Introduction to Fourier optics. Textbook: Goodman (chapters 2-4) Overview:
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1 Introuton to ourr opts Ttbook: Goon (ptrs -) Ovrv: nr n nvrnt ssts T ourr trnsor Slr rton rsnl n runor pprotons.
2 . nr ssts n ourr trnsor tutorl (rnr) sst onnts n nput to n output su tt: It s s to b lnr t stss: S S S S T rspons o lnr sst n b prss b ts rspons to (bss) st o lntr untons. In prtulr sn n nvokn lnrt n tt: S Ts ns t puls rspons (or pont spr unton) o t lnr sst: S n portnt subst o lnr ssts r nvrnt ssts n t puls rspons s onl pnnt on t rn n oornts: In ts s t sst rspons s spl onvoluton btn t nput n t trnsr unton: Ts pls tt n rqun on t output o n nvrnt lnr sst s rl ultplton o t nput b t trnsr unton H Wr G GH H p To urtr suss ts lt us rbr so o t proprts o t ourr trnsor:
3 Dnton o t D ourr trnsor p Slrl t nvrs trnsor s: G G p T ontons or ts bn nt r tt t unton s "rulr" ppls or ll psl rltons n t optl rl. T ourr trnsor oprton n b v s oposton o unton n p r tlt n bss o lntr untons o t or pln vs t nl tn. to t X-s n sptl pro So portnt proprts o t T (qut vnt ro t nton) r: nrt: Slrt: b G b b St tor (lnr st trnsor to ps st): b p b Prsvl Tor ("onsrvton o nr"): G Proo n D (D proo ollos rtl t s prour)
4 G ' ' ' Convoluton tor: H G Proo n D (D proo ollos rtl t s prour) H G utoorrlton tor: G Proo n D (D proo ollos rtl t s prour) G G G ourr ntrl tor: ourr trnsor o sprbl untons: I
5 tn It s usul to onsr so T p p rt sn p s ll s to onsr T n lnrl oornts (spll or untons r onl pnnt on r). Drvton o t rsnl rton orul ts strt n ro Mll's qutons: D H t B E t D B Sn s nrll sp nvrnt n b usn: W t: E E E
6 B B E n E t t t t E E E E E ln E ln n E t ro t ovrll: E n E t E ln n n or oonous u t t slr quton: n E E t W s vl nr pt t bounrs I t slr l s ssu to b o t or (onorot): u P t Pos t P n t slr Mll quton ols t or P Pp P Hlolt quton: k t t k n. Ts quton n b solv b ns o t us o Grn's tor rltn t vlus o unton ns t volu to sur ntrl. Brl ts stts tt or n to "rsonbl" untons G: G v G G s G I ll not o trou t tls but lon srpton n b oun n Goon. Intutvl t Grn's unton osn or t Hlolt quton s sprl v: or : G R kr n n
7 k G R Wr R s t stn ro n rbtrr pont P Prorn t ntrton on t lt n s r lt t t vlu o t t orn (sn t lpln oprtor lsr s just k ro Hlolt quton). Tus t ntrton s on ovr sur oprsn pln n spr strtn to nnt. nr so ontons t vlu o t lttr vnss lvn onl t ntrton ovr t pln. Ts rsults n t Rl-Sorrl soluton s rll ttl orulton o t Huns prnpl. r kr P P oss prtur Wr t osounts or t t o o t Grn's unton n n n s vnss or sll nls. Rltn ts to t s prvousl ts n b ntrprt s t puls rspons o propton ro t prtur r: r P P os kr No lt us onsr t sll nl pprotons or ts. W t t ntrl: / k os / prtur P T pnn on t ntrton oornts pprs bot n t nurtor (ps tr) n n t nontor (pltu tr). or lr obvousl t pltu ontrbuton s sll. T osn tr s lso n pltu tr s sll t or lr. Ps trs n v o ours u r t so ll kp t. W tus t: rsnl rton: k P prtur / t us no onsr to or rstrtv solutons to t probl o rton ro n prtur. T rst s tvl sll nl soluton rtnn onl t rst orr (n ) trs o:
8 8 r Sn nt to nrlt t r orr trs rqur tt 8 or: 3 R Wr R s tpl nson o obsrvton n bot plns. or R= n = ts ounts to: 3 3 In ts s r sp propton n b srb b: k prtur k runor rton: Ts s or rstrtv pproton. t us rst pn t rsnl orul. W t: k prtur k I s so lr tt: Tn t qurt ps trs on n n b nlt n r lt t: k prtur k k
