Goos-Hänchen effect

Curator and Contributors

1.00 - Paul R. Berman

The Goos-Hänchen effect refers to the lateral displacement that a wave having finite cross section undergoes when it is totally internally reflected at an interface of two media having different indices of refraction. This lateral shift can be explained in the simplest sense as resulting from the propagation of an evanescent wave parallel to the interface, or as a displacement of the wave in a time interval that can be interpreted as the time delay associated with the scattering process.

History

The Goos-Hänchen effect is illustrated in Fig. 1 for a beam of light having finite cross section. For an angle of incidence $$\theta _{i}$$ greater than the critical angle required for total internal refection, there is a displacement $$s$$ along the surface. The Goos-Hänchen Shift (GHS) is usually defined as the beam's lateral shift $$D=s\cos \theta _{i}$$ indicated in the diagram. The existence of a lateral shift in total internal reflection is often attributed to Newton, based on Proposition 94 of the Principia or Observation 1 of Book 2, Part 1 of Newton's Optiks. Newton gave both a theoretical basis and experimental evidence for penetration of light into medium 2 under conditions of total internal reflection. However, Newton's picture is associated with a mechanical model, whereas the Goos-Hänchen effect is intrinsically linked to the wave nature of the radiation field or, in the case of quantum mechanics, to the wave nature of matter. In fact, there can be a Goos-Hänchen effect associated with most any form of wave-like phenomena.

Figure 1: An beam of electromagnetic radiation (blue beam) is totally internally reflected (red beam) at the interface between two media having indices of refraction $$n_{1}$$ and $$n_{2}\ ,$$ respectively. There is a displacement $$s$$ along the surface. The Goos-Hänchen shift is the lateral shift $$D$$ indicated in the diagram. The angle of incidence is denoted by $$\theta _{i}\ .$$

An extensive list of references and an historical overview of both theory and experiment related to the Goos-Hänchen effect is given in the four-part article by Lotsch (Lotsch KV, 1970). Puri and Birman (Puri and Birman, 1986) also provide a survey of early work on the Goos-Hänchen effect, as well as more recent extensions involving spatially dispersive media and the role played by surface polaritons. Theories of a lateral shift in total internal reflection of electromagnetic waves were developed by Picht (Picht J, 1929) and by Schaefer and Pich (Schaefer and Pich, 1937). Goos and Hänchen (Goos and Hänchen, 1947) measured this lateral displacement for the first time. Their experimental work inspired new theoretical work by Artmann (Artmann K, 1948) and v. Fragstein (v. Fragstein C, 1949), in which expressions for the lateral shift were obtained, with different shifts predicted for field polarization parallel to or perpendicular to the plane of incidence. Although Goos and Hänchen did not find such a polarization dependence in their original work, they were able to measure the effect in 1949 (Goos and Lindberg-Hänchen, 1949). Wolter (Wolter H, 1950) carried out a measurement of the Goos-Hänchen shift that was in good agreement with theory. The Goos-Hänchen effect continues to attract the interest of researchers. It is but one of a large class of non-specular processes, including the Imbert-Federov effect and reflection by multi-layered media [see, for example, the articles by Tamir (Tamir T, 1986), Li (Li C-F, 2007) , and Krayzel et al. (Krayzel et al., 2010) ].

Theory

Goos-Hänchen Shift in Optical Total Internal Reflection

The GHS was first discussed in the context of total internal reflection of electromagnetic radiation. Although the GHS occurs only for beams having finite cross section, the GHS can be related to the phase of the amplitude reflection coefficient calculated for plane waves incident on the interface with an angle of incidence equal to $$\theta _{i}\ .$$ The relationship between the GHS and the phase of the reflection coefficient can be established by considering the scattering of an electromagnetic having finite cross section by the interface. In effect, the GHS is connected with the propagation parallel to the surface of an evanescent wave in medium 2. Alternatively, one can view the GHS in terms of the time delay time $$\tau$$ associated with the scattering of a radiation pulse incident on the interface from medium 1; in this picture, the GHS results from the propagation of the pulse parallel to the interface in medium 2 during the time interval $$\tau$$ (Chiu and Quinn., 1972).

