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Spherical Aberration: the Next Frontier Only by Understanding the Math Can Ophthalmologists Target the Ideal Correction for Their Patients

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Spherical Aberration: the Next Frontier Only by Understanding the Math Can Ophthalmologists Target the Ideal Correction for Their Patients COVER STORY Spherical Aberration: The Next Frontier Only by understanding the math can ophthalmologists target the ideal correction for their patients. BY JACK T. HOLLADAY, MD, MSEE pherical aberration is the property of a single spheri- faces are referred to as conic sections, because they can be cal surface to refract rays too strongly as one moves created by taking a section of a cone. If we allow these from the center of the pupil peripherally compared surfaces to have a toric apex (astigmatism), then the re- with the paraxial (central) focus (Figure 1). The sulting surfaces are hyperboloid, paraboloid, prolate ellip- Simage formed when spherical aberration is present causes a soid, spheroid, and oblate ellipsoid. Each of these surfaces halo (or blur) around the paraxial image (Figure 2). The ter- can be determined by three variables: horizontal and ver- minology is the same in an optical system with multiple tical apical radii (Rxz and Ryz) and an asphericity quotient surfaces such as the human eye, which has four refracting (Q-value) in a three-dimensional Cartesian coordinate surfaces (the anterior/posterior cornea and the anterior/ system (X, Y, and Z). posterior crystalline lens). If the system has more optical Equation 1. power away from the optical axis, toward the periphery, there is positive spherical aberration. If the opposite is true, then it is negative. Spherical aberration is radially symmet- ric, meaning that all semimeridians are identical and have where Equation 2. the same ray-tracing diagram as if it were the only aberra- tion in the system. This article analyzes spherical aberration mathemati- & cally in order to identify the optimal amount to target Table 1 shows the values of Q and their corresponding for the human eye. surfaces. Figure 3 provides a cross-section (two-dimen- sional plot) of these surfaces. THE MATHEMATICS OF ASPHERIC Figure 4 illustrates a colored ray tracing of a spherical SURFACES AND SPHERICAL ABERRATION: surface. The longitudinal spherical aberration (LSA’) is the THE ANTERIOR CORNEA difference between the marginal focus (L’m) and the Biological single surfaces are rarely spherical. More paraxial focus (L’p) along the optical axis. Transverse often they are aspheric, as are the surfaces of the cornea spherical aberration (TSA’) is the height of the marginal and crystalline lens. Some of the simplest aspheric sur- ray at the paraxial focal plane. Figure 2. Spherical aberration is perceived as halos around Figure 1. This diagram of positive spherical aberration illus- lights that cause the symptoms of glare. Increasing amounts trates longitudinal and lateral spherical aberration. of spherical aberration create larger halos. NOVEMBER/DECEMBER 2006 I CATARACT & REFRACTIVE SURGERY TODAY I 95 COVER STORY Figure 4. This diagram of positive spherical aberration shows marginal focus (L’m), paraxial focus (L’p), and longitudinal spherical aberration (LSA’). When only primary (third-order spherical aberration) is present, the best focus is midway between the marginal and paraxial foci. The longitudinal spherical aberration can also be expressed as a series Equation 4. LSA’ = a Y2 + b Y4 + c Y6 + … where the first term is referred to as third-order LSA, the second term as fifth-order LSA, etc., and the a is the Figure 3. These conicoids have various Q-values. Blue coefficient of the third-order term, the b is the coefficient (Q = +5) is an oblate ellipsoid, magenta (Q = +1) is an oblate of the fifth-order term, etc. In most simple optical sys- ellipsoid, yellow (Q = 0) is a sphere, aqua (Q = -0.26, mean tems, including the normal human eye, it is rare to have human cornea) is a prolate ellipsoid, red (Q = -0.50) is a pro- more than third-order LSA, so the first term is all that is late ellipsoid, brown (Q = -1) is a paraboloid, and green used clinically. In optical design, the term zonal SAz is (Q = -5) is a hyperboloid. used. Instead of the marginal ray, it is customary to use a ray of height that is 0.707 of the marginal ray (Ym) such Written as formulas, it is that the areas inside and outside this diameter are equal. Equation 3a. LSA’ = L’m – L’p The resulting equations for third-order LSA are Equation 3b. TSA’ = H’ = (LSA’)tan U’m Equation 5a. Marginal Third-Order where U’m is the slope of the marginal ray after refrac- tion. When the marginal ray intercept is anterior to the paraxial focus, the spherical aberration is positive. If pos- Equation 5b. Zonal Third-Order terior to the paraxial focus, it is negative. TABLE 1. Q-VALUES AND THEIR CORRESPONDING SURFACES where