1Introduction and Overview
For many optical systems, selecting good combinations of polarization elements may be difficult. Similarly, understanding and controlling the optical system’s polarization properties can present substantial challenges and, for some systems, requires man-years of dedicated polarization engineering. Such systems can be called polarization critical optical systems because they present polarization challenges and have specifications that are difficult to meet. Liquid crystal displays and optics for microlithography are just two examples of polarization critical systems. Polarization engineering is the task of designing, fabricating, testing, and often mass producing with high yield such polarization critical optical systems.
Of course, the polarization properties of many other optical systems are small and not always significant for their operation. For example, many lenses make relatively small changes to the polarization state, of the order of a percent or less. Even if the polarization effects are small, they may still be interesting and it may be necessary to ensure that some polarization specification is met.
This book is dedicated to understanding these polarization effects or polarization aberrations, large and small. One of the principal techniques will be polarization ray tracing. Ray tracing is a set of algorithms for calculating the paths of light rays through optical systems. Polarization ray tracing adds calculations to follow the evolution of the polarization state and will display information on how the distribution of polarization and the polarization properties are for the ray paths.
1.1Polarized Light
Light is a transverse electromagnetic wave, a moving electric and magnetic field. The light’s electric and magnetic fields oscillate in a direction transverse to the direction of light propagation. Light is generated when charges, electrons and protons, accelerate and oscillate. Then, the light’s forces cause charges to oscillate in return. Polarization refers to the properties of the light in this transverse plane, describing whether the light is polarized or unpolarized and the orientation of this polarization. The polarization state of light can be controlled with polarization elements. The polarization state also changes at lenses, mirrors, optical coatings, diffraction gratings, crystals, liquid crystals, and many other interfaces and materials. The book’s goal is to develop efficient and general methods for understanding the changes of polarization of light with polarization elements, through optical systems, and in the natural environment.
The human eye is not sensitive to the polarization of light. We sense the brightness and color of light but cannot tell if light is polarized or unpolarized, or how the polarization is oriented. Since humans are polarization blind, we are unaware of the many polarization effects around us, indoors and outdoors, as shown in Figures 1.1 and 1.2. Our eyes cannot tell that the light from rainbows and the glare reflecting from the road are highly polarized, nor can they sense the polarized light from liquid crystal displays. We do not see how polarized light is scrambled by the stress in our eyeglasses (Figure 1.1) and automobile windows (Figure 1.2). Polarizers and polarimeters can make such polarization effects visible. Many animals do see the polarization of light, such as bees, ants, and octopus. They use polarization for a variety of purposes including to navigate, communicate, and find prey.
Figure 1.1(Left top) A leptocephalus eel larva seen in visible light is nearly transparent to avoid predators. (Left bottom) When viewed with polarizing filter, the eel is more visible due to polarization change associated with birefringence in parts of its body. Thus, polarization vision is useful in finding such prey. (Photo from NOAA.1) (Right) Eyeglasses under crossed-circular polarizers.
Figure 1.2Moonroof photographed under a sunny day with (top) and without (bottom) a polarizer filter on the camera. The rainbow checkerboard pattern reveals the stress birefringence of the tempered glass.
1.2Polarization States and the Poincaré Sphere
Monochromatic light is light with a single pure frequency, a single wavelength, and a cosinusoidal electric field. Monochromatic light is an idealization, the limit as the wavelength spread from a light source approaches zero. Light from single-frequency lasers is nearly monochromatic, but the light has some small spectral bandwidth. Chapter 2 (Polarized Light) presents the mathematical description of light. Here, a few of these concepts are introduced.
The polarization ellipse is the figure traced by the tip of the electric field vector for monochromatic light, repeating each period. Monochromatic light must be polarized, either linearly, elliptically, or circularly polarized, because it is periodic. When the light’s electric field only oscillates in a single direction, such as along the positive and negative y-axis as in Figure 1.3, the light is linearly polarized. The direction of oscillation is the plane of polarization or angle of polarization.
Figure 1.3Electric field associated with a monochromatic light field linearly polarized in the y-direction.
For linearly polarized light propagating along the z-axis, the light’s electric field has x- and y-components that are in phase; both components go to zero at the same time, twice per period. Otherwise, the light is elliptically polarized, as in Figure 1.4, and the x- and y-components are out of phase. The tip of the electric field vector traces an ellipse, once per period, creating the polarization ellipse, one of the iconic figures of polarization optics. The polarization ellipse will always be drawn looking into the beam. Monochromatic circularly polarized light has a constant electric field amplitude whose orientation uniformly rotates in the transverse plane. Circularly polarized light occurs in two forms, right circularly polarized and left circularly polarized. By convention, right circularly polarized light rotates clockwise and left circularly polarized light rotates counterclockwise in time looking into the beam. Similarly, elliptically polarized light has right or left helicity depending on the direction of rotation.
Figure 1.4(Left) The polarization ellipse for an elliptical polarization state is the ellipse drawn by the tip of the electric field vector during one period. (Middle) The electric field as a function of time is shown with the bounding ellipse. (Right) Usually, just the ellipse is drawn to indicate the polarization state. The location of the arrow indicates the phase of the wave.
Light also consists of a magnetic field with a vector pointing perpendicular to the electric field in the transverse plane, oscillating in phase with the electric field. Figure 1.5 shows monochromatic light’s electric field (red) and magnetic field (blue) oscillating in space for several polarization states. The interaction of light with matter is dominated by the electric field in most types of light–matter interactions; thus, by convention, the light’s polarization state is described by its electric field. The magnetic field is important for calculating the interaction of light with several types of media including magnetic materials, birefringent materials, diffraction gratings, and liquid crystals.
Figure 1.5Electric (red) and magnetic (blue) fields for (left) 90° polarized, (center) 45° polarized, and (right) left circularly polarized light in space.
The family of polarization ellipses can be conveniently represented on the surface of a unit sphere, the Poincaré sphere, shown in Figure 1.6, shown in front and back views. Right circularly polarized light is at the top of the sphere, the north pole. Left circularly polarized light is at the bottom of the sphere, the south pole. Linearly polarized states lie on the equator. Note that for a complete circuit around the equator, the polarization state rotates by 180°, which returns x-polarized light into x-polarized light. The rest of the Poincaré sphere’s surface describes elliptically polarized light, with a right circular helicity in the northern hemisphere and a left circular helicity in the southern hemisphere. The ellipticity is nearly linear close to the equator; the ellipticity approaches circular near the poles. The surface of the Poincaré sphere continuously represents all possible polarized ellipses. It is useful to represent the Poincaré sphere on a flat surface as shown in Figure 1.7.