Two distinct cases need be considered, polarization of the electric field perpendicular to the plane of incidence (TE or transverse electric polarization) and polarization parallel to the plane of incidence (TM or transverse magnetic polarization). By matching boundary conditions at the interface, one obtains the standard Fresnel equations for transmission and reflection at an interface. If $$\sin \theta _{i}>n_{2}/n_{1}\equiv n<1,$$ there is total internal reflection; the magnitude of the reflection coefficient is unity and its phase determines the GHS. Assuming that the magnetic permeability constant of each medium is approximately equal to unity, the phase of the (amplitude) refection coefficient is

$\left( \phi _{R}\right) _{\perp }=-2\tan ^{-1}\left( \frac{\sqrt{\sin ^{2}\theta _{i}-n^{2}}}{\cos \theta _{i}}\right) ,$

for electric field polarization perpendicular to the plane of incidence and

$\left( \phi _{R}\right) _{\parallel }=-2\tan ^{-1}\left( \frac{\sqrt{\sin ^{2}\theta _{i}-n^{2}}}{n^{2}\cos \theta _{i}}\right)$

for electric field polarization parallel to the plane of incidence.

If the phase is considered as a function of $$k_{1y}=k_{1}\sin \theta _{i},$$ where $$k_{1}=2\pi /\lambda _{1}$$ is the wavelength in medium 1, the displacement along the surface can be calculated as

$\tag{1} s=-\frac{\partial \phi _{R}}{\partial k_{1y}}=-\frac{1}{k_{1}\cos \theta _{i} }\frac{d\phi _{R}}{d\theta _{i}}.$

For electric field polarization perpendicular to the plane of incidence, the corresponding GHS is $\tag{2} D_{\perp }=s_{\bot }\cos \theta _{i}=-\frac{1}{k_{1}}\frac{d\left( \phi _{R}\right) _{\bot }}{d\theta _{i}}=\frac{\lambda _{1}}{\pi }\frac{\sin \theta _{i}}{\sqrt{\sin ^{2}\theta _{i}-n^{2}}}.$

For electric field polarization parallel to the plane of incidence, the GHS is $\tag{3} D_{\parallel }=s_{\Vert }\cos \theta _{i}=-\frac{1}{k_{1}}\frac{d\left( \phi _{R}\right) _{\Vert }}{d\theta _{i}}= \frac{n^{2}}{\sin ^{2}\theta _{i}\left( 1+n^{2}\right) -n^{2}} D_{\perp},$

These equations were derived by Artmann (Artmann K, 1948) (the form of Eq. (3) given by Artmann was valid only near the critical angle). Near the critical angle, $$D_{\parallel }\approx D_{\perp }/n^{2}>D_{\perp }$$. There is energy flow parallel to the surface in medium 2, but no energy flow perpendicular to the surface; the transmitted wave is an evanescent wave. A numerical simulation of the Goos-Hänchen effect is shown in Fig. 2.

Figure 2: Numerical simulation of the Goos-Hänchen effect. The incident TE beam is taken to have a Gaussian cross-section with waist equal to $$35\lambda _{1}$$. The relative index of refraction is $$n=0.913$$ and the average angle of incidence is $$\theta _{i}=67^{\circ }$$. For these parameters, $$s_{\bot }=2.54\lambda _{1}$$ and $$D_{\bot }=0.99\lambda _{1}$$. The orange line is the interface and there is an evanescent wave in medium 2, above the interface. (Image provided by A. Moreau).

An alternative explanation of the GHS can be given in terms of the time delay associated with the scattering of a radiation pulse at the interface. In this case, the phase of the reflection coefficient is considered to be a function of $k_{1x}=k_{1}\cos \theta _{i}$. The incident radiation pulse is not scattered instantaneously by the surface, but reemerges into medium 1 after a time delay given by

$\tau =\frac{1}{v_{1}\cos \theta _{i}}\frac{\partial \phi _{R}}{\partial k_{1x}}=-\frac{1}{k_{1}v_{1}\sin \theta _{i}\cos \theta _{i}}\frac{d\phi _{R} }{d\theta _{i}},$

where $$v_{1}$$ is the speed of light in medium 1. During this time delay, the pulse propagates parallel to the surface and is displaced by a distance $s=v_{1}\sin \theta _{i}\tau =-\frac{1}{k_{1}\cos \theta _{i}}\frac{d\phi _{R} }{d\theta _{i}},$

in agreement with Eq. (1).

