Development of a Digital Offset Laser Lock

Transcript

1 College of William and Mary W&M ScholarWorks Undergraduate Honors Theses Theses, Dissertations, & Master Projects Development of a Digital Offset Laser Lock Ian W. Hage College of William and Mary Follow this and additional works at: Part of the Atomic, Molecular and Optical Physics Commons, and the Controls and Control Theory Commons Recommended Citation Hage, Ian W., "Development of a Digital Offset Laser Lock" (2016). Undergraduate Honors Theses. Paper This Honors Thesis is brought to you for free and open access by the Theses, Dissertations, & Master Projects at W&M ScholarWorks. It has been accepted for inclusion in Undergraduate Honors Theses by an authorized administrator of W&M ScholarWorks. For more information, please contact scholarworks@wm.edu.

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3 t t s t s r str t s t s r s t tr s2st st 3 s r s t r q 2 t r s t r t r st s r s r q 2 s2st s r t s r rt r s t r q 2 r s t t t t s rs t t r 1 t r q 2 r r s t t t s t s s s r s t str t r r rt t r P tr r t t r t s r s r q 2 s t rr r s r t 2 t r s t t t r q 2 s s t t r t r r 2 t r t ts t t s r st r s t r r r t t t P r t t s r s2st tr t r r2 s r s2st s t 1 r st 3 s r s r q 2 s rs st t t r t r r ts s s rr r r t s t s r t2 rt r r t s r s 1tr 2 s s t t t t s ts t rr t t s tr s r r st r q 2 r r t t s t st r r r s tr s r r rt tr s t 1 s r r s t s t t rr s2st 2 s r rr st t2 r t r tr s2st s r t r r q 2 s t s r 2 t t r r q s s2st t s t s s s s t s t r 3 s r ss s t ss 2 r s tt st t s ts 2 s t 1 r ts s rs s s t t t s r s r s t r2 tr 1 t 1 rs s s r 3 t r q 2 t t s r s st t tt r t 3 st s r r t t r t s t s t r r r t tr s t t 5 2 S 1/2 5 2 P 3/2 s s t r S 1/2 s t r s 2 r s t t s r t 3 r s t r 2 r s t 1 t st t s 2 3 r r s t t r 3 r

4 r 2 r s tt s r 2 s s t t t r s t 3 s r t t t s st s t 3 r t s2st r s r t r q 2 s t r r 2 r s t t t t s2st r s t t r t 2 tr s t r s t s rt r 2 s t s t t 2 r s t s t t tr s t s r s t t t t t2 2 t r r 3 r s t ss s s t r s t s t s r r q 2 t tr t 2 r s 2 s 2 t s t r rt 2 t t t s s t t s t t r s s r r 2 s 2 t t t r s r t t s r q 2 s t q s t r s t r t tr s t t tr r r tr t s r 1 r ts s r s2st s t t r t ss 2 s t t t r s r s t t r t r t t st st s rs t t t r r q 2 s s s t t t t t r t r s r tr t s t ts t t s t2 t r t s r s t t s t2 s

5 2 t I < E 2 total > τ =< [E 1 cos(ω 1 t)+e 2 cos(ω 2 t+ϕ)] 2 > τ r E 1 E 2 r t tr t s s t r q s ω 1 ω 2 s r ϕ s t t s r t r t t t t r 2 s r r q s r r s t r rt3 s t 1 r r s t r t t t t r s t r r rt3 1 q t s I < E 2 1cos 2 (ω 1 t)+e 2 2cos 2 (ω 2 t+ϕ)+2e 1 E 2 cos(ω 1 t)cos(ω 2 t+ϕ) > τ r ss t r 1 t t tr tr t t2 t t r s r q s t t r t s r t r r q s r s t 2 s t r 2 s cos(u)cos(v) = 1 2 [cos(u v)+cos(u+v)] I < E1cos 2 2 (ω 1 t) > τ + < E2cos 2 2 (ω 2 t+ϕ) > τ + < E 1 E 2 cos((ω 1 +ω 2 )t+ϕ) > τ + < E 1 E 2 cos((ω 1 ω 2 )t ϕ) > τ t s r r q s t t r t st t r st r tr 2 s t r s t sq r s t r s t s r q 2 t r r r 2 t t t t r cos 2 (x) s 1 2 t r cos(x) s r t st t r s r r q 2 t r s s ts r s r t r t t t r t t t tr s I 1 2 E E2 2 +E 1 E 2 cos((ω 1 ω 2 )t ϕ) 2 r st t t r s r t 2 r r q 2 s t t t r q 2 r t t t s s s 1 t 2 t q t t2 t t s r tr t t s r s 2 r rts t r t r 2 r t t s tr s t t s r q 2 s 3 s t s r q 2 tr s r r q r t s s s

