Handbook of optical systems. Vol. 2: Physical image formation / W.Singer, M.Totzeck, H.Gross / ed. by H.Gross. - Weinheim: Wiley-VCH, 2005. - xxiv, 690 p.: ill. - Incl. bibl. ref. - Ind.: p.685-690. - ISBN 3-527-40378-7  Место хранения:   02 | Отделение ГПНТБ СО РАН | Новосибирск

Оглавление / Contents

Introduction .................................................. XIX 17 The Wave Equation ......................................... 1 17.1 Introduction .............................................. 2 17.2 From Maxwell to Helmholtz ................................. 2 17.2.1 Maxwell's Equations and the Inhomogeneous Wave Equation .......................................... 17.2.2 Wave Equation in Homogeneous Media and the Scalar Wave Equation .............................. 4 17.2.3 The Dispersion Relation of the Harmonic Wave Solution .......................................... 6 17.3 Elementary Waves in Free Space ............................ 9 17.3.1 The Electromagnetic Plane Wave .................... 9 17.3.2 Spherical Wave ................................... 11 17.3.3 Dipole Wave ...................................... 11 17.3.4 Radiated Field of a Harmonic Current Distribution ..................................... 13 17.3.5 A Note on Plane and Spherical Waves .............. 13 17.4 Energy, Irradiance and Intensity ......................... 14 17.5 The Angular Spectrum ..................................... 17 17.5.1 Spatial Frequency Representation ................. 17 17.5.2 Transformation of the Three-dimensional Spectrum into Two Dimensions ..................... 19 17.5.3 Free-space Propagation of Transverse Fields ...... 20 17.5.4 Periodic Fields with Discrete Spectra ............ 22 17.5.5 Boundary Conditions and the Spatial Frequency Spectrum ......................................... 23 17.5.6 Vector Field Representation by Spatial Frequencies ...................................... 24 17.6 Evanescent Waves ......................................... 26 17.7 Approximative Solutions to the Wave Equation ............. 28 17.7.1 Geometrical Optics and the Eikonal Equation ...... 28 17.7.2 Paraxial Wave Equation ........................... 29 17.7.3 Transport of Intensity ........................... 30 17.7.4 Gaussian Beams ................................... 31 17.7.5 Ray Equivalent of Gaussian Beams ................. 36 17.7.6 Gaussian Beams in Two Dimensions ................. 37 17.8 Literature ............................................... 39 18 Scalar Diffraction ....................................... 41 18.1 Introduction ............................................. 42 18.2 KirchhofF Diffraction Integral ........................... 44 18.2.1 Inconsistency of the KirchhofF Diffraction Integral ......................................... 48 18.3 1st and 2nd Rayleigh-Sommerfeld Diffraction Integral ..... 48 18.4 Two-dimensional Diffraction .............................. 50 18.5 Muygens Principle ........................................ 52 18.6 Fourier Space Formulation ................................ 54 18.7 Examples of Scalar Diffraction Patterns .................. 57 18.7.1 Diffraction Fields Behind Slits .................. 57 18.7.2 Diffraction by a Rectangular Aperture ............ 59 18.8 Fresnel Diffraction ...................................... 60 18.8.1 Computation ...................................... 61 18.8.2 Validity ......................................... 62 18.9 Coffin's Fresnel Diffraction Integral .................... 64 18.9.1 Definition ....................................... 64 18.9.2 Example .......................................... 67 18.10 Fraunhofer Diffraction ................................... 69 18.11 Grating Diffraction ...................................... 71 18.11.1 Ronchi Grating .................................. 71 18.11.2 The Sinusoidal Phase Grating and Surface Fabrication Errors .............................. 76 18.12 Scalar Diffraction at Dielectric Objects ................. 79 18.13 Babinet's Principle ...................................... 82 18.14 Scalar Scattering ........................................ 85 18.15 Boundary Diffraction Waves ............................... 89 18.15.1 Geometrical Theory of Diffraction ................ 90 18.15.2 An Empirical Boundary Diffraction Wave ........... 94 18.16 Literature ............................................... 96 19 Interference and Coherence ............................... 99 19.1 Basic Principles ........................................ 100 19.1.1 Introduction .................................... 100 19.1.2 Two-beam Interference and Double Slit Diffraction ..................................... 102 19.1.3 Contributions of Different Points of the Light Source .......................................... 305 19.1.4 The High-frequency Term ......................... 