EE243 Advanced Electromagnetic Theory Lec # 24 Imaging as Diffraction

Transcription

1 43 Advaced lectromagetic Theory Lec # 4 Imagig as Diffractio Fresel Zoes ad Les lae Wave Spectra, Les Capture ad Image Resolutio ad Depth of Focus N-wave imagig Resolutio hacemet: Off axis Illumiatio hase-shiftig Masks Readig: (This lecture is self cotaied ad is based o excerpts from Bor ad Wolf Chapters 8 ad 9 plus Chapter by Smith i Sheats ad Smith) 1

2 Overview Optical imagig is a applicatio of diffractio i which the plaewaves that make up the spatial diffractio spectrum are capture ad reprocessed. The equivalece theorem allows a plaewave represetatio to be built from ad H tagetial o a plae. Waves that travel at agles captured by the les are redirected ad rephased by the les to arrive at the focal poit with the phase at which they left object plae. Resolutio ehacemet modifies plaewave spectrum The itra-wave phase is all importat Describes image behavior i space about the focal plae Accouts for les imperfectios (Aberratios) Used to develop iterferometric istrumetatio

3 Fresel Zoes Huyge s Sphere Object (oit Source or illumiated Mask) Zoe 1 Zoe Image (Mathematical Surface) Field at a poit ca be viewed as addig ad subtractig cotributios from λ/ phase Zoes A Fresel les cosists of small rig leses that map ray directio from small to large circle ad flip phase of egative zoes (so all add) 3

4 Imagig as a Form of Diffractio x o x y o θ Object z o θ Image y Object Image Mask Les lae Object seds out plaewave spatial spectrum (rays) Les is i far field (Fourier Trasform) ad low pass filters the spatial harmoics Les redirects plae wave directios (chage θ Object to θ Image ) to coverge ad to each have its origial phase at z o the image plae z 0. Image is i far field of les (Fourier Trasform) ad is thus the iverse trasform of spatial spectrum after low pass filterig z 4

5 eˆ * s ikr e ( x) F ( k, k ) F r laewave xpasio i ik x * * ( k, k0 ) e [ ωeˆ ( Bs ) + eˆ ( k ( s ))] da 4π xample for a mask with period i x directio. S 1 0 TOTAL TOTAL N N Start from Trasverse compoets of ad B o plae Make plaewave spectrum expasio betwee mask ad les (assume periodic ad switch to e jwt ) Les the low pass filters ad apodizes/phases trasmitted spectrum ropagatio to image plae is thus the Fourier Trasform of the filtered/phased spectrum j A e k Jackso 10.7 ( θ ) x + k cos( o) ( 0 si o o 0 θ jφ( θ ) j ( k ( i ) xi + k ( i) i 0 si θ 0 cos θ ( θ ) e e i z o ) 5 z i )

6 lectric Field as Sum of lae Waves Simplify to (x,z) plae, i y-directio, A 1, Φ 0 Mask with period Bragg Coditio Implies ( ) si θ λ cos ( θ ) λ 1 k k x ( θ ) xˆ + k cos( )zˆ xˆ + k zˆ k si x 0 z 0 θz TOTAL e j k ( θ ) x+ k cos( ) ( 0 si 0 θ z) e j ( k x) Three wave case for o-axis illumiatio of mask with period TOTAL π π π π j ( x+ cos j 0 1e λ + 0e λ + ( θ ) z) (0 x+ cos( θ ) z) ( x cos( θ ) ) z + 1 e j π π λ 6

7 Optical System oit Spread Fuctio Mask Les Wafer Image of a pi hole (Diffractio limited) Relatioship for electric fields The small pihole due to its size diffracts uiformly over all agles. i hole This diffractio uiformly fills the les pupil. The les re-phases the remaiig emergig rays so that they re-coverge at the wafer with the same relative phases ad uiform magitude. The electric field at the waver is thus the iverse Fourier trasform of a disk Airy Fuctio. The itesity is the time average of the square of the electric field (Airy fuctio) The patter shape is idepedet of the shape of the pi hole with diameter 1.λ/NA. The peak is proportioal to pi hole area the peak I is proportioal to Area or (dimesio) 4. 7

8 Resolutio i rojectio ritig f focal distace d les diameter oit spread fuctio Null positio F# f/d f f 1.λ 0.61λ d d λ NA Miimum separatio of a star to be visible. DG Fig. Ch 5 8

9 Optical rojectio ritig arameters #0 Key arameters: λ, NA, σ Wavelegth λ 48 m) Numerical Aperture NA si (θ) 0.5 artial Coherece Factor σ (NAc/NAo) 0.3 9

10 Resolutio ~ Trasverse Variatio Larger agles give higher resolutio φ #1 Resolutio / λ/( siφ) 0.5(λ/NA)) λ 48 m Assumes oe wave is oaxis ad the other off-axis λ TRANS λ/siφ 3.λ 800m The most useful rays i formig a image are those with the same pitch as the patter Wave graphic by Ogi glader ad Kie Lam 10

11 Depth of Focus i rojectio ritig # Depth of Focus λ/(na ) Result must be modified for a) High NA, ad b) Two waves at arbitrary agles. DG Fig. Ch 5 11

12 arameters for Microlab rojectio riters Workig Resolutio Tool λ m NA σ k 1 θ LN deg θ ILL deg k 1 λ/na m λ/(4na) m TFR m Μ Caogh GCA-g GCA-i ASML- DUV TFR Total focus rage x Rayleigh Depth of Focus DOF λ M is the demagificatio factor DOF k L LINWIDTH k NA 1 λ ( ) NA 1