6 r s2st s t s t t r s r t s r t rr s s t s r ts t tr s s r t r t st st s rs t rs t t r r 2 r t tr s s r t s t t r q 2 t r r q 2 s t s r r t t s r r q 2 s t r q 2 r s rt 2 t t r rt t t 1t r P tr r t t r t s r r q 2 r t t r t s t t t r q 2 r 2 t t r q 2 s t s r r s tr s t r t t t s rt t tr s 2 t t s t tt t P t r r t s t sq r t r sq r s s t t r q 2 r t rt r r t s r q 2 s r s s r t r r r t s tr t 3 s r q 2 s s t t s t t t P t s s tr s s t t t s2st s r q 2 t r Pr r t rr 2 P s s t r r t r t r r t t t s s r s t r s r q 2 t s r t s r r q 2 s t t 2 t s r r t s r q s s r rt t t t rr r t s r r q 2 t r s tt P t s t s rr r s s t r t s s r ts t t t rt r t ts s t ts t t s r rt t t s rr r st r q r s st rr r s t t t rs tr s s t s t s ss t r t r t r 2s r t s str t r t

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8 r P t rr s t t ss t r t r t P ts r r s t t ts r tr t t t s rst t r r t t s2st 3 t r t t s t t r t r s s t s t t s t t r t s t s s r ts t t r q 2 s s t t r t r s s sq r s tr t t r ts t s s s s s s s r t r t r rt r s s t r q 2 t s ss r r t t s r 3 r t t s rr t 2 s t t t r t s t t s s t t t t t s s t 1t r t r q 2 t s s t s t r t t r t t s tr ts t t r q 2 t t r t r q 2 t rr t t r q 2 rr r r t t s t r s t t r s 1 s 2 t s 2 s t t r q 2 t r t s t t r t rr r r q s s 2 2 r ss t s tt s t tt r t t r s t s t t rr r r t s t t r t t s s t r t s r t t s r t t t r t t s t r s s r t t ts r t r s s t 2 t t t s s t rr r t r s s r ts t t s s t s t s t t r ts r st r s r ss s r t 2 s s t t r q r 3 2 r 3 s tr rs s r t s t r t P t s s s t 2 str ts t t r r t s t s ts

9 r2 t r rs t r q 2 r r t s t t s s r t r t r r 2 t s r s t r t t t s s s rr r 2 rr t st s t t t t r r s s t t t s t s t t t rr t s tt t st rt t t rr t t t s t 1 t t rs s s r t t rs s s r ts r t ts t rs t rs t rs tr t t t t r t t t 2s t r tt t s r q 2 s t t rs r st tt t t r r q s r t r t s t r t s t s r t t t s t t r r s t t t r t r t t t rs s rr t 2 r r 2 t s t r t r t t ts t t r rt t t t t r t r s r t P s ts t 1 r s t r t r s 2s t s r r t s s t ts

10 r r t P r t t rs r s r t r s 2 s r t t r s r t t 2 t r t r s r ts s s t t s r r t r t P t t r r s t s t s t s t t r s r t r s r t t t rs s2 r ss s r s t s s r s t r q 2 s t t 2 r s t t s t t t s r s s tr t t r r 2 s t s r s r ss 2s t r s s P t tt t s s 3 r t r ts r s t st r r t r s st rt s s t t t s s s ss t t t st rr r s t s st s r tt t t s s ss 2 t r s t t t t 2 t r

11 s sts t rs t rs r t 2 s s r r ts st r r t r s s r t st P r t t2 t rr t t r t 1 ss st rt t ss r r t r r r t s t r r t r t r t r r t r s tr ts t s r t r t t r 2 s t t s t t s r r t s r s t r t t r s2st s t 1 t r t rs t t s r r t r 2 t t t str t s s r t s t r ss t r t r q 2 t r t s t rs r t t rt r r s r t 1 s t t rst t s t rs t r r s st r t r t r r ss t t s r t r t r 2 t t r 2 s t t s t t s t r s t r s s r s t t r t r ss t r t r t t t s t t t r r t r r r 2 ± r t rs

12 s s t r t r st s r t t r s r t ts t s r s t r tr t t r r s rr t ts r s r s t r s t r t 1 s t st 2 ss s r r t t 2 t s 3 r t t s2st s t r P s t r t r q t r s 2 r r q s r 3 t 3 r t t 1 t t r t t t t s r r t r t r s s r t r s r r r t s t t r q 2 t t t t rr t P r s tt s r2 3 s t r q 2 r s t t t 1 s r t r t r s r 3 r q 2 r 1 sts s t r rr r t t s s r t t t t t t t P s ts t r r 2 r q 2 r 3 s rr r s s ss t 3 t r s r q 2 t t r 2 t r s tt r 2 t s r s t s 2 t s t t s t t t

13 r r s s t t t rs s P r q 2 rr r r t s s t t t r 3 t r q 2 tt s s t r t s s r r 3 t r s P r q 2 3 s r t t r s s t t t t s 2 r t r q 2 r 3 t 3 st t t s s s r s t s t s r r 2 t r r t t t t t s t t 2 t t s t t t P

14 r r s s t r q 2 s st r 3 t 3 r s s t t s t s s st s r t t t P t r s s t t s r t t ss t t r s s r t s tr s t t t r t r t r r r r 2 t t t r s t r s t r t r s s s r r r t 1 t s tr r s t t t st s s r 2 3 s 2 t t t s t s r t P t s s s t rt r r t r s r t t t s t s r r r ss s r t r t t s2st s t t r s t r t t t t s r t r t t t s t r t rs r s r 1 t s 3 sq r s s s 2 t t P t

