J = 1 J = 1 0 J J =1 J = Bout. Bin (1) Ey = 4E0 cos(kz (2) (2) (3) (4) (5) (3) cos(kz (1) ωt +pπ/2) (2) (6) (4) (3) iωt (3) (5) ωt = π E(1) E = [E e

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Download "J = 1 J = 1 0 J J =1 J = Bout. Bin (1) Ey = 4E0 cos(kz (2) (2) (3) (4) (5) (3) cos(kz (1) ωt +pπ/2) (2) (6) (4) (3) iωt (3) (5) ωt = π E(1) E = [E e"

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1 ) ) Cov&o for rg h of olr&o for gog o&v r&o: - Look wv rog&g owr ou (look r&o). - F r wh o&o of fil vor. - I h CCWLHCP CWRHCP - u &l & hv oo g, h lr- fil vor r ou rgh- h orkrw for RHCP! 3) For h followg fil: - r olr&o of h wv (lr, rulr, ll&l h) () l45 ( fi ). - rw h fu&o of & ( fi ) - wr ow hr o vor rr&o or hr ol lookg h r&o () l - () ω 6 5 () 6 5 [ () ω ] () ω π/ () S CW Rgh h () () 4 o(k ω π/) () ω π S 4 o(k ω) Crulr olr&o () ω ()() ω π () ω] ( o [ ) o(k ω) vor RHCP () 4 o(k ω) ω π - () ω 4 () [() ] () ω π/ 4 o(k ω) () k 4 o(k ω π/) ( ) 3- Norl o vor h u oo) 4 o(k (uull ω π/) ω π/ 4- ω () ω [ ] ) o(k 4 o(k ω) ( ω) 4 Bou: D r h olr&o oo(k f of ω π/) ω ω π () R g wv of wo our- rog&g [ ] 5- () ω w [v ] l ω 4 o(k ω π/) ou () ω () k ω [ ] ) π/ ou.5 ( ) ω π/ ) ω Bou B

2 - RHCP ro. log - () (ˆ ŷ) k [ () ω ] (ˆ ŷ) o(k ω) Th lrl olr 45 gr wr o - o vor Norl o RHCP 3- H: o vor u ou r rog6g h r6o wh rgh- h oor 4-5- Bou: Dr h olr&o of of g wv of wo our- rog&g l wv

3 - RHCP r&g lrl olr 45 gr wr o - lr- fl vor r ou rgh- h orkrw for 3- () (ˆ π ŷ) k (ˆ ŷ) k Th lo lrl olr 45 gr wr o -. Howvr, o h () h rog&o h r&o! You gh o wr h o vor Bu h woul NOT orr!! o vor u ou r rog&g h r&o wh rgh- h oor 4-5- Bou: Dr h olr&o of of g wv of wo our- rog&g l wv

4 - RHCP r&g lrl olr 45 gr wr o - lr- fl vor r ou rgh- h orkrw for 3- () (ˆ π ŷ) k (ˆ ŷ) k Th lo lrl olr 45 gr wr o -. Howvr, o h () h rog&o h r&o! You gh o wr h o vor Bu h woul NOT orr!! o vor u ou r rog&g h r&o wh rgh- h oor rl I h w oor rog log (ˆ ŷ ) k o k RH oor To vul h: hk ou ro3g h,, rou o rg o - o vor Norl o vor

5 - RHCP r&g lrl olr 45 gr wr o - lr- fl vor r ou rgh- h orkrw for 3- () (ˆ π ŷ) k (ˆ ŷ) k Th lo lrl olr 45 gr wr o -. Howvr, o h () h rog&o h r&o! You gh o wr h o vor Bu h woul NOT orr!! I h w oor o vor (ˆ ŷ ) k Norl o vor 4-5- Bou: Dr h olr&o of of g wv of wo our- rog&g l wv

6 4- Ro CCW w L h S ll&l, l- h olr&o Cov&o for rg h of olr&o: - look f, r - look wv rog&g owr ou (look r&o) - h CCWLHCP CWRHCP Norl o vor - look f, r - look f, r

