Light and X-Ray Optics (eBook)

Refraction, Reflection, Diffraction, Optical Devices, Microscopic Imaging

Emil Zolotoyabko (Autor)

eBook Download: EPUB

2023
298 Seiten
De Gruyter (Verlag)
978-3-11-114089-6 (ISBN)

Infos

Contemporary optics is the foundation of many of today's technologies including various focusing and defocusing devices, microscopies and imaging techniques. Light and X-ray Optis for Materials Scientists and Engineers offers a guide to basic concepts and provides an accessible framework for understanding this highly application-relevant branch of science for materials scientists, physicists, chemists, biologists, and engineers trained in different disciplines. The text links the fundamentals of optics to modern applications, especially for promotion of nanotechnology and life science, such as conventional, near-field, confocal, phase-contrast microscopies and imaging schemes based on interference and diffraction phenomena. Written by a noted expert and experienced instructor, the book contains numerous worked examples throughout to help the reader gain a thorough understanding of the concepts and information presented. The text covers a wide range of relevant topics, including reflection, refraction, and focusing phenomena, wave polarization and birefringence in crystals, optics in negative materials, metamaterials, and photonic structures, holography, light and X-ray interferometry, extensive description of diffraction optics, including dynamical X-ray diffraction, and more.

Emil Zolotoyabko is Professor Emeritus in the Department of Materials Science and Engineering of the Technion-Israel Institute of Technology. For six years he served as Faculty Dean and for four years as a member of the Technion's Standing Committee. He was awarded the Schlesinger Prize for Encouragement of Research, the Henry Taub Prize for Excellence in Research, the Technion Excellence in Teaching, and held the Abraham Tulin Academic Chair. Emil Zolotoyabko has authored more than 200 scientific publications, four books and four chapters in books devoted to the development of new X-ray diffraction methods and their applications for studying the structure and dynamical characteristics of different materials systems. Main scientific achievements include: direct wave summation method in X-ray diffraction simulations; development of Mössbauer diffraction to study low-frequency lattice dynamics by inelastic scattering in crystals at phase transitions; fast stroboscopic X-ray diffraction and imaging techniques for investigating crystal dynamics on a nanosecond time scale; energy-variable X-ray diffraction for microstructural characterization with sub-micron spatial resolution; measurement of the speed of X-rays; discovery by high-resolution X-ray diffraction of the protein-associated anisotropic lattice distortions in biocomposites grown by living organisms.

Textauszug

Chapter 1 Foundations of geometrical optics

1.1 Electromagnetic waves in a homogeneous medium: the refractive index

The refractive index n of the medium, in which light propagates, is of crucial importance to the vast variety of optical phenomena. Further, we calculate the refractive index using a general approach based on Maxwell’s equations:

(1.1)divD=ρf

(1.2)divB=0

(1.3)rotE=−∂B∂t

(1.4)rotH=J+∂D∂t

Here ρf is the density of free (noncompensated) charges, which for nonintentionally charged materials equals zero:

(1.5) ρf=0

Differential operators in eqs. (1.1.) and (1.2) are called divergence (∇⋅), whereas those in eqs. (1.3) and (1.4) are rotor or curl (∇×). The displacement field D and magnetic induction B are connected, correspondingly, to the electric field E and magnetic field H as follows:

(1.6)D=εE=εmε0E

(1.7)B=μH=μmμ0H

where ε0  and μ0 are, respectively, the dielectric permittivity and magnetic permeability of a vacuum, while εm and μm are, respectively, the dielectric and magnetic constants of a medium. Note that in anisotropic media, parameters εm and μm are, in fact, tensors of second rank. We will discuss this issue in more detail in Chapter 4, in relation to light polarization in anisotropic crystals. Here we consider isotropic materials such as glass, which is often used for fabricating conventional optical components. In this case, parameters εm and μm are scalars, though frequency dependent.

To proceed, we recall the linear relationship between the vectors of current density J and electric field E (Ohm’s law), which are interconnected via the specific electrical conductivity σ:

(1.8) J=σE 

Together with eqs. (1.5)–(1.8), this yields:

(1.9)divD=divE= 0

(1.10)divB=divH=0

(1.11)rotE=−μ0μm∂H∂t

(1.12)rotH=J+∂D∂t=σE+εmε0∂E∂t

Applying a rot (curl) vector operator (∇×) to both sides of eq. (1.11) and using eq. (1.12), one obtains:

(1.13)rotrotE=−μ0μmσ∂E∂t+εmε0∂2E∂t2

Since rotrotE=graddivE−grad⋅grad E, then with the aid of eq. (1.9) we find:

