Optimized configuration for liquid crystal Stokes polarimeters in the presence of fast-axis orientation errors.

We present an optimal configuration for Stokes polarimeters based on liquid crystal variable retarders, with the minimum number of measurements. Due to the inherent variations of the director orientation of the liquid crystal molecules, we propose a configuration that minimizes the sensibility of the polarimeter to fast-axis variations. For the optimization we consider a scheme that maximizes the volume of a tetrahedron inscribed in the Poincare sphere, to address additive and Poisson noise, with one of the vertices invariant to changes in the axis positions. We provide numerical simulations, considering misalignment errors, to analyze the robustness of the configuration. The results show that the proposed configuration helps to maintain the volume enclosed by the tetrahedron with high tolerance to fast-axis orientation errors. The condition number will remain below 3.07 for common misalignment errors and below 1.88 for more controlled liquid crystals. This optimization will improve the performance of liquid crystals polarimeters, with a more robust configuration that also considers misalignment errors, beyond additive and Poisson noise.

Caustics and wavefronts produced by multiple refraction-reflections through plane-parallel plates.

We study the formation of caustics and wavefronts produced by multiple refraction-reflections through a plane-parallel transparent plate, assuming a point source placed at an arbitrary position along the optical axis. The caustic surfaces are obtained by using the envelope's method. Subsequently, the wavefronts are directly related to the involutes, which are associated with the envelopes for all the rays. Hence by using the Malus-Dupin theorem, we obtain their respective wavefronts produced by multiple refraction-reflections through a plane-parallel transparent plate. On the other hand, we implement Huygens' principle to obtain the wavefronts leaving the plate after undergoing multiple reflections inside the plate, which we have called zero-distance phase wavefronts. Finally, we establish the correspondence between the wavefronts obtained by Huygens' principle and the involutes associated with caustic surfaces; they are brought in coincidence assuming parallel curves from each other.

Equations for designing a dialyte with minimal TSA based on the caustic surfaces.

A method to design a dialyte based on the caustic surfaces is presented, with correction at the third-order spherical aberration. We predefined, a priori, the value for the back focal length of the separated doublet for two different wavelengths to calculate their radii of curvature, which depend on all the parameters involved in the refraction process. Additionally, we approached the caustic surface by applying a Taylor series, which provided the analytical spherical aberration coefficient at the third order. Two alternatives to design achromatic separated doublets are provided. Finally, we extended the method to design achromatic cemented triplets.

Effects of typical liquid-crystal retarder errors on optimized Stokes polarimeters.

Liquid crystal variable retarders (LCVRs) are often used in Stokes polarimeters as they allow the measurement of different polarization components by applying an electric field that manipulates the induced retardance. However, the optical retardance introduced by these devices is in general not homogenous across the aperture. Another problem with this type of devices is that the fast-axis orientation is not homogenous, and it changes with the applied voltage. For the optimization of polarimeters, in terms of the noise amplification from the intensity measurements to the polarimetric data, the condition number (CN) is often used, but the effects of LCVR spatial variations are not considered. This paper analyzes the impact of errors in LCVRs in a set of optimized Stokes polarimeters simulated by adding errors in the induced retardance and fast-axis orientation. Then, the CN is calculated to observe the effect of these errors on the optimization. We show how errors in the LCVRs lead to different impacts in the polarimetric measurements for different optimized polarimeters, depending on their experimental parameters. Furthermore, we present the propagation error theory to choose the best experimental parameters to reduce the nonideal effects in optimized polarimeters.

Approach for designing thick cemented doublet lenses based on the caustic surface.

A new, to the best of our knowledge, method for designing a thick-lens achromatic doublet based on the concept of a caustic surface to correct both third- and fifth-order spherical aberration is presented. We consider two different wavelengths brought into coincidence at the back focal length instead of the effective focal length as it is usually done, to calculate the radii of curvature assuming predefined values for axial thicknesses and their indices of refraction for both lenses. Alternatively, we apply Taylor's series around the optical axis, and to vanish the approximate caustic surface, we obtain the values for the conic constants, which reduce at third- and fifth-order spherical aberration. Two designs for cemented doublets are provided assuming that the lenses are cemented. Finally, we propose a method to qualitatively test doublet lenses by using null screens, considering to place the detection plane at arbitrary positions.

