Lo´pez-Gil et al. Vol. 15, No. 9/September 1998/J. Opt. Soc. Am. A 2563 Generation of third-order spherical and coma aberrations by use of radially symmetrical fourth-order lenses N. Lo´pez-Gil Section of Neurobiology and Behavior, Cornell University, Ithaca, New York 14853-2702 and Laboratorio de O´ ptica, Departamento de Fı´sica, Universidad de Murcia, 30071 Murcia, Spain H. C. Howland Section of Neurobiology and Behavior, Cornell University, Ithaca, New York 14853-2702 B. Howland 4913 Regent Street, Madison, Wisconsin 53705 N. Charman Department of Optometry and Vision Science, University of Manchester Institute of Science and Technology, P.O. Box 88, Manchester, M60 1QD, UK R. Applegate Department of Ophthalmology, University of Texas Health Science Center at San Antonio, San Antonio, Texas 78284-6230 Received December 24, 1997; revised manuscript received April 28, 1998; accepted May 1, 1998 We have extended the method of Alvarez [J. Am. Optom. Assoc. 49, 24 (1978)] to generate a variable magni- tude of third-order spherical and/or coma aberration by using a combination of fourth-order plates with a mag- nification system. The technique, based on the crossed-cylinder aberroscope, is used to measure the wave- front aberration generated by the plates. The method has been applied to correct the third-order spherical aberration generated by an artificial eye as well as the coma produced by a progressive addition ophthalmic lens. The simplicity of the method and its relatively low cost make it attractive for partial correction of the aberrations of the eye. © 1998 Optical Society of America [S0740-3232(98)03709-0] OCIS codes: 220.1000, 220.1010, 220.4830, 220.0220, 170.4460, 170.4470. 1. INTRODUCTION develop optical systems that generate the appropriate ab- Several methods have been employed in the past two de- erration to correct the one present in the optical system of cades to measure the monochromatic aberrations of the the eye. In the past, much effort has been directed at human eye. Most are objective methods based on the eliminating high-order aberrations from optical systems analysis of light that leaves the eye after reflection by the by optimizing design, often with the help of computers. retina. The subjective method, using a crossed-cylinder As an example, the Schmidt negative fourth-order correc- aberroscope as described by Howland and Howland,1 has tor plate was designed to remove the spherical aberration 5 been made objective by the introduction of an ophthalmo- from a spherical mirror. Generally, optical engineers scopic pathway that records the image of a distorted grid strive to remove monochromatic aberrations from their on the retina.2 The information given by the distortion of optical systems by refining the (usually spherical) shapes the grid when the light traverses the eye toward the of their surfaces, the refractive indices of their lenses, and retina is used in this method. The double-pass method the placement of the optical elements. However, several analyzes the image of a point source, taking into account researchers have used optical devices such as lenses, to the fact that it is affected in both passes,3 while the tech- generate a specific amount of chromatic aberration,6,7 or nique that employs a Hartmann–Shack detector4 bases contact lenses, to produce third-order spherical its analysis on the returned wave front of the light that aberration.8 They have analyzed the effects of these passes through the ocular media when the light is leaving lenses on vision and, in particular, on dynamic events the eye. such as accommodation. Although those techniques have With the advent of techniques for measuring the wave been successfully applied with interesting results, they aberration of the human eye, it has become desirable to are capable of inducing only constant amounts of aberra- 0740-3232/98/092563-09$15.00 © 1998 Optical Society of America 2564 J. Opt. Soc. Am. A/Vol. 15, No. 9/September 1998 Lo´pez-Gil et al. 4 4 tion. A variable amount of second- and third-order aber- W~x, y! 5 1000~n 2 1!$K1@x 1 ~ y 1 h1! rations can be generated by arrangements composed of 1 2x2 y 1 h 2 1 K x4 1 y 1 h 4 plano–spherical or cylindrical lenses.9,10 The advantage ~ 1! # 2@ ~ 2! of those setups is the inexpensiveness of their compounds, 2 2 1 2x ~ y 1 h2! #%, (1) but, because of the large number of elements, their per- fect alignment is rather complicated. Two relatively where it has been assumed that both plates are in prox- simple techniques have been applied recently to compen- imity. This expression can be rewritten as sate the wave aberration. The first is based on the use of 2 2 2 deformable mirrors, typically used in astronomy, which W~x, y! 5 1000~n 2 1!