Anamorphic Optical Systems Research Articles

Polarization aberration analysis of catadioptric anamorphic optical systems and its effect on the point spread function

Polarization aberration analysis of catadioptric anamorphic optical systems and its effect on the point spread function

Ghost reflections of Gaussian beams in anamorphic optical systems with an application to Michelson interferometer.

In this paper, a methodology is developed to model and analyze the effect of undesired (ghost) reflections of Gaussian beams that are produced by anamorphic optical systems. The superposition of these beams with the nominal beam modulates the nominal power distribution at the recording plane. This modulation may cause contrast reduction, veiling parts of the nominal image, and/or the formation of spurious interference fringes. The developed methodology is based on synthesizing the beam optical paths into nominal and ghost optical beam paths. Similar to the nominal beam, we present the concept that each ghost beam is characterized by a beam size, wavefront radius of curvature, and Gouy phase in the paraxial regime. The nominal and ghost beams are sequentially traced through the system and formulas for estimating the electric field magnitude and phase of each ghost beam at the recording plane are presented. The effective electric field is the addition of the individual nominal and ghost electric fields. Formulas for estimating Gouy phase, the shape of the interference fringes, and the central interference order are introduced. As an application, the theory of the formation of the interference fringes by Michelson interferometer is presented. This theory takes into consideration the ghost reflections that are formed by the beam splitter. To illustrate the theory and to show its wide applicability, simulation examples that include a Mangin mirror, a Michelson interferometer, and a black box optical system are provided.

Shaping propagation invariant laser beams

Propagation-invariant structured laser beams possess several unique properties and play an important role in various photonics applications. The majority of propagation invariant beams are produced in the form of laser modes emanating from stable laser cavities. Therefore, their spatial structure is limited by the intracavity mode formation. We show that several types of anamorphic optical systems (AOSs) can be effectively employed to shape laser beams into a variety of propagation invariant structured fields with different shapes and phase distributions. We present a propagation matrix approach for designing AOSs and defining mode-matching conditions required for preserving propagation invariance of the output shaped fields. The propagation matrix approach was selected, as it provides a more straightforward approach in designing AOSs for shaping propagation-invariant laser beams than the alternative technique based on the Gouy phase evolution, especially in the case of multielement AOSs. Several practical configurations of optical systems that are suitable for shaping input laser beams into a diverse variety of structured propagation invariant laser beams are also presented. The laser beam shaping approach was applied by modeling propagation characteristics of several input laser beam types, including Hermite–Gaussian, Laguerre–Gaussian, and Ince–Gaussian structured field distributions. The influence of the Ince–Gaussian beam semifocal separation parameter and the azimuthal orientation between the input laser beams and the AOSs onto the resulting shape of the propagation invariant laser beams is presented as well.

无遮拦三反射镜变形光学系统设计

无遮拦三反射镜变形光学系统设计

Orthonormal aberration polynomials for anamorphic optical imaging systems with circular pupils

In a recent paper, we considered the classical aberrations of an anamorphic optical imaging system with a rectangular pupil, representing the terms of a power series expansion of its aberration function. These aberrations are inherently separable in the Cartesian coordinates (x,y) of a point on the pupil. Accordingly, there is x-defocus and x-coma, y-defocus and y-coma, and so on. We showed that the aberration polynomials orthonormal over the pupil and representing balanced aberrations for such a system are represented by the products of two Legendre polynomials, one for each of the two Cartesian coordinates of the pupil point; for example, L(l)(x)L(m)(y), where l and m are positive integers (including zero) and L(l)(x), for example, represents an orthonormal Legendre polynomial of degree l in x. The compound two-dimensional (2D) Legendre polynomials, like the classical aberrations, are thus also inherently separable in the Cartesian coordinates of the pupil point. Moreover, for every orthonormal polynomial L(l)(x)L(m)(y), there is a corresponding orthonormal polynomial L(l)(y)L(m)(x) obtained by interchanging x and y. These polynomials are different from the corresponding orthogonal polynomials for a system with rotational symmetry but a rectangular pupil. In this paper, we show that the orthonormal aberration polynomials for an anamorphic system with a circular pupil, obtained by the Gram-Schmidt orthogonalization of the 2D Legendre polynomials, are not separable in the two coordinates. Moreover, for a given polynomial in x and y, there is no corresponding polynomial obtained by interchanging x and y. For example, there are polynomials representing x-defocus, balanced x-coma, and balanced x-spherical aberration, but no corresponding y-aberration polynomials. The missing y-aberration terms are contained in other polynomials. We emphasize that the Zernike circle polynomials, although orthogonal over a circular pupil, are not suitable for an anamorphic system as they do not represent balanced aberrations for such a system.