Null screens to evaluate the shape of freeform surfaces: progressive addition lenses.

We propose a method for measuring the shape of freeform surfaces such as Progressive Addition Lenses (PAL). It is based on optical deflectometry by considering a non-uniform pattern of spots computed by using the null-screen method. This pattern is displayed on a flat LCD monitor being reflected on the freeform under test and whose image is recorded on a CCD camera placed at a predefined off-axis position. We use one image to calibrate the experimental setup and another to measure the freeform surface. We develop an iterative algorithm to retrieve the surface under test and calculate the spherical and cylindrical dioptric powers of the frontal freeform of a commercial PAL under test.

Ronchi-Hartmann type null screens for testing a plano-freeform surface with a detection plane inside a caustic surface.

We have implemented an exact ray trace through a plano-freeform surface for an incident plane wavefront. We obtain two caustic surfaces and provide the critical points related to the ray tracing process. Additionally, we study the propagation of the refracted wavefronts through the plane-curved surface. Finally, by using the Ronchi-Hartmann type null screen and placing the detection plane within the caustic region, we have evaluated the shape of a plano-freeform optical surface under test, obtaining an RMS difference in sagitta value of 6.3 µm.

Exact equations to measure highly aberrated wavefronts with the Hartmann test.

An exact vector expression for the deformations of a wavefront from any chosen reference surface, as a function of the directions of the real and reference rays, is deduced. It can be used with slope measuring test methods, such as Hartmann or Ronchi tests, but the need for a spherical reference is removed. We present simulated and experimental results to show the feasibility of this proposal.

Nonimaging metaoptics.

So far, metalenses have only been studied in imaging optics, where a point from the object space is mapped to a corresponding point in the image space. Here we explore metalenses and metamirrors for dealing with the optimal transfer of light energy. Owing to its compactness and high design flexibility, metasurface-based flat optics may open new opportunities in the nonimaging field, which deals with light concentration and illumination. The building blocks of metalenses are subwavelength-spaced scatterers. By suitably adjusting their shape, size, position, and orientation, one can control the light spatial distribution, as is desired in nonimaging problems. In this Letter, we introduce nonimaging metaoptics, review its basics, and briefly explore three cases: the compound-metasurface concentrator [analogous to the compound parabolic concentrator (CPC)], the total internal reflection (TIR) metalens (analogous to the TIR lens), and a simple condensing metalens.

Singlet lens for generating aberration-free patterns on deformed surfaces.

We introduce a general closed-form analytic formula to design special lenses that generate spherical aberration-free extended images specified previously by the user. The formula considers arbitrary and non-conventional patterns. The formalism is tested with well-established ray tracing techniques.

Sagittal and tangential foci produced by tilted plane wavefronts refracted through simple lenses.

We study the formation of caustic and wavefront surfaces produced by a tilted plane wavefront propagating through spherical positive lenses. The shape of the caustic surface is a function of the indices of refraction, the geometrical parameters of the lens involved in the process of refraction, and the obliquity angle with respect to the optical axis, as we expect. We provide exact and approximate analytic equations for tangential and sagittal focal surfaces and also for Petzval field curvature considering arbitrary lenses.

Huygens principle: exact and approximate wavefronts propagated through conic lenses.

Exact and approximate formulae for refracted wavefronts through singlet lenses are obtained by considering an incident plane wavefront propagating along the optical axis. We provide two different approaches for the wavefronts approximated at the second order based on the Huygens principle and the Malus-Dupin theorem. We have in the first method found a way to use iterative wavelets instead of the usual evaluated integral to arrive at these formulae, showing a physical and mathematical correspondence between both methods. Finally, we introduce a parabolic wavefront into the irradiance transport equation in order to provide an analytical solution for the distribution of illumination.

Analytic conic constants to reduce the spherical aberration of a single lens used in collimated light.

We study the formation of caustic surfaces produced by conic lenses, considering a plane wavefront propagating parallel to the optical axis. The shape of the caustic can be modified by changing the parameters of the lens in such a way that if we are able to vanish the caustic, the optical system produces the sharpest diffraction-limited images. Alternatively, caustic surfaces with a large area can be applied to the design of non-imaging optical systems, with potential applications such as diffusers of light for illumination or solar concentrators. Here, we provide analytic equations for the conic constants, principal surfaces, and caustic surfaces, and also approximations at the third and fifth orders formed by conic lenses, in order to reduce the spherical aberration at these orders.