Figure 1.6The Poincaré sphere contains a representation of all polarized states on its surface, with linearly polarized states arranged around the equator, and the two circularly polarized states at the poles.
Figure 1.7Two representations of the Poincaré sphere on a plane: (left) Mollweide projection and (right) equi-rectangular projection, where the entire top row is right circularly polarized and the entire bottom row is left circularly polarized.
Several methods are used to describe polarization states depending on the application. Two element complex vectors, Jones vectors, and three element polarization vectors that are covered in Chapter 2 are particularly well suited for optical design, diffraction, and interferometry, cases where the phase of the light is essential. The Stokes parameters covered in Chapter 3 are well suited for laboratory measurements and descriptions of the natural light outdoors; these are polychromatic and incoherent applications where phase has less meaning.
1.3Polarization Elements and Polarization Properties
A polarization element is any optical element used to alter or control the polarization state of light and to transform light between polarization states. Polarization elements are classified into three broad categories—polarizers, retarders, and depolarizers—based on whether they change the amplitudes, phases, or coherence of the light. Mirrors, lenses, thin films, and nearly all optical elements alter polarization to some extent, but are not usually considered as polarization elements because that is not their primary role, but a side effect.
Polarizers transmit a known polarization state independent of the incident polarization state. Most common are linear polarizers that transmit linearly polarized light along their transmission axis. A linear polarizer is a device that, when placed in an incident unpolarized beam, produces a beam of light where the electric field vector is oscillating primarily in one plane with only a small component in the perpendicular plane. An ideal polarizer has a transmission of one for the specified polarization state and transmission of zero for the orthogonal polarization state. Polarizers are an example of diattenuators, or partial polarizers, which have two transmissions Tmax and Tmin for two orthogonal polarization states. Diattenuators can be characterized by their diattenuation
1.1which varies from one for a polarizer to zero for an optical element that transmits all incident polarization states equally. Sheet polarizers, or Polaroid, which come in large plastic sheets, absorb one polarization state transmitting the orthogonal state. Polarizing beam splitters direct two orthogonal polarization components into different directions, as shown in Figure 1.8.
Figure 1.8A polarizing beam splitter splits orthogonal polarization components into two directions.
Retarders have two different optical path lengths associated with two special polarization states, the fast state and the slow state. The slow state is delayed, or retarded, with respect to the fast state. The retardance is the difference in optical path lengths, which will describe the relative phase change between the two states. Arbitrary incident polarization states divide into the fast state and slow state when entering a retarder, and these two components emerge from the retarder with different optical path lengths, as shown in Figure 1.9. Linear retarders divide the light into two linear polarized components, 90° apart, retarding one of the states. A quarter wave linear retarder introduces a relative phase delay of a quarter of a wavelength of the light, and is useful for converting linearly polarized light into circularly polarized light. A half wave linear retarder delays one linearly polarized component by half a wavelength, and is useful for changing the orientation of linearly polarized light.
Figure 1.9Three states of polarized light, propagating from the left in air, enter a half wave linear retarder (birefringent waveplate) with a vertical fast axis (lower refractive index). The two planes are the entrance and exit faces. The vertical polarized mode (top, lower refractive index) has a longer wavelength inside the retarder, so its optical path length is less, 2 waves, than the horizontally polarized mode (center) by ½ wave. (Bottom) Right circularly polarized light divides into the two modes that propagate separately and then combine exiting the retarder. This exiting beam is now left circularly polarized due to the half wave optical path difference (retardance) between the two modes.
Depolarizers scramble the state of polarization and convert polarized light into unpolarized light. Depolarization is usually associated with scattering, particularly multiple scattering. Integrating spheres will readily depolarize a beam of light. Thin slabs of opal, a gem consisting of closely packed spheres of quartz, are sold as depolarizers. Most projection screens commonly used in classrooms and meeting rooms will effectively depolarize a beam of light. Try illuminating a screen with polarized light and observe that the scattered light cannot be extinguished with a polarizer. Lenses, mirrors, filters, and other typical optical elements exhibit very small amounts of depolarization, typically less than a few tenths of a percent. Hence, in the majority of optical systems, the magnitude of depolarization is small and not significant. Optical surfaces are carefully fabricated and coated to minimize scattering; thus, depolarization is generally very small from high-quality optical surfaces. Figure 1.10 shows a spatially depolarized set of polarization ellipses.
Figure 1.10A depolarized field with spatially varying polarization states.
1.4Polarimetry and Ellipsometry
The polarization properties of a beam of light are measured with a polarimeter, which is a camera, radiometer, or spectrometer configured to measure the flux through a set of polarizers and calculate the polarization state; these are light-measuring polarimeters. Figure 1.11 shows the layout of one type of light-measuring polarimeter, a rotating retarder imaging polarimeter. Figure 1.12 shows a polarization image of the Meinel Optical Sciences building at the University of Arizona. One interesting feature is the polarization of the windows along the front side. Partially polarized skylight reflecting from the windows has its polarization changed different amounts depending on the angle of the windows. Thus, polarimetery can be used to help recover information on the orientations of objects. One application of imaging polarimetry is the measurement of aerosols. Light scattered from small particles undergoes a significant change in polarization state, which can then be used to determine the size and density of aerosol particles. Aerosol polarimeters measure sunlight scattered from the atmosphere to study the properties of atmospheric aerosols. Imaging polarimeters in earth orbit can map the aerosol content of the atmosphere globally.
Figure 1.11A rotating retarder imaging polarimeter measures the polarization state by passing the light through a retarder, a linear polarizer, and into a camera. The retarder steps to several angles and the measurements from several images are used to calculate the degree of polarization, the angle of polarization, and the ellipticity of the light as images.
Figure 1.12Polarization images of the front of the Meinel Optical Sciences building at the University of Arizona. (Top row) Intensity images. (Middle row) Degree of linear polarization images. (Bottom row) Angle of linear polarization images. (Left column) Red/Green/Blue images (RGB). (Second column) 470-nm images. (Third column) 550-nm images. (Right column) 660-nm images. The north side of the Meinel building is all glass panes at a variety of different angles, reflecting skylight with a range of degrees of polarization and angles of polarization (ranging from orange ~−80° to green ~50°. The right side of the building is copper metal with few windows and an AoLP of ~80°. The sky is mostly cloudy with a low DoLP (dark) except for a few blue patches showing white in the middle row. At 880 nm, the cloudy sky mostly has the same AoLP as the blue sky, while at 470 nm, there is considerable variation of AOLP in the cloudy sky. (Taken by Karlton Crabtree and Narantha Balagopal.)
