Reduction of the effect of aberrations in a joint-transform correlator

We report the study of the influence of optical aberrations in a joint-transform correlator: The wave aberration of the optical system is computed from data obtained by ray tracing. Three situations are explored: We consider the aberration only in the first diffraction stage (generation of power spectrum), then only in the second (transformation of the power spectrum into correlation), and finally in both stages simultaneously. The results show that the quality of the correlation is determined mostly by the aberrations of the first diffraction stage and that we can optimize the setup by moving the cameras along the optical axis to a suitable position. The good agreement between the predicted data and the experimental results shows that the method explains well the behavior of optical diffraction systems when aberrations are taken into account.     

Paraxial theory for birefringent lenses

The generalized ray tracing for the extraordinary ray through uniaxial crystals developed by Avenda?o-Alejo and Stavroudis [J. Opt. Soc. Am. A 19, 1674 (2002)] has been applied to derive paraxial refracting equations. Paraxial equations are derived for three cases where the incident, ordinary, and extraordinary rays lie in the incident plane: (a) the crystal axis is parallel to the optical axis, (b) the crystal axis is orthogonal to the optical axis and lies in the plane of incidence, and (c) the crystal axis is orthogonal to both the optical axis and the incident plane. The paraxial ray-tracing equations for the extraordinary ray are represented by matrix operators. The elements of the matrix system give all the information of the focal points and of the principal points. Gaussian formulas are derived, and some examples are presented.     

Propagation of a decentered elliptical Gaussian beam through apertured aligned and misaligned paraxial optical systems

By expanding the hard aperture function into a finite sum of complex Gaussian functions, approximate analytical formulas for a decentered Gaussian beam (DEGB) passing through apertured aligned and misaligned paraxial apertured paraxial optical systems are derived in terms of a tensor method. The results obtained by using the approximate analytical expression are in good agreement with those obtained by using the numerical integral calculation. Furthermore, approximate analytical formulas for a decentered elliptical Hermite-Gaussian beam (DEHGB) through apertured paraxial optical systems are derived. As an application example, approximate analytical formulas for a decentered elliptical flattened Gaussian beam through apertured paraxial optical systems are derived. Our results provide a convenient way for studying the propagation and transformation of a DEGB and a DEHGB through apertured paraxial optical systems.     

Optimization method for ultra-wide-angle and panoramic optical systems

To an ultra-wide-angle and panoramic optical system, the aberrations of point object at any field angle are separated into two types: the aperture-ray aberrations of off-axis point object and the chief-ray aberrations. A simple form of the triangular formulae of tracing an oblique-incidence ray is derived to calculate the chief-ray parameters and their aberrations; moreover, the aperture-ray aberrations of an off-axis point object are analyzed with the plane-symmetric aberration theory. Based on the two types of aberrations, we present a merit function for ultra-wide-angle and panoramic optical systems; the optimization program with the differential-evolution algorithm is then developed. To validate the optimization method we finally optimize a fish-eye lens and a catadioptric omnidirectional imaging system.     

Generalized Raytracing and Applications

The paper deals with an enlargement of the common raytracing algorithm. It allows a simultaneous handling of all rays in an infinitesimal neighbourhood of a given ray. The method can be used for calculating the astigmatic foci of wavefronts and for direct computation of derivatives of transverse ray aberrations and wave aberrations with respect to parameters of the system. The formulae are sufficiently general to handle optical systems without rotational symmetry and with aspheric lenses and mirrors.     

Fractional Fourier transformers through reflection

We show that an arbitrary paraxial optical system, compounded with its reflection in an appropriately warped mirror, is a pure fractional Fourier transformer between coincident input and output planes. The geometric action of reflection on optical systems is introduced axiomatically and is developed in the paraxial regime. The correction of aberrations by warp of the mirror is briefly addressed.     

Astigmatic foci along a skew ray

Abstract

A method for astigmatism calculations along any skew ray is presented. This is useful in the case of extremely wide angle optical systems (and not only imaging systems), where skew ray aberrations are most dominant, but it is essential in the case of systems which are not rotationally symmetric, where all the rays may be skew. Equations relating the position of the astigmatic foci before and after refraction or reflection are derived from the analysis of infinitesimal wave fronts in the immediate neighbourhood of the point of incidence of any ray, but particularly the central ray of any pencil on an optical surface. The method is then illustrated and checked with representative numerical examples.     

