- •Lens Design Fundamentals
- •Contents
- •Preface to the Second Edition
- •Preface to the First Edition
- •A Special Tribute to Rudolf Kingslake
- •1.1. Relations Between Designer and Factory
- •1.1.1 Spherical versus Aspheric Surfaces
- •1.1.2 Establishment of Thicknesses
- •1.1.3 Antireflection Coatings
- •1.1.4 Cementing
- •1.1.5 Establishing Tolerances
- •1.1.6 Design Tradeoffs
- •1.2. The Design Procedure
- •1.2.1 Sources of a Likely Starting System
- •1.2.2 Lens Evaluation
- •1.2.3 Lens Appraisal
- •1.2.4 System Changes
- •1.3. Optical Materials
- •1.3.1 Optical Glass
- •1.3.2 Infrared Materials
- •1.3.3 Ultraviolet Materials
- •1.3.4 Optical Plastics
- •1.4. Interpolation of Refractive Indices
- •1.4.1 Interpolation of Dispersion Values
- •1.4.2 Temperature Coefficient of Refractive Index
- •1.5. Lens Types to be Considered
- •2.1. Introduction
- •2.1.1 Object and Image
- •2.1.2 The Law of Refraction
- •2.1.3 The Meridional Plane
- •2.1.4 Types of Rays
- •2.1.5 Notation and Sign Conventions
- •2.2. Graphical Ray Tracing
- •2.3. Trigonometrical Ray Tracing at a Spherical Surface
- •2.3.1 Program for a Computer
- •2.4. Some Useful Relations
- •2.4.1 The Spherometer Formula
- •2.4.2 Some Useful Formulas
- •2.4.3 The Intersection Height of Two Spheres
- •2.4.4 The Volume of a Lens
- •2.5. Cemented Doublet Objective
- •2.6. Ray Tracing at a Tilted Surface
- •2.6.1 The Ray Tracing Equations
- •2.6.2 Example of Ray Tracing through a Tilted Surface
- •2.7. Ray Tracing at an Aspheric Surface
- •3.1. Tracing a Paraxial Ray
- •3.1.1 The Standard Paraxial Ray Trace
- •3.1.2 The (y – nu) Method
- •3.1.3 Inverse Procedure
- •3.1.4 Angle Solve and Height Solve Methods
- •3.1.6 Paraxial Ray with All Angles
- •3.1.7 A Paraxial Ray at an Aspheric Surface
- •3.1.9 Matrix Approach to Paraxial Rays
- •3.2. Magnification and the Lagrange Theorem
- •3.2.1 Transverse Magnification
- •3.2.2 Longitudinal Magnification
- •3.3. The Gaussian Optics of a Lens System
- •3.3.1 The Relation between the Principal Planes
- •3.3.2 The Relation between the Two Focal Lengths
- •3.3.3 Lens Power
- •3.3.4 Calculation of Focal Length
- •3.3.5 Conjugate Distance Relationships
- •3.3.6 Nodal Points
- •3.3.7 Optical Center of Lens
- •3.3.8 The Scheimpflug Condition
- •3.4. First-Order Layout of an Optical System
- •3.4.1 A Single Thick Lens
- •3.4.2 A Single Thin Lens
- •3.4.3 A Monocentric Lens
- •3.4.4 Image Shift Caused by a Parallel Plate
- •3.4.5 Lens Bending
- •3.4.6 A Series of Separated Thin Elements
- •3.4.7 Insertion of Thicknesses
- •3.4.8 Two-Lens Systems
- •3.5. Thin-Lens Layout of Zoom Systems
- •3.5.1 Mechanically Compensated Zoom Lenses
- •3.5.2 A Three-Lens Zoom
- •3.5.4 A Four-Lens Optically Compensated Zoom System
- •3.5.5 An Optically Compensated Zoom Enlarger or Printer
- •Endnotes
- •4.1. Introduction
- •4.2. Symmetrical Optical Systems
- •4.3. Aberration Determination Using Ray Trace Data
- •4.3.1 Defocus
- •4.3.2 Spherical Aberration
- •4.3.3 Tangential and Sagittal Astigmatism
- •4.3.4 Tangential and Sagittal Coma
- •4.3.5 Distortion
- •4.3.6 Selection of Rays for Aberration Computation
- •4.3.7 Zonal Aberrations
- •4.3.8 Tangential and Sagittal Zonal Astigmatism
- •4.3.9 Tangential and Sagittal Zonal Coma
- •4.3.10 Higher-Order Contributions
- •4.4. Calculation of Seidel Aberration Coefficients
