11/15/13 Goos-Hanchen shift - Jens Nöckel

pages.uoregon.edu/noeckel/gooshanchen/ 1/9

Goos-Hänchen effect in

microcavities

Microcavity modes created by non-specular

reflections

This page is primarily motivated by our paper, "Goos-Hänchen induced vectoreigenmodes in a dome cavity", David H. Foster, J. U. Nöckel and Andrew K. Cook, Opt.Lett. 32, 1764 (2007) (PDF).

My main goal here is to give a basic informal introduction to the phenomenon thatforms the basis of our paper. At the time of this writing, this page certainly seems tobe more explanatory than the Wikipedia entry. Could I edit the Wikipedia page? Yes,but so could you after reading this page... (at Wikipedia, self-promotion is rightlyconsidered distasteful, though not uncommon; on this page, that rule doesn't apply).There's a more formal discussion of this phenomenon at Scholarpedia.

"Non-specular" reflections refer to deviations from the "ideal" behavior of a mirror,which is to bounce an incoming light beam back in the shape of a V with a well-definedvertex. There has been a recent surge of interest in such effects, and our maincontribution was to show for the first time that some unexpected modes in opticalcavities can be explained by incorporating Goos-Hänchen shifts into a classical raymodel, giving rise to unique bifurcations. We validated this claim in a fully vectorial,three-dimensional solution of Maxwell's equations for a dome cavity with a dielectricmirror. Vectorial field calculations are important for this purpose because reflections atdielectric surfaces are polarization-dependent.

What is the Goos-Hänchen effect?

What better way to learn about science than televesion, right?Yes indeed, there is in fact a TV appearance of the Goos-Hänchen effect! So just watch episode 403 of Numb3rs:"Hollywood Homicide" and you'll be educated.

Actually, don't waste your time. The Goos-Hänchen effect in theshow is completely bogus. Come to think of it, so is most of theseries. "Formulaic Formula Forensics." The image on the right isfrom a web site called The math behind Numb3rs, where youcan glimpse the variety and fun of science without the "middle man" (coincidentally,that page appears to date back to the same year when our paper on the effectappeared – 2007).

The Goos-Hänchen effect is a phenomenon of classical optics in which a light beamreflecting off a surface is spatially shifted as if it had briefly penetrated the surfacebefore bouncing back. The interface has to be between different dielectric materials(such as glass or water), and absorption or transmission should be small enough toallow a recognizable reflected beam to form.

The reason why the show "Numb3rs" is unlikely to have a justifiable need for thiseffect is simply that the spatial shifts are of order of the wavelength of light (i.e., inthe micron range, optimistically). The above image captures part of the physicsbehind this effect, but it also obscures part of the physics because it doesn't simulatean actual light beam.

11/15/13 Goos-Hanchen shift - Jens Nöckel

pages.uoregon.edu/noeckel/gooshanchen/ 2/9

But what is the Goos-Hänchen effect, really?

There are a couple of ingredients thatwork together in the Goos-Häncheneffect, and I am going to illustrate thisusing a movie of electromagnetic wavereflection at a boundary between air(topmost part of the picture) and amaterial of high refractive index (n =3.3, as for Gallium Arsenide at awavelength of 8 μm), occupying thebulk of the image.

The angle of incidence with respect

to the vertical is χ = 32°, which in this material is enough to guarantee total internal

reflection.

The two-dimensional calculation behind this movie is exact, it does not make theapproximation of "Gaussian beams" that is commonly used in laser physics. This way,we can rule out that the Goos-Hänchen effect is an artifact of a misguidedapproximation or a numerical error.

The animation reminds us that light beams canbe described as waves that propagate in a well-defined direction but also have a well-definedwidth. This makes beams different from planewaves, which form wave fronts of infinite width.For comparison, two nearly plane water wavesare shown in the movie on the right. They arevery wide, and cross each other at a well-defined angle, forming an interference pattern over a large area (it was MemorialDay weekend, but the lake was virtually undisturbed, thanks to the Oregonclimate).

In the total-internal reflection geometry, the incident wave is coming from thebottom left, travels upward inside the material and is ultimately reflected at the

surface if χ is large enough. An interference pattern develops only in the small

region where the beams overlap. Outside that region, the wave can clearly becalled a "beam."

The dielectric-air interface near the top of the image is not a perfect mirror. Thewave penetrates into the air and appears to travel parallel to it (from left toright) until the reflection forces it back into the dielectric (heading toward thebottom right).

