Books

Henning Hoeber et al.

Classical and Modern Diffraction Theory

$Classical and Modern Diffraction Theory$ Society of Exploration Geophysics
ISBN: 978-1-56080-322-5

1. Edition: 2016
Soft Cover, 352 pages

Description

Providing geophysicists with an in-depth understanding of the theoretical and applied background for the seismic diffraction method, "Classical and Modern Diffraction Theory" covers the history and foundations of the classical theory and the key elements of the modern diffraction theory.

Chapters include an overview and a historical review of classical theory, a summary of the experimental results illustrating this theory, and key principles of the modern theory of diffraction; the early cornerstones of classical diffraction theory, starting from its inception in the 17th century and an extensive introduction to reprinted works of Grimaldi, Huygens, and Young; details of the classical theory of diffractions as developed in the 19th century and reprinted works of Fresnel, Green, Helmholtz, Kirchhoff, and Rayleigh; and the cornerstones of the modern theory including Keller's geometrical theory of diffraction, boundary-layer theory, and super-resolution. Appendices on the Cornu spiral and Babinet's principle are also included.

Preface

Diffraction is the response of waves to local heterogeneity in the medium in which they propagate. The phenomenon applies to all wave types and is the basis of applications in numerous scientific disciplines. The maturity level of applications varies from area to area. For example, in materials science, diffraction has been used for a long time in nondestructive testing to detect and describe fractures. Applications in civil engineering use diffraction acoustics extensively. In aerospace and defense, diffraction plays a prominent role in radar and stealth technologies. In the oil and gas industry, diffraction to complement seismic reflection methods is an emerging technology. Application of seismic diffraction methods to detect smallscale reservoir heterogeneity (i.e., fractures, stratigraphic pinchouts, and channel edges) is likely to increase dramatically, however. The industry is moving toward exploiting smaller and more complex conventional reservoirs and extensive new unconventional resource plays, where fracture-controlled permeability is the important factor for recovery.

Classical and Modern Diffraction Theory is concerned primarily with the diffraction forward problem. We address the inverse problem only briefly, in the context of the modern concept of superresolution. A treatment of the progression from early work on diffraction to the theory's leading edge is designed to assist the reader in understanding the physics of rigorous mathematical diffraction theory.

Although this volume was motivated by the need to provide geophysicists with an in-depth understanding of the diffraction phenomena from first principles, it is suitable equally to scientists in any discipline. Because classical diffraction theory and the transition to the modern theory were associated largely with the diffraction of light, this volume deals almost exclusively with diffraction in the context of optics. For the geophysicist, this volume provides a solid foundation for the companion volume Seismic Diffraction, which deals with the forward and inverse problems in seismic wave propagation.

The first chapter covers the foundations of diffraction theory. This includes a historical review of the classical theory, a summary of the experimental results illustrating this theory, and key principles of the modern theory of diffraction. The paper by Arkadiew (translated from German for this volume) provides an excellent reproduction of the classical experiments and introduces the similarity law of diffraction.

Chapter 2 provides the founding papers of classical diffraction theory, beginning in the 17th century. The editors offer an extensive introduction to the reprinted works by Francesco Grimaldi, Christiaan Huygens, and Thomas Young. A new translation from Latin of the defining work on diffraction by Grimaldi is presented. It relates a set of experiments which led Grimaldi to conclude that a new, fourth mode of light propagation exists — he called it diffraction.

Still in the 17th century, Huygens developed an early form of wave theory, now known as Huygens' principle (or construction). This theory allowed him to derive the laws of reflection and refraction, but, as he makes no reference to the phase of wave motion, his theory is unable to explain interference and diffraction effects. Young was the first who understood that diffraction patterns were produced by wavefield interference. The combination of Huygens' principle with the phenomenon of interference allows for a theory that can explain rectilinear propagation of light, the laws of reflection and refraction, and the laws of diffraction. Thus, the earliest scientific solution to the diffraction problem is based profoundly on Huygens' wave-theory construction and the understanding of interference provided by Young. Young's second significant contribution to diffraction theory is the idea of a boundary diffraction wave, the precursor to the modern geometrical theory of diffraction (the subject of Chapter 4).

Chapter 3 details the classical theory of diffraction, with reprinted works by Augustin Jean Fresnel, George Green, Hermann von Helmholtz, Gustav Kirchhoff, and Lord Rayleigh (John William Strutt). New translations of the papers by Helmholtz and Kirchhoff from German were prepared expressly for this volume.

Chapter 3 begins with Fresnel's mathematical formalization of Huygens' principle and definition of the Fresnel integral. Fresnel modified the classical principle of Huygens and proposed that the secondary spherical waves interfere amongst themselves. Fresnel's idea influenced all subsequent developments of the theory of diffraction. The major step in its development was carried out by Kirchhoff in 1882, based on the mathematical developments by Green and Helmholtz. Chapter 3, therefore, includes parts of Green's classic essay covering the original derivation of Green's theorem, which served as a foundation of the integral theorems by Helmholtz and Kirchhoff. Kirchhoff showed that the Young-Fresnel principle is an approximate form of Helmholtz's rigorous integral theorem and based Fresnel's source theory on this. In the case of diffraction from the aperture of a plane screen, Kirchhoff described the field on the screen in the form of an integral over the aperture (Kirchhoff approximation), that is, resulting from the interference of Huygens' secondary sources. Rayleigh later used Kirchhoff's integral to show that the wavefield can be determined satisfactorily by the boundary conditions on the screen. From Rayleigh's work (and later that of Sommerfeld), it became clear that the study of diffraction of waves could be reduced formally to an ordinary boundary-value problem of mathematical physics.

Chapter 4 addresses the cornerstones of modern theory. The use of high-frequency asymptotic approximations of the wavefield is a common theme in this chapter. The centerpiece of Chapter 4 is the geometrical theory of diffraction. This chapter includes defining works by Keller and others, along with the experimental results by Becknell and Coulson, which inspired Keller's theory.

Boundary-layer approximation is a second area of focus in Chapter 4. This deals with wavefield behavior in the penumbra at the shadow boundary where ray theory is not applicable. Here, the wave phenomenon is governed by Fock's equation of transverse diffusion (Fock et al., 1945), describing the energy flow from the illuminated zone into the shadow zone. Chapter 4 addresses the question of resolution for the inverse problem. Traditionally, this is defined in the classic paper by Rayleigh (included in Chapter 3), defining a measure of resolution known as the Rayleigh criterion. Other papers in this chapter deal with the quest toward superresolution, which aims at exceeding the Rayleigh resolution limit.

In Appendix A, we include a new translation of M. Alfred Cornu's work, defining what is now known as the Cornu spiral. Early modeling of the diffraction response was facilitated by Cornu's graphical representation of the Fresnel integral. The Cornu spiral is found in modern textbooks on diffraction because it provides an invaluable conceptual representation of the forward problem.

Appendix B features a new translation of a portion of Me´moires d'optique me´te´orologique by Jacques M. Babinet. A fundamental symmetry of the diffraction response is given by Babinet's principle — an intriguing phenomenon in which, for example, the diffraction response of an aperture is complementary to that of an opaque object of the same dimensions. Appendix B also features Louis Albert Necker, describing the author's paradoxical observations that inspired Babinet's principle.