Design and Correction of Optical Systems

Transcription

1 Design and Correction of Optical Systems Lecture 7: PSF and Optical transfer function Herbert Gross Summer term 017

2 Preliminary Schedule - DCS Basics Materials and Components Paraxial Optics Optical Systems Geometrical Aberrations Wave Aberrations PSF and Transfer function Further Performance Criteria Optimization and Correction Correction Principles I Correction Principles II Optical System Classification Law of refraction, Fresnel formulas, optical system model, raytrace, calculation approaches Dispersion, anormal dispersion, glass map, liquids and plastics, lenses, mirrors, aspheres, diffractive elements Paraxial approximation, basic notations, imaging equation, multi-component systems, matrix calculation, Lagrange invariant, phase space visualization Pupil, ray sets and sampling, aperture and vignetting, telecentricity, symmetry, photometry Longitudinal and transverse aberrations, spot diagram, polynomial expansion, primary aberrations, chromatical aberrations, Seidels surface contributions Fermat principle and Eikonal, wave aberrations, expansion and higher orders, Zernike polynomials, measurement of system quality Diffraction, point spread function, PSF with aberrations, optical transfer function, Fourier imaging model Rayleigh and Marechal criteria, Strehl definition, -point resolution, MTF-based criteria, further options Principles of optimization, initial setups, constraints, sensitivity, optimization of optical systems, global approaches Symmetry, lens bending, lens splitting, special options for spherical aberration, astigmatism, coma and distortion, aspheres Field flattening and Petzval theorem, chromatical correction, achromate, apochromate, sensitivity analysis, diffractive elements Overview, photographic lenses, microscopic objectives, lithographic systems, eyepieces, scan systems, telescopes, endoscopes Special System Examples Zoom systems, confocal systems

3 3 Contents 1. Ideal point spread function. PSF with aberrations 3. Strehl ratio 4. Two-point-resolution 5. Optical transfer function 6. Resolution and contrast

4 4 Huygens Principle Every point on a wave is the origin of a new spherical wave (green) The envelope of all Huygens wavelets forms the overall wave front (red) Light also enters the geometrical shadow region The vectorial superposition principle requires full coherence propagation direction stop Huygens- Wavelets wave fronts geometrical shadow

5 Diffraction at the System Aperture Self luminous points: emission of spherical waves Optical system: only a limited solid angle is propagated, the truncaton of the spherical wave results in a finite angle light cone In the image space: uncomplete constructive interference of partial waves, the image point is spreaded The optical systems works as a low pass filter spherical wave truncated spherical wave point spread function object point x = 1. / NA object plane image plane

6 PSF by Huygens Principle Huygens wavelets correspond to vectorial field components: - represented by a small arrow - the phase is represented by the direction - the amplitude is represented by the length Zeros in the diffraction pattern: destructive interference Ideal point spread function: point spread function zero intensity closed loop single wavelets sum central peak maximum constructive interference wave front pupil stop side lobe peak 1 ½ round trips

7 PSF by Huygens Principle Apodization: variable lengths of arrows r p point spread function homogeneous pupil: same length of all arrows I(x p ) apodization: decreasing length of arrows wave front pupil stop Aberrations: variable orientation of arrows real wave front point spread function real wavefront with aberrations central peak reduced ideal spherical wavefront central peak maximum ideal wave front pupil stop

8 Fraunhofer Point Spread Function 8 Rayleigh-Sommerfeld diffraction integral, Mathematical formulation of the Huygens-principle ik r r ' i e EI ( r) E( r') r r' cos dx' dy' d Fraunhofer approximation in the far field for large Fresnel number N F r p z 1 Optical systems: numerical aperture NA in image space Pupil amplitude/transmission/illumination T(x p,y p ) Wave aberration W(x p,y p ) complex pupil function A(x p,y p ) Transition from exit pupil to image plane E( x', y' ) Point spread function (PSF): Fourier transform of the complex pupil function iw ( x p, y p A( x, y ) T( x, y ) e p p AP T p i xpx' y p y' iw xp, y p RAP x, y e e p p p ) dx p dy p

