International Journal of Optics

Volume 2016 (2016), Article ID 8103891, 5 pages

http://dx.doi.org/10.1155/2016/8103891

## Components of Lens Power That Regulate Surface Principal Powers and Relative Meridians Independently

^{1}School of Electrical and Information Engineering, University of the Witwatersrand, Johannesburg, Private Bag 3, Wits 2050, South Africa^{2}School of Computer Science and Applied Mathematics, University of the Witwatersrand, Johannesburg, Private Bag 3, Wits 2050, South Africa

Received 3 February 2016; Revised 17 April 2016; Accepted 29 May 2016

Academic Editor: Nicusor Iftimia

Copyright © 2016 H. Abelman and S. Abelman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Paraxial light rays incident in air on alternate refracting surfaces of a thick lens can yield complementary powers. This paper aims to test when these powers are invariant as surface refractive powers interchange in the expression. We solve for relevant surface powers. Potential anticommutators yield the nature of surface principal refractions along obliquely crossing perpendicular meridians; commutators yield meridians that align with those on the next surface. An invariant power component orients relative meridians or the nature of the matrix power on each noncylindrical surface demands that the other component varies. Another component of lens power aligns relative meridian positions for distinct principal powers. Interchanging surface power matrices affects this component. A symmetric lens power results if perpendicular principal meridians are associated with meridians on an opposite rotationally symmetric surface. For thin lenses, meridian alignment may be waived. An astigmatic contact lens can be specified by symmetric power despite having separated surfaces.

#### 1. Introduction

Curved lens surface elements close to an optical axis relative to the radii of the surfaces receive paraxial rays. Such elements and the pole of the surface are in sensibly equivalent planes transverse to the axis. Each refracting surface of a lens approximates a plane transverse to the axis, and the thickness of a lens is the separation of these planes, constant everywhere for rays, and equal to the axial thickness of the lens. We assume a uniform refractive index in air. Light rays encounter elements on separated surfaces say 1 and 2, with rotational symmetry for which meridians chosen for their distinct powers do not exist. Such a lens has the scalar power given by known as Gullstrand’s equation [1]. Rays from air may enter on either face without affecting this lens power. Meridians (none is preferred) on one surface necessarily align with those on the other surface. Power of surface 1 is conjugate to power of surface 2 and powers are called conjugate variables since when these scalars interchange, the expression for is left invariant by the transposition. Physical conjugates are ubiquitous in techniques and principles of optics and eye care [2].

In the next sections, preferred meridians of lesser symmetry introduced to each lens surface moderate rotational symmetry. Both refracting surfaces may have principal powers along meridians that may be perpendicular or oblique, aligning or crossing obliquely. Let light from air be refracted by a “back” surface of a stationary nonflipped lens. Then in expressions like (1) transposed powers denote that rays meet surfaces in a new order. Matrix surface powers that yield a symmetric or asymmetric invariant power are found. Previous work considered systems with asymmetric powers [3]. Independent coefficients of lens thickness measuring the effect of transposition may be commutators or anticommutators of lens surface powers and are made explicit in the expression for lens power. We also show that principal meridians on the respective lens surfaces are aligned as real surface powers commute [4]. Further, if surface powers anticommute, possible principal powers on lens surfaces are equal-and-opposite powers that cross obliquely at 45°. Matrices in commutators have coincident eigenvectors and coincident eigenvectors are those of commuting matrices. Surface principal powers along oblique meridians [5] or perpendicular meridians that are not aligned are reasons for lenses to have antisymmetric dioptric power matrices. Symmetry of matrices serves as a frame of reference and leads to knowledge about the problem that can be identified with the eigenvectors often measured by instruments in the consulting area.

#### 2. Method

Let surface 1 of a lens have matrix power and let the opposite surface 2 have power . Suppose principal meridians on lens surfaces cross obliquely. From surfaces 1 to 2, a power matrix of the lens may be [6]called the Gullstrand equation, generalized in that it follows from rays traced through toric surfaces with matrix powers and [4]. A lens is stationary and as surface powers in expressions are interchanged, this transposition represents light first incident on a “back” refracting surface. If the first two components of an asymmetric power are symmetric, this work shows that the component may be asymmetric with the principal powers of the lens generally along oblique meridians. Explicit answers are available for when the principal meridians for separate lens surfaces are aligned and the nature then of the power of the lens. Does anticommutation of surface powers (see (4)) confirm this? How are powers and related for them to interchange in components of (2) such that is invariant or not (the lens has not been flipped)?

If the thickness of the lens is neglected, simultaneous equations and are satisfied in (2). The square matrices and with the same dimensions are conjugate surface powers that are added in any order (associative) leaving the expression for power of the thin lens invariant. A surface with oblique principal meridians contributes an antisymmetric component in (3) so that the matrix is asymmetric [5] and can be expressed as four components: sphere, cylinder, axis, and asymmetry [7].

In lens power equation (2) the coefficient of benefits from the decomposition of the lens surface power products into bracketed terms with noteworthy distinct clinical meanings seen as likely commutators and anticommutators in the identitythat may each contribute to the symmetry of lens power. If the lens thickness is neglected, these independent meaningful potential symmetry components in (2) and (4) play no role in the lens power as in (3).

We select the left bracket in (4) to write another independent component of lens power asThe power in (5) becomes when and interchange so that surface power is not conjugate to . This is the only power component of the lens with this property. The reader can confirm that is the nonzero antisymmetric component of the power of a lens whose surfaces are toric but only if surface meridians are obliquely crossed. In (5) commuting matrices () imply alignment of principal meridians of surfaces. We return to this point after (8).

The remaining simultaneous contrasting matrix component of follows from the term in the right bracket in (4):where in (3) is included in in which surface power is seen to be a conjugate of since is invariant when interchanging powers in (6) and it is immaterial whether rays first encounter the lens on surface 1 or 2. In addition is the symmetric matrix component of the power of a thick lens whose surfaces are toric and is closest to antisymmetric in that the Frobenius norm is a minimum [8]. With reference to (2), (5), and (6) two components of the matrix power of the lens areAs and trade places in (7), changes sign and power of the lens becomes [6] Powers of surfaces with preferred meridians have been interchanged for matrices from (7) to (8). For lens powers not to change, surface matrix power becomes the conjugate matrix power of and invariant of (6) is the power for a thick lens. For this oror the powers of the refracting surfaces commute. Equation (9) can be shown to be four dependent scalar equations. The solution for the surface powers and of a thick lens requires arbitrary real constants and and the identity matrix. Matrices and can be shown to each have distinct eigenvalues. They represent distinct principal powers that can differ on respective lens surfaces. Different matrices and have a common set of eigenvectors if and only if matrices commute as in (9) [9]. Eigenvectors represent aligning principal meridian directions on respective lens surfaces. Surface powers that commute have principal values along meridians that align from surface to surface and conversely. Equation (9) for thick lenses is valid for the following reason [10]. Pre- and postmultiply by :so that these matrices commute as in (9). Thus surface principal powers along arbitrary aligning meridians of a thick lens satisfy and have a power matrix that remains unchanged. A lensometer measures back and front surface vertex powers and may not be the general detector for lens power . Since and have a common set of linearly independent eigenvectors [9], (9) is valid and and commute. Only one of and may represent the power of a surface without preferred meridians (such matrices always commute, (1) and Table 1). For a thick lens, commuting powers and in are conjugates that may be multiplied in any order in their product in (2) and (6).