107 19.1.5 The Low-frequency Term .......................... 108 19.1.6 Different Light Source Points with Statistical Phase ........................................... 209 19.2 Mathematical Description of Coherence ................... 113 19.2.1 Coherence Function .............................. 113 19.2.2 Wigner Distribution Function .................... 116 19.2.3 Moments of the Wigner Distribution Function ..... 120 19.2.4 Smoothing of the Wigner Distribution Function and Diffraction Focus ........................... 121 19.2.5 Wigner Distribution Function of Coherent Fields .......................................... 122 19.2.6 Ambiguity Function .............................. 123 19.2.7 The Characterizing Functions in their Context ... 125 19.3 Temporal Coherence ...................................... 126 19.3.1 Superposition of Signals with Different Frequency ....................................... 326 19.3.2 Spectral Distribution of a Light Source ......... 327 19.3.3 Bandwidth-limited Signals ....................... 328 19.3.4 Axial Coherence Length .......................... 330 19.3.5 Thermal Light Sources ........................... 333 19.3.6 Temporal Coherence in the Michelson Interferometer .................................. 334 19.4 Spatial Coherence ....................................... 335 19.4.1 Introduction .................................... 335 19.4.2 Propagation of the Coherence Function ........... 338 19.4.3 Van Cittert-Zernike Theorem ..................... 240 19.4.4 The Coherence Function of a Circular Source ..... 240 19.4.5 Coherence Function behind a Double Slit ......... 243 19.4.6 Propagation of the Wigner Distribution Function ........................................ 246 19.5 Gaussian Schell Beams ................................... 249 19.5.1 Definition of Gaussian Schell Beams ............. 249 19.5.2 Coherence and Wigner Functions of Gaussian" Schell Beams .................................... 154 19.5.3 Basis Mode Expansion of Partial Coherent Fields .......................................... 256 19.6 Statistical Optics and Speckle .......................... 159 19.6.1 Photon Statistics ............................... 259 19.6.2 The Speckle Effect .............................. 262 19.6.3 Speckle Parameters and Surface Structure ........ 263 19.6.4 Computation of Speckle Effects .................. 265 19.6.5 Speckle Reduction ............................... 269 19.7 Array Homogenizer ....................................... 272 19.7.1 Setup of the System ............................. 272 19.7.2 Pupil Filling ................................... 275 19.7.3 Coherence Effects ............................... 276 19.7.4 Example Calculation ............................. 277 19.8 Miscellaneous ........................................... 279 19.8.1 General Coherence Length ........................ 279 19.8.2 General Degree of Coherence ..................... 282 19.8.3 Coherence and Polarization ...................... 283 19.9 Literature .............................................. 284 20 The Geometrical Optical Description and Incoherent Imaging ................................................. 287 20.1 Introduction ............................................ 288 20.2 Characteristic Functions ................................ 289 20.2.1 Geometrical Optics and the Wave Equation ........ 289 20.2.2 The Characteristic Functions .................... 291 20.2.3 Geometrical-optical imaging ..................... 294 20.2.4 The Canonical Pupil ............................. 296 20.2.5 A Note on Diffractive Optical Elements .......... 299 20.3 The Ideal Wave-optical Image of a Point and Geometrical-optical Image Formation ..................... 200 20.3.1 The Scalar Lьneburg Integral .................... 200 20.3.2 Energy Discussions for Optical Imaging .......... 204 20.3.3 The Airy Disc ................................... 206 20.3.4 Incoherent Resolution ........................... 210 20.4 Aberrations of Optical Systems .......................... 211 20.4.1 The Small-aberration Limit: The Strehl Ratio .... 211 20.4.2 Expansion of the Wave-front Error into Zernike Polynomials ..................................... 212 20.4.3 Point Images for Different Aberrations .......... 217 20.4.4 Distortion, Defocus and Astigmatism ............. 219 20.4.5 Spherical Aberrations Zg, Coma Z7 and Z8 ........ 220 20.4.6 Line of Sight ................................... 221 20.4.7 Wave Aberrations for Annular Pupils ............. 224 20.4.8 Extended Zernike Expansion ...................... 227 20.5 Helmholtz-Lagrange Invariant and Phase-space Description ............................................. 231 20.5.1 The Phase Space ................................. 231 20.5.2 The Resolution Limit in the Space Dorrfliin and in the Spatial Frequency Domain ............. 