Chapter 1: Introduction and Overview surveys polarized light and polarization optics and introduces several polarization issues in optical systems. The electromagnetic nature of light is introduced. The polarization elements such as polarizers, retarders, depolarizers, and associated properties such as birefringence are explained. A series of polarization issues are treated graphically: the Maltese cross pattern of uncoated lenses, the field of view dependence of polarizers and retarders, polarization aberrations due to thin film optical coatings, stress birefringence, and the angle dependence of liquid crystals.
Chapter 2: Polarized Light covers the mathematical treatment of monochromatic light and plane waves in two dimensions with Jones vectors and in three dimensions with the polarization vector. The polarization ellipse is explored in detail. Basic vector mathematics is reviewed along the way. The chapter concludes with a discussion of spherical wavefronts and the polarization of light sources.
Chapter 3: Stokes Parameters and the Poincaré Sphere treats polychromatic light, partially polarized light, and incoherent light. The Stokes parameters have an unusual non-orthogonal coordinate system that works wonderfully for problems in radiometry and remote sensing. The three-dimensional representation of the Stokes parameters, the Poincaré sphere simplifies the analysis of many problems with polarization elements.
Chapter 4: Interference of Polarized Light studies polarization fringes as well as intensity fringes in interference. Polarization issues can compromise interferometers and prevent good holograms from being recorded. Interference of partially polarized and polychromatic light leads naturally to the Stokes parameters description of light.
Chapter 5: Jones Matrices and Polarization Properties develops the powerful model of polarization elements as matrices. Each polarization element and polarization property has a corresponding family of Jones matrices. The propagation of light through a series of polarization elements is considered, with valuable examples for the optical engineer. The Jones matrix is considered for the reflection of light at normal incidence from mirrors; this Jones matrix presents a paradox that will be resolved with the polarization ray tracing matrix. The concept of Jones matrices where the incident and exiting beams are not parallel, very important in optical design, is developed.
Chapter 6: Mueller Matrices provides a powerful method for performing incoherent light calculations for polarization elements: polarizers, diattenuators, retarders, as well as depolarizers. Reflection and refraction Mueller matrices are introduced to extend the method to problems containing optical elements. Depolarization is a rich phenomenon with nine degrees of freedom.
Chapter 7: Polarimetry applies the Mueller matrix methods to the measurement of Stokes parameters and Mueller matrices. Stokes polarimeters have many applications in remote sensing including characterizing aerosols in the atmosphere and finding man-made objects in clutter. Mueller matrices are used to test polarization elements and photonic devices, and as ellipsometers to measure film thicknesses and refractive indices.
Chapter 8: Fresnel Equations describes polarization changes that occur at dielectric interfaces, total internal reflection, and reflection from metal mirrors. Incident light is analyzed into s- and p-polarized components that are studied separately as eigenpolarizations. Very large polarization effects occur at the critical angle where the derivative of the retardance becomes infinite.
Chapter 9: Polarization Ray Tracing Calculus develops the 3 × 3 polarization ray tracing matrix and associated algorithms. This calculus systematizes polarization ray tracing with a three-dimensional polarization ray tracing matrix, the P matrix, a generalization of the Jones matrix into three dimensions. A major advantage of the P matrix is its definition in global coordinates; it solves deep problems with Jones matrices and local coordinates due to singularities and non-uniqueness, a theme developed throughout this book. As a result, anyone who ray traces an optical system with P will get the same matrix, unlike a Jones or Mueller matrix calculation where the answer depends on the sequence of local coordinates selected. Algorithms are provided to calculate diattenuation and retardance using P. The Jones matrix for reflection at normal incidence from a mirror is the same as the Jones matrix for a half wave retarder for transmission; this paradox is resolved. Ray tracing through a polarization interferometer (shown) provides an important example of the calculus.
