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Parastigmatic corneal surfaces

Applied Optics

May 2014
53(16):3438-3447

DOI:10.1364/AO.53.003438

Source
PubMed

Authors:

Camilo Valencia-Estrada

OLEDCOMM

Daniel Malacara-Doblado

Centro de Investigaciones en Optica

Principal meridians of the corneal vertex of the human ocular system are not always orthogonal. To study these irregular surfaces at the vertex, which have principal meridians with an angle different from 90°, we attempt to define so-called parastigmatic surfaces; these surfaces allow us to correct several classes of irregular astigmatism, with nonorthogonal principal meridians, using a simple refractive surface. We will create a canonical surface to describe the surfaces of the human cornea with a short and simple formula, using two additional parameters to the current prescription: the angle between principal meridians and parharmonic variation of curvatures between them.

Polar representation (a) of the function r = sin 2 α . This curve is named rhodonea, which is a name given by the Italian mathematician Guido Grandi between 1723 and 1728 and which resembles the petals of roses. If the function there is r = sin α for 0 ≤ α ≤ 2 π , the graph is obtained only in the superior half-plane (quadrants I and II), as there are no negative radii, so sign convention provides that radii are positive if they are in the superior half-plane, and negative if they are in the inferior half-plane (quadrants III and IV). (b) Polar representation of the function r = psin 2 ( α , β , e ) , which we named elliptical rhodonea for 0 ≤ α < 2 π , with the same sign convention for radii according to the half-plane, with β = π / 4 and e = 1 / 2 . The same rules for signs displayed in (a) remain.

…

Polar representation of function r = psin 2 ( α , β , e ) , for 0 ≤ α < 2 π , with the sign convention for radii, with β = π / 3 and e = 0.7 , 0.75, 0.8, 0.85, 0.9, 0.95, and 0.98.

…

Temporal representation of function psin ( α , β , e ) , for 0 ≤ α < 4 π , with the sign convention for radii, with β = π / 3 and e = 0.8 , 0.9 and 0.999. When e = 1 , the function converges to a zero amplitude pulse, which is useful for modeling other physical and engineering problems. The canonical function sin ( α ) is drawn using dashed lines.

…

Temporal representation of function psin ( α ) , for 0 ≤ α < 2 π , with the sign convention for radii, with β = π / 2 and e = 0.45 , 0.55, 0.65, 0.75, 0.85, 0.92, and 0.98. The canonical function sin ( α ) is drawn using dashed lines.

…

Eccentricity area allowed for the parharmonic parastigmatism appears saturated in gray tone. For one parastigmatism β in degrees, a valid eccentricity must be chosen within the shaded region.

…

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Parastigmatic corneal surfaces

Juan Camilo Valencia-Estrada1,* and Daniel Malacara-Doblado1,2

1Centro de Investigaciones en Óptica A.C., CIO, Col. Lomas del Campestre, León de los Aldama, GTO, 37150, Mexico

2e-mail: dmalacdo@cio.mx

*Corresponding author: camilo@cio.mx

Received 29 January 2014; accepted 29 March 2014;

posted 15 April 2014 (Doc. ID 205703); published 26 May 2014

Principal meridians of the corneal vertex of the human ocular system are not always orthogonal. To study

these irregular surfaces at the vertex, which have principal meridians with an angle different from 90°,

we attempt to define so-called parastigmatic surfaces; these surfaces allow us to correct several classes of

irregular astigmatism, with nonorthogonal principal meridians, using a simple refractive surface. We

will create a canonical surface to describe the surfaces of the human cornea with a short and simple

formula, using two additional parameters to the current prescription: the angle between principal meri-

OCIS codes: (080.0080) Geometric optics; (080.3630) Lenses; (220.0220) Optical design and fabrica-

tion; (330.4460) Ophthalmic optics and devices; (330.7326) Visual optics, modeling; (330.7333) Visual

optics, refractive anomalies.

http://dx.doi.org/10.1364/AO.53.003438

1. Introduction

In a canonical fashion, astigmatic surfaces are those

that have, at all points, two different principal

(orthogonal) curvatures, cxand cy, maximum and

minimum, in any order, such that they satisfy Euler’s

formula at a regular (mathematically) point to

determine curvature cθin any direction θwith

cθcxcos2θcysin2θ. Numerous scientists have

extensively studied astigmatic surfaces in many

ways [1,2].

Then, based on the canonical definition, parastig-

matic surfaces are not strictly astigmatic at the

vertex because this point is always irregular

(mathematically) [3].

There is an infinite number of astigmatic surfaces,

but only a few of them are of ophthalmic interest, es-

pecially those having at least two meridional planes

of symmetry and infinite transverse planes of sym-

metry. The best known are toric surfaces, which

have the characteristic shape of a doughnut, with

or without a central hole, and can be expressed math-

ematically as having the vertex at the origin of the

Cartesian system, with

zc−1

x−

c−1

x−c−1

y

c−2

y−y2

q2

−x2

sor

zc−1

y−

c−1

y−c−1

x

c−2

x−x2

q2

−y2

s;(1)

for its different principal curvatures, cxand cy.

The most popular astigmatic surfaces in the opti-

cal and ophthalmic industry are spherocylindrical

surfaces, which differ (subtly) from toric surfaces.

These surfaces can be represented by the Cartesian

formula [4]:

zcxx2cyy2

1

1−cxx2cyy22x2y2−1

q;(2)

and even other authors [2] simplify the approach

with

1559-128X/14/163438-10$15.00/0

3438 APPLIED OPTICS / Vol. 53, No. 16 / 1 June 2014

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