OTHER TYPES of WAVES Ion-sound waves (eletostati low fequeny waves) ae longitudinal waves simila lassial sound in gas s kt k M B plasma sound is slow fo eletons, but fast fo ions Eleton density is in eah moment in equilibium with instantaneous potential: e e ne nexp n... kbte kbte e n n kt eletons ae isothemal ( ) B e Eleti field may be also deived fom quasi-neutality ondition sum of foes ating on eletons = Vlny
kt ee e p n ikk T B e e B e n n n hydodynami equation fo ion fluid i n i i i p i Mv ZeE i i ni E n ikv ik Poisson equation is not needed Zen en e Zn n i i n E slow motion quasineutality i kbte kbte n Zni we expess en en equations of motion n i Zn n Z n Vlny
imn ivi ikikbtn i i ikkbte Zni - ion motion ini ni ikvi ontinuity equation dispesion equation usually ions ae adiabati i = 5/3 ZT T k T Zk T M i B i B e s k If e i, then thee is stong ollisionless damping by ions, phase veloity s ion themal veloity Ion-sound waves fo ZTe Ti ae weakly damped ion-sound veloity s ZkBT M e Vlny 3
The above applied plasmati appoximation does not hold fo lage k ompaable with De. Consequently, dispesion elation will be deived without plasmati appoximation: E k e Zn n n i e e n e kt B e / Density petubation is inluded into Poisson equation n e k kt Zen i Zen B e i De k De potential is inluded into ion equation of motion Vlny 4
ZkBTe ikbt i k M k De M Dispesion elation of ion-sound waves diffes due to inlusion of deviation fom quasineutality only by the tem The most simple elation k k De, a T De n Ze i n Z e i pi M M ion plasma fequeny Vlny 5
Elmg. waves in plasmas without extenal magneti field B Maxwell s equation E B t E B t j we tansfom to wave equations E E t j t We expess high-fequeny uent v ee ene e j i E ime me eleton density does not hange (ontinuity n = ) we use identity A gaddiva A Vlny 6
phase fo E j E en E E e i t t t me t p t i k E E k p k k ( / ) / t p t v / k k p k p goup p d dk t v g k / / v wave does not popagate, it penetates into plasma only via skin-effet fo p k and wave is totally efleted (ut off fequeny) Vlny 7
me n e ne Re( ) n n itial density n = m -3 λ=,6 μm (Nd-lase) n = 9 m -3 λ=,6 μm (CO -lase) n = 3 m -3 λ=,6 m (m waves) Eleton-ion ollisions ause absoption ollisional absoption is also alled invese bemsstahlung absoption If ei is effetive eleton-ion ollision fequeny elative pemittivity is t with positive imaginay pat p ei p i ei ei Vlny 8
Waveveto has imaginay pat leading to wave damping in popagation dietion When eleton ollides with ion, the odeed osillation veloity of eleton tansfoms to stohasti themal veloity and wave must missing osillation enegy to the eleton and enegy of wave is thus absobed by eletons Nomal inidene of eletomagneti wave on plana plasma Pemittivity depends only on x ( = (x)) and waveveto is in x dietion B ot E t ot ot gad div D ot B t if time sale of density vaiations is E E t div D div E E div E x, t x i t E and i t e Vlny 9
E E k x stationay wave equation if ε is slowly vaying in spae WKB appoximation d d '' d i k x i k x E k E E E x e E x e E k E ik i E e i k x... x x x. řád. řád.řád. ode k E E fulfilled.ode E k ik i E x x 4 E k Vlny
E i k dx E E e e 4 4 i k dx WKB solution (no efletion!!) Suh density (pemittivity) pofiles do exist whee WKB solves the task exatly Citial point neighbohood - WKB appoximation is not valid Solution is found fo linea pofile of eleton density n e elative pemittivity ax is whee S in itial sufae Plasma density is gowing in dietion of x axis field must go to fo x 3 E d E ax is E ax is E x a d Vlny
Thee exists exat solution that fulfills the bounday ondition E 3CAi Ai = Aiy funtion 3 3 CJ J 3 3 3 3 3 3 C I I 3 3 3 3 Re() > Re() < S= (no absoption) - minima = Vlny
