Aberrations
Aberrations determine the amount of detail that can be observed in the image. High quality imagery is of little value unless sufficient power or energy reaches the sensor. This is where basic radiometric concepts are introduced, particularly as applied to lens behavior. Power-related measurements such as lens transmission, relative illumination falloff, and veiling glare are discussed. Image formation is accomplished via glass and/or mirrored surfaces that are curved (spherical for the most part). Consequently, it is important to be able to measure the basic parameters of refractive index and radius of curvature. Depending on the application, test objects observed with the imaging system could either be finite size resolution targets or point 'star' sources.
The aberrations measured on an optical bench are recorded as image plane coordinates, i.e., axial and lateral displacements relative to a reference location. But aberrations can also be described in the exit pupil in terms of wavefronts. Hence the book also explores aberrations as described in the pupil and how pupil aberrations are connected to the point spread function. Then the phenomenon of interference is introduced (with a practical application involving the measurement of optical windows).
The specific property carrying information about pupil aberration is the optical path difference (OPD) between a reference wavefront and the aberrated wavefront. Interferometry is the methodology used to measure OPD. Variations in OPD are presented to the metrologist as fringe patterns which must be reduced and analyzed to extract aberration content.
The reality of pupil aberrations is visualized next through the auspices of the point diffraction interferometer, with an industry workhorse, the Fizeau interferometer, described and its operating principle discussed in some detail.
Using the Fizeau as our principle metrology instrument we examine a broad range of test configurations for measuring a wide variety of optical components and systems — mirror flats, spheres, aspheres, and Cassegrain telescopes. The main asphere of interest is the parabolic mirror (both on- and off-axis). The latter also includes an in-depth discussion of the null lens. Test configurations for refractors such as singlets, and multi-element camera systems are considered. Testing non-rotationally symmetric optics such as rectangular aperture cylindrical lenses and mirrors is reviewed. Alternative interferometer types such as the Twyman-Green are also examined. Finally, an exploration is made of the Fizeau interferometer as an imaging system in the context of a null lens test.
Once fringe patterns have been acquired they need to be reduced to provide salient information such as peak-to-valley, rms wavefront, and aberration content. Both manual and automatic techniques are discussed. Interferogram analysis via phase shifting is introduced. Shop tests of optical surfaces via the use of localized fringe techniques (test plates) are reviewed. A reflection from a thin parallel glass plate broaches the topic of multiple ray interference. Next, numerous indirect aberration measurement techniques are explored. Included are the classical knife-edge test, lateral shear interferometry, the Hartmann and Shack-Hartmann test. The use of a radiometric technique known as the axial intensity (or Strehl) scan is also explored.
The effect aberrations have on image quality of finite scenes can best be described by use of the modulation transfer functions (MTF). The basic MTF concept is introduced and expanded with the help of Linear Optics and Fourier Optics analytic approaches. Various MTF measurement methods are examined, including slit and knife-edge scan techniques. Also discussed is a radiometric approach known as pupil auto-correlation. For telescopes such as the Cassegrain the MTF is also influenced by light scatter from mirror surfaces due to micro-roughness. Consequently, the measurements of surface roughness by both direct and indirect methods are discussed.
Up to this point the book has concentrated on making measurements on optical systems using light based instrumentation. It concludes with a discussion of direct measurements on light fields themselves. Included are discussions on wavefront sensors, polarization and coherence, and broadband and spectral radiometrics.
Table of contents
Preface . . . . . . . . . .. . . . . . . . . . . . . . . . . . xv
Chapter 1 Optical Testing . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Working Definition . . . . . . . . . . . . . . . . . . . 1
1.3 Hands-On . . . . . . . . . . . . . . . . . . . . . . . . 1
