### Spie Press Book

Optical Imaging and Aberrations, Part III: Wavefront AnalysisFormat | Member Price | Non-Member Price |
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*Wavefront Analysis*is Part III of a series of books on

*Optical Imaging and Aberrations*. It has evolved out of the author's work and lectures over the years on wavefront analysis as applied to optical design and testing. Its focus is on the use of orthonormal polynomials that represent balanced classical aberrations in optical imaging systems with pupils of various shapes. After a brief introduction to optical imaging, aberrations, and orthonormalization of a set of polynomials over a certain domain to obtain polynomials that are orthonormal over another domain, this book describes in detail the polynomials appropriate for various shapes of the system pupil. Starting with the system that is most common in imaging, namely, the one with a circular pupil, systems with annular, hexagonal, elliptical, rectangular, square, and slit pupils are considered. Included in this list are also systems with circular and annular pupils with Gaussian illumination, anamorphic systems with square and circular pupils, and those with circular and annular sector pupils. These chapters start with a brief discussion of aberration-free imaging that includes both the PSF and the OTF of a system. A separate chapter is devoted to a discussion of the pitfalls of using the Zernike circle polynomials for systems with noncircular pupils by applying them to systems with annular and hexagonal pupils. Similarly, a chapter is devoted to the calculation of orthonormal aberration coefficients from the wavefront or the wavefront slope data. Each chapter ends with a brief summary that describes the essence of its content.

