View source for Goos–Hänchen effect

[[Image:Goos-Hanchen-Shift.svg|right|thumbnail|upright=1.5|Ray diagram illustrating the physics of the Goos–Hänchen effect]]

The '''Goos–Hänchen effect''' (named after [[Fritz Goos|Hermann Fritz Gustav Goos]] (1883&ndash;1968) and [[Hilda Hänchen]] (1919&ndash;2013) is an [[optical phenomenon]] in which [[Linear polarization|linearly polarized]] light undergoes a small lateral shift when [[Total internal reflection|totally internally reflected]]. The shift is perpendicular to the direction of propagation in the plane containing the incident and reflected beams. This effect is the linear-polarization analog of the [[Imbert–Fedorov effect]].

[[Acoustics|Acoustic]] analog of the Goos–Hänchen effect is known as Schoch displacement.<ref>{{cite journal |last1=Atalar |first1=Abdullah |author1-link=Abdullah Atalar |title=An angular‐spectrum approach to contrast in reflection acoustic microscopy |journal=[[Journal of Applied Physics]] |date=1978 |volume=49 |pages=5130–5139 |doi=10.1063/1.324460|hdl=11693/50717 |hdl-access=free }}</ref>

== Description ==
This effect occurs because the reflections of a finite-sized beam interferes along a line transverse to the average propagation direction. As shown in the figure, the superposition of two plane waves with slightly different angles of incidence but with the same frequency or wavelength is given by
: <math>
 \mathbf{\underline{E}}(x, z, t) = \mathbf{\underline{E}}^\text{TE/TM} \left(e^{j\mathbf{k}_1 \cdot \mathbf{r}} + e^{j\mathbf{k}_2 \cdot \mathbf{r}}\right) \cdot e^{-j \omega t},
</math>
where
: <math>
 \mathbf{k}_1 = k [\cos(\theta_0 + \Delta \theta) \hat{\mathbf{x}} + 
  \sin(\theta_0 + \Delta \theta) \hat{\mathbf{z}}],
</math>
and
: <math>
 \mathbf{k}_2 = k [\cos(\theta_0 - \Delta \theta) \hat{\mathbf{x}} + 
  \sin(\theta_0 - \Delta \theta) \hat{\mathbf{z}}],
</math>
with 
: <math>
 k = \frac{\omega }{c} n_1.
</math>

It can be shown{{cn|date=December 2024}} that the two waves generate an interference pattern transverse to the average propagation direction, 
: <math>
 \mathbf{k}_0 = k (\cos\theta_0 \hat{\mathbf{x}} + \sin\theta_0 \hat{\mathbf{z}}),
</math>
and on the interface along the <math>(y, z)</math> plane.

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.

== Research ==
This effect continues to be a topic of scientific research, for example, in the context of [[nanophotonics]] applications. A negative ''Goos–Hänchen shift'' was shown by Walter J.&nbsp;Wild and [[Lee Giles]].<ref>{{cite journal |last1=Wild |first1=Walter J. |first2=C. Lee |last2=Giles |title=Goos-Hänchen shifts from absorbing media |journal=Physical Review A |volume=25 |issue=4 |year=1982 |doi = 10.1103/physreva.25.2099|url=http://clgiles.ist.psu.edu/papers/PhysRevA-1982.pdf |pages=2099–2101 |bibcode=1982PhRvA..25.2099W }}</ref> Sensitive detection of biological molecules is achieved based on measuring the Goos–Hänchen shift, where the signal of lateral change is in a linear relation with the concentration of target molecules.<ref>{{cite journal |author=Jiang, L. |title=Multifunctional hyperbolic nanogroove metasurface for submolecular detection |journal=Small |pages=1–7 |year=2017 |doi=10.1002/smll.201700600 |url=https://www.researchgate.net/publication/317591412 |display-authors=etal |volume=13 |issue=30 |pmid=28597602}}</ref> The work by M.&nbsp;Merano et al.<ref>{{cite journal |author1=M. Merano |author2=A. Aiello |author3=G. W. ‘t Hooft |author4=M. P. van Exter |author5=E. R. Eliel |author6=J. P. Woerdman |author-link6=Han Woerdman |journal=Optics Express |volume=15 |pages=15928–15934 |year=2007 |title=Observation of Goos Hänchen Shifts in Metallic Reflection |issue=24 |bibcode=2007OExpr..1515928M |doi=10.1364/OE.15.015928 |pmid=19550880 |arxiv=0709.2278 |s2cid=5108819 }}</ref> studied the Goos–Hänchen effect experimentally for the case of an optical beam reflecting from a metal surface (gold) at 826&nbsp;nm. They report a substantial negative lateral shift of the reflected beam in the plane of incidence for a ''p''&nbsp;polarization and a smaller positive shift for the ''s''&nbsp;polarization case.