11/15/13 Goos-Hanchen shift - Jens Nöckel

pages.uoregon.edu/noeckel/gooshanchen/ 3/9

Hover over the image to discover the Goos-Hänchen effect. The image representsexactly the same situation as shown in the grayscale movie above, only plotteddifferently so as to emphasize the highest intensity portions of the beam (essentially,I'm plotting the time averaged energy density of the moving wave pattern on anonlinear color scale, and this eliminates the wave trains except where they formstanding-wave patterns in the region where incident and reflected waves overlap).

When your mouse is over the image, you see the reflection process as it would havebeen expected to occur without the Goos-Hänchen shift. The red lines forming aninverted V follow the corresponding specular-reflection path without the shift. As yourmouse leaves the image, notice how the reflected beam moves over to the right, sothe red reference line is no longer in the center of the beam. That's the Goos-Hänchen shift as it occurs in the actual numerical experiment. It's such a small effectthat you need this kind of image comparison to appreciate what's going on.

The most common explanation for the Goos-Hänchen shift starts withthe observation that spatially confined beams as above can bethought of as superpositions of infinitely extended plane waves. Planewaves reflected by an interface (assumed flat and infinite for ourpurposes), experience a phase shift which may depend on the angle of

incidence, χ. But looking closely at our incident beam, one finds that the plane waves

from which it is constructed don't all have the same value of χ. The more focused we

want our beam to be, the larger the range of χ for the required plane-wave

components. You can see this in the first movie, and in the image on the left: thebeams fan out slightly as they recede from the focus, and their wave fronts arecurved like segmens of concentric circles.

The shape of the beam transverse to its propagation direction depends on theamplitude and phase with which the plane-wave components are superimposed. So ifthe phases of different plane waves are shifted differently upon reflection, thetransverse shape of the reflected beam will be modified. Measuring the intensitymaximum of this reflected beam, one then observes a transverse displacement.

The size of the shift can only be obtained from a wave calculation. This becomes evenmore important when the reflecting interface is not between a homogeneous dielectricand empty space. The Goos-Hänchen effect can also be observed when the "outer"medium is a more complex, structured material. Most high-quality mirrors in opticsare in fact of the latter type: they are made of multiple alternating layers withthickness smaller than the wavelength (i.e., they require nano-scale fabrication), andtheir reflectivity is based purely on interference, which is of course a wavephenomenon. The work to be described below and in [4] relies on such mirrors.

No matter what the reflection mechanism is (total internal reflection ornanostructured materials), a calculation of the Goos-Hänchen shift always starts by

calculating the plane-wave reflection phase shifts as a function of χ, the way I

described it above. In the simplest analytical treatments, that's all you need to know— it doesn't even matter what the precise shape of the incident beam is.

Relation to the ray picture

The displacement of the reflected intensity pattern due to reflection phase shifts canbe re-interpreted when talking about the scattering process in the ray picture,because the resulting beam shape allows us to define a reflected ray. Essentially, rayscan be understood as fictitious particles that are pushed forward by the wave frontslike surfers riding toward the beach.

The Goos-Hänchen shift is the oldest confirmed example of a whole

Goos-Hänchen shift in total internal reflection.

11/15/13 Goos-Hanchen shift - Jens Nöckel

pages.uoregon.edu/noeckel/gooshanchen/ 4/9

An incident ray bundle is reflected by adielectric interface with a lateral offset,

the Goos-Hänchen shift Δ.

class of so-called non-specular effects. Some argue that it datesback to 1718, because Isaac Newton speculated (no pun intended)about the possibility of non-specular deflection of light in his famouswork, "Opticks, or, A Treatise of the Reflections, Refractions,Inflections and Colours of Light." This story is told for example inPhysics World — but I cannot find any evidence that it's true. WhatNewton actually said goes in a rather different direction:

Query 4. Do not the Rays of Light which fall upon Bodies, and are reflected or refracted,begin to bend before they arrive at the bodies; and are they not reflected, refracted, andinflected, by one and the same Principle, acting variously in various Circumstances?

This should be viewed in the context of Query 1 in which Newton speculates whetherlight rays can be bent by objects "at a distance:" He proposes a non-specular effectbased on forces experienced by the corpuscles of light, much like what we now knowgravity is capable of doing (and Newton's theory does indeed predict that light isdeflected by stars, albeit by half as much as what was later calculated from Einstein'sgeneral theory of relativity)! But unfortunately for Newton, corpuscles of light turnedout not to be able to explain all optical phenomena.