Sample measuring polarimeters, such as Mueller matrix polarimeters, determine a material’s polarization properties by illuminating a sample with a sequence of polarization states and measuring the exiting polarization states (Figure 1.13). Mueller matrix polarimeters are used for measuring polarization elements, liquid crystal cells, retinal imaging, and other forms of biological imaging. One form of a sample-measuring polarimeter is the ellipsometer, originally developed in the 1880s by Paul Drude, who was seeking a method to measure the refractive index of materials, particularly metals. He realized that if he reflected linearly polarized light with the angle of polarization at 45° from the plane of incidence, he could measure the refractive index of a surface from the orientation of the major axis and ellipticity of the reflected light’s polarization. The name ellipsometry followed from this characterization of the polarization ellipse.2 Later, ellipsometric technology evolved to be able to very precisely measure the thickness and refractive index of single-layer thin films. Thus, ellipsometry aided the development of antireflection coatings, such as quarter wave thick magnesium fluoride coatings in the 1920s. By measuring at multiple angles of incidence and wavelengths, the properties of multilayer thin films are accurately obtained by ellipsometry.3 Ellipsometry is now an essential technique in the fabrication of integrated circuits and microelectronics, as well as other applications in industrial metrology.
Figure 1.13(Top) A Mueller matrix imaging polarimeter illuminates and measures a sample with many combinations of illuminating polarizers and analyzing polarizers to measure the Mueller matrix, diattenuation, retardance, and depolarization of samples. This particular Mueller matrix polarimeter illuminates the sample through a linear polarizer and a retarder that steps in angle. The light interacts with a sample and exits with its polarization state changed. The light may reflect from, transmit through, diffract from, or scatter from a sample depending on the measurement. This change is analyzed by a retarder that steps in angle and a linear polarizer before entering a camera or radiometer. (Bottom) A commercial Mueller matrix polarimeter configured for transmission Mueller matrix measurements. This polarimeter measures thirty Mueller matrices per second. (Courtesy of Axometrics, Huntsville, Alabama.) The light source and polarization generator are in the top head and the analyzer and detector are in the lower head.
1.5Anisotropic Materials
Anisotropic materials have a refractive index that varies with the direction of the light’s electric field. Many anisotropic materials are crystals, for example, calcite and quartz. Within a single crystal calcite, all of the calcium-to-carbon bonds are oriented in one direction, called the optic axis. The three carbon oxygen bonds in the carbonate radical are oriented in the perpendicular plane. Light polarized along the optic axis interacts with a different set of chemical bonds compared to light polarized in the orthogonal plane, resulting in two refractive indices for two polarizations associated with one incident direction of propagation. When light refracts into calcite or other anisotropic materials, it refracts into two modes with orthogonal polarization, ordinary and extraordinary, which, in general, propagate in different directions. This is clearly seen in Figure 1.14.
Figure 1.14Light propagating through calcite divides into two modes, which follow different paths and generates two images.
Hence, during polarization ray tracing, each ray entering an anisotropic crystal results in two exiting rays that need to be traced to the output of the optical system. As shown in Figure 1.15, when there is a second anisotropic optical element, the rays double again and four rays need to be traced through the remainder of the system. In general, a system containing N anisotropic elements produces 2N separate rays, all of which need to be traced to simulate the light through the system. Each of these rays takes a separate path and has its own amplitude, polarization, and optical path length. Thus, the light in the exit pupil may be described by 2N separate wavefront aberration functions; each of these partial waves can have different amplitude aberration, defocus, spherical aberration, coma, astigmatism, and so on.
Figure 1.15Polarization ray trace showing a ray propagates into calcite and splits into two rays. Entering a second anisotropic material, titanium oxide, each ray doubles again. Each incident wavefront generates four exiting wavefronts.
1.6Typical Polarization Problems in Optical Systems
Polarization provides the basis of operation for many types of optical systems. Examples include liquid crystal displays and the ellipsometers used in the microlithography industry, polarization instruments that test the composition and thicknesses of the many layers that are deposited during chip fabrication. But polarization is also a problem in many systems and is thus frequently analyzed during the optical design process. A quick survey of the polarization aberrations of some simple optical systems will help understand the goals and methods for polarization ray tracing and analysis.
1.6.1Angle Dependence of Polarizers
One of the issues optical designers face is the dependence of polarizers on the angle of incidence of light. Both dichroic polarizers (sheet polarizers) and wire grid polarizers absorb the polarization component that projects onto their absorption axis. The extinction is high when the angle of incidence varies in two planes: the plane perpendicular to the transmission axes and the parallel plane. But for fundamental geometrical reasons, this high extinction is reduced when the light propagates at other directions. The leakage is worst for propagation in planes at 45° to the extinction axes. Figure 1.16 is a 3D view of two polarizers, one polarizer with its absorption aligned along the y-axis (green) followed by a polarizer aligned along the x-axis. The on-axis direction is represented by a short black line. As an incident beam of light moves off-axis in the diagonal direction as shown in Figure 1.17, the absorption axes are no longer orthogonal and an increasing amount of light leaks through. Thus, if a 45° cone of light is incident, the transmitted intensity appears as in Figure 1.18.
Figure 1.16A schematic 3D view through two polarizers from several angles in the x–z plane. The transmission axis of the front polarizer is vertical and indicated with green lines. The transmission axis of the back polarizer is horizontal indicated with red lines. The thick black line indicates the z-axis, normal to the two polarizers. (Left) At normal incidence, the front and back transmission axes are perpendicular. Viewing at 15° (second), 30° (third), and 45° (right), the green and red lines remain perpendicular. The extinction of the polarizer pair is good when illuminated in these directions.
Figure 1.17A view through the same two polarizers as the light propagation direction varies in the diagonal plane. (Left) At 0° angle of incidence, along the z-axis, the lines are perpendicular. As the angle of incidence in the plane at 45° to x and y increases from (second) 15°, to (third) 30°, to (right) 45°, the projection of the polarizers’ transmission axes no longer appears orthogonal and the polarizer pair leaks light accordingly.
Figure 1.18Contour plot of the fractional leakage through a crossed pair of polarizers for a 45° cone of light.
1.6.2Wavelength and Angle Dependence of Retarders
Simple birefringent waveplates, the most common retarders, have properties that vary with wavelength and angle of incidence. Since the ordinary and extraordinary refractive indices of a uniaxial crystal vary with wavelength, known as dispersion, the birefringence is also a function of wavelength. Figure 1.19 shows the wavelength dependence of retardance for a quartz quarter wave retarder and MgF2 quarter wave retarder.