15 r 1 t r s tr t s s t s t r t r t r r s t s t 3 s rs sts t t t t r t rs t s s r ss P tr r rt t t s s r s t str t t Pr rt t r P tr r t tr r t s s s r t 2 r t t r ts ts s r r t P r t s t tr s r t t rr r s r t 2 t P t t s t t t t s r t tr ts t r q 2 s s r tr s s t s s r rt t t rr r s r rt t t t t r t rr r s rs s r s2st s P t t t P tr t 2 t s s t t r r r2 ts s rr t s r s tr t t t st s t t s r t r s r tr r ts t tr r r 2 r t r r t P r t s s r r rt P s s r t 2 s rt r t t r t s r r 2 t r t r q t s r t s r t s r t r s t r s s t r t r r s t rt r s tr 2 t r t t r s st t t r s st r s t t r t r s r rt t t rs r t t t t r s st

16 r P r t r s r r P t r rt t r t P 3 s tr 2 t r rt t r t rr t tr rs t tt r t t r s 1 t tr t r t r s t s t t P r t s r s s s s s r r t t s 2 s r s t r q 2 2 tr 2 t P r t rst s t s 2 t r t tr rr t r t s r s s t 1 r r t t t s s t s r r q 2 s t tr s r q rr t s s t tr s 1 t r 2 tr 2 r r r r q 2 1 rs s rr t t r s t t r rr t r q 2 t rr t r r s 1 rs s t r rr t s t t r t t t 1t r2 P r t t2 3 tr s r t tr s s s r rt t2 t t t r t s r r q 2 s r ss s r 2 r t rr t tr t s s r t ts t r P tr r t r s t rr t tr 3 tr s s t r t r s r ts s t s t rr r s s t t s r ts ss t r s r s t t 2 st 1 rs s 3 r t rr t tr r s t rr t tr s t t s rr t s t s ss r2 t t r t r s 2 rr rs t t rs st r t 2 t 3

17 tr r 3 tr r s r str t r s t t t 1t r s r s r s s r r q 2 s s s t s r t t t r t2 t 2 tr rr r r P tr r tr2 t P r t t t t r r s s 2 r 2 r s rr2 t s s t st t t r ss s r s2st s t tr t t st 2 t rt rs t r t r t tr r tr2 r str t t t t t rs t r s t 2 tr t t t r s r s str t t r t t t rs t t ts r t s t r s s t r s t t t t t s 2 r r r s s t t P r t s t t r s tr s 1tr 2 rt t r st t r t r t rs s r t r r t r t t s r t r s 2 P s r s r t r s r t r st s 1 t s t s rs t t r ts t t r str t r r t rs r t r r t r t2 r r t t rr t t tr ss ss t t P r t s t ss r q r s t st t t s r r r2 t st t r t s t sq r rr r s s t t r t t t t s r r s s r sq r rr r s t r rt tr s s 2 sq r t t r tr s s 2 tr r s r r s t s 1 t tt r s s r t s s tr s

18 t s r s t t t r t r rt r t r s t s r P r s s 1 s r t t r t t r t s r rt s 1 s s r s s tt s r t 1 t r s s r P tr r t st s tr r 2 1 t r r rt r t s t t t rr t P s s r t t t s r t s t s t t rs t r ts str t t t rts t r t r t t s ss s P tr r s 1 t r t t r t s r t t r r t 2 t t r r t r t t s r

19 r P r t s t r t ts t rr r s t t t P r t P t t t rs t t r t t s t r st t t r s rs t r2 t r s t r s t r t r t s t t t s t r s s t st s t s t r t t r s tr 2st ss 2 1 t t r r 2 t t t r r s rt tr s s t r t t s r t s2st r t st t r s r s r r r t r ss t s r r s r s s r t st ts ts 1 1 s s t t s t s r s2st r s 2s t rr t s2st st t 1 r t P r s s t t 1 1t r s r s r s str t t r t r t r s t s t 1 r st t s t t t s r s s r t t r r q 2 s2st s t t r r q 2 r rs r t r s t s r r 2 r s r t t s s t r s2st s t rs r q r ± s 2 t s 2 t rt r s r q r

20 r s2st s 2 r t ts r s s s t r t t 1 r t r t t t rts r s r s r t r t t t rs t t r t t r rs s r r t s r t P P t t s rt t 2 t t t 2 t t t t rs r s t s r t 2 t rt r s r s r tr t t r r s t r s r s t s r t r rt r r s t t t t rt r s q t P t r r t 2 tt t s s t t rt r s s t r q 2 t t s st t 2 r t s st r t r rs t s r s 1tr s r t r r t rs t r t ts 1 s tr t t t t r t ts t r t rs r t r t t t t 1 t s t r t 1 r ts t r s r s t t rs r ts r t t t 1 ts t 1 r s t r t r t t s r t r r s r s s ss rt r t 1t s t

