Vectorial diffraction theory

Transcription

1 -9- Vectorial diffraction theor Xlab Group Seminar March 8 th, 3 Jeongmin Kim PhD student in Mechanical Engineering Imaging theor Object Imaging sstem Image dimension, transmittance, scattering, fluorescent?, coherence, wavelength, polariation Transfer characteristics such as PSF and OTF, determined b diffractions & aberrations An actual imaging process is too comple to full describe. Yet, PSF (point spread function) and OTF (optical transfer function) are still fundamental concepts and ke properties of imaging sstems.

2 -9- Hugens-Fresnel Priple to obtain PSF P For a monochromatic case, Σ θ r cos where, P Observation plane ( ) + ( ) + Paraial approimation! Object plane Pupil plane ik t(, ) = P(, ) ep ( + ) f Image plane PSF of a microscope objective h,,, In optical coordinate,,,, sin, v Normalied Intensit PSF (u = ) h,, h,,, How accurate is this model? v 3

3 -9- Limitation of the scalar theor M. Mansuripur, J. Opt. Soc. Am. A, Vol. 3, No. (986) C. J. R. Sheppard, J. Opt. Soc. Am. A, Vol. 4, No. 8 (987) Flagello et al., J. Opt. Soc. Am. A, Vol. 3, No. (996) The scalar theor neglects the three critical factors ) Vector nature of the electromagnetic radiation (polariation) ) No Fresnel approimation! (a.k.a. small-angle or paraial approimation) 3) Intensit distribution over the eit pupil (apodiation) Vectorial diffraction theor takes these into account! 4 Content Vectorial diffraction theor Scalar diffraction theor Kirchhoff approimation Debe approimation Numerical calculation eamples Optical transfer function (OTF) Vectorial point spread function (PSF) and OTF in Aial-Plane Optical Microscop (APOM) Vectorial PSF in oblique-plane imaging Conclusion 3

4 -9- Scalar diffraction theor Governing equations Mawell s equations B E B H (Gauss s law) (Farada s law) (Ampere s law) EM wave equations In linear, isotropic, homogeneous, nondispersive, nonmagnetic, and source-free medium E E H H Helmholt equation For a monochromatic scalar wave E, How to calculate? It is the solution of electrostatic boundar-value problems. The Green s theorem acts as a mathematical tool to handle the BCs. 6 Scalar diffraction theor (/3) Divergence theorem For an vector field A defined in the volume V bounded b the closed surface S Green s theorem (a.k.a. Green s nd identit), 4 Kirchhoff diffraction integral (a.k.a. Integral theorem of Helmholt and Kirchhoff) Observation point Radiation sources outside the volume 4,,, 4 Infinite-space Green function (diverging spherical wave) 7 4

5 -9- Scalar diffraction theor (3/3) Σ cos Screen with openings Kirchhoff BCs (or Kirchhoff approimations) ) & vanish ecept the openings. ) Eopenings ( r ') = Eunpurturbed ( r ') ' E ( r ') = ' E ( r ') openings Kirchhoff diffraction formula 4 unpurtubed (R>>λ) Hugens-Fresnel Priple b the st Raleigh-Sommerfeld diffraction formula, 4, cos Dirichlet problem 8 Transition to the Vectorial Imaging Theor 4 A vector analog of Green s theorem ( ), vector identities, and Mawell equations 4 4 (SI) (SI) J. D. Jackson, Classical Electrodnamics, 3 rd ed. (John Wile & Sons, 999) J. A. Stratton, Electromagnetic Theor, McGraw-Hill (94) : The aberrated wavefront over the reference sphere of the eit pupil : The normal vector to the wavefront, : Electric and magnetic field after the refraction at the eit pupil G : Green function Gauss s law, vector identities, and R λ Vectorial Kirchhoff Integral Vectorial Kirchhoff diffraction integral Time-averaged electric energ densit 9

6 -9- Apodiation Energ projection factor at the eit pupil - how a ra densit is distributed matters! E ident h E S h = fg( θ ) Eit pupil (or reference sphere) Geometrical focus = S E ( h, ψ ) hdhdψ E ( θ, φ) f sinθ dθdφ ident E ( θ ) = E ( h) S ident g( θ ) g '( θ ) sinθ In Abbe s sine condition, h = f sinθ J. J. Stamnes, Waves in Focal Regions, IOP Publishing (986) E ( θ ) = E ( h) cosθ S ident Depolariation E, k E S =? Eit pupil (or reference sphere) Geometrical focus g( θ ) g '( θ) ( N k ) E N k (( N k ) k ) E ( N k ) N ES = + sin θ ( N k ) E N k (( N k ) k ) E ( N k ) N T. D. Visser, S. H. Wiersma, J. Opt. Soc. Am. A, Vol. 8, No. 9 (99) 6