1.4 Measurement Error . . . . . . . . . . . . . . . . . . . .2
1.4.1 Accuracy and Repeatability. . . . . . . . . . . . . . .2
1.4.2 Calculating Error . . . . . . . . . . . . . . . . . . .2
1.4.3 Calibration . . . . . . . . . . . . . . . . . . . . . .3
1.5 The Lesson of HST . . . . . . . . . . . . . . . . . . . .3
1.6 The Journey . . . . . . . . . . . . . . . . . . . . . . .4
1.7 Instructor Note . . . . . . . . . . . . . . . . . . . . .8
Part I Optical Bench Testing . . . . . . . . . . . . . . . . . . 9
Chapter 2 Geometric Optics Review. . . . . . . . . . . . . . . . 11
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Paraxial Ray-Trace Equations (PRTE) . . . . . . . . . . .11
2.3 Optical Power. . . . . . . . . . . . . . . . . . . . . . 13
2.3.1 Power of Single Optical Surface . . . . . . . . . . . .13
2.3.2 Power of Two-Component System . . . . . . . . . . . . .15
2.4 Optical Power and Effective Focal Length . . . . . . . . 16
2.5 Gaussian Lens Formula . . . . . . . . . . . . . . . . . .17
2.6 Magnification. . . . . . . . . . . . . . . . . . . . . . 17
2.7 Stops and Pupils. . . . . . . . . . . . . . . . . . . . .18
2.8 F-number . . . . . . . . . . . . . . . . . . . . . . . . 19
2.9 Marginal and Chief Rays . . . . . . . . . . . . . . . . .20
2.10 Homework . . . . . . . . . . . . . . . . . . . . . . . .21
Chapter 3 Collimators . . . . . . . . . . . . . . . . . . . . . .23
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Collimator Structures . . . . . . . . . . . . . . . . . .23
3.3 Collimator Light Sources. . . . . . . . . . . . . . . . .26
3.4 Setting up and Aligning a Collimator . . . . . . . . . . 30
3.4.1 Alignment Process. . . . . . . . . . . . . . . . . . . 30
3.4.2 Establishing Collimation: Autocollimation Procedure . .32
3.5 Alternative Techniques . . . . . . . . . . . . . . . . . 34
3.5.1 Beam Diameter Versus Distance. . . . . . . . . . . . . 34
3.5.2 Shear plate . . . . . . . . . . . . . . . . . . . . . .35
3.6 Aligning a Multi-element Lens to a Collimator. . . . . . 36
3.7 Homework . . . . . . . . . . . . . . . . . . . . . . . . 38
Chapter 4 EFL Measurements . . . . . . . . . . . . . . . . . . . 41
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 41
4.2 Nodal Points. . . . . . . . . . . . . . . . . . . . . . .41
4.3 T-Bar Nodal Slide (TBNS) Description . . . . . . . . . . 42
4.4 Measuring EFL via Nodal Method . . . . . . . . . . . . . 43
4.5 T-Bar Nodal Slide Calibration . . . . . . . . . . . . . .43
4.6 Measuring EFL via Lateral Magnification . . . . . . . . .45
4.7 Measuring EFL (λ) — Axial Color . . . . . . . . . . . . .46
4.8 Measuring the Entrance Pupil and F-number . . . . . . . .47
4.9 Measuring EFL Change with Temperature . . . . . . . . . .48
4.10 Some Thoughts on Measurement. . . . . . . . . . . . . . 49
Chapter 5 Aberrations Part I. . . . . . . . . . . . . . . . . . .51
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 51
5.2 Positional Aberrations: Field Curvature and Distortion . 51
5.3 Measuring Positional Aberrations on TBNS . . . . . . . . 51
5.3.1 Measuring Field Curvature . . . . . . . . . . . . . . 51
5.3.2 Measuring Distortion. . . . . . . . . . . . . . . . . 55
5.4 Aberrations Affecting Quality of "Point Image" . . . . . 57
5.4.1 Spherical Aberration . . . . . . . . . . . . . . . . . 57
5.4.2 Coma . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.4.3 Astigmatism. . . . . . . . . . . . . . . . . . . . . . 58
5.5 Linear Hartmann Screen and Ray-Fan Plots . . . . . . . . 59
5.6 Measuring the “Point” Degrading Aberrations on TBNS . . .62
5.6.1 Measuring Spherical Aberration . . . . . . . . . . . . 62
5.6.1.1 Annular Zone Method . . . . . . . . . . . . . . .62
5.6.1.2 Minimum Blur Method . . . . . . . . . . . . . . .63
5.6.1.3 Linear Hartmann Screen . . . . . . . . . . . . . 63
5.6.1.4 Axial Intensity Method . . . . . . . . . . . . . 65
5.6.2 Coma . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.6.3 Astigmatism. . . . . . . . . . . . . . . . . . . . . . 66
5.7 Homework . . . . . . . . . . . . . . . . . . . . . . . . 67
Chapter 6 Refractive Index . . . . . . . . . . . . . . . . . . . 69
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 69
6.2 Glass Properties . . . . . . . . . . . . . . . . . . . . 69
6.2.1 Refractive Index . . . . . . . . . . . . . . . . . . . 69
6.2.2 Dispersion . . . . . . . . . . . . . . . . . . . . . . 69
6.2.3 Glass Chart . . . . . . . . . . . . . . . . . . . . . .73
6.2.4 Partial Dispersion . . . . . . . . . . . . . . . . . . 74
6.3 Measuring Refractive Index. . . . . . . . . . . . . . . .74
6.3.1 Microscope Method. . . . . . . . . . . . . . . . . . . 74
6.3.2 Critical Angle . . . . . . . . . . . . . . . . . . . . 78