21 t r r s t r r t s2st s t 2 t st r t st t s s t t s2st t st t s r t t r ts t ss r t t t ts t s s s st r s 2 r q r s r s t t t s s s r tr s s r t s 1 st t r s t s2st s t t2 s s r r r st t rr t s t t t P s r r t r s 1 rs s t t 2 s r q 2 t r t t s P s t r r r t t rr t t r t t st r t t s st r t s t t s r t t r t st r rr r s t t s t t t P tr r s r t 2 r t s t2 t r s t s s t t r t s t ss s r s t t s2st s r t s rr t 2 t r q 2 s t st t 2 2 t t r t r q 2 s t r t 2 t s t r tr s2st s s 2 t r t t r r tr r r r t tr s t t r q 2 s t s t t r t t t s t s r t P rt r t t s r s t r s t2 t s 2 r t r t s s t r t r q 2 s r r q 2 t r t t 1 s s r t s 2 r tr r2 s r t t t r s tt t s t t s t r r s t r s t r 2 t 1 r t s ss s s t s t r t t 1 r s r r2 t r s tt P t t s s t 2 1t r s t s t r tr r t s t rs s 2 t t t s2st r ts t r t r q 2 s 1t r r r s t s r q 2 P s r 2 q t t r q 2 s t s rt t t t t t t r t r q 2 s s rt tr s t s r r ts 2 t t s r t t2 t 1t r r r r s t t t t P s t s r s s s t t s r t 2 r t t 2 t t s r r s t t s rts s 1 s t r t r t 2s s 1 r ts

22 r s r t tr s s r s t s str t t s r t s 2 s t r t r r t tr s tr r r tr r str t s tt r r s P t s Pr t r str

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24 vv6vv6vv6vvvvvv vv 6v666vv6vv6vv6vvvv v266 6v vv66vv6vv6vv6vvvv ) ) ))) ))))))) A A A ) ) ) AAA) ) A ) ) ) ) ) ) ) ) v ) v v) ) ) ) ) A) A ) ) )A ) ) \ ka ) kvk k A) A ) ) A ) ) ) A ) A ) v ) )A ) v v) ) ) ) v Av A ) ) v v f )) A A ) ) ) ) ) ) ) )) ) ) ) ) A l A) A A ) ) A A A ) ) ) ) v A ) ) v ) )A ) A ) ) v v) ) ) A ) ) ) v f )) ) )l A l ) ) k) A ) ) kvk k kvk kvk k) ) ) ))) ) A ) A ) v ) ) ) ) y A ) ) ))) ) A ) ) )A ) ))) A f k) ) A) A ka A A A) A ) A ) A v Av A f)))a )))A ka A ); [ A); ) A) ; ) ) ) v v ) ) ) k; ) )kvk k ) k [ A); A ; ) )kvk ) ))kvka ) )? A) ; ) ) ) [ A); A ; ) ) ))) ))))))) ) ) ))) [ A); ) A ) ; ) v v ) v ) ) A [)A); v Av ; ) v v ) ) A ; ))) )v))a ) ) )))A k; ))) k))va ) f))); ) ))) ) )f))) ) [ A); A ; ) [ A); ) A ; )) ) A l A ; ) A [ A); A ; [ A); A A ; )) A A l A ; ) ) ) [ A); A ; )) A l A ; ))) ) [ A); A ;

25 vv6vv6vv6vvvvvv vv 6v666vv6vv6vv6vvvv v266 6v vv66vv6vv6vv6vvvv ) [ A); A A ; )) A A l A ; ) ) ) v v A ; ) A ) ; )) ) A ) l v A ; ) k A ) ) ; )) ) l v A ; A k; )) ka ; ) )) ka l A k; ) AAA) v ))) ) ) [ A); A A))); ) [ A); A ) ; ) )) A ) l A A))); ) ) ) y )) ) A ) ) ka A) A A ); ) ) " ) " ) ) ) ) A A ) ) A A ) A A A A v A ); ) ) ) ) v ) ) ) ) ) A) A A A ) A A A A A A ); ) v v) ) ) v vy ) A A A ) A ) A v Av A ) A ) ); ) ) k) ka k) ka )))A ka A k); kvk ) ) ) A ) A A ) A A A)))); ) ))) ) y A ) A ) A ka A A A f)))); ) ) ) ))) )

26 vvvv66v6v6v6vv 6v v6 v vvvv v6 v sssssssss sssssss sss sss sssss sssssssssss ss sssssss sss sssss ssssss sss sssss ssssssssss sss sssss ssssssssss sss sss ssssss sss ssssss sssss ;ssssss;sssss;sssssss;ssssss ssssss;sss ssssss;sss sss ;ssss;ssssss sssss;ssss;; sssss [ : ; sssssss;ssssss; sssss s ssss ss ssssss sssssss ssssss sssss [ 7: ; ssssss;sss ; sssssss ss sss ssess sssss sssss [ 9: ; ssssss;sss ; sssssss ss sss ssess sssss 7 sssss [ : ; sss ;ssss;ssssss; sssss ssss ssssssssss ss ssssss; ssss 9 ssssss sss [ 9: ; sssss;ssss; ss sss sssssssss ssssss sss sssssss ssssss ss ss sssssssss ss sss ;sssssss sssssss sssss sssss;ssss 0s ; sss ss sss sssss 7 ssss ;sssssss;ssssss; 0s : sssss;ssss ; ssssss;sss ; 9 0s : sssss;ssss ; ssssss;sss ; 0s : sssss;ssss ; sss ;ssss;ssssss; sssssss sss sssss ss ss sss ssssk sss sssss

