Handbook of Optical Systems

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1 Handbook of Optical Systems Edited by Herbert Gross Volume 3: Aberration Theory and Correction of Optical Systems Herbert Cross, Hannfried Zügge, Martin Peschka, Fritz Blechinger BICENTENNIAL BICENTENNIA WILEY-VCH Verlag GmbH & Co. KGaA

Preface XIX Introduction XXI 29 Aberrations 1 29.1 Introduction 2 29.2 Power Series Expansions 8 29.3 Chromatic Aberrations 13 29.4 Primary Aberrations 16 29.4.1 Aperture and Field Dependence 16 29.4.2 Symmetry and Periodicity Properties 18 29.

2 Preface XIX Introduction XXI 29 Aberrations Introduction Power Series Expansions Chromatic Aberrations Primary Aberrations Aperture and Field Dependence Symmetry and Periodicity Properties Presentation of Aberrations and their Impact on Image Quality Calculation of the Seidel Sums Stop Shift Formulae Several Aberration Expressions from the Seidel Sums Thin Lens Aberrations Pupil Aberrations High-order Aberrations Fifth-order Aberrations Seventh and Higher-order Aberrations Zernike Polynomials Special Aberration Formulae Sine Condition and the Offence against the Sine Condition Herschel Condition Aplanatism and Isoplanatism Aldis Theorem Spherical Aberration, a Surface Contribution Formula Aplanatic Surface and Aplanatic Lens Literature Image Quality Criteria Introduction Geometrical Aberrations 76 Handbook ofoptical Systems: Vol. 3. Aberration Theory and Correction ofoptical Systems. Edited by Herbert Gross Copyright 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN:

3 XI Transverse Aberrations Spot Diagrams Wave Aberrations 80 Introduction PV and RMS Value of the Wavefront PV and RMS Values of Simple Aberrations Influence of the Spatial Frequency Strehl Ratio Introduction Simple Analytical Relations Approximations of the Strehl Ratio Special Criteria Rayleigh Criterion Marechal Criterion % Strehl Criterion Criteria for PSF and Intensity Distributions Introduction Apodization Spatial Moments Kurtosis Parameter Beam Quality M Relation between M 2 and the Conventional Criteria Spot or Beam Diameter Point Resolution Introduction Incoherent Two-point Resolution Coherent Two-point Resolution DepthofFocus Best Receiving Plane Defocus Criterion of Fisher Depth of Focus for Visual Detection MTF Criteria Introduction Connection with other Criteria MTF for Ideal and Defocused Systems MTF for Aberrations Sagittal and Tangential Structures Polychromatic OTF Geometrical Approximated Transfer Function GTF Phase Transfer Function PTF Argand Diagramm Contrast Versus Resolution Threshold Modulation Hopkins Factor Area Criteria of the MTF 155

30.9.14 Coherent Transfer Function 156 30.9.15 Test Charts 157 30.9.16 Image Examples 161 30.10 Edge Criteria 163 30.10.1 EdgeWidth 163 30.10.2 Edge Steepness 267 30.10.3 Acutance and Edge Defect 268 30.

4 Coherent Transfer Function Test Charts Image Examples Edge Criteria EdgeWidth Edge Steepness Acutance and Edge Defect Line Criteria Resolution of Lines LSFCriterionofStruve Bossung Plots Encircled Energy Introduction Energy Curve of the Airy Pattern Ensquared Energy Displaced Energy Criterion Special Criteria Relative Ceiling Fidelity Structural Content Correlation Relations and Comparison between the Criteria Distortion Color Aberrations Transverse Color Longitudinal Color Transmission and Illumination Illumination Fall-off Special Illumination Profiles FieldDependenceoftheQuality Statistical Aberrations Introduction Statistical Surfaces Statistical Wave Aberrations Point Spread Function in the Presence of Statistical Aberrations Transfer Function in the Presence of Statistical Aberrations Atmospheric Perturbations Special Aspects Complete Chain of Image Formation Discretizarion Problems Motion Blur Special Imaging Modes Parasitic Light Polarization Literature 222 I XI

$1.7 Refractive Index 222 31.1.8 Dispersion 222 31.1.9 Relative Partial Dispersion 222 31.1.10 GRIN, Gradient Index Material 222 31.1.11 Cemented Surface 223 31.1.12 Aplanatic Surface 223 31.1.13 Aspherical Surface 223 31.$