6.3.3 Brewster’s Angle . . . . . . . . . . . . . . . . . . . 80
6.3.4 Prism Refractometer . . . . . . . . . . . . . . . . . .82
6.3.4.1 Governing Equation. . . . . . . . . . . . . . . .82
6.3.4.2 Procedure for Making n(λ) Measurements . . . . . 84
6.4 Dispersion . . . . . . . . . . . . . . . . . . . . . . . 87
6.5 Homework . . . . . . . . . . . . . . . . . . . . . . . . 88
Chapter 7 Radius of Curvature . . . . . . . . . . . . . . . . . .89
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 89
7.2 Surface Sagitta ("SAG") . . . . . . . . . . . . . . . . .89
7.2.1 General Spheric Expansion . . . . . . . . . . . . . . .89
7.2.2 Surface Sag . . . . . . . . . . . . . . . . . . . . . .90
7.3 Measuring Radius of Curvature . . . . . . . . . . . . . .90
7.3.1 Eyeball Estimate . . . . . . . . . . . . . . . . . . . 90
7.3.2 Via Microscope . . . . . . . . . . . . . . . . . . . . 91
7.3.3 Via Interferometer . . . . . . . . . . . . . . . . . . 92
7.3.4 Test Plates . . . . . . . . . . . . . . . . . . . . . .94
7.3.5 Sag and the Spherometer . . . . . . . . . . . . . . . .94
7.4 Derivation of Ball Radius Correction . . . . . . . . . . 96
7.5 Measuring RoC of an Asphere. . . . . . . . . . . . . . . 99
7.6 Homework . . . . . . . . . . . . . . . . . . . . . . . . 100
Chapter 8 Image Resolution . . . . . . . . . . . . . . . . . . . 101
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 101
8.2 Air Force Resolution Target . . . . . . . . . . . . . . .101
8.3 Aerial Resolution Measured on TBNS . . . . . . . . . . . 103
8.3.1 Aerial Resolution on Best Focal Surface. . . . . . . . 104
8.3.2 Aerial Resolution on Flat Observation Surface. . . . . 104
8.4 Recorded Resolution on Flat Formats . . . . . . . . . . .105
8.5 Tangential and Radial Corrections. . . . . . . . . . . . 106
8.6 Area Weighted Average Resolution (AWAR) . . . . . . . . .110
8.7 Sensor Example: Photographic Film . . . . . . . . . . . .110
8.7.1 Film Basics . . . . . . . . . . . . . . . . . . . . . .112
8.7.2 Lens Resolution Test on Film . . . . . . . . . . . . . 114
8.8 Homework . . . . . . . . . . . . . . . . . . . . . . . . 116
Chapter 9 Radiometry and Lenses . . . . . . . . . . . . . . . . .117
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 117
9.2 Radiometry Basics . . . . . . . . . . . . . . . . . . . .117
9.2.1 Intensity . . . . . . . . . . . . . . . . . . . . . . .117
9.2.2 Radiance . . . . . . . . . . . . . . . . . . . . . . . 118
9.2.3 Lambertian Radiator . . . . . . . . . . . . . . . . . .119
9.2.4 Power through a Lens . . . . . . . . . . . . . . . . . 120
9.2.5 Image Irradiance . . . . . . . . . . . . . . . . . . . 120
9.3 Image Irradiance and Lens F-number . . . . . . . . . . . 121
9.4 Diffusers. . . . . . . . . . . . . . . . . . . . . . . . 123
9.4.1 Plate Diffusers . . . . . . . . . . . . . . . . . . . .123
9.4.2 Integrating Spheres . . . . . . . . . . . . . . . . . .123
9.5 Lens Transmission Measurements. . . . . . . . . . . . . .124
9.5.1 How to Measure Transmission . . . . . . . . . . . . . .124
9.5.2 Spectral Transmission: Double-Pass Method . . . . . . .126
9.6 Relative Illumination Falloff (RIF) . . . . . . . . . . .129
9.6.1 Power Through Tilted and Shifted Apertures . . . . . . 129
9.6.2 Lenses and RIF . . . . . . . . . . . . . . . . . . . . 129
9.6.3 Measuring RIF on TBNS. . . . . . . . . . . . . . . . . 130
9.7 Veiling Glare . . . . . . . . . . . . . . . . . . . . . .133
9.7.1 Sources of Veiling Glare . . . . . . . . . . . . . . . 133
9.7.2 Measuring Veiling Glare . . . . . . . . . . . . . . . .133
9.8 Homework . . . . . . . . . . . . . . . . . . . . . . . . 134
Chapter 10 Star Tests . . . . . . . . . . . . . . . . . . . . . 137
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . .137
10.2 The PSF . . . . . . . . . . . . . . . . . . . . . . . . 137
10.3 Physical Translation of Sampling Aperture. . . . . . . .139
10.3.1 Profile Measurements . . . . . . . . . . . . . . . . 139
10.3.2 LSF Measurement . . . . . . . . . . . . . . . . . . . 143
10.3.3 Knife-Edge Distribution (KED) . . . . . . . . . . . . 143
10.4 Sampling Aperture Arrays (CCD) . . . . . . . . . . . . .145
10.5 Scans and Convolution . . . . . . . . . . . . . . . . . 146
10.6 Axial PSF Scans. . . . . . . . . . . . . . . . . . . . .151
10.7 Radial Energy Distribution (RED). . . . . . . . . . . . 153
10.8 Star Image Resolution . . . . . . . . . . . . . . . . . 153
10.9 Homework . . . . . . . . . . . . . . . . . . . . . . . .155
Part II Pupil Aberrations and Interference. . . . . . . . . . . .157
Chapter 11 Aberrations Part II . . . . . . . . . . . . . . . . . 159
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . .159
11.2 Description of a Wavefront . . . . . . . . . . . . . . .159