Extended ABCD matrix formalism for the description of femtosecond diffraction patterns; application to femtosecond digital in-line holography with anamorphic optical systems

We present a new model to predict diffraction patterns of femtosecond pulses through complex optical systems. The model is based on the extension of an ABCD matrix formalism combined with generalized Huygens-Fresnel transforms (already used in the CW regime) to the femtosecond regime. The model is tested to describe femtosecond digital in-line holography experiments realized in situ through a cylindrical Plexiglas pipe. The model allows us to establish analytical relations that link the holographic reconstruction process to the experimental parameters of the pipe and of the incident beam itself. Simulations and experimental results are in good concordance. Femtosecond digital in-line holography is shown to allow significant coherent noise reduction, and this model will be particularly efficient to describe a wide range of optical geometries. More generally, the model developed can be easily used in any experiment where the knowledge of the precise evolution of femtosecond transverse patterns is required.

Orthonormal aberration polynomials for anamorphic optical imaging systems with rectangular pupils

The classical aberrations of an anamorphic optical imaging system, representing the terms of a power-series expansion of its aberration function, are separable in the Cartesian coordinates of a point on its pupil. We discuss the balancing of a classical aberration of a certain order with one or more such aberrations of lower order to minimize its variance across a rectangular pupil of such a system. We show that the balanced aberrations are the products of two Legendre polynomials, one for each of the two Cartesian coordinates of the pupil point. The compound Legendre polynomials are orthogonal across a rectangular pupil and, like the classical aberrations, are inherently separable in the Cartesian coordinates of the pupil point. They are different from the balanced aberrations and the corresponding orthogonal polynomials for a system with rotational symmetry but a rectangular pupil.

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.

Aberrations of anamorphic optical systems II Primary aberration theory for cylindrical anamorphic systems

By applying a new method for aberration derivation to anamorphic systems made from cylindrical surfaces, we obtain a complete set of primary aberration coefficients. This set is in a form similar to the well-known Seidel aberrations for rotationally symmetrical optical systems (RSOS). The aberration coefficients include first-order quantities only. By tracing four nonskew paraxial marginal and chief rays in the two associated RSOS of the anamorphic system, the aberration coefficients can be derived.

Aberrations of anamorphic optical systems I: the first-order foundation and method for deriving the anamorphic primary aberration coefficients

We suggest that, in the paraxial region, a double-plane symmetric optical system (anamorphic system) can be treated as two associated rotationally symmetric optical systems (RSOS). We find that paraxial quantities in the anamorphic system can be expressed as linear combinations of the paraxial marginal and chief rays traced in the two associated RSOS. As a result, we provide a set of equations that are key to derive the primary aberration coefficients for various anamorphic optical system types. By applying the generalized Aldis theorem to anamorphic optical systems, we build up the anamorphic total ray aberration equations. These equations can be reduced to third-order form, that is, the anamorphic primary ray aberration equations. We find that the terms in the anamorphic primary ray aberration equations can be expressed as paraxial marginal and chief ray-trace data in the two associated RSOS, together with normalized object and stop coordinates. More importantly, we build up a novel method for deriving the anamorphic primary aberration coefficients for anamorphic optical systems of various types.

Third-order aberration coefficients of anamorphic GRIN optical systems: a one-dimensional refractive index distribution

Third-order geometrical aberration coefficients of anamorphic optical systems with a one-dimensional refractive index distribution are obtained in an explicit form.

Lensless anamorphic Fourier transform hologram recorded with prism systems

A novel configuration for recording a lensless anamorphic Fourier transform hologram of a given object's light distribution is proposed. The method is based on the use of prism anamorphic optical systems coupled with phase cancellation at the hologram plane. Anamorphic systems with cylindrical lenses and prisms are critically evaluated through computer simulations for their suitability in anamorphic Fourier transform holographic configurations. A complete theoretical analysis and experimental demonstration of the recording and reconstruction of a lensless anamorphic Fourier transform hologram are presented.

Anamorphic optical systems using programmable spatial light modulators

Anamorphic optical systems with different horizontal and vertical magnifications are constructed by using programmable anamorphic lenses written on a magneto-optic spatial light modulator.

Holographic optical generation of prism anamorphs

Analyses of prism anamorphic optical systems with single and multiple prisms in plane and spherical wave illumination geometries are presented. A theoretical evaluation and experimental demonstration of a new technique for recording holographic anamorphic elements by using prism anamorphic optical systems is discussed. It is seen that this technique offers the advantage of generating elements with low f/number. Another advantage of this recording is the easy generation of elements with different anamorphic factors by using a single configuration.

One-dimensional optical correlator based on outer product architecture

Optical correlators of 1D signals implemented with anamorphic optical systems are described. The outer product matrix of two input signals is calculated optically and the elements of the matrix are summed up along the lines to the diagonal direction so as to obtain the cross-correlation of the signals. An experimental verification of the concept is presented.