Diffraction Gratings

Transcription

1 ECE st Cntury Elctromagntics Instructor: Offic: Phon: E Mail: Dr. Raymond C. Rumpf A 337 (915) rcrumpf@utp.du Lctur #9 Diffraction Gratings Lctur 9 1 Lctur Outlin Fourir sris Diffraction from gratings Th plan wav spctrum Plan wav spctrum for crossd gratings Th grating spctromtr Littrow gratings Pattrnd fanout gratings Diffractiv optical lmnts Lctur 9 Slid 2 1

2 Fourir Sris Born: Did: March 21, 1768 in Yonn, Franc. May 16, 183 in Paris, Franc. Jan Baptist Josph Fourir 1D Compl Fourir Sris If a function f() is priodic with priod, it can b pandd into a compl Fourir sris. f a m m 2 m j 2 2 m j 1 am f d 2 Typically, w rtain only a finit numbr of trms in th pansion. M f am mm 2 m j Lctur 9 Slid 4 2

3 2D Compl Fourir Sris For 2D priodic functions, th compl Fourir sris gnralizs to 2 p 2qy 2 p 2qy j j y 1 y,,,, f y a p q a p q f y da A p q A Lctur 9 Slid 5 Diffraction from Gratings 3

4 Filds in Priodic Structurs Wavs in priodic structurs tak on th sam priodicity as thir host. k Lctur 9 Slid 7 Diffraction Ordrs Th fild must b continuous so only discrt dirctions ar allowd. Th allowd dirctions ar calld th diffraction ordrs. Th allowd angls ar calculatd using th famous grating quation. Allowd Not Allowd Allowd Lctur 9 Slid 8 4

5 Fild in a Priodic Structur Th dilctric function of a sinusoidal grating can b writtn as r Kr r r,avg cos A wav propagating through this grating taks on th sam symmtry. jkinc r Er Ar jkinc r A r,avg coskr jk jkinc K r jkinc K r inc r A A A r,avg 2 2 wav 1 wav 2 wav 3 Lctur 9 9 Grating Producs Nw Wavs Th applid wav splits into thr wavs. jkinc r jkinc r jkinc Kr jkinc Kr Each of thos splits into thr wavs as wll. jkinc r jkinc r jkinc Kr jkinc Kr jkinc Kr jkinc Kr jkinc 2K r And ach of ths split, and so on. jkinc r jkinc Kr jk inc Kr jkinc r jkinc 2K r This quation dscribs k m k mk m inc,..., 2, 1,,1,2,..., th total st of allowd diffraction ordrs. Lctur 9 1 5

6 Wav Incidnt on a Grating inc K Boundary conditions rquird th tangntial componnt of th wav vctor b continuous. k,trn? k,inc Th wav is ntring a grating, so th phas matching condition is,inc k m k mk Th longitudinal vctor componnt is calculatd from th disprsion rlation z avg k m k n k m For larg m, k z,m can actually bcom imaginary. This indicats that th highst diffraction ordrs ar vanscnt. Lctur 9 11 Th Grating Equation k inc inc n inc 1 Th Grating Equation navg sin m ninc sininc m sin Not, this rally is just,inc k m k mk Proof: k m k,inc mk 2 kn avgsin m kn incsininc m K n avg navg sin m ninc sininc m navg sin m ninc sininc m n trn navg sin m ninc sininc m sin Lctur 9 z 12 6

7 Grating Equation in Diffrnt Rgions Th angls of th diffractd mods ar rlatd to th wavlngth, rfractiv ind, and grating priod through th grating quation. Th grating quation only prdicts th dirctions of th mods, not how much powr is in thm. nrf Rflction Rgion nrf sin m ninc sininc m n inc Transmission Rgion ntrn sin m ninc sininc m n trn Lctur 9 Slid 13 Diffraction in Two Dimnsions W know vrything about th dirction of diffractd wavs just from th grating priod. Diffraction tnds to occur along th lattic plans. Th grating quation says nothing about how much powr is in th diffractd mods. W nd to solv Mawll s quations for that! Lctur 9 Slid 14 7

8 Effct of Grating Priodicity Subwavlngth Grating Subwavlngth Grating Low Ordr Grating High Ordr Grating n inc n inc n inc n inc n avg n avg n avg n avg n avg n n n avg ninc inc inc Lctur 9 Slid 15 Animation of Grating Diffraction at Normal Incidnc Lctur 9 Slid 16 8

