Imaging

Treating an imaging system as a linear system and use llinear system properties to d iscuss both coherent and incoherent imaging. Use a one dimensional pin hole camera to study the theory of incoherent imaging. Two different criteria, Rayleigh and Sparrow, are used to define the resolution limits of the camera. From the simple theory define the optical transfer function and the modulation transfer function as appropriate characterizations of complex imaging systems. A review of the human imaging system emphasizes tits idfferences with man made cameras. Coherent imaging is based on Abbe’s theory of microscopy. A simple 4f imaging system can be used to understand how spatial resolution is limited by the optical aperture and by controlling the aperture, we can enhance the edges of an image or remove noise intensity noise on a plane wave. Apodizing the aperture allows astronomers to locate planents orbiting distant stars.

Aberrations

Using simple ray tracinig technliques presented in Chapter 6, we demonstrate that a general ray is not focused to the position predicted by paraxial theory. The aberration displayed is spherical aberration. Two methods of measuring aberration: the use of optical path difference to characterize wavefront aberration. The transverse ray coefficients to generate a ray intercept plot. Experimental examples of all the third order aberrations are given. In addition to spherical aberration, they include coma, astigmatism, field curvature, and distortion Only two types of aberration correction are discussed, removal of spherical aberration in the Hubble Space telescope and chromatic aberration. A detailed example of chromatic aberration is given.

Geometrical Optics

Discuss the limits imposed by the paraxial approximation. Define the sign convention based on the cartesian coordinate system, the foiundation of analytic geometery. Demonstrate ray tracing technique to derive the ABCD maxtrix which will generate both the gaussian and Newtonian form of the thin lens equation and the lens maker’s equation. The cardinal points of a lens are also derived. The ABCD matrix is used to explore the methods used in ray tracing to locate the aperture stop of a Cooke’s triplet lens system. In the problem set, the student is asked to use the aperture stop to locate the entrance and exit pupil of a Cooke’s triplet.

Interference and Coherence

The superposition principle states that the sum of solutions of the wave equation is also a solution. Linearity forbids scattering of one photon by another and waves with orthogonal polarization will not interfer and if no interference is observed the light is said to be incoherent. There is no record carried by any of the waves to indicate outside of the overlap region that the waves ever commingled. Interference involve dielectric layers can produce antireflection coatings. Michelson interferometer, temporal coherence and its use in spectroscopy. The interferometer coupled with a light source with well defined coherence properties are currently used in optical coherence imaging. With multiple reflections a Fabry Perot interferometer can be created to provide optical feedback in a laser. Young’s two slits interferometer can be used to measure the size of distant stars.

Guided Waves

Use a one-dimension theory based on the normal to a plane wave to evaluate the propagation of light in a dielectric layer. If the ray strikes the walls of the guide it must meet the condition for total reflection. The theory, called the zigzag model, is not a “pure” geometric theory because we use the properties of interference and of phase change upon total reflection that we derived using wave theory. The theory predicts that the rays propagating down the guide can only strike the walls at discrete angles. These angles are the modes of the guide. The theory allows us to calculate the number of bounces/unit length the ray can make. We show limitaitons on coupling energy into the guided modes and the numerical aperture of a mode. We discuss losses and dispersion in the guide and introduce photonic crystals that allow hollow guides to be constructed.

Reflection and Refraction

We explore the propagation of a light wave when the propagation velocity of the medium change in a discontinuous way. We rely only on the wave properties of light to obtain the fundamental laws of reflection and refraction and use boundary conditions developed for Maxwell’s Equations to obtain the Fresnel formula for the amount of light transmitted or reflected. We alos turn to the more fundamental Principle of Fermat to derive the two laws but limit detailed examination to the law of reflection. We look at the effects of polarization on reflection and consider the physical cause of Brewter’s angle. The topic of total reflection is covered in more detail because of its importance inf fiber optics. We have ignored any loss in the propagation of light but now we examine reflection from metals and discover the effects on reflection of using a protective coating on metals

Polarizers

Introduction to how to measure polarization and the degree of polarization, The various physical processes used to control polarization including: absorption, reflection, interference, and birefringence. An explaination of the operation of a wire grid and its molecular equivalent, a polaroid plastic are given. Fresnel formula for reflection is used to calculate the Stokes component of the reflected light. A brief introduction to the concept of birefringence is given and the various designs of birefringent prism polarizers are listed with their advantanges and shortcomings. A sample design of a prism polarizer is given. Quarter wave and half wave retarders and their use are discussed. An optional section is devoted to optical activity and the use of chiral measurements in chemistry and drug manufacturing are discussed.

Fraunhofer Diffraction

We will develop a simple derivation of the Huygens-Fresnel integral based on an application of Huygens’ Principle and on the addition of waves to calculate an interference field starting with two apertures as in Young’s two slit experiment extending to N apertures and then a continuum distribution. A detailed look is made of the obliquity factor and a constant value is derived to be used in the simple derivation. We use one dimensional theory for most of the discussion but do present the results of diffraction from a circular aperture as it will be needed in our later discussion of imaging. A second description of the propagation of light, useful when using laser light sources, the gaussian wave, is introduced and examples are given of the use of the theory in geometrical optics and laser design.

Electromagnetic Theory

The theory of light is described by Maxwell’s Equationsand they provide information about the fundamental properties of light. The wave equation is contained within Maxwell’s equations and proof is provided but is an example of a topic that can be skipped. The electromagnetic wave is a transverse wave of both the electric and magnetic field which are also mutually perpendicular. We discuss some of the differences between classical and quantum theory of light but restrict the use of classical wave theory in this text. The classical electromagnetic wave has a momentum that has led to the development of optical twezzers of great use in biological motors. Because the amplitude of the electromagnetic wave is a vector quantity we introduce the concept of polarization to describe the vector properties. We will need the capability in our discussion of reflection.

Fresnel Diffraction

Fresnel diffraction is discussed in terms of a description of waves traveling near the stationary point. That is the point that lies on a line connecting the source and the observation point. We discuss a rectangular aperture using Fresnel integrals or graphicly using the Cornu Spiral We discuss a circular aperture in terms of Fresnel zones and we develop a simple formula for the calculation of the radius of a Fresnel zone. Using the concept of Fresnel zones we develop an expalination of Fermat’s Principle and explain the origin of Poisson’t spot. The Fresnel zones generate an understanding of the operation of the pinhole camera.