With the increasing interest in using freeform surfaces in optical systems due to the novel application opportunities and manufacturing techniques, new challenges are constantly emerging. Optical systems have traditionally been using circular apertures, but new types of freeform systems call for different aperture shapes. First non-circular aperture shape that one can be interested in due to tessellation or various folds systems is the rectangular one. This paper covers the comparative analysis of a simple local optimization of one design example using different orthogonalized representations of our freeform surface for the rectangular aperture. A very simple single surface off-axis mirror is chosen as a starting system. The surface is fitted to the desired polynomial representation, and the whole system is then optimized with the only constraint being the effective focal length. The process is repeated for different surface representations, amongst which there are some defined inside a circle, like Forbes freeform polynomials, and others that can be defined inside a rectangle like a new calculated Legendre type polynomials orthogonal in the gradient. It can be observed that with this new calculated polynomial type there is a faster convergence to a deeper minimum compared to “defined inside a circle” polynomials. The average MTF values across 17 field points also show clear benefits in using the polynomials that adapted more accurately to the aperture used in the system.
In this work, we present a novel imaging design formed by two optical surfaces with rotational symmetry. In these designs, both object and image shapes are given but mapping from object to image is obtained through the design process. In the examples considered, the image from a planar object surface is virtual and located at infinity and is seen from a known pupil, which can emulate a human eye. The differential equation method is used to provide single optical surface imaging designs by considering the local properties of the imaging surface and the wavefronts. In the first introductory part, both the rotational symmetrical and the freeform single surface imaging designs are presented using the differential equation method. In these designs, not only the mapping is obtained in the design process, but also the shape of the object is found. In the second part, the method is extended to two surface designs with rotational symmetry and the astigmatism of the image has been studied. By adding one more optical surface to the system, the shape of the rotational symmetrical object can be designed while controlling the tangential rays and sagittal rays simultaneously. As a result, designs without astigmatism (at the small pupil limit) on a planar object surface have been obtained.
In this work, we demonstrate how it is possible to sharply image multiple object points. The Simultaneous Multiple Surface (SMS) design method has usually been presented as a method to couple N wave-front pairs with N surfaces, but recent findings show that when using N surfaces, we can obtain M image points when N<M under certain conditions. We present the evolution of SMS method, from its basics, to imaging two object points through one surface, the transition from two to three objet points obtained by increasing the parallelism, and getting to the designs of six surfaces imaging up to eight object points. These designs are limited with the condition that the surfaces cannot be placed at the aperture stop. In the process of maximizing the object points to sharp image, we try to exhaust the degrees of freedom of aspherics and free-forms. We conjecture that maximal SMS designs are very close to a good solution, hence using them as a starting point for the optimization will lead us faster to a final optical system. We suggest here different optimization strategies which combined with the SMS method are proven to give the best solution. Through the example of imaging with the high aspect ratio, we compare the results obtained optimizing the rotational lens and using a combination of SMS method and optimization, showing that the second approach is giving significantly smaller value of overall RMS spot diameter.
In this work, novel imaging designs with a single freeform optical surface (either refractive or reflective) are presented. In these designs, not only the mapping is obtained in the design process, but also the shape of the object is found. In the examples considered, the image is virtual and located at infinity and is seen from known pupil, which can emulate a human eye. In the first introductory part, 2D designs and 3D designs by rotation using the differential equation method for the limit case of small pupil have been reviewed. Furthermore, the differential equation method is used to provide the freedom to control the tangential rays and sagittal rays simultaneously. In the second part, according to the study of astigmatism of different types of design with rotational symmetry, the differential equation method for 3D rotational design without astigmatism (at the small pupil limit) on a curved object surface has been extended to 3D freeform design. The result of this extended method has been proved to coincide with the former 3D design by rotation which is a special case of 3D freeform design. Finally, the initial condition has been used as an additional freedom to control the shape of the object surface. As a result, a reflective design with a much flatter object surface has been obtained.
In this work, novel imaging designs with a single optical surface (either refractive or reflective) are presented. In some of
these designs, both object and image shapes are given but mapping from object to image is obtained as a result of the
design. In other designs, not only the mapping is obtained in the design process, but also the shape of the object is found.
In the examples considered, the image is virtual and located at infinity and is seen from known pupil, which can emulate
a human eye.
In the first introductory part, 2D designs have been done using three different design methods: a SMS design, a
compound Cartesian oval surface, and a differential equation method for the limit case of small pupil. At the point-size
pupil limit, it is proven that these three methods coincide.
In the second part, previous 2D designs are extended to 3D by rotation and the astigmatism of the image has been
studied. As an advanced variation, the differential equation method is used to provide the freedom to control the
tangential rays and sagittal rays simultaneously. As a result, designs without astigmatism (at the small pupil limit) on a
curved object surface have been obtained. Finally, this anastigmatic differential equation method has been extended to
3D for the general case, in which freeform surfaces are designed.
While multichannel configurations are well established for non-imaging applications, they have not been used yet
for imaging applications. In this paper we present for the first time some of multichannel designs for imaging
systems. The multichannel comprises discontinuous optical sections which are called channels. The phase-space
representation of the bundle of rays going from the object to the image is discontinuous between channels. This
phase-space ray-bundle flow is divided in as many paths as channels there are but it is a single wavefront both at the
source and the target. Typically, these multichannel systems are at least formed by three optical surfaces: two of
them have discontinuities (either in the shape or in the shape derivative) while the last is a smooth one. Optical
surfaces discontinuities cause at the phase space the wave front split in separate paths. The number of discontinuities
is the same in the two first surfaces: Each channel is defined by the smooth surfaces in between discontinuities, so
the surfaces forming each separate channel are all smooth. Aplanatic multichannel designs are also shown and used
to explain the design procedure.
New ultra-thin optical designs are presented. They are formed by optical sections (called channels) working in parallel
(multichanneling) to provide the desired optical function. The phase-space representation of the bundle of rays going
from the source to the target is discontinuous between channels. This phase-space ray-bundle flow is divided in as many
branches as channels there are but it is a single trunk at the source and at the target. Typically, these multichannel
devices are at least formed by three optical surfaces: two of them have discontinuities (in the shape or in the shape
derivative) while the last one is smooth. The discontinuities of the optical surfaces are causing the separation of the flow
in branches (in the phase space). The number of discontinuities is the same in the two first surfaces: Each channel is
defined by the smooth surfaces in between discontinuities, so the surfaces forming each separate channel are all smooth.
No diffractive analysis is done.
New ultra-thin optical designs are presented that comprise discontinuous optical sections (called channels) working in
parallel (multichanneling) to provide the desired optical function. Aplanatic (a particular case of SMS-design)
multichannel designs are also shown and used to explain more easily the design procedure. Typically, these multichannel
devices are at least formed by three optical surfaces: one of them has discontinuities in the shape, a second one may have
discontinuities in its derivative while the third one is smooth. The number of discontinuities is the same in the two first
surfaces: Each channel is defined by the smooth surfaces in between the discontinuities, so that the surfaces forming
each separate channel are all smooth. No diffractive effects are considered.
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