Design of Hartmann type null screens for testing a plano-convex aspheric lens with a CCD sensor inside the caustic.

A new method to design Hartmann type null screens to test either qualitatively or quantitatively fast plano-convex aspherical lenses is presented. We design both radial and square null screens that produce arrays of circular spots uniformly distributed at predefined planes, considering that the CCD sensor is solely placed inside the caustic region. The designs of these null screens are based on knowledge of the caustic by refraction and on exact ray tracing. The null screens also serve to improve the alignment in optical systems.

Exact wavefronts and caustic surfaces produced by planar ripple lenses.

We obtain simple exact formulas for both caustic and refracted wavefronts through smooth surfaces by considering an incident plane wavefront propagating along the optical axis, providing a condition for total internal reflection (TIR). On the other hand, a formula to provide the maximum slopes for refracted rays outside of the lens is directly related to the condition to obtain the inflection points on the refracting surface. Demanding simultaneously both conditions can potentially provide the shape for an optimized surface which reduces the gap produced by TIR and to refract efficiently all the light outside of the lens. This has a wide potential in applications on the field of non-imagining systems and illumination.

Invisibility assessment: a visual perception approach.

Determining the invisibility of an optical cloak intended to hide something has become a complex problem in recent years. There are many invisibility mechanisms, the performance is quite different from technique to technique, and it is desirable to have a precise metric for their comparison. Here, we propose a simple metric that assesses the perceived invisibility. This invisibility index is based on the fact that the human visual system (HVS) is highly sensitive to spatial frequencies, and then uses the Fourier transform and the contrast sensitivity function of the HVS to assess invisibility.

Modeling LED street lighting.

LED luminaires may deliver precise illumination patterns to control light pollution, comfort, visibility, and light utilization efficiency. Here, we provide simple equations to determine how the light distributes in the streets. In particular, we model the illuminance spatial distribution as a function of Cartesian coordinates on a floor, road, or street. The equations show explicit dependence on the luminary position (pole height and arm length), luminary angle (fixture tilt), and the angular intensity profile (radiation pattern) of the LED luminary. To achieve this, we propose two mathematical representations to model the sophisticated intensity profiles of LED luminaries. Furthermore, we model the light utilization efficiency, illumination uniformity, and veiling luminance of glare due to one or several LED streetlamps.

Analytic aspheric coefficients to reduce the spherical aberration of lens elements used in collimated light.

We provide closed-form formulas for aspheric terms for either plano-convex or convex-plano aspheric lenses as functions of the paraxial parameters involved in the process of refraction. These formulas are obtained through an expansion in Taylor's series from the exact caustic equation produced by aspheric lenses considering a plane wavefront propagating parallel to the optical axis and impinging on the refracting surface. A comparison of the aspheric coefficients obtained through our analytic formulas and commercial optical design software is presented, showing good agreement. This is useful in reducing spherical aberration.

Huygens' Principle: Exact wavefronts produced by aspheric lenses.

We obtain simple exact formulas for the refracted wavefronts through plano-convex aspheric lenses with arbitrary aspheric terms by considering an incident plane wavefront propagating along the optical axis. We provide formulas for the zero-distance phase front using the Huygens' Principle and the Malus-Dupin theorem. Using the fact that they are equivalent, we have in the second method found a way to use an improper integral, instead of the usual evaluated integral, to arrive at these formulas. As expected, when the condition of total internal reflection is satisfied, there is no contribution to the formation of the refracted wavefront.

Caustics in a meridional plane produced by plano-convex aspheric lenses.

We study the formation of caustic surfaces formed in both convex-plano and plano-convex aspheric lenses by considering a plane wave incident on the lens along the optical axis. Using the caustic formulas and a paraxial approximation we derive expressions to evaluate the spherical aberration at third-order and also provide a formula to reduce it, where the first-order aspheric term is given in a simply analytic equation. Furthermore, we redefine the method to evaluate the circle of least confusion for a positive lens as a function of all parameters involved in the process of refraction through the aspheric lenses.