Ординатура / Офтальмология / Английские материалы / Optics Learning by Computing with Examples using MATLAB_Dieter Moller_2007

5.3.1Differentiation “Time” ∂/∂t . . . . . . . . . . . . . . . . . . . 208

5.3.2Differentiation “Space” i∂/∂x + j∂/∂y + k∂/∂z . . . . . . 208

5.4 Poynting Vector in Vacuum . . . . . . . . . . . . . . . . . . . . . . . . 209

5.5Electromagnetic Waves in an Isotropic Nonconducting Medium . . . . . 210

5.6Fresnel’s Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

5.6.1Electrical Field Vectors in the Plane of Incidence

(Parallel Case) . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

5.6.2Electrical Field Vector Perpendicular to the Plane of Incidence

(Perpendicular Case) . . . . . . . . . . . . . . . . . . . . . . . 214

5.6.3Fresnel’s Formulas Depending on the

Angle of Incidence . . . . . . . . . . . . . . . . . . . . . . . . 215

5.6.4Light Incident on a Denser Medium, n1 < n2, and the

Brewster Angle . . . . . . . . . . . . . . . . . . . . . . . . . . 216

5.6.5Light Incident on a Less Dense Medium, n1 > n2, Brewster and

Critical Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

5.6.6Reflected and Transmitted Intensities . . . . . . . . . . . . . . . 222

5.6.7 Total Reflection and Evanescent Wave . . . . . . . . . . . . . . 228

5.7Polarized Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 5.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

5.7.2 Ordinary and Extraordinary Indices of Refraction . . . . . . . . 231

5.7.3Phase Difference Between Waves Moving in the Direction of or Perpendicular to the Optical Axis . . . . . . . . . . . . . . . . . 232

5.7.4 Half-Wave Plate, Phase Shift of π . . . . . . . . . . . . . . . . 233

5.7.5Quarter Wave Plate, Phase Shift π/2 . . . . . . . . . . . . . . . 235

5.7.6Crossed Polarizers . . . . . . . . . . . . . . . . . . . . . . . . . 238

5.7.7General Phase Shift . . . . . . . . . . . . . . . . . . . . . . . . 240

6.2.1Two Interfaces at Distance d . . . . . . . . . . . . . . . . . . . 253

6.2.2Plate of Thickness d (λ/2n2) . . . . . . . . . . . . . . . . . . 255

8.2Optical Constants of Dielectrics . . . . . . . . . . . . . . . . . . . . . . 316

8.2.1The Wave Equation, Electrical Polarizability, and

Refractive Index . . . . . . . . . . . . . . . . . . . . . . . . . . 316

8.2.2Oscillator Model and the Wave Equation . . . . . . . . . . . . . 317

8.3Determination of Optical Constants . . . . . . . . . . . . . . . . . . . . 320

8.3.1Fresnel’s Formulas and Reflection Coefficients . . . . . . . . . . 320

8.3.4 Sellmeier Formula . . . . . . . . . . . . . . . . . . . . . . . . . 324

8.4Optical Constants of Metals . . . . . . . . . . . . . . . . . . . . . . . . 326

8.4.1 Drude Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 326

8.4.2Low Frequency Region . . . . . . . . . . . . . . . . . . . . . . 327

8.4.3High Frequency Region . . . . . . . . . . . . . . . . . . . . . . 328

8.4.4 Skin Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

8.4.5Reflectance at Normal Incidence and Reflection Coefficients

with Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . 333

8.4.6Elliptically Polarized Light . . . . . . . . . . . . . . . . . . . . 334 A8.1.1 Analytical Expressions and Approximations for the

9.1.5Fourier Transformation of a Product of Two Functions and the

9.2.3The Fourier Transform Integral . . . . . . . . . . . . . . . . . . 355

9.2.4Discrete Length and Frequency Coordinates . . . . . . . . . . . 356

10.2Spatial Waves and Blackening Curves, Spatial Frequencies, and

Fourier Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 376

10.3Object, Image, and the Two Fourier Transformations . . . . . . . . . . . 382

10.3.1Waves from Object and Aperture Plane and Lens . . . . . . . . . 382

10.3.2Summation Processes . . . . . . . . . . . . . . . . . . . . . . . 383

10.3.3 The Pair of Fourier Transformations . . . . . . . . . . . . . . . 385

10.4Image Formation Using Incoherent Light . . . . . . . . . . . . . . . . . 386 10.4.1 Spread Function . . . . . . . . . . . . . . . . . . . . . . . . . . 386