9 intensity Perfect Point Spread Function Circular homogeneous illuminated aperture: intensity distribution transversal: Airy scale: D Airy 1. NA 1,0 n axial: sinc R E scale NA Resolution transversal better than axial: x < z 0,8 0,6 vertical lateral r optical axis 0,4 z image plane aperture cone lateral 0, 0, J1 v Iu 0 I 0, v I v 0 u / v sin u / 4, I u / 4 Scaled coordinates according to Wolf : axial : u = z n / NA transversal : v = x / NA 0 axial Ref: M. Kempe

10 10 Ideal Psf I(r,z) focal point spread spot z axial sinc optical axis aperture cone r lateral Airy image plane

11 11 Abbe Resolution and Assumptions Abbe resolution with scaling to /NA: Assumptions for this estimation and possible changes A resolution beyond the Abbe limit is only possible with violating of certain assumptions Assumption Resolution enhancement 1 Circular pupil ring pupil, dipol, quadrupole Perfect correction complex pupil masks 3 homogeneous illumination dipol, quadrupole 4 Illumination incoherent partial coherent illumination 5 no polarization special radiale polarization 6 Scalar approximation 7 stationary in time scanning, moving gratings 8 quasi monochromatic 9 circular symmetry oblique illumination 10 far field conditions near field conditions 11 linear emission/excitation non linear methods

12 1 Perfect Lateral Point Spread Function: Airy Airy function : Perfect point spread function for several assumptions Distribution of intensity: r J1 NA ( ) I r r NA Normalized transverse coordinate x ar R akr krsin u' ak sin' R Airy diameter: distance between the two zero points, diameter of first dark ring I(r) D Airy / D Airy n' sin u' r

13 13 Perfect Lateral Point Spread Function: Airy Airy distribution: Gray scale picture Zeros non-equidistant Logarithmic scale Encircled energy log I(r) E circ (r) 1 r D Airy ring 1.48%. ring.79% 1. ring 7.6% peak 83.8% r / r Airy

14 14 Perfect Axial Point Spread Function Axial distribution of intensity Corresponds to defocus Normalized axial coordinate NA z u z 4 Scale for depth of focus : Rayleigh length I(z) z sin sin u / 4 I( z) I0 Io z u / 4 R E n' sin u' n' NA Zero crossing points: 0. equidistant and symmetric, 0.1 Distance zeros around image plane 4R E z/ R E z = R E 4R E

15 15 Defocussed Perfect Psf Perfect point spread function with defocus Representation with constant energy: extreme large dynamic changes z = -R E z = -1R E focus z = +1R E z = +R E normalized intensity I max = 5.1% I max = 9.8% I max = 4% constant energy

16 16 Psf with Aberrations Psf for some low oder Zernike coefficients The coefficients are changed between c j = The peak intensities are renormalized trefoil coma 5. order astigmatism 5. order spherical 5. order c = 0.0 c = 0.1 c = 0. c = 0.3 c = 0.4 c = 0.5 c = 0.7 coma astigmatism spherical defocus

17 17 Axial and Lateral Ideal Point Spread Function Comparison of both cross sections Ref: R. Hambach

18 18 Annular Ring Pupil Generation of Bessel beams r p r z Ref: R. Hambach

19 19 Spherical Aberration Axial asymmetrical distribution off axis Peak moves Ref: R. Hambach

20 0 Gaussian Illumination Known profile of gaussian beams Ref: R. Hambach

21 1 Strehl Ratio Important citerion for diffraction limited systems: Strehl ratio (Strehl definition) Ratio of real peak intensity (with aberrations) referenced on ideal peak intensity D S I I ( real) PSF ( ideal PSF ) 0,0 0,0 D D S takes values between D S = 1 is perfect S A( x, y) e iw ( x, y) A( x, y) dxdy dxdy I( x ) 1 Critical in use: the complete information is reduced to only one number The criterion is useful for 'good' systems with values D s > 0.5 distribution broadened peak reduced Strehl ratio ideal, without aberrations real with aberrations r

22 Approximations for the Strehl Ratio Approximation of Marechal: ( useful for D s > 0.5 ) but negative values possible Bi-quadratic approximation Exponential approach D s Wrms 1 4 Wrms D 1 s D s e 4 W rms D S Marechal defocus exponential biquadratic exac t c 0 Computation of the Marechal approximation with the coefficients of Zernike D s 1 N n1 cn0 1 n 1 N n n1 m0 c nm n 1