27 v6v6vvvv66vvv vvvv yy yyy yyyyyyyyyy yyyyyy yyyyyyyy y yyyyy yyyyyy yyyyyy yyyy y yyyyyyy yyyyyyy yyyyyyy yyyyyy yyyyyyyyyyy;yyyyy;yyyy; yyyyyyy;yyyyyyyy; yyyyy;y;;; yyyy;; yyyyy [ y:0; yyyyy;yyyy; yyyyy yyyy; yyyyyy yf fyyf yyff y yyyyyy yyy [ 8:0; yyyyyyy;yyyyyyyy; yyyyyy fyy fyyyy yyyyy yyy yyyy y y yyyyy yyyyyy yyyyyy yyy [y:0; yyyyy;y;;; yyyyyy yy yyy yyyyy yyyyyyyy yff ; )yff; 0)yy; 8 yyy [y:0; yyyyyyy0; y yyy [y:0; yyyyyyy ; 0 yyy [y:0; yyyyyyy ; yyy [y:0; yyyyyyy ; yyy [y:0; yyyyyyy ; yyy [y:0; yyyyyyy ; yyy [y:0; yyyy;yyyyyyy; y yyy [ :0; yyy;yyyyyyyy; yyyyyyy yyy 8 yyy [ :0; yyy;yyyyyyyyy0; yyy;yyyyyyyyy ; yyy;yyyyyyyyy ; yyy;yyyyyyyyy ; yyy;yyyyyyyyy ; yyy;yyyyyyyyy ; y 0 yy yyyyyyyy)y yyy yyyyyyy ;y)y) yyyy yyy yyy );yy yff: yyyy ) yff; )y) ) yy; yyyyyyy yyyyy yyyyyyy;yyyyyyyy ) y1y ; yyyyy;y;; ) 1y ; yyy y yy)yyy yyyyy yyyy ;yyyy; 8 1y0: yyyy;yyyyyyy =) 1y ; yyyyyyyyy yyyyyyy fyy yyyyyyyy yyyyyy y 1y : yyyy;yyyyyyy =) 1y0 ; yyyyyyyyy yyyyy yyyy fyy yyyyyyyy yyyyyy 0 yyyyyyy yyy yyyyy yy yy)yyy yyyye yyyyyy fyy yyyyyyyyyy yyyy yf fyyf;yyffyyyyyy yy)yyy yyyyy yy yyyyyyy yyyyy;yyyy yy yyyyy yf yyyyyyy ) yy yy zy) yyyyy;yyyyyyyy ) ;;yyyyy;yyyy; 0;y ;; y yyyyyyyyy yyyyy;yyyy yy yyyyy yf yyyyyyy 0yy yy zy) 8 yy yyyyyyyyyy yyyyy yy yyyyyyy yyyyyy; yyy;yyyyyyy yy) ryyy yyyyy yyy y yyy;yyyyyyyy ) yyyyy;yyyy; 0 yy y yyyyyyy yyy;yyyyyyyyy0 ) yyy;yyyyyyyy % 1y 0 0; yyyyyyy0 ) yyyyyy;yyyyyyyyyyyyy; yyy;yyy;yyyyyyyyy0;; yy 0y yyyyyyy y yyy;yyyyyyyy ) yyy;yyyyyyyy y 1y 0 0; yyy;yyyyyyyyy ) yyy;yyyyyyyy % 1y 0 0; 8 yyyyyyy ) yyyyyy;yyyyyyyyyyyyy; yyy;yyy;yyyyyyyyy ;; y 0 yy 00y yyyyyyy yyy;yyyyyyyy ) yyy;yyyyyyyy y 1y 0 0; yyy;yyyyyyyyy ) yyy;yyyyyyyy % 1y 0 0; yyyyyyy ) yyyyyy;yyyyyyyyyyyyy; yyy;yyy;yyyyyyyyy ;; yy 000y yyyyyyy y yyy;yyyyyyyy ) yyy;yyyyyyyy y 1y 0 0; yyy;yyyyyyyyy ) yyy;yyyyyyyy % 1y 0 0; 8 yyyyyyy ) yyyyyy;yyyyyyyyyyyyy; yyy;yyy;yyyyyyyyy ;; y y0 yy 0;000y yyyyyyy y yyy;yyyyyyyy ) yyy;yyyyyyyy y 1y 0 0;

