Editing Group velocity (section)

== In lossy or gainful media ==
The group velocity is often thought of as the velocity at which [[energy]] or information is conveyed along a wave. In most cases this is accurate, and the group velocity can be thought of as the [[signal velocity]] of the [[wave]]form. However, if the wave is travelling through an absorptive or gainful medium, this does not always hold. In these cases the group velocity may not be a well-defined quantity, or may not be a meaningful quantity.

In his text "Wave Propagation in Periodic Structures",<ref>{{cite book |author=Brillouin, L. |title=Wave Propagation in Periodic Structures |url=https://archive.org/details/in.ernet.dli.2015.166889 |page=75 |publisher=McGraw Hill |place=New York |year=1946}}</ref> [[Léon Brillouin|Brillouin]] argued that in a lossy medium the group velocity ceases to have a clear physical meaning. An example concerning the transmission of electromagnetic waves through an atomic gas is given by Loudon.<ref>{{cite book |author=Loudon, R. |title=The Quantum Theory of Light |publisher=Oxford |year=1973}}</ref> Another example is mechanical waves in the [[solar photosphere]]: The waves are damped (by radiative heat flow from the peaks to the troughs), and related to that, the energy velocity is often substantially lower than the waves' group velocity.<ref>{{cite journal |author=Worrall, G. |journal=Solar Physics |title=On the Effect of Radiative Relaxation on the Flux of Mechanical-Wave Energy in the Solar Atmosphere |volume=279 |issue=1 |pages=43–52 |year=2012 |doi=10.1007/s11207-012-9982-z|bibcode = 2012SoPh..279...43W |s2cid=119595058 }}</ref>

Despite this ambiguity, a common way to extend the concept of group velocity to complex media is to consider spatially damped [[plane wave]] solutions inside the medium, which are characterized by a ''complex-valued'' wavevector. Then, the imaginary part of the wavevector is arbitrarily discarded and the usual formula for group velocity is applied to the real part of wavevector, i.e.,
:<math>v_{\rm g} = \left(\frac{\partial \left(\operatorname{Re} k\right)}{\partial \omega}\right)^{-1} .</math>

Or, equivalently, in terms of the real part of complex [[refractive index]], {{math|1=<u>''n''</u> = ''n'' + ''iκ''}}, one has<ref name=Boyd1170885>{{Cite journal |pmid = 19965419| year = 2009 |last1 = Boyd| first1 = R. W. |title = Controlling the velocity of light pulses |journal = Science |volume = 326| issue = 5956 |pages = 1074–7 |last2 = Gauthier |first2 = D. J. |doi = 10.1126/science.1170885 |bibcode = 2009Sci...326.1074B |url = http://www.phy.duke.edu/~qelectron/pubs/Science326_1074_2009.pdf |citeseerx = 10.1.1.630.2223 |s2cid = 2370109}}</ref>
:<math>\frac{c}{v_{\rm g}} = n + \omega \frac{\partial n}{\partial \omega} .</math>

It can be shown that this generalization of group velocity continues to be related to the apparent speed of the peak of a wavepacket.<ref>{{cite web |title=Dispersion |url=https://scholar.harvard.edu/files/david-morin/files/waves_dispersion.pdf |archive-url=https://web.archive.org/web/20120521232240/http://www.people.fas.harvard.edu/~djmorin/waves/dispersion.pdf |archive-date=2012-05-21 |url-status=live |website=people.fas.harvard.edu |first=David |last=Morin |date=2009 |access-date=2019-07-11}}</ref> The above definition is not universal, however: alternatively one may consider the time damping of standing waves (real {{mvar|k}}, complex {{mvar|ω}}), or, allow group velocity to be a complex-valued quantity.<ref>{{cite journal |doi=10.1063/1.860877 |title=Real group velocity in a medium with dissipation |journal=Physics of Fluids B: Plasma Physics |volume=5 |issue=5 |pages=1383 |year=1993 |last1=Muschietti |first1=L. |last2=Dum |first2=C. T. |bibcode = 1993PhFlB...5.1383M }}</ref><ref>{{cite journal |doi=10.1103/PhysRevE.81.056602 |pmid=20866345 |title=Complex group velocity and energy transport in absorbing media |journal=Physical Review E |volume=81 |issue=5 |pages=056602 |year=2010 |last1=Gerasik |first1=Vladimir |last2=Stastna |first2=Marek |bibcode = 2010PhRvE..81e6602G }}</ref> Different considerations yield distinct velocities, yet all definitions agree for the case of a lossless, gainless medium.