11.3 Interaction of Wavefronts with Optical Systems . . . . .160
11.4 Wavefront Description. . . . . . . . . . . . . . . . . .161
11.5 Image Plane and Exit Pupil Aberrations Relationship. . .166
11.6 Defocus . . . . . . . . . . . . . . . . . . . . . . . . 168
11.6.1 Derivation via Sag Equation . . . . . . . . . . . . . 168
11.6.2 Difference between W020 and Wd. . . . . . . . . . . . 169
11.6.3 Defocus and Minimum Blur . . . . . . . . . . . . . . .171
11.7 Wavefronts and Diffraction . . . . . . . . . . . . . . .171
11.8 Homework . . . . . . . . . . . . . . . . . . . . . . . .172
Chapter 12 Interference and Optical Windows . . . . . . . . . . .175
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . .175
12.2 General Interference Equation . . . . . . . . . . . . . 175
12.3 Fringe Contrast. . . . . . . . . . . . . . . . . . . . .179
12.4 Interference of Tilted Beams . . . . . . . . . . . . . .179
12.5 Window Wedge . . . . . . . . . . . . . . . . . . . . . .181
12.6 Effect of Window Wedge on Image . . . . . . . . . . . . 185
12.7 Non-Linear Windows. . . . . . . . . . . . . . . . . . . 186
12.8 Homework . . . . . . . . . . . . . . . . . . . . . . . .186
Chapter 13 Visualizing Pupil Aberrations . . . . . . . . . . . . 189
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . .189
13.2 Optical Windows and Entrance Pupil . . . . . . . . . . .189
13.2.1 Fringe Contours due to Window Reflections . . . . . . 189
13.2.2 Window Single-Pass Wavefront . . . . . . . . . . . . .190
13.2.3 Window Wavefront and Imaging . . . . . . . . . . . . .192
13.3 Exit Pupil Aberrations and the Point Diffraction .. . . 194
13.3.1 PDI Structure . . . . . . . . . . . . . . . . . . . . 194
13.3.2 Operational Principle. . . . . . . . . . . . . . . . .195
13.3.3 PDI System . . . . . . . . . . . . . . . . . . . . . .195
13.4 PDI Reference and Test-Beam Relationship . . . . . . . .197
13.4.1 Pinhole Size . . . . . . . . . . . . . . . . . . . . .197
13.4.2 PDI Transmission and Metal Film Thickness . . . . . . 198
Chapter 14 The Fizeau Interferometer. . . . . . . . . . . . . . .203
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . .203
14.2 The Fizeau Interferometer . . . . . . . . . . . . . . . 204
14.2.1 Basic Layout . . . . . . . . . . . . . . . . . . . . .204
14.2.2 Align and View Modes . . . . . . . . . . . . . . . . .204
14.2.3 Test Optic Imaging . . . . . . . . . . . . . . . . . .205
14.2.4 The Transmission Flat . . . . . . . . . . . . . . . . 206
14.2.5 Wavefront versus Optical Surface. . . . . . . . . . . 207
14.2.6 Fringe Contrast . . . . . . . . . . . . . . . . . . . 208
14.3 Sensors in the Observation Plane. . . . . . . . . . . . 209
14.4 Factory Fizeau . . . . . . . . . . . . . . . . . . . . .209
14.5 Practical Concerns . . . . . . . . . . . . . . . . . . .211
14.5.1 Mechanical and Acoustical Vibration . . . . . . . . . 212
14.5.2 Air Currents . . . . . . . . . . . . . . . . . . . . .212
14.5.3 Mounting . . . . . . . . . . . . . . . . . . . . . . .212
14.5.4 Transmission Flat Costs. . . . . . . . . . . . . . . .213
Part III Interferometer Test Configurations . . . . . . . . . . .215
Chapter 15 Test Configurations I . . . . . . . . . . . . . . . . 217
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . .217
15.2 The Null Fringe . . . . . . . . . . . . . . . . . . . . 217
15.3 Testing Flats. . . . . . . . . . . . . . . . . . . . . .217
15.3.1 Fringe Order and Piston. . . . . . . . . . . . . . . .218
15.3.2 Hill or Valley? . . . . . . . . . . . . . . . . . . . 221
15.3.3 Determining Radius of Curvature . . . . . . . . . . . 223
15.4 The Reference or Retro Flat. . . . . . . . . . . . . . .224
15.5 Window Testing in Cavity Mode. . . . . . . . . . . . . .224
15.6 Testing Spherical Surfaces. . . . . . . . . . . . . . . 226
15.6.1 The Transmission Sphere . . . . . . . . . . . . . . . 226
15.6.2 Concave Spheres . . . . . . . . . . . . . . . . . . . 226
15.6.3 The Retro Sphere. . . . . . . . . . . . . . . . . . . 228
15.6.4 Convex Spheres . . . . . . . . . . . . . . . . . . . .228
15.6.5 Radius of Curvature Determination. . . . . . . . . . .229
15.7 The Test Point . . . . . . . . . . . . . . . . . . . . .229
15.8 Homework . . . . . . . . . . . . . . . . . . . . . . . .229
Chapter 16 Test Configurations II . . . . . . . . . . . . . . . .231
16.1 Introduction . . . . . . . . . . . . . . . . . . . . . .231
16.2 Refractors and Dispersion . . . . . . . . . . . . . . . 231
16.3 Afocal Systems . . . . . . . . . . . . . . . . . . . . .232
16.4 Testing Lenses . . . . . . . . . . . . . . . . . . . . .234
16.4.1 Object at Infinity: Photographic Lenses . . . . . . . 234
16.4.1.1 Multi-Element . . . . . . . . . . . . . . . . . 234