Chapter 10: Optical Ray Tracing presents algorithms for ray tracing optical systems and calculating the wavefront and polarization aberration functions. Polarization ray tracing algorithms account for the polarization effects of coated and uncoated interfaces, for example, Fresnel coefficients, during the ray trace. Polarization ray tracing matrices are used to calculate the transmittance (apodization), diattenuation, and retardance properties of rays. A grid of rays across a wavefront form the basis for determination of the polarization aberration function. A cell phone lens with biaspheric surfaces is included as an example for the ray tracing concepts. The chapter concludes with a historical review of polarization ray tracing.
Chapter 11: The Jones Pupil and Local Coordinate Systems analyzes the difficult issues of converting from ray trace results defined on spherical surfaces in three dimensions into flat surface representations as a Jones pupil. The Jones pupil is commonly used in industry for the representation of polarization aberration. In order to use Jones pupils properly, the subtleties of local coordinate systems are explained and optimal methods are presented. Two principal local coordinate systems are developed: dipole coordinates and double pole coordinates. For high numerical aperture wavefronts, double pole coordinate systems become more convenient since this coordinate system more closely approximates the natural behavior of lenses. Double pole coordinates also contain a fascinating doubly degenerate singular point. The cell phone lens example is continued to illustrate how P arrays are converted into the Jones pupil.
Chapter 12: Fresnel Aberrations The Fresnel equations are applied to several example optical systems and the resulting polarization aberrations are surprising. An uncoated lens between crossed polarizers leaks light in the Maltese cross pattern. The metal coatings in a Cassegrain telescope introduce a small amount of astigmatism into the on-axis beam! The telescope’s point spread function between crossed polarizers is dark in the center with four islands of light in a square. One clever application of the Fresnel equations is the Fresnel rhomb, a total internal reflection-based quarter wave retarder.
Chapter 13: Thin Films covers several of the most important classes of optical thin films and their polarization properties:
Antireflection coatings
Enhanced reflection coatings
Metal beam splitting coatings
Polarization beam splitting coatings
Chapter 14: Jones Matrix Data Reduction with Pauli Matrices deals with the interpretation of Jones pupils as diattenuation and retardance aberration functions. The Jones matrix chapter presents the forward problemof calculating Jones matrices from the polarization properties, diattenuation and retardance. Here, Pauli matrices and matrix exponentials are used to convert Jones matrices into diattenuation and retardance components by finding a canonical form for Jones matrices.
Chapter 15: Paraxial Polarization Aberrations examines the form of polarization aberrations in radially symmetric systems, starting from the angle of incidence function, integrating typical diattenuation and aberration functions, leading to the second-order polarization aberration patterns, which resemble defocus, tilt, and piston. The aberrations associated with the paraxial ray trace are examined as a launching point for a full aberration expansion.
Chapter 16: Image Formation with Polarization Aberration treats diffraction and the calculation of the point spread function and optical transfer function in the presence of polarization aberrations. Variations in the polarization state exiting an optical system cause interesting variations in the image and its polarization structure. The polarization aberrations of corner cubes are very large due to skew aberration and the retardance arising from total internal reflection.
Chapter 17: Parallel Transport and the Calculation of Retardance describes retarders and the calculation of retardance. Retardance is a particularly subtle and sometimes paradoxical concept, typically described as an optical path difference between two orthogonal polarization states. However, when a system has more than two interfering beams, other conceptual issues arise, issues that complicate measuring and interpreting the properties of birefringent films for displays. A paradox occurs in the calculation of retardance in three dimensions, when the incident ray, the intermediate ray segments, and the exiting ray are not parallel to each other. It is shown how, with detailed knowledge of the ray path through an optical system, the retardance paradox is resolved simply.
Chapter 18: A Skew Aberration applies the resolution of the retardance paradox to rotations of polarization state that occur in non-polarizing optical systems. A new type of polarization aberration, skew aberration, is explained using the Pancharatnam/Berry phase. A unique characteristic of skew aberration is its presence in ideal non-polarizing optical systems. This effect is significant for high-NA optical systems with large fields of view; microlithography systems are good examples of such optical systems. The effects of skew aberration on polarization point spread functions and optical transfer functions are derived.