28 v6v6vvvv66vvv vvvv y yyy;yyyyyyyyy ) yyy;yyyyyyyy % 1y 0 0; y yyyyyyy ) yyyyyy;yyyyyyyyyyyyy; yyy;yyy;yyyyyyyyy ;; y y yy 00;000y yyyyyyy yy yyy;yyyyyyyy ) yyy;yyyyyyyyy 1y 0 0; y yyy;yyyyyyyyy ) yyy;yyyyyyyy % 1y 0 0; y8 yyyyyyy ) yyyyyy;yyyyyyyyyyyyy; yyy;yyy;yyyyyyyyy ;; yy 0 yy yyyyyyyyy )yyyy yy yyy yyfy yf;yyy;yyyyyyyyy )) 1y0000; yyyyy yyyyyyy ) 1y ; yf ;yyy;yyyyyyyyy )) 1y0000; yyyyy y yyyyyyy ) 1y ; yf ;yyy;yyyyyyyyy )) 1y0000; 8 yyyyy y yyyyyyy ) 1y ; 80 yf ;yyy;yyyyyyyyy )) 1y0000; 8 yyyyy 8 yyyyyyy ) 1y ; 8 yf ;yyy;yyyyyyyyy )) 1y0000; 8 yyyyy 8 yyyyyyy ) 1y ; 8y yyy 8 yyy 88 yyy 8y yyy y0 yyy y yy yyyyyyy yyyyyyy yyyyyy y yyyyyyy;yyyyyyyy ) 0yyyy;yyyyyyy;yyyyyyy ;yyyyyyy ;yyyyyyy ;yyyyyyy ;yyyyyyy ;yyyyyyy0 ;; y y yyy y yy fyyyyyyy [y:0; yyyyyy;yyyyyyyyyyyyy; yyy; y yyyyy [ :0; yyyyy;yyyyyy; yyy; y8 yyyyy yy yyyyyy;yyyyyyyyyyyyy; yyy ) 1y ; 00 0 yf ;yyyyy;yyyyyy; yyy )) 1y0000; 0 yyyyyy;yyyyyyyyyyyyy; yyy ) 1y ; 0 0 yf ;yyyyy;yyyyyy; yyy )) 1y000 ; 0 yyyyyy;yyyyyyyyyyyyy; yyy ) 1y 00 ; 0y 0 yf ;yyyyy;yyyyyy; yyy )) 1y00 0; 08 yyyyyy;yyyyyyyyyyyyy; yyy ) 1y ; 0y 0 yf ;yyyyy;yyyyyy; yyy )) 1y00 ; yyyyyy;yyyyyyyyyyyyy; yyy ) 1y0 0000; yf ;yyyyy;yyyyyy; yyy )) 1y0 00; yyyyyy;yyyyyyyyyyyyy; yyy ) 1y00 00 ; y yf ;yyyyy;yyyyyy; yyy )) 1y0 0 ; yyyyyy;yyyyyyyyyyyyy; yyy ) 1y ; 8 y yf ;yyyyy;yyyyyy; yyy )) 1y0 0; 0 yyyyyy;yyyyyyyyyyyyy; yyy ) 1y ; yf ;yyyyy;yyyyyy; yyy )) 1y0 ;

29 v6v6vvvv66vvv vvvv yyyyyy;yyyyyyyyyyyyy; yyy) 1y 000; yf ;yyyyy;yyyyyy; yyy )) 1y 000; y yyyyyy;yyyyyyyyyyyyy; yyy ) 1y ; 8 yf ;yyyyy;yyyyyy; yyy )) 1y 00 ; y yyyyyy;yyyyyyyyyyyyy; yyy ) 1y00 000; 0 yyy yyyfyyyyyyy y yyyyyyyyy 8

30 vvvvv6vvvv6vvv vv 6v666 v6 6 v6 vvv v266 6vv vv 66 v6 6 v6 vvv e ee eeeeeee ee eee eeeee eeee eee e eee eeeeee eeee e eeeee eeeeeeeeee /ee eee /// eeeee/ eee eeeee eee eeee e/ee/e ee/ee/ ee /eee//e/e ee /ee/ee /eeee/eeeeee;eeeee/ ///;eee/ eeee;eee;; eee/e eeeee; ee eeeeeeee eeeee eeeee e/ee/e ee/ ///;eee; eeeeee e/e eeee eeeee e/ee/e ee/ eeee;eee; eee eee eeeee eeee eeeee eeeeee ee ee/e eeee eeeey ee/ [ :e; ee/eeee; eeee/eeeee eeeee eeeeeeee ee/ee/ eee eeee eeeee e e e e e e e e eeeeeee ne/ee ee/eeee = 0neeee; ///;eee = 0ne; eeee;eee = 0ne; eee eeeeye ( ;eeeee/e eeeee; ne/ee ee/eeee = ee/eeee + ; ee ;ee/eeee < ; ne/ee ///;eee <= ; eeee;eee <=e; eee ee ;;ee/eeee <e; == ;ee/eeee == ;; ne/ee eeee;eee <= ; ///;eee <= ; eee ee ;;ee/eeee == e;==;ee/eeee< ;; ne/ee ///;eee <=e; eeee;eee <= ; eee ee ;;ee/eeee == ; == ;ee/eeee < ;;ne/ee eeee;eee <= e; ///;eee <=e; eee eeee ee ;ee/eeee == ; ne/ee ee/eeee <= e; eeeeeeee ee/eeee eee eee eeeee eeeeye neeee eee/ee/ee ee/ee/ee /eeee/eeeeee;eeeee/ ///;eee/ eeee;eee;; eee/e eeeee; ee eeeeeeee eeeee eeeee e/ee/e ee/ ///;eee; eeeeee e/e eeee eeeee e/ee/e ee/ eeee;eee; eee eee eeeee eeee eeeee eeeeee ee ee/e eeee eeeey ee/ [ :e; ee/eeee; eeee/eeeee eeeee eeeeeeee ee/ee/ eee eeee eeeee eeeeeee ne/ee ee/eeee = e0neeeee; ///;eee = 0ne; ee eeee;eee = 0ne; e eee e e eeeeye ( ;eeeee/e eeeee; ne/ee e ee/eeee <= ee/eeee + ; ee ee ;ee/eeee < e; ne/ee e ///;eee <= ; e ee ;ee/eeee == ; ne/ee e eeee;eee <= ; e eee e eee eeee ee ;ee/eeee == e; ne/ee ///;eee <=e;