Clearly it's not justified to say that the Goos-Hänchen was predicted by Newton,because it's a wave effect. However, we do know that Newton's ideas about light areby no means obsolete:

When we speak of "fictitious particles," you should keep in mind that there is a dualitybetween wave and particle description of light, and the particles of light (photons) canmanifest themselves in very real ways if we decide to measure them. Conversely, whatwe learn here about light waves is equally applicable to quantum particles, because allof them can also act like waves. The Physics World story linked above is a case inpoint, reporting on the Goos-Hänchen effect in neutron scattering.

Putting the Goos-

Hänchen effect into the

ray picture

The ray representation of the lateral shiftis illustrated in the figure on the right.

It is possible to think about the lateralshift as resulting from from a penetrationof the beam into the outside medium.But what determines the "penetrationdepth" of the ray? The answer is, as wenoted above, that it's a wavephenomenon and hence outside the scope of ray physics. That's why we didn't drawthis penetrating ray in the adjacent picture. This emphasizes the fact that the Goos-Hänchen shift is not the same as the classical-mechanics problem of a particlereflected by a smooth potential-energy gradient: our shift produces a discontinuity inthe ray trajectory which isn't governed by a potential energy (in the sense of classicalmechanics).

In such a seemingly pathological situation it's especially interesting to ask what therelation between the ray and wave description of the system looks like. Many standardoptical setups (in particular when Gaussian beams are involved) can be described fullyby identifying one or a few rays, and decorating them with suitable wave patterns(i.e., phase information) in well-known ways.

When rays form families with well-defined caustics, one can often describe the wavesolution succesfully using the WKB approximation or a generalization of it, theEinstein-Brillouin-Keller method. The simplest illustration is the plane wave, whichcorresponds to an infinitely extended bundle of parallel rays. When such a treatmentis possible, it is often unnecessary to look at individual rays in the family, and instead

The moral ofthe story:don't believeanything yousee on TV oron theinternet...

Search

Home

Vitae

News

Papers

Microlasers

Research

Teaching

Computing

Address

11/15/13 Goos-Hanchen shift - Jens Nöckel

pages.uoregon.edu/noeckel/gooshanchen/ 5/9

Scattering spectrum of an elliptical resonator (mean radius R)

one works with the eikonal which describes the wave fronts to which all the rays mustbe perpendicular. See examples for the circular dieletric on a separate page. In theeikonal, the scattering phase can be incorporated as just one of several contributionsto the phase that accumulates as the wave fronts evolve.

The importance of individual rays increases drastically in systems where the WKBmethod breaks down, because that corresponds to the scenario where ray chaos mayappear. In chaotic dynamics, new periodic ray patterns can form by bifurcation ofexisting simpler orbits.

So there are situations where isolated ray trajectories are needed to describe thewave patterns. An important type of wave that can be put into correspondence withan isolated ray is the (fundamental) Gaussian beam. It is for this type of wave thatthe Goos-Hänchen effect was first described.

Why modify the ray picture by an effect that goes

away in the short-wavelength limit?

I first learned about therelevance of the Goos-Hänchen effect indielectric microresonatorsfrom Richard Chang andDipak Chowdhury [1].This was in the mid-nineties, and in theseresonators (see image onthe right) I wasobserving reasonableagreement between thelifetimes of quasiboundstates from my numericalwave calculations, and aray model in which I hadincorporated curvature-corrected Fresnelreflectivity formulas. Soinitially, I decided not tolook for additionalcorrections due to theGoos-Hänchen shiftbecause it didn't seemquantitatively necessary.

However, the Goos-Hänchen shift has a strange fascination, so I decided to explore itsome more with Martina Hentschel at the Max-Planck Institute for Physics of ComplexSystems in Dresden [2]. One motivation came from an apparently boring test casethat I originally only studied to validate my numerical computations of quasiboundstates: the mathematical ellipse (see figure), for which the wave equation can be"solved by hand" under Dirichlet boundary conditions, i.e., if its perimeter were anideal mirror. One says that the Dirichlet problem in the ellipse is integrable. Thismeans in particular that the internal dynamics of the ellipse should not display anytraces of chaotic ray orbits. An ellipse is a very particular type of oval: there are manyovals (e.g., the egg!) which aren't ellipses. Compared to general ovals of identicaleccentricity, it is typically a good approximation to consider the dielectric ellipse asnon-chaotic, as can be seen on this page discussing dynamical eclipsing.