23 3 Strehl Ratio Criterion In the case of defocus, the Rayleigh and the Marechal criterion delivers a Strehl ratio of D S The criterion D S > 80 % therefore also corresponds to a diffraction limit This value is generalized for all aberration types aberration type coefficient Marechal approximated Strehl exact Strehl 8 defocus Seidel a defocus Zernike c spherical aberration Seidel a spherical aberration Zernike c astigmatism Seidel a astigmatism Zernike c coma Seidel a coma Zernike c

24 4 Quality Criteria for Point Spread Function Criteria for measuring the degradation of the point spread function: 1. Strehl ratio. width/threshold diameter 3. second moment of intensity distribution 4. area equivalent width 5. correlation with perfect PSF 6.power in the bucket a) Strehl ratio b) Standard deviation c) Light in the bucket d) Equivalent width SR / D s STDEV LIB EW e) Second moment f) Threshold width g) Correlation width h) Width enclosed area SM FWHM Ref P=50% WEA CW

25 5 Point Resolution According to Abbe Transverse resolution of an image: - Detection of object details / fine structures - basic formula of Abbe x k Fundamental dependence of the resolution from: 1. wavelength. numerical aperture angle 3. refractive index 4. prefactor, depends on geometry, coherence, polarization, illumination,... nsin Basic possibilities to increase resolution: 1. shorter wavelength (DUV lithography). higher aperture angle (expensive, 75 in microscopy) 3. higher index (immersion) 4. special polarization, optimal partial coherence,... Assumptions for the validity of the formula: 1. no evanescent waves (no near field effects). no non-linear effects (-photon)

26 6 Incoherent -Point Resolution : Rayleigh Criterion Rayleigh criterion for -point resolution Maximum of psf coincides with zeros of neighbouring psf x 1 D Airy 0.61 nsin u Contrast: V = 0.15 Decrease of intensity between peaks I = I I(x) 1 sum of PSF PSF 1 PSF x / r airy

27 7 Incoherent -Point-Resolution: Sparrow Criterion Criterion of Sparrow: vanishing derivative in the center between two point intensity distribution, corresponds to vanishing contrast d I( x) d x x0 0 Modified formula x Sparrow D nsin u x Usually needs a priory information Applicable also for non-airy distributions Used in astronomy Rayleigh Airy I(x) x / r airy

28 8 Incoherent -point Resolution Criterions Visual resolution limit: Good contrast visibility V = 6 % : 0.83 x nsin u D Airy Total resolution: Coincidence of neighbouring zero points of the Airy distributions: V = 1 Extremly conservative criterion x D Airy 1. nsin u Contrast limit: V = 0 : Intensity I = 1 between peaks 0.51 x nsin u D Airy

29 9 -Point Resolution Distance of two neighboring object points Distance x scales with / sinu Different resolution criteria for visibility / contrast V x = 1. / sinu total V = 1 x = 0.68 / sinu visual V = 0.6 x = 0.61 / sinu Rayleigh V = 0.15 x = / sinu Sparrow V = 0

30 30 -Point Resolution Intensity distributions below 10 % for points with different x (scaled on Airy) x =.0 x = 1. x = 1.0 x = 0.83 x = 0.61 x = x = x = 0.5

31 31 Incoherent Resolution: Dependence on NA Microscopical resolution as a function of the numerical aperture NA = 0. NA = 0.3 NA = 0.45 NA = 0.9

32 3 Resolution of Fourier Components object detail high spatial frequencies numerical aperture resolved frequencies object point low spatial frequencies object sum image for low NA decomposition of Fourier components (sin waves) image for high NA object high spatial frequencies Ref: D.Aronstein / J. Bentley

33 Optical Transfer Function: Definition Normalized optical transfer function (OTF) in frequency space: Fourier transform of the Psf- intensity Absolute value of OTF: modulation transfer function MTF Gives the contrast at a special spatial frequency of a sine grating OTF: Autocorrelation of shifted pupil function, Duffieux-integral Interpretation: interference of 0th and 1st diffraction of the light in the pupil H OTF ( v ) x f v f v P x P x dx x * x ( p ) ( p ) p P( x ) p dx p y L y o H object OTF x o ( v, v ) Fˆ I ( x, y) x objective pupil y y' p direct light PSF x' p at object diffracted light in 1st order x L y condenser conjugate to object pupil x light source

