Abstract

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.

© 2014 Optical Society of America

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References

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  1. Z. Malacara and D. Malacara Hernández, Handbook of Optical Design (Marcel Dekker, 2004).
  2. J. García Márquez, D. Malacara Doblado, and D. Malacara Hernández, “Axially astigmatic surfaces: different types and their properties,” Opt. Eng. 12, 3422–3426 (1996).
    [CrossRef]
  3. A. Gray, “The definition of a regular surface in Rn,” in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. (CRC Press, 1997), pp. 281–285
  4. C. Menchaca and D. Malacara, “Toroidal and sphero-cylindrical surfaces,” Appl. Opt. 25, 3008–3009 (1986).
    [CrossRef]
  5. , “Optics and photonics—preparation of drawings for optical elements and systems—Part 12: aspheric surfaces,” International Organization for Standardization, ICS: 01.100.20; 37.020. 15 (2007).
  6. J. Einighammer, “The individual virtual eye,” Ph.D. dissertation (University of Tübingen, 2008).
  7. V. Lakshminarayanan and A. Fleck, “Zernike polynomials: a guide,” J. Mod. Opt. 58, 545–561 (2011).
    [CrossRef]
  8. A. Prata and W. V. T. Rusch, “Algorithm for computation of Zernike polynomials expansion coefficients,” Appl. Opt. 28, 749–754 (1989).
    [CrossRef]
  9. C. Valencia and A. Bedoya, “Modelo matemático de la cámara anterior del ojo humano,” Master’s thesis (EAFIT University, 2010).
  10. C. Valencia and D. Malacara, “Parastigmatic surfaces and lenses,” Patent pending. WIPOMX2012/000127, WO/2013/089548 (14December2012).

2011 (1)

V. Lakshminarayanan and A. Fleck, “Zernike polynomials: a guide,” J. Mod. Opt. 58, 545–561 (2011).
[CrossRef]

1996 (1)

J. García Márquez, D. Malacara Doblado, and D. Malacara Hernández, “Axially astigmatic surfaces: different types and their properties,” Opt. Eng. 12, 3422–3426 (1996).
[CrossRef]

1989 (1)

1986 (1)

Bedoya, A.

C. Valencia and A. Bedoya, “Modelo matemático de la cámara anterior del ojo humano,” Master’s thesis (EAFIT University, 2010).

Einighammer, J.

J. Einighammer, “The individual virtual eye,” Ph.D. dissertation (University of Tübingen, 2008).

Fleck, A.

V. Lakshminarayanan and A. Fleck, “Zernike polynomials: a guide,” J. Mod. Opt. 58, 545–561 (2011).
[CrossRef]

García Márquez, J.

J. García Márquez, D. Malacara Doblado, and D. Malacara Hernández, “Axially astigmatic surfaces: different types and their properties,” Opt. Eng. 12, 3422–3426 (1996).
[CrossRef]

Gray, A.

A. Gray, “The definition of a regular surface in Rn,” in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. (CRC Press, 1997), pp. 281–285

Lakshminarayanan, V.

V. Lakshminarayanan and A. Fleck, “Zernike polynomials: a guide,” J. Mod. Opt. 58, 545–561 (2011).
[CrossRef]

Malacara, D.

C. Menchaca and D. Malacara, “Toroidal and sphero-cylindrical surfaces,” Appl. Opt. 25, 3008–3009 (1986).
[CrossRef]

C. Valencia and D. Malacara, “Parastigmatic surfaces and lenses,” Patent pending. WIPOMX2012/000127, WO/2013/089548 (14December2012).

Malacara, Z.

Z. Malacara and D. Malacara Hernández, Handbook of Optical Design (Marcel Dekker, 2004).

Malacara Doblado, D.

J. García Márquez, D. Malacara Doblado, and D. Malacara Hernández, “Axially astigmatic surfaces: different types and their properties,” Opt. Eng. 12, 3422–3426 (1996).
[CrossRef]

Malacara Hernández, D.

J. García Márquez, D. Malacara Doblado, and D. Malacara Hernández, “Axially astigmatic surfaces: different types and their properties,” Opt. Eng. 12, 3422–3426 (1996).
[CrossRef]

Z. Malacara and D. Malacara Hernández, Handbook of Optical Design (Marcel Dekker, 2004).

Menchaca, C.

Prata, A.