13 Immersio Lithography Cocept Imagig i a liquid medium with refractive idex offers a a factor of reductio i resolutio 193 m 1.44 to m 1. to 1.5? Implemetatio: Drop ad Drag Dispese water from frot side of les, use the surface tesio to make the drop follow the les, ad suck i the liquid o the back of the les. 13

14 UV rojectio (X-Ray ) Lithography Absorbig Mask used off-axis 13.5 m S 13.4 m wavelegth NA λ/NA 31 m TFR λ/na 149 m λ/ layer pairs 14

15 Bragg Coditio Icidet ray with wave frots L S Quartz L+S si λ ϕ λ φ Trasmitted ray Chrome Wavefrots Ray of Light Diffracted ray #3 The Bragg coditio sets the diffractio agles 15

16 16 10 Applied M Fall 006, Neureuther Lecture #4 Ver 11/19/06 lectric Field Spectrum M(u) from (x) x comb x rect A x m / ) ( { } x comb F x rect A F x m F u M / ) ( ) ( Sheats ad Smith ( ) ( ) u u u u c A u M 0 0 si δ Values are ½, 1/π, 1/3π, 1/5π u 0 1/

17 lectric Field: Siusoids Biary Mask with period ad opeig space s s Whe filtered to three waves (0, +1, ad -1) ( x) πx cos k x1 π/ A [ ( )] π s [ ( s )] π si Whe s / ( x) cos π πx 17

18 upil Wave Traffic: artial Coherece #4 Les is a low pass filter of mask diffractio at Bragg agles Siθ Y NA Shifted by siφ l/ Les upil Siθ MAX NA Coe of Icidet Light Some misses pupil Diffracted Orders from a mask with period Siθ X + otetial for eterig the pupil 18

19 Itesity as Square of lectric Field The eergy carried by a wave ad the work doe o a material are proportioal to the time average of the square of the electric field. Thus the itesity is proportioal to whe the field is real ad * whe phasors are used ad is complex. Itesity * gives I πx πx * ( x) + cos( ) + 4 cos Sice the Fourier trasform coverges to the average at a discotiuity, the electric field at the mask edge will be about 0.5, ad the itesity at a mask edge will be about 0.5. #5 The itesity at a mask edge is oly 30% of the clear field itesity regardless of feature type ad size. 1 19

20 C I I max max + I I mi mi Cotrast vs. eriod Cotrast 1.0 First Order Max Dead Zoe Third Order Max L S 0.5 Ubalaced 3rd λ/na Cutoff 0

21 1 10 Applied M Fall 006, Neureuther Lecture #4 Ver 11/19/06 lectric Field ad Itesity: Defocus ( ) ( ) ( ) ) cos ( 1 ) cos (0 0 ) cos ( z x j z x j z x j TOTAL e e e θ λ π π θ λ π π θ λ π π cos( ) ) ( 1 ) ( 0 1 ) cos( z j z j TOTAL e x e θ λ π λ π π Note that cos(θ -1 ) cos(θ +1 ) ad cos(θ 0 ) 1 ( ) 1 cos λ θ Ad that for a biary (real trasmissio fuctio) mask The Rayleigh defocus z value to give λ/4 is ( ) 1 cos θ λ z As expected the o-axis ad off-axis parts are 90 o out of phase.

22 olarizatio ffects at High NA arallel Orietatio erpedicular Orietatio Vector Additio I MAX Issue X ~cosθ I MAX ~(cosθ) I MIN Issue Z ~siθ I MIN ~(siθ) total total

23 Resolutio hacemet Techiques Resolutio hacemet mphasizes High Frequecies Covetioal Illumiatio Biary Mask Les Capture 0 Frequecy 0 Frequecy hase Mask 0 0 Frequecy Frequecy 0 0 Modified Illumiatio Frequecy I-Les Filter Frequecy Bokor, Neureuther, Oldham, Circuits ad Devices,

24 Two Ray Ifiite DOF Ray # 1 θ 1 θ k y Ray # k x itch eriod itch Whe θ 1 θ the cotributios from Ray #1 ad Ray # track each exactly with axial distace ad a INFINIT depth of focus is produced. k Trasverse λ si ( θ ) k 0 π k Trasverse si( θ ) si( θ ) NA λ NA Doubled Resolutio! With ifiite DOF 4

25 Top Hat Geeral Shapes Illumiatio Schemes Aular DOF, Cotacts k k 1.3 k k 1.7 σ IN σ OUT 0.85 Quadruple H,V lies, DOF k k.0 upil -1 1 upil Dipole V lies, DOF k k upil H lies ad cotacts formed 1 via a double exposure upil The k1 factor is iversely proportioal to the lateral separatio of the illumiatio k 1 1/( x separatio) 5

26 Quadrapole Optimizatio Dipole illumiatio ca prit smaller lies but lies oly. 6

27 hase-shiftig Mask Types Alteratig (Strog) Atteuatig (Weak) Used for Cotacts 6% to 10% gives slope improvemet of 30%. Sidelobe issue. hase dge Requires secod trim mask exposure. Chromeless (Oly 0 order) 7

28 Atteuatig hase-shiftig Masks Itesity of 6% comes from a electric field of -0.5 Goig from positive electric fields to egative electric fields icreases edge slope ad creates darker itesity ear edge. 8

29 10 Applied M Fall 006, Neureuther Lecture #4 Ver 11/19/06 Alteratig hase-shiftig Mask erfect Null / Frequecy doubled Sheats ad Smith 9

30 hase-shiftig Mask: lectric Fields Sheats ad Smith 30