16.4.1.2 Testing Singlets and Achromats . . . . . . . . .235
16.4.2 Testing Finite Conjugate Lenses . . . . . . . . . . . 237
16.5 Testing Lenses Off-Axis . . . . . . . . . . . . . . . . 238
16.6 Telescopes . . . . . . . . . . . . . . . . . . . . . . .239
16.7 Retrace Error (Ray Mapping Error). . . . . . . . . . . .240
16.8 Homework . . . . . . . . . . . . . . . . . . . . . . . .243
Chapter 17 Test Configurations IIIA . . . . . . . . . . . . . . .245
17.1 Introduction . . . . . . . . . . . . . . . . . . . . . .245
17.2 Basics of Aspheric Surfaces. . . . . . . . . . . . . . .246
17.3 Departure from Sphere. . . . . . . . . . . . . . . . . .248
17.4 Surface Normals. . . . . . . . . . . . . . . . . . . . .249
17.5 W040 versus W040N . . . . . . . . . . . . . . . . . . . 251
17.5.1 Spherical Aberration Generated by a Parabolic Mirror. 251
17.5.2 Axial and Angular Distribution . . . . . . . . . . . .252
17.6 The Null Lens . . . . . . . . . . . . . . . . . . . . . 252
17.6.1 The Offner Null Lens . . . . . . . . . . . . . . . . .253
17.6.2 Physical Embodiment of an Offner Null Lens . . . . . .254
17.6.3 Testing Parabolic Mirror at the Focal Point . . . . . 255
17.7 Homework . . . . . . . . . . . . . . . . . . . . . . . .257
Chapter 18 Test Configurations IIIB . . . . . . . . . . . . . . .259
18.1 Introduction . . . . . . . . . . . . . . . . . . . . . .259
18.2 Null Lens Test of an OAP . . . . . . . . . . . . . . . .259
18.3 Alignment . . . . . . . . . . . . . . . . . . . . . . . 260
18.4 Test Pallet. . . . . . . . . . . . . . . . . . . . . . .260
18.5 Coarse Alignment Steps. . . . . . . . . . . . . . . . . 261
18.5.1 Coarse Interferometer Alignment . . . . . . . . . . . 264
18.5.2 Fine Alignment of OAP. . . . . . . . . . . . . . . . .268
18.6 Image Distortion . . . . . . . . . . . . . . . . . . . .270
18.7 Homework . . . . . . . . . . . . . . . . . . . . . . . .270
Chapter 19 Test Configurations IV. . . . . . . . . . . . . . . . 271
19.1 Introduction . . . . . . . . . . . . . . . . . . . . . .271
19.2 Cylindrical Optics . . . . . . . . . . . . . . . . . . .272
19.2.1 Fiber Optic Reference (FOR) . . . . . . . . . . . . . 272
19.2.2 Cylindrical Lenses . . . . . . . . . . . . . . . . . .272
19.2.3 FOR Operating Principle. . . . . . . . . . . . . . . .273
19.2.4 Cylindrical Mirrors . . . . . . . . . . . . . . . . . 275
19.3 Convex Compound Mirrors . . . . . . . . . . . . . . . . 276
19.4 Free Electron Laser Grazer . . . . . . . . . . . . . . .276
19.4.1 X-Ray Mandrels . . . . . . . . . . . . . . . . . . . .278
19.5 Concave Compound Mirrors (Wolter Telescopes) . . . . . .282
19.6 Homework . . . . . . . . . . . . . . . . . . . . . . . .283
Chapter 20 Other Interferometers for Optical Testing. . . . . . .285
20.1 Introduction . . . . . . . . . . . . . . . . . . . . . .285
20.2 Twyman-Green Interferometer . . . . . . . . . . . . . . 285
20.2.1 Internal Arrangement. . . . . . . . . . . . . . . . . 285
20.2.2 Quality of Internal Components . . . . . . . . . . . .287
20.2.3 LUPI Twyman-Green Interferometer . . . . . . . . . . .288
20.2.4 The Insides of a Commercial (Boxed) TGI. . . . . . . .289
20.2.5 TGI Test Configurations . . . . . . . . . . . . . . . 289
20.2.5.1 Testing Very Small Lenses with TGI . . . . . . .289
20.2.6 The MIC-1 . . . . . . . . . . . . . . . . . . . . . . 291
20.3 Shack Cube Interferometer . . . . . . . . . . . . . . . 293
20.4 Point Diffraction Interferometer (PDI) . . . . . . . . .296
20.5 Koster’s Prism Interferometer (KPI) . . . . . . . . . . 296
20.6 Homework . . . . . . . . . . . . . . . . . . . . . . . .299
Chapter 21 Interferometer as an Imaging System. . . . . . . . . .301
21.1 Introduction . . . . . . . . . . . . . . . . . . . . . .301
21.2 Building the Fizeau Model. . . . . . . . . . . . . . . .302
21.3 An Interferometer/Null Lens/Parabolic Mirror Model . . .303
21.4 Imaging Model. . . . . . . . . . . . . . . . . . . . . .304
21.5 Revised Double-Pass Model . . . . . . . . . . . . . . . 306
21.6 Interferometer Imaging . . . . . . . . . . . . . . . . .307
21.6.1 Retrace Error . . . . . . . . . . . . . . . . . . . . 307
21.6.2 Distortion . . . . . . . . . . . . . . . . . . . . . .307
21.7 Interferometer/Null Lens/OAP System . . . . . . . . . . 308
21.8 An Empirical Test . . . . . . . . . . . . . . . . . . . 309
Part IV Collecting and Analyzing Fringe Data . . . . . . . . . . 313
Chapter 22 Fringe Analysis . . . . . . . . . . . . . . . . . . . 315
22.1 Introduction . . . . . . . . . . . . . . . . . . . . . .315
22.2 Peak-to-Valley, Average, Variance, and RMS . . . . . . .315
22.3 OPD Profiles . . . . . . . . . . . . . . . . . . . . . .318
22.4 Full Pupil Variance and RMS . . . . . . . . . . . . . . 321