7 -9- Numerical calculation: Vectorial Kirchhoff integral cos cos, sin, Eit pupil Geometrical focus sos sinsin cos For an aplanatic sstem sin cos sin sin cos,,, cos sin sin sin cos cos cos cos sin cos sin cos sin cos Helpful references: W. Hsu, R. Barakat, J. Opt. Soc. Am. A, Vol., No., (994) Yajun Li, J. Opt. Soc. Am. A, Vol., No. () S. Hell et al, J. Microscop, Vol. 69 (993) D. G. Flagello, T. Milster, A. E. Rosenbluth, J. Opt. Soc. Am. A, Vol. 3, No. (996) Peter Torok, J. Opt. Soc. Am. A, Vol., No. (998) Debe approimation Geometrical focus If P is ver close to O, R = r r ' r N N r ' ( ) 4 i e = λ nf iknf iknn r E( r ) E S Se ds Vectorial Debe Integral Eit pupil This approimation leads to aiall smmetric intensit distribution, and valid if nf sin fna Fresnel number: N = α F λ = λn,,, cos sin sin sin cos cos cos cos sin cos sin cos sin cos 3 7

8 -9- Kirchhoff vs. Debe E. Wolf, Proc. Ro. Soc. A, 3, 349 (99) 4 A chronicle of diffraction theor 69 Hugens Hugen s priple 83 Air Air disk for circular aperture sstem: [J (v)/v] 88, 886 Lommel, Struve Near focus regions (out-of-focus) 879 and later Raleigh, Strehl Monochromaticall aberrated images (/4λ rule) 898 Schwartschild Near focus regions 99 Debe Plane wave approach 99-9 Ignatowsk Polariation effects in focusing 94s Nijboer Using Zernike s polnomials (934) Hopkins Polariation effects in focusing (detailed investigation) Nienhuis, Nilboer Larger aberration effects 949 Kampen Method of stationar phase 9s Wolf Encircled energ 97 Farnell focal shift found 99 96, 967 Richards, Wolf Wolf, Boivin 98 Hopkins OTF pioneer Polariation effects in focusing (detailed investigation) 97-98s Various numerical techniques J. J. Stamnes, Waves in Focal Regions, IOP Publishing (986) 8

9 -9- Comparative calculation Intensit PSF of each component (NA:.3, f:4λ, -pol.) I I I Power.976 :.9*-4 :.3 M. Mansuripur, J. Opt. Soc. Am. A, Vol. 3, No. (986) Intensit ( comp) Intensit ( comp) * Intensit ( comp) (um) - - (um) (um) PSF at the focal plane N=3, λ=. um -pol. NA=.3 NA=.9 Scalar vs. Vectorial theor (IPSF at NA=.3, N=3) Scalar vs. Vectorial theor (IPSF at NA=.9, N=3) Scalar Vectorial (-slice, -pol.) Vectorial (-slice, -pol.) Vectorial (unpol.) Radial coordinate (um) log scale Scalar vs. Vectorial theor (IPSF at NA=.3, N=3) Radial coordinate (um).3.4. Scalar vs. Vectorial theor (IPSF at NA=.9, N=3) Normalied intensit Normalied intensit Scalar Vectorial (-slice, -pol.) Vectorial (-slice, -pol.) Vectorial (unpol.) Scalar Vectorial (-slice, -pol.) Vectorial (-slice, -pol.) Vectorial (unpol.) Scalar Vectorial (-slice, -pol.) Vectorial (-slice, -pol.) Vectorial (unpol.).9 Normalied intensit Normalied intensit Radial coordinate (um) Radial coordinate (um)