Oblique inidene and esonane absoption Pemittivity depends only on x ( = (x)) and waveveto sin x y x k k k k p-polaization E div E Re sin efletion point E B TE wave = s-polaization B E TM wave = p-polaization Vlny 3
d B d db dx dx dx E x kb sin y B sin B singulaity in the itial sufae Resonane absoption t l (tansvese elmg. wave tansfes into longitudinal) - l plasma wave annot esape fom plasma ollisional o ollisionless absoption in piniple linea poess it exists even at small intensities I when A f q 3 sin q k L Ex x nomal inidene no E x L Ex x efletion point fa fom x Vlny 4
os B 3 kl B x 3 6 (fo small q) 3 A q q q << 3 q >> 4 3 8 3 A exp q exp q 3 3 maximum A.5 when q.65 ollisions x i E x z B x sin Vlny 5
The width of field maximum n n n L L Absobed enegy / ei sin W E dz Ez x B x L / W B x L sin independent of Vlny 6
Wam plasma (spatial dispesion of longitudinal field) t 3v Te D E gad dive dive 3 Plasma wave popagates fom itial sufae to aified plasma, in deeasing density wavenumbe k gows and thus v deeases. When it beomes ompaable to themal veloity Landau damping (it aeleates eletons out of plasma) At highe intensities plasma wave damps via nonlinea mehanism of wavebeaking enegy is tansfeed to a small goup of so alled hot (fast) eletons. Eletons ae aeleated both into taget, and to plasma-vauum, whee most of eletons ae efleted in eletostati field of sheath bak into taget. Vlny 7
Elmg. waves in plasma with extenal magneti field B A. k B (popagation nomal to magneti field) if E B B does not influene wave odinay wave-o E B extaodinay wave - X influened by field B v v B v j E => y y x x x Eleti field is not nomal to the popagation dietion Vlny 8
im v e E v B e x x y im v e E v B e y y x j en v x x j en v y y Nomal uent also depends on the longitudinal omponent of eleti field Wave equation is witten fo E y and fo E x equation fo x-omponent of ot(h) en k Ey i jy i ie y Ex me i en Ex ijx iex Ey me Afte E x is inopoated into wave equation, we expess p the plasma pemittivity fo wave X p h Vlny 9
fo h pemittivity pemittivity fo L 4 p and fo R 4 p po h thee is esonane ( k ) total absoption of wave points L, R ae utoffs ( k ) total efletion of the wave B. k B (popagation along magneti field) Left-handed iula wave (L wave) p L - utoff Popagation fo L pemittivity fo R wave Vlny
Right-handed iula wave (R wave) yloton otation of eletons is ight-handed - esonane!! p popagates fo R and (whistles) esonane fo CMA diagam (eletomagneti waves) It ontains boundaies between patiula aeas of popagation types vetially popagation along B, hoizontally nomally to B dependene of phase veloity on dietion of popagation tansition between types of waves if popagation dietion hanges Vlny
Ion eletomagneti waves (hydomagneti waves) B exist only in pesene of a) k B Alfvén wave ik E ib eleton density ik B ie ne vi v e v ex E B v ey ee ev B e im v ZeE Zev B B i i i EB dift equation of motion Vlny
ZeB ZeB M viy i vix i vix Mi v ix i M ZeE i x pi Ze n M We substitute uent into wave eq. dispesion el. fo Alfvén veloity i k v A v i ion plasma fequeny A ion yloton fequeny k pi whee v B A pi M B z, t B k Y z t dz i Y B E z y y B, vb B y x B k B Y B shift of field line position, v B veloity of field line motion Vlny 3
Ex vb vd B Plasma moves with the same veloity as the field line Magneti field line ae fozen in plasma Alfvén wave tansvese waves on sting Sting magneti field line (Hannes Alfvén Nobel pize 97) b) k B magnetosound wave E B dispesion elation fo old plasma k v k A v A va + fo Te additional ation of pessue v A s k k v A s va + Vlny 4