( ) 4. Jones Matrix Method 4.1 Jones Matrix Formulation A retardation plate with azimuth angle y. V û ë y û. év ù év ù év. ë y û.
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1 4. Jons Mati Mthod 4. Jons Mati Foulation A tadation plat with aziuth angl Yh; 4- Linal polaizd input light é = ë û Dcoposd into th slow and ast noal ods és é cos sin é = sin cos ë- û ë û R ( ), otation ati At th output plan, dint phas dlas o th slow and ast ods -ins ( p / l) d / é i s 0 - G s -i é 0 s = Þ ( / ) i / -in p l d G 0 é é é 0, û Th output light is givn b é ( ) o ( ) = R - W R é ë û º W o p G = ( ns - n ) d, phas tadation l p = ( ns + n ) d l º W, Jons ati o wav plat. W W =, : unita dt W =, : uniodula
2 A. Hal-wav Rtadation Plat and Quat-wav Plat Ÿ Hal-wav Rtadation Plat G = p Yh; 4- A vtical polaization light. é0 = Th Jons ati o th plat with é - é-i 0 é é 0 -i W = 0 i = - -i 0 = 45 o (btwn slow- and -ais) Th output light is hoizontall polaizd é-i = 0 Rotation o th polaization b 90 o In gnal Th hal-wav plat with otats th lina polaization b. It convts LHC polaization into RHC polaization and vic vsa, gadlss o.
3 l Quat-wav Plat G = p / Yh; 4-3 Th Jons ati o th plat with = 45 o (btwn slow- and -ais) -ip /4 é - é 0 é é -i W = ip /4 = 0 - -i ë û Fo a vticall polaizd incidnt light, th output light is é-i = LHC polaization In gnal tical polaization Lt-hand cicula polaization Hoizontal polaization Right-hand cicula polaization
4 B. Gnal Poptis o Jons Mati Fo a light popagating o z=0 plan to z=l plan. é L é = M () L û û Yh; 4-4 Fo a light popagating o z=l plan to z=0 plan. é é L = () û L û Ti Rvsal St Fo () é é L = û L û é = * û é 0 é 0 * = M 0 û 0 û é = * M = (3) * û Pincipl o Rcipocit Tanspos o () Appl = é é ë L L û ë û M% é û on both sids é L é L é ë L L û = é L M % û L û é é = * û é = ë û M % 443 û It should b = M % (4) * 0 0 Þ = 0 0 é é é L û û L û Fo (3) and (4) = M M = : Unita
5 Ÿ Rtolction Th incidnt wav using th pola and aziuth angls k q, = k sinq cos ˆ + sinq sin ˆ + cosq zˆ Th tolctd wav k = -k q and Yh; 4-5 Ÿ Mio lction o -plan k = k sinq cos ˆ + sinq sin ˆ - cosq zˆ It is notd k = k ( q, + p ) Jons atics o k, k, k M,, M Jons atics o (5) M ( q, ) = ( q, + p ) Fo cipocit ( q, ) = M % ( q, ) (5) Jons ati o io lction M (, ) = M % (, + ) q q p 4. Intnsit Tansission Spctu Th lctic ilds o th incidnt and ging wavs ée E =, E û ée E = E û Th intnsit is dind as I = E E = E + E Hitian conjugat Th tansittanc is dind as T = E E + E + E
6 4.3 Optical Poptis o a Twistd atic Liquid Cstals(T-LC) T-LC is subdividd into wav-plats with dint aziuth angls. Yh; 4-6 Aziuth angl o th -th plat :., total twist angl. G p d = - o. l Th phas tadation o a plat : ( n n ) Th Jons ati o T-LC = M = P W M = R (- ) é æ ö o ç = W R ë è øû Jons ati o th -th plat. R æ ö æ ö ç - o ç è ø W R è ø W -ig/ é 0 = ë 0 û o ig/ Th Jons ati bcos é -ig/ -ig/ cos sin M = R (- ) ig/ ig/ - sin cos û Us Chbshv s idntit and lt é G sin sin cos - i écos - sin M = sin cos, ë û sin G sin - cos + i û wh æ G ö = + ç è ø (6) Us s-coodinats o o-coodinats (slow/ast, o taodina/odina) é G sin sin cos - i é é = ë sin sin o û G o - cos + i û Paalll to output dicto. Paalll to input dicto. (7)
7 A. Adiabatic Following (Wavguiding in T-LC) Fo E-od opation é G sin cos - i é é é = 0 = ë o û ë sin o û - û (8) Yh; 4-7 In ost T-LC, << G (LC o E7: d = 0, D n = 0.3, = p /, l = 500 n / G = /37 ) Th output Jons ati is appoiatd -ig/ é é = ë o û ë 0 û Input polaization is paalll to th dicto at th ntanc suac. Output polaization is also paalll to th dicto at th it suac. (Polaization otation ss to ollow th twist o LC olculs) This also happns o th O-od opation. B. 90 o Twistd atic Liquid Cstal 90 o T-LC btwn paalll polaizs. Onl o o (8) tansits though th scond polaiz. sin sin + u æ ö T = ç = è ø + u G u = = n - no d, = p / l, wh T 0 o << G Th sidual tansission is du to th sall llipticit in th output ba. Polaization Ellips Eq. (7) psnts a polaization llips, not a lina polaization. Th llipticit and th angl o th ajo ais a,
8 C. Tansission Poptis o a Gnal T-LCD Yh; 4-8 Th input and output polaization stats é écos Fnt é écos Fit =, sin = nt sin û ë F û û ë Fit û Th tansittanc is % * T = M Changing th vaiabls a = F, b = F - nt it Th aiu tansittanc T= whn tan a = tan b = -a
9 D. oal Mods o Popagation in a Gnal T-LC Th ignvalu quation o (7) Yh; 4-9 é G sin sin cos - i é é = g sin G sin o o - cos + i û Th two solutions é é - i + u =, g = : twistd taodina od ë - i ( + u - u o û ) ( ) + i + u, g : od, wh é é- i + u - u G = = to u = = n - no d ë o û l Two noal ods a llipticall polaizd and othogonal to ach oth. Th ajo as a paalll to ith -ais o o-ais. Th phas tadation is G = + u = Linal Polaizd Ba W want a linal polaizd output ba o a linal polaizd input ba. W din c = o, al o a linal polaizd ba Fo (7) th polaization stat o th output ba is sin æ G sin ö - + ç cos + i c c = è ø æ G sin ö sin ç cos - i + c è ø Th solution o al c and c c n n, c c = - ± + = - : wh n = tan Th aziuth angls o th two input stats a p tan = + n - n, = + : Two stats a othogonal. Th aziuth angls o th two output stats a = -, = -
10 4.4 Phas Rtadation at Obliqu incidnc 4.5 Conoscop 4.6 Rlction Popt o a Gnal T-LCD with a Ra Mio Yh; 4-0
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