**Preface****Acknowledgments****Symbols and Notation****1. OPTICAL IMAGING**- 1.1 Introduction
- 1.2 Diffraction Image
- 1.2.1 Pupil Function
- 1.2.2 PSF
- 1.2.3 OTF
- 1.3 Strehl Ratio
- 1.3.1 General Expression
- 1.3.2 Approximate Expression in Terms of Aberration Variance
- 1.4 Aberration Balancing
- 1.5 Summary
- References
**2. OPTICAL WAVEFRONTS AND THEIR ABERRATIONS**- 2.1 Introduction
- 2.2 Optical Imaging
- 2.3 Wave and Ray Aberrations
- 2.4 Defocus Aberration
- 2.5 Wavefront Tilt
- 2.6 Aberration Function of a Rotationally Symmetric System
- 2.7 Observation of Aberrations: Interferograms
- 2.8 Summary
- References
**3. ORTHONORMAL POLYNOMIALS AND GRAM-SCHMIDT ORTHONORMALIZATION**- 3.1 Introduction
- 3.2 Orthonormal Polynomials
- 3.3 Equivalence of Orthogonality-Based Coefficients and Least-Squares Fitting
- 3.4 Orthonormalization of Zernike Circle Polynomials over Noncircular Pupils
- 3.5 Unit Pupil
- 3.6 Summary
- References
**4. SYSTEMS WITH CIRCULAR PUPILS**- 4.1 Introduction
- 4.2 Pupil Function
- 4.3 Aberration-Free Imaging
- 4.3.1 PSF
- 4.3.2 OTF
- 4.4 Strehl Ratio and Aberration Tolerance
- 4.4.1 Strehl Ratio
- 4.4.2 Defocus Strehl Ratio
- 4.4.3 Approximate Expressions for Strehl Ratio
- 4.5 Balanced Aberrations
- 4.6 Description of Zernike Circle Polynomials
- 4.6.1 Analytical Form
- 4.6.2 Circle Polynomials in Polar Coordinates
- 4.6.3 Polynomial Ordering
- 4.6.4 Number of Circle Polynomials through a Certain Order
*n* - 4.6.5 Relationships among the Indices
*n*,*m*, and*j* - 4.6.6 Uniqueness of Circle Polynomials
- 4.6.7 Circle Polynomials in Cartesian Coordinates
- 4.7 Zernike Circle Coefficients of a Circular Aberration Function
- 4.8 Symmetry Properties of Images Aberrated by a Circle Polynomial Aberration
- 4.8.1 Symmetry of PSF
- 4.8.2 Symmetry of OTF
- 4.9 Isometric, Interferometric, and Imaging Characteristics of Circle Polynomial Aberrations
- 4.9.1 Isometric Characteristics
- 4.9.2 Interferometric Characteristics
- 4.9.3 PSF Characteristics
- 4.9.4 OTF Characteristics
- 4.10 Circle Polynomials and Their Relationships with Classical Aberrations
- 4.11 Zernike Coefficients of a Scaled Pupil
- 4.11.1 Theory
- 4.11.2 Application to a Seidel Aberration Function
- 4.11.3 Numerical Example
- 4.12 Summary
- References
**5. SYSTEMS WITH ANNULAR PUPILS**- 5.1 Introduction
- 5.2 Aberration-Free Imaging
- 5.2.1 PSF
- 5.2.2 OTF
- 5.3 Strehl Ratio and Aberration Balancing
- 5.4 Orthonormalization of Circle Polynomials over an Annulus
- 5.5 Annular Polynomials
- 5.6 Annular Coefficients of an Annular Aberration Function
- 5.7 Strehl Ratio for Annular Polynomial Aberrations
- 5.8 Isometric, Interferometric, and Imaging Characteristics of Annular Polynomial Aberrations
- 5.9 Summary
- References
**6. SYSTEMS WITH GAUSSIAN PUPILS**- 6.1 Introduction
- 6.2 Gaussian Pupil
- 6.3 Aberration-Free Imaging
- 6.3.1 PSF
- 6.3.2 Optimum Gaussian Radius
- 6.3.3 OTF
- 6.4 Strehl Ratio and Aberration Balancing
- 6.5 Orthonormalization of Zernike Circle Polynomials over a Gaussian Circular Pupil
- 6.6 Gaussian Circle Polynomials Representing Balanced Primary Aberrations for a Gaussian Circular Pupil
- 6.7 Weakly Truncated Gaussian Pupils
- 6.8 Aberration Coefficients of a Gaussian Circular Aberration Function
- 6.9 Orthonormalization of Annular Polynomials over a Gaussian Annular Pupil
- 6.10 Gaussian Annular Polynomials Representing Balanced Primary Aberrations for a Gaussian Annular Pupil
- 6.11 Aberration Coefficients of a Gaussian Annular Aberration Function
- 6.12 Summary
- References
**7. SYSTEMS WITH HEXAGONAL PUPILS**- 7.1 Introduction
- 7.2 Pupil Function
- 7.3 Aberration-Free Imaging
- 7.3.1 PSF
- 7.3.2 OTF
- 7.4 Hexagonal Polynomials
- 7.5 Hexagonal Coefficients of a Hexagonal Aberration Function
- 7.6 Isometric, Interferometric, and Imaging Characteristics of Hexagonal Polynomial Aberrations
- 7.7 Seidel Aberrations, Standard Deviation, and Strehl Ratio
- 7.7.1 Defocus
- 7.7.2 Astigmatism
- 7.7.3 Coma
- 7.7.4 Spherical Aberration
- 7.7.5 Strehl Ratio
- 7.8 Summary
- References
**8. SYSTEMS WITH ELLIPTICAL PUPILS**- 8.1 Introduction
- 8.2 Pupil Function
- 8.3 Aberration-Free Imaging
- 8.3.1 PSF
- 8.3.2 OTF
- 8.4 Elliptical Polynomials
- 8.5 Elliptical Coefficients of an Elliptical Aberration Function
- 8.6 Isometric, Interferometric, and Imaging Characteristics of Elliptical Polynomial Aberrations
- 8.7 Seidel Aberrations and Their Standard Deviations
- 8.7.1 Defocus
- 8.7.2 Astigmatism
- 8.7.3 Coma
- 8.7.4 Spherical Aberration
- 8.8 Summary
- References
**9. SYSTEMS WITH RECTANGULAR PUPILS**- 9.1 Introduction
- 9.2 Pupil Function
- 9.3 Aberration-Free Imaging
- 9.3.1 PSF
- 9.3.2 OTF
- 9.4 Rectangular Polynomials
- 9.5 Rectangular Coefficients of a Rectangular Aberration Function
- 9.6 Isometric, Interferometric, and Imaging Characteristics of Rectangular Polynomial Aberrations
- 9.7 Seidel Aberrations and Their Standard Deviations
- 9.7.1 Defocus
- 9.7.2 Astigmatism
- 9.7.3 Coma
- 9.7.4 Spherical Aberration
- 9.8 Summary
- References
**10. SYSTEMS WITH SQUARE PUPILS**- 10.1 Introduction
- 10.2 Pupil Function
- 10.3 Aberration-Free Imaging
- 10.3.1 PSF
- 10.3.2 OTF
- 10.4 Square Polynomials
- 10.5 Square Coefficients of a Square Aberration Function
- 10.6 Isometric, Interferometric, and Imaging Characteristics of Square Polynomial Aberrations
- 10.7 Seidel Aberrations and Their Standard Deviations
- 10.7.1 Defocus
- 10.7.2 Astigmatism
- 10.7.3 Coma
- 10.7.4 Spherical Aberration
- 10.8 Summary
- References
**11. SYSTEMS WITH SLIT PUPILS**- 11.1 Introduction
- 11.2 Aberration-Free Imaging
- 11.2.1 PSF
- 11.2.2 Image of an Incoherent Slit
- 11.3 Strehl Ratio and Aberration Balancing
- 11.3.1 Strehl Ratio
- 11.3.2 Aberration Balancing
- 11.4 Slit Polynomials
- 11.5 Standard Deviation of a Primary Aberration
- 11.3 Summary
- References
**12. USE OF ZERNIKE CIRCLE POLYNOMIALS FOR NONCIRCULAR PUPILS**- 12.1 Introduction
- 12.2 Relationship Between the Orthonormal and the Corresponding Zernike Circle Coefficients
- 12.3 Use of Zernike Circle Polynomials for the Analysis of an Annular Wavefront
- 12.3.1 Zernike Circle Coefficients in Terms of the Annular Coefficients
- 12.3.2 Interferometer Setting Errors
- 12.3.3 Wavefront Fitting
- 12.3.4 Application to an Annular Seidel Aberration Function
- 12.4 Use of Zernike Circle Polynomials for the Analysis of a Hexagonal Wavefront
- 12.4.1 Zernike Circle Coefficients in Terms of Hexagonal Coefficients
- 12.4.2 Interferometer Setting Errors
- 12.4.3 Numerical Example
- 12.5 Aberration Coefficients from Discrete Wavefront Data
- 12.6 Summary
- References
**13. ANAMORPHIC SYSTEMS**- 13.1 Introduction
- 13.2 Gaussian Imaging
- 13.3 Classical Aberrations
- 13.4 Strehl Ratio and Aberration Balancing for a Rectangular Pupil
- 13.5 Aberration Polynomials Orthonormal over a Rectangular Pupil
- 13.6 Expansion of a Rectangular Aberration Function in Terms of Orthonormal Rectangular Polynomials
- 13.7 Anamorphic Imaging System with a Circular Pupil
- 13.7.1 Balanced Aberrations
- 13.7.2 Orthonormal Polynomials Representing Balanced Aberrations
- 13.8 Comparison of Polynomials for Rotationally Symmetric and Anamorphic Imaging Systems
- 13.9 Summary
- References
**14. NUMERICAL WAVEFRONT ANALYSIS**- 14.1 Introduction
- 14.2 Zernike Coefficients from Wavefront Data
- 14.2.1 Theory
- 14.2.2 Numerical Example
- 14.3 Zernike Coefficients from Wavefront Slope Data
- 14.3.1 Theory
- 14.3.2 Alternative Approach for Obtaining Zernike Coefficients from Wavefront Slope Data
- 14.3.3 Numerical Example
- 14.4 Summary
- References
**APPENDIX: SYSTEMS WITH SECTOR PUPILS**

