Physical Optics. Lecture 3: Fourier optics Herbert Gross.

Transcription

1 Phsical Optics Lecture 3: Fourier optics Herbert Gross

2 Phsical Optics: Content No Date Subject Ref Detailed Content.4. Wave optics G Comple fields, wave equation, k-vectors, interference, light propagation, interferometr 8.4. Diffraction G Slit, grating, diffraction integral, diffraction in optical sstems, point spread function, aberrations Fourier optics G Plane wave epansion, resolution, formation, transfer function, phase imaging 4.5. Qualit criteria and Raleigh and Marechal criteria, Strehl ratio, coherence effects, two-point G resolution resolution, criteria, contrast, aial resolution, CTF Photon optics K Energ, momentum, time-energ uncertaint, photon statistics, fluorescence, Jablonski diagram, lifetime, quantum ield, FRET Coherence K Temporal and spatial coherence, Young setup, propagation of coherence, speckle, OCT-principle Polarization G Introduction, Jones formalism, Fresnel formulas, birefringence, components Laser K Atomic transitions, principle, resonators, modes, laser tpes, Q-switch, pulses, power Nonlinear optics K Basics of nonlinear optics, optical susceptibilit, nd and 3rd order effects, CARS microscop, photon imaging 3.6. PSF engineering G Apodization, superresolution, etended depth of focus, particle trapping, confocal PSF.6. Scattering L Introduction, surface scattering in sstems, volume scattering models, calculation schemes, tissue models, Mie Scattering 7.6. Gaussian beams G Basic description, propagation through optical sstems, aberrations Generalized beams G Laguerre-Gaussian beams, phase singularities, Bessel beams, Air beams, applications in superresolution microscop 4.7. Miscellaneous G Coatings, diffractive optics, fibers K = Kempe G = Gross L = Lu

3 3 Contents Introduction Optical transfer function Resolution Image formation Phase imaging

4 Definitions of Fourier Optics Phase space with spatial coordinate and. angle. spatial frequenc in mm - 3. transverse wavenumber k v k v Fourier spectrum k k A( v, v ) Fˆ E(, ) structure k / g diffracted ra direction k corresponds to a wave epansion i k k,, (,, ) A k k z E z e d d g = / Diffraction at a grating with period g: deviation angle of first diffraction order varies linear with = /g sin g v

5 5 Grating Diffraction and Resolution a) resolved incident light b) not resolved diffracted orders optical sstem Arbitrar epaneded into a spatial frequenc spectrum b Fourier transform Ever frequenc component is considered separatel To resolve a spatial detail, at least two orders must be supported b the sstem off-ais illumination g g sin m sin NA g Ref: M. Kempe NA

6 Number of Supported Orders A structure of the is resolved, if the first diffraction order is propagated through the optical imaging sstem The fidelit of the increases with the number of propagated diffracted orders. / +. / -. order. / +. / / -. order. / / +. / -. / +3. / -3. order

7 Resolution of Fourier Components detail high spatial frequencies numerical aperture resolved frequencies point low spatial frequencies sum for low NA decomposition of Fourier components (sin waves) for high NA high spatial frequencies Ref: D.Aronstein / J. Bentle

8 8 Propagation of Plane Waves Phase of a wave The spectral component is simpl multiplied b a phase factor in during propagation the function h is the phase function Back-transforming this into the spatial domain: Propagation corresponds to a convolution with the impulse response function Fresnel approimation for propagation: z z z z z h e e e e n z i z i n z i i z ;, cos,,,,, ;,, z i z E z E z e h z E z... ;, ;, d d e e U e z i z U i z i z z z i P,,, ;,, E z H z E z i d d e z h z H ;, ;,

