Design and Correction of Optical Systems Lecture 11: Correction principles II 2014-06-18 Herbert Gross Summer term 2014 www.iap.uni-jena.de 2 Preliminary Schedule Law of refraction, Fresnel formulas, optical system model, raytrace, calculation 1 09.04. Basics approaches Dispersion, anormal dispersion, glass map, liquids and plastics, lenses, mirrors, 2 16.04. Materials and Components aspheres, diffractive elements Paraxial approximation, basic notations, imaging equation, multi-component 3 23.04. Paraxial Optics systems, matrix calculation, Lagrange invariant, phase space visualization Pupil, ray sets and sampling, aperture and vignetting, telecentricity, symmetry, 4 30.04. Optical Systems photometry Longitudinal and transverse aberrations, spot diagram, polynomial expansion, 5 07.05. Geometrical Aberrations primary aberrations, chromatical aberrations, Seidels surface contributions Fermat principle and Eikonal, wave aberrations, expansion and higher orders, 6 14.05. Wave Aberrations Zernike polynomials, measurement of system quality Diffraction, point spread function, PSF with aberrations, optical transfer function, 7 21.05. PSF and Transfer function Fourier imaging model Rayleigh and Marechal criteria, Strehl definition, 2-point resolution, MTF-based 8 28.05. Further Performance Criteria criteria, further options Principles of optimization, initial setups, constraints, sensitivity, optimization of 9 04.06. Optimization and Correction optical systems, global approaches Symmetry, lens bending, lens splitting, special options for spherical aberration, 10 11.06. Correction Principles I astigmatism, coma and distortion, aspheres Field flattening and Petzval theorem, chromatical correction, achromate, 11 18.06. Correction Principles II apochromate, sensitivity analysis, diffractive elements Overview, photographic lenses, microscopic objectives, lithographic systems, 12 25.06. Optical System Classification eyepieces, scan systems, telescopes, endoscopes 13 02.07. Special System Examples Zoom systems, confocal systems 14 09.07. Further Topics New system developments, modern aberration theory,... 3 Contents 1. Field flattening and Petzval theorem 2. Field lenses 3. Chromatical correction 4. Achromate and apochromate 5. Diffractive elements 6. Miscellaneous 4 Petzval Theorem for Field Curvature . Petzval theorem for field curvature: 1 n 'n n ' k k 1. formulation for surfaces m Rptz k nk nk 'rk 2. formulation for thin lenses (in air) 1 1 Rptz j n j f j . Important: - no dependence on bending - no dependence on stop location . Natural behavior: image curved object towards system plane . Problem: collecting systems with f > 0: If only positive lenses: R always negative ptz R ideal optical system real image . Typical scaling for single lens: image plane R shell ptz n 1.6 f 5 Petzval Theorem for Field Curvature . Goal: vanishing Petzval curvature 11 h 1 1 and positive total refractive power j R j n f f ptz j h1 f j j for multi-component systems . Solution: General principle for correction of curvature of image field: 1. Positive lenses with: - high refractive index - large marginal ray heights - gives large contribution to power and low weighting in Petzval sum 2. Negative lenses with: - low refractive index - samll marginal ray heights - gives small negative contribution to power and high weighting in Petzval sum 6 Flattening Meniscus Lenses . Possible lenses / lens groups for correcting field curvature . Interesting candidates: thick mensiscus shaped lenses r 2 2 1 n 'n 1 n 1 d k k d Rptz k nk nk 'rk n f n r1r2 r1 1. Hoeghs mensicus: identical radii 2 (n 1) d - Petzval sum zero F' nr 2 - remaining positive refractive power 2. Concentric meniscus, r r d - Petzval sum negative 2 1 - weak negative focal length 1 (n 1)d (n 1)d F' - refractive power for thickness d: R n r r d ptz 1 1 nr1(r1 d) 3. Thick meniscus without refractive power n 1 1 (n 1)2 d Relation between radii 0 r2 r1 d n Rptz nr1 nr1 d (n 1) 7 Correcting Petzval Curvature r r . Triplet group with + - + 1 2 r 3 d/2 collimated n2 n1 . Effect of distance and 1/Rpet [1/mm] -1 refractive indices 10 SF66 / FK3 / SF66 10-2 BK7 -3 10 r1 50 70 100 [mm] From : H. Zügge 8 Field Curvature 1 F . Correction of Petzval field curvature in lithographic lens j R j n for flat wafer j h j . Positive lenses: Green hj large F Fj j h1 . Negative lenses : Blue hj small . Correction principle: certain number of bulges Microscope Objective Lens a) . Possible setups for flattening single meniscus the field lense . Goal: b) - reduction of Petzval sum two meniscus lenses - keeping astigmatism corrected c) symmetrical triplet d) achromatized meniscus lens e) two meniscus lenses achromatized f) modIfied achromatized triplet solution Field Flatness one waist two waists . Principle of multi-bulges to reduce Petzval sum 1 1 n' r k n f p k k . Seidel contributions show principle Petz Petz 