It is a known fact that resonator shapes that would be integrable for Dirichletboundary conditions generally do not retain that property in their wave equation whenthey are made of dielectric material. This is true not only for the ellipse, but even forthe mundane rectangle. Although the interface between an elipse and the surroundingmedium coincides with the coordinate lines of the elliptic cylinder coordinate system,

11/15/13 Goos-Hanchen shift - Jens Nöckel

pages.uoregon.edu/noeckel/gooshanchen/ 6/9

A dome resonator whose bottom mirror induces a Goos-Hänchen shift. Theheight of the dome is slightly less than that of a perfect hemisphere, which

implies that a specularly reflected V-shaped ray would not be possible on purelygeometrical grounds. The observed V-shaped wave field can only exist because

of the Goos-Hänchen shift.

the wave field on the boundary cannot be assumed to have a constant value (orconstant derivatives, for that matter). The reason is that the light will attempt to leakout preferentially near spots of highest curvature, and the curvature is not constantunless the ellipse degenerates to a circle. Some visualizations of how light penetratesthe surface of a circular dielectric are shown on a spearate page.

It is this penetration into the interfacial region that underlies the physics of theGoos-Hänchen effect. However, the effect has (for good reason) been studied mostlyin the context of planar interfaces, and not curved ones such as ellipses. Although itis also possible to make a convincing argument for the existence of the effect incircular cavities [1], there are some confusing questions that arise when generalizingto shapes like the ellipse.

In [3], we mention the Goos-Hänchen effect as the explanation for the non-separability of the dielectric ellipse. Considering the ray limit of a dielectric cavity,the internal dynamics of the ellipse is strictly integrable when specular reflection atthe interface is assumed, whereas the wave equation is not integrable. Couldthere be a corrected, non-integrable ray dynamics that describes the internalcavity fields better?

Goos-

Hänchen-

induced

cavity

modes

The fundamentaldifficulty with theray picture isthat all wavesare plagued byuncertaintyrelations whichmake itimpossible tosimultaneouslypin downquantities thatare related toeach other byFourier transformation. This is what I was referring to above, when I said that a range

of χ is contained in the plane-wave decomposition of a focused beam.

In the ellipse, the spatially varying curvature makes it necessary to specify theposition at which a reflection occurs, while the Goos-Hänchen shift depends on theincident beam's direction of propagation. This incident angle cannot be pinned downsimultaneously with the position, but that's just what has to be done in the raypicture. As a result, reliable analytical formulas for the shift in the presence of curvedinterfaces have so far not been derived.

There is by definition not a lot of room in a microcavity, but one can, so to speak,make more room by shrinking the wavelength in comparison to the cavity dimensions.In this semiclassical limit, the uncertainty relations become less uncertain, and theray picture becomes more accurate. Then it becomes possible to observe the Goos-Hänchen shift in new ways that are peculiar to the resonator scenario. In contrast tofree-space experiments with incident and reflected beams, we can for example lookfor the resonance frequencies at which certain modes appear.

This is what's depicted in the figure above.

11/15/13 Goos-Hanchen shift - Jens Nöckel

pages.uoregon.edu/noeckel/gooshanchen/ 7/9

As we proved in [4], the Goos-Hänchen shift does indeed manifest itself in thespectra and wave functions of cavities with dielectric boundary conditions. Thegeometry we considered is nearly integrable, and in that sense it is similar to theexample of the ellipse. However, we did the calculations for a three-dimensional dome.The dome is a hemispherical shell whose height is just slightly shorter than its radius.This acts as an ideal curved mirror, and the cavity is closed off by a planar, dielectricmultilayer (Bragg) mirror.

As with the ellipse, this dielectric mirror is partly penetrable. The leakage from theresonator gives all its modes a finite lifetime, but in the context of the Goos-Hächenshift we were more interested in the resonance wavelengths and the spatial modestructure than in the lifetimes. Bragg mirrors are a type of dielectric mirror that caninduce reflection phase shifts in a more controllable, engineered way than the simpledielectric boundary shown in the movie earlier.

If we now lookfor the ray onwhich this wavepattern could bebased, we haveto draw agemoetry thatincludes the shift

Δ, as shown on

the left. We write

Δ(χ) here

because theGoos-Hänchenshift depends onthe angle of

incidence χ, as

mentionedearlier.