9 Optical Transfer Function: Definition Normalized optical transfer function (OTF) in frequenc space: Fourier transform of the Psf- intensit Absolute value of OTF: modulation transfer function MTF Gives the contrast at a special spatial frequenc of a sine grating OTF: Autocorrelation of shifted pupil function, Duffieu-integral Interpretation: interference of th and st diffraction of the light in the pupil H OTF ( v ) f v f v P P d * ( p ) ( p ) p P( ) p d p L o H OTF o ( v, v ) Fˆ I (, ) ive pupil ' p direct light PSF ' p at diffracted light in st order L condenser conjugate to pupil light source

10 Contrast / Visibilit The MTF-value corresponds to the intensit contrast of an d sin grating Visibilit The maimum value of the intensit is not identical to the contrast value since the minimal value is finite too V I I I() ma ma I I min min peak decreased slope decreased Concrete values:.9 I ma I I ma V I min minima.5 increased

11 Duffieu Integral and Contrast Separation of pupils for. and +-. Order patern period 8 /NA /NA pupil MTF function contrast % difraction order spatial frequenc NA/ NA/ Image contrast for sin- I V=% V=5% min I=.33 V=33% I=.5 min V=% I=.66 min Ref: W. Singer

12 Optical Transfer Function of a Perfect Sstem Loss of contrast for higher spatial frequencies.9.8 ideal MTF / ma contrast / ma

13 3 Optical Transfer Function of a Perfect Sstem Aberration free circular pupil: Reference frequenc na n sin u ' v o f D Cut-off frequenc: Air.5 g MTF v na nsin u ' v f ma Analtical representation.5 / ma H MTF v ( v) arccos v v v v v Separation of the comple OTF function into: - absolute value: modulation transfer MTF - phase value: phase transfer function PTF H OTF ( v, v PTF ( v, v ) H ( v, v ) e MTF ih )

14 Calculation of MTF Some more eamples 4 -dim case MTF circular pupil Ring pupil = central obscuration (75%),5 Apodization = reduced transmission at pupil edge (Gauss to 5%) The transfer of frequencies depends on transmission of pupil Ring pupil higher contrast near the diffraction limit Apodisation increase of contrast at lower frequencies Ref: B. Böhme

15 Sagittal and Tangential MTF Due to the asmmetric geometr of the psf for finite field sizes, the MTF depends on the azimuthal orientation of the structure Generall, two MTF curves are considered for sagittal/tangential oriented structures g MTF tangential ideal.5 sagittal tangential sagittal tangential.5 / ma arbitrar rotated tangential sagittal sagittal

16 Ref: B. Dube, Appl. Opt. 56 (7) Tangential vs Sagittal Resolution Asmmetr of the PSF Coma 3rd creates a :3 diameter pattern Usuall coma oriented towards the ais, then MTF S > MTF T (lower row) spatial frequenc spatial frequenc

17 7 Contrast and Resolution High frequent structures : contrast reduced Low frequent structures: resolution reduced contrast brillant blurred sharp milk resolution

18 8 Contrast and Resolution Contrast vs contrast as a function of spatial frequenc Tpical: contrast reduced for increasing frequenc V Compromise between resolution and visibilt is not trivial and depends on application H MTF Contrast sensitivit H CSF / c

19 Contrast / Resolution of Real Images Degradation due to. loss of contrast. loss of resolution resolution, sharpness contrast, saturation

20 Test: Siemens Star Determination of resolution and contrast with Siemens star test chart: Central segments b/w Growing spatial frequenc towards the center Gra ring zones: contrast zero Calibrating spatial feature size b radial diameter Nested gra rings with finite contrast in between: contrast reversal, pseudo resolution

21 Phase Space: 9 -Rotation Transition pupil- : 9 rotation in phase space Planes Fourier inverse Marginal ra: space coordinate ---> angle ' Chief ra: angle ---> space coordinate ' Fourier pupil location marginal ra ' ' chief ra f

22 Resolution More incoherent points more independent self luminous points: emission of N spherical waves summation of intensities First point (green) p, p truncated spherical wave D Air Object wave aperture Ref: B. Böhme