1. bulge 1. waist 2. bulge 2. waist 3. bulge 0.2 0 -0.2 10 20 30 40 50 60 Size Reduction by Aspheres 1. bulge 2. bulge 3. bulge 10 8 . Sensitivity of refractive lenses 6 spherical 4 for aspheres 2 0 1.5 1 coma 0.5 0 0.5 0.4 0.3 astigmatism 0.2 0.1 0 0.4 0.3 0.2 distortion 0.1 0 5 10 15 20 25 30 35 40 45 50 55 sum pupil aspherical 12 Flattening Field Lens Effect of a field lens for flattening the image surface 1. Without field lens 2. With field lens curved image surface image plane image flat shell image field lens 13 Field Lenses . Field lens: in or near image planes . Influences only the chief ray: pupil shifted . Critical: conjugation to image plane, surface errors sharply seen marginal ray chief ray field lens in intermediate original lens L image plane 2 shifted pupil pupil lens L1 14 Field Lens im Endoscope Ref : H. Zügge 15 Chromatical Aberrations . Various cases of chromatical aberration correction a) axial and lateral color corrected b) axial color corrected F FC C MR CR FC FC c) lateral color corrected d) no color corrected F C C F F C F C 16 Axial Colour: Achromate (a) (b) (c) . Compensation of axial colour by appropriate glass choice . Chromatical variation of the spherical aberrations: spherochromatism (Gaussian aberration) . Therefore perfect axial color correction BK7 BK7 F2 BK7 N-SSK8F2 (on axis) are often not feasable n = 1.5168 n = 1.5168 1.6200 n = 1.5168 1.6177 = 64.17 = 64.17 36.37 = 64.17 49.83 F= 1 F = 2.31 -1.31 F = 4.47 -3.47 r p r p r p 1 1 1 486 nm 588 nm 656 nm z z z -2.5 0 -0.20 0 0.20 -100 0 100 Ref : H. Zügge Achromate . Achromate: - Axial colour correction by cementing two Crown in different glasses front - Bending: correction of spherical aberration at the full aperture - Aplanatic coma correction possible be clever choice of materials . Four possible solutions: - Crown in front, two different bendings - Flint in front, two different bendings Flint in front . Typical: - Correction for object in infinity - spherical correction at center wavelength with zone - diffraction limited for NA < 0.1 - only very small field corrected solution 1 solution 2 Achromate: Realization Versions . Advantage of cementing: solid state setup is stable at sensitive middle surface with large curvature . Disadvantage: a) flint in front loss of one degree of freedom . Different possible realization forms in practice edge contact cemented b) crown in front edge contact cemented contact on axis broken, Gaussian setup Achromate : Basic Formulas . Idea: 1. Two thin lenses close together with different materials 2. Total power F F1 F2 F F 3. Achromatic correction condition 1 2 0 1 2 . Individual power values 1 1 F1 F F F 2 1 2 1 1 1 . Properties: 2 1. One positive and one negative lens necessary 2. Two different sequences of plus (crown) / minus (flint) 3. Large -difference relaxes the bendings 4. Achromatic correction indipendent from bending 5. Bending corrects spherical aberration at the margin 6. Aplanatic coma correction for special glass choices 7. Further optimization of materials reduces the spherical zonal aberration Achromate: Correction sMR' . Cemented achromate: 6 degrees of freedom: aplanatic 3 radii, 2 indices, ratio 1/2 case . Correction of spherical aberration: diverging cemented surface with R positive spherical contribution 1 for nneg > npos case with 2 solutions . Choice of glass: possible goals case with 1. aplanatic coma correction one solution and coma 2. minimization of spherochromatism case without solution, correction 3. minimization of secondary spectrum only spherical minimum . Bending has no impact on chromatical correction: is used to correct spherical aberration at the edge . Three solution regions for bending 1. no spherical correction 2. two equivalent solutions 3. one aplanatic solution, very stable Achomatic solutions in the Glass Diagram flint crown negative lens positive lens Achromat Achromate . Achromate . Longitudinal aberration . Transverse aberration . Spot diagram 486 nm 587 nm 656 nm y' = 486 nm axis rp 1 = 656 nm sinu' = 587 nm 1.4° 486 nm 587 nm 656 nm 2° s' [mm] 0 0.1 0.2 Coma Correction: Achromate Image height: y’ = 0 mm y’ = 2 mm Achromat . Bending of an achromate Pupil section: meridional meridional sagittal bending Transverse y' y' y' - optimal choice: small residual spherical Aberration: 0.05 mm 0.05 mm 0.05 mm aberration (a) - remaining coma for finite field size . Splitting achromate: additional degree of freedom: - better total correction possible (b) - high sensitivity of thin air space . Aplanatic glass choice: vanishing coma (c) . Cases: a) simple achromate, Achromat, splitting sph corrected, with coma (d) b) simple achromate, coma corrected by bending, with sph Achromat, aplanatic glass choice c) other glass choice: sph better, coma reversed (e) d) splitted achromate: all corrected e) aplanatic glass choice: all corrected Wave length: Ref : H.