7 4- Ro CCW w L h S ll&l, l- h olr&o Norl o vor 5- (ˆ 3ŷ) k (ˆ 3ŷ) o(k ω) Lrl olr gl w.r.. o - lookg h r&o 7 gr

8 Polr&o Dv: rfor&o of o vor ( r) oo T T 3 T T () () ou T ou T 3 T T ou ou B ou Ouu ol. B Iu ol.

9 Polr&o Dv: rfor&o of o vor ( r) oo T T 3 T T () () ou T ou T 3 T T 4) Cor h followg u: olrr wh log olrr wh log ou 4) wr h o TRIX for h fr o olrr. 4) k h rou o f h oo r how h o lgh of olr&o g hrough.

10 Polr&o Dv: rfor&o of o vor ( r) oo T T 3 T T () () ou T ou T 3 T T 4) Cor h followg u: olrr wh log olrr wh log H for 4) ou ou B Ouu ol. Iu ol. 4) wr h o TRIX for h fr o olrr. Solv for r l 4) k h rou o f h oo r how h o lgh of olr&o g hrough.

11 /3 Polr&o Dv: rfor&o of o vor () () () oo ( r) () T T3 T T () Fourr rfor uff ou T() ou T3 TT T () ou H for 4) ou () ou () ou T 4B Ouu ol. Iu ol. () 4) wr h o TRIX for h fir o olrr. Solv for k r l g hrough. olr&o 4) k () r how h o l gh o f h rou o fi h oo k k ( ) RHCPol ω π k ( ) RHCPol 5) rfrg RHCPol rl (wh ffr k k Th k ( ) rfr&v : ) wll rou ou k ω π/ ol () () k ( ) ffr rog&o h h for h k h r : r oo r of h o vor. k ω lorhcp (3/) () ou λ )k ( ) () RHCPol ( B q λ q k ou k ( 3 ) ( 3 ) o(k ω) q,,,... ou k q,,,... o π r r o(k ω) (3/) ou () 4 () 4) Cor h followg u: olrr w h log T T3 T T olrr wh log ()

12 /3 Polr&o Dv: rfor&o of o vor () () () oo ( r) () T T3 T T () Fourr rfor uff ou T() ou T3 TT T () ou H for 4) ou () ou () ou T 4B Ouu ol. Iu ol. /3 () 4) wr h o TRIX for h fir o olrr. Solv for k r l g hrough. () 4) k fi h oo r how h o l gh o f olr&o h rou o k k ( ) RHCPol ω π k Fourr rfor ( ) uff RHCPol 5) rfrg RHCPol rl ( wh ffr k k Th k ( ) rfr&v : ) wll rou ou k ω π/ ol () () k ( ) ffr rog&o h h for h k h r : r oo r of h o vor. l ) h r (h ll qurr- wv Suo h h hk () ho o h k ω lorhcp (3/) () ou λ )k ( ) () RHCPol ( B q λ q () krhcpol k ou ( 3 ) ( 3 ) o(k ω) q,,,... ou k r q,,,... o π r r o(k ω) (3/) ou () 4 () 4) Cor h followg u: olrr w h log T T3 T T olrr wh log ()

13 /3 Polr&o Dv: rfor&o of o vor () () () oo ( r) () T T3 T T () Fourr rfor uff ou T() ou T3 TT T () ou H for 4) ou () ou () ou T 4B Ouu ol. Iu ol. /3 () 4) wr h o TRIX for h fir o olrr. Solv for k r l g hrough. () 4) k fi h oo r how h o l gh o f olr&o h rou o k k ( ) RHCPol ω π k Fourr rfor ( ) uff RHCPol 5) rfrg RHCPol rl ( wh ffr k k Th k ( ) rfr&v : ) wll rou ou k ω π/ ol () () k ( ) ffr rog&o h h for h k h r : r oo r of h o vor. l ) h r (h ll qurr- wv Suo h h hk () ho o h k ω lorhcp (3/) 5) W h wll h o&l l o o lrl k () ou u olr λ ) ( ) () RHCPol ( B () q 45 gr λw.r.. h? l45 lr q () krhcpol k ou ( 3 ) ( 3 ) o(k ω)?, () q,... () k, ou 6 5 r q,,,... o r r o(k ω) (3/) π ou () 4 () 4) Cor h followg u: olrr w h log T T3 T T olrr wh log ()