23 Resolution More incoherent points more independent self luminous points: emission of N spherical waves summation of intensities First point (green) Second point (blue) p, p truncated spherical wave D Air D Air Object. wave aperture Ref: B. Böhme

24 Resolution More incoherent points more independent self luminous points: emission of N spherical waves summation of intensities First point (green) Second point (blue) Third point (red) truncated spherical wave p, p D Air D Air D Air Object 3. wave aperture Ref: B. Böhme In the aperture (pupil ) we observe a wave for each point For N points N independent waves wit different directions Diffraction for all waves Superposition of the point s

25 Fourier Optics Point Spread Function Optical sstem with magnification m Pupil function P, Pupil coordinates p, p g psf (,, ', ') N P p, p e ik z p ' m ' m p d p d p PSF is Fourier transform of the pupil function (scaled coordinates) source point g (, ) N Fˆ P, psf p p point distribution

26 Fourier Theor of Incoherent Image Formation Transfer of an etended distribution I(,) In the case of shift invariance (isoplanas): incoherent convolution Intensities are additive I I I inc inc ( ', ') ( ', ') g ( ',, ', ) I(, ) dd g ( ', ' ) I(, ) dd psf psf ( ', ') I (, )* I (, ) psf obj intensit intensit single psf

27 Fourier Theor of Incoherent Image Formation Transfer of an etended distribution I obj (,) I ima ( ', ') I (, ',, ') I (, ) d d psf obj In the case of shift invariance (isoplanatism): incoherent convolution Intensities are additive In frequenc space: - product of spectra - linear transfer theor - spectrum of the psf works as low pass filter onto the spectrum - Optical transfer function H I otf ( v, v ) FT I (, ) PSF I I intensit ( v, v ) H otf ( v, v ) Iobj ( v, v ) ima ima ( ', ') I ( ', ') I (, ) d d ( ', ') I (, )* I (, ) psf psf obj obj intensit single psf

28 Incoherent Image Formation Eample: incoherent imaging of bar pattern near the resolution limit resolved not resolved

29 9 Fourier Theor of Image Formation Coherent Imaging Incoherent Imaging amplitude U(,) Fourier transform amplitude spectrum u(v,v ) intensit I(,) Fourier transform intensit spectrum I(v,v ) convolution product convolution produkt PSF amplituderesponse H psf ( p, p ) Fourier transform coherent transfer function h CTF (v,v ) squared PSF, intensitresponse I psf ( p, p ) Fourier transform optical transfer function H OTF (v,v ) result result result result amplitude U'(',') Fourier transform amplitude spectrum u'(v',v' ) intensit I'(',') Fourier transform intensit spectrum I'(v ',v ') 4. Image simulation

30 Fourier Theor of Coherent Image Formation Transfer of an etended distribution E obj (,) Eima ( ', ') Apsf,, ', ' E(, ) dd In the case of shift invariance (isoplanasie): coherent convolution of fields Comple fields additive In frequenc space: - product of spectra - linear transfer theor with fields - spectrum of the psf works as low pass filter onto the spectrum - Coherent optical transfer function H E I ctf ima ima ( v, v ) FT E (, ) PSF amplitude distribution ( v, v ) Hctf ( v, v ) Eobj ( v, v ) ( v, v ) Hctf ( v, v ) Eobj ( v, v ) Eima ( ', ') Apsf ', ' Eobj (, ) dd E ima, E (, ) ( ', ') A psf obj amplitude distribution single point

31 Comparison Coherent Incoherent Image Formation incoherent coherent bars resolved bars not resolved bars resolved bars not resolved

32 Partial Coherent Imaging Complete description of an optical sstem:. Light source. Illumination sstem, amplitude response h ill 3. Transmission 4. Observation / imaging sstem with amplitude response h obs illumination field s, s p, p i, i source illumination sstem observation sstem pupil P h obs I s h ill I i I o sensor