### Preface

This book is Part III of a series of books on Optical Imaging and Aberrations. Part I on *Ray Geometrical Optics* and Part II on *Wave Diffraction Optics* were published earlier. Part III is on *Wavefront Analysis*, which is an integral part of optical design, fabrication, and testing. In optical design, rays are traced to determine the wavefront and thereby the quality of a design. In optical testing, the fabrication errors and, therefore, the associated aberrations are measured by way of interferometry. In both cases, the quality of the wavefront is determined from the aberrations obtained at an array of points. The aberrations thus obtained are used to calculate the mean, the peak-to-valley, and the standard deviation values. While such statistical measures of the wavefront are part of wavefront analysis, the purpose of this book is to determine the content of the wavefront by decomposing the ray-traced or test-measured data in terms of polynomials that are orthogonal over the expected domain of the data. These polynomials must include the basic aberrations of wavefront defocus and tilt, and represent balanced classical aberrations.

We start Part III with an outline of optical imaging in the presence of aberrations in Chapter 1, i.e., on how to obtain the point-spread and optical transfer functions of an imaging system with an arbitrary shaped pupil. The Strehl ratio of a system as a measure of image quality is introduced in this chapter, and shown to be dependent only on the aberration variance when the aberration is small. It is followed in Chapter 2 with a brief discussion of the wavefronts and aberrations. This chapter introduces the nomenclature of aberrations. How to obtain the orthogonal polynomials over a certain domain from those over another is discussed in Chapter 3. For systems with a circular pupil, the Zernike circle polynomials are well known for wavefront analysis. They are discussed at length in Chapter 4. These polynomials are orthogonalized over an annular pupil in Chapter 5, and over a Gaussian pupil in Chapter 6. They are obtained similarly for systems with hexagonal, elliptical, rectangular, square, and slit pupils in the succeeding chapters. For each pupil, the polynomials are given in their orthonormal form so that an expansion coefficient (with the exception of piston) represents the standard deviation of the corresponding polynomial aberration term. The standard deviation of a Seidel aberration with and without aberration balancing is also discussed in these chapters.

Since the Zernike circle polynomials form a complete set, a wavefront over any domain can be expanded in terms of them. However, the pitfalls of their use over a domain other than circular and resulting from the lack of their orthogonality over the chosen domain are discussed in Chapter 12. Finally, the aberrations of anamorphic systems are discussed, and polynomials suitable for their aberration analysis are given in Chapter 13 for both rectangular and circular pupils. The use of the orthonormal polyonomials for determining the content of a wavefront is demonstrated in Chapter 14 by computer simulations of circular wavefronts. The determination of the aberrations coefficients from the wavefront slope data, as in a Shack-Hartmann sensor, is also discussed in this chapter.

Virendra N. Mahajan

El Segundo, California

June 2013

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