34 34 Interpretation of the Duffieux Iintegral Interpretation of the Duffieux integral: overlap area of 0th and 1st diffraction order, interference between the two orders objective pupil y' p direct light The area of the overlap corresponds to the information transfer of the structural details x' p Frequency limit of resolution: areas completely separated y o at object diffracted light in 1st order y L object x o q y p x L shifted pupil areas p y condenser conjugate to object pupil f x f y x p x light source area of integration

35 8 35 Duffieux Integral and Contrast Separation of pupils for 0. and +-1. Order pattern period /NA MTF function pupil image contrast 100 % Image contrast for sin-object I 1 0 NA/ V=10% Ref: W. Singer V=50% V=3% I min =0.3 min I =0.5

36 36 Optical Transfer Function of a Perfect System Aberration free circular pupil: Reference frequency v o a f sinu' Cut-off frequency: g MTF v G na nsin u' v 0 f Analytical representation / max H MTF v ( v) arccos v v v v v 0 Separation of the complex OTF function into: - absolute value: modulation transfer MTF - phase value: phase transfer function PTF H OTF ( v, v PTF x y ( v, v ) H ( v, v ) e x y MTF x y ih )

37 37 Contrast / Visibility The MTF-value corresponds to the intensity contrast of an imaged sin grating Visibility The maximum value of the intensity is not identical to the contrast value since the minimal value is finite too Concrete values: I I max V V I I max max I(x) 1 I I min min peak decreased slope decreased object image I max I min minima increased x

38 Modulation Transfer Convolution of the object intensity distribution I(x) changes: 1. Peaks are reduced. Minima are raised 3. Steep slopes are declined 4. Contrast is decreased I(x) I max -I min original image high resolving image low resolving image x

39 39 Sagittal and Tangential MTF Due to the asymmetric geometry of the psf for finite field sizes, the MTF depends on the azimuthal orientation of the object structure Generally, two MTF curves are considered for sagittal/tangential oriented object structures g MTF tangential plane y 1 ideal 0.5 sagittal tangential sagittal tangential / max arbitrary rotated tangential sagittal x sagittal plane

40 40 Real MTF Real MTF of system with residual aberrations: 1. contrast decreases with defocus. higher spatial frequencies have stronger decrease g MTF (z,f) g MTF z = Zernike coefficients: c 5 = 0.0 c 7 = 0.05 c 8 = 0.03 c 9 = 0.05 max = 0.05 max = 0.1 max = 0. max = 0.3 max = 0.4 max = 0.5 max = 0.6 max = 0.7 max = z = 0.1 R u z = 0. R u z = 0.3 R u z = 1.0 R u z = 0.5 R -0.5 u z in R U

41 41 Hopkins Factor Resolution/contrast criterion: Ratio of contrasts with/without aberrations for one selected spatial frequency g MTF ( real) gmtf ( v) ( v) ( ideal) g ( v) MTF 1 g MTF Real systems: Choice of several application relevant frequencies e.g. photographic lens: 10 Lp/mm, 0 Lp/mm, 40 Lp/mm 0.5 real ideal g MTF ideal real g MTF 0

42 MTF [%] bei 10, 0, 40 Lp/mm... tan sag 4 Modulation Transfer Function Photographic lenses with different performance Objektiv 1 f/ 3.5 Objektiv Lens 1 f/3.5 Lens max. MTF Bildhöhe [mm] max. MTF Image height 10 c/mm 0 c/mm 40 c/mm

43 Calculation of MTF Some more examples 43 1-dim case MTF circular pupil Ring pupil = central obscuration (75%) 1 0,5 Apodization = reduced transmission at pupil edge (Gauss to 50%) 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

44 44 Resolution Test Chart: Siemens Star a. original b. good system c. defocus d. spherical e. astigmatism f. coma

45 45 Contrast and Resolution Contrast vs contrast as a function of spatial frequency Typical: contrast reduced for increasing frequency V Compromise between resolution and visibilty is not trivial and depends on application 1 H MTF Contrast sensitivity H CSF c

46 46 Optical Transfer Function of a Perfect System Loss of contrast for higher spatial frequencies ideal MTF / max contrast / max

47 47 Contrast / Resolution of Real Images Degradation due to 1. loss of contrast. loss of resolution resolution, sharpness contrast, saturation