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
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