Figure 1.19The wavelength dependence of birefringent (red) quartz, and (blue) magnesium fluoride waveplate retarders with a quarter wave of retardance at 600 nm.
A retarder’s retardance also varies with the angle of incidence. Because the path length through the retarder increases with angle of incidence, and the extraordinary refractive index varies with angle, the retarder’s retardance usually has a toroid-like variation with angle; a calculation for a quartz quarter wave retarder is shown in Figure 1.20. For y-polarized light, the exiting polarization state is unchanged for light incident along the x–z plane and the y–z plane but changes in all other directions. The variation of retardance is easily seen when the retarders or other birefringent crystals are placed in a conoscope, an optical instrument that focuses polarized light through a sample and analyzes the light with a second polarizer. Figure 1.21 (left) shows a conoscopic image of a thick A-plate of calcite in a conoscope and Figure 1.21 (right) shows the image for a thick crystal of calcite with its optic axis along the plate normal, a configuration known as a C-plate.
Figure 1.20The variation of a quarter wave retarder’s retardance (1.571 rad) with angle of incidence decreases with angle along the plane containing the optical axis (vertical) and increases with angle along the orthogonal plane (horizontal). Along the diagonals, the retardance is nearly constant with angle of incidence.
Figure 1.21Conoscopic images through (left) an A-plate and (right) a C-plate of calcite, showing the variation of exiting polarization with angle and wavelength as colored fringes.
1.6.3Stress Birefringence in Lenses
Another issue with polarized light and lenses, particularly injection-molded lenses, is stress birefringence. Stress birefringence is a spatially varying birefringence resulting from forces within the lens that compress or stretch the material’s atoms, causing birefringence. Stress birefringence can become frozen into a glass blank or molded lens during fabrication as the lens material cools unevenly. Stress can also arise from externally applied stresses, such as forces on the lens from lens mounts or even gravity. Thus, stress birefringence causes unwanted retardance variation. Stress birefringence can be made visible by placing the lens between crossed polarizers in a system called a polariscope. The stress-induced retardance will change the polarization state of some of the light, and this changed component will pass the second polarizer. By viewing the polarization leakage, as shown in Figure 1.22, the stress can be visualized. Stress birefringence can often be reduced by annealing, heating a lens close to the glass transition temperature, and then cooling it slowly to reduce internal stresses. Annealing is a routine process for high-quality optical glass. Figure 1.23 (left) shows a piece of glass under mechanical stress in the polariscope. The intensity distribution reveals regions where the polarization state has been modified. Figure 1.23 (right) shows a piece of glass with high stress from rapid cooling. Stress birefringence in lenses reduces their image quality.
Figure 1.22An injection-molded lens with stress birefringence (left) between crossed polarizers (at the top) and (right) after rotating the lens by 45°.
Figure 1.23(Left) Pressure-induced stress birefringence in the polariscope. A piece of glass supported by two points on the upper right side is being stressed by a screw on the lower left side. (Right) A square piece of glass with large stresses frozen into the glass viewed between crossed polarizers showing the wavelength variation of the stress-induced retardance.
1.6.4Liquid Crystal Displays and Projectors
Liquid crystals are a thick soup of birefringent molecules, typically rod-shaped molecules chosen for their dipole moment. These molecules rotate in response to applied electric fields that modulate the retardance of the cell. Figure 1.24 shows the molecular orientation in a typical twisted nematic liquid crystal cell for several voltages. When placed between polarizers, the liquid crystal functions as a voltage-controlled intensity modulator. To create a liquid crystal display, a thin layer of liquid crystal is placed in a cell between two glass plates. An array of electrodes is fabricated in an addressable array to adjust the electric field to each pixel and packaged with an array of tiny red, green, and blue filters.
Figure 1.24Orientation of liquid crystal molecules in a twisted nematic cell for several voltages, increasing from left to right. Light propagates in the vertical direction. Liquid crystal molecules are anchored to top and bottom horizontal planes. At 0 V (far left), the molecules twist in the horizontal plane about the z-axis (vertical). As the voltage increases (moving to the right), the molecules begin to tip out of the horizontal plane in the center of the cell, reducing their contribution to the retardance. At high voltages, the molecules at the center are rotated to the vertical plane, while molecules at the top and bottom remain anchored to the substrates. (Far right) Retardance variation (fast axis orientation and retardance magnitude) of the example liquid crystal cell across the field.
Liquid crystal cells, displays, and projectors present some of the most challenging polarization aberration problems. Undesired polarization variation in displays ends up as undesired color variation in the display. The eye is very sensitive to color variations; hence, undesired polarization variation must be kept to a minimum. One major cause of color variation in liquid crystal cells is a large variation of retardance with angle of incidence. The retardance magnitude can change with angle, the fast axis can change, and the ellipticity of the fast axis can change. An example of such retardance variation is shown in Figure 1.24 on the right. The ellipses denote the fast state and the magnitude of the ellipses indicates the retardance magnitude. Such variations, if not compensated, cause the color and brightness of displays to vary with angle of incidence, which is very distracting. This angle-dependent retardance is commonly fixed by adding another film of birefringent molecules, a biaxial multilayer film, typically fabricated with disk-shaped molecules, which can effectively compensate the rod-shaped liquid crystal molecules birefringence with angle as shown in Figure 1.25 (right).
Figure 1.25In all displays, liquid crystal cells are matched with compensating films to reduce the polarization aberrations. Here, the film on the top compensated the top half of the liquid crystal cell, and the compensating film on the bottom compensates the lower half, greatly improving the color quality of the display by reducing undesired polarization state variation and leakage through polarizers.
1.7Optical Design
Optical design is the engineering practice of finding good and useful combinations of optical elements. The subject is also referred to as lens design, because the design of lenses, telescopes, and microscopes and the understanding of their aberrations were one of the central research areas as optical design became formalized in the late 1800s.
Optical systems can be divided into imaging systems and other types of optical systems, such as illumination systems. Imaging systems are designed to take input spherical waves and transform them into spherical output waves. However, it is not possible with lenses and mirrors to transform the input waves from a finite area of the object into perfectly spherical output waves.1 Some deviation from sphericity for the exiting waves is inevitable with combinations of lenses and mirrors. These deviations from spherical wavefronts are the aberrations. The wavefronts are surfaces of constant phase and constant optical path length from the source. The optical path length can be thought of as the number of wavelengths along a path through an optical system, although its value is usually given in meters. Variations of the optical path length of a small fraction of a wavelength have significant impact on image quality. The priority in conventional optical design, by which we mean optical design without consideration of polarization, is the calculation of optical path length by ray tracing. Variations of optical path length have a much larger effect on image quality than variations of amplitude or polarization state. One of the most important tasks in optical design is to minimize the optical system’s aberrations over the desired range of wavelengths and object positions by optimizing the system. Control of wavefront aberrations is exquisite in many types of optical systems such as lenses for television and movie production and lenses for microlithography.
