Lens Design II. Lecture 1: Aberrations and optimization Herbert Gross. Winter term

Transcription

1 Lens Design II Lecture 1: Aberrations and optimization Herbert Gross Winter term 18

2 Preliminary Schedule Lens Design II Aberrations and optimization Repetition 4.1. Structural modifications Zero operands, lens splitting, lens addition, lens removal, material selection Aspheres Correction with aspheres, Forbes approach, optimal location of aspheres, several aspheres Freeforms Freeform surfaces, general aspects, surface description, quality assessment, initial systems Field flattening Astigmatism and field curvature, thick meniscus, plus-minus pairs, field lenses Chromatical correction I Achromatization, axial versus transversal, glass selection rules, burried surfaces Chromatical correction II Secondary spectrum, apochromatic correction, aplanatic achromates, spherochromatism Special correction topics I Symmetry, wide field systems, stop position, vignetting Special correction topics II Telecentricity, monocentric systems, anamorphotic lenses, Scheimpflug systems Higher order aberrations High NA systems, broken achromates, induced aberrations Further topics Sensitivity, scan systems, eyepieces Mirror systems special aspects, double passes, catadioptric systems Zoom systems Mechanical compensation, optical compensation Diffractive elements Color correction, ray equivalent model, straylight, third order aberrations, manufacturing

3 3 Contents 1. Geometrical aberrations. Wave aberrations and Zernikes 3. Point spread function 4. Modulation transfer function 5. Principles of optimization

4 4 Optical Image Formation Perfect optical image: All rays coming from one object point intersect in one image point Real system with aberrations: 1. tranverse aberrations in the image plane. longitudinal aberrations from the image plane 3. wave aberrations in the exit pupil object plane wave aberrations image plane optical system transverse aberrations longitudinal aberrations

5 5 Notations for an Optical System x, y object coordinates, especially object height x', y' image coordinates, especially image height x p,y p coordinates of entrance pupil x' p, y' p coordinates of exit pupil s object distance form 1st surface s' image distance form last surface p entrance pupil distance from 1st surface p' exit pupil distance from last surface Dx' sagittal transverse aberration Dy' meridional transverse aberration y p x' P y' p p' x' s' y' P' y' Dx' P' Dy' image plane z x P y' p x y system surfaces s p x P y p entrance pupil x' P exit pupil object plane y P

6 6 Polynomial Expansion of Aberrations Representation of -dimensional Taylor series vs field y and aperture r Selection rules: checkerboard filling of the matrix Constant sum of exponents according to the order Image location Primary aberrations / Seidel y r 5 Field y Spherical Coma Astigmatism y y 1 y y 3 y 4 y 5 y cos 3 cos y 5 cos Distortion r Tilt Distortion Distortion primary secondary r 1 y r 1 cos y 4 r 1 cos r 1 Defocus y r 1 Aperture Astig./Curvat. y 4 r 1 y r cos y 3 r r r cos 3 Coma primary y 3 r cos r 3 y r 3 cos r 3 Spherical primary y r 3 y r 4 cos r 4 Coma secondary Spherical r 5 secondary Secondary aberrations

7 7 Pupil Sampling Ray plots Spot sagittal ray fan tangential aberration Dy sagittal aberration Dx diagrams x p x p tangential ray fan Dy y p whole pupil area

8 Spherical Aberration 8 Typical chart of aberration representation Reference: at paraxial focus Primary spherical aberration at paraxial focus Wave aberration tangential sagittal Transverse ray aberration Dy Dx Dy.1 mm.1 mm.1 mm Pupil: y-section x-section x-section Modulation Transfer Function MTF MTF at paraxial focus MTF through focus for 1 cycles per mm cyc/mm -... z/mm Ref: H. Zügge Geometrical spot through focus. mm z/mm

9 Surface Contributions: Example 9 Seidel aberrations: representation as sum of surface contributions possible Gives information on correction of a system Example: photographic lens Retrofocus F/.8 Field: w= S I Spherical Aberration S II Coma S III Astigmatism S IV Petzval field curvature S V Distortion C I Axial color C II Lateral color -4 Surface Sum