- •Endnotes
- •5.1. Introduction
- •5.2. Spherochromatism of a Cemented Doublet
- •5.2.4 Secondary Spectrum
- •5.2.5 Spherochromatism
- •5.3. Contribution of a Single Surface to the Primary Chromatic Aberration
- •5.4. Contribution of a Thin Element in a System to the Paraxial Chromatic Aberration
- •5.5. Paraxial Secondary Spectrum
- •5.7.1 Secondary Spectrum of a Dialyte
- •5.7.2 A One-Glass Achromat
- •5.8. Chromatic Aberration Tolerances
- •5.8.1 A Single Lens
- •5.8.2 An Achromat
- •5.9. Chromatic Aberration at Finite Aperture
- •5.9.1 Conrady’s D – d Method of Achromatization
- •5.9.3 Tolerance for the D – d Sum
- •5.9.5 Paraxial D – d for a Thin Element
- •Endnotes
- •6.1. Surface Contribution Formulas
- •6.1.1 The Three Cases of Zero Aberration at a Surface
- •6.1.2 An Aplanatic Single Element
- •6.1.3 Effect of Object Distance on the Spherical Aberration Arising at a Surface
- •6.1.4 Effect of Lens Bending
- •6.1.6 A Two-Lens Minimum Aberration System
- •6.1.7 A Four-Lens Monochromat Objective
- •6.2. Zonal Spherical Aberration
- •6.3. Primary Spherical Aberration
- •6.3.1 At a Single Surface
- •6.3.2 Primary Spherical Aberration of a Thin Lens
- •6.4. The Image Displacement Caused by a Planoparallel Plate
- •6.5. Spherical Aberration Tolerances
- •6.5.1 Primary Aberration
- •6.5.2 Zonal Aberration
- •Endnotes
- •7.1. The Four-Ray Method
- •7.2. A Thin-Lens Predesign
- •7.2.1 Insertion of Thickness
- •7.2.2 Flint-in-Front Solutions
- •7.3. Correction of Zonal Spherical Aberration
- •7.4. Design Of an Apochromatic Objective
- •7.4.1 A Cemented Doublet
- •7.4.2 A Triplet Apochromat
- •7.4.3 Apochromatic Objective with an Air Lens
- •Endnotes
- •8.1. Passage of an Oblique Beam through a Spherical Surface
- •8.1.1 Coma and Astigmatism
- •8.1.2 Principal Ray, Stops, and Pupils
- •8.1.3 Vignetting
- •8.2. Tracing Oblique Meridional Rays
- •8.2.1 The Meridional Ray Plot
- •8.3. Tracing a Skew Ray
- •8.3.1 Transfer Formulas
- •8.3.2 The Angles of Incidence
- •8.3.3 Refraction Equations
- •8.3.4 Transfer to the Next Surface
- •8.3.5 Opening Equations
- •8.3.6 Closing Equations
- •8.3.7 Diapoint Location
- •8.3.8 Example of a Skew-Ray Trace
- •8.4. Graphical Representation of Skew-Ray Aberrations
- •8.4.1 The Sagittal Ray Plot
- •8.4.2 A Spot Diagram
- •8.4.3 Encircled Energy Plot
- •8.4.4 Modulation Transfer Function
- •8.5. Ray Distribution from a Single Zone of a Lens
- •Endnotes
- •9.1. The Optical Sine Theorem
- •9.2. The Abbe Sine Condition
- •9.2.1 Coma for the Three Cases of Zero Spherical Aberration
- •9.3. Offense Against the Sine Condition
- •9.3.1 Solution for Stop Position for a Given OSC
- •9.3.2 Surface Contribution to the OSC
- •9.3.3 Orders of Coma
- •9.3.4 The Coma G Sum
- •9.3.5 Spherical Aberration and OSC
- •9.4. Illustration of Comatic Error
- •Endnotes
- •10.1. Broken-Contact Type
- •10.2. Parallel Air-Space Type
- •10.3. An Aplanatic Cemented Doublet
- •10.4. A Triple Cemented Aplanat
- •10.5. An Aplanat with A Buried Achromatizing Surface
- •10.6. The Matching Principle
- •Endnotes
- •11.1. Astigmatism and the Coddington Equations
- •11.1.1 The Tangential Image
- •11.1.2 The Sagittal Image
- •11.1.3 Astigmatic Calculation
- •11.1.5 Astigmatism for the Three Cases of Zero Spherical Aberration