234 20.5.3 The Space-Bandwidth Product ..................... 236 20.6 Literature .............................................. 237 21 The Abbe Theory of Imaging .............................. 239 21.1 Introduction ............................................ 240 21.2 Phenomenological Description of Imaging ................. 244 21.2.1 The Explanation of Image Formation According to Abbe and the Abbe Resolution ................. 244 21.2.2 The Information About an Object Contained in an Image ........................................ 249 21.2.3 Koehler Illumination and the Visibility ......... 252 21.2.4 The Siedentopf Illumination Principle ........... 255 21.2.5 Imaging with Different Colours .................. 259 21.2.6 Aplanatic Correction and Geometrical Optics ..... 260 21.3 The Mathematical Description of Fourier Optical Imaging ................................................. 262 21.3.1 Imaging with Uncorrelated Light Sources ......... 262 21.3.2 Consideration of Magnification .................. 267 21.4 Coherence in Imaging .................................... 269 21.4.1 The Coherent Image .............................. 269 21.4.2 Incoherent Imaging .............................. 272 21.4.3 One-Dimensional Incoherent Imaging .............. 273 21.4.4 Systems with Rotational Symmetry ................ 275 21.4.5 Conditions for Incoherent, Partially Coherent and Coherent Imaging ............................ 277 21.4.6 Imaging with Correlated Light Sources ........... 280 21.5 Literature .............................................. 281 22 Coherence Theory of Optical Imaging ..................... 283 22.1 Introduction ............................................ 284 22.2 Theoretical Description of Partially Coherent Image Formation ............................................... 284 22.2.1 Hopkins Transmission Cross Coefficient .......... 284 22.2.2 Image Fidelity .................................. 287 22.2.3 Hopkins Formalism for Periodic Objects .......... 288 22.2.4 Aberrations in the Linear Grating Image ......... 293 22.3 The Coherence Function and the Coherence Transfer Function ................................................ 296 22.4 The Phase Space Description ............................. 300 22.4.1 Transformation of Coherence and Wigner Distribution Function ........................... 300 22.4.2 Propagation of the Wigner Distribution Function in Free Space .......................... 303 22.4.3 Compilation of the Transformations .............. 307 22.5 Optical Imaging in the Presence of Aberrations .......... 309 22.5.1 Linear Systems and Classification of Aberrations ..................................... 309 22.5.2 Random Non-stationary Aberrations: Stray Light and Flare ................................. 314 22.6 Literature .............................................. 317 23 Three-dimensional Imaging ............................... 319 23.1 Introduction ............................................ 320 23.2 The Ewald Sphere and the Generalized Pupil .............. 321 23.2.1 The Ewald Sphere ................................ 321 23.2.2 The Generalized Aperture and the Three- dimensional Point-spread Function ............... 322 23.3 The Three-dimensional Transfer Function ................. 327 23.3.1 Born Approximation and the Laue Equation ........ 327 23.3.2 Dдndliker's Representation and the Shape of the Three-dimensional Transfer Function ......... 330 23.3.3 Resolution, Depth Resolution and Depth of Focus ........................................... 335 23.3.4 3D-Transfer Functions in Microscopy ............. 338 23.3.5 Magnification and a Comment on Absolute Instruments ..................................... 340 23.4 Selected Examples of the Three-Dimensional Transfer Function ................................................ 343 23.4.1 Transfer Function for Incoherent Imaging with a = 1 ........................................... 343 23.4.2 Partial Coherent Image Examples ................. 344 23.4.3 'Tayloring' of the 3D-Transfer Function ......... 346 23.4.5 Influence of Aberrations ........................ 351 23.5 Literature .............................................. 352 24 Image Examples of Selected Objects ...................... 355 24.1 Introduction ............................................ 356 24.2 Two-point Resolution .................................... 356 24.2.1 Incoherent Versus Coherent Two-point Resolution ...................................... 356 24.2.2 Image of a Double Slit for Coherent and Incoherent Illumination ......................... 360 24.2.3 Phase Shift and Oblique Illumination ............ 364 24.3 The Image of an Edge .................................... 365 24.3.1 The Coherent Image of an Amplitude and Phase Edge ............................................ 365 24.3.2 The Incoherent Image of an Amplitude Edge ....... 