$@K1 1 K2#~x 1 y ! 11 allow the recording of a high-quality fundus image. 2 2 The second uses a liquid-crystal device, which locally var- 1 @4~K1h1 1 K2h2!# y~x 1 y ! ies the phase of the wave front and, in so doing, compen- 2 2 2 2 1 @2~K1h1 1 K2h2 !#~x 1 3y ! sates the wave-front error by transmission.12 Although 3 3 4 4 these technologies have a great potential in generating 1 @4~K1h1 1 K2h2 !# y 1 ~h1 1 h2 !%, and compensating different amounts of wave aberrations, (2) they are rather expensive, at least at the present time. In this paper we extend the method of Alvarez13 to gen- where the first, second, third, and fourth coefficients, in erate third-order spherical and coma aberrations as a brackets, are related to third-order spherical aberration, coma along the Y axis, astigmatism with axis at 0° or 90°, simple and more economical means of correcting some of 14 the aberrations of the eye. The solutions presented here and tilt along the X axis and piston, respectively. employ wave aberration plates of fixed value, which, Equation (2) can also be expressed in orthogonal func- when combined with other wave aberration plates or op- tions, which are especially interesting if one wishes to cor- tical elements, give variable amounts of coma and/or rect a specific amount of aberration to minimize the wave- front variance. We can then convert the wave front in spherical aberration. The method is applied to compen- 15 sate the third-order spherical aberration of an artificial Eq. (1) to the equivalent Zernike expression to obtain eye and the third-order coma of a progressive addition 2 2 2 2 W~x, y! 5 z0 1 z2 y 1 z4~2x 1 2y 2 1! 1 z5~y 2 x ! ophthalmic lens. 2 3 4 4 1 z8~3x y 1 3y 2 2y! 1 z12~6x 1 6y 2. METHODS 1 12x2y2 2 6x2 2 6y2 1 1!, (3) In the plates that we used, the first surface is planar and where x and y are coordinates normalized to the pupil ra- the other has a radial symmetrical shape with the form dius, r, over which the wave front is defined and the 4 K(r ). At the edge of a pupil with a 1-mm radius, a Zernike coefficients, zi , are related to K1,2 and h1,2 as 4 4 2 2 2 4 z0 5 A@~K1h1 1 K2h2 ! 1 2~K1h1 1 K2h2 !r 1 1/3~K1 1 K2!r #~piston!, 3 3 3 z2 5 4A@2/3~K1h1 1 K2h2!r 1 ~K1h1 1 K2h2 !r#~tilt!, 2 2 2 4 z4 5 A@2~K1h1 1 K2h2 !r 1 1/2~K1 1 K2!r #~defocus!, 2 2 2 z5 5 2A~K1h1 1 K2h2 !r ~astigmatism!, 3 z8 5 ~4A/3!~K1h1 1 K2h2!r ~third-order coma!, 4 z12 5 ~A/6!~K1 1 K2!r ~third-order spherical!, single plate generates a third-order spherical aberration where A is the constant 1000(n 2 1). In analyzing these (in terms of the Zernike expansion) of (1000)(n coefficients, one can avoid the contribution of the astigma- 2 1)K/6 mm (see the corresponding Zernike coefficient tism and certain amounts of defocus by using h1 5 h2 below). A variable amount of coma can be generated by 5 0or displacement of just one of these plates perpendicular to 2 the optical axis, but the spherical aberration remains, K1/K252~h2/h1! , (4) and tilt, astigmatism, and defocus will also appear. Moreover, the values of the latter aberrations increase as which forces K1 and K2 to have opposite signs. Under the second (astigmatism and defocus) or the third power this condition, it is also possible to avoid the spherical ab- (tilt) of the displacement. These undesired aberrations, erration by giving K1 and K2 opposite values. In this especially the remaining third-order spherical, can be to- particular case, the amount of third-order coma will vary tally or partially avoided by a combination of two plates. linearly with the displacement of the plates, producing a 3 As a general example, when two plates with shapes value of (8/3)AK1h1(r ). The tilt produces a displace- 4 4 K1(r ) and K2(r ) are combined and displaced a distance ment only in the image plane, which does not pose a sig- of h1 and h2 , respectively, in the same direction (e.g., y) nificant problem for isoplanatic systems. The plates in- 4 with respect to the optical axis (z direction), they produce duce a residual amount of defocus, A(K1 1 K2)r /2, that a wave aberration can be avoided by use of the appropriate spherical lens, by Lo´pez-Gil et al.