(1.14)rotrotE=−grad⋅grad E=−∇2E

where ∇2=∂2∂x2+∂2∂y2+∂2∂z2 is the Laplace operator. Substituting eq. (1.14) into eq. (1.13) yields

(1.15)∇2E=μ0μmσ∂E∂t+εmε0∂2E∂t2

In a vacuum, μm= εm= 1, whereas the conductivity σ = 0. Therefore, eq. (1.15) converts to the standard wave equation

(1.16)∇2E−μ0ε0∂2E∂t2=0

with the light velocity in a vacuum equal to

(1.17)c=1μ0ε0

In fact, substituting the plane wave

(1.18)E=E0expikr−ωt

into eq. (1.16) yields the dispersion law, that is, the relationship between the light frequency ω and wave vector k

(1.19) ω2=1μ0ε0k02

where k0 is the wave vector in vacuum. One can rewrite the dispersion law (1.19) in its more familiar linear form

(1.20)ω=1μ0ε0k0=ck0=ck0

justifying definition (1.17).

In the case of a homogeneous medium, differing from a vacuum (i.e., with μm≠1, εm≠ 1, and generally nonzero conductivity σ > 0), a similar procedure leads to the modified wave equation:

(1.21)c2(∇2E)=μmεm∂2E∂t2+σε0∂E∂t

Substituting the plane wave (1.18) into eq. (1.21) yields

(1.22)c2k2=μmεmω2+iσωε0

and

(1.23) |k|=k=ωcμmεm+iμmσε0ω1/2

Expression (1.23) is, in fact, a new dispersion law for an electromagnetic wave in a medium. This dispersion law (1.23) is very different from that of an electromagnetic wave in a vacuum (1.20):

(1.24)k0=ωc

Using eqs. (1.23) and (1.24), one can calculate the refractive index n=k/k0, which determines the key optical phenomena for electromagnetic waves interacting with a material

(1.25)n=kk0=μmεm+iμmσε0ω1/2

Note that the refractive index comprises the real and imaginary parts, being responsible, respectively, for light refraction and attenuation. We elaborate further on this issue in Chapter 6.

In dielectrics (e.g., in a glass), the conductivity σ = 0; correspondingly, the wave equation (1.21) transforms into

(1.26) ∂2E∂t2=Vp2∇2E

with the phase velocity of light in a medium equal to

(1.27)Vp=cμmεm=cnd

In other words, the refractive index

(1.28)nd=μmεm

shows in what proportion the phase velocity of light propagation in a medium (Vp) is decreased with respect to that in a vacuum c.

For nonmagnetic materials (μm = 1), eqs. (1.25) and (1.28) are transformed into

(1.29)n=kk0=εm+iσε0ω1/2

and further to the well-known expression for nonmagnetic dielectrics (σ = 0):

(1.30)nd=εm

In most materials used for visible light optics, nd is around 1.3−1.8. In contrast, the refractive index for X-rays is a bit less than 1, that is, n=1−δ, with δ≈10−5 – 10−6. Despite the smallness of δ, the fact that n<1 leads to some interesting optical phenomena that are absent in the optics of visible light. We discuss the related consequences in more detail in the following chapters (e.g., in Chapters 2 and 3).

1.2 Inhomogeneous medium: eikonal equation and mirage formation

As we will learn later in this book, the wave aspects in light propagation through material objects are not important (or play a little role) when the light wavelength λ=2π/k is much smaller than the characteristic sizes of these objects. In this limit (λ → 0), which is called geometrical optics, the basic phenomena in light propagation, such as reflection and refraction, are completely determined by the refractive index n only.

In this context, let us consider the more general case of light propagation through an inhomogeneous medium in which the refractive index is the coordinate-dependent function, nx,y,z. In this case, the solution of the wave equation (1.26) is not more the plane wave. For simplicity, we use the wave equation for the scalar function fx,y,z:

(1.31) ∂2f∂t2=Vp2x,y,z∂2f∂x2+∂2f∂y2+∂2f∂z2

and considering the...

Produkt-Details

• Erscheint lt. Verlag: 7.9.2023
• Reihe/Serie: De Gruyter STEM
• Reihe/Serie: De Gruyter STEM
• Zusatzinfo: 7 b/w and 203 col. ill., 0 b/w and 0 col. tbl.
• Sprache: englisch
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Schlagworte: Light Difraction • Lumineszenz • microscopic imaging • Mikroskopie • optical devices • Optik • Photochemie • Wave interference • X-ray diffraction
• ISBN-10: 3-11-114089-X / 311114089X
• ISBN-13: 978-3-11-114089-6 / 9783111140896