# Fifth-Order Theory of Astigmatism of Thin Spherical Spectacle Lenses

Purpose. To demonstrate and analyze the fifth-order theory of oblique astigmatism of a thin spherical spectacle lens and make a comparison with the third-order theory and exact ray tracing.

Methods. Fifth-order equations were derived and used for analysis of oblique astigmatism of a spherical spectacle lens to calculate analytically the shape of the lens with corrected oblique astigmatism for large angles of field of view. These results were compared with those of finite ray tracing and the third-order aberration theory.

Results. Formulas for the calculation of oblique astigmatism of a thin spherical spectacle lens were derived. These formulas analytically express oblique astigmatism of the third and fifth order. The theory presented generalizes the third-order description of astigmatism of the spherical spectacle lens and derived equations enable calculation of the shape of the spectacle lens with corrected astigmatism even for a large field of view. The fifth-order solution is compared with the third-order theory and the exact solution found by ray tracing. Differences between the third- and fifth-order theory are <0.05 D for spherical lenses, which is negligible clinically.

Conclusions. The presented fifth-order equations, which are a generalization of the third-order formulas for the description of oblique astigmatism, can be used for the analytical expression of the fifth-order astigmatism of the spherical lens. They can simply be applied for the initial design of lenses with corrected astigmatism for large angles of view, something not possible using the third-order theory. We conclude that astigmatism of the fifth order has little effect on the image quality of the spectacle lens, and the third-order theory is satisfactory for practical calculations in optometry.

*PhD

Department of Physics, Faculty of Civil Engineering, Czech Technical University in Prague, Prague, Czech Republic.

Received April 5, 2011; accepted June 17, 2011.

**Jiri Novak**; Czech Technical University in Prague; Faculty of Civil Engineering; Department of Physics; Thakurova 7; 166 29 Prague 6; Czech Republic; e-mail: novakji@fsv.cvut.cz

Spectacle lenses are used in practice to correct ametropia of the human eye. The main goal of spectacle lens design is to correct major lens aberrations for the eye, which rotates to look at an object. Considering the fact that a diameter of the entrance pupil of the human eye is small with respect to the focal length of the spectacle lens, then the principal aberration, which affects the image quality of the lens, is oblique astigmatism.^{1â€“6} It is possible to minimize residual astigmatism by a proper choice of the shape of the spectacle lens. The shape of the lens can be analytically determined using the third-order theory of aberrations,^{7,8} but this theory is restricted to small angles of field of view, and it is not correct for larger angles. Third-order equations for the correction of oblique astigmatism and other aberrations in thin spherical lenses have been described and used several times in literature.^{2â€“6,9â€“12}

This theory was also used for the description and design of aspheric spectacle lenses with 0 oblique astigmatism when either surface of the lens is a conicoid aspheric.^{4,10â€“12} Other articles have also focused on exact calculations using ray tracing for spherical, aspheric, and spherocylindrical lenses.^{13â€“15} The third-order theory is quite useful for an initial design of the spectacle lens and a fundamental view on relationships between basic parameters of the lens. The spectacle lens with corrected astigmatism, which is designed using the third-order aberration theory, has really minimum oblique astigmatism for small angles of field of view. Astigmatism of the lens increases for larger angles of field of view and the third-order aberration theory is not able to describe this deviation. The fifth-order theory of aberrations, which gives more accurate results for larger angles of field of view and larger numerical apertures of optical systems has been described.^{16â€“20} But the fifth-order formulas are very complicated, and those authors had not used them for design of any particular optical system or optical element.

The aim of this work is to describe a theory of oblique astigmatism of the thin spherical spectacle lens, which will be capable of expressing analytically oblique astigmatism of the third and fifth order in a relatively simple form and allow calculation of the shape of the spectacle lens. We focus on a theory of a spherical spectacle lens for correction of spherical ametropia. Such a generalized theory enables an analytical calculation of the shape of the spectacle lens with corrected astigmatism even for large angles of field of view. The fifth-order solution is also compared with the third-order theory and the exact solution found by ray tracing. The deviations are discussed in relation to clinical standards in optometry. The influence of a finite thickness of the lens on astigmatism and distortion is also studied using the third- and fifth-order aberration theories.