14 /3 Polr&o Dv: rfor&o of o vor () () () oo ( r) () T T3 T T () Fourr rfor uff ou T() ou T3 TT T () ou H for 4) ou () ou () ou T 4B Ouu ol. Iu ol. /3 () 4) wr h o TRIX for h fir o olrr. Solv for k r l g hrough. () 4) k fi h oo r how h o l gh o f olr&o h rou o k k ( ) /3 RHCPol ω π k Fourr rfor ( ) uff RHCPol 5) rfrg RHCPol rl ( wh ffr k k Th k ( ) rfr&v : ) wll rou ou k ω π/ ol () () k ( ) ffr rog&o h h for h k h r : r oo r of h o vor. Fourr rfor uff l ) h r (h ll qurr- wv Suo h h hk h () ho o k ω lorhcp () (3/) 5) W h wll h o&l l o o lrl k () ou u olr λ ) ( ) () RHCPol ( B () q 45 gr λw.r.. h? l45 RHCP LHCP lr q () krhcpol 5) Wh wll h o&l l o o lrl olr u k ou ( 3 ) ( 3 ) o(k ω)? () () R w.r.. h? Wh w () q,hcp... gr ll, o, o L() HCP lgh? k ou 6 5 r q,,,... o π r r o(k ω) (3/) () ou4 () 4 () LCP 4) Cor h followg u: olrr w h log T T3 T T olrr wh log ()

15 /3 Polr&o Dv: rfor&o of o vor () () () oo ( r) () T T3 T T () Fourr rfor uff ou T() ou T3 TT T () ou H for 4) ou () ou () ou T 4B Ouu ol. Iu ol. /3 () 4) wr h o TRIX for h fir o olrr. Solv for k r l g hrough. () 4) k fi h oo r how h o l gh o f olr&o h rou o k k ( ) /3 RHCPol ω π k Fourr rfor ( ) uff RHCPol 5) rfrg RHCPol rl ( wh ffr k k Th k ( ) rfr&v : ) wll rou ou k ω π/ ol () () k ( ) ffr rog&o h h for h k h r : r oo r of h o vor. Fourr rfor uff l ) h r (h ll qurr- wv Suo h h hk h () ho o k ω lorhcp () (3/) 5) W h wll h o&l l o o lrl k () ou u olr λ ) ( ) () RHCPol ( B () q 45 gr λw.r.. h? l45 RHCP LHCP lr q () krhcpol 5) Wh wll h o&l l o o lrl olr u k ou ( 3 ) ( 3 ) o(k ω)? () () R w.r.. h? Wh w () q,hcp... gr ll, o, o L() HCP lgh? k ou 6 5 how o 3D ov gl 6) If h g r roj wh rulr olr&o, work? r q,,,... o π r r o(k ω) (3/) () ou4 () 4 () LCP 4) Cor h followg u: olrr w h log T T3 T T olrr wh log ()

16 /3 Polr&o Dv: rfor&o of o vor () () () oo ( r) () T T3 T T () Fourr rfor uff ou T() ou T3 TT T () ou H for 4) ou () ou () ou T 4B Ouu ol. Iu ol. /3 () 4) wr h o TRIX for h fir o olrr. Solv for k r l g hrough. () 4) k fi h oo r how h o l gh o f olr&o h rou o k - olrr - olrr k ( ) /3 RHCPol ou φ ω π B k Fourr ( ) rfor uff RHCPol 5) rfrg RHCPol rl ( wh ffr k k Th k ( ) rfr&v T 3) T wll rou :ou T ou k ω π/ ol () () k ( ) ffr rog&o h h for h k h r : r oo r of h o vor. Fourr rfor uff l ) h r (h ll qurr- wv Suo h h hk ho o h () T T3 T T k ω lorhcp () (3/) 5) W h wll h o&l l o o lrl k () ou u olr λ ) ( ) () RHCPol ( B () q 45 gr λw.r.. h? l45 RHCP LHCP lr q () krhcpol T ou 5) Wh wll h o&l l o o lrl olr u k ou ( 3 ) ( 3 ) o(k ω)? () () R w.r.. h? Wh w () q,hcp... gr ll, o, o L() HCP lgh? k ou 6 5 how o 3D ov gl h rulr olr&o, 6) If h g r roj w work? r q,,,... o π r r o(k ω) (3/) ou4 () 4 () () LCP 4) Cor h followg u: olrr w h log T T3 T T olrr wh log ()

