Chapter 6 Polarization and Crystal Optics

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1 EE 485, Winter 4, Lih Y. Lin Chpter 6 Polriztion nd Crstl Optics - Polriztion Time course of the direction of E ( r, t - Polriztion ffects: mount of light reflected t mteril interfces. bsorption in some mterils. Scttering. Refrctive inde (thus velocit of nisotropic mterils. Opticll ctive mterils to rotte polriztion. 6. Polriztion of Light Consider E ( z, t Re{ ep( ω( t z } c (6.- With comple envelope ˆ + ˆ (6.- Trce the endpoint of E ( z, t t ech position z s function of time. The polriztion ellipse ep( ϕ, ep( ϕ E z, t E ˆ + E ˆ (6.-3 ( [ ω( t z + ϕ ] c [ ω( t z + ϕ ] E cos (6.-4 E cos c (6.-4b E E EE + cos ϕ sin ϕ (6.-5 n ellipse. The shpe of the ellipse depends on nd ϕ. The size + of the ellipse determines intensit I (η: impednce of the medium. η

2 EE 485, Winter 4, Lih Y. Lin Rottion direction (viewed from the direction towrds which the wve is pproching: ϕ > ϕ : Clockwise rottion ϕ < : Counter-clockwise rottion ϕ Linerl-polrized light E onl E onl Or ϕ or π E E for ϕ E E for ϕ π ±45 polriztion Circulrl-polrized light

3 EE 485, Winter 4, Lih Y. Lin ϕ π Right circulrl-polrized ϕ π Left circulrl-polrized B. Mtri Representtion The Jones Vector Comple envelopes for E ( r, t : ep( ϕ, ep( ϕ J Orthogonl polriztions * * J, J + (6.-7 ( Epnsion of rbitrr polriztion If J nd J re normlized nd orthogonl to ech other, then n rbitrr polriztion J αj + α J α ( J, J, α ( J, J 3

4 EE 485, Winter 4, Lih Y. Lin 4 Emple: + Mtri representtion of polriztion devices Input:, Output: T T T T T (6.-9 T: Jones mtri TJ J (6.- Liner polrizers T (for ˆ polrizer (6.-, Wve retrders Γ ep( T (6.- Γ ep( component is deled b phse Γ. : fst is, : slow is. Emples: ( π / Γ (qurter-wve retrder

5 EE 485, Winter 4, Lih Y. Lin 5 ( π Γ (hlf-wve retrder Polriztion rottors cos sin sin cos T ( sin( cos( sin cos Cscded polriztion devices... T T T T M Coordinte trnsformtion J R J ( ' (6.-4 cos sin sin cos ( R (6.-5 ( ( ' TR R T (6.-6

6 EE 485, Winter 4, Lih Y. Lin T R( T' R( (6.-7 Norml modes (of polriztion sstem Sttes of polriztion tht re not chnged when the wve is trnsmitted through the opticl sstem. TJ µj (6.-9 µ: eigenvlue J: eigenvector If T is Hermitin, i.e., T T *, the norml modes re orthogonl to ech other, nd cn be used s n epnsion bsis. The response to the sstem cn be evluted more esil if the input wve is decomposed into the two norml modes: J αj + α J TJ T( α J + α J α µ J + α µ J Emples (Eercise 6.-4: ( The norml modes of the liner polrizer re linerl polrized wves. (b The norml modes of the wve retrder re linerl polrized wves. (c The norml modes of the polriztion rottor re right nd left circulrl polrized wves. 6. Reflection nd Refrction E E E3 J, J, J3 E E E3 J tj, t : Jones mtri for trnsmission J 3 rj, r : Jones mtri for reflection t r t, r t r 6

7 EE 485, Winter 4, Lih Y. Lin E te, E t E E3 r E, E3 r E (6.- (6.-3 For trnsverse electric (TE polriztion, E E ˆ, H H n cos n cos r n cos + n cos (6.-4 t + r (6.-5 For trnsverse mgnetic (TM polriztion, E E, H Hˆ n cos n cos r n cos + n cos (6.-6 n t ( + r n (6.-7 r, t, r, t cn be comple numbers r r ep( ϕ, r r ep( ϕ TE polriztion Eternl reflection ( n < n ϕ π r n n n + t º ( º n 7

8 EE 485, Winter 4, Lih Y. Lin Internl reflection ( n > n For < c, ϕ t º ( º, r n n n + n t c, r For > c, totl internl reflection, r, ϕ tn sin sin c (6.-9 cos TM polriztion Eternl reflection ( n < n r is rel t º ( º, r n n n + is positive n n t B tn, r ( B : Brewster ngle n For > B, r becomes negtive, π t 9º, r -, ϕ π ϕ 8

9 EE 485, Winter 4, Lih Y. Lin Internl reflection ( n > n t º ( º, r is negtive, r n n n +, ϕ π n For < B, r decreses with, ϕ π t B, r For > B, r becomes positive nd increses with, ϕ t c, r For > c, totl internl reflection, r, sin sin ϕ c tn (6.- cos sin c 9