Renard (Renard RH, 1964) questioned the validity of Eqs. (2) and (3) since they lead to a finite shift at grazing incidence, $$\theta _{i}=\pi /2\ .$$ Using a model in which energy is transported from one side of the beam to the other, he arrived at modified expressions that vanished at $$\theta _{i}=\pi /2\ .$$ (Lotsch KV, 1968) also found expressions that vanished at $$\theta _{i}=\pi /2\ ,$$ but these were brought into question by Carnaglia (Carnaglia CK, 1976), who suggested that a corrected version of Lotsch's results agrees with the Artmann result. The existence of a nonvanishing GHS at grazing incidence was further supported by the work of Lai et al. (Lai et al., 1986), McGuirk and Carnaglia (McGuirk and Carnaglia, 1967), and Lai et al. (Lai et al., 2000), who argued that the Artmann result is valid provided the incident beam subtends angles of incidence that are between $$\pi /2-\epsilon$$ and $$\theta _{c}+\epsilon \ ,$$ where $$\epsilon$$ is typically on the order of a hundredth of a radian or so. Further validation of the Artmann result was provided by the numerical simulations of Shi et al. (Shi et al., 2007).

The expressions for the GHS given above diverge at the critical angle where the approximations used in their derivation break down. Early generalizations to include angles near the critical angle were given by Artmann (Artmann K, 1948) and by Wolter (Wolter H, 1950). More recently, Horowitz and Tamir (Horowitz and Tamir, 1971) and Lai et al. (Lai et al., 2000) developed theories involving uniform approximations near the critical angle. In these theories, the maximum GHS actually occurs for $$\theta _{i}>\theta _{c}$$ and there are nonvanishing shifts for $$\theta _{i}<\theta _{c}$$ as well (even though, in the plane wave approximation, there is no GHS for $$\theta _{i}<\theta _{c}$$ since the reflection coefficient is real for $$\theta _{i}<\theta _{c}$$). Both these results reflect the fact that a beam having finite width contains a range of angles of incidence about some average angle of incidence. Hence at angles near the critical angle, there are components in the incident beam that undergo both normal as well as total internal reflection. The maximum value of $$D_{\perp }$$ is typically on the order of a few $$\lambda _{1}$$ for a beam whose width is on the order of a hundred $$\lambda _{1}\ .$$

Goos-Hänchen Shift in Quantum Scattering

There is a quantum analogue to the GHS, in which medium 1 is replaced by the vacuum, medium 2 by a constant potential, and the optical field pulse by a quantum-mechanical wave packet. The potential energy function in the $$x$$ direction is given by $$V(x)=V_{0}\Theta (x)\ ,$$ where $$V_{0}>0$$ and $$\Theta (x)$$ is the Heaviside step function that vanishes for $$x<0$$ and is equal to unity for $$x\geq 0.$$ If $$E>V_{0}$$ and the angle of incidence is greater than some critical angle, an expression for the GHS can be obtained using standard methods of quantum mechanics. As in the optical case, the GHS can be related to the phase of the reflection coefficient of the corresponding plane wave problem. The eigenfunctions for a plane wave associated with a particle having mass $$m$$ and energy $$E$$ that is incident on the interface with an angle if incidence $$\theta _{i}$$ can be written as

$\psi_{E} \left( x,y\right) =\left\{ \begin{array}{c} e^{ik_{y}y}\left( e^{ik_{x}x}+Re^{-ik_{x}x}\right) \; \; x<0 \\ Te^{ik_{y}y}e^{-\sqrt{\kappa ^{2}-k_{x}^{2}}x} \; \; x>0 \end{array} \right. ,$

where $$k_{x}=k\cos \theta _{i}\ ,$$ $$k_{y}=k\sin \theta _{i}\ ,$$ $$k=\sqrt{ 2mE/\hbar ^{2}}\ ,$$ $$\kappa =\sqrt{2mV_{0}/\hbar ^{2}}\ ,$$ $$\hbar$$ is Planck's constant divided by $$2\pi \ ,$$ $$R$$ is the (amplitude) reflection coefficient, and $$T$$ is the (amplitude) transmission coefficient. It is assumed that $$\left( \kappa ^{2}-k_{x}^{2}\right) >0$$ (or, equivalently, $$\sin ^{2}\theta _{i}>1-V_{0}/E$$), since this corresponds to total internal reflection. The amplitude reflection coefficient is $$R=e^{i\phi _{R}}\ ,$$ where