Coating-induced wave-front aberrations: on-axis astigmatism and chromatic aberration in all-reflecting systems

The coatings used on telescope mirrors and other optical interfaces can have a marked effect on an optical system's image quality. We describe the wave-front aberrations, particularly the defocus and on-axis astigmatism, that are induced by the s- and p-phase shifts of coatings. These coating-induced wave-front aberrations are very small, particularly near the design wavelengths of the coatings, but they can under certain circumstances overshadow the geometric wave-front aberrations of the system. The wave-front aberrations that are induced by reflection-enhanced coatings on an à/1.5 Cassegrain telescope are numerically evaluated as an example. A theory of coating-induced on-axis astigmatism and chromatic aberration is presented, and a simple algorithm that uses polarization ray tracing to calculate coating-induced defocus and astigmatism coefficients in radially symmetric systems is provided.     

Polarization aberrations. 1. Rotationally symmetric optical systems

The polarization in isotropic radially symmetric lens and mirror systems in the paraxial approximation is examined. Polarized aberrations are variations in the phase, amplitude, and polarization state of the electromagnetic field across the exit pupil. Some are dependent on the incident polarization state and some are not. Expressions through fourth order for phase, amplitude, linear diattenuation, and linear retardance aberrations are derived in terms of the chief and marginal ray angles of incidence and the Taylor series expansion coefficients of the Fresnel equations for reflection and transmission at uncoated and thin-film-coated interfaces. Applications to polarization ray tracing are discussed.     

Variant of the anastigmatic telescope with three mirrors for back focal length

In this paper, an optical design is presented for an anastigmatic telescope with back focal length corrected with exact ray tracing to eliminate spherical, coma, and astigmatism aberrations. The telescope is formed of three conical mirrors, two of them polished on the same substratum. The optical design is divided into three stages: we began the design obtaining the Gaussian parameters in a first-order solution; posteriorly, were obtained analytically the three mirrors' asphericity in a third-order design. The final design stage consists of the implementation of the Fermat's principle, the Abbe sine condition, and the Coddington equations for the exact correction for the three aforementioned aberrations.     

Propagation of hollow Gaussian beams through apertured paraxial optical systems

On the basis of the generalized Collins formula and the expansion of the hard-aperture function into a finite sum of complex Gaussian functions, an approximate analytical formula for a hollow Gaussian beam propagating through an apertured paraxial stigmatic (ST) ABCD optical system is derived. Some numerical examples are given. Furthermore, by using a tensor method, we derive approximate analytical formulas for a hollow elliptical Gaussian beam propagating through an apertured paraxial general astigmatic ABCD optical system and an apertured paraxial misaligned ST ABCD optical system. Our results provide a convenient way for studying the propagation and transformation of a hollow Gaussian beam and a hollow elliptical Gaussian beam through an apertured general optical system.     

Hollow elliptical Gaussian beam and its propagation through aligned and misaligned paraxial optical systems

A new mathematical model called hollow elliptical Gaussian beam (HEGB) is proposed to describe a dark-hollow laser beam with noncircular symmetry in terms of a tensor method. The HEGB can be expressed as a superposition of a series of elliptical Hermite-Gaussian modes. By using the generalized diffraction integral formulas for light passing through paraxial optical systems, analytical propagation formulas for HEGBs passing through paraxial aligned and misaligned optical systems are obtained through vector integration. As examples of applications, evolution properties of the intensity distribution of HEGBs in free-space propagation were studied. Propagation properties of HEGBs through a misaligned thin lens were also studied. The HEGB provides a convenient way to describe elliptical dark-hollow laser beams and can be used conveniently to study the motion of atoms in a dark-hollow laser beam.     

Aberrations of anamorphic optical systems III: the primary aberration theory for toroidal anamorphic systems

We apply a new method for optical aberration derivation to anamorphic systems made from toroidal surfaces and obtain a complete set of primary aberration coefficients. This set is written in a form similar to the well-known Seidel aberrations for rotationally symmetrical optical systems and includes first-order quantities only, thus it can be easily applied to anamorphic lens design practice. By tracing four nonskew paraxial marginal and chief rays, the 16 anamorphic primary aberration coefficients can be easily calculated.     