5 XII 31 Correction of Aberrations Strategies Introduction Lens Bending Power Splitting Power Combination Distances Stop Position Refractive Index Dispersion Relative Partial Dispersion GRIN, Gradient Index Material Cemented Surface Aplanatic Surface Aspherical Surface Mirror Diffractive Surface Symmetry Principle Field Lens Monochromatic Aberrations Spherical Aberration Coma Astigmatism Petzval Curvature Distortion High-order Aberrations Chromatic Aberrations Axial Color and Secondary Spectrum Lateral Color Spherochromatism Coexistence of Aberrations Literature Principles of Optimization Introduction Numerics of Optimization Notation Linear Matrix Algebra Local Expansion of the Error Function The Control Function One-dimensional Minimum Search Significance of the Result Termination of the Iteration Efficiency ofvariables and Weighting Factors Performance of an Algorithm 307

XIII 32.2.10 Numerical Calculation of Derivatives 307 32.3 Constraints 309 32.3.1 Introduction 309 32.3.2 Kuhn-Tucker Conditions 312 32.3.3 Penalty Function 313 32.3.4 Barrier Methods 325 32.

6 XIII Numerical Calculation of Derivatives Constraints Introduction Kuhn-Tucker Conditions Penalty Function Barrier Methods Local Solution Methods Introduction Method of Steepest Descent Method of Newton-Raphson without Constraints Damped Least-squares Method without Constraints Damped Least-squares Method with Constraints Conjugate Gradient Method Method of Davidon, Fletcher and Powell Method of Levenberg-Marquardt Orthogonalization ofthe System Matrix Derivative-free Simplex Methods Comparison of Algorithms Global Optimization Methods Introduction Simulated Annealing Genetic Optimization Optimization ofoptical Systems Introduction Example 1: Bending of a Thin Lens Example 2: Achromatic Doublet Parameters ofoptical Systems Constraints of Optical Systems Merit Function Special Aspects Starting Systems in Lens Design Introduction Thin Lens Start System tructural Approach according to Shafer Controlling the Optimization Process The Complete Design Process Structural Changes in the System Expert Systems Global Optimization in Optical Design The Saddle Point Method of Bociort Isshikis Method ofthe Global Explorer Adaptive Correction Method According to Glatzel Literature 369

XIV 33 Optimization Process 371 33.1 General Aspects 372 33.1.1 Introduction 372 33.1.2 Addition and Removal of a Lens 372 33.1.3 Methods of Improving a Design 377 33.1.4 Zero Power Operations 378 33.

7 XIV 33 Optimization Process General Aspects Introduction Addition and Removal of a Lens Methods of Improving a Design Zero Power Operations Substitution of Standard Radii Properties of Microscope Objective Lenses General Discussion of Micro-objective Lenses Setup of a Micro-objective Lens Performance and Qualityof Micro-objective Lenses Special Aspects Analysis of Several Existing Design Solutions Development ofa Monochromatic High NA Microscope Lens Specification and Strategy Initial Lens Setup Increasing the Numerical Aperture Improving the On-axis Performance Extending the Field of View Improvement of the Performance I Removing Unnecessary Surfaces Improvement of the Performance II Improving the Field Uniformity Obtaining the Desired Working Distance Making the System Telecentric Documentation of the Final Design Overview of the Design Stages Literature Special Correction Features Aspherical Surfaces Introduction Classification of Aspherical Surfaces Exact Mirror Aspheres for Stigmatic Imaging Refracting Surface Corrected for Spherical Aberration Polynomial Aspherical Surfaces Scaling and Conversion of Aspherical Coefficients The Higher-order Problem Aplanatic Imaging with Aspheres Seidel Contributions of Aspheres Best Location for an Asphere Inside a System Choice of the Expansion Order Realization Aspects for Aspheres Free-form Aspheres Gradient Index Media 463

