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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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
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t t t s r ss t s s st tr s t s r t Pr rt t r P tr r t t t s s t rr r s t r rr t s s t t tr t s r st ts r q 2 s s2st s s r t s t t r s r s2st s 2st s t rs t2 t rs t2 st r t rs t2 r s t s s r t s r tr s s2st s s st t s s r 2 t s r q 2 s r t rs 2st s t t rs t2 st r t rs t2 r t 3 t t s t t t t s r 2 ts s2st s t t 1 r t t t rs s2st t t rs t2 r s rt r 2 t rt 2 r ts s P s P s r q 2 s r t s2st t s t s s tr sts t t s2st s 2 r 2 s 2 r q 2 t t t s s t 2 t t s r t t s2st t rs t2 P P s t s r s2st s t t r 2 t t t r s 3 r2 2 t r s s s r r t t P r s r r t t r s r r t r2 s t r rt s r r r tt t r r r s r t r s t P t r r ts r q 2 t t s r rts t t t s r r r s r s r r t r t t P r r 2 tr t t t r r s st ts t s t t
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
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
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
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
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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
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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
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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;
666666666666666666 vvvvv6vvvv6vvv vv 6v666 v6 6 v6 vvv6 626666v266 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
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
666666666666666666 vvvv66vvv vv 6v666 v6 v6vv6 vv 6 626666v266 6v vv66 v6 v6vv6 vv k lle lllle llllle llklll klllk lleklelll
666666666666666666 6666vv6vv 6 666 6 v6 6 6 616666 166 6v 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/////
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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