10 EE 485, Winter 4, Lih Y. Lin Power reflectnce nd trnsmittnce R r (6.- T R (6.-3 t norml incidence, for both TE nd TM, n n R n + n Emples: ( Glss (n.5 nd ir (n interfce R.4 for norml incidence (b Gs (n 3.6 nd ir (n interfce R.3 for norml incidence

11 EE 485, Winter 4, Lih Y. Lin 6.3 Optics of nisotropic Medi. Refrctive Indices Permittivit tensor D ε E, i,,, 3 (6.3- i { } ε i ε i : Electric permittivit tensor (3 3 D εe ε is smmetricl, ε ε onl 6 independent numbers. i i Principl es nd principl refrctive indices Choose coordinte sstem such tht ε ε ε ε3 D εe, D εe, D3 ε3e3 (6.3- This coordinte sstem defines the principl es of the crstl. Principl refrctive indices: ε ε ε3 n, n, n3 (6.3-3 ε ε ε Isotropic, uniil, nd biil crstls Isotropic: n n n3 Uniil: n n no (ordinr inde, ordinr es n 3 n e (etrordinr inde, etrordinr is Positive uniil: n e > no Negtive uniil: n e < no Biil: n n n3 The inde ellipsoid n n n3 (6.3-7 Direction of D determines the refrctive inde from the inde ellipsoid.

12 EE 485, Winter 4, Lih Y. Lin B. Propgtion long Principl is Consider plne wve propgting long z-direction. Norml modes For E Eˆ, D εeˆ, c k n k c n ˆ, D εeˆ For E E, c k n k c n Polriztion long n rbitrr direction Decompose the electric field: E E ˆ + Eˆ The - nd -components trvel with different speeds. The phse retrdtion ϕ ( n n kd Linerl-polrized wve becomes n ellipticll-polrized wve. The crstl cts like wve retrder. C. Propgtion in n rbitrr Direction

13 EE 485, Winter 4, Lih Y. Lin Determine the polriztions nd refrctive indices n nd n b of the norml modes of wve trveling in û direction. ( Drw plne pssing thru the origin of the inde ellipsoid, norml to û. The intersection of the plne with the ellipsoid is n ellipse, clled the inde ellipse. ( The hlf-width of the mor nd minor es of the inde ellipse re the refrctive indices n nd n b of the two norml modes. (3 The directions of the mor nd minor es of the inde ellipse re the directions of the vectors D nd D b for the norml modes. These directions re orthogonl. (4 The vectors E nd E b m be determined from D nd D b b D εe. k H ωd (6.3-8 E ωµ H (6.3-9 k Emple: Uniil crstl For wve trveling t n ngle with the opticl is (z-is, the inde ellipse hs hlf-length n o nd n (. cos sin + (6.3-5 n ( n o n e The norml modes hve refrctive indices n n for ordinr wve, nd n b n( for etrordinr wve. n ( no when º, nd n ( ne when 9º. For ordinr wve, D is long the principl is, D nd E re prllel. D is perpendiculr to k nd ẑ. For etrordinr wve, D nd E re not prllel (unless D is long ẑ. D is in the k-z plne. o 3

14 EE 485, Winter 4, Lih Y. Lin D. Double Refrction In the crstl, the opticl wve cn be decomposed into the two norml-mode components, propgting in two different directions: For ordinr wve (TE, E opticl is, sin no sin o For etrordinr wve (TM, sin n( sin e e 6.4 Opticl ctivit nd Frd Effect. Opticl ctivit Certin mterils ct nturll s polriztion rottors. The norml modes re circulrl-polrized wves. The phse velocit for the right-circulrl-polrized wve is c n+, nd the phse velocit of the left-circulrl-polrized wve is c. n 4

15 EE 485, Winter 4, Lih Y. Lin The mteril rottes linerl-polrized wve b propgtion distnce of d. π( n n λ ρ + π( n n λ φ + is clled rottor power. d fter Medium eqution: D εe ε ξ E (6.4- For plne wve E( r Eep( k r, G ξk, D εe + εg E (6.4-3 Depends on k direction. Therefore reversl of the propgtion reverses the sense of rottion of the polriztion plne. B. Frd Effect The mteril cts s polriztion rottor when plced in sttic mgnetic field. The rottor power is proportionl to the component of B in k-direction. Medium eqution: D εe + εγb E (6.4-8 Independent of k, but dependent on B. Therefore reversl of the propgtion does not reverse the sense of rottion of the polriztion plne. Opticl isoltors cting s one-w vlve for the opticl wve. 5

16 EE 485, Winter 4, Lih Y. Lin 6.5 Liquid Crstls The molecules re elongted, with n e for the long is nd n o for the short is. The twist ngle of the molecules: αz (6.5- Phse retrdtion coefficient: β ( ne no k (6.5- In generl, β >> α. If the incident wve t z is linerl polrized, the wve mintins its liner polriztion, but the plne of polriztion rottes in lignment with the moleculr twist. Output wve polrized t αd. 6

17 EE 485, Winter 4, Lih Y. Lin pplictions: Opticl switch nd displ (LCD. 7

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