3. Electromagnetic Propagation in Anisotropic Media 3.1 Maxwell s Equations and Dielectric Tensor _

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1 3. ltrmagti Prpagati i Aistrpi Mdia 3. Mawll s quatis ad Diltri Tsr _ b t a J + t a ρ _ Yh; 3- Th stitutiv rlatis a εb εb + m _ μ μ + j ε ad μ : Prmittivity ad prmability tsrs f rak. Thy dpd b ad i strg filds. m ad j : ltri ad magti plariatis. Rwrit a ε b : Summati vr rpatd idis i ij j Diltri tsr I magti ad traspart mdia, th tsr is ral ad symmtri, ε ε ij ji Th tsr bms diagal i a wll hs rdiats(priipal diltri as), ε ε εy ε y. ε priipal diltri stats priipal rfrativ idis. If th diltri stats dpd frquy, th mdium is disprsiv. At lw frquis (H-GH) Th diltri stats ar rspsibl fr th fft f applid ltri fild. At high frquis Th rfrativ id is rspsibl fr th prpagatig wavs.

2 3. Pla Wavs i Hmgus Mdia ad Nrmal Surfa A pla wav is giv by i( ωt k r) iωt ( r, t) R i i R ( t k r) b, ( r, t) R H ω i Cmpl amplitud Phasr Yh; 3- Isrt ths it Mawll s qs. k ωμh (a) k H ωd H s, μ D s H (b) Frm (a) k ( k ) + ω με ω k sˆ ω με ky k kky kk kk y ωμεy k k kk y y kk kk y ω με k k y dt.. fr trivial sluti () Th sluti rprsts a 3-D surfa i k-spa. Nrmal surfa It sists f tw shlls havig fur pits i mm. Tw lis frm th rigi t th mm pits ar ptial as. (a) O tat fr < y < (b) Fr y <. A sphr ad a llipsid. Tw diffrt k valus fr a giv prpagati dirti.

3 ω Slv () usig dt.., k sˆ, s s y s + + y ε ε ad s. is ukw., y, ar fid i th mdium : Frsl quati f wavrmals Yh; 3-3 Th ltri fild is giv as s sy s y (3) Orthgality f Nrmal Mds Th tw slutis, D ad D a D ˆ ˆ s, D s ( ) R i t k r D ω i It a b prvd D D (a) (b) Frm (3b) H s, D s H. () μ D s ( s ) s s i μ μ trasvrs. μ ( ), D ad s ar i th sam pla. (d) Cmbiig (a)-(d) Nt that D D, but i gral. igmds ar rthgal ad thy satisfy s i : Orthgality diti (4) ( H )

4 Yh; 3-4 A. k i th y-pla k i q. () ω με ky kk y kk y ω μεy k y ω με k k y (a) Th sluti is with k + k ωμε ωμε y ω k k + ky Th thr sluti frm (3), Frm dt.. i X prti f (a) s s y y / y s θ + y si θ θ : agl btw sˆ ad ˆ ω k B. k i th y-pla Similarly, th tw rmal mds ar btaid as with k ω s y y s with k θ ω y, y s θ + si / C. k i th -pla Th tw rmal mds ar btaid as with k y ω s s with k ω θ, s θ + si /

5 D. Classifiati f Mdia Thr priipal idis, y, ar all diffrt Biaial rystal, tw ptial as. Yh; 3-5 Lablig vti < < Optial as i -pla y Wh ε ε y ε,. Uiaial rystal, ptial ais. ε ε ε rdiary id. trardiary id > < psitiv uiaial (mst LC) gativ uiaial quati f th rmal surfa i this as : k + ky k ω k ω + : A sphr ad a llipsid tuhig at tw pits -ais. Wh y, th rmal surfa bms a sphr. Istrpi rystal Optial symmtry is lsly rlatd t th pit grup f th rystal. [Tabl 3.]. Pwr Flw i Aistrpi Mdia Th ltri fild with tw rmal mds + Th magti fild is H H + H s s μ + μ Th Pytig vtr S H. Usig rhgality diti (4) S isˆ sˆi H + sˆi H : Pwr rthgality thrm ( ) ( ) Th ttal pwr alg th bam dirti is a sum f idividual md pwrs

6 3.3 Light Prpagati i Uiaial Mdia Mst LCs ar uiaial, frm () k + ky k ω k ω + : Nrmal surfa f uiaial rystal A llipsid. A sphr : Thy mt at tw pits -ais Rfrativ id f -wav. Rfrativ id f -wav. s θ si θ + : θ btw k ad pti ais (5) A futi f θ Yh; 3-6 Th ltri fild frm () with ε ε ε ad ε ε y s ssy ss ss y sy ss y y ss ss y s sy O sluti is -wav giv by sy s ω with k ˆ s Pla mad by k ad ˆ. Th ltri displamt is giv frm D ε k ˆ D k ˆ Frm (3), -wav is btaid as s s y ω with k ( ) ˆ θ s s Th ltri displamt is giv by D k D D k Sphrial Crdiats siθ sφ sˆ siθ siφ, sθ siφ siφ s φ, D sφ, sθ sφ sθ sφ sθ si φ, D sθsiφ siθ siθ (6)

