Microlens quality assessment using the Extended Nijboer-Zernike diffraction theory

Transcription

1 Mcrolens qualty assessment usng the Extended Njboer-Zernke dracton theory J.J.M. Braat, S. van Haver, S.F. Perera Delt Unversty o Technology Department o Imagng Scence and Technology Optcs Research Group Lorentzweg, 68 CJ Delt, The Netherlands E-mal: j.j.m.braat@tudelt.nl

2 Contents General approach to lens qualty measurement Through-ocus ntensty measurements and aberraton retreval Extended Njboer-Zernke theory o dracton usng the Debye ntegral The Raylegh and Debye dracton ntegral, small-numercal-aperture case Requred modcatons o the Debye ntegral Numercal tests and error analyss Conclusons Braat, van Haver, Perera Lens qualty assessment EOS - Optcal Mcrosystems 9

3 Lens qualty assessment Methods: ntererometry, pont-spread uncton analyss, Shack-Hartmann test, requency transer etc. Pont-spread uncton analyss rom through-ocus ntensty plots - Gerchberg-Saxton algorthm (pupl mage plane) - requency transer recovery (electron mcroscopy) - Extended Njboer-Zernke (ENZ) aberraton retreval ENZ-retreval based on Debye dracton ntegral or pontspread ormaton. For small NA and/or small pupl dameter, Raylegh dracton ntegral s requred! Braat, van Haver, Perera Lens qualty assessment EOS - Optcal Mcrosystems 9

4 Contents General approach to lens qualty measurement Through-ocus ntensty measurements and aberraton retreval Extended Njboer-Zernke theory o dracton usng the Debye ntegral The Raylegh and Debye dracton ntegral, small-numercal-aperture case Requred modcatons o the Debye ntegral Numercal tests and error analyss Conclusons Braat, van Haver, Perera Lens qualty assessment EOS - Optcal Mcrosystems 9 4

5 Phase retreval rom through-ocus ntensty patterns propagaton drecton through ocus Ext pupl (waveront + obstructon ) Braat, van Haver, Perera Lens qualty assessment EOS - Optcal Mcrosystems 9 5

6 Dracton geometry (ext pupl mage) A s (mage o) dractng aperture R s dstance ext pupl to ocus F Debye ntegral s an approxmaton o the eld n a general pont P R O Integraton s over all ponts Q n the dractng aperture plane A or over the ext pupl sphere through the orgn O. z Deocusng range Δ<<R Braat, van Haver, Perera Lens qualty assessment EOS - Optcal Mcrosystems 9 6

7 CCD detecton o through-ocus mages For a slghtly aberrated lens system, the almost dracton-lmted spot should be well resolved by the detector pxels pxels (5 µm each) or a typcal dracton unt o.6 λ/na Wth λ6 nm, NA choce n range.5<na<. Detecton o source pont at low aperture conjugate sde! α NAsnα Captured CCD-mages Braat, van Haver, Perera Lens qualty assessment EOS - Optcal Mcrosystems 9 7

8 Contents General approach to lens qualty measurement Through-ocus ntensty measurements and aberraton retreval Extended Njboer-Zernke theory o dracton usng the Debye ntegral The Raylegh and Debye dracton ntegral, small-numercal-aperture case Requred modcatons o the Debye ntegral Numercal tests and error analyss Conclusons Braat, van Haver, Perera Lens qualty assessment EOS - Optcal Mcrosystems 9 8

9 The ENZ-ormalsm based on the Debye ntegral (, φ; ) m m m (, ) exp{ φ} U r nvn r m mn, m m { } n n ( ) { } P( ρ, θ) A( ρ, θ) exp Φ ( ρ, θ) R ρ exp mθ mn, Detecton o I(r,φ,) U(r,φ,), nonlnear operaton! Braat, van Haver, Perera Lens qualty assessment EOS - Optcal Mcrosystems 9 9

10 Lnearzed retreval method or Zernke coecents Basc prncple Observed ntensty analytc expresson lnearzed analytc expresson (m,n) x basc-unctons Match experment to theory by solvng a lnear system o equatons n (m,n) wth the ad o recorded mages Through-ocus mages Experment Theory Match Braat, van Haver, Perera Lens qualty assessment EOS - Optcal Mcrosystems 9

11 Contents General approach to lens qualty measurement Through-ocus ntensty measurements and aberraton retreval Extended Njboer-Zernke theory o dracton usng the Debye ntegral The Raylegh and Debye dracton ntegral, small-numercal-aperture case Requred modcatons o the Debye ntegral Numercal tests and error analyss Conclusons Braat, van Haver, Perera Lens qualty assessment EOS - Optcal Mcrosystems 9

12 Raylegh and Debye dracton ntegral; small-na expressons or axal ntensty Analytc expresson or Raylegh on-axs ntensty gven by H. Osterberg and L.W. Smth, J. Opt. Soc. Am. 5, 5-54 (96) Concept o optcal axal coordnate ntroduced by Y. L, J. Opt. Soc. Am. 7, (98). Further rened by C.J.R. Sheppard and P. Török, J. Opt. Soc. Am. A, 56-6 (). Optcal axal coordnate (small NA) : Raylegh on - axs ntensty (small NA) : I( Δ) Debye approxmaton or optcal axal coordnate : Debye on - axs ntensty I( Δ) Δ k sn snc α snc. ( Δ / R kδsn 4 ) : α Braat, van Haver, Perera Lens qualty assessment EOS - Optcal Mcrosystems 9. krδsn α. ( R + Δ) kδsn α.