Q-Value Surface Q < -1 Hyperboloid Q = -1 Paraboloid When only third-order spherical aberration is present, -1 < Q < 0 Prolate ellipsoid the “best” focus (circle of least confusion) is midway between the paraxial focus and marginal focus (one half Q = 0 Spheroid of LSAm). If third- and fifth-order spherical aberration are Q > 0 Oblate ellipsoid present, then the best focus moves to three quarters of LSAm. 96 I CATARACT & REFRACTIVE SURGERY TODAY I NOVEMBER/DECEMBER 2006 COVER STORY aberration for any object and image, the optical path dif- ference must be the same for any rays traced through the system at height Y and must satisfy the following two- dimensional equation (Y and Z as defined in Equation 1 with X = 0): Equation 8. Equation 8 is a form of the famous Descartes equation. Unfortunately, for all objects at any distance, the aspheric surface is not a conic section, so none of the aforemen- tioned conic surfaces (Equation 2) can eliminate spheri- Figure 5. The surface of the Z(4,0) Zernike term for primary cal aberration for any object at any finite distance. If the (third-order) spherical aberration is shown. object is at infinity, then the formula reduces to the sim- ple equation The spherical aberration may also be described using Equation 9. wave aberration form, which expresses the aberration of the wave at the refracting surface. Equation 6. This equation shows that, for an object at infinity, the negative of the ratio of the anterior index of refraction Similar to Equation 4, the wave aberration for third- and squared divided by the posterior index of refraction order spherical aberration is the first term, fifth-order squared is the perfect Q-value (conic surface) to elimi- spherical aberration is the second term, etc. For an nate spherical aberration. For the cornea, the index of aspheric surface, the primary spherical aberration (third- refraction is 1.376, and the anterior medium is air. The order) may be determined with the following equation1: ideal asphericity for the anterior surface of the cornea to Equation 7. eliminate spherical aberration for a distant object would therefore be a prolate ellipsoid with Equation 10. Q= - (1.0002/1.3762) = -0.528. Several investigators have shown that the mean Q-value where n is the index of refraction of the media of the of the anterior cornea over a 6-mm zone is -0.260 (0.00 to object, n’ is the index of refraction of the media of the -0.50) with a wide variance.2 The normal human cornea image, L is the distance to the object, and L’ is the dis- with a Q-value of -0.260 has less positive spherical aberra- tance to the image from the vertex of the surface for the tion than a sphere (Q = 0), but it is rarely free of spherical marginal ray. aberration (Q = -0.53). An important clinical point is that, A number of observations can be made when looking even if the anterior cornea did have a Q-value of -0.528, it at this equation. The first term (divided by eight) is the would only be free of spherical aberration for an object at spherical aberration of a spherical surface, and the sec- infinity. An object at any distance other than infinity will ond term (after the plus sign) is the additional term nec- result in spherical aberration. essary to describe the primary spherical aberration of an Another common method of describing a wavefront ρ aspheric surface with a Q-value of QSEQ. In this equation, aberration over a circular pupil with radius is the the primary spherical aberration coefficient is a propor- Zernike polynomials. They are easily related to the classi- tional Q-value (as Q becomes more positive, the spheri- cal aberrations, but two terms are required for aberra- cal aberration increases) and inversely related to the api- tions that are not rotationally symmetric such as astig- cal radius (as the apical radius decreases or K-reading matism and coma. Such terms are decomposed into an x increases, the spherical aberration increases). For any sin- and a y component, but they can be easily combined gle object and apical spheroequivalent radius, there is a into one term with a single magnitude and orientation. spheroequivalent Q-value that will give zero primary The term for primary (third-order) spherical aberration spherical aberration (note: this is only true for a single is given by the following Zernike polynomial: object at a given distance, not for an object at any other Equation 11. distance). If we want to find the surface that has no spherical NOVEMBER/DECEMBER 2006 I CATARACT & REFRACTIVE SURGERY TODAY I 97 COVER STORY All of the Zernike terms above Z(0,0) have an average value of zero, and all terms are orthogonal and therefore mutually exclusive. Because each term has an average value of zero over the entire surface, the reference plane of zero (Figure 5 where zero is green) is not the paraxial focus but the best focus (circle of least confusion), as seen in Figure 4.
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