369 24.3.3 Partially Coherent Edge Image ................... 370 24.3.4 The Determination of the Optical Transfer Function from the Edge Image .................... 375 24.4 The Line Image .......................................... 376 24.4.1 The Line Image of a Rotational-symmetrical Lens ............................................ 376 24.4.2 Coherent Line or Slit Image ..................... 377 24.4.3 Incoherent Line or Slit Image ................... 380 24.5 The Grating Image ....................................... 381 24.5.1 The Coherent Linear Grating Image ............... 381 24.5.2 The Coherent Grating Image with Aberrations ..... 384 24.5.3 The Influence of the Coherence Parameter a on the Grating Image ............................... 386 24.5.4 Influence of the Shape of the Effective Light Source on the Grating Image ..................... 389 24.5.5 Wigner Distribution Function for Gratings, Talbot Effect and Propagation-invariant Fields .. 394 24.6 Pinhole Imaging and Quasi-point Sources ................. 399 24.6.1 Introduction .................................... 399 24.6.2 Incoherent Image of a Circular Object ........... 400 24.6.3 Quasi-point Source .............................. 402 24.6.4 Pinhole with Coherent Illumination .............. 404 24.6.5 Pinhole with Partial Coherent Illumination ...... 405 24.6.6 Defocusing Planes and Deconvolution ............. 406 24.7 Literature .............................................. 407 25 Special System Examples and Applications ................ 409 25.1 Introduction ............................................ 410 25.2 Point-spread Functions for Annular Pupils ............... 410 25.2.1 Introduction .................................... 410 25.2.2 Annular Pupils, Central Obscuration and Pupil Filters ......................................... 412 25.3 Point-spread Functions of Non-uniform Illuminated Pupils .................................................. 426 25.3.1 Introduction .................................... 416 25.3.2 General Gaussian Apodization .................... 427 25.3.3 Gaussian Profile with Truncation ................ 428 25.4 Engineering of the Point-spread Function by Pupil Masks ................................................... 423 25.4.1 Introduction .................................... 423 25.4.2 Characterization of the Three-dimensional Point-spread Function ........................... 423 25.4.3 Characterization of Extended Depth of Focus ..... 426 25.4.4 Relation Between Axial and Transverse Resolution ...................................... 427 25.4.5 Ambiguity Function as Defocussed Transfer Function ........................................ 429 25.4.6 Image Multiplexing .............................. 430 25.4.7 Fundamental Relationships ....................... 432 25.4.8 Calculation of Masks ............................ 432 25.5 Special Pupil Masks ..................................... 433 25.5.1 Introduction .................................... 433 25.5.2 Phase Masks According to Toraldo ................ 434 25.5.3 Logarithmic Phase Mask .......................... 435 25.5.4 Chirped Ring Pupil .............................. 437 25.5.5 Complex Filter Described by Zernike Expansions .. 439 25.5.6 Cubic Phase Plates for Extended Depth of Focus .. 442 25.5.7 Structured Illumination ......................... 447 25.6 Selected Practical Applications for Pupil Filtering Techniques .............................................. 450 25.6.1 Phase Contrast Filtering, Dark-field Illumination .................................... 450 25.6.2 Frequency Doubling .............................. 453 25.6.3 Defect Filtering ................................ 455 25.6.4 Ronchi Test ..................................... 456 25.7 Literature .............................................. 463 26 Polarization ............................................ 465 26.1 Introduction ............................................ 467 26.2 Polarization States ..................................... 467 26.2.1 Representation of Polarization States ........... 468 26.2.2 Jones Vector .................................... 468 26.2.3 Ellipse of Polarization ......................... 470 26.2.4 Orthogonal Jones Vectors ........................ 472 26.2.5 Jones Vectors in Different Bases ................ 472 26.2.6 Unpolarized Light ............................... 472 26.2.7 Partial Polarization ............................ 473 26.2.8 Polarization Matrix ............................. 473 26.2.9 Stokes Vector ................................... 475 26.2.10 Poincare Sphere ................................. 478 26.3 Jones Matrix ............................................ 479 26.3.1 Definition ...................................... 479 26.3.2 Jones Matrix Acting on a Jones Vector ........... 