7 A. Prpagati Prpdiular t th -ais θ 9 sφ sˆ siφ Yh; 3-7 Frm (5), rfrativ idis f -wav ad -wav ar giv by ad. Th ltri fild vtrs frm (6) siφ sφ : Prpdiular t s ad ˆ : Paralll t ĉ Th ttal ltri fild i phasr tati ik ir ik ir ω ω +, : k ˆ, ˆ s k s ampl Cirular plariati wh 9 rtati f ttal ω π ω ( ) d, ( ) d π wh

8 B. Prpagati i -pla φ siθ sˆ sθ Yh; 3-8 Rfrativ idis fr -wav ad -wav frm (5), / s θ si θ, ( θ ) + Th ltri filds frm (6), sθ, siθ C. Prpagati alg -ais θ sˆ Oly rfrativ id Oly rmal md. ( ). 3.4 Dubl Rfrati at a Budary Sll s law is applid at th budary siθ siθ But i i t t t is a futi f th trasmissi agl θ t i gral.

9 3.5 Aistrpi Absrpti ad Plarirs Crtai matrials hav aistrpi absrpti as wll as birfrig. Yh; 3-9 Plarid sht sists f aligd dl-lik hrapathit mluls. ltri fild paralll t th absrpti ais Strg absrpti prpdiular Wak absrpti -typ plarir trasmits -wav ad attuats -wav. i( ω/ ) sˆ ir wav :, i( ω/ ) sˆir ( ω/ ) sˆir wav : i, Attuati f th ltri fild. -typ plarir trasmits -wav ad attuats -wav. i( ω/ ) sˆir ( ω/ ) 3sˆir wav : i3, i( ω/ ) ( θ) sˆ ir wav : θ, Frm (5), ( θ ) ( ) / s θ si θ + with mpl ( θ ) is als mpl. I this as -wav is als slightly attuatd. If sˆ -ais, θ π / θ is ral. N attuati f -wav ad ( ) A. titi Rati ad Ral Sht Plarirs Th titi rati is T trasmissi with plariati tras. ais T trasmissi with plariati tras. ais Tp T + T Trasmitta f uplarid light thrugh a pair f rssd plarirs : T TT Trasmitta f uplarid light thrugh a plarir : T ( T + T ) Trasmitta f uplarid light thrugh a pair f paralll plarirs : ( ) B. Fild f Viw f Crssd Plarirs Crssd plarirs ˆ ˆ -wavs t ths plarirs k ˆ ˆ k ˆ ˆ, NOT ˆ ˆ i gral. k ˆ k ˆ Th lakag f uplarid light thrugh a pair f rssd plarirs 4 si θ si φ s φ T ˆi ˆ ( si θ si φ)( si θs φ) k sˆ siθ s φ, siθ si φ, sθ Us ( ) N lakag at φ r 9 Larg lakag at larg θ with φ 45

10 3.6 Optial Aitivity ad Faraday Rtati Optially ativ rystal a rtat th pla f plariati f th trasmittd light. Du t mlul strutur ad arragmt Yh; 3- I liquid, mlul strutur dtrmi th ptial rtatry pwr. Mstly hlial strutur. Dgrs pr timtr. Quart urs i bth right-hadd ad lft-hadd rystalli frms. Frsl bsrvd that th ptial ativity ariss frm irular dubl rfrati. igmds ar right ad lft irularly plarid wavs Dtrrtatry, r right-hadd. (Th ppsit is lvrtatry, r lft-hadd) At th iput pla, th wav is liarly plarid alg -ais Aftr th prpagati vr a dista, th ltri fild is D ( / ) ( / ) iωt i ω r i ω l ( R + L ) ( iy), ( + iy ) ( l r ) ( l r ) iωt i( ω/ ) ( ) ω ω l + r D s + y si Liar plariati makig a agl f ω( )/ with rspt t -ais. π l r. λ Rtatry pwr is ρ ( ) l r Thry f ptial ativity Th idud dipl p α H Du t hiral strutur f mluls Chagig magti fild Idud urrt Tim-varyig sparati arud H. f hargs alg H. β Th ltri flu dsity is giv by D ε + iε G ε + iε G ( [ ]) Gyrati tsr Gyrati vtr, paralll t th prpagati dirti

11 Yh; 3- A. Faraday Rtati Th pla f plariati rtats i a magti fild applid paralll t th bam dirti. Th gyrati vtr is giv prprtial t B G γ B, γ : matgyrati ffiit Th spifi rtati (Rtati pr uit lgth) ρ VB, V : Vrdt stat f th iidt bam Displamt f ltr Lrt fr i B Latral displamt f ltr D ε + iε γb 3.7 Light Prpagati i Biaial Mdia skip