What follows is a very simple trigonometric calculation that determines the angle of

incidence χ of the ray: We know a V-shaped ray must be self-retracing, i.e., hit the

dome mirror perpendicularly so as to be reflected back onto itself. The vertex of the Vmust thus coincide with the center of curvature of the dome, which lies a smalldistance z1 below the bottom mirror surface. Now the dashed lines around the center

0 describe a right triangle whose catheti are z1 and Δ/2, which leads to the equation

above the figure. This can be solved for χ because Δ(χ) is a known function.

Once you have χ, you know the shape and length of the unique stable ray that will

support a v-shaped wave pattern in the cavity. Its characteristics (such as thefrequencies at which it will appear in the spectrum, and its spot size on the mirrorplane) can then be predicted. To re-iterate: if we eliminate the horizontal Goos-Hänchen displacement from our geometric argument, there exists no solution for a v-shaped ray!

The predictions of our Goos-Hänchen-augmented ray model agree very well withnumerical simulations of the wave fields, with no adjustable parameters. This showsthat resonant modes in optical microcavities can be used to detect the Goos-Häncheneffect. This is a qualitatively new approach to measuring this effect, because it doesn'trely on directly observing the tiny lateral shift of an incident beam. We infer all theproperties of the shift from the very presence or absence of individual resonatormodes, and their properties — i.e., the opening angle of the V and the wavelength,which is determined by the length of the ray. To find out more about the details ofthe system, please refer to [4].

Goos-Hänchen billiards

To conclude, it is worth going one step further with a ray-based analysis. The self-

11/15/13 Goos-Hanchen shift - Jens Nöckel

pages.uoregon.edu/noeckel/gooshanchen/ 8/9

retracing orbit justdiscussed is absent ifwe switch off theGoos-Hänchen effect(as can be done byreplacing the dielectricmirror at the bottomwith an ideal metalreflector). It is createdby a bifurcation inwhich a pair of neworbits are "born." Butthe second member ofthat pair is a periodicorbit which doesn'tcorrespond to anobserved wavepattern. If one new ray orbit creates a new mode, why doesn't the other?

If we really want to understand the wave patterns in our simulations, it is importantto analyze the stability of the ray on which the mode is based. This requires lookingnot just at single orbits but at their neighborhoods. In other words, I end up doing anew type of billiard-ball simulation in which reflections are not specular, but to whichthe same methods can be applied that are known from our previous work [3].

That's what's shown in the last image. It represents a Poincaré surface of sectionof the Goos-Hänchen billiard, i.e., a ray billiard where reflections at the dielectric mirrorexperience the Goos-Hänchen shift. The horizontal axis is the angle of incidence of aray at the planar mirror, and the vertical axis measures the "rate of change" of that

angle between bounces. The labels u, s refer to unstable and stable trajectories. In a

Poincaré section, stable trajectories are surrounded by closed (oval) curves that

represent rays oscillating around them. The pair u, s in the middle is the one created

in the bifurcation mentioned above. But only stable rays can be used to constructtransversally confined wave modes of the cavity.

Compared to the dielectric ellipse, our dome cavity simulations are actually much moredifficult to do because we're dealing with a 3D structure with mixed boundaryconditions, leakiness and polarization-dependent effects. But despite all these real-world complications, the Goos-Hänchen effect lets us say: billards are back! And inthe future, other non-specular effects will likely enter the game as well, enriching boththe repertoire of billiard physics and our understanding of realistic opticalmicrocavities.

1. Chowdhurry, D. Q., Leach, D. H., and Chang, R. K."Effect of the Goos-Hänchen shift on the geometrical-optics model for spherical-cavity mode spacing"J. Opt. Soc. Am. A 11: 1110-1116, 1994

2. "Resonance widths and curvature corrections of Fresnel's formulas in dielectriccylinders" by M. Hentschel and J.U. Nöckel, poster presented at the GermanPhysical Society Spring Meeting, Bonn (2000)

3. Nöckel, J. U. and Chang, R. K."2d microcavities: theory and experiment"Experimental Methods in the Physical Sciences 40, Academic Press (2002), pp.185-226

4. Foster, D. H., Cook, A. K., and Nöckel, J. U."Goos-Hänchen induced vector eigenmodes in a dome cavity"Opt. Lett. 32: 1764-1766, 2007

11/15/13 Goos-Hanchen shift - Jens Nöckel

pages.uoregon.edu/noeckel/gooshanchen/ 9/9

This page © Copyright Jens Uwe Nöckel, 05/2011

Last modified: Sat Nov 17 20:56:42 PST 2012