1 January 2010.qxd:1 11/12/09 09:38 Page 4 dispensingoptics 4 January 2010 Wavefront aberrations and spectacle lenses In response to requests for more challenging CET, part one Darryl Meister ABOM, explains the fundamentals of wavefront aberrations and image quality in part one of this two-part series CompetencIes covered: Optical appliances Target group: Dispensing opticians, optometrists Although evaluating the ‘wavefront’ point can be represented theoretically corresponding wavefront as both aberrations of a lens or optical system using rays emanating outward from propagate away from the object or has become commonplace in fields the object or waves emanating toward the image. such as astronomy, in which adaptive outward that are perpendicular to optical systems minimise the wavefront those rays. Just as rays of light diverge Aberrations are focusing errors that aberrations of telescopes caused by from an object point, waves of light prohibit proper image formation. There atmospheric turbulence, there is spread out like ripples of water are several ways to characterise the increasing interest in the ophthalmic travelling away from a stone that has aberrations produced by a lens or applications of this technology. This been dropped into a pond. At any optical system. In geometrical optics, interest has been driven in no small given distance from the original object ray tracing is often utilised to calculate part by recent advances in laser point, a wavefront exists that the path of a bundle of rays from an refractive surgery that in theory could represents the envelope bounding object point as the rays are refracted allow surgeons to reduce the ‘high- waves of light that have travelled an at the various surfaces of each lens or order’ wavefront aberrations of the equal distance from the object. optical element. Aberrations are then eye, in addition to the traditional determined by calculating the spherical and cylindrical refractive As the distance from the object distance of these refracted rays from errors, using a procedure known as increases, the curvature of these the intended focal point. Alternatively, wavefront-guided ablation. Eliminating wavefronts becomes progressively the deformation of the corresponding the high-order aberrations of the eye flatter, eventually appearing flat wavefront of light as it passes through may allow us to achieve supernormal beyond ‘optical infinity’. The object the optical system may be also vision, with better than ‘normal’ visual point serves as the common centre of determined. acuity and contrast sensitivity1. curvature of these wavefronts (Figure 1). Conversely, light converging to a point In a perfect optical system, spherical Wavefront aberrations focus can be described using wavefronts of light from an object The propagation of ‘light’ has been spherical wavefronts that become point should converge to a single described as a rapid movement of progressively smaller, converging to an point focus at the desired image energy particles (photons) that travel infinitely small image point. Further, a location, such as the retina of the eye. in a wave-like manner. The ray of light from the same object point In the presence of optical aberrations, propagation of light from an object remains perpendicular to the however, these wavefronts become This article has been approved for 1 CET point by the GOC. It is open to all FBDO members, including associate member optometrists. Insert your answers to the six multiple choice questions (MCQs) on the answer sheet inserted in this issue and return by 11 February 2010 to ABDO CET, Courtyard Suite 6, Braxted Park, Great Braxted, Witham CM8 3GA OR fax to 01621 890203, or complete online at www.abdo.org.uk. Notification of your mark and the correct answers will be sent to you. If you complete online, please ensure that your email address and GOC number are up-to-date. The pass mark is 60 per cent. The answers will appear in our March 2010 issue. C-12500 1 January 2010.qxd:1 11/12/09 09:38 Page 5 Continuing Education and Training either too steep, too flat, or distorted IDEAL WAVEFRONT Y (Figure 2) T from their ideal shape . I N I RAY F N I Accordingly, rays of light corresponding T WAVE A SINGLE W A T N POINT to these wavefronts are spread out or V I E O P FOCUS F R T smeared at the plane of the desired O C E N J T B focus by these aberrations, instead of O intersecting at a single, sharp point. ABERRATED WAVEFRONT Y Wavefront aberrations represent a T I N I F N convenient way to characterise I T A SPREAD T complex optical errors in focus N OF I O P FOCUS produced by an optical system. At T DISTANCE FROM OBJECT TO IMAGE POINT C E J B any point across the aperture of the EQUALS RADIUS OF WAVEFRONT