Consider an example cell phone lens (Figure 1.26; U.S. Patent 7,453,654 embodiment #3) and a set of ray paths calculated by ray tracing. Five collimated (parallel rays) beams of light are shown entering on the left in object space and are represented by rays, lines normal to the wavefront. An optical analysis program, Polaris-M2 in the case of this example, calculates the intersection of each ray with the first surface; these are the ray intercepts. Using Snell’s law, the ray directions are calculated inside the first lens; these are the propagation vectors. Then, the refracted ray is propagated until it intersects the second surface and the length of the ray is calculated between the first and second ray intercept. The product of the ray length and the refractive index is the optical path length for the ray segment. The process repeats, finding a ray intercept and then refracting the ray, until the ray exits the last surface into image space. The aperture stop for this lens is located at the first surface, where the different colored rays from each field intersect. More details on the process of ray tracing are found in Chapter 10 (see discussion on geometrical and polarization ray tracing).
Figure 1.26An example of a ray trace through a four-element cell phone camera lens. On-axis rays are in red. Rays from the off-axis fields are in green, blue, purple, and brown. A plane parallel IR-rejecting filter, blue, is located to the left of the image plane at the right.
To evaluate the lens’ image quality and aberrations, a set of rays are traced through the system to the image plane. The rays for the on-axis beam are seen to converge to a small area, almost to a single point, while the rays from the off-axis beams do not converge as well. To evaluate the aberrations, the optical path length for a grid of rays is calculated on a spherical surface, the reference sphere, centered on the image point, as shown in Figure 1.27. Figure 1.28 shows the wavefront aberration expressed in fractions of a wavelength for the on-axis object point. Figure 1.29 shows the wavefront aberration for the green-colored off-axis beam.
Figure 1.27The incident wavefront (purple) at an example lens (green) is spherical. The exiting wavefront converges to the image point (right). Intersecting the center of the exit pupil (lavender), a reference sphere (blue) is constructed centered on the image point. The exiting aberrated wavefront (purple, yellow lines) is a surface of constant optical path length from the object and entrance pupil. The separation along the ray paths between the reference sphere and aberrated wavefront is the wavefront aberration. When these surfaces coincide, the wavefront is spherical and aberration free. A “diffraction-limited wavefront” is generally regarded as remaining within one-quarter wavelength of the reference sphere.
Figure 1.28The wavefront aberration for the on-axis beam of Figure 1.26 in two presentation formats, an oblique plot (left) and a colored contour plot (right), is shown. About a quarter of a wavelength of spherical aberration, a fourth-order bowl-shaped aberration, is visible. An ideal spherical wavefront would have a flat wavefront aberration plot.
Figure 1.29The wavefront aberration for the off-axis beam of Figure 1.26 in two presentation formats has about four-tenths of a wave of coma and one wave of astigmatism and a quarter of a wave of spherical aberration.
The effect of these aberrations on the image is calculated by the methods of Fourier optics as described in Chapter 16 (Image Formation with Polarization Aberration). The image of a point source is called the point spread function or PSF, which is calculated by taking a Fourier transform of the wavefront at the exit pupil. Figure 1.30 shows two representations of the on-axis PSF, an oblique projection plot and a colored contour plot. This PSF has a form close to the PSF of an ideal wavefront, known as the Airy disk, but is enlarged by the spherical aberration. The peak intensity has been reduced to about 40% of the intensity of the PSF formed without aberration; thus, the image’s Strehl ratio is 0.4. Figure 1.31 shows two views of the off-axis PSF, where the PSF is much broader and the peak intensity is even further reduced because of the larger aberration.
Figure 1.30The on-axis PSF, the distribution of flux in the image of a point source, in (left) oblique perspective plot and (right) false-colored contour plot. Arbitrary flux units.
Figure 1.31The off-axis PSF in (left) oblique perspective plot and (right) false-colored contour plot. Because of the larger aberration for this beam, the peak flux is lower and the distribution of light is much broader (producing lower resolution) than the on-axis PSF of Figure 1.30.
In conventional optical design, the assumptions used to calculate the PSF in these examples are the following: (1) the transmission of all the rays is equal; (2) the output polarization state is constant across the exit pupil. The calculations consider only the effects of optical path length variation, the effect of wavefront aberration. These for Figures 1.28 to 1.31 are the assumptions that this book refers to as the assumptions of conventional optical design. The polarization aberrations have been neglected, which, since they are often small, is a fine approximation for many systems.
In fact, the transmission of rays does vary. Each ray has a different set of angles of incidence leading to a variation of transmittance at each interface. Further, the polarization state of the light is slightly changed upon refraction so the polarization state is not uniform in the exit pupil. These amplitude and polarization changes depend on the antireflection coatings used on each surface. Thus, to calculate the effect of the coatings with polarization ray tracing, the coatings must be specified as well as the lens shapes and refractive indices. Some coatings will cause much larger amplitude and polarization changes than others. For a system like this cell phone lens, the effect of the coatings on the wavefront aberration and PSF can be quite small. In these cases when the effects of the wavefront aberration are much larger than the effects of the amplitude and polarization aberration, the assumptions of conventional optical design are justified. To find out when these assumptions are justified, it is necessary to perform the appropriate extra polarization calculations, determining the amplitude, optical path length, and polarization changes at each ray intercept, and cascading these effects together into a polarization ray trace.
From the early 1960s through the mid-1990s, commercial optical design programs were based only on optical path length calculations. During this period, the conventional ray tracing assumptions were adequate for a majority of optical design calculations. But by the beginning of the twenty-first century, polarization calculations were needed in many optical design problems to accurately simulate advanced optical systems with high numerical aperture, to perform tolerance analyses on such polarization sensitive systems, and to understand the effects of optical coatings on the wavefront aberrations and polarization aberrations. Now, full-featured optical design programs allow coatings to be specified on optical surfaces, the output polarization states to be calculated, and the polarization properties of ray paths to be determined.
1.7.1Polarization Ray Tracing
The objective of polarization ray tracing is to calculate the polarization states exiting from optical systems and to determine the polarization properties, the diattenuation, retardance, and depolarization, associated with the ray paths. It is very useful to understand the light paths through optical systems in terms of the equivalent polarization elements. What are the polarization properties, the diattenuation, retardance, and depolarization, of the light paths? What would be the equivalent polarization elements, diattenuators and retarders, which reproduce the polarization state changes?
