## Hilda Hänchen and the Goos-Hänchen effect

Today, the United Nations declared February 11 to be the “International Day of Women and Girls in Science,” starting a new effort to get more women into science and keep them there.  In honor of this new day, I thought I would talk about the major discovery of Hilda Hänchen (1919-2013) who, for her PhD, co-discovered what is now known as the Goos-Hänchen effect, a peculiar and strange effect in optics that was first observed some seventy years ago and has presented many surprises since its discovery!

I suspect that most optics researchers aren’t even aware that the “Hänchen” of the Goos-Hänchen effect is a woman, even those who are familiar with her eponymous effect.  I only came across this because I was researching the effect for my upcoming textbook on singular optics!

Hilda Hänchen, born in what was then the Weimar Republic, worked on her PhD at the University of Hamburg under Fritz Goos, and her 1943 dissertation was on what became known as the Goos-Hänchen effect.  World War II broke out in full force not long after, and Hänchen worked as a managing research assistant at the State Physics Institute of Hamburg.  This role was only temporary, as women were only allowed to work as assistants to leave academic spots open for men returning from war.  I suspect that her options were limited in war-torn and post-war Germany, but she appears to have remained somewhat active as an academic at least through the 1970s.

So what is the Goos-Hänchen effect?  This will require a little bit of explanation.  We start with the law of refraction, usually referred to as Snell’s law, which describes how a light ray bends as it passes across an interface between two transparent media.  Snell’s law is written as

$\displaystyle n_1\sin\theta_1 = n_2\sin\theta_2$,

though we won’t worry about the math right now!  The image below illustrates the idea.

When light passes from a “rarer” medium (like air) with a smaller refractive index $n_1$ to a “denser” medium (like water) with a larger refractive index $n_2$, it deflects towards the perpendicular to the surface (dashed line).  When light goes from a denser medium to a rarer medium, the reverse process happens.

The bend of light from a denser to rarer medium must be accounted for by spear fishermen, who have to aim where the fish actually is, not where the light makes it appear to be!

But there is a curious asymmetry in light going rare to dense versus dense to rare. If we look at the rare to dense case, we can color all the possible angles of incidence and angles of refraction in red, as shown below.

But going the other way, from dense to rare, there are rays which have no companion, such as the one illustrated in red below.

So what happens to a ray such as this, which apparently cannot satisfy Snell’s law at all?  It turns out that the light will be completely reflected, and none of it will transmit into the rarer medium.  This is the phenomenon of total internal reflection; light hitting a dense to rare interface at a sufficiently steep angle will simply reflect, and is essentially trapped inside.  This phenomenon is the foundation of the field of fiber optics, in which light is transmitted long distances without loss through glass fibers.  A large amount of internet traffic is now conveyed through fiber optics, including probably this post!

A crude illustration of how fiber optics works.

But there is more to the story of total internal reflection!  It was first studied in detail by none other than the great Isaac Newton, who used a prism to study the effect.  An illustration of how this might have worked is shown below.

The thickness of the rays is supposed to indicate their brightness.  In the normal reflection case, part of the light gets reflected, part gets transmitted.  In the total internal reflection case, all gets reflected.

But Newton noticed something else — when a second prism is brought really close and parallel to the first one, light starts to get transmitted again!

This process, known as frustrated total internal reflection (FTIR), only occurs when the second prism is brought within a distance comparable to the wavelength of the light being used.  In the case of visible light, this wavelength is about 0.0005 mm, a very small number!

So what causes light to be transmitted across the seemingly “empty” region between the prisms?  Newton didn’t really have a good explanation for this effect, largely because he viewed light as being a stream of particles.  We now know that light is a wave, and it is the wave properties of light that result in FTIR.  What Newton saw as empty space outside the prism in total internal reflection, we now know is filled with localized light wave that “clings” very closely to the surface.

This “creeping” wave is known as an evanescent wave, the word “evanescent” invoking something that is fleeting and nearly imperceptible, which such waves are!  Evanescent waves are of crucial importance in the modern field of nano-optics, though this is a story for another time.

This brings us back to the discovery of Goos and Hänchen!  If you look at the images I have drawn of total internal reflection above, I have shown the reflected ray bouncing off of the glass at the same point where the incident ray hits.  However, when one does a rigorous theoretical analysis of the reflection, one finds that the picture should really look as follows.

A beam of light, reflected off of a surface in total internal reflection, ends up being reflected from a point further along than where the incident field hits! This is the Goos-Hänchen shift.  Goos and Hänchen reported the definitive experimental observation of this effect in their 1947 paper¹, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” or “A new and fundamental test for total reflection.”

