Optical lens element with a prescription zone, suitable for use in wraparound or protective type eyewear. The element may also include a peripheral vision zone, with no prismatic jump between the zones. Design methods for the prescription zone include temporally rotating a prescription section about a vertical axis through the optical center thereof, and/or decentring the optical axis of said prescription section relative to the geometric axis thereof, and providing partial surface correction for astigmatic and/or mean power errors. For prescription powers in the range −6.0 to +6.0 diopters with 0 to 3 cyl, the optical lens element may be designed such that its front surface can be mounted in a frame of constant curvature of at least 5.0 diopters, with its back surface providing good clearance from temples and eyelashes. Applications include ophthalmic sunglass lenses.

Single vision lenses

The propagation of any general astigmatic laser beam through any paraxial optical system can be characterized to second order by a 4 × 4 matrix of second moments in the transverse spatial and angular coordinates, with this matrix having at most ten independent elements. We show how to interpret the two existing beam invariants that contain these elements in terms of an overall beam quality factor Meff4 and an intrinsic antigmatism factor a. We describe a general procedure for determining all ten parameters for any arbitrary beam by measuring only second-order moments of the beam irradiance in a single transversal plane after the beam has passed through at most four simple astigmatic optical systems. A number of such systems containing, in particular, only one or two rotating cylindrical lenses and a CCD camera are proposed and analyzed in more detail.

Measurement of all ten second-order moments of an astigmatic beam by the use of rotating simple astigmatic (anamorphic) optics

The two-dimensional (2D) nonseparable linear canonical transform (NS-LCT) is a unitary, linear integral transform that relates the input and output monochromatic, paraxial scalar wave fields of optical systems characterized by a 4×4 ray tracing matrix. In addition to the obvious generalizations of the 1D LCT (which are referred to as separable), the 2D-NS-LCT can represent a variety of nonaxially symmetric optical systems including the gyrator transform and image rotation. Unlike the 1D LCT, the numerical approximation of the 2D-NS-LCT has not yet received extensive attention in the literature. In this paper, (1) we develop a sampling theorem for the general 2D-NS-LCT which generalizes previously published sampling theorems for the 1D case and (2) we determine which sampling rates may be chosen to ensure that the obvious discrete transform is unitary.

Two-dimensional nonseparable linear canonical transform: sampling theorem and unitary discretization

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.

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

A new approach to the representation of nonsymmetrical optical systems by matrices is introduced. In the paraxial approximation each component of an optical system is represented by a 4 × 4 unitary matrix, and the product of those matrices yields the transfer matrix of the system. The transfer matrix that represents the propagation between two arbitrary planes through the system containing two independently rotated cylindrical lenses is decomposed into the product of three matrices. The eigenvalues of the submatrices in this factorized form determine the focal lengths of the equivalent system and the localization of the foci of the system with respect to these arbitrarily chosen planes.

Matrix decompositions for nonsymmetrical optical systems

A matrix representation of non-symmetrical optical systems is introduced. Each component of an optical system containing sphero-symmetrical lenses and cylindrical lenses rotated by arbitrary angles about the optical axis may be represented in the paraxial approximation by a 4*4 unitary matrix. The powers of such lenses and lens systems are matrices instead of scalars. When partitioned matrices are used, the resulting 2*2 matrices are formally identical to the matrices of the normal matrix representation, except that the elements of the matrices are 2*2 matrices instead of scalars.

H H Arsenault

Papers

A matrix representation for non-symmetrical optical systems