Polarization effects occur due to s- and p-component differences in reflection and refraction. The s- and p-components of the incident light are defined in Figure 1.32 (left and middle). Figure 1.32 (right) plots the intensity transmission coefficients, the fraction of light transmitted as a function of angle of incidence, calculated from the Fresnel equations, for refraction from air into uncoated surfaces with refractive indices of 1.5, 2, and 4. Antireflection-coated interfaces have similar curves but with generally improved transmission, closer to one.
Figure 1.32(Left) The definition of s and p components of a light beam at an interface. (Right) The Fresnel coefficients for transmission from air into lenses of different refractive indices for s light (green) and p light (blue) are a function of the angle of incidence. The difference in transmission is a source of diattenuation or partial polarization.
The principal polarization ray tracing method is the polarization matrix propagation method. A polarization matrix is calculated for each ray intercept and ray segment. Matrix multiplication cascades the polarizing interactions. Finally, a polarization matrix, such as a Jones matrix or Mueller matrix, is calculated for each ray path from object space to image space. This information is combined with the optical path length from conventional ray tracing and a variety of additional analyses performed.
The polarization matrix propagation method can determine the output polarization state for all incident polarization states and describe the diattenuation and retardance for the ray paths. Then, it is useful to understand why the polarization state changed and, if the magnitude is troublesome, what might be done about it. The simplest way to describe the ray paths is with Jones matrices, 2 × 2 matrices with complex elements, shown in Equation 1.2; here, matrix elements are expressed in both Cartesian complex number form and polar form,
1.2Stop and contemplate the consequences of this polarization matrix propagation method for the polarization optical designer and the other engineers who need to use and understand his work. Conventional optical design describes the aberration with the wavefront aberration function, a scalar function with one value at each point on the wavefront in the exit pupil. The polarization matrix propagation method replaces this scalar function with a Jones matrix at each point on the wavefront. This function is called the polarization aberration function or the Jones pupil. Going from a representation with one variable, optical path length, for each ray to a matrix with eight variables at each ray is a very substantial complexification! This book goes one small step at a time, taking several chapters to elaborate on all these degrees of freedom and provide guidance on how to use this additional information. Thus, we will learn to interpret the polarization aberrations and understand their effect on image formation and various measurements. It is no wonder that the early optical designers did not include the calculations for uncoated or coated lens surfaces and mirrors in their image quality calculations; it’s not easy.
And it gets more complicated!
1.7.2Polarization Aberrations of Lenses
The polarization aberration of lenses arises due to the effect of the Fresnel equations and thin film equations at their surfaces. For an on-axis spherical wave at a spherical surface, the angle of incidence increases approximately linearly from the center of the pupil and the plane of incidence is radially oriented as shown in Figure 1.33 (left). For most antireflection coatings, the difference between the transmission for light polarized in the p-plane (radially) and s-plane (tangentially) increases approximately quadratically, as is seen near the origin (left side) of Figure 1.32. Thus, an uncoated lens surface actually acts as a weak linear polarizer with a radially oriented transmission axis with an approximately quadratically increasing diattenuation, as shown in Figure 1.33 (right). Figure 1.34 shows the polarization pupil maps, series of polarization ellipses sampled around the exit pupil of an uncoated lens when 90° linear, 45° linear, and left circularly polarized beams enter the lens. In the left figure, the beam is brighter at the top and bottom, dimmer on the right and left, and the polarization is rotated toward the radial direction at the edge of the pupil along the diagonals. The middle pattern has the same form but is rotated about the center by 45°. When circularly polarized light is incident, the light becomes steadily more elliptical and less circular toward the edge of the pupil, and the ellipse’s major axis is oriented radially. If the uncoated lens is placed between crossed polarizers, light is extinguished along the two polarizer axes but leaks along the diagonals of the pupil as shown in Figure 1.35, a pattern known as the Maltese cross. An interesting pattern is observed when a Maltese cross beam is brought to focus. Because of this polarization aberration, the PSF, in the absence of any wavefront aberration, is dark in the center and along the x- and y-axes, but has four islands of light in one ring and dimmer islands of light further from the center as shown in Figure 1.36.
Figure 1.33(Left) The plane of incidence and angle of incidence functions for an on-axis spherical wave incident at a spherical surface are shown. The angle of incidence is radially oriented and increases linearly from the center. (Right) The diattenuation orientation and magnitude at a lens surface for an on-axis source are shown. The diattenuation is radially oriented and the magnitude increases quadratically. Diattenuation aberrations will be represented with brown lines.
Figure 1.34The effect of the diattenuation aberration of Figure 1.33 on incident (left) 90° linear, (center) 45° linear, and (right) left circularly polarized beams.
Figure 1.35When an uncoated lens is observed between crossed (x- and y-oriented) polarizers, light leaks; this is a Maltese cross pattern.
Figure 1.36The PSF of an uncoated lens between crossed (x- and y-oriented) polarizers in (left) an intensity plot and (right) an oblique perspective plot.
Most lens surfaces have antireflection coatings. For a lens surface with an antireflection layer of a quarter of a wavelength thick MgF2, the diattenuation is typically reduced to 1/5 of its uncoated value, providing a substantial reduction in polarization aberration. Typically, a very small amount of retardance is also introduced.
For a multi-element lens, the diattenuation and retardance contributions accumulate. Both positive and negative lenses introduce diattenuation of the same sign. Consider the pair of microscope objectives in Figure 1.37 where collimated light enters the first objective, comes to a focus between the two objectives, and is collimated by the second objective. This pair of low polarization microscope objectives has a numerical aperture of 0.55. Figure 1.38 shows the measured polarization aberrations of the low polarization microscope objective pair. The diattenuation, which here has a larger effect than the retardance, reaches 0.09 at the edge of the pupil. Even at this level of low polarization aberration, an almost 10% polarizer, when the first lens is illuminated with collimated linearly polarized light, and the exiting light is blocked with an orthogonal polarizer, the pupil-averaged leakage has a Maltese cross pattern.
Figure 1.37The optical layout for a polarization aberration measurement in an imaging polarimeter where collimated light enters one microscope objective, comes to focus, and exits the second objective collimated.
Figure 1.38Polarization aberration measurement of a pair of microscope objectives for an on-axis beam shows the (left) diattenuation aberration distribution, which reaches a maximum of 0.09, and the (right) retardance aberration distribution, which reaches a maximum of 3° or 0.09 waves. Both aberrations have small polarization aberration near the center of the pupil where angles of incidence are small. The measurement is performed with a Mueller matrix polarimeter.