Rusch, W. V. T.

Valencia, C.

C. Valencia and A. Bedoya, “Modelo matemático de la cámara anterior del ojo humano,” Master’s thesis (EAFIT University, 2010).

C. Valencia and D. Malacara, “Parastigmatic surfaces and lenses,” Patent pending. WIPOMX2012/000127, WO/2013/089548 (14December2012).

Appl. Opt. (2)

J. Mod. Opt. (1)

V. Lakshminarayanan and A. Fleck, “Zernike polynomials: a guide,” J. Mod. Opt. 58, 545–561 (2011).
[CrossRef]

Opt. Eng. (1)

J. García Márquez, D. Malacara Doblado, and D. Malacara Hernández, “Axially astigmatic surfaces: different types and their properties,” Opt. Eng. 12, 3422–3426 (1996).
[CrossRef]

Other (6)

A. Gray, “The definition of a regular surface in Rn,” in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. (CRC Press, 1997), pp. 281–285

, “Optics and photonics—preparation of drawings for optical elements and systems—Part 12: aspheric surfaces,” International Organization for Standardization, ICS: 01.100.20; 37.020. 15 (2007).

J. Einighammer, “The individual virtual eye,” Ph.D. dissertation (University of Tübingen, 2008).

Z. Malacara and D. Malacara Hernández, Handbook of Optical Design (Marcel Dekker, 2004).

C. Valencia and A. Bedoya, “Modelo matemático de la cámara anterior del ojo humano,” Master’s thesis (EAFIT University, 2010).

C. Valencia and D. Malacara, “Parastigmatic surfaces and lenses,” Patent pending. WIPOMX2012/000127, WO/2013/089548 (14December2012).

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Figures (10)

Fig. 1.
Fig. 1.

Polar representation (a) of the function r=sin2α. 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=sinα for 0α2π, 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=psin2(α,β,e), which we named elliptical rhodonea for 0α<2π, 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.

Fig. 2.
Fig. 2.

Polar representation of function r=psin2(α,β,e), for 0α<2π, 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.

Fig. 3.
Fig. 3.

Temporal representation of function psin(α,β,e), for 0α<4π, 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.

Fig. 4.
Fig. 4.

Temporal representation of function psin(α), for 0α<2π, 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.

Fig. 5.
Fig. 5.

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.

Fig. 6.
Fig. 6.

Parharmonic sine astigmatism with β=π/2, e=0.45 (a), 0.9 (b), and 0.98 (c), of three apical sectors of parastigmatic tori, with the same radii of curvature rX=10 and rY=8. The meridian of highest curvature is 90° from the horizontal axis (meridian of lowest curvature).

Fig. 7.
Fig. 7.

Parharmonic sine parastigmatism with β=70° and e=0.45 (a), 0.9 (b), and 0.98 (c), of three apical sectors of parastigmatic tori, with the same radii of curvature rX=10 and rY=8. The meridian of highest curvature is 70° from the horizontal axis (meridian of lowest curvature). Its position can be inferred by observing the pattern of distortion.

Fig. 8.
Fig. 8.

Parharmonic cosine astigmatism with β=π/2, e=0.45 (a), 0.9 (b), and 0.98 (c), of three apical sectors of parastigmatic tori, with the same radii of curvature rX=10 and rY=8. The meridian of highest curvature is 90° from the horizontal axis (meridian of lowest curvature).

Fig. 9.
Fig. 9.

Parharmonic cosine parastigmatism with β=70° and e=0.45 (a), 0.9 (b), and 0.98 (c), of three apical sectors of parastigmatic tori, with the same radii of curvature rX=10 and rY=8. The meridian of highest curvature is 70° from the horizontal axis (meridian of lowest curvature). Its position can be inferred by observing the pattern of distortion.

Fig. 10.
Fig. 10.

Corneal topography simulated with parharmonic-cosine parastigmatism, with parameters cX=41/337.5, cY=44/337.5, KX=0.06, KY=0.1, ϕ=10° and e=0.7, for three different values of β=70°, 80°, 90° respectively.