9 Animation of Grating Diffraction at an Angl of Incidnc Lctur 9 Slid 17 Wood s Anomalis Robrt W. Wood obsrvd rapid variations in th spctrum of light diffractd by gratings which h could not plain. Latr, Hssl plaind thm an idntifid two diffrnt typs. Typ 1 Rayligh Singularitis Rapid variation in th amplituds of th diffractd mods that corrspond to th onst or disapparanc of othr diffractd mods. Typ 2 Rsonanc Effcts A rsonanc condition arising from laky wavs supportd by th grating. Today, w call this guidd mod rsonanc. Robrt Williams Wood R. W. Wood, Phil. Mag. 4, 396 (192) A. Hssl, A. A. Olinr, A Nw Thory of Wood s Anomalis Lctur 9 on Optical Gratings, Appl. Opt., Vol. 4, No. 1, 1275 (1965). 18 9

10 Grating Cutoff Wavlngth Whn m bcoms imaginary, th mth mod is vanscnt and cut off. Assuming normal incidnc (i.. inc = ), th grating quation rducs to nsin m m Th first diffractd mods to appar ar m = 1. Th cutoff for th first ordr mods happns whn (±1) = sin91 n n To prvnt th first ordr mods, w nd or n To nsur w hav first ordr mods, w nd or n Lctur 9 Slid 19 Total Numbr of Diffractd Mods Givn th grating priod and th wavlngth, w can dtrmin how many diffractd mods ist. Again, assuming normal incidnc, th grating quation bcoms m m m m sin sin 1 navg navg Thrfor, a maimum valu for m is m ma navg Th total numbr of possibl diffractd mods M is thn 2m ma +1 M 2navg 1 Lctur 9 Slid 2 1

11 Dtrmining Grating Cutoff Conditions Condition ordr mod No 1 st ordr mods Ensur 1 st ordr mods No 2 nd ordr mods Ensur 2 nd ordr mods No m th ordr mods Ensur m th ordr mods Rquirmnts Always ists unlss thr is total intrnal rflction Grating priod must b shortr than what causs 1) = 9 Grating priod must b largr than what causs 1) = 9 Grating priod must b shortr than what causs 2) = 9 Grating priod must b largr than what causs 2) = 9 Grating priod must b shortr than what causs m) = 9 Grating priod must b largr than what causs m) = 9 Lctur 9 Slid 21 Thr Mods of Opration for 1D Gratings Bragg Grating Coupls nrgy btwn countr propagating wavs. Diffraction Grating Coupls nrgy btwn wavs at diffrnt angls. Long Priod Grating Coupls nrgy btwn copropagating wavs. Applications Thin film optical filtrs Fibr optic gratings Wavlngth division multipling Dilctric mirrors Photonic crystal wavguids Applications Bam splittrs Pattrnd fanout gratings Lasr locking Spctromtry Snsing Anti rflction Frquncy slctiv surfacs Grating couplrs Applications Snsing Dirctional coupling Lctur 9 Slid 22 11

12 Analysis of Diffraction Gratings Dirction of th Diffractd Mods nsin m ninc sininc m sin Diffraction Efficincy and Polarization of th Diffractd Mods W must obtain a rigorous solution to Mawll s quations to dtrmin amplitud and polarization of th diffractd mods. E jh H j E E H Lctur 9 Slid 23 Applications of Gratings Subwavlngth Gratings Only th zro ordr mods may ist. Littrow Gratings Gratings in th littrow configuration ar a spctrally slctiv rtrorflctor. Pattrnd Fanout Gratings Gratings diffract lasr light to form imags. Holograms Holograms ar stord as gratings. Applications Polarizrs Artificial birfringnc Form birfringnc Anti rflction Effctiv ind mdia Applications Snsors Lasrs Spctromtry Gratings sparat broadband light into its componnt colors. Lctur 9 Slid 24 12

13 Th Plan Wav Spctrum Priodic Functions Can B Epandd into a Fourir Sris A Wavs in priodic structurs oby Bloch s quation j r, E y A A Th nvlop A() is priodic along with priod so it can b pandd into a Fourir sris. m S m 2 m j 2 m j S m A d Lctur 9 Slid 26 13

14 Rarrang th Fourir Sris (1 of 2) A priodic fild can b pandd into a Fourir sris. j r A, E y m m S m S m 2 m j j r 2 m j j r 2 m j j j y y S m Hr th plan wav trm j is brought insid of th summation. m Lctur 9 Slid 27 Rarrang th Fourir Sris (2 of 2) can b combind with th last compl ponntial. j j y y, E y Sm m m S m j y y 2 m Now lt k and ky,m m, y, E y m S m 2 m j 2 m j jk m r 2 m km aˆ aˆ y y Lctur 9 Slid 28 14