The presence of retardance in an optical system indicates that the system has polarization-dependent optical path lengths and thus will have different interferograms in different polarization states. The retardance polarization aberration pattern of spherical, parabolic, ellipsoidal, and hyperbolic mirrors with metal coatings seen on-axis and of associated optical systems like Cassegrain telescopes has a tangentially oriented fast axis with a retardance magnitude that increases from the center of the pupil. Figure 1.39 (left) shows the form of retardance aberration of on-axis spherical and conic mirrors. The phase advances for polarization states parallel to the lines and decreases for polarization states perpendicular to the lines. Thus, for a spherical wavefront with 90° linearly polarized light, the wavefront becomes deformed like Figure 1.39 (right). For on-axis sources, because of the retardance, these metal mirrors, Cassegrain telescopes, and similar optics have a different quadratic phase variation along the two axes; the mirrors have introduced style="font-weight: bold; color: #EB0024"astigmatism into the on-axis beam, something that is not calculated by conventional ray tracing! This astigmatism is oriented with the plane of polarization of the incident light; when the polarization state is rotated, the astigmatic wavefront aberration rotates with it. Fortunately, common optical systems like Cassegrain telescopes usually have less than a tenth of a wave of this metal coating-induced astigmatism and so this source of astigmatism is not a high priority to optical designers. Nonetheless, this source of astigmatism should be understood because it is easily seen in interferometric tests and begs for an explanation when it appears.
Figure 1.39(Left) Retardance aberration of spherical or parabolic mirrors illuminated on-axis; lines indicate the orientation of the retardance fast axis and magnitude of the retardance over the pupil. (Right) An astigmatic wavefront results when a spherical wavefront of 90° linearly polarized light interacts with the retardance aberration (left). White indicates the optical path length of the chief ray. Violet shows shorter optical path lengths and green shows longer optical path lengths.
1.7.3High Numerical Aperture Wavefronts
Numerical aperture characterizes the range of angles over which an optical system can accept or transmit light; F-number or F/# describes the same property. Beams with high numerical aperture, a large cone angle, are valuable in optics because they can focus light into smaller images. Hence, there is a constant push for systems with still higher and higher numerical aperture.
High numerical aperture beams must have polarization variations, because the polarization state, which is transverse to the wavefront, cannot remain uniform in three dimensions; it must curve around the sphere. These intrinsic high numerical aperture polarization state variations are frequently detrimental, broadening the image from the ideal diffraction-limited patterns.
Consider a hemispherical light beam, which corresponds to a numerical aperture of one, which subtends a solid angle of 2π steradians. For example, when x-polarized light is incident on such a high numerical aperture lens, the exiting polarization is of the form shown in Figure 1.40. Near the z-axis, the optical axis in the center of the beam, the polarization is nearly uniformly polarized. Along the y-axis, the light can remain polarized in the x-direction all the way to the edge of the pupil, since these vectors along x are tangent to the sphere. Along the x-axis, the light must tip upward, with a negative z-component, and downward, with a positive z-component, to remain on the surface of the sphere. The polarization state of the light continues to rotate until, at the right and left sides of the pupil in Figure 1.40 (left), the light becomes polarized in the ±z-direction, since here the light is propagating in the ±x-direction. Around the edge of the pupil, the polarization varies as shown in Figure 1.41. The result of this polarization variation is a PSF that becomes elongated in one direction, much like astigmatism. Figure 1.40 shows only one way that the polarization might vary in a high numerical aperture beam.
Figure 1.40A high numerical aperture (NA) spherical wave linearly polarized in the x-direction (left) viewed along the z-axis, (center) viewed along the y-axis, and (right) viewed along the x-axis. This polarization is aligned with the double pole coordinates of Section 11.4 with a double pole located on the -z-axis.
Figure 1.41The distribution of linearly polarized light around the edge of the hemispherical wavefront of Figure 1.40.
There is great interest in microlithography and microscopy for other polarization distributions with useful imaging properties, particularly the radial and tangentially polarized beams shown in Figure 1.42. Note that these light states cannot be extended to the origin without discontinuity and thus are created with a dark spot in the center.
Figure 1.42The polarization distributions in (left) a tangentially polarized wavefront and (right) a radially polarized wavefront.
1.8Comment on Historical Treatments
The transverse nature of light waves and the properties of polarization have played a central role in the development of optics and physics. A very nice summary of the history of polarized light is contained in Goldstein’s Polarized Light4 before Chapter 1. Another summary is Brosseau’s Fundamentals of Polarized Light, A Statistical Optics Approach in Part 1 (“Historical Survey of Understanding of Polarized Light”). The understanding of polarized light and diffraction advanced rapidly from 1800 through 1830, when consensus finally emerged that light was a transverse wave. A compelling account of the scientific controversies during this formative period is found in Buchwald’s book, The Rise of the Wave Theory of Light.4 An 1842 book, Lectures on Polarized Light by Pereira, available online, documents the sophisticated understanding of polarized light in the first half of the nineteenth century.5
Polarized light optics took a giant step forward with the invention of Polaroid plastic sheet polarizer. Before the invention of Polaroid, polarizers tended to be small and expensive, such as the Nichol prism. The availability of large inexpensive sheet polarizers and retarders helped propel a rapid advance in polarization optics and related fields. The history of dichroic polarizers is described by Land and West6 and by Grabau.7
1.9Reference Books on Polarized Light
The following is a short list of books on polarized light or texts with significant polarized light sections that the authors feel students would find helpful. Goldstein contains a thorough treatment of polarization mathematics, discussion of Fresnel equations, ellipsometry, and many other topics. Können is a good starting point for introductory users such as high school and undergraduate students with a nonmathematical discussion of polarized light. A free version of Können8 is available online. Mansuripur provides a nonmathematical discussion of many polarization effects at the graduate level.9
Shurcliff, W. A., Polarized Light. Production and Use, Cambridge, MA: Harvard University Press, 1966.
Azzam, R. M. A. and Bashara, N. M., Ellipsometry and Polarized Light, 2nd edition Amsterdam: Elsevier, 1987.
Kliger, D. S. and Lewis, J. W., Polarized Light in Optics and Spectroscopy, Elsevier, 1990.
Können, G. P., Polarized Light in Nature, CUP Archive, 1985.
Hecht, E., Optics, 4th edition, Addison Wesley Longman, 1998.
Brosseau, C., Fundamentals of Polarized Light: A Statistical Optics Approach, New York: Wiley, 1998.