17 /3 Polr&o Dv: rfor&o of o vor () () () oo ( r) () T T3 T T () Fourr rfor uff ou T() ou T3 TT T () ou H for 4) ou () ou () ou T 4B Ouu ol. Iu ol. /3 () 4) wr h o TRIX for h fir o olrr. Solv for k r l g hrough. () 4) k fi h oo r how h o l gh o f olr&o h rou o k - olrr - olrr k ( ) /3 RHCPol ou φ ω π B k Fourr ( ) rfor uff RHCPol 5) rfrg RHCPol rl ( wh ffr k k Th k ( ) rfr&v T 3) T wll rou :ou T ou k ω π/ ol () () k ( ) ffr rog&o h h for h k h r : r oo r of h o vor. Fourr rfor uff l ) h r (h ll qurr- wv Suo h h hk ho o h () T T3 T T k ω lorhcp () (3/) 5) W h wll h o&l l o o lrl k () ou u olr λ ) ( ) () RHCPol ( B () q 45 gr λw.r.. h? l45 RHCP LHCP lr q () krhcpol T ou 5) Wh wll h o&l l o o lrl olr u k ou ( 3 ) ( 3 ) o(k ω)? () () R w.r.. h? Wh w () q,hcp... gr ll, o, o L() HCP lgh? k ou 6 5 how o 3D ov gl h rulr olr&o, 6) If h g r roj w work? r q,,,... o π r r o(k ω) (3/) ou4 () 4 () () LCP 4) Cor h followg u: olrr w h log T T3 T T olrr wh log ()

18 /3 Polr&o Dv: rfor&o of o vor () () () oo ( r) () T T3 T T () Fourr rfor uff ou T() ou T3 TT T () ou H for 4) ou () ou () ou T 4B Ouu ol. Iu ol. /3 () 4) wr h o TRIX for h fir o olrr. Solv for k r l g hrough. () 4) k fi h oo r how h o l gh o f olr&o h rou o k u - olrr - olrr rrr k ( ) /3 RHCPol ou φ ω π B k Fourr ( ) rfor uff RHCPol 5) rfrg RHCPol rl ( wh ffr k k Th k ( ) rfr&v T 3) T wll rou :ou T ou k ω π/ ol () () k ( ) ffr rog&o h h for h k h r : r oo r of h o vor. Fourr rfor uff l ) h r (h ll qurr- wv Suo h h hk ho o h () T T3 T T k ω lorhcp () (3/) 5) W h wll h o&l l o o lrl k () ou u olr λ ) ( ) () RHCPol ( B () q 45 gr λw.r.. h? l45 RHCP LHCP lr q () krhcpol T ou 5) Wh wll h o&l l o o lrl olr u k ou ( 3 ) ( 3 ) o(k ω)? () () R w.r.. h? Wh w () q,hcp... gr ll, o, o L() HCP lgh? k ou 6 5 how o 3D ov gl h rulr olr&o, 6) If h g r roj w work? r q,,,... o π r r o(k ω) (3/) ou4 () 4 () () LCP 4) Cor h followg u: olrr w h log T T3 T T olrr wh log ()

