then Eq. 13 gets the form
Equation 14 can be expressed as
Substituting formulas 15 and 16 into Eq. 12, we obtain the equation of the fourth order for the power D1 of the front surface of the lens. We can set this equation equal to 0 and calculate the power of the spectacle lens with eliminated oblique astigmatism of the third and fifth order. We can write the following equation for the front surface power
Using the previous equation of the fourth order, we can calculate the power of the front surface D1 of the thin spherical spectacle lens. Other parameters of the lens can be determined from Eq. 8. Equation 12, which makes it possible to calculate oblique astigmatism of the lens, includes both the third-order astigmatism and the fifth-order astigmatism. If we restrict ourselves to the third-order aberration theory, we obtain the quadratic equation for the calculation of the front surface power. As far as we know such equations for calculation of the shape of the spectacle lens with corrected fifth-order astigmatism had never been published in literature.
Now, we focus on the analysis of the difference between the derived solution, the third-order calculations, and the exact ray tracing. Formulas exist for exact ray tracing astigmatic pencils through a general optical surface. One can use the approach based on the so-called generalized Coddington equations.22,23 However, these equations do not offer the possibility of explicit determination of formulas for the calculation of the spectacle lens shape. These formulas are valuable for ray tracing of astigmatic pencils through the lens. It is not possible to calculate analytically the shape of the spectacle lens using these equations and one must use numerical methods for ray tracing, which have been described in detail.7,8,16,20–24 One can also use another approach introduced by Feder.24 Detailed analysis of spot diagrams is performed, for example, in ref. 7. Commercially offered software Zemax25 and Oslo26 are very frequently used for the design and analysis of optical systems. The formula for the fifth-order astigmatism derived in our work has a relatively simple form and makes it possible to perform the analytical calculation of the shape of the spectacle lens by solving the fourth-order equation; this is not possible by the generalized Coddington equations.
Examples and Discussion
Now, we present a comparison of the third- and fifth-order aberration theories for examples of thin spherical spectacle lenses. We assume two values of refractive index of the spectacle lens (n = 1.5254 and n = 1.80), the angle of field of view w′ = 30° and the case of the rotating eye, i.e., the effective pupil of the lens coincides with the center of rotation of the eye and x2=25 mm. By using Eqs. 12, 15, 16, and 17, we calculated the lens parameters for the anastigmatic lenses (with 0 oblique astigmatism) with different values of power and refractive index. The results are presented in Table 1.
Fig. 1 presents a comparison of the dependence of oblique astigmatism on the field of view angle w′ for the anastigmatic spherical spectacle lens with power D = 5 dpt calculated using Eq. 17. We can see that for larger values of field of view, there is a difference between the third- and the fifth-order calculations and fifth-order formulas offer much better elimination of astigmatism than classical third-order formulas. Differences between the third- and fifth-order theories are <0.05 D for spherical lenses, which is a negligible value in relation to clinical standards in optometry. Thus, the contribution of the fifth-order astigmatism need not be considered for the design of the spectacle lenses and the lens shape can be calculated only using the third-order theory with satisfactory results. However, derived formulas are more generally valid and are not limited to spectacle lens design. For example, lenses used for laser scanning can be characterized by a relatively large difference between the third- and fifth-order theories, which has a significant effect on the image quality.
The exact calculation using ray tracing shows a small deviation of the fifth-order solution from the exact solution. Table 2 presents a comparison of values of residual oblique astigmatism of the positive spherical lens D=+5 dpt) for the third- and fifth-order theories and different values of the angle of field of view w′. One can see that the fifth-order equations result in better elimination of oblique astigmatism of the spectacle lens for larger angles.
Furthermore, we performed an analysis of the influence of a finite thickness of the lens on its astigmatism and distortion. The results can be found in Table 3 for a −5 D lens and refractive index, n = 1.5254. The meaning of the particular parameters in Table 3 is as follows: r1 and r2 are radii of curvature, d is the lens thickness, ds′ is sagittal field curvature, dt′ is tangential field curvature, astig is astigmatism, D is the lens power, n is the refractive index of the lens, x2′ is the position of the exit pupil, and w′ is the angle of view. All linear dimensions in Table 3 are in millimeters.
The fifth-order effect on distortion is practically negligible. Distortion of the −5D lens is ∼3.3% in both cases (third- and fifth order theories). The influence of individual aberrations and refractive index of the lens is discussed in more detail in refs. 5 and 6, and hence we did not focus on this problem in our present work. The third-order Petzval curvature is equal to 1/(nf), where n is the refractive index of the lens and f is its focal length. The fifth-order Petzval curvature is practically identical with the third-order Petzval curvature.7
It is well known that the possible solutions for zero oblique astigmatism for thin spherical spectacle lenses using the third-order aberration theory can be presented as the dependence between the front or back surface power and the equivalent lens power. This dependence is an elliptic curve known as the Tscherning ellipse.2,4,5,13,14 There are two solutions of front surface power for thin spherical lenses. Those lenses, which are the least curved, are represented by the lower portion of the Tscherning ellipse (Oswalt branch). Lenses, which are most curved, are represented by the upper portion of the ellipse (Wollaston branch). The design of spectacle lenses is based on the Ostwalt branch, which corresponds to lens forms more cosmetically attractive and less difficult to produce. Outside the Tscherning ellipses, there is no solution for anastigmatic lenses. The range of equivalent lens power depends on the lens refractive index, the distance of the spectacle lens from the center of rotation of the eye, and the distance of the object from the lens. The analysis of the influence of refractive index of the spectacle lens on its properties is performed in ref. 5 within the validity of the third-order aberration theory. It is advantageous to use a higher index lens material because of smaller edge thickness, lower overall volume, and weight. Higher refractive index of the spectacle lens material enables a theoretical realization of spherical lenses with higher negative power that can be used for the correction of high myopia.
The range of possible designs of anastigmatic spherical spectacle lenses can be also determined by solving Eq. 17 within the validity of the fifth-order aberration theory. This can similarly be derived with the third-order theory an elliptic curve (Tscherning ellipse) for different values of power D. There exist two possible real solutions of the front surface power calculated from the fourth-order Eq. 17. The ellipse again has the lower part (Oswalt branch) and the upper part (Wollaston branch), which correspond to both real roots of Eq. 17. The difference between the third- and the fifth-order theories is shown in Fig. 2 for spherical lenses with refractive index, n = 1.5254. One can see that the elliptic curve is slightly changed because of considering higher order equations. However, these differences are very small from the point of view of their influence on visual acuity and one can ignore them clinically. Our analysis has shown that the contribution of the fifth-order theory can be ignored in optometric practice.
The theory we presented is a generalization of the third-order equations for the description of oblique astigmatism of the spectacle lens. Derived formulas can be used to calculate analytically the shape of the spectacle lens with corrected astigmatism even for large angles of field of view, which was not possible using the third-order theory. The derived formulas are generally valid and can be applied beyond spectacle lenses. We believe that this is a new contribution to the theory of aberrations and optical imaging.
This work was supported by grant MSM6840770022 from Ministry of Education of Czech Republic.
Czech Technical University in Prague
Faculty of Civil Engineering
Department of Physics
166 29 Prague 6
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Keywords:© 2011 American Academy of Optometry
oblique astigmatism; fifth-order theory; spectacle lens; aberrations