Born, M. and Wolf, E., Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th expanded edition, Cambridge University Press, 1999.
Mansuripur, M., Classical Optics and Its Applications, 2nd edition, Cambridge University Press, 2009.
Collette, E., Field Guide to Polarization, SPIE Press, 2006.
Cloude, S., Polarisation: Applications in Remote Sensing, Oxford University Press, 2009.
Goldstein, D., Polarized Light, Revised and expanded 3rd edition, Vol. 83, Boca Raton, FL: CRC Press, 2011.
Horváth, G., Polarized Light and Polarization Vision in Animal Sciences, Springer Series in Vision Research (2), 2014.
1.10Problem Sets
Why do monochromatic waves have periodic electric fields? Why must the electric field of a monochromatic plane wave trace an ellipse in the transverse plane?
Draw the polarization ellipse for linearly polarized light oriented at 30°. Plot Ex(t) and Ey(t), the x- and y-components in which plane does the magnetic field oscillate?
In Figure 1.34, how does the major axis of the polarization ellipses vary moving around the edge of the pupil for 90° incident polarization? For 45° polarization? For left circular polarization?
Using a sheet polarizer, rotate a pair of polarizing sunglasses and estimate the alignment of the transmission axis for the left and right lenses. Is the axis horizontal or vertical? Is the axis exactly the same or can a small difference be observed?
Illuminate a projection screen with a liquid crystal projector with polarized output.
How is the light from the projector polarized in the red, green, and blue bands?
Examine the light scattered from the screen with a rotating polarizer. Project a red, green, and then blue scene and visually estimate the degree of linear polarization of the scattered light.
Does the polarization depend on the angle of scatter?
Study the polarization properties of an artificial rainbow with a linear polarizer. Face away from the sun. Create a mist of water with a hose, lawn sprinkler, or mister. Working from a balcony or place where you can look downward into the mist against a dark background provides the best results. View the rainbow through the polarizer. Produce a diagram of the polarization of the rainbow across the arc, showing the orientation of the linear polarization. The degree of polarization is very difficult to estimate visually but is probably near 90%. Does the degree of linear polarization appear to vary for the different colors?
For a lens spherical surface illuminated from infinity with a collimated beam on-axis, does the angle of incidence increase approximately linearly or quadratically from the center?
Consider the linearly polarized hemispherical wavefront in Figure 1.40.
In which plane through the origin are all the E-fields pointing in the same direction?
At which opposite points in the pupil are the electric fields in opposite directions?
How does the electric field rotate as one moves around the edge of this hemispherical wavefront?
Where in Figure 1.22 is the stress the greatest?
A particular diattenuator, or partial polarizer, has a maximum transmittance Tmax = 0.7 and a diattenuation D = 0.999. Find Tmin.
Contrast ratio is defined as
. Find an expression for the diattenuation as a function of the contrast ratio. When the diattenuation is 0.999, what is the contrast ratio?At 600 nm, a particular quartz plate has three waves of retardance and a MgF2 plate has four waves or retardance, yielding a one-wave retarder. Referring to Figure 1.19, at what wavelength will the combination be a 3/4 wave retarder? A 5/4 wave retarder?
Take two sheet polarizers and cross their transmission axes. Take pictures as they rotate about the horizontal axis, vertical axis, and diagonal axes. Describe how the leakage, the lack of extinction varies.
Consider light passing through two crossed polarizers aligned along x = (1, 0, 0) and y = (0, 1, 0) propagating along the direction k = (sinϕ cosθ, sinϕ sinθ, cosϕ). By projecting the two polarizers onto the transverse plane, find the apparent angle χ between their absorption axes. The transmission through the polarizer pair is T = cos2χ. Perform a Taylor series of T in the angle of incidence θ along the diagonal ϕ = 45° to determine the lowest-order polynomial variation in transmittance.
Malus’s law states that when a perfect linear polarizer is placed in a linearly polarized beam, the fraction of light transmitted (F) is given by: F = cos2 θ, where θ is the angle between the incident polarization and the transmission axis of the polarizer. Consider N polarizers placed one after another (a cascade) in a beam polarized at 0°. The first polarizer is oriented horizontally (0°), and each subsequent polarizer is rotated a fixed amount relative to the previous polarizer such that the last polarizer is always vertically oriented (90°). For example, if N = 4, the orientations of the polarizers would be (0°, 30°, 60°, 90°). Using Malus’s law, what is the fraction of light transmitted through 2, 4, and 8 polarizers arranged in this fashion? How does the transmission behave as N increases? (If light passes through a linear polarizer at β, the exiting light is linearly polarized at β with an attenuated magnitude.) If the incident light is linearly polarized at α, through the linear polarizer at β, the exiting light is linearly polarized at β with an attenuated magnitude cos2 (α − β).
References
1S. Johnsen and T. Frank, Polarization Vision, Operation Deep Scope (2005). S. Johnsen using images from E. Widder, NOAA (http://oceanexplorer.noaa.gov/explorations/05deepscope/background/polarization/media/eel.html, accessed on July 15, 2017).
2R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, Elsevier Science (1987).
3I. Ohlidal and D. Franta, Ellipsometry of thin film systems, in Progress in Optics, Vol. 41, ed. E. Wolf, Elsevier (2000), pp. 181–282.
4J. Z. Buchwald, The Rise of the Wave Theory of Light, Optical Theory and Experiment in the Early Nineteenth Century, The University of Chicago Press (1989).
5J. Pereira, On the polarization of light, and its useful applications, Pharm. J 2 (1842): 619–637. (https://play.google.com/store/books/details?id=OylbAAAAcAAJ&rdid=book-OylbAAAAcAAJ&rdot=1, accessed October 25, 2016.)
6E. H. Land and C. D. West, Dichroism and dichroic polarizers, Colloid Chemistry 6 (1946): 160–1905.
7M. Grabau, Polarized light enters the world of everyday life, Journal of Applied Physics 9.4 (1938): 215–225.
8G. P. Können, Polarized light in nature, CUP Archive (1985). (http://s3.amazonaws.com/guntherkonnen/documents/249/1985_Pol_Light_in_Nature_book.pdf?1317929665, accessed October 25, 2016.)
9M. Mansuripur, Classical Optics and Its Applications, Cambridge: Cambridge University Press (2002).
1 There are a few exceptions such as Maxwell’s fisheye lens and the Luneburg lens with a curved object and curved image surface. These exceptions are not suitable for camera lenses, cell phone lenses, and most imaging applications.
2 Polaris-M is a polarization ray tracing program available from Airy Optics, Tucson, AZ.