$\phi _{R}=-2\tan ^{-1}\left( \frac{\sqrt{\kappa ^{2}-k^{2}+k_{y}^{2}}}{\sqrt{ k^{2}-k_{y}^{2}}}\right) ,$

considered as a function of $$k_{y}$$, and the GHS is equal to

$D=s\cos \theta _{i}=-\cos \theta _{i}\frac{\partial \phi _{R}}{\partial k_{y} }=\frac{2\cos \theta _{i}k_{y}}{k_{x}\sqrt{\kappa ^{2}-k_{x}^{2}}}=\frac{2}{k }\frac{\sin \theta _{i}}{\sqrt{\sin ^{2}\theta _{i}-\left( 1-V_{0}/E\right) } }.$

If we define an effective index of refraction for this problem by $$n^{2}=\left( 1-V_{0}/E\right)$$ and use the fact that $$\lambda _{dB}=2\pi /k$$ is the de Broglie wavelength of the particle, we arrive at

$D=\frac{\lambda _{dB}}{\pi }\frac{\sin \theta _{i}}{\sqrt{\sin ^{2}\theta _{i}-n^{2}}},$

a result that was obtained by Hora (Hora H, 1960). Since the GHS is proportional to the de Broglie wavelength $$\lambda _{dB}$$ of the particle, it is a purely wave-like effect. There is a probability current density parallel to the surface in medium 2, but no probability flow perpendicular to the surface; the transmitted wave is an evanescent wave. The quantum GHS has the same form as that of the optical GHS for the case of electric field polarization perpendicular to the plane of incidence.

Experiment

The definitive experiment establishing the lateral shift was carried out by Goos and Hänchen (Goos and Hänchen, 1947). They compared total internal reflection from the back surface of a prism with the reflection from a silver strip that was deposited on the back of the prism. Since the effect is very small, on the order of several optical wavelengths, they multiplied the relative shift between the light that was totally internally reflected and the light that was reflected from the silver by using an "optical waveguide" (parallel surfaces between which many reflections occurred) that allowed them to increase the relative shift by a factor of 70 or so, limited mainly by losses in reflections from the silver strip. The experiment was further refined in 1949 to the point where they could distinguish the difference in the shift between $$D_{\parallel }$$ and $$D_{\perp }\ .$$ Wolter (Wolter, 1950) repeated the Goos-Hänchen experiment with increased resolution and obtained excellent agreement between his experimental results and theory over a small range of angles about the critical angle. He coined the name Goos-Hänchen Effect for the lateral shift, a label which has remained to this day. Thus, by 1950, the GHS had been firmly established.

The GHS continued to attract attention as new technologies became available. Cowan and Anicin (Cowan and Anicin, 1977) observed the GHS shifts for both TE and TM polarizations for microwave radiation incident on a paraffin prism using a single reflection of the beam. Bretenaker (Bretenaker F, 1992) measured the difference between $$D_{\perp }$$ and $$D_{\parallel }$$ in a single reflection optical experiment using a He-Ne laser cavity field. Schwefel (Schwefel HGL, 2008) used the same method as that employed by Goos and Hänchen in their 1947 experiment, but, by using a glass half-cylinder and a partially coherent LED light source, they were able to measure the effect for a single reflection of both TE and TM waves for all angles of incidence. Their results are consistent with a finite shift at $$\theta _{i}=\pi /2$$ [earlier evidence for a finite shift at $$\pi /2$$ was obtained by Rhodes and Carnaglia (Rhodes and Carnaglia, 1977)].

Extensions to other Domains

The papers of Lotsch (Lotsch KV, 1970) and Puri and Birman (Puri and Birman, 1986) contain references to a large number of both theoretical and experimental studies of the Goos-Hänchen effect in acoustics, nonlinear optics, absorbing media, spatially dispersive media, plasmas, semiconductors, superlattices, etc. Much of this work is motivated by the possibility that the GHS can serve as a probe of scattering and excitations that occur at and near the interface of two bulk materials. More recently, there have been theoretical proposals for measuring a GHS in neutron scattering (Mâaza and Pardo, 1997, Ignatovitch VK, 2004), in negatively refracting material (Berman PR, 2004), and in graphene (Bretenaker F, 1992). An experiment has been carried out in which evidence for the GHS in neutron scattering was claimed (deHaan et al., 2010). It looks as though the Goos-Hänchen effect will continue to attract the attention of researchers for many years to come.

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