33 Coherence Parameter Finite size of source : aperture cone with angle u ill Observation sstem: aperture angle u obs Definition of coherence parameter : Ratio of numerical apertures Limiting cases: coherent = u ill << u obs sin u sin u ill obs incoherent = u ill >> u obs source o, o lens i, i u ill u obs illumination observation

34 Coherence Parameter Heuristic eplanation of the coherence parameter in a sstem:. coherent: Psf of illumination large in relation to the observation coherent illumination etended source small stop of condenser condenser ive lens Psf of observation inside psf of illumination. incoherent: Psf of illumination small in comparison to the observation incoherent illumination etended source large stop of condenser Psf of observation contains several illumination psfs

35 4 - f - Setup Simple setup with smmetrical sstem with accessible Fourier Different focal lengths of the subsstems allows a get a magnification m > Fourier rear sstem ' p ' front sstem p f f f f

36 Fourier Filtering Imaging of a crossed grating Spatial frequenc filtering b a slit: pupil complete open Case : - pupil open - Cross grating d Case : - truncation of the pupil b a split - onl one direction of the grating is resolved pupil truncated b slit

37 Fourier Filtering Imaging of part of the moon b scanning scanned with stripes filtered without artefacts Artefacts b scanning stripes visible Stripes corresponds to a pointed line in the Fourier spectrum If the diffraction orders are blocked / filtered out, the qualit is improved spectrum of the scanned filtered spectrum false orders suppressed

38 Abbes Diffraction Theor of Image Formation Wave optical interpretation of the optical formation The objekt is considered as a superposition of several gratings (Fourier picture) Ever grating diffracts the light in the various orders Onl those orders are contributing to the, which lie inside the cone of the numerical aperture of the optical sstem There is a limiting spatial frequenc and a minimum feature size, that is resolved b the sstem.6 nsin u obj There are special assumptions for the validit of Abbe's formula There are three options for improving the resolution:. lowering the wavelength. increasing the inde of refraction 3. using larger aperture cone angles grating ive lens diffraction spectrum

39 39 Abbe Theor of Microscopic Resolution Location of diffraction orders in the back focal th order ive lens back focal Increasing grating angle with spatial frequenc of the grating +st -st +st -st +st -st

40 Fouriertheorie of Image Formation One illumination point generates a wave in the space Diffraction of the wave at the structure Diffraction orders occur in the pupil Constructive interference of all supported diffraction orders in the Too high spatial frequencies are blocked light source f f pupil f f f f s s() s U () T() u() h() U () Ref: W. Singer

41 Microscopic Imaging 4 contrast patern period spatial frequenc % NA/ NA/ /NA /NA 8 pupil difraction order - source pupil imaging lens ' ' p p Fourier f f f f front sstem rear sstem

42 4 Resolution and Spatial Frequencies Grating pupil Imaging with NA =.8 Imaging with NA =.3 Ref: L. Wenke

43 Imaging of Phase Structures Eample Pure phase Intensit in the. stack for 5 defocus positions. Image after phase retrieval 3.difference

44 44 Zernike Phase Contrast Principle Illumination and ring. decentered. centered Ref.: M. Kempe

45 Zernike Phase Contrast Phase contrast Conventional bright field

46 Phase Contrast Microscop Adjustment of phase ring and illumination Comparison: brightfield phase contrast Ref: M. Kempe Quelle:

47 Differential Interference Contrast Contrast of phase s can also be obtained b interference of sheared beams In DIC the sheared beams can be created and the beams can be overlapped b Wollaston prisms (orthogonal polarization) Interference of two beams with displacement δ b analzer phase gradient imaging I r, r, r, r r epi,, r, r I, r Ref: M. Kempe

48 Differential Interference Contrast Lateral shift : preferred direction Visisbilit depends on orientation of details shift 45 -shift ( ) -shift (9 )

49 49 DIC Phase Imaging Orientation of prisms and shift size determines the anisotropic formation Ref.: M. Kempe