22.4.1 Average Volumes under a Mountain. . . . . . . . . . . 321
22.4.2 Average Volumes under an OPD Mountain . . . . . . . . 322
22.4.3 Calculation Example . . . . . . . . . . . . . . . . . 322
22.5 Strehl Ratio . . . . . . . . . . . . . . . . . . . . . .324
22.5.1 Definition. . . . . . . . . . . . . . . . . . . . . . 324
22.5.2 Relationship to Variance . . . . . . . . . . . . . . .325
22.5.3 Numerical Example . . . . . . . . . . . . . . . . . . 326
22.5.4 Axial Location of δs . . . . . . . . . . . . . . . . .326
22.5.5 Shape of OPD Plot at δds. . . . . . . . . . . . . . . 327
22.6 Homework . . . . . . . . . . . . . . . . . . . . . . . .327
Chapter 23 Fringe Analysis II . . . . . . . . . . . . . . . . . .329
23.1 Introduction . . . . . . . . . . . . . . . . . . . . . .329
23.2 Inputting Data . . . . . . . . . . . . . . . . . . . . .329
23.2.1 Fringe Following . . . . . . . . . . . . . . . . . . .329
23.2.2 Phase Shift Interferometry (PSI) . . . . . . . . . . .332
23.2.3 Discontinuities. . . . . . . . . . . . . . . . . . . .335
23.3 Fitting Data using Polynomials . . . . . . . . . . . . .337
23.3.1 Example using Fourier Series . . . . . . . . . . . . .337
23.3.2 Zernike Polynomials . . . . . . . . . . . . . . . . . 337
23.4 Fitting Interferometric Data using Zernikes . . . . . . 340
23.4.1 Qualitative Explanation . . . . . . . . . . . . . . . 340
23.4.2 Quantitative Explanation . . . . . . . . . . . . . . .341
23.5 Sample Analysis. . . . . . . . . . . . . . . . . . . . .342
23.6 Calculating Seidel Magnitudes from the Zernikes. . . . .344
23.7 Obscured, and Non-Circular Apertures . . . . . . . . . .344
23.8 Instantaneous PSI. . . . . . . . . . . . . . . . . . . .347
Chapter 24 Test Plates and Multiple Beam Interference . . . . . .349
24.1 Introduction . . . . . . . . . . . . . . . . . . . . . .349
24.2 Optical Testing and Localized Fringes . . . . . . . . . 349
24.2.1 Understanding Localized Fringes . . . . . . . . . . . 349
24.2.2 Local Irradiance and Fringe Spacing. . . . . . . . . .350
24.2.3 Optical Contact . . . . . . . . . . . . . . . . . . . 353
24.2.4 Test Plates . . . . . . . . . . . . . . . . . . . . . 354
24.3 Interference between Multiple Rays . . . . . . . . . . .355
24.3.1 Optical Path and Phase Changes . . . . . . . . . . . .355
24.3.2 Reflection and Transmission Amplitudes at Interface . 357
24.3.3 Stokes Relations . . . . . . . . . . . . . . . . . . .359
24.3.4 Final Form for ER . . . . . . . . . . . . . . . . . . 360
24.3.5 Reflected Irradiance. . . . . . . . . . . . . . . . . 361
24.4 Homework . . . . . . . . . . . . . . . . . . . . . . . .362
Part V Indirect Test Methods . . . . . . . . . . . . . . . . . . 363
Chapter 25 Foucault Knife-Edge Test I . . . . . . . . . . . . . .365
25.1 Introduction . . . . . . . . . . . . . . . . . . . . . .365
25.2 Foucault Knife-Edge Test: Basic Description . . . . . . 365
25.3 Relating the Parameters . . . . . . . . . . . . . . . . 366
25.3.1 Example . . . . . . . . . . . . . . . . . . . . . . . 367
25.4 The Knife-Edge and Spherical Aberration . . . . . . . . 369
25.5 Rotationally Symmetric Optic and Knife-Edge. . . . . . .372
25.6 Asymmetric Aberrations . . . . . . . . . . . . . . . . .376
25.7 Ronchi Test . . . . . . . . . . . . . . . . . . . . . . 377
Chapter 26 Foucault Knife-Edge Test II . . . . . . . . . . . . . 379
26.1 Introduction . . . . . . . . . . . . . . . . . . . . . .379
26.2 Departure from Sphere . . . . . . . . . . . . . . . . . 379
26.3 Departure from Parabola . . . . . . . . . . . . . . . . 380
26.4 Participatory Example/Problem . . . . . . . . . . . . . 381
26.4.1 Finding the Slope Profile. . . . . . . . . . . . . . .381
26.4.2 Finding the DFP Profile. . . . . . . . . . . . . . . .381
26.4.2.1 Stacking Local Tilts. . . . . . . . . . . . . . 382
26.4.2.2 Stacking Local Sticks . . . . . . . . . . . . . 383
26.4.2.3 Adding Up Areas. . . . . . . . . . . . . . . . .384
26.5 Homework . . . . . . . . . . . . . . . . . . . . . . . .385
Chapter 27 Lateral Shear Techniques . . . . . . . . . . . . . . .387
27.1 Introduction . . . . . . . . . . . . . . . . . . . . . .387
27.2 Principle of Pupil Shear . . . . . . . . . . . . . . . .388
27.3 Defocus: 1-D Math Analysis . . . . . . . . . . . . . . .388
27.4 Diffraction Gratings . . . . . . . . . . . . . . . . . .391
27.5 Pupil Shear via Grating . . . . . . . . . . . . . . . . 393
27.6 Ronchi Test as Lateral Shear . . . . . . . . . . . . . .394
27.6.1 Moiré . . . . . . . . . . . . . . . . . . . . . . . . 395
27.6.2 Orders Generated for a Focused Beam on a Ronchi . . . 397
27.6.3 Discussion . . . . . . . . . . . . . . . . . . . . . .399
