Goos-Hänchen effect

The Goos-Hänchen effect (also Goos-Hänchen shift ) describes an optical phenomenon of wave refraction during total reflection : If a linearly polarized electromagnetic wave (e.g. light) from an optically denser medium falls at the critical angle on an interface with one another optically thinner medium, the light beam experiences a longitudinal offset in the plane of incidence. The wave is not reflected at the interface, but at a virtual parallel plane in the optically thinner medium. ${\displaystyle \theta _{\mathrm {T} }}$

This effect was already predicted (or assumed) by Isaac Newton , but only experimentally demonstrated in 1943 by Fritz Goos (1883–1968) and Hilda Hänchen (1919–2013).

In the case of total reflection of circular or elliptically polarized light, the Goos-Hänchen effect (longitudinal displacement) occurs together with the Imbert-Fedorov effect , a displacement transverse to the plane of incidence.

An analogous effect with acoustic waves was called the Schoch effect by Lotsch .

description

Ray diagram illustrating the Goos-Hänchen shift

According to the law of reflection , the incident ray is reflected at the interface at the same angle with which it entered the interface. However, the closer you get to the critical angle , the further apart the points of incident and reflected light at the interface. If the rays are lengthened further, one finds that they are reflected at the angle predicted by the law of reflection; only that the level of reflection already lies in the optically thinner medium. ${\displaystyle \theta _{T}}$

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The transverse offset results from the evanescent behavior of the wave in the thinner medium and is defined as the location at which the field amplitude has dropped to 1 / e times its maximum (see ATR spectroscopy ). This is defined as:

${\displaystyle d_{\mathrm {p} }={\frac {\lambda }{2\pi n_{1}{\sqrt {\sin ^{2}\theta _{1}-\left({\frac {n_{2}}{n_{1}}}\right)^{2}}}}}}$

Consequently, the Goos-Hänchen shift is calculated

${\displaystyle L=\tan \theta \cdot 2d_{\mathrm {p} }}$

proof

For a long time, attempts have been made to prove the stay of the wave in the optically thinner medium by considering the energy flow in the media. The problem arose, however, that energy can only be measured if it is diverted in some way, which would interfere with the refraction. Since the Goos-Hänchen effect has only minor effects, these would have been quickly disrupted.

Ultimately, Goos and Hänchen's idea was to apply a strip of reflective silver layer to a prism . A light beam that fell on the interface at the critical angle within the prism - and thus totally reflected - was directed partly onto the silver and partly onto the free glass. Because silver has a significantly higher refractive index than air, the conditions for total reflection no longer exist for this part of the light, because the critical angle depends on the refractive indices of the media involved (see total reflection ). Instead, the light is usually reflected here. This beam was used as a reference measurement and the difference to the really totally refracted beam - in which the Goos-Hänchen effect occurred - was observed. In order to intensify the effect, the light was reflected several times over silvered or non-silvered surfaces.

If a non-linear medium is used as an optically thin medium, the effect can be observed somewhat better for angles close to the critical angle.

Explanation

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The waves that are obtained from the Maxwell equations as solutions of the wave equation must be continuous (i.e. continuous ). After this, it cannot be that a wave simply breaks off and disappears directly at the interface . There must therefore be so-called evanescent waves at the edge of a medium : continuous continuations of the waves that drop exponentially and can therefore usually be ignored because they only penetrate the other materials slightly. As a consequence of Maxwell's equations, there is a wave in the optically thinner medium.