- •11.1.6 Astigmatism at a Tilted Surface
- •11.2. The Petzval Theorem
- •11.2.1 Relation Between the Petzval Sum and Astigmatism
- •11.2.2 Methods for Reducing the Petzval Sum
- •11.3. Illustration of Astigmatic Error
- •11.4. Distortion
- •11.4.1 Measuring Distortion
- •11.4.2 Distortion Contribution Formulas
- •11.4.3 Distortion When the Image Surface Is Curved
- •11.5. Lateral Color
- •11.5.1 Primary Lateral Color
- •11.6. The Symmetrical Principle
- •11.7. Computation of the Seidel Aberrations
- •11.7.1 Surface Contributions
- •11.7.2 Thin-Lens Contributions
- •11.7.3 Aspheric Surface Corrections
- •11.7.4 A Thin Lens in the Plane of an Image
- •Endnotes
- •12.1.1 Distortion
- •12.1.2 Tangential Field Curvature
- •12.1.3 Coma
- •12.1.4 Spherical Aberration
- •12.2. Simple Landscape Lenses
- •12.2.1 Simple Rear Landscape Lenses
- •12.2.2 A Simple Front Landscape Lens
- •12.3. A Periscopic Lens
- •12.4. Achromatic Landscape Lenses
- •12.4.1 The Chevalier Type
- •12.4.2 The Grubb Type
- •12.5. Achromatic Double Lenses
- •12.5.1 The Rapid Rectilinear
- •12.5.3 Long Telescopic Relay Lenses
- •12.5.4 The Ross “Concentric” Lens
- •Endnotes
- •13.1. The Design of a Dagor Lens
- •13.2. The Design of an Air-Spaced Dialyte Lens
- •13.4. Double-Gauss Lens with Cemented Triplets
- •13.5. Double-Gauss Lens with Air-spaced Negative Doublets
- •Endnotes
- •14.1. The Petzval Portrait Lens
- •14.1.1 The Petzval Design
- •14.1.2 The Dallmeyer Design
- •14.2. The Design of a Telephoto Lens
- •14.3. Lenses to Change Magnification
- •14.3.1 Barlow Lens
- •14.3.2 Bravais Lens
- •14.4. The Protar Lens
- •14.5. Design of a Tessar Lens
- •14.5.1 Choice of Glass
- •14.5.2 Available Degrees of Freedom
- •14.5.3 Chromatic Correction
- •14.5.4 Spherical Correction
- •14.5.5 Correction of Coma and Field
- •14.5.6 Final Steps
- •14.6. The Cooke Triplet Lens
- •14.6.2 The Thin-Lens Predesign of the Bendings
- •14.6.3 Calculation of Real Aberrations
- •14.6.4 Triplet Lens Improvements
- •Endnotes
- •15.1. Comparison of Mirrors and Lenses
- •15.2. Ray Tracing a Mirror System
- •15.3. Single-Mirror Systems
- •15.3.1 A Spherical Mirror
- •15.3.2 A Parabolic Mirror
- •15.3.3 An Elliptical Mirror
- •15.3.4 A Hyperbolic Mirror
- •15.4. Single-Mirror Catadioptric Systems
- •15.4.1 A Flat-Field Ross Corrector
- •15.4.2 An Aplanatic Parabola Corrector
- •15.4.3 The Mangin Mirror
- •15.4.4 The Bouwers–Maksutov System
- •15.4.5 The Gabor Lens
- •15.4.6 The Schmidt Camera
- •15.4.7 Variable Focal-Range Infrared Telescope
- •15.4.8 Broad-Spectrum Afocal Catadioptric Telescope
- •15.4.9 Self-Corrected Unit-Magnification Systems
- •15.5. Two-Mirror Systems
- •15.5.1 Two-Mirror Systems with Aspheric Surfaces
- •15.5.2 A Maksutov Cassegrain System
- •15.5.3 A Schwarzschild Microscope Objective
- •15.5.4 Three-Mirror System
- •15.6. Multiple-Mirror Zoom Systems
- •15.6.2 All-Reflective Zoom Optical Systems
- •15.7. Summary
- •Endnotes
- •16.1. Design of a Military-Type Eyepiece
- •16.1.1 The Objective Lens
- •16.1.2 Eyepiece Layout
- •16.2. Design of an Erfle Eyepiece
- •16.3. Design of a Galilean Viewfinder
- •Endnotes
- •17.1. Finding a Lens Design Solution
- •17.1.1 The Case of as Many Aberrations as There Are Degrees of Freedom
- •17.1.2 The Case of More Aberrations Than Free Variables
- •17.1.3 What Is an Aberration?