15 Th Plan Wav Spctrum, E y m S m jk m r W rarrangd trms and now w s that a priodic fild can also b thought of as an infinit sum of plan wavs at diffrnt angls. This is th plan wav spctrum of a priodic fild. Fild Plan Wav Spctrum Lctur 9 Slid 29 Longitudinal Wav Vctor Componnts of th Plan Wav Spctrum Th wav incidnt on a grating can b writtn as E, y E inc k k n sin jk,inc ky,inc y,inc inc inc k k n cos y,inc inc inc inc y Phas matching into th grating lads to 2 km k,inc m m, 2, 1,,1,2, Not: k is always ral. Each wav must satisfy th disprsion rlation. y y grat k m k m k n k m k n k m grat 2 W hav two possibl solutions hr. 1. Purly ral k y 2. Purly imaginary k y. Lctur 9 Slid 3 15

16 Visualizing Phas Matching into th Grating Th wav vctor pansion for th first 11 mods can b visualizd as k inc k 5 k 4 k 3 k 2 k 1 k k 1 k 2 k 3 k 4 k 5 Each of ths is phas matchd into matrial 2. Th longitudinal componnt of th wav vctor is calculatd using th disprsion rlation in matrial 2. n 1 k y is imaginary. Th fild in matrial 2 is vanscnt. Lctur 9 Slid 31 n 2 k y is ral. A wav propagats into matrial 2. Not: Th vanscnt filds in matrial 2 ar not compltly vanscnt. Thy hav a purly ral k so powr flows in th transvrs dirction. k y is imaginary. Th fild in matrial 2 is vanscnt. y Conclusions About th Plan Wav Spctrum Filds in priodic mdia tak on th sam priodicity as th mdia thy ar in. Priodic filds can b pandd into a Fourir sris. Each trm of th Fourir sris rprsnts a spatial harmonic (plan wav). Sinc thr ar in infinit numbr of trms in th Fourir sris, thr ar an infinit numbr of spatial harmonics. Only a fw of th spatial harmonics ar actually propagating wavs. Only ths can carry powr away from a dvic. Tunnling is an cption. Lctur 9 Slid 32 16

17 Plan Wav Spctrum from Crossd Gratings Grating Trminology 1D grating Ruld grating 2D grating Crossd grating Rquirs a 2D simulation Rquirs a 3D simulation Lctur 9 Slid 34 17

18 Diffraction from Crossd Gratings Doubly priodic gratings, also calld crossd gratings, can diffract wavs into many dirctions. Thy ar dscribd by two grating vctors, K and K y. Two boundary conditions ar ncssary hr. km k,inc mk m..., 2, 1,,1, 2,... k n k nk n..., 2, 1,,1,2,... k inc y y,inc y K K y 2 ˆ 2 yˆ y K K y Lctur 9 Slid 35 Transvrs Wav Vctor Epansion Crossd gratings diffraction in two dimnsions, and y. To quantify diffraction for crossd gratings, w must calculat an pansion for both k and k y. 2 m kmk,inc m,, 2, 1,,1, 2, 2 n kynky,inc n,, 2, 1,,1, 2, y k m n k m k n yˆ, ˆ t y % TRANSVERSE WAVE VECTOR EXPANSION M = [-floor(n/2):floor(n/2)]'; N = [-floor(ny/2):floor(ny/2)]'; k = kinc - 2*pi*M/L; ky = kyinc - 2*pi*N/Ly; [ky,k] = mshgrid(ky,k); W will us this cod for 2D PWEM, 3D RCWA, 3D FDTD, 3D FDFD, 3D MoL, and mor. Lctur 8 Slid 36 18

19 Visualizing th Transvrs Wav Vctor Epansion k m ky n ktran m, n k ˆ ˆ m ky n y Lctur 9 Slid 37 Longitudinal Wav Vctor Epansion Th longitudinal componnts of th wav vctors ar computd as, k m n k n k m k n rf 2 2 z rf y, k m n k n k m k n trn 2 2 z trn y Th cntr fw mods will hav ral k z s. Ths corrspond to propagating wavs. Th othrs will hav imaginary k z s and corrspond to vanscnt wavs that do not transport powr. k m n tran, Lctur 9 Slid 38 kz m, n 19