480 26.3.3 Succession of Jones Matrices .................... 480 26.3.4 Jones Matrix Acting on a Polarization Matrix .... 482 26.3.5 Examples of Jones Matrices ...................... 481 26.3.6 Rotated and Mirrored Jones Matrix ............... 482 26.3.7 Jones Matrix for Different Basis Polarization States .......................................... 483 26.3.8 Eigenpolarizations of a Jones Matrix ............ 483 26.3.9 Jones Matrix of a Retarder ...................... 484 26.3.10 Jones Matrix of a Partial Polarizer ............. 487 26.3.11 Pauli's Spin Matrices ........................... 489 26.3.12 Jones Matrix Decomposition ...................... 489 26.4 Mьller Matrix ........................................... 491 26.4.1 Definition ...................................... 492 26.4.2 Examples ........................................ 492 26.5 Mьller-Jones Matrix ..................................... 493 26.6 Light in Anisotropic Media .............................. 494 26.6.1 Anisotropic Media ............................... 494 26.5 Principal Refractive Indices of an Anisotropic Medium Without Spatial Dispersion and Optical Activity ......... 495 26.6.3 Fresnel Ellipsoid ............................... 496 26.6.4 Index Ellipsoid ................................. 497 26.6.5 Types of Birefringent Media ..................... 497 26.7 Eigenwaves in Anisotropic Media ......................... 502 26.7.1 Plane Waves in Anistropic Media ................. 501 26.7.2 Eigenwaves and their Polarization ............... 502 26.7.3 Properties of the Eigenpolarizations ............ 506 26.7.4 The Intersection Ellipse ........................ 506 26.8 Jones Matrix of Propagation ............................. 507 26.9 Jones Matrices of Propagation for Common Media .......... 508 26.9.1 Eigenpolarizations and-values ................... 508 26.9.2 Coordinate Systems .............................. 509 26.9.3 Uniaxial Crystal ................................ 509 26.9.4 Biaxial Crystal ................................. 510 26.9.5 CaF2 with Spatial Dispersion at λ = 193 nm ...... 511 26.10 Beam-splitting in an Anisotropic Medium ................. 522 26.11 Examples of Polarization-optical Elements ............... 516 26.11.1 Quarter-wave and Half-wave Retarder ............. 526 26.11.2 Babinet-Soleil Compensator ...................... 526 26.11.3 Faraday Rotator ................................. 528 26.11.4 Brewster Plate .................................. 519 26.12 Literature .............................................. 520 27 Vector Diffraction ...................................... 523 27.1 Introduction ............................................ 524 27.2 Focus Computation for Polarized Fields .................. 525 27.2.1 Geometry for Focus Computation .................. 525 27.2.2 Richards-Wolf integral .......................... 526 27.2.3 Plane Wave Expansion ............................ 531 27.2.4 Focus Fields for Various Input Polarizations .... 533 27.3 Vector Kirchhoff Diffraction Integral ................... 538 27.4 Analytical Solutions .................................... 538 27.4.1 Plane Interface: Fresnel's Equations ............ 540 27.4.2 Diffraction at a Circular Cylinder .............. 542 27.4.3 Mie Scattering .................................. 547 27.5 Numerical Methods for Arbitrary Structures .............. 553 27.6 Coupled Dipole Method ................................... 553 27.7 Integral Equation Approach and Moment Method ............ 555 27.7.1 The Moment Method ............................... 555 27.7.2 Form of Scattering Operator ..................... 556 27.7.3 Scattering in Three-layer Medium ................ 557 27.8 Fourier Modal Method .................................... 563 27.8.1 Theory .......................................... 563 27.8.2 Diffraction Efficiency .......................... 568 27.9 Finite-difference Method ................................ 568 27.9.1 Boundary Conditions ............................. 570 27.9.2 Implicit Paraxial Wave Equation in Two Dimensions ...................................... 572 27.9.3 Paraxial Wave Equation in Cylindrical Coordinates ..................................... 572 27.9.4 AD I-formulation of the Paraxial Wave Equation in Three Dimensions ............................. 575 27.9.5 Split-step-beam Propagation Method .............. 576 27.10 Rigorous Diffraction in Optical Imaging ................. 579 27.10.1 Dielectrics and Metals .......................... 579 27.11 Simulation of Polarized Imaging by use of Rigorous Diffraction ............................................. 583 27.12 Literature .............................................. 587 28 Polarization and Optical Imaging ........................ 589 28.1 Introduction ............................................ 590 28.2 The Image-forming Field ................................. 