10 -9- Aial PSF N=3, λ=. um NA=.3 NA= Scalar Vectorial (-pol.) Vectorial (unpol.).9.8 Scalar Vectorial (-pol.) Vectorial (unpol.).7.7 Normalied intensit Normalied intensit Aial coordinate log scale Aial coordinate (um) Normalied intensit Normalied intensit Scalar -7 Vectorial (-pol.) Vectorial (unpol.) Aial coordinate (um) Scalar -7 Vectorial (-pol.) Vectorial (unpol.) Aial coordinate (um) 8 Intensit PSF s FWHM Radial direction Aial direction Lateral PSF sie with NA (at λ=.um, N=3) 3 Aial PSF sie with NA (at λ=.um, N=3) FWHM (um). Scalar Vectorial (-slice, -pol.) Vectorial (-slice, -pol.) Vectorial (unpol.) FWHM (um) Scalar Vectorial (-pol.) Numerical Aperture, NA Numerical Aperture, NA 4 FWHM deviation from the scalar theor FWHM deviation from the scalar theor Percentage (%) 3 Vectorial (-slice, -pol.) Vectorial (-slice, -pol.) Vectorial (unpol.) Percentage (%) Vectorial (-pol.) Numerical Aperture, NA Numerical Aperture, NA Sheppard and Matthews, J. Opt. Soc. Am. A, 4, 34-36, 987 u = k sin α (Pseudo-paraial approimation) 9

11 -9- Degree of depolariation with NA E = E ˆ Relative power of each polariation comp. at focus..8 Eit pupil (or reference sphere) Geometrical focus Normalied power.6.4. E E E NA The portion of aial polariation component(i ) reases up to 4.3% of the total power at.9 NA. Aial shift at low Fresnel number!.4,., 8, 7 Lateral coordinate, (um) Aial coordinate, (um) On-ais intensit profile Intensit Aial coordinate, (um) Strong dependence on Fresnel number (N F ) NA =. N = N = N = N = N = NA =. N = N = N = N = N =.4. NA =.9 N = N = N = N = N = Intensit..8 Intensit.8 Intensit Aial coordinate, (um) Aial coordinate, (um) Aial coordinate, (um)

12 -9- Resolution enhancement with annular pupils NA=.9 N=3 λ=. um OD = ID = In-focus IPSF (@ NA=.9, λ=.9, N=3) FWHM : nm FWHM : 36 nm In-focus IPSF (@ NA=.9, λ=.9, N=3) FWHM : 9 nm FWHM : 394 nm pol. (um) (um) % () 8% () In-focus IPSF (@ NA=.9, λ=.9, N=3) FWHM: 3 nm In-focus IPSF (@ NA=.9, λ=.9, N=3) FWHM: 68 nm Unpolaried source (um) (um) % Comparison between each approimation Normalied intensit Normalied intensit Scalar paraial Scalar Debe Vectorial Debe Vectorial Kirchhoff (R - neglected) Vectorial Kirchhoff Radial coordinate (um) NA=.4, n=., N=3, λ=. um (unpolaried) ikn r E( r ) E e ds S s ik r r f E( r ) E (r ) e ds S ikn r E( r ) E cosθ e ds S ikr ( ) ( ˆ) ( ˆ e E r Es Es R N + N R) E S s ds R Scalar paraial Scalar Debe Vectorial Debe Vectorial Kirchhoff (R - neglected) Vectorial Kirchhoff Radial coordinate (um) Vectorial Debe approimation is simple, fast, and accurate in high NA sstem. 3

13 -9- Optical transfer function (OTF) Real space () ρ Frequenc space () m n s l = m + n D Amplitude PSF Intensit PSF D FFT h( ρ ) CTF c( l) = P( l) For paraial, circular-aperture, linear, space-invariant sstems, D FFT h( ρ) OTF C( l) = c( l) c ( l) = P( l) P( l) 3D Amplitude PSF Intensit PSF h(r) 3D FFT CTF c( l, s) = P( l) δ (s l / ) h( r 3D FFT ) OTF C( l, s) = c( l, s) 3 c ( l, s) Back to D: C( l) = A C( l, s) ds 4 OTF in a circular lens Paraial (low NA sstem) Scalar Debe (high NA + Sine condition) Vectorial? D CTF OTF c( l) = P( l) l l l C( l) = cos π CTF OTF No analtical epression found! How is it defined? 3D CTF OTF c l s P l δ s l (, ) = ( ) ( / ) s l C( l, s) = Re + l l CTF OTF c l s P( l) δ s l l (, ) = ( + ) s C( l, s) = p E, pɶ π l + s p / ( ɶ ) β / ( ɶ ) + ( α α ) if l s l sin s cos How is it defined? 6 4 3D OTF D OTF (α=67º) l l + s pɶ = s l + s 4 cosα β = cos + p s ɶ /.... s l s l Min Gu, Advanced optical imaging theor, Springer (999) 3