Chapter 19: Birefringent Ray Trace presents ray tracing methods for anisotropic materials and addresses the handling of ray doubling. Polarization ray tracing through anisotropic materials requires tracking a large number of parameters for all split rays:
Propagation direction k
Poynting vector S
Mode refractive indices n
Complex Fresnel coefficients a
Electric field orientations Ê
The anisotropic algorithm handles double refraction and reflection, coated anisotropic interfaces, evanescent rays, including total internal reflection and inhibited reflection. The book’s website www.polarizedlight.org contains animations of light beams through biaxial materials as well as the evolution of polarization states.
Chapter 20 Beam Combination with Polarization Ray Tracing Matrices analyzes interactions of multiple wavefronts with similar and different propagation directions. As part of the ray tracing processes, the interaction of multiple exiting beams must be simulated to analyze birefringent devices and interferometers. One of the many issues combining multiple wavefronts is the relative positions of ray grids and the necessity for interpolation.
Chapter 21: Uniaxial Materials and Components explores light propagation and ray tracing in common uniaxial devices. The index ellipsoid helps explain wavefront propagation through birefringent interface. In uniaxial materials, the extraordinary mode’s birefringent aberrations are complicated due to an angularly varying refractive index. It is important to understand such aberrations in analyzing waveplates.
Chapter 22: Crystal Polarizers presents new analyses on common, but misunderstood, optical components—the crystal polarizers, including Glan–Taylor and Glan–Thompson. This treatment results in new insights into the polarizer’s field of view, the apodization of its beams, and their aberrations. The numerous minor beams generated from crystal polarizers are described and their paths are explained. The analysis becomes more complicated, but fascinating for pairs of parallel and crossed crystal polarizers with incident spherical wavefronts.
Chapter 23: Diffractive Optical Elements Diffractive optical elements are simulated using rigorous coupled wave analysis. The polarization properties of reflection gratings, wire grid polarizers, and subwavelength structures for antireflection coatings are explored. Incorporating diffractive optical elements accurately in polarization ray tracing by integration of their amplitude coefficients (Fresnel coefficients) is explained.
Chapter 24: Liquid Crystal Cells Liquid crystal cells manipulate the polarization of light by rotating liquid crystal molecules to create electrically controllable retarders, polarization controllers, and spatial light modulators. The most common and historically important configurations are analyzed. One of the major issues in liquid crystal cells is the variation of retardance with angle. These angular polarization aberrations are compared between designs and studied as a basis for pairing liquid crystal cells with field correcting biaxial multilayer films to make high-performance liquid crystal displays. To attain their position of dominance in the display market, liquid crystal technology overcame many obstacles, including absorption, scattering, low contrast, switching time, uniformity, limited viewing angle, disclinations, and polarization aberration.
Chapter 25: Stress-Induced Birefringence Stressed-induced birefringence is a widespread problem in optics that frequently occurs in injection-molded plastic optics due to molding processes and in glass lenses as a result of poor opto-mechanical mounting techniques. Stress, internal forces in optical elements, changes the distances between atoms, generating a spatially varying birefringence. Stress can arise during glass forming, during the injection molding of plastic lenses, or from the mounting of optical elements. This chapter’s algorithms simulate the propagation of polarized light through optical elements with stress birefringence. Common data structures for storing stress information in CAD files are discussed. Methods are included for interpreting colored polariscope images of stress birefringence.
Chapter 26: Multi-Order Retarders and the Mystery of Discontinuities Multi-order retarders are retarders with retardance greater than one wave. The retardance of a compound retarder (formed from several birefringent plates) can, for example, vary continuously from 1½ waves to 2½ waves of retardance without ever passing through 2 waves of retardance! Retardance can simultaneously assume multiple values when the fast axes of birefringent components are not parallel or perpendicular to each other. Measured data verify this complex but fascinating issue.
Chapter 27: Summary and Conclusions brings all the book’s issues into perspective to get the big picture. Critical issues such as optical path length, phase, retardance, and coordinate systems are reviewed and tolerancing of polarization is discussed. Polarized Light and Optical Systems wraps up by discussing the issues in communicating polarization effects and aberrations. How can the optical designers and engineers best communicate this complex information with colleagues, vendors, and production peers?