27.7 Homework . . . . . . . . . . . . . . . . . . . . . . . .400
Chapter 28 Hartmann Tests . . . . . . . . . . . . . . . . . . . .401
28.1 Introduction . . . . . . . . . . . . . . . . . . . . . .401
28.2 Determining Mirror Surface Profile. . . . . . . . . . . 401
28.2.1 Single Hole . . . . . . . . . . . . . . . . . . . . . 401
28.2.2 Linear Hartmann Plate. . . . . . . . . . . . . . . . .403
28.2.3 Rectilinear Hole Array. . . . . . . . . . . . . . . . 407
28.3 Measuring TRA using a Hartmann Plate . . . . . . . . . .408
28.4 The Shack-Hartmann Approach . . . . . . . . . . . . . . 411
28.5 Scanning Hartmann Sensor (SHAPE) . . . . . . . . . . . .414
28.6 Homework . . . . . . . . . . . . . . . . . . . . . . . .416
Chapter 29 Axial Intensity . . . . . . . . . . . . . . . . . . . 419
29.1 Introduction . . . . . . . . . . . . . . . . . . . . . .419
29.2 AIP for Unaberrated System . . . . . . . . . . . . . . .419
29.3 Diffractive Depth of Focus . . . . . . . . . . . . . . .424
29.4 AIP of a Slightly Spherically Aberrated System . . . . .425
29.5 AIP for Significant Spherical Aberration . . . . . . . .425
29.6 Using AIP to Measure Spherical Aberration . . . . . . . 426
29.7 Things That Can Affect AIP Symmetry . . . . . . . . . . 426
29.7.1 Sampling Aperture. . . . . . . . . . . . . . . . . . .426
29.7.2 F-number . . . . . . . . . . . . . . . . . . . . . . .427
29.7.3 Gaussian Beams. . . . . . . . . . . . . . . . . . . . 429
29.8 Homework . . . . . . . . . . . . . . . . . . . . . . . .430
Chapter 30 Modulation Transfer Function I . . . . . . . . . . . .431
30.1 Introduction . . . . . . . . . . . . . . . . . . . . . .431
30.2 What is Modulation? . . . . . . . . . . . . . . . . . . 431
30.3 Imaging and Convolution . . . . . . . . . . . . . . . . 433
30.4 Convolution and Fourier Transform . . . . . . . . . . . 434
30.4.1 The Delta Function . . . . . . . . . . . . . . . . . .434
30.5 The Optical Transfer Function. . . . . . . . . . . . . .436
30.6 Homework . . . . . . . . . . . . . . . . . . . . . . . .439
Chapter 31 MTF Measurement I . . . . . . . . . . . . . . . . . . 441
31.1 Introduction . . . . . . . . . . . . . . . . . . . . . .441
31.2 Slit Scanning of Finite-Area Sinusoidal Images . . . . .441
31.3 Slit and Knife-Edge Images . . . . . . . . . . . . . . .444
31.3.1 Image of a Slit . . . . . . . . . . . . . . . . . . . 444
31.3.2 Image of Knife-Edge . . . . . . . . . . . . . . . . . 446
31.4 MTF via LSF . . . . . . . . . . . . . . . . . . . . . . 448
31.5 Acquiring LSF Data. . . . . . . . . . . . . . . . . . . 449
31.6 Square Bar MTF (MTFS) . . . . . . . . . . . . . . . . . 449
31.6.1 Alternate Scan Implementation . . . . . . . . . . . . 453
31.7 Homework . . . . . . . . . . . . . . . . . . . . . . . .454
Chapter 32 MTF Measurement II. . . . . . . . . . . . . . . . . . 455
32.1 Introduction . . . . . . . . . . . . . . . . . . . . . .455
32.2 Variable Lateral Shear Interferometry . . . . . . . . . 455
32.3 Total Power within Pupil Overlap . . . . . . . . . . . .456
32.4 Auto-Correlation . . . . . . . . . . . . . . . . . . . .457
32.4.1 Basic Auto-Correlation . . . . . . . . . . . . . . . .457
32.4.2 Auto-Correlation of Circular Pupil . . . . . . . . . .458
32.5 Pupil Auto-Correlation (PAC) . . . . . . . . . . . . . .459
32.6 Connecting Pupil Auto-Correlation with Total Power Φ . .460
32.7 Pupil Auto-Correlation Interferometer (PACI) . . . . . .462
32.7.1 Kelsall Interferometer . . . . . . . . . . . . . . . .462
32.8 Pupil Auto-Correlation and the Optical Transfer Function463
32.8.1 Short Derivation . . . . . . . . . . . . . . . . . . .463
32.8.2 Example Optical Transfer Function on Circular Pupil . 465
32.8.3 Reinterpretation of Pupil Auto-Correlation. . . . . . 466
32.8.4 The Modulation Transfer Function Connection . . . . . 467
32.9 Homework . . . . . . . . . . . . . . . . . . . . . . . .467
Chapter 33 Surface Roughness . . . . . . . . . . . . . . . . . . 469
33.1 Introduction . . . . . . . . . . . . . . . . . . . . . .469
33.2 Direct Measurement of Surface Roughness. . . . . . . . .469
33.2.1 Mechanical Profilometer . . . . . . . . . . . . . . . 469
33.2.2 Optical Profilometry . . . . . . . . . . . . . . . . .470
33.3 Analysis of Profilometer Data . . . . . . . . . . . . . 472
33.3.1 Peak-to-Valley and Average . . . . . . . . . . . . . .472
33.3.2 RMS Roughness . . . . . . . . . . . . . . . . . . . . 474
33.3.3 Histogram and Gaussian Fits. . . . . . . . . . . . . .474
33.4 Surface Periodicity. . . . . . . . . . . . . . . . . . .477
33.5 Mechanical versus Optical Profilometry . . . . . . . . .481
33.6 Measuring Scattered Light. . . . . . . . . . . . . . . .482
33.6.1 Total Integrated Scatter . . . . . . . . . . . . . . .482