### Astigmatism of Thin Spherical Spectacle Lens

When a narrow pencil of light is incident obliquely on the spectacle lens, it is refracted differently in diverse meridians. The focus of refracted rays in the tangential and sagittal section is not the same. The deviation between tangential and sagittal power errors is called oblique astigmatism. The tangential and sagittal power errors are defined by differences between the tangential or sagittal focus and the far point sphere. We focus on the case of oblique astigmatism of the spherical spectacle lens. The following formulas are valid for tracing the narrow pencil through the spherical surface. We can write for the sagittal plane^{5,6}

and for the tangential plane

where n is the index of refraction of the object space, nâ€² is the index of refraction of the image space, s is the object distance of the sagittal pencil, sâ€² is the image distance of the sagittal pencil, t is the object distance of the tangential pencil, tâ€² is the image distance of the tangential pencil, Îµ is the angle of incidence, Îµâ€² is the angle of refraction, and r is the radius of curvature of the spherical surface.

If we use Eqs. 1 and 2 for tracing the narrow pencil through the thin spherical spectacle lens with radii of curvature r_{1} and r_{2}, we obtain

and

where

whereas n is the index of refraction of the lens, T is the power of the spectacle lens in the tangential section, and S is the power of the spectacle lens in the sagittal section. Angles of reflection and refraction at individual surfaces of the lens can be determined by tracing the meridional ray through the lens. We obtain the following formulas

where w is the angle between the incident ray and the optical axis, wâ€² is the angle between the outgoing ray and the optical axis, x_{1} is the object distance, and xâ€²^{2} is the image distance. Astigmatism of the thin spherical lens can be then calculated using the formula:

### Calculation of Astigmatism Coefficients of the Third and Fifth Order

Equation 7 allows calculation of the astigmatism of the thin spherical spectacle lens, but it cannot be used for the determination of the shape of the lens with corrected astigmatism in practice. We now modify Eq. 7 to enable the calculation of the shape of the spectacle lens with corrected astigmatism. Thus,

where D_{1} is the power of the front surface, D_{2} is the power of the back surface, and D is the power of the whole lens. It holds

where x_{2} is the position of the exit pupil of the lens and x_{1} is the position of the entrance pupil of the lens. Usually two positions of the exit pupil of the lens have been investigated in the literature. The first corresponds to the position of the effective pupil of the lens identical with the center of the rotation of the eye, i.e., it describes the foveal vision of the rotating eye. The second position describes peripheral vision of the stationary eye, i.e., the effective pupil of the lens is identical with the entrance pupil of the eye. Further, we obtain

Denoting

we can express

Considering the following mathematical relationships

where O(6) denotes terms of the sixth and higher order in sin Î±, which we neglected for the following calculation. We can write

where

By substitution of previous formulas into Eq. 7, we obtain for oblique astigmatism of the lens

We can express oblique astigmatism in the form

where

Denoting

then Eq. 13 gets the form

where

Equation 14 can be expressed as

where

Substituting formulas 15 and 16 into Eq. 12, we obtain the equation of the fourth order for the power D_{1} 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 D_{1} 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.^{7,8,16,20â€“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 Zemax^{25} and Oslo^{26} 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 wâ€² = 30Â° 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 x_{2}=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 wâ€² 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 wâ€². 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 âˆ’5 D lens and refractive index, n = 1.5254. The meaning of the particular parameters in Table 3 is as follows: r_{1} and r_{2} are radii of curvature, d is the lens thickness, dsâ€² is sagittal field curvature, dtâ€² is tangential field curvature, astig is astigmatism, D is the lens power, n is the refractive index of the lens, x_{2}â€² is the position of the exit pupil, and wâ€² 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 âˆ’5D lens is âˆ¼3.3% 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.

## ACKNOWLEDGMENTS

This work was supported by grant MSM6840770022 from Ministry of Education of Czech Republic.

Jiri Novak

Czech Technical University in Prague

Faculty of Civil Engineering

Department of Physics

Thakurova 7

166 29 Prague 6

Czech Republic

e-mail: novakji@fsv.cvut.cz

## REFERENCES

**Keywords:**

oblique astigmatism; fifth-order theory; spectacle lens; aberrations