Electric Anisotropy, Magnetic Anisotropy, Uniaxial and Biaxial Materials, Bianisotropic Media (Definitions)

Transcription

1 leti nisotop agneti nisotop Uniaial and iaial ateials ianisotopi edia efinitions medium is alled eletiall anisotopi if tenso Note that and ae no longe paallel medium is magnetiall anisotopi if tenso Note that and ae no longe paallel whee is the pemittiit μ whee μ is the pemeabilit medium an be both eletiall and magnetiall anisotopi onside the ase of eletiall anisotopi medium fo whih z z z z zz stals in geneal ae desibed b a smmeti pemittiit tenso hen thee alwas eist a oodinate tansfomation that tansfoms the smmeti mati to a diagonal mati as gien b his new oodinate sstem is alled the inipal Sstem and the zz thee oodinate aes ae alled the inipal es Fo ubi stal and the stal is isotopi zz Fo tetagonal heagonal and hombohedal stals two of the thee ae equal Suh stal is alled uniaial zz he pinipal ais that is diffeent displas the anisotop is alled the optial ais Fo the aboe z-ais is the optial ais Fo the aboe stal thee is a two dimensional degenea If zz > we sa that the medium has positie uniaial behaio and if zz < we sa that the medium has negatie uniaial behaio

2 If zz we sa that the stal is biaial amples of biaial stals ae othohombi monolini and tilini bianisotopi medium poides a oupling between eleti and magneti fields he onstitutie elations fo a bianisotopi medium is gien b ξ ς μ bianisotopi medium plaed in an eleti o magneti field beomes both polaized and magnetized lmost an media in motion beomes bianisotopi he fist ases of bianisotopi mateials wee indeed moing dieletis and magneti mateials in the pesene of eleti o magneti fields In 888 Roentgen disoeed that moing dieletis beome magnetized when plaed in an eleti field In 95 Wilson showed that a moing dieleti beomes eletiall polaized when plaed in a unifom magneti field he topis of moing mateials and thei onstitutie elations ae the subjet of the elatiisti eletomagneti theo Speial elatiit equies that all phsial laws to be haateized b mathematial equations that ae fom-inaiant fom one obsee to the othe independent of the elatie motions of the two obsees hat is to sa that the phsial laws emain fominaiant unde oentz tansfomation awell s equations ae fom-inaiant; howee onstitutie elations ae onl fom-inaiant when the ae witten in the bianisotopi fom agnetoeleti ateials: al isto agnetoeleti mateials wee fist poposed b andau and ifshitz [957] and zaloshinsii [959] he wee fist obseed b sto in 96 in antifeomagneti homium oide he onstitutie elations poposed b zaloshinsii was of the fom ξ ξ and zz ξ zz

3 ξ ξ μ ξ zz μ μ zz ate it was shown b Indenbom [96] and iss [96] that 58 magneti stal lasses an ehibit magnetoeleti effets In 948 ellegen intodued a new element alled gato whih in addition to esisto induto apaito and tansfome was used to desibe an eleti netwo o ealize this new element ellegen had imagined a new medium fo whih the onstitutie elations wee gien b ξ and ξ μ whee ξ μ ellegen had assumed that the medium had pemanent eleti dipole p and magneti dipole m that wee anti-paallel to eah othe suh that an applied whih aligned the p also aligned the m o similal an applied whih aligned the m also aligned the p ellegen also onsideed the geneal onstitutie elations ξ and ς μ and studied the smmet popeties b onsideing the eneg onseation hial edia Fo hial mateials the onstitutie elations ae gien b t μ t whee is alled the hial paamete amples of hial mateials ae suga solutions amino aids N et hial mateials ae bi-isotopi onstitutie aties he onstitutie elations in the most geneal fom ae witten as