Observing this effect presented significant experimental challenges.  Though I have shown a really exaggerated shift in the image above, typically the shift is roughly a few wavelengths in size at most, and much smaller than the width of the light beam being reflected.  So how does one actually spot this shift?  Simply by using a very long piece of glass and reflecting the light beam² multiple times before measuring its shift!

Figure showing the use of multiple reflections to measure the Goos-Hanchen shift, from their original paper.

Each reflection causes the light beam to shift a little further, so by using multiple reflections, the effect can be emphasized dramatically.  In the full experiment, they reflected the beam a lot of times to get a really big, observable effect:

Geometrically, one can predict where the beam should end up in the absence of the Goos-Hänchen shift, and compare it with where it actually ends up!

The Goos-Hänchen shift is particularly interesting because it is easy to see in the mathematics how the shift arises on reflection, and it is possible to see it experimentally, but it is not so easy to explain why the beam shifts on reflection.  The primary way to interpret it, apparently championed by Goos and Hãnchen themselves, involves evanescent waves.

The hypothesis is that the light, instead of immediately reflecting at the surface of the glass, instead converts to an evanescent wave and “creeps” along the outer surface before becoming a reflected wave. Because evanescent waves only exist in the case of total internal reflection, this seemed like a plausible explanation.

Trying to prove it, however, led to additional surprises.  In 1950, Hans Wolter³ came up with the clever idea to study, theoretically, the power flow of light near the surface of total internal reflection.  He expected that the flow of power would look something like Goos and Hänchen’s figure 1 (Abb. 1) above: power flowing in from the dense medium, “creeping” along the rare medium, and reflecting again in the dense medium.  He did not find so simple a picture of the Goos-Hänchen effect, however.  What he found instead was truly bizarre: light going in a circle!  My reproduction of his original figure is shown below.

Wolter’s vortex of power flow. Though the image doesn’t make it look like it, the light is actually incident from the upper left; in this small region near the vortex, it appears to be coming from below!

This shows the flow of energy (white lines) near the interface between the dense medium (above) and the rare medium (below); the interface is the dashed line.  Wolter found that, at certain points near the interface but within the dense medium, the light ends up doing a “loop-the-loop,” swirling around a central point like water circling a drain.  What Wolter found, in fact, is the first clear example of what is now known as an optical vortex, which I have written about before.

The Goos-Hänchen shift has led to other surprises.  In 1982, researchers ¹¹ looked at the shift when light reflects from materials that are partially opaque, and found that it can be negative.  If we draw this in a similar manner to Goos and Hänchen’s figure 1, it would look as follows.

In this case, it should be noted that we do not have total internal reflection — instead, the absorbing properties of the medium force the light in the absorbing medium to “creep” near the surface, leading to an evanescent shift similar to that discussed above.

Even more recently, it has been shown that one can get “large” ²² or “giant” ³³ negative shifts when the absorbing medium is a metal with an appropriate structure.  The most recent paper I found is from 2014, which shows that Hilda Hänchen’s PhD work continues to remain relevant today, some seventy years later!

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¹ F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. (436) 7–8, 333–346 (1947).

² I should note that I have not yet translated the original paper from German, so I may make some errors in interpretation of their results.  I’m 95% confident I know what’s going on, though!

³ H. Wolter, “Zur Frage des Lichtweges bei Totalreflexion,” Z. Naturforsch. A 5 (1950), 276-283.

¹¹ W.J. Wild and C.L. Giles, “Goos-Hänchen shifts from absorbing media,” Phys. Rev. A 25 (1982), 2099-2101.

²² P.T. Leung, C.W. Chen, H.-P. Chiang, “Large negative Goos–Hanchen shift at metal surfaces,” Opt. Commun. 276 (2007), 206-208.

³³ J. Soni, S. Mansha, S. Dutta Gupta, A. Banerjee, and N. Ghosh, “Giant Goos–Hänchen shift in scattering: the role of interfering localized plasmon modes,” Opt. Lett. 39 (2014), 4100-4103.

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### 5 Responses to Hilda Hänchen and the Goos-Hänchen effect

1. kaleberg says:

Since this shift seems to depend on the wavelength of the light, does it also account partially for the spreading of the spectrum in a prism?

2. JimM says:

Evanescent waves are the basis for a new strategy to improve radiative heat transfer.

Because I’d read this blog post I was able to guess that before I read the linked article.

3. Both waves are reflected from the surface and undergo different phase shifts, which leads to a lateral shift of the finite beam. Therefore the Goos–Hänchen effect is a coherence phenomenon.

4. mantobhi says:

Nice Info

5. S.danny says:

Is there any relationship between Goos-Hänchen shift and the phase shift of totally reflected light?

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