10 1 Wave Aberration in Optical Systems Definition of optical path length in an optical system: Reference sphere around the ideal object point through the center of the pupil Chief ray serves as reference Difference of OPL : optical path difference OPD Practical calculation: discrete sampling of the pupil area, real wave surface represented as matrix y y p y' p y' upper coma ray optical system W wave aberration chief ray w' image point z chief ray wave front object point lower coma ray reference sphere Object plane Op Entrance pupil EnP Exit plane ExP Image plane Ip

11 11 Pupil Sampling All rays start in one point in the object plane The entrance pupil is sampled equidistant In the exit pupil, the transferred grid may be distorted In the image plane a spreaded spot diagram is generated object plane point entrance pupil equidistant grid optical system exit pupil transferred grid image plane spot diagram y o y p y' p y' x o x p x' p x' z

12 1 Zernike Polynomials Expansion of wave aberration surface into elementary functions / shapes m W ( r, ) c Z ( r, ) n n mn nm n m = cos Zernike functions are defined in circular coordinates r, Z m n ( r, ) R m n sin ( m) for ( r) cos( m) for 1 for m m m Ordering of the Zernike polynomials by indices: n : radial m : azimuthal, sin/cos Mathematically orthonormal function on unit circle for a constant weighting function Direct relation to primary aberration types n = sin

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

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

15 15 Psf with Aberrations Zernike coefficients c in Spherical aberration Astigmatism Coma Spherical aberration, Circular symmetry Astigmatism, Split of two azimuths Coma, Asymmetric c =. c =.3 c =.5 c =.7 c = 1.

16 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 th 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

17 17 Optical Transfer Function of a Perfect System Aberration free circular pupil: Reference frequency v o a f sinu' 1.5 g MTF Maximum cut-off frequency: v max v na f nsin u' Analytical representation.5 1 / max H MTF v ( v) arccos p v v v 1 Separation of the complex OTF function into: - absolute value: modulation transfer MTF - phase value: phase transfer function PTF v v H OTF ( v, v PTF x y ( v, v ) H ( v, v ) e x y MTF x y ih )

18 18 Basic Idea of Optimization Topology of the merit function in dimensions Iterative down climbing in the topology topology of meritfunction F start iteration path x 1 x

19 19 Nonlinear Optimization Mathematical description of the problem: n variable parameters m target values Jacobi system matrix of derivatives, Influence of a parameter change on the various target values, sensitivity function Scalar merit function Gradient vector of topology x f (x) J i j f x m F( x) w g j i1 F x j i j i y f ( x) i Hesse matrix of nd derivatives H jk x F x j k

20 Optimization in Optical Design Merit function: Weighted sum of deviations from target values Formulation of target values: 1. fixed numbers. one-sided interval (e.g. maximum value) 3. interval g f ist j j f j1, m soll j Problems: 1. linear dependence of variables. internal contradiction of requirements 3. initail value far off from final solution Types of constraints: 1. exact condition (hard requirements). soft constraints: weighted target Finding initial system setup: 1. modification of similar known solution. Literature and patents 3. Intuition and experience

21 1 Optimization Merit Function in Optical Design Goal of optimization: Find the system layout which meets the required performance targets according of the specification Formulation of performance criteria must be done for: - Apertur rays - Field points - Wavelengths - Optional several zoom or scan positions Selection of performance criteria depends on the application: - Ray aberrations - Spot diameter - Wavefornt description by Zernike coefficients, rms value - Strehl ratio, Point spread function - Contrast values for selected spatial frequencies - Uniformity of illumination Usual scenario: Number of requirements and targets quite larger than degrees od freedom, Therefore only solution with compromize possible

22 Lens bending und shift of principal plane Ray path at a lens of constant focal length and different bending Quantitative parameter of description X: The ray angle inside the lens changes X R R 1 R R 1 The ray incidence angles at the surfaces changes strongly The principal planes move For invariant location of P, P the position of the lens moves P P' F' X = -4 X = - X = X = + X = +4

23 3 Correcting Spherical Aberration: Lens Splitting Transverse aberration Correction of spherical aberration: Splitting of lenses (a) 5 mm Distribution of ray bending on several surfaces: - smaller incidence angles reduces the effect of nonlinearity - decreasing of contributions at every surface, but same sign Last example (e): one surface with compensating effect (b) (c) 5 mm 5 mm Improvement (a)à(b) : 1/4 Improvement (b)à(c) : 1/ 5 mm (d) Improvement (c)à(d) : 1/4.5 mm (e) Improvement (d)à(e) : 1/75 Ref : H. Zügge