4 and whee is the speed of light in auum and and ae maties whih thei elements ae alled the onstitutie paametes Note that and elate the eleti and magneti fields togethe When and the medium is alled bianisotopi When thee is no oupling between eleti and magneti fields ie and we hae and In this ase the medium is alled anisotopi If I and I μ whee I is the identit mati then medium is said to be isotopi he elations and an be witten as ee is a 6 6 onstitutie mati boe is alled - pesentation he eason fo hoog the aboe fom is that onstitutie elations witten as ae fom inaiant unde oentz tansfomation he ae so alled oentz-oaiant and eah fom a gle tenso in fou dimensional spae Othe epesentations ae also possible Fo eample o o whee the ae alled - - o - pesentation espetiel eise: Find the mati elements fo in tems of and nisotopi edium and K sstem We onside awell s equations in a soue fee egion e i i J J ρ he time hamoni awell s equations ae gien b j ω j ω and he assumption that thee ae no soues within a gien egion of spae does not mean that thee ae no soues anwhee else In fat if this was the ase thee will be no field anwhee We assume that fields ae geneated at a gien point in spae and now we ae studing thei dnamial eolution awa fom the soue

5 We hae seen that fo a plane wae j ep we hae ω ω Fom the last two equations we see that is pependiula to the plane ontaining both and et us all this plane the --plane If μ μ is a sala funtion then also lies in the --plane Fo a medium with we see that ma not lie on the --plane Fo this eason in anisotopi medium we define the polaization in tems of instead of whih Reall that onting eto and hene the powe flow is along t t is not neessail in the same dietion as the popagation eto inside an anisotopi medium In othe wods the dietion of powe flow fo a plane wae inside an anisotopi medium is not neessail the same as the dietion of the wae eto K oodinate Sstem o mae ou stud of anisotopi medium easie we will tansfom ou z oodinate sstem to the K oodinate sstem Wheeas the unit eto in z ae aˆ aˆ aˆ z in the K we designate them b e ˆ ˆ e We will tae the to be along ê ie ê Fom the figue we an see e aˆ os aˆ aˆ os ˆ z ê lies in the --plane and is pependiula to the pojetion of ê to the --plane It is gien b π π e aˆ aˆ ˆ os e aˆ os aˆ ˆ ê an be alulated fom e ˆ e aˆ os os aˆ os aˆ ˆ K sstem an be obtained fom the z sstem b multiple otations z

6 ansfoming a Veto Fom z to K and Vie Vesa et the eto in z sstem to be gien b z and in K sstem b then and whee os os os os os os Note that is unita eise: Show that indeed is gien b aboe and alulate onstitutie Relations in the K sstem: he Fomulation Reall that in z sstem the fomulation of onstitutie elations was gien b With the help of tansfomation and we will find the equialent elations in the K sstem Note that in the K sstem the and will tae a simple fom than and e [eall ê ] In long hand an be witten as Ug the fat that and and an be witten as 4 5 ultipling 4 and 5 with and eaanging tems we hae 6

7 7 he last two equations an be witten as whee the definition of and ae eident ispesion Relation fo ianisotopi edium In the K sstem simila to z sstem we hae ω ω 4 Fom and 4 and e in K ê then and Fom ω ω ˆ e ω ˆ e ω ˆ e ω ˆ e ω and ω

8 Now eall that and Ug and in ω and ω we hae [ ] ω and [ ω 4 ] and 4 an futhe be witten as ω and 5 ω 7 5 and 7 an be gien in mati fom aoding to ω ω 8 ω ied both side of 8 b and let u we hae u u We aied at aboe b ug ω and We do the same steps but now with ω and to aie at u u Now 9 and an be used to eliminate the o Fo eample let us eliminate the b ug We hae u u 9

9 u u u u he aboe an be finall witten as u u u u Fo aboe equation to hae nontiial solutions the deteminant of the mati multipling must be zeo his ondition will poide us with the equied dispesion elation eise: Find eoe the dispesion elation fo an isotopi homogeneous medium haateized b and fom ou peious disussion