31 vvvvv6vvvv6vvv vv 6v666 v6 6 v6 vvv v266 6vv vv 66 v6 6 v6 vvv e e ee ;ee/eeee == ; ne/ee eeee;eee <= e; eee ee ;ee/eeee == ; ne/ee ee/eeee <= e; eeeeeeee ee/eeee eee eee eee eeeee eeeeye neeee eee/ee/eeee

32 vvvv66vvv vv 6v666 v6 v6vv6 vv v266 6v vv66 v6 v6vv6 vv llllllllll lllllllll ll lllll llllll llll lkllll llklll llklk lkllll klllll ll ll llklll klllk klelll lllllll;llllkk lllll;llllllk lllllll;llllll;; lllll llllk; lllllllllll llllk kk lll llk ll lllllk lll llllllll kk lkk llllk k lllll lllll;llllll; llklll llll lllllllll lllll ll klll llll lllllll 9 k k 9 k ll:ll kkl llklll llkl: llllll [ 9:k; lllllll;llllll; ll klll llll lllllll llllllk kkklll klllll lll [ :k; lllllllkl;llll;ll; llklll llll lllll lllllllk llkll llkkll ll lllllll kklk lll [ 9:k; lllllllkl;llll;lll; llklll llll llllll lllllllk kkklll klllll llllkll llkkll ll lllllll kkl lllllle kl lllllllkl;llll;ll kklll klllll lll [ 9:k; lllllllkl;lll; lllllllllll llllk lllllllk llkll llkkll ll lllllll kklk kkklll klllll lll lll;lllll;llklllll; lllll llllk lllllll kll llkllllle lllll ll kklk ll k)k llllllllllll ; 9klll klllll; kll klllll k llllll lllllll;llllll[ 9:k; [ lllllllkl;llll;lll[ 9:k;; lllllllk ll klll llll lllll lllll ll llll ll llllll ellllll lllllll 9 kllll k lllllllkl;llll;ll [ 0kkkkkkkkkkkkkkkkkkkkkkkk; lllllllkl;llll;lll [ k0kkkkkkkkkkkkkkkkkkkkk; lllllllkl;lll [ k0kkkkkkkkkkkkkkkkkkkkk; lll;lllll;llklllll [ 0kk; lle k llklll ( ;llllell llllk; kllll llkllll ll llk ll kllk llllllllll ll kk lkk llllk lllllllkl;lll [ lllllllkl;lll + ; 9 lll;lllll;llklllll [ k; k llll ;lllllllkl;lll [[ 0k ; ll llll lllll ll kkk k kllll ll lll llllk llll ;lllllllkl;lll [[ 90k k kkkk kk kkkkk; ll llll lllll ll k)k kllll ll lll llllk ; kkl; ll ;lllllllkl;lll [[ kkkkkk; lllllll ll k kl kllk kk lkk llllk kllll lllllllkl;llll;lll /[ ;lllllllkl;llll;lll k;; lllllllllkl;llll;lll llkll llkkll ll llllll kl lllllllkl;llll;lllkl k lllllllkl;lll /[ k; lll;lllll;llklllll /[ ; lllll kllk ll llelllll llkl lllllll ll llll lle lle lllle ll llklll klllk 9 kk llklll ( ;llllell lllll;llllll ll llllell lll;lllll;llklllll; kllll k llkllll ll llk ll kllk llllllllll ll klll llll llllll k ll ll kllll ll llk ll kllk llllllllll ll llklllll lllllllll lllll llellllll k k ll ;lll;lllll;llklllll [[ ; kllll kk lllllllkl;llll;ll /[ k; k lle k llll kllll k lllllllkl;llll;ll /[ lllllllkl;llll;ll + ; k9 lle

33 vvvv66vvv vv 6v666 v6 v6vv6 vv v266 6v vv66 v6 v6vv6 vv k lle lllle llllle llklll klllk lleklelll

34 vv6vv v v 66 6 v6 6 /////// //// /// // / ////// // / ////// /// ////// /// //// //// //// // //// //// // e///// ////;// ;////; ////; ;/;;////; //////; ///; ////;; ///// [ 5:0; ////; ////; ///// ;/;;////; 5 / ////// /// //////; ///; ////// /// ////; 0 /// [ :0; //e; //5 //// ///// /////// ///// //////; /// [ ////; ////[ 5;; //e [ 50/00000; 5 //// [ 0/0; / /// ////// ;/;;////;//// [ ;/;;////; //// / ( ;/////// ;/;;////; ///// 0 //e [ //e + ; // ;//e =[ /; ///// //// =[ 0; //////; /// =[ ////; ////[ / //e;; //; ;/// /////// //// // ///e BBB // /BB /// 5 //// // ;//e > /; ///// / //////; /// =[ 0; // //0/ /////;//// // // //// ;/////;/ / ////// /e// 0//e/// :0////; ////[0;0 /////// // 0 //// =[ ; // ;//e [[ ; ///// //e =[ 0; 0 /// /// /// ///// //// / ///;; 5 ///e/////