Excellent rays and Poynting vectors for explanation

An important effect that is usually disregarded in calculations is the finite width of the incident waves: The wave, as a bundle of light, has outer edge rays and inner rays. It can now be shown mathematically that those marginal rays at the interface which first come into contact with the interface (left in the figure) release part of their energy into the optically thinner medium. This energy transport is elegantly described by the Poynting vector ; it indicates the direction in which energy flows . On the left-hand side, the Poynting vector points slightly diagonally into the optically thinner medium. However, this only applies to a small margin area; the angle of the Poynting vectors decreases continuously from the edge towards the inside, so the energy conduction in the middle areas of the wave is parallel to the interface. At the other edge of the light beam, on the other hand, the Poynting vector points from the optically thinner to the optically denser medium and thus energy is pumped into the optically denser medium again , which represents the reflected wave. ${\displaystyle {\vec {S}}}$

This transport of energy is responsible for ensuring that there is no net energy flow at or across the interface ; that is, the energy absorbed and emitted is exactly balanced. The waves required for this, which cause energy to oscillate back and forth , were demonstrated by J. Picht .

Another interpretation (by H. Wolter ) is that the energy flowing into the optically thinner medium on the one hand establishes the evanescent wave there and this evanescent wave gives its energy back to the reflected beam. This would explain why the shifting of the reflection plane happens in similar distances as the propagation of the evanescent wave extends.

Interestingly, in order to be able to mathematically describe the above-described reciprocating and energy-transporting waves, at least two incoming waves are required . In a normal calculation, one actually incorrectly simplifies the incoming wave (as mentioned above) to a single, plane wave. This alone is not enough to explain the waves running back and forth. Since the Fresnel formulas (with which one can predict the reflection and absorption of waves depending on the angle) are based on plane waves, one cannot adequately describe the Goos-Hänchen effect with them.

Analogue in quantum mechanics

In quantum mechanics, when considering probability currents at potential barriers, there is an analogue to the optical Goos-Hänchen offset.

Let the potential be

${\displaystyle V(x,y)={\begin{cases}0&{\text{falls }}x<0{\text{ (Gebiet I)}}\\V_{0}&{\text{falls }}x\geq 0{\text{ (Gebiet II)}}\end{cases}}}$

The overall wave function can be represented as the product of an x ​​and y wave function, which both describe plane waves and each satisfy the Schrödinger equation .

${\displaystyle \phi (x,y)=\phi _{1}(x)\phi _{2}(y)}$

So that the incoming wave from the negative is totally reflected, must be. In area II, the wave vector of the x-wave function becomes imaginary. It is an evanescent wave. ${\displaystyle x}$ ${\displaystyle V_{0}>E_{1}}$

${\displaystyle k_{x,II}={\frac {1}{\hbar }}{\sqrt {2m(E_{1}-V_{0})}}={\frac {i}{\hbar }}{\sqrt {2m|E_{1}-V_{0}|}}=:i\kappa _{x}\quad {\text{mit}}\quad \kappa _{x}\in \mathbb {R} }$

One defines

${\displaystyle \vartheta :=\arctan {\frac {\kappa _{x}}{k_{x,I}}}}$

and calculates the probability current density in both areas. Let A be the amplitude of the incoming plane wave.

${\displaystyle {\vec {J}}(x,y)={\begin{pmatrix}J_{x}\\J_{y}\end{pmatrix}}={\begin{cases}{\begin{pmatrix}0\\{{\frac {2\hbar k_{y}}{m}}|A|^{2}\left(1+\cos(2k_{x,I}+2\vartheta )\right)}\end{pmatrix}}&{\text{falls }}x<0{\text{ (Gebiet I)}}\\{\begin{pmatrix}0\\{\frac {4\hbar k_{y}}{m}}|A|^{2}(\cos \vartheta )^{2}e^{-2\kappa _{x}x}\end{pmatrix}}&{\text{falls }}x\geq 0{\text{ (Gebiet II)}}\end{cases}}}$

