Redistribution of light from a given source in order to create a prescribed intensity pattern on a given set in 3D space is a task arising in numerous applications. Designing optical systems with capabilities to perform such tasks reliably in a wide variety of applications is the overall goal of much of research in optics. An important practical case is the “laser beam shaping problem” which requires transformation of a Gaussian beam from a laser into a beam with “flat top” intensity profile. It has been shown by John A. Hoffnagle and C. Michael Jefferson that a pair of plano-aspheric lenses can be used to transform a collimated, radially symmetric, Gaussian beam to a radially symmetric, collimated beam with “flat-top” intensity profile. Here we discuss this design problem when a priori requirement of radial symmetry is not imposed neither on the input/output radiances nor on the geometry of the input/output beams. The solution in this case is sought among “freeform” optical surfaces.
Design of freeform refractive lenses is known to be a difficult inverse problem. But solutions, if available, can
be very useful, especially in devices required to redirect and reshape the radiance of the source into an output
irradiance redistributed over a given target according to a prescribed pattern. In this report we present results
of theoretical and numerical analysis of refractive lenses designed with the Supporting Quadric Method. It is
shown that such freeform lenses have a particular simple geometry and qualitatively their diffractive properties
are comparable with rotationally symmetric lenses designed with classical methods.
Freeform optics is a relatively new field; it uses the methods necessary to describe surfaces lacking symmetry, and/or surfaces that create non-symmetrical irradiance distributions. The Supporting Quadrics Method (SQM) developed by Oliker is a superb for generating any desired irradiance distribution. The SQM uses an envelope of quadrics to create prescribed irradiance distributions. These optical systems are tested in ray trace software, where diffraction effects are not taken into account. It is important to understand the diffraction effects present in an optic, when moving from the ray trace stage to the prototype stage. Here we study the diffraction effects of Supporting Quadrics Method.
KEYWORDS: Optical design, Lens design, Freeform optics, Geometrical optics, Collimation, Near field optics, Lenses, Nonimaging optics, Near field, Ray tracing
Structural color is produced when nanostructures called schemochromes alter light reflected from a surface through different optic principles, in contrast with other types of colors that are produced when pigments selectively absorb certain wavelengths of light. Research on biogenic photonic nanostructures has focused primarily on bird feathers, butterfly wings and beetle elytra, ignoring other diverse groups such as spiders. We argue that spiders are a good model system to study the functions and evolution of colors in nature for the following reasons. First, these colors clearly function in spiders such as the tarantulas outside of sexual selection, which is likely the dominant driver of the evolution of structural colors in birds and butterflies. Second, within more than 44,000 currently known spider species, colors are used in every possible way based on the same sets of relatively simple materials. Using spiders, we can study how colors evolve to serve different functions under a variety of combinations of driving forces, and how those colors are produced within a relatively simple system. Here, we first review the different color-producing materials and mechanisms (i.e., light absorbing, reflecting and emitting) in birds, butterflies and beetles, the interactions between these different elements, and the functions of colors in different organisms. We then summarize the current state of knowledge of spider colors and compare it with that of birds and insects. We then raise questions including: 1. Could spiders use fluorescence as a mechanism to protect themselves from UV radiation, if they do not have the biosynthetic pathways to produce melanins? 2. What functions could color serve for nearly blind tarantulas? 3. Why are only multilayer nanostructures (thus far) found in spiders, while birds and butterflies use many diverse nanostructures? And, does this limit the diversity of structural colors found in spiders? Answering any of these questions in the future will bring spiders to the forefront of the study of structural colors in nature.
The design equations of a freeform single lens optical systems for redistributing the input radiation in a prescribed manner over a given target set are presented in explicit form. These equations are derived here and their analysis is carried out in geometrical optics approximation. The qualitative properties of solutions to such problems are discussed in detail and illustrated by examples. Difficulties arising in numerical computations of solutions in the rotationally symmetric case are also noted.
Canonical illumination control problems are studied through the calculation of an appropriate ray mapping. We show that when the problem enjoys certain symmetries, the ray mapping can be calculated independently of the lens design. Once the ray mapping is known, the lens can be constructed. Such a separation of ray mapping and lens determination greatly simplifies the design task. We provide a few examples to illustrate this concept.
Problem definition: Transfer maximum monochromatic flux from a 1-mm-square Lambertian source in air to an equal-étendue nonimmersed
target. The target surface is rectangular with a 16:9 aspect ratio. The surface area of the target must be at least 4 mm2.
The target is defined such that only rays incident on the target surface at angles of θmax or less, relative to the surface
normal, are considered to be within the phase space of the target, where the value of θmax is determined by the equal-étendue
requirement.
The design equations of freeform two-lens optical systems for redistributing the input radiation in a prescribed
manner over a given target set are presented in explicit form. These equations are applied to derive sensitivity measures of the output radiation to figure errors. The general results are applied in several special cases validated against known results for rotationally symmetric lenses.
The problem of design of a reflector system for redirecting and reshaping the irradiance distribution of a point source over a prescribed target set in the near-field is considered. The solution of this problem is developed under the geometrical optics (GO) approximation. The novelty is that our method is not limited to the usual input and/or output irradiance profiles which are rotational or rectangular symmetric. Consequently, the designer has more flexibility in satisfying the requirements of a practical design problem.
Numerous optical and electromagnetic applications require numerical design of reflecting surfaces in 3D with capabilities to redirect the input energy flow and reshape the energy radiation intensity of a source into a prescribed output irradiance distribution over a specified target surface. In the geometrical optics approximation, a systematic application of the ray tracing equations and energy conservation law reduces the problem, in many cases, to finding numerical solutions to nonlinear, second order partial differential equations. If the severe limitation of rotational symmetry is not assumed then the resulting equations are very far from being standard and require significant efforts for their theoretical investigation and reliable numerical solution. In recent years a quite general approach combining geometric techniques with methods from calculus of variations has been developed and applied to a rigorous and unified investigation of several classes of such equations. Moreover, this approach allows implementations in provably convergent numerical algorithms. In this paper I outline this approach in the problem of designing a reflecting surface capable of redirecting the energy flow from a point source so that the reflected rays have directions specified in advance as a subset on the far-sphere and the output irradiance density is also pre-specified in advance as a function of the reflected direction. A numerical example illustrating the solution is also presented.
The problem of design of a two-mirror optical systems for reshaping the irradiance distribution of a laser beam in a prescribed manner is considered. These designs are developed under the geometrical optics approximation. The novelty is that our method is not limited to input and/or output radiance profiles which are rotational or rectangular symmetric. Moreover, the method leads to two beam-shaping systems for one of which the first mirror is concave and the second is convex.
KEYWORDS: Mirrors, Color center lasers, Beam shaping, Reflectors, Ray tracing, Crystals, Laser systems engineering, Wavefronts, Color centers, Laser crystals
A two-mirror asymmetrical system has been designed to reshape a laser beam into an elliptic uniform beam profile for use in a new superbroadband color center laser system. The surface points of the two mirrors are obtained by solving a second order partial differential equation of the Monge-Ampeere type which is based on conversation of energy and the law of reflection. The technique is illustrated by presenting the design of a two-mirror system which converts a circular Gaussian beam into an elliptic beam with a uniform profile. Such a reshaper system is planned for use in the development of superbroadband or control generation spectrum color center lasers, which have a significant potential for optical communications, signal multiplexing, and information coding as well as laser spectroscopy, photochemistry, medicine, color microscopy, and holography.
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