Meetings Calendar

Abstract

Reduced (fractional) coordinates (σ, τ; x y) are defined for the object and image planes and for the paraxial entrance and exit pupils. These coordinates are defined to give exact values of the direction cosines and the coordinates of suitable points on the corresponding rays in the initial object space. For a ray in any other space of the optical system, its direction cosines and the coordinates of its point of incidence at a surface are represented by power series in (σ, τ; x y), and the corresponding paraxial quantities are defined to be the coefficients of the linear terms. The customary paraxial refraction and transfer equations for rays then appear as exact algebraic relations between these coefficients. The customary ambiguity in the significance of the paraxial approximations and formulae is thereby removed, and paraxial theory is put on the rigorous basis necessary for its applications, using methods that are immediately applicable to (for example) systems containing media of non-uniform refractive index.     

Optical tolerances of an actively tilted high-numerical-aperture two-lens objective for optical recording

Analytical expressions for the primary wave-front aberrations of an actively tilted two-lens objective are derived, and expressions for the higher-order wave-front aberrations for disk tilt of this lens system are presented. This analysis is important because the two-lens objective opens the way to achieving higher-numerical-aperture systems for optical recording with acceptable tolerances that cannot be achieved with a single-lens objective. To test whether the conclusions drawn from the analytically derived results remain valid for high numerical apertures, we compare the results with those obtained by ray tracing: It is shown not only that the two-lens system is tolerant of disk-thickness variations and decentering of the lenses but that it can also be made tolerant of disk tilt when the lens facing the disk is actively tilted.     

Three-dimensional polarization ray-tracing calculus I: definition and diattenuation

A three-by-three polarization ray-tracing matrix method for polarization ray tracing in optical systems is presented for calculating the polarization transformations associated with ray paths through optical systems. The method is a three-dimensional generalization of the Jones calculus. Reflection and refraction algorithms are provided. Diattenuation of the optical system is calculated via singular value decomposition. Two numerical examples, a three fold-mirror system and a hollow corner cube, demonstrate the method.     

Decentered Gaussian beams, ray bundles, and Bessel-Gauss beams

The concept of ray parameters for decentered Gaussian beams is developed on a formal basis. When the beam propagates through first-order optical systems, these parameters are transformed as the ray parameters of geometrical optics. It is shown how this feature helps one to understand the behavior of more sophisticated beams that can be considered as bundles of decentered Gaussian beams. In particular, the case of Bessel-Gauss beams and their recently introduced generalizations is analyzed, and simple transformation formulas are obtained.     

Optical phase space, Wigner representation, and invariant quality parameters

Wigner's quasi probability and related functional and operator methods of quantum mechanics have recently played an important role in optics. We present an account of some of these developments. The symmetry structures underlying the ray and wave approaches to paraxial optics are explored in some detail, and the manner in which the Wigner phase-space representation captures the merits of both approaches is brought out. A fairly self-contained analysis of the second or intensity moments of general astigmatic partially coherent beams and of their behavior under transmission through astigmatic first-order optical systems is presented. Geometric representations of the intensity moments that render the quality parameters or polynomial invariants manifest are discussed, and the role of the optical uncertainty principle in assigning unbeatable physical bounds for these invariants is stressed. Measurement of the ten intensity moments of an astigmatic partially coherent beam is considered.     

Telecentric confocal optics for aberration correction of acousto-optic tunable filters

A telecentric confocal optical arrangement is presented that greatly reduces the diffraction aberrations of the acousto-optic tunable filter (AOTF). Analytical expressions for the aberrations were identified based on the fundamental properties of Bragg diffraction, and additional aberrations due to focusing through the AOTF were also included. The analysis was verified by use of a geometrical ray trace optical code, and an experimental AOTF system was analyzed. Considerable improvement in the potential spatial resolution is predicted with confocal optics, which could accommodate large pixel-limited image fields of greater than 10(6) pixels. When the image quality of the experimental system was assessed, the resolution was found to be improved by the confocal optics and was diffraction limited. Higher resolution could have been obtained with the use of larger optics to increase the throughput before being limited by the aberrations.