8 Introduction GRIN Lenses with Radial Parabolic Index Profile Perfect Solutions Wood or Rod Lenses Axial GRIN Media Seidel Aberrations ofa GRIN Lens with Rotational Symmetry Seidel Aberrations of Radial GRIN Media Seidel Aberrations of Axial GRIN Media Aberrations of Radial GRIN Media Aberrations of Axial GRIN Lenses GRADIUM Media Examples of Radial and Axial Gradient Systems Examples of GRADIUM Systems Chromatic Aberrations Principles for Correction of GRIN Systems Correction With GRADIUM Lenses General Application Aspects of GRIN Lenses Systems with Diffractive Elements Introduction Working Principle of Diffractive Elements Types of Diffractive Element Equivalent Aspherical Phase Mask Sweatt Model Dispersion Fresnel Zone Lens Achromatic Hybrid Lens Multi-order Diffractive Lenses Seidel Aberrations Diffractive Singlet Diffraction Efficiency Diffractive Optics for Broad Spectral Ranges Athermalization with Diffractive Elements Transition between Refractive and Diffractive Surfaces Optical Design with Diffractive Elements Practical Aspects and Tolerances Applications Further Examples Non-axisymmetrical Systems Introduction Axis Ray and 3D Geometry Image Tilt and Anamorphism Second-order Environmental Propagation Around the Axis Ray Vector Aberration Theory of Tilted Axisymmetrical Components Third-order Aberrations for Tilted Component Systems General Distortion 562

$4.12 Example of a General 3D System 579 34.4.13 Example with Refractive Component 584 34.5 Literature 587 35 Tolerancing 595 35.1 Introduction 597 35.$

9 XVI Generalized Aberration Theory for Plane Symmetrie Systems Systems With General 3D Geometry Examples of Anamorphotic Systems Schiefspiegier Telescopes Example of a General 3D System Example with Refractive Component Literature Tolerancing Introduction Tolerances for Optical Elements and Optical Systems Introduction to Tolerances in the International Standard ISO Stress Birefringence Bubbles and Inclusions Inhomogeneity and Striae Surface-form Tolerances Spatial Frequencies of Surface Errors Surface-form Tolerances for Aspherical Surfaces Surface Imperfection Tolerances Surface Texture Centering Tolerances Decenter and Tilt Tolerances Centering of Lenses with Spherical Surfaces Centering Errors in Aspherical Lenses Typical Types of Centering Errors in Practical Tolerancing Control of Centering Errors in Bonding Processes Centering Errors of Lenses in Mounts Tolerance Costs Tolerances, Compensators and Adjustment Compensators for Typical Aberrations Modeling of Adjustment Example: Adjustment of Spherical Aberration, On-axis Astigmatism and On-axis Coma Tolerance Distributions Practical Tolerancing Assigning Tolerances by Sensitivity Analysis Sensitivity Analysis Statistical Simulations Inverse Tolerancing Prism Tolerances Introduction Angle Errors of Prisms in the Principal Plane Principal Angle Errors of Special Prisms Pyramidal Error Pyramidal Errors in Special Prisms 701

XVII 35.8.6 Calculation of Image Rotation 70 35.8.7 Error in the Orientation ofa Prism 706 35.8.8 Roof-angle Tolerances 706 35.8.9 Angle Errors in a Corner Cube Prism 710 35.8.10 Astigmatism Tolerance of Prisms 712 35.

10 XVII Calculation of Image Rotation Error in the Orientation ofa Prism Roof-angle Tolerances Angle Errors in a Corner Cube Prism Astigmatism Tolerance of Prisms Literature 714 A2 Optical Design Software OptaliX 717 A2.1 Introduction 718 A2.2 Program User Interface 718 A2.2.1 Command Line 719 A2.2.2 Functions and Arithmetic Expressions 721 A2.2.3 Lens Database Items 721 A2.3 Configuration and System Data 722 A2.3.1 Fields 722 A2.3.2 Wavelengths 723 A2.3.3 Apertures 724 A2.4 Surface Data 724 A2.4.1 Surface Editor 725 A2.4.2 Surface Types 726 A2.4.3 Surface Apertures 728 A2.5 Worked Examples 731 A2.5.1 Tilted Surfaces Example 731 A2.5.2 Aspherical Surfaces Example 733 A2.5.3 Zoom Example 734 A2.5.4 Global Surface References 737 A2.6 Optical Design Import and Export 739 A2.7 OpTaliX-PRO Capabilities 742 A2.8 Obtaining OpTaliX-LT 747 Index 749

j Jacobi matrix 295 Index flattening mirror 258 flint glass 231 form tolerance 598, 605 ff free-form aspheres 456 Fresnel zone plate 499, 503 f

j Jacobi matrix 295 Index flattening mirror 258 flint glass 231 form tolerance 598, 605 ff free-form aspheres 456 Fresnel zone plate 499, 503 f 749 a Abbe number 41, 222, 269, 490, 502 aberrations 2, 216 astigmatism 13, 28 axial chromatic aberration 13, 269 axial color 13, 269 chromatic aberrations 2, 13, 187, 268, 280 chromatic difference in