19 /3 Polr&o Dv: rfor&o of o vor () () () oo ( r) () T T3 T T () Fourr rfor uff ou T() ou T3 TT T () ou H for 4) ou () ou () ou T 4B Ouu ol. Iu ol. /3 () 4) wr h o TRIX for h fir o olrr. Solv for k r l g hrough. () 4) k fi h oo r how h o l gh o f olr&o h rou o k u - olrr - olrr rrr k No o uu! ( ) /3 RHCPol () ou ou φ φ ω π B B k Fourr ( ) rfor uff RHCPol 5) rfrg RHCPol rl ( wh ffr k k Th k ( ) rfr&v T 3) T wll rou :ou T () ou ω π/ ol () ou () φ k k ( B ffr rog&o h h for h k h ) r : r oo r of h o vor. Fourr rfor uff l ) h r (h ll qurr- wv Suo h h hk ho o h () T T3 T T T T T k ω lorhcp () (3/) 5) W h wll h o&l l o o lrl ou k 3 () ou u olr λ ) ( ) () RHCPol ( B () q 45 gr λw.r.. h? l45 RHCP LHCP lr q () RHCPol T ou k u o o lrl T T 5) Wh wll h o&l l olr 3T T k ou ( 3 ) ( 3 ) o(k ω)? () () R w.r.. h? Wh w () q,hcp... gr ll, o, o L() HCP lgh? k ou 6 5 how o 3D ov gl h rulr olr&o, 6) If h g r roj w work? r q,,,... ou T o (3/) π r r ou4 LCP () 4 o(k ω) () () 4) Cor h followg u: olrr w h log T T3 T T olrr wh log ()

20 uff /3 ********* *************** () ************************************************** Lr log (h ll qurr- wv l ) RHCP Lr 45 for uff ************************************************** *************** () () () () rfor uff Fourr Fourr rfor uff lr LHCP ************************************************** RHCPol Lr log () ou ou 4 LHCP () () Lr ou - 45 ou Bou B Bou B r () () () () (ν)i(ν)ν () (ν)i(ν)ν LCP I I () λ q () () LCP ou ou Ig fro: U U ( r )U U U ( r )U ω π () () ouhl://rkw.logo.//3/holor- offr- 35- ou fil- - 3.hl I I () l45 B B B B ou ou ( r r ) ( r r)r ) ( r ) ( ( ( r )r ) () q,,,... work 3D ov l45 LCP π(ν)i(ν)ν /λ ω π/ ki / k /λ I / π(ν)i(ν)ν g ffr ou g π r r r F. Th o h k k rffr k k k ou k r l45 h rr U ( U U ru)u ) I (, ) I (, r )U ω ( I I π 4 4 π 4 4 w h g ( rr)r ) ( r ) ( r r ) ( ( ( r )r ) U k I h kk k( )/ k k( rovg )/ 4 ou ( r r ) (, ) (k /4) (k /4) (, ) /4) /λ /4) / π(k rul&g / π (k k /λ π k π k k for&o k k,, h ou r h νlluo k q νk q q FSR k k k k k k k( k k k( )/ 3D )/ vwg. T h (, ) F [r(/)] F [(δ( /4) δ( /4))] (, ) F [r(/)] F [(δ( /4) δ( /4))] I (, ) I (, ) 4 o(k ω) 4 k k π π π 4 4 k k π 4 4 RHCPol, qu&o r, h ow r h, λq k kk k h wo f k( )/ k( )/ (, q ) ωh (, )If (k /4) (k (k /4) (k /4) h /4) w π o 6) h g r roj rulr olr&o, ow () 4 o(k ω k k π k π/) RHCPol π k k k k( k, k, k( )/ )/ δ( /4))] (,r) 3D F ov Fw[r(/) (δ( /4) (, ) [r(/) (δ( /4) δ( /4))] g l ork? k k λ/ / π k π k q k,, kk k( )/ k RHCPol k( )/ () ω (, ) F [r(/)] [(δ( [ /4) rδ( (, ) F [r(/)] F [(δ( /4) Fδ( /4))] ω kπ/ () ]/4))] k λ π π k k ω π,,

Overview. Splay trees. Balanced binary search trees. Inge Li Gørtz. Self-adjusting BST (Sleator-Tarjan 1983).

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