33.6.2 Angle Resolved Scatter (ARS) . . . . . . . . . . . . .484
33.7 Homework . . . . . . . . . . . . . . . . . . . . . . . .487
Part VI Measurement of Light Fields . . . . . . . . . . . . . . .489
Chapter 34 Measuring Light Fields I . . . . . . . . . . . . . . .491
34.1 Introduction . . . . . . . . . . . . . . . . . . . . . .491
34.2 Wavefront Sensors . . . . . . . . . . . . . . . . . . . 491
34.2.1 Direct WFS . . . . . . . . . . . . . . . . . . . . . .491
34.2.2 Indirect Wavefront Sensors . . . . . . . . . . . . . .492
34.2.2.1 Lateral Shear Interferometer WFS . . . . . . . . . .492
34.2.2.2 Hartmann-Based WFS. . . . . . . . . . . . . . . . . 494
34.2.3 Indirect WFS (Radiometric) . . . . . . . . . . . . . .494
34.2.3.1 Axial Intensity . . . . . . . . . . . . . . . . 496
34.2.3.2 Curvature Sensing . . . . . . . . . . . . . . . 497
34.2.3.3 Phase Retrieval . . . . . . . . . . . . . . . . 501
34.2.3.4 Light Level . . . . . . . . . . . . . . . . . . 503
34.3 Wavefront Calibration . . . . . . . . . . . . . . . . . 505
Chapter 35 Measuring Light Fields II: Polarization . . . . . . . 507
35.1 Introduction . . . . . . . . . . . . . . . . . . . . . .507
35.2 Measuring Pure Polarization States . . . . . . . . . . .508
35.3 The Polarization Ellipse. . . . . . . . . . . . . . . . 512
35.4 Ellipsometers Proper . . . . . . . . . . . . . . . . . .513
35.5 The Quarter-Wave Plate . . . . . . . . . . . . . . . . .515
35.6 Polarization and Wavefront Sensors . . . . . . . . . . .516
35.7 Coherence Measurements . . . . . . . . . . . . . . . . .521
35.7.1 Spatial Coherence . . . . . . . . . . . . . . . . . . 521
35.7.2 Temporal Coherence . . . . . . . . . . . . . . . . . .522
35.8 Homework . . . . . . . . . . . . . . . . . . . . . . . .525
Chapter 36 Measuring Light Fields III: Radiometrics . . . . . . .527
36.1 Introduction . . . . . . . . . . . . . . . . . . . . . .527
36.2 The Standard Lamp . . . . . . . . . . . . . . . . . . . 527
36.3 Radiometer. . . . . . . . . . . . . . . . . . . . . . . 528
36.3.1 Broadband Radiometry . . . . . . . . . . . . . . . . .529
36.3.2 Sensor Head Calibration . . . . . . . . . . . . . . . 530
36.3.3 Photometry. . . . . . . . . . . . . . . . . . . . . . 531
36.4 Gratings and Monochrometers. . . . . . . . . . . . . . .533
36.4.1 Reflection Gratings . . . . . . . . . . . . . . . . . 533
36.4.2 Blazed Grating. . . . . . . . . . . . . . . . . . . . 534
36.4.3 Monochrometers . . . . . . . . . . . . . . . . . . . .537
36.5 Spectroradiometers . . . . . . . . . . . . . . . . . . .538
36.5.1 Basic System . . . . . . . . . . . . . . . . . . . . .538
36.5.2 Order Overlap . . . . . . . . . . . . . . . . . . . . 539
36.5.3 Spectroradiometers Proper . . . . . . . . . . . . . . 540
36.5.4 Spectroradiometer Calibration. . . . . . . . . . . . .541
36.6 Energy Measurements . . . . . . . . . . . . . . . . . . 543
36.7 Radiance Measurements . . . . . . . . . . . . . . . . . 543
Chapter 37 Spectrometry . . . . . . . . . . . . . . . . . . . . .547
37.1 Introduction . . . . . . . . . . . . . . . . . . . . . .547
37.2 Spectrometer . . . . . . . . . . . . . . . . . . . . . .547
37.3 Fabry-Perot Interferometer . . . . . . . . . . . . . . .548
37.3.1 Multiple Ray Interference Transmission . . . . . . . .549
37.3.2 Fringe Sharpness . . . . . . . . . . . . . . . . . . .550
37.3.3 Spectral Resolution . . . . . . . . . . . . . . . . . 554
37.3.4 Interference (Spike) Filter . . . . . . . . . . . . . 556
37.4 Spectrophotometry. . . . . . . . . . . . . . . . . . . .557
37.4.1 Visible Spectrophotometer . . . . . . . . . . . . . . 558
37.4.2 Fourier Transform Infrared Spectrophotometer (FTIR) . 560
Chapter 38 Energy and Photographic Film in the Digital Age . . . 567
38.1 Introduction . . . . . . . . . . . . . . . . . . . . . .567
38.2 Generating the Characteristic Curve . . . . . . . . . . 568
38.2.1 Sensitometer . . . . . . . . . . . . . . . . . . . . .568
38.2.2 Densitometer . . . . . . . . . . . . . . . . . . . . .569
38.2.3 The Plot . . . . . . . . . . . . . . . . . . . . . . .570
38.3 The Nutting Model. . . . . . . . . . . . . . . . . . . .570
38.4 The Microdensitometer . . . . . . . . . . . . . . . . . 572
38.5 Film Speed and RMS Granularity . . . . . . . . . . . . .574
38.6 Film MTF. . . . . . . . . . . . . . . . . . . . . . . . 574
38.7 Film Spectral Response . . . . . . . . . . . . . . . . .575
38.8 IR Presensitization Photography . . . . . . . . . . . . 576
38.9 Format Sizes. . . . . . . . . . . . . . . . . . . . . . 578
Appendix A Answers To Selected Homework Problems . . . . . . . . 579
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 583
Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585
