 (1909), 114-115.

6.  C. Davisson and L.H. Germer, “Reflection of electrons by a crystal of nickel”. Nature  119 (1927), 558–560.

7.  C. Jönsson, Zeitschrift für Physik 161 (1961), 454.  Reprinted in English as “Electron diffraction at multiple slits,”  Am. J. Phys. 42 (1974), 4- 11.

8. R. Crease, The Prism and the Pendulum: The Ten Most Beautiful Experiments in Science (Random House, New York, 2003).

9.  P.G. Merli, G.F. Missiroli and G. Pozzi, “On the statistical aspect of electron interference phenomena,” Am. J. Phys. 44 (1976), 306-307.

10. A. Tonomura, J. Endo, T. Matsuda, T. Kawasaki and H. Ezawa, “Demonstration of single-electron buildup of an interference pattern,” Am. J. Phys. 57 (1989), 117-120.

11. T.L. Dimitrova and A. Weis, “The wave-particle duality of light: A demonstration experiment,” Am. J. Phys. 76 (2008), 137-142.

12. Full disclosure: I know very well the authors of the following papers, which is of course why I am aware of them!

13.  H.F. Schouten, T.D. Visser and E. Wolf, “New effects in Young’s interference experiment with partially coherent light,” Opt. Lett. 28 (2003), 1182-1184.

14.  H.F. Schouten, G. Gbur, T.D. Visser and E. Wolf, “Phase singularities of the coherence functions in Young’s interference pattern,” Opt. Lett. 28 (2003), 968-970.

15. H.F. Schouten, N. Kuzmin, G. Dubois, T.D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. ‘t Hooft, D. Lenstra and E.R. Eliel, “Plasmon-assisted two-slit transmission: Young’s experiment revisited,” Phys. Rev. Lett. 94 (2005), 053901.

16. 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.