20 Visualizing th Ovrall Wav Vctor Epansion k m n tran, k rf z m, n k m n rf, k m n trn, Lctur 9 Slid 39 Th Grating Spctromtr 2

21 What is a Grating Spctromtr Diffraction Grating Sparatd Colors Input Light Lctur 9 41 Spctral Snsitivity W start with th grating quation. navg sin m ninc sininc m W dfin spctral snsitivity as how much th diffractd angl changs with rspct to wavlngth m. m m m cos m navg cos m navg m This quation tlls us how to maimiz snsitivity. 1. Diffract into highr ordr mods ( m). 2. Us short priod gratings ( ). 3. Diffract into larg angls ( m)). 4. Diffract into air ( n avg ). Lctur

22 Ocan Optics HR4 Grating Spctromtr Fibr Optic Input 1 SMA Connctor 4 Collimating Mirror 7 Collction Lnss 1 Optional UV Dtctor 2 Entranc Slit 5 Diffraction Grating 8 Dtctor Array 3 Optional Filtr 6 Focusing Mirror 9 Optional Filtr Lctur 9 43 Littrow Gratings 22

23 Littrow Configuration In th littrow configuration, th +1 ordr rflctd mod is paralll to th incidnt wav vctor. This forms a spctrally slctiv mirror. Incidnt +1 Lctur 9 45 Conditions for th Littrow Configuration Th grating quation is nsin m nsininc m Th littrow configuration occurs whn 1 inc Th condition for th littrow configuration is found by substituting this into th grating quation. 2nsin inc Lctur

24 Spctral Slctivity Typically only a con of angls rflctd from a grating is dtctd. W wish to find d/d by diffrntiating our last quation. d 2n cos d Typically this is usd to calculat th rflctd bandwidth. 2n cos f n f c 2 2 cos Linwidth (optics and photonics) Bandwidth (RF and microwav) Lctur 9 47 Eampl (1 of 2) Dsign a mtallic grating in air that is to b opratd in th littrow configuration at 1 GHz at an angl of 45. Solution Right away, w know that n 1. inc 45 8 m c 31 s 3. cm f 1 GHz Th grating priod is thn found to b 3. cm 2.12 cm 2nsininc 21.sin45 Lctur

25 Eampl (2 of 2) Solution continud Assuming a 5 con of angls is dtctd upon rflction, th bandwidth is cm 1 GHz cos 45 f 8 m s 5.87 GHz Lctur 9 49 Pattrnd Fanout Gratings 25

26 Nar-Fild to Far-Fild Aftr propagating a long distanc, th fild within a plan tnds toward th Fourir transform of th initial fild. L FFT E, y, E, y, L Lctur 9 51 What is a Pattrnd Fanout Grating? Diffraction grating forcs th fild to tak on th profil of th invrs Fourir transform of an imag. Aftr propagating vry far, th fild taks on th profil of th imag. Lctur

27 Grchbrg-Saton Algorithm: Initialization Nar Fild amplitud phas Stp 3 Rplac amplitud FFT Far Fild amplitud phas Stp 4 Calculat far fild amplitud phas amplitud phas FFT -1 Stp 2 Calculat nar fild Stp 1 Start with dsird far fild imag. Lctur 9 53 Grchbrg-Saton Algorithm: Itration Nar Fild amplitud phas Stp 7 Rplac amplitud FFT Far Fild amplitud phas Stp 8 Calculat far fild amplitud phas FFT -1 Stp 6 Calculat nar fild Stp 5 Rplac amplitud with dsird imag. Lctur

28 Grchbrg-Saton Algorithm: End Nar Fild Far Fild amplitud phas amplitud phas FFT This is th phas function of th diffractiv optical lmnt. This is what th final imag will look lik. Aftr svral dozn itrations Lctur 9 55 Th Final Fanout Grating Phas Function (rad) A surfac rlif pattrn is tchd into glass to induc th phas function onto th bam of light. W could also print an amplitud mask using a high rsolution lasr printr. Lctur

29 Diffractiv Optical Elmnts What is a Diffractiv Optical Elmnt Convntional Lns Diffractiv Optical Lns (Frsnl Zon Plat) If th dvic is only rquird to oprat ovr a narrow band, dvics can b flattnd. Th flattnd dvic is calld a diffractiv optical lmnt (DOE). Lukas Chrostowski, Optical gratings: Nano-nginrd lnss, Natur Photonics 4, (21). Lctur