The y-component of the probability current density in area I describes a standing wave that results from the superposition of incident and reflected waves. The Goos-Hänchen offset of the evanescent wave can be read off in area II. With ( is the refractive index) the size given above can be derived from this . ${\displaystyle J_{y}}$ ${\displaystyle k\propto n}$ ${\displaystyle n}$ ${\displaystyle d_{p}}$

literature

• F. Goos, H. Hänchen: About the penetration of totally reflected light into the thinner medium . In: Annals of Physics . tape 435 , no. 5 , 1943, pp. 383-392 , doi : 10.1002 / andp.19434350504 .
• F. Goos, H. Hänchen: A new and fundamental attempt at total reflection . In: Annals of Physics . tape 436 , no. 7–8 , 1947, pp. 333-346 , doi : 10.1002 / andp.19474360704 .
• F. Goos, Hilda Lindberg-Hänchen: New measurement of the beam displacement effect with total reflection . In: Annals of Physics . tape 440 , no. 3–5 , 1949, pp. 251-252 , doi : 10.1002 / andp.19494400312 .
• Rémi H. Renard: Total Reflection: A New Evaluation of the Goos-Hänchen Shift . In: Journal of the Optical Society of America . tape 54 , no. October 10 , 1964, p. 1190-1196 , doi : 10.1364 / JOSA.54.001190 .
• Helmut KV Lotsch: Reflection and refraction of a beam of light at a plane interface. In: Journal of the Optical Society of America . tape 58 , no. April 04 , 1968, p. 551-561 , doi : 10.1364 / JOSA.58.000551 .
• K. Tamasaku, T. Ishikawa: The Goos-Hänchen effect at Bragg diffraction. In: Acta Cryst. A58, 2002, pp. 408-409, doi: 10.1107 / S0108767302006700 .
• WT Dong, Lei Gao, CW Qiu: Goos-Hänchen shifts of an electromagnetic wave reflected from a chiral metamaterial slab . 2009, arxiv : 0906.4587v1 .
• Frederique de Fornel: Evanescent waves: from Newtonian optics to atomic optics . Springer-Verlag, Berlin 2001, ISBN 3-540-65845-9 , pp. 12-18 .
• PR Berman: Goos-Hänchen shift in negatively refractive media . In: Physical Review E . tape 66 , no. 6 , 2002, pp. 67603 , doi : 10.1103 / PhysRevE.66.067603 .

Web links

• Paul R. Berman Goos-Hänchen Effect , Scholarpedia

Individual evidence

1. ↑ ^{a b} Frédérique de Fornel: Evanescent waves: from Newtonian optics to atomic optics . Springer-Verlag, Berlin 2001, ISBN 3-540-65845-9 , pp. 12-18 .
2. ↑ F. Goos, H. Hänchen: A new and fundamental attempt at total reflection . In: Annals of Physics . tape 436 , no. 7–8 , 1947, pp. 333-346 , doi : 10.1002 / andp.19474360704 .
3. ↑ ^{a b c} Helmut KV Lotsch: Beam displacement at total reflection: The Goos-Hänchen effect, Pt.III . In: Optics . tape 32 , no. 4 , 1971, ISSN  0030-4026 , p. 299-319 .
4. ↑ Joachim Schubert: Physical Effects: Applications; Descriptions . Physik-Verlag, Weinheim 1982, ISBN 3-87664-053-9 .
5. ↑ Helmut K. solder: Beam Displacement at Total Reflection: The Goos-Hänchen Effect I . In: Optics . tape 32 , no. 2 , 1970, ISSN  0030-4026 , pp. 116-137 .
6. ↑ Helmut KV Lotsch: Beam Displacement at Total Reflection: The Goos-Hänchen Effect II . In: Optics . tape 32 , 1970, ISSN  0030-4026 , pp. 189-204 .
7. ↑ Helmut KV Lotsch: Beam Displacement at Total Reflection: The Goos-Hänchen Effect IV . In: Optics . tape 32 , no. 6 , 1971, ISSN  0030-4026 , p. 553-569 .
8. ^ Claude Cohen-Tannoudji, Bernard Diu, Franck Laloë: Quantum Mechanics . Volume 1. Hermann, 1977, ISBN 2-7056-8392-5 , pp. 282 ff . ( Limited preview in Google Book Search [accessed March 2, 2013]).