14 -9- OTF comparison FFT(D PSF) Normalied intensit Scalar paraial Scalar Debe Vectorial Debe NA=.4, λ =.um OTF Scalar paraial appro. (analtical) Scalar paraial appro. Scalar Debe appo. (analtical) Scalar Debe appro. Vectorial Debe appro. (unpolaried).. Normalied intensit Radial coordinate (um) Scalar paraial Scalar Debe Vectorial Debe Radial coordinate (um). NA=.4, λ =.um 3 4 Spatial frequenc (ccles/um) NA λ 6 Optical diffraction theor for PSF Scalar Debe Vectorial Kirchhoff diffraction formula (Kirchhoff BCs) Vectorial Debe Vectorial Kirchhoff integral Scalar Paraial λ, n, NA Polchromatic, partial coherence Apodiation Polariation, N for an N, R λ 7 4

15 -9- Content Vectorial diffraction theor Scalar diffraction theor Kirchhoff approimation Debe approimation Numerical calculation eamples Optical transfer function (OTF) Vectorial point spread function (PSF) and OTF in Aial-Plane Optical Microscop (APOM) Vectorial PSF in oblique-plane imaging Conclusion 8 Aial Plane Optical Microscop (APOM) Tongcang, Sadao sample Objective lens Tube lens Tube lens Objective lens f f f f f f f f Lens Mirror at 4º Real focus line Virtual objects line - - um 3f Detector Assumptions for PSF derivation: - No aberrations - Monochromatic 9

16 -9- APOM pupil function,, Object space mirror Spherical eit pupil Incident beam mirror Reflected beam Effective pupil APOM s normalied pupil function P( θ, φ), θ ( φ) θ θ, φ φ φ min ma min ma, otherwise, h = ( ) min ( ) = sin csc ( NA / n ), θ φ φ ( ) θ ma = sin NA/ n, φ min = sin (n/ NA), ma = π sin (n/ NA) φ 3 Effective pupil function of APOM With different NA NA.3 NA.3 NA.4 NA.4 Carl Zeiss X objectives NA n.... Aperture angle (θ ma ) 7.8º 8.79º 6.64º 67.8º 7.4º 3 6

17 -9- D Intensit PSF NA =.3, n =., λ =. um (unpolaried), f =.64 mm.8 APOM D PSF.8 Conventional microscop D PSF (um) (um) FWHM = 83 nm FWHM = 93 nm FWHM = 949 nm (aial).33x FWHM = nm 4.4X FWHM = nm.x (aial) FWHM = 64 nm The partial spherical pupil makes the APOM s resolving power worse than that of conventional optical microscop, along all the,, and directions. 3 D IPSF with NA n =., λ =. um (unpolaried), f =.64 mm APOM D PSF (NA =.3) APOM D PSF (NA =.4) APOM D PSF (NA =.) (um). (um). (um) Calculated FWHM 9 FWHM ratios FWHM (nm) APOM (, lateral) APOM (, lateral) APOM (, aial) CM (&, lateral) CM (, aial) CM: conventional microscop Ratio APOM()/CM() APOM()/CM() APOM()/CM() APOM()/APOM() CM: conventional microscop NA NA 33 7

18 -9- (um) D OTF of APOM (@ NA=.4, λ=.um) (um) D PSF (CM) D PSF (APOM) Log Log Spatial frequenc, n (ccles/um) Spatial frequenc, m (ccles/um) - - D OTF (CM) - Spatial frequenc, m (ccles/um) D OTF (APOM) - Spatial frequenc, m (ccles/um) Cut-off frequenc: m = 4.97 ccles/um (-%), m =.6 ccles/um (-7%) Modulation transfer function, OTF D OTF slices APOM -ais (FFT) APOM -ais (FFT) CM, Vectorial Debe (FFT) CM, Scalar Debe (analtical) CM, Scalar Debe (FFT) CM, Paraial theor (analtical) Spatial frequenc (ccles/um) Calculation errors of APOM s OTF can be as big as %. 34 Image of a straight edge NA =.4, n =., λ =. um (unpolaried), f =.64 mm. Edge object.8. Edge object.8 (um) -. FWHM: ~8 nm FWHM: 93 nm Edge response (um) Edge response (um) Normalied intensit at = um Edge response Estimated LSF FWHM: 76 nm (um).8.6 Normalied intensit at = um Edge response Estimated LSF FWHM: nm (um) The Line Spread Function (LSF) of APOM is oherentl estimated from a differentiation of the edge response, which can also be predicted from the eperimental edge response. 3 8