590 28.3 Interference of Electromagnetic Waves ................... 592 28.3.1 Two-beam Vector Interference .................... 592 28.3.2 Contrast for High-NA, s- and p-polarization ..... 593 28.3.3 Influence of Recording Medium ................... 594 28.3.4 Vector Effect in Optical Microscopy ............. 595 28.3.5 Vector Effect in Optical Lithography ............ 595 28.4 Polarized Ray Trace ..................................... 596 28.4.1 Definition of Ray, Beam and Path ................ 597 28.4.2 Ray-splitting at Anisotropic Elements ........... 597 28.4.3 Refraction and Reflection at Birefringent Interfaces ...................................... 598 28.4.4 The Single-path Approximation ................... 599 28.5 Optical Systems with Polarization Effects ............... 604 28.6 Polarized Imaging Model ................................. 605 28.6.1 Scalar Image .................................... 606 28.6.2 Vector Image for Completely Polarized Illumination .................................... 607 28.6.3 Vector Image for Partially Polarized Illumination .................................... 609 28.7 Vector Point-spread Function ............................ 610 28.7.1 VPSF for Complete Polarization .................. 610 28.7.2 VPSF for Unpolarized Illumination ............... 611 28.8 Polarized Optical Transfer Function ..................... 612 28.8.1 Polarized Illumination .......................... 612 28.8.2 Unpolarized Illumination ........................ 622 28.9 Jones Matrix Pupil ...................................... 612 28.9.1 Definition for Completely Polarized Illumination .................................... 613 28.9.2 Separation of a Scalar Factor ................... 614 28.9.3 Decomposition into Retardance and Diattenuation ................................... 615 28.9.4 Example ......................................... 616 28.10 Jones Matrix Pupils in the Polarization Matrix Calculus ................................................ 617 28.11 Jones-matrix-based System Optimization .................. 619 28.10 Aberrations of the Transmitted Wavefront ................ 620 28.13 Jones-Zernike Wavefront Aberrations ..................... 621 28.13.1 Principle of the Modal Characterization of a Jones Pupil ................................... 621 28.13.2 Jones-Zernike Expansion ......................... 622 28.13.3 Properties of the Jones-Zernike Polynomials ..... 623 28.14 Literature .............................................. 625 Mathematical Appendix ................................... 627 A.l Linear Systems .......................................... 629 A.2 Fourier Series and Fourier Integral ..................... 631 A.2.1 Compilation of Basic Properties of the Fourier Transform ........................................ 632 A.2.2 Special Functions and their Fourier Transforms ... 634 A.3 Convolution and Correlation ............................. 637 A.3.1 Convolution ...................................... 637 A.3.2 Correlation ...................................... 637 A.3.3 Power Spectral Density and RMS Value ............. 638 A.4 Discrete Signals ........................................ 639 A.4.1 The Sampling Theorem ............................. 639 A.4.2 Leakage .......................................... 642 A.4.3 Indexing of the Numerical Discrete Fast Fourier Transform ........................................ 642 A.5 z-Transform ............................................. 644 A.5.1 Definition ....................................... 644 A.5.2 Numerical Evaluation of the z-transform .......... 646 A.5.3 Sine Interpolation ............................... 648 A.6 Hankel Transform ........................................ 648 A.6.1 Definition ....................................... 648 A.6.2 Numerical Computation ............................ 649 A.7 Practical Calculation of Diffraction Integrals .......... 655 A.7.1 The Oscillation Problem .......................... 655 A.7.2 Spatial and Spectral Resolution .................. 660 A.7.3 Periodic Boundary Conditions ..................... 662 A.7.4 x-z Sampling of the Ewald Sphere ................. 663 A.7.5 Equivalent Diffraction Setups .................... 663 A.7.6 Optimal Conditioning of the Fresnel Diffraction .. 666 A.7.7 Numerical Algorithms ............................. 669 A.7.8 Fresnel Integrals ................................ 672 A.8 Orthogonal Polynomials on Rectangular Domains ........... 675 A.8.1 Chebyshev Polynomials ............................ 675 A.8.2 One-dimensional Legendre Polynomials ............. 677 A.8.3 Two-dimensional Chebyshev Polynomials ............ 678 A.8.4 Legendre Polynomials in Two Dimensions ........... 679 A.9 Literature .............................................. 683 Index ......................................................... 685