The above generalization of group velocity for complex media can behave strangely, and the example of [[anomalous dispersion]] serves as a good illustration.
At the edges of a region of anomalous dispersion, <math>v_{\rm g}</math> becomes infinite (surpassing even the [[speed of light]] in vacuum), and <math>v_{\rm g}</math> may easily become negative 
(its sign opposes Re{{mvar|k}}) inside the band of anomalous dispersion.<ref name="DEWSL06"/><ref name="BLSB06"/><ref>{{Citation |doi=10.1109/JPROC.2010.2052910 |volume=98 |issue=10 |pages=1775–1786 |last1=Withayachumnankul |first1=W. |first2=B. M. |last2=Fischer |first3=B. |last3=Ferguson |first4=B. R. |last4=Davis |first5=D. |last5=Abbott |title=A Systemized View of Superluminal Wave Propagation |journal=Proceedings of the IEEE | year=2010|s2cid=15100571 }}</ref>

=== Superluminal group velocities ===
Since the 1980s, various experiments have verified that it is possible for the group velocity (as defined above) of [[laser]] light pulses sent through lossy materials, or gainful materials, to significantly exceed the [[speed of light in vacuum]] {{mvar|c}}. The peaks of wavepackets were also seen to move faster than {{mvar|c}}.

In all these cases, however, there is no possibility that signals could be carried [[faster than light|faster than the speed of light in vacuum]], since the high value of {{mvar|v}}<sub>{{mvar|g}}</sub> does not help to speed up the true motion of the sharp wavefront that would occur at the start of any real signal. Essentially the seemingly superluminal transmission is an artifact of the narrow band approximation used above to define group velocity and happens because of resonance phenomena in the intervening medium. In a wide band analysis it is seen that the apparently paradoxical speed of propagation of the signal envelope is actually the result of local interference of a wider band of frequencies over many cycles, all of which propagate perfectly causally and at phase velocity. The result is akin to the fact that shadows can travel faster than light, even if the light causing them always propagates at light speed; since the phenomenon being measured is only loosely connected with causality, it does not necessarily respect the rules of causal propagation, even if it under normal circumstances does so and leads to a common intuition.<ref name=Boyd1170885/><ref name="DEWSL06">{{Citation |first1=Gunnar |last1=Dolling |first2=Christian |last2=Enkrich |first3=Martin |last3=Wegener |first4=Costas M. |last4=Soukoulis |first5=Stefan |last5=Linden |title=Simultaneous Negative Phase and Group Velocity of Light in a Metamaterial |journal=Science |volume=312 |pages=892–894 |year=2006 |doi=10.1126/science.1126021 |pmid=16690860 |issue=5775 |bibcode = 2006Sci...312..892D |s2cid=29012046 |url=https://publikationen.bibliothek.kit.edu/1000010972/901157 }}</ref><ref name="BLSB06">{{Citation |first1=Matthew S. |last1=Bigelow |first2=Nick N. |last2=Lepeshkin |first3=Heedeuk |last3=Shin |first4=Robert W. |last4=Boyd |title=Propagation of a smooth and discontinuous pulses through materials with very large or very small group velocities |journal=Journal of Physics: Condensed Matter |volume=18 |pages=3117–3126 |year=2006 |doi=10.1088/0953-8984/18/11/017 |issue=11 |bibcode = 2006JPCM...18.3117B |s2cid=38556364 }}</ref><ref name="GSBKB06">{{Citation |first1=George M. |last1=Gehring |first2=Aaron |last2=Schweinsberg |first3=Christopher |last3=Barsi |first4=Natalie |last4=Kostinski |first5=Robert W. |last5=Boyd |title=Observation of a Backward Pulse Propagation Through a Medium with a Negative Group Velocity |journal=Science |volume=312 |pages=895–897 |year=2006 |doi=10.1126/science.1124524 |pmid=16690861 |issue=5775 |bibcode = 2006Sci...312..895G |s2cid=28800603 }}</ref><ref name="SLBBJ05">{{Citation |first1=A. |last1=Schweinsberg |first2=N. N. |last2=Lepeshkin |first3=M.S. |last3=Bigelow |first4=R. W. |last4=Boyd |first5=S. |last5=Jarabo |title=Observation of superluminal and slow light propagation in erbium-doped optical fiber |journal=Europhysics Letters |volume=73 |issue=2 |pages=218–224 |year=2005 |doi=10.1209/epl/i2005-10371-0 |bibcode=2006EL.....73..218S |url=http://www.hep.princeton.edu/%7Emcdonald/examples/optics/schweinsberg_epl_73_218_06.pdf |citeseerx=10.1.1.205.5564 |s2cid=250852270 }}{{Dead link|date=June 2024 |bot=InternetArchiveBot |fix-attempted=yes }}</ref>