==Generation of giant Goos–Hänchen shift ==
It is known that the value of lateral position Goos–Hänchen shift is only 5–10&nbsp;μm at a total internal reflection interface of water and air, which is very difficult to be experimentally measured.<ref>{{cite journal |author=A. W. Snyder |title=Goos-Hänchen shift |journal=Applied Optics |volume=15 |pages=236–238 |date=1976 |issue=1 |doi=10.1364/AO.15.000236 |pmid=20155209 |bibcode=1976ApOpt..15..236S |url=https://www.osapublishing.org/ao/abstract.cfm?uri=ao-15-1-236}}</ref><ref>{{cite journal |author=R. H. Renard |title=Total reflection: A new evaluation of the Goos–Hänchen shift |journal=Journal of the Optical Society of America |volume=54 |pages=1190–1197 |date=1964 |issue=10 |doi=10.1364/JOSA.54.001190 |url=https://www.osapublishing.org/josa/abstract.cfm?uri=josa-54-10-1190}}</ref> In order to generate a giant Goos–Hänchen shift up to 100&nbsp;μm, surface plasmon resonance techniques were applied based on an interface between metal and dielectric.<ref>{{cite journal |author=X. Yin |title=Goos-Hänchen shift surface plasmon resonance sensor |journal=Applied Physics Letters |volume=89 |pages=261108 |date=2006 |issue=26 |doi=10.1063/1.2424277 |bibcode=2006ApPhL..89z1108Y |url=https://aip.scitation.org/doi/10.1063/1.2424277 |doi-access=free}}</ref><ref>{{cite journal |author=A. D. Parks |title=Weak value amplification of an off-resonance Goos–Hänchen shift in a Kretschmann–Raether surface plasmon resonance device |journal=Applied Optics |volume=54 |pages=5872–5876 |date=2015 |issue=18 |doi=10.1364/AO.54.005872 |pmid=26193042 |bibcode=2015ApOpt..54.5872P |url=https://www.osapublishing.org/ao/viewmedia.cfm?uri=ao-54-18-5872&seq=0}}</ref><ref>{{cite journal |author=S. Zeng |title=Plasmonic metasensors based on 2D hybrid atomically thin perovskite nanomaterials |journal=Nanomaterials |volume=19 |pages=1289–1296 |date=2020 |issue=7 |doi=10.3390/nano10071289 |pmid=32629982 |pmc=7407500 |doi-access=free}}</ref> The electrons on the metallic surface are strongly resonant with the optical waves under specific excitation condition. The light has been fully absorbed by the metallic nanostructures, creating an extreme dark point the resonance angle. Thus, a giant Goos–Hänchen position shift is generated by this singular dark point at the [[Total internal reflection|totally internally reflected]] interface.<ref>{{cite journal |author=Y. Wang |title=Targeted sub-attomole cancer biomarker detection based on phase singularity 2D nanomaterial-enhanced plasmonic biosensor |journal=Nano-Micro Letters |volume=13 |pages=96–112 |date=2021 |issue=1 |doi=10.1007/s40820-021-00613-7 |pmid=34138312 |pmc=7985234 |arxiv=2012.07584 |bibcode=2021NML....13...96W |s2cid=229156325}}</ref> This giant Goos–Hänchen shift has been applied not only for highly sensitive detection of biological molecules, but also for the observation of photonic spin Hall effect, which is important in quantum information processing and communications.<ref>{{cite journal |author=K. Y. Bliokh |title=Spin–orbit interactions of light |journal=Nature Photonics |volume=9 |pages=796–808 |date=2015 |issue=12 |doi=10.1038/nphoton.2015.201 |arxiv=1505.02864 |bibcode=2015NaPho...9..796B |s2cid=118491205 |url=https://www.nature.com/articles/nphoton.2015.201}}</ref><ref>{{cite journal |author=X. Yin |title=Photonic spin Hall effect at metasurfaces |journal=Science |volume=339 |pages=1405–1407 |date=2015 |issue=6126 |doi=10.1126/science.1231758 |pmid=23520105 |s2cid=5740891 |url=https://www.science.org/doi/10.1126/science.1231758}}</ref>

==References==
{{Reflist}}
;Books
* {{cite book |last1=de Fornel |first1=Frédérique |title=Evanescent Waves: From Newtonian Optics to Atomic Optics |date=2001 |publisher=Springer |location=Berlin |isbn=9783540658450 |pages=12–18 |language=en}}
* {{cite journal |doi=10.1002/andp.19474360704|title=Ein neuer und fundamentaler Versuch zur Totalreflexion |year=1947 |last1=Goos |first1=F. |last2=Hänchen |first2=H. |journal=Annalen der Physik |volume=436 |issue=7–8 |pages=333–346 |bibcode=1947AnP...436..333G |doi-access=free }}
* {{cite journal |doi=10.1078/0030-4026-00205|title=Evaluation of a total reflection set-up by an interface geometric model |year=2003 |last1=Delgado |first1=M. |last2=Delgado |first2=E. |journal=Optik |volume=113 |issue=12 |pages=520–526 |bibcode=2003Optik.113..520D }}

{{DEFAULTSORT:Goos-Hanchen effect}}
[[Category:Optical phenomena]]