10.4.4Examples of Convolution with Spread Function . . . . . . . . . 388

10.4.5Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . 392

10.4.6Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395

10.5 Image Formation with Coherent Light . . . . . . . . . . . . . . . . . . 398

10.5.1Spread Function . . . . . . . . . . . . . . . . . . . . . . . . . . 398

10.5.2Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

10.5.3Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . 401

10.6 Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403

10.6.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403

10.6.2Recording of the Interferogram . . . . . . . . . . . . . . . . . . 403

10.6.3Recovery of Image with Same Plane Wave Used

11.5Coma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423

11.6Aplanatic Lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

11.7Astigmatism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427

11.7.1Astigmatism of a Single Spherical Surface . . . . . . . . . . . . 427

11.7.2Astigmatism of a Thin Lens . . . . . . . . . . . . . . . . . . . . 428

11.8Chromatic Aberration and the Achromatic Doublet . . . . . . . . . . . . 430

11.9Chromatic Aberration and the Achromatic Doublet with

Separated Lenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432

C H A P T E R

Geometrical

Optics

1.1 INTRODUCTION

Geometrical optics uses light rays to describe image formation by spherical surfaces, lenses, mirrors, and optical instruments. Let us consider the real image of a real object, produced by a positive thin lens. Cones of light are assumed to diverge from each object point to the lens. There the cones of light are transformed into converging beams traveling to the corresponding real image points. We develop a very simple method for a geometrical construction of the image, using just two rays among the object, the image, and the lens. We decompose the object into object points and draw a line from each object point through the center of the lens. A formula is developed to give the distance of the image point, when the distance of the object point and the focal length of the lens are known. We assume that the line from object to image point makes only small angles with the axis of the system. This approximation is called the paraxial theory. Assuming that the object and image points are in a medium with refractive index 1 and that the lens has the focal length f, the simple mathematical formula

gives the image position xi when the object position x0 and the focal length are known.

Formulas of this type can be developed for spherical surfaces, thin and thick lenses, and spherical mirrors, and one may call this approach the thin lens model.

For the description of the imaging process, we use the following laws.

1.Light propagates in straight lines.

2.The law of refraction,

1

21. GEOMETRICAL OPTICS

The light travels through the medium of refractive index n1 and makes the angle θ1 with the normal of the interface. After traversing the interface, the angle changes to θ2, and the light travels in the medium with refractive index n2.

3. The law of reflection

The law of reflection is the limiting case for the situation where both refraction indices are the same and one has a reflecting surface. The laws of refraction and reflection may be derived from Maxwell’s theory of electromagnetic waves, but may also be derived from a “mechanical model” using Fermat’s Principle.

The refractive index in a dielectric medium is defined as n c/v, where v is the speed of light in the medium and c is the speed of light in a vacuum. The speed of light is no longer the ratio of the unit length of the length standard over the unit time of the time standard, but is now defined as 2.99792458 × 108m/s for vacuum. For practical purposes one uses c 3 × 108m/s, and assumes that in air the speed v of light is the same as c. In dielectric materials, the speed v is smaller than c and therefore, the refractive index is larger than 1.

Image formation by our eye also uses just one lens, but not a thin one of fixed focal length. The eye lens has a variable focal length and is capable of forming images of objects at various distances without changing the distance between the eye lens and the retina. Optical instruments, such as magnifiers, microscopes, and telescopes, when used with our eye for image formation, can be adjusted in such a way that we can use a fixed focal length of our eye. Image formation by our eye has an additional feature. Our brain inverts the image arriving on the retina, making us think that an inverted image is erect.

1.2FERMAT’S PRINCIPLE AND THE LAW OF REFRACTION

In the seventeenth century philosophers contemplated the idea that nature always acts in an optimum fashion. Let us consider a medium made of different sections, with each having a different index of refraction. Light will move through each section with a different velocity and along a straight line. But since the sections have different refractive indices, the light does not move along a straight line from the point of incidence to the point of exit.

The mathematician Fermat formulated the calculation of the optimum path as an integral over the optical path

P2

P1