Scalar Diffraction from a Circular Aperture

Scalar diffraction from a circular aperture is a ubiquitous problem that arises in a variety of disciplines, such as optics (lenses), acoustics (speakers), electromagnetics (dish antennas), and ultrasonics (piston transducers). The problem endures despite centuries of research because each new generation of researchers rediscovers it and adds some novel insight or new result to the existing literature. Scalar Diffraction from a Circular Aperture promises a few new results and several novel insights, particularly with regard to spatial averaging. Although the text emphasizes ultrasonic diffraction, the results and insights developed are general and may be applied to the many practical problems involving scalar diffraction from a circular aperture.
Included are novel insights on mirror-image diffraction, autoconvolution diffraction, and coherent and incoherent averaging. Examples from ultrasonic imaging, a coherent imaging modality, are used to develop a fairly general theory that connects over a century of research on scalar diffraction from a circular aperture. The material is based on a synthesis of mathematics, physical optics, linear systems theory, and scalar diffraction theory. Thus, engineers, scientists, mathematicians, and students working in optics, acoustics, antenna design, biomedical engineering, non-destructive testing, and astronomy will find Scalar Diffraction from a Circular Aperture interesting, provocative, and useful.

Inhalt

1. Introduction.- 1. Ultrasonic Reflection Imaging.- 2. Diffraction from a Circular Aperture.- 3. The Arccos & Lommel Diffraction Formulations.- 4. One-way and Two-way Diffraction.- 5. Spatial Averaging.- 6. The Need for Diffraction Correction.- 7. Mathematical Definitions.- 8. Scope and Assumptions.- 9. Preview.- 10. Criticism and Counter.- 2. Literature Review.- 1. Ultrasonic Reflection Imaging.- 2. Diffraction from a Circular Aperture.- 3. Spatially Averaged Diffraction Corrections.- 4. Short-Time Fourier Techniques.- 5. Short-Time Fourier Techniques in Ultrasound.- 6. Chapter Summary.- 3. Two Diffraction Formulations.- 1. The Lommel Diffraction Formulation.- 2. Discussion of the Lommel Diffraction Formulation.- 3. The Arccos Diffraction Formulation.- 4. Discussion of the Arccos Diffraction Formulation.- 5. Similarities and Differences.- 6. An Approximate Fourier Transform Pair.- 7. Verification.- 8. Computational Considerations.- 9. The Focused Case.- 10. Chapter Summary.- 4. Spatially Averaged one-way Diffraction.- 1. Spatially Averaged Arccos Diffraction Formulation.- 2. Analysis of Time-Domain Results.- 3. Spatially Averaged Lommel Diffraction Formulation.- 4. Analysis of Frequency-Domain Results.- 5. Extending Fourier Equivalence.- 6. Verification.- 7. Computational Considerations.- 8. Chapter Summary.- 5. Spatially Averaged two-way Diffraction.- 1. Spatially Averaged Arccos Diffraction Formulation.- 2. Spatially Averaged Lommel Diffraction Formulation.- 3. Analysis of Frequency-Domain Results.- 4. Extending Fourier Equivalence.- 5. Verification.- 6. Computational Considerations.- 7. Chapter Summary.- 6. Experimental Investigation.- 1. A Computational Consideration.- 2. Equipment and Processing.- 3. Experiments, Images, and Centroids.- 4. Discussion of Results.-5. Chapter Summary.- 7. Analytical Investigation.- 1. Diffraction and Linear Models.- 2. Harmonic Imaging and Non-Linear Ultrasound.- 3. Focused One-Way Results.- 4. Coherent vs. Incoherent Averaging.- 5. Mirror-Image vs. Autoconvolution Diffraction.- 6. Chapter Summary.- 8. Recommendations for Further Research.- 1. General.- 2. Fourier Equivalence.- 3. Spatially Averaged One-Way Diffraction.- 4. Spatially Averaged Autoconvolution Diffraction.- 5. More Experiments and Analysis.