35 v6vv6v6vv 6v v6 v v6v v v6 v CCCCCCCCCCC CCCC CCC CCC CCC CCC CC C C CC CC C CCC C CCCCCC CC CCC CC C CC CCC C C CCCCC C CCC CCC C C C CCCC C CC C CC CCC CCCCCCC CC C C CCCCC CCC CCC C C C CCC CCC CCC C CCC C C;CCCC; C C; C C ; CCCC;CC;CCC;; C C [ : ; CCCC; C C; CC s C CsCC CCCC Cs C CCC C C C C ; CCCCCCC s C CCC C C CC C C C CC C CCC [ : ; CCCC;CC;CCC; CCC [ : ; C C C;CCCC; CCCCC CCCC CCC CC C C C CCCC; C C CCC [ : ; CCCC C C; C ; CCCCCCCC C C C CCC ; CCC CC C C C CC CC C CC CC CC s CC s CC CC C C C CC CsC C C C scc CCCC;CC;CCC = 0s ; C C C;CCCC = 0s ; C C C sc C scc C ;CCCC; C C > ; scc CCCCC C CC C C CC CCC CCC CCCCCCC C C C CC ; ; C) ; CCCC C C CCC C CC C C C C;CCCC == ; CCC CC C CsC C s CCC CCC C CC C ;CsCC CC C C; CCC;C C ; 0s : CCCC;CC;CCC == CCC C C; C CCC C ;C C C;CCCC;; CCC C C CCC C C CsCC 0s : CCCC;CC;CCC == CCCC C; C CCC C ;C C C;CCCC;; CCC C C CCCC C CsCC C C CCC C C CCC C C CCCCCCC C C CC C ;CCCC; C C == ; scc C C C;CCCC == CCCC; C C; CCC;C C ; 0s : CCCC;CC;CCC == CCC C C; C CCC C ;C C C;CCCC;; CCC C C CCC C C CsCC 0s : CCCC;CC;CCC == CCCC C; C CCC C ;C C C;CCCC;; CCC C C CCCC C CsCC C C CCC C C CCC C C CC CCCCCCC C C C CCC C CkkCkk s C k C C C [ : ; CCC C C; C CCC C ; C C [ : ; CCCC; C C; scc CCCC C C; C = ;CCCC; C C* ;C + ; CCC C C; C CCC C = CCCC C C; C [ : ;; C C C CC C C C C C [ : ; CCCC C; C CCC C ; C C [ : ; CCCC; C C; scc CCCC C C; C = ; *; / CCCC; C C;;C ; CCCC C; C CCC C = CCCC C C; C [ : ;; C C C CC C C C CCCC C

36 v6vv66vv66vvvv v 6v666 v6 6 v6 v 6 6v v 66 v6 6 v6 v yyyyyyyyyyyy yyyyyyyyyy yyyyyyy yyyyyy yyyyyyyyy yyy yyyyyyyy yyyyyy yyyyyyyyy yeyyyy ;yyyyy;yyyy;;yyyyyy;yyyy; yyyyyy;yyyy; yyyy;yyyy; yyyy;; yytyy [ 7:0; yyyyyy;yyyy; yyyyyyyy yyyyyyyyy yey yyyyy; yytyy y yyyyyyyy yyy; yyyyy;yy; rryyy yyyyyy yyyyyy yytyy [ 9:0; yyyyyy;yyyy; yyyyyyyyyy yyyyyyyyy ey yyyyy yy yeyyyyy yeyyyy eyytyy [ :0; yyyy;yyyy; 7 eyytyy yyyy; r yyy [ :0; yyyyyyyyyey; yyyyyyyyyy yey yyyyyyyyyey etyyyyyey 9 yyy y yyy;yyy; yy ryyy yyyyyyyy ye yyt yyyy ey yyyy ey y;y yyyyyyyyy yyyyyyyyyy ;0)teyyyy)y; )yyyyyy)y; 0 yyyyyyy yyyyy yyyyyyyyyey ) 0y ; y yyy;yyy ) 0y0; yyy yyyyyy yyyy;yyyy[ :0; ) yyyyyyyyyey[ :0;; yyyyyyy; eyytyy yeyyyyy ye eyytyy yyyyy 7 yyyyyy yyyy ) y yyy;yyy; r 9 yyyyyy yyyyy 0 yy ;yyyyyy;yyyy )) yyyyyy;yyyy; yyyyy yyyyyyyyyey ;) yyyyyy;yyyy r yyyyyy;yyyy; y yyy;yyy ;) 0; yyy yyyy yyyyy yyyyyyyyyey ;) yyyyyy;yyyy r yyyyyy;yyyy; yy;ey ye yyy yy yyyy yyyeyyyyyey? y yyy;yyy ;) ; 7 yyy r 9 yyy yyyyy yyyyyy yyey 0 yyyyeyyyy