- •17.1.4 Solution of the Equations
- •17.2. Optimization Principles
- •17.3. Weights and Balancing Aberrations
- •17.4. Control of Boundary Conditions
- •17.5. Tolerances
- •17.6. Program Limitations
- •17.7. Lens Design Computing Development
- •17.8. Programs and Books Useful for Automatic Lens Design
- •17.8.1 Automatic Lens Design Programs
- •17.8.2 Lens Design Books
- •Endnotes
- •Index
306 |
The Oblique Aberrations |
11.3 ILLUSTRATION OF ASTIGMATIC ERROR
As has been observed, our dutiful cemented doublet (Section 2.5) suffers from astigmatism, as is evident by examination of the ray fans plots in Figure 11.13, the focusing ray bundles in Figure 11.14a, and the field curves shown in Figure 11.14b. Also, both the tangential and sagittal fields are inward curving, and the maximum zonal spherical aberration is less than 10% of the peak astigmatic error at 5 . By comparing the 3.5 plot with the 5 plot, we can see that the aberration plots are linear with r and the ratio of the errors between these plots is about 2:1.
From our study, we recognize that the aberration is primary linear astigmatism since ðH 05 =H03:5 Þ2 2 and the linear behavior with r. Consequently, only s3 and s4 of the field-dependent aberration coefficients have significant values. Recalling Eqs. (4-6) and (4-7), it is easy to compute that s3 0:79s4, which means the Petzval curve is also inward curving but lies between the sagittal curve and the image plane. Computing the transverse astigmatism using real rays was discussed in Section 4.3.3. The tangential component is given by
TASTðr; HÞ ¼ Yðr; 0 ; HÞ Yðr; 180 ; HÞ 2Yðr; 0 ; 0Þ ¼ 0:053159
and the sagittal component by
SASTðr; HÞ ¼ 2½Xðr; 90 ; HÞ Yðr; 0 ; 0Þ& ¼ 0:025650
which compare favorably to Figure 11.14. Coma is insignificant.
To observe what degradation in image formation the astigmatism in our lens will cause, we can generate a simulated image of a photograph using an analysis feature available in some lens design programs. Figure 11.15a shows the original and Figure 11.15b the resultant image (see page 311). The quadratic growth of the blur as a function of field angle is demonstratively illustrated. Compare this image with the linear blur growth due to coma shown in Figure 9.9b. Notice that the fine detail is observable over a larger central area than coma as a consequence of the quadratic growth of the blur. However, the image degradation at the top/bottom center and left/right sides is similar since the blur sizes are roughly the same although the shapes differ. The blurring in the corners is worse for Figure 11.15b than for Figure 9.9b.
11.4 DISTORTION
Distortion is a peculiar aberration in that it does not cause any loss of definition but merely a radial displacement of an image point toward or away from the lens axis. Distortion is calculated by determining the height Hpr0 at which the
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Figure 11.13 Monochromatic ray fans for Section 2.5 cemented doublet.
308 |
The Oblique Aberrations |
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–0.20 |
0.00 |
0.20 |
(b)
Figure 11.14 (a) Monochromatic focusing behavior for Section 2.5 cemented doublet. (b) Field curves for the tangential and sagittal foci.
11.4 Distortion |
309 |
(a) |
(b) |
Figure 11.15 (a) Original photograph. (b) Image formed by Section 2.5 cemented doublet showing the effect of astigmatism.
principal ray intersects the image plane, and comparing that height with the ideal Lagrangian or Gaussian image height calculated by paraxial formulas. Thus
distortion ¼ H0pr h0
where h0 for a distant object is given by ( f tan Upr), or for a near object by (Hm), where m is the image magnification.
As discussed in Section 4.3.5, distortion is aperture-independent coma and can be resolved into a series of powers of H0, namely,
distortion ¼ s5H 03 þ m12H 05 þ t20H 07 þ . . . |
(11-8) |
However, very few lenses exhibit much distortion beyond the first cubic term. Because of the cubic law, distortion increases rapidly once it begins to appear, and this makes the corners of the image of a square, for example, stretch out for positive (pincushion) distortion, or pull in with negative (barrel) distortion.
The magnitude of distortion is generally expressed as a percentage of the image height, at the corners of a picture. Figure 11.16 shows two typical cases of moderate amounts of pincushion distortion, namely, 4% and 10%, respectively. The diagrams represent images that should be 50 mm squares, the quantity d beneath each figure being the lateral displacement of the midpoints of the