Equations (42)

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z=cx1(cx1cy1+cy2y2)2x2orz=cy1(cy1cx1+cx2x2)2y2,
z=cxx2+cyy21+1(cxx2+cyy2)2(x2+y2)1,
z=cxx2+cyy21+1(cx2x2+cy2y2),
z=c(x2+y2)1+1c2(x2+y2)=cρ21+1c2ρ2
z=cxx2+cyy21+1(cxx2Kx+1+cyy2Ky+1)2(x2+y2)1+j=2(A2jx2j+B2jy2j),
z=cxx2+cyy21+1(Kx+1)cx2x2(Ky+1)cy2y2+j=2C2j(cxx2j+cyy2j).
y=±b1x2;
r=b(b2cos2α+sin2α)1/2.
r=b(b2cos2(αθ)+sin2(αθ))1/2.
dydx|xmin=0rsinα+rcosαrcosαrsinα|αmin=γ=0.
(b2+1)cosγ+(b21)cos(γ2θ)=0.
γ=cos1(e2sin(2θ)222e2+e4+e2(e22)cos(2θ)).
θ=12(γ+cos1(2e2e2cosγ)).
[r(γ)γ]=[2e22(e21)(2+e2(1+cos(2θ)))1cos1(e2sin(2θ)222e2+e4+e2(e22)cos(2θ))],
[hk]=[e2sin(2θ)22e2(1+cos(2θ))222e2(1+cos(2θ))].
r=42(1e2)|sinα|(2e2(1+cos(2(αθ))))2e2(1+cos(2θ)).
drdα|β=02(2e2)cosβe2(3cos(β2θ)cos(3β2θ))=0,
θ=cos1(Re6+7e2+4(12e2)cos(2β)(2e2)cos(4β)821+e2+e4+(1+e2e4)cos(2β)sin3β4e2(53cos(2β))).
(2sinβ)15+2cos(2β)3cos2(2β)1cos(2β)<e<1.
psinα=±(e2cos(2(βθ))+e22e2cos(2(αθ))+e22)sinαsinβ,
psinαβπ/2=±2(1e2)sinα2e2(1cos(2α)).
pcosα=(e2cos(2(βθ))+e22e2cos(2(αβ+θ))+e22)sin(αβ)sinβ,
pcosαβπ/2=±2(1e2)cosα2e2(1+cos(2α)).
cθ=cX1psin2α+cYpsin2α,
cθ=cXpcos2α+cY1pcos2α.
cθ=cY1psin2α+cXpsin2α,
cθ=cYpcos2α+cX1pcos2α.
{x=rcosα=r1sin2αx=R1psin2α=X2+Y21psin2(tan1(Y/X))y=rsinαy=Rpsinα=X2+Y2psin(tan1(Y/X)),
x=±X2+(1[2+e2(1+cos(2(βθ)))(2+e2(1+cos(2(θtan1(Y/X)))))sinβ]2)Y2,
y=±2+e2(1+cos(2(βθ)))(2+e2(1+cos(2(θtan1(Y/X)))))sinβY,
x=±X((X2+Y2)(X2+(1e4)Y2)X2+(1e2)Y2)
y=±Y((1e2)(X2+Y2)X2+(1e2)Y2).
{x=rcosαx=Rpcosα=X2+Y2pcos(tan1(Y/X))y=rsinα=r1cos2αy=R1pcos2α=X2+Y21pcos2(tan1(Y/X)),
x=±(2+e2(1+cos(2(βθ))))(XYcotβ)2+e2(1+cos(2(βθtan1(Y/X))))
y=±X2+Y2((2+e2(1+cos(2(βθ))))(XYcotβ)2+e2(1+cos(2(β+θ+tan1(Y/X)))))2.
x=±X((1e2)(X2+Y2)(1e2)X2+Y2)
y=±Y((X2+Y2)((1e4)X2+Y2)(1e2)X2+Y2).
z=cxx2+cyy21+1(cxx2Kx+1+cyy2Ky+1)2(x2+y2)1,
z(r,α)=r2(cY+(cXcY)w(α))1+1r2(cXKX+1w(α)+cYKY+1(1w(α)))2,
w(α)=((2+e2(1+cos(2(βθ))))sin(αβϕ)(2+e2(1+cos(2(α+βθ+ϕ))))sinβ)2,
K(r)=337.5d2zdr2(1+(dzdr)2)3/2,
K(r)=337.5(cY+(cXcY)w(α))(1r2(cY2KY(w(α)1)22cXcY((Kx+1)(Ky+1)1)w(α)(w(α)1)+cX2KXw(α)2))3/2.

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