19 -9- Oblique plane microscop (OPM) Object space mirror Given a mirror tilt of α, an image plane is α tilted OPM pupil function if,, θ θs, φ π or P( θ, φ) = θs θ θma, φ( θ ) φ φ( θ ), otherwise if >,,where NA NA S sin cos( ) sin( ), θ = n α n α NA θ = n ma sin, φ θ = α θ + α θ n φ ( θ ) = π φ ( θ). ( ) sin cot( ) cot NA csc( ) csc, A,, ),,, ),,, ), θs θ θma, φ( θ ) φ φ( θ ) P( θ, φ) =, otherwise cos sin 37 9

20 -9-3D intensit PSF formula derived I(, p, q) M = A + B + C + D, where ( φ + θ φ ) ( k θ φ ) ( ) ( ) i sinθ cosφ sin ( ksinθ cosφ ) sin cos cos cos sin cos p( sin( α )cosθ cos( α )sinθ sinφ ) + θma π / ik q ( cos( α )cosθ sin( α )sinθ sinφ ) + A = E cosθ sinθ i cosθ sinφcosφsin ksinθ cosφ e dφdθ θ S φ ( θ ) ( ( ( α ) )) ( ( α ) ) ( ( α ) ) I + I cos tan p cos / B π I p ( ) ( ) = sin tan cos / ii cos tan p cos / ( ) ( ) ( ( ) ) ( α ) ikpsin ( ) ( α ) ikpsin θs ikpsin α I = E cos sin ( cos ) cos ( ) sin θ θ + θ J k + p α θ e dθ θs = + I E cosθ sin θ J k p cos α sinθ e dθ θs I = E cos sin ( cos ) cos ( ) sin θ θ θ J k + p α θ e dθ i ( cosθ ) sinφcosφsin ( ksinθ cosφ ) p( sin( α )cosθ cos( α )sinθ sinφ ) + ik ( ) ( ) ( ) q + θ θ φ θ φ θ φ φ θ sinθ sinφcos ( ksinθ cosφ ) θma π / cos( α )cosθ sin( α )sinθ sinφ C = E cos sin cos cos sin cos ksin cos e d d θ + S φ ( θ ) ( ( ( α ) )) ( ( α ) ) ( ( α ) ) I sin tan p cos / D = π I I cos( tan p cos / ) ii sin ( tan p cos / ) 38 D PSF at different mirror angle NA =.4, n =., λ =. um (unpol.).8 OPM D PSF (α = ).8 OPM D PSF (α = ).8 OPM D PSF (α = ).8 OPM D PSF (α = 3 ) p (um) p (um) p (um) p (um) p (um) OPM D PSF (α = 4 ) FWHM (nm) Predicted FWHMs in oblique plane imaging -ais p-ais Mirror tilt (degree) 39

21 -9- (ideas) Resolution enhancement in oblique-plane imaging Diffractive optical element (DOE) Beam splitter, another objective, reflector ra Diffractive element Objective lens reference sphere Beam splitter Objective lens Corner cube reflector (mirror or prism) ra n ra ra Vignetting Both schematics ensures more effective aperture area, thus enhancing the resolving power. More through analsis is required to predict theoretical resolution. Eperimental verification of the proposed ideas can be a research topic. 4 Conclusion Vectorial diffraction theor was introduced. The theor covers polariation, apodiation, non-paraial assumption. Depolariation leads to broader PSF. Aial shift occurs with Fresnel number around or below. Vectorial Debe approimation gives accurate PSFs for high NA objective lens. PSF and OTF in APOM was calculated with Vectorial Debe approimation and FFT approach, respectivel. 3D PSF formula in oblique plane imaging was derived for the first time. D OTF calculation is still in progress. Finding out 3D OTF shape and their cutoff frequenc boundar ma also be interesting. The anisotropic lateral resolution in oblique/aial plane imaging would be mostl compensated with the proposed schematics. Eperimental verification in a low NA & a small tilt angle could be performed. 4