Development of a Digital Offset Laser Lock

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College of William and Mary W&M ScholarWorks Undergraduate Honors Theses Theses, Dissertations, & Master Projects 5-2016 Development of a Digital Offset Laser Lock Ian W. Hage College of William and Mary Follow this and additional works at: https://scholarworks.wm.edu/honorstheses 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 953. https://scholarworks.wm.edu/honorstheses/953 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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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

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 E2 1 + 1 2 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

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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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

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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

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 2 3 3 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

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

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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

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

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

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

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

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

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

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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666666666666666666 vvvvv6vvvv6vvv vv 6v666 v6 6 v6 vvv6 626666v266 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;

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666666666666666666 vvvv66vvv vv 6v666 v6 v6vv6 vv 6 626666v266 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

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666666666666666666 v6vv6v6vv 6v666 6 6 v6 v 6 616666 166 v6v v 66 6 6 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

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