9 W s just (sl) ourr trnsor o t prtur l or R= n = ts ounts to:
10 Spl runor rton pttrns: Rtnulr n rulr prturs or trnssson trou rtnulr prtur t: rt rt t Ts bn sprbl unton t ourr trnsor s just ultplton o t to nvul ourr trnsors vn: k k sn sn n or t ntnst I I sn sn T t o t ntrl lob n rton s: or trnssson trou rulr prtur t: q r q t or t ourr trnsor s Jn unton kr kr J r kr k Wt t orrsponn ntnst strbuton: kr kr J r I Ts s ll n r pttrn. T t o t ntrl lob n ts s s. pltu rtn Consr rtnulr ontnn n pltu rtn:
11 os rt rt t In t rton ts ls to t "stnr" sn pnn. In t rton n ppl t onvoluton tor to t t rspons usn t ourr trnsor o t rtn: os To t: k k sn sn sn sn I orrsponn to n rtn lns n t prtur t pks r ll sprt n t ntnst strbuton s pprotl t su o t squr pltus: I sn sn sn sn Tus t orr trnssson s / n o t slobs (tr + n - orrs) v l n o /6 or u o ~6%. To bo or nt tr ps rtns or rltn rtns v to b us. Ts ll b suss or lbortl n t nt tutorl. nulr sptru o Wvs n ltrntv soluton to t probl o rton s prsnt b onsrn rsp propton s lnr trnsorton o t solutons o t r-sp Hlolt quton: k T solutons o r pln vs. t us onsr t l strbuton t = s suton o rtns o rnt sptl rquns ( ourr trnsor n rl sp): Tn: r t nvrs trnsor s:
12 Psll n ntrprt ts s su o pln vs sn pln v o t or: t r k t p t: k ˆ ˆ ˆ s n ntnst pttrn s pro n t pln t t pro nrss s bos sllr. Tus usn: ; ; s tt t ourr trnsor s just oposton to pln vs. t us no onsr t propton o ts n sp. Sn: Tkn t pln o t: usn t Hlolt quton bos: Sn ts ntrl quton s ulll or n strbuton ovr t ntrton vrbls t quton ust lso ol or t ntrn not just t ntrl lvn:
13 Wos soluton s: Psll ts ns tt r sp propton just ntrous rltv ps st btn ntrrns o vrous pln v oponnts. T rsnl pproton orrspons to t pproton or to lost orr sll nl rton. Evnsnt vs Clrl sotn s ron t ts ptur n t sptl rqun s rtr tn /. In ts s ns tt s nr. T soluton n tt s s not o proptn v but rtr o n ponntll n v: Ts vs o not propt n ovr lnt sl o t nvrs sptl rqun. ll t norton on ts rprsnts t n tl o t prtur s lost n t r l. In t r sp propton n t r-l n b onsr s rulr ltr n sptl rqun sp. r T t o rtn prtur on t SPW or rtn prtur t trnstt snl s: t t Sn ultplton s rl sp s onvoluton n ourr sp tn:
14 T t t T or n non pln v t outon nulr sptru s just t ourr trnsor o t prtur. Tus t sllr t prtur t bror t nulr strbuton (r nl rton). rsnl rton o n pltu rtn (Tlbot s). t us onsr n n pltu rtn but no onl n t rsnl pproton n usn t SPW ppro. or splt lts onsr n nnt rtn n D. W strt t os t T trnsr unton or n vn oponnt o t SPW s: r H Su tt or =/ t: H Sn t = t ourr trnsor s: os T t n t: j j W s spl to: os j To l t ntnst strbuton:
15 I os os os No tr r tr spl ss r ts prsson t urtr spl. or n : os I W s n t o t on t =. or n : os I W s n nvrt o t on t =. or n : os I W s n pltu rtn t t oubl rqun. Ts pnonon s rlt to n rvvl n pro sst. or t s o nrl pro sst t ols onl n t rsnl pproton sn tn t vrous rqun oponnts propt n ntr ultpls o t s ps tor.
Having a glimpse of some of the possibilities for solutions of linear systems, we move to methods of finding these solutions. The basic idea we shall
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