13 Raylegh and Debye dracton ntegral, small NA eatures NA. NA.5 Braat, van Haver, Perera Lens qualty assessment EOS - Optcal Mcrosystems 9

14 Contents General approach to lens qualty measurement Through-ocus ntensty measurements and aberraton retreval Extended Njboer-Zernke theory o dracton usng the Debye ntegral The Raylegh and Debye dracton ntegral, small-numercal-aperture case Requred modcatons o the Debye ntegral Numercal tests and error analyss Conclusons Braat, van Haver, Perera Lens qualty assessment EOS - Optcal Mcrosystems 9 4

15 Requred modcaton o the Debye ntegral to approxmate the Raylegh ntegral: axal and lateral coordnate transormaton Raylegh U R ( x, y, z) λ A z R QP exp z' R QF U ( x', y') { k( R R )} QP QF dx' dy' O Debye U D ( x, y, z) λ A z z' U RQF exp ( x', y') { k ( r r )} dx' dy' QP QF z Braat, van Haver, Perera Lens qualty assessment EOS - Optcal Mcrosystems 9 5

16 Braat, van Haver, Perera Lens qualty assessment EOS - Optcal Mcrosystems 9 6 Debye Raylegh: axal and lateral coordnate transormaton { } { } { } ( ) { } { } α π λ α π λ α π λ α π λ ϕ ϑ ρ π ϕ ϑ ρ π ρ ρ α π λ ρ ϑ ρ ρ ϕ ϑ ρ π ρ ϑ ρ λ α sn : Normalzed ampltude correcton actor. dstance rom aperture uncton o pattern as a lateral expanson dracton sn sn 4 sn ' wth ) cos( ' exp ) cos( exp. towards aperture zeros ntensty between axal dstance compacton o ' exp sn exp exp. ) cos( exp )exp, ( sn ),, ( n normalzed coordnates : Debye ntegral R r R R R r r r r R d d U z y x U p D unt crcle r

17 Debye Raylegh: regon o applcablty o the coordnate transormaton U D ( x, y, z) sn λ α U p unt crcle ( ρ, ϑ)exp { ρ } exp{ π rρ cos( ϑ ϕ) } ρdρdϑ. In adaptng the Debye ntegral to the Raylegh ntegral by an axal and lateral coordnate transormaton ( r exponental that are proportonal to r r', '), hgher order terms n the ρ and rρ could not be ncluded. Wth the requrement that these exponental actors nowhere exceed the value o. radan, we nd the ollowng regon o applcablty o the coordnate transormaton a) pupl dameter ρ 6λ b) axal dstance ext pupl to ocus R λ Braat, van Haver, Perera Lens qualty assessment EOS - Optcal Mcrosystems 9 7

18 Debye Raylegh: resdual errors ater transormaton Braat, van Haver, Perera Lens qualty assessment EOS - Optcal Mcrosystems 9 8

19 Contents General approach to lens qualty measurement Through-ocus ntensty measurements and aberraton retreval Extended Njboer-Zernke theory o dracton usng the Debye ntegral The Raylegh and Debye dracton ntegral, small-numercal-aperture case Requred modcatons o the Debye ntegral Numercal tests and error analyss Conclusons Braat, van Haver, Perera Lens qualty assessment EOS - Optcal Mcrosystems 9 9

20 Pupl uncton retreval based on adapted Debye ntegral Input pupl uncton (ampltude / phase) Standard retreval (Debye ntegral) Retreval usng adapted Debye ntegral Braat, van Haver, Perera Lens qualty assessment EOS - Optcal Mcrosystems 9

21 Mcrolens pupl uncton retreval Mcrolens pupl uncton (ampltude / phase) Standard retreval (Debye ntegral) Retreval usng adapted Debye ntegral Braat, van Haver, Perera Lens qualty assessment EOS - Optcal Mcrosystems 9

22 Pupl uncton retreval, numercal data NA.5, dameter pupl. mm, λ66 nm Input pupl uncton (complex Zernke coecents) Standard retreval wth Debye ntegral Retreval wth adapted Debye ntegral Braat, van Haver, Perera Lens qualty assessment EOS - Optcal Mcrosystems 9

23 Mcrolens pupl uncton retreval, numercal data NA., dameter pupl.4 mm, λ4 nm Input pupl uncton (complex Zernke coecents) Standard retreval wth Debye ntegral Retreval wth adapted Debye ntegral Braat, van Haver, Perera Lens qualty assessment EOS - Optcal Mcrosystems 9

24 Concluson Small-NA magng (NA<.) ntroduces asymmetry around ocus The ENZ pupl uncton retreval method based on the Debye ntegral has to be adapted by an axal and lateral coordnate transormaton that scales wth the deocusng (NA.) The regon o applcablty o the adapted retreval method s adequate or mcroscope objectves and mcrolenses provded the physcal dameter o the lens aperture exceeds typcally 5 wavelengths Numercal experments showed the valdty o the adapted ENZ retreval method or small NA systems and mcrolenses Braat, van Haver, Perera Lens qualty assessment EOS - Optcal Mcrosystems 9 4