CURVATURE O optical system, such as the pupil of the ERROR eye, the wavefront error is the effective optical separation, or Figure 1: A wavefront represents the envelope Figure 2: In the presence of optical aberrations, difference in optical path length, that bounds waves of light at a given distance wavefronts of light do not converge to a sharp from an object point or an image point point focus at the desired image plane between the actual aberrated wavefront and the ideal reference wavefront. These errors are usually these fundamental shapes or modes Zernike aberration modes are expressed in micrometres or microns (Figure 3). commonly represented using a double ( m), which are equal to one- indexing scheme: thµousandth of a millimetre (0.001mm). Further, each Zernike mode is m n associated with a particular type of Z For a given object point, the overall optical error, or wavefront aberration, where n is a whole number indicating shape of the wavefront aberration allowing the wavefront errors to be the radial order and m is a positive or after it has been refracted by the described as a combination of negative integer indicating the optical system can be modelled and quantities of more basic optical meridional frequency of the mode. For quantified mathematically, if a aberrations that are also more familiar instance, the polynomial function of 1 sufficient number of wavefront error to eye care professionals. Each Zernike the Zernike mode expressed as Z3 , measurements are taken across the mode includes two components: varies as a cubic function away the aperture of the system. The shape of 1. The radial order component centre of the pupil (n = 3) and varies an aberrated wavefront can be indicates the exponential variation of sinusoidally one time around the modelled from wavefront error the polynomial function away from circumference of the pupil (m = 1). measurements by ‘fitting’—or closely the centre of the pupil—or how many approximating—the measurements peaks and troughs occur away from Zernike aberration modes are with a series of polynomials. Using the centre. commonly grouped by their radial mathematical curve fitting techniques, 2. The meridional frequency order (n), which indicates how much the measurement data is ‘expanded’ component indicates the number of the contribution of a given range of or expressed as a sum of terms of sinusoidal repetitions of the function modes to the overall shape of the polynomial functions. Increasing the around the meridians of the pupil—or wavefront depends upon pupil size. number of terms used in these how often the peaks and troughs Each radial order contains one or polynomials results in successively repeat around the circumference. Continued overleaf more accurate approximations of the original wavefront. The most common polynomial series in ophthalmic use is TH the Zernike polynomial series, which 0 Z 0 has been shown to characterise the 0 PISTON optical aberrations of the eye effectively2. 1ST Z -1 Z 1 Zernike polynomial series 1 1 VERTICAL TILT HORIZONTAL TILT R The Zernike polynomial series is used to E D R R TRADITIONAL E O breakdown or decompose complex ND D L EYEGLASS 2 R A O I PRESCRIPTION D -2 0 2 wavefronts into a collection of W A Z 2 Z 2 Z 2 O L R polynomial basis functions called OBLIQUE ASTIGMATISM DEFOCUS WTR/ATR ASTIGMATISM R E modes. These basis functions allow the D R O shape of even complex wavefronts to RD H 3 G I H be broken down, or decomposed, Z -3 Z -1 Z 1 Z 3 Figure 3: Displaying the 3 3 3 3 into an assortment of more basic OBLIQUE TREFOILVERTICAL COMA HORIZONTAL COMA WTR/ATR TREFOIL Zernike polynomial component shapes. Just as you can functions by their reconstruct a model of a house by radial order produces 4TH a pyramid of -4 -2 0 2 4 combining various simple shapes, such Z 4 Z 4 Z 4 Z 4 Z 4 as triangles and rectangles, you can aberration modes OBLIQUE QUATREFOILOBLIQUE SECONDARY ASTIG SPHERICAL ABERRATION WTR/ATR SECONDARY ASTIG NORMAL QUATREFOIL (modes to the fourth MERIDIONAL FREQUENCY also reconstruct a given wavefront order have been -4-3 -2 -1 0 +1 +2 +3 +4 from an appropriate combination of shown) 1 January 2010.qxd:1 11/12/09 09:38 Page 6 dispensingoptics 6 January 2010 with polynomial terms of lower or equal order to minimize the variance of the Zernike polynomial. These ABERRATED WAVEFRONT WTR/ATR TREFOIL VERTICAL COMA SPHERICAL ABERRATION features result in some useful properties3: • The contribution of an individual Zernike aberration mode to the total =++ wavefront error is given by the magnitude of its corresponding YXX Y YXX Y coefficient. • The variance of each aberration RMS ERROR = √3 µM C 3 = 1 µM C -1 